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[ "Exact resolution function for double-disk chopper neutron time-of-flight spectrometers : Application to reflectivity", "Exact resolution function for double-disk chopper neutron time-of-flight spectrometers : Application to reflectivity" ]	[ "Didier Lairez \nLaboratoire Léon Brillouin\nCEA-CNRS\nUniversité Paris-Saclay\n91191Gif-sur-YvetteFrance\n\nLaboratoire des solides irradiés\nCEA-École polytechnique-CNRS-Institut Polytechnique de Paris\n91128PalaiseauFrance\n", "Alexis Chennevière \nLaboratoire Léon Brillouin\nCEA-CNRS\nUniversité Paris-Saclay\n91191Gif-sur-YvetteFrance\n", "Frédéric Ott \nLaboratoire Léon Brillouin\nCEA-CNRS\nUniversité Paris-Saclay\n91191Gif-sur-YvetteFrance\n" ]	[ "Laboratoire Léon Brillouin\nCEA-CNRS\nUniversité Paris-Saclay\n91191Gif-sur-YvetteFrance", "Laboratoire des solides irradiés\nCEA-École polytechnique-CNRS-Institut Polytechnique de Paris\n91128PalaiseauFrance", "Laboratoire Léon Brillouin\nCEA-CNRS\nUniversité Paris-Saclay\n91191Gif-sur-YvetteFrance", "Laboratoire Léon Brillouin\nCEA-CNRS\nUniversité Paris-Saclay\n91191Gif-sur-YvetteFrance" ]	[]	The exact resolution function in transfer vector for the reflectometer HERMÈS at Laboratoire Léon Brillouin is calculated as an example of neutron time-of-fligh spectrometer with double-disk chopper. Calculation accounts for wavelength distribution of the incident beam, tilt of the chopper axis, collimation and gravity, without approximation of gaussian distributions or independence of these different contributions. Numerical implementation is provided that matches the sections of the paper. We show that data fitting using this exact resolution function allows us to reach much better results than its usual approximation by a gaussian profile.	10.1107/s1600576720001764	[ "https://arxiv.org/pdf/1906.09577v2.pdf" ]	195,345,377	1906.09577	22b4d0c56df0a6c40cccd67229e428d36f8792ee	Exact resolution function for double-disk chopper neutron time-of-flight spectrometers : Application to reflectivity Didier Lairez Laboratoire Léon Brillouin CEA-CNRS Université Paris-Saclay 91191Gif-sur-YvetteFrance Laboratoire des solides irradiés CEA-École polytechnique-CNRS-Institut Polytechnique de Paris 91128PalaiseauFrance Alexis Chennevière Laboratoire Léon Brillouin CEA-CNRS Université Paris-Saclay 91191Gif-sur-YvetteFrance Frédéric Ott Laboratoire Léon Brillouin CEA-CNRS Université Paris-Saclay 91191Gif-sur-YvetteFrance Exact resolution function for double-disk chopper neutron time-of-flight spectrometers : Application to reflectivity submitted for publication to J. Appl. Cryst. (Dated: June 25, 2019) The exact resolution function in transfer vector for the reflectometer HERMÈS at Laboratoire Léon Brillouin is calculated as an example of neutron time-of-fligh spectrometer with double-disk chopper. Calculation accounts for wavelength distribution of the incident beam, tilt of the chopper axis, collimation and gravity, without approximation of gaussian distributions or independence of these different contributions. Numerical implementation is provided that matches the sections of the paper. We show that data fitting using this exact resolution function allows us to reach much better results than its usual approximation by a gaussian profile. I. INTRODUCTION It is quite common for physical measurements to attempt to reach the limit of a given technique. In many cases, this amounts to measure a quantity with an accuracy better than the resolution of the apparatus. Said like that, it seems impossible. For instance, how to discriminate the position of two peaks which are closer from each other than their width ? It is possible if we expect a given shape for each peak, do the convolution of expectation with the resolution function of the apparatus and compare the result with the measurement (this is commonly called "data fitting"). Another example : neutron specular reflectivity allows us to gets structural informations on thin layers at an interface i.e. layers thicknesses and densities (for an introduction to reflectivity see for instance [1]). The latters are determined relative to the scattering length density difference ∆ρ between the two infinite media separated by the interface and consequently from the value of the critical transfer vector q c below which total reflexion occurs; e.g. for the air/silicon interface q c = 0.0102Å −1 yields to ∆ρ = q 2 c /16π = 2.07 × 10 10 cm −2 , which is the correct result. A shift of 3% for q c = 1.03 × 0.0102Å −1 gives ∆ρ = 2.20 × 10 10 cm −2 , which is not acceptable for many users of reflectometers. But 3% is the order of magnitude of the resolution. This barrier can be bypass if we know that the reflectivity curve should obey to a given function. To reach these limits, the calculation of the resolution function has to be as accurate as possible. In particular the approximation that all random variables that contribute to the resolution have a gaussian probability density is likely not satisfactory [2] especially in case the resolution is relaxed to gain flux. This approximation, which was legitimate when the means of calculation were insufficient, is no longer justified. Neutrons diffraction, scattering or reflectivity measurements take a "picture" of a sample in the reciprocal space * Electronic address: [email protected] for which the conjugate variable of distance is the transfer vector q that is practically proportional to the ratio of the incident angle θ to the wavelength λ. Basically the resolution σ q is such as (σ q /q) 2 (σ θ /θ) 2 +(σ λ /λ) 2 (where σ holds for standard deviations). On the other hand, if we do not account for the transfer function of the sample, the signal is proportional to the incident neutron-flux i.e. to the product σ θ σ λ . Thus for a given flux, the minimum for σ q /q is obtained for σ θ /θ = σ λ /λ. Time-of-flight techniques do the work at constant θ (thus constant σ θ /θ) as a function of λ. If we require a constant flux, since σ θ /θ is constant, σ λ /λ should also be constant. This is achieved with double-disk choppers [3] and this argument is the major reason of their wide spread. The counterpart is a broad resolution σ λ at large wavelength, precisely in the region where a good accuracy is often needed (edge of the total reflexion plateau). Hence the interest of making an exact calculation of the resolution function. In this paper we present the calculation of the overall and exact resolution function for the reflectometer HERMÈS at Laboratoire Léon Brillouin. In spite its current neutron source is continuous (Orphée reactor), this spectrometer is based on the time-of-flight principle that is to become generalized with the increase of pulsed neutron sources, it is equipped with a double-disk chopper that is now a standard. So, the step-by-step calculation developed here has a broad scope. For the most part, the different contributions to the final resolution have already been mentioned separately in the literature : wavelength distribution of the incident beam, tilt of the chopper axis, collimation and gravity. But, here they are all presented in a uniform and comprehensive manner that makes the procedure likely easier to implement and adapt in each specific case and even to extend for other techniques such as small-angle scattering or diffraction. We write analytically the whole resolution function that accounts for all these contributions without any approximation of gaussian distributions or independence of the different contributions. We show that data fitting using this exact resolution function allows us to reach much more accurate results than its usual approximation by a gaussian profile. Also, a python code that implements these cal- culations is provided. II. BRIEF DESCRIPTION OF THE REFLECTOMETER In Fig.II, a schematic diagram of the reflectometer is shown. Neutron-pulses are produced by a three-disk chopper of same characteristics from Airbus company (numbered 1, 2 and 3 with respect to neutrons trajectory). In the standard configuration these disks rotate in the same direction at a pulsation ω around the same axis. Also, they have a fixed angular aperture of same width ϕ a = 165 • allowing neutrons to pass. Disk 2 and 3 are in a fixed position 2 m from each other, whereas disk 1 can be placed at three different distances from disk 2 (0.1, 0.35 and 1 m, respectively). Essentially, disk 1 and 2 control the wavelength resolution whereas disk 3 is mainly devoted to avoid the time-overlap of the slowest neutrons of a given pulse with the fastest of the next one. The collimator is basically made of two horizontal slits of half-width r 1 and r 2 (numbered 1 and 2 with respect to neutrons trajectory) located between disk 2 and 3, at the same height and spaced by d c = 1.8 m. The width of the slits are tuned so that the angular resolution remains consistent with the wavelength resolution resulting from the disk chopper parameters. In the following, we will consider three different typical configurations summarized in Table I that correspond to high (HR), medium (MR) and low resolution (LR), respectively. Specular reflexion at the desired angle θ 0 is obtained by rotation of the sample. In case a non-horizontal beam is needed (e.g. for the study of horizontal liquid surfaces), two plane mirrors are placed in the collimator to deviate the beam. These mirrors have no incidence on the resolution and will be ignored in the following. Specular reflexion is measured in a vertical plane at angle 2θ 0 with a detector whose wide aperture allows all reflected neutrons to be captured whatever their trajectory. I: Typical configurations corresponding to high (HR), medium (MR) and low resolution (LR). x(1), x(2), x(3) and x(4) are the positions of disk 1, 2, 3 and of the detector, respectively. 2r1 and 2r2 are the widths of the first and second slit of collimator at fixed distance dc = 1800 mm apart. All lengths are in mm HR MthatR LR x(4) − x(3) 2375 x(3) − x(2) 2000 x(2) − x(1) 100 350 1000 2r1 1 2 4.6 2r2 1 1 1 III. WAVELENGTH RESOLUTION The wavelength resolution results basically from the incident beam distribution and from the transfer function of the chopper. The latter is mainly controlled by the phases of the first two-disks (hence our title), but in a general way the third also should be accounted for at long wavelength. In this section we examine these different points and present the way to calibrate the phases in question. A. Resolution of a double-disk chopper We first consider a chopper made only of the first two disks. Let us denote x(i) the positions (as in Table I), ϕ o (i) the phase for disk opening and ϕ c (i) the phase for disk closure. By convention, we denote the actual phase as ωt − ϕ, so that ϕ > 0 states for a delay. The measurement consists in recording the number of neutron arrivals at time t 0 over a timebase that is periodically restarted (triggered) at each revolution of the chopper, so that to t 0 corresponds a phase of arrival ϕ(4) = ωt 0 . Fig.2 shows the corresponding time-of-flight diagram using ωt as abscissa. From the de Broglie's equation, in this diagram the kinematics curve of a neutron of wavelength λ and velocity v is a straight-line with the reverse slope ω v = ω h/m × λ(1) with h the Planck's constant, m the mass of neutron and h/m = 3956Å m/s. Simple geometrical considerations (see Fig.2) show that neutrons with phase of arrival ϕ(4) = ωt 0 have a velocity between v 1 and v 2 such as ω v 1 = ϕ(4) − ϕ c (1) x(4) − x(1) and ω v 2 = ϕ(4) − ϕ o (2) x(4) − x(2)(2) which are symbolized by the two red lines in Fig.2. The bisector of these two lines (red dashed line) has a reverse slope that writes ω v 0 = ϕ(4) − ϕc(1)+ϕo(2) 2 x(4) − x(1)+x(2) 2(3) The choice for the origine of phase just between opening of disk 2 and closure of disk 1 and the origin of distance between disk 1 and 2 is quite natural. With these coordinates one has ω v 0 = ϕ(4) x(4)(4) that defines for the double-disk chopper the nominal wavelength that is simply proportional to the time-offlight. Let us denote = ϕ o (2) − ϕ c (1) 2 and h = x(2) − x(1) 2(5) Then Eq.2 rewrites ω v 1 = ϕ(4) − x(4) + h and ω v 2 = ϕ(4) − x(4) − h(6) To these boundaries correspond the wavelengths λ 1 and λ 2 that allow us to define the half range of transmitted wavelength ∆λ and the median wavelength λ 0 as : ∆λ = λ 2 − λ 1 2 ; λ 0 = λ 2 + λ 1 2(7) From Eq.6, one gets : ∆λ = h/m ω × ϕ(4)h + x(4) x(4) 2 − h 2(8) and λ 0 = h/m ω × ϕ(4)x(4) + h x(4) 2 − h 2 (9) Note that in the general case ( = 0) the median λ 0 is not simply proportional to the time-of-flight. The relative resolution of the chopper is ∆λ λ 0 = ϕ(4)h + x(4) ϕ(4)x(4) + h(10) In the case = 0, one obtains : ∆λ λ 0 = h x(4)(11) so that the relative resolution of the chopper is constant whatever the time of arrival t 0 . This is the main reason for using double-disk chopper as it allows us to optimize the resolution with respect to neutron-flux [3]. From Table I, the three standard configurations correspond to ∆λ/λ 0 = 1, 4 and 10%, respectively. B. Three-disk chopper and time-of-flight channel The general case of a three-disk chopper is a bit more complicated because for long time-of-flight the third disk comes into play. This implies a modification of Eq.2. In standard configurations ϕ c (1) ϕ o (2) ϕ o (3), so that the velocity v 1 of fastest neutrons that reach the detector at phase ϕ(4) is limited either by the closure of disk 1 or disk 3 (see Fig.3) and the velocity v 2 of slowest neutrons is limited by the opening of disk 2 or disk 3. This can be formalized in the general case as : ω v 1 = max i∈{1,2,3} ϕ(4) − ϕ c (i) x(4) − x(i) ; ω v 2 = min i∈{1,2,3} ϕ(4) − ϕ o (i) x(4) − x(i) (12) The last point is that neutron counters record the number of neutron-arrivals in the time interval between t 0 − ∆t and t 0 + ∆t referred to as a "time-of-flight channel". The phases for the opening and closure of this channel are ϕ o (4) = ω(t 0 − ∆t) and ϕ c (4) = ω(t 0 + ∆t) and Eq.12 rewrites : Fig.4, the relative half range of transmitted wavelengths calculated from these two boundaries is plotted for 256 time-of-flight channels covering 360 • of a chopper revolution for the three typical configurations of Table I. The curves show a plateau corresponding to the "doubledisk" chopper regime described by Eq.11. At large wavelength, the decreasing of the resolution is due to the third disk that comes into play, whereas at short wavelength departure from the plateau is due to the channel width. (1)) delimits two regimes for the velocity v1 of fastest neutrons that is limited either by the closure of disk 1 (red), or by the one of disk 3 (green). The same occurs for slow neutrons of velocity v2 (case not shown). Table I with ϕc(1) = ϕo(2) = ϕo (3). The top x-axis is the median wavelength λ0 for ω = 30 Hz. The dashed lines mark the separation ωtc of the two regimes delimited by the dashed line in Fig.3. ω v 1 = max i∈{1,2,3} ϕ o (4) − ϕ c (i) x(4) − x(i) ; ω v 2 = min i∈{1,2,3} ϕ c (4) − ϕ o (i) x(4) − x(i) (13) In(x(3) − x(1))/(ϕc(3) − ϕc(1)) = (x(4) − x(1))/(ωtc − ϕc C. Phase calibration and tilt of the chopper axis The time resolved neutron counting is triggered at each revolution by the chopper electronics. On this subject, the only unknown parameter is the phase shift φ trig of this trigger with respect to the physical origin of phases. This point is resolved by measuring the transmission of a cristalline material (e.g. a graphite crystal) that displays a characteristic attenuation at wavelength λ * [4]. No matter the value of λ * , the corresponding time-of-flight t * is constant and the related phase writes φ * = ωt * + φ trig . Measuring φ * as a function of ω and extrapolating to ω = 0 gives φ trig . Taking the rotation of disk 1 as a reference, the phase shift of disk 2 and 3 are chosen by the experimentalist and kept constant by the electronics with a control loop feedback, which ensures that no variation or drift occurs during measurements. However, possible differences between phases setpoints and their actual values have to be measured. Also, a vertical tilt or misalignment of the centers of the three disks with the spectrometer axis affects the kinematics line of neutrons allowed to pass through the chopper in the same manner as a phase difference. Hence the need of disk-phase calibration. The phase of disk 2 affects the short-wavelength cutoff (λ 1 ) and the chopper resolution in the first regime of Fig.4, whereas a the phase of disk 3 affects the largewavelength cutoff (λ 2 ) in the second regime and the timeof-flight channel at which this regime begins. Let us first consider the former. From Fig.2, one can see that the fastest neutrons passing through the chopper are such that ω/(h/m)×λ min = ω/v max = /h. In Fig.5, the measured λ min is plotted as a function of ϕ o (2) − ϕ c (1) = 2 , the linear behavior of the cut-off does not passes through the origin : a constant δ (here equal to 1 • for the three configurations) should be added to ϕ o (2) − ϕ c (1) in order to obtain the correct value for λ min . In practice, the phase of disk 2 in Eq.13 have to be increased by δ. Note that the method here reported to measure ϕ o (2) is equivalent to measuring the intensity of a monochromatic beam as a function of the phase ϕ o (2) − ϕ c (1) [5], but is probably simpler to perform routinely because no monochromator is needed. In the same way, the large-wavelength cut-off of the chopper transmission allows us to determine the actual phase of disk 3. The phase ωt c of the cut-off (see Fig.3), is such as : ωt c − ϕ c (1) (x(4) − x(1))/(x(3) − x(1)) = ϕ c (3)(14) From the measurement of ωt c the actual value of ϕ c (3) can be deduced (Fig.6). We found that a constant value η = −3.5 • has to be added to the phase of disk 3. D. Incident beam and detector efficiency For a white beam of uniform distribution, the probability density D(λ) of wavelength for neutrons passing through the chopper with phase of arrival ωt 0 is a rectangular function that writes : D(λ) = 0 , if λ ∈ [λ 1 , λ 2 ] 1/(λ 2 − λ 1 ) , if λ ∈ [λ 1 , λ 2 ](15) where λ 1 and λ 2 are the wavelength boundaries of the time-of-flight channel given by Eq.13. Thermalized neutrons have a Maxwell-Boltzmann distribution of velocity that is altered by neutron guides. Also, detectors efficiency exponentially decays with wavelength. Let us denote H B (λ) the effective wavelength distribution that is the product of the actual distribution by the detector efficiency. H B (λ) is the only observable variable. For a given time-of-flight channel, chopping a neutron beam amounts to applying the rectangular function D(λ) of Eq.15 to H B (λ) [2] (see Fig.7). The probability density of wavelength for neutrons recorded in this channel is Table I. of Table I. The strong asymmetry of H(λ) results in an mean wavelengthλ = λH(λ) dλ significantly different from the median λ 0 . We plotted in Fig.9 the ratioλ/λ 0 as a function of the median λ 0 for the three different chopper configurations of Table I. The wider the resolution, the moreλ deviates from λ 0 . H(λ) = H B (λ)D(λ)(16) IV. ANGLULAR RESOLUTION In this section, we focus on beam collimation and gravity, which both contribute to the final distribution of incidence angle of neutrons on the sample. A. Beam divergence Let us consider the divergence in the vertical plane of a neutron-beam collimated with two horizontal slits of halfwidth r 1 and r 2 , respectively, spaced by d c . The distribution function of the angle α of neutron trajectory results from the convolution of two rectangular functions centered on 0 and half-width r 1 /d c and r 2 /d c , respectively. This is a trapezoidal function that writes for r 1 > r 2 : P (α) =          d c 2r 1 for \| α \|< α 1 d c 2r 1 1 − d c 2r 2 (\| α \| −α 1 ) for α 1 <\| α \|< α 2 0 for α 2 <\| α \| with α 1 = r 1 − r 2 d c and α 2 = r 1 + r 2 d c (17) As convolution is commuall thetative, the two rectangular functions are fully interchangeable as far as only the beam divergence is concerned. So that, in case r 1 < r 2 Table I. their values can be exchanged in order Eq.17 to be applied. However for some reasons related to the beam size (the "footprint" 2r 2 sin(θ) has to be smaller than the sample and also r 2 restrains the accuracy of the alignment of the sample), r 1 > r 2 is preferred and r 2 is kept constant. In Fig.10 typical curves for P (α) are plotted for standard collimations of Table I. B. Deviation due to gravity Due to gravity, neutrons deviate from a straight line and horizontal trajectory. This deflection increases with the time-of-flight (i.e. with the wavelength) and must be taken into account for specular reflectivity measured in the vertical plane [6]. Let us assume that the detector and the collimator slits are properly aligned to the neutron guide using a criterion of maximum neutron flux measured by integrating over the whole wavelengthdistribution. As neutrons of short wavelength are majority (see Fig.7) and neglecting their deviation due to gravity, the line joining the two slits of the collimator is almost horizontal and the slits are at the same height. The apex of parabolic trajectories of all neutrons is thus at the middle between the two slits of the collimator. Let us denote v x = λ/(h/m) and v z the horizontal and vertical components of neutrons velocity, respectively. At the apex, v z = 0. At any point beyond the apex v z < 0 and the deflection angle is tan −1 (v z /v x ) v z /v x . If d 1 is the flight distance from the middle of the collimator to the sample, the time-of-flight is d 1 /v x , thus v z = −gd 1 /v x , with g the gravitational acceleration. The deviation angle (which is always negative) on the sample is: γ = −gd 1 v 2 x = −(cλ) 2 with c = (gd 1 ) 1/2 h/m(18) The distribution J(γ) of the deviation is related to the distribution of wavelength H(λ) via the general relation J(γ) = 1 2c(−γ) 1/2 H((−γ) 1/2 /c)(19) Basically, J(γ) has a shape comparable to H(λ) (i.e. a rectangular function applied to the wavelength distribution of the incident beam). Let us denote [γ 1 , γ 2 ] the support interval of J(γ) and ∆γ = (γ 2 − γ 1 )/2. From Eq.18, one obtains ∆γ = 2(cλ 0 ) 2 × (∆λ/λ 0 ). The quantity c is typically of the order of 8×10 −4Å −1 for d 1 = 1 m, thus in a standard configuration ∆γ is small compared to the beam divergence (support of P (α)). However, in case the relative angle resolution is very small compared to the wavelength resolution (e.g. binning of tof-channels to increase statistics [7], narrow collimation due to small size of the sample...) gravity should affect the width of the angular resolution. In any case, the main effect of gravity is due to the mean valueγ = γJ(γ) dγ that is non-zero and depends on the time-of-flight channel (i.e. on λ 0 ). In Fig.11, the mean deviationγ is plotted as a function of λ 0 . C. Distribution of incidence angle Let us denote θ 0 the nominal angle (i.e. the angle calculated from the inclination of the sample) of the neutron beam with respect to the interface under study. This angle is negative in case neutrons come from below the interface and positive if they come from over (this sign is imposed by gravity). The actual incidence angle θ of neutrons on the sample is θ = θ 0 + α − γ(20) The distribution for θ is thus given by Table I. G(θ) = P (θ − θ 0 + γ)J(γ) dγ(21) As J(γ) is much narrower than P (α) its shape affects only slightly the one of G(θ) but mainly induces an increase in the mean valueθ compared to the nominal value θ 0 . Note that as γ is negative and due to the sign in Eq.20, gravity increases \| θ \| for θ 0 > 0 but decreases \| θ \| for θ 0 < 0. In Fig.12, the angular resolution function G(θ) is plotted for θ 0 = 1 • for the different time-of-flight channels and the three typical configurations of the reflectometer. V. RESOLUTION OF TRANSFER VECTOR The physical parameter related to structural informations on the measured samples is the transfer vector q (the conjugate variable of distance) that is defined as q = 4π sin(θ)/λ(22) However, the only directly adjustable parameters of the spectrometer are θ and λ, both being distributed around θ 0 and λ 0 . For a given time-of-flight channel, the resolution function R(q) of the transfer vector should be thus a generalized convolution of the probability densities H(λ) (Fig.8) and G(θ) (Fig.12) which corresponding random variables are combined following Eq.22. However due to gravity, θ and λ are not strictly independent and "convoluting" directly their densities is not correct even if it is a good approximation (because J(γ) is narrow compared to P (α) in most cases). As moreover this approximation does not save time, we prefer an exact treatment that will remain valid even in case the width of J would become significant (see section IV B). For the sake of simplicity, let us first deal with the "small angle approximation" sin(θ) θ. Then, using Eq.20 the transfer vector rewrites : q = 4πθ λ = 4π θ 0 + α − γ λ(23) where λ and α are random variables with densities H(λ) and P (α), respectively, whereas γ = −(cλ) 2 . Let us denote M (q) the measurement of the physical quantity m(q). Quite generally, for one time-of-flight channel at q 0 = 4πθ 0 /λ 0 one can write M (q 0 ) = dλH(λ) dαP (α) m 4π θ 0 + α − γ λ (24) Using α = (λq/4π) − θ 0 − (cλ) 2 and dα = (λ/4π) dq, one gets: M (q 0 ) = dλH(λ) dq λ 4π P λq 4π − θ 0 − (cλ) 2 m(q) = dqm(q) dλH(λ) λ 4π P λq 4π − θ 0 − (cλ) 2 (25) By definition, the last integral is the resolution function of the transfer vector q : R(q) = dλH(λ) λ 4π P λq 4π − θ 0 − (cλ) 2(26) The same procedure can be used for the exact expression q = 4π sin(θ 0 + α − γ)/λ leading to α = sin −1 (λq/4π) − θ 0 − (cλ) 2 and dα = λ dq/ (4π) 2 − (λq) 2 . One obtains : R(q) = dλ H(λ)λP (sin −1 (λq/4π) − θ 0 − (cλ) 2 ) 4π 1 − (λq/4π) 2(27) Eq.27 can be numerically calculated as is (see python code https://bitbucket.org/LLBhermes/pytof/) from Eq.16 (using Eq.15 and 13) and 17. In Fig.13, the whole resolution R(q) is plotted for the three typical configurations as a function of the time-of-flight channel for θ 0 = 1 • . Firstly, notice the shift of the mean transfer vector valueq = qR(q) dq compared to q 0 . This shift results from the wavelength distribution of the incident beam (Fig.9) and from gravity (Fig.11). Both effects contribute to the same result for θ 0 > 0 but oppose if θ 0 < 0. To properly account for this effect, the simplest and more accurate way is to compute the exact resolution function. Secondly, notice that the profile of the exact resolution function differs from its gaussian approximation (see Fig.14). In any case, the exact resolution function has a compact support (i.e. R is non-zero inside a closed and bounded set of q-values), unlike gaussian curve that should be numerically cut beyond an arbitrary number of standard deviation. For the configuration allowing a wider resolution, the differences between exact resolution and gaussian approximation are much more important and even the modal-values differ. All these differences cannot reasonably be taken into account summarizing the resolution with a simple standard deviation. VI. APPLICATION Let us consider the measurement M i (performed with statistical errors bars σ i with i ∈ {1, 2, . . . N }) of the reflectivity of a sample over N time-of-flight channels. We denote m(q, p) the theoretical reflectivity that depends on n physical parameters that make the coordinates of the vector p. For a time-of-flight channel at q i of resolution function R i , the expected theoretical measurement M th,i is : M th,i (p) = m(q, p)R i (q) dq(28) Let us define the relative distance χ 2 per channel between M and M th as χ 2 (p) = 1 N N i=1 M i − M th,i (p) σ i 2(29) The parameter-vector p can be experimentally determined by minimizing χ 2 following a standard numerical optimization procedure (curve fitting). The correctness of the resolution function can thus be evaluated : 1) from the correctness of the so determined parameters values; and 2) from the correctness of the final matching between M and M th (low value for χ 2 and no correlation in the residual (M − M th )/σ). For such evaluation, sample-candidate should display a strong variation of reflectivity in the accessible q-range, in order to maximize the effect of convolution by the resolution function (Eq. 28), with a minimum number of unknown parameters (coordinates of p). From this point of view, the reflectivity near the total reflexion plateau of the interface between air and a smooth and pur solid is likely the most adequate. We have choosen an amorphous silica block with a polished surface, which reflectivity writes : m(q) = q − q 2 − q 2 c q + q 2 − q 2 c 2 × e −h 2 q √ q 2 −q 2 c(30) The first term of this product is the Fresnel's reflectivity of a perfectly flat surface, where q c = (16πρ) 1/2 is the edge of the total reflectivity plateau and ρ the scattering length density of amorphous silica. Whereas the second term accounts for the surface roughness of characteristic height h. Measurements were done in the three configurations of Table I for θ 0 = 1 • . Results and data fitting are plotted in Fig.15. For the three configurations, the value determined for q c are in very good agreement and also fully consistent with the density of amorphous silica (hereq c = 1.307 × 10 −2Å−1 leads to the density d = 2.15 g/cm 3 for amorphous silica). Also, the values for χ 2 at the optimum are very small. The increase of χ 2 with the broadening of resolution is simply due to the gain of flux resulting in smaller statistical error bars σ i in Eq.29. The use of the exact resolution function as plotted in Fig.13, should appear unnecessarily tricky compared to the use of a gaussian curve of same average and standard deviation. This is not the case for two reasons: 1) the calculation of the full resolution function is anyway needed to determine the average and standard deviation value; 2) once that is achieved, convolution by the full resolution or its gaussian approximation requires the same computing power. Also, convolution by the full resolution function will always produces better results in particular at low resolution. Best fits obtained using the full resolution or its gaussian approximation (computed over a supporthalf-width equal to 3 times the standard deviation) are compared in Fig.16-top for the low resolution configuration of Table I. Differences are subtle but are emphasized by plotting the residual (M − M th )/σ (Fig.16-bottom) that clearly shows correlations in the case of the gaussian approximation. By definition, using the exact resolution function is more accurate than using its gaussian approximation, but it also saves time. Computation of the exact resolution function is done only once during data reduction. The time spent is negligible compared to the time needed for data fitting that consists in many iterations of convolution of a theoretical model by the resolution. The exact resolution function has a compact support whereas its gaussian approximation has not. Thus for a given high percentile, the gaussian approximation needs a more extended sampling and thus is more time consuming during the data fitting stage than the exact function. This removes a lot of interest in the approximation. VII. CONCLUSION In this paper we present the calculation of the exact and comprehensive resolution function for a timeof-flight neutron reflectometer in a way that accounts for all contributions without any assumption of the gaussian distribution or independence of the corresponding variables. The step-by-step procedure matches with a fully documented Python module(https://bitbucket.org/ LLBhermes/pytof/) that can be easily used for numerical applications to any specific case, from the computation of the resolution to data-fitting. We have shown that in case the resolution is relaxed, the resulting resolution function departs strongly from a gaussian profile and that using the exact function provides much more accurate results. This point will be highly relevant with the emergence of compact and low-flux neutron sources (see e.g. [8]) which will likely require such relaxed resolutions. FIG. 1 : 1Schematic diagram of HERMÈS reflectometer. FIG. 2 : 2Double chopper: flight distance vs. phase ωt. In blue the closed sector of disks. Neutrons that reach the detector at phase ϕ(4) = ωt0 have a kinematics that lies between the two red lines which slopes are given by Eq.2. The dashed line in red is the bisector and states for the nominal wavelength. FIG. 3 : 3Three-disk chopper: compared to Fig.2, the 3rd disk comes into play for long time-of-flight. The dashed line in black with slope FIG. 4 : 4Relative half range of transmitted wavelengths ∆λ/λ0 versus phase ωt0 for 256 time-of-flight channels covering 360 • of data-acquisition for the three standard configurations of FIG. 5 : 5Minimum wavelength λmin as a function of ϕo(2) − ϕc(1), measured for the three different configurations of Table I; at ω = 30 Hz. Extrapolation to λmin = 0 gives the angle δ = 1 • that should added to the phase of disk 2. FIG. 6 : 6Value of the phase ϕc(3) deduced from Eq.14 and the measurement of the phase ωtc of the large wavelength cut-off for a setpoint ϕc(3) = 165 • . The difference with this setpoint gives η = −3.5 • that should be added to the phase of disk 3. FIG. 7 :FIG. 8 : 78Effective wavelength distribution HB(λ) (blue) of the incident beam provided by the reactor Orphée on the guide G6-2 and wavelength distributions H(λ) (not normalized) for one time-of-flight channels centered on λ0 = 15Å (Wavelength distributions H(λ) (Eq.16) vs. λ/λ0 (xaxis) and vs. time-of-flight channel centered on the median wavelength λ0 (y-axis). Calculated for the three configurations ofTable Iwith ∆t = 0, = 0.H B (λ) can be measured using a single-disk chopper with a small ∆λ (i.e. much smaller than the threedisk chopper) independent of the time-of-flight channel. For further calculations, H B (λ) can then be properly parametrized using an ad hoc function. InFig.8, the wavelength distribution H is plotted for the different time-of-flight channels and for the three configurations FIG. 9 : 9Ratio of the mean wavelengthλ to the median wavelength λ0 vs. λ0 for the three configurations of FIG. 10 : 10Typical angular distribution P (α) due to beam divergence for the three standard collimations of FIG. 11 : 11Mean deviationγ of neutrons due to gravity (for d1 = 1 m) vs. λ0 for the three configurations ofTable.I. J(γ) = H(λ)× \| dλ/ dγ \|, with λ = (−γ) 1/2 /c (the inverse relation of Eq.18). One obtains : FIG. 12 : 12Angular resolution G(θ) vs. θ/θ0 (x-axis, where θ0 = 1 • is the nominal angle) and vs. time-of-flight channel centered on the median wavelength λ0 (y-axis), for the three configurations of FIG. 14 : 14. 13: Total resolution R(q) of transfer vector q vs. q/q0 (x-axis, where q0 = 4π sin(θ0)/λ0, θ0 = 1 • is the nominal angle) and vs. time-of-flight channel centered on the median wavelength λ0 (y-axis), for the configurations ofTable I. Exact resolution R(q) (full line) vs. transfer vector q for θ0 = 1 • and λ0 = 15Å (crosssection of 3D plot ofFig.13), for the configurations ofTable I. Dotted lines are gaussian curves of same mean and standard deviation. FIG. 15 : 15Reflectivity for an amorphous silica block in the region of the edge of the total reflection. Top : measurement M , best fit M th (Eq. 28) and model m(q) (Eq. 30) vs. 1/q (because channels are almost regularly spaced in wavelength). Bottom: same data with y-coordinates multiplied by q 4 . For these best fits qc = 1.306, 1.314, 1.300 × 10 −2Å−1 and χ 2 = 0.59, 1.8 and 2.4 for the high-resolution (HR), mediumresolution (MR) and low-resolution (LR), respectively. FIG. 16 : 16Reflectivity for an amorphous silica block in the region of the edge of the total reflection plateau. Top : measurement M and best fits M th using the full-resolution or its gaussian approximation (computed over a support-half-width equal to 3 times the standard deviation). Bottom : residual (M − M th )/σ. TABLE . F Cousin, A Chennevière, 10.1051/epjconf/201818804001EPJ Web of Conferences. 1884001F. Cousin and A. Chennevière, EPJ Web of Confer- ences 188, 04001 (2018), URL https://doi.org/10. 1051/epjconf/201818804001. . A R J Nelson, 10.1107/S0021889813021936Journal of Applied Crystallography. 461338A. R. J. Nelson, Journal of Applied Crystallogra- phy 46, 1338 (2013), URL https://doi.org/10.1107/ S0021889813021936. . A A Van Well, 10.1016/0921-4526(92)90521-SPhysica B: Condensed Matter. 1809290521A. A. van Well, Physica B: Condensed Matter 180- 181, 959 (1992), URL https://doi.org/10.1016/ 0921-4526(92)90521-S. . E Fermi, W J Sturm, R G Sachs, 10.1103/PhysRev.71.589Phys. Rev. 71E. Fermi, W. J. Sturm, and R. G. Sachs, Phys. Rev. 71, 589 (1947), URL https://doi.org/10.1103/PhysRev. 71.589. . P Gutfreund, T Saerbeck, M A Gonzalez, E Pellegrini, M Laver, C Dewhurst, R Cubitt, 10.1107/S160057671800448XJournal of Applied Crystallography. 51P. Gutfreund, T. Saerbeck, M. A. Gonzalez, E. Pellegrini, M. Laver, C. Dewhurst, and R. Cubitt, Journal of Applied Crystallography 51, 606 (2018), URL https://doi.org/ 10.1107/S160057671800448X. . I Bodnarchuk, S Manoshin, S Yaradaikin, V Kazimirov, V Bodnarchuk, 10.1016/j.nima.2010.12.074Nuclear Instr. Meth. Phys. Res. A. 631121I. Bodnarchuk, S. Manoshin, S. Yaradaikin, V. Kazimirov, and V. Bodnarchuk, Nuclear Instr. Meth. Phys. Res. A 631, 121 (2011), URL https://doi.org/10.1016/j. nima.2010.12.074. . R Cubitt, T Saerbeck, R A Campbell, R Barker, P Gutfreund, 10.1107/S1600576715019500Journal of Applied Crystallography. 48R. Cubitt, T. Saerbeck, R. A. Campbell, R. Barker, and P. Gutfreund, Journal of Applied Crystallogra- phy 48, 2006 (2015), URL https://doi.org/10.1107/ S1600576715019500. . F Ott, CEA Paris SaclayTechnical ReportF. Ott, Technical Report, CEA Paris Saclay (2018), URL https://hal-cea.archives-ouvertes.fr/ cea-01873010.	[]
[ "Conceptual Inadequacy of the Shannon Information in Quantum Measurementš", "Conceptual Inadequacy of the Shannon Information in Quantum Measurementš" ]	[ "Caslav Brukner \nInstitute for Experimentalphysics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria\n", "Anton Zeilinger \nInstitute for Experimentalphysics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria\n" ]	[ "Institute for Experimentalphysics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria", "Institute for Experimentalphysics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria" ]	[]	In a classical measurement the Shannon information is a natural measure of our ignorance about properties of a system. There, observation removes that ignorance in revealing properties of the system which can be considered to preexist prior to and independent of observation. Because of the completely different root of a quantum measurement as compared to a classical measurement conceptual difficulties arise when we try to define the information gain in a quantum measurement using the notion of Shannon information. The reason is that, in contrast to classical measurement, quantum measurement, with very few exceptions, cannot be claimed to reveal a property of the individual quantum system existing before the measurement is performed. PACS number(s): 03.65.-w, 03.65.Bz, 03.67.-a	10.1103/physreva.63.022113	[ "https://export.arxiv.org/pdf/quant-ph/0006087v3.pdf" ]	119,381,924	quant-ph/0006087	7431030e9b3cdb90fbfb4643e2013c1b08ca8808	Conceptual Inadequacy of the Shannon Information in Quantum Measurementš Caslav Brukner Institute for Experimentalphysics University of Vienna Boltzmanngasse 5A-1090ViennaAustria Anton Zeilinger Institute for Experimentalphysics University of Vienna Boltzmanngasse 5A-1090ViennaAustria Conceptual Inadequacy of the Shannon Information in Quantum Measurementš (March 31, 2022)arXiv:quant-ph/0006087v3 5 Apr 2002 In a classical measurement the Shannon information is a natural measure of our ignorance about properties of a system. There, observation removes that ignorance in revealing properties of the system which can be considered to preexist prior to and independent of observation. Because of the completely different root of a quantum measurement as compared to a classical measurement conceptual difficulties arise when we try to define the information gain in a quantum measurement using the notion of Shannon information. The reason is that, in contrast to classical measurement, quantum measurement, with very few exceptions, cannot be claimed to reveal a property of the individual quantum system existing before the measurement is performed. PACS number(s): 03.65.-w, 03.65.Bz, 03.67.-a In a classical measurement the Shannon information is a natural measure of our ignorance about properties of a system. There, observation removes that ignorance in revealing properties of the system which can be considered to preexist prior to and independent of observation. Because of the completely different root of a quantum measurement as compared to a classical measurement conceptual difficulties arise when we try to define the information gain in a quantum measurement using the notion of Shannon information. The reason is that, in contrast to classical measurement, quantum measurement, with very few exceptions, cannot be claimed to reveal a property of the individual quantum system existing before the measurement is performed. In classical physics information is represented as a binary sequence, i.e a sequence of bit values, each of which can be either 1 or 0. When we read out information that is carried by a classical system we reveal a certain bit value that exists even before the reading of information is performed. For example, when we read out a bit value encoded as a pit on a compact disk, we reveal a property of the disk existing before the reading process. This means that in a classical measurement the particular sequence of bit values obtained can be considered to be physically defined by the properties of the classical system measured * . The information read is then measured by the Shannon measure of information [1] which can operationally be defined as the number of binary questions (questions with "yes" or "no" answers only) needed to determine the actual sequence of 0's and 1's. In quantum physics information is represented by a sequence of qubits, each of which is defined in a twodimensional Hilbert space. If we read out the information carried by the qubit, we have to project the state of the qubit onto the measurement basis {\|0 , \|1 } which will give us a bit value of either 0 or 1. Only in the exceptional case of the qubit in an eigenstate of the measurement apparatus the bit value observed reveals a property already carried by the qubit. Yet in general the value obtained by the measurement has an element of irreducible randomness and therefore cannot be assumed to reveal the bit value or even a hidden property of the system existing before the measurement is performed. This implies that in a sequence of measurements on qubits in a superposition state a\|0 +b\|1 (\|a\|, \|b\| = {0, 1}) the particular sequence of bit values 0 and 1 obtained cannot, not even in principle † , be considered in any way to be defined before the measurements are performed. The non-existence of well-defined bit values prior to and independent of observation suggests that the Shannon measure, as defined by the number of binary questions needed to determine the particular observed sequence 0's and 1's, becomes problematic and even untenable in defining our uncertainty as given before the measurements are performed. Here we will critically analyze the applicability of the axiomatic derivation of the Shannon measure for the case of quantum measurement. We will also show that Shannon information is not useful in defining the informa- † As theorems like those of Kochen-Specker [2] show, it is fundamentally not possible to assign to a quantum system (noncontextual) properties corresponding to all possible measurements. The theorems assert that for a quantum system described in a Hilbert space of dimension equal to or larger than three, it is possible to find a set of n projection operators which represent the yes-no questions about an individual system, such that none of the 2 n possible sets of answers is compatible with the sum rule of quantum mechanics for orthogonal decomposition of identity [3] (i.e. if the sum of a subset of mutually commuting projection operators is the identity one and only one of the corresponding answers ought to be "yes"). This means that it is not possible to assign a definite unique answer to every single yes-no question represented by a projection operator independent of which subset of mutually commuting projection operators one might consider it with together. If there are no definite (context-independent) answers to all possible yes-no questions that can be asked about the system then the operational concept of the Shannon measure of information itself, defined as the number of yes-no questions needed to determine the particular answers the system gives, becomes highly problematic. tion content in a quantum system. In fact we will see that when we try to apply Shannon's postulate in quantum measurements or when we try to define the information content by the Shannon information a certain element emerges that escapes complete and full description in quantum mechanics. This element is always associated with the objective randomness of individual quantum events and with quantum complementarity. In the end we will briefly discuss a novel and more suitable measure of information [4]. Yet at first we will return to a discussion in more detail of the operational definition of Shannon information to quantum measurements. II. DISCUSSION OF THE OPERATIONAL DEFINITION FOR A SEQUENCE OF MEASUREMENTS For classical observations Shannon's measure of information can conceptually be motivated through an operational approach to the question. We will follow the introduction of Shannon's measure of information as given by Uffink [6]. Consider an urn filled with N colored balls. There are n 1 , n 2 , ..., n m balls with various different colors: black, white, ..., red. Now the urn is shaken, and we draw one after the other all balls from the urn. To what extent can we predict the particular color sequence drawn? Certainly, if all the balls in the urn are of the same color, we can completely predict the color sequence. On the other hand, if the various colors are present in equal proportions and if we have no knowledge about the arrangement of the balls after shaking the urn, we are maximally uncertain about the color sequence drawn. As noticed in [6] one can think of these situations as extreme cases on a varying scale of predictability. This suggest that the uncertainty we have before drawing about the particular color sequence that will be drawn is defined by the total number of different possible color sequences that are in accordance with the given number of balls with their respective colors in the urn. Consider now a situation where a long sequence of N balls are drawn from an infinite "sea" of balls with proportions p 1 , p 2 , ..., p m for the different colours in the sea. Then a long sequence contains with high probability about p 1 N balls of the first colour, p 2 N balls of the second colour etc. (such a sequence is called typical sequence). The probability to obtain a particular typical sequence (particular colour sequence) is given by [1] p(sequence) = p p1N 1 p p2N 2 ...p pmN m = 1 2 N H(1) where H = − m i=1 p i log p i(2) is the Shannon information expressed in bits with the logarithm taken to base 2. Consequently, the total number of distinct typical sequences is given by W ≃ 2 N H .(3) Suppose now that one wishes to identify a specific color sequence of the drawn balls from the complete set of possible color sequences by asking questions to which only "yes" or "no" can be given as an answer. Of course, the number of questions needed will depend on the questioning strategy adopted. In order to make this strategy the most optimal, that is, in order that we can expect to gain maximal information from each yes-or-no question, we evidently have to ask questions whose answers will strike out always half of the possibilities. Since there are W = 2 N H possible different (typical) color sequences (all of them have equal probability to be drawn), the minimal number of yes-no questions needed is just N H. Or equivalently, the Shannon information expressed in bits is the minimal number of yes-no questions necessary to determine which particular sequence of outcomes occurs, divided by N [5][6][7]. A particular color sequence is specified by writing down, in order, the yes's and no's encountered in traveling from the root to the specific leaf of the tree as schematically depicted in Fig. 1 for an explicit example with an urn containing black and white balls only. If instead of balls with pre-assigned colors we consider quantum systems whose individual properties are not defined before the measurements are performed, does the Shannon measure of information still define the information gain in the measurements appropriately? More precisely, we ask here the question whether the total number W = 2 N H of different possible (typical) sequences of outcomes is suitable as a measure of our uncertainty before the sequence of quantum measurements is performed. In classical physics the behavior of the whole ensemble follows from the behavior of its intrinsic different individual constituents which can be thought of as being defined to any precision. This is not the case in quantum mechanics. The principal indefiniteness, in the sense of fundamental nonexistence of a detailed description of and prediction for the individual quantum event resulting in the particular measurement result, implies that the particular sequence of outcomes specified by writing down, in order, the yes's and no's encountered in a row of yes/no questions asked is not defined before the measurements are performed. No definite outcomes exists before measurements are performed and therefore the number of different possible sequence of outcomes does not characterize our uncertainty about the individual system given before measurements are performed. However, once the sequence of quantum measurements is performed and the measurement results are obtained, the measure of information needed to specify the particular sequence of outcomes realized is defined appropriately by the Shannon measure. In the sense that an individual quantum event manifests itself only in the measurement process and is not precisely defined before measurement is performed, we may speak of "generation" of that specific information in the measurement. III. INAPPLICABILITY OF SHANNON'S POSTULATES IN QUANTUM MEASUREMENTS As observed by Uffink [6], an important reason for preferring the Shannon measure of information lies in the fact that it is uniquely characterized by Shannon's intuitively reasonable postulates. This has been expressed strongly by Jaynes [8] : "One ... important reason for preferring the Shannon measure is that it is the only one that satisfies ... [Shannon's postulates]. Therefore one expects that any deduction made from other information measures, if carried far enough, will eventually lead to contradiction." A good way to continue our discussion is by reviewing how Shannon, using his postulates, arrived at his famous expression. He writes [1]: "Suppose we have a set of possible events whose probabilities of occurrence are p 1 , p 2 , ..., p n . These probabilities are known but that is all we know concerning which event will occur. Can we find a measure of how much "choice" is involved in the selection of the event or how uncertain we are of the outcome? If there is such a measure, say H(p 1 , p 2 , ..., p n ), it is reasonable to require of it the following properties: 1 = 1 2 , p 2 = 1 3 , p 3 = 1 6 . On the right we first choose between two possibilities each with probability 1 2 , and if the second occurs make another choice with probabilities 2 3 , 1 3 . The final results have the same probabilities as before. We require, in this special case, that H 1 2 , 1 3 , 1 6 = H 1 2 , 1 2 + 1 2 H 2 3 , 1 3 . The coefficient 1 2 is the weighing factor introduced because this second choice occurs half the time." Shannon then shows that only the function (2) satisfies all three postulates. It is the third postulate which determines the logarithm form of the function and, as we will argue, it is this postulate which leads to problems when quantum measurements are involved. We now turn to the discussion of Shannon's postulates. While the first two postulates are natural for every meaningful measure of information, the last postulate might deserve more justification The third Shannon postulate originally formulated as an example was reformulated as an exact rule by Faddeev [9,6]: For every n ≥ 2 H(p 1 , .., p n−1 , q 1 , q 2 ) = H(p 1 , .., p n−1 , p n )+p n H q 1 p n , q 2 p n ,(4) where p n = q 1 + q 2 . Without physical interpretation the recursion postulate (the name was suggested in [6]) (4) is merely a mathematical expression which is certainly necessary for the uniqueness of the function (2) but has no further physical significance. We adopt the following well-known interpretation [6,10]. Assume the possible outcomes of the experiment to be a 1 , ..., a n and H(p 1 , ..., p n ) to represent the amount of information that is gained by the performance of the experiment. Now, decompose event a n into two distinct events a n ∧b 1 and a n ∧b 2 ("∧" denotes "and", thus a ∧ b denotes a joint event). Denote the probabilities of outcomes a n ∧ b 1 and a n ∧ b 2 by q 1 and q 2 , respectively. Then the left-hand side H(p 1 , ..., p n−1 , q 1 , q 2 ) of Eq. (4) represents the amount of information that is gained by the performance of the experiment with outcomes a 1 , ..., a n−1 , a n ∧ b 1 , a n ∧ b 2 . When the outcome a n occurs, the conditional probabilities for b 1 and b 2 are q1 pn and q2 pn respectively and the amount of information gained by the performance of the conditional experiment is H q1 pn , q2 pn . Hence the recursion requirement states that the information gained in the experiment with outcomes a 1 , ..., a n−1 , a n ∧b 1 , a n ∧b 2 equals the sum of the information gained in the experiment with outcomes a 1 , ..., a n and the information gained in the conditional experiment with outcomes b 1 or b 2 , given that the outcome a n occurred with probability p n . This interpretation implies that the third postulate can be rewritten as H (p(a 1 ), ..., p(a n−1 ), p(a n ∧ b 1 ), p(a n ∧ b 2 )) ( = H(p(a 1 ), ..., p(a n−1 ), p(a n )) + p(a n )H( p(b 1 \|a n ), p(b 2 \|a n )) where p(a n ) = p(a n ∧ b 1 ) + p(a n ∧ b 2 ) p(a n ∧ b 1 ) = p(a n )p(b 1 \|a n ) and (6) p(a n ∧ b 2 ) = p(a n )p(b 2 \|a n ). Here p(b i \|a n ) i = 1, 2 denotes the conditional probability for outcome a n given the outcome b i occurred and p(a n ∧ b i ) denotes the joint probability that outcome a n ∧ b i occurs. If we analyze the generalized situation with n outcomes a i of the first experiment A, m outcomes b j of the conditional experiment B and mn outcomes a i ∧ b j of the joint experiment A∧B, we may then rewrite the recursion postulate in a short form as H(A ∧ B) = H(A) + H(B\|A)(7) where H(B\|A) = n j p(a j )H(b 1 \|a j , ..., b m \|a j ) is the average information gained by observation B given that the conditional outcome a j occurred weighted by probability p(a j ) for a j to occur. It is essential to note that the recursion postulate is inevitably related to the manner in which we gain information in a classical measurement. In fact, in classical measurements it is always possible to assign to a system simultaneously attributes corresponding to all possible measurements, here a i , b j and a i ∧ b j . Also, the interaction between measuring apparatus and classical system can be thought to be made arbitrarily small so that the experimental determination of A has no influence on our possibility to predict the outcomes of the possible future experiment B. In conclusion, the information expected in a classical experiment from the joint experiment A ∧ B is simply the sum of the information expected from the first experiment A and the conditional information of the second experiment B with respect to the first, as expressed in Eq. (7). Therefore, only for the special case of commuting, i.e. simultaneously definite observables, the axiomatic derivation of the Shannon measure of information is applicable and the use of the Shannon information is justified to define the uncertainty given before quantum measurements are performed. However, in general, if A and B are noncommuting observables, the joint probabilities on the left-hand side of Eq. (5) cannot in principle be assigned to a system simultaneously, and consequently Shannon's crucial third postulate which is necessary for the uniqueness of Shannon's measure of information ceases to be well-defined. Having seen that the third Shannon postulate in general is not applicable in quantum measurements we next introduce two requirements that are immediate consequences of Shannon's postulates and in which all the probabilities that appear are well-defined in quantum mechanics. We will show that the two requirements are violated by the information gained in quantum measurements implying that the Shannon measure loses its preferential status with respect to alternative expressions when applied to define information gain in quantum measurements. 1. Every new observation reduces our ignorance and increases our knowledge. In his work Shannon [1] offers a list of properties to substantiate that H is a reasonable measure of information. He writes: "It is easily shown that H(A ∧ B) ≤ H(A) + H(B) with equality only if the events are independent (i.e., p(a i ∧ b j ) = p(a i )p(b j )). The uncertainty of a joint event is less than or equal to the sum of the individual uncertainties". He continues further in the text: "... we have H(A) + H(B) ≥ H(A ∧ B) = H(A) + H(B\|A). Hence, H(B) ≥ H(B\|A).(8) The uncertainty of B is never increased by knowledge of A. It will be decreased unless A and B are independent events, in which case it is not changed" (we have changed Shannon's notation to coincide with that of our work). 2. Information is indifferent on the order of acquisition. The total amount of information gained in successive measurements is independent of the order in which it is acquired, so that the amount of information gained by the observation of A followed by the observation of B is equivalent to the amount of information gained from the observation of B followed by the observation of A H(A) + H(B\|A) = H(B) + H(A\|B).(9) This is an immediate consequence of the recursive postulate which can be obtained when we write the recursion postulate in two different ways depending on whether the observation of A is followed by the observation of B or vice versa. An explicit example for a sequence of classical measurements is given in Fig. 3. Are these two requirements satisfied by information gained in quantum measurements? Consider a beam of randomly polarized photons. Filters F , F 45 • and F ↔ are oriented vertically, at +45 • , and horizontally respectively, and can be placed so as to intersect the beam of photons (Fig. 4). If we insert filter F the number of photons observed at the detection plate will be approximately half of the number in the incoming beam. The outgoing photons now all have vertical polarization. Notice that the function of filter F cannot be explained as a "sieve" that only lets those photons pass that are already of vertical polarization in the incoming beam. If that were the case, only a certain small number of the randomly polarized incoming photons would have vertical polarization, so we would expect a much larger attenuation of the beam of photons as they pass the filter. Denote with A and B properties of the photon to have polarization at +45 • and horizontal polarization, respectively. If F ↔ is inserted behind the filter F we are certain that none of the photons will pass through (Fig. 4a). For a photon with vertical polarization we have complete knowledge of the property B, i.e. H(B) = 0. Notice that a "sieve" model could explain this behaviour. If we now insert F 45 • between F and F ↔ we observe an effect which cannot be explained by a sieve model where the filter does not change the object. However we now observe a certain number of photons at the detection plate (about 1 4 of the number of photons in the beam passed through F ) as shown in Fig. 4b. In this case our knowledge of the property B is not complete anymore. The acquisition of information about property A therefore leads to a decrease of our knowledge about property B, i.e. H(B\|A) > 0. Note that on the photons absorbed by the filter F 45 • we cannot measure property B subsequently. However already for the subensemble of the photons passing through the filter our uncertainty about property B becomes larger than 0 implying 0 = H(B) < H(B\|A) which clearly violates requirement (8). Another example of sequence of quantum measurements where requirement (9) is violated is given in Fig. 5. Clearly, violation of the requirements (8) and (9) What is the origin of the violation of the requirements (8) and (9) in quantum measurements? In contrast to a classical measurement which just adds some new knowledge to our knowledge at hand from the previous measurements, in a quantum measurement the gain of the new knowledge is always at the expense of irrecoverable loss of complementary classes of knowledge. This originates from the distinction between "total" and "complete" information in quantum physics. In classical physics the total information about a system is complete. In quantum physics the total information of a system, represented by the state vector, is never complete in the sense that all possible future measurement results are precisely defined ‡ . In fact, the total information of a quantum system suffices to specify the eigenstate of one nondegenerate (with one-dimensional eigenspaces only) observable only. For example, the state of a photon passing through filter F is specified by the complete knowledge about the property A of vertical polarization. If we let a photon in this state pass through filter F 45 • as given in Fig. 4b, our knowledge of the photon changes, and therefore its representation, the quantum state, also changes. The total information of a photon in the new state is completely exhausted in specifying property B of polarization at 45 • and no further information is left to also specify property A, thus implying unavoidable loss of the previous knowledge about this property. This further implies that the set of future probabilistic predictions specified by the new projected state is indifferent to the knowledge collected from the previous measurements in the whole history of the system. Such a view was assumed by Pauli [12] who writes § : "Bei Unbestimmtheit einer Eigenschaft eines ‡ Yet, we do not hesitate to emphasize that it certainly is complete in the sense that it is not possible to have more information about a system than what can be specified in its quantum state. In fact, the state vector represents that part of our knowledge about the history of a system which is necessary to arrive at the maximum possible set of probabilistic predictions for all possible future observations of the system. For example, a set of complex amplitudes of a ψ-function is a specific representation of the catalog of our knowledge of the system. This view was assumed by Schrödinger [11] who wrote: "Sie ((die ψ-Funktion )) ist jetzt das Instrument zur Vorausage der Wahrscheinlichkeit von Maßzahlen. In ihr ist die jeweils erreichte Summe theoretisch begründeter Zukunfterwartungen verkörpert, gleichsam wie in einem Katalog niedergelegt. Translated: "It (the ψ-function) is now the means for predicting the probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog." § In translation: "In the case of indefiniteness of a property of a system for a certain experimental arrangement (for a certain state of the system) any attempt to measure that property Systems bei einer bestimmten Anordnung (bei einem bestimmten Zustand des Systems) vernichtet jeder Versuch, die betreffende Eigenschaft zu messen, (mindestend teilweise) den Einfluß der früheren Kenntnisse vom System auf die (eventuell statistischen) Aussagenüber spätere mögliche Messungsergebnisse." This clearly makes possible to violate requirements (8) and (9) in quantum measurements. FIG. 4. The gain of knowledge in a new observation reduces our knowledge at hand from a previous observation. Filters F , F 45 • and F↔ are oriented vertically, at +45 • and horizontally, respectively. If filter F↔ is inserted behind the filter F , no photons are observed at the detector plate (Fig. a). In this case our knowledge about horizontal polarization (property B) of a photon passing through filter F is complete. If filter F 45 • is inserted between F and F↔, a certain number of photons (1/4 of the number of photon passing through F ) will be observed at the detection plate (Fig. b). Now acquisition of information about the polarization at +45 • (property A) leads to the decrease of our previous knowledge about horizontal polarization of the photon. Here a certain misconception might be put forward that arises from a certain practical point of view. According to that view, for example, complementarity between interference pattern and information about the path of the particle in the double-slit experiment is considered to arise from the fact that any attempt to observe the particle path would be associated with an uncontrollable disturbance of the particle. Such a disturbance in itself would then be the reason for the loss of the interference pattern. In such of view it would be possible to define Shannon's information for all attributes of the system sidestroys (at least partially) the influence of earlier knowledge of the system on (possibly statistical) statements about later possible measurement results." multaneously, and the third Shannon postulate, as well as the requirements (8) and (9), would be violated because of the unavoidable disturbance of the system occurring whenever the subsequently measured property B is incompatible with the previous one A. Yet, this is a misconception for two reasons. FIG. 5. Dependence of information on the order of its acquisition in successive quantum measurements. A spin-1/2 particle is in the state \|z+ spin-up along the z-axis. Spin along the x-axis and spin along the direction in the x-z plane tilted at an angle α from the z-axes are successively measured, in one order in (a) and in the opposite order in (b). Whereas we obtain an equal portion H(cos 2 (π/4 − α/2), sin 2 (π/4 − α/2)) of information in the conditional (subsequent) measurement both in (a) and in (b), the amounts of information H(cos 2 α/2, sin 2 α/2) and H( 1 2 , 1 2 ) = 1 we gain in the first measurement in (a) and in the first measurement in (b) respectively, can be significantly different. Specifically for α → 0 we have complete knowledge about spin along the direction at the angle α in (a) but absolutely no knowledge about the spin along the x-axis in (b). We emphasize that we do not assume any specific functional dependence for the measure of information H. Firstly, as theorems like those of Bell [13] or Greenberger, Horne and Zeilinger [14] show, it is not possible, not even in principle, to assign to a quantum system simultaneously observation-independent properties which in order to be in agreement with special relativity have to be local. We therefore cannot speak of a "disturbance" in the measurement process if there are no objective properties to disturb. Secondly, over the last few years experiments were considered and some already performed, where the reason why no interference pattern arises is not due to any uncontrollable disturbance of the quantum system or the clumsiness of the apparatus. Rather the lack of interference is due to the fact that the quantum state is prepared in such a way as to permit path information to be obtained, in principle, independent of whether the experimenter cares to read it out or not. One line of such research considers the use of micromasers in atomic beam experiments [15], another one concerns experiments on correlated photon states emerging from nonlinear crystals through the process of parametric-down conversion [16]. The view that complementarity must be based on the much more fundamentally property of mutual exclusiveness of different classes of information of a quantum system was assumed by Pauli [12] in the analysis of the uncertainty relations * * : " ... diese Relationen enthalten die Aussage, daß jede genaue Kenntnis des Teilchenortes zugleich eine prinzipielle Unbestimmtheit, nicht nur Unbekanntheit des Impulses zur Folge hat und umgekehrt. Die Unterscheidung zwischen (prinzipieller) Unbestimmtheit und Unbekanntheit und der Zusammenhang beider Begriffe sind für die ganze Quantentheorie entscheidend." IV. DIFFICULTIES IN DEFINING THE INFORMATION CONTENT OF A QUANTUM SYSTEM To define the information content of a physical system one might consider different measures of information. However only those measures of information have physical significance according to which the defined information content of the system possesses properties which naturally follow from the physical situation considered. These properties are, for example, invariance under changes of the modes of observation of the system and conservation in time if there is no information exchange with an environment. We show now that the information content of a quantum system, if it is assumed to be measured by the Shannon measure of information, cannot be defined in any way to have these properties. The classical world appears to be composed of particles and fields, and the properties of each one of these constituents could be specified quite independently of the particular phenomenon discussed or of the experimental procedure a physicist chooses to determine these proper- * * In translation:"... these relations contain the statement that any precise knowledge of the position of a particle implies a fundamental indefiniteness, not just an unknownness, of the momentum for a consequence and vice versa. The distinction between (fundamental) indefiniteness and unknownness, and the relation between these two notions is decisive for the whole quantum theory." ties. In other words the properties of constituents of the classical world are noncontextual. In particular, the total lack of information about a classical pointlike system (with no rotational and internal degrees of freedom) defined as Shannon's information associated with the probability distribution over the phase space is independent of the specific set of variables chosen to describe the system completely (such as position and momentum, or bijective functions of them) and conserved in time if there is no information exchange with an environment (i.e. if the system is dynamically independent from the environment and not exposed to a measurement) † † . Operationally the total information content of a classical system can be obtained in the joint measurement of position and momentum, or in successive measurements in which the observation of position is followed by the observation of momentum or vice versa ‡ ‡ . Contrary to the classical concepts most quantummechanical concepts are limited to the description of phenomena within some well-defined experimental context, that is, always restricted to a specific experimental procedure the physicist chooses. In particular the amount of information gained in an individual quantum measurement depends strongly on the specific experimental context. In the optimal experiment when the measurement basis \|i coincides with the eigenbasis of the density matrixρ of the system:ρ\|i = w i \|i the amount of information gained is maximal (See for example [3]). Since in the basis corresponding to the optimal experiment the density operator is represented by a diagonal matrix with elements w i , the information gain defined by the Shannon measure equals the von Neumann entropy as given by § § † † Given the probability distribution ρ( r, p, t) over the phase space the total lack of information of a classical system is defined as [17] H total (t) = − d 3 rd 3 pρ( r, p, t) log ρ( r, p, t) µ( r, p) , where a background measure µ( r, p) is an additional ingredient that has to be added to the formalism to ensure invariance under change of variables when we consider continuous probability distributions. The conservation of H total in time for a system with no information exchange with an environment is implied by the Hamiltonian evolution of a point in phase space. ‡ ‡ In full analogy with (9) we may write H total ( r, p) = H( r)+ H( p\| r) = H( p) + H( r\| p). § § For a given density matrixρ the von Neumann entropy S(ρ) = −T r(ρ logρ)(11) is widely accepted as a suitable definition for the information content of a quantum system. For a system described in H = − i w i log w i = −T r(ρ logρ).(12) This has the important property to be invariant under unitary transformationsρ →ÛρÛ + . The invariance under unitary transformations implies invariance under the change of the representation (basis) ofρ and conservation in time if there is no information exchange with an environment. The later precisely means that if we perform the optimal experiments both at time t 0 and at some future time t, Shannon's information measures associated to the optimal experiments at the two times will be the same, i.e. H(t) = − i w i (t) log w i (t) = − i w i log w i = H(t 0 ).(13) Here, the eigenvalues of the density matrix at time t are w i (t) = w i . However, without the additional knowledge of the eigenbasis of the density matrixρ we cannot find the optimal experiment and obtain directly the Shannon information associated. Also, all the statistical predictions that can be made for the optimal measurement are the same as if we had an ordinary (classical) mixture, with fractions w i of the systems giving with certainty results that are associated to the eigenvectors \|i . In this sense the optimal measurement is a classical type measurement and therefore in this particular case, and only then, Shannon's measure defines the information gain in a measurement appropriately * * * . Considering also our previous discussion it is therefore not surprising that Shannon's N -dimensional Hilbert space this ranges from log N for a completely mixed state to 0 for a pure state. The von Neumann entropy has the important property to be invariant under unitary transformations. However, we observe that any function of the form T r(f (ρ)) (the operator f (ρ) is identified by having the same eigenstates asρ and the eigenvalues f (wj ), equal to the function values taken at the eigenvalues wj ofρ.) possesses this invariance property. We also observe that the von Neumann entropy is a property of the quantum state as a whole without explicit reference to information contained in individual measurements. * * * Consider a situation where instead using of single systems to send information to the receiver a sender uses a sequence of N systems where each individual system is drawn from an ensemble of pure states {\|ψ1 , ..., \|ψn }, with frequency of occurrence {w1, ..., wn} respectively. It was shown in [18] that for sufficiently large N there are 2 NS(ρ) highly distinguishable sequences of pure states which become mutually orthogonal as N → ∞. Here S(ρ) = −T r(ρ logρ) is the von Neumann entropy andρ = n i wi\|ψi ψi\|. This means that if the sender uses a sequence consisting of a choice of states that respects the a priori frequencies wi, and the receiver distin-measure is useful only when applied to measurements which can be understood as classical measurements. Which set of individual measurements should we perform and how to combine individual measures of information obtained in the set in order to arrive at the information content of a quantum system if we do not know the eigenbasis of the density matrix? Quantum complementarity implies that the total information content of the system might be partially encoded in different mutually exclusive (complementary) observables. These have the property that complete knowledge of the eigenvalue of any one of the observables excludes any knowledge about the eigenvalues of all other observables. Such a set of observables for a spin-1/2 particle can for example be spin components along orthogonal directions. We consider now a quantum system described in n dimensional Hilbert space and we denote a complete set of m mutually complementary observables † † † by {Â,B, ...}. The property of mutual expansiveness implies that if the system is in an eigenstate of one of the observables, for example, in the eigenstate \|a j of the observableÂ and we measure any other observable from the set, sayB, projecting the system onto states {\|b 1 , ..., \|b i , ..., \|b n }, the individual outcome is completely random (all measurement results are equally probable) \| a j \|b i \| 2 = 1 n ∀i, j.(14) It was shown in [19] that the density matrix of the system can fully be reconstructed if one performs a complete set of mutually complementary observations. This suggest that the total information content of a quantum system represented by a density matrixρ is all obtainable guishes whole sequences rather than individual states, then the (Shannon) information transmitted per system can be made arbitrary close to S(ρ). Here again the total density matrixρ N of N systems can be made arbitrary close to the one as if we had a classical mixture of the 2 NS(ρ) sequences of states. † † † To specify a system described by a n × n density matrix completely one needs n 2 − 1 independent real numbers. Any individual, complete measurement (we consider here only complete measurements, i.e., where operators associated to the measurements are without degeneracy) with n possible outcomes defines n − 1 independent probability values (the sum of all probabilities for all possible outcomes in an individual experiment is one). Therefore, just on the basis of counting the number of independent variables, we expect that the number of different measurements we need in order to determine the density matrix completely is n 2 −1 n−1 = n + 1. Ivanovic [19], and Wootters and Fields [20] demonstrated the existence of exactly n + 1 mutually complementary observables by an explicit construction in the cases of n prime and n = 2 k . from a complete set of mutually complementary measurements. To obtain the total information one however cannot perform the set of measurements successively because, unlike the classical case, the information obtained in successive quantum measurements depends on the order of its acquisition (see Fig. 5 and discussion above). Instead it seems that any attempt to obtain the total information content of a quantum system has to be related to the complete set of mutually complementary experiments performed on systems that are all in the same quantum state. We suggest that it is therefore natural to require that the total information content in a system in the case of quantum systems is sum of the individual amounts of information over a complete set of m mutually complementary observables. As already mentioned above, for a spin-1/2 particle these are three spin projections along orthogonal directions. If we define the information gain in an individual measurement by the Shannon measure the total information encoded in the three spin components is given by H total := H 1 (p + x , p − x ) + H 2 (p + y , p − y ) + H 3 (p + z , p − z ) . (15) Here, e.g. p + x is the probabilities to find the particle with spin up along direction x. Considering now an explicit example we will show that the total information H total based on the Shannon measure is in general not invariant under unitary transformations. We calculate (15) for a spin-1/2 particle in the state \|ψ = cos θ/2\|z+ + sin θ/2\|z− and we find that H total = (16) − 1 − sin θ 2 log 1 − sin θ 2 − 1 + sin θ 2 log 1 + sin θ 2 − cos 2 θ 2 log cos 2 θ 2 − sin 2 θ 2 log sin 2 θ 2 + 1 depends on the parameter θ, thus being not invariant under unitary transformations. This associates a number of highly counter-intuitive properties to H total : 1) it can be different for states of the same purity (e.g. it takes its maximal value of 2 bits of information for θ = 0 and it takes its minimal value of 1.36 bits for θ = π/4); 2) it changes in time even for a system completely isolated from the environment where no information can be exchanged with environment; 3) it can take different values for different sets of the three orthogonal spin projections. These unnatural properties we see again as a strong indications for inadequacy of the Shannon measure to define the information gain in an individual quantum measurement. V. A SUGGESTED ALTERNATIVE MEASURE OF INFORMATION We suggest that it is natural to require that the information content of the quantum system defined as a sum of individual measures over a complete set of mutually complementary measurements is invariant under unitary transformations. Having shown that this cannot be achieved with the Shannon measure of information we now introduce a new measure of information that differs both mathematically and conceptually from Shannon's measure of information and according to which the information content has invariance property. The new measure of information is a quadratic function of probabilities ‡ ‡ ‡ I(p 1 , ..., p n ) = n i=1 p i − 1 n 2 ,(18) and it takes into account that for quantum systems the only features known before an experiment is performed are the probabilities for various events to occur (See [4] for discussion; there a specific normalization factor in expression (18) was used resulting in maximally k bits for n = 2 k possible outcomes). The measure I(p 1 , ..., p n ) takes its maximal value of (n − 1)/n if one p i = 1 and it takes its minimal value of 0 when all p i are equal. The important property of the new measure of information is that the total information defined with respect to it is invariant under unitary transformations. Using Eq. (14) one obtains that the sum over individual measures of information of mutually complementary observations results in [24] I total := m j=1 I(p j 1 , ..., p j n ) = m j=1 n i=1 p j i − 1 n 2 = T rρ 2 − 1 n ,(19) for a system described by the density matrixρ. Here p j i denotes the probability to observe the i-th outcome of ‡ ‡ ‡ Expressions of the general type of Eq. (18) were studied in detail by Hardy, Littlewood and Pólya [21]. They introduced a general class of mathematical expressions Mα = n i=1 p α i α−1 for 0 ≤ α ≤ ∞(17) that from the point of view of information theory all can be assumed to quantify information properly. These expressions are also closely related to Tsallis's [22] nonextensive entropy Sα = 1 1−α n i=1 (p α i − 1) and Rányi's [23] entropy Hα = 1 1−α log n i=1 p α i . the j-th observable. The total information content of the system therefore might all be encoded in one single observable or, alternatively it might be partially encoded in all m mutually complementary observables. For a composite system in a product state the total information can all be encoded in individual systems constituting the composite system or, alternatively in the extreme case of maximally entangled states it can all be encoded in joint properties of the systems with no information left in individual systems [4]. Independent of the various possibilities to encode information the total information content of the system cannot fundamentally exceed the maximal possible amount of information that can be encoded in an individual observable [= (n − 1)/n]. This upper limit is reached when the system is in the pure state. When the system is in a completely mixed state the total information takes its minimal value of 0. The property of invariance under unitary transformations implies that the total information content of the system does not dependent of the particular set of mutually complementary observables; it is a characterization of the state of the system alone, not of the specific reference set of complementary observables. Furthermore, since evolution in time is described by a unitary operation the total information of the system is conserved in time if there is no information exchange with the environment. We would like to note that the total information (19) was used in [25] to study the transfer of entanglement and information for quantum teleportation of an unknown entangled state through noisy quantum channels. The total information (19) belongs to the set of quantum counterparts of nonextensive entropies finding its application in increasing number of problems in quantum physics, e.g. description and controlling of laser cooling [26], a nonextensive approach to the decoherence problem [27], description and quantifying of entanglement, and deducing criterions for separability of density matrices [28,29]. CONCLUSIONS In this work we have stressed some conceptual difficulties arising when Shannon's notion of information is applied to define information gain in a quantum measurement. In particular we find that the axiomatic derivation of Shannon's measure of information is not applicable in quantum measurements in general. We also show that the information content of a quantum system defined according to Shannon's measure possesses some strongly non-physical properties. We argue that these difficulties in defining the information gain in quantum measurement by the Shannon measure of information arise whenever it is not possible, not even in principle, to assume that attributes observed are assigned to the quantum system before the observation is performed. Having critized Shannon's measure of information as being not appropriate for identifying the information gain in quantum measurement we proposed a new measure of information in quantum mechanics that both mathematically and conceptually differs from Shannon's measure of information. While Shannon's information is applicable when measurement reveals a preexisting property, the new measure of information takes into account that for quantum systems the only features known before an experiment is performed are the probabilities for various events to occur. In general, which specific event occurs is objectively random. The total information content of a quantum system defined according to the new measure of information as the sum of the individual measures of information for mutually complementary observations is invariant under unitary transformations. This implies that the total information content of the system is invariant under transformation from one complete set of complementary variables to another and is conserved in time if there is no information exchange with an environment. PACS number(s): 03.65.-w, 03.65.Bz, 03.67.-a I. INTRODUCTION For example, for N=4, there is only one color sequence •••• if all balls are white, 4 possible color sequences ••••, ••••, ••••, ••••, if there are three black and one white ball in the urn, yet 6 possible color sequences ••••, ••••, ••••, ••••, •••• and •••• if there are two black and two white balls in the urn. FIG. 1 . 1Binary question tree to determine the specific sequence of outcomes (color of the drawn balls) in a sufficiently large number N of experimental trials (number of drawings). An urn is filled with black and white balls with proportions p1 and p2, respectively. The expected number of questions needed to determine the actual (typical) sequence of outcomes is N H, where H = −p1 log p1 − p2 log p2. 1. H should be continuous in the p i . 2. If all the p i are equal, p i = 1 n , then H should be a monotonically increasing function of n. With equally likely events there is more choice, or uncertainty, when there are more possible events. 3. If a choice be broken down into two successive choices, the original H should be the weighted sum of the individual values of H. The meaning of this is illustrated in Fig. 2. At the left we have three possibilities p FIG. 2 . 2Decomposition of a choice from three possibilities. Figure taken from [1]. occurs when the corresponding operators A and B do not commute. FIG. 3 . 3Indifference of information to the order of its acquisition in classical measurements. A box is filled with balls of different compositions (plastic and wooden balls) and different colors (black and white balls). Now, the box is shaken. In (a) we first draw a ball asking about the color of the drawn ball and gain H(color) = 1 bit of information. Subsequently, we put the black and white balls in separate boxes, draw a ball from each box separately and ask about the composition of the drawn ball. We gain H bl (comp.) = 0 bits for the black balls and H wh (comp.) = 1 bit for the white balls. In (b) we pose the two questions in the opposite order. We firstly ask about the composition of the drawn ball and gain H(comp.) = 0.81 bits. In a conditional drawing we ask about the color of the drawn ball and gain Hwo(color) = 0 bits for wooden balls and H pl (color) = 0.92 bits for plastic balls. The total information gained is independent of the particular order the two questions are posed, i.e. H(color) + 1/2H bl (comp.) + 1/2H wh (comp.) = H(comp.)+1/4Hwo(color)+3/4H pl (color) = 1.5. ACKNOWLEDGMENTIn the previous version we did not make proper full reference to the work of J. Uffink[6]. We would like to thank J. Uffink for pointing out this inadequancy as well as an error in our previous Eq. (1). We also thank C. Simon for helpful comments and discussions. This work has been supported by the Austrian Science Foundation FWF, Project No. F1506 and the US National Science Foundation NSF Grant No. PHY 97-22614. A copy can be found at www.math.washington. C E Shannon, Bell Syst. Tech. J. 27379C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948). A copy can be found at www.math.washington.edu/ ∼ hillman/ Entropy/infcode.html . S Kochen, E P Specker, J. Math. and Mech. 1759S. Kochen and E. P. Specker, J. Math. and Mech. 17, 59 (1967). A Peres, Quantum Theory: Concepts and Methods. 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C Tsallis, S Lloyd, M Baranger, quant-ph/0007112Chaos, Solitons, and Fractals. 13431Chaos, Solitons, and Fractals, 13, 431 (2002), C. Tsallis, S. Lloyd, and M. Baranger, e-print: quant-ph/0007112.	[]
[ "Massive graviton dark matter with environment dependent mass: A natural explanation of the dark matter-baryon ratio", "Massive graviton dark matter with environment dependent mass: A natural explanation of the dark matter-baryon ratio" ]	[ "Katsuki Aoki \nDepartment of Physics\nWaseda University\n169-8555ShinjukuTokyoJapan\n", "Shinji Mukohyama \nCenter for Gravitational Physics\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan\n\nKavli Institute for the Physics and Mathematics of the Universe (WPI)\nUTIAS\nThe University of Tokyo\n277-8583KashiwaChibaJapan\n" ]	[ "Department of Physics\nWaseda University\n169-8555ShinjukuTokyoJapan", "Center for Gravitational Physics\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan", "Kavli Institute for the Physics and Mathematics of the Universe (WPI)\nUTIAS\nThe University of Tokyo\n277-8583KashiwaChibaJapan" ]	[]	We propose a scenario that can naturally explain the observed dark matter-baryon ratio in the context of bimetric theory with a chameleon field. We introduce two additional gravitational degrees of freedom, the massive graviton and the chameleon field, corresponding to dark matter and dark energy, respectively. The chameleon field is assumed to be non-minimally coupled to dark matter, i.e., the massive graviton, through the graviton mass terms. We find that the dark matter-baryon ratio is dynamically adjusted to the observed value due to the energy transfer by the chameleon field. As a result, the model can explain the observed dark matter-baryon ratio independently from the initial abundance of them.	10.1103/physrevd.96.104039	[ "https://arxiv.org/pdf/1708.01969v2.pdf" ]	119,220,247	1708.01969	fcfb0627ccbd5659c59c80b2fc666d477f698419	Massive graviton dark matter with environment dependent mass: A natural explanation of the dark matter-baryon ratio 15 Dec 2017 Katsuki Aoki Department of Physics Waseda University 169-8555ShinjukuTokyoJapan Shinji Mukohyama Center for Gravitational Physics Yukawa Institute for Theoretical Physics Kyoto University 606-8502KyotoJapan Kavli Institute for the Physics and Mathematics of the Universe (WPI) UTIAS The University of Tokyo 277-8583KashiwaChibaJapan Massive graviton dark matter with environment dependent mass: A natural explanation of the dark matter-baryon ratio 15 Dec 2017(Dated: December 18, 2017) We propose a scenario that can naturally explain the observed dark matter-baryon ratio in the context of bimetric theory with a chameleon field. We introduce two additional gravitational degrees of freedom, the massive graviton and the chameleon field, corresponding to dark matter and dark energy, respectively. The chameleon field is assumed to be non-minimally coupled to dark matter, i.e., the massive graviton, through the graviton mass terms. We find that the dark matter-baryon ratio is dynamically adjusted to the observed value due to the energy transfer by the chameleon field. As a result, the model can explain the observed dark matter-baryon ratio independently from the initial abundance of them. I. INTRODUCTION Cosmological observations have confirmed the big bang cosmology and determined the cosmological parameters precisely [1]. The matter contents of the Universe may be phenomenologically given by the standard model particles, the cosmological constant Λ, and cold dark matter (CDM). However, the theoretical explanation of the origin of the extra ingredients, dark matter and dark energy, is still lacked. The theoretically expected value of the cosmological constant is too large to explain the present accelerating expansion. An alternative idea is that the acceleration is obtained by a potential of a scalar field instead of Λ, and this idea is often called the quintessence model [2]. This scalar field could be originated from the gravity sector [3]. A large class of scalar-tensor theories and f (R) theories can be recast in the form of a theory of a canonical scalar field with a potential after the conformal transformationg µν = A 2 (φ)g µν and the field redefinition Φ = Φ(φ) where Φ is the canonically normalized field. The metricg µν is called the Jordan frame metric which the standard model particles are minimally coupled to whereas g µν is the Einstein frame metric in which the gravitational action is given by the Einstein-Hilbert action. In this case, the scalar field has the nonminimal coupling to the matter fields via the coupling function A. Dark matter is also one of the biggest mystery of the modern cosmology. Although many dark matter candidates have been proposed in the context of the particle physics, any dark matter particles have not been discovered yet [4][5][6][7][8][9]. The existence of dark matter is confirmed via only gravitational interactions. Hence, exploring dark matter candidate in the context of gravity is also a considerable approach. Not only dark energy but also dark matter could be explained by modifications of gravity. For instance, a natural extension of general relativity is a theory with a massive graviton (see [10] for a review). If a graviton obtains a mass, the massive graviton can be a dark matter candidate [11][12][13][14][15][16]. A viable dark matter scenario has to explain the present abundance of dark matter which usually leads to a constraint on a production scenario. However, a question arises: why are the energy densities of dark matter and baryon almost the same? If baryon and dark matter are produced by a common mechanism, almost the same abundance could be naturally obtained. On the other hand, if productions of the two are not related but independent, the coincidence might indicate that two energy densities are tuned to be the same order of the magnitude by a mechanism after the productions. In the present paper, we shall combine two ideas of the modifications of gravity by using the proposal of [17]: the non-minimal coupling of φ and the existence of the massive graviton. We call this theory the chameleon bigravity theory which contains three types of gravitational degrees of freedom: the massless graviton, the massive graviton, and the chameleon field φ. We identify the massive graviton with dark matter. Since dark matter is originated from the gravity sector, the coupling between φ and the dark matter may be given by a different way from the matter sector. We promote parameters in the graviton mass terms to functions of φ [18,19], giving rise to a new type of coupling between φ and dark matter. In this case, as discussed in [17], the field value of φ depends on the environment due to the non-minimal coupling as with the chameleon field [20,21], which makes the graviton mass to depend on the environment. We find that the ratio between energy densities of dark matter and baryon is dynamically adjusted to the observed value by the motion of φ and then the ratio at the present is independent of the initial value. Hence, our model can explain the coincidence of the abundance of dark matter and baryon. Furthermore, if the potential of φ is designed to be dark energy, the chameleon field φ can give rise to the present acceleration of the universe. Both dark energy and dark matter are explained by the modifications of gravity in our model. The paper is organized as follows. We introduce the chameleon bigravity theory in Sec. II. In Sec. III, we show the Friedmann equation regarding the massive graviton is dark matter. We also point out the reason why the dark matter-baryon ratio can be naturally explained in the chameleon bigravity theory if we consider the massive graviton as dark matter. Some analytic solutions are given in Sec. IV and numerical solutions are shown in Sec. V. These solutions reveal that the observed dark matter-baryon ratio is indeed dynamically obtained independently from the initial ratio. We summarize our results and give some remarks in Sec. VI. In Appendix, we detail the derivation of the Friedmann equation. II. CHAMELEON BIGRAVITY THEORY We consider the chameleon bigravity theory in which the mass of the massive graviton depends on the environment [17]. The action is given by S = d 4 x √ −g M 2 g 2 R[g] − 1 2 K 2 (φ)g µν ∂ µ φ∂ ν φ + M 2 g m 2 4 i=0 β i (φ)U i [s] + M 2 f 2 d 4 x −f R[f ] + S m [g, ψ] ,(2.1) where φ is the chameleon field and S m is the matter action. The functions K(φ) and β i (φ) are arbitrary functions of φ. The matter fields universally couple to the Jordan frame metricg µν = A 2 (φ)g µν with a coupling function A(φ). The potentials U i [s] (i = 0, · · · , 4) are the elementary symmetric polynomials of the eigenvalues of the matrix s µ ν which is defined by the relation [22][23][24] s µ α s α ν = g µα f αν . (2.2) The potential of φ is not added explicitly since the couplings between φ and the potentials U i yield the potential of φ and thus an additional potential is redundant. Note that the field φ is not a canonically normalized field. The canonical field Φ is given by the relation dΦ = K(φ)dφ ,(2.3) by which the function K does not appear explicitly in the action when we write down the theory in terms of Φ. Since β i and A are arbitrary functions, we can set K = 1 by the redefinitions of β i and A without loss of generality. Nevertheless, we shall retain K and discuss the general form of the action (2.1). In general, the functions β i (φ) can be chosen independently. In the present paper, however, we consider the simplest model such that β i (φ) = −c i f (φ) where c i are constant while f (φ) is a function of φ. As we will see in next section, the graviton mass and the potential of φ around the cosmological background are given by m 2 T (φ) := 1 + κ κ m 2 f (φ)(c 1 + 2c 2 + c 3 ) , (2.4) V 0 (φ) := m 2 M 2 p f (φ)(c 0 + 3c 1 + 3c 2 + c 3 ) , (2.5) with κ = M 2 f /M 2 g and M 2 p = M 2 g + M 2 f . In this case, both the potential form of φ and the φ-dependence of the graviton mass are determined by f (φ) only. 1 Note that V 0 is the bare potential of φ. The effective potential of φ is given by not only V 0 but also the amplitude of the massive graviton as well as the energy density of matter due to the non-minimal couplings (see Eq. (3.8)). III. BASIC EQUATIONS In this section, we derive the basic equations to discuss the cosmological dynamics in the model (2.1) supposing that the massive graviton is dark matter. We assume the coherent dark matter scenario in which dark matter is obtained from the coherent oscillation of the zero momentum mode massive gravitons [16]. Since the zero momentum mode of the graviton corresponds to the anisotropy of the spacetime, we study the Bianchi type I universe instead of the Friedmann-Lemaître-Robertson-Walker (FLRW) universe. The ansatz of the spacetime metrics are ds 2 g = −dt 2 + a 2 [e 4σg dx 2 + e −2σg (dy 2 + dz 2 )] , (3.1) ds 2 f = ξ 2 −c 2 dt 2 + a 2 {e 4σ f dx 2 + e −2σ f (dy 2 + dz 2 )} ,(3.2) where {a, ξ, c, σ g , σ f } are functions of the time t. We assume the matter field is a perfect fluid whose energymomentum tensor is given by T µ ν = A 4 (φ) × diag[−ρ(t), P (t), P (t), P (t)] ,(3.3) where ρ and P are the energy density and the pressure in the Jordan frame, respectively. The conservation law of the matter field iṡ ρ + 3 (Aa) · Aa (ρ + P ) = 0 ,(3. 4) 1 Since we have absorbed the potential of φ in the mass term of the graviton, m 2 T Mp and V 0 seem to be a same order of magnitude. However, m 2 T M 2 p and V 0 are not necessary to be the same order because they represent different physical quantities. Indeed, we will assume V 0 ≪ m 2 T M 2 p . where a dot is the derivative with respect to t. As shown in [16,25], the small anisotropies σ g and σ f can be a dark matter component of the universe in the bimetric model without the chameleon field φ. We generalize their calculations to those in the present model (2.1). All equations under the ansatz (3.1) and (3.2) are summarized in Appendix. Here, we only show the Friedmann equation and the equations of motion of the massive graviton and the chameleon field because other equations are not important for the following discussion. We assume the graviton mass m T is larger than the Hubble expansion rate H :=ȧ/a. After expanding the equations in terms of anisotropies and a small parameter ǫ := H/m T , the Friedmann equation is given by 3M 2 p H 2 = ρA 4 + 1 2 K 2φ2 + V 0 + 1 2φ 2 + 1 2 m 2 T ϕ 2 , (3.5) where ϕ is the massive graviton which is given by a combination of the anisotropies σ g and σ f (see Eqs. (A20) and (A21)). The equations of motion of the massive graviton ϕ(t) and the chameleon field φ(t) arë ϕ + 3Hφ + m 2 T (φ)ϕ = 0 , (3.6) K φ + 3Hφ +Kφ + ∂V eff ∂φ = 0 , (3.7) where the effective potential of the chameleon field is given by V eff := V 0 (φ) + 1 2 m 2 T (φ)ϕ 2 + 1 4 A 4 (φ)(ρ − 3p) . (3.8) Note that, although the bigravity theory contains the degree of freedom of the massless graviton (see Eq. (A19)), we neglect the contribution to the Friedmann equation from the massless graviton because the energy density of the massless graviton decreases faster than those of other fields. The effect of the massless graviton is not important for our discussions. We notice that the basic equations (3.5), (3.6) and (3.7) are exactly the same as the equations in the theory with two scalar fields given by the action S = d 4 x √ −g M 2 p 2 R[g] − 1 2 K 2 (φ)(∂φ) 2 − V 0 (φ) − 1 2 (∂ϕ) 2 − 1 2 m 2 T (φ)ϕ 2 +S m [g, ψ] . (3.9) The cosmological dynamics in (2.1) with H ≫ m T can be reduced into that in (3.9). Our results obtained below can be straightforward generalized even in the case of (3.9) up to the discussion about the cosmological dynamics. The action (3.9) gives a toy model of the chameleon bigravity theory. However, the equivalence between (2.1) and (3.9) holds only for the background dynamics of the universe in H ≫ m T . The equivalence between the two actions does not hold for small-scale perturbations around the cosmological background [16]. We first consider a solution φ = φ min = constant which is realized when ∂V eff ∂φ = α f V 0 + 1 2 m 2 T ϕ 2 + α A (ρ − 3P )A 4 = 0 ,(3.10) where α A := 1 K d ln A dφ = d ln A dΦ , α f := 1 K d ln f dφ = d ln f dΦ . (3.11) The equation (3.10) is not always compatible with φ = constant since each term in (3.10) has different time dependence in general. Nonetheless, as we shall see below, they can be compatible with each other if ǫ ≪ 1. In other words, a common constant value of φ min can be a solution all the way from the radiation dominant (RD) epoch to the matter dominant (MD) epoch of the universe. When the chameleon field is constant, the bare potential V 0 acts as a cosmological constant which has to be subdominant in the RD and the MD eras. The constant φ implies that the graviton mass does not vary and thus we obtain φ 2 T = m 2 T ϕ 2 T ∝ a −3 , (3.12) where · · · T represents the time average over an oscillation period. The massive gravitons behave like a dark matter component of the universe. When we focus on the time scales much longer than m −1 T , m 2 T ϕ 2 in Eq. (3.10) can be replaced with m 2 T ϕ 2 T , which scales as (3.12). Since ρ − 3P also scales as ∝ a −3 in the RD and the MD, the decaying laws of ρ − 3P and m 2 T ϕ 2 in (3.10) are the same in this case. Hence, when the oscillation timescale of the massive graviton is much shorter than the timescale of the cosmic expansion, i.e., ǫ ≪ 1, φ = constant can be a solution all the way from the RD to the MD. The value of φ min is determined by simply solving Eq. (3.10). Supposing that the massive graviton is the dominant component of dark matter, Eq. (3.10) in the RD and MD eras is replaced with (α f ρ G + 2α A ρ b ) A 4 = 0 , (3.13) where ρ b is the baryon energy density and we have ignored V 0 . The energy density of massive graviton in the Jordan frame is defined by ρ G := 1 2 A −4 φ 2 + m 2 T ϕ 2 T = A −4 m 2 T ϕ 2 T , (3.14) which depends on the chameleon φ. Therefore, if α A and α f are assumed to be α A /α f ≃ −5/2, the ratio between dark matter and baryon is automatically tuned to be the observational value. The dark matter-baryon ratio could be naturally explained without any fine-tuning of the productions of dark matter and baryon. Needless to say, the initial value of φ must not be at the bottom of the effective potential (φ = φ min ). We shall study the dynamics of φ and discuss whether φ approaches φ min before the MD era of the universe. Although we do not assume φ is constant, we assume φ does not rapidly move so that the graviton mass varies adiabaticallyṁ T m 2 T ≪ 1 . (3.15) Under the adiabatic condition (3.15) we can take the adiabatic expansion for the massive graviton: ϕ = u(t) cos m T [φ(t)]dt + · · · , (3.16) with a slowly varying function u(t). The adiabatic condition (3.15) is indeed viable for ǫ ≪ 1 since we will see the time dependence of m T is given by a power law of a (see Eq. (4.8) for example). The time average over an oscillation period yields φ 2 T = m 2 T ϕ 2 T = m 2 T u 2 /2. After taking the time average over an oscillation period under the adiabatic condition, the equations are reduced into 3M 2 p H 2 = A 4 ρ r + A 4 ρ b + 1 2 K 2φ2 + V 0 + 1 2 m 2 T u 2 ,(3.17) and K φ + 3Hφ +Kφ + α f V 0 + 1 4 α f m 2 T u 2 + α A A 4 ρ b = 0 , (3.18) 4u + 6Hu + α f uKφ = 0 ,(3.19) where ρ r and ρ b are the energy densities of radiation and baryon which decrease as ρ r ∝ (aA) −4 and ρ b ∝ (aA) −3 because of the conservation equation. The dynamics of the scale factor a, the chameleon field φ, and the amplitude of the massive graviton u are determined by solving these three equations. By using the density parameters, the Friedmann equation is rewritten as 1 = Ω r + Ω b + Ω φ + Ω G ,(3.20) with Ω r := A 4 ρ r 3M 2 p H 2 , (3.21) Ω b := A 4 ρ b 3M 2 p H 2 , (3.22) Ω φ :=φ 2 + 2V 0 6M 2 p H 2 ,(3.23)Ω G := m 2 T u 2 6M 2 p H 2 . (3.24) We also introduce the total equation of state parameter in the Einstein frame w E := −1 − 2Ḣ 3H 2 . (3.25) The above quantities are defined in the Einstein frame. Since the matter fields minimally couple with the Jordan frame metric, the observable universe is expressed by the Jordan frame metric. Hence, we also define the Hubble expansion rate and the effective equation of state parameter in the Jordan frame as H J := (Aa) · A 2 a , (3.26) w tot := −1 − 2Ḣ J 3AH 2 J . (3.27) IV. ANALYTIC SOLUTIONS In this section we show some analytic solutions under the simplest case K = 1 , A = e βφ/Mp , f = e −λφ/Mp , (4.1) with the dimensionless constants β and λ. This model yields that the coupling strengths α A and α f are constant. We consider four stages of the universe: the radiation dominant era, around the radiation-matter equality, the matter dominant era, and the accelerate expanding era. The analytic solutions are found in each stages of the universe as follows. A. Radiation dominant era We first consider the regime when the contributions to the Friedmann equation from baryon and dark matter are subdominant, that is, Ω b , Ω G ≪ 1. The Hubble expansion rate is then determined by the energy densities of radiation and φ. Since the effective potential of φ are determined by the energy densities of baryon and dark matter, in this situation the potential force can be ignored compared with the Hubble friction term (V 0 is assumed to be always ignored during both radiation and matter dominations). Then, we obtaiṅ φ ∝ a −3 , (4.2) which indicates that the field φ loses its velocity due to the Hubble friction and then φ becomes a constant φ i . We can ignore Ω φ and then find the standard RD universe. At some fixed time deep in the radiation dominant era, we therefore set φ = φ i as the initial condition of φ. We shall then denote the initial values of the energy densities of baryon and the massive graviton as ρ b,i and ρ G,i , respectively. Note that this constant initial value of φ is not necessary to coincide with the potential minimum φ = φ min , i.e., φ i = φ min . The ratio ρ G,i /ρ b,i is not tuned to be five at this stage. B. Following-up era We then discuss the era just before radiation-matter equality in which we cannot ignore the potential force for φ. As discussed in the previous subsection, we find φ = φ i in the RD universe. When the potential force for φ becomes relevant, the chameleon field φ starts to evolve into the potential minimum φ = φ min . Due to the motion of φ, the smaller one of ρ G and ρ b follows up the larger one. We obtain ρ G /ρ b = −2α A /α f when the chameleon field reaches the minimum φ min . We call this era of the universe the following-up era. If the initial value φ i is close to the potential minimum φ min , the dark matter-baryon ratio is already tuned to be almost the value −2α A /α f , which we set to ∼ 5, and thus we do not need to discuss this case. We therefore study the case with φ i < φ min and the case with φ i > φ min (which correspond to ρ G,i ≫ ρ b,i and ρ b,i ≫ ρ G,i , respectively). We shall discuss them in order. 1. ρG,i ≫ ρ b,i before the equal time If dark matter (i.e., massive gravitons) is overproduced, the equations are reduced tö φ + 3Hφ − λ 4M p m 2 T u 2 = 0 , (4.3) 3M 2 p H 2 = A 4 ρ r + 1 2φ 2 + 3m 2 T u 2 ,(4.4) and (3.19), where we have ignored the contributions from baryon. The system admits a scaling solution φ = M p λ ln t + constant , u ∝ t −1/2 , a ∝ t 1/2 ,(4.5) where the density parameters in the Einstein frame are given by Ω G = 4 3λ 2 , Ω φ = 2 3λ 2 , Ω r = 1 − 2 λ 2 . (4.6) The effective equation of state parameter in the Jordan frame is given by w tot = λ − 2β 3(λ + 2β) ,(4.7) and then w tot = −2/9 if 2β = 5λ. This solution exists only when λ 2 > 2 since the density parameter has to be 0 < Ω r < 1. For this scaling solution, the graviton mass decreases as m 2 T ∝ a −2 ,(4.8) which guarantees the adiabatic condition (3.15) when ǫ ≪ 1. The energy density of massive gravitons in the Einstein frame decreases as A 4 ρ G = 1 2 m 2 T u 2 ∝ a −4 . (4.9) On the other hand, the energy density of baryon in the Einstein frame "increases" as A 4 ρ b ∝ Aa −3 ∝ a −3+2 β λ ,(4.10) (For example, we obtain A 4 ρ b ∝ a 2 when 2β = 5λ). Therefore, even if baryon is negligible at initial, the baryon energy density grows and then it cannot be ignored when the energy density of baryon becomes comparable to that of dark matter. Note that the Jordan frame energy density of baryon, ρ b , always decays as a −3 J where a J = Aa is the scale factor of the Jordan frame metric. The quantity A 4 ρ b is the energy density in the Einstein frame. In the Einstein frame, the interpretation of the peculiar behavior of A 4 ρ G and A 4 ρ b is that the energy density of massive gravitons is converted to that of baryon through the motion of the chameleon field φ. Although we have considered the non-relativistic massive gravitons, the energy density of that in the Einstein frame behaves as radiation which implies that the field φ removes the energy of massive gravitons (indeed, the graviton mass decreases due to the motion of φ). The removed energy is transferred into baryon via the non-minimal coupling. During the scaling solution, the massive graviton never dominates over radiation because both energy densities of the massive graviton and radiation obey the same decaying law A 4 ρ r , A 4 ρ G ∝ a 4 . Hence, the field φ can reach the bottom of the effective potential before the MD era. After reaching the bottom of the effective potential, the standard decaying laws for matters A 4 ρ r ∝ a −4 and A 4 ρ G , A 4 ρ b ∝ a −3 are recovered, then the usual dynamics of the universe is obtained with the observed dark matter-baryon ratio. We note that the following-up of the baryon energy density can be realized even if the scaling solution does not exist (λ 2 < 2). The dynamics of this case is numerically studied in Sec. V. ρ b,i ≫ ρG,i before the equal time In this case, the equations for the scale factor and φ form a closed system given bÿ φ + 3Hφ + β M p A 4 ρ b = 0 , (4.11) 3M 2 p H 2 = A 4 ρ r + A 4 ρ b + 1 2φ 2 . (4.12) The scaling solution is then found as φ = − M p 2β ln t + constant , a ∝ t 1/2 ,(4.13) in which the density parameters are Ω b = 1 3β 2 , Ω φ = 1 6β 2 , Ω r = 1 − 1 2β 2 ,(4.14) where β has to satisfy β 2 > 1/2. During this scaling solution, the universe does not expand in the Jordan frame. Although the Einstein frame scale factor expands as the RD universe, a ∝ t 1/2 , the Jordan frame scale factor is given by a J = aA = constant . (4.15) The solution for u is found by substituting the scaling solution into (3.19). We obtain u ∝ a − 3 2 + λ 4β , m 2 T ∝ a λ/β ,(4.16) and then the energy density of massive graviton varies as A 4 ρ G ∝ a −3+λ/2β . (4.17) The adiabatic condition (3.15) is guaranteed when ǫ ≪ 1. When 2β ≃ 5λ, the graviton mass roughly increases as m 2 T ∝ a 2/5 and the energy density of massive gravitons in the Einstein frame decreases as A 4 ρ G ∝ a −14/5 . Therefore, even if the energy density of massive gravitons is significantly lower than that of baryon, the correct dark matter-baryon ratio is realized in time since the energy density of massive gravitons decreases slower than that of baryon. C. Matter dominant era After φ reaches the potential minimum φ min , the chameleon field φ does not move during the MD universe. As shown in [16], when φ is constant, the massive graviton behaves as CDM and then the standard MD universe is obtained. D. Accelerating expanding era After the MD era, the universe must show the accelerating expansion due to dark energy. Although one can introduce a new field to obtain the acceleration, we consider a minimal scenario such that the chameleon field itself is dark energy, i.e., the accelerating expansion is realized by the potential V 0 . When V 0 becomes relevant to the dynamics of φ, the chameleon field again rolls down which leads to a decreasing of m T . As a result, the energy density of massive gravitons rapidly decreases and then we can ignore the contributions from massive gravitons. The basic equation during the accelerating expansion is thus given by 3M 2 p H 2 = A 4 ρ b + 1 2φ 2 + V 0 , (4.18) φ + 3Hφ − λ M p V 0 + β M p A 4 ρ b = 0 , (4.19) which yield a scaling solution 20) in which φ = 2M p λ ln t + constant , a ∝ t 2 3 (1+β/λ) ,(4.Ω b = λ 2 + βλ − 3 (β + λ) 2 , Ω φ = β 2 + βλ + 3 (β + λ) 2 ,(4.21) and w tot = − 2β 4β + λ . (4.22) The scaling solution exists when For 2β = 5λ, we find w tot = −5/11 and the inequality (4.23) is reduced into λ 2 > 6/7. The amplitude of the massive graviton is given by 24) and then the density parameter of massive graviton decreases as λ(β + λ) > 3 .u ∝ t − 1 2 (1+2β/λ) ,(4.Ω G ∝ t −1−2β/λ . (4.25) The energy density of massive graviton gives just a negligible contribution during this scaling solution which guarantees the equations (4.18) and (4.19). On the other hand, when λ 2 < 6/7, the non-minimal coupling is small so that the field φ can be approximated as a standard quintessence field. As a result, the acceleration is obtained by the slow-roll of φ and then the dark energy dominant universe is realized. V. COSMIC EVOLUTIONS In this section, we numerically solve the equations (3.17)- (3.19). We discuss two cases, the over-produced case (ρ G,i ≫ ρ b,i ) and the less-produced case (ρ G,i ≪ ρ b,i ), in order. A. Over-produced case First, we consider the over-produced case ρ G,i ≫ ρ b,i . We assume (4.1) which we call Model A. A cosmological dynamics is shown in Fig. 1. We set ρ G,i /ρ b,i = Ω G,i /Ω b,i = 200 at the initial of the numerical calculation. Although dark matter is initially over-produced, the energy density of baryon follows up that of dark matter and then we obtain ρ G /ρ b ≃ 5 when a J = Aa ∼ 10 −4 where we normalize the Jordan frame scale factor a J so that Ω φ \| aJ =1 = 0.7. We note the following-up of ρ b is obtained even if λ 2 > 2 is not satisfied (In Fig. 1, we set λ 2 = (6/5) 2 < 2). The dynamics of the universe is precisely tested by the CMB observations after the decoupling time a J ≃ 10 −3 . The evolutions of the total equation of state parameters are shown in Fig. 1. The dynamics of the observable universe is represented by the Jordan frame quantity w tot because the visible matters couple with the Jordan frame metric. On the other hand, since the dark matter (i.e., massive gravitons) is originated from the the gravity sector, dark matter feels the dynamics of the Einstein frame whose equation of state parameter is denoted by w E . Although the large deviation of dynamics from the standard cosmological one appears before the decoupling time a J 10 −3 , the standard dust dominant universe is recovered around the decoupling time. When we increase the values of β and λ, the deviation from the standard evolution is amplified which is caused by the oscillation of φ around φ min as shown in Fig, 2. Since the Jordan frame scale factor is given by a J = Aa = ae βφ/Mp , the oscillation of φ yields the oscillation of a J which is amplified by increasing of β. Fig. 1 does not show the dark energy "dominant" universe even in the accelerating phase. Instead, the acceleration is realized by the scaling solution as explained in Sec. IV. If this scaling solution can pass the observational constraints, it might give an answer for the other coincidence problem of dark energy: why the present dark energy density is almost same as that of matter? However, the cosmological dynamics after the decoupling time is strongly constrained by the observations. Thus, the dark energy model with the scaling solution should have a severe constraint (see [26,27] for examples). Furthermore, the large coupling α A M −1 p leads to that the Compton wavelength of the chameleon field has to be less than Mpc to screen the fifth force in the Solar System [28]; however, the coupling functions (4.1) require the Gpc scale Compton wavelength to give the current accelerating expansion. We then provide a model in which the couplings α A and α f are initially large but they become small in time. This behavior is realized by the model K 2 = (1 − φ 2 /M 2 ) −1 , A = e βφ/Mp , f = e −λφ/Mp , (5.1) which we call Model B. The only difference from Model A is that K is a function of φ. If the amplitude of the field φ is small at initial (φ ≪ M ), Model B gives the same behavior as Model A. After φ starts to roll and then \|φ\| → M , the kinetic function K increases which causes the decreasing of the non-minimal couplings α A , α f → 0 (see Figs. 2 and 3). Note that the field value is restricted in the range −M < φ < M in Model B which gives a constraint \|φ min \| < M to obtain ρ G /ρ b ≃ 5. A numerical solution of Model B is shown in Fig. 3. The evolutions of α f and α A are shown in the bottom of Fig. 3. As we expected, the couplings become weak in time. In particular, if φ min ≃ M in Model B, the cosmological dynamics is quite similar to that in the ΛCDM model after the decoupling time. The evolutions in Model A and Model B are divided into four regimes: the radiation dominant era (a J 10 −8 ), the following-up era (10 −8 a J 10 −3 ), the dust dominant era (10 −3 a J 10 −1 ), and the accelerating era (10 −1 a J ). In Model A, the deviations from the ΛCDM model appear in the following-up era and the accelerating era. On the other hand, the deviation appears only in the following-up era in Model B which is after the Big Bang Nucleosynthesis (BBN) a J ≃ 10 −8 and before the CMB last scattering surface a J ≃ 10 −3 . Since the relation between the temperature and the Hubble expansion rate during the BBN era is essentially the same as in the standard cosmology, Model B is compatible with the standard BBN. In Model B, evolution of perturbations at the CMB scales is also expected to be the same as in the standard cosmology. On the other hand, the non-standard evolution before the CMB last scattering surface may change the evolution of perturbations at smaller scales. These deviations may give observational constraints on our models or may help addressing some of the tensions between the standard ΛCDM and observational data at small scales. B. Less-produced case Next, we discuss the case of the less-produced dark matter ρ G,i ≪ ρ b,i . A numerical solution is shown in Fig. 4. Although the energy density of dark matter is initially smaller than that of baryon, the correct abundance (ρ G /ρ b ≃ 5) is obtained. Since the non-minimal coupling is not so large (β = 1 in the case of Fig. 4), the universe evolves into the dark energy dominant universe in the future. The following-up era is realized when Ω b ≃ 1/3β 2 in the scaling solution in which the dynamics of the universe is deviated from the standard one. Hence, the small value of β yields that the following-up era is close to the decoupling time and then the deviation may give a large effect on the CMB physics. For instance, Fig. 4 indicates that the deviation still exists at a J ≃ 10 −3 . In Model B, the coupling strength is time-dependent. Hence, we can obtain a scenario in which the nonminimal couplings are initially large whereas the couplings turn to be weak after the ratio ρ G /ρ b is dynamically adjusted to the observed value as shown in Fig. 5. In this case, the standard cosmological dynamics is re- When the bare potential V 0 becomes relevant to φ, the field value increases as shown in Fig. 6. In Model B, this increasing leads to the increasing of the non-minimal couplings α A and α f in the future a J ≫ 1. Needless to say, if the form of the bare potential V 0 is modified in order thatφ < 0 in the dark energy dominant era (for example, λ < 0), the coupling will be weak in the future as with Fig. 3. As a result, we can obtain a viable cosmological dynamics even in the less-produced case. Although the initial abundance of dark matter is much smaller than the observed value, the chameleon field provides the energy transfer from baryon to dark matter via the non-minimal couplings. The correct abundance of dark matter can be realized without the fine-tuning of the initial condition. VI. CONCLUDING REMARKS In the present paper, we provide a cosmological scenario by which the observed dark matter-baryon ratio can be naturally explained. We have added two new ingredients to the standard model: the massive graviton The evolution of the chameleon field φ with ρG,i/ρ b,i = 0.01 and φi = 0. We set β = 1, λ = 2/5 for Model A and β = 5, λ = 2, M = Mp for Model B. and the chameleon field corresponding to dark matter and dark energy, respectively. The matter fields are minimally coupled to the Jordan frame metric which leads to the non-minimal coupling of the matter fields to the chameleon field. On the other hand, the chameleon field may have a different coupling to dark matter since dark matter, the massive graviton, is originated from the gravity sector. We have assumed that the chameleon field has a non-minimal coupling to the massive graviton via the mass terms of the graviton. Two different non-minimal couplings of the chameleon field realize that the ratio ρ G /ρ b is dynamically relaxed to ∼ 5 without the tuning of the initial condition. We have studied two simple models: Eq. (4.1) (Model A) and Eq. (5.1) (Model B). 2 In these models, we have an additional era in the universe before the radiationmatter equality which we have called the following-up era. Even if dark matter is initially over-produced or less-produced, the chameleon field transfers the energy of the larger one of dark matter and baryon into that of the smaller one via the non-minimal couplings. After the ratio ρ G /ρ b is tuned, the standard cosmological dynamics can be recovered. To realize the following-up era, the non-minimal couplings should be at least comparable to the usual gravitational interaction, \|α A \|, \|α f \| ∼ M −1 p . This "large" non-minimal couplings may lead to a deviation from the ΛCDM cosmology. However, in Model B, the coupling strengths are time-dependent and then the present nonminimal couplings can be small as shown in Fig. 3 and Fig. 5. The chameleon bigravity theory (2.1) was originally proposed in [17] in order to avoid the Higuchi instability of cosmological solutions [29][30][31][32][33][34][35][36][37] and to circumvent the low cutoff scale of the standard bigravity theory. When the graviton mass is smaller than the Hubble expansion rate, the scalar mode of the massive graviton exhibits the ghost instability or the gradient instability, in general. However, the chameleon bigravity can yield a cosmological solution in which m T /H is constant during the radiation dominant universe and then the homogeneous and isotropic solution does not suffer from the instability even in the early universe [17,38]. The cutoff scale of the gravity sector also becomes high in the early universe and thus this mechanism make it possible for us to apply the bigravity theory to the early universe. Since we have considered only the cases with m T ≫ H, the Higuchi instability is not problematic for the present discussion and thus we have not discussed the stability issue of the homogeneous spacetimes (3.1) and (3.2). Never-theless, it would be interesting to study whether or not the chameleon bigravity also gives a stable homogeneous but anisotropic solution even in the early epoch of the universe. As discussed in [13,16], the massive graviton with a constant mass is a viable dark matter candidate in the wide range of the mass 10 −23 eV m 10 7 (M g /M f ) 2/3 eV. The lower bound is imposed so that dark matter halos form in dwarf galaxy scales, while the upper bound is given by the requirement that the lifetime of the massive graviton be longer than the age of the universe. In the chameleon bigravity theory studied in the present paper, by the same argument as before, we obtain the lower bound on the present value of m T , i.e., m T \| aJ =1 10 −23 eV. The lifetime of the massive graviton also constrains the present value of the mass as follows. Since the abundance of dark matter is automatically tuned to the observed value by the scaling solution, we need to discuss the lifetime of the massive graviton only after the following-up era in our scenario 3 . The graviton mass remains almost constant during this epoch. As a result, the requirement of a long enough lifetime gives the upper bound on the present value of the mass as m T \| aJ =1 10 7 (M g /M f ) 2/3 eV. Since we have not discussed concrete observational constraints on our models, the observational viability of the models is an open question. Theoretically, our model can explain the dark matter-baryon ratio for any initial condition except for ρ G,i = 0 or ρ b,i = 0 because the following-up era can be the scaling solution and then it never ends unless the smaller one of ρ g and ρ b catches up the larger one. However, for example, ρ G,i /ρ b,i ≃ 200 should be an observational upper bound on the initial ratio in the over-produced case with β = 3, λ = 6/5. If ρ G,i /ρ b,i 200, the dynamics of the universe at BBN is changed from the radiation dominant universe and then it may give an observational constraint. Furthermore, we should take into account local gravity constraints of the fifth force since the following-up of dark matter or baryon requires a large non-minimal coupling of the chameleon field. Although the current non-minimal coupling can be small in Model B, there still exists a small fifth force and then the local gravity experiments may give a constraint on our model as well. 4 We leave the details of the observational constraints of our models for a future work. FIG. 1 . 1The evolution of the density parameters and the total equation of state parameters in terms of the Jordan frame scale factor aJ = Aa which is normalized to be Ω φ \|a J =1 = 0.7. We set β = 3 and λ = 2β/5 = 6/5 in Model A (4.1). We assume the initial ratio between dark matter and baryon as ρG,i/ρ b,i = ΩG,i/Ω b,i = 200 with φi = 0.FIG. 2. The evolution of the chameleon field φ in Model A and Model B with ρG,i/ρ b,i = 200. FIG. 3 . 3The same figures asFig. 1and the evolution of αA and α f in Model B. We set β = 3, λ = 6/5 and M = Mp. We assume the same initial condition asFig. 1. FIG. 4 . 4The evolution of the density parameters and the total equation of state parameter. We set β = 1 and λ = 2/5 in Model A. We assume the initial ratio between dark matter and baryon as ρG,i/ρ b,i = 0.01 with φi = 0. covered around a J ≃ 10 −4 . Although the dynamics is deviated from the standard one in 10 −6 a J 10 −4 , it should have little effect on CMB. FIG. 5 . 5The same figures as Fig. 3 in the less-produced case. We set β = 5, λ = 2 and M = Mp, and assume ρG,i/ρ b,i = 0.01 with φi = 0. FIG. 6. Although we chose exponential forms of A and f , other types of the non-minimal coupling can be discussed. Even in this case, the dark matter-baryon ratio could be explained when α A /α f ≃ −5/2. Decay of massive graviton during and after the nucleosynthesis but before the end of the following-up era may have some impacts on the primordial abundances of the light elements. We leave studies of such effects to future publications.4 The massive graviton also yields a fifth force. We should also discuss the Vainshtein screening mechanism of the massive graviton. However, if the graviton mass is high enough (e.g., m T \| a J =1 ≫ 10 −4 eV), the fifth force propagated by the massive graviton does not exist in the observed scales due to the Yukawa suppression. ACKNOWLEDGMENTSThe work of K.A. was supported in part by Grantsin-Aid from the Scientific Research Fund of the Japan Society for the Promotion of Science (No. 15J05540). The work of S.M. was supported in part by JSPS Grant-in-Aid for Scientific Research No. 17H02890, No. 17H06359, and by World Premier International ResearchCenter Initiative (WPI), MEXT, Japan.Appendix A: Bianchi I universe We consider the axisymmetric Bianchi I universe(3.1)and(3.2). We consider the matter field minimally coupled withg µν whose energy-momentum tensor is given by the form(3.3)and the conservation law is given by(3.4). We find following equations: the Friedmann equationsthe equations for the shearsthe equation for the chameleon fieldand the constraintwhere we have definedWe then expand the equations in terms of σ. Note that when the anisotropies are dominant components of the universe the amplitude of σ is given bywith ǫ = H/m T . Hence, the small anisotropy can be the dominant component when the graviton mass is larger than the Hubble expansion rate (ǫ ≪ 1). 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[ "A Richer Theory of Convex Constrained Optimization with Reduced Projections and Improved Rates", "A Richer Theory of Convex Constrained Optimization with Reduced Projections and Improved Rates" ]	[ "Tianbao Yang \nThe University of Iowa\n52242Iowa CityIAUSA\n", "Qihang Lin \nThe University of Iowa\n52242Iowa CityIAUSA\n", "Lijun Zhang \nNational Key Laboratory for Novel Software Technology\nNanjing Univer-sity\n210023NanjingChina\n" ]	[ "The University of Iowa\n52242Iowa CityIAUSA", "The University of Iowa\n52242Iowa CityIAUSA", "National Key Laboratory for Novel Software Technology\nNanjing Univer-sity\n210023NanjingChina" ]	[ "Proceedings of the 34 th International Conference on Machine Learning" ]	This paper focuses on convex constrained optimization problems, where the solution is subject to a convex inequality constraint. In particular, we aim at challenging problems for which both projection into the constrained domain and a linear optimization under the inequality constraint are time-consuming, which render both projected gradient methods and conditional gradient methods (a.k.a. the Frank-Wolfe algorithm) expensive. In this paper, we develop projection reduced optimization algorithms for both smooth and non-smooth optimization with improved convergence rates under a certain regularity condition of the constraint function. We first present a general theory of optimization with only one projection. Its application to smooth optimization with only one projection yields O(1/ǫ) iteration complexity, which improves over the O(1/ǫ 2 ) iteration complexity established before for nonsmooth optimization and can be further reduced under strong convexity. Then we introduce a local error bound condition and develop faster algorithms for non-strongly convex optimization at the price of a logarithmic number of projections. In particular, we achieve an iteration complexity of O(1/ǫ 2(1−θ) ) for non-smooth optimization and O(1/ǫ 1−θ ) for smooth optimization, where θ ∈ (0, 1] appearing the local error bound condition characterizes the functional local growth rate around the optimal solutions. Novel applications in solving the constrained ℓ 1 minimization problem and a positive semi-definite constrained distance metric learning problem demonstrate that the proposed algorithms achieve significant speed-up compared with previous algorithms.	null	[ "https://arxiv.org/pdf/1608.03487v2.pdf" ]	35,608,031	1608.03487	9b7e2faf94310aaadc4f14d15b8ba3836914ee7c	A Richer Theory of Convex Constrained Optimization with Reduced Projections and Improved Rates 2017 Tianbao Yang The University of Iowa 52242Iowa CityIAUSA Qihang Lin The University of Iowa 52242Iowa CityIAUSA Lijun Zhang National Key Laboratory for Novel Software Technology Nanjing Univer-sity 210023NanjingChina A Richer Theory of Convex Constrained Optimization with Reduced Projections and Improved Rates Proceedings of the 34 th International Conference on Machine Learning the 34 th International Conference on Machine LearningSyd-ney, Australia2017Correspondence to: Tianbao Yang <[email protected]>. This is the long version of our paper appearing in the This paper focuses on convex constrained optimization problems, where the solution is subject to a convex inequality constraint. In particular, we aim at challenging problems for which both projection into the constrained domain and a linear optimization under the inequality constraint are time-consuming, which render both projected gradient methods and conditional gradient methods (a.k.a. the Frank-Wolfe algorithm) expensive. In this paper, we develop projection reduced optimization algorithms for both smooth and non-smooth optimization with improved convergence rates under a certain regularity condition of the constraint function. We first present a general theory of optimization with only one projection. Its application to smooth optimization with only one projection yields O(1/ǫ) iteration complexity, which improves over the O(1/ǫ 2 ) iteration complexity established before for nonsmooth optimization and can be further reduced under strong convexity. Then we introduce a local error bound condition and develop faster algorithms for non-strongly convex optimization at the price of a logarithmic number of projections. In particular, we achieve an iteration complexity of O(1/ǫ 2(1−θ) ) for non-smooth optimization and O(1/ǫ 1−θ ) for smooth optimization, where θ ∈ (0, 1] appearing the local error bound condition characterizes the functional local growth rate around the optimal solutions. Novel applications in solving the constrained ℓ 1 minimization problem and a positive semi-definite constrained distance metric learning problem demonstrate that the proposed algorithms achieve significant speed-up compared with previous algorithms. Introduction In this paper, we aim at solving the following convex constrained optimization problem: min x∈R d f (x), s.t. c(x) ≤ 0, (1) where f (x) is a smooth or non-smooth convex function and c(x) is a lower-semicontinuous and convex function. The problem can find applications in machine learning, signal processing, statistics, marketing optimization, and etc. For example, in distance metric learning one needs to learn a positive semi-definite (PSD) matrix such that similar examples are close to each other and dissimilar examples are far from each other (Weinberger et al., 2006;Xing et al., 2003), where the positive semi-definite constraint can be cast into a convex inequality constraint. Another example arising in compressive sensing is to minimize the ℓ 1 norm of high-dimensional vector subject to a measurement constraint (Candès & Wakin, 2008). Although general interior-point methods can be applied to solve the problem with linear convergence, they suffer from exceedingly high computational cost per-iteration. Another solution is to employ the projected gradient (PG) method (Nesterov, 2004a) or the conditional gradient (CG) method (Frank & Wolfe, 1956), where the PG method needs to compute the projection into the constrained domain at each iteration and CG needs to solve a linear optimization problem under the constraint. However, for many constraints (e.g., PSD, quadratic constraints) both projection into the constrained domain and the linear optimization under the constraint are time-consuming, which restrict their capabilities to solving these problems. Recently, there emerges a new direction towards addressing the challenge of expensive projection that is to reduce the number of projections. In the seminal paper (Mahdavi et al., 2012), the authors have proposed two algorithms with only one projection at the end of iterations for non-smooth convex and strongly convex optimization, respectively. The idea of both algorithms is to move the constraint function into the objective function and to control the violation of constraint for intermediate solutions. While their developed algorithms enjoy an optimal convergence rate for non-smooth optimization (i.e., O(1/ǫ 2 ) iteration complexity) and a close-to-optimal convergence rate for strongly convex optimization (i.e., O(1/ǫ) 1 ), there still lack of theory and algorithms with reduced projections and faster rates for smooth convex optimization and for convex optimization without strong convexity assumptions. In this paper, we make significant contributions by developing a richer theory of convex constrained optimization with reduced projections and faster rates. To be specific, • we develop a general framework and theory of optimization with only one projection, where any favorable smooth or non-smooth convex optimization algorithms can be employed to solve the intermediate augmented unconstrained objective function. We discuss in full details the applicability of the proposed algorithms to problems with polyhedral, quadratic or PSD constraints. • Applying the general theory to smooth convex optimization 2 with Nesterov's accelerated gradient methods yields an iteration complexity of O(1/ǫ) with only one projection. In addition, when equipped with an optimal algorithm for strongly convex optimization the general theory implies the optimal iteration complexity of O(1/ǫ) for strongly convex optimization with only one projection. For smooth and strongly convex optimization, the general theory implies an iteration complexity of O(1/ǫ β ) where β ∈ (1/2, 1) with only one projection and a sufficiently large number of iterations. • Building on the general framework and theory, we further develop an improved theory with faster convergence rates for non-strongly convex optimization at the price of a logarithmic number of projections. In particular, we show that under a mild local error bound condition, the iteration complexities can be reduced to O(1/ǫ 2(1−θ) ) for non-smooth optimization and O(1/ǫ 1−θ ) for smooth optimization, where θ ∈ (0, 1] is a constant in the local error bound condition that characterizes the local growth rate of functional values. To our knowledge, these are the best convergence results with only a logarithmic number of projections for non-strongly convex optimization. We also demonstrate their effectiveness for solving compressive sensing and distance metric learning problems. Related Work The issue of high projection cost in projected gradient descent has received increasing attention in recent years. Most studies are based on the Frank-Wolfe technique that eschews the projection in favor of a linear optimization over the constrained domain (Jaggi, 2013;Hazan & Kale, 2012;Lacoste-Julien et al., 2013;Garber & Hazan, 2015). It happens that for many bounded domains (e.g., bounded balls for vectors and matrices, a PSD constraint with a 1 where O() suppresses a logarithmic factor. 2 where the constraint function is assumed to be smooth. bounded trace norm) the linear optimization over the constrained domain is much cheaper than projection into the constrained domain (Jaggi, 2013). However, there still exist many constraints that render both projection into the constrained domain and linear optimization under the constraint are comparably expensive. Examples include polyhedral constraints, quadratic constraints and a PSD constraint 3 . To tackle these complex constraints, the idea of optimization with a reduced number of projections was explored in several studies since (Mahdavi et al., 2012). In a recent paper (Chen et al., 2016), the authors show that for stochastic strongly convex optimization, the optimal convergence rate can be achieved using a logarithmic number of projections. In contrast, the developed theory in this paper implies that only one projection is sufficient to achieve the optimal convergence rate for strongly convex optimization, and a logarithmic number of projections can be used to accelerate convergence rates for non-strongly convex optimization. Cotter et al. (2016) proposed a stochastic algorithm for solving heavily constrained problems with many constraint functions by extending the work of (Mahdavi et al., 2012). Nonetheless, their focus is not to improve the convergence rates. Zhang et al. (2013) studied the smooth and strongly convex optimization and they proposed a stochastic algorithm with O(κ log(T )) projections and proved an O(1/T ) convergence rate, where κ is the condition number and T is the total number of iterations. Nonetheless, if the condition number is high the number of projections could be very large. In addition, their algorithm utilizes the mini-batch to avoid frequent projections in stochastic optimization, which is different from the present paper. We note that several recent works also exploit different forms of error bound conditions to improve the convergence (Wang & Lin, 2014;So, 2013;Hou et al., 2013;Zhou et al., 2015;Yang & Lin, 2016;Xu et al., 2016). Most notably, the technique used in our work is closely related to (Yang & Lin, 2016). However, for constrained optimization problems the methods in (Yang & Lin, 2016) still need to conduct projections at each iteration. Finally, we comment on the differences between the proposed methods and the classical penalty methods that also move the constraint into the objective using a penalty function (Bertsekas, 1996). The major differences are that (i) the classical penalty methods typically require solving each subproblem exactly while our methods do not require that; and (ii) the classical penalty methods typically guarantee asymptotic convergence while our methods have explicit convergence rates. Preliminaries Let Ω = {x ∈ R d : c(x) ≤ 0} denote the constrained domain, Ω * denote the optimal solution set and f * denote the optimal objective value. We denote by ∇f (x) the gradient and by ∂f (x) the subgradient of a smooth or non-smooth function, respectively. When f (x) is a non-smooth function, we consider the problem as non-smooth constrained optimization. When both f (x) and c(x) are smooth, we consider the problem as smooth constrained optimization. A function f (x) is L-smooth if it has a Lipschitz continu- ous gradient, i.e., ∇f (x) − ∇f (y) ≤ L x − y , where · denotes the Euclidean norm. A function f (x) is µ- strongly convex if it satisfies f (x) ≥ f (y) + ∂f (y) ⊤ (x − y) + µ 2 x − y 2 . In the sequel, dist(x, Ω) denotes the distance of x to a set Ω, i.e., dist(x, Ω) = min u∈Ω x − u . Let [s] + be a hinge operator that is defined as [s] + = s if s ≥ 0, and [s] + = 0 if s < 0. Throughout the paper, we make the the following assumptions to facilitate the development of our algorithms and theory. Assumption 1. For a convex minimization problem (1), we assume (i) there exists a positive value ρ > 0 such that min c(x)=0 v∈∂c(x),v =0 v ≥ ρ,(2) or more generally there exists a constant ρ > 0 for any x ∈ R d , such that x ♮ = arg min u∈R d ,c(u)≤0 u − x 2 satisfies x ♮ − x ≤ [c(x)] + /ρ.(3) (ii) there exists a strictly feasible solution such that c(x) < 0; (iii) both f (x) and c(x) are defined everywhere and are Lipschitz continuous with their Lipschitz constants denoted by G and G c , respectively. We make several remarks about the assumptions. The inequality in (2) is introduced in (Mahdavi et al., 2012), which is to ensure the distance from the final solution before projection to constrained domain Ω is not too large. Note that the inequality in (3) is a more general condition than (2) as seen from the following lemma. Lemma 1. For any x ∈ R d , let x ♮ = arg min c(u)≤0 u − x 2 . If (2) holds, then (3) holds. The above lemma is implicit in the proof of (Mahdavi et al., 2012). We will provide more discussions about Assumption 1(i) -the key assumption, and exhibit the value of ρ for a number of commonly seen constraints (e.g., polyhedral, quadratic and PSD constraints). To make the presentation more fluent, we postpone these discussions to Section 6. The strict feasibility assumption (ii) allows us to explore the KKT condition of the projection problem shown below. Assumption (iii) imposes mild Lipschitz continuity conditions on both f (x) and c(x). Traditional projected gradient descent methods need to solve the following projection at each iteration Π Ω [x] = arg min c(u)≤0 u − x 2 . Conditional gradient methods (a.k.a. the Frank-Wolfe technique) need to solve the following linear optimization at each iteration min u∈R d ,c(u)≤0 u ⊤ ∇f (x). For many constraint functions (see Section 6), solving the projection problem and the linear optimization could be very expensive. A General Theory of Optimization with only one projection In this section, we extend the idea of only one projection proposed in (Mahdavi et al., 2012) to a general theory, and then present optimization algorithms with only one projection for non-smooth and smooth optimization, respectively. To tackle the constraint, we introduce a penalty function h γ (x) parameterized by γ, which obeys the following certificate: there exist constants C ≥ 0 and λ > G/ρ such that h γ (x) ≥ λ[c(x)] + , ∀x h γ (x) ≤ Cγ, ∀x such that c(x) ≤ 0.(4) From the above condition, it is clear that γ ≥ 0. It is notable that the penalty function h γ (x) will also depend on λ; however it will be set to a constant value, thus the dependence on λ is omitted. We will construct such a penalty function h γ (x) for non-smooth and smooth optimization in next two subsections. We propose to optimize the following augmented objective function min x∈R d F γ (x) = f (x) + h γ (x).(5) We can employ any applicable optimization algorithms to optimize F γ (x) pretending that there is no constraint, and finally obtain a solution x T that is not necessarily feasible. In order to obtain a feasible solution, we perform one projection to get x T = Π Ω ( x T ). The following theorem allows us to convert the convergence of x T for F γ (x) to that of x T for f (x). Theorem 1. Let A be any iterative optimization algorithm applied to min x F γ (x) with T iterations, which starts with x 1 and returns x T as the final solution, such that the following convergence of x T holds for any x ∈ R d F γ ( x T ) − F γ (x) ≤ B T (γ; x, x 1 ),(6) where B T (γ; x, x 1 ) → 0 when T → ∞. Suppose that Assumption 1 hold, then f ( x T ) − f (x * ) ≤ λρ λρ − G (Cγ + B T (γ; x * , x 1 )),(7) where x T = Π Ω [ x T ] and x * is an optimal solution to (1). Remark: It is worth mentioning that we omit some constant factors in the convergence bound B T (γ; x, x 1 ) that are irrelevant to our discussions. The notation B T (γ; x, x 1 ) emphasizes that it is a function of γ and depends on x 1 and a target solution x and it will be referred to as B T . In the next several subsections, we will see that by carefully choosing the penalty function h γ (x) we are able to provide nice convergence for smooth and non-smooth optimization with only one projection. In the above theorem, we assume the optimization algorithm A is deterministic. However, a similar result can be easily extended to a stochastic optimization algorithm A. Proof. First, we consider c( x T ) ≤ 0, which implies that x T = x T . Due to the certificate of h γ (x), F γ ( x T ) ≥ f ( x T ) and F γ (x * ) ≤ f (x * ) + Cγ. Hence f ( x T ) ≤ F γ ( x T ) ≤ F γ (x * ) + B T (γ; x 1 , x * ) ≤ f (x * ) + Cγ + B T (γ; x 1 , x * ) . Then (7) follows due to λρ/(λρ − G) ≥ 1. Next, we assume c( x T ) > 0. Inequality (6) implies that f ( x T ) + λ[c( x T )] + ≤ f (x * ) + Cγ + B T (γ; x * , x 1 ). (8) By Assumption 1(i), we have [c( x T )] + ≥ ρ x T − x T . Combined with (8) we have λρ x T − x T ≤ f (x * ) − f ( x T ) + Cγ + B T (γ; x * , x 1 ) ≤ G x T − x T + Cγ + B T (γ; x * , x 1 ), where the last inequality follows that fact f ( x * )−f ( x T ) ≤ f (x * ) − f ( x T ) + f ( x T ) − f ( x T ) ≤ G x T − x T because the Lipschitz property and f (x * ) ≤ f ( x T ). Therefore we have x T − x T ≤ Cγ + B T (γ; x * , x 1 , ) λρ − G . Finally, we obtain f ( x T ) − f (x * ) ≤ f ( x T ) − f ( x T ) + f ( x T ) − f (x * ) ≤ G x T − x T + Cγ + B T (γ; x * , x 1 ) ≤ λρ λρ − G (Cγ + B T (γ; x * , x 1 )). Non-smooth Optimization Since an optimal convergence rate for general non-smooth optimization with only one projection has been attained in (Mahdavi et al., 2012), in this subsection we present an optimal convergence result for strongly convex problems. For non-smooth optimization, we can choose h(x) = λ[c(x)] + , and hence γ = 0. We will use deterministic subgradient descent as an example to demonstrate the convergence for f (x), though many other optimization algorithms designed for non-smooth optimization are applicable (e.g., the stochastic subgradient method). The update of subgradient descent method is given by the following x t+1 = x t − η t ∂F (x t ), t = 1, . . . , T,(9) where η t is an appropriate step size. If f (x) is µ-strongly convex, the step size can be set as η t = 1/(µt) and the final solution can be computed by the α-suffix averaging Rakhlin et al., 2012), or by the polynomial decay averaging with x T = 1 αT T t=(1−α)T +1 x t with α > 0 (x t = (1− s+1 s+t ) x t−1 + s+1 s+t x t and s ≥ 1 (Shamir & Zhang, 2013). Both schemes can attain B T = O(1/(µT )) for the conver- gence of F (x) when f (x) is µ-strongly convex. Combining this with Theorem 1, we have the following convergence result with the proof omitted due to its simplicity. Corollary 2. Suppose that Assumption 1 holds and f (x) is µ-strongly convex. Set F (x) = f (x) + λ[c(x)] + with λ ≥ G/ρ. Let (9) run for T iterations with η t = 1/(µt) . Let x T be computed by α-suffix averaging or the polynomial decay averaging. Then with only one projection x T = Π Ω ( x T ), we achieve f ( x T ) − f * ≤ λρ λρ − G (G + λG c ) 2 O(1) µT . Remark: We note that the O(1/(µT )) is also achieved for strongly convex optimization in (Zhang et al., 2013;Chen et al., 2016) but with a logarithmic number of projections. In contrast, Corollary 2 implies only one projection is sufficient to achieve the optimal convergence for strongly convex optimization. Smooth Optimization For smooth optimization, we consider both f (x) and c(x) to be smooth 4 . Let the smoothness parameter of f (x) and c(x) be L f and L c , respectively. In order to ensure the augmented function F γ (x) to be still a smooth function, we construct the following penalty function h γ (x) = γ ln (1 + exp (λc(x)/γ)) .(10) The following proposition shows that h γ (x) is a smooth function and obeys the condition in (4). Proposition 1. Suppose c(x) is L c -smooth and G c - Lipschitz continuous. The penalty function in (10) is a (λL c + λ 2 G 2 c 4γ )-smooth function and satisfies (i) h γ (x) ≥ λ[c(x)] + and (ii) h γ (x) ≤ γ ln 2, ∀x such that c(x) ≤ 0. Then F γ (x) is a smooth function and its smoothness parameter is given by L F = L f + λL c + λ 2 G 2 c 4γ . Next, we will establish the convergence for f (x) using Nesterov's optimal accelerated gradient (NAG) methods. The update of one variant of NAG can be written as follows x t+1 = y t − ∇F γ (y t )/L F y t+1 = x t+1 + β t+1 (x t+1 − x t ),(11) where the value of β t can be set to different values depending on whether f (x) is strongly convex or not (see Corollary 3). Previous work have established the conver- gence of x T = x T for F γ (x), in particular B T = O( LF T 2 ) for smooth non-strongly convex optimization and B T = O L F exp(−T µ LF ) for smooth and strongly convex optimization. By combining these results with Theorem 1 and appropriately setting γ, we can achieve the following convergence of x T for f (x). (10). Let (11) run for T iterations and Corollary 3. Suppose that Assumption 1 holds, dist(y 0 , Ω * ) ≤ D, f (x) is L f -smooth and c(x) is L c -smooth. Set F γ (x) = f (x) + h γ (x) with λ > G/ρ and h γ (x) beingx T = Π Ω (x T ). • If f (x) is convex, we can set γ = λGcD (T +1) √ 2 ln 2 , β t = τt−1−1 τt , where τ t = 1+ √ 1+4τ 2 t−1 2 with τ 0 = 1, and achieve f ( x T )−f * ≤ λρ λρ − G λG c D √ 2 ln 2 T + 1 + (L f + λL c )D 2 (T + 1) 2 • If f (x) is µ-strongly convex, we can set γ = 1 T 2α with α ∈ (1/2, 1) and β t = √ LF − √ µ √ LF + √ µ , and achieve f ( x T ) − f * ≤ O 1 T 2α + 1 T 4α , as long as T ≥ L f +λLc+λ 2 G 2 c /4 µ 1 2(1−α) (4α ln T ) 1 1−α . Remark: The convergence results above indicate an O(1/ǫ) iteration complexity for smooth optimization and O(1/ǫ 1/(2α) ) with α ∈ (1/2, 1) for smooth and strongly convex optimization with only one projection. Omitted proofs can be found in appendix. Improved Convergence for Non-strongly Convex Optimization In this section, we will develop improved convergence for non-strongly convex optimization at a price of a logarithmic number of projections by considering an additional condition on the target problem. To facilitate the presentation, we first introduce some notations. The ǫ-sublevel set S ǫ and ǫ-level set L ǫ of the problem (1) are denoted by S ǫ = {x ∈ Ω : f (x) ≤ f * + ǫ}, and L ǫ = {x ∈ Ω : f (x) = f * + ǫ}, respectively. Let x † ǫ denote the closest point in the ǫ-sublevel set S ǫ to x ∈ Ω, i.e., x † ǫ = arg min u∈Ω u − x 2 , s.t. f (u) ≤ f * + ǫ.(12) Let x * denote the closest optimal solution in Ω * to x, i.e., x * = arg min u∈Ω * u − x 2 . In this section, we will make the following additional assumption about the problem (1). Assumption 2. For a convex minimization problem (1), we (1) satisfies a local error bound condition, i.e., there exist θ ∈ (0, 1] and σ > 0 such that for any assume (i) there exist x 0 ∈ Ω and ǫ 0 ≥ 0 such that f (x 0 )− min x∈Ω f (x) ≤ ǫ 0 ; (ii) Ω * is a non-empty convex compact set; (iii) the optimization problemx ∈ S ǫ we have dist(x, Ω * ) ≤ σ(f (x) − f * ) θ where Ω * denotes the optimal set and f * denotes the optimal value. Remark: we would like to remark that the new assumption only imposes mild conditions on the problem. In particular, Assumption 2 (i) supposes there is a lower bound of the optimal value f * , which usually holds in machine learning applications where the objective function if non-negative; Assumption 2 (ii) ensures that S ǫ is also bounded (Rockafellar, 1970), therefore the σ in the local error bound is finite, which can be easily satisfied for a norm regularized or constraint problems; the local error bound condition holds for a broad family of functions (e.g., semi-algebraic functions or real subanalytic functions (Jerome Bolte, 2015; Yang & Lin, 2016)). In Section 7, we will also demonstrate several applications of the improved algorithms proposed in this section by establishing the local error bound condition. Although the local error bound condition is much weaker than the strong convexity assumption, below we will propose novel algorithms leveraging this condition with faster convergence and only a logarithmic number of projections. Non-smooth Optimization To establish an improved convergence for non-smooth optimization, we develop a new algorithm shown in Algorithm 1 based on subgradient descent (GD) method, to which we refer as LoPGD. The algorithm runs for K epochs and each epoch employs GD for minimizing F (x) = f (x) + λ[c(x)] + with a feasible solution x k−1 ∈ Ω as a starting point and t iterations of updates. At the end of each epoch, the averaged solution x k is projected into the constrained domain Ω and the solution x k will be used as the starting point for next epoch. The step size η k is decreased by half every epoch starting from a given value η 1 . The theorem below establishes the iteration complexity of LoPGD and also exhibits the values of K, t and η 1 . To simplify notations, we let p = λρ λρ−G andḠ = G + λG c . Theorem 4. Suppose Assumptions 1 and 2 hold. Let η 1 = ǫ0 2pḠ 2 , K = ⌈log 2 (ǫ 0 /ǫ)⌉ and t = 4σ 2 p 2Ḡ2 ǫ 2(1−θ) in Algorithm 1, where θ and σ are constants appearing in the local error bound condition. Then f (x K ) − f * ≤ 2ǫ. Algorithm 1 LoPGD 1: INPUT: K ∈ N + , t ∈ N + , η 1 2: Initialization: x 0 ∈ Ω, ǫ 0 3: for k = 1, 2, . . . , K do 4: Let x k 1 = x k−1 5: for s = 1, 2, . . . , t − 1 do 6: Update x k s+1 = x k s − η k ∂F (x k s ) 7: end for 8: Let x k = t s=1 x k s /t 9: Let x k = Π Ω [ x k ] and η k+1 = η k /2 10: end for Algorithm 2 LoPNAG 1: INPUT: K ∈ N + , t 1 , . . . , t K ∈ N + , γ 1 2: Initialization: x 0 ∈ Ω, ǫ 0 3: for k = 1, 2, . . . , K do 4: Let y k 0 = x k−1 5: for s = 0, 1, 2, . . . , t k − 1 do 6: Update x k s+1 = y k s − 1 L k ∇F γ k (x k s ) 7: Update y k s+1 = x k s+1 + β s+1 (x k s+1 − x k s ) 8: end for 9: Let x k = x k t k , x k = Π Ω [ x k ] and γ k+1 = γ k /2 10: end for Remark: Since the projection is only conducted at the end of each epoch and the total number of epochs is at most K = ⌈log 2 (ǫ 0 /ǫ)⌉, so the total number of projections is only a logarithmic number K. The iteration complexity in Theorem 4 is O(1/ǫ 2(1−θ) ) that improves the standard result of O(1/ǫ 2 ) without strong convexity. With θ = 1/2, we can achieve O(1/ǫ) iteration complexity with only O(log(1/ǫ)) projections. Smooth Optimization Similar to non-smooth optimization, we also develop a new algorithm based on NAG shown in Algorithm 2, where F γ (x) is defined using h γ (x) in (10), L k = L Fγ k is the smoothness parameter of F γ k and β s = τs−1−1 τs , s = 1, . . . , is a sequence with τ s updated as in Corollary 3. We refer to this algorithm as LoPNAG. The key idea is to use to a sequence of reducing values for γ k instead of using a small value as in Corollary 3, and solve each augmented unconstrained problem F γ k (x) approximately with one projection. The theorem below exhibits the iteration complexity of LoPNAG and reveals the values of K, γ 1 and t 1 , . . . , t K . To simplify notations, we letL = L f + λL c . Remark: It is not difficult to show that the total number of iterations is bounded by O(1/ǫ 1−θ ), which improves the one in Corollary 3 without strong convexity. If f (x) is a simple non-smooth function whose proximal mapping can be easily computed (e.g., ℓ 1 norm), we can replace step 6 in Algorithm 2 by a proximal mapping to handle f (x), which gives the same convergence result in Theorem 5. An example is presented in Section 7 for compressive sensing with θ = 1/2. Discussion of Assumption 1 (i) One might note that a key condition for developing the theory with reduced projections is Assumption 1 (i). Although Mahdavi et al. (2012) has briefly mentioned that the condition can be satisfied for a PSD cone or a Polytope (a bounded polyhedron), their discussion lacks of details in particular on the value of ρ in (2) or (3). Below, we discuss the condition in details about three types of constraints. Polyhedral constraints. First, we show that when c(x) is a polyhedral function, i.e., its epigraph is a polyhedron (not necessarily bounded), the inequality (3) is satisfied. To this end, we explore the polyhedral error bound (PEB) condition (Gilpin et al., 2012;Yang & Lin, 2016). In particular, if we consider an optimization problem, min x∈R d h(x), where the epigraph of h(x) is polyhedron. Let H * denote the optimal set and h * denote the optimal value of the problem above. The PEB says that there exists ρ > 0 such that for any x ∈ R d dist(x, H * ) ≤ (h(x) − h * )/ρ.(13) To show that the inequality (3) holds for a polyhedral function c(·), we can consider the optimization problem min x∈R d [c(x)] + . The optimal set of the above problem is given by H * = {x ∈ R d : c(x) ≤ 0}. For any x such that c(x) > 0, let x ♮ = arg min c(u)≤0 u − x 2 be the closest point in the optimal set to x. Therefore if c(·) is a polyhedral function so does [c(x)] + , by the PEB condition (13) there exists a ρ > 0 such that x − x ♮ ≤ ([c(x)] + − min x [c(x)] + )/ρ = [c(x)] + /ρ. Let us consider a concrete example, where the problem has a set of affine inequalities c ⊤ i x − b i ≤ 0, i = 1, . . . , m. There are two methods to encode this into a single constraint function c(x) ≤ 0. The first method is to use (3) is then guaranteed by Hoffman's bound and the parameter ρ is given by the minimum non-zero eigenvalue of C ⊤ C (Wang & Lin, 2014). Note that the projection onto a polyhedron is a linear constrained quadratic programming problem, and the linear optimization over a polyhedron is a linear programming problem. Both have polynomial time complexity that would be high if m and d are large (Karmarkar, 1984;Kozlov et al., 1980). Quadratic constraint. A quadratic constraint can take the form of Ax − y 2 ≤ τ , where A ∈ R m×d and y ∈ R m . Such a constraint appears in compressive sensing (Candès & Wakin, 2008) 5 , where the goal is to reconstruct a sparse high-dimensional vector x from a small number of noisy measurements y = Ax + ε ∈ R m with m ≪ d. The corresponding optimization problem is c(x) = max 1≤i≤m c ⊤ i x − b i ,min x∈R d x 1 , s.t. Ax − y 2 ≤ τ.(14) where τ ≥ ε 2 is an upper bound on the magnitude of the noise. To check the Assumption 1(i), we note that c(x) = Ax− y 2 − τ and ∇c(x) = A ⊤ (Ax− y). Let us consider that A has a full row rank 6 and denote by v = Ax − y, then on the boundary c(x) = 0 we have v = √ τ and A ⊤ v ≥ τ λ min (AA ⊤ ), where λ min (AA ⊤ ) > 0 is the minimum eigenvalue of AA ⊤ ∈ R m×m . Therefore the Assumption 1(i) is satisfied with ρ = τ λ min (AA ⊤ ). It is notable that the projection and the linear optimization under the quadratic constraint require solving a quadratic programming problem and therefore could be expensive. PSD constraint. A PSD constraint X 0 for X ∈ R d×d can be written as an inequality constraint −λ min (X) ≤ 0, where λ min (X) denotes the minimum eigen-value of X. The subgradient of c(X) = −λ min (X) when λ min (X) = 0 is given by Conv{−uu ⊤ \| u = 1, Xu = 0}, i.e., the convex hull of the outer products of normalized vectors in the null space of the matrix X. In appendix, we show that if the dimension of the null space of X is r with 1 ≤ r ≤ d, the norm of the subgradient of c(X) on the boundary c(X) = 0 is lower bounded by ρ = 1 √ r ≥ 1 √ d . Finally, we note that computing a subgradient of [c(X)] + only needs to compute one eigen-vector corresponding to the smallest eigen-value. In contrast, both projection and linear optimization under a PSD constraint could be very expensive for high-dimensional problems. In particular, the projection onto a PSD domain needs to conduct a singular value decomposition. The linear optimization over a PSD cone is ill-posed due to that PSD cone is not compact (the solution is either 0 or infinity). One may add an artificial constraint on the upper bound of the eigen-values. According to (Jaggi, 2013), the time complexity for solving this linear optimization problem approximately up to an accuracy level ǫ ′ is O(N d 1.5 /ǫ ′ 2.5 ) with N being the number of non-zeros in the gradient and ǫ ′ decreasing iteratively 5 Here we use the square constraint to make it a smooth function so that the proposed algorithms for smooth optimization are applicable by using proximal gradient mapping to handle the ℓ1 norm. 6 which is reasonable because m ≪ d. required in the Frank-Wolfe method, which could be much more expensive especially for high-dimensional problems and in later iterations than computing the first eigen-pairs at each iteration in our methods. Applications Compressive Sensing We first consider a compressive sensing problem in (14). Becker et al. (2011) proposed an optimization algorithm based on the Nesterov's smoothing and the Nesterov's optimal method for the smoothed problem, known as NESTA. It needs to perform the projection into the domain Ax − y 2 ≤ τ at every iteration and has an iteration complexity of O(1/ǫ). In contrast, the presented algorithm with only one projection in Section 4.2 using Nesterov's accelerated proximal gradient method (Beck & Teboulle, 2009) to solve the unconstrained problem enjoys an iteration complexity of O(1/ǫ). Moreover, we present a theorem below showing that the problem (14) satisfies the local error bound condition with θ = 1/2, and hence the presented LoPNAG enjoys an O(1/ √ ǫ) iteration complexity with only a logarithmic number of projections. Theorem 6. Let f (x) = x 1 , c(x) = Ax − y 2 − τ , Ω * denote the optimal set and f * be the optimal solution to (14). Assume that there exists x 0 such that Ax 0 −y 2 < τ and 0 ∈ Ω * . Then for any ǫ > 0, x ∈ R d such that c(x) ≤ 0 and f (x) ≤ f * + ǫ, there exists 0 < σ < ∞ such that dist(x, Ω * ) ≤ σ(f (x) − f * ) 1/2 . Hence, LoP-NAG can have an iteration complexity of O(1/ √ ǫ) with only O(log(1/ǫ)) projections. Next, we demonstrate the effectiveness of the LoPNAG for solving the compressive sensing problem in (14) by comparing with NESTA. We generate a synthetic data for testing. In particular, we generate a random measurement matrix A ∈ R m×d with m = 1000 and d = 5000. The entries of the matrix A are generated independently with the uniform distribution over the interval [−1, +1]. The vector x * ∈ R d is generated with the same distribution at 100 randomly chosen coordinates. The noise ε ∈ R m is a dense vector with independent random entries with the uniform distribution over the interval [−ζ, ζ], where ζ is the noise magnitude and is set to 0.01. Finally the vector y was obtained as y = Ax * + ε. We use the Matlab package of NESTA 7 . For fair comparison, we also use the projection code in the NESTA package for conducting projection. To handle the unknown smoothness parameter in the proposed algorithm, we use the backtracking technique (Beck & Teboulle, 2009). The parameter γ is initially set to 0.001 and decreased by half every 5000 iterations after a projection and the target smoothing parameter in NESTA is set to 10 −5 . For the value of λ in LoPNAG, we tune it from its theoretical value to several smaller values and choose the one that yields the fastest convergence. We report the results in Table 7.1, which include different number of iterations, the corresponding number of projections, the recovery error of the found solution compared to the underlying true sparse solution, the objective value (i.e., the ℓ 1 norm of the found solution) and the running time. Note that each iteration of NESTA requires two projections because it maintains two extra sequence of solutions. From the results, we can see that LoP-NAG converges significantly faster than NESTA. Even with only one projection, we are able to obtain a better solution than that of NESTA after running 10000 iterations. High-dimensional Distance Metric Learning Consider the following distance metric learning problem: min A 0 1 2\|E\| (i,j)∈E (1 − y ij − x i − x j 2 A ) 2 + τ A off 1 ,(15) where E denotes all pairs of training examples, y ij = 1 indicates x i , x j belong to the same class and y ij = −1 indicates they belong to different classes, z 2 A = z ⊤ Az and A off 1 = i =j \|A ij \|. We note that such a formulation is useful for high dimensional problems due to the ℓ 1 regularizer. A similar formulation with different forms of loss function has been adopted in literature (Qi et al., 2009). We consider the square loss because it gives us faster convergence with a logarithmic number of projections by LoPGD. Due to the presence of the non-smooth PSD constraint and the ℓ 1 regularizer, Nesterov's accelerated proximal gradient methods can not be applied efficiently to solving (15) and the augmented unconstrained problem. Nevertheless, we can apply the proposed LoPGD method for solving the problem with a logarithmic number of projections. Regarding the constant θ in the local error bound condition for (15), it still remains an open problem. Nonetheless, a local error bound condition with θ = 0.5 might be established under certain regularity condition of the problem (Zhou & So, 2015;Cui et al., 2017). For example, Cui et al. (2017) provided a direct analysis of a local error bound condition with θ = 0.5 for a class of constrained convex symmetric matrix optimization problems regularized by nonsmooth spectral functions (including the indicator function of a PSD constraint). They established sufficient conditions (Theorem 16) for a local error bound condition with θ = 0.5 to hold, which reduces to a regularity condition for (15) depending on the optimal solutions of the problem. A thorough analysis of the regularity condition is much more involved and left as an open problem. Next, we demonstrate the empirical performance of LoPGD for solving (15). We use the colon-cancer data available on libsvm web portal, which has 2000 features and 62 examples. Fourty examples are used as training examples to generate 780 pairs to learn the distance metric. The regularization parameter is set to τ = 0.001. We compare LoPGD, gradient descent method with only one projection (referred to as OPGD), and standard projected GD (referred to PGD). The step size in PGD and OPGD is set to η 0 / √ t, where t is the iteration index. We use the same tuned initial step size for all algorithms. The number of iterations per-epoch in LoPGD is set to 1000. The penalization parameter λ in both OPGD and LoPGD is tuned and set to 10. In Table 2, we report the objective values, the #of iterations/projections, and running time across the first 8000 iterations. We can see that LoPGD converges dramatically faster than PGD and also much faster than OPGD. Conclusion We have developed a general theory of optimization with only one projection for a family of inequality constrained convex optimization problems. It yields an improved iteration complexity for smooth optimization compared with non-smooth optimization. By exploring the local error bound condition, we further develop new algorithms with a logarithmic number of projections and achieve better convergence for both smooth and non-smooth optimization without strong convexity assumption. Applications in compressive sensing and distance metric learning demonstrate the effectiveness of the proposed improved algorithms. A. Proof of Lemma 1 When c(x) ≤ 0, x ♮ = x. There is nothing to prove. Therefore we consider c(x) > 0 and x ♮ = x. By KKT conditions, there exists ζ ≥ 0 and v ∈ ∂c(x ♮ ) such that x ♮ − x + ζv = 0, and ζc(x ♮ ) = 0 Since x ♮ = x, then ζ > 0, c(x ♮ ) = 0 and v = 0. There- fore, x − x ♮ is the same direction as v. On the other hand, c(x) = c(x) − c(x ♮ ) ≥ (x − x ♮ ) ⊤ v = v x − x ♮ ≥ ρ x − x ♮ where the second equality uses the fact that x − x ♮ is the same direction as v and the last inequality uses the inequality (2). B. Proof of Proposition 1 The two inequalities are straightforward to prove. We prove the smoothness property. Let q(z) = exp(z)/(1 + exp(z)). It is not difficult to see that q(z) is 1/4-Lipschtiz continuous function. The gradient of h γ (x) is given by ∇h γ (x) = exp(λc(x)/γ) 1 + exp(λc(x)/γ) λ∇c(x) = q(λc(x)/γ)∇c(x) Then ∇h γ (x) − ∇h γ (y) = q(λc(x)/γ)λ∇c(x) − q(λc(y)/γ)λ∇c(y)) ≤ q(λc(x)/γ)λ∇c(x) − q(λc(y)/γ)λ∇c(x) + q(λc(y)/γ)λ∇c(x) − q(λc(y)/γ)λ∇c(y)) ≤ λG C 4 \|λc(x)/γ − λc(y)/γ\| + λL c x − y ≤ λ 2 G 2 c 4γ + λL c x − y C. Proof of Corollary 3 The following proposition shows the convergence of F (x). Proposition 2. (Beck & Teboulle, 2009;Nesterov, 2004b) Assume F (x) is L F -smooth. Let (11) run for T iterations. If F (x) is convex, we can set β t = τt−1−1 τt , where τ t = 1+ √ 1+4τ 2 t−1 2 with τ 0 = 1. Then for any x ∈ R d we have F (x T ) − F (x) ≤ 2L F y 0 − x 2 (T + 1) 2 If F (x) is µ-strongly convex, we can set β t = √ L f − √ µ √ L f + √ µ . Then for any x ∈ R d we have F (x T ) − F (x) ≤ exp −T µ L F F (y 0 ) − F (x) + µ 2 y 0 − x 2 We first prove the convergence for a smooth convex function f (x). From Theorem 1 and the construction of h γ (x) in (10), we have f ( x T ) − f (x * ) ≤ p γ ln 2 + 2L F y 0 − x * 2 (T + 1) 2 where p = λρ λρ−G . By Proposition 1, we have L F = L f + λL c + λ 2 G 2 c 4γ . Then f ( x T ) − f (x * ) ≤ pγ ln 2 + p λ 2 G 2 c y 0 − x * 2 2γ(T + 1) 2 + 2(L f + λL c ) y 0 − x * 2 (T + 1) 2 ≤ p γ ln 2 + λ 2 G 2 c D 2 2γ(T + 1) 2 + 2(L f + λL c )D 2 (T + 1) 2 = p √ 2 ln 2λG c D (T + 1) + 2(L f + λL c )D 2 (T + 1) 2 where the last equality is due to the value of γ. Next, we prove the convergence for a smooth and strongly convex function f (x). First, we have F (x T ) − F (x * ) ≤ exp −T µ L F · ∇F (x * ) ⊤ (y 0 − x * ) + L F + µ 2 y 0 − x * 2 ≤ exp −T µ L F Ḡ y 0 − x * + L F y 0 − x * 2 ≤ exp −T µ L F Ḡ D + exp −T µ L F L F D 2 Note that x * is not the optimal solution to F (x), hence ∇F (x * ) = 0 and we use its Lipschitz continuity property whereḠ = G + λG c . Following Theorem 1, we have f ( x T ) − f (x * ) ≤ pγ ln 2 + p exp −T µ L F Ḡ D + exp −T µ L F L F D 2 ≤ pγ ln 2 + p exp −T µ L F Ḡ D + p exp −T µ L F λ 2 G 2 c D 2 4γ + p exp −T µ L F (L f + λL c )D 2 To avoid clutter, we will consider the dominating term. Consider γ = 1 T 2α ≤ 1. To bound the second term, we have exp −T µ L F λ 2 G 2 c D 2 4γ = O exp −T µ L F T 2α = O   exp   −T µ L f + λL c + λ 2 G 2 c 4γ   T 2α   ≤ O exp −T µγ L f + λL c + λ 2 G 2 c /4 T 2α = O exp −T 1−α µ L f + λL c + λ 2 G 2 c /4 T 2α Consider T to be sufficiently larger such that T ≥ L f +λLc+λ 2 G 2 c /4 µ 1 2(1−α) (4α ln T ) 1 1−α , then exp −T 1−α µ L f + λL c + λ 2 G 2 c /4 ≤ 1 T 4α Therefore exp −T µ L F λ 2 G 2 c D 2 4γ ≤ O 1 T 2α Similarly we can show the last two terms are dominated by O(1/T 4α ). As a result, f ( x T ) − f (x * ) ≤ O 1 T 2α + 1 T 4α D. Proof of Theorem 4 We first present a key lemma. Lemma 2. Let D ǫ = max x∈Lǫ dist(x, Ω * ). Then for any x ∈ Ω and ǫ > 0 we have x − x † ǫ ≤ D ǫ ǫ (f (x) − f (x † ǫ )).(16) The above lemma was established in (Yang & Lin, 2016). Proof. We consider x ∈ S ǫ , otherwise the conclusion holds trivially. By the first-order optimality conditions of (12), we have for any u ∈ Ω, there exists ζ ≥ 0 (the Lagrangian multiplier of problem (12)) (x † ǫ − x + ζ∂f (x † ǫ )) ⊤ (u − x † ǫ ) ≥ 0(17) Let u = x in the first inequality we have ζ∂f (x † ǫ ) ⊤ (x − x † ǫ ) ≥ x − x † ǫ 2 We argue that ζ > 0, otherwise x = x † ǫ contradicting to the assumption x ∈ S ǫ . Therefore f (x) − f (x † ǫ ) ≥ ∂f (x † ǫ ) ⊤ (x − x † ǫ ) ≥ x − x † ǫ 2 ζ = x − x † ǫ ζ x − x † ǫ(18) Next we prove that ζ is upper bounded. Since −ǫ = f (x * ǫ ) − f (x † ǫ ) ≥ (x * ǫ − x † ǫ ) ⊤ ∂f (x † ǫ ) where x * ǫ is the closest point to x † ǫ in the optimal set. Let u = x * ǫ in the inequality of (17), we have (x † ǫ − x) ⊤ (x * ǫ − x † ǫ ) ≥ ζ(x † ǫ − x * ǫ ) ⊤ ∂f (x † ǫ ) ≥ ζǫ Thus ζ ≤ (x † ǫ − x) ⊤ (x * ǫ − x † ǫ ) ǫ ≤ D ǫ x † ǫ − x ǫ Therefore x − x † ǫ ζ ≥ ǫ D ǫ Combining the above inequality with (18) we have f (x) − f (x † ǫ ) ≥ ǫ D ǫ x − x † ǫ which completes the proof. Proposition 3 (Zinkevich (2003)). Let x t+1 = x t − η∂F (x t ) run for T iterations. Assume ∂F (x) ≤Ḡ. Then for any x ∈ R d F ( x T ) − F (x) ≤ ηḠ 2 2 + x 1 − x 2 2ηT where x T = T t=1 x t /T . D.1. Proof of Theorem 4 Let ǫ k = ǫ0 2 k . We assume x 1 , . . . , x K−1 ∈ S 2ǫ ; otherwise the result holds trivially. Let x † k,ǫ ∈ Ω denote the closest point to x k in the sublevel set S ǫ of f (x). Then f (x † k,ǫ ) = f * + ǫ, k = 1, . . . , K − 1, which is because we assume that x 1 , . . . , x K−1 ∈ S 2ǫ . We will prove by induction that f (x k ) − f * ≤ ǫ k + ǫ. It holds for k = 0 because of Assumption (i). We assume it holds for k − 1 and prove it is true for k ≤ K. We consider the k-th epoch of LoPGD. Applying Proposition 3, we have for any x ∈ R d F ( x k ) − F (x) ≤ η kḠ 2 2 + x k−1 − x 2 2η k t Let x = x † k−1,ǫ ∈ Ω. Then F ( x k ) − F (x † k−1,ǫ ) ≤ η kḠ 2 2 + x k−1 − x † k−1,ǫ 2 2η k t Since F (x † k−1,ǫ ) = f (x † k−1,ǫ ) + λ[c(x † k−1,ǫ )] + = f (x † k−1,ǫ ) F ( x k ) = f ( x k ) + λ[c( x k )] + Then f ( x k ) + λ[c( x k )] + − f (x † k−1,ǫ ) ≤ η kḠ 2 2 + x k−1 − x † k−1,ǫ 2 2η k t Bt Then f ( x k ) + λ[c( x k )] + ≤ f (x † k−1,ǫ ) + B t Then λρ x k − x k ≤ f (x † k−1,ǫ ) − f ( x k ) + B t ≤ f (x † k−1,ǫ ) − f (x k ) + f (x k ) − f ( x k ) + B t Assume f (x k ) − f * > ǫ (otherwise the proof is done), thus f (x † k−1,ǫ ) ≤ f (x k ). Then λρ x k − x k ≤ G x k − x k + B t leading to x k − x k ≤ B t λρ − G Then f (x k ) − f (x † k−1,ǫ ) ≤ f (x k ) − f ( x k ) + f ( x k ) − f (x † k−1,ǫ ) ≤ G x k − x k + B t = λρ λρ − G B t = p η kḠ 2 2 + x k−1 − x † k−1,ǫ 2 2η k t ≤ p η kḠ 2 2 + D 2 ǫ (f (x k−1 ) − f (x † k−1,ǫ )) 2 2η k tǫ 2 ≤ p η kḠ 2 2 + σ 2 (f (x k−1 ) − f (x † k−1,ǫ )) 2 2η k tǫ 2(1−θ)) where the third inequality uses Lemma 2 and the last inequality uses the local error bound condition. Since we assume f (x k−1 ) − f * ≤ ǫ k−1 + ǫ. Thus f (x k−1 ) − f (x † k−1,ǫ ) ≤ ǫ k−1 . Then f (x k ) − f (x † k−1,ǫ ) ≤ pη kḠ 2 2 + pσ 2 ǫ 2 k−1 2η k tǫ 2(1−θ)) By noting the values of η k = ǫ k−1 2pḠ 2 and t = 4σ 2 p 2Ḡ2 ǫ 2(1−θ) , we have f (x k ) − f (x † k−1,ǫ ) ≤ ǫ k−1 4 + ǫ k−1 4 = ǫ k . Therefore f (x k ) − f * ≤ ǫ k + ǫ due to the assumption f (x k−1 ) ≥ f * +2ǫ and f (x † k−1,ǫ ) = ǫ. By induction, we therefore show that with at most K = log 2 (ǫ 0 /ǫ) epochs, we have f (x K ) − f * ≤ ǫ K + ǫ ≤ 2ǫ E. Proof of Theorem 5 Following a similar analysis and using the convergence of NAG and Proposition 1, we have f (x k ) − f (x † k−1,ǫ ) ≤ pγ k ln 2 + p λ 2 G 2 c x k−1 − x † k−1,ǫ 2 2γ k t 2 k + p 2(L f + λL c ) x k−1 − x † k−1,ǫ 2 t 2 k By using Lemma 2 and the local error bound condition, we have f (x k ) − f (x † k−1,ǫ ) ≤ p γ k ln 2 + λ 2 G 2 c σ 2 ǫ 2 k−1 2γ k t 2 k ǫ 2(1−θ) + p 2(L f + λL c )σ 2 ǫ 2 k−1 t 2 k ǫ 2(1−θ) Plugging the values of γ k = ǫ k−1 6p ln 2 , t k = σ ǫ 1−θ max{λG c p √ 18 ln 2, 12(L f + λL c )ǫ k−1 } into the above inequality yields f (x k ) − f (x † k−1,ǫ ) ≤ ǫ k−1 6 + ǫ k−1 6 + ǫ k−1 6 = ǫ k Then f (x k ) − f * ≤ ǫ k + ǫ Therefore the total number of iterations is σ ǫ 1−θ max λGcp √ 18 ln 2 log 2 (ǫ0/ǫ), 12L K k=1 ǫ 1/2 k−1 ≤ σ ǫ 1−θ max λGcp √ 18 ln 2 log 2 (ǫ0/ǫ), 12L √ 2ǫ0 √ 2 − 1 F. Lower bound of the subgradient of the constraint function for a PSD constraint We first show that Conv{−uu ⊤ \| u = 1, Xu = 0} = Conv{−U \|U 0, Tr(X ⊤ U ) = 0, rank(U ) = 1, Tr(U ) = 1}. In fact, given any u ∈ R d with u = 1 and Xu = 0, we can show uu ⊤ 0, Tr(X ⊤ uu ⊤ ) = u ⊤ Xu = 0, rank(uu ⊤ ) = 1 and Tr(uu ⊤ ) = u 2 = 1, which means uu ⊤ belongs to the set on the right. Since the set on the left is the convex hull of all such uu ⊤ , the set on the left is included in the set of the right. On the other hand, given any element U from the set of the right, we can represent it as U = K k=1 λ k U k where K k=1 λ k = 1, λ k ≥ 0, U k = u k u ⊤ k for some u k , Tr(X ⊤ U k ) = 0 and Tr(U k ) = 1 for k = 1, . . . , K. These three properties of U k imply Xu k = 0 and u k 2 = 1 so that U is a convex combination of some elements of the set on the left. Therefore, the set on the right is included in the set of the left. It is easy to see that the set on the left is a subset of the set on the right. To show the opposite, given any element U from the set of the right, we consider its eigenvalue decomposition U = K k=1 λ k u k u ⊤ k where K ≤ d and λ k > 0 and u k are the eigenvalue and the corresponding eigenvector with u k = 1. Since X is PSD, the property Tr(X ⊤ U ) = 0 implies K k=1 λ k u ⊤ k Xu k = 0 so that Xu k must be zero for k = 1, . . . , K. As a result, U = K k=1 λ k U k with U k = u k u ⊤ k being an element in the set on the left. Note that Tr(U ) = K k=1 λ k u ⊤ k u k = K k=1 λ k = 1. This means U is in the set on the left also. If the dimension of the null space of X is r with 1 ≤ r ≤ d then we can write X = V ΣV ⊤ , where Σ = Σ r 0 0 0 is a diagonal matrix with Σ r ∈ R d−r,d−r . We can set the constant ρ to be the solution of the following optimization problem. To simplify the problem, we note that Tr(X T U ) = Tr(V ΣV ⊤ U ) = Tr(ΣV ⊤ U V ) = 0. Let V ⊤ U V = U 11 , U 12 U 21 , U 22 where U 11 ∈ R (d−r)×(d−r) and U 22 ∈ R r×r . Because Σ is a diagonal matrix with nonnegative entries, it then leads to that the diagonal entries of U 11 are all zeros, as a result U 11 = 0 and consequentially U 21 = U 12 = 0 due to that V ⊤ U V 0. As a result, U F = V ⊤ U V F = U 22 F and Tr(U ) = Tr(V ⊤ U V ) = Tr(U 22 ). Therefore, we get ρ = arg min U22∈R r×r U F , s.t. U 22 0, Tr(U 22 ) = 1 As a result, ρ = 1 √ r ≥ 1 √ d . G. Proof of Theorem 6 The proof follows similarly to that of Theorem 3.5 in (Li & Pong, 2016) but tailored to the our problem at hand. Let x * denote an optimal solution to min c(x)≤0 f (x). Let Ω = {x : c(x) ≤ 0} and F (x) = f (x) + I Ω (x), where I Ω (x) denotes the indicator function associated with Ω. Due to the strict feasibility condition, i.e., there exists x 0 such that Ax 0 − y 2 < τ , then by the Lagrangian theory there exists ζ ≥ 0 such that F (x * ) = min x∈R d F (x) = min x∈R d f (x)+ζc(x) = f (x * )+ζc(x * ) The KKT condition implies 0 ∈ ∂f (x * ) + ζ∇c(x * ) and ζc(x * ) = 0. Since x * = 0, thus 0 ∈ ∂f (x * ), and consequentially ζ > 0. As a result, c(x * ) = 0. In view of (Rockafellar, 1970)[Theorem 28.1], we have x * ∈ arg min F (x) = {x : c(x) = 0}∩arg min f (x)+ζc(x) By noting the form of f (x) + ζc(x) = x 1 + ζ( Ax − y 2 − τ ) = h(Ax) + x 1 , where h(·) is a strongly convex function, it must hold that Ax is a constant for x ∈ arg min f (x) + ζc(x), i.e., c(x) is a constant over arg min x f (x) + ζc(x). Then for any x ∈ arg min f (x) + ζc(x), we have c(x) = c(x * ) = 0. It follows that arg min F (x) = {x : c(x) = 0} ∩ arg min f (x) + ζc(x) = arg min f (x) + ζc(x) Next, we consider the problem min x f ζ (x) = f (x) + ζc(x) = x 1 + ζ( Ax − y 2 − τ ). It has been shown that there exists σ > 0 (e.g. for any u ∈ R d such that f ζ (u) − min u f ζ (u) ≤ ǫ. Next, consider any x such that c(x) ≤ 0 and f (x) ≤ f * + ǫ. We have f ζ (x) − min u f ζ (u) = f (x) + ζc(x) − f (x * ) ≤ f (x) − f * ≤ ǫ. As a result, (20) holds for any x such that c(x) ≤ 0 and f (x) ≤ f * + ǫ, i.e., dist(x, arg min f ζ (u)) ≤ σ(f ζ (x) − min u f ζ (u)) 1/2 = σ(f (x) + ζc(x) − f * ) 1/2 ≤ σ(f (x) − f * ) 1/2 In view of (19), we can finish the proof. Theorem 5 . 5Suppose Assumptions 1 and 2 hold and f (x) is L f -smooth and c(x) is L c -smooth. Let γ 1 = ǫ0 6p ln 2 , K = ⌈log 2 (ǫ 0 /ǫ)⌉ and t k = σ ǫ 1−θ max{λG c p √ 18 ln 2, 12(L f + λL c )ǫ 0 /2 k−1 } in Algorithm 2, where θ and σ are constants appearing in the local error bound condition. Then f (x K ) − f * ≤ 2ǫ. which is a polyhedral function and therefore satisfies (3). The second method is to use c(x) = [Cx − b] + , where [a] + = max(0, a) and C = (c 1 , . . . , c m ) ⊤ . Thus [c(x)] + = [Cx − b] + . The inequality Next, we want to showConv{−U \|U 0, Tr(X ⊤ U ) = 0, rank(U ) = 1, Tr(U ) = 1} = {−U \|U 0, Tr(X ⊤ U ) = 0, Tr(U ) = 1} t. U 0, Tr(X ⊤ U ) = 0, Tr(U ) = 1 u, arg min f ζ (u)) ≤ σ(f ζ (u) − min u f ζ (u)) 1/2 Table 1 . 1LoPNAG vs. NESTA for solving the compressive sensing problem.LoPNAG NESTA Iters -Projs Rec. Err. Objective Time (s) Iters -Projs Rec. Err. Objective Time (s) 5000 -1 0.018017 52.042878 18.04 1000 -2000 0.137798 52.703275 48.49 10000 -2 0.018038 52.042418 35.88 3000 -6000 0.018669 52.050051 93.84 15000 -3 0.018043 52.042358 53.09 5000 -10000 0.018659 52.050046 245.23 20000 -4 0.018043 52.042358 70.24 8000 -16000 0.018657 52.050045 404.72 25000 -5 0.018043 52.042358 87.32 10000 -20000 0.018657 52.050044 501.65 Table 2. LoPGD vs. OPGD and PGD for solving the considered distance metric learning problem. LoPGD OPGD PGD Iters -Projs Objective Time (h) Iters -Projs Objective Time (h) Iters -Projs Objective Time (h) 1000 -1 0.0953 0.22 1000 -1 0.1707 0.20 1000 -1000 0.1491 7.97 2000 -2 0.0695 0.43 2000 -1 0.1583 0.40 2000 -2000 0.1278 15.46 4000 -4 0.0494 0.87 4000 -1 0.1469 0.80 4000 -4000 0.1072 29.39 6000 -6 0.0428 1.33 6000 -1 0.1398 1.22 6000 -6000 0.0957 43.36 8000 -8 0.0405 1.89 8000 -1 0.1343 1.64 8000 -8000 0.0879 57.43 Indeed, a linear optimization over a PSD constraint is illposed because the PSD domain is unbounded. it can be extended to when f (x) is non-smooth but its proximal mapping can be easily solved. http://statweb.stanford.edu/˜candes/nesta/ AcknowledgementsWe are grateful to all anonymous reviewers for their helpful comments. T. Yang is partially supported by National Science Foundation (IIS-1463988, IIS-1545995). L. Zhang thanks the support from NSFC (61603177) and JiangsuSF (BK20160658). A fast iterative shrinkagethresholding algorithm for linear inverse problems. Amir Beck, Marc Teboulle, SIAM J. Img. Sci. 2Beck, Amir and Teboulle, Marc. A fast iterative shrinkage- thresholding algorithm for linear inverse problems. SIAM J. Img. Sci., 2:183-202, 2009. A fast and accurate first-order method for sparse recovery. Stephen Becker, Jérôme Bobin, Emmanuel J Candès, Nesta, 1936- 4954SIAM J. Img. Sci. 4Becker, Stephen, Bobin, Jérôme, and Candès, Emmanuel J. Nesta: A fast and accurate first-order method for sparse recovery. SIAM J. Img. Sci., 4:1-39, 2011. ISSN 1936- 4954. Constrained Optimization and Lagrange Multiplier Methods (Optimization and Neural Computation Series). Dimitri P Bertsekas, Athena Scientific. 18865290431 editionBertsekas, Dimitri P. Constrained Optimization and La- grange Multiplier Methods (Optimization and Neural Computation Series). Athena Scientific, 1 edition, 1996. ISBN 1886529043. An introduction to compressive sampling. Emmanuel J Candès, Michael B Wakin, IEEE Signal Processing Magazine. 252Candès, Emmanuel J. and Wakin, Michael B. An introduc- tion to compressive sampling. IEEE Signal Processing Magazine, 25(2):21 -30, 2008. Optimal stochastic strongly convex optimization with a logarithmic number of projections. Jianhui Chen, Yang, Tianbao, Lin, Qihang, Lijun Zhang, Yi Chang, Proceedings of the Thirty-Second Conference on Uncertainty in Artificial Intelligence (UAI). the Thirty-Second Conference on Uncertainty in Artificial Intelligence (UAI)Chen, Jianhui, Yang, Tianbao, Lin, Qihang, Zhang, Lijun, and Chang, Yi. Optimal stochastic strongly convex op- timization with a logarithmic number of projections. In Proceedings of the Thirty-Second Conference on Uncer- tainty in Artificial Intelligence (UAI), 2016. A light touch for heavily constrained SGD. Andrew Cotter, Maya R Gupta, Pfeifer , Proceedings of the 29th Conference on Learning Theory (COLT). the 29th Conference on Learning Theory (COLT)Cotter, Andrew, Gupta, Maya R., and Pfeifer, Jan. A light touch for heavily constrained SGD. In Proceedings of the 29th Conference on Learning Theory (COLT), pp. 729-771, 2016. Quadratic growth conditions for convex matrix optimization problems associated with spectral functions. Ying Cui, Chao Ding, Xinyuan Zhao, Cui, Ying, Ding, Chao, and Zhao, Xinyuan. Quadratic growth conditions for convex matrix optimization prob- lems associated with spectral functions. CoRR, 2017. An algorithm for quadratic programming. Marguerite Frank, Philip Wolfe, Naval Research Logistics (NRL). 3Frank, Marguerite and Wolfe, Philip. An algorithm for quadratic programming. Naval Research Logistics (NRL), 3:149-154, 1956. Faster rates for the frankwolfe method over strongly-convex sets. 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[ "On The Painlevé Property For A Class Of Quasilinear Partial Differential Equations", "On The Painlevé Property For A Class Of Quasilinear Partial Differential Equations" ]	[ "Stanislav Sobolevsky [email protected] \nNew York University\n\n" ]	[ "New York University\n" ]	[]	The last decades saw growing interest across multiple disciplines in nonlinear phenomena described by partial differential equations (PDE). Integrability of such equations is tightly related with the Painlevé property -solutions being free from moveable critical singularities. The problem of Painlevé classification of ordinary and partial nonlinear differential equations lasting since the end of XIX century saw significant advances for the equation of lower (mainly up to fourth with rare exceptions) order, however not that much for the equations of higher orders.Recent works of the author have completed the Painlevé classification for several broad classes of ordinary differential equations of arbitrary order, advancing the methodology of the Panlevé analysis. This paper transfers one of those results on a broad class of nonlinear partial differential equationsquasilinear equations of an arbitrary order three or higher, algebraic in the dependent variable and including only the highest order derivatives of it. Being a first advance in Painlevé classification of broad classes of arbitrary order nonlinear PDE's known to the author, this work highlights the potential in building classifications of that kind going beyond specific equations of a limited order, as mainly considered so far.	null	[ "https://arxiv.org/pdf/1809.03640v1.pdf" ]	119,138,142	1809.03640	94a2911105d9a409ebe45ae24c5875b13530799e	On The Painlevé Property For A Class Of Quasilinear Partial Differential Equations 11 Sep 2018 Stanislav Sobolevsky [email protected] New York University On The Painlevé Property For A Class Of Quasilinear Partial Differential Equations 11 Sep 2018 The last decades saw growing interest across multiple disciplines in nonlinear phenomena described by partial differential equations (PDE). Integrability of such equations is tightly related with the Painlevé property -solutions being free from moveable critical singularities. The problem of Painlevé classification of ordinary and partial nonlinear differential equations lasting since the end of XIX century saw significant advances for the equation of lower (mainly up to fourth with rare exceptions) order, however not that much for the equations of higher orders.Recent works of the author have completed the Painlevé classification for several broad classes of ordinary differential equations of arbitrary order, advancing the methodology of the Panlevé analysis. This paper transfers one of those results on a broad class of nonlinear partial differential equationsquasilinear equations of an arbitrary order three or higher, algebraic in the dependent variable and including only the highest order derivatives of it. Being a first advance in Painlevé classification of broad classes of arbitrary order nonlinear PDE's known to the author, this work highlights the potential in building classifications of that kind going beyond specific equations of a limited order, as mainly considered so far. Introduction Nonlinear Partial Differential Equations and especially evolutionary PDE's, describing dynamics of nonlinear phenomena saw increasing interest across different fields of physics, such as statistical mechanics, fiber optics, fluid dynamics, condensed matter, elementary particle physics, astrophysics as well as reactiondiffusion systems in chemistry and competition of species in biology [1]. Painlevé property plays important role in the analysis of PDE's as being tightly connected with their integrability. Equations with Painlevé property often happen to be integrable either analytically either through inverse scattering transform [2]. Moreover, famous Ablowitz-Ramani-Segur conjecture [3] suggests that all reductions of PDE's integrable by means of inverse scattering transform should possess the Painlevé property. On the other hand, when one can not find the way to integrate equation having Painlevé property, it often warrants even higher interest. This is because solutions of such an equation could be seen as a source for new single-valued or meromorphic functions, such as second-order Painlevé transcendents (see for instance [4], chapter XIV) and their higher-order generalizations [5]. Additional practical interest to the equations with Painlevé property is warranted by the fact that their solutions often demonstrate interesting properties, such as solutions of Korteweg-de-Vries [6] or Schrödinger [7] equations being able to describe propagation of bell-shaped solitary waves [1,2,8] or even collisions of multiple solitons [9]. The Painlevé classification of ordinary differential equations is one of the long-lasting problems of analytic theory of differential equations rooted in the end of XIX century [4]. For PDE's the Painlevé has been defined in the 1980's by Weiss, Tabor And Carnevale in [10]. In spite of numerous achievements on the classification of the equations of some limited order (mainly up to fourth) over the last more than hundred years, the general problem for the higher order equations remains unsolved. This paper contributes towards solution of the arbitrary-order problem by proving absence of the Painlevé property classification for a broad class of quasilinear PDE's. Recent advances in Painlevé classification of ordinary and partial differential equations The Painlevé classification of the non-linear ordinary differential equations (ODE) w (n) = F (w (n−1) , w (n−2) , ..., w, z),(1) is mainly known for the order n ≤ 4. For the order n = 1 the necessary and sufficient condition of the Painlevé property in case of the right-hand side F algebraic in w is a well-known classical result (see for instance [4], chapter XIII). For the order n = 2 classification of the equations with the rational right-hand side F has been built in the classical works of Painlevé and Gambier (see for instance [4], chapter XIV). For the order n = 3 classification has been started in the famous work of Chazy [11] and recently completed by C.Cosgrove [12] in case of a polynomial right-hand side. Finally for the order n = 4 the polynomial problem was completed by C.Cosgrove [13,14]. And although Painlevé classification has been successfully completed by the author for certain algebraic classes of equations of the arbitrary order, such as binomial-type equations [15][16][17] or arbitrary order differential equations with quadratic right-hand side [18], the classification of ODE's of order n ≥ 5 is not yet accomplished in any general enough case. In particular it was found that nonlinear equations w (n) = F (w, z),(2) where F is algebraic in w and locally analytic in z never possess Painlevé property for n ≥ 3 [19][20][21][22]. Painlevé classification of the PDE's largely goes along the same lines as the classification of the ODE's. Second order semilinear PDE's A(z, t)w zz + B(z, t)w zt + C(z, t)w tt = F (z, t, w, w z , w t ) with the right-hand side F rational in w, w z , w t with the coefficients locally analytic in z, t have been classified in [23][24][25]. Third order equations with a polynomial right-hand side considered by [26]. Certain partial results were obtained for particular classes of equations of the fourth and some higher orders [27][28][29][30][31]. So far there are no results on Painlevé classification known to the author that would concern broad classes of PDE's of an arbitrary order. While the methodology of Painlevé analysis for PDE's has its own specifics, in some cases it could be sufficient to simply consider their correspondent reductions to ODE's, enabling transfer of the corresponding classifications for the ODE's into the PDE domain. The present work is a first attempt of transferring recent results of the author on the arbitrary order ODE's to the PDE's. Quasilinear PDE depending only on the highest order derivatives Consider an autonomous quasilinear partial differential equation containing only the highest order derivatives ν, i νi=n A ν (w) ∂ n w ∂x ν1 1 ∂x ν2 2 ...∂x νm m = P (w),(3) where x = (x 1 , x 2 , ..., x m ) is a vector of m independent complex variables, ν = (ν 1 , ν 2 , ..., ν m ) are multi-indexes, and P as well as A ν are polynomials in w without common roots P (w * ) = A ν (w * ) = 0, ∀ν (i.e. equation is non-reducible), while either degP > 1 or ∃ν : degA ν > 0 (i.e. equation is essentially nonlinear). Example of such an equation of the order n = 3 with m = 2 independent variables z, t might look like w ttt + w zzz = w 2 .  ν   j c νj j   A ν (w)   d n w dz n = P (w),(4) i.e. takes the form w (n) = P (w) Q(w) ,(5) where w = w(z) and its derivatives stand for ordinary derivatives in z, while Q = ν j c νj j A ν (w) is a polynomial in w. It can be seen that under certain choice of the constants c 1 , c 2 , ..., c m the polynomial Q could be guaranteed not to have common roots with P , while degQ = max degA ν (its major coefficient as well as values in all the roots of P are linear combinations of non-zero major coefficients/values of A ν and could be guaranteed non-zero for certain c 1 , c 2 , ..., c m ). This way equation (5) is essentially non-linear and belongs to the class (2). According to [19,21] non-linear equations of the class (5) admit moveable critical singularities. Then so does the corresponding solution w = w(c 1 x 1 + c 2 x 2 + ... + c m x m ) of the original equation (3). The proof of the theorem is now complete. Conclusions For a broad class of autonomous quasilinear nonlinear partial differential equations of the arbitrary order n ≥ 3, including only the highest order derivatives of order n, it has been proven that the solutions admit moveable critical singularities, i.e. no such equation could possess the Painlevé property. The present work provides an example and opens up potential for transferring recent results in the Painlevé classification of higher order ordinary differential equations into the partial differential equation domain. And while direct reduction to ordinary differential equations used in this paper is not always sufficient for the Painlevé classification of partial differential equations, we believe this approach could largely help excluding the classes of equations that certainly do not possess Painlevé property, enabling other methods of Painlevé analysis for the remaining classes. Resulting Painlevé classifications could benefit multiple applied areas concerned with integrability of nonlinear evolutionary partial differential equations. Theorem 1 . 1Solutions of the equations of the class (3) always admit moveable critical singularities, so such equations do not possess Painlevé property. Proof. One can see that all possible reductions of the equation (3) along the one-dimensional linear manifolds z = c 1 x 1 + c 2 x 2 + ... + c m x m could be written as  Painlevé analysis for nonlinear partial differential equations. M Musette, The Painlevé property. SpringerMusette M (1999) Painlevé analysis for nonlinear partial differential equa- tions. In: The Painlevé property, Springer. pp. 517-572. Solitons, nonlinear evolution equations and inverse scattering. 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[ "Adaptive Mesh Refinement for Characteristic Grids", "Adaptive Mesh Refinement for Characteristic Grids" ]	[ "Jonathan Thornburg " ]	[]	[]	I consider techniques for Berger-Oliger adaptive mesh refinement (AMR) when numerically solving partial differential equations with wave-like solutions, using characteristic (double-null) grids. Such AMR algorithms are naturally recursive, and the bestknown past Berger-Oliger characteristic AMR algorithm, that of Pretorius & Lehner (J. Comp. Phys. 198 (2004), 10), recurses on individual "diamond" characteristic grid cells. This leads to the use of fine-grained memory management, with individual grid cells kept in 2-dimensional linked lists at each refinement level. This complicates the implementation and adds overhead in both space and time. Here I describe a Berger-Oliger characteristic AMR algorithm which instead recurses on null slices. This algorithm is very similar to the usual Cauchy Berger-Oliger algorithm, and uses relatively coarse-grained memory management, allowing entire null slices to be stored in contiguous arrays in memory. The algorithm is very efficient in both space and time. I describe discretizations yielding both 2nd and 4th order global accuracy. My code implementing the algorithm described here is included in the electronic supplementary materials accompanying this paper, and is freely available to other researchers under the terms of the GNU general public license.Adaptive mesh refinement (AMR) algorithms are now a vital part of computational science and are particularly valuable in the numerical solution of partial differential equations (PDEs) whose solutions have a wide dynamic range across the problem domain. Here I focus on explicit finite difference methods and PDEs which have propagating-wave solutions.	10.1007/s10714-010-1096-z	[ "https://arxiv.org/pdf/0909.0036v3.pdf" ]	16,024,782	0909.0036	a0c692c4a86b6d2f8847fd0318b7022aac50eae3	Adaptive Mesh Refinement for Characteristic Grids 14 May 2010 13 May 2010 Jonathan Thornburg Adaptive Mesh Refinement for Characteristic Grids 14 May 2010 13 May 2010General Relativity and Gravitation manuscript No. (will be inserted by the editor) This paper is dedicated to the memory of Thomas Radke, my late friend, colleague, and partner in many computational adventures.0425Dm0270-c0270Bf0260Lj Keywords adaptive mesh refinementfinite differencingcharacteristic coordinatescharacteristic gridsBerger-Oliger algorithm I consider techniques for Berger-Oliger adaptive mesh refinement (AMR) when numerically solving partial differential equations with wave-like solutions, using characteristic (double-null) grids. Such AMR algorithms are naturally recursive, and the bestknown past Berger-Oliger characteristic AMR algorithm, that of Pretorius & Lehner (J. Comp. Phys. 198 (2004), 10), recurses on individual "diamond" characteristic grid cells. This leads to the use of fine-grained memory management, with individual grid cells kept in 2-dimensional linked lists at each refinement level. This complicates the implementation and adds overhead in both space and time. Here I describe a Berger-Oliger characteristic AMR algorithm which instead recurses on null slices. This algorithm is very similar to the usual Cauchy Berger-Oliger algorithm, and uses relatively coarse-grained memory management, allowing entire null slices to be stored in contiguous arrays in memory. The algorithm is very efficient in both space and time. I describe discretizations yielding both 2nd and 4th order global accuracy. My code implementing the algorithm described here is included in the electronic supplementary materials accompanying this paper, and is freely available to other researchers under the terms of the GNU general public license.Adaptive mesh refinement (AMR) algorithms are now a vital part of computational science and are particularly valuable in the numerical solution of partial differential equations (PDEs) whose solutions have a wide dynamic range across the problem domain. Here I focus on explicit finite difference methods and PDEs which have propagating-wave solutions. The most powerful and general AMR algorithms for problems of this type are those based on the pioneering work of Berger and Oliger (1984) (see also Berger (1982Berger ( , 1986; Berger and Colella (1989)). These algorithms use locally uniform grids, refined in space and time as needed, with fine grids (which generally cover only a small part of the problem domain) overlaying coarse grids. At each time step, coarse grids are integrated first and spatial boundary conditions for fine-grid integrations are obtained by time-interpolation from the coarse grids. This whole process is applied recursively at each of the possibly-many levels of mesh refinement. Berger and Oliger's original work, as well as most following work, used Cauchy-type coordinates and grids, where initial data is given on a spacelike hypersurface and the solution is then computed one spacelike slice at a time within a numerical problem domain with (typically) timelike boundaries. For problems where the propagating-wave PDEs are naturally posed on an infinite domain, these finite-domain timelike boundaries require radiation boundary conditions. For many problems of interest these boundary conditions can only be approximate, and for the Einstein equations or similar constrained PDE systems they may render the evolution system ill-posed, generate significant boundary reflections, and/or generate significant constraint violations. In practice it's often difficult and/or computationally expensive to reduce these boundary-condition errors to an acceptably low level. 1 As an alternative to Cauchy formulations, here I consider characteristic formulations, where the numerical problem domain's boundaries are null geodesics. This makes it very easy to impose boundary conditions on the continuum PDEs in a well-posed and constraintpreserving manner, and to approximate these boundary conditions very accurately in the finite differencing scheme. While Cauchy-type AMR is now widely used in numerical relativity, and characteristic formulations are also not uncommon, there has been much less study of Berger-Oliger AMR using characteristic formulations. This is the topic of this paper. The best-known work on Berger-Oliger characteristic AMR is that of Pretorius and Lehner (2004), who describe an algorithm which treats the two null coordinates symmetrically, and whose fundamental unit of recursion is the "diamond" double-null characteristic grid cell. This leads to their code using very fine-grained memory management, with each individual grid point at each refinement level containing linked-list pointers to its neighboring grid points in each null direction. This makes the programming more complicated and adds some space and time overhead. Their algorithm has O(∆ 2 ) global accuracy, where ∆ is the grid resolution. In contrast, the AMR algorithm I describe here is much closer to the earlier work of Hamadé and Stewart (1996), treating the two null coordinates asymmetrically and only recursing on null slices. (In Cauchy-evolution terms, the slice-recursion algorithm treats one null coordinate as a "time" coordinate labelling null slices and the other as a "space" coordinate labelling events on a null slice.) My algorithm uses relatively coarse-grained memory management, with all the grid points in a single null slice level stored in a single set of arrays which can easily be stored contiguously in memory. This leads to relatively simple programming with only a small loss of efficiency from the coarser-grained adaptivity. I describe finite differencing schemes and interpolation operators which yield O(∆ 4 ) global accuracy, as well as the usual O(∆ 2 ). By using C++ templates, my code is able to support both cases with no run-time overhead. To demonstrate the slice-recursion AMR algorithm I use a simple model problem, the spherically symmetric real or complex scalar wave equation on a Schwarzschild-spacetime background, with a time-dependent Dirac δ -function source term. This problem is generally representative of a wide range of black-hole perturbation problems and, more generally, of PDEs where characteristic AMR algorithms may be appropriate. The remainder of this paper is organized as follows: the remainder of this section outlines the notation used in this paper. Section 2 describes the model problem. Section 3 gives a brief outline of the unigrid finite differencing schemes I use for globally 2nd and 4th order accuracy. Section 4 describes how the local truncation error of the finite differencing scheme can be estimated. Section 5 describes the slice-recursion AMR algorithm and compares it to other Cauchy and characteristic Berger-Oliger algorithms. Section 6 presents tests of the AMR algorithm to demonstrate that it is accurate and efficient. Section 7 draws general conclusions. Appendix A gives a detailed description of the unigrid finite differencing schemes I use. Appendix B discusses some implementation aspects of the AMR algorithm. Notation I generally follow the sign and notation conventions of Wald (1984), with a (−, +, +, +) metric signature. I use the Penrose abstract-index notation, with Latin indices abc running over spacetime coordinates. ∇ a is the covariant derivative operator associated with the 4metric. I use upper-case sans-serif letters A, B, C, . . . to label grid points and (in section 5 and appendix A) finite difference grids. I describe my notation for finite difference grids in detail in section 5.1. I use SMALL CAPITALS for the names of software packages and (in appendix B.4) major data structures in my AMR code. ⌈x⌉ denotes the smallest integer ≥ x. I use a pseudocode notation to describe algorithms: Lines are numbered for reference, but the line numbers are not used in the algorithm itself. # marks comment lines, while keywords are typeset in bold font and most variable names in typewriter font (a few variable names are mathematical symbols, such as "ℓ max "). "X ← Y " means that the variable X is assigned the value of the expression Y . Variables are always declared before use. The declaration of a variable explicitly states the variable's type and may also be combined with the assignment of an initial value, as in "integer j ← 0". The looping construct "for integer X from A to B by C" is inspired by BASIC but also includes a declaration of the loop variable (with scope limited to the loop body, as in C++ and Perl). The looping semantics are the same as Fortran's "do X = A, B, C", with the increment C defaulting to 1 if omitted. Conditional statements use PL/I-inspired syntax (if-then-else). { and } delimit the scope of procedures, loop bodies, and either of the branches of conditional statements. Procedures (subroutines) are marked with the keyword procedure, and are explicitly invoked with a call statement. Procedure names are typeset in typewriter font. When referring to a procedure as a noun in a figure caption or in the main text of this paper, the procedure name is suffixed with "()", as in "foo()". Model Problem The basic AMR algorithm presented here is quite general, but for ease of exposition I present it in the context of a simple model problem. This model problem derives from the calculation of the radiation-reaction "self-force" on a scalar particle orbiting a Schwarzschild black hole, but for purposes of this paper the model problem may be considered by itself, divorced from its physical context. Thus, consider Schwarzschild spacetime of mass M and introduce ingoing and outgoing null coordinates u and v respectively, so the line element is ds 2 = − f (r) du dv + r 2 dΩ 2 ,(1) where r is (thus defined to be) the usual areal radial coordinate, f (r) ≡ 1 − 2M/r, and dΩ 2 is the line element on a 2-sphere of constant r. It's also useful to define the Schwarzschild time coordinate t Schw = 1 2 (v + u) and the "tortise" radial coordinate r * = 1 2 (v − u) = r + 2M log r 2M − 1 .(2) In this paper I only consider the region outside the event horizon, r > 2M, so the coordinates t Schw , r, and r * are always nonsingular, t Schw is always timelike, and both r and r * are always spacelike. My computational scheme requires numerically inverting (2) to obtain r(r * ); I discuss this inversion in appendix B.1. The model problem is the spherically symmetric scalar wave equation on this background spacetime, with a time-dependent Dirac δ -function source term (stationary in space), φ +V (r)φ ≡ ∂ 2 φ ∂ u ∂ v +V (r)φ = S(t Schw )δ (r − r p ) ,(3) where = ∇ a ∇ a is the usual curved-space wave operator, φ is a real or complex scalar field, r p > 2M is a specified "particle" radius giving the spatial position of the source-term worldline, V (r) is a specified (smooth) position-dependent potential which varies on a typical spatial scale M and which vanishes at spatial infinity, and S(t Schw ) is a specified time-dependent (typically highly-oscillatory) source term defined along the source worldline r = r p . I define · to be a pointwise norm on the scalar field φ . For reasons discussed in section 4.2, in the complex-scalar-field case · should be the complex magnitude rather than (say) the L 1 norm φ 1 = Re[φ ] + Im[φ ] , even though the latter is slightly cheaper to compute. As shown in figure 1, the problem domain is a square in u max ]. I take the particle worldline r = r p to symmetrically bisect the domain. For a given domain, it's convenient to introduce "relative v" (rv) and "relative u" (ru) coordinates rv = v − v min and ru = u − u min respectively. Initial data must be specified along the "southwest" and "southeast" faces of the domain, v = v min and u = u min respectively. The "northwest" and "northeast" faces of the problem domain, u = u max and v = v max respectively, are ingoing null characteristics with respect to the problem domain, so no boundary conditions need be posed there. This is a key advantage of a characteristic evolution scheme. In contrast, using a Cauchy evolution scheme an outgoing-radiation boundary condition is generally required on each timelike boundary. (v, u) space, (v, u) ∈ [v min , v max ]× [u min , Assuming smooth initial data on the southwest and southeast grid faces, φ is C ∞ everywhere in the problem domain except at the particle worldline. In practice, φ and its spacetime gradients display high dynamic ranges across the problem domain, varying rapidly near the particle worldline but only slowly far from the worldline. This makes a unigrid scheme quite inefficient and is the primary motivation for using AMR. Because of the δ -function source term, at the particle worldline φ is generically only C 1 , i.e., φ is continuous but its gradient generically has a jump discontinuity across the particle worldline. Any finite differencing or interpolation operators which include grid points on both sides of the particle worldline must take the discontinuity into account. Unigrid Finite Differencing The foundation of any mesh-refinement scheme is a stable and locally consistent unigrid discretization. To describe this, I introduce a uniform finite-difference grid over the problem domain, with equal grid spacing ∆ in v and u. I use j and i as the integer grid-point coordinates corresponding to v and u respectively, v j ≡ v min + j∆ (4a) u i ≡ u min + i∆ ,(4b) where j and i range from 0 to N ≡ D/∆ inclusive, where D = v max −v min = u max −u min is the problem-domain size. As is common in finite-differencing computations, I use the notation that subscripting a grid function denotes its value at the specified grid point, for example φ j,i ≡ φ (v=v j , u=u i ) .(5) To simplify the finite differencing near the particle worldline, I require that the grid be placed such that if the particle worldline passes through a grid cell, it does so symmetrically, bisecting through the center of the cell. 2 This assumption considerably simplifies the finite differencing near the particle worldline, and makes it easy to represent the effects of the δfunction source term accurately. (See Tornberg and Engquist (2004) for a general discussion of the numerical treatment of δ -function terms in PDEs.) The finite difference schemes I consider here are all explicit, with molecules summarized in figure 2. Briefly, for 2nd order global accuracy I use a standard diamond-cell integration scheme (Gómez and Winicour (1992); Gómez et al. (1992); Gundlach et al. (1994); Burko and Ori (1997); Lousto and Price (1997); Lousto (2005); Winicour (2009)), while for 4th order global accuracy I use a modified version of the scheme described by Lousto (2005);Haas (2007). I describe the finite differencing schemes in detail in appendix A. With one exception discussed in appendix A.3, these finite differencing schemes are stable. 2 This symmetric passage is only possible because r p is time-independent. For the more generic case where r p is time-dependent, then in general the particle worldline would pass obliquely through the cell. As discussed by, for example, Martel and Poisson (2002); Lousto (2005); Haas (2007), this would considerably complicate the finite differencing of cells intersecting the particle worldline. However, it wouldn't alter the overall character of the AMR algorithm. It's important to know the domain of dependence of the finite difference computation of φ j,i , i.e., the set of grid points ( j+β , i+α) where φ j+β ,i+α is used as an input in computing φ j,i . (I define the "radius" of such a finite difference computation as the maximum of all such \|β \| and \|α\|, the radius in the j direction as the maximum of all such \|β \|, and the radius in the i direction as the maximum of all such \|α\|.) For 2nd order global accuracy, the domain of dependence is precisely that shown in figure 2, i.e., it comprises the 3 grid points {( j, i−1), ( j−1, i), ( j−1, i−1)}. For 4th order global accuracy, this set depends on the position of ( j, i) relative to the particle worldline, but it never includes points outside the 4 slices { j, j−1, j−2, j−3} or outside the 4 u positions {i, i−1, i−2, i−3}. r * t Schw u v r = r p v = v m i n v = v m a x u = u m i n u = u m a x Given this domain of dependence, there are many possible orders in which the φ j,i may be computed in a unigrid integration. However, the algorithms I consider here all integrate the grid points in "raster-scan" order, i.e., using an outer loop over j (so that each iteration of the outer loop integrates a single v = constant null slice) and an inner loop over i. Local versus Global Truncation Errors for Characteristic Schemes Recall the (standard) definitions of the local and global truncation error of a finite differencing scheme (Kreiss and Oliger (1973); Choptuik (1991); Richtmyer and Morton (1994); LeVeque (2007)): The local truncation error (LTE) is a pointwise norm of the discrepancy that results when the exact solution of the PDE is substituted into the finite difference equations at a grid point. The global truncation error (GTE) is a pointwise norm of the difference between the exact solution of the PDE and the result of solving the finite difference equations using exact arithmetic (i.e., without floating-point roundoff errors). The LTE and GTE are both grid functions. For a stable and consistent Cauchy evolution scheme, the GTE and LTE are normally of the same order in the grid spacing ∆ (Kreiss and Oliger (1973); Choptuik (1991); Richtmyer and Morton (1994); LeVeque (2007)). However, in a characteristic evolution, errors can build up cumulatively over many grid points, so the GTE is generically worse than the LTE. For a 1+1 (space + time) dimensional problem such as that considered here, errors can accumulate over O(N 2 ) grid points where N = O(1/∆ ), so generically an O(∆ n+2 ) LTE is required to guarantee an O(∆ n ) GTE. Corresponding to this, the finite differencing schemes I present here for 2nd and 4th order global accuracy (GTE) actually have 4th and 6th order LTE respectively, except near the particle worldline where one order lower GTE is acceptable because there are only O(N) near-worldline grid cells. i i − 1 j j − 1 2nd order global accuracy i i − 1 i − 2 i − 3 j j − 1 j − 2 j − 3 4th order global accuracy point (j, i) where φ j,i is computed point where φ is used as an input in computing φ j,i Initial Data The globally-4th-order finite differencing scheme summarized in figure 2 and described in detail in appendix A.2 is a 4-level scheme: the finite-difference molecule for computing φ j,i includes points on 3 previous slices and at 3 previous u positions. Physical boundary conditions are (only) given on the southeast and southwest grid boundaries, so starting the numerical integration requires that "extended initial data" somehow be computed, comprising the physical boundary data, the next 2 slices after the southwest grid boundary, and the next 2 grid points on each succeeding slice after the southeast grid boundary. I describe several different schemes for constructing the extended initial data in appendix A.2.4. Estimation of the Local Truncation Error As is common in AMR algorithms, I implement the "adaptive" part of AMR using an estimate of the numerical solution's local truncation error (LTE). I compute this via the standard technique (LeVeque (2007, section A.6)) of comparing the main numerical integration with that from a coarser-resolution integration. In the context of Berger-Oliger style AMR, where a hierarchy of grids are being integrated concurrently at varying resolutions, there are several possible choices for the coarser-resolution comparison solution. It can come from a separate "shadow" AMR hierarchy (each level of which is coarser than the corresponding level of the main AMR hierarchy), or it can come from the next-coarsest level of the (single) AMR hierarchy (the "self-shadow hierarchy" technique of Pretorius (2002b); Pretorius and Lehner (2004)). Here I use a different technique, also used by Hamadé and Stewart (1996), where the coarser-resolution comparison is obtained locally within the (unigrid) evolution at each level of the grid hierarchy, by simply subsampling every 2nd grid point of every 2nd slice. That is, suppose we have a finite differencing scheme with O(∆ n ) LTE, whose domain of dependence for computing φ j,i is the set of K grid points {( j+β k , i+α k ) 1 ≤ k ≤ K} for some constants {β k } and {α k }. For any ( j, i) and ∆ , let φ (∆ ) j,i be the value of φ j,i obtained from the usual numerical integration with grid spacing ∆ . Let φ (∆ →2∆ ) j,i be the value of φ j,i obtained by taking a single step of size 2∆ using as inputs the φ (∆ ) j,i values at the set of K grid points {( j+2β k , i+2α k ) 1 ≤ k ≤ K}. Then I estimate the LTE in computing φ (∆ ) j,i as LTE ≈ 1 2 n − 4 φ (∆ ) j,i − φ (∆ →2∆ ) j,i(6) where the normalization factor is obtained by comparing the LTE of a single 2∆ -sized step with that accumulated in 4 separate ∆ -sized steps covering the same region of spacetime. This scheme is easy to implement and works well, although it does limit the locations where the LTE can be estimated to those where (i) the data for a 2∆ -sized step is available, and where (ii) both the ∆ -and 2∆ -sized steps use the same finite differencing scheme. Constraint (i) implies, for example, that when a new refinement level is created, LTE estimates aren't available for it until a few v = constant slices have been integrated on that level. For the finite differencing schemes described here, constraint (ii) implies that the LTE estimate isn't available for cells within a few grid points of the particle worldline. I haven't found either of these constraints to be a problem in practice. Cost of Computing the LTE Estimate The cost in space (memory usage) associated with this LTE-estimation scheme is the requirement that sufficiently many adjacent slices be kept in memory simultaneously for the 2∆ -sized steps. For the globally-2nd-order finite differencing scheme described in section 3, the LTE estimator requires data from 3 adjacent slices. As discussed in section 5.3, for 2nd order global accuracy my AMR algorithm uses interpolation operators which use data from up to 4 adjacent slices, so the LTE estimator is "free" in the sense that it doesn't increase the number of slices needing to be kept in memory beyond what the rest of the computation already requires. For the globally-4th-order finite differencing scheme described in section 3, the LTE estimator requires 7 adjacent slices, while the interpolation operators only need 6 adjacent slices, so the LTE estimator adds a 1 6 ≈ 17% fractional overhead in memory usage. I discuss the CPU-time cost of computing the LTE estimate in footnote 9 in section 5.3. Smoothing the LTE Estimates In an AMR scheme, mesh-refinement boundaries and regridding operations tend to introduce small amounts of interpolation noise into the numerical solution, which tends to be amplified in the LTE estimate. In particular, for the scheme described here, there are often isolated points with anomolously high LTE estimates. To avoid having these falsely trigger (unwanted) mesh refinements, I smooth the LTE estimates on each slice with a moving median-of-3 filter before comparing them to the error threshold. Pretorius and Lehner (2004) describe the use of a moving-average smoothing of the LTE estimate to address a slightly different problem in their characteristic AMR algorithm: The point-wise TE [truncation error] computed using solutions to wave-like finitedifference equations is in general oscillatory in nature, and will tend to go to zero at certain points within the computational domain . . . , even in regions of relatively high truncation error. We do not want such isolated points of (anomalously) small TE to cause temporary unrefinement, . . . For the model problem I consider here, "temporary unrefinement" doesn't seem to be a problem in practice so long as the norm · is chosen to be the complex magnitude of φ . This appears to be because while the complex phase of φ oscillates rapidly along the particle worldline, φ 's complex magnitude tends to remain relatively constant. (In an early version of my AMR code where I used the L 1 norm φ 1 = Re[φ ] + Im[φ ] , I found that temporary unrefinement did indeed tend to occur, as the rapidly changing complex phase of φ translated into corresponding changes in φ 1 .) For other physical systems, further smoothing of the estimated LTE might be necessary. Adaptive Mesh Refinement The Berger-Oliger Algorithm The Berger-Oliger AMR algorithm for Cauchy evolutions of hyperbolic or hyperbolic-like PDEs (Berger and Oliger (1984); see also Berger (1982Berger ( , 1986; Berger and Colella (1989)) was first used in numerical relativity by Choptuik (1986Choptuik ( , 1989Choptuik ( , 1992Choptuik ( , 1993, and is now widely used for a variety of problems. Schnetter et al. (2004) give a nice summary of some of the considerations involved in using the Berger-Oliger algorithm with evolution systems which contain 2nd spatial derivatives but only 1st time derivatives. Lehner et al. (2006); Brügmann et al. (2008); Husa et al. (2008) discuss adjustments to the algorithm (particularly interpolation and prolongation operators) necessary to obtain higher-than-2ndorder global finite differencing accuracy in a Berger-Oliger scheme. Pretorius and Choptuik (2006) discuss refinements to the standard Berger-Oliger algorithm to accomodate coupled elliptic-hyperbolic systems of PDEs. MacNeice et al. (2000); Burgarelli et al. (2006) discuss quadtree/octtree grid structures and their use with Berger-Oliger mesh refinement. Many individual codes also have published descriptions, including (among many others), AD (Choptuik (1989(Choptuik ( , 1992(Choptuik ( , 1994), AMRD/PAMR (Pretorius (2002b,a,c)), BAM (Brügmann (1996); Brügmann et al. (2004Brügmann et al. ( , 2008), CARPET/CACTUS (Schnetter et al. (2004); Schnetter (2001); Goodale et al. (2003Goodale et al. ( , 1999), CHOMBO/AMRLIB/BOXLIB (Colella et al. (2009b); Su et al. Olson and MacNeice (1999)), and SAMRAI Kohn (2002a, 1999); Hornung et al. (2006); Hornung and Kohn (2002b)). Although the focus of this paper is on characteristic Berger-Oliger AMR, it's useful to begin with a brief review of the Cauchy Berger-Oliger algorithm. I will only present a few of the algorithm's properties that are particularly relevant here; see the references cited in the previous paragraph for more extensive discussions of the algorithm, its rationale (i.e., why the algorithm is constructed in the way that it is), and how it may be modified to meet various situations. To describe the Berger-Oliger algorithm it's convenient to consider a generic PDE with propagating-wave solutions in 1+1 (space+time) dimensions, and define global timelike and spacelike coordinates t and x respectively. 3 I consider uniform finite difference grids in these coordinates, with indices j and i indexing the t and x dimensions respectively. In contrast to the characteristic-grid case discussed in section 3, I don't assume that the grid cells are square, i.e., I don't assume anything about the Courant number (the ratio of the time step to the spatial resolution). The basic data structure of the Berger-Oliger algorithm is that of a hierarchy of such uniform grids, each having a different resolution. The grids are indexed by an integer "refinement level" ℓ in the range 0 ≤ ℓ ≤ ℓ max . (In general ℓ max is time-dependent, but for convenience I don't explicitly show this in the notation.) I refer to the grid at refinement level ℓ as G (ℓ) , to the time level ("slice") j (i.e., the slice t = t j ) of G (ℓ) as G (ℓ) j , to the grid point (t=t j , x=x i ) as ( j, i), and to the grid function value(s) at this grid point as G x ∈ [G (ℓ) j .x min , G (ℓ) j .x max ], with corresponding grid-point indices i ∈ [G (ℓ) j .i min , G (ℓ) j .i max ]. As suggested by the notation, in general the domain of G (ℓ) isn't rectangular, i.e., in general G (ℓ) j .x min , G (ℓ) j .x max , G (ℓ) j .i min , and G (ℓ) j .i max all vary with j. The coarsest or "base" grid G (0) covers the entire problem domain. For each integer ℓ with 0 < ℓ ≤ ℓ max , G (ℓ) has a resolution 2 ℓ times finer than that of G (0) , 5 and typically covers only some proper subset of the problem domain. Here I consider only "vertex-centered" grids, where every 2nd G (ℓ+1) point coincides with a G (ℓ) point. 6 The Berger-Oliger algorithm requires that the grids always be maintained such that for any ℓ, on any slice common to both G (ℓ) and G (ℓ+1) , the region of the problem domain covered by G (ℓ+1) is a (usually proper) subset of that covered by G (ℓ) . I refer to this property as the "proper nesting" of the grids. 7 Each grid G (ℓ) maintains its own current slice for the integration, denoted G (ℓ) current j ; the j coordinate of this slice is denoted G (ℓ) .current j. Each slice is integrated with the same finite differencing scheme and Courant number. 8 Although I describe each G (ℓ) here as containing the entire time history of its integration, in practice only the most recent few time slices need to be stored in memory. The precise number of time slices needed is set by the larger of the number needed by the unigrid finite differencing scheme, the LTE estimation, and by the interpolations used in the Berger-Oliger algorithm. This is discussed further in section 5.3. Figure 3 (ignoring the lines marked with •, whose purpose will be discussed in section 5.3) gives a pseudocode outline of the Berger-Oliger algorithm. Notice that the algo-(ℓ) j to have multiple connected components for good efficiency. This would somewhat complicate the data structures, but it wouldn't alter the overall character of the AMR algorithm. 5 This can easily be generalized to any other integer refinement ratio > 1 (including having the refinement ratio vary from one level to another). Larger refinement ratios reduce the ℓ max needed for a given total dynamic range of resolution in the refinement hierarchy and thus reduce some of the "bookkeeping" overheads in the computation. However, smaller refinement ratios give a smoother variation of the grid resolution (i.e., a variation with smaller jumps) across the problem domain, allowing the resolution to be better matched to that needed to just obtain the desired LTE at each event, which improves the overall efficiency of the computation. For this latter reason I use a refinement ratio of 2 :1 in my AMR algorithm and code. 6 An alternative approach uses "cell-centered" grids, where grid points are viewed as being at the center of grid cells and it is these grid cells which are refined (so that the G (ℓ+1) grid points are located 1 4 and 3 4 of the way between adjacent G (ℓ) grid points). This approach is particularly useful with finite volume discretizations (LeVeque (2002)) and is used by, for example, the PARAMESH mesh-refinement framework (MacNeice et al. (2000); Olson and MacNeice (2005); Olson (2006); Olson and MacNeice (1999)) and the BAM code (Brügmann et al. (2004). 7 The proper-nesting requirement is actually slightly stronger: each G (ℓ+1) grid point must be far enough inside the region covered by G (ℓ) to allow interpolating data from G (ℓ) . In practice, in the v direction this requirement is enforced implicitly by the Berger-Oliger algorithm, while in the u direction this requirement must be enforced explicitly in the regridding process (procedure shrink to ensure proper nesting() in figure 4). 8 Using the same Courant number at each refinement level (i.e., scaling the time step on each grid proportional to the spatial resolution) is known as "subcyling in time" and is widely, though not universally, used in Berger-Oliger AMR codes. Dursi and Zingale (2003) disuss some of the tradeoffs determining whether or not subcycling is worthwhile. rithm is recursive, and that this recursion is at the granularity of an entire slice. That is, the algorithm integrates an entire slice (lines 11-19 in figure 3) before recursing to integrate the finer grids (if any). A key part of the Berger-Oliger algorithm is regridding (lines 23-28 in figure 3), where the grid hierarchy is updated so that each G (ℓ) covers the desired spatial region for the current time. As shown in more detail in figure 4, if this requires adding a new G (ℓ) to the hierarchy, or moving an existing G (ℓ) to cover a different set of spatial positions than it previously covered, then data must be interpolated from coarser refinement levels to initialize the new fine-grid points. Finer grids may also need to be updated to maintain proper nesting. Because of this, and because of the data copying discussed below, regridding is moderately expensive, typically costing O(1) times as much (at each level of the refinement hierarchy) as integrating a single time step. To prevent this cost from dominating the overall computation, regridding is only done on "selected" slices; in practice a common choice is to regrid on every kth slice at each level of the refinement hierarchy, for some (constant) parameter k ∼ 4. For similar reasons, the LTE estimate is often only computed at every kth grid point. 9 Figure 5 shows an example of the operation of the slice-recursion algorithm discussed in section 5.3. However, parts (a)-(e) of this figure can also be interpreted as an example of the operation of the (Cauchy) Berger-Oliger algorithm discussed in this section, with u as the spatial coordinate and v as the time coordinate. [The example shown is unrealistic in one way: to allow the figure to show a relatively small number of grid points (and thus be at a larger and more legible scale), the figure ignores the limits on regridding discussed in section 5.3.1, which my code actually enforces.] Because the globally-4th-order finite differencing scheme illustrated in figure 5 is a 4level scheme, the extended initial data for the base grid comprises 3 slices and 3 points on each succeeding slice. Figure 5a shows the base grid just after the computation of the first evolved point of its first evolved slice (i.e., the first slice which isn't entirely part of the extended initial data). Figure 5b shows an example of the LTE being checked at several points, and (after smoothing) exceeding the error threshold at one of these. The regridding procedure thus creates a new fine grid (lines 18-22 in figure 4). Figure 5c shows this new fine grid just after the computation of the first evolved point of its first evolved slice. Notice that the actual spatial extent of the newly-created fine grid is larger than just the set of points where the (smoothed) LTE exceeds the error threshold, for two reasons: -If the (smoothed) LTE estimate exceeds the error threshold at some location on the current slice, then logically we don't know which point(s) in the LTE-estimate molecule have inaccurate data. The algorithm thus includes all the LTE-estimate molecule's points in the region-to-be-refined (line 18 of figure 3). -The algorithm also uses "buffer zones" to further enlarge the region-to-be-refined in the spatial direction beyond the set of points just described (lines 20-21 in figure 3). The buffer zones are 2 grid points on each side of this set for the example shown in figure 5c. The buffer zones are used for two reasons: -The buffer zones ensure that moving solution features and their finite-difference domains of dependence will remain within the refined region -and thus be wellresolved -throughout the time interval before the next regridding operation. -The buffer zones also help to ensure that if there are any finer grids in the grid hierarchy, the finite-difference domains of dependence for interpolating the nextfiner grid's spatial boundary data from the current grid (line 9 in figure 3; figure 5c,d) will avoid regions where the solution is not well-resolved by the current grid (and hence the interpolation would be inaccurate). When a new fine grid G (ℓ+1) is created, at what time level should it be placed relative to the next-coarser grid G (ℓ) ? There are a number of possible design choices here, ranging from the time level of the most recent all-points-below-the-error-threshold G (ℓ) LTEestimate check up to the time level of the some-points-above-the-error-threshold G (ℓ) LTEestimate check which triggered the creation of the new fine grid. [Placing the fine grid at an earlier time level makes the total integration slightly more expensive, but lessens the use of insufficiently-accurate coarse-grid data (i.e., G (ℓ) data whose LTE estimate exceeds the error threshold) in interpolating the fine grid's initial data.] As shown in the example of figure 5c, I have (somewhat arbitrarily) chosen to place the newly-created fine grid G (ℓ+1) with the last (most-future) of its initial-data slices (those which are entirely interpolated from the nextcoarser grid G (ℓ) ; line 20 of figure 4) at the time level G (ℓ) current j−1 , one coarse-grid time step before the time level on which the over-threshold LTE estimates were computed. Each slice G (ℓ) j is a standard 1-dimensional grid function or set of grid functions, and so may be stored as a contiguous array or set of arrays in memory. This is easy to program, and allows the basic time integration (lines 11-19 in figure 3) to be highly efficient. 10,11 It also means that the amount of additional "bookkeeping" information required to organize the computation is very small, proportional only to the maximum number of distinct grids at any time. The one significant disadvantage of contiguous storage is that when regridding requires expanding G (ℓ) j then the existing data must be copied to a new (larger) set of arrays. However, the cost of doing this is still relatively small (less than the cost of a single time step for all the grid points involved). After the recursive calls to integrate finer grids (lines 33 and 34 in figure 3), G (ℓ+1) has been integrated to the same time level as G (ℓ) . Figure 5d shows an example of this. Since G (ℓ+1) has twice as fine a resolution as G (ℓ) , it presumably represents the solution more accurately at those events common to both grids. To prevent the coarser grids from gradually becoming more and more inaccurate as the integration proceeds (which would eventually contaminate the finer grids via coarse-to-fine interpolations in regridding), the algorithm copies ("injects") G (ℓ+1) back to G (ℓ) at those events common to both grids. This is done at lines 36-41 in figure 3; figure 5e shows an example of this. The Pretorius-Lehner Algorithm Pretorius and Lehner (2004) discuss modifications to the standard Berger-Oliger algorithm to accomodate characteristic evolution. Their algorithm treats the two null directions sym-10 Because the algorithm primarily sweeps sequentially through contiguously-stored grid functions, it should have fairly low cache miss rates. Moreover, many modern computer systems have special (compiler and/or hardware) support for automatically prefetching soon-to-be-used memory locations in code of this type, further reducing the average memory-access time and thus increasing performance. 11 In fact, with appropriate software design the basic integration routine can often be reused intact, or almost intact, from an existing unigrid code. For example, this is common when using the CARPET (Schnetter et al. (2004); Schnetter (2001) 1 # integrate G (ℓ) forward by one time step 2 procedure time step (integer ℓ) 3 { 4 G (ℓ) .current j ← G (ℓ) .current j + 1 5 integer j ← G (ℓ) .current j 6 if (ℓ = 0) 7 then set G (ℓ) j spatial boundary data from the physical boundary conditions 8 and (if used) the extended initial data 9 else set G (ℓ) j spatial boundary data by spacetime-interpolating from the next-coarser grid G (ℓ−1) 10 11 # main integration (includes local-truncation-error estimation at selected points of selected slices) 12 region where to refine ← ∅ 13 for integer ii from G (ℓ) 1 # add, delete, and/or move G (ℓ) so it covers the region R 2 # (may also move or delete finer grids G (k) for k > ℓ in order to maintain proper nesting) 3 procedure regrid (integer ℓ, region R) 4 { 5 if (the next coarser grid G (ℓ−1) does not have enough time levels stored 6 to allow spacetime-interpolation from G (ℓ−1) ) 7 j .i min to G (ℓ) j .i max 14 { 15 G (ℓ) j, then return # ignore the regridding request for now; if the need for regridding persists, 8 # the Berger-Oliger algorithm will keep requesting regridding, until eventually 9 # G (ℓ−1) will have enough time levels stored for interpolation to be possible 10 11 if (R = ∅) 12 then { 13 destroy any G (k) with k ≥ ℓ 14 ℓ max ← ℓ − 1 15 } 16 17 else if (G (ℓ) does not exist) 18 then { 19 create G (ℓ) covering the region R 20 initialize G (ℓ) by spacetime-interpolating data from the next coarser grid G (ℓ−1) 21 ℓ max ← ℓ 22 } 23 24 else { 25 move G (ℓ) so it covers the region R, initializing any new points 26 by spacetime-interpolating data from the next coarser grid G (ℓ−1) 27 call shrink to ensure proper nesting (ℓ+1) 28 } 29 } 30 31 # shrink G (ℓ) and any finer grids as necessary so as to ensure proper nesting 32 procedure shrink to ensure proper nesting (integer ℓ min ) 33 { 34 max radius ← the largest radius of any interpolation molecule in the i direction 35 for integer ℓ from ℓ min to ℓ max 36 { 37 integer j ← G (ℓ−1) .current j 38 region R shrunken ← [G (ℓ−1) j .i min+max radius, G (ℓ−1) j .i max−max radius] 39 if (G (ℓ) ⊆ R shrunken) 40 then { 41 region R intersection ← R shrunken ∩ region covered by G (ℓ) 42 if (R intersection = ∅) 43 then { 44 destroy any G (k) with k ≥ ℓ 45 ℓ max ← k − 1 46 return # there are now no finer levels, so this procedure is done 47 } 48 shrink G (ℓ) to just cover the region R intersection 49 } 50 } 51 } metrically, and instead of using a separate regridding step, interleaves the integration, injection, LTE estimation, and updating of the mesh-refinement hierarchy at a very fine granularity (essentially that of individual diamond cells). This gives an elegant algorithm where the integration can proceed simultaneously in both null directions, "flowing" across the problem domain in a way that's generally not known in advance. Because of this unpredictable flow, Pretorius and Lehner don't use contiguous arrays to store the grid functions. Rather, they use a fine-grained linked-list data structure, where each grid point at each refinement level ℓ stores explicit pointers to its 4 (null) neighboring points at that refinement level, and also to the grid points at that same event at refinement levels ℓ±1. The Pretorius-Lehner algorithm explicitly walks these pointer chains to locate neighboring points for the (unigrid) integration at each refinement level, to create finer grids, and to inject fine-grid results back into coarser grids at each level of the refinement hierarchy. In comparison to the contiguous storage possible with the standard (Cauchy) Berger-Oliger algorithm, this linked-list storage allocation avoids data copying when grids must be grown. However, the per-grid-point pointers require extra storage, and following the pointer chains adds some programming complexity and extra execution time. 12,13 The Slice-Recursion Algorithm Here I describe a different variant of the Cauchy Berger-Oliger algorithm for characteristic evolution. The basic concept of this algorithm is to treat one null direction (v) as a "time" and the other (u) as a "space", then apply the standard Berger-Oliger algorithm as discussed in section 5.1 (with one significant modification discussed below). Figure 3 (now including the lines marked with •) gives a pseudocode outline of this "slice-recursion" algorithm, and figure 5 shows an example of the algorithm's operation for the globally-4th-order finite differencing scheme described in section 3. For a Cauchy evolution, the future light cone of a grid point contains only O(1) grid points on the next t = constant slice. In contrast, for a characteristic evolution, the future light cone of a grid point ( j * , i * ) contains all points ( j, i) with j ≥ j * and i ≥ i * . To see the impact of this difference on the Berger-Oliger algorithm, suppose that on some v = constant slice we have a coarser grid G (ℓ) overlaid by a finer grid G (ℓ+1) covering the coarse-grid coordinate region i ∈ [i 1 , i 2 ]. Then the injection of the fine-grid results back to the coarse grid (lines 36-41 in figure 3; figure 5e) restores the coarse-grid solution to the fine-grid accuracy for i ∈ [i 1 , i 2 ]. However, unlike in the Cauchy case, the coarse-grid solution for the slice "tail" i > i 2 remains inaccurate, because its computation was affected by the (inaccurate) pre-injection coarse-grid region i ∈ [i 1 , i 2 ]. The solution to this problem is to re-integrate the "tail" i > i 2 of the slice after the injection (lines 43-48 of figure 3; figure 5f). Depending on the placement of the fine grid relative to the coarse grid, the cost of the re-integration may vary from negligible up to roughly the cost of a single time step for the coarse grid (i.e., a factor of 2 increase in the cost of the coarse-grid part of the computation). This overhead only affects grids with 0 ≤ ℓ < ℓ max ; the finest grid (ℓ = ℓ max ) never needs to be re-integrated. In practice the re-integration overhad is generally modest; I present numerical test results quantifying this in section 6. The slice-recursion algorithm I present here is quite similar to that outlined by Hamadé and Stewart (1996). Their algorithm shares the basic Berger-Oliger mesh-hierarchy structure, uses the same LTE estimator (section 4), imposes the same nesting requirements on the grid hierarchy, and does the same "tail" re-integration (lines 43-48 of figure 3; figure 5f). However, their algorithm uses a 4 : 1 refinement ratio between adjacent levels in the mesh-refinement hierarchy, whereas I use a 2:1 ratio in the slice-recursion algorithm for the reasons outlined in footnote 5. They discuss only globally-2nd-order finite differencing. When the grid hierarchy contains 3 or more refinement levels, the evolution and regridding schemes described by Hamadé and Stewart (1996) are somewhat different than those presented here: Their algorithm integrates child grids at all levels of the grid hierarchy up to the same time level before doing any fine-to-coarse-grid injections (lines 36-41 of figure 3; figure 5e) or regridding (figure 4), whereas the algorithms presented here follow the standard Berger-Oliger pattern where injections and regridding are interleaved with the evolution of different refinement levels in an order corresponding to the depth-first traversal of a complete binary tree. Unlike the algorithm of Hamadé and Stewart (1996), the slice-recursion algorithm presented here is purely recursive, treating all levels of the refinement hierarchy in exactly the same way except for the setup of the base grid's extended initial data and the details of how the spatial boundary data are determined at the start of each slice's integration (lines 6-9 of figure 3). An important consequence of the algorithm being organized this way is that for any integer k ≥ 1, the algorithm's computations on a grid hierarchy with k refinement levels are identical (apart from the initial-data setup just noted) to those in each of the recursive calls (lines 33 and 34 of figure 3) for a problem with k+1 refinement levels. More generally, the treatment of the k finest refinement levels in the grid hierarchy is independent of the presence of any coarser level(s). I find that this simplifies debugging, by making the algorithm's behavior on small test problems with only a few refinement levels very similar to its behavior on large "physics" problems with many refinement levels. Like any Berger-Oliger algorithm, the slice-recursion algorithm needs to interpolate data from coarse to fine grids (line 9 of figure 3 and lines 20 and 25-26 of figure 4, figure 5c). I use a mixture of 1-dimensional and 2-dimensional Lagrange polynomial interpolation for this, in all cases chosen so as to avoid crossing the particle worldline. Appendix B.3 describes the interpolation operators in detail. These operators may use data from up to 4 [6] adjacent slices for the 2nd [4th] order GTE schemes, which sets a lower bound on the number of slices of each G (ℓ) which must be kept in memory. For 2nd order GTE, my code keeps only the minimum (4) number of slices in memory; for 4th order GTE, it keeps one extra slice in memory (for a total of 7) to accomodate the LTE estimator (section 4; figure 5b,c). Avoiding Undesired Mesh Refinement When adding a new refinement level to the mesh-refinement hierarchy, the interpolation of initial data (line 20 of figure 4) tends to introduce low-level noise into the field variables. The same is true when an existing fine grid is moved to a new position. In either case, this noise can cause the LTE estimate to be inaccurate. It's thus useful to allow this noise to decay (i.e., be damped out by the inherent dissipation in the finite differencing scheme) before using the LTE estimate to determine the placement of another new refinement level. To this end, my code suppresses regridding operations for the first 8 [16] time steps of a new or newly-moved grid, for the 2nd [4th] order GTE finite differencing scheme respectively. My code also suppresses creating a new fine grid if insufficient data is available for the interpolation of all 4 [7] slices kept in memory for the 2nd [4th] order GTE finite differencing scheme respectively. When using the slice-recursion algorithm for the self-force computation, the arbitrary initial data on the southwest and southeast grid faces induces spurious radiation near these grid faces. This radiation is of no physical interest, so there's no need for the mesh-refinement algorithm to resolve it. Moreover, as discussed in footnote 4, not resolving the spurious radiation also allows a significant simplification of the code's data structures. Thus my code suppresses mesh refinement for the first ∼ 100M of the integration, and for the first ∼ 100M of each slice thereafter. To reduce the effects of the interpolation noise when adding new refinement levels, my code also turns on the mesh refinement gradually, adding new refinement levels only at the rate of one each 10M of evolution. More precisely, the code limits the maximum refinement level to ℓ max ≤    0 if rvu < 100M rvu − 100M 10M if rvu ≥ 100M ,(7) where rvu = min(rv, ru) is the distance from the closest point on the southeast or southwest grid faces. Numerical Tests of the AMR Algorithm As a test case for the slice-recursion AMR algorithm, I consider a particular example of the model problem of section 2, which arises in the course of calculating the radiationreaction "self-force" on a scalar particle orbiting a Schwarzschild black hole (Barack and Ori (2002), see Barack (2009) for a general review). I take φ to be a complex scalar field, with the potential V (r) and source term S(t Schw ) given by V ℓ (r) = f (r) 4 2M r 3 + ℓ(ℓ + 1) r 2 (8) S ℓm (t Schw ) = πq f 2 (r p )a ℓm r p E(r p ) exp −imω(r p )t Schw ,(9) where ω(r) = M r 3 and E(r) = f (r) 1 − 3M r −1/2 .(11) are respectively the orbital frequency and energy per unit mass of a particle in circular orbit at areal radius r in Schwarzschild spacetime. The coefficients a ℓm are defined such that the spherical harmonic Y ℓm (θ = π 2 , ϕ) = a ℓm e imϕ , i.e., a ℓm =      (−1) (ℓ+m)/2 2ℓ + 1 4π (ℓ + m − 1)!! (ℓ − m − 1)!! (ℓ + m)!! (ℓ − m)!! if ℓ−m is even 0 if ℓ−m is odd ,(12) where the "double factorial" function is defined by n!! = n · (n − 2)!! if n ≥ 2 1 if n ≤ 1 .(13) For this test, I take ℓ = m = 10, r p = 10M, and use a problem domain size of D = 200M on a side. Here v min = 12.773M, u min = −12.773M, and the grid extends from rv = 0M to rv = 200M and from ru = 0M to ru = 200M. The physical boundary data is zero along the southwest and southeast grid faces, and the extended initial data is computed using the 2-level subsampling scheme described in appendix A.2.4. The base grid has a resolution of 0.25M, the finite differencing is the globally-4th-order scheme, and the error tolerance for the LTE estimate is 10 −16 . As discussed in section 5.3.1, mesh refinement is suppressed for the first 100M of the evolution and the first 100M of each slice thereafter, and then turned on gradually at the rate of one refinement level for each further 10M of evolution. Figure 6 shows a "map" of the mesh refinement, giving the highest refinement level at each event in the problem domain. Notice that the highest-refinement grids only cover small regions close to the particle worldline. The online supplemental materials which accompany this paper include a movie (online resource 1) showing the spacetime-dependence of φ and the placement of the refined grids; figure 7 shows several sample frames and explains them in more detail. Because of the adaptive placement of refined grids, it's difficult to do a standard convergence test (Choptuik (1991)). Instead, I have followed Choptuik (1992) in adding an option Fig. 6 This figure shows a "map" of the mesh refinement, giving the highest refinement level at each event in the problem domain. The vertical dashed line labelled "particle" shows the particle worldline. The diagonal dashed line labelled "slice" shows the rv = 150M slice; figure 8 shows convergence tests on that slice. to my code to write out a "script" of the position and grid spacing of each grid generated by the AMR algorithm, and a related option to "play back" such a script. For a convergence test, I first run a "record" evolution, then generate a "playback×N" script by refining each grid in the script by a chosen (small integer) factor N, and finally "play back" the refined script. Figure 8 shows an example of such a convergence test for the field φ on the rv = 150M slice. The code shows excellent 4th order convergence in the interior of each grid, across mesh-refinement boundaries, and near to the particle. Profiling the code shows that almost all of the CPU time is consumed in the basic diamond-cell integration code, and the code's overall running time is closely proportional to the number of diamond cells integrated. In other words, the AMR bookkeeping consumes only a negligible fraction of the CPU time. For this evolution, the AMR evolution integrates a total of 35.6 × 10 6 diamond cells. Of these, 16% are accounted for by the re-integration of coarse slices after fine-grid recursion (lines 43-48 of figure 3). A hypothetical unigrid evolution covering the entire problem domain at the resolution of the finest AMR grid (level 5) would require integrating 655 ×10 6 diamond cells, 18.4 times as many as the AMR evolution. That is, for this problem the AMR evolution was approximately a factor of 18 faster than an equivalent-resolution (and thus equivalent-accuracy) unigrid evolution. To further characterize the performance of the slice-recursion algorithm, I consider a sample of 295 separate evolutions of the same model problem, with each evolution having a different (ℓ, m) in the range 0 ≤ ℓ ≤ 40 and 0 ≤ m ≤ ℓ, and a problem-domain size between 400M and 30 000M. 14 The globally-4th-order scheme is used for all of these evolutions, with an error tolerance for the LTE estimate of 10 −16 . Figure 9 shows histograms of the re-integration overhead (the fraction of all diamond-cells integrations which occur as part of coarse-grid re-integrations) and the AMR speedup factor for this sample of evolutions. The re-integration overhead is usually between 20% and 25%, and never exceeds 30%. For this sample the AMR speedup factor varies over a much wider range, from as low as 8.9 to as high as 400. The median speedup factor is 19, and 95% of the speedup factors are between 12 and 51. Conclusions The main result of this paper is that only a small modification (the tail re-integration, i.e., the lines marked with • in figure 3) is needed to adapt the standard Cauchy Berger-Oliger AMR algorithm to characteristic coordinates and grids. The resulting "slice-recursion" algorithm uses relatively coarse-grained control for the recursion and the adaptivity part of AMR, which greatly simplifies the memory management, allowing entire null slices to be stored in contiguous arrays in memory. The algorithm can readily accomodate any order of finite differencing scheme; I present schemes for both 2nd and 4th order global truncation error. The numerical tests results presented here demonstrate that the slice-recursion algorithm is highly efficient and displays excellent 4th order convergence. This algorithm readily accomodates a moving "particle" Dirac δ -function source term (which leads to a solution which is generically only C 1 at the particle position); there is no loss of convergence there. Fig. 8 This figure shows φ and its convergence on the rv = 150M slice. In each subfigure, the solid vertical line at ru = 150M shows the particle position. The upper subfigure (which shows the entire slice) plots the real and imaginary parts of φ on the right (linear) scale, and the complex norm φ on the left (logarithmic) scale. It also plots one of the convergence differences record − playback×2 on the left (logarithmic) scale. The horizontal lines show the portion of this slice covered by each refinement level. The lower subfigure (which shows an expanded view of a ±8M region centered on the particle position) plots the real and imaginary parts of φ on the right (linear) scale, and the four convergence differences (scaled by the 4th power of the resolution) on the left (logarithmic) scale. Notice that the three higher-resolution scaled-convergence-difference curves are almost superimposed, indicating excellent 4th-order convergence, and that this convergence is not degraded either across the mesh-refinement boundaries at rv = 143.3M and rv = 153.1M, or near the particle at rv = 150M. The main potential disadvantage of the slice-recursion algorithm over the standard Cauchy Berger-Oliger algorithm is the cost of the tail re-integration. This cost depends on the placement of fine grids relative to their coarser parent grids; the worst possible case is a factor of 2 overhead in the coarse-grid part of the integration. There is no overhead for the finest level of the mesh-refinement hierarchy. In practice I find the re-integration cost to be quite modest, typically about 20-25% of the total computation, and (in my tests) never exceeding 30%. Compared to an equivalent-resolution unigrid evolution, I find that the slice-recursion algorithm is always much faster, with the exact speedup factor varying widely from one problem to another: for a sample of 295 test cases, 95% of the speedup factors are between 12 and 51. The largest practical obstacle to the use of Berger-Oliger mesh refinement algorithms, including the slice-recursion algorithm presented here, is probably their implementation (programming) complexity. This is much less of an obstacle if codes can be shared across projects and researchers. To this end, my code implementing the slice-recursion algorithm is included in the electronic supplementary materials accompanying this paper (online resource 2), and is freely available to other researchers under the terms of the GNU general public license. This code uses C++ templates to support both 2nd and 4th order global truncation error, and both real and complex scalar fields, with no run-time overhead. It would be fairly easy to adapt the code to other finite differencing schemes and/or PDEs. Pretorius and Lehner (2004, sections 4.2 and 4.3) describe how their characteristic Berger-Oliger algorithm can be extended to problems of higher dimensionality than 1+1 (i.e., problems whose domains have multiple spacelike dimensions), and how the algorithm may be parallelized. The techniques they describe should all apply equally to the slicerecursion algorithm presented here. Acknowledgements I thank Leor Barack, Darren Golbourn, and Norichika Sago for introducing me to the self-force problem, and both they and Ian Hawke for many valuable conversations. I thank Virginia J. Vitzthum for useful comments on this manuscript. I thank the University of Southampton and the Max-Planck-Institut für Gravitationsphysik for their generous support during various stages of the research described in this manuscript. Fig. 9 This figure shows histograms of the re-integration overhead (the fraction of all diamond-cell integrations which occur as part of coarse-grid re-integrations, i.e., those in lines 43-48 of figure 3) and of the AMR speedup factor (the ratio of the number of cells that would have been integrated in hypothetical unigrid evolution covering the entire problem domain at the resolution of the finest AMR grid, to the number of cells actually integrated in the AMR evolution) for a sample of 295 evolutions. The re-integration overhead is usually between 20% and 25%, and never exceeds 30%, while the speedup factor varies much more widely. A Details of the Unigrid Finite Differencing Scheme In this appendix, I describe the finite differencing schemes in detail for the model problem (3), for both 2nd and 4th order GTE. In this appendix only, when describing the computation of φ at a particular grid point ( j,i), I sometimes use an abbreviated notation for grid-point indexing, denoting the grid point ( j − n,i − m) by a subscript nm. Such subscripts can be distinguished from the usual grid-point indices by the absence of a comma between the two indices. These abbreviated subscripts may be either integral or half-integral, and in this latter context (only) I also use the abbreviation h ≡ 1 2 . For example, φ 12 denotes φ j−1,i−2 , while φ 0h denotes φ j,i−1/2 . A.1 Second Order Global Accuracy To discretize the wave equation (3) to 2nd order global accuracy in the grid spacing ∆ , I use a standard diamond-cell integration scheme (Gómez and Winicour (1992); Gómez et al. (1992); Gundlach et al. (1994); Burko and Ori (1997); Lousto and Price (1997); Lousto (2005); Winicour (2009)): Consider a double-null "diamond" grid cell of side ∆ , with "north", "west", "east", and "south" vertices (grid points) N, W, E, and S respectively, and central point C, as shown in figure 10. Consider first the vacuum case, where the particle worldline r = r p doesn't intersect the cell and hence the right hand side of the wave equation (3) vanishes everywhere in the cell. Integrating this equation over the cell then gives φ N + φ S − φ E − φ W + ∆ 2 V C φ E + φ W 2 + O(∆ 4 ) = 0 ,(14) or equivalently φ N = φ E + φ W − φ S − ∆ 2 V C φ E + φ W 2 + O(∆ 4 ) ,(15a) where subscripts denote the value of the field at the corresponding grid point. If the particle worldline r = r p intersects the cell, then as discussed in section 3, I assume it does so symmetrically. Integrating the right-hand-side source term in (3) over the grid cell then adds an extra term cell S(t Schw )δ (r − r p ) dvdu = 2 f (r p ) t C +∆/2 t C −∆/2 S(t Schw ) dt Schw (15b) to the right hand side of (15a), where t C ≡ (t Schw ) C . For the test case considered in section 6, this source-term integral becomes 2 f (r p ) t C +∆/2 t C −∆/2 S ℓm (t Schw ) dt Schw = 2πq f (r p )a ℓm r p E(r p ) exp −imω(r p )t C ∆ sinc 1 2 mω(r p )∆ + O(∆ 3 ) . (16) A.2 Fourth Order Global Accuracy To discretize the wave equation (3) to 4th order global accuracy in the grid spacing ∆ , I use a scheme based on that of Haas (2007) (see also Lousto (2005)), but modified in its treatment of cells near the particle. The modification makes the scheme fully explicit, removing the need for an iterative computation at each cell intersecting the particle. However, the scheme is only valid with the assumption noted above, that if the particle intersects a cell it does so symmetrically. In practice (cf. footnote 2), this means that the scheme is only valid for the case where r p is constant, i.e., the particle is in a circular orbit around the central black hole. Consider the computation of φ j,i , and suppose the particle worldline intersects the v = v j slice at the grid point i = i p . There are several different cases for the finite differencing scheme, depending on the sign and magnitude of i − i p . Figure 10 shows all of these cases. A.2.1 \|i − i p \| ≥ 3 (Cell far from the particle) If \|i − i p \| ≥ 3 (so that the particle worldline doesn't intersect any part of the finite difference molecule), then I use the finite differencing scheme of Haas's equations (2.7), (2.10), and (4.10). That is (following Lousto (2005) and Haas (2007)) I first define the new grid function G ≡ V φ . I then interpolate G hh via Haas's equation (2.7), which in my notation reads G hh = 1 16 (8G 10 + 8G 01 ) + (−4G 20 + 8G 11 − 4G 02 ) + (G 30 − G 21 − G 12 + G 03 ) + O(∆ 3 ) .(17) I then compute G Σ ≡ G h0 + G 0h + G 1h + G h1 via Haas's equation (2.10), which in my notation reads G Σ = 2 1 − 1 2 ∆ 2 2 V hh G hh + 1 − 1 2 ∆ 2 2 V h0 V h0 φ 10 + 1 − 1 2 ∆ 2 2 V 0h V 0h φ 01 + 1 2 V h0 − 2V hh +V 0h (φ 10 + φ 01 ) ,(18) where ∆ 2 ≡ ∆ /2. Finally, I compute φ 00 via Haas's equation (4.7), which in my notation reads φ 00 = −φ 11 + 1 − 1 4 ∆ 2 3 − 1 16 ∆ 4 3 V hh (V hh +V 10 ) φ 10 + 1 − 1 4 ∆ 2 3 − 1 16 ∆ 4 3 V hh (V hh +V 01 ) φ 01 − 1 − 1 4 ∆ 2 3 V hh ∆ 2 3 (G Σ + 4G hh ) ,(19) where ∆ 3 ≡ ∆ /3. Taylor-series Approximation For more general purposes, it's usually desirable to approximate the PDE to full accuracy everywhere in the problem domain, i.e., to compute the extended initial data to O(∆ 4 ) accuracy. Lousto (2005, section 4.1) describes a Taylor-series approximation scheme to do this. Two-Level Subsampling Another scheme for computing extended initial data is to define an auxiliary "subsampling" grid with spacing ∆ ss ≪ ∆ , which only covers the extended-initial-data region. Choose ∆ ss such that it integrally divides ∆ and such that ∆ ss ∝ ∆ 2 for sufficiently high resolution. The extended initial data can now be computed by integrating the auxiliary grid using the globally-2nd-order scheme, then subsampling data from the auxiliary grid to the main grid. Because ∆ ss ∝ ∆ 2 at sufficiently high resolution, O(∆ ss 2 ) = O(∆ 4 ), i.e., the global accuracy remains 4th order with respect to the main-grid resolution ∆ . The auxiliary grid only needs to cover the first 3 slices of the main grid, together with the first 3 grid points of each later main-grid slice. However, this corresponds to O(1/∆ ) auxiliary-grid spacings, so integrating all the auxiliary-grid points requires O(1/∆ 3 ) CPU time and O(1/∆ 2 ) memory. This is much more expensive than the main-grid computation, which only requires O(1/∆ 2 ) CPU time and O(1/∆ ) memory. i i − 1 j j − 1 2nd order N W E S C i i − 1 i − 2 i − 3 j j − 1 j − 2 j − 3 4th order: molecule far from the particle (G hh computed via (9) ) i i − 1 i − 2 i − 3 j j − 1 j − 2 j − 3 4th order: molecule near to, and to the left of, the particle (G hh computed via (13)) i i − 1 i − 2 i − 3 j j − 1 j − 2 j − 3 4th order: molecule near to, and to the right of, the particle (G hh computed via (12)) point (j, i) where φ is computed cell-center point (φ is not used) point where φ is used in the globally-2nd-order scheme (8) (either directly or via the Richardson-extrapolation scheme (15)) point where φ is used in interpolating G hh via (9), (12), and/or (13) point where φ is used in computing G Σ via (10) and/or φ 00 via (11) i i − 1 i − 2 i − 3 j j − 1 j − 2 j − 3 4th order: molecule symmetrically bisected by the particle Recursive Doubling A more efficient approach is to use a recursive-doubling scheme. Here we use a sequence of q auxiliary grids A (0) , A (1) , A (2) , . . . , A (q−1) , with A (k) having spacing 2 k−q ∆ . We choose q such that 2 −q ∆ ≤ ∆ * < 2 1−q ∆ for some ∆ * ∝ ∆ 2 for sufficiently high resolution. The extended initial data for the main grid can now be computed as follows (see figure 11 for an example): First integrate A (0) using the globally-2nd-order scheme for 4 slices, and for 4 points on each succeeding slice. Then for each k = 1, 2, 3, . . . , q−1, subsample from A (k−1) to obtain the extended initial data to integrate A (k) using the globally-4th-order scheme for 2 slices, and for 2 points on each succeeding slice. Finally, subsample from A (q−1) to obtain the extended initial data to integrate the main grid using the globally-4th-order scheme. δv δu point in auxiliary grid A (0) (spacing 1) computed using globally-2nd-order scheme point in auxiliary grid A (1) (spacing 2) computed using globally-4th-order scheme (extended initial data subsampled from auxiliary grid A (0) ) point in auxiliary grid A (2) (spacing 4) computed using globally-4th-order scheme (extended initial data subsampled from auxiliary grid A (1) ) point in main grid (spacing 8) computed using globally-4th-order scheme (extended initial data subsampled from auxiliary grid A (2) ) Fig. 11 This figure shows an example of the recursive-doubling technique for constructing extended initial data for the globally-4th-order initial data scheme. The grid axes are plotted in terms of δ u = u − u min and δ v = v − v min , and the grid spacings are in units of finest (∆ (0) rd ) auxiliary grid spacing. In this example q = 3 auxiliary grids are used. Each auxiliary grid only has to cover 2 or 4 slices, and 2 or 4 points on each succeeding slice, so this scheme is much more efficient than the two-level subsampling scheme: in the high-resolution limit the total cost of all the auxiliary-grid integrations is O(1/∆ 2 ) CPU time, the same order as the main-grid computation. Unfortunately, even with careful memory management (discarding grid points as soon as they're no longer needed) the auxiliary grids still require O(1/∆ 2 ) memory, much more than the main-grid computation's O(1/∆ ) memory requirement. It's relatively easy to implement the recursive-doubling scheme in a code using the type of fine-grained linked-list data structures described by Pretorius and Lehner (2004), where the integration can "flow" in either the v direction or the u direction at any stage in the computation. However, for the slice-recursion algorithm we typically require that the integration proceed sequentially in the v direction and that any data reuse or subsampling take place within the small (typically 4-7) number of slices kept in memory at each refinement level. The recursive-doubling initial-data scheme thus requires interleaving the integration of the different auxiliary grids along the southeast face of the grid. Figure 12 gives a pseudocode outline of an algorithm to generate the appropriate sequence of integrations, subsamplings, and other grid operations for the recursivedoubling scheme. A.3 The Coarse-Grid Instability The finite differencing schemes discussed here become unstable at very low resolutions, in a manner somewhat resembling the classic Courant-Friedrichs-Lewy instability of Cauchy finite differencing. I have not mathematically analyzed this instability, 17 but empirically it only occurs for very low resolutions (large ∆ ), with the instability threshold (the smallest ∆ for which the instability appears) depending on ℓ, but not on m. Figure 13 shows the ℓ and ∆ for which the instability occurs. Notice that the instability threshold decreases gradually with ℓ, and is somewhat smaller for the globally-4th-order scheme than for the globally-2nd-order scheme. In practice, it's rare for this "coarse-grid instability" to be a significant problem because reasonable accuracy requirements normally force much higher resolutions than those where the instability would occur. The one exception to this is the base grid, which might otherwise be made very coarse (allowing the AMR to refine it as needed); the coarse-grid instability prevents this by requiring the base grid to be finer than the instability threshold. B Implementation Details B.1 Computing r(r * ) The finite differencing schemes discussed here use finite-difference grids which are locally uniform in v and u, so it's trivial to compute the r * coordinate of any grid point. However, the coefficients in the wave equation (3) are all given as explicit functions of r, so the code needs to know the r coordinate of each grid point (and, for the globally-4th-order scheme, also of the center of each grid zone). My code computes this as follows: Define y ≡ ln r 2M − 1 (24a) x * ≡ r * 2M ,(24b) so that r = 2M (1 + e y ) and the definition (2) becomes x * = 1 + y + e y . Then y(x * ) (and hence r(r * )) can be found by using Newton's method to find a zero of the function h(y) = 1 + y + e y − x (25) An initial guess for Newton's method can be obtained by neglecting either y or e y in (25), giving y initial guess = log(x − 1) if x * > 1 (y −0.577) x * − 1 if x * ≤ 1 (y −0.577)(26) 1 global integer ℓ max # maximum ℓ for which we have created A (ℓ) 2 3 procedure recursive doubling initial data (integer q) 4 { 5 call create grid(0) 6 7 for integer j from 1 to j max 8 { 9 if (j = 5 · 2 ℓmax ) 10 then A (ℓmax) .i max ← 4 · 2 ℓmax # discard A (ℓmax) grid points which are no longer needed 11 if ((ℓ max < q) and (j = 6 · 2 ℓmax )) 12 then j,δ ← subsample from A (ℓ−1) 49 A (ℓ) j,2δ ← subsample from A (ℓ−1) 50 for integer i from 3δ to A (ℓ) .i max by δ 51 { 52 A (ℓ) j,i ← update using the globally-4th-order scheme 53 } 54 } 55 } Fig. 12 This figure gives an outline of the recursive-doubling algorithm for constructing extended initial data for the globally-4th-order initial data scheme. Coordinates for all grids are measured in units of the finest (A (0) ) auxiliary grid spacing, relative to the south corner of the problem domain. Thus, for example, the grid A (3) has spacing 2 3 = 8 and uses grid-point indices {0,8,16,24,32,40, .. . }. The Newton's-method solution is moderately expensive for a computation which (logically) is needed at each grid point: it typically requires 3-10 iterations, with each iteration needing an exp() computation and several other floating-point arithmetic operations. Fortunately, within any single grid r depends only on j − i, so in a unigrid code it's easy to precompute r for all possible values of j−i (of which there are only O(N) for an N×N grid) when the grid is first set up. For the slice-recursion algorithm a somewhat more dynamic "radius cache" of r coordinates is needed, with updates each time regridding grows, shrinks, or relocates a grid. However, the set of j−i involved is still always a contiguous interval, so the cache bookkeeping overhead (over and above the storage arrays for the r coordinates themselves) is only O(1) per grid. B.2 Local Coordinates for each Refinement Level Consider a single slice, and a pair of adjacent grid points in it at some refinement level ℓ, say G (ℓ) j,i and G (ℓ) j,i+1 , viewed as events in spacetime. Since the grid spacing of G (ℓ+k) is 2 k times finer than that of G (ℓ) , these same two events are necessarily 2 k grid points apart in G (ℓ+k) . This means that it's impossible to define integer gridpoint coordinates which simultaneously (a) have adjacent grid points separated by 1 in the integer coordinates at each refinement level, and (b) assign a given event the same integer coordinates at each refinement level. In my AMR code I keep property (a), but discard property (b): each mesh-refinement level has its own local integer coordinate system for indexing grid points, and the code maintains explicit fine-to-coarse and coarse-to-fine coordinate transformations between each pair (ℓ,ℓ+1) of adjacent refinement levels. These transformations are used when interpolating data from coarse to fine grids (discussed in detail in section B.3), when injecting fine-grid results back to coarse grids (lines 36-41 in figure 3), in setting up the "tail" reintegration (lines 43-48 of figure 3), and in checking the proper-nesting condition in regridding (lines 31-48 of figure 4). This scheme has worked very well, and I recommend its use to others implementing Berger-Oliger mesh-refinement codes. However, for the extended-initial-data algorithm of figure 12, it's convenient to use integer coordinates which keep property (b), but discard property (a). The CARPET code (Schnetter et al. (2004);Schnetter (2001)) also uses integer coordinates of this latter type. Fig. 13 This figure shows the stability behavior of unigrid evolutions with varying ℓ and ∆ , using Gaussian initial data, no source term, and a problem domain size D = 100M. Notice that for each ℓ, the evolutions are always stable for ∆ less than some threshold value. B.3 Interpolation Operators As discussed in sections 5.1 and 5.3, the slice-recursion algorithm needs to interpolate data from coarse to fine grids in several situations: -When creating a new grid G (ℓ) , the first few slices (1 [3] slices for globally 2nd [4th] order finite differencing) of the newly-created grid are initialized by interpolating from the next coarser grid G (ℓ−1) (line 20 of figure 4; figure 5c). -When time-integrating any grid G (ℓ) finer than the base grid, the extended initial data on each new G (ℓ) slice (i.e., the first 1 [3] points on the slice for globally 2nd [4th] order finite differencing), must be interpolated from the next coarser grid G (ℓ−1) (line 9 of figure 3; figure 5c,d). before the main integration of the slice can be started. -When moving an existing grid G (ℓ) to a new position in the current v = constant slice, newly-created points are initialized by interpolating from the next coarser grid G (ℓ−1) (lines 25-26 of figure 4). As shown in figure 14, the precise choice of interpolation operator (which is made independently at each G (ℓ) grid point) depends on the relative position of the G (ℓ) interpolation point with respect to the next coarser grid G (ℓ−1) . All the interpolation operators considered here are Lagrange polynomial interpolants, which assume smoothness, so if the interpolation position is within a few grid points of the particle worldline (where φ is only C 1 ), then a different interpolation operator needs to be chosen so as to avoid crossing the particle worldline: -If the interpolation point coincides with a coarse-grid (G (ℓ−1) ) point, then the "interpolation" is just a copy of the data. -Otherwise, if the interpolation point's time (v) coordinate coincides with that of a coarse-grid (G (ℓ−1) ) v = constant slice, then the interpolation is a 1-dimensional Lagrange polynomial interpolation in space (u) within this slice, using 4 [6] points for globally 2nd [4th] order finite differencing. The interpolation is constrained not to cross the particle worldline and not to use data from outside the spatial (u) extent of the coarse slice. The interpolation is centered if this is possible within these constraints, otherwise it's as minimally off-centered as is necessary to satisfy them. -Otherwise, if the interpolation point's spatial (u) coordinate coincides with that of a coarse-grid (G (ℓ−1) ) u = constant line of grid points, then depending on the relative position of the interpolation point and the particle worldline, there are two cases: 1. If the interpolation point is not close to the particle worldline, then the interpolation is a 1-dimensional Lagrange polynomial interpolation in time (v) within the u = constant line of coarse-grid (G (ℓ−1) ) points, again using 4 [6] points for globally 2nd [4th] order finite differencing. The set of input points for this interpolation is always the most recent 4(6) slices of the coarse grid (G (ℓ−1) ). 2. Alternatively, if the interpolation point is too close to the particle worldline (i.e., if the 1-dimensional Lagrange polynomial interpolation molecule of case 1 would cross the particle worldline), then the interpolation is a 2-dimensional Lagrange polynomial interpolation in spacetime, chosen so as to not cross the particle worldline. This case is described further below. -Otherwise (i.e., if the interpolation point lies in the center of a coarse-grid (G (ℓ−1) ) cell), the interpolation is a 2-dimensional Lagrange polynomial interpolation in spacetime, again chosen so as to not cross the particle worldline. This case is described further below. While it is straightforward to construct Lagrange polynomial interpolation operators in 1 dimension, doing so in 2 (or more) dimensions is more difficult. The basic concept is the same -an interpolating polynomial is matched to the known grid function values at some set of molecule points, then evaluated at the interpolation point -but there are several complications. In 1 dimension the choice of interpolating polynomial is obvious, but in multiple dimensions different choices are possible. That is, let (v * ,u * ) be a fixed reference point somewhere near the interpolation point (v,u), and define the relative coordinates x ≡ v * − v and y ≡ u * − u. Then an nth degree interpolating polynomial in x and y might reasonably be defined as either f (x,y) = ∑ 0≤p+q≤n p≥0,q≥0 a pq x p y q (27) or as f (x,y) = ∑ 0≤p≤n 0≤q≤n a pq x p y q . Given the choice of an interpolating polynomial, there are still many different molecules possible, even given the requirement that the values of the interpolating polynomial at the molecule points uniquely determine the polynomial coefficients. I have used the Maple symbolic algebra system (Char et al. (1983, version 11, http://www.maplesoft.com/)) to experiment with different interpolation molecules and to compute their coefficients. Figure 15 summarizes the set of spacetime-interpolation molecules used in my code. The actual coefficients may be obtained from the Maple output files in the sfevol/coeff/ directory of the source code included in the electronic supplementary materials accompanying this article (online resource 2). B.4 Data Structures As noted earlier in this paper, the largest practical obstacle to the use of Berger-Oliger mesh refinement algorithms is the complexity of programming, debugging, and testing them. To help reduce this complexity for other researchers, here I briefly outline the main data structures and debugging/testing strategies I have found useful in implementing the slice-recursion algorithm. coarse-grid point fine-grid point where data is copied from the coarse grid fine-grid point where data is space-interpolated from the coarse grid fine-grid point where data is time-interpolated from the coarse grid (spacetime-interpolated when near the particle worldline) fine-grid point where data is spacetime-interpolated from the coarse grid (using variant molecules when near the particle worldline) A SLICE object represents a single G (ℓ) j slice at a single refinement level, i.e., it stores all the grid functions needed to represent the solution of the PDEs on that slice. In my code, SLICE is a C++ template with the template parameter selecting the PDE system (e.g., real or complex scalar field) to be supported. A CHUNK object stores enough adjacent slices at a single refinement level to be able to take time steps, i.e., it stores 4 [7] adjacent SLICE objects for the 2nd [4th] order GTE finite differencing schemes described in this paper. CHUNK also maintains the radius cache discussed in appendix B.1. In my code, CHUNK is a C++ template with 2 template parameters, one selecting the PDE system and the other selecting the finite differencing scheme (2nd versus 4th order GTE) and thus implicitly the number of adjacent slices to be stored. To avoid unnecessary data copying, at each time step CHUNK circularly rotates pointers to a fixed set of SLICE objects. When debugging the code, CHUNK (and SLICE) can be thoroughly tested using unigrid evolutions. A MESH object represents an entire grid hierarchy as described in section 5.1. That is, a MESH object stores a stack of CHUNK objects, one for each refinement level, together with the necessary bookkeeping information to compute the fine-to-coarse and coarse-to-fine coordinate transformations described in appendix B.2. When debugging the code, MESH can be tested by manually creating a grid hierarchy and testing that the expected results are obtained for various operations on it such as adding a new refinement level, dropping a refinement level, moving the chunk at some refinement level to a new location, interpolating or copying data from one refinement level to another, or transforming the per-refinement-level coordinates from one refinement level to another. Finally, the actual slice-recursion Berger-Oliger and regridding algorithms are implemented in terms of the various MESH operations. I found it difficult to thoroughly test the Berger-Oliger and regridding logic, but this comprises a relatively small body of code -most of the overall complexity of the software lies in MESH and lower-level code, which is relatively straightforward to test. Fig. 1 1This figure shows the overall problem domain, and the (u,v) and (t Schw ,r * ) coordinates. The vertical dashed line marks the particle worldline. Fig. 2 2This figure summarizes the unigrid finite differencing molecules. The left subfigure shows the molecule used for 2nd order global accuracy. The right subfigure shows the molecule used for 4th order global accuracy far from the particle worldline; near the worldline the molecules are more complicated, and are shown infigure 10. Open circles show the point where φ is being computed; solid circles show points where φ is used as an input in the computation of φ j,i . (2006);Colella et al. (2009a);Rendleman et al. (2000);Lijewski et al. (2006)), the Choptuik et al. axisymmetric code(Choptuik et al. (2003a,b);Pretorius and Choptuik (2006)), HAD(Liebling ( , 2004;Anderson et al. (2006)), GRACE/HDDA/AMROC/DAGH(Parashar and Li (2009);Parashar and Browne (2000);Deiterding (2006Deiterding ( , 2005b;Mitra et al. (1995)), OVERTURE(Brown et al. (1999a,b);Henshaw et al. (2002)), PARAMESH(MacNeice et al. (2000);Olson and MacNeice (2005);Olson (2006); ),PARAMESH (MacNeice et al. (2000);Olson and MacNeice (2005);Olson (2006);Olson and MacNeice (1999)), and SAMRAI(Hornung and Kohn (2002a);Hornung et al. (2006);Hornung and Kohn (2002b)) mesh-refinement infrastructures. Fig. 3 3This figure gives an outline of the standard Berger-Oliger AMR algorithm (if the lines marked with • are omitted), and of the slice-recursion algorithm (if the lines marked with • are included). See figure 4 for an outline of the procedure regrid() which is called at line 25. Fig. 4 Fig. 5 45This figure gives an outline of the regridding procedure regrid() which is called by the main Berger-Oliger algorithm (figure 3), and of the auxiliary procedure shrink to ensure proper nesting() which is called by regrid(). grid initial and spatial-boundary data fine-grid evolved point coarse-grid extended initial data coarse-grid evolved point coarse-grid reintegrated point point with LTE ≤ threshold point with LTE > threshold LTE-estimate molecule This figure shows an example of the operation of the slice-recursion algorithm, using the globally-4th-order finite differencing scheme described in section 3. Parts (a)-(e) can also be interpreted as an example of the operation of the (Cauchy) Berger-Oliger algorithm, with u as the spatial coordinate and v as the time coordinate. See the main text for further discussion. Fig. 7 7This figure shows sample frames from the movie accompanying this paper (online resource 1). Each frame of the movie shows a single rv = constant slice, and plots Re[φ ], Im[φ ], and φ on the left scale, and the spatial extent of each refinement level (on the right scale) as the horizontal lines. The u, ru, and rv coordinates are all shown in units of the Schwarzschild spacetime mass M. Each frame of the movie also includes a subplot (in the lower right corner) showing the location of that plot's rv = constant slice within the entire problem domain. The vertical line in the subplot shows the particle worldline. Fig. 10 10This figure shows the unigrid finite differencing molecules. For clarity, the 2nd order molecule is enlarged relative to the 4th order molecules. In the lower 3 subfigures the vertical dashed lines show possible positions for the particle worldline. Fig. 14 14This figure shows the type of interpolation operator used for each possible relative position of a finegrid point with respect to the next coarser grid. The space-interpolation and time-interpolation operators are described in the text. The spacetime-interpolation molecules are shown infigure 15. create the auxiliary grid A (ℓ) and update ℓ max 23 procedure create grid(integer ℓ) 24 { 25 Create the auxiliary grid A (ℓ) with spacing δ = 2 ℓ and size A (ℓ) .i max = the size of the main grid 26 A 2δ ← subsample from A (ℓ−1) 31 } 32 ℓ max ← ℓ 33 } 34 35 # integrate the auxiliary grid A (ℓ) on the slice j 36 procedure integrate slice(integer ℓ, integer j)call create grid(ℓ max +1) 13 14 for integer ℓ from 0 to ℓ max 15 { 16 if (j mod 2 ℓ = 0) 17 then call integrate slice(ℓ, j) 18 } 19 } 20 } 21 22 # (ℓ) 0 ← physical boundary data on the southwest grid face 27 if (ℓ > 0) 28 then { 29 A (ℓ) δ ← subsample from A (ℓ−1) 30 A (ℓ) 37 { 38 A (ℓ) j,0 ← physical boundary data on the southeast grid face 39 if (ℓ = 0) 40 then { 41 for integer i from 1 to A (0) .i max 42 { 43 A (ℓ) j,i ← update using the globally-2nd-order scheme 44 } 45 } 46 else { 47 integer δ ← 2 ℓ 48 A (ℓ) In my code I (somewhat arbitrarily) always use interpolating polynomials of the form (27). I use n = 3 [5] for 4th [6th] order LTE (corresponding to 2nd [4th] order GTE). See Givoli (1991) for a general review of numerical radiation boundary conditions, and Kidder et al. (2005); Rinne (2006); Buchman and Sarbach (2006, 2007); Rinne et al. (2007); Ruiz et al. (2007); Seiler et al. (2008); Rinne et al. (2009) for recent progress towards non-reflecting and constraint-preserving radiation boundary conditions for the Einstein equations. For the model problem of section 2 these coordinates may be taken to be t = t Schw and x = r * . For the model problem of section 2 this isn't a significant restriction, since apart from the (ignorable) spurious radiation discussed in section 5.3.1, the resolution required to adequately represent φ tends to decrease monotonically with distance from the particle worldline, so that for a given LTE threshold, the region of a slice needing a given resolution is just a single closed interval. For other problems this might not be the case, requiring some or all of the G The LTE estimate discussed in section 4 roughly doubles the cost of integrating a single grid point, so estimating the LTE at every kth grid point of every kth slice adds a fractional overhead of roughly 1/k 2 to the computation. For k = 4 (the example shown infigure 5b) this is only about 6%. The largest execution-time cost is probably that due to (nearby) grid points which are accessed in succession not being in contiguous, or even nearby, memory locations. This leads to increased cache miss rates and thus poorer performance.13 The execution time of the dynamic storage allocation routines themselves (C's malloc() and free(), or other languages' equivalents) may also be substantial. However, this can be greatly reduced by using customized storage-allocation routines that allocate grid points in large batches. These and other optimization techniques for dynamic storage allocation are discussed by, for example,Wilson et al. (1995);Lea (2000). This set of evolutions arises naturally as part of a calculation of the radiation-reaction "self-force" on a scalar particle orbiting a Schwarzschild black hole. Details of this calculation will be reported elsewhere; for present purposes these evolutions provide a useful set of test cases for the AMR algorithm. Notice that once G hh is known, (18) and (19) use only the grid-function values φ 10 , φ 01 , and φ 11 , and thus require only that the particle worldline doesn't pass between these points; this condition is satisfied whenever i = i p .16 There's no problem in starting the integration of other grids, since the extended initial data for any G (ℓ) with ℓ > 0 is interpolated from its parent grid G (ℓ−1) (line 20 offigure 4). Gómez and Winicour (1992);Gómez et al. (1992) discuss the stability of diamond-cell integration schemes in spherical symmetry;Welling (1983); Winicour (2009, section 3.3) discuss subtleties in applying the CFL condition to a more general null-cone evolution algorithm in axisymmetry. A.2.2 i = i p ± 1 or i = i p ± 2 (Cell near the particle)If i − i p = −1 or −2 (so we are computing φ at a grid point near to, but to the right of, the particle), then the interpolation (17) would use data from both sides of the particle's worldline (hereinafter I refer to this as "crossing" the worldline), violating the smoothness assumptions used in the interpolation's derivation. Instead, I use the "right" interpolationwhich (since i < i p ) doesn't cross the worldline. If i − i p = +1 or +2 (so we are computing φ at a grid point near to, but to the left of, the particle), then again the interpolation (17) would cross the particle's worldline, so I instead use the "left" interpolationwhich (since i > i p ) doesn't cross the worldline.[Each of the interpolations(17),(20), and(21)is actually valid for any grid function which is smooth (has a convergent Taylor series in v and u) throughout the region spanned by the interpolation molecule.]Once G hh has been computed, G Σ and then G 00 can be computed in the same manner as before, i.e., via(18)and(19)respectively. 15A.2.3 i = i p (Cell symmetrically bisected by the particle)If i = i p , then for each k ∈ {1,2,3} I first compute an estimate φ (k) j,i for φ j,i using the globally-2nd-order scheme (15) applied to the diamond cell of size k∆ defined by the four grid points. 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[ "New Trends in Parallel and Distributed Simulation: from Many-Cores to Cloud Computing 1", "New Trends in Parallel and Distributed Simulation: from Many-Cores to Cloud Computing 1" ]	[ "Gabriele D'angelo \nDepartment of Computer Science and Engineering\nUniversity of Bologna\nItaly\n", "Moreno Marzolla \nDepartment of Computer Science and Engineering\nUniversity of Bologna\nItaly\n" ]	[ "Department of Computer Science and Engineering\nUniversity of Bologna\nItaly", "Department of Computer Science and Engineering\nUniversity of Bologna\nItaly" ]	[]	Recent advances in computing architectures and networking are bringing parallel computing systems to the masses so increasing the number of potential users of these kinds of systems. In particular, two important technological evolutions are happening at the ends of the computing spectrum: at the "small" scale, processors now include an increasing number of independent execution units (cores), at the point that a mere CPU can be considered a parallel shared-memory computer; at the "large" scale, the Cloud Computing paradigm allows applications to scale by offering resources from a large pool on a pay-as-you-go model. Multi-core processors and Clouds both require applications to be suitably modified to take advantage of the features they provide. Despite laying at the extreme of the computing architecture spectrum -multi-core processors being at the small scale, and Clouds being at the large scale -they share an important common trait: both are specific forms of parallel/distributed architectures. As such, they present to the developers well known problems of synchronization, communication, workload distribution, and so on. Is parallel and distributed simulation ready for these challenges? In this paper, we analyze the state of the art of parallel and distributed simulation techniques, and assess their applicability to multi-core architectures or Clouds. It turns out that most of the current approaches exhibit limitations in terms of usability and adaptivity which may hinder their application to these new computing architectures. We propose an adaptive simulation mechanism, based on the multi-agent system paradigm, to partially address some of those limitations. While it is unlikely that a single approach will work well on both settings above, we argue that the proposed adaptive mechanism has useful features which make it attractive both in a multi-core processor and in a Cloud system. These features include the ability to reduce communication costs by migrating simulation components, and the support for adding (or removing) nodes to the execution architecture at runtime. We will also show that, with the help of an additional support layer, parallel and distributed simulations can be executed on top of unreliable resources.	10.1016/j.simpat.2014.06.007	[ "https://arxiv.org/pdf/1407.6470v1.pdf" ]	15,343,688	1407.6470	4a8e3f9774f4e506fccdfb7e312d1c6d512b3b08	New Trends in Parallel and Distributed Simulation: from Many-Cores to Cloud Computing 1 24 Jul 2014 Gabriele D'angelo Department of Computer Science and Engineering University of Bologna Italy Moreno Marzolla Department of Computer Science and Engineering University of Bologna Italy New Trends in Parallel and Distributed Simulation: from Many-Cores to Cloud Computing 1 24 Jul 201410.1016/j.simpat.2014.06Preprint submitted to Simulation Modelling Practice and Theory July 25, 20141 The publisher version of this paper is available at http://dx.doi.org/10.1016/j.simpat.2014.06. 007. Please cite as: Gabriele D'Angelo, Moreno Marzolla. New trends in parallel and distributed simulation: From many-cores to Cloud Computing. Simulation Modelling Practice and Theory, Elsevier. An early version of this work appeared in [1]. This paper is an extensively revised and extended version in which significantly more than 30% is new material.SimulationParallel and Distributed SimulationCloud ComputingAdaptive SystemsMiddlewareAgent-Based Simulation Recent advances in computing architectures and networking are bringing parallel computing systems to the masses so increasing the number of potential users of these kinds of systems. In particular, two important technological evolutions are happening at the ends of the computing spectrum: at the "small" scale, processors now include an increasing number of independent execution units (cores), at the point that a mere CPU can be considered a parallel shared-memory computer; at the "large" scale, the Cloud Computing paradigm allows applications to scale by offering resources from a large pool on a pay-as-you-go model. Multi-core processors and Clouds both require applications to be suitably modified to take advantage of the features they provide. Despite laying at the extreme of the computing architecture spectrum -multi-core processors being at the small scale, and Clouds being at the large scale -they share an important common trait: both are specific forms of parallel/distributed architectures. As such, they present to the developers well known problems of synchronization, communication, workload distribution, and so on. Is parallel and distributed simulation ready for these challenges? In this paper, we analyze the state of the art of parallel and distributed simulation techniques, and assess their applicability to multi-core architectures or Clouds. It turns out that most of the current approaches exhibit limitations in terms of usability and adaptivity which may hinder their application to these new computing architectures. We propose an adaptive simulation mechanism, based on the multi-agent system paradigm, to partially address some of those limitations. While it is unlikely that a single approach will work well on both settings above, we argue that the proposed adaptive mechanism has useful features which make it attractive both in a multi-core processor and in a Cloud system. These features include the ability to reduce communication costs by migrating simulation components, and the support for adding (or removing) nodes to the execution architecture at runtime. We will also show that, with the help of an additional support layer, parallel and distributed simulations can be executed on top of unreliable resources. Introduction In the last decade, computing architectures were subject to a significant evolution. Improvements happened across the whole spectrum of architectures, from individual processors to geographically distributed systems. In 2002, Intel introduced the Hyper-threading (HT) technology in its processors [2]. An HT-enabled CPU has a single execution unit but can store two architecture states at the same time. HT processors appear as two "logical" CPUs, so that the Operating System can schedule two independent execution threads. This allows a marginal increase of execution speed due to the more efficient use of the shared execution unit. Improvements of miniaturization and chip manufacturing technologies naturally led to the next step, where multiple execution units ("cores") are put on the same processor die. Multi-core CPUs can actually execute multiple instructions at the same time. These types of CPUs are now quite ubiquitous, being found on devices ranging from high-end servers to smartphones and tablets; CPUs with tens or hundreds of processors are already available [3]. The current generation of desktop processors often combines a multi-core design with HT. This means that each physical core will be seen by the Operating System as two virtual processors. While many multi-core processors are homogeneous shared-memory architectures, meaning that all cores are identical, asymmetric or heterogeneous multi-core systems also exist. A notable example is the Cell Broadband Engine [4], that contains two PowerPC cores and additional vector units called Synergistic Processing Elements (SPE). Each SPE is a programmable vector co-processor with a separate instruction set and local memory for instructions and data. Other widely used asymmetric multi-core systems include General-Purpose Graphics Processing Units (GP-GPU) [5], which are massively parallel devices containing up to thousands of simple execution units connected to a shared memory. The memory is organized in a complex hierarchy, since RAM chips do not provide enough bandwidth to feed all cores at maximum speed. GP-GPUs are generally implemented as add-on cards which are inserted into a host computer. The host CPU then sends data and computation to the GPU through data transfers from host to device memory; results are fetched back to host memory when computation completes. In modern multi-core processors, the level of complexity introduced by the hardware is quite high; unfortunately, this complexity can not be ignored, and instead it has to be carefully exploited at the application level. In fact, writing efficient applications for multi/many-core processors requires a detailed knowledge of the processor internals to orchestrate communication and computation across the available cores. Not only hardware is undergoing the major changes just described: at the software layer the Everything as a Service (EaaS) is gaining momentum thanks to Cloud Computing. According to NIST [6], Cloud Computing is a model for enabling on-demand network access to a shared pool of resources. Depending on the type of resource which is provided, we can identify different Cloud service models. In a Software as a Service (SaaS) Cloud, customer receives access to application services running in the Cloud infrastructure;"Google Apps" is an example of a SaaS Cloud. A Platform as a Service (PaaS) Cloud provides programming languages, tools and a hosting environment for applications developed by the Cloud customer. Examples of PaaS solutions are AppEngine by Google, Force.com from SalesForce, Microsoft's Azure and Amazon's Elastic Beanstalk. Finally, an Infrastructure as a Service (IaaS) Cloud provides its customers with low level computing capabilities such as processing, storage and networks where the customer can run arbitrary software, including operating systems and applications. The architectural evolutions described above are not completely transparent to existing applications, which therefore should be modified to take full advantage of new features. In this paper, we focus on a specific class of software applications, namely simulation tools. Simulation can be very demanding in terms of computational resources, for example when dealing with large and detailed models [7]. Parallel and Distributed Simulation (PADS) [7] aims at studying methodologies and techniques for defining and executing simulation models on parallel and distributed computing architectures. PADS can be considered a specialized sub-field of parallel computing and simulation; as such, it shares many common issues with other types of parallel application domains (e.g., load balancing, partitioning, optimizing communication/computation ratio, and so on). After a brief introduction on PADS, we review the most important challenges faced by PADS on multi-core processors and Clouds, and argue that it needs ad-hoc solutions which are quite different from those commonly used to develop parallel or distributed applications. We see that, the most common PADS technologies are not adequate for coping with such architectural evolution. In particular, we analyze the limitations of current PADS approaches in terms of performance, functionality and usability. Moreover, we discuss the problem of finding new metrics for the evaluation of the PADS performance. In fact, in our view, such metrics have to consider many different aspects such as the execution speed, the cost of resources and the simulation reliability. Finally, we suppose that an adaptive approach can overcome many of these problems and therefore we describe a new parallel/distributed simulation technique that is based on the dynamic partitioning of the simulation model. This paper is organized as follows. In Section 2 we give some background notions on Parallel and Distributed Simulation. Section 3 deals with the challenges presented to PADS by many-core architectures and Clouds. In Section 4 we describe and discuss our proposal aimed to obtain more adaptable PADS. Finally, concluding remarks will be discussed in Section 5. Background A computer simulation is a software program that models the evolution of some real or abstract system over time. Simulation can be useful to evaluate systems before they are built, to analyze the impact of changes on existing systems without the need to physically apply the changes, or to explore different design alternatives. In this paper, we focus on Discrete Event Simulation (DES) [8]: in a DES, the model evolution happens at discrete points in time by means of simulation events. Hence, the simulation model is evolved through the creation, delivery and execution of events. In its simplest form, a DES is implemented using a set of state variables, an event list, and a global clock representing the current simulation time [8]. 3 A simulator capable of exploiting a single execution unit (e.g., a single CPU core) is called a sequential (monolithic) simulator. In a sequential simulation the execution unit processes events in non-decreasing timestamp order, and updates the model state variables and the global clock accordingly. The main advantage of this approach is its simplicity, but is clearly inappropriate for large and complex models. First, complex models might generate a large number of events, putting a significant workload on the CPU. Furthermore, the memory space required to hold all state information may exceed the RAM available on a single processor [9]. Parallel and Distributed Simulation (PADS) relies on partitioning the simulation model across multiple execution units. Each execution unit manages only a part of the model; in PADS each execution unit handles its local event list, but locally generated events may need to be delivered to remote execution units. Partitioning of the simulation model allows each processor to handle a portion of the state space and a fraction of the events; the parallel execution of concurrent events can also lead to a reduction of the simulation run time. Additional advantages of PADS include: the possibility to integrate simulators that are geographically distributed, the possibility to integrate a set of commercial off-the-shelf simulators, and to compose different simulation models in a single simulator [7]. A PADS is obtained through the interconnection of a set of model components, usually called Logical Processes (LPs). Therefore, each LP is responsible for the management of the evolution of a subset of the system and interacts with the other LPs for all the synchronization and data distribution operations [7]. In practice, each LP is usually executed by a processor core. The scientific literature distinguishes between parallel and distributed simulation although the difference is quite elusive. Usually, the term parallel simulation refers to simulation modeling techniques using shared-memory multiprocessors or tightly coupled parallel machines. Conversely, distributed simulation refers to simulation techniques using loosely coupled (e.g., distributed memory) architectures [10]. It should be observed that, in practice, execution platforms are often hybrid architectures with a number of shared-memory multiprocessors connected through a LAN. The lack of a global state and the presence of a network that interconnects the different parts of the simulator has some important consequences: • the simulation model must be partitioned in components (the LPs) [11]. In some cases, the partitioning is guided by the structure and the semantics of the simulated system. For example, if the simulation involves groups of objects such that most interactions happen among objects of the same group, then it is natural to partition the model by assigning all the objects in the same group to the same LP. However, in the general case, it may be difficult if not impossible to identify an easy way to partition the model. As a general rule, partitioning should guarantee that the workload is balanced across the available LPs, and the communication between the LPs is reduced [12,13,14,15]. Therefore, the partitioning criteria strictly depends both on the simulation model to be partitioned [16], on the underlying hardware architecture on which the model will be executed, and the characteristics of the synchronization algorithm that is implemented [17]; • the results of a parallel/distributed simulation must be identical to those of a 4 sequential execution. This means that causal consistency [18,7] among simulation events must be guaranteed (see Section 2.1). While causal consistency can be trivially achieved in sequential simulations by simply executing events in nondecreasing timestamp order, it is much harder to achieve in PADS, especially on distributed-memory systems where LPs do not share a global view of the model. Therefore, PADS must resort to some kind of synchronization among the different parts that compose the simulator. Specific algorithms are needed for the synchronization of the LPs involved in the execution process; • each component of the simulator will produce state updates that are possibly relevant for other components. The distribution of such updates in the execution architecture is called data distribution. For obvious reasons, sending all state updates to all LPs is impractical. A better approach is to match the data production and consumption using some publish-subscribe method in which LPs can declare which type of data they are interested in and what they produce [19]. Partitioning, synchronization and data distribution are important problems in the context of PADS. Synchronization is of particular interest since parallel and distributed simulations tend to be communication bound rather than computation bound. Therefore, the choice of the synchronization algorithm, as well as the type of communication network used by the execution host, plays the most important role in determining the performance and scalability of the simulation model. Synchronization Implementing a parallel simulation requires that all simulation events are delivered following a message-based approach. Two events are said to be in causal order if one of them may depend on the other [18]. Enforcing causal order in a PADS requires that the different LPs are synchronized. That is because every LP can proceed at different speed, and therefore each partition of the simulation model may evolve at a different rate. Different solutions to this problem have been proposed, which can be broadly summarized in three main families: time-stepped, optimistic and conservative simulations. In a time-stepped synchronization, the simulated time is divided in fixed-size intervals called timesteps. All LPs are required to complete the current timestep before moving to the next one [20]. The implementation of this approach requires a barrier synchronization between all LPs at the end of each step. Time-stepped synchronization is quite easy to implement, and it is most appropriate for models which already exhibit some form of lockstep synchronization (e.g., VLSI circuit simulation, where the clock signal is used to synchronize the functional blocks of the circuit). For other applications, it may be difficult to define the correct timestep size; furthermore, time-stepped mechanisms require that all newly generated events must be scheduled for future timesteps only (i.e., it is not possible to generate a new event to be executed during the current timestep), and this requirement can be too limiting for some applications. The goal of the conservative synchronization approach is to prevent causality violations while allowing the global simulation time to advance at an arbitrary rate. Therefore, a LP can process an event with timestamp t only if no other event with timestamp t < t will be received in the future. The Chandy-Misra-Bryant (CMB) [21] algorithm can be used to guarantee causal delivery of simulation events in a distributed system. The CMB algorithm requires that each LP i has a separate incoming message queue Q ji for each LP j it receives events from. Each LP is required to generate events in non-decreasing timestamp order, so that LP i can identify the next "safe" event to process by simply checking all incoming queues Q ji : if all queues are nonempty, then the event with lowest timestamp t is safe and can be processed. If there are empty queues, this mechanism can lead to deadlock, which can be avoided by introducing null messages with no semantic content. The goal of these messages is to break the circular chain that is a necessary condition for deadlock. The main drawback of the CMB synchronization mechanism is the high overhead introduced by null messages in terms of both network load and computational overhead. The optimistic synchronization approach does not impose the execution of safe messages only: each LP can process the events as soon as they are received. Causality violations can happen if some message (called a straggler ) with timestamp in the past is received by some LP. When a causality violation is detected, the affected LP must roll back its local state to a simulation time prior to the straggler timestamp. The roll back mechanism must be propagated to all other LPs whose simulation time has advanced past that of the straggler [22,23]. This may trigger a rollback cascade that brings back the whole simulation to a previous state, from where it can be re-executed and process the straggler in the appropriate order. Optimistic synchronization mechanisms require that each LP maintains enough state data and a log of sent messages, in order to be able to perform rollbacks and propagate them to other affected LPs. A suitable snapshot mechanism must be executed periodically in order to compute a global state that is "safe", meaning that it can not be rolled back. This is essential to reclaim memory used for events and state variables that become no longer necessary. This problem is known as the Global Virtual Time (GVT) computation, where the GVT denotes the earliest timestamp of an unprocessed message in an optimistic simulation. Computing the GVT is not unlikely taking a snapshot of a distributed computation, for which the well known Chandy-Lamport algorithm [24] can be used. However, in the context of PADS, more efficient ad-hoc algorithms have been proposed [25]. Software tools Many tools have been developed to support the implementation of PADS, some of which are described below. µsik [26] is a multi-platform micro-kernel for the implementation of parallel and distributed simulations. The micro-kernel provides a rich set of advanced features such as the support for reverse computation and some kind of automated-load balancing. The last version of µsik was released in 2005 and now the development seems to have stopped. SPEEDES [27] and the WarpIV Kernel [28] have been used as testbeds for investigating new technologies, such as the Qheap, a new data structure for event list management. Furthermore, SPEEDES has been used for many seminal works on load-balancing in optimistic synchronization [29,30]. Finally, PRIME [31] and PrimoGENI [32] have specific focus on very high scalability and real-time constraints, mainly in complex networking environments. One important advance in the field of PADS is the IEEE 1516-High Level Architecture (HLA) standard [33] [40] Georgia Tech C/C++ Partial Non-free MAK RTI [41] VT MAK C/C++, Java Full Commercial Pitch pRTI [42] Pitch Technologies Multiple Full Commercial CERTI [43] ONERA Multiple Partial GPL/LGPL OpenSkies RTI [44] Cybernet Systems C++ Partial Commercial Chronos RTI [45] Magnetar Games C++/.NET Unknown Commercial The Portico Project [46] -C++, Java Partial CDDL SimWare RTI [47] Nextel Aerospace C++ Partial Commercial Open HLA [48] -Java Partial Apache OpenRTI [49] FlightGear C++,Python Unknown LGPL Table 1: Some implementations of the HLA specification. ture for distributed simulation that allows computer simulations to interact with other simulations using standard interfaces. The interest in IEEE 1516 has been initially very strong and it is still fundamental for interoperability and reusability of simulators. Despite of this, some drawbacks have been reported: the software architecture required by HLA [35], the lack of interoperability among different implementations [36], its complexity and steep learning curve [37] and, in some cases, poor execution time [38]. Table 1 shows some of the available (full or partial) implementations of the HLA specification. In general, as expected, commercial proprietary implementations are more compliant with the specification and feature-rich. However, the Portico Project implementation is very promising even if it still does not support the new HLA-evolved interface specification. PADS on Many-core and Cloud Computing Platforms The usage of many-core chips for running PADS is not a novel topic. Most of the existing works deal with the usage of optimistic Time Warp-based tools and their performance evaluation [50,51,52]. For example, in [50] it is shown that the architecture of many-core processors is rather complex and that some techniques are needed for increasing the execution speed (e.g., multi-level Time Warp, multi-level memory-aware algorithms, detailed frequency control of cores). The opportunity to control the speed of each CPU core is further investigated in [51]. In this case, the idea is that by controlling the execution speed of each core, it is possible to limit the number of roll-backs in optimistic synchronization. This approach is not different from adjusting the speed of CPUs, a widely investigated topic, but in this case the adjustment is done on different regions of the same chip. In [52] a multi-threaded implementation of an optimistic simulator is studied on a 48 core computing platform. This implementation, avoiding multiple message copying and reducing the communication latency, is able to reduce the synchronization overhead and improve performance. Finally, [53] shows how the Godson-T Architecture Simulator (GAS) has been ported to a many-core architecture with very good performance results but this has been possible mainly thanks to the loosely coupled nature of events in the specific simulation model of the GAS. In [54] the authors evaluate the performance of a conservative simulator on a multi-core processor. Since the performance of conservative synchronization protocols can be affected by the large number of null messages generated to avoid deadlock conditions, the authors evaluate a number of known optimizations to reduce this overhead. A new hybrid protocol is finally proposed in order to combine the effectiveness of existing solutions. A growing number of papers address the problem of running PADS on Cloud execution architectures. The early works on this topic have mainly considered the usage of Private Clouds [55,56,57]; more recent works deal with the usage of Public Clouds and the related problems [58,59,60]. In [61,62] the authors describe the state of the art of PADS on Cloud and propose a specific architecture based on a two-tier process partitioning for workload consolidation. In [63] an optimistic HLA-based simulation is considered and a mechanism for advancing the execution speed of federates at comparable speed is proposed. In [64], an approach based on concurrent simulation (that we have investigated in the past [65]) has been ported to the Cloud. Finally, it is worth to mention [66] in which the authors propose to build simulations on the Cloud using handheld devices and a web-based simulation approach. Even if many-core CPUs and Cloud Computing environments are very different execution architectures, with specific problems, it should be clear that the state of the art is, in both cases, made of quite specific solutions. Solutions that are tailored for a specific problem or a given simulation mechanism. In all cases with its own benefits and drawbacks. Challenges As already observed in the introduction, recent technological evolutions of computing systems require software developers to adapt their applications to new computing paradigms. Simulation developers, in particular those working on PADS, are no exception. In this section we discuss more specifically how multi-core processors and Cloud Computing affect simulation users. The following issues that needs to be addressed by future PADS systems: 1. Transparency: parallel simulation systems should provide the possibility to hide low level details (e.g., synchronization and state-management issues) to the modeler, should the modeler be unwilling to deal manually with these issues. 2. Simulation as a Service: what are the main challenges that should be addressed in order to provide simulation "as a service" using a Cloud? 3. Cost Models: Cloud resources are usually provisioned on a pay-per-use pricing model, with "better" resources (e.g., more powerful virtual processors, more memory) having higher cost. Is this pricing model suitable for PADS? 4. Performance: Cloud Computing allows applications to dynamically shape the underlying computing infrastructure by requesting resources to be added/removed at run time. Cloud-enabled PADS systems should be extended to make use of this opportunity. Transparency Two of the main goals of the research work on PADS in the last decades were: i) make it fast; ii) make it easy to use [67]. Today, PADS can be very fast when executed in the right conditions [68], e.g., when the simulation model is properly partitioned, the appropriate synchronization algorithm is used and the execution architecture is fast, reliable and possibly homogeneous. In terms of usability, however, PADS does not yet work "out of the box". In principle, the user of simulation tools should focus on modeling and analysis of the results. In practice, the choice of a specific tool (e.g., a sequential simulator of some kind) will preclude a future easy migration of the model towards another sequential simulator of a different kind, or towards a PADS system. For example, if a user wishes to migrate from a sequential to a parallel simulation, he will likely need to cope with such details as the choice of synchronization algorithm (optimistic or conservative), the allocation of simulated entities on physical execution units, details of the messaging system that allows simulated entities to communicate, and so on. Specifically, in PADS it is necessary to partition the simulation model across the available execution units. A good partitioning strategy requires that the communication among the partitions are minimized (cluster interactions) and the workload is evenly distributed across the execution units (load balancing). In case of static simulation models, i.e., models where the communication patterns among partitions do not change significantly over time, it is possible to statically define the partitions at model definition time. It should be observed that, even in the static scenario, identifying an optimal simulation model partitioning where inter-partition communication is minimized is a known NP-complete problem [69]. If the communication pattern changes over time, then some form of adaptive load balancing should be employed [70]. Adaptive load balancing is appealing since it can be applied also to static models to allow the "best" partitioning to be automatically identified without any specific application-level knowledge. Adaptive load balancing is difficult to implement (see [71] for an heuristic approach based on game theory), and is still subject of active research A better way to deal with the problems above would be to separate the simulation model from the underlying implementation mechanisms, which in the case of PADS means that low level details such as synchronization, data distribution, partitioning, load balancing and so on should be hidden to the user. This proved to be quite difficult to achieve in practice: an example is again the High Level Architecture (IEEE 1516) [33] that supports optimistic synchronization, but the low-level implementation of all the support mechanisms (such as rollbacks, see Section 2) is left to the simulation modeler [72]. Therefore, an existing conservative HLA simulation can hardly be ported to an optimistic simulation engine. Simulation as a Service As already described in Section 2, some work on Cloud-based PADS has already been done [55,56,57]. In many of these works, Cloud technology is used for executing simulations in private Clouds. The goal is to enjoy the typical benefits of Cloud Computing (elasticity, scalability, workload consolidation) without giving up the guarantees offered by an execution environment over which the user has a high degree of control. However, the vast majority of simulation users do not own a Private Cloud, therefore it would be much more interesting to run parallel simulations on a Public Cloud, using resources rented with a pay-per-usage model. Obviously, the partitioning problem described above still holds; additionally, due to its nature, a Public Cloud provides a somewhat less predictable environment to applications. For example, execution nodes provided to users may be located in different data centers, have slightly different raw processing power, and be based on physical resources (e.g., processors) that are shared with other Cloud customers through virtualization techniques. Considering the non-technical aspects, one of the potential advantages of the Public Cloud is the existence of multiple competing providers, that could -at least in principle -lead to a a "market of services". In other words, the same service (e.g., computation) could be provided by many vendors and therefore the price is the result of a market economy of supply and demand. To take advantage of the competition among vendors, and the resulting price differences, the user must be able to compose services from different providers. The result is an heterogeneous architecture where performance and reliability aspects must be carefully taken into account. However, the scenario above is unlikely, due to the lack of interoperability and the use of different APIs by different vendors. While in the short term vendor lock-in is beneficial to service providers, there already is pressure towards the use of open standards in Cloud Computing, especially in the academic/research community, culminated in the Open Cloud Computing Interface (OCCI) specifications [73]. It remains to be seen if and when open Cloud standards will be able to carve into the corporate community. Cost Models The performance of a simulator is usually evaluated on the basis of the time needed to complete a simulation run (Wall-Clock-Time, WCT). However, this is not the only metric, and in some scenario it may also not be the most relevant one. For example, if the simulation is going to be executed on resources acquired from a Cloud provider, another important metric is the total cost of running the simulation, which is related both to to the WCT and also to the amount of computation and storage resources acquired. Therefore, the user of simulation tools will have to decide: • How much time he can wait for the results; • How much he wants to pay for running the simulation. The first constraint (how much time the user is willing to wait) should be always taken into consideration in any cost model. If no upper bound is set on the total execution time, then the cost of running the simulation essentially drops to zero, since that would allow the use of extremely cheap (if not totally free) resources of proportionally low performance. In practice, nobody is really willing to wait an arbitrary long time; hence a maximum waiting time is always defined, and this influences the cost of the resources that must be allocated to run the simulation: faster (and therefore expensive) resources are needed if the simulation result has to be provided quickly; slower (cheaper) resources can be employed if the user can tolerate a longer waiting time. It is clear that, in a Public Cloud scenario, every computation and communication overhead should be minimized or at least carefully scrutinized. However, the two traditional synchronization mechanisms used in PADS (the Chandy-Misra-Bryant algorithm and the Time Warp protocol) have not been developed with these considerations in mind. 10 The Chandy-Misra-Bryant (CMB) algorithm [21] is one of the most well-known mechanisms used for conservative synchronization. Due to its nature, it requires the definition of some artificial events with the aim of making the simulation proceed. The number of such messages introduced by the synchronization algorithm can be very large [74,75]. Obviously, this communication overhead has a big effect on the WCT. Over the years, many variants have been proposed to reduce the number of such messages [76] and therefore reduce the communication cost. Despite of this, in some cases, the CMB synchronization can still be prohibitive under the performance viewpoint. On the other hand, the consideration that computation is much faster and cheaper than communication is at the basis of optimistic synchronization, e.g., the Time Warp protocol [22]. Both the conservative and optimistic synchronization mechanisms described above are not well suited for execution environments based on a "pay per use" pricing policy. Conservative synchronization is in general communication-bound and does not make effective use of CPUs. On the other hand, in a optimistic simulation, a very large part of the computation can be thrown away due to roll-backs. Very often the roll-backs do not remain clustered in a part of the simulator as they spread to the whole distributed execution architecture. This diffusion is usually implemented using specially crafted messages (called anti-messages) that consume a non-negligible amount of bandwidth. Finally, to implement the roll-back support mechanism it is necessary to over-provision many parts of the execution architecture (e.g., volatile memory). In a Public Cloud environment, all these aspects can have a very large impact on the final cost paid for running the simulation. To summarize, the Public Cloud comes with a new cost model. Up to now most of the budget was for the hardware, the simulation software tools and writing the simulation model. Now it is no longer necessary to buy hardware for running the simulations, as computation is one of the many services that can be rented. If the goal is to have simulations that really follow the "everything as a service" paradigm, then we need some better way for building PADS, with mechanisms that need to be less "expensive", both in terms of computation and communication requirements. In this case, the main evaluation metric should not be the execution speed but the execution cost (or more likely a combination of both). A deeper look at the cost model of Public Clouds (e.g. Amazon [77]) reveals some interesting facts. The pricing is often complex, with many different choices, configurations and options. For example, a new customer willing to implement a new service on top of the Amazon EC2 Cloud has to choose among several options: "On-Demand Instances 2 ", "Reserved Instances" (Light, Medium or Heavy Utilization versions) and "Spot Instances". Furthermore, there are many instance types to choose from (e.g., General Purpose, Compute Optimized, GPU, Memory Optimized, Storage Optimized, Micro), each of them with many available options. Finally, for the dismay of many costumers, the price of each instance changes in the different regions (i.e., zones of the world). It is clear that each price and the detailed service specifications (e.g., the amount of free data transfer provided to each virtual instance) are part of a business strategy and therefore can change very quickly. Focusing more on the technical aspects, many of the "elastic" features (the possibility to dynamically scale up or down the resources (e.g. computation, memory, storage) provided by the Cloud have been built specifically for Web applications and therefore cannot be easily used for building simulators. In [58], the authors analyze the cost optimization of parallel/distributed simulations run on the Amazon EC2 platform. More in detail, a cost to performance metric is defined and it used to find what EC2 instance type delivers the highest performance per dollar. In the paper, the most common EC2 instance types are analyzed under some assumptions (e.g. it is not considered that each partial instance-hour consumed is billed as a full hour). More in detail, the authors find that, under their metric, the best strategy is to use large (and costly) instances. This happens because in this kind of instance it is possible to cluster together many LPs and therefore minimize the network load. Performance We now focus our attention on the performance of a PADS implemented on a Public Cloud, when then goal is to obtain the results as fast as possible (i.e. the minimizing the WCT). As usual, even if there are many aspects that could be considered, we focus on synchronization. Limiting the discussion on synchronization is not a concern given that if synchronization has poor performances then the rest of the simulator will not do much better. What would happen if the synchronization algorithms described in Section 2.1 are run on a Public Cloud without any modification? What performance can be expected? Our analysis starts with the simplest synchronization algorithm: the time-stepped. As said before, the simulation time is divided in a sequence of steps and it is possible to proceed to the next timestep only when all the LPs have completed the current one. It is clear that the execution speed is bounded by the slowest component. This can be very dangerous in execution environments in which the performance variability is quite high. The whole simulation could have to stop due to a single LP that is slow in responding. This could happen for an imbalance in the model partitioning or for some network issues. What about the Chandy-Misra-Bryant algorithm? We have already said that this algorithm is very demanding in terms of communication resources and that, in this case too, a slow LP could be the bottleneck of the whole simulation. The last possibility is to use an optimistic synchronization algorithm such as the Jefferson's timewarp [22]. This algorithm is not very promising either: timewarp is well-known to have very good performance when all LPs can proceed with an execution speed that is almost the same. This usually means that all LPs have to be very homogeneous in terms of hardware, network performance and load. Otherwise, the whole simulation would be slowed down by the roll-backs caused by the slow LPs. A requirement that is hard to satisfy in a Public Cloud environment. All these synchronization approaches have some characteristics that do not fit well with the Public Cloud architecture. A more detailed analysis shows that each approach has some pros and cons but the key problem is that all of them are not adaptable. It is just like working on dynamic problems with a static methodology. In [58], the authors assess the performance and cost efficiency of different conservative time synchronization protocols (i.e. null-message sending strategies) when a distributed simulation is run on a range of Cloud resource types that are available on Amazon EC2 (i.e., different instance types). This work demonstrates that the simulation execution time can be significantly reduced using synchronization algorithms that are tailored for this specific execution environment and furthermore it shows that the performance variability, which is typical in low price instance types, has a noticeable impact on performance. Moreover, the authors foresee the adoption of dynamic forms of partitioning of the simulation model. An interesting aspect of Cloud Computing is that it allows the application to shape the underlying execution infrastructure, rather than the other way around. This means that the application may require more resources (e.g., instantiate other computing nodes) or release them at run time. Focusing on computing power, a Cloud application may require a larger/lower number of computing nodes of the same type of those already running (horizontal scaling), or request to upgrade some or all the current computing nodes to a higher configuration, e.g., by asking for a faster processor or more memory (vertical scaling). PADS could benefit from the above scaling opportunities, since they could be used to overcome the load balancing issues. Specifically, a simulator could request more computing nodes to reduce the granularity of the model partitioning (horizontal scaling), with the goal of reducing the CPU utilization of processor-bound simulations. On the other hand, for communication-bound models the simulator could consolidate highly interacting LPs on the same processor to replace remote communications with local ones. In this case, the simulator may request the Cloud to upgrade the nodes where the higher number of LPs are located (vertical scaling) in order to avoid the introduction of a CPU-bound bottleneck. Deciding when and how to scale is still an open problem. For the when part, the user needs to define a decision procedure that tells when additional resources must be requested (upscaling) or when some of the resources being used can be relinquished to the Cloud provider (downscaling). Upscaling may be required to speed up the simulation, e.g., to meet some deadline on the completion time; upscaling may also be used during interactive simulations to focus on some interesting phenomena with a higher level of detail. Upscaling brings the issue of how to scale, i.e., choosing between horizontal and vertical scaling. The decision ultimately depends on the cost model being used (refer again to Section 3.3). If the user is only interested in maximizing the performance of PADS, then vertical scaling is more likely to provide benefits since it consolidates the workload on a lower number of more powerful nodes, thus reducing the impact of communication costs. If the cost model takes into consideration both performance and cost, then the choice between vertical and horizontal scaling becomes less obvious. The Quest for Adaptivity So far, we have analyzed some of the limitations of current PADS approaches. Before attempting to address these limitations, it is important to realize that there is no "silver bullet", i.e., no single solution that addresses them all in a comprehensive and coherent way. The last attempt to obtain a "one size fits all" approach to PADS resulted in the the IEEE 1516 standard [33] that, as we mentioned, attracted several criticisms due to its complexity and limitations [35,36,37,38]. In this section we describe an approach to build scalable and adaptive simulation models by addressing the partitioning problem [15], that is the decomposition of the simulation model into a number of components and their allocation among the execution units. Our approach aims at achieving two goals: balance the computation load in the execution architecture and minimize the communication overhead [11]. If both 13 these requirements are satisfied, then the simulation execution is likely to be carried out efficiently. The difficult part is that the balancing procedure should be transparent to users and adaptive. Adaptivity is of extreme importance given that both the behavior of the simulation model and the state of the underlying execution architecture can not be accurately predicted. In our view, the adaptive partitioning of the simulation model is a prerequisite for a solution to most of the PADS problems described in the previous sections. Model Decomposition A complex simulation model should be partitioned in smaller parts referred to as Simulated Entities (SEs). Each SE interacts through message exchanges with other SEs to implement the expected behavior. We assume that the execution environment consists of a set of interconnected Physical Execution Units (PEUs). For example, each PEU can be a core in a modern multi-core CPU, a processor in a shared memory multiprocessor, a node in a LAN-based cluster or even a Cloud instance. Following the PADS approach, the simulated model is partitioned among all the PEUs and each PEU is responsible for the execution of only a part of the model. In a traditional PADS, the model is partitioned in a set of Logical Processes (LPs) and each LP runs on a different PEU. Instead, in our case the LPs act as containers of SEs. In other words, the simulation model is partitioned in basic components (i.e., the SEs) that are allocated within the LPs. The SEs does not need to be statically allocated on a specific LP; indeed, each SE can migrate to improve the runtime efficiency of the simulator [78], as will be described shortly. For better scalability, new LPs can be allocated during the simulation, and idle ones can be disposed. In practice, the simulation is organized as a Multi Agent System (MAS) [79]. The MAS paradigm has the potential to enhance usability and transparency of simulation tools (as discussed in Section 3.1). Choosing the proper granularity of the SEs can be problematic. Having a large number of very simple entities will probably increase the communication and management overhead. Conversely, having only a few big entities means that the workload can be spread less effectively across the available PEUs, resulting in bad load balancing. The appropriate granularity of entities can sometimes be "suggested" by the simulation model itself; for example, it is pretty natural to model each node in a wireless network as a single SE. Dynamic Partitioning To properly partition the model, we have to consider two main aspects. First, a PADS has to deal with a significant communication cost due to network latency and bandwidth limitations. Second, the execution speed of the simulator is bounded by its slowest component and therefore effective load balancing strategies must be implemented to quickly identify and remove performance "hot spots". As seen in Section 3.4, both the communication overhead and the computation bottlenecks have a big influence on the simulators performance (e.g., synchronization). An effective strategy to reduce communication costs is to cluster the strongly interacting SEs together within the same LP [80]. However, a too aggressive clustering may result in poor load balancing if too many SEs are brought inside the same LP. Since communication patterns may change over time, the optimal partitioning is in general time-dependent, and is the result of a dynamic optimization problem with multiple conflicting goals and unknown, time-varying parameters. Figure 1: A PADS with three LPs and ten SEs. 1a) Initial situation; 1b) After migration. Figure 1a shows an example where a distributed simulation runs on two hosts connected by a local area network. Host 1 has a single PEU (e.g., a processor with a single core) and executes one LP, Host 2 has two PEUs executing one LP each. An LP is the container of a set of SEs that interact through message exchanges (depicted as dotted lines in the figure). The colors used to draw the SEs represent the different interaction groups, that is, groups of SEs that interact the most. The interaction groups are {SE 1 , SE 3 , SE 7 , SE 10 }, {SE 2 , SE 5 , SE 6 } and {SE 4 , SE 8 , SE 9 }. If these interaction groups are expected to last for some amount of time, a more efficient allocation that reduces the communication overhead would be to migrate SEs as shown in Figure 1b, so that each interaction group lies within the same LP. In this simple example, interaction groups are of approximately the same size, and therefore the new allocation is well balanced. Since the interaction pattern between SEs may change unpredictably, it is necessary to rely on heuristics that monitor the communication and the load of each LP, and decide if and when a migration should happen. Obviously migrations have a non-negligible cost that includes the serialization of state variables of the SE to be migrated, the network 15 transfer delay, and the de-serialization step required before normal operations can be restored at the destination LP. The overall execution speed of the whole simulation is therefore the result of two competing forces: one that tries to aggregate SEs to reduce communication costs, and the other one that tries to migrate SEs away from overloaded LPs. Our experience with the practical implementation of adaptive migration strategies is reported in the following. In the last years, we pursued the approach described so far with the realization of a new simulation middleware called Advanced RTI System (ARTÌS) and the companion GAIA+ framework (Generic Adaptive Interaction Architecture) [15,78,80]. The high level structure of our simulator is shown in Figure 2. The upper layer (labeled Simulation Model ) is responsible for the allocation of the state variables to represent the evolution of the modeled system, the implementation of the model behavior and the necessary event handlers. The simulation model is built on top of the GAIA+ software framework, that provides the simulation developer with a set of services such as high level communication primitives between SEs, creation of SE instances and so on. GAIA+ can analyze the interactions (i.e., message exchanges) between SEs to decide if and how SEs should be clustered together, also providing the necessary support services. Given the distributed nature of a PADS, each LP analyzes the interactions of all SEs it hosts, and takes migration decisions based on local information and information received from other LPs, so that adaptivity can be achieved without resorting to any centralized component. Finally, in the layered architecture shown in the figure, the ARTÌS middleware [81] provides common PADS functionalities such as synchronization, low level communication between LPs, and simulation management services. ARTÌS and GAIA+ To validate ARTÌS and GAIA+, a number of models have been implemented, including wired and wireless communication environments [82,83]: using the ideas previously described, it has been possible to manage the fine grained simulation of complex communication protocols such as IEEE 802.11 in presence of a huge number of nodes (up to one million) [15]. In the wired case, we are working on the design and evaluation of gossip 16 protocols in unstructured networks (e.g., scale-free, small-world, random) [84,85]. Good scalability has been achieved, using off-the-shelf hardware, thanks to the use of adaptive migration and load balancing techniques. Both this software tools have been designed to be platform-independent. In particular, the dynamic partitioning feature provided by GAIA+ fits very well with the horizontal and vertical scalability provided by Cloud Computing. The ARTÌS middleware, the GAIA+ framework, sample simulation models, and the support scripts and scenario definition files are freely available for research purposes [81]. A large part of the software is provided in both binary and source form. We expect to make the source code for all components available through an open source license. Case study A complete evaluation of GAIA+/ARTÌS is outside the scope of this paper; however, a simple case study can be useful to better understand the proposed mechanism, and will be illustrated in this section. For the sake of simplicity, we consider only the adaptive clustering approach used to reduce the communication overhead. Advanced forms of load-balancing and reaction to background load have been implemented as well, but will not be examined here. We Our analysis starts with a scenario in which simulated entities do not migrate across LPs (the GAIA+ migration engine is turned off). To balance the workload evenly across the three LPs, each one should receive approximately 9999/3 = 3333 MHs. A simple metric that can be used to assess the ability to properly cluster the simulated components is the Local Communication Ratio (LCR), defined as the percentage of local messages with respect to the total messages sent or received by a simulated component (higher is better). A random assignment of MHs to LPs results in a LCR of 100 #LP (%), where #LP as the number of LPs in the PADS. By taking into consideration the model semantics, we know that the communication between entities exhibits a strong spatial locality within the simulation area, since each MH only interacts with a local neighborhood. Therefore, a better assignment of MHs to LPs can be obtained by partitioning the simulated area, e.g., in vertical stripes, and to allocate all MHs in each stripe to the same LP, as shown in Figure 3a. However, such allocation works well for the initial simulation steps, but then quickly degenerates as the spatial position of MHs change as the results of their movement. The situation can be observed in Figures 3b through 3d, where the initial spatial locality of MHs is rapidly destroyed. Figure 4 shows what happens when the adaptive migration facility of GAIA+ is turned on, starting from a completely random allocation of MHs to LPs. GAIA+ analyzes the 17 (Figure 4b) shows that the MHs have been clustered in groups, and that each group is determined by its position in the simulated area. It is interesting to observe that Figure 4b is similar to what would be produced by Schelling's segregation model [86]; indeed, each MH has a preference towards the other MHs with which it communicates the most, and GAIA+ tries to cluster those MHs together within the same LP. The performance of the static allocation versus the dynamic allocation implemented by GAIA+ is shown in Figure 5. The green line shows the mean LCR as a function of the simulated time step if MHs are statically assigned to the LPs at the beginning, while the red line shows the mean LCR when the adaptive migration of GAIA+ is enabled. Figure 5a corresponds to the situation in which all MHs are initially randomly assigned to the LPs. As discussed above, the expected LCR in this situation is about 33%, which is the initial value at step t = 0. As the simulation proceeds, the LCR provided by the static allocation remains at the initial value. On the other hand, if the adaptive migration feature (GAIA+) is activated then the LCR rapidly increases. In Figure 5b we use the "sliced" allocation at t = 0, where MHs are assigned to LPs according to their initial position. In the case of static mapping (green line), the LCR drops to 33% as the movement of MHs destroys the initial statically imposed locality. If adaptive migration is used, the LCR remains stable at a nearly optimal value. It is important to remark that a high LCR value does not guarantee, by itself, that a PADS scales well. In fact, migrating simulation entities does have a cost that can be significant if a large amount of state information needs to be moved from one PEU to another. However, previous studies have shown that good scalability can indeed be achieved even for models that do not exhibit the simple spatial locality of wireless ad-hoc networks [87]. It is important to remark that the particular metric considered in this section (LCR) is independent from the execution environment (e.g., private clusters, Private or Public Clouds). In fact, a high LCR value demonstrates that the dynamic clustering of the simulated entities performed by GAIA+ leads to a reduction in the communication overhead since most communications are local. The actual impact of this reduction on the Fault Tolerance So far, we have described our effort in building adaptive PADS. In our opinion, this is a first step towards simulations that are able to run efficiently on modern computing infrastructures. A key point is still missing: the support for fault tolerance in distributed simulation. Usually, if a LP crashes (e.g., due to hardware failure of the underlying PEU), then the whole simulation aborts. In the case of a long-running simulation that is executed on hundreds or thousands of PEUs, the probability of a hardware failure during execution can not be neglected. Therefore, fault tolerance is essential, especially if we wish to run large-scale simulations on cheap but unreliable execution systems such as resources provided by a Cloud infrastructure. A new software layer called GAIA Fault-Tolerance (GAIA-FT) extends the interface provided by GAIA+ with provisions for fault tolerant execution. A GAIA-FT simulation is made by a set of interacting Virtual Simulated Entities (VSEs). A VSE is implemented as a set of identical SEs running on different LPs, therefore achieving fault tolerance through replication. By selecting the number of replicas it is possible to decide how many crashes we want to tolerate, and also cope with byzantine failures. From the user's point of view, a VSE is just a special type of SE; in other words, the replication introduced by GAIA-FT is totally transparent and is built on the lower layers of the GAIA+ architecture. This means that the adaptive strategies described so far are still available and can be used to migrate replicas within a VSE to improve load balancing and communication. The basic implementation of GAIA-FT has been recently completed, and we are in the process of evaluating the replication mechanism. Of course, fault tolerance comes at a cost, since replication increases the number of SEs to be partitioned. Therefore requiring more resources to execute the simulation, both in term of execution nodes and wall-clock time. This means that, a larger number of LPs will be used and that they will need to be kept synchronized in order for the simulation to advance, so the issues already described in Section 3.4 apply. Finally, when multiple copies of the same SE are created, the simulation middleware must ensure that all copies reside in LPs that are run on different execution nodes, to avoid that a failed processor brings down multiple replicas of the same SE. This makes the load balancing task more complex, since a new constraint is introduced (namely, multiple copies of the same SE must never reside on the same LP or on LPs that are run on the same execution node). Identifying a reasonable trade-off between reliability and performance in a parallel simulation is an interesting research topic that we are pursuing. Conclusions In this paper, we have reviewed the main ideas behind Parallel and Distributed Simulation (PADS), and we observed that current PADS technologies are unable to fulfill many requirements (e.g. usability, adaptivity) in the context of the new parallel computing architectures. Most computing devices are already equipped with multi-core CPUs, and Cloud-based technologies, where computation, storage and communication resources can be acquired "on demand" using a pay-as-you go model, are gaining traction very quickly. The "simulation as a service" paradigm has already been proposed as a possible application of Cloud technologies to enhance simulation tools. We have shown that, the current PADS techniques are unable to fit well with these architectures, and that more work is needed to increase usability and performance of simulators in such conditions. We claim that a solution for such problems is required to deal with model partitioning across the execution nodes. To this aim, we propose an approach based on multi-agent systems. Its main characteristic is the adaptive migration of the simulated entities between the execution units. Heuristics can be used to reducing the communication cost and to achieve better load balancing of the simulation workload. Our proposal has been implemented in the ARTÌS/GAIA+ simulation middleware and tested with different models, with promising results. Finally, the horizontal and vertical scaling possibilities provided by Cloud systems (see Section 3.4) deserve further investigations in the context of PADS. The Cloud Computing paradigm allows applications to request the type and amount of resources of their choice at run time, enabling dynamic sizing of the execution environment. How to enable a larger class of applications to take advantage of these possibilities is yes to be understood. Figure 2 : 2High level structure of the simulator and its main components consider a time-stepped, ad hoc wireless network model of 9999 Mobile Hosts (MHs). The simulated scenario is a two-dimensional area (10000 × 10000 units) with periodic boundary conditions, in which each MH follows a random waypoint mobility model (maxspeed = 10 space-units/time-unit, 70% of nodes move at each timestep). The communication model is very simple and does not consider the details of low level medium access control. At each timestep, a random subset of 20% of the MHs broadcasts a ping message to all nodes that are within the transmission radius of 250 space units. This model captures both the dynamicity of the simulated systems and the space locality aspects of wireless communication. The MHs have been partitioned in 3 PEUs, each one running a single LP. Figure 3 : 3Spatial position and allocation in LPs of the mobile wireless nodes simulation at different time steps. The color and shape of dots shows in which LP each node is allocated. Adaptive migration is not active (GAIA+ off). communication pattern of each MH and clusters the interacting hosts in the same LP. A snapshot at the end of the run Figure 4 : 4Spatial position and allocation in LPs of the mobile wireless nodes at different time steps. The color and shape of dots shows in which LP each node is allocated. Adaptive migration is activated (GAIA+ ON). Figure 5 : 5Local Communication Ratio (LCR) evolution with different initial allocations. simulation wall-clock time depends on the (absolute) communication costs of the specific execution environment. that has been approved in 2000 and recently revised under the SISO HLA-Evolved Product Development Group [34]. 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[ "The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle/ZF", "The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle/ZF" ]	[ "Lawrence C Paulson \nComputer Laboratory\nUniversity of Cambridge\n15 JJ Thomson AvenueCB3 0FDCambridgeEngland\n" ]	[ "Computer Laboratory\nUniversity of Cambridge\n15 JJ Thomson AvenueCB3 0FDCambridgeEngland" ]	[]	The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle/ZF. The proof builds upon a previous mechanization of the reflection theorem[18]. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kunen's Set Theory[9], and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized.	10.1112/s1461157000000449	[ "https://arxiv.org/pdf/2104.12674v1.pdf" ]	121,778,086	2104.12674	5793562da9a765066a2f8604646643d820a80b57	The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle/ZF 26 Apr 2021 Lawrence C Paulson Computer Laboratory University of Cambridge 15 JJ Thomson AvenueCB3 0FDCambridgeEngland The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle/ZF 26 Apr 20211 CONTENTS 2 The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle/ZF. The proof builds upon a previous mechanization of the reflection theorem[18]. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kunen's Set Theory[9], and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized. Introduction In 1940, Gödel [5] published his famous monograph proving that the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are consistent with respect to the other axioms of set theory. This theorem addresses the first of Hilbert's celebrated list of mathematical problems. I have attempted to reproduce this work in Isabelle/ZF. When so much mathematics has already been checked mechanically, what is the point of checking any more? Obviously, the theorem's significance makes it a challenge, as does its size and complexity, but the real challenge comes from its reliance on metamathematics. As I have previously noted [18], some theorems seem difficult to formalize even in their statements, let alone in their proofs. Gödel's work is not a single formal theorem. It consists of several different theorems which, taken collectively, can be seen as expressing the relative consistency of the axiom of choice. At the end of Chapter VII, Gödel remarks that given a contradiction from the axioms of set theory augmented with AC, a contradiction in basic set theory "could actually be constructed" [5, p. 87]. This claim is crucial: logicians prefer consistency proofs to be constructive. Gödel's idea [4,6] is to define a very lean model, called L, of set theory. L contains just the sets that must exist because they can be defined by formulae. Then, prove that L satisfies the ZF axioms and the additional axiom "every set belongs to L," which is abbreviated V = L. We now know that V = L is consistent with ZF, and can assume this axiom. (The conjunction of ZF and V = L is abbreviated ZFL.) We conclude by proving that AC and GCH are theorems of ZFL and therefore are also consistent with ZF. A complication in Gödel's proof is its use of classes. Intuitively speaking, a class is a collection of sets that is defined by comprehension, { \| ( )}. Every set is trivially a class, namely { \| ∈ }, but a proper class is too big to be a set. Formal set theories restrict the use of classes in order to eliminate the danger of paradoxes. Modern set theorists use Zermelo-Fraenkel (ZF) set theory, where classes exist only in the metalanguage. That is, the class { \| ( )} is just an alternative notation for the formula ( ), and ∈ { \| ( )} is just an alternative notation for ( ). The universal class, V, corresponds to the formula True. An "equation" like V = ∈ON stands for ∀ . ∃ . ON( ) ∧ ∈ . (Here, ON denotes the class of ordinal numbers.) Gödel's monograph [5] uses von Neumann-Bernays-Gödel (NBG) set theory, which allows quantification over classes but restricts their use in other ways. With either axiom system, classes immensely complicate the reasoning, making it highly syntactic. Why did Gödel use classes? Working entirely with sets, he could have used essentially the same techniques to prove that if is a model of ZF then there ex-ists a model ( ) of ZFC. (ZFC refers to the ZF axioms plus AC.) Therefore, if ZFC has no models, then neither does ZF. But with this approach, he can no longer claim that if he had a contradiction in ZFC then a contradiction in ZF "could actually be constructed." 1 For the sake of this remark, which is not part of any theorem statement, Gödel chose a more difficult route. Classes create more difficulties for formal proof checkers (which have to face foundational issues squarely) than they do for mathematicians writing in English. The proof uses metatheoretic reasoning extensively. Gödel writes [5, p. 34], However, the only purpose of these general metamathematical considerations is to show how the proofs for theorems of a certain kind can be accomplished by a general method. And, since applications to only a finite number of instances are necessary . . ., the general metamathematical considerations could be left out entirely, if one took the trouble to carry out the proofs separately for any instance. I decided to take the trouble, not using metatheory but relying instead on a mechanical theorem prover. This paper describes the Isabelle/ZF proofs. It indicates the underlying mathematical ideas and sometimes discusses practical issues such as proof length or machine resources used. It necessarily omits much material that would be too long or too repetitious. The paper concerns how existing mathematics is formalized; it contains no original mathematics. Overview. The paper begins by outlining Gödel's relative consistency proof ( §2). After a brief overview of Isabelle/ZF, the paper describes the strategy guiding the formalization ( §3) and presents some elementary absoluteness proofs ( §4). It then discusses relativization issues involving well-founded recursion ( §5). Turning away from absoluteness, the paper proceeds to describe the formalization of the constructible universe and the proof that L satisfies the ZF axioms ( §6); then, it describes how the reflection theorem is used to prove that L satisfies the separation axiom ( §7). Absoluteness again takes centre stage as the paper presents the relativization of two essential datatypes ( §8) and finally presents the absoluteness of L itself ( §9). Finally, the paper presents the Isabelle proof that AC holds in L ( §10), and offers some conclusions ( §11). Proof Outline Recall that Gödel's idea is to define a lean model of set theory, the class L of the constructible sets. Figure 1 shows L (shaded) as a subclass of the universe, V. The vertical line represents the class ON of the ordinals. Gödel's proof involves four main tasks: 1. defining the class L within ZF 2. proving that L satisfies the ZF axioms 3. proving that L satisfies V = L 4. proving that V = L implies the axiom of choice As we shall see, Isabelle is well-suited for completing the first and last parts. Both the definition of L and the proof of the axiom of choice are straightforward exercises in mechanized set theory. The second task cannot quite be completed: all of the ZF axioms can be verified apart from separation, which is an axiom scheme, and each instance requires its own proof. As for the third task, the Isabelle proof that L satisfies V = L is much longer than I would like because the metamathematical techniques that abbreviate textbook proofs are not available. Once we have completed the first three tasks, we should be able to conclude that if ZF is consistent, then so is ZFL. (And from the fourth task, if ZF is consistent, then so is ZFC.) This inference requires reasoning in the metatheory, which is not possible using Isabelle/ZF, so the machine formalization omits it. Standard treatments also gloss over this step, regarding it as obvious. Section 2.4 below expands on this issue. 7 The Problem With Class Models Because L is a proper class, we cannot adopt the usual notion of satisfaction. To formalize the standard Tarski definition of truth [10, p. 60] requires first defining, in set theory, a set to represent the syntax of first-order formulae. is easily defined, either using Gödel-numbering or as a recursive data structure. If is a set, ∈ represents a formula with free variables, and 1 , . . . , ∈ then \|= ( 1 , . . . , ) can be defined by recursion on the structure of . If M is a proper class, then the obvious definition of M \|= ( ì ) cannot be formalized in set theory; the environments that hold the bindings of free variables would have to belong to a function space whose range was all of M. Tarski's theorem on nondefinability of truth [9, p. 41] asserts that no formula ( ) expresses V \|= . If for each formula we write for the corresponding element of , then ↔ ¬ ( ) is a theorem for some sentence . Satisfaction cannot be defined, at least if M = V. Relativization Gödel instead expressed satisfaction for class models syntactically. This approach abandons the set of formula representatives in favour of real formulae. Set theory uses a first-order language with no constant symbols, no function symbols and no relation symbols other than ∈ and =. Variables are the only terms. Gödel's key concept is relativization. 2 If M is a class and is a formula, define M recursively as follows: The Formal Treatment of Terms Despite the lack of terms in their formal language, set theorists use elaborate notational conventions. In other branches of mathematics, an expression like ( ) ( ) − ℎ( , ) means what it says: functions , and ℎ are applied and the results combined by multiplication and subtraction. But in set theory, each expression ( ) abbreviates a formula ( , ), which reduces the meaning of = ( ) to a combination of ∈ and =. For example, we can express the meaning of = ∪ by the predicate union( , , ), defined by ∀ . ∈ ↔ ∈ ∨ ∈ . We can similarly define inter( , , ) to express = ∩ . Combining these predicates gives meaning to more complex terms; for example, = ( ∪ ) ∩ abbreviates ∃ . union( , , ) ∧ inter( , , ). Variable binding notation, ubiquitous in set theory, causes complications. In ∈ ( ), what is ? Syntactically, ( ) is a term with parameter , so we can take it as an abbreviation for some formula ( , ). But then becomes an operation on formulae rather than one on sets. An equally legitimate alternative [7, p. 34] is to regard as a function in set theory -formally, the set of pairs { , ( ) \| ∈ }. Set theorists generally say little about these notational conventions and act as if terms were meaningful in themselves. But relativization forces us to make the translation from terms to formulae explicit. In the Isabelle formalization, I have defined relational equivalents of dozens of term formers. I have included a class argument in each one to perform relativization at the same time; we can express the relativized term (( ∪ ) ∩ ) M as ∃ ∈ M. union(M, , , ) ∧ inter(M, , , ) The hardest tasks were (1) to define relational equivalents of the complicated expressions generated by Isabelle/ZF for recursively defined sets and functions and (2) to cope with the sheer bulk of the definitions. Gödel's Claim Viewed Proof-Theoretically The purpose of relativization is to express claims of the form " is true in M." To prove that L satisfies the ZF axioms and V = L, we must prove L for each ZF axiom , and we must prove (V = L) L . Now we can consider Gödel's claim that from a contradiction in ZFL a contradiction in ZF "could actually be constructed." His claim is proof-theoretic. A contradiction in ZFL is a proof, Π, of ⊥ from finitely many ZF axioms and V = L: 1 . . . V = L Π ⊥ Once we have proved that L satisfies the axioms of ZFL, we have the + 1 proofs ZF L 1 . . . ZF L (V = L) L . Verifying Gödel's claim reduces to showing that we can always construct a proof Π L of ⊥ L from the relativized premises: L 1 . . . L (V = L) L Π L ⊥ L For then we get a proof of ZF ⊥ L , which is just ZF ⊥. So how we obtain Π L from Π? To be concrete, suppose we are working with a natural deduction formalization of first-order logic. By the normal form theorem [20], since the conclusion of the proof is atomic, we can assume that Π applies only elimination rules. We must modify Π so that it accepts relativized versions of its premises and delivers a relativized version of its conclusion. The only hard cases involve quantifiers. Where Π applies the existential elimination rule to ∃ . ( ), it delivers the formula ( ) to the rest of the proof. (Assume that has already been renamed, if necessary.) At the corresponding position, Π L should apply the existential and conjunction elimination rules to ∃ . ∈ L ∧ ( ), delivering the formulae ∈ L and ( ) to the rest of the proof. Universal quantifiers require a bit more work. First, recall that the language of set theory has no terms other than variables. Where Π applies the universal elimination rule to ∀ . ( ), it delivers the formula ( ) to the rest of the proof, where is a variable. At the corresponding position, Π L should apply the existential and conjunction elimination rules to ∀ . ∈ L → ( ). But before it can deliver the formula ( ), it requires a proof of ∈ L. We will indeed have ∈ L if the variable is obtained by a previous existential elimination, but what if was chosen arbitrarily? We can handle such cases by inserting at this point an application of the empty set axiom, which will yield a new variable (say ) and the assumption ∈ L. Intuitively, we are replacing all free variables in Π by 0. The sketchy argument above cannot be called a rigorous proof of Gödel's claim. But it is more detailed than standard expositions of Gödel's proof. Kunen relegates the relevant lemma to an appendix, and for the proof he merely remarks "Similar to the easy direction of the Gödel Completeness Theorem" [9, p. 141]. To Gödel, it was all presumably trivial. I have not formalized the argument in Isabelle/ZF because that would require formalizing the metatheory. Defining the Class L The equation V = ∈ON expresses the universe of sets as the union of the cumulative hierarchy { } ∈ON , which is recursively defined by 0 = 0, +1 = P ( ) and = < when is a limit ordinal. We obtain L by a similar construction, replacing the powerset operator P by the definable powerset operator, D. Essentially, D ( ) yields the set of all subsets of that can be defined by a formula taking parameters over . If we define the set of formulae and the satisfaction relation \|= as outlined above, then we can make the definition D ( ) = { ∈ P ( ) \| ∃ ∈ . ∃ 1 . . . ∈ . = { ∈ \| \|= ( , 1 , . . . , )}}. (The ellipsis can be eliminated in favour of lists over .) Finally, we define the constructible universe: L = ∈ON , where 0 = 0, +1 = D ( ) and = < when is limit. Kunen proves that L satisfies the ZF axioms, remarking [9, p. 170] "only the Comprehension Axiom required any work." His remark applies to the Isabelle/ZF proofs. L inherits most of the necessary properties from V. Even the axiom scheme of replacement can be proved as the theorem replacement(L,P); the proof is independent of the formula P. However, the proof of comprehension for the formula requires an instance of the reflection theorem for , which requires recursion over the structure of . Each instance of comprehension therefore has a different proof from the ZF axioms. At the metalevel, of course, all these proofs are instances of a single algorithm. For Isabelle/ZF, this means that each instance of comprehension must be proved separately, although the proof scripts are nearly identical. Absoluteness: Proving (V = L) L Proving that L satisfies V = L is a key part of the proof, and despite first appearances, it is far from trivial. It amounts to saying that the construction of L is idempotent: L L = L. The underlying concept is called absoluteness, which expresses that a given operator or formula behaves the same in a class model M as it does in V, the universe. A class M is transitive if ∈ M implies ⊆ M, and we shall only be concerned with transitive models below. Most constructions are absolute. The empty set can only be a set having no elements, and ⊆ can only mean that every element of belongs to . If and are sets then their union can only be the set containing precisely the elements of those sets. Many complicated notions are also absolute: domains and ranges of relations, bijections, well-orderings, order-isomorphisms, ordinals. With some effort, we can show the absoluteness of recursively defined data structures and functions. Powersets, except in trivial cases, are not absolute. For example, P ( ) might contain subsets of the natural numbers that cannot be shown to exist. The function space → is not absolute because of the obvious connection between P ( ) and → {0, 1}. More subtly, cardinality is not absolute: if is a countable model of set theory, and is an uncountable cardinal according to , then obviously must be really be countable, with the bijections between and lying outside . This situation is called Skolem's paradox [9, p. 141]. Metamathematical arguments are an efficient means of proving absoluteness. For example, any concept that is provably equivalent (in ZF) to a formula involving only bounded quantifiers is absolute [9, p. 119]. This is the class of Δ ZF 0 formulae. The larger class of Δ ZF 1 formulae can also be shown to be absolute. Unfortunately, all such arguments are beyond our reach unless we formalize the metatheory. The Consequences of V = L Once we have proved that L is absolute, we obtain ZF (V = L) L . We can then investigate the consequences of assuming V = L. To prove the axiom of choice, it suffices to prove that every set can be well-ordered. The key step, given a well-ordering of , is to construct a well-ordering of D ( ). It comes from the lexicographic ordering on tuples , 1 , . . . , for ∈ and 1 , . . . , ∈ . So if is well-ordered, so is +1 . By transfinite induction, each level of the construction of L is well-ordered. The axiom V = L is very strong. Gödel proved that it implies the generalized continuum hypothesis. Jensen later proved that it implies the combinatorial principle known as ♦, and it has many additional consequences. But it is important to note that such proofs are entirely separate from that of ZF (V = L) L . We prove ZFL AC, ZFL GCH and ZFL ♦, but we do not prove ZF AC L , ZF GCH L and ZF ♦ L . Those results, if we want them, are most easily obtained in the metatheory, using the general fact that if then L L . Introduction to the Isabelle/ZF Formalization Isabelle [11,14] is an interactive theorem prover that supports a variety of logics, including set theory and higher-order logic. Isabelle provides automatic tools for simplification and logical reasoning. They can be combined with single-step inferences using a traditional tactical style or as structured proof texts. The Proof General user interface provides an effective interactive environment. Isabelle has been applied to a huge number of verification tasks, including the semantics of the Java language [21] and the correctness of cryptographic protocols [16]. Most of these proofs use Isabelle/HOL, the version of Isabelle for higher-order logic. Isabelle/HOL's polymorphic type system is ideal for modelling problems in computer science. Isabelle also supports Zermelo-Fraenkel set theory. Formalized material includes the traditional concepts of functions, ordinals, order types and cardinals. Isabelle/ZF also accepts definitions of recursive functions and data structures; in this it resembles other computational logics, with the important difference of being typeless. Some problems do call for a typeless logic. Isabelle/ZF is also good for investigating foundational issues, and, of course, for formalizing proofs in axiomatic set theory. Previous published work on Isabelle/ZF describes its basic development [13] and its treatment of recursive functions [13] and inductive definitions [17]. Another paper describes proofs drawn from set theory textbooks [19]. Particularly noteworthy are the proofs of equivalence between various formulations of the axiom of choice. Those proofs, formalized by Grabczewski, are highly technical, demonstrating that advanced set theory proofs can be replicated in Isabelle/ZF given enough time and effort. That is precisely why we should investigate Gödel's proof of the relative consistency of AC: much of the reasoning takes place outside set theory. The previous section has presented many reasons why we should formalize Gödel's proof directly in the metatheory. That strategy does not require a set theory prover. We could use any system that lets us define the first-order formulae, the set theory axioms, and the set of theorems derivable from any given axioms. We would enjoy a number of advantages. • Relativization could easily be defined by recursion on the structure of formulae. • Metatheorems about absoluteness -for example, that all Δ ZF 0 formulae are absolute -could be proved and used to obtain simple proofs of many absoluteness results. • The constructiveness of the consistency result could be stated and proved. However, the metatheoretical strategy also presents difficulties. We would have to work in the pure language of set theory, which reduces all concepts to membership and equality, and is unreadable; an alternative would be to formalize the familiar term language. We would constantly be reasoning about an explicitly formalized inference system for ZF rather than using our prover's built-in reasoning tools. I believe this strategy would involve as much work as the strategy I adopted, although the work would be distributed differently. The choice resembles the standard one we face when we model a formal language: shall we adopt a deep or a shallow embedding? A shallow embedding maps phrases in the language to corresponding phrases in the prover's logic. It works well for reasoning about specific examples, but does not allow metareasoning (proofs about the language). A deep embedding involves formalizing the language's syntax and semantics in the prover's logic. The extra mechanism allows metareasoning but complicates reasoning about specific examples. Compared with a shallow embedding, the strengths and weaknesses are exchanged. I have chosen to formalize Gödel's theorem in set theory, minimizing any excursions into the metatheory. This strategy still requires defining relational equivalents for each element of set theory's term language, while limiting my exposure to unreadable relational formulae. After all, the critical proofs involve showing that various concepts are absolute, which means that they do not vary from one model of set theory to another. Each absoluteness proof justifies replacing some primitive of the relational language by its counterpart in the term language. Thus Isabelle's simplifier can transform relational formulae into ones using terms, exploiting the existing formalization of set theory. This plan worked well for basic concepts such as union, intersection, relation, function, domain, range, image, inverse image and even ordinal. The absoluteness proofs for well-orderings, recursive functions and recursive data types were harder: • If a concept is defined in terms of non-absolute primitives, such as powerset, it must be proved equivalent to a suitable alternative definition. • Much of the theory of well-founded recursion must be formalized from scratch in the relational language. • Higher-order functions complicate the relational language. • Recursive functions generate complicated fixedpoint definitions that must be converted into relational form manually. Relativization and Absoluteness: Basics The first step is to define the relational language, introducing predicates for all the basic concepts of set theory. Each predicate takes a class as an argument so that it can express relativization. This relational language will later allow appeals to the reflection theorem. Space permits only a few of the predicates to appear below. Note that the class quantifications ∀ ∈ M and ∃ ∈ M are written ∀ From the Empty Set to Functions We begin with definitions of trivial concepts such as the empty set and the subset relation. A set z is empty if it has no elements: "empty(M,z) == ∀ x[M]. x ∉ z" "subset(M,A,B) == ∀ x[M]. x ∈ A −→ x ∈ B" All Isabelle definitions in this paper are indicated by a vertical line, as shown. A set z is the unordered pair of a and b if it contains those two sets and no others. The Kuratowski definition of ordered pairs , = {{ , }, { , }} is then expressed using the predicate upair: A set z is the union of a and b if it contains their elements and no others. The general union ( ), also written as { \| ∈ }, has an analogous definition. "union(M,a,b,z) == ∀ x[M]. x ∈ z ←→ x ∈ a \| x ∈ b" "big union(M,A,z) == ∀ x[M]. x ∈ z ←→ (∃ y[M]. y ∈A & x ∈ y)" A set z is the domain of the relation r if it consists of each element x such that x,y ∈ r for some y. "is domain(M,r,z) == ∀ x[M]. x ∈ z ←→ (∃ w[M]. w ∈r & (∃ y[M] . pair(M,x,y,w)))" Relativizing the Ordinals Now we can define relational versions of ordinals and related concepts. The formalization is straightforward. An ordinal is a transitive set of transitive sets. A limit ordinal is a non-empty, successor-closed ordinal. A successor ordinal is any ordinal that is neither empty nor limit. The set of natural numbers, , is a limit ordinal that contains no limit ordinals. Defining the Zermelo-Fraenkel Axioms Formally defining the ZF axioms relative to a class M lets us express that M satisfies those axioms. Each axiom is relativized so that all quantified variables range over M . We begin with extensionality: "extensionality(M) == ∀ x[M]. ∀ y[M]. (∀ z[M]. z ∈ x ←→ z ∈ y) −→ x=y" The separation axiom is also known as comprehension: "separation(M,P) == ∀ z[M]. ∃ y[M]. ∀ x[M]. x ∈ y ←→ x ∈ z & P(x)" This only yields a valid instance of separation if the formula P obeys certain syntactic restrictions. All quantifiers in P must be relativized to M, and the free variables in P must range over elements of M . These restrictions prevent us from assuming separation as a scheme by leaving P as a free variable. We must separately note every instance of separation that we need. If it meets the syntactic restrictions, then later we shall be able to prove that L satisfies it. That looks bad when we recall that the native separation axiom in Isabelle/ZF, and the theorems using it, are schematic in P. But if we formalize Bernays-Gödel set theory as a new Isabelle logic (creating the system Isabelle/BG) then the same problem occurs elsewhere. The analogue of separation in BG set theory is the General Existence Theorem, which is a metatheorem: proving each instance requires a separate construction. To compensate, at least BG has no axiom schemes. The axioms of unordered pairs, unions and powersets all state that M is closed under the given operation: The foundation axiom states that every non-empty set has a ∈-minimal element: Call a formula univalent over a set if it describes a class function on that set. The replacement axiom holds for univalent formulae: "univalent(M,A,P) == ∀ x[M]. x∈A −→ (∀ y[M]. ∀ z[M]. P(x,y) & P(x,z) −→ y=z)" "replacement(M,P) == ∀ A[M]. univalent(M,A,P) −→ (∃ Y[M]. ∀ b[M]. (∃ x[M]. x∈A & P(x,b)) −→ b ∈ Y)" Intuitively, if F is a class function and and is a set, then replacement says that F " (the image of under F ) is a set. However, the axiom formalized above is weaker: it merely asserts (relative to the class M) that F " ⊆ for some set . To get the set we really want, namely F " , we must apply the axiom of separation to . The weak form of replacement can be proved schematically for L. The strong form cannot be proved schematically because of its reliance on separation. Introducing a Transitive Class Model The absoluteness proofs are carried out with respect to an arbitrary class model M, although they are only needed for L. Generalizing the proofs over other models has two advantages: it separates the absoluteness proofs from reasoning about L and it allows the proofs to be used with other class models. Isabelle's locale mechanism [8] makes the generalization possible. A locale packages the many properties required of M, creating a context in which they are implicitly available. A proof within a locale may refer to those properties and to other theorems proved in the same locale. A locale can extend an older one, creating a context that includes everything available in the ancestor locales. The class M is assumed to be transitive (transM ) and to satisfy some relativized ZF axioms, such as unordered pairing (upair ax) and replacement. It contains the set of natural numbers, nat (which is also the ordinal ). This locale does not assume any instances of separation. Easy Absoluteness Proofs Here is a canonical example of an absoluteness result. The phrase in M trivial includes the lemma in the locale. The proof refers to the definition of empty set (empty def) and to the transitivity of M (the locale assumption transM ); it uses blast, an automatic prover. The attribute [simp] declares empty abs as a simplification rule: the simplifier will replace any occurrence of empty(M,z) by z=0 provided it can prove M(z). From now on, usually just the statements of theorems will be shown, not header lines and proofs. Here are some similar absoluteness results, also proved in locale M trivial and declared to the simplifier. Most have trivial proofs like the one shown above. These theorems express absoluteness because the class M disappears from the righthand side: the meaning of subset, image, etc., is the same as its meaning in V. Each theorem also expresses the correctness of an element of the relational language, for example that big union captures the meaning of Union. Absoluteness results involving ordinals are also easily proved: Thus we see that the simplifier can rewrite relational formulae into term notation, provided we are able to prove that they refer to elements of M . For this purpose, there are many results showing that M is closed under the usual set-theoretic constructions. In particular, we can use the separation axiom for a specific formula P: "M(A) =⇒ M(Union(A))" "[[M(A); M(B)]] =⇒ M(A ∪ B)" "[[separation(M,P); M(A)]] =⇒ M({x∈A. P(x)})" Also useful are logical equivalences to simplify assertions involving M : "M({a,b}) ←→ M(a) & M(b)" "M( a,b ) ←→ M(a) & M(b)" Absoluteness Proofs Assuming Instances of Separation All the theorems shown above are proved without recourse to the axiom of separation. Obviously many set-theoretic operators are defined using separation -possibly in the guise of strong replacement -so we now extend locale M trivial accordingly. Only a few of the 11 instances of separation appear above. Omitted are the more complicated ones, for example concerning well-founded recursion. By Inter separation it follows that M is closed under intersections. From the lemma declaration, you can see that the proof takes place in locale M basic. All results proved in locale M trivial remain available. By cartprod separation it follows that the class M is closed under Cartesian products. The proof is complicated because the powerset operator (which is not absolute) occurs in the definition. A trivial corollary is that M is closed under disjoint sums. I devoted some effort to minimizing the number of instances of separation required. For example, the inverse image operator is expressed in terms of the image and converse operators. Then the domain and range operators can be expressed in terms of inverse image and image. We obtain five closure theorems from the two assumptions image separation and converse separation: "[[M(A); M(r)]] =⇒ M(r''A)" "[[M(A); M(r)]] =⇒ M(r-''A)" "M(r) =⇒ M(converse(r))" "M(r) =⇒ M(domain(r))" "M(r) =⇒ M(range(r))" These five operators are also absolute. Here is the result for domain: "[[M(r); M(z)]] =⇒ is domain(M,r,z) ←→ z = domain(r)" Although we assume that M satisfies the powerset axiom, we cannot hope to prove M(A) =⇒M(Pow(A)). The powerset of A relative to M is smaller than the true powerset, containing only those subsets of A that belong to M . Similarly, we cannot show that M contains all functions from A to B. However, it holds for a finite case, essentially the set of -tuples: "[[n∈nat; M(B)]] =⇒ M(n->B)" This lemma will be needed later to prove the absoluteness of transitive closure. Some Remarks About Functions In set theory, a function is a single-valued relation and thus is a set of ordered pairs. Operators such as powerset and union, which apply to all sets, are not functions. (Strictly speaking, there are no operators in the formal language of set theory, since the only terms are variables.) Isabelle/ZF distinguishes functions from operators syntactically. • The application of the function f to the argument x is written f'x. On the other hand, application of an operator to its operand is written using parentheses, as in Pow(X), or using infix notation. • Function abstraction over a set A is indicated by x∈A, and yields a set of pairs. For instance, x∈A. x denotes the identity function on A. Operators are essentially abstractions over the universe, as in x. Pow(Pow(x)). Abstraction can also express predicates; for instance, x. P(x) & Q(x) is the conjunction of the two predicates P and Q. Kunen [9, p. 14] defines function application in the usual way: ' is "the unique such that , ∈ ." Isabelle/ZF originally adopted a formal version of this definition, using a description operator [13, §7.5]. The relational version of the operator, namely fun apply(M,f,x,y), held if the pair x,y belongs tof for that unique y. My original definitions of function application, in its infix and relational forms, both followed Kunen's definition. However, the absoluteness theorem relating them was conditional on the function application's being well-defined. That made it harder to simplify fun apply(M,f,x,y) to f'x = y and often forced proofs to include what was essentially type information. Redefining function application by ' = ( "{ }) solved these problems by eliminating the definite description. The new definition looks peculiar, but it agrees with the old one when the latter is defined. Its relational version is straightforward: Thus it follows that M is closed under function application, which is also absolute: Well-Founded Recursion The hardest absoluteness proofs concern recursion. Well-founded recursion is the most general form of recursive function definition. The proof that well-founded relations are absolute consists of several steps. Well-orderings, which are wellfounded linear orderings, are somewhat easier to prove absolute. Absoluteness of Well-orderings The concept of well-ordering is the first we encounter whose absoluteness proof is hard. One direction is easy: if relation well-orders , then it also well-orders relative to M. For if every nonempty subset of has an -minimal element, then trivially so does every nonempty subset of that belongs to M; this is Lemma IV 3.14 in Kunen [9, p. 123]. For proving the converse direction, Kunen (Theorem IV 5.4, page 127) reasons that "every well-ordering is isomorphic to an ordinal." We can obtain this result by showing that order types exist in M and are absolute. The proof requires some instances of separation and replacement for M. The theory defines various properties of relations, relative to a class M . Transitivity, linearity, and other simple properties have the obvious definitions and are easily demonstrated to be absolute. The definition of well-founded refers to the existence of r-minimal elements, as discussed above. "wellfounded on(M,A,r) == ∀ x[M]. x≠0 −→ x ⊆ A −→ (∃ y[M]. y ∈x &˜(∃ z[M]. z∈x & z,y ∈ r))" A well-ordering is a well-founded relation that is also linear and transitive. Kunen's lemma IV 3.14 takes the following form: "well ord(A,r) =⇒ wellordered(M,A,r)" The definition of order types is standard; see Theorem I 7.6 of Kunen [9, p. 17]. We use replacement to construct a function that maps elements of A to ordinals, proving that its domain is the whole of A and that each element of its range is an ordinal. Its range is the desired order type. But the construction must be done relative to M . In particular, when we need well-founded induction on r, we must apply a relativized induction rule: "[[a∈A; wellfounded on(M,A,r); M(A); separation(M, x. x∈A −→˜P(x)); ∀ x∈A. M(x) & (∀ y ∈A. y,x ∈ r −→ P(y)) −→ P(x)]] =⇒ P(a)" One premise is an instance of the separation axiom involving the negation of the induction formula. Each time we apply induction, we must assume another instance of separation. After about 250 lines of proof script, we arrive at Kunen's Theorem IV 5.4. The notion of well-ordering is absolute: "[[M(A); M(r)]] =⇒ wellordered(M,A,r) ←→ well ord(A,r)" Order types are absolute. That is, if f is an order-isomorphism from between (A,r) and some ordinal i, then i is the order type of (A,r). These results are not required in the sequel, but I found their proofs useful preparation for tackling the more general problem of well-founded recursion. Functions Defined by Well-founded Recursion Are Absolute It is essential to show that functions can be defined by well-founded recursion in M and that such functions are absolute. This is Kunen's theorem IV 5.6, page 129. Let be a well-founded relation. If is recursively defined over then ( ) is derived from and from various ( ) where ranges over the set of -predecessors of . This set is just −1 "{ }, the inverse image of { } under , more explicitly { \| , ∈ }. Writing the body of as ( , ), with free variables and , we get the recursion equation: ( ) = ( , ( −1 "{ }))(1) Note that ( −1 "{ }) denotes the function obtained by restricting topredecessors of . If and are given, then the existence of a suitable function follows by well-founded induction over , as I have described in previous work [15]. I have had to repeat some of these proofs relative to M . The theorems may assume only the relativized assumption wellfounded(M,r), which for the moment is weaker than wf(r). About 200 lines of proof script are necessary, but fortunately much of this material is based on earlier proofs. We reach a key result concerning the existence of recursive functions: The predicate is recfun(r,a,H,f) expresses that f satisfies the recursion equation (1) for the given relation r and body H for all r-predecessors of a. So the theorem states that if r is well-founded and transitive then there exists f in M satisfying the recursion equation below a. Obviously r and a must belong to the class M, which moreover must be closed under H. Two additional premises list instances of separation and replacement, which depend upon r and H. Before we can assume such instances, we must express them relative to M . That in turn requires a relativized version of is recfun: "M is recfun(M,MH,r,a,f) == ∀ z[M]. z ∈ f ←→ (∃ x[M]. ∃ y[M]. ∃ xa[M]. ∃ sx[M]. ∃ r sx[M]. ∃ f r sx[M]. pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) & pre image(M,r,sx,r sx) & restriction(M,f,r sx,f r sx) & xa ∈ r & MH(x, f r sx, y))" This definition is the translation of equation (1) into relational language. (Observe how quickly this language becomes unreadable.) In particular, the binary operator H becomes the ternary relation MH. The argument H makes is recfun a higher-order function, which complicates subsequent work. We cannot relativize is recfun once and for all, but if MH is expressed in relational language, then so is M is recfun. The predicate relation2 expresses that is f is the relational form of f over M : "relation2(M,is f,f) == ∀ x[M]. ∀ y[M]. ∀ z[M]. is f(x,y,z) ←→ z = f(x,y)" The predicate is wfrec expresses that z is computed from a and MH by wellfounded recursion over r. The body of the definition expresses the existence of a function f satisfying equation (1) , with z = H(a,f). "is wfrec(M,MH,r,a,z) == ∃ f[M]. M is recfun(M,MH,r,a,f) & MH(a,f,z)" We now reach two lemmas, stating that M is recfun and is wfrec behave as intended. The first result is absoluteness of is recfun. Among the premises are that M is closed under H and that MH is the relational form of H: Under identical premises, we get the corollary "is wfrec(M,MH,r,a,z) ←→ (∃ g [M]. is recfun(r,a,H,g) & z=H(a,g))" Making Well-founded Recursion Available Mathematically speaking, we have already proved the absoluteness of wellfounded recursion. Pragmatically speaking, unfortunately, more work must be done to package the results so that they can be used in formal proofs. In particular, we need a theorem relating the predicate is wfrec defined above with the function wfrec provided by Isabelle/ZF [15, §3.1]; wfrec(r,a,H) denotes the result of f(a), where f is the function with body H defined by recursion over r. The development of well-founded recursion assumes r to be transitive. To apply well-founded recursion to other relations requires a theory of transitive closure. Isabelle/ZF defines the transitive closure of a relation inductively [15, §2.5]. Inductive definitions are abstract and elegant, but they do not lend themselves to absoluteness proofs because they use the powerset operator. We must find an alternative definition, and an obvious one is based on the intuition ≺ * ⇐⇒ = 0 ≺ 1 ≺ · · · ≺ = . The sequence 0 , 1 , . . . , can be modelled as a finite function: as noted in Section 4.6, finite functions are absolute. From ≺ * it is trivial to define the transitive closure, ≺ + . In the definition below, f is the sequence and A is intended to represent the field of r: "rtrancl alt(A,r) == {p ∈ A * A. ∃ n∈nat. ∃ f ∈ succ(n) -> A. (∃ x y. p = x,y & f'0 = x & f'n = y) & (∀ i∈n. f'i, f'succ(i) ∈ r)}" It is easy to prove that this definition coincides with Isabelle/ZF's inductive one: "rtrancl alt(field(r),r) = rˆ" Since every concept used in the new definition is absolute, we merely have to relativize this definition to M, defining rtran closure mem(M,A,r,p) to hold when p is an element of rtrancl alt(A,r). I omit the definition because the relational language is unreadable. We cannot even use the constant 0 but must introduce a variable zero and constrain it by empty(M,zero). The next two predicates relativize the reflexive-transitive and transitive closure of a relation: Once we assume an instance of separation involving rtran closure mem, closure and absoluteness results follow directly: "M(r) =⇒ M(rtrancl(r))" "[[M(r); M(z)]] =⇒ rtran closure(M,r,z) ←→ z = rtrancl(r)" "M(r) =⇒ M(trancl(r))" "[[M(r); M(z)]] =⇒ tran closure(M,r,z) ←→ z = trancl(r)" If a relation is well-founded then so is its transitive closure. The following lemma use useful because at this point we do not know that wellfounded(M,r) is equivalent to wf(M,r). After about 130 lines of proof script, we arrive at some important theorems. One asserts absoluteness, relating the predicate is wfrec with the operator wfrec: The theorems fortunately require identical instances of replacement. Both theorems assume trans(r); omitted are more general theorems that relax the assumption of transitivity. Both theorems use the predicate wfrec replacement to express a necessary instance of replacement. Its arguments are the ternary predicate MH, which represents the body of the recursive function, and the well-founded relation r. 6 Defining First-Order Formulae and the Class L We pause from proving absoluteness results in order to consider our main objective, namely the class L and its properties. The most logical order of presentation might have been to develop L first and then to prove that constructibility is absolute. The order of presentation adopted here better represents how I actually carried out the proofs. Kunen similarly presents general absoluteness results before he introduces L. Internalized First-Order Formulae The idea of L is to introduce, at each stage, the sets that can be defined from existing ones by a first-order formula with parameters. Neither Gödel [5] nor Kunen actually use first-order formulae, preferring more abstract constructions that achieve the goal more easily. However, Isabelle/ZF's recursive datatype package automates the task of defining the set of first-order formulae and the satisfaction relation on them. Gödel's earlier proof [6] also uses first-order formulae. The obvious representation of first-order formulae is de Bruijn's [2], where there are no variable names. Instead, each variable reference is a non-negative integer, where zero refers to the innermost quantifier and larger numbers refer to enclosing quantifiers. If the integer is greater than the number of enclosing quantifiers, than it is a free variable. This representation eliminates the danger of name confusion. It is particularly useful for formulae with parameters, since their order is determined numerically rather than by name. datatype "formula" = Member ("x ∈ nat", "y ∈ nat") \| Equal ("x ∈ nat", "y ∈ nat") \| Nand ("p ∈ formula", "q ∈ formula") \| Forall ("p ∈ formula") Having only four cases simplifies the relativization of functions on formulae. All propositional connectives are expressed in terms of Nand. "Neg(p) == Nand(p,p)" "And(p,q) == Neg(Nand(p,q))" "Or(p,q) == Nand(Neg(p),Neg(q))" "Implies(p,q) == Nand(p,Neg(q))" "Iff(p,q) == And(Implies(p,q), Implies(q,p))" "Exists(p) == Neg(Forall(Neg(p)))" The Satisfaction Relation Satisfaction is a primitive recursive function on formulae. Thanks to the nameless representation, interpretations are simply lists rather than functions from variable names to values. The familiar list function nth, defined below, looks up variables in interpretations: "nth(0, Cons(a, l)) = a" "nth(succ(n), Cons(a,l)) = nth(n,l)" "nth(n, Nil) = 0" The second of these equations is subject to the condition n ∈ nat. Note that element zero is the head of the list. Another useful function is bool of o, which converts a truth value to an integer: "bool of o(P) == (if P then 1 else 0)" This conversion is necessary because Isabelle/ZF is based on first-order logic. Formulae are not values, so we encode them using integers. We thus define a recursive predicate as a recursive integer-valued function. We are now able to define the function satisfies, which takes a set (the domain of discourse), a formula and an interpretation (written env for environment). It returns 1 or 0, depending upon whether or not the formula evaluates to true or false: "satisfies(A,Member(x,y)) = ( env ∈ list(A). bool of o (nth(x,env) ∈ nth(y,env)))" "satisfies(A,Equal(x,y)) = ( env ∈ list(A). bool of o (nth(x,env) = nth(y,env)))" "satisfies(A,Nand(p,q)) = ( env ∈ list(A). not ((satisfies(A,p)'env) and (satisfies(A,q)'env)))" "satisfies(A,Forall(p)) = ( env ∈ list(A). bool of o (∀ x∈A. satisfies(A,p)'(Cons(x,env)) = 1))" The abstraction and explicit function applications involving environments are necessary because the environments can vary in the recursive calls. The last line of satisfies deserves attention. The universal formula Forall(p) evaluates to 1 just if p evaluates to 1 in every environment obtainable from env by adding an element of A. Such environments have the form Cons(x,env) for x∈A. The satisfaction predicate, sats, is a macro that refers to the function satisfies. translations "sats(A,p,env)" == "satisfies(A,p)'env = 1" The satisfaction predicate enjoys a number of properties that relate the internalized formulae to real formulae. All the equivalences are subject to the typing condition env ∈ list(A). For example, the membership and equality relations behave as they should: "sats(A, Member(x,y), env) ←→ nth(x,env) ∈ nth(y,env)" "sats(A, Equal(x,y), env) ←→ nth(x,env) = nth(y,env)" The propositional connectives also work: "sats(A, Neg(p), env) ←→˜sats(A,p,env)" "(sats(A, And(p,q), env)) ←→ sats(A,p,env) & sats(A,q,env)" "(sats(A, Or(p,q), env)) ←→ sats(A,p,env) \| sats(A,q,env)" Quantifiers work too. Notice how the environment is extended: "sats(A, Exists(p), env) ←→ (∃ x∈A. sats(A, p, Cons(x,env)))" The Arity of a Formula The arity of a formula is, intuitively, its set of free variables. In sats(A,p,env), if the arity of p does not exceed the length of env, then the environment supplies values to all of p's free variables. Take each de Bruijn reference, adjusted for the depth of quantifier nesting at that point; the arity is the maximum of the resulting values. The recursive definition of function arity is simpler than this description. "arity(Member(x,y)) = succ(x) ∪ succ(y)" "arity(Equal(x,y)) = succ(x) ∪ succ(y)" "arity(Nand(p,q)) = arity(p) ∪ arity(q)" "arity(Forall(p)) = Arith.pred(arity(p))" Note that ∪ = max{ , } in set theory and that Arith.pred denotes the predecessor function. Trivial corollaries of this definition tell us how to compute the arities of other connectives: "arity(Neg(p)) = arity(p)" "arity(And(p,q)) = arity(p) ∪ arity(q)" The following result is more interesting. Extra items in the environment (exceeding the arity) are ignored. Here @ is the list "append" operator, so env @ extra is env with additional items added. Our aim is to regard the conjunction ∧ as having the free variables , 1 , . . . , . The occurrences of in both formulae must be identified, while the parameter lists of the two formulae must be kept disjoint. To achieve our aim may require renaming one of the formula's free variables. The de Bruijn representation refers to variables by number rather than by name. The variables shown as above always have the de Bruijn index zero, so they will be identified automatically. We keep the parameter lists disjoint by renumbering the free variables in one of the formulae. Since must be left alone, we only renumber the variables having an index greater than zero. Renumbering functions are often necessary with the de Bruijn approach, though normally they rename variables during substitution. When efficiency matters, the renumbering functions take an argument specifying what number should be added to the variables. Here, the definitions are for reasoning about rather than for execution, so renaming for us means adding one; repeating this allows renaming by larger integers. In the following definitions, nq refers to the number of quantifiers enclosing the current point. Any de Bruijn index smaller than nq must not be renamed. The Renaming Function First, we need a one-line function that renames a de Bruijn variable: "incr var(x,nq) == if x<nq then x else succ(x)" Now we can define the main renaming function. As with satisfies above, abstraction and explicit function applications are necessary: the argument nq ("nesting of quantifiers") varies in the recursive calls. In the Member and Equal case, the variables are simply renamed. The Nand case recursively renames the subformulae using the same nesting depth, while the Forall case renames its subformula using an increased nesting depth. "incr bv(Member(x,y)) = ( nq ∈ nat. Member (incr var(x,nq), incr var(y,nq)))" "incr bv(Equal(x,y)) = ( nq ∈ nat. Equal (incr var(x,nq), incr var(y,nq)))" "incr bv(Nand(p,q)) = ( nq ∈ nat. Nand (incr bv(p)'nq, incr bv(q)'nq))" "incr bv(Forall(p)) = ( nq ∈ nat. Forall (incr bv(p) ' succ(nq)))" Recall the example at the start of this section, concerning a set defined by the conjunction ∧ . If we are to conjoin the formulae and and combine their sets of parameters, then we need to ensure that some of the parameters are only visible to and the rest are only visible to . The following lemma makes this possible: and thus the renaming allows an additional value to be put into the environment at position m. The renamed formula will ignore the new value. By repeated renaming, we can construct a formula that will ignore a section of the parameter list that is intended for another formula. The next result describes the obvious relationship between arity and renaming. Renaming increases a formula's arity by one, unless the variable being renamed does not exist, when renaming has no effect. "[[p ∈ formula; n ∈ nat]] =⇒ arity (incr bv(p) ' n) = (if n < arity(p) then succ(arity(p)) else arity(p))" Considering how trivial the notion of arity is, many proofs about it (including this one) are complicated by innumerable case splits. Getting the simplifier to prove most of them automatically requires some ingenuity. Many other tiresome proofs about arities are omitted. Renaming all but the first bound variable One more thing is needed before we can define sets using conjunctions. As discussed at the beginning of Sect. 6.4, when a formula defines a set, the variable with de Bruijn index zero gives the extension of that set, while the remaining free variables serve as parameters. Therefore, our basic renaming operator must only rename variables having a de Bruijn index of one or more: "incr bv1(p) == incr bv(p)'1" Finally we reach a lemma justifying our intended use of renaming. If the environment has an initial segment bvs of length n and if we apply the incr bv1 n times, then the modified formula ignores the bvs part. But the renamed and original formulae agree on the first element of the environment, shown above as x. The Definable Powerset Operation The definable powerset operator is called DPow: "DPow(A) == {X ∈ Pow(A). ∃ env ∈ list(A). ∃ p ∈ formula. arity(p) ≤ succ(length(env)) & X = {x∈A. sats(A, p, Cons(x,env))}}" A set X belongs to DPow(A) provided there is an environment env (a list of values drawn from A) and a formula p. The constraint arity(p) ≤ succ(length(env)) indicates that the environment should interpret all but one of p's free variables. The variable whose de Bruijn index is zero determines the extension of X via the satisfaction relation: sats(A, p, Cons(x,env)). You may want to compare this with the informal discussion in the previous section, or with Definition VI 1.1 of Kunen [9, p. 165]. Some consequences of this definition are easy to prove. The empty set is defined by the predicate . ≠ , and singleton sets by . = . "0 ∈ DPow(A)" "a ∈ A =⇒ {a} ∈ DPow(A)" The complement of a set X is defined by negating the formula used to define X. Intersection is done by conjoining the defining formulae, using the renaming techniques developed in the previous section. Union is then trivial by de Morgan's laws. "X ∈ DPow(A) =⇒ (A-X) ∈ DPow(A)" "[[X ∈ DPow(A); Y ∈ DPow(A)]] =⇒ X Int Y ∈ DPow(A)" "[[X ∈ DPow(A); Y ∈ DPow(A)]] =⇒ X Un Y ∈ DPow(A)" And thus DPow coincides with Pow (the real powerset operator) for finite sets: Proving that the Ordinals are Definable In order to show that DPow is closed under other operations, we must be able to code their defining formulae as elements of the set formula. The treatment of the subset relation is typical. We begin by encoding the formula ∀ . ∈ → ∈ . Below, x and y are de Bruijn indices, which are incremented to succ(x) and succ(x) because the quantifier introduces a new variable binding. "subset fm(x,y) == Forall(Implies(Member(0,succ(x)), Member(0,succ(y))))" The arguments are just de Bruijn indices because internalized formulae have no terms other than variables. It is trivial to prove that subset fm maps a pair of de Bruijn indices to a formula: "[[x ∈ nat; y ∈ nat]] =⇒ subset fm(x,y) ∈ formula" The arity of the formula is the maximum of those of its operands: "[[x ∈ nat; y ∈ nat]] =⇒ arity(subset fm(x,y)) = succ(x) ∪ succ(y)" The following equivalence involves absoluteness, since it relates subset fm to the real subset relation, ⊆. To reach this conclusion requires the additional assumption Transset(A), saying that A is a transitive set. The premise x < length(env) puts a bound on x (which is a de Bruijn index), ensuring that nth(x,env) belongs to A. We must repeat this exercise (details omitted) for the concepts of transitive set and ordinal. This lets us prove that ordinals are definable, leading to a result involving ordinals and DPow. This lemma ultimately leads to a proof that L contains all the ordinals. Defining L, The Constructible Universe The constant Lset formalizes the family of sets { } ∈ON . Its definition in Isabelle/ZF uses a standard operator for transfinite recursion. We also define L = ∈ON : "Lset(i) == transrec(i, %x f. y ∈x. DPow(f'y))" "L(x) == ∃ i. Ord(i) & x ∈ Lset(i)" Some effort is required before we can transform the cryptic definition of Lset into the usual recursion equations. First, we prove Kunen's [9, p. 167] lemma VI 1.6, which states the transitivity and monotonicity of the : "Transset(A) =⇒ Transset(DPow(A))" "Transset(Lset(i))" "i≤j −→ Lset(i) ⊆ Lset(j)" Then we reach the 0, successor and limit equations for the : "Lset(0) = 0" "Lset(succ(i)) = DPow(Lset(i))" "Limit(i) =⇒ Lset(i) = ( y ∈i. Lset(y))" The basic properties of L, as presented in Kunen's IV §1, are not hard to prove. For example, L contains the ordinals: "Ord(i) =⇒ i ∈ Lset(succ(i))" "Ord(i) =⇒ L(i)" Eliminating the Arity Function The function arity can be surprisingly hard to reason about, particularly when we try to encode higher-order operators. Once we have established the basic properties of L, we can prove its equivalence to a new definition that does not involve arities. Here is another form of definable powerset: "DPow'(A) == {X ∈ Pow(A). ∃ env ∈ list(A). ∃ p ∈ formula. X = {x∈A. sats(A, p, Cons(x,env))}}" This version omits the constraint arity(p) ≤ succ(length(env)) but is otherwise identical to DPow. The point is that if the environment is too short, attempted variable lookups will yield zero; recall the properties of nth from Sect. 6.2. If the set A is transitive, then it contains zero as an element. So the too-short environment can be padded to the right with zeroes. "Transset(A) =⇒ DPow(A) = DPow'(A)" Each Lset(i) is a transitive set, so they can be expressed using DPow' rather than DPow: "Lset(i) = transrec(i, %x f. y ∈x. DPow' (f ' y))" The equation above, proved by transfinite induction, lets us relativize Lset without having to formalize the functions arity and length. That eliminates a lot of work. The following lemma is helpful for proving instances of separation. The first, quantified, premise asks for an equivalence between the real formula P and the internalized formula p. Often we can derive p from P automatically by supplying a set of suitable inference rules. "[[∀ x∈Lset(i). P(x) ←→ sats(Lset(i), p, Cons(x,env)); env ∈ list(Lset(i)); p ∈ formula]] =⇒ {x∈Lset(i). P(x)} ∈ DPow(Lset(i))" Also, the lemma makes no reference to arity, thanks to the equivalence between DPow' and DPow. 6.9 The Zermelo-Fraenkel Axioms Hold in L Following Kunen VI §2, it is possible to prove that L satisfies the Zermelo-Fraenkel axioms. Separation is the most difficult case and is considered later. Basic Properties of L We begin with simple closure properties. Many of them involve exhibiting an element of formula describing the required set. We typically begin by starting in Lset(i) and proving that the required set belongs to Lset(succ(i)). L is closed under unions: "X ∈ Lset(i) =⇒ Union(X) ∈ Lset(succ(i))" "L(X) =⇒ L(Union(X))" L is closed under unordered pairs. More work is necessary because the sets a and b may be introduced at different ordinals: "a ∈ Lset(i) =⇒ {a} ∈ Lset(succ(i))" "[[a ∈ Lset(i); b ∈ Lset(i)]] =⇒ {a,b} ∈ Lset(succ(i))" "[[a ∈ Lset(i); b ∈ Lset(i); Limit(i)]] =⇒ {a,b} ∈ Lset(i)" "[[L(a); L(b)]] =⇒ L({a, b})" Also, is closed under ordered pairs provided is a limit ordinal. This result is needed in order to apply the reflection theorem to L. Specifically, it is needed because my version of the reflection theorem [18] uses ordered pairs to cope with the possibility of a formula having any number of free variables. "[[a ∈ Lset(i); b ∈ Lset(i); Ord(i)]] =⇒ a,b ∈ Lset(succ(succ(i)))" "[[a ∈ Lset(i); b ∈ Lset(i); Limit(i)]] =⇒ a,b ∈ Lset(i)" A Rank Function for L Some proofs require the L-rank operator. Kunen (VI 1.7) defines ( ) to denote the least such that ∈ +1 : "lrank(x) == i. x ∈ Lset(succ(i))" Here is one consequence of this definition: "Ord(i) =⇒ x ∈ Lset(i) ←→ L(x) & lrank(x) i" A more important result, whose proof involves lrank, states that every set of constructible sets is included in some Lset: This theorem is useful in proving that L satisfies the separation axiom. However, note that ⊆ L does not imply ∈ L, not even if is a set of natural numbers. The lrank operator is useful for proving that L satisfies the powerset axiom: "L(X) =⇒ L({y ∈ Pow(X). L(y)})" Note that the powerset of X in L comprises all subsets of X that belong to L. It is potentially a superset of DPow(X). The lrank operator also assists in the proof that L satisfies the replacement axiom. The idea is to use replacement on the ranks of the members of L: "[[L(X); univalent(L,X,Q)]] =⇒ ∃ Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) ⊆ Y" The proof of replacement is schematic, and therefore independent of the formula Q. But it is the weak form of replacement. It concludes that the range of Q (viewed as a class function) is included in some constructible set Y. Strong replacement, which is the version we really want, asserts that the range itself is constructible. Each instance of strong replacement requires proving an instance of the axiom of separation. Instantiating the Locale M trivial Now we are ready to show that L satisfies all the properties we assumed of the class M, which we used to develop the general theory of absoluteness. The class L is transitive: "[[y ∈x; L(x)]] =⇒ L(y)" The facts about L proved above can be summarized using the relativized forms of the ZF axioms: "Union ax(L)" "upair ax(L)" "power ax(L)" "replacement(L,P)" We do not need L to satisfy the foundation axiom. However, this fact is a trivial consequence of the foundation axiom: "foundation ax(L)" The theorems above are all we need to prove "PROP M trivial(L)". This theorem makes all the results proved in locale "M trivial" available as theorems about L. In particular, the absoluteness and closure results listed in Sect. 4.5 above apply to L. Comprehension in L It remains to show that L satisfies the axiom of separation. The proof requires the reflection theorem. As described elsewhere [18], my formalization of that theorem applies to any class M = ∈ON , where the family { } ∈ON is increasing and continuous. An additional condition is that if is a limit ordinal then must be closed under ordered pairing. Isabelle's locale mechanism captures these requirements, and we can now instantiate the locale with the class L = ∈ON . However, making it ready for practical use requires additional work. The Reflection Relation The reflection theorem states that if ( 1 , . . . , ) is a formula in variables then there exists a closed and unbounded class C such that for all ∈ C and 1 ,. . . , In fact, we only need the weaker conclusion that C is unbounded, which enables us to find a suitable > given any ordinal . Applying the reflection theorem yields an Isabelle formula describing the class C. These formulae may be interesting in the case of small examples [18], but in typical applications they are huge. The trivial proofs, which merely refer to other instances of reflection, take minutes of computer time; the resulting theorems amount to pages of text. The obvious solution is to express the reflection theorem using an existential quantifier, but classes cannot be quantified over: they are formulae. Fortunately, Isabelle makes a distinction between the object-logic (here firstorder logic) and the metalogic (a fragment of higher-order logic) [12]. I was able to formalize a metaexistential quantifier. It lies outside of first-order logic -in particular, Isabelle will reject any attempt to use it in comprehensions. However, it can be used in top-level assertions, which is all we need. We can now define the reflection relation between two formulae P and Q: Q(a,x))))" "REFLECTS[P,Q] == (??C. Closed Unbounded(C) & (∀ a. C(a) −→ (∀ x ∈ Lset(a). P(x) ←→ It relates the formulae just if there exists a class C satisfying the conclusion of the reflection theorem [18]. That is, C is a closed, unbounded class of ordinals such that P and Q agree on . The existential quantifier, ??C, hides the prohibitively large formula describing this class. The following lemma illustrates the use of the reflection relation. Note that the quantification over classes has disappeared. "[[REFLECTS[P,Q]; Ord(i)]] =⇒ ∃ j. i<j & (∀ x ∈ Lset(j). P(x) ←→ Q(j,x))" If REFLECTS[P,Q] and i is an ordinal then there exists a larger ordinal j for which P and Q agree. Our choice of i can make j arbitrarily large. The general form of the reflection theorem uses the relativization operator, which cannot be expressed in Isabelle/ZF. However, given a specific formula , we can generate an instance of the reflection theorem relating L and . Here is the base case, where normally P should have the form ∈ or = : In the conclusion, a quantification over L is related to one over , as suggested by the general form of the reflection theorem. The premise uses the projection operators for ordered pairs to introduce the new variable, z; syntactically, x. P(fst(x),snd(x)) is a unary formula. Internalized Formulae for Some Set-Theoretic Concepts Every operator or concept that is used in an instance of the axiom of separation must be internalized. If the defining formula is complicated, then writing the corresponding element of formula requires a manual (and error-prone) translation into de Bruijn notation. The Isabelle/ZF development of constructibility theory contains about 100 such encodings. A typical example resembles that shown in Sect. 6.6 above for subset fm. First to be internalized are elementary concepts such as the empty set, unordered and ordered pairs, unions, intersections, domain and range. The union predicate was defined in Sect. 4.1 as ∀ . ∈ ↔ ∈ ∨ ∈ . In the corresponding formula, the variables x, y and z range over de Bruijn indices. "union fm(x,y,z) == Forall(Iff(Member(0,succ(z)), Or(Member(0,succ(x)), Member(0,succ(y)))))" As for subset fm above, we can prove that union fm yields an element of the set formula. The theorem about satisfaction now takes the following form: "[[x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)]] =⇒ sats(A, union fm(x,y,z), env) ←→ union( * A, nth(x,env), nth(y,env), nth(z,env))" Here, A is Isabelle syntax for the class given by the set A, that is, { \| ∈ }. The theorem above simply expresses the equivalence between the relational formula union and union fm, which is its translation into an element of set formula. Such equivalences are usually trivial: they simply relate two syntaxes for formulae. They do not express the equivalence between union fm and ∪, which would be an instance of absoluteness. After internalizing a predicate such as union, it makes sense to prove its instance of the reflection theorem too, since both results will be needed when proving instances of separation. Most reflection proofs are trivial two-line scripts: 1. Unfold the concept's definition (here union). Repeatedly apply existing reflection theorems. Each predicate is internalized similarly. Parts of the declarations and proofs can be copied from those of another predicate. However, getting the definition right requires careful attention to the original first-order definition. Higher-Order Syntax Higher-order syntax is ubiquitous in naive set theory.In the union ∈ ( ), the higher-order variable represents an indexed family of sets. In the function abstraction ∈ ( ), the higher-order variable represents the function's body. Isabelle/ZF additionally uses higher-order syntax to express many forms of recursion, and so forth. Although this syntax is indispensable, it is also illegitimate: formal set theory has no non-trivial terms, let alone higher-order ones. We must formalize the conventions governing higher-order syntax into the language of set theory. Converting a higher-order operator such as x∈A. b(x) into relational form yields a higher-order predicate. Among its arguments is a predicate is b that expresses the function body, b, in relational form. If is b is purely relational, then so is the definiens of is lambda. "is lambda(M, A, is b, z) == ∀ p[M]. p ∈ z ←→ (∃ u[M]. ∃ v[M]. u∈A & pair(M,u,v,p) & is b(u,v))" This definition states that z is a -abstraction provided its elements are ordered pairs that satisfy is b and whose first component belongs to A. The following predicate expresses that is f represents the relational version of f for arguments ranging over A: "Relation1(M,A,is f,f) == ∀ x[M]. ∀ y[M]. x∈A −→ is f(x,y) ←→ y = f(x)" This abbreviation, and similarly Relation2, etc., are useful for expressing absoluteness results. If is b is the relational equivalent of b, and if the class M contains each b(m) for m∈A, then is lambda(M,A,is b,z) is the relational version of x∈A. b(x). And thus -abstraction is absolute: "[[Relation1(M,A,is b,b); M(A); ∀ m[M]. m∈A −→ M(b(m)); M(z)]] =⇒ is lambda(M,A,is b,z) ←→ (z = x∈A. b(x))" Showing that M is closed under -abstraction requires a separate instance of strong replacement for each b. Internalizing is lambda is not completely straightforward. The predicate argument, is b, becomes a variable ranging over the set formula. "lambda fm(p,A,z) == Forall(Iff(Member(0,succ(z)), Exists(Exists(And (Member(1,A#+3), And(pair fm(1,0,2), p))))))" Given a formula and two de Bruijn indices, lambda fm yields another formula: "[[p ∈ formula; x ∈ nat; y ∈ nat]] =⇒ lambda fm(p,x,y) ∈ formula" But there is no binding mechanism for expressing predicates that take arguments or refer to local variables. The formula p must refer to its first argument using the de Bruijn index 1 and to its second using the index 0 (both to be increased in the usual way if p contains quantifiers). If we are lucky, then we can arrange matters such that the actual arguments have the right indices, and otherwise we can force the indices to agree by introducing quantifiers and equalities: in the internalization of ∀ . ∀ . = ∧ = → , the variable with de Bruijn index 1 will refer to and similarly the index 0 will refer to . If p contains free references to other variables, their de Bruijn indices must be increased by 3 because p is inserted into a context enclosed by three quantifiers. The satisfaction theorem for is lambda formalizes the remarks above: lemma sats lambda fm: assumes is b iff sats ∈ "!!a0 a1 a2. [[a0∈A; a1∈A; a2∈A]] =⇒ is b(a1,a0) ←→ sats(A, p, Cons(a0,Cons(a1,Cons(a2,env))))" shows "[[x ∈ nat; y ∈ nat; env ∈ list(A)]] =⇒ sats(A, lambda fm(p,x,y), env) ←→ is lambda( A, nth(x,env), is b, nth(y,env))" The assumes-shows syntax eases the use of the complicated assumption, which states that is b agrees with p for the fixed environment env extended with three additional elements of A. I have not been able to simplify the form of this theorem while retaining its generality. It gets more complicated when one higher-order operator refers to another. One such operator has a quantifier nesting depth of 12. When an operator uses its higher-order argument more than once, we must ensure that the two contexts are similar, adding quantifiers if necessary to make the nesting depths agree. Instances of the reflection theorem for higher-order operators must take into account the possibility of the higher-order argument's referring to local variables. Although is lambda expects is b to have only two arguments, below we formalize it with three arguments (plus its class argument). The extra argument is bound by the REFLECTS operator, allowing direct reference to elements of L or Lset(i). theorem is lambda reflection: assumes is b reflection: "!!f g h. REFLECTS[ x. is b(L, f(x), g(x), h(x)), i x. is b( Lset(i), f(x), g(x), h(x))]" shows "REFLECTS[ x. is lambda(L, A(x), is b(L,x), f(x)), i x. is lambda( Lset(i), A(x), is b( Lset(i),x), f(x))]" The arity of a higher-order function naturally depends upon that its function argument. I found the properties so unintuitive and their proofs so vexing that I undertook the work described in Sect. sec:no-arity, which eliminates the need for theorems concerning arities. Proving Instances of Separation The set comprehension { ∈ \| ( )} comes from the separation axiom scheme instantiated to the formula . The axiom of replacement yields a set that may be bigger than we want, again requiring an appeal to separation. Because I have not formalized the metatheory, the Isabelle/ZF development cannot express the proof that the separation scheme holds for L. Each instance has to be proved individually. Fortunately, the proof scripts are nearly identical. Given , the first step is to prove instance of the reflection theorem for that formula. The next step is to run a proof script corresponding to the sketch in Kunen [9, p. 169]. The formula will of course be expressed using the relational language, using predicates such as union. Executing the proof script will automatically generate an internalized formula, with union fm in the corresponding place. The lemmas outlined on the preceding pages suffice to prove many instances of separation. Consider the instance that justifies the existence of the intersection Inter(A). We must first prove the corresponding instance of the reflection theorem: "REFLECTS[ x. ∀ y[L]. y ∈A −→ x ∈ y, i x. ∀ y ∈Lset(i). y ∈A −→ x ∈ y]" Such instances are written manually. A text editor can replace quantification over L by quantification over in the second formula. The proof, almost always, is a one-line appeal to previous reflection theorems. The statement of each instance of separation comes from the corresponding locale assumption. The locale refers to an arbitrary class M, so we must replace M by L. The proof scripts are typically three lines long and follow a regular pattern. Note that any parameters used in the separation formula (here ) must be elements of L. The following instance of separation justifies relational composition. I leave the corresponding instance of reflection to your imagination. After proving ten or so instances of separation, we arrive at a cryptic theorem: "PROP M basic(L)" This asserts that L satisfies the conditions of the locale M basic, namely all the instances of separation needed to derive well-founded recursion. The absoluteness and closure results proved in that locale (described in Sect. 4.6) -now become applicable to L. Automatic Internalization of Formulae Isabelle's ability to translate formulae written in the relational language into members of formula simplifies the proofs of separation. Here is an example, from the proof of the instance shown above (about relational composition). The first proof step applies a lemma for proving instances of separations. It yields a subgoal that has the assumptions r ∈ Lset(j) and s ∈ Lset(j), where j is arbitrary. We have to prove that the comprehension belongs to the next level of the constructible hierarchy, namely DPow(Lset(j)): {xz ∈ Lset(j) . ∃ x∈Lset(j). ∃ y ∈Lset(j). ...} ∈DPow(Lset(j)) The second proof step applies a lemma for proving membership in DPow(Lset(j)). It yields three subgoals (Fig. 2). The first is to show the equivalence between the real formula (∃ xa∈Lset(j). ∃ y ∈Lset(j). ...) and sats(Lset(j), ?p3(j), [x,r,s]). This is the satisfaction relation applied to ?p3(j), a "logical variable" that can be replaced by any expression, possibly involving the bound variable j. The third subgoal in Fig. 2, namely ?p3(j) ∈ formula, checks that the chosen expression is an internalized formula. The second subgoal verifies that the environment, [r,s], is well-typed -namely, that it belongs to list(Lset(j)). The third proof step is this: apply (rule sep rules \| simp)+ It applies some theorem of sep rules, then simplifies, then repeats if possible. This finishes the proof. All separation proofs have this form, save only that sometimes sep rules needs to be augmented with additional theorems. Formula synthesis works in a way familiar to all Prolog programmers. Essentially, the theorems in sep rules comprise a Prolog program for generating internalized formulae. Most of the "program clauses" relate real formulae to internal ones and are derived from the basic properties of the satisfaction relation. For example, this one relates the real conjunction P&Q with the term And(p,q). The first two subgoals concern the synthesis of p and q. The third subgoal expresses a type constraint on env. 1. j x. [[L(r); L(s); r ∈ Lset(j); s ∈ Lset(j); x ∈ Lset(j)]] =⇒ (∃ xa∈Lset(j). ∃ y ∈Lset(j). ∃ z∈Lset(j). pair( Lset(j), xa, z, x) ∧ (∃ xy ∈Lset(j). pair( Lset(j), xa, y, xy) ∧ (∃ yz∈Lset(j). pair( Lset(j), y, z, yz) ∧ xy ∈ s ∧ yz ∈ r))) ←→ sats(Lset The environment, which initially contains the parameters of the separation formula, gets longer with each nested quantifier. Each higher-order operator can add several elements to the environment; as mentioned above in Sect. 7.3. A base case of synthesis relates the formula x∈y with the term Member(i,j). The first two subgoals concern the synthesis of the de Bruijn indices i and j: Other base cases concern predicates of the relational language. This theorem, which relates the formula union( A,x,y,z) with the term union fm(i,j,k), is just a reworking of a theorem shown in Sect. 7.2 above. "[[nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)]] =⇒ union( A, x, y, z) ←→ sats(A, union fm(i,j,k), env)" Given the subgoal nth(?i,env) = x, Isabelle can synthesize ?i. This de Bruijn index is determined by x, which comes from the original formula, and env, which is given in advance. If x matches the head of the environment, then ?i should be zero: "nth(0, Cons(a, l)) = a" And if it does not match, then we should discard the head and attempt to synthesize a de Bruijn index using the tail: "[[nth(n,l) = x; n ∈ nat]] =⇒ nth(succ(n), Cons(a,l)) = x" The automatic synthesis of internalized formulae saves much work in proofs of separation. In principle, we could rewrite every relational formula into its primitive constituents of membership and equality, removing the need for union fm and 100 similar constants. But if too few internalized primitives have been defined, formula synthesis takes many minutes. Absoluteness of Recursive Datatypes The Isabelle/ZF proofs discussed up to now include the construction of the class L and the proof that it is a model of the Zermelo-Fraenkel axioms. The next step is to show that L satisfies V = L. That fact follows by the absoluteness of constructibility, which follows by the absoluteness of satisfaction. Consulting the definition of satisfies reveals that we must still prove the absoluteness of lists, formulae, the function nth, and several other notions. Isabelle/ZF defines the sets list(A) and formula automatically from their user-supplied descriptions [17]. These fixedpoint definitions have advantages, but their use of the powerset operator is an obstacle to proving absoluteness. For a start, Pow(D) must be eliminated from this definition: We can prove that the least fixedpoint of a monotonic, continuous function h can be expressed as the union of the finite iterations of h. This equation not only eliminates Pow(D), but every occurrence of D, which is the "bounding set" [15, §2.2] and is itself typically defined using powersets. In order to apply this equation, we must prove that standard datatype constructions preserve continuity. The case bases are that the constant function and the identity function are continuous: "contin( X. A)" "contin( X. X)" Sums and products preserve continuity: These four lemmas cover all finitely-branching datatypes, including lists and formulae. Absoluteness for Function Iteration In the equation above for least fixed points, the term hˆn(0) abbreviates iterates(h,n,0). Isabelle/ZF defines iterates(F,n,x) by the obvious primitive recursion on n∈nat. Absoluteness of datatype definitions will follow from the absoluteness of iterates. Recall that a well-founded function definition consists of a relation and function body ; recall equation (1) of Sect. 5.2. Relativizing such a function definition, requires relativizing by an Isabelle/ZF relation, say MH. So to relativize iterates, we declare is iterates in terms of another predicate iterates MH, representing the body of the recursion. Incidentally, is nat case(M,a,isb,n,z) expresses case analysis on the natural number n. Note that we again work in the general setting of a class M satisfying certain conditions. Later, we shall prove that L meets those conditions. The absoluteness theorem for well-founded recursion requires an instance of strong replacement for each function being defined. But iterates is a higherorder function, so technically iterates(F,n,x) involves a separate instance of well-founded recursion for each F. The function iterates replacement can express each required instance of replacement; its argument isF is the relational form of F. Assuming such an instance of replacement, and given that isF is the relational version of F, the absoluteness of iterates is a corollary of the general theorem about well-founded recursion. We similarly find that M is closed under function iteration. Absoluteness for Lists and Formulae The formal treatment of continuity and iterates enables us to prove that lists and formulae are absolute. The definition of lists generated by the Isabelle/ZF datatype [17] is too complicated to relativize easily. Instead, we prove its equivalence to a more abstract (and familiar) definition. The function given to lfp continuous by construction, which lets us replace the the least fixed point by iteration and eliminate the non-absolute set univ(A): "contin( X. {0} + A * X)" "list(A) = ( n∈nat. ( X. {0} + A * X)ˆn (0))" Now the absoluteness of list(A) is obvious. But each element of this equation must be formalized in order to prove absoluteness. We begin by introducing an abbreviation for finite iterations of X. {0} + A * X -that is, for finite stages of the list construction. Formulae are proved absolute in just the same way. We express the set formula as an abstract least fixed point of a suitable function, prove that function to be continuous, and eliminate the lfp operator: "formula = lfp(univ(0), X. ((nat * nat) + (nat * nat)) + (X * X + X))" "contin( X. ((nat * nat) + (nat * nat)) + (X * X + X))" "formula = ( n∈nat. ( X. ((nat * nat) + (nat * nat)) + (X * X + X))ˆn (0))" Proceeding as for lists, we define the predicates is formula functor, is formula N, mem formula and finally is formula. We obtain the desired theorems: "M(formula)" "M(Z) =⇒ is formula(M,Z) ←→ Z = formula" Recursion over Lists and Formulae We have already seen (Sect. 5) that functions defined by well-founded recursion are absolute. For mathematicians, that is enough to justify the absoluteness of functions defined recursively on lists or formulae. Proof tool users, however, must work through the details for each instance. Usually automation makes it easy to apply general results to particular circumstances. However, the Isabelle/ZF translation of recursive function definitions is rather complicated. 3 There are good reasons for this complexity, such as support for a form of polymorphism. However, it makes the absoluteness proofs more difficult: the complications have to be taken apart and relativized one by one. At least there is no need to treat recursion over lists. Defining the class L involves only one list function, namely nth. Given a natural number and a list , this function returns the th element of , counting from 0. Obviously this amounts to taking the tail of the list times and returning the head of the result. The recursion in nth is an instance of iterates. Isabelle/ZF defines the head and tail functions hd and tl. The absoluteness proofs use modified versions called hd' and tl', which extend hd and tl to return 0 if their argument is ill-formed (the details are unimportant). Relativization is simpler when a function's behaviour is fully specified. Now we can prove an equivalence for nth: "[[xs ∈ list(A); n ∈ nat]] =⇒ nth(n,xs) = hd' (tl'ˆn (xs))" Its relational equivalent, is nth, has an obvious definition in terms of the relational equivalents of iterates, tl and hd: Recursion over lists is absolute in general. Proving this claim would require much work, and is unnecessary for proving that V = L is absolute. The function satisfies involves recursion over the datatype of formulae, and its absoluteness proof consists of several stages. Isabelle/ZF expresses recursion on datatypes in terms of ∈-recursion, which is recursion on a set's rank [15, §3.4]. Absoluteness for ∈-recursion will follow from that of well-founded recursion once we have established the absoluteness of ∈-closure. Then we shall be in a position to consider recursion over formulae. Five instances of strong replacement are necessary for the proofs sketched above. There are two each for the absoluteness of list(A) and formula, and one for the absoluteness of nth(n,l). The locale M datatypes encapsulates these additional constraints on the class M . It is one of several locales used to keep track of instances of separation and replacement in this development. Absoluteness for ∈-Closure If is a set, then its ∈-closure is the smallest transitive set that includes . Formally, the ∈-closure of is ∈ ( ). Here ( ) denotes the -fold union of , defined by 0 ( ) = and +1 ( ) = ( ( )). This is just another instance of iterates, as we can prove: Absoluteness for transrec The Isabelle/ZF operator transrec expresses ∈-recursion, which includes transfinite recursion as a special case: transrec( , ) = ( , ∈ .transrec( , )). Its definition is a straightforward combination of the operators eclose, wfrec (which expresses well-founded recursion), and Memrel (which encodes the membership relation as a set). Thus the definition of the relational version, is transrec, is also straightforward. Our previous results lead directly to a proof of absoluteness: In these theorems, transrec replacement abbreviates a specific use of wfrec replacement, which justifies this particular recursive definition (recall Sect. 5.3). Recursion over Formulae The Isabelle/ZF treatment of recursive functions on datatypes involves nonabsolute concepts, namely the cumulative hierarchy { } ∈ON and the rank function [15, §3.6]. For proving absoluteness, I proved an equation stating that recursion over formulae could be expressed differently. The new formulation refers to the depth of a formula, defined by "depth(Member(x,y)) = 0" "depth(Equal(x,y)) = 0" "depth(Nand(p,q)) = succ(depth(p) ∪ depth(q))" "depth(Forall(p)) = succ(depth(p))" Introducing depth seems to be a step backwards, since it requires relativizing another recursive function on formulae. But we can express the depth of a formula in terms of is formula N, which we need anyway (Sect. 8.2); is formula N(M,n,F) holds just if F is the set of formulae generated by n unfoldings of the datatype definition -which is all formulae of depth less than n. Working from this definition, we find that the depth of a formula is absolute: "[[p ∈ formula; n ∈ nat]] =⇒ is depth(M,p,n) ←→ n = depth(p)" For relativization, I modified the standard Isabelle/ZF treatment of recursion over formulae, replacing the set by formula and the rank of a set by the depth of a formula. If f is a recursive function on formulae, then the evaluation of f(p) begins by determining the depth of p, say . Then the recursion equation for f is unfolded + 1 times, using transfinite recursion. The resulting nonrecursive function is finally applied to p. This approach unfortunately needs an explicitabstraction over formulae and another instance of the replacement axiom. With the benefit of hindsight, I might have saved much work by seeking simpler ways of expressing recursion over formulae, such as by well-founded recursion on the subformula relation. The recursive definition of a function f is specified by four parameters a, b, c and d, corresponding to the four desired recursion equations: f(Member(x,y)) = a(x,y) f(Equal(x,y)) = b(x,y) f(Nand(p,q)) = c(p,f(p),q,f(q)) f(Forall(p)) = d(p,f(p)) Given the datatype definition of formula, Isabelle/ZF automatically defines the operator formula rec for expressing recursive functions. The term formula rec(a,b,c,d,p) denotes the value of the function f above applied to the argument p. More concisely, formula rec(a,b,c,d) denotes the the function f itself. The details of the definitions are illustrated elsewhere, using the example of lists [15, §4.3]. In order to express the recursion theorem, it helps to have first defined an abbreviation for its case analysis on formulae. "formula rec case(a,b,c,d,h) == formula case (a, b, u v. c(u, v, h ' succ(depth(u)) ' u, h ' succ(depth(v)) ' v), u. d(u, h ' succ(depth(u)) ' u))" Now we can express recursion on formulae in terms of absolute concepts: "p ∈ formula =⇒ formula rec(a,b,c,d,p) = transrec (succ(depth(p)), x h. Lambda(formula, formula rec case(a,b,c,d,h))) ' p" The proof is by structural induction on p. Note that the argument h of formula rec case is a partially unfolded recursive function taking two curried arguments. The second argument is some subformula u and the first is succ(depth(u)). The intuition behind this theorem may be obscure, but that is no obstacle to proving absoluteness. Many routine details must be taken care of, including relativization and absoluteness for the formula constructors Member, Equal, Nand and Forall and for the operator formula case. Obviously formula rec is a higher-order function. Its absoluteness proof depends upon absoluteness assumptions for the function arguments a, b, c and d. Its relational version needs those arguments to be expressed in relational form as predicates is a, is b, is c and is d. The absoluteness theorem depends upon 10 assumptions in all: two for each of is a, is b, is c and is d and two instances of replacement. After many intricate but uninteresting details, we arrive at two key theorems. If the class M is closed under the parameters a, b, c and d then it is closed under the corresponding recursion: Recursion over formulae is absolute: "[[p ∈ formula; M(z)]] =⇒ is formula rec(M,MH,p,z) ←→ z = formula rec(a,b,c,d,p)" In this theorem, MH abbreviates the relativization of the argument of transrec shown above: "MH(u::i,f,z) == ∀ fml[M]. is formula(M,fml) −→ is lambda (M, fml, is formula case (M, is a, is b, is c(f), is d(f)), z)" 9 Absoluteness for L In order to prove V = L, we must prove the absoluteness of three main functions: 1. satisfies, the satisfaction function on formulae 2. DPow, the definable powerset function 3. Lset, which expresses the levels of the constructible hierarchy. Of these functions, Lset is defined by transfinite recursion from DPow, which in turn has a straightforward definition in terms of satisfies. But proving the absoluteness of satisfies is very complicated. Absoluteness of satisfies is merely an instance of the absoluteness of recursion over formulae, and is therefore trivial. That does not relieve us of the task of formalizing the details. The file containing the satisfies absoluteness proof is one of the largest in the entire development. This file divides into two roughly equal parts. The first half contains internalizations and reflection theorems for operators such as depth and formula case. It expresses the four cases of satisfies in both functional and relational form, and proves absoluteness for each case. Six instances of strong replacement are required: one for each case of the recursion (because each contains a -abstraction), another to justify the use of transrec, and yet another to justify the -abstraction in formula rec. These axioms are assumed to hold of an arbitrary class model M . They are used to show that the formalization satisfies the conditions of the absoluteness theorem for formula rec described in the previous section. The second half of the file is devoted to proving that the six instances of replacement hold in L. The four cases of the recursion (in their relational form) must each be internalized. This tiresome task involves, as always, translating a definition involving real formulae into one using internalized formulae. Then, the six instances of replacement are justified. Finally, the pieces are put together. Proving that satisfies is Absolute Working in the class M, we assume additional instances of the replacement axiom and apply them to the definition of satisfies, which is reproduced here: "satisfies(A,Member(x,y)) = ( env ∈ list(A). bool of o (nth(x,env) ∈ nth(y,env)))" "satisfies(A,Equal(x,y)) = ( env ∈ list(A). bool of o (nth(x,env) = nth(y,env)))" "satisfies(A,Nand(p,q)) = ( env ∈ list(A). not ((satisfies(A,p)'env) and (satisfies(A,q)'env)))" "satisfies(A,Forall(p)) = ( env ∈ list(A). bool of o (∀ x∈A. satisfies(A,p)'(Cons(x,env)) = 1))" Many additional concepts must be internalized. Consider the predicate is depth, which formalizes the depth of a formula: "depth fm(p,n) == Exists(Exists(Exists( And(formula N fm(n#+3,1), And(Neg(Member(p#+3,1)), And(succ fm(n#+3,2), And(formula N fm(2,0), Member(p#+3,0))))))))" We prove the usual theorem relating the satisfaction of depth fm to the truth of is depth "[[x ∈ nat; y < length(env); env ∈ list(A)]] =⇒ sats(A, depth fm(x,y), env) ←→ is depth( A, nth(x,env), nth(y,env))" And we generate yet another instance of the reflection theorem: "REFLECTS[ x. is depth(L, f(x), g(x)), i x. is depth( Lset(i), f(x), g(x))]" The internalization of is formula case is omitted, but its definition is 15 lines long and contains 11 quantifiers. The theorem statements relating is formula case to formula case are also long and complicated. And of course they are higher-order, requiring the methods of Sect. 7.3. In order to relativize satisfies, we must first define constants corresponding to formula rec's parameters a, b, c and d. Here are the two base cases: Each of these functions is then re-expressed in relational form. Here is the first: Once we have done the other three, we can define an instance of MH for satisfies, expressing the body of the recursion as a predicate: At this point we must assume (by declaring a locale) the six instances of replacement mentioned above. That enables us to prove absoluteness for the parameters a, b, c and d used to define satisfies. For example, the class M is closed under satisfies a: This theorem states that satisfies is a(M,A,x,y,zz) is the relational equivalent of satisfies a(A,x,y) provided x and y belong to the set nat. It can be seen as an absoluteness result subject to typing conditions on x and y. Proofs are obviously easier if the absoluteness results are unconditional, but sometimes typing conditions are difficult to avoid. Analogous theorems are proved for satisfies is b, satisfies is c and satisfies is d. Thus we use the first four instances of replacement. The last two instances, which are specific to satisfies, let us discharge the more general instances of replacement that are conditions of formula rec's absoluteness theorem. We ultimately obtain absoluteness for satisfies: Proving the Instances of Replacement for L Now we must justify those six instances of strong replacement by proving that they hold in L. Recall that strong replacement is the conjunction of replacement (which holds schematically in L, but may yield too big a set) and an appropriate instance of separation (Sect. 4.3). As always, proving instances of separation requires internalizing many formulae. Isabelle can do this automatically, but unless it is given enough internalized formulae to use as building blocks, the translation requires much time and space. I internalized many concepts manually, declaring their internal counterparts as constants and proving their correspondence with the original concepts. Here is the internal equivalent of satisfies is a: "satisfies is a fm(A,x,y,z) == Forall( Implies(is list fm(succ(A),0), lambda fm( bool of o fm(Exists( Exists(And(nth fm(x#+6,3,1), And(nth fm(y#+6,3,0), Member(1,0))))), 0), 0, succ(z))))" Obviously, the same task must be done for the other satisfies relations and for the concepts used in their definitions. We finally can internalize the body of satisfies: "satisfies MH fm(A,u,f,zz) == Forall( Implies(is formula fm(0), lambda fm( formula case fm(satisfies is a fm(A#+7,2,1,0), satisfies is b fm (A#+7,2,1,0), satisfies is c fm(A#+7,f#+7,2,1,0), satisfies is d fm(A#+6,f#+6,1,0), 1, 0), 0, succ(zz))))" Now, we can prove the six instances of replacement. Here is the first one, for the Member case of satisfies: The theorem statement may look big, but the proof has only four commands. The corresponding instances of the reflection theorem (not shown) is twice as big, but its proof has only one command. We proceed to prove the fifth instance of replacement: Our reward for this huge effort is that the absoluteness of satisfies now holds for L: "[[L(A); L(z); p ∈ formula]] =⇒ is satisfies(L,A,p,z) ←→ z = satisfies(A,p)" Absoluteness of the Definable Powerset Conceptually, the absoluteness of DPow is trivial, since it is just a comprehension involving satisfies. The formal details require a modest effort. There are more internalizations, such as that of is formula rec. Note that concepts only have to be internalized if they appear in an instance of separation, which may only happen long after the concept is first relativized. Unfortunately, formula rec is a complex higher-order function; in its relational form, one argument gets enclosed within 11 quantifiers. Completing this task enables us to internalize is satisfies: "satisfies fm(x) == formula rec fm(satisfies MH fm(x#+5#+6,2,1,0))" Recall that DPow is the definable powerset operator. It has a variant form, DPow', that does not involve the function arity. The two operators agree on transitive sets, so in particular we can use DPow' to construct L. Now we must relativize DPow'. Its definition refers to the powerset operator, which is not absolute. It can equivalently be expressed using a set comprehension, which here represents an appeal to the replacement axiom: "DPow'(A) = {z . ep ∈ list(A) × formula, ∃ env ∈ list(A). ∃ p ∈ formula. ep = env,p & z = {x∈A. sats(A, p, Cons(x,env))}}" Within the comprehension is another comprehension, which appeals to separation. The formula sats(A, p, Cons(x,env)) needs to be relativized (as the predicate is DPow sats) and internalized. Then, we again extend the list of assumptions about the class M to include these instances of replacement and separation. Using them, we can prove that M is closed under definable powersets: "M(A) =⇒ M(DPow'(A))" We can also express the equation for DPow' shown above in relational form, defining the predicate is DPow', and prove absoluteness: To make these results available for L, we must first prove that L satisfies the new instances of replacement and separation. Here is the latter: "[[L(A); env ∈ list(A); p ∈ formula]] =⇒ separation(L, x. is DPow sats(L,A,env,p,x))" Absoluteness of Constructibility The proof that L satisfies V = L nearly finished. Only the operator Lset, which denotes the levels of the constructible hierarchy, remains to be proved absolute. Recall that it can be expressed using DPow': "Lset(i) = transrec(i, %x f. y ∈x. DPow' (f ' y))" So now we must internalize the predicate is DPow'. First we must internalize the operators used in its definition. Among those are the predicate is Collect, which recognizes set comprehensions. The equation for Lset above involves two further instances of replacement: one for the use of transrec and another for the indexed union. Adding them to our list of constraints on M allows us to prove that that class is closed under the Lset operator: "[[Ord(i); M(i)]] =⇒ M(Lset(i))" We can also define its relational version: "is Lset(M,a,z) == is transrec(M, %x f u. u = ( y ∈x. DPow' (f ' y)), a, z)" Notice that this definition is not purely relational. That is all right because is Lset is not used in any instance of separation and thus need not be internalized. We can now prove that the constructible hierarchy is absolute: "[[Ord(i); M(i); M(z)]] =⇒ is Lset(M,i,z) ←→ z = Lset(i)" As remarked earlier, results such as this express absoluteness because the class model M drops out of the right-hand side. The left-hand side refers to our formalization of in M, which by the theorem is equivalent to itself. As always, making this result available to L requires proving the new instances of replacement. I omit the details, which contain nothing instructive. We can finally formalize L M , the relativization of L. A set x is constructible (with respect to any class M satisfying the specified ZF axioms) provided there exists an ordinal i and a level of the constructible hierarchy Li such that x ∈ Li. The following theorem is a trivial consequence of the absoluteness results and the definitions of constructible and L. "L(x) =⇒ constructible(L,x)" This theorem expresses our goal, namely that V = L holds in L or more formally (V = L) L . For this statement is equivalent to (∀ . L( )) L and thus to ∀ . L( ) → L L ( ). We can drop the universal quantifier. The antecedent of the implication is formalized as L(x) and the consequent as constructible(L,x). This proof ends the most difficult part of the development. The Axiom of Choice in L The formalization confirms that V = L is consistent with the axioms of set theory. Obviously any consequence of V = L, such as the axiom of choice, is consistent with those axioms too. Proving consequences of V = L involves working in an entirely different way, and a much pleasanter one. Dispensing with the relational language, relativization, internalization and absoluteness, we can instead work in native set theory with the additional axiom V = L. Assuming V = L, the proof of the axiom of choice is simple [9, p. 173]. It suffices to prove that every set can be well-ordered. In fact, we can well-order the whole of L. The set of internalized formulae is countable, and therefore wellordered. The well-ordering of L derives from its cumulative construction and from the well-ordering of formulae. For , ∈ L, say that precedes if • originates earlier than in the constructible hierarchy -that is, there is some such that ∈ and ∉ . • and originate at the same level , but the combination of defining formula and parameters for lexicographically precedes the corresponding combination for . Each element of +1 is a subset of that can be defined by a formula, possibly involving parameters from . We can assume the induction hypothesis that is well-ordered. Before we can undertake this transfinite induction, we must complete several tasks: 1. exhibiting a well-ordering on lists, for the parameters of a definable subset 2. exhibiting a well-ordering on formulae 3. combining these to obtain a well-ordering of the definable powerset 4. show how to extend our well-ordering to the limit case of the transfinite induction A Well-Ordering for Lists First we inductively define a relation on lists: the lexicographic extension of a relation on the list's elements. Let r denote a relation over the set A. Then the relation rlist(A,r) is the least set closed under the following rules: There are several well-known injections from × into , but defining one of them and proving it to be injective would involve some effort. Instead we can appeal to a corollary of ⊗ = , which is already available [19, §5] in Isabelle/ZF: Thus we have × ≈ : there is a bijection, which is also an injection, between × and . However, although an injection exists, we have no means of naming a specific bijection. Therefore, we conduct the entire proof of the axiom of choice under the assumption that some injection exists. The final theorem is existential, which will allow the assumption to be discharged. We declare a locale to express this new assumption, calling the injection fn. Recall that nat is Isabelle/ZF's name for the ordinal : locale Nat Times Nat = fixes fn assumes fn inj: "fn ∈ inj(nat * nat, nat)" Proving that enum(fn,p) defines an injection from formulae into the naturals requires a straightforward double induction over formulae: "( p ∈ formula. enum(fn,p)) ∈ inj(formula, nat)" Using the enumeration as a measure function, we find that the set of formulae is well-ordered: "well ord(formula, measure(formula, enum(fn)))" The functions defined below all have an argument f, which should range over injections from × into . In proofs, this injection will always be fn from locale Nat Times Nat. The definiens of a constant definition cannot refer to fn because it is a variable. Defining the Well-ordering on DPow(A) The set DPow(A) consists of those subsets of A that can be defined by a formula, possibly using elements of A as parameters (Sect. 6.5). We can define a wellordering on DPow(A) from one on A. We get a well-ordering on formulae from their injection into the natural numbers. To handle the parameters, we define a wellordering for environments -lists over A -and combine it with the well-ordering of formulae. A subset of A might be definable in more than one way; to make a unique choice, we map environment/formula pairs to ordinals. The well-ordering on environment/formula pairs is the lexicographic product (given by rmult) of the well-orderings on lists (rlist) and formulae (measure). "env form r(f,r,A) == rmult(list(A), rlist(A, r), formula, measure(formula, enum(f)))" Well-Ordering in the Limit Case The proof that is well-ordered appeals to transfinite induction on the ordinal . The induction hypothesis is that is well-ordered if < . In the limit case, = < . Recall (Sect. 6.9.2) that L-rank ( ) of is the least such that ∈ +1 . If is a limit ordinal then we order elements of first by their L-ranks; if two elements have the same L-rank, say , then we order them using the existing well-ordering of +1 . In the Isabelle formalization, i is the limit ordinal and r(j) denotes the wellordering of Lset(j): "rlimit(i,r) == if Limit(i) then {z ∈ Lset(i) × Lset(i). ∃ x' x. z = x',x & (lrank(x') < lrank(x) \| (lrank(x') = lrank(x) & x',x ∈ r(succ(lrank(x)))))} else 0" We can prove that the limit ordering is linear provided the orderings of previous stages are also linear: Under analogous conditions, the rlimit(i,r) is a well-ordering of Lset(i). The proofs are straightforward, and I have omitted many details. "[[Limit(i); ∀ j<i. well ord(Lset(j), r(j))]] =⇒ well ord(Lset(i), rlimit(i,r))" Transfinite Definition of the Well-Ordering for L The well-ordering on L is defined by transfinite recursion. The Isabelle definition refers to the cryptic transrec operator, so let us pass directly to the three immediate consequences of that definition. For the base case, the well-ordering is the empty relation: "L r(f,0) = 0" For the successor case, the well-ordering is given by applying DPow r to the previous level. "L r(f, succ(i)) = DPow r(f, L r(f,i), Lset(i))" For the limit case, the well-ordering is given by rlimit. "Limit(i) =⇒ L r(f,i) = rlimit(i, L r(f))" Thanks to the results proved above, a simple transfinite induction proves that L r(fn,i) well-orders the constructible level Lset(i). "Ord(i) =⇒ well ord(Lset(i), L r(fn,i))" Note that this theorem refers to fn, an injection from × into . Recall (Sect. 10.2) that we know such that such functions exist but have not defined a specific one. We have been able to prove our theorems by working in a locale that assumes the existence of fn. Now, we can eliminate the assumption. We use an existential quantifier to hide the well-ordering in the previous theorem, so that fn no longer appears. Then, by the mere existence of such an injection, it follows that every Lset(i) can be well-ordered: "Ord(i) =⇒ ∃ r. well ord(Lset(i), r)" To wrap things up, let us package the axiom V = L as a locale: locale V equals L = assumes VL: "L(x)" The axiom of choice -in the guise of the well-ordering theorem -is a trivial consequence of the previous results. theorem (in V equals L) AC: "∃ r. well ord(x,r)" Conclusions What has been accomplished? I have mechanized the proof of the relative consistency of the axiom of choice, largely following a standard textbook presentation. The formal proof is much longer than the textbook version because it is complete in all details and uses no metatheoretical reasoning. As noted in Section 2, Gödel's proof comprises four tasks, which we can now express more precisely: The proof that L satisfies V = L is by far the largest and most difficult part of the development. It involves proving L to be absolute, which requires converting every concept used in its definition into relational form and proving absoluteness. The sheer number of concepts is an obstacle, and some of them are hard to express in relational form, especially those involving recursion. Most of the relations have to be re-expressed using an internal datatype of formulae. My formalization has two limitations. First, I am not able to prove that L satisfies the axiom scheme of comprehension. Although Isabelle/ZF handles schematic proofs easily, the proof of comprehension for the formula requires an instance of the reflection theorem for . Each instance of comprehension therefore has a different proof and must be proved separately. The reflection theorem is proved by induction (at the metalevel) on the structure of ; thus, all these proofs are instances of one algorithm, and they are generated by nearly identical proof scripts [18]. The inability to prove the comprehension scheme makes the absoluteness proofs harder: every necessary instance of comprehension is listed. Instantiating these proofs to L has required proving that each of those instances held in L. There are about 35 such instances. My formalization has another limitation. The proof that L satisfies V = L cannot be combined with the proof that V = L implies the axiom of choice in order to conclude that L satisfies the axiom of choice. The reason is that the two instances of V = L are formalized differently: one is relativized and the other is not. Here I have followed the textbook proofs, which prove V = L, declare that the axiom of constructibility can be assumed, and proceed to derive the consequences of that axiom. We could remedy both limitations by tackling the whole problem in a quite different way, by formalizing set theory as a proof system and working entirely in the metatheory. I leave this as a challenge for the theorem-proving community. A by-product of the work is a general theory of absoluteness for arbitrary class models of ZF. It could be used for other formal investigations of inner models. Future investigators might also try formalizing the proof that L satisfies the generalized continuum hypothesis and the combinatorial principle ♦. 6 Figure 1 : 61The Constructible Universe, L ( = ) M abbreviates = ( ∈ ) M abbreviates ∈ ( ∧ ) M abbreviates M ∧ M (¬ ) M abbreviates ¬( M ) (∃ . ) M abbreviates ∃ . ∈ M ∧ M Dually (∀ . ) M abbreviates ∀ . ∈ M → M , if universal quantifiers are defined as usual. (When working in ZF, we should write M( ) instead of ∈ M above.) Relativization bounds all quantifiers in by M. It is intuitively clear that M expresses that is true in M. But while the satisfaction relation (\|=) can be defined within set theory, relativization can only be defined in the metalanguage: it combines two arguments, and M, which lie outside ZF. x[M] and ∃ x[M] in Isabelle. For example, ∀ x[M]. P(x) is definitionally equivalent to ∀ x. M(x) −→P(x). " upair(M,a,b,z) == a ∈ z & b ∈ z & (∀ x[M]. x∈z −→ x=a \| x=b)" "pair(M,a,b,z) == ∃ x[M]. upair(M,a,a,x) & (∃ y[M]. upair(M,a,b,y) & upair(M,x,y,z))" " transitive set(M,a) == ∀ x[M]. x∈a −→ subset(M,x,a)" "ordinal(M,a) == transitive set(M,a) & (∀ x[M]. x∈a −→ transitive set(M,x))" x[M]. x∈a −→ (∃ y[M]. y ∈a & successor(M,x,y)))" " successor ordinal(M,a) == ordinal(M,a) &˜empty(M,a) &˜limit ordinal(M,a)" " omega(M,a) == limit ordinal(M,a) & (∀ x[M]. x∈a −→˜limit ordinal(M,x))" " upair ax(M) == ∀ x[M]. ∀ y[M]. ∃ z[M]. upair(M,x,y,z)" "Union ax(M) == ∀ x[M]. ∃ z[M]. big union(M,x,z)" "power ax(M) == ∀ x[M]. ∃ z[M]. powerset(M,x,z)" " foundation ax(M) == ∀ x[M]. (∃ y[M]. y ∈x) −→ (∃ y[M]. y ∈x &˜(∃ z[M]. z∈x & z∈y))" " strong replacement(M,P) == ∀ A[M]. univalent(M,A,P) −→ (∃ Y[M]. ∀ b[M]. b ∈ Y ←→ (∃ x[M]. x∈A & P(x,b)))" locale M trivial = fixes M assumes transM: "[[y ∈x; M(x)]] =⇒ M(y)" and upair ax: "upair ax(M)" and Union ax: "Union ax(M)" and power ax: "power ax(M)" and replacement: "replacement(M,P)" and M nat [iff]: "M(nat)" lemma (in M trivial) empty abs [simp]: "M(z) =⇒ empty(M,z) ←→ z=0" apply (simp add: empty def) apply (blast intro: transM) done " M(A) =⇒ subset(M,A,B) ←→ A ⊆ B" "M(z) =⇒ upair(M,a,b,z) ←→ z={a,b}" "M(z) =⇒ pair(M,a,b,z) ←→ z= a,b " "[[M(r); M(A); M(z)]] =⇒ image(M,r,A,z) ←→ z = r''A" "[[M(A); M(B); M(z)]] =⇒ cartprod(M,A,B,z) ←→ z = A×B" "[[M(a); M(b); M(z)]] =⇒ union(M,a,b,z) ←→ z = a ∪ b" "[[M(A); M(z)]] =⇒ big union(M,A,z) ←→ z = Union(A)" " M(a) =⇒ ordinal(M,a) ←→ Ord(a)" "M(a) =⇒ limit ordinal(M,a) ←→ Limit(a)" "M(a) =⇒ successor ordinal(M,a) ←→ Ord(a) & (∃ b[M]. a = succ(b))" locale M basic = M trivial + assumes Inter separation: "M(A) =⇒ separation(M, x. ∀ y[M]. y ∈A −→ x∈y)" and Diff separation: "M(B) =⇒ separation(M, x. x ∉ B)" and cartprod separation: "[[M(A); M(B)]] =⇒ separation(M, z. ∃ x[M]. x∈A & (∃ y[M]. y ∈B & pair(M,x,y,z)))" and image separation: "[[M(A); M(r)]] =⇒ separation(M, y. ∃ p[M]. p∈r & (∃ x[M]. x∈A & pair(M,x,y,p)))" and converse separation: "M(r) =⇒ separation(M, z. ∃ p[M]. p∈r & (∃ x[M]. ∃ y[M]. pair(M,x,y,p) & pair(M,y,x,z)))" lemma (in M basic) Inter closed: "M(A) =⇒ M(Inter(A))" " [[M(A); M(B)]] =⇒ M(A×B)" "[[M(A); M(B)]] =⇒ M(A+B)" "fun apply(M,f,x,y) == (∃ xs[M]. ∃ fxs[M]. upair(M,x,x,xs) & image(M,f,xs,fxs) & big union(M,fxs,y))" " [[M(f); M(a)]] =⇒ M(f'a)" "[[M(f); M(x); M(y)]] =⇒ fun apply(M,f,x,y) ←→ f'x = y" " wellordered(M,A,r) == transitive rel(M,A,r) & linear rel(M,A,r) & wellfounded on(M,A,r)" " [[wellordered(M,A,r); f ∈ ord iso(A, r, i, Memrel(i)); M(A); M(r); M(f); M(i); Ord(i)]] =⇒ i = ordertype(A,r)" " [[wellfounded(M,r); trans(r); separation(M, x.˜(∃ f[M]. is recfun(r,x,H,f))); strong replacement(M, x z. ∃ y[M]. ∃ g[M]. z= x,y & is recfun(r,x,H,g) & y = H(x,g)); M(r); M(a); ∀ x[M]. ∀ g[M]. function(g) −→ M(H(x,g))]] =⇒ ∃ f[M]. is recfun(r,a,H,f)" " [[∀ x[M]. ∀ g[M]. function(g) −→ M(H(x,g)); M(r); M(a); M(f); relation2(M,MH,H)]] =⇒ M is recfun(M,MH,r,a,f) ←→ is recfun(r,a,H,f)" "rtran closure(M,r,s) == ∀ A[M]. is field(M,r,A) −→ (∀ p[M]. p ∈ s ←→ rtran closure mem(M,A,r,p))" "tran closure(M,r,t) == ∃ s[M]. rtran closure(M,r,s) & composition(M,r,s,t)" " [[wf(r); trans(r); relation(r); M(r); M(a); M(z); wfrec replacement(M,MH,r); relation2(M,MH,H); ∀ x[M]. ∀ g[M]. function(g) −→ M(H(x,g))]] =⇒ is wfrec(M,MH,r,a,z) ←→ z=wfrec(r,a,H)" Another states that the class M is closed under well-founded recursion: "[[wf(r); trans(r); relation(r); M(r); M(a); wfrec replacement(M,MH,r); relation2(M,MH,H); ∀ x[M]. ∀ g[M]. function(g) −→ M(H(x,g))]] =⇒ M(wfrec(r,a,H))" " wfrec replacement(M,MH,r) == strong replacement(M, x z. ∃ y[M]. pair(M,x,y,z) & is wfrec(M,MH,r,x,y))" " [[arity(p) ≤ length(env); p ∈ formula; env ∈ list(A); extra ∈ list(A)]] =⇒ sats(A, p, env@extra) ←→ sats(A, p, env)" 6.4 Renaming (Renumbering) Free Variables If is a set, then the subset { ∈ \| ( , 1 , . . . , )} is determined by the choice of and of the parameters 1 , . . . , , which are elements of . These are the definable subsets of . Now, consider the problem of showing that the definable sets are closed under intersection. Suppose another subset of is defined by a formula and parameters +1 , . . . , + : { ∈ \| ( , +1 , . . . , + )} Then, their intersection can presumably be defined by { ∈ \| ( , 1 , . . . , ) ∧ ( , +1 , . . . , + )} " [[p ∈ formula; bvs ∈ list(A); env ∈ list(A); x ∈ A]] =⇒ sats(A, incr bv(p) ' length(bvs), bvs @ Cons(x,env)) ←→ sats(A, p, bvs@env)"For the intuition, suppose that bvs is the list [ 0 , . . . , −1 ] (and therefore has length m). Then the conclusion essentially sayssats(A, incr bv(p) ' m, [ 0 , . . . , −1 , , , . . . , ]) ←→ sats(A, p, [ 0 , . . . , −1 , , . . . , ])" "[[p ∈ formula; bvs ∈ list(A); x ∈ A; env ∈ list(A); length(bvs) = n]] =⇒ sats(A, iterates(incr bv1, n, p), Cons(x, bvs@env)) ←→ sats(A, p, Cons(x,env))" " Finite(A) =⇒ DPow(A) = Pow(A)" " [[x < length(env); y ∈ nat; env ∈ list(A); Transset(A)]] =⇒ sats(A, subset fm(x,y), env) ←→ nth(x,env) ⊆ nth(y,env)" " Transset(A) =⇒ {x ∈ A. Ord(x)} ∈ DPow(A)" M ( 1 , . . . , ) ⇐⇒ ( 1 , . . . , ). Reflection relationships can be formed over the propositional connectives, here negation, conjunction and biconditionals:"REFLECTS[P,Q] =⇒ REFLECTS[ x.˜P(x), a x.˜Q(a,x)]" "[[REFLECTS[P,Q]; REFLECTS[P',Q']]] =⇒ REFLECTS[ x. P(x) ∧ P'(x), a x. Q(a,x) ∧ Q'(a,x)]" "[[REFLECTS[P,Q]; REFLECTS[P',Q']]] =⇒ REFLECTS[ x. P(x) ←→ P'(x), a x. Q(a,x) ←→ Q'(a,x)]"Reflection relationships can be formed over the quantifiers:"REFLECTS[ x. P(fst(x),snd(x)), a x. Q(a,fst(x),snd(x))] =⇒ REFLECTS[ x. ∃ z[L]. P(x,z), a x. ∃ z∈Lset(a). Q(a,x,z)]" " REFLECTS[ x. union(L,f(x),g(x),h(x)), i x. union( ** Lset(i),f(x),g(x),h(x))]" " [[strong replacement(M, x y. x∈A & y = x, b(x) ); M(A); ∀ m[M]. m∈A −→ M(b(m))]] =⇒ M( x∈A. b(x))" " L(A) =⇒ separation(L, x. ∀ y[L]. y ∈A −→ x∈y)" " [[L(r); L(s)]] =⇒ separation(L, xz. ∃ x[L]. ∃ y[L]. ∃ z[L]. ∃ xy[L]. ∃ yz[L]. pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & xy ∈s & yz∈r)" . [[L(r); L(s); r ∈ Lset(j); s ∈ Lset(j)]] =⇒ [r, s] ∈ list(Lset(j)) 3. j. [[L(r); L(s); r ∈ Lset(j); s ∈ Lset(j)]]=⇒ ?p3(j) ∈ formula Figure 2 : 2Subgoals ready for automatic synthesis of a formula "[[P ←→ sats(A,p,env); Q ←→ sats(A,q,env); env ∈ list(A)]] =⇒ (P & Q) ←→ sats(A, And(p,q), env)"This "program clause" relates the real quantification ∀ x∈A. P(x) with the term Forall(p). The first subgoal concerns the synthesis of p in an environment augmented with an arbitrary x∈A: "[[!!x. x∈A =⇒ P(x) ←→ sats(A, p, Cons(x, env)); env ∈ list(A)]] =⇒ (∀ x∈A. P(x)) ←→ sats(A, Forall(p), env)" " [[nth(i,env) = x; nth(j,env) = y; env ∈ list(A)]] =⇒ (x∈y) ←→ sats(A, Member(i,j), env)" " lfp(D,h) == Inter({X ∈ Pow(D). h(X) ⊆ X})" We proceed by formalizing standard concepts from domain theory [1, pp. 51-56]. A set is directed if it is non-empty and closed under least upper bounds. A function is continuous if it preserves the unions of directed sets. " directed(A) == A≠0 & (∀ x∈A. ∀ y ∈A. x∪y ∈ A)" "contin(h) == (∀ A. directed(A) −→ h( A) = ( X∈A.h(X)))" " [[contin(F); contin(G)]] =⇒ contin( X. F(X) + G(X))" "[[contin(F); contin(G)]] =⇒ contin( X. F(X) × G(X))" "iterates MH(M,isF,v,n,g,z) == is nat case(M, v, m u. ∃ gm[M]. fun apply(M,g,m,gm) & isF(gm,u), n, z)" "is iterates(M,isF,v,n,Z) == ∃ sn[M]. ∃ msn[M]. successor(M,n,sn) & membership(M,sn,msn) & is wfrec(M, iterates MH(M,isF,v), msn, n, Z)" " iterates replacement(M,isF,v) == ∀ n[M]. n∈nat −→ wfrec replacement(M, iterates MH(M,isF,v), Memrel(succ(n)))" " [[iterates replacement(M,isF,v); relation1(M,isF,F); n ∈ nat; M(v); M(z); ∀ x[M]. M(F(x))]] =⇒ is iterates(M,isF,v,n,z) ←→ z = iterates(F,n,v)" " [[iterates replacement(M,isF,v); relation1(M,isF,F); n ∈ nat; M(v); ∀ x[M]. M(F(x))]] =⇒ M(iterates(F,n,v))" " list N(A,n) == ( X. {0} + A * X)ˆn (0)"Next, we relativize the function X. {0} + A * X. The predicate number1 recognizes the number 1, which equals the set {0}. " is list functor(M,A,X,Z) == ∃ n1[M]. ∃ AX[M]. number1(M,n1) & cartprod(M,A,X,AX) & is sum(M,n1,AX,Z)" Next, we relativize the function list N, the finite iterations: "is list N(M,A,n,Z) == ∃ zero[M]. empty(M,zero) & is iterates(M, is list functor(M,A), zero, n, Z)" We relativize membership in list(A) as membership in list N(A,n) for some n. The predicate finite ordinal recognizes the natural numbers. "mem list(M,A,l) == ∃ n[M]. ∃ listn[M]. finite ordinal(M,n) & is list N(M,A,n,listn) & l ∈ listn" Finally, we can relativize the set of lists itself: "is list(M,A,Z) == ∀ l[M]. l ∈ Z ←→ mem list(M,A,l)" After proving absoluteness of list N(A,n), we obtain the absoluteness of list(A) and prove that M is closed under list formation. "M(A) =⇒ M(list(A))" "[[M(A); M(Z)]] =⇒ is list(M,A,Z) ←→ Z = list(A)" " is nth(M,n,l,Z) == ∃ X[M]. is iterates(M, is tl(M), l, n, X) & is hd(M,X,Z)" Absoluteness is proved with no effort: "[[M(A); n ∈ nat; l ∈ list(A); M(Z)]] =⇒ is nth(M,n,l,Z) ←→ Z = nth(n,l)" " eclose(A) = ( n∈nat. Unionˆn (A))" Relativization proceeds as it did for lists. The details are omitted, but they culminate in the definition of a relational version of eclose(A): "is eclose(M,A,Z) == ∀ u[M]. u ∈ Z ←→ mem eclose(M,A,u)" The standard membership and absoluteness results follow: "M(A) =⇒ M(eclose(A))" "[[M(A); M(Z)]] =⇒ is eclose(M,A,Z) ←→ Z = eclose(A)" " [[transrec replacement(M,MH,i); relativize2(M,MH,H); Ord(i); M(i); M(z); ∀ x[M]. ∀ g[M]. function(g) −→ M(H(x,g))]] =⇒ is transrec(M,MH,i,z) ←→ z = transrec(i,H)" We similarly find that M is closed under ∈-recursion: "[[transrec replacement(M,MH,i); relativize2(M,MH,H); Ord(i); M(i); ∀ x[M]. ∀ g[M]. function(g) −→ M(H(x,g))]] =⇒ M(transrec(i,H))" A formula p has depth n if it satisfies is formula N(M,succ(n),F) and not is formula N(M,n,F): "is depth(M,p,n) == ∃ sn[M]. ∃ formula n[M]. ∃ formula sn[M]. is formula N(M,n,formula n) & p ∉ formula n & successor(M,n,sn) & is formula N(M,sn,formula sn) & p ∈ formula sn" . env ∈list(A). bool of o (nth(x,env) ∈ nth(y,env))" "satisfies b(A) ==x y. env ∈list(A). bool of o (nth(x,env) = nth(y,env))"In the two recursive cases, the variables rp and rq denote the values returned on the recursive calls for p and q, respectively:"satisfies c(A) == p q rp rq. env ∈list(A). not(rp ' env and rq ' env)""satisfies d(A) == p rp. env ∈list(A). bool of o (∀ x∈A. rp ' (Cons(x,env)) = 1)" " satisfies is a(M,A) == x y zz. ∀ lA[M]. is list(M,A,lA) −→ is lambda(M, lA, env z. is bool of o(M, ∃ nx[M]. ∃ ny[M]. is nth(M,x,env,nx) & is nth(M,y,env,ny) & nx∈ny, z), zz)" ∀ fml[M]. is formula(M,fml) −→ is lambda (M, fml, is formula case (M, satisfies is a(M,A), satisfies is b(M,A), satisfies is c(M,A,f), satisfies is d(M,A,f)), z)" Finally, satisfies itself can be relativized: "is satisfies(M,A) == is formula rec (M, satisfies MH(M,A))" This lemma relates the fragments defined above to the original primitive recursion in satisfies. Induction is not required: the definitions are directly equal! "satisfies(A,p) = formula rec (satisfies a(A), satisfies b(A), satisfies c(A), satisfies d(A), p)" " M(A) =⇒ Relation2(M, nat, nat,satisfies is a(M,A), satisfies a(A))" " [[M(A); M(z); p ∈ formula]] =⇒ is satisfies(M,A,p,z) ←→ z = satisfies(A,p)" " [[L(A); x ∈ nat; y ∈ nat]] =⇒ strong replacement (L, env z. ∃ bo[L]. ∃ nx[L]. ∃ ny[L]. env ∈ list(A) & is nth(L,x,env,nx) & is nth(L,y,env,ny) & is bool of o(L, nx ∈ ny, bo) & pair(L, env, bo, z))" " [[n ∈ nat; L(A)]] =⇒ transrec replacement(L, satisfies MH(L,A), n)" Finally, we prove the sixth instance of replacement: "[[L(g); L(A)]] =⇒ strong replacement (L, x y. mem formula(L,x) & (∃ c[L]. is formula case(L, satisfies is a(L,A), satisfies is b(L,A), satisfies is c(L,A,g), satisfies is d(L,A,g), x, c) & pair(L, x, c, y)))" " [[M(A); M(Z)]] =⇒ is DPow'(M,A,Z) ←→ Z = DPow'(A)" " constructible(M,x) == ∃ i[M]. ∃ Li[M]. ordinal(M,i) & is Lset(M,i,Li) & x ∈ Li" [ [well ord(A,r); InfCard(\|A\|)]] =⇒ A × A ≈ A " [[Limit(i); ∀ j<i. linear(Lset(j), r(j)) ]] =⇒ linear(Lset(i), rlimit(i,r))" See Gödel[5, p. 76] or for a modern treatment Kunen[9, p. 112]. INTRODUCTION TO THE ISABELLE/ZF FORMALIZATION RELATIVIZATION AND ABSOLUTENESS: BASICS RELATIVIZATION AND ABSOLUTENESS: BASICS WELL-FOUNDED RECURSION WELL-FOUNDED RECURSION "[[wellfounded(M,r); M(r)]] =⇒ wellfounded(M,rˆ+)" DEFINING FIRST-ORDER FORMULAE AND THE CLASS L "(∀ x∈A. L(x)) =⇒ ∃ i. Ord(i) & A ⊆ Lset(i)" "[[bnd mono(D,h); contin(h)]] =⇒ lfp(D,h) = ( n∈nat. hˆn(0))" See § §3.4 and 4.3.1 of Paulson[15]. "p ∈ formula =⇒ M(formularec(a,b,c,d,p))" "[[M(A); x∈nat; y ∈nat]] =⇒ M(satisfies a(A,x,y))" "well ord(A,r) =⇒ well ord(DPow(A), DPow r(fn,r,A))" Acknowledgements. Krzysztof Grabczewski devoted much effort to an earlier, unsuccessful, attempt to formalize this material. Isabelle work is supported by the U.K.'s Engineering and Physical Sciences Research Council, grant GR/M75440. Markus Wenzel greatly improved Isabelle's locale construct to support these proofs. Kenneth Kunen gave advice that helped in my formalization of the reflection theorem. The referee made a number of valuable comments on this paper. Cons(a,l) ∈ rlist(A,r). =⇒ Cons. length(l') = length(l); a',a ∈ r=⇒ Cons(a,l'), Cons(a,l) ∈ rlist(A,r)" "[[length(l') = length(l); a',a ∈ r; Cons(a,l) ∈ rlist(A,r. =⇒ Cons, =⇒ Cons(a',l'), Cons(a,l) ∈ rlist(A,r)" This theorem has a 14-line proof script involving a double structural induction on lists. If the element ordering is linear, then so is the list ordering. linear(A,r) =⇒ linear(list(A),rlist(A,r)If the element ordering is linear, then so is the list ordering. This theorem has a 14-line proof script involving a double structural induction on lists. "linear(A,r) =⇒ linear(list(A),rlist(A,r))" This theorem is proved by induction on the length of the list followed by inductions over the element ordering and the list ordering. The proof script is under 20 lines, but the argument is complicated. If the element ordering is well-founded, then so is the list ordering. well ord(A,r) =⇒ well ord(list(A), rlist(A,r)If the element ordering is well-founded, then so is the list ordering. This theo- rem is proved by induction on the length of the list followed by inductions over the element ordering and the list ordering. The proof script is under 20 lines, but the argument is complicated. "well ord(A,r) =⇒ well ord(list(A), rlist(A,r))" An injection from the set of formulae into the set of natural numbers is easily defined by recursion on the structure of formulae. However, it requires an injection from pairs of natural numbers to natural numbers. The enumeration function for formulae takes this injection as its first argument, f: "enum(f, Member(x,y)) = f ' 0, f ' x,y. Gödel-numbering is the obvious way to well-order the set of formulae. enum(f, Equal(x,y)) = f ' 1, f ' x,y " "enum(f, Nand(p,qGödel-numbering is the obvious way to well-order the set of formulae. An injec- tion from the set of formulae into the set of natural numbers is easily defined by recursion on the structure of formulae. However, it requires an injection from pairs of natural numbers to natural numbers. The enumeration function for formulae takes this injection as its first argument, f: "enum(f, Member(x,y)) = f ' 0, f ' x,y " "enum(f, Equal(x,y)) = f ' 1, f ' x,y " "enum(f, Nand(p,q)) = f ' 2, f ' enum(f,p), enum(f,q). enum(f, Forall(p)= f ' 2, f ' enum(f,p), enum(f,q) " "enum(f, Forall(p)) . = , = f ' succ(2), enum(f,p) " The order type of the resulting well-ordering yields a map (given by ordermap) from environment/formula pairs into the ordinals. For each member of DPow(A), the minimum such ordinal will determine its place in the well-ordering. "env form map(f,r,A,z) == ordermap(list(A) × formula, env form r(f,r,A)) ' z" If r well-orders A and X is a definable subset of A, then let us define DPow ord(f,r,A,X,k) to hold if k corresponds to some definition of X -informally, k defines X. A) × Formula, well ord(A,r) =⇒ well ord(list(A) × formula, env form r(fn,r,A)). DPow ord(f,r,A,X,k) == ∃ env ∈ list(A). ∃ p ∈ formula. arity(p) ≤ succ(length(envUsing existing theorems, it is trivial to prove that this construction well-orders the set list(A) × formula: "well ord(A,r) =⇒ well ord(list(A) × formula, env form r(fn,r,A))" The order type of the resulting well-ordering yields a map (given by ordermap) from environment/formula pairs into the ordinals. For each member of DPow(A), the minimum such ordinal will determine its place in the well-ordering. "env form map(f,r,A,z) == ordermap(list(A) × formula, env form r(f,r,A)) ' z" If r well-orders A and X is a definable subset of A, then let us define DPow ord(f,r,A,X,k) to hold if k corresponds to some definition of X -infor- mally, k defines X: "DPow ord(f,r,A,X,k) == ∃ env ∈ list(A). ∃ p ∈ formula. arity(p) ≤ succ(length(env)) & sats(A, p, Cons(x,env))} & env form map(f,r,A, env,p ) = k. X = {x∈a, X = {x∈A. sats(A, p, Cons(x,env))} & env form map(f,r,A, env,p ) = k" . Similarly, let us define DPow least(f,r,A,X) to be the smallest ordinal defining X: "DPow least(f,r,A,X) == k. DPow ord(f,r,A,X,kSimilarly, let us define DPow least(f,r,A,X) to be the smallest ordinal defining X: "DPow least(f,r,A,X) == k. DPow ord(f,r,A,X,k)" Since k determines env and p, we find that an ordinal can define at most one element of DPow(A). DPow ord(fn,r,A,X,k)Since k determines env and p, we find that an ordinal can define at most one element of DPow(A): "[[DPow ord(fn,r,A,X,k); . A Dpow Ord(fn,R, Y , well ord(A,r)DPow ord(fn,r,A,Y,k); well ord(A,r)]] We also find that every element of DPow(A) is defined by some ordinal, given by DPow least: "[[X ∈ DPow(A). =⇒ X=y, well ord(A,r)=⇒ X=Y" We also find that every element of DPow(A) is defined by some ordinal, given by DPow least: "[[X ∈ DPow(A); well ord(A,r)]] . A =⇒ Dpow Ord ; Fn, R, X , A , =⇒ DPow ord(fn, r, A, X, DPow least(fn,r,A,X))" Now DPow least can serve as a measure function to define the well-ordering on DPow(A). Now DPow least can serve as a measure function to define the well-ordering on DPow(A). DPow r(f,r,A) == measure(DPow(A), DPow least(f,r,A). "DPow r(f,r,A) == measure(DPow(A), DPow least(f,r,A))" Using general facts about relations defined by measure functions, we easily find that DPow(A) is well-ordered: 1. defining the class L within ZF 2. proving, for every ZF axiom. that L is a ZF theorem 3. proving (V = L) L in ZF 4. proving that ZF + V = L implies the axiom of choiceUsing general facts about relations defined by measure functions, we easily find that DPow(A) is well-ordered: 1. defining the class L within ZF 2. proving, for every ZF axiom , that L is a ZF theorem 3. proving (V = L) L in ZF 4. proving that ZF + V = L implies the axiom of choice Introduction to Lattices and Order. B A Davey, H A Priestley, Cambridge University PressB. A. Davey and H. A. Priestley. Introduction to Lattices and Order. Cambridge University Press, 1990. Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser Theorem. N G De Bruijn, Indagationes Mathematicae. 34N. 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[ "BALANCING POLYHEDRA", "BALANCING POLYHEDRA" ]	[ "Gábor Domokos ", "Flórián Kovács ", "Zsolt Lángi ", "Krisztina Regős ", "Péter T Varga " ]	[]	[]	We define the mechanical complexity C(P ) of a convex polyhedron P, interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria, and the mechanical complexity C(S, U ) of primary equilibrium classes (S, U ) E with S stable and U unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class (S, U ) E with S, U > 1 is the minimum of 2(f + v − S − U ) over all polyhedral pairs (f, v), where a pair of integers is called a polyhedral pair if there is a convex polyhedron with f faces and v vertices. In particular, we prove that the mechanical complexity of a class (S, U ) E is zero if, and only if there exists a convex polyhedron with S faces and U vertices. We also give asymptotically sharp bounds for the mechanical complexity of the monostatic classes (1, U ) E and (S, 1) E , and offer a complexity-dependent prize for the complexity of the Gömböc-class (1, 1) E .	10.26493/1855-3974.2120.085	[ "https://arxiv.org/pdf/1810.05382v3.pdf" ]	119,294,613	1810.05382	bfd1bda8586961a55481bb2b62ec7ec4b2ea4d08	BALANCING POLYHEDRA Gábor Domokos Flórián Kovács Zsolt Lángi Krisztina Regős Péter T Varga BALANCING POLYHEDRA We define the mechanical complexity C(P ) of a convex polyhedron P, interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria, and the mechanical complexity C(S, U ) of primary equilibrium classes (S, U ) E with S stable and U unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class (S, U ) E with S, U > 1 is the minimum of 2(f + v − S − U ) over all polyhedral pairs (f, v), where a pair of integers is called a polyhedral pair if there is a convex polyhedron with f faces and v vertices. In particular, we prove that the mechanical complexity of a class (S, U ) E is zero if, and only if there exists a convex polyhedron with S faces and U vertices. We also give asymptotically sharp bounds for the mechanical complexity of the monostatic classes (1, U ) E and (S, 1) E , and offer a complexity-dependent prize for the complexity of the Gömböc-class (1, 1) E . 1. Introduction 1.1. Basic concepts and the main result. Polyhedra may be regarded as purely geometric objects, however, they are also often intuitively identified with solids. Among the most obvious sources of such intuition are dice which appear in various polyhedral shapes: while classical, cubic dice have 6 faces, a large diversity of other dice exist as well: dice with 2, 3,4,6,8,10,12,16,20,24,30 and 100 faces appear in various games [33]. The key idea behind throwing dice is that each of the aforementioned faces is associated with a stable mechanical equilibrium point where dice may be at rest on a horizontal plane. Dice are called fair if the probabilities to rest on any face (after a random throw) are equal [9], otherwise they are called loaded [8]. The concept of mechanical equilibrium may also be defined in purely geometric terms: Definition 1. Let P be a convex polyhedron, let int P and bd P denote its interior and boundary, respectively and let c ∈ int P . We say that q ∈ bd P is an equilibrium point of P with respect to c if the plane H through q and perpendicular to [c, q] supports P at q. In this case q is nondegenerate, if H ∩ P is the (unique) face of P that contains q in its relative interior. A nondegenerate equilibrium point q is called stable, saddle-type or unstable, if dim(H ∩ P ) = 2, 1 or 0, respectively. Throughout this paper we deal only with equilibrium points with respect to the center of mass of polyhedra, assuming uniform density. A support plane is a generalization of the tangent plane for non-smooth objects. While it is a central concept of convex geometry its name may be related to the mechanical concept of equilibrium. If c coincides with the center of mass of P , then equilibrium points gain intuitive interpretation as locations on bd P where P may be balanced if it is supported on a horizontal surface (identical to the support plane) without friction in the presence of uniform gravity. Equilibrium points may belong to three stability types: faces may carry stable equilibria, vertices may carry unstable equilibria and edges may carry saddle-type equilibria. Denoting their respective numbers by S, U, H, by the Poincaré-Hopf formula [20] for a convex polyhedron one obtains the following relation for them: (1) S + U − H = 2, which is strongly reminiscent of the well-known Euler formula (2) f + v − e = 2, relating the respective numbers f , v and e of the faces, vertices and edges of a convex polyhedron. In the case of regular, homogeneous, cubic dice the formulae (1) and (2) appear to express the same fact, however, in case of irregular polyhedra the connection is much less apparent. While the striking similarity between (1) and (2) can only be fully explained via deep topological and analytic ideas [20], our goal in this paper is to demonstrate an interesting connection at an elementary, geometric level. To this end, we define The numbers S, U, H may serve, from the mechanical point of view, as a firstorder characterization of P and via (1) the triplet (S, U, H) may be uniquely represented by the pair (S, U ), which is called primary equilibrium class of P [30]. Based on this, we denote by (S, U ) E the family of all convex polyhedra having S stable and U unstable equilibrium points with respect to their centers of mass. In an analogous manner, the numbers (v, e, f ) (also called the f -vector of P ) serve as a first-order combinatorial characterization of P , and via (2) they may be uniquely represented by the pair (f, v). Here, we call the the family of all convex polyhedra having v vertices and f faces the primary combinatorial class of P , and denote it by (f, v) C . The face structure of a convex polyhedron P permits a finer combinatorial description of P . In the literature, the family of convex polyhedra having the same face lattice is called a combinatorial class; here we call it a secondary combinatorial class, and discuss it in Section 5. In an entirely analogous manner, one can define also secondary equilibrium classes of convex bodies, for more details the interested reader is referred to [14]. While it is immediately clear that for any polyhedron P we have (4) f ≥ S, v ≥ U, inverse type relationships (e.g. defining the minimal number of faces and vertices for given numbers of equilibria) are much less obvious. (3) N = S + U + H, n = f + v + e. A trivial necessary condition for any die to be fair can be stated as f = S and it is relatively easy to construct a polyhedron with this property. The opposite extreme case (when a polyhedron is stable only on one of its faces) appears to be far more complex and several papers [1,4,23] are devoted to this subject to which we will return. Motivated by this intuition we define the mechanical complexity of polyhedra. Definition 2. Let P be a convex polyhedron and let N (P ), n(P ) denote the total number of its equilibria and the total number of its k-faces (i. e., faces of k dimensions) for all values k = 0, 1, 2, respectively. Then C(P ) = n(P ) − N (P ) is called the mechanical complexity of P . Mechanical complexity may not only be associated with individual polyhedra but also with primary equilibrium classes. Definition 3. If (S, U ) E is a primary equilibrium class, then the quantity C(S, U ) = inf{C(P ) : P ∈ (S, U ) E } is called the mechanical complexity of (S, U ) E . Our goal is to find the values of C(S, U ) for all primary equilibrium classes. For S, U > 1 we will achieve this goal while for S = 1 or U = 1 we provide some partial results. To formulate our main results, we introduce the following concept: Definition 4. Let x, y be positive integers. We say that (x, y) is a polyhedral pair if and only if x ≥ 4 and x 2 + 2 ≤ y ≤ 2x − 4. The combinatorial classification of convex polyhedra was established by Steinitz [26,27], who proved, in particular, the following. The geometric interpretation of R(S, U ) is given in the left panel of Figure 2. Since (4) holds for any polyhedron P ∈ (S, U ) E , we immediately have the trivial lower bound for mechanical complexity: (5) C(S, U ) ≥ 2R(S, U ). Based on Definition 4, the function R(S, U ) can be expressed as (6) R(S, U ) =          S 2 − U + 2, if S > 4 and S > 2U − 4, U 2 − S + 2, if U > 4 and U > 2S − 4, 8 − S − U, if S, U ≤ 4, 0 otherwise. Our main result is Theorem 2, stating that this bound is sharp if S, U > 1: Theorem 2. Let S, U ≥ 2 be positive integers. Then C(S,U) = 2R(S,U). We remark that, as a consequence of Theorem 2, C(S, U ) = 0 if and only if (S, U ) is a polyhedral pair. For monostatic equilibrium classes (S = 1 or U = 1) we cannot provide a sharp value for their mechanical complexity. However, we will provide an upper bound for their complexity, which differs from 2R(S, U ) only by a constant: Theorem 3. If S ≥ 4 then C(S, 1) ≤ 59 + (−1) S + 2R(S, 1); if U ≥ 4 then C(1, U ) ≤ 90 + 2R(1, U ). We also improve the lower bound (5) in some of these classes by generalizing a theorem of Conway [5] about the non-existence of a homogeneous tetrahedron with only one stable equilibrium point. We state our result in the following form: Theorem 4. Any homogeneous tetrahedron has S ≥ 2 stable and U ≥ 2 unstable equilibrium points. We summarize all results (including those about monostatic classes) in Figure 2. 1.2. Sketch of the proof. The main idea of the proofs of Theorems 2 and 3 is to provide explicit constructions for at least one polyhedron P in each class (S, U ) E , S, U > 1 with mechanical complexity C(P ) = 2R(S, U ), in class (S, 1) E , S ≥ 4 with C(P ) = 59 + (−1) S + 2R(S, 1), and in class (1, U ) E , U ≥ 4 with C(P ) = 90 + 2R(1, U ). By Definition 3, such a construction establishes an upper bound for C(S, U ). In case of S > 1 and U > 1, by Remark 1, this coincides with the lower bound while for S = 1 or U = 1 the bounds remain separate. Our proof consists of five parts: (d) for non-polyhedral classes with U > S ≥ 6, we construct examples by recursive, local manipulations starting with polyhedral classes containing simple polyhedra (Subsection 3.2). (e) for non-polyhedral classes with 6 ≤ U < S we provide examples by using the polyhedra obtained in (d) and the properties of polarity proved in Section 2. We also show how to modify the construction in (d) for this case (Subsection 3.2). In column U = 1 and row S = 1 we give bounds. If two integers are given in square brackets then they are the lower and upper bounds for C(S, U ), if only one integer is given in square brackets then it is the lower bound (and no upper bound is available). (f) for monostatic classes with S = 1 or U = 1 we provide examples using Conway's polyhedron P C in class (1, 4) E , we also construct a polyhedron P 3 in class (3, 1) and subsequently we apply recursive, local truncations (Section 4). In Section 2, we prove a number of lemmas which help us keep track of the change of the center of mass of a convex polyhedron under local deformations and establish a connection between equilibrium points of a convex polyhedron and its polar. The local manipulations in our proof may be regarded as generalizations of the algorithm of Steinitz [16]. Figure 3 summarizes the steps outlined above. Preliminaries Before we prove some lemmas that we need for Theorem 2, we make a general remark about small truncations: Remark 2. Observe that (i) a nondegenerate (stable) equilibrium point s F on face F of a convex polyhedron P exists iff the orthogonal projection s F of c(P ) (the center of mass of P ) onto F is in the relative interior of F ; (ii) a vertex q is a nondegenerate (unstable) equilibrium point of P iff the plane perpendicular to q − c(P ) and containing q contains no other point of P ; (iii) a nondegenerate equilibrium point s E on an edge E of P exists iff the orthogonal projection s E of c(P ) onto E is in the relative interior of E, and the angle between c(P ) − c E and any of the two faces of P containing E is acute. In the paper, we deal only with a convex polyhedron P which has only nondegenerate equilibria. Then the following observation is used many times in the paper: (a) if a vertex q of P is slightly perturbed such that the directions of the edges starting at q change only slightly, then the new vertex is a nondegenerate equilibrium iff q is a nondegenerate equilibrium; (b) if an edge E of P is slightly perturbed such that the normal vectors of the two faces containing E change only slightly, then the new edge contains a nondegenerate equilibrium iff E contains a nondegenerate equilibrium; (c) if a face F of P is slightly perturbed, then the new face contains a nondegenerate equilibrium iff F contains a nondegenerate equilibrium. It is worth noting that since unstable vertices correspond to local maxima of the Euclidean distance function measured from the center of mass, any local perturbation of P yields at least one unstable vertex near q in (a). A similar observation can be made for the face F in (c). In the following, conv X, aff X, int X and cl X denote the convex hull, the affine hull, the interior and the closure of the set X ⊂ R d , respectively. The origin is denoted by o. For any convex polytope P in R d , we denote by V (P ) the set of vertices of P , and the volume and the center of mass of P by w(P ) and c(P ), respectively. The polar of the set X is denoted by X • . The first three lemmas investigate the behavior of the center of mass of a convex polyhedron under local deformations. Lemma 1. Let P be a convex polyhedron and let q be a vertex of P . Let P ε be a convex polyhedron such that P ε ⊂ P , and every point of P \ P ε is contained in the ε-neighborhood of q. Let c = c(P ) and c ε = C(P ε ). Then there is a constant γ > 0, independent of ε, such that \|c ε − c\| ≤ γε 3 holds for every polyhedron P ε satisfying the above conditions. Proof. Without loss of generality, let c = o,c ε = c(cl(P \ P ε )), w = w(P ) and w ε = w(P ε ). Then o = w ε c ε + (w − w ε )c ε , implying that c ε = − w−wε wεc ε . Note that for some γ > 0 independent of ε, we have 0 ≤ w−wε wε < 2 w−wε w ≤ γ ε 3 . Furthermore, for some γ > 0, \|q −c ε \| ≤ γ ε, which yields that \|c ε \| is bounded. Thus, the assertion readily follows. Lemma 2. Let F be a triangular face of the convex polyhedron P , and assume that each vertex of P lying in F has degree 3. Let q 1 , q 2 and q 3 be the vertices of P on F , and for i = 1, 2, 3, let L i denote the line containing the edge of P through q i that is not contained in F . For i = 1, 2, 3 and τ ∈ R, let q i (τ ) denote the point of L i at the signed distance τ from q i , where we orient each L i in such a way that q i (τ ) is a point of P for any sufficiently small negative value of τ . Let U be a neighborhood of o, and for any t = (τ 1 , τ 2 , τ 3 ) ∈ U , let W (t) = w(P (t)) and C(t) = c(P (t)), where P (t) = conv ((V (P ) \ {q 1 , q 2 , q 3 }) ∪ {q 1 (τ 1 ), q 2 (τ 2 ), q 3 (τ 3 )}). Then the Jacobian of the function W (t)C(t) is nondegenerate at t = o. Proof. It is sufficient to show that the partial derivatives of the examined function span R 3 . Without loss of generality, we may assume that q 1 , q 2 and q 3 are linearly independent. Consider the polyhedron P (τ 1 , 0, 0) for some τ 1 > 0, and let T ( τ 1 ) = conv{q 1 , q 2 , q 3 , q 1 (τ 1 )}, W (τ 1 ) = w(T (τ 1 )) andC(τ 1 ) = c(T (τ 1 )). Let A be the area of the triangle conv{q 1 , q 2 , q 3 }. If τ 1 > 0 is sufficiently small, then ∂ ∂τ 1 W (t)C(t) t=(0,0,0) = sin α 1 A 12 (2q 1 + q 2 + q 3 ), W (τ 1 , 0, 0)C(τ 1 , 0, 0) = w(P )c(P ) +W (τ 1 )C(τ 1 ). SinceC(τ 1 ) = 1 4 (q 1 + q 2 + q 3 + q 1 (τ 1 )), it follows that ∂ ∂τ 1 W (t)C(t) t=(0,0,0) = sin α 1 A 12 (2q 1 + q 2 + q 3 ), where α i denotes the angle between L i and the plane through q 1 , q 2 , q 3 . Using a similar consideration, we obtain the same formula if τ 1 < 0, and similar formulas, where q 2 or q 3 plays the role of q 1 , in the partial derivatives with respect to τ 2 or τ 3 , respectively. Note that 0 < α 1 , α 2 , α 3 ≤ π 2 . Thus, to show that the three partial derivatives are linearly independent, it suffices to show that the vectors 2q 1 + q 2 + q 3 , q 1 + 2q 2 + q 3 and q 1 + q 2 + 2q 3 are linearly independent. To show it under the assumption that q 1 , q 2 , q 3 are linearly independent can be done using elementary computations, which we leave to the reader. Remark 3. We remark that Lemma 2 can be 'dualized' in the following form: Assume that q is a 3-valent vertex of P , and each face of P that q lies on is a triangle. Furthermore, let U be a neighborhood of q, and for any x ∈ U , let W (x) = w (conv ((V (P ) \ {q}) ∪ {x})), and C(x) = c (conv ((V (P ) \ {q}) ∪ {x})). Then the Jacobian matrix of the function W (·)C(·) : U → R 3 is nondegenerate at q. Remark 4. If the Jacobian of a smooth vector-valued function in R 3 is nondegenerate, by the Inverse Function Theorem it follows that the function is surjective. Thus, a geometric interpretation of Lemma 2 and Remark 3 is that under the given conditions, by slight modifications of a vertex or a face of P the function w(P )c(P ) moves everywhere within a small neighborhood of its original position. In the forthcoming two lemmas we investigate the connection between polarity and equilibrium points. Lemma 3. Let S be a nondegenerate simplex in the Euclidean space R d such that o ∈ int S. Then o = c(S • ) if, and only if o = c(S). Proof. Let the vertices of S be denoted by p 1 , p 2 , . . . , p d+1 . For i = 1, 2, . . . , d + 1, let n i denote the orthogonal projection of o onto the facet hyperplane H i of S not containing p i , and let H i be the hyperplane through o and parallel to H i . We remark that since o ∈ int S, none of the p i s and the n i s is zero. Finally, let α i denote the angle between p i and n i . Lemma 4. Let P be a convex d-polytope in the Euclidean space R d such that o ∈ int P , and let P • be its polar. Let F be a k-face of P , where 0 ≤ k ≤ d − 1, and let F denote the corresponding (d − k − 1)-face of P • . Then F contains a nondegenerate equilibrium point of P with respect to o if, and only if F contains a nondegenerate equilibrium point of P • with respect to o. Assume that o = c(S). Then for all values of i, we have dist(p i , H i ) = d dist(H i , H i ), where dist(A, B) = inf{\|a − b\| : a ∈ A, b ∈ B} is the distance Proof. Let F = conv{p i : i ∈ I}, where I is the set of the indices of P such that p i is contained in F , and let p be the orthogonal projection of o onto aff F . Let L = aff(F ∪ {o}), and let L c denote the orthogonal complement of L passing through o. For any facet hyperplane of P containing F , let n j , j ∈ J denote the projection of o onto this hyperplane. Let H + j be the closed half space {q ∈ R d : q, n j ≤ n j , n j }. LetH + i = H + i ∩ H for any i / ∈ I. Finally, letn i be the component of n i parallel to H. Before proving the lemma, we observe that for any given vectors n 1 , n 2 , . . . , n k spanning R d , the following are equivalent: (a) o is an interior point of a polytope Q in R d with outer facet normals n 1 , n 2 , . . . , n k . (b) There are some λ 1 , λ 2 , . . . , λ k > 0 such that o ∈ int Q , where Q = conv{λ 1 n 1 , λ 2 n 2 , . . . , λ k n k }. (c) We have o ∈ int conv{λ 1 n 1 , λ 2 n 2 , . . . , λ k n k } for any λ 1 , λ 2 , . . . , λ k > 0. We note that if a polytope Q satisfies the conditions in (a), then its polar Q = Q • satisfies the conditions in (b), and vice versa. Finally, observe that if F contains an equilibrium point, then by exclusion it is p. We show that p is a nondegenerate equilibrium point of F if, and only if it is contained in the relative interiors of the conic hulls of the p i s as well as those of the n j s. First, let p be a nondegenerate equilibrium point. Then p ∈ relint F , that is, it is in the relative interior of the conic hull (in particular, the convex hull) of the p i s. Observe that since the projection of o onto aff F is p, for any j ∈ J, the projection of n j onto aff F is p. In other words, n j ∈ L = aff(L c ∪{p}) for all j ∈ J. Since p is a vertex of the polytope P ∩ L , the vectors n j , j ∈ J span this linear subspace, or equivalently, the vectorsn j span L c . Observe that the intersection of P with the affine subspace (1 − ε)p + L c , for sufficiently small values of ε > 0, is a (d − k − 1)-polytope, with outer facet normalsn j , j ∈ J, which contains (1 − ε)p in its relative interior. By the observation in the previous paragraph, it follows that o is contained in the relative interior of the convex hull of then j s, which implies that p is contained in the relative interior of the conic hull of the n j s. On the other hand, if p is contained in the relative interior of the conic hull of the p i s, then the fact that p ∈ aff F implies that p ∈ relint F . Furthermore, if p is contained in the relative interior of the conic hull of the n j s, then o is contained in the relative interior of the convex hull of then j s. Thus, the only solution for q ∈ L c of the system of linear inequalities q,n j ≤ 0, where j ∈ J, is q = p, which implies that the only point of P in p + L c is p. This means that p is a nondegenerate equilibrium point of P . Finally, observe that the vertices of F are the points nj \|nj \| 2 , and the projections of o onto the facet hyperplanes of P • containing F are the points pi \|pi\| 2 . Furthermore, aff F = p \|p\| 2 + L c , which yields that the projection of o onto aff F is p \|p\| 2 . Combining it with the consideration in the previous paragraph, this yields the assertion. The next corollary is an immediate consequence of Lemmas 3 and 4 and, together with the result of Conway [5], implies Theorem 4. Corollary 1. Every homogeneous tetrahedron has at least two vertices which are equilibrium points. Furthermore, there are inhomogeneous tetrahedra with exactly one vertex which is an equilibrium point. 3. Polyhedra with many stable or unstable equilibria: proof of Theorem 2 3.1. Proof of Theorem 2 for polyhedral pairs. We need to show that if the class (S, U ) E is defined by a polyhedral pair, then there is a polyhedron with S faces and U vertices. For brevity, we call such a polyhedron a minimal polyhedron in class (S, U ) E . We dothe construction separately in several cases. 3.1.1. Case 1. S = U ≥ 4. Let S ≥ 4, and consider a regular (S − 1)-gon R S in the (x, y)-plane, centered at o and with unit inradius. Let P v (h) be the pyramid with base R v and apex (0, 0, h). By its symmetry properties, P S (h) is a minimal polyhedron in the class (S, S) E for all h > 0. 3.1.2. Case 2. S > 4 and S < U ≤ 2S − 4. In this case the proof is based on Lemma 5. Lemma 5. Assume that P is a minimal polyhedron in class (S, U ) E having a vertex of degree 3. Then there is a minimal polyhedron in class (S + 1, U + 2) E having a vertex of degree 3. Proof. Let P be a minimal polyhedron in class (S, U ) E with a vertex q of degree 3. For sufficiently small ε > 0, let P ε ⊂ P be the intersection of P with the closed half space with inner normal vector c − q, at the distance ε from q. We show that if ε is sufficiently small, then P ε satisfies the conditions in the lemma. If ε is sufficiently small, the boundary of this half space intersects only those edges of P that start at q. Thus, P ε has one new triangular face F , and three new vertices q 1 , q 2 , q 3 on F . Since q is not a vertex of P ε , P ε has S + 1 faces and U + 2 vertices. Furthermore, q 1 , q 2 and q 3 have degree 3, which means that we need only to show that P ε is a minimal polyhedron. To do it, we set c = c(P ) and c ε = c(P ε ). Note that by (2) and (1), every edge of a minimal polyhedron contains an equilibrium point. Thus, by Remark 2, if ε is sufficiently small, then every edge of P ε , apart from those in F , contains an equilibrium point with respect to c ε . We intend to show that if ε is sufficiently small, then the edges of P ε in F also contain equilibrium points with respect to c ε , which, by (2) and (1) clearly implies that P ε is a minimal polyhedron. Consider, e.g. the edge E = [q 1 , q 2 ], and let F 3 be the face of P ε different from F and containing E. Let h and s be the equilibrium point on E and on F 3 , respectively, with respect to c. Let α and β denote the dihedral angles between the planes aff(E ∪ {c}) and aff F , and the planes aff(E ∪ {c}) and aff F 3 , respectively. The fact that h is an equilibrium point with respect to c is equivalent to saying that the orthogonal projection of c onto the line of E is h, and that 0 < α, β < π 2 . Since h is contained in the plane aff{q, c, s} for all values of ε, and it is easy to see that there is some constant γ > 0 independent of ε such that \|q 1 − h\|, \|q 2 − h\| ≥ γ ε. Similarly, an elementary computation shows that for some constant γ > 0 independent of ε, we have 0 < α, β ≤ π 2 − γ ε. Thus, Lemma 1 implies that for small values of ε, E contains an equilibrium point with respect to c ε , implying that P ε is a minimal polyhedron. Now, consider some class (S, U ) E with S > 4 and S < U ≤ 2S − 4. Then, if we set k = U − S and S 0 = S − k, we have 0 < k ≤ S − 4 and 4 ≤ S 0 . In other words, (S, U ) E = (S 0 + k, S 0 + 2k) E for some S 0 ≥ 4 and k > 0. Now, by the proof in Case 1, the class (S 0 , S 0 ) E contains a minimal polyhedron, e.g. a right pyramid P S0 (h) with a regular (S 0 − 1)-gon as its base, where h > 0 is arbitrary. Note that the degree of every vertex of P S0 (h) on its base is 3, and thus, applying Lemma 5 yields a minimal polyhedron in class (S 0 + 1, S 0 + 2) E having a vertex of degree 3. Repeating this argument (k − 1) times, we obtain a minimal polyhedron in class (S, U ) E . Case 3. S > 4 and S 2 + 2 ≤ U < S. Note that these inequalities are equivalent to U > 4 and U < S ≤ 2U − 4. For the proof in this case we need Lemma 6. Lemma 6. Assume that there is a minimal polyhedron P in class (S, U ) E having a triangular face. Then there is a minimal polyhedron P in class (S + 2, U + 1) E having a triangular face F . Proof. Let c = c(P ), and let c F be its orthogonal projection on the plane of F . Since P is a minimal polyhedron, c F is a relative interior point of F , and an equilibrium point with respect to c (see also Fig. 5 for illustration). Letc be the centroid of F and define the vector u asc − c F . Let v be the outer unit normal vector of F , and for any 0 < ε and 0 ≤ α ≤ 1, let T εα denote the tetrahedron with base F and apex q = c F + εv + αu such that T εα ∩ P = F . Let P εα = T εα ∪ P , c = c(P εα ), and c F be the orthogonal projection of c on the plane of F . By Remark 2, for a sufficiently small ε, equilibrium points on all vertices of P εα except q, as well as on all edges and faces of P εα not containing q will be preserved. It is also easy to see from simple geometric considerations that for small values of ε, every face and vertex of P εα contains an equilibrium point with respect to c if q, c F and c are collinear. In the special case of u = 0, those points are obviously collinear. In any other case, it is also straightforward to see that c F ∈ relint conv{c F , c F /4 + 3c/4}. Let us define d(α) = (c F + αu − c F ) · u. Since d continuously varies with α and d(0) < 0, d(1) > 0, for any small ε > 0 there is an α 0 such that apex q and all edges and faces it is contained in have equilibrium points. Remark 5. Note that the argument also yields a polyhedron P such that there is an equilibrium point on each face and at every vertex of P with respect to the original reference point: in this case we may choose the value of α in the proof simply as α = 0. Proof. Table 1 contains an example for a tetrahedron in each of the 9 classes (illustrated in Figure 6) and for the tetrahedron we have n = f +v +e = 14, consequently an upper bound for complexity can be computed as (6) we have the same for the lower bound we proved the claim. Proof. Table 2 contains an example for a pentahedron in each of the 6 classes (illustrated in Figure 6) and for the pentahedron we have n = f + v + e = 18, consequently an upper bound for complexity can be computed as (6) we obtain the same lower bound for all 6 classes so we proved the claim. C(S, U ) ≤ 14 − S − U − H = 16 − 2S − 2U . Since fromC(S, U ) ≤ 18 − S − U − H = 20 − 2S − 2U . From 3.2.3. Case 3. S ≥ 5 and U > 2S − 4, or 2 ≤ S ≤ 4 and U ≥ 6. First, we prove the following lemma. Figure 6. The 8 tetrahedra in Table 1 and the 6 pentahedra in Table 2, the regular tetrahedron and the symmetrical pyramid in equilibrium classes (S, U ) E , S, U ∈ {2, 3, 4, 5} produced by 3D printing. = A y = A z = B x = C z = D z = 0, B y = 1. Then there is convex polyhedron P ∈ (S, U +2) E with f +1 faces and v +2 vertices. Proof. Let the saddle points on E a and E b be denoted by x a and x b . In the proof, based on Remark 2, we show that there is an arbitrarily small truncation of P by a plane that intersects F in a line close to x a and x b that results in two new unstable vertices u a and u b . We choose a suitable truncation from a 2-parameter family of truncations defined as follows: For any t ∈ [0, 1], set y a (t) = tq 2 + (1 − t)q 1 and y b (t) = tq j−1 + (1 − t)q j . Let G(s, t) be the plane that intersects [q 1 , q 2 ] at y a (s) and [q j−1 , q j ] at y b (t), whose angle with the plane of F is a sufficiently small value ε > 0 (the term 'sufficiently small' is explained in the next paragraph) and truncates the vertices q 2 , q 3 , . . . , q j−1 . For i = 2, 3, . . . , j − 1, let q i (s, t) be the intersection of G(s, t) with the edge of P starting at q i and not contained in F . Finally, let P (s, t) be the truncation of P by G(s, t), that is, P (s, t) = cl(P \ conv{y a (s), y b (t), q 2 , ..., q j−1 , q 2 (s, t), . . . , q j−1 (s, t)}. We denote the center of mass of P (s, t) by c(s, t), and the projection of c and c(s, t) onto the plane of F by c F and c F (s, t), respectively. Furthermore, we denote the new edge of P (s, t) starting at y a (s) and different from [y a (s), y b (t)] by Y a (s, t), and define Y b (s, t) similarly. We choose some ε > 0 to satisfy the following conditions: with respect to any point c ∈ V , the original polyhedron P has equilibrium points on the same faces and edges, and at the same vertices, as with respect to the center of mass c of P , where V is the locus of the centers of mass of all truncations of P by the plane G(s, t), s, t ∈ [0, 1]. Furthermore, we assume also that G(s, t) truncates no vertex or equilibrium point of P other than those on F , and that there is some arbitrarily small, fixed value δ > 0 (independent of (s, t)) such that c(s, t) is a Lipschitz function at every (s, t) with Lipschitz constant δ, i.e. \|c(s + ∆s, t + ∆t) − c(s, t)\| ≤ δ (∆s) 2 + (∆t) 2 for all s, t ∈ [0, 1]. First, we show that for some suitable choice of s and t, the orthogonal projections of c(s, t) onto the lines of E a and E b are y a (s) and y b (t), respectively. To do this, we use a consequence of Brouwer's fixed point theorem, the so-called Cube Separation Theorem from [22], which states the following: Let the pairs of opposite facets of a d-dimensional cube K be denoted by F i and F i , i = 1, 2, . . . , d, and let C i , i = 1, 2, . . . , d be compact sets such that C i 'separates' F i and F i , or in other words, K \ C i is the disjoint union of two open sets Q i , Q i such that F i ⊂ Q i , and F i ⊂ Q i . Then d i=1 C i = ∅. To apply this theorem, we set K = {(s, t) : 0 ≤ s, t ≤ 1}, and define Q 1 , C 1 and Q 1 as the set of pairs (s, t) such that the orthogonal projection of c(s, t) onto the line of E a is a relative interior point of [y a (s), q 1 ], coincides with y a (s), or does not belong to [y a (s), q 1 ], respectively. We define Q 2 , C 2 and Q 2 similarly. Then these sets satisfy the conditions of theorem, and we obtain a pair (s,t) with the desired property. Note that by the choice of ε > 0, it holds that in a neighborhood of (s,t), the orthogonal projection of c(s, t) onto the line of Y a (s, t) is in the relative interior of Y a (s, t), and the same holds also for the projection onto the line of Y b (s, t). Now we choose some (s , t ) sufficiently close to (s,t) such that the intersections of G(s , t ) and G(s,t) with F are parallel, and that of G(s , t ) is closer to q 2 and q j−1 than that of G(s,t). By the Lipschitz property of c(s, t), we have that the distance of the two intersection lines is greater than \|c(s , t )−c(s,t)\|, and hence, the projections of c(s , t ) onto the lines of E a and E b lie in the relative interior of the segments [y a (s ), q 1 ] and [y b (t ), q j ], respectively. From this it readily follows that both these edges of P = P (s , t ) and also Y a (s , t ) and Y b (s , t ) contain saddle points with respect to c(s , t ). This implies also that y a (s ) and y b (t ) are vertices of P carrying unstable equilibrium points, and the assertion follows. ii) q 1 contains an unstable and E b = [q j−1 , q j ] contains a saddle-type equilibrium point. Then there exists a polyhedron P ∈ (S, U + 1) E with f + 1 faces and v + 1 vertices. Remark 6. A simplified version of the proof of Lemma 9 can be used to prove the same statement for a fixed reference point c. To prove Theorem 2 in Case 3, we construct a simple polyhedron with U vertices that has S stable and U unstable points. Since any polyhedron in class (S, U ) E has at least U vertices, and among polyhedra with U vertices those with a minimum number of faces are the simple ones, such a polyhedron clearly has minimal mechanical complexity in class (S, U ) E . First, consider the case that S ≥ 5 and U > 2S − 4. Let U 0 = 2S − 4. By the construction in Subsection 3.1, class (S, U 0 ) E contains a simple polyhedron P 0 with U 0 vertices and S faces. Remember that to construct P 0 we started with a tetrahedron T in class (4, 4) E , and in each step we truncated a vertex of the polyhedron sufficiently close to this vertex. Throughout the process, the vertex can be chosen as one of those created during the previous step. Since in this case the conditions of Lemma 9 are satisfied for any face of P 0 , applying Lemma 9 to it we obtain a polyhedron P 1 with two more vertices, one more face, two more unstable and the same number of stable points. By subsequently applying the same procedure, we can construct a convex polyhedron in class (S, U ) E for every even value of U . To obtain a polyhedron in class (S, U ) E where U is odd, we can modify a polyhedron in class (S, U − 1) E according to Corollary 3. Now, consider the case that 2 ≤ S ≤ 4, and U ≥ 6. Then, starting with a tetrahedron in class (S, 4) E (based on the data of Table 1, all three tetrahedra meet the conditions of Lemma 9) we can repeat the argument in the previous paragraph. Theorem 2 in Case 4 can be deduced from Case 3 using direct geometric properties of polarity. Nevertheless, also the proof in Case 3 via Lemma 9 can be dualized as well. In Lemma 10 and Corollary 3 we prove dual versions of Lemma 9 and Corollary 2, respectively, which we are going to use also in Section 4, in our investigation of monostatic polyhedra. Since Theorem 2 follows from Lemma 10 and Corollary 3 similarly like in the proof of Case 3, we leave it to the reader. 3.2.4. We start with the proof using polarity. Considering a tetrahedron T centered at o, a straightforward modification of the construction in Lemma 5 and by Remark 6, we may construct a simple polyhedron P with U vertices that has S stable and U unstable equilibrium points with respect to o. Using small truncations, we may assume that P is arbitrarily close to T measured in Hausdorff distance. Furthermore, without loss of generality, we may assume that a face of T , and all vertices of this face have degree 3 in P . Let this face of T be denoted by F . Recall that P • denotes the polar of P . By Lemma 3, c(T • ) = o, and by the continuity of polar and the center of mass, c(P • ) is 'close' to o. On the other hand, since the vertex q of P • corresponding to F has degree 3, and each face containing q is a triangle, Lemma 2 implies that by a slight modification of q we obtain a polyhedron Q such that c(Q) = o, and a face/edge/vertex of Q contains an equilibrium point with respect to o if, and only if the corresponding vertex/edge/face of P contains an equilibrium point with respect to o. Thus, Q satisfies the required properties. As we mentioned, an alternative way to prove Theorem 2 in Case 4 is using Lemma 10 and Corollary 3. Then there exists a polyhedron P ∈ (S + 2, U ) E with f + 2 faces and v + 1 vertices. Proof. In the proof, we show that for a sufficiently small pyramid erected over the triangle T = conv{q 1 , q j−1 , q j } (which is contained by F and carries a stable equilibrium point) followed by a truncation of P by the plane of the three new faces of the pyramid, results in three new faces instead of F all carrying stable equilibrium and two new edges both carrying saddle-type equilibrium, see Fig. 8a. Let the intersection point of the line through q 1 and c F with E be denoted by x. We choose the apex q of the pyramid from a fixed, sufficiently small neighborhood V of x. Let U be the set of the centers of mass of the modified convex polyhedra, which we denote by P (q). We choose V in such a way that, apart from the three new faces and edges, and the new vertex, P (q) and P have equilibrium points on the same faces and edges, and at the same vertices. Furthermore, we choose V such that for all q ∈ V , the face structure of the resulting polyhedron P (q) is the one described in the previous paragraph, and for any y ∈ U , the Euclidean distance function from y on [q j−1 , q] ∪ [q, q j ] has a unique local minimum, and this point is different from q, for all q ∈ V . Note that the latter condition implies that the new Figure 8. Increasing the number of stable equilibria by two. Schematic view of the pyramid with three light faces instead of the original dark one denoted as F (a); view perpendicular to face F : full circles mean stable equilibrium points, the empty circle is the projection of c(α, β, γ) onto F (b); Illustration for the application of the Cube Separation Theorem for compact sets X α and X β (c). vertex q is not an unstable equilibrium point. Thus, we need to prove only that, with a suitable choice of q, all the three new faces contain a new stable equilibrium point. We parametrize q using the following parameters: • the angle α of the plane of conv{q j−1 , q j , q} and the plane of F . Here we assume that 0 ≤ α ≤ α 0 , where the sum of α 0 and the dihedral angle of P at E is π. • the angle β between two rays, both starting at q 1 , and containing q j−1 and the orthogonal projection q F of q onto the plane of F , respectively. Here we set β 1 ≤ β ≤ β 2 , where [β 1 , β 2 ] is a sufficiently small interval containing the angle ∠q j−1 q 1 c F . • the angle γ between the ray starting at q 1 and containing q, and the plane of F . Here we assume that 0 < γ < γ 0 for some small, fixed value γ 0 . We choose the values of β 1 , β 2 , γ 0 such that in the permitted range of the parameters, q ∈ V . For brevity, we may refer to P (q(α, β, γ)) as P (α, β, γ), c(P (α, β, γ)) as c(α, β, γ) and observe that these three quantities determine q. Note that, using the idea of the proof of Lemma 1, we have that \|c(P (q))−c(P )\| = O(γ), and for some constant C > 0 independent of α, β, γ, if \|α − α\| ≤ γ, then \|c(α , β, γ) − c(α, β, γ)\| ≤ Cγ 2 . Fix some γ > 0, and let X α be the set of pairs (α, β) ∈ [0, α 0 ] × [β 1 , β 2 ] such that the planes through E, and containing c(α, β, γ) and q(α, β, γ), respectively, are perpendicular. Furthermore, let X β be the set of pairs (α, β) ∈ [0, α 0 ] × [β 1 , β 2 ] such that q 1 , and the projections of c(α, β, γ) and q(α, β, γ) onto the plane of F are collinear. If γ > 0 is sufficiently small, the property \|c(P (q)) − c(P )\| = O(γ) implies that X α strictly separates the sets {(0, β) : β ∈ [β 1 , β 2 ]} and {(α 0 , β) : β ∈ [β 1 , β 2 ]}, and X β strictly separates the sets {(α, β 1 ) : α ∈ [0, α 0 ]} and {(α, β 2 ) : α ∈ [0, α 0 ]}. Since X α and X β are compact, we may apply the Cube Separation Theorem [22] as in the proof of Lemma 9. From this, it follows that there is some (α γ , β γ ) ∈ X α ∩X β . It is easy to see that (α γ , β γ ) ∈ X α implies that for sufficiently small values γ, P (α γ , β γ , γ) has stable equilibrium points on both faces containing the new edge [q 1 , q(α γ , β γ , γ)]. Furthermore, the orthogonal projection of c(α γ , β γ , γ) onto the plane containing E and q = q(α γ , β γ , γ) lies on E. Now, let us replace α γ by α = α γ −γ. Then, since in this case \|c(α , β γ , γ)−c(α γ , β γ , γ)\| ≤ Cγ 2 , we have that if γ is sufficiently small, then the orthogonal projection of c(α , β γ , γ) onto the face conv{q j−1 , q j , q(α , β γ , γ)} lies inside the face; that is, P has a stable equilibrium point on this face. This yields the assertion. Corollary 3. If all conditions (i)-(iv) of Lemma 10 hold, then there is a polyhedron P ∈ (S + 1, U ) E with f + 2 faces and v + 1 vertices. Monostatic polyhedra: proof of Theorem 3 Our theory of mechanical complexity highlights the special role of polyhedra in the first row and first column of the (S, U ) grid. These objects have either only one stable equilibrium point (first row) or just one unstable equilibrium point (first column) and therefore they are called collectively monostatic. In particular, the first row is sometimes referred to as mono-stable and the first column as mono-unstable. Our theory provided only a rough lower bound for their mechanical complexity. While no general upper bound is known, individual constructions provide upper bounds for some particular classes; based on these values one might think that the mechanical complexity of these classes, in particular when both S and U are relatively low, is very high. Monostatic objects have peculiar properties, apparently the overall shape in these equilibrium classes is constrained. In [30] the thinness T and the flatness F of convex bodies is defined (1 ≤ T, F ≤ ∞) and it is shown that, for nondegenerate convex bodies, T = 1 if and only if U = 1 and F = 1 if and only if S = 1. This constrained overall geometry may partly account for the high mechanical complexity of monostatic polyhedra. 4.1. Known examples. The first (and probably best) known such object is the monostatic polyhedron P C constructed by Conway and Guy in 1969 [4](cf. Figure 9) having mechanical complexity C(P C ) = 96. Recently, there have been two additions: the polyhedron P B by Bezdek [1] (cf. Figure 10) and the polyhedron P R by Reshetov [23] with respective mechanical complexities C(P B ) = 64 and C(P R ) = 70. It is apparent that all of these authors were primarily interested in minimizing the number of faces on the condition that there is only one stable equilibrium, so, if one seeks minimal complexity in any of these classes it is possible that these constructions could be improved. Also, as we show below, the same ideas can be used to construct examples of mono-unstable polyhedra. The construction in [4] relies on a delicate calculation for a certain discretized planar spiral, defining a planar polygon P , serving as the basis of a prism which is truncated in an oblique manner (cf. Figure 9). The spiral consists of 2m similar right triangles, each having an angle β = π/m at the point o. The cathetus of the smallest pair of triangles has length r 0 , and this will be the vertical height of o when the solid stands in stable equilibrium. We denote the height of the center of mass c by r in the same configuration. It is evident from the construction that if P is a homogeneous planar disc then we have r > r 0 since such a disc cannot be monostatic [15]. However, it is also clear that for a non-uniform mass distribution resulting in r < r 0 , P would be monostatic (cf. Figure 9). In the construction of Conway and Guy we can regard r as a function r(a, b) of the geometric parameters a, b (cf. Figure 9). Apparently, r(0, b) = r 1 and r(a, 0) = r 2 are constants. If P is the aforementioned homogeneous disc then we have r = r 2 > r 0 . Next we state a corollary to the main result of [4]: Corollary 4. If m ≥ 9 then r 1 < r 0 . Fig. 9. Now erect a mirrorsymmetric pyramid over the polygon with its apex close to the bottom edge: the vertical coordinate of the body centre of the pyramid will then be close to 3r 3 /4. It can be shown that for a sufficiently flat pyramid (we call it P 3 ) will be in classes (3, 1) E and (18, 18) C . Introducing a small asymmetry to P 3 by moving the apex off the symmetry plane, a polyhedron P 2 is obtained which belongs to classes (2, 1) E and (18, 18) C . These 'mono-unstable' polyhedra are illustrated in Figure 11. An overview of the discussed monostatic polyhedra is shown in Figure 12 on an overlay of the (f, v) C and (S, U ) E grids. Proof of Theorem 3. Proof. Consider the case C(1, U ) first. The polyhedron P C has a narrow rectangular face with a stable point and two saddle points on opposite short edges of the same face. They do not satisfiy condition (i) of Lemma 9 because of being collinear, but both 17-gonal faces of P C can slightly be rotated to get P C according to Remark 2 in a way that no equilibrium points appear or disappear but the two edges with saddle points become nonparallel, and thus Lemma 9 turns to be applicable. Since the same face of P C contains four unstable points as well (and none of them is collinear with the stable and any saddle point), Corollary 2 can directly be applied to get P D with C(P D ) = C(P C )+2 = 98. It means that C(1, 4) ≤ 2R(1, 4)+90 and C(1, 5) ≤ 2R(1, 5) + 90. Applying now Lemma 9 on both P C and P D successively, P 3 P 2 Figure 11. Schematic view of two polyhedra P 3 ∈ (3, 1) E , (18, 18) C and P 2 ∈ (2, 1) E , (18, 18) C , obtained by using the ideas of the Conway and Bezdek constructions. Stable, unstable and saddle-type equilibria are marked with s i , u j , h k . In case of P 3 we have i = 1, 2, 3, j = 1, k = 1, 2 and in case of P 2 we have i = 1, 2, j = 1, k = 1. Complexity can be computed as C(P 3 ) = 2(18 + 18 − 3 − 1) = 64, C(P 2 ) = 2(18 + 18 − 2 − 1) = 66 the assertion readily follows. Note that P B could not be used as departure instead of P C , since its saddle points are not on edges of the same face. A similar path is taken for the case C(S, 1). Depart now P 3 with C(P 3 ) = 64: that polyhedron has a 17-gonal face with a stable equilibrium and there is a vertex and an edge on its perimeter having an unstable and a saddle point, respectively. Now it is possible again to slightly rotate the plane of the symmetric triangular face about an axis which is perpendicular to the 17-gon and runs through the apex of the pyramid, making the stable (s 3 ) and saddle (h 1 ) point to move off the symmetry axis of the 17-gon, so that they become non-collinear with u 1 (Remark 2 guarantees that it can always be done without changing the number of equilibrium points of any kind). Applying or not Corollary 3 first then Lemma 10 successively gives C(S, 1) ≤ S + 61 and C(S, 1) ≤ S + 62 for odd and even S, respectively, which is equivalent to the second statement of the theorem. 4.4. Gömböcedron prize. While the construction of monostatic polyhedra with less than 34 edges appears to be challenging (cf. Figure 12), the only case which has been excluded is the tetrahedron with e = 6 edges. It also appears to be very likely that Gömböc-like polyhedra in class (1, 1) E do exist, however, based on this chart and the previous results, one would expect polyhedra with high mechanical complexity. To further motivate this research we offer a prize for establishing the mechanical complexity C(1, 1), the amount p of the prize is given in US dollars as (9) p = 10 6 C(1, 1) . Generalizations and applications 5.1. Complexity of secondary equilibrium classes. A special case of Theorem 2 states that for any polyhedral pair (f, v) one can construct a homogeneous polyhedron P with f faces and v vertices in such a manner that C(P ) = 0. In other words, in any primary combinatorial class there exist polyhedra with zero complexity. A natural generalization of this statement is to ask whether this is also true for any secondary combinatorial class of convex polyhedra. While we do not have this result, we present an affirmative statement for the inhomogeneous case: Proof. By (1) and (2) it is sufficient to show that every edge of P contains an equilibrium point with respect to o. Let E be an edge of P that touches S 2 at a point q, and let H be the plane touching S 2 at q. Clearly, H is orthogonal to q, and since every face of P intersects the interior of the sphere, we have H ∩ P = E. Thus, q is an equilibrium point of P with respect to o. Since a variant of the Circle Packing Theorem [3] states that every combinatorial class contains a Koebe polyhedron, it follows that every combinatorial class contains an inhomogeneous polyhedron with zero mechanical complexity. To find a homogeneous representative appears to be a challenge. In [19], the author strengthened the result in [3] by showing the existence of a Koebe polyhedron P in each combinatorial class such that the center of mass of the k-dimensional skeleton of P , where k = 0, 1 or 2, coincides with o. This result and Proposition 1 imply that replacing c(P ) by the center of mass of the k-skeleton of a polyhedron with 0 ≤ k ≤ 2, every combinatorial class contains a polyhedron with zero mechanical complexity. 5.2. Inverse type questions. The basic goal of this paper is to explore the nontrivial links between the combinatorial (f, v) C and the mechanical (S, U ) E classification of convex polyhedra. The concept of mechanical complexity (Definition 2) helps to explore the (S, U ) E → (f, v) C direction of this link. Inverse type questions may be equally useful to understand this relationship: for example, a natural question to ask is the following: Is it true that any equilibrium class (S, U ) E intersects all but at most finitely many combinatorial classes (f, v) C ? Here it is worth noting that it is easy to carry out local deformations on a polyhedron that increase the number of faces and vertices, but not the number of equilibria. Alternatively, one may ask to provide the list of all (S, U ) E classes represented by homogeneous polyhedra in a given combinatorial class (f, v) C . A similar question may be asked for a secondary combinatorial class of polyhedra. In general, we know little about the answers, however we certainly know that (4) holds and we also know that S = f, U = v is a part of this list. The minimal values for S and U are less clear. In particular, based on our previous results it appears that the values S = 1 and U = 1 can be only achieved for sufficiently high values of f, v. On the other hand, Theorem 4 and Lemma 7 resolve this problem at least for the (4, 4) C class. The latter is based on a global numerical search and this could be done at least for some polyhedral classes, although the computational time grows with exponent (f + v). 5.3. Inhomogeneity and higher dimensions. While here we described only 3D shapes, the generalization of Definitions 2 and 3 to arbitrary dimensions is straightforward. While the actual values of mechanical complexity are trivial in the planar case (class (2) E has mechanical complexity 2 and every other equilibrium class has mechanical complexity zero), the d > 3 dimensional case appears an interesting question in the light of the results of Dawson et al. on monostatic simplices in higher dimensions [5,6,7]. We formulated all our results for homogeneous polyhedra, nevertheless, some remain valid in the inhomogeneous case which also offers interesting open questions. In particular, the universal lower bound (4) is independent of the material density distribution so it remains valid for inhomogeneous polyhedra and as a consequence, so does Theorem 2. However, our other results (in particular the bounds for monostatic equilibrium classes) are only valid for the homogeneous case. In the latter context it is interesting to note that Conway proved the existence of inhomogeneous, monostatic tetrahedra [5]. Applications. Here we describe some problems in mineralogy, geomorphology and industry where the concept of mechanical complexity could potentially contribute to the efficient description and the better understanding of the main phenomena. 5.4.1. Crystal shapes. Crystal shapes are probably the best known examples of polyhedra appearing in Nature and the literature on their morphological, combinatorial and topological classification is substantial [17]. However, as crystals are not just geometric objects but also (nearly homogeneous) 3D solids, their equilibrium classification appears to be relevant. The number of static balance points has been recognized as a meaningful geophysical shape descriptor [11,13,29] and it has also been investigated in the context of crystal shapes [28]. The theory outlined in our paper may help to add new aspects to their understanding. While the study of a broader class of crystal shapes is beyond the scope of this paper, we can illustrate this idea in Figure 13 by two examples of quartz crystals with identical number of faces displaying a large difference in mechanical complexity. The length a of the middle, prismatic part of the hexagonal crystal shape (appearing on the left side of Figure 13) is not fixed in the crystal. As we can observe, for sufficiently small values of the length a the crystal will be still in the same combinatorial class (18,14) C , however, its mechanical complexity will be reduced to zero. b b a Figure 13. Quartz crystals. Left: Hexagonal habit in classes (18,14) C and (6, 2) E , C(P ) = 48. Right: Cumberland habit [32] in classes (18,32) 5.4.2. Random polytopes, chipping models and natural fragments. There is substantial literature on the shape of random polytopes [25] which are obtained by successive intersections of planes at random positions. Under rather general assumptions on the distribution of the intersecting planes it can be shown [25] that the expected primary combinatorial class of such a random polytope is (6, 8) C , however, there are no results on the mechanical complexity. A very special limit of random polytopes can be created if we use a chipping model [24,18] where one polytope is truncated with planes in such a manner that the truncated pieces are small compared to the polytope. Although not much is known about the combinatorial properties of these polytopes, it can be shown [12] that under a sufficiently small truncation the mechanical complexity either remains constant or it increases (this is illustrated in Figure 1). Apparently, random polytopes can be used to approximate natural fragments [10]. There is data available on the number and type of static equilibria of the latter, so any result on the mechanical complexity of random polytopes could be readily tested and also used to identify fragmentation processes. 5.4.3. Assembly processes. In industrial assembly processes parts are processed by a feeder and often these parts can be approximated by polyhedra. These polyhedra arrive in a random orientation on a horizontal surface (tray) and end up ultimately on one of their faces carrying a stable equilibrium. Based on the relative frequency of this position, one can derive face statistics and the throughput of a part feeder is heavily influenced by the face statistics of the parts processed by the feeder. Design algorithms for feeders are often investigated from this perspective [2,31]. It is apparent that one key factor determining the entropy of the face statistics is the mechanical complexity of the polyhedron, in particular, higher mechanical complexity leads to better predictability of the assembly process so this concept may add a useful aspect to the description of this industrial problem. Concluding remarks. We showed an elementary connection between the Euler and Poincaré-Hopf formulae (1) and (2): the mechanical complexity of a polyhedron is determined jointly by its equilibrium class (S, U ) E and combinatorial class (f, v) C . Mechanical complexity appears to be a good tool to highlight the special properties of monostatic polyhedra and offers a new approach to the classification of crystal shapes. We defined polyhedral pairs (x, y) of integers (cf. Definition 4) and showed that they play a central role in both classifications: they define all possible combinatorial classes (f, v) C while in the mechanical classification they correspond to classes with zero complexity. Figure 1 Figure 1 . 11shows three polyhedra where the values for all these quantities can be compared. (a) Three polyhedra interpreted as homogeneous solids with given numbers for faces (f ), vertices (v), edges (e), stable equilibria (S), unstable equilibria (U ) and saddle-type equilibria (H), their respective sums n = f + v + e, N = S + U + H and mechanical complexity C = n−N (given in Definition 2). (b) Polyhedron in column a3 shown on the overlay of the (S, U ) and (f, v) grids, complexity obtained from distance between corresponding diagonals. Theorem 1 . 1For any positive integers f, v, there is a convex polyhedron P with f faces and v vertices if and only if (f, v) is a polyhedral pair. Remark 1 . 1Let (S, U ) E be a primary equilibrium class with S, U ≥ 1, and let R(S, U ) = inf{f + v − S − U : (f, v) is a polyhedral pair and f ,v satisfy (4)}. (a) for classes (S, S) E with S ≥ 4, suitably chosen pyramids have zero mechanical complexity (Section 3).(b) for classes 1 < S ≤ 5 and 1 < U ≤ 5, (S, U ) E = (4, 4) E , (5, 5) E , we provide examples found by computer search (Subsection 3.2, Tables 1 and 2). (c) for polyhedral classes with S = U , we construct examples by recursive, local manipulations of the pyramids mentioned in (a) (Subsection 3.1). Figure 2 . 2Summary of results for S, U ≤ 10. Left panel: the (S, U ) grid with some selected polyhedra as examples. Polyhedral pairs on the (S, U ) grid have white background. The function R(S, U ) illustrated for classes (2, 2) E , (2, 9) E , (10, 3) E . Right panel: Mechanical complexity of equilibrium classes (S, U ) E . Polyhedral pairs on the (S, U ) grid have white background. Sharp values for mechanical complexity C(S, U ) are given as integers without brackets. Figure 3 . 3Summary of the proof. Left panel: Symbols on the (S, U ) grid indicate how polyhedra in the given equilibrium class (S, U ) E have been constructed. Dark background corresponds to classes where polyhedra have been identified by computer search. Light grey background corresponds to polyhedral pairs. Symbols are explained in the right panel. For S, U > 1 the indicated constructions provide minimal complexity and thus the complexity of the class itself. Zero indicates that no polyhedron is known in that class. Right panel: Symbols in the left panel explained briefly with reference to sections, subsections and sub-subsections of the paper. Figure 4 . 4Summary of the proof. (a)-(b) Upper row: schematic picture of local manipulations L1-L6, showing local face structure and equilibria on original and manipulated polyhedra P and P , respectively. Lower rows: Original and manipulated polyhedra P and P shown on the (f, v) and (S, U ) grids. of the sets A and B. This implies that the projection of p i onto the line through o and n i is −dn i for all values of i, or in other words, (7) cos α i \|p i \| = −d\|n i \| for all values of i. On the other hand, it is easy to see that if (7) holds for all values of i, then o = c(S). The vertices of S • are the points p i = ni \|ni\| 2 , where i = 1, 2, . . . , d + 1, and the projection of o onto the facet hyperplane of P • not containing p i is n i = pi \|pi\| 2 . Hence, the angle between p i and n i is α i . Similarly like in the previous paragraph, o = c(S • ) if, and only if (8) cos α i \|p i \| = −d\|n i \| holds for all values of i. On the other hand, if cos α i \|p i \| = −d\|n i \| for some value of i, then cos α i \|p i \| = cos αi \|ni\| = − d \|pi\| = −d\|n i \|, and vice versa. Thus, (7) and (8) are equivalent, implying Lemma 3. Figure 5 . 5Building a tetrahedron on a triangular face of a convex polyhedron Now we prove Theorem 2 for Case 3. Like in Case 2, if we set k = S − U and S 0 = U − k, then S 0 ≥ 4, k > 0, and (S, U ) E = (S 0 + 2k, S 0 + k) E . Consider the right pyramid P S0 (h) in Case 1. This pyramid has S 0 faces consisting of S 0 − 1 triangles and one regular (S 0 −1)-gon shaped face. Thus, applying the construction in Lemma 6 k times subsequently yields the desired polyhedron.3.2. Proof of Theorem 2 for non-polyhedral pairs. 3.2.1. Case 1. 2 ≤ S ≤ 4 and 2 ≤ U ≤ 4. Lemma 7. Let S, U ∈ {2, 3, 4}. Then C(S, U ) = 2R(S, U ). 1 . 1Examples for tetrahedra in equilibrium classes (S, U ) E , S, U ∈ {2, 3, 4}, (S, U ) = 4, 4. Constant vertex coordinates for all tetrahedra are A x = A y = A z = B y = C z = 0, B x = 1. 3.2.2. Case 2. 2 ≤ S ≤ 4, U = 5 or 2 ≤ U ≤ 4, S = 5 . This case follows from Lemma 8. Lemma 8. Let 2 ≤ S ≤ 4, U = 5 or 2 ≤ U ≤ 4, S = 5. Then C(S, U ) = 2R(S, U ). Lemma 9 . 9Let P ∈ (S, U ) E be a convex polyhedron with f faces and v vertices. Let q i , i = 1...j be successive vertices of an m-gonal (m ≥, j ≥ 3) face F of P such that i) the lines aff({q 1 , q 2 }) and aff({q j−1 , q j }) intersect at some point q with the property \|q − q 1 \| > \|q − q 2 \|; ii) both edges E a = [q 1 , q 2 ] and E b = [q j−1 , q j ] contain saddle points; iii) the vertices q i , i = 2...j − 1 are trivalent. ( Figure 7 . 7Increasing the number of unstable equilibria by two. Views perpendicular to the plane F (a) and edge [u a , u b ] (b). Corollary 2 . 2Let conditions (i) and (iii) of Lemma 9 hold and (ii) be modified as follows: Case 4 . 4U ≥ 5 and S > 2U − 4, or 2 ≤ U ≤ 4 and S ≥ 6. Lemma 10 . 10Let P ∈ (S, U ) E be a convex polyhedron with f faces and v vertices.Let q i , i = 1...j − 1, j, ...m(j ≥ 3) be successive vertices of an m-gonal (m ≥ 3) face F of P such thati) P has a stable equilibrium point c F on F , which is contained in the relative interior of the triangle T = conv{q 1 , q j−1 , q j }; ii) the edge E = [q j−1 q j ] contains a saddle-type equilibrium point c E ; iii) the vertices q i , i = 2, . . . , j − 1 and i = j + 1, . . . , m are trivalent; iv) q 1 , c F and c E are not collinear. Figure 9 . 9Schematic view of the monostatic polyhedron P C ∈ (1, 4) E , (19, 34) C constructed by Conway and Guy in 1969[4]. Stable, unstable and saddle-type equilibria are marked with s i , u j , h k , i = 1, j = 1, 2, 3, 4, k = 1, 2, 3, respectively. Complexity can be computed as C(P C ) = 2(19 + 34 − 1 − 4) = 96 Figure 10 . 10Schematic view of the monostatic polyhedron P B ∈ (1, 3) E , (18, 18) C constructed by Bezdek in 2011 [1]. Stable, unstable and saddle-type equilibria are marked with s i , u j , h k , i = 1, j = 1, 2, 3k = 1, 2, respectively. Complexity can be computed as C(P B ) = 2(18 + 18 − 1 − 3) = 64 4.2. Examples in (3, 1) E and (2, 1) E . Consider a Conway construction with b = 0 and denote its vertical centroidal coordinate by r 3 : it equals the centroidal coordinate of a plane polygon depicted on the right of Figure 12 . 12Polyhedra with a single stable or unstable equilibrium point. The grid shown is an overlay of the (f, v) and the (S, U ) grids. White squares correspond to polyhedral pairs. Location of monostatic polyhedra is shown with black capital letters on the (f, v) grid and white capital letters on the (S, U ) grid. Abbreviations: P C : Conway and Guy, 1969[4], P B : Bezdek, 2011[1], P R : Reshetov, 2014[23]. P 2 , P 3 : current paper,Figure 11. Complexity for these polyhedra can be readily computed as C(P C ) = 96, C(P B ) = 64, C(P R ) = 70, C(P 3 ) = 64, C(P 2 ) = 66. 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[ "Build Scripts with Perfect Dependencies", "Build Scripts with Perfect Dependencies" ]	[ "Sarah Spall \nSAM TOBIN-HOCHSTADT\nIndiana University\nIndiana University\n\n", "FacebookNeil Mitchell \nSAM TOBIN-HOCHSTADT\nIndiana University\nIndiana University\n\n" ]	[ "SAM TOBIN-HOCHSTADT\nIndiana University\nIndiana University\n", "SAM TOBIN-HOCHSTADT\nIndiana University\nIndiana University\n" ]	[]	Build scripts for most build systems describe the actions to run, and the dependencies between those actionsbut o en build scripts get those dependencies wrong. Most build scripts have both too few dependencies (leading to incorrect build outputs) and too many dependencies (leading to excessive rebuilds and reduced parallelism). Any programmer who has wondered why a small change led to excess compilation, or who resorted to a "clean" step, has su ered the ill e ects of incorrect dependency speci cation. We outline a build system where dependencies are not speci ed, but instead captured by tracing execution. e consequence is that dependencies are always correct by construction and build scripts are easier to write. e simplest implementation of our approach would lose parallelism, but we are able to recover parallelism using speculation. change, saving signi cant work when only util.c changes. M can also run multiple commands in parallel when neither depends on the other, such as the two invocations of gcc -c. With a shell script, these are both challenging research problems in incremental computing and automatic parallelization, respectively, and unlikely to be solvable for arbitrary programs such as gcc.Builds without dependenciesIn this paper we show how to take the above shell script and gain most of the bene ts of a M build ( §2). Firstly, we can skip those commands whose dependencies haven't changed by tracing which les they read and write ( §2.4) and keeping a history of such traces. Secondly, we can run some commands in parallel, using speculation to guess which future commands won't interfere with things already running ( §2.8). e key to speculation is a robust model of what "interfering" means -we call a problematic interference a hazard, which we de ne in §2.6.We have implemented these techniques in a build system called R 1 , introduced in §3, which embeds commands in a Haskell script. A key part of the implementation is the ability to trace commands, whose limitations we describe in §3.4. We show that our design decisions produce correct builds in by formalizing hazards and demonstrating important properties about the safety of speculation in §4. To evaluate our claims, and properly understand the subtleties of our design, we converted existing M scripts into R scripts, and discuss the performance characteristics and M script issues uncovered in §5. We also implement two small but non-trivial builds from scratch in R and report on the lessons learned. Our design can be considered a successor to the M build system (McCloskey 2008), and we compare R with it and other related work in §6. Finally, in §7 we conclude and describe future work.BUILD SCRIPTS FROM COMMANDSOur goal is to design a build system where a build script is simply a list of commands. In this section we develop our design, starting with the simplest system that just executes all the commands in order, and ending up with the bene ts of a conventional build system.Executing commandsGiven a build script as a list of commands, like in §1, the simplest execution model is to run each command sequentially in the order they were given. Importantly, we require the list of commands is ordered, such that any dependencies are produced before they are used. We consider this sequential execution the reference semantics, and as we develop our design further, require that any optimised/cached implementation gives the same results.Value-dependent commandsWhile a static list of commands is su cient for simple builds, it is limited in its expressive power. Taking the build script from §1, the user might really want to compile and link all .c les -not just those explicitly listed by the script. A more powerful script might be: FILES=$(ls *.c) for FILE in $FILES; do gcc -c $FILE done gcc -o main.exe ${FILES% 1 h ps://github.com/ndmitchell/ra le ,	10.1145/3428237	[ "https://arxiv.org/pdf/2007.12737v1.pdf" ]	220,793,362	2007.12737	891139139d05496165ba9b9b0bf025bf884a4fba	Build Scripts with Perfect Dependencies Sarah Spall SAM TOBIN-HOCHSTADT Indiana University Indiana University FacebookNeil Mitchell SAM TOBIN-HOCHSTADT Indiana University Indiana University Build Scripts with Perfect Dependencies 1 Build scripts for most build systems describe the actions to run, and the dependencies between those actionsbut o en build scripts get those dependencies wrong. Most build scripts have both too few dependencies (leading to incorrect build outputs) and too many dependencies (leading to excessive rebuilds and reduced parallelism). Any programmer who has wondered why a small change led to excess compilation, or who resorted to a "clean" step, has su ered the ill e ects of incorrect dependency speci cation. We outline a build system where dependencies are not speci ed, but instead captured by tracing execution. e consequence is that dependencies are always correct by construction and build scripts are easier to write. e simplest implementation of our approach would lose parallelism, but we are able to recover parallelism using speculation. change, saving signi cant work when only util.c changes. M can also run multiple commands in parallel when neither depends on the other, such as the two invocations of gcc -c. With a shell script, these are both challenging research problems in incremental computing and automatic parallelization, respectively, and unlikely to be solvable for arbitrary programs such as gcc.Builds without dependenciesIn this paper we show how to take the above shell script and gain most of the bene ts of a M build ( §2). Firstly, we can skip those commands whose dependencies haven't changed by tracing which les they read and write ( §2.4) and keeping a history of such traces. Secondly, we can run some commands in parallel, using speculation to guess which future commands won't interfere with things already running ( §2.8). e key to speculation is a robust model of what "interfering" means -we call a problematic interference a hazard, which we de ne in §2.6.We have implemented these techniques in a build system called R 1 , introduced in §3, which embeds commands in a Haskell script. A key part of the implementation is the ability to trace commands, whose limitations we describe in §3.4. We show that our design decisions produce correct builds in by formalizing hazards and demonstrating important properties about the safety of speculation in §4. To evaluate our claims, and properly understand the subtleties of our design, we converted existing M scripts into R scripts, and discuss the performance characteristics and M script issues uncovered in §5. We also implement two small but non-trivial builds from scratch in R and report on the lessons learned. Our design can be considered a successor to the M build system (McCloskey 2008), and we compare R with it and other related work in §6. Finally, in §7 we conclude and describe future work.BUILD SCRIPTS FROM COMMANDSOur goal is to design a build system where a build script is simply a list of commands. In this section we develop our design, starting with the simplest system that just executes all the commands in order, and ending up with the bene ts of a conventional build system.Executing commandsGiven a build script as a list of commands, like in §1, the simplest execution model is to run each command sequentially in the order they were given. Importantly, we require the list of commands is ordered, such that any dependencies are produced before they are used. We consider this sequential execution the reference semantics, and as we develop our design further, require that any optimised/cached implementation gives the same results.Value-dependent commandsWhile a static list of commands is su cient for simple builds, it is limited in its expressive power. Taking the build script from §1, the user might really want to compile and link all .c les -not just those explicitly listed by the script. A more powerful script might be: FILES=$(ls .c) for FILE in $FILES; do gcc -c $FILE done gcc -o main.exe ${FILES% 1 h ps://github.com/ndmitchell/ra le , INTRODUCTION Every non-trivial piece of so ware includes a "build system", describing how to set up the system from source code. Build scripts (Mokhov et al. 2018) describe commands to run and dependencies to respect. For example, using the M build system (Feldman 1979), a build script might look like: is script contains three rules. Looking at the rst rule, it says main.o depends on main.c, and is produced by running gcc -c main.c. What if we copy the commands into a shell script? We get: gcc -c main.c gcc -c util.c gcc -o main.exe main.o util.o at's shorter, simpler and easier to follow. Instead of declaring the outputs and dependencies of each command, we've merely given one valid ordering of the commands (we could equally have put gcc -c util.c rst). is simpler speci cation has additional bene ts. First, we've xed some potential bugs -these commands depend on the undeclared dependency gcc, and whatever header les are used by main.c and util.c. ese bugs could be xed in the Make le by adding a dependency on the version of gcc used, as well as by listing every header le transitively included by main.c and util.c Furthermore, as the les main.c and util.c evolve, and their dependencies change (by changing the #include directives), the shell script remains correct, while the M script must be kept consistent or builds will become incorrect. Why dependencies? Given the manifest simplicity of the shell script, why write a Makefile? Build systems such as M have two primary advantages, both provided by dependency speci cation: incrementality and parallelism. M is able to re-run only the commands needed when a subset of the les is script now has a curious mixture of commands (ls, gcc), control logic (for) and simple manipulation (changing le extension 2 ). Importantly, there is no xed list of commands -the future commands are determined based on the results of previous commands. Concretely, in this example, the result of ls changes which gcc commands are executed. e transition from a xed list of commands to a dynamic list matches the Applicative vs Monadic distinction of Mokhov et al. (2018, §3.5). ere are three approaches to modelling a dynamic list of commands: (1) We could consider the commands as a stream given to the build system one by one as they are available. e build system has no knowledge of which commands are coming next or how they were created. In this model, a build script supplies a stream of commands, with the invariant that dependencies are produced before they are used, but provides no further information. e main downside is that it becomes impossible to perform any analysis that might guide optimisation. (2) We could expose the full logic of the script to the build system, giving a complete understanding of what commands are computed from previous output, and how that computation is structured. e main downside is that the logic between commands would need to be speci ed in some constrained domain-speci c language (DSL) in order to take advantage of that information. Limiting build scripts to a speci c DSL complicates writing such scripts. (3) It would be possible to have a hybrid approach, where dependencies between commands are speci ed, but the computation is not. Such an approach still complicates speci cation (some kind of dependency speci cation is required), but would allow some analysis to be performed. In order to retain the desired simplicity of shell scripts, we have chosen the rst option, modelling a build script as a sequence of commands given to the build system. Future commands may depend on the results of previous commands in ways that are not visible to the build system. e commands are produced with "cheap" functions such as control logic and simple manipulations, for example, using a for loop to build a list of object les. We consider the cheap commands to be xed overhead, run on every build, and not cached or parallelised in any way. If any of these cheap manipulations becomes expensive, they can be replaced by a command, which will then be visible to the build system. e simple list of commands from §1 is a degenerate case of no interesting logic between commands. An important consequence of the control logic not being visible to the build system is that the build system has no prior knowledge of which commands are coming next, or if they have changed since it last executed them. As a result, even when the build is a simple static script such as from §1, when it is manually edited, the build will execute correctly. e build system is unaware if you edited the script, or if the commands were conditional on something that it cannot observe. erefore, this model solves the problem of self-tracking from Mokhov et al. (2018, §6.5). Dependency tracing For the rest of this section we assume the existence of dependency tracing which can tell us which les a command accesses. Concretely, we can run a command in a special mode such that when the command completes (and not before) we can determine which les it read and wrote; these les are considered to be the command's inputs and outputs respectively. We cannot determine at which point during the execution these les were accessed, nor which order they were accessed in. 1:4 Sarah Spall, Neil Mitchell, and Sam Tobin-Hochstadt We cannot prevent or otherwise redirect an in-progress access. We discuss the implementation of dependency tracing, and the reasons behind the (frustrating!) limitations, in §3.4. Skipping unnecessary commands When running a command, we can use system call tracing to capture the les that command reads and writes, and then a er the command completes, record the cryptographic hashes of the contents of those les. If the same command is ever run again, and the inputs and outputs haven't changed (have the same hashes), it can be skipped. is approach is the key idea behind both M (McCloskey 2008) and F (Hoyt et al. 2009). However, this technique makes the assumption that commands are pure functions from their inputs to their outputs, meaning if a command's input les are the same as last time it executed, it will write the same values to the same set of les. Below are four ways that assumption can be violated, along with ways to work around it. Non-deterministic commands Many commands are non-deterministic -e.g. the output of ghc object les contains unpredictable values within it (a consequence of the technique described by Augustsson et al. (1994)). We assume that where such non-determinism exists, any possible output is equally valid. Incorporating external information Some commands incorporate system information such as a timestamp, so a cached value will be based on the rst time the command was run, not the current time. For compilations that embed the timestamp in metadata, the rst timestamp is probably ne. For commands that really want the current time, that step can be li ed into the control logic (as per §2.2) so it will run each time the build runs. Similarly, commands that require unique information, e.g. a GUID or random number, can be moved into control logic and always run. Reading and writing the same le If a command both reads and writes the same le, and the information wri en is fundamentally in uenced by the le that was read, then the command never reaches a stable state. As an example, echo x >> foo.txt will append the character x every time the command is run. Equally, there are also commands that read the existing le to avoid rewriting a le that hasn't changed (e.g. ghc generating a .hi le) and commands that can cheaply update an existing le in some circumstances (the Microso C++ linker in incremental mode). We make the assumption that if a command both reads and writes to a le, that the read does not meaningfully in uence the write, otherwise it is not really suitable as part of a build system because the command never reaches a stable state and will re-run every time the build is run. Simultaneous modi cation If a command reads a le, but before the command completes something else modi es the le (e.g. a human or untracked control logic), then the nal hash will not match what the command saw. It is possible to detect such problems with reads by ensuring that the modi cation time a er computing the hash is before the command was started. For simultaneous writes the problem is much harder, so we require that all les produced by the build script are not simultaneously wri en to. In general we assume all commands given to the build system are well behaved and meet the above assumptions. Cloud builds We can skip execution of a command if all the les accessed have the same hashes as any previous execution ( §2.4). However, if only the les read match a previous execution, and the les that were wri en have been stored away, those stored les can be copied over as outputs without rerunning the command. If that storage is on a server, multiple users can share the results of one compilation, resulting in cloud build functionality. While this approach works well in theory, there are some problems in practice. Machine-speci c outputs Sometimes a generated output will only be applicable to the machine on which it was generated -for example if a compiler auto-detects the precise chipset (e.g. presence of AVX2 instructions) or hardcodes machine speci c details (e.g. the username). Such information can o en be li ed into the command line, e.g. by moving chipset detection into the control logic and passing it explicitly to the compiler. Alternatively, such commands can be explicitly tagged as not being suitable for cloud builds. Relative build directories O en the current directory, or user's pro le directory, will be accessed by commands. ese directories change if a user has two working directories, or if they use di erent machines. We can solve this problem by having a substitution table, replacing values such as the users home directory with $HOME. If not recti ed, this issue reduces the reusability of cloud results, but is otherwise not harmful. Non-deterministic builds If a command has non-deterministic output, then every time it runs it may generate a di erent result. Anything that transitively depends on that output is likely to also vary on each run. If the user temporarily disconnects from the shared storage, and runs a non-deterministic command, even if they subsequently reconnect, it is likely anything transitively depending on that command will never match again until a er a clean rebuild. ere are designs to solve this problem (e.g. the modi cation comparison mechanism from Erdweg et al. (2015)), but the issue only reduces the e ectiveness of the cloud cache, and usually occurs with intermi ent network access, so can o en be ignored. Build consistency As stated in Mokhov et al. (2018, §3.6), a build is correct provided that: If we recompute the value of the key (…), we should get exactly the same value as we see in the nal store. Speci ed in terms more applicable to our design, it means that a er a build completes, an immediate subsequent rebuild should have no e ect because all commands are skipped (assuming the commands given to the build system are the same). However, there are sequences of commands, where each command meets our assumptions separately (as per §2.4), but the combination is problematic: echo 1 > foo.txt echo 2 > foo.txt is program writes 1 to foo.txt, then writes 2. If the commands are re-executed then the rst command reruns because its output changed, and a er the rst command reruns, now the second commands output has changed. More generally, if a build writes di erent values to the same le multiple times, it is not correct by the above de nition, because on a rebuild both commands would re-run. But even without writing to the same le twice, it is possible to have an incorrect build: sha1sum foo.txt > bar.txt sha1sum bar.txt > foo.txt Here sha1sum takes the SHA1 hash of a le, rst taking the SHA1 of foo.txt and storing it in bar.txt, then taking the SHA1 of bar.txt and storing it in foo.txt. e problem is that the script rst reads from foo.txt on the rst line, then writes to foo.txt on the second line, meaning that when the script is rerun the read of foo.txt will have to be repeated as its value has changed. Writing to a le a er it has already been either read or wri en is the only circumstance in which a build, where every individual command is well-formed (as per §2.4), is incorrect. We de ne such a build as hazardous using the following rules: Read then write If one command reads from a le, and a later command writes to that le, on a future build, the rst command will have to be rerun because its input has changed. is behaviour is de ned as a read-write hazard. We assume the build author ordered the commands correctly, if not the author can edit the build script. Write then write If two commands both write to the same le, on a future build, the rst will be rerun (its output has changed), which is likely to then cause the second to be rerun. is behaviour is de ned as a write-write hazard. Using tracing we can detect hazards and raise errors if they occur, detecting that a build system is incorrect before unnecessary rebuilds occur. We prove that a build system with deterministic control logic, given the same input the control logic will produce the same output, and with no hazards always results in no rebuilds in §4.2. e presence of hazards in a build does not guarantee that a rebuild will always occur, for example if the write of a le a er it is read does not change the le's value. But, such a build system is malformed by our de nition and if the write de nitely can't change the output, then it should not be present. Explicit Parallelism A build script can use explicit parallelism by giving further commands to the build system before previous commands have completed. For example, the script in §2.2 has a for loop where the inner commands are independent and could all be given to the build system simultaneously. Such a build system with explicit parallelism must still obey the invariant that the inputs of a command must have been generated before the command is given, requiring some kind of two-way feedback that an enqueued command has completed. Interestingly, given complete dependency information (e.g. as available to M ) it is possible to infer complete parallelism information. However, the di culty of specifying complete dependency information is the a raction of a tracing based approach to build systems. Implicit Parallelism (Speculation) While explicit parallelism is useful, it imposes a burden on the build script author. Alternatively we can use implicit parallelism, where some commands are executed speculatively, before they are required by the build script, in the hope that they will be executed by the script in the future and thus skipped by the build system. Importantly, such speculation can be shown to be safe by tracking hazards, provided we introduce a new hazard speculative-write-read, corresponding to a speculative command writing to a le that is later read by a command required by the build script (de ned precisely in §4.1). Given a script with no hazards when executed sequentially, we show in §4.2: 1) that any ordering of those commands that also has no hazards will result in an equivalent output, see Claim 3; 2) that any parallel or interleaved execution without hazards will also be equivalent, see Claim 4; and 3) if any additional commands are run that don't cause hazards, they can be shown to not change the results the normal build produces, see Claim 5. Finally, we prove that if a series of commands contains hazards, so will any execution that includes those required commands, see Claim 6. As a consequence, if we can predict what commands the build script will execute next, and predict that their execution will not cause hazards, it may be worth speculatively executing them. E ective speculation requires us to predict the following pieces of data. Future commands e bene t of speculatively executing commands is that they will subsequently be skipped, which only happens if the speculative command indeed occurs in the build script. e simplest way to predict future commands is to assume that they will be the same as they were last time. It is possible to predict in a more nuanced manner given more history of which commands run. Given more information about the build, e.g. all the control logic as per §2.2 choice 2, it would be possible to use static analysis to re ne the prediction. Absence of hazards If a hazard occurs the build is no longer correct, and remediation must be taken (e.g. rerunning the build without speculation, see §4.3). erefore, performance can be signi cantly diminished if a speculative command leads to a hazard. Given knowledge of the currently running commands, and the les all commands accessed in the last run, it is possible to predict whether a hazard will occur if the access pa erns do not change. If tracing made it possible to abort runs that performed hazardous accesses then speculation could be unwound without restarting, but such implementations are di cult (see §3.4). Recovering from Hazards caused by speculation If a build using speculative execution causes a hazard, it is possible that the hazard is entirely an artefact of speculation. ere are a few actions the build system could take to recover and these are discusses in section §4.3. IMPLEMENTING RATTLE We have implemented the design from §2 in a build system called R . We use Haskell as the host language and to write the control logic. Copying the design for S (Mitchell 2012), a R build script is a Haskell program that uses the R library. 3.1 A R example A complete R script that compiles all .c les like §2.2 is given in Figure 1, with the key API functions in Figure 2. Looking at the example, we see: • A R script is a Haskell program. It makes use of ordinary Haskell imports, and importantly includes Development.Rattle, o ering the API from Figure 2. • e rattle function takes a value in the Run monad and executes it in IO. e Run type is the IO monad, enriched with a ReaderT (Jones 1995) containing a reference to shared mutable state (e.g. what commands are in ight, where to store metadata, location of shared storage). • All the control logic is in Haskell and can use external libraries -e.g. System.FilePath for manipulating FilePath values and System.FilePattern for directory listing. Taking the example of replacing the extension from .c to .o, we are able to abstract out this pa ern as toO and reuse it later. Arbitrary Haskell IO can be embedded in the script using liftIO. All of the Haskell code (including the IO) is considered control logic and will be repeated in every execution. • Commands are given to the build system part of R using cmd. We have implemented cmd as a variadic function (Kiselyov 2015) which takes a command as a series of String (a series of space-separated arguments), [String] (a list of arguments) and CmdOption (command execution modi ers, e.g. to change the current directory), returning a value of type Run (). e function cmd only returns once the command has nished executing (whether that is by actual execution, skipping, or fetching from external storage). • We have used forP in the example, as opposed to forM, which causes the commands to be given to R in parallel ( §2.7). We could have equally used forM and relied on speculation for parallelism ( §2.8). Looking at the functions from Figure 2, there are two functions this example does not use. e cmdWriteFile and cmdReadFile functions are used to perform a read/write of the le system through Haskell code, causing hazards to arise if necessary. Apart from these functions, it is assumed that all Haskell control code only reads and writes les which are not wri en to by any commands. 3.2 Alternative R wrappers Given the above API, combined with the choice to treat the control logic as opaque, it is possible to write wrappers that expose R in new ways. For example, to run a series of commands from a le, we can write: main = rattle $ do [x] <-liftIO getArgs src <-readFile x forM_ (lines src) cmd Here we take a command line argument, read the le it points to, then run each command sequentially using forM . We use this script for our evaluation in §5. An alternative API could be provided by opening up a socket, and allowing a R server to take command invocations through that socket. Such an API would allow writing R scripts in other languages, making use of the existing R implementation. While such an implementation should be easy, we have not yet actually implemented it. Specific design choices and consequences Relative to the reference design in §2 we have made a few speci c design choices, mostly in the name of implementation simplicity: • All of our predictions (see §2.8) only look at the very last run. is approach is simple, and in practice, seems to be su cient -most build scripts are run on very similar inputs most of the time. • We run a command speculatively if 1) it hasn't been run so far this build; 2) was required in the last run; and 3) doesn't cause a hazard relative to both the completed commands and predicted le accesses of the currently running commands. Importantly, if we are currently running a command we have never run before, we don't speculate anything -the build system has changed in an unknown way so we take the cautious approach. • We never a empt to recover a er speculation (see §4.3), simply aborting the build and restarting without any speculation. • We treat command lines as black boxes, never examining them to predict their outputs. For many programs a simple analysis (e.g. looking for -o ags) might be predictive. • We use a shared drive for sharing build artefacts, but allow the use of tools such as NFS or Samba to provide remote connectivity and thus full "cloud builds". • R can go wrong if a speculated command writes to an input le, as per §4.3. is problem hasn't occurred in practice, but dividing les into inputs and outputs would be perfectly reasonable. Typically the inputs are either checked into version control or outside the project directory, so that information is readily available. • It is important that traces (as stored for §2.4) are only recorded to disk when we can be sure they were not e ected by any hazards (see §4.3). at determination requires waiting for all commands which ran at the same time as the command in question to have completed. • We model queries for information about a le (e.g. existence or modi cation time) as a read for tracing purposes, thus depending on the contents of the le. For queries about the existence of a le, we rerun if the le contents changes, which may be signi cantly more frequent than when the le is created or deleted. For queries about modi cation time, we don't rerun if the modi cation time changes but the le contents don't, potentially not changing when we should. In practice, most modi cation time queries are to implement rebuilding logic, so can be safely ignored if the le contents haven't changed. Tracing approaches In §2.3 we assume the existence of dependency tracing which can, a er a command completes, tell us which les that command read and wrote. Unfortunately, such an API is not part of the POSIX standard, and is not easily available on any standard platform. We reuse the le access tracing features that have been added to S (Mitchell 2020), which in turn are built on top of F (Acereda 2019). e techniques and limitations vary by OS: • On Linux we use LD LIBRARY PRELOAD to inject a di erent C library which records accesses. It can't trace into programs that use system calls directly (typically Go programs (Donovan and Kernighan 2015)) or statically linked binaries. In future we plan to integrate B B (Roundy 2019a) as an alternative, which uses ptrace and xes these issues. • On Mac we also use LD LIBRARY PRELOAD, but that technique is blocked for security reasons on system binaries, notably the system C/C++ compiler. Installing a C/C++ compiler from another source (e.g. N (Dolstra et al. 2004)) overcomes that limitation. • On Windows we can't trace 64bit programs spawned by 32bit programs, but such a pa ern is rare (most binaries are now 64bit), o en easily remedied (switch to the 64bit version), and possible to x (although integrating the x is likely di cult). In practice, none of the limitations have been overly problematic in the examples we have explored. We designed R to work with the limitations of the best cross-platform tracing easily available -but that involves trade-o s. An enhanced, portable system would be a signi cant enabler for R . Our wishlist for tracing would include a cross-platform API, precise detection times, detection as access happens, and executing code at interception points. CORRECTNESS R 's design relies on taking a sequence of commands from the user, and additionally running some commands speculatively. In this section we argue for the correctness of this approach with respect to the reference semantics of sequential evaluation. roughout, we assume the commands themselves are pure functions from the inputs to their outputs (as reported by tracing), without this property no build system more sophisticated than the shell script of §1 would be correct. Hazards, formally In §2.6 we introduced the notion of hazards through intuition and how they are used. Here we give more precise de nitions: Read-write hazard A build has a read-write hazard if a le is wri en to a er it has been read. Given the limitations of tracing (see §3.4) we require the stronger property that for all les which are both read and wri en in a single build, the command which did the write must have nished before any commands reading it start executing, because we cannot assume the command nished writing to the le anytime before it completed. Write-write hazard A build has a write-write hazard if a le is wri en to a er it has already been wri en. Stated equivalently, a build has a write-write hazard if more than one command in a build writes to the same le. Speculative-write-read hazard A build has a speculative-write-read hazard if a le is wri en to by a speculated command, then read from by a command in the build script, before the build script has required the speculated command. Correctness properties We now state and prove (semi-formally) several key correctness properties of our approach, particularly in the treatment of hazards. We de ne the following terms: Command A command c reads from a set of les (c r ), and writes to a set of les (c w ). We assume the set of les are disjoint (c r ∩ c r ≡ ∅), as per §2.4. Deterministic command A command is deterministic if for a given set of les and values it reads, it always writes the same values to the same set of les. We assume commands are deterministic. Fixed point A deterministic command will have no e ect if none of the les it reads or writes have changed since the last time it was run. Since the command is deterministic, and its reads/writes are disjoint, the same reads will take place, the same computation will follow, and then the original values will be rewri en as they were before. We describe such a command as having reached a xed point. Correct builds If a build containing a set of commands C has reached a xed point for every command c within the set, then it is a correct build system as per the de nition in §2.6. No hazards A build has no hazards if no command in the build writes to a le that a previous command has read or wri en to. Input les We de ne the input les as those the build reads from but does not write to, namely: c ∈C c r \ c ∈C c w Output les We de ne the output les as those the build writes to, namely: c ∈C c w C 1 (C ) . If a set of commands C is run, and no hazards arise, then every command within C has reached a xed point. Proof: A command c has reached a xed point if none of the les in c r or c w have changed. Taking c r and c w separately: (1) None of the les in c w have changed. Within a set of commands C, if any le was wri en to by more than one command it would result in a write-write hazard. erefore, if a command wrote to a le, it must have been the only command to write to that le. (2) None of the les c r have changed. For a read le to have changed would require it to be wri en to a erwards. If a le is read from then wri en to it is a read-write hazard, so the les c r can't have changed. As a consequence, all commands within C must have reached a xed point. C 2 (U ). A build with deterministic control logic that completes with no hazards has reached a xed point. Proof: e build's deterministic control logic means that beginning with the rst command in the build, which is unchanged, given the same set of input les, the command will write to the same set of output les and produce the same results. It follows that the subsequent command in the build will be unchanged as well since the results of the previous command did not change. It follows by induction that each of the following commands in the build will remain unchanged. Because the commands in the build will not have changed, it follows from the proof of Claim 1 that the commands will not change the values of any les, therefore, the build is at a xed point. C 3 (R ). Given a script with no hazards when executed sequentially, with the same initial le contents, any other ordering of those commands that also has no hazards will result in the same terminal le contents. Proof: e proof of Claim 1 shows that any script with no hazards will result in a xed point. We can prove the stronger claim, that for any lesystem state where all inputs have identical contents, there is only one xed point for any ordering of the same commands that has no hazards. For each command, it can only read from les that are in the inputs or les that are the outputs of other commands. Taking these two cases separately: (1) For read les that are in the inputs, they aren't wri en to by the build system (or they wouldn't be in the inputs), and they must be the same in both cases (because the lesystem state must be the same for inputs). erefore, changes to the inputs cannot have an e ect on the command. (2) For read les that are the outputs of other commands, they must have been wri en to before this command, or a read-write hazard would be raised. erefore, the rst command that performs writes cannot access any writes from other commands, and so (assuming determinism) its writes must be a consequence of only inputs. Similarly, the second command can only have accessed inputs and writes produced by the rst command, which themselves were consequences of inputs, so that commands writes are also consequences of the input. By induction, we can show that all writes are consequences of the inputs, so identical inputs results in identical writes. C 4 (P ). Given a script with no hazards when executed sequentially, any parallel or interleaved execution without hazards will also be equivalent. Proof: None of the proof of Claim 3 relies on commands not running in parallel, so that proof still applies. C 5 (A ). Given a script with no hazards when executed, speculating unnecessary commands that do not lead to hazards will not have an e ect on the build's inputs or outputs. Proof: Provided the inputs are unchanged, and there are no hazards, the proof from Claim 3 means the same outputs will be produced. If speculated commands wrote to the outputs, it would result in a write-write hazard. If speculated commands wrote to the inputs before they were used, it would result in a speculative-write-read hazard. If speculated commands wrote to the inputs a er they were used, it would result in a read-write hazard. erefore, assuming no hazards are raised, the speculated commands cannot have an e ect on the inputs or outputs. C 6 (P ). If a sequence of commands leads to a hazard, any additional speculative commands or reordering will still cause a hazard. Proof: All hazards are based on the observation of le accesses, not the absence of le accesses. erefore, additional speculative commands do not reduce hazards, only increase them, as additional reads/writes can only cause additional hazards. For reordering, it cannot reduce write-write hazards, which are irrespective of order. For reordering of read-write hazards, it can only remove such a hazard by speculating the writing command before the reading command. However, such speculation is a speculative-read-write hazard, ensuring a hazard will always be raised. Hazard Recovery If a build involving speculation reaches a hazard there are several remedies that can be taken. In this section we outline these approaches and when each approach is safe to take. Restart with no speculation. Restarting the entire build that happened with speculation, with no speculation, is safe and gives an equivalent result provided the inputs to the build have not changed (Claims 3, 5). Unfortunately, while writing to an input is guaranteed to give a hazard, restarting does not change the input back. As a concrete example, consider the build script with the single command gcc -c main.c. If we speculate the command echo 1 >> main.c rst, it will raise a speculative-write-read hazard, but restarting will not put main.c back to its original value, potentially breaking the build inde nitely. We discuss consequences and possible remediations from speculative commands writing to inputs in §3.3, but note such a situation should be very rare and likely easily reversible with minor user intervention. e build system only executes commands that were previously part of the build script, making it unlikely a build author would include commands that would irreversibly break their build. For all other recovery actions, we assume that restarting with no speculation is the baseline state, and consider them safe if it is equivalent to restarting with no speculation. is is the approach we currently take in R ; below we describe other approaches also justi ed in our model. Restart with less speculation. Given the proofs in this section, running again with speculation is still safe. However, running with speculation might repeatedly cause the same speculation hazard, in which case the build would never terminate. erefore, it is safe to restart with speculation provided there is strictly less speculation, so that (in the worst case) eventually there will be a run with no speculation. As a practical ma er, if a command is part of the commands that cause a hazard, it is a good candidate for excluding from speculation. Raise a hazard. Importantly, if there is no speculation (including reordering and parallelism), and a hazard occurs, the hazard can be raised immediately. is property is essential as a base-case for restarting with no speculation. If two required commands, that is commands in the users build script, cause a write-write hazard, and there have been no speculative-write-read hazards, then the same hazard will be reraised if the command restarts. For write-write, this inevitability is a consequence of Claim 3, as the same writes will occur. If two required commands cause a read-write hazard, and the relative order they were run in matches their required ordering, then the same hazard will be reraised if the command restarts, because the relative ordering will remain the same, Claim 6. However, if the commands weren't in an order matching their required ordering, it is possible the write will come rst avoiding the hazard, so it cannot be immediately raised. Continue. If the hazard only a ects speculated commands, and those commands have no subsequent e ect on required commands, it is possible to continue the build. However, it is important that the commands have no subsequent e ect, even on required commands that have not yet been executed, or even revealed to the build system. Consequently, even if a build can continue a er an initial hazard, it may still fail due to a future hazard. A build can continue if the hazards only a ect speculated commands, that is either a write-write hazard where both commands were speculated, or a read-write hazard where the read was speculated. For other hazards, continuing is unsafe. If a speculated command is a ected by an initial hazard, and that command is later required by the build script, a new non-continuable hazard will be raised. EVALUATION In this section we evaluate the design from §2, speci cally our implementation from §3. We show how the implementation performs on the example from §1 in §5.1, on microbenchmarks in §5.2, and then on real projects that currently use M -namely FSATrace ( §5.3), Redis ( §5.4), Vim ( §5.5), tmux ( §5.6), and Node.js ( §5.7). We implement two sample projects using R in §5.9. We have focused our real-world comparison on M projects using C because such projects o en have minimal external dependencies (making them easier to evaluate in isolation), and despite using the same programming language, have an interesting variety of dependency tracking strategies and size (ranging from one second to three hours) and . Most newer programming languages (e.g. Rust, Haskell etc) have dedicated built tools (e.g. Cargo, Cabal) so small projects in such languages tend to not use general-purpose build tools. For benchmarks, the rst four ( §5.1- §5.4) were run on a 4 core Intel i7-4790 3.6GHz CPU (16Gb RAM). e remaining benchmarks were run on a 32 core Intel Xeon E7-4830 2.13GHz (64Gb RAM). Validating Ra le's suitability In §1 we claimed that the following build script is "just as good" as a proper M script. ere are two axes on which to measure "just as good" -correctness and performance. Performance can be further broken down into how much rebuilding is avoided, how much parallelism is achieved and how much overhead R imposes. Correctness R is correct, in that the reference semantics is running all the commands, and as we have shown in §2 (see §4 for detailed arguments), and tested for with examples, R obeys those semantics. In contrast, the M version may have missing dependencies which causes it not to rebuild. Examples of failure to rebuild include if there are changes in gcc itself, or changes in any headers used through transitive #include which are not also listed in the M script. Rebuilding too much R only reruns a command if some of the les it reads or writes have changed. It is possible that a command only depends on part of a le, but at the level of abstraction R works, it never rebuilds too much. As a ma er of implementation, R detects changes by hashing the le contents, while M uses the modi cation time. As a consequence, if a le is modi ed, but its contents do not change (e.g. using touch), M will rebuild but R will not. Parallelism e script from §1 has three commands -the rst two can run in parallel, while the the third must wait for both to nish. M is given all this information by dependencies, and will always achieve as much parallelism as possible. In contrast, R has no such knowledge, so it has to recover the parallelism by speculation (see §2.8). During the rst execution, R has no knowledge about even which commands are coming next (as described in §2.2), so it has no choice but to execute each command serially, with less parallelism than M . In subsequent executions R uses speculation to always speculate on the second command (as it never has a hazard with the rst), but never speculate on the third until the rst two have nished (as they are known to con ict). Interestingly, sometimes R executes the third command (because it got to that point in the script), and sometimes it speculates it (because the previous two have nished)it is a race condition where both alternatives are equivalent. While R has less parallelism on the rst execution, if we were to use shared storage for speculation traces, that can be reduced to the rst execution ever, rather than the rst execution for a given user. Overhead e overhead inherent in R is greater than that of M , as it hashes les, traces command execution, computes potential hazards, gures out what to speculate and writes to a shared cloud store (in the rest of this section we use a locally mounted shared drive). To measure the overhead, and prove the other claims in this section, we created a very simple pair of les, main.c and util.c, where main.c calls printf using a constant returned by a function in util.c. We then measured the time to do: (1) An initial build from a clean checkout; (2) a rebuild when nothing had changed; (3) a rebuild with whitespace changes to main.c, resulting in the same main.o le; (4) a rebuild with meaningful changes to main.c, resulting in a di erent main.o le; and (5) a rebuild with meaningful changes to both C les. We performed the above steps with 1, 2 and 3 threads, on Linux. To make the parallelism obvious, we modi ed gcc to sleep for 1 second before starting. e numbers are: As expected, we see that during the initial build R doesn't exhibit any parallelism, but M does (1). In contrast, R bene ts when a le changes in whitespace only and the resulting object le doesn't change, while M can't (3). We see 3 threads has no bene t over 2 threads, as this build contains no more parallelism opportunities. Comparing the non-sleep portion of the build, M and R are quite evenly matched, typically within a few milliseconds, showing low overheads. We focus on the overheads in the next section. Measuring overhead In order to determine what overhead R introduces, we ran a xed set of commands with increasingly more parts of R enabled. R command execution builds on the command execution from S (Mitchell 2012), which uses F for tracing and the Haskell process library for command execution. We ran the commands in a clean build directory in 7 ways: (1) Using make -j1 (M with 1 thread), as a baseline. (2) Using System.Process from the Haskell process library. (3) Using cmd from the S library (Mitchell 2012), which builds on the process library. (4) Using cmd from S , but wrapping the command with F for le tracing. (5) Using cmd from S with the Traced se ing, which runs F and collects the results. (6) Using R with no speculation or parallelism, and without shared storage. (7) Using R with all features, including shared storage. To obtain a set of commands typical of building, we took the latest version of F 3 and ran make -j1 with maximum verbosity, recording the commands that were printed to the standard output. On Windows F runs 25 commands (21 compiles, 4 links). On Linux F runs 9 commands (7 compiles, 2 links). On Linux the list of commands produces write-write hazards, because it compiles some les (e.g. shm.c) twice, once with -fPIC (position independent code), and once without. However, both times it passes -MMD to cause gcc to produce shm.d at the same location (shm.d is used to track dependencies). at write-write hazard is a genuine problem, and in an incredibly unlucky build F would end up with a corrupted shm.d le. To remedy the problem we removed the -MMD ag as it doesn't impact the benchmark and isn't required by R . We ran all sets of commands on both Windows and Linux. Commands Windows Linux 1) Make 9.96s 100% 1.19s 100% 2) process 10.26s 103% +3% 1.18s 99% -1% 3) S 10.58s 106% +3% 1.17s 98% -1% 4) S + F 12.66s 127% +21% 1.23s 103% +5% 5) S + Traced 13.06s 131% +4% 1.23s 103% +0% 6) R 14.43s 145% +14% 1.25s 105% +2% 7) R + everything 14.53s 146% +1% 1.27s 107% +2% In the above chart both Windows and Linux have three columns -the time taken (average of the three fastest of ve runs), that time as a percentage of the M run, and the delta from the row above. e results are signi cantly di erent between platforms: On Windows, we see that the total overhead of R makes the execution 46% slower. 21% of the slowdown is from F (due to hooking Windows kernel API), with the next greatest overhead being from R itself. Of the R overhead, the greatest slowdown is caused by canonicalizing paths. Using the default NTFS le system, Windows considers paths to be case insensitive. As a result, we observe paths like C:\windows\system32\KERNELBASE.dll, which on disk are called C:\Windows\System32\KernelBase.dll, but can also be accessed by names such as C:\Windows\System32\KERNEL˜1.DLL. Unfortunately, Windows also supports case sensitive le systems, so using case-insensitive equality is insu cient. On Windows, enabling the default anti-virus (Windows Defender) has a signi cant impact on the result, increasing the M baseline by 11% and the nal time by 29%. ese results were collected with the anti-virus disabled. On Linux, the total overhead is only 7%, of which nearly all (5%) comes from tracing. ese results show that tracing has a minor but not insigni cant e ect on Linux, whereas on Windows can be a substantial performance reduction. Additionally, the Windows results had far more variance, including large outliers. us, our remaining benchmarks were all run on Linux. F To compare M and R building F we took the commands we extracted for §5.2 and ran the build script for the 100 previous commits in turn, starting with a clean build then performing incremental builds. To make the results readable, we hid any commits where all builds took ¡ 0.02s, resulting in 26 interesting commits. We ran with 1 to 4 threads, but omit the 2 and 3 thread case as they typically fall either on or just above the 4 thread results. As we can see, the rst build is always ¿ 1s for R , but M is able to optimise it as low as 0.33s with 4 threads. Otherwise, R and M are competitive -users would struggle to see the di erence. e one commit that does show some variation is commit 2, where both are similar for 1 thread, but R at 4 threads is slightly slower than 1 thread, while M is faster. e cause is speculation leading to a write-write hazard. Concretely, the command for linking fsatrace.so changed to include a new le proc.o. R starts speculating on the old link command, then gets the command for the new link. Both commands write to fsatrace.so, leading to a hazard, and causing R to restart without speculation. Redis is an in-memory NoSQL database wri en in C, developed over the last 10 years. Redis is built using recursive M (Miller 1998), where the external dependencies of Redis (e.g. Lua and Linenoise) have their own M scripts which are called by the main Redis M script. Using stamp les, on successive builds, the external dependencies are not checked -meaning that users must clean their build when external dependencies change. Redis When benchmarking R vs M we found that R was about 10% slower due to copying les to the cloud cache (see §2.5), something M does not do. erefore, for a fairer comparison, we disabled the cloud cache. For local builds, the consequence of disabling the cloud cache is that if a le changes, then changes back to a previous state, we will have to rebuild rather than get a cache hit -something that never happens in this benchmark. With those changes R is able to perform very similarly to M , even while properly checking dependencies. Di erences occur on the initial build, where R has no information to speculate with, and commit 1, where R correctly detects no work is required but M rebuilds some dependency information. (1) For the 20th commit in the chart, M did much more work than R . is commit changed the script which generates osdef.h, in such a way that it did not change the contents of osdef.h. M rebuilt all of the object les which depend on this header le, while R skipped them as the le contents had not changed. Vim (2) For the 29th commit in the chart, R did much more work than M . is commit changed a prototype le that is indirectly included by all object les. is prototype le is not listed as a dependency of the object les, so M did not rebuild them, but R noticed the changed dependency. is missing dependency could have resulted in an incorrect build or a build error, but in this case (by luck) seems not to have been signi cant. A future prototype change may result in incorrect builds. tmux is a terminal multiplexer in development for the last 12 years. It is built from source using a combination of a conventional sh autogen.sh && ./configure && make sequence, making use of A , A , , and M . To compare a R based tmux build to the M based one, we rst ran sh autogen.sh && ./configure, and then used the resulting M scripts to generate R build scripts. tmux was built over a series of 38 commits with both M and R . For the 11th, 22nd and 23rd commits R and M took noticeably di erent amounts of time. In all cases the cause was a build script change, preventing R from speculating, and thus taking approximately as long as a full single-threaded build. For those three commits where the build script changed, it was possible for a write-write hazard to occur, and we occasionally observed it for the 11th and 22nd commits (but not in the 5 runs measured above). tmux Node.js Node.js is a JavaScript runtime built on Chrome's V8 Javascript engine that has been in development since 2009. e project is largely wri en in JavaScript, C++, Python and C, and is built using M and a meta-build tool called Generate Your Projects (GYP) (Google 2009). Node.js build system. To build Node.js from source, a user rst runs ./configure, which con gures the build and runs GYP, generating Make les. From then on, a user can build with M . GYP generates the majority of the Make les used to build Node.js, starting from a series of source .gyp les specifying metadata such as targets and dependencies. GYP generates a separate .mk Make le for each target, all of which are included by a generated top-level Make le. at top-level Make le can be run by M , but through a number of tricks, GYP controls the dependency calculation: (1) All tracked commands are run using the do cmd function from the top-level Make le. e do cmd function checks the command, and if anything has changed since the last run, reruns it. To ensure this function is called on every run, targets whose rules use do cmd have a fake dependency on the phony target FORCE DO CMD, forcing M to rerun them every time. is mechanism operates much like R command skipping from §2.4. (2) Tracked commands create dependency les for each target, recording observed dependencies from the last run. Typically, this information is produced by the compiler, e.g. gcc -MMD. It appears the build of Node.js has tried to take an approach closer to that of R , namely, tracking dependencies each time a command runs and making sure commands re-run when they have changed. e implementation is quite complicated and relies on deep knowledge of precisely how M operates, but still seems to not match the intention. To compare to R , we built Node.js over a series of 38 commits with both M and R , with 1 thread and 4 threads. e R build was created for each commit by recording every command executed by the original build, excluding the commands exclusively dealing with dependency les. Due to the large variations in time, the above plot uses a log scale. For building from scratch, M and R took approximately the same amount of time with a single thread. However, with 4 threads M was able to build Node nearly four times faster, since R had no speculation information. For many commits R and M give similar results, but for those with the least amount of work, where only a few les change, R is able to beat M . e fastest observed M time is 35 seconds, while R goes as low as 12 seconds. e major cause is that M always rebuilds the generated les but R (correctly) does not. For three commits, 2, 12, and 29, the R build with 4 threads was slower than single threaded R . e cause was read-write hazards occurring, forcing R to restart the build with no speculation. e build script changed at each of these commits, and so R speculating commands from the previous script led to hazards with the changed commands in the new script. Summary comparing with M Looking across the projects in this section, a few themes emerge. • For most commits, R and M give similar performance. e one case that is not true is the initial build at higher levels of parallelism, where M outperforms R , because R has no speculation information. If R used a shared speculation cache for the initial build that di erence would disappear. Such a cache would only need to provide approximate information to get most of the bene ts, so the nearest ancestor commit with such information should be su cient -there would be no need to maintain or curate the list. • e di erence between 1 thread and 4 threads can be quite signi cant, and typically M and R do equally well with parallelism. e implicit parallelism in R is an important feature which works as designed. • In rare cases R su ers a hazard, which can be observed on the build time plot. • Every M project has a minor variation on how dependencies are managed. Most make use of gcc to generate dependency information. Some provide it directly (e.g. F ), some also use stamp les (e.g. Redis) and some build their own mechanisms on top of M (e.g. Node.js, but also Mokhov et al. (2016, §2)). • Many projects give up on dependencies in less common cases, requiring user action, e.g. Redis requires a clean build if a dependency changes. • R correctly rebuilds if the build le itself updates, but most of the M projects do not. e one exception is Node.js, through its use of its dependency encoding in the Make le. Writing R scripts In order to see how di cult it is to write build scripts for R , we tried writing two examplesnamely F , and a program to fetch Haskell dependencies and install them. Building F . We implemented a R script to build F . Unlike §5.3, where we took the outputs from M , this time we took the source Make les and tried to port their logic to R . We did not port the testing or clean phony target, but we did port both Windows and Linux support. e script rst de nes 11 variables, following much the same pa ern as the Make le de nes the same variables -some of which are conditional on the OS. Next we compile all the C sources, and nally we do a few links. e total code is 19 non-comment lines, corresponding to 46 non-comment lines in the original Make les. Given the setup of the code, it was trivial to use explicit parallelism for compiling all C les, but doing so for the linking steps, and arranging for the links to be performed as soon as possible, would have been tedious. Taking some representative pieces from the build, we have: e overall experience of porting the Make le was pleasant. e explicit isWindows is much nicer than the Make le approach. e availability of lists and a rich set of operations on them made certain cases more natural. e use of functions, e.g. objects, allowed simplifying transformations and giving more semantically meaningful names. e powerful syntax of cmd, allowing multiple arguments, makes the code look relatively simple. e only downside compared to M was the use of quotes in more places. Haskell Dependencies. As a second test, we wrote a program to fetch and build a Haskell library and its dependencies. For Haskell packages, Cabal is both a low-level library/interface for con guring, building and installing, and also a high-level tool for building sets of packages. Stackage is a set of packages which are known to work together. Stack is a tool that consumes the list of Stackage packages, and using low-level Cabal commands, tries to install them. We decided to try and replicate Stack, with the original motivation that by building on top of R we could get cloud builds of Haskell libraries for free. We were able to write a prototype of something Stack-like using R in 71 non-comment lines. Our prototype is capable of building some projects, showing the fundamental mechanisms work, but due to the number of corner cases in Haskell packages, is unlikely to be practical without further development. e main logic that drives the process is reproduced below. Reading this code, given a Stackage resolver (e.g. nightly-2012-10-29), we download the resolver to disk, read it, then recursively build projects in an order satisfying dependencies. e download uses curl, which has no meaningful reads, so will always be skipped. Importantly, the URL includes the full resolver which will never change once published. e parsing of the resolver is Haskell code, not visible to R , repeated on every run (the time taken is negligible). We then build a recursive memoisation pa ern using memoRec, where each package is only built once, and while building a package you can recursively ask for other packages. We then nally demand the packages that were requested. Writing the program itself was complex, due to Haskell packaging corner cases -but the R aspect was pleasant. Having Haskell available to parse the con guration and having e cient Map data structures made it feasible. Not worrying about dependencies, focusing on it working, vastly simpli ed development. ere were lots of complications due to the nature of Haskell packages (e.g. some tools/packages are special, parsing metadata is harder than it should be), but only a few which interacted closely with R : (1) ere are some lock les, e.g. package.cache.lock, which are both read and wri en by many commands, but not in an interesting way. We explicitly ignored these les from being considered for hazards. (2) Haskell packages are installed in a database, which by default is appended to with each package registration. at pa ern relies on the order of packages and is problematic for concurrency. Many build systems, e.g. B , use multiple package databases, each containing one package. We follow the same trick. (3) By default, Cabal produces packages where some elds (but not all) include an absolute path. To make the results generally applicable for cloud builds, we remove the absolute paths. To ensure the absolute paths never make it to the cloud, we run the command that produces and the command that cleans up as a single command from the point of view of R . (4) Since cachable things have to be commands, that required pu ing logic into commands, not just shell calls. To make that pleasant, we wrote a small utility to take Template Haskell commands (Sheard and Peyton Jones 2002) and turn them into Haskell scripts that could then be run from the command line. 6 RELATED WORK e vast majority of existing build systems are backward build systems -they start at the nal target, and recursively determine the dependencies required for that target. In contrast, R is a forward build system-the script executes sequentially in the order given by the user. Comparison to forward build systems e idea of treating a script as a build system, omi ing commands that have not changed, was pioneered by M (McCloskey 2008) and popularised by F (Hoyt et al. 2009). In both cases the build script was wri en in Python, where cheap logic was speci ed in Python and commands were run in a traced environment. If a traced command hadn't changed inputs since it was last run, then it was skipped. We focus on F , which came later and o ered more features. F uses strace on Linux, and on Windows uses either le access times (which are either disabled or very low resolution on modern Windows installations) or a proprietary and unavailable tracker program. Parallelism can be annotated explicitly, but o en omi ed. ese systems did not seem to have much adoption -in both cases the original sources are no longer available, and knowledge of them survives only as GitHub copies. R di ers from an engineering perspective by the tracing mechanisms available (see §3.4) and the availability of cloud build (see §2.5) -both of which are likely just a consequence of being developed a decade later. R extends these systems in a more fundamental way with the notion of hazards, which both allows the detection of bad build scripts, and allows for speculation -overcoming the main disadvantage of earlier forward build systems. Stated alternatively, R takes the delightfully simple approach of these build systems, and tries a more sophisticated execution strategy. Recently there have been three other implementations of forward build systems we are aware of. (1) S (Mitchell 2012) provides a forward mode implemented in terms of a backwards build system. e approach is similar to the F design, o ering skipping of repeated commands and explicit parallelism. In addition, S allows caching custom functions as though they were commands, relying on the explicit dependency tracking functions such as need already built into S . e forward mode has been adopted by a few projects, notably a library for generating static websites/blogs. e tracing features in R are actually provided by S as a library (which in turn builds on F ). (2) F (Roundy 2019b) is based on the B B tracing library. Commands are given in a custom le format as a static list of commands (i.e. no monadic expressive power as per §2.2), but may optionally include a subset of their inputs or outputs. e commands are not given in any order, but the speci ed inputs/outputs are used to form a dependency order which F uses. If the speci ed inputs/outputs are insu cient to give a working order, then F will fail but record the actual dependencies which will be used next timea build with no dependencies can usually be made to work by running F multiple times. (3) S (Mokhov 2019) takes a static set of commands, without either a valid sequence or any input/output information, and keeps running commands until they have all succeeded. As a consequence, S may run the same command multiple times, using tracing to gure out what might have changed to turn a previous failure into a future success. S also reuses the tracing features of S and F . ere are signi cantly fewer forward build systems than backwards build systems, but the interesting dimension starting to emerge is how an ordering is speci ed. e three current alternatives are the user speci es a valid order (F and R ), the user speci es partial dependencies which are used to calculate an order (F ) or the user speci es no ordering and search is used (S and some aspects of F ). 6.2 Comparison to backward build systems e design space of backward build systems is discussed in (Mokhov et al. 2018). In that paper it is notable that forward build systems do not naturally t into the design space, lacking the features that a build system requires. We feel that omission points to an interesting gap in the landscape of build systems. We think that it is likely forward build systems could be characterised similarly, but that we have yet to develop the necessary variety of forward build systems to do so. ere are two dimensions used to classify backward build systems: Ordering Looking at the features of R , the ordering is a sequential list, representing an excessively strict ordering given by the user. e use of speculation is an a empt to weaken that ordering into one that is less linear and more precise. Rebuild For rebuilding, R looks a lot like the constructive trace model -traces are made, stored in a cloud, and available for future use. e one wrinkle is that a trace may be later invalidated if it turns out a hazard occurred (see §3.3). ere are also two features available in some backward build systems that are relevant to R : Sandboxing Some backward build systems (e.g. B (Google 2016)) run processes in a sandbox, where access to les which weren't declared as dependencies are blocked -ensuring dependencies are always su cient. A consequence is that o en it can be harder to write a B build system, requiring users to declare dependencies like gcc and system headers that are typically ignored. e sandbox doesn't prevent the reverse problem of too many dependencies. Remote Execution Some build systems (e.g. B and B XL (Microso 2020)) allow running commands on a remote machine, usually with a much higher degree of parallelism than is available on the users machine. If R was able to leverage remote execution then speculative commands could be used to ll up the cloud cache, and not involve local writes to disk, eliminating all speculative hazards -a very a ractive property. Remote execution in B sends all les alongside the command, but R doesn't know the les accessed in advance, which makes that model infeasible. Remote execution in B XL sends the les it thinks the command will need, augments the le system to block if additional les are accessed, and sends further requests back to the initial machine -which would t nicely with R . Analysis of existing build systems We aren't the rst to observe that existing build systems o en have incorrect dependencies. Bezemer et al. (2017) performed an analysis of the missing dependencies in M build scripts, nding over 1.2 million unspeci ed dependencies among four projects. To detect missing dependencies, Licker and Rice (2019) introduced a concept called build fuzzing, nding race-conditions and errors in 30 projects. It has also been shown that build maintenance requires as much as a 27% overhead (McIntosh et al. 2011), a substantial proportion of which is devoted to dependency management. Our anecdotes from §5 all reinforce these messages. Speculation Speculation is used extensively in many other areas of computer science, from processors to distributed systems. If an action is taken before it is known to be required, that can increase performance, but undesired side-e ects can occur. Most speculating systems a empt to block the side-e ects from happening, or roll them back if they do. e most common use of speculation in computer science is the CPU - Swanson et al. (2003) found that 93% of useful CPU instructions were evaluated speculatively. CPUs use hazards to detect incorrect speculation, with similar types of read/write, write/write and write/read hazards (Pa erson and Hennessy 2013) -our terminology is inspired by their approaches. For CPUs many solutions to hazards are available, e.g. stalling the pipeline if a hazard is detected in advance or the Tomasulo algorithm (Tomasulo 1967). We also stall the pipeline if we detect potential hazards, although have to do so with incomplete information, unlike a CPU. e Tomasulo algorithm involves writing results into temporary locations and the moving them over a erwards -use of remote execution ( §6.2) might be a way to apply similar ideas to build systems. Looking towards so ware systems, Welc et al. (2005) showed how to add speculation to Java programs, by marking certain parts of the program as worth speculating with a future. Similar to our work, they wanted Java with speculation to respect the semantics of the sequential version of the program, which required two main techniques. First, all data accesses to shared state are tracked and recorded, and if a dependency violation occurs the o ending code is revoked and restarted. Second, shared state is versioned using a copy-on-write invariant to ensure threads write to their own copies, preventing a future from seeing its continuation's writes. read level speculation (Ste an and Mowry 1998) is used by compilers to automatically parallelise programs, o en by executing multiple iterations of a loop body simultaneously. As before, the goal is to maintain the semantics as single-threaded execution. Techniques commonly involve bu ering speculative writes (Ste an et al. 2000) and ensuring that a read re ects the speculative writes of threads that logically precede it. Speculation has also been investigated for distributed systems. Nightingale et al. (2005) showed that adding speculation to distributed le systems such as NFS can make some benchmarks over 10 times faster, by allowing multiple le system operations to occur concurrently. A model allowing more distributed speculation, even in the presence of message passing between speculated distributed processes, is presented by Tapus (2006). Both these pieces of work involve modifying the Linux kernel with a custom le system to implement roll backs transparently. All these approaches rely on the ability to trap writes, either placing them in a bu er and applying them later or rolling them back. Unfortunately, such facilities, while desirable, are currently di cult to achieve in portable cross-platform abstractions ( §3.4). We have used the ideas underlying speculative execution, but if the necessary facilities became available in future, it's possible we could follow the approaches more directly. CONCLUSION AND FUTURE WORK In this paper we present R , a build system that takes a sequence of actions and treats them as a build script. From the user perspective, they get the bene ts of a conventional build system (incrementality, parallelism) but at lower cost (less time thinking about dependencies, only needing to supply a valid ordering). Compared to conventional build systems like M , R presents a simpler user interface (no dependencies), but a more complex implementation model. Our evaluation in §5 shows that for some popular real-world projects, switching to R would bring about simplicity and correctness bene ts, with negligible performance cost. e two places where builds aren't roughly equivalent to M are the initial build (which could be solved with a global shared cache) and when speculation leads to a hazard. ere are several approaches to improving speculation, including giving R a list of commands that should not be speculated (which can be a perfect list for non-monadic builds), or giving R a subset of commands' inputs/outputs (like F does), or implementing be er recovery strategies from speculation errors. Our evaluation focuses on projects whose build times are measured in seconds or minutes, with only one project in the hours range. It is as yet unclear whether similar bene ts could be achieved on larger code bases, and whether the R approach of "any valid ordering" is easy to describe compositionally for large projects. Our next steps are scaling R and incorporating feedback from actual users. Fig. 1 1CmdOption = Cwd FilePath \| ... forP :: [a] -> (a -> Run b) -> Run [b] cmd :: CmdArguments args => args Fig. 2 . 2The R API ( 3 ) 3Generated source les are produced and managed using .intermediate targets. By relying on a combination of do cmd, FORCE DO CMD and a special prerequisite .INTERMEDIATE that depends on all .intermediate targets, the intention appears to be to regenerate the generated code whenever something relevant has changed. However, in our experiments, those intermediate targets seem to run every time. We take some liberties with shell scripts around replacing extensions, so as not to obscure the main focus. , Vol. 1, No. 1, Article 1. Publication date: January 2016. , Vol. 1, No. 1, Article 1. Publication date: January 2016. h ps://github.com/jacereda/fsatrace/commit/41 ba17da580f81ababb32ec7e6e5fd49f11473 , Vol. 1, No. 1, Article 1. Publication date: January 2016. 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Alan Donovan, Brian Kernighan, e Go Programming Language. Addison-Wesley Professional1st ed.Alan Donovan and Brian Kernighan. 2015. e Go Programming Language (1st ed.). Addison-Wesley Professional. A sound and optimal incremental build system with dynamic dependencies. Sebastian Erdweg, Moritz Lichter, Manuel Weiel, ACM SIGPLAN Notices. 50Sebastian Erdweg, Moritz Lichter, and Manuel Weiel. 2015. A sound and optimal incremental build system with dynamic dependencies. ACM SIGPLAN Notices 50, 10 (2015), 89-106. Make -A program for maintaining computer programs. Stuart Feldman, 9Stuart Feldman. 1979. Make -A program for maintaining computer programs. So ware: Practice and experience 9, 4 (1979), 255-265. Generate Your Projects. Google, Google. 2009. Generate Your Projects. (2009). h ps://gyp.gsrc.io/. . Google, Google. 2016. Bazel. (2016). h p://bazel.io/. Fabricate: e be er build tool. Berwyn Hoyt, Bryan Hoyt, Ben Hoyt, Berwyn Hoyt, Bryan Hoyt, and Ben Hoyt. 2009. Fabricate: e be er build tool. (2009). h ps://github.com/SimonAl e/ fabricate. Functional programming with overloading and higher-order polymorphism. Mark Jones, Advanced Functional Programming, First International Spring School on Advanced Functional Programming Techniques. Mark Jones. 1995. Functional programming with overloading and higher-order polymorphism. In Advanced Functional Programming, First International Spring School on Advanced Functional Programming Techniques. 97-136. Polyvariadic functions and keyword arguments: pa ern-matching on the type of the context. Oleg Kiselyov, Oleg Kiselyov. 2015. Polyvariadic functions and keyword arguments: pa ern-matching on the type of the context. (2015). h p://okmij.org/ p/Haskell/polyvariadic.html. Detecting incorrect build rules. Nándor Licker, Andrew Rice, 2019 IEEE/ACM 41st International Conference on So ware Engineering (ICSE. Nándor Licker and Andrew Rice. 2019. Detecting incorrect build rules. In 2019 IEEE/ACM 41st International Conference on So ware Engineering (ICSE). 1234-1244. . Bill Mccloskey, Bill McCloskey. 2008. Memoize. (2008). h ps://github.com/kgaughan/memoize.py. An empirical study of build maintenance e ort. Shane Mcintosh, Bram Adams, Anh Nguyen, Yasutaka Kamei, Ahmed Hassan, Proc. ICSE '11. ICSE '11ACM PressShane McIntosh, Bram Adams, anh Nguyen, Yasutaka Kamei, and Ahmed Hassan. 2011. An empirical study of build maintenance e ort. In Proc. ICSE '11. ACM Press, 141-150. Recursive Make considered harmful. Journal of AUUG Inc. Microso . 2020. Build Accelerator.19Peter MillerMicroso . 2020. Build Accelerator. (2020). h ps://github.com/microso /BuildXL. Peter Miller. 1998. Recursive Make considered harmful. Journal of AUUG Inc 19, 1 (1998), 14-25. Shake before building: Replacing Make with Haskell. Neil Mitchell, ACM SIGPLAN Notices. ACM47Neil Mitchell. 2012. Shake before building: Replacing Make with Haskell. In ACM SIGPLAN Notices, Vol. 47. ACM, 55-66. File access tracing. Neil Mitchell, Neil Mitchell. 2020. File access tracing. (2020). h ps://neilmitchell.blogspot.com/2020/05/ le-tracing.html. Stroll: an experimental build system. Andrey Mokhov, Andrey Mokhov. 2019. Stroll: an experimental build system. (2019). h ps://blogs.ncl.ac.uk/andreymokhov/stroll/. Build systemsà la carte. Andrey Mokhov, Neil Mitchell, Simon Peyton Jones, 10.1145/3236774Proceedings ACM Programing Languages. 2ArticleAndrey Mokhov, Neil Mitchell, and Simon Peyton Jones. 2018. Build systemsà la carte. Proceedings ACM Programing Languages 2, Article 79 (24 September 2018), Article 79, 79:1-79:29 pages. DOI:h p://dx.doi.org/10.1145/3236774 Non-recursive Make considered harmful: build systems at scale. Andrey Mokhov, Neil Mitchell, Simon Peyton Jones, Simon Marlow, Proceedings of the 9th International Symposium on Haskell. the 9th International Symposium on HaskellACMAndrey Mokhov, Neil Mitchell, Simon Peyton Jones, and Simon Marlow. 2016. 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IEEE. 1998 Fourth International Symposium on High-Performance Computer Architecture. IEEEJ Gregory Ste an and Todd C Mowry. 1998. e potential for using thread-level data speculation to facilitate automatic parallelization. In Proceedings 1998 Fourth International Symposium on High-Performance Computer Architecture. IEEE, 2-13. An Evaluation of Speculative Instruction Execution on Simultaneous Multithreaded Processors. Steven Swanson, Luke K Mcdowell, Michael M Swi, Susan J Eggers, Henry M Levy, 10.1145/859716.859720ACM Trans. Comput. Syst. 21340Steven Swanson, Luke K. McDowell, Michael M. Swi , Susan J. Eggers, and Henry M. Levy. 2003. An Evaluation of Speculative Instruction Execution on Simultaneous Multithreaded Processors. ACM Trans. Comput. Syst. 21, 3 (Aug. 2003), 314 340. DOI:h p://dx.doi.org/10.1145/859716.859720 Distributed speculations: providing fault-tolerance and improving performance. Cristian Tapus, California Institute of TechnologyPh.D. DissertationCristian Tapus. 2006. Distributed speculations: providing fault-tolerance and improving performance. Ph.D. Dissertation. California Institute of Technology. An E cient Algorithm for Exploiting Multiple Arithmetic Units. M Tomasulo, IBM Journal of Research. M. Tomasulo. 1967. An E cient Algorithm for Exploiting Multiple Arithmetic Units. IBM Journal of Research and	[]
[ "A Graph-Matching Approach for Cross-view Registration of Over-view and Street-view based Point Clouds", "A Graph-Matching Approach for Cross-view Registration of Over-view and Street-view based Point Clouds" ]	[ "Xiao Ling \nGeospatial Data Analytics Laboratory\nThe Ohio State University\n218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA\n\nDepartment of Civil\nEnvironmental and Geodetic Engineering\nThe Ohio State University\n218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA\n", "Rongjun Qin \nGeospatial Data Analytics Laboratory\nThe Ohio State University\n218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA\n\nDepartment of Civil\nEnvironmental and Geodetic Engineering\nThe Ohio State University\n218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA\n\nDepartment of Electrical and Computer Engineering\nThe Ohio State University\n205 Dreese Lab, 2036 Neil Avenue43210ColumbusOHUSA\n\nTranslational Data Analytics Institute\nThe Ohio State University\n\n" ]	[ "Geospatial Data Analytics Laboratory\nThe Ohio State University\n218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA", "Department of Civil\nEnvironmental and Geodetic Engineering\nThe Ohio State University\n218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA", "Geospatial Data Analytics Laboratory\nThe Ohio State University\n218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA", "Department of Civil\nEnvironmental and Geodetic Engineering\nThe Ohio State University\n218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA", "Department of Electrical and Computer Engineering\nThe Ohio State University\n205 Dreese Lab, 2036 Neil Avenue43210ColumbusOHUSA", "Translational Data Analytics Institute\nThe Ohio State University\n" ]	[]	Wide-area 3D data generation for complex urban environments often needs to leverage a mixed use of data collected from both air and ground platforms, such as from aerial surveys, satellite, and mobile vehicles, etc. On one hand, such kind of data with information from drastically different views (ca. 90° and more) forming cross-view data, which due to very limited overlapping region caused by the drastically different line of sight of the sensors, is difficult to be registered without significant manual efforts. On the other hand, the registration of such data often suffers from non-rigid distortion of the street-view data (e.g., non-rigid trajectory drift), which cannot be simply rectified by a similarity transformation. In this paper, based on the assumption that the object boundaries (e.g., buildings) from the over-view data should coincide with footprints of faç ade 3D points generated from street-view photogrammetric images, we aim to address this problem by proposing a fully automated geo-registration method for cross-view data, which utilizes semantically segmented object boundaries as view-invariant features under a global optimization framework through graph-matching: taking the over-view point clouds generated from stereo/multi-stereo satellite images and the street-view point clouds generated from monocular video images as the inputs, the proposed method models segments of buildings as nodes of graphs, both detected from the satellite-based and street-view based point clouds, thus to form the registration as a graph-matching problem to allow non-rigid matches; to enable a robust solution and fully utilize the topological relations between these segments, we propose to address the graph-matching problem on its conjugate graph solved through a belief-propagation algorithm. The matched nodes will be subject to a further optimization to allow precise-registration, followed by a constrained bundle adjustment on the street-view image to keep 2D-3D consistencies, which yields well-registered street-view images and point clouds to the satellite point clouds. Our proposed method assumes no or little prior pose information (e.g. very sparse locations from consumer-grade GPS (global positioning system)) for the street-view data and has been applied to a large cross-view dataset with significant scale difference containing 0.5 m GSD (Ground Sampling Distance) satellite data and 0.005 m GSD street-view data, 1.5 km in length involving 12 GB of data. The experiment shows that the proposed method has achieved promising results (1.27 m accuracy in 3D), evaluated using collected LiDAR point clouds. Furthermore, we included additional experiments to demonstrate that this method can be generalized to process different types of over-view and street-view data sources, e.g., the open street view maps and the semantic labeling maps.Related workAlgorithms addressing point cloud registration vary with the 3D data including their resolution, level of overlap, accuracy, assumed outlier/error models and heterogeneities between two 3D scans. There are a few widely-practiced algorithms in 3D point clouds registration for generally well-collected 3D point clouds, e.g., those from single-sources, being noiseless, and with sufficient overlaps (at least 20%) (Stechschulte et al., 2019), such as the Iterative Closest Point (ICP) methods (	null	[ "https://arxiv.org/pdf/2202.06857v1.pdf" ]	246,822,528	2202.06857	4ecf5934acd065f9d68c2b97771e030fcb0356ef	A Graph-Matching Approach for Cross-view Registration of Over-view and Street-view based Point Clouds Xiao Ling Geospatial Data Analytics Laboratory The Ohio State University 218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA Department of Civil Environmental and Geodetic Engineering The Ohio State University 218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA Rongjun Qin Geospatial Data Analytics Laboratory The Ohio State University 218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA Department of Civil Environmental and Geodetic Engineering The Ohio State University 218B Bolz Hall, 2036 Neil Avenue43210ColumbusOHUSA Department of Electrical and Computer Engineering The Ohio State University 205 Dreese Lab, 2036 Neil Avenue43210ColumbusOHUSA Translational Data Analytics Institute The Ohio State University A Graph-Matching Approach for Cross-view Registration of Over-view and Street-view based Point Clouds Preprint of the accepted version, with only minor editorial differences from the final version 1 corresponding author: Tel: +1 614 2924356, Email address: [email protected] (Rongjun Qin)cross-view registrationglobal optimizationmulti-view satellite image Wide-area 3D data generation for complex urban environments often needs to leverage a mixed use of data collected from both air and ground platforms, such as from aerial surveys, satellite, and mobile vehicles, etc. On one hand, such kind of data with information from drastically different views (ca. 90° and more) forming cross-view data, which due to very limited overlapping region caused by the drastically different line of sight of the sensors, is difficult to be registered without significant manual efforts. On the other hand, the registration of such data often suffers from non-rigid distortion of the street-view data (e.g., non-rigid trajectory drift), which cannot be simply rectified by a similarity transformation. In this paper, based on the assumption that the object boundaries (e.g., buildings) from the over-view data should coincide with footprints of faç ade 3D points generated from street-view photogrammetric images, we aim to address this problem by proposing a fully automated geo-registration method for cross-view data, which utilizes semantically segmented object boundaries as view-invariant features under a global optimization framework through graph-matching: taking the over-view point clouds generated from stereo/multi-stereo satellite images and the street-view point clouds generated from monocular video images as the inputs, the proposed method models segments of buildings as nodes of graphs, both detected from the satellite-based and street-view based point clouds, thus to form the registration as a graph-matching problem to allow non-rigid matches; to enable a robust solution and fully utilize the topological relations between these segments, we propose to address the graph-matching problem on its conjugate graph solved through a belief-propagation algorithm. The matched nodes will be subject to a further optimization to allow precise-registration, followed by a constrained bundle adjustment on the street-view image to keep 2D-3D consistencies, which yields well-registered street-view images and point clouds to the satellite point clouds. Our proposed method assumes no or little prior pose information (e.g. very sparse locations from consumer-grade GPS (global positioning system)) for the street-view data and has been applied to a large cross-view dataset with significant scale difference containing 0.5 m GSD (Ground Sampling Distance) satellite data and 0.005 m GSD street-view data, 1.5 km in length involving 12 GB of data. The experiment shows that the proposed method has achieved promising results (1.27 m accuracy in 3D), evaluated using collected LiDAR point clouds. Furthermore, we included additional experiments to demonstrate that this method can be generalized to process different types of over-view and street-view data sources, e.g., the open street view maps and the semantic labeling maps.Related workAlgorithms addressing point cloud registration vary with the 3D data including their resolution, level of overlap, accuracy, assumed outlier/error models and heterogeneities between two 3D scans. There are a few widely-practiced algorithms in 3D point clouds registration for generally well-collected 3D point clouds, e.g., those from single-sources, being noiseless, and with sufficient overlaps (at least 20%) (Stechschulte et al., 2019), such as the Iterative Closest Point (ICP) methods ( Introduction Coupled street-view and over-view images are regarded as a typical cross-view dataset, the perspective view differences of which are ca. 90° and more. Co-registering such dataset is scientifically challenging yet practically very useful, as it allows to spatially reference a large amount and a mixture of heterogeneous data for purposes of 3D modeling, data conflation and georeferencing. Often the over-view dataset comes with accurate or at least approximated geo-referenced information, and streetview with partial (often inaccurate) or no geo-referencing information but reflect high resolution and detailed information on faç ades of buildings and other objects, thus being naturally complementary. In addition, over-view datasets, particularly those from high-resolution satellite sensors, are consistently injecting contents to data archives with global coverage, and street-view datasets can be nowadays more cheaply captured with mobile agents running along the streets (mobile vehicles, crowdsourcing videos and images), which may lead to much lower cost when these data are being used. Having data from both views geometrically consistent will greatly benefit 3D data collection applications as it provides an alternative solution in cases where high quality data (e.g. decimeter or centimeter level oblique imageries, LiDAR (Light detection and ranging) point clouds) are not available or too expensive to acquire, and cases that data scanned from drastically different views need to be combined for practical applications such as for full three dimensional modeling for smart cities (Heinly et al., 2015). Turning well-captured over-view images (e.g. photogrammetric image blocks) into 3D geometry are nowadays standard practices for typical urban and suburban areas, which yields reasonably accurate 3D point clouds or meshes that present the scene (Colomina and Molina, 2014). However, cases for processing street-view images are more variable with respect to cameras to be used (e.g. low-cost camera with large lens distortions), disturbances from moving objects, convergence images and scene complexities, thus often yield complicated and non-rigid distortions (Remondino et al., 2017;Wu, 2014). For example, a typical case is the use of single-trajectory video frames for 3D street-view scene reconstruction, which theoretically presents sub-optimal camera networks which might lead to topographical distortion (Dall'Asta et al., 2015) and trajectory drifts (Nobre et al., 2017) in the resulting geometry. Therefore, because 3D geometry generated from the over-view datasets is more metrically correct, it can serve as the base dataset to which 3D geometry generated from the street-view can align. Although there have been attempts to address similar problems such as cross-view localization (Castaldo et al., 2015;Hu et al., 2018;Liu and Li, 2019) (i.e. localization on an over-view image given a ground-view) and cross-view synthesis (Regmi and Borji, 2018;Zhai et al., 2017;Lu et al., 2020;Toker et al., 2021) (simulating over-view using ground-view or vice versa), robust cross-view data co-registration appears to be more challenging and has not yet been well investigated. In general, coregistration of such over-view and street-view data encounters three major challenges: 1. Almost all the sensors collecting the 3D information are bearing-only and produce information only at line of sight, therefore the over-view and street-view dataset, due to the significant view differences, share very limited areas in common for extracting textural or geometric correspondences to build observations (Shan et al., 2014); 2. For cases that no georeferencing information is available, searching for coarse positional alignment to initiate the coregistration can be extremely difficult given the complexity of geometry in the street view; 3. The non-rigid topographical distortions often presented in street-view 3D geometry generated from a single-trajectory data collection hinder the use of simple transformation models such as a similarity or rigid transformation, in which more complicated and potentially non-parametric models are needed (Bruno and Roncella, 2019;Lee, 2009). These three challenges are essentially difficult cases in which common solutions used for 3D data co-registration often failed (Cheng et al., 2018;Pomerleau et al., 2015). For example, for challenge 1), none of the existing image feature extraction and matching methods are capable of processing cross-view images without overlapping texture (Morel and Yu, 2009) and 3D feature extraction & matching method are rather immature even for well-captured and high-quality 3D data (Hana et al., 2018). Although there exists learning based methods that learn feature correspondences in a cross-view scenario (Hu et al., 2018;Tian et al., 2017), such methods are usually sensitive to availability of training samples and are often subjected to a lack of transferability and generalizability for regions that are geographically and contextually different (Bengio et al., 2013). Solutions for challenge 2) often based on local and structural information to build correspondences; a typical application scenario is on navigation in GPS-denied environments, in which one needs to incrementally evaluate the built trajectory from video frames and map overlays, and formulate the matching through graphs to build approximate correspondences (Mourikis and Roumeliotis, 2007;Vaca-Castano et al., 2012). For challenge 3), although there exist a number of approaches for coregistration of deformable 3D shapes (Papazov and Burschka, 2011), these methods mostly focus on fitting pre-existing parameterized models (e.g. human body with underlying skeleton models) (Marin et al., 2020). However, the case of topological distortion in our context, is as a result of inaccurate camera poses, interior orientations and lens distortions, which may vary with the terrain and cannot be simply addressed using parametric models for correction. The intent of this work is to provide an accurate solution to co-register the street-view and over-view 3D data in a large urban region, in which we assume very little or no geo-referencing information for street-view data. Given that the over-view and street-view data are collected in an approximated 90• view difference, it is reasonable to assume that the object boundaries (e.g., buildings) from the over-view data should coincide with footprints of faç ade points from street-view. Therefore, in this paper, we propose to take advantage of the semantic information (e.g., buildings) for cross-view registration. Specifically, we model segments of buildings as nodes of graphs, both detected from the satellite-based and street-view based point clouds, thus to form the registration as a graph-matching problem to allow non-rigid matches; to enable a robust solution and fully utilize the topological relations between these segments, we propose to address the graph-matching problem on its conjugate graph solved through a belief-propagation algorithm. The matched nodes will be subject to a further optimization to gain precise piece-wise smooth rigid 3D transformations, followed by a constrained bundle adjustment on the street-view image to correct non-rigid distortions over the entire trajectory, to yield well-registered street-view images and point clouds at least with the same order of magnitude of the over-view data resolution (or, GSD (ground sampling distance)). The rest of this paper is organized as follows: Section 2 presents a literature survey related to relevant methodologies in 2D/3D cross-view registrations. Section 3 describes the proposed methodology for data preprocessing and cross-view registration. Experimental results and comparative studies on a cross-view dataset consisting of satellite and ground-view images are presented in Section 4. Section 5 concludes this paper by analyzing the advantages and drawbacks of our methodology that inform our planned future works. (Rusinkiewicz, 2019), fast robust ICP (Zhang et al., 2021)) and classic least-squares surface matching (Gruen and Akca, 2005). In particular, the ICP algorithm (and potentially its variants) is widely implemented into open-source / commercial software packages (Girardeau-Montaut, 2020; Rusu and Cousins, 2011), that often serve as the "first-trial" baseline when attempting a point cloud registration problem. The ICP algorithm and its variants assume the point clouds to first start with good initial positions (coarsely aligned), usually guided by manually collected 3D correspondences or external observations such as from GPS/IMU (Inertial Measurement Unit), and then the algorithm iteratively optimize transformation parameters (rigid or affine) by minimizing sum of point-wise distance metrics between assumed correspondences. Research focuses on these topics are to achieve better convergences of optimization given coarse initializations, for example, the GoICP (J. algorithm is capable of finding a global solution for point cloud registration, which as an improved version of the classic ICP method, utilized the BnB search (Breuel, 2003) to step out of local minimum that ICP trapped in using an iterative solution, while the work only assumed a rigid transformation while leaving more complicated models undiscussed. Besides the effort in achieving more robust and accurate matching with two pieces of 3D point clouds, only a few studies have been published for cases where point clouds are scanned from drastically different views and with potentially different resolution, the level of occlusions and lack of overlap easily fail foregoing algorithms, an example is the work proposed by B. , which extracted building outlines from airborne and terrestrial laser scanning point clouds respectively and then applied a general graph-matching approach specifically targeted on these outlines to obtain building correspondences, however it relied on well-delineated and complete line features which are luxury for noisy data like us. Furthermore, if any of the 3D point clouds present non-rigid distortions (e.g. due to systematic errors introduced by data capture, such as point clouds generated by images captured by uncalibrated data), these type of well-defined registration algorithms assuming parametric transformation models may not work. The classic 3D registration solutions (e.g. ICP) assume structured dense correspondences (i.e. one-to-one correspondence for all points in the overlapping region), while to accommodate more general scenario where 3D pieces are incomplete and noisy, solutions are sought by using pattern recognition techniques adapted to 3D structures for interest feature detection and matching, with the goal to independently locate 3D corresponding points that are distinctive and with high confidence. Popular 3D features include 3D points (Castellani et al., 2008;Sipiran and Bustos, 2011) and 3D line segments (Lin et al., 2017). The identification and matching of the 3D interest points are based on either handcraft features descriptors such as spin image (Johnson and Hebert, 1999), shape context (Belongie et al., 2000), and distribution histogram (Anguelov et al., 2005), or learnable descriptors such as local volumetric patch descriptor (Zeng et al., 2017), fully-convolutional geometric features (FCGF) (Choy et al., 2019) and combined multi-Layer Perception (CMLP) . However, these descriptors of 3D key points and lines utilize local structures of the 3D data, which can hardly be built equivalent on scans of 3D data coming from completely different views, and even more challenging when there exist large resolution difference and very little overlap between scans. Non-rigid and unknown distortions of the point clouds present as the largest hurdle when performing registration between two 3D point clouds scans, since on one hand, selecting the appropriate transformation model can be challenging, and on the other hand, the optimization of transformation parameters may get more difficult as the model is getting more complex. Existing solutions assume simple shaped models, by considering this unknown transformation to be approached by using mixture models, such as the work of Ma et al. (2018); Myronenko and Song (2010), where the potential over-parametrization was addressed by incorporating manifold regularization through the well-known Expectation Maximization (EM) algorithm (Myronenko and Song, 2010); a similar attempt solved these non-rigid parameters through kernel correlation maximization problems through Gaussian Mixture models (Jian and Vemuri, 2010). However, these methods worked with assumed nonrigid transformation model and compact and simple shaped geometry with good overlaps, thus can hardly be applied directly to cross-view registration cases where non-rigid distortion are more challenging. For point clouds generated from monocular video frames collected from a moving platform, as to be in this work, the resulting long and thin point clouds along the street may subject to severe trajectory drift owing to inaccurate camera interior orientations and the suboptimal camera networks, thereby the geometry may be inconsistently piecewise rigid/non-rigid, and cannot be simply modeled using a few transformation parameters. To sum, these multiple challenges, i.e. the lack of overlap for reliable 3D features, drastically view difference, and nonparametric distortions of geometry, have positioned the cross-view point cloud registration a unique problem to solve. Despite there are a few efforts designated to address the "cross-view" aspects by using deep learning methods to correlate ground-level and over-view images (Liu and Li, 2019;Hu et al., 2018;Tian et al., 2017;Castaldo et al., 2015), these work mainly focused on the "localization" aspects that meant to provide an image-level correspondence and so far were incapable of achieving pixel/point level correspondence identification cross different views. This paper aims to address the challenges specifically related to cross-view point cloud registration by 1) taking semantic object (e.g. buildings) boundaries as view-invariant feature for street-view to over-view matching, and 2) converting the global registration into a graph-matching and energy minimization problem and solved by belief propagation to obtain segment-level correspondences for registration. The rationale for using building boundary is that the boundaries from the over-view data should coincide with footprints of faç ade points from streetview, and the theoretical registration accuracy of building boundaries should be in the same magnitude as the GSD of the overview satellite images, which should be much more accurate than start-of-the-art image localization methods. Moreover, the using of graph matching can largely reduce the search space for the non-rigid registration, which enables the global solution for registering the cross-view point clouds with arbitrary initialization. Finally, the segment-level registration matches the assumption of piecewise rigid transformation well and can be performed robustly and efficiently. Methodology Given a point cloud generated by over-view sources and a point cloud collected from the street-view images by a monocular camera (with very limited or no geo-referencing information), we define the cross-view 3D point cloud registration as to precisely register the street-view point clouds to the over-view point clouds at the point level. Our solution, as mentioned above, relies on object boundaries casted on the ground, and performs segment-level matching through a global optimization to address piecewise rigid transformation, followed by point-cloud level fine registration. Figure 1 presents the general workflow of the proposed co-registration method: starting from producing the point clouds from both the over-view source (here we used multi-view satellite images at 0.5 GSD) and the street-view images (video frames from a single GoPro camera) as a preprocessing step; in a second step, semantic information (i.e., building segments) are extracted from both street-view and over-view point clouds using methods in Subsection 3.2. With the extracted buildings from both views, the segment-level correspondences are then formulated as a graph-matching problem solved by belief-propagation to yield globally consistent 2D segment matches, followed by point cloud level fine registration. After the 3D registration, we further apply a constrained bundle adjustment on the street-view images to enable the image poses to be consistent with the registrations as well, with which the dense point clouds can be optionally reproduced. Details of those steps are introduced in the following sub-sections. Preprocessing The input over-view images consist of 0.5m resolution Worldview (I/II) images, and the street-view images are collected by a single GoPro video camera (more details about the data will be introduced in Section 4). The preprocessing for separately converting the over-view and street-view images follow standard and existing photogrammetry and dense matching techniques: The multi-view satellite images are processed following a multi-stereo approach as described in (Qin, 2017(Qin, , 2016, which performs a pair-wise reconstruction to DSM, followed by a DSM (Digital Surface Model) fusion. The core matching algorithm uses a hierarchical Semi-Global Matching (Hirschmuller, 2008) with modifications to accommodate large-format images (Qin, 2014), and the approach for selecting and ranking the pairs follows the approach as described in (Qin, 2019) based on the available images and their metadata (DigitalGlobe, 2020). The street-view video frames are processed through an enhanced structure from motion (SfM)/photogrammetry approach which introduces a few strategies from the SLAM (Simultaneous Localization and Mapping) community (Mur-Artal and Tardós, 2017), i.e. with velocity models for feature tracking to enable robust incremental relative orientation and bundle adjustment, and it should be noted that without these strategies, a standard SfM may unlikely to succeed for a longer trajectory of video frames due to the unstable feature correspondences detected merely by standard feature operators (Lowe, 2004). Dense matching over these oriented frames is performed using a multi-view approach implemented in the open-source software OpenMVS (Cernea, 2015), the results of which are shown in multiple figures in this paper (e.g. Building segment extraction We assume the registration model from the street-view point clouds to the over-view data to be piece-wise rigid, meaning that registration from a section of the street-view point clouds to the over-view data follows a rigid transformation, while different sections of the point clouds might not share the same rigid transformations. Therefore, the idea here is to firstly segment both the over-view and street-view point clouds, and potentially correspond them to facilitate matching to accommodate our assumption on the piece-wise rigid transformation model. Therefore, in a first step, segments from the buildings or any offterrain object (since they are primarily buildings, we call them building segments for simplicity hereafter) need to be extracted for segment-level matching. Building segment extraction from over-view data. To take the boundary of buildings as the cue for registration, we first extract building segments from the over-view data. Here we assume orthophoto and DSM are available (can be easily converted from the dense point clouds if not). The extraction of building segments in our experiments follows a heuristic approach based on a grey-level top-hat based detector (Vincent, 1993;Qin and Fang, 2014) on the DSM, followed by a few cascade filtering using indices such as NDVI (normalized difference vegetation index) (Carlson and Ripley, 1997) and shadow index (Huang and Zhang, 2011). This extracts a building mask and individual building segments are extracted using a connected component analysis (Grana et al., 2010). An example of results is shown in Figure 2, and it can be seen that artifacts exist such as those detected segments in the river, and our approach in matching these segments necessarily consider such erroneous detections (introduced in Section 3.3). It should be noted that the building segment extraction algorithm is replaceable by other and possibly more advanced algorithms to obtain more accurate building segments, such as deep learning based detectors (Guo et al., 2020;Cheng and Fu, 2020), or by any existing and accurate building footprint data (e.g. some high-quality OpenStreetMap data (OpenStreetMap, 2021)) if well-aligned with the over-view images. Note when 3D data are not available, our approach may be able to register the street-view point clouds to be horizontally aligned with the building footprints, given the nature of our method intending to align the street-view point clouds to the over-view data through the building boundaries. Building segment extraction from street-view data. To identify building segments from the street-view point clouds, we first separate faç ade points from the ground points based on their normals. Here we assume that the Z plane of the street-view point clouds approximately aligns with the ground coordinate frame, thereby points with normals orthogonal to the vertical direction can be identified as faç ade points. To estimate the point-wise normal, we built a KD-tree structure (Muja and Lowe, 2014) for each point to find their k-nearest-neighboring point set , which will be used to calculate its normal vector . A histogram of orientations of all points are built and the most frequent direction is further validated and determined as the vertical direction , and this will allow a correction of the to be aligned with the vertical direction of the satellite point clouds (whose vertical direction can be extracted following the same method) through a rotation applied to the street-view point clouds. Points with their normal direction approximately perpendicular to the vertical direction are considered to be faç ade points. We specifically define the following two criteria to be met for determining a point with normal direction as a faç ade point: 1) the intersection angle between and is bigger than 75 degrees; 2) more than 75% neighbors in meet 1) to robustify the decision. In practice we find this process to be very effective in determining faç ade points, which produces primarily building faç ade point clouds with a few other small objects such as trees. We apply a standard region-growing segmentation (Rabbani et al., 2006) method to identify individual building segment as well as eliminating small segments of points, here in this work we eliminate any point clusters whose lengths are less than 5m in diameter horizontally. Figure 3 shows the extracted point clouds segments and it can be observed that almost all of these segments remains to be part of man-made/building objects. These dense points can be then further projected onto the Z plane to yield 2D building boundaries. It should be noted that Figure 3(a) shows a street-view point clouds of a closed loop, while the errors caused by trajectory drift lead to non-rigid distortions of the whole point cloud. Moreover, although faç ade points and their projected segments in 2D are extracted and used to serve the 2D building segment registration, the ground points are still associated with the nearest building segments, thus for any subsequent transformation, these points will be transformed with along with the building segment they belong to. (a) Point clouds (b) Individual segments Cross-view registration With the preprocessing steps mentioned in previous sections (Section 3.1-3.2), complete building segments from the overview data (Figure 2(b)) and partial building segments from the street-view point clouds (Figure 3(b)) projected to the Z-plane serve as the objects of interest for matching, in which the over-view building segments follow the geodetic coordinate frame derived from the satellite images and the street-view building segments are in arbitrary coordinate frame (with the vertical direction aligned with that of the satellite point clouds and the scale known). Given the non-rigid distortion of the street-view point clouds as shown in Figure 3(a), there is obviously not a single rigid transformation that transforms these 2D street-view building segments to be aligned with the over-view building segments, and with our piece-wise rigid assumption, the first task is to correspond the street-view building segments to the over-view segments (segment-level correspondence), and then perform rigid registrations per segment, with smoothness constraints over the individual rigid model to avoid the registration being too aggressive at a single segment-pair (due to errors), thus to yield global and point cloud level registration. Segment-level correspondence through graph-matching To correspond the 2D segments derived from the over-view and street-view data, the approach must abide by the following possible challenges: 1) there may not exist an initial correspondence; 2) the street-view segments are partial in shape and features can be hardly extracted to correspond to the complete shaped over-view building segments; 3) there might be one-tomany, or many-to-one correspondences or no correspondences. We consider this correspondence problem as graph-matching task, meaning to assign for each street-view segment, an over-view segment as the correspondence. However, given the first two challenges, extracting information that represent distinctive features for matching can be hardly achievable. Therefore, our solution seeks for the use of topological relationships between neighboring segments as a relatively more robust metric to corresponding segments. A concept is shown in Figure 4, where pairs of neighboring segments tend to provide more information to match incomplete street-view segments to the over-view segments. Therefore, we consider formulating the problem of matching building segments (segment as a node) into a conjugate graph (an edge linking a pair of segments as the node), which has more information to explore per node, such as orientations of the edge, length of the edge etc. As a result, our solution performs a graph-matching on this conjugate graph, that for each street-view edge ( , ) (defined as a pair of street-view segments and , where , are segment indices), we seek for its corresponding over-view edge ( , ) (defined as a pair of over-view segments and , where , are segment indices), where the label space is effectively represented as = { 1, 1 , 2, 2 , ⋯ , , } and , refers to an over-view edge (a node in the conjugated graph). Note that in this conjugate graph, although the number of nodes increases as compared to the original graph (one segment per node), we only enable nodes representing a pair of neighboring segments (through K-nearest criterion, here we use = 4), thus to greatly reduce the complexity of ( ( − 1) reduced to 4 , refers to the number of over-view segments). Given a street-view edge (a pair of street-view segments) _ , we evaluate the potential matchiness of a candidate over-view edge _ (a pair of over-view segments) by computing their minimal Euclidean distance at the 2D level. This is performed following three steps, first, we abstract both of the edges as lines ( _ and _ ), the end points of which are centers of the segments, and compute a rigid 2D transformation _2_ transforming _ to _ ; second, apply this 2D transformation to the street-view edge _ (or the pair of street-view segments), yielding _ ′ = _2_ ( _ ) to be approximately aligned with candidate over-view edge _ . Third, the distance between the _ ′ and _ , since they are close enough in the 2D plane, can be computed with the aid of distance map (Fabbri et al., 2008): Figure 4 as an example of the distance map that used to compute 2D shape matches. Here we offset _ ′ within a certain window (i.e. the bounding box of _ ) and take the local minimal of its distance to _ as the matchiness of the shapes (denoted as ). The offset that achieves the minimal distance can be incorporated into _2_ , denoted as _2_ . Apparently a "winner-takes-all" strategy based merely on the potential of matchiness between the street-view and over-view edge will be unlikely to yield good registration results, as the quality of segments vary, and such a local solution will generate noisy results. We consider a smoothness constraint that enforces the transformations _2_ obtained at the segment-level to be consistent for neighboring edges (or neighboring nodes in the conjugate graph), thereby to formulate an energy minimization problem for the conjugated graph that considers the matchiness described above as the data terms and the smoothness constraint as the smooth term in below: ( ) = ∑ ( ) + ∑ ∑ ( , ) ∈ (1) where ( ) refers to the data term, being the matchiness computed as described above for a node (a street-view edge) and the over-view edge ∈ = { 1, 1 , 2, 2 , ⋯ , , } as the potential candidate. denotes the neighborhood of , here defined as the over-view edges that share a node with . ( , ), refers to the smooth term, that punishes inconsistent transformation between neighboring nodes (over-view edges). Details about these terms are further given in below. 1. Data term: the costs for each node are normalized to [0, 1]; note here we define a null label 0 , to represent the case that when a street-view edge does not correspond to any over-view edge, in this case the cost ( 0 ) is directly set to 1. Smooth term ( , ) is defined in a truncated form that separately evaluating the consistency of rotation and translations from _2_ for neighboring nodes: ( , ) = { 1, ( , ) ∈ ℕ( ), ‖ − ‖ < ℎ ‖ − ‖ < ℎ 1, = 0 = 0 2, ℎ(2) where ℕ( ) is defined as a neighborhood set that contains over-view edges that shares a common node. Here ( , ) ∈ ℕ( ) means that the smooth term favors (i.e., with a low cost) neighboring street-view edges to have neighboring overview edges as correspondences. is the rotation angle in 2D, and is the translation derived from _2_ , ℎ is the angle threshold which is set to 10°, and ℎ is the translation threshold set to 100 meters. The second condition = 0 or = 0 means the neighborhood of a street-view edge is allowed to not correspond to any over-view edge. 1 is set as a smaller value to favor consistent transformation between neighboring edges, otherwise 2 (as a bigger value). Note this energy minimization problem assumes the conjugate graph a Markov-random field (Li, 1994), while the marginal distribution of the statistical variables of the graph (if considering the label for each node) is not independent. Thus, traditional graph-cut solver (Kolmogorov and Zabin, 2004) that aims to maximize marginal distribution will unlikely work. Thus, we use the loopy belief propagation algorithm (Coughlan and Ferreira, 2002), which iteratively minimize the energy through message passing and inference: given a pair of neighboring nodes (i.e. two street-view edges that shares a common node), the directed messages , from to , which is initialized to 1, is updated by considering all message flowing into ( The graph-matching result of the sample trajectory (Figure 5(a)) is shown in Figure 5(b), where the street-view segments (colored in isolated solid line) are visualized on the top-view building segments (white mask). Center of gravity for the overview building segments are in solid green line and that of street-view segments is in blue dash line, which are topologically corresponded well. It should be noted that these are fragmental and noisy street-view segments in the red-rectangle region, which failed to find corresponding over-view segments, which are marked as null correspondences by our algorithm. (a) street-view trajectory (b) matching results to find corresponding over-view segments, which are marked as null correspondences by our algorithm. There two tunable but dependent parameters 1, 2, the difference of which control the smoothness of the transformation parameters between neighboring buildings. Therefore we fixed 2 to 0.6 (as an empirical mid-value (in a scale of 1)) and test how the algorithm performed according to difference 1 on a sample trajectory as shown in Table 1. It shows different c1, when set between 0.1 and 0.5 will lead to convergence with differed iteration number. According to the table we thus take 1 = 0.1, 2 = 0.6 as a good empirical value throughout the experiment. Point-cloud level fine registration The algorithm described in Section 3.3.1 provides a solution for matching 2D building segment, here we perform a fine registration at the building segment level by 1) further refining 2D segment-level alignment, and 2) performing a 3D rotation to align the street-view point clouds to the over-view point clouds, and both of these two approaches are rather heuristic: Fine 2D registration at the building segment level: We further refine the 2D transformation parameters by performing an fine-level exhaustive search, where the range of the search for the 2D rotation stays within [−10°, 10°] with a predefined interval (e.g. 1°) as the searching step, and the offset range is within the bounding box of the over-view segments, with one pixel per step. The process can be speed up using a dichotomy search, while in practice this 2D level matching is operated in the image grid with the same resolution as the over-view image (i.e. satellite orthophoto in our case), thus can be processed efficiently. This will yield a 2D rotation matrix 2 and offset 2 . Fine-level 3D registration: To advance the registration to 3D, a fine-level registration is performed to align the Z direction of the street-view point clouds to the over-view point clouds. Although the Z-plane of the entire street-view point clouds have been roughly aligned with the over-view point clouds (as introduced Section 3.2), these may still exist rotational errors (primarily roll and pitch). We consider the same approach used for the whole street-view point clouds to correct segment level rotation error: for each building segment and its associated ground points, we first separate the ground points based on the normals (those whose intersection angle with the vertical direction is smaller than 15°, followed by a median filtering to remove noises), and then compute a rotation matrix based on the normal of the segment-level ground points and the normal computed from the over-view point clouds to achieve segment level 3D registration; this is then followed by a Z-plane correction by computing the Z-offset between the street-view and over-view ground points. With the resulting transformation parameters after the heuristic searching process, we obtain for each street-view segment, the 2D offset 2 , 2D rotation 2 , 3D rotation and Z offset . Therefore, the final 3D transformation 3 = [ 3 \| 3 ] from the raw street-view dense point to the over-view point cloud for each street-view segment can be expressed as: 3 = [ 2 0 0 1 ](5) = [ 2 \| ] Although this process is heuristic while it practically has yield promising results, and an example is shown in Figure 6 that depicts a building segment with and without the point cloud level refinement. This process will output a list of 3D transformations for extracted street-view building segments (as introduced in Section 3.3.1) and their associated point clouds Bundle adjustment With the registered street-view point clouds with the rigid transformations for each segment 3 = { 3 , = 1,2, … , }, we can re-optimize the camera poses such that these are consistent with the registered street-view point clouds, so to benefit applications such as texture mapping. The re-optimization requires the 3D and 2D correspondences to be pre-recorded during the preprocessing stage, thus when the 3D points are transformed based on 3 , the correspondences can be reused as observations to readjust the poses. Here we assume the 3D and 2D correspondences are kept for the street-view SfM reconstruction, and we employ a robust transformation procedure using a distance-weighted averaging when applying the piece-wise rigid transformation 3 : for each 3D point belonging to a segment , the transformation not only applies the 3 , but also 3 (where ∈ are neighboring segments) to this point, and the transformed point takes a weighted average of 3D points produced by applying these transformations, where the weight is inversely proportional to the distance between the point and the segment , as follows: where ( \| , ) is the projection of a 3D point onto the image {( , )} following the collinearity equations (Thompson et al., 1966) and , is the corresponding feature point on the image, Δ is difference between the estimated and the registered pose. is a constant parameter to leverage the contribution of the constraint and it is set to 20 in our experiments. The first term of this equation minimizes the reprojection error and the second penalize the poses to be too different from the registered poses. The constrained bundle adjustment problem in Equation 7 can be easily implemented and solved using the Ceres Solver (Agarwal and Mierle, 2012) and the yielded image poses can be used to optionally to reproduce the dense street-view point clouds resulting in globally consistent data with respect to the over-view point clouds. This process of re-optimizing the image poses and reproducing the dense point clouds using the new poses, may correct minor registration errors of individual building segments in previous processing. 3 = ∑ 1≤ Experiments The testing region is in the main campus of The Ohio State University in Columbus, Ohio, USA. covering an area of ca. 16 2 . Twelve WorldView I/II satellite images covering the testing region are collected and processed using the approach described in Qin et al. (2019b) based on the satellite stereo processing software RSP (Qin, 2017(Qin, , 2016 to generate the DSM, orthophoto and point clouds for this region. For the street-view data, we mounted a GoPro Hero 7 camera on the top of a car and drive around the campus covering a trajectory equivalent to 33 km (resulting in approximately 300GB of video data). The speed of the vehicle is 25 mph on average, which was used to estimate the scale . We quantitatively analyzed the accuracy of the proposed cross-view geo-registration algorithm in both 3D and 2D. For 3D analyses, a Lidar point cloud is collected from a mobile platform covering a 1.5 km street in length in our testing region, data shown in Figure 7(a). The 2D analyses evaluates the proximity of the projected building boundaries from street-view to the derived building boundaries of the over-view DSM. (Figure 3(a)). (a) LiDAR point clouds used for 3D evaluation (b) 2D reference DSM boundaries used for evaluation (c) registered street-view points Registration accuracy The accuracy of the geo-registration in 3D is evaluated by comparing the street-view point clouds (Figure 7(c)) to the LiDARbased 3D scan (Figure 7(a)). The symmetric cloud-to-cloud distance metric -the Chamfer distance is used for evaluation: here we compute for each point 3 in the street-view point clouds its closest distance to the LiDAR point clouds as where (·,·) refers to 3D Euclidean distance. The symmetric Chamfer distance averages the distances of these points as the cloud-to-cloud distance. To accelerate and remove outliers of the point clouds, we disregard any point-to-point distance that are larger than 10 m, which has yielded the 1.27 meters of error and its point level distribution is shown in Figure 8(a), which can be seen that most of the per point errors are less than 2 meters and the computed standard deviation (std.) of this error distribution is 1.34 meters. Both the metrics (Chamfer distance and the std.) at the range of 1-1.5 meters indicates the registration has matched to the over-view dataset at the level of 2-3 pixels considering its GSD being 0.5 meters. (a) 3D registration accuracy (b) 2D registration accuracy The similar evaluation is performed at the 2D level, which uses the same distance metric (simply reduced to 2D), and gives a horizontal metric accuracy (in Chamfer distance) of 0.99 meters with a standard deviation of 1.04, with distribution show in Figure 8(b) indicating errors are mostly less than 2 meters. This is equivalent to two pixels of accuracy in the over-view data. Figure 9 shows a few detailed results at the building level for visual assessment, LiDAR point clouds drawn to serve as a reference. It can be seen that these faç ade points (from the street-view) and the roof points are consistently aligned well and form the complete point clouds for the buildings. Registration of street-view point clouds to OpenStreetMap and semantic labels Lidar Street-view point clouds + Lidar Street-view + over-view point clouds Since our proposed methods take the building boundaries/segments as the major source for registration, it can be flexibly applied to any kind of 2D vector data. Here we assume only 2D over-view information are available, in the form of OpenStreetMap (OSM) (OpenStreetMap, 2021) building segments, or segments detected from the Orthophoto through semantic segmentation. An example of these two types of data are shown in Figure 10(a) and (b), respectively the OSM data and the semantic labels detected using a U-Net detector described in (Qin et al., 2019a,b). We simply regard these two types of data as our over-view building segments as apply the algorithms described in Section 3.3.1-3.3.2. Note for the OSM, we intentionally eliminated building segments that are under 5 meters in diameter, to match the detectable buildings in satellite data for fair comparison. Results of the registration in a small test region is shown in Figure 10(c) and ( Large-scale experiments We further expand our evaluation scope by applying the proposed method to larger region that includes 10 trajectories of data involving approximately 150 K street-view images, covering an area of 16 this region is shown in Figure 11(a) and (b), respectively showing the street-view point clouds and the combined street-view and over-view point clouds, particularly in Figure 11(b) the large-scale point clouds shows that both the street-view and overview point clouds are well aligned. Since there is no mobile LiDAR data available this experimental region, we only evaluate the 2D accuracy based on the over-view building segments as the ground truth. An average 2D registration error (Chamfer distance as described in Section 4.1 of 1.24 m is obtained for the 10 trajectories. If ranked based on the registration accuracy, the top 5 trajectories with most buildings in this area has obtained an average 2D registration error of 1.05 m, while the rest obtained 1.52 m on average. Figure 12(a) and (b) show two examples where both point clouds are registered extremely well (with the registration errors are less than 0.5m, i.e. less than a pixel). (a) street-view points (b) combined street-view + over-view point clouds Conclusions In this paper, we present a solution to address a novel and challenging problem in 3D data registration, where we aim to automatically register the street-view point clouds generated from monocular video cameras, to over-view point clouds generated from multi-view satellite images (with 0.5 meters). The problem is challenging in that 1) the over-view and streetview point clouds are compensatory in terms of their line of sight, and share very little common area for registration; 2) there exist a huge difference in terms of the resolution of two types of data (a factor of a hundred); 3) the street-view trajectory presents non-rigid, non-parametric topographical distortions that are not suitable for simple and parametric registration models (e.g. rigid transformation). To this end, we observe that the most informative information to tie these two datasets are building boundaries which are extractable from both of the two datasets, thus the solution we developed evolved around this observation by firstly perform 2D building boundary matching, where challenges of matching incomplete building segments are tackled using a global optimization framework, and the 3D registration are further refined through a heuristically exhaustive search. The camera poses are then readjusted based on the registration results to generate consistent and well-registered street-view point clouds. The proposed method is evaluated with a rich dataset that involves as many as ten trajectories of video frame data (150 K in total) with 33 in length, covering an region of 16 2 , we show that our proposed method achieves 0.99 m registration accuracy in horizontal direction, a 1.27 m in 3D; we have also demonstrated that the proposed method is flexible enough to register to other type of dataset including OSM and building segments extracted from semantic segmentation results. The proposed method assumes the scale of the street-view data is known, while it is designed to be GPS agonistic and does not require any initial correspondences between the over-view and street-view data. In occasions where coarse correspondences are known, they can be easily adapted to our proposed framework. However, since the proposed method heavily depends on boundaries of buildings or off-terrain objects, we found that in practice that the registration accuracy can be dependent on the building boundaries, and apparently in scene with very sparse buildings/off-terrain objects, the proposed method may fail, in which coarse GPS information might be used to improve the robustness of the algorithm, and this will be attempted in our future work. Disclaimer: Mention of brand names in this paper does not constitute an endorsement by the authors. 6 Figure 1 : 1Workflow of the proposed cross-view registration algorithm. Details of each component are introduced in the subsections. Figure 3 (a), Figure 7 (c) , Figure 11 ( 37,11a) etc.). Note in our work, we assume the scale of the model is known based on measurements of known dimensional objects or coarse GPS information. Figure 2 : 2An example of building segment extraction from the over-view multispectral orthophoto and DSM. (a) The orthophoto with buildings highlighted in red, (b) Individual building segments shown in different colors. Figure 3 : 3Extracted façade segment (b) from the street-view point clouds (a). These segments are colorized to indicate their independence, and each segment will be served as the candidates to match with the over-view building segments. Note the point cloud shows street sides of a closed loop while is open due to error caused by trajectory drifts (non-rigid and non-parametric distortion). Figure 4 : 4A pair of segments serving as a node in the conjugate graph. Given two over-view building segments _ = { , } and two street-view building segments _ = { , }, the coarse aligned street-view building segments _ ′ is firstly obtained by transforming centers of _ to centers of _ and the distance map for over-view segments is calculated where the intensity of pixel denotes the distance to closest point in segments, then the final transformation is obtained by moving the _ ′ back and forth to achieve the minimal distance. except for message from ) via Equation 3, where ℕ( )\ is the set of neighboring street-view edges except j for , the cost of each label (over-view edge) for are computed from Equation 4 and the smallest cost is taken as its selected label, this procedure iterates until the computed energy Equation 1 converges, and labels with the smallest cost for each street-view edges are taken as final solutions. Figure 5 : 5Illustration of the result of the segment-level matching for the sample trajectory. (a) building segments from street-view data; (b) matching results. The solid green line connects the centers of buildings from over-view building segments, while the blue dash line connects the centers of building segments, colorized to differentiate different street-view segments. Street-view segments in the red-rectangle region failed Figure 6 : 6An example depicting the combined point clouds with and without the point-cloud level fine registration. the distance between a 3D point to the street-view building segment, and the similar transformation can be performed directly on the original poses of the image, by applying the appropriate 3 to the translation of the pose and the 3 on the rotation matrix, resulting in the registered image poses {( , )}.To ensure the registered image poses follow the collinearity equation and the epipolar geometry, we re-optimize the poses through a constrained bundle adjustment, the constraint being that the estimated poses should be close to the registered pose while satisfy collinearity equations. Hereby the formulation is: for the registered street-view 3D points { } and the 2D observations , , as well as the registered pose of each image {( , )} we minimize: Figure 7 ( 7b) shows the DSM building boundaries of an area as the reference (manually edited to ensure the fidelity of the evaluation). The registered street-view dense point clouds (reproduced using the optimized pose described in Section 3.3.3) is shown in Figure 7(c), which present no visible trajectory drift as compared to the original street-view point clouds directly computed through SfM Figure 7 : 7The 3D Lidar point clouds (a) and 2D DSM building boundaries (b) are utilized as ground truth for the accuracy evaluation of the registered street-view dense point clouds (c) in our experiment. The length for this close looped area is 1.5 km, and buildings on both sides are scanned. Segments shown in (b) colorized to differentiate individual segments. for each LiDAR point 3 in , its closest distance to the street-view point clouds as( 3 ,). Here we only consider points are symmetrically consistent in these two clouds to form the Chamfer distance: 3 from and 3 from are mutually the closest point to each other: Figure 8 : 8Registration accuracy in 3D and 2D by comparing to the ground-truth Lidar point cloud and the over-view building boundary points. For the 3D case, the average and mean squared errors are 1.27m and 1.34m, respectively. For the 2D case, those values are 0.99m and 1.04m, respectively. Figure 9 : 9Examples of registered street-view point clouds to the over-view point clouds. From left to right: LiDAR point cloud, the street-view point clouds as compared to LiDAR, combined street-view and over-view point clouds. Figure 11 :Figure 12 : 1112Registration results in a large region covering 16 km 2 . (a) is the registered street-view point clouds and (b) is the merged cross-view point clouds. Two well-registered examples. (a) and (b) are two examples showing buildings with good street-view and over-view alignment (registration error less than 0.5 m). The over-view point clouds are in the first column while the cross-view point clouds are in second column. Table 1 . 1Iterationnumbers used in graph matching when 2 = 0.6 on a sample trajectory. 1 0 0.1 0.2 0.3 0.4 0.5 0.6 Iteration number diverged 8 7 10 16 42 diverged d), which shows that the proposed method works equivalently well for different types of data. The Chamfer distance and the std. of the error (as used for accuracy evaluation in Section 4.1) are 1.21 meters & 1.34 meters for OSM data, and 1.46 meters & 1.59 meters for semantic labeling dataset, which similarly match the accuracy concluded in Section 4.1. Note that all these experiments were performed using the same parameters throughout this work without tuning.Figure 10: Registration results on the OSM and semantic labels dataset. Building segments from street-view are colored in blue. The Chamfer distance and the std. of the error are 1.21 meters & 1.34 meters for OSM data, and 1.46 meters & 1.59 meters for semantic labeling dataset.(a) OSM building segments (b) semantic labels (c) OSM registration result (d) semantic labels registration result = { 3 , = 1,2, … , } for subsequent processing. , with the cumulated trajectory 33 in length.The dataset has yielded over 1 billion street-view points and around 10 million over-view points for registration. Results of AcknowledgementThe study is supported by the ONR grant (Award No. N000141712928). 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[ "Charge-density wave induced by combined electron-electron and electron-phonon interactions in 1T -TiSe 2 : A variational Monte Carlo study", "Charge-density wave induced by combined electron-electron and electron-phonon interactions in 1T -TiSe 2 : A variational Monte Carlo study" ]	[ "Hiroshi Watanabe \nComputational Quantum Matter Research Team\nRIKEN Center for Emergent Matter Science (CEMS)\n351-0198WakoSaitamaJapan\n", "Kazuhiro Seki \nComputational Condensed Matter Physics Laboratory\nRIKEN\n351-0198WakoSaitamaJapan\n", "Seiji Yunoki \nComputational Quantum Matter Research Team\nRIKEN Center for Emergent Matter Science (CEMS)\n351-0198WakoSaitamaJapan\n\nComputational Condensed Matter Physics Laboratory\nRIKEN\n351-0198WakoSaitamaJapan\n\nComputational Materials Science Research Team\nRIKEN Advanced Institute for Computational Science (AICS)\n650-0047KobeHyogoJapan\n" ]	[ "Computational Quantum Matter Research Team\nRIKEN Center for Emergent Matter Science (CEMS)\n351-0198WakoSaitamaJapan", "Computational Condensed Matter Physics Laboratory\nRIKEN\n351-0198WakoSaitamaJapan", "Computational Quantum Matter Research Team\nRIKEN Center for Emergent Matter Science (CEMS)\n351-0198WakoSaitamaJapan", "Computational Condensed Matter Physics Laboratory\nRIKEN\n351-0198WakoSaitamaJapan", "Computational Materials Science Research Team\nRIKEN Advanced Institute for Computational Science (AICS)\n650-0047KobeHyogoJapan" ]	[]	To clarify the origin of a charge-density wave (CDW) phase in 1T -TiSe2, we study the ground state property of a half-filled two-band Hubbard model in a triangular lattice including electronphonon interaction. By using the variational Monte Carlo method, the electronic and lattice degrees of freedom are both treated quantum mechanically on an equal footing beyond the mean-field approximation. We find that the cooperation between Coulomb interaction and electron-phonon interaction is essential to induce the CDW phase. We show that the "pure" exciton condensation without lattice distortion is difficult to realize under the poor nesting condition of the underlying Fermi surface. Furthermore, by systematically calculating the momentum resolved hybridization between the two bands, we examine the character of electron-hole pairing from the viewpoint of BCS-BEC crossover within the CDW phase and find that the strong-coupling BEC-like pairing dominates. We therefore propose that the CDW phase observed in 1T -TiSe2 originates from a BEC-like electron-hole pairing.	10.1103/physrevb.91.205135	[ "https://arxiv.org/pdf/1505.02207v2.pdf" ]	119,295,961	1505.02207	837cd332f5f2c58c5f2c2cea750a92be4bef4cc3	Charge-density wave induced by combined electron-electron and electron-phonon interactions in 1T -TiSe 2 : A variational Monte Carlo study 29 May 2015 (Dated: June 1, 2015) Hiroshi Watanabe Computational Quantum Matter Research Team RIKEN Center for Emergent Matter Science (CEMS) 351-0198WakoSaitamaJapan Kazuhiro Seki Computational Condensed Matter Physics Laboratory RIKEN 351-0198WakoSaitamaJapan Seiji Yunoki Computational Quantum Matter Research Team RIKEN Center for Emergent Matter Science (CEMS) 351-0198WakoSaitamaJapan Computational Condensed Matter Physics Laboratory RIKEN 351-0198WakoSaitamaJapan Computational Materials Science Research Team RIKEN Advanced Institute for Computational Science (AICS) 650-0047KobeHyogoJapan Charge-density wave induced by combined electron-electron and electron-phonon interactions in 1T -TiSe 2 : A variational Monte Carlo study 29 May 2015 (Dated: June 1, 2015)numbers: 7110-w7145Lr7135Lk7127+a To clarify the origin of a charge-density wave (CDW) phase in 1T -TiSe2, we study the ground state property of a half-filled two-band Hubbard model in a triangular lattice including electronphonon interaction. By using the variational Monte Carlo method, the electronic and lattice degrees of freedom are both treated quantum mechanically on an equal footing beyond the mean-field approximation. We find that the cooperation between Coulomb interaction and electron-phonon interaction is essential to induce the CDW phase. We show that the "pure" exciton condensation without lattice distortion is difficult to realize under the poor nesting condition of the underlying Fermi surface. Furthermore, by systematically calculating the momentum resolved hybridization between the two bands, we examine the character of electron-hole pairing from the viewpoint of BCS-BEC crossover within the CDW phase and find that the strong-coupling BEC-like pairing dominates. We therefore propose that the CDW phase observed in 1T -TiSe2 originates from a BEC-like electron-hole pairing. I. INTRODUCTION Charge-density wave (CDW) is widely observed in lowdimensional solids and has been extensively studied both experimentally and theoretically [1,2]. Transition metal dichalcogenides MX 2 (M=transition metal, X=S, Se, Te) are one of the typical CDW materials with a layered triangular lattice structure. They show various CDW patterns, depending on the combination of M and X [3][4][5], and quite often superconductivity (SC) is observed next to the CDW phase by applying pressure [6,7], doping [8,9], or intercalation [10]. However, the origin of CDW and SC has not been fully understood and it has been still under debate in spite of the long and extensive studies so far. Recently, 1T -TiSe 2 , one of the old transition metal dichalcogenides, has again attracted much interests in the context of exciton condensation. This material is a semimetal or a semiconductor in room temperature [11][12][13] and shows a commensurate CDW transition with a 2×2×2 superstructure below T c ∼ 200K. The Fermi surface (FS) partially remains even in the CDW phase due to the imperfect opening of the energy gap. The origin of the CDW phase is still controversial and the usual nesting mechanism seems unlikely due to its poor FS nesting [14]. Alternatively, exciton condensation has been proposed as a possible mechanism for the CDW phase [15][16][17]. Indeed, the large spectral weight transfer between Ti 3d and Se 4p bands and the flat energy spectrum just below the Fermi energy have been observed [16], strongly sug- * [email protected] gesting the possibility of exciton condensation. On the other hand, another possible mechanism for the CDW transition is a band Jahn-Teller effect which results from electron-phonon interaction [18][19][20][21]. A large lattice distortion of several percent observed below T c [4] indicates the strong coupling between electronic and lattice degrees of freedom [22]. Very recently, it is proposed that both mechanisms work cooperatively for the CDW transition [23][24][25][26][27]. Exciton condensation is a quantum state expected in low carrier density systems such as a semimetal or a semiconductor and has been extensively studied since 1960s [28][29][30][31]. The exciton is a bound pair of an electron and a hole in different bands mediated by the interband Coulomb interaction. When the binding energy of an exciton exceeds the band gap, the system has an instability toward condensation of excitons, namely, a spontaneous hybridization between different bands. Since the repulsive Coulomb interaction is attractive between an electron and a hole, the exciton condensation is expected to occur in principle without considering any additional "glue" of the electron-hole pair. Although extensive efforts have been devoted for half a century and several candidates have been proposed [16,[32][33][34][35], the generally accepted materials for the exciton condensation are still absent. Therefore, any conclusive evidence for the exciton condensation in real materials is highly desired for further progress and 1T -TiSe 2 would be one of the promising examples. In the pioneering studies for exciton condensation [29,30], an isotropic band dispersion with perfect FS nesting is assumed, for which the excitonic instability is always present in a semimetallic case. The extension to anisotropic band dispersions shows that the degree of FS nesting greatly affects the instability of exciton condensation [36]. Moreover, the mean-field approximation generally overestimates the instability toward ordered states, including the exciton condensation. Therefore, for discussion of the exciton condensation in real materials, the realistic band dispersion and the appropriate method beyond the mean-field approximation are both required. In this paper, a half-filled two-band Hubbard model with electron-phonon interaction in a triangular lattice is studied to understand the origin of the CDW phase in 1T -TiSe 2 . The ground state properties are calculated using the variational Monte Carlo (VMC) method for multiorbital systems [37]. We find that the cooperation between Coulomb interaction and electron-phonon interaction is essential to induce the CDW phase. The CDW phase is observed between the normal metal and band insulator phases with intermediate interband Coulomb interaction. We show that the "pure" exciton condensation without lattice distortion is difficult to realize under the poor FS nesting condition in a triangular lattice. Furthermore, we systematically calculate the momentum resolved hybridization between the two bands to show that the strong-coupling BEC-like pairing dominates in the CDW phase. Our results therefore suggest that the CDW phase observed in 1T -TiSe 2 originates from the strong-coupling BEC-like electron-hole pairing. The rest of this paper is organized as follows. In Sec. II, a two-band Hubbard model in a two-dimensional triangular lattice is introduced as a low energy effective model for 1T -TiSe 2 . The detailed explanation of the VMC method and the variational wave functions are also given in Sec. II. The numerical results are then provided in Sec. III. Finally, the implication of our results for 1T -TiSe 2 is discussed in Sec. IV, followed by the summary in Sec. V. II. MODEL AND METHOD We consider a two-band Hubbard model in a twodimensional triangular lattice defined as H = k,σ ε c k c † kσ c kσ + k,σ ε f k f † kσ f kσ + U c i n c i↑ n c i↓ + U f i n f i↑ n f i↓ + U ′ i n c i n f i + 1 √ N k,q,σ g(k, q)(b q + b † −q )c † k+qσ f kσ + H.c. + q ω(q) b † q b q + 1 2 ,(1) where c † kσ (f † kσ ) creates an electron in c (f ) band with momentum k and spin σ (=↑, ↓). The band dispersions ε c k and ε f k are given as ε c k = 2t c cos k x + 2 cos 1 2 k x cos √ 3 2 k y + 2t ′ c cos √ 3k y + 2 cos 3 2 k x cos √ 3 2 k y + µ c (2) and ε f k = 2t f cos k x + 2 cos 1 2 k x cos √ 3 2 k y ,(3) respectively. We introduce the next-nearest-neighbor hopping t ′ c to locate the bottom of c band at M points. U c (U f ) is an on-site intraband Coulomb interaction within c (f ) band and U ′ is an on-site interband Coulomb interaction between c and f bands. n α iσ is a number operator of α (= c, f ) electron at site i with spin σ and n α i = n α i↑ +n α i↓ . g(k, q) is an electron-phonon coupling constant and b † q is a creation operator of phonon with momentum q and frequency ω(q). In this model, the lattice distortion changes the c-f bond length as shown in Fig. 1(d), which couples to the c-f hybridization modulated with wave vector q through g(k, q). The total number of sites is indicated by N . The noninteracting tight-binding parameters, (t f , t c , t ′ c , µ c ) = (1.0, 0.2, 0.15 , 6.0) t,(4) are set to mimic the electronic structure of 1T -TiSe 2 with a hole pocket at the Γ point and electron pockets at the M points as shown in Figs. 1(a) and 1(b). The ordering wave vectors connecting Γ and M points are denoted as q 1 , q 2 , and q 3 . Although the ordering wave vectors observed in 1T -TiSe 2 connect Γ and L points with a finite k z component [38], here we consider a pure two-dimensional model for simplicity and the limitation of the model is discussed later. The effect of Coulomb interaction and electron-phonon interaction is treated on an equal footing using a VMC method. We consider the trial wave function as follows: \|Ψ = P e-ph \|Ψ ph \|Ψ e .(5)\|Ψ e = P(2) G P J c \|Φ is an electron wave function consisting of three parts. \|Φ is a Slater determinant constructed by diagonalizing the one-body part of Hamiltonian H including the variational tight-binding parame- ters (t f = 1,t c ,t ′ c ,μ c ) and the off-diagonal element V which induces the c-f hybridization. Here, we assume V = V 0 exp[−A(ε c k+qi −ε f k ) 2 ] , and V 0 and A are both variational parameters. V 0 is an amplitude of the c-f hybridization and A controls the internal extent of exciton in k space. We have found that the variational energy is improved by introducing A and the behavior of A is related to the BCS-BEC crossover of exciton condensation [39]. P (2) G is a Gutzwiller factor extended for two-band systems [39,40]. In P The trial wave function for phonon is assumed to be a Gaussian in the normal coordinate {Q q } representation [41,42], Ψ ph ≡ {Q q } Ψ ph = exp − q 1 2 (Q q − β q ) 2 α 2 q ,(6) where Q q = i u i e −iq·ri / √ N is Fourier transform of real space lattice distortion {u i } at site i. Since the ordering wave vectors q 1 , q 2 , and q 3 are exactly half of the reciprocal lattice vectors of the normal phase [see Fig. 1(a)], the corresponding normal coordinate Q qi (i = 1, 2, 3) are real numbers. Therefore, we can take the trial wave function and variational parameters α q and β q all real. Notice that α q controls the extent of the Gaussian wave function, i.e., the amplitude of lattice vibration, and that The remaining part is an electron-phonon projection operator: P e-ph = exp γ i u i n c i (2 − n f i ) . This operator controls the attraction between c electrons and f holes which results from the electron-phonon interaction and γ is a variational parameter. The variational parameters in \|Ψ are thereforet c ,t ′ c , µ c , V 0 , A, {g Γ }, {v αβ ij }, {α q }, {β q }, and γ, and they are simultaneously optimized using stochastic reconfiguration method [43]. The system sizes are varied from L × L = 12 × 12 to 24 × 24 with antiperiodic boundary conditions in both directions of primitive lattice vectors for the triangular lattice. Figure 2(a) shows the ground state phase diagram where U c = U f = U and U ′ are varied for fixed g(k, q)/t = g/t = 0. 19 and ω(q)/t = ω/t = 0.1 [44]. We find that there are three distinct phases in the phase diagram: normal metal (NM), charge-density-wave insulator (CDWI), and band insulator (BI). When U ′ is large enough, the c band is lifted above the Fermi energy and the BI phase with the empty c band and the fullyoccupied f band is stabilized. No static lattice distortion is observed in both NM and BI phases. III. RESULT Between the NM and BI phases, the CDWI phase emerges where the c-f hybridization parameter ∆ q = k,σ c † k+qσ f kσ + H.c.(7) is finite [45] for q corresponding to the three ordering wave vectors q 1 , q 2 , and q 3 , simultaneously, implying a triple-q CDW state. Here, O = Ψ\|O\|Ψ / Ψ\|Ψ with the optimized \|Ψ . Thus, the first Brillouin zone is folded as indicated in Fig. 1(a). It leads to the charge disproportionation in 2×2 unit cell [ Fig. 1(c)] with one charge rich A site and three charge poor B, C, and D sites [see Fig. 3(a)]. In the CDWI phase, a static lattice distortion always occurs through the electron-phonon interaction and the "pure" exciton condensation without lattice distortion is never found. Note that the CDWI phase is limited to a narrow region in Fig. 2(a) especially for small U/t in spite of a finite g/t. This is in sharp contrast with the case of a square lattice where the NM phase appears only at U ′ = 0 and the exciton condensation phase is widely observed even without the electron-phonon interaction [39,47,48]. This difference is caused by the different FS nesting condition: the FS nesting is better (perfect if only with the nearest-neighbor hopping) in the square lattice but poor in the triangular lattice. Therefore, the electron-phonon interaction is indispensable to manifest the CDWI phase under the poor FS nesting [2]. We also show the phase diagram in Fig. 2(b) where U and g are varied with U ′ = U/2. The CDWI region is enlarged with increasing U and g, implying that both Coulomb interaction and electron-phonon interaction stabilize the CDWI phase. This result is thus qualitatively consistent with previous study [26]. We also find that the CDWI phase is not stabilized, but only the NM and BI phases appear, when g = 0, at least, in a realistic parameter region. This suggests that the "pure" exciton condensation induced by the Coulomb interaction alone, the original idea of exciton condensation [28][29][30], is difficult to realize in our model. The pure exciton condensation certainly occurs in particular models such as one-dimensional models [23,49] or twodimensional models with perfectly nested electron and hole FSs [39,47,48,50]. Therefore, the stability of the pure exciton condensation depends strongly on the lattice structure and the underlying FS. Let us now examine the detailed properties of the CDWI phase. Figure 3(a) shows the distribution of average c-electron density n c X and f -hole density 2− n f X in 2×2 unit cell (X=A, B, C, and D). It is found in Fig. 3(a) that c electrons and f holes, i.e., mobile carriers, are concentrated mostly at A site with n c A + n f A > 2 to gain the c-f hybridization energy coupled with the lattice distortion, while the number of these mobile carriers are small and n c X + n f X < 2 at B, C, and D sites. Therefore, the system clearly exhibits the charge disproportionation. Notice that the mobile carrier densities at B, C, and D sites are the same within the statistical errors simply because the three ordering wave vectors q 1 , q 2 , and q 3 are equivalent in a hexagonal lattice structure. Next, let us discuss the lattice degrees of freedom in the CDWI phase. In the VMC calculation, the bond length u i always fluctuates around the average value during the Monte Carlo steps [41]. As shown in Fig. 3(b), we find that the bond length at A site is shortened from the original one (u A < 0), while those at B, C, and D sites are lengthened (only u B > 0 is shown). Figs. 3(a) and 3(b) thus confirm that the electronic and lattice degrees of freedom are strongly coupled and cooperatively induce the CDWI phase. Furthermore, we study the character of electron-hole pairing from the viewpoint of BCS-BEC crossover, which has been often discussed in the exciton problems [39,48,[50][51][52][53]. For this purpose, we calculate the momentum resolved c-f hybridization φ(k) defined as Brillouin zone, indicating the strong-coupling BEC-like pairing due to the large Coulomb interaction. Although φ(k) becomes less extended with decreasing the Coulomb interaction, it still has a broad structure away from the Fermi momentum k F , as shown in Fig. 4(b) for (U/t, U ′ /t)=(3.0, 1.5). Indeed, the CDWI region rapidly decreases with decreasing U [see Fig. 2(a)] and our systematic calculations do not find a clear BCS-like region in the CDWI phase shown in Fig. 2. Because of i) the poor nesting between c-electron and f -hole FSs and ii) the small density of states around the Fermi energy for low carrier densities, the energy gain due to the gap opening induced by the c-f hybridization in the vicinity of k F is small and hence the weak-coupling BCS-like pairing is not favored. Even in such a case, the BEC-like tightly-bounded electron-hole pairing in real space can be induced by the electron-phonon interaction with the help of Coulomb interaction and dominates the CDWI phase. In contrast, we have found a clear and wide BCSlike region for the same model but in a square lattice with perfectly nested FSs [39]. Therefore, the FS nesting is essential for the BCS-like pairing. φ(k) = q,σ c † k+qσ f kσ + H.c. .(8) IV. DISCUSSION Finally, let us discuss the implication of our results for 1T -TiSe 2 . In our model, Ti 3d and Se 4p bands are simplified as c and f bands, respectively, and the orbital characters are ignored. Moreover, our model only includes the change of c-f bond length which couples with exciton condensation. Even with these simplifications, our model captures the important energy scales of 1T -TiSe 2 . The electron-phonon coupling used here is 4g 2 /ω = 0 − 3.6t ≈ 0 − 1.8 eV, which is relevant for 1T -TiSe 2 [5] if we take t ≈ 0.5 eV. The lattice distortion obtained in our calculation is 0.05 − 0.2Å, consistent with the observed value ∼ 0.085Å [4,54]. Our results thus suggest that the CDW phase observed in 1T -TiSe 2 is due to the strong-coupling BEC-like electron-hole pairing. Indeed, the BEC-like character is indicated by several theoretical works [13,27] and experimental observations such as a short coherence length estimated by Kohn anomaly [5], lack of incommensurate CDW phase, relatively high electrical resistivity above T c , and a large value of 2∆/k B T c (∆: the CDW gap) [38]. On the other hand, the chiral CDW phase observed in 1T -TiSe 2 [55,56] is beyond our model. In the chiral CDW phase, the charge density is modulated with clockwise or anticlockwise pattern. The proper descrip-tion of this phase requires three dimensionality [57] or higher-order electron-phonon and phonon-phonon interactions [26] which induce the phase difference between the three ordering wave vectors. The origin of the SC induced by applying pressure or intercalation of Cu atoms is also an interesting unresolved issue. The relation between the CDW and the SC is still controversial [58,59] and both conventional [20,23,60] and unconventional SC [61] have been proposed. Our results suggest that both electronic and lattice degrees of freedom are crucial to understand the origin of the SC. Our study will be a first step toward the unified understanding of various quantum phases observed in 1T -TiSe 2 . V. SUMMARY In summary, we have studied the two-band Hubbard model in a triangular lattice for 1T -TiSe 2 with the electron-phonon interaction. The VMC method is employed to treat the electronic and lattice degrees of freedom on an equal footing beyond the mean-field approximation. We have shown that both Coulomb and electronphonon interactions stabilize the CDW phase. We have found that the "pure" exciton condensation without the lattice distortion is difficult to realize and the electronphonon interaction is essential for the CDW phase. The character of electron-hole pairing within the CDW phase has also been examined by calculating the momentum resolved c-f hybridization. We have shown that the strongcoupling BEC-like pairing dominates the CDW phase. Under the poor FS nesting condition and with small density of states around Fermi energy, the energy gain due to the gap opening in the vicinity of k F is small and hence the weak-coupling BCS-like pairing is not favored. Our results thus conclude that the CDW phase observed in 1T -TiSe 2 originates from the strong-coupling BEC-like electron-hole pairing due to the cooperative Coulomb and electron-phonon interactions. FIG. 1 . 1(color online) (a) Fermi surfaces (blue circle and red ellipses) and (b) energy dispersions in the noninteracting limit for the two-band Hubbard model with electron density n = 2. A set of tight-binding parameters used is (t f , tc, t ′ c , µc) = (1.0, 0.2, 0.15, 6.0) t with t = t f as an energy unit. (c) Schematic real-space figure of a 2×2 superstructure (dotted lines) with each supercell containing 4 sites (A, B, C, and D). The corresponding ordering wave vectors q1, q2, and q3 are indicated in (a). Dashed lines in (a) represent the folded Brillouin zone due to the formation of the 2×2 superstructure in (c). (d) Schematic real-space figure of the lattice distortion considered in the c-f plane along one particular direction, e.g., AB direction indicated in (c). ui represents the displacement of c atom at site i from its original position. of charge and spin configuration at each site \|Γ , i.e., \|0 = \|0 0 , \|1 = \|0 ↑ , · · · , \|15 = \|↑↓ ↑↓ , are differently weighted and their weight {g Γ } are optimized as variational parameters. P J c = exp[− i =j αβ v αβ ij n α i n β j ] is a charge Jastrow factor which controls long-range charge correlations. Here, v αβ ij = v αβ (\|r i − r j \|) is assumed and r i is the position of site i. FIG. 2 . 2(color online) Ground state phase diagram of the two-band Hubbard model in (a) U -U ′ plane (g/t = 0.19) and (b) U -g plane (U ′ = U/2). We set Uc = U f = U and ω/t = 0.1. NM, CDWI, and BI denote normal metal, charge-densitywave insulator, and band insulator, respectively. The electron density is n = 2, i.e., at half filling. β q corresponds to the average value of Q q and thus there exists a static lattice distortion with u i = 0 for a finite β q . The Monte Carlo update scheme for {Q q } and the estimation of phonon energy are the same as in Ref.42. FIG. 3 . 3(color online) (a) Average c-electron density n c X (red bars) and f -hole density 2 − n f X (blue bars) in 2 × 2 unit cell for X = A, B, C, and D [see Fig. 1(c)]. (b) Snap shots of c-f bond length uA and uB at A and B sites, respectively, as a function of Monte Carlo (MC) step. The model parameters used are (U/t, U ′ /t, g/t, ω/t)=(4.0, 2.0, 0.19, 0.1) for L = 24 in the CDWI phase. FIG. 4 . 4(color online) Momentum resolved c-f hybridization φ(k) for (a) (U/t, U ′ /t)=(8.0, 4.0) and (b) (U/t, U ′ /t)=(3.0, 1.5). The noninteracting Fermi momentum k c F (folded around Γ point at the center) and k f F are shown with white solid and black dashed curves, respectively [see also Fig. 1(a)]. g/t = 0.19 and ω/t = 0.1 are fixed for L = 24. Figure 4 4shows φ(k) for (U/t, U ′ /t)=(8.0, 4.0) and (3.0, 1.5), both being located in the CDWI phase in Fig. 2. For (U/t, U ′ /t)=(8.0, 4.0), φ(k) is extended in the whole ACKNOWLEDGMENTThe authors thank Y. Fuseya, T. Shirakawa, T. Kaneko, and K. Imura for useful discussions. The computation has been done using the RIKEN Cluster of Clusters (RICC) facility and the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. This work has been supported by JSPS KAKENHI Grant No. 26800198 and in part by RIKEN iTHES Project. . M D Johannes, I I Mazin, Phys. Rev. B. 77165135M. D. Johannes and I. I. Mazin, Phys. Rev. B 77, 165135 (2008). . J A Wilson, F J Di Salvo, S Mahajan, Phys. Rev. Lett. 32882J. A. Wilson, F. J. Di Salvo, and S. 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[ "Identification of jump Markov linear models using particle filters", "Identification of jump Markov linear models using particle filters" ]	[ "Andreas Svensson ", "Thomas B Schön ", "Fredrik Lindsten " ]	[]	[]	Jump Markov linear models consists of a finite number of linear state space models and a discrete variable encoding the jumps (or switches) between the different linear models. Identifying jump Markov linear models makes for a challenging problem lacking an analytical solution. We derive a new expectation maximization (EM) type algorithm that produce maximum likelihood estimates of the model parameters. Our development hinges upon recent progress in combining particle filters with Markov chain Monte Carlo methods in solving the nonlinear state smoothing problem inherent in the EM formulation. Key to our development is that we exploit a conditionally linear Gaussian substructure in the model, allowing for an efficient algorithm.Solving (2) is challenging and there are no closed form solutions available. Our approach is to derive an expectation maximization (EM) [10] type of solution, where the strategy is to separate the original problem into two closely linked problems. The first problem is a challenging, but manageable nonlinear state smoothing problem and the second problem is a tractable optimization problem. The nonlinear smoothing problem we can solve using a combination of sequential Monte Carlo (SMC) methods (particle filters and particle smoothers)[11]and Markov chain Monte Carlo (MCMC) methods[27]. More specifically we will make use of particle MCMC (PMCMC), which is a systematic way of exploring the strengths of both approaches by using SMC to construct the necessary high-dimensional Markov kernels needed in MCMC[1,19].Our main contribution is a new maximum likelihood estimator that can be used to identify jump Markov linear models on the form (1). The estimator exploits the conditionally linear Gaussian substructure that is inherent in (1) via Rao-Blackwellization. More specifically we derive a Rao-Blackwellized version of the particle stochastic approximation expectation maximization (PSAEM) algorithm recently introduced in [18].Jump Markov linear models, or switching linear models, is a fairly well studied class of hybrid systems. For recent overviews of existing system identification methods for jump Markov linear models, see[13,22]. Existing approaches considering the problem under study here include two stage methods, where the data is first segmented (using e.g. change detection type of methods) and the individual models are then identified for each segment, see e.g.[23,6]. There has also been approximate EM algorithms proposed for identification of hybrid systems[5,15]and the very recent [3] (differing from our method in that we use stochastic approximation EM and Rao-Blackwellization). There are also relevant relationships to the PMCMC solutions introduced in [33] and the SMC-based on-line EM solution derived in[34].There are also many approaches considering the more general problem with an unknown number of modes K and an unknown state dimension n z , see e.g.[12] and [4], making use of Bayesian nonparametric models and mixed integer programming, respectively.	10.1109/cdc.2014.7040409	[ "https://arxiv.org/pdf/1409.7287v1.pdf" ]	18,028,396	1409.7287	63c05a53b5fe8bafce88601277ca334db60b2536	Identification of jump Markov linear models using particle filters September 26, 2014 Andreas Svensson Thomas B Schön Fredrik Lindsten Identification of jump Markov linear models using particle filters September 26, 2014 Jump Markov linear models consists of a finite number of linear state space models and a discrete variable encoding the jumps (or switches) between the different linear models. Identifying jump Markov linear models makes for a challenging problem lacking an analytical solution. We derive a new expectation maximization (EM) type algorithm that produce maximum likelihood estimates of the model parameters. Our development hinges upon recent progress in combining particle filters with Markov chain Monte Carlo methods in solving the nonlinear state smoothing problem inherent in the EM formulation. Key to our development is that we exploit a conditionally linear Gaussian substructure in the model, allowing for an efficient algorithm.Solving (2) is challenging and there are no closed form solutions available. Our approach is to derive an expectation maximization (EM) [10] type of solution, where the strategy is to separate the original problem into two closely linked problems. The first problem is a challenging, but manageable nonlinear state smoothing problem and the second problem is a tractable optimization problem. The nonlinear smoothing problem we can solve using a combination of sequential Monte Carlo (SMC) methods (particle filters and particle smoothers)[11]and Markov chain Monte Carlo (MCMC) methods[27]. More specifically we will make use of particle MCMC (PMCMC), which is a systematic way of exploring the strengths of both approaches by using SMC to construct the necessary high-dimensional Markov kernels needed in MCMC[1,19].Our main contribution is a new maximum likelihood estimator that can be used to identify jump Markov linear models on the form (1). The estimator exploits the conditionally linear Gaussian substructure that is inherent in (1) via Rao-Blackwellization. More specifically we derive a Rao-Blackwellized version of the particle stochastic approximation expectation maximization (PSAEM) algorithm recently introduced in [18].Jump Markov linear models, or switching linear models, is a fairly well studied class of hybrid systems. For recent overviews of existing system identification methods for jump Markov linear models, see[13,22]. Existing approaches considering the problem under study here include two stage methods, where the data is first segmented (using e.g. change detection type of methods) and the individual models are then identified for each segment, see e.g.[23,6]. There has also been approximate EM algorithms proposed for identification of hybrid systems[5,15]and the very recent [3] (differing from our method in that we use stochastic approximation EM and Rao-Blackwellization). There are also relevant relationships to the PMCMC solutions introduced in [33] and the SMC-based on-line EM solution derived in[34].There are also many approaches considering the more general problem with an unknown number of modes K and an unknown state dimension n z , see e.g.[12] and [4], making use of Bayesian nonparametric models and mixed integer programming, respectively. Introduction Consider the following jump Markov linear model on state space form s t+1 \| s t ∼ p(s t+1 \|s t ),(1a)z t+1 = A st+1 z t + B st+1 u t + w t ,(1b)y t = C st z t + D st u t + v t ,(1c) where ∼ means distributed according to and the (discrete) variable s t takes values in {1, . . . , K} (which can be thought of as different modes which the model is jumping between) and the (continuous) variable z t lives in R nz . Hence, the state variable consists of x t (z t , s t ). Furthermore, v t ∈ R ny and w t ∈ R nz are zero mean white Gaussian noise and Ew t w T t = Q st+1 , Ev t v T t = R st and Ew t v T t ≡ 0. The output (or measurement) is y t ∈ R ny , the input is u t ∈ R nu . As K is finite, p(s t+1 \|s t ) can be defined via a matrix Π ∈ R K×K with entries π mn p(s t+1 = n\|s t = m). We are interested in off-line identification of jump Markov linear models on the form (1) for the case of an unknown jump sequence, but the number of modes K is known. More specifically, we will formulate and solve the Maximum Likelihood (ML) problem to compute an estimate of the static parameters θ of a jump Markov linear model based on a batch of measurements y 1:T {y 1 , . . . , y T } and (if available) inputs u 1:T by solving, θ ML = arg max θ∈Θ p θ (y 1:T ). ( Here θ {{A n , B n , C n , D n , Q n , R n } K n=1 , Π}, i.e., all unknown static parameters in model (1). Here, and throughout the paper, the dependence on the inputs u 1:T is implicit. Expectation maximization algorithms The EM algorithm [10] provides an iterative method for computing maximum likelihood estimates of the unknown parameters θ in a probabilistic model involving latent variables. In the jump Markov linear model (1) we observe y 1:T , whereas the state x 1:T is latent. The EM algorithm maximizes the likelihood by iteratively maximizing the intermediate quantity Q(θ, θ ) log p θ (x 1:T , y 1:T )p θ (x 1:T \| y 1:T )dx 1:T . More specifically, the procedure is initialized in θ 0 ∈ Θ and then iterates between computing an expected (E) value and solving a maximization (M) problem, (E) Compute Q(θ, θ k−1 ). (M) Compute θ k = arg max θ∈Θ Q(θ, θ k−1 ). Intuitively, this can be thought of as 'selecting the new parameters as the ones that make the given measurements and the current state estimate as likely as possible'. The use of EM type algorithms to identify dynamical systems is by now fairly well explored for both linear and nonlinear models. For linear models, there are explicit expressions for all involved quantities, see e.g. [14,30]. For nonlinear models the intermediate quantity Q(θ, θ ) is intractable and we are forced to approximate solutions; see e.g. [18,29,21,7]. This is the case also for the model (1) under study in this work. Indeed, the maximization step can be solved in closed form for the model (1), but (3) is still intractable in our case. It is by now fairly well established that we can make use of sequential Monte Carlo (SMC) [11] or particle Markov chain Monte Carlo (PMCMC) [1] methods to approximate the joint smoothing distribution for a general nonlinear model arbitrarily well according to p(x 1:T \| y 1:T ) = N i=1 w i T δ x i 1:T (x 1:T ),(4) where x i 1:T are random samples with corresponding importance weights w i T , δ x is a point-mass distribution at x and we refer to {x i 1:T , w i T } N i=1 as a weighted particle system. The particle smoothing approximation (4) can be used to approximate the integral in (3). Using this approach within EM, we obtain the particle smoothing EM (PSEM) method [21,29]. PSEM can be viewed as an SMC-analogue of the well known Monte Carlo EM (MCEM) algorithm [32]. However, it has been recognized that MCEM, and analogously PSEM, makes inefficient use of the generated samples [9]. This is particularly true when the simulation step is computationally expensive, which is the case when using SMC or PMCMC. To address this shortcoming, [9] proposed to use a stochastic approximation (SA) [26] Q k (θ) = (1 − γ k ) Q k−1 (θ) + γ k log p θ (y 1:T , x 1:T [k]),(5) with {γ k } ∞ k=1 being a sequence of step sizes which fulfils ∞ k=1 γ k = ∞ and ∞ k=1 γ 2 k < ∞. In the above, x 1:T [k] is a sample state trajectory, simulated from the joint smoothing distribution p θ k (x 1:T \| y 1:T ). It is shown by [9] that the SAEM algorithm-which iteratively updates the intermediate quantity according to (5) and computes the next parameter iterate by maximizing this stochastic approximation-enjoys good convergence properties. Indeed, despite the fact that the method requires only a single sample x 1:T [k] at each iteration, the sequence {θ k } k≥1 will converge to a maximizer of p θ (y 1:T ) under reasonably weak assumptions. However, in our setting it is not possible to simulate from the joint smoothing distribution p θ k (x 1:T \| y 1:T ). We will therefore make use of the particle SAEM (PSAEM) method [18], which combines recent PMCMC methodology with SAEM. Specifically, we will exploit the structure of (1) to develop a Rao-Blackwellized PSAEM algorithm. We will start our development in the subsequent section by considering the smoothing problem for (1). We derive a PMCMC-based Rao-Blackwellized smoother for this model class. The proposed smoother can, principally, be used to compute (3) within PSEM. However, a more efficient approach is to use the proposed smoother to derive a Rao-Blackwellized PSAEM algorithm, see Section 4. Smoothing using Monte Carlo methods For smoothing, that is, finding p θ (x 1:t \|y 1:t ) = p θ (s 1:T , z 1:T \|y 1:T ), various Monte Carlo methods can be applied. We will use an MCMC based approach, as it fits very well in the SAEM framework (see e.g. [2,17]), which together shapes the PSAEM algorithm. The aim of this section is therefore to derive an MCMC-based smoother for jump Markov linear models. To gain efficiency, the jump sequence s 1:T and the linear states z 1:T are separated using conditional probabilities as p θ (s 1:T , z 1:T \|y 1:T ) = p θ (z 1:T \|s 1:T , y 1:T )p θ (s 1:T \|y 1:T ). This allows us to infer the conditionally linear states z 1:T using closed form expressions. Hence, it is only the jump sequence s 1:T that has to be computed using approximate inference. This technique is referred to as Rao-Blackwellization [8]. 3.1 Inferring the linear states: p(z 1:T \|s 1:T , y 1:T ) State inference in linear Gaussian state space models can be performed exactly in closed form. More specifically, the Kalman filter provides the expressions for the filtering PDF p θ (z t \|s 1:t , y 1:t ) = N (z t \| z f ;t , P f ;t ) and the one step predictor PDF p θ (z t+1 \|s 1:t+1 , y 1:t ) = N (z t \| z p;t+1 , P p;t+1 ). The marginal smoothing PDF p θ (z t \|s 1:T , y 1:T ) = N (z t \| z s;t , P s;t ) is provided by the Rauch-Tung-Striebel (RTS) smoother [25]. See, e.g., [16] for the relevant results. Here, we use N (x \| µ, Σ) to denote the PDF for the (multivariate) normal distribution with mean µ and covariance matrix Σ. 3.2 Inferring the jump sequence: p(s 1:T \|y 1:T ) To find p(s 1:T \|y 1:T ), an MCMC approach is used. First, the concept of using Markov kernels for smoothing is introduced, and then the construction of the kernel itself follows. MCMC makes use of ergodic theory for statistical inference. Let K θ be a Markov kernel (to be defined below) on the T -fold product space {1, ..., K} T . Note that the jump sequence s 1:T lives in this space. Furthermore, assume that K θ is ergodic with unique stationary distribution p θ (s 1:T \|y 1:T ). This implies that by simulating a Markov chain with transition kernel K θ , the marginal distribution of the chain will approach p θ (s 1:T \|y 1:T ) in the limit. Specifically, let s 1:T [0] be an arbitrary initial state with p θ (s 1:T [0]\|y 1:T ) > 0 and let s 1:T [k] ∼ K θ (·\|s 1:T [k− 1])) for k ≥ 1, then by the ergodic theorem [27]: 1 n n k=1 h(s 1:T [k]) → E θ [h(s 1:T )\|y 1:T ] ,(7) as n → ∞ for any function h : {1, ..., K} T → R. This allows a smoother to be constructed as in Algorithm 1. Algorithm 1 MCMC smoother 1: Initialize s 1:T [0] arbitrarily 2: for k ≥ 1 do 3: Generate s 1:T [k] ∼ K θ (·\|s 1:T [k − 1]) 4: end for We will use the conditional particle filter with ancestor sampling (CPF-AS) [19] to construct the Markov kernel K θ . The CPF-AS is similar to a standard particle filter, but with the important difference that one particle trajectory (jump sequence), s 1:T , is specified a priori. The algorithm statement for the CPF-AS can be found in, e.g., [19]. Similar to an auxiliary particle filter [11], the propagation of p θ (s 1:t−1 \|y 1:t−1 ) (approximated by {s i 1:t−1 , w i t−1 } N i=1 ) to time t is done using the ancestor indices {a i t } N i=1 . To generate s i t , the ancestor index is sampled according to P a i t = j ∝ w j t−1 , and s i t as s i t ∼ p θ (s t \|s a i t t−1 ). The trajectories are then augmented as s i 1:t = {s a i t 1:t−1 , s i t }. This is repeated for i = 1, . . . , N − 1, whereas s N t is set as s N t = s t . To 'find' the history for s N t , the ancestor index a N t is drawn with probability P a N t = i ∝ p θ (s i 1:t−1 \|s t:T , y 1:T ).(8) The probability density in (8) is proportional to p θ (y t:T , s t:T \|s i 1:t−1 , y 1:t−1 )p θ (s i 1:t−1 \|y 1:t−1 ),(9) where the last factor is the importance weight w i t−1 . By sampling s 1: T [k + 1] = s J 1:T from the rendered set of trajectories {s i 1:T , w i T } N i=1 with P (J = j) = w j T , a Markov kernel K θ mapping s 1:T [k] = s 1:T to s 1:T [k + 1] is obtained. For this Markov kernel to be useful for statistical inference we require that (i) it is ergodic, and (ii) it admits p θ (s 1:T \|y 1:T ) as its unique limiting distribution. While we do not dwell on the (rather technical) details here, we note that these requirements are indeed fulfilled; see [19]. Rao-Blackwellization Rao-Blackwellization of particle filters is a fusion of the Kalman filter and the particle filter based on (6), and it is described in, e.g., [28]. However, Rao-Blackwellization of a particle smoother is somewhat more involved since the process x t \|y 1:T is Markovian, but not s t \|y 1:T (with z t marginalized, see, e.g., [33] and [20] for various ways to handle this). A similar problem as for the particle smoothers arises in the ancestor sampling (8) in the CPF-AS. In the case of a non-Rao-Blackwellized CPF-AS, (8) reduces to w i t−1 p(x t \|x i t−1 ) [19] . This does not hold in the Rao-Blackwellized case. To handle this, (8) can be rewritten as w i t−1 p(y t:T , s t:T \|s i 1:t−1 , y 1:t−1 ). Using the results from Section 4.4 in [20] (adapted to model (1)), this can be written (omitting w i t−1 , and with the notation z 2 Ω z T Ωz, P ΓΓ T , i.e. the Cholesky factorization, Q t F t F T t and A t A s t etc.) p(y t:T , s t:T \|s i 1:t−1 , y 1:t−1 ) ∝ Z t−1 \|Λ t−1 \| −1/2 exp(− 1 2 η t−1 ),(11a)with Λ t = Γ i,T f ;t Ω t Γ i f ;t + I,(11b)η t = z i f ;t 2 Ωt − 2λ T t z i,T f ;t − Γ n f ;t (λ t − Ω t z n f ;t ) 2 M −1 t ,(11c) where Ω t = A T t+1 I − Ω t+1 F t+1 M −1 t+1 F T t+1 Ω t+1 A t+1 ,(11d)Ω t = Ω t + C T t R −1 t C t ,(11e)M t = F T t ΩF t + I,(11f)λ t = A T t+1 I − Ω t+1 F t+1 M −1 t+1 F T t+1 m t ,(11g)λ t = λ t + C T t R −1 t (y t − D t u t ),(11h)m t = ( λ t+1 − Ω t+1 B t+1 u t+1 ).(11i) and Ω T = 0 and λ T = 0. The Rao-Blackwellization also includes an RTS smoother for finding p θ (z 1:T \|s 1:T , y 1:T ). Summarizing the above development, the Rao-Blackwellized CPF-AS (for the jump Markov linear model (1)) is presented in Algorithm 2, where p θ (y t \|s i 1:t , y 1: t−1 ) = N (y t ; C s i t z n p;t + D s i t u t , C s i t P p;t C T s i t + R s i t )(12) is used. Note that the discrete state s t is drawn from a discrete distribution defined by Π, whereas the linear state z t is handled analytically. The algorithm implicitly defines a Markov kernel K θ that can be used in Algorithm 1 for finding p(s 1:T \|y 1:T ), or, as we will see, be placed in an SAEM framework to estimate θ (both yielding PMCMC [1] constructions). Identification of jump Markov linear models In the previous section, an ergodic Markov kernel K θ leaving p θ (s 1:T \|y 1:T ) invariant was found as a Rao-Blackwellized CPF-AS summarized in Algorithm 2. This will be used together with SAEM, as it allows us to make one parameter update at each step of the Markov chain smoother in Algorithm 1, as presented as PSAEM in [18]. (However, following [18], we make use of all the particles generated by CPF-AS, and not only Draw a i t with P a i t = j = w j t−1 for i = 1, . . . , N − 1. 8: Draw s i t with P s i t = n = π s i t−1 ,n for i = 1, . . . , N − 1. 9: Compute {Λ i t−1 , η i t } according to (11b)-(11c). 10: Draw a N t with P a N t = i ∝ w i t−1 π s i t−1 ,s N t \|Λ i t−1 \| −1/2 exp(− 1 2 η i t−1 ). 11: Set s i 1:t = {s 13: Compute z i p;t , P i p;t , z i f ;t and P i f ;t for i = 1, . . . , N . 14: Set w i t ∝ p θ (y t \|s i t , y 1:t−1 ) for i = 1, . . . , N s.t. i w i t = 1. Q k (θ) = (1 − γ k ) Q k−1 (θ)+ γ k N i=1 w i T E θ where the expectation is w.r.t. z 1:T . Putting this together, we obtain a Rao-Blackwellized PSAEM (RB-PSAEM) algorithm presented in Algorithm 3. Note that this algorithm is similar to the MCMC-based smoother in Algorithm 1, but with the difference that the model parameters are updated at each iteration, effectively enabling simultaneous smoothing and identification. 4: Compute Q k (θ) according to (13). 5: Compute θ k = arg max θ∈Θ Q k (θ) 6: end for (For notational convenience, the iteration number k is suppressed in the variables related to {s i 1:T , w i T } N i=1 .) With a strong theoretical foundation in PMCMC and Markovian stochastic approximation, the RB-PSAEM algorithm presented here enjoys very favourable convergence properties. In particular, under certain smoothness and ergodicity conditions, the sequence of iterates {θ k } k≥1 will converge to a maximizer of p θ (y 1:T ) as k → ∞, regardless of the number of particles N ≥ 2 used in the internal CPF-AS procedure (see [18,Proposition 1] together with [17] for details). Furthermore, empirically it has been found that a small number of particles can work well in practice as well. For instance, in the numerical examples considered in Section 5, we run Algorithm 3 with N = 3 with accurate identification results. For the model structure (1), there exists infinitely many solutions to the problem (2); all relevant involved matrices can be transformed by a linear transformation matrix and the modes can be re-ordered, but the input-output behaviour will remain invariant. The model is therefore over-parametrized, or lacks identifiability, in the general problem setting. However, it is shown in [24] that the Cramér-Rao Lower Bound is not affected by the over-parametrization. That is, the estimate quality, in terms of variance, is unaffected by the over-parametrization. Maximizing the intermediate quantity When making use of RB-PSAEM from Algorithm 3, one major question arises from Step 5, namely the maximization of the intermediate quantity Q k (θ). For the jump Markov linear model, the expectation in (13) can be expressed using sufficient statistics, as will be shown later, as an inner product N i=1 w i T E θ k log p θ (y 1:T , z 1:T , s i 1:T )\|s i 1:T , y 1:T = S k , η(θ) ,(14) for a sufficient statistics S and corresponding natural parameter η(θ). Hence Q k can be written as Q k (θ) = (1 − γ k ) Q k−1 (θ) + γ k S k , η(θ) = S k , η(θ)(15) if the transformation S k = (1 − γ k )S k−1 + γ k S k(16) is used. In detail, neglecting constant terms in the last expression. This can be verified to be an inner product (as indicated in (14)) in S = {S (1) , S (2) , S (3) }. Here the sufficient statistics S (1) n,m = N i=1 w i T T t=1 1 s i t = m, s i t−1 = n ,(17b)S (2) n = N i=1 w i T T t=1 1 s i t = n ,(17c)S (3) n = N i=1 w i T T t=1 and H θ n =         I A T n B T n Q −1 n   I A n B n   0 0 I C T n D T n R −1 n   I C n D n           (17f) have been used. Further notation introduced is 1 (·) as the indicator function, and M i t\|T =         P i s;t P i s;t,t−1 0 0 P i s;t 0 P i s;t,t−1 P i s;t−1 0 0 P i s;t,t−1 0 0 0 0 0 0 0 0 0 0 0 0 0 P i s;t P i s;t,t−1 0 0 P i s;t−1 0 0 0 0 0 0 0         .(17g) For computing this, the RTS-smoother in step 17 in Algorithm 2 has to be extended by calculation of P s;t+1,t Cov z s,t+1 z T s;t , which can be done as follows [31, Property P6.2] P s;t,t−1 = P f ;t J T t−1 + J t (P s;t+1,t − A t+1 P f ;t )J T t−1 ,(18) initialized with P T, T −1\|T = (I − K T C T )A T P f ;t−1 . For notational convenience, we will partition S n as S (3) n =     Φ n Ψ n Ψ T n Σ n Ω n Λ n Λ T n Ξ n     .(19)A n B n = Ψ n Σ −1 n ,(20a)C n D n = Λ n Ξ −1 n ,(20b)Q n = (S (2),k n ) −1 Φ n − Ψ n Σ −1 n Ψ T n ,(20c)R n = (S (2),k n ) −1 Ω n − Λ n Ξ −1 n Λ T n ,(20d) for n, m = 1, . . . , K. Remark: If B ≡ 0, the first square bracket in (17e) can be replaced by z i,T s;t−1 , and (20b) becomes A n = Ψ n Σ −1 n . The case with D ≡ 0 is fully analogous. Proof. With arguments directly from [14,Lemma 3.3], the maximization of the last part of (17a) for a given s t = n (for any sufficient statistics Z in the inner product, and in particular Z = S k ), is found to be (20b)-(20e). [18] using N = 20 particles (blue) and PSEM [29] using N = 100 particles and M = 20 backward trajectories (red). Using Lagrange multipliers and that i π n,m = 1, the maximum w.r.t. Π of the first part of (17a) is obtained as π n,m = S (1),k n,m l S (1),k n,l .(21) Computational complexity Regarding the computational complexity of Algorithm 3, the most important result is that it is linear in the number of measurements T . It is also linear in the number of particles N . Numerical examples Some numerical examples are given to illustrate the properties of the Rao-Blackwellized PSAEM algorithm. The Matlab code for the examples is available via the homepage of the first author. Example 1 -Comparison to related methods The first example concerns identification using simulated data (T = 3 000) for a one-dimensional (n z = 1) jump Markov linear model with 2 modes (K = 2) (with parameters randomly generated according to A n ∼ U [−1,1] , B n ∼ U [−5,5] , C n ∼ U [−5,5] , D n ≡ 0, Q n ∼ U [0.01,0.1] , R n ∼ U [0.01,0.1] ) with low-pass filtered white noise as u t . The following methods are compared: 1. RB-PSAEM from Algorithm 3, with (only) N = 3 particles, 2. PSAEM as presented in [18] with N = 20, 3. PSEM [29] with N = 100 forward particles and M = 20 backward simulated trajectories. The initial parameters θ 0 are each randomly picked from [0.5θ , 1.5θ ], where θ is the true parameter value. The results are illustrated in Figure 1, which shows the mean (over all modes and 7 runs) H 2 error for the transfer function from the input u to the output y. From Figure 1 (note the log-log scale used in the plot) it is clear that our new Rao-Blackwellized PSAEM algorithm has a significantly better performance, both in terms of mean and in variance between different runs, compared to the previous algorithms. Example 2 -Identification of multidimensional systems Let us now consider a two-dimensional system (n z = 2) with K = 3 modes. The eigenvalues for A n are randomly picked from [−1, 1]. The other parameters are randomly picked as B n ∼ U [−5,5] , C n ∼ U [−5,5] , D n ≡ 0, Q n ∼ I 2 · U [0.01,0.1] , R n ∼ U [0.01,0.1] , and the system is simulated for T = 8 000 time steps with input u t being a low-pass filtered white noise. The initialization of the Rao-Blackwellized PSAEM algorithm is randomly picked from [0.6θ , 1.4θ ] for each parameter. The number of particles used in the particle filter is N = 3. Figure 2a shows the mean (over 10 runs) H 2 error for each mode, similar to Figure 1. Figure 2b shows the estimated Bode plots after 300 iterations. As is seen from Figure 2b, the RB-PSAEM algorithm has the ability to catch the dynamics of the multidimensional system fairly well. CONCLUSION AND FUTURE WORK We have derived a maximum likelihood estimator for identification of jump Markov linear models. More specifically an expectation maximization type of solution was derived. The nonlinear state smoothing problem inherent in the expectation step was solved by constructing an ergodic Markov kernel leaving the joint state smoothing distribution invariant. Key to this development was the introduction of a Rao-Blackwellized conditional particle filter with ancestor sampling. The maximization step could be solved in closed form. The experimental results indicate that we obtain significantly better performance both in terms of accuracy and computational time when compared to previous state of the art particle filtering based methods. The ideas underlying the smoother derived in this work have great potential also outside the class of jump Markov linear models and this is something worth more investigation. Indeed, it is quite possible that it can turn out to be a serious competitor also in finding the joint smoothing distribution for general nonlinear state space models. s 1:T [k + 1], to compute the intermediate quantity in the SAEM.) Draw s i 1 ∼ p 1 (s 1 \|y 1 ) for i = 1, . . . , N − 1. 2: Compute {Ω t , λ t } T t=1 for s 1:T according to (11d) -(11i). 3: Set (s N 1 , . . . , s N T ) = (s 1 , . . . , s T ). 4: Compute z i f,1 and P i f,1 i = 1, . . . , N . 5: Set w i 1 ∝ p θ (y 1 \|s i 1 ) (12) for i = 1, . . . , N s.t. for i = 1, . . . , N . 15: end for 16: for t = T to 1 do 17:Compute z i s;t , P i s;t for i = 1, . . . , N 18: end for 19: Set s 1:T [k + 1] = s J 1:T with P (J = j) = w j T This leads to the approximation (cf. (5)) k log p θ (y 1:T , z 1:T , s i 1:T )\|s i 1:T , y 1:T , : Initialize θ 0 and s 1:T [0], and Q 0 (θ) ≡ 0. 2: for k ≥ 1 do 3: Run Algorithm 2 to obtain {s i 1:T , w i T } N i=1 and s 1:T [k]. θ k log p θ (y 1:T , z 1:T , s i 1:T )\|s i 1:T , y log(\|Q n \|\|R n \|) + Tr(H θ n S (3) n ) (17a) Lemma 1 . 1Assume for all modes n = 1, . . . , K, that all states z are controllable and observable and t 1 (s t = n) u T t u t > 0. The parameters θ maximizing Q k (θ) for the jump Markov linear model(1) Φ n , Ψ n , . . . are the partitions of S (3),k n indicated in (19), and S (i) are the 'SA-updates' (16) of the sufficient statistics (17b)-(17d). Figure 1 : 1Numerical example 1. Mean (lines) and 0.5 standard deviation (fields) H 2 error for 7 runs of our RB-PSAEM using N = 3 particles (black) PSAEM H2 error for each mode. plots of the estimates (black), true (dashed grey) and the initializations (dotted red). 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An approach to time series smoothing and forecasting using the EM algorithm. Journal of Time Series Analysis, 3(4):253-264, 1982. Time series analysis and its applications with R examples. H Robert, David S Sumway, Stoffer, SpringerNew YorkRobert H Sumway and David S Stoffer. Time series analysis and its applications with R examples. Springer, New York, 2006. A Monte Carlo implementation of the EM algorithm and the poor man's data augmentation algorithms. C G Greg, Martin A Wei, Tanner, Journal of the American Statistical Association. 85411Greg CG Wei and Martin A Tanner. A Monte Carlo implementation of the EM algorithm and the poor man's data augmentation algorithms. Journal of the American Statistical Association, 85(411):699-704, 1990. Efficient Bayesian inference for switching state-space models using discrete particle Markov chain Monte Carlo methods. 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[ "ON THE ALMOST EVERYWHERE CONTINUITY", "ON THE ALMOST EVERYWHERE CONTINUITY" ]	[ "Joël Blot " ]	[]	[]	The aim of this paper is to provide characterizations of the Lebesguealmost everywhere continuity of a function f : [a, b] → R. These characterizations permit to obtain necessary and sufficient conditions for the Riemann integrability of f .	null	[ "https://arxiv.org/pdf/1411.3582v1.pdf" ]	119,492,612	1411.3582	41383bcaea230838cf7464a6e1b2604319ea4f95	ON THE ALMOST EVERYWHERE CONTINUITY 13 Nov 2014 Joël Blot ON THE ALMOST EVERYWHERE CONTINUITY 13 Nov 2014continuityfunction of one real variableRiemann integrability MSC 2010: 26A1526A42 The aim of this paper is to provide characterizations of the Lebesguealmost everywhere continuity of a function f : [a, b] → R. These characterizations permit to obtain necessary and sufficient conditions for the Riemann integrability of f . Introduction The main aim of this paper is to establish the following theorem. Theorem 1.1. Let a > b be two real numbers, and f : [a, b] → R be a function. We assume that f admits a finite right-hand limit at each point of [a, b) except on a Lebesgue-negligible set (respectively on a at most countable set). Then f is continuous at each point of [a, b] except on a Lebesgue-negligible set (respectively on a at most countable set). The origine of this work is a paper of Daniel Saada [4] which states that a real function defined on a real segment which right-hand continuous possesses at most an at most countable subset of discontinuity points. Saada attributes the proof of this result to Alain Rémondière. Studying this result and its proof, we see that it contains a central argument that we have described in our Lemma 4.1 and Lemma 4.2, and we use this argument to obtain other results. And so the present work is a continuation of the work of Rémondière and Saada. In Section 2 we precise our notation and we give comments on them. In Section 3, we establish lemmas which are useful for the proof of Theorem 1.1. In Section 4, we provide results on the left-hand continuity and on the right-hand continuity. In Section 5 we give the proof of Theorem 1.1. In Section 6 we establish corollaries of Theorem 1.1. Notation We use the left-hand oscillation of f at x ∈ (a, b] defined by ω L (x) := lim h→0+ ( sup y∈[x−h,x] f (y)) − lim h→0+ ( inf y∈[x−h,x] f (y)) and also the righ-hand oscillation of f at x ∈ [a, b) defined by ω R (x) := lim h→0+ ( sup y∈[x,x+h] f (y)) − lim h→0+ ( inf y∈[x,f (y)) ≥ f (x) > −∞. Also note that we have inf y∈[x−h,x] f (y) ≤ f (x) and consequently lim h→0+ ( inf y∈[x−h,x] f (y)) ≤ f (x) < +∞. And so, ω L (x) is a sum of two elements of (−∞, +∞] and therefore it is well-defined in (−∞, +∞]; more precisely its belongs to [0, +∞]. For similar reasons, ω R (x) is well-defined in [0, +∞]. We use the following notation when g : [a, b] → (−∞, +∞] and r ∈ R: Remark 2.1. The following equivalence hold. {g = 0} := {x ∈ [a, b] : g(x) = 0}, {g > r} := {x ∈ [a, b] : g(x) > r}, {g ≤ r} := {x ∈ [a, b] : g(x) ≤ r}. A subset N ⊂ [a, b] is called Lebesgue-negligible ( A) When x ∈ (a, b], ω L (x) = 0 if and only if f is left-hand continuous at x. (B) When x ∈ [a, b), ω R (x) = 0 if and only if f is right-hand continuous at x. (C) When x ∈ (a, b), (ω L (x) = 0 and ω R (x) = 0) if and only if f is continuous at x. These equivalences are easy to prove. One important fact is that x belongs to the neighborhoods [x − h, x] and [x, x + h]. Remark 2.2. When we will speak of the left-hand limit (respectively of the righthand limit) of the function f at x, we speak of the limit of f (y) when y → x, y > x (respectively y → x, y < x); the point x is not included into his "'neighborhoods"'. The situation is different in the definition of the oscillations ω L and ω R . We denote f (x−) := lim preliminaries We establish lemmas which are useful to the proof of Theorem 1.1. Lemma 3.1. Let f : [a, b] → R be a function, and z ∈ [a, b). We assume that f admits a finite right-hand limit at z. Then we have : ∀ǫ > 0, ∃λ(z, ǫ) > 0, ∀x ∈ (z, z + λ(z, ǫ)], ω L (x) ≤ ǫ. Proof. We arbitrarily fix ǫ > 0. Using the assumption, there exists d z ∈ R such that ∃η(z, ǫ) > 0, ∀x ∈ [a, b], z < x ≤ z + η(z, ǫ) =⇒ \|f (x) − d z \| ≤ ǫ. (3.1) When x ∈ (z, z + η(z, ǫ 4 )] and when y ∈ (z, x], we have y ∈ (z, z + η(z, ǫ 4 )], and using (3.1) we obtain \|f (x) − d z \| ≤ ǫ 4 and \|f (x) − d z \| ≤ ǫ 4 that implies \|f (x) − f (y)\| ≤ \|f (x) − d z \| + \|f (y) − d z \| ≤ 2 ǫ 4 = ǫ 2 =⇒ f (x) − ǫ 2 ≤ f (y) ≤ f (x) + ǫ 2 . Then, for all h ∈ (0, x − z], we obtain f (x) − ǫ 2 ≤ inf y∈[x−h,x] f (y) ≤ sup y∈[x−h,x] f (y)) ≤ f (x) + ǫ 2 =⇒ f (x) − ǫ 2 ≤ lim h→0+ ( inf y∈[x−h,x] f (y)) ≤ lim h→0+ ( sup y∈[x−h,x] f (y)) ≤ f (x) + ǫ 2 =⇒ 0 ≤ ω L (x) ≤ f (x) + ǫ 2 − (f (x) − ǫ 2 = ǫ. And so it suffices to take λ(z, ǫ) := η(z, ǫ 4 ). Using similar arguments we can prove the following result. be a function, and z ∈ (a, b]. We assume that f admits a finite left-hand limit at z. Then we have : Proof. Since a positive measure is additive, for all finie subset J ⊂ I, we have µ(∪ j∈J S j ) = j∈J µ(S j ). Since a positive measure is monotonic, Lemma 3.2. Let f : [a, b] → R∀ǫ > 0, ∃ν(z, ǫ) > 0, ∀x ∈ [z − ν(z, ǫ), z), ω R (x) ≤ ǫ.∪ j∈J S j ⊂ [a, b] implies µ(∪ j∈J S j ) ≤ µ([a, b]) = b − a, and so we have j∈J µ(S j ) ≤ b − a < +∞ for all finite subset J of I. Therefore the family of non negative real numbers (µ(S i )) i∈I is summable in [0, +∞), and consequently the set {i ∈ I : µ(S i ) = 0} is at most countable (Corrolary 9-9, p. 220 in [2]). Since µ(S i ) > 0 for all i ∈ I, we obtain that I is at mot countable. Remark 3.4. We can also prove Lemma 3.3 by building a function ϕ : I → Q in the following way: since Q is dense into R, for each i ∈ I, there exists ϕ(i) ∈ Q∩S i . Since I i ∩I j = ∅ when i = j, we have ϕ(i) = ϕ(j) when i = j. And so ϕ is injective. Since Q is countable, ϕ(I) ⊂ Q is at most countable, and using an abridgement of ϕ, we build a bijection between ϕ(I) and I. Limits on one side, continuities on the other side The following results establish that the existence of left-hand (respectively righthand) limits implies the right-hand (respectively left-hand) continuity. Proof. We arbitrarily fix ǫ > 0. Using Lemma 3.1, denoting λ z := λ(z, ǫ), we obtain the following assertion. ∀z ∈ {ω L > ǫ} ∩ ([a, b] \ N ), ∃λ z > 0, (z, z + λ z ] ⊂ {ω L ≤ ǫ}. (4.1) Let z 1 , z 2 ∈ {ω L > ǫ} ∩ ([a, b] \ N ), z 1 = z 2 . We can assume that z 1 < z 2 . After (4.1), we cannot have z 2 into (z 1 , z 1 + λ z1 ], therefore we have z 2 > z 1 + λ z1 , and we have proven: ∀z 1 , z 2 ∈ {ω L > ǫ} ∩ ([a, b] \ N ), z 1 = z 2 =⇒ (z 1 , z 1 + λ z1 ] ∩ (z 2 , z 2 + λ z2 ] = ∅. We have also µ((z, z + λ z ]) = λ z > 0. Then using Lemma 3.3, we can assert that ∀ǫ > 0, {ω L > ǫ} ∩ ([a, b] \ N ) is at most countable. (4.2) Note that {ω L > 0} = ω −1 L ((0, +∞]) = ω −1 L ( n∈N * ( 1 n , +∞]) = n∈N * ω −1 L (( 1 n , +∞]) = n∈N * {ω L > 1 n } =⇒ {ω L > 0} ∩ ([a, b] \ N ) = n∈N * ({ω L > 1 n } ∩ ([a, b] \ N )). Using (4.2), since a countable union of at most countable subsets is at most countable, we ontain the following assertion. {ω L > 0} ∩ ([a, b] \ N ) is at most countable. (4.3) Note that {ω L > 0} = ({ω L > 0} ∩ ([a, b] \ N )) ∪ ({ω L > 0} ∩ N ). Since ({ω L > 0} ∩ N ) ⊂ N and since N is Lebesgue-negligible (respectively at most countable), ({ω L > 0} ∩ N ) is Lebesgue-negligible (respectively at most countable). Recall that an at most countable subset of R is Lebesgue-negligible. And so when N is Lebesgue-negligible, {ω L > 0} is Lebesgue-negligible as a union of two Lebesguenegligible susbets, and when N is at most countable, {ω L > 0} is at most countable as a union of two at most countable subsets. Using (A) of Remark 2.1, the lemma is proven. Proceegings as in the proof of Lemma 4.1, we obtain the following result. Note that {ω L = 0} ∩ {ω R = 0} is exactly the set of the points of (a, b) where f is continuous. We have [a, b]\({ω L = 0}∩{ω R = 0}) = [a, b]∩({ω L > 0}∪{ω R > 0}) = {ω L > 0}∪{ω R > 0}. This set is Lebesgue-negligible (respectively at most countable) as a union of two Lebesgue-negligible (respectively at most countable) sets. Note that {a, b} is Lebesgue-negligible (respectively at most countable) and so set of the discontinuity points of f is Lebesgue-negligible (respectively at most countable). Consequences A first consequence of Theorem 1.1 is the following result. Theorem 6.1. Let a > b be two real numbers, and f : [a, b] → R be a function. Then the following assertions are equivalent. (α) The set of the discontinuity points of f is Lebesgue-negligible (respectively at most countable). (β) The set of the left-hand discontinuity points of f is Lebesgue-negligible (respectively at most countable). (γ) The set of the right-hand discontinuity points of f is Lebesgue-negligible (respectively at most countable). (δ) The set of the points where f does not admit a finite left-hand limit is Lebesgue-negligible (respectively at most countable). (ǫ) The set of the points where f does not admit a finite right-hand limit is Lebesgue-negligible (respectively at most countable). Proof. The implications (α) =⇒ (β) =⇒ (δ) are easy, and (δ) =⇒ (α) is Theorem 1.1. The implications (α) =⇒ (γ) =⇒ (ǫ) are easy. we can do a proof which is similar to this one of Theorem 1.1 to prove (ǫ) =⇒ (α). when there exists B, a borelian subset of [a, b], such that N ⊂ B and µ(B) = 0 where µ denotes the Lebesgue measure of R. Such a vocabulary is used for instance in[1]. Lemma 3. 3 . 3Let I be a nonempty set, and (S i ) i∈I be a family of subintervals of [a, b] such that S i ∩ S j = ∅ when i = j, and such µ(S i ) > 0 for all i ∈ I, where µ denotes the Lebesgue measure of R. Then I is at most countable. Lemma 4. 1 . 1let f : [a, b] → R be a function, and N be a Lebesgue-negligible (respectively at most countable) subset of[a, b]. We assume that f admits a finite right-hand limit at each x ∈ [a, b) \ N . Then the set of the points of [a, b] where f is not left-hand continuous is Lebesguenegligible (respectively at most countable). Lemma 4. 2 . 2let f : [a, b] → R be a function, and M be a Lebesgue-negligible (respectively at most countable) subset of [a, b]. We assume that f admits a finite left-hand limit at each x ∈ (a, b] \ M . Then the set of the points of [a, b] where f is not right-hand continuous is Lebesguenegligible (respectively at most countable). 5. Proof of Theorem 1.1 Using Lemma 4.1 and (A) of Remark 2.1, {ω L > 0} is Lebesgue-negligible (respectively at most countable) since {ω L > 0} is exactly the set of the points of (a, b] where f is not left-hand continuous. Now, setting M = {ω L > 0}, for all x ∈ [a, b] \ M , f (x−) = f (x) ∈ R, and the assumption of Lemma 4.2 is fulfilled. Consequently we obtain that {ω R > 0} is Lebesgue-negligible (respectively at most countable) after (B) of Remark 2.1. x+h] f (y)).Note that we have sup y∈[x−h,x] f (y) ≥ f (x) and consequentlyDate: November 4, 2014. lim h→0+ ( sup y∈[x−h,x] Acknowledgements. I thanks my colleagues B. Nazaret, M. Bachir and J.-B. Baillon for interesting discussions on these topics.About the Riemann-integrability we recall a famous theorem of Lebesgue,[3]p. 29,[5]p. 20.Theorem 6.2. Let a > b be two real numbers, and let f : [a, b] → R be a bounded function. Then the following assertions are equivalent.(i) f is Riemann integrable on [a, b].(ii) The set of the discontinuity points of f is Lebesgue-negligible.As a consequence of Theorem 6.1 and of the previous classical theorem ofLebesgue,we obtain the following result on the Riemann-integrability. An easy consequence of this result is the following one. integration and topological vectors spaces. G Choquet, Lectures on analysis. 1W.A. Benjamin, IncG. Choquet, Lectures on analysis; Volume 1: integration and topological vectors spaces, W.A. Benjamin, Inc., Reading, Massachussets, 1969. . G Choquet, Topologie , Masson et Cie. second revised editionG. Choquet, Topologie, second revised edition, Masson et Cie, Paris, 1973. H Lebesgue, Leçons sur l'intégration et la recherche des fonctions primitives. Paris; Bronx, New YorkChelsea Publishing Companysecond edition. reprinted byH. Lebesgue, Leçons sur l'intégration et la recherche des fonctions primitives, second edition, Gauthier-Villars, Paris, 1928; reprinted by Chelsea Publishing Company, Bronx, New York, 1973 Fonctions continues presque-partout. D Saada, D. Saada, Fonctions continues presque-partout, www.daniel-saada.eu Integral, measure and derivative: a unified approach. G E L Shilov & B, Gurevitch, Dover Publications, IncUpper Saddle River, NJ; New Yorkreprinted byG.E. Shilov & B.L. Gurevitch, Integral, measure and derivative: a unified approach, Prentice- Hall, Inc., Upper Saddle River, NJ, 1966; reprinted by Dover Publications, Inc., New York, 1977. Joël Blot: Laboratoire SAMM EA 4543, Université Paris 1 Panthéon-Sorbonne, centre P.M.F., 90 rue de Tolbiac. 75634 Paris cedex 13, France. E-mail address: [email protected]ël Blot: Laboratoire SAMM EA 4543, Université Paris 1 Panthéon-Sorbonne, centre P.M.F., 90 rue de Tolbiac, 75634 Paris cedex 13, France. E-mail address: [email protected]	[]
[ "The WISSH quasars project VIII. The impact of extreme radiative field in the accretion disk -X-ray corona interplay", "The WISSH quasars project VIII. The impact of extreme radiative field in the accretion disk -X-ray corona interplay" ]	[ "L Zappacosta \nINAF -Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly\n", "E Piconcelli \nINAF -Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly\n", "M Giustini \nDep. de Astrofísica\nCentro de Astrobiología (CSIC-INTA)\nCamino Bajo del Castillo s/n, Villanueva de la CañadaE-28692MadridSpain\n", "G Vietri \nINAF -IASF Milano\nvia A. Corti 1220133MilanoItaly\n", "F Duras \nLAM\nAix Marseille Univ\nCNRS\nCNES\nMarseilleFrance\n", "G Miniutti \nDep. de Astrofísica\nCentro de Astrobiología (CSIC-INTA)\nCamino Bajo del Castillo s/n, Villanueva de la CañadaE-28692MadridSpain\n", "M Bischetti \nINAF -Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly\n", "A Bongiorno \nINAF -Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly\n", "M Brusa \nDipartimento di Fisica e Astronomia\nUniversità degli Studi di Bologna\nvia Gobetti 93/240129BolognaItaly\n\nINAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nVia Gobetti 93/340129BolognaItaly\n", "M Chiaberge \nDepartment of Physics and Astronomy\nJohns Hopkins University\nBloomberg Center21218BaltimoreMDUSA\n", "A Comastri \nINAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nVia Gobetti 93/340129BolognaItaly\n", "C Feruglio \nINAF -Osservatorio Astronomico di Trieste\nVia G. Tiepolo 11I-34143TriesteItaly\n", "A Luminari \nINAF -Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly\n\nDepartment of Physics\nUniversity of Rome \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomeItaly\n", "A Marconi \nDipartimento di Fisica e Astronomia\nUniversità di Firenze\nVia G. Sansone 1I-50019\n\nSesto Fiorentino (Firenze)\nItaly\n\nINAF-Osservatorio Astrofisico di Arcetri\nLargo E. Fermi 550125FirenzeItaly\n", "C Ricci \nNúcleo de Astronomía de la Facultad de Ingeniería\nUniversidad Diego Portales\nAv. Ejército Libertador 441SantiagoChile\n\nKavli Institute for Astronomy and Astrophysics\nPeking University\n100871BeijingChina\n", "C Vignali \nDipartimento di Fisica e Astronomia\nUniversità degli Studi di Bologna\nvia Gobetti 93/240129BolognaItaly\n", "F Fiore \nINAF -Osservatorio Astronomico di Trieste\nVia G. Tiepolo 11I-34143TriesteItaly\n" ]	[ "INAF -Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly", "INAF -Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly", "Dep. de Astrofísica\nCentro de Astrobiología (CSIC-INTA)\nCamino Bajo del Castillo s/n, Villanueva de la CañadaE-28692MadridSpain", "INAF -IASF Milano\nvia A. Corti 1220133MilanoItaly", "LAM\nAix Marseille Univ\nCNRS\nCNES\nMarseilleFrance", "Dep. de Astrofísica\nCentro de Astrobiología (CSIC-INTA)\nCamino Bajo del Castillo s/n, Villanueva de la CañadaE-28692MadridSpain", "INAF -Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly", "INAF -Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly", "Dipartimento di Fisica e Astronomia\nUniversità degli Studi di Bologna\nvia Gobetti 93/240129BolognaItaly", "INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nVia Gobetti 93/340129BolognaItaly", "Department of Physics and Astronomy\nJohns Hopkins University\nBloomberg Center21218BaltimoreMDUSA", "INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nVia Gobetti 93/340129BolognaItaly", "INAF -Osservatorio Astronomico di Trieste\nVia G. Tiepolo 11I-34143TriesteItaly", "INAF -Osservatorio Astronomico di Roma\nvia di Frascati 3300078Monte Porzio CatoneItaly", "Department of Physics\nUniversity of Rome \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomeItaly", "Dipartimento di Fisica e Astronomia\nUniversità di Firenze\nVia G. Sansone 1I-50019", "Sesto Fiorentino (Firenze)\nItaly", "INAF-Osservatorio Astrofisico di Arcetri\nLargo E. Fermi 550125FirenzeItaly", "Núcleo de Astronomía de la Facultad de Ingeniería\nUniversidad Diego Portales\nAv. Ejército Libertador 441SantiagoChile", "Kavli Institute for Astronomy and Astrophysics\nPeking University\n100871BeijingChina", "Dipartimento di Fisica e Astronomia\nUniversità degli Studi di Bologna\nvia Gobetti 93/240129BolognaItaly", "INAF -Osservatorio Astronomico di Trieste\nVia G. Tiepolo 11I-34143TriesteItaly" ]	[]	Hyperluminous quasars (L bol 10 47 erg s −1 ) are ideal laboratories to study the interaction and impact of extreme radiative field and the most powerful winds in the AGN nuclear regions. They typically exhibit low coronal X-ray luminosity (L X ) compared to the UV and MIR radiative outputs (L UV and L MIR ) with a non-negligible fraction of them reporting even ∼1 dex weaker L X compared to the prediction of the well established L X -L UV and L X -L MIR relations followed by the bulk of the AGN population. We report in our WISE/SDSS-selected Hyperluminous (WISSH) z = 2 − 4 broad-line quasar sample, the discovery of a dependence between the intrinsic 2-10 keV luminosity (L 2−10 ) and the blueshifted velocity of the CIV emission line (v CIV ) indicative of accretion disc winds. In particular, sources with fastest winds (v CIV 3000 km s −1 ) possess ∼0.5-1 dex lower L 2−10 than sources with negligible v CIV . No similar dependence is found on L UV , L MIR , L bol , photon index and absorption column density. We interpret these findings in the context of accretion disc wind models. Both magnetohydrodynamic and line-driven models can qualitatively explain the reported relations as a consequence of X-ray shielding from the inner wind regions. In case of line-driven winds, the launch of fast winds is favoured by a reduced X-ray emission, and we speculate that these winds may play a role in directly limiting the coronal hard X-ray production.	10.1051/0004-6361/201937292	[ "https://arxiv.org/pdf/2002.00957v3.pdf" ]	211,020,904	2002.00957	8a2aeca4044bd2e3a460b43efccf884216eaccd2	The WISSH quasars project VIII. The impact of extreme radiative field in the accretion disk -X-ray corona interplay 3 Feb 2020 February 5, 2020 February 5, 2020 L Zappacosta INAF -Osservatorio Astronomico di Roma via di Frascati 3300078Monte Porzio CatoneItaly E Piconcelli INAF -Osservatorio Astronomico di Roma via di Frascati 3300078Monte Porzio CatoneItaly M Giustini Dep. de Astrofísica Centro de Astrobiología (CSIC-INTA) Camino Bajo del Castillo s/n, Villanueva de la CañadaE-28692MadridSpain G Vietri INAF -IASF Milano via A. Corti 1220133MilanoItaly F Duras LAM Aix Marseille Univ CNRS CNES MarseilleFrance G Miniutti Dep. de Astrofísica Centro de Astrobiología (CSIC-INTA) Camino Bajo del Castillo s/n, Villanueva de la CañadaE-28692MadridSpain M Bischetti INAF -Osservatorio Astronomico di Roma via di Frascati 3300078Monte Porzio CatoneItaly A Bongiorno INAF -Osservatorio Astronomico di Roma via di Frascati 3300078Monte Porzio CatoneItaly M Brusa Dipartimento di Fisica e Astronomia Università degli Studi di Bologna via Gobetti 93/240129BolognaItaly INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna Via Gobetti 93/340129BolognaItaly M Chiaberge Department of Physics and Astronomy Johns Hopkins University Bloomberg Center21218BaltimoreMDUSA A Comastri INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna Via Gobetti 93/340129BolognaItaly C Feruglio INAF -Osservatorio Astronomico di Trieste Via G. Tiepolo 11I-34143TriesteItaly A Luminari INAF -Osservatorio Astronomico di Roma via di Frascati 3300078Monte Porzio CatoneItaly Department of Physics University of Rome "Tor Vergata" Via della Ricerca Scientifica 1I-00133RomeItaly A Marconi Dipartimento di Fisica e Astronomia Università di Firenze Via G. Sansone 1I-50019 Sesto Fiorentino (Firenze) Italy INAF-Osservatorio Astrofisico di Arcetri Largo E. Fermi 550125FirenzeItaly C Ricci Núcleo de Astronomía de la Facultad de Ingeniería Universidad Diego Portales Av. Ejército Libertador 441SantiagoChile Kavli Institute for Astronomy and Astrophysics Peking University 100871BeijingChina C Vignali Dipartimento di Fisica e Astronomia Università degli Studi di Bologna via Gobetti 93/240129BolognaItaly F Fiore INAF -Osservatorio Astronomico di Trieste Via G. Tiepolo 11I-34143TriesteItaly The WISSH quasars project VIII. The impact of extreme radiative field in the accretion disk -X-ray corona interplay 3 Feb 2020 February 5, 2020 February 5, 2020Astronomy & Astrophysics manuscript no. zappacosta_arxiv c ESO 2020 Letter to the EditorX-rays: galaxies -Galaxies: active -quasars: emission lines -quasars: supermassive black holes -Galaxies: high- redshift Hyperluminous quasars (L bol 10 47 erg s −1 ) are ideal laboratories to study the interaction and impact of extreme radiative field and the most powerful winds in the AGN nuclear regions. They typically exhibit low coronal X-ray luminosity (L X ) compared to the UV and MIR radiative outputs (L UV and L MIR ) with a non-negligible fraction of them reporting even ∼1 dex weaker L X compared to the prediction of the well established L X -L UV and L X -L MIR relations followed by the bulk of the AGN population. We report in our WISE/SDSS-selected Hyperluminous (WISSH) z = 2 − 4 broad-line quasar sample, the discovery of a dependence between the intrinsic 2-10 keV luminosity (L 2−10 ) and the blueshifted velocity of the CIV emission line (v CIV ) indicative of accretion disc winds. In particular, sources with fastest winds (v CIV 3000 km s −1 ) possess ∼0.5-1 dex lower L 2−10 than sources with negligible v CIV . No similar dependence is found on L UV , L MIR , L bol , photon index and absorption column density. We interpret these findings in the context of accretion disc wind models. Both magnetohydrodynamic and line-driven models can qualitatively explain the reported relations as a consequence of X-ray shielding from the inner wind regions. In case of line-driven winds, the launch of fast winds is favoured by a reduced X-ray emission, and we speculate that these winds may play a role in directly limiting the coronal hard X-ray production. Introduction The most luminous active galactic nuclei (AGN) are expected to exhibit the strongest and clear-cut manifestations of winds (Menci et al. 2008;Faucher-Giguère & Quataert 2012). Indeed the fastest and most energetic winds have been reported in hyperluminous quasars (i.e. with bolometric luminosity L bol 10 47 ergs −1 ; Wu et al. 2011;Fiore et al. 2017;Vietri et al. 2018;Meyer et al. 2019;Perrotta et al. 2019). Luminous quasars are typically characterised by their low coronal X-ray luminosity (L X ) compared to the disc UV, and larger-scale dust-reprocessed Mid-Infrared (MIR) luminosities (L UV and L MIR , respectively) as parametrized by the α OX 1 and L MIR /L X parameters (e.g. Vignali et al. 2003;Just et al. 2007;Lusso & Risaliti 2016;Stern 2015;Martocchia et al. 2017;Chen et al. 2017). Past studies on 1 α OX is the X-ray to optical spectral index between the rest-frame luminosities at 2500 Å and 2 keV; i.e. α OX = 0.3838 log(L 2keV /L 2500 Å ) large quasar samples over a wide luminosity range (L bol ≈ 10 45 − 10 48 erg s −1 ) have also reported indications of a further weak dependence of the α OX parameter on the velocities of the broad emission line (BEL) winds, as parametrized by the CIV blueshift; i.e. negative velocity shifts (v CIV ) of the CIV emission line (e.g. Richards et al. 2011;Kruczek et al. 2011;Vietri et al. 2018). However once removed the α OX luminosity dependence by adopting ∆α OX , i.e. the difference between the measured α OX and the one predicted by the α OX − L 2500 Å relation (e.g. Just et al. 2007), the dependence with v CIV resulted to be much less significant (e.g. Gibson et al. 2008;Ni et al. 2018;Timlin et al. 2019). Notice that these dependences on v CIV may be regarded as representative for the bulk of the quasar population since CIV BEL winds are a common feature in quasars (e.g. Shen et al. 2011). A similar dependence has been also reported for the maximum velocity measured in line-of-sight detected winds such as those reported in the broad-absorption line (BAL) quasars (e.g. A&A proofs: manuscript no. zappacosta_arxiv Gallagher et al. 2006;Gibson et al. 2009;Stalin et al. 2011). Nonetheless a peculiar category of z = 2 − 3 optically luminous quasars selected to have weak broad emission lines (with rest-frame equivalent width of the CIV, REW CIV < 15 Å) and fast BEL winds (v CIV −2000 km s −1 Richards et al. 2011), revealed mostly 1-2 dex weaker X-ray emission (Gibson et al. 2008;Wu et al. 2011) compared to the α OX − L 2500 Å expectations. Sources with v CIV −2000 km s −1 and REW CIV 20 Å have been reported in our sample of 86 broad-line unlensed highly accreting (λ Edd > 0.4) MIR/optically bright z = 2 − 4 hyperluminous quasars. These sources have been selected to be the MIR-brightest WISE/SDSS sources with z > 1.5 and flux density S 22µm > 3 mJy (WISSH; Bischetti et al. 2017). The WISSH quasars exhibit widespread evidence of winds at all scales from nuclear BAL (Bruni et al. 2019), to ionised [OIII]/Ly α galactic-/circumgalactic-scale outflows (Bischetti et al. 2017;Travascio et al. 2020). In particular in Vietri et al. (2018) we reported a surprisingly high fraction (∼ 70%) of sources with weak UV/optical BEL spectra (e.g. REW CIV 20 Å) and extreme CIV blueshifts (v CIV < −2000 km s −1 ) in a sub-sample of the WISSH quasars. Furthermore Martocchia et al. (2017) found a large spread in the L X with a non-negligible fraction having ∼0.5-1 dex fainter values than the average. In this Letter we explore the relation between the extreme radiative field of the hyperluminous WISSH quasars and their Xray coronal properties, and interpret it in the context of accretion disc wind scenarios. We adopt a ΛCDM cosmology with Ω Λ = 0.73 and H 0 = 70 km s −1 Mpc −1 throughout the paper. Errors and upper/lower limits are quoted at 68.3% and 90% confidence level, respectively. Sample presentation and X-ray data reduction and analysis In this work we consider the radio-quiet hyperluminous WISSH sources with (i) reported v CIV measures relative to their systemic redshift and (ii) available X-ray data. The selected sample of thirteen sources is reported in Table A.1 and has been mainly drawn from the Vietri et al. (2018) WISSH sub-sample of 18 quasars for which CIV emission line properties have been derived. We also complement our work by including five Type 1 radio-quiet hyperluminous sources at similar z with published v CIV (Coatman et al. 2017, hereafter C17; see Table A.1) and available X-ray archived data. Further details on the sample selection are reported in Appendix A. We consider both Chandra and XMM-Newton observations available for each source (see Table B.1). For each dataset we perform standard data reduction as detailed in Appendix B. The X-ray spectral modelling is performed in the 0.2-10 keV and 0.3-8 keV bands for XMM-Newton and Chandra, respectively. We employ an intrinsically absorbed power-law model further modified by the Galactic absorption. Further details on the modelling and the derived parameters are reported in Appendix C and Results Fig . 1 shows the unabsorbed (i.e. intrinsic) 2-10 keV luminosity (L 2−10 ; left panel) and the X-ray unabsorbed and UV deextincted α OX (middle panel) as a function of v CIV for the WISSH quasars (black). Both quantities strongly correlate with v CIV 2 with Spearman correlation coefficient ρ ≈ 0.6 and two-sided nullhypothesis probability of p ≈ 0.02. This is a consequence of the lack of significant correlation between L 2500Å and v CIV (right panel; ρ = 0.32 and p ≈ 0.29) and the limited dispersion of L 2500Å . Hence the sources with the largest negative v CIV (i.e. larger blueshifts) are 0.5−1 dex X-ray weaker and exhibit steeper α OX than the sources with the lowest v CIV . No significant correlation between v CIV and L bol and L 6µm is found (see left panels in Fig. 2 and Table 1). Therefore their ratio with L 2−10 , i.e. the X-ray bolometric correction k bol,X = L bol /L 2−10 and L 6µm /L 2−10 , result to be strongly dependent on v CIV (see right panels in Fig. 2 and Table 1). The inclusion of the hyperluminous quasars from C17 in the relations involving L 2−10 , L bol and k bol,X further confirms the strength and significance of the dependence with v CIV . Notice that Vietri et al. (2018) and C17 adopt slightly different definitions of v CIV (see notes on Table A.1). This difference does not change our result. Indeed for the two WISSH quasars reported in both samples (i.e. J1106+6400 and J1201+1206) the v CIV reported by C17 are 400 − 500 km s −1 larger than the values reported by Vietri et al. (2018). This small systematic offset makes little difference in our correlations and if we correct the v CIV of the C17 sub-sample by 500 km s −1 we obtain a slightly stronger L 2−10 −v CIV correlation with ρ = 0.63 and p = 0.005. 3 The ∆α OX values (based on the α OX − L 2500Å relation from Just et al. 2007) appear to be strongly dependent on v CIV (see Table 1). Similarly to past works, we also calculate α pow OX , i.e. derived by performing spectral fitting with a unabsorbed power-law model for the 2 keV luminosity estimation (see Appendix C) and the relative ∆α pow OX (see Table C.1). Both quantities exhibit strong and significant correlation with v CIV , similarly to α OX and ∆α OX . We find that 4 and 5 sources (∼ 30 − 40% of the considered WISSH sub-sample) are X-ray weak (red circles in Fig. 1), as they have respectively ∆α OX and ∆α pow OX smaller than −0.2, a threshold value typically adopted (e.g. Luo et al. 2015) to identify X-ray weak sources. Notice that, given the small number of sources and the sample selection function (i.e. only X-ray detected sources for which basic X-ray spectral analysis can be performed), the fraction of X-ray weak sources must be considered with caution. For all the significant relations (i.e. with p 0.02) we also compute ρ accounting for the uncertainties in the measurements. We generate 10000 random realizations of the sample by Gaussian distributing each considered quantity according to its bestfit value and error. We obtain ρ sim and its uncertainty by adopting the mean value and standard deviation of the distribution of ρ for the simulated datasets. We also calculate p sim,90 , the 90% percentile on the distribution of the p values. The derived ρ sim and p sim,90 confirm the significance of the relations. For all quantities we derive linear relations with v CIV by employing the BCES(Y\|X) method (Akritas & Bershady 1996). We show them in Fig. 1 and Table 1 reports ρ, p, ρ sim , p sim,90 and the slope (α) and y-intercept (β) of the linear relation. Fig 2. As for the derived X-ray spectral parameters, we do not report a significant dependence of the photon index (Γ) on v CIV (see Table 1) but please bear in mind that we have weak constraints on Γ for the X-ray weaker sources (see top panel of Fig. 3). As for the column density (N H ) we find the WISSH quasars to exhibit moderate values of absorption which are compatible with their Type 1 nature. Indeed seven sources have measured N H 10 23 cm −2 with a mean log(N H /cm −2 ) ≈ 22.3. The other sources have upper limits in the range log(N H /cm −2 ) = 21−23.7, with the largest values mainly driven by poor statistics in the spectra ( 50 net-counts). Lower panel of Fig. 3 reports N H as a function of v CIV . We compute the Spearman's rank correlation coefficient accounting for upper limits 4 . We find a weak but not significant anti-correlation both including or excluding the C17 data (see Table 1). Finally we mention a lack of correlation between λ Edd (as reported in Vietri et al. 2018) and v CIV for the WISSH quasars. Discussion We report a relation between L 2−10 and α OX with v CIV in a sample of MIR/optically-selected hyperluminous quasars and the lack of a similar dependence in the UV, MIR and in L bol . The use of good-quality X-ray data and the well-defined quasar selection, resulting in a narrow L bol (≈ L UV ) range, allows to measure a marked α OX -v CIV (∆α OX -v CIV ) dependence (i.e. stronger in terms of correlation coefficient than those reported to date in samples probing the bulk of the quasar population; Richards et al. 2011;Vietri et al. 2018;Timlin et al. 2019) and reveals a clearcut dependence on L 2−10 . This suggests that at these luminosity regimes, L 2−10 is the main driver of the α OX -v CIV relation. Notice that the large average v CIV for our quasars of ∼ −2900 km s −1 agrees with the increasing trend of v CIV ∝ L 0.28 bol relation reported e.g. in Vietri et al. (2018) for an extended sample spanning more than three decades in L bol (see also Timlin et al. 2019 for a similar result). Accordingly the lack of significant correlation for log L bol -v CIV in our sample is the result of the restricted luminosity range spanned by our quasars. Theoretical and observational arguments suggest that BEL winds in AGN are produced at accretion disc scales (e.g. Elvis 2000), where the AGN luminosity output is large in the UV/Xray bands. Accretion disc winds in AGN can be sustained by either magnetic or radiative forces; the observational results of this Letter imply that whatever is the driving mechanism of the disc wind, the fastest UV winds appear in the X-ray weakest sources. In MHD-driven scenarios, the presence of the disc wind does not depend on the X-ray radiative ouput; however the existence of observable ions does. Indeed, a general correlation between α OX and, for example, v CIV is expected to exist from simple photoionization arguments. A larger X-ray luminosity is generally effective in stripping the bound electrons off the CIV atoms up to large disc radii, allowing the formation of low-velocity winds. Y−X relation is Y = α × X + β. Lower X-ray luminosities allows instead the presence of CIV at distances closer to the SMBH where larger terminal velocities are expected. In particular, in order not to get over-ionized, the UV wind must be accompanied by a inner shield of partially ionized gas absorbing the X-ray flux, which is itself part of the wind driven by the magnetic forces (see e.g. Fukumura et al. 2010). In radiation-driven scenarios, the wind is driven by radiation pressure on spectral lines, and the relative contribution of the UV and X-ray emission is instead very important to determine the existence of the wind itself (Murray et al. 1995;Proga et al. 2000;Proga & Kallman 2004). In particular, for a given UV luminosity, a weak X-ray emission (i.e. steep α OX ) is crucial for the existence of fast radiation-driven winds. Indeed, in the inner disc regions the gas opacity to UV transitions drops as the atoms are over-ionised by an intense X-ray flux. In this case, the wind cannot be efficiently accelerated beyond escape velocity and eventually falls back to the disc as a failed wind (FW). The FW contains clumps of dense gas in the proximity of the X-ray emitting corona. The FW effectively shields the gas located farther away from the ionising X-ray photons, and allows the acceleration of disc material at all radii where the radiation pressure is large enough to overcome the gravitational pull of the SMBH (Proga et al. 2000;Proga & Kallman 2004;Risaliti & Elvis 2010). A high X-ray flux would produce a vast inner zone of FW and would allow the launch of disc winds only at large radii. Conversely a lower X-ray flux would produce a reduced inner FW zone, and would allow the formation of disc winds on scales closer to the SMBH. As the terminal velocity of the wind is inversely proportional to its launching distance from the central SMBH, a lower X-ray emission favours in general the launch of faster radiation-driven accretion disc winds, compared to a higher X-ray emission (see e.g. Giustini & Proga 2019, for a recent review). Interestingly, because of their high density, FW clumps can further cause an efficient cooling of the corona via bremsstrahlung emission, therefore leading to a weakening (quenching) of the inverse Compton X-ray emission (Proga 2005;Laor & Davis 2014). Notice that, recent post-processing radiative transfer calculations suggest that the FW is not able to efficiently prevent over-ionization and, therefore, may not be so crucial for the wind acceleration. Indeed, the FW is found to have a higher ionization state (than previous estimates) and a much limited X-ray shielding power (Higginbottom et al. 2014). Evidence against the Xray shielding scenario comes from X-ray observations of semirelativistic BAL winds reporting weak/moderate X-ray absorption (∼ 0.5 − 5 × 10 22 cm −2 ; Hamann et al. 2013). A similar evidence seems to be in place also for hyperluminous quasars. Indeed, our result would support a high degree of intrinsic Xray quenching (Fig. 1, left panel) rather than a dependence on nuclear X-ray shielding (Fig. 3, lower panel). Further support to the hypothesis of coronal X-ray weakness comes from NuSTAR estimates that at least ∼ 1/3 of luminous BAL quasars may exhibit significant X-ray weakness (Luo et al. 2014). However, the Luo et al. sample consists of heavily obscured (N H ≈ 10 24 cm −2 ) BAL quasars, which in principle may have part of the X-ray emission further suppressed by nuclear shielding. Studies focusing on samples of weak-line quasars explain their properties in the context of simple orientation-dependent nuclear shielding of the X-ray emission without invoking coronal quenching (Wu et al. 2011;Luo et al. 2015;Ni et al. 2018). In such a model the shield may likely be produced by the geometrically thick inner accretion disc regions, and it would lead to a significant dependence on N H and a lack of correlation with the intrinsic L X . In our MIR/optically selected WISSH quasars we instead find a marked L 2−10 -v CIV dependence and a lack of correlation with N H . Notice though that so far only simple cold absorbers have been adopted in modelling their X-ray spectrum. Probably the inner shielding disc regions would require a more complex and ionized absorber. In this sense the findings by Hamann et al. (2013) of semi-relativistic BAL winds in sources with low/moderate observed cold N H may imply the existence of such an ionized absorber which would act as a ionization shield. The current X-ray data quality for our sample is not sufficient enough to add further free parameters to account for ionized absorption. We mention that orientation may also play a role in producing the large scatter in the L 2−10 -v CIV relation. Indeed it may reflect the projection of the wind velocity field depending on the line-of-sight inclination of the disc-corona structure. For instance, in the scenario envisaged by Elvis (2000) which predicts an extreme funnel-shaped geometry of the wind for luminous quasars (see Fig. 7 in Elvis 2000), we qualitatively expect that at a given L 2−10 the highest emission line blueshifts are reported by sources seen at a relatively large inclinations (compatible with their Type 1 nature), while the lowest blueshifts are seen in sources viewed pole-on. X-ray observations of well-monitored Ultra-Fast Outflows (UFO; e.g. Tombesi et al. 2010;Gofford et al. 2013) seem to support the important role of radiation in driving AGN disc winds. For example, the hyperluminous local quasar PDS 456, known to display recurrent and variable UFO (e.g. Reeves et al. 2009;Nardini et al. 2015), fits well in the L 2−10 -v CIV relation as it is reported to have a CIV blueshift of ∼ 5200 km s −1 (O'Brien et al. 2005) and L 2−10 ≈ (0.3−1)×10 45 erg s −1 . Interestingly, Matzeu et al. (2017) reported on PDS 456 a positively correlated variability (over a period of 13 years) between the UFO velocity and the X-ray luminosity 5 . If PDS 456 is representative of the high-redshift hyperluminous quasars, this level of X-ray 5 Notice that variable UFO have been observed also in other four high-z bright quasars (APM 08279+5255, PG 1115+080, H 1413+117 and HS 1700+641; Chartas et al. 2003Chartas et al. , 2007Saez & Chartas 2011;Lanzuisi et al. 2012). A similar correlation between the quasar X-ray lu-variability could account for the scatter in the relation. As for the opposite signs of correlation for CIV BEL winds and UFOs velocities with L 2−10 , if proven to be a common characteristics in luminous quasars, they may as well be qualitatively explained in the context of the radiation-driven accretion disc wind scenario. Indeed, the X-ray luminosity, acting on X-ray line transitions, is responsible in one case for the radiative acceleration of the UFO and in the other case as ionisation state regulator of UV line-driven BEL winds. Notice though that recent photoionization and radiative transfer calculations by Dannen et al. (2019) suggest that the line driving mechanisms may not be relevant in plasma with high ionization parameters typical of UFOs (i.e. ξ > 1000; e.g. Tombesi et al. 2011;Nardini et al. 2015). In this case MHD-driven winds may be a viable mechanism to explain the positive correlation exhibited by UFOs as recently reported by Fukumura et al. (2018). Dedicated deep X-ray observations, performed also at lower luminosity regimes, will be crucial to test and shed light on the origin of the v CIV dependence by better constraining the spectral parameter for the X-ray weak sources. In this regard ATHENA will further enable us to investigate the properties of the accretion disc-scale absorber (e.g. kinematics and ionisation state), discriminate between competing disc wind scenarios and further investigate the role of UFOs (e.g. Martocchia et al. 2017;Barret & Cappi 2019). Future studies on α OX , k bol,X , L 6µm /L 2−10 will need to take into account for the reported marked dependences on v CIV ( Fig. 1 and 2) at these high luminosity regimes in order to obtain better constrainted relations and remove possible systematics due to the inclusion of highly blueshifted and hence X-ray weak sources. The sources considered in this work are reported in Table A.1. They are selected starting from the 18 WISSH quasars analysed by Vietri et al. (2018) and for which CIV emission line properties have been estimated. We include only the twelve radio-quiet sources with available archived X-ray Chandra and XMM-Newton data (see Table B.1). To this sample we added an additional WISSH source (J1441+0454) with available X-ray archived data and for which constraints for the CIV emission line have been obtained and will be reported in Vietri et al. in preparation. In Fig. A.1 we report the L MIR −z for our sub-sample (red points) together with the entire WISSH sample. The sources in our sub-sample are mainly clustered at z ≈ 2.1 and z ≈ 3.4, therefore encompassing the large z range of the WISSH sources, and exhibit L MIR values representative of the entire WISSH sample. X-ray data for eleven sources have already been analysed by Martocchia et al. (2017). In this work additional and new proprietary and publicly available X-ray data are considered for seven sources (J0958+2827, J1201+0116, J1236+6554, J1326-0005, J0900+4215, J1106+6400, J2123-0050). Two of these sources (J0958+2827, J1326-0005) are not in the Martocchia et al. (2017) sample. The final WISSH sample studied here consists of thirteen sources. Chandra and XMM-Newton X-ray data are available for eleven and six sources, respectively. Three sources have both Chandra and XMM-Newton coverage. The log of the X-ray observations for these sources is reported in Table B.1. In order to increase the number of hyperluminous sources with available X-ray data and CIV spectral properties, we consider an additional sample starting from the sources analysed in C17 with L bol > 10 47 erg s −1 and available pointed X-ray data. The C17 sample is a heterogeneous compilation of 230 quasars with available infrared and optical spectroscopy. Our final sample consists of the five sources reported in Table A.1. Chandra and XMM-Newton data are available for two and three sources, respectively (see Table B.1). Finally, the considered sources are all undetected in the FIRST/NVSS radio surveys with the only exception of three sources (J0900+4215, J1441+0454, J1549+1245) with 1.4 GHz integrated flux densities of ∼ 1.7 mJy. From the integrated FIRST flux-densities at 1.4 GHz we calculated the radio loudness parameter R defined as the ratio between the rest-frame 6 cm, and 2500 Å flux densities 6 . We obtained R < 10 for all the sources and therefore we conclude that they are not radioloud (Kellermann et al. 1989). all the source counts. For the background we adopted annular source-free regions centered on the quasar with inner and outer radii of 6 and 30 arcsec, respectively. For the source j0958 we reduced two observations and then added spectral and response files with the FTOOLS script addascaspec. The reduction of the XMM-Newton data (both pn and MOS 7 ) was performed with SAS v16.0.0. All the observations were performed in full-window mode with the thin filter applied. The light curves for pn and MOS exposures were screened at energies > 10 keV (10 − 12 keV for the pn) for high background flaring periods. We adopted a count-rate threshold filtering criterion with typical values of 0.3-0.5 and 0.1-0.2 counts s −1 for pn and MOS, respectively. The resulting net-exposure times are reported in Table B.1. We selected X-ray events corresponding to patterns 0-4 and 0-12 for pn and MOS, respectively. The source extraction was performed using the same circular apertures for both pn and MOS detectors. In order to include all the source counts and simultaneously minimise the background contribution, for each source we adopted different apertures ranging from 12 to 30 arcsec. The background spectrum was extracted in the chip including the source from circular (1-2 arcmin radius) and annular (inner and outer radii up to 0.7 and 4 arcmin) source-free regions for pn and MOS, respectively. The same data reduction steps have been followed in case of the sources belonging to the C17 sample (see Table B.1 for details on their X-ray observations). We obtained for Chandra spectra with backgroundsubtracted counts ranging from 24 to ∼ 200 in the 0.3-8 keV band, reaching in one case ∼ 800 counts (J2123−0050). As for XMM-Newton we have obtained spectra with backgroundsubtracted counts ranging from 70 (126) to 1700 (1600) for pn (MOS1+MOS2) detectors. We consistently grouped all Chandra and XMM-Newton spectra at a minimum of 1 backgroundsubtracted count per bin. The X-ray modelling was performed A&A proofs: manuscript no. zappacosta_arxiv Notes. a : for the WISSH quasars the redshifts are derived from the narrow component of the Hβ emission line (see Bischetti et al. 2017;Vietri et al. 2018); b : peak of the CIV emission line in units of km s −1 relative to the Hβ emission line as reported by Vietri et al. (2018) and C17 for the WISSH and C17 samples, respectively. Notice that Vietri et al. (2018) and C17 adopt slightly different definitions of v CIV . Vietri et al. (2018) estimates v CIV as relative to the wavelength of the peak of the modelled CIV line while C17 takes as a reference the wavelength that bisects the cumulative flux distribution of the modelled line; c : extinction-corrected derived from spectral energy distribution modelling (Duras et al. 2017;Duras et al. in prep.); d : from Vietri et al. (2018) and C17; e : detected in the FIRST survey with 1.4 GHz integrated flux density of 1.7-1.8 mJy ; f : from Vietri et al. in prep.; g : from 5100Å assuming a bolometric correction from Runnoe et al. (2012). with XSPEC v.12.9.0i by adopting the Cash statistic (C-stat) with the implemented direct background subtraction (Cash 1979;Wachter et al. 1979). A&A proofs: manuscript no. zappacosta_arxiv Fig. 1 . 1Left panel: L 2−10 as a function of v CIV for the WISSH quasars (black circles). We also report hyperluminous quasars at z = 2 − 3 (C17 sample, blue triangles). Middle panel: intrinsic α OX as a function of v CIV for the WISSH quasars. Red circles indicates X-ray weak sources (see Sect. 3 for details). Right panel: L UV as a function of v CIV . Solid and dashed lines report the best-fit linear relation and 1σ uncertainties on the normalization for the WISSH-only relation. Dotted line in the left panel report the best-fit relation with the addition of the C17 sample. Orange lines in the right panel report the best-fit L 2−10 -v CIV relation renormalized to the mean values of L UV . Fig. 2 . 2Top-panels: log L 6µm (left) and L 6µm /L 2−10 (right) as a function of v CIV . Bottom panels: log L bol (left) and k bol,X (right) as a function of v CIV . The meaning of the data points and lines is the same ofFig. 1. Fig. 3 . 3Γ and N H as a function of v CIV (top and bottom panels). Data points and linear fits are reported as inFig. 1. Fig . A.1. L 6µm vs z for our WISSH sub-sample (filled red circles) and the entire WISSH sample (empty black circles). Table C .1. C Table 1 . 1Correlation and linear regression coefficientsRelation sample a ρ p b ρ c sim p d sim,90 α e β log L 2−10 -v CIV W 0.63 0.0210 0.68±0.06 0.0340 0.13 ± 0.04 45.7 ± 0.2 W+C17 0.53 0.0244 0.56±0.04 0.0346 0.14 ± 0.04 45.8 ± 0.1 log L 2500 -v CIV W 0.32 0.2872 - - 0.01 ± 0.03 32.2 ± 0.1 log L 6µm -v CIV W 0.27 0.3680 - - 0.01 ± 0.02 47.1 ± 0.1 log L bol -v CIV W 0.38 0.1976 - - 0.02 ± 0.02 47.8 ± 0.1 W+C17 0.15 0.5546 - - 0.00 ± 0.02 47.7 ± 0.1 k bol,X -v CIV W -0.69 0.0111 -0.69±0.05 0.0269 −0.11 ± 0.02 2.1 ± 0.1 W+C17 -0.61 0.0087 -0.61±0.04 0.0163 −0.13 ± 0.03 2.0 ± 0.1 L 6µm /L 2−10 -v CIV W -0.67 0.0150 -0.71±0.05 0.0223 −0.11 ± 0.02 1.5 ± 0.1 α ox -v CIV W 0.68 0.0139 0.68±0.04 0.0252 0.05 ± 0.01 −1.7 ± 0.1 ∆α ox -v CIV W 0.68 0.0139 0.68±0.04 0.0253 0.05 ± 0.01 0.08 ± 0.05 α pow ox -v CIV W 0.69 0.0094 0.71±0.04 0.0171 0.08 ± 0.01 −1.7 ± 0.1 ∆α pow ox -v CIV W 0.71 0.0088 0.71±0.03 0.0160 0.08 ± 0.01 0.13 ± 0.06 Γ-v CIV W 0.24 0.4258 - - 0.03 ± 0.04 1.95 ± 0.11 W+C17 0.28 0.2564 - - 0.04 ± 0.03 2.01 ± 0.08 log N H -v CIV W -0.14 g 0.6295 g - - −0.10 ± 0.13 h 21.5 ± 0.4 h W+C17 -0.07 g 0.7645 g - - −0.12 ± 0.14 h 21.2 ± 0.5 h Notes. a : W (WISSH), W+C17 (WISSH + C17 sample) ; b : null-hypothesis probability; c : mean and standard deviation of the ρ accounting through simulations for errors in the two quantities (see text for details). This has been computed only for the relations with p 0.02; d : 90% percentile of the distribution of the p values obtained from the simulated random datasets (see text for details). This has been computed only for the relations with p 0.02; e : units of 10 −3 ; g : generalized Spearman's ρ computed with the survival analysis package ASURV v. 1.2; h : linear regression based on EM algorithm with normal distribution computed with the survival analysis package ASURV v. 1.2. The functional form for a Table A A.1. Properties of the WISSH quasars and on the additional sources from the C17 sampleName z a v CIV b L 2500 Å c νL 6µm c L bol λ Edd d (SDSS) (km s −1 ) log(erg s −1 Hz −1 ) log(erg s −1 ) log(erg s −1 ) WISSH sample J0801+5210 3.257 −2720 +180 −180 32.3 47.2 47.8 c 0.7 J0900+4215 e 3.294 −560 +130 −110 32.4 47.3 48.0 c 3.1 J0958+2827 3.434 −5640 +180 −180 32.1 46.9 47.6 c 0.8 J1106+6400 2.221 −2870 +150 −170 32.3 47.1 47.8 c 0.5 J1111+1336 3.490 −1920 +130 −130 32.2 47.1 47.7 c 0.5 J1201+0116 3.248 −3390 +130 −130 32.1 47.1 47.6 c 1.0 J1236+6554 3.424 −3360 +110 −110 32.2 47.0 47.7 c 0.8 J1326-0005 3.303 50 +180 −130 32.2 47.1 47.8 c 2.1 J1421+4633 3.454 −5010 +290 −130 32.2 47.0 47.7 c 0.6 J1441+0454 e 2.075 −4210 +70 −140 f 31.9 46.8 47.3 c 0.7 f J1521+5202 2.218 −7420 +200 −200 32.4 47.2 47.9 c 0.7 J1549+1245 e 2.365 −370 +250 −200 32.3 47.1 47.8 c 0.4 J2123-0050 2.282 −2330 +160 −150 32.2 47.0 47.7 c 1.1 C17 sample J0304-0008 3.296 −583 ± 16 - - 47.4 g 1.6 J0929+2825 3.407 −2080 ± 264 - - 47.6 g 0.4 J0942+0422 3.284 −860 ± 117 - - 47.8 g 1.5 J1426+6025 3.197 −3503 ± 45 - - 48.0 g 0.5 J1621-0042 3.729 −713 ± 124 - - 47.9 g 1.0 Table C C.1. Properties of the WISSH and C17 samples derived from the X-ray spectral analysis Name C-stat dof Γ log N H log f 0.5−2 a log f 2−10 b log L 2−10c α OX (α pow OX ) d ∆α OX (∆α pow OX ) e (SDSS) log(cm −2 ) log(erg s −1 ) WISSH sample J0801+5210 90.5 87 1.79 +0.14 −0.14 < 21.9 1.5 2.7 45.30 +0.03 −0.03 −2.01 +0.01 −0.01 (−2.01) −0.20 (−0.20) J0900+4215 1014.8 1089 1.86 +0.04 −0.03 < 21.3 8.5 13.1 46.04 +0.01 −0.01 −1.76 +0.01 −0.00 (−1.76) +0.07 (0.07) J0958+2827 8.5 9 1.87 +0.79 −0.64 < 23.7 0.2 0.6 44.73 +0.13 −0.13 −2.14 +0.05 −0.05 (−2.27) −0.36 (−0.48) J1106+6400 208.7 203 2.20 +0.09 See Table A.1 for its definition.3 We mention that the relation between L 2−10 and the CIV BEL wind velocity is still in place (ρ ≈ 0.77 and p = 0.003) even if we replace v CIV , reporting the peak of the CIV emission line, with the velocity of the CIV outflow component as estimated in the two component CIV spectral modelling byVietri et al. (2018). We used the ASURV package v. 1.2 to account for censored data(Lavalley et al. 1992;Feigelson & Nelson 1985;Isobe et al. 1986).Indeed the fact that all these sources shines at λ Edd ≈ 1 and the reported large uncertainties on the λ Edd (which are mainly driven by the constraints on SMBH mass;Vietri et al. 2018) prevent an accurate investigation of a possible trend. In order to derive the K-corrected rest-frame 6 cm flux densities from the 1.4 GHz observed ones we assumed a power-law synchrotron spectrum S (ν) ∝ ν −α , with α = 0.5 (e.g.Jiang et al. 2007). Compared toMartocchia et al. (2017) which analysed pn-only spectra we further improved the statistics of the XMM-Newton spectra by adding the MOS data. Acknowledgements. We thank the referee for his/her comments and suggestions. We thank Silvia Martocchia for useful discussions. LZ, EP, AB and MB acknowledge financial support under ASI/INAF contract 2017-14-H.0. CR acknowledges support from the CONICYT+PAI Convocatoria Nacional subvencion a instalacion en la academia convocatoria año 2017 PAI77170080. MG is supported by the "Programa de Atracción deTalentoAppendix A: Sample selectionAppendix B: X-ray data reduction Data from sources listed inTable B.1 whose OBSID is marked by an asterisk have been also analysed inMartocchia et al. (2017). We performed the reduction of the new Chandra data with CIAO 4.9 (with version 4.7.3 of the Chandra Calibration Database). We reprocessed each data set with the script chan-dra_repro with the default script settings in order to generate level 2 data with the most updated calibration. We inspected the light-curves on each observation in order to check for possible high background periods and did not find any observation affected by such contaminations. The spectral extraction and the creation of the relative response files were performed using the CIAO script specextract. We used source extraction circular regions with radii in the range 1.5-3 arcsec in order to includeAppendix C: X-ray spectral analysisThe X-ray modelling, carried out with the same model consistently for all the sources, is performed in the 0.2-10 keV and 0.3-8 keV for the XMM-Newton and Chandra detectors, respectively. The adopted model is a simple power-law model absorbed by the Galactic interstellar medium and by the obscuring medium (i.e. nuclear absorber+interstellar matter) at the redshift of the source. The obscuration is parametrized as uniform cold absorbers adopting the wabs and zwabs multiplicative models for Galactic and intrinsic absorbers, respectively. In the cases of sources with spectra from more than one detector (i.e. either for XMM-Newton or XMM-Newton+Chandra) a joint fit is performed with the addition in the modelling of a constant term to account for cross-calibration and possible source flux variation (in case of observations taken at different epochs). During the modelling we tie the pn and MOS constants whenever the MOS one exceeds by more than 7% the pn constant 8 . For all the sources except J0900+4215 the calibration constants have been linked among the XMM-Newton detectors. For J0900+4215 (the 8 The MOS detectors have been reported to have at most 7% higher fluxes than the pn as detailed in the XMM-SOC-CAL-TN-0052 Issue 6.0 available in http://xmm2.esac.esa.int/docs/documents/CAL-TN-0052.ps.gz. source with the best quality spectra) the MOS spectra were found to have constants ∼ 3% higher than the PN. For three sources a joint fit analysis with XMM-Newton and Chandra has been performed (i.e. J0900+4215, J1106+6400, J2123−0050). We find J0900+4215 and J1106+6400 to exhibit Chandra calibration constants which significantly differ from the XMM-Newton ones by a factor ∼ 0.6 ± 0.1 and 1.8 ± 0.3 (errors are 90% level), respectively. This indicates that the sources have varied their flux between the two observations. For J2123−0050 the Chandra constant is a factor > 2 higher than the XMM-Newton ones. This is likely due to flux loss in the XMM-Newton spectra due to the presence of a source at ∼ 20 arcsec from the quasar which forced us to shrink the quasar spectral extraction region to a circle with radius 12 arcsec in order to minimise the contamination. Apart from the calibration constants, the modelling was performed linking all other parameters and leaving three parameters free to vary, i.e. the source column density (N H ), photon index (Γ) and power-law normalization. Error estimation for N H and Γ was performed leaving all the previously discussed parameters free to vary during the calculation. The uncertainty on the unabsorbed X-ray luminosities (i.e. 2 keV and 2-10 keV) were estimated by freezing Γ to its best-fit value.Table C.1 reports for each source the fit statistic (C-stat) and degrees of freedom (dof) of the modelling along with Γ, N H , the observed fluxes at 0.5-2 keV and 2-10 keV ( f 0.5−2 and f 2−10 respectively), L 2−10 , the corrected (for extinction/absorption) α OX and ∆α OX (based onJust et al. 2007). Determinations of α OX from literature are usually derived based on 2 keV luminosity estimated under the assumption of a power-law X-ray spectral model without ac-L. Zappacosta et al.: Impact of extreme radiative field in quasar disk/corona counting for intrinsic absorption. Hence, for consistency we derived also α pow OX and ∆α pow OX by estimating the 2 keV luminosity from simple unabsorbed power-law spectral modelling (we only accounted for Galactic absorption). As for the quantities regarding fluxes and luminosities (i.e. f 0.5−2 , f 2−10 , L 2−10 , α OX and ∆α OX ) we adopt and report inTable C.1 the pn-derived values for all the sources observed by XMM-Newton-only. For the sources for which we performed joint XMM-Newton and Chandra modellings we adopt the XMM-Newton-derived quantities for J1106+6400 (the Chandra observations are of lower quality) and the Chandra ones for J0900+4215 and J2123−0050. Indeed for J0900+4215 the Chandra observation has been performed closest in time to the SDSS spectrum in which the relative v CIV has been measured. For J2123−0050 the Chandra spectrum does not suffer from contamination from the nearby X-ray source.We tested for possible X-ray model-dependent systematics on the derived L 2−10 . For each WISSH source with best-fit values or upper limits on N H larger than 10 22 cm −2 , we parametrized the intrinsic absorption with a partially ionised absorber by adopting the model zxipcf with covering fraction f c = 1. We left free to vary both N H and the ionisation parameter (ξ). We obtain an estimated L 2−10 which is on average 0.15 dex larger than the luminosities derived by using a cold absorption model. In this case we still have a significant L 2−10 -v CIV relation (ρ = 0.74 and p = 0.006).We mention that restricting the energy range of the WISSH spectral modellings by limiting the low energy bound to 0.5 keV for all the instruments does not make a significant difference in our result. Indeed L 2−10 varies by not more than ±0.05 dex. This level of variation does not change the strength of the reported correlation with v CIV (i.e. ρ = 0.75 and p = 0.004).--Notes. a : 0.5-2 keV observed flux in units of 10 −14 erg s −1 cm −2 ; b : 2-10 keV observed flux in units of 10 −14 erg s −1 cm −2 ; c : 2-10 keV unabsorbed luminosity; d : α OX computed using extinction-corrected 2500 Å and unabsorbed 2 keV luminosities. In parenthesis is reported the α pow OX with 2 keV luminosity from unabsorbed power-law modelling.; e : difference between the measured α OX and predicted one based on the α OX − L 2500 Å relation fromJust et al. (2007). In this luminosity regime the recently derived α OX − L 2500 Å relation byLusso & Risaliti (2016), derived for the X-ray detected quasars with E(B − V) < 0.1 and 1.6 ≤ Γ ≤ 2.8, predict α OX values ∼ 0.03 smaller (i.e. more negative) than theJust et al. (2007)ones. 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[ "Title : will be set by the publisher A NUMERICAL STUDY OF PENROSE-LIKE INEQUALITIES IN A FAMILY OF AXIALLY SYMMETRIC INITIAL DATA", "Title : will be set by the publisher A NUMERICAL STUDY OF PENROSE-LIKE INEQUALITIES IN A FAMILY OF AXIALLY SYMMETRIC INITIAL DATA" ]	[ "José Luis Jaramillo ", "Nicolas Vasset ", "Marcus Ansorg " ]	[]	[ "EAS Publications Series" ]	Our current picture of black hole gravitational collapse relies on two assumptions: i) the resulting singularity is hidden behind an event horizon -weak cosmic censorship conjecture -and ii) spacetime eventually settles down to a stationarity state. In this setting, it follows that the minimal area containing an apparent horizon is bound by the square of the total ADM mass (Penrose inequality conjecture). Following Dain et al. 2002, we construct numerically a family of axisymmetric initial data with one or several marginally trapped surfaces. Penrose and related geometric inequalities are discused for these data. As a by-product, it is shown how Penrose inequality can be used as a diagnosis for an apparent horizon finder numerical routine.	10.1051/eas:0830039	[ "https://arxiv.org/pdf/0712.1741v1.pdf" ]	118,725,656	0712.1741	ac85297a2480da60d15836b795332712eddfc637	Title : will be set by the publisher A NUMERICAL STUDY OF PENROSE-LIKE INEQUALITIES IN A FAMILY OF AXIALLY SYMMETRIC INITIAL DATA 2008 José Luis Jaramillo Nicolas Vasset Marcus Ansorg Title : will be set by the publisher A NUMERICAL STUDY OF PENROSE-LIKE INEQUALITIES IN A FAMILY OF AXIALLY SYMMETRIC INITIAL DATA EAS Publications Series 2008Editors : will be set by the publisher Our current picture of black hole gravitational collapse relies on two assumptions: i) the resulting singularity is hidden behind an event horizon -weak cosmic censorship conjecture -and ii) spacetime eventually settles down to a stationarity state. In this setting, it follows that the minimal area containing an apparent horizon is bound by the square of the total ADM mass (Penrose inequality conjecture). Following Dain et al. 2002, we construct numerically a family of axisymmetric initial data with one or several marginally trapped surfaces. Penrose and related geometric inequalities are discused for these data. As a by-product, it is shown how Penrose inequality can be used as a diagnosis for an apparent horizon finder numerical routine. Introduction and methodology. Our present goal is the study, using numerical techniques, of certain geometric inequalities conjectured to hold in asymptotically flat Cauchy slices containing an apparent horizon (AH). This is an ambitious objective, since numerical tools can offer at best a counterexample -not a general proof -and experience with these inequalities has revealed the difficulty of this task. Precisely because of this confidence in the inequalities, the line of reasoning can be reversed in appropriate settings: inequalities can be (tentatively) taken for granted and used as a diagnosis to test specific geometric and/or numerical constructions. At the end of the day, we aim at gaining geometric insight about these inequalities in regimes that are difficult to probe by standard analytic techniques. We formulate here our strategy and present some preliminary results. 1.1. Geometric inequalities. Penrose inequality is our prototype of geometric inequality. It follows from a chain of heuristic arguments (Penrose 1973) in the context of black hole gravitational collapse, in particular probing weak cosmic censorship conjecture and the assumption of an evolution towards a final stationary state. Penrose inequality conjectures: A ≤ 16πM 2 ADM , where M ADM is the total ADM mass and A is the minimal area enclosing the (possibly non-connected) AH. It has been proved in the Riemannian case K ij = 0 (Huisken & Ilmanen 2001, Bray 2001, an equality is conjectured to hold only for Schwarzschild. Here we focus on axisymmetric data, for which an angular momentum J can be unambiguously defined and Penrose inequality can be strenghtened (Dain et al. 2002) to A ≤ 8π M 2 ADM + M 4 ADM − J 2 , where equality is conjectured to hold only for Kerr data. We shall refer to this latter point as the Dain's rigidity conjecture. Rhs expression only makes sense if \|J\| ≤ M 2 ADM , an angular momentum-mass inequality recently proved by Dain for vacuum, maximal (K = 0), asymptotically flat, axisymmetric data with a connected AH (Dain 2007 and references therein). Equality holds only for extremal Kerr data. Petroff and Ansorg have proposed a quasi-local bound for \|J\| in terms of the area A 8π\|J\| ≤ A , in the restricted setting of stationary, axisymmetric configurations of black holes surrounded by matter. This conjecture has been extended to include charges, and equality has been shown to exactly correspond to the extremal case (Ansorg & Pfister 2007). Simultaneously, it has been argued (Booth & Fairhurst 2007) the non-validity of this quasi-local inequality for generic axisymmetric data. Here we consider this latter non-stationary generic situation. We rewrite previous inequalities in terms of bounded dimensionless parameters ǫ P , ǫ A , ǫ D , ǫ P A : ǫ P := A 16πM 2 ADM ≤ 1 , ǫ D := \|J\| M 2 ADM ≤ 1 ǫ A := A 8π(M 2 ADM + √ M 4 ADM −J 2 ) ≤ 1 , ǫ P A := 8π\|J\| A ≤ 1 . (0.1) In particular, Dain's rigidity conjecture reads: ǫ A = 1 ⇔ (γ ij , K ij ) are Kerr data. 1.2. Axisymmetric Initial Data: deformations on Kerr. A conformal construction of vacuum, maximal and axisymmetric data -parametrized by two free functions q and ω -is presented in Dain et al. 2002. By further restricting q and ω, we have studied: i) binary black hole data and ii) deformations of Kerr. We discuss here Kerr deformations and will present the binary case elsewhere: 1. Choice of q. Fixed by a choice of conformal metric as the representative with unit determinant in the conformal class of Kerr in quasi-isotropic coordinates. 2. Choice of ω as: ω(J, M Kerr , λ) = ω BY (J) − λ · ω ∆ (J, M Kerr ). Here ω BY (J) is associated with the Bowen-York extrinsic curvature in Dain et al. 2002 method, whereas ω ∆ (J, M Kerr ) is such that ω(J, M Kerr , λ = 1) is compatible with Kerr. 3. Marginally Trapped Outer Surface (MOTS) inner boundary condition at an excised sphere of coordinate radius r = 1, when solving the Hamiltonian constraint for the conformal factor. The introduced scale fixes M Kerr in terms of J. In sum, we work with data [γ ij (J, λ), K ij (J, λ)] parametrised by J and a deformation parameter λ -Kerr data correspond to λ = 1. Data are numerically implemented using both the spectral methods in the Meudon C++ Lorene library and the spectral methods developed by one of the authors in Ansorg et al. 2005. 1.3. Extraction of geometric information: AH-finders. The assessment of the discussed geometric inequalities primarily concerns AHs properties. First, we need to know their location. By construction, data in the considered family contain a MOTS at the inner excised sphere. In the generic case, we need an AH-finder routine to locate the outermost MOTS. In this case, we make use of the spectral 3D spectral integral-iteration AH-finder presented in Lin & Novak 2007. 2. Results. Fixing J and screening different values of λ, we monitor the dimensionless quantities ǫ P , ǫ A , ǫ D , ǫ P A . First we check that, as λ departs from λ = 1, the data indeed move away from Kerr -i.e. ǫ A departs from 1, as they should according to Dain's rigidity conjecture. Increasing λ, a critical λ o exists for each J at which a second outer horizon detaches from the inner one. For sufficiently large λ, ǫ A grows over 1 for the inner horizon, whereas the strenghtend Penrose inequality still holds since the exterior horizon satisfies ǫ A ≤ 1 -cf. Fig. 1. The other inequalities are also satisfied, and no surprises appear. At this point we reverse the line of reasoning. First, we use the inequalities to test the Lin & Novak AH-finder. As Fig. 1-left shows, outer ǫ A grows with λ. Penrose inequality sets a definite geometric limit to this: ǫ A ≤ 1 should hold for all λ's if the AH-finder is properly working. Pushing λ to very large values (bounded by numerical limits) we have checked the validity of the inequalitycf. Fig. 1-right. The AH-finder is not producing spurious solutions and is actually converging to a MOTS, otherwise there is no reason for ǫ A ≤ 1 to hold. Most importantly, this asymptotic behaviour indicates the outermost character of the outer MOTS. Additional quantitative tests have shown (spectral) exponential convergence, an accuracy of δA/A ∼ 10 −12 , and the robustness of the AH-finder. Second, Fig. 1-right suggests lim λ→∞ ǫ A = 1. According to Dain's rigidity con-jecture this limit corresponds to Kerr data. As a necessary condition for this, the AH should approach a Non-Expanding Horizon (NEH) as λ diverges, namely the outer null normal shear σ + should vanish in this limit. Fig. 1-right confirms this vanishing asymptotic behaviour, reached after an intermediate stage in which the AH definitely departs from a NEH -similar results are obtained using a dimensionless S \|σ + \| 2 dA. In this sense, our preliminary numerical results do support Dain's conjecture. Before concluding, we note that a mass-angular momentum inequality -milder than Dain's -follows from Penrose and Petroff-Ansorg bounds: 8π\|J\| ≤ A ≤ 8π M 2 ADM + M 4 ADM − J 2 . An optimal fitting among all three inequalities would be obtained with a strong Petroff-Ansorg inequality: 8π \|J\| + M 4 ADM − J 2 ≤ A. Our numerical experiments show its violation for J large enough, whereas standard Petroff-Ansorg inequality is not disproved. This is indeed a doubtful manner of proposing new inequalities, that could be referred to as geometric numerology. 3. Conclusions and perspectives. We have presented the elements of an approach to the numerical study of Penrose-like geometric inequalities. Potential applications to the assessment of numerics have been illustrated by performing a geometric test to the Lin & Novak AH-finder. More generally, Penrose inequality has been proposed as a practical diagnosis in the determination of the outermost character of a given MOTS. This can be useful in settings where little intuition is available about the properties of the AH -this has indeed proved to be crucial in our studies of the binary case, where employed coordinates (Ansorg 2005) strongly affect the form and location of the AH. Finally, following a strategy that resembles a canonical-conjugate version of Bray's flow of conformal metrics approach to Penrose inequality (Bray 2001) -by employing a flow of conformal extrinsic curvatures instead -we have found support to Dain's proposal of using ǫ A = 1 as a characterization of Kerr data. Given its cheap cost -evaluation of a single real number -this has a clear interest for the numerical community. Next step consists in fully developing the outlined approach. We will pursue the study of the binary case, much richer and less understood -e.g. Dain's massangular momentum inequality is not proved in the non-connected case. We will further assess Dain's rigidity conjecture, by implementing more general deformations of Kerr, and will explore the general validity of Petroff-Ansorg inequality. Fig. 1 . 1Left: (1 − ǫ A ) against λ showing the emergence of a second horizon, the violation of ǫ A ≤ 1 for the inner horizon, whereas Penrose inequality still holds. Right: (1 − ǫ A ) -dashed line-and the maximum of the outgoing shear σ+, -bold line-against large λ's. . M Ansorg, Phys. Rev. D. 7224018Ansorg, M. 2005, Phys. Rev. D 72, 024018. . M Ansorg, H Pfister, arXiv:0708.4196Preprintgr-qcAnsorg, M. & Pfister, H. 2007, Preprint: arXiv:0708.4196 [gr-qc]. . I Booth, S Fairhurst, arXiv:0708.2209Preprintgr-qcBooth, I. & and Fairhurst, S. 2007, Preprint: arXiv:0708.2209 [gr-qc]. . H L Bray, S Dain, arXiv:0707.3118v1J. Diff. Geom. 59Preprintgr-qcBray, H.L. 2001, J. Diff. Geom 59, 177. Dain, S. 2007, Preprint: arXiv:0707.3118v1 [gr-qc]. . S Dain, C Lousto, R Takahashi, J. Diff. Geom. 65353Phys. Rev. DDain, S., Lousto, C. & Takahashi, R. 2002, Phys. Rev. D 65, 104038. Huisken G. & Ilmanen, T. 2001, J. Diff. Geom. 59, 353. . L M Lin, J Novak, Annals N. Y. Acad. Sci. 24Class. Quantum. Grav.Lin, L.M. & Novak, J. 2007, Class. Quantum. Grav. 24, 2665. Penrose, R. 1973, Annals N. Y. Acad. Sci. 224, 125-134.	[]
[ "Cones over pseudo-Riemannian manifolds and their holonomy", "Cones over pseudo-Riemannian manifolds and their holonomy" ]	[ "D V Alekseevsky ", "V Cortés ", "A S Galaev ", "T Leistner " ]	[]	[]	By a classical theorem ofGallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone is para-Kählerian. For Lorentzian cones we get a complete description of the possible (local) holonomy algebras in terms of the metric of the base manifold.MSC: 53C29; 53C50	10.1515/crelle.2009.075	[ "https://arxiv.org/pdf/0707.3063v2.pdf" ]	14,940,821	0707.3063	1e1dff0d0fb1c1573e06640bab9decc167928a73	Cones over pseudo-Riemannian manifolds and their holonomy 25 Jul 2007 D V Alekseevsky V Cortés A S Galaev T Leistner Cones over pseudo-Riemannian manifolds and their holonomy 25 Jul 2007Holonomy groupspseudo-Riemannian conesdoubly warped productspara-Sasaki and para-Kähler structures By a classical theorem ofGallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone is para-Kählerian. For Lorentzian cones we get a complete description of the possible (local) holonomy algebras in terms of the metric of the base manifold.MSC: 53C29; 53C50 Introduction Let (M, g) be a (connected) pseudo-Riemannian manifold of signature (p, q) = (−, · · · , −, +, · · · , +). We denote by H the holonomy group of (M, g) and by h ⊂ so(V ) (V = T p M , p ∈ M ) the corresponding holonomy algebra. We say that h is decomposable if V contains a proper non-degenerate h-invariant subspace. By Wu's theorem [Wu64] this means that M is locally decomposed as a product of two pseudo-Riemannian manifolds. In the opposite case h is called indecomposable. We say that h is reducible if it preserves a (possibly degenerate) proper subspace of V . The holonomy algebra h is of exactly one of the following types: (i) decomposable, (ii) reducible indecomposable or (iii) irreducible. Let ( M = R + × M, g = dr 2 + r 2 g) be the (space-like) metric cone over (M, g). We denote by H the holonomy group of ( M , g) and by h ⊂ so( V ) ( V = T p M , p ∈ M ) the corresponding holonomy algebra. In the present article we shall describe the geometry of the base (M, g) for each of the three possibilities (i-iii) for the holonomy algebra of the cone M . Our first result describes the holonomy algebra and local structure of a manifold (M, g) with decomposable holonomy h of the cone. Theorem 4.1. Let (M, g) be a pseudo-Riemannian manifold with decomposable holonomy algebra h of the cone M . Then the manifold (M, g) has full holonomy algebra so(p, q), where (p, q) is the signature of the metric g. Furthermore, there exists an open dense submanifold M ′ ⊂ M such that any point p ∈ M ′ has a neighborhood isometric to a pseudo-Riemannian manifold of the form (a, b) × N 1 × N 2 with the metric given either by g = ds 2 + cos 2 (s)g 1 + sin 2 (s)g 2 or g = −ds 2 + cosh 2 (s)g 1 + sinh 2 (s)g 2 , where g 1 and g 2 are metrics on N 1 and N 2 respectively. Let us recall the following fundamental theorem of Gallot which settles the problem for Riemannian cones over complete Riemannian manifolds. Theorem 1 (S. Gallot, [Gal79]). Let (M, g) be a complete Riemannian manifold of dimension ≥ 2 with decomposable holonomy algebra h of the cone M . Then (M, g) has constant curvature 1 and the cone is flat. If, in addition, (M, g) is simply connected, then it is equal to the standard sphere. (2) a pseudo-Riemannian manifold M 2 = R + × N 1 × N 2 with the metric −ds 2 + cosh 2 (s)g 1 + sinh 2 (s)g 2 , where (N 1 , g 1 ) and (N 2 , g 2 ) are pseudo-Riemannian manifolds and (N 2 , g 2 ) has constant sectional curvature −1 or dim N 2 ≤ 1. Moreover, the cone M 2 is isometric to the open subset {r 1 > r 2 } in the product of the space-like cone (R + × N 1 , dr 2 + r 2 g 1 ) over (N 1 , g 1 ) and the time-like cone (R + × N 2 , −dr 2 + r 2 g 2 ) over (N 2 , g 2 ). For compact and complete pseudo-Riemannian manifolds (M, g) we are able to establish the same conclusion as in Theorem 1: Theorem 6.1. Let (M, g) be a compact and complete pseudo-Riemannian manifold of dimension ≥ 2 with decomposable holonomy group H of the cone M . Then (M, g) has constant curvature 1 and the cone is flat. We remark that for indefinite pseudo-Riemannian manifolds compactness does not imply completeness, see for example [O'N83, p. 193] for a geodesically incomplete Lorentz metric on the 2-torus (the so-called Clifton-Pohl torus). Since there is no simply connected compact indefinite pseudo-Riemannian manifold of constant curvature 1, we obtain the following corollary. Corollary 1. If (M, g) is a simply connected compact and complete indefinite pseudo-Riemannian manifold, then the holonomy algebra of the cone ( M , g) is indecomposable. Now we consider the case (ii) when the holonomy algebra h of the cone M is indecomposable but reducible. We completely analyse the situation in the following two cases: (ii.a) h preserves a decomposition T p M = V ⊕ W (p ∈ M ) into two complementary subspaces V and W . (ii.b) M is Lorentzian. In the case (ii.a) one can show that M admits (locally) a para-Kähler structure, which means that the holonomy algebra h preserves two complementary isotropic subspaces. The following theorem characterises para-Kählerian cones as cones over para-Sasakian manifolds. Theorem 8.1. Let (M, g) be a pseudo-Riemannian manifold. There is a one-to-one correspondence between para-Sasakian structures (M, g, T ) on (M, g) and para-Kähler structures ( M , g, J) on the cone ( M , g). The correspondence is given by T → J := ∇T . Similarly, we have the following characterisation of the case when the cone M admits (locally) a para-hyper-Kähler structure, which means that the holonomy algebra h preserves two complementary isotropic subspaces T ± and a skew-symmetric complex structure J such that JT + = T − . In particular, it preserves the para-hyper-complex structure ( J 1 , J 2 , J 3 J 1 J 2 ), where J 1 \| T ± = ±Id and J 2 = J. Theorem 8.2. Let (M, g) be a pseudo-Riemannian manifold. There is a one-to-one correspondence between para-3-Sasakian structures (M, g, T 1 , T 2 , T 3 ) on (M, g) and parahyper-Kähler structures ( M , g, J 1 , J 2 , J 3 = J 1 J 2 ) on the cone ( M , g). The correspondence is given by T α → J α := ∇T α . Finally, we consider the case (ii.b) when the cone is Lorentzian with indecomposable reducible holonomy algebra. Theorem 9.1. Let (M, g) be a Lorentzian manifold of signature (1, n − 1) or a negative definite Riemannian manifold and ( M = R + × M, g) the cone over M with Lorentzian signature (1, n) or (n, 1) respectively. If the holonomy algebra h of M is indecomposable reducible (i.e. preserves an isotropic line) then it annihilates a non-zero isotropic vector. The next theorem treats the case of a Lorentzian cone M over a negative definite Riemannian manifold M . Theorem 9.1'. Let (M, g) be a negative definite Riemannian manifold and ( M , g) the cone over M equipped with the Lorentzian metric of signature (+, −, · · · , −). If M admits a non-zero parallel isotropic vector field then M is locally isometric to a manifold of the form (M 0 = (a, b) × N, g = −ds 2 + e −2s g N ), (1.1) where a ∈ R ∪ {−∞}, b ∈ R ∪ {+∞}, a < b and (N, g N ) is a negative definite Riemannian manifold. Furthermore, if hol( M 0 ) is indecomposable then hol( M 0 , g) ∼ = hol(N, g N ) ⋉ R dim N . If the manifold (M, g) is complete then the isometry is global, (N, g N ) is complete and (a, b) = R. The last theorem treats a Lorentzian cone M over Lorentzian manifold M . Let us conclude this introduction with some brief remarks about applications of these. The theorem of Gallot was used by C. Bär in the classification of Riemannian manifolds admitting a real Killing spinor [Bär93]. In general, a pseudo-Riemannian manifold admits a real/imaginary Killing spinors if and only if its space-like/time-like cone admits a parallel spinor (details in [Boh03]). Hence, a strategy for studying manifolds with Killing spinors is to study their cones with parallel spinors. Now, in order to classify manifolds with parallel spinors the knowledge of their holonomy group is essential. In the Riemannian situation Gallot's result reduces the problem to irreducible holonomy groups of cones. With our results this strategy becomes applicable to arbitrary signature. , where (N, g N ) is a positive definite Riemannian manifold. Furthermore, if hol( M 0 ) is indecomposable then hol( M 0 , g) ∼ = hol(N, g N ) ⋉ R dim N . Another applications in the same spirit -solutions to an overdetermined system of PDE's correspond to parallel sections for a certain connection -comes from conformal geometry. Here, to a conformal class of a metric one can assign the so-called Tractor bundle with Tractor connection. Parallel sections of this connection correspond to metrics in the conformal class which are Einstein. For conformal classes which contain an Einstein metric the holonomy of the conformal Tractor connection reduces to the Levi-Civita holonomy of the Fefferman-Graham ambient metric [FG85]. For conformal classes containing proper Einstein metrics with positive/negative Einstein constant the ambient metric reduces to the space-like/time-like cone over a metric in the conformal class [Leit05,Arm05,AL06]. Again, our results enable us to describe the holonomy of the conformal Tractor connection by the holonomy of the cone. To carry out the details of both applications lies beyond the scope of this paper and will be subject to future research. f ′ 1 f 1 2 − ε κ 1 f 2 1 = c if dim N 1 > 1, f ′ 2 f 2 2 − ε κ 2 f 2 2 = c if dim N 2 > 1. Solving these equations we get the following corollary. We will denote by g k , g ′ k , g ′′ k pseudo-Riemannian metrocs of constant curvature k ∈ {±1, 0}. Corollary 2.3. Then the following doubly warped product metrics g k have constant curvature k: g −ε = εds 2 + cosh 2 (s)g ′ −ε + sinh 2 (s)g ′′ ε g ε = εds 2 + cos 2 (s)g ′ ε + sin 2 (s)g ′′ ε g −ε = εds 2 + e 2s g ′ 0 g 0 = εds 2 + s 2 g ′ ε + g ′′ 0 g −ε = εds 2 + cosh 2 (s)dt 2 + sinh 2 (s)g ′′ ε g −ε = εds 2 + cosh 2 (s)g ′ −ε + sinh 2 (s)du 2 g ε = εds 2 + cos 2 (s)dt 2 + sin 2 (s)g ′′ ε g ε = εds 2 + cos 2 (s)g ′ ε + sin 2 (s)du 2 g 0 = εds 2 + s 2 dt 2 + g ′′ 0 g −ε = εds 2 + sinh 2 (s)g ′ ε g −ε = εds 2 + cosh 2 (s)g ′ −ε g ε = εds 2 + sin 2 (s)g ′ ε g 0 = εds 2 + g ′ 0 g −ε = εds 2 ± cosh 2 (s)dt 2 ± sinh 2 (s)du 2 g ε = εds 2 ± cos 2 (s)dt 2 ± sin 2 (s)du 2 g −ε = εds 2 ± sinh 2 (s)dt 2 g −ε = εds 2 ± cosh 2 (s)dt 2 g ε = εds 2 ± sin 2 (s)dt 2 Any doubly warped product of pseudo-Riemannian manifolds (N 1 , g 1 ), (N 2 , g 2 ) which has constant curvature ±1 or 0 belongs to the above list up to a shift s → s + s 0 and rescaling (f 2 i , g i ) → (λ 2 i f 2 i , 1 λ 2 i g i ). Geometric realisation of doubly warped products of constant curvature Now we give a realisation of the above doubly warped products in terms the pseudo-sphere models of the spaces of constant curvature. The standard pseudo-spheres as models of spaces of curvature ±1 Let R t,s = (R t+s , ·, · − t i=1 dx 2 i + t+s i=t+1 dx 2 i ) be the standard pseudo-Euclidian vector space of signature (t, s). We denote by S t,s + := {x ∈ R t,s+1 \| x, x = +1} S t,s − := {x ∈ R t+1,s \| x, x = −1} the two unit pseudo-spheres. The induced metric g ± = g S t,s ± of S t,s ± has signature (t, s) and constant curvature ±1. More precisely the curvature tensor is given by R(X, Y )Z = ±( Y, Z X − X, Z Y ). Notice that S 0,n + = S n is the standard unit n-sphere, S 0,n − = H n is hyperbolic n-space, S 1,n−1 + = dS n is de Sitter n-space and S 1,n−1 = AdS n is anti de Sitter n-space. Flat space as cone over the pseudo-spheres The domains R t,s ± := {x ∈ R t,s \| ± x, x > 0} ⊂ R t,s are isometrically identified via the map (r, x) → rx with the space-like or time-like cone over S t,s−1 + or S t−1,s − endowed with the metric ±dr 2 + r 2 g ± , respectively. In particular, the space-like cone over a space of constant curvature 1 and the time-like cone over a space of constant curvature −1 are flat. Realisation of doubly warped products by double polar coordinates Now we show that any splitting of a pseudo-Euclidian vector space as an orthogonal sum of two pseudo-Euclidian subspaces induces local parametrisations of the pseudo-spheres. Using these 'double polar' parametrisations (more precisely, polar equidistant parametrisations [AVS93]) we will show that the spaces of constant curvature can be locally presented as doubly warped products with trigonometric or hyperbolic warping functions over spaces of appropriate constant curvature. We consider the pseudo-spheres S + (V ) = S t,s + ⊂ V = R t,s+1 and S − (V ) = S t,s − ⊂ V = R t+1,s . Any orthogonal decomposition V = V 1 ⊕ V 2 = {v = x + y\|x ∈ V 1 , y ∈ V 2 }, ·, · = ·, · 1 + ·, · 2 defines a diffeomorphism (s,x,ȳ) → x + y, x = cos(s)x, y = sin(s)ȳ, (2.4) of (0, π 2 ) × S ε (V 1 ) × S ε (V 2 ) onto the (not necessarily connected) domain D = {v = x + y ∈ S ε (V )\|0 < ε x, x 1 < 1}. Similarly the map (s,x,ȳ) → x + y, x = cosh(s)x, y sinh(s)ȳ, (2.5) is a diffeomorphism of R + × S ε (V 1 ) × S −ε (V 2 ) onto the domain D ′ = {v = x + y ∈ S ε (V )\|ε x, x 1 > 1}. Proposition 2.3. With respect to the diffeomorphisms (2.4) and (2.5) the metric g ε of S ε (V ) is given by g ε \| D = εds 2 + cos 2 (s)g Sε(V 1 ) + sin 2 (s)g Sε(V 2 ) g ε \| D ′ = −εds 2 + cosh 2 (s)g Sε(V 1 ) + sinh 2 (s)g S −ε (V 2 ) . Horospherical coordinates and corresponding warped products Let (V, ·, · ) be an indefinite pseudo-Euclidian vector space, p, q ∈ V two isotropic vectors such that p, q = 1 and W = span{p, q} ⊥ . Then R + ×W ∋ (s, ξ) → y = up+vq+x ∈ S ε (V ), u = ± 1 2 e −s (ε−e 2s ξ, ξ ), v = ±2e s , x = e s ξ is a diffeomorphism onto the domain S ε (V ) ∩ {y ∈ V \| ± v > 0}. In the coordinates (s, ξ) the hypersurfaces s = const correspond to the hyperplane sections (horospheres) {y ∈ S ε (V )\| y, p = ±e s } and the curves ξ = ξ 0 = const are geodesics perpendicular to the horospheres. A direct calculation shows that: Proposition 2.4. The induced metric of the pseudo-sphere S ε (V ) in horospherical coordinates (s, ξ) is given by: g ε = εds 2 + e 2s g 0 , where g 0 = dξ 2 is the induced pseudo-Euclidian metric on W . Completeness of some doubly warped products Proposition 2.5. Let (N 1 , g 1 ), (N 2 , g 2 ) be pseudo-Riemannian manifolds and (M = I × N 1 ×N 2 , g = εds 2 +f 1 (s) 2 g 1 +f 2 (s) 2 g 2 ) a doubly warped product with non-constant warping functions as in Corollary 2.3. Then (M, g) is complete only in the following cases: (i) g = εds 2 + cosh 2 (s)g 1 , where I = R and g 1 is complete, and (ii) g = εds 2 + e 2s g 1 where I = R and εg 1 is complete and positive definite. Proof. The system (2.2-2.3) has solutions u i = u 0 i = const, s = at + b, which are complete if and only if I = R. This excludes all the warping functions which have a zero. It remains to check that the metric (i) is complete for any complete metric g 1 and that (ii) is complete only if εg 1 is complete and positive definite. In fact, in both cases the squared velocity l = g(γ,γ) is constant. In the second case, for instance, it is given by l = εṡ 2 + ε 1 e 2su2 , u := u 1 , which yieldsÿ = εly after the substitution y = e s . The differential equation y = εly admits solutions which are positive on the real line if and only if εl > 0. The positivity is necessary since y = e s . This shows that g is positive or negative definite, i.e. εg 1 is positive definite. The other case is similar, see [Boh03], where the case of Lorentzian signature is considered. Examples of cones with reducible holonomy Let g = cdr 2 + r 2 g be the cone metric on M := R + × M , where (M, g) is a pseudo-Riemannian manifold. Depending on the sign of the constant c the cone is called space-like (c > 0) or time-like (c < 0). Later on we will assume, without restriction of generality, that c = 1. In fact, as we allow g to be of any signature we can rescale g by 1 c ∈ R * . We denote by ∂ r the radial vector field. The Levi-Civita connection of the cone ( M , g) is given by ∇ ∂r ∂ r = 0, ∇ X ∂ r = 1 r X, ∇ X Y = ∇ X Y − r c g(X, Y )∂ r ,      (3.1) for all vector fields X, Y ∈ Γ(T M ) orthogonal to ∂ r . The curvature R of the cone is given by the following formulas including the curvature R of the base metric g: ∂ r R = 0, R(X, Y )Z = R(X, Y )Z − 1 c (g(Y, Z)X − g(X, Z)Y ) , or R(X, Y, Z, U ) = r 2 R(X, Y, Z, U ) − 1 c (g(Y, Z)g(X, U ) − g(X, Z)g(Y, U )) ,      (3.2) for X, Y, Z, U ∈ T M . This implies that if (M, g) is a space of constant curvature κ, i.e. R(X, Y, Z, U ) = κ (g(X, U )g(Y, Z) − g(X, Z)g(Y, U )) , the cone has the curvature r 2 κ − 1 c (g(X, U )g(Y, Z) − g(X, Z)g(Y, U )). In particular, if κ = 1 c , then the cone is flat, as it is the case for the c = 1 cone over the standard sphere of radius 1 or the c = −1 cone over the hyperbolic space. From now on we assume c = ±1. We denote by ( M = R + × M, g = dr 2 + r 2 g), the space-like cone over (M, g) and by ( M − = R + × M, g − = −dr 2 + r 2 g) the time-like cone. Notice that the metric g − of the time-like cone M − over (M, g) is obtained by multiplying the metric dr 2 − r 2 g of the space-like cone over (M, −g) by −1. Thus it is sufficient to consider only space-like cones. We will now present some examples which illustrate that Gallot's statement is not true in arbitrary signature, and that the assumption of completeness is essential even in the Riemannian situation. Example 3.1. Let (F, g F ) be a complete pseudo-Riemannian manifold of dimension at least 2 and which is not of constant curvature 1. Then the pseudo-Riemannian manifold (M = R × F, g = −ds 2 + cosh 2 (s)g F ) is complete, the restricted holonomy group of the cone over (M, g) is non-trivial and admits a non-degenerate invariant proper subspace. Proof. The manifold (M, g) is complete by Proposition 2.5. The non-vanishing terms of the Levi-Civita connection ∇ of (M, g) are given by ∇ X ∂ s = tanh (s)X, ∇ ∂s X = ∂ s X + tanh (s)X, ∇ X Y = ∇ F X Y + cosh (s)sinh (s)g F (X, Y )∂ s ,    (3.3) where X, Y ∈ T F are vector fields depending on the parameter s and ∇ F is the Levi-Civita connection of the manifold (F, g F ). Consider on M the vector field X 1 = cosh 2 (s)∂ r − 1 r sinh (s)cosh (s)∂ s . We have g(X 1 , X 1 ) = cosh 2 (s) > 0. It is easy to check that the distribution generated by the vector field X 1 and by the distribution T F ⊂ T M is parallel. For the curvature tensor R of (M, g) we have R(X, Y )Z = R F (X, Y )Z + tanh 2 (s) (g F (Y, Z)X − g F (X, Z)Y ) , (3.4) where X, Y, Z, U ∈ T F and R F is the curvature tensor of (F, g F ). This shows that (M, g) cannot have constant sectional curvature, unless F has constant curvature 1 (see Corollary 2.3). Thus the cone ( M , g) is not flat. Example 3.2. Let M be a manifold of the form R×N with the metric g = −(dt 2 +e −2t g N ), where (N, g N ) is a pseudo-Riemannian manifold. Then 1. The light-like vector field e −t (∂ r + 1 r ∂ t ) on the space-like cone M is parallel. 2. If (N = N 1 × N 2 , g N = g 1 + g 2 ) is a product of a flat pseudo-Riemannian manifold (N 1 , g 1 ) and of a non-flat pseudo-Riemannian manifold (N 2 , g 2 ), then M is locally decomposable and not flat 1 . In fact, there is a parallel non-degenerate flat distribution of dimension dim N 1 on M . The manifold (M, g) in Example 3.2 is complete if and only g N is complete and positive definite, see Proposition 2.5. Notice that g is the hyperbolic metric in horospherical coordinates if (N, g N ) is Euclidian space. Example 3.3. Let (M 1 , g 1 ) and (M 2 , g 2 ) be two pseudo-Riemannian manifolds. Then the product of the cones ( M 1 × M 2 = (R + × M 1 ) × (R + × M 2 ), g = (dr 2 1 + r 2 1 g 1 ) + (dr 2 2 + r 2 2 g 2 )) is the space-like cone over the manifold (M = 0, π 2 × M 1 × M 2 , g = ds 2 + cos 2 (s)g 1 + sin 2 (s)g 2 ). Proof. Consider the functions r = r 2 1 + r 2 2 ∈ R + , s = arctg r 2 r 1 ∈ 0, π 2 . Since r 1 , r 2 > 0, the functions r and s give a diffeomorphism R + × R + → R + × 0, π 2 . For M 1 × M 2 we get M 1 × M 2 ∼ = R + × 0, π 2 × M 1 × M 2 and g 1 + g 2 = dr 2 + r 2 (ds 2 + cos 2 (s)g 1 + sin 2 (s)g 2 ). Suppose that the manifolds (M 1 , g 1 ) and (M 2 , g 2 ) are Riemannian. Then the manifold (M, g) is Riemannian and incomplete. The cone over M is decomposable. Moreover, it is not flat, unless the manifolds (M 1 , g 1 ), (M 2 , g 2 ) are of dimension less than 2 or of constant curvature 1, see Corollary 2.3. Example 3.3 shows that the completeness assumption in Theorems 1 and 6.1 is necessary. Example 3.4. Let (M 1 , g 1 ) and (M 2 , g 2 ) be two pseudo-Riemannian manifolds. Then the space-like cone over the manifold (M = R + × M 1 × M 2 , g = −ds 2 + cosh 2 (s)g 1 + sinh 2 (s)g 2 ) is isometric to the open subset Ω = {r 1 > r 2 } in the product of the cones ( M 1 × M 2 = (R + × M 1 ) × (R + × M 2 ), g 1 + g 2 − = (dr 2 1 + r 2 1 g 1 ) + (−dr 2 2 + r 2 2 g 2 )). Proof. Consider the functions r = r 2 1 − r 2 2 ∈ R + , s = artanh r 2 r 1 ∈ R + . The functions r and s give a diffeomorphism {(r 1 , r 2 ) ∈ R 2 \|0 < r 2 < r 1 } → R + × R + . For M 1 × M 2 we get Ω ∼ = R + × R + × M 1 × M 2 and g 1 + g 2 − = dr 2 + r 2 (−ds 2 + cosh 2 (s)g 1 + sinh 2 (s)g 2 ). Example 3.5. Let (t, x 1 , . . . , x n , x n+1 , . . . , x 2n ) be coordinates on R 2n+1 . Consider the metric g given by g =   −1 0 u t 0 0 H t u H G   where • u = (u 1 , . . . , u n ) is a diffeomorphism of R n , depending on x 1 , . . . , x n , • H = 1 2 ∂ ∂x j (u i ) n i,j=1 its non-degenerate Jacobian, and • G the symmetric matrix given by G ij = −u i u j . Then the space-like cone over (R 2n+1 , g) is not flat but its holonomy representation decomposes into two totally isotropic invariant subspaces. For the proof of this see Proposition 8.3 in Section 8. Local structure of decomposable cones In this section we assume that the holonomy group of the cone ( M , g) is decomposable and we give a local description of the manifold (M, g), independently of completeness. Suppose that the holonomy group Hol x of ( M , g) at a point x ∈ M is decomposable, that is T x M is a sum T x M = (V 1 ) x ⊕(V 2 ) x of two non-degenerate Hol x -invariant orthogonal subspaces. They define two parallel non-degenerate distributions V 1 and V 2 . Denote by X 1 and X 2 the projections of the vector field ∂ r to the distributions V 1 and V 2 respectively. We have ∂ r = X 1 + X 2 . (4.1) We decompose the vectors X 1 and X 2 with respect to the decomposition T M = T R + ⊕ T M , X 1 = α∂ r + X, X 2 = (1 − α)∂ r − X, (4.2) where α is a function on M and X is a vector field on M tangent to M . We have Proof. Suppose that α = 1 on an open subspace V ⊂ M . We claim that ∂ r ∈ V 1 on V . Indeed, on V we have g(X, X) = α − α 2 , g(X 1 , X 1 ) = α, g(X 2 , X 2 ) = 1 − α.X 1 = ∂ r + X, X 2 = −X and g(X, X) = 0. We show that X = 0. Let Y 2 ∈ V 2 . We have the decomposition Y 2 = l∂ r + Y, where l is a function on M and Y ∈ T M . It is ∇ Y 2 X 1 = ∇ l∂r+Y (∂ r + X) = 1 r Y + ∇ Y 2 X. Note that Y = Y 2 − lX 1 + lX. Hence, ∇ Y 2 X 1 = 1 r (Y 2 − lX 1 + lX) + ∇ Y 2 X. Since X, ∇ Y 2 X, Y 2 ∈ V 2 and ∇ Y 2 X 1 ∈ V 1 , we see that 1 r (Y 2 + lX) + ∇ Y 2 X = 0. From g(X, X) = 0 it follows that g( ∇ Y 2 X, X) = 0. Thus we get g(Y 2 , X) = 0 for all Y 2 ∈ V 2 . Since V 2 is non-degenerate, we conclude that X = 0. Thus ∂ r ∈ V 1 . Let Y 2 ∈ V 2 , then ∇ Y 2 ∂ r = 1 r Y 2 . Since the distribution V 1 is parallel and ∂ r ∈ V 1 , we see that Y 2 = 0 and V 2 = 0. Contradiction. We now consider the dense open submanifold U ⊂ M . The vector fields X 1 , X 2 and X are nowhere isotropic on U . For i = 1, 2 let E i ⊂ V i be the subdistribution of V i orthogonal to X i . Denote by L the distribution of lines on U generated by the vector field X. We get on U the orthogonal decomposition T M = T R ⊕ L ⊕ E 1 ⊕ E 2 . Lemma 4.2. Let Y 1 ∈ E 1 and Y 2 ∈ E 2 , then on U we have 1. Y 1 α = Y 2 α = ∂ r α = 0. 2. ∇ Y 1 X = 1−α r Y 1 , ∇ Y 2 X = − α r Y 2 . 3. ∇ ∂r X = ∂ r X + 1 r X = 0. 4. ∇ X X = (1−α) 2 r − Xα X 1 + α 2 r − Xα X 2 . Proof. Using (4.2), we have ∇ Y 1 X 1 = (Y 1 α)∂ r + α r Y 1 + ∇ Y 1 X, ∇ Y 1 X 2 = −(Y 1 α)∂ r + 1−α r Y 1 − ∇ Y 1 X. Since Y 1 ∈ E 1 ⊂ V 1 and the distributions V 1 , V 2 are parallel, projecting these equations onto V 2 and adding them yields ∇ Y 1 X 2 = 0. Then, from the second equation, we see that Y 1 α = 0 and ∇ Y 1 X = ∇ Y 1 X = 1−α r Y 1 . The other claims can be proved similarly. Since ∂ r α = 0, the function α is a function on M . Note that U = R + × U 1 , where U 1 = {x ∈ M \|α(x) = 0, 1} ⊂ M. Claim 3 of Lemma 4.2 shows that X = 1 rX , whereX is a vector field on the manifold M . Hence the distributions L and E = E 1 ⊕ E 2 do not depend on r and can be considered as distributions on M . Claim 2 of Lemma 4.2 shows that the distributions E 1 and E 2 also do not depend on r. We get on U 1 the orthogonal decompositions T M = L ⊕ E, E = E 1 ⊕ E 2 . Lemma 4.3. The function α satisfies on U 1 the following differential equatioñ Xα = 2(α − α 2 ). Proof. From r 2 ∇ X X = ∇XX = ∇XX − rg(X,X)∂ r = ∇XX − r(α − α 2 )∂ r and Claim 4 of Lemma 4.2 we conclude that ∇XX ∈ T M is a linear combination of X 1 and X 2 and hence proportional to X = (1 − α)X 1 − αX 2 . This implies (2α − 1) Xα − 2 r (α − α 2 ) = 0. If α = 1 2 , then ∇ X X = 1 4r ∂ r and ∇XX = r 2 ∂ r . The last equality is impossible. Corollary 4.1. On M we haveX = 1 2 grad(α). Lemma 4.3 implies that if t is a coordinate on M corresponding to the vector fieldX, then α(t) = e 2t e 2t + c , where c is a constant. From Lemmas 4.2 and 4.3 it follows that ∇ X X = − α − α 2 r ∂ r + 1 − 2α r X. On the subset U 1 ⊂ M we get the following ∇XX = (1 − 2α)X, ∇ Y 1X = (1 − α)Y 1 , ∇ Y 2X = −αY 2 , (4.4) for any Y 1 ∈ Γ(E 1 ) and Y 2 ∈ Γ(E 2 ). Theorem 4.1. Let (M, g) be a pseudo-Riemannian manifold. If the holonomy group of the metric cone over (M, g) admits a non-degenerate invariant subspace, then hol(M, g) = so(p, q), where (p, q) is the signature of the metric g. Proof. Since the distributions V 1 and V 2 are parallel, for any Y 1 ∈ V 1 and Y 2 ∈ V 2 we have R(Y 1 , Y 2 ) = 0. From (3.2) it follows that R(X, Y 2 ) = R(X, Y 1 ) = 0. Hence, R(X, Y ) = X ∧ Y for all vector fields Y on M . By Lemma 4.1, there exists y ∈ M such that g y (X,X) = 0. The holonomy algebra of the manifold M at the point y contains the subspaceX y ∧ T y M . Since g y (X,X) = 0, this vector subspace generates the whole Lie algebra so(T y M, g y ). (1.) For W we have a decomposition W = (a, b) × N 1 × N 2 , (a, b) ⊂ 0, π 2 and for the metric g\| W we have g\| W = ds 2 + cos 2 (s)g 1 + sin 2 (s)g 2 , where (N 1 , g 1 ) and (N 2 , g 2 ) are pseudo-Riemannian manifolds; Moreover, any point (r, x) ∈ R + × W ⊂ M has a neighborhood of the form ((a 1 , b 1 ) × N 1 ) × ((a 2 , b 2 ) × N 2 ), (a 1 , b 1 ), (a 2 , b 2 ) ⊂ R + with the metric (dt 2 1 + t 2 1 g 1 ) + (dt 2 2 + t 2 2 g 2 ). (2.) For W we have a decomposition W = (a, b) × N 1 × N 2 , (a, b) ⊂ R + and for the metric g\| W we have g\| W = −ds 2 + cosh 2 (s)g 1 + sinh 2 (s)g 2 , where (N 1 , g 1 ) and (N 2 , g 2 ) are pseudo-Riemannian manifolds. Moreover, any point (r, x) ∈ R + × W ⊂ M has a neighborhood of the form ((a 1 , b 1 ) × N 1 ) × ((a 2 , b 2 ) × N 2 ), (a 1 , b 1 ), (a 2 , b 2 ) ⊂ R + with the metric (dt 2 1 + t 2 1 g 1 ) + (−dt 2 2 + t 2 2 g 2 ). Proof. We need the following Lemma 4.4. (i) The distributions E 1 , E 2 , E = E 1 ⊕ E 2 ⊂ T M defined on U 1 ⊂ M are involutive and the distributions E 1 ⊕ L, E 2 ⊕ L ⊂ T M are parallel on U 1 . (ii) Let x ∈ U 1 and M x ⊂ U 1 the maximal connected integral submanifold of the distri- bution E. Then the distributions E 1 \| Mx , E 2 \| Mx ⊂ T M x = E\| Mx are parallel. Proof. (i) On U ⊂ M , the distribution E i = V i ∩ T M is the intersection of two involutive distributions and hence involutive, for i = 1, 2. The corresponding distributions E 1 , E 2 of U 1 ⊂ M are therefore involutive. The involutivity of E follows from Corollary 4.1. Next we prove that E i ⊕ L is involutive. The formulas (4.4) show that ∇ E iX = E i . Now we check that ∇ Y 1 Y ′ 1 ∈ Γ(E 1 ⊕ L) for all Y 1 , Y ′ 1 ∈ Γ(E 1 ). Calculating the scalar product with Y 2 ∈ Γ(E 2 ) we get g(∇ Y 1 Y ′ 1 , Y 2 ) = −g(Y ′ 1 , ∇ Y 1 Y 2 ) = −g(Y ′ 1 , ∇ Y 1 Y 2 ) = 0, since the distribution V 2 is parallel. (ii) The fact that E i ⊕ L ⊂ T M is parallel implies that E i ⊂ T M x is parallel. Now we return to the Examples 3.3 and 3.4. In Example 3.3 we have V 1 = T M 1 , V 2 = T M 2 , E 1 = T M 1 , E 2 = T M 2 , X 1 = cos(s)∂ r 1 , X 2 = sin(s)∂ r 2 , α = cos 2 (s), X = − 1 r sin(s) cos(s)∂ s . Note that 0 < α < 1. In Example 3.4 we have V 1 = T M 1 , V 2 = T M 2 − , E 1 = T M 1 , E 2 = T M 2 , X 1 = cosh (s)∂ r 1 , X 2 = sinh (s)∂ r 2 , α = cosh 2 (s), X = − 1 r sinh (s)cosh (s)∂ s . Note that α > 1. Let x ∈ U 1 . we have two cases: (1.) 0 < α(x) < 1; (2.) α(x) < 0 or α(x) > 1. Case (1.) Suppose that 0 < α(x) < 1. Then 0 < α < 1 on some open subset W ⊂ U 1 containing the point x. Thus g(X,X)ĝ(X, X) = α − α 2 > 0 on W . Recall thatX is a gradient vector field, see Corollary 4.1. Hence we can assume that W has the form (a, b) × N , where (a, b) ⊂ R and N is the level set of the function α. Note also that the level sets of the function α are integral submanifolds of the involutive distribution E. SinceX is orthogonal to E and Z(g(X,X)) = 0 for all Z ∈ E, the metric g\| W can be written as g\| W = ds 2 + g N , where g N is a family of pseudo-Riemannian metrics on N depending on the parameter s. We can assume that ∂ s = −X √ g(X,X) . By Lemma 4.4 and the Wu theorem, the manifold W is locally a product of two pseudo-Riemannian manifolds. For Y 1 , Z 1 ∈ E 1 and Y 2 , Z 2 ∈ E 2 in virtue of Lemma 4.2 we have (LX g)(Y 1 , Z 1 ) = 2(1 − α)g(Y 1 , Z 1 ), (LX g)(Y 1 , Y 2 ) = 0, (LX g)(Y 2 , Z 2 ) = −2αg(Y 2 , Z 2 ). This means that the one-parameter group of local diffeomorphisms of W generated by the vector fieldX preserves the Wu decomposition of the manifolds W . Hence the manifold (N, g\| N ) can be locally decomposed into a direct product of two manifolds N 1 and N 2 which are integral manifolds of the distributions E 1 and E 2 such that g N = h 1 + h 2 , where h i , i = 1, 2 is a metric on N i which depends on s. From Lemmas 4.2,4.3 it follows that the function α depends only on s and satisfies the following differential equation ∂ s α = −2 α − α 2 . Hence, α = cos 2 (s + c 1 ), where c 1 is a constant. We can assume that c 1 = 0. Since on W we have 0 < α < 1 and ∂ s α < 0, we see that (a, b) ⊂ 0, π 2 . Let Y 1 , Z 1 ∈ E 1 be vector fields on W such that [Y 1 , ∂ s ] = [Z 1 , ∂ s ] = 0. From (4.4) it follows that ∇ Y 1 ∂ s = − √ α−α 2 α Y 1 . The Koszul formula implies that 2g(∇ Y 1 ∂ s , Z 1 ) = ∂ s g(Y 1 , Z 1 ). Thus we have −2tan (s)g(Y 1 , Z 1 ) = ∂ s g(Y 1 , Z 1 ). This means that h 1 = cos 2 (s)g 1 , where g 1 does not depend on s. Similarly, h 2 = sin 2 (s)g 2 , where g 2 does not depend on s. For the cone over W we get R + × W = R + × (a, b) × N 1 × N 2 and g\| R + ×W = dr 2 + r 2 (ds 2 + cos 2 (s)g 1 + sin 2 (s)g 2 ). Consider the functions t 1 = r cos(s), t 2 = r sin(s). They define a diffeomorphism from where (a 1 , b 1 ), (a 2 , b 2 ) ⊂ R + and r ∈ (a 1 , b 1 ). R + × (a, b) onto a subset V ⊂ R + × R + . Let (r, y) ∈ R + × W ⊂ M , then there exist a subset (a 1 , b 1 ) × (a 2 , b 2 ) ⊂ V, On the subset ((a 1 , b 1 ) × N 1 ) × ((a 2 , b 2 ) × N 2 ) ⊂ R + × W the metric g has the form (dt 2 1 + t 2 1 g 1 ) + (dt 2 2 + t 2 2 g 2 ). Case (2.) Suppose that α(x) > 1. Now ∂ s = −X √ α 2 −α and the function α satisfies ∂ s α = 2 α 2 − α. Hence, α = cosh 2 (s + c 1 ). Again we can assume c 1 = 0 and from ∂ s α > 0 we get (a, b) ⊂ R + . For the metric g\| W on W = (a, b) × N 1 × N 2 we have g\| W = −ds 2 + g N = −ds 2 + cosh 2 (s)g 1 + sinh 2 (s)g 2 . The case α(x) < 0 is equivalent to the case α(x) > 1 by interchanging the roles of V 1 and V 2 , which interchanges α with 1 − α and X with −X. Theorem 4.2 is proved. Geodesics of cones Using Proposition 2.1, we now calculate the geodesics on the cone. Suppose Γ(t) = (r(t), γ(t)) is a geodesic on the cone ( M , g), where γ(t) is a curve on (M, g). Suppose we have the initial conditions Γ(0) = (r, x) andΓ(0) = (ρ, v), for some x ∈ M, v ∈ T x M . Then r(t) and γ(t) satisfy 0 =r(t) − r(t)g (γ(t),γ(t)) , (5.1) 0 = 2ṙ(t)γ(t) + r(t)∇γ (t)γ (t). (5.2) Now one makes the following ansatz. Suppose that γ is given as a reparametrisation of a geodesic β : R → M of g: γ(t) = β(f (t)) where β is a geodesic of g with initial condition β(0) = x andβ(0) = v = 0, implying the initial conditions for f : f (0) = 0 andḟ (0) = 1. Asγ(t) =ḟ (t)·β(f (t)), g (γ(t),γ(t)) =ḟ (t) 2 g β (f (t)) ,β(f (t)) and ∇γ (t)γ (t) =f (t)β(f (t)), we get from (5.1) and (5.2) 0 =r(t) − r(t)ḟ (t) 2 g (v, v) , (5.3) 0 = 2ṙ(t)ḟ (t) + r(t)f (t) (5.4) with initial conditions r(0) = r , f (0) = 0 , r(0) = ρ ,ḟ (0) = 1. The solution to these equations is straightforward by distinguishing several cases. From now on we assume that ρ = 0 and v = 0 and consider the remaining cases for v being light-like, space-like, or time-like. i.e. r(t) = ρt + r on the one hand, and f (t) = rt ρt+r on the other. This implies that f and thus Γ is defined for t ∈ [0, − r ρ ) if ρ < 0, and for t ≥ 0 otherwise. 2.) v is not light-like, g(v, v) = 0, i.e. β is a space-like or time-like geodesic. Then we set g(β(t),β(t)) = g(v, v) =: ±L 2 with L > 0. The equations (5.3) and (5.4) become 1.) v is light-like, g(v, v) = 0, i.e. β0 =r(t) ∓ r(t)ḟ (t) 2 L 2 , 0 = 2ṙ(t)ḟ (t) + r(t)f (t). The solutions of these equations are the following r ± (t) = (ρt + r) 2 ± L 2 r 2 t 2 , f ± (t) = 1 L arctan ± Lrt ρt + r , in which we have introduced the notation arctan + := arctan and arctan − := artanh. Obviously r + is defined for all t ∈ R whereas f + is defined for t ∈ [0, − r ρ ) if ρ < 0, and for t ≥ 0 otherwise. The functions r − and f − are defined on an interval [0, T ), where T is the first positive zero of the polynomial ((Lr − ρ)t − r)((L + rρ)t + r) or T = ∞ if the polynomial has no positive zero. More explicitly, T = r Lr−ρ if ρ < Lr and T = ∞ if Lr ≤ ρ. Cones over compact complete manifolds Here we generalise the proof of Gallot [Gal79] for metric cones over compact and geodesically complete pseudo-Riemannian manifolds. We obtain the following result. Proof. In the Riemannian case, the values of the function α defined in 4.2 are trivially restricted to the interval [0, 1], since α =ĝ(X 1 , X 1 ) ≥ 0 and 1 − α =ĝ(X 2 , X 2 ) ≥ 0. We shall now establish the same result for compact complete pseudo-Riemannian manifolds (M, g). Example 3.1 shows that completeness does not suffice. where c 1 is a constant. Since α(γ(0)) < 0, we see that c 1 < 0. If c 1 ≤ −1, then for all s > 0 we have γ(s) ∈ U x and α(γ(s)) tends to −∞ as s tends to +∞. If c 1 > −1, then for all s < 0 we have γ(s) ∈ U x and α(γ(s)) tends to −∞ as s tends to −∞. Since M is compact, we get a contradiction. The case α(x) > 1 is similar. Now we can prove the theorem completely analogously to Gallot by verifying the same lemmas as in his proof. Lemma 6.2. Let Γ(t) = (r(t), γ(t)) be a geodesic in ( M , g). Then the vector field along Γ defined by H(t) := r(Γ(t))∂ r (Γ(t)) − tΓ(t) is parallel along Γ(t). Proof. The lemma follows directly from (3.1): ∇Γ (t) H(t) = ∇Γ (t) r(t)∂ r − tΓ(t) =ṙ(t)∂ r + r(t) ∇Γ (t) ∂ r −Γ(t) − t ∇Γ (t)Γ (t) =0 =ṙ(t)∂ r + r(t) ∇γ (t) ∂ r = 1 r(t)γ (t) −Γ(t) =ṙ(t)∂r +γ(t) = 0, where r(t) := r(Γ(t)). For each point q ∈ M we denote by M 1 q and M 2 q the integral manifolds of the distributions V 1 and V 2 passing through the point q. For i = 1, 2 we define the following subsets of M : C i := p ∈ M \| ∂ r (p) ∈ V i . Then we can prove the following lemma. Lemma 6.3. Let p 1 ∈ C 1 (respectively, p 2 ∈ C 2 ). Then M 2 p 1 (respectively, M 1 p 2 ) is totally geodesic and flat. Proof. The leaves of the foliations induced by V 1 and V 2 are totally geodesic, since both distributions are parallel. It suffices to show that M 1 p 2 is flat. Consider a geodesic Γ of M 1 p 2 starting at p 2 = (r, x). Then the vector field along this geodesic H(t) defined as in Lemma 6.2 is parallel. We have H(0) = r∂ r ∈ V 2 andΓ(0) ∈ V 1 which implies H(t) = r(t)∂ r − tΓ(t) ∈ (V 2 ) Γ(t) anḋ Γ(t) ∈ (V 1 ) Γ(t) , as the distributions V i are invariant under parallel transport. Since ∂ r R = 0, we have R(., .)H(t) = −t R(., .)Γ(t). Since R(., .) are elements of the holonomy algebra leaving V 1 and V 2 invariant this implies that R(., .)Γ(t) = 0. From this we see that the Jacobi fields along Γ are those of a flat manifold, which implies that M 1 p 2 is flat. Recall that we have a dense open subset U = {x ∈ M \|α(x) = 0, 1} ⊂ M . Lemma 6.4. Any point p ∈ U has a flat neighbourhood. Proof. Fix a point p ∈ U . Note that for i = 1, 2 we have C i ∩ U = ∅. Consider the geodesic Γ(t) starting at p and satisfying the initial conditionΓ(0) = −r(p)X 1 (p). Let H(t) be the vector field along Γ as in Lemma 6.2. We claim that if the geodesic Γ(t) exists for t = 1, then Γ(1) ∈ C 2 . Indeed, suppose that Γ(t) exists for t = 1. Denote by τ : T p M → T Γ(1) M the parallel displacement along Γ(t). Since H(1) = r(Γ(1))∂ r (Γ(1)) −Γ(1), we have r(Γ(1))∂ r (Γ(1)) = H(1) +Γ(1). From Lemma 6.2 and the fact that Γ(t) is a geodesic it follows that r(Γ(1))∂ r (Γ(1)) = τ (H(0))+τ (Γ(0)) = τ (r(p)∂ r (p)−r(p)X 1 (p)) = r(p)τ (X 2 (p)) ∈ V 2 (Γ(1)). This shows that Γ(1) ∈ C 2 . Now we prove that the geodesic Γ(t) exists for t = 1. We can apply the results of the previous section. In the notations of the previous section we have v = −r(p)X(p) and ρ = −r(p)α(p). Since 0 < α(p) < 1, we have 0 < L 2 = g(v, v) = α(p) − α 2 (p) and r − \|ρ\| > 0. Then the function r(t) defining the geodesic Γ(t) is defined on R. The other defining function f (t) is given by f (t) = 1 L arctan Lr(p)t ρt + r(p) = 1 L arctan Lt 1 − α(p)t . We see that f is defined for t ∈ [0, 1] as α(p) < 1. Thus the geodesic Γ(t) is defined for t ∈ [0, 1]. Since the integral manifolds of the distribution V 1 are totally geodesic andΓ(0) ∈ V 1 (p), we have M 1 p = M 1 Γ(1) . From Lemma 6.3 it follows that M 1 p is flat. Similarly we show that M 2 p is also flat. Hence, any point p has a flat neighbourhood. From Lemma 6.4 it follows that the dense subset U ⊂ M is flat. Thus M is flat and (M, g) has constant sectional curvature 1. This finishes the proof of Theorem 6.1. Corollary 2. If (M, g) is a simply connected compact and complete indefinite pseudo-Riemannian manifold. Then the holonomy algebra of the cone ( M , g) is indecomposable. Proof. This follows from the fact that simply connected indefinite pseudo-Riemannian manifolds of constant curvature are never compact. (2) a pseudo-Riemannian manifold M 2 = R + × N 1 × N 2 with the metric −ds 2 + cosh 2 (s)g 1 + sinh 2 (s)g 2 , where (N 1 , g 1 ) and (N 2 , g 2 ) are pseudo-Riemannian manifolds and (N 2 , g 2 ) has constant sectional curvature −1 or dim N 2 ≤ 1. Cones over complete manifolds Moreover, the cone M 2 is isometric to the open subset {r 1 > r 2 } in the product of the space-like cone (R + × N 1 , dr 2 + r 2 g 1 ) over (N 1 , g 1 ) and the time-like cone (R + × N 2 , −dr 2 + r 2 g 2 ) over (N 2 , g 2 ). Proof. By going over to the universal covering, if necessary, we can assume that (M, g) is simply connected. Then M is simply connected and decomposable. Let ∪ i∈I W i = U 1 be the representation of the open subset U 1 ⊂ M as the union of disjoint connected open subsets. For each W i we have two possibilities: (1.) 0 < α < 1 on W i ; (2.) α < 0 or α > 1 on W i . Consider these two cases. (1.) Suppose that 0 < α < 1 on W i . Similarly to the proof of Theorem 6.1 we can show that the cone over W i is flat. (2.) Suppose that α > 1 on W i . As in the proof of Lemma 6.1 we can show that α(W i ) = (1, +∞). To proceed we need the following statement which is a generalisation of an argument used in the proof of Theorem 27 in [Boh03]. Proof. First we notice that 0 = g(Z, Z) = (f 2 · h) • α. As M is connected, we may assume that g(Z, Z) > 0 and thus h > 0. As the sign of f plays no role in what follows we also assume that f > 0. Furthermore, α satisfies the following differential equation on M , X(α) = 1 f • α Z(α) = (f • α)g(X, X) = (f · h) • α. (7.1) Let φ : F c × I c → M , (p, t) → φ t (p) be the flow of the vector field X. The proof is now based on the observation that if p and q are in the same level set of α, then α(φ t (p)) = α(φ s (q)) ⇐⇒ t = s (7.2) for all t, s ∈ R. To verify this, for each p ∈ F c we consider the real function ϕ c : I c ∋ t → α(φ t (p)) ∈ Im α which satisfies the ordinary differential equation ϕ ′ c (t) = dα φt(p) (X(φ t (p)) = f (ϕ c (t)) · h(ϕ c (t)) > 0. (7.3) Hence, for each p ∈ F c the function ϕ c (t) = α(φ t (p)) is subject to the ordinary differential equation (7.3) with the same initial condition ϕ c (0) = α (φ 0 (p)) = α (φ 0 (q)) = c. Uniqueness of the solution implies that α (φ t (p)) = α ((φ t (q)) for all t and all q ∈ F c . This proves (⇐=) of (7.2), and shows that ϕ c does not depend on the starting point p ∈ F c . Having this, (7.3) also shows that ϕ c is strictly monotone, and thus injective which gives (=⇒) of (7.2). (7.2) shows that the flow φ of X sends one level set F c of α to another one F d , i.e. α(φ t (p)) = α(φ t (q)) for all t ∈ I c and p, q in the same level set F c . Next, we show that two level sets that are joint by an integral curve of X are diffeomorphic. In fact, if p ∈ F c and q = φ t (p), φ t is a local diffeomorphism between F c and F d . (=⇒) of (7.2) implies that φ t \| fc is injective. To verify that it is surjective we notice that φ −t \| F d is also an injective local diffeomorphism. Hence, φ −t • φ t = id Fc , which implies that φ t : F c → F d is a global diffeomorphism. Finally, we show that for two level sets there is at least one flow line connecting them. To this end, we set φ(F c ) := {φ t (p) \| p ∈ F c , t ∈ I c } and write M = c∈Im α φ(F c ). We have seen that, if F c and F d are connected by an integral curve, then they are diffeomorphic under φ t . But the maximality of I c and I d implies that φ(F c ) = φ(F d ). If, on the other hand, F c and F d are not joined by an integral curve then, by maximality of I c and I d , a common point of φ(F c ) and φ(F d ) would lie on an integral curve joining F c and F d , i.e. φ(F c ) ∩ φ(F d ) = ∅. In the latter case M can be written as disjoint union of open sets φ(F c ) which is not possible as M was supposed to be connected. Hence, each integral curve meets each level set once, they are all diffeomorphic, i.e. M is diffeomorphic to I c × F c . But this implies that M is diffeomorphic to (Im α) × F where F is a level set of α. Resuming the proof of the theorem we notice that the vector field X =X √ α 2 −α is geodesic and proportional to the gradient of α. Since (M, g) is complete and X is geodesic its integral curves are defined for all t. As in the proof of Proposition 7.1 one shows that level sets are mapped onto level sets under the flow of X. This shows that ( * ) in Proposition 7.1 is satisfied for the vector field X\| W i ∈ Γ(T W i ) on the manifold W i : for F c the interval I c is limited by the real number a for which φ a (F c ) ⊂ F 1 . Hence, we can apply Proposition 7.1 to the manifolds W i and the vector field X\| W i ∈ Γ(T W i ). Combining the result with the proof of Case (2.) from Theorem 4.2 yields a decomposition W i = R + × N 1 × N 2 . For the metric g\| W i we obtain that g\| W i = −ds 2 + cosh 2 (s)g 1 + sinh 2 (s)g 2 , where (N 1 , g 1 ) and (N 2 , g 2 ) are pseudo-Riemannian manifolds. The fact that the cone ( W i , g\| W i ) is isometric to an open subset of the product of a space-like cone over (N 1 , g 1 ) and of a time-like cone over (N 2 , g 2 ) is shown by Example 3.4. By a variation of the proof of Theorem 6.1 we will show now that the time-like cone over the manifold (N 2 , g 2 ) is flat and we will explain why it is not the case for the manifold (N 1 , g 1 ). Fix a point p ∈ W i . Consider the geodesics Γ 1 (t) and Γ 2 (t) starting at p and satisfying the initial conditionsΓ 1 (0) = −r(p)X 1 (p) andΓ 2 (0) = −r(p)X 2 (p). Now we prove that the geodesic Γ 2 (t) exists for t = 1 and the geodesic Γ 1 (t) does not exist for t = 1. We can apply the results of section 5. For Γ 1 we have v 1 = −r(p)X(p) and ρ 1 = −r(p)α(p); for Γ 2 we have v 2 = r(p)X(p) and ρ 2 = r(p)(α(p) − 1). From Section 5 it follows that the functions r 1 (t) and f 1 (t) defining the geodesic Γ 1 (t) are defined on the interval 0, 1 √ α 2 (p)−α(p)+α(p) ⊂ [0, 1). The functions r 2 (t) and f 2 (t) defining the geodesic Γ 2 (t) are defined on the interval 0, 1 √ α 2 (p)−α(p)−α(p)+1 ⊃ [0, 1]. Thus the geodesic Γ 2 (t) is defined for t ∈ [0, 1] and the geodesic Γ 1 (t) is not defined for all t ∈ [0, 1]. As in the proof of Theorem 6.1 we get that the manifold M 2 p is flat. This means that the induced connection on the distribution V 2 \| W i is flat and the time-like cone over the manifold (N 2 , g 2 ) is flat, i.e. (N 2 , g 2 ) has constant sectional curvature −1 or dim N 2 ≤ 1. Note that as in Example 3.1 it can be α > 1 on M , then C 2 = ∅ and the induced connection on V 1 need not be flat. The case α\| W i < 0 is similar, with the roles of V 1 and V 2 interchanged. Lemma 1 (cf. Thm. 14.4 [Kra07]). Let E be a pseudo-Euclidian vector space and h ⊂ so(E) an indecomposable Lie subalgebra. It E admits a non-trivial h-invariant decomposition E = V ⊕ W then it admits an h-invariant decomposition E = V ′ ⊕ W ′ into a sum of totally isotropic subspaces. By the lemma, we can assume that V, W are totally isotropic of the same dimension, which implies that the metric has neutral signature. In this section we use a similar approach as in the previous sections but with different structures coming up. These structures are related to a para-complex structure, and to a para-Sasakian structure. We recall the basic definitions given in [CMMS04] and [CLS06]. Definition 8.1. 1. Let V be a real finite dimensional vector space. A para-complex structure on V is an endomorphism J ∈ End(V ), such that J 2 = Id and the two eigenspaces V ± := ker(Id ∓ J) of J have the same dimension. The pair (V, J) is called a para-complex vector space. 2. Let V be a distribution on a manifold M . An almost para-complex structure on V is a field J ∈ Γ(End V) of paracomplex structures in V. It is called integrable or paracomplex structure on V if the eigen-distributions V ± := ker(Id ∓ J) are involutive. 3. A manifold M endowed with a para-complex structure on T M is called a paracomplex manifold. Similar to the complex case, the integrability of J is equivalent to the vanishing of the Nijenhuis tensor N J defined by N J (X, Y ) := J ([JX, Y ] + [X, JY ]) − [X, Y ] − [JX, JY ], X, Y ∈ ΓV. (8.1) Definition 8.2. 1. Let (V, J) be a para-complex vector space equipped with a scalar product g. (V, J, g) is called para-hermitian vector space if J is an anti-isometry for g, i.e. J * g := g(J., J.) = −g. (8.2) 2. A (almost) para-hermitian manifold (M, J, g) is an (almost) para-complex manifold (M, J) endowed with a pseudo-Riemannian metric g such that J * g = −g. The two-form ω := g(J, ) = −g(, J) is called the para-Kähler form of (M, J, g). (M, J, g) is a para-hermitian manifold (M, J, g) such that J is parallel with respect to the Levi-Civita-connection ∇ of g. A para-Kähler manifold As in the complex case, the condition ∇J is equivalent to N J = 0 and dω = 0. In contrary to the complex case, a 2n-dimensional para-hermitian manifold has to be of neutral signature (n, n). Note that eigen-distributions V ± of J are totally isotropic and auto-orthogonal, i.e. (V ± ) ⊥ = V ± . For a para-Kähler manifold the condition ∇J = 0 means that the ±1-eigen-distributions V ± are parallel. We get Proposition 8.1. A pseudo-Riemannian manifold (M, g) is a para-Kähler manifold if and only if the holonomy group preserves a decomposition of the tangent space into a direct sum of two totally isotropic subspaces. In the following we will show that metric cones with para-Kähler structure are precisely cones over para-Sasakian manifolds. Definition 8.3. A para-Sasakian manifold is a pseudo-Riemannian manifold (M, g) of signature (n + 1, n), where n + 1 is the number of time-like dimensions, endowed with a time-like geodesic unit Killing vector field T such that ∇T defines an integrable paracomplex structure J = ∇T \| E : E → E on E = T ⊥ . The pair (g, T ) is called a para-Sasakian structure. Note that the eigen-distributions E ± of J = ∇T \| E are totally isotropic and J is an anti-isometry of g\| E . Indeed, using the condition that T is a Killing field for X ± and Y ± in Γ(E ± ), we get 0 = (L T g)(X ± , Y ± ) = g(∇ X ± T, Y ± ) + g(∇ Y ± T, X ± ) = 2g(X ± , Y ± ). (8.3) A para-Sasakian manifold carries several other structures. First of all it has contact structure given by the contact form θ := g(T, .). Indeed, for dθ we get that dθ(X + , X − ) = −g(T, [X + , X − ]) = 2g(X + , X − ), (8.4) with X ± ∈ Γ(E ± ) . Since E ± are dual to each other, this implies that θ ∧ dθ n = 0, hence θ is a contact form. The Reeb vector field of this contact structure is T , because dθ(T, X) = −g(T, [T, X]) = −g(T, ∇ T X − ∇ X T ) = 0. (8.5) It also admits a para-CR structure (see for example [AMT05]), which is defined on a (2n + 1)-dimensional manifold M as an n-dimensional subbundle E of T M together with a para-complex structure J on E. For a para-Sasakian manifold this para-CR structure is given by the one-form θ. From (8.4) and from the assumption that E ± are involutive we see that the Levi-form L θ ∈ Γ(S 2 E) of this para-CR structure, defined by L θ (X, Y ) := dθ\| E (X, JY ), is given by the metric, L θ (X, Y ) = dθ(X, JY ) = −2g(X, Y ) (8.6) and is thus non-degenerate. Hence for a para-Sasakian manifold, the metric g can be expressed in terms of the contact form θ and its Levi form: g = −θ 2 − 1 2 L θ . (8.7) This is in analogy to strictly pseudo-convex pseudo-Hermitian structures (see for example [Bau99b] and [Bau99a]). Although the definition of a para-Sasakian structure seems rather weak, it entails the following properties. Lemma 8.1. Let (M, g, T ) be a para-Sasakian manifold with E = T ⊥ and eigen-distributions E ± . Then: 1. E ± are auto-parallel and N J \| E ± = ∇J\| E ± = 0. For X ± ∈ Γ(E ± ) it holds that ∇ X − X + = −g(X + , X − )T mod E + and ∇ X + X − = g(X + , X − )T mod E − . 3. For X ± ∈ Γ(E ± ) it holds [T, X ± ] ⊂ Γ(E ± ). Proof. 1. Let X ± and Y ± be in E ± . (8.3) implies that g(∇ X ± Y ± , T ) = −g(X ± , Y ± ) = 0, which ensures that ∇ X ± Y ± ∈ E. Now, E ± are integrable, which implies on the one hand the relation for N J , and gives on the other hand, using the Koszul formula, that g(∇ X ± Y ± , Z ± ) = 0 for all Z ± ∈ E ± . Hence, E ± are auto-parallel, which yields the relation for ∇J. 2. First of all we have that g(∇ X − X + , T ) = −g(X + , ∇ X − T ) = g(X + , X − ). Next we show that ∇ X − X + is orthogonal to E + . In the following equations g(Y − i , Y + j ) = δ ij and the lower indices +, −, 0 denote the corresponding component in E ± and RT : 2g(∇ Y − i Y + j , Y + k ) Koszul = g [Y − i , Y + j ], Y + k + g [Y + k , Y + j ], Y − i + g [Y + k , Y − i ], Y + j = g [Y − i , Y + j ] − , Y + k + g [Y + k , Y + j ], Y − i + g [Y + k , Y − i ] − , Y + j (8.4) = − 1 2 g T, Y + k , [Y − i , Y + j ] − + [Y + k , Y + j ], Y − i + Y + j , [Y + k , Y − i ] − = − 1 2 g T, Y + k , [Y − i , Y + j ] − + Y − i , [Y + j , Y + k ], + Y + j , [Y + k , Y − i ] − = − 1 2 g T, Y + k , [Y − i , Y + j ] + Y − i , [Y + j , Y + k ], + Y + j , [Y + k , Y − i ] =0 Jacobi identity + 1 2 g T, Y + k , [Y − i , Y + j ] + ∈E + + Y + j , [Y + k , Y − i ] + ∈E + =0 + 1 2 g T, Y + k , [Y − i , Y + j ] 0 + Y + j , [Y + k , Y − i ] 0 = − 1 2 g T, Y + k , g(T, [Y − i , Y + j ]) =−2δ ij T + Y + j , g(T, [Y + k , Y − i ]) =2δ ik T = g T, δ ij Y + k , T − δ ik Y + j , T = 0 This implies that ∇ X − X + ∈ RT ⊕ E + , which proves the second statement. The last point follows from the general fact: If T is a Killing vector field, and θ = g(T, .), then L T ∇θ = 0. (8.8) Indeed, the Killing equation for T is equivalent to ∇θ = 1 2 dθ ∈ Ω 2 M . This implies for arbitrary tangent vectors X and Y using the skew symmetry of ∇θ that 0 = 1 2 ddθ(T, X, Y ) = T (∇θ(X, Y )) − X (∇θ(T, Y )) + Y (∇θ(T, X)) −∇θ([T, X], Y ) + ∇θ([T, Y ], X) − ∇θ([X, Y ], T ) = (L T ∇θ) (X, Y ) − X (∇θ(T, Y )) + Y (∇θ(T, X)) − ∇θ([X, Y ], T ) = (L T ∇θ) (X, Y ) − X (θ(∇ Y T )) + Y (θ(∇ X T )) + θ(∇ [X,Y ] T ) = (L T ∇θ) (X, Y ) − θ(R(X, Y )T ) = (L T ∇θ) (X, Y ). This can easily be applied to our situation, where we have that ∇θ = g(J., .). For X ± ∈ E ± and Y ∈ T M , (8.8) implies that 0 = (L T ∇θ)(X ± , Y ) = T (g(JX ± , Y )) − g(J([T, X ± ]), Y ) − g(JX ± , [T, Y ]) = ± (L T g)(X ± , Y ) =0 ±g([T, X ± ], Y ) − g(J([T, X ± ]), Y ), which gives [T, X ± ] ∈ E ± . Using these properties we obtain a description of para-Sasakian manifolds which might look more familiar. φ 2 = id + g(., T )T (8.9) (∇ U φ)(V ) = −g(U, V )T + g(V, T )U, ∀U, V ∈ T M (8.10) Proof. First, let (M, g) be a pseudo-Riemannian manifold of signature (n + 1, n) with a time-like geodesic unit Killing vector field T satisfying (8.9) and (8.10). The fact that T is geodesic means that φT = 0 and implies that φ preserves E := T ⊥ . Putting J := φ\| E , the equation (8.9) shows that J 2 = id E , i.e. φ is a skew-symmetric involution and therefore a para-complex structure. Finally (8.10) ensures that J = φ\| E is integrable because ∇J\| E ± = 0. For the converse statement we assume that (M, g, T ) is a para-Sasakian manifold. Setting φ := ∇T we get φ 2 \| E = J 2 = id and φ 2 (T ) = 0 which gives (8.9). We have to check (8.10): For U = V = T both sides of (8.10) are zero. For U = X ∈ T ⊥ and V = T the right hand side is given by g(T, T )X = −X, but also the left hand side which is (∇ X φ)(T ) = −φ(∇ X T ) = −φ 2 (X) = −X. For U = T and V = X ± ∈ E ± the right hand side vanishes, and the left hand side as well because of [T, E ± ] ⊂ E ± : (∇ T φ)(X ± ) = ∇ T JX ± − J(∇ T X ± ) = ±[T, X ± ] + X ± − J([T, X ± ]) − J 2 (X ± ) = ±[T, X ± ] − J([T, X ± ]) = 0. For U and V both in E ± both sides vanish because of the integrability of the para-complex structure. For U = X + ∈ E + and V = X − ∈ E − the right hand side of (8.10) is equal to −g(X + , X − )T and the left hand side is given by Proof. First assume that (M, g, T ) is a para-Sasakian manifold with para-complex structure J = ∇T on E := T ⊥ , which splits into eigen-distributions E ± . The para-complex structure on the metric cone ( M , g) is defined by (∇ X + φ)X − = −∇ X + X − − φ(∇ X + X − ) = −g(X + ,J := ∇T. Because of the formula for the covariant derivative of the cone, J is given by J(∂ r ) = ∇ ∂r T = 1 r T J(T ) = ∇ T T = ∇ T T − rg(T, T )∂ r = r∂ r J(X) = ∇ X T = ∇ X T − rg(X, T )∂ r = J(X), X ∈ E, which implies that J is an almost para-complex structure, and also an almost parahermitian structure with respect to the cone metric g. The eigen-distributions of J are given by V ± = R(r∂ r ± T ) ⊕ E ± . They are involutive because the distributions E ± are involutive and [r∂ r ± T, X ± ] = ± [T, X ± ] ∈ E ± for X ± ∈ Γ(E ± ). Hence, ∇T defines a para-Kähler structure on the cone. Now assume that the cone ( M , g) over (M, g) is a para-Kähler manifold with paracomplex structure J. We consider the decomposition T M = V + ⊕ V − into the totally isotropic eigen-distributions of J. Then the radial vector field decomposes as follows, ∂ r = ρ∂ r + X :=X + ∈V + + (1 − ρ)∂ r − X :=X − ∈V − , (8.11) where X ∈ Γ( M ) is a global vector field tangent to M . This vector field defines a para-Sasakian structure. First of all, we prove Proof. As V + and V − are totally isotropic, we get for X defined in (8.11) 0 = ρ 2 + r 2 g(X, X) = (1 − ρ) 2 + r 2 g(X, X). This implies ρ = 1 2 and g(X, X) = − 1 4r 2 . By the holonomy invariance of the distributions V ± , stated in Proposition 8.1, we get V + ∋ ∇ ∂r 1 2 ∂ r + X = ∇ ∂r X and similar V − ∋ ∇ ∂r 1 2 ∂ r − X = − ∇ ∂r X, which implies ∇ ∂r X = 0. Hence, [∂ r , X] = − 1 r X, and thus X = 1 2r T where T is a vector field on M with g(T, T ) = −1. It follows that T is a geodesic vector field because: V ± ∋ ∇ T (r∂ r ± T ) = T ± (∇ T T + r∂ r ) = ± (r∂ r ± T ) =2rX ± ∈V ± ±∇ T T, i.e. ∇ T T ∈ V + ∩V m = {0}. Hence, the vector fields X ± , defined in (8.11) are given by X ± = 1 2 ∂ r ± 1 r T for T a time-like geodesic unit vector field on M . We consider now the orthogonal complement of X ∓ in V ± . Lemma 8. 3. Let E ± := {Y ∈ V ± \| g(Y, X ∓ ) = 0} ⊂ V ± be the orthogonal complement of X ∓ in V ± . Then E ± are tangential to M , orthogonal to T , totally isotropic, and E := E + ⊕ E − is the orthogonal complement of T in T M . Proof. As V + is totally isotropic any U = a∂ r + Y ∈ E + (Y ∈ T M ) is orthogonal to X + and X − , which is equivalent to 0 = a ± rg(Y, T ). Hence, a = g(Y, T ) = 0. The same holds for U ∈ E − . Both are totally isotropic with respect to g as V ± are totally isotropic with respect to g. This gives the following decomposition of the tangent bundle into three non-degenerate distributions T M = R · ∂ r ⊕ ⊥ R · T ⊕ ⊥ E + ⊕ E − , where E + and E − are totally isotropic. Lemma 8.4. The vector field T satisfies ∇T \| E ± = ±id. Proof. The holonomy invariance of V ± implies that ∇X ± : T M → V ± . But the formulae for ∇ imply that ∇X ± \| T M = 1 2r id T M ± ∇T . Applying this to E ± gives that ∇T leaves E + and E − invariant. Hence, ∇X ± is zero on E ∓ , and thus ∇T \| E ∓ = ∓id. As T is a vector field on M , its orthogonal complement E does not depend on the radial coordinate r and defines a distributions on M. The same holds for E ± because ∇T is an endomorphism on E which does not depend on r and is given as ±id on E ± , which are also denoted by E and E ± . Thus, ∇T \| E± = ∇T \| E ± = ±id defines an almost paracomplex structure J on E = T ⊥ ⊂ T M . As its eigen-spaces E ± are totally isotropic, J is an anti-isometry, g(J., J.) = −g. This implies that T is a Killing vector field: Proof. For Y + and Z + in E + by the holonomy invariance of V + , it is [Y + , Z + ] ∈ V + . Hence, it suffices to show that [Y + , Z + ] ⊥ X − . But this is true because ∇X − \| E + = 0 (see the proof of Lemma 8.4): g([Y + , Z + ], X − ) = − g(Z + , ∇ Y + X − ) + g(Y + , ∇ Z + X − ) = 0. We get the same for E − . Summarising we get that T is a geodesic, time-like unit Killing vector field, and ∇T is an integrable para-complex structure on T ⊥ . Hence, (M, g, T ) is a para-Sasakian manifold. Examples of para-Sasakian manifolds Now we construct a family of para-Sasakian manifolds (M, g, T ) of positive non constant curvature which implies that the associated coneM is not flat. We will describe (g, T ) in terms of coordinates. Let (M, g, T ) be a para-Sasaki manifold. Consider a filtration of T M by integrable distributions E + ⊂ R · T ⊕ E ⊂ T M . The Frobenius Theorem implies existence of local coordinates on M adapted to this filtration and a hypersurface which contains the leaves of E + and is transversal to T . We choose local coordinates on this hypersurface adapted to E + . Since T is a Killing vector field, its flow can be used to extend these coordinates to coordinates (t, x 1 , . . . , x n , x n+1 , . . . , x 2n ) on some open subset U ⊂ M such that ∂ ∂t = T \| U and ∂ ∂x i ∈ Γ(E + \| U ). Obviously, ith respect to these coordinates the metric g is given by the matrix of the form   −1 0 u t 0 0 H t u H G   . Here u = (u 1 , . . . , u n ) ∈ C ∞ (U, R n ), H a non-degenerate matrix of real functions on U and G a symmetric matrix of real functions on U . We choose a basis Y − i of of vector fields on E − which such that 0 = g(T, Y − i ), δ ij = g( ∂ ∂x i , Y − j ), and 0 = g(Y − i , Y − j ) , First of all, these orthogonality relations imply that Y − i := H ij u j · T + b ij ∂ ∂x j + H ij ∂ ∂x j+n where H ij is the inverse matrix to H ij and b ij + b ji = −H ik (u k u l + G kl ) H jl . (8.12) As T is a Killing vector field and Y − i ∈ Γ(E − ), we get that [T, Y − i ] = 0 which implies that H, u, and b do not depend on t. Now we consider the condition (8.6) which can be written as g\| E = − 1 2 L θ or −2δ ij = g([ ∂ ∂x i , Y − j ], T ) = − ∂ ∂x i (H jk u k ) + ∂ ∂x i (H jk )u k = −H jk ∂ ∂x i (u k ). It implies H ij = 1 2 ∂ ∂x j (u i ). (8.13) Then we evaluate the condition that ∇T acts as −id on E − . Note that the inverse matrix of the metric is given by   −1 v t 0 v F H −1 0 (H t ) −1 0   , where v = H −1 u and F ij = b ij + b ji . We calculate ∇ Y − i T = b ij ∂ ∂x j + H ij ∇ ∂ ∂x j+n T = −v i T + −H ij ∂ ∂x j+n + b ij − F ji + 1 2 H ik H jl T (G kl ) + ∂ ∂x k+n (u l ) − ∂ ∂x l+n (u k ) ∂ ∂x j . Hence, ∇ Y − i T = −Y − i is equivalent to 2b ij = F ji − 1 2 H ik H jl T (G kl ) + ∂ ∂x k+n (u l ) − ∂ ∂x l+n (u k ) . which gives 2(b ij + b ji ) = 2F ji − H ik H jl T (G kl ). This implies that also G does not depend on t, but together with (8.12) it also gives a formula for b ij , namely b ij = − 1 2 H ik H jl u k u l + G kl + 1 2 ∂ ∂x k+n (u l ) − ∂ ∂x l+n (u k ) . (8.14) Finally, we evaluate the integrability of E − := span(Y − i ) n i=1 . We write this condition as ∇ [Y − i ,Y − j ] T = −[Y − i , Y − j ] and obtain after a lengthy but straightforward calculation that this is equivalent to 0 = Λ ij b iq ∂ ∂x q (b jp ) − H lr b rp ∂ ∂x q (H jl ) + H iq ∂ ∂x q+n (b jp ) − H lr b rp ∂ ∂x q+n (H jl ) , in which Λ ij denotes the skew symmetrization with respect to the indices i and j. Although we do not find the general solution of this equation we will construct solutions with b ij ≡ 0. We make the following ansatz. We assume that ∂ ∂x i+n (u j ) = 0, and set G ij := −u i · u j . This implies that b ij = 0 which gives that Y − i = H ij u j · T + ∂ ∂x j+n with H ij = 1 2 ∂ j u i , and ensures that E − = span(Y − i ) is the (−1)-eigen-space of ∇T and integrable. In fact, we get for the Levi-Civita connection of this metric: ∇ T T = 0 ∇ T ∂ i = ∂ i ∇ T ∂ i+n = −u i T − ∂ i+n , i.e. ∇ T Y − i = −Y − i ∇ ∂ i ∂ j = H kl ∂ i (H lj )∂ k ∇ ∂ i+n ∂ j+n = 2u i u j T + u i ∂ j+n + u j ∂ i+n ∇ ∂ i ∂ j+n = −H ij T − u j ∂ i , which implies ∇ ∂ i Y − j = δ ij T + H kl ∂ i (H jk )Y − l ∇ ∂ i+n Y − j = u i Y − j ∇ Y − i Y − j = 0 Now we check that the curvature of the metric is not constant. Calculating the curvature and denoting Y + i by ∂ i , we get R(T, Y ± i ) : T → Y ± i Y ∓ i → T R(Y ± i , Y ± j ) : Y ∓ k → δ jk Y ± i − δ ik Y ± j R(Y ± i , Y ∓ j ) : Y k ± → −δ jk Y ± i − 2δ ij Y ± k Y k ∓ → δ ik Y ∓ i + 2δ ij Y ∓ k and the remaining terms being zero. The last terms show that (M, g) does not have constant sectional curvature. This can also be seen by calculating the derivatives of the curvature which are zero apart from one term: (∇ ∂ i R) (T, Y − j , ∂ k , Y − l ) = − R(∂ i , Y − j , ∂ k , Y − l ) =−δ kj δ il −2δ ij δ kl + R(T, Y − j , ∂ k , δ il T ) =−δ il δ kj = 2 (δ kj δ il + δ ij δ kl ) . Hence, (M, g) is not locally symmetric. Note that the curvature R of this metric is of the form R = R 1 − 2ω ⊗ J, where R 1 is the curvature of a space of constant curvature, J the para-complex structure and w = g(., J.) is the para-Kähler form. Altogether we have proven: Proposition 8. 3. Let (t, x 1 , . . . , x n , x n+1 , . . . , x 2n ) be coordinates on R 2n+1 and consider the metric g given by where R 1 is the curvature tensor of a space of constant curvature 1 in dimension 2n + 1, J the para-complex structure and w = g(J., .) is the para-Kähler form. In particular, the space-like cone over (R 2n+1 , g) is para-Kähler and non-flat, i.e. its holonomy representation is non-trivial and decomposes into two totally isotropic invariant subspaces. Remark 8.1. 1. It is obvious that the Abelian group R n+1 acts isometrically on (R 2n+1 , g) via R n+1 ∋ (c, c 1 , . . . , c n ) :   t x i x i+n   →   t + c x i + c i x i+n   As these isometries also fix the para-Sasaki vector field T = ∂ ∂t , they are automorphisms of the para-Sasaki structure (g, T ). Hence, we can consider a lattice Γ ⊂ R n+1 and compactify (R 2n+1 , g) along these directions in order to obtain a para-Sasakian structure on R 2n+1 /Γ = T n+1 × R n , where T n+1 denotes the (n + 1)-torus. We do not know under which conditions on the u i 's there are more automorphisms, and if one can find enough in order to compactify the manifold by this method. 2. The manifolds obtained in this way are curvature homogeneous. More examples of para-Kähler cones are given in [CLS06] by conical special para-Kähler manifolds defined by a holomorphic prepotential of homogeneity 2. Further results on the holonomy of para-Kähler manifolds can be found in [BBI97]. (v) the tensors ∇T α define a para-hyper-complex structure on E := span{T 1 , T 2 , T 3 } ⊥ . (Here we are using that the conditions (ii-iii) imply ∇ E T α ⊂ E.) The assumption that h preserves two complementary isotropic subspaces T ± ⊂ T p M and a skew-symmetric complex structure J ∈ End T p M with JT + = T − can be now reformulated by saying that M locally admits a para-hyper-Kähler structure ( g, J 1 , J 2 , J 3 = J 1 J 2 ), where J 1 \| T ± = ±Id and ( J 3 ) p = J. The corresponding geometry of the base manifold (M, g) is para-3-Sasakian: Theorem 8.2. Let (M, g) be a pseudo-Riemannian manifold. There is a one-to-one correspondence between para-3-Sasakian structures (M, g, T 1 , T 2 , T 3 ) on (M, g) and parahyper-Kähler structures ( M , g, J 1 , J 2 , J 3 = J 1 J 2 ) on the cone ( M , g). The correspondence is given by T α → J α := ∇T α . Proof. By Theorem 8.1 the para-Sasakian structures (g, T 1 ) and (g, T 2 ) induce two para-Kähler structures ( g, J 1 ) and ( g, J 2 ) on the space-like cone ( M , g). Similarly, the pseudo-Sasakian structure (g, T 3 ) induces a pseudo-Kähler structure ( g, J 3 ) on M . It suffices to show that J 1 J 2 = − J 2 J 1 = J 3 . We recall that the vector fields T α (considered as vector fields on M ) are related to T 0 := r∂ r by T α = J α T 0 . Using J α = ∇T α , we show that the conditions (iii-iv) in Definition 8.4 imply that the structures J α preserve the four-dimensional distribution H := span{T i \|i = 0, 1, 2, 3} and act as the standard para-hyper-complex structure on H. In fact, first it is clear that J α acts in the standard way on the plane P α spanned by T 0 and T α . Second the relations (iii-iv) easily imply that J α preserves the plane P ′ α = P ⊥ α ∩ H. Since J 2 α = ±Id, the action of J α on P ′ α is completely determined by: J 1 T 2 = ∇ T 2 T 1 = ∇ T 2 T 1 (iii) = T 3 J 2 T 1 = ∇ T 1 T 2 (iv) = −2T 3 + ∇ T 2 T 1 = −T 3 J 3 T 1 = ∇ T 1 T 3 (iv) = −2T 2 + ∇ T 3 T 1 = −2T 2 − g(∇ T 3 T 1 , T 2 )T 2 = −2T 2 + g(T 3 , ∇ T 2 T 1 )T 2 − T 2 . This shows that the endomorphisms J α act as the standard para-hyper-complex structure on H. Finally, the condition (v) in Definition 8.4 shows that the J α act also as a parahyper-complex structure on E = H ⊥ . g\| M 0 = dr 2 + r 2 (−ds 2 + e −2s g N ). Lorentzian cones Define the manifold M 1 = R + × R × N and extend the metric g\| M 0 to the metric g 1 on M 1 . Consider the diffeomorphism R + × R + → R + × R given by (x, y) → 2xy, ln 2x y . The inverse diffeomorphism has the form (r, s) → r 2 e s , re −s . We have ∂ x = e −s ∂ r + e −s r ∂ s , ∂ y = e s 2 ∂ r − e s 2r ∂ s . We get the decomposition M 1 = R + × R + × N, and the metric g 1 has the form g 1 = 2dxdy + y 2 g N . of the open subset U 1 ⊂ M as a union of disjoint connected open subsets. At each x ∈ W i we have g x (Z,Z) = 0. Hence for each W i we can use that arguments of the proof of Claim 2 of the theorem. Suppose that (M, g) is complete. As in proof of Claim 2 of the theorem we can show that U i = W i for each i ∈ I. From Claim 2 of Lemma 9.1 it follows that the vector fieldZ α is a geodesic vector field on U 1 . Let x ∈ U 1 and let γ(s) be the geodesic such that γ(0) = x andγ(s) =Z (γ(s)) α(γ(s)) if γ(s) ∈ U 1 . Along the set {γ(s)\|γ(s) ∈ U 1 } we have α(γ(s)) = e −s . Hence, γ(s) is defined for all s ∈ R, γ(R) ⊂ U 1 and α(γ(R)) = R + , i.e. (a, b) = R. For pseudo-Riemannian manifolds (M, g) the completeness assumption yields only the following generalisation of Gallot's result: Theorem 7.1. Let (M, g) be a complete pseudo-Riemannian manifold of dimension ≥ 2 with decomposable holonomy h of the cone M . Then there exists an open dense submanifold M ′ ⊂ M such that each connected component of M ′ is isometric to a pseudo-Riemannian manifold of the form (1) a pseudo-Riemannian manifold M 1 of constant sectional curvature 1 or If the manifold (M, g) is complete then each connected component of M ′ is isometric to a manifold of the form (1.1), where (N, g N ) is a positive definite Riemannian manifold and (a, b) = R. . 1 . 1The open subset U = {x\|α(x) = 0, 1} ⊂ M is dense. Theorem 4. 2 . 2Let (M, g) be a pseudo-Riemannian manifold and ( M = R + × M, g = dr 2 + r 2 g) the cone over M . Suppose that the holonomy group of ( M , g) admits a nondegenerate proper invariant subspace. Then there exists a dense open submanifold U 1 ⊂ M such that each point x ∈ U 1 has an open neighborhood W ⊂ U 1 that satisfies one of the following conditions is a light-like geodesic. Then the equations become 0 =r(t), 0 = 2 ρḟ (t) + (ρt + r)f (t), Theorem 6. 1 . 1Let (M, g) be a compact and complete pseudo-Riemannian manifold of dimension ≥ 2 with decomposable holonomy group H of the cone M . Then (M, g) has constant curvature 1 and the cone is flat. Lemma 6. 1 . 1Under the assumptions of Theorem 6.1, the function α on M satisfies 0 ≤ α ≤ 1.Proof.On the open dense subset U 1 ⊂ M we define the vector fieldX =X \|g(X,X)\| =X \|α 2 − α\| .From (4.4) and Lemma 4.3 it follows that ∇XX = 0, i.e.X is a geodesic vector field. Let x ∈ M and suppose that α(x) < 0. Denote by U x ⊂ U 1 the connected component of the set U 1 containing the point x. Since M is complete, we have a geodesic γ(s) such that γ(0) = x andγ(s) =X(γ(s)) if γ(s) ∈ U x . From Lemma 4.3 it follows thaṫ γ(s)α =X(γ(s))α = −2 α 2 − α for all s such that γ(s) ∈ U x . Hence along the curve {γ(s)\|γ(s) ∈Ū x } we have α(s) = (e −2s + c 1 ) 2 4e −2s c 1 , Theorem 7. 1 . 1Let (M, g) be a complete pseudo-Riemannian manifold of dimension ≥ 2 with decomposable holonomy h of the cone M . Then there exists an open dense submanifold M ′ ⊂ M such that each connected component of M ′ is isometric to a pseudo-Riemannian manifold of the form (1) a pseudo-Riemannian manifold M 1 of constant sectional curvature 1 or Proposition 7 . 1 . 71Let (M, g) be a connected pseudo-Riemannian manifold and α ∈ C ∞ (M ) with gradient Z such that g(Z, Z) = 0 and Z = (f • α) · X for a vector field X such that g(X, X) = h • α, where f and h are smooth functions on the open interval Im α. If the flow φ of X satisifies the following condition, ( * ) For all c ∈ Im α exists an open interval I c such that for all p ∈ F c := α −1 (c) the interval I c is the maximal intervall on which the flow t → φ t (p) is defined, then M is diffeomorphic to the product of the image of the function α and a level set of α.In particular, if the manifold (M, g) is geodesically complete and the vector field X is a geodesic vector field, then M is diffeomorphic to Im(α) × level set. Kähler cones and para-Sasakian manifoldsNow we consider the case where the holonomy algebra h of the space-like cone M over (M, g) is indecomposable and preserves a decomposition T p M = V ⊕ W (p ∈ M ) into two complementary (necessarily degenerate) subspaces V and W . The next lemma 2 reduces the problem to the case V = V ⊥ , W = W ⊥ . Proposition 8.2. (M, g, T ) is a para-Sasakian manifold if and only if (M, g) is a pseudo-Riemannian manifold of signature (n + 1, n) and T a time-like geodesic unit Killing vector field, such that the endomorphism φ := ∇T ∈ Γ(End(T M )) satisfies: X − )T because of the second point of the lemma. Now we can formulate the main theorem of this section. Theorem 8.1. Let (M, g) be a pseudo-Riemannian manifold. There is a one-to-one correspondence between para-Sasakian structures (M, g, T ) on (M, g) and para-Kähler structures ( M , g, J) on the cone ( M , g). The correspondence is given by T → J := ∇T . Lemma 8. 2 . 2The vector field 2rX = J(r∂ r ) on M is tangent to M and r-independent. Its restriction to the submanifold M ∼ = {1} × M ⊂ R + × M = M defines is a time-like geodesic unit vector field T on (M, g). Lemma 8.5. T is a Killing vector field on M . Proof. (L T g) (U, V ) = g(JU, V ) + g(JV, U ) = 0, because J is an anti-isometry. Lemma 8.6. E + and E − are involutive. • u = (u 1 , . . . , u n ) is a diffeomorphism of R n , depending on x 1 , . . . , x n , its non-degenerate Jacobian, and• G is the symmetric matrix given by G ij = −u i u j ,i.e. u j dx i+n dx j+n .(8.15)Then the manifold (R 2n+1 , g) is para-Sasakian, not locally symmetric, and its curvature is given by the following formulasR\| T ⊥ ×T ⊥ ×T ⊥ = R 1 (J., J.) − 2ω ⊗ J \| T ⊥ ×T ⊥ ×T ⊥ , andR(T, .) = R 1 (T, .), . Let (M, g) be a Lorentzian manifold of signature (+, · · · , +, −) or a negative definite Riemannian manifold and ( M = R + × M, g) the cone over M Thus we get the decompositions M 0 = R + × (a, b) × N and equipped with the Lorentzian metric. If M admits a non-zero parallel isotropic vector field then there exists an open dense submanifold M ′ ⊂ M such that any point of M ′ has a neighborhood isometric to a manifold of the form (1.1)Theorem 9.1". Let (M, g) be a Lorentzian manifold and ( M , g) the cone over M We learned this from Helga Baum communicated to us by Lionel Bérard Bergery Acknowledgements. The authors thank the International Erwin Schrödinger Institute for Mathematical Physics in Vienna for the hospitality during the Special Research Semester Geometry of Pseudo-Riemannian Manifolds with Applications in Physics.Para-3-Sasakian manifolds and para-hyper-Kähler conesNow we study the case when the holonomy algebra h of the cone M preserves two complementary isotropic subspaces T ± and a skew-symmetric complex structure J such that JT + = T − . Let us provide the definitions needed to formulate a result analogous to Theorem 8.1 in this case.Definition 8.4.1. Let V be a real finite dimensional vector space. A para-hypercomplex structure on V is a triple (J 1 , J 2 , J 3 = J 1 J 2 ), where (J 1 , J 2 ) is a pair of anticommuting para-complex structures on V .2. Let M be a smooth manifold and V be a distribution on M . An almost para-hypercomplex structure on V is a triple J α ∈ Γ(End V), α = 1, 2, 3, such that, for all p ∈ M , (J 1 , J 2 , J 3 ) p is a para-hyper-complex structure on V p . It is called integrable if the J α are integrable.3.A para-hyper-Kähler manifold is a pseudo-Riemannian manifold (M, g) endowed with a parallel para-hyper-complex structure (J 1 , J 2 , J 3 ) consisting of skew-symmetric endomorphisms J α ∈ Γ(End T M ).A para-3-Sasakian manifoldis a pseudo-Riemannian manifold (M, g) of signature (n + 1, n) endowed with three orthogonal unit Killing vector fields (T 1 , T 2 , T 3 ) such that (i) the vector fields T 1 , T 2 are time-like and define para-Sasakian structures (g, T 1 ), (g, T 2 ), see Definition 8.3, (ii) T 3 is space-like and defines a (pseudo-)Sasakian structure (g, T 3 ),(iv) the vector fields satisfy the following sl 2 (R) commutation relations:equipped with the Lorentzian metric g = dr 2 + r 2 g (of signature (+, · · · , +, −) or (+, −, · · · , −)). Suppose that the cone ( M , g) admits a parallel distribution of isotropic lines. If M is simply connected, then ( M , g) admits a non-zero parallel light-like vector field.2. Let (M, g) be a negative definite Riemannian manifold and ( M , g) the cone over M equipped with the Lorentzian metric of signature (+, −, · · · , −). Suppose that the cone ( M , g) admits a non-zero parallel light-like vector field, then each point x ∈ M has a neighbourhood of the formand for the metric g\| M 0 we haveand for the metric g\| U i we haveSince p 1 is recurrent,where β is a 1-form on M . Hence, g(Y, Z) = 0 on V for all Y ∈ T M . Thus, Z = 0 and p 1 = 0 on V . Let y ∈ U . We have R y (X, Y )∂ r = 0 for all X, Y ∈ T M . Hence, R y (X, Y )p 1 = R y (X, Y )Z. On the other hand, R y (X, Y ) takes values in the holonomy algebra hol y and hol y preserves the line Rp 1y . Hence,Let x ∈ U and let p x = p 1x . Consider any curve γ(t), t ∈ [a, b] such that γ(a) = x and denote by τ γ :where c ∈ R. From this and the Ambrose-Singer theorem it follows that hol x annihilates the vector p x . Since M = R + × M is simply connected, we get a parallel light-like vector field p on M . Claim 1 of the theorem is proved. Now suppose that we have a light-like parallel vector field p on M . Consider the decompositionwhere α is a function on M and Z ∈ T M ⊂ T M . Note that g(Z, Z) = −α 2 , (9.1) and Z is nowhere vanishing. As above we can prove that the open subset U = {x ∈ M \|α(x) = 0} is dense in M .3. ∇ ∂r Z = ∂ r Z + 1 r Z = 0, i.e. Z = 1 rZ , whereZ is a vector field on M .Proof. Claims 1-3 follow from the fact that ∇p = 0. Claim 4 follows from (9.1) and Claim 1 of the lemma.From Claim 1 of Lemma 9.1 it follows that α can be considered as a function on M and α is constant in the directions orthogonal to the vector fieldZ.Let x ∈ M , α(x) = 0 and let γ(t) be the curve of the vector fieldZ passing through the point x. From Claim 1 of Lemma 9.1 it follows that along γ(t) we havewhere c ∈ R is a constant. 2. Suppose that (M, g) is a negative definite Riemannian manifold. In this case the vector fieldZ is nowhere light-like. From Lemma 9.1 it follows that the gradient of the function α is equal to the vector fieldZ. Hence each point x ∈ M has an open neighborhood M 0 diffeomorphic to the product (a, b) × N , where N is a manifold diffeomorphic to the level sets of the function α\| M 0 . Note also that the level sets of the function α\| M 0 are orthogonal to the vector fieldZ. Consequently the metric g\| M 0 must have the following formwhere g 1 is a family depending on the parameter s of Riemannian metrics on the level sets of the function α\| M 0 , and ∂ s =Z α .From Lemma 9.1 it follows that the function α\| M 0 satisfies the following differential equation ∂ s α = −α.Hence, α(s) = c 1 e −s , where c 1 ∈ R is a constant. Changing s, we can assume that c 1 = ±1. Both cases are similar and we suppose that c 1 = 1. Note that (a, b) = − ln(inf α\| M 0 , sup α\| M 0 ). Let Y 1 , Y 2 ∈ T M be vector fields orthogonal toZ and such that [Y 1 , ∂ s ] = [Y 2 , ∂ s ] = 0. From Lemma 9.1 it follows that ∇ Y 1 ∂ s = −Y 1 . From the Koszul formula it follows that 2g(∇ Y 1 ∂ s , Y 2 ) = ∂ s g(Y 1 , Y 2 ). Thus we have −2g 1 (Y 1 , Y 2 ) = ∂ s g 1 (Y 1 , Y 2 ). This means that g 1 = e −2s g N , where the metric g N does not depend on s. ) is the holonomy algebra of the Riemannian manifold. ) × N we have M 2 ⊂ M 0 ⊂ M 1 . Let g 2 = g\| M 2 . Applying Theorem 4.2 in [Lei06] to the Lorentzian situation it follows that hol(M 1 , g 1 ) = hol. M 2 , g 2 ) ∼ = hol(N, g N ) ⋉ R dim N , where hol(N, g NThus, hol( M 0 , g\| M 0 ) ∼ = hol(N, g N ) ⋉ R dim N . If the manifold (M, g) is complete, then the global decomposition follows from Propo-. sition 7.1. From Proposition 2.5 it follows that (a, b) = R and that (N, g N ) is complete. Claim 2 of the theorem is provedthere exist two intervals (a 1 , b 1 ), (a 2 , b 2 ) ⊂ R + such that 1 ∈ (a 2 , b 2 ) and for M 2 = (a 1 , b 1 ) × (a 2 , b 2 ) × N we have M 2 ⊂ M 0 ⊂ M 1 . Let g 2 = g\| M 2 . Applying Theorem 4.2 in [Lei06] to the Lorentzian situation it follows that hol(M 1 , g 1 ) = hol(M 2 , g 2 ) ∼ = hol(N, g N ) ⋉ R dim N , where hol(N, g N ) is the holonomy algebra of the Riemannian manifold (N, g N ). Thus, hol( M 0 , g\| M 0 ) ∼ = hol(N, g N ) ⋉ R dim N . If the manifold (M, g) is complete, then the global decomposition follows from Propo- sition 7.1. From Proposition 2.5 it follows that (a, b) = R and that (N, g N ) is complete. Claim 2 of the theorem is proved. Suppose that (M, g) is a Lorentzian manifold. Consider the open subset U 1 = {x ∈ M \|α(x) = 0} ⊂ M . Obviously. Suppose that (M, g) is a Lorentzian manifold. Consider the open subset U 1 = {x ∈ M \|α(x) = 0} ⊂ M . Obviously, U 1 Maximally homogeneous para-CR manifolds of semisimple type. 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Twistor spinors on Lorentzian manifolds, CR-geometry and Feffer- man spaces. In Kolář, Ivan (ed.) et al., Differential geometry and applications. Proceedings of the 7th international conference, DGA 98, and satellite confer- ence of ICM in Berlin, Brno, Czech Republic, August 10-14, 1998. Brno: Masaryk University. 29-37 . 1999. Sur l'holonomie des variétés pseudoriemanniennes de signature (n, n). L , Bérard Bergery, A Ikemakhen, Bull. Soc. Math. France125L. Bérard Bergery and A. Ikemakhen. Sur l'holonomie des variétés pseudo- riemanniennes de signature (n, n). Bull. Soc. Math. France, 125(1):93-114, 1997. Killing spinors on Lorentzian manifolds. C Bohle, J. Geom. Phys. 453-4C. Bohle. Killing spinors on Lorentzian manifolds. J. Geom. Phys., 45(3- 4):285-308, 2003. Affine hyperspheres associated to special para-Kähler manifolds. V Cortés, M.-A Lawn, L Schäfer, Int. J. Geom. Methods Mod. Phys. 3V. Cortés, M.-A. Lawn, and L. Schäfer. 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[ "SPECTRAL SYNTHESIS IN THE MULTIPLIER ALGEBRA OF A C 0 (X)-ALGEBRA", "SPECTRAL SYNTHESIS IN THE MULTIPLIER ALGEBRA OF A C 0 (X)-ALGEBRA" ]	[ "Robert J Archbold ", "Douglas W B Somerset " ]	[]	[]	Let A be a C 0 (X)-algebra with continuous map φ from Prim(A), the primitive ideal space of A, to a locally compact Hausdorff space X. Then the multiplier algebra M (A) is a C(βX)-algebra with continuous map φ :Thus H x is strictly closed if and only if H x =J x , and the 'spectral synthesis' question asks when this happens. In this paper it is shown that, for σ-unital A, H x is strictly closed for all x ∈ Im(φ) if and only if J x is locally modular for all x ∈ Im(φ) and φ is a closed map relative to its image. Various related results are obtained.2000 Mathematics Subject Classification: 46L05, 46L57.General C 0 (X)-algebrasIn this section we gather some information about C 0 (X)-algebras and establish the basic facts about spectral synthesis (Proposition 2.6). For C 0 (X)-algebras we follow the terminology of[6].Let A be a C 0 (X)-algebra with base map φ : Prim(A) → X, and recall that X φ = Im(φ). Then X φ is completely regular; and if A is σ-unital, X φ is σ-compact and hence normal [7, Section 2]. For x ∈ X φ , set J x = {P ∈ Prim(A) : φ(P ) = x}, and for x ∈ X \ X φ , set J x = A. For a ∈ A, the function x → a + J x (x ∈ X) is upper semi-continuous [35, Proposition C.10]. The C 0 (X)-algebra A is said to be continuous if, for all a ∈ A, the norm	10.1093/qmath/has048	[ "https://arxiv.org/pdf/1210.3251v1.pdf" ]	56,218,406	1210.3251	efc458cef0b0587614cf2eb8b9ca22368d221425	SPECTRAL SYNTHESIS IN THE MULTIPLIER ALGEBRA OF A C 0 (X)-ALGEBRA 11 Oct 2012 Robert J Archbold Douglas W B Somerset SPECTRAL SYNTHESIS IN THE MULTIPLIER ALGEBRA OF A C 0 (X)-ALGEBRA 11 Oct 20122000 Mathematics Subject Classification: 46L05, 46L57. Let A be a C 0 (X)-algebra with continuous map φ from Prim(A), the primitive ideal space of A, to a locally compact Hausdorff space X. Then the multiplier algebra M (A) is a C(βX)-algebra with continuous map φ :Thus H x is strictly closed if and only if H x =J x , and the 'spectral synthesis' question asks when this happens. In this paper it is shown that, for σ-unital A, H x is strictly closed for all x ∈ Im(φ) if and only if J x is locally modular for all x ∈ Im(φ) and φ is a closed map relative to its image. Various related results are obtained.2000 Mathematics Subject Classification: 46L05, 46L57.General C 0 (X)-algebrasIn this section we gather some information about C 0 (X)-algebras and establish the basic facts about spectral synthesis (Proposition 2.6). For C 0 (X)-algebras we follow the terminology of[6].Let A be a C 0 (X)-algebra with base map φ : Prim(A) → X, and recall that X φ = Im(φ). Then X φ is completely regular; and if A is σ-unital, X φ is σ-compact and hence normal [7, Section 2]. For x ∈ X φ , set J x = {P ∈ Prim(A) : φ(P ) = x}, and for x ∈ X \ X φ , set J x = A. For a ∈ A, the function x → a + J x (x ∈ X) is upper semi-continuous [35, Proposition C.10]. The C 0 (X)-algebra A is said to be continuous if, for all a ∈ A, the norm Introduction Let A be a C * -algebra with multiplier algebra M(A) [2,9]. The ideal structure of M(A) is typically much more complicated than that of A and has been widely studied by a number of authors [1,16,23,26,27,31,32]. In [6,7] we investigated certain aspects of the ideal structure of M(A) which arise from a C 0 (X)-algebra structure on the algebra A. The notion of a C 0 (X)-algebra was introduced by Kasparov [22] following extensive earlier work on continuous and upper semi-continuous fields (see, for example, [10,17,19,25,34]). As noted in [7, Section 1], every C * -algebra is a C 0 (X)-algebra, typically in many ways. C 0 (X)algebras have been studied in [3,8,13,14,15,20,28]. In [6, Section 1], we saw that if A is a C 0 (X)-algebra then corresponding to each x ∈ X there are two natural ideals H x and J x in M(A) which one might hope would be equal but in fact need not be so. The aim of this paper is to characterize the equality H x =J x in terms of A and X (without reference to M(A)). Recall that A is a C 0 (X)-algebra if there is a continuous map φ, called the base map, from Prim(A), the primitive ideal space of A with the hull-kernel topology, to the locally compact Hausdorff space X [35, Proposition C.5]. We will use X φ to denote the image of φ in X. If A is a C 0 (X)-algebra then M(A) is a C(βX)-algebra with continuous map φ : Prim(M(A)) → βX (the Stone-Čech compactification of X) extending φ (see Section 2). 3). The 'spectral synthesis' question asks for conditions on A and X characterizing when H x is strictly closed. It was shown in [7], for instance, that if A is stable and σ-unital then for x ∈ X φ , H x is strictly closed if and only if x is a P-point in X φ . In this paper we return to the question in a more general context. The main result (Corollary 3.7) is that if A is a σ-unital C 0 (X)-algebra with base map φ then H x is strictly closed for all x ∈ X φ if and only if J x is locally modular for all x ∈ X φ and φ is a closed map relative to its image. Here J x is said to be locally modular if whenever Q lies in the boundary in Prim(A) of H(x) = {P ∈ Prim(A) : P ⊇ J x } then there exists a neighbourhood V of Q in Prim(A) \ U(x) (where U(x) is the interior of H(x)) such that A/ ker V is a unital C * -algebra. If A is separable then the same characterization is valid for spectral synthesis at a point x ∈ X φ (Corollary 3.10), namely H x is strictly closed if and only if J x is locally modular and φ is a locally closed map at x. For general σ-unital C 0 (X)-algebras this condition is close to characterizing spectral synthesis at a point (see Theorem 3.6) but does not quite succeed (Example 4.5), and we have to leave the problem open. The structure of the paper is as follows. Section 2 gives basic information on C 0 (X)algebras and spectral synthesis. Section 3 contains some of the main results of the paper, described above. Section 4 looks more closely at pointwise spectral synthesis for a σ-unital C 0 (X)-algebra A and identifies the points in X φ which are difficult to deal with. The main result is the characterization of pointwise spectral synthesis in the case when A is a continuous C 0 (X)-algebra (Theorem 4.6). In Sections 5 and 6, we restrict to the important special case when φ is the complete regularization map for Prim(A) and the connecting order Orc(A) is finite. In Section 5, we show that, in this case, if A is σ-unital then the local modularity of J x implies that the complete regularization map φ is locally closed at x. Hence if A is also separable then H x is strictly closed if and only if J x is locally modular (Corollary 5.4). In Section 6, we show that if J x is locally modular then either J x does not contain the centre of A or the hull H(x) of J x in Prim(A) must have non-empty interior-an unusual property unless H(x) is a clopen set (Corollary 6.3). Note added in revision. After this paper was submitted for publication, we learned that David McConnell (private communication) had independently obtained Proposition 2.6(iii), Proposition 2.6(i) (in the case where φ is the complete regularization map), and versions of Corollary 4.3, Corollary 4.4 and Theorem 4.6 with somewhat stronger hypotheses. We are grateful to the referee for a number of helpful comments and for pointing out an error in the original proof of Theorem 3.6. function x → a + J x (x ∈ X) is continuous. By Lee's theorem [ In view of Proposition 2.1(ii), (A +J)/J is canonically isomorphic to A/J. If A/J is unital then (A +J)/J is a unital essential ideal of M(A)/J and therefore equal to M(A)/J . Now suppose that A is a C 0 (X)-algebra, x ∈ X and a ∈ A. If A/J x is unital, the spectrum of a+J x (in A/J x ) coincides with the spectrum of a+J x in M(A)/J x by the previous remark. If A/J x is non-unital, the spectrum of a + J x (in the unitization of A/J x ) is equal to the spectrum of a +J x in (A +J x )/J x + C(1 +J x ) and hence in M(A)/J x [12, 1.3.10(ii)]. The following proposition was proved in [6, Proposition 1.2]. Proposition 2.2. Let A be a C 0 (X)-algebra with base map φ. Then φ has a unique extension to a continuous map φ : Prim(M(A)) → βX such that φ(P ) = φ(P ) for all P ∈ Prim(A). Hence M(A) is a C(βX)-algebra with base map φ and Im(φ) = cl βX (X φ ). Now let A be a C 0 (X)-algebra with base map φ and let φ : Prim(M(A)) → βX be as in Proposition 2.2. For x ∈ βX, we define H x = {Q ∈ Prim(M(A)) : φ(Q) = x}, a closed two-sided ideal of M(A). Thus H x is defined in relation to (M(A), βX, φ) in the same way that J x (for x ∈ X) is defined in relation to (A, X, φ). It follows that for each b ∈ M(A), the function x → b + H x (x ∈ βX) is upper semi-continuous. The next proposition was proved in [7, Proposition 1.3]. Proposition 2.3. Let A be a C 0 (X)-algebra with base map φ, and set X φ = Im(φ). (i) For all x ∈ X, J x ⊆ H x ⊆J x and J x = H x ∩ A. (ii) For all x ∈ X, H x is strictly closed if and only if H x =J x . (iii) For all b ∈ M(A), b = sup{ b +J x : x ∈ X φ } = sup{ b + H x : x ∈ X φ }. We now turn to the subject of spectral synthesis and our first proposition seeks to justify the use of the name. Recall that in the the theory of commutative Banach algebras, spectral synthesis holds at a point x in the maximal ideal space provided that each element of the algebra whose Gelfand transform vanishes at x can be approximated in (the original) norm by elements whose transforms vanish in a neighbourhood of x. At this stage we need some more notation. For a C * -algebra A, let Z(A) denote the centre of A. Now let A be a C 0 (X)-algebra with base map φ. For b ∈ M(A), let Z(b) = {x ∈ X φ : b ∈J x } and let Int Z(b) be the interior of Z(b) relative to X φ . Recall that the Dauns-Hofmann isomorphism θ A : C b (Prim(A)) → Z(M(A)) has the property that θ A (f )a + P = f (P )(a + P ) for f ∈ C b (Prim(A)), a ∈ A, and P ∈ Prim(A) (equivalently, θ A (f ) +P = f (P )(1 +P )). Proposition 2.4. Let A be a C 0 (X)-algebra with base map φ and let x ∈ X φ . Set H alg x = {b ∈ M(A) : x ∈ Int Z(b)}. Then H alg x ⊆ H x and H x is the norm-closure of H alg x . Proof. Let b ∈ H alg x . Then x lies in the interior U of Z(b) in X φ . There exists f ∈ C b (X φ ) such that f (x) = 0 and f (X φ \ U) = {1}. Let z = θ A (f • φ) ∈ Z(M(A)). Suppose that Q ∈ Prim(M(A)) and Q ⊇ H x . Let (P α ) be a net in Prim(A) such thatP α → Q. Since z is central, z + Q = lim z +P α = lim \|f (φ(P α ))\| = \|f (φ(Q))\| = \|f (x)\| = 0. Thus z ∈ Q and hence z ∈ H x . For P ∈ Prim(A), zb +P = f (φ(P ))(b +P ) = b +P because b ∈J φ(P ) ⊆P whenever f (φ(P )) = 1. Hence zb = b and therefore b ∈ H x . For the second part of the proof, let b ∈ H x and ǫ > 0. By upper semi-continuity, there is an open neighbourhood U of x in X such that b + H y < ǫ for all y ∈ U. Let V = U ∩ X φ . There exists a continuous function f : X φ → [0, 1] such that f (x) = 0 and f (X φ \ V ) = {1}. Define g : [0, 1] → [0, 1] by g(t) = 0 (0 ≤ t ≤ 1/2) and g(t) = 2t − 1 (1/2 < t ≤ 1). Let h = g • f . Then h(X φ \ V ) = {1} and there exists an open neighbourhood W of x in X φ such that h\| W = 0. Let z = θ A (h • φ) ∈ Z(M(A)). Suppose that y ∈ W , P ∈ Prim(A) and φ(P ) = y. Then zb +P = h(y)(b +P ) = 0. It follows that zb ∈J y and hence zb ∈ H alg x . Finally, for P ∈ Prim(A), (b − bz) +P = (1 − h(φ(P ))) b +P ≤ (1 − h(φ(P ))) b + H φ(P ) < ǫ since h(φ(P )) = 1 if φ(P ) / ∈ U. Hence b − zb < ǫ. It follows from Proposition 2.3 and Proposition 2.4 that, for x ∈ X φ , the ideal H x is strictly closed if and only if every element b ∈ M(A) which vanishes at x can be approximated in norm by elements vanishing in a neighbourhood of x in X φ . Thus we think of H x being strictly closed as corresponding to 'spectral synthesis holding at x'. As in [6], we define µ : C 0 (X) → Z(M(A)) by µ(f ) = θ A (f • φ) (f ∈ C 0 (X)) Now set Z ′ (A) = µ(C 0 (X)) ∩ A and note that Z ′ (A) ⊆ Z(A). The next lemma was proved in [6, Lemma 2.1]. Lemma 2.5. Let A be a C 0 (X)-algebra with base map φ and let x ∈ X φ . Then J x ⊇ Z ′ (A) if and only if there exists R ∈ Prim(M(A)) with R ⊇ A such that φ(R) = x. We can now give the basic results on spectral synthesis for general C 0 (X)-algebras. Proposition 2.6. Let A be a C 0 (X)-algebra with base map φ and let x ∈ X φ . (i) If J x ⊇ Z ′ (A) then H x is strictly closed in M(A). (ii) If x is an isolated point in X φ then H x is strictly closed in M(A). (iii) If J x ⊇ Z ′ (A) and A/J x is unital then H x is not strictly closed in M(A). Proof. (i) This was proved in [6, Proposition 2.2]. (ii) Since x is an isolated point in X φ , W = φ −1 (x) is a clopen subset of Prim(A). Set Y = Prim(A) \ W . Suppose that R ∈ Prim(M(A)) and R ⊇J x . Then R is not in the closure of the set {P : P ∈ W }. Since the set {P : P ∈ Prim(A)} is dense in Prim(M(A)), R must lie in the closure of the set {P : P ∈ Y }. Hence the continuity of φ implies that φ(R) lies in the closure of {φ(P ) : P ∈ Y }. Thus φ(R) ∈ X \ {x} and so R ⊇ H x . (iii) Let p ∈ A such that p+J x is the identity for A/J x . Then ap−a, pa−a ∈ J x for all a ∈ A and hence 1 − p ∈J x . On the other hand, by Lemma 2.5 there exists R ∈ Prim(M(A)) with R ⊇ A such that φ(R) = x. Since R ⊇ A, p ∈ R, and hence 1 − p / ∈ R. Thus 1 − p / ∈ H x . If J x ⊇ Z ′ (A) then A/J x is unital [6, Proposition 2.2]. So the three cases of Proposition 2.6 cover all possibilities for x except when x is a non-isolated point of X φ with A/J x non-unital. This is the case of interest which will occupy us for the rest of the paper. We will use the following notation. Let U φ = {x ∈ X φ : J x ⊇ Z ′ (A)}, an open subset of X φ (see [6, Section 2]); and let W φ = X φ \ U φ . Let ∂U φ denote the boundary of U φ in X φ . Finally in this section, we consider spectral synthesis for a closed subset E of X φ . Define In the following analogue of Proposition 2.4, we restrict to the case where A is σ-unital in order to ensure that X φ is normal. Proposition 2.7. Let A be a σ-unital C 0 (X)-algebra with base map φ and let E be a closed subset of X φ . Then H alg E ⊆ H E and H E is the norm-closure of H alg x . Proof. Let b ∈ H alg E . Then E is contained in the interior U of Z(b) in X φ . Since X φ is normal, there exists f ∈ C b (X φ ) such that f (E) = {0} and f (X φ \ U) = {1}. Let z = θ A (f • φ) ∈ Z(M(A)). As in the proof of Proposition 2.4, z ∈ H E and b = zb ∈ H E . For the second part of the proof, let b ∈ H E and ǫ > 0. By upper semi-continuity, there is an open neighbourhood U of E in X such that b + H y < ǫ for all y ∈ U. Let V = U ∩ X φ . Since X φ is normal, there exists an open neighbourhood W of E in X φ , with closure contained in V , and a continuous function f : X φ → [0, 1] such that f (W ) = {0} and f (X φ \ V ) = {1}. Let z = θ A (f • φ) ∈ Z(M(A)). As in the proof of Proposition 2.4, zb ∈ H alg x and b − zb < ǫ. Let A be a σ-unital C 0 (X)-algebra with base map φ and let E be a closed subset of X φ . If P ∈ Prim(A) and P ⊇ J E then φ(P ) ∈ E because φ is continuous and E is closed. Hence J E = x∈E J x ⊆ x∈E H x = H E ⊆ x∈EJ x =J E , by Proposition 2.1(iv). It follows from Proposition 2.1(i) and Proposition 2.7 that H alg E is dense in x∈EJ x (which may be thought of as spectral synthesis for E) if and only if H E is strictly closed in M(A). We therefore define spectral synthesis for E to mean that H E is strictly closed in M(A). If E and F are closed subsets of X φ then H E∪F = H E ∩ H F . In contrast to the theory of commutative Banach algebras, it follows that if E and F have spectral synthesis then so does E ∪ F . We shall see in Proposition 2.9 that the question of spectral synthesis for a closed subset E of X φ (i.e. whether H E is strictly closed in M(A)) can be reduced to the question of spectral synthesis for a singleton, but at the expense of changing the base map φ. We will need the following standard topological lemma, where X/E is the quotient space of a topological space X obtained by identifying all of the points in a given subset E. Lemma 2.8. Let X be a normal, Hausdorff space and let E be a non-empty closed subset. Set Y = X/E. Then Y is normal and Hausdorff and hence completely regular. Proof. Let q : X → Y be the quotient map. For y ∈ Y , q −1 (y) is closed and so Y is a T 1 space. Let B and C be disjoint, non-empty closed subsets of Y . Then G = q −1 (B) and H = q −1 (C) are disjoint closed sets in the normal space X, and hence there exist disjoint open sets U and V such that G ⊆ U and H ⊆ V . Without loss of generality, we may assume that G ⊇ E and hence G ∩ E = ∅. Thus, replacing U by U \ E, we may assume that U does not meet E. Hence U is saturated with respect to the equivalence relation corresponding to Note that if X is Hausdorff but non-normal then X has a closed set E such that X/E is not completely regular. To see this, let E and F be disjoint closed subsets of X which cannot be separated by disjoint open sets. Let Y = X/E with quotient map q and set q(E) = e. Then e and the closed set q(F ) cannot be separated by disjoint open sets, so Y is not even regular. Now let A be a C 0 (X)-algebra with base map φ. Let E be a non-empty closed subset of X φ . Set Y = X φ /E and let q : X φ → Y be the quotient map. If A is σ-unital then Y is completely regular by Lemma 2.8 and so A is a C(βY )-algebra with base map ψ = q • φ. Let e = q(E). The next proposition relates J E and J e , and H E and H e . Note that if f ∈ C b (Prim(A)) then the element θ A (f ) ∈ Z(M(A)) induces a function f ∈ C(Prim(M(A)) such that θ A (f ) + Q = f (Q)(1 + Q) (Q ∈ Prim(M(A))). In particular, f (P ) = f (P ) for all P ∈ Prim(A). Proposition 2.9. Let A be a σ-unital C 0 (X)-algebra with base map φ and let E be a nonempty closed subset of X φ . Set Y = X φ /E and let q : X φ → Y be the quotient map. Let ψ = q • φ. Then A is a C(βY )-algebra with base map ψ, and J e = J E and H e = H E , where {e} = q(E). Proof. For P ∈ Prim(A), φ(P ) ∈ E if and only if ψ(P ) = e and therefore J E = J e . For P ∈ Prim(A), q • φ(P ) = (q • φ)(P ) = q(φ(P )) and hence q • φ = q • φ by continuity. Thus if Q ∈ Prim(M(A)) and φ(Q) ∈ E then ψ(Q) = e. Hence H E ⊇ H e . For the reverse inclusion, suppose that Q ∈ Prim(M(A)) with Q / ∈ W := {R ∈ Prim(M(A)) : R ⊇ H E }. Let D be a compact neighbourhood of Q in Prim(M(A)) with D disjoint from W . Then F := φ(D) is a compact subset of βX φ and hence L := F ∩ X φ is closed in X φ and disjoint from E, and φ(Q) lies in the closure of L in βX φ . Since X φ is normal, φ(Q) does not lie in the closure of E in βX φ and hence there is a continuous, real-valued function f on βX φ such that f (φ(Q)) = 1 and f (E) = {0}. There is a well-defined function g : Y → R such that g • q = f \| X φ . Let U be an open subset of R. Then q −1 (g −1 (U)) = f −1 (U) ∩ X φ , an open subset of X φ . Thus g −1 (U) is open in Y and so g is continuous. Let g be the extension of g to a continuous function on βY . Then g • q = f by continuity and so g • q • φ(Q) = 1. Hence ψ(Q) = q • φ(Q) = e. Thus Q ⊇ H e , and hence H E = H e . Global and local spectral synthesis In this section we characterize 'global spectral synthesis' by showing that, for a σ-unital C 0 (X)-algebra A, H x is strictly closed for all x ∈ X φ if and only if J x is locally modular for all x ∈ X φ and φ is a closed map relative to its image (Corollary 3.7). For separable A we can also characterize 'spectral synthesis at a point': if A is separable then for x ∈ X φ , H x is strictly closed if and only if J x is locally modular and φ is locally closed at x (Corollary 3.10). We begin by analyzing the property of H x being strictly closed into two separate subproperties (Proposition 3.2). For this, we need the following lemma. Proof. Set J = ker(Y ∪ Z) and let B = A/J. Let π : A → B be the quotient map. Then the map P → π(P ) (P ∈ Y ∪ Z) carries Y ∪ Z homeomorphically onto Prim(B). Thus there exists b ∈ Z(M(B)) such that b + π(P ) ∼ = 1 + π(P ) ∼ for P ∈ Y and b + π(P ) ∼ = 0 for P ∈ Z. By [30, Theorem 10] the canonical map M(A) → M(B) is surjective, and hence there exists a ∈ M(A) such that a +P = 1 +P for P ∈ Y and a +P = 0 for P ∈ Z. Thus, by definition of the hull-kernel topology, a + Q = 0 for all Q in the closure ofZ in Prim(M(A)), and by considering (1 − a) we see that likewise a + Q = 1 + Q for all Q in the closure ofỸ in Prim(M(A)). HenceỸ andZ have disjoint closures in Prim(M(A)). The first of the two sub-properties into which we analyze the property of strict closure is as follows. Recall that H(x) = φ −1 (x) (x ∈ X φ ). For x ∈ X φ we say that the base map φ is locally closed at x [24, §13.XIV] if whenever Y is a closed subset of Prim(A) such that x lies in the closure of φ(Y ) then x ∈ φ(Y ), that is, Y ∩ H(x) is non-empty. For example, if x is an isolated point in X φ (which implies that H(x) is a clopen subset of Prim(A)) then φ is trivially locally closed at x. Note, however, that H(x) could be clopen, yet x non-isolated in X φ . In this case, φ would not be locally closed at x, see Example 4.8(ii). We say that φ is relatively closed if φ(Y ) is closed in X φ for all closed subsets Y of Prim(A). Clearly φ is relatively closed if and only if φ is locally closed at each x ∈ X φ . Proposition 3.2. Let A be a C 0 (X)-algebra with base map φ and let x ∈ X φ . Consider the following properties: (i) H x is strictly closed; (ii) φ is locally closed at x; (iii) for each b ∈J x and ǫ > 0 there is an open set V ⊆ Prim(A) with H(x) ⊆ V such that b +P < ǫ for all P ∈ V . Then (i)⇒(iii) and (ii)+(iii)⇒(i). If A is σ-unital then (i)⇒(ii) and hence (i) is equivalent to (ii)+(iii). Proof. (i)⇒(iii) Suppose that H x =J x and let b ∈J x and ǫ > 0. Then by the upper semi- continuity of norm functions there is an neighbourhood U of x in X φ such that b + H y < ǫ for all y ∈ U. Set V = φ −1 (U). Then H(x) ⊆ V and for all P ∈ V b +P ≤ b +J φ(P ) ≤ b + H φ(P ) < ǫ. (ii)+(iii)⇒(i) Let b ∈J x and let ǫ > 0 be given. Then by assumption there is an open set V ⊆ Prim(A) with H(x) ⊆ V such that b +P < ǫ for all P ∈ V . Set Y = Prim(A) \ V . Then Y is closed and Y ∩ H(x) is empty. Since φ is locally closed at x, it follows that x does not belong to the closure of φ(Y ) in X φ . Hence there exists g ∈ C b (X φ ) with 0 ≤ g ≤ 1 such that g(φ(Y )) = {1} and g(x) = 0. Let z = θ A (g • φ) ∈ Z(M(A)) so that 0 ≤ z ≤ 1 and z +Q = g(φ(Q))(1 +Q) for all Q ∈ Prim(A). Let R ∈ Prim(M(A)) with φ(R) = x. There is a net (P α ) in Prim(A) such thatP α → R. Then x = φ(R) = lim φ(P α ) and so g(φ(P α )) → g(x) = 0. Thus z +P α → 0 and so, since z ∈ Z(M(A)), z + R = 0. It follows that z ∈ H x and hence zb ∈ H x . But zb − b = sup{ (zb − b) +Q : Q ∈ Prim(A)} = sup{ (zb − b) +Q : Q ∈ V } < ǫ. Since ǫ was arbitrary, b ∈ H x and henceJ x = H x . (i)⇒(ii) (assuming that A is σ-unital). Suppose that φ is not locally closed at x and let Y be a closed subset of Prim(A) such that x lies in the closure of φ(Y ) but H(x) ∩ Y is empty. Let W be the closure ofỸ in Prim(M(A)). Then φ(W ) is a compact and hence closed subset of βX containing φ(Y ). Hence there exists R ∈ W such that φ(R) = x and therefore R ⊇ H x . By Proposition 2.1(iv), the closure ofH( x) = {P : P ∈ φ −1 (x)} in Prim(M(A)) is equal to Prim(M(A)/J x ) (where the latter is identified with the hull ofJ x in Prim(M(A))). But, since A is σ-unital, W is disjoint from Prim(M(A)/J x ) by Lemma 3.1 Thus R ⊇J x and soJ x = H x . In particular, we notice that if A is σ-unital then a necessary condition for spectral synthesis at x ∈ X φ is that φ should be locally closed at x. To understand the second sub-property (property (iii) of Proposition 3.2) into which we have analyzed the property of being strictly closed, we introduce the idea of local modularity. To define this it is helpful to have the following notation. For x ∈ X φ , let ∂H(x) be the boundary and U(x) the interior of H(x) in Prim(A). We say that J x is locally modular if for each P ∈ ∂H(x) there exists a relatively open neighbourhood V of P in Prim(A) \ U(x) such that A/ ker V is a unital C * -algebra. For instance, if x is an isolated point of X φ then H(x) is clopen in Prim(A) and so ∂H(x) is empty and hence J x is vacuously locally modular. Secondly, if J x ⊇ Z ′ (A) (that is, x ∈ U φ ) then by upper semi-continuity of norm functions and functional calculus we may find z ∈ Z ′ (A) such that z + J y = 1 A/Jy for all y in an open neighbourhood V of x in X φ (cf. the proof of [6, Proposition 2.3]). Hence z + P = 1 A/P for all P ∈ W = φ −1 (V ), and H(x) ⊆ W , so again J x is locally modular. On the other hand, if there exists P ∈ ∂H(x) such that A/P is non-unital then clearly J x is not locally modular. The definition of local modularity is intrinsic to A, in that it does not mention M(A), and it is a condition that should be possible to check in concrete cases. The following equivalent condition, however, seems easier to work with, although it does involve M(A). To describe this, we need a slight variant of the definition of ∼. Recall from [33] that for P, Q ∈ Prim(A) we write P ∼ Q if P and Q cannot be separated by disjoint open subsets of Prim(A) (for a fuller discussion, see Section 5 below). For the multiplier algebra M(A) of a C 0 (X)-algebra A, we define ∼ x as follows. For x ∈ X φ and Q, R ∈ Prim(M(A)) \Ũ (x) we say that Q ∼ x R if there is a net (P α ) in Prim(A) \ U(x) such that (P α ) converges to both Q and R. If H(x) has empty interior then ∼ x coincides with the relation ∼ on Prim(M(A)) because the canonical image of Prim(A) is dense in Prim(M(A)). Otherwise Q ∼ x R ⇒ Q ∼ R, but the converse need not hold. For the next lemma, we need the definition of a primal ideal and of the topology τ s . An ideal J in a C * -algebra A is primal if whenever I 1 , . . . , I n is a finite collection of ideals of A with the product I 1 . . . I n = {0} then I i ⊆ J for at least one i ∈ {1, . . . , n} [5]. Every primitive ideal is prime and hence primal. The set of proper primal ideals of A is denoted Primal ′ (A). The τ s topology on Primal ′ (A) is defined to be the weakest topology for which all the norm functions I → a + I (a ∈ A, I ∈ Primal ′ (A)) are continuous (see [17,Section II] ). If A is unital then Primal ′ (A) is τ s -compact [4, Proposition 4.1]. Lemma 3.3. Let A be a C 0 (X)-algebra with base map φ and let x ∈ X φ . Consider the following conditions: (i) J x is locally modular; (ii) for all P ∈ ∂H(x) and R ∈ Prim(M(A)/A),P ∼ x R. Then (i)⇒(ii), and (i) and (ii) are equivalent if A is σ-unital. Proof. Suppose first that (i) holds, and let P ∈ ∂H(x). Let V be an open neighbourhood of P in Prim(A) \ U(x) such that A/ ker V is unital. Let W be the closure of V in Prim(A), so that ker V = ker W . Write J = ker W (so that W = Prim(A/J)) and recall that the quotient map q J : A → A/J has a canonical extensionq J : M(A) → M(A/J) = A/J. For each b ∈ M(A) there exists a ∈ A such that b − a ∈ kerq J =J, so A +J = M(A). Thus if R ∈ Prim(M(A)/A) then R ⊇J. It follows thatW = Prim(M(A)/J ), a closed subset of Prim(M(A)). Hence if (P α ) is a net in Prim(A) \ U(x) with P α → P then eventually P α ∈ V , so all the limits of the net (P α ) in Prim(M(A)) lie inW . ThusP ∼ x R for any R ∈ Prim(M(A)/A). Conversely, suppose that (ii) holds and that A is σ-unital. Let u be a strictly positive element in A with u = 1. Let Q ∈ ∂H(x) and suppose that A +Q = M(A). Then there exists a maximal ideal M of M(A) with M ⊇ A +Q, and hence M ∼ xQ contradicting (ii). Thus A +Q = M(A), so A/Q is unital and (1 − u) +Q < 1. Let P ∈ ∂H(x) and suppose, for a contradiction, that there is a net (P α ) in Prim(A)\U(x) with P α → P and (1 − u) +P α → 1. By the τ s -compactness of Primal ′ (M(A)), and by passing to a subnet if necessary, we may assume that there exists J ∈ Primal ′ (M(A)) such thatP α → J (τ s ). Hence (1 − u) + J = 1 and so there exists R ∈ Prim(M(A)/J) such that (1−u)+R = 1 [12, 3.3.6]. SinceP α → J (τ s ),P α → R and so R ⊇ A by (ii). Then R =Q, where Q := R ∩ A ∈ Prim(A), and so P α → Q. Since φ is continuous, φ(Q) = φ(P ) = x and so Q ∈ H(x). But P α ∈ Prim(A) \ U(x) for all α and so Q ∈ ∂H(x), contradicting the fact that (1 − u) +Q = 1. Thus no such net (P α ) exists and so there exists ǫ > 0 and a neighbourhood V of P in Prim(A) \ U(x) such that (1 − u) +Q < 1 − ǫ for all Q ∈ V . Now let f be a continuous function on [0, 1] with f (0) = 0, f ([ǫ, 1]) = {1}, and f = 1, and set v = f (u). Then (1 − v) +Q = 0 for all Q ∈ V and so v + ker V is the identity in A/ ker V . Hence J x is locally modular, as required. The next proposition shows part of the connection between local modularity and property (iii) of Proposition 3.2. Proposition 3.4. Let A be a C 0 (X)-algebra with base map φ and let x ∈ X φ . If J x is locally modular then property (iii) of Proposition 3.2 holds, namely for each b ∈J x and ǫ > 0 there is an open set V ⊆ Prim(A) with H(x) ⊆ V such that b +P < ǫ for all P ∈ V . Hence if J x is locally modular and φ is locally closed at x then H x is strictly closed. Proof. Suppose that J x is locally modular and let b ∈J x and ǫ > 0. Let P ∈ ∂H(x) and suppose for a contradiction that there is a net (P α ) in Prim(A) \ U(x) such that P α → P and b +P α ≥ ǫ. Then by compactness of the set W = {R ∈ Prim(M(A)) : b + R ≥ ǫ}, and by passing to a subnet of (P α ) if necessary, there exists R ∈ W such thatP α → R. Since J x is locally modular, it follows from Lemma 3.3 that R =Q for some Q ∈ Prim(A). Since P α → Q, φ(Q) = φ(P ) = x and so b ∈Q, contradicting the fact that b +Q ≥ ǫ. Therefore no such net (P α ) exists. Thus there is an open set V P containing P such that b +Q < ǫ for all Q ∈ V P . Taking V = U(x) ∪ P ∈∂H(x) V P gives the required set V . The final statement now follows from Proposition 3.2 ((ii)+(iii)⇒(i)). It will follow from Theorem 3.6 that the converse to Proposition 3.4 holds if A is separable. For a general σ-unital C 0 (X)-algebra A, however, it is possible for property (iii) of Proposition 3.2 to hold for a particular x ∈ X φ without J x being locally modular, see Example 4.5. Nevertheless the relation between the two properties is very close, as we shall see. To show this, we need the following theorem from [7,Theorem 2.5]. For an element a in a C * -algebra A, let sp(a) denote the spectrum of a; and for a ≥ 0, let min sp(a) be the smallest number in sp(a). The function g from the unit interval [0, 1] to the space C[0, 1] is as follows (where for r ∈ [0, 1], g r is the continuous function on [0, 1] corresponding to r): g 0 (t) = 1 for all t ∈ [0, 1]; for 0 < r ≤ 1/2, g r (t) =      0 (0 ≤ t ≤ r/2) (2t/r) − 1 (r/2 ≤ t ≤ r) 1 (r ≤ t ≤ 1); g r = g 1/2 for r ≥ 1/2. Theorem 3.5. Let A be a σ-unital C 0 (X)-algebra with base map φ and set X φ = Im(φ). Let u be a strictly positive element in A with u = 1. Let f ∈ C b (X φ ) with 0 ≤ f ≤ 1, let U beb ≤ 1 such that (i) b +J x = g f (x) (u +J x ) (x ∈ X φ ); (ii) b ∈ A + H x ⊆ A +J x for all x ∈ U; (iii) 1 − b ∈J x for all x ∈ X φ \ U and 1 − b ∈ H x for all x ∈ X φ \ cl(U); (iv) (1 − b) +J x = 1 for all x ∈ V and (1 − b) + H x = 1 for all x ∈ cl(V ). Furthermore, (v) H x is not strictly closed in M(A) for all x ∈ cl(V ) \ U. In the context of Theorem 3.5, note that if x ∈ U and A/J x is non-unital then 0 ∈ sp(u + J x ) and hence x ∈ V . Theorem 3.6. Let A be a σ-unital C 0 (X)-algebra with base map φ. Let x ∈ X φ and let Z be a zero set of X φ with x ∈ Z. Suppose that J x is not locally modular. Then there exists y ∈ Z for which property (iii) of Proposition 3.2 fails, that is, for which there exists c ∈J y and ǫ > 0 such that there is no open set V ⊆ Prim(A) with H(y) ⊆ V and c +P < ǫ for all P ∈ V . In particular, H y is not strictly closed. Proof. Since J x is not locally modular, it follows from Lemma 3. U 1 ∩Ṽ 1 ∩ (Int D) ∼ ∩ (Prim(A) \ U(x)) ∼ is a non-empty, relatively open, subset of (Prim(A) \ U(x)) ∼ . Since Prim(A) \ H(x) is dense in Prim(A) \ U(x), we may choose Q 1 ∈ Prim(A) \ H(x) withQ 1 ∈ U 1 ∩Ṽ 1 ∩ (Int D) ∼ . Set x 1 = φ(Q 1 ) . Then x 1 = x and so there exists a continuous function f 1 : X φ → [0, 1/4] with f 1 (x 1 ) = 1/4 and f 1 (x) = 0. For n ≥ 2, we will inductively define points x n ∈ X φ and continuous functions f n : X φ → [0, 1/2 n+1 ] with f n (x n ) = 1/2 n+1 and f n (x) = 0 = f n (x m ) for 1 ≤ m ≤ n − 1. Note that x 1 and f 1 satisfy these conditions. Suppose that n ≥ 2 and that x 1 , . . . , x n−1 and f 1 , . . . , f n−1 satisfy the required conditions. Let W n = {Q ∈ Prim(A) : n−1 i=1 f i (φ(Q)) < 1/2 n+1 }. Then W n is an open neighbourhood of P and hence, sinceP ∼ x R, it follows that U n ∩Ṽ n ∩ (Int D) ∼ ∩W n ∩ (Prim(A) \ U(x)) ∼ is a non-empty, relatively open, subset of (Prim(A) \ U(x)) ∼ . Thus we may choose Q n ∈ Prim(A) \ H(x) withQ n ∈ U n ∩Ṽ n ∩ (Int D) ∼ ∩W n . Set x n = φ(Q n ). Then x n = x and x n = x m for 1 ≤ m ≤ n − 1 because n−1 i=1 f i (x m ) ≥ f m (x m ) = 1 2 m+1 > 1 2 n+1 > n−1 i=1 f i (x n ). Thus there exists a continuous function f n : X φ → [0, 1/2 n+1 ] with f n (x n ) = 1/2 n+1 and f n (x) = 0 = f n (x m ) for 1 ≤ m ≤ n − 1. Set f = ∞ n=1 f n . Then f is continuous and, for each n ≥ 1, 1 2 n+1 = f n (x n ) ≤ f (x n ) = n−1 i=1 f i (x n ) + f n (x n ) < 1 2 n+1 + 1 2 n+1 = 1 2 n . Thus f (x n ) → 0 as n → ∞. Furthermore, since (1 − u) +Q n > 1 − 1/2 n+2 , it follows that min sp(u +Q n ) < 1 2 n+2 ≤ f (x n ) 2 and hence 0 ∈ sp(g f (xn) (u +Q n )) for n ≥ 1. Since Q n ∈ D for all n, the compactness of D implies that there exists Q ∈ D and a subnet (Q nα ) of (Q n ) such that Q nα → Q. Set y = φ(Q). Since Q n ∈ V n (n ≥ 1), h(x n ) → 0 as n → ∞. It follows that h(y) = 0 and hence y ∈ Z. Furthermore, f (x nα ) → f (y) and so f (y) = 0. Let b be an element of M(A) corresponding to f as in Theorem 3.5. Then 1 − b ∈J y by Theorem 3.5(iii). Since φ(Q n ) = x n (n ≥ 1), it follows thatQ n ⊇J xn and hence that b +Q n = g f (xn) (u +Q n ) by Theorem 3.5(i). Then 1 ≥ (1 − b) +Q n ≥ (1 +Q n ) − g f (xn) (u +Q n ) = 1 − min sp(g f (xn) (u +Q n )) = 1. Hence (1 − b) +Q nα = 1 for all α. Suppose that V is an open subset of Prim(A) such that H(y) ⊆ V . Then Q ∈ V and so eventually Q nα ∈ V . Thus property (iii) of Proposition 3.2 fails at y for c = 1−b ∈J y and ǫ = 1/2. Hence H y is not strictly closed by Proposition 3.2. Regarding the hypotheses of Theorem 3.6, we note that the complete regularity of X φ implies that every neighbourhood of x contains a zero set containing x. In the proof of Theorem 3.6, we could have checked that y ∈ cl(V ) \ U (in the terminology of Theorem 3.5) and used Theorem 3.5(v) to deduce that H y is not strictly closed. We have preferred, however, to obtain the stronger result that property (iii) of Proposition 3.2 fails at y. Armed with Theorem 3.6, we can now prove some of the main results of the paper. Corollary 3.7 (global spectral synthesis). Let A be a C 0 (X)-algebra with base map φ. Consider the following two properties. (i) H x is strictly closed for all x ∈ X φ . (ii) J x is locally modular and φ is locally closed at x for all x ∈ X φ . Then (ii)⇒(i), and (i) and (ii) are equivalent if A is σ-unital. Proof. (ii)⇒(i). This follows from Proposition 3.4. (i)⇒(ii) (assuming that A is σ-unital). This follows from (i)⇒((ii)+(iii)) of Proposition 3.2 together with Theorem 3.6 (taking Z = X φ , for example). Corollary 3.8 (spectral synthesis at a point). Let A be a σ-unital C 0 (X)-algebra with base map φ and let x be a G δ -point in X φ . Then H x is strictly closed if and only if J x is locally modular and φ is locally closed at x. Proof. This follows from Proposition 3.4 and from Proposition 3.2 and Theorem 3.6 taking the zero set Z in Theorem 3.6 to be Z = {x}. Recall that a completely regular (Hausdorff) space X is perfectly normal if every closed subset of X is the zero set of a continuous real-valued function on X. Every metric space is perfectly normal. Lemma 3.9. Let A be a separable C 0 (X)-algebra with base map φ. Then X φ is perfectly normal. Proof. Let V be an open subset of X φ . Set Y = φ −1 (V ) and let J be the ideal of A such that Prim(J) is canonically homeomorphic to Y . Then J is separable. Let u be a strictly positive element in J. For each n ∈ N, set Y n = {P ∈ Prim(A) : u + P ≥ 1/n}. Then Y n is compact and so φ(Y n ) is a compact (hence closed) subset of X φ . Since V = ∞ n=1 φ(Y n ), we see that V is an F σ subset of X φ . But X φ is normal, and in a normal space an open F σ subset is a cozero set. Thus V is a cozero set. Corollary 3.10. Let A be a separable C 0 (X)-algebra with base map φ and let x ∈ X φ . Then H x is strictly closed if and only if J x is locally modular and φ is locally closed at x. Proof. This follows as for Corollary 3.8, noting that {x} is a zero set of X φ for each x ∈ X φ by Lemma 3.9. Pointwise spectral synthesis and P-points Recall from Section 2 that if A is a C 0 (X)-algebra with base map φ then U φ = {x ∈ X φ : J x ⊇ Z ′ (A)}, an open subset of X φ , ∂U φ denotes the boundary of U φ in X φ and W φ = X φ \U φ . Here Z ′ (A) = µ(C 0 (X)) ∩ A. We saw in Proposition 2.6 that if x ∈ U φ or if x is an isolated point of X φ then H x is strictly closed, while if A/J x is unital but x / ∈ U φ then H x is not strictly closed. We remarked that the remaining points to consider are those for which A/J x is non-unital and x ∈ W φ and is non-isolated in X φ . From Section 3 we now have a complete characterization of global spectral synthesis for σ-unital A, and a complete characterization of pointwise spectral synthesis for separable A, but only a near-characterization of pointwise spectral synthesis for the more general case when A is σ-unital. In this section we approach the σ-unital case from another angle. We characterize the points x in the interior of W φ in X φ for which H x is closed (these turn out to be precisely the P-points) and make partial progress for the difficult case of points x ∈ ∂U φ (cf. for example [6, Proposition 3.5]). We also give a complete characterization of pointwise spectral synthesis for the important case when the base map φ is open (Theorem 4.6). We begin with the following lemma. Lemma 4.1. Let A be a σ-unital C 0 (X)-algebra with base map φ and let x ∈ W φ . Let u ∈ A be strictly positive with u ≤ 1. Then there is a net (x α ) in X φ with x α → x and (1 − u) +J xα → 1. Proof. By Lemma 2.5 there exists R ∈ Prim(M(A)/A) with R ⊇ H x . Let (P α ) be a net in Prim(A) such thatP α → R. Then φ(P α ) → φ(R) = x. Hence x α := φ(P α ) → x. On the other hand, since R ⊇ A, 1 = (1 − u) + R ≤ lim inf (1 − u) +P α ≤ lim inf (1 − u) +J xα ≤ lim sup (1 − u) +J xα ≤ 1. Hence (1 − u) +J xα → 1. Next we recall the definition of a P-point. Let X be a completely regular (Hausdorff) space. A point x ∈ X is a P-point if every continuous real-valued function vanishing at x vanishes in a neighbourhood of x [18, 4L]. Equivalently, x is a P-point if x does not lie in the boundary of any cozero set. If the space X is perfectly normal then every singleton is a zero set and so a P-point is necessarily an isolated point. A space in which every point is a P-point is a P-space. We are now ready for the first main result of this section. Theorem 4.2. Let A be a σ-unital C 0 (X)-algebra with base map φ. If x ∈ X φ is a P-point in X φ then H x is strictly closed. Conversely, if x ∈ W φ and H x is strictly closed then x is a P-point in W φ . Proof. Let x ∈ X φ and suppose that H x is not strictly closed. We show that x is not a Ppoint in X φ . By [ X φ , \|\|b + H x \|\| ≤ sup{\|\|b +J y \|\| : y ∈ X φ ∩ W }. Hence for every neighbourhood V of x in X φ there exists y ∈ V such that bu / ∈J y . Thus x lies in the closure of ∞ n=1 W n . Since Conversely, suppose that x ∈ W φ is not a P-point in W φ . Let f be a continuous function on W φ with f (x) = 0 such that x lies in the closure of the cozero set of f . Replacing f by min{\|f \|, 1}, we may assume that 0 ≤ f ≤ 1. Since X φ is normal and W φ is closed in X φ , we may extend f to a continuous function f on X φ with 0 ≤ f ≤ 1. Let y ∈ W φ ∩ coz(f ). By Lemma 4.1 there is a net (y α ) in X φ with y α → y and (1 − u) +J yα → 1. Hence eventually 2 min sp(u + J yα ) = 2(1 − (1 − u) +J yα ) ≤ f (y α ), since f (y α ) → f (y) > 0. It follows that the set V of Theorem 3.5 (associated with the cozero set coz(f )) has closure cl(V ) containing W n = {y ∈ X φ : \|\|bu + J y \|\| ≥ 1/n} = φ({P ∈ Prim(A) : \|\|bu + P \|\| ≥ 1/n}),W φ ∩ coz(f ). Hence x ∈ cl(V ) since x lies in the the closure of W φ ∩ coz(f ) = coz(f ). Thus J x = H x by Theorem 3.5(v). Theorem 4.2 has some useful consequences. Let Int W φ denote the interior of W φ relative to X φ . Corollary 4.3. Let A be a C 0 (X)-algebra with base map φ and let x ∈ Int W φ . If A is σ-unital then H x is strictly closed if and only if x is a P-point in X φ . If A is separable then H x is strictly closed if and only if x is an isolated point in X φ . Proof. Suppose that A is σ-unital and that x is not a P-point in X φ . Then there exists f ∈ C b (X φ ) such that f (x) = 0 and x lies in the closure of coz(f ). Let (x α ) be a net in coz(f ) such that x α → x, and set f = f \| W φ . Then f is continuous and f (x) = 0 but eventually x α ∈ Int W φ and so x lies in the closure of the cozero set of f . Thus x is not a P-point in W φ and so H x is not strictly closed by Theorem 4.2. If A is separable then {x} is a zero set in X φ by Lemma 3.9 and hence x is P-point in X φ if and only if it is isolated in X φ . ( i) Let x ∈ X φ . If A is σ-unital then H x is strictly closed if and only if x is a P-point in X φ . If A is separable then H x is strictly closed if and only if x is an isolated point in X φ . (ii) If A is σ-unital then H x is strictly closed for all x ∈ X φ if and only if X φ is discrete. Proof. Since Z ′ (A) = {0}, U φ = ∅ and W φ = X φ . Thus part (i) follows from Corollary 4.3. Part (ii) now follows because the space X φ is σ-compact, and a σ-compact P-space is discrete (see, for example, the proof of [7,Lemma 4.4]). If A is a stable C 0 (X)-algebra then Z It follows from Theorem 4.2 and Proposition 3.2 that if A is a σ-unital C 0 (X)-algebra and x ∈ X φ is a P-point in X φ then φ is locally closed at x and that property (iii) of Proposition 3.2 holds. On the other hand, J x need not be locally modular as the following example shows. Example 4.5. Let X be a compact Hausdorff space with a non-isolated P-point x (e.g. take X to be ω 1 + 1, where ω 1 is the first uncountable ordinal, with the usual topology) and let A = C(X) ⊗ K(H) (where K(H) is the algebra of compact operators on a separable, infinite-dimensional Hilbert space H). There is a homeomorphism φ : Prim(A) → X such that φ({f ∈ C 0 (X) : f (y) = 0} ⊗ K(H)) = y (y ∈ X) . Then J x is not locally modular because ∂H(x) is non-empty but A/P is non-unital for all P ∈ Prim(A). Nevertheless, H x is strictly closed by Theorem 4.2. In Theorem 4.2 we saw a characterization, for A σ-unital, of when H x is strictly closed for x ∈ Int W φ and we know that H x is always strictly closed if x ∈ U φ (Proposition 2.6(i)). The remaining points to consider are those in ∂U φ . For general σ-unital C 0 (X)-algebras we are not able to characterize the points x ∈ ∂U φ for which H x is strictly closed (though we have seen a necessary condition in Theorem 4.2), but if we make the further assumption that the base map φ is open then we can show that there are no such points. Theorem 4.6. Let A be a continuous, σ-unital C 0 (X)-algebra with base map φ and let x ∈ X φ . Then the following are equivalent: (i) H x is strictly closed; (ii) either x ∈ U φ , or x ∈ Int W φ and x is a P-point in X φ . Proof. (ii)⇒(i) If x ∈ U φ , or if x ∈ Int W φ with x a P-point in X φ , then H x is strictly closed by Proposition 2.6(i) and Corollary 4.3. (i)⇒(ii) Corollary 4.3 shows that if x ∈ Int W φ with H x strictly closed then x must be a P-point in X φ . It is enough, therefore, to show that if x ∈ ∂U φ then H x is not strictly closed. If A/J x is unital then this follows from Proposition 2.6(iii). Hence we may assume that A/J x is non-unital. Let u be a strictly positive element of A with u = 1. Then (1−u)+J x = 1. For each n ≥ 1, there exists P n ∈ Prim(A/J x ) such that (1 − u) +P n > 1 − 1/2 n+1 . Hence the set V n = {Q ∈ Prim(A) : (1 − u) +Q > 1 − 1/2 n+1 } is an open neighbourhood of P n and so, since φ is open, the set φ(V n ) is an open neighbourhood of φ(P n ) = x in X φ . Thus is closed in X φ and x / ∈ X n . If y ∈ U φ then A/J y is unital [6, Proposition 2.2] and therefore (1 − u) +J y < 1. Hence ∞ n=1 X n ⊇ U φ and it follows that x lies in the closure of ( ∞ n=2 X n ) \ X 1 . Since x / ∈ X n there exists f n ∈ C b (X φ ) with 0 ≤ f n ≤ 1/2 n such that f n (x) = 0 and f n (X n ) = {1/2 n }. Set f = ∞ n=1 f n . Then f ∈ C b (X φ ) with 0 ≤ f ≤ 1 and f (x) = 0. Let W be the cozero set of f and let V = {y ∈ W : 2 min sp(u + J y ) ≤ f (y)}. Suppose that y ∈ X φ with y ∈ X n+1 \ X n for some n ≥ 1. Then (1 − u) +J y > 1 − 1/2 n+1 and so min sp(u + J y ) < 1/2 n+1 . Since y ∈ X m for all m ≥ n + 1, f (y) ≥ 1/2 n and so f (y) > 2 min sp(u + J y ). Hence y ∈ V , and thus V ⊇ ( ∞ n=2 X n ) \ X 1 . Hence x ∈ cl(V ) \ W and so it follows from Theorem 3.5(v) thatJ x = H x , as required. Corollary 4.7. Let A be a continuous, σ-unital C 0 (X)-algebra with base map φ. Then the following are equivalent: (i) for all x ∈ X φ , H x is strictly closed; (ii) U φ and W φ are clopen in X φ , and W φ is discrete. Proof. (ii)⇒(i). This follows immediately from Theorem 4.6. (i)⇒(ii). Theorem 4.6 implies that ∂U φ is empty and hence X φ has the required decomposition into clopen sets U φ and W φ . Since X φ is σ-compact, W φ must be σ-compact as well. But by Theorem 4.6, W φ is a P-space, and a σ-compact P-space is discrete, see [7,Lemma 4.4]. It is interesting to compare Corollary 4.7 with [6, Theorem 3.8] which characterizes, for a continuous σ-unital C 0 (X)-algebra A, when M(A) is a continuous C(βX)-algebra. The conditions in Corollary 4.7 are markedly stronger than those in [6, Theorem 3.8], notably the requirement that W φ be discrete as against being a basically disconnected space. On the other hand, if A is separable then there is a much closer fit with [6, Corollary 3.9]: indeed, if A is continuous and separable and X φ = X then H x is strictly closed for all x ∈ X φ if and only if M(A) is continuous for µ. We conclude this section with a couple of elementary abelian examples, part of whose significance will appear in the next two sections. (ii) An abelian C 0 (X)-algebra A with x ∈ X φ such that H x is not strictly closed in M(A). Let A = C 0 ([0, 1)∪ [2,3]) and set X = [0, 1]. Then we may identify Prim(A) with [0, 1)∪ [2,3]. Let φ : Prim(A) → X be given by φ(x) = x (0 ≤ x < 1) and φ(x) = 1 (2 ≤ x ≤ 3). Then J x is locally modular for all x ∈ X, but φ is not locally closed at x = 1. Hence H 1 is not strictly closed in M(A). Example 4.8. (i) An abelian C 0 (X)-algebra A with x ∈ W φ such that H x is strictly closed in M(A). Let Y = {(x, y) ∈ R 2 : y ≥ 0} C 0 (X)-algebras where the base map φ is the complete regularization map In this section we investigate C 0 (X)-algebras A where the base map φ is the complete regularization map on Prim(A) [18,Theorem 3.9]. Thus X may be taken to be the complete regularization space (in cases where this is locally compact) or its Stone-Čech compactification. Restricting φ in this way places a considerable constraint on its behaviour, as we shall see. This case is of special interest for two reasons: firstly, the complete regularization map interacts with the topology on Prim(A) in a way that is lacking with more general continuous maps, and secondly, every continuous map from Prim(A) to a locally compact Hausdorff space factors through the complete regularization map. Under the hypotheses that φ is the complete regularization map and that Orc(A) < ∞ (a technical assumption which is usually satisfied), we show that if A is σ-unital and J x is locally modular then φ is locally closed at x and H x is strictly closed (Theorem 5.3). Thus the 'locally closed at x' part of the hypothesis in the final sentence of Proposition 3.4 is automatically satisfied in this case (contrast with Example 4.8(ii)). We begin by explaining the notation Orc(A) [33]. Recall that for a C * -algebra A and for P, Q ∈ Prim(A) we write P ∼ Q if P and We also need the following topological lemma characterizing separation by open sets. We say that a topological space is locally compact if every point has a neighbourhood base of compact sets. Lemma 5.1. Let X be a locally compact topological space and let Y and Z be subsets of X which are Lindelof in the relative topology. Then the following are equivalent: (i) The closure of Y 1 does not meet Z and the closure of Z 1 does not meet Y . (ii) There exist disjoint open subsets U and V of X with Y ⊆ U and Z ⊆ V . Proof. Suppose first that (ii) holds. Then X \ V is disjoint from Z and is a closed set containing the neighbourhood U of Y and hence containing Y 1 . Similarly X \ U is disjoint from Y and is a closed set containing Z 1 . Hence (i) holds. Conversely, suppose that (i) holds. Let x ∈ Y . Since the closure of Z 1 does not meet Y , x has an open neighbourhood disjoint from Z 1 . Hence, by the local compactness of X, x has a compact neighbourhood U x ⊆ X \ Z 1 . Then U 1 x is closed because U x is compact, and U 1 x does not meet Z because U x does not meet Z 1 . Similarly for each x ∈ Z there exists a neighbourhood V x of x such that the closure of V x does not meet Y . Since Y and Z are Lindelof, we may obtain a countable collection, say U 1 , U 2 , . . ., of the sets Int U x such that ∞ i=1 U i covers Y , and likewise a countable collection V 1 , V 2 , . . . of the sets Int V x such that ∞ i=1 V i covers Z. For each i ≥ 1, set U ′ i = U i \ i j=1 V j and V ′ i = V i \ i j=1 U j (where V i denotes the closure of V i , etc). Set U = ∞ i=1 U ′ i and V = ∞ i=1 V ′ i . Then it is easily checked that Y ⊆ U and Z ⊆ V . If x ∈ U ∩ V then there exist U ′ i and V ′ j such that x ∈ U ′ i ∩ V ′ j . Without loss of generality we may suppose that i ≥ j. But then V ′ j ⊆ V j which is disjoint from U ′ i , a contradiction. Hence U and V are disjoint, and thus (ii) holds. Now let A be a σ-unital C 0 (X)-algebra. Let E be a non-empty closed subset of X φ , set Y = X φ /E and let q : X φ → Y be the quotient map. Set ψ = q • φ and {e} = q(E). We saw in Proposition 2.9 that the question of spectral synthesis for the set E can be reduced to that of the point e, and for this reason we have previously confined ourselves to considering spectral synthesis at points. If we restrict φ to be the complete regularization map, however, then we can no longer make this reduction (because the reduction changes the base map), and we will therefore have to work with closed sets in this section. With this in mind, and with the notation above, we say that φ is locally closed at E if ψ is locally closed at e, and that J E is locally modular if J e is locally modular. Elementary topological arguments show that φ is locally closed at E if and only if whenever Y is a closed Proposition 5.2. Let A be a σ-unital C 0 (X)-algebra with base map φ and let E be a nonempty closed subset of X φ . Consider the following conditions (i) H E is strictly closed. subset of Prim(A) such that φ(Y ) ∩ E = ∅ (i.e. Y ∩ φ −1 (E) = ∅ ) then φ(Y ) ∩ E = ∅. Note that H(e) = (q • φ) −1 (e) = φ −1 (E), (ii) J E is locally modular and φ is locally closed at E. Then (ii)⇒(i), and (i) and (ii) are equivalent if A is separable. Proof. With the notation above, we have J e = J E and H e = H E by Proposition 2.9. Hence (ii)⇒(i) follows immediately from Proposition 3.4. Conversely, suppose that A is separable and that (i) holds. Then H e is strictly closed by Proposition 2.9 and it follows from Corollary 3.10 that J e = J E is locally modular and q • φ is locally closed at e. Hence φ is locally closed at E by definition. We are now ready for the main theorem of the section. Theorem 5.3. Let A be a σ-unital C 0 (X)-algebra and suppose that φ is the complete regularization map for Prim(A) and that Orc(A) < ∞. Let E be a non-empty closed subset of X φ and suppose that J E is locally modular. Then φ is locally closed at E and H E is strictly closed. Proof. As before, let q : X φ → X φ /ER β ) of (R α ) with R β → R ∈ Prim(M(A)/A). ThenQ ∼ e R and so Q ∈ Y , as required. Next we note that Y 1 is closed in Prim(A). To see this, let (Q α ) be a net in Y 1 with Q α → Q ∈ Prim(A). Then Q / ∈ U(e), for otherwise eventually Q α ∈ U(e) ⊆ H(e), which is impossible since H(e) is a ∼-saturated set disjoint from Y . Let (P α ) be a net in Y such that P α ∼ Q α for each α. Since P α , Q α / ∈ H(e),P α ∼ eQα . By the compactness of Prim(M(A)) there exists R ∈ Prim(M(A)) and a subnet (P β ) of (P α ) such thatP β → R. Hence R ∼ eQ . Note that the disjoint sets Z k and H(e) are ∼-saturated and hence so is the set F := Prim(A) \ (Z k ∪ H(e)). We now define an equivalence relation ⋄ on Prim(A) as follows. The ⋄-equivalence classes are: Z k , H(e), and the ∼-components of F . Set W = Prim(A)/⋄, equipped with the quotient topology, and let p : Prim(A) → W be the quotient map. We show that p is a closed map. Let T be a closed subset of Prim(A) and set T ′ = p −1 (p(T )). Let (Q α ) be a net in T ′ with a limit Q ∈ Prim(A). We must show that Q ∈ T ′ . For each Q α there exists R α ∈ T such that Q α ⋄ R α , and by the compactness of Prim(M(A)) and by passing to subnets of (Q α ) and (R α ), if necessary, we may assume that there exists R ∈ Prim(M(A)) withR α → R. If R =S for some S ∈ Y then Q ∈ Y 1 as required. Otherwise R ∈ Prim(M(A)/A) and hence Q ∈ Y ⊆ Y 1 . Thus Y 1 is closed. Since A is σ-unital, We consider various cases. If (Q α ) is frequently in Z k then Q ∈ Z k , since Z k is closed. Hence Q ⋄ Q α for Q α ∈ Z k and so Q ∈ T ′ since T ′ is ⋄-saturated. A similar argument shows that Q ∈ T ′ if (Q α ) is frequently in H(e). Hence we may restrict attention to the case when Q α ∈ F for all α. This implies that d A (Q α , R α ) ≤ k and hence that for each α there is a walk Q α ∼ Q 1 α ∼ . . . ∼ Q k α = R α (possibly with repetitions) of length k between Q α and R α . HenceQ α ∼ eQ 1 α ∼ e . . . ∼ eQ k α =R α . Using the compactness of Prim(M(A)) \Ũ (e), and passing to successive subnets, we obtain a walkQ ∼ e Q 1 ∼ e . . . ∼ e Q k = R of length k in Prim(M(A)) \Ũ (e) such thatQ, Q 1 , . . . , Q k all lie in (F ) − , the closure ofF in Prim(M(A)). Suppose that P ∈ F − (the closure of F in Prim(A)) and thatP ∼ e P ′ for some P ′ ∈ (F ) − . Then P / ∈ U(e) and P / ∈ Y and hence, by the definition of Y , P ′ =S for some S ∈ Prim(A). Furthermore, S ∈ F − becauseS ∈ (F ) − . Since Q ∈ F − , it follows by induction that Q i =S i (1 ≤ i ≤ k) for some S i ∈ Prim(A), and hence that Q ⋄ S k . But S k ∈ T , since T is closed in Prim(A), and hence Q ∈ T ′ as required. Thus we have shown that p is a closed map. r(H))). Applying this characterization in the present case to each of the points of C and D relative to E ′ and F ′ we obtain ⋄-saturated open sets E ′′ and F ′′ such that C ′ ⊆ E ′′ ⊆ E ′ and D ′ ⊆ F ′′ ⊆ F ′ . Hence p(E ′′ ) and p(F ′′ ) are disjoint open sets of W containing C and D respectively. Thus W is normal. (d) ⊆ H ⊆ O (where H is saturated if H = r −1 ( It follows that there is a positive continuous function f on W with f = 1 such that f (p(Z k )) = {1} and f (p(H(e))) = {0}. Then g = f • p is a continuous function on Prim(A) with g(Z k ) = {1} and g(H(e)) = {0}. Since Prim(A) \ V ⊆ Z k , g has the property required at the start of the proof. Now let A be a C 0 (X)-algebra with base map φ. Then φ factors as φ = ψ • φ A where φ A is the complete regularization map and ψ is continuous. The advantage of the following result over Proposition 5.2 is that ψ is a map between completely regular spaces and should therefore be simpler to analyze. Let X φ A denote the image of Prim(A) under the complete regularization map φ A . Then ψ is a map from X φ A → X φ . By analogy with our earlier definition, we say that ψ is locally closed at a non- empty subset E ⊆ X φ if whenever W is a closed subset of X φ A with ψ(W ) ∩ E = ∅ (i.e. W ∩ ψ −1 (E) = ∅) then ψ(W ) ∩ E = ∅. Corollary 5.4. Let A be a σ-unital C 0 (X)-algebra with base map φ and suppose that Orc(A) < ∞. Write φ = ψ • φ A where φ A is the complete regularization map for Prim(A). Let E be a non-empty closed subset of X φ . Consider the following conditions: (i) H E is strictly closed; (ii) J E is locally modular and ψ : X φ A → X φ is locally closed at E. Then (ii)⇒(i), and (i) and (ii) are equivalent if A is separable. Proof. Set H(E) = φ −1 (E). First, suppose that (ii) holds and let Y be a closed subset of Prim(A) with Y ∩ H(E) empty. Let W be the closure of φ A (Y ) in X φ A . Then W does not meet φ A (H(E)) = ψ −1 (E) by Theorem 5.3. Hence ψ(W ) ∩ E = ∅ and so the closure of ψ(W ) in X φ does not meet E, since ψ is locally closed at E. But φ(Y ) ⊆ ψ(W ) and hence φ(Y ) ∩ E = ∅. Thus φ is locally closed at E and so J E is strictly closed by Proposition 5.2. Conversely, suppose that (i) holds and that A is separable. Then J E is locally modular and φ is locally closed at E by Proposition 5.2. Let W be a closed set in X φ A such that W ∩ψ −1 (E) is empty. Set Y = φ −1 A (W ) . Then Y is closed in Prim(A) and Y ∩H(E) is empty. Hence the closure of φ(Y ) does not meet E. But φ(Y ) = ψ(W ) and hence ψ(W ) ∩ E = ∅. Thus ψ is locally closed at E. In particular, if A in Corollary 5.4 is separable and ψ is a closed map (for example, the identity map when φ = φ A ) then H E is strictly closed if and only if J E is locally modular. Locally modular ideals In this final section we look at locally modular ideals in the case when φ is the complete regularization map and Orc(A) < ∞. We saw immediately after the definition of local modularity that there are two 'easy' ways for J x to be locally modular: if x ∈ U φ or if x is an isolated point in X φ . Example 4.8 gave two examples where J x is locally modular with x ∈ ∂U φ , and in the first of these H(x) has empty interior in Prim(A). We will show that such behaviour cannot occur when φ is the complete regularization map and Orc(A) < ∞. In this case, if J x is locally modular then either x ∈ U φ or H(x) has non-empty interior (Corollary 6.3). Recall that for a C * -algebra A, we say that P, Q ∈ Prim(A) belong to the same Glimm class if f (P ) = f (Q) for all continuous, bounded, real-valued functions f on Prim(A) (equivalently, φ A (P ) = φ A (Q), where φ A is the complete regularization map on Prim(A)). The algebra A + Z(M(A)) in the next result was introduced by Dixmier [11]. Proof. First note that A is an essential ideal in C, so that Prim(A) is (homeomorphic to) a dense open subset of Prim(C). Suppose that Q 1 , Q 2 are distinct elements of Prim(C) \ Prim(A). Then Q i = M i + A where M i is a maximal ideal of Z(M(A)) containing A ∩ Z (i = 1, 2). It follows that Z(M(A)) separates Q 1 and Q 2 . Thus each Glimm class in Prim(C) contains at most one element of Prim(C) \ Prim(A). Hence if P 1 ∼ P 2 ∼ . . . ∼ P n is a path in Prim(C) then at most one element from Prim(C) \ Prim(A) can occur among the P i 's. It follows that d C (P 1 , P n ) ≤ 2 Orc(A) + 2 as required. The next theorem is a general result giving a dichotomy for ∼-components in Prim(A) for any C * -algebra A for which Orc(A) < ∞. Theorem 6.2. Let A be a C 0 (X)-algebra and suppose that φ is the complete regularization map for Prim(A) and that Orc(A) < ∞. Let T be a ∼-component of Prim(A), so that T ⊆ H(x) for some x ∈ X φ . Then either (i) J x ⊇ Z(A) and T = H(x); or (ii) J x ⊇ Z(A) and there exist P ∈ T and R ∈ Prim(M(A)) with R ⊇ A such thatP ∼ R. Proof. Suppose that P ∈ T and R ∈ Prim(M(A)) with R ⊇ A andP ∼ R. Then φ(R) = φ(P ) = φ(P ) = x and it follows from Lemma 2.5 that J x ⊇ Z(A). We must show, therefore, that if there do not exist P ∈ T and R ∈ Prim(M(A)) with R ⊇ A such thatP ∼ R then alternative (i) holds. Suppose, then, that for all P ∈ T and R ∈ Prim(M(A)) with R ⊇ A,P ∼ R. Set k = Orc(A), and let Q ∈ T . Note thatT = {R ∈ Prim(M(A)) : d M (A) (Q, R) ≤ k}, by the supposition that for all P ∈ T and R ∈ Prim(M(A)) with R ⊇ A,P ∼ R. HenceT is a closed (and thus compact) subset of Prim(M(A)) by [33, Corollary 2.3] applied k times. Set L = kerT , a closed ideal of M(A). If A + L were a proper ideal of M(A) there would exist R ∈ Prim(M(A)) such that R ⊇ A + L. Hence R ⊇ L and so R ∈T sinceT is a closed subset of Prim(M(A)), but also R ⊇ A. This is a contradiction, and hence A + L = M(A). Now set C = A + Z(M(A)), and for each P ∈ Prim(A) let P ′ be the unique primitive ideal in C such that P ′ ∩ A = P . Let T ′ := {P ′ : P ∈ T }. We claim that K ∼ P ′ whenever P ∈ T and K ∈ Prim(C) with K ⊇ A. Supposing otherwise, there exist P ∈ T , K ∈ Prim(C) with K ⊇ A, and a net (P α ) in Prim(A) such that P ′ α → P ′ and P ′ α → K. Since K ⊇ A and C ⊆ M(A) = A + L, K ⊇ L ∩ C. Hence there exists c ∈ L ∩ C such that c + K = 1. By lower semi-continuity, c + P ′ α ≥ 1/2 eventually. Hence c +P α ≥ 1/2 eventually (because both c + P ′ α and {(x, y) ∈ Y : ker θ x,y ∈ W } ∩ Y is closed in Y (with the usual topology), and (ii) if G ∈ W then ker θ x,0 ∈ W for all x ∈ R. In particular {G} is an open subset of Prim(A). Set X φ = Y /L and let q : Y → X φ be the quotient map. Set X = βX φ . Define φ : Prim(A) → X φ ⊆ X by φ(θ x,y ) = q(x, y) ((x, y) ∈ Y ) and φ(G) = q(0, 0). Then φ is the complete regularization map for Prim(A) and φ(G) is non-isolated in X φ . For (x, y) ∈ Y \ L, J q(x,y) = ker θ x,y while J q(0,0) = G. Each point of Y has a compact neighbourhood in Y and hence J x is locally modular for each x ∈ X φ , although A/G is non-unital. Since Y is normal, the map φ is easily seen to be relatively closed and hence H x is strictly closed for each x ∈ X φ by Proposition 3.4. Taking z = q(0, 0) = q(G), however, we have that H(q(0, 0)) = {ker θ x,0 : x ∈ R} ∪ {G} and this has non-empty interior {G}. (ii) A C 0 (X)-algebra with x ∈ ∂U φ such that J x is locally modular, φ is locally closed at x, and H(x) has empty interior. Let A be the C * -algebra defined as follows (see [33,Example 2.8]). Let B be the C * -algebra consisting of all continuous functions from the interval [0, 1] into the 2 × 2 complex matrices. Let B(1) be the C * -subalgebra of B consisting of those functions f ∈ B satisfying f (2 −n ) = diag(λ 2n−1 (f ), λ 2n (f )), (n ≥ 1), and f (0) = diag(λ(f ), λ(f )), for some complex numbers λ(f ), λ n (f ) (n ≥ 1). Let A = {f ∈ B(1) : λ 2n (f ) = λ 2n+1 (f ) (1 ≤ n < ∞) and λ(f ) = 0}. Then A is separable. Set X φ = (0, 1]/ * where for r, s ∈ (0, 1], r * s if r = s or if r, s ∈ {2 −n : n ≥ 1} and let q : (0, 1] → X φ denote the quotient map. Set X = βX φ and ∞ = q(1/2). Define φ : Prim(A) → X by φ(P y ) = y (y ∈ (0, 1] \ {2 −n : n ≥ 1}) and φ(λ i ) = ∞ (i = 1, 3, 5, . . .). Then φ is the complete regularization map for Prim(A). If x ∈ X φ \ {∞} then J x ⊇ Z(A) = Z ′ (A) and hence x ∈ U φ and J x is locally modular. It is easy to see directly that J ∞ is also locally modular. But A/J ∞ is non-unital, since Prim(A/J ∞ ) = {ker(λ i ) : i = 1, 3, 5, . . .} is non-compact, and hence J ∞ ⊇ Z(A). Thus U φ = X φ \{∞} and W φ = {∞}. We show that φ is a relatively closed map. Let Y be a closed subset of Prim(A) and set Y ′ = φ −1 (φ(Y )). Then Y ′ = Y if Y ∩ {ker(λ i ) : i = 1, 3, 5, . . .} is empty, and Y ′ = Y ∪ {ker(λ i ) : i = 1, 3, 5, . . .} otherwise. In either case Y ′ is closed, and hence φ is relatively closed. It follows, therefore, from Proposition 3.4 that H x is strictly closed for every x ∈ X φ . For x ∈ X φ , let J x be the closed ideal of A defined by J x = {P ∈ Prim(A) : φ(P ) = x} and let H x be the closed ideal of M(A) defined by H x = {Q ∈ Prim(M(A)) : φ(Q) = x}. LetJ x be the strict closure of J x in M(A). Then J x ⊆ H x ⊆J x and hence H x is strictly closed if and only if H x =J x (see Proposition 2. 35, Proposition C.10 and Theorem C.26], this happens if and only if the base map φ is open. Let J be a proper, closed, two-sided ideal of a C * -algebra A. The quotient map q J : A → A/J has a canonical extensionq J : M(A) → M(A/J). We define a proper, closed, two-sided idealJ of M(A) byJ = kerq J = {b ∈ M(A) : ba, ab ∈ J for all a ∈ A}. The following proposition was proved in [6, Proposition 1.1]. Proposition 2.1. Let J be a proper, closed, two-sided ideal of a C * -algebra A. Then (i)J is the strict closure of J in M(A); (ii)J ∩ A = J; (iii) if P ∈ Prim(A) thenP is primitive (and hence is the unique ideal in Prim(M(A)) whose intersection with A is P ); (iv)J = {P : P ∈ Prim(A) and P ⊇ J} and for all b ∈ M(A) b +J = sup{ b +P : P ∈ Prim(A) and P ⊇ J}; (v) (A +J)/J is an essential ideal in M(A)/J . Furthermore, the map P →P (P ∈ Prim(A)) maps Prim(A) homeomorphically onto a dense, open subset of Prim(M(A)) [29, 4.1.10]. For S ⊆ Prim(A), we writeS = {P : P ∈ S}. J E = {P ∈ Prim(A) : φ(P ) ∈ E}, H E = {Q ∈ Prim(M(A)) : φ(P ) ∈ E} and H alg E = {b ∈ M(A) : E ⊆ Int Z(b)}. q and so q(U) is an open neighbourhood of B and is disjoint from q(V ). If H ⊇ E then V is saturated and so q(V ) is open, and if H ⊇ E then V \ E is a saturated open set containing H and so q(V \ E) is an open neighbourhood of C. Hence Y is normal, and being T 1 , it is also Hausdorff and completely regular. Lemma 3 . 1 . 31Let A be a σ-unital C * -algebra and let Y and Z be disjoint closed subsets of Prim(A). Then the closures ofỸ andZ are disjoint in Prim(M(A)). the cozero set of f and let V = {x ∈ U : 2 min sp(u + J x ) ≤ f (x)}. Let cl(U) and cl(V ) be the closures of U and V respectively in X φ . Then there exists b ∈ M(A) with 0 ≤ 3 that there exist P ∈ ∂H(x) and R ∈ Prim(M(A)/A) such thatP ∼ x R. Let D be a compact neighbourhood of P in Prim(A). Let u ∈ A be a strictly positive element with u = 1. Since u ∈ A, it follows that (1 − u) + R = 1. Hence by [12, 3.3.2], for each n the set U n = {Q ∈ Prim(M(A)) : (1 − u) + Q > 1 − 1/2 n+2 } is an open neighbourhood of R in Prim(M(A)). Let h : X φ → [0, 1] be a continuous function such that Z(h) = Z and, for n ≥ 1, let V n = {Q ∈ Prim(A) : h(φ(Q)) < 1/n}. Then V n is an open neighbourhood of P in Prim(A). SinceP ∼ x R, it follows that W n is a compact subset of X φ . But x / ∈ W n and hence there exists a continuous functionf n : X φ → [0, 1] such that f n (x) = 0 and f n (W n ) = {1}. Set f = ∞ n=1 f n /2 n . Then f (x) = 0, but x lies in the closure of the cozero set of f . Hence x is not a P-point. Corollary 4 . 4 . 44Let A be a C 0 (X)-algebra with base map φ and suppose that Z ′ (A) = {0}. (A) = {0} and hence Z ′ (A) = {0}. Thus Corollary 4.4 extends [7, Theorem 4.5]. be the upper half-plane, and let L = {(x, y) ∈ Y : y = 0} be the x-axis. Let Y /L be the quotient space (which is completely regular by Lemma 2.8). Set A = C 0 (Y ). Then we may identify Prim(A) with Y in the usual way and define φ : Prim(A) → β(Y /L) by φ((x, y)) = [(x, y)] ∈ Y /L. Thus X φ = Y /L. Then J [(0,0)] is locally modular (since every point in L has a compact neighbourhood in Y ) and φ is locally closed at [(0, 0)] (since Y is normal). However, [(0, 0)] / ∈ U φ because if f ∈ C(βY /L) and f • φ ∈ A = C 0 (Y ) then f ([(0, 0)]) = 0. Q cannot be separated by disjoint open sets in Prim(A). The relation ∼ on Prim(A) induces a graph structure on Prim(A) whereby P and Q are adjacent if P ∼ Q. The distance d A (P, Q) between P and Q is then defined as the length of the shortest path from P to Q (and is ∞ if no such path exists). The diameter of a ∼-component of Prim(A) is the supremum of the distances between primitive ideals in the component (with the convention that a singleton component, such as when Prim(A) is Hausdorff, has diameter 1). The connecting order, Orc(A), is the supremum of the diameters of ∼-components of Prim(A). Clearly Orc(A) is an integer between 1 and ∞, and all possibilities occur, including ∞ [33, Example 2.8] (see also Example 6.4(ii) below). The smaller that Orc(A) is, the nearer Prim(A) is to being Hausdorff, with the case Orc(A) = 1 corresponding to ∼ being an equivalence relation on Prim(A). For a subset Y ⊆ Prim(A) and for n ≥ 0, let Y n = {P ∈ Prim(A) : ∃ Q ∈ Y, d A (P, Q) ≤ n}. the hull of J E in Prim(A), and recall that U(e) is the interior of H(e) in Prim(A). For Q, R ∈ Prim(M(A)) \Ũ (e), recall that we write Q ∼ e R if there is a net (P α ) in Prim(A) \ U(e) such thatP α → Q, R. be the quotient map and {e} = q(E). By Proposition 5.2, it suffices to show that φ is locally closed at E. Let V be a proper open subset of Prim(A) with V ⊇ H(e). It suffices to produce a continuous function g on Prim(A) with g(Prim(A) \ V ) = {1} and g(H(e)) = {0} (for then g induces a continuous function on φ(Prim(A)) separating φ(Prim(A) \ V ) and E, so that φ(Prim(A) \ V ) ∩ E = ∅). Set Y = {Q ∈ Prim(A) \ U(e) : ∃R ∈ Prim(M(A)/A) withQ ∼ e R}. Then Y ∩ H(e) is empty by Lemma 3.3. We claim that Y is a closed subset of Prim(A). To see this, let (Q α ) be a net in Y with Q α → Q ∈ Prim(A). Then Q / ∈ U(e) since Prim(A) \ U(e) is closed. By definition of Y , there is a net (R α ) in Prim(M(A)/A) withQ α ∼ e R α for each α. By the compactness of Prim(M(A)/A), there is a subnet ( Prim(A) is σ-compact and hence every closed subset of Prim(A) is a Lindelof space. Thus, since H(e) is closed and ∼-saturated, we may apply Lemma 5.1 to H(e) and Y to obtain disjoint open subsets V ′ and V ′′ of Prim(A) with H(e) ⊆ V ′ and Y ⊆ V ′′ . By intersecting V ′ with the open set V of the first paragraph, we may assume that V ′ ⊆ V . Set Z = Prim(A) \ V ′ . Then Z ⊇ V ′′ ⊇ Y and the boundary of Z in Prim(A) does not meet V ′′ . Let k = Orc(A). Then the same argument that showed that Y 1 is closed also shows, inductively, that Z 1 , . . . , Z k are closed. Note that if Q belongs to the boundary of Z k in Prim(A) then Q / ∈ V ′′ and so Q / ∈ Y . Thus there does not exist R ∈ Prim(M(A)/A) withQ ∼ e R. Now let C and D be non-empty, disjoint closed subsets of W . Then C ′ := p −1 (C) and D ′ := p −1 (D) are disjoint closed ∼-saturated subsets of Prim(A). Thus C ′ and D ′ are Lindelof and so Lemma 5.1 implies the existence of disjoint open sets E ′ and F ′ containing C ′ and D ′ respectively. We now use a standard characterization (see e.g. [24, §13.XIV Theorem 3]): a quotient map r : M → N is closed if and only if whenever d ∈ N and O is an open set containing r −1 (d) then there exists a saturated open set H such that r −1 Lemma 6 . 1 . 61Let A be a C * -algebra with Orc(A) < ∞ and let C = A + Z(M(A)). Then Orc(C) ≤ 2 Orc(A) + 2. For y ∈ (0, 1] \ {2 −n : n ≥ 1}, set P y = {f ∈ A : f (y) = (0)}. Then Prim(A) = {P y : y ∈ (0, 1] \ {2 −n : n ≥ 1}} ∪ {ker(λ i ) : i = 1, 3, 5, . . .}. 21, Theorem 10.1.7], there existsb ∈J x \ H x with b = b + H x = 1.Let u be a strictly positive element of A with u = 1, and recall that for y ∈ X φ , b ∈J y if and only if bu ∈J y (cf.[6, Section 2]). For each n ≥ 1, set W n = {y ∈ X φ : bu +J y ≥ 1/n}. By[7, Lemma 4.2], for every neighbourhood W of x in cl βX X n := X φ \ φ(V n ) = {y ∈ X φ : (1 − u) +J y ≤ 1 − 1/2 n+1 } By the supposition of the second paragraph, it must be that R =S for some S ∈ Prim(A). Hence S ∈ T . Thus we have c ∈ L ⊆S = R, contradicting the fact that c+R ≥ 1/2. It follows, then, that. K ∈ K ∼ P ′ Whenever P ∈ T, Prim, compactness of the set {S ∈ Prim(M(A)) : c + S ≥ 1/2}, we may assume, without loss of generality, that P α → R in Prim(M(A)). where c + R ≥ 1/2. Since P ′ α → P ′ we have P α → P andP α →P. C) with K ⊇ A, and hence that T ′ is a ∼-component in Prim(C)compactness of the set {S ∈ Prim(M(A)) : c + S ≥ 1/2}, we may assume, without loss of generality, that P α → R in Prim(M(A)), where c + R ≥ 1/2. Since P ′ α → P ′ we have P α → P andP α →P . ThusP ∼ R. By the supposition of the second paragraph, it must be that R =S for some S ∈ Prim(A). Hence S ∈ T . Thus we have c ∈ L ⊆S = R, contradicting the fact that c+R ≥ 1/2. It follows, then, that K ∼ P ′ whenever P ∈ T and K ∈ Prim(C) with K ⊇ A, and hence that T ′ is a ∼-component in Prim(C). Thus T = H(x). Now, let P ∈ T , and let φ C : Prim(C) → Y be the complete regularization map for Prim(C), where Y is the space of Glimm ideals of C with the induced completely regular topology. By Lemma 6.1, Orc(C) ≤ 2 Orc(A) + 2 < ∞, and since Prim33C) is compact it follows that T ′ is a Glimm class in Prim(C). Then the set W = φ C (Prim(C) \ Prim(A)By Lemma 6.1, Orc(C) ≤ 2 Orc(A) + 2 < ∞, and since Prim(C) is compact it follows that T ′ is a Glimm class in Prim(C) [33, Corollary 2.7]. It follows at once that T is a Glimm class in Prim(A). Thus T = H(x). Now, let P ∈ T , and let φ C : Prim(C) → Y be the complete regularization map for Prim(C), where Y is the space of Glimm ideals of C with the induced completely regular topology. Then the set W = φ C (Prim(C) \ Prim(A)) . ∈ , ′ , P ′ ⊇ P ⊇ J X Thus Z / ∈ J X, Hence J x ⊇ Z(A). ∈ P ′ , and P ′ ⊇ P ⊇ J x , and thus z / ∈ J x . Hence J x ⊇ Z(A). By Theorem 6.2 there exists P ∈ T and R ∈ Prim(M(A)/A) such thatP ∼ R. Since H(x) has empty interior, P ∈ ∂H(x) andP ∼ x R. By Lemma 3.3, J x is not locally modular. We conclude with two examples of x ∈ ∂U φ with J x locally modular. The first has Orc(A) < ∞ and H(x) with non-empty interior. The second has Orc(A) = ∞ and H(x) with empty interior. Corollary 6.3. Let A be a C 0 (X)-algebra and suppose that φ is the complete regularization map for Prim(A) and that Orc(A) < ∞. Let x ∈ X φ with J x locally modular. Then either x ∈ U φ or H(x) has non-empty interior. Proof. Suppose that x / ∈ U φ and that H(x) has empty interior. Let T be a ∼-component of H(x). showing that the condition Orc(A) < ∞ in Corollary 6.3 is not redundantCorollary 6.3. Let A be a C 0 (X)-algebra and suppose that φ is the complete regularization map for Prim(A) and that Orc(A) < ∞. Let x ∈ X φ with J x locally modular. Then either x ∈ U φ or H(x) has non-empty interior. Proof. Suppose that x / ∈ U φ and that H(x) has empty interior. Let T be a ∼-component of H(x). By Theorem 6.2 there exists P ∈ T and R ∈ Prim(M(A)/A) such thatP ∼ R. Since H(x) has empty interior, P ∈ ∂H(x) andP ∼ x R. By Lemma 3.3, J x is not locally modular. We conclude with two examples of x ∈ ∂U φ with J x locally modular. The first has Orc(A) < ∞ and H(x) with non-empty interior. The second has Orc(A) = ∞ and H(x) with empty interior, showing that the condition Orc(A) < ∞ in Corollary 6.3 is not redundant. As in Example 4.8(i), let Y = {(x, y) ∈ R 2 : y ≥ 0} be the upper half-plane, and let L = {(x, y) ∈ Y : y = 0} be the x-axis. Set B = C 0 (Y ) and C = C 0 (L), and let π : B → C be the surjective * -homomorphism given by π(b) = b\| L (b ∈ B). Let H be a separable, infinite-dimensional Hilbert space, B(H) the algebra of bounded operators on H, and K(H) the algebra of compact operators on H. Let ρ : C → B(H) be a * -monomorphism such that ρ(C) ∩ K(H) = {0}. Set D = ρ(C) + K(H), a C * -subalgebra of B(H). )-algebra with z ∈ ∂U φ such that J z is locally modular, φ is locally closed at z, and H(z) has non-empty interior. Example 6.4. (i) A C 0 (X. Note that each element d ∈ D can be uniquely expressed in the form d = g + T where g ∈ ρ(C) and T ∈ K(H)Example 6.4. (i) A C 0 (X)-algebra with z ∈ ∂U φ such that J z is locally modular, φ is locally closed at z, and H(z) has non-empty interior. As in Example 4.8(i), let Y = {(x, y) ∈ R 2 : y ≥ 0} be the upper half-plane, and let L = {(x, y) ∈ Y : y = 0} be the x-axis. Set B = C 0 (Y ) and C = C 0 (L), and let π : B → C be the surjective * -homomorphism given by π(b) = b\| L (b ∈ B). Let H be a separable, infinite-dimensional Hilbert space, B(H) the algebra of bounded operators on H, and K(H) the algebra of compact operators on H. Let ρ : C → B(H) be a * -monomorphism such that ρ(C) ∩ K(H) = {0}. Set D = ρ(C) + K(H), a C * -subalgebra of B(H). Note that each element d ∈ D can be uniquely expressed in the form d = g + T where g ∈ ρ(C) and T ∈ K(H). For (x, y) ∈ Y , let θ x,y be the character on A given by θ x,y (b, d) = b((x, y)). Set G = {(b, d) ∈ A : π(b) = 0, T = 0}. Since any irreducible representation of A extends to an irreducible representation of B ⊕ D (on a possibly larger Hilbert space), Prim(A) = {ker θ x,y : (x, y) ∈ Y } ∪ {G}. Note that G ⊆ ker θ x,0 for all x ∈ R. Set A = {(b, d) ∈ B ⊕ D : ρ(π(b)) = g}. Then A is separable. It follows that a subset W of Prim(A) is closed if and only if (i) ReferencesSet A = {(b, d) ∈ B ⊕ D : ρ(π(b)) = g}. Then A is separable. For (x, y) ∈ Y , let θ x,y be the character on A given by θ x,y (b, d) = b((x, y)). 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[ "Effects of partial measurements on teleportation of quantum resources and quantum Fisher information in a relativistic scenario", "Effects of partial measurements on teleportation of quantum resources and quantum Fisher information in a relativistic scenario" ]	[ "M Jafarzadeh \nPhysics Department\nFaculty of Sciences\nUrmia University\nP.B. 165UrmiaIran\n", "H Rangani Jahromi \nPhysics Department\nFaculty of Sciences\nJahrom University\nJahromIran ARTICLE HISTORY\n", "M Amniat-Talab \nPhysics Department\nFaculty of Sciences\nUrmia University\nP.B. 165UrmiaIran\n" ]	[ "Physics Department\nFaculty of Sciences\nUrmia University\nP.B. 165UrmiaIran", "Physics Department\nFaculty of Sciences\nJahrom University\nJahromIran ARTICLE HISTORY", "Physics Department\nFaculty of Sciences\nUrmia University\nP.B. 165UrmiaIran" ]	[]	Quantum resources (QRs) and quantum Fisher information (QFI) of an input state, usually degrade during the teleportation under the Unruh effect. In this paper, we address the teleportation of a single and a two-qubit quantum state through the Unruh effect experienced by a mode of a free Dirac field, and study the effects of the partial measurement (PM) and partial measurement reversal (PMR) on the QRs and QFI of the teleported states. We investigated how the teleported QRs and QFI can be improved with the combined effect of PM and PMR for both single-qubit and two-qubit teleportation. We also consider how we can control the behavior of the QFI, quantum coherence (QC) as well as fidelity, fixing the acceleration, in single-qubit state teleportation. Our results show that QFI with respect to weight parameter (F P M out (θ)) enhances with the increase in measurements strength. In addition, we discuss in detail the optimal behavior of the QFI associated with the phase parameter (F P M out (ϕ)), QC as well as fidelity with respect to the PM and PMR strength and examine the Unruh effect on optimal estimation. In particular, it is found that in the single-qubit scenario, the PM strength (PMR strength), with which the optimal estimation of phase parameter occurs, is the same strength with which the teleportation quality and the QC of the output single-qubit state reaches to its maximum value. On the other hand, generalizing the results for two-qubit teleportation and comparing the teleported QFIs in both single and two-qubit scenarios, we find that the encoded information in the weight parameter is better protected against the Unruh effect in the process of two-qubit teleportation. However, extraction of information encoded in the phase parameter is more efficient in single-qubit teleportation than the two-qubit one.	null	[ "https://arxiv.org/pdf/1902.02089v1.pdf" ]	119,196,050	1902.02089	a99e16df5c0f4742d60d4005f343de2beb62f675	Effects of partial measurements on teleportation of quantum resources and quantum Fisher information in a relativistic scenario 6 Feb 2019 M Jafarzadeh Physics Department Faculty of Sciences Urmia University P.B. 165UrmiaIran H Rangani Jahromi Physics Department Faculty of Sciences Jahrom University JahromIran ARTICLE HISTORY M Amniat-Talab Physics Department Faculty of Sciences Urmia University P.B. 165UrmiaIran Effects of partial measurements on teleportation of quantum resources and quantum Fisher information in a relativistic scenario 6 Feb 2019Compiled February 7, 2019CLASSIFICATION 0367-a, 0367Hk, 0367Lx, 0365Yz, 0365Ta, 0365Ud KEYWORDS TeleportationUnruh effectpartial measurementquantum resourcequantum Fisher informationfidelity Quantum resources (QRs) and quantum Fisher information (QFI) of an input state, usually degrade during the teleportation under the Unruh effect. In this paper, we address the teleportation of a single and a two-qubit quantum state through the Unruh effect experienced by a mode of a free Dirac field, and study the effects of the partial measurement (PM) and partial measurement reversal (PMR) on the QRs and QFI of the teleported states. We investigated how the teleported QRs and QFI can be improved with the combined effect of PM and PMR for both single-qubit and two-qubit teleportation. We also consider how we can control the behavior of the QFI, quantum coherence (QC) as well as fidelity, fixing the acceleration, in single-qubit state teleportation. Our results show that QFI with respect to weight parameter (F P M out (θ)) enhances with the increase in measurements strength. In addition, we discuss in detail the optimal behavior of the QFI associated with the phase parameter (F P M out (ϕ)), QC as well as fidelity with respect to the PM and PMR strength and examine the Unruh effect on optimal estimation. In particular, it is found that in the single-qubit scenario, the PM strength (PMR strength), with which the optimal estimation of phase parameter occurs, is the same strength with which the teleportation quality and the QC of the output single-qubit state reaches to its maximum value. On the other hand, generalizing the results for two-qubit teleportation and comparing the teleported QFIs in both single and two-qubit scenarios, we find that the encoded information in the weight parameter is better protected against the Unruh effect in the process of two-qubit teleportation. However, extraction of information encoded in the phase parameter is more efficient in single-qubit teleportation than the two-qubit one. Introduction Quantum teleportation (1 ) is undoubtedly one of the most striking implications predicted by quantum mechanics and it is an important ingredient for quantum communication and quantum information processing (QIP) (2 , 3 ). In the last decades theoretical and experimental consideration of quantum teleportation has attracted many researchers' attention (4 -13 ). Quantum teleportation is described as a process by which an arbitrary unknown quantum state can be transmitted faithfully from one object to another, without physical traveling of the object itself. The system is isolated from the external forces in the original form of the teleportation (1 ), and a maximally entangled pair is used as the resource. However, decoherence (14 , 15 ) is an inevitable phenomenon in open quantum systems which takes place due to the interaction between the system and environment. This leads to the degradation of quantum correlations, a fundamental resource for QIP, and therefore influences the fidelity in quantum state teleportation (16 -18 ). Relativistic quantum information (RQI) (19 , 20 ) aims to realize the relationship between relativity as well as quantum information, and combine relativistic effects to amend quantum information tasks, e.g., quantum teleportation. Moreover, in RQI we try to understand how these protocols may be realized in curved space time. Unruh effect (21 , 22 ), a significant prediction in quantum field theory, proposes that a uniformly accelerated observer in Minkowski spacetime (Rindler observer) associates a thermal bath of Rindler particles to the no particle state of inertial observer (called Minkowski vacuum). The decoherence effect, produced by the Unruh effect, suppresses the quantum resources (QR) such as quantum coherence (23 ), quantum discord (24 , 25 ) and entanglement (25 ) in the case of bosonic or Dirac field modes. The degradation of QRs unavoidably decreases the confidence of some quantum information tasks like quantum teleportation. In this context it is really important to preserve QRs from decoherence during the teleportation process. Here we investigate the teleportation of a single and a two-qubit quantum state through the Unruh effect experienced by a mode of a free Dirac field, considered as a noise channel which we name it Unruh channel In addition to the teleportation of the whole quantum state, we also investigate the teleportation of the information encoded into a particular parameter. In contrast to quantum state teleportation where the quality of teleportation is characterized by fidelity, the credibility of teleportation of specific information is usually determined by quantum Fisher information (QFI) (26 -29 ). QFI, representing the sensitivity of the state with respect to changes in a parameter, plays an important role in parameter estimation theory and is extensively employed in QIP. In particular, QFI has many applications in quantum information tasks such as entanglement detection (30 , 31 ), specifying the non-Markovianity (32 -34 ), and consideration of uncertainty relations (35 -37 ). Hence it is of interest to study QFI in relativistic framework. Nevertheless, it is shown that the QFI is fragile and can be broken easily because of unavoidable decoherence effects (38 -42 ). This is the most restricting factor in QFI applications for quantum teleportation. Therefore, protecting the QFI from decoherence is a fundamental subject. In weak or partial measurement (PM), associated with a positive-operator valued measure (POVM), the system state does not completely collapse such that the initial state could be reversed with some operations. Recently, PMs together with partial measurement reversals (PMR) have been exploited as a practical method to protect quantum correlations of two-qubits as well as two-qutrits and the fidelity of a singlequbit, from amplitude damping (AD) decoherence (43 -47 ). In ref. (48 ) the effect of partial measurements on the teleportation of QFI for a single-qubit state under the amplitude damping noise has been studied and it has been illustrated that the combination of PM and PMR could totally eliminate the influence of decoherence. The effects of PM and PMR on the enhancement of quantum coherence and QFI, transmitted under a quantum spin-chain channel, have been considered in ref. (49 ). Moreover it has been shown that PM and PMR are able to improve the fidelity of teleportation when one or both qubits of the maximally entangled state shared between Alice and Bob suffer from the AD decoherence (50 ). It was also shown in ref (50 ) that this protocol works for the Werner states. However limited attention has been paid to protect the QRs and QFI against Unruh decoherence during the procedure of teleportation. Motivated by this, we study the enhancement effect of PM and PMR on teleportation of QRs and QFI through the Unruh noise channel for both single and two-qubit input quantum states. In this paper, we have investigated the following scenario: the system consists of an inertial observer Alice and a uniformly accelerated observer Rob. Two PMs are performed before and after Robs acceleration, which are called PM and PMR, respectively. Then we use the above mentioned system as a resource in order to teleport a single and a two-qubit state, and consider how the degradation effect of the Unruh channel on the teleportation of QRs and QFI as well as teleportation fidelity can be improved by PM or PMR. According to our results, the combined effect of PM and PMR with the same strengths (p = q) may improve the teleportation of QRs and QFI with respect to phase parameter ϕ, and also teleportation fidelity in both single-qubit and two-qubit scenarios. This paper is organized as follows: In Sec. II we give a brief description about teleportation, PM, PMR, QRs and QFI. The physical model is presented in Sec. III. We study the single-qubit teleportation as well as two-qubit teleportation under the Unruh noise channel in Sec. IV and Sec. V, respectively. Finally, Sec. VI is devoted to conclusion. PRELIMINARIES Teleportation The main idea of quantum teleportation is transferring quantum information about an unknown quantum state to another location where it is spatially separated. An important factor in quantum teleportation is the channel connecting sender and receiver. In standard teleportation protocol T 0 , local quantum operations, used to teleport the input state, includes Bell measurements and Pauli rotations. According to Bowen and Bose results, the standard teleportation protocol T 0 with mixed states as resource is tantamount to a generalized depolarizing channel (51 ). Single-qubit teleportation As mentioned above, teleportation protocol using a two-qubit mixed state as a resource, acts as a generalized depolarizing channel, therefore the output state for a teleported single-qubit state is obtained as follows (51 ) ρ out = Λ (ρ ch ) ρ in , = 3 i=0 Tr (B i ρ ch ) σ i ρ in σ i(1) where B i are the Bell states associated with the Pauli matrices σ i , B i = (σ 0 ⊗ σ i ) B 0 (σ 0 ⊗ σ i ) , i = 1, 2, 3(2) in which σ 0 = I, σ 1 = σ x , σ 2 = σ y and σ 3 = σ z . Moreover, we have B 0 = 1 2 (\|00 + \|11 ) ( 00\| + 11\|), without loss of the generality. Two-qubit teleportation Teleportation of an unknown entangled state via two independent, equally entangled quantum channels has been studied by Lee and Kim (52 ). Actually, their protocol may be carried out by doubling the standard teleportation protocol T 0 . Figure 1 displays the schematic drawing of entanglement teleportation. An unknown entangled state ρ in is generated by source S, and its particles are dispensed into A 1 and A 2 . Besides, two independent entangled pairs (one of them numbered 3 and 5, the other pair numbered 4 and 6) are produced from source E. These pairs, each characterized by density matrix ρ ch , play the role of the quantum channel. The measurement result at A i (i = 1, 2) is transmitted through the classical channel C i to B i . Based on the information received by the classical communication, the unitary transformations are done on the particles 5 and 6 received at B i (i = 1, 2) to complete the teleportation. Generalizing equation (1), the output state of the entanglement teleportation is found as follows ρ out = ij p ij (σ i ⊗ σ j ) ρ in (σ i ⊗ σ j ) , i, j = 0, x, y, z.(3) where p ij = Tr E i ρ ch Tr E j ρ ch and p ij = 1. Here E 0 = \|ψ − ψ − \|, E 1 = \|φ − φ − \|, E 2 = \|φ + φ + \|, E 3 = \|ψ + ψ + \| and \|ψ ± = \|01 ±\|10 2 as well as \|φ ± = \|00 ±\|11 2 are Bell states. Partial measurement (PM) and partial measurement reversal(PMR) We first give a brief introduction about the PM and PMR. In contrast with the standard von Neumann projective measurement, which completely collapses the measured system, PM, as a generalization of standard von Neumann projective measurement, does not totally collapse the initial state into the eigenstates, and hence it is reversible. For a single-qubit, the PM is described by the following pair of measurement operators: M 0 = 1 − p\|0 0\| + \|1 1\|,(4)M 1 = √ p\|0 0\|,(5) where p (0 ≤ p ≤ 1) is the strength of PM and M † 0 M 0 + M † 1 M 1 = I. M 1 is identical to von Neumann projective measurement and is irreversible, while M 0 is a PM that we are interested in this study. In order to reverse the effect of the PM, i.e., in order to recover the primary state, we need to use the inverse of M 0 , M −1 0 = 1 √ 1 − q 0 1 1 0 √ 1 − q 0 0 1 0 1 1 0 = 1 √ 1 − q XM 0 X,(6) where X = \|0 1\| + \|1 0\| is the bit-flip operation. Last term of Eq.(6) implies that the reverse procedure M 0 , can be implemented physically by the sequence of a bit-flip operation, another PM with measurement strength q, and a second bit-flip operation. Quantum Fisher information QFI is an important concept in parameter estimation theory. QFI of an unknown parameter λ encoded in quantum state ρ (λ) is defined as (26 , 57 ) F Q (λ) = Tr ρ (λ) L 2 = Tr [(∂ λ ρ (λ)) L] ,(7) where L, the symmetric logarithmic derivative (SLD), is given by ∂ λ ρ (λ) = 1 2 (Lρ (λ) + ρ (λ) L) , with ∂ λ = ∂/∂λ. Using the spectrum decomposition of ρ (λ), ρ (λ) = i p i \|φ i φ i \|, where \|φ i and p i are eigenvectors and eigenvalues of the matrix ρ (x), respectively; one can rewrite the QFI as follows (58 ) F Q (λ) = i,j 2 p i + p j \| φ i \|∂ λ ρ (λ) \|φ j \| 2 = i (∂ λ p i ) 2 p i + 2 i =j (p i − p j ) 2 p i + p j \| φ i \|∂ λ φ j \| 2 ,(8) A simple and explicit expression can be acquired for the single-qubit state. Any qubit state can be expressed in the Bloch sphere representation as ρ = 1 2 (I + ω · σ)(9) where ω = (ω x , ω y , ω z ) T is the Bloch vector and σ = (σ x , σ y , σ z ) indicates the Pauli matrices. Hence the QFI of the single-qubit state can be formulated as follows (59 ) F Q (λ) = \|∂ λ ω\| 2 + (ω·∂λω) 2 1−\|ω\| 2 , \|ω\| < 1, \|∂ λ ω\| 2 , \|ω\| = 1.(10) where \|ω\| < 1 is used for a mixed state while \|ω\| = 1 is applicable for a pure state. Quantum resources Quantum coherence. Quantum coherence (QC) arising from the superposition principle is an important resource in quantum information and quantum computation processing. It plays a fundamental role in quantum mechanics. Various measures are expressed to quantify the coherence such as, trace norm distance coherence (53 ), l 1 norm, and relative entropy of coherence (54 ). For a quantum state with the density matrix ρ, the l 1 norm measure of quantum coherence (54 ) quantifying the coherence through the off diagonal elements of the density matrix in the reference basis, is given by C l1 (ρ) = i,j i =j \|ρ ij \|(11) Entanglement. Entanglement is recognized as a resource in quantum information processing (QIP) and is accountable to the advantage of many quantum computation and communication tasks. Actually, entanglement indicates correlations regarding non separability of the state of a composite quantum system. Entanglement of a bipartite system is quantified conveniently by concurrence (55 ) which can be computed analytically for a X state as follows C(ρ) = 2max {0, C 1 (ρ), C 2 (ρ)} ,(12) where C 1 (ρ) = \|ρ 14 \| − √ ρ 22 ρ 33 , C 2 (ρ) = \|ρ 23 \| − √ ρ 11 ρ 44 , and ρ ij 's are the elements of density matrix. Concurrence equals unity for maximally entangled states and vanishes for separable states. Quantum discord. Quantum discord representing quantumness of the state of quantum system is a resource for certain quantum technologies. It can be preserved for a long time even when entanglement shows a sudden death. QD for any bipartite system is defined as difference between total correlations (i.e., quantum mutual information) and classical correlations. Computation of QD for general states is not usually a convenient task since it involves the optimization of the classical correlations. However, for a two-qubit X state system, the analytical expression of QD can be obtained as (56 ) QD(ρ AB ) = min (Q 1 , Q 2 ) ,(13) where Q j = H (ρ 11 + ρ 33 ) + 4 i=1 λ i log 2 λ i + D j , (j = 1, 2) , D 1 = H   1 + [1 − 2 (ρ 33 + ρ 44 )] 2 + 4 (\|ρ 14 \| + \|ρ 23 \|) 2 2   ,(14)D 2 = − i ρ ii log 2 ρ ii − H (ρ 11 + ρ 33 ) , H (x) = −xlog 2 x − (1 − x) log 2 (1 − x) , and λ i 's denote the eigenvalues of density matrix ρ AB . Physical model First we consider a free Minkowski Dirac field ψ in 3+1 dimensions iγ µ ∂ µ ψ − mψ = 0,(15) where γ µ are the Dirac gamma matrices, m is the particle mass and ψ is a spinor wave function. The field can be represented from the perspective of inertial and uniformly accelerated observers (see details in (60 )). We investigate a system including an inertial observer Alice (A) and a uniformly accelerated observer Rob (R). For the inertial observer, Minkowski coordinates (t, z) are the most proper coordinates to describe the field. The field can be expanded in terms of positive and negative frequency Minkowski modes ψ + k and ψ − k , which they form a complete orthonormal set, ψ = dk a k ψ + k + b † k ψ − k(16) where the wave vector k represents the modes of massive Dirac fields. Moreover, a † k , b † k and a k , b k denote, respectively, the creation and annihilation operators for positive and negative frequency modes of momentum k. Since Rob is the uniformly accelerated observer, in order to describe what he sees, Rindler coordinates (τ, ξ) should be used. As it can be seen in Fig. 2, Rindler spacetime manifests two regions I and II, causally disconnected. Because of the eternal acceleration, Rob travels on a hyperbola compelled in the region I. Compared with Eq. (16), Dirac field can be expanded in terms of positive and negative frequency Rindler modes ψ = dk c I k ψ I+ k + d I † k ψ I− k + c II k ψ II+ k + d II † k ψ II− k(17) where c n k , c n † k represent the annihilation and creation operators for Rindler particle and d n k , d n † k denote those of the antiparticle, in the region n with n = I, II. The Minkowski and Rindler creation and annihilation operators are connected via the Bogoliubov transformation a k = cos rc I k − sin rd II † −k , b † −k = sin rc I k + cos rd II † −k ,(18)where r = arccos 1 + e −2πω a , ω is the Dirac particle frequency and a is the acceleration. Since 0 < a < ∞, therefore r ∈ [0, π/4]. Let Alice and Rob initially share the following entangled state at a point in Minkowski spacetime \|Ψ (0) = sin ϑ 2 \|0 A \|0 R + cos ϑ 2 \|1 A \|1 R .(19) As it can be seen in Fig. 2, Rindler spacetime manifests two regions I and II, which are causally disconnected. Because of the eternal acceleration, Rob travels on a hyperbola compelled in the region I. With the single-mode approximation, the Minkowski vacuum state can be expressed in terms of the Rindler regions I and II states (60 ): \|0 M = cos r\|0 I \|0 II + sin r\|1 I \|1 II ,(20) and the excited state is given by \|1 M = \|1 I \|0 II(21) Note that the observers in regions I and II are causally disconnected. Since the mode corresponding to II in not observable, it should be traced out. We assume that Rob first performs a PM of the form (4) on his particle, and then uniformly accelerates. In the next step, a PMR is carried out by Rob in region I. Since Rob is restricted to region I due to the causality condition, we trace the state over region II. Provided that the PM and PMR are successfully accomplished, the following mixed state between Alice and Rob is obtained(61 ) ρ A,I = 1 N 2     sin 2 ϑ 2 p cos 2 r 0 0 sin ϑ 2 cos ϑ 2 √ pq cosr 0 sin 2 ϑ 2 pq sin 2 r 0 0 0 0 0 0 sin ϑ 2 cos ϑ 2 √ pq cosr 0 0 cos 2 ϑ 2 q     . (22) where N 2 = sin 2 ϑ 2 p cos 2 r + sin 2 ϑ 2 pq sin 2 r + cos 2 ϑ 2 q is the normalization factor, p = 1 − p and q = 1 − q, in which p and q represent the first and the second PM strengths. Preparation probability The probability of preparing the system in state (22) via the above prescription is obtained as P = P 1 .P 2 , = p sin 2 ϑ 2 cos 2 r + q p sin 2 ϑ 2 sin 2 r + cos 2 ϑ 2 .(23) where Fig. 3, the probability is plotted versus acceleration parameter r, for ϑ = π/2. We see that the probability of preparing the state of the system decreases with increase in p or q. Moreover, we see that P(p, q) is larger than P(q, p) provided that p > q, i.e., when the strength of the first measurement is larger than the second measurement the probability becomes greater. The same results are obtain for 0 < ϑ < π 2 . Next we discuss how PM and PMR affect the degradation of QRs and QFI teleportation through the Unruh noise channel (Eq. (22)). P 1 = tr M † 0 M 0 ρ (0) in which ρ (0) = \|Ψ (0) Ψ (0) \| and P 2 = tr M † 0 M 0 ρ (2) where ρ (2) = tr II (\|Ψ Ψ\|), with \|Ψ = 1 √ N1 sin ϑ 2 √ p (cos r\|0 A \|1 I \|0 II + sin r\|0 A \|0 I \|1 II ) + cos ϑ 2 \|1 A \|0 I \|0 II and N 1 = sin 2 ϑ 2 p + cos 2 ϑ 2 . In Single-qubit teleportation under the Unruh noise channel In this section, we investigate the teleportation of QFIs and QC related to singlequbit state, through the Unruh noise channel. We consider \|ψ in = cos θ/2\|0 + e iϕ sin θ/2\|1 , 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π as the input state in the process of teleportation, where θ and ϕ are the weight and phase parameters, respectively. We use the shared state between Alice and Rob, Eq. (22), as the resource (ρ A,I = ρ ch ) to teleport the single-qubit input state. Using Eq. (1), the output state can be obtained as follows ρ P M out = 1 N 2 Acos 2 θ 2 + Dsin 2 θ 2 Fe −iϕ sin θ Fe iϕ sin θ Asin 2 θ 2 + Dcos 2 θ 2 ,(24) where A = sin 2 ϑ 2 p cos 2 r + cos 2 ϑ 2 q, D = sin 2 ϑ 2 pq sin 2 r,(25)F = sin ϑ 2 cos ϑ 2 pq cos r, For input state \|ψ in = cos θ/2\|0 + e iϕ sin θ/2\|1 , the QFIs with respect to parameters θ and ϕ are easily found to be F in (θ) = 1 and F in (ϕ) = sin 2 θ, respectively. It is seen that F in (ϕ) is dependent on θ and is maximized for θ = π 2 while F in (θ) is independent of weight parameter θ and has a constant value. Therefore, the balance-weighted input state is preferable. Using Eqs. (10) and (24), QFI with respect to weight and phase parameters are found, respectively, as follows F P M out (θ) = 1 N 2 2   (A − D) 2 sin 2 θ + 4F 2 cos 2 θ + 1 4 (A − D) 2 − 4F 2 2 sin 2 2θ N 2 2 − (A − D) 2 cos 2 θ − 4F 2 sin 2 θ    .(26)F P M out (ϕ) = 4\| Fsin θ N 2 \| 2(27) In Fig. 4, QFI with respect to weight parameter, F P M out (θ), for single-qubit state teleportation through the pure Unruh decoherence and for the case that the combination of PM and PMR have been applied, is plotted as a function of acceleration parameter r. It can be seen that after teleportation under pure Unruh channel (i.e., p = q = 0), when the acceleration increases, QFI decays monotonously for all values of the initial parameter ϑ. Studying the behavior of F P M out (θ), when the PM and PMR are applied on the channel, we observe that applying either PM (i.e., q = 0) or PMR (i.e., p = 0) may improve F P M out (θ) for all initial states of the channel (see Figs. 4(a) and 4(b) ). For sufficiently strong measurement strength (p → 1 or q → 1), the precision of estimating weight parameter can be enhanced remarkably and it is almost robust against the Unruh decoherence. The important question that comes up is that if the acceleration is constant, how one can control the QFI by applying PM and PMR. In Fig. 5 we consider the F P M out (θ) behavior versus p. It is observed that in the absence of PMR (q = 0), F P M out (θ) enhances with increase in p (space dashed purple line) for all values of the channel parameter ϑ. It is also seen that with the combined effect of PM and PMR, estimation precision of weight parameter is also improved. We obtain the same results investigating the behavior of F P M out (θ) versus p. In Fig. 6, F P M out (ϕ) for single-qubit state teleportation, is plotted as functions of PM as well as PMR strength for fixed value of acceleration parameter r = 0.6 and the maximally entangled input state (θ = π/2, ϕ = 0). It is seen from Fig. 6(a) that for π 2 ≤ ϑ < π, with increase in PM strength F P M out (ϕ) increases to reach a maximum value and then it decreases with more increase of p. Moreover, comparing the behavior of F P M out (ϕ) for different values of PMR strength, we see that with increase in q, optimal estimation of the phase parameter occurrs for larger values of p. Nevertheless, increase of the PMR strength interestingly raises the optimal value of the QFI, leading to enhancement of the phase parameter estimation. We also see, in that range of θ, while for small values of p, the QFI may fall with an increase in q, it can enhance as q increases for larger values of p. We obtain the same results investigating the behavior of F P M out (ϕ) versus q for 0 < ϑ ≤ π 2 . In particular, in this range, the QFI may decrease with an increase in p for small values of q, while it can exhibit increasing behavior as p increases for large values of q (see Fig.6(b)). Considering the optimal behavior of single-qubit teleported QFI with respect to phase parameter as functions of p and q, we obtain p opt and q opt as follows Figure 7. shows how optimal values of p and q vary in terms of acceleration parameter r. Decrease of the optimal value of p opt with increase in acceleration (see Fig. 7(a).) indicates when r increases the optimal estimation of phase parameter can be realized by weaker PM. However, Fig. 7(b) shows that more strong PMR is required for attaining the optimal QFI when the accelerated observer moves with more larger acceleration. p opt = qcos 2 r sin 2 ϑ 2 − cosϑ(1 − q) sin 2 ϑ 2 (1 − qsin 2 r) , q opt = sin 2 ϑ 2 (p + 2(1 − p)cos 2 r) − 1 sin 2 ϑ 2 (p + (1 − p)cos 2 r) − 1 .(28) Behavior of the QFI with respect to phase parameter, F P M out (ϕ), as a function of ac- Figure 8. Single-qubit teleported QFI with respect to phase parameter, ϕ, as functions of acceleration parameter r by fixing θ = π 2 for (a) 0 < ϑ < π 2 , (b) ϑ = π 2 and (c) π 2 < ϑ < π. (a) (b) (c) celeration parameter, r, for different ranges of the channel parameter, ϑ, is investigated in Fig. 8. It is seen that for teleportation under pure Unruh channel (i.e., p = q = 0), there is monotonous degradation in F P M out (ϕ) with increase in r. However, we find that the combined effect of PM and PMR with the same strength, (p = q), leads to partially improvement of the the estimation precision of the phase parameter. Besides, when this common measurement strength increases F P M out (ϕ) is protected much better for π 2 ≤ ϑ < π; it even increases surprisingly with increase in acceleration for π 2 < ϑ < π, in the limit p → 1 and q → 1. In addition, our numerical calculation shows that in order to protect the QFI with respect to ϕ and QR of the teleported state against the Unruh effect, we can use the following special choice for PMR strength (61 ) q s = 1 − (1 − p) cos 2 r.(29) In fact, the Unruh noise may be approximately eliminated provided that the PM strength is sufficiently strong (p → 1) and the above choice for the PMR is applied (see blue dashed lines in Fig. 8) If we intend to teleport only the information encoded into the phase parameter, we can manage the input state by choosing the weight parameter as θ = π 2 , to estimate the phase parameter with the best precision; i.e., the best estimation of phase parameter is obtained if the input state is maximally entangled (see Fig. 9). In the following, the effect of PM or PMR on QC teleportation of single-qubit are studied. Using the l 1 -norm measure (Eq. (11)), QC for the density matrix (24), can be obtained as follows Figure 9. Single-qubit teleported QFI with respect to phase parameter ϕ as functions of θ, fixing the acceleration parameter r = 0.6. C l1 ρ P M out = \| sinϑ √ pqcos rsin θ N 2 \|(30) (a) (b) (c) Figure 10. Quantum coherence of the teleported single-qubit state as functions of acceleration parameter r by fixing θ = π 2 for (a) 0 < ϑ < π 2 , (b) ϑ = π 2 and (c) π 2 < ϑ < π. In the case of teleportation without application of PM or PMR on the Unruh channel, i.e., p = q = 0 and then N 2 = 1, we find C l1 (ρ out ) = \|sinϑ cos rsin θ\| (31) which is the teleported quantum coherence under the pure Unruh decoherence. Investigating QC of the teleported single-qubit state as functions of r or studying its behavior versus PM and PMR strength for fixed value of the acceleration parameter, one can see that the results, qualitatively similar to F P M out (ϕ), are observed (see Figs. (c) Figure 12. Fidelity of the single-qubit teleportation as functions of acceleration parameter r by fixing θ = π 2 and ϕ = 0 for (a) 0 < ϑ < π 2 , (b) ϑ = π 2 and (c) π 2 < ϑ < π. 10 and 11). In order to determine the quality of teleportation, i.e., closeness of the teleported state to the input state, the fidelity (62 ) between ρ in and ρ out defined as f (ρ in , ρ out ) = Tr (ρ in ) = ψ in \|ρ out \|ψ in , should be computed. Therefore, we obtain f = 1 N 2 A − D 2 + Fcos 2ϕ sin 2 θ + D ,(32) In Fig. 12, the teleportation fidelity versus acceleration parameter r has been plot- ted. It is seen that the results obtained for fidelity is the same as the obtained results for QC and F P M out (ϕ), i.e., fidelity degrades with increase in r under pure Unruh effect. However, the combined effect of PM and PMR for channel parameter lying in the region π 2 ≤ ϑ < π, can improve the quality of teleportation and it may even enhance with increase in acceleration for π 2 < ϑ < π in the limit p, q → 1. Moreover, Unruh decoherence is approximately eliminated for all values of ϑ, with q = q opt and in the limit p → 1, consequently the teleportation process may be implemented with better quality. Now we investigate the fidelity of the single-qubit teleportation as functions of PM as well as PMR strength. As it is seen in Fig. 13(a), similar to F P M out (ϕ) and QC, with proper selection of the channel parameter ϑ the quality of teleportation may be enhanced with increase in p or q to reach a maximum value. Besides, in the range 0 < ϑ ≤ π 2 ( π 2 < ϑ < π), analyzing the QFI behavior as a function of q (p), we see that the QFI may be decreased (improved) with an increase in p (q) for small values of q (p), while it can exhibit increasing behavior as p (q) increases for large values of q(p). In addition, optimal teleportation fidelity becomes greater with increase in q or q, hence the teleportation process is done more successfully. Finally, in Figs. 14(a) and 14(b) we compare and illustrate the harmonic behavior of F P M out (ϕ), QC and teleportation fidelity as functions of PM strength for π 2 < ϑ < π, and PMR strength for 0 < ϑ ≤ π 2 , in the case of single-qubit teleportation. We can conclude that for both q = 0 and q = 0, the PM strength which optimizes the estimation precision of the phase parameter, is the strength at which the quality of teleportation is the best and the coherence of the output single-qubit state reaches to its maximum value. Investigating the behavior of the above mentioned quantities as functions of PMR strength, we achieve the same results (see Figs. 14(c) and 14(d)). Two-qubit teleportation under the Unruh noise channel In order to study QRs and QFIs teleportation of a two-qubit state through the Unruh channel, \|ψ in = cos θ/2\|10 + e iϕ sin θ/2\|01 , 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π is considered as the input state in the teleportation process. We follow Kim and Lee's two-qubit teleportation protocol (52 ), and use two copies of the shared state between Alice and Rob as the quantum channel. Using Eq. (3), we obtain the output state as ρ P M out = 1 N 2 2     AD 0 0 0 0 A 2 cos 2 θ 2 + D 2 sin 2 θ 2 2F 2 e −iϕ sin θ 0 0 2F 2 e iϕ sin θ A 2 sin 2 θ 2 + D 2 cos 2 θ 2 0 0 0 0 AD     ,(33) where A, D and F are determined by Eq. (25). Now we apply PM or PMR on Rob's particle, before teleporting the two-qubit state. Then, we study the influence of PM or PMR on the degradation effect of the Unruh noise channel on QRs and QFI teleportation. Using density matrix (33), the corresponding quantum coherence is obtained as follows C l1 ρ P M out = 4\| F 2 sin θ N 2 2 \|,(34) The results, obtained for teleportation of two-qubit QC under Unruh noise channel, are similar to single-qubit teleportation. According to the Eqs. (12) and (33), the entanglement of the teleported two-qubit state is obtained as Figure 15. Entanglement of the teleported two-qubit state as functions of acceleration parameter r by fixing θ = π 2 for (a) 0 < ϑ ≤ π 2 , and (b) π 2 < ϑ < π. C ρ P M out = 2M ax 0, 2\| F 2 sin θ N 2 2 \| − \| AD N 2 2 \| ,(35) In Fig. 15, we plot the concurrence of teleported two-qubit state as a function of acceleration parameter r for different strengths of PM and PMR. It is clear that the entanglement absolutely decreases with increase in acceleration under the pure Unruh decoherence. However, it can be amplified with combined action of PM and PMR for all values of initial channel parameter ϑ. In fact, when the strength of PM increases, the entanglement degradation decreases, especially in the limit p = q → 1, entanglement is approximately protected against the Unruh decoherence. Surprisingly, as seen in Fig. 15, in that limit, the teleported entanglement may increase under the Unruh effect for initial channel parameter lying in the region π 2 < ϑ < π. In addition, it is seen that the entanglement is also improved by applying q s even without first PM (i.e., p = 0) for 0 < ϑ ≤ π 2 . Considering the behavior of QD as a function of acceleration parameter r for different PMs strength, we see that combined action of PM and PMR can raise QD for ϑ lying in the range π 2 ≤ ϑ < π (see Fig.16). In particular, in the limit p, q → 1, QD may increase with applying PMs for π 2 < ϑ < π. Moreover, if we choose q = q opt , QD can increase even in the absence of first PM (i.e., p = 0) for 0 < ϑ ≤ π 2 . Using Eqs. (8) and (33), we find the two-qubit teleported QFIs with respect to weight and phase parameters as follows F P M out (θ) = 1 N 2 2   ζ + 8A 2 D 2 ζ 2 − 16F 4 ζ (A 2 − D 2 ) 2 − 16F 4 cos 2θ − (ζ 2 + 4 (A 2 D 2 − 4F 4 ))   ,(36)F P M out (ϕ) = 16F 4 sin 2 θ ζN 2 2 .(37) where ζ = A 2 + D 2 . Surprisingly, we obtain the results similar to single-qubit teleportation, investigating the teleportation of two-qubit QFI under Unruh channel. In Figs. 17 and 18, we compare teleportation of QFI in both single and two-qubit cases (supposing that θ or ϕ carries the same information in both cases). In Fig. 17, we see that the information encoded in the weight parameter θ is better protected against Unruh effect during teleportation of two-qubit state, comparing it with the single- Figure 16. QD of the teleported two-qubit state as functions of acceleration parameter r by fixing θ = π 2 and ϕ = 0 for (a) 0 < ϑ < π 2 , (b) ϑ = π 2 , and (c) π 2 < ϑ < π (a) (b) Figure 17. Comparing teleported F P M out (θ) for single and two-qubit states, fixing θ = π 2 and ϑ = π 2 (a) in the absence of measurements, (b) in the presence of measurements. qubit scenario. Nevertheless, extraction of information encoded into phase parameter ϕ, is more efficient in single-qubit teleportation than the two-qubit one (see Fig. 18). Therefore, depending on what parameter we want to teleport, we use single or twoqubit state to encode the required information. Finally, fidelity for the two-qubit teleportation under the Unruh channel, are found to be f = 1 N 2 2 A 2 − D 2 4 + F 2 cos 2ϕ 2sin 2 θ + D 2 ,(38) We get the results similar to single-qubit teleportation fidelity, investigating the (a) (b) Figure 18. Comparing teleported F P M out (ϕ) for single and two-qubit states by fixing θ = π 2 and ϑ = π 2 (a) in the absence of measurements, (b) in the presence of measurements. (a) (b) Figure 19. Comparing the fidelity of single and two-qubit teleportation, fixing θ = π 2 , ϕ = 0 and ϑ = π 2 for (a) in the absence of measurements, (b) in the presence of measurements. behavior of two-qubit teleportation fidelity under the Unruh noise channel with and without applying the measurements. Comparing the fidelity of single and two-qubit teleportation in Fig. 19, we observe that quality of teleportation is better in single-qubit case than the two-qubit one. It means that single-qubit teleportation is more robust against the Unruh decoherence. Summary and conclusions Teleportation of QRs and QFI of single and two-qubit states, under the Unruh effect experienced by a mode of a free Dirac field, was discussed in this paper. We investigated the conditions under which the degradation effect of the Unruh effect on QRs and QFI teleportation can be improved by PMs, and found that the value of initial parameter of the channel ϑ plays a key role in this scenario. Moreover, we examined how the partial measurements can be performed to eliminate the Unruh effect or how they may be designed such that the Unruh effect can be used to enhance the quantum communication. Besides, fixing the acceleration and considering the behavior of the QFI, QC and teleportation fidelity as functions of PM strength (PMR strength), we found that F P M out (ϕ), QC and teleportation fidelity harmonically increase to reach a maximum value and then decrease with more increase in p (q). We also analytically analysed the optimal behavior of the QFI associated with the phase parameter. Finally, comparing the teleportation of QFI for single and two-qubit cases as functions of acceleration, we showed that the information encoded in the weight parameter θ is better protected against the Unruh effect in the case of two-qubit teleportation. However, in the case of single-qubit teleportation, encoding information in the phase parameter ϕ is more efficient. Therefore, we encode the information into either the weight or phase parameter, depending on either the two or single-qubit scenario, respectively, is used for the teleportation. Figure 1 . 1Schematic drawing of entanglement teleportation. Figure 2 . 2Minkowski spacetime. Figure 3 . 3The probability of preparing the state of the channel, for ϑ = π/2. Figure 4 . 4Single-qubit teleported QFI with respect to weight parameter, θ, as functions of acceleration parameter r by fixing θ = π 2 and for 0 < ϑ < π for (a) q = 0, (b) p = 0. Figure 5 . 5Single-qubit teleported QFI with respect to weight parameter, θ, as functions of PM strength, p, for r = 0.6, and different values of PMR strength. Figure 6 . 6Single-qubit teleported QFI with respect to phase parameter, ϕ, as functions of (a) PM strength, p, fixing ϑ = 3π 4 and (b) PMR strength, q, fixing ϑ = π 4 ; where we have chosen the acceleration parameter r = 0.6. Figure 7 . 7The optimal value of PM and PMR strengths as functions of acceleration parameter r for (a) q = 0.6 and ϑ = 3π 4 and (b) p = 0.6 and ϑ = π 4 . Figure 11 . 11Quantum coherence of the teleported single-qubit state as functions of (a) PM strength, p, fixing ϑ = 3π 4 and (b) PMR strength, q, fixing ϑ = π 4 ; for fixed value of the acceleration parameter r = 0.6. Figure 13 . 13Fidelity of the single-qubit teleportation as functions of (a) PM strength, p, fixing ϑ = 3π 4 and (b) PMR strength, q, fixing ϑ = π 4 ; for fixed value of the acceleration parameter r = 0.6. Figure 14 . 14Comparing teleported F P M out (ϕ), QC as well as teleportation fidelity for single-qubit state by fixing θ = π 2 , ϕ = 0, (a) versus p in the absence of PMR, q = 0, (b) versus p in the presence of PMR, q = 0.6, (c) versus q in the absence of PM, p = 0, and (d) versus q in the presence of PM, p = 0.6. Acknowledgement(s)We wish to acknowledge the financial support of Urmia University and Jahrom University. . C H Bennett, G Brassard, C Crepeau, R Jozsa, A Peres, W K Wootters, Phys. Rev. Lett. 701895C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. 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[ "Theoretical investigation of the more suitable rare earth to achieve high gain in waveguide based on silica containing silicon nanograins doped with either Nd 3+ or Er 3+ ions", "Theoretical investigation of the more suitable rare earth to achieve high gain in waveguide based on silica containing silicon nanograins doped with either Nd 3+ or Er 3+ ions" ]	[ "Alexandre Fafin ", "Julien Cardin [email protected]*[email protected] ", "Christian Dufour ", "Fabrice Gourbilleau \nboulevard Maréchal Juin Gourbilleau\n14050Caen cedex 4, DaldossoD. Navarro-Urrios, M. Melchiorri, C. Garcia, P. Pellegrino, B. Garrido, C. Sada, G. Battaglin, FFrance\n", "A A N Macdonald ", "Quan Hryciw ", "A Li ", "" Meldrum ", "Luminescence ", "\nCIMAP\nCNRS/CEA\nENSICAEN\nUCBN\n" ]	[ "boulevard Maréchal Juin Gourbilleau\n14050Caen cedex 4, DaldossoD. Navarro-Urrios, M. Melchiorri, C. Garcia, P. Pellegrino, B. Garrido, C. Sada, G. Battaglin, FFrance", "CIMAP\nCNRS/CEA\nENSICAEN\nUCBN" ]	[]	We present a comparative study of the gain achievement in a waveguide whose active layer is constituted by a silica matrix containing silicon nanograins acting as sensitizer of either neodymium ions (Nd 3+ ) or erbium ions (Er 3+ ). By means of an auxiliary differential equation and finite difference time domain (ADE-FDTD) approach that we developed, we investigate the steady states regime of both rare earths ions and silicon nanograins levels populations as well as the electromagnetic field for different pumping powers ranging from 1 to 10 4 mW/mm 2 . Moreover, the achievable gain has been estimated in this pumping range. The Nd 3+ doped waveguide shows a higher gross gain per unit length at 1064 nm (up to 30 dB/cm) than the one with Er 3+ doped active layer at 1532 nm (up to 2 dB/cm). Taking into account the experimental background losses we demonstrate that a significant positive net gain can only be achieved with the Nd 3+ doped waveguide. Study of optical losses of Nd 3+ doped silicon rich silicon oxide for laser cavity," Thin Solid Films 520, 4026-4030 (2012).	10.1364/oe.22.012296	[ "https://arxiv.org/pdf/1405.5338v1.pdf" ]	23,308,973	1405.5338	6adde990fcdd7932a7a4ddc069a01bd736a89cc3	Theoretical investigation of the more suitable rare earth to achieve high gain in waveguide based on silica containing silicon nanograins doped with either Nd 3+ or Er 3+ ions Alexandre Fafin Julien Cardin [email protected]*[email protected] Christian Dufour Fabrice Gourbilleau boulevard Maréchal Juin Gourbilleau 14050Caen cedex 4, DaldossoD. Navarro-Urrios, M. Melchiorri, C. Garcia, P. Pellegrino, B. Garrido, C. Sada, G. Battaglin, FFrance A A N Macdonald Quan Hryciw A Li " Meldrum Luminescence CIMAP CNRS/CEA ENSICAEN UCBN Theoretical investigation of the more suitable rare earth to achieve high gain in waveguide based on silica containing silicon nanograins doped with either Nd 3+ or Er 3+ ions References and links 1. N. Daldosso and L. Pavesi, "Nanosilicon photonics," Laser Photonics Rev. 3, 508-534 (2009). 2. G. P. Agrawal, Fiber-optic communication systems (John Wiley & Sons, 2010). 3. O. Lumholt, A. Bjarklev, T. Rasmussen, and C. Lester, "Rare earth-doped integrated glass components: modeling and optimization," J. Lightwave Technol. 13, 275-282 (1995). 4. A. Podhorodecki, J. Misiewicz, F. Gourbilleau, J. Cardin, and C. Dufour, "High energy excitation transfer from silicon nanocrystals to neodymium ions in silicon-rich oxide film," Electrochem. Solid-State Lett. 13, K26-K28 (2010). 5. A. Polman and F. C. J. M. van Veggel, "Broadband sensitizers for erbium-doped planar optical amplifiers: review," J. Opt. Soc. Am. B 21, 871-892 (2004).OCIS codes: (0501755) Computational electromagnetic methods(1605690) Rare-earth- doped materials(2304480) Optical amplifiers(2307370) Waveguides(2305590) Quantum- well, -wire and -dot devices We present a comparative study of the gain achievement in a waveguide whose active layer is constituted by a silica matrix containing silicon nanograins acting as sensitizer of either neodymium ions (Nd 3+ ) or erbium ions (Er 3+ ). By means of an auxiliary differential equation and finite difference time domain (ADE-FDTD) approach that we developed, we investigate the steady states regime of both rare earths ions and silicon nanograins levels populations as well as the electromagnetic field for different pumping powers ranging from 1 to 10 4 mW/mm 2 . Moreover, the achievable gain has been estimated in this pumping range. The Nd 3+ doped waveguide shows a higher gross gain per unit length at 1064 nm (up to 30 dB/cm) than the one with Er 3+ doped active layer at 1532 nm (up to 2 dB/cm). Taking into account the experimental background losses we demonstrate that a significant positive net gain can only be achieved with the Nd 3+ doped waveguide. Study of optical losses of Nd 3+ doped silicon rich silicon oxide for laser cavity," Thin Solid Films 520, 4026-4030 (2012). Introduction The development of waveguide optical amplifiers based on rare earth (RE) doped silicon based matrix is of great interest for the semiconductor research community [1]. Erbium ions are particularly interesting due to an optical transition at 1532 nm which coincides with the maximum of transmission in optical glass fiber [2]. However, in the case of erbium this transition involves the ground level, which may limit the achievable gain due to reabsorption mechanisms [3]. To overcome this critical issue, neodymium ion has been recently proposed because its emission scheme makes it more suitable for achieving higher gain [4]. In such a system, the amplification is commonly achieved through RE level population inversion by an appropriate optical pumping. One main drawback of RE ions is their low absorption cross section. However this can be overcome by the use of sensitizers that are characterized by a larger absorption cross section. Those sensitizers have shown an efficient transfer of energy to RE ions in their vicinity. Several sensitizers of RE have been proposed in literature. Polman et al [5] shows the sensitization of Er 3+ ions by different kinds of sensitizers such as ytterbium ions, metal ions and silicon nanograins (Si-ng). The work of MacDonald et al [6] presents the sensitization of Nd 3+ ions by Si-ng. In this paper we present a comparative study of waveguides with an active layer containing Si-ng and doped either with erbium or neodymium ions. The typical composition and structure of such waveguides is presented in section 1.1. The electromagnetic field and level populations of Si-ng and RE ions have been computed using an algorithm published in a previous paper [7] and briefly detailed in section 1.2. In section 2, we describe levels populations equations associated with Si-ng, erbium ions and neodymium ions. For the two RE ions, due to their different transition time properties, two particular ways of calculation will be detailed. In section 3, we present for both RE population inversions, population map and optical gain as a function of optical pump power. We conclude by the comparison of the optical gain of two waveguides doped either with erbium or neodymium ions as a function of the pump power. The waveguide is composed of three layers (Fig. 1). The bottom cladding layer is composed of pure silica. In order to ensure optical confinement of modes, this layer is about 5 to 8 µm thick in a typical experimental waveguide doped either with Nd 3+ ions [8] or Er 3+ ions [9]. In this modeling method the thickness of bottom cladding layer was taken equal to 3.5 µm in order to limit the use of memory. The 2 µm active layer constituted of silicon rich silicon oxide (SRSO) contains Si-ng and RE ions. A pure silica strip layer is stacked on the top of the SRSO layer. The static refractive index (i.e. refractive index which remains constant with wavelength) of the active layer (1.5) has been chosen greater than the one of the strip and bottom cladding layers (1.448) to ensure the guiding conditions. In order to investigate which is the most suitable RE between Nd 3+ and Er 3+ for achieving high gain, both waveguides are pumped continuously (CW) by the propagation in the active layer of a pump mode at 488 nm. A signal mode is co-propagated in the active layer in order to investigate the achievability of amplification by stimulated emission. This signal corresponds to a transition occurring between electronics levels of RE, either at 1532 nm for erbium ions or at 1064 nm for neodymium ions [10,11]. The waveguide dimensions are identical for erbium or neodymium ions and according to the experimental conditions we propagate the fundamental transverse electric mode (TE 00 ) for the pump and signal along z direction. The calculation and the injection of mode profiles in ADE-FDTD method for all wavelengths considered here are described in our previous paper [7]. Calculation method The electromagnetic fields (E, H) and Poynting vector (R = E × H) as well as RE and Si-ng populations in steady state are computed by the algorithm described by Fafin et al [7]. This algorithm is based on finite-difference time-domain method (FDTD) and auxiliary differential equations (ADE). The FDTD method consists in the discretization in time and space of the electromagnetic fields [12] according to the K. Yee algorithm [13]. The ADE makes the link between electromagnetic fields and absorption and emission processes by the use of a set of polarisation densities P i j following a Lorentz electronic oscillator model [12,14]. A typical polarisation equation (Eq. (1)) between two level populations i and j is described below: ∂ 2 P i j ∂t 2 + ∆ω i j ∂ P i j ∂t + ω 2 i j P i j = κ i j (N i − N j )E(1) where ∆ω i j is the transition linewidth including radiative, non-radiative and de-phasing processes [15], and ω i j is the resonance frequency of this transition. κ i j used in [7] depends on the transition lifetime τ i j and on the optical index n. The time evolution of levels populations for each emitter (RE, Si-ng) is described by a rate equation which depends on polarisation densities of considered transitions, lifetimes, transfer coefficient and levels populations. Since in visible wavelength range, the electromagnetic field has a characteristic time of the order of 10 −15 s and the levels populations of emitters have characteristic lifetimes as long as a few milliseconds [16, 17], a classical ADE-FDTD calculation is impossible in a reasonable time [7]. Indeed with the classical ADE-FDTD method the equations of populations are calculated simultaneously with the electromagnetic field leading to about 10 15 iterations. We have recently overcome this multiscale issue [7] after splitting the classical ADE-FDTD single loop into two interconnected loops separating short time and long time processes. The electromagnetic fields and polarisation densities are calculated in the short time loop and the rate equations in the long time one. This method allows us to reduce drastically the number of iterations from 10 15 to 10 5 and consequently reduces the calculation time to more reasonable duration (7 days for the present waveguide at bi-processors quad-core Intel Nehalem EP @ 2.8 GHz). Moreover, in order to minimize phase velocity error and velocity anisotropy errors inherent to the FDTD method, the algorithm was set up with the time and space steps introduced in our previous paper [7]. Description of populations In this section we describe the excitation mechanism of the erbium and neodymium ions and the numerical calculation of the levels populations at the steady state. Silicon nanograins We model silicon nanograins (Si-ng) as a two levels system (Fig. 2) where the ground and excited levels populations (respectively N Si0 and N Si1 ) are given by the rate equations Eq. (2) and Eq. (3). Due to a low probability of multi-exciton generation in a single Si-ng [18], we assume the excitation of one single exciton by Si-ng, therefore the Si-ng population will correspond to the exciton population. After non-radiative transitions in the conduction band, the exciton may either radiatively recombine or excite an emitter in its vicinity. This energy transfer may occur if the energy gap between conduction and valence band of Si-ng matches the energy gap between the RE fundamental level and an upper level leading to a possible emission. According to literature [19] the lifetime of excited level N Si1 is chosen at τ Si 10 \| r nr = 50 µs. Since few papers studied the energy transfer coefficient in Si-ng and erbium ions [20,21] and to our knowledge no study was made on this coefficient in Si-ng and neodymium, this energy transfer coefficient between Si-ng and both RE is assumed identical and taken equal to K = 10 −14 cm 3 /s. We took the same concentration of Si-ng equal to 10 19 cm −3 for both active layers doped either with Er 3+ or Nd 3+ ions. In order to simulate a realistic absorption and emission cross section equal to 10 −16 cm 2 the linewidth ∆ω i j and the number of polarizations N p are respectively fixed to 10 14 rad/s and 2756 according to the method explained in [7]. These parameters are reported in Table 1. dN Si 1 (t) dt = + 1 hω Si 10 E (t) dP Si 10 (t) dt − N Si 1 (t) τ Si 10 \| r nr − KN Si 1 (t) N 0 (t) (2) dN Si 0 (t) dt = − 1 hω Si 10 E (t) dP Si 10 (t) dt + N Si 1 (t) τ Si 10 \| r nr + KN Si 1 (t) N 0 (t)(3) Erbium ions Erbium ions are modelled by four levels: 0 ( 4 I 15/2 ), 1 ( 4 I 13/2 ), 2 ( 4 I 11/2 ) and 3 ( 4 I 9/2 ). We consider two non-radiative transitions 3→2, 2→1 and one radiative transition from level 1 to level 0 at 1532 nm. The emission cross section of this transition is equal to 6 × 10 −21 cm −2 [20, 21] and corresponds to a linewidth ∆ω 10 equal to 0.15 × 10 15 rad/s with one polarisation (N p = 1) [7]. Moreover there is an up-conversion process from level 1 to level 0 and 3 which can be modelled by a coefficient C up = 5 × 10 −17 cm 3 /s. The concentration of erbium ions is equal to 10 20 cm −3 . The time evolution of Er 3+ levels populations is described by the rate equations Eq. (4) to Eq. (7). All parameters of erbium ions transitions are taken from [20, 21] and reported in Table 2. dN 3 (t) dt = − N 3 (t) τ 32 \| nr + KN Si 1 (t) N 0 (t) +C up N 2 1 (4) dN 2 (t) dt = + N 3 (t) τ 32 \| nr − N 2 (t) τ 21 \| nr − N 2 (t) τ 20 \| nr (5) dN 1 (t) dt = + 1 hω 10 E (t) dP 10 (t) dt + N 2 (t) τ 21 \| nr − N 1 (t) τ 10 \| r nr − 2C up N 2 1 (6) dN 0 (t) dt = − 1 hω 10 E (t) dP 10 (t) dt (7) + N 2 (t) τ 20 \| nr + N 1 (t) τ 10 \| r nr − KN Si 1 (t) N 0 (t) +C up N 2 1 Neodymium ions Neodymium ions are modelled by five levels: 0 ( 4 I 9/2 ), 1 ( 4 I 11/2 ), 2 ( 4 I 13/2 ), 3 ( 4 F 3/2 ) and 4 ( 4 F 5/2 + 2 H 9/2 ). We consider three non-radiative transitions 4→3, 2→1, 1→0 and three radiative transitions 3→2 at 1340 nm, 3→1 at 1064 nm and 3→0 at 945 nm. The emission cross section of these transitions is equal to 10 −19 cm −2 [24] and correspond to linewidths ∆ω i j reported in Table 3. The up-conversion coefficient C up ranging from 1 × 10 −17 to 5 × 10 −17 cm 3 /s was found in [22,23]. This value leads to an equivalent lifetime τ up = 1 C up N 1 that remains 10 times larger than the longest level lifetime of Nd 3+ ions which allows neglecting the up-conversion process [7]. The concentration of neodymium ions is equal to 10 20 cm −3 . The time evolution of Nd 3+ levels populations is described by the rate equations Eq. (8) to Eq. (12). Parameters of neodymium ions transitions are taken from [11, 24] and reported in Table 3. dN 4 (t) dt = − N 4 (t) τ 43 \| nr + KN Si 1 (t) N 0 (t) (8) dN 3 (t) dt = + 1 hω 30 E (t) dP 30 (t) dt + 1 hω 31 E (t) dP 31 (t) dt + 1 hω 32 E (t) dP 32 (t) dt (9) + N 4 (t) τ 43 \| nr − N 3 (t) τ 30 \| r nr − N 3 (t) τ 31 \| r nr − N 3 (t) τ 32 \| r nr dN 2 (t) dt = − 1 hω 32 E (t) dP 32 (t) dt + N 3 (t) τ 32 \| r nr − N 2 (t) τ 21 \| nr (10) dN 1 (t) dt = − 1 hω 31 E (t) dP 31 (t) dt + N 3 (t) τ 31 \| r nr − N 1 (t) τ 10 \| nr + N 2 (t) τ 21 \| nr (11) dN 0 (t) dt = − 1 hω 30 E (t) dP 30 (t) dt + N 3 (t) τ 30 \| r nr + N 1 (t) τ 10 \| nr − KN Si 1 (t) N 0 (t)(12) Difference in calculation method of populations We aim at calculating levels populations in steady states in a fast and convenient manner. Consequently, the preferred method would be an analytical calculation of populations by setting dN i /dt = 0. However, this is only applicable in case of neodymium levels equations. Moreover, the up-conversion term in erbium rate equations leads to equations that are hardly analytically solvable. In that case, the steady states of population levels are reached by a finite difference method. This calculation was possible using a reasonable time step of 0.01 µs ten times lower than the shortest lifetime (0.1 µs) considered in this model. The calculation time is then no longer negligible but does not rise up significantly the global calculation time of our ADE-FDTD method. Results and discussion The solution of Eq. (2) to Eq. (12) gives the levels populations in their steady states. We define for i → j transition the population inversion (section 3.1) as the ratio (N i − N j ) over the total population number (N tot = 10 20 at/cm 3 for RE or N Si tot = 10 19 at/cm 3 for Si-ng). We deduced also the optical gain (section 3.2) at 1064 nm for Nd 3+ and 1532 nm for Er 3+ . These values are computed for a pump power ranging from 1 to 10 4 mW/mm 2 . Populations inversion We present in this section the spatial distribution of population inversion in waveguides doped either with erbium or neodymium ions for a pump power of 1000 mW/mm 2 in a longitudinal section view along the propagation axis (Fig. 3). For both waveguides the plots show a decrease of population inversion with direction of propagation which can be attributed to the coupling between rare earth ions and silicon nanograins. For erbium ions, the population inversion remains positive over a length of 1.5 µm. Beyond this length, the population inversion becomes negative witnessing the threshold effect occurring with three-level system. For neodymium ions the population inversion remains positive along the whole structure. Indeed, in this four-level system, the level 1 is depopulated quickly to the ground level leading to N 3 >> N 1 . We now consider the population inversion distribution of Si-ng presented in longitudinal section view in Fig. 4 for both RE. This decrease of population inversion of Si-ng shows an identical behaviour with direction of propagation as the one observed with RE. This is characteristic of the pump strong absorption due to the presence of the nanograins as shown in our previous paper [7]. (Fig. 5). On this figure, we observe along the whole length of the waveguide a larger Si-ng population inversion with erbium than with neodymium. Since the Si-ng modeling is the same with both RE, the population inversion difference is due to each specific Si-ng/RE interaction. This interaction is governed by the transfer coefficient K, which is the same for both RE, but also by specific transitions lifetimes of each RE. The larger population inversion observed for erbium than with neodymium is consequently due to the difference of lifetimes between these RE (Table 2 and 3). This observation leads us to the conclusion that RE excitation occurring from Si-ng is more efficient in the case of neodymium than in the case of erbium ions due to the specific time dynamics of transitions in different RE. Figure 6 shows the influence of pump power on population inversion for the erbium and the neodymium ions as well as for the Si-ng. We observe that the population inversion occurs above a threshold pump power (600 mW/mm 2 ) for erbium. This behaviour is typical of a three-level system. For neodymium ions, we observe a positive population inversion for the whole pump power range which is characteristic of a four-level system. For high pump power (1000 mW/mm 2 ), both populations inversions reach a comparable value witnessing the saturation of the excitation mechanism. Finally, we plot (dotted lines) the population inversion of Si-ng as a function of pump power. Whatever the pump power, this inversion is higher for erbium than for neodymium ions. This evidences a more efficient transfer from Si-ng to neodymium ions than to erbium ions. Optical gain From population levels N i (x, y, z) we deduced the local gross gain per unit length g dB/cm (Eq. (13)) at the signal wavelength: g dB/cm (x, y, z) = 10 ln10 σ em N high (x, y, z) − σ abs N low (x, y, z)(13) where N high and N low are respectively the higher and lower levels of the considered transition and σ abs and σ em are absorption and emission cross sections. For erbium ions, we make the link N high = N 1 and N low = N 0 and for neodymium ions N high = N 3 and N low = N 1 and we assume equal emission and absorption cross sections for one RE. The local gross gain per unit length recorded at x = 4.5 µm and y = 8.55 µm (center of the XY section of the active layer) and z = 0 (beginning of the waveguide) is plotted in Fig. 7. For Er 3+ doped waveguide, we find that, above a threshold pump power of 1550 mW/mm 2 , a positive gross gain is reached which increases up to 2 dB/cm for the highest pump power simulated in our study. In order to estimate the net gain, we must account for the background losses such as those found by Navarro-Urrios et al [27] on comparable samples (3.0 dB/cm at 1532 nm). We can conclude that it is not possible to reach a positive net gain in this range of pump power. However Navarro-Urrios et al found a positive net gain equal to 0.3 dB/cm, this small difference with our modelling result may be explained by higher pump power that was used in the experiment (4.10 5 mW/mm 2 to 6.10 6 mW/mm 2 ). For Nd 3+ doped waveguide, we find that the optical gain remains positive over the whole power range. It increases up to 30 dB/cm for the highest pump power of 10 4 mW/mm 2 . We can also estimate a net gain taking into account background losses of 0.8 dB/cm found by Pirasteh et al in a similar system [8]. Figure 7 shows that for a pump power above 130 mW/mm 2 the losses (dashed line) can be compensated leading to a net gain. The gross gain per unit length obtained for Nd 3+ remains higher that the one obtained for Er 3+ whatever the pump power range. The gross gain per unit length difference (between Nd 3+ and Er 3+ doped waveguide) is all the higher as the pump power increases. This feature may be due to the difference in Si-ng/RE transfer efficiency (section 3.1) linked to the levels dynamics as well as to the difference in absorption/emission cross section (σ Er = 6×10 −21 cm 2 against σ Nd = 1 × 10 −19 cm 2 ). The three categories of commercially available optical amplifier on C-band (EDFA, EDWA and SOA) present a gain level about 20 to 25 dB with a working length ranging from cm to few meters for EDFA and for a power consumption mainly dedicated to optical or electrical pumping of about few watts [28]. Since we found a low gain and a short length of positive population inversion by modeling of Er 3+ doped based waveguide on the broad range of pump power. We conclude to the impossibility of achievement of an optical amplifier with this configuration of co-propagating pump and signal which would compete with commercially available systems. The gross gain per unit length reachable with Nd 3+ , about one order of magnitude larger than the one obtained with Er 3+ , could lead a significant amplification. To our knowledge, there is no commercially available comparable optical amplifier based on Nd3+ emission bands. However, Nd 3+ doped Aluminum oxide channel waveguide amplifiers developed by Yang et al show a maximal internal gain of 6 dB/cm for a pump power of 45 mW [23]. Conclusion Our algorithm based on ADE-FDTD method previously used for modeling steady states of fields, population levels and gross gain per unit length for Nd 3+ doped waveguide has been extended with success to the case of erbium ions doped waveguide. We have demonstrated that the neodymium ions are more suitable than the erbium ions to obtain a net positive gain per unit length in silica based waveguide containing silicon nanograins. The theoretical maximum gross gain per unit length of 2 dB/cm at 1532 nm (10 4 mW/mm 2 ) does not compensate background losses experimentally estimated to 3 dB/cm. On the contrary, the use of neodymium ions leads to a gross gain per unit length of 30 dB/cm at 1064 nm (10 4 mW/mm 2 ). Moreover the background losses are compensated above a pump power threshold of 130 mW/mm 2 . This theoretical demonstration of a large gross gain per unit length for a Nd 3+ doped active layer may justify further experimental work in order to achieve Nd 3+ doped silicon based waveguide optical amplifier or laser. In order to investigate the possibility of achieving larger gain further studies may be performed with other concentrations of rare earth and Si-ng, other rare earth and other pumping configurations. This method may be applied to study the electromagnetic fields and levels populations distribution in steady states of systems with other kind of emitters (quantum dots, quantum wells...) and in other configuration (VCELs, down-converting layers...). Fig. 1 . 1Transverse section view of the waveguide constituted by bottom and strip cladding layers of silica surrouding the active layer constituted by silicon rich silicon oxide (SRSO) matrix doped with silicon nanograins (Si-ng) and Nd 3+ or Er 3+ ions. Fig. 2 . 2Excitation mechanism of (a) erbium ions and (b) neodymium ions Fig. 3 . 3Population inversion along the direction of propagation for the erbium ions (on the left) and the neodymium ions (on the right) for a pump power equal to 1000 mW/mm 2 Fig. 4 . 4Population inversion of Si-ng along the direction of propagation in the case of erbium ions (on the left) and neodymium ions (on the right) for a pump power equal to 1000 mW/mm 2 From Fig. 4 we extract a particular set of values of the population inversion of Si-ng along the direction of propagation recorded at x = 4.5 µm and y = 8.55 µm (center of the XY section of the active layer) Fig. 5 .Fig. 6 . 56Population inversion of Si-ng along the direction of propagation in the case of erbium and neodymium recorded at x = 4.5 µm and y = 8.55 µm (center of the XY section of the active layer) for a pump power equal to 1000 mW/Population inversion for neodymium, erbium ions and silicon nanograins divided by the rare earth ions concentration (10 20 at/cm 3 ) recorded at x = 4.5 µm and y = 8.55 µm (center of the XY section of the active layer) and z=0 (beginning of the waveguide) Fig. 7 . 7Local gross gain per unit length at the center of the active layer and in the beginning of the waveguide as a function of the pumping power for a waveguide doped with Nd 3+ (open circle) and a waveguide doped with Er 3+ (open square) recorded at x = 4.5 µm and y = 8.55 µm (center of the XY section of the active layer) and z = 0 (beginning of the waveguide). Losses found by Pirasteh et al [8] are represented (dashed line). Table 1 . 1Parameters levels of silicon nanograinsj → i Lifetime (s) ω i j (10 15 rad/s) ∆ω i j (10 14 rad/s) N p 1 → 0 50.10 −6 3.682 1 2756 Table 2 . 2Parameters levels of erbium ionsj → i 3→2 2→1 2→0 1→0 Lifetime (s) 0.1 × 10 −6 2.4 × 10 −6 710 × 10 −6 8.5 × 10 −3 ω i j (10 15 rad/s) 1.23 ∆ω i j (10 15 rad/s) 0.15 N p 1 Table 3 . 3Parameters levels of neodymium ionsj → i 4→3 3→2 3→1 3→0 2→1 1→0 Lifetime (s) 230 × 10 −12 1000 × 10 −6 200 × 10 −6 250 × 10 −6 970 × 10 −12 510 × 10 −12 ω i j (10 15 rad/s) 1.34 1.77 1.99 ∆ω i j (10 15 rad/s) 0.67 0.18 0.11 N p 1 1 1 AcknowledgmentsThe authors are grateful to the French Nation Research Agency, which supported this work through the Nanoscience and Nanotechnology program (DAPHNES project ANR-08-NANO-005) and Centre de ressources informatiques de Haute-Normandie, (CRIHAN France) for computing facilities. Nanosilicon photonics. N Daldosso, L Pavesi, Laser Photonics Rev. 3N. Daldosso and L. Pavesi, "Nanosilicon photonics," Laser Photonics Rev. 3, 508-534 (2009). G P , Fiber-optic communication systems. John Wiley & SonsG. P. Agrawal, Fiber-optic communication systems (John Wiley & Sons, 2010). Rare earth-doped integrated glass components: modeling and optimization. O Lumholt, A Bjarklev, T Rasmussen, C Lester, J. Lightwave Technol. 13O. Lumholt, A. Bjarklev, T. Rasmussen, and C. Lester, "Rare earth-doped integrated glass components: modeling and optimization," J. Lightwave Technol. 13, 275-282 (1995). High energy excitation transfer from silicon nanocrystals to neodymium ions in silicon-rich oxide film. A Podhorodecki, J Misiewicz, F Gourbilleau, J Cardin, C Dufour, Electrochem. Solid-State Lett. 13A. Podhorodecki, J. Misiewicz, F. Gourbilleau, J. Cardin, and C. Dufour, "High energy excitation transfer from silicon nanocrystals to neodymium ions in silicon-rich oxide film," Electrochem. Solid-State Lett. 13, K26-K28 (2010). Broadband sensitizers for erbium-doped planar optical amplifiers: review. A Polman, F C J M Van Veggel, J. Opt. Soc. Am. B. 21A. Polman and F. C. J. M. van Veggel, "Broadband sensitizers for erbium-doped planar optical amplifiers: review," J. Opt. Soc. Am. B 21, 871-892 (2004). Luminescence of Nd-enriched silicon nanoparticle glasses. A N Macdonald, A Hryciw, Quan Li, A Meldrum, Opt. Mater. 28A. N. MacDonald, A. Hryciw, Quan Li, and A. Meldrum, "Luminescence of Nd-enriched silicon nanoparticle glasses," Opt. Mater. 28, 820-824 (2006). Modeling of the electromagnetic field and level populations in a waveguide amplifier: a multi-scale time problem. A Fafin, J Cardin, C Dufour, F Gourbilleau, Opt. Express. 21A. Fafin, J. Cardin, C. Dufour, and F. Gourbilleau, "Modeling of the electromagnetic field and level populations in a waveguide amplifier: a multi-scale time problem," Opt. Express 21, 24171-24184 (2013).	[]
[ "BRS symmetry from renormalization group flow", "BRS symmetry from renormalization group flow" ]	[ "M. DM Bonini \nDipartimento di Fisica\nDipartimento di Fisica, Università di Milano and INFN\nUniversità di Parma\nINFN\nGruppo Collegato di Parma\nSezione di MilanoItaly\n", "' Attanasio \nDipartimento di Fisica\nDipartimento di Fisica, Università di Milano and INFN\nUniversità di Parma\nINFN\nGruppo Collegato di Parma\nSezione di MilanoItaly\n", "G Marchesini \nDipartimento di Fisica\nDipartimento di Fisica, Università di Milano and INFN\nUniversità di Parma\nINFN\nGruppo Collegato di Parma\nSezione di MilanoItaly\n" ]	[ "Dipartimento di Fisica\nDipartimento di Fisica, Università di Milano and INFN\nUniversità di Parma\nINFN\nGruppo Collegato di Parma\nSezione di MilanoItaly", "Dipartimento di Fisica\nDipartimento di Fisica, Università di Milano and INFN\nUniversità di Parma\nINFN\nGruppo Collegato di Parma\nSezione di MilanoItaly", "Dipartimento di Fisica\nDipartimento di Fisica, Università di Milano and INFN\nUniversità di Parma\nINFN\nGruppo Collegato di Parma\nSezione di MilanoItaly" ]	[]	By using the exact renormalization group formulation we prove perturbatively the Slavnov-Taylor (ST) identities in SU(2) Yang-Mills theory. This results from two properties: locality, i.e. the ST identities are valid if their local part is valid; solvability, i.e. the local part of ST identities is valid if the couplings of the effective action with non-negative dimensions are properly chosen.	10.1016/0370-2693(94)01676-4	[ "https://arxiv.org/pdf/hep-th/9412195v1.pdf" ]	16,158,412	hep-th/9412195	df9de999d6252c2d2795c5ffcd6545f769b0276e	BRS symmetry from renormalization group flow arXiv:hep-th/9412195v1 22 Dec 1994 November 1994 M. DM Bonini Dipartimento di Fisica Dipartimento di Fisica, Università di Milano and INFN Università di Parma INFN Gruppo Collegato di Parma Sezione di MilanoItaly ' Attanasio Dipartimento di Fisica Dipartimento di Fisica, Università di Milano and INFN Università di Parma INFN Gruppo Collegato di Parma Sezione di MilanoItaly G Marchesini Dipartimento di Fisica Dipartimento di Fisica, Università di Milano and INFN Università di Parma INFN Gruppo Collegato di Parma Sezione di MilanoItaly BRS symmetry from renormalization group flow arXiv:hep-th/9412195v1 22 Dec 1994 November 1994 By using the exact renormalization group formulation we prove perturbatively the Slavnov-Taylor (ST) identities in SU(2) Yang-Mills theory. This results from two properties: locality, i.e. the ST identities are valid if their local part is valid; solvability, i.e. the local part of ST identities is valid if the couplings of the effective action with non-negative dimensions are properly chosen. Introduction A renormalized theory is defined by giving the "relevant part" of the effective action, i.e. its local part involving couplings which have non-negative mass dimension 1 . If the fields have non-zero mass these relevant couplings could be given by the first coefficients of the Taylor expansion of vertex functions around zero momenta. If the fields have zero mass the expansion must be done around some non-vanishing Euclidean subtraction point µ = 0. For the massless Φ 4 4 theory there are three relevant couplings corresponding to the physical mass, wave function normalization and interaction strength g at the subtraction point µ. These three physical couplings define completely the theory. In a gauge theory the effective action contains more couplings than physical parameters. In the SU(2) Yang-Mills theory for instance the effective action Γ[φ, γ] (with φ = (A µ , c) the vector and ghost fields and γ = (u µ , v) the BRS sources) contains nine relevant couplings but only three are fixed by the vector and ghost field normalizations and by the interaction strength g at a subtraction point µ. After fixing these three physical parameters, the relevant part of the Yang-Mills effective action can be written Γ rel [φ, γ] ≡ T (µ) 4 Γ[φ, γ] = S BRS [φ, γ] +hΓ rel [φ, γ; ρ i ] , where we denote by T (µ) d the operator which extracts from a given functional of dimension d − 4 its relevant part (in four space-time dimension) with a non-vanishing subtraction point µ. For SU (2) the BRS classical action [1], in the Feynman gauge, is given by S BRS = d 4 x − 1 4 F 2 µν − 1 2 (∂ µ A µ ) 2 + 1 g w µ · D µ c − 1 2 v · c ∧ c andΓ rel [φ, γ; ρ i ] ≡ d 4 x ρ 1 1 2 A 2 µ + ρ 2 1 2 (∂ µ A µ ) 2 + ρ 3 w µ · c ∧ A µ + ρ 4 1 2 v · c ∧ c + ρ 5 g 2 4 (A µ ∧ A ν ) 2 + ρ 6 g 2 4 (A µ · A ν ) 2 ,(1)with F µν = ∂ µ A ν − ∂ ν A µ + gA µ ∧ A ν , D µ c = ∂ µ c + gA µ ∧ c and w µ = u µ + g∂ µc . The six couplings ρ i inΓ rel vanish at tree level and should be constrained by the gauge symmetry. For instance ρ 1 is the vector field mass and we expect it must vanish. The gauge symmetry requires that the effective action satisfies the ST identities ∆ Γ [φ, γ] ≡ S Γ ′ Γ ′ [φ, γ] = 0 , Γ ′ [φ, γ] = Γ[φ, γ] + 1 2 d 4 x(∂ µ A µ ) 2 ,(2) where S Γ ′ is the usual Slavnov operator (see for instance [2]). Also for ∆ Γ [φ, γ] one defines the relevant part. Since this functional has dimension one we have ∆ Γ,rel [φ, γ; δ i ] = T (µ) 5 ∆ Γ [φ, γ] , where δ i are the relevant parameters, i.e. the coefficients of monomials in the fields, sources and momenta of dimension not greater than five. For the SU (2) ∆ Γ,rel [φ, γ; δ i ] = d 4 x δ 1 A µ · ∂ µ c − δ 2 A µ · ∂ 2 ∂ µ c + δ 3 A µ · (∂ 2 A µ ) ∧ c + δ 4 A µ · (∂ µ ∂ ν A ν ) ∧ c + 1 2 δ 5 (∂ µ w µ ) · c ∧ c + 1 2 δ 6 (w µ ∧ A µ ) · (c ∧ c) + δ 7 ((∂ µ A µ ) · A ν )(A ν · c) + δ 8 ((∂ µ A µ ) · c)(A ν · A ν ) + δ 9 ((∂ ν A µ ) · A ν )(A µ · c) + δ 10 ((∂ ν A µ ) · A µ )(A ν · c) + δ 11 ((∂ ν A µ ) · c)(A µ · A ν ) . (3) The important question is then whether it is possible to fix the couplings ρ i in such a way to ensure that the full set of ST identities (2) is satisfied. In perturbation theory this question is solved in the positive sense since almost two decades. In dimensional regularization with minimal subtraction [3]- [5] of gauge theories without chiral fermions both the bare Lagrangian and the regularization procedure do not break the BRS symmetry so that the effective action (and then also the couplings ρ i ) automatically satisfies ST identities. In chiral gauge theories dimensional regularization breaks the symmetry [3,5,6]. However, if no anomalies are present, one can implement the ST identities by introducing non-invariant local counterterms [7]. This is done by a so called "fine tuning procedure". This fact is independent of the regularization since the classification of all possible anomalies is a purely algebraic problem [2,8], i.e. anomalies are associated to the existence of non-trivial cohomology classes of the Slavnov operator. Becchi has recently [9] shown that exact renormalization group (RG) flow [10]- [12] can be used to deduce the ST identities. By taking advantage of the fact that in the RG flow one introduces an infrared (IR) cutoff Λ, he has defined the relevant couplings at the nonphysical point Λ = 0. This allowed him to use a vanishing subtraction point µ = 0. The connection with the physical couplings defined in (1) is then indirect but can be obtained perturbatively. The ST identities are then analyzed at Λ = 0 and then, because of the IR problem, one cannot continue to the physical point Λ = 0. In this note we follow the same analysis but work at the physical point Λ = 0 and prove directly the ST identities (2) for the effective action Γ[φ, γ]. We are then able to discuss the symmetry in terms of the relevant couplings defined in (1). In a more detailed paper [13] there will be the full calculations for the SU(2) case and the precise expression of the values of the six couplings ρ i . We denote by ∆ (ℓ) Γ [φ, γ] and ∆ Γ,rel [φ, γ; δ (ℓ) i ] the ST functional and its relevant part at loop ℓ. We prove the following two properties: Locality: if for any loop ∆ Γ,rel [φ, γ; δ (ℓ) ] = 0 ,(4) then for any loop ∆ (ℓ) Γ [φ, γ] = 0 .(5) Solvability: The set of equations (4) can be solved perturbatively by appropriately fixing the couplings ρ (ℓ) i at loop ℓ as functions of the couplings ρ (ℓ ′ ) i at lower loops ℓ ′ < ℓ. At zero loop one has Γ (ℓ=0) [φ, γ] = S BRS [φ, γ] which implies ∆ (ℓ=0) Γ [φ, γ] = 0. From this and the above properties one concludes, by induction on the number of loops, that the ST identities are satisfied perturbatively. We shall call the condition (4), i.e. δ i = 0, the fine tuning equations. The second property is a consequence of the consistency condition S Γ ′ ∆ Γ = S 2 Γ ′ Γ ′ = 0 .(6) To show that (6) ensures the solvability of the fine tuning equations is relatively simple for µ = 0. In this case one can check by inspection that the relevant part of (6) gives enough relations among the parameters δ i to make solvable the fine tuning equations (for example in the SU(2) case only six δ i are independent). Here we work at µ = 0 and use RG flow method to prove solvability. The difficulty arising from the presence of a non-vanishing subtraction point is due to the fact that by applying T (µ) 6 to S Γ ′ ∆ Γ , i.e. by taking the relevant part, one introduces not only the relevant part of ∆ Γ but also some irrelevant vertices evaluated at the subtraction points. Then one has to study also the vanishing of irrelevant parts. This will be shown by using the RG flow. Renormalization group flow In order to prove these properties by using the RG flow method, we recall first how the effective action Γ[φ, γ] and the functional ∆ Γ [φ, γ] are obtained within the RG formulation. For more details see [9,13]. (i) One introduces the Wilsonian effective action S eff [φ, γ; Λ, Λ 0 ] which is obtained by path integration over the fields with frequencies Λ 2 < p 2 < Λ 2 0 . In this functional one uses propagators with IR and UV cutoff function K ΛΛ 0 (p) equal to one in the above region and rapidly vanishing outside. From this definition one has that the coefficients of the monomials quadratic in the fields in S eff [φ, γ; Λ, Λ 0 ] are proportional toh, thus at zero loop order S (ℓ=0) eff [φ, γ; Λ, Λ 0 ] does not contain quadratic monomials. This property will play a key rôle in the perturbative proof of locality and solvability by RG flow method. At the physical point Λ = 0 and Λ 0 → ∞ the functional S eff [φ, γ; 0, ∞] generates the amputated connected Green functions. The physical effective action Γ[φ, γ] is then obtained from S eff [φ, γ; 0, ∞] by Legendre transform so that the relevant parameters ρ i in (1) From the definition of S eff [φ, γ; Λ, Λ 0 ] one finds that this functional satisfies an evolution equation [10,11] in the IR cutoff Λ Λ∂ Λ S ef h = (2π) 8h p 1 p 2 Λ∂ Λ K 0Λ (p) e −iS eff /h 1 2 δ δA a µ (−p) δ δA a µ (p) + δ δc a (−p) δ δc a (p) e iS eff /h ,(7) where p ≡ d 4 p (2π) 4 and K 0Λ (p) is the cutoff function vanishing for p 2 > Λ 2 . This equation can be integrated by giving boundary conditions in Λ. As boundary conditions for the RG flow one assumes that at the UV point Λ = Λ 0 → ∞ the irrelevant part of the Wilsonian action vanishes S eff,irr [φ, γ; Λ 0 , Λ 0 ] ≡ (1 − T (µ) 4 ) S eff [φ, γ; Λ 0 , Λ 0 ] → 0 , for Λ 0 → ∞ .(8) As far as the point where to fix S eff,rel [φ, γ; Λ, Λ 0 ] is concerned, one may choose the physical point Λ = 0 and Λ 0 → ∞ where S eff,rel [φ, γ; 0, ∞] is given in terms of the couplings ρ i in (1). In this formulation the usual loop expansion of the effective action can be obtained by solving iteratively the RG evolution equation (7) with these boundary conditions and the zero loop input Γ (ℓ=0) [φ, γ] = S BRS [φ, γ]. As shown by Polchinski [11], the RG formulation provides a very simple method to prove perturbative renormalizability, i.e. the limit Λ 0 → ∞ can be taken. The proof can be applied to the gauge theory case [14]. RG supplies also a simple method to prove [14] that a massless theory is IR finite in perturbation theory, i.e. the limit Λ → 0 can be taken, provided a non-vanishing subtraction point µ = 0 is introduced. (ii) One then considers the ST identities in (2). It can be shown that these identities can be formulated directly for the Wilsonian action S eff [φ, γ; Λ, Λ 0 ] for any Λ and Λ 0 even away from the physical point Λ = 0 and Λ 0 → ∞ . One introduces the generalized BRS transformation δA a µ (p) = − i g ηp µ c a (p) + K 0Λ (p)η δS eff δu a µ (−p) , δc a (p) = K 0Λ (p)η δS eff δv a (−p) , δc a (p) = i g ηp µ A a µ (p) , with η a Grassmann parameter and a the gauge index. The cutoff function is cancelled by an inverse cutoff function entering in the sources. From this transformation one finds the following form of the ST identities ∆ eff [φ, γ; Λ, Λ 0 ] = 0 (9) where ∆ eff [φ, γ; Λ, Λ 0 ] ≡ p K 0Λ (p)e −iS eff δ δA a µ (p) δ δu a µ (−p) + δ δc a (p) δ δv a (−p) e iS eff +i p p 2 A a µ (p) δ δu a µ (p) + i g p µ c a (p) δ δA a µ (p) + i g p µ w a µ (p) δ δv a (p) − i g p µ A a µ (p) δ δc a (p) S eff . For Λ = Λ 0 → ∞ we have K 0Λ 0 (p) → 1 and S eff becomes local (see (8)). Therefore in this limit we have ∆ eff,irr [φ, γ; Λ 0 , Λ 0 ] ≡ (1 − T (µ) 5 ) ∆ eff [φ, γ; Λ 0 , Λ 0 ] → 0 , as Λ 0 → ∞ .(10) At the physical point Λ = 0 and Λ 0 → ∞ , ∆ eff [φ, γ; 0, ∞] is related to ∆ Γ [φ, γ] via Legendre transform, i.e. the vertices of ∆ Γ [φ, γ] are obtained from the vertices of ∆ eff [φ, γ; 0, ∞] by neglecting the one particle (A, c) reducible contributions. (iii) The RG flow for the functional ∆ eff is given by the following linear evolution equation Λ∂ Λ ∆ eff [φ, γ; Λ, Λ 0 ] = {M 1 [S eff ] +hM 2 } · ∆ eff [φ, γ; Λ, Λ 0 ] ,(11) where M 1 and M 2 are M 1 [S eff ] = −(2π) 8 p 1 p 2 Λ∂ Λ K 0Λ (p) δS eff δA a µ (p) δ δA a µ (−p) + δS eff δc a (p) δ δc a (−p) − c ↔c , M 2 = i 2 (2π) 8 p 1 p 2 Λ∂ Λ K 0Λ (p) δ δA a µ (p) δ δA a µ (−p) + δ δc a (p) δ δc a (−p) − c ↔c . The boundary conditions for (11) are obtained from the ones of S eff [φ, γ; Λ, Λ 0 ]. At the UV point the irrelevant part vanishes (see (10)) and at the physical point Λ = 0 and Λ 0 → ∞ the relevant part is given by the parameters δ i . By taking advantage of (10) and perturbative renormalizability in the following we will consider the limit Λ 0 → ∞ . Locality We now prove the property of locality (4), (5). To do this we show that from the hypothesis (4) we have ∆ (ℓ) eff [φ, γ; Λ, ∞] = 0(12) for any loop and any value of Λ and this implies the identity (5). Indeed at the physical point Λ = 0 the condition ∆ (11) becomes Λ∂ Λ ∆ (ℓ) eff [φ, γ; Λ, ∞] = M 1 [S (ℓ=0) eff ] · ∆ (ℓ) eff [φ, γ; Λ, ∞] ,(13) where M 1 [S (ℓ=0) eff ] does not contain monomials linear in the fields or sources (as observed before S (ℓ=0) eff [φ, γ] does not contain quadratic monomials). First we consider the case ℓ = 0 and prove that ∆ eff,n (· · · ; Λ, ∞) the vertices of ∆ eff at loop ℓ with n fields or sources. The dots denote momenta, internal and Lorentz indices. For n = 2, i.e. for the coefficient of the Ac monomial, ∆ (ℓ=0) eff,2 (p; Λ, ∞), we have from (13) Λ∂ Λ ∆ (ℓ=0) eff,2 (p; Λ, ∞) = 0 . By using the boundary condition (10) we conclude that ∆ (ℓ=0) eff,2 (p; Λ, ∞) is given by its relevant part at the physical point. Since in ∆ eff,2 there are no one particle reducible contributions, this vertex at the physical point is the same as the Ac-vertex of ∆ Γ which, according to the hypothesis (4), has no relevant part. Therefore also ∆ eff,2 has no relevant part and we have that the full vertex vanishes, ∆ (ℓ=0) eff,2 (p; Λ, ∞) = 0. Now we proceed by iteration on n, namely we suppose that ∆ (ℓ=0) eff,n ′ (· · · ; Λ, ∞) = 0 for all n ′ < n and show that ∆ (ℓ=0) eff,n (· · · ; Λ, ∞) = 0. Using again the fact that S (0) eff has no quadratic monomials, we have from (13) Λ∂ Λ ∆ (ℓ=0) eff,n (· · · ; Λ, ∞) = 0 . By using the boundary condition (10), we have that ∆ (ℓ=0) eff,n (· · · ; Λ, ∞) is given by its relevant part at the physical point, which is zero as a consequence of the hypothesis (4). This is due to the fact that the n-vertices of ∆ eff and ∆ Γ are the same. Indeed the one particle reducible contributions of the n-vertices of ∆ eff involve vertices with n ′ < n which are zero because of the inductive hypothesis. Therefore we have in general ∆ (ℓ=0) eff [φ, γ; Λ, ∞] = 0. Consider now the case ℓ > 0. We assume ∆ (ℓ ′ ) eff [φ, γ; Λ, ∞] = 0 for any ℓ ′ < ℓ and we want to prove (12) at loop ℓ. The proof consists in repeating exactly the same procedure of induction on n which has been followed above for the zero loop case. This concludes the proof that ∆ (ℓ) eff [φ, γ; Λ, ∞] = 0 for any ℓ. Solvability at µ = 0 We show here that the fine tuning equations (4) can be solved for µ = 0 as a consequence of the consistency condition (6). Namely for SU(2) one can fix the six couplings ρ i of the effective action (1) in such a way that the eleven relevant parameters δ i vanish. For µ = 0 the proof of the solvability of (4) is a direct consequence of the consistency condition (6) which gives [9] gδ 2 = δ 3 + δ 4 , δ 6 = −gδ 5 , δ 7 = δ 9 = δ 11 , 2δ 8 = δ 10 . The general relevant functional ∆ Γ which satisfies the above relations can be expressed as the BRS variation of a local functional, namely ∆ Γ,rel [φ, γ; δ i ] = S Γ ′(0) Γ rel [φ, γ;ρ i ] .(14) This expresses the triviality of the ghost number one cohomology class of the Slavnov operator for the SU(2) Yang-Mills theory. If this representation holds, then the eleven parameters δ i are the following functions of the six parametersρ i δ 1 = 1 gρ 1 , δ 2 = 1 gρ 2 , δ 3 =ρ 5 , δ 4 = gδ 2 − δ 3 =ρ 2 −ρ 5 , δ 6 = −gδ 5 =ρ 5 −ρ 6 , δ 7 = δ 9 = δ 11 = g(ρ 3 −ρ 4 −ρ 5 ) , 2δ 8 = δ 10 = 2g(ρ 5 −ρ 3 ) .(15) For µ = 0 the representation (14) is not valid a priori, since the consistency condition (6) gives relations among the δ i which involve also irrelevant contributions of the ∆ Γ vertices evaluated at some subtraction point µ = 0. However we have seen that the irrelevant parts can be set to zero iteratively. Before showing how this procedure works we recall how, even in the case of a non-vanishing subtraction point µ = 0, the fine tuning equations can be solved loopwise if the representation (14) holds. From (2) we have ∆ (ℓ) Γ [φ, γ] = 2 S Γ ′(0) Γ ′ (ℓ) + ℓ−1 k=1 S Γ ′(k) Γ ′ (ℓ−k) . By applying T (µ) 5 , we obtain the relevant part ∆ Γ,rel [φ, γ; δ (ℓ) i ] = 2 S Γ ′(0) Γ ′ rel [φ, γ; ρ (ℓ) i ] + Ω (ℓ) [φ, γ] ,(16) where Ω (ℓ) [φ, γ] = T (µ) 5 ℓ−1 k=1 S Γ ′(k) Γ ′ (ℓ−k) + 2 T (µ) 5 S Γ ′(0) − S Γ ′(0) T (µ) 4 Γ ′ (ℓ) . The crucial observation now is that Ω (ℓ) depends only on the relevant parameters ρ (ℓ ′ ) i at lower loops ℓ ′ < ℓ. This is obvious, since the product of two relevant vertices is a relevant vertex so that T (µ) 5 S Γ ′(0) T (µ) 4 = S Γ ′(0) T (µ) 4 . As a consequence (T (µ) 5 S Γ ′(0) − S Γ ′(0) T (µ) 4 ) Γ ′ rel = 0 and Ω (ℓ) does not receive contribution from the couplings ρ (ℓ) i . From eqs. (14) and (16) one has that Ω (ℓ) must be of the form Ω (ℓ) = S Γ ′(0) Γ ′ rel [φ, γ; ρ ′ (ℓ) i ] , ρ ′ (ℓ) i =ρ (ℓ) i − 2ρ (ℓ) i , where ρ ′ (ℓ) i are given in terms of the couplings ρ (ℓ ′ ) i at lower loops ℓ ′ < ℓ. Therefore one has ρ i = 0, i.e. ∆ (ℓ) Γ [φ, γ] = 0 if one sets ρ (ℓ) i = − 1 2 ρ ′ (ℓ) i .(17) This ends the proof of the fact that if (14) holds then the fine tuning equations can be solved even for µ = 0. We now come to discuss whether the representation (14) can be used also for µ = 0. As recalled before (14) is not valid if there are irrelevant contributions in the various vertices of ∆ Γ . To show how to use (14) in this case one proceeds as follows by exploiting the perturbative RG results obtained in the previous section. (i) The Ac-vertex ∆ Γ,2 is given only by its relevant part (see proof of locality), i.e. given by the parameters δ 1 and δ 2 in (3). Therefore for this vertex the representation (14) holds and from (17) one can fix ρ 1 and ρ 2 in such a way the full vertex ∆ Γ,2 vanishes. One shows [13] that the equations δ 1 = δ 2 = 0 are solved by ρ 1 = ρ 2 = 0. (ii) Once ∆ Γ,2 = 0, the AAc− and wcc−vertices ∆ Γ,3 are given only by their relevant parts (see proof of locality), i.e. given by the parameters δ 3 , δ 4 and δ 5 in (3). Therefore, for these relevant vertices the consistency condition (6) gives δ 3 + δ 4 = 0 . Then the representation (14) holds and one can fix the couplings ρ 3 , ρ 4 in such a way that ∆ Γ,3 = 0. One shows that the equations δ 3 = δ 4 = δ 5 = 0 are solved by ρ 3 = 0 and ρ 4 given by an irrelevant part of the wcA-vertex of Γ[φ, γ] evaluated at some Euclidean subtraction point. For the exact evaluation of ρ 4 in SU(2) see ref. [13]. (iii) Once ∆ Γ,3 = 0, the AAAc− and wccA−vertices ∆ Γ,4 are given only by their relevant parts (see again proof of locality), i.e. given by the parameters δ 6 , . . . , δ 11 in (3). For these relevant vertices the consistency condition gives δ 6 = 0 , δ 7 = δ 9 = δ 11 , 2δ 8 = δ 10 , so that the representation (14) holds and one can fix ρ 5 and ρ 6 in such a way ∆ Γ,4 = 0. One shows that the equations δ 6 = . . . = δ 11 = 0 are solved by ρ 5 and ρ 6 given by irrelevant parts of wcA−, wcAA−, AAA− and AAAA−vertices of Γ[φ, γ] evaluated at some Euclidean subtraction points. For the exact evaluation of ρ 5 and ρ 6 in SU(2) see ref. [13]. Comments By using the exact RG flow we have proved perturbatively that the ST identities (2) are valid provided that the relevant part of the effective action is properly chosen. In the SU (2) case one has Γ rel [φ, γ; ρ i ] = S BRS +h d 4 x ρ 4 2 v · c ∧ c + ρ 5 g 2 4 (A µ ∧ A ν ) 2 + ρ 6 g 2 4 (A µ · A ν ) 2 , where the only non-vanishing couplings ρ 4 , ρ 5 and ρ 6 are given in terms of appropriate irrelevant vertices of Γ evaluated at the subtraction point [13]. This form allows one to perform the perturbative expansion since irrelevant vertices at loop ℓ involve relevant couplings at lower loops ℓ ′ < ℓ. The method is general. As shown in ref. [14] it can be applied for instance to SU(2) gauge theory with fermions. The application to the case of chiral gauge theories without anomalies should be also possible along the same lines. The RG method provides in principle a non-perturbative formulation thus it could be used to extend the proof of locality and solvability beyond perturbation theory. It is then important to pin down the points where perturbation theory was needed in this work. Here the proof of locality is essentially based on the following two facts: 1) in the RG flow (11) for ∆ eff [φ, γ; Λ, ∞] we used only the linear operator M 1 and neglected M 2 . This is possible only in perturbation theory because of the inductive hypothesis; 2) we used the fact that M 1 does not contain monomials linear in the fields, which is true only if one uses, via induction, the tree approximation of S eff . For the proof of solvability one uses the locality of ∆ Γ (see second part of sect. 4) to set to zero irrelevant contributions in the consistency condition (6). Due to this fact also the proof of this property is restricted to the perturbative framework. Thus, it seems that the crucial point is the non-perturbative extension of locality. We have benefited greatly from discussions with C. Becchi and M. Tonin. are simply related to the relevant parameters of S eff [φ, γ; 0, ∞]. is obtained by induction on the number of loops and on the number of fields n in the vertices of ∆ eff [φ, γ; Λ, ∞]. A crucial point in the inductive proof is the fact that if ∆ (ℓ ′ ) eff [φ, γ; Λ, ∞] = 0 for any ℓ ′ < ℓ, then at loop ℓ, the RG flow [φ, γ; Λ, ∞] = 0 by induction on n. We denote by ∆ (ℓ) case, there are eleven relevant parameters δ i and we have1 In this paper the relevant parts include also what is usually called marginal. . C Becchi, A Rouet, R Stora, Phys. Lett. 52344C. Becchi, A. Rouet and R. Stora, Phys. Lett. 52B (1974) 344. C Becchi, Lectures on the renormalization of gauge theories. B.S. DeWitt and R. 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[ "THE COMPLEXITY OF REACHABILITY IN AFFINE VECTOR ADDITION SYSTEMS WITH STATES", "THE COMPLEXITY OF REACHABILITY IN AFFINE VECTOR ADDITION SYSTEMS WITH STATES" ]	[ "Michael Blondin [email protected] ", "Mikhail Raskin [email protected] ", "\nUniversité de Sherbrooke\nCanada\n", "\nTechnische Universität München\nGermany\n" ]	[ "Université de Sherbrooke\nCanada", "Technische Universität München\nGermany" ]	[ "Logical Methods in Computer Science" ]	Vector addition systems with states (VASS) are widely used for the formal verification of concurrent systems. Given their tremendous computational complexity, practical approaches have relied on techniques such as reachability relaxations, e.g., allowing for negative intermediate counter values. It is natural to question their feasibility for VASS enriched with primitives that typically translate into undecidability. Spurred by this concern, we pinpoint the complexity of integer relaxations with respect to arbitrary classes of affine operations.More specifically, we provide a trichotomy on the complexity of integer reachability in VASS extended with affine operations (affine VASS). Namely, we show that it is NP-complete for VASS with resets, PSPACE-complete for VASS with (pseudo-)transfers and VASS with (pseudo-)copies, and undecidable for any other class. We further present a dichotomy for standard reachability in affine VASS: it is decidable for VASS with permutations, and undecidable for any other class. This yields a complete and unified complexity landscape of reachability in affine VASS. We also consider the reachability problem parameterized by a fixed affine VASS, rather than a class, and we show that the complexity landscape is arbitrary in this setting. CC Creative Commons 3:2 M. Blondin and M. Raskin Vol. 17:3is the reachability problem: given configurations x and y, is it possible to reach y starting from x? Such queries allow, e.g., to verify whether unsafe states can be reached in concurrent programs. The notorious difficulty of the reachability problem led to many proofs of its decidability over the last decades[ST77,May81,Kos82,Lam92,Ler10,Ler11,Ler12]. While the problem has been known to be EXPSPACE-hard since 1976 [Lip76], its computational complexity has remained unknown until very recently, where it was shown to be TOWERhard [CLL + 19] and solvable in Ackermannian time [LS15, LS19]. Given the potential applications on the one hand, and the tremendously high complexity on the other hand, researchers have investigated relaxations of VASS in search of a tradeoff between expressiveness and algorithmic complexity. Two such relaxations consist in permitting either: (a) transitions to be executed fractionally, and consequently counters to range over Q ≥0 (continuous reachability); or (b) counters to range over Z (integer reachability). In both cases, the complexity drops drastically: continuous reachability is P-complete and NP-complete for Petri nets and VASS respectively [FH15, BH17], while integer reachability is NP-complete for both models [HH14, CHH18]. Moreover, these two types of reachability have been used successfully to prove safety of real-world instances like multithreaded program skeletons, e.g., see [ELM + 14, ALW16, BFHH17].Although VASS are versatile, they are sometimes too limited to model common primitives. Consequently, their modeling power has been extended with various operations. For example, (multi-)transfers, i.e., operations of the formx ← x + n i=1 y i ; y 1 ← 0; y 2 ← 0; · · · ; y n ← 0, allow, e.g., for the verification of multi-threaded C and Java program skeletons with communication primitives [KKW14, DRV02]. Another example is the case of resets, i.e., operations of the form x ← 0, which allow, e.g., for the validation of some business processes [WvdAtHE09], and the generation of program loop invariants [SK19]. Many such extensions fall under the generic family of affine VASS, i.e., VASS with instructions of the form x ← A · x + b. As a general rule of thumb, reachability is undecidable for essentially any class of affine VASS introduced in the literature; in particular, for transfers and resets [AK76, DFPS98].Given the success of relaxations for the practical analysis of (standard) VASS, it is tempting to employ the same approach for affine VASS. Unfortunately, continuous reachability becomes undecidable for mild affine extensions such as resets and transfers. However, integer reachability was recently shown decidable for affine operations such as resets (NP-complete) and transfers (PSPACE-complete)[HH14,BHM18]. While such complexity results do not translate immediately into practical procedures, they arguably guide the design of algorithmic verification strategies.Contribution. Thus, these recent results raise two natural questions: for what classes of affine VASS is integer reachability decidable? And, whenever it is decidable, what is its exact computational complexity? We fully answer these questions in this paper by giving a precise trichotomy: integer reachability is NP-complete for VASS with resets, PSPACE-complete for VASS with (pseudo-)transfers and VASS with (pseudo-)copies, and undecidable for any other class. In particular, this answers a question left open in [BHM18]: integer reachability is undecidable for any class of affine VASS with infinite matrix monoid. Vol. 17:3 THE COMPLEXITY OF REACHABILITY IN AFFINE VASS 3:3This clear complexity landscape is obtained by formalizing classes of affine VASS and by carefully analyzing the structure of arbitrary affine transformations; which could be of independent interest. In particular, it enables us to prove a dichotomy on (standard) reachability for affine VASS: it is decidable for VASS with permutations, and undecidable for any other class. To the best of our knowledge, this is the first proof of the folkore rule of thumb stating that "reachability is undecidable for essentially any class of affine VASS".We further complement these trichotomy and dichotomy by showing that the (integer or standard) reachability problem has an arbitrary complexity when it is parameterized by a fixed affine VASS rather than a fixed class.Related work. Our work is related to [BHM18] which shows that integer reachability is decidable for affine VASS whose matrix monoid is finite (refined to EXPSPACE by [BHK + 20]); and more particularly PSPACE-complete in general for VASS with transfers and VASS with copies. While it is also recalled in [BHM18] that integer reachability is undecidable in general for affine VASS, the authors do not provide any necessary condition for undecidability to hold. Moreover, the complexity landscape for affine VASS with finite monoids is left blurred, e.g., it does not give necessary conditions for PSPACE-hardness results to hold, and the complexity remains unknown for monoids with negative coefficients. This paper completes the work initiated in [BHM18] by providing a unified framework, which includes the notion of matrix class, that allows us to precisely characterize the complexity of integer reachability for any class of affine VASS.Our work is also loosely related to a broader line of research on (variants of) affine VASS dealing with, e.g., modeling power [Val78], accelerability [FL02], formal languages [CFM12], coverability [BFP12], and the complexity of integer reachability for restricted counters [FGH13] and structures [IS16].	10.46298/lmcs-17(3:3)2021	[ "https://arxiv.org/pdf/1909.02579v5.pdf" ]	237,646,907	1909.02579	e806c536d1205fb8a86b03f76525fae04f6458dc	THE COMPLEXITY OF REACHABILITY IN AFFINE VECTOR ADDITION SYSTEMS WITH STATES 2021 Michael Blondin [email protected] Mikhail Raskin [email protected] Université de Sherbrooke Canada Technische Universität München Germany THE COMPLEXITY OF REACHABILITY IN AFFINE VECTOR ADDITION SYSTEMS WITH STATES Logical Methods in Computer Science 17331202110.46298/LMCS-17(3:3)2021Submitted Oct. 30, 2020 Vector addition systems with states (VASS) are widely used for the formal verification of concurrent systems. Given their tremendous computational complexity, practical approaches have relied on techniques such as reachability relaxations, e.g., allowing for negative intermediate counter values. It is natural to question their feasibility for VASS enriched with primitives that typically translate into undecidability. Spurred by this concern, we pinpoint the complexity of integer relaxations with respect to arbitrary classes of affine operations.More specifically, we provide a trichotomy on the complexity of integer reachability in VASS extended with affine operations (affine VASS). Namely, we show that it is NP-complete for VASS with resets, PSPACE-complete for VASS with (pseudo-)transfers and VASS with (pseudo-)copies, and undecidable for any other class. We further present a dichotomy for standard reachability in affine VASS: it is decidable for VASS with permutations, and undecidable for any other class. This yields a complete and unified complexity landscape of reachability in affine VASS. We also consider the reachability problem parameterized by a fixed affine VASS, rather than a class, and we show that the complexity landscape is arbitrary in this setting. CC Creative Commons 3:2 M. Blondin and M. Raskin Vol. 17:3is the reachability problem: given configurations x and y, is it possible to reach y starting from x? Such queries allow, e.g., to verify whether unsafe states can be reached in concurrent programs. The notorious difficulty of the reachability problem led to many proofs of its decidability over the last decades[ST77,May81,Kos82,Lam92,Ler10,Ler11,Ler12]. While the problem has been known to be EXPSPACE-hard since 1976 [Lip76], its computational complexity has remained unknown until very recently, where it was shown to be TOWERhard [CLL + 19] and solvable in Ackermannian time [LS15, LS19]. Given the potential applications on the one hand, and the tremendously high complexity on the other hand, researchers have investigated relaxations of VASS in search of a tradeoff between expressiveness and algorithmic complexity. Two such relaxations consist in permitting either: (a) transitions to be executed fractionally, and consequently counters to range over Q ≥0 (continuous reachability); or (b) counters to range over Z (integer reachability). In both cases, the complexity drops drastically: continuous reachability is P-complete and NP-complete for Petri nets and VASS respectively [FH15, BH17], while integer reachability is NP-complete for both models [HH14, CHH18]. Moreover, these two types of reachability have been used successfully to prove safety of real-world instances like multithreaded program skeletons, e.g., see [ELM + 14, ALW16, BFHH17].Although VASS are versatile, they are sometimes too limited to model common primitives. Consequently, their modeling power has been extended with various operations. For example, (multi-)transfers, i.e., operations of the formx ← x + n i=1 y i ; y 1 ← 0; y 2 ← 0; · · · ; y n ← 0, allow, e.g., for the verification of multi-threaded C and Java program skeletons with communication primitives [KKW14, DRV02]. Another example is the case of resets, i.e., operations of the form x ← 0, which allow, e.g., for the validation of some business processes [WvdAtHE09], and the generation of program loop invariants [SK19]. Many such extensions fall under the generic family of affine VASS, i.e., VASS with instructions of the form x ← A · x + b. As a general rule of thumb, reachability is undecidable for essentially any class of affine VASS introduced in the literature; in particular, for transfers and resets [AK76, DFPS98].Given the success of relaxations for the practical analysis of (standard) VASS, it is tempting to employ the same approach for affine VASS. Unfortunately, continuous reachability becomes undecidable for mild affine extensions such as resets and transfers. However, integer reachability was recently shown decidable for affine operations such as resets (NP-complete) and transfers (PSPACE-complete)[HH14,BHM18]. While such complexity results do not translate immediately into practical procedures, they arguably guide the design of algorithmic verification strategies.Contribution. Thus, these recent results raise two natural questions: for what classes of affine VASS is integer reachability decidable? And, whenever it is decidable, what is its exact computational complexity? We fully answer these questions in this paper by giving a precise trichotomy: integer reachability is NP-complete for VASS with resets, PSPACE-complete for VASS with (pseudo-)transfers and VASS with (pseudo-)copies, and undecidable for any other class. In particular, this answers a question left open in [BHM18]: integer reachability is undecidable for any class of affine VASS with infinite matrix monoid. Vol. 17:3 THE COMPLEXITY OF REACHABILITY IN AFFINE VASS 3:3This clear complexity landscape is obtained by formalizing classes of affine VASS and by carefully analyzing the structure of arbitrary affine transformations; which could be of independent interest. In particular, it enables us to prove a dichotomy on (standard) reachability for affine VASS: it is decidable for VASS with permutations, and undecidable for any other class. To the best of our knowledge, this is the first proof of the folkore rule of thumb stating that "reachability is undecidable for essentially any class of affine VASS".We further complement these trichotomy and dichotomy by showing that the (integer or standard) reachability problem has an arbitrary complexity when it is parameterized by a fixed affine VASS rather than a fixed class.Related work. Our work is related to [BHM18] which shows that integer reachability is decidable for affine VASS whose matrix monoid is finite (refined to EXPSPACE by [BHK + 20]); and more particularly PSPACE-complete in general for VASS with transfers and VASS with copies. While it is also recalled in [BHM18] that integer reachability is undecidable in general for affine VASS, the authors do not provide any necessary condition for undecidability to hold. Moreover, the complexity landscape for affine VASS with finite monoids is left blurred, e.g., it does not give necessary conditions for PSPACE-hardness results to hold, and the complexity remains unknown for monoids with negative coefficients. This paper completes the work initiated in [BHM18] by providing a unified framework, which includes the notion of matrix class, that allows us to precisely characterize the complexity of integer reachability for any class of affine VASS.Our work is also loosely related to a broader line of research on (variants of) affine VASS dealing with, e.g., modeling power [Val78], accelerability [FL02], formal languages [CFM12], coverability [BFP12], and the complexity of integer reachability for restricted counters [FGH13] and structures [IS16]. Introduction Vector addition systems with states (VASS), which can equivalently be seen as Petri nets, form a widespread general model of infinite-state systems with countless applications ranging from the verification of concurrent programs to the modeling of biological, chemical and business processes (see, e.g., [GS92,KKW14,EGLM17,HGD08,van98]). They comprise a finite-state controller with counters ranging over N and updated via instructions of the form x ← x + c which are executable if x + c ≥ 0. The central decision problem concerning VASS • d ≥ 1 is the number of counters of V; • Q is a finite set of elements called control-states; • T ⊆ Q × Z d×d × Z d × Q is a finite set of elements called transitions. For every transition t = (p, A, b, q), let src(t) := p, M (t) := A, ∆(t) := b and tgt(t) := q. A configuration is a pair (q, v) ∈ Q × Z d written q(v). For all t ∈ T and D ∈ {Z, N}, we write p(u) t − → D q(v) if u, v ∈ D d , src(t) = p, tgt(t) = q, and v = M (t) · u + ∆(t). The relation − → D is naturally extended to sequences of transitions, i.e., for every w ∈ T k we let w − → D := w k − − → D • · · · • w 2 −→ D • w 1 −→ D . Moreover, we write p(u) − → D q(v) if p(u) t − → D q(v) for some t ∈ T , and p(u) * − → D q(v) if p(u) w − → D q(v) for some w ∈ T * . As an example, let us consider the affine VASS of Figure 1, i.e., where d = 2, Q = {p, q, r} and T is as depicted graphically. We have: p(3, 1) s − → Z q(1, 0) t − → Z r(1, 0) u − → Z q(1, 1) t − → Z r(2, 0) u − → Z q(2, 2) t − → Z r(4, 0). More generally, p(x, 1) * − → Z r(2 k , 0) for all x ∈ Z and k ∈ N >0 . However, p(3, 0) s − → N q(1, −1) does not hold as counters are not allowed to become negative under this semantics. Classes of matrices. Let us formalize the informal notion of classes of affine VASS, such as "VASS with resets", "VASS with transfers", "VASS with doubling", etc., used throughout the literature. Such classes depend on the extra operations they provide, i.e., by their affine transformations, and more precisely by their linear part (matrices) rather than their additive part (vectors). Affine VASS extend standard VASS since they always include the identity matrix, which amounts to not applying any extra operation. Moreover, as transformations can be composed along sequences of transitions, their matrices are closed under multiplication, i.e., if matrices M (s) and M (t) are allowed on transitions s and t of two affine VASS of a given class, then M (s) · M (t) is typically also allowed in an affine VASS of the class as it is understood that s and t can be composed. In other words, matrices form a monoid. In addition, classes of affine VASS typically considered do not pose restrictions on the number of counters that can be used, or on the subset of counters on which operations can be applied. In other words, their affine transformations can be extended to arbitrary dimensions and can be applied on any subset of counters, e.g., general "VASS with resets" allow to reset any counter, not just say the first one. We formalize these observations as follows. For every k ≥ 1, let I k be the k × k identity matrix and let S k denote the set of permutations over [k]. Let P σ ∈ {0, 1} k×k be the permutation matrix of σ ∈ S k . For every matrix A ∈ Z k×k , every permutation σ ∈ S k and every n ≥ 1, let σ(A) := P σ · A · P σ −1 , and let A n ∈ Z (k+n)×(k+n) be the matrix such that: A n := A 0 0 I n . A class (of matrices) is a set of matrices C ⊆ k≥1 Z k×k that satisfies {σ(A), A n , I n , A · B} ⊆ C for every A, B ∈ C, every σ ∈ S dim A and every n ≥ 1. In other words, C is closed under counter renaming; each matrix of C can be extended to larger dimensions; and C ∩ Z k×k is a monoid under matrix multiplication for every k ≥ 1. Note that "counter renaming" amounts to choosing a set of counters on which to apply a given transformation, i.e., it renames the counters, applies the transformation, and renames the counters back to their original names. Let us illustrate this. Consider the classical case of transfer VASS, i.e., where the contents of a counter can be transferred onto another counter with operations of the form "x ← x + y; y ← 0". In matrix notation, this amounts to: O := 1 1 0 0 . Now, consider a system with three counters c 1 , c 2 and c 3 . This system should be able to compute "c 1 ← c 1 + c 2 + c 3 ; c 2 ← 0; c 3 ← 0", but matrix O cannot achieve this on its own. However, it can be done with the following matrix: O :=   1 1 0 0 0 0 0 0 1   ·   1 0 1 0 1 0 0 0 0   =   1 1 1 0 0 0 0 0 0   . We have O = O 1 ·σ(O 1 ) where σ := (2; 3). Thus, the operation can be achieved by any class containing O. The symmetric operation "c 3 ← c 1 + c 2 + c 3 ; c 1 ← 0; c 2 ← 0", e.g., can also be achieved with appropriate permutations. Hence, this corresponds to the usual notion of transfers: we are allowed to choose some counters and apply transfers in either direction. Note that requiring P σ · A ∈ C for classes would be too strong as it would allow to permute the contents of counters even for classes with no permutation matrix, such as resets. Classes of interest. We say that a matrix A ∈ Z k×k is a pseudo-reset, pseudo-transfer or pseudo-copy matrix if A ∈ {−1, 0, 1} k×k and if it also satisfies the following: • pseudo-reset matrix: A is a diagonal matrix; • pseudo-transfer matrix: A has at most one nonzero entry per column; • pseudo-copy matrix: A has at most one nonzero entry per row. We omit the prefix "pseudo-" if A ∈ {0, 1} k×k . Note that the sets of (pseudo-)reset matrices, (pseudo-)transfer matrices, and (pseudo-)copy matrices all form classes. Moreover, (pseudo-)reset matrices are both (pseudo-)transfer and (pseudo-)copy matrices. Note that the terminology of "reset", "transfer" and "copy" comes from the fact that such matrices implement operations like "x ← 0", "x ← x + y; y ← 0" and "x ← x; y ← x", 3:6 M. Blondin and M. Raskin Vol. 17:3 as achieved respectively by the matrices of transitions s (reset), t (transfer ) and u (copy) illustrated in Figure 1. Reachability problems. We say that an affine VASS V = (k, Q, T ) belongs to a class of matrices C if {M (t) : t ∈ T } ⊆ C, i.e., if all matrices appearing on its transitions belong to C. The reachability problem and integer reachability problem for a fixed class C are defined as: Reach C Input: an affine VASS V that belongs to C, and two configurations p(u), q(v); Decide: p(u) * − → N q(v) in V? Z-Reach C Input: an affine VASS V that belongs to C, and two configurations p(u), q(v); Decide: p(u) * − → Z q(v) in V? A complexity trichotomy for integer reachability This section is devoted to the proof of our main result, namely the trichotomy on Z-Reach C : Theorem 3.1. The integer reachability problem Z-Reach C is: (i) NP-complete if C only contains reset matrices; (ii) PSPACE-complete, otherwise, if either C only contains pseudo-transfer matrices or C only contains pseudo-copy matrices; (iii) Undecidable otherwise. It is known from [HH14, Cor. 10] that NP-hardness holds for affine VASS using only the identity matrix (recall that our definition of reset matrices include the identity matrix), and that NP membership holds for any class of reset matrices. Hence, (i) follows immediately. Thus, the rest of this section is dedicated to proving (ii) and (iii). 3.1. PSPACE-hardness. For the rest of this subsection, let us fix some class C that either only contains pseudo-transfer matrices or only contains pseudo-copy matrices. We prove PSPACE-hardness of Z-Reach C by first proving that PSPACE-hardness holds if either: • C contains a matrix with an entry equal to −1; or • C contains a matrix with entries from {0, 1} and a nonzero entry outside of its diagonal. For these two cases, we first show that C can implement operations x ← −x or (x, y) ← (y, x) respectively, i.e., sign flips or swaps. Essentially, each of these operations is sufficient to simulate linear bounded automata. Before investigating these two cases, let us carefully formalize what it means to implement an operation: Definition 3.2. Let f : Z k → Z k and let τ ∈ {0, ?}. Given a set of counters X ⊆ [m], let V X :=    v ∈ Z m : j ∈X v(j) = 0    if τ = 0, and let V X := Z m otherwise. We say that C τ -implements f if for every n ≥ k, there exist counters X = {x 1 , x 2 , . . . , x n }, matrices {F σ : σ ∈ S k } ⊆ C and m ≥ n such that the following holds for every σ ∈ S k and v ∈ V X : (a) dim F σ = m; (b) (F σ · v)(x σ(i) ) = f (x σ(1) , x σ(2) , . . . , x σ(k) )(i) for all i ∈ [k]; (c) (F σ · v)(x σ(i) ) = v(x σ(i) ) for every i ∈ [k]; (d) F σ · v ∈ V X . We further say that C implements f if it either 0-implements or ?-implements f . Definition 3.2 (b) and (c) state that it is possible to obtain arbitrarily many counters X such that f can be applied on any k-subset of X, provided that the counter values belong to V X . Moreover, (d) states that vectors resulting from applying operation f also belong to V X , which ensures that f can be applied arbitrarily many times. Note that (a) allows for extra auxiliary counters whose values are only restricted by V X . Informally, ?-implementation means that we use additional counters that can hold arbitrary values, while 0-implementation requires the extra counters to be initialized with zeros but promises to keep them in this state. It turns out that pseudo-transfer matrix classes 0-implement the functions we need, while pseudo-copy matrix classes ?-implement them. Proof. Let n ≥ 1 and A ∈ C be such that A a,b = −1 for some counters a and b. Let d := dim A. We extend A with n + 2 counters X := X ∪ {y, z}, where X := {x i : i ∈ [n]} are the counters for which we wish to implement sign flips, and {y, z} are auxiliary counters. More formally, let A := A n+2 where X = [d + 1, d ] and d := d + n + 2. For every s, t ∈ X such that s = t, let B s,t := π s,t (A ) and let C t := σ t (A ) where π s,t := (a; t)(b; s) and σ t := (a; t). For every x ∈ X, let F x := B z,x · B y,z · B x,y if a = b, C x otherwise. Intuitively, B s,t (resp. C t ) flips the sign from source counter s (resp. t) to target counter t. If a = b, then matrix F x implements a sign flip in three steps using auxiliary counters y and z, as illustrated in Figure 2. Otherwise, F x implements sign flip directly in one step. ? depicts the case where A is a pseudo-transfer (resp. pseudo-copy) matrix. A solid or dashed edge from s to t represents operation s ← t or s ← −t respectively. B x,y B y,z B z,x Filled nodes indicate counters that necessarily hold 0. Symbol "?" stands for an integer whose value is irrelevant and depends on A and the counter values. Let us consider the case where A is a pseudo-transfer matrix. From the definition of B s,t and C t , it can be shown that for every s, t, u ∈ X such that s = t and u ∈ {s, t}, the following holds: (i) B s,t · e s = C t · e t = −e t , and (ii) B s,t · e u = C t · e u = e u . Let us show that we 0-implement sign flips, so let V :=    v ∈ Z d : j ∈X v(j) = 0    . Let v ∈ V and x ∈ X. By definition of V , v = y∈X v(y) · e y . Let v := j∈X\{x} v(j) · e j . Items (b), (c) and (d) of Definition 3.2 are satisfied since: F x · v = F x · v(x) · e x + F x · v = v(x) · F x · e x + v (by (ii) and def. of F x ) = v(x) · −e x + v (by (i), (ii) and def. of F x ) = −v(x) · e x + v . The proof of (i) and (ii), and the similar proof for the case where A is a pseudo-copy matrix, are analogous (see Appendix A). Proposition 3.4. Z-Reach C is PSPACE-hard if C has a matrix with an entry equal to −1. Proof. We give a reduction, partially inspired by [BHM18, Thm. 10], from the membership problem for linear bounded automata, which is PSPACE-complete (e.g., see [HU79,Sect. 9.3 and 13]). Let w ∈ {0, 1} k and let A = (P, Σ, δ, p init , p acc ) be a linear bounded automaton where: • P is its finite set of control-states; • Σ = {0, 1} is its input and tape alphabet; • δ : P × Σ → P × Σ × {Left, Right} is its transition function; and • p init and p acc are its initial and accepting control-states, respectively. We construct an affine VASS V = (d, Q, T ) and configurations p(u), q(v) such that V belongs to C, and p(u) * − → Z q(v) ⇐⇒ A accepts w. For every control-state p and head position j of A, there is a matching control-state in V, i.e., Q := {q p,j : p ∈ P, 1 ≤ j ≤ k} ∪ Q, where Q will be auxiliary control-states. We associate two counters to each tape cell of A, i.e., d := 2 · k. For readability, let us denote these counters {x j , y j : j ∈ [k]}. We represent the contents of tape cell i by the sign of counter y j , i.e., y j > 0 represents 0, and y j < 0 represents 1. We will ensure that y j is never equal to 0, which would otherwise be an undefined representation. Since V cannot directly test the sign of a counter, it will be possible for V to commit errors during the simulation of A. However, we will construct V in such a way that erroneous simulations are detected. The gadget depicted in Figure 3 simulates a transition of A in three steps: • x i is incremented; • y i is incremented (resp. decremented) if the letter a to be read is 0 (resp. 1); • the sign of y i is flipped if the letter b to be written differs from the letter a to be read. Let u ∈ Z d be the vector such that for every j ∈ [k]: u(x j ) := 1 and u(y j ) := (−1) w j . Provided that V starts in vector u, we claim that: q p,i q p ,i+1 x i ← x i + 1 y i ← y i + (−1) a y i ← (−1) b−a · y i Figure 3: Gadget of V simulating transition δ(p, a) = (p , b, Right) of A. The gadget for direction Left is the same except for q p ,i+1 which is replaced by q p ,i−1 . Note that a and b are fixed, hence expressions such as (−1) a are constants; they do not require exponentiation. • k j=1 (\|x j \| ≥ \|y j \| > 0) is an invariant; • V has faithfully simulated A so far if and only if k j=1 (\|x j \| = \|y j \|) holds; • if V has faithfully simulated A so far, then the sign of y j represents w j for every j ∈ [k]. Let us see why this claim holds. Let i ∈ [k]. Initially, we have \|x i \| = \|y i \| and the sign of y i set correctly. Assume we execute the gadget of Figure 3, resulting in new values x i and y i . Let λ ≥ 0 be such that \|x i \| = \|y i \| + λ. Let c ∈ {0, 1} be the letter represented by y i . If c = a, then \|x i \| = \|y i \| + λ and the sign of y i represents b as desired. If c = a, then \|x i \| = \|y i \| + (λ + 1). Thus, we have \|x i \| = \|y i \| if and only if no error was made before and during the execution of the gadget. From the above observations, we conclude that A accepts w if and only if there exist i ∈ [k] and v ∈ Z d such that q p init ,1 (u) * − → Z q pacc,i (v) and \|v(x j )\| = \|v(y j )\| for every j ∈ [k]. This can be tested using the gadget depicted in Figure 4, which: • detects nondeterministically that some control-state of the form q pacc,i has been reached; • attempts to set y j to its absolute value for every j ∈ [k]; • decrements x j and y j simultaneously for every j ∈ [k]. r q p acc ,i q p acc ,1 q p acc ,k y 1 ← −y 1 Figure 4: Gadget of V for tesing whether A was faithfully simulated and has accepted w. x 1 ← x 1 − 1 y 1 ← y 1 − 1 y k ← −y k x k ← x k − 1 y k ← y k − 1 Due to the above observations, it is only possible to reach r(0) if \|x j \| = \|y j \| for every j ∈ [k] before entering the gadget of Figure 4. Thus, we are done proving the reduction since A accepts w if and only if q p init ,1 (u) * − → Z r(0). Sign flips. The above construction considers sign flips as a "native" operation. However, this is not necessarily the case, and instead relies on the fact that class C either 0-implements 3:10 M. Blondin and M. Raskin Vol. 17:3 or ?-implements sign flips, by Proposition 3.3. Thus, the reachability question must be changed to q p init ,1 (u, 0) * − → Z r(0, 0) to take auxiliary counters into account. Moreover, if C ?-implements sign flips, then extra transitions (r, I, e j , r) and (r, I, −e j , r) must be added to T , for every auxiliary counter j, to allow counter j to be set back to 0. Note that control-state r can only be reached after the simulation of A, hence it plays no role in the emulation of sign flips. Moreover, if there is an error during the simulation of A and the extra transitions set the auxiliary counters to zero, we will stil detect it as the configuration will be of the form r(w, 0) where w = 0. In the two forthcoming propositions, we prove PSPACE-hardness of the remaining case. Proposition 3.5. If C contains a matrix with entries from {0, 1} and a nonzero entry outside of its main diagonal, then it implements swaps, i.e., the operation f : Z 2 → Z 2 such that f (x, y) := (y, x). Proof. Let n ≥ 2 and let A ∈ C be a matrix with entries from {0, 1} and a nonzero entry outside of its main diagonal. Let d : = dim A. There exist a, b ∈ [d] such that A a,b = 1 and a = b. Let us extend A with n + 1 counters X := X ∪ {z} where X := {x i : i ∈ [n]}. More formally, let A := A n+1 , X = [d + 1, d ] and d := d + n + 1. For all s, t ∈ X such that s = t, let B s,t := π s,t (A ) where π s,t := (b; s)(a; t). For every distinct counters x, y ∈ X, let F x,y := B z,x · B x,y · B y,z . x y z B y,z B x,y B z,x x y z ? ? ? B y,z B x,y B z,x right) diagram depicts the case where A is a transfer (resp. copy) matrix. An edge from counter s to counter t represents operation s ← t. Filled nodes indicate counters that necessarily hold 0. Symbol "?" stands for an integer whose value is irrelevant and depends on A and the counter values. Intuitively, B s,t moves the contents from some source counter s to some target counter t, and F x,y implements a swap in three steps using an auxiliary counter z as depicted in Figure 5. In the case where A is a transfer matrix, B s,t resets s, provided that t held value 0. Let us consider the case where A is a transfer matrix. From the definition of B s,t , it can be shown that for every s, t, u ∈ X such that s = t and u ∈ {s, t}, the following holds: (i) B s,t · e s = e t , and (ii) B s,t · e u = e u . Let us show that we 0-implement swaps, so let Let v ∈ V X and let x, y ∈ X be such that V X :=    v ∈ Z d : j ∈X v(j) = 0    .x = y. By definition of V X , v = j∈X v(j) · e j . Let v := j∈X\{x,y} v(j) · e j . Items (b), (c) and (d) of Definition 3.2 are satisfied since we obtain the following by applications of (i) and (ii): F x,y · v = F x,y · (v(x) · e x + v(y) · e y ) + F x,y · v (by def. of v ) = F x,y · (v(x) · e x + v(y) · e y ) + v (by (ii) and def. of F x,y ) = B z,x · B x,y · B y,z · (v(x) · e x + v(y) · e y ) + v (by def. of F x,y ) = B z,x · B x,y · (v(x) · e x + v(y) · e z ) + v (by (ii) and (i)) = B z,x · (v(x) · e y + v(y) · e z ) + v (by (i) and (ii)) = v(x) · e y + v(y) · e x + v (by (ii) and (i)). The proof of (i) and (ii), and the similar proof for the case where A is a copy matrix, are analogous (see Appendix A). Proposition 3.6. Z-Reach C is PSPACE-hard if C contains a matrix with entries from {0, 1} and a nonzero entry outside of its main diagonal. Proof. It is shown in [BHMR19] that Z-reachability is PSPACE-hard for affine VASS with swaps, using a reduction from the membership problem for linear bounded automata. Here, we may not have swaps as a "native" operation. However, by Proposition 3.5, class C implements swaps. Thus, as in the proof of Proposition 3.4, if the reachability question is of the form p(u) * − → Z q(v), then it must be changed to p(u, 0) * − → Z q(v, 0) . Moreover, if the class C ?-implements swaps, then new transitions must be introduced to allow auxiliary counters to be set back to 0. Recall that under ?-implementation, there is no requirement on the value of the auxiliary counters, hence these new transitions do not interfere with the emulation of swaps. We now proceed to prove the main result of this subsection, namely Theorem 3.1 (ii): Proof of Theorem 3.1 (ii). Let M k := C ∩ Z k×k for every k ≥ 1. Theorem 7 of [BHM18] shows that Z-Reach C belongs to PSPACE if each M k is a finite monoid of at most exponential norm and size in k. Let us show that this is the case. First, since C is a class that contains only pseudo-transfer (reps. pseudo-copy) matrices, and since the product of two such matrices remains so, M k is a monoid which is finite as M k ⊆ {−1, 0, 1} k×k . Moreover, by definitions of pseudo-transfer and pseudo-copy matrices, each such matrix can be described by cutting it into k lines and specifying for each line either the position of the unique nonzero entry (which is −1 or 1), or the lack of such entry. Therefore, for every k ≥ 1, it is the case that M k ≤ 1 and \|M k \| ≤ (2k + 1) k ≤ (4k) k = 2 2k+k log k ≤ 2 poly(k) . It remains to show PSPACE-hardness. By assumption, C contains a nonreset matrix A. Since C ≤ 1, we have A = 1 as no class can be such that C = 0. If A contains an 3:12 M. Blondin and M. Raskin Vol. 17:3 entry equal to −1, then we are done by Proposition 3.4. Otherwise, A only has entries from {0, 1}, and hence we are done by Proposition 3.6. 3.2. Undecidability. In this subsection, we first show that any class C, that does not satisfy the requirements for Z-Reach C ∈ {NP-complete, PSPACE-complete}, must be such that C ≥ 2. We then show that this is sufficient to mimic doubling, i.e., the operation x → 2x, even if C does not contain a doubling matrix. In more details, we will (a) construct a matrix C that provides a sufficiently fast growth; which will (b) allow us to derive undecidability by revisiting a reduction from the Post correspondence problem which depends on doubling. Proposition 3.7. Let C be a class that contains some matrices A and B which are respectively not pseudo-copy and pseudo-transfer matrices. It is the case that C ≥ 2. Proof. By assumption, A and B respectively have a row and a column with at least two nonzero entries. We make use of the following lemma shown in Appendix A: if C contains a matrix which has a row (resp. column) with at least two nonzero entries, then C also contains a matrix which has a row (resp. column) with at least two nonzero entries with the same sign. Since C is a class, we can assume that dim A = dim B = d for some d ≥ 2, as otherwise the smallest matrix can be enlarged. Thus, there exist i, i , j, k ∈ [d] and a, b, a , b = 0 such that: • j = k; • A i,j = a and A i,k = b; • B j,i = a and B k,i = b ; and • a > 0 ⇐⇒ b > 0 and a > 0 ⇐⇒ b > 0. Note that the reason we can assume A and B to share counters j and k is due to C being closed under counter renaming. We wish to obtain a matrix with entry A i,j · B j,i + A i,k · B k,i = a · a + b · b . We cannot simply pick A · B as (A · B) i,i may differ from this value due to other nonzero entries. Hence, we rename all counters of B, except for j and k, with fresh counters. This way, we avoid possible overlaps and we can select precisely the four desired entries. More formally, Let i := i + d if i ∈ {j, k} and i := i otherwise. We have: A and B , and by C i,i = ∈[2d] A i, · B σ( ),i (by def. of C and by σ(i ) = i ) = ∈[d] s.t. σ( )∈[d] A i, · B σ( ),i (by def. ofi, i ∈ [d]) = A i,j · B j,i + A i,k · B k,i = a · a + b · b . Since a and b (resp. a and b ) have the same sign, and since a, b, a , b = 0, we conclude that \|C i,i \| ≥ 2 and consequently that C ≥ C ≥ 2. To avoid cumbersome subscripts, we write e for e 1 in the rest of the section. Moreover, let λ (C) := (C · e)(1) for every matrix C and ∈ N. The following technical lemma will be key to mimic doubling. It shows that, from any class of norm at least 2, we can extract a matrix with sufficiently fast growth. Lemma 3.8. For every class of matrices C such that C ≥ 2, there exists C ∈ C with λ n+1 (C) ≥ 2 · λ n (C) for every n ∈ N. Proof. Let A ∈ C be a matrix with some entry c such that \|c\| ≥ 2. We can assume that c ≥ 2. Indeed, if it is negative, then we can multiply A by a suitable permutation of itself to obtain an entry equal to c · c. We can further assume that c is the largest positive coefficient occurring within A, and that it lies on the first column of A, i.e., A k,1 = c for some k ∈ [d] where d := dim A. We consider the case where k = 1. The case where k = 1 will be discussed later. For readability, we rename counters {1, 2, . . . , d} respectively by X := {x 1 , x 2 , . . . , x d }. Note that (A · e)(x 1 ) = c ≥ 2 · e(x 1 ) as desired. However, vector A · e may now hold nonzero values in counters x 2 , . . . , x d . Therefore, if we multiply this vector by A, some "noise" will be added to counter x 1 . If this noise is too large, then it may cancel the growth of x 1 by ≈ c. We address this issue by introducing extra auxiliary counters replacing x 2 , . . . , x d at each "iteration". Of course, we cannot have infinitely many auxiliary counters. Fortunately, after a sufficiently large number m of iterations, the auxiliary counters used at the first iteration will contain sufficiently small noise so that the process can restart from there. More formally, let A : = A \|Y \| where Y := {y i,j : 0 ≤ i < m, j ∈ [2, d] } is the set of auxiliary counters, and m ≥ 1 is a sufficiently large constant whose value will be picked later. Let V be the set of vectors v ∈ Z \|X\|+\|Y \| satisfying v(x 1 ) > 0 and \|v(y i,j )\| ≤ 3c 4 i · v(x 1 ) 4d for every y i,j ∈ Y. Let us fix some vector v 0 ∈ V . For every 0 ≤ i < m, let B i := σ i (A ) and v i+1 : = B i · v i where σ i is the permutation σ i := j∈[2,d] (x j ; y i,j ). We claim that: v m (x 1 ) ≥ 2 · v 0 (x 1 ) and v m ∈ V. The validity of this claim proves the lemma. Indeed, C·v 0 = v m where C := B m−1 · · · B 1 ·B 0 . Hence, an application of C yields a vector whose first component has at least doubled. Since e ∈ V and the resulting vector also belong to V , this can be iterated arbitrarily many times. Let us first establish the following properties for every 0 ≤ i < m and j ∈ [2, d]: (a) v i (y i,j ) = v i−1 (y i,j ) = · · · = v 0 (y i,j ) and v i+1 (y i,j ) = v i+2 (y i,j ) = · · · = v m (y i,j ); (b) v i+1 (x 1 ) ∈ 3c 4 · v i (x 1 ), 5c 4 · v i (x 1 ) ; and (c) \|v i+1 (y i,j )\| ≤ 2c · v i (x 1 ). Property (a), which follows from the definition of B i , essentially states that the contents of counter y i,j is only altered from v i to v i+1 . Properties (b) and (c) bound the growth of the counters in terms of x 1 . Let us prove these two latter properties by induction on i. By definition of v i+1 , we have v i+1 (x 1 ) = c · v i + δ where δ := z =x 1 (B i ) x 1 ,z · v i (z).Therefore, v i+1 (x 1 ) ∈ [c · v i (x 1 ) − \|δ\|, c · v i (x 1 ) + \|δ\|] , and hence property (b) follows from: \|δ\| ≤ z∈X∪Y z =x 1 \|(B i ) x 1 ,z \| · \|v i (z)\| = j∈[2,d] \|A x 1 ,x j \| · \|v i (y i,j )\| (by def. of B i and σ i ) = j∈[2,d] \|A x 1 ,x j \| · \|v 0 (y i,j )\| (by (a)) ≤ dc · (3c/4) i · v 0 (x 1 ) 4d (by v 0 ∈ V and by maximality of c) ≤ dc · v i (x 1 ) 4d (by v i (x 1 ) ≥ (3c/4) i · v 0 (x 1 ) from (b)) = c 4 · v i (x 1 ). Similarly, property (c) holds since, for every j ∈ [2, d]: \|v i+1 (y i,j )\| ≤ \|A x j ,x 1 \| · v i (x 1 ) + ∈[2,d] \|A x j ,x \| · \|v i (y i, )\| (by def. of B i and σ i ) ≤ c · v i (x 1 ) + dc · (3c/4) i · v 0 (x 1 ) 4d (by (a) and v 0 ∈ V ) ≤ c · v i (x 1 ) + dc · v i (x 1 ) 4d (by (b)) ≤ 2c · v i (x 1 ). We may now prove the claim. Let m be sufficiently large so that (3c/4) m ≥ 8cd. We have v m (x 1 ) ≥ (3c/4) m · v 0 (x 1 ) ≥ 8cd · v 0 (x 1 ) by (b) and definition of m. Hence, since c ≥ 2 and d ≥ 1, we have v m (x 1 ) ≥ 2 · v 0 (x 1 ), which satisfies the first part of the claim. Moreover, the second part of the claim, namely v m ∈ V , holds since for every y i,j ∈ Y , we have: \|v m (y i,j )\| = \|v i+1 (y i,j )\| (by (a)) ≤ 2c · v i (x 1 ) (by (c)) ≤ 2c · v m (x 1 ) (3c/4) m−i (by v m (x 1 ) ≥ (3c/4) m−i · v i (x 1 ) from (b)) = 2c · (3c/4) i · v m (x 1 ) (3c/4) m ≤ 2c · (3c/4) i · v m (x 1 ) 8cd (by def. of m) = (3c/4) i · v m (x 1 ) 4d . We are done proving the lemma for the case A k,1 = c ≥ 2 with k = 1. This case is slightly simpler as c lies on the main diagonal of A which means that v i+1 (x 1 ) ≈ c · v i (x 1 ). If k = 1, then we have v i+1 (x k ) ≈ c · v i (x 1 ) instead, which breaks composability for the next iteration. However, this is easily fixed by swapping the names of counters x k and x 1 . Let us fix a class C such that C ≥ 2 and the matrix C obtained for C from Lemma 3.8. For simplicity, we will write λ instead of λ (C). We prove two intermediary propositions that essentially show that C can encode binary strings. Let f b (v) := C · v + b · e for both b ∈ {0, 1} and every v ∈ Z dim C . Let f ε be the identity function, and let f x := f xn • · · · • f x 2 • f x 1 for every x ∈ {0, 1} n . Let γ x := f x (e)(1) for every x ∈ {0, 1} * . Let ε := ∅ and w := {i ∈ [k] : w i = 1} be the "support" of w for every sequence w ∈ {0, 1} + of length k > 0. Proposition 3.9. For every x ∈ {0, 1} * , the following holds: γ x = λ \|x\| + i∈ x λ \|x\|−i . Proof. It suffices to show that f x (e) = C \|x\| · e + i∈ x C \|x\|−i · e for every x ∈ {0, 1} * . Let us prove this by induction on \|x\|. If \|x\| = 0, then x = ε, and hence f x (e) = e = C 0 · e. Assume that \|x\| > 0 and that the claim holds for sequences of length \|x\| − 1. There exist b ∈ {0, 1} and w ∈ {0, 1} * such that x = wb. We have: Proof. Let < lex denote the lexicographical order over {0, 1} * . It is sufficient to show that for every x, y ∈ {0, 1} * the following holds: if x < lex y, then γ x < γ y . Indeed, if this claim holds, then for every x, y ∈ {0, 1} * such that x = y, we either have x < lex y or y < lex x, which implies γ x = γ y in both cases. Let us prove the claim. Let x, y ∈ {0, 1} * be such that x < lex y. We either have \|x\| < \|y\| or \|x\| = \|y\|. If the former holds, then the claim follows from: It remains to prove the case where \|x\| = \|y\| = k for some k > 0. Since x < lex y, there exist u, v, w ∈ {0, 1} * such that x = u0v and y = u1w. Let := k − \|u\| − 1. Note that = \|v\| = \|w\|. The proof is completed by observing that: f x (e) = C · f w (e) + b · e = C ·   C \|w\| · e + i∈ w C \|w\|−i · e   + b · e (by induction hypothesis) =   C \|w\|+1 · e + i∈ w C \|w\|+1−i · e   + b · e =   C \|x\| · e + i∈ x \{\|x\|} C \|x\|−i · e   + b · C \|x\|−\|x\| · e = C \|x\| · e + i∈ x C \|x\|−i · e (by def. of x ).γ x = λ \|x\| + i∈ x λ \|x\|−i (by Proposition 3.9) ≤ λ \|x\| + \|x\| i=1 λ \|x\| /2 i (by Lemma 3.8) = λ \|x\| ·   1 + \|x\| i=1 1/2 i   = λ \|x\| · (2 − 1/2 \|x\| ) < 2 · λ \|x\| (since λ \|x\| > 0) ≤ λ \|y\|(γ y − γ x = λ + i∈ w λ −i − i∈ v λ −i (by Proposition 3.9) ≥ λ − \|v\| i=1 λ −i ≥ λ − \|v\| i=1 λ /2 i (by λ ≥ 2 i · λ −i from Lemma 3.8) = λ · 1 − i=1 1/2 i (by \|v\| = ) = λ /2 > 0. We may finally prove the last part of our trichotomy. Theorem 3.11. Z-Reach C is undecidable if C ≥ 2. Proof. We give a reduction from the Post correspondence problem inspired by [Rei15]. There, counter values can be doubled as a "native" operation. Here, we adapt the construction with our emulation of doubling. Let us consider an instance of the Post correspondence problem over alphabet {0, 1}: Γ := u 1 v 1 , u 2 v 2 , . . . , u v . We say that Γ has a match if there exists w ∈ Γ + such that the underlying top and bottom sequences of w are equal. Let C be the matrix obtained for C from Lemma 3.8, let d := dim C, and let e be of size d. For every x ∈ {0, 1} * , let g x and h x be the linear mappings over Z 2d defined as f x , but operating on counters 1, 2, . . . , d and counters d + 1, d + 2, . . . , 2d respectively. Let V := (2d, Q, T ) be the affine VASS such that Q and T are as depicted in Figure 6. Note that V belongs to C. Indeed, g x and h x can be obtained from matrix C ∈ C and the fact that C is a class, and hence closed under counter renaming and larger dimensions. We claim that p(e, e) * − → Z r(e, e) if and only if Γ has a match. Note that any sequence w ∈ T + from p to p computes g wx • h wy for some word p q 1 q i q r g u 1 h v 1 g u h v (−e, −e) (−e, −e) g u i h v iw x w y ∈ Γ + . Therefore: p(e, e) * − → Z r(e, e) ⇐⇒ ∃w ∈ T + , v ∈ Z 2d : p(e, e) w − → Z p(v) * − → Z r(e, e) ⇐⇒ ∃w ∈ T + , v ∈ Z 2d : p(e, e) w − → Z p(v) and v(1) = v(d + 1) ⇐⇒ ∃w ∈ T + : γ wx = γ wy (by def. of g, h and γ) ⇐⇒ ∃w ∈ T + : w x = w y (by Proposition 3.10) ⇐⇒ Γ has a match. We conclude this section by proving Theorem 3.1 (iii) which can be equivalently formulated as follows: Corollary 3.12. Z-Reach C is undecidable if C does not only contain pseudo-transfer matrices and does not only contain pseudo-copy matrices. Proof. By Proposition 3.7, C ≥ 2, hence undecidability follows from Theorem 3.11. A complexity dichotomy for reachability This section is devoted to the following complexity dichotomy on Reach C , which is mostly proven by exploiting notions and results from the previous section: Theorem 4.1. The reachability problem Reach C is equivalent to the (standard) VASS reachability problem if C only contains permutation matrices, and is undecidable otherwise. 4.1. Decidability. Note that the (standard) VASS reachability problem is the problem Reach I where I := n≥1 I n . Clearly Reach I ≤ Reach C for any class C. Therefore, it suffices to show the following: Proposition 4.2. Reach C ≤ Reach I for every C that only contains permutation matrices. Proof. Let V = (d, Q, T ) be an affine VASS that belongs to C. We construct a (standard) VASS V = (d, Q , T ) that simulates V. Recall that a (standard) VASS is an affine VASS that only uses the identity matrix. For readability, we omit the identity matrix on the transitions of V . We assume without loss of generality that each transition t ∈ T satisfies either ∆(t) = 0 or M (t) = I. Indeed, since permutation matrices are nonnegative, every transition of T can be splitted in two parts: first applying its matrix, and then its vector. The control-states and transitions of V are defined as Q := {q σ : q ∈ Q, σ ∈ S d } and T := S ∪ S vec , which are to be defined shortly. Intuitively, each control-state of V stores the current control-state of V together with the current renaming of its counters. Whenever 3:18 M. Blondin and M. Raskin Vol. 17:3 a transition t ∈ T such that ∆(t) = 0 is to be applied, this means that the counters must be renamed by the permutation M (t). This is achieved by: S := {(p σ , 0, q π•σ ) : (p, P π , 0, q) ∈ T, σ ∈ S d }. Similarly, whenever a transition t ∈ T such that M (t) = I is to be applied, this means that ∆(t) should be added to the counters, but in accordance to the current renaming of the counters. This is achieved by: S vec := {(p σ , P σ · b, q σ ) : (p, I, b, q), σ ∈ S d }. A routine induction shows that p(u) * − → N q(v) in V iff p ε (u) * − → N q σ (P σ · v) in V , where ε denotes the identity permutation. Since this amounts to finitely many reachability queries, i.e., \|S d \| = d! queries, this yields a Turing reduction 1 . 4.2. Undecidability. We show undecidability by considering three types of classes: (1) classes with matrices with negative entries; (2) nontransfer and noncopy classes; and (3) transfer or copy classes. In each case, we will argue that an "undecidable operation" can be simulated, namely: zero-tests, doubling and resets. Proposition 4.3. Reach C is undecidable for every class C that contains a matrix with some negative entry. Proof. Let A ∈ C be a matrix such that A i,j < 0 for some i, j ∈ [d] where d := dim A. We show how a two counter Minsky machine M can be simulated by an affine VASS V belonging to C. Note that we only have to show how to simulate zero-tests. The affine VASS V has 2d counters: counters j and j + d which represent the two counters x and y of M; and 2d − 2 auxiliary counters which will be permanently set to value 0. x ← x + c y ← y + c x = 0? Observe that for every λ ∈ N, the following holds: y = 0? (c · e j , 0) (0, c · e j )A · λe j = 0 if λ = 0, λ · A ,j otherwise. Thus, A simulates a zero-test as it leaves all counters set to zero if counter j holds value zero, and it generates a vector with some negative entry otherwise, which is an invalid 1 Although it is not necessary for our needs, the reduction can be made many-one by weakly computing a matrix multiplication by P σ −1 onto d new counters, from each control-state qσ to a common state r. configuration under N-reachability. Figure 7 shows how each transition of M is replaced in V. We are done since (m, n) * − → N (m , n ) in M ⇐⇒ (m · e j , n · e j ) * − → N (m · e j , n · e j ) in V. Proposition 4.4. Reach C is undecidable if C does not only contain transfer matrices and does not only contain copy matrices. Proof. If C contains a matrix with some negative entry, then we are done by Proposition 4.3. Thus, assume C only contains nonnegative matrices. By Proposition 3.7, we have C ≥ 2. Let C be the matrix obtained for C from Lemma 3.8. Since C ≥ 0, we have C · v ≥ 0 for every v ≥ 0. Hence, multiplication by C is always allowed under N-reachability. Thus, the reduction from the Post correspondence problem given in Theorem 3.11 holds here under N-reachability, as the only possibly (relevant) negative values arose from C. We may finally prove the last part of our dichotomy: Theorem 4.5. Reach C is undecidable for every class C with some nonpermutation matrix. Proof. Let A ∈ C be a matrix which is not a permutation matrix. By Propositions 4.3 and 4.4, we may assume that A is either a transfer or a copy matrix. Hence, A must have a column or a row equal to 0, as otherwise it would be a permutation matrix. Thus, we either have A ,i = 0 or A i, = 0 for some i ∈ [d] where d := dim A. We show that C implements resets, i.e., the operation f : Z → Z such that f (x) := 0. This suffices to complete the proof since reachability for VASS with resets is undecidable [AK76]. Let X := {d + 1, d + 2, . . . , d + n} be the counters for which we wish to implement resets. Let A := A n and let B x := σ x (A ) where σ x := (x; i). Let x, y ∈ X be such that x = y. Case A ,i = 0. We have: B x · e x = (B x ) ,x = A ,i = 0. Similarly, it can be shown that B x · e y = e y . Hence, class C 0-implements resets. Case A i, = 0. The following holds for every v ∈ Z d+n : (B x · v)(x) = ∈[d+n] B x, · v( ) = ∈[d+n] A i,σx( ) · v( ) = 0. Similarly, (B x · v)(y) = v(y). Hence, class C ?-implements resets (see Appendix A). Parameterization by a system rather than a class In this section, we consider the (integer or standard) reachability problem parameterized by a fixed affine VASS, rather than a matrix class. We show that in contrast to the case of classes, this parameterization yields an arbitrary complexity (up to a polynomial). More formally, this section is devoted to establishing the following theorem: There exists an affine VASS V such that L and the following problem are interreducible under polynomial-time many-one reductions: D-Reach V Input: two configurations p(u), q(v); Decide: p(u) * − → D q(v) in V? In order to show Theorem 5.1, we will prove the following technical lemma: iff ϕ(f (w)) holds. Let V be the affine VASS given by Lemma 5.2 for ϕ, and let p, q be its associated control-states. To reduce L to Reach V , we construct, on input w, the pair I w := p(f (w), 0), q(0, 0) . We have w ∈ L iff ϕ(f (w)) holds iff I w ∈ Reach V by Lemma 5.2. The reduction from D-Reach V to L is as follows. On input I = r(v), r (v ) : • If r = p or r (v ) = q(0), then we check whether r(v) * − → D r (v ) in polynomial time and return a positive (resp. negative) instance of L if it holds (resp. does not hold). This is possible by Lemma 5.2 and by nontriviality of L. • Otherwise, I = p(x, u), q(0, 0) for some value x and some vector u, so it suffices to return f −1 (x). Indeed, by Lemma 5.2, I ∈ D-Reach V iff ϕ(x) holds iff f −1 (x) ∈ L. In order to prove Lemma 5.2, we first establish the following proposition. Informally, it states that although an affine VASS cannot evaluate a polynomial P with the mere power of affine functions, it can evaluate P "weakly". Moreover, its structure is simple enough to answer any reachability query in polynomial time. Proof. We adapt and simplify a contruction of the authors which was given in [BHMR19] for other purposes. Let us first consider the case of a single monomial P (y 1 , . . . , y k ) = cy d 1 1 · · · y d k k with c ≥ 1. We claim that the affine VASS V depicted in Figure 8 satisfies the claim. Its Figure 8: Affine VASS evaluating cy d 1 1 · · · y d k k , where "x++", "x−−", "x += x " and "x −= x " stand respectively for "x ← x + 1", "x ← x − 1", "x ← x + x " and "x ← x − x ". A := {a 0 } ∪ {a i,j : i ∈ [k], j ∈ [d i ]} ∪ {a out } serve as accumulators. p r 0 r 1,1 r 1,d1 r k,1 r k,dk q a 0 ← c y 1,1 ← y 1 y k,dk ← y k a out ← a k,dk a 0 ← 0 a i,j ← 0 (∀i, j) y 1,1 −− a 1,1 += a 0 y 1,1 ++ a 1,1 −= a 0 y 1,d1 −− a 1,d1 += a 1,d1−1 y 1,d1 ++ a 1,d1 −= a 1,d1−1 y k,1 −− a k,1 += a k−1,dk−1 y k,1 ++ a k,1 −= a k−1,dk−1 y k,dk −− a k,dk += a k,dk−1 y k,dk ++ a k,dk −= a k,dk−1 y 1,d1 ← y 1 y k,1 ← y k By construction, counters from Y are never altered, merely copied onto Y . Moreover, if counters from Y ∪ A initially hold zero, and if loops are executed so that counters from Y reach zero, then a out contains P (y 1 , . . . , y k ) when reaching control-state q, and every other auxiliary counter holds zero (due to the final resets). It remains to argue that Item (b) holds. Let us consider a query "r(v) * − → D r (v )". Observe that, although V has nondeterminism in its loops and uses nonreversible operations (copies and resets), it is still "reversible" since the inputs Y are never altered and since each counter from Y can only be altered at a single copy or via the two loops next to it. This provides enough information to answer the reachability query. Indeed, we can: • Answer "false" if v and v disagree on Y ; • Pretend the accumulators do not exist and traverse V backward from r (v ) to r by undoing the loops (either up only or down only), ensuring each counter from Y reaches its correct value; and answer "false" if not possible; • Traverse V forward from r(v) by running through the loops the number of times identified by the previous traversal; this now allows to determine the value of the accumulators A; • If the values of A are incorrect in r (v ), or if we ever drop below zero in the case of D = N, then we answer "false", and otherwise "true". Observe that there is no need to execute the loops one step at a time, e.g., if v(y i ) = 10 ≥ 3 = v (y i,j ), then we can compute 7 directly rather than in seven steps. Let us now consider the general case. We can write the polynomial P as P (y 1 , . . . , y k ) = 1≤i≤m Q i (y 1 , . . . , y k ) − m<i≤n Q i (y 1 , . . . , y k ), where each Q i is a monomial with positive coefficient. Thus, we can compose the above construction sequentially with n distinct sets of auxiliary counters (only Y is shared). Let a out,i be output counter for Q i . We add a last transition that computes into an extra counter and resets a out,1 , . . . , a out,n . This preserves all of the above properties. Indeed, each copy variable is still only affected locally by a single copy and the next two loops. Note that ordering the monomials only matters for D = N as summing, e.g., −2 + 5 blocks, while 5 − 2 does not. We may now conclude by proving Lemma 5.2, which we recall: Lemma 5.2. Let D ∈ {Z, N} and let ϕ : D → {0, 1} be a nontrivial computable predicate. There exists an affine VASS V = (d, Q, T ) and control-states p, q ∈ Q such that: (a) For every vector u, it is the case that p(x, u) * − → D q(0, 0) in V iff ϕ(x) holds; (b) Any query "r(v) * − → D r (v )" can be answered in polynomial time if r = p or r (v ) = q(0). Proof. Since ϕ is decidable, by Matiyasevich's theorem 3 [Mat71], there exists a polynomial P , with integer coefficients and k variables, such that for every x ∈ D: ϕ(x) holds ⇐⇒ ∃y ∈ D k−1 : P (x, y) = 0. Let V be the affine VASS obtained from Proposition 5.3 for P , and let p and q be its associated control-states. Let us show that the affine VASS V depicted in Figure 9 satisfies the claim. It uses counters C := Y ∪ Y where Y := {y 1 , . . . , y k } are the k first counters of V corresponding to the variables of P ; and where Y forms the other auxiliary counters of V . We let m := \|C\| and sometimes refer to the counters C as {c 1 , . . . , c m }. For the rest of the proof, let us write u X to denote the vector obtained by restricting u to counters X ⊆ C. The affine VASS V is divided into two parts connecting p to q: the upper gadget made of control-states {a, b 1 , . . . , b m }, and the lower gadget made of V . Figure 9: Affine VASS V of Lemma 5.2, where "x++", "x−−" and "reset {x 1 , x 2 , . . .}" stand respectively for "x ← x + 1", "x ← x − 1" and "x 1 ← 0; x 2 ← 0; · · · ". V p p q q reset Y y 2 ++ y k ++ y 2 −− y k −− reset Y a b 1 . . . b m c 1 ← 1 c m ← 1 c 1 ← −c 1 c m ← −c m c 1 ++ c m ++ c 1 ++ c m ++ c 1 −− c m −− The purpose of the lower gadget is to satisfy Item (a) by checking whether ϕ(y 1 ) holds. This is achieved by (1) guessing an arbitrary assignment to counters Y \ {y 1 }; (2) resetting auxiliary counters Y ; (3) evaluating P (y 1 , . . . , y k ); and (4) resetting counters Y . 3 There are two variants of Matiyasevich's theorem: stated either over N or Z. This follows, e.g., from Lagrange's four-square theorem, i.e., any number from N can be written as a 2 +b 2 +c 2 +d 2 where a, b, c, d ∈ Z. The purpose of the upper gadget is to satisfy Item (b) by simplifying other queries from p to q. More precisely, the upper gadget allows to generate an arbitrary nonzero vector. This is achieved by (1) setting each counter c i ∈ C to an arbitrary value; (2) nondeterministically setting a counter c j to 1; (3) setting c j to an arbitrary positive value; (4) nondeterministically keeping or flipping the sign of c j (the latter only works if D = Z). These steps ensure that all counter values are possible, provided that some counter c j = 0. Let us first show Item (a). Suppose p(x, u) * − → D q(0, 0). Clearly, the target is not reached through the upper gadget. So, the following holds for some value x and some vectors y, y : p(x, u Y \{y 1 } , u Y ) * − → D p(x, y, u Y ) − → D p (x, y, 0) * − → D q (x , y , 0) − → D q(0, 0, 0). Since V does not alter counters Y , we have (x, y) = (x , y ) and consequently: p (x, y, 0) * − → D q (x, y, 0). Hence, P (x, y) = 0, which implies that ϕ(x) holds, as desired. Conversely, if ϕ(x) holds, then P (x, y) = 0 holds for some vector y. Clearly, we can achieve the following: p(x, u Y \{y 1 } , u Y ) * − → D p(x, y, u Y ) − → D p (x, y, 0) * − → D q (x, y, 0) by Prop. 5.3 − → D q(0, 0, 0). Let us now show Item (b). Recall that we want to show that queries not covered by Item (a) can be answered in polynomial time. These are queries of the form r(v) * − → D r (v ) where ¬(r = p ∧ r (v ) = q(0)), which amounts to either r = p or r (v ) = q(0). We assume w.l.o.g. that r can reach r in the underlying graph, as it can be tested in linear time. Let Q be the control-states of V . We make a case distinction on r and r , and explain each time how to answer the query. Lower part: • Case r = r = p. Amounts to v (c) = v(c) for every c ∈ C \ {y 2 , . . . y k }. • Case r = p, r ∈ Q . Recall that V does not alter Y , that p can generate any values within Y \ {y 1 }, and that the transition from p to p resets Y . Hence, we have: p(v) * − → D r (v ) ⇐⇒ v(y 1 ) = v (y 1 ) ∧ p (v Y , 0) * − → D r (v Y , v Y ). The latter can be answered in polynomial time by Proposition 5.3. • Case r = p, r = q. Since r = p, we must have v = 0 by assumption. Thus, the answer is "true" since the upper gadget allows to reach any nonzero vector in q. • Case r, r ∈ Q . Can be tested in polynomial time by Proposition 5.3. • Case r ∈ Q , r = q. Recall that V does not alter Y , and that the transition from q to q resets Y , but not Y . Hence, we have: r(v) * − → D q(v ) ⇐⇒ r(v Y , v Y ) * − → D q (v Y , v Y ). The latter can be answered in polynomial time by Proposition 5.3. Upper part: • Case r ∈ {p, a} and r = a. The answer is always "true". • Case r ∈ {p, a} and r = b i . Amounts to v (c i ) ≥ 1. • Case r = a, r = q. Amounts to i∈[m] v (c i ) = 0. • Case r = r = b i . Amounts to v (c i ) ≥ v(c i ) and v (c j ) = v(c j ) for all j = i. • Case r = b i , r = q. Amounts to \|v (c i )\| ≥ \| max(v(c i ), 0)\| and v (c j ) = v(c j ) for j = i. Conclusion and further work Motivated by the use of relaxations to alleviate the tremendous complexity of reachability in VASS, we have studied the complexity of integer reachability in affine VASS. Namely, we have shown a trichotomy on the complexity of integer reachability for affine VASS: it is NP-complete for any class of reset matrices; PSPACE-complete for any class of pseudo-transfers matrices and any class of pseudo-copies matrices; and undecidable for any other class. Moreover, the notions and techniques introduced along the way allowed us to give a complexity dichotomy for (standard) reachability in affine VASS: it is decidable for any class of permutation matrices, and undecidable for any other class. This provides a complete general landscape of the complexity of reachability in affine VASS. We further complemented these trichotomy and dichotomy by showing that, in contrast to the case of classes, the (integer or standard) reachability problem has an arbitrary complexity when it is parameterized by a fixed affine VASS. A further direction of study is the range of possible complexities for integer reachability relations for specific matrix monoids, which is entirely open. Another direction lies in the related coverability problem which asks whether p(u) * − → D q(v ) for some v ≥ v rather than v = v. As shown in the setting of [HH14,CHH18], this problem is equivalent to the reachability problem for D = Z. Indeed, given an affine VASS V, it is possible to construct an affine VASS V such that p(u) * − → Z q(v) in V ⇐⇒ p(u, −u) * − → Z q(v, −v) in V .() This can be achieved by replacing each affine transformation Ax + b of V by A 0 0 A x x + b −b . Furthermore, note that classes are closed under this construction, which shows that integer coverability and integer reachability are equivalent w.r.t. classes. However, () does not hold for D = N. In this case, it is well-known, e.g., that the coverability problem is decidable for VASS with resets or transfers, while the reachability problem is not. Hence, a precise characterization of the complexity landscape remains unknown in this case. More formally, we wish to obtain a matrix D with positive entries a 2 and b, and more precisely such that D i,j = a 2 and D i,k = b for some counter j . Let B := A d , C := σ(B) and D := B · C where σ : [2d] → [2d] is the following permutation: σ := (j; i; i + d) · ∈[d]\{i,j} ( ; + d) if j = i, ∈[d]\{i} ( ; + d) if j = i. We claim that D is as desired. First, observe that: D i,k = ∈[2d] B i, · B σ( ),k+d (by def. of D and σ(k) = k + d) = B i,k · B k+d,k+d (since B σ( ),k+d = 0 ⇐⇒ σ( ) = k + d) = b (since B k+d,k+d = 1). Thus, D has a positive entry on row i. It remains to show that D has another positive entry on row i. We make a case distinction on whether j = i. Case j = i. Note that k = i + d. Hence, we are done since: Case j = i. Note that k = j = i. Hence, we are done since: = a 2 (since i = j). D i,i = ∈[2d] B i, · B σ( We are done proving the proposition for the case of rows. For the case of columns, we can instead assume that A T ∈ C, i.e., the transpose of A belongs to C. Since D T is as desired, we simply have to show that D T ∈ C. This is the case since: D T = (B · C) T = C T · B T = (P σ · B · P σ −1 ) T · B T (since C = σ(B)) = (P σ · B T · P σ −1 ) · B T (since P π −1 = P T π for every perm. π) = σ(B T ) · B T = σ((A d ) T ) · (A d ) T = σ((A T ) d ) · (A T ) d (since (A d ) T = (A T ) d ). ∈ C (since A T ∈ C). A.4. Details for the proof of Theorem 4.5. We prove the missing details for both cases: Case A ,i = 0. We have B x · e y = e y since: (B x · e y )(k) = (B x )k, y = A σx(k),y (since y = x) = 1 ⇐⇒ k = y (by def. of A and σ x ). Case A i, = 0. We have: (B x · v)(y) = ∈[d+n] B y, · v( ) = A y,y · v(y) (by y = x and def. of A and σ x ) = v(y). Figure 1 : 1Example of an affine VASS. Proposition 3 . 3 . 33If C contains a matrix with some entry equal to −1, then it implements sign flips, i.e., the operation f : Z → Z such that f (x) := −x. Figure 2 : 2Effect of applying F x for the case a = b, where the left (resp. right) diagram Figure 5 : 5Effect of applying F x,y , where the left (resp. let A := A d , B := B d and C := A · σ(B ), where σ : [2d] → [2d] is the permutation: σ := ∈[d]\{j,k} ( ; + d). Proposition 3 . 10 . 310For every x, y ∈ {0, 1} * , it is the case that x = y if and only if γ x = γ y . Figure 6 : 6Affine VASS V for the Post correspondence problem. Arcs labeled by mappings of the form g x and h x stand for sequences of \|x\| transitions implementing g x and h x . Figure 7 : 7Top: example of a counter machine M. Bottom: an affine VASS V simulating M. . Let ∅ ⊂ L ⊂ {0, 1} * be a decidable decision problem and let D ∈ {Z, N}. Lemma 5. 2 . 2Let D ∈ {Z, N} and let ϕ : D → {0, 1} be a nontrivial computable predicate. There exists an affine VASS V = (d, Q, T ) and control-states p, q ∈ Q such that:(a) For every vector u, it is the case that p(x, u) * − → D q(0, 0) in V iff ϕ(x) holds; (b) Any query "r(v) * − → D r (v )"can be answered in polynomial time if r = p or r (v ) = q(0). Before proving Lemma 5.2, let us see why it implies the validity of Theorem 5.1: Proof of Theorem 5.1. Let f : {0, 1} * → D be a polynomial-time computable and invertible bijection. 2 Let ϕ : D → {0, 1} be the characteristic function of L encoded by f , i.e., w ∈ L Proposition 5. 3 . 3Let D ∈ {Z, N}, and let P be a polynomial with integer coefficients and k variables over D. There exist an affine VASS V = (d, Q, T ) and control-states p, q ∈ Q such that P (x) = 0 iff p(x, 0) * − → D q(x, 0) in V. Moreover, the following properties hold:(a) No transition of V alters its k first counters (corresponding to x); (b) Any reachability query "r(v) * − → D r (v )" can be answered in polynomial time. 2 If D = N, we can set f (w) := val(1w) − 1, i.e., f = {ε → 0, 0 → 1, 1 → 2, 00 → 3, . . .}. If D = Z, we can set f (ε) := 0 and f (w1 · · · wn) := −1 wn · val(1w1 · · · wn−1), i.e., f = {ε → 0, 0 → 1, 1 → −1, 00 → 2, 01 → −2, . . .}. are C := Y ∪ Y ∪ A, where Y := {y 1 , . . . , y k } corresponds to the input variables; Y := {y i,j : i ∈ [k], j ∈ [d i ]} are auxiliary counters that allow to copy Y ; and B i, · B σ( ),j (by def. of D and σ(i + d) = j)= : ,σ( )∈[d] B i, · B σ( ),j (since B = A d and i, j ∈ [d]) = B i,j · B i,j (by def. of σ) = a 2 . ),i (by def. of D and σ(i) = i) = : ,σ( )∈[d] B i, · B σ( ),i (since B = A d and i ∈ [d]) = B i,i · B i,i (by def. of σ) Vol. 17:3 THE COMPLEXITY OF REACHABILITY IN AFFINE VASS 3:23 Vol. 17:3 THE COMPLEXITY OF REACHABILITY IN AFFINE VASS 3:31 This work is licensed under the Creative Commons Attribution License. To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, 171 Second St, Suite 300, San Francisco, CA 94105, USA, or Eisenacher Strasse 2, 10777 Berlin, Germany Appendix A. AppendixA.1. Details for the proof of Proposition 3.3.Pseudo-transfer matrix. Let us first prove properties (i) and (ii) stated within the proof of Proposition 3.5 for the case where A is a pseudo-transfer matrix. The validity of (i) for B s,t follows from:(B s,t · e s )(k) = (B s,t ) k,s = A πs,t(k),b(since π s,t (s) = b)= −1 if π s,t (k) = a else 0 (A.1) = −1 if k = t else 0 (since π s,t (t) = a)where (A.1) follows from A a,b = −1 and the fact that A is a pseudo-transfer matrix. The validity of (ii) B s,t follows from:The same proofs apply mutatis mutandis for C t .Pseudo-copy matrix. Let us now prove Proposition 3.3 for the case where A is a pseudocopy matrix. For this case, we consider ?-implementation and hence V X = Z d . Similarly to the case of pseudo-transfer matrices, we claim that for every v ∈ V X and every s, t, u ∈ X such that s = t and u ∈ {s, t}, the following holds:Indeed, we have: where the last equality follows from A a,b = −1 and the fact that A is a pseudo-copy matrix. Moreover, we have:Again, the same proofs apply mutatis mutandis for C t . We now prove the proposition. Let x ∈ X and v ∈ V X . We obviously have F x · v ∈ V X . If a = b, we have:and if a = b, we have:Similarly, by applying (2) repeatedly, we derive (F x · v)(y) = v(y) for every y ∈ X \ {x}.A.2. Details for the proof of Proposition 3.5. The details of the proof are similar to those of the proof of Proposition 3.3.Transfer matrix. Let us first prove properties (i) and (ii) stated within the proof of Proposition 3.5 for the case where A is a transfer matrix. The validity of (i) follows from:The validity of (ii) follows from:Copy matrix. Let us now prove Proposition 3.5 for the case where A is a copy matrix. For this case, we consider ?-implementation and hence V X = Z d . Similarly to the case of transfer matrices, we claim that for every v ∈ V X and every s, t, u ∈ X such that s = t and u ∈ {s, t}, the following holds:(1) (B s,t · v)(t) = v(s);(2) (B s,t · v)(u) = v(u). Indeed, we have:where the last equality follows from A a,b = 1 and the fact that A is a copy matrix. Morever, we have:We may now prove the proposition. Let v ∈ V X and let x, y ∈ X be such that x = y. We obviously have F x,y · v ∈ V X . Moreover, we have:(by def. of F x,y ) = (B x,y · (B y,z · v))(z) (by (1)) = (B y,z · v)(z) (by(2))= v(y) (by (1)), and symmetrically (F x,y · v)(y) = v(x). Similarly, by applying (2) three times, we derive (F x,y · v)(u) = v(u) for every u ∈ X \ {x, y}.A.3. Details for the proof of Proposition 3.7. Let us prove the technical lemma invoked within the proof of Proposition 3.7:Lemma A.1. For every class C, if C contains a matrix which has a row (resp. column) with at least two nonzero elements, then C also contains a matrix which has a row (resp. column) with at least two nonzero elements with the same sign.Proof. We first consider the case of rows. Let A ∈ C, i, j, k ∈ [d] and a, b = 0 be such that A i,j = a, A i,k = b and j = k. If a and b have the same sign, then we are done. Thus, assume without loss of generality that a < 0 and b > 0. Let us first give an informal overview of the proof where we see A as an operation over some counters. We have two counters x and y that we wish to sum up (with some positive integer coefficients), using a supply of counters set to zero. We apply A to x and some zero counters to produce a · x in some counter (while discarding extra noise into some other ones), and we then apply A again to a · x, y and some zero counters in such a way that we get a 2 · x + b · y. 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[ "Active Contour with A Tangential Component", "Active Contour with A Tangential Component" ]	[ "Junyan Wang \nSchool of Electrical and Electronic Engineering\nNanyang Technological University\n639798Singapore\n", "Kap Luk Chan \nSchool of Electrical and Electronic Engineering\nNanyang Technological University\n639798Singapore\n" ]	[ "School of Electrical and Electronic Engineering\nNanyang Technological University\n639798Singapore", "School of Electrical and Electronic Engineering\nNanyang Technological University\n639798Singapore" ]	[]	Conventional edge-based active contours often require the normal component of an edge indicator function on the optimal contours to approximate zero, while the tangential component can still be significant. In real images, the full gradients of the edge indicator function along the object boundaries are often small. Hence, the curve evolution of edge-based active contours can terminate early before converging to the object boundaries with a careless contour initialization. We propose a novel Geodesic Snakes (GeoSnakes) active contour that requires the full gradients of the edge indicator to vanish at the optimal solution. Besides, the conventional curve evolution approach for minimizing active contour energy cannot fully solve the Euler-Lagrange (EL) equation of our GeoSnakes active contour, causing a Pseudo Stationary Phenomenon (PSP). To address the PSP problem, we propose an auxiliary curve evolution equation, named the equilibrium flow (EF) equation.Based on the EF and the conventional curve evolution, we obtain a solution to the full EL equation of GeoSnakes active contour. Experimental results validate the proposed geometrical interpretation of the early termination problem, and they also show that the proposed method overcomes the problem.	10.1007/s10851-014-0519-y	[ "https://arxiv.org/pdf/1204.6458v1.pdf" ]	13,100,077	1204.6458	126693b4b910796eb642cba3a9bdc542ee22351d	Active Contour with A Tangential Component 29 Apr 2012 Junyan Wang School of Electrical and Electronic Engineering Nanyang Technological University 639798Singapore Kap Luk Chan School of Electrical and Electronic Engineering Nanyang Technological University 639798Singapore Active Contour with A Tangential Component 29 Apr 2012Preprint submitted to Elsevier May 10, 2014arXiv:1204.6458v1 [cs.CV] Euler-Lagrange equation, pseudo stationary phenomenon, equilibrium flowObject segmentationactive contourcurve evolution* Conventional edge-based active contours often require the normal component of an edge indicator function on the optimal contours to approximate zero, while the tangential component can still be significant. In real images, the full gradients of the edge indicator function along the object boundaries are often small. Hence, the curve evolution of edge-based active contours can terminate early before converging to the object boundaries with a careless contour initialization. We propose a novel Geodesic Snakes (GeoSnakes) active contour that requires the full gradients of the edge indicator to vanish at the optimal solution. Besides, the conventional curve evolution approach for minimizing active contour energy cannot fully solve the Euler-Lagrange (EL) equation of our GeoSnakes active contour, causing a Pseudo Stationary Phenomenon (PSP). To address the PSP problem, we propose an auxiliary curve evolution equation, named the equilibrium flow (EF) equation.Based on the EF and the conventional curve evolution, we obtain a solution to the full EL equation of GeoSnakes active contour. Experimental results validate the proposed geometrical interpretation of the early termination problem, and they also show that the proposed method overcomes the problem. Euler-Lagrange equation, pseudo stationary phenomenon, equilibrium flow Introduction Energy minimization provides a principled framework for various fundamental computer vision problems. Active contour was proposed for object segmentation based on energy minimization. The essential idea of the active contour is to model the object boundaries by the contour curves that minimize the functional energy which measures the error of boundary detection. The active contour has been adopted in many application domains of computer vision, such as surveillance video analysis [1] [2] and medical image analysis [3] [4] etc. There are both edge-based active contours [5] [6] [3] [7] [8] and region-based active contours [9] [10] [11] [12]. There are also some recent attempts on improving the region based methods [13], and improving the edge based active contours [14,8,15]. More recently, some efforts have been devoted to convex relaxations and global optimization of region based active contour models such as [16]. However, it is arguable that neither the region-based nor the edge-based model is superior to the other in general. There are also recent efforts on integrating the region based and edge based methods [17,18]. Generally speaking, the edgebased active contours are capable of achieving more accurate boundary extraction comparing with region-based active contours, but they generally require careful initializations. This paper revisits a classic problem with edge-based active contours and provides some new insights to the problem. In the literature, e.g. [19], [20], [14], [21] and [15], the problem is often stated as: when the curve is initialized rela- 3. We obtain a curve evolution method to solve the EL equations containing both the normal and tangential components. The rest of the paper is organized as follows. In Section 2, we introduce the geodesic active contour and the works relevant to the problem of early termination of curve evolution. In Section 3, we present our problem statement followed by our formulation and solution. In Section 4, experiments are conducted to evaluate our proposed method and to also valid our theory. In Section 5, we conclude the paper and present some further possibilities beyond this paper. This work is based on the preliminary observations presented at BMVC 2008 [22]. Background The Geodesic Active Contour Our discussions are mostly based on the general Geodesic Active Contour (GAC) model. The energy functional of GAC is as follows. C * = arg min C L(C) L(C) = 1 0   C p T   g(∇I) 0 0 g(∇I)   C p   1 2 dp = 1 0 g C p dp = C gds(1) where ds = C p dp is the arclength parameterization and the edge indicator function g can be defined below. g(∇I) = 1 1 + G σ * ∇I q (2) in which G σ is a Gaussian filter of width σ, and we may assume q = 2. The justifications of this choice of edge indicator can be found in [5] [23] [6]. We rewrite the Euler-Lagrange equation of GAC as follows. δL δC = ∇g, N N − gκ N = 0 (3) where κ is the curvature of curve C, and N is the normal vector of C. Empirically, the smoothing term gκ is close to zero on a converged smooth curve. Hence, ∇g, N is also close to zero on the curve. Previous works on the early termination problem in curve evolution This subsection will review some pioneering works on this early termination problem. Their methods have been proven effective for addressing this problem to various degrees. Our approach differs from them in that we aim to not only remedy the problem but also investigate the cause of the problem. The early termination of local gradient based active contours was first reported in [19]. He proposed the Balloon term to push the curve either to shrink or to expand before reaching the object boundary. Generally, the Balloons can be applied when there is little attraction sensed by the evolving curve. Xu and Prince [20] observed that the Snakes might not converge to concave boundary even when there is large attraction. We quote their statement below. Although the external forces 1 correctly point toward the object boundary, within the boundary concavity the forces point horizontally in opposite directions. Therefore, the active contour is pulled apart to-1 the gradient vectors We also reproduce a U-shape as well as the associated normalized velocity field in Figure 1 to visualize the statement above. Based on this observation, Xu and Prince [20] also proposed the Gradient Vector Flow (GVF) to extend and smooth the gradient field in order to address the problem for extracting objects with moderately concave boundaries. This method is tested to be effective for moderate boundary concavity. Paragios et al. [14] applied the GVF to GAC with the level set method for extracting multiple objects. However, they discovered a problem which we quote below. ...the proposed flow does not perform propagation when the NGVF 2 is close to orthogonal to the inward normal ... propagation will not take place, as well as change of the topology, even if they are supported by the level set technique. normalized gradient vector flow We can compare this statement above with the observations by Xu and Prince [20] to conclude that both the original Snakes and the level set based active contours suffer from the same problem. Paragios and his colleagues propose the adaptive balloon to force the active contour to evolve when the normal projection of the gradient is close to zero. This idea works for images of relatively simple topology, as shown in our experiments. For images of complex topology, the contour might be able to avoid converging to the positions where the normal projection of the gradient is close to zero, but it can oscillate near these positions since the Adaptive Balloon is formulated to be in the same direction of the external force. Li et al. [21] proposed to pre-segment the image before curve evolution. The segmentation is based on the geometry of the GVF. By segmenting the image, the closed curve can also be cut into smaller ones. However, the segmentation of In that paper, the authors conjectured that the early termination of gradient based curve evolution can be due to the undesired critical points, such as saddle points or maxima. The authors also demonstrated that MAC can surpass the undesired critical points and converge to boundaries with severe concavities or boundaries of multiple objects for many images while having less constraints on the initializations. However, MAC is a curve evolution framework, and the curve evolution does not necessarily minimize an energy functional. Hence, MAC is not formulated under energy minimization framework. Besides, MAC requires the edge detection to produce little spurious edge as a preprocessing step. The detected edge helps producing an indicator of region homogeneity, and the resultant curve evolution behaves like region based active contours as observed in the experiments presented in this paper. This problem with GAC may be considered as a problem of local optimality of the curve evolution based method, but the globally optimal solution of GAC can be a dot, because GAC tries to find the contour having minimal weighted contour length. In the experiments presented later, we will show that our method compares favorably to the related methods for object segmentation in images containing relatively complex structures. Active contour with a tangential component An interpretation of the early termination problem The object boundaries perceived by human are located at the peaks of the magnitude of image gradients according to the theory of edge detection [24]. Based on the magnitude of image gradients, an edge indicator function can be formulated as in Eq. (2). Variants of the edge indicator functions have now been commonly used in the formulations of active contours, e.g. [7]. A prototypical edge-based active contour is the GAC. In GAC model, the boundary is considered as the contour corresponding to the minimal GAC energy. The GAC energy is small, if the edge indication on the contour is strong. The Euler-Lagrange (EL) equation for minimizing the GAC energy requires the gradients of the edge indicator in the normal direction of the optimal curve to approach zero. However, this requirement is insufficient. It has been observed that the curve evolution for solving the EL equation often converges at the non-boundary positions where the tangential component is still significant [20] [14]. In fact, the full gradients of the edge indicator along the boundaries are often small, i.e. both the normal and tangential components of the gradients of the edge indicator along the boundary are close to zero, such as Figure 2. In practice, the gradient field of the edge indicator functions of images containing noise or spurious edges are topologically complex. In such gradient fields, we can easily find the non-boundary curves on which the normal component of the gradients is zero but the tangential is non-zero. Some more examples can be found in the experiment section. Note that the assumption of small gradients along object boundary is not always valid. For example, the inhomogeneous region may lead to non-zero gradients on the boundary. The intensity inhomogeneity in medical images can often be corrected by preprocessing [25]. Without requiring the tangential component to be close to zero, we should not expect to obtain correct object boundaries. In practice, this leads to the early termination problem. Figure 3 shows a possible situation of early termination of the curve evolution in GAC. Let the curve segments at A and B be the same curve segment from two successive iterations of the curve evolution in a vector field. The black arrows in Figure 3 (b) denote the actual attraction velocity on the contour curves. The attraction velocity is the projection of the vector field, visualized by the red arrows, in the normal direction of the curve. As the curve evolves, the curve at A at first moves towards position B in the first iteration. Then, in the second iteration, the curve at B will move towards A. By using a sufficiently small time step, the curve will finally stop at some place where the gradients are nearly orthogonal to the inward/outward normal, but not necessarily zero. The curve evolution terminates too early since the curve converges to the place in-between the boundaries where the normal components vanishes but not the tangential ones. The GeoSnakes and the Pseudo Stationary Phenomenon In the following, we propose a new active contour model that requires the full gradients of an edge indicator to approach zero at the optimal solution. The model is given as follows. C * = argmin C L(C) = C (ln •g • C)dp + C C p dp(4) where • is the function composition operator, ln •g • C = ln(g(C)) is the composition of the three functions, g is the edge indicator function defined in Eq. (2). The corresponding EL equation is the following. δL(C) δC = 1 g ∇g − κ N = 0 ⇔ ∇g − gκ N = 0 ⇔ ∇g, T T + ∇g, N N − gκ N = 0 (5) where T is the tangential of the contour curve and N is the normal. The first equivalence holds since g is guaranteed positive. Comparing (5) with (3), we can observe that (5) contains an additional tangential component, which is also the tangential component of the gradient of the edge indicator. There exists geometric and parametric active contours. A discussion regarding the relationship between the geometric active contours and the parametric active contours is in [26]. The major difference between the two kinds of active contours ∂ t C = gκ N − ∇g, T T − ∇g, N N(6) Unfortunately, it is commonly believed that the tangential component in the curve evolution equations will only automatically reparameterize the contour curve, and the tangential component is conventionally omitted in the implementation of the curve evolution. This behavior can be explained with the help of the Lemma of curve evolution stated in [28]. ∂ t C(p, t) = α(p, t) T (C(p, t)) + β(p, t) N(C(p, t))(7) If β does not depend on the parametrization, meaning that β is a geometric intrinsic characteristic of the curve, then the image of C(p, t) that satisfies Equation (7) is identical to the image of the family of curves C(p, t), parameterized by p ∈ B, that satisfies      ∂ t C(p, t) = β(p, t) N(p, t) ∂p ∂t = − α C p (8a)(8b) The above lemma states that the geometry of the curve evolution only depends on the velocity in the normal direction of the curve, although the reparametrization by ∂p ∂t may last forever due to the possibly non-vanishing α. A brief proof of this lemma can be found in the book of [29]. The omission of the tangential component in the original EL equation can also be seen in the level set method for implementing general curve evolution.Recall the level set method for curve evolution [30]. ∂ t Φ\| Φ=0 = ∇Φ, ∂ t C \| Φ=0 = ∇Φ, β N + α T Φ=0 , by Equation (7) = β ∇Φ All the above says that the curve evolution is independent of the tangential velocity on the closed curve. It also implies that the converged solution of Eq. (8) may not necessarily be the stationary solution of Eq. (7). To be specific, there may exists a non-vanishing reparametrization velocity, i.e. the tangential velocity, on the converged curve. This can be ascertained by rephrasing Lemma 3.1 to be the following. Corollary 3.2 (Pseudo Stationary Phenomenon). The converged curve of Eq. (8), denoted by C(p, ∞) which satisfies ∂ t C(p, ∞) = β(C(p, ∞)) N(C(p, ∞)) = 0(10) can lead to ∂ t C(p, ∞) = α(C(p, ∞)) T (C(p, ∞)) + β(C(p, ∞)) N(C(p, ∞)) = 0(11) where p = p(p, t), α = − C p ∂p ∂t , ∂p ∂t is the parametrization and α(C(p, ∞)) can be large. C(p, ∞) is the converged curve. The above property is termed the Pseudo Stationary Phenomenon (PSP), since the stationarily converged curve may not be a solution of the original EL equation such as (5). Interestingly, the early termination of GAC is corresponded to the PSP of GeoSnakes. The Equilibrium Flow and the Alternating Curve Evolution ∂ t C = β(p, t) N(p, t) + F (p, t), T (p, t) T (p, t)(12) where F is a vector field. The objective is to ensure both the normal component, β, and the tangential component, F (p, t), T (p, t) , to vanish when the curve evolution stops. Since the curve evolution is independent of the tangential velocity, the tangential component plays no role in the whole process of curve evolution. Therefore, an auxiliary flow termed the Equilibrium Flow (EF) is proposed to ensure the tangential velocity to be zero when the curve evolution stops. The EF equation is defined as follows. ∂ t C(p, t) = F (p, t), T (p, t) N(p, t)(13) This equations is named the Equilibrium Flow due to the following property. The proof is given in Appendix. Another more interesting property of EF is the following. Proposition 3.4. The curve evolution of (13) and the reparametrization dp dt = − F (p,t), N (p,t) Cp where F = −∇g, leads to g(C(p, t)) = g(C(p, t + τ )) for any τ before termination of the curve evolution. The proof is given in Appendix. This proposition implies that the function g for a fixed p would not change during the curve evolution driven by EF. With a proper contour parametrization, the curve evolution driven by EF can solve for the tangential component of EL while the energy on the contour will not be changed. By using this EF flow, we obtain a solution to the EL equation involving both the normal and tangential components, which is formally stated below. Proposition 3.5. Given an arbitrary initial curve C 0 , the convergent curve C(p, ∞) driven by the following system of flows is the stationary solution of the original curve evolution (12) as well as the corresponding EL equation. ∂ t C k 1 = β(p, t) N, C k 1 (p, 0) = C k−1 2 (p, ∞) (14a) ∂ t C k 2 = F , T N , C k 2 (p, 0) = C k 1 (p, ∞) (14b)            k = 1, 2, . . . C 0 1 (p, 0) = C 0 2 (p, ∞) = C 0 C(τ, ∞) = lim k→∞ C k 1 (τ, ∞) = lim k→∞ C k 2 (τ, ∞) (14c) This claim is true since Eq. (12) can be satisfied only when both the β and α are zero, and the curve evolutions (14a) and (14b) defines a process to ensure the β and α to be zero respectively. This proposition tells that the alternation of the two curve evolutions can provide a solution to the original EL equation containing both normal and tangential component, which is previously unknown. The derived new curve evolution framework can have the following interpretation. Starting from an initial contour curve, the curve evolution of (14a) may converge to a pseudo stationary position, where the PSP occurs. Then the curve evolution of (14b) can help the curve evolution to escape from the pseudo stationary position. At the convergence of (14b), the value of the edge indicator at every contour point of C(p) can be guaranteed unchanged (by Proposition 3.4), and the curve evolution of (14a) has been reactivated because ∇g, N tends to be large according to Proposition 3.3. The process continues until the system of equations in (14) converges. The curve evolution of (14b) cannot be directly implemented in a level set framework in which the tangent T cannot be conveniently used. Fortunately, in 2D case, there is a useful relation that N = R T , where R is a matrix for ±90 degree rotation. Therefore, the EF can be rewritten as follows. ∂ t C(p, t) = F , T N = ( F T T ) N = ( F T R N ) N = ((R T F ) T N ) N = R T F , N N(15) The EF (14b) in the system Eqs. The algorithm The problem we discussed previously is general, but we still focus on analyzing and solving problems with the GeoSnakes where β = gκ − ∇g, N . We adopted the level set method [31] for implementation. The pseudo code is shown in Algorithm 1. Note that the maximum iteration is reached if the contour has little motion. The maximum cycle is chosen to be 3 in the experiment. Experimental results In this section, we present the experimental results that validate our interpretation and formulation for solving the early termination problem. The proposed method is also compared with other related methods to show the practical usefulness of this work. Experiment settings We describe the details of the parameter settings for implementation of the methods for preprocessing and comparison. The detailed implementation can be found in the original papers that are cited below. Edge map [5] [6]. The σ of the Gaussian kernel that is used to generate edge map g in Eq. (2) is empirically chosen to be 3 for all the images; GVF [20]. The diffusion of the edge map g is done by GVF as a preprocessing for all the gradient based methods that are evaluated. We compute the GVF by using if mod(cycle, 2) = 0 then 10: C k+1 ⇐ C k by the direct gradient decent flow of Eq. (14a) 11: else 12: C k+1 ⇐ C k by the EF, i.e., Eq. (15). 13: end if 14: k = k + 1; 15: end while 16: c = c + 1; 17: end while the source code provided in the GVF web site 3 . We select the parameter µ = 0.1; Balloon [19] [6]. The Balloon used for comparison is chosen to be positive to shrink the curve; Adaptive Balloon [32] [14]. For comparison, we adopt the version in [32], which is simpler for analysis comparing to [14], while almost equivalent. In our imple- mentation λ = 1, β = 0.1; Chan-Vese model [10]. We set λ 1 , λ 2 = 1, µ = 0.1, and υ = 0 which is a Balloon term. The δ ǫ and H ǫ are chosen as in [10]; MAC [15]. We use Canny edge detection as a preprocessing to specify the current along the boundaries of possible objects, and we set the α = 0.5 in the paper for all the images; Approximated Dirac delta. The approximated delta function δ ǫ for level set implementation, excluding Chan-Vese model, are chosen to be as, The Balloon is chosen to be 1. The algorithms are implemented in MATLAB. δ 1 (x) =    0, \|x\| > 1 The coding of the active contours is largely inspired by Chunming Li's implementation 5 . Segmentation results Before presenting the main results of the segmentation, the proposed method is demonstrated for segmentation of the U-shape in Figure 1. The result is shown is also stationary to the curve evolution by gradient descent. in Figure 5. The other images used for evaluating the proposed method are shown in Figure 7 overlaid with the red initial curves. The same initializations are used in all the comparisons. The initial curves are partially inside the objects and partially outside, which means that solely shrinking or solely expanding the initial curves cannot extract the objects. The image sizes can be found in Table 1 The curve evolution process of the system of alternating curve evolutions following (14), called the GAC+EF, are shown in Figure 9. The objects are correctly segmented. Figure 10 shows The tangential and normal velocities In the following, a quantitative analysis of the proposed method is provided. From the quantitative results, the stable convergence of the curve evolution algorithms can also be observed. One major contribution in this paper is about the proposed solution, namely the EF, to the Pseudo Stationary Phenomenon that occurs if the tangential velocity on the converged curve does not vanish. The quantitative results demonstrate that the proposed method can reduce the tangential velocity effectively. The tangential velocity in the curve evolutions of the proposed method is shown in the second row from top in Figure 12. The tangential velocity is quite large when GAC converges (0-k1), but due to the EF (k2-k4) the tangential velocity approaches zero, where k3 is the iteration of convergence. The velocities do not reach zero exactly because of the errors in discretization. The normal velocity in the third row from top in Figure 12 shows that the normal velocity drops significantly during the direct curve evolutions of the gradient descent flow. The iterations corresponding to k1, k2, k3, and k4 in Figure 3.12 are given in Table 1. The computational times for the curve evolutions are also presented. The computational cost can be reduced by using fast implementations, e.g. [33] [34] [35]. Discussions and conclusion The PSP problem may happen not only in object segmentation. It may also exist in other computer vision or computer graphics tasks involving optimizing closed curves. Hence, the proposed method may be extended to address the PSP in the different contexts. The proposed method is derived based on geometrical observations and analysis of the PSP problem. Experimentally, it shows that the EF repositions the curve to escape from the PSP. This suggests that one may also approach the PSP from the perspective of repositioning a pseudo-stationary curve. The assumption of small gradients along object boundary is not always valid. For example, the inhomogeneous region may leads to inhomogeneous gradients. This paper addresses the issue of early termination of curve evolution for images containing moderately complex structures and relatively homogeneous regions. The intensity inhomogeneity in the medical images can be corrected by preprocessing [25]. This work provides new geometric insights of the problem of early termination of the curve evolution for general edge-based active contours, giving rise to new criteria and solution of edge-based active contours. Appendix A. Proofs Appendix A.1. Proof of Proposition 3.3 Proof Since F (C * ), T (C * ) = 0, we can directly obtain N (C * ) = F (C * ) F (C * ) (A.1) or, F (C * ) = 0 (A.2) which is one of the definitions of level set, and this completes the proof. Appendix A.2. Proof of Proposition 3.4 Proof Substituting (13) and dp dt into dC dt , we obtain the following. where R is a 90 o rotation matrix of size 2 × 2. Thus, taking derivative of g w.r.t. t we obtain the following. shows the total magnitude of the tangential velocity along the curves; The second row shows the total magnitude of the normal velocity along the curves. k1,k2,k3,k4 area some key iterations in the curve evolutions. 0-k2 is the curve evolution by GAC; k2-k4 is by EF; from k4 onwards is by GAC tively arbitrarily, the active contour can stop early and some part of the converged curve can still be far from the boundaries of the objects of interest. We investigate the cause of the early termination of curve evolution in general edge-based active contours. We observe that the full gradients of the edge indicator function along the object boundaries are often small. However, conventional edge-based active contours, such as the Geodesic Active Contour (GAC), often only require the normal component of the edge indicator function on the optimal contours to approximate zero, while the tangential component can be still significant. Based on this observation, we propose a novel active contour model: the Geodesic Snakes (GeoSnakes) active contour model. The derived Euler-Lagrange (EL) equation of the GeoSnakes model requires the full gradients of the edge indicator to be close to zero on the optimal contours. However, the conventional curve evolution method does not fully solve the EL equation of the GeoSnakes model, although the curve evolution can still converge stationarily. This phenomenon is named the Pseudo Stationary Phenomenon (PSP). To address the PSP, we propose an auxiliary curve evolution equation, named the Equilibrium Flow (EF). The full EL equation of the GeoSnakes for boundary extraction is solved by alternating the Equilibrium Flow and the conventional gradient descent curve evolution. From our point of view, our contributions are as follows. 1. We elucidate importance of the tangential component of the gradient of the edge indicator along the boundary curve for boundary locating. This observation contradicts the conventional view that the tangential component is merely a useless reparameterization force. 2. We obtain a new active contour model of which the EL equation can be satisfied by a smooth contour curve if both the normal and tangential components of the gradient of the edge indicator along the contour approach zero. Figure 1 : 1The U-shape and the opposite gradient vectors ward each of the fingers of the U-shape, but not made to progress downward into the concavity GVF is not in the active contour framework. The cause of the early termination problem of the curve evolution of general edge-based active contours was not addressed there. More recently, Xie and Mirmehdi [15] formulated a novel edge-based curve evolution equation motivated by the mathematical formulations of magnetostatic / Lorentz force, as an alternative to the conventional edge-based active contours. The curve evolution model was named the Magnetostatic Active Contour (MAC). Figure 2 : 2The full gradients of the edge indicator near the object boundary are close to zero. (a) is an image containing an object of interest. In (b), the center is the edge indicator function for (a). The intensity of the edge indicator is visualized by the darkness in figure (b). The surrounding patches in (b) show the gradient vector field of the edge indicator along the object boundary (dashed curve). Figure 3 : 3The curve stops (or oscillates) when the gradient is nearly orthogonal to the inward normal of the curve. (a) is a 3D visualization of a typical edge indication function overlayed by the negative gradient field; (b) demonstrates the curve evolution in the negative gradient field within the gray-shaded box region in (a). See text for details. is that the curve evolution equation of parametric active contours may contain a tangential component whereas the other one contains only the normal component.GAC is a typical geometric active contour. A typical parametric active contour containing tangential component is the original Snakes active contour[27] [26].Note that the normal component in the EL equation of GeoSnakes is accidentally the same as that of the GAC. The proposed model is named the Geodesic Snakes (GeoSnakes) since it is related to both the GAC and Snakes. Now let us turn to the solution to the GeoSnakes model. The conventional solution to active contour is the gradient descent curve evolution. The corresponding curve evolution equation of GeoSnakes is as follows. Lemma 3. 1 . 1Given the closed curve C(p, t) parameterized by arbitraryp ∈B at an artificial time t with the normal N , the tangent T of the curve, and given the geometric flow of a curve evolution by On the one hand, the curve evolutions with only the normal component is insufficient for boundary location and the tangential component should also be considered. On the other hand, the curve evolutions can only solve for the normal component of the curve evolution equation, even when the curve evolution equation contains a tangential component. Thus, we find ourselves in a dilemma. In the following, we propose a solution to the curve evolution equation with a tangential component, i.e., a solution to the PSP problem. The curve evolution equation concerned can be written in the most general form as follows. Proposition 3. 3 . 3The stationary solution C * of the Eq.(13)is either on stationary points in F or on level sets of the potential g corresponding to the gradient field F . (14a) and (14b) can be understood as a repositioning process of the curve. The repositioning process can help the curve to escape from the pseudo stationary positions. The visualization of the idea of EF is presented in Figure 4. The rotation of gradient field can reactivate the curve evolution when it is trapped by PSP, while the true stationary positions remain stationary. Figure 4 : 4The practical effect of rotating gradient field. The top left is a typical gradient field, visualized by dark blue arrows, in which the green line segment cannot be moved by the local gradients due to the PSP. The top right is the vector field generated by the rotation, visualized by the arrows in red, in which the line segment can now be moved by this vector field. The bottom visualizes both the gradient field and the rotated vector field. It shows that the stationary positions, i.e., the light blue curve, remain stationary after the rotation Algorithm 1 1Alternating Curve Evolutions Input: Input Image I, MaximumCycle, MaximumIteration Output: C k 1: I sm ⇐ I, by anisotropic smoothing 2: g ⇐ ∇I sm , ; Re-initialization. We adopted the 1 st order accurate Essentially Non-Oscillatory (ENO1) based on Baris Sumengen's MATLAB toolbox of the Level Set method4 ; Figure 5 : 5Segmentation of the U-shape.(a) is the initialization. (b) is the PSP due to the curve evolution by gradient descent. (c)-(f) is the curve evolution by EF. The curve in (f) . The chosen images are of different characteristics. The image of two rectangles inFigure 7(a) is relatively simple for segmentation since the gray level of both foreground and background are homogeneous and distinct, the edges are strong too. Besides, there are only two objects, hence the topology is simple. InFigure 7(b), the image is also simple except that there are three objects, which is a little more difficult than the former one. InFigure 7(c), there are nine curve segments forming three circles. The edge-based active contours may see three circles of disconnected boundaries in the image, and the region-based method may see the objects in the image as composed of nine narrow and curved regions. Figures 7(d) and 7(e) are MRI medical images of human brain with multiple tumor regions. The original brain scans are in Figure 6. From Figures 7(d) and (e), we see that there are many spurious curved-edges in the images. Besides, there are some regions with intensities similar to the targeted tumor regions. These make the region-based active contour to be unsuitable for such images. For edge-based active contours, there also exists the Pseudo Stationary Phenomenon. Figure 6 :Figure 7 :Figure 8 :Figure 9 : 6789The original real images and the tumor regions (in boxes) Inputs and initializations In Figure 8, we visualize the normalized gradient field of the edge indication functions of the images in Figures 7 (a), (b) and (c) respectively to demonstrate the PSP. We also visualize the saddle points by the blue dots on the red lines of which the gradients are in the tangential direction. Later we shall see the early termination problem of GAC, i.e. the PSP problem of the GeoSnakes, near those lines. Gradient fields, PSP and Saddle points Object Extraction by proposed method. The top row is the curve evolution by GAC; The middle is the curve evolution by EF; The bottom row is the final convergence. the segmentation results produced by other methods, including the GAC, GAC with Balloon, GAC with adaptive Balloon, MAC and the Chan-Vese model, with the same initialization shown in 7. It can be observed that none of these compared methods can extract all the objects accurately like GAC+EF. GAC suffers from the early stop of the curve evolution. GAC with Balloon can surpass edges and converge to the interior boundaries or vanish; GAC with adaptive Balloon can be applied for extracting 2 objects of simple topology as shown in Figure 10, yet it still suffers from the PSP in other images. The gradient field used for the former three methods are extended by Gradient Vector Flow. Interestingly, both the GAC and the GAC with the adaptive Balloon might stop quite close to the Pseudo Stationary Positions as shown in Figure 3, while GAC with adaptive Balloon can escape from some Pseudo Stationary Positions of relatively simple topology. It can also be observed that both MAC and Chan-Vese model can extract the narrow curved regions in the image of three disconnected circles shown in Figure 10. It implies that MAC shares certain characteristics of the region based methods. The edge detection process in MAC does not produce a local gradient field but the indication of intensity homogeneity. The Chan-Vese active contour is not sensitive to initialization but it often captures other undesired regions. The MAC generally fails to converge to the object boundaries if there are too many edges and corners distributed everywhere in the images. Figure 11 11visualizes the 3D tumor model that is obtained by the segmentation of the entire MRI sequence ofFigure 7(e) using GAC+EF. The cutting plate corresponds to the segmentation result shown in the bottom right corner inFigure 9 . = F , T N − F , N T = F , R T N N + F , R T T T = R F , N N + R F , T T = −R∇g(p, t) (A.3) Figure 10 : 10Results by other methods. First column from the left is the object extraction by GAC; Second column is the object extraction by GAC with Balloon; Third column is the object extraction by GAC with adaptive Balloon; Fourth column is the object extraction by MAC; Last column is the object extraction by Chan-Vese active contour. 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[ "Possibilities of MgB 2 /Cu Wires Fabricated by the in-situ Reaction Technique", "Possibilities of MgB 2 /Cu Wires Fabricated by the in-situ Reaction Technique" ]	[ "E Martínez ", "R Navarro ", "\nInstituto de Ciencia de Materiales de Aragón (CSIC\nDepartamento de Ciencia\nUniversidad de Zaragoza\n\n", "\nC/ María de Luna 3\nTecnología de Materiales y Fluidos CPS\n50018ZaragozaSpain\n" ]	[ "Instituto de Ciencia de Materiales de Aragón (CSIC\nDepartamento de Ciencia\nUniversidad de Zaragoza\n", "C/ María de Luna 3\nTecnología de Materiales y Fluidos CPS\n50018ZaragozaSpain" ]	[]	The superconducting properties of copper-sheathed MgB 2 wires fabricated by conventional powder-in-tube techniques and the in-situ reaction procedure are analysed. The influence of the processing conditions and initial (1+x)Mg + 2B (x = 0, 0.1, 0.2) proportions of the precursors on the critical current values of the wires have been studied. In particular, the limits of the available temperatures and times for heat treatments imposed by the chemical reaction between Mg and Cu, and their effect on the superconducting properties of the wires, are discussed. The analysis includes the study of the sample microstructure and phase composition as well as of the critical current temperature and field dependences. The wires show high thermal stability during direct transport measurements and carry a critical current density of 1.3 × 10 9 A/m 2 at 15 K in the self-field for optimised processing conditions.	null	[ "https://arxiv.org/pdf/cond-mat/0306090v1.pdf" ]	119,035,936	cond-mat/0306090	50a9527f2bb7721d762d6425757429c23298f29c	Possibilities of MgB 2 /Cu Wires Fabricated by the in-situ Reaction Technique Date of submission: 27-May-2003 E Martínez R Navarro Instituto de Ciencia de Materiales de Aragón (CSIC Departamento de Ciencia Universidad de Zaragoza C/ María de Luna 3 Tecnología de Materiales y Fluidos CPS 50018ZaragozaSpain Possibilities of MgB 2 /Cu Wires Fabricated by the in-situ Reaction Technique Date of submission: 27-May-20031 * To be published in "Horizons in Superconductivity Research" (Nova Science Publishers, Inc., NY, 2003)MgB 2critical currentsmagnetizationPowder-in-Tubewires PACS: 7470Ad8525Kx7425Ha7460Jg The superconducting properties of copper-sheathed MgB 2 wires fabricated by conventional powder-in-tube techniques and the in-situ reaction procedure are analysed. The influence of the processing conditions and initial (1+x)Mg + 2B (x = 0, 0.1, 0.2) proportions of the precursors on the critical current values of the wires have been studied. In particular, the limits of the available temperatures and times for heat treatments imposed by the chemical reaction between Mg and Cu, and their effect on the superconducting properties of the wires, are discussed. The analysis includes the study of the sample microstructure and phase composition as well as of the critical current temperature and field dependences. The wires show high thermal stability during direct transport measurements and carry a critical current density of 1.3 × 10 9 A/m 2 at 15 K in the self-field for optimised processing conditions. Introduction The discovery of MgB 2 superconductor [1] has opened great expectations as it has been revelled as a suitable candidate for practical largescale applications. Key properties of such interest are: a) the relatively high critical temperature, T c~3 9 K, allowing working temperatures on the range of 20-30 K, nowadays easily reachable with cryocoolers; b) the low cost and rather easy preparation of the rough materials; and c) the good grain connectivity and the absence of week links [2]. Fortunately, powder-in-tube (PIT) technologies, widely searched at laboratory and fully developed at industrial scale in recent years for the fabrication of PbBi-2223 tapes, have been proved to be useful in the conformation of MgB 2 /metal composite wires. This is, indeed, one of the most attractive technologies for long-wire fabrication of hard and brittle materials because of its potential scalability and production flexibility. In the PIT methods, metallic tubes, with the precursor powders packed inside, are drawn to wires or rolled to tapes. The precursors may be mixtures of unreacted Mg and B powders, the so called in-situ reaction approach [3][4][5][6]; pre-reacted MgB 2 powders [4,5,[7][8][9][10], called ex-situ reaction technique; or partially reacted methods containing mixtures of MgB 2 , Mg and B powders [5,11]. In most cases, heat treatments in absence of oxygen are used either to sinter or to react the precursor. Although unsintered conductors made from reacted MgB 2 powders already have appreciable J c values up to 10 9 Am -2 at 4.2 K in the self-field, the magnetic field dependence is improved notably by sintering [4,[8][9][10]. All these methods have their own advantages and drawbacks. Thus, ex-situ procedures usually yield wires and tapes with smaller grains and more homogeneous microstructures, resulting in improved J c (B) dependences, probably because of grain boundary pinning. On the contrary, it has the drawback of being very sensitive to the MgB 2 precursor impurities [12]. On the other hand, the in-situ procedure uses lower heat treatment temperatures, allowing a larger range of metallic sheaths, being then more flexible in the material selection. Nevertheless, since the density of the initial Mg+2B mixtures are significantly lower than that of the MgB 2 phase, this method has the intrinsic disadvantage of a low final density. The selection of adequate sheaths, giving thermal, electrical and mechanical stability without deterioration of the superconductor, is crucial and constitutes nowadays an open issue to be addressed prior to reach technologically useful composite MgB 2 /metal conductors. Different metals sheaths have already been used: iron [5,10], copper [3,6,7], nickel [7,8,10], silver [3,6,7], cupro nickel alloys [9] and stainless steel [9,13], as well as different metal combinations such as: (from inside to outside) Ta/Cu [14] and mechanically enforced Fe/Cu/SS [4]. Up to now the highest J c (B,T) values are obtained with hard metal sheaths that do not react with Mg or MgB 2 , such as steel and iron, but these wires, as consequence of a poor thermal stability, easily quench at low fields. On the other hand, silver sheathed wires [3,6,7] present very poor results compared with copper or nickel sheathed wires and therefore have been disregarded. The use of copper as sheath would be an excellent choice for this purpose due to its high thermal conductivity. Copper is a low cost and ductile metal very suitable for the PIT fabrication of MgB 2 composite wires and tapes. Moreover, as in low temperature superconducting wires, it might give good mechanical support and excellent thermal stability. In addition, the good soldability of Cu would facilitate the achievement of low resistance electric contacts to external current feeding sources. Nevertheless, the chemical reaction of Mg and MgB 2 with Cu, during the compulsory heat treatments, causes a reduction of the wire J c values. This reactivity, which is produced by the diffusion of Mg into the Cu, together with a strong reduction of the melting temperature of the produced Cu-Mg alloys, may be avoided using costly and more complex to process diffusions barriers. Trying to improve the thermal stability of MgB 2 /metal composite wires, which is a major technological challenge, it is worthwhile a deeper study to determine the limits of the superconducting properties of MgB 2 / Cu wires associated to the reactivity of copper and Mg. Cu may be also an interesting material since it alloys well with Ni, which has also been used with quite good results [8], and thus the use of different Cu-Ni proportions would allow the control of thermal, mechanical and electrical properties of the sheath. With this aim here we present results on PIT monofilamentary wires fabricated from Cu tubes with unreacted mixtures of Mg and B powders using stoichiometric as well as Mg-rich compositions. The influence of the processing conditions and the initial Mg: B proportions on the final wire J c values have been studied. Special attention is paid to the limits of the thermal treatments temperatures and times imposed by the reactivity of Mg in Cu and their effect on the superconducting properties of the wires. With this aim, a complete analysis of the microstructure and phase compositions of the wires by SEM and EDX is presented, together with measurements of the temperature and field dependences of the critical currents, obtained directly by transport and from the M-B hysteresis loops. Experimental details Single filament Cu-sheathed MgB 2 wires were prepared using the conventional PIT method. The initial powder was a mixture of Mg (Goodfellow 99.8% purity with grain size lower than 250 µm) and amorphous boron (Alfa -Aesar, 99.99% purity and 355 mesh). Powders of atomic stoichiometry and with excess of Mg were prepared with three different proportions (1+x)Mg + 2B, with x = 0, 0.1 and 0.2. These precursor powders were well mixed in a vibratory mill for 10 minutes and packed inside copper tubes of 4.0 mm outer diameter and 2.5 or 3.0 mm inner ones. Subsequently, the tubes were cold drawn in round dyes down to 1.1 or 1.2 mm final diameters in 0.1-mm reduction steps. The final wires have core diameters of 0.6 or 0.8 mm, typically, depending on the initial inner diameter of the copper tubes. The wire was then cut in 6 to 7 cm long pieces and their ends mechanically sealed with soft copper to avoid leaking of Mg during the annealing. To prevent oxidation, the in-wire reaction was done in sealed quartz tubes with argon at temperatures ranging from 620 to 700 ˚C and times from 15 minutes to 48 h. Fig. 1 shows a typical temperature ramp of the sample heattreatment followed by slow cooling inside the furnace or room temperature quenching. The microstructure and phase composition of the wires were analysed by SEM and X-ray energy dispersive spectroscopy (EDX) techniques, respectively. A commercial Quantum Design SQUID magnetometer was used to perform AC and DC magnetic measurements over 4 mm long wire samples, keeping their axis perpendicular to the applied field. The critical temperature, T c , and the superconductor-normal transition were analysed by AC susceptibility measurements, χ ac (T), using AC field with amplitudes of 0.1 mT and low frequency of 1 Hz, to minimise the contribution of the eddy currents induced in the metallic sheaths. Isothermal magnetisation DC loops, M(B), up to fields of 5 T, were also measured. Electric DC transport measurements were performed on 6 cm long wires at temperatures ranging from 15 to 40 K using a cryocooler system and sinusoidal current ramps of typically 1 to 3 seconds scan time. The critical currents, I c (T), were determined by the standard four-probe using the 1 µV/cm criterion. Results and discussion Microstructure and reactivity A typical longitudinal cross-section of MgB 2 /Cu composite wires after annealing is shown in Figs. 2(a) and 2(b), which correspond to a wire with x = 0, annealed at 700 ˚C during 30 minutes and afterwards cooled down inside the furnace. The superconducting core has a typical irregular shape, partially produced during the mechanical conformation because of the lower hardness of copper and also during the heat treatment by the reactivity of Mg from the core with the inner sheath wall. A reaction layer adjacent to the superconducting core with darker contrast and 20 to 30 µm thickness is easily observed in Fig. 2(b). EDX and X-Ray analyses [6], have indicated that this layer corresponds to MgCu 2 while the rest of the sheath remains pure copper. Inside the core there are also some homogeneously dispersed MgCu 2 grains, with sizes typically ranging from 20 to 80 µm. Moreover, as point out previously, it should be remarked that all samples show larger porosity than those made from reacted precursor. The predominant superconducting MgB 2 phase, observed by the darker grey contrast in the SEM pictures of Figs. 2(a) and 2(b) have eutecticlike structure [6]. An enlarged view of this structure, which contains 8±2 atm.% of Cu, is show in Fig. 2(c), where the darker grey contrast corresponds to the superconducting MgB 2 phase and the lighter phase to a Mg-Cu intermetallic compound, probably Mg 2 Cu. The formation of these intermetallic compounds causes an overall Mg deficit, which may induce the formation of boron-rich phases at given points (darker areas in Fig. 2(d)). This is more likely on samples annealed at lower temperatures and longer times, while is not observed in samples with Mg excess (x = 0.2). . The existence of finely dispersed Cucontaining phases in the eutectic-like structure as well as larger MgCu 2 grains inside the core, may enhance the thermal stability and the mechanical performance of the wires, but it would also result in a reduction of the superconducting volume and therefore in a decrease of the overall J c of the wires. On the other hand, the presence of relatively large MgCu 2 grains inside the core may also affect negatively the superconducting performance of the wires and their homogeneity In order to analyse the effect of the cooling rate on the microstructure of the samples, two wires with x = 0.2 annealed at 700 ˚C during 18 minutes, but cooled in the furnace or quenched, were compared. A very similar microstructure was observed in both samples with MgCu 2 grains of the same size, although in the quenched wire fewer areas with boron-rich phases appear. Independently of the annealing temperatures, above and below the Mg melting (T m (Mg) = 649 ºC, [15]) and the treatment times, the same phases are present in the wires. This indicates that some amount of Mg diffuses in Cu, which at the range of annealing temperatures used here 620 -700 ºC and according with the Cu-Mg equilibrium phase diagram [15], may produce liquid phases. Moreover, it has to be noted that at higher annealing temperatures, 780 -800 ºC, there is a large increase of the Mg reactivity with Cu, which results in drastic reductions of the critical current of the wires, even for short annealing times (5 minutes). where the ratio χ'/χ'(5 K) changes from 0.9 to 0.1, are ∆T = 1.5 K. These values are just slightly higher than the ones observed in wires fabricated by similar methods but using harder and nonreactive sheaths, such as iron [5]. Transport current at self-field: Influence of heat treatments and initial precursors The influence of the heat treatment conditions on the transport critical current density values, J c,t may be analysed in Fig. 4(a). These values correspond to 15 K, self-field measurements on a wire fabricated from precursor powders of stoichiometric proportions and cooled down inside the furnace. The error bars of the 700 ºC results are typical bounds of measurements on different pieces of the same wire treated under the same conditions. By annealing at temperatures above T m (Mg), in the range between 680 and 700 ºC, values up to J c,t = 9×10 8 A/m 2 are obtained for short times (15-30 minutes), while for longer ones, there are important reductions of J c,t . On the contrary, for heat treatments at temperatures bellow T m (Mg) (620 ºC, for example) the critical current density increases with the annealing time, and much longer times (48 hours) are needed to reach J c,t values similar to the wires annealed at temperatures above T m . Finally, as pointed out in the above section, for higher annealing temperatures, 800 ºC, although the wires are still superconducting, the critical current values decrease drastically, even for very short annealing times (5 minutes) and sharp heating ramps. From these results it is clear that the formation of MgB 2 is relatively fast, and short annealing times at relatively low temperatures (700 ºC) are sufficient. On the other hand, the observed fast decrease of the critical current density with the annealing time, becomes faster at higher temperatures and is certainly due to the higher diffusivity of Mg in Cu resulting on a higher reactivity. Due to this reactivity with the Cu sheaths, wires from stoichiometric powders develop cores with an overall Mg deficit upon annealing, causing the formation of boron-rich phases and therefore a decrease of the critical current values. In order to analyse this effect, the superconducting properties of wires made from precursor powders with Mg excess have been studied. Fig. 4(b) shows the dependence of the self-field critical current density values at 15 K on the initial Mg excess, J c,t (x), for different annealing times at 700 ºC. Results for slow-cooling (full symbols and crosses) and Our results also indicate that quenched samples have considerable lower critical current values than those cooled in the furnace. For self-field conditions, best results of J c,t (15 K) = 1.3×10 9 A/m 2 , which correspond to a critical current of 700 A, were obtained by annealing 18 minutes at 700 ºC and afterwards slow cooling. The temperature dependences of the self-field transport critical currents of different samples are shown in Fig. 5. The values have been scaled with the measured ones at 15 K. Most samples show a similar behaviour, independently of the annealing temperatures and times, giving an almost linear temperature decay and currents densities close to zero at the 36-37 K range. The wires with poor critical currents (circles) as well as the quenched wires (×), show stronger J c,t (T) dependences and becomes zero at lower temperatures (~35 K), in agreement with the χ'(T) measurements displayed in Fig. 3. The extrapolation of the above data would give estimates of J c,t = 2×10 9 A/m 2 at 5 K in the self-field, for the wires with best properties. All samples show high thermal stability and do not quench during transport measurements, even though these were performed without cryogenic liquid or exchange gas surrounding the sample, being cooled by the good thermal contact of their ends to the cold stage of the cryocooler. For critical current values of 500-700 A, the temperature of the sample, measured by a thermocouple soldered to the sheath, rises between 2 and 3 K during the measurements when the critical current is surpassed up to measured voltages of 10 µV/cm, and has fast recovery to the set-up temperature after the current is switched off. Magnetic field dependence of J c The magnetic field dependence of the critical current density have been estimated form the magnetization hysteresis curves measured under perpendicular applied fields (J c,m ), using critical state models. Fig. 6 shows the results obtained from the width of the hysteresis loop measurements, ∆M as J c,m = (3π/8)∆M/R, in S.I. units [16], for different wires at 5 and 20 K. Critical current density values estimated from magnetization measurements, although slightly higher, are in good agreement with those obtained by DC transport (see Fig. 4). Note that on transport self-field measurements, for J c,t =10 9 A/m 2 , the magnetic field at the surface of the wire core is between 0.2 to 0.3 T. Although the J c,m values change among wires, all of them share the same field dependence, indicating that the annealing and cooling conditions or the excess of Mg in the precursor powders essentially do not change the pinning mechanisms of these samples. As it was been already observed [3][4][5][6][8][9][10][11]13,14,17], this material has a sharp field dependence, in our case with a decrease of J c,m down to 10 8 A/m 2 for magnetic fields of 3.5 T at 5 K and 2 T at 20 K. Studies reported so far in the literature [17][18][19], have demonstrated a wide range of J c (B) behaviours for different polycrystalline MgB 2 materials, depending on the geometry and manufacturing techniques. Strong differences among the results on bulk, thin films, wires, tapes, etc, have been observed. A review of such results may be seen for instance in ref. [17]. Among undoped MgB 2 materials studied so far, thin films have shown the best superconducting behaviour under magnetic fields, with transport J c exceeding 10 9 A/m 2 at 4.2 K and 10 T [18]. Although the pinning mechanisms are still not clear, grain boundaries have been suggested to play an important role on polycrystalline materials. In this sense, the typical MgB 2 grain size of thin films (less than 10 nm) is much smaller than those of bulk samples, wires and tapes (0.1-1 µm). This difference, together with other defects at the nanoscale range, would explain the larger pinning and therefore the smoother J c (B) dependencies measured in thin films [18,19]. Furthermore, other possibilities, such as pinning by oxygen and MgO particles incorporated in thin films have been also proposed. For wires and tapes, the field decay depends on the metallic sheath, as well as on the precursor powders. The slowest J c field decays, reported so far, correspond to wires fabricated from reacted MgB 2 powders [4,5]. Again, this better behaviour has been ascribed to vortex pinning along grain boundaries, since typical grain sizes of wires made from unreacted Mg+2B mixtures tend to be larger than those from reacted powders. It must be emphasise that, frequently, optima J c values at low magnetic fields does not give best results at high magnetic fields [5,8], because small grains give a larger number of junctions and normally have less current capability at low fields. The hardness of the sheath as well as the reactivity with the precursors are also key factors. The use of harder and less-reactive metallic sheaths as iron [5], nickel [8] or combinations of different metals such as Nb/Cu/SS or Ta/Cu/SS [4] give rise to different J c (B) decays depending on the reaction method, in-situ or ex-situ, as well as on the heat treatment conditions. J c values higher than 10 8 A/m 2 have been obtained for applied magnetic fields ranging from 3.5 to 6.5 T at 4.2 K [4,8], and from 2 to 3.5 T at 20 K [5]. The J c (B) behaviour of the Cu sheathed wires here reported, coincides with results obtained by other groups [19] using the in-situ reaction and copper sheaths. These results are of the order, but in the lower limits, of those obtained for wires of harder sheaths as iron, stainless steel and nickel, using the ex-situ technique. These differences are due to the in-situ reaction it-self and to the reactivity between the Cu sheath and the Mg precursors together with the insufficient hardness of copper. On the other hand, main advantages of these MgB 2 /Cu wires are certainly related to the good thermal stability given by the high thermal conductivity of the sheath. This allows carrying out transport measurements up to high currents 500-700 A without quenching, which are very likely on samples with others sheaths such as iron, SS, etc [4,5], even without surrounding cryogenic liquid or gas to thermalise the sample. Further efforts are needed to obtain conductors thermally stable and with high performance at fields between 5 to 10 T, both necessary for technical applications. Conclusions We have proved the feasibility of Cu/MgB 2 composite monocore wires by the in-situ reaction of boron and Mg powders by annealing at temperatures below 700 ºC. A reaction MgCu 2 layer of about 20µm thickness surrounding the superconducting core and MgCu 2 grains dispersed homogeneously inside the core are formed during the heat treatments. Due to the reactivity of Mg with the Cu sheath, wires with initial stoichiometric proportions develop an overall Mg deficit during annealing. Therefore, wires prepared from precursor powders with Mg excess, (1+x)Mg + 2B, were also analysed. An overall increase of the transport critical current densities, J c,t , with the Mg excess on the precursors are observed. For wires with x = 0.2, J c,t increases between 50% to 70% the values of wires fabricated from stoichiometric proportions for most annealing conditions. The MgB 2 formation is fast, so that shorts annealing times (15 minutes) at 680-700 ºC are sufficient to obtain self-field J c,t (15 K) values of the order of 1.3×10 9 A/m 2 , corresponding to critical currents, I c~ 700 A. Longer annealing times reduce J c values due to the reactivity of Mg and Cu. Our results also indicate that quenched samples have considerable lower critical current values than those cooled down inside the furnace. The field dependence has been also obtained from magnetic measurements, observing a decrease of the J c,m values down to 10 8 A/m 2 for applied magnetic fields of 3.5 T at 5 K and of 2 T at 20 K. The analysed wires show high thermal stability and do not quench to normal state during transport measurements even without the presence of any cryogenic liquid or gas surrounding the sample. Nevertheless, the porosity of the superconducting core that would affect negatively on the conductors homogeneity, and J c (B) decays sharper than those obtained on wires prepared by ex-situ methods with harder metals, constitute their main disadvantages. Further efforts are needed to obtain conductors which combine both, thermal stabilisation with high performance at high fields (5-10 T) necessary for technical applications. Figure 1 1Typical annealing conditions and cooling rates used in the in-situ fabrication of MgB 2 / Cu composite wires. Figure 2 2SEM micrographs of longitudinal polished wires with x = 0 and cooled down in furnace. Figs. (a), (b) and (c) correspond to a wire annealed 30 minutes at 700 ºC with different enlargements, being (c) a detail of the main phase containing the MgB 2 superconductor -circle in (b)-. Fig. (d) Micrograph of a wire annealed 48 hours at 620 ºC showing a rich-boron phase (dark contrast). The in-phase component, χ', of the χ ac (T) measurements of several wires are displayed in Fig. 3, where for comparative purposes, the results have been scaled using the \|χ'(5 K)\| values. In all samples, χ'(T) at low enough temperatures reach values close to the corresponding ones to perfect diamagnetism, χ'(0 K) = -1/(1-N), N being the demagnetisation factor. For cylinders of high L/R aspect ratio (12 < L/R < 14 in our case) under perpendicular fields N tends to 1/2. All samples show very similar T c values, ranging from 38.0 to 38.5 K, but with different transition broadening. Wires fabricated with Mgexcess precursors (x = 0.2) show sharper transitions than stoichiometric ones. Moreover, in both cases, quenched samples have a worse behaviour, that is, wider transitions and slighter lower T c values. The better transition widths, ∆T, experimentally defined by the temperature range Figure 3 3Results of χ'(T) under perpendicular ac fields of 1 Hz and field amplitude of 0.1 mT on wires annealed at 700 ºC during different times and for two precursor compositions (x = 0 and 0.2). Symbols joined by dashed lines correspond to quenched samples. Figure 4 4(a) Transport critical current density, J c,t , at 15 K in the self-field of a wire made from stoichiometric precursors for different heat treatments and cooled inside the furnace. (b) J c,t dependence on the initial Mg excess of the precursors, x, for wires annealed at 700 ºC and different times, cooled in the furnace (filled symbols and crosses) and quenched (open symbols). quenched samples (open symbols) show overall increases of the J c,t values with the Mg excess. For the wires with x = 0.2, J c,t improves by 50-70% with respect to the ones fabricated from stoichiometric proportions for most annealing conditions, while the properties for the x = 0.1 analogue do not differ significantly from those of x = 0. Note that although, for all wires, J c,t decreases with the 700 ºC annealing times, there are slight differences on the time decays depending on the Mg excess of the precursors. 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[ "Nielsen equalizer theory ", "Nielsen equalizer theory " ]	[ "P Christopher Staecker " ]	[]	[]	We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k − 1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface.As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori. *	10.1016/j.topol.2011.05.032	[ "https://arxiv.org/pdf/1008.2154v2.pdf" ]	54,999,598	1008.2154	018134e42d204ae1c5c8db0689567243fe23be96	Nielsen equalizer theory * 8 Jul 2011 July 11, 2011 P Christopher Staecker Nielsen equalizer theory * 8 Jul 2011 July 11, 2011 We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k − 1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface.As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori. * Introduction The goal of this paper is to generalize the basic definitions and results of Nielsen coincidence theory to a theory of equalizer sets for sets of (possibly more than two) mappings. For spaces X and Y and maps f 1 , . . . , f k : X → Y , the equalizer set is defined as Eq(f 1 , . . . , f k ) = {x ∈ X \| f 1 (x) = · · · = f k (x)}. This generalizes the coincidence set Coin(f 1 , f 2 ) = {x ∈ X \| f 1 (x) = f 2 (x)} for two mappings. Nielsen coincidence theory, see [3], estimates the number of coincidence points of a pair of maps in a homotopy invariant way. Most of the techniques are a generalization of ideas from fixed point theory, see [5]. In coincidence theory, one defines the Nielsen number N (f 1 , f 2 ) of a pair of maps, which is a lower bound for the minimal coincidence number MC (f 1 , f 2 ): N (f 1 , f 2 ) ≤ MC (f 1 , f 2 ) = min{# Coin(f ′ 1 , f ′ 2 ) \| f ′ i ≃ f i }. The above quantities are in fact equal when X and Y are compact n-manifolds of the same dimension n = 2. In this paper we extend this theory to equalizer sets. The typical setting for Nielsen coincidence theory is for maps X → Y of compact manifolds of the same dimension. For maps f 1 , f 2 : X → Y in this setting, transversality arguments show that we can change the maps by homotopy so that Coin(f 1 , f 2 ) is a set of finitely many points. Proceeding in the same setting with more than two maps immediately gives: Theorem 1.1. If X and Y are compact manifolds of the same dimension, and f 1 , . . . , f k : X → Y are maps with k > 2, then these maps can be changed by homotopy so that the equalizer set is empty. Proof. Well-known transversality arguments show that we can change f 2 by a homotopy to f ′ 2 so that Coin(f 1 , f ′ 2 ) is a finite set of points. Similarly we obtain f ′ 3 ≃ f 3 such that Coin(f 1 , f ′ 3 ) is a finite set of points. These homotopies can be arranged so that Coin(f 1 , f ′ 2 ) and Coin(f 1 , f ′ 3 ) are disjoint. Thus Eq(f 1 , f ′ 2 , f ′ 3 , f 4 , . . . , f k ) ⊂ Coin(f 1 , f ′ 2 ) ∩ Coin(f 1 , f ′ 3 ) = ∅. Thus there is no interesting theory for counting the minimal number of equalizer points between compact manifolds of the same dimension, since this number is always zero. In this sense, the equalizer equation f 1 (x) = · · · = f k (x) is "overdetermined" when the dimensions of the domain and codomain are equal. In order to obtain an interesting theory we must increase the dimension of the domain space. Unlike in coincidence theory, where allowing maps X → Y of spaces of different dimension requires very different techniques, the positive codimension setting is the appropriate one for producing a Nielsen equalizer theory which closely resembles that for coincidences. In particular, for equalizers of k maps, we will require X and Y to be of dimensions (k − 1)n and n, respectively, for any n. Consider the following example: Example 1.2. We will examine the equalizer set of three maps f, g, h : T 2 → S 1 from the 2 dimensional torus to the circle. Viewing the torus as the quotient of R 2 by the integer lattice, and S 1 as the quotient of R by the integers, we will specify our maps by integer matrices of size 1 × 2. Let the maps be given by matrices: A f = (3 1), A g = (0 2), A h = (−1 − 1). Let C f g = Coin(f, g), with C f h and C gh defined similarly, and we have Eq(f, g, h) = C f g ∩ C gh ∩ C f h . (Actually the equalizer set is the intersection of any two of these coincidence sets.) / / / / / / / C f g : C gh : It is straightforward to compute these sets. For example, C f g is the set of points (x, y) with 3x + y = 2y mod Z 2 , which is to say y = 3x mod Z 2 . Similarly computing the sets C f h and C gh produces the picture in Figure 1, where the torus is drawn as [0, 1] × [0, 1] with opposite sides identified. We see in the picture that Eq(f, g, h) consists of 10 points (the nine points where the lines visibly intersect, plus the intersection at the identified corners of the diagram). C f h : _ _ _ In this paper we will define the Nielsen number N (f, g, h) which is a lower bound for the minimum number of equalizer points when the maps are changed by homotopy. In Theorem 4.4 we give a simple formula for computing this quantity on tori, which in this example gives N (f, g, h) = 0 2 −1 −1 − 3 1 3 1 = 10. Thus these maps cannot be changed by homotopy to have fewer than 10 equalizer points. The construction of the theory is facilitated by a fundamental correspondence between Eq(f 1 , . . . , f k ) and the coincidence set of a pair of related maps. Let F, G : X → Y k−1 be given by F (x) = (f 1 (x), . . . , f 1 (x)), G(x) = (f 2 (x), . . . , f k (x)). (⋆) Since X and Y are compact with dimensions (k − 1)n and n respectively, the above F and G are maps between compact manifolds of the same dimension, and Coin(F, G) = Eq(f 1 , . . . , f k ). This correspondence is well-behaved under homotopy, since changing the maps f i by homotopies corresponds in a natural way to a change of F and G by homotopies. As we shall see, the homotopyinvariant behavior of Eq(f 1 , . . . , f k ) is the same as that of Coin(F, G), and we may define Nielsen-type invariants for the equalizer set in terms of the same invariants from the coincidence theory of (F, G). In Section 2 we define the Reidemeister and equalizer classes which form the building blocks for our theory. In Section 3 we define the Nielsen number and in Section 4 we give some computational results for maps into Jiang spaces and maps of tori. In Section 5 we give an application to Nielsen coincidence theory in positive codimensions, giving a full computation of the "geometric Nielsen number" on tori. We would like to thank Robert F. Brown for helpful comments. Reidemeister and equalizer classes Let X and Y be spaces with universal covering spaces (connected, locally pathconnected, and semilocally simply connected), and let X and Y be the universal covering spaces with projection maps p X : X → X and p Y : Y → Y . For maps f 1 , . . . , f k : X → Y , we wish to construct a Reidemeister-type theory for the equalizer points Eq(f 1 , . . . , f k ), so that each point has an algebraic Reidemeister class, and two equalizer points can be combined by homotopy only when their classes are equal. Our basic result is a generalization of a well-known result from coincidence theory which is stated in part (without proof) as Lemma 2.3 of [2]. For the sake of completeness we give a full proof. The proof is similar to that of Theorem 1.5 in [5], which is the corresponding statement in fixed point theory. Throughout, elements of the fundamental group are viewed as deck transformations on the universal covering space. Theorem 2.1. Let f 1 , . . . , f k : X → Y be maps with lifts f i : X → Y and induced homomorphisms φ i : π 1 (X) → π 1 (Y ). We have Eq(f 1 , . . . , f k ) = α2,...,α k ∈π1(Y ) p X Eq( f 1 , α 2 f 2 , . . . , α k f k ). 2. For α i , β i ∈ π 1 (X), the sets p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) and p X Eq( f 1 , β 2 f 2 , . . . , β k f k ) are disjoint or equal. The above sets are equal if and only if there is some z ∈ π 1 (X) with β i = φ 1 (z)α i φ i (z) −1 for all i. Proof. For the first statement, take some x ∈ Eq(f 1 , . . . , f k ) and some x ∈ p −1 X (x). We have p Y ( f i ( x)) = f i (x) = f 1 (x) for all i, and thus the values f i ( x) all differ by deck transformations. That is, there are α i ∈ π 1 (Y ) with f 1 ( x) = α 2 f 2 ( x) = · · · = α k f k ( x), which is to say that x ∈ Eq( f 1 , α 2 f 2 , . . . , α k f k ), and so x ∈ p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) as desired. Now we prove statement 3. First, let us assume that p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) = p X Eq( f 1 , β 2 f 2 , . . . , β k f k ). This means that for any point x ∈ Eq( f 1 , α 2 f 2 , . . . , α k f k ), there is some deck transformation z ∈ π 1 (X) with z x ∈ Eq( f 1 , β 2 f 2 , . . . , β k f k ). Then we have β i f i (z x) = f 1 (z x) = φ 1 (z) f 1 ( x) = φ 1 (z)α i f i ( x) = φ 1 (z)α i φ i (z) −1 f i (z x) Since the two lifts β i f i and φ 1 (z)α i φ i (z) −1 f i agree at a point, they are the same lift, and thus β i = φ 1 (z)α i φ i (z) −1 as desired. For the converse in statement 3, assume that β i = φ 1 (z)α i φ i (z) −1 for all i, and take x ∈ p X Eq( f 1 , α 2 f 2 , . . . , α k f k ). Then we have φ 1 (z)α i = β i φ i (z) for all i, and so f 1 (z x) = φ 1 (z) f 1 ( x) = φ 1 (z)α i f i ( x) = β i φ i (z) f i ( x) = β i f i (z x). Thus z x = Eq( f 1 , β 2 f 2 , . . . , β k f k ), and so x ∈ p X Eq( f 1 , β 2 f 2 , . . . , β k f k ), and we have shown p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) ⊂ p X Eq( f 1 , β 2 f 2 , . . . , β k f k ). A symmetric argument shows the converse inclusion, and so the above sets are equal. For statement 2, it suffices to show that if there is a point x ∈ p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) ∩ p X Eq( f 1 , β 2 f 2 , . . . , β k f k ), then the two sets of the above intersection are equal. For such a point x, there are x 0 , x 1 ∈ p −1 X (x) with x 0 ∈ Eq( f 1 , α 2 f 2 , . . . , α k f k ), x 1 ∈ Eq( f 1 , β 2 f 2 , . . . , β k f k ). Let z ∈ π 1 (X) with z x 0 = x 1 . Then we have β i f i (z x 0 ) = β i f i ( x 1 ) = f 1 ( x 1 ) = f 1 (z x 0 ) = φ 1 (z) f 1 ( x 0 ) = φ 1 (z)α i f i ( x 0 ) = φ 1 (z)α i φ i (z) −1 f i (z x 0 ). The above equality shows two lifts of f i agreeing at the point z x 0 , and so we have β i = φ 1 (z)α i φ i (z) −1 , which by statement 3 implies that p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) = p X Eq( f 1 , β 2 f 2 , . . . , β k f k ) as desired. Let R(φ 1 , . . . , φ k ) = π 1 (Y ) k−1 / ∼ be the quotient of π 1 (Y ) k−1 by the following relation, inspired by statement 3 above: (α 2 , . . . , α k ) ∼ (β 2 , . . . , β k ) if and only if there is some z ∈ π 1 (X) with β i = φ 1 (z)α i φ i (z) −1 for all i ∈ {2, . . . , k}. We call R(φ 1 , . . . , φ k ) the set of Reidemeister classes for φ 1 , . . . , φ k . Then the theorem above gives the following disjoint union Eq(f 1 , . . . , f k ) = (αi)∈R(φ1,...,φ k ) p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) The above union partitions the equalizer set into Nielsen equalizer classes (or simply equalizer classes). That is, C ⊂ Eq(f 1 , . . . , f k ) is an equalizer class if and only if there are α i with C = p X Eq( f 1 , α 2 f 2 , . . . , α k f k ). Note that an equalizer class can be empty. The equalizer classes are related to the coincidence classes of the pair (F, G) from equation (⋆) in the following way: Theorem 2.2. A subset C ⊂ Eq(f 1 , . . . , f k ) is an equalizer class if and only if C is a coincidence class when regarded as a subset of Coin(F, G). That is, C is an equalizer class if and only if there is a deck transformation A ∈ π 1 (Y k−1 ) with C = p X Coin( F , A G) for some lifts F and G of F and G. Proof. First we assume that C is an equalizer class, and so we have lifts f i of f i and α i ∈ π 1 (Y ) with C = p X Eq( f 1 , α 2 f 2 , . . . , α k f k ). Let F and G be given by: F ( x) = ( f 1 ( x), . . . f 1 ( x)), G( x) = ( f 2 ( x), . . . , f k ( x)), and let A : Y k−1 → Y k−1 be A( y 2 , . . . y k ) = (α 2 y 2 , . . . , α k y k ). Then we have Eq( f 1 , α 1 f 2 , . . . , α k f k ) = Coin( F , A G), and so C = p X Coin( F , A G) as desired. Now for the converse we assume that C is a coincidence class of (F, G), which means there are lifts F and G of F and G with a deck transformation A ∈ π 1 (Y k−1 ) such that C = p X Coin( F , A G). Since F and G are lifts of F and G, we can write F ( x) = ( f 2 1 ( x), . . . , f k 1 ( x)), G( x) = ( f 2 ( x), . . . , f k ( x)) where each f i 1 is a lift of f 1 , and f j is a lift of f j for j ≥ 2. Similarly we may factor A as A = α 1 × · · · × α k for α i ∈ π 1 (Y ). Each of the f i 1 may be different, but there is a single lift f 1 of f 1 with deck transformations β i such that β i f 1 = f i 1 . Then we have Coin( F , A G) = Coin((β 2 f 1 , . . . , β k f 1 ), (α 2 f 2 , . . . , α k f k )) = Coin(( f 1 , . . . , f 1 ), (β −1 2 α 2 f 2 , . . . , β −1 k α k f k ) = Eq( f 1 , β −1 2 α 2 f 2 , . . . , β −1 k α k f k ) and so C = p X Coin( F , A G) is an equalizer class. The equalizer classes can be described nicely in terms of paths in X and their images under the f i : Proof. Our points x, x ′ are in the same equalizer class if and only if they are in the same coincidence class of the pair (F, G). A standard result in coincidence theory shows that this is equivalent to the existence of a path γ in X from x to x ′ with F (γ) ≃ G(γ). This is equivalent to (f 1 , . . . , f 1 )(γ) ≃ (f 2 , . . . , f k )(γ), which is equivalent to f 1 (γ) ≃ f i (γ) for each i. The equalizer index and the Nielsen number Let Eq(f 1 , . . . , f k , U ) = Eq(f 1 , . . . , f k ) ∩ U , and let Coin(f, g, U ) = Coin(f, g) ∩ U . Our index for equalizer sets will be defined in terms of the coincidence index i. We first review some properties of the coincidence index. Let f, g : M → N be maps between compact orientable manifolds of the same dimension. The coincidence index i(f, g, U ) is an integer valued function defined for open sets U with Coin(f, g, U ) compact. It satisfies the following properties: • Homotopy: Let f ′ ≃ f and g ′ ≃ g, by homotopies F t and G t , such that the set {(x, t) \| x ∈ Coin(F t , G t , U )} ⊂ M × [0, 1] is compact (such a pair of homotopies is called admissible). Then i(f, g, U ) = i(f ′ , g ′ , U ). • Additivity: If U 1 ∩ U 2 = ∅ and Coin(f, g, U ) ⊂ U 1 ∪ U 2 , then i(f, g, U ) = i(f, g, U 1 ) + i(f, g, U 2 ). • Solution: If i(f, g, U ) = 0, then Coin(f, g, U ) is not empty. We wish to define a similar index in the equalizer setting. Let X and Y be compact orientable manifolds of dimensions (k − 1)n and n, respectively, with maps f 1 , . . . , f k : X → Y . We call (f 1 , . . . , f k , U ) admissible when Eq(f 1 , . . . , f k , U ) is compact. Let F, G : X → Y k−1 be the maps as in (⋆). These are maps between compact orientable manifolds of the same dimension. When (f 1 , . . . , f k , U ) is admissible, then Coin(F, G, U ) = Eq(f 1 , . . . , f k , U ) is compact, and thus the coincidence index i(F, G, U ) is defined. We define the equalizer index ind(f 1 , . . . , f k , U ) to be i(F, G, U ). This equalizer index satisfies the appropriate homotopy, additivity, and solution properties. Then the equalizer index ind(f 1 , . . . , f k , U ) is defined and satisfies the following properties: If (f 1 , . . . , f k , U ) and (f ′ 1 , . . . , f ′ k , U ) are admissable and f i ≃ f ′ i with homotopy H i , we say that (H i ) is an admissible homotopy of (f 1 , . . . , f k , U ) to (f ′ 1 , . . . , f ′ k , U ) when the set {(x, t) \| x ∈ Eq(H 1 t , . . . , H k t , U )} ⊂ X × I is compact.• Homotopy: If (f 1 , . . . , k k , U ) is admissibly homotopic to (f ′ 1 , . . . , f ′ k , U ), then ind(f 1 , . . . , f k , U ) = ind(f ′ 1 , . . . , f ′ k , U ). • Additivity: If U 1 ∩ U 2 = ∅ and Eq(f 1 , . . . , f k , U ) ⊂ U 1 ∪ U 2 , then ind(f 1 , . . . , f k , U ) = ind(f 1 , . . . , f k , U 1 ) + ind(f 1 , . . . , f k , U 2 ). • Solution: If ind(f 1 , . . . , f k , U ) = 0, then Eq(f 1 , . . . , f k , U ) is not empty. Proof. The proofs of these properties all follow from the same properties of the coincidence index of the pair F, G as in (⋆). For an equalizer class C, we define the index of C, written ind(f 1 , . . . , f k , C), as ind(F, G, U ), where U is an open set with Coin(F, G, U ) = C (such an open set will always exist because coincidence classes are closed and X is compact). At this point we take a slight diversion to give a note on the computation of the index of differentiable maps in terms of their derivatives. When each of f i is differentiable, the maps F and G will also be differentiable, and the derivative maps DF x , DG x : R (k−1)n → R (k−1)n are defined at each point x ∈ X. Let x ∈ Eq(f 1 , . . . , f k ) be an equalizer point. We say that x is nondegenerate when DG x − DF x is nonsingular. In this case there is a neighborhood U around x containing no other coincidence points of F and G, and thus no other equalizer points, and the index can be computed by the well-known formula from coincidence theory: ind(f 1 , . . . , f k , U ) = i(F, G, U ) = sign det(DG x − DF x ). The definitions of F and G give the following formula in terms of the f i . Theorem 3.2. Let f 1 , . . . , f k : X → Y be maps of compact orientable manifolds of dimensions (k − 1)n and n respectively, and let x ∈ Eq(f 1 , . . . , f k ) be nondegenerate. Then there is a neighborhood U of x with Eq(f 1 , . . . , f k , U ) = {x} such that ind(f 1 , . . . , f k , U ) = sign det    df 2 − df 1 . . . df k − df 1    where all derivatives are taken at the point x (each row in the above is a n × (k − 1)n block matrix, so that the whole matrix has size (k − 1)n × (k − 1)n). Now we discuss the index theory for the non-orientable case. For the coincidence theory of maps f, g : M → N of compact (perhaps non-orientable) manifolds of the same dimension, the coincidence index cannot in general be defined. There is a related semi-index (see [2]) which plays a similar role. The semi-index, which we denote \|i\|, is defined not for arbitrary open sets, but only for coincidence classes, and satisfies properties similar to those of the coincidence index. Let C ⊂ Coin(f, g) be a coincidence class with C = p Coin( f , α g). Then if f ≃ f ′ and g ≃ g ′ , these homotopies will lift, producing maps f ′ ≃ f and g ′ ≃ g which are lifts of f ′ and g ′ respectively. Thus D = p Coin( f ′ , α g ′ ) is a coincidence class of (f ′ , g ′ ), and we say that D is "related to C" with respect to the pair of homotopies. If f, g : M → N are maps of compact manifolds of the same dimension and C is a (possibly empty) coincidence class, then \|i\|(f, g, C) is defined and satisfies: • Homotopy: If f ′ ≃ f and g ′ ≃ g, and D is the coincidence class of (f ′ , g ′ ) which is related to C with respect to these homotopies, then \|i\|(f, g, C) = \|i\|(f ′ , g ′ , D). • Solution: If \|i\|(f, g, C) = 0, then C is not empty. • Naturality: If M and N are orientable, then \|i\|(f, g, C) = \|i(f, g, C)\|, the absolute value of the usual coincidence index. In the setting of equalizer theory for maps f 1 , . . . , f k : X → Y of compact (possibly nonorientable) manifolds with an equalizer class C, we define the equalizer semi-index as in the orientable case: let (F, G) be as in (⋆), and we define \| ind \|(f 1 , . . . , f k , C) = \|i\|(F, G, C). Given homotopies f ′ i ≃ f i , the "rela- tion" between equalizer classes of (f 1 , . . . , f k ) and (f ′ 1 , . . . , f ′ k ) is defined exactly as in coincidence theory. The following has routine proofs similar to those for Theorem 3.1. Theorem 3.3. Let f 1 , . . . , f k : X → Y be maps of compact (possibly nonorientable) manifolds of dimensions (k − 1)n and n respectively, and let C ⊂ Eq(f 1 , . . . , f k , U ) be an equalizer class. Then the equalizer semi-index \| ind \|(f 1 , . . . , f k , C) is defined and satisfies the following properties: • Homotopy: If f i is homotopic to f ′ i for each i and D is the equalizer class of (f ′ 1 , . . . , f ′ k ) which is related to C, then \| ind \|(f 1 , . . . , f k , C) = \| ind \|(f ′ 1 , . . . , f ′ k , D). • Solution: If \| ind \|(f 1 , . . . , f k , C) = 0, then C is not empty. • Naturality: If X and Y are orientable, then \| ind \|(f 1 , . . . , f k , C) = \| ind(f 1 , . . . , f k , C)\|, the absolute value of the usual equalizer index. An equalizer class is called essential if its index (or semi-index in the nonorientable case) is nonzero. Definition 3.4. The Nielsen [equalizer] number N (f 1 , . . . , f k ) is defined to be the number of essential equalizer classes of (f 1 , . . . , f k ). From Theorem 2.2 and the definition of the index of a class, we see that N (f 1 , . . . , f k ) is equal to the Nielsen coincidence number of the pair (F, G). Since the Nielsen equalizer number is so closely related to a coincidence number, we can obtain a Wecken-type theorem for the minimal number of equalizer points. Let ME (f 1 , . . . , f k ) be the minimal number of equalizer points, defined as ME (f 1 , . . . , f k ) = min{# Eq(f ′ 1 , . . . , f ′ k ) \| f ′ i ≃ f i }. By the solution properties of the index and semi-index, every essential equalizer class must contain an equalizer point, and so N (f 1 , . . . , f k ) ≤ ME (f 1 , . . . , f k ). These two quantities are in fact equal in most cases, as the following theorem shows. Theorem 3.5. Let f 1 , . . . , f k : X → Y be maps of compact manifolds of dimensions (k − 1)n and n respectively. If (k − 1)n = 2, then ME(f 1 , . . . , f k ) = N (f 1 , . . . , f k ). In the case of "proper" equalizer theory (when k > 2), the result holds for all k and n except (k, n) = (3, 1), which is to say equalizer theory of three maps from a compact surface to the circle. Proof. The second statement is simply a consequence of k, n being natural numbers with (k − 1)n = 2, so we focus on the first statement. Let (F, G) be defined as in (⋆), and we have N (f 1 , . . . , f k ) = N (F, G). The maps F, G are maps between compact manifolds of dimension (k − 1)n. By our hypothesis this dimension is not 2, and so the Wecken theorem for coincidences (see [3]) gives maps F ′ ≃ F and G ′ ≃ G with # Coin(F ′ , G ′ ) = N (F, G). A result of Brooks in [1] shows that in fact there is a single map G ′′ ≃ G with Coin(F ′ , G ′ ) = Coin(F, G ′′ ), and thus # Coin(F, G ′′ ) = N (F, G). Our map G ′′ is a map of X → Y k−1 , so it can be written as G ′′ (x) = (g 2 (x), . . . , g k (x)) with g i ≃ f i . Now we have # Eq(f 1 , g 2 , . . . , g k ) = # Coin(F, G ′′ ) = N (F, G) = N (f 1 , . . . , f k ), and so ME (f 1 , . . . , f k ) ≤ N (f 1 , . . . , f k ) as desired. 4 Some computations 4 .1 Jiang spaces One setting in which the fixed point and coincidence Nielsen numbers are easily calculated is for maps on Jiang spaces. See [5] for the definition and basic results in fixed point theory. The class of Jiang spaces includes topological groups, generalized lens spaces and certain other homogeneous spaces, and is closed under products. The main result (see [3]) from coincidence theory concerning Jiang spaces is the following: Our theorem concerning Jiang spaces is the following result, which is facilitated by the coincidence theory of the maps (F, G) as in (⋆). Proof. Let F, G : X → Y k−1 be given as in (⋆): F (x) = (f 1 (x), . . . f 1 (x)), G(x) = (f 2 (x), . . . , f k (x)). Since Y is a Jiang space, then Y k−1 is a Jiang space. Thus by Theorem 4.1 all coincidence classes of F, G will have the same coincidence index. But the equalizer classes of f 1 , . . . , f k are the same as the coincidence classes of F, G, with the same indices, so all equalizer classes of f 1 , . . . , f k will have the same equalizer index. Define the Reidemeister number and Lefschetz number as: R(f 1 , . . . , f k ) = #R(φ 1 , . . . , φ k ) (this quantity may be infinite) and L(f 1 , . . . , f k ) = ind(f 1 , . . . , f k , X). Then we obtain: Corollary 4.3. If f 1 , . . . , f k : X → Y are maps of compact orientable manifolds of dimensions (k − 1)n and n respectively, and Y is a Jiang space, then: • If L(f 1 , . . . , f k ) = 0 then N (f 1 , . . . , f k ) = 0. • If L(f 1 , . . . , f k ) = 0 then N (f 1 , . . . , f k ) = R(f 1 , . . . , f k ). Proof. By the additivity property, L(f 1 , . . . , f k ) is the sum of the indices of each equalizer class. By Theorem 4.2 all classes have the same index, thus L(f 1 , . . . , f k ) = 0 means that all classes are inessential and so N (f 1 , . . . , f k ) = 0. If the Lefschetz number is not zero then all classes are essential and so the Nielsen number is simply the number of classes, which is the Reidemeister number. Tori We can give a very specific formula for the Nielsen number of maps f 1 , . . . , f k : T (k−1)n → T n on tori. We will view T m as the quotient of R m by the integer lattice, and consider maps which are induced by linear maps on R (k−1)n → R n taking Z (k−1)n to Z n . We can think of such maps as n × (k − 1)n matrices with integer entries. We now prove the formula which was used in the computation of Example 1.2. The result generalizes the well known formula for the Nielsen coincidence number on tori which was proved in Lemma 7.3 of [4]: if f 1 , f 2 are given by square matrices A 1 and A 2 , then N (f 1 , f 2 ) = \| det(A 2 − A 1 )\|. Theorem 4.4. If f 1 , . . . , f k : T (k−1)n → T n are maps on tori given by matrices A i , then N (f 1 , . . . , f k ) = det    A 2 − A 1 . . . A k − A 1    Proof. Let F, G : T (k−1)n → T (k−1)n be as in (⋆). Then F and G will be given by block matrices A F =    A 1 . . . A 1    , A G =    A 2 . . . A k    , and so the formula for the Nielsen coincidence number on tori gives N (F, G) = det    A 2 − A 1 . . . A k − A 1    But N (F, G) = N (f 1 , . . . , f k ), and so the result is proved. We further note that since tori have the Wecken property for coincidence theory, we can drop the dimension assumption of Theorem 3.5. Theorem 4.5. Let f 1 , . . . , f k : T (k−1)n → T n be maps of tori. Then ME(f 1 , . . . , f k ) = N (f 1 , . . . , f k ). Proof. Let (F, G) be as in (⋆), and then since tori have the Wecken property there is a map G ′′ ≃ G with # Coin(F, G ′′ ) = N (F, G). We finish the argument as in the last paragraph of the proof of Theorem 3.5. Coincidence theory with positive codimension We end with an application to coincidence theory with positive codimension, which typically requires much more difficult techniques than those of this paper. In this setting we consider maps f 1 , f 2 : X → Y of compact manifolds of dimensions m and n with m > n and try to minimize by homotopies the quantity #π 0 (Coin(f 1 , f 2 )), the number of path components of Coin(f 1 , f 2 ). There is no coincidence index in the positive codimension setting, and so the problem of judging essentiality of classes is more complicated. A coincidence class C ⊂ Coin(f 1 , f 2 ) is removable by homotopy when there is some pair of homotopies f i ≃ f ′ i such that C is "related" (in the sense of Theorem 3.3) to the empty class. When a class is not removable by homotopy, it is called geometrically essential. The number of geometrically essential classes is called the geometric Nielsen number, which we denote N G (f 1 , f 2 ). Any two coincidence points which can be connected by a path of coincidence points will be in the same coincidence class. Thus each class is a union of path components of Coin(f 1 , f 2 ), and so N G (f 1 , f 2 ) ≤ #π 0 (Coin(f 1 , f 2 )). Since N G (f 1 , f 2 ) is homotopy invariant, in fact it is a lower bound for the minimal number of path components of the coincidence set when f 1 and f 2 are changed by homotopies. We begin with a simple result which in some cases can demonstrate that a coincidence class is geometrically essential. Theorem 5.1. Let f 1 , . . . , f k : X → Y be maps of spaces of dimension (k − 1)n and n respectively, and let f i , f j be any two of these maps. Then each equalizer class of (f 1 , . . . , f k ) is a subset of some coincidence class of (f i , f j ), and any coincidence class containing an essential equalizer class is geometrically essential. Proof. To show that each equalizer class is a subset of a coincidence class, let C be an equalizer class. Then there are lifts f i and deck transformations α i with C = p X Eq(α i f 1 , α 2 f 2 , . . . , α k f k ) ⊂ p X Coin(α i f i , α j f j ), and the right side above is a coincidence class. Now let D ⊂ Coin(f i , f j ) be a coincidence class containing some essential equalizer class C ⊂ D. If D were removable by a homotopy as a coincidence class, then necessarily C would be removable by a homotopy as an equalizer class, which is impossible since C is essential. Thus D is geometrically essential. We can state the above in terms of Nielsen numbers: Corollary 5.2. Let f 1 , f 2 : X → Y be maps of spaces of dimension (k − 1)n and n respectively. If there are maps f 3 , . . . , f k with N (f 1 , . . . , f k ) = 0, then N G (f 1 , f 2 ) = 0. Proof. If N (f 1 , . . . , f k ) = 0 then there is an essential equalizer class of (f 1 , . . . , f k ), which by Theorem 5.1 is contained in a geometrically essential coincidence class of (f 1 , f 2 ). The existence of this coincidence class means that N G (f 1 , f 2 ) = 0. Now we focus on tori, for which we can be much more specific about the value of N G (f 1 , f 2 ). As we will see, Corollary 5.2 is strong enough to give a complete computation of N G (f 1 , f 2 ) based on the matrices which specify the maps, even in the case where the domain dimension is not a multiple of the codomain dimension. Many of the ideas in the following proofs are taken from similar proofs in the codimension zero coincidence theory on tori from [4]. Recall that a map f : T m → T n on tori has an induced homomorphism on the fundamental groups which is given by an n × m integer matrix. We will refer to this matrix as the "induced matrix" of f . Lemma 5.3. Let f 1 , f 2 : T (k−1)n → T n be maps of tori with induced matrices A 1 , A 2 . Then N G (f 1 , f 2 ) = 0 if and only if A 2 − A 1 has rank n. Proof. First we show that N G (f 1 , f 2 ) = 0 implies that A 2 − A 1 has rank n. Equivalently, we assume that A 2 − A 1 does not have rank n, and we will show N G (f 1 , f 2 ) = 0. In this case, the image of the matrix A 2 − A 1 is not R n . Take some v ∈ R n \ im(A 2 − A 1 ) with v ∈ Z n , and let A ′ 2 : R (k−1)n → R n be given by A ′ 2 (x) = A 2 (x) + v. This induces a map f ′ 2 : T (k−1)n → T n with f ′ 2 ≃ f 2 and Coin(f 1 , f ′ 2 ) = ∅. Thus N G (f 1 , f 2 ) = 0 as desired. When A 2 − A 1 does have rank n, we can choose matrices A 3 , . . . , A k so that det    A 2 − A 1 . . . A k − A 1    = 0, and so there are maps f 3 , . . . , f k with N (f 1 , . . . , f k ) = 0. By Corollary 5.2 this implies that N G (f 1 , f 2 ) = 0. In fact we can remove the requirement that the domain dimension is a multiple of the codomain dimension. Proof. The fact that N G (f 1 , f 2 ) = 0 implies A 2 − A 1 has rank n was a step of the proof in Lemma 5.3 which did not require the dimension assumption. Thus it remains to show that if A 2 − A 1 has rank n, then N G (f 1 , f 2 ) = 0. Let k > 2 be an integer with (k−1)n ≥ m. Then define g 1 , g 2 : T (k−1)n → T n as g i = f i • σ, where σ : T (k−1)n → T m is the projection onto the first m coordinates (viewing the torus as a product of circles). Let B i be the (k−1)n×n integer matrix representing g i . As a matrix, B i is simply A i with columns of zeros added, and so the rank of A 2 − A 1 is the same as that of B 2 − B 1 . Our assumption that A 2 −A 1 has rank n, means that B 2 −B 1 has rank n, and so by Lemma 5.3 we have N G (g 1 , g 2 ) = 0, and we have a geometrically essential coincidence class D ⊂ Coin(g 1 , g 2 ). Let f i be lifts of f i , and let g i = f i • σ, where σ is the projection onto the first m coordinates of T (k−1)n = R (k−1)n . Then g i is a lift of g i , and so there is some α ∈ π 1 (T n ) with D = p Coin( g 1 , α g 2 ). Let x ∈ σ(D), so there is some y with σ(y) = x and a lift y = p −1 (y) with g 1 ( y) = α g 2 ( y), and thus f 1 ( σ( y)) = α f 2 ( σ( y)). Since p( σ( y)) = σ(y) = x, we have x ∈ p Coin( f 1 , α f 2 ). This set C = p Coin( f 1 , α f 2 ) is a coincidence class of (f 1 , f 2 ), and we have shown that σ(D) ⊂ C. Recall that we are trying to show that N G (f 1 , f 2 ) = 0. For the sake of a contradiction, assume that N G (f 1 , f 2 ) = 0, so that each class (in particular the class C) is removable by homotopy. This means means there are maps f ′ i ≃ f i with lifts f ′ i ≃ f i such that p Coin( f ′ 1 , α f ′ 2 ) = ∅.(1) Let g ′ i = f ′ i • σ and g ′ i = f ′ i • σ. Then g ′ i ≃ g i and g ′ i ≃ g i . Since D is geometrically essential, the related class p Coin( g ′ 1 , α g ′ 2 ) must be nonempty. Take some y in this class and a point y ∈ p −1 (y) with g ′ 1 ( y) = α g ′ 2 ( y). Then we have f ′ 1 ( σ( y)) = α f ′ 2 ( σ( y)), and so σ(y) ∈ p Coin( f ′ 1 , α f ′ 2 ), which contradicts (1). The above showed that if A 2 −A 1 has does not have rank n, then N G (f 1 , f 2 ) = 0. We conclude with a precise computation of N G (f 1 , f 2 ) in the case where A 2 − A 1 does have rank n. Theorem 5.5. Let f 1 , f 2 : T m → T n be maps of tori with induced matrices A 1 , A 2 . If A 2 − A 1 has rank n, then N G (f 1 , f 2 ) = #π 0 (Coin(f 1 , f 2 )) = # coker(A 2 − A 1 ), where coker(A 2 − A 1 ) = Z n / im(A 2 − A 1 ), the cokernel of A 2 − A 1 when viewed as a homomorphism Z m → Z n . Proof. By Theorem 5.4, the assumption that A 2 − A 1 has rank n is equivalent to N G (f 1 , f 2 ) = 0. Since T n is a Jiang space, this means that N G (f 1 , f 2 ) = R(f 1 , f 2 ), the number of coincidence classes. (See [3], Theorem 4.14, for results concerning coincidence theory of maps into Jiang spaces with positive codimension.) We prove the theorem by showing that the number of classes is equal to the number of path components of Coin(f 1 , f 2 ), and also equal to the order of the quotient group Z n / im(A 2 − A 1 ). First we establish the equality for the number of path components of Coin(f 1 , f 2 ). Each coincidence class is a union of path components, and so it suffices to show that each class is connected. Given some class C, we have C = p Coin(A 1 , αA 2 ), where α is a covering transformation α : R n → R n . The covering transformations are all given by addition by elements of Z n . Thus C is the projection of the set C = {x ∈ R (k−1)n \| A 1 x = A 2 x + v} for some v ∈ Z n . Since A 1 and A 2 are linear, C will be connected, and thus p( C) = C is connected. Now we show that R(f 1 , f 2 ) is equal to the order of Z n / im(A 2 − A 1 ). We compute R(f 1 , f 2 ) as the number of Reidemeister classes in π 1 (T n ). Two elements v, w ∈ π 1 (T n ) ∼ = Z n are in the same Reidemeister class when there is some z ∈ π 1 (T m ) with v = A 1 z + w − A 2 z. That is to say, v, w ∈ Z n have the same Reidemeister class when w−v ∈ im(A 2 − A 1 ). Thus the number of Reidemeister classes is the order of Z n / im(A 2 − A 1 ), as desired. Theorems 5.4 and 5.5 together give a full computation of the Nielsen coincidence number for tori. This is a direct generalization of the formula from coincidence theory given in Lemma 7.3 of [4] (recall that for square integral matrices we have # coker( A 2 − A 1 ) = \| det(A 2 − A 1 )\|). To illustrate the above, we compute the geometric Nielsen coincidence numbers for the maps f, g, h : T 2 → S 1 from Example 1.2. Example 1.2, Continued. Recall our maps were given by matrices: A f = (3 1), A g = (0 2), A h = (−1 − 1). For each pair of matrices the rank assumption of Theorem 5.5 holds. It is straightforward to compute the required cokernels. We have A g − A f = (3 − 1), and so im(A g − A f ) = Z, since gcd(3, −1) = 1. Thus the cokernel is trivial and so N G (f, g) = 1. A similar computation shows that N G (g, h) = 1. For (f, h), we have A h − A f = (2 − 2), and so im(A h − A f ) = 2Z. Thus the cokernel is Z/2Z, and so N G (f, h) = 2. By Theorem 5.5 these Nielsen numbers should agree with the number of path components of the coincidence sets. Counting components in Figure 1 indeed gives #π 0 (Coin(f, g)) = #π 0 (Coin(g, h)) = 1 and #π 0 (Coin(f, h)) = 2. Introduction The goal of this paper is to generalize the basic definitions and results of Nielsen coincidence theory to a theory of equalizer sets for sets of (possibly more than two) mappings. For spaces X and Y and maps f 1 , . . . , f k : X → Y , the equalizer set is defined as Eq(f 1 , . . . , f k ) = {x ∈ X \| f 1 (x) = · · · = f k (x)}. This generalizes the coincidence set Coin(f 1 , f 2 ) = {x ∈ X \| f 1 (x) = f 2 (x)} for two mappings. Nielsen coincidence theory, see [4], estimates the number of coincidence points of a pair of maps in a homotopy invariant way. Most of the techniques are a generalization of ideas from fixed point theory, see [7]. In coincidence theory, one defines the Nielsen number N (f 1 , f 2 ) of a pair of maps, which is a lower bound for the minimal coincidence number MC (f 1 , f 2 ): The above quantities are in fact equal when X and Y are compact n-manifolds of the same dimension n = 2. In this paper we extend this theory to equalizer sets. The typical setting for Nielsen coincidence theory is for maps X → Y of compact manifolds of the same dimension. For maps f 1 , f 2 : X → Y in this setting, transversality arguments show that we can change the maps by homotopy so that Coin(f 1 , f 2 ) is a set of finitely many points. At each of these points we define a coincidence index which is then used to define the Nielsen number. In the case of differentiable manifolds we can define the index in terms of the determinant of the derivative maps at each coincidence point (see [8] for this approach). N (f 1 , f 2 ) ≤ MC (f 1 , f 2 ) = min{# Coin(f ′ 1 , f ′ 2 ) \| f ′ i ≃ f i }. When the dimension of X is greater than that of Y , the typical approach to the coincidence index breaks down. In this case the derivative maps cannot be linear isomorphisms, and so their determinants cannot be used. A modified approach based on determinants is given by Jezierski in [6] which applies for maps into tori, but this is a fairly restrictive setting. In this paper we will show that the typical approach, expressible in terms of determinants, does indeed succeed in the positive codimension setting when we admit more mappings to our theory, i.e. when we move from coincidence theory to equalizer theory. In this sense equalizer theory would seem to be the most natural and straightforward Nielsen-type theory in positive codimensions. Compared to the various approaches to positive codimension Nielsen coincidence theory (many are surveyed in [4]), our equalizer theory is substantially simpler and much more closely resembles classical Nielsen fixed point and coincidence theory. Nielsen equalizer theory will require a specific codimensional setting. Attempting a homotopy-invariant study of equalizers in codimension zero immediately gives: Theorem 1.1. If X and Y are compact manifolds of the same dimension, and f 1 , . . . , f k : X → Y are maps with k > 2, then these maps can be changed by homotopy so that the equalizer set is empty. Proof. Well-known transversality arguments show that we can change f 2 by a homotopy to f ′ 2 so that Coin(f 1 , f ′ 2 ) is a finite set of points. Similarly we obtain f ′ 3 ≃ f 3 such that Coin(f 1 , f ′ 3 ) is a finite set of points. These homotopies can be arranged so that Coin(f 1 , f ′ 2 ) and Coin(f 1 , f ′ 3 ) are disjoint. Thus Eq(f 1 , f ′ 2 , f ′ 3 , f 4 , . . . , f k ) ⊂ Coin(f 1 , f ′ 2 ) ∩ Coin(f 1 , f ′ 3 ) = ∅. Thus there is no interesting theory for counting the minimal number of equalizer points between compact manifolds of the same dimension, since this number is always zero. In this sense, the equalizer equation f 1 (x) = · · · = f k (x) is "overdetermined" when the dimensions of the domain and codomain are equal. In order to obtain an interesting theory we must increase the dimension of the domain space. In particular, for equalizers of k maps, we will require X and Y to be of dimensions (k − 1)n and n, respectively, for any n. Consider the following example: / / / / / C f g : C gh : Example 1.2. We will examine the equalizer set of three maps f, g, h : T 2 → S 1 from the 2-dimensional torus to the circle. Viewing the torus as the quotient of R 2 by the integer lattice, and S 1 as the quotient of R by the integers, we will specify our maps by integer matrices of size 1 × 2. Let the maps be given by matrices: C f h : _ _ _A f = (3 1), A g = (0 2), A h = (−1 − 1) . Let C f g = Coin(f, g) , with C f h and C gh defined similarly, and we have Eq(f, g, h) = C f g ∩ C gh ∩ C f h . (Actually the equalizer set is the intersection of any two of these coincidence sets.) It is straightforward to compute these sets. For example, C f g is the set of points (x, y) with 3x + y = 2y mod Z 2 , which is to say y = 3x mod Z 2 . Similarly computing the sets C f h and C gh produces the picture in Figure 1, where the torus is drawn as [0, 1] × [0, 1] with opposite sides identified. We see in the picture that Eq(f, g, h) consists of 10 points (the nine points where the lines visibly intersect, plus the intersection at the identified corners of the diagram). In this paper we will define the Nielsen number N (f, g, h) which is a lower bound for the minimum number of equalizer points when the maps are changed by homotopy. In Theorem 4.4 we give a simple formula for computing this quantity on tori, which in this example gives N (f, g, h) = 0 2 −1 −1 − 3 1 3 1 = 10. Thus these maps cannot be changed by homotopy to have fewer than 10 equalizer points. The construction of the theory is facilitated by a fundamental correspondence between Eq(f 1 , . . . , f k ) and the coincidence set of a pair of related maps. Let F, G : X → Y k−1 be given by (f 1 (x), . . . , f 1 (x)), G(x) = (f 2 (x), . . . , f k (x)). F (x) = (⋆) Since X and Y are compact with dimensions (k − 1)n and n respectively, the above F and G are maps between compact manifolds of the same dimension, and Coin(F, G) = Eq(f 1 , . . . , f k ). This correspondence is well-behaved under homotopy, since changing the maps f i by homotopies corresponds in a natural way to a change of F and G by homotopies. As we shall see, the homotopyinvariant behavior of Eq(f 1 , . . . , f k ) is the same as that of Coin(F, G), and we may define Nielsen-type invariants for the equalizer set in terms of the same invariants from the coincidence theory of (F, G). In Section 2 we define the Reidemeister and equalizer classes which form the building blocks for our theory. In Section 3 we define the Nielsen number and in Section 4 we give some computational results for maps into Jiang spaces and maps of tori. In Section 5 we give an application to Nielsen coincidence theory in positive codimensions, giving a full computation of the "geometric Nielsen number" on tori. We would like to thank Robert F. Brown for helpful comments, and Philip Heath for bringing the reference [6] to our attention. Reidemeister and equalizer classes Let X and Y be spaces with universal covering spaces (connected, locally pathconnected, and semilocally simply connected), and let X and Y be the universal covering spaces with projection maps p X : X → X and p Y : Y → Y . For maps f 1 , . . . , f k : X → Y , we wish to construct a Reidemeister-type theory for the equalizer points Eq(f 1 , . . . , f k ), so that each point has an algebraic Reidemeister class, and two equalizer points can be combined by homotopy only when their classes are equal. Our basic result is a generalization of a well-known result from coincidence theory which is stated in part (without proof) as Lemma 2.3 of [2]. For the sake of completeness we give a full proof. The proof is similar to that of Theorem 1.5 in [7], which is the corresponding statement in fixed point theory. Throughout, elements of the fundamental group are viewed as deck transformations on the universal covering space. Let f 1 , . . . , f k : X → Y be maps with lifts f i : X → Y and induced homomorphisms φ i : π 1 (X) → π 1 (Y ). Theorem 2.1. We have Eq(f 1 , . . . , f k ) = α2,...,α k ∈π1(Y ) p X Eq( f 1 , α 2 f 2 , . . . , α k f k ). 2. For α i , β i ∈ π 1 (X), the sets p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) and p X Eq( f 1 , β 2 f 2 , . . . , β k f k ) are disjoint or equal. The above sets are equal if and only if there is some z ∈ π 1 (X) with β i = φ 1 (z)α i φ i (z) −1 for all i. Proof. For the first statement, take some x ∈ Eq(f 1 , . . . , f k ) and some x ∈ p −1 X (x). We have p Y ( f i ( x)) = f i (x) = f 1 (x) for all i, and thus the values f i ( x) all differ by deck transformations. That is, there are α i ∈ π 1 (Y ) with f 1 ( x) = α 2 f 2 ( x) = · · · = α k f k ( x), which is to say that x ∈ Eq( f 1 , α 2 f 2 , . . . , α k f k ), and so x ∈ p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) as desired. Now we prove statement 3. First, let us assume that p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) = p X Eq( f 1 , β 2 f 2 , . . . , β k f k ). This means that for any point x ∈ Eq( f 1 , α 2 f 2 , . . . , α k f k ), there is some deck transformation z ∈ π 1 (X) with z x ∈ Eq( f 1 , β 2 f 2 , . . . , β k f k ). Then we have β i f i (z x) = f 1 (z x) = φ 1 (z) f 1 ( x) = φ 1 (z)α i f i ( x) = φ 1 (z)α i φ i (z) −1 f i (z x) Since the two lifts β i f i and φ 1 (z)α i φ i (z) −1 f i agree at a point, they are the same lift, and thus β i = φ 1 (z)α i φ i (z) −1 as desired. For the converse in statement 3, assume that β i = φ 1 (z)α i φ i (z) −1 for all i, and take x ∈ p X Eq( f 1 , α 2 f 2 , . . . , α k f k ). Then we have φ 1 (z)α i = β i φ i (z) for all i, and so f 1 (z x) = φ 1 (z) f 1 ( x) = φ 1 (z)α i f i ( x) = β i φ i (z) f i ( x) = β i f i (z x). Thus z x = Eq( f 1 , β 2 f 2 , . . . , β k f k ), and so x ∈ p X Eq( f 1 , β 2 f 2 , . . . , β k f k ), and we have shown p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) ⊂ p X Eq( f 1 , β 2 f 2 , . . . , β k f k ). A symmetric argument shows the converse inclusion, and so the above sets are equal. For statement 2, it suffices to show that if there is a point x ∈ p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) ∩ p X Eq( f 1 , β 2 f 2 , . . . , β k f k ), then the two sets of the above intersection are equal. For such a point x, there are x 0 , x 1 ∈ p −1 X (x) with x 0 ∈ Eq( f 1 , α 2 f 2 , . . . , α k f k ), x 1 ∈ Eq( f 1 , β 2 f 2 , . . . , β k f k ). Let z ∈ π 1 (X) with z x 0 = x 1 . Then we have β i f i (z x 0 ) = β i f i ( x 1 ) = f 1 ( x 1 ) = f 1 (z x 0 ) = φ 1 (z) f 1 ( x 0 ) = φ 1 (z)α i f i ( x 0 ) = φ 1 (z)α i φ i (z) −1 f i (z x 0 ). The above equality shows two lifts of f i agreeing at the point z x 0 , and so we have β i = φ 1 (z)α i φ i (z) −1 , which by statement 3 implies that p X Eq( f 1 , α 2 f 2 , . . . , α k f k ) = p X Eq( f 1 , β 2 f 2 , . . . , β k f k ) as desired. Let R(φ 1 , . . . , φ k ) = π 1 (Y ) k−1 / ∼ be the quotient of π 1 (Y ) k−1 by the following relation, inspired by statement 3 above: (α 2 , . . . , α k ) ∼ (β 2 , . . . , β k ) if and only if there is some z ∈ π 1 (X) with β i = φ 1 (z)α i φ i (z) −1 for all i ∈ {2, . . . , k}. We call R(φ 1 , . . . , φ k ) the set of Reidemeister classes for φ 1 , . . . , φ k . Then the theorem above gives the following disjoint union Eq(f 1 , . . . , f k ) = (αi)∈R(φ1,...,φ k ) p X Eq( f 1 , α 2 f 2 , . . . , α k f k ). The above union partitions the equalizer set into Nielsen equalizer classes (or simply equalizer classes). That is, C ⊂ Eq(f 1 , . . . , f k ) is an equalizer class if and only if there are α i with C = p X Eq( f 1 , α 2 f 2 , . . . , α k f k ). Note that an equalizer class can be empty. The equalizer classes are related to the coincidence classes of the pair (F, G) from equation (⋆) in the following way: Theorem 2.2. A subset C ⊂ Eq(f 1 , . . . , f k ) is an equalizer class if and only if C is a coincidence class when regarded as a subset of Coin(F, G). That is, C is an equalizer class if and only if there is a deck transformation A ∈ π 1 (Y k−1 ) with C = p X Coin( F , A G) for some lifts F and G of F and G. Proof. First we assume that C is an equalizer class, and so we have lifts f i of f i and α i ∈ π 1 (Y ) with C = p X Eq( f 1 , α 2 f 2 , . . . , α k f k ). Let F and G be given by F ( x) = ( f 1 ( x), . . . f 1 ( x)), G( x) = ( f 2 ( x), . . . , f k ( x)), and let A : Y k−1 → Y k−1 be A( y 2 , . . . y k ) = (α 2 y 2 , . . . , α k y k ). Then we have Coin( F , A G), and so C = p X Coin( F , A G) as desired. Eq( f 1 , α 1 f 2 , . . . , α k f k ) = Now for the converse we assume that C is a coincidence class of (F, G), which means there are lifts F and G of F and G with a deck transformation A ∈ π 1 (Y k−1 ) such that C = p X Coin( F , A G). Since F and G are lifts of F and G, we can write F ( x) = ( f 2 1 ( x), . . . , f k 1 ( x)), G( x) = ( f 2 ( x), . . . , f k ( x)) where each f i 1 is a lift of f 1 , and f j is a lift of f j for j ≥ 2. Similarly we may factor A as A = α 1 × · · · × α k for α i ∈ π 1 (Y ). Each of the f i 1 may be different, but there is a single lift f 1 of f 1 with deck transformations β i such that β i f 1 = f i 1 . Then we have Coin( F , A G) = Coin((β 2 f 1 , . . . , β k f 1 ), (α 2 f 2 , . . . , α k f k )) = Coin(( f 1 , . . . , f 1 ), (β −1 2 α 2 f 2 , . . . , β −1 k α k f k )) = Eq( f 1 , β −1 2 α 2 f 2 , . . . , β −1 k α k f k ) and so C = p X Coin( F , A G) is an equalizer class. The equalizer classes can be described nicely in terms of paths in X and their images under the f i : Proof. Our points x, x ′ are in the same equalizer class if and only if they are in the same coincidence class of the pair (F, G). A standard result in coincidence theory shows that this is equivalent to the existence of a path γ in X from x to x ′ with F (γ) ≃ G(γ). This is equivalent to (f 1 , . . . , f 1 )(γ) ≃ (f 2 , . . . , f k )(γ), which is equivalent to f 1 (γ) ≃ f i (γ) for each i. The equalizer index and the Nielsen number Let Eq(f 1 , . . . , f k , U ) = Eq(f 1 , . . . , f k ) ∩ U , and let Coin(f, g, U ) = Coin(f, g) ∩ U . Our index for equalizer sets will be defined in terms of the coincidence index i. We first review some properties of the coincidence index. Let f, g : M → N be maps between compact orientable manifolds of the same dimension. The coincidence index i(f, g, U ) is an integer valued function defined for open sets U with Coin(f, g, U ) compact. It satisfies the following properties: • Homotopy: Let f ′ ≃ f and g ′ ≃ g, by homotopies F t and G t , such that the set {(x, t) \| x ∈ Coin(F t , G t , U )} ⊂ M × [0, 1] is compact (such a pair of homotopies is called admissible). Then i(f, g, U ) = i(f ′ , g ′ , U ). • Additivity: If U 1 ∩ U 2 = ∅ and Coin(f, g, U ) ⊂ U 1 ∪ U 2 , then i(f, g, U ) = i(f, g, U 1 ) + i(f, g, U 2 ). • Solution: If i(f, g, U ) = 0, then Coin(f, g, U ) is not empty. We wish to define a similar index in the equalizer setting. Let X and Y be compact orientable manifolds of dimensions (k − 1)n and n, respectively, with maps f 1 , . . . , f k : X → Y . We call (f 1 , . . . , f k , U ) admissible when Eq(f 1 , . . . , f k , U ) is compact. Let F, G : X → Y k−1 be the maps as in (⋆). These are maps between compact orientable manifolds of the same dimension. When (f 1 , . . . , f k , U ) is admissible, then Coin(F, G, U ) = Eq(f 1 , . . . , f k , U ) is compact, and thus the coincidence index i(F, G, U ) is defined. We define the equalizer index ind(f 1 , . . . , f k , U ) to be i(F, G, U ). This equalizer index satisfies the appropriate homotopy, additivity, and solution properties. If (f 1 , . . . , f k , U ) and (f ′ 1 , . . . , f ′ k , U ) are admissable and f i ≃ f ′ i with homotopy H i , we say that (H i ) is an admissible homotopy of (f 1 , . . . , f k , U ) to (f ′ 1 , . . . , f ′ k , U ) when the set . . , f k , U ) is defined and satisfies the following properties: . . . , f k , U 2 ). {(x, t) \| x ∈ Eq(H 1 t , . . . , H k t , U )} ⊂ X × I is compact.• Homotopy: If (f 1 , . . . , k k , U ) is admissibly homotopic to (f ′ 1 , . . . , f ′ k , U ), then ind(f 1 , . . . , f k , U ) = ind(f ′ 1 , . . . , f ′ k , U ). • Additivity: If U 1 ∩ U 2 = ∅ and Eq(f 1 , . . . , f k , U ) ⊂ U 1 ∪ U 2 , then ind(f 1 , . . . , f k , U ) = ind(f 1 , . . . , f k , U 1 ) + ind(f 1 , • Solution : If ind(f 1 , . . . , f k , U ) = 0, then Eq(f 1 , . . . , f k , U ) is not empty. Proof. The proofs of these properties all follow from the same properties of the coincidence index of the pair F, G as in (⋆). For an equalizer class C, we define the index of C, written ind(f 1 , . . . , f k , C), as ind(F, G, U ), where U is an open set with Coin(F, G, U ) = C (such an open set will always exist because coincidence classes are closed and X is compact). At this point we take a slight diversion to give a note on the computation of the index of differentiable maps in terms of their derivatives. When X and Y are differentiable manifolds and each of f i is differentiable, the maps F and G will also be differentiable, and the derivative maps DF x , DG x : R (k−1)n → R (k−1)n are defined at each point x ∈ X. Let x ∈ Eq(f 1 , . . . , f k ) be an equalizer point. We say that x is nondegenerate when DG x − DF x is nonsingular. In this case there is a neighborhood U around x containing no other coincidence points of F and G, and thus no other equalizer points, and the index can be computed by the well-known formula from coincidence theory: ind(f 1 , . . . , f k , U ) = i(F, G, U ) = sign det(DG x − DF x ). The definitions of F and G give the following formula in terms of the f i . df 2 − df 1 . . . df k − df 1    where all derivatives are taken at the point x (each row in the above is an n × (k − 1)n block matrix, so that the whole matrix has size (k − 1)n × (k − 1)n). Now we discuss the index theory for the nonorientable case. For the coincidence theory of maps f, g : M → N of compact (perhaps nonorientable) manifolds of the same dimension, an integer-valued coincidence index cannot in general be defined. There is a related semi-index (see [2]) which plays a similar role. The semi-index, which we denote \|i\|, is defined not for arbitrary open sets, but only for coincidence classes, and satisfies properties similar to those of the coincidence index. Let C ⊂ Coin(f, g) be a coincidence class with C = p Coin( f , α g). Then if f ≃ f ′ and g ≃ g ′ , these homotopies will lift, producing maps f ′ ≃ f and g ′ ≃ g which are lifts of f ′ and g ′ respectively. Thus D = p Coin( f ′ , α g ′ ) is a coincidence class of (f ′ , g ′ ), and we say that D is "related to C" with respect to the pair of homotopies. If f, g : M → N are maps of compact manifolds of the same dimension and C is a (possibly empty) coincidence class, then \|i\|(f, g, C) is defined and satisfies: • Homotopy: If f ′ ≃ f and g ′ ≃ g, and D is the coincidence class of (f ′ , g ′ ) which is related to C with respect to these homotopies, then \|i\|(f, g, C) = \|i\|(f ′ , g ′ , D). • Solution: If \|i\|(f, g, C) = 0, then C is not empty. • Naturality: If M and N are orientable, then \|i\|(f, g, C) = \|i(f, g, C)\|, the absolute value of the usual coincidence index. In the setting of equalizer theory for maps f 1 , . . . , f k : X → Y of compact (possibly nonorientable) manifolds with an equalizer class C, we define the equalizer semi-index as in the orientable case: let (F, G) be as in (⋆), and we define \| ind \|(f 1 , . . . , f k , C) = \|i\|(F, G, C). Given homotopies f ′ i ≃ f i , the "relation" between equalizer classes of (f 1 , . . . , f k ) and (f ′ 1 , . . . , f ′ k ) is defined exactly as in coincidence theory. The following has routine proofs similar to those for Theorem 3.1. . . , f k , C) is defined and satisfies the following properties: • Homotopy: If f i is homotopic to f ′ i for each i and D is the equalizer class of (f ′ 1 , . . . , f ′ k ) which is related to C, then \| ind \|(f 1 , . . . , f k , C) = \| ind \|(f ′ 1 , . . . , f ′ k , D). • Solution: If \| ind \|(f 1 , . . . , f k , C) = 0, then C is not empty. An equalizer class is called essential if its index (or semi-index in the nonorientable case) is nonzero. Definition 3.4. The Nielsen [equalizer] number N (f 1 , . . . , f k ) is defined to be the number of essential equalizer classes of (f 1 , . . . , f k ). From Theorem 2.2 and the definition of the index of a class, we see that N (f 1 , . . . , f k ) is equal to the Nielsen coincidence number of the pair (F, G). Since the Nielsen equalizer number is so closely related to a coincidence number, we can obtain a Wecken-type theorem for the minimal number of equalizer points. Let ME (f 1 , . . . , f k ) be the minimal number of equalizer points, defined as ME (f 1 , . . . , f k ) = min{# Eq(f ′ 1 , . . . , f ′ k ) \| f ′ i ≃ f i }. By the solution properties of the index and semi-index, every essential equalizer class must contain an equalizer point, and so N (f 1 , . . . , f k ) ≤ ME (f 1 , . . . , f k ). These two quantities are in fact equal in most cases, as the following theorem shows. Theorem 3.5. Let f 1 , . . . , f k : X → Y be maps of compact manifolds of dimensions (k − 1)n and n respectively. If (k − 1)n = 2, then ME(f 1 , . . . , f k ) = N (f 1 , . . . , f k ). In the case of "proper" equalizer theory (when k > 2), the result holds for all k and n except (k, n) = (3, 1), which is to say equalizer theory of three maps from a compact surface to the circle. Proof. The second statement is simply a consequence of k, n being natural numbers with (k − 1)n = 2, so we focus on the first statement. Let (F, G) be defined as in (⋆), and we have N (f 1 , . . . , f k ) = N (F, G). The maps F, G are maps between compact manifolds of dimension (k − 1)n. By our hypothesis this dimension is not 2, and so the Wecken theorem for coincidences (see [4]) gives maps F ′ ≃ F and G ′ ≃ G with # Coin(F ′ , G ′ ) = N (F, G). A result of Brooks in [1] shows that in fact there is a single map G ′′ ≃ G with Coin(F ′ , G ′ ) = Coin(F, G ′′ ), and thus # Coin(F, G ′′ ) = N (F, G). Our map G ′′ is a map of X → Y k−1 , so it can be written as G ′′ (x) = (g 2 (x), . . . , g k (x)) with g i ≃ f i . Now we have # Eq(f 1 , g 2 , . . . , g k ) = # Coin(F, G ′′ ) = N (F, G) = N (f 1 , . . . , f k ), and so ME (f 1 , . . . , f k ) ≤ N (f 1 , . . . , f k ) as desired. 4 Some computations 4 .1 Jiang spaces One setting in which the fixed point and coincidence Nielsen numbers are easily calculated is for maps on Jiang spaces. See [7] for the definition and basic results in fixed point theory. The class of Jiang spaces includes topological groups, generalized lens spaces and certain other homogeneous spaces, and is closed under products. The main result (see [4]) from coincidence theory concerning Jiang spaces is the following: Our theorem concerning Jiang spaces is the following result, which is facilitated by the coincidence theory of the maps (F, G) as in (⋆). If f 1 , . . . , f k : X → Y are maps of compact orientable manifolds of dimensions (k − 1)n and n respectively and Y is a Jiang space, then every equalizer class has the same index. Proof. Let F, G : X → Y k−1 be given as in (⋆): F (x) = (f 1 (x), . . . f 1 (x)), G(x) = (f 2 (x), . . . , f k (x)). Since Y is a Jiang space, then Y k−1 is a Jiang space. Thus by Theorem 4.1 all coincidence classes of F, G will have the same coincidence index. But the equalizer classes of f 1 , . . . , f k are the same as the coincidence classes of F, G, with the same indices, so all equalizer classes of f 1 , . . . , f k will have the same equalizer index. Define the Reidemeister number and Lefschetz number as: R(f 1 , . . . , f k ) = #R(φ 1 , . . . , φ k ) (this quantity may be infinite) and L(f 1 , . . . , f k ) = ind(f 1 , . . . , f k , X). Then we obtain: Tori We can give a very specific formula for the Nielsen number of maps f 1 , . . . , f k : T (k−1)n → T n on tori. We will view T m as the quotient of R m by the integer lattice, and consider maps which are induced by linear maps on R (k−1)n → R n taking Z (k−1)n to Z n . We can think of such a map as an n × (k − 1)n matrix with integer entries, which we call the "induced matrix". We now prove the formula which was used in the computation of Example 1.2. The result generalizes the well-known formula for the Nielsen coincidence number on tori which was proved in Lemma 7.3 of [5]: if f 1 , f 2 are given by square matrices A 1 and A 2 , then N (f 1 , f 2 ) = \| det(A 2 − A 1 )\|. . . , f k : T (k−1)n → T n are maps on tori with induced matrices A i , then N (f 1 , . . . , f k ) = det    A 2 − A 1 . . . A k − A 1    Proof. Let F, G : T (k−1)n → T (k−1)n be as in (⋆). Then the induced matrices of F and G will be given by block matrices A F =    A 1 . . . A 1    , A G =    A 2 . . . A k    , and so the formula for the Nielsen coincidence number on tori gives . . . , f k ), and so the result is proved. N (F, G) = det    A 2 − A 1 . . . A k − A 1    But N (F, G) = N (f 1 , We further note that since tori have the Wecken property for coincidence theory, we can drop the dimension assumption of Theorem 3.5. Proof. Let (F, G) be as in (⋆), and then since tori have the Wecken property there is a map G ′′ ≃ G with # Coin(F, G ′′ ) = N (F, G). We finish the argument as in the last paragraph of the proof of Theorem 3.5. Coincidence theory with positive codimension We end with an application to coincidence theory with positive codimension, which typically requires much more difficult techniques than those of this paper. In this setting we consider maps f 1 , f 2 : X → Y of compact manifolds of dimensions m and n with m > n and try to minimize by homotopies the quantity #π 0 (Coin(f 1 , f 2 )), the number of path components of Coin(f 1 , f 2 ). There is no coincidence index in the positive codimension setting, and so the problem of judging essentiality of classes is more complicated. A coincidence class C ⊂ Coin(f 1 , f 2 ) is removable by homotopy when there is some pair of homotopies f i ≃ f ′ i such that C is "related" (in the sense of Theorem 3.3) to the empty class. When a class is not removable by homotopy, it is called geometrically essential. The number of geometrically essential classes is called the geometric Nielsen number, which we denote N G (f 1 , f 2 ). Any two coincidence points which can be connected by a path of coincidence points will be in the same coincidence class. Thus each class is a union of path components of Coin(f 1 , f 2 ), and so N G (f 1 , f 2 ) ≤ #π 0 (Coin(f 1 , f 2 )). Since N G (f 1 , f 2 ) is homotopy invariant, in fact it is a lower bound for the minimal number of path components of the coincidence set when f 1 and f 2 are changed by homotopies. We begin with a simple result which in some cases can demonstrate that a coincidence class is geometrically essential. Let f 1 , . . . , f k : X → Y be maps of spaces of dimension (k − 1)n and n respectively, and let f i , f j be any two of these maps. Then each equalizer class of (f 1 , . . . , f k ) is a subset of some coincidence class of (f i , f j ), and any coincidence class containing an essential equalizer class is geometrically essential. Proof. To show that each equalizer class is a subset of a coincidence class, let C be an equalizer class. Then there are lifts f i and deck transformations α i with C = p X Eq(α 1 f 1 , α 2 f 2 , . . . , α k f k ) ⊂ p X Coin(α i f i , α j f j ), and the right side above is a coincidence class. Now let D ⊂ Coin(f i , f j ) be a coincidence class containing some essential equalizer class C ⊂ D. If D were removable by a homotopy as a coincidence class, then necessarily C would be removable by a homotopy as an equalizer class, which is impossible since C is essential. Thus D is geometrically essential. We can state the above in terms of Nielsen numbers: Proof. If N (f 1 , . . . , f k ) = 0 then there is an essential equalizer class of (f 1 , . . . , f k ), which by Theorem 5.1 is contained in a geometrically essential coincidence class of (f 1 , f 2 ). The existence of this coincidence class means that N G (f 1 , f 2 ) = 0. Now we focus on tori, for which we can be much more specific about the value of N G (f 1 , f 2 ). As we will see, Corollary 5.2 is strong enough to give a complete computation of N G (f 1 , f 2 ) based on the matrices which specify the maps, even in the case where the domain dimension is not a multiple of the codomain dimension. Theorem 5.4 below, computing the value of N G (f 1 , f 2 ) on tori, is proved by Jezierski in [6]. Most of the argument follows exactly the proof in the codimension zero case given in [5]. The key novel step in the positive codimension setting is the following lemma. Lemma 5.3. Let f 1 , f 2 : T m → T n be maps of tori with induced matrices A 1 , A 2 . If A 2 − A 1 has rank n, then N G (f 1 , f 2 ) = 0. Jezierski proves this by using the fact that, for m > n, we can consider T n as a subspace of T m and then note that the restrictions of f 1 , f 2 will have a nonremovable coincidence class. Jezierski's approach effectively decreases the domain dimension in order to apply the classical codimension zero theory. We take the opposite approach of increasing the domain dimension by taking a product with circles and introducing additional maps f 3 , . . . , f k which allow us to apply Corollary 5.2. While Jezierski's approach is simpler for this particular argument, it relies strongly on the fact that there is a standard embedding of T n inside T m . Since we do not use this fact, we hope that our strategy may be useful in other settings. Proof of Lemma 5.3. First we consider the case where m = (k − 1)n for some k. In this case we can choose matrices A 3 , . . . , A k so that det    A 2 − A 1 . . . A k − A 1    = 0, and so there are maps f 3 , . . . , f k with N (f 1 , . . . , f k ) = 0. By Corollary 5.2 this implies that N G (f 1 , f 2 ) = 0. For general m, let k > 2 be an integer with (k − 1)n ≥ m. Then define g 1 , g 2 : T (k−1)n → T n as g i = f i • σ, where σ : T (k−1)n → T m is the projection onto the first m coordinates (viewing the torus as a product of circles). Let B i be the (k − 1)n × n integer matrix representing g i . As a matrix, B i is simply A i with columns of zeros added, and so the rank of A 2 − A 1 is the same as that of B 2 − B 1 . Our assumption that A 2 −A 1 has rank n means that B 2 −B 1 has rank n, and so by our first case we have N G (g 1 , g 2 ) = 0, and we have a geometrically essential coincidence class D ⊂ Coin(g 1 , g 2 ). Let f i be lifts of f i , and let g i = f i • σ, where σ is the projection onto the first m coordinates of T (k−1)n = R (k−1)n . Then g i is a lift of g i , and so there is some α ∈ π 1 (T n ) with D = p Coin( g 1 , α g 2 ). Let x ∈ σ(D), so there is some y with σ(y) = x and a lift y = p −1 (y) with g 1 ( y) = α g 2 ( y), and thus f 1 ( σ( y)) = α f 2 ( σ( y)). Since p( σ( y)) = σ(y) = x, we have x ∈ p Coin( f 1 , α f 2 ). This set C = p Coin( f 1 , α f 2 ) is a coincidence class of (f 1 , f 2 ), and we have shown that σ(D) ⊂ C. Recall that we are trying to show that N G (f 1 , f 2 ) = 0. For the sake of a contradiction, assume that N G (f 1 , f 2 ) = 0, so that each class (in particular the class C) is removable by homotopy. This means there are maps f ′ i ≃ f i with lifts f ′ i ≃ f i such that p Coin( f ′ 1 , α f ′ 2 ) = ∅.(1) Let g ′ i = f ′ i • σ and g ′ i = f ′ i • σ. Then g ′ i ≃ g i and g ′ i ≃ g i . Since D is geometrically essential, the related class p Coin( g ′ 1 , α g ′ 2 ) must be nonempty. Take some y in this class and a point y ∈ p −1 (y) with g ′ 1 ( y) = α g ′ 2 ( y). Then we have f ′ 1 ( σ( y)) = α f ′ 2 ( σ( y)), and so σ(y) ∈ p Coin( f ′ 1 , α f ′ 2 ), which contradicts (1). The above provides the key step in the proof of the following complete computation of the geometric Nielsen number on tori. The argument in codimension zero given in [5] carries without modification in arbitrary codimension, except for this step. Jezierski presents the details, along with a different argument substituting for Lemma 5.3 in [6]. Theorem 5.4. Let f 1 , f 2 : T m → T n be maps of tori with induced matrices A 1 , A 2 . If A 2 − A 1 has rank n, then N G (f 1 , f 2 ) = #π 0 (Coin(f 1 , f 2 )) = # coker(A 2 − A 1 ), where coker(A 2 − A 1 ) = Z n / im(A 2 − A 1 ), the cokernel of A 2 − A 1 when viewed as a homomorphism Z m → Z n . As a brief illustration, we compute the geometric Nielsen coincidence numbers for the maps f, g, h : T 2 → S 1 from Example 1.2. Example 1.2, Continued. Recall our maps were given by matrices: A f = (3 1), A g = (0 2), A h = (−1 − 1). For each pair of matrices the rank assumption of Theorem 5.4 holds. It is straightforward to compute the required cokernels. We have A g − A f = (3 − 1), and so im(A g − A f ) = Z, since gcd(3, −1) = 1. Thus the cokernel is trivial and so N G (f, g) = 1. A similar computation shows that N G (g, h) = 1. For (f, h), we have A h − A f = (2 − 2), and so im(A h − A f ) = 2Z. Thus the cokernel is Z/2Z, and so N G (f, h) = 2. By Theorem 5.4 these Nielsen numbers should agree with the number of path components of the coincidence sets. Counting components in Figure 1 indeed gives #π 0 (Coin(f, g)) = #π 0 (Coin(g, h)) = 1 and #π 0 (Coin(f, h)) = 2. Figure 1 : 1Coincidence sets and equalizer points for Example 1.2. Theorem 2 . 3 . 23Two points x, x ′ ∈ Eq(f 1 , . . . , f k ) are in the same equalizer class if and only if there is some path γ : [0, 1] → X from x to x ′ such that f 1 (γ) and f i (γ) are homotopic as paths with fixed endpoints for all i. Theorem 3 . 1 . 31Let f 1 , . . . , f k : X → Y be maps of compact orientable manifolds of dimensions (k − 1)n and n respectively, and let U ⊂ X be open with (f 1 , . . . , f k , U ) admissable. Theorem 4 . 1 . 41If f, g : M → N are maps of compact orientable manifolds of the same dimensions and N is a Jiang space, then every coincidence class has the same index. Theorem 4 . 2 . 42If f 1 , . . . , f k : X → Y are maps of compact orientable manifolds of dimensions (k − 1)n and n respectively and Y is a Jiang space, then every equalizer class has the same index. Theorem 5. 4 . 4Let f 1 , f 2 : T m → T n be maps of tori with induced matrices A 1 , A 2 . Then N G (f 1 , f 2 ) = 0 if and only if A 2 − A 1 has rank n. Figure 1 : 1Coincidence sets and equalizer points for Example 1.2. Theorem 2 . 3 . 23Two points x, x ′ ∈ Eq(f 1 , . . . , f k ) are in the same equalizer class if and only if there is some path γ : [0, 1] → X from x to x ′ such that f 1 (γ) and f i (γ) are homotopic as paths with fixed endpoints for all i. Theorem 3. 1 . 1Let f 1 , . . . , f k : X → Y be maps of compact orientable manifolds of dimensions (k − 1)n and n respectively, and let U ⊂ X be open with (f 1 , . . . , f k , U ) admissable.Then the equalizer index ind(f 1 , . Theorem 3 . 2 . 32Let f 1 , . . . , f k : X → Y be maps of compact orientable manifolds of dimensions (k − 1)n and n respectively, and let x ∈ Eq(f 1 , . . . , f k ) be nondegenerate. Then there is a neighborhood U of x with Eq(f 1 , . . . , f k , U ) = {x} such that ind(f 1 , . . . , f k , U ) Theorem 3. 3 . 3Let f 1 , . . . , f k : X → Y be maps of compact (possibly nonorientable) manifolds of dimensions (k − 1)n and n respectively, and let C ⊂ Eq(f 1 , . . . , f k , U ) be an equalizer class.Then the equalizer semi-index \| ind \|(f 1 , . • Naturality: If X and Y are orientable, then \| ind \|(f 1 , . . . , f k , C) = \| ind(f 1 , . . . , f k , C)\|, the absolute value of the usual equalizer index. Theorem 4 . 1 . 41If f, g : M → N are maps of compact orientable manifolds of the same dimensions and N is a Jiang space, then every coincidence class has the same index. Corollary 4 . 3 . 43If f 1 , .. . , f k : X → Y are maps of compact orientable manifolds of dimensions (k − 1)n and n respectively, and Y is a Jiang space, then:• If L(f 1 , . . . , f k ) = 0 then N (f 1 , . . . , f k ) = 0. • If L(f 1 , . . . , f k ) = 0 then N (f 1 , . . . , f k ) = R(f 1 , . . . , f k ).Proof. By the additivity property, L(f 1 , . . . , f k ) is the sum of the indices of each equalizer class. By Theorem 4.2 all classes have the same index, thus L(f 1 , . . . , f k ) = 0 means that all classes are inessential andso N (f 1 , .. . , f k ) = 0. If the Lefschetz number is not zero then all classes are essential and so the Nielsen number is simply the number of classes, which is the Reidemeister number. Theorem 4 . 4 . 44If f 1 , . Theorem 4 . 5 . 45Let f 1 , . . . , f k : T (k−1)n → T n be maps of tori. Then ME(f 1 , . . . , f k ) = N (f 1 , . . . , f k ). Corollary 5 . 2 . 52Let f 1 , f 2 : X → Y be maps of spaces of dimension (k − 1)n and n respectively. If there are maps f 3 , . . . , f k with N (f 1 , . . . , f k ) = 0, then N G (f 1 , f 2 ) = 0. * MSC2000: 54H25, 55M20 † Address: Department of Mathematics and Computer Science, Fairfield University, Fairfield CT, USA ‡ Email: [email protected] § Web: http://faculty.fairfield.edu/cstaecker ¶ Keywords: Nielsen theory, equalizer theory, positive codimension coincidence theory AddendumThe author was fortunate to attend the International Conference on Nielsen Theory and Related Topics at Capitol Normal University in Beijing in Summer 2011 after this article was initially published. There he learned from Peter Wong that a Nielsen type equalizer theory in the category of smooth orientable manifolds was obtained as a special case of a more general theory by Dobreńko and Kucharsky in[3].For smooth orientable manifolds M and N and a submanifold B ⊂ N with dim X + dim B = dim Y , Dobreńko and Kucharsky define a general Nielsen theory for counting the cardinality of f −1 (B) for a map f : M → N . In a brief parenthetical, they remark that applying this theory to a tuple map (f 1 , . . . , f k ) : M → N k and letting B be the diagonal in N k gives a "Nielsen number of coincidences of two or more maps" which essentially matches the theory presented in this paper.The author would like to thank Peter Wong for this reference. On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy. R Brooks, Pacific Journal of Mathematics. 139R. Brooks. On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy. Pacific Journal of Mathematics, 139:45-52, 1971. The coincidence Nielsen number on nonorientable manifolds. R Dobreńko, J Jezierski, The Rocky Mountain Journal of Mathematics. 23R. Dobreńko and J. Jezierski. The coincidence Nielsen number on nonori- entable manifolds. The Rocky Mountain Journal of Mathematics, 23:67-85, 1993. The Handbook of Topological Fixed Point Theory. D L Gonçalves, R.F. BrownSpringerCoincidence theoryD. L. Gonçalves. Coincidence theory. In R.F. Brown, editor, The Handbook of Topological Fixed Point Theory, pages 3-42. Springer, 2005. The Nielsen number product formula for coincidences. Fundamenta Mathematicae. J Jezierski, 134J. Jezierski. The Nielsen number product formula for coincidences. Funda- menta Mathematicae, 134:183-212, 1990. B Jiang, Lectures on Nielsen fixed point theory. American Mathematical Society14B. Jiang. Lectures on Nielsen fixed point theory. Contemporary Mathematics 14, American Mathematical Society, 1983. On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy. R Brooks, Pacific Journal of Mathematics. 139R. Brooks. On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy. Pacific Journal of Mathematics, 139:45-52, 1971. The coincidence Nielsen number on nonorientable manifolds. R Dobreńko, J Jezierski, The Rocky Mountain Journal of Mathematics. 23R. Dobreńko and J. Jezierski. The coincidence Nielsen number on nonori- entable manifolds. The Rocky Mountain Journal of Mathematics, 23:67-85, 1993. On the generalization of the Nielsen number. R Dobreńko, Z Kucharsky, Fundamenta Mathematicae. 134R. Dobreńko and Z. Kucharsky. On the generalization of the Nielsen number. Fundamenta Mathematicae, 134:1-14, 1990. The Handbook of Topological Fixed Point Theory. D L Gonçalves, R.F. BrownSpringerCoincidence theoryD. L. Gonçalves. Coincidence theory. In R.F. Brown, editor, The Handbook of Topological Fixed Point Theory, pages 3-42. Springer, 2005. The Nielsen number product formula for coincidences. Fundamenta Mathematicae. J Jezierski, 134J. Jezierski. The Nielsen number product formula for coincidences. Funda- menta Mathematicae, 134:183-212, 1990. The Nielsen coincidence number of maps into tori. J Jezierski, Quaestiones Mathematicae. 24J. Jezierski. The Nielsen coincidence number of maps into tori. Quaestiones Mathematicae, 24:217-223, 2001. B Jiang, Lectures on Nielsen fixed point theory. American Mathematical Society14B. Jiang. Lectures on Nielsen fixed point theory. Contemporary Mathematics 14, American Mathematical Society, 1983. On the uniqueness of the coincidence index on orientable differentiable manifolds. P C Staecker, Topology and Its Applications. 154arxiv eprint math.GN/0607751P. C. Staecker. On the uniqueness of the coincidence index on orientable differentiable manifolds. Topology and Its Applications, 154:1961-1970, 2007. arxiv eprint math.GN/0607751.	[]
[ "The twistorial structure of loop-gravity transition amplitudes", "The twistorial structure of loop-gravity transition amplitudes" ]	[ "Simone Speziale \nCentre de Physique Théorique\nCampus de Luminy, Case 90713288Marseille, FranceEU\n", "Wolfgang M Wieland \nCentre de Physique Théorique\nCampus de Luminy, Case 90713288Marseille, FranceEU\n" ]	[ "Centre de Physique Théorique\nCampus de Luminy, Case 90713288Marseille, FranceEU", "Centre de Physique Théorique\nCampus de Luminy, Case 90713288Marseille, FranceEU" ]	[]	The spin foam formalism provides transition amplitudes for loop quantum gravity. Important aspects of the dynamics are understood, but many open questions are pressing on. In this paper we address some of them using a twistorial description, which brings new light on both classical and quantum aspects of the theory. At the classical level, we clarify the covariant properties of the discrete geometries involved, and the role of the simplicity constraints in leading to SU(2) Ashtekar-Barbero variables. We identify areas and Lorentzian dihedral angles in twistor space, and show that they form a canonical pair. The primary simplicity constraints are solved by simple twistors, parametrized by SU(2) spinors and the dihedral angles. We construct an SU(2) holonomy and prove it to correspond to the (lattice version of the) Ashtekar-Barbero connection. We argue that the role of secondary constraints is to provide a non trivial embedding of the cotangent bundle of SU(2) in the space of simple twistors. At the quantum level, a Schrödinger representation leads to a spinorial version of simple projected spin networks, where the argument of the wave functions is a spinor instead of a group element. We rewrite the Liouville measure on the cotangent bundle of SL(2,C) as an integral in twistor space. Using these tools, we show that the Engle-Pereira-Rovelli-Livine transition amplitudes can be derived from a path integral in twistor space. We construct a curvature tensor, show that it carries torsion off-shell, and that its Riemann part is of Petrov type D. Finally, we make contact between the semiclassical asymptotic behaviour of the model and our construction, clarifying the relation of the Regge geometries with the original phase space. * Unité Mixte de Recherche (UMR 7332) du CNRS et des Univ. Aix-Marseille et Sud Toulon Var. Unité affiliée à la FRUMAM. arXiv:1207.6348v2 [gr-qc] 27 Sep 2012 1 Itself constructed from the 4-dimensional Levi-Civita density˜ dabc via dt˜ abc =˜ dabc ∂ d t, where t is the time coordinate. 2 Completing the canonical analysis[35,39]shows that also the remaining components of the 4-dimensional torsion 2-form vanish, some of them implying evolution equations for the triad e i on the spatial hypersurface, the others fixing the boost component of the Lagrange multiplier Λ i = A i (∂t) to the value Im(Λ i ) = N a K i a + e ia ∂aN , where K i a = Im(A i a) is the extrinsic curvature and N , N a and ∂t denote lapse, shift and the time flow vector-field. 3 These are defined in the time-gauge, but fully covariant formulations exist[40,[44][45][46]]. An alternative parametrization is suggested by Alexandrov[44].	10.1103/physrevd.86.124023	[ "https://arxiv.org/pdf/1207.6348v2.pdf" ]	59,406,729	1207.6348	9d26f60a4b52418808b5e5515b5e73b95898a534	The twistorial structure of loop-gravity transition amplitudes (Dated: May 2, 2014) Simone Speziale Centre de Physique Théorique Campus de Luminy, Case 90713288Marseille, FranceEU Wolfgang M Wieland Centre de Physique Théorique Campus de Luminy, Case 90713288Marseille, FranceEU The twistorial structure of loop-gravity transition amplitudes (Dated: May 2, 2014) The spin foam formalism provides transition amplitudes for loop quantum gravity. Important aspects of the dynamics are understood, but many open questions are pressing on. In this paper we address some of them using a twistorial description, which brings new light on both classical and quantum aspects of the theory. At the classical level, we clarify the covariant properties of the discrete geometries involved, and the role of the simplicity constraints in leading to SU(2) Ashtekar-Barbero variables. We identify areas and Lorentzian dihedral angles in twistor space, and show that they form a canonical pair. The primary simplicity constraints are solved by simple twistors, parametrized by SU(2) spinors and the dihedral angles. We construct an SU(2) holonomy and prove it to correspond to the (lattice version of the) Ashtekar-Barbero connection. We argue that the role of secondary constraints is to provide a non trivial embedding of the cotangent bundle of SU(2) in the space of simple twistors. At the quantum level, a Schrödinger representation leads to a spinorial version of simple projected spin networks, where the argument of the wave functions is a spinor instead of a group element. We rewrite the Liouville measure on the cotangent bundle of SL(2,C) as an integral in twistor space. Using these tools, we show that the Engle-Pereira-Rovelli-Livine transition amplitudes can be derived from a path integral in twistor space. We construct a curvature tensor, show that it carries torsion off-shell, and that its Riemann part is of Petrov type D. Finally, we make contact between the semiclassical asymptotic behaviour of the model and our construction, clarifying the relation of the Regge geometries with the original phase space. * Unité Mixte de Recherche (UMR 7332) du CNRS et des Univ. Aix-Marseille et Sud Toulon Var. Unité affiliée à la FRUMAM. arXiv:1207.6348v2 [gr-qc] 27 Sep 2012 1 Itself constructed from the 4-dimensional Levi-Civita density˜ dabc via dt˜ abc =˜ dabc ∂ d t, where t is the time coordinate. 2 Completing the canonical analysis[35,39]shows that also the remaining components of the 4-dimensional torsion 2-form vanish, some of them implying evolution equations for the triad e i on the spatial hypersurface, the others fixing the boost component of the Lagrange multiplier Λ i = A i (∂t) to the value Im(Λ i ) = N a K i a + e ia ∂aN , where K i a = Im(A i a) is the extrinsic curvature and N , N a and ∂t denote lapse, shift and the time flow vector-field. 3 These are defined in the time-gauge, but fully covariant formulations exist[40,[44][45][46]]. An alternative parametrization is suggested by Alexandrov[44]. I. INTRODUCTION An intriguing relation between loop quantum gravity and twistors has recently emerged in the literature [1][2][3][4]. It relies on a new parametrization of the classical phase space of holonomies and fluxes. In this paper we push this representation further and study dynamical properties of the theory, addressing a number of open questions. Among these, we show that transition amplitudes for loop quantum gravity can be written as path integrals in twistor space, and that torsion is present off-shell. In the first part of the paper, we focus on classical aspects of the twistorial representation. We review the results appeared in [1][2][3][4], and put them together in a coherent picture. Using the twistorial parametrization, we identify a pair of canonically conjugated variables, that corresponds geometrically to an area and a Lorentzian dihedral angle. We study the algebra of primary simplicity constraints in twistor space, and show that the constraint surface is parametrized by SU(2) holonomies and fluxes, plus the dihedral angle. The simple twistors solution of the constraints are determined by SU (2) spinors and the dihedral angle. We then argue that solving the secondary constraints leads to a non-trivial embedding of SU(2) variables in the covariant space, a structure that reproduces at the discrete level what is achieved by the Ashtekar-Barbero variables. In fact, we also prove that the reduced SU(2) connection is the (lattice version of the) parallel transport with respect to the Ashtekar-Barbero connection. Finally, we give the map from twistors to covariant twisted geometries, and show that the role of the simplicity constraints is to match the left and right metric structures, precisely as in the continuum theory. In the second part of the paper, we use these structures to derive a series of results for the quantum theory. In Section III, we show that the classical phase space and its algebra of constraints can be quantized, leading to a Hilbert space of quantum twistor networks. This is achieved choosing a Schrödinger quantization of twistor space, as in [2]. The resulting states are wave functions on spinors, instead of cylindrical functions on the group. A basis is given by the homogeneous functions appearing in the unitary irreducible representations of the Lorentz group. They carry a representation of the spinorial Heisenberg algebra, that includes the holonomy-flux algebra, and introduce a new framework for covariant loop quantum gravity. Treating the diagonal simplicity constraints as first class, and the second class off-diagonal constraints via a master constraint technique, we derive what we call simple quantum twistor networks. These are related to the simple projected spin networks of [5], which appear [6,7] as boundary states of the EPRL spin foam model [8] and its generalizations to arbitrary cellular decompositions [9,10] (see also [11][12][13][14][15][16][17][18]26], and [19] for a recent introduction). The change of representation between the spinorial wave functions and the cylindrical functions can be given explicity, and appears naturally in the construction of the spin foam dynamics. The existence of an SU(2)-invariant scalar product and the equivalence with the Hilbert space of loop quantum gravity are natural results in our formalism. In Section IV, we use the twistorial parametrization to rewrite the Liouville measure of the cotangent bundle T * SL(2, C). This requires a Faddeev-Popov procedure for the area-matching constraint, which we provide explicitly, and prove gauge-invariance of the path integral. The result has applications to the spin foam formalism in general. In Section V we derive the EPRL transition amplitudes as a path integral in twistor space, using the previous results: the quantized phase space, the Liouville measure, plus a discretization of the BF action which is bilinear in the spinors. The result is an independent derivation of the model, where the wedge amplitude is like the infinitesimal step of a Feynman path integral, with the intermediate position eigenstates given by the constrained states, and a "straight" evolution given by the BF action. The calculation is based on the properties of the homogeneous functions in the boundary states, and involves the evaluation of a certain complex integral. The final amplitude perfectly coincides with the (generalized) EPRL model, up to additional face factors that can be independently specified, for instance as argued in [20,21]. The EPRL model has the important property of reproducing the Regge action in the large spin limit [17] (see also [18,[22][23][24][25]). In Section VI, we present the relation between the semiclassical Regge behaviour and the covariant phase space. In doing so, we explain some key aspects of the large spin asymptotics, which have so far gone unnoticed. First, part of the saddle point equations determine a certain subset of the primary simplicity constraint surface. Second, the secondary constraints are solved by the remaining saddle point equations. This key step rests significantly on the restriction to triangulations and the assumption of flatness of the 4-simplices, a case in which a Levi-Civita connection is known from Regge calculus. We show that the dihedral angles entering the Regge action are precisely those of our phase space. We discuss the "semi-coherence" of the model, in the sense that not the whole phase space structure plays a dynamical role. This shows up, among other things, in the fact that the areas are purely quantum numbers. We also introduce a curvature tensor, and show that off-shell it has non-vanishing torsional components, consistently with the continuum formalism. In the conclusions we summarize our results and discuss some of the new lines of research that this program proposes. There are two appendices. The first gives a list of conventions on spinor calculus and useful formulae for the irreducible unitary representations of the Lorentz group, while the second appendix contains the explicit evaluation of an integral entering the derivation of the amplitude. Further concerning conventions, the papers [1,3,4] use an index-free notation, whereas [2] uses the standard spinorial notation. Here, primi juvenes, we work with explicit indices. In the Appendix, we provide a translation to the index-free notation. Accordingly, A, B, C, . . . are spinor indices in the left-handed representations, their complex conjugate (i.e. their right handed counterparts) carry bars, i.e. we writeĀ,B,C, . . . . Also, I, J, K, . . . label internal Minkowski vectors, a, b, c, . . . are abstract indices in tangent space, and i, j, k, . . . run from one to three. The metric signature is (−, +, +, +), resulting in Minkowski vectors corresponding to anti-Hermitian matrices in the irreducible ( 1 2 , 1 2 ) representation of SL(2, C). Brackets (· · · ) and [· · · ] surrounding the indices denote their normalised symmetrisation and antisymmetrisation respectively, and 0123 = 1 fixes the normalisation of the internal Levi-Civita tensor. II. FROM TWISTORS TO TWISTED GEOMETRIES A. Twistors and T * SL (2, C) Following our previous works [1][2][3][4], we begin with an abstract, oriented graph decorated by a pair of twistors (Z, Z ) ∈ T 2 C 8 on each link, and associate Z and Z to the source and target points respectively. This will be a general rule for us, "tilded" quantities always belong to the final point. Each twistor is described as a pair of spinors, Z = (ω A ,πĀ) ∈ C 2 ⊕C 2 * =: T, where ω A is left-handed, and the right-handed partπĀ lies in the complex-conjugate dual vector space. We equip the space T 2 with an SL(2, C)-invariant symplectic structure, whose non-vanishing Poisson brackets are π A , ω B = δ B A = − π A , ω B , and πĀ,ωB = δB A = − π Ā,ω B . We have chosen opposite signs between tilded and untilded Poisson brackets, but symmetric brackets would work as well. As it is well known from the literature [27][28][29], T and T 2 carry a representation of the Lorentz group, preserving the symplectic structure. The generators of the action are the left-handed bispinors Π AB = − 1 2 ω (A π B) , Π AB = 1 2 ω (A π B) ,(2) and their right-handed complex conjugates. That is, the Hamiltonian vector fields of (Π,Π) and (Π ,Π ) generate the canonical SL(2, C) action on Z and Z . The proof is a straightforward application of the brackets (1), but also needs some additional SL(2, C) structures. The first is the invariant antisymmetric -tensor (its components fixed by requiring 01 = 01 = 1) mapping contravariant spinors to their algebraic duals and vice versa, ω A = BA ω B , ω A = AB ω B , AC BC = B A = δ A B .(3) The second are the anti-Hermitian sl(2, C) generators τ A Bi , related to the Pauli matrices σ i through 2iτ i := σ i . These matrices induce a map between sl(2, C) and C 3 : Π ∈ sl(2, C) : Π A B = Π i τ A Bi , Π i ∈ C 3 .(4) Then, Π i , Π j = − ij k Π k , Π i , Π j = − ij k Π k , Π i , Π j = 0,(5) and the same for their conjugated. These are the Poisson brackets for (two copies of) the Lorentz algebra, in the chiral splitting. The chiral generators Π i = 1 2 L i + iK i , Π i = 1 2 L i + iK i(6) are complex, with real and imaginary parts, L and K, generating rotations and boosts respectively. We will later identify the sl(2, C) elements Π and Π with the gravitational fluxes, i.e. the Plebanski 2-form smeared over 2-dimensional submanifolds in the hypersurface of initial data. Be this hypersurface space-like, we require Π AB Π AB = 0, implying linear independence of the spinors: πω := AB π A ω B = π A ω A = 0, π ω = 0.(7) With this restriction, the pair (π A , ω A ) forms a complete basis of C 2 , just as well as (π A , ω A ). We introduce the linear map translating one to the other, h A B ω B = ω A , h A B π B = π A ,(8) which we will later identify with the holonomy along the link. For this map to be unimodular, i.e. h ∈ SL(2, C), it must preserve the bilinear generated by AB , hence C := πω − π ω = 0.(9) In the following, we will refer to (9) as the (complex) area-matching constraint. Furthermore, thanks to the restriction (7), we can uniquely parametrise the holonomy in terms of the basis spinors as h A B = ω A π B − π A ω B √ π ω √ πω .(10) The functions Π, Π and h are related by Π = − πω π ω hΠh −1 ,(11) and span 14 out of 16 dimensions of T 2 . They obey the Poisson brackets Π i , h = −hτ i , Π i , h = τ i h, (12a) h A B , h C D = − 2C (πω)(π ω ) AC Π BD + BD Π AC .(12b) On the constraint hypersurface C = 0, two key properties hold: Firstly, the adjoint representation relates the fluxes, Π = −hΠh −1 . Secondly, the components of the holonomy commute. Hence, we recover the Poisson algebra of T * SL(2, C) with Π and Π as the (chiral) left-and right-invariant Hamiltonian vector fields on the group manifold: Π i , Π j = − ij k Π k , Π i , h = −hτ i , Π i , h = τ i h, h A B , h C D C=0 = 0.(13) In fact, we can easily see that this procedure amounts exactly to a symplectic reduction T * SL(2, C) SL(2, C) × sl(2, C) T 2 / /C. On the C = 0 constraint hypersurface, the Hamiltonian vector field X C = {C, ·} generates the orbits exp (zX C +zXC) : (ω, π, ω , π ) → (e z ω, e −z π, e z ω , e −z π ), z ∈ C. The functions Π, Π and h are invariant under such gauge transformations, and thus span the space obtained by symplectic reduction. Let us add two more remarks to complete the analysis. First, the map between twistors and holonomy-flux variables is 2-to-1, since exchanging spinors as (ω, π, ω , π ) → (π, ω, π , ω ) leaves both holonomy (10) and flux (2) unchanged. Hence, to arrive at the reduced phase space, we also have to divide out this residual Z 2 symmetry (15). Second, because of the restriction πω = 0, the spinorial parametrization cannot cover the submanifold of (h, Π) : Π AB Π AB = −Tr Π 2 = 0, and what we truly find is T * SL(2, C) removed from all its null configurations. The complete isomorphism could be defined through a suitable treatment of the degenerate configurations, see e.g. the analogue situation in the SU(2) case [30]. However, below we will identify Π with the Plebanski field smeared over 2-dimensional surfaces in a t = const. slice of initial data. Be this hypersurface space-like, the restriction is automatically fulfilled, and has no physical consequence for the following. B. Twistors from the LQG action The interest in the phase space of SL(2, C) holonomies and fluxes comes from loop quantum gravity. We work in the first-order tetrad formalism, and start from the often-called Holst action for general relativity. In terms of chiral variables, it is S Holst [A, e] = P 2 β + i iβ M Σ A B (e) ∧ F B A (A) + cc.,(16) where P = 8π G N /c 3 is the Planck length, β > 0 is the Barbero-Immirzi parameter, and "cc." denotes complex conjugation of everything preceding. See e.g. [31][32][33][34] for the case with boundary terms. The action (16) is a non-analytic functional of the left-handed sl(2, C) connection A, and the four soldering forms e transforming in the irreducible ( 1 2 , 1 2 ) representation of SL(2, C). Curvature F A B and Plebanski 2-form Σ A B (e) are uniquely determined by the equations F A B (A) = dA A B + A A C ∧ A C B , Σ A B (e) = e AC ∧ e BC = i 2 τ A Bi i 2 i lm e l ∧ e m + e 0 ∧ e i ,(17) where in the last expression, we have explicitly put the projector onto the left-handed variables. In order to read off the symplectic structure, a 3+1 split M = Σ × R is needed. We take the pullback of the sl(2, C) connection onto the spatial hypersurface Σ t = Σ × {t}, call this (by a little abuse of notation) A i a and find its conjugate momentum to be Π i a = − P 2 β + i 4iβ˜ abc Σ ibc ,(18) where˜ abc is the spatial Levi-Civita density 1 on Σ t . The only non-vanishing Poisson brackets are Π i a (p), A j b (q) = δ j i δ a bδ (p, q) = Π i a (p),Ā j b (q) ,(19) whereδ(p, q) is the three-dimensional Dirac distribution (a scalar density) on Σ t , and δ a b , δ j i are spatial Kronecker symbols. The Cauchy hypersurface Σ t = Σ × {t} of initial data carries a time normal n a , allowing us to define a Hermitian metric for C 2 . We work in the time gauge, in which this normal gives (or rather: is represented by) the identity matrix, that is the one corresponding to the canonical SU(2) subgroup of the Lorentz group: δ AĀ := −i √ 2n AĀ = −i √ 2e AĀ a n a , time-gauge: δ 00 = 1 = δ 11 , δ 01 = 0 = δ 10 .(20) It is this metric with respect to which the Pauli matrices (4) are Hermitian, and it is this normal with respect to which the real and imaginary parts of Π correspond to rotations and boosts (6) respectively. The chiral splitting has led us to a complex phase space. To guarantee that the metric is real, reality conditions must be imposed. Using our gauge condition (20) equation (17) constrains in fact all components of˜ abc Σ ibc to be real. This imposes the reality conditions 1 β + i Π i + cc. = 0 ⇔ K i + βL i = 0 ⇔ Π i = −e iϑΠ i ,(21) on the momentum Π i . Here we have introduced the angle e iϑ = β + i β − i , β = cot ϑ 2 .(22) The intermediate form K + βL = 0 shows explicitly, as highlighted in [35], that the reality conditions amount to the canonical version of the primary simplicity constraints, in their linear version introduced in [8]. The final form further shows that they match the two chiral metric structures induced by SL(2, C), as discussed in [36][37][38]. A complete canonical analysis of the action can be found e.g. in [35,[39][40][41][42][43]. Preserving the primary constraints (21) under Hamiltonian time evolution leads to secondary constraints, given by the vanishing of the spatial projection˜ abc D b e i c of the torsion 2-form. Once the primary constraints are solved, the secondary ones imply that the spatial part of the Lorentz connection is Levi-Civita. 2 The system of primary and secondary simplicity constraints is second class, and canonical coordinates on the reduced phase space are provided by Ashtekar-Barbero variables 3 [47][48][49]. The crucial step towards loop quantisation [50,51] is a certain "covariant" smearing of the continuous gravitational phase space. One introduces a graph Γ in the spatial manifold, consisting of oriented links γ, γ , . . . , to each of which we assign a dual, i.e. an oriented surface t, t , . . . transversally intersecting the corresponding links. We may think of the graph Γ as being dual to a cellular decomposition of the spatial manifold, each node of Γ dual to a 3-cell. The elementary phase space variables are then smeared over these lower dimensional objects, obtaining a collection of holonomies and fluxes: SL(2, C) h[t] = h γ = Pexp − γ A , sl(2, C) Π[t] = p∈t h q→γ(0) Π p h γ(0)→q .(23) Here Pexp is the path ordered exponential, and h q→γ(0) denotes the holonomy parallely transporting from the integration variable q ∈ t along γ towards the initial point γ(0) of the link dual to the surface t. Since each surface t carries an orientation there is also the oppositely oriented element t −1 which come along with h[t −1 ] = h[t] −1 , and Π[t −1 ] = −h[t]Π[t]h[t] −1 ≡ Π [t].(24) Under this smearing, the Poisson structure (19) reduces precisely to (13) on each link, while variables at different links commute. See [50,52] and also [53] for a more recent discussion. The smeared phase space is thus the Cartesian product of T * SL(2, C) associated to each link, which we derived from a network of twistors supplemented by the complex area-matching condition (9). Local Lorentz invariance is imposed as in lattice gauge theories by a closure condition on the SL(2, C) generators at each node, namely G n = t∈n Π[t] = 0 and its complex conjugate. Then, we have to realize the primary and secondary simplicity constraints, and find the analogue of the Ashtekar-Barbero variables for the space of solutions. This is a challenging problem, and a general solution is still unknown. As pointed out long ago [54][55][56], a successful discretization can be found restricting attention to a 4-simplex, and further assuming that the interior is flat. In this case we are in the framework of Regge calculus: the geometry is uniquely described by edge lengths, and the Levi-Civita connection is known. In the general case, the situation is different. The primary constraints remain tractable, as they do not involve the connection, and can be defined on the boundary of any 4-cell, without assumptions on its interior. The secondary constraints, on the other hand, lack so far a general treatment. The main difficulty is that outside the framework of Regge geometries we do not know how to define the Levi-Civita connection. And in general, even on shell of the primary constraints, the data on the boundary graph do not describe Regge geometries, but a generalization going under the name of twisted geometries [1,30,57], which we will review below. These (i) are defined for a general cellular decomposition, (ii) allow for discontinuities in the metric (possibly related to torsion), and (iii) do not impose conditions on the interior curvature of the 4-cells. The Levi-Civita connection is so far not understood in this framework. We will see in the rest of this Section how the twistor description of the phase space brings new light to many of these questions. C. Reality conditions and reduction to SU(2) spinors We will now rewrite the reality conditions in terms of the spinorial representation, and solve them explicitly. The result will reduce twistors down to SU(2) spinors, with the emergence of the SU(2) holonomy of the β-dependent Ashtekar-Barbero connection. We discretize (21) on both source and target variables of each link, ∀t : Π[t] = e iϑ Π † [t], Π [t] = e iϑ Π † [t],(25) where the Hermitian conjugate is taken according to (Π † ) A B = δ AĀ δ BBΠBĀ .(26) In the spinorial parametrization, the first equation in (25) reads ω (A π B) = −e iϑ δ AĀ δ BBω (ĀπB) .(27) It apparently gives two equivalent decompositions of Π AB in terms of spinors and their complex conjugate. But the decomposition of a symmetric bispinor is unique up to exchange and complex rescaling of the constituents, therefore π and ω must be linearly related. Furthermore, part of the complex rescaling is fixed by the phase appearing explicitly in (27), leaving only the freedom to real rescalings. Hence, we can parametrize the solutions as π A = re i ϑ 2 δ AĀωĀ , ω A = − 1 r e i ϑ 2 δ AĀπĀ , r ∈ R − {0}.(28) The matching of left and right geometries as implied by (25) immediately translates into the left and right spinors being proportional. The same conclusion holds in a general gauge, with a generic normal replacing the identity matrix, as in (20). Remarkably, the simplicity equations then take up the same form as Penrose's incidence relation. It would be intriguing to explore the existence of a deeper connection between these two notions. That simplicity implies proportionality of the spinors is a key result, and was also derived in [3]. It means that a simple twistor, i.e. a twistor satisfying the simplicity constraints, is determined by a single spinor, plus a real number, whose meaning will become clear below. By contractions with ω and π, equation (27) can be conveniently separed in two parts, F 1 = i β + i ω A π A + cc. = 0, F 2 = i √ 2 δ AĀ π AωĀ = n AĀ π AωĀ = 0.(29) Here, F 1 is real and Lorentz-invariant, while F 2 is complex but only SU(2) invariant. Following the literature, we will refer to F 1 as the diagonal simplicity constraint, and F 2 as off-diagonal. The constraints F i , i = 1, 2, the corresponding F i for the tilded spinors, and the area matching C, form a system of constraints on the link space T 2 ∼ = C 8 . The algebra can be easily checked to give {F 1 , F 2 } = − 2iβ β 2 + 1 F 2 , {F 2 ,F 2 } = i Im(πω), {C, F 1 } = 0, {C, F 2 } = −F 2 = −{C, F 2 },(30) and the same for tilded quantities. The system should be supplemented with secondary constraints coming from a suitable Hamiltonian, and we will come back to this point below, because it plays an important role in the identification of the extrinsic curvature. Neglecting any secondary constraints for the moment, we conclude that the diagonal simplicity constraints F 1 and F 1 are of first class, as well as C, whereas F 2 and F 2 are second class. That some constraints are second class even in the absence of secondary constraints is a well-known consequence of the non-commutativity (13) of the fluxes. The first class constraints generate orbits inside the constraint hypersurface. The orbits of C are given in (14), whereas those generated by the diagonal simplicity constraints are found from {F 1 , ω A } = i β + i ω A , {F 1 , π A } = − i β + i π A .(31) We also remark that the system is reducible, since only three of the four constraints F 1 , F 1 Re(C) and Im(C) are linearly independent. We thus have three independent first class constraints, and two, complex, second class constraints. The reduced phase space has 16 − 3 × 2 − 2 − 2 = 6 real dimensions, and we will now prove it to be T * SU (2). To that end, it is convenient to treat separately the area matching and the simplicity constraints, the order being irrelevant. There are two convenient choices of independent constraints, depending on the order in which one solves them. If solving the simplicity first, we can choose C red = C β + i + cc., F 1 , F 1 , F 2 , F 2 F .(32) If instead we solve C first, we can take Re(C), Im(C), D := F 1 + F 1 , F 2 , F 2 F red .(33) The situation is summarised in Figure 1. Let us proceed solving the simplicity constraints first. For the untilded quantities, (28) solves all four F = 0 constraints, however the expression is not F 1 -gauge-invariant. For each half-link, gauge-invariant quantities live on the reduced space T/ /F C 2 , and are parametrized by a single spinor, say z A ∈ C 2 . Since the simplicity constraints introduce a Hermitian metric, we have a norm ω 2 = δ AĀ ω AωĀ , and use it to define J = ω 2 1 + β 2 r,(34) which satisfies {F 1 , J } = 0. In terms of J , equation (28) gives Then, the reduced spinor, F 1 -gauge-invariant, can be taken to be π A = (β + i) J ω 2 δ AĀωĀ , πω = (β + i)J .(35)T 2 C 2 × C 2 T * SL(2, C) T * SU(2) F C F red C redz A = √ 2J ω A ω iβ+1 , z = √ 2J .(36) Since we are assuming π A ω A = 0, this implies J = 0. We can further always assume J > 0: In the case J < 0, the sign is flipped by simultaneously exchanging π with ω and π with ω , and we have already seen this operation to be a symmetry (15) of our spinorial parametrisation. Hence, selecting the sign of J removes the Z 2 symmetry of the reduction. The same results apply to the tilded quantities. The reduced space T 2 / /F C 2 × C 2 is parametrised by the following spinors, z A = √ 2J ω A ω iβ+1 , z A = 2J ω A ω iβ+1 .(37) Notice that they transform linearly under rotations, but not under boosts: they are SU(2) spinors. The Lorentzian structures are partially eliminated by the gauge-choice needed to define the linear simplicity constraints. To get the Dirac brackets for the reduced SU(2) spinors, we introduce the embedding ι of the F = 0 constraint hypersurface into the original twistorial phase space, and compute the pullback of the symplectic potential. This gives ι * Θ = ι * π A dω A − π A dω A + cc. = ι * β(J + J ) d ln ω ω + β(J − J ) d ln( ω ω ) + i J ω 2 δ AĀωĀ dω A − i J ω 2 δ AĀω Ā dω A + cc. = i 2 δ AĀ zĀdz A −z Ā dz A − cc. .(38) The induced Dirac brackets are the canonical brackets of four harmonic oscillators, zĀ, z A D = −iδĀ A = − z Ā , z A D .(39) This reduction is illustrated in the top horizontal line of Figure 1. The next step is to implement the areamatching condition. As anticipated, part of C = 0 is automatically satisfied on the surface of F 1 = F 1 = 0. Using (37), the independent part C red can be seen to give the real-valued SU(2) version of the area-matching condition introduced in [1], that is C red = z 2 − z 2 = 0.(40) The gauge orbits generated by C red are U(1) phase transformations z → e iϕ z, for some angle ϕ. As proven in [1], canonical variables on the reduced phase space (C 2 × C 2 )/ /C red are SU(2) holonomies and fluxes, satisfying their canonical Poisson algebra. We are thus left with the phase space T * SU(2), with elements (U, Σ) ∈ SU(2) × su (2) parametrized as 4 [1] U A B (z, z ) = z A δ BBzB + δ AĀz Ā z B z z , Σ AB (z, z ) = β P 2 i 2 z (A δ B)BzB .(41) This proves that the symplectic reduction of T 2 by the area-matching and simplicity constraints gives T * SU(2). Let us conclude this Section with two important remarks. The first is the identification of an abelian pair of canonically conjugated variables on T * SL(2, C). We introduce the quantity Ξ := 2 ln ω ω .(42) An explicit calculation shows that at C = 0 {Re(πω), Ξ} = 1.(43) Also, from the second line of the symplectic potential (38), we immediately see that {J , Ξ} = 1 β(44) on the surface of F i = F i = 0. The conjugated pair corresponds to the (oriented) area and (boost) dihedral angle associated with the dual face t. In fact, from (18), the squared area equals A 2 [t] := δ ij Σ i [t]Σ j [t] = P 4 β 2 2 J 2 .(45) Notice also that the quantity Re(πω) appearing in (43) reduces to the area when the simplicity constraints are satisfied. As for the dihedral angle, it is defined by the scalar product between the time-like normals of the two 3-cells sharing the face, that is n and n . These are both related to the identity matrix by the time gauge (20). The non-trivial information is then carried by the SO(1,3) holonomy Λ(h γ ) between the two, needed to evaluate the scalar product in the same frame. A short calculation then gives n I Λ(h γ ) I J n J = n AĀ h γ A BhγĀB n BB = − 1 2 1 \|πω\| 2 δ AĀ δ BB ω A π B − π A ω B ω Āπ B −π Āω B = = − 1 2 ω 2 ω 2 + ω 2 ω 2 = −ch(Ξ),(46) valid on the constraint surface (32). The dihedral angle between 3-cells describes the extrinsic curvature in Regge calculus, therefore this abelian pair captures a scalar part of the ADM Poisson brackets, as we'll make clearer in the next Section. The second remark concerns the orbits generated by D. Let us define the hypersurface of T * SL(2, C) solution of the simplicity constraints. From (33), we see that on the space reduced by C = 0, that is T * SL(2, C), the independent simplicity constraints are D = F 2 = F 2 = 0. These equations characterize a 7-dimensional constraint hypersurface within T 2 . From the previous construction, we know that six dimensions are spanned by the SU(2) holonomy-flux variables, or equivalently by the SU(2) spinors reduced by (40). Since {D, z A } = 0 = {D, z A }, {D, Ξ} = 4 1 + β 2 ,(47) the seventh dimension spreads along the orbits of D, each of which can be parametrized by the angle Ξ. Accordingly, we denote the constraint surface T Ξ , and T Ξ T * SU(2) × R. This means that a pair of simple twistors, solutions of the area-matching and the simplicity constraints, are parametrized by the SU(2) spinors, plus the dihedral angle. On T Ξ , the Lorentz fluxes already coincide with the su(2) Lie algebra elements introduced in (41), providing a discrete counterpart of the continuum equation (18). For the Lorentz holonomy we find, plugging (37) and (42) into (10), h red A B ≡ h A B F =0 = e − 1 2 (iβ+1)Ξ z A δ BBzB + e 1 2 (iβ+1)Ξ δ AĀz Ā z B z z .(48) This is still a completely general SL(2, C) group element. If we now choose the specific Ξ = 0 section through the orbits of D, it reduces to an SU(2) holonomy, and coincides with the D-invariant holonomy U . The constraint hypersurface T Ξ plays an important role, because there we can distinguish the reduced Lorentz holonomy (48) from the SU(2) holonomy (41). The difference is captured by the orbits of the diagonal simplicity constraint. D. Ashtekar-Barbero holonomy and extrinsic curvature Consider the constraint hypersurface T Ξ , and the two holonomies U (z, z ) and h red (z, z , Ξ). While h red describes the Lorentzian parallel transport, we now show that the SU(2) holonomy U (z, z ) equals the holonomy of the real-valued Ashtekar-Barbero connection A (β) = Γ + βK (here Γ and K are the real and imaginary components of the selfdual SL(2, C) connection A = Γ + iK). Namely, that U (z, z ) = U γ := Pexp − γ Γ + βK .(49) This identification is very important for the spin foam formalism, and the understanding of the relation between covariant and canonical structures. It is needed to match the boundary states appearing in spin foam models with the SU (2) spin network states found from the canonical approach, see e.g. the discussions in [6,7,59,60]. To prove (49), let us first recall (see equation (23)) that h is a left-handed group element corresponding to the parallel transport by the left-handed part of the Lorentz connection, A = Γ + iK, where Γ represents the intrinsic covariant 3-derivative. This 3-derivative defines the SU(2) parallel transport G γ := Pexp − γ Γ i τ i ∈ SU(2).(50) The intrinsic and extrinsic contributions to the holonomies can be disentangled via an "interaction picture" for the path-ordered exponentials, 5 h γ = Pexp − γ Γ + iK = G γ Pexp − i 1 0 dt G −1 γ(t) K γ(t) (γ)G γ(t) ≡ G γ V K ,(51)U γ = Pexp − γ Γ + βK = G γ Pexp − β 1 0 dt G −1 γ(t) K γ(t) (γ)G γ(t) ≡ G γ V β K .(52) Both holonomies provide maps C 2 → C 2 between tilded and untilded spinors, but while h transports the covariant ω A -spinors, U transports the reduced spinors z A . Let us introduce a short-hand ket notation, \|0 ≡ z A z , \|1 ≡ δ AĀzĀ z , \|0 ≡ z A z , \|1 ≡ δ AĀz Ā z .(53) The holonomies can be thus characterized as the unique solutions to the equations \|0 = e (iβ+1)Ξ/2 h\|0 = U \|0 , \|1 = e (−iβ+1)Ξ/2 (h † ) −1 \|1 = U \|1 .(54) Next, we recall that the source and target generators of the Lorentz algebra are related via the holonomy, see (11). This relation, together with the simplicity constraints, implies that Π = e iϑ Π † = −e iϑ (h −1 Π h) † = −h † Π (h −1 ) † = h † hΠ(h † h) −1 .(55) We see that the simplicity constraints automatically lead to a certain "alignment" between the holonomy and the generators, that immediately translates into an equation for the spinors: (h † h) A B ω B = e −Ξ ω A , (h † h) A B π B = e Ξ π A ,(56) with Ξ given in (42). Inserting (51) in (56), we find V † K V K \|0 = e −Ξ \|0 , V † K V K \|1 = e Ξ \|1 .(57) For small extrinsic curvature, we have that V K > 0 and V † K = V K such that this eigenvalue equation has just one solution, given by 6 V K = e −Ξ/2 \|0 0\| + e Ξ/2 \|1 1\|.(58) Within the same approximation, we also have V β K = e iβΞ/2 \|0 0\| + e −iβΞ/2 \|1 1\|.(59) Finally, using the interaction picture in (55), as well as properties (58) and (59), we find U \|0 = e (iβ+1)Ξ/2 h\|0 = GV β K \|0 , U \|1 = e (−iβ+1)Ξ/2 (h † ) −1 \|1 = GV β K \|1 ,(60) and since \|0 and \|1 are a complete basis, this proves the desired result (49). We remark that what we have proved here is valid as an approximation for small curvature. That is, the precise statement is that the SU(2) holonomy U provides the lattice version of the Ashtekar-Barbero connection. As the phase space on a fixed graph only carries a notion of holonomy, and not of point-wise connection, our result is perfectly satisfying. If on the other hand one were interested in an exact continuous equivalence, this would require a projection on the simplicity constraint surface performed at every point of the graph [6,60]. As pointed out above, it would be true also in the case of covariantly constant extrinsic curvature. The equation (60) provides a discrete counterpart to A (β) = Γ + βK = A + (β − i)K, with Ξ playing the role of the extrinsic curvature. In this respect, notice also that from the linearized form of (58), and the continuum interpretation of V K , we deduce Ξ ≈ 1 0 ds R (ad) (G −1 γ(s) ) i j K j γ(s) (γ)n i [t],(61) where R (ad) (G) i j ∈ SO(3) is the SU(2) element G in the adjoint representation. That is, the dihedral angle approximates the extrinsic curvature smeared over the dual link, projected down onto the direction n i [t] normal to the surface. As anticipated earlier, the canonical pairing (44) between Ξ and the area A[t] nicely describes the scalar part of the ADM phase space of general relativity, where [50] flux Σ i a and extrinsic curvature K i a are canonical conjugated. We conclude that the SU(2) spinors z and z obtained from the symplectic reduction parametrise holonomies and fluxes of the SU(2) Ashtekar-Barbero variables. To prove this identification, it has been necessary to work on the covariant phase space, or at least on the constraint hypersurface T Ξ ∼ = T * SU(2) × R, where we could disentangle extrinsic and intrinsic parts of the SU(2) holonomy. Therefore, to have a full geometric meaning, the SU(2) variables need to be embedded in T Ξ . This should not come as a surprise: from the continuum theory we know that one needs to embed the Ashtekar-Barbero connection into the space of Lorentzian connections in order to distinguish intrinsic from extrinsic contributions, and that the secondary constraints provide this embedding. Similarly in the discrete theory, we expect the secondary constraints to provide a non-trivial embedding of T * SU(2) in T Ξ . More precisely, we expect the secondary constraints, and thus the embedding, to be defined only at the level of the complete graph, and not link by link, hence it is more correct to speak of an embedding of T * SU(2) L in T Ξ L . Let us discuss this in more details. In the continuum theory, Ashtekar-Barbero variables, (Σ, A (β) = Γ + βK), are canonical coordinates on the reduced phase space, but are well-defined everywhere as functions on the original phase space. Then, solving the secondary constraints gives Γ = Γ(Σ), and provides a specific embedding of the SU(2) variables into the original phase space. If one forgets about secondary constraints, and treats the linear primary constraints as a first-class system, one ends up with a quotient space of orbits A (β) = const. (because of the brackets between the Lorentz connection and the reality conditions (21)), intersecting the constraint hypersurface transversally (because the Hamiltonian flow of second-class constraints always points away from the constraint hypersurface). Then, restoring the secondary constraints provides a non-trivial section, i.e. a gaugefixing through these orbits, that is an embedding mapping any pair (Σ, A (β) ) towards a point (Π, A = Γ + iK) in the original phase-space. Such treatment of second-class constraints resonates with the gauge-unfixing ideas [61,62] recently applied to the framework of loop quantum gravity in [63,64]. At the discrete level, whatever the correct representation of the secondary constraints may be, it is reasonable to assume that they have the same effect on the constraint algebra, making D second class. Solving them, which typically can not be done link by link but requires knowing the graph, should provide a non-trivial section 7 through the orbits (47) of D, that is a non-local function Ξ t (z t , z t ) where each link dihedral angle is determined by spinors all over the graph. This idea can be made explicit with the ubiquitous example of the flat 4-simplex. In this case, a metric geometry is defined by the ten edge lengths e . Then, all spinors are functions of these data (modulo gauges), and in particular, for each link, Ξ t = Ξ t ( e ) give the dihedral angles, while G t ( e ) equal the holonomies of the Levi-Civita connection. Hence, on the graph phase space T Ξ L there is a functional dependence Ξ t (z t , z t ) between the ten dihedral angles and the twenty spinors, which provides the desired nontrivial section of the bundle T Ξ L . The dependence includes all the spinors and thus non-local on the graph, because each dihedral angle depends in general on all edge lengths. The constraint structure including the role of the secondary constraints is illustrated in Figure 2. Concerning the explicit form of the secondary constraints, we do not investigate them here, but hope to come back to it in future research. In particular, it has been argued in [65,66] that these constraints should be identified with the shape matching conditions of [67], the ones reducing twisted geometry to Regge geometry. A direct test of this claim would require commuting the primary constraints with a suitable Hamiltonian, but this has not been attempted yet. On the other hand, as mentioned at the end of section IIB, we expect that there should exist a notion of Levi-Civita connection (that is a solution of the secondary constraints) even in the absence of shape-matching conditions. If this is the case, then the situation would be different from the one advocated in [65]. T 2 C 2 × C 2 × C C 2 × C 2 T * SL(2, C) T * SU(2) × R T * SU(2) T * SU(2) F F1, F 1 orbits C C red C red F red D orbits Γ-dependent secondary E. Twisted geometries To complete the classical analysis, let us give the mapping between SL(2, C) holonomy-fluxes and the variables of the twisted geometries parametrization [1,4,30]. These variables consists of areas and angles associated to a cellular decomposition dual to the graph, and permit to interpret the classical phase space in terms of discrete geometries. In what follows, we always assume to be on-shell of the area matching, so πω = π ω . We first notice that we can write the holonomy (10) as h = g(ω , π )g(ω, π) −1 , where g(ω, π) = 1 √ πω ω 0 π 0 ω 1 π 1 .(62) Following [4], we use the Iwasawa decomposition for SL(2, C) group elements, g = n(ζ)T α e Φτ3 , (ζ, α, Φ) ∈ C 3 , n(ζ) = 1 1 + \|ζ\| 2 1 ζ −ζ 1 , T α = 1 α 0 1 .(63) Comparing (62) and (63), we identify −ζ −1 = ω 0 ω 1 , Φ = −2 arg(ω 0 ) + i ln ω 2 πω , α = ω\|π πω e 2i arg(ω 0 ) ,(64) where ω\|π := δ AĀ π AωĀ . It is also convenient to define the angle ξ := 2 arg(ω 0 ) − 2 arg(ω 0 ) + βΞ.(65) We then find h = n(ζ )T α e (−ξ+(β−i)Ξ)τ3 T −1 α n −1 (ζ). (66a) Similarly, it is easy to verify that the fluxes (2), in their matricial form (4), read Π = − i 2 πω n(ζ)T α τ 3 T −1 α n −1 (ζ), Π = i 2 πω n(ζ )T α τ 3 T −1 α n −1 (ζ ),(66b) where the opposite sign is inherited directly from the opposite sign in (2). The parametrizations (66) of the covariant holonomy-flux variables gives a map (Π, h) → (ζ, ζ , α, α , ξ, Ξ, πω),(67) in terms of the area of the face, πω, and a collection of angles, which we dub covariant twisted geometries as in [4]. In these variables, the simplicity constraints (25) are solved on the family of hypersurfaces α = α = 0, πω = (β + i)J ,(68) which corresponds to T Ξ . On the trivial section Ξ = 0, we recover T * SU(2) parametrized by twisted geometries (see [1,30], adapted to the conventions of this paper), U = n(ζ )e −ξτ3 n −1 (ζ), Σ = β P 2 i 2 J n(ζ)τ 3 n −1 (ζ).(69) The map between the spinorial parametrization (41) and the twisted geometry parametrization is J = z 2 2 , ζ = −z 1 z 0 , ξ = 2 arg(z 0 ) − 2 arg(z 0 ),(70) consistently with (37) and (64). The twisted geometry variables show explicitly the nature of the discrete geometries associated with the holonomy-flux algebra. First of all, one should consider a cellular decomposition dual to the graph, and assign a 3d Cartesian frame within each 3-cell. Gauge invariance at the nodes, imposed by the closure condition t∈n Π i [t] = 0 and its complex conjugate, guarantees that we can apply Minkowski's theorem to infer the existence of a unique convex polyhedron around the node [68]. The collection of polyhedra defines twisted geometries, a generalization of Regge geometries allowing discontinuous metrics [30,68,69]. However, the left and right generators Π andΠ identify two different geometries. They coincide only when the simplicity conditions hold, (68). This is precisely the role of the constraints also in the continuum theory: they match the left and right metric structures induced by the two Urbantke metrics [36][37][38]. Twisted geometries show that the constraints have exactly the same role also at the discrete level. At the level of reduced Dirac brackets, the area becomes conjugated to the angle ξ, {J , ξ} = 1,(71) which is now independent of the Immirzi parameter. Notice the remarkable analogy between (65), relating the dihedral angle to the class angle of the SU(2) holonomy, and the canonical transformation in the continuum theory from the extrinsic curvature to the Ashtekar-Barbero connection, (66a) and (69) make manifest the Ashtekar-Barbero interpretation of U established in the previous Section. A (β) = A + (β − i)K. Equations III. EPRL QUANTIZATION OF TWISTED GEOMETRIES The classical phase space of twistors on a link can be quantized, leading to quantum twistor networks and quantized twisted geometries. In the following, we choose a procedure inspired by the EPRL model. We take a Schrödinger representation, and follow a Dirac procedure, quantizing first the unconstrained algebra, and then implementing the constraints. Our starting point is an auxiliary Hilbert space carrying a unitary representation of the canonical Poisson algebra (1), π A ,ω B ] = −i δ B A = − π A ,ω B .(72) Since the constraint structure is commutative, see Fig. 1, let us first study the reduction by the simplicity constraints. That allows us to focus on a single half-link, and consider only the untilded quantities. The Schrödinger representation is given by wave functions f (ω) ∈ L 2 (C 2 , d 4 ω), where d 4 ω = (1/16)(dω A ∧ dω A ∧ cc.) is the canonical SL(2, C)-invariant integration measure 8 on C 2 , and operators ω A f (ω A ) = ω A f (ω A ), π A f (ω A ) = i ∂ ∂ω A f (ω A ).(73) A "momentum" representationπ A = π A ,ω A = i ∂/∂π A is also possible. The two representations are related by the Fourier transform f (π) = 1 π 2 C 2 d 4 ω e − i πω−cc. f (ω), f (ω) = 1 π 2 C 2 d 4 π e + i πω−cc. f (π),(74) whose properties are reviewed in Appendix A. With the usual physicist's abuse of notation, we denote the Fourier transform in the same way as the original function. Since the constraints involve the Euler dilatation operator, ω A ∂/∂ω A , a convenient (generalized) basis is provided by its eigenfunctions. These are homogeneous functions f (ρ,k) (ω), parametrised by a pair (ρ ∈ R, 2k ∈ Z), such that f (ρ,k) (λω) = λ −k−1+iρλk−1+iρ f (ρ,k) (ω).(75) In particular, it follows that ω A ∂ ∂ω A f (ρ,k) (ω) = (−k − 1 + iρ)f (ρ,k) (ω), (76a) ωĀ ∂ ∂ωĀ f (ρ,k) (ω) = (+k − 1 + iρ)f (ρ,k) (ω). (76b) The auxiliary Hilbert space L 2 (C 2 , d 4 ω) carries a unitary, reducible action of SL(2, C) with generators L i and K i . The homogeneous functions span irreducible (infinite dimensional) representations, with Casimirs L 2 − K 2 = k 2 − ρ 2 − 1, L i K i = −kρ. It is a generalized basis, since the homogeneous functions are distributions in L 2 (C 2 , d 4 ω), and not square integrable. This is similar to what happens for e.g. the action of the Euclidean group on L 2 (R 3 , d 3 x), where the irreducible representations are labelled by the total energy ∝ p 2 , but the basis-elements, being plane waves exp i p · x, are not square-integrable. A finite, SL(2, C)-invariant Hermitian inner product is provided by a 2- dimensional surface integral on PC 2 ⊂ C 2 , with measure d 2 ω := i/2ω A dω A ∧ωĀdωĀ. An orthonormal basis is then provided by elements {f (ρ,k) j,m }, labelled by spins j = k, k + 1; . . . and magnetic numbers m = −j, . . . , j corresponding to the canonical SU(2) subgroup of SL(2, C), f (ρ,k) j,m , f (ρ,k) j ,m = i 2 PC 2 ω A dω A ∧ωĀdωĀf (ρ,k) j,m (ω A )f (ρ,k) j ,m (ω A ) = δ jj δ mm .(77) The basis diagonalises L 2 and L 3 . See Appendix A and [70][71][72] for more details. Notice that thanks to the homogeneity of the integrand, (77) is independent of the way PC 2 is embedded into C 2 . The existence of this Hermitian product plays a key role in the constraint reduction. The representations with (ρ, k) and (−ρ, −k) are unitary equivalent. We can thus always restrict ourselves to k > 0, which agrees with what we have done earlier: The half integers k are the quanta of J , choosing them positive fits our gauge condition J > 0 introduced in above. Reasons to consider both are given in [73]. A. Simplicity constraints Since F 1 is first class, it can be imposed strongly. However, the complex constraint F 2 is of second class, making a different procedure necessary. Motivated by deriving the EPRL model, we implement F 1 strongly and F 2 weakly. This can be done introducing a master constraint, a procedure [50] quite common in LQG: F 2 = 0 ⇔ M =F 2 F 2 , and F 1 , M = 0.(78) The new constraint algebra is abelian, and both M and F 1 can be imposed strongly. Choosing a normal ordering as in [2], we find the following quantum constraints, F 1 = β 2 + 1 (β − i)ω A ∂ ∂ω A − (β + i)ωĀ ∂ ∂ωĀ − 2i ,(79a)F 2 = i n AĀωĀ ∂ ∂ω A ,(79b)M = F † 2 F 2 = 2 4 ω A ∂ ∂ω A ∂ ∂ωĀωĀ − L 2 − K 2 + 2 L 2 .(79c) Both F 1 and M are diagonal on our canonical basis (77), and the action can be easily evaluated to give F 1 f (ρ,k) j,m = 2 β 2 + 1 − βk + ρ f (ρ,k) j,m ! = 0 ⇔ ρ = βk,(80)Mf (ρ,k) j,m = 1 2 j(j + 1) − k(k + 1) f (ρ,k) j,m ! = 0 ⇔ j = k.(81) That is, the non-Lorentz-invariant master constraint selects the lowest spin j labels. The resulting wavefunctions are f (βj,j) j,m (ω A ). From Appendix A, we find them to be f (βj,j) j,m (ω A ) = 2j + 1 π 2j j + m ω 2(iβj−j−1) (ω0) j+m (ω1) j−m .(82) These functions are orthonormal with respect to the inner product on the Riemann sphere (77), and provide a map, often denoted Y -map in the literature, from the j-th irrep of SU(2) to the unitary irreducible (βj, j)representation of SL(2, C): Y : H j \|j, m → f (βj,j) j,m ∈ H (βj,j) .(83) In the following, we also use the notation \|β; j, m for the SL(2, C) ket, 9 and write ωω\|β; j, m ≡ f (βj,j) j,m (ω A ). At this point, we would like to discuss two subtle aspects of the quantization. As the action generated by F 1 is non-compact, Dirac's quantization does not lead to a proper subspace of the auxiliary Hilbert space L 2 (C 2 , d 4 ω). Accordingly, the solution space spanned by (82) only makes sense in terms of distributions, to be integrated over the previsouly defined PC 2 inner product (77). The distribution by itself is not a function on the fully reduced phase space C 4 / /F C 2 , but depends also on the F 1 orbits, {F 1 , f (βj,j) j,m } = 0.(84) We can see this explicitly if we insert the parametrization (37) in the definition of the wave function (82), which gives f (βj,j) j,m (ω A ) = 2j + 1 π 2j j + m (z0) j+m (z1) j−m ω 2 z 2j ,(85) where the non-F 1 -invariant quantity ω 2 appears. On the other hand, the half-density [39,50,74] √ d 2 ωf (βj,j) j,m (ω A ) = √ d 2 z 2j + 1 π 2j j + m (z0) j+m (z1) j−m z 2j+2 ,(86) is F 1 -invariant, thanks to the homogeneity of the measure d 2 ω on PC 2 . Hence, it is the half-densities that are properly defined on the reduced phase space. 10 The second remark concerns the case of j = 0. In fact, j = 0 corresponds classically to the degenerate configurations J = 0, for which the twistorial description of the phase space breaks down. To complete the quantization, we need to provide independently the missing state. If we extrapolate (82) to j = 0 we would find a non-trivial state, given by π −1/2 ω −2 . This choice could pose problems with cylindrical consistency, so we fix it by hand to be the trivial state, 9 In the literature, the alternative notation \|βj, j; j, m ≡ \|β; j, m is often found. 10 A toy model may further illustrate this subtlelty. Consider a particle in R 3 . Our constraint be just the radial momentum constrained to vanish, that is F = pr = x · p/\| x\| = 0. The auxiliary Hilbert space is just the familiar L 2 (R 3 , d 3 x). To makepr self-adjoint its quantisation is given in terms of the covariant derivative operatorpr := −i (∂r + r −1 ). Solutions in the kernel of the constraint look like Ψ(r, ϑ, ϕ) = r −1 ψ(ϑ, ϕ). As functions on the original phase-space, they fail to be gauge invariant, since {pr, Ψ} = 0, while the half densities Ψ f (0,0) (ω A ) ≡ 1.(87)√ d 3 x = ψ √ sin ϑdrdϑdϕ are. An argument in favour of this choice is that the primary simplicity constraints are all first class when \|L i \| = 0. Hence, one can identify (87) as the unique invariant state satisfying (21) as strong operator equations. A final comment before moving on. The reader may have also noticed a similarity between the quantum state (82) and the SU(2) coherent states. In fact, the state \|j, z A = 2j + 1 π j m=−j 2j j + m (z 0 ) j+m (z 1 ) j−m z 2j \|j, m ,(88) in the SU(2) j-th irreducible representation, represents a coherent state peaked on the direction identified by z 0 /z 1 on PC 2 ∼ = S 2 , normalized with respect to the measure d 2 z on PC 2 . It is then easy to see that √ d 2 ωf (βj,j) j,m (ω A ) = √ d 2 z z 2 j, z A \|j, m ,(89) which implies that the Y -map endows SU(2) coherent states with the interpretation of functions on the solution space of the simplicity constraints. For later convenience, we also give the overlap ωω\|Y \|j, z A = j m=−j f (βj,j) j,m (ω A ) j, m\|j, z A = 2j + 1 π ω 2(iβj−1) ω\|z ω z 2j .(90) where ω\|z := δ AĀ z AωĀ . Although we are labeling the SU(2) coherent states with spinors, as e.g. in [17], it is only the Hopf section z 0 /z 1 that carries a semiclassical meaning. The norm and the overall phase of the spinor have no physical counterparts from the point of view of SU(2). B. Area-matching constraints For the tilded quantities, since we have an opposite sign in the Poisson brackets (72), it is convenient to take wave functionsf (π ) ∈ L 2 (C 2 , d 4 π ), and π Af (π ) = π Af (π ), ω Af (π ) = i ∂ ∂π Af (π ). Solutions to the constraints can be obtained as before, restricting to the homogeneous functions f (βj,j) j,m (π ). Later aspects of the quantum theory make it convenient to work with a slightly different basis, given by a dual map and an extra phase η β,j , f (βj,j) ,m (π ) := η β,j (−1) j+m f (βj,j) j,−m (π ) = η β,j 2j + 1 π 2j j + m π 2(iβj−j−1) (π 0 ) j−m (−π 1 ) j+m ,(92) where η β,j := ( i 2 ) iρ−j (− i 2 ) iρ+j Γ(j + 1 − iρ) Γ(j + 1 + iρ) .(93) See Appendix A for details on the SL(2, C) dual map. We will also use a ket notation, as for the untilded wave-functions, and write β; j,m\|π π ] =f (βj,j) ,m (π ), where the square bracket keeps track of the fact that we are working with the dual basis. The advantage of this choice of basis is to be related to the following Fourier transform, β; j, m\|π π ] ≡f (βj,j) ,m (π A ) = 1 (2π) 2 C 2 d 4 ω e i 2 π A ω A −cc. f (βj,j) j,m (ω A ),(94) which plays a role in the construction of the spin foam model. Accordingly, we quantize the phase space T 2 by means of a "mixed" Schrödinger representation, with wave functions G(ω, π ) ∈ L 2 (C 2 × C 2 , d 4 ωd 4 π ), and operators (73) and (91). Since homogeneous functions diagonalize the quantum area-matching constraint, a solution is immediately provided by a restriction on the labels, Ĉ f (ρ,k) ⊗ f (ρ ,k ) (ω, π ) = 0 ⇒ (ρ , k ) = (ρ, k).(95) We recover here the reducibility of the system at the quantum level, since the constraint is redundant once we impose both diagonal simplicities. It then amounts to = j. The space of solutions of both simplicity and area-matching quantum constraints is spanned by the functions G (j) m ,m (π A , ω A ) :=f (βj,j) ,m (π A )f (βj,j) j,m (ω A ) ≡ β; j, m \|π π ] ωω\|β; j, m .(96) The argument contains position variables at the source, and momentum variables at the target. They satisfy all constraints, F 1 G (j) m ,m (π A , ω A ) = 0 = F 1 G (j) m ,m (π A , ω A ),(97a)MG (j) m ,m (π A , ω A ) = 0 = M G (j) m ,m (π A , ω A ), (97b) CG (j) m ,m (π A , ω A ) = i ω A ∂ ∂ω A − π A ∂ ∂π A G (j) m ,m (π A , ω A ) = 0,(97c) and depend only on the reduced phase space variables, namely {C, G (j) m ,m } = 0 = {C, G (j) m ,m }, D, √ d 2 ω d 2 π G (j) m ,m = 0.(98) The first brackets can be established thanks to the property G (j) m ,m (λω A , λ −1 π A ) = G (j) m ,m (π A , ω A ) ∀λ ∈ C − {0}, whereas the second requires the use of half-density as described in the previous section. As before, the states are restricted to j = 0, and for j = 0 we independently fix G (0,0) ≡ 1 for later cylindrical consistency of the spin foam model. The boundary state functions carry an irreducible, unitary representation of the Lorentz group, with scalar product induced from (77), and the Y -map is explicitly implemented by the restriction on the irreps. The functions are very similar to projected spin networks [75], but not identical: the difference is that we are quantizing on the twistor vector space, and not on functions on the Lorentz group. The latter representation, and its basis of simple projected spin networks will appear below when we study the EPRL spin foam model. IV. PATH INTEGRAL MEASURE IN TERMS OF TWISTORS In this Section, we use the above framework to express the Liouville measure on the symplectic manifold T * SL(2, C) in a simple and straightforward way in twistor space. This measure plays an important role in the spin foam formalism, and will be used below in deriving the EPRL model. A. Definition of the integration measure On twistor space T 2 there is a natural integration measure given by the symplectic volume, d 16 µ = Φ ∧Φ ∧ Φ ∧Φ , Φ := dπ A ∧ dπ A ∧ dω B ∧ dω B .(99) We are interested in projecting this measure to the reduced phase-space T 2 / /C of gauge orbits generated by the complex area-matching condition C = 0. The constraint being first class, this can be done following the Faddeev-Popov method. However, since the gauge transformations generated by C are just rescalings (14), the Faddeev-Popov determinant is trivial, thus a C-gauge-invariant 12-dimensional integral can be written as d 14 µ gf δ C (C) G. The gauge-fixed measure appearing on the right-hand side is obtained by taking the interior product of the Liouville measure (99) with the generator X C = {C, ·} of the gauge orbits, d 14 µ gf (Z, Z ) := iΥ ∧Ῡ, Υ := ι X C Φ ∧ Φ .(100) We now prove that integrating any function on the reduced phase space T 2 / /C against this measure gives a C-gauge invariant quantity. Let G be defined on T 2 / /C, thus in particular constant along the orbits, {C, G} = 0 = {C, G}. Next, consider two gauge-fixing surfaces D 1 and D 2 , that can continuously be deformed into one another, and intersect all gauge orbits exactly once. 11 Applying Stokes' theorem to the region R bounded by D 1 and D 2 , we find D1 d 14 µ gf G − D2 d 14 µ gf G = i R d ∧ ι X C Φ ∧ ι XCΦ G = i R (∂ +∂) ∧ ι X C Φ ∧ ι XCΦ G ,(101) where ∂ = dω A ∂ ∂ω A + dπ A ∂ ∂π A + dω A ∂ ∂ω A + dπ A ∂ ∂π A(102) is the analytic part of the exterior derivative. But since Φ is already an analytic form of highest degree and L X C Φ = 0 = L X CῩ together with ι X CΦ = 0, we immediately get that i R ∂ ∧ ι X C Φ ∧ ι XCΦ G = i R ∂ ∧ ι X C ∧ Φ ∧ ι XCΦ G = i R ∂ ∧ ι X C + ι X C ∧ ∂ ∧ Φ ∧ ι XCΦ G = = i R L X C Φ ∧ ι XCΦ G = i R Φ ∧ ι XCΦ L X C G = i R Φ ∧ ι XCΦ C, G = 0, This concludes the proof that (100) is a measure on the reduced phase space T 2 / /C. Since we have already recalled the space is isomorphic to T * SL(2, C) T 2 / /C, the measure can also be shown to be equivalent to the standard measure on T * SL(2, C) given by the product of the Lesbegue measure on the algebra, times the Haar measure on the group. It is useful to prove this equivalence explicitly, which we do next. B. Equivalence with canonical measure on T * SL(2, C) To that end, we use the parametrizations (2) and (10). Then, an explicit calculation shows that the analytic part of the Lesbegue measure on the sl(2, C) reads d 3 Π := − 2 3 Tr dΠ ∧ dΠ ∧ dΠ = 1 4 ω A dω A ∧ π B dπ B ∧ d(π C ∧ ω C ).(104) The spinorial decomposition of the analytic part of the Haar measure is more elaborate. We first introduce a complex basis in sl(2, C): Π AB = − 1 2 π (A ω B) , Q AB + = ω A ω B , Q AB − = π A π B ,(105) to work out the analytic part of the Maurer-Cartan form in terms of spinors: ι * (h −1 dh) =(πω) −2 ι * (ω A dω A − ω A dω A )Q − + (π A dπ A − π A dπ A )Q + + 4(π A dω A − π A dω A )Π ,(106) where we used C = 0, and ι is the embedding of the C = 0 hypersurface into T 2 . With this decomposition, the left-handed part of the Haar measure gives d 3 h := − 2 3 Tr h −1 dh ∧ h −1 dh ∧ h −1 dh = = 4 (πω) 3 ι * (ω A dω A − ω A dω A ) ∧ (π B dπ B − π B dπ B ) ∧ (π C dω C − π C dω C ) .(107) Putting the two quantities together one recovers 1 2 9 T 2 gf d 14 µ gf (Z, Z ) δ C C(Z, Z ) G h(Z, Z ), Π(Z) = T * SL(2,C) d 3 Π ∧ d 3 h ∧ d 3Π ∧ d 3h G(h, Π).(108) Here Π(Z) and h(Z, Z ) are short-hand notations for the twistorial parametrisation (2,10). Notice that (107) provides a definition of the Haar measure in terms of spinors. Alternative definitions have been given in [76] for SU (2), and [4] for SL(2, C). They involve an unconstrained integration, with Gaussian measures instead of δ functions and gauge-fixing. Both approaches work as well, since one is integrating functions which do not have a dependence along the Gaussian slope. The result shows that the basic Liouville measure on T * SL(2, C) can be expressed in terms of twistors. Therefore, any gauge invariant added to it fits into this framework. In the following, we will just consider the basic BF measure to derive the EPRL model, but the reader should keep in mind that any further non-trivial term, such as those induced by secondary constraints [77], could also be described in this language. V. CONSTRUCTING THE EPRL SPINFOAM MODEL In this Section we study a specific way to provide quantum dynamics to this system, which leads us to the EPRL spin foam model [8]. The dynamics can be derived in three steps. The first step is a decomposition of the spacetime manifold into 4-cells, and the assignement of the Hilbert space of states (96) to the boundary graph of each cell. The second step is to give dynamics in the bulk with exponentials of the BF action, suitably discretized in terms of twistors. This is the framework of the Plebanski action that describes gravity as BF plus simplicity constraints [19,[36][37][38]. Finally, we integrate the boundary states weighted by the BF action against the measure previously defined. The result reproduces the transition amplitudes of the EPRL model, and thus provides a new and independent derivation thereof, based on the twistorial representation of loop quantum gravity. We will refer specifically to triangulations, with 4-simplices as fundamental cells, but the results immediately generalize to arbitrary cellular decompositions, and what we recover is the generalized EPRL model of [9,10]. A. Discretizing the action The Holst action (16) is equivalent to the following Plebanski action, S Plebanski−Holst [Σ, A, Ψ] = P 2 β + i iβ M Σ A B ∧ F B A [A] + Ψ · S(Σ) + cc.,(109) where Ψ is a Lagrangian multiplier imposing the simplicity constraints S(Σ). See [19,[36][37][38] for details. In particular, the phase space structure is the same [35,41]. This action is particularly advantageous to discretize, since it does not involve the tetrad, but bivectors which we know already how to treat as fluxes. The first step to build the model is a cellular decomposition C of the spacetime manifold. For simplicity, we will refer to a simplicial decomposition, but our construction immediately generalize to arbitrary decompositions, in which case we recover the generalization of the EPRL models appeared in [9,10]. For the case of a simplicial decomposition, C is made of 4-simplices, each of which consists of five tetrahedra and 10 triangles. Every triangle t hinges several 4-simplices. Its dual face f subdivides in wedges w tv [78], each of which intersect one of those simplices. In every 4-simplex a triangle t separates two tetrahedra in the corresponding 3-boundary. The wedge w tv is now bounded by a line, that starts at one of these tetrahedra, intersects t transversally, reaches the other adjacent tetrahedron and passes through the center of the 4-simplex before finally closing to a loop. See Figures 3 and 4 for illustrations. The orientation of the loop ∂w tv is fixed by requiring the relative orientation (t, w tv ) to be positive. The wedge allows us to bridge between the boundary and the bulk of each 4-simplex. Half of the wedge boundary coincides with a link in the boundary graph of the 4-simplex, and carries the T 2 phase space. We have smeared fluxes Σ[t w ] and Σ [t w ] associated with the two tetrahedra sharing the triangle t w . The wedge label keeps track of the 4-simplex to which the triangle belongs. Reintroducing physical constants, we have, hiding the wedge labels, Σ AB = − P 2 2iβ β + i Π AB = P 2 2 iβ β + i ω A π B + ω B π A ,(110) and the same for Σ . The other half of the phase space variables is the holonomy (10) along the boundary link. Let us also introduce two new holonomies, g and g paralelly transporting from the center of the 4-simplex to the two boundary tetrahedra incident to the wedge. In this way, we can also form a wedge loop holonomy, given by the product of the phase space holonomy (10), and the auxiliary bulk holonomies, h B A [∂w] = −g B C g D C ω D π A − π D ω A π E ω E π F ω F .(111) This holonomy extends from the boundary to the bulk, and it is a mixed quantity depending on the phase space variables and the additional bulk holonomies. Notice that we have introduced the phase space individually for each wedge, i.e. for each 4-simplex. Hence, the same link viewed from adjacent 4-simplices carries independent copies of the phase space. Using these variables, we can discretize the BF part of the action as a sum of wedge contributions, S BF [Σ, A] = P 2 β + i iβ M Σ A B ∧ F B A [A] + cc. = ≈ P 2 β + i iβ w Σ A B [t w ]h B A [∂w] + cc. ≡ w S wedge [g w , g w ; Z w , Z w ](112) Taking into account the presence of the two fluxes Σ[t w ] and Σ [t w ] = Σ[t −1 w ], each wedge contribution can be written as S w [g , g; Z , Z] = P 2 β + i 2iβ Σ A B [t w ]h B A [∂w] + Σ A B [t −1 w ]h B A [∂w −1 ] + cc. = = N w 2 gg −1 A B ω B π A + π B ω A + cc.(113) where N w := 1 2 √ πω √ π ω + √ π ω √ πω (114) equals 1 on the C = 0 constraint surface. Notice the factor 1/2 in (113). Its presence will lead to an extra phase in the spin foam amplitude, which we decided to reabsorb in the boundary states via the phase η β,j . B. The EPRL amplitude We construct the quantum amplitude taking the BF action S wedge to propagate the constrained boundary states (96) along the bulk of the wedge. This leads to the following path integral, G (j) m ,m (g , g) := 1 2 15 π 6 T 2 gf d 14 µ gf δ C (C) β; j, m \|π π ]e iSw(g ,g;Z ,Z) ωω\|β; j, m ,(115) where the numerical factors have been chosen for later convenience, and we dropped the subscript w in the variables since we are considering a single wedge at a time. The integral is over a hypersurface T 2 gf of T 2 defining a gauge section of C, and it is invariant of the gauge chosen thanks to the invariance properties (98) and (103). The integral expression is only defined for j = 0, since at the degenerate value the twistorial description of T * SL(2, C) does not work. Therefore, we need to independently fix the wedge amplitude for j = 0. We do so by requiring cylindrical consistency of the final amplitude, which is satisfied by G (0) 00 (g , g) ≡ 1. In the following, we assume that this is the definition at j = 0. The structure of (115) resembles the infinitesimal step of a Feynman path integral, where the position eigenstates are represented by the constrained boundary states, propagated by the BF action. The main result of this section is that the integrals can be explicitly performed leading to the Wigner matrices of simple projected spin networks, that is, G (j) m ,m (g , g) = µ(j) D (βj,j) jm jm (g g −1 ),(116) where µ(j) is an o(1/j) overall function. The results also provides the transformation between quantum twistor network states and SL(2, C) cylindrical functions. Before proving the result, let us briefly review how this quantity leads to the EPRL spin foam model. First, we define amplitudes A f , associated with the faces of the 2-complex. These are obtained taking the product of wedge amplitudes belonging to the face, and summing over the intermediate states. This gives an SU(2) trace Tr j on the magnetic quantum numbers m, which automatically implies that all spins in the face match, and a sum over the overall j, A(g) = j µ f (j) Tr j   w∈f D (βj,j) (g w g −1 w )   . (117a) Finally, we multiply the face amplitudes together and integrate over the remaining connection variables, Z C = v,τ dg vτ f A f (g) (117b) where dg denotes the Haar measure on SL(2, C), and redundant integrations should be dropped to guarantee finiteness of the 4-simplex amplitude [79]. This specific way to express the EPRL partition function through face amplitudes can be found for instance in [80], together with the other equivalent formulations. In the rest of this section, we prove (116). As a first step, let us write the C constraint using a Lagrange multiplier z, and the integral representation of the complex δ-function δ C (C) = i 8π 2 C dz ∧ dz e i 2 zC−cc. ,(118)thus G (j) m ,m (g , g) = i 2 18 π 8 T 2 gf d 14 µ gf C dz ∧ dz exp i 2 zC + i 2 N w gg −1 A B ω B π A + π B ω A − cc. G (j) m ,m (π , ω). Next, we introduce spinors at the center of the 4-simplex obtained parallel transporting with g and g , v ω A = (g −1 ) A B ω B , v π A = (g −1 ) A B π B , v ω A = (g −1 ) A B ω B , v π A = (g −1 ) A B π B .(119) Thanks to the invariance under SL(2, C) transformations of C and the measure, we can change variables in the integral, from the wedge spinors to the v spinors, and, dropping the supscript v, we find G (j) m ,m (g , g) = i 2 18 π 8 C dz ∧ dz T 2 gf d 14 µ gf e i 2 π A (ω A +zω A )− i 2 π A (ω A +zω A )−cc. G (j) m ,m (g π ) A , (gω) A .(120) To perform the integrals on T 2 gf , we now specify which particular gauge section the 14-dimensional surface T 2 gf corresponds to. It has to intersect all gauge orbits (14) exactly once, and the final result is independent of this choice thanks to the manifest gauge invariance of the integrand. We set: T 2 gf ⊂ T 2 : π A , ω A , π A ∈ C 2 arbitrary , but: ω A ∈ PC 2 :      ω 0 = e iϕ cos ϑ 2 ω 1 = sin ϑ 2 for : ϕ ∈ (0, 2π), ϑ ∈ (0, π). (121) In the coordinates chosen the integration measure decomposes according to 1 2 15 T 2 gf d 14 µ gf G = i 2 S ω A dω A ∧ωĀdωĀ C 2 d 4 π C 2 d 4 ω C 2 d 4 π G.(122) We see in (120) the appearence of a δ-function from the d 4 π integration, δ C 2 (ω A + zω A ) = 1 (2π) 4 C 2 d 4 π e i 2 π A (ω A +zω A )−cc. .(123) Writing this δ explicitly, and inserting the expression (96) of G, (120) reads G (j) m ,m (g , g) = − 1 π 4 C dz ∧ dz S ω A dω A ∧ωĀdωĀ C 2 d 4 ω C 2 d 4 π δ C 2 (ω A + zω A ) e − i 2 π A (ω A +zω A )−cc.f (βj,j) ,m (g π ) A f (βj,j) j,m (gω) A = − 4 π 2 C dz ∧ dz S ω A dω A ∧ωĀdωĀ C 2 d 4 ω δ C 2 (ω A + zω A )f (βj,j) j,m (g (ω + zω )) A f (βj,j) j,m (gω) A = − 4 π 2 C dz ∧ dz PC 2 ω A dω A ∧ωĀdωĀf (βj,j) j,m (1 − z 2 )(g ω) A f (βj,j) j,m (gω) A ,(124) where in the second line we used the Fourier transform (74), and in the final line we used the δ function to eliminate an integral. Thanks to the homogeneity property of the canonical basis functions, we can rewrite this expression as G (j) m ,m (g , g) = µ(j) i 2 PC 2 ω A dω A ∧ωĀdωĀf (βj,j) j,m (g ω) A f (βj,j) j,m (gω) A ,(125) where µ(j) := 8i π 2 C dz ∧ dz(1 −z 2 ) −j−1−iβj (1 − z 2 ) +j−1−iβj .(126) The integral on the complex projective plane gives precisely the Wigner matrices for the irreducible unitary representations of the Lorentz group, as a consequence of the scalar product (77), i 2 PC 2 ω A dω A ∧ωĀdωĀf (βj,j) j,m (g ω A )f (βj,j) j,m (gω A ) = D (βj,j) m jm (g g −1 ).(127) The non-analytic complex integral is explicitly computed in the Appendix B, and gives µ(j) = 4 1 + β 2 1 πj e i∆(β,j) ,(128) where e i∆(β,j) is a phase and quickly converges to 1 as j becomes large. This completes the proof of (116), with µ(j) given in (128). Up to this factor, we recover the fundamental structure of the EPRL model. Our derivation shows that its building block, the specific Wigner matrices appearing in (116), have the interpretation of path integrals over twistor space. Concerning the µ(j) factors, these come as a direct consequence of the integral over the z Lagrange multiplier, and are thus absent in derivations of the model not based on twistor space. On the other hand, there is the freedom to assign extra holonomy-independent face amplitudes, and this is typically exploited to recover a factor of 2j + 1 needed to guarantee the convolution property of the transition amplitudes at fixed graphs [20,21]. If one so wishes, the same freedom can be exploited here to introduce extra face weights given by (2j + 1)/µ(j), thus completing the matching with the EPRL model. VI. SEMICLASSICAL PROPERTIES In this Section, we study some semiclassical properties of the model, using the twistorial formalism. We introduce a notion of curvature and torsion tensors written in terms of the spinors. Next, we will relate our twistors to the spinors used in the asymptotic analysis of Barrett et al. [17], thereby embedding the large spin behaviour in the original phase space. A. Curvature tensor in terms of spinors An important application of our construction is that it allows us to introduce a curvature tensor, and study its decomposition in terms of irreducible components and its Petrov classification. To that end, we work with the wedge spinors at the center of the 4-simplex defined in equation (119), and reintroduce the v supscript. Once we have performed in (115) the integration over v π, there is a Lagrange multiplier z in a δ-function on C 2 , imposing v ω A = −z v ω B .(129) The transformation mapping ( v π, v ω) to ( v π , v ω ) must be a proper SL(2, C) element, but v π A v ω A = 0 implying that this is a complete basis in C 2 . We can thus decompose v π into the v π and v ω spinors to get: v π A = −z −1 v π A − u v ω A ,(130) where u ∈ C is the component of v π with respect to v ω, hence depends on the bulk holonomies g and g . If we now look at the wedge holonomy, we get h A B [∂w] = − 1 πω z v ω A v π B − (z −1 v π A + u v ω A ) v ω B ≈ δ A B + F A B [w],(131) at first order in the coordinate area of the wedge. Lowering one index, F AB [w] ≈ − 1 πω (z − z −1 ) v ω (A v π B) − u v ω (A v ω B) .(132) At this point, we can use the spinors to introduce a (complex) null tetrad in the internal Minkowski space, analogue to the Newman-Penrose tetrad. This is localised at the center of the 4-simplex, and can be defined as follows: I ≡ i v π A v πĀ, k I ≡ i v ω A v ωĀ, m I ≡ v ω A v πĀ,m I ≡ − v ωĀ v π A ,(133)2 = k 2 = m 2 = 0, I k I = m Im I = −\| v π v ω\| 2 . Here we used abstract index notation, and X AĀ ≡ X I : X AĀ = i √ 2 σ AĀ I X I ∈ C 2 ⊗C 2 (see Appendix) . We took capital latin letters starting at I for internal Lorentz indices, as customary in spin foam literature. These should not be confused with spinorial indices, starting at the beginning of the alphabet. We remark that m and m identify the plane of the triangle in the reference frame of the 4-simplex. In fact, Λ(g) I K Λ(g) J L Σ KL [t] ∝ im [ImJ] ,(134) where Σ IJ ∝ (1 − β )Π IJ , and Λ(g) I J = g A BḡĀB denotes the proper Lorentz transformation associated to the bulk holonomy g ∈ SL(2, C) previously introduced. Consequently, the bivector [I k J] spans a time-like plane orthogonal to the triangle in the flat reference metric. By construction, the wedge is parallel to this plane. Then, we can define the following curvature tensor in the internal Lorentz indices, F CD IJ [w] := a w F CD [w] [I k J] ,(135) where the overall scale factor a w measures the area of the wedge and depends a priori on all the details of the geometry of the 4-simplex. The expression (135) defines a curvature tensor whose only non-vanishing components lie in the plane of the wedge. This is the usual set-up of Regge calculus and loop quantum gravity, and it is consistent with the tetrahedra bounding the 4-simplex being flat. In this way, we have achieved something new for the spin foam formalism, that is a description of the full curvature tensor. Having introduced a (chiral) curvature tensor, 12 we can decompose it into its SL(2, C) irreducible parts, 1), using the tensor: (Ψ, T, Φ, Φ 0 ) ∈ (2, 0) ⊕ (1, 0) ⊕ (0, 0) ⊕ (1,F CD IJ ≡ F CD AĀBB = Ψ ABCD + (T DB CA + T CA DB ) + Φ 0 ( CA DB + DA CB ) ¯ ĀB + AB Φ CDĀB . (136) Notice that here the (0, 0) ⊕ (1, 1) components are themselves chiral, in the sense that they contain the lefthanded projector as in (17). For instance, Φ 0 contains both δ I[K δ L]J and IJKL traces. From the continuum theory, we know that if the connection is Levi-Civita the curvature coincides with the Riemann tensor. In this case, Ψ is the (chiral) Weyl tensor, (Φ, Φ 0 ) give the Ricci tensor with Φ 0 as its trace, and the algebraic Bianchi identities guarantee that the (1, 0) component vanishes. Conversely, any non-zero contorsion contributes to all components. Applying this decomposition to (135), expressed in terms of spinors through (132) and (133), we obtain a spinorial description of the various irreducible components. Explicitly, Φ 0 = − a w 24 (z − z −1 )\|πω\| 2 , (137a) Φ CDĀB = a w 2 (z − z −1 ) v ω (C v π D) − u v ω (C v ω D) v π (Ā v ωB ) ,(137b)Ψ ABCD = a w 2πω πω (z − z −1 ) v ω (A v ω B v π C v π D) − u v ω (A v ω B v ω C v π D) ,(137c)T AB = a w 8 uπω v ω A v ω B .(137d) We see that all components are non-vanishing, hence the off-shell wedge curvature carries contorsion. This is rather welcomed: what we have defined is an off-shell quantity, and in loop quantum gravity the contorsion is an independent component of the connection, thus can take any value off-shell. To further interpret the equations, consider the case of vanishing torsion. Then the algebraic Bianchi identities impose Φ 0 ! =Φ 0 , T AB ! = 0, Φ ABCD ! = Φ CDĀB .(138) Looking at (137), we see that the conditions are fulfilled provided z ! ∈ R, u ! = 0.(139) In this way we are able to identify a component of the bulk holonomy, u, which describes contorsion. At the same time, the Lagrange multiplier z picks up a geometric interpretation on the C = 0 constraint surface, where Re(z) is related to the Riemann tensor, and Im(z) to contorsion. Finally, let us discuss the algebraic classification of the resulting Riemann tensor. When (139) are satisfied, we see from (137c) that the Weyl part has two principal null directions. It is thus of Petrov type D, precisely as posited in Regge calculus [81]. Unlike in ordinary Regge calculus, we are able to describe additional contorsion components. In the EPRL model, the on-shell wedge is flat, thus all components of this tensor vanish. One should then look at a curvature tensor associated to faces, which can be also constructed with our methods. They are in fact rather general, all is required is the existence of a Lorentzian phase space structure. B. Wedge flatness and large spin limit A key property of the EPRL model is to reproduce exponentials of the Regge action in the large spin limit [17]. The proof heavily rests upon the use of spinors, and an extra input of wedge (i.e. 4-simplex) flatness. What we want to show in this Section is the relation between the spinors appearing in the semiclassical analysis and the twistor phase space description of our paper. The starting point of the semiclassical analysis is the "propagator" (127), associated to each wedge. This can be written either in terms of the original phase space spinors, or those transported to the center of the 4-simplex via (119), P = PC 2 d 2 v ω f (βj,j) j,m (g v ω A )f (βj,j) j,m (g v ω A ) = PC 2 d 2 ω f (βj,j) j,m (g g −1 ω A )f (βj,j) j,m (ω A ).(140) By direct comparison, we identify v ω with the spinor denoted z in [17]. The Hermitian scalar product appearing in (127) can be written as a double integral, over both ω and π, if one uses the dual map given by the complex structure as shown in Appendix A. Upon doing so, we can identify ( v ω, v π) with the spinor pair denoted (z, w) in [17], and bring in the full phase space structure. However, π plays no role in the semiclassical analysis in the literature, an aspect we will comment upon below. Instead, the expression (140) is used, with the help of a different type of additional spinors. Following [13], the traces over the magnetic indices in (117a) are replaced by resolutions of the identity written in terms of the spinorial coherent states, that is m j=−m \|jm jm\| = S 2 d 2 ξ \|j, ξ A j, ξ A \|,(141) where S 2 is parametrised in stereographic coordinates by the Hopf section ξ 0 /ξ 1 , and −2id 2 ξ = ξ A dξ A ∧ cc. is the canonical measure. Here we took the SU(2) spinors of unit norm, which we can do without any loss of information in the following. The only semiclassical information contained in these states concerns directions in R 3 , identified by the Hopf section, or equivalently n i [t]τ AB i = 1 2i ξ (A δ B)BξB ,(142) and interpreted as unit vectors normal to the triangle in the frame of the source tetrahedron. With the resolution of the identity in terms of coherent states, (140) contains the overlap ωω\|j, ξ A , computed in (90). Thanks to the factorization property of this overlap, (140) can be rewritten as the exponential of an action where the spin appears linearly, P = 2j + 1 π d 2 v ω exp(s w ) g v ω 2 g v ω 2 , s w [ v ω, g, g ; ξ, ξ ] = βj   ln g v ω 2 g v ω 2 + Φ   ,(143) where Φ := 2 β ln   g v ω\|ξ g v ω ξ \|g v ω g v ω   .(144) Now comes the key input of wedge flatness. If the wedge is flat, the bulk holonomies add up to the phase space SL(2, C) holonomy, h(ω, ω , π, π ) = g g −1 , and thus ω A = g A B v ω B , ω A = g A B v ω B .(145) This immediately leads to g v ω 2 g v ω 2 = ω 2 ω 2 = e Ξ .(146) We have recovered the quantity identified with the extrinsic curvature in the classical study of the phase space, Section II D. Hence, the action in (143) cointains the area-angle term βjΞ. This is a quite satisfying state of affairs, as it shows that the model contains the correct structure of the Regge action with areas βj. Notice however that the equations of motion of this action have little in common with discrete general relativity. What is needed at this point are constraints relating the connection and area-angle variables among each other, so to recover the edge lengths as fundamental variables, and the specific functional dependence of areas and dihedral angles upon them. Crucial help in this direction comes from the large spin limit. The action (143) is complex and with negative real part, and in the large spin limit the path integral (143) is dominated by configurations where the real part of the action vanishes. By inspection of (144), these are located at ω proportional to ξ, that is ξ A = e iϕ ω A ω , ξ A = e iϕ ω A ω .(147) Using again the wedge flatness, these equations imply ξ A = e Ξ/2 e −i(ϕ−ϕ ) (g g −1 ) A B ξ B .(148) Remarkably, the saddle point equations relate the phase space spinors to the SU(2) boundary spinors in the same way as the primary simplicity constraints relate them to the reduced SU(2) spinors (36). However, there is an important catch, in that the relative phase is not fixed, as it is instead in (36). Therefore, we can identify the Hopf sections of the ξ's with our z's, but not the phases. This is consistent with the fact that the phases of the SU(2) coherent states carry no semiclassical information. Notice also from (148) that the ξ's are not parallel transported by the Ashtekar-Barbero holonomy U , as it is for the z's. The information about the extrinsic curvature is not in the boundary spinors, but in the bulk holonomies, through (146). The actual dependence of the action (143) on Ξ, a D-dependent quantity, may seem puzzling at first sight, since the boundary states implement strongly the D constraint. It is the wedge BF action that breaks the D-symmetry, and so reintroduces a dependence on Ξ in the path integral. 13 This is an important point, because as we discussed at length, a non-trivial embedding of SU (2) in the larger space is necessary to properly talk about extrinsic curvature and Ashtekar-Barbero holonomy. So although the boundary states are insensitive to the extrinsic curvature, it enters the picture via the bulk dynamics. We have shown that in the large spin limit we can embed the boundary data in the holonomy-flux phase space. The Hopf sections of the SU(2) coherent states are mapped to their equivalent of the reduced spinors (36), and the Ξ is mapped in the integration variables through (146). On the other hand, the classical areas z t do not appear in the action (143) nor in the semiclassical analysis. What is interpreted as the areas are purely quantum numbers, the spins j t . To complete the saddle point analysis, it remains to impose the vanishing of the gradient of (143). A crucial ingredient at this point is to restrict attention to foams given by duals to triangulations. Then, the wedge flatness implies the flatness of each 4-simplex in the triangulation. In this case, it was shown in [17] that the vanishing of the gradient of (143) can only be satisfied if the boundary data (j t , ξ A t ) correspond to a Regge 4-simplex, (j t ( e ), ξ A t ( e )). At the saddle point, the group elements g are the Regge-Levi-Civita holonomies, and Ξ = θ( e ) is the dihedral angle between 4-normals. Notice in particular that as a consequence, the h holonomies solve the secondary constraints at the saddle point. This gives the right functional dependence in (143), and the correct Regge dynamics is recovered if the term in Φ does not contribute. This was proved in [17], where it was put to zero as a phase choice on the boundary of the 4-simplex. In fact, that this term does not contribute can be shown more generally, face by face on the foam. From (117a), the action on a face is simply s f = w s w . If g g −1 is a pure boost, as at the saddle point, then we immediately have w Φ w = 0, and so on each face, s f = βj( e )Ξ f ( e ), Ξ f := w∈f Ξ w .(149) The fact that the second term of s w vanishes exactly if the connection is Levi-Civita makes the expression look very much like a discrete form of the Holst action. The discussion shows that the classical interpretation of the large spin asymptotics is consistent with our phase space structure. Furthermore, it provides an instance of how a physical connection, solution of both primary and secondary simplicity constraints, provides a non-trivial embedding of SU (2) in the twistor space. The embedding depends on the spins, not on the norms of the SU(2) spinors, because in this model the areas are quantized. In this sense, the model is "semi-coherent". The directions are represented by classical quantities in phase space, but the areas are quantum numbers. This semi-coherence shows up in the way the ππ half of the phase space immediately drops out of the analysis. Their completely auxiliary role is evident also from the starting point (120), where we showed that the EPRL wedge amplitude is a path integral on twistor space: If we interpret the integrand as the exponential of an action, the latter has no interesting dynamics, only trivial solutions. Technically, this is due to the lack of gluing of the phase space spinors, and it is not a problem for the semiclassical analysis of [17] because as explained, it is the spins and the boundary data spinors that carry the gluing and the classical interpretation of tetrahedra. The lack of gluing is in turn inherited from imposing the simplicity constraints as restrictions on the spin labels of the boundary states, so again it is a sign of the "semi-coherence" of the model. A face amplitude respecting all gluing conditions would not involve any boundary spin labels but be just an integral over twistors, one for each tetrahedron adjacent to the spin foam face, and would not use independent variables on each wedge. Our formalism provides the means to formulate the LQG dynamics directly in these more covariant terms. Amplitudes obtained in this way should not deviate much from the EPRL model, but we leave this open for future work. The basic question here is whether it is possible to formulate a spin foam model as a path integral with an action manifestly discretizing general relativity, thus making the naive semiclassical limit immediate. This idea has been addressed in various ways in the literature [65,66,77,[82][83][84][85], and a similar approach has been developed for the Euclidean theory in [86]. We think our twistorial framework provides tools rich enough to make progress in this direction. VII. CONCLUSIONS AND PERSPECTIVES The twistorial description of loop quantum gravity is a powerful tool to investigate both classical and quantum aspects of the theory. As previously shown in [2][3][4], twistors can be used to describe the theory's covariant phase space on a given graph, that is holonomies and fluxes of SL(2, C). This is achieved assigning a pair of twistors with equal norms to each link of the graph. In this way, we embed the non-linear holonomy-flux algebra in a much simpler algebra of canonical Darboux form. The first advantage of doing so shows up in dealing with the simplicity constraints. In the usual path to the quantum theory, one solves the (primary and secondary) simplicity constraints at the continuum level, and then smears the resulting SU(2) variables. Here we have shown that swapping reduction and smearing is also possible. One smears the covariant SL(2, C) variables, and the SU(2) variables are recovered solving the discretized simplicity constraints. As in the continuum, the process requires solving the primary and secondary constraints in successive steps. The primary constraint surface is a 7-dimensional hypersurface in T * SL(2, C), parametrized by SU(2) holonomies and fluxes, plus the dihedral angle Ξ between the normals to the source and target 3-cells, in the time gauge. In a general gauge, the picture is unchanged, with an additional boost on each 3-cell transforming the normal out of its canonical time gauge. From the twistorial perspective, the constraint surface is parametrized by what we call simple twistors, parametrized by SU(2) spinors and the dihedral angle, through equations (35) and (42). The familiar notion of simple bivectors translates elegantly into simplicity of twistors. We also found that the dihedral angle is a good coordinate along the orbits generated by the diagonal simplicity constraint. This has important consequences: Assuming that the secondary constraints turn the diagonal simplicity constraints into second class, their solution is then provided by a specific, physical, gaugefixing section through the orbits. Whatever the gauge section is, we proved that the SU(2) holonomy corresponds to the Ashtekar-Barbero connection, with Ξ measuring the extrinsic curvature projected along the normal to the face. The proof introduces a nice discrete counterpart to the continuum formula A (β) = A + (β − i)K. It is given by the relation between the SU(2) holonomy and the initial Lorentzian holonomy, equations (49) and (60), or equivalently by the relation between the SU(2) class angle and the dihedral angle, equation (65). The results show that a consistent symplectic reduction can be obtained after smearing, without any outsourcing from the continuum theory, and have important applications for the interpretation of the theory and the construction of spin foam models. It remains to formulate an explicit discretization of the secondary constraints, and study the gauge sections they identify. This has been an important open question in the field for many years. The twistorial formalism offers a way to address it, and we hope to come back to this in future research. For the moment, we verified our treatment of the secondary constraints using the simple case of a flat 4-simplex, which is also the one relevant for the EPRL spin foam model. Unlike the case of primary constraints, the solution to the secondary constraints involves a non-local graph structure, and can not be found on each link separately. Twistors lead to significant insights also in the quantum theory. We quantize the phase space and its Poisson algebra with a Schrödinger picture, and obtain quantum twistor networks, instead of cylindrical functions on the group. The new states are the homogeneous functions appearing as the canonical basis of the unitary representations of the Lorentz group. They still carry a representation of the holonomy-flux algebra as a sub algebra, and achieve a separation of the source and target structures of the node, which are entangled in the usual holonomy representation. Proceeding with Dirac, implementing the diagonal simplicity constraints strongly and the off-diagonal constraints weakly, selects the subspace of "simple" irreps (ρ = γj, k = j). In this way we obtain a representation for the Hilbert space of loop quantum gravity, where the argument of the wave-function is a pair of spinors instead of a group element. The representation is related to the simple projected spin networks [5,75] that appear as boundary states of the EPRL spin foam model [5][6][7]. In fact, we show that the translation between the spinorial wave functions and the cylindrical functions is provided precisely by the spin foam wedge amplitude. A future goal of our research is to evaluate radiative corrections of quantum gravity transition amplitudes. These have so far provided very hard to compute, and the hope is to improve computational power through the complex analysis methods made available by the twistor language. To that end, we established some preliminary results in this paper. The first concerns rewriting the Liouville measure of T * SL(2, C) in terms of twistors. This is given by equation (108). The second is a discretization of the BF action as a bilinear in the spinors, see (113). Using these tools we gave an independent derivation of the EPRL model, where each individual wedge amplitude is a path integral in twistor space resembling the infinitesimal step of a Feynman path integral, with the position eigenstates replaced by the quantum states solution to the constraints, propagated with the BF action along the bulk of the wedge. We then investigated some of the semiclassical properties of the EPRL model. We defined a Newman-Penrose frame through the spinors, and used it to compute the curvature tensor. We computed its irreducible components, and found two interesting results. The first is the presence, in the off-shell formalism, of torsional components. This is important for the consistency of the theory, because the models are meant to be a quantization of first-order gravity. After imposing a condition of vanishing torsion through the Bianchi identities, we identified the Weyl and Ricci components of the (now Riemannian) curvature tensor, and showed the latter to be of Petrov type D, as it is in Regge calculus. In the EPRL model, the on-shell value of the wedge curvature is zero, and it is the face curvature tensor that carries the dynamical information. This can also be studied with our techniques. The flatness of the wedge is also crucial in deriving the wellknown asymptotic behaviour of the EPRL model [17,18]: In the large spin limit, the amplitude on a 4-simplex reproduces exponentials of the Regge action. An interesting question concerns the relation between the Regge behavior and the phase space. We answered the question showing explicitly that: (i) the saddle point equations capture the spinorial version of the simplicity constraints, and (ii) the secondary constraints are solved by the Levi-Civita connection of Regge calculus. This defines an embedding of the SU(2) variables in the covariant phase space, with a non-trivial section of the dihedral angles, Ξ(j t ). The embedding is a function of the Regge data j t , that is the areas of the triangles. As a particular feature of the model, these areas are quantized, and not classical quantities. This reflects the semi-coherence of the boundary states, which are semiclassical in the directions, but sharp on the areas, and it also shows up in the lack of a real dynamical role for the momentum π. In this respect, it would be interesting to look at a version of the path integral where the areas are classical data. The construction of such path integral, and its precise relation to the EPRL model, we leave for future work. ACKNOWLEDGMENTS Discussions with Sergey Alexandrov, Eugenio Bianchi, Maïté Dupuis, Jurek Lewandowski, Alejandro Perez, Carlo Rovelli and Mingyi Zhang are gratefully acknowledged. We would also like to thank the Perimeter Institute, and the Physics Departments of the Universities of Warsaw and Erlangen for hospitality during part of the time this project was developed. This work is partially funded by the ANR Program Blanc LQG09. Appendix A: Spinors, the Lorentz group and its unitary representations In this Appendix we review and collect properties of the Lorentz group and its representations that were used in the paper. Further details can be found in [70][71][72] . Index-free notation We used explicit spinorial indices in most of the formulas. It is also convenient to introduce a ket notation and dispose of the indices. We define \|ω = ω A , ω\| = \|ω † = δ AĀωĀ , ω 2 = ω\|ω ,(A1) where, again, the Hermitian conjugate is taken with respect to the time normal introduced in (20) and it is SU(2) invariant but not SL(2, C) invariant. A further SU(2) quantity is the complex structure J (or "parity"), which allows us to introduce the dual spinor (Jω) A , which we denote \|ω] for brevity: \|ω] := − \|ω = −δ AB¯ BĀωĀ, [ω\| = ω A = ω B BA , [π\|ω = AB π A ω B = πω.(A2) The latter bi-linear is related to the SU(2)-invariant norm by the SU(2) complex structure, [π\|ω = Jπ\|ω .(A3) In the index-free notation, the holonomy-flux variables are Π i = [ω\|τ i \|π , h = \|ω [π\| − \|π [ω\| [π\|ω [π \|ω ,(A4) and the simplicity constraints [π\| = re i θ 2 ω\|.(A5) The papers [3,4,58] use the index-free notation, but with different conventions. Among these, the left-handed spinor ω A is denoted as \|t , as here, while π A is written as u\|, so that the Lorentz bilinear reads u\|t . Spinors and the Lorentz group Consider the group of special orthochronous Lorentz transformations L ↑ + and its universal cover, that is SL(2, C). The intertwining σ-matrices provide the relation between them two, that is the map Λ : SL(2, C) g → Λ(g) ∈ L ↑ + : g A BḡĀB σ BB I = Λ(g) J I σ AĀ J .(A6) These intertwiners consist of the identity and the three Pauli matrices σ A Bi , we set σ AĀ 0 = δ AĀ , σ AĀ i = σ A Bi δ BĀ .(A7) But there is another important invariant structure, given by the -tensors AB = − BA , AB = − BA , AB CB = C A = δ A C , 01 = 01 = 1.(A8) Being SL(2, C) invariant, we use them to move spinor indices: π A = π B BA , π A = AB π B .(A9) For if X I be a point in Minkowski space the intertwiners (A7) provide an antilinear bijection into the space of anti-Hermitian 2 × 2 matrices: X AĀ ≡ X I : X AĀ = i √ 2 σ AĀ I X I ∈ C 2 ⊗C 2 .(A10) This anti-isomorphism respects the rules for index moves for both spinors and Minowski vectors: X AĀ Y AĀ = −X 0 Y 0 + X 1 Y 1 + X 2 Y 2 + X 3 Y 3 = X I Y I .(A11) The differential map introduced in (A6), that is the push-forward Λ * gives another isomorphism. It maps sl(2, C) into so(1, 3) according to Λ * : sl(2, C) Ω A B = 1 2 Σ A B IJ Ω IJ → Ω I J ∈ so(1, 3),(A12) where we have introduced the Σ-matrices Σ A B IJ = − 1 2 σ AC [IσCB J] ,(A13) which are selfdual 1 2 M N IJ Σ M N = Σ IJ = iΣ IJ (A14) for 0123 = 1. In (A13) the anti-symmetrisation is meant to be over Minkowski indices I, J only. These generators are sometimes a bit cumbersome to work with, if we introduce the anti-Hermitian matrices τ A Bi = 1 2i σ A Bi we can write, 1 2 Ω IJ Σ IJ = τ i 1 2 l i m Ω lm + iΩ i o =: τ i Ω i ,(A15) and call Ω i ∈ C 3 the selfdual components of Ω I J ∈ so(1, 3). The group elements of L ↑ + are generated by the exponential of its Lie algebra: Λ(Ω) = exp 1 2 Σ IJ Ω IJ ∈ L ↑ + , where Σ IJ M N = 2δ M [I η J]N (A16) are the generators of the algebra, and η IJ denotes the internal Minkowski metric. We introduce the generators of rotations and boosts respectively L i = i 2 l i m Σ lm , K i = iΣ io ,(A17) together with their complex combinations Π i := 1 2 L i + iK i ,Π i := 1 2 L i − iK i .(A18) The isomorphism (A12) maps them towards the selfdual generators: Λ * Π i = iτ i , Λ * Πi = 0 (A19) The commutation relations split into two sectors of opposite chirality: [Π i , Π j ] = i ij m Π m , [Π i ,Π j ] = i ij mΠ m , [Π i ,Π j ] = 0. (A20) The two Casimirs of the Lorentz algebra are: 1 4 IJM N Σ IJ Σ M N = 1 2 Σ IJ Σ IJ = −2L i K i , and 1 2 Σ IJ Σ IJ = K i K i − L i L i (A21) Unitary representations We will give here a short overview over the unitary irreducible representations of the Lorentz group, key for understanding the EPRL model. Further reading may be found in [70][71][72]. The representation SL(2, C) g : ω ∈ C 2 → gω ≡ g A B ω B .(A22) of SL(2, C) on C 2 is already irreducible, but not unitary. The induced representations on functions f : C 2 → C, with the natural L 2 (C 2 , d 4 ω) inner product is unitary though reducible. This immediately follows from the homogeneity and unimodularity of the transformation. Irreducible unitary representations are then built just from homogenous functions on C 2 . For the principle series, the weights of homogeneity are parametrised by a half integer 2k ∈ Z and some ρ ∈ R. That is we are dealing with functions ∀λ = 0, ω A ∈ C 2 − {0} : f (λω A ) = λ −k−1+iρλ+k−1+iρ f (ω A ).(A23) From this formula we can easily see for if the pair (ρ, k) label an irreducible unitary representation, its complex conjugate is labelled by (−ρ, −k). A canonical basis in this infinite-dimensional space is given by the following functions, f (ρ,k) j,m = 2j + 1 π ω 2(iρ−1) R (j) (U −1 (ω)) j m ,(A24) where j ≥ k and m = −j, . . . , j, and R j (U ) m n = j, m\|R (j) (U )\|j, n , for U (ω) = 1 ω ω 0 −ω1 ω 1ω0 ∈ SU(2),(A25) are the entries of the spin j Wigner matrix for the SU(2) element U (ω) constructed from the spinor. The basis elements (A24) diagonalise a complete set of commuting operators: L 2 − K 2 f where L i and K i are the quantisation of the generators earlier. It is quite convenient to introduce a multi-index notation to group the pair (j, m) into a single index µ. We will also use the notationμ to keep track of the complex conjugate representation, and use Einstein's summation convention for the µ indices. With our choices, the matrix representation of the group is the right action, defined according to: D(g)f (ρ,k) µ (ω A ) := f (ρ,k) µ (g −1 ) A B ω B = f (ρ,k) µ (−ω B g B A ) = D (ρ,k) (g) ν µ f (ρ,k) ν (ω A ).(A27) Since the representation is unitary, it admits an SL(2, C)-invariant Hermitian inner product. This is defined as a surface integral on PC 2 ⊂ C 2 , 14 f (ρ,k) µ \|f (ρ,k) ν = i 2 PC 2 ω A dω A ∧ωĀdωĀf (ρ,k) µ (ω A )f (ρ,k) ν (ω A ) = δμ ν ,(A28) its value being independent of the way PC 2 is embedded into C 2 thanks to the homogeneity of the integrand. The SL(2, C) group locally represents the group of special orthochronous transformations. To recover the full Lorentz group we also need parity P f (ρ,k) µ (ω A ) = f (ρ,k) µ (δ AĀωĀ ),(A29) and 2j ∈ N + . Although the integrand is not holomorphic, the integral can be manipulated to a contour integral using Stokes' theorem. To do that, we write (1 − z 2 ) a = 1 2 ∂ ∂z zF (−a, 1 2 ; 3 2 ; z 2 ) ,(B2) where [87] F (a, b; c; z) = Γ(c) Γ(b)Γ(c − b) 1 0 dt t b−1 (1 − t) c−b−1 (1 − tz) −a , Re(c) > Re(b) > 0,(B3) is the hypergeometric function. (B2) can be verified using x 0 dt(1 − t 2 ) a = x 2 1 0 dt t − 1 2 (1 − x 2 t) a .(B4) Hence, I β,j = i 4 D dz ∧ dz ∂ ∂z zF (−a, 1 2 ; 3 2 ; z 2 ) (1 −z 2 ) b = i 4 D d ∧ dz zF (−a, 1 2 ; 3 2 ; z 2 )(1 −z 2 ) b .(B5) The hypergeometric function F (·, ·; ·; z 2 ) is analytic (hence differentiable) unless z 2 ∈ {x ∈ R\|x ≥ 1}, where there is a brach cut, so the integration domain D denotes the complex plane cut along the real x-axis from −∞ to x = −1 and x = 1 to x = ∞. Applying Stoke's theorem, we find I β,j = i 4 lim 0 lim R→∞ h + R,ε + h − R,ε + k R,ε dz zF (−a, 1 2 ; 3 2 ; z 2 )(1 −z 2 ) b ,(B6) where h ± ∞,ε are Hankel contours encircling the points ±1, while k R,0 denotes a circle of radius R around the origin. Orientation and shape of the contour are fixed in Fig.5. The contribution from the integral around the Therefore, in the limit R → ∞ only the Hankel contours h + ε,∞ and h − ε,∞ can contribute to the integral I β,j . A moment of reflection reveals them to be equal: h − ε,R dz zF (−a, 1 2 ; 3 2 ; z 2 )(1 −z 2 ) b = h + ε,R dz zF (−a, 1 2 ; 3 2 ; z 2 )(1 −z 2 ) b .(B7) The problem when trying to evaluate these integrals is that F (·, ·; ·; z) is generally not single valued for z real and z > 1. However there is now a remarkable identity relating the hypergeometric function around the branch cut to those regions where it behaves perfectly regular, it reads [87]: F (a, b; c; z) = Γ(c)Γ(c − a − b) Γ(c − a)Γ(c − b) F (a, b; a + b − c + 1; 1 − z)+ + (1 − z) c−a−b Γ(c)Γ(a + b − c) Γ(a)Γ(b) F (c − a, c − b; c − a − b + 1; 1 − z)(B8) We insert this identity into I β,j , and eventually get: I β,j = i 2 lim ε 0 h + ,∞ dz z(1 −z 2 ) b Γ( 3 2 )Γ(a + 1) Γ(a + 3 2 ) F (−a, 1 2 ; −a; 1 − z 2 )+ − 1 2a + 2 (1 − z 2 ) a+1 F ( 3 2 + a, 1; a + 2; 1 − z 2 )(B9) In the limit → 0 the second part of this equation vanishes: For if a − b ∈ Z, the function (1 −z) b (1 − z) a+1 F ( , ; ; 1 − z) is single valued (though not analytic) around the branch cut. But we can split the Hankel contour in two parts, one lying in the upper half of the complex plane, the other one in the lower half. Since the integrand is single-valued around the cut, and both contributions appear with opposite signs they cancel in the limit of → 0. Concerning the first part of (B9), We have thus reduced our original integral (B1) to an ordinary analytic line integral. This we calculate by the usual methods and find lim ε 0 h + ε,∞ dz(1 − z 2 ) b = −2i sin(πb) Γ(− 1 2 − b)Γ(b + 1) Γ( 1 2 ) .(B12) With Γ(z)Γ(1 − z) sin πz = π we further simplify and eventually get for j > 0: I β,j = − iπ β − i 1 4j Γ(−iβj + j)Γ(+iβj + j + 1 2 ) Γ(+iβj + j)Γ(−iβj + j + 1 2 ) = π 1 + β 2 1 4j e i∆(β,j) . (B13) In the large spin limit we can use Stirling's asymptotic formula to approximate the ratio of gamma functions by (1 + iβ) 1/2 /(1 − iβ) 1/2 , and I β,j ≈ π 1 + β 2 1 4j , for j 1. The integral (B1) is ill-defined for j = 0. This is not directly relevant for the spin foam model, where the amplitude at j = 0 is assigned independently requiring cylindrical consistency. Nonetheless, the integral can be regularized to vanish at this singular value. Let us first introduce polar coordinates, and perform a partial fraction expansion to find: The last two integrals can readily be performed. Set x = cos ϕ to get: I β,0 = i 2 C dz ∧ dz 1 (1 − z 2 )(1 −z 2 ) = π −π dϕπ −π dϕ sign(ϕ) e iϕ − e −iϕ = −2i 1 0 dx 1 − x 2 .(B16) The second integral goes around the unit circle. We recognise the integrand is analytic along this path unless it reaches the point z = −1, and can thus smoothly deform the integration domain to find: π −π dϕ ϕ e iϕ − e −iϕ = − \|z\|=1 dz ln z z 2 − 1 = = − lim ε 0 1 0 dx ln(−1 + x − iε) (1 − x) 2 − 1 − 1 0 dt ln(−1 + x + iε) (1 − x) 2 − 1 + i π −π dϕεe iϕ ln(εe iϕ ) ε 2 e 2iϕ − 1 = = 2iπ 1 0 dx 1 (1 − x) 2 − 1 = −2iπ 1 0 dx 1 − x 2 .(B17) We put the branch cut for the complex logarithm on the negative real axis, got for any 0 < x < 1 that and used that the integral around the small semicircle vanishes due to lim ε→0 ε ln ε = 0. Hence, in both (B16) and (B17) there appears the very same integral. Each integral diverges logarithmically at its upper bound x = 1: 2 1 0 dx x 2 − 1 = ln \|x − 1\| − ln \|x + 1\| 1 x=0 = −∞.(B19) But, if we were to take the limit x → 1 in both (B16) and (B17) equally fast, both contributions would sum up to zero. We can thus put the regularised integral to vanish: I β,0 reg = 0.(B20) FIG. 1 . 1Primary constraint structures between twistor and holonomy-flux spaces. F and C schematically denote the simplicity and area matching constraints, and arrows include division by gauge orbits, when relevant. FIG. 2 . 2More detailed constraint structures and the role of the secondary constraints. In the presence of secondary constraints, the rightmost part of the diagram becomes irrelevant, as the orbits of D are no longer pure gauges. In the final step, we have reintroduced the graph structure, as a proper definition of the secondary constraints should not be local on the links. and equally for the anti-analytic part. Therefore the integral is independent of the gauge-section chosen, if {G, C} = 0 = {G,C} : D1 d 14 µ gf G = D2 d 14 µ gf G. FIG. 3 .FIG. 4 . 34In a given 4-simplex two tetrahedra share a triangle t, that is in turn dual to a wedge wtv. The wedge is bounded by a loop. Half of the loop lies in the boundary of the 4-simplex, and connects the two tetrahedra τ and τ , piercing through the triangle t. The other half enters the bulk and passes through the center v of the 4-simplex. Left: a triangle t in the spatial hypersurface bounds two tetrahedra. Twistors Z and Z are attached to the underlying spinnetwork graph. Right: The same triangle seen from a 4-dimensional perspective. The triangle t is dual to a spinfoam face f consisting of several wedges wtv one for each adjacent vertex v. FIG. 5 . z→0 FSince 5z→0The integration domain is bounded by a Hankel contour around the branch cuts in the complex plane.circle at infinity vanishes. To prove it, we use[87] F (a, b; c; z) = Γ(c)Γ(b − a) Γ(b)Γ(c − a) (−z) −a F (a, 1 − c + a; 1 − b + a; z −1 )+ + Γ(c)Γ(a − b) Γ(a)Γ(c − b) (−z) −b F (b, 1 − c + b; 1 − a + b; z −1 ).For j > 0 and a = −iβj + j − 1 we find the limits lim (Re(a + b + 1) = −1 < 0 and Re(2b + 1) = −2j − 1 < 0 we getlim R→∞ k R,0 dz zF (−a, 1 2 ; 3 2 ; z 2 )(1 −z 2 ) b ≤ lim \|2a + 1\| \|R 2a \| F (−a,− 1 2 − a; 1 2 − a, R −2 e −2iϕ ) + Γ( 3 2 )\|Γ(−a − 1 2 )\| \|Γ(−a)\| R −1 F ( 1 2 , 0; a + 3 2 , R −2 e −2iϕ ) = 0 F (−a, 1 2 ; −a; 1 − z 2 ) = F ( 1 2 , −a; −a; 1 − z 2 ) = z 1 − z 2 ) b . − e −iϕ (e iϕ − r) −1 − (e −iϕ − r) iϕ − e −iϕ ln(e iϕ − r) − ln(e −iϕ − r) iϕ − e −iϕ 2iπsign(ϕ) − 2iϕ . (B15) x ± iε) = ±iπ, With respect to the literature[1,58], we have added the dimensional coefficents of the physical flux induced from the action. Also, the holonomy appearing here does not flip the spinors along the link, consistently with the definition of h. The alternative choice is to swap ω and π in(8). This allows to eliminate the opposite sign of the initial Poisson brackets, and induces an extra tensor in the the twisted geometries parametrization given below. This can be explicitly proven by looking at the defining differential equation for the holonomy, which admits a unique solution for the inital conditions U γ(0) = 1 = h γ(0) . It is the same type of equality that appears in the interaction picture used in time-dependent perturbation theory, with Γ being the free Hamiltonian, and K the potential. The solution is exact if the extrinsic curvature is covariantly constant along the link, i.e. G −1 γ(t) K γ(t) (γ)G γ(t) is t-independent. The trivial section being Ξ = 0. We define the complex Lesbegue measure as d 2 x = (i/2)dζ ∧ dζ. This normalization is responsible for the various powers of 2 appearing in later formulas. This is always possible thanks to the abelianity of the transformation. The remaining right-handed components are obtained by complex conjugation. Accordingly, the saddle point equations define an hypersurface in (the ωω -polarization of) the original phase space. This hypersurface depends on the relative phases ϕ and ϕ , and it is not invariant under D. Because of the homogeneity, integrating over all of C 2 would lead to divergences. and time reversalboth of which have recently gained[73]some interest in LQG. From (A23) we can realize parity and time reversal map the irreducible unitary representation of labeles (ρ, k) to those of (ρ, −k) and (−ρ, k) respectively. In each representation space there are two invariants, the first one is the Hermitian inner product (A28) introduced in above, the second one is the -invariantIts matrix elements arewhere Euler's Γ function appears. Though infinite dimensional, each of the invariants comes with an inverse, andThanks to the completeness of the basis, (A31) and (A28) imply for each irreducible subspace (ρ, k) a relation between the the ket and its dual,Since both δ νν and µν are invariant this map commutes with the group action, and implicitly shows the representation labelled by (ρ, k) is unitarily equivalent to its complex conjugate, that is the (−ρ, −k) representation. The map (A34) allows us to relate the bilinear invariant (A31) to the Hermitian inner product (A28), which we used in section VI B to compare our formulas to those of[17]. This is analogue to the finite dimensional case, see (A3), except that now both quantities are SL(2, C) invariant.The dual vector can be obtained also by Fourier transform, up to a phase, as was used in the main text in (94). In fact, we haveand defines an antilinear map from the (ρ, k) representation onto itself, whereas complex conjugation maps the (ρ, k) towards the (−ρ, −k) representation, implicitly showing that (ρ, k) and (−ρ, −k) are unitarily equivalent. We will give a detailed proof of this integral elsewhere, but let us mention the basic strategy behind: First, thanks to the SL(2, C) invariance of the integral, one can realize the left hand side equals the right hand side up to a constant. This constant can only depend on the labels ρ and k. Next, one shows, this constant has unit norm. Calculating the integral for the states of spin labels k = j = m, eventually gives the phase appearing in (A35).Appendix B: The integral determining µ(j)In section V we encountered an integral over the complex plane that contributes to the loop quantum gravity face amplitude. In this Appendix, we present the details of its evaluation, as well as the explicit form of the phase ∆(β, j). 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[ "Particle Transport in Graphene Nanoribbon Driven by Ultrashort Pulses", "Particle Transport in Graphene Nanoribbon Driven by Ultrashort Pulses" ]	[ "D Babajanov \nTurin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan\n", "D U Matrasulov \nTurin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan\n", "R Egger \nInstitut für Theoretische Physik\nHeinrich-Heine-Universität\nD-40225DüsseldorfGermany\n" ]	[ "Turin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan", "Turin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan", "Institut für Theoretische Physik\nHeinrich-Heine-Universität\nD-40225DüsseldorfGermany" ]	[]	We study charge transport in a graphene zigzag nanoribbon driven by an external time-periodic kicking potential. Using the exact solution of the time-dependent Dirac equation with a delta-kick potential acting in each period, we study the time evolution of the quasienergy levels and the time-dependent optical conductivity. By variation of the kicking parameters, the conductivity becomes widely tunable.PACS. 72.80.	10.1140/epjb/e2014-50610-6	[ "https://arxiv.org/pdf/1409.2719v2.pdf" ]	118,658,331	1409.2719	3d59dffd3c8ac22af41f03ccee3bb59a912c6a36	Particle Transport in Graphene Nanoribbon Driven by Ultrashort Pulses 3 Oct 2014 D Babajanov Turin Polytechnic University 17. Niyazov Str100095Tashkent, TashkentUzbekistan D U Matrasulov Turin Polytechnic University 17. Niyazov Str100095Tashkent, TashkentUzbekistan R Egger Institut für Theoretische Physik Heinrich-Heine-Universität D-40225DüsseldorfGermany Particle Transport in Graphene Nanoribbon Driven by Ultrashort Pulses 3 Oct 2014EPJ manuscript No. (will be inserted by the editor) the date of receipt and acceptance should be inserted laterPACS 7280Vp -7867Wj -7322Pr We study charge transport in a graphene zigzag nanoribbon driven by an external time-periodic kicking potential. Using the exact solution of the time-dependent Dirac equation with a delta-kick potential acting in each period, we study the time evolution of the quasienergy levels and the time-dependent optical conductivity. By variation of the kicking parameters, the conductivity becomes widely tunable.PACS. 72.80. Introduction Ever since its experimental discovery a decade ago, the physics of graphene has been a hot topic in condensed matter physics, see Refs. [1,2,3,4,5,6,7] for reviews. One of the most intensely studied class of problems concerns electronic transport in bulk or confined graphene monolayers. We here focus on the particle dynamics in externally driven graphene samples, where a general goal is to achieve tunability of charge transport. A rich variety of predicted and observed phenomena due to time-dependent fields have been reported in recent publications [8,9,10,11,12,13,14,15]. In particular, Ref. [8] argues that time-periodic spinorbit interactions lead to an interesting time evolution of the spin polarization and of the optical conductivity. Particle transport can also be induced by a time-dependent elastic deformation field [9], or in a.c. driven graphene nanoribbons, where by adopting a tight-binding model, the authors of Ref. [13] found a strong dependence of transport properties on the geometry of the ribbon edges. Furthermore, Ishikawa [14] studied electron transport in graphene perturbed by a time-periodic vector potential, which results in an enhancement of interband transitions. Finally, electron transport and current resonances in the presence of a time-dependent scalar potential barrier have been studied in Ref. [15], where a resonant enhancement of both electron backscattering and the currents across and along the barrier was reported when the modulation frequencies satisfy certain resonance conditions. Ultrafast dynamics and particle transport in graphene driven by ultrashort optical pulses have also been studied recently [16,17,18,19,20,21,22]. The experimental observation of a bright broadband photoluminescence in graphene Send offprint requests to: [email protected] interacting with femtosecond laser pulses was reported in Ref. [18]. Moreover, the authors of Ref. [22] have studied the modification of the bandstructure under ultrashort optical pulses and the carrier dynamics caused by the optical response of graphene, arguing that the electron dynamics in the time-dependent electric field of the laser pulse becomes irreversible, with a large residual conduction band population. In addition, the formation of a laser-induced band gap was discussed in Ref. [20]. In this paper, we study electron transport in graphene nanoribbons interacting with an external time-periodic scalar potential represented by a sequence of δ-kicks. The setup is schematically shown in Fig. 1. Such a potential could be created by applying laser pulses to free-standing samples. Using the exact solution of the time-dependent Dirac equation within one kicking period, we compute the transport properties of the system, such as the probability current and the optical conductivity, as a function of time. As we have discussed above, periodically driven graphene could be realized through the interaction with a.c. voltages [13], pulsed laser fields [23], surface acoustic waves [24], or time-periodic straining [9]. Here we focus on the case of ultrashort optical pulses [16,17,18,19]. Let us mention at this stage that some time ago, both the classical and the quantum dynamics of systems interacting with a delta-kicking potential have been extensively studied in the context of nonlinear dynamics and quantum chaos theory [25,26,27]. A remarkable feature of periodically driven quantum systems is the quantum localization phenomenon, which implies a suppression of the growth of the average kinetic energy with time; for the corresponding classical system, this energy grows linearly in time. However, the case of delta-kicked graphene nanoribbons is more complicated due to the kicking-field-induced modification of the graphene bandstructure. Such a modification of the bandstructure is the underlying reason for the interesting electronic and transport phenomena found in driven graphene. The main effect of the driving force is to cause inter-and intra-band transitions, leading to excitation and "ionization" of valenceband electrons to the conduction band. Another effect caused by driving fields in graphene is a band-gap opening or widening [20], which allows one to tune the electronic properties using an external time-dependent field. Below, we analyze the time evolution of the quasi-bandstructure and of the optical conductivity in delta-kicked graphene nanoribbons. We find that the quasienergy levels exhibit intra-band crossings and inter-band anticrossings, where the time-dependent effective density of states reaches a local maximum when the anticrossings take place. This may increase the number of charge carriers in the conduction band, with a subsequent increase of the current and of the optical conductivity. Indeed, as is shown by our analysis of the time-dependent conductivity in Sec. 3, depending on the kicking parameters, the conductivity may monotonically grow in time, while in other kicking regimes, such a growth is suppressed. The remainder of this paper is organized as follows. In Sec. 2 we briefly recall the Dirac equation for graphene zigzag nanoribbons, basically following the approach of Brey and Fertig [28], and discuss the solution of the timedependent Dirac equation in the presence of the δ-kicking potential. This solution is then utilized to compute the time evolution of the quasi-bandstructure, and, in Sec. 3, the optical conductivity in different kicking regimes. Finally, Sec. 4 contains some concluding remarks. Below we often use units whereh = 1. Kicked graphene nanoribbon 2.1 Unperturbed Hamiltonian In this work we study the electronic behavior of kicked zigzag graphene nanoribbons, see Fig. 1 for an illustration, within the Dirac equation approach [28]. It is well established that low-energy quasiparticles in an extended graphene sheet are accurately described by the massless two-dimensional (2D) Dirac Hamiltonian [1] H 0 = v F 0 p x − ip y p x + ip y 0 ,(1) where p x = −i∂ x , p y = −i∂ y , and v F ≈ 10 6 m/sec denotes the Fermi velocity. The 2 × 2 matrix structure of H 0 is with respect to sublattice space, corresponding to the (A/B-type) basis atoms of graphene's honeycomb lattice [1]. Since we do not take into account electron-electron interaction effects here, the two different valleys (K points) as well as the two spin projections decouple, and we can focus on a single-valley spinless system in Eq. (1). The spinor eigenstates of the zigzag nanoribbon with periodic boundary conditions along the longitudinal y-direction, see Fig. 1, are written as ψ(x, y) = e ikyy L y Φ(x), Φ(x) =   φ A (x) φ B (x)   ,(2) where k y is the conserved wave number along the y-direction. Periodic boundary conditions yield k y = 2πn y /L y with integer n y . To take into account the zigzag edges at x = 0 and x = L x , where L x is the width of the nanoribbon, we impose the boundary conditions [28] φ A (L x ) = φ B (0) = 0,(3) and put 0 ≤ x ≤ L x henceforth. Next we summarize the spinor solutions ψ n (x, y) solving the stationary Dirac equation for eigenenergy E n in the absence of the kicking potential, H 0 ψ n (x, y) = E n ψ n (x, y), n = (n x , n y ),(4) where the integer n x serves as a band index and n y parametrizes k y . The boundary conditions (3) imply that the eigenvalues of Eq. (4) are obtained from the transcendental equation [28] k y − z k y + z = e −2Lxz ,(5) which admits two type of solutions, namely (i) confined modes (standing waves), and (ii) surface states. We start by discussing solutions of type (i), which are purely imaginary, z nx = ik x , and lead to the eigenenergies E n = ±v F k 2 x + k 2 y ,(6) where the upper (lower) sign corresponds to the conduction (valence) band. Equation (5) now simplifies to k y = k x tan(k x L x ) ,(7) and solutions for k x (labeled by n x = 1, 2, . . .) correspond to confined modes. The respective eigenstate (2), with the energy E n in Eq. (6), has the transverse wavefunction with normalization constant N n . Φ n (x) = N n sin(k x x) ± i En [−k x cos(k x x) + k y sin(k x x)] ,(8) Next we turn to surface state [type (ii)] solutions. These correspond to purely real z = k x in Eq. (5), where the eigenstate reads Φ n (x) = N ′ n 2 sinh(k x x) 1 En [k y sinh(k x x) − k x cosh(k x x)] . (9) The eigenenergy is now given by E n = ± k 2 y − k 2 x ,(10) and N ′ n is another normalization constant. The surface state energies equal zero for sufficiently large positive k y , but they are absent for k y < 0. In Fig. 2, the resulting bandstructure of a typical graphene zigzag nanoribbon is plotted. Including the kicking potential We are now ready to include the external driving potential. We consider a periodic sequence of delta-kicks of kicking strength ε and period T . (No confusion with the symbol for temperature should arise here; we always consider the zero-temperature limit.) Writing H = H 0 + V diag(1, 1), the additional term is given by the time-periodic scalar potential, V (x, t) = ε cos (2πx/λ) ∞ l=0 δ(t − lT ),(11) where λ is the wavelength of the kicking pulse. Experimentally, such delta-kicks could be realized by standing-wave laser pulses [29,30,31], or by half-cycle laser pulses [32]. For instance, the delta-kicked quantum rotor, representing a well-known paradigm of quantum chaos theory, can be experimentally realized in ultracold atoms that interact with the periodic standing wave of a near-resonant laser field [29]. Significant progress concerning the experimental realization of graphene interacting with ultrashort laser pulses has also been reported recently [17,18,19]. Combining the experimental methods in Refs. [29,30,31,32] with those in Refs. [17,18,19] could allow to implement the delta-kicked graphene nanoribbon discussed here in the lab. The dynamics of a state Ψ = Ψ (x, y, t) is then governed by the time-dependent 2D Dirac equation, i∂ t Ψ = HΨ . To solve this equation, we expand Ψ (x, y, t) in terms of the complete set of eigenfunctions of the unperturbed graphene zigzag nanoribbon discussed in Sec. 2.1, Ψ (x, y, t) = n A n (t)ψ n (x, y),(12) where n = (n x , n y ) and the summation implicitly includes the ± sign for the conduction and valence band, respectively. To ensure normalization, the initial values (at time t = 0) of the complex-valued expansion coefficients A n (t) in Eq. (12) satisfy the condition n \|A n (0)\| 2 = 1. Within one time period, the amplitude A n then follows the time evolution A n (t + T ) = n ′ V nn ′ e −iE n ′ T A n ′ (t)(14) where E n is the unperturbed eigenenergy of the respective mode, see Sec. 2.1, and we define the matrix elements V nn ′ = Lx 0 dx Ly 0 dy ψ † n ′ (x, y)e iε cos(2πx/λ) ψ n (x, y),(15) where nonzero matrix elements exist only for n y = n ′ y due to the translation invariance in y-direction. In calculating these matrix elements, we use a well-known Bessel function expansion formula for the exponential term. In numerical calculations, one may choose only a few non-zero initial coefficients A n (0) subject to Eq. (13). In particular, we tested the impact of using different choices for A n (0), such as randomly chosen or equally distributed values. All choices were found to give qualitatively similar results for the time-evolved state Ψ (t) after many kicks. For the calculations presented below, we chose a random distribution for the coefficients A n (0) containing ≈ 15 non-zero entries, where we take into account only states with energy E n below the Fermi level. The Fermi level is here assumed at the neutrality point, i.e., E F = 0, in order to maximally emphasize the Dirac fermion nature of the graphene nanoribbon. Our procedure for choosing the initial values for the A n coefficients mimics the zerotemperature average over the filled Fermi sea. We have carefully checked that different initial values lead to the same physical results after a short initial transient. Given the wave function, one can compute different characteristics of the carrier dynamics and, in particular, investigate charge transport in a kicked graphene nanoribbon. In Fig. 3, the time evolution of quasienergy levels is shown for a few selected states. We observe several crossings of the levels within the conduction (or within the valence) band. However, levels coming from different bands exhibit anticrossing, where levels closely approach each other up to some time when they start to separate again. After a certain number of kicks, one can then again observe crossings or anticrossings, where intraand interband transitions become more frequent. Since initially the valence band is filled, this can lead to an increase in the number of electrons in the conduction band, and thereby to current flow. A related enhancement of intra-and inter-band transitions has also been reported in Ref. [22] for graphene subject to ultrashort laser pulses. When the quasienergy levels separate from each other again after the crossing or anticrossing, intra-and interband transitions become less frequent, and one can expect a decrease in the current. Such features indeed appear in the conductivity, as we study next. Optical conductivity The interaction of external electromagnetic fields with solids generally causes a modification of their electronic proper-ties and, in particular, of the bandstructure [22,33]. Using the solution of the time-dependent Dirac equation for delta-kicking potential discussed in Sec. 2, one can compute such modifications in the bandstructure, see Fig. 3. In this section, we focus on the optical conductivity of our system, which represents an important observable of experimental interest and can provide precious insights about the transport mechanisms at play in kicked graphene nanoribbons. Within linear response theory, the Kubo formula yields for the diagonal elements of the time-dependent conductivity tensor (α = x, y) [34] σ αα (x, y; t, ω) = e 2 ω ∞ 0 dτ e −iωτ(16)× [J α (x, y, t), J α (x, y, t − τ )] , where [, ] denotes the commutator and the particle current density along the α-direction is [1] J α (x, y, t) = v F Ψ † (x, y, t)σ α Ψ (x, y, t),(17) with standard Pauli matrices σ α=x,y acting in sublattice space. The average in Eq. (16) is taken with respect to the filled Fermi sea at the initial time t = 0, present before the kicking potential is switched on. In Eq. (16), we focus on the long-wavelength limit by probing the two current operators appearing in the Kubo formula at the same point in space. In terms of the expansion coefficients A n (t) appearing in the expansion (12), Eq. (17) takes the form J α (x, y, t) = v F nn ′ A * n ′ (t)A n (t)ψ † n ′ (x, y)σ α ψ n (x, y),(18) where A * denotes the complex conjugate of A. Inserting Eq. (18) into Eq. (16), the conductivity at time t follows as a lengthy (but straightforwardly obtained) expression involving the time-dependent coefficients {A n (t ′ )} for 0 < t ′ < t. As described in Sec. 2.2, the zero-temperature average over the filled Fermi sea is implemented by choosing suitable initial values for those coefficients. Below we discuss the time-dependence of the conductivity averaged over the sample area and taken in the ω → 0 limit, Figure 4 shows numerical results for σ xx (t) for representative kicking potential parameters. We observe that σ xx (t) grows during some initial time interval, followed by a suppression of the growth along with the development of oscillatory behavior. These features can be linked to the presence of crossings and anticrossings in the bandstructure, as discussed above. σ αα (t) = lim ω→0 Re Ly 0 dy L y Lx 0 dx L x σ αα (x, y; t, ω). (19) Next, Fig. 5 presents σ xx (t) for a fixed kicking period, T = 1 fs, but now for different values of the kicking strength ε. For the shown regime of rather short times, the conductivity monotonically grows in time. Clearly, for larger kicking strength, σ xx (t) grows more rapidly, where the slope of the growth is approximately proportional to ε. However, as seen in Fig. 4, after a longer time span, the conductivity will be suppressed again. Nonetheless, the variation of the kicking strength allows for a considerable tunability of the conductivity. The dependence of the conductivity on a variation of the kicking period T (with fixed strength ε) is illustrated by Fig. 6. We find that σ xx (t) monotonically grows within the considered time interval, although at larger times (not shown) the conductivity decreases and becomes quasi-oscillatory again, as displayed before in Fig. 4. The inset of Fig. 6 also shows the time-dependence of the average kinetic energy, E(t) = Lx 0 dx Ly 0 dy Ψ * (x, y, t)H 0 Ψ (x, y, t) = n \|A n (t)\| 2 E n .(20) This quantity exhibits time-periodic behavior, again reflecting the periodic appearance of band crossings as illustrated by Fig. 3. Such a behavior is different from a localization-induced saturation expected from quantum chaos theory but also differs from the simple monotonic growth expected on classical grounds [25,26,27]. Next, in Fig. 7, we compare the different components of the conductivity, namely σ xx (t) and σ yy (t). Although the kicking force acts along x-direction, we find very similar values for the conductivities in both directions. The spectral rearrangement caused by the kicking force is thus quite efficient in also inducing current fluctuations in the transverse (y) direction. Finally, Fig. 8 presents the spatio-temporal evolution of the probability density, \|Ψ (x, y, t)\| 2 . Since the result is homogeneous along the y-direction, we show it as 2D color-scale plot in the (x, t) plane; note the symmetry under spatial reflection with respect to the ribbon midpoint x = L x /2. Figure 8 provides additional information about the possibility of spatio-temporal quantum localization of electrons in the kicked graphene nanoribbon. In fact, we conclude from Fig. 8 that the carriers are not fully localized inside the ribbon, although signatures of localization near the midpoint are visible for finite time spans. Concluding remarks In this paper, we have studied time-dependent particle transport in graphene zigzag nanoribbons driven by an external time-periodic δ-kicking potential. The time-dependent Dirac equation can be solved exactly within a single kicking period, and numerical iteration of this solution provides access to the wave function at arbitrary time. Using this wave function, we have computed the time-dependent optical conductivity (and other quantities). The conductivity is observed to initially grow as a function of time up to certain time, after which the conductivity decreases and ultimately shows quasi-oscillatory behavior. We find it rather remarkable that by judiciously choosing the strength ε and the time period T of the kicking field, one can achieve almost arbitrary results for the oscillation period and for the amplitude of the conductivity. In particular, it is possible to choose parameters such that the initial increase extends for very long time. The described behavior of σ αα (t) can be linked to the existence of (anti-)crossings in the quasienergy bands of the driven system. The model studied in our work could be realized in zigzag ribbons made of monolayer graphene samples that are exposed to standing-wave ultrashort laser pulses, such as those discussed in Refs. [29,30,31]. The above results also may help in solving the problem of tunable charge transport in graphene-based electronic devices. Acknowledgments This work has been supported by the Volkswagen-Stiftung. Fig. 1 . 1Sketch of a kicked zig-zag graphene nanoribbon of width Lx and length Ly. Ultrashort periodic pulses applied to the nanoribbon act as kicking potential. Fig. 2 . 2Band structure of unperturbed graphene nanoribbon. The red dots indicate eigenenergies of confined modes, and the blue dots correspond to surface states. The nanoribbon has width Lx = 4.92 nm and length Ly = 12.3 nm, and a0 = 0.246 nm is the lattice unit. Fig. 3 . 3The upper panel shows the time evolution of a few selected quasienergy levels. The lower panel shows the same but on a magnified scale near the first level (anti-)crossing. The ribbon width Lx and length Ly were chosen as inFig. 2. The kicking strength is ε = 2.7 × 10 −13 eVs, with period T = 0.1 fs and wavelength λ = 2.46 nm. Fig. 4 .Fig. 5 . 45Conductivity σxx(t) in units of e 2 /h, see Eq. (19), as a function of time for a kicked graphene nanoribbon. 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[ "Maximum Dispersion and Geometric Maximum Weight Cliques ", "Maximum Dispersion and Geometric Maximum Weight Cliques " ]	[ "Sándor P Fekete ", "Henk Meijer " ]	[]	[]	We consider a facility location problem, where the objective is to "disperse" a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected locations is maximized. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme. * A preliminary version of this	10.1007/s00453-003-1074-x	[ "https://arxiv.org/pdf/cs/0310037v1.pdf" ]	3,263,869	cs/0310037	7f15e45eac117e558591b68983aaf17674fa7892	Maximum Dispersion and Geometric Maximum Weight Cliques * 17 Oct 2003 Sándor P Fekete Henk Meijer Maximum Dispersion and Geometric Maximum Weight Cliques * 17 Oct 2003 We consider a facility location problem, where the objective is to "disperse" a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected locations is maximized. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme. * A preliminary version of this Introduction A common problem in the area of facility location is the selection of a given number of k locations from a set P of n feasible positions, such that the selected set has optimal distance properties. Natural objective functions are the maximization of the minimum distance, or of the average distance between selected points; dispersion problems of this type come into play whenever we want to minimize interference between the corresponding facilities. Examples include oil storage tanks, ammunition dumps, nuclear power plants, hazardous waste sites, and fast-food outlets (see [14,5]). In the latter paper, the problem of maximizing the average distance is called the Remote Clique problem. Formally, problems of this type can be described as follows: given a graph G = (V, E) with n vertices, and non-negative edge weights w v1,v2 = d(v 1 , v 2 ) for (v 1 , v 2 ) ∈ E. Given k ∈ {2, . . . , n}, find a subset S ⊂ V with \|S\| = k, such that w(S) := (vi,vj )∈E(S) d(v i , v j ) is maximized. (Here, E(S) denotes the edge set of the subgraph of G induced by the vertex set S.) From a graph theoretic point of view, this problem has been called a heaviest subgraph problem. Being a weighted version of a generalization of the problem of deciding the existence of a k-clique, i.e., a complete subgraph with k vertices, the problem is strongly NP-hard [16]. It should be noted that Håstad [11] showed that the problem Clique of maximizing the cardinality of a set of vertices with a maximum possible number of edges is in general hard to approximate within n 1−ε . For the heaviest subgraph problem, we want to maximize the number of edges for a set of vertices of given cardinality, so Håstad's result does not imply an immediate performance bound. Related Work Over recent years, there have been a number of approximation algorithms for various subproblems of this type. Feige and Seltser [7] have studied the graph problem (i.e., edge weights are 0 or 1) and showed how to find in time n O((1+log n k )/ε) a k-set S ⊂ V with w(S) ≥ (1 − ε) k 2 , provided that a k-clique exists. They also gave evidence that for k ≃ n 1/3 , semidefinite programming fails to distinguish between graphs that have a k-clique, and graphs with densest k-subgraphs having average degree less than log n. Kortsarz and Peleg [12] describe a polynomial algorithm with performance guarantee O(n 0.3885 ) for the general case where edge weights do not have to obey the triangle inequality. A newer algorithm by Feige, Kortsarz, and Peleg [8] gives an approximation ratio of O(n 1/3 log n). For the case where k = Ω(n), Asahiro, Iwama, Tamaki, and Tokuyama [3] give a greedy constant factor approximation, while Srivastav and Wolf [15] use semidefinite programming for improved performance bounds. For the case of dense graphs (i.e., \|E\| = Ω(n 2 )) and k = Ω(n), Arora, Karger, and Karpinski [1] give a polynomial time approximation scheme. On the other hand, Asahiro, Hassin, and Iwama [2] show that deciding the existence of a "slightly dense" subgraph, i.e., an induced subgraph on k vertices that has at least Ω(k 1+ε ) edges, is NP-complete. They also showed it is NPcomplete to decide whether a graph with e edges has an induced subgraph on k vertices that has ek 2 n 2 (1 + O(v ε−1 )) edges; the latter is only slightly larger than ek 2 n 2 (1 − v−k vk−k ) , which is the the average number of edges in a subgraph with k vertices. For the case where edge weights fulfill the triangle inequality, Ravi, Rosenkrantz, and Tayi [14] give a heuristic with time complexity O(n 2 ) and prove that it guarantees a performance bound of 4. (See Tamir [17] with reference to this paper.) Hassin, Rubinstein, and Tamir [10] give a different heuristic with time complexity O(n 2 + k 2 log k) with performance bound 2. On a related note, see Chandra and Halldórsson [5], who study a number of different remoteness measures for the subset k, including total edge weight w(S). If the graph from which a subset of size k is to be selected is a tree, Tamir [16] shows that an optimal weight subset can be determined in O(nk) time. In many important cases there is even more known about the set of edge weights than just the validity of triangle inequality. This is the case when the vertex set V corresponds to a point set P in geometric space, and distances between vertices are induced by geometric distances between points. Given the practical motivation for considering the problem, it is quite natural to consider geometric instances of this type. In fact, it was shown by Ravi, Rosenkrantz, and Tayi in [14] that for the case of Euclidean distances in two-dimensional space, it is possible to achieve performance bounds that are arbitrarily close to π/2 ≈ 1.57 For other metrics, however, the best performance guarantee is the factor 2 by [10]. Despite of these approximation results, it should be noted that the complexity status of the problem is still open, i.e., it is known known whether the problem is NP-hard. An important application of our problem is data sampling and clustering, where points are to be selected from a large more-dimensional set. Different metric dimensions of a data point describe different metric properties of a corresponding item. Since these properties are not geometrically related, distances are typically not evaluated by Euclidean distances. Instead, some weighted L 1 metric is used. (See Erkut [6].) For data sampling, a set of points is to be selected that has high average distance. For clustering, a given set of points is to be subdivided into k clusters, such that points from the same cluster are close together, while points from different clusters are far apart. If we do the clustering by choosing k center points, and assigning any point to its nearest cluster center, we have to consider the same problem of finding a set of center points with large average distance, which is equivalent to finding a k-clique with maximum total edge weight. For results on maximizing the minimum L 1 distance within a selected set of n points see Baur and Fekete [4], who showed that finding such a set within a given polygon cannot be approximated arbitrarily well, unless P=NP. Finally, Gritzmann, Klee, and Larmann [9] have studied a somewhat related geometric selection problem: Given a set v of m points in n-dimensional space, choose a subset of n + 1 points, such that the total volume of the resulting simplex is maximum. They showed that this problem is NP-hard when n is part of the input. Main Results In this paper, we consider point sets P in d-dimensional space, where d is some constant. For the most part, distances are measured according to the rectilinear "Manhattan" norm L 1 . Our results include the following: • A linear time (O(n)) algorithm to solve the problem to optimality in case where k is some fixed constant. This is in contrast to the case of Euclidean distances, where there is a well-known lower bound of Ω(n log n) in the computation tree model for determining the diameter of a planar point set, i.e., the special case d = 2 and k = 2 (see [13]). • A polynomial time approximation scheme for the case where k is not fixed. This method can be applied for arbitrary fixed dimension d. For the case of Euclidean distances in two-dimensional space, it implies a performance bound of √ 2 + ε, for any given ε > 0. Preliminaries For the most part of this paper, all points are assumed to be points in the plane. Spaces of arbitrary fixed dimension will be discussed in the end. Distances are measured using the L 1 norm, unless noted otherwise. The x-and y-coordinates of a point p are denoted by x p and y p . If p and q are two points in the plane, then the distance between p and q is d(p, q) = \|x p − x q \| + \|y p − y q \|. We say that q is above p in direction of a vector c, if the inner products satisfy q, c ≥ p, c We say that a point p is maximal in direction c with respect to a set of points P if it maximizes the inner product { c, x \| x ∈ P }. For example, if p is an element of a set of points P and p has a maximal y-coordinate, then p is maximal in direction (0,1) with respect to P , and a point p with minimal x-coordinate is maximal in direction (-1,0) with respect to P . If the set P is clear from the context, we simply state that p is maximal in direction c. The weight of a set of points P is the sum of the distances between all pairs of points in this set, and is denoted by w(P ). Similarly, w(P, Q) denotes the total sum of distances between two sets P and Q. For L 1 distances, w x (P ) and w x (P, Q) denote the sum of x-distances within P , or between P and Q. Cliques of Fixed Size Let S = {s 0 , s 1 , . . . , s k−1 } be a maximum weight subset of P , where k is a fixed integer greater than 1. We will label the x-and y-coordinates of a point s ∈ S by some (x a , y b ) with 0 ≤ a < k and 0 ≤ b < k such that x 0 ≤ x 1 ≤ . . . ≤ x k−1 and y 0 ≤ y 1 ≤ . . . ≤ y k−1 . (Note that in general, a = b for a point s = (x a , y b ).) Then w(S) = 0≤i<j<k (x j − x i ) + 0≤i<j<k (y j − y i ). Now we can use local optimality to reduce the family of subsets that we need to consider: Lemma 1 There is a maximum weight subset S ′ of P of cardinality k, such that each point in S ′ is maximal in direction (2i + 1 − k, 2j + 1 − k) with respect to P \ S ′ for some values of i and j with 0 ≤ i, j < k. Proof: Consider a maximum weight subset S ⊂ P of cardinality k. Let s = (x i , y i ′ ) be a point in S, such that there are i points s l = (x l , y l ′ ) ∈ S \ {s} with x l ≤ x i (i.e., to the left of s) and k − i − 1 points s l = (x l , y l ′ ) ∈ S \ {s} with x l > x i (i.e. , strictly to the right of s). Similarly let there be j points below s and k − j − 1 points strictly above s. We claim that s is maximal in direction (2i + 1 − k, 2j + 1 − k) with respect to P − S. Consider replacing s by a point s ′ = s + (x h , y h ) in P − S. Let δ = (2i + 1 − k)x h + (2j + 1 − k)y h . Let S ′ = S \ {s} ∪ {s ′ }. Assume first that point s ′ = (x ′ i , y ′ i ′ ) is such that x ′ i and y ′ i ′ have the same rank in S ′ as x i and y i ′ have in S, i.e., there are i points s l = (x l , y l ) ∈ S ′ \ {s ′ } with x l ≤ x ′ i and j points s l = (x l , y l ′ ) ∈ S ′ \ {s ′ } with y l ′ ≤ y ′ i ′ . Replacing s by s ′ changes the x-distances to the points left of s by ix h , and the x-distances to the points right of s by (k − i − 1)(−x h ). Similarly, the y-distances change by jy h and (k − j − 1)y h . So w(S ′ ) = w(S) + (2i + 1 − k)x h + (2j + 1 − k)y h = w(S) + δ. Since w(S) is maximum, we derive that δ ≤ 0, i.e., no point in P − S is above any point in S in direction (2i + 1 − k, 2j + 1 − k). If the x-and y-coordinates of s ′ do not have the same rank in S ′ as the x-and y-coordinates of s in S, then it is not hard to show that w(S ′ ) > w(S)+δ, so δ < 0. Therefore in this case, s ′ is strictly below s in direction (2i + 1 − k, 2j + 1 − k). We can also conclude that if s = (x i , y i ′ ) and s ′ = (x ′ i , y ′ i ′ ) are at the same level in direction (2i + 1 − k, 2j + 1 − k), i.e., if δ = 0, then the x-and ycoordinates of s ′ do have the same rank in S ′ as the x-and y-coordinates of s in S and w(S) = w(S ′ ). 2 Theorem 1 Given a constant value for k, a maximum weight subset S of a set of n points P , such that S has cardinality k, can be found in linear time. Proof: Consider all directions of the form (2i+1−k, 2j+1−k) with 0 ≤ i, j < k. For each direction (a, b), find S k (a, b), a set of k points that are maximal in direction (a, b) with respect to P − S k (a, b). Compute the set ∪S k (a, b) and try all possible subsets of size k of this set until a subset of maximum weight is found. Correctness follows from the fact that Lemma 1 implies that S ⊂ ∪S k (a, b). Since k is a constant, each set S k (a, b) can be found in linear time. Since the cardinality of ∪S k (a, b) is less than or equal to k 3 , the result follows. From the discussion at the end of the proof of Lemma 1 we can conclude that if the set of k points maximal in a direction (a, b) is not unique, any set of k points maximal in this direction will work equally well. 2 Note that in the above estimate, we did not try to squeeze the constants in the O(n) running time. A closer look shows that for k = 2, not more than 2 subsets of P need to be evaluated for possible optimality, for k = 3, 8 subsets are sufficient. Cliques of Variable Size In this section we consider the scenario where k is not fixed, i.e., k is part of the input. We show that there is a polynomial time approximation scheme (PTAS), i.e., for any fixed positive ε, there is a polynomial approximation algorithm that finds a solution that is within (1 + ε) of the optimum. The basic idea is to use for each of the d coordinates a suitable subset of m ε coordinate values that subdivide an optimal solution into subsets of equal cardinality. More precisely, we describe the case d = 2; we find (by enumeration) a subdivision of an optimal solution into m ε × m ε rectangular cells C ij , each of which must contain a specific number k ij of selected points. From each cell C ij , the points are selected in a way that guarantees that the total distance to all other cells except for the m ε − 1 cells in the same "horizontal" strip or the m ε − 1 cells in the same "vertical" strip is maximized. As it turns out, this can be done in a way that the total neglected distance within the strips is bounded by a fraction of (5m ε − 9)/(2(m ε − 1)(m ε − 2)) of the weight of an optimal solution, yielding the desired approximation property. See Figure 1 for the overall picture. For ease of presentation we assume that k is a multiple of m ε and m ε > 2. Approximation algorithms for other values of k can be constructed in a similar fashion. Consider an optimal solution of k points, denoted by OPT. Furthermore consider a division of the plane by a set of m ε + 1 x-coordinates ξ 0 ≤ . . . ≤ ξ 1 ≤ ξ mε . Let X i := {p = (x, y) ∈ ℜ 2 \| ξ i ≤ x ≤ ξ i+1 , 0 ≤ i < m ε } be the vertical strip between coordinates ξ i and ξ i+1 . By enumeration of possible choices of ξ 0 , . . . , ξ mε we may assume that the ξ i have the property that, for an optimal solution, from each of the m ε strips X i precisely k/m ε points of P are chosen. (A small perturbation does not change optimality or approximation properties of solutions. This shows that in case of several points sharing the same coordinates, ties may be broken arbitrarily; in that case, points on the boundary between two strips may be considered belonging to one or the other of those strips, whatever is convenient to reach the appropriate number of points in a strip.) In a similar manner, suppose we know m ε + 1 y-coordinates η 0 ≤ η 1 ≤ . . . ≤ η mε such that from each horizontal strip Y i := {p = (x, y) ∈ ℜ 2 \| η i ≤ y ≤ η i+1 , 0 ≤ i < m ε } a subset of k/m ε points are chosen for an optimal solution. Let C ij := X i ∩ Y j , and let k ij be the number of points in OPT that are chosen from C ij . Since 0≤i<mε k ij = 0≤j<mε k ij = k/m ε , we may assume by enumeration over the O(k m ε ) possible partitions of k/m ε into m ε pieces that we know all the numbers k ij . Finally, define the vector ∇ ij := ((2i + 1 − m ε )k/m ε , (2j + 1 − m ε )k/m ε ). Now our approximation algorithm is as follows: from each cell C ij , choose some k ij points that are maximal in direction ∇ ij . (Overlap between the selections from different cells is avoided by proceeding in lexicographic order of cells, and choosing the k ij points among the candidates that are still unselected.) Let HEU be the point set selected in this way. It is clear that HEU can be computed in polynomial time. We will proceed by a series of lemmas to determine how well w(HEU) approximates w(OPT). In the following, we consider the distances involving points from a particular cell C ij . Let HEU ij be the set of k ij points that are selected from C ij by the heuristic, and let OPT ij be a set of k ij points of an optimal solution that are attributed to C ij . Let S ij = OPT ij ∩ HEU ij . Furthermore we define S ij = HEU ij \ OPT ij , andS ij = OPT ij \ HEU ij . Let HEU i• , OPT i• , HEU •j and OPT •j be the set of k/m ε points selected from X i and Y j by the heuristic and an optimal algorithm respectively. Finally HEU i• := HEU \ HEU i• , HEU •j := HEU \ HEU •j , OPT i• := OPT \ OPT i• and OPT •j := OPT \ OPT •j . Lemma 2 w x (HEU ij , HEU i• ) + w y (HEU ij , HEU •j ) ≥ w x (OPT ij , OPT i• ) + w y (OPT ij , OPT •j ). Proof: Consider a point p ∈S ij . Thus, there is a point p ′ ∈ S ij that was chosen by the heuristic instead of p. Now we can argue like in Lemma 1: Let h = (h x , h y ) = p ′ −p. When replacing p in OPT by p ′ , we increase the x-distance to the ik/m ε points "left" of C ij by h x , while decreasing the x-distance to (m ε −i−1)k/m ε points "right" of C ij by h x . In the balance, this yields a change of ((2i + 1 − m ε )k/m ε )h x . Similarly, we get a change of ((2j + 1 − m ε )k/m ε )h y for the y-coordinates. By definition, we have assumed that the inner product h, ∇ ij ≥ 0, so the overall change of distances is nonnegative. Performing these replacements for all points in OPT \ HEU, we can transform OPT to HEU, while increasing the sum of distances w x (OPT ij , OPT i• ) + w y (OPT ij , OPT •j ) to the sum w x (HEU ij , HEU i• ) + w y (HEU ij , HEU •j ). 2 In the following three lemmas we show that the total difference between the weight of an optimal solution w(OPT) and the total value of all the right hand sides (when summed over i) of the inequality in Lemma 2 is a small fraction of w(OPT). Lemma 3 0<i<mε−1 w x (OPT i• ) ≤ w x (OPT) 2(m ε − 2) . Proof: Let δ i = ξ i+1 − ξ i . Since i(m ε − i − 1) ≥ m ε − 2 for 0 < i < m ε − 1, we have for 0 < i < m ε − 1 w x (OPT i• ) ≤ k 2 2m 2 ε δ i ≤ ik m ε (m ε − i − 1)k m ε δ i 1 2(m ε − 2) . Since OPT has ik/m ε and (m ε − i − 1)k/m ε points to the left of ξ i and right of ξ i+1 respectively, we have w x (OPT) ≥ 0<i<mε−1 ik m ε (m ε − i − 1)k m ε δ i so 0<i<mε−1 w x (OPT i• ) ≤ 1 2(m ε − 2) w x (OPT). 2 Lemma 4 For i = 0 and i = m ε − 1 we have w x (OPT i• ) ≤ w x (OPT) m ε − 1 Proof: Without loss of generality assume i = 0. Let x 0 , x 1 , · · · , x (k/mε)−1 be the x-coordinates of the points p 0 , p 1 , . . . , p (k/mε)−1 in OPT 0• . So w x (OPT 0• ) = ( k m ε − 1)(x k mε −1 − x 0 ) + ( k m ε − 3)(x k mε −2 − x 1 ) + . . . ≤ ( k m ε − 1)(ξ 1 − x 0 ) + ( k m ε − 3)(ξ 1 − x 1 ) + . . . ≤ k m ε (ξ 1 − x 0 ) + k m ε (ξ 1 − x 1 ) + . . . Since ξ 1 − x j ≤ x − x j where 0 ≤ j < k/m ε and x is the x-coordinate of any point in OPT 0• and since there are (m ε − 1)k/m ε points in OPT 0• , we have ξ 1 − x j < m ε (m ε − 1)k w x (p j , OPT 0• ) so w x (OPT 0• ) ≤ k m ε m ε (m ε − 1)k 0≤i< k 2mε w x (p i , OPT 0• ) ≤ 1 m ε − 1 0≤i< k mε w x (p i , OPT 0• ) = 1 m ε − 1 w x (OPT 0• , OPT 0• ) ≤ 1 m ε − 1 w x (OPT). 2 This proves the main properties. Now we only have to combine the above estimates to get an overall performance bound: Lemma 5 0≤i<mε w x (OPT i• , OPT i• ) + 0≤j<mε w y (OPT •j , OPT •j ) ≥ (1 − 5m ε − 9 2(m ε − 1)(m ε − 2) )w(OPT)). Proof: From Lemmas 3 (applied for indices 0 < i < m − 1) and 4 (applied twice, once for i = 0, and once for i = m − 1), we derive that 0≤i<mε w x (OPT i• ) ≤ 5m ε − 9 2(m ε − 1)(m ε − 2) w x (OPT) and similarly 0≤i<mε w y (OPT •j ) ≤ 5m ε − 9 2(m ε − 1)(m ε − 2) w y (OPT). Since w(OPT) = w x (OPT) + w y (OPT) = 0≤i<mε w x (OPT i• , OPT i• ) + 0≤i<mε w x (OPT i• ) + 0≤j<mε w y (OPT •j , OPT •j ) + 0≤j<mε w y (OPT •j ), the result follows. 2 Putting together Lemma 2 and the error estimate from Lemma 5, the approximation theorem can now be proven. Theorem 2 For any fixed m, HEU can be computed in polynomial time, and w(HEU) ≥ (1 − 5m ε − 9 2(m ε − 1)(m ε − 2) )w(OPT). The running time is exponential in 1 ε . Proof: The claim about the running time is clear. (The only step that is exponential in 1 ε is the enumeration over all O(k mε ) possible partitions of k/m ε into m ε pieces.) Using Lemmas 2 and 5 we derive w(HEU) ≥ 0≤i<mε w x (HEU i• , HEU i• ) + 0≤j<mε w y (HEU •j , HEU •j ) ≥ 0≤i<mε w x (OPT i• , OPT i• ) + 0≤j<mε w y (OPT •j , OPT •j ) ≥ (1 − 5m ε − 9 2(m ε − 1)(m ε − 2) )w(OPT). 2 Implications It is straightforward to modify our above arguments to point sets under L 1 distances in an arbitrary d-dimensional space, with fixed d. Theorem 3 Given a constant value for k and d, a maximum weight subset S of a set of n points in d-dimensional space, such that S has cardinality k, can be found in linear time. If d and ε are constants, but k is not fixed, then there is a polynomial time algorithm that finds a subset whose weight is within (1 + ε) of the optimum. For the case of fixed k, it is straightforward to generalize the argument from Section 3 to see that there are at most (2k) d interesting directions to consider. For k being part of the input, the approximation scheme can be generalized in a straightforward manner by using an m ε -subdivision in each coordinate direction. Again the complexity ends up being exponential in 1 ε , as well as in d. For the case of L ∞ distances in the plane, the results for L 1 distances can be applied by a standard argument: A rotation by π/4 transforms L ∞ distances into L 1 distances and vice versa. Furthermore, we can use the approximation scheme from the previous section to get a √ 2(1 + ε) approximation factor for the case of Euclidean distances in two-dimensional space, for any ε > 0: In polynomial time, find a k-set S 1 such that L 1 (S) is within (1 + ε) of an optimal solution OPT 1 with respect to L 1 distances. Let OPT 2 be an optimal solution with respect to L 2 distances. Then L 2 (S) ≥ 1 √ 2 L 1 (S) ≥ 1 √ 2(1 + ε) L 1 (OPT 1 ) ≥ 1 √ 2(1 + ε) L 1 (OPT 2 ) ≥ 1 √ 2(1 + ε) L 2 (OPT 2 ), and the claim follows. Similarly, any norm has its characteristic approximation factor ρ with respect to L 1 or L ∞ distances; this factor immediately yields a (ρ + ε)-approximation for geometric dispersion. Conclusions We have presented algorithms for geometric instances of the maximum weighted k-clique problem. Our results give a dramatic improvement over the previous best approximation factor of 2 that was presented in [10] for the case of general metric spaces. This underlines the observation that geometry can help to get better algorithms for problems from combinatorial optimization. Furthermore, the algorithms in [10] give better performance for Euclidean metric than for Manhattan distances. We correct this anomaly by showing that among problems involving geometric distances, the rectilinear metric may allow better algorithms than the Euclidean metric. It remains an interesting open problem to show NP-hardness of a geometric version of the problem for spaces of fixed dimension. In particular, the case of Manhattan distances in the plane may actually turn out to be polynomially solvable. Figure 1 : 1Subdividing the plane into cells AcknowledgmentsWe would like to thank Katja Wolf and Magnús Halldórsson for helpful discussions, Rafi Hassin for several relevant references, and three anonymous referees for useful comments. Polynomial time approximation schemes for dense NP-hard problems. 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Approximation algorithms for dispersion problems. Barun Chandra, Magnus M Halldórsson, J. Algorithms. 382Barun Chandra and Magnus M. Halldórsson. Approximation algorithms for dispersion problems. J. Algorithms, 38(2):438-465, 2001. The discrete p-dispersion problem. E Erkut, European Journal of Operational Research. 46E. Erkut. The discrete p-dispersion problem. European Journal of Opera- tional Research, 46:48-60, 1990. On the densest k-subgraph problems. U Feige, M Seltser, Weizmann InstituteTechnical Report CS97-16U. Feige and M. Seltser. On the densest k-subgraph problems. Technical Re- port CS97-16, Weizmann Institute, http://www.wisdom.weizmann.ac.il, 1997. The dense k-subgraph problem. Uriel Feige, David Peleg, Guy Kortsarz, Algorithmica. 293Uriel Feige, David Peleg, and Guy Kortsarz. The dense k-subgraph prob- lem. Algorithmica, 29(3):410-421, 2001. Largest j-simplices in n-polytopes. Discrete and Computational Geometry. P Gritzmann, V Klee, D Larman, 13P. Gritzmann, V. Klee, and D. Larman. Largest j-simplices in n-polytopes. Discrete and Computational Geometry, 13:477-515, 1995. Approximation algorithms for maximum dispersion. R Hassin, S Rubinstein, A Tamir, Operations Research Letters. 21R. Hassin, S. Rubinstein, and A. Tamir. Approximation algorithms for maximum dispersion. Operations Research Letters, 21:133-137, 1997. Clique is hard to approximate within n 1−ε. J Håstad, Acta Mathematica. 182J. Håstad. Clique is hard to approximate within n 1−ε . Acta Mathematica, 182:105-142, 1999. On choosing a dense subgraph. G Kortsarz, D Peleg, Proceedings of the 34th IEEE Annual Symposium on Foundations of Computer Science. the 34th IEEE Annual Symposium on Foundations of Computer SciencePalo Alto, CAG. Kortsarz and D. Peleg. On choosing a dense subgraph. In Proceedings of the 34th IEEE Annual Symposium on Foundations of Computer Science, pages 692-701, Palo Alto, CA, 1993. Computational Geometry: An Introduction. 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[ "Magneto-electrical orientation of lipid-coated graphitic micro-particles in solution †", "Magneto-electrical orientation of lipid-coated graphitic micro-particles in solution †" ]	[ "Johnny Nguyen ", "Sonia Contera ", "Isabel Llorente García " ]	[]	[]	We demonstrate, for the first time, confinement of the orientation of micron-sized graphitic flakes to a welldefined plane. We orient and rotationally trap lipid-coated highly ordered pyrolytic graphite (HOPG) microflakes in aqueous solution using a combination of uniform magnetic and AC electric fields and exploiting the anisotropic diamagnetic and electrical properties of HOPG. Measuring the rotational Brownian fluctuations of individual oriented particles in rotational traps, we quantitatively determine the rotational trap stiffness, maximum applied torque and polarization anisotropy of the micro-flakes, as well as their dependency on the electric field frequency. Additionally, we quantify interactions of the micro-particles with adjacent glass surfaces with various surface treatments. We outline the various applications of this work, including torque sensing in biological systems. † Electronic supplementary information (ESI) available: Videos of HOPG micro-akes in solution in the presence of a vertical orienting magnetic eld, rotating to align with a horizontal AC electric eld. See	10.1039/c6ra07657b	[ "https://pubs.rsc.org/en/Content/ArticlePDF/2016/RA/C6RA07657B" ]	101,502,060	1604.07239	b8b60db61edeb678c8c088133ee02a327ffe4903	Magneto-electrical orientation of lipid-coated graphitic micro-particles in solution † Johnny Nguyen Sonia Contera Isabel Llorente García Magneto-electrical orientation of lipid-coated graphitic micro-particles in solution † 10.1039/c6ra07657b We demonstrate, for the first time, confinement of the orientation of micron-sized graphitic flakes to a welldefined plane. We orient and rotationally trap lipid-coated highly ordered pyrolytic graphite (HOPG) microflakes in aqueous solution using a combination of uniform magnetic and AC electric fields and exploiting the anisotropic diamagnetic and electrical properties of HOPG. Measuring the rotational Brownian fluctuations of individual oriented particles in rotational traps, we quantitatively determine the rotational trap stiffness, maximum applied torque and polarization anisotropy of the micro-flakes, as well as their dependency on the electric field frequency. Additionally, we quantify interactions of the micro-particles with adjacent glass surfaces with various surface treatments. We outline the various applications of this work, including torque sensing in biological systems. † Electronic supplementary information (ESI) available: Videos of HOPG micro-akes in solution in the presence of a vertical orienting magnetic eld, rotating to align with a horizontal AC electric eld. See Introduction Micro-and nano-particles of carbon such as graphene/graphite platelets and carbon nanotubes have unique electrical, magnetic, optical and mechanical properties that, combined with their biocompatible nature and their chemical and biological functionalization capability, make them very attractive for numerous applications. The controlled manipulation and orientation of such micro/nano-particles over macroscopic length-scales has interesting applications for batteries and energy storage devices, 1 for opto-electronic devices, including recently developed magnetically and electrically switched graphene-based liquid crystals, [2][3][4][5] and for the creation of novel articial composite materials with tailored anisotropic properties such as conductive polymers and gels, 6 material reinforcements, 7 materials for thermal management solutions, 8 hydrophobic coatings, infra-red absorbing coatings, etc. The controlled orientation of graphene-based micro/nano-particles can aid the synthesis of large-scale mono-crystalline graphene composites 9,10 and can lead to advances in template-mediated synthesis, for instance to induce the ordered deposition of organic molecules. 11 In this paper, we demonstrate, for the rst time, connement of the orientation of graphitic micro-akes to a well-dened plane using a novel magneto-electrical approach for fully controlling micro-particle orientation. We orient and rotationally trap soluble, biocompatible, lipid-coated micro-particles of highly ordered pyrolytic graphite (HOPG) in aqueous solution. Our scheme takes advantage of the anisotropic diamagnetic and electrical properties of HOPG micro-akes (graphene layer stacks) 12 to orient them parallel to a plane dened by two perpendicular elds: a vertical static magnetic eld ($240 mT); and a horizontal, linearly polarized electric eld ($2 Â 10 4 V m À1 ) oscillating at frequencies above 10 MHz. Our inexpensive set-up is made from one permanent magnet and two thin wire electrodes, involving no microfabrication. While we demonstrate our scheme on HOPG, the impact of our results extends to other forms of carbon-based micro/nano-particles. Current methods for large-scale vertical orientation of graphene-based particles such as, for instance, vacuum ltration 8 and chemical vapour deposition, 13 are involved and costly and do not provide full control of graphene plane orientation. As for magnetic and electrical orientation methods, only a handful of publications have reported the magnetic alignment of graphite/graphene micro/nano-particles in solution, 1,2,6,14 and the use of electric elds to orient these particles has turned out to be more challenging than originally expected. Only very recently, the use of alternating-current (AC) electric elds has been reported as a successful method for the orientation of graphite/graphene akes 3-5 and carbon nanotubes. 15,16 However, these methods also lack full control of particle orientation, since the use of a single orienting eld (either magnetic or electric) still allows particles to freely rotate around the eld direction. Our results present a number of key novelties with respect to previous work. By simultaneously applying magnetic and AC electric elds in different directions, we restrict micro-particle rotations and conne particle orientation to the plane dened by these elds. Furthermore, we track and analyse the rotational motion of individual micro-particles, as opposed to monitoring ensemble averages. We detect micro-particle orientation using robust image processing techniques which improve the detection accuracy in comparison to indirect methods reported in previous studies. [3][4][5] By tracking the rotation of the particles and analyzing their Brownian orientational uctuations around equilibrium in the rotational traps, we quantitatively determine the rotational trap stiffness, maximum applied orientational torques and relevant polarization anisotropy factor for the micro-akes, as well as how these depend on the frequency of the AC electric eld. This is the rst report of such measurements for lipid-coated HOPG micro-akes. Additionally, we quantitatively characterize the interaction of the micro-akes with nearby functionalized glass surfaces, reliably discriminating this interaction from the effect of the orienting elds in our analysis, and potentially opening a new route to measuring the strength of interfacial interactions. Crucially, rotational trapping (connement to a plane) allows the powerful use of Brownian uctuation analysis methods which enable precise quantitative measurements that cannot be found in previous studies. Our achievement of controlled orientation paves the way for advances in a wide range of applications. As well as the applications mentioned above, the manipulation of individual biocompatible graphitic micro-particles can open up new exciting possibilities for biological and chemical sensing, [17][18][19] and for fundamental biophysical and biochemical studies. Indeed, our ability to detect weak torques shows that our scheme could be applied to the precise sensing of biologically relevant torques. Our experiments contribute to the currently growing interest in the diamagnetic manipulation of micro/ nano-particles, 20 particularly within a biological context, where most matter is diamagnetic. The recent demonstration of magnetically controlled nano-valves 21 is one excellent example of the enormous potential of diamagnetic manipulation. Principles of magnetic and electrical orientation In our scheme, we rst apply a vertical magnetic eld in order to align the HOPG micro-akes parallel to the eld direction. Once aligned, the micro-akes are still free to rotate around the magnetic eld direction. We then apply an AC electric eld perpendicular to the magnetic eld to rotate the micro-akes and constrain their orientation to the plane containing both the magnetic and electric elds (see Fig. 1). We dene a xed laboratory frame of reference with axes (X, Y, Z), where Z is the vertical direction (Fig. 1). The static magnetic eld B 0 is along Z and the electric eld E 0 oscillates along the horizontal X direction. We also dene a particle frame of reference with axes (x, y, z) xed to the HOPG particle. The x-y plane corresponds to the HOPG graphene planes and z is normal to the graphene planes. Magnetic alignment of lipid-coated HOPG The magnetic properties of graphite/graphene have rarely been exploited despite the well known diamagnetic nature of graphite. 22,23 HOPG is highly magnetically anisotropic and one of the most strongly diamagnetic materials known, particularly along the direction perpendicular to the graphene planes (outof-plane direction). Its out-of-plane (t) and in-plane (k) volume magnetic susceptibilities are c t ¼ À4.5 Â 10 À4 and c k ¼ À8.5 Â 10 À5 , respectively, as measured by Simon et al. 24 For comparison, the magnetic susceptibility of water (also diamagnetic) is c water z À9 Â 10 À6 . The magnetic manipulation of micro-particles in solution presents the advantages of being contactless, non-invasive, biocompatible, largely insensitive to the solvent's conductivity, ionic strength and pH, not accompanied by undesirable effects in solution (such as electrophoretic migration or electrochemical reactions in electro-manipulation), and cheap and simple thanks to the availability of strong NdFeB permanent magnets. 20 The effective induced magnetic moment that a particle with magnetic susceptibility tensor c 2 , immersed in a uid with magnetic susceptibility tensor c 1 , experiences in the presence of a uniform, static magnetic eld, B 0 , is: 25 m eff z V 2 m 0 ðc 2 À c 1 Þ$B 0 ;(1) where m 0 is the permeability of free space and V 2 is the volume of the particle. Eqn (1) results from the fact that, for diamagnetic particles, the magnetic susceptibilities are very small (<10 À3 in absolute value) so that demagnetising elds inside the particle are negligibly small. As a consequence of this, particle shape and geometry have a negligible effect on the effective magnetic moment induced on the particle. In the particle frame of reference, the volume magnetic susceptibility tensors for the anisotropic HOPG particles (c 2 ) and for the isotropic uid (c 1 ) are expressed as: c 2 ¼ 0 @ c k 0 0 0 c k 0 0 0 c t 1 A ; c 1 ¼ 0 @ c 1 0 0 0 c 1 0 0 0 c 1 1 A ;(2) where c 1 z c water is the isotropic magnetic susceptibility of the aqueous solution and c k and c t are the in-plane and out-ofplane magnetic susceptibilities for HOPG, given in the rst paragraph of this sub-section. The direction of the magnetic eld B 0 in the particle frame is specied by the spherical polar angles (q, f), with q being the angle with respect to z and f being the angle that the projection of B 0 onto the x-y plane makes with the x axis [see Fig. 1(a)]. We can therefore write B 0 ¼ B 0 (sin q cos fx + sin q sinfŷ + cos qẑ), where B 0 is the amplitude of the applied magnetic eld. In the presence of this eld, the particle experiences a magnetic torque T m ¼ m eff Â B 0 . The only non-zero components of this torque are T m x and T m y , so that the particle will rotate around an inplane axis until it reaches orientational equilibrium. The net magnetic torque around the in-plane (k) axis is given by: T m k ¼ V 2 B 0 2 2m 0 À c t À c k Á sinð2qÞ:(3) Note that this torque relies on the intrinsic magnetic anisotropy of the particles [(c t À c k ) s 0] and is independent of the magnetic susceptibility of the surrounding uid. The torque depends on the orientation of the graphene planes with respect to the applied eld and on particle volume, but is independent of particle shape. This key aspect makes diamagnetic manipulation extremely versatile and powerful. Interestingly, the absolute value \|c t À c k \| for HOPG increases with decreasing temperature. 22,26,27 The magnetic potential energy can be expressed as: U m ¼ À ð B 0 0 m eff ðBÞ$dB ¼ À V 2 B 0 2 2m 0 ÂÀ c k À c 1 Á þ À c t À c k Á cos 2 q Ã :(4) The orientations of stable rotational equilibrium which satisfy vU m /vq and v 2 U m /vq 2 > 0 depend on the sign of the particle's magnetic anisotropy (c t À c k ), so that for diamagnetic HOPG, with (c t Àc k ) < 0 and c t < c k < 0, the equilibrium orientations correspond to q ¼ AE90 . Therefore, the HOPG akes rotate until their graphene planes align parallel to the direction of the applied magnetic eld, in order to minimize the magnetic interaction energy, as depicted in Fig. 1(a). The surrounding uid has the effect of opposing the rotation of the particles through the contributions of rotational viscous drag and thermal rotational Brownian uctuations. The magnetic eld in our experiments (B 0 $ 240 mT) is strong enough to overcome these contributions so that HOPG microparticles can be quickly magnetically aligned and stably maintained into a vertical orientation. Our HOPG micro-akes are coated with a phospholipid layer to facilitate dispersion in aqueous solution. Phospholipid molecules are also diamagnetically anisotropic and tend to align perpendicular to applied magnetic elds, as evidenced by the magnetic deformation of phospholipid bilayers and liposomes in strong magnetic elds (>4 T). 20,[28][29][30] We neglect magnetic effects on our lipid layers given that we use modest magnetic elds up to 0.3 T, that the magnetic anisotropy of similar phospholipid molecules 29,30 is two orders of magnitude lower than the value of (c t À c k ) for HOPG, and that lipid layers on our particles are $100 times thinner than the HOPG particle size. Note that, by contrast, the lipid layer plays a major role in the AC electrical orientation of the HOPG particles, as detailed in the following section. Electrical alignment of lipid-coated HOPG The theory describing the electro-orientation of lipid-coated HOPG micro-particles is more complex than that for diamagnetic orientation because the electric de-polarization effects cannot be ignored, i.e., the electric eld inside the particle cannot be approximated to be equal to the applied external electric eld (this could be done in the diamagnetic case because c ( 1). Lipid-coated HOPG micro-akes can be modeled as oblate, layered, anisotropic ellipsoids and the theoretical frameworks by Jones 25 and Asami 31 can be used to calculate the non-trivial full expression for the effective induced electric moment on the particles. The dependency on electric eld frequency f is introduced via complex permittivities e that describe the relevant dielectric and conducting properties of the HOPG core, lipid layer and solution (e ¼ 3 À is u3 0 , where 3 and s are the relevant static relative permittivity and conductivity, respectively, 3 0 is the permittivity of free space and u ¼ 2pf is the angular frequency of the applied AC electric eld). The reader can refer to Asami et al. 31 for details of how Laplace's equation can be solved to nd out the electrical potential outside a layered ellipsoid in order to derive the effective dipole moment of the submerged particle and its dependency on the frequency of the AC electric eld (see also Section 3.5 for an intuitive explanation of how this frequency dependency arises). The effective electric dipole moment components for a layered ellipsoid can be expressed as: p eff,k (t) ¼ V 2 3 1 3 0 K k E 0k (t),(5) where V 2 is the particle's volume, 3 1 is the relative static permittivity of the uid, the sub-index k indicates the x, y, z directions in the particle frame of reference, K k are the complex effective polarization factors and E 0k (t) are the components of the external AC electric eld. Similarly to the magnetic case, the orientational electric torque can be obtained as T e ¼ p eff Â E 0 . In our experiments, E 0 is applied once the micro-akes have been pre-aligned with the vertical magnetic eld. E 0 is linearly polarised along the horizontal X direction and makes an angle a to the z axis [see Fig. 1 (b)], so that E 0 ¼ E 0 (sin ax + cos aẑ), where E 0 is the eld amplitude. In the presence of this eld, the particles feel a time-averaged electric torque around the inplane y direction, of the form: T e k ¼ 1 4 V 2 3 1 3 0 E 0 2 sinð2aÞRe Â K t À K k Ã ;(6) where K k h K x,y and K t h K z are the effective complex polarization factors for the in-plane and out-of-plane particle directions, respectively. These factors include the dependency on electric-eld frequency, on particle shape and on dielectric and conductive properties of HOPG, lipids and solution. The polarization factors have non-trivial forms, particularly for layered particles with an anisotropic core, as is the case here. 25 Note that eqn (6) has a very similar form to the previous eqn (3) for the magnetic orientational torque. The maximum amplitude of the electrical orientational torque, which occurs at angles a ¼ AE45 , is given by: T e max ¼ V 2 3 1 3 0 E 0 2 4 Re Â K k À K t Ã :(7) Hence, we can write T e k ¼ ÀT e max sin(2a). Analogously to eqn (4), the electric interaction potential energy is: U e ¼ À V 2 3 1 3 0 E 0 2 4 Re Â K k þ À K t À K k Á cos 2 a Ã :(8) For the parameters in our experiments, with Re[K t À K k ] < 0 and Re[K k ] > Re[K t ] > 0, stable rotational equilibrium takes place for orientation angles a ¼ AE90 , i.e., particles rotate until their graphene planes align parallel to the applied electric eld direction. Once aligned, the particles are rotationally trapped as long as the electric eld is on. For small-angle deviations from orientational equilibrium, the rotational trap can be considered approximately harmonic, i.e., T e k (a) ¼ ÀT e max sin(2a) z Àk e a, where k e ¼ 2T e max is the electrical rotational trap stiffness. Experimental demonstration Solubilized lipid-coated HOPG micro-particles HOPG micro-particles are very hydrophobic and strongly aggregate in aqueous solution. Amphiphilic lipid molecules, which form biological cell membranes, are good biocompatible candidates to coat HOPG micro-particles and disperse them in aqueous saline solution, as predicted by simulation for graphene. 32 Lipid monolayers have been shown to coat graphene oxide 33,34 and graphene sheets. 35 The protocol that we have developed for functionalization can be found in Appendix A. Experimental set-up To generate a static, near-uniform magnetic eld along the vertical Z direction at the sample region, we use a NdFeB permanent magnet (grade N50, 25 Â 25 Â 10 mm, $0.4 T at magnet surface, from Magnet Sales UK). The magnet is placed $6 mm below the sample (see Fig. 2), resulting in a magnetic eld strength $240 mT at the sample (measured with a gaussmeter). The magnetic eld gradient along Z is $0.03 mT mm À1 so that variations in eld strength and direction over the mm length scales relevant to our experiments are negligible. The horizontal time-varying electric eld is generated by two parallel horizontal wires (lying along Y) glued onto the sample glass slide with nail varnish. These thin insulated copper wires (50 mm diameter) are placed at a centre-to-centre distance d $ 150 mm (see Fig. 2). An applied voltage of $4.6 V pp (peak-topeak) results in electric eld magnitudes E 0 z 2 Â 10 4 V m À1 at the sample region between the wires, with the eld linearly polarized along the horizontal X direction. We use AC electric eld frequencies in the range 1-70 MHz. A sample ($20 ml) of lipid-coated HOPG micro-particles in 20 mM NaCl aqueous solution is placed onto the glass slide with the two wires and sealed with a glass coverslip on top. The sample is imaged from above with a custom-made microscope (total magnication of 40) onto a CCD camera, at acquisition rates up to 150 frames per second. A small white-light LED illuminates the sample from below, as shown in Fig. 2. We have carried out measurements of magneto-electrical orientation on 10 individual lipid-coated HOPG particles with a narrow spread of particle sizes and shapes and with average dimensions $2 mm Â 4 mm Â 7 mm (0.5 mm standard deviations, see Appendix B). The vertical Z position of all micro-akes imaged in experiments is approximately the same, with variations of order 0.5 mm. Measurement sequence In the absence of applied elds, the HOPG micro-akes in the sample solution tend to lie horizontally, with their graphene planes parallel to the slide surface, due to their oblate shape, as shown in Fig. 3(a). When the magnet is placed under the Fig. 2 Schematic of experimental set-up. The magnet generates a vertical magnetic field along Z. The wires generate a horizontal electric field along X. HOPG micro-flakes are vertically oriented parallel to the X-Z plane by the magnetic and electric fields as shown in the zoomed-in view and in more detail in Fig. 1. Micro-flake rotation is imaged from above the sample with a microscope onto a CCD camera (image plane parallel to X-Y plane). sample, the vertical magnetic eld aligns the micro-akes onto vertical planes, as shown in Fig. 3(b), following the principles explained in Section 2.1. At this point, the micro-akes are still free to rotate around the vertical Z axis. When the horizontal AC electric eld is turned on, the micro-akes rotate around Z until their graphene planes align with the applied electric eld direction as shown in the image sequence in Fig. 3(c) and as described in Section 2.2. The micro-ake orientation is thereby conned to the X-Z plane. The rotation of the micro-akes when the electric eld is applied is recorded with the microscope and camera (see ESI Videos †). Individual isolated particles are imaged in order to avoid interaction effects. No heating effects are observed throughout the experiments. Image analysis: tracking rotational motion The use of the strong magnetic eld to keep the micro-akes vertically aligned allows us to restrict our analysis to a single rotational degree of freedom, 4, for rotations around Z, where we dene 4 as the angle between the HOPG graphene planes and the positive X direction [see Fig. 3(c)]. For our xed electric eld polarization along X, 4 ¼ (90 À a) and T e k ¼ ÀT e max sin (24) [see eqn (6)], with orientational equilibrium corresponding to 4 ¼ 0 . The particle-rotation video sequences are analyzed using automated image processing algorithms in Matlab, in order to extract 4 and the particle dimensions as described in Appendix B. This reproducible detection of the particle's orientation represents an improvement with respect to alternative indirect methods reported in the literature 3-5 based on light-transmission levels through graphitic micro-particle dispersions. Frequency dependencyeffect of the lipid coating The use of AC elds and insulated wires as electrodes is essential to avoid undesired electrochemical effects in ionic and biocompatible solutions. Furthermore, given that phospholipid membranes are insulating, 25 MHz AC electric elds are required to observe the electro-orientation of the particles. This is because the lipid layer effectively insulates the HOPG at low and medium frequencies and only becomes electrically transparent at high frequencies. Maxwell-Wagner interfacial polarization effects arise due to the presence of several interfaces (solutionlipids-HOPG) with characteristic charge build-up and relaxation time constants which depend on the electrical properties at either side of the boundary. 25 These time-varying interfacial free charges introduce a time dependency of the effective induced electrical dipole moment and give rise to a frequency dependency of the electric torque experienced by the lipid-coated HOPG particles, as explained in the theory Section 2.2 and as evidenced by the measurements presented in Section 4. Actual electric eld amplitude at the sample For a full quantitative discussion of the electric torque exerted on the micro-akes, we need to know the exact magnitude of the electric eld between the sample electrodes. At the high frequencies employed (1-70 MHz), electrical impedancemismatch effects in the circuit connections can give rise to reections at junction boundaries and cavity build-up effects. These lead to frequency-dependent variations of the effective electric eld amplitude at the sample. The input voltage signal fed to our sample wires also presents some variation with frequency. These effects are measured and taken into consideration in order to correct our measured values of rotational trap stiffness and maximum electric torque as a function of frequency, as detailed in Appendix C. Results and discussion Below, we initially present data for the observed rotation of individual lipid-coated HOPG micro-akes upon application of the AC electric eld, including ts to the solution of the rotational equation of motion neglecting thermal uctuations. We then use measurements of rotational Brownian uctuations around the equilibrium orientation for aligned particles (rotationally trapped) to determine the rotational trap stiffness k e , maximum electric torque T e max and polarization anisotropy factor Re[K k À K t ] for the HOPG micro-akes as a function of frequency. This analysis reveals the presence of weak interactions between the micro-particles and the nearby glass surface, which we also quantify to discriminate their effect from that of the electric torque. All experiments are carried out in the presence of the vertical magnetic eld B 0 z 0.24 T. The maximum orienting magnetic torque [V 2 B 0 2 (c t À c k )/(2m 0 ) in eqn (3)] applied to the particles is $2.4 Â 10 À16 Nm, calculated for HOPG particles with average size $2 mm Â 4 mm Â 7mm using the values of c k and c t given in Section 2.1. This strong magnetic torque is two orders of magnitude larger than the applied electric torque (see measurements below) and is crucial to ensuring our micro-akes stay vertically aligned throughout the course of electroorientation experiments. also ESI Videos †). Particle rotation is only observed for electric-eld frequencies above $10 MHz (for E 0 $ 2 Â 10 4 V m À1 and V 2 $30 mm 3 on average). Initially, 4 z 70-80 ; the AC electric eld is turned on at time zero and the micro-ake rotates until it reaches orientational equilibrium at 4 z 0 within a few seconds. Fig. 4 shows that particles align faster with increasing electric eld frequency, indicating stronger electric torques. The equation of motion for the electro-orientation of the micro-particles in solution is given by the balance of torques: Rotation of micro-akes I k 4 :: ¼ ÀC k _ 4 þ T e k þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k B TC k q W ðtÞ;(9) where I k is the moment of inertia of the micro-ake around its in-plane axis, C k is the corresponding rotational friction coefficient and T e k is the applied electric torque [T e k ¼ ÀT e max sin (24), see eqn (6) and (7)]. The last term in eqn (9) corresponds to the stochastic rotational Brownian uctuations at temperature T, where k B is Boltzmann's constant and W(t) is a random variable with Gaussian distribution of zero mean and unity variance. The inertial term I k € 4 in eqn (9) can be neglected in comparison to the viscous drag term ÀC k _ 4. As a rst approximation, we can neglect rotational Brownian uctuations and solve C k _ 4 ¼ ÀT e max sin (24) to nd an analytical expression for the evolution of the micro-ake orientation in time: 4ðtÞ ¼ arctan tanð4 0 Þ Â exp À 2 T e max C k t ! ;(10) where 4 0 is the orientation angle at time t ¼ 0. Fits of our data to eqn (10) are presented in Fig. 4, showing good agreement. The value of T e max could be extracted from these ts using calculated values for C k (see Appendix D). However, we use a more accurate method which does not rely on pre-calculated C k values, as described in the following sub-section. Orientational uctuations for aligned micro-akes By analysing orientational uctuations of micro-akes around equilibrium (4 z 0 ) in electrical rotational traps, we extract measurements of T e max , k e and Re[K k À K t ] at different electric eld frequencies. Considering the effect of rotational Brownian uctuations is essential in order to detect weak torques in environments dominated by thermal uctuations. The Brownian motion of ellipsoidal particles was rst theoretically described in 1934 by Perrin, 36 but was not experimentally veri-ed until 2006. 37 The orientational uctuations of the micro-akes as they stay aligned with the electric eld are monitored for 20-30 seconds. The electric eld is then turned off and particles are recorded for a further 20-30 seconds to monitor their free rotational uctuations in the absence of the electric eld. Angular uctuations derived from the measured video recordings are shown in Fig. 5 for a single HOPG micro-ake at $23 C. Data corresponding to electric eld frequencies 20 MHz, 40 MHz and 60 MHz are presented, together with data in the absence of the electric eld. Fluctuations are clearly smaller in amplitude when the electric eld is on, and decrease with increasing eld frequency, as shown by the histograms on the right-hand side, evidencing rotational trapping and successful alignment. The mean square angular displacement (angular MSD) is dened as h[D4(s)] 2 i, where D4(s) ¼ 4(t 0 + s) À 4(t 0 ) are the orientational uctuations over a time interval s and t 0 is the initial time. The angle brackets indicate averaging over all initial instants. For the case of free (untrapped) rotational diffusive behaviour, the angular MSD should depend linearly on 36 In the presence of rotational trapping, the angular MSD deviates from this linear behaviour. This is shown in Fig. 6(a), where the measured angular MSD corresponding to the orientational uctuation data in Fig. 5 is presented. s as h[D4(s)] 2 i ¼ 2D 4 s, where D 4 ¼ k B T/C k is the corresponding rotational diffusion coefficient. The non-linearity of the angular MSD due to rotational trapping is observed even when the electric eld is off [ Fig. 6(a)]. This is due to the fact that particles fall under gravity and interact non-negligibly with the nearby glass surface in a way that opposes their rotation. Both the small effect of these interactions with the glass and the strong alignment in the presence of the AC electric eld are described theoretically as rotational traps with trap stiffness k glass and k e , respectively. For small-angle uctuations, the electrical rotational trap is approximately harmonic so that T e k (4) ¼ ÀT e max sin (24) z Àk e 4, with k e ¼ 2T e max . A similar assumption is made for the trap due to interactions with the glass. Hence, the full equation of rotational motion becomes: ÀC k _ 4 À k total 4 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k B TC k q W ðtÞ ¼ 0;(11) where k total is equal to either (k e + k glass ) when the electric eld is on, or k glass when it is off. The well-known theory of Brownian motion in a harmonic trap in solution [38][39][40] can be applied so that the angular MSD is given by: D ½4ðt 0 þ sÞ À 4ðt 0 Þ 2 E ¼ 2k B T k total 1 À exp À k total C k s ! ;(12) and the auto-correlation of the uctuations around the equilibrium orientation in the rotational trap is: RðsÞ ¼ h4ðt 0 Þ4ðt 0 þ sÞi ¼ k B T k total exp À k total C k s :(13) The angular MSD data in Fig. 6(a) shows how the measured asymptotic values at long lag times [pre-factor 2k B T/k total in eqn (12)] decrease with increasing electric eld frequency, indicating increased rotational trap stiffness and electric torques at higher frequencies. The corresponding measured autocorrelation R(s) is shown in Fig. 6(b). The same trend of increasing trap stiffness with frequency is evidenced by the faster exponential decay and lower R(s ¼ 0) values observed at higher frequencies, as given by eqn (13). 4.2.1 Frequency dependency of maximum electric torque, rotational trap stiffness and polarisation anisotropy. In order to extract quantitative values of k total , we t the auto-correlation of the measured orientational uctuations as a function of time lag to the exponential function in eqn (13). This choice is motivated by the fact that the equivalent method for translational uctuations is considered one of the most reliable ones out of several available methods for calibrating the trap stiffness in optical tweezers. 39 From the t, k total can be obtained from the autocorrelation at zero lag time R(s ¼ 0), using T ¼ 23 C. For each frequency, we extract k total,on ¼ (k e + k glass ) from the uctuations recorded with the electric eld on, and k total,off ¼ k glass from the uctuations recorded with the electric eld off to isolate interactions with the glass surface. The electrical trap stiffness, k e , is obtained from the subtraction k total,on À k total,off . The value of the maximum electric torque is obtained as T e max ¼ k e /2. For a given micro-ake, approximately ve measurements are taken at a given frequency, which are averaged to obtain mean and standard deviation values at that frequency. The standard errors of the mean are used as error bars. The process is repeated for different frequencies in order to investigate the frequency dependency of the electric torque, which follows from eqn (7). Results are then corrected for the voltage variations with frequency that originate from the RF source and impedance mismatch effects explained in Section 3.6 and Appendix C. Fig. 7(a) shows values of k e and k glass versus frequency for the electro-orientation of an individual lipid-coated HOPG micro-ake in solution near a glass surface passivated with polyethylene glycol (PEG)-silane. PEGylation, i.e., the coating of a surface with largely non-interacting PEG polymer chain brushes, is a well known technique for surface passivation. 41 The resulting k glass is low and approximately constant in time (independent of frequency, as expected), which enables the reliable characterisation of interactions between micro-particle and glass. The electric rotational trap stiffness k e , and therefore T e max , decrease with decreasing frequency due to the electrically insulating effect of the lipid coating at low to medium frequencies. Data for the maximum electrical torque T e max measured for 10 different micro-akes are presented in Fig. 7(b), including the overall mean and standard error for all measurements. Given the narrow spread of particle sizes and shapes chosen for our experiments, the trend with frequency remains clearly visible. From the 10 particles measured, 3 were on PEG-silane passivated glass slides, 4 on plasma cleaned slides and 3 on untreated glass. In Fig. 7(b), the measured average T e max is in the range 0-0.6 Â 10 À18 Nm, depending on the applied electric eld frequency, corresponding to an average torsional stiffness (k e ) range of 0-1.2 Â 10 À18 Nm rad À1 . In the theoretical expression for T e max in eqn (7), we have a frequency-independent pre-factor, V 2 3 1 3 0 E 0 2 /4, multiplying the frequency-dependent polarization anisotropy factor Re[K k À K t ]. The pre-factor evaluates to $2 Â 10 À18 Nm, calculated using V 2 ¼ 4pabc/3 for our particles modeled as ellipsoids with semi-axis lengths a ¼ 1 mm, b ¼ 2 mm and c ¼ 3.5 mm on average, 3 0 ¼ 8.85 Â 10 À12 F m À1 , 3 1 z 80 for 20 mM NaCl at 23 C (ref. 42) and E 0 ¼ 2 Â 10 4 V m À1 . Using the T e max measurements in Fig. 7(b) and this calculated pre-factor, we obtain a polarization anisotropy factor in the range $0-0.3 for our lipid-coated HOPG micro-akes, for frequencies in the range 10-70 MHz. This is indicated by the right-hand-side axis in Fig. 7(b) and is a useful measurement for a number of HOPG electro-manipulation experiments. 25 The reproducible frequency dependence of the orientational torque on the micro-akes is a useful feature which enables frequency-control of micro-particle orientation for numerous applications. The turnover frequency at which electric torque values begin to increase can be altered, for instance, by modifying the thickness of the insulating lipid layer on the micro-akes, the conductivity of the solution or the micro-ake aspect ratio, opening interesting control possibilities. 4.2.2 Interactions with the glass surface. Additionally, we have used our measurements to characterize the interactions of individual lipid-coated HOPG micro-akes with glass slides with various surface treatments, given that micro-akes fall under gravity coming into contact with the glass slide. The quantitative characterization of surface sticking effects and glass-lipid interactions is of particular interest to applications in biology. k glass is determined from the above mentioned measurements of rotational Brownian uctuations acquired in the absence of the electric eld as explained in the previous paragraphs in Section 4.2. Results for k glass from 282 measurements for 10 different particles on untreated glass, plasma-cleaned glass and PEG-silane passivated glass are compared in the histograms shown in Fig. 8. The histograms present clear differences, with PEG-silane passivation leading to the lowest k glass mean and spread values, i.e., to the weakest rotational trapping due to surface sticking. Untreated glass shows a wider distribution of k glass values than PEG-silane treated glass. Plasma-cleaned slides show an even larger k glass spread, as expected by the generation of charged groups on the glass surface during the plasma cleaning process. While measurements have been carried out using 10 different particles with a narrow spread of particle sizes and shapes (see Appendix B), differences in particle geometry and the various possible values of glassparticle contact area throughout the measurements contribute to the spread of the measured k glass values. Conclusions and outlook We have presented a detailed study of the magneto-electrical orientation and rotational trapping of lipid-coated HOPG micro-particles in aqueous solution, including a solid theoretical framework and quantitative experimental results. Measurements of the maximum magnetic and electric orientational torques, rotational trap stiffness and polarisation anisotropy of the micro-particles have been presented and their dependency on the frequency of the applied electric eld has been investigated. These measurements are the rst reported for lipid-coated HOPG micro-akes, with the observed frequency dependency opening the door to the implementation of frequency controlled electro-orientation using this technique. The interactions of the lipid-coated particles with glass surfaces with different surface treatments have been characterized by analyzing their weak rotational trapping effect on the micro-akes, with this effect being reliably discriminated from the stronger rotational trapping generated by the electric eld. Our results demonstrate, for the rst time, connement to a well-dened plane of the orientation of graphitic micro-akes in solution, with this new method exploiting the electrical and diamagnetic anisotropy of the particles via the application of simultaneous perpendicular magnetic and electric elds. The combination of magnetic and electric elds in different congurations can open up new ways of manipulating graphitic micro/nano-particles for their orientation, connement and transport, 25,43 with the use of time-varying elds enabling frequency-control of the magneto-electrical manipulation. The principles of our method extend to other carbon-based micro/ nano-particles, such as graphene platelets or carbonnanotubes, and our scheme has great potential for being scaled down via micro-fabrication. Alternative schemes could be devised to orient and conne graphitic micro-akes to a plane making use of rotating electric/magnetic elds, or of fast time-switching between elds in two perpendicular directions, as demonstrated e.g. for the orientation of diamagnetic, anisotropic polymer and cellulose bers. 44,45 Applications in biochemistry and biological and medical physics are particularly relevant to our experiments. Our lipidcoated HOPG micro-particles are biocompatible and our results in NaCl aqueous solution easily extend to biocompatible solutions. Our particles can be functionalized and specically bound to biomolecules such as antibodies, protein complexes and nucleic acids, and the lipids can be uorescently labeled or conjugated to polymers, for instance to change the electrical properties of the coating. 46 HOPG micro-particles of regular sizes could be generated using current micro-fabrication techniques, such as ion-beam milling. Our particles hence constitute a promising tool with the potential to function as carriers, labels or specic targets for biological and chemical sensing applications. [17][18][19]47 Cells could be attached onto HOPG microplates which could be manipulated for versatile cell-to-cell interaction experiments, such as those involving immune response, virus transfer, neuron activity, etc. As another example, given their capability of absorbing infra-red light, graphitic micro-particles could also be good candidates for photo-thermal cell therapy (e.g. for cancer treatment) or for temperature-jump studies in vitro/in vivo. Furthermore, our HOPG micro-particles in calibrated rotational traps of known torsional trap stiffness could be used for sensing biologically relevant torques. These are typically in the range 0.01-1 Â 10 À18 Nm (ref. 48) (e.g. 0.02-0.08 Â 10 À8 Nm for ATP synthase (F1-ATPase) 49 or $0.01 Â 10 À18 Nm for RNA polymerase 48 ). The rotational trap stiffnesses we have measured are 0-1.2 Â 10 À18 Nm rad À1 and can be controlled by varying particle size and electric eld magnitude and frequency. These stiffness values are within the range of those in magnetic tweezer experiments with superparamagnetic micro-beads tethered to a surface. 50 The torque resolution for sensing depends on the uncertainty of the angle detection via rotational tracking. We achieve $ 0.2 degrees ($0.004 rad) in our experiments (see Appendix B), meaning that our reduced rotational trap stiffnesses would allow sufficient resolution for biological torque sensing. Most current single-molecule force and torque spectroscopy techniques suffer from the disadvantage of coupled torque and force sensing. 48 The use of our HOPG microparticles together with appropriately applied magnetic and electric elds to independently control rotational trapping and translational connement, could result in the future in extremely useful, decoupled force and torque sensing schemes. Appendices A Preparation of lipid-coated, dispersed HOPG microparticles We have developed the following protocol to coat and solubilize micron-sized HOPG particles with lipid bilayers. Briey, 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC, Avanti Polar Lipids) is dissolved in chloroform (CHCl 3 ) and the resulting solution is dried under a N 2 gas stream. The solventfree lipid lm is hydrated at a temperature ($60 C) well above the lipid main phase transition (À2 C) with 20 mM aqueous solution of NaCl, resulting in a nal lipid concentration of 0.6 mg ml À1 . To obtain small unilamellar lipid vesicles, this dispersion is briey sonicated with a probe sonicator. Then the sample is centrifugated (30 minutes at 8000 rpm) to pellet down titanium probe particles and residual multilamellar vesicles. Small pieces ($1 mm 3 ) of a HOPG crystal (ZYH grade, 3.5 mosaic spread, from MikroMasch) are introduced in the NaCl aqueous lipid solution and sonicated in a small bath at 60 C for 5-30 minutes to induce the coating of the particles. Longer sonication times lead to a larger population of solubilised HOPG particles. The result is a clearly dispersed solution of HOPG particles. For our experiments, micron-sized akes with average dimensions $2 mm Â 4 mm Â 7 mm are selected. Fig. 9 shows atomic force microscopy (AFM) images of lipidcoated HOPG micro-particles of different sizes. The AFM image in solution [ Fig. 9(a)] shows very round particles, corresponding to the external lipid bilayers forming a vesicle-like structure. Force versus indentation curves [ Fig. 9(c)] reveal 4-5 indentation steps, suggesting that the particles are coated with several lipid bilayers. B Image processing Particles are rst automatically detected on each frame in the sequence by means of a thresholding operation. This results in a connected region representing the particle mask, surrounded by a background region. The ellipse that best ts the particlemask shape is then found [particles are approximated as ellipsoids, see last frame in Fig. 3(c)] and the orientation angle 4 of the particle is obtained. The uncertainty of our angle measurements is $0.2 degrees. This is determined using samples of xed HOPG micro-akes and calculating the standard deviation of the distribution of angles extracted with our image processing algorithms over 20 000 repetitions. Particle-dimension estimates are obtained for each micro-ake from the lengths of the minor and major axes of the tted ellipses. For the 10 different individual lipid-coated HOPG micro-particles we have used in our measurements, the average dimensions are $2 mm Â 4 mm Â 7 mm (ellipsoid axis lengths with standard deviations AE0.5 mm). C Correction for the variation with frequency of the actual voltage amplitude at the sample It is important to take into consideration impedance-mismatch effects in our electrical connections at the high frequencies (1-70 MHz) of the electric elds employed in our experiments. Our circuit components (radio frequency (RF) signal generator, amplier, switch, coaxial cables) are all specied for an impedance Z 0 ¼ 50 U. However, the impedance we measure for our electrode wires is Z w $100 U, consistent with that of a parallel-wire transmission line made out of enamel-coated wires (50 mm wire diameter, 150 mm wire distance). It is actually not possible to match this to Z 0 ¼ 50 U due to geometrical constraints. We minimise abrupt changes in impedance by carefully tapering connecting wire distances and use twisted wires where possible to avoid RF-noise pick up. The unavoidable impedance mismatch results in some reections at connection junctions and cavity build-up effects which vary with the frequency of the signal. Additionally, our amplied input voltage signal presents some variation with frequency. We can measure the input alternating voltage amplitude, V in , into the electrode wires and the output voltage, V out , transmitted through them [see Fig. 10(a)], as a function of frequency, and dene V r ¼ V out /V in . By considering voltage signal reection and transmission through two subsequent boundaries with impedances Z 0 / Z w and Z w / Z 0 at either side [see Fig. 10(a)], we can calculate the voltage amplitude at the sample as V w ¼ V in [2Z w /(Z 0 + Z w )]. The factor [2Z w /(Z 0 + Z w )] corresponds to the voltage transmission coefficient through the input Z 0 / Z w boundary. Z w can be calculated as Z w ¼ ðZ 0 =V r Þ½ð2 À V r Þ þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À V r p using the measured value of V r at each frequency. We nd no appreciable difference between voltage measurements in solution and in dry samples, within our measurement uncertainties, owing to the fact that our thin sample wires are insulated with an enamel coating. Fig. 10(b) shows the measured V in and V out amplitudes and calculated V w as a function of frequency, normalised to the value of the input voltage amplitude at zero frequency, V in,0 (typically $2.3 V). Five measurements are taken at each frequency, re-soldering the thin electrode wires each time, to check reproducibility upon re-connection. Mean values are shown as data points, with standard errors as error bars. Since, as given by eqn (7) with E 0 ¼ V w /d and as shown by the data in Fig. 11, the electric torque is proportional to the square of the electric eld modulus and, therefore, proportional to V w 2 , a correction factor (V w /V in,0 ) À2 is used in our data analysis to multiply and scale our measured values of electric torque as a function of frequency, in order to correct for the abovementioned effects. D Formulas for oblate ellipsoids Approximating our HOPG micro-akes as oblate ellipsoids with semi-axes a < b z c, with a perpendicular to the graphene planes and b and c parallel to them, we can use analytical expressions to calculate the rotational friction coefficient, C k , for rotations of the ellipsoid around any of its in-plane axes: 36 C k ¼ 32ph 3 ða 4 À b 4 Þ ð2a 2 À b 2 ÞS obl À 2a ; (14) where h is the dynamic viscosity of the solution and the geometrical factor S obl for oblate ellipsoids takes the form: S obl ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi b 2 À a 2 p arctan ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi b 2 À a 2 p a ! :(15) Fig. 1 1Schematic of magneto-electrical orientation of HOPG microflakes represented as oblate ellipsoids. (a) HOPG particle rotation upon application of a vertical magnetic field B 0 . (b) HOPG particle rotation upon additional application of a horizontal electric field E 0 . Fig. 3 3Microscope brightfield (transmission) images of lipid-coated HOPG micro-flakes: (a) with no applied fields; (b) vertically aligned in the presence of an applied vertical magnetic field $240 mT; (c) sequence of rotation around Z upon turning on a horizontal electric field oscillating at 30 MHz. Fig. 4 Fig. 4 44shows the measured evolution in time of the orientation angle 4 for a single micro-ake during electro-orientation (see Measured rotation of a single HOPG micro-flake during electro-orientation. The electric field is turned on at t ¼ 0. The micro-flake rotates until it aligns with the electric field direction (4 z 0 ). Data shown for electric field frequencies: 20 MHz, 40 MHz and 60 MHz, with E 0 $ 2 Â 10 4 V m À1 . Solid lines: fits to eqn(10).This journal is © The Royal Society of Chemistry 2016 RSC Adv., 2016, 6, 46643-46653 \| 46647 Paper RSC Advances Open Access Article. Published on 20 April 2016. Downloaded on 9/23/2020 6:26:41 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. Fig. 5 5Fluctuations in HOPG micro-particle orientation 4 as a function of time in the absence and presence of the orienting electric field, for field frequencies 20 MHz, 40 MHz and 60 MHz. Histograms of 4 values shown on the right. Fig. 6 6(a) Angular MSD (data points and envelope for error bars) for the orientational fluctuation data shown inFig. 5.(b) Auto-correlation [data points and fits to eqn(13)] for the same fluctuation data. Fig. 7 7(a) Measured rotational trap stiffness versus frequency for an individual lipid-coated HOPG micro-flake: k e for the electrical rotational trap and k glass for the trap due to glass-particle interactions. Data points: mean value from 5-10 measurements per frequency. Error bars: standard error of the mean (SE). (b) Data for the maximum electric torque T e max ¼ k e /2 measured for 10 different micro-flakes. Data points from all particles shown together with overall mean value (solid line) and SE (light blue envelope) for each frequency. The axis on the right hand side shows the polarisation anisotropy factor Re[K k À K t ]. Fig. 8 8Characterization of interactions of individual lipid-coated HOPG micro-flakes with glass surfaces with various treatments. Histograms of trap stiffness k glass due to glass-particle interactions for 10 different particles on untreated glass slides (48 measurements), plasma cleaned glass slides (145 measurements) and PEG-silane passivated glass slides (89 measurements). Fig. 9 9(a) Amplitude modulation (AM) AFM image of dispersed graphite flakes of different sizes coated with lipids in solution. They take a characteristic round shape. The height range is 20 nm. (b) AM-AFM image of a lipid-coated graphite flake imaged in air. (c) Force vs. indentation curves on two large flakes of figure (a), the arrows indicate the points where the AFM tip seems to penetrate a lipid bilayer, suggesting that the flakes are coated with several lipid bilayers. Fig. 10 10(a) Schematic of transmission line representing the thin electrode wires at the sample and connections to either side. (b) Measured values versus frequency of the input voltage amplitude into the wires, V in , voltage amplitude transmitted through them, V out , and calculated voltage amplitude at the sample, V w , all normalised to the input voltage at zero frequency, V in,0 . Fig. 11 11Measured maximum electric torque T e max and rotational trap stiffness k e as a function of the squared voltage amplitude applied to the sample electrode wires, V w 2 , for an electric field frequency of 20 MHz, for an individual micro-flake. AcknowledgementsThe authors thank Jonathan G. Underwood, Peter Barker and Phil H. Jones for very useful discussions. 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[]	[]	[]	[]	Leopold Vietoris and Guido Hoheisel showed how the existence of lim x→0 sin x x can be derived from the trigonometric addition formulas. In this article two new proofs for this result are given. In addition it is discussed how this limit is related to the definition of π.	null	[ "https://arxiv.org/pdf/1302.1167v1.pdf" ]	119,127,175	1302.1167	2f6904a357730db94d1c7704daf0968829db2bcb	3 Feb 2013 3 Feb 2013arXiv:1302.1167v1 [math.HO] lim x→0 sin x x and the definition of π Helmut ZeiselAMS Subject Classification: 26A0939B2297D40 Keywords: Trigonometric functionsMathematical inductionJensen's inequality Leopold Vietoris and Guido Hoheisel showed how the existence of lim x→0 sin x x can be derived from the trigonometric addition formulas. In this article two new proofs for this result are given. In addition it is discussed how this limit is related to the definition of π. Introduction The computation of lim x→0 sin x x is the fundamental step for the differentiation of the trigonometric functions. Leopold Vietoris (1957) discussed the usual approaches how to derive this limit and explained the drawbacks of these approaches. As conclusion he showed how the existence of this limit can be derived from the trigonometric addition formulas. A similar result was found by Hoheisel (1947). These results are also discussed in Aczél (1966). Current textbooks, however, still either use the same old problematic proofs for lim x→0 sin x x = 1 or use this equation as an axiom without further explanation. For example, Heuser [3] defines sin and cos in axiomatic way using the addition theorems as functional equations and lim x→0 sin x x = 1 as one of the axioms. He states that the given axioms are not completely independent from each other but does not say in particular how this applies to the limit axiom. In this article the approach of Heuser is used. The used axioms to define sine and cosine are, however, slightly different from Heuers' axioms. Then the ideas of Vietoris and Hoheisel are applied to show how the existence of lim x→0 sin x x can be derived from these axioms. Finally it is discussed how the value of this limit is related to the definition of π. The definition of the trigonometric functions In this section the trigonometric functions are analytically defined by some axioms which can easily be shown geometrically. These axioms can be considered as a system of functional equations for sine and cosine. cos c (x ± y) = cos c x cos c y ∓ sin c x sin c y (1) sin c (x ± y) = sin c x cos c y ± cos c x sin c y The subscript c is used to point out that it is not specified whether the angles are measured in degrees or radians: For the moment c > 0 is just the angle of the complete circle and the right angle has a measure of c/4. Additionally the following normalizations are used: cos c (c/4) = 0 (3) sin c (c/4) = 1(4) The particular choice of c is not important because a change of c is only a change of the angle measurement unit and can easily be done by the transformations sin c 1 (x) = sin c 2 c 2 c 1 x and cos c 1 (x) = cos c 2 c 2 c 1 x .(5) The final axiom is that sin c is invertible and monotonically increasing and cos c is invertible and monotonically decreasing in the interval [0, c/4]. In particular, this implies that both functions are continuous. If functions satisfying the geometric properties of sine and cosine exist, they must fulfil these axioms. There are some simple implications from this axioms: The continuous function f (x) = cos 2 c (x) + sin 2 c (x) satisfies the functional equation f (x + y) = f (x)f (y) . This is one of the functional equations discussed by Cauchy [2] and the only continuous solution with f (c/4) = 1 is the function with constant value 1. sin c (0) = sin c (x − x) = sin c x cos c x − cos c x sin c x = 0,(6)cos c (0) = cos c (x − x) = cos 2 c x + sin 2 c x = 1,(7)sin c (0 − x) = sin c 0 cos c x − cos c 0 sin c x = − sin c x,(8)cos c (0 − x) = cos c 0 cos c x + sin c 0 sin c x = cos c x.(9) Additionally tan c x is defined as tan c x = sinc x cosc x for cos c x = 0, which is in particular fulfilled for x ∈ (−c/4, c/4). Then the law of addition for tan c is tan c (x + y) = tan c x + tan c y 1 − tan c x tan c y .(10) A proof based on induction From the above properties, for x < c/(8n) the following inequalities can be derived by induction: sin c (nx) ≤ n sin c x (11) because sin c ((n + 1)x) = sin c (nx) cos c x + cos c (nx) sin c x ≤ sin c (nx) + sin c x ≤ (n + 1) sin c x. tan c (nx) ≥ n tan c x (12) because tan c ((n + 1)x) = tan c (nx) + tan c x 1 − tan c (nx) tan c x ≥ n tan c x + tan c x = (n + 1) tan c x. Combining these two equations gives n sin c ((n + 1)x) ≤ (n + 1) sin c (nx) because n sin c ((n + 1)x) = n tan c x cos c x cos c (nx) + n sin c (nx) cos c x ≤ tan c (nx) cos c x cos c (nx) + n sin c (nx) cos c x ≤ (n + 1) sin c (nx). and n tan c ((n + 1)x) ≥ (n + 1) tan c (nx) because tan c ((n + 1)x) = n sin c (nx) cos c x + n sin c x cos c (nx) n cos c x cos c (nx) − n sin c x sin c (nx) ≥ n sin c (nx) cos c x + sin c (nx) cos c (nx) n cos c x cos c (nx) − sin 2 c (nx) = tan c (nx) 1 + 1 n cos c x cos c (nx) − sin 2 c (nx) ≥ tan c (nx) (1 + 1/n) . By induction one gets for m ≥ n sin c (mx) m ≤ sin c (nx) n ≤ tan c (nx) n ≤ tan c (mx) m(15) Now consider two arbitrary positive rational numbers p q ≤ r s ≤ c/8, i.e. ps ≤ qr and set x := 1 qs , n := ps, and m := qr. Then sin c (r/s) qr ≤ sin c (p/q) ps ≤ tan c (p/q) ps ≤ tan c (r/s) qr(16) and multiply by qs gives sin c (r/s) r/s ≤ sin c (p/q) p/q ≤ tan c (p/q) p/q ≤ tan c (r/s) r/s .(17) By continuity, for every real x, y with 0 < x ≤ y ≤ c/8 sin c y y ≤ sin c x x ≤ tan c x x ≤ tan c y y .(18) So if x approaches 0 from the right, then sinc x x is increasing and bounded from above by the decreasing function tanc x x ; in particular lim x→0+ A proof based on convex functions Another proof for the differentiability is based on the observation that sin x is concave in [0, c/4] and that concave functions have a derivative everywhere except on at most countable many points, see e.g. [8], p. 304. To prove that sin x is concave, it is sufficient to show that it is midpoint concave: If f is continuous and midpoint convex, i.e. if f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) holds for λ = 1 2 then it holds for every λ ∈ [0, 1], and similarly for concave functions. This was shown by Jensen [5]. A simple proof of Jensen's result can be found in [6], Theorem 1.1.4. For 0 ≤ x, y ≤ c/8 sin c x + sin c y = 2 sin c x + y 2 cos c x − y 2 ≤ 2 sin c x + y 2 , sin c is midpoint concave, tan c (2x) + tan c (2y) 2 − tan c (x + y) = (tan c x − tan c y) 2 (tan c x + tan c y) (1 − tan c x tan c y) (1 − tan 2 c x) (1 − tan 2 c y) ≥ 0, tan c is midpoint convex. Now for f (0) = 0 and f convex, one has for 0 < x ≤ y f (x) x = f (x) − f (0) x − 0 ≤ f (y) − f (0) y − 0 = f (y) y , see e.g. [8], p. 303, and similiar for concave functions, which implies (18). The definition of π Up to now it has been shown that lim x→0 sinc x x exists; the value of this limit, however, has not yet been computed. Hoheisel only mentioned that there exists a value for c such that lim x→0 sinc x x = 1. Vietoris used the limit of some "well known" recursive sequence for π to show lim x→0 sin 2π x x = 1. Whether this limit is well known or not depends, however, on the used definition of π. The classic geometric definition of π is the area of the unit circle, and this area can be computed as the limit of the areas of inscribed regular n-gons. Now the inscribed regular n-gon has an area of a n = n 2 sin c c n Substituting x = c/n gives π = lim n→∞ a n = c 2 lim x→0 sin c x x , and, as has been shown, this limit exists. Since sin c 1 (x) = sin c 2 c 2 c 1 x(19) this limit is independent of the particular choice of c. So actually there is no need to compute the value of lim x→0 sinc x x ; it simply can be used as the analytic definition of π. Alternatively one could define π using the area of the circumscribed regular n-gons, which is A n = n tan c c 2n and leads by defining x = c 2n to π = lim n→∞ A n = c 2 lim x→0 tan c x x , which is the same value as above. If one chooses c = 2π, one gets lim x→0 sin 2π x x = 1, and sin 2π and cos 2π can easily be expanded as a Taylor series. This corresponds to the often used definition of π/2 as the first positive zero of cos 2π . From the presented point of view, however, this definition of π is already implicitly contained in lim x→0 sin 2π x x = 1, Summary It is possible to prove the exisitence of lim x→0 sin x x essentially just from trigonometric theorems of addition. This limit can be interpreted as an implicit definition of π. This is more intuitive than assuming lim x→0 sin x x = 1 just as an axiom and, in contrast to using geometric proofs, is exact from the point of analysis. Both limits are nonzero, and, since their quotient is cos c x, they are equal. Since sinc x 1947) proved the same inequality (18) in a different way. Lectures on Functional Equations and Their Applications. J Aczél, Academic PressNew YorkAczél, J. (1966) Lectures on Functional Equations and Their Applica- tions. New York: Academic Press. Cours d'analyse de L'École Royale Polytechnique. Premiere Partie. Analyse Algébrique. A.-L Cauchy, Chez Debure fréresParisCauchy, A.-L. (1821) Cours d'analyse de L'École Royale Polytechnique. Premiere Partie. Analyse Algébrique. Paris: Chez Debure fréres. . H Heuser, Lehrbuch der Analysis. Teil. B. G. Teubner1Heuser, H.(1993) Lehrbuch der Analysis. Teil 1. Stuttgart: B. G. Teub- ner. Funktionalgleichung und Differenzierbarkeit bei den trigonometrischen. G Hoheisel, Funktionen Math. Ann. 120Hoheisel, G.(1947) Funktionalgleichung und Differenzierbarkeit bei den trigonometrischen Funktionen Math. Ann. 120, 10-11. Sur les fonctions convexes et les inégalités entre les valeurs moyennes. J L W V Jensen, Acta Math. 30Jensen, J. L. W. V. (1906) Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 30, 175-193. C Niculescu, L E Perrsson, Convex Functions and their Applications. A Contemporary Approach. BerlinSpringerNiculescu, C., and Perrsson, L.E. (2005) Convex Functions and their Applications. A Contemporary Approach. Berlin: Springer. Vom Grenzwert lim x→0 sin x x. L Vietoris, Elemente Math. 12Vietoris, L. (1957) Vom Grenzwert lim x→0 sin x x . Elemente Math. 12. . W Walter, Analysis. 1SpringerWalter, W. (2004) Analysis 1. Berlin: Springer	[]
[ "On the rate of convergence of Berrut's interpolant at equally spaced nodes", "On the rate of convergence of Berrut's interpolant at equally spaced nodes" ]	[ "Walter F Mascarenhas " ]	[]	[]	We extend the recent work by professors G. Mastroianni and J. Szabados regarding the barycentric interpolant introduced by J.-P. Berrut in 1988, for equally spaced nodes. We prove their two conjectures and present a sharp description of the asymptotic error incurred by the interpolants when the derivative of the interpolated function is absolutely continuous. We also contribute to the solution of the broad problem they raised regarding the order of approximation of these interpolants, by showing that such interpolants have order of approximation of order 1/n for functions with derivatives of bounded variation.	null	[ "https://arxiv.org/pdf/1708.04235v3.pdf" ]	119,582,069	1708.04235	0cf01f29db40d3ca1508b3bb02cb87cd85ef80d6	On the rate of convergence of Berrut's interpolant at equally spaced nodes 3 Nov 2017 July 30, 2018 Walter F Mascarenhas On the rate of convergence of Berrut's interpolant at equally spaced nodes 3 Nov 2017 July 30, 2018 We extend the recent work by professors G. Mastroianni and J. Szabados regarding the barycentric interpolant introduced by J.-P. Berrut in 1988, for equally spaced nodes. We prove their two conjectures and present a sharp description of the asymptotic error incurred by the interpolants when the derivative of the interpolated function is absolutely continuous. We also contribute to the solution of the broad problem they raised regarding the order of approximation of these interpolants, by showing that such interpolants have order of approximation of order 1/n for functions with derivatives of bounded variation. Introduction In a recent article [6], professors G. Mastroianni and J. Szabados discuss barycentric interpolation of functions f : [−1, 1] → R at equally spaced nodes x k,n := 2k/n − 1 for k = 0, . . . , n. They analyze the order of approximation of the barycentric interpolant introduced by J.-P. Berrut [1]: B n ( f , x) := N n ( f , x) D n (x) for x ∈ {x 0,n , . . . , x n,n } and B n f , x k,n = f x k,n , (1) with N n ( f , x) := n ∑ k=0 (−1) k f x k,n x − x k,n and D n (x) := n ∑ k=0 (−1) k 1 x − x k,n .(2) They proved some results and stated two conjectures and a broad open problem about the rate at which B n ( f ) approximates f for some classes of functions. For instance, they showed that the error B n ( f ) − f ∞ is of order 1/n for functions with derivatives in the class Lip 1 of functions with continuity modulus ω(t) ≤ κt. In this article we extend their work, by presenting a detailed analysis of the asymptotic behavior of the interpolation error for functions with absolutely continuous derivatives. We denote the class of such functions by AC 1 , and emphasize that, unlike the definition of the Sobolev space W 2,1 ([−1, 1]), we require that f ′ (x) is defined for all x ∈ [− 1,1] in order for f to belong to AC 1 (we consider directional derivatives at x ∈ {−1, 1}.) We also analyze functions with derivatives of bounded variation, and denote their class by BV 1 , with the same requirement on the derivatives. We prove the two conjectures by Mastroianni and Szabados and show that the order of convergence of the interpolants B n ( f ) above is also of order 1/n for f ∈ BV 1 . Their first conjecture, which we state below, is about the interpolation error for functions f ∈ Lip 1, and we prove it in Section 2. n (B n ( f , x) − f (x)) .(3) This description is given by Theorem 1 below and uses the functions O( f , x) := f (x) − f (1) 2 (x − 1) − f (x) − f (−1) 2 (x + 1) ,(4)E( f , x) := f (1) − f (x) 2 (x − 1) + f (−1) − f (x) 2 (x + 1) .(5) (Throughout the article, O stands for odd and E stands for even.) We must be careful when analyzing the sequences in Equation (3) when x = x k,n is a node, because both the denominator and the numerator of B n ( f , x) are discontinuous at such x, and the interpolant is defined in a different way for them in Equation (1). As a result, the error has more favourable properties at the nodes and this may confuse our analysis of the convergence for a general x. For instance, if x ∈ {−1, 1} then the error B n ( f , x) − f (x) is zero for all n, and the same holds for x = 0 when n is even. In order to handle this issue precisely, we state the following definitions: Definition 1 (Sequence) We say that an increasing function n : N → N with n( j) = n j is "a sequence n j ." The sequence is odd if n j is odd for all j, and it is even if n j is even for all j. Definition 2 (Regular point) We say that x ∈ [−1, 1] is regular for the sequence n j if there exists j 0 such that j ≥ j 0 ⇒ x ∈ x 0,n j , x 1,n j , . . . , x n j ,n j . Definition 3 (The compactification of R) In order to handle infinite limits, we write R := R {+∞, −∞} as the two point compactification of R, endowed with the usual topology and extension of the operators < and ≤. In particular, R and its subset [π/2, +∞], which are relevant to our discussion, are compact in our topology for R. All irrational points are regular for every sequence n j ; the points ±1 are not regular for any sequence, and 0 is regular for odd sequences and irregular for even ones. Given x ∈ [−1, 1], we can decompose any sequence n j in at most three parts: one in which x is a node for all j, so that B n j ( f , x) = f (x) for all j, an two other sequences for which x is regular, one even and another odd (of course, some parts may not be necessary.) Therefore, by understanding the regular points for odd and even sequences we can get the full picture regarding the pointwise convergence of the interpolation error. We now state our first formal result. Theorem 1 (The limits of n (B n ( f , x) − f (x)) for f in AC 1 ) Let f be a function in AC 1 , n j an odd sequence, and x ∈ [−1, 1] such that lim j→∞ n j B n j ( f , x) − f (x) = L ∈ R.(6) If x is irrational then, for the function O( f , x) in Equation (4), L ∈ O( f , x) := − 2 \|O( f , x)\| π , 2 \|O( f , x)\| π ,(7) and if x is rational then there exists a finite set O(x) ⊂ R \ {0}, defined in Equation (86) in Section 6, such that and if x is a regular rational point for n j then L ∈ O( f , x) := {O( f , x) /y, y ∈ O(x)}. Conversely, if L ∈ O( f , x) then there exists an odd sequence n j for which x is regular and Equation (6) holds. Similarly, if n j is an even sequence, Equation (6) holds and x is irrational then L ∈ E( f , x) := − 2 \|E( f , x)\| π , 2 \|E( f , x)\| π , and if x is rational then there exists a finite set E(x) ⊂ R \ {0}, defined in Equation (87) in Section 6, such that and if x is a regular rational point for n j then L ∈ E( f , x) := {E( f , x) /y, y ∈ E(x)}. Conversely, if L ∈ E( f , x) then there exists an even sequence n j for which x is a regular point and Equation (6) B n ( f ) − f ∞ = o(1/n) if and only if f is constant (when n = 2, 4, ...), or f is linear (when n = 1, 3...). In fact, if B n ( f ) − f ∞ = o(1/n) and z ∈ (−1, 1) is irrational then Theorem 1 implies that O( f , z) = {0} and Equation (4) leads to O( f , z) = f (z) − f (1) 2 (z − 1) − f (z) − f (−1) 2 (z + 1) = 0,(8) and by the continuity of f Equation (8) must hold for all x ∈ [−1, 1]. Therefore, f (x) = f (1) + f (−1) 2 + f (1) − f (−1) 2 x, and f is linear. This proves the second conjecture for odd sequences. The same argument using the part of Theorem 1 for even sequences leads to E( f , x) = f (1) − f (x) 2 (x − 1) + f (−1) − f (x) 2 (x + 1) = 0. For x = 0 this equation implies that f (x) = (x − 1) f (−1) + (x + 1) f (1) 2x ,(9) the continuity of f at x = 0 yields f (1) = f (−1), and Equation (9) shows that f is constant. This finishes the proof of the second conjecture for f ∈ AC 1 . Besides the second conjecture above, we can prove other interesting results using Theorem 1. For instance, if x is rational then 0 ∈ O(x) E(x) and the reader will be able to prove the following corollary: Theorem 2 (Uniform convergence for f ∈ AC 1 ) If f ∈ AC 1 and n j is an odd sequence then, for the function O( f ) defined in Equation (4), lim j→∞ n j B n j ( f ) − f − O( f ) /D n j ∞ = 0 and if n j is an even sequence then lim j→∞ n j B n j ( f ) − f − E( f ) /D n j ∞ = 0, for E( f ) defined in Equation (5), Lemma 6 in Section 3 yields n/D n ∞ ≤ 1, and it is clear that O( f ) ∞ ≤ f ′ ∞ and E( f ) ∞ ≤ f ′ ∞ . These observations combined with Theorem 2 lead to an uniform upper bound of order 1/n in the interpolation error for f ∈ AC 1 , but we can derive this bound under the weaker assumption of derivatives of bounded variation: Theorem 3 (Uniform convergence when f ∈ BV 1 ) If f ∈ BV 1 then n B n ( f ) − f ∞ ≤ T f ′ [−1, 1]/2 + max{ O( f ) ∞ , E( f ) ∞ },(10) where T f ′ [−1, 1] is the total variation of f ′ in [−1, 1]. We prove the results above in the next sections. In Section 2 we prove the first conjecture. In Section 3 we discuss the denominator of the interpolant defined in Equation (1). In Section 4 we analyze the numerator of the error B n ( f , x) − f (x) for functions in AC 1 . In Section 5 we analyze the numerator for f ∈ BV 1 . Finally, in Section 6 we combine the results in Sections 3. 4 and 5 to prove Theorems 1, 2 and 3. We would like to mention that André Pierro de Camargo suggested another proof of the second conjecture for functions with continuous third derivatives. For odd n, Theorem 5 in Section 4 indicates that N n ( f , x) − f (x) D n (x) ≈ f (x) − f (1) 2 (x − 1) − f (x) − f (−1) 2 (x + 1) , and by solving this expression for f (x) we derive the interpolant f (x) ≈B n ( f , x) := N n ( f , x) + f(−1) 2(x+1) − f(1) 2(x−1) D n (x) + 1 2(x+1) − 1 2(x−1) . Note thatB n is obtained by changing the absolute value of the first and last weights of the interpolant in Equation (1) from 1 to 1/2. A similar argument applies to even n and the resulting barycentric interpolantB n has better convergence properties than Berrut's interpolant. In fact,B n is the interpolant corresponding to d = 1 in the Floater-Hormann family [4], and using the theory presented in [4] we could prove the second conjecture for f ∈ C 3 by analyzing the asymptotic behavior of B n −B n . In summary, the present article shows that actually, from the perspective of order of approximation, Berrut's interpolants are biased by the functions O( f ) and E( f ), and we see little reason for using them instead of the interpolantB n above. In fact, in his latter work [2] prof. Berrut himself has mentioned that using half integer weights at the endpoints instead of ±1 leads to a better convergence rate. Theorem 2 shows that the interpolantB n has order of approximation o(1/n), and the most relevant questions in this subject are not the ones raised by professors Mastroianni and Szabados, and which we discuss in detail here. It is our opinion that it is more important to understand how we should choose the weights in the barycentric interpolants in order to improve them, so that we can justify the expensive 2n + 3 divisions per evaluation required by these interpolants. This will be the subject of our next article about barycentric interpolation. Proof of the first conjecture In this section we prove Conjecture 1 by presenting f ∈ Lip 1 such that, for t n := 1/n and n j : = 2 2 j ,(11) we have B n j f ,t n j − f t n j = B n j f ,t n j ≥ ln(n j ) 20 n j .(12) The function f is given by f (x) := ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ F 0 3 m 4 m ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ 5 m 6 m ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ H p = hat p = R p F p F p = fall p R p = raise p 7 m 8 m ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ √ m−5 m ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ √ m−4 m H √ m−8 4 √ m−3 m ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ R √ m−4 4 Figure 1: The function f m . The support of f m is [1/m, ( √ m − 3)/m]. The plot is divided in raise and fall regions, with R p starting at x = 4p/m and F p starting at x = (4p + 2)/m. By joining R p and F p we obtain the hat H p . Note that the series in Equation (13) converges to f ∈ Lip 1 because n j = 2 2 j and the identities n 2 j = 2 2 j+1 = n j+1 ⇒ √ n j+1 − 3 n j+1 < 1 √ n j+1 = 1 n j(19) imply that the support of the functions f n j are disjoint, and f m ∈ Lip 1. Equation (12) follows from Equation (13) and the following Lemmas: Lemma 1 (The error for the first terms) If 100 ≤ i < j then f n i t n j = 0 and B n j f n i ,t n j ≥ − 9 8n j .(20) Lemma 2 (The error for the main term) For j ≥ 100 we have that f n j t n j = 0 and B n j f n j ,t n j ≥ ln(n j ) 16n j . Lemma 3 (The error for the last terms) For i > j ≥ 100 and 0 ≤ k ≤ n j we have f n i t n j = 0 and f n i x k,n j = 0. The lemmas above show that f t n j = 0 for j ≥ 100, and the second part of Equation (12) follows from these lemmas because B n j f ,t n j = n j ∑ k=0 (−1) k ∑ ∞ i=100 f n i x k,n j t n j − x k,n j n j ∑ k=0 (−1) k 1 t n j − x k,n j = n j ∑ k=0 (−1) k ∑ j i=100 f n i x k,n j t n j − x k,n j n j ∑ k=0 (−1) k 1 t n j − x k,n j = j ∑ i=100 n j ∑ k=0 (−1) k f n i x k,n j t n j − x k,n j n ∑ i=0 (−1) k 1 t n j − x k,n j = j ∑ i=100 B n j f n i ,t n j ≥ B n j f n j ,t n j − 9 j − 100 8n j ≥ ln(n j ) 16 n j − 9 j 8n j(23) and, finally, the reader can verify that for j ≥ 100 j < ln 2 2 j /1000 = ln(n j ) /1000 and Equation (12) follows from Equation (23). We end this section presenting a proof of the lemmas above and one more lemma: Lemma 4 (Shifted harmonic sums) If a > 0 and ℓ ≥ 1 is an integer then ℓ−1 ∑ j=0 1 a + j ≥ ln(a + ℓ) − ln(a) + 1 2a − 1 2 (a + ℓ) .(24) Proof of Lemma 1. If i < j then n i < n j , t n j = 1/n j < 1/n i and Equation (14) implies that f n i t n j = 0. This proves the first part of Equation (20). Let N n j f n i ,t n j and D n j t n j be as in Equation (2). Lemma 6 in Section 3 shows that D n j t n j ≥ n j and this reduces the proof of Lemma 1 to the verification of the equation N n j f n i ,t n j ≥ −3/4,(25) as we do below. Note that the definition of n j in Equation (11) implies that if 100 ≤ i < j then, for m = n i , n j = 4qm with q ≥ 16, t n j = 1 n j = 1 4qm and x k,n j = 2k 4qm − 1. Equation (14) shows that f m x k,n j = f m 2k 4qm − 1 = 0 if k < 2qm+2q or k ≥ 2qm+2q √ m − 3 , and Equation (2), with the index k replaced by k + 2qm, leads to N n j f n i ,t n j = 2q( √ m−3)−1 ∑ k=2q (−1) k f m 2k 4qm 1 4qm − 2k 4qm = 4qm 2q( √ m−3)−1 ∑ k=2q (−1) k+1 f m 2k 4qm 2k − 1 . Motivated by Figure 1, we split the parcels of N n j f n i ,t n j in h := ( √ m − 8)/4 hats plus the last half of R 0 , which we call by R − , the part F 0 , and the first half of R ( √ m−4)/4 , which we call by R + . Formally we have N n j f n i ,t n j 4qm = R − + F 0 + h ∑ p=1 (F p + R p ) + R + = R − + F 0 + h ∑ p=1 H p + R + , for R − := 4q−1 ∑ k=2q (−1) k+1 f m 2k 4qm 2k − 1 ,(26)F p := 8(p+1)q−1 ∑ k=8pq+4q (−1) k+1 f m 2k 4qm 2k − 1 ,(27)R p := 8pq+4q−1 ∑ k=8pq (−1) k+1 f m 2k 4qm 2k − 1 ,(28)R + := q( √ m−3)−1 ∑ k=2q(2 √ m−4) (−1) k+1 f m 2k 4qm 2k − 1 ,(29)H p := R p + F p , and to prove Equation (25) it suffices to show that R − , R + , H p > 0 and F 0 ≥ −3 16qm ,(30) and this is done from this point to the end of this proof. Let us start by writing R p as a sum of positive terms. In raising ranges f m is defined by Equations (15), (17) and (18), and Equation (28) yields R p = 8pq+4q−1 ∑ k=8pq (−1) k+1 2k 4qm − 4p+1 m 2k − 1 = 1 4qm 8pq+4q−1 ∑ k=8pq (−1) k+1 (2k − 4q − 16pq) 2k − 1 . Splitting the indexes k in even and odd groups we obtain R p = −1 4qm 4pq+2q−1 ∑ ℓ=4pq 4ℓ − 4q − 16pq 4ℓ − 1 − 4ℓ + 2 − 4q − 16pq 4ℓ + 1 = −1 4qm 4pq+2q−1 ∑ ℓ=4pq 1 − 4q − 16pq 4ℓ − 1 − 1 − 4q − 16pq 4ℓ + 1 , and R p = 16pq + 4q − 1 2qm 4pq+2q−1 ∑ ℓ=4pq 1 16ℓ 2 − 1 > 0.(31) The same argument using Equations (26) and (29) shows that R − , R + > 0. Similarly, for F p Equations (16) and (27) lead to F p = 8(p+1)q−1 ∑ k=8pq+4q (−1) k+1 4p+3 m − 2k 4qm 2k − 1 = 1 4qm 8( j+1)q−1 ∑ k=8pq+4q (−1) k+1 (16pq + 12q − 2k) 2k − 1 . As before, F p = 1 4qm 4(p+1)q−1 ∑ ℓ=4pq+2q 4ℓ − 16pq − 12q 4ℓ − 1 − 4ℓ + 2 − 16pq − 12q 4ℓ + 1 = 1 4qm 4(p+1)q−1 ∑ ℓ=4pq+2q 1 − 16pq − 12q 4ℓ − 1 − 1 − 16pq − 12q 4ℓ + 1 , and F p = − 16pq + 12q − 1 2qm 4(p+1)q−1 ∑ ℓ=4pq+2q 1 16ℓ 2 − 1 .(32) In particular, for p = 0 we have F 0 = − 12q − 1 2qm 4q−1 ∑ ℓ=2q 1 16ℓ 2 − 1 ≥ − 12q − 1 2qm 2q 64q 2 − 1 = − 1 2qm 24q 2 − 2q 64q 2 − 1 ≥ −3 16qm , and this proves Equation (30). We now show that, for p ≥ 1, H p = R p + F p > 0. Replacing ℓ by k + 2q in Equation (32) and ℓ by k in Equation (31) we obtain H p = 1 2qm 4pq+2q−1 ∑ k=4pq a k , for a k = 16pq + 4q − 1 16k 2 − 1 − 16pq + 12q − 1 16 (k + 2q) 2 − 1 , and our final goal is to show that a k > 0. We can write a k as u k /v k for u k := (16pq + 4q − 1) 16 (k + 2q) 2 − 1 − (16pq + 12q − 1) 16k 2 − 1 and v k := 16k 2 − 1 16 (k + 2q) 2 − 1 . The denominator v k is clearly positive, and in order to analyze u k we replaced k by 4pq + ξ q, with ξ ∈ [0, 2), and used Wolfram Alpha to obtain u k = 8q 256p 2 q 2 + 256pq 2 − 32pq − 16q 2 ξ 2 + 32q 2 + 32q 2 ξ − 8qξ − 8q + 1 . Since we are concerned with q ≥ 16, p ≥ 1 and ξ ∈ [0, 2), it is clear that u k > 0 and the proof of Lemma 1 is complete. ✷ Proof of Lemma 2. Let us write m = n j . According to Equation (11), t m = 1/m, and Equation (15) yields f m (t m ) = 0. We have that B m ( f m ,t m ) = N m ( f m ,t m ) /D m (t m ) for N m ( f m ,t m ) and D m (t m ) in Equation (2). Since 1 2m = x m 2 ,m + 1 2m < t m = 1 m < x m 2 +1,m − 1 2m = 3 2m and m is a multiple of four and we have that t m = 2 × (2m) + 0 + 1 4m − 1. Equation ( N m ( f m ,t m ) := m/2+ √ m−4 2 ∑ k=m/2+1 (−1) k f m (2k/m − 1) 1/m − 2k/m + 1 . Making the change of indexes k = m/2 + i and noting that m/2 is even we obtain N m ( f m ,t m ) = √ m−4 2 ∑ i=1 (−1) i (−1) i+1 /m 1/m − 2i/m = √ m−4 2 ∑ i=1 1 2i − 1 = 1 2 √ m−6 2 ∑ i=0 1 i + 1/2 , and Lemma 4 with a = 1/2 and ℓ = ( √ m − 4)/2 yields 2N m ( f m ,t m ) ≥ ln √ m − 3 2 −ln(1/2)+1− 1 2 1 2 + √ m−4 2 = ln √ m − 3 +1− 1 √ m − 3 . Therefore, N m ( f m ,t m ) = ln √ m /2 + δ m /2 = ln(m) /4 + δ m /2(34)for δ m = 1 + ln 1 − 3 √ m − 1 √ m − 3 . Since 4m ≥ 2 2 100 we have that δ m > 0. Therefore, Equation (34) implies Equation (33) and this proof is complete. ✷ Proof of Lemma 3. Equation (19) implies that if i > j then √ n i − 3 n i < 1 √ n i ≤ 1 n j = t n j , and Equation (14) implies that f n i t n j = 0. In order to show that f n i x k,n j = 0 we recall that x k,n j = 2k/n j − 1 and analyze two possibilities: (i) If k ≤ n j /2 then x k,n j ≤ 0, and Equation (14) implies that f n i x k,n j = 0. (ii) If k > n j /2 then x k,n j ≥ 2 n j ≥ 2 √ n i > √ n i − 3 n i , and Equation (14) shows that f n i x k,n j = 0. Therefore, f n i x k,n j = 0 in both cases we have proved Lemma 3. b + 1 b + 1 − 1 b t − 1 b + t = 1 b − t b (b + 1) − 1 b + t . Since h b (0) = b b (1) = 0 and h b is concave we have that h b ≥ 0. Therefore, 0 ≤ 1 0 h b (t) dt = 1 b − 1 2b (b + 1) − ln(b + 1) + ln(b) and 1 b ≥ 1 2b (b + 1) + ln(b + 1) − ln(b) = 1 2b − 1 2 (b + 1) + ln(b + 1) + ln(b) . It follows that ℓ−1 ∑ j=0 1 a + j ≥ ℓ−1 ∑ j=0 1 2 (a + j) − 1 2(a + j + 1) + ln(a + j + 1) − ln(a + j) = ln(a + ℓ) − ln(a) + 1 2a − 1 2 (a + ℓ) and we are done. ✷ The denominator In this section we analyse the denominator D n (x) of the interpolant in Equation (1), using the function A : [0, 1) → R given by A(x) := ∞ ∑ k=0 (−1) k 4k + 2 (2k + 1) 2 − x .(35) This function is increasing and can be extended to a homeomorphism between [0, 1] and [π/2, +∞] ⊂ R, with the topology in the introduction, as shown by the next lemma. In the rest of the article we work with this extension of A and its inverse A −1 . (35) is increasing, A(0) = π/2 and Lemma 5 (The function A) The function A defined in Equation −1/2 ≤ A(x) − 2 1 − x ≤ π − 4 2 < −0.42.(36) In particular, A can be extended to a homeomorphism between [0, 1] and [π/2, +∞]. The section is based upon the observation that for a regular x, as j tends to infinity the denominator D n j (x) can be accurately described by the expression D n j (x) ≈ (−1) ι n j (x) n j A ρ 2 n j (x) , where A is the function defined in Equation (35) Formally, we have the following lemma: Lemma 6 (The size and sign of the denominator) If x ∈ (−1, 1)\{x 0,n , . . . , x n,n } then ρ n (x) ∈ (−1, 1), sign(D n (x)) = (−1) ι n (x) , \|D n (x) /n\| − A ρ 2 n (x) ≤ 1 4 (1 + ι n (x)) + 1 4 (n − ι n (x)) ≤ 1 2 ,(39) and \|D n (x) /n\| ≥ 1 and \|D n (x) /n\| ≥ A ρ 2 n (x) /2 ≥ 3 4 (1 − ρ 2 n (x)) . (40) In particular, if x is regular for the sequence n j then lim j→∞ D n j (x) n j A ρ 2 n j (x) = 1.(41) The last two lemmas imply that the possible values for lim j→∞ D n j (x) /n j can be found by analysing the limits lim j→∞ ι n j (x) and lim j→∞ ρ 2 n j (x). This corollary leads to a clean description of the limits lim j→∞ D n j (x) /n j when x is irrational, due to the following theorem by S. Hartmann: Theorem 4 (Hartmann's Theorem [5]) For every irrational number ξ , and integers s, a, b, with s ≥ 1, there are infinitely many integers u and v > 0 such that ξ − u v ≤ 2s 2 v 2 with u ≡ a mod s and v ≡ b mod s.(44) Using Hartmann's theorem we can prove the following Lemma: Lemma 7 (Convergence of the denominator for irrational x) If x ∈ (−1, 1) is irrational then for each r n ∈ {0, 1} and y with \|y\| ∈ [π/2, +∞] there exists a sequence n j such that n j ≡ r n mod 2 and lim j→∞ 1 n j D n j (x) = y. In words, Lemma 7 shows that if x is irrational then we can obtain all elements in the extended intervals [−∞, −π/2] and [π/2, +∞] as limits for D n j (x) /n j , for sequences n j with the same parity, be this parity odd or even. Unfortunately things are more complex when x is rational and we must consider a few cases, as we do in the next lemmas. The first one shows that the set of possible limits for D n (x) /n is finite in this case. Lemma 8 (Finitely many limits D n (x) /n for x rational) For p, q ∈ N, with q = 0. If x = p/q − 1 ∈ (−1, 1) is regular for the sequence n j and lim j→∞ 1 n j D n j (x) = L ∈ R(46) then L is finite and \|L\| = A m 2 /q 2 for some m ∈ Z with 0 ≤ m < q. Moreover, there exists j 0 such that if j ≥ j 0 then (−1) ι n j (x) = sign(L) and ρ n j (x) = m/q. The hypothesis of Lemma 8 accounts for L = ±∞, but its thesis states that this case is actually not possible. In particular, this Lemma implies that if x is regular for n j then the D n j (x) /n j are bounded. Lemma 8 also shows that if the sequence D n j (x) /n j converges in R and x is rational and regular then ρ n j (x) and the parity of ι n j (x) become eventually constant, and L belongs to one of the two finite sets O(p/q) := L ∈ R such that Equation (46) holds for some odd sequence n j for which x = p/q − 1 is regular} (47) and E(p/q) := L ∈ R such that Equation (46) holds for some even sequence n j for which x = p/q − 1 is regular} . (48) The description of the sets of limits O(p/q) and E(p/q) is a tedious exercise in elementary number theory, but we present it below for completeness. The possible cases are listed in the next three corollaries. After the statement of these corollaries we end this section with the proofs of the result stated in it. and E(p/q) = ±A (2ℓ + 1) 2 /q 2 with ℓ ∈ {0, 1, . . . , (q − 3)/2} .(50) Corollary 4 (O(2p/q) and E(2p/q)) If gcd(p, q) = 1 and q is odd then O(2p/q) = (−1) s A (4ℓ + 2p − 2s − q) 2 /q 2 for s ∈ {0, 1} and ℓ ∈ Z with s − p + 1 ≤ 2ℓ ≤ s − p + q − 1}(51) and E(2p/q) = (−1) s A (4ℓ − 2s − q) 2 /q 2 for s ∈ {0, 1} and ℓ ∈ Z with s + 1 ≤ 2ℓ ≤ s + q − 1}.(52) Finally, Corollary 5 (O(p/2q) and E(p/2q)) If gcd(p, q) = 1 and p is odd then O(p/2q) = ±A (2ℓ + 1) 2 4q 2 with ℓ ∈ {0, 1, . . . , q − 1}(53) and E(p/2q) = ±A ℓ 2 /q 2 with ℓ ∈ {0, 1, . . . , q − 1} .(54) Proof of Lemma 5. The derivative of A A ′ (x) = ∞ ∑ k=0 (−1) k 4k + 2 (2k + 1) 2 − x 2 has parcels of alternating signs and decreasing absolute values, with a positive first term. Therefore A ′ (x) > 0 for all x ∈ [0, 1), and A is a increasing function of x. Moreover, executing the command Sum[ 2 (-1)^k / (2 k + 1), k = 0 to Infinity ] in the software Wolfram Alpha we obtain that A(0) = π/2. The same argument used above shows that the function h : [0, 1] → R given by h(x) = A(x) − 2 1 − x = − ∞ ∑ k=1 (−1) k 4k + 2 (2k + 1) 2 − x This proves Equation (36). ✷ Proof of Lemma 6. We have that x ι n (x),n < x < x ι n (x)+1,n ⇒ 0 < θ n (x) := n (x + 1)/2 − ι n (x) < 1. Equation (37) defines ρ n (x) := n x − x ι n (x),n − 1 and ρ n (x) = n (x − 2 (n (x + 1)/2 − θ n (x)) /n + 1) − 1 = 2θ n (x) − 1 ∈ (−1, 1). Therefore ρ 2 n (x) < 1, and the definition of D n in Equation (2) leads to D n (x) = ι n (x) ∑ k=0 (−1) k 1 x − x k,n + n ∑ k=ι n (x)+1 (−1) k 1 x − x k,n = ι n (x) ∑ k=0 (−1) ι n (x)−k 1 x − x ι n (x)−k,n + n−ι n (x)−1 ∑ k=0 (−1) ι n (x)+k 1 x ι n (x)+k+1,n − x = (−1) ι n (x) ι n (x) ∑ k=0 (−1) k 1 x − x ι n (x),n + 2k/n + n−ι n (x)−1 ∑ k=0 (−1) k 1 x ι n (x)+1,n − x + 2k/n = (−1) ι n (x) n ι n (x) ∑ k=0 (−1) k 1 n x − x ι n (x),n + 2k + n−ι n (x)−1 ∑ k=0 (−1) k 1 n x ι n (x),n − x + 2k + 2 = (−1) ι n (x) n ι n (x) ∑ k=0 (−1) k 1 2k + 1 + ρ n (x) + n−ι n (x)−1 ∑ k=0 (−1) k 1 2k + 1 − ρ n (x) . Therefore, D n (x) = (−1) ι n (x) n (U n (x) + V n (x)) (55) for U n (x) := ι n (x) ∑ k=0 (−1) k 1 2k + 1 + ρ n (x)(56) and V n (x) := n−ι n (x)−1 ∑ k=0 (−1) k 1 2k + 1 − ρ n (x) .(57) Since ρ n (x) ∈ (−1, 1) the absolute values of the parcels of the sum U n (x) and V n (x) decrease with k, their sign alternate, and the first parcel is positive. Therefore, U n (x) and V n (x) are positive and Equation (55) shows that D n (x) has the sign claimed by Lemma 6. Moreover, the definition (35) of the function A shows that A(x) = ∞ ∑ k=0 (−1) k 1 2k + 1 − √ x + 1 2k + 1 + √ x and Equation (55) yields \|D n (x)\| /n−A ρ 2 n (x) = ∞ ∑ k=ι n (x)+1 (−1) k 1 2k + 1 + ρ n (x) + ∞ ∑ k=n−ι n (x) (−1) k 1 2k + 1 − ρ n (x) . It follows that \|D n (x)\| /n − A ρ 2 n (x) ≤ G n (x) + H n (x) , for G n (x) := ∞ ∑ k=0 (−1) k 1 2k + 2ι n (x) + 3 + ρ n (x) and H n (x) := ∞ ∑ k=0 (−1) k 1 2k + 2 (n − ι n (x)) + 1 − ρ n (x) . Replacing k by 2ℓ and 2ℓ + 1 in the expression of G n above we obtain ι n (x)) . G n (x) = ∞ ∑ ℓ=0 1 4ℓ + 2ι n (x) + 3 + ρ n (x) − 1 4ℓ + 2ι n (x) + 5 + ρ n (x) = ∞ ∑ k=0 2 (4ℓ + 2ι n (x) + 3 + ρ n (x)) (4ℓ + 2ι n (x) + 5 + ρ n (x)) ≤ ∞ t=0 2 (4t + 2ι n (x) + 3 + ρ n (x)) (4t + 2ι n (x) + 5 + ρ n (x)) dt = 1 4 ln 1 + 2 2ι n (x) + 3 + ρ n (x) ≤ 1 2 (2ι n (x) + 3 + ρ n (x)) ≤ 1 4 (1 + The integral above was computed with Wolfram Alpha, and a similar computation shows that H n (x) ≤ 1 4 (n − ι n (x)) , and the second part of Equation (39) holds. It follows that \|D n (x) /n\| ≥ A ρ 2 n (x) − 1/2 ≥ π 2 − 1/2 > 1.07 > 1, because A ρ 2 n j (x) ≥ π/2. This proves the first part of bound (40). We also have \|D n (x) /n\| ≥ A ρ 2 n (x) − 1/πA ρ 2 n (x) > A ρ 2 n (x) /2,and 2/(1 − ρ 2 n (x)) ≥ 2 because ρ 2 n (x) ∈ [0, 1). Equation (36) shows that A ρ 2 n (x) ≥ 2 1 − ρ 2 n (x) − 1 2 ≥ 2 1 − ρ 2 n (x) − 1 4 × 2 1 − ρ 2 n (x) = 3 4 (1 − ρ 2 n (x)) , and this proves the second Equation in (40). Finally, for every x ∈ (−1, 1) regular we have that lim n j →∞ ι n j (x) = lim n j →∞ n − ι n j (x) = +∞. This observation and the equations above imply Equation (41). ✷ Proof of Corollary 2. Let us assume Equation (42) and prove Equation (43). Lemma 6 shows that \|D n (x) /n\| ≥ 1 for all x and n. Therefore, L = 0 and for j large enough we must have sign D n j (x) = sign(L) , and Equation (39) shows that this is also the sign of (−1) ι n j (x) . Therefore, lim j→∞ (−1) ι n j (x) = sign(L) . Moreover, Equation (41) implies that lim j→∞ A ρ 2 n j (x) = L.(58) Since A is continuous and [0, 1] is compact, this implies that L ∈ A([0, 1]) = [π/2, +∞] and L ≥ π/2. Finally, since A −1 is continuous Equation (58) implies that lim j→∞ ρ 2 n (x) = A −1 (L) and the proof of Equation (43) is complete. Let us now assume Equation (43) so that if z j is a sequence such that lim j→∞ z j = z then y = lim j→∞ (−1) r i A z 2 j . Lemma 6 shows that to prove Lemma 7 it suffices to define a sequence n j such that n j ≡ r n mod 2, ι n j (x) ≡ r i mod 2 and lim j→∞ ρ n j (x) = z. Since the image of A −1 is [0, 1] we have that z ∈ [0, 1], and there exist sequences p j , q j ∈ N with lim j→∞ p j /q j = z and 0 < p j /q j < 1. We start with an empty set of integers n j , and build them by induction. At the jth step we use Hartmann's Theorem with ξ = x + 1, s = 4q j , a = 2q j r i + p j + q j and b = q j r n , and conclude that there exist infinitely many numbers u and v such that x + 1 − 4q j u + 2q j r i + p j + q j 4q j v + q j r n < 32q 2 j (4q j v + q j r n ) 2 ≤ 2 v 2 . This implies that x + 1 = 4u + 2r i + 1 + p j /q j 4v + r n + θ j 1 v 2(60) for some θ j ∈ [−2, 2]. Taking a pair (u j , v j ) with v j so large that 0 < 1 + p j /q j − 2 (4v j + r n ) /v 2 j < 1 + p j /q j + 2 (4v j + r n ) /v 2 j < 2,(61) and for which n j := 4v j + r n is larger than the previous n j , we obtain a n j which satisfies the parity requirement in Lemma 7 and n j (x + 1)/2 = 2u j + r i + 1 + p j /q j + θ j n j /v 2 j /2. The definition (37) of ι n and Equation (61) implies that ι n j (x) = ⌊n j (x + 1)/2⌋ = 2u j + r i , and this ι n j (x) has the parity claimed by Equation (59), and Equation (60) yields x − x ι n j (x),n j = x + 1 − 2 2u j + r i 4v j + r n = 1 + p j /q j 4v j + r n + θ j 1 v 2 j , and the definition (37) of ρ yields ρ n j (x) = n j x − x ι n j (x),n j − 1 = p j /q j + θ j (4v j + r n ) /v 2 j . Since θ j ≤ 2 and r n ∈ {0, 1}, we have that lim j→∞ ρ n j (x) = lim j→∞ p j /q j = z, and the proof of Lemma 7 is complete. ✷ Proof of Lemma 8. Corollary 2 shows that lim j→∞ ρ n j (x) = M := A −1 (\|L\|) ,(62) and there exists j 0 such that if j ≥ j 0 then (−1) ι n j (x) = sign(L) and ρ n j (x) = M + ε j with ε j ≤ 1 − (qM − ⌊qM⌋) 2q .(63) Equation (38) and the hypothesis x = p/q − 1 imply that p/q = 2ι n j (x) + σ j (M + ε j ) + 1 /n j ,(64) with σ j ∈ {−1, 1}, and pn j − 2qι n j (x) − σ j ⌊qM⌋ − q = σ j (qε j + (qM − ⌊qM⌋)) .(65) Since σ j = 1, Equation (63) yields σ j (qε j + (qM − ⌊qM⌋)) ≤ (1 − (qM − ⌊qM⌋)) /2 + qM − ⌊qM⌋ = (1 + (qM − ⌊qM⌋)) /2 < 1. Since the left hand side of Equation (65) is integer, we have that qε j + (qM − ⌊qM⌋) = 0 ⇒ ε j = ⌊qM⌋/q − M, Equation (63) yields ρ n j (x) = ⌊qM⌋/q, and Equation (62) implies that ⌊qM⌋ = qM. It follows that qM ∈ Z and M = m/q for some m ∈ Z. Therefore, ρ n j (x) = m/q, and the proof of Lemma 8 is complete. ✷ Proof of Corollary 3. For a regular x = p/q − 1, with lim j→∞ D n j (x) /n j = L, Lemma 8 implies that there exist i j ∈ N, m ∈ Z with \|m\| < q, and s ∈ {0, 1} such that p/q = (2 (2i j + s) + m/q + 1)/n j , (−1) s = sign(L) and \|L\| = A m 2 /q 2 , and the first Equation above is equivalent to p n j = 2 (2i j + s) q + m + q.(66) When n j is odd, this equation implies that ℓ := m/2 ∈ Z, and \|ℓ\| ≤ (q − 1)/2. Therefore, \|L\| = A 4ℓ 2 /q 2 and the set in Equation (49) p m j = (2i j + s) q + ℓ + (q − p)/2 = (2q) i j + (sq + ℓ + (q − p)/2) . For every s and ℓ this equation has infinitely many solutions (m j , i j ) ∈ N × N because gcd(p, 2q) = 1. Therefore, for every m = 2ℓ, and s ∈ {0, 1} there exist infinitely many n j = 2m j + 1 which satisfy Equation (66), and all elements in the set O(p/q) in Equation (49) are indeed limits of sequences D n j (x) /n j with odd n j . This completes the verification of Equation (49). When n j is even, Equation (66) implies that ℓ := (m − 1)/2 ∈ Z, \|2ℓ + 1\| < q and \|L\| = A (2ℓ + 1) 2 /q 2 , and the set in Equation (50) does contain all the relevant limits L. Moreover, for m j = n j /2 ∈ Z and m = 2ℓ + 1, Equation (66) reduces to p m j = (2i j + s) q + ℓ + (q + 1)/2 = (2q) i j + (sq + ℓ + (q + 1)/2) and, as before, we can find infinitely many (m j , i j ) which satisfy this equation, and use then to generate sequences n j with all the limits in the set in Equation (50). As a result, Equation (50) is valid, and this proof is complete. ✷ Proof of Corollary 4. If x = 2p/q − 1 is regular and lim j→∞ D n j (x) /n j = L then Lemma 8 implies that there exist i j ∈ N, m ∈ Z with \|m\| < q and s ∈ {0, 1} such that 2p/q = (2 (2i j + s) + m/q + 1)/n j , (−1) s = sign(L) and \|L\| = A m 2 /q 2 . The first Equation above is equivalent to 2pn j = 2 (2i j + s) q + m + q, and it implies that h := (m − 1)/2 ∈ Z. Therefore, p n j = (2i j + s) q + h + (q + 1)/2.(67) If n j is odd then ℓ := (s + h − p + (q + 1)/2) ∈ Z, and m = 4ℓ + 2p − 2s − q. Since \|m\| < q we have that −q + 1 ≤ 4ℓ + 2p − 2s − q ≤ q − 1 and 1 − 2p + 2s ≤ 4ℓ ≤ −2p + 2s + 2q − 1 ⇒ s − p + 1 ≤ 2ℓ ≤ s − p + q − 1, and the set in Equation (51) contains all the relevant limits. Conversely, for m j := (n j + 1)/2 and h = 2ℓ + p − s − (q + 1)/2, Equation (67) reduces to pm j = i j q + ℓ + s (q − 1)/2, and since gcd(p, q) = 1 there exist infinitely many m j and i j which satisfy this equation, and all elements of the set O(2p/q) in Equation (51) are indeed limits corresponding to conveniently chosen odd sequences. If n j is even then Equation (67) yields ℓ := (s + h + (q + 1)/2) ∈ Z. Since m = 2h + 1, we obtain h = 2ℓ − s − (q + 1)/2 ⇒ m = 4ℓ − 2s − q, and the bound \|m\| < q leads to 1 + s ≤ 2ℓ < s + q − 1, and Equation (52) is correct. Finally, with m j := n j /2 ∈ Z and h above, Equation (67) reduces to pm j = i j q + ℓ + s (q − 1)/2, and since gcd(p, q) = 1 there exist infinitely many (m j , i j ) which satisfy this equation. ✷ Proof of Corollary 5. If x = p/2q − 1 is regular and lim j→∞ D n j (x) /n j = L then Lemma 8 implies that there exist i j ∈ N, m ∈ Z with \|m\| < 2q, and s ∈ {0, 1}, such that p 2q = (2 (2i j + s) + m/2q + 1)/n j , (−1) s = sign(L) and \|L\| = A m 2 4q 2 . The first equation above is equivalent to p n j = 4 (2i j + s) q + m + 2q.(68) When n j is odd, ℓ := (m − 1)/2 ∈ Z and the bound \|m\| < 2q implies that \|ℓ\| ≤ q − 1 and Equation (53) is correct. Conversely, for m = 2ℓ + 1 and m j = (n j − 1)/2 Equation (68) reduces to p m j + = (4q)i j + sq + ℓ + 1 − p 2 + q, and since gcd(p, 4q) = 1, for each s and ℓ this equation has infinitely many solutions (n j , i j ), which we can use to build sequences with the limits in the set in Equation (53). When n j is even, Equation (68) implies that ℓ := m/2 ∈ Z and the bound \|m\| < 2q implies that \|ℓ\| ≤ q − 1 and Equation (54) is correct. Conversely, for m = 2ℓ above and m j := n j /2 ∈ Z, Equation (68) reduces to p m j = 4i j + 2sq + ℓ + q, and since gcd(p, 4q) = 1, for each s and ℓ this equation has infinitely many solutions (n j , i j ), from which we can obtain sequences with the limits in Equation (54). ✷ 4 The numerator of the error for f in AC 1 In this section we explore the consequences of the observation in the introduction that Berrut's interpolants are biased. After we remove the bias, the relevant quantity for understanding the convergence of the interpolants B n is defined as ∆ n ( f , x) := f (−1) − f (x) 2 (x + 1) + (−1) n f (1) − f (x) 2 (x − 1) + 1 n n−1 ∑ k=1 (−1) k f x k,n − f (x) x − x k,n(69) for x ∈ {x 0,n , . . . , x nn }, and ∆ n f , x k,n := 0. We can then express the combination of ∆ n ( f , x) and the bias O( f , x) for n j = 2 j + 1 odd as B 2n+1 ( f , x) − f (x) = (∆ 2n+1 ( f , x) + O( f , z)) /D 2n+1 (x) .(70) For n j = 2n the bias is E( f , x) and we have B 2n ( f , x) − f (x) = (∆ 2n ( f , x) + E( f , x)) /D 2n (x) .(71) The expression for ∆ n ( f , x) for both parities is the same, that is, the bias is related to parity, but the mean term ∆ n ( f , x) is not. We can then obtain a clean result regarding the convergence of the numerator of the error, which we prove in the end of this section. 3. Finally, if i ≥ n − √ δ n/4 then we define m as in Equation (77), ℓ = 0, M and F as in Equations (76) and (78), and L := 0. We now bound M. Splitting each parcel in two parts, and grouping consecutive halves and using the Mean Value Theorem we obtain 2 \|M\| = n−2ℓ−1 ∑ k=2m f x k,n − f (x) x − x k,n − f x k+1,n − f (x) x − x k+1,n ≤ n−2ℓ−1 ∑ k=2m f ′ (ξ k ) − f ′ (ξ k+1 ) with \|ξ k − x\| ≤ 2 max {i − 2m, n − 2ℓ − i}/n. The indexes ℓ and m were defined in Equations (74) and (77) so that 0 < i − 2m ≤ √ δ n/4, 0 < n − 2ℓ − i ≤ √ δ n/4 and \|ξ k − x\| ≤ √ δ n 2n This implies that \|ξ k − ξ k+1 \| ≤ δ /n, n−2ℓ ∑ k=2m \|ξ k − ξ k+1 \| ≤ (n − 2ℓ − 2m) δ /n ≤ √ δ n/2 × δ /n ≤ δ , and Equation (72) implies that M ≤ ε/3. We now show that L ≤ ε/3 in the case in which it is different from zero (By symmetry, the same bound applies to F.) Defining y k = x n−2ℓ+k,n , we can group the terms of L as −2 (−1) n L = ℓ−1 ∑ j=0 f y 2 j+1 − f (x) y 2 j+1 − x − f y 2 j+2 − f (x) y 2 j+2 − x − f y 2 j − f (x) y 2 j − x − f y 2 j+1 − f (x) y 2 j+1 − x = 2 n ℓ−1 ∑ j=0 [y 2 j , x, y 2 j+1 , f ] − [y 2 j+2 , x, y 2 j+1 , f ], where [x 1 , x 2 , x 3 , f ] denotes the divided difference of second order corresponding to x 1 , x 2 , x 3 and f , because y 2 j − y 2 j+1 = y 2 j+1 − y 2 j+2 = −2/n. Since f ′ is absolutely continuous, the Genocchi-Hermite formula [3] yields f ′′ x + t z 2 j+2 + s z 2 j+1 ds. Therefore, \|L\| ≤ 2 \|O( f , x)\| /π and Equation (7) is correct. Conversely, if z ∈ O( f , x) then either z = 0 or \|z\| ∈ (0, 2O( f , x) /π]. In the first case, Lemma 7 yields an odd sequence n j such that lim j→∞ 1 n j D n j (x) = +∞ and we have that lim j→∞ n j B n j ( f , x) − f (x) = ∆ n ( f , z) n j D n j (x) = O( f , x) × 0 = z. Otherwise, when z = 0, y = z/O( f , x) ∈ [−2/π, 2/π] \ {0}(85) and Lemma 7 yields an odd sequence n j such that lim j→∞ 1 n j D n j (x) = 1/y, and lim j→∞ n j B n j ( f , x) − f (x) = ∆ n ( f , z) n j D n j (x) = O( f , x) × z/O( f , x) = z. Therefore, we have proved the converse part of Theorem 1 for an irrational x and an odd sequence n j . The same argument applies for an irrational x and an even sequence n j , replacing O( f , x) by E( f , x) and O( f , x) by E( f , x). Let us then analyze a rational x. Since x is regular, we must have x ∈ (−1, 1), and there exist positive integers p and q with gcd(p, q) = 1 such that x = p/q − 1, and we can use the argument applied in the irrational case replacing the interval [−2/π, 2/π] in Equation (85) by the set O(p/q) or E(p/q) in Corollaries 3, 4 and 5 in Section 3 corresponding to the parity of p and q, and replacing the intervals n B n ( f ) − f ∞ ≤ ∆ n ( f ) ∞ + max{ O( f ) ∞ , E( f ) ∞ } D n ∞ /n , and Equation (40) and Corollary 6 imply Equation (10). ✷ Conjecture 1 ( 1First conjecture by Mastroianni and Szabados) There exists a function f ∈ Lip 1 such thatlim sup n→∞ n log n B n ( f ) − f ∞ > 0.In their second conjecture, Mastroianni and Szabados do not specify a particular class of functions, but we have found that if f ∈ BV 1 then we can bound the sequencen B( f ) − f ∞by a constant depending on f . Moreover, if f ∈ AC 1 and x ∈ [−1, 1] then we can describe exactly all possible accumulation points of the sequence Corollary 1 ( 1Large errors for rational x) If f ∈ AC 1 and x ∈ [−1, 1] is rational and regular for the sequence n j , O( f , x) = 0 and E( f , x) = 0 then lim inf j→∞ n j B n j ( f , x) − f (x) > 0. However, Theorem 1 has a serious limitation: it is only a pointwise result, and it does not imply the more interesting bound lim sup n→∞ n B n ( f ) − f ∞ < +∞ considered by Mastroianni and Szabados in their open problem. Fortunately, we can also prove uniform convergence results for f ∈ AC 1 : 39) in Section 3 with ρ n (x) = 0 shows that 0 < D m (t m ) ≤ A(0) + 1/2 = πm/2 + 1/2 < 4m, and in order to prove Lemma 2 it suffices to show that N m ( f m ,t m ) ≥ ln(m) /4. (33) This is our goal now. Equations (14)-(18) imply that f m (2k/m − 1) = (−1) k+1 /m for k = m/2 + 1, . . ., m/2 + √ m − 4 /2 and f m (2k/m − 1) = 0 for the remaining ks (see Figure 1.) Therefore, ✷ Proof of Lemma 4 . 4For b > 0, let h b : [0, 1] → R be the function h b (t) := 1 , ι n (x) := ⌊n (x + 1)/2⌋ and ρ n (x) := n x − x ι n (x),n − 1,(37)so that ι n (x) ∈ {0, . . . , n − 1},x = 2ι n (x) + ρ n (x) + 1 n − 1 and ρ n (x) ∈ (−1, 1). Corollary 2 ( 2Convergence of D n (x) /n) If x is regular for the sequence n (x) = A −1 (\|L\|) . (43) Corollary 3 ( 3O(p/q) and E(p/q) for odd p and q) If gcd(p, q) = 1 and p and q are odd then O(p/q) = ±A 4ℓ 2 /q 2 with ℓ ∈ {0, 1, . . . , (q − 1)/2} (49) (39) implies Equation (42) and we are done. ✷ Proof of Lemma 7. Let r i ∈ {0, 1} be such that (−1) r i = sign(y) and z := A −1 (\|y\|), does contain all relevant limits L. Conversely, with m = 2ℓ and m j = (n j − 1)/2, Equation (66) is equivalent to ff ′′ ((1 − t − s) u + s v + t w) ds dt, ′′ x + t z 2 j + s z 2 j+1 ds − [ −2 \|O( f , x)\| /π, 2 \|O( f , x)\| /π ] and [ −2 \|E( f , x)\| /π, 2 \|E( f , x)\| /π ] by the sets O( f , x) = {O( f , x) /y, y ∈ O(p/q)}(86)andE( f , x) = {E( f , x) /y, y ∈ E(p/q)}.(87) ✷ Proof of Theorem 2. Theorem 2 follows from Lemma 6 and Theorem 5. ✷ Proof of Theorem 3. Equations (70) and (71) show that holds . holdsTheorem 1 has far reaching implications for f ∈ AC 1 . For instance, it yields a simple proof of the second conjecture by Mastroianni and Szabados.Conjecture 2 (Second conjecture by Mastroianni and Szabados) We have Theorem 5 (The uniform convergence of the numerator) If f ∈ AC 1 then lim n→∞ ∆ n ( f ) ∞ = 0.Proof of Theorem 5. Given ε ∈ (0, 1), by the absolute continuity of f ′ (x) there exists δ ∈ (0, 1) for whichand we now definetake n ≥ n 0 and x ∈ [−1, 1] and show that \|∆ n ( f , x)\| ≤ ε. If x ∈ {x 0n , . . . , x n,n } then ∆ n ( f , x) = 0 by definition and we are done. For x ∈ {x 0n , . . . , x n,n }, let i be the index such that x i,n is the node closest to x. We split the sum which defines ∆ n ( f , x) in Equation(69)in at most three parts: F (first), M (middle) and L (last), according to the distance of x to ±1. When x is too close to −1 we leave the First region empty, and if x is too close to 1 then the Last range is left empty. When not empty, the First range corresponds to parcels with indexes from 0 to 2m. The Middle range spans the indexes from 2m to n − 2ℓ, and contains of the order of √ δ n parcels (the parcel corresponding to k = 2m is split between the First and Middle ranges.) When not empty, the Last range starts at index n − 2ℓ and ends a index n, and the parcel of index n − 2ℓ is split between the Middle and Last ranges.Formally, we define 1. If i < √ δ n/4 then, since n ≥ n 0 , Equation (73) implies that n > i + √ δ n/4 and we define m := 0,F := 0,anddefine M and L as in Equations (76) and (77), andThe changes of variableshave the same Jacobian z 2 j+1 with respect to s andSince z k+1 − z k = 2/n,The boundIt follows that In this section we analyze the function ∆ n ( f , x) defined in Equation(69)for functions f with derivatives of bounded variation. In summary, we show that in this case ∆ n is bounded by half the total variation of f ′ . Our proof follows from this version of the Mean Value Theorem:Theorem 6 (A monotone Mean Value Theorem) Let a, b ∈ R be such that a < b, and let f : [a, b] → R be a continuous function, which is differentiable in(a, b). If c and ξ c are such that a < ξ c < c < b andWe prove Theorem 6 at the end of this section. By induction, we conclude from this theorem that given an increasing sequence b 0 , . . . , b m , with b 0 > a, we can find a non decreasing sequence ξ i , with ξ i ∈ (a, b i ), such thatUsing this observation, it is easy to prove the following corollary:Corollary 6 (The numerator of the error for f in BV 1 ) If f ∈ BV 1 and its derivative has total variation T f ′ [−1, 1] < +∞ then the function ∆ n in Equation(69)satisfiesand Theorem 6 yields an increasing sequence ξ 0 , . . . , ξ n ∈ [−1, 1] such thatIt then follows thatThis proves Corollary 6, and we now present the proof of Theorem 6. Proof of Theorem 6. Let us start the proof with the particular case in whichBy the traditional Mean Value Theorem, there exists µ ∈ (c, b) such thatEquations (80) and (82) imply that f ′ (ξ c ) = 0, and sinceand we have verified Equation (81) assuming that (82) holds. To handle the general case it suffices to apply the argument above toIn fact, g(c) = f (a) = g(a) and Equation (80) implies thatAs a result, the argument above yields ξ b ∈ [ξ c , b) such thatIt then follows from Equation(84)thatand we are done with the general case. ✷Combining the numerator with the denominatorIn this section we combine the results from the previous sections to prove Theorems 1, 2 and 3.Proof of Theorem 1.We start with an odd sequence n j and an irrational x for which lim j→∞ n j B n j ( f , x) − f (x) converges to L ∈ R. According to Equation(69) Since derivatives have the intermediate value property, there exists ξ b ∈ [ξ c , µ] ⊂ [ξ , b) such that f ′ (ξ b ) = v. As a result. have that v lies between 0 = f ′ (ξ c ) and f ′ (µ). Equations (82) and (83) lead tohave that v lies between 0 = f ′ (ξ c ) and f ′ (µ). Since derivatives have the interme- diate value property, there exists ξ b ∈ [ξ c , µ] ⊂ [ξ , b) such that f ′ (ξ b ) = v. As a result, Equations (82) and (83) lead to Rational functions for guaranteed and experimentally wellconditioned global interpolation. J.-P Berrut, Comput. Math. Appl. 151Berrut, J.-P., (1988) Rational functions for guaranteed and experimentally well- conditioned global interpolation, Comput. Math. Appl., 15 (1), 1-16. Linear Barycentric Rational Interpolationwith Guaranteed Degree of Exactness. J.-P Berrut, Springer Proceedings in Mathematics & Statistics. G.E. Fasshauer and L.L. SchumakerSan Antonio201Approximation Theory XVBerrut, J.-P., (2017) Linear Barycentric Rational Interpolationwith Guaranteed De- gree of Exactness, in Approximation Theory XV:San Antonio 2016, Springer Pro- ceedings in Mathematics & Statistics 201, G.E. Fasshauer and L.L. Schumaker (eds.), 1-20. Divided differences. C De Boor, Surv. Approx. Theory. 1de Boor, C., (2005) Divided differences, Surv. Approx. Theory, 1, 46-69. Barycentric rational interpolation with no poles and high rates of approximation. M S Floater, K Hormann, K , Numer. Math. 1072Floater, M. S. and K. Hormann, K., (2007) Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math., 107 (2), 315-331. Sur une condition supplementaire dans les approximations diophantiques. S Hartman, Colloq. Math. 2Hartman, S., (1951) Sur une condition supplementaire dans les approximations diophantiques, Colloq. Math.,(2), 48-51. Barycentric interpolation at equidistant nodes. G Mastroianni, J Szabados, Jaen Journal on Approximation. 91Mastroianni G., and J. Szabados (2017) Barycentric interpolation at equidistant nodes, Jaen Journal on Approximation 9(1)	[]
[ "On Clutter Ranks of Frequency Diverse Radar Waveforms", "On Clutter Ranks of Frequency Diverse Radar Waveforms" ]	[ "Member, IEEEYimin Liu ", "Le Xiao ", "Xiqin Wang ", "Fellow, IEEEArye Nehorai " ]	[]	[]	Frequency diverse (FD) radar waveforms are attractive in radar research and practice. By combining two typical FD waveforms, the frequency diverse array (FDA) and the steppedfrequency (SF) pulse train, we propose a general FD waveform model, termed the random frequency diverse multi-input-multioutput (RFD-MIMO) in this paper. The new model can be applied to specific FD waveforms by adapting parameters. Furthermore, by exploring the characteristics of the clutter covariance matrix, we provide an approach to evaluate the clutter rank of the RFD-MIMO radar, which can be adopted as a quantitive metric for the clutter suppression potentials of FD waveforms. Numerical simulations show the effectiveness of the clutter rank estimation method, and reveal helpful results for comparing the clutter suppression performance of different FD waveforms.	null	[ "https://arxiv.org/pdf/1603.08189v1.pdf" ]	17,811,287	1603.08189	cb3dc8748573360f490a22927fc06e5a506f2ffa	On Clutter Ranks of Frequency Diverse Radar Waveforms Member, IEEEYimin Liu Le Xiao Xiqin Wang Fellow, IEEEArye Nehorai On Clutter Ranks of Frequency Diverse Radar Waveforms 1Index Terms-Frequency diverse waveformradar cluttermoving target indicationMIMO radar Frequency diverse (FD) radar waveforms are attractive in radar research and practice. By combining two typical FD waveforms, the frequency diverse array (FDA) and the steppedfrequency (SF) pulse train, we propose a general FD waveform model, termed the random frequency diverse multi-input-multioutput (RFD-MIMO) in this paper. The new model can be applied to specific FD waveforms by adapting parameters. Furthermore, by exploring the characteristics of the clutter covariance matrix, we provide an approach to evaluate the clutter rank of the RFD-MIMO radar, which can be adopted as a quantitive metric for the clutter suppression potentials of FD waveforms. Numerical simulations show the effectiveness of the clutter rank estimation method, and reveal helpful results for comparing the clutter suppression performance of different FD waveforms. I. INTRODUCTION W AVEFORM diversity has led to many interesting and promising concepts in the research and practice of the radar community in the past decade. By exploring waveform adaptivity in different domains, such as the spatial (antenna beampattern), temporal, spectral, code, and polarization domains, remarkable improvements have been realized in radar abilities, such as high resolution imaging, target recognition, clutter suppression, and electronic-counter-countermeasures (ECCM) [1]. Among the different kinds of diverse waveforms, frequency diverse (FD) waveforms are attractive due to their ease of use in system implementation [2], efficiency in widebandwidth synthesis [3], and robust in spectral compatibility and resilience [4]. In 2006, a new array antenna, named the frequency diverse array (FDA), was introduced in [5]. By linearly [5] or randomly [6] (called LFDA or RFDA, respectively) assigning the carrier frequencies of array elements, an FDA can provide a beampattern which depends on both direction and range, and brings important benefits like transmit beamforming [7], target range-direction estimation [8], and jamming resistance [9], to list a few. Moreover, FDA-based algorithms enable advantages in clutter or interference discrimination, and in Y. Liu, L. Xiao, and X. Wang moving target detection. By using an FDA, the clutter in forward-looking radar was alleviated [10]. In [11], clutter whose delay was outside of one pulse repetition interval (PRI) was successfully discriminated, hence the target detection performance of an airborne multi-input-multi-output (MIMO) radar was improved. However, quantitive metrics of FDA radars' clutter suppression performance are still inadequate. As defined in the IEEE Radar Standard P686/D2 (January 2008), frequency diversity radar is "a radar that operates at more than one frequency, using either parallel channels or sequential groups of pulses". Following this definition, the FDA can be regarded as one kind of FD waveform that is distributed in the spatial-spectral domain. Similarly, another kind of widely-used waveform, stepped-frequency (SF) pulse train [12], can be regarded as FD waveforms distributed in the temporal-spectral domain. In the SF pulse train, the carrier frequencies of pulses in a coherent processing interval (CPI) shift linearly [13] or randomly [14] (termed linear SF (LSF), or random SF (RSF)). Basic LSF pulse train can synthesize a large bandwidth to achieve a very high range resolution [3]. Furthermore, if the carrier frequencies of successive pulses shift randomly, as in the RSF pulse trains, the ambiguity functions will become thumbtack-like, which implies an uncoupled high resolution in both range and velocity [3]. Moreover, the ECCM performances of RSF radars are also outstanding, due to its frequency agility [2]. Although some algorithms have been proposed for clutter suppression [15] [16], the corresponding performance evaluations for the SF, especially RSF radars, still need research. In this paper, we try to provide general quantitive information about the clutter suppression potential of the FD radar waveforms. A new FD waveform model, named the random-frequency-diverse-MIMO (RFD-MIMO), is proposed by integrating the FDA and SF pulse train. Because the carrier frequencies of the array elements and pulses in RFD-MIMO vary agilely, the new waveform is diverse in a 3D (spatial-temporal-spectral) domain. Moreover, this model can be applied to existing FD waveforms by adapting the model parameters. Furthermore, the clutter rank, defined as the rank of a radar's clutter covariance matrix (CCM), is evaluated based on this model. As introduced in [17] and [18], the clutter rank can quantify the averaged clutter suppression performance of a radar. Hence the results of this paper can be regarded as a quantitive metric of the clutter suppression potential that can be achieved by the coherent processing of a frequency diversity radar. It should be noted that this study focuses on coherent approaches to clutter suppression, and the intention is different from conventional works which arXiv:1603.08189v1 [cs.IT] 27 Mar 2016 The rows and columns of the bricks represent the pulses and transmitting elements, respectively. employed the de-correlated radar cross section (RCS) response properties. In addition, unlike traditional clutter rank studies, such as [18] and [19], the main challenge encountered by this work was the disordered phase relationships between array elements and pulses, caused by the frequency diversity. The main contributions of this paper are as follows. 1) We propose a general model, named RFD-MIMO, of FD radar waveforms. In this model, the carrier frequency of each transmitting array element and each pulse can be assigned an arbitrary value, and each receiving array element can receive all the possible carrier frequencies simultaneously. 2) We construct the target and clutter model of the RFD-MIMO waveform. Based on this model, we derive the expressions of the FDA, SF, and FD-MIMO radars' clutter. 3) By exploring the features of an RFD-MIMO's CCM, we derive an approximation of the new FD waveform's clutter rank. We find that the frequency diversity radar's CCM is sparse, and can be permuted to a block diagonal matrix, which notably reduces the complexity of the clutter rank estimation. 4) We substantiate the clutter rank estimation of RFD-MIMO to specific FD radar waveforms, and quantify the clutter suppression potentials of different frequency diversity radars. The results reveal that, first, in radars using FDA or SF pulse trains, random carrier frequency assignments have advantages in clutter suppression over their linear counterparts. Second, wideband pulses and MIMO antennas are more suitable for target detection in heavy clutter scenarios. The rest of this paper is organized as follows. Section II presents the radar schematic, and constructs the system and signal models of the RFD-MIMO waveform. In Section III, by exploring the CCM, we derive an estimation approach for the clutter rank of the new waveform. In Section IV, discussion and numerical results for radars with specific FD waveforms are provided as substantiations of the provided method. Conclusions are drawn in the last section. Notations: The important and frequently used notations are Notation Discription c The speed of light t The time variable Z The set of all integers C The set of all complex numbers fc The initial carrier frequency ∆f The frequency increment d T The distance between transmitting antenna elements d R The distance between receiving antenna elements T The pulse repetition interval (PRI) L The number of transmitting antenna elements R The number of receiving antenna elements P The number of pulses in a pulse train G The frequency diverse code matrix G Q The augmented frequency diverse code matrix with pulse bandwidth Q∆f M Q The set composed of all the unique entries in G Q l The transmitting array element index vector r The receiving array element index vector p The pulse index vector q The sub-band index vector D The clutter range region V The clutter velocity region A The clutter direction sine region C The clutter rank L FD The frequency diversity loss (FDL) R{·} The rank of a matrix vec{·} Column vectorization of a matrix [·] a,b The ath row, bth column entry of a matrix [·]a The ath entry(column) of a vector(matrix) \| · \| The number of unique elements of a vector/matrix/set \| · \| 2 The l 2 -norm of a scalar/vector · The difference between the maximum and minimum entries in a vector/matrix/set · The largest integer which is no larger than an argument · The smallest integer which is no less than an argument (·) T , (·) H The transpose and Hermitian of an argument (·) * The element-wise complex conjugation of an argument I m {·} The column vector composed of row indices corresponding to a vector/matrix's entries which equal to m I = m {·} The column vector composed of column indices corresponding to a vector/matrix's entries which equal to m 1 A×B An A × B matrix (vector) with all-one entries ⊗, The Kronecker and Hadamard products The Khrati-Rao product ⊕ The stretched sum (defined in Section II) listed in Table I. II. SYSTEM AND SIGNAL MODELS In this section, we first formulate the RFD-MIMO radar waveform as a general model of FD waveforms, and then give the expression of the echoes from targets and clutter for the RFD-MIMO and specific frequency diversity radars. A. General Model As introduced in [6], the FDA can be regarded as an FD waveform which is distributed in the spatial-spectral domain. Beyond this, we introduce a pulse train into the waveform to expand it from 2D (spatial-spectral) diversity to 3D (spatial-temporal-spectral) diversity. The system schematic of a radar with RFD-MIMO waveform is shown in Fig. 1. In the RFD-MIMO radar, the transmitting and receiving antennas are colocated. There are L array elements, indexed by l = 0, 1, . . . , L − 1, in the transmitting antenna, and R array elements, indexed by r = 0, 1, . . . , R − 1, in the receiving antenna. The elements in both antennas are equally separated, with the inter-element distances of the transmitting and receiving antennas being d T and d R , respectively. In the RFD-MIMO radar, every transmitting element can be assigned an arbitrary carrier frequency which is chosen from a candidate frequency set. The carrier frequency of every element can (but is not necessarily required to) vary from pulse to pulse. Thus a carrier frequency in the waveform is equal to an initial frequency (f c ) plus an integral multiple of a frequency increment (∆f ). The integer is named the frequency diverse code (FDC), and for all the P pulses (indexed by p = 0, 1, . . . , P − 1) in the pulse train, there are P L integers, which can be arranged into a frequency diverse code matrix (FDCM), G ∈ Z P ×L . Then the carrier frequency transmitted by the lth array element in the pth pulse is f p,l = f c + ∆f [G] p,l .(1) For the signal of each pulse, both narrowband and wideband cases are considered. In narrowband cases, each pulse is assumed as monotone, with the same frequency as its carrier frequency. In wideband cases, following the convention in [20], it is supposed that all the pulses have the same bandwidth B, which can be divided into Q sub-bands (Q is an integer, and B = Q∆f ); the signal of every sub-band can be regarded as a monotone multiplied by a modulation coefficient. Then the transmitted signal of the pth pulse, lth array element, and qth (q = 0, 1, . . . Q − 1) sub-band is s p,l,q (t) = β p,l,q · exp j2π f c + ∆f ([G] p,l + q) t , (2) where β p,l,q is the modulation coefficient. According to (2), the FDCM can be expanded to an augmented-FDCM (a-FDCM), given by G Q = G ⊕ q,(3) where q = [0, 1, . . . , Q − 1] T . In (3), ⊕ is the stretched sum operator, where A ⊕ B = A ⊗ 1 size(B) + 1 size(A) ⊗ B 1 . 1 The definition of size(·) follows the eponymous MATLAB routine which returns the size of a matrix. Fig. 2. An illustration of the data arrangement format of an RFD-MIMO waveform. The data matrix contains P Q rows, where the (pQ + q)th row corresponds to the qth sub-band in the pth pulse. There are LR columns, where the (lR+r)th column corresponds to the echo from the lth transmitting to the rth receiving array element. Then the RFD-MIMO waveform has \|M Q \| frequency points, and (2) can be rewritten as s p,l,q (t) = β p,l,q · exp j2π(f c + ∆f [G Q ] pQ+q,l )t . (4) In addition, due to the colocated assignment of the MIMO antenna, the direction 2 α = sin θ and radial velocity v of a point scatterer can be regarded as identical with respect to (w.r.t.) every transmitting and receiving array element [21]. Therefore, at the pth pulse, the time delay from the lth transmitting element to the scatterer and back to the rth receiving element is τ p,l,r (D, v, α) = 1 c (2D + 2vpT + αld T + αrd R ),(5) where T is the pulse repetition interval (PRI), and D is the initial range (when t = 0) between the scatterer and the 0th array element. Disregarding both the energy divergence in the wave propagation paths and the variation of the target's reflection factor, the received echo from a unit point scatterer can be seen as the time-delayed version of the transmitted signal. Thus the echo of the qth sub-band from the lth transmitting to the rth receiving array element is r p,l,r,q (t; D, v, α) = β p,l,q · exp j2π(f c + ∆f [G Q ] pQ+q,l ) · t − τ p,l,r (D, v, α) .(6) At each receiving array element, echoes from all the transmitting elements are demodulated and then match-filtered respectively by their own carrier frequencies. For the qth subband, this procedure can be expressed by b p,l,r,q (t; D, v, α) = r p,l,r,q (t; D, v, α) · exp − j2π(f c + ∆f [G] p,l )t ·β * p,l,q exp(−j2πq∆f t).(7) Substituting (6) into (7) gives the match-filtered sub-band echo: b p,l,r,q (D, v, α) = β p,l,q 2 2 · exp − j 2π c (f c + ∆f [G Q ] pQ+q,l ) ·(2D + 2vpT + αld T + αrd R ) ,(8) which is time-invariant. In one pulse train, the number of baseband samples (termed the measurement dimension) is P QLR. All the P QLR samples can be arranged into a (P Q)×(LR) data matrix, whose (pQ+q)th row and (lR+r)th column entry is b p,l,r,q (D, v, α). This data arrangement format is illustrated in Fig. 2. As shown in (8), the baseband echoes of an RFD-MIMO radar depend simultaneously on the scatterer's range, velocity, and direction. Thus by vectorizing the data matrix, u(D, v, α) ∈ C (P QLR)×1 can be denoted as the rangevelocity-direction steering vector, where [u(D, v, α)] (lR+r)P Q+pQ+q = b p,l,r,q (D, v, α).(9) Furthermore, (8) also shows that the range-velocitydirection steering vector u(D, v, α) can be decomposed into the Hadamard products of the modulation vector β, the range steering vector u D (D), the velocity steering vector u V (v), and the direction steering vector u A (α): u(D, v, α) = β u D (D) u V (v) u A (α),(10) where [β] (lR+r)P Q+pQ+q = β p,l,q 2 2 , ∀r = 0, 1, . . . , R − 1, (11) u D (D) = vec exp −j 4π c D(f c +∆f G Q )⊗1 1×R , (12) u V (v) = vec exp − j 4π c T v f c + ∆f G Q (p ⊗ 1 Q×L ) ⊗ (1 1×R ) ,(13) and u A (α) = vec exp − j 2π c α(f c + ∆f G Q ⊗ 1 1×R ) (d T 1 P Q×1 ⊗ l T ) ⊕ (d R r T ) .(14) In the above equations, p = [0, 1 . . . , P − 1] T , l = [0, 1, . . . , L − 1] T , and r = [0, 1, . . . , R − 1] T . In moving target indication (MTI) [2], clutter, which is defined as the unwanted echo, is usually regarded as a superimposition of received echoes from scatterers whose ranges, velocities, and directions are in a certain region (the clutter region). In this work, we denote D, V, and A as the clutter range, velocity, and direction regions, respectively. Hence the clutter echo vector can be calculated by r C = D V A ρ(D, v, α) · u(D, v, α)dDdvdα,(15) where ρ(D, v, α) is the clutter reflection density of the rangevelocity-direction coordinate {D, v, α}, as shown in Fig. 3. The three-fold integral interval in (15) is determined as follows. 1) Clutter range region. Due to the clutter distribution features in practice [2] and the narrowband assumption of each sub-band, clutter is usually combined with echoes from scatterers located over a large range. However, as can be seen in (12), the values of the range steering vectors are identical 3 for scatterers with range differences that are multiples of c/(2∆f ). Thus the clutter range region is D = 0, c 2∆f . 2) Clutter velocity region. Because the unambiguous target velocity is inversely proportional to the PRI in pulsed radars [2], the clutter velocity region should be a subset of the unambiguous interval of velocity: V ⊆ − c 4f c T , c 4f c T . 3) Clutter direction region. As introduced in [21], for a collocated MIMO radar, the clutter direction region should be a subset of the unambiguous interval of direction: A ⊆ [− c 2d R f c , c 2d R f c ]. Moreover, the integral in (15) can be approximated in a discrete mode by summing up the echoes from all the voxels in the clutter region. Assuming there are N C voxels, each of which has a reflection amplitude ρ n (n = 1, 2 . . . , N C ), then the clutter echo vector can be re-written as r C = C · ρ,(16) where ρ = [ρ 1 , ρ 2 , . . . , ρ NC ] T , and C is the clutter steering matrix (CSM), whose nth column is the steering vector of the range-velocity-direction coordinate {D n , v n , α n }. B. Application to Specific Frequency Diverse Waveforms The system and signal model presented in last subsection can be applied to specific FD waveforms by adapting corresponding parts of the model, such as P , Q, L, R, d R , and G Q . Applications to FDA, SF, and frequency diverse MIMO (FD-MIMO) radar waveforms will be derived in this subsection. 1) FDA: In radars using FDA antennas, different array elements transmit and receive different carrier frequencies which can be shifted linearly [5] (linear FDA, LFDA ) or randomly [6] (random FDA, RFDA). The range-direction dependent beampatterns are synthesized by processing the received echo. In the FDA, the carrier frequencies of array elements are kept invariant throughout the operation. Thus the RFD-MIMO can be applied by the following steps. First, reduce the FDCM, G, to a row vector of L entries, g T . Second, if the signal of a single pulse is wideband, the a-FDCM will be expressed by G Q = q ⊕ g T .(17) Because most research works about the FDA has focused on the beampatterns, the system models are usually formulated with one pulse, which makes the target velocity unobservable. Thus in (13), the variable P should set to 1, and the velocity steering vector u V (v) should be trivialized as an all-one vector. Moreover, each array element in an FDA transmits and receives with its own carrier frequency, thus in (5), l = r and d T = d R . In addition, the clutter direction region should be adapted to A ⊆ [−c/(4d R f c ), c/(4d R f c )]. Hence the direction steering vector can be expressed by u A (α) = vec exp − j 4π c α f c + ∆f (q ⊕ g T ) (d T 1 Q×1 ⊗ l T ) .(18) Finally, the integral in (15) should be calculated on only the clutter range region and clutter direction region. Therefore, the clutter model of an FDA radar is r C = D A ρ(D, α) · β u D (D) u A (α)dDdα. 2) SF pulse train: The applications to LSF and RSF pulse trains are straightforward. In radars with LSF or RSF pulse trains, the antennas are usually configured as single-inputsingle-output (SISO). Thus the FDCM should be reduced to a column vector g with P entries, which represents the carrier frequencies of all pulses in a pulse train. For pulses with bandwidth Q∆f , the a-FDCM is given by G Q = g ⊕ q.(19) The single element antenna makes the target's direction unobservable in this instance. Hence in the waveform model, L = R = 1, d R = d R = 0, and the direction steering vector u D (D) should be replaced by an all-one vector, 1 (P Q)×1 . With the above steps, the steering vectors of LSF and RSF pulse trains are range-velocity dependent: u(D, v) = β u D (D) u V (v),(20) where u D (D) = vec exp − j 4π c D(f c + ∆f g ⊗ q) ,(21)u V (v) = vec exp − j 4π c T v(f c + ∆f g ⊗ q) (p ⊗ 1 Q×1 ) .(22) Because it differs from that of an FDA radar, the clutter echo vector should be calculated by integrals on the clutter range region and clutter velocity region: r C = D V ρ(D, v) · β u D (D) u V (v)dDdv. 3) FD-MIMO and its space-time adaptive processing (STAP) applications: The MIMO technique has been applied to the FDA waveform to improve the measurement dimension [20]. The FD-MIMO waveform is quite similar to the original RFD-MIMO model. However, most researches on FD-MIMO is focused on the range-direction dependent beampattern. Therefore, the a-FDCM of an FD-MIMO waveform can be written as G Q = q ⊕ g T .(23) As the steering vector, the application can be accomplished by changing the velocity steering vector into an all-one vector, and removing the variable v from the parameters of the steering vector. However, in ground moving target indication (MTI) applications [17], such as space-time adaptive processing for airborne radars [11], the pulse train is introduced in the waveform, and the ground clutter's spatial frequency is assumed to be linearly proportional to the temporal frequency [17]. For the most commonly studied side-looking mode antennas, the relationship between the spatial and temporal frequencies is v = α · v p ,(24) where v p is the platform velocity. Thus the velocity steering vector can be embedded into the direction steering vector, and the clutter model of an FD-MIMO radar with STAP applications can be formulated by substituting (24) into (15): r C = D A ρ(D, 0, α)·β u D (D) u V (αv p ) u A (α)dDdα, where A is the direction region covered by the array element. III. CLUTTER RANK ESTIMATION Clutter rank, defined as the rank of a radar's CCM, is an important parameter for the quantification of target detection performance in clutter environments [17], [18]. A small clutter rank relative to the whole measurement dimension means that the radar has a greater ability to suppress the clutter [18]. In this section, we explore the futures of the RFD-MIMO's CCM and CSM, and then give a theorem for the clutter rank estimation of the new waveform. A. Features of the CCM and CSM According to the definition of the CCM, we have that R C = E r C r H C = CE ρρ H C H .(25) From the basic properties of matrices [22], the clutter rank is given by C R{R C } ≤ min R{C}, R E{ρρ H } ≤ R{C},(26) which means the clutter rank of a radar is no larger than the rank of its CSM. The equality in the third row of (26) is valid if the covariance matrix of the clutter reflection amplitudes, E{ρρ H }, is full rank. Furthermore, the rank of the CSM is R{C} = R{CC H } = R NC n=1 u(D n , v n , α n )u H (D n , v n , α n ) . Then, the clutter rank estimation is relaxed to the rank estimation of the Gramian matrix of the CSM's Hermitian. Moreover, the summation in the above equation can be calculated by integrals on the clutter range, velocity, and direction regions: CC H = D V A u(D n , v n , α n )u H (D n , v n , α n )dDdvdα.(27) By substituting (10) into (27), CC H can be decomposed into the Hadamard products of four matrices: CC H = (ββ H ) D u D (D)u H D (D)dD RD V u V (v)u H V (v)dv RV A u A (α)u H A (α)dα RA = (ββ H ) R D R V R A ,(28) where ββ H is rank-1, and R D , R V , and R A are all (P QLR) × (P QLR) matrices. Moreover, with the following lemma, it can be shown that the second component of (28), R D , has good features which can simplify the clutter rank estimation. where I r (x) = x − (P Q) · x/(P Q) , I c (x) = x/(P Q) . Proof: According to the definition of R D in (28), [R D ] a,b = D [u D (D)] a [u D (D)] * b dD = D e −j 4π c Dz dD, where z ∈ Z and z = [G Q ] Ir(a),Ic(a) − [G Q ] Ir(b),Ic(b) . If the condition given in (29) is satisfied, [R D ] a,b = c 2∆f 0 e −j 4π∆f c D·0 dD = c 2∆f . Otherwise, if [G Q ] Ir(a),Ic(a) = [G Q ] Ir(b),Ic(b) ,(30) z becomes a non-zero integer. According to the Cauchy's integral theorem [23], [R D ] a,b = c 2∆f 0 e −j 4π∆f c D·z dD = 0. Lemma 1 is proven. Lemma 1 and (28) mean that many entries of CC H are zero. Meanwhile, because R D is symmetric, the rest of the non-zero entries can be permuted into a block diagonal matrix by row and column swapping, where each diagonal block corresponds to a frequency point, f c +m∆f (m ∈ M Q ). Moreover, because the rank of a matrix remains unchanged during the row and column swapping, R{CC H } can be decomposed to the sum of several smaller matrices' ranks: R{CC H } = m∈M Q R R Cm ,(31) where R Cm is the diagonal block corresponding to the frequency point f c + m∆f . Fig. 4 shows the entries' magnitudes in the original CC H , and the lower half displays those of the permuted matrix. It can be clearly seen that after the row and column swapping, the permuted CC H has a block diagonal structure of four diagonal blocks. In accordance with the row and column swapping, each R Cm can be decomposed to a Hadamard product of two matrices: R Cm = R Vm R Am ,(32) where R Vm = V u Vm (v)u H Vm (v)dv,(33)R Am = A u Am (α)u H Am (α)dα.(34) In (33-34), u Vm and u Am are sub-velocity and sub-direction steering vectors, given by u Vm (v) = exp − j 4π c v(f c + m∆f ) · (T I m (G Q ) Q ) ⊗ 1 R×1 ,(35)u Am (α) = exp − j 2π c α(f c + m∆f ) · (d T I = m (G Q )) ⊕ (d R r) .(36) With above derivations, CC H can be expressed by small matrices, such as R Vm and R Am , whose dimensions are notably reduced from the original ones. In addition, each R Vm and R Am depends only on a single frequency point, f c + m∆f . These features allow the clutter rank estimation of a diverse waveform to be accomplished in a "frequency point by frequency point" manner, which greatly reduces the complexity of the original problem. B. Rank Estimation The integrals in (33) and (34) can be approximated in matrix form, by discretizing the clutter velocity region and clutter direction region into N V velocity grids and N A direction grids, respectively: R Vm = C Vm C H Vm , R Am = C Am C H Am , where [C Vm ] n = u Vm (v n ), n = 1, 2, . . . , N V ,(37) [C Am ] n = u Am (α n ), n = 1, 2, . . . , N A . Take the rank estimation of R Vm as an example. According to the basic properties of matrices [22], R{R Vm } = R{C Vm }. Furthermore, (35) and (37) show that the columns of C Vm can be regarded as complex sinusoids sampled on the mth temporal sampling aperture, which is defined as the set composed of all the unique entries in vector T I m (G Q )/Q , in increasing order. Some characteristics of the sampling apertures should be noted. First, if the a-FDCM has multiple non-identical columns, the sampling apertures corresponding to different frequency points may overlap. Second, each sampling aperture can be divided into sub-apertures by splitting two successive sampling instants into two sub-apertures when the gap between these two instants is larger than the Nyquist sampling interval, c/(2(f c + m∆f ) V ). All the sub-apertures corresponding to the frequency point, f c + m∆f , are gathered as a set, T m (V). From the above discussion, an approximation of R{R Vm } can be provided with the help of prolate spheroidal wave function (PSWF) theory [24]. According to PSWF theory, complex sinusoids, whose energies are mostly confined in a certain "time(T )-frequency(W )" region, can be well approximated by linear combinations of W T + 1 orthogonal functions. In our case, the "frequencies" of the complex sinusoids, u Vm (v n ), vary in an extent of 2(f c + m∆f ) V /c. The "time" should be counted separately for every sub-aperture in the T m (V). Then we have the following lemma. Lemma 2: The rank of R Vm can be approximated by R{R Vm } ≈ U Vm min \|I m {G Q }\|, R Vm (V),R Vm (V) ,(39)where R Vm (V) = 2 c (f c + m∆f ) V T I m {G Q } Q + 1,(40) andR Vm (V) = T ∈Tm(V) 2 c (f c + m∆f ) V T + 1.(41) In equation (39), the first and second terms in the function min{·} are needed because the maximal number of linearly dependent signals with identical sampling instants is no larger than the number of the sampling instants, and cannot be reduced by introducing new sampling instants. An example of R{R Vm } and U Vm is provided in Fig. 5. In this example, we formulated an RFD-MIMO waveform with 32 transmitting array elements, 8 receiving array elements, and 128 monotone pulses. Both linear and random carrier frequency assignments were considered. The numbers of carrier frequencies were set as \|G\| = 1 (the fixed-frequency case, expressed similarly hereinafter), 4, 8, and 16. In this simulation, the extent of the clutter velocity region varied from zero to one times the unambiguous extent of velocity. Both R{R Am } and U Vm were normalized by the numbers of unique sampling instants in the temporal sampling aperture, and are indicated by symbols and dotted/dashed lines, respectively. The different colors indicate to different carrier frequency numbers. The circles and triangles represent linear and random carrier frequency assignments, respectively. In this simulation, the approximations given by Lemma 2 matched the true ranks well. In addition, other phenomena could be seen: The fixedfrequency waveforms had the smallest rank; the larger the carrier frequency number, the faster the normalized rank grew; and the ranks of random carrier frequency assignments were smaller than those of the linear ones. The rank estimation of R Am can be derived in a similar way. As shown in (36) and (38), the columns of C Am can be regarded as the discrete samples of complex sinusoids whose "frequencies" vary in an extent of (f c + m∆f ) A /c. Moreover, the sampling instants are distributed on the mth spatial sampling aperture, which is determined by (d T I = m {G Q }) ⊕ (d R r) . The Nyquist sampling interval in the spatial domain is c/((f c + m∆f ) Ad R ). Thus the sampling instants corresponding to f c + m∆f can be divided to sub-apertures, all of which are gathered as a set, S m (A). With the above definitions, we have the following lemma. Lemma 3: The rank of R Am can be approximated by R{R Am } ≈ U Am min \|(d T I = m {G Q }) ⊕ (d R r)\|, R Am (A), R Am (A) ,(42) where R Am (A) = 1 c (f c + m∆f ) A (d T I = m {G Q }) ⊕ (d R r) + 1,(43) andR Am (A) = S∈Sm(A) 1 c (f c + m∆f ) A S + 1.(44) An example of R{R Am } and U Am is given in Fig. 6. The simulation setup and legends are the same as those in Fig. 5, except the pulse number is 32 and the transmitting elements number is 128. Although the system configurations were similar to those of the simulation for R{R Vm }, the results were different. In this example, the clutter ranks corresponding to different carrier frequency numbers had only small differences between each other, for both linear and random carrier frequency assignments. The reason can be briefly explained as follows: In the MIMO antenna, the spatial sampling aperture of each fre-quency point is the stretched sum of the transmitting aperture (d T I = m {G Q }) and the receiving aperture (d R r). Because d T is usually several times larger than d R , the discontinuity of entries in I = m {G Q } leads to a wide gap between spatial sampling instants. Hence when A is small, the spatial sampling aperture begins to divide into small sub-apertures. These sub-apertures are composed of integer multiples of R sampling instants, and the distances between neighboring sampling instants are d R . Therefore, the rank is approximately proportional to the number of sampling instants, regardless of the number and the assignment of carrier frequencies. With Lemma 2 and Lemma 3, we have the following lemma for the rank of R Cm . Lemma 4: The ranks of the diagonal blocks, R Cm , satisfy the following inequalities: U Cm ≤ R{R Cm } ≤ min{K m , U Vm U Am },(45) where U Cm min K m , U Vm + U Am − 1 ,(46) and K m is the dimension of R Cm . Proof: The proof can be found in Appendix A. With the above preparation, we deduce the following theorem for the clutter rank estimation of the RFD-MIMO waveform: Theorem 1: If the covariance matrix of the clutter reflection amplitudes is full rank, then the clutter rank of an RFD-MIMO radar, C, satisfies the following inequalities: m∈M Q U Cm ≤ C ≤ m∈M Q min{K m , U Vm U Am },(47) where U Vm , U Am , and U Cm are defined in (39), (42), and (46), respectively. Proof: The proof of Theorem 1 can be accomplished straightforwardly by combining (31) and Lemma 4. IV. APPLICATIONS AND DISCUSSION Due to the flexible configuration of the a-FDCM, the RFD-MIMO waveform is a general model for FD waveforms whose carrier frequencies can vary for different array elements and/or pulses. In this section, we will give the rank estimation for specific kinds of FD waveforms by the corollaries of Theorem 1. A. Metrics of Clutter Suppression Performance As concluded in [18], a higher clutter rank relative to the measurement dimension means that the clutter spreads over a larger portion of the whole signal space, leaving fewer clutterfree dimensions for the target detection. In this section, we will compare the normalized clutter rank (NCR, defined as the clutter rank, C, normalized by the measurement dimension, P QLR) between different FD radar waveforms, to show the corresponding clutter suppression potentials. The reasons for choosing the NCR are as follows. First, as explained in Appendix B, the SCNR opt , defined as the optimal output signal-to-clutter-noise-ratio which can be achieved by linear filtering, is approximately proportional to the target energy which is spread in the orthogonal complement of the CCM's eigenspace: SCNR opt ≈ 1 σ 2 P ⊥ RC · u(D, v, α) 2 2 ,(48) where P ⊥ RC is the projection matrix. Numerical investigations (not analytically proven yet) showed that the averaged (w.r.t. all the unambiguous extents of target range, velocity, and direction) target power distributed on the CCM's eigenspace is linearly proportional to the normalized clutter rank (NCR): mean D,v,α P ⊥ RC · u(D, v, α) 2 2 ∝ 1 − C P QLR .(49) The result in (49) implies that the averaged SCNR opt can be predicted by the difference between the quantity 1 and the NCR. Furthermore, the frequency diversity loss (FDL), defined as the ratio between the SCNR opt of an FD waveform and the SCNR opt of a fixed-frequency waveform, can be expressed by the NCR: L FD ≈ 1 − C FD /(P QLR) 1 − C 0 /(P QLR) ,(50) where L FD is the FDL, and C FD and C 0 are the clutter ranks of the fixed-frequency and the FD waveforms for the same clutter environment 4 . B. Frequency Diverse Array As introduced in subsection II-B1, the applications of the RFD-MIMO to FDA can be accomplished by adapting the matrix G to a row vector g T . Thus according to Theorem 1, the clutter rank of an FDA radar can be evaluated by the following corollary. Corollary 1: (Frequency Diverse Array) For an FDA radar with a clutter direction region A, the clutter rank is C ≈ m∈M Q min \|I = m {q ⊕ g T }\|, 2 c (f c + m∆f ) d T I = m {q ⊕ g T } · A + 1, S∈Sm(A) 2 c (f c + m∆f ) A S + 1 .(51) The NCRs of both LFDA and RFDA are illustrated in Fig. 7. In this example, there were L = 256 array elements, and the carrier frequency of each element was selected from a set of \|G\| = 1, 4, and 8 integers. Both monotone (Q = 1) and wideband (Q = 16) pulses were simulated. The clutter rank C and its approximation given in (51) were calculated and then normalized by the measurement dimension P QLR. In the figure, the NCRs for different system configurations are indicated by different symbols, and the approximations are plotted by dashed and dotted lines for the LFDA and the RFDA, respectively. It is shown that the fixed-frequency (\|G\| = 1) waveforms have the lowest NCRs, and that the lower the carrier frequency number, the smaller the NCR. In addition, the NCRs of wideband pulse waveforms (Q = 16) are much smaller than those of the monotone (Q = 1) ones. Furthermore, the LFDA has a higher NCR than the RFDA, especially when A is in the intermediate portion of the normalized extent of the clutter direction region. This phenomenon will be further discussed in subsection IV-E, together with the SF pulse trains. C. Stepped-Frequency Pulse Train Coherent clutter suppression and moving target indication are long-term problems for the SF, especially for the RSF (also known as frequency agile coherent [25]) radars [2]. With Theorem 1, we can provide quantitative predictions of the clutter suppression performance for SF radars. According to the appliaction steps given in subsection II-B2, we have the following corollary. Corollary 2: (Stepped-Frequency) For an SF radar with clutter velocity region V, the clutter rank is C ≈ m∈M Q min I m {g ⊕ q} , 2 c (f c + m∆f ) V T I m {g ⊕ q} Q + 1, T ∈Tm(V) 2 c (f c + m∆f ) V T + 1 .(52) The NCRs and their approximations for both the LSF and the RSF radars are illustrated in Fig. 8. The pulse number was 256, and the carrier frequency of each pulse was selected from a set of \|G\| = 1, 4, and 8 integers. Both monotone (Q = 1) and wideband (Q = 16) pulse waveforms were simulated. Results are shown with legends similar to those in Fig. 7. The results revealed by Fig. 8 are analogous to Fig. 7: Fixedfrequency radars have the lowest NCRs; the higher the carrier frequency number, the larger the NCR; the NCRs of wideband pulse waveforms are much smaller than those of monotone pulse waveforms; and comparing to the LSF, the advantages of RSF pulse trains are apparent. D. FD-MIMO and its STAP Applications As introduced in [20], the MIMO technique will remarkably increase the measurement dimension of an FDA radar. Moreover, the results in this section will show that compared to the FDA , the MIMO antenna can also alleviate the increases in NCRs caused by frequency diversity. Following the steps given in subsection II-B3, the RFD-MIMO can be easily applied to FD-MIMO, and the clutter rank can be evaluated as in Corollary 3. Corollary 3: (FD-MIMO) For an FD-MIMO radar with clutter direction region A, the clutter rank is C ≈ m∈M Q min (d T I = m {G Q }) ⊕ (d R r) , 1 c (f c + m∆f )) A (d T I = m {G Q }) ⊕ (d R r) + 1, S∈Sm(A) 1 c (f c + m∆f ) A S + 1 ,(53) where G Q is formulated as in (23). Simulations of the FD-MIMO radar were conducted with a virtual array of 512 elements, which was synthesized by 64 transmitting and 8 receiving array elements. The distance between the receiving elements was half the wavelength, and d T = 8d R . Similarly, \|G\| = 1, 4, and 8 carrier frequencies were respectively simulated with both linear and random carrier frequency assignments. However, only monotone pulses were considered, due to the huge computer memory consumption of the wideband pulse cases. It can be seen in Fig. 9 that, unlike in the FDA or SF radar, the NCRs and the corresponding approximations changed very little for all the carrier frequency numbers. This result means that the MIMO antenna structure leads to lower NCRs for the FD waveforms, and consequently higher clutter suppression potential. For the STAP applications of an airborne FD-MIMO radar, the ground clutter is usually supposed to be stable, and its velocity relative to the radar antenna is caused by the platform's speed. Therefore, the temporal sampling apertures can be embedded into the spatial sampling apertures to form a new group of sampling apertures, given by (d T I = m {G Q }) ⊕ (d R r) ⊕ (2v p T p), m ∈ M Q .(54) Thus the clutter rank of an airborne FD-MIMO with sidelooking mode can be evaluated by the following corollary. Corollary 4: (FD-MIMO STAP) For an airborne FD-MIMO radar with side-looking mode, if the beam coverage of the array element is A, the clutter rank is C ≈ m∈M Q min (d T I = m {G Q }) ⊕ (d R r) ⊕ (2v p T p) , 1 c (f c + m∆f )) A E m (O) + 1, E∈Em(A) 1 c (f c + m∆f ) A E + 1 ,(55) where E m (A) is the set of sub-apertures defined on the embedded sampling aperture, and G Q is formulated as in (23). The simulation results for the FD-MIMO STAP are given in Fig. 10. The radar had 16 transmitting, and eight receiving array elements, and a pulse train with 16 monotone pulses, which led to a measurement dimension of P QLR = 2048. A lower NCR than in the above three instance can be expected due to the embedding of the temporal sampling apertures within the spatial ones. As shown in Fig. 10, for a certain A , the difference between NCRs of different carrier frequency numbers (\|G\| = 4 and 8) and different carrier frequency assignments are small, as for the original FD-MIMO waveforms. However, the NCRs of an FD-MIMO STAP are much lower than the other three kinds of FD waveforms, and the FDL is around 0 ∼ −0.4 dB for this system configuration. E. Discussion In this subsection, we give intuitive explanations and further results for the clutter suppression performance of the FDA and SF radars. As shown in Fig. 7 and Fig. 8, in the FDA or SF radars using monotone pulses, the NCRs of waveforms with linear carrier frequency assignments increase faster than those of their random counterparts. In the FDA or SF radar, the frequency diversity is in a 2D (spatial-spectral or temporalspectral) domain. In this case, the linear carrier frequency assignment makes the sampling aperture of each frequency point periodic, and the inter-sampling-instant gaps \|G\| times larger than those of the fixed-frequency waveforms. Thus when the extents of clutter regions satisfy A < c 2d R f c \|G\| , or V < c 2T f c \|G\| , aperture splitting will not happen, and S ≈ (L − 1)d T where \|S m \| = 1, T ≈ (P − 1)T where \|T m \| = 1. Hence according to Theorem 1, the clutter ranks will grow \|G\| times faster than in the fixed-frequency cases, until they touch the ceiling determined by the measurement dimensions. However, in the spatial or temporal sampling apertures of random carrier frequency assignments, there must be gap(s) between successive sampling instants which is(are) larger than the splitting threshold. The sampling apertures begin dividing into sub-apertures when the extents of clutter regions are small, which makes the clutter ranks grow more moderately, as shown by the triangles and dotted lines in Fig. 7 and Fig. 8. Through the above analysis, we conclude that in FDA or SF radars with monotone pulses, random frequency assignments can give a lower NCR, and provide a better clutter suppression potential than linear frequency assignments. However, for a waveform with wideband pulses, if the pulse bandwidth is much larger than the range of the carrier frequencies, then Q∆f ∆f G , the sub-bands of each pulse will not only increase the measurement dimensions, but also make the the sampling apertures of each frequency point densely and evenly distributed. Thus according to Theorem 1, the clutter rank will approximately increase at a speed that is direct proportion to the measurement dimension, which keeps the NCRs constant. Finally, we provide a target detection performance comparison between FDA radars with fixed, linearly-assigned, and randomly-assigned carrier frequencies. The waveform parameters used in this simulation were as follows: The number of array elements L = 256; the inter-element distance d R = c/(4f c ); the clutter direction region A = [−c/(32d R f c ), c/(32d R f c )]; the numbers of carrier frequencies \|G\| = 1, 4, and 8; and the pulse bandwidths Q = 1∆f and Q = 16∆f . A point target was placed outside the clutter direction region, and the match-filted SNR varied from 6 dB to 24 dB. The detection probabilities (P d ) of the different waveform configurations were simulated, while the false alarm rates (P fa ) were kept constant at P fa = 10 −5 . The results given in Fig. 11 show that the fixed-frequency waveforms have the best detection performance, and the detection probabilities of the wideband pulses are higher than those of the monotone pulses. Moreover, for the RFDA of monotone pulses, the waveforms with a smaller number of carrier frequencies perform better. For the LFDA of monotone pulses, when \|G = 8\|, the clutter spreads over the whole signal space, which makes P d always equal to zero. However, the results of \|G = 4\| were ignored, because the P d was analogous to the fixed-frequency in some part of the non-clutter region (otherwise zero), which made the averaged detection probabilities pointless. In addition, the simulation results of SF pulse trains were similar to those of FDA, hence we omitted them. V. CONCLUSION In this paper, we constructed a new FD radar waveform, named RFD-MIMO, by combining two FD waveforms, the FDA and the SF pulse train. The RFD-MIMO can be adopted as a general model, and applied to specific FD waveforms by easy adaptions. Furthermore, by exploring the block diagonal features of the CCM, we proposed an effective approach to estimate the clutter rank of the new waveform model. Then the clutter suppression performances of typical FD waveforms were quantitively evaluated by the corollaries of the approach. Numerical results verified the estimation approach, and revealed two properties of the frequency diversity radars's clutter: The random carrier frequency assignments are more advantageous than their linear counterparts in the coherent clutter suppression of FDA or SF radars, and wideband pulses and MIMO antennas are more suitable for frequency diversity radars to detect targets in clutter. APPENDIX A For the first inequality in (45), the integrals in (33) and (34) can be approximated by summations: R Cm = V A (u Vm u Am )(u Am u Am ) H dvdα ≈ (V H A H ) H (V H A H ),(56) where V = C H Vm , A = C H Am , and R{R Cm } = R{V A}. (57) In the case that R{R Vm } + R{R Am } − 1 ≤ K m , if R{V A} = R{R Cm } < R{R Vm } + R{R Am } − 1 U, (58) every subset of U columns in V A is linearly dependent. Thus ∃c ∈ C U ×1 , which satisfies that According to the formulations of the columns in V, each entry of the column index set, {0, 1, . . . , K m −1}, corresponds to a sampling instant in the temporal sampling aperture. Thus V's column index set can be divided into two subsets. The first one, termedV, corresponds to all the unique sampling instants; the other, termedṼ, corresponds to the redundant instants, each of which is a replica of an entry inV. Subsets A andÃ are defined similarly. Furthermore, from the format properties of the temporal and spatial sampling apertures, we have that \|V ∩Ā\| ≥ 1. (Λ U {V} Λ U {A}) · c = 0,(59) Moreover, in the approximation in (56), the clutter direction and velocity regions are divided into uniform grids, which makes V and A Vandermonde matrices. It has been proven in [26] that, for a Vandermonde matrix, its Kruskal-rank (determined when every subset of Kruskal-rank columns in the matrix is linearly independent and at least one subset of Kruskal-rank+1 columns is linearly dependent) equals its rank. Thus every subset of R{V} columns in ΛV{V} and every subset of R{A} columns in ΛĀ{A} are linearly independent. Then it can be verified that the column index set, U (V ∩Ā) ∪V ∪Â,(62) and R{Λ U {V}} = R{V}, R{Λ U {A}} = R{A}.(64) Equation (63) and (64) violate the result in (60), which is an inference of the assumptions in (58). Hence the first inequality in (45) in proven. The second inequality in (45) is straightforward, due to the property that the rank of a Hadamard product of two matrices is no larger than the product of the two matrices' ranks [22]. Lemma 4 is proven. APPENDIX B According to the optimal receiver theory [27], the optimal linear receiver filtering is the one which achieves the highest output SCNR for a specific waveform and clutter. For a point target with range D, velocity v, and direction α, if the receiver noise is additive white Gaussian with a noise power σ 2 , the optimal clutter suppression performance can be achieved by optimizing the filtered SCNR w.r.t. the filter coefficient vector w: w = arg max w w H u(D, v, α) 2 2 w H (R C + σ 2 I) w ,(65) where w ∈ C (P QLR)×1 . By minimum variance distortionless response (MVDR) beamforming [27], (65) can be solved analytically, where the optimized SCNR is expressed by SCNR opt = u H (D, v, α)(R C + σ 2 I) −1 u(D, v, α). (66) In (66), the inversion of the matrix R C + σ 2 I can be calculated by eigen-decomposition. Because the eigenvectors of R C are also those of R C + σ 2 I, the eigenspace of R C + σ 2 I can be divided into the clutter-subspace and the noise-subspace, where the clutter-subspace is spanned by the eigenvectors of R C , {v i } C i=1 : R C = C i=1 λ i v i v H i .(67) In (67), {λ i , v i } is the ith eigenvalue-eigenvector pair of R C . Moreover, the dimension of noise-subspace is P QLR−C, and its orthogonal bases, termed {v i } P QLR i=C+1 , can be constructed via Gram-Schmidt orthogonalization: R C + σ 2 I = C i=1 (λ i + σ 2 )v i v H i + P QLR i=C+1 σ 2 v i v H i . (68) Substituting (68) into (66), the maximized output SCNR is SCNR opt = C i=1 1 λ i + σ 2 v H i u(D, v, α) 2 2 + 1 σ 2 P QLR i=C+1 v H i u(D, v, α) 2 2 .(69) In (67), λ i equals the eigen-spectral distribution of the clutter power, which can be considered as dominantly large, compared to the noise power in practice [17]. Thus 1/(λ i + σ 2 ), the coefficient of the first part in the right side of (69), approaches zero. Therefore, SCNR opt can be approximated by SCNR opt ≈ 1 σ 2 P QLR i=C+1 v H i u(D, v, α) 2 2 = 1 σ 2 P ⊥ RC · u(D, v, α) 2 2 .(70) Fig. 1 . 1A brief schematic of an RFD-MIMO radar. Each brick corresponds to a pulse in the waveform, and different colors signify different carrier frequencies. Fig. 3 . 3A simple illustration of the clutter range-velocity-direction region, and the clutter reflection density. Lemma 1 : 1The ath, bth entry of R D , [R D ] a,b , is non-zero, if and only if [G Q ] Ir(a),Ic(a) = [G Q ] Ir(b),Ic(b) , Fig. 4 . 4An example of the entries' magnitudes in CC H , before (upper half) and after (lower half) row and column swapping. Fig. 4 4shows an example for the sparse and symmetric features of CC H . In this example, the RFD-MIMO waveform had four different carrier frequencies and 16 monotone pulses in one CPI. The numbers of transmitting and receiving array elements were four and eight, respectively. The upper half of Fig. 5 . 5The normalized U Vm and the ranks of R Vm , for different normalized extents of clutter velocity region. Fig. 6 . 6The normalized U Am and the normalized ranks of R Am , for different normalized extents of clutter direction region. = 1 , 8 Fig. 7 . 187\|G\| = 4 Linear, Q = 1, \|G\| = 8 Linear, Q = 16, \|G\| = 4 Linear, Q = 16, \|G\| = 8 Random, Q = 1, \|G\| = 4 Random, Q = 1, \|G\| = 8 Random, Q = 16, \|G\| = 4 Random, Q = 16, \|G\| = NCRs and their approximations for the FDA radar waveforms. = 1 , 8 Fig. 8 . 188\|G\| = 4 Linear, Q = 1, \|G\| = 8 Linear, Q = 16, \|G\| = 4 Linear, Q = 16, \|G\| = 8 Random, Q = 1, \|G\| = 4 Random, Q = 1, \|G\| = 8 Random, Q = 16, \|G\| = 4 Random, Q = 16, \|G\| = NCRs and their approximations for the SF radar pulse trains. Fig. 9 . 9NCRs and their approximations for the FD-MIMO radar waveforms. Fig. 10 . 10The NCRs and their approximations of the airborne FD-MIMO radars with side-looking mode. = 1 , 8 Fig. 11 . 1811\|G\| = 4 Random, Q = 1, \|G\| = 8 Random, Q = 16, \|G\| = 4 Random, Q = 16, \|G\| = 8 Linear, Q = 1, \|G\| = 8 Linear, Q = 16, \|G\| = 4 Linear, Q = 16, \|G\| = Detection probability comparison of FDA radars with different waveform configurations. P fa = 10 −5 . ∀U ⊆ {0, 1, . . . , K m − 1}, and\|U\| = U, where Λ U {V} and Λ U {A} are subsets of U columns in V and A with a same column index set, U. Because (59) equals to that vec{Λ U {V}(diag{c}Λ U {A} H )} = 0, 0 = R{Λ U {V}(diag{c}Λ U {A} H )} ≥ R{Λ U {V}} + R{Λ U {A}} − U, which means R{Λ U {V}} + R{Λ U {A}} ≤ R{V} + R{A} − 1. (60) V ⊆V \ (V ∩Ā), \|V\| = R{V} − \|V ∩Ā\|, A ⊆Ā \ (V ∩Ā), \|Â\| = R{A} − \|V ∩Ā\|, satisfies \|U \| ≤ U, are with the Department of Electronic Engineering, Tsinghua University, Beijing, 100084, China. e-mail: [email protected], [email protected], wangxq [email protected]. A. Nehorai is with the Department of Electrical and Systems Engineering, Washington University, St. Louis, MO 63130 USA. e-mail: [email protected]. The work of Y. Liu was supported by the National Natural Science Foundation of China (Grant No. 61571260 and 61201356). The work of A. Nehorai was supported by the AFSOR(Grant No. FA9550-11-1-0210). Corresponding e-mail: [email protected]. TABLE I GLOSSARY IOF NOTATIONS For conciseness, we call both θ and α = sin θ "direction", because they can be easily distinguished within their context. The phase factor −j4πfcD/c can be regarded as a part of the scatterer's reflection amplitude, because it remains constant w.r.t. different p, q, r, and l. It should be noted that in this work, the clutter suppression is accomplished by coherent processing, thus the results defer from the traditional incoherent cases. In addition, clutter whose delay is larger than one CPI is unconsidered. 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[ "Probing the Higgs Boson via VBF with Single Jet Tagging at the LHC", "Probing the Higgs Boson via VBF with Single Jet Tagging at the LHC" ]	[ "Amanda Kruse \nDepartment of Physics\nUniversity of Wisconsin\n53706MadisonWIUSA\n", "Alan S Cornell \nNational Institute for Theoretical Physics\nSchool of Physics\nUniversity of the Witwatersrand\n2050WitsSouth Africa\n", "Mukesh Kumar \nNational Institute for Theoretical Physics\nSchool of Physics\nUniversity of the Witwatersrand\n2050WitsSouth Africa\n", "Bruce Mellado \nSchool of Physics\nUniversity of the Witwatresrand\nPrivate Bag 32050WitsSouth Africa\n", "Xifeng Ruan \nSchool of Physics\nUniversity of the Witwatresrand\nPrivate Bag 32050WitsSouth Africa\n" ]	[ "Department of Physics\nUniversity of Wisconsin\n53706MadisonWIUSA", "National Institute for Theoretical Physics\nSchool of Physics\nUniversity of the Witwatersrand\n2050WitsSouth Africa", "National Institute for Theoretical Physics\nSchool of Physics\nUniversity of the Witwatersrand\n2050WitsSouth Africa", "School of Physics\nUniversity of the Witwatresrand\nPrivate Bag 32050WitsSouth Africa", "School of Physics\nUniversity of the Witwatresrand\nPrivate Bag 32050WitsSouth Africa" ]	[]	The signature produced by the Standard Model Higgs boson in the Vector Boson Fusion (VBF) mechanism is usually pinpointed by requiring two well separated hadronic jets, one of which (at least) of them tends to be in the forward direction. With the increase of instantaneous luminosity at the LHC, the isolation of the Higgs boson produced with the VBF mechanism is rendered more challenging. In this paper the feasibility of single jet tagging is explored in a high-luminosity scenario. It is demonstrated that the separation in rapidity between the tagging jet and the Higgs boson can be effectively used to isolate the VBF signal. This variable is robust from the experimental and QCD stand points. Single jet tagging allows us to probe the spin-CP quantum numbers of the Higgs boson.	10.1103/physrevd.91.053009	[ "https://arxiv.org/pdf/1412.4710v1.pdf" ]	118,795,729	1412.4710	978c174c59fa337bcda2802c5d9d55aaab52a52b	Probing the Higgs Boson via VBF with Single Jet Tagging at the LHC 15 Dec 2014 Amanda Kruse Department of Physics University of Wisconsin 53706MadisonWIUSA Alan S Cornell National Institute for Theoretical Physics School of Physics University of the Witwatersrand 2050WitsSouth Africa Mukesh Kumar National Institute for Theoretical Physics School of Physics University of the Witwatersrand 2050WitsSouth Africa Bruce Mellado School of Physics University of the Witwatresrand Private Bag 32050WitsSouth Africa Xifeng Ruan School of Physics University of the Witwatresrand Private Bag 32050WitsSouth Africa Probing the Higgs Boson via VBF with Single Jet Tagging at the LHC 15 Dec 2014numbers: 1480Cp1360-r1130Er The signature produced by the Standard Model Higgs boson in the Vector Boson Fusion (VBF) mechanism is usually pinpointed by requiring two well separated hadronic jets, one of which (at least) of them tends to be in the forward direction. With the increase of instantaneous luminosity at the LHC, the isolation of the Higgs boson produced with the VBF mechanism is rendered more challenging. In this paper the feasibility of single jet tagging is explored in a high-luminosity scenario. It is demonstrated that the separation in rapidity between the tagging jet and the Higgs boson can be effectively used to isolate the VBF signal. This variable is robust from the experimental and QCD stand points. Single jet tagging allows us to probe the spin-CP quantum numbers of the Higgs boson. I. INTRODUCTION The discovery of a Higgs boson [1] by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) is a major milestone for the history of particle physics [2,3]. It is also a new opportunity for deeper understanding of the fundamental interactions. A new sector is now available for exploration in the Standard Model (SM) [4] and physics beyond the SM (BSM). With the observation of the Higgs-like particle, the first measurements of the observables sensitive to its couplings to SM particles have become possible. The LHC experiments are expected to collect a sizable amount of Higgs boson candidates in the next few years. Together with searches for additional Higgs-like resonances, the exploration of couplings with increased statistics has become a focus. The exploration of couplings at the LHC suffers from a number of limitations. The measurement of the total width and life time is not possible. The width is too small to be measurable and a number of potential decay products would remain undetectable at the LHC. As a result, the LHC can only measure ratios of couplings in a quasi-model independent way [5]. The isolation of the Higgs boson with the Vector Boson Fusion (VBF) mechanism is of great importance for the exploration of the coupling strength. The VBF mechanism is also critical to exploring the tensor structure of the HV V couplings, where V = W, Z. The isolation of the VBF mechanism with a large signal to background ratio and high purity is reliant on the ability to tag forward hadron jets. The ATLAS and CMS experiments have demonstrated the feasibil-ity of forward jet tagging [34] in the challenging conditions of proton-proton collisions at the LHC. However, with the increase of the instantaneous luminosity, necessary to reach O(100) fb −1 -O(1) ab −1 integrated luminosity, the probability of fake jet tagging increases considerably. This leads experimentalists to increase the transverse momentum (p T ) threshold, resulting in significant loss of signal acceptance. In a recent study for the assessment of the sensitivity to the VBF signal in the High-Luminosity (HL) LHC p T thresholds ranging from 50 GeV to 77 GeV have been considered, depending on the jet rapidity [6]. In this setup the expected accuracy of the VBF signal strength lags behind that of other measurements, such as the W H and ttH production mechanisms. In order to ameliorate this problem it is suggested to revisit some of the ideas pertaining to isolating the Higgs boson with a single jet tag [7][8][9]. Single jet tagging was explored with the intention to identify regions of the phase-space where the Higgs boson could be isolated from non-resonant backgrounds. Here single jet tagging is re-evaluated with the primary intention to separate VBF from the gluon-gluon fusion (ggF) production mechanism. In this paper the rapidity difference between the leading jet and the Higgs boson is considered as a means of achieving the necessary signal to background ratio with single jet tagging. Here it is demonstrated that this observable is robust from the QCD stand point for both VBF production and the ggF mechanisms. A perturbative analysis is performed to understand the stability of this observable against scale variations. Effects related to multiple soft gluon radiation are also investigated. The discriminating power of the observable studied here is evaluated in the context of the di-photon decay channel. The ability to study the spin-CP quantum numbers of the Higgs boson in the presence of a single jet tag is discussed. The article is organized as follows: Section II gives a brief overview of the Higgs boson production in association with high p T hadronic jets; Section III discusses ratios relevant to the H + 1j final state; Section IV gives a brief account of the tools used; Section V reports the perturbative analysis of the observable under study; Section VI quantifies the discriminating power of the observable under study; Section VII discusses the ability to probe the spin-CP quantum numbers of the Higgs boson produced via VBF with single jet tagging; Section VIII summarizes the conclusions of the paper. II. THE HIGGS BOSON AND JET PRODUCTION The phenomenology pertaining to the production of the Higgs boson at hadron colliders is vast and well understood [10]. The leading production mechanism for Higgs bosons in association with high p T hadronic jets is the ggF mechanism, which occurs via a quark loop. In this process the production of jets involve radiative corrections. In the limit that the top quark is very heavy, the cross-section can be computed via an effective Lagrangian (see Ref. [11] and references therein) as: L ef f = − 1 4 AΦG A µν G A,µν ,(1) where Φ stands for the scalar Higgs boson field and G A µν is the field strength of the SU(3) color gluon field. The effective coupling A = α s /(3πν), where ν = (G F √ 2) −1 = (246 GeV) 2 is the vacuum expectation value (VEV). The effective Lagrangian generates vertices leading to the production of the Higgs boson in association with gluons (the Feynman rules can be found in Ref. [12], for instance). The leading process for the production of H + 1j emerges mainly from the partonic process gg → gH. (2) The cross-section for H + 1j is known at α 4 s [11,[13][14][15]. The production of H + 1j from gg → H + j is known at α 5 s [16]. Scale-driven variations of the cross-section are typically calculated by taking the largest variations by changing the renormalization (µ R ) and factorization (µ F ) scales by factors of two. In this setup the crosssection varies within 20% in a wide range of the p T of the leading parton relevant to Higgs boson searches at the LHC. The cross-section variation obtained by setting up the nominal scales to the Higgs boson mass, or to a dynamic choice of the Higgs boson transverse energy, are very similar. Using the effective Lagrangian approach, significant differences in the radiation patterns are observed with respect to Drell-Yan production [7,8]. The leading and subleading partonic processes for the production of H + 2j with ggF at the LHC are gg → ggH, qg → qgH. ( The cross-secton for H + 2j with the ggF production mechanism is known at α 5 s [17]. The lower order amplitudes for H + 1j and H + 2j scattering are available exactly, without the use of the effective coupling approach. These calculations are quite complex and one does not expect that higher order corrections in QCD will be calculated for the exact top mass. It needs to be argued that the two loop calculations performed for H + 1j and H + 2j scattering are valid for the Higgs mass, m H , and transverse momentum, p T H , smaller than the top mass. The Higgs boson production via the VBF is a subleading process that provides high p T hadronic jets at leading order (LO). The impact of the QCD higher order corrections on the production cross-section and the jet kinematics are known to be small. In order to appreciate the unique kinematics of the VBF process it is most intuitive to express the cross-section in a factorized form. Consider a fermion f of a center-of-mass (c.m.) energy E radiating a gauge boson V (s ≫ M 2 V ), the cross-section of the scattering f a → f ′ X via V exchange can be expressed as: σ(f a → f ′ X) ≈ dx dp 2 T P V /f (x, p 2 T ) σ(V a → X),(4) where σ(V a → X) is the cross-section of the V a → X scattering and P V /f can be viewed as the probability distribution for a weak boson V of energy xE and transverse momentum p T . The dominant kinematical feature is a nearly collinear radiation of V off f , often called the "Effective W -Approximation" (see Ref. [18] and references therein) when the center of mass energy is much greater than the mass of the weak bosons, the probability distributions of the weak bosons with different polarizations can be approximated by: P T V /f (x, p 2 T ) ∝ 1 + (1 − x) 2 x p 2 T (p 2 T + (1 − x)M 2 V ) 2 (5) P L V /f (x, p 2 T ) ∝ 1 − x x (1 − x)M 2 V (p 2 T + (1 − x)M 2 V ) 2 .(6) These expressions lead to the following observations: 1 Unlike the QCD partons that scale like 1/p 2 T at the low transverse momentum, the final state quark f ′ typically has p T ∼ √ 1 − xM V ≤ M W . 2 Due to the 1/x behavior for the gauge boson distribution, the out-going parton energy (1 − x) E tends to be high. Consequently, it leads to an energetic forward jet with small, but finite, angle with respect to the beam. 3 At high p T , P T V /f ∼ 1/p 2 T and P L V /f ∼ 1/p 4 T , and thus the contribution from the longitudinally polarized gauge bosons is relatively suppressed at high p T to that of the transversely polarized. In conclusion, the production of jets in association with the Higgs boson displays significant differences with respect to the production of jets in association with other particles in the SM. These differences are exploited when exploring the phase-space to isolate the Higgs boson signal. These features are also prominent in the production of the Higgs boson in association with one high transverse momentum jet. III. PRODUCTION MECHANISM RATIOS AND SINGLE JET TAGGING In this section the role of single jet tagging for the exploration of some of the properties of the Higgs boson is discussed. Given the limitations imposed by the inability to measure branching ratios in proton-proton collisions, it is convenient to define appropriate ratios. By defining ratios, where the rate of the Higgs boson decaying into the same flavor of particles is considered, uncertainties related to the total decay width cancel out. The ggF, VBF and VH production mechanisms are sensitive to different couplings. In searching for physics beyond one can consider two groups of ratios: • Ratios of rates of the same decay modes involving the production of ggF to VBF. If the VBF signal is isolated with the help of the H + 2j category, then we encounter a difficulty. QCD-related uncertainties of the contamination of the ggF process in the phase-space of the H + 2j category, used for the isolation of the VBF mechanism, would not cancel out. This would lead to approximately 15% of theoretical uncertainty on the ratio. To estimate it one would need to add experimental uncertainties, which are significant here too. In this paper a ratio based on the H + 1j final state is suggested instead, as a means to secure strong cancelation of these effects. • Ratios of rates of the same decay modes involving the production of ggF to V H are used. The isolation of V H with a dedicated H + 2j category is hindered by the large contamination from the ggF mechanism. In order to effectively pursue a similar approach as suggested in Ref. [19], where the di-jet system is required to be boosted. This ratio would still suffer from similar theoretical uncertainties, as in the case discussed above. Another important ratio emerges from final states with leptons. Despite the reduced rate, the ratio involving leptons provides for an excellent opportunity to isolate the V H production mechanism without concerns about contamination from the ggF mechanism. Based on this discussion let's consider the following experimental ratio: R ggF V BF (1j) = g 2 + V 2 g 1 + V 1 = ξ g (1j)g 1 + V 2 g 1 + V 1 ≈ ξ g (1j) + V 2 g 1 ,(7) where g 1 (V 1 ) and g 2 (V 2 ) correspond to the rate of the Higgs boson via the ggF (VBF) mechanism in the region of the phase-space enriched with the ggF (VBF) mechanism. Here g 1 is the experimental measurement of the rate of the ggF+1j, whereas g 2 and V 2 would be estimates extracted with the procedure.The region of the phase-space where the ggF mechanism dominates over the VBF is where QCD-like radiation patterns are characteristic. Theory uncertainties from ξ g (1j) = g 2 /g 1 and V 2 would need to be considered. The dominant theory uncertainty would emerge from the QCD uncertainties of the rate of ggF+1j, whereas experimental uncertainties would tend to cancel out. It is important to note that the ratio R ggF V BF (1j) is robust against pile-up effects. The ratio V 1 /g 1 is expected to be small, hence the approximation in Eq. (7). The following expression is used for the relative uncertainty of the extraction of the VBF signal in the H + 1j category: g 2 + V 2 ⊕ g1ξ g (1j) ⊕ δξ g (1j)g 1 /V 2 ,(8) where δξ g (1j) is the scale variation obtained with the next-to-leading-order (NLO) matrix element of the ggF+1j process. The first two terms in Eq.(8) are related to the statistical error of the measurement. It is found that the best discriminator to disentangle the ggF and VBF processes is the rapidity difference between the Higgs boson and the leading jet, ∆y Hj ; the separation in rapidity between the Higgs boson and the leading jet. For a quantitative statement see Sec. VI. This approach is valid for the extraction of the VBF signal that later can be related to the quasi-inclusive rate of ggF, g 0 : R ggF V BF = g 2 + V 2 g 0 + V 0 = ξ g (1j)g 1 + V 2 g 0 + V 0 ≈ ξ g (1j) g 1 g 0 + V 2 g 0 ,(9) where g 0 = g incl − g 1 and g incl would be the total inclusive cross-section for the ggF mechanism. The dominant theoretical error in this case would be the QCD uncertainty in the total cross-section of the ggF mechanism. In this approach the theory uncertainty on the ggF+1j rate would cancel out, except for its contribution to g 0 , which is small. Experimental uncertainties related to hadronic jet reconstruction would not cancel out in this approach. That said, since the VBF signal is extracted with a H +1j category, these uncertainties are not expected to be large. Third approach would be to use the H + 2j category for the extraction of the VBF signal. Here g 2 = ξ g (2j)g ′ 2 , where g ′ 2 would be measured in the QCD-like region. This can be achieved by applying requirements on the rapidity difference between the tagging jets (∆y jj ) in order to define a region with dominant ggF contribution and another with dominant VBF contribution. IV. SETUP AND TOOLS Monte-Carlo events were generated for the two production modes: ggF and VBF. Two versions of the MINLO [20,21] generator were used for the production of ggF+jets: HJ and HJJ. The first incorporates NLO matrix elements up to one parton, whereas the second is up to two partons. The VBF events were also produced with POWHEG [22] and are also accurate at NLO. Both samples were produced for a Higgs mass of 126.8 GeV using the CT10 parton distribution functions at NLO [23]. The samples were then interfaced with Pythia8 [24] which adds the showering and hadronization of the events as well as the underlying event. Within Pythia8, stable particles are clustered into jets using the anti-k t algorithm [25] with a cone size of ∆R = 0.4. A number of fiducial cuts were applied to the samples. The p T of the leading (subleading) photon is required to be greater than 40 (30) GeV and within a rapidity \|y\| < 2.4. In addition, the photons are isolated, which is accomplished by requiring the amount of transverse energy within ∆R = 0.4 to be less than 14 GeV. Jets are then required to have p T > 30 GeV and to be within a rapidity \|y\| < 4.4. An overlap removal is applied on jets, where any jet within ∆R < 0.4 of a photon is removed, and any jet within ∆R < 0.2 of an electron is removed. The package MCFM [17,26] is used for the evaluation of the scale uncertainties of ggF+jets and VBF in the cor-ners of the phase-space of interest here (see Section V). It is worth noting that while MCFM is a parton-level generator, studies performed with POWHEG and MINLO are at particle level. V. PERTURBATIVE ANALYSIS The scale variations of ggF+1j and VBF are evaluated with MCFM at NLO as a function of the rapidity difference between a Higgs boson and the leading jet at parton level. The K-factors [35] for ggF+1j are remarkably flat up to ∆y Hj ≈ 5, beyond which statistical fluctuations become a limiting factor. The scale variations are evaluated for the renormalization and factorization scales both at the same time, and separately. The size of the scale variations is also stable for ∆y Hj < 5. Scale variations for the VBF process are well behaved. However, unlike the ggF+1j case, the K-factors are not flat with ∆y Hj . The K-factors behave almost linearly with ∆y Hj ranging from 0.9 at ∆y Hj ≈ 0 to 1.35 at ∆y Hj = 7. It is important to note that the observable ∆y Hj displays similar features to the invariant mass of the Higgs boson and the leading jet in terms of the flatness scale uncertainties. That said, experimentally, ∆y Hj is robust with respect to hadronic energy scale uncertainties. Figure 1 displays the ∆y Hj distribution for the VBF and ggF+1j production mechanisms at particle level (see Section IV) . Changes in the differential cross-sections due to scale variations are shown in the form of bands around the central values. The scale variations for VBF are well behaved, as expected. The situation with ggF+jets requires some discussion. At low values of ∆y Hj the cross-section variations due to scale variations are larger for HJ than for HJJ, which is expected. However for ∆y Hj > 3.5 the cross-section variations are larger for HJ than for HJJ. This seems an indication that the calculation may not be particularly reliable for large values of ∆y Hj . Fortunately, this region of the phase-space does not play a critical role in the separation between ggF and VBF. In Section VI it will be seen that a cut of ∆y Hj > 1.4 is an optimal requirement to separate ggF and VBF. This requirement is far enough from what seems to be a problematic region. To obtain an estimate of the scale uncertainty for ggF events which fall into the H + 1j category, the ratio of events with ∆y Hj > 1.4 to events with ∆y Hj < 1.4 is studied. The scale uncertainties for the ggF cross-section were found by varying the factorization and renormalization scales up and down by a factor of 2. This uncertainty on the cross-section was then propagated to the ∆y Hj ratio of events, and was found to be 6.5%. VI. DISCRIMINATION In this Section a qualitative statement is made about the relevance of ∆y Hj as a discriminator to extract the VBF signal. A generic corner of the phase-space is identified in order to evaluate loss of VBF signal acceptance as a result of the increase of the jet p T thresholds imposed by the pile-up conditions. The region defined by the following requirements assumes the presence of two hadronic jets with p T > 30 GeV, and in the pseudorapidity range \|η j \| < 4.5: pseudorapidity separation between the tagging jets ∆η jj > 2.8, η γγ − ηj1+ηj2 2 < 2.4, where the indices indicate the object for which pseudorapidity is calculated, the azimuthal angle difference between the system of the tagging jets and the di-photon system, ∆φ γγ,jj > 2.6 rad, and the di-jet invariant mass, m jj > 560 GeV. This region is best suited for the extraction of the VBF signal in the presence of two high p T jets. Two classes of Higgs boson events are identified: double tag, or events that pass the requirements specified above; single tag, or events that fall outside the region but that display a jet in the event with ∆y Hj above a certain threshold. Events classified as single tag appear in a region of the phase-space currently not explored for the extraction of the VBF signal by the ATLAS and CMS experiments. It is important to evaluate the evolution of the Higgs boson signal cross-section with the jet p T threshold and its correlation with the ∆y Hj . Figure 2 shows the dependence of the effective cross-section for the ggF+jets (upper plot) VBF (lower plot) and processes as a function of the sub-leading jet p T threshold, p T j2 and ∆y Hj for double tag events. The correlation between ∆y Hj and p T j2 is significantly different for both processes: whereas ∆y Hj decreases with the p T j2 threshold for ggF+jets, one observes a weak correlation in VBF. This is an important feature for the effectiveness of ∆y Hj as a discriminator to extract the VBF signal: as the jet threshold increases the separation becomes stronger. Table I displays the results of a one-dimensional optimization using ∆y Hj as a discriminator, for different values of the jet p T threshold. The value of the threshold on ∆y Hj depends little on the p T threshold and it is fixed at ∆y Hj > 1.4. Shown are the signal VBF and ggF+1j background rates and the expected accuracy on the measurement of the VBF signal strength. For the evaluation of the latter a 5% uncertainty on the ggF+1j background extraction is assumed (see Section III for a discussion on the subject). The optimal value of the threshold of ∆y Hj depends weakly on the jet p T threshold. For thresholds on p T j2 above 50 GeV the sensitivity of the single tag category becomes dominant. The signal rate for both VBF and ggF+jets mechanisms for double tag events decrease rapidly with the jet p T threshold. This effect is further quantified in Tab. I, where event yields for the VBF signal and ggF+jets rates for 300 fb −1 integrated luminosity are given as a function of the jet p T threshold. When shifting the threshold from 30 GeV to 55 GeV, the rate of VBF signals classified as double tag events drops by a factor of two. This effect seriously affects the sensitivity of the experiments to the extraction of the VBF in high instantaneous luminosity scenarios. The mild increase in the VBF signal to ggF+jets rate does not compensate the strong loss of VBF signal. It is important to note that the effects discussed here are only applicable to the SM (see Section VII). Table I also displays the expected yield for the VBF production mechanism using single tag events. The rate of this class of events evolves with the increase of the p T threshold as a result of two competing effects: increase of yield that do not pass the double tag requirements and the decrease of yield because of increase of threshold on the p T of the leading jet. This leads to a significantly milder decrease in the yield with the p T threshold for single tag events compared to that of double tag events. The rate of VBF to ggF yields is significantly poorer for single tag events compared to double tag events. This is partially alleviated by the large VBF signal yield produced by the single tag category. A study of the ∆y Hj spectrum displayed by the diphoton non-resonant production was studied with the SHERPA package [27]. The shape of the ∆y Hj spectrum follows closely that of ggF+jets. VII. EXPLORATION OF SPIN-CP QUANTUM NUMBERS In Ref. [28] it was suggested to explore the spin-CP quantum numbers of the Higgs boson in VBF via the study of the azimuthal angle correlation of the scattered quarks. Experimentally this implies reconstructing two well separated hadronic jets. It is difficult to gain indirect access to this observable in the final state considered here. The azimuthal angle separation between the Higgs boson and the leading jet does not have sufficient sensitivity to the information of interest. In Refs. [29,30] it was pointed out that the tensor structure of the HV V vertex (V = Z, W ) manifests itself through other observables in addition to the one considered in Ref. [28]. The sensitivity to new physics in the HV V couplings in the ∆y Hj distribution is evaluated here. In the SM, the couplings of the Higgs boson to the massive electroweak gauge bosons are precisely formulated and come out as g HV V ∝ gM V V µ V µ where g is the SU(2) coupling constant. However, this is not the most general form of the Higgs-gauge boson vertex. Parametrising the coupling of a scalar state to two vector bosons in the form iΓ µν (p, q)ǫ µ (p)ǫ * ν (q), one can write down the most general form of the HV V vertex as Γ µν (p, q) = Γ SM µν +Γ BSM µν (p, q), with the SM and the beyond SM components given by: Γ SM µν = −gM V g µν ,(10)Γ BSM µν (p, q) = g M V [λ (p · q g µν − p ν q µ ) + λ ′ ǫ µνρσ p ρ q σ ] ,(11) where λ and λ ′ are effective coupling strengths, respectively for higher dimension CP-even and CP-odd operators, and we will assume that they are the same for W and Z bosons. These operators may be generated within the SM at higher orders of perturbation theory, although the resulting couplings are likely to be very small. In general, λ and λ ′ can be treated as momentum dependent form factors that may also be complex valued. However, we take the approach that BSM vertices can be generated from an effective Lagrangian, which treats λ and λ ′ as coupling constants [28]. The most striking difference between the SM and BSM vertices of Eqs. (10) and (11) is that the latter has an explicit dependence on the momentum of the gauge bosons. It is this feature that is the source of the differences that the BSM vertices generate in the kinematic distributions of tagging jets in the VBF and V H processes, compared to the SM case. In our analysis, the vertices for the Lagrangians in the SM and in BSM with spin-0 bosons are calculated in FeynRules [31] and passed to the event-generator MadGraph [32], which is used for the generation of the matrix elements for Higgs production in VBF. To obtain the cross-sections and distributions at parton-level, the CTEQ6L1 parton distribution functions are used [33]. The factorization and re-normalization scales are set on an event-by-event basis to the transverse energy of the Higgs boson. For the selection cuts, partons are required to have transverse momentum p T > 10 GeV, rapidity \|y\| < 5 and be separated by ∆R > 0.4. Figure 3 displays the rapidity separation between the Higgs boson and the leading parton in the event. Results are shown at parton level. The solid black curve corresponds to the SM case, when λ = λ ′ = 0. The dotted and dashed line include admixtures of the SM and BSM contributions with λ ′ = 0 and λ = 1, 0.5, respectively. As pointed out in Refs. [29,30], the BSM vertexes in Eq.(11) introduce dependence on the particle momenta. This feature distorts the kinematics of the scattered quarks with respect to the prediction of the SM. One of the relevant effects is the reduction of the rapidity separation between the scattered quarks. Figure 3 illustrates the effect on the rapidity separation between the Higgs boson and the leading jet. With the inclusion of spin-0 + BSM admixtures, the ∆y Hj distribution is pushed towards lower values. The jet transverse momentum distribution is also a potential discriminant to explore the tensor structure of the HV V coupling. VIII. CONCLUSIONS With the increase of the number of soft proton-proton collisions at the LHC, the probablity for fake forward jets will increase significantly. As a result jet transverse momentum thresholds will need to be increased, strongly reducing the phase-space to isolate the Higgs boson produced with the VBF mechanism using two well separated hadronic jets. The prospects of isolating the VBF mechanism with single jet tagging is explored here by using the difference in rapidity between the leading jet and the Higgs boson as a discriminator. It is demonstrated that this observable is robust from the QCD standpoint for both the VBF and ggF production mechanisms. For thresholds of the jet transverse momenta greater than 50 GeV, the sensitivity to the VBF mechanism of the single tag final state may become dominant. The combination of the single and double tagged final states provides enhanced stability of the measurement of the Higgs boson rate produced via VBF against stringent pileup conditions. The exploration of the Higgs boson spin-CP quantum numbers via VBF is not only possible with double jet tagging. Here it is demonstrated that the spin-CP quantum numbers can also be explored with the VBF mechanism using single jet tagging. [1] F. Englert FIG. 1 : 1Distributions of the rapidity separation between the Higgs boson and the tagging jet. The VBF and ggF+1j processes are described with POWHEG and MINLO, respectively. The ggF+1j production mechanism is described with the HJ and HJJ versions of MINLO (see text). Variations in the differential cross-sections due to scale variations are shown in the form of bands around the central values. FIG. 2 : 2Effective cross-section (in pb) of ggF+jets and VBF production as a function of the jet pT and ∆yHj thresholds for double tag events with the di-photon decay channel. The upper and lower plots correspond to ggF+jets and VBF production, respectively. j S ggF S/ggF ∆µ S ggF S/ggF ∆µ ∆µ T ot FIG. 3 : 3Distribution of the rapidity separation between the Higgs boson and the leading jet in VBF. Results are shown at parton level for the SM case (λ = 0) and non-zero BSM contributions (see text). TABLE I : IVBF signal and ggF+jets rates for 300 fb −1 inte- grated luminosity with the di-photon decay channel Thresh- olds on the jet pT are given in GeV. Results for different values of the threshold on pT j2 are given and are obtained for an optimal requirement of ∆yHj > 1.4. 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[ "Rotation induced superfluid-normal phase separation in trapped Fermi gases", "Rotation induced superfluid-normal phase separation in trapped Fermi gases" ]	[ "M Iskin \nJoint Quantum Institute\nNational Institute of Standards and Technology\nand University of Maryland\n20899-8423GaithersburgMarylandUSA\n", "E Tiesinga \nJoint Quantum Institute\nNational Institute of Standards and Technology\nand University of Maryland\n20899-8423GaithersburgMarylandUSA\n" ]	[ "Joint Quantum Institute\nNational Institute of Standards and Technology\nand University of Maryland\n20899-8423GaithersburgMarylandUSA", "Joint Quantum Institute\nNational Institute of Standards and Technology\nand University of Maryland\n20899-8423GaithersburgMarylandUSA" ]	[]	We use the Bogoliubov-de Gennes formalism to analyze the effects of rotation on the ground state phases of harmonically trapped Fermi gases, under the assumption that quantized vortices are not excited. We find that the rotation breaks Cooper pairs that are located near the trap edge, and that this leads to a phase separation between the nonrotating superfluid (fully paired) atoms located around the trap center and the rigidly rotating normal (nonpaired) atoms located towards the trap edge, with a coexistence (partially paired) region in between. Furthermore, we show that the superfluid phase that occurs in the coexistence region is characterized by a gapless excitation spectrum, and that it is distinct from the gapped phase that occurs near the trap center.	10.1103/physreva.79.053621	[ "https://arxiv.org/pdf/0811.3010v2.pdf" ]	119,205,841	0811.3010	d6bf87f2439b8e83397aa1bcfd45bd33341d2fe0	Rotation induced superfluid-normal phase separation in trapped Fermi gases 21 Jan 2009 M Iskin Joint Quantum Institute National Institute of Standards and Technology and University of Maryland 20899-8423GaithersburgMarylandUSA E Tiesinga Joint Quantum Institute National Institute of Standards and Technology and University of Maryland 20899-8423GaithersburgMarylandUSA Rotation induced superfluid-normal phase separation in trapped Fermi gases 21 Jan 2009(Dated: January 21, 2009) We use the Bogoliubov-de Gennes formalism to analyze the effects of rotation on the ground state phases of harmonically trapped Fermi gases, under the assumption that quantized vortices are not excited. We find that the rotation breaks Cooper pairs that are located near the trap edge, and that this leads to a phase separation between the nonrotating superfluid (fully paired) atoms located around the trap center and the rigidly rotating normal (nonpaired) atoms located towards the trap edge, with a coexistence (partially paired) region in between. Furthermore, we show that the superfluid phase that occurs in the coexistence region is characterized by a gapless excitation spectrum, and that it is distinct from the gapped phase that occurs near the trap center. With the ultimate success of techniques for trapping and cooling atomic gases developed and improved gradually since the 1980s, atomic Fermi gases have emerged as unique testing grounds for many theories of exotic matter in nature, allowing for the creation of complex yet very accessible and controllable many-body quantum systems. For instance, evolution from the weakly attracting Bardeen-Cooper-Schrieffer (BCS) limit to the weakly repulsive molecular Bose-Einstein condensation (BEC) have been observed in a series of remarkable experiments. The ground state phases of such atomic Fermi gases have since been the subject of intense theoretical and experimental research worldwide [1,2]. To verify the superfluid ground state of atomic Fermi gases, it is essential to analyze their response to rotation. For instance, it has been theoretically predicted that sufficiently fast rotation of superfluid Fermi gases excites quantized vortices in the form of hexagonal vortex lattices [3,4,5]. Such vortex lattices have recently been observed for both population balanced [6] and imbalanced [7] systems. These vortex experiments have not only complemented previously found signatures, but also provided the ultimate evidence to support the superfluid nature of the ground state. Recently, the effects of rotation on the ground state phases of harmonically trapped Fermi gases has been theoretically studied at unitarity, under the assumption that quantized vortices are not excited [8] (see also Ref. [9]). It has been argued that the rotation causes a complete phase separation between a nonrotating superfluid core and a rigidly rotating normal gas, with a discontinuous density at the interface. Subsequently, this problem has been analyzed using the mean-field BCS framework within the semi-classical local density approximation (LDA) [10]. In addition to the superfluid and normal regions, a coexistence region is found, the possibility of which is not considered in Ref. [8]. In this manuscrript, we go beyond the semi-classical LDA and develop a quantum mechanical Bogoliubov-de Gennes (BdG) formalism to analyze the effects of rotation on the ground state phases of harmonically trapped Fermi gases. We discuss both population balanced and imbalanced mixtures throughout the BCS-BEC evolution. Our main results are in qualitative agreement with those of Ref. [10]. We find that the rotation breaks Cooper pairs that are located near the trap edge, and that this leads to a phase separation between the nonrotating superfluid (fully paired) atoms located around the trap center and the rigidly rotating normal (nonpaired) atoms located towards the trap edge, with a coexisting (partially paired) region in between. This leads to a continuous density and superfluid order parameter as a function of radial distance. Furthermore, we show that the superfluid phase that occurs in the coexistence region is characterized by a gapless excitation spectrum, and that it is distinct from the gapped phase that occurs near the trap center. We obtain these results by solving the mean-field BdG equations in the rotating frame (in units of = k B = 1) K ↑ (r) − ΩL z ∆(r) ∆ * (r) −K * ↓ (r) + ΩL * z u η (r) v η (r) = ǫ η u η (r) v η (r) ,(1)where K σ (r) = −∇ 2 /(2M ) − µ σ (r) , Ω is the rotation frequency around the z-axis of the trapping potential, L z is the z-component of the angular momentum operator, and the off-diagonal self-consistency field ∆(r) = g ψ ↑ (r)ψ ↓ (r) is the local superfluid order parameter. Here, σ ≡ {↑, ↓} labels the trapped hyperfine states, M is the mass, µ σ (r) = µ σ − V (r) is the local chemical potential, µ σ is the global chemical potential, V (r) = M ω 2 r 2 /2 is the trapping potential which is assumed to be spherically symmetric, ω is the trapping frequency, g > 0 is the strength of the zeroranged attractive interactions between ↑ and ↓ atoms, and ... is a thermal average. The quasiparticle wavefunctions u η (r) and v η (r) are related to the particle annihilation operator ψ σ (r) via the Bogoliubov-Valatin transformation ψ σ (r) = η [u η,σ (r)γ η,σ − s σ v * η,σ (r)γ † η,−σ ], where γ † η,σ and γ η,σ are the quasiparticle creation and annihilation operators, respectively, and s ↑ = +1 and s ↓ = −1. Since the BdG equations are invariant under the transformation v η,↑ (r) → u * η,↑ (r), u η,↓ (r) → −v * η,↓ (r) and ǫ η,↓ → −ǫ η,↑ , it is sufficient to solve only for u η (r) ≡ u η,↑ (r), v η (r) ≡ v η,↓ (r) and ǫ η ≡ ǫ η,↑ as long as we keep all of the solutions with positive and negative eigenvalues. We assume ∆(r) = −g η u η (r)v * η (r)f (ǫ η ) is real without loosing generality, where f (x) = 1/[exp(x/T ) + 1] is the Fermi function and T is the temperature. Furthermore, we can relate g to the two-body scattering length a F via 1/g = −M/(4πa F ) + M k c (r)/(2π 2 ) where k 2 c (r) = 2M [ǫ c + µ(r)] and µ(r) = [µ ↑ (r) + µ ↓ (r)]/2. Here, ǫ c is the energy cutoff to be specified below, and our results depend weakly on the particular value of ǫ c provided that it is chosen sufficiently high. The order parameter equation has to be solved self-consistently with the number equation N σ = drn σ (r), where n σ (r) = ψ † σ (r)ψ σ (r) is the local density of fermions, leading to n ↑ (r) = η \|u η (r)\| 2 f (ǫ η ) and n ↓ (r) = η \|v η (r)\| 2 f (−ǫ η ). Next, we expand u η (r) and v η (r) in the complete basis of the harmonic trapping potential eigenfunctions given by K σ (r)φ n,ℓ,m (r) = ξ σ n,ℓ φ n,ℓ,m (r), where ξ σ n,ℓ = ω(2n + ℓ + 3/2) − µ σ is the eigenvalue and φ n,ℓ,m (r) = R n,ℓ (r)Y ℓ,m (θ r , ϕ r ) is the eigenfunction. Here, n is the radial quantum number, and ℓ and m are the orbital angular momentum and its projection, respectively. The angular part Y ℓ,m (θ r , ϕ r ) is a spherical harmonic and the radial part is R n,ℓ (r) = √ 2(M ω) 3/4 [n!/(n + ℓ + 1/2)!] 1/2 e −r 2 /2rℓ L ℓ+1/2 n (r 2 ), wherer = √ M ωr is dimen- sionless and L j i (x) is an associated Laguerre polynomial. We choose L z = −i∂/∂ϕ r and η ≡ {ℓ, m, γ}, leading to u ℓ,m,γ (r) = n c ℓ,m,γ,n φ n,ℓ,m (r) and v ℓ,m,γ (r) = n d ℓ,m,γ,n φ n,ℓ,m (r). This expansion reduces the BdG equations to a 2(n ℓ + 1) × 2(n ℓ + 1) matrix eigenvalue problem for a given {ℓ, m} state n ′ K n,n ′ ↑,ℓ − mΩδ n,n ′ ∆ n,n ′ ℓ ∆ n ′ ,n ℓ −K n,n ′ ↓,ℓ − mΩδ n,n ′ c ℓ,m,γ,n ′ d ℓ,m,γ,n ′ = ǫ ℓ,m,γ c ℓ,m,γ,n d ℓ,m,γ,n .(2) Here, n ℓ = (n c − ℓ)/2 is the maximal radial quantum number and n c is the radial quantum number cutoff, such that we include only the single particle states with ω(2n + ℓ + 3/2) ≤ ǫ c = ω(n c + 3/2). In Eq. 2, the diagonal matrix element is K n,n ′ σ,ℓ = ξ σ n,ℓ δ n,n ′ where δ i,j is the Kronecker delta, and the off-diagonal matrix element is ∆ n,n ′ ℓ ≈ r 2 dr∆(r)R n,ℓ (r)R n ′ ,ℓ (r). Although ∆(r) becomes axially symmetric when Ω = 0, it is convenient to define ∆(r) = dΩ r ∆(r)/(4π) leading to ∆(r) = − g 4π ℓ,m,γ,n,n ′ R ↑ ℓ,m,γ,n (r) R ↓ ℓ,m,γ,n ′ (r)f (ǫ ℓ,m,γ ),(3) where we introduced R ↑ ℓ,m,γ,n (r) = c ℓ,m,γ,n R n,ℓ (r) and R ↓ ℓ,m,γ,n (r) = d ℓ,m,γ,n R n,ℓ (r). Similarly, we define n σ (r) = dΩ r n σ (r)/(4π) leading to n σ (r) = 1 4π ℓ,m,γ,n,n ′ R σ ℓ,m,γ,n (r) R σ ℓ,m,γ,n ′ (r)f (s σ ǫ ℓ,m,γ ). (4) Lastly, N σ reduces to N ↑ = ℓ,m,γ,n c 2 ℓ,m,γ,n f (ǫ ℓ,m,γ ) and N ↓ = ℓ,m,γ,n d 2 ℓ,m,γ,n f (−ǫ ℓ,m,γ ). When Ω → 0, the eigenfunction coefficients c ℓ,m,γ,n and d ℓ,m,γ,n and the eigenvalues ǫ ℓ,m,γ become independent of m, and Eqs. (2), (3) and (4) reduce to the usual ones, see e.g. Ref. [11,12,13,14]. Therefore, due to the coupling between different m states, the rotating case is numerically much more involved compared to the nonrotating case. In addition to n σ (r), we want to calculate the local density of normal (rotating) fermions n σ,N (r). For this purpose, we use the local current density J σ (r) = J σ (r) , where J σ (r) = [1/(2M i)][ψ † σ (r)∇ψ σ (r) − H.c.] is the quantum mechanical probability current operator and H.c. is the Hermitian conjugate. This leads to J ↑ (r) = [1/(2M i)] η [u * η (r)∇u η (r)f (ǫ η ) − H.c.] and J ↓ (r) = [1/(2M i)] η [v η (r)∇v * η (r)f (−ǫ η ) − H.c.]. The current, similar to the classical case, can be written as J σ (r) = n σ,N (r)v(r), where n σ,N (r) is the local density and v(r) = Ω z × r is the local velocity of normal fermions corresponding to a rigid-body rotation. Since the normal fermions are expelled towards the trap edge, we approximate n σ,N (r) = dΩ r n σ,N (r)/(4π) as n σ,N (r) ≈ s σ 4πM Ωr 2 ℓ,m,γ,n,n ′ m R σ ℓ,m,γ,n (r) R σ ℓ,m,γ,n ′ (r)f (s σ ǫ ℓ,m,γ ),(5) such that J σ (r) = dΩ r J σ (r)/(4π) ∼ Ωrn σ,N (r). Having discussed the BdG formalism, next, we analyze the ground state (T = 0) phases for both population balanced (N ↑ = N ↓ or µ ↑ = µ ↓ ) and imbalanced (N ↑ = N ↓ or µ ↑ = µ ↓ ) Fermi gases. This is achieved by solving the BdG equations (2), (3) and (4) self-consistently as a function of the dimensionless parameter 1/(k F a F ) where k F is the Fermi momentum defined via the Fermi energy (6) and N = N ↑ + N ↓ = (n F + 1)(n F + 2)(n F + 3)/3. Here, n F and r F are the corresponding Fermi level and Thomas-Fermi radius, respectively. In our numerical calculations, we choose n F = 15 and n c = 65, which corresponds to a total of N = 1632 fermions and ǫ c ≈ 4ǫ F , respectively. Here, it is important to emphasize that we do not expect any qualitative change in our results with higher values of n F and/or n c , except for minor quantitative variations. In Fig. 1, we consider a weakly interacting Fermi gas on the BCS side with 1/(k F a F ) = −0.5, and show n σ (r) and ∆(r) for nonrotating (Ω = 0) and rotating (Ω = 0.5ω) cases. When N ↑ = N ↓ , we find that ∆(r) depletes everywhere inside the trap and especially around the trap edge. This is because it is easier to break the Cooper pairs that are located towards the trap edge in comparison to the ones that occupy the center (see below). Our result is quantitatively different from the LDA one where ∆(r) depletes only around the trap edge [10]. When N ↑ = 3N ↓ , in addition to such an effect, the spatial modulation of ∆(r) disappears in the rotating case as shown in Fig. 1(b). The depletion of ∆(r) leads to a decrease (increase) in n σ (r) around the trap center (edge) due to the centrifugal force caused by the rotation. It also leads to a phase separation between the nonrotating fully paired superfluid (FPS) atoms located around the trap center and the rigidly rotating normal (nonpaired) ones located towards the trap edge, with a coexistence region in between, which is in agreement with the LDA result [10]. We characterize the FPS, coexistence and normal phases by n σ (r) ≫ n σ,N (r) = 0, n σ (r) n σ,N (r) = 0 and n σ (r) = n σ,N (r) = 0, respectively. Notice that ∆(r) decreases smoothly as a function of r from the FPS to the normal region where it vanishes, which is in contrast with the LDA result where ∆(r) is nonanalytic at the transition [10]. Since ∆(r) is finite in the coexistence region, this region corresponds to a partially paired superfluid (PPS), and it occupies a larger region compared to the LDA resuls [10]. In addition, the trap center becomes a PPS for Ω 0.5ω when 1/(k F a F ) = −0.5. ǫ F = ω(n F + 3/2) = k 2 F 2M = 1 2 M ω 2 r 2 F ≈ ω(3N ) 1/3 In Fig. 2, we consider a strongly interacting Fermi gas at unitarity with 1/(k F a F ) = 0, and show n σ (r) and ∆(r) for nonrotating (Ω = 0) and rotating (Ω = 0.7ω) cases. The main effects of rotation are qualitatively similar to the weakly interacting case. However, since the Cooper pairs become more strongly bound as a function of the interaction strength, it requires much faster Ω to break them. For instance, at unitarity, the entire superfluid is robust for Ω 0.3ω, and the trap center stays as an FPS even for Ω ∼ ω (not shown). Therefore, the effects of rotation become weaker as the interaction strength increases, and both the PPS and the normal regions eventually disappear in the molecular limit (not shown), i.e. rotation can not break any Cooper pair in the molecular limit. Another important observable is the local angular momentum defined by L z,σ (r) = ψ † σ (r)L z ψ σ (r) . This leads to L z,↑ (r) = −i η u * η (r)∂u η (r)/∂ϕ r f (ǫ η ) and L z,↓ (r) = −i η v η (r)∂v * η (r)/∂ϕ r f (−ǫ η ). Since the superfluid atoms do not carry angular momentum, L z,σ (r) is directly related to the local density of normal fermions via L z,σ (r) = dΩ r L z,σ (r)/(4π) ≈ M Ωr 2 n σ,N (r). Therefore, L z,σ (r) can be easily extracted from Figs. 1 and 2. For completeness, the total angular momentum L z,σ = drL z,σ (r) becomes L z,↑ = ℓ,m,γ,n,n ′ mc 2 ℓ,m,γ,n f (ǫ ℓ,m,γ ) and L z,↓ = − ℓ,m,γ,n,n ′ md 2 ℓ,m,γ,n f (−ǫ ℓ,m,γ ), and it increases with increasing Ω. In the weakly attracting limit, L z,σ becomes its rigid-body value when Ω increases high enough so that ∆(r) → 0 everywhere (not shown). The microscopic mechanism responsible for the pair breaking effects can be understood analytically within the semiclassical LDA, i.e. each component of the Fermi gas is considered as locally homogenous at each position r with a local chemical potential µ σ (r). In this approximation, the local quasiparticle and quasihole excitation branches are [10,15] E 1,2 (p, r) = µ ↑ − µ ↓ 2 + v(r) · p ± E 0 (p, r),(7) where E 0 (p, r) = [p 2 /(2M ) − µ(r)] 2 + ∆ 2 (r) is the usual spectrum for nonrotating and population balanced mixtures, v(r) = Ω z × r is the velocity and p is the momentum. In Fig. 3, we present three schematic diagrams showing E 1 (p, r) and E 2 (p, r) as a function of p for fixed values of r. At each r, the many-body ground state wavefunction fills up all of the states with negative energy, and excitations correspond to removing a quasiparticle or a quasihole from a filled state and adding it to the one that is not filled. In order to show that these excitation spectra correspond to three topologically distinct superfluid phases that can be observed in atomic sys-tems, next we discuss E 1,2 (p, r) in the z = 0 plane such that r ≡ (x, y, 0). First, we consider the population balanced case where µ ↑ = µ ↓ . In the FPS phase, the excitation spectrum is symmetric around the zero energy axis, i.e. E 1 (p, r) = −E 2 (p, r), leading to a gapped spectrum as shown in Fig. 3(a). However, in the rotating case, there is a local asymmetry between the pairing states {r, p, ↑; r, −p, ↓} and {r, −p, ↑; r, p, ↓}. When this asymmetry becomes sufficiently large, there exists a momentum space region p − ≤ p ≤ p + where E 1 (−p, r) ≤ 0 and E 2 (p, r) ≥ 0. This occurs for position space region r ≥ r T when the condition A(r) = 2M Ω 2 r 2 [µ(r) + M Ω 2 r 2 /2] − ∆ 2 (r) ≥ 0 is satisfied, where r T is defined through A(r T ) = 0 and p ± = 2M [µ(r) + M Ω 2 r 2 ] ± 2M A(r). Therefore, both E 1,2 (p, r) have two zeros at p = p ± , and the excitation spectrum becomes gapless at these momenta. We characterize this phase as the PPS, and its excitation spectrum is shown in Fig. 3(b). For a weakly attracting gas, the condition A(r) ≥ 0 can only be satisfied near the trap edge when Ω ≪ ω, but it can also be satisfied near the trap center when Ω ∼ ω. However, in the strongly interacting limit where µ is small yet positive, this condition can only be satisfied for sufficiently small values of ∆(r) near the trap edge even when Ω ∼ ω. Finally, in the molecular limit where µ is negative, this condition can not be satisfied anywhere inside the trap, leading to a superfluid phase with a gapped excitation spectrum, i.e. the Cooper pairs are robust in the molecular limit. As one may expect, the asymmetric pairing caused by the rotation does not lead to a local population imbalance at any position r since E 1 (−p, r) ≤ 0 when E 2 (p, r) ≥ 0 and vice versa. However, this asymmetry prevents the formation of Cooper pairs in the phase space region when both E 1,2 (p, r) ≥ 0 or both E 1,2 (p, r) ≤ 0, and thus it is responsible for the creation of the PPS and the normal phases. We find that the local excitation spectrum changes from gapped (FPS) to gapless (PPS) at position r = r T . In homogenous (infinite) systems such a change is classified as a topological quantum phase transition [16]. Since this change occurs in the momentum space, it does not leave any strong signature in the momentum averaged observables such as ∆(r), n(r), etc. However, its direct consequences can be observed via a recently developed position and momentum resolved spectrocopy [17]. For instance, the local density distributions [15] n ↑,↓ (p, r) = u 2 (p, r)f [±E 1,2 (p, r)] + v 2 (p, r)f [±E 2,1 (p, r)], are nonanalytic at p = p ± when r ≥ r T . Here, u 2 (p, r) = For population imbalanced superfluids (PIS) where µ ↑ = µ ↓ , the excitation branches shift upwards (downwards), and one of them cross zero energy axis when N ↑ > N ↓ (N ↑ < N ↓ ), leading to a gapless excitation spectrum. This is expected since the excess fermions can only exist in regions with both E 1,2 (p, r) ≥ 0 or both E 1,2 (p, r) ≤ 0. In the absence of rotation [15], for a weakly attracting PIS only one of the excitation branches has four zeros as shown in Fig. 3(c) for the N ↑ > N ↓ case. However, in the strongly attracting limit, this branch has only two zeros (not shown). When the system is rotating, the excitation spectra tilt similar to that shown in Fig. 3(b) (not shown). We remark in passing that a similar quantum phase transition with its experimental signatures has recently been discussed in the context of trapped p-wave superfluids [18]. To conclude, we used the BdG formalism to analyze the effects of adiabatic rotation on the ground state phases of harmonically trapped Fermi gases. We found that the rotation breaks Cooper pairs that are located near the trap edge, and that this leads to a phase separation between the nonrotating superfluid (fully paired) atoms located around the trap center and the rigidly rotating normal (nonpaired) atoms located towards the trap edge with a coexistence (partially paired) region in between. We also showed that the rotation reveals a topological quantum phase transition in the momentum space as a function of radial distance. An interesting extention of our work is to study emergence of dynamic instabilities for fast enough rotation [8,19]. FIG. 1 : 1We compare the density nσ(r) and the superfluid order parameter ∆(r) for nonrotating (Ω = 0) and rotating (Ω = 0.5ω) Fermi gases. Here, 1/(kF aF ) = −0.5, and N ↑ = N ↓ in (a) and N ↑ = 3N ↓ in (b). We also show the density nσ,N (r) of normal fermions (squared-dotted line) for the rotating case. FIG. 2 : 2We compare the density nσ(r) and the superfluid order parameter ∆(r) for nonrotating (Ω = 0) and rotating (Ω = 0.7ω) Fermi gases. Here, 1/(kF aF ) = 0.0, and N ↑ = N ↓ in (a) and N ↑ = 3N ↓ in (b). We also show the density nσ,N (r) of normal fermions (squared-dotted line) for the rotating case. FIG. 3 : 3Schematic diagrams showing the excitation spectrum of (a) a gapped fully paired superfluid (FPS) phase at r = 0, (b) a gapless partially paired superfluid (PPS) phase at r ≥ rT , and (c) a gapless population imbalanced superfluid (PIS) phase at r = 0 as a function of momentum p. {1 + [ǫ(p) − µ(r)]/E 0 (p, r)]}/2 and v 2 (p, r) = {1 − [ǫ(p) − µ(r)]/E 0 (p, r)]}/2 are the usual coherence factors with ǫ(p) = p 2 /(2M ). . Ultra-Cold Fermi, Gases, Procs. of the Int. Sch. of Phys. 'Enrico Fermi. M. Inguscio, W. Ketterle, and C. 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[ "Quasinormal modes of the BTZ black hole are generated by surface waves supported by its boundary at infinity", "Quasinormal modes of the BTZ black hole are generated by surface waves supported by its boundary at infinity" ]	[ "Yves Décanini \nFaculté des Sciences\nUMR CNRS 6134 SPE\nEquipe Physique Semi-Classique (et) de la Matière Condensée Université de Corse\nBP 5220250CorteFrance\n", "Antoine Folacci \nFaculté des Sciences\nUMR CNRS 6134 SPE\nEquipe Physique Semi-Classique (et) de la Matière Condensée Université de Corse\nBP 5220250CorteFrance\n" ]	[ "Faculté des Sciences\nUMR CNRS 6134 SPE\nEquipe Physique Semi-Classique (et) de la Matière Condensée Université de Corse\nBP 5220250CorteFrance", "Faculté des Sciences\nUMR CNRS 6134 SPE\nEquipe Physique Semi-Classique (et) de la Matière Condensée Université de Corse\nBP 5220250CorteFrance" ]	[]	We develop the complex angular momentum method in the context of the BTZ black hole physics. This is achieved by extending a formalism introduced a long time ago by Arnold Sommerfeld, which allows us to define and use the Regge pole concept in a framework where the notion of an S matrix does not exist. The Regge poles of the BTZ black hole are exactly obtained and from the associated Regge trajectories we determine its quasinormal mode complex frequencies. Furthermore, our approach permits us to physically interpret them: they appear as Breit-Wigner-type resonances generated by surface waves supported by the black hole boundary at infinity which acts as a photon sphere.	10.1103/physrevd.79.044021	[ "https://arxiv.org/pdf/0901.1642v2.pdf" ]	118,550,580	0901.1642	f18cde1bc9fcb146e08fae408bef3f90d6464b08	Quasinormal modes of the BTZ black hole are generated by surface waves supported by its boundary at infinity 8 Feb 2009 (Dated: February 8, 2009) Yves Décanini Faculté des Sciences UMR CNRS 6134 SPE Equipe Physique Semi-Classique (et) de la Matière Condensée Université de Corse BP 5220250CorteFrance Antoine Folacci Faculté des Sciences UMR CNRS 6134 SPE Equipe Physique Semi-Classique (et) de la Matière Condensée Université de Corse BP 5220250CorteFrance Quasinormal modes of the BTZ black hole are generated by surface waves supported by its boundary at infinity 8 Feb 2009 (Dated: February 8, 2009)PACS numbers: 0470-s We develop the complex angular momentum method in the context of the BTZ black hole physics. This is achieved by extending a formalism introduced a long time ago by Arnold Sommerfeld, which allows us to define and use the Regge pole concept in a framework where the notion of an S matrix does not exist. The Regge poles of the BTZ black hole are exactly obtained and from the associated Regge trajectories we determine its quasinormal mode complex frequencies. Furthermore, our approach permits us to physically interpret them: they appear as Breit-Wigner-type resonances generated by surface waves supported by the black hole boundary at infinity which acts as a photon sphere. I. INTRODUCTION By using complex angular momentum (CAM) techniques, we showed some years ago that the quasinormal mode (QNM) complex frequencies of the Schwarzschild black hole of mass M are Breit-Wigner-type resonances generated by a family of "surface waves" lying on its photon sphere at r = 3M [1]. More precisely, by noting that each surface wave is associated with a Regge pole of the S matrix of the Schwarzschild black hole [1,2,3], we have been able to construct the spectrum of the QNM complex frequencies from the Regge trajectories, i.e., from the curves traced out in the CAM plane by the Regge poles as a function of the frequency. In this way, we have established, on a "rigorous" basis, an appealing and physically intuitive interpretation of the Schwarzschild black hole QNMs suggested by Goebel in 1972 [4], i.e., that they could be interpreted in terms of gravitational waves in spiral orbits close to the unstable circular photon orbit at r = 3M which decay by radiating away energy (see also Refs. [5,6,7,8] for alternative implementations of the Goebel interpretation). Now, it is natural to use the CAM approach in order to physically understand the resonant aspects of more general black holes. We believe that, mutatis mutandis, it should be fairly easy to generalize the CAM analysis developed in Refs. [1,2,3] for all the asymptotically flat black holes with a photon sphere. By contrast, for black holes immersed in a non-asymptotically flat background, i.e., in a framework where the notion of an S matrix does not exist, we can expect to encounter new difficulties. However, because such black holes play a central * Electronic address: [email protected] † Electronic address: [email protected] role in the context of superstring theory and quantum gravity, their CAM analysis seems to be an interesting and important task which could, in particular, shed light on AdS/CFT correspondence and holography from a new point of view. In the present paper, our aim is more modest even if we try to make some steps in this direction. We shall consider the most simple black hole immersed in an asymptotically anti-de Sitter space-time, namely the (2+1)dimensional Bañados-Teitelboim-Zanelli (BTZ) black hole [9]. It seems to us very interesting in order to test our approach because, in that particular space-time, the wave equation can be solved exactly [10,11,12,13]. We shall revisit the QNM problem for this black hole from the point of view of the CAM approach in order to analyze it semiclassically, i.e., in term of surface waves. In Sec. II, we shall define the Regge poles and the associated Regge modes for a massless scalar field defined on the BTZ black hole by extending a formalism introduced a long time ago by Sommerfeld [14] as an alternative to the usual Watson approach of scattering [15]. It permits us to use the Regge pole concept in a framework where the notion of an S matrix does not exist. The Regge poles and the Regge modes of the BTZ black hole are exactly obtained and physically interpreted in terms of surface waves supported by its boundary, this boundary furthermore playing the role of a photon sphere. In Sec. III, we shall first construct from the Regge modes the diffractive part of the Feynman propagator associated with the scalar field. We shall then show that the poles of its temporal Fourier transform are the QNM complex frequencies of the BTZ black hole and prove that they are generated by the surface waves supported by the black hole boundary. In a short conclusion, we shall make some remarks concerning the AdS/CFT correspondence and the possibility to consider black hole photon spheres as holographic screens. The metric of the spinless BTZ black hole with mass M > 0 "immersed" into AdS 3 with length scale ℓ is given by ds 2 = − r 2 − r 2 h ℓ 2 dt 2 + r 2 − r 2 h ℓ 2 −1 dr 2 + r 2 dφ 2 (1) where t ∈] − ∞, +∞[, r > r h and φ has period 2π. Here r h = ℓ √ M denotes the horizon radius of the BTZ black hole. Propagating on this gravitational background, we consider a massless minimally coupled scalar field Φ solution of the wave equation Φ = 0.(2) By inserting the metric (1) into (2), the wave equation provides − r 2 − r 2 h ℓ 2 −1 ∂ 2 Φ ∂t 2 + r 2 − r 2 h ℓ 2 ∂ 2 Φ ∂r 2 + 3r 2 − r 2 h rℓ 2 ∂Φ ∂r + 1 r 2 ∂ 2 Φ ∂φ 2 = 0 (3) and we can look for its solutions by separation of variables or, more precisely, by using the ansatz Φ ν,ω (t, r, φ) = 1 √ r f ν,ω (r)e i(νφ−ωt) .(4) The radial equation satisfied by f ν,ω (r) takes the form r 2 − r 2 h ℓ 2 d 2 f ν,ω (r) dr 2 + 2r ℓ 2 d f ν,ω (r) dr + r 2 − r 2 h ℓ 2 −1 ω 2 − ν 2 r 2 − r 2 h 4ℓ 2 r 2 − 3 4ℓ 2 f ν,ω (r) = 0.(5) This differential equation can be solved exactly (see [10,11,12,13]) in terms of hypergeometric functions [16]. We then obtain for the general form of its solution f ν,ω (r) = A ν,ω r r h 1 2 1 − r 2 h r 2 −i ℓ 2 ω 2r h F − iℓ 2r h (ℓω − ν), − iℓ 2r h (ℓω + ν); 1 − i ℓ 2 ω r h ; 1 − r 2 h r 2 +B ν,ω r r h 1 2 1 − r 2 h r 2 +i ℓ 2 ω 2r h F + iℓ 2r h (ℓω − ν), + iℓ 2r h (ℓω + ν); 1 + i ℓ 2 ω r h ; 1 − r 2 h r 2(6) where A ν,ω and B ν,ω are arbitrary complex constants. From now on, we consider more particularly the mode solutions defined by (4) and (6) which correspond to the so-called QNMs and the so-called Regge modes. They are both defined as mode solutions which are purely ingoing at the horizon (i.e., for r = r h ) and vanish at infinity (i.e., for r = +∞). The first condition selects B ν,ω = 0. By noting that F (a, b; c; 1) = [Γ(c)Γ(c − a − b)]/[Γ(c − a)Γ(c − b)] if c = 0, −1, −2, . . . and Re(c − a − b) > 0, the second one imposes 1 − iℓ 2r h (ℓω − ν) = −n with n ∈ N (7a) or 1 − iℓ 2r h (ℓω + ν) = −n with n ∈ N.(7b) Let us first consider the QNMs. Because they are periodic in φ, we must have ν = m ∈ Z. They are therefore only defined for the discrete complex values of the frequency ω (see also Refs. [12,13]) ω ± mn = ± m ℓ − i 2r h ℓ 2 (n + 1) with m ∈ Z and n ∈ N (8) [here, the plus sign correspond to (7a) and the minus one to (7b)] and they are given by Φ ± mn (t, r, φ) = 1 √ r f ± mn (r)e i(mφ−ω ± mn t) (9) with f ± mn (r) = A ± mn r r h 1 2 1 − r 2 h r 2 −i ℓ 2 ω ± mn 2r h ×F −n − 1, − iℓ 2r h (ℓω ± mn ± m); 1 − i ℓ 2 ω ± mn r h ; 1 − r 2 h r 2 .(10) It should be noted that ∀m ∈ Z, ω + −mn = ω − mn but Φ + −mn (t, r, φ) = Φ − mn (t, r, φ). As a consequence, each QNM complex frequency is two-fold degenerated. Let us now consider the Regge modes. They are defined on the covering space of the BTZ black hole obtained by relaxing the condition of periodicity in the coordinate φ, i.e., by considering that φ ∈] − ∞, +∞[, and by furthermore assuming that ω > 0. The covering space of the BTZ black hole considered here is, in fact, nothing other than one of the 12 charts which permits us to provide a global covering of AdS 3 in BTZ coordinates [17]. However, it should be noted that we do not work on AdS 3 : relaxing the condition of periodicity in the coordinate φ is just a trick which will permit us to consider multivalued mode solutions of the wave equation (2) which describe surface waves propagating around the BTZ black hole and to take into account their multiple circumnavigations as well as the associated radiation damping due to their attenuations. Such a trick has been invented by Sommerfeld in order to analyze scattering by spheres and to emphasize the role of surface waves (see Ref. [14] and, more particularly, Appendix II of Chapter V as well as the appendix of Chapter VI). The Sommerfeld approach is an alternative to the usual CAM approach of scattering developed by Watson [15]. It allows us to use the Regge pole concept in a framework where the notion of an S matrix does not exist, and therefore to extend our CAM analysis of the Schwarzschild black hole [1] to the BTZ one. The Regge modes of the BTZ black hole are only defined for particular complex values of the parameter ν which we shall call Regge poles even if they are not the poles of an S matrix. These Regge poles are exactly given by ν ± n (ω) = ±ωℓ ± i 2r h ℓ (n + 1) with ω > 0 and n ∈ N (11) [here, the plus sign correspond to (7a) and the minus one to (7b)] and the corresponding Regge modes are given by Φ ± νn(ω) (t, r, φ) = 1 √ r f ± νn(ω) (r)e i[ν ± n (ω)φ−ωt] (12a) = 1 √ r f ± νn(ω) (r)e ∓ 2r h ℓ (n+1)φ e i[±ωℓφ−ωt] (12b) with f ± νn(ω) (r) = A ± νn(ω) r r h 1 2 1 − r 2 h r 2 −i ℓ 2 ω 2r h ×F −n − 1, n + 1 − i ℓ 2 ω r h ; 1 − i ℓ 2 ω r h ; 1 − r 2 h r 2 .(13) It should be noted that, even if the radial parts of Φ + νn(ω) and Φ − νn(ω) are identical, these mode solutions are different. In fact, because we have ν + n (ω) = −ν − n (ω), they respectively describe waves with identical properties but propagating counterclockwise and clockwise with an exponential decay around the BTZ black hole, with \|Re ν ± n (ω)\| = ωℓ representing their azimuthal propagation constant and \|Im ν ± n (ω)\| = (2r h /ℓ)(n + 1) their damping constant. B. More on the physical interpretation of the Regge modes The Regge poles and the Regge modes obtained in the previous subsection can be semiclassically interpreted in terms of surface waves supported by the boundary at infinity of the BTZ black hole, this boundary furthermore playing the role of a photon sphere. We shall now discuss more precisely these two important results which permit us to establish some interesting analogies between the BTZ and the Schwarzschild black holes. The propagative behavior in exp (12)-(13) permits us to note that the waves they describe circle the BTZ black hole in time (i[Re ν ± n (ω)φ − ωt]) = exp(i[±ωℓφ − ωt]) of the Regge modesT = 2π ω \|Re ν ± n (ω)\| = 2πℓ.(14) Furthermore, a scalar photon (associated with the massless scalar field Φ) on the circular orbit with constant radius R takes the time T ′ = 2πR R 2 −r 2 h ℓ 2 = 2πR R 2 ℓ 2 − M(15) to circle the BTZ black hole. Such result can be easily found by solving ds 2 = 0 with ds 2 given by (1), i.e. by integrating the equation of a circular null geodesic. By equating T and T ′ , we obtain that necessarily R → +∞. In other words, the circular orbit of the scalar photon lies on the BTZ black hole boundary and we can consider that the set of Regge modes constitutes a family indexed by n ∈ N of surface waves supported by this boundary. Let us now consider the photon sphere of the BTZ black hole. We recall that, for a static spherically symmetric black hole of dimension d with metric of the form ds 2 = −g tt (r)dt 2 + g rr (r)dr 2 + r 2 dΩ 2 d−2 ,(16) the photon sphere is defined by the greater positive solution of the equation g ′ tt (r) g tt (r) = 2 r(17) [see Ref. [18] for a general definition and an extension of the photon sphere concept in an arbitrary space-time and Eq. (54) of this paper]. For the BTZ black hole, admits a unique formal solution for r → +∞. In that sense, the boundary of the BTZ black hole can be formally considered as its photon sphere which then acts as the support of the Regge surface waves (12)- (13). III. DIFFRACTED FEYNMAN PROPAGATOR AND QUASINORMAL FREQUENCIES The Feynman propagator associated with the scalar field Φ as well as the corresponding retarded and advanced Green functions satisfy the wave equation x G(x, x ′ ) = −δ 3 (x, x ′ )(18) which, in the BTZ black hole space-time defined by (1), takes the form − r 2 − r 2 h ℓ 2 −1 ∂ 2 ∂t 2 + r 2 − r 2 h ℓ 2 ∂ 2 ∂r 2 + 3r 2 − r 2 h rℓ 2 ∂ ∂r + 1 r 2 ∂ 2 ∂φ 2 G(t, r, φ; t ′ , r ′ , φ ′ ) = − 1 r δ(t − t ′ )δ(r − r ′ )δ(φ − φ ′ ).(19) All these Green functions can be constructed by Fourier transform from the Green function G ω (r, φ; r ′ , φ ′ ) with ω > 0 defined as the symmetric solution of the Helmholtz-type equation r 2 − r 2 h ℓ 2 ∂ 2 ∂r 2 + 3r 2 − r 2 h rℓ 2 ∂ ∂r + 1 r 2 ∂ 2 ∂φ 2 + r 2 − r 2 h ℓ 2 −1 ω 2 G ω (r, φ; r ′ , φ ′ ) = − 1 2πr δ(r − r ′ )δ(φ − φ ′ )(20) which vanishes at infinity. For example, we have for the Feynman propagator G F (t, r, φ; t ′ , r ′ , φ ′ ) = 1 2π +∞ −∞ [Θ(+ω)G +ω (r, φ; r ′ , φ ′ ) +Θ(−ω)G −ω (r, φ; r ′ , φ ′ )]e i[ω(t−t ′ )] dω.(21) Here, Θ denotes the Heaviside step function and it should be noted that, in order to construct the Feynman propagator G F (t, r, φ; t ′ , r ′ , φ ′ ), we need the Green function G ω (r, φ; r ′ , φ ′ ) also for ω < 0. In fact, as we shall see later, we will also need G ω (r, φ; r ′ , φ ′ ) in the full complex ω-plane in order to discuss the resonant aspects of the BTZ black hole. It can be obtained by analytic continuation from its expression for ω > 0. From now on, we shall mainly focus our attention on the Green function G ω (r, φ; r ′ , φ ′ ). Following Sommerfeld [14] (see also Sec. 2.1 and Appendix A.1 of Ref. [19] for a pedagogical introduction to the Sommerfeld method and the full paper for its application to cylinders), we construct G ω (r, φ; r ′ , φ ′ ) from the Regge modes defined in the previous section or, more precisely, we seek it in the form G d ω (r, φ; r ′ , φ ′ ) = 1 √ r +∞ n=0 [f ν + n (ω) (r)V + n,ω (φ; r ′ , φ ′ ) +f ν − n (ω) (r)V − n,ω (φ; r ′ , φ ′ )].(22) Here we consider that the functions f ν ± n (ω) (r) are given by (13) with A ± νn(ω) = 1 and we assume that the functions V ± n,ω (φ; r ′ , φ ′ ) are such that G d ω (r, φ; r ′ , φ ′ ) is a solution of (20) symmetric under the exchange (r, φ) ↔ (r ′ , φ ′ ). Furthermore, by working in Sec. II with the Regge modes, we have deferred the imposition of periodicity in the coordinate φ on the solutions of the wave equation but now we shall impose this condition on (22) and therefore on (21). It is important to note that the Green function constructed by inserting (22) into (21) differs from the Feynman propagator which could be obtained from the exact normalized mode solutions of the wave equation (2). Indeed, the Regge modes do not constitute a complete system of solution of this equation. In fact, the Sommerfeld method permits us to only consider that part of the exact Feynman propagator which describe more particularly "diffraction" by the BTZ black hole. Such a result was noted by Sommerfeld for its analysis of scattering by spheres and remains valid in the present context. This drawback is not too serious because, in fact, it is the diffractive part of the Feynman propagator which contains all the information about the resonant aspects of the problem. By inserting (22) into (20) and by using (5) with ν = ν ± n (ω), we find that the functions V ± n,ω (φ; r ′ , φ ′ ) must satisfy 1 r 2 +∞ n=0 f ν + n (ω) (r) ∂ 2 ∂φ 2 + ν + n (ω) 2 V + n,ω (φ; r ′ , φ ′ ) +f ν − n (ω) (r) ∂ 2 ∂φ 2 + ν − n (ω) 2 V − n,ω (φ; r ′ , φ ′ ) = − 1 2π √ r δ(r − r ′ )δ(φ − φ ′ ).(23) We then multiply (23) by f ν ± p (ω) (r) and integrate over the radial domain r ∈ [r h , +∞[ which contains r ′ . The orthonormalization relation +∞ r h 1 r 2 f ν ± n (ω) (r)f ν ± p (ω) (r) dr = ±N ± n (ω)δ np (24) with N ± n (ω) = i n!(n + 1)! 2ℓ ν ± n (ω) × Γ 1 − i ℓ 2 ω r h 2 Γ n + 1 − i ℓ 2 ω r h Γ n + 2 − i ℓ 2 ω r h(25) for the Regge radial modes which can be obtained from (5) by generalizing, mutatis mutandis, the calculation displayed in Appendix A.1 of Ref. [19], permits us to write ∂ 2 ∂φ 2 + ν ± n (ω) 2 V ± n,ω (φ; r ′ , φ ′ ) = ∓ 1 2πN ± n (ω) f ν ± n (ω) (r ′ ) √ r ′ δ(φ − φ ′ ).(26) The general solution of (26) can be sought in the form V ± n,ω (φ; r ′ , φ ′ ) = Θ(φ ′ − φ)[A ± n,ω (r ′ , φ ′ ) cos(ν ± n (ω)φ) +B ± n,ω (r ′ , φ ′ ) sin(ν ± n (ω)φ)] + Θ(φ − φ ′ )[C ± n,ω (r ′ , φ ′ ) cos(ν ± n (ω)φ) +D ± n,ω (r ′ , φ ′ ) sin(ν ± n (ω)φ)].(27) Then, by inserting (27) into (26) and by using the condition that the functions V ± n,ω (φ; r ′ , φ ′ ) and their derivatives must be single-valued (now, we impose periodicity in the coordinate φ), we determine the functions A ± n,ω (r ′ , φ ′ ), B ± n,ω (r ′ , φ ′ ), C ± n,ω (r ′ , φ ′ ) and D ± n,ω (r ′ , φ ′ ) and we obtain V ± n,ω (φ; r ′ , φ ′ ) = ∓ f ν ± n (ω) (r ′ )/ √ r ′ 4πN ± n (ω)ν ± n (ω) sin[πν ± n (ω)] ×[Θ(φ ′ − φ) cos[ν ± n (ω)(π − φ ′ + φ)] +Θ(φ − φ ′ ) cos[ν ± n (ω)(π + φ ′ − φ)]]. (28) Now, by inserting (28) into (22) and by taking into account the expression (25) of the normalization factor N ± n (ω), we obtain the final result G d ω (r, φ; r ′ , φ ′ ) = + iℓ 2π +∞ n=0 a n (ω) sin[πν + n (ω)] f ν + n (ω) (r) f ν + n (ω) (r ′ ) √ rr ′ Θ(φ ′ − φ) cos[ν + n (ω)(π − φ ′ + φ)] +Θ(φ − φ ′ ) cos[ν + n (ω)(π − φ + φ ′ )] − iℓ 2π +∞ n=0 a n (ω) sin[πν − n (ω)] f ν − n (ω) (r) f ν − n (ω) (r ′ ) √ rr ′ Θ(φ − φ ′ ) cos[ν − n (ω)(π − φ + φ ′ )] +Θ(φ ′ − φ) cos[ν − n (ω)(π − φ ′ + φ)](29) with a n (ω) = Γ n + 1 − i ℓ 2 ω r h Γ n + 2 − i ℓ 2 ω r h n!(n + 1)! Γ 1 − i ℓ 2 ω r h 2 = n + 1 − i ℓ 2 ω r h 1 − i ℓ 2 ω r h n 2 n!(n + 1)! .(30) Here the symmetry of G d ω (r, φ; r ′ , φ ′ ) under the exchange (r, φ) ↔ (r ′ , φ ′ ) appears explicitly. G d ω (r, φ; r ′ , φ ′ ) has been constructed for ω > 0. By analytic continuation from the positive real ω-axis, it is obtained in the full complex ω-plane. Before exploiting the expression of G d ω (r, φ; r ′ , φ ′ ) given by (29) and (30) in order to recover the resonant aspects of the BTZ black hole, it seems to us interesting to make a digression which provide a physical interpretation of G d ω (r, φ; r ′ , φ ′ ). By using 1 sin πν = −2i +∞ k=0 e +iπ(2k+1)ν for Im ν > 0 and 1 sin πν = +2i +∞ k=0 e −iπ(2k+1)ν for Im ν < 0, we can write G d ω (r, φ; r ′ , φ ′ ) = + ℓ 2π +∞ n=0 a n (ω) f ν + n (ω) (r) f ν + n (ω) (r ′ ) √ rr ′ × Θ(φ ′ − φ) +∞ k=0 e i[ν + n (ω)(φ ′ −φ+k2π)] + e i[ν + n (ω)(2π−φ ′ +φ+k2π)] + φ ↔ φ ′ + ℓ 2π +∞ n=0 a n (ω) f ν − n (ω) (r) f ν − n (ω) (r ′ ) √ rr ′ × Θ(φ − φ ′ ) +∞ k=0 e −i[ν − n (ω)(φ−φ ′ +k2π)] + e −i[ν − n (ω)(2π−φ+φ ′ +k2π)] + φ ′ ↔ φ .(32) By reinstating the temporal dependance in e iω(t−t ′ ) into Eq. (32) [see Eq. (21)], this expression provides a physi-cal interpretation of the diffractive Feynman propagator. It appears as a sum over n ∈ N, i.e., over all the surface waves supported by the BTZ black hole boundary. In that sum, terms like e i[ν ± n (ω)(φ ′ −φ+k2π)] correspond to the contributions of surface waves propagating around the black hole and the sums over the index k take into account their multiple circumnavigations. We shall now complete this section by considering the analytic structure of G d ω (r, φ; r ′ , φ ′ ) in the complex ωplane and, more precisely, by looking for its poles in this plane because they correspond to the resonance frequencies of the scalar field Φ propagating in the BTZ black hole space-time. The only poles of G d ω (r, φ; r ′ , φ ′ ) are the zeros of the functions sin[πν ± n (ω)] with n ∈ N. They are therefore obtained by solving ν ± n (ω) = m with m ∈ Z(33) which provides the exact results given by (8) for the QNM complex frequencies and which furthermore clarifies the meaning of the indices n and m introduced to denote them: the QNM complex frequencies are grouped into families labeled by the indices n ∈ N, each family being associated with a given surface wave or equivalently with a Regge pole, and the members of a given family are indexed by m ∈ Z. It is worth noting that the method we have developed in this article should not only be regarded as a new way to calculate the QNM complex frequencies ω ± mn . It is, above all, an approach which permits us to physically interpret them: they appear as Breit-Wigner-type resonances generated by the family of surface waves supported by the BTZ black hole boundary at infinity. Indeed: -In the immediate neighborhood of a QNM complex frequency ω ± mn , G d ω (r, φ; r ′ , φ ′ ) given by (29) has the Breit-Wigner form, i.e., we can write G d ω (r, φ; r ′ , φ ′ ) ≈ N ± ω (r, φ; r ′ , φ ′ ) ω − ω ± (o) mn + iΓ ± mn /2 .(34) with ω ± (o) mn = ±m/ℓ and Γ ± mn /2 = (2r h /ℓ 2 )(n + 1). This result is a direct consequence of the formula sin(πx) ≈ (−1) m π(x − m) for x → m with m ∈ Z. -For a given value of n, a term like 1/ sin[πν ± n (ω)] is produced by interference between the different components of the n-th Regge surface wave supported by the black hole boundary [see Eq. (31) and compare (29) with (32)], each component corresponding to a different number of circumnavigations. Furthermore, a constructive interference between its different components occurs when the quantity Re ν ± n (ω) coincides with an integer, i.e., for real resonance frequencies ω ± (o) mn obtained from the Bohr-Sommerfeld-type quantization conditions Re ν ± n ω ± (o) mn = m m ∈ Z.(35) This equation provides again the real parts ω ± (o) mn of the family of QNM complex frequencies ω ± mn generated by the n-th Regge surface wave. IV. CONCLUSION AND PERSPECTIVES In the present paper, by considering the Regge poles of the spinless BTZ black hole, we have provided a new interpretation for its QNMs: they can be considered as generated by surface waves propagating on its boundary and the associated complex frequencies are Breit-Wignertype resonances. This interpretation can be easily extended to the rotating BTZ black hole [20]. In that case, the Regge modes are associated with the Regge poles ν + n (ω) = +ωℓ + i 2(r + − r − ) ℓ (n + 1) (36a) ν − n (ω) = −ωℓ − i 2(r + + r − ) ℓ (n + 1) (36b) with ω > 0 and n ∈ N. Here r + and r − denote the outer and inner horizon radii of the rotating BTZ black hole which are linked with its mass M and its angular momentum J by M = (r 2 + + r 2 − )/ℓ 2 and J = 2r + r − /ℓ. The Regge modes correspond again to surface waves supported by the boundary of the black hole but, now, it should be noted that they describe surface waves with different properties. Indeed, even if they have the same azimuthal propagation constant \|Re ν ± n (ω)\| = ωℓ, their attenuations are different: the damping constant \|Im ν + n (ω)\| = [2(r + − r − )/ℓ](n + 1) of the wave which propagates counterclockwise around the BTZ black hole (i.e., which is in co-rotation with the black hole) is lesser than the damping constant \|Im ν − n (ω)\| = [2(r + + r − )/ℓ](n + 1) of the wave which propagates clockwise. Such a behavior leads directly to the splitting of the QNM complex frequencies: when we insert (36a) and (36b) into the resonance condition (33), we obtain the exact results [13] ω + mn = + m ℓ − i 2(r + − r − ) ℓ 2 (n + 1) (37a) ω − mn = − m ℓ − i 2(r + + r − ) ℓ 2 (n + 1) (37b) with m ∈ Z and n ∈ N. Now we have ω + −mn = ω − mn and the two-fold degeneracy of the QNM complex frequencies noted in Sec. II is removed due to the rotation of the BTZ black hole which induces different damping for the surface waves propagating counterclockwise and clockwise. Because the BTZ black hole geometry frequently appears as a factor in the near horizon geometry of higher dimensional black holes of string theories (see, e.g., Ref. [21]), our Regge poles analysis could be directly generalized and analogous results could be obtained in a more general context. Similarly, it would be interesting to explore the more general situation of quantum corrected BTZ black holes (see, e.g., Ref. [22]). We believe that all these results could be helpful in order to shed light on AdS/CFT correspondence from a new point of view. Unfortunately, we have been unable to make important steps in this direction. In particular, we do not actually have at our disposal a clear CFT 2 -interpretation of the BTZ Regge poles analogous to the interpretation of the QNM complex frequencies provided in Refs. [23,24]. It is finally important to recall that the boundary of the BTZ black hole can also be considered as its photon sphere. It is therefore quite tempting to wonder if the photon sphere of all the other black holes might play a central role in the context of holography. In particular, could one holographically map a quantum field theory (or a string theory) defined on the Schwarzschild black hole of mass M on a conformally invariant quantum field theory defined on its photon sphere at r = 3M ? And more generally, could one not systematically consider photon spheres as holographic screens? II. REGGE MODES OF THE BTZ BLACK HOLEA. Quasinormal modes and Regge modes of the BTZ black hole AcknowledgmentsWe thank Stéphane Ancey, Denis Bernard, Paul Gabrielli, Bruce Jensen and Bernard Raffaelli for various discussions concerning some of the topics considered in this article. . Y Décanini, A Folacci, B P Jensen, Phys. Rev. D. 67124017Y. Décanini, A. Folacci, and B. P. Jensen, Phys. Rev. D 67, 124017 (2003). . N Andersson, K.-E Thylwe, Class. Quantum Grav. 112991N. Andersson and K.-E. Thylwe, Class. 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[ "Robust supervised learning with coordinate gradient descent", "Robust supervised learning with coordinate gradient descent" ]	[ "Stéphane Gaïffas ", "Ibrahim Merad " ]	[]	[]	This paper considers the problem of supervised learning with linear methods when both features and labels can be corrupted, either in the form of heavy tailed data and/or corrupted rows. We introduce a combination of coordinate gradient descent as a learning algorithm together with robust estimators of the partial derivatives. This leads to robust statistical learning methods that have a numerical complexity nearly identical to non-robust ones based on empirical risk minimization. The main idea is simple: while robust learning with gradient descent requires the computational cost of robustly estimating the whole gradient to update all parameters, a parameter can be updated immediately using a robust estimator of a single partial derivative in coordinate gradient descent. We prove upper bounds on the generalization error of the algorithms derived from this idea, that control both the optimization and statistical errors with and without a strong convexity assumption of the risk. Finally, we propose an efficient implementation of this approach in a new Python library called linlearn, and demonstrate through extensive numerical experiments that our approach introduces a new interesting compromise between robustness, statistical performance and numerical efficiency for this problem.	null	[ "https://arxiv.org/pdf/2201.13372v1.pdf" ]	246,431,198	2201.13372	51a04b55bc0e21434a11895a9616d13b6c53ac1b	Robust supervised learning with coordinate gradient descent February 1, 2022 Stéphane Gaïffas Ibrahim Merad Robust supervised learning with coordinate gradient descent February 1, 2022Robust methodsHeavy-tailed dataOutliersRobust gradient descentCoordinate gradient descentGeneralization error This paper considers the problem of supervised learning with linear methods when both features and labels can be corrupted, either in the form of heavy tailed data and/or corrupted rows. We introduce a combination of coordinate gradient descent as a learning algorithm together with robust estimators of the partial derivatives. This leads to robust statistical learning methods that have a numerical complexity nearly identical to non-robust ones based on empirical risk minimization. The main idea is simple: while robust learning with gradient descent requires the computational cost of robustly estimating the whole gradient to update all parameters, a parameter can be updated immediately using a robust estimator of a single partial derivative in coordinate gradient descent. We prove upper bounds on the generalization error of the algorithms derived from this idea, that control both the optimization and statistical errors with and without a strong convexity assumption of the risk. Finally, we propose an efficient implementation of this approach in a new Python library called linlearn, and demonstrate through extensive numerical experiments that our approach introduces a new interesting compromise between robustness, statistical performance and numerical efficiency for this problem. Introduction Outliers and heavy tailed data are a fundamental problem in supervised learning. As explained by Hawkins (1980), an outlier is a sample that differs from the data's "global picture". A ruleof-thumb is that a typical dataset may contain between 1% and 10% of outliers (Hampel et al., 2011), or even more than that depending on the considered application. For instance, the inherently complex and random nature of users' web browsing makes web-marketing datasets contain a significant proportion of outliers and have heavy-tailed distributions (Gupta and Kohli, 2016). Statistical handling of outliers was already considered in the early 50's (Dixon, 1950;Grubbs, 1969) and motivated in the 70's the development of robust statistics (Huber, 1972(Huber, , 1981. Setting. In this paper, we consider the problem of large-scale supervised learning, where we observe possibly corrupted samples (X i , Y i ) n i=1 of a random variable (X, Y ) ∈ X × Y with distribution P , where X ⊂ R d is the feature space and Y ⊂ R is the set of label values. We focus on linear methods, where the learning task corresponds to finding an approximation of an optimal parameter θ ∈ argmin θ∈Θ R(θ) where R(θ) := E (X θ, Y ) , where Θ is a convex compact subset of R d with diameter ∆ containing the origin and : R × Y → R + is a loss function satisfying the following. We denote (z, y) := ∂ (z, y)/∂z. Assumption 1. The loss z → (z, y) is convex for any y ∈ Y, differentiable and γ-smooth in the sense that \| (z, y) − (z , y)\| ≤ γ\|z − z \| for all z, z ∈ R and y ∈ Y. Moreover, there exist q ∈ [1, 2], which we will call the asymptotic polynomial degree, and positive constants C ,1 , C ,2 , C ,1 and C ,2 such that \| (z, y)\| ≤ C ,1 + C ,2 \|z − y\| q and \| (z, y)\| ≤ C ,1 + C ,2 \|z − y\| q−1 for all z ∈ R and y ∈ Y. Note that Assumption 1 holds for the majority of loss functions used both for regression and classification, such as the square loss (z, y) = (z − y) 2 /2 with q = 2 or the Huber loss (Huber, 1964) (z, y) = r τ (z − y) for z, y ∈ R with γ = 1 and q = 1, where r τ (u) = 1 2 u 2 1 \|u\|≤τ + τ (\|u\| − 1 2 τ )1 \|u\|>τ with τ > 0 and the logistic loss (z, y) = log(1 + e −yz ) for z ∈ R and y ∈ {−1, 1} with γ = 1/4 and q = 1. We will see shortly that a smaller degree q associated to the loss entails looser requirements on the data distribution. If P were known, one could approximate θ using a first-order optimization algorithm such as gradient descent (GD), using iterations of the form θ t+1 ← θ t − η∇R(θ t ) with ∇R(θ) = E[ (X θ, Y )X](2) for t = 1, 2, . . . where η > 0 is a learning rate. Empirical risk minimization. With P unknown, most supervised learning algorithms rely on empirical risk minimization (ERM) (Vapnik, 1999;Geer and van de Geer, 2000), which requires (a) the fact that samples are independent and with the same distribution P and (b) that P has sub-Gaussian tails, as explained below. Such assumptions are hardly ever met in practice, and entail implicitly that, for real-world applications, the construction of a training dataset requires involved data preparation, such as outlier detection and removal, data normalization and other issues related to feature engineering (Zheng and Casari, 2018;Kuhn and Johnson, 2019). An implicit 1 ERM estimator of θ is a minimizer of the empirical risk R n given by θ erm n ∈ argmin θ∈Θ R n (θ) where R n (θ) := 1 n n i=1 (X i θ, Y i ),(3) for which one can prove sub-Gaussian deviation bounds under strong hypotheses such as boundedness of or sub-Gaussian concentration (Massart and Nédélec, 2006;Lecué and Mendelson, 2013). In the general case, ERM leads to poor estimations of θ whenever (a) and/or (b) are not met, corresponding to situations where (a) the dataset contains outliers and (b) the data distribution has heavy tails. This fact motivated the theory of robust statistics (Huber, 1964(Huber, , 2004Hampel, 1971;Hampel et al., 2011;Tukey, 1960). The poor performance of ERM stems from the loose deviation bounds of the empirical mean estimator. Indeed, as explained by Catoni (2012) for the estimation of the expectation of a real random variable, the Chebyshev inequality provably provides the best concentration bound for the empirical mean estimator in the general case, so that the error is Ω(1/ √ nδ) for a confidence 1 − δ. Gradient Descent (GD) combined with ERM leads to an explicit algorithm using iterations (2) with gradients estimated by an average over the samples ∇ erm R(θ) := ∇R n (θ) = 1 n n i=1 (X i θ, Y i )X i ,(4) which is, as explained above, a poor estimator of ∇R(θ) beyond (a) and (b). Robust gradient descent. A growing literature about robust GD estimators Liu et al., 2019;Holland, 2019;Geoffrey et al., 2020) suggests to perform GD iterations with ∇ erm R(θ) replaced by some robust estimator of ∇R(θ). An implicit estimator is considered by Lecué et al. (2020), based on the minimization of a robust estimate of the risk objective using median-of-means. Robust estimators of ∇R(θ) can be built using several approaches including geometric median-of-means ; robust coordinate-wise estimators (Holland and Ikeda, 2019a) based on a modification of Catoni (2012); coordinate-wise median-of-means or trimmed means (Liu et al., 2019) or robust vector means through projection and truncation . Other works achieve robustness by performing standard training on disjoint subsets of data and aggregating the resulting estimators into a robust one (Minsker et al., 2015;Brownlees et al., 2015). We discuss such alternative methods in more details in Section 4 below. These procedures based on GD require to run costly subroutines (at the exception of Lecué et al. (2020); Geoffrey et al. (2020)) that induce a considerable computational overhead compared to the non-robust approach based on ERM. The aim of this paper is to introduce robust and explicit learning algorithms, with performance guarantees under weak assumptions on (X i , Y i ) n i=1 , that have a computational cost comparable to the non-robust ERM approach. As explained in Section 2 below, the main idea is to combine coordinate gradient descent with robust estimators of the partial derivatives ∂R(θ)/∂θ j , that are scalar (univariate) functionals of the unknown distribution P . We denote \|A\| as the cardinality of a finite set A and use the notation k = {1, . . . , k} for any integer k ∈ N \ {0}. We denote x j as the j-th coordinate of a vector x. We will work under the following assumption. Assumption 2. The indices of the training samples n can be divided into two disjoint subsets n = I ∪ O of outliers O and inliers I for which we assume the following: (a) we have \|I\| > \|O\|; (b) the pairs (X i , Y i ) i∈I are i.i.d with distribution P and the outliers (X i , Y i ) i∈O are arbitrary; (c) there is α ∈ (0, 1] such that E \|X j \| max(2,q(α+1)) < +∞, E \|Y q−1 X j \| 1+α < +∞ and E \|Y \| q < +∞ (5) for any j ∈ d where q ∈ [1, 2] is the loss' asymptotic polynomial degree from Assumption 1. Assumption 2 is purposely vague about \|I\| and \|O\| and the value of α ∈ (0, 1]. Indeed, conditions on \|O\| and α will depend on the considered robust estimator of the partial derivatives, as explained in Section 3 below, including theoretical guarantees with α < 1 and cases with E[Y 2 ] = +∞ (for the Huber loss for instance). The existence of a second moment for X is indispensable for the objective R(θ) to be Lipschitz-smooth, see Section 2.2 below. Square loss. For the square loss we have q = 2 and E[Y 2 ] < +∞ is required for the risk R(θ) and its partial derivatives to be well-defined. Note that we have E[\| (X θ, Y )X j \| 1+α ] = E[\|Y X j \| 1+α ] for θ = 0 ∈ Θ, which makes (5) somewhat minimal in order to ensure the existence of the moment we need for the loss derivative for all θ ∈ Θ. Huber loss. For the Huber loss, we have q = 1 and the only requirement on Y is E\|Y \| < +∞ and we have max(2, q(α + 1)) = 2 ensuring that E[\|X j \| 2 ] < +∞, a requirement for the Lipschitzsmoothness of R(θ), as detailed in Section 2.2. Logistic loss. For the logistic loss we have \|Y \| ≤ 1 and q = 1 so that the only assumption is once again E[\|X j \| 2 ] < +∞. Main contributions. We believe that this paper introduces a new interesting compromise between robustness, statistical performance and numerical efficiency for supervised learning with linear methods through the following main contributions: • We introduce a new approach for robust supervised learning with linear methods by combining coordinate gradient descent (CGD) with robust estimators of the partial derivatives used in its iterations (Section 2). We explain that this simple idea turns out to be very effective experimentally (Section 6), and amenable to an in-depth theoretical analysis (see Section 2.2 for guarantees under strong convexity and Section 5 without it). • We consider several estimators of the partial derivatives using state-of-the-art robust estimators (Section 3) and provide theoretical guarantees for CGD combined with each of them. For some robust estimators, our analysis requires only weak moments (allowing E[Y 2 ] = +∞ in some cases) together with strong corruption (large \|O\|). We provide guarantees for several variants of CGD namely random uniform sampling, importance sampling and deterministic sampling of the coordinates (Section 2.2). • We perform extensive numerical experiments, both for regression and classification on several datasets (Section 6). We compare many combinations of gradient descent, coordinate gradient descent and robust estimators of the gradients and partial derivatives. Some of these combinations correspond to state-of-the-art algorithms (Lecué et al., 2020;Holland and Ikeda, 2019a;, and we consider also several supplementary baselines such as Huber regression (Owen, 2007), classification with the modified Huber loss (Zhang, 2004), Least Absolute Deviation (LAD) (Edgeworth, 1887) and RANSAC (Fischler and Bolles, 1981). Our experiments provide comparisons of the statistical performances and numerical complexities involved in each algorithm, leading to an in-depth comparison of state-of-the-art robust methods for supervised linear learning. • All the algorithms studied and compared in the paper are made easily accessible in a few lines of code through a new Python library called linlearn, open-sourced under the BSD-3 License on GitHub and available here 2 . This library follows the API conventions of scikit-learn (Pedregosa et al., 2011). 2 Robust coordinate gradient descent CGD is well-known for its efficiency and fast convergence properties based on both theoretical and practical studies (Nesterov, 2012;Shevade and Keerthi, 2003;Genkin et al., 2007;Wu and Lange, 2008) and is the de-facto standard optimization algorithm used in many machine learning libraries. In this paper, we suggest to use CGD with robust estimators g j (θ) of the partial derivatives g j (θ) := ∂R(θ)/∂θ j ∈ R of the true risk given by Equation (1), several robust estimators g j (θ) are described in Section 3 below. Iterations At iteration t + 1, given the current iterate θ (t) , CGD proceeds as follows. It chooses a coordinate j t ∈ d (several sampling mechanisms are possible, as explained below) and the parameter is updated using θ (t+1) j ← θ (t) j − β j g j (θ (t) ) if j = j t θ (t+1) j ← θ (t) j otherwise(6) for all j ∈ d , where β j > 0 is a step-size for coordinate j. A single coordinate is updated at each iteration of CGD, and we will designate d iterations of CGD as a cycle. The CGD procedure is summarized in Algorithm 1 below, where we denote by X ∈ R n×d the features matrix with rows X 1 , . . . , X n and where X j • ∈ R n stands for its j-th column. Algorithm 1 Robust coordinate gradient descent 1: Inputs: Learning rates β 1 , . . . , β d > 0; estimators ( g j (·)) d j=1 of the the partial derivatives; initial parameter θ (0) ; distribution p = [p 1 · · · p d ] over d and number of iterations T . 2: Compute I (0) ← Xθ (0) 3: for t = 0, . . . , T − 1 do 4: Sample a coordinate j t ∈ {1, . . . , d} with distribution p independently of j 1 , . . . , j t−1 5: Compute g jt (θ (t) ) using I (t) and put D (t) ← −β jt g jt (θ (t) ) 6: Update the inner products using I (t+1) ← I (t) + X jt • D (t) 7: Apply the update θ (t+1) jt ← θ (t) jt + D (t) 8: end for 9: return The last iterate θ (T ) A simple choice for the distribution p is the uniform distribution over d , but improved convergence rates can be achieved using importance sampling, as explained in Theorem 1 below, where the choice of the step-sizes (β j ) d j=1 is described as well. The partial derivatives estimators ( g j (·)) d j=1 described in Section 3 will determine the statistical error of this explicit learning procedure. Note that line 6 of Algorithm 1 uses the fact that I (t+1) = Xθ (t+1) = j =jt X j • θ (t+1) j + X jt • θ (t+1) jt = j =jt X j • θ (t) j + X jt • θ (t) jt + D (t) = I (t) + X jt • D (t) . This computation has complexity O(n), and we will see in Section 3 that the complexity of the considered robust estimators g jt (θ (t) ) at line 5 is also O(n), so that the overall complexity of one iteration of robust CGD is also O(n). This makes the complexity of one cycle of robust CGD O(nd), which corresponds to the complexity of one iteration of GD using the non-robust estimator ∇ erm R(θ), see Equation (4). A more precise study of these complexities is discussed in Section 3, see in particular Table 1. Moreover, we will see experimentally in Section 6 that our approach is indeed very competitive computationally in terms of the compromise between computations and statistical accuracy, compared to all the considered baselines. Comparison with robust gradient descent. Robust estimators of the expectation of a random vector (such as the geometric median by Minsker et al. (2015)) require to solve a d-dimensional optimization problem at each iteration step while, in the univariate case, a robust estimator of the expectation can be obtained at a cost comparable to that of an ordinary empirical average. Of course, one can combine such univariate estimators into a full gradient: this is the approach considered for instance by Holland and Ikeda (2019a); Holland (2019); Holland and Ikeda (2019b); Liu et al. (2019); Tu et al. (2021), but this approach accumulates errors into the overall estimation of the gradient. This paper introduces an alternative approach, where univariate estimators of the partial derivatives are used immediately to update the current iterate. We believe that this is the main benefit of using CGD in this context: even if our theoretical analysis hardly explains this, our understanding is that one iteration of CGD is impacted by the estimator error of a single partial derivative, that can be corrected straight away in the next iteration, while one iteration of GD is impacted by the accumulated estimation errors of the d partial derivatives, when using d univariate estimators for efficiency, instead of a computationally involved d-dimensional estimator (such as geometric median). Theoretical guarantees under strong convexity In this Section, we provide theoretical guarantees in the form of upper bounds on the risk R(θ (T ) ) (see Equation (1)) for the output θ (T ) of Algorithm 1. These upper bounds are generic with respect to the considered robust estimators g j (·) d j=1 and rely on the following definition. Definition 1. Let δ ∈ (0, 1) be a failure probability. We say that a partial derivatives estimator g has an error vector (δ) ∈ R d + if it satisfies P sup θ∈Θ g j (θ) − g j (θ) ≤ j (δ) ≥ 1 − δ (7) for all j ∈ d . In Section 3 below, we specify a value of j (δ) for each considered robust estimator which will lead to upper bounds on the risk. Recall that g j (θ) = ∂R(θ)/∂θ j and let us denote as e j the j-th canonical basis vector of R d . We need the following extra assumptions on the optimization problem itself. Assumption 3. There exists θ ∈ Θ satisfying the stationary gradient condition ∇R(θ ) = 0. Moreover, we assume that there are Lipschitz constants L j > 0 such that g j (θ + he j ) − g j (θ) ≤ L j \|h\| for any j ∈ d , h ∈ R and θ ∈ Θ such that θ + he j ∈ Θ. We also consider L > 0 such that g(θ + h) − g(θ) ≤ L h for any h ∈ Θ and θ ∈ Θ such that θ + h ∈ Θ. We denote L max := max j∈ d L j and L min := min j∈ d L j . Under Assumptions 1 and 2, we know that the Lipschitz constants (L j ) j∈ d and L do exist. Indeed, the Hessian matrix of the risk R(θ) is given by ∇ 2 R(θ) = E (θ X, Y )XX , where (z, y) := ∂ 2 (z, y)/∂z 2 , so that L j = sup θ∈Θ E (θ X, Y )(X j ) 2 and L = sup θ∈Θ ∇ 2 R(θ) op ,(8) where H op stands for the operator norm of a matrix H. Assumption 1 entails L j ≤ γE (X j ) 2 , which is finite because of Equation (5) from Assumption 2. In order to derive linear convergence rates for CGD, it is standard to require strong convexity (Nesterov, 2012;Wright, 2015). Here, we require strong convexity on the risk R(θ) itself, as described in the following. Assumption 4. We assume that the risk R given by Equation (1) is λ-strongly convex, namely that R(θ 2 ) ≥ R(θ 1 ) + ∇R(θ 1 ), θ 2 − θ 1 + λ 2 θ 2 − θ 1 2 (9) for any θ 1 , θ 2 ∈ Θ. Assumption 4 is satisfied whenever λ min ∇ 2 R(θ) ≥ λ for any θ ∈ Θ, where λ min (H) stands for the smallest eigenvalue of a symmetric matrix H. For the least-squares loss, this translates into the condition λ min E[XX ] ≥ λ. Note that one can always make the risk λ-strongly convex by considering ridge penalization, namely by replacing R(θ) by R(θ) + λ 2 θ 2 2 , but we provide also guarantees without this Assumption in Section 5 below. The following Theorem provides an upper bound over the risk of Algorithm 1 whenever the estimators g j (·) have an error vector (δ), as defined in Definition 1. We introduce for short R = R(θ ) = min θ∈Θ R(θ). Theorem 1. Grant Assumptions 1, 3 and 4. Let θ (T ) be the output of Algorithm 1 with step-sizes β j = 1/L j , an initial iterate θ (0) , uniform coordinates sampling p j = 1/d and estimators of the partial derivatives with error vector (·). Then, we have E R(θ (T ) ) − R ≤ R(θ (0) ) − R 1 − λ L max d T + L max 2λL min (δ) 2 2(10) with probability at least 1−δ, where the expectation is w.r.t. the sampling of the coordinates. Now, if Algorithm 1 is run as before, but with an importance sampling distribution p j = L j / k∈ d L k , we have E R(θ (T ) )] − R ≤ R(θ (0) ) − R 1 − λ j∈ d L j T + 1 2λ (δ) 2 2(11) with probability at least 1 − δ. The proof of Theorem 1 is given in Appendix B. It adapts standard arguments for the analysis of CGD (Nesterov, 2012;Wright, 2015) with inexact estimators of the partial derivatives. The statistical error (δ) 2 2 is studied in Section 3 for each considered robust estimator of the partial derivatives. Both (10) and (11) are upper bounds on the excess risk with exponentially vanishing optimization errors (called linear rate in optimization) and a constant statistical error. The optimization error term of (11), given by R(θ (0) ) − R 1 − λ j∈ d L j T , goes to 0 exponentially fast as the number of iterations T increases, with a contraction constant better than that of (10) since j∈ d L j ≤ dL max . This can be understood from the fact that importance sampling better exploits the knowledge of the Lipschitz constants L j . Also, note that T is the number of iterations of CGD, so that T = Cd where C is the number of CGD cycles. Therefore, defining L := 1 d j∈ d L j , we have 1 − λ dL Cd ≤ 1 − λ L C , for d ≥ 1, which leads to a linear rate at least similar to the one of GD (Bubeck, 2015). Theorem 1 proves an upper bound on the excess risk R(θ (T ) ) − R of the iterates of robust CGD directly, without using an intermediate upper bound on θ (T ) − θ 2 2 . This differs from the approaches used by ; Holland and Ikeda (2019a) that consider robust GD (while we introduce robust CGD here) to bound the excess risk of the iterates. This allows us to obtain a better contraction factor for the optimization error and a better constant in front of the statistical error. Note that we can derive also an upper bound on θ (T ) − θ 2 2 , see Theorem 4 in Appendix B. Note that the iterations considered in Algorithm 1 do not perform a projection in Θ. Indeed, one can show that θ (t) − θ is also subject to a contraction and is therefore decreasing w.r.t. t. Thus, if θ (0) = 0, iterates θ (t) naturally belong to the 2 ball of radius 2 θ . Step-sizes. The step-sizes β j = 1/L j are unknown, since they are functionals of the unknown distribution P . So, we provide, in Appendix A.1, theoretical guarantees similar to that of Theorem 1 using step-sizes β j = 1/ L j , where L j is a robust estimator of the upper bound L j := γE (X j ) 2 ≥ L j of the Lipschitz constant L j . A deterministic result. The previous Theorem 1 provides upper bounds on the expectation of the excess risk with respect to the sampling of the coordinates used in CGD. In Theorem 2 below, we provide an upper bound similar to the one from Theorem 1, but with a fully deterministic variant of CGD, where we replace line 4 of Algorithm 1 with a deterministic cycling through the coordinates. Theorem 2. Grant Assumptions 1, 3 and 4. Let θ (T ) be the output of Algorithm 1 with step-sizes β j = 1/L j , an initial iterate θ (0) , deterministic cycling over d such that {j td+1 , j td+2 , . . . , j (t+1)d−1 } = d for any t and estimators of the partial derivatives with error vector (·). Then, we have R(θ (T ) ) − R ≤ R(θ (0) ) − R 1 − 2λκ T + 3 8λκL min (δ) 2 2 with probability at least 1 − δ, where we introduced the constant κ = 1 8L max (1 + d(L max /L min )) . The proof of Theorem 2 is given in Appendix B and uses arguments from Beck and Tetruashvili (2013) and Li et al. (2017). It provides an extra guarantee on the convergence of CGD, for a very general choice of coordinates cycling, at the cost of degraded constants compared to Theorem 1, both for the optimization and statistical error terms. Robust estimators of the partial derivatives We consider three estimators of the partial derivatives g j (θ) = ∂R(θ) ∂θ j = E (X θ, Y )X j that can be used within Algorithm 1: Median-of-Means in Section 3.1, Trimmed mean in Section 3.2 and an estimator that we will call "Catoni-Holland" in Section 3.3. We provide, for each estimator, a concentration inequality for the estimation of g j (θ) for fixed θ under a weak moments assumption (Lemmas 2, 3 and 4). We derive also uniform versions of the bounds in each case (Propositions 1, 2, 3 and 4) which define the error vectors to be plugged into Theorems 1 and 2. We also discuss in details the numerical complexity of each estimator and explain that they all are, in their own way, an interpolation between the empirical mean and the median. We wrap up these results in Table 1 The deviation bound optimality in Table 1 is meant in terms of the dependence, up to a constant, on the sample size n, required confidence δ ∈ (0, 1) and distribution variance 3 . An estimator's deviation bound is deemed optimal if it fits the lower bounds given by Theorems 1 and 3 in Lugosi and Mendelson (2019a). Let us introduce the centered moment of order 1 + α of the partial derivatives and its maximum over Θ, given by m α,j (θ) := E (X θ, Y )X j − E[ (X θ, Y )X j ] 1+α and M α,j = sup θ∈Θ m α,j (θ) (12) for α ∈ (0, 1]. Note that m 1,j (θ) = V (X θ, Y )X j and we know that m α,j (θ) exists, as explained in the next Lemma. Lemma 1. Under Assumptions 1 and 2 the risk R(θ) is well defined for all θ ∈ Θ and we have E (X θ, Y )X j 1+α < +∞ for any j ∈ d and θ ∈ Θ. The proof of Lemma 1 involves simple algebra and is provided in Appendix B. Let us introduce g i j (θ) := (X i θ, Y i )X j i ,(13) the sample i ∈ n partial derivative for coordinate j ∈ d . Median-of-Means The Median-Of-Means (MOM) estimator is the median g MOM j (θ) := median g (1) j (θ), . . . , g (K) j (θ)(14) of the block-wise empirical means g (k) j (θ) := 1 \|B k \| i∈B k g i j (θ)(15) within blocks B 1 , . . . , B K of roughly equal size that form a partition of n and that are sampled uniformly at random. This estimator depends on the choice of the number K of blocks used to compute it, which can be understood as an "interpolation" parameter between the ordinary mean (K = 1) and the median (K = n). It is robust to heavy-tailed data and a limited number of outliers as explained in the following lemma. Lemma 2. Grant Assumptions 1 and 2 with α ∈ (0, 1]. If \|O\| ≤ K/12, we have: P g MOM j (θ) − g(θ) j > (24m α,j (θ)) 1/(1+α) K n α/(1+α) ≤ e −K/18 for any fixed j ∈ d and θ ∈ Θ. If we fix a confidence level δ ∈ (0, 1) and choose K := 18 log(1/δ) , we have g MOM j (θ) − g(θ) j ≤ c α m α,j (θ) 1/(1+α) log(1/δ) n α/(1+α) ≤ c α M 1/(1+α) α,j log(1/δ) n α/(1+α)(16) with a probability larger than 1 − δ, where c α := 2 (3+α)/(1+α) 3 (1+2α)/(1+α) . The proof of Lemma 2 is given in Appendix B and it adapts simple arguments from Lugosi and Mendelson (2019a) and Lecué et al. (2020). Compared to Lugosi and Mendelson (2019a), it provides additional robustness with respect to \|O\| ≥ 1 outliers and compared to Lecué et al. (2020) it provides guarantees with weak moments α < 1. An inspection of the proof of Lemma 2 shows that it holds also under the assumption \|O\| ≤ (1 − ε)K/2 for any ε ∈ (0, 1) with an increased constant c α = 8 × 3 1/(1+α) /ε (1+2α)/(1+α) . This concentration bound is optimal under the (1 + α)-moment assumption (see Theorems 1 and 3 in Lugosi and Mendelson (2019a)) and is sub-Gaussian when α = 1 (finite variance). The next proposition provides a uniform deviation bound over Θ for g MOM j (θ). Proposition 1. Grant Assumptions 1 and 2 with α ∈ (0, 1] and \|O\| ≤ K/12. We have P sup θ∈Θ g MOM j (θ) − g j (θ) ≤ MOM j (δ) ≥ 1 − δ for any j ∈ d , with MOM j (δ) := c α M j,α + m L,α n α 1/(1+α) log(d/δ) + d log(3∆n α/(1+α) /2) n α/(1+α) + (L + L j ) 1 n α/(1+α) where L = γE X 2 , m L,α = E\|γ X 2 − L\| 1+α and c α = 2 (3+2α)/(1+α) 3 (1+3α)/(1+α) . The proof of Proposition 1 is given in Appendix B and uses methods similar to Lemma 2 with an ε-net argument. This defines the error vector MOM (δ) of the MOM estimator of the partial derivatives in the sense of Definition 1, that can be combined directly with the convergence results from Theorems 1 and 2 from Section 2. Since the optimization error decreases exponentially w.r.t. the number of iterations T in these theorems, while the estimator error (δ) 2 is fixed, one only needs T = O( (δ) 2 ) to make both terms of the same order. About uniform bounds. What is necessary to obtain a control of the excess risk of robust CGD is a control of the noise terms \| g j (θ (t) ) − g j (θ (t) )\|, where both iterates θ (t) and estimators g j (·) of the partial derivatives depend on the same data. This forbids the direct use of a deviation such a the one from Lemma 2 (and Lemmas 3 and 4 below) where θ must be deterministic. We use in this paper an approach based on uniform deviation bounds (Propositions 1, 3 and 4) in order to bypass this problem, similarly to Holland and Ikeda (2019b) and many other papers using empirical process theory. This is of course pessimistic, since θ (t) goes to θ as t increases. Another approach considered in is to split data into segments of size n/T and to compute the gradient estimator using a segment independent of the ones used to compute the current iterate. This approach departs strongly from what is actually done in practice, and leads to controls on the excess risk expressed with δ = δ/T and n = n/T instead of δ and n, hence a deterioration of the control of the excess risk. Our approach based on uniform deviations also suffers from a deterioration, due to the use of an ε-net argument, observed in Proposition 1 through the extra d α/(1+α) factor when compared to Lemma 2. Avoiding such deteriorations is an open difficult problem, either using uniform bounds or data splitting. In addition to Proposition 1, we propose another uniform deviation bound for g MOM j (θ) using the Rademacher complexity, which is a fundamental tool in statistical learning theory and empirical process theory (Ledoux and Talagrand, 1991;Koltchinskii, 2006;Bartlett et al., 2005). Let us introduce R j (Θ) = E sup θ∈Θ i∈I ε i g i j (θ) for j ∈ d , where (ε i ) i∈I are i.i.d Rademacher variables and where we recall that I contains the inliers indices (see Assumption 2). Proposition 2. Grant Assumptions 1 and 2 with α ∈ (0, 1]. If \|O\| ≤ K/12, we have P sup θ∈Θ g MOM j (θ) − g j (θ) ≥ max 36M α,j (n/K) α 1/(1+α) , 64R j (Θ) n ≤ e −K/18 for any j ∈ d . If we fix a confidence level δ ∈ (0, 1) and choose K := 18 log(1/δ) , we have sup θ∈Θ g MOM j (θ) − g(θ) j ≤ max c α M 1/(1+α) α,j log(d/δ) n α/(1+α) , 64R j (Θ) n(17) with a probability larger than 1 − δ for all j ∈ d , where c α : = 2 (2+α)/(1+α) 3 2 . Moreover, if µ 2(1+α) X,j := E[(X j ) 2(1+α) ] < +∞ for all j ∈ d we have R j (Θ) ≤ γ∆C α nµ 1+α X,j k∈ d µ 1+α X,k 1/(1+α) = O((nd) 1/(1+α) ), where C α is a constant depending only on α. The proof of Proposition 2 is given in Appendix B and borrows arguments from Lecué et al. (2020); Boucheron et al. (2013). For α = 1, the bound (17) leads to a O( √ nd) bound similar to that of Theorem 2 from Lecué et al. (2020), although we consider here a different quantity (Rademacher complexity of the partial derivatives, towards the study of the explicit robust CGD algorithm, while implicit algorithms are studied therein). Note also that we do not prove similar uniform bounds using the Rademacher complexity for the TM and CH algorithms considered below, an interesting open question. Comparison with ; Holland and Ikeda (2019a). A first distinction of our results compared to ; Holland and Ikeda (2019a) is the use and theoretical study of robust CGD instead of robust GD. A second distinction is that we work under 1 + α moments on the partial derivatives of the risk, while ; Holland and Ikeda (2019a) require α = 1. Our setting is similar but more general than the one laid out in Holland and Ikeda (2019a) since the latter does not consider the presence of outliers. Theorem 5 from Holland and Ikeda (2019a) states linear convergence of the optimization error thanks to strong convexity similarly to our Theorem 1. Their management of the statistical error is quite similar and leads to the same rate. However, our bound involves the sum of the coordinatewise moments of the gradient thanks to Proposition 1, an improvement over the bound from Holland and Ikeda (2019a) which is only stated in terms of a uniform bound on the coordinate variances. Another reference point is the heavy-tailed setting of , which deals with heavy-tails independently from the problem of corruption and requires α = 1. More importantly, the approach considered in relies on data-splitting, which departs significantly from what is done in practice, while we do not perform data-spitting but use uniform bounds, as discussed above. Complexity of g MOM j (θ). The computation of g MOM j (θ) requires (a) to sample a permutation of n to sample the blocks B 1 , . . . , B K , (b) to compute averages within the blocks and (c) to compute the median of K numbers. Sampling a permutation of n has complexity O(n) using the Fischer-Yates algorithm (Knuth, 1997), and so does the computation of the averages, so that (a) and (b) have complexity O(n). The computation of the median of K numbers can be done using the quickselect algorithm (Hoare, 1961) with O(K) average complexity, leading to a complexity O(n + K) = O(n) since K < n. Trimmed Mean estimator The idea of the Trimmed Mean (TM) estimator is to exclude a proportion of data in the tails of their distribution to achieve robustness. We are aware of two variants: (1) one in which samples in the tails are removed, the remaining samples being used to compute an empirical mean and (2) another variant in which samples in the tails are clipped but not removed from the empirical mean. Variant (1) is robust to η-corruption 4 whenever the data distribution is sub-exponential (Liu et al., 2019) or sub-Gaussian (Diakonikolas et al., 2019, 019a,b). Variant (2), also known as Winsorized mean, enjoys a sub-Gaussian deviation (Lugosi and Mendelson, 2019a) for heavy-tailed distributions. Both robustness properties are shown simultaneously (sub-Gaussian deviations under a heavy-tails assumption and η-corruption) in Lugosi and Mendelson (2021) (see Theorem 1 therein). We consider below variant (2), which proceeds as follows. First, the TM estimator splits n = n/2 ∪ n/2 where n/2 = n \ n/2 , assuming without loss of generality that n is even, and it computes the sample derivatives g i j (θ) given by (13) for all i ∈ n . Then, given a proportion ∈ [0, 1/2), it computes the and 1 − quantiles of (g i j (θ)) i∈ n/2 given by q := g ([ n/2]) j (θ) and q 1− := g ([(1− )n/2]) j (θ), where g (1) j (θ) ≤ · · · ≤ g (n/2) j (θ) is the order statistics of (g i j (θ)) i∈ n/2 and where [x] is the lower integer part of x ∈ N. Finally, the estimator is computed as g TM j (θ) = 2 n i∈ n/2 q ∨ g i j (θ) ∧ q 1− ,(18) where a ∧ b := min(a, b) and a ∨ b := max(a, b), namely it is the average of the partial derivatives from samples in n/2 clipped in the interval [q , q 1− ]. Note that g TM j (θ) is also some form of "interpolation" between the average and the median through : it is the average of the partial derivatives for = 0 and their median for = 1/2. As explained in the next lemma, the TM estimator is robust both to a proportion of corrupted samples and heavy-tailed data. Lemma 3. Grant Assumptions 1 and 2 with α ∈ (0, 1] and assume that \|O\| ≤ ηn with η < 1/8. If we fix a confidence level δ ∈ (0, 1) and choose = 8η + 12 log(4/δ)/n, we have \| g TM j (θ) − g j (θ)\| ≤ 7m α,j (θ) 1/(1+α) 4η + 6 log(4/δ) n α/(1+α) ≤ 7M 1/(1+α) α,j 4η + 6 log(4/δ) n α/(1+α) with a probability larger than 1 − δ. The proof of Lemma 3 is given in Appendix B and extends Theorem 1 from Lugosi and Mendelson (2021) to α ∈ (0, 1] instead of α = 1 only. It shows that the TM estimator has the remarkable quality of being simultaneously robust to heavy-tailed and a fraction of corrupted data, as opposed to MOM which is only robust to a limited number of outliers. Note that for the computation of the TM estimator, the splitting n = n/2 ∪ n/2 is a technical theoretical requirement used to induce independence between q , q 1− and the sample partial derivatives (g i j (θ)) i∈ n/2 involved in the average (18). Our implementation does not use this splitting. Comparison with . A comparison between Lemma 3 and the results by pertaining to the corrupted setting is relevant here. We first point out that corruption in is modeled as receiving data from the "η-contaminated" distribution (1 − η)P + ηQ with Q an arbitrary distribution. On the other hand, Lemma 3 considers the more general η-corrupted setting where an η-proportion of the data is replaced by arbitrary outliers after sampling. In this case, Lemma 3 results in a statistical error with a dependence of order √ ηd in the corruption (on the vector euclidean norm). On the other hand, Lemma 1 in yields a better dependence of order √ η log d in the corresponding case. Keep in mind, however, that Algorithm 2 from which achieves this rate requires recursive SVD decompositions to compute a robust gradient making it computationally heavy and impractical for moderately high dimension. Additionally, the relevant results in require a stronger moment assumption on the gradient and impose additional constraints on the corruption rate η. We also mention Algorithm 5 from which yields an even better dependence on the dimension (see their Lemma 2), although it involves a computationally costly procedure as well. Besides, knowledge of the trace and operator norm of the covariance matrix of the estimated vector is required which makes the algorithm more difficult to use in practice. Proposition 3. Grant Assumptions 1 and 2 with α ∈ (0, 1] and \|O\| ≤ ηn. We have P sup θ∈Θ g TM j (θ) − g j (θ) ≤ TM j (δ) ≥ 1 − δ for any j ∈ d with TM j (δ) := 28 M j,α + m L,α n α(1+α) 1/(1+α) 2η + 3 log(4d/δ) + d log(3∆n α/(1+α) /2) n α/(1+α) + L + L j n α/(1+α) where L and m L,α are as in Proposition 1. The proof of Proposition 3 is given in Appendix B and uses an ε-net argument to obtain a uniform bound. The error vector TM (δ) can be plugged into Theorem 1 for example. Similarly to MOM, the resulting statistical error has optimal dependence on the (1 + α)-moments of the partial derivatives (12). Complexity of g TM j (θ). The most demanding part for the computation of g TM j (θ) is the computation of q and q 1− . A naive idea is to sort all n values at an average cost O(n log n) with quicksort for example (Hoare, 1961) and to simply retrieve the desired order statistics afterwards. Of course, better approaches are possible, including the median-of-medians algorithm (not to be confused with MOM), which remarkably manages to keep the cost of finding an order statistic with complexity O(n) even in the worst case (see for instance Chapter 9 of Cormen et al. (2009)). However, the constant hidden in the previous big-O notations seriously impact performances in real-world implementations: we compared several implementations experimentally and concluded that a variant of the quickselect algorithm (Hoare, 1961) was the fastest for this problem. Catoni-Holland estimator This estimator is a variation of the robust mean estimator by Catoni (2012) introduced by Holland and Ikeda (2019a) for robust statistical learning, hence the name "Catoni-Holland", that we will denote g CH j (θ). It is defined as an M-estimator which consists in solving n i=1 ψ g i j (θ) − ζ s j (θ) = 0(19) with respect to ζ, where ψ is an uneven function satisfying ψ(0) = 0, ψ(x) ∼ x when x ∼ 0 and ψ(x) = o(x) when x → +∞ and where s j (θ) > 0 is a scale estimator. An approximate solution can be found using the fixed-point iterations ζ k+1 = ζ k + s j (θ) n n i=1 ψ g i j (θ) − ζ k s j (θ) , which can easily be shown to converge to the desired value thanks to the monotonicity and Lipschitz-property of ψ. Following Holland and Ikeda (2019a), we use the function ψ( Catoni (2012). As explained in Holland and Ikeda (2019a), the scale estimator is given by x) = 2 arctan(exp(x)) − π/2, while functions satisfying − log(1 − x + x 2 /2) ≤ ψ(x) ≤ log(1 + x + x 2 /2) are considered ins j (θ) := σ j (θ) n 2 log(4/δ) ,(20) for a confidence level δ ∈ (0, 1), where σ j (θ) is an estimator of the standard deviation of the partial derivative σ j (θ) := m 1, j (θ) 1/2 = V[ (X θ, Y )X j ] 1/2 , see (12). The estimator σ j (θ) is defined through another M-estimator solution to n i=1 χ g i j (θ) −ḡ j (θ) σ = 0(21) with respect to σ, whereḡ j (θ) = 1 n n i=1 g i j (θ) and χ is an even function satisfying χ(0) < 0 and χ(x) > 0 as x → +∞. We use the same function as in Holland and Ikeda (2019a) given by χ(u) = u 2 /(1 + u 2 ) − c where c is such that Eχ(Z) = 0 for Z a standard Gaussian random variable. To compute σ j (θ) we use also fixed-point iterations σ k+1 = σ k 1 − χ(0) n n i=1 χ g i j (θ) −ḡ j (θ) σ k .(22) We refer to the supplementary material of Holland and Ikeda (2019a) for further details on this procedure. The CH estimator can be understood, once again, as an interpolation between the average and the median of the partial derivatives. Indeed, whenever s is large, the function ψ(·/s) is close to the sign function, which, if used in (19), leads to an M -estimator corresponding to the median ( Van der Vaart, 2000). For s small, ψ(·/s) is close to the identity, so that minimizing (19) leads to an ordinary average. As explained in the next lemma, this estimator is robust to heavytailed data (with α = 1). Lemma 4. Grant Assumptions 1 and 2 with α = 1 and assume that O = ∅ (no outliers). For some failure probability δ > 0, assume that we have, with probability at least 1 − δ/2, that σ j (θ)/C ≤ σ j (θ) ≤ C σ j (θ) for some constant C > 1. Then, we have \| g CH j (θ) − g j (θ)\| ≤ C σ j (θ) 8 log(4/δ) n ≤ C Σ j 8 log(4/δ) n with probability at least 1 − δ, where Σ j = M 1,j = sup θ∈Θ σ j (θ). The proof of Lemma 4 is given in Appendix B and is an almost direct application of the deviation bound from Holland and Ikeda (2019a). If C ≈ 1, the deviation bound of g CH j (·) is better than the ones given in Lemmas 2 and 3 with α = 1. This stems from the fact that the analysis of Catoni's estimator (Catoni, 2012) results in a deviation with the best possible constant (Devroye et al., 2016). However, contrary to MOM and TM, an estimator of the scale is necessary: it makes CH computationally much more demanding (see Figure 1 below), since it requires to perform two fixed-point iterations to approximate both σ j (θ) and g CH j (θ) and it requires Assumption 2 with α = 1 so that σ j (θ) < +∞. Moreover, there is no guaranteed robustness to outliers, a fact confirmed by the numerical experiments performed in Section 6 below. = ∅. Denote L = E[γ X 2 ], σ 2 L = V[γ X 2 ] and assume that for all θ, θ ∈ Θ such that θ − θ ≤ 1/ √ n we have 1 2 σ 2 j ( θ) ≤ σ 2 j (θ) ≤ 2σ 2 j ( θ) and σ j (θ) σ L ≥ 1 √ n . Furthermore, assume that for all θ ∈ Θ, the variance estimator σ j (θ) defined by (21) satisfies σ j (θ)/C ≤ σ j (θ) ≤ C σ j (θ) for some constant C > 1 with probability at least 1 − δ/2. Then, we have P sup θ∈Θ g CH j (θ) − g j (θ) ≤ CH j (δ) ≥ 1 − δ for any j ∈ d with CH j (δ) := 4C 2Σ j + σ L √ n log(4d/δ) + d log(3∆ √ n/2) n + L + L j √ n where L is as in Proposition 1. The proof of Proposition 4 is given in Appendix B. It uses again an ε-net argument combined with a careful control of the variations of g CH j (θ) with respect to θ. Compared with Holland and Ikeda (2019a), we make a different use of the CH estimator: while it is used therein to estimate the whole gradient ∇R(θ) during the robust GD iterations, we use it here to estimate the partial derivatives g j (θ) during iterations of robust CGD. The numerical experiments from Section 6 confirm, in particular, that our approach leads to a considerable speedup and improved statistical performances when compared to Holland and Ikeda (2019a). The statements of Lemma 4 and Proposition 4 require α = 1, while a very recent extension of Catoni's bound is available for α ∈ (0, 1). However, the necessity to estimate the centered (1 + α)-moment subsists (standard-deviation for α = 1). Although iteration (22) may be adapted to this case, theoretical guarantees for it do lack. Note that even for α = 1, the statements of Lemma 4 and Proposition 4 require assumptions on σ 2 j (θ) and σ j (θ): an extension to α ∈ (0, 1] would lead to a set of even more intricate assumptions. Complexity of g CH j (θ). It is not straightforward to analyze the complexity of this estimator, since it involves fixed-point iterations with a number of iterations that can vary from one run to the other. However, each iteration has complexity O(n) and we observe empirically that the number of iterations is of constant order (usually smaller than 10) independently from the required confidence. Therefore, the overall complexity remains in O(n) as demonstrated also by Figure 1 below. The latter also shows that the numerical complexity of CH is larger than that of MOM and TM, which later impacts the overall training time. A comparison of the numerical complexities As explained above, all the considered estimators of the partial derivatives have a numerical complexity O(n). However, they perform different computations and have very different running times in practice. So, in order to compare their actual computational complexities we perform the following experiment. We consider an increasing sample size n between 10 2 and 10 6 on a logarithmic scale and run all the estimators: MOM, TM, CH and ERM, which is the average of the per-sample partial derivatives g i j (θ). We fix their parameters so as to obtain deviation bounds with confidence 1 − δ = 99%: this corresponds to 82 blocks for MOM, = 72/n for TM and δ = 0.01 for CH, but the conclusion is similar with different combinations of parameters. We use random samples with student t(2.1) distribution (a finite variance distribution but with heavy tails, although run times do not differ by much when using different distributions). This leads to the display proposed in Figure 1, where we display the averaged timings over 100 repetitions (together with standard-deviations). We observe that the run times of the estimators increase with a similar slope (on a logarithmic scale) against the sample size, confirming the O(n) complexities. However, their timings differ significantly. MOM and TM share similar timings (TM becomes faster than MOM for large samples) and are about 10 times slower than ERM. CH is the slowest of all and is roughly 50 times slower than ERM. This is of course related to the fact that CH requires to perform the fixed-point iterations each of which roughly costing Θ(n). In all cases, the estimators' complexities remain in O(n) so that the complexity of a single iteration of robust CGD (see Algorithm 1) using either of them is O(n), which is identical to the complexity of a non-robust ERM-based CGD. This means that Algorithm 1 achieves robustness at a limited cost, where the computational difference lies only in the constants in front of the big O notations. Related works Robust statistics have received a longstanding interest and started in the 60s with the pioneering works of Tukey (1960) and Huber (1964). Since then, several works pursued the development of robust statistical methods including non-convex M -estimators (Huber, 2004), 1 tournaments (Devroye and Györfi, 1985;Donoho and Liu, 1988) and methods based on depth functions (Chen et al., 2018;Gao et al., 2020;Mizera et al., 2002), the latter being difficult to use in practice because of their numerical complexity. A renewal of interest has manifested recently, related, on the one hand, to the increasing need for algorithms able to learn from large non-curated datasets and on the other hand, to the de-velopment of robust mean estimators with good theoretical guarantees under weak moment assumptions, including Median-of-Means (MOM) (Nemirovskij and Yudin, 1983;Alon et al., 1999;Jerrum et al., 1986) and Catoni's estimator (Catoni, 2012). Under adversarial corruption (Charikar et al., 2017), several statistical learning problems related to robustness are studied, such as parameter estimation (Lai et al., 2016;Minsker et al., 2018;Diakonikolas et al., 019a;Lugosi and Mendelson, 2021), regression (Klivans et al., 2018;Liu et al., 2020;Cherapanamjeri et al., 2020;Bhatia et al., 2017), classification (Lecué et al., 2020Klivans et al., 2009;Liu and Tao, 2015), PCA (Li, 2017;Candès et al., 2011;Paul et al., 2021) and most recently online learning (van Erven et al., 2021). In the heavy-tailed setting, a robust learning approach introduced in Brownlees et al. (2015) proposes to optimize a robust estimator of the risk based on Catoni's mean estimator (Catoni, 2012) resulting in an implicit estimator for which they show near-optimal guarantees under weak assumptions on the data. However, the new risk may not be convex (even if the considered loss is), so that its minimization may be expensive and lead to an estimator unrelated to the one theoretically studied, potentially making the associated guarantees inapplicable. More recently, an explicit variant was proposed in Zhang and Zhou (2018) which applies Catoni's influence function to each term of the sum defining the empirical risk for linear regression. The associated optimum enjoys a sub-Gaussian bound on the excess risk, albeit with a slow rate since the 1 loss was used. A follow-up extended this result under weaker distribution assumptions . The main drawback of this approach is that the unconventional use of the influence function introduces a considerable amount of bias which appears in the excess risk bounds. Another approach proposed in Minsker et al. (2015); Hsu and Sabato (2016) aims at obtaining a robust estimator by computing standard ERMs on disjoint subsets of the data and aggregating them using a multidimensional MOM. This approach has recently been used as well in Holland (2021) with various aggregation strategies in order to perform robust distributed learning. Although the previous works use easily implementable aggregation procedures, the associated deviation bounds are sub-optimal (see for instance Lugosi and Mendelson (2019a)). Moreover, dividing the data into multiple subsets makes the method impractical for small sample sizes and may introduce bias coming from the choice of such a subdivision. In the setting where an η-proportion of the data consist of arbitrary outliers, a robust metaalgorithm is introduced in Diakonikolas et al. (019b), which repeatedly trains a given base learner and filters outliers based on an eccentricity score. The method reaches the target σ √ η error rate with σ the gradient standard deviation, although the requirement of multiple training rounds may be computationally expensive. More recently, robust solutions to classification problems were proposed in Lecué et al. (2020) by using MOM to estimate the risk and computing gradients on trustworthy data subsets in order to perform descent. A variant was also proposed by the same authors in Lecué et al. (2020) where a pair of parameters is alternately optimized for a min-max objective. The resulting algorithm is efficient numerically, though it requires a vanishing step-size to converge due to the variance coming from gradient estimation. Moreover, the provided theoretical guarantees concern the optimum of the formulated problem but not the optimization algorithm put to use. Several recent papers Holland and Ikeda, 2019b;Holland, 2019;Holland and Ikeda, 2019a;Chen et al., 2017) perform a form of robust gradient descent, where learning is guided by various robust estimators of the true gradient ∇R(θ). Two robust gradient estimation algorithms are proposed in . The first one is a vector analog of MOM where the scalar median is replaced by the geometric median GMed(g 1 , . . . , g K ) := argmin g∈R d K j=1 g − g j 2 ,(23) which can be computed using the algorithm given in Vardi and Zhang (2000). This vector mean estimator enjoys improved concentration properties over the standard mean as shown in Minsker et al. (2015) although these remain sub-optimal (see also Lugosi and Mendelson (2019a)). A line of works (Lugosi and Mendelson, 2019b;Hopkins, 2018;Cherapanamjeri et al., 2019;Depersin and Lecué, 2019;Lugosi and Mendelson, 2021;Lei et al., 2020) specifically addresses the issue of devising efficient procedures with optimal deviation bounds. Supervised learning with robustness to heavy-tails and a limited number of outliers is thus achieved but at a possibly high computational cost. The second algorithm called "Huber gradient estimator" is intended for Huber's -contamination setting. It uses recursive SVD decompositions followed by projections and truncations in order to filter out corruption. The method proves to be robust to data corruption but its computational cost becomes prohibitive as soon as the data has moderately large dimensionality. Theoretical guarantee without strong convexity In this section we provide an upper bound similar to that of Theorem 1, but without the strong convexity condition from Assumption 4. As explained in Theorem 3 below, without strong convexity, the optimization error shrinks at a slower sub-linear rate when compared to Theorem 1 (a well-known fact, see Bubeck (2015)). In order to ensure that robust CGD, which uses "noisy" partial derivatives, remains a descent algorithm, we assume that the parameter set can be written as a product Θ = j∈ d Θ j and replace the iterations (6) (corresponding to Line 5 in Algorithm 1) by θ (t+1) j ← proj Θ j θ (t) j − β j τ j g j (θ (t) ) if j = j t θ (t+1) j ← θ (t) j otherwise,(24) where proj Θ j is the projection onto Θ j and τ is the soft-thresholding operator given by τ (x) = sign(x)(\|x\| − ) + with (x) + = max(x, 0). In Theorem 3 below we use j = j (δ), the j-th coordinate of the error vector from Definition 1, which is instantiated for each robust estimator in Section 3. Since it depends on the moment m α,j , it is not observable, so we propose in Lemma 6 from Appendix A.2 an observable upper bound deviation for it based on MOM. This use of soft-thresholding of the partial derivatives can be understood as a form of partial derivatives (or gradient) clipping. However, note that it is rather a theoretical artifact than something to use in practice (we never use τ in our numerical experiments from Section 6 below). Indeed, the operator τ naturally appears for the following simple reason: consider a convex Lsmooth scalar function f : R → R with derivative g(x) := f (x). An iteration of gradient descent from x 0 uses an increment δ that minimizes the right-hand side of the following inequality: f (x 0 + δ) ≤ Q(δ, x 0 ) := f (x 0 ) + δg(x 0 ) + L 2 δ 2 , namely argmin δ Q(δ, x 0 ) = −g(x 0 )/L leading to the iterate x 0 −g(x 0 )/L with ensured improvement of the objective. In our context, g(x) is unknown and we use an estimator g(x) satisfying \| g(x) − g(x)\| ≤ with a large probability. Taking this uncertainty into account leads to the upper bound f (x 0 + δ) ≤ Q(δ, x 0 ) := f (x 0 ) + δ g(x 0 ) + L 2 δ 2 + \|δ\|, and, after projection onto the parameter set, to the iteration (24) since argmin δ Q(δ, x 0 ) = x 0 − τ ( g(x 0 ))/L, with guaranteed decrease of the objective. The clipping of partial derivatives is unnecessary in the strongly convex case since each iteration translates into a contraction of the excess risk, so that the degradations caused by the gradient errors remain controlled (see the proof of Theorem 1). No such contraction can be established without strong convexity, and clipping prevents gradient errors to accumulate uncontrollably. Theorem 3. Grant Assumptions 1 and 3 with Θ = j∈ d Θ j . Let θ (T ) be the output of Algorithm 1 where we replace iterations (6) by (24) with step-sizes β j = 1/L j , an initial iterate θ (0) ∈ Θ, uniform coordinates sampling p j = 1/d and estimators of the partial derivatives with error vector (·). Then, we have with probability at least 1 − δ E R(θ (T ) )] − R ≤ d T + 1 j∈ d L j 2 θ (0) j − θ j 2 + R(θ (0) ) + 2 (δ) 2 T + 1 T t=0 θ (t) − θ 2 , where the expectation is w.r.t the sampling of the coordinates. Moreover, we have θ (t) − θ 2 ≤ θ (t−1) − θ 2 with the same probability, for all t ∈ T . The proof of Theorem 3 is given in Appendix B and is based on the proof of Theorem 5 from Nesterov (2012) and Theorem 1 from Shalev-Shwartz and Tewari (2011) while managing noisy partial derivatives. The optimization error term vanishes at a sublinear 1/T rate and is initially of order R(θ (0) ) plus the potential Φ(θ) = d j=1 L j (θ j − θ j ) 2 /2 which is instrumental in the proof. Notice that (δ) 2 appears without the square which translates into "slow" 1/ √ n rates instead of "fast" 1/n rates stated achieved by the bounds from Section 2. This degradation is an unavoidable consequence of the loss of strong convexity of the risk (Srebro et al., 2010). Numerical Experiments The theoretical results given in Sections 2, 3 and 5 can be applied to a wide range of linear methods for supervised learning, with guaranteed robustness both with respect to heavy-tailed data and outliers. We perform below experiments that confirm these robustness properties for several tasks (regression, binary classification and multi-class classification) on several datasets including a comparison with many baselines including the state-of-the-art. Algorithms The algorithms introduced in this paper are compared with several baselines among the following large set of algorithms. For all algorithms, we use, unless specified otherwise, the least-squares loss for regression, and the logistic loss for classification (both for binary and multiclass problems, using the multiclass logistic loss). The algorithms studied and compared below can be used easily in a few lines of Python code with our library called linlearn, open-sourced under the BSD-3 License on GitHub and available here: https://github.com/linlearn/linlearn. This library follows the API conventions of scikit-learn (Pedregosa et al., 2011). CGD algorithms: MOM, CH, TM and CGD ERM. The MOM, CH and TM algorithms are the different variants of robust CGD (Algorithm 1) introduced in this paper, respectively based on medianof-means, trimmed mean and Catoni-Holland estimators of the partial derivatives introduced in Section 3. We include also CGD ERM which is CGD using a non-robust estimation of the partial derivatives based on a mean. GD algorithms: ERM, LLM, HG, GMOM, CH GD and Oracle. These are all GD algorithms using different estimators of the gradients. ERM uses a non-robust gradient based on a simple mean. LLM corresponds to Algorithm 1 from Lecué et al. (2020). It uses a MOM estimation of the risk and performs GD using gradients computed as the mean of the gradients from the block corresponding to the median of the risk. HG is Algorithm 2 from , called Huber Gradient Estimator, which uses recursive SVD decompositions and truncations to compute a robust gradient. GMOM is Algorithm 3 from , which estimates gradients using a geometric MOM (based on the geometric median). CH GD is the robust GD algorithm from Holland and Ikeda (2019a), which uses gradients computed as coordinate-wise CH estimators. We consider also Oracle, which is GD performed with "oracle" gradients, namely the gradient of the unobserved true risk (only available for linear regression experiments using simulated data). Extra algorithms: RANSAC, HUBER and LAD. We consider also the following extra algorithms. For regression, we consider RANSAC (Fischler and Bolles, 1981), using the implementation available in the scikit-learn library (Pedregosa et al., 2011). HUBER stands for ERM learning with the modified Huber loss (Zhang, 2004) for classification and Huber loss (Owen, 2007) for regression. LAD is ERM learning using the least absolute deviation loss (Edgeworth, 1887), namely regression using the mean absolute error instead of least-squares. Regression on simulated datasets We consider the following simulation setting for linear regression with the square loss. We generate features X ∈ R d with d = 5 with a non-isotropic Gaussian distribution with covariance matrix Σ and labels Y = X θ + ξ for a fixed θ ∈ R d and simulated noise ξ. Since all distributions are known in such simulated data, we can compute the true risk and true gradients (used in Oracle). We consider the following simulation settings: (a) ξ is centered Gaussian; (b) ξ is Student with ν = 2.1 degrees of freedom (heavy-tailed noise). We consider then settings (c), (d), (e) and (f) where ξ is the same in (b) but 1% of the data is replaced by outliers as follows. For case (c), X ∈ R 5 is replaced by a constant equal to λ max (Σ) (largest eigenvalue of Σ) and labels are replaced by 2y max with y max = max i∈I \|y i \|; for (d) we do the same as (c) and additionally multiply labels by −1 with probability 1/2; for (e) we sample X = 10λ max (Σ)v + Z where v ∈ R 5 is a fixed unit vector chosen at random and Z is a standard Gaussian vector and labels are i.i.d. Bernoulli random variables; finally for (f) we sample X = 10λ max (Σ)V where V is uniform on the unit sphere and labels y = y max × (ε + U ) where ε is a Rademacher variable and U is uniform in [−1/5, 1/5]. For this experiment on simulated datasets, we fix the parameters of the robust estimators of the partial derivatives using the confidence level δ = 0.01 and the number of outliers for MOM and TM. We report, for all considered simulation settings (a)-(f), the average over 30 repetitions of the excess risk for the square loss (y-axis) of all considered algorithms along their iterations (x-axis, corresponding to cycles for CGD and iterations for GD) in Figure 2. We observe that CGD-based algorithms generally converge faster than GD-based ones, independently of the quality of the optimum found. For (a) with Gaussian noise and no outliers, the final performance of all algorithms is roughly similar to that of ERM (as expected since the sample mean has optimal deviation guarantees for sub-Gaussian distributions) except for LLM and HG that converge slowly. For (b) with heavy-tailed noise and no outliers, ERM clearly degrades when compared to robust methods, with HG reaching the best result and LLM the worst. For settings (c)-(f) with heavy-tailed noise and outliers, we observe different behaviours. We observe that ERM and CH are the most sensitive to outliers, especially in setting (c) and (e) (single-direction corruption of the gradients), where GMOM and HG (based on robust gradient estimation) perform best while MOM and TM are close competitors. For settings (d) and (f) where gradients can be corrupted in multiple directions, the performance difference between GMOM/HG and MOM/TM is small. A remarkable property to keep in mind is that MOM/TM always converge faster. Finally, while we observe that LLM is robust to heavy tails and outliers, its use of a median mini-batch and vanishing descent steps makes it unstable and prevents it from converging to a good minimum, compared to other algorithms. Classification on several datasets We consider classification tasks (binary and multiclass) on several datasets from the UCI Machine Learning Repository (Dua and Graff, 2017), see Appendix A.3 for more details. We use the logistic loss for binary classification and the multiclass logistic loss for multiclass problems. For k-class problems with k > 2, the iterates are d × k matrices and CGD is performed block-wise along the class axis. In this case, a CGD cycle performs again d iterations (one for each feature coordinate) and each iteration updates the k corresponding model weights (a form of block coordinate gradient descent, see Blondel et al. (2013) for arguments in favor of this approach). For each considered dataset, we corrupt an increasing random fraction of samples with uninformative outliers or a heavy-tailed noise. Each algorithm is hyper-optimized using cross-validation over an appropriate grid of hyper-parameters. See Appendix A.3 for further details about the experiments. Then, each algorithm is trained again using the full training dataset 10 times over to account for the randomness lying within each method (although most procedures remain very stable across runs) and we finally report in Figure 3 the median accuracy obtained on a 15% test-set (y-axis) for each dataset, corruption level (x-axis) and algorithm. We observe that, as expected, the accuracy of each algorithm deteriorates with an increasing proportion of corrupted samples. We observe that the robust CGD algorithms introduced in this paper are almost always superior, to all the considered baselines, and only suffer from a reasonable decrease in accuracy along the x-axis (from 0% to 40% corrupted samples) compared to all baselines. In particular, TM and MOM are generally the best with GMOM being the closest competitor, and as expected from the theory, CH is less robust to corruption than both TM and MOM. Finally, note that the mere use of CGD instead of GD can give a significant advantage in order to find better optima as can be seen for the gas and statlog datasets. In order to illustrate the computational performance of each method, we report in Figure 4 the test accuracy (y-axis) against the training time (x-axis) along iterations of each algorithm for two datasets (rows) and 0%, 15% and 30% corruption (resp. first, middle and last column). With 0% corruption (first column), most algorithms reach their final accuracy within few iterations for the two considered datasets and our algorithms are somewhat slower than standard methods such as ERM and HUBER. When corruption is present, our robust CGD algorithms reach a better accuracy, and they do so faster than other robust algorithms, such as CH GD and GMOM. Also, we can observe on this display, once again, the lack of stability of LLM. Regression on several datasets We consider the same experimental setting (data corruption, hyper-optimization of algorithms) as in Section 6.3 but on different datasets from the UCI Machine Learning Database for regression tasks, see Appendix A.3 for details. We use the square loss for training and use the mean squared error (MSE) as a test metric, excepted for HUBER, RANSAC and LAD which proceed differently. We report the results in Figures 5 and 6. Figure 5 shows the test MSE (y-axis) against the proportion of corrupted samples (x-axis) for several datasets and algorithms while Figure 6 displays the test MSE against the training time analogously to Figure 4. Note that RANSAC, HUBER and LAD appear through vertical lines only in Figure 6 since these use the scikit-learn implementations that do not give access to the training history. We observe that TM and MOM are, once again, clear favorites. Despite the fact that HG and GMOM prove to be very robust and are able to improve MOM and TM in certain instances by a small margin, their running times is slower and for some datasets orders of magnitude larger, as observed in Figure 6. This confirms the results observed as well on classification problems, that our robust CGD algorithms (TM and MOM) offer an excellent compromise : Test accuracy (y-axis) against computation time (x-axis) along training iterations on two datasets (rows) for 0% corruption (first column), 15% corruption (middle column) and 30% corruption (last column). between statistical accuracy, robustness and computational effort. Note also that we observe again the strong sensitivity of CH to outliers and the unstable performance of LLM. Conclusion In this paper, we introduce new robust algorithms for supervised learning by combining two ingredients: robust CGD and several robust estimators of the partial derivatives. We derive convergence results for several variants of CGD with noisy partial derivatives and prove deviation bounds for all the considered robust estimators of the partial derivatives under somewhat minimal moment assumptions, including cases with infinite variance, and the presence of arbitrary outliers (except for the CH estimator). This leads to very robust learning algorithms, with a numerical cost comparable to that of non-robust approaches based on empirical risk minimization, since it lets us bypass the need of a robust vector mean estimator and allows to update model weights immediately using a robust estimator of a single partial derivative only. This is substantiated in our numerical experiments, that confirm the fact that our approach offers an excellent compromise between statistical accuracy, robustness and computational effort. Perspectives include robust learning algorithms in high dimension, achieving sparsity-aware generalization bounds, which is beyond the scope of this paper, since it would require different algorithms based on methods such as mirror descent with an appropriately chosen divergence, see for instance Shalev-Shwartz and Tewari (2011) A Supplementary theoretical results and details on experiments A.1 The Lipschitz constants L j are unknown The step-sizes (β j ) j∈ d used in Theorems 1 and 2 are given by β j = 1/L j , where the Lipschitz constants L j are defined by (8). This makes them non-observable, since they depend on the unknown distribution of the non-corrupted features P X i for i ∈ I. We cannot use linesearch (Armijo, 1966) here, since it requires to evaluate the objective R(θ), which is unknown as well. In order to provide theoretical guarantees similar to that of Theorem 1 without knowing (L j ) d j=1 , we use the following approach. First, we use the upper bound U j := γE (X j ) 2 ≥ L j ,(25) which holds under Assumption 1 and estimate E[(X j ) 2 ] to build a robust estimator of U j . In order to obtain an observable upper bound and to control its deviation with a large probability, we introduce the following condition. Definition 2. We say that a real random variable Z satisfies the L ζ -L ξ condition with constant C ≥ 1 whenever it satisfies E \|Z − EZ\| ζ 1/ζ ≤ C E \|Z − EZ\| ξ 1/ξ .(26) Using this condition, we can use the MOM estimator to obtain a high probability upper bound on E[(X j ) 2 ] as stated in the following lemma. Lemma 5. Grant Assumption 2 with α ∈ (0, 1] and suppose that for all j ∈ d , the variable (X j ) 2 satisfies the L (1+α) -L 1 condition with a known constant C. For any fixed j ∈ d , let σ 2 j be the MOM estimator of E[(X j ) 2 ] with K blocks. If \|O\| ≤ K/12, we have P 1 − 12 1/(1+α) C K n α/(1+α) −1 σ 2 j ≤ E[(X j ) 2 ] ≤ exp(−K/18). If we fix a confidence level δ ∈ (0, 1) and choose K := 18 log(1/δ) , we have 1 − 216 1/(1+α) C log(1/δ) n α/(1+α) −1 σ 2 j > E[(X j ) 2 ] with a probability larger than 1 − δ. The proof of Lemma 5 is given in Appendix B. Denoting U j the upper bounds it provides on E[(X j ) 2 ], we can readily bound the Lipschitz constants as L j ≤ γ U j which leads to the following statement. Corollary 1. Grant the same assumptions as in Theorem 1 and Proposition 1. Suppose additionally that for all j ∈ d , the variable (X j ) 2 satisfies the L (1+α) -L 1 condition with a known constant C and fix δ ∈ (0, 1). Let θ (T ) be the output of Algorithm 1 with step-sizes β j = 1/L j where L j := γ U j and U j are the upper bounds from Lemma 5 with confidence δ/2d, an initial iterate θ (0) , importance sampling distribution p j = L j / k∈ d L k and estimators of the partial derivatives with error vector (·). Then, we have E R(θ (T ) )] − R ≤ (R(θ (0) ) − R ) 1 − λ j∈ d L j T + 1 2λ (δ/2) 2 2(27) with probability at least 1 − δ. The proof of Corollary 1 is given in Appendix B. It is a direct consequence of Theorem 1 and Lemma 5 and shows that an upper bound similar to that of Theorem 1 can be achieved with observable step-sizes. One may argue that the L (1+α) -L 1 condition simply bypasses the difficulty of deriving an observable upper bound by arbitrarily assuming that a ratio of moments is observed. However, we point out that a hypothesis of this nature is indispensable to obtain bounds such as the one above (alternatively, consider a real random variable with an infinitesimal mass drifting towards infinity). In fact, the L (1+α) -L 1 condition is much weaker than the requirement of boundedness (with known range) common to most known empirical bounds (Maurer and Pontil, 2009;Audibert et al., 2009;Mnih et al., 2008). A.2 Observable upper bound for the moment m α,j Since the moment m α,j , it is not observable, so we propose in Lemma 6 below an observable upper bound deviation for it based on MOM. Let us introduce now a robust estimator m MOM α,j (θ) of the unknown moment m α,j (θ) using the following "two-step" MOM procedure. First, we compute g MOM j (θ), the MOM estimator of g j (θ) with K blocks given by (14). Then, we compute again a MOM estimator on \|g i j (θ) − g MOM j (θ)\| 1+α for i ∈ n , namely m MOM α,j (θ) := median m (1) α,j (θ), . . . , m (K) α,j (θ) ,(28) where m (k) α,j (θ) := 1 \|B k \| i∈B k g i j (θ) − g MOM j (θ) 1+α , using uniformly sampled blocks B 1 , . . . , B K of equal size that form a partition of n . Lemma 6. Grant Assumptions 1 and 2 with α ∈ (0, 1] and suppose that for all j ∈ d and θ ∈ Θ the partial derivatives (X θ, Y )X j satisfy the L (1+α) 2 -L (1+α) condition with known constant C for any j ∈ d (see Definition 2). Then, if \|O\| ≤ K/12, we have P m MOM α,j (θ) ≤ (1 − κ)m α,j (θ) ≤ 2 exp(−K/18) where κ = + 24(1 + α) (1+ )K n α/(1+α) and = (24(1 + C (1+α) 2 )) 1/(1+α) K n ) α/(1+α) . The proof of Lemma 6 is given in Appendix B. A.3 Experimental details We provide in this section supplementary information about the numerical experiments conducted in Section 6. A.3.1 Datasets The main characteristics of the datasets used from the UCI repository are given in Table 2 and their direct URLs are given in Table 3. A.3.2 Data corruption For a given corruption rate η, we obtain a corrupted version of a dataset by replacing an η-fraction of its samples with uninformative elements. For a dataset of size n we choose O ⊂ n which satisfies \|O\| = ηn up to integer rounding. The corruption is applied prior to any preprocessing except in the regression case where label scaling is applied before. The affected subset is chosen (Candanedo and Feldheim, 2016) 20,560 5 0 2 gas (Vergara et al., 2012) 13,910 128 0 6 drybean (Koklu and Ozkan, 2020) 13,611 16 0 7 energy (Candanedo et al., 2017) 19,735 27 0 bike (Fanaee-T and Gama, 2014) 17 uniformly at random. Since many datasets contain both continuous and categorical data features, we distinguish two different corruption mechanisms which we apply depending on their nature. The labels are corrupted as continuous or categorical values when the task is respectively regression or classification. Denote X ∈ R n×(d+1) the data matrix with the vector of labels added to its columns. Let J ⊂ d + 1 denote the index of continuous columns, we compute µ j and σ j their empirical means and standard deviations respectively for j ∈ J. We also sample a random unit vector u of size \| J\|. • For categorical feature columns, for each corrupted index i ∈ O, we replace X i,j with a uniformly sampled value among {X •,j } i.e. among the possible modalities of the categorical feature in question. • For continuous features, for each corrupted index i ∈ O, we replace X i, J with equal probability with one of the following possibilities: a vector ξ sampled coordinatewise according to ξ j = r j + 5 σ j ν where r j is a value randomly picked in the column X •,j and ν is a sample from the Student distribution with 2.1 degrees of freedom. a vector ξ sampled coordinatewise according to ξ j = µ j + 5 σ j u j + z where z is a standard gaussian. a vector ξ sampled according to ξ = µ + 5 σ ⊗ w where w is a uniformly sampled unit vector. A.4 Preprocessing We apply a minimal amount of preprocessing to the data before applying the considered learning algorithms. More precisely, categorical features are one-hot encoded while centering and standard scaling is applied to the continuous features. A.5 Parameter hyper-optimization We use the hyperopt library to find optimal hyper-parameters for all algorithms. For each dataset, the available samples are split into training, validation and test sets with proportions 70%, 15%, 15%. Whenever corruption is applied, it is restricted to the training set. We run 50 rounds of hyper-parameter optimization which are trained on the training set and evaluated on the validation set. Then, we report results on the test set for all hyper-optimized algorithms. For each algorithm, the hyper-parameters are tried out using the following sampling mechanism (the one we specify to hyperopt): • MOM, GMOM, LLM: we optimize the number of blocks K used for the median-of-means computations. This is done through a block size = K/n hyper-parameter chosen with loguniform distribution over [10 −5 , 0.2] • CH and CH GD: we optimize the confidence δ used to define the CH estimator's scale parameter (see Equation (20) This proof follows, with minor modifications, the proof of Theorem 1 from Wright (2015). Using Definition 1 , we obtain P[E] ≥ 1 − δ where E := ∀j ∈ d , ∀t ∈ [T ], g j (θ (t) ) − g j (θ (t) ) ≤ j (δ) . (29) Let us recall that e j stands for the j-th canonical basis of R d and that, as described in Algorithm 1, we have θ (t+1) = θ (t) − β jt g t e jt , where we use the notations g t = g jt (θ (t) ) and g t = g jt (θ (t) ) and where we recall that j 1 , . . . , j t is a i.i.d sequence with distribution p. We introduce also j := j (δ). Using Assumption 3, we obtain R(θ (t+1) ) = R θ (t) − β jt g t e jt ≤ R(θ (t) ) − g(θ (t) ), β jt g t e jt + L jt 2 β 2 jt g 2 t = R(θ (t) ) − β jt g 2 t − β jt g t ( g t − g t ) + L jt β 2 jt 2 g 2 t + ( g t − g t ) 2 + 2g t ( g t − g t ) = R(θ (t) ) − β jt g t (1 − L jt β jt )( g t − g t ) − β jt 1 − L jt β jt 2 g 2 t + L jt β 2 jt 2 ( g t − g t ) 2 = R(θ (t) ) − 1 2L jt g 2 t + 1 2L jt ( g t − g t ) 2 ≤ R(θ (t) ) − 1 2L jt g 2 t + 2 jt 2L jt(30) on the event E, where we used the choice β jt = 1/L jt and the fact that \| g t − g t \| ≤ jt on E. Since j 1 , . . . , j t is a i.i.d sequence with distribution p, we have for any (j 1 , . . . , j t−1 )-measurable and integrable function ϕ that E t−1 ϕ(j t ) = j∈ d ϕ(j)p j , where we denote for short the conditional expectation E t−1 [·] = E t−1 [·\|j 1 , . . . , j t−1 ]. So, taking E t−1 [·] on both sides of (30) leads, whenever p j = L j / d k=1 L k , to E t−1 R(θ (t+1) ) ≤ R(θ (t) ) − 1 2 k L k g(θ (t) ) 2 + 1 2 k L k Ξ, where we introduced Ξ := (δ) 2 2 , while it leads to E t−1 R(θ (t+1) ) ≤ R(θ (t) ) − 1 2L max d g(θ (t) ) 2 + 1 2dL min Ξ whenever p j = 1/d, simply using L min ≤ L j ≤ L max . In order to treat both cases simultaneously, considerL = k=1 L k and¯ = Ξ/(2 k L k ) whenever p j = L j / d k=1 L k andL = dL max and /(2dL min ) whenever p j = 1/d and continue from the inequality E t−1 R(θ (t+1) ) ≤ R(θ (t) ) − 1 2L g(θ (t) ) 2 +¯ . Introducing φ t := E R(θ (t) ) − R and taking the expectation w.r.t. all j 1 , . . . , j t we obtain φ t+1 ≤ φ t − 1 2L E g(θ (t) ) 2 +¯ .(31) Using Inequality (9) with θ 1 = θ (t) gives R(θ 2 ) ≥ R(θ (t) ) + ∇R(θ (t) ), θ 2 − θ (t) + λ 2 θ 2 − θ (t) 2 for any θ 2 ∈ R d , so that by minimizing both sides with respect to θ 2 leads to R ≥ R(θ (t) ) − 1 2λ g(θ (t) ) 2 namely φ t ≤ 1 2λ E g(θ (t) ) 2 , by taking the expectation on both sides. Together with (31) this leads to the following approximate contraction property: φ t+1 ≤ φ t 1 − λ L +¯ , and by iterating t = 1, . . . , T to φ T ≤ φ 0 1 − λ L T +¯ L λ , which allows to conclude the Proof of Theorem 1. B.2 Proof of Theorem 2 This proof reuses ideas from Li et al. (2017) and Beck and Tetruashvili (2013) and adapts them to our context where the gradient coordinates are replaced with high confidence approximations. Without loss of generality, we initially assume that the coordinates are cycled upon in the natural order. We condition on the event (29) which holds with probability ≥ 1 − δ as in the proof of Theorem 1 and denote j = j (δ) and Euc = (δ) . Let the iterations be denoted as θ (t) for t = 0, . . . , T and θ (t) i+1 = θ (t) i − β i+1 g(θ (t) i ) i+1 e i+1 for i = 0, . . . , d − 1 with β i = 1/L i , θ (t) 0 = θ (t) and θ (t) d = θ (t+1) . With these notations we have R(θ (t) ) − R(θ (t+1) ) = d−1 i=0 R(θ (t) i ) − R(θ (t) i+1 ). Similarly to (30) in the proof of Theorem 1 we find: R(θ (t) i ) − R(θ (t) i+1 ) ≥ 1 2L i+1 g(θ (t) i ) 2 i+1 − 2 i+1 , leading to R(θ (t) ) − R(θ (t+1) ) ≥ d−1 i=0 1 2L i+1 g(θ (t) i ) 2 i+1 − 1 2L min d−1 i=0 2 i+1 .(32) The following aims to find a relationship between d−1 i=0 1 2L i+1 g(θ (t) i ) 2 i+1 and g(θ (t) ) 2 2 which we do by comparing coordinates. For the first step in a cycle we have g(θ (t) ) 1 = g(θ (t) 0 ) 1 because θ (t) = θ (t) 0 . Let j ∈ {1, . . . , d − 1}, by the Mean Value Theorem, there exists γ (t) j ∈ R d such that we have: g(θ (t) ) j+1 = g(θ (t) ) j+1 − g(θ (t) j ) j+1 + g(θ (t) j ) j+1 = ∇g j+1 (γ (t) j ) θ (t) − θ (t) j + g(θ (t) j ) j+1 = ∂R(γ (t) j ) ∂ j+1 ∂ 1 , . . . , ∂R(γ (t) j ) ∂ j+1 ∂ j , 0, . . . , 0 (θ (t) − θ (t) j ) 1 , . . . , (θ (t) − θ (t) j ) j , 0, . . . , 0 + g(θ (t) j ) j+1 = [H j+1,1 , . . . , H j+1,j , 0, . . . , 0] g 1 (θ (t) 0 ) L 1 , . . . , g j (θ (t) j−1 ) L j , 0, . . . , 0 + g(θ (t) j ) j+1 = [H j+1,1 , . . . , H j+1,j , 0, . . . , 0] g 1 (θ (t) 0 ) + δ (t) 1 L 1 , . . . , g j (θ (t) j−1 ) + δ (t) j L j , 0, . . . , 0 + g(θ (t) j ) j+1 = H j+1,1 √ L 1 , . . . , H j+1,j L j , L j+1 , 0, . . . , 0 h j+1 g 1 (θ (t) 0 ) √ L 1 , . . . , g d (θ (t) d−1 ) √ L d gt + [H j+1,1 , . . . , H j+1,j , 0, . . . , 0] h j+1 δ (t) 1 L 1 , . . . , δ (t) d L d = h j+1 g t + h j+1 A −1 δ (t) , where we introduced the following quantities: A ∈ R d equal to A = diag(L j ) d j=1 , the vector δ (t) ∈ R d is such that δ (t) j = g(θ (t) j−1 ) j − g(θ (t) j−1 ) j which satisfies \|δ (t) j \| ≤ j , the matrix H = (h 1 , . . . , h d ) and H = A 1/2 +HA −1/2 = ( h 1 , . . . , h d ) . In the case j = 0 the vector h j+1 = h 1 is simply zero. This allows us to obtain the following estimation: g(θ (t) ) 2 = d j=1 g(θ (t) ) 2 j = d j=1 ( h j g t + h j A −1 δ (t) ) 2 ≤ d j=1 2( h j g t ) 2 + 2(h j A −1 δ (t) ) 2 = 2 H g t 2 + 2 HA −1 δ (t) 2 ≤ 2 H 2 g t 2 + 2 L 2 min H 2 2 Euc = 2 H 2 d−1 i=0 1 L i+1 g(θ (t) i ) 2 i+1 + 2 L 2 min H 2 2 Euc .(33) We can bound the spectral norm H as follows: H 2 = A 1/2 + HA −1/2 2 ≤ 2 A 1/2 2 + 2 HA −1/2 2 ≤ 2 L max + H 2 L min . For H , we use the coordinate-wise Lipschitz-smoothness in order to find H 2 ≤ H 2 F = d j=1 h j 2 ≤ d j=1 ∇g j (γ (t) j−1 ) 2 ≤ d j=1 L 2 j ≤ dL 2 max . Combining the previous inequality with (32) and (33), we find: R(θ (t) ) − R(θ (t+1) ) ≥ 1 8L max (1 + d Lmax L min ) g(θ (t) ) 2 − 2 Euc 2 ( 1 L min + d Lmax L min 2 2L max (1 + d Lmax L min ) ≥ 1 8L max (1 + d Lmax L min ) g(θ (t) ) 2 − 2 Euc 2 1 L min + 1 2L min dL max /L min 1 + d Lmax L min ≥ 1 8L max (1 + d Lmax L min ) =:κ g(θ (t) ) 2 − 3 4L min 2 Euc , where the last step uses that dLmax/L min 1+d Lmax L min ≤ 1. Using λ-strong convexity by choosing θ 1 = θ (t) in inequality (9) and minimizing both sides w.r.t. θ 2 we obtain: R(θ (t) ) − R ≤ 1 2λ g(θ (t) ) 2 , which combined with the previous inequality yields the contraction inequality: R(θ (t+1) ) − R ≤ (R(θ (t) ) − R )(1 − 2λκ) + 3 4L min 2 Euc , and after T iterations we have: R(θ (T ) ) − R ≤ (R(θ (0) ) − R )(1 − 2λκ) T + 3 2 Euc 8L min λκ , which concludes the proof of Theorem 2. To see that the proof still holds for any choice of coordinates satisfying the conditions in the main claim, notice that the computations leading up to Inequality (33) work all the same if one were to apply a permutation to the coordinates beforehand. B.3 Convergence of the parameter error We state and prove a result about the linear convergence of the parameter under strong convexity. Theorem 4. Grant Assumptions 1, 3 and 4. Let θ (T ) be the output of Algorithm 1 with constant step-size β = 2 λ+L , an initial iterate θ (0) , uniform coordinates sampling p j = 1/d and estimators of the partial derivatives with error vector (·). Then, we have E θ (T ) − θ 2 ≤ θ (0) − θ 2 1 − 2βλL d(λ + L) T + √ d(λ + L) λL (δ) 2(34) with probability at least 1 − δ, where the expectation is w.r.t. the sampling of the coordinates. Proof. As in the proof of Theorem 1, let ( g j (θ)) d j=1 be the estimators used and introduce the notations g t = g jt (θ (t) ) and g t = g jt (θ (t) ). We also condition on the event (29) which holds with probability 1 − δ and use the notations Euc = (δ) 2 and j = j (δ). We denote · L 2 the L 2 -norm w.r.t. the distribution over j t i.e. for a random variable ξ we have ξ L 2 = E jt ξ 2 . We compute: θ (t+1) − θ L 2 = θ (t) − β jt g t e jt − θ L 2 ≤ θ (t) − β jt g t e jt − θ L 2 + β jt ( g t − g t ) L 2 . (35) We first treat the first term of (35), in the case of uniform sampling with equal step-sizes β j = β we have: θ (t) − βg t e jt − θ 2 = θ (t) − θ 2 + β 2 g 2 t − 2β g t e jt , θ (t) − θ . By taking the expectation w.r.t. the random coordinate j t we find: θ (t) − βg t e jt − θ 2 L 2 = E θ (t) − βg t e jt − θ 2 = E θ (t) − θ 2 + β 2 d E g(θ (t) ) 2 − 2 β d E g(θ (t) ), θ (t) − θ = E θ (t) − θ 2 + β d 2 E g(θ (t) ) 2 − 2 β d E g(θ (t) ), θ (t) − θ + β 2 d E g(θ (t) ) 2 1 − 1 d ≤ E θ (t) − θ 2 1 − 2βλL d(λ + L) + β d β d − 2 λ + L E g(θ (t) ) 2 + β 2 d E g(θ (t) ) 2 1 − 1 d = E θ (t) − θ 2 1 − 2βλL d(λ + L) + β d β − 2 λ + L E g(θ (t) ) 2 ≤ E θ (t) − θ 2 1 − 2βλL d(λ + L) =:κ 2 . The first inequality is obtained by applying inequality (2.1.15) from Nesterov (2014) (see also Bubeck (2015) Lemma 3.11) and the second one is due to the choice of β. We can bound the second term as follows: g t − g t 2 L 2 = E jt g t − g t 2 = 1 d d j=1 g j (θ (t) ) − g j (θ (t) ) 2 ≤ 2 Euc d . Combining the latter with the former bound, we obtain the approximate contraction: θ (t+1) − θ L 2 ≤ κ θ (t) − θ L 2 + β Euc √ d . By iterating this argument on T rounds we find that: θ (T ) − θ L 2 ≤ κ T θ (0) − θ L 2 + β Euc √ d(1 − κ) . Finally, the following inequality yields the result in the case of uniform sampling: 1 1 − κ ≤ 1 + 1 − 2βλL d(λ+L) 2βλL d(λ+L) ≤ d(λ + L) βλL . B.4 Proof of Lemma 1 Let θ ∈ Θ, using Assumption 1 we have: \| (θ X, Y )\| ≤ C ,1 + C ,2 \|θ X − Y \| q ≤ C ,1 + 2 q−1 C ,2 (\|θ X\| q + \|Y \| q ). Taking the expectation and using Assumption 2 shows that the risk R(θ) is well defined (recall that q ≤ 2). Next, since 1 ≤ q ≤ 2, simple algebra gives (θ X, Y )X j 1+α ≤ C ,1 + C ,2 \|θ X − Y \| q−1 X j 1+α ≤ 2 α C ,1 X j 1+α + (C ,2 (\|(θ X) q−1 X j \| + \|Y q−1 X j \|)) 1+α ≤ 2 α C ,1 X j 1+α + C ,2 d k=1 \|θ k \| q−1 \|(X k ) q−1 X j \| + \|Y q−1 X j \| 1+α ≤ 2 α C ,1 X j 1+α + 2 α (C ,2 ) 1+α d α d k=1 \|θ k \| (q−1)(1+α) \|(X k ) q−1 X j \| 1+α + \|Y q−1 X j \| 1+α . Given Assumption 2, it is straightforward that E\|X j 1+α < ∞ and E\|Y q−1 X j \| 1+α < ∞. Moreover, using a Hölder inequality with exponents a = q(1+α) (q−1)(1+α) and b = q (the case q = 1 is trivial) we find: E (X k ) q−1 X j 1+α ≤ E X k q(1+α) 1/a E X j q(1+α) 1/b , which is finite under Assumption 2. This concludes the proof of Lemma 1. B.5 Proof of Lemma 2 This proof follows a standard argument from Lugosi and Mendelson (2019a); Geoffrey et al. (2020) in which we use a Lemma from Bubeck et al. (2013) in order to control the (1+α)-moment of the block means instead of their variance. Indeed, we know from Lemma 1 that under Assumptions 1 and 2, the gradient coordinates have finite (1+α)-moments, namely E[\| (X θ, Y )X j \| 1+α ] < +∞ for any j ∈ d . Recall that ( g (k) j (θ)) k∈ K stands for the block-wise empirical mean given by Equation (15) and introduce the set of non-corrupted block indices given by K = {k ∈ K : B k ∩ O = ∅}. We will initially assume that the number of outliers satisfies \|O\| ≤ (1 − ε)K/2 for some 0 < ε < 1. Note that since samples are i.i.d in B k for k ∈ K, we have E g (k) j (θ) = g j (θ). We use the following Lemma from Bubeck et al. (2013). Lemma 7 (Lemma 3 from Bubeck et al. (2013)). Let Z, Z 1 , . . . , Z n be a i.i.d sequence with m α = E[\|Z − EZ\| 1+α ] < +∞ for some α ∈ (0, 1] and putZ n = 1 n i∈ n Z i . Then, we havē Z n ≤ EZ + 3m α δn α 1/(1+α) for any δ ∈ (0, 1), with a probability 1 − δ. Lemma 7 entails that g (k) j (θ) − g j (θ) ≤ 3m j,α (θ) δ (n/K) α 1/(1+α) =: η j,α,δ (θ) with probability larger than 1 − 2δ , for each k ∈ K, since we have n/K samples in block B k . Now, recalling that g j (θ) is the median (see (14)), we can upper bound its failure probability as follows: P g MOM j (θ) − g j (θ) ≥ η j,α,δ (θ) ≤ P k∈ K 1 g (k) j (θ) − g j (θ) ≥ η j,α,δ (θ) > K/2 ≤ P k∈K 1 g (k) j (θ) − g j (θ) ≥ η j,α,δ (θ) > K/2 − \|O\| , since at most \|O\| blocks contain one outlier. Since the blocks B k are disjoint and contain i.i.d samples for k ∈ K, we know that k∈K 1 g (k) j (θ) − g j (θ) ≥ η j,α,δ (θ) follows a binomial distribution Bin(\|K\|, p) with p ≤ 2δ . Using the fact that Bin(\|K\|, p) is stochastically dominated by Bin(\|K\|, 2δ ) and that E[Bin(\|K\|, 2δ )] = 2δ \|K\|, we obtain, if S ∼ Bin(\|K\|, 2δ ), that P g MOM j (θ) − g j (θ) ≥ η j,α,δ (θ) ≤ P S > K/2 − \|O\| = P S − ES > K/2 − \|O\| − 2δ \|K\| ≤ P S − ES > K(ε − 4δ )/2 ≤ exp − K(ε − 4δ ) 2 /2 , where we used the fact that \|O\| ≤ (1 − ε)K/2 and \|K\| ≤ K for the second inequality and the Hoeffding inequality for the last. This concludes the proof of Lemma 2 for the choice ε = 5/6 and δ = 1/8. B.6 Proof of Proposition 1 Step 1. First, we fix θ ∈ Θ and try to bound g MOM j (θ) − g j (θ) in terms of quantities only depending on θ which is the closest point to θ in an ε-net. Recall that ∆ is the diameter of the parameter set Θ and let ε > 0 be a positive number. There exists an ε-net covering Θ with cardinality no more than (3∆/2ε) d i.e. a set N ε such that for all θ ∈ Θ there exists θ ∈ N ε such that θ − θ ≤ ε. Consider a fixed θ ∈ Θ and j ∈ d , we wish to bound the quantity g MOM j (θ) − g j (θ) . Using the ε-net N ε , there exists θ such that θ − θ ≤ ε which we can use as follows: g MOM j (θ) − g j (θ) ≤ g MOM j (θ) − g j ( θ) + g j ( θ) − g j (θ) ≤ g MOM j (θ) − g j ( θ) + L j ε,(36) where we used the gradient's coordinate Lipschitz constant to bound the second term. We now focus on the second term. Introducing the notation g i j (θ) = (θ X i , Y i )X j i , we have g i j (θ) = ( θ X i , Y i )X j i + ( (θ X i , Y i ) − ( θ X i , Y i ))X j i =:∆ i . Let (B k ) k∈ K be the blocks used to compute the MOM estimator and associated block means g (k) j (θ) and g (k) j ( θ). Notice that the MOM estimator is monotonous non decreasing w.r.t. to each of the entries g i j (θ) when the others are fixed. Without loss of generality, assume that g MOM j (θ) − g j ( θ) ≥ 0 then we have: g MOM j (θ) − g j ( θ) ≤ q g MOM j ( θ) − g j ( θ) ,(37) where q g MOM j ( θ) is the MOM estimator obtained using the entries θ X i , Y i X j i + εγ X i 2 = g i j ( θ) + εγ X i 2 instead of g i j (θ) . Note that q g MOM j ( θ) no longer depends on θ except through the fact that θ is chosen in N ε so that θ − θ ≤ ε. Indeed, using the Lipschitz smoothness of the loss function and a Cauchy-Schwarz inequality we find that: \|∆ i \| ≤ γ θ − θ · X i · \|X j i \| ≤ εγ X i 2 . Step 2. We now use the concentration property of MOM to bound the quantity which is in terms of θ. The samples (g i j ( θ) + εγ X i 2 ) i∈ n are independent and distributed according to the ran- dom variable ( θ X, Y )X j + εγ X 2 . Denote L = γE X 2 and for k ∈ K let g (k) j ( θ) = K n i∈B k g i j ( θ) and L (k) = K n i∈B k γ X i 2 . We use Lemma 7 for each of these pairs of means to obtain that with probability at least 1 − δ /2: g (k) j ( θ) − g j ( θ) ≤ 6m j,α ( θ) δ (n/K) α 1/(1+α) =: η j,α,δ /2 ( θ), and with probability at least 1 − δ /2 L (k) − L ≤ 6m L,α δ (n/K) α 1/(1+α) =: η L,α,δ /2 , where m L,α = E\|γ X 2 − L\| 1+α . Hence for all k ∈ K P g (k) j ( θ) + ε L (k) − g j ( θ) > η j,α,δ /2 ( θ) + ε(L + η L,α,δ /2 ) ≤ P g (k) j ( θ) − g j ( θ) > η j,α,δ /2 ( θ) + P L (k) − L > η L,α,δ /2 ≤ δ /2 + δ /2 = δ . Now defining the Bernoulli variables U k := 1 g (k) j ( θ) + δ L (k) − g j ( θ) > η j,α,δ /2 ( θ) + ε L + η L,α,δ /2 , we have just seen they have success probability ≤ δ , moreover P q g MOM j ( θ) − g j ( θ) ≥ η j,α,δ /2 ( θ) + ε(L + η L,α,δ /2 ) ≤ P k∈ K U k > K/2 ≤ P k∈K U k > K/2 − \|O\| , since at most \|O\| blocks contain one outlier. Since the blocks B k are disjoint and contain i.i.d samples for k ∈ K, we know that k∈K U k follows a binomial distribution Bin(\|K\|, p) with p ≤ δ . Using the fact that Bin(\|K\|, p) is stochastically dominated by Bin(\|K\|, δ ) and that E[Bin(\|K\|, δ )] = δ \|K\|, we obtain, if S ∼ Bin(\|K\|, δ ), that P q g MOM j ( θ) − g j ( θ) ≥ η j,α,δ /2 ( θ) + ε L + η L,α,δ /2 ≤ P S > K/2 − \|O\| = P S − ES > K/2 − \|O\| − δ \|K\| ≤ P S − ES > K(ε − 2δ )/2 ≤ exp − K(ε − 2δ ) 2 /2 , where we used the condition \|O\| ≤ (1 − ε )K/2 and \|K\| ≤ K for the second inequality and the Hoeffding inequality for the last. To conclude, we choose ε = 5/6 and δ = 1/4 and combine (36), (37) and the last inequality in which we take K = 18 log(1/δ) and use a union bound argument to obtain that with probability at least 1 − δ for all j ∈ d q g MOM j ( θ)−g j ( θ) ≤ (24m j,α ( θ)) 1/(1+α) +ε(24m L,α ) 1/(1+α) 18 log(d/δ) n α/(1+α) +εL. (38) Step 3. We use the ε-net to obtain a uniform bound. For θ ∈ Θ denote θ(θ) ∈ N ε the closest point in N ε satisfying in particular θ(θ) − θ ≤ ε, we write, following previous arguments sup θ∈Θ g MOM j (θ) − g j (θ) ≤ sup θ∈Θ g MOM j (θ) − g j ( θ(θ)) + g j ( θ(θ)) − g j (θ) ≤ sup θ∈Θ q g MOM j ( θ(θ)) − g j ( θ(θ)) + εL j = max θ∈Nε q g MOM j ( θ) − g j ( θ) + εL j . Here, we make a union bound argument over θ ∈ N ε for the inequality (38) and choose ε = n −α/(1+α) to obtain the final result concluding the proof of Proposition 1. B.7 Proof of Proposition 2 This proof reuses arguments from the proof of Theorem 2 in Lecué et al. (2020). We wish to bound g MOM j (θ) − g j (θ) with high probability and uniformly on θ ∈ Θ. Fix θ ∈ Θ and j ∈ d , we have g MOM j (θ) = median g (1) j (θ), . . . , g (K) j (θ) with g (k) j (θ) = K n i∈B k g i j (θ) where the blocks B 1 , . . . , B K constitute a partition of n . Define the function φ(t) = (t − 1)1 1≤t≤2 + 1 t>2 , let K = {k ∈ K , B k ∩ O = ∅} and J = k∈K B k . Thanks to the inequality φ(t) ≥ 1 t≥2 , we have: sup θ∈Θ K k=1 1 g (k) j (θ) − g j (θ) > x ≤ sup θ∈Θ k∈K E φ 2 g (k) j (θ) − g j (θ) /x + \|O\| + sup θ∈Θ k∈K φ 2 g (k) j (θ) − g j (θ) /x − E φ 2 g (k) j (θ) − g j (θ) /x . Besides, the inequality φ(t) ≤ 1 t≥1 , an application of Markov's inequality and Lemma 7 yield: E φ 2 g (k) j (θ) − g j (θ) /x ≤ P g (k) j (θ) − g j (θ) ≥ x/2 ≤ 3m α,j (θ) (x/2) 1+α (n/K) α . Therefore, recalling that we defined M α,j := sup θ∈Θ m α,j (θ) we have sup θ∈Θ K k=1 1 g (k) j (θ) − g j (θ) > x ≤ K 3M α,j (x/2) 1+α (n/K) α + \|O\| K + sup θ∈Θ 1 K k∈K φ 2 g (k) j (θ) − g j (θ) /x − E φ 2 g (k) j (θ) − g j (θ) /x . Now since for all t we have 0 ≤ φ(t) ≤ 1, McDiarmid's inequality says with probability ≥ 1 − exp(−2y 2 K) that: sup θ∈Θ 1 K k∈K φ 2 g (k) j (θ) − g j (θ) /x − E φ 2 g (k) j (θ) − g j (θ) /x ≤ E sup θ∈Θ 1 K k∈K φ 2 g (k) j (θ) − g j (θ) /x − E φ 2 g (k) j (θ) − g j (θ) /x + y. Using a simple symmetrization argument (see for instance Lemma 11.4 in Boucheron et al. (2013)) we find: E sup θ∈Θ 1 K k∈K φ 2 g (k) j (θ) − g j (θ) /x − E φ 2 g (k) j (θ) − g j (θ) /x ≤ 2E sup θ∈Θ 1 K k∈K ε k φ 2 g (k) (θ) − g(θ) /x , where the ε k s are independent Rademacher variables. Since φ is 1-Lipschitz and satisfies φ(0) = 0 we can use the contraction principle (see Theorem 11.6 in Boucheron et al. (2013)) followed by another symmetrization step to find 2E sup θ∈Θ 1 K k∈K ε k φ 2 g (k) j (θ) − g j (θ) /x ≤ 4E sup θ∈Θ 1 K k∈K ε k g (k) j (θ) − g j (θ) /x ≤ 8 xn E sup θ∈Θ i∈J ε i g i j (θ) ≤ 8R j (Θ) xn . Taking \|O\| ≤ (1 − ε)K/2, we found that with probability ≥ 1 − exp(−2y 2 K) sup θ∈Θ K k=1 1 g (k) j (θ) − g j (θ) > x ≤ K 3M α,j (x/2) 1+α (n/K) α + \|O\| K + 8R j (Θ) xn . Now by choosing y = 1/4 − \|O\|/K and x = max 36M α,j (n/K) α 1/(1+α) , 64R j (Θ) n , we obtain the deviation bound: P sup θ∈Θ g MOM j (θ) − g j (θ) ≥ max 36M α,j (n/K) α 1/(1+α) , 64R j (Θ) n ≤ P sup θ∈Θ K k=1 1 g (k) j (θ) − g j (θ) > x > K/2 ≤ exp(−2(ε − 1/2) 2 K/4) ≤ exp(−K/18), where the last inequality comes from the choice ε = 5/6. A simple union bound argument lets the previous inequality hold for all j ∈ d with high probability. Finally, assuming that X j has finite fourth moment for all j ∈ d , we can control the Rademacher complexity. In this part, we assume without loss of generality that I = n , we first write R j (Θ) = E sup θ∈Θ n i=1 ε i (θ X i , Y i )X j i = E n i=1 ε i (0, Y i )X j i + sup θ∈Θ n i=1 ε i ( (θ X i , Y i ) − (0, Y i ))X j i . Denote φ i (t) = ( (t, Y i ) − (0, Y i ))X j i and notice that E n i=1 ε i (0, Y i )X j i = 0. Notice also that φ i (0) = 0 and φ i is γ\|X j i \|-Lipschitz for all i. We use a variant of the contraction principle adapted to our case in which functions with different Lipschitz constants appear. We use Lemma 11.7 from Boucheron et al. (2013) and adapt the proof of their Theorem 11.6 to make the following estimations: E sup θ∈Θ n i=1 ε i φ i (θ X i ) = E E sup θ∈Θ n−1 i=1 ε i φ i (θ X i ) + ε n φ n (θ X n ) (ε i ) n−1 i=1 , (X i , Y i ) i∈ n ≤ E E sup θ∈Θ n−1 i=1 ε i φ i (θ X i ) + ε n γ\|X j n \|θ X n (ε i ) n−1 i=1 , (X i , Y i ) i∈ n = E sup θ∈Θ n−1 i=1 ε i φ i (θ X i ) + ε n γ\|X j n \|θ X n . By iterating the previous argument n times we find: E sup θ∈Θ n i=1 ε i φ i (θ X i ) ≤ E sup θ∈Θ n−1 i=1 ε i γ\|X j i \|θ X i . Now recalling that the diameter of Θ is ∆, we use Lemma 8 below with p = 1 to bound the previous quantity as: E sup θ∈Θ n i=1 ε i γ\|X j i \|θ X i = γE sup θ∈Θ θ, n i=1 ε i X i \|X j i \| ≤ γ∆E E n i=1 ε i X i \|X j i \| 1 (X i ) i∈ n ≤ γ∆C α E n i=1 X i 1+α \|X j i \| 1+α 1/(1+α) ≤ γ∆C α nE (X j ) 2(1+α) 1/2 k∈ d E (X k ) 2(1+α) 1/2 1/(1+α) , where we used a Cauchy-Schwarz inequality in the last step, which concludes the proof of Proposition 2. Lemma 8 (Khintchine inequality variant). Let α ∈ (0, 1] and (x i ) i∈ n be real numbers with n ∈ N and p > 0 and (ε i ) i∈ n be i.i.d Rademacher random variables then we have the inequality: E n i=1 ε i x i p 1/p ≤ B p,α n i=1 \|x i \| 1+α 1/(1+α) with the constant B p,α := 2p 1+α α αp/(1+α)−1 Γ αp 1+α . Moreover, for p = 1 the constant B 1,α is bounded for any α ≥ 0. Proof. This proof is a generalization of Lemma 4.1 from Ledoux and Talagrand (1991) and uses similar methods. For all λ > 0 we have: E exp λ i ε i x i = i E exp(λε i x i ) = i cosh(λx i ) ≤ i exp \|λx i \| 1+α 1 + α = exp i \|λx i \| 1+α 1 + α , where we used the inequality cosh(u) ≤ exp \|u\| 1+α 1+α valid for all u ∈ R which can be quickly proven. Since both functions are even, fix u > 0 and define f u (α) = exp \|u\| 1+α 1+α − cosh(u), we can show that f u is monotonous on [0, 1] separately for u ∈ (0, √ e) and (e, +∞) and notice that f u (0) and f u (1) are both non-negative for all u > 0 thanks to the famous inequality cosh(u) ≤ e u 2 /2 . Therefore, the inequality holds for u ∈ (0, √ e) and (e, +∞). Finally, for u ∈ ( √ e, e), the function f u (α) reaches a minimum at f u (1/ log(u) − 1) = u e − cosh(u) and by taking logarithms we have u e ≥ cosh(u) ⇐⇒ log(1 + e 2u ) ≤ u + log(2) + e log(u) but since the derivatives verify 2 1+e −2u ≤ 2 ≤ 1 + e/u for u ∈ ( √ e, e) and e e/2 ≥ cosh( √ e) the desired inequality follows by integration. By homogeneity, we can focus on the case n i=1 \|x i \| 1+α 1/(1+α) = 1, we compute: E i ε i x i p = +∞ 0 P i ε i x i p > t dt ≤ 2 +∞ 0 exp λ 1+α 1 + α − λt 1/p dt = 2 +∞ 0 exp − α 1 + α u (1+α)/α du p = 2p 1 + α α αp/(1+α)−1 Γ αp 1 + α = B p p,α , where we used the previous inequality and chose λ = (t 1/p ) 1/α in the last step. This proves the main inequality. Finally, it is easy to see that B 1,α is bounded for high values of α while for α ∼ 0 it is consequence of the fact that Γ(x) ∼ 1/x near 0 and the limit x x → 0 when x → 0 + . B.8 Proof of Lemma 3 As previously, Lemma 1 along with Assumptions 1 and 2 guarantee that the gradient coordinates have finite (1 + α)-moments. From here, Lemma 3 is a direct application of Lemma 9 stated and proved below. In the following lemma, for any sequence (z i ) N i=1 of real numbers, (z * i ) N i=1 denotes a non-decreasing reordering of it. Lemma 9. Let X 1 , . . . , X N , Y 1 , . . . , Y N denote an η-corrupted i.i.d sample with rate η from a random variable X with expectation µ = EX and with finite 1 + γ centered moment E\|X − µ\| 1+γ = M < ∞ for some 0 < γ ≤ 1. Denote µ the -trimmed mean estimator computed as (α, min(x, β)) and the thresholds α = Y * N and β = Y * (1− )N . Let 1 > δ ≥ e −N /4, taking = 8η + 12 log(4/δ) n , we have µ = 1 N N i=1 φ α,β ( X i ) with φ α,β (x) = max\| µ − µ\| ≤ 7M 1 1+γ ( /2) γ 1+γ(39) with probability at least 1 − δ. Proof. This proof goes along the lines of the proof of Theorem 1 from Lugosi and Mendelson (2021) with the main difference that only the (1 + γ)-moment is used instead of the variance. Denote X the random variable whose expectation µ = EX is to be estimated and X = X − µ. Let X 1 , . . . , X N , Y 1 , . . . , Y N the original uncorrupted i.i.d. sample from X and let X 1 , . . . , X N , Y 1 , . . . , Y N denote the corrupted sample with rate η. We define the following quantity which will intervene in the proof: E( , X) := max E X − Q /2 (X) 1 X≤Q /2 (X) , E X − Q 1− /2 (X) 1 X≥Q 1− /2 (X) . (40) Step 1. We first derive confidence bounds on the truncation thresholds. Define the random variable U = 1 X≥Q 1−2 (X) . Its standard deviation satisfies σ U ≤ P 1/2 (X ≥ Q 1−2 (X)) = √ 2 . By applying Bernstein's inequality we find with probability ≥ 1 − exp(− N/12) that: i : Y i ≥ µ + Q 1−2 (X) ≥ 3 N/2, a similar argument with U = 1 X>Q 1− /2 (X) yields with probability ≥ 1 − exp(− N/12) that: i : Y i ≤ µ + Q 1− /2 (X) ≥ (1 − (3/4) )N, and similarly with probability ≥ 1 − exp(− N/12) we have: i : Y i ≤ µ + Q 2 (X) } ≥ 3 N/2, and with probability ≥ 1 − exp(− N/12): i : Y i ≥ µ + Q /2 (X) ≥ (1 − (3/4) )N, so that with probability ≥ 1 − 4 exp(− N/12) ≥ 1 − δ/2 the four previous inequalities hold simultaneously. We call this event E which only depends on the variables Y 1 , . . . , Y N . Since η ≤ /8, if 2ηN samples are corrupted we still have: i : Y i ≥ µ + Q 1−2 (X) } ≥ ((3/2) − 2η)N ≥ N and i : Y i ≤ µ + Q 1− /2 (X) ≥ (1 − (3/4) − 2η)N ≥ (1 − )N consequently, the two following bounds hold Q 1−2 (X) ≤ Y * (1− )N − µ ≤ Q 1− /2 (X) and similarly Q /2 (X) ≤ Y * N − µ ≤ Q 2 (X). This provides guarantees on the truncation levels used which are α = Y * N and β = Y * (1− )N . Step 2. We first bound the deviation 1 N N i=1 φ α,β (X i ) − µ in the absence of corruption. W e write: 1 N N i=1 φ α,β (X i ) ≤ 1 N N i=1 φ µ+Q 2 (X),µ+Q 1− /2 (X) (X i ) = E φ µ+Q 2 (X),µ+Q 1− /2 (X) (X) + 1 N N i=1 φ µ+Q 2 (X),µ+Q 1− /2 (X) (X i ) − E φ µ+Q 2 (X),µ+Q 1− /2 (X) (X) .(41) The first term is dominated by: The case Q 1− /2 (X) ≤ 0 is similarly handled. Moreover, a simple calculation yields E X 2 1 Q 2 (X)≤X≤Q 1− /2 (X) ≤ M max \|Q 2 (X)\|, \|Q 1− /2 (X)\| 1−γ ≤ 2 2M 2/(1+γ) . All in all, we have shown the inequality: E φ µ+Q 2 (X),µ+Q 1− /2 (X) (X) − E[φ µ+Q 2 (X),µ+Q 1− /2 (X) (X)] 2 ≤ 6 2M 2/(1+γ) , which we now use to apply Bernstein's inequality on the sum in (41) to find, conditionally on Y 1 , . . . , Y n , with probability at least 1 − δ/4: 1 N N i=1 φ α,β (X i ) ≤ µ + E(4 , X) + 6 log(4/δ) N 2M 1/(1+γ) + log(4/δ) 3N (Q 1− /2 (X) + E( , X)) ≤ µ + 2E(4 , X) + 6 log(4/δ) N 2M 1/(1+γ) + log(4/δ) 3N Q 1− /2 (X) ≤ µ + 2E(4 , X) + (3/2)M 1/(1+γ) ( /2) γ/(1+γ) , where we used (43), the fact that log(4/δ) N ≤ /12 and the assumption that δ ≥ e −N /4. Using the same argument on the lower tail, we obtain, on the event E, that with probability at least 1 − δ/2 1 N N i=1 φ α,β (X i ) − µ ≤ 2E(4 , X) + (3/2)M 1 1+γ ( /2) γ/(1+γ) . Step 3. Now we show that 1 N N i=1 φ α,β (X i ) − 1 N N i=1 φ α,β ( X i ) is of the same order as the previous bounds. There are at most 2ηN indices such that X i = X i and for such differences we have the bound: φ α,β (X i ) − φ α,β ( X i ) ≤ \|Q /2 (X)\| + \|Q 1− /2 (X)\|, and since we have η ≤ /8 then 1 N i=1 φ α,β (X i ) − 1 N i=1 φ α,β ( X i ) ≤ 2η \|Q /2 (X)\| + \|Q 1− /2 (X)\| ≤ 2 max \|Q /2 (X)\|, \|Q 1− /2 (X)\| ≤ M 1/(1+γ) ( /2) γ/(1+γ) , where the last step follows from (42) and (43). Finally, using similar arguments along with Hölder's inequality, we show that: E \|X − Q /2 (X)\|1 X≤Q /2 (X) ≤ E \|X\|1 X≤Q /2 (X) + E \|Q /2 (X)\|1 X≤Q /2 (X) ≤ M 1/(1+γ) ( /2) γ/(1+γ) + \|Q /2 (X)\|( /2) ≤ 2M 1/(1+γ) ( /2) γ/(1+γ) , and a similar computation for E \|X − Q 1− /2 (X)\|1 X≥Q 1− /2 (X) leads to E(4 , X) ≤ 2M 1/(1+γ) (2 ) γ/(1+γ) . This completes the proof of Lemma 9. B.9 Proof of Proposition 3 Step 1. Notice that the TM estimator is also a monotonous non decreasing function of each of its entries when the others are fixed. This allows us to replicate Step 1 of the proof of Proposition 1. We define an ε-net N ε on the set Θ, fix θ ∈ Θ and let θ be the closest point in N ε . We obtain, for all j ∈ d , the inequalities: g TM j (θ) − g j (θ) ≤ g TM j (θ) − g j ( θ) + g j ( θ) − g j (θ) ≤ q g TM j ( θ) − g j ( θ) + εL j ,(44) where q g TM j ( θ) is the TM estimator obtained for the entries θ X i , Y i X j i + εγ X i 2 = g i j ( θ) + εγ X i 2 . Step 2. We use the concentration property of the TM estimator to bound the previous quantity which is in terms of θ. The terms g i j ( θ) + εγ X i 2 i∈ n are independent and distributed according to Z := θ X, Y X j + γε X 2 . Obviously we have E θ X, Y X j = g j (θ). Furthermore, let L = Eγ X 2 , so that E g i j ( θ) + εγ X i 2 = g j (θ) + εL. We will apply Lemma 9 for q g TM j ( θ). Before we do so, we need to compute the centered (1 + α)-moment of Z. Let m j,α ( θ) and m L,α be the centered (1 + α)-moments of (θ X, Y )X j and γ X 2 respectively, we have: E Z − EZ 1+α ≤ 2 α m j,α (θ) + ε 1+α m L,α . Now applying Lemma 9 we find with probability no less than 1 − δ q g TM j ( θ) − g j ( θ) − εL ≤ 7 m j,α ( θ) + ε 1+α m L,α 1/(1+α) (2 ) α/(1+α) , with δ = 8η + 12 log(4/δ) n . By combining with (44) and using a union bound argument, we deduce that with the same probability, we have for all j ∈ d q g TM j ( θ) − g j ( θ) ≤ 7 m j,α ( θ) + ε (1+α) 2 m L,α 1/(1+α) (4 dδ ) α/(1+α) + εL. Step 3. We use the ε-net to obtain a uniform bound. We proceed similarly as in the proof of Proposition 1. For θ ∈ Θ denote θ(θ) ∈ N ε the closest point in N ε satisfying in particular θ(θ) − θ ≤ ε, we write, following previous arguments sup θ∈Θ g TM j (θ) − g j (θ) ≤ sup θ∈Θ g TM j (θ) − g j ( θ(θ)) + g j ( θ(θ)) − g j (θ) ≤ sup θ∈Θ q g TM j ( θ(θ)) − g j ( θ(θ)) + εL j = max θ∈Nε q g TM j ( θ) − g j ( θ) + εL j . Taking union bound over θ ∈ N ε for the inequality (45) and choosing ε = n −α/(1+α) concludes the proof of Proposition 3. B.10 Proof of Lemma 4 Similarly to the proof of Lemma 2, the assumptions, this time taken with α = 1, imply that the gradient has a second moment so that the existence of σ 2 j = V(g j (θ)) is guaranteed. We apply Lemma 1 from Holland and Ikeda (2019a) with δ/2 to obtain: 1 2 \| g CH j (θ) − g j (θ)\| ≤ Cσ 2 j s + s log(4δ −1 ) n with probability at least 1 − δ/2, where C is a constant such that we have: − log(1 − u + Cu 2 ) ≤ ψ(u) ≤ log(1 + u + Cu 2 ), and one can easily check that our choice of ψ, the Gudermannian function, satisfies the previous inequality for C = 1/2. This, along with the choice of scale s according to (20) and our assumption on σ j yields the announced deviation bound by a simple union bound argument. B.11 Proof of Proposition 4 In this proof, for a scale s > 0 and a set of real numbers (x i ) i∈ n , we letx = 1 n i∈ n x i be their mean and define the function ζ s (x i ) i∈ n as the unique x satisfying i∈ n ψ x −x s = 0. Since the function ψ is increasing the previous equation has a unique solution. Moreover, for fixed scale s, the function ζ s (x i ) i∈ n is monotonous non decreasing w.r.t. each x i when the others are fixed. Step 1. We proceed similarly as in the proof of Proposition 1 except that we only use the monotonicity of the CH estimator with fixed scale. Let N ε be an ε-net for Θ with ε = 1/ √ n. We have \|N ε \| ≤ (3∆/2ε) d with ∆ the diameter of Θ. Fix a coordinate j ∈ d , a point θ ∈ Θ and let θ be the closest point to it in the ε-net. We wish to bound the difference g CH j (θ) − g j (θ) ≤ g CH j (θ) − g j ( θ) + g j ( θ) − g j (θ) ≤ g CH j (θ) − g j ( θ) + εL j , where we have the CH estimator g CH j (θ) = ζ s(θ) (g i j (θ)) i∈ n with scale s(θ) computed according to (20) and (21). Assume, without loss of generality that g CH j (θ) − g j ( θ) ≥ 0. Using the nondecreasing property of the CH estimator at a fixed scale, we find that g CH j (θ) − g j ( θ)\| = ζ s(θ) (g i j (θ)) i∈ n − g j ( θ) ≤ ζ s(θ) (g i j ( θ) + εγ X i 2 ) i∈ n − g j ( θ) . Indeed, one has g i j (θ) = g i j ( θ) + g i j (θ) − g i j ( θ) ≤ g i j ( θ) + γ θ − θ · X i · \|X j i \| ≤ g i j ( θ) + εγ X i 2 . We introduce the notation q g CH j ( θ) := ζ s(θ) (g i j ( θ) + εγ X i 2 ) i∈ n so that: g CH j (θ) − g j ( θ) ≤ q g CH j ( θ) − g j ( θ) . Step 2. We now use the concentration property of CH to bound the previous quantity which is in terms of θ. We apply Lemma 1 from Holland and Ikeda (2019a) with δ/2 and scale s(θ) to the samples (g i j ( θ) + εγ X i 2 ) i∈ n which are independent and distributed according to the random variable θ X, Y X j + εγ X 2 with expectation g j ( θ) + εL. Using our assumptions on σ L , σ j (θ), σ j ( θ), σ j (θ) and the definition of the scale s(θ) according to (20) we find: 1 2 q g CH j ( θ) − g j ( θ) − εL = 1 2 ζ s(θ) (g i j ( θ) + εγ X i 2 ) i∈ n − g j ( θ) − εL ≤ CV(g i j ( θ) + εγ X i 2 ) s(θ) + s(θ) log(4/δ) n ≤ CC V(g i j ( θ) + εγ X i 2 ) σ j (θ) 2 log(4/δ) n + C σ j (θ) 2 log(4/δ) n ≤ CC 2(σ 2 j ( θ) + ε 2 σ 2 L ) σ j (θ) 2 log(4/δ) n + C σ j (θ) 2 log(4/δ) n ≤ CC 2 √ 2σ j ( θ) + εσ L 2 log(4/δ) n + 2C σ j ( θ) log(4/δ) n ≤ 4C σ j ( θ) log(4/δ) n + 2C εσ L log(4/δ) n ≤ 2C (2σ j ( θ) + εσ L ) log(4/δ) n . A simple union bound yields that for all j ∈ d q g CH j ( θ) − g j ( θ) ≤ 4C (2σ j ( θ) + εσ L ) log(4d/δ) n + εL.(46) Step 3. We use the ε-net to obtain a uniform bound. We proceed similarly to the proof of Proposition 1. For θ ∈ Θ denote θ(θ) ∈ N ε the closest point in N ε satisfying in particular θ(θ) − θ ≤ ε, we write, following previous arguments sup θ∈Θ g CH j (θ) − g j (θ) ≤ sup θ∈Θ g CH j (θ) − g j ( θ(θ)) + g j ( θ(θ)) − g j (θ) ≤ sup θ∈Θ q g CH j ( θ(θ)) − g j ( θ(θ)) + εL j = max θ∈Nε q g CH j ( θ) − g j ( θ) + εL j . Taking union bound over θ ∈ N ε for the inequality (46) and using the choice ε = 1/ √ n concludes the proof of Proposition 4. B.12 Proof of Corollary 1 Under the assumptions made, the constants (L j ) j∈ d are estimated using the MOM estimator and we obtain the bounds (L j ) j∈ d which hold with probability at least 1 − δ/2 by a union bound argument. The rest of the proof is the same as that of Theorem 1 using a failure probability δ/2 instead of δ and replacing the constants (L j ) j∈ d by their upperbounds accordingly. The result then follows after a simple union bound argument. B.13 Proof of Lemma 5 Let B 1 , . . . , B K be the blocks used for the estimation so that B 1 ∪· · ·∪B K = n and B k 1 ∩B k 2 = ∅ for k 1 = k 2 . Let K denote the uncorrupted block indices K = {k ∈ K such that B k ∩ O = ∅} and assume \|O\| ≤ (1−ε)K/2. For k ∈ K let σ 2 k = K n i∈B k X 2 i be the block means computed by MOM. Denote N = n/K, by using (a slight generalization of) Lemma 7 and the L (1+α) -L 1 condition satisfied by X 2 with a known constant C, we obtain that with probability at least 1 − δ we have \| σ 2 k − σ 2 \| ≤ 3E\|X 2 − σ 2 \| 1+α δN α 1 1+α ≤ 3 δN α 1 1+α CE\|X 2 − σ 2 \| ≤ 3 δN α 1 1+α Cσ 2 , which implies the inequality σ 2 ≤ 1 − C 3 δN α 1 1+α −1 σ 2 k . Define the Bernoulli random variables U k = 1 σ 2 > 1 − C 3 δN α 1 1+α −1 σ 2 k for k ∈ K which have success probability ≤ δ. Denote S = k U k , we can bound the failure probability of the estimator as follows: P 1 − C 3 δN α 1 1+α −1 σ 2 < σ 2 ≤ P S > K/2 − \|O\| = P S − ES > K/2 − \|O\| − δ\|K\| ≤ P S − ES > K(ε − 2δ)/2 ≤ exp − K(ε − 2δ) 2 /2 , where we used the fact that \|O\| ≤ (1 − ε)K/2 and \|K\| ≤ K for the second inequality and Hoeffding's inequality for the last. The proof is finished by taking ε = 5/6 and δ = 1/4. B.14 Proof of Lemma 6 Lemma 6 is a direct consequence of the following result. Lemma 10. Let X 1 , . . . , X n an i.i.d sample of a random variable X with expectation EX = µ and (1 + α)-moment E\|X − µ\| 1+α = m α < ∞. Assume that the variable X satisfies the L (1+α) 2 -L (1+α) condition with constant C > 1. Let µ be the median-of-means estimate of µ with K blocks and m α a similarly obtained estimate of m α from the samples (\|X i − µ\| 1+α ) i∈ n . Then, with probability at least 1 − 2 exp(−K/18) we have m α ≥ (1 − κ)m α , with κ = + 24(1 + α) 1+ n/K α 1+α and = 3×2 2+α (1+C (1+α) 2 ) (n/K) α 1 1+α . Proof. Let µ be the MOM estimate of µ with K blocks, using Lemma 2, we have with probability at least 1 − exp(−K/18), \|µ − µ\| > (24m α ) 1 1+α K n α 1+α .(47) Let m α be the MOM estimate of m α obtained from the samples \|X i − µ\| 1+α i∈ n . Denote B 1 , . . . , B K the blocks we use, we have: m α = median K n i∈B j \|X i − µ\| 1+α j∈ K for any i ∈ n . Let N = n/K, using the convexity of the function f (x) = \|x\| 1+α we find that: 1 N i∈B j X i − µ 1+α = 1 N i∈B j (X i − µ) + (µ − µ) 1+α ≥ 1 N i∈B j \|X i − µ\| 1+α + 1 N (1 + α) i∈B j \|X i − µ\| α sign(X i − µ)(µ − µ) ≥ 1 N i∈B j \|X i − µ\| 1+α − (1 + α)\|µ − µ\| 1 N i∈B j \|X i − µ\| α ≥ 1 N i∈B j \|X i − µ\| 1+α − (1 + α)\|µ − µ\| 1 N i∈B j \|X i − µ\| 1+α α 1+α ,(48) where the last step uses Jensen's inequality. Using Lemma 7 we have, for δ > 0, the concentration bound P 1 N i∈B j X i − µ 1+α − m α > 3E \|X − µ\| 1+α − m α 1+α δN α 1 1+α ≤ δ which, using that X satisfies the L (1+α) 2 -L (1+α) condition, translates to P 1 N i∈B j X i − µ 1+α − m α > ≤ 3E \|X − µ\| 1+α − m α 1+α 1+α N α ≤ 3 × 2 α E\|X − µ\| (1+α) 2 + m 1+α α 1+α N α ≤ 3 × 2 α m 1+α α 1 + C (1+α) 2 1+α N α . Replacing with m α we find P 1 N i∈B j \|X i − µ\| 1+α − m α > m α ≤ 3 × 2 α 1 + C (1+α) 2 N α 1+α . Now conditioning on the event (47) and using the previous bound with = 3×2 α 1+C (1+α) 2 N α δ 1 1+α in (48), we obtain that P 1 N i∈B j X i − µ 1+α ≤ (1 − )m α − (1 + α) 24m α N α 1 1+α ((1 + )m α ) α 1+α ≤ δ =⇒ P 1 N i∈B j X i − µ 1+α ≤ 1 − − 24(1 + α) 1 + N α 1+α =:(1−κ) m α ≤ δ. Now define U j as the indicator variable of the event in the last probability. We have just seen it has success rate less than δ. We can use the MOM trick, assuming the number of outliers satisfies \|O\| ≤ K(1 − ε)/2 for ε ∈ (0, 1), we have for S = j U j P( m α ≤ (1 − κ)m α ) ≤ P(S > K/2 − \|O\|) = P S − ES > K/2 − \|O\| − δ\|K\| ≤ P S − ES > K(ε − 2δ)/2 ≤ exp − K(ε − 2δ) 2 /2 . Taking ε = 5/6 and δ = 1/4 yields that the previous probability is ≤ exp(−K/18). Finally, recall that we conditioned on the event where the deviation \|µ − µ\| is bounded as previously stated and that this event holds with ≥ 1 − exp(−K/18). Taking this conditioning into account and using a union bound argument leads to the fact that the bound m α ≥ (1 − κ)m α holds with probability at least 1 − 2 exp(−K/18). B.15 Proof of Theorem 3 This proof is inspired from Theorem 5 in Nesterov (2012) and Theorem 1 in Shalev-Shwartz and Tewari (2011) while keeping track of the degradations caused by the errors on the gradient coordinates. We condition on the event (29) and denote j = j (δ) and Euc = (δ) 2 . We define for all θ ∈ Θ u j (θ) = argmin ϑ∈Θ j g j (θ)(ϑ − θ j ) + L j 2 (ϑ − θ j ) 2 + j \|ϑ − θ j \| = proj Θ j θ j − β j τ j g j (θ) and denote θ (t) the optimization iterates for t = 0, . . . , T and j t the random coordinate sampled at step t and let g t = g jt (θ (t) ) for brevity. We have that u jt (θ (t) ) satisfies the following optimality condition ∀ϑ ∈ Θ jt g t + L jt u jt (θ (t) ) − θ (t) jt + jt ρ t ϑ − u jt (θ (t) ) ≥ 0, where ρ t = sign u jt (θ (t) ) − θ (t) jt . Using this condition for ϑ = θ (t) jt and the coordinate-wise Lipschitz smoothness property of R we find R(θ (t+1) ) ≤ R(θ (t) ) + g jt (θ (t) ) u jt (θ (t) ) − θ (t) jt + L jt 2 u jt (θ (t) ) − θ (t) jt 2 ≤ R(θ (t) ) + ( g t + jt ρ t ) u jt (θ (t) ) − θ (t) jt + L jt 2 u jt (θ (t) ) − θ (t) jt 2 (49) ≤ R(θ (t) ) − L jt 2 u jt (θ (t) ) − θ (t) jt 2 .(50) Defining the potential Φ(θ) = d j=1 L j (θ j − θ j ) 2 , we have: Φ(θ (t+1) ) = Φ(θ (t) ) + 2L jt u jt (θ (t) ) − θ (t) jt θ (t) jt − θ jt + L jt u jt (θ (t) ) − θ (t) jt 2 = Φ(θ (t) ) + 2L jt u jt (θ (t) ) − θ (t) jt u jt (θ (t) ) − θ jt − L jt u jt (θ (t) ) − θ (t) jt 2 ≤ Φ(θ (t) ) − 2( g t + jt ρ t ) u jt (θ (t) ) − θ jt − L jt u jt (θ (t) ) − θ (t) jt 2 = Φ(θ (t) ) + 2( g t + jt ρ t ) θ jt − θ (t) jt − 2 ( g t + jt ρ t ) u jt (θ (t) ) − θ (t) jt + L jt 2 u jt (θ (t) ) − θ (t) jt 2 ≤ Φ(θ (t) ) + 2( g t + jt ρ t ) θ jt − θ (t) jt + 2 R(θ (t) ) − R(θ (t+1) ) ≤ Φ(θ (t) ) + 2g jt (θ (t) ) θ jt − θ (t) jt + 2 R(θ (t) ) − R(θ (t+1) ) + 4 jt θ jt − θ (t) jt , 1 : 1Properties of some robust estimators, where ERM = Empirical Risk Minimizer (ordinary mean), MOM = Median-of-Means, CH = Catoni-Holland and TM = Trimmed Mean. We recall that n = sample size and \|O\| = number of outliers. The parameters of each estimators are: the number of blocks K in MOM, a scale parameter s > 0 in CH and a proportion of samples in TM. Proposition 4 . 4Grant Assumptions 1 and 2 with α = 1 and O Figure 1 : 1Average running time (y-axis) of all the considered estimators against an increasing sample size (x-axis). The run times increase with a similar slope (on a logarithmic scale), confirming O(n) complexities, but differ significantly: ERM is of course the fastest, followed by TM and MOM (both are close) and finally CH, which is the slowest. Figure 2 : 2Excess-risk for the square loss (y-axis) against iterations (x-axis) for all the considered algorithms in the simulation settings (a)-(c) (top row) and (d)-(f) (bottom row). We zoom-in the last iterations for simulation settings (a) and (b) to improve readability. Figure 3 : 3Test accuracy (y-axis) against the proportion of corrupted samples (x-axis) for six datasets and the considered algorithms. Figure 4 4Figure 4: Test accuracy (y-axis) against computation time (x-axis) along training iterations on two datasets (rows) for 0% corruption (first column), 15% corruption (middle column) and 30% corruption (last column). Figure 5 :Figure 6 : 56Mean squared error (y-axis) against the proportion of corrupted samples (x-axis) for six datasets and the considered algorithms. Mean squared error (y-axis) against computation time (x-axis) along training iterations on two datasets (rows) for 0% corruption (first column), 15% corruption (middle column) and 30% corruption (last column). ;Juditsky et al. (2020). ) chosen with log-uniform distribution over [e −10 , 1] • TM, HG: we optimize the percentage used for trimming uniformly in [10 −5 , 0.3] • RANSAC: we optimize the value of the min samples parameter in the scikit-learn implementation, chosen as 4 + m with m an integer chosen uniformly in 100 • HUBER: we optimize the epsilon parameter in the scikit-learn implementation chosen uniformly in [1. below.Optimal deviation bound Robustness to outliers Numerical complexity Hyper- parameter ERM No None O(n) None MOM Yes Yes for \|O\| < K/2 O(n + K) K ∈ n CH Yes None O(n) Scale s TM Yes Yes for \|O\| < n/8 O(n) Proportion ∈ [0, 1/2) Table Table 2 : 2Main characteristics of the datasets used in experiments, including number of samples, number of features, number of categorical features and number of classes. Table 3 : 3The URLs of all the datasets used in the paper, giving direct download links and supplementary details. 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[ "How to Run a Campaign: Optimal Control of SIS and SIR Information Epidemics", "How to Run a Campaign: Optimal Control of SIS and SIR Information Epidemics" ]	[ "Kundan Kandhway \nDepartment of Electronic Systems Engineering\nIndian Institute of Science\n560012BangaloreIndia\n", "Joy Kuri \nDepartment of Electronic Systems Engineering\nIndian Institute of Science\n560012BangaloreIndia\n" ]	[ "Department of Electronic Systems Engineering\nIndian Institute of Science\n560012BangaloreIndia", "Department of Electronic Systems Engineering\nIndian Institute of Science\n560012BangaloreIndia" ]	[]	Information spreading in a population can be modeled as an epidemic. Campaigners (e.g. election campaign managers, companies marketing products or movies) are interested in spreading a message by a given deadline, using limited resources. In this paper, we formulate the above situation as an optimal control problem and the solution (using Pontryagin's Maximum Principle) prescribes an optimal resource allocation over the time of the campaign. We consider two different scenarios-in the first, the campaigner can adjust a direct control (over time) which allows her to recruit individuals from the population (at some cost) to act as spreaders for the Susceptible-Infected-Susceptible (SIS) epidemic model. In the second case, we allow the campaigner to adjust the effective spreading rate by incentivizing the infected in the Susceptible-Infected-Recovered (SIR) model, in addition to the direct recruitment. We consider time varying information spreading rate in our formulation to model the changing interest level of individuals in the campaign, as the deadline is reached. In both the cases, we show the existence of a solution and its uniqueness for sufficiently small campaign deadlines. For the fixed spreading rate, we show the effectiveness of the optimal control strategy against the constant control strategy, a heuristic control strategy and no control. We show the sensitivity of the optimal control to the spreading rate profile when it is time varying.Information can be communicated to the population directly by the campaigner (direct recruitment of individuals to spread the message). However, recruiting individuals and direct information communication comes with a cost (such as placing advertisement in the mass media). A campaigner may also provide incentives to individuals for spreading the message. Such an incentive is termed as a word-of-mouth incentive and it rewards an individual who refers a product or a piece of information to others.The campaigner possesses limited resources and is unable to communicate information to the entire population. Not only resource allocation among different strategies, but also the timing of direct recruitment of individuals and giving out word-of-mouth incentives are crucial for maximizing the information epidemic. We model the information spreading process as a Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered (SIR) epidemic process with time varying information spreading rate, and formulate an optimal control problem which aims to minimize the campaign cost over a given period of time.SIS and SIR epidemic processes are suitable for modeling information epidemics due to the similarities in the ways disease spreads in a biological network and this information spreads in social networks. When susceptible and infected individuals interact, the topic of interest may come up with some probability, which will lead to transfer of information from infected to susceptible individual. This process is very similar to the way in which a communicable disease spreads in a population. The SIS model is suitable for cases when we are trying to engage the population in a conversation about some topic. Such a scenario can be encountered in political campaigns. SIS allows infected nodes to 'recover' back to the susceptible state, so that it can receive a different message about the same topic. The SIR model is suitable for situations where nodes participate in message spreading for random amounts of time and then recover (and stop message dissemination). Such a scenario may be encountered in viral marketing of a newly launched product or promotion of a movie, where information about the product is transmitted by enthusiastic individuals who gradually lose interest in promoting the product.Related Work: Although there are a lot of studies on preventing the spread of disease and computer viruses in human and computer networks through optimal control[1,2,3,4,5,6,7,8,9,10], information epidemics have attracted less attention (see for example[11]). Apart from differences in epidemic models, our objective is to maximize the spread of information while the studies discussed above aim to contain the epidemic. To be more precise, the above studies aim to remove individuals from the infected class through the application of control signal(s), while our aim is the opposite. In addition, the cost functions used by [2, 6] and [7] are linear in control, while our cost is quadratic in control.[8] and [9] assumed a time-varying state variable in the cost function, while our formulation considers only the final system state.[1] considered a metapopulation model of epidemics which is different from the models used in this paper. The SIR model is used in [3] and [5], but the controls (level of vaccination and treatment) are specific to biological epidemics and unsuitable for information epidemics. The work in [10] aims to prevent the epidemic and the author uses educational campaigns as controls. The educational campaigns encourage susceptibles to protect themselves from the disease and increase the removal rate of infected individuals.[12] and [13] analyze the so called Push, Pull and Push-pull algorithms for rumor spreading on technological and social networks. The connection graph of the nodes is known. The authors fix the strategy for information spreading: nodes may either 'Push' the information to their neighbors, 'Pull' the information from them or do both. The aim is to compute bounds on the number of communication rounds required to distribute the message to almost all the nodes for the given connection graph. Similarly, the authors in[14]aim to compute bounds on the number of communication rounds required for distributed computation of the average value of sensor readings (e.g. temperature) for sensors deployed in a field. Note that finding optimal strategies for information spreading is not the aim of[12,13,14].The study in[15]defines an information or a joke spreading in a population as a rumor. It aims to maximize the spread of rumor in the Daley-Kendall and Maki-Thompson models, which are different from the Kermack-McKendrick SIS/SIR models used in this paper. We believe that certain scenarios like political campaigns are better modeled by Kermack-McKendrick SIS model (recovery is independent of interaction between individuals) than the Daley-Kendall/Maki-Thompson models (when two infected meet, one or both recover). Moreover, [15] used impulse control, while our models assume the system can be controlled over the whole campaign period. The Authors in[16]and [17] devise optimal advertisement and pricing strategies for newly launched products. But they do not consider viral message propagation, where individuals in the population interact with one another to spread a piece of information, as is the case in this paper.Optimal control of spreading software security patches in technological networks is discussed in[18]. The system model used by[18]is tailored to technological networks and is different from the one used in our study. The study uses	10.1016/j.amc.2013.12.164	[ "https://arxiv.org/pdf/1401.6702v1.pdf" ]	18,617,814	1401.6702	8c8bcab9c4ce75eaa80adf35629c592bf1e63f45	How to Run a Campaign: Optimal Control of SIS and SIR Information Epidemics 26 Jan 2014 Kundan Kandhway Department of Electronic Systems Engineering Indian Institute of Science 560012BangaloreIndia Joy Kuri Department of Electronic Systems Engineering Indian Institute of Science 560012BangaloreIndia How to Run a Campaign: Optimal Control of SIS and SIR Information Epidemics 26 Jan 2014Information EpidemicsOptimal ControlPontryagin's Maximum principleSocial NetworksSusceptible-Infected-Recovered (SIR)Susceptible-Infected-Susceptible (SIS) Information spreading in a population can be modeled as an epidemic. Campaigners (e.g. election campaign managers, companies marketing products or movies) are interested in spreading a message by a given deadline, using limited resources. In this paper, we formulate the above situation as an optimal control problem and the solution (using Pontryagin's Maximum Principle) prescribes an optimal resource allocation over the time of the campaign. We consider two different scenarios-in the first, the campaigner can adjust a direct control (over time) which allows her to recruit individuals from the population (at some cost) to act as spreaders for the Susceptible-Infected-Susceptible (SIS) epidemic model. In the second case, we allow the campaigner to adjust the effective spreading rate by incentivizing the infected in the Susceptible-Infected-Recovered (SIR) model, in addition to the direct recruitment. We consider time varying information spreading rate in our formulation to model the changing interest level of individuals in the campaign, as the deadline is reached. In both the cases, we show the existence of a solution and its uniqueness for sufficiently small campaign deadlines. For the fixed spreading rate, we show the effectiveness of the optimal control strategy against the constant control strategy, a heuristic control strategy and no control. We show the sensitivity of the optimal control to the spreading rate profile when it is time varying.Information can be communicated to the population directly by the campaigner (direct recruitment of individuals to spread the message). However, recruiting individuals and direct information communication comes with a cost (such as placing advertisement in the mass media). A campaigner may also provide incentives to individuals for spreading the message. Such an incentive is termed as a word-of-mouth incentive and it rewards an individual who refers a product or a piece of information to others.The campaigner possesses limited resources and is unable to communicate information to the entire population. Not only resource allocation among different strategies, but also the timing of direct recruitment of individuals and giving out word-of-mouth incentives are crucial for maximizing the information epidemic. We model the information spreading process as a Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered (SIR) epidemic process with time varying information spreading rate, and formulate an optimal control problem which aims to minimize the campaign cost over a given period of time.SIS and SIR epidemic processes are suitable for modeling information epidemics due to the similarities in the ways disease spreads in a biological network and this information spreads in social networks. When susceptible and infected individuals interact, the topic of interest may come up with some probability, which will lead to transfer of information from infected to susceptible individual. This process is very similar to the way in which a communicable disease spreads in a population. The SIS model is suitable for cases when we are trying to engage the population in a conversation about some topic. Such a scenario can be encountered in political campaigns. SIS allows infected nodes to 'recover' back to the susceptible state, so that it can receive a different message about the same topic. The SIR model is suitable for situations where nodes participate in message spreading for random amounts of time and then recover (and stop message dissemination). Such a scenario may be encountered in viral marketing of a newly launched product or promotion of a movie, where information about the product is transmitted by enthusiastic individuals who gradually lose interest in promoting the product.Related Work: Although there are a lot of studies on preventing the spread of disease and computer viruses in human and computer networks through optimal control[1,2,3,4,5,6,7,8,9,10], information epidemics have attracted less attention (see for example[11]). Apart from differences in epidemic models, our objective is to maximize the spread of information while the studies discussed above aim to contain the epidemic. To be more precise, the above studies aim to remove individuals from the infected class through the application of control signal(s), while our aim is the opposite. In addition, the cost functions used by [2, 6] and [7] are linear in control, while our cost is quadratic in control.[8] and [9] assumed a time-varying state variable in the cost function, while our formulation considers only the final system state.[1] considered a metapopulation model of epidemics which is different from the models used in this paper. The SIR model is used in [3] and [5], but the controls (level of vaccination and treatment) are specific to biological epidemics and unsuitable for information epidemics. The work in [10] aims to prevent the epidemic and the author uses educational campaigns as controls. The educational campaigns encourage susceptibles to protect themselves from the disease and increase the removal rate of infected individuals.[12] and [13] analyze the so called Push, Pull and Push-pull algorithms for rumor spreading on technological and social networks. The connection graph of the nodes is known. The authors fix the strategy for information spreading: nodes may either 'Push' the information to their neighbors, 'Pull' the information from them or do both. The aim is to compute bounds on the number of communication rounds required to distribute the message to almost all the nodes for the given connection graph. Similarly, the authors in[14]aim to compute bounds on the number of communication rounds required for distributed computation of the average value of sensor readings (e.g. temperature) for sensors deployed in a field. Note that finding optimal strategies for information spreading is not the aim of[12,13,14].The study in[15]defines an information or a joke spreading in a population as a rumor. It aims to maximize the spread of rumor in the Daley-Kendall and Maki-Thompson models, which are different from the Kermack-McKendrick SIS/SIR models used in this paper. We believe that certain scenarios like political campaigns are better modeled by Kermack-McKendrick SIS model (recovery is independent of interaction between individuals) than the Daley-Kendall/Maki-Thompson models (when two infected meet, one or both recover). Moreover, [15] used impulse control, while our models assume the system can be controlled over the whole campaign period. The Authors in[16]and [17] devise optimal advertisement and pricing strategies for newly launched products. But they do not consider viral message propagation, where individuals in the population interact with one another to spread a piece of information, as is the case in this paper.Optimal control of spreading software security patches in technological networks is discussed in[18]. The system model used by[18]is tailored to technological networks and is different from the one used in our study. The study uses Introduction Use of social networks by political campaigners and product marketing managers is increasing day by day. It gives them an opportunity to influence a large population connected via the online network, as well as the human network where two individuals interacting with each other in daily life are connected. A piece of information, awareness of brands, products, ideas and political ideologies of candidates spreads through such a network much like pathogens in the human network, and this phenomenon is called an "information epidemics". The goal of the campaigner is to 'infect' as many individuals or nodes as possible with the message by the campaign deadline. Such an effort incurs advertisement cost. Some campaigns aim to create a 'buzz' about some topic by engaging people in a conversation about some topic (a scenario encountered in political campaigns, for example). Others are more focused: for example, advertisements to maximize the sale of a product or promotion of a movie. Resource limitations (monetary, manpower or otherwise) indicate the need to formulate optimal campaigning strategies which can achieve these goals at minimum cost. Such an optimization problem can be formulated as an optimal control problem. An optimal control problem aims to optimize a cost functional (a function of state and control variables) subject to state equation constraints that govern system evolution. In this paper, we aim to address the problems described above. We assume a homogeneously mixed population. Individuals communicate and exchange messages with one another on their own, giving rise to information epidemics. a four-compartment model (instead of the two and three compartment models used in our study) with an additional compartment for representing the number of nodes destroyed by the malware (apart from susceptibles, infected and recovered nodes). In addition, the controls used are the fraction of disseminators and the dissemination rate of the security patches, which are different from the controls used in this paper. Cost minimization for marketing campaigns is explored in [11]. The cost functional used there is linear in control, while our model assumes a quadratic cost. Although [11] uses SIS and SIR models like we do, but the objective functional is different from ours. The objective functionals in our models are weighted sums of the fraction of nodes which have received the message by the campaign deadline and the costs of applying controls. In contrast, the objective functional in [11] for the SIS case is to minimize the campaign cost (subject to a constraint on the fraction of infected nodes at the deadline) and for the SIR case, the objective is to maximize the fraction of nodes who have received the message (subject to a budget constraint for running the campaign). Moreover, we discuss the uniqueness of the solutions to our models; this is not discussed by [11] and [18]. Furthermore, we study the effect of word-ofmouth control on epidemic spreading, which is not explored in the above studies. A major difference between an information epidemic and a biological epidemic is that in the case of a biological epidemic, the infection rate and recovery rate are constant throughout the season (assuming that pathogens do not mutate within a season). On the other hand, the interest level of the population during the campaign period (for elections or promotions of upcoming movies) changes as we approach the deadline (poll date or movie release date). We have modeled this by making the effective information spreading rate a time varying quantity. Previous studies have ignored this characteristic of information epidemics. The following are the primary contributions of this paper: (i) Formulation of optimal control problems for maximizing information spread in two different models. In the first, information spreads through an SIS process and in the next, through an SIR process. The control signal in the SIS model is the intensity with which direct recruitment of spreaders from the population can be done. The SIR model has an additional control signal-the word-of-mouth control, which controls the spreading rate of the information epidemic. Our formulation involves a quadratic cost function and a time dependent effective information spreading rate. (ii) We show the existence of a solution in both the problems. (iii) We establish uniqueness of the solution in both the cases, when the campaign deadline is sufficiently small. (iv) We compare the overall cost incurred by the optimal control strategy with the constant control strategy, a heuristic control strategy and no control, when the information spreading rate is constant throughout the campaign period. (v) When the effective spreading rate varies over the campaign duration, we demonstrate the sensitivity of the optimal control with respect to different effective spreading rate profiles. The rest of the paper is organized as follows: Sections 2 and 3 formulate the optimal control problem for SIS and SIR information epidemics, respectively. Sections 4 and 5 analyze the respective problems. Results are shown in Section 6 and conclusions are drawn in Section 7. System Model and Problem Formulation: SIS Epidemic We consider a system of N nodes (or individuals) which is fixed throughout the campaign time 0 ≤ t ≤ T . In the case of SIS epidemics, individuals are divided into two compartments-susceptible (those who are yet to receive a tagged message) and infected (those who have already received the message). A susceptible node becomes infected at a certain rate if it comes in contact with an infected individual. The idea is to create a 'buzz' in the population about some topic (e.g. a political campaign). An infected node can 'recover' back to the susceptible state which allows it to receive a different message about the same topic. This makes SIS a suitable model for such scenarios. By the campaign deadline T , the number of infected individuals (who are engaged in some sort of discussion about the topic) is the quantity of interest. Let S (t), I(t) denote the number of susceptible and infected nodes at time t. Let s(t) = S (t)/N ≥ 0 and i(t) = I(t)/N ≥ 0, therefore s(t) + i(t) = 1. The information spreading rate at time t is denoted by β ′ (t), which is assumed to be a bounded quantity. Practical considerations will impose such a restriction on β ′ (t). In a small interval dt at time t, a susceptible node that is in contact with a single infected node, changes its state to "infected" with probability β ′ (t)dt. We assume that each node in the population is in contact with an average of k m others, chosen randomly, at any time instant t. Thus, any susceptible node will have an average of k m i(t) infected neighbors and will acquire information with probability 1 − (1 − β ′ (t)dt) k m i(t) ≃ β ′ (t)k m i(t)dt. Since a fraction s(t) of population is susceptible, the rate of increase of infected nodes in the population due to susceptible-infected contact is β ′ (t)k m i(t)s(t). We define the effective spreading rate at time t as β(t) = β ′ (t)k m . An infected node "falls back" to being susceptible with a rate γ. In the limit of large N, the mean field equations for the evolution of s(t) and i(t) in the (uncontrolled) SIS process is given by [19, equations adapted for time varying β(t)], s(t) = −β(t)s(t)i(t) + γi(t), i(t) = β(t)s(t)i(t) − γi(t). The objective functional is chosen to be J = −i(T ) + T 0 bu 2 (t)dt. The rationale behind such a choice is as follows. Applications like poll campaigns only care about the final number of infected individuals on the polling day, i(T ), and not on the evolution history, i(t), 0 ≤ t < T . This is captured in the first term of the objective functional. The cost of running the campaign accrues over time and this is represented by the integral. The choice of quadratic cost function is consistent with the literature [3,9]. Let u-a bounded Lebesgue integrable function-denote the recruitment control applied by the campaigner, with u(t) representing its value at time t. We define U as follows: Definition 1. u ∈ U {u : u is Lebesgue integrable, 0 ≤ u(t) ≤ u max }. Thus, u max uniformly bounds all functions in U. The control signal u denotes the rate at which nodes are directly recruited from the population to act as spreaders. Practical constraints on executing the control will impose the property of boundedness on the control signal. Thus the optimal control problem can be formulated as: min u∈U J = −i(T ) + T 0 bu 2 (t)dt (1) subject to:ṡ(t) = −β(t)s(t)i(t) + γi(t) − u(t)s(t) i(t) = β(t)s(t)i(t) − γi(t) + u(t)s(t) (2) i(t) ≥ 0, s(t) ≥ 0 i(t) + s(t) = 1 i(0) = i 0 , s(0) = 1 − i 0 . Here, i 0 denotes the initial fraction of infected nodes who act as seeds of the epidemic. System Model and Problem Formulation: SIR Epidemics An SIR epidemic model has an additional compartment-recovered-in addition to the susceptible and infected classes discussed before. It is suitable in modeling situations where nodes participate in message spreading for a random amount of time and then "recover" (and stop message dissemination). Such a scenario may be encountered in viral marketing of a newly launched product or promotion of a movie, where enthusiastic individuals gradually lose interest in promoting the product. Let R(t) and r(t) = R(t)/N ≥ 0 denote the number and fraction of recovered nodes at time t so that s(t) +i(t) +r(t) = 1. The effective information spreading rate at time t is β(t). Simultaneously, infected nodes switch to "recovered" at a rate γ, independent of others. The mean field equations governing the SIR process in the limit of large N are [19, adapted for variable β(t)]:ṡ (t) = −β(t)s(t)i(t) i(t) = β(t)s(t)i(t) − γi(t) r(t) = γi(t). In this case we assume that the campaigner can allocate her resources in two ways. At time t, she can directly recruit individuals from the population with rate u 1 (t), to act as spreaders (via advertisements in mass media). In addition, she can incentivize infected individuals to make further recruitments (e.g. monetary benefits, discounts or coupons to current customers who refer their friends to buy services/products from the company). This effectively increases the spreading rate of the message at time t from β(t) to β(t)+u 2 (t) where u 2 (t) denotes the "word-of-mouth" control signal which the campaigner can adjust at time t. The controls, u 1 ∈ U 1 and u 2 ∈ U 2 , where the sets are defined as follows: Definition 2. U 1 {u : u is Lebesgue integrable, 0 ≤ u(t) ≤ u 1max }, U 2 {u : u is Lebesgue integrable, 0 ≤ u(t) ≤ u 2max }. The cost of applying the control is quadratic over the time horizon of the campaign, 0 ≤ t ≤ T , and the reward is the total fraction of population which received the message at some point in time, i.e., i(T ) + r(T ) = 1 − s(T ). The optimal control problem can then be stated as: min u 1 ∈U 1 ,u 2 ∈U 2 J = −1 + s(T ) + T 0 bu 2 1 (t) + cu 2 2 (t) dt (3) subject toṡ(t) = − β(t) + u 2 (t) s(t)i(t) − u 1 (t)s(t) i(t) = β(t) + u 2 (t) s(t)i(t) + u 1 (t)s(t) − γi(t) r(t) = γi(t) i(t) ≥ 0, s(t) ≥ 0, r(t) ≥ 0 i(t) + s(t) + r(t) = 1 i(0) = i 0 , s(0) = 1 − i 0 , r(0) = 0. Analysis of the Controlled SIS Epidemic Substituting s(t) = 1 − i(t), problem (1) can be rewritten as, min u∈U J = −i(T ) + T 0 bu 2 (t)dt (4) subject toi(t) = −β(t)i 2 (t) + β(t) − γ − u(t) i(t) + u(t) (5) 0 ≤ i(t) ≤ 1 i(0) = i 0 .(6) Existence of a Solution Theorem 4.1. There exist an optimal control signal u ∈ U and a corresponding solution i * (t) to the initial value problem (5) and (6) such that u ∈ argmin u∈U {J(u)} in problem (4). Proof. The theorem can be proved by application of the Cesari Theorem [20, pg 68]. Let the right hand side (RHS) of (5) be denoted by f (u(t), i(t)). The following requirements of the Theorem are met: f (u(t), i(t)) satisfies the required bound \| f (u(t), i(t))\| ≤ C 0 (1 + \|i(t)\| + \|u(t)\|) (with C 0 = sup\|β(t) + γ + u(t)\|; note that β(t) is bounded and 0 ≤ u(t) ≤ u max ). The set U and the set of solutions to initial value problem (5) and (6) (u(t), i(t)) is linear in u(t). In the cost functional (4), the integrand, bu 2 (t) ≥ C 1 \|u(t)\| C 2 − C 3 . It is required that C 1 > 0, C 2 > 1 which is satisfied if we choose C 1 = b, C 2 = 1.5, C 3 = 0. Solution to the SIS Optimal Control Problem We use Pontryagin's Maximum principle [22] to solve the optimal control problem (4). Through this technique, we get a system of ordinary differential equations (ODEs) in terms of state and adjoint variables (with initial and boundary conditions, respectively) which are satisfied at the optimum. The system of ODEs can be solved numerically using boundary value ODE solvers. Let λ(t) denote the adjoint variable. At time t, let u * (t) denote the optimum control and, i * (t) and λ * (t) the state and adjoint variables evaluated at the optimum. Hamiltonian: The objective function in (4) has been multiplied by −1 to convert the problem to a maximization problem. H(i(t), u(t), λ(t), t) = −bu 2 (t) + λ(t) − β(t)i 2 (t) + (β(t) − γ − u(t))i(t) + u(t) . Adjoint equation:λ * (t) is − ∂ ∂i(t) H(i(t), u(t), λ(t), t) evaluated at the optimum. λ * (t) = − ∂ ∂i(t) H(i(t), u(t), λ(t), t) i(t)=i * (t),u(t)=u * (t), λ(t)=λ * (t) = 2β(t)i * (t)λ * (t) − β(t) − γ − u * (t) λ * (t).(7) Hamiltonian Maximizing Condition: At the interior points ∂ ∂u(t) H(i(t), u(t), λ(t), t) i(t)=i * (t),u(t)=u * (t),λ(t)=λ * (t) = −2bu * (t) − λ * (t)i * (t) + λ * (t) = 0. Hence the Hamiltonian maximizing condition leads to u * (t) =              0 if λ * (t)(1−i * (t)) 2b < 0, λ * (t)(1−i * (t)) 2b if 0 ≤ λ * (t)(1−i * (t)) 2b ≤ u max , u max if λ * (t)(1−i * (t)) 2b > u max , ⇒ u * (t) = min max λ * (t)(1 − i * (t)) 2b , 0 , u max . (8) Transversality condition: From the transversality condition we get λ * (T ) = 1.(9) Substituting (8) in (5) and (7) and using the initial condition (6) and boundary condition (9) , we get a system of ODEs which can be solved using standard boundary value problem ODE solving techniques. We have implemented the shooting method [23] in MATLAB to solve the boundary value problem. Equations (5) and (7) are solved using MATLAB's initial value problem solver ode45() with Equation (7) initialized arbitrarily. Naturally, the solution will not satisfy the required boundary condition (9); hence the estimation of the initial condition of Equation (7) is improved using the optimization routine fminunc() until the boundary condition (9) is met with desired accuracy. Another option is to use forward-backward sweep method explained in [1,3]. Uniqueness of the Solution to the SIS Optimal Control Problem Theorem 4.2. For a sufficiently small campaign deadline, T , the state and adjoint trajectories at the optimum and the optimal control to problem (4) are unique. Proof. The proof technique is same as in [24], details are in Appendix A. Analysis of the Controlled SIR Epidemic After removing the redundant information, problem (3) can be rewritten in terms of two state variables and two controls as follows (sets U 1 and U 2 are according to Definition 2): min u 1 ∈U 1 ,u 2 ∈U 2 J = −1 + s(T ) + T 0 bu 2 1 (t) + cu 2 2 (t) dt(10) subject to: ṡ(t) = − β(t) + u 2 (t) s(t) 1 − s(t) − r(t) − u 1 (t)s(t)(11)r(t) = γ 1 − s(t) − r(t) (12) 0 ≤ s(t), r(t) ≤ 1 s(0) = 1 − i 0 , r(0) = 0.(13) Existence of a Solution Theorem 5.1. There exist an optimal control signals u 1 ∈ U 1 , u 2 ∈ U 2 and corresponding solutions s * (t), r * (t) to the initial value problem (11), (12) and (13) such that (u 1 , u 2 ) T ∈ argmin (10). u 1 ∈U 1 ,u 2 ∈U 2 {J(u 1 , u 2 )} in problem Proof. The theorem can be proved by application of the Cesari Theorem [20, pg 68]. The details are omitted. Solution to the SIR Optimal Control Problem with Direct and Word-of-mouth Control Let λ s (t) and λ r (t) be the adjoint variables. At time t, let u * 1 (t), u * 2 (t) denote the optimum controls and, s * (t), r * (t) and λ * s (t), λ * r (t) the state and adjoint variables evaluated at the optimum. Using Pontryagin's Maximum Principle [22] we get the following equations. Hamiltonian: The objective function in (10) has been multiplied by −1 to convert the problem to maximization problem. H(s(t), r(t), u 1 (t), u 2 (t), λ s (t), λ r (t), t) = − bu 2 1 (t) − cu 2 2 (t) + λ s (t) − β(t) + u 2 (t) s(t) + β(t) + u 2 (t) s 2 (t) + β(t) + u 2 (t) s(t)r(t) − u 1 (t)s(t) + λ r (t) γ − γs(t) − γr(t) . Adjoint Equations: λ * s (t) = β(t)λ * s (t) − 2β(t)λ * s (t)s * (t) − β(t)λ * s (t)r * (t) + λ * s (t)u * 1 (t) + λ * s (t)u * 2 (t) − 2λ * s (t)u * 2 (t)s * (t) −λ * s (t)u * 2 (t)r * (t) + γλ * r (t) (14) λ * r (t) = −β(t)λ * s (t)s * (t) + λ * s (t)u * 2 (t)s * (t) + γλ * r (t)(15) Hamiltonian Maximizing Condition: Derivative of the Hamiltonian evaluates to zero at interior points, hence the Hamiltonian maximizing condition leads to u * 1 (t) = min max λ * s (t)s * (t) −2b , 0 , u 1max .(16) and, u * 2 (t) = min max λ * s (t)s * (t) 1 − 2s * (t) − r * (t) −2c , 0 , u 2max .(17) Transversality condition: λ * s (T ) = −1 and λ * r (T ) = 0. Substituting the values of u * 1 (t) and u * 2 (t) from (16) and (17) to (11), (12), (14) and (15) and solving the system of ODEs numerically using the technique described in Section 4.2, we can compute the state and adjoint variables and hence the optimal control. In addition, plots in Section 6.3 show variation of the cost functional J with respect to various parameters in the SIS and SIR models. Uniqueness of the Solution to the SIR Optimal Control Problem with Direct Recruitment and Word-of-mouth Control Theorem 5.2. For a sufficiently small campaign deadline, T , the state and adjoint trajectories at the optimum and the solution to the optimal control problem (10) are unique. Proof. The proof technique is same as in [24], details are in Appendix B. Results We divide this section into three parts. Section 6.1 studies the control signal and corresponding state evolution for constant spreading rate and Section 6.2 for variable spreading rate. The tree in Fig. 1 shows how the results are organized in these two subsections. In Section 6.3 we study the role played by various parameters (β, γ, T, b, c) on the cost functional J for both SIS and SIR models. an infected node in the early stages of epidemic outbreak. For the (uncontrolled) SIS and SIR epidemic considered in this paper, R 0 = β/γ, the ratio of effective spreading rate to the recovery rate [19]. Give a campaign deadline T , basic reproductive number R 0 captures how viral the information epidemic is. Qualitatively, increasing β or decreasing γ, while holding the other constant, is expected to have same result. If R 0 > 1 for the uncontrolled system, the epidemic, on the average will become endemic and for R 0 < 1, the epidemic dies out with probability 1 [19]. We discuss these cases separately: (i) With reference to Figs. 2a (for SIS) and 2b (for SIR), where R 0 > 1, direct control signal is strong when the targeted population (susceptibles) are in abundance (at beginning of the campaign period) and vice versa. Early infection increases the extent of information spreading as the system has a tendency to sustain the population in the infected state (because infection is faster than recovery). Also, the word-of-mouth control in Fig. 2b switches on from zero when populations of both susceptible and infected individuals reach significant levels. Providing word-of-mouth incentive is effective only when there are substantial number of infected nodes as well as enough number of susceptibles to convince. (ii) When R 0 < 1 for the uncontrolled system, the uncontrolled information epidemic dies out. The control and state evolutions for this case are shown in Figs. 3a (for SIS) and 3b (for SIR). We find the direct control signal to be less variable over time. In fact, in the SIS case, it increases with time. Fast recovery (compared to infection) of the nodes makes a strong control at the beginning stages of the epidemic ineffective. Also, notice that in the SIR case (Fig. 3b), the optimal strategy advocates not using word-of-mouth control throughout the campaign duration. (iii) We make an additional observation with reference to the control signals and the state evolution curves plotted for R 0 = 10 and R 0 = 20 in Fig. 2a. Given a campaign deadline T , as R 0 = β/γ increases, (a) the control effort (measured by area under the control curve) decreases and (b) control signal has limited effect on system evolution. Thus, campaigns which are less viral will benefit more from the application of optimal control than the campaigns which are more viral. Such an observation has implications on marketing strategies for new products launched by a reputed company compared to a newbie in the market, or publicity of a movie by a famous director compared to a newcomer. Similar observations were made for direct and word-of-mouth controls in case of the SIR model, but the curves corresponding to R 0 = 20 are omitted for brevity. Variable Effective Spreading Rate Over Time To model the varying interest of a population in spreading the information during the campaign period, we consider three different functions β 1 (t), β 2 (t) and β 3 (t). We model the cases of increasing, decreasing and fluctuating interests (18), (19) and (20) as we approach the deadline through these functions. The functions are increasing sigmoid, decreasing sigmoid and cosine (plotted in Fig. 4) and are defined as: β 1 (t) = β m + β M − β m 1 + e −a 1 (t−c 1 ) ,(18)β 2 (t) = (β M − β m ) 1 − 1 1 + e −a 2 (t−c 2 ) ,(19)β 3 (t) = c m + c a cos(2πt/T ),(20) where the values of the parameters used are: β m = .01, β M = 2, T = 5, a 1 = 2, c 1 = 3, a 2 = 2, c 2 = 2, c m = 1, c a = 1 and t ∈ [0, 5]. Wherever β i (t), i = 1, 2, 3 are used, the recovery rate is set to γ = 0.1. The increasing effective spreading rate, β 1 (t) may represent the increasing interest of people to talk about election candidates as we approach the polling date. The decreasing effective spreading rate, β 2 (t) may represent gradual loss of interest of people in talking about some newly launched product (e.g. a computer game) after its release. Fluctuating effective spreading rate β 3 (t) may represent changes in demand of a product/service with time (e.g., movie tickets for weekend shows may have higher demand than tickets for weekday shows). Depending on the application, other profiles for β(t) are possible. The controls and state evolutions for the SIS and SIR models discussed in this paper for the time-varying effective spreading rates defined in Equations (18), (19) and (20) are plotted in Figs. 5-7. The shape of the optimal control may be different from the case when the effective spreading rate β is a constant over time. This shows the need to determine the interest level of the population (and hence β(t)) before deciding on the optimal control strategy. The figures also show the effectiveness of the optimal control strategy over the case when no control is used, in increasing the number of infected nodes in the SIS model, and number of infected and recovered nodes in the SIR model. Thus, optimal campaigning is beneficial in real world scenarios where the effective information spreading rate may be variable. The recovery rate may also be a time dependent quantity in real world applications. The framework developed in this paper can be easily modified to include both time dependent β(t) and γ(t). Comparison Between Optimal, Constant and a Heuristic Control The effective spreading rate is again constant in this section, β(t) = β, ∀t ∈ [0, T ]. The aim of this section is to quantify the effectiveness of the optimal control strategy over simple and intuitive or "common sense" control strategies which do not involve any optimization. We first introduce a simple heuristic control strategy which requires no knowledge of optimal control theory. Let s nc (t), i nc (t) and r nc (t) be the fractions of susceptible, infected and recovered individuals at time t when no control is applied. Since the direct control targets susceptibles, in both the models, its effectiveness depends on the proportion of the susceptible population at the time it is applied. A reasonable heuristic direct control signal could be u 1max · s nc (t) where u 1max is the maximum allowed direct control in the given model. It adjusts the strength of the direct control signal according to the fraction of susceptibles in the no control scenario at any time instant. Word-of-mouth control requires infected individuals to convince susceptibles; hence, it is effective when the numbers of both susceptibles and infected individuals are significant. A heuristic word-of-mouth control signal for the model in Section 3 could be u 2max · s nc (t) · i nc (t) , where u 2max is the maximum allowed word-of-mouth control. We name these controls 'follow s nc (t)' for the SIS model and 'follow s nc (t), s nc (t)i nc (t)' for the SIR model. Please note that these controls are decided and fixed at the beginning of the campaign period (open loop strategies) and are obtained by referring to the quantities for the uncontrolled system. Another simple control strategy applies constant control throughout the campaign period; the control signal is set to half of the maximum allowed signal strength. Thus constant control has values 1 2 u max in the SIS model and 1 2 u 1max (direct), 1 2 u 2max (word-of-mouth) in the SIR model. Fig. 8 shows the shapes of different control signals and the corresponding state evolutions for the SIS model. The cost functional involves weighted sums of the total control effort (area under the control curve) and the final fraction of infected individuals, i(T ). Notice that i(T ) is similar for the three strategies; however, the total control efforts are considerably different. For a better idea of the performance of the optimal control compared to the other strategies, we plot the cost functional J with respect to one of the parameters β, γ, T, b or c for various strategies for both SIS and SIR models in Figs. 9-12. We make following observations from the plots in this section: However, J for the optimal control strategy is always smaller than J for no control. This shows the advantage of optimal control over heuristic controls and illustrates the fact that (for some parameter vectors), an ill-planned campaign may prove more costly than no campaign at all. (ii) With reference to Figs. 9 and 11 (for both the models), from the difference between the curves corresponding to no control and optimal control cases (for large values of β and T respectively), one can conclude the following: If the uncontrolled system is capable of achieving a high value of (1 − s(t)) (either due to high β or high T , given fixed values of other parameters), application of an optimal control strategy does not improve (decrease) costs too much compared to no control. Note that the cost of applying control is zero for 'no control' case and J in that case is nothing but −(1 − s(T )). In other cases, application of the optimal control decreases the value of J compared to the no control strategy. (iii) From Figs. 9 and 10: As R 0 = β/γ increases, J corresponding to the optimal control strategy decreases. In other words, more viral campaigns are less costly to run than less viral campaigns. (iv) From Fig. 12: J increases as one of the weight parameters b or c increases (other parameters held fixed). We incur more costs as application of control becomes dearer. (v) From Fig. 12: The relative increase in J with respect to increase in b or c for the optimal control strategy is less than that for the 'follow s nc (t)' or the 'follow s nc (t), s nc (t)i nc (t)' strategy and constant control strategy. Thus, the optimal control strategy is less sensitive to changes in b or c than other control strategies. Conclusion In this paper, we have studied optimal control strategies for running campaigns on a homogeneously mixed population when the information spreading rate is a function of time. The change in the spreading rate over time reflects the change in the interest level of the population in the subject of the campaign. The first model assumes that information spreads through an SIS process and the campaigner can directly recruit members of the population, at some cost, to act as spreaders. The second model allows the campaigner to incentivize infected individuals (leading to increased effective spreading rate), in addition to the direct recruitment in the SIR epidemic process. We have shown the existence of solutions for the two models, and uniqueness of the solutions for sufficiently small campaign deadline. For both the cases, for constant spreading rate, we have showed the effectiveness of the optimal control strategy over the constant control strategy, a heuristic control strategy and no control. We have shown the sensitivity of control to the time varying spreading rate profile. Our study can provide useful insights to campaign managers working to disseminate a piece of information in the most cost effective manner. Estimating parameter values such as information spreading rate and recovery rate are nontrivial for real world campaigns, and studying sensitivity of control strategies to estimation errors forms an interesting future research direction. Appendix A. Proof for Theorem 4.2 (Uniqueness of the Solution to the SIS Model) For sufficiently small deadline, T , the uniqueness of the solution to the SIS model can be established using techniques similar to [24]. If the solution to the optimal control problem is non-unique, consider two solutions (i, λ) and (î,λ). The time variable t is dropped for notational brevity. Without loss of generality, for 0 ≤ t ≤ T , let, i = e at x, λ = e −at y,î = e atx andλ = e −atŷ , (A.1) where a is a positive real number. Note that x, y,x,ŷ are functions of t, 0 ≤ x,x ≤ 1 and 0 ≤ y,ŷ ≤ y max , for 0 ≤ t ≤ T . We have used (xy −xŷ) = (xy −xy +xy −xŷ) and m 2 + n 2 ≥ 2\|mn\|. Using (A.1) in (5) e atẋ + axe at = −β(t)e 2at x 2 + β(t) − γ − u e at x + u. Writing similar equation for dî dt and subtracting from above we get, e at ẋ −ẋ + ae at (x −x) = − β(t)e 2at (x 2 −x 2 ) + β(t) − γ e at (x −x) − e at (ux −ûx) + (u −û). Substituting (A.1) in (7) we get two equations for i, λ andî,λ, subtracting them we get, (ẏ −ẏ) − a(y −ŷ) =2β(t)e at (xy −xŷ) − (β(t) − γ)(y −ŷ) + (uy −ûŷ). Multiplying both sides by y −ŷ we get, (y −ŷ)(ẏ −ẏ) − a(y −ŷ) 2 =2β(t)e at x(y −ŷ) 2 + 2β(t)e atŷ (x −x)(y −ŷ) − (β(t) − γ)(y −ŷ) 2 + y(u −û)(y −ŷ) +û(y −ŷ) 2 . Integrating both sides with respect to t from 0 to T , we get, Thus the solution to state and costate equations and hence the optimal control is unique for sufficiently small deadline, T . Notice that C i > 0, i = 8, ..., 11; otherwise it can be estimated as 0 due to inequality in (A.5). Appendix B. Proof for Theorem 5.2 (Uniqueness of the Solution to the SIR Model with Direct Recruitment and Word-of-mouth Control) For sufficiently small deadline, T , the uniqueness of the solution can be shown. If the solution to the optimal control problem is non-unique, consider the two solutions (r, s, λ s , λ r ) and (r,ŝ,λ s ,λ r ). The time variable t is dropped for notational brevity. Without loss of generality, for 0 ≤ t ≤ T , let, s = e at x, r = e at y, λ s = e −at p, λ r = e −at q and s = e atx ,r = e atŷ ,λ s = e −atp ,λ r = e −atq . The technique used is same as in Appendix A, hence only the final form of the estimations are shown here. (u 1 −û 1 ) 2 ≤ A 1 (x −x) 2 + A 2 (p −p) 2 , (u 2 −û 2 ) 2 ≤ (A 3 e aT + A 4 )(x −x) 2 + A 5 e aT (y −ŷ) 2 + (A 6 e aT + A 7 )(p −p) 2 . From the state equations (11) and (12) we get, are non empty (due to Lipschitz continuity of f (u(t), i(t)) [21, pg. 185]). The control signal takes values in a closed set [0, u max ]. The cost due to the terminal state in the cost functional (4) takes values in a compact interval [0, 1]. The function f Figure 1 : 1Organization of plots related to the shapes of the control signals and state evolutions in Sections 6.1 and 6.2. , i 0 = 0 0) SIS epidemic, β = 1 (R 0 = 10) and β = 2 (R 0 = 20) Parameter values: γ = 0.1, T = 5, b = 15, u max = 0.06(right Y−axis) wom control (right Y−axis) (b) SIR epidemic, β = 1 or R 0 = 10 Parameter values: γ = 0.1, T = 5, b = 15, c = 1, u 1max = 0.06, u 2max = 0.3, s 0 = 0.99, i 0 = 0.01. Figure 2 := 0 20Optimal control, state evolutions with control and state evolutions without control for the SIS epidemic (model in Section 2) and the SIR epidemic (model in Section 3). Note that state variables are plotted with respect to the left Y-axis and control signals are plotted with respect to the right Y-axis.6.1. Constant Effective Spreading Rate Over TimeFirst we consider the case when the effective spreading rate is constant over time. Thus, β(t) = β, ∀t ∈ [0, T ]. The basic reproductive number, R 0 , for an epidemic is defined as the expected number of secondary infections caused by epidemic, β = .03 or R 0 = 0.3 Parameter values: γ = 0.1, T = 5, b = 15, u max = 0.06, i 0 epidemic, β = .03 or R 0 = 0.3 Parameter values: γ = 0.1, T = 5, b = 15, c = 1, u 1max = 0.06, u 3max = 0.3, s 0 = 0.99, i 0 = .01 Figure 3 : 3Optimal control, state evolution with control and state evolution without control for the SIS epidemic (model in Section 2) and the SIR epidemic (model in Section 3). Note that state variables are plotted with respect to the left Y-axis and control signals are plotted with respect to the right Y-axis. Figure 4 : 4Time varying effective spreading rate, β 1 (t), β 2 (t) and β 3 (t) defined in equations respectively. Parameter values: β m = .01, β M = 2, T = 5, a 1 = 2, c 1 = 3; a 2 = 2, c 2 = 2, c m = 1, c a = 1 and t ∈ [0, 5]. ) SIS epidemic. Parameter values: γ = 0.1, T = 5, b = 15, u max = 0.06, i 0 = 0.01. ) SIR epidemic. Parameter values: γ = 0.1, T = 5, b = 15, c = 1, u 1max = 0.06, u 2max = 0.3, s 0 = 0.99, i 0 = 0.01. Figure 5 : 5Optimal control, state evolution with control and state evolution without control for the SIS and SIR epidemic, for time-varying spreading rate β 1 (t). Note that state variables are plotted with respect to the left Y-axis and control signals are plotted with respect to the right Y-axis. epidemic. Parameter values: γ = 0.1, T = 5, b = 15, u max = 0.06, i 0 = 0.01. epidemic. Parameter values: γ = 0.1, T = 5, b = 15, c = 1, u 1max = 0.06, u 2max = 0.3, s 0 = 0.99, i 0 = 0.01. Figure 6 : 6Optimal control, state evolution with control and state evolution without control for the SIS and SIR epidemics for time-varying spreading rate β 2 (t). Note that state variables are plotted with respect to the left Y-axis and control signals are plotted with respect to the right Y-axis. ) SIS epidemic. Parameter values: γ = 0.1, T = 5, b = 15, u max = 0.06, i 0 = 0.01. (right Y−axis) wom control (right Y−axis) (b) SIR epidemic. Parameter values: γ = 0.1, T = 5, b = 15, c = 1, u 1max = 0.06, u 2max = 0.3, s 0 = 0.99, i 0 = 0.01. Figure 7 : 7Optimal control, state evolution with control and state evolution without control for the SIS and SIR epidemics for time-varying spreading rate β 3 (t). Note that state variables are plotted with respect to the left Y-axis and control signals are plotted with respect to the right Y-axis. of the proportion of infected individuals for different control strategies. Figure 8 : 8Shapes of different control signals and the corresponding state evolutions for the SIS model. Parameter values: β = 1, γ = 0.1, (R 0 = 10), T = 5, b = 15, u max = 0.06, i 0 = 0.1. Note: both of the Figures contribute to the objective function which is optimized, see Figs. 9a-12a for objective function values for different strategies with varying parameter values. ) SIS epidemic. Parameter values: γ = 0.1, T = 5, b = 15, i 0 = 0.01, u max = 0.06. ) SIR epidemic. Parameter values: γ = 0.1, T = 5, b = 15, c = 1, u 1max = 0.06, u 2max = 0.3, s 0 = 0.99, i 0 = 0.01. Figure 9 : 9Objective functional J vs spreading rate β for different control strategies. epidemic. Parameter values: β = 1, T = 5, b = 15, i 0 = 0.01, u max = 0.06. epidemic. Parameter values: β = 1, T = 5, b = 15, c = 1, u 1max = 0.06, u 2max = 0.3, s 0 = 0.99, i 0 = 0.01. Figure 10 : 10Objective functional J vs recovery rate γ for different control strategies. ) SIS epidemic. Parameter values: β = 1, γ = 0.1, b = 15, i 0 = 0.01, u max = 0.06. ) SIR epidemic. Parameter values: β = 1, γ = 0.1, b = 15, c = 1, u 1max = 0.06, u 2max = 0.3, s 0 = 0.99, i 0 = 0.01. Figure 11 : 11Objective functional J vs campaign deadline T for different control strategies. epidemic. Parameter values: β = 1, γ = 0.1, T = 5, i 0 = 0.01, u max = 0.06. epidemic. Parameter values: β = 1, γ = 0.1, T = 5, u 1max = 0.06, u 2max = 0.3, s 0 = 0.99, i 0 = 0.01. Figure 12 : 12Cost functional J vs weight parameters (b, c) (i) From Figs. 9-12: for some parameter vectors, the cost functional J for the constant control strategy and 'follow s nc (t)' (for SIS model) or 'follow s nc (t), s nc (t)i nc (t)' (for SIR model) strategy is more than J for no control. u = min max e −at y(1 − e at x) 2b , 0 , u max , u = min max e −atŷ (1 − e atx ) 2b , 0 , u max .Thus we can estimate (u −û) Multiplying both sides by(x −x) e at (x −x)(ẋ −ẋ) + ae at (x −x) 2 = − β(t)e 2at (x +x)(x −x) 2 + e at β(t) − γ (x −x) 2 − e at u(x −x) 2 + (1 − e atx )(u −û)(x −x) ≤e aT C 1 (x −x) 2 + C 2 \|(u −û)(x −x)\| ≤e aT C 3 (x −x) 2 + C 4 (u −û) 2 . C 2 = 2max(\|1 − e atx \|). Integrating both sides with respect to t from 0 to T , we get,1 2 (x −x) 2 (T ) + (a − C 3 e aT ) T 0 (x −x) 2 dt ≤ C 4 T 0 (u −û) 2 dt (A.3) t)e atŷ \|\|(x −x)(y −ŷ)\|dt + (β max − γ) −ŷ) 2 (0) + (a − C 5 e aT ) T 0 (x −x) 2 + (a − C 6 e aT − C 7 ) −ŷ) 2 (0) + (a − C 8 e aT − C 9 ) T 0 (x −x) 2 + (a − C 10 e aT − C11 ) to the conclusion that x =x and y =ŷ for, 1 −û 1 ) 2 dt + (C 4 e aT + C 5 ) T 0 (u 2 −û 2 ) 2 dt and, Costate equations(14)and(15)lead to,and,Finally the estimates from state and costate equations are added and the estimates of (u 1 −û 1 ) 2 and (u 1 −û 1 ) 2 are used in the right hand side to get an inequality of the form,Notice that D i > 0, i = 1, ..., 8 otherwise it can be estimated to 0 due to above inequality. Finally, x =x, y =ŷ, p = p, q =q for,Thus the solution is unique for sufficiently small deadline, T . Optimal Control of Vaccine Distribution in a Rabies Metapopulation Model. E Asano, L J Gross, S Lenhart, L A , Mathematical Bioscience and Engineering. 5E. 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[ "Optimum Control of Flow around a Circular Cylinder with Non-Uniform Suction", "Optimum Control of Flow around a Circular Cylinder with Non-Uniform Suction" ]	[ "James Ramsay \nDepartment of Mechanical Engineering\nUniversity of Canterbury\nChristchurchNew Zealand\n", "Mathieu Sellier \nDepartment of Mechanical Engineering\nUniversity of Canterbury\nChristchurchNew Zealand\n", "Wei Hua Ho \nDepartment of Mechanical and Industrial Engineering\nUniversity of South Africa\nPretoriaSouth Africa\n" ]	[ "Department of Mechanical Engineering\nUniversity of Canterbury\nChristchurchNew Zealand", "Department of Mechanical Engineering\nUniversity of Canterbury\nChristchurchNew Zealand", "Department of Mechanical and Industrial Engineering\nUniversity of South Africa\nPretoriaSouth Africa" ]	[]	In the present study, numerical investigations were performed to determine the optimum non-uniform suction profiles to control the flow around a circular cylinder in the range of Reynolds numbers 4 < < 200. To investigate how the characteristics of the optimal control and the resulting flow change depending on the optimisation objective, several objectives were explored, namely: minimising the separation angle, total drag, and pressure drag. A variety of suction control implementations were investigated and compared to the performance of uniform suction. It was determined that the optimal non-uniform suction profiles consisted of a distribution with compact support, and a single locus. The location of the optimum suction region and the amount of suction necessary to achieve each objective varied substantially with Reynolds number, but with a predictable relationship. It is also shown that these parameters can alternatively be considered as related to the separation angle of the uncontrolled flow (the initial separation angle). Depending on the objective, the control parameters varied greatly: less suction was necessary to minimise total drag than to eliminate separation. Nonuniform suction profiles were much more efficient at eliminating boundary layer separation, requiring the removal of less than half the volume of fluid as uniform control to achieve the same objective. Total elimination of boundary layer separation did not always result in an improvement in total drag, and in some circumstances increased it. An analysis of the drag components showed that, although the pressure drag was substantially improved by boundary layer suction, the disappearance of the separated region downstream resulted in faster flow over the cylinder and consequently a higher skin friction drag. The drag-optimised flows had characteristics very close to those of a steady cylinder within the unsteady regime. These results show that the balance of drag components must be an important consideration when designing flow control systems and that, when done appropriately, substantial improvement can be seen in the flow characteristics.	null	[ "https://arxiv.org/pdf/1903.12326v1.pdf" ]	88,522,961	1903.12326	82d0035ee24f3b67905202aac782e739ece9c137	Optimum Control of Flow around a Circular Cylinder with Non-Uniform Suction James Ramsay Department of Mechanical Engineering University of Canterbury ChristchurchNew Zealand Mathieu Sellier Department of Mechanical Engineering University of Canterbury ChristchurchNew Zealand Wei Hua Ho Department of Mechanical and Industrial Engineering University of South Africa PretoriaSouth Africa Optimum Control of Flow around a Circular Cylinder with Non-Uniform Suction 1 In the present study, numerical investigations were performed to determine the optimum non-uniform suction profiles to control the flow around a circular cylinder in the range of Reynolds numbers 4 < < 200. To investigate how the characteristics of the optimal control and the resulting flow change depending on the optimisation objective, several objectives were explored, namely: minimising the separation angle, total drag, and pressure drag. A variety of suction control implementations were investigated and compared to the performance of uniform suction. It was determined that the optimal non-uniform suction profiles consisted of a distribution with compact support, and a single locus. The location of the optimum suction region and the amount of suction necessary to achieve each objective varied substantially with Reynolds number, but with a predictable relationship. It is also shown that these parameters can alternatively be considered as related to the separation angle of the uncontrolled flow (the initial separation angle). Depending on the objective, the control parameters varied greatly: less suction was necessary to minimise total drag than to eliminate separation. Nonuniform suction profiles were much more efficient at eliminating boundary layer separation, requiring the removal of less than half the volume of fluid as uniform control to achieve the same objective. Total elimination of boundary layer separation did not always result in an improvement in total drag, and in some circumstances increased it. An analysis of the drag components showed that, although the pressure drag was substantially improved by boundary layer suction, the disappearance of the separated region downstream resulted in faster flow over the cylinder and consequently a higher skin friction drag. The drag-optimised flows had characteristics very close to those of a steady cylinder within the unsteady regime. These results show that the balance of drag components must be an important consideration when designing flow control systems and that, when done appropriately, substantial improvement can be seen in the flow characteristics. INTRODUCTION The flow around a circular cylinder has been the subject of much interest for over a century and, for almost as long, so has the problem of controlling it. This bluff body flow can produce high drag and lift characteristics as well as a variety of undesirable flow phenomenaespecially vortex shedding. To complicate the search for optimal control parameters, the behaviour of the flow and its impact on the aerodynamic characteristics of the cylinder vary widely with relatively small changes in Reynolds number ( = ). Although much focus has been paid to minimising the drag or eliminating vortex shedding by flow control, the phenomenon of boundary layer separationwhich plays an important role in the arising of both these features -has not attracted as much scrutiny. Despite the sustained interest in this field, the problem of optimal control for this flow is a matter that is far from settled. The flow around a circular cylinder offers a complex problem for the study of flow control because the features of the flow can be changed substantially only by varying the Reynolds number. In the range of Reynolds numbers from 10 0 to 10 7 , Williamson 1 distinguishes the resulting flows into nine regimes with unique features and phenomena. Zdravkovich 2 divides them more finely, with four additional regimes. At low , the flow is unseparated and steady ( < 6), but in the range of 6 < < 47 the boundary layer separates and a pair of vortices form behind the cylinder; the separation point moves toward the front of the cylinder with increasing . At about = 47, the well-known phenomenon of vortex shedding begins and continues in a two-dimensional manner up until = 188.5 where the vortex shedding begins to exhibit 3D features. With further increasing the wake becomes even more complex as different areas of the flow transition to turbulence, beginning with the wake, moving up the shear layers, and finally transitioning in the boundary layer, which marks the onset of the well-known 'drag crisis'. At this point, the separation point is delayed significantly, jumping much further aft and providing a substantial decrease in pressure drag. Clearly, any attempt to provide optimal flow control over this entire range would be exceedingly difficult if not impossible. However, by investigating small regions of the range, relationships between the flow and control parameters can be established which may shed light on other regimes (or to entirely different body-fluid flows). Much research has been performed on the cylinder at high , particularly near the critical point ≈ 2 × 10 5 (sub-critical regime), because many engineering situations in real life experience such conditions. For investigating the physics of flow control, though, the low range offers the best conditions. The small range of < 200, which was used for this study, includes three very different flow regimes with major shifts in phenomena. The flow remains fully laminar and two-dimensional over the entire range, though, which means it can be easily modelled and experimented upon. One of the important features of bluff body flows is the separation angle, , which marks the point where the boundary layer separates from the surface as measured from the leading or trailing edge of the body (trailing edge for this paper). It is commonly held that once the boundary layer has separated and become sufficiently established, the separation angle does not change significantly until the drag crisis at ≈ 2 × 10 5 where it jumps from about 100° to 60° 3,4 . Unfortunately, this parameter has not been considered to be a particularly important feature for measurement or control. In many papers exploring the flow around circular cylinders, the separation angle is only described as a variable of secondary importanceusually only in discussion of the features of the pressure profiles. In actuality, however, the separation angle is in many ways the key mover of the flow phenomena, not their object. The pressure drag, , which makes up almost all of the total drag on the cylinder at high is predominantly caused by the separation of the boundary layer. Vortex shedding occurs when vortical structures developed in the shear layers grow from the surface. Furthermore, experiments over a wide range by Weidman 5 , and separately by Achenbach 6 , show that the separation angle is a more active feature than commonly held, perpetually moving across the entire range, and not always monotonically. In 2004 Wu et al. 7 performed a review of experiments which had measured the separation angle of the cylinder at low , and found that the results between studies deviated by as much as 10°. By their own experimentation and numerical simulation, the researchers found accurate values and determined the causes of the deviationmostly due to different measurement techniques. This demonstrates one of the many reasons that the separation angle has been relatively unexplored in the literaturedifficulty in measuring it accurately. An approximation that the separation point corresponds to the inflection point of the pressure curve over the cylinder is commonly used, as by Fransson et al 8 , but may not be entirely accurate. However, the improvement of experimental techniques and equipment has now eliminated many of these problems, and these difficulties are not present in computational fluid dynamics (CFD). Therefore, there is no reason that the separation angle should not be a feature of particular interest-both in terms of its importance in the arising of other phenomena like vortex shedding, but also as a parameter that itself might be directly controlled. To date, many active and passive methods of control have been investigated to control the flow around the circular cylinder. These range from simple geometric features such as splitter plates 9 and helical strakes 10 to complex active measures like plasma actuators 11 or magnetic fields 12 . Many of these are described in the Annual Review by Choi, Jeon and Kim 13 or in a 2016 review by Rashidi et al. 14 Each of these methods have achieved some success at reducing drag or weakening vortex shedding, though often at significant cost. One of the simplest forms of flow control is boundary layer suction. This method removes the low momentum fluid particles at the surface, thus entraining higher momentum particles from the free-stream to reinvigorate the boundary layer, and delaying separation. This method of flow control is as old as the boundary layer concept itselfwith Prandtl testing his theory by experimenting on slot suction of a cylinder 15 . Nevertheless, it is still not a settled matter how this method can optimally control the flow around a circular cylinder, i.e. achieve the control objective with the least suction/fluid removal. Boundary layer suction has many advantages compared to other control methods. For one, the geometry of the body does not have to be changed (although the materials of the surface may have to). Further, the method is simple and practicalits parameters can be adjusted easily and have a wide range; there is no multi-physicality to this control. And of particular importance to this study, the method is well researched both experimentally and numerically, but there is no clear consensus on its optimal parameters. Experiments on suction control began in the early 20 th Century, and much interest was paid to this subject for the improvement of aerodynamic characteristics for aircraft during the Second World War 16 . Over this time, two main applications of suction control were investigated: uniform suction over the entire surface of a cylinder with porous wallsas researched experimentally by Thwaites and his colleagues 17,18and slot suction, where only part of the boundary layer is removed through a slot or series of slots in the cylinder surface. Pankhurst and Thwaites 17 found that sufficient uniform suction resulted in the flow approaching that seen for potential flow, with improvements in drag and elimination of vortex shedding. More recently, Fransson et al. 8 determined a relationship between the controlled flow using uniform suction on the cylinder and the uncontrolled flow at a different Reynolds number, and that these could be linked via the Strouhal number. However, the effective Reynolds relationship found by Fransson et al. is only applicable if the control does not entirely suppress vortex shedding which is a common objective for bluff body flow control. Uniform suction has its disadvantages, namely the inefficiency of removing material at all areas of the cylindereven where it may not be necessaryand its limited control parameters. Slot suction is similarly disadvantaged, being limited in its location of application, the distribution profile of the suction, and the discontinuity of its nature. A study of distributed slot suction (varying in the spanwise direction) by Kim and Choi 19 found that the objective of control (total drag in this case) was very sensitive to the location of the slots, and at = 100 the best results were achieved with the slots located between 80-100°. A better approach to suction control combines the benefits of each of these methodsnon-uniform suction. This method is applied similarly to uniform suction by using a porous surface, however the suction is applied unevenly, with the possibility of any potential distribution over the surfacecontinuous or otherwise. Theoretically, this allows much more precise control of the flow, with the possibility to concentrate the suction control at critical areas of the surface and apply no control where it is unnecessary. Because there are so many potential profiles for this method, determining its "optimum" is not straightforward. Some papers on this subject have been published, particularly with the focus of utilising it in conjunction with feedback from the flow in order to mitigate the wide-range of potential implementations. Min and Choi 20 employed sub-optimal feedback to control the parameters for non-uniform suction and blowing. The researchers successfully reduced the pressure drag of the surface, and achieved pressure profiles closer to those predicted by potential flow theory. However, as the focus of this study was the development of suboptimal feedback using limited time-steps of control, this paper only offers an introduction to what the true optimal control might look like with this method. Li et al. performed a similar numerical experiment and carried out a complete adjoint optimisation procedure with unsteady, time-dependent simulations. The researchers achieved a complete control of vortex shedding for up to = 110. They also found that the optimal controls were insensitive to initial conditions if the control was applied for time-scales longer than the vortex shedding period. The researchers used objectives for the error between the flow field and potential flow field, the enstrophy of the flow (to suppress vortex shedding), and the minimisation of drag. The separation angle of the flow was not investigated. Very recently, the optimum spanwise-varying suction/blowing control of a 3D circular cylinder in 2D flow was determined using eigenmode analysis by Boujo et al. 21 Due to the nature of this method, though, it can only be used to optimise for stabilisation or frequency modification which does not necessarily coincide with minimised drag. All of these studies were performed by numerical methods; non-uniform suction has not been explored substantially by physical experimentation. These studies suggest that non-uniform suction offers a control method that can be used to optimally achieve a variety of objectives for the flow around circular cylinders. Despite this interest though, the optimal profiles for this type of control are not clear, nor has a comparison with the other implementations of suction control (uniform and slot suction) been performed to any depth. Throughout most investigations into the control of flow around a circular cylinder, the importance and impact of the separation angle appears to have been under-investigated. Therefore, the objective of the present study is to comprehensively investigate the variety of suction control methods for a circular cylinder by numerical investigation, with particular focus on the separation angle as an output parameter and control objective. A variety of optimisation studies with differing setups and objectives were performed in the Reynolds range of 4 < < 200 in order to best determine the relationships between the three factors of controluncontrolled flow, control parameters, and the resulting flow. This range offers a variety of flow regimes (unseparated flow, separated steady flow, and vortex shedding) while offering a simple, two-dimensional and fully laminar flow to carry out many simulations. Of particular interest were the impacts of different control setups and objectives on the features of separation angle, drag components, pressure profiles, and the general features of the flow. METHODS AND MODELLING Geometry, mesh and solver methods The governing equations for unsteady incompressible and isothermal viscous flow are described by the following: + = − + 1 2 , = 0, where represents the velocity components, gives the directional vectors, and is the pressure. For some of the simulations, the flow was modelled as steady, and so the time derivatives vanished. The laminar flow module of COMSOL Multiphysics was used to solve the system for all the studies in the present investigation. Fluid properties for water at 20°C were used for the model, employing the above equations. For the time-dependent cases, an implicit time-stepping method was employed with 'Intermediate' stepping to reduce the CFL restrictions while preventing numerical smearing and enabling the instabilities that induce vortex shedding. A time-step giving 30 steps per vortex shedding period resulted in accurate and sufficiently precise results for the mesh described in Figure 1 and Table I. The upper and lower boundaries were modelled as no-slip moving walls with the same velocity profile as the inlet. This minimises any potential blockage effects that may result from the artificially limited domain. The actual domain has blockage ratio, = 1 50 = 0.02 which was shown to reduce the error in measured separation angle to below 0.2% by Wu et al. A pressure outlet condition was imposed on the outlet (right-boundary) with zero relative pressure. The cylinder walls were modelled as a fixed-velocity outlet with defined normal outflow velocity, = , = 0, where and are the normal and tangential velocity components at the wall, respectively. This boundary condition made it possible to define any suction or blowing profile on the cylinder wall by only changing the function that defines , the suction velocity. In keeping with the terminology typically used in the literature, the non-dimensional suction coefficient, = × 100, was used as the control parameter, from which was defined. In this paper with a lower case 'c' refers to the local suction coefficient at any particular point on the cylinder, while refers to the global suction coefficient of the cylinder as a whole, = 1 2 ∮ . The two definitions will be useful given that nonhomogeneous suction profiles are the subject of this investigation. With this boundary condition, locations where no suction is applied have the same definition as a no-slip wall. Suction control setup One of the objectives of this study was to investigate how non-uniform suction profiles might more efficiently control the flow around the circular cylinder than uniform suction. To explore this, three methods for applying nonuniform suction profiles were devised. These are summarised in Figure 3. The values underneath each setup show the number of control parameters for each distribution. Since a theoretically infinite variety of non-uniform suction profiles might be generated, and since it was desirable to be able to model any shape of these profiles, the six-segment profile was employed. Of the potential suction profiles, these may be divided into two categories: those with one locus of suction and those with multiple. The six-segment control method can model both. For this setup, the cylinder is divided into twelve equal segments and the suction coefficient at the centre of each is used as the control variables, . As we were not concerned with the lift or lateral forces of the cylinder, the lower half segments were set to mirror the upper half, so there were only six segments for control. To create a continuous suction profile from the six discrete values, pointwise interpolation with a continuous second-derivative constraint was applied. In this way, a suction profile with multiple loci of suction could be modelled. Preliminary investigations found that six-segment control, as opposed to using more or less segments, gave the best control at a Reynolds number of 180. Preliminary investigations also found that the optimal suction profiles typically had only one locus of suction. The "moving distribution" was defined to generate this type of distribution using fewer parameters. Here three control parameters are used: the maximum local suction coefficient, , location of suction, , and spread of suction, . To create a smooth suction profile, a cubic curve was defined from these parameters and the condition of zero-gradient was applied at the edges and centre of the profile. Thus, a distribution with compact support can be generated, and by changing the parameter, it can be moved over the cylinder surface. The flexi-moving distribution was identical to the moving distribution profile except for the addition of a fourth control parameter: a bias factor, . This allowed the suction profile to be asymmetrical, though it introduces the risk of a profile that is so steep it creates separation bubbles in the flow. In addition to these non-uniform suction profiles, uniform suction was also investigated. Here, only one parameter was necessary, , which was held constant at all locations on the cylinder surface. Optimisation methods To best understand how the optimal flow control changes depending on the objective of the control, three different cost functionals were defined for the present study: 1 = = ( \| =1.001 = 0, − < 0) (1) 2 = = + = 2 2 ∮ (− ( )\| = + − ( ) \| = ) cos( ) (2) 3 = = 2 2 ∮− ( )\| = cos( )(3) 1 is the angle of separation as measured from the trailing edge. For this study, the separation point was located by determining where the flow reverses direction very near to the wall (measured along a curve with a diameter of 1.002 ). In some cases, such as in the unsteady vortex shedding regime, there are several points of separation and reattachment. The angle of separation we are concerned with is that for the first point of separation, i.e. that closest to the front stagnation point (the leading edge, LE). This identification was carried out using a custom MATLAB function that was implemented into the COMSOL model. It was verified by comparison to the point that the skin friction drag reduces to zero, another feature of separation points. Control that prevents separation will minimise 1 asymptoting to zero as the separation point reaches the trailing edge. 2 is the total drag coefficient, the sum of the pressure drag, , and skin friction drag, , coefficients. The total drag of the uncontrolled cylinder is quite non-linear in relation to so we would not expect the prevention of separation to necessarily correspond to a minimisation of drag due to the several impacting factors contributing to this force. 3 is the pressure drag. It is typically assumed that the pressure drag arises solely from the separation of the boundary layer. The loss of momentum to the vortical structures that form as a consequence means that the pressure is not fully recovered on the lee side of the body. However, pressure drag can arise even in an unseparated flow as there is still momentum lost through the no-slip interaction with the wall. Because of this, the minimisation of pressure drag was investigated as its own objective to see how it compares to the separation objective results, 1 . In addition to these three, an additional objective was included in each of the studies. The optimal suction control was defined for this study as that which would achieve the objective with least effort, therefore a secondary objective is necessary to measure the controller effort, i.e. the net suction. This objective is defined as the cost function, , below. The overall cost functional for the studies investigated here, is therefore the sum of and (where is one of 1 , 2 or 3 dependent on the present study). As the efficiency of control is of secondary concern to achieving the actual flow characteristic objective, a scaling factor of 0.01 was employed in the addition of to the global objective as shown in Equation 5. = = 1 2 ∮ ( )(4) = + 1 100 (5) The 'Optimization Module' of COMSOL Multiphysics includes several optimisation methods that can be used to solve problems such as those presented here. In earlier investigations, the Coordinate Search (CO) method had been used to some success, however it is very sensitive to the number of parameters and their order. The Nelder-Mead 22 (NM) method, on the other hand, was less sensitive to the parameter order though often taking much longer to converge to a final solution. It was less likely to converge to a local minimum instead of the global, however. In this investigation, both methods were used at different times. For the separation angle objective with the six-segment control a variety of the methods were tested, as well as a variety of orderings for the control parameters. For the drag objectives, however, the NM method was used almost exclusively as it was the most robust, and for the moving distribution setupwith only a few parametersit was not too slow. Here, the typical optimisation process required about 100 iterations, and could take between 20-40 minutes total depending on the Reynolds number. Each iteration consists of solving the steady-state flow for the input parameters, evaluating the objective function, and adjusting the control parameters for the next iteration. In this paper, where a study has been repeated with different optimisation methods or parameter ordering, the best results will be shown, except where comparisons between the methods/setup are made. RESULTS Validation of model The model was validated by comparing the measured separation angles for the uncontrolled flows to those found in experiments and other numerical studies, as reviewed by Wu et al 7 . This is shown in Figure 3 (a). It can be seen in this figure, that the time-dependent simulations model the flow accuratelythe instantaneous behaviour is accurate too, although these values from the literature are not shown in the plot. The steady-state results diverge from the expected values within the vortex shedding regime ( > 47). These values do follow the same trend as the actual behaviour, however, and result in earlier separation angles so provide a conservative approximation. Since it is known that sufficient boundary layer suction stabilises an unsteady flow, and it is anticipated that a control that eliminates boundary layer separation will necessarily be stable, it was decided that the steady-state solver would be satisfactory for the optimisation process. This assumption may follow for the separation angle objective, but the final state of the controlled flow is less well-defined for the drag objectives, and therefore the results may not be as accurate. For this reason, time-dependent simulations were also used to validate the optimal results for each objective. As will be discussed later, this assumption of steady flow, indeed, was not accurate for the drag objectives at all . In this study, the aerodynamic characteristics of the cylinder are of particular interest, therefore it was necessary to validate the measurements of the drag components also. Again, this was carried out by comparing the uncontrolled flow from time-dependent simulations with historical data [23][24][25] . These results are shown in Figure 3 (b) and show a good fit. The data from Wieselsberger 25 and Tritton 23 come from physical experiments, while the results from Henderson are from direct numerical simulations (DNS). The values from Henderson are used as the benchmark in this paper as the author provided good fits to his data. They are limited, however, in that Henderson only modelled the cylinder at > 25 so the fits may not be valid for the steady regimes. Again, in the vortex shedding regime, the steady-state model diverges from the actual values. It is important to note that the steady solver underestimates the drag, despite the separation angle being determined to be closer to the leading edge of the cylinder. One might have expected the pressure drag, and therefore the total drag, to be larger with an earlier separation angle but that is not the case here. Without the modelling of the unsteady vortices and the large lateral movements in the wake, the effects on the pressure and drag are not appropriately captured. Separation objective In this section, we present results from the optimisation studies with the objective of minimising the separation angle, 1 . We investigate the effectiveness of each suction control setup, with particular focus on the total suction coefficient, , the location of suction, , the effect on drag and its components, and the pressure profile over the cylinder with and without control. A. Best control methods For all control setups, the objective of eliminating boundary layer separation was successfully achieved. A sample of the resulting flows can be seen in Figure 4 along with the instantaneous flow field of the uncontrolled case for comparison. It can be seen in these figures that the controlled flows all have a similar structure, with a much smaller, symmetrical wake. The streamlines illustrate how the freestream fluid is entrained as it passes the cylinder to replace the fluid removed through the suctioned surface. An important feature to note is that the velocity vectors near the top and bottom of the cylinder are much larger in the controlled cases. This is because there is no longer stagnated and separated flow downstream, so the fluid can flow more quickly over the cylindermore like potential flow. This may be necessary for eliminating separation, but it can have adverse effects on the drag characteristics. It may be difficult to tell by the flow fields in Figure 4 alone that the amount of suction required to eliminate separation is much greater for the uniform case than the non-uniform methods of control. The plots in Figure 5 show the suction quantity coefficient, , against the Reynolds number and the initial separation angle before control is applied. It is clear from this figure that uniform suction requires much more control effort to eliminate separation compared to any of the other methods. In all instances (except the trivial non-separated cases), the control effort is at least twice that of the non-uniform suction. These plots show interesting features regarding the relationship between the flow features and the necessary control effort. For all methods, the amount of suction required to eliminate separation increases in the vortex-pair regime ( < 47) up to a point, after which it decreases with increasing . Figure 5 (a) provides rational fits for the uniform and moving distribution data each with a 2 nd degree numerator and denominator. For the relationships with the initial separation angle in Figure 5 (b), polynomial fits were more appropriate. It is important to note that these are only fits for the data, and there is no physical justification for their form or parameter values. Nevertheless, clear trends can be seen between the uncontrolled flow features and the necessary control parameters. Extrapolating these trends to higher Reynolds numbers suggests that the suction control effort would plateau near = 23 and = 10 for uniform and moving distribution suction respectively. However, given what we know of the uncontrolled flow, and how substantially the characteristics of the flow change with increasing alone, it may be unreasonable to try and apply these formulae outside the regimes they were tested on. Pankhurst & Thwaites 17 found that a suction quantity of √ ≥ 30 , was required to eliminate separation on a cylinder fitted with a splitter plate in the range of 10 4 − 10 5 . Extrapolating the present results to this regime, a suction coefficient of √ ≈ 14 is necessary a . Likewise, Pankhurst & Thwaites' relationship cannot be extended to the present regime, < 200, as it would suggest a suction coefficient of = 150.9 would be necessary at = 40 which is obviously not the case. The Reynolds number does not contain sufficient information about the drastic changes in flow features to allow for these relationships to be extended. On the other hand, the separation angle is a feature of the uncontrolled flow that itself is altered with the regime changes. Since, there is a clear trend between the uncontrolled separation angle, 0 , and the required controland given the uncontrolled separation angle reflects the conditions of the flow regimeit may be possible to extend the relationship with 0 into higher ranges. After all, the mechanism by which the suction reinvigorates the boundary layer should also remain the same whenever the boundary layer and the shear layers directly adjacent to it are still laminar, in other words, almost up to the transition to turbulence at ≈ 2 × 10 5 . It can be seen in Figure 5 that the best results are usually achieved with the flexi-moving distribution. The improvement in compared to the moving distribution profile is marginal thougha maximum difference of 40% at = 10, but less than 3% at all higher Re, and in some instances the moving distribution giving a better result ( = 80, 100). Furthermore, the flexi-moving distribution has potential issues with discontinuities in the model due to the steep suction profiles it can generate (as seen at = 120 in Figure 4). Finally, as discussed earlier, the optimisation process is most stable for setups with the least number of control parameters. Therefore, since there seems to be little advantage in using the more complex flexi-moving distribution over its symmetrical variety, the rest of the results will be concerned with the moving distribution profiles only. a Pankhurst &Thwaites defined as the flow rate through the porous wall divided by ( ), i.e. = according to our notation, hence the introduction of the term to their equation in this text. Figure 5 (a) demonstrates how important the appropriate setup of control parameters and optimisation methods are. The 'Other Solutions' shown for the six-segment setup were mostly generated when the Coordinate Search method was employed, with differing orders of control parameters. Some of these solutions were generated when testing other optimisation methods such as BOBYQA 26 or Nelder-Mead 22 . The sensitivity to the optimisation setup was particularly high near the regime change at = 47. These solutions highlight one of the major downsides of using optimisation procedures: the convergence to local minima instead of the global minimum for the objective functional. With carefully selected methods and appropriate setups, however, this risk can be minimised. Reducing the number of independent control parameters reduces the chance of convergence to a local minimum. Similar tests of different methods and parameter orders for the other control setups (moving and flexi-moving distributions) did not generate such divergent results. B. Optimal suction profiles An interesting result from the optimisation studies for this objective was how much the optimal profiles move and morph depending on the Reynolds number. At low the suction profiles are narrowly spread and positioned near the leading edge of the cylinder. As increases, the profile spreads out and moves further leeward on the cylinder. This shift can be seen in Figure 6 where a sample of the results at different are shown for the moving distribution control. In addition, lines marking the uncontrolled separation angle and the angle of suction are shown for each profile. Evidently, there is a relationship between the suction angle, , and the Reynolds number, and by extension the uncontrolled separation angle also. Figure 7 From this figure, it is clear that there is a strong relationship between , 0 and . Similar discussion can be made about these relations as for in the earlier section. These points will not be repeated beyond stating that the results confirm a dependence on the Reynolds number and on the uncontrolled separation angle for the application of suction control; that it is not constant at all ; and that it would be of interest to see if these results could be extended to higher . There appears to be less correlation with the spread of suction, , however, which has an average value of 75°. C. Effect on drag The aerodynamic characteristics of the cylinder were of particular interest in this study, so the drag components for the optimally controlled cases were measured. These are plotted in Figure 8 alongside the values for the uncontrolled case using the relationships taken from the numerical analysis by Henderson 24 . There are several features to note here, in particular: the general trend of the total drag, , and the behaviour of its two components, & . . However, where one might have expected the elimination of boundary layer separation to improve the drag, it is evident that this is not always the case. For all < 100, the drag is worse for the controlled case than the uncontrolled case; only above = 100 is improvement seen. The explanation for this behaviour can be found by analysing the components of the drag, shown in the same figure. The first thing to note is the shift in pressure drag. Since the pressure drag is predominately contributed to by the loss of momentum to the boundary layer and the vortices that form in the separated region, one might have expected the pressure drag to be eliminated entirely, along with the boundary layer separation. This is clearly not the case. The optimisation studies searched for control parameters that eliminated separation with minimal suction effort, therefore the boundary layer is not entirely removed. With this objective, despite being successfully reduced to zero, momentum from upstream is still lost to the boundary layer, so the pressure is not fully recovered over the leeward side. This can be seen visually by the slow velocity region in the wakes of the controlled flows in Figure 4. Nevertheless, the pressure drag is substantially reduced, particularly in the vortex shedding regime. Counteracting the improvement of this one component, is a worsening of the other: the skin friction drag. With the removal of the separated region of flow, the boundary layer has a higher velocity across the entire surface of the cylinder, as was highlighted by the increased velocity vectors in Figure 4. This higher boundary layer velocity results in a stronger shear force and, correspondingly, a greater skin friction drag. At low , where the viscous effects of the flow are more important, this increase in skin friction drag can overwhelm the improvement in pressure drag to result in a worsening of the total drag. In this case, for the objective of eliminating separation using moving distribution control, this occurs for all < 100. Only above = 100, where the inertial effects are sufficiently dominant and the improvement in pressure drag is more substantial, does the control work in the favour of reducing total drag. This balance of drag components will be discussed to further depth in the section on drag objectives. D. Pressure profiles To complete the analysis of the controlled flow behaviour and characteristics, the pressure coefficient profiles are provided in Figure 9. In the first plot, Figure 9 (a), the time-averaged pressure coefficients for the uncontrolled flow are compared to values from experiments in the literature. These values, labelled 'Zdravkovich', in the plot are taken from experimental values from Thom 27 , and Homann 28 in the range 36 < < 107 and compiled by Zdravkovich in his textbook 2 . Hence, why at low Re values, ≤ 20, the pressure coefficients are seen to differ quite substantially. At none of the investigated values does the leading edge pressure coefficient equal unity as determined by potential flow theory, however it does trend towards that value with increasing . It is common in the literature to approximate the location of the separation point as corresponding to the inflection point on the pressure profile 5,6 . Looking at this figure, it can be seen that the inflection point roughly corresponds with the measured though not always, as with = 4. In Figure 9 (b), the controlled flow is compared to the uncontrolled flow for > 40. There are several features to highlight here. Firstly, the pressure profile of the controlled flow fills out more to become similar to the profile given by potential flow theory. The minimum pressure coefficient is much lower, though, with values between -3.75 and -4.5. This is similar to what was seen in the experiments by Pankhurst and Thwaites 17 . The lower pressure coefficient likely arises due to the flow being accelerated more than it would in the inviscid case, as the suction removes the low momentum particles and providing a pressure gradient to entrain higher velocity free-stream particles, accelerating them as they replenish the boundary layer. In addition, as the flow is no longer inhibited by the separated region, the pressure at the leading edge moves closer to = 1. Finally, it is important to note that the plateau of near the trailing edge is not eradicatedmomentum is still lost to the boundary layer. and objectives So far, we have investigated the optimal control for eliminating separation on the circular cylinder. As has been discussed earlier, however, much of the present literature is more concerned with the vortex shedding and drag coefficients than the boundary layer behaviour. Now, we consider the effect of changing the objective of the optimisation, to minimise the total drag or pressure drag using the moving distribution control method. Figure 10 below shows the resulting flow fields at = 120 for the optimal control found for each objective using the moving distribution method. It is clear from this figure that the control required to achieve each objective differs significantly, as does the behaviour of the resulting flow. While the uncontrolled flow figure was taken from a timedependent simulation, the other figures were taken from the final stage of the optimisation process and thus with a steady-state condition. Naturally, the steady-state flows are symmetrical, therefore, and there are no lateral movements in the wake as with the uncontrolled flow. A. Comparison with objective The first feature to note in Figure 10 is the difference in suction profiles for each optimum control. As was shown earlier, the suction to eliminate separation at = 120 was spread wide and focussed near the trailing edge, with a relationship close to = 180 −0.25 . The suction profiles for the drag objectives are narrower and closer to the top and bottom of the cylinder (90° and 270°). It can be seen visually, that the amount of suction, , is much smaller for these objectives alsoparticularly for the total drag objective in Figure 10 (c). This makes sense given what the earlier results revealed about the balance of drag components: while boundary layer suction can reduce the pressure drag, it comes at the cost of increasing the skin friction drag. The separation angle objective often resulted in a net increase in drag because the suction was too strong, so it is appropriate that the optimal suction profile to minimise total drag employs less suction. This balance can be observed in the velocity vectors near the wall for the separation objective and the total drag objective. For the former, the vectors are concentrated and large, whereas for the total drag control a more balanced profile is observed. The resulting flow fields for each objective differ significantly. The most obvious feature is the wake sizeboth its length and width. The wake in these figures can be considered the paler blue region centred on the trailing edge, bordered by the dark blue lines where the flow is stagnant. This delineates the two shear layers of reversed flow in the wake, and forward flow outside. With this definition, the separation objective flow in Figure 10 (b) has no real wake as it has no reversed flow, only stagnant fluid. On the other hand, the total drag objective has the longest wake. The pressure drag objective is about halfway between in terms of length, but has a similar wake-width to that for the objectivea separation angle corresponding to about 45° from the trailing edge. Figure 11 shows how the optimal control differs depending on the objective of the optimisation. The amount of suction and the location of suction are plotted against the Reynolds number, as in Figure 5 and Figure 7. As with the objective, clear trends can be seen in the optimal control parameters for the other objectives. Contrary to the trend seen for minimising , however, Figure 11 (a) shows that the amount of suction required to achieve the drag objectives decreases with increasing in all flow regimes. This figure also shows a levelling off at = 5 for the amount of suction to minimise total drag within the vortex shedding regime. Figure 11 (b) shows that the location of optimal suction for the drag objectives follows a similar trend to that for the separation angle. A power law is seen for each, and approximate fits are given in the legend of that figure. These fits have been rounded, so are not necessarily the best fits for the data, but help to make comparisons easier. It can be seen that the drag objectives result in suction profiles located closer to the leading edge, and begins to level off near 90°. This fits with what Kim and Choi 19 found for = 100, with the drag on a cylinder improving most by slot suction and blowing when the slots were located between 80° and 100°. The data for the pressure drag objective is more scattered here. (a) (b) Figure 11: Effect of objective on optimal suction characteristics: (a) the amount of suction, and (b) the location of suction. In addition to the control characteristics, the features of the resulting flow are of equal interest. Since the drag objectives result in optimally controlled flows that still have boundary layer separation, Figure 12 shows the separation angle for each controlled case in comparison to the time-averaged value for the uncontrolled case taken from Wu et al 7 . Again, we see a levelling off of the data for the total drag objective. Here the separation angle plateaus near the 45° mark, thus the difference in separation angle to the uncontrolled case increases in the vortex shedding regime. This suggests that in the vortex shedding regime there is a particular optimal control case that scales appropriately with the changes that occur as increases in this region. The 45° mark may bear some significance as the location at which the separation is restrained to by the optimal control for . Figure 12: Separation angle for the optimally controlled flow for each objective. B. Drag components The effect of control on the components of drag is of great interest for these objectives, particularly how the skin friction and pressure drag changes are balanced to achieve the minimum possible total drag. The total drag of the optimally controlled flows using the moving distribution control method are displayed in Figure 13. Here, the relationships from Henderson 24 are also plotted for comparison to the uncontrolled case. From this figure, it can be seen that the improvement to drag is much more substantial for these objectives than when eliminating separationparticularly in the vortex shedding regime. Before the von Karman street begins forming at = 47, there is little improvement in the total drag from either the or objectives. Whereas the separation objective in many instances resulted in a worsened total drag coefficient, the objective resulted in controlled flows that were in no case worse than the uncontrolled flow. On the surface this is unsurprising, as the optimisation algorithm would return zeroed control parameters if no suction setup could improve upon the uncontrolled flowthus the total drag should never be higher than the uncontrolled case for this objectivehowever, as can be seen in Figure 11 and Figure 12, the optimal flow was achieved with significant control applied in every instance. This shows that the drag can always be improved, or at least matched, by the application of suction control in the entire investigated Reynolds range. It also suggests that very different control parameters may result in flows with near-identical macroscale features. The objective, on the other hand, had slightly worse drag characteristics in the unseparated and vortex pair regimes. This fits with what was learned from the separation objective results. The particularly important result shown in Figure 13 is the vast improvement in drag that occurs in the vortex shedding regime. Here, where the uncontrolled drag line begins to curve upwards, the drag-optimised cylinder characteristics to decrease at the same rate as in the vortex-pair regime. This means that, once vortex shedding has begun, as continues to increase, the maximum improvement in drag will also increase. This bodes well for practical implementations, as typical engineering scenarios of bluff body flow occur at much higher . It is interesting that the characteristics for the total drag objective continue along the same trend as the 'subcritical' uncontrolled flow (subcritical here meaning before the onset of vortex shedding, < 47). Bearing in mind that the results for the objective represent the best possible improvement in drag (at least using suction control), this suggests some physical significance about the flow. The transition from steady separated flow to transient vortex shedding has a big impact on the aerodynamic characteristics of the uncontrolled cylinder: the pressure drag stops decreasing with and begins to increase, though the skin friction drag coefficient continues to decrease at near its prior rate, resulting in the curved profile in the range 47 < < 188. 5. The fact that the optimally controlled flow is unaffected by this transition, and the total drag continues to decrease at its prior rate, suggests that the significant change imposed by the control is the elimination of vortex sheddingand therefore, that the elimination of vortex shedding provides the maximum possible improvement in drag. This would be an important finding, because much of the literature regarding the control of flow around a circular cylinder has already been focussed on the elimination of vortex shedding (typically because of the detrimental lateral forces), or separately on minimising the total drag. If the present hypothesis is correct, and the best drag reduction is achieved by the elimination of vortex shedding alone, then this would go some way in connecting the findings in the literature and helping to clarify the best aims for potential flow control methods. Figure 14 goes some way to verifying this hypothesis. In this figure three sources of data are shown for each of the drag components: the actual drag coefficients for the uncontrolled flow (dashed lines), the drag coefficients for the uncontrolled flow with the subcritical trend continued into the vortex shedding regime (dotted lines), and finally the results of the optimally controlled flow from the present study (solid lines). In addition, some manipulation to the data has been performed. As before, the suction control resulted in a decrease in pressure drag and an increase in skin friction drag. To test the vortex shedding hypothesis, the pressure drag curve was shifted up 0.38 to line up with the uncontrolled value at the regime change, correspondingly the skin friction curve was shifted down an equal amount. The total drag curve was not shifted. This figure has many interesting features. First of all, the dramatic improvement on total drag by the optimal nonuniform suction is more evident on this standard grid. Here it can be seen that it is not only substantially improved compared to the uncontrolled cylinder, but it is even better than for a cylinder with the subcritical trend continued. The second important feature, is the alignment of the pressure drag coefficient. This characteristic lines up almost perfectly with the extended subcritical relationship. It is reasonable to consider the rise in pressure drag of the uncontrolled cylinder to be due entirely to the onset of dynamic vortex shedding, therefore these results suggest that the absolute best characteristics that can be achieved are those that minimally eliminate vortex shedding. The characteristics defined by the dotted lines in this figure could be considered to be those of an idealised cylinder that, through extreme smoothness or some other mechanism, is somehow able to maintain steady flow in the vortex shedding regime, i.e. the instability jump to vortex shedding never occurs as is increased on the cylinder. Fornberg [29][30][31] has long highlighted the importance of the 'steady though unstable' cylinder in typically unsteady regions as a potential aim for flow control. Here we see that it may represent the optimal controlled flow for minimising dragat least for control by boundary layer suction. The optimal flow achieved with non-uniform suction control here is even better than this idealised cylinder. The skin friction drag coefficient comes in under both uncontrolled cylinder characteristics, hence why the total drag is improved even beyond the subcritical trend. This may be due to the non-uniform suction minimising the increase in skin friction by only accelerating the boundary layer over a portion of the surface, rather than the entirety as the idealised cylinder would encounter. Verification by time-dependent studies One of the major limiting factors of the optimisation procedure in this study was the use of steady-state solvers for flows that, when not appropriately controlled, are unsteady. To test the validity of the assumptions in the optimisation process, time-dependent studies were performed with the optimal control found for each objective at = 60 and = 180testing the two extremes of the 2D vortex shedding regime. The key results of these studies are shown in Figure 15 below, along with the steady-state results which are plotted as markers on the right vertical of the grid. In each case, the vortex shedding was allowed to fully develop on the uncontrolled cylinder, before the control was ramped up from 0% to 100% of its strength in a linear fashion over the course of = 5 , where is the period of vortex shedding given by Roshko's equation = = 0.212 − 4.5 . The controlled flow was then allowed to fully develop. The time was non-dimensionalised according to * = 0 , where 0 is the distance from inlet to the centre of the cylinder (see Figure 1). Figure 15 shows the results of these verification studies, with the steady-state results shown as markers on the right vertical axis. As can be seen in Figure 15, the results actually fit quite well between the time-dependent and steadystate controlled results. However, it is evident for the results at = 180, the assumption that the optimally controlled flows are steady was incorrect for the and objectives as anticipated. In Figure 15 (b) it can be seen by the oscillation of the separation angles on the upper and lower surfaces that the flow is not steady. Vortex shedding has not been eliminated as was hypothesised earlier. Despite the assumption breaking down here, the steady-state results still predict the actual values reasonably well as shown by the agreement to the final time-step. Even for the oscillating separation angles, the steady-state result gives the maximum separation angle for the uncontrolled case. The drag values are mostly accurate also, underestimating the values for the objective only slightly, as was shown to be the behaviour for the steady-state approximations in Figure 3 (b). The only values that are significantly inaccurate are those for the drag components for minimising pressure drag at = 180. This can be seen in Figure 15 (d), where the pressure drag and skin friction drag components are underestimated, resulting in the steady-state total drag being lower than the reality by 0.106 (-3.29%). In addition to some of the transient details being missed in the steady-state simulations, another feature was absent for the flow at = 60 with the objective control applied. That feature is a small separation bubble which formed where the suction was applied in the time-dependent study. The impact of this can be seen in Figure 15 (a) where the separation angle was detected first at 93.39° where the separation bubble formed. This bubble was very small, as can be seen in the inset of this figure, and had no significant impact on the flow and the later major separation point (which is also plotted on this figure). It does highlight, however, that making the assumption of steady-state flow presents some risk of small or transient features being missed. In the bigger picture, though, the steady-state optimisation process achieved valuable results with much smaller time requirements for the simulations. The major characteristics of the flow were accurately modelled. It is safe to say, therefore that the results from the steady-state optimisation studies are sufficiently accurate to be relied upon in this Reynolds rangeparticularly, when only concerned with the general behaviour of the flow: the aerodynamic characteristics, the separation point, etc. The steady-state studies failed to show transient behaviour in the flow, such as the oscillation of the near wake and vortex shedding in the far wake. This verification by timedependent studies validates the key results of the optimally controlled flows, but does not necessarily confirm that they are the optimum. Such a validation would require full parametric studies of transient flows that makes it computationally prohibitive. For these results, the optimal control for the separation angle objective can be considered correct, while the optimal controls for the other objectives may be assumed to be true, with some caution. CONCLUDING REMARKS In this study, we have presented a method for determining the optimum control parameters of boundary layer suction in an effort to achieve a variety of objectives for the flow around a circular cylinder: eliminating separation, , minimising total drag, , and minimising pressure drag . Numerical simulations were performed on the flow around a circular cylinder in the Reynolds range 4 < < 200, and optimisation algorithms were used to determine the best parameters for the control. To thoroughly investigate the numerous potential applications of non-uniform suction, several models were tested. The moving distributiona suction distribution with compact support and single locusproved to be superior. It provided efficient control, while being simple to implement and achieve optimality. The results of this study provide a variety of insights into the problem of flow around a circular cylinder and flow control generally. Firstly, the approach of carrying out optimisation processes on unsteady flow using a steady-state approximation provided results that were sufficiently accurate while enabling wide-ranging simulations to be carried out in far less time than corresponding time-dependent simulations. This approach may be applied at higher Reynolds numbers for this flowor for other types of external and internal flowwith the caveat that at higher , where more of the flow becomes unsteady and turbulent, its reliability will likely be diminished. The assumption that the optimally controlled flow will be steady holds less well for objectives not directly related to vortex shedding (as seen by the and objective results), but can likely be as successfully employed for parameters directly related to the unsteady aspects of the flow (e.g. the lift coefficient or certain vorticity measures) as for the separation angle parameter used in the present study. Secondly, clear relationships have been derived between characteristics of the uncontrolled flow and the optimal control for a variety of important objectives. The substantial improvement in efficiency of non-uniform suction in comparison to uniform suction, suggest this is the best form of boundary layer suction control. However, the dependency of its location on the parameters of the flow may make it impractical in many circumstances. Since and change with predictable trends though, such as = 180 −0.25 for the objective, this issue may be mitigated. Another important finding was the balance of drag components in the suction-controlled flows. Eliminating separation did not always reduce the total drag because the skin friction component is increased due to the fuller boundary layer profile. Above = 100, however, the controlled attached flow has a reduced drag as skin friction becomes less significant. It may be reasonably expected that for all > 100, if separation is prevented by optimal suction control, the total drag will be reduced. The controlled flow with minimised total drag almost always required less suction, in order to balance the pressure and skin friction drag components. The drag-optimised flows followed trends similar to that seen for a cylinder with steady flow in the unsteady regime (steady, unstable flows as described by Fornberg 31 ). The pressure drag component followed almost the same relationship with as an idealised steady flow, but shifted down due to the delayed separationthe total drag was similar. This may present an opportunity for a computationally cheap method for predicting the maximum potential improvement in drag that optimum suction control can providesimply evaluating the flow in question with the steady Navier-Stokes equations. Finally, the investigation into the importance of the separation angle, both as an uncontrolled parameter of the flow and as an objective of control, has yielded valuable results. It has been determined that the relationships between the flow and the optimal control could equally be expressed as a relationship with 0 than with . In many cases this results in less convoluted connections; the − 0 relationship was linear, even. This may have the advantage of making it easier to apply the lessons of optimal control in one flow regime to anotheror indeed to flows with bodies of totally different shapes. Less dependence on the initial separation angle was seen for the drag objectives, however it plays a significant role as an outcome of the controlled flows, as seen by the repeated = 45° for the drag-optimised flows. The outcomes found here improve the current understanding of the control of flow around bluff bodiesand separated flows generallywith the importance of the separation angle and the relationships of the drag components seen to be significant. These results, and the methods used to obtain them, bear great promise for application in broader Reynolds ranges or for different types of flow, and provide a potential method for economically estimating the lowest possible drag by boundary layer suction control. Figure 1 : 1the range investigated here. The computational domain and mesh are shown in Figure 1. This domain was based on that employed successfully by Wu et al. Sketch of (a) computational domain and (b) close-up view of element mesh around the cylinder The inlet (left-boundary) was assigned a uniform flow boundary condition with velocity, = = , = 0. Figure 2 : 2Schematic of the types of control methods investigated. Figure 3 : 3Validation of model by comparison to historical data for (a) the separation angle and (b) the time-averaged total drag coefficient. Here, the error-bars indicate the span of instantaneous values, while the points are time-averaged values. Figure 4 : 4Instantaneous flow fields for controlling separation angle at = 120: (a) uncontrolled case, (b) uniform suction, (c) six-segment control, (d) moving distribution, and (e) flexi-moving distribution. The streamlines, velocity vectors, and velocity surface are shown. The paler streamline in (a) originates on the curve used for detecting the separation angle. Figure 5 : 5Global suction coefficient for optimal control to prevent separation plotted against (a) the Reynolds number, (b) the separation angle of the uncontrolled flow. The dashed vertical lines indicate regime changes of the uncontrolled flow. Figure 6 :Figure 7 : 67Variations in optimal suction profiles for moving distribution control at various Re. The inner dotted circle marks where the local suction coefficient is 100, i.e.= . The uncontrolled (initial) separation angle, 0 , and resulting centre of suction, , are also plotted and their values displayed. How the non-uniform suction profile with moving distribution control moves with (a) Reynolds number and (b) initial separation angle. The vertical lines mark the regime changes of the uncontrolled flow. Figure 8 : 8Drag components for the optimally controlled (solid lines) and uncontrolled (dashed lines) flows. The vertical lines delineate the uncontrolled flow regime changes. Figure 8 8shows that all the values follow similar trends to the uncontrolled case: beginning very high and decreasing with a power-law relationship with increasing Figure 9 : 9Pressure coefficient profiles over surface of cylinder for (a) uncontrolled case, and (b) both uncontrolled (solid line) and controlled (dot-dashed line) cases. Figure 10 : 10Instantaneous flow field at = 120 showing velocity vectors and surfaces, as well as streamlines from the inlet for the (a) uncontrolled case, and controlled for minimising (b) separation angle, (c) total drag and (d) pressure drag objectives. Figure 13 : 13Total drag of the optimally controlled cases compared with the time-averaged value for the uncontrolled flow. Figure 14 : 14Drag components for optimally controlled flow with total drag objective, 2 . Note, and have been shifted up and down 0.38 respectively. Here the characteristics are shown for the present total drag-optimised results (solid lines), uncontrolled cylinder (dashed lines), and the extended sub-critical trends (dotted lines). The vertical lines mark the regime changes for the uncontrolled flow. Figure 15 : 15Results from time-dependent studies of the optimum moving distribution control for each objective, showing (a-b) the instantaneous separation angles, and (c-d) the drag components. Inset on (a) is a tangential velocity surface for the objective at = 60 showing the separation bubble and main separation, that give rise to two sets of measurement for . 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[ "Tracer Diffusion on a Crowded Random Manhattan Lattice", "Tracer Diffusion on a Crowded Random Manhattan Lattice" ]	[ "Carlos Mejía-Monasterio [email protected] \nSchool of Agricultural, Food and Biosystems Engineering\nTechnical University of Madrid\nAv. Complutense s/n28040MadridSpain\n", "Sergei Nechaev [email protected] \nInterdisciplinary Scientific Center J.-V. Poncelet (UMI CNRS 2615)\nBolshoy Vlasyevskiy Lane 11119002MoscowRussia\n\nLebedev Physical Institute RAS\n119991MoscowRussia\n", "Gleb Oshanin [email protected] \nInterdisciplinary Scientific Center J.-V. Poncelet (UMI CNRS 2615)\nBolshoy Vlasyevskiy Lane 11119002MoscowRussia\n\nLaboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600)\nSorbonne Université\nCNRS\n4 Place Jussieu75252, Cedex 05ParisFrance\n", "Oleg Vasilyev [email protected] \nMax-Planck-Institut für Intelligente Systeme\nHeisenbergstr. 3D-70569StuttgartGermany\n\nIV. Institut für Theoretische Physik\nUniversität Stuttgart\nPfaffenwaldring 57D-70569StuttgartGermany\n" ]	[ "School of Agricultural, Food and Biosystems Engineering\nTechnical University of Madrid\nAv. Complutense s/n28040MadridSpain", "Interdisciplinary Scientific Center J.-V. Poncelet (UMI CNRS 2615)\nBolshoy Vlasyevskiy Lane 11119002MoscowRussia", "Lebedev Physical Institute RAS\n119991MoscowRussia", "Interdisciplinary Scientific Center J.-V. Poncelet (UMI CNRS 2615)\nBolshoy Vlasyevskiy Lane 11119002MoscowRussia", "Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600)\nSorbonne Université\nCNRS\n4 Place Jussieu75252, Cedex 05ParisFrance", "Max-Planck-Institut für Intelligente Systeme\nHeisenbergstr. 3D-70569StuttgartGermany", "IV. Institut für Theoretische Physik\nUniversität Stuttgart\nPfaffenwaldring 57D-70569StuttgartGermany" ]	[]	We study by extensive numerical simulations the dynamics of a hardcore tracer particle (TP) in presence of two competing types of disorderfrozen convection flows on a square random Manhattan lattice and a crowded dynamical environment formed by a lattice gas of mobile hard-core particles. The latter perform lattice random walks, constrained by a single-occupancy condition of each lattice site, and are either insensitive to random flows (model A) or choose the jump directions as dictated by the local directionality of bonds of the random Manhattan lattice (model B). We focus on the TP disorder-averaged meansquared displacement, (which shows a super-diffusive behaviour ∼ t 4/3 , t being time, in all the cases studied here), on higher moments of the TP displacement, and on the probability distribution of the TP position X along the x-axis. Our analysis evidences that in absence of the lattice gas particles the latter has a Gaussian central part ∼ exp(−u 2 ), where u = X/t 2/3 , and exhibits slower-than-Gaussian tails ∼ exp(−\|u\| 4/3 ) for sufficiently large t and u. Numerical data convincingly demonstrate that in presence of a crowded environment the central Gaussian part and non-Gaussian tails of the distribution persist for both models.	10.1088/1367-2630/ab7bf1	[ "https://arxiv.org/pdf/1912.03169v3.pdf" ]	208,857,619	1912.03169	ae2f7741eb987391651a69ad4b694d4111081d78	Tracer Diffusion on a Crowded Random Manhattan Lattice 12 Dec 2019 Carlos Mejía-Monasterio [email protected] School of Agricultural, Food and Biosystems Engineering Technical University of Madrid Av. Complutense s/n28040MadridSpain Sergei Nechaev [email protected] Interdisciplinary Scientific Center J.-V. Poncelet (UMI CNRS 2615) Bolshoy Vlasyevskiy Lane 11119002MoscowRussia Lebedev Physical Institute RAS 119991MoscowRussia Gleb Oshanin [email protected] Interdisciplinary Scientific Center J.-V. Poncelet (UMI CNRS 2615) Bolshoy Vlasyevskiy Lane 11119002MoscowRussia Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600) Sorbonne Université CNRS 4 Place Jussieu75252, Cedex 05ParisFrance Oleg Vasilyev [email protected] Max-Planck-Institut für Intelligente Systeme Heisenbergstr. 3D-70569StuttgartGermany IV. Institut für Theoretische Physik Universität Stuttgart Pfaffenwaldring 57D-70569StuttgartGermany Tracer Diffusion on a Crowded Random Manhattan Lattice 12 Dec 2019random Manhattan latticetracer diffusionhard-core lattice gassimple exclusion processquenched versus dynamical disorder ‡ Corresponding author We study by extensive numerical simulations the dynamics of a hardcore tracer particle (TP) in presence of two competing types of disorderfrozen convection flows on a square random Manhattan lattice and a crowded dynamical environment formed by a lattice gas of mobile hard-core particles. The latter perform lattice random walks, constrained by a single-occupancy condition of each lattice site, and are either insensitive to random flows (model A) or choose the jump directions as dictated by the local directionality of bonds of the random Manhattan lattice (model B). We focus on the TP disorder-averaged meansquared displacement, (which shows a super-diffusive behaviour ∼ t 4/3 , t being time, in all the cases studied here), on higher moments of the TP displacement, and on the probability distribution of the TP position X along the x-axis. Our analysis evidences that in absence of the lattice gas particles the latter has a Gaussian central part ∼ exp(−u 2 ), where u = X/t 2/3 , and exhibits slower-than-Gaussian tails ∼ exp(−\|u\| 4/3 ) for sufficiently large t and u. Numerical data convincingly demonstrate that in presence of a crowded environment the central Gaussian part and non-Gaussian tails of the distribution persist for both models. Introduction In many realistic systems encountered across several disciplines -e.g., physics, chemistry, molecular and cellular biology, -random motion of tracer particles takes place in presence of disorder, either temporal or spatial, which may originate from a variety of different factors [1][2][3][4][5][6][7][8][9][10][11][12][13]. Understanding the impact of disorder on dynamics is thus a challenging issue, which has important conceptual and practical implications. Quenched (frozen) spatial disorder which entails a temporal trapping of a tracer particle (TP) at some positions, often produces an anomalous sub-diffusive behaviour, especially in low-dimensional systems. Here, the TP trajectories are spatially more confined than the trajectories of a standard Brownian motion. As a consequence, the disorder-averaged mean-squared displacement (DA MSD) behaves as < X 2 (t) >∼ t γ , with t being time and γ -the dynamical exponent which is less than unity. Here and henceforth, the bar denotes averaging over thermal histories while the angle brackets stand for averaging over disorder. Striking examples of such a dynamical behaviour are provided by, e.g., the so-called Sinai diffusion in one-dimensional systems [14] (see also Refs. [3][4][5][6]) in which the DA MSD grows as < X 2 (t) >∼ ln 4 t (i.e., formally, γ = 0), Sinai diffusion in presence of a constant external bias [15,16] or migration of excited states along a one-dimensional array of randomly placed donor centres [1,6]. In this latter example the dynamical exponent γ is non-universal and equals the mean density of donor centres times the characteristic length-scale of the distance-dependent (exponential) transfer rate. If this product is less than unity, a sub-diffusive motion takes place. Two other examples concern diffusion in the "impurity band" [17] and the so-called Random Trap model [18][19][20]. Here, as well, γ is non-universal and is less than unity in some region of the parameter space. In higher-dimensional systems, diffusion in presence of such a disorder typically becomes normal (see, however, Refs. [17,21,22]) and the disorder affects only the value of the diffusion coefficient. Diffusion is also normal in the asymptotic large-t limit in one-dimensional systems with a periodic disorder. Here, however, the value of the diffusion coefficient may exhibit strong sample-to-sample fluctuations and thus have non-trivial statistical properties, such that the averaged diffusion coefficient will not be representative of the actual behaviour (see, e.g., Ref. [23]). The large-t relaxation of the diffusion coefficient to its asymptotic value may shed some light on the kind of disorder one is dealing with [24]. Random frozen convection (velocity) flows most often produce a super-diffusion with γ > 1. To name just two such situations, we mention a model in which a TP is passively advected by quenched, layered, randomly-oriented flows (say, along the x-axis) and undergoes a normal diffusion in the direction perpendicular to them (i.e., along the y-axis), as well as its generalisation -a random Manhattan lattice (see Fig. 1), in which the orientation of convection flows randomly fluctuates both along the streets and avenues (i.e., along both x-and y-axes). The former model was introduced originally for the analysis of conductivity of inhomogeneous media in a strong magnetic field [25] and of the dynamics of solute in a stratified porous medium with flow parallel to the bedding [26]. In such a setting, usually referred to as the Matheron -de Marsily (MdM) model according to the names of authors of Ref. [26], the TP dynamics in the flow direction (along the x-axis) is characterised by a super-diffusive law of the form < X 2 (t) >∼ t 3/2 , i.e., γ = 3/2. Many interesting generalisations and more details on the available analytical and numerical results can be found in Refs. [27][28][29][30][31][32][33][34][35]. Diffusion of a single TP on a square random Manhattan lattice has been analysed in Refs. [27,28]. It was shown, by using simple analytical arguments and a numerical analysis, that in this case the DA MSD also exhibits a super-diffusive behaviour, but with a somewhat smaller dynamical exponent γ = 4/3, i.e., the DA MSD of the x-component of the TP position obeys < X 2 (t) >∼ t 4/3 . This model has been also widely studied in different contexts in mathematical literature (see, e.g., Ref. [36]). A generalisation of a random Manhattan lattice was invoked as an example of a plausible geometric disorder in a recent analysis of the localisation length exponent for plateau transition in quantum Hall effect [37]. This latter setting, however, is clearly more complicated than the MdM model with the layered flows and the theoretical progress here is rather limited; the behaviour beyond the temporal evolution of a DA MSD is still largely unknown. Dynamical disorder emerges naturally when the TP's transition rates fluctuate randomly in time, as it happens, for instance, in physical processes underlying the socalled diffusing-diffusivity models [38][39][40][41][42][43][44] or the dynamic percolation [45][46][47]. Another pertinent case concerns the situations when the TP evolves in a dynamical environment of mobile steric obstacles -interacting crowders which impede its dynamics (see, e.g., Refs. [10][11][12]). A paradigmatic example of such a situation is provided by a TP diffusion in lattice gases of hard-core particles, which undergo the so-called simple exclusion process (see Ref. [13] for a recent review), i.e., perform lattice random walks subject to the constraint that each lattice site can be at most singly occupied. It is well-known that in such an environment the particles' dynamics is strongly correlated. These correlations are especially important and cause an essential departure from standard diffusive motion in two cases: a) in one-dimensional geometry -the socalled single-files, in which the particles cannot bypass each other and the initial order of particles is preserved at all times; and b) on ramified comb-like structures consisting of an infinitely long single-file backbone with infinitely long single-file side branches, which permit for some re-ordering of particles. In single-files, the TP meansquared displacement exhibits an anomalous sub-diffusive behaviour X 2 (t) ∼ t 1/2 . This striking result was first obtained analytically by Harris [48] (see Refs. [49,50] for a review), and holds also for all the cumulants of X(t) [51,52] and in case of multiple TPs [53][54][55]. On crowded comb-like structures, the TP mean-squared displacement exhibits a variety of sub-diffusive transients and, in some cases, an ultimate subdiffusive behaviour [56]. On higher-dimensional lattices, the TP dynamics becomes diffusive in the large-t limit with the effective diffusion coefficient being a non-trivial function of the density of crowders and other pertinent parameters [57][58][59][60][61][62][63]. This nontrivial behaviour of the diffusion coefficient is associated with the enhanced probability of backward jumps -in a crowded environment, for any particle it is more probable to return back to the site it just left vacant, than to keep on going farther away [57][58][59][60][61][62][63]. Meanwhile, a considerable knowledge is accumulated through case-by-case theoretical and numerical analyses of the TP dynamics in a variety of model systems with either quenched or dynamical disorder (see, e.g., Refs. [1][2][3][4][5][6][7][8][9][10][11][12][13] and references therein). On contrary, still little is known about the TP diffusion in situations in which several types of disorder are acting simultaneously. To the best of our knowledge, the only work addressing specifically this question is recent Ref. [50], which focused on the TP random motion in single-files of hard-core particles having a broad scale-free distribution of waiting times, e.g., due to a temporal trapping of particles. Using some subordination arguments and numerical analysis, it was shown that here a combined effect of the disorder in transition rates and of the dynamical environment leads to a severe slowing-down of the TP random motion. Namely, the DA MSD of the TP follows X 2 (t) ∼ ln 1/2 (t), i.e., exhibits an essentially slower growth with time than the one taking place in systems in which either type of disorder is present alone. In case when a characteristic mean waiting time exists, i.e., the distribution is not scalefree, but the second moment diverges, the DA MSD grows faster than logarithmically, X 2 (t) ∼ t γ with γ < 1/2, but still slower than the above mentioned Harris' law. This paper is devoted to a question of the TP dynamics in presence of two interspersed types of disorder, which act concurrently and compete with each other. We consider the TP random motion subject to quenched random convection flows, which prompt a super-diffusive behaviour of the TP, in a dynamical environment which is damping its random motion. More specifically, we study here by extensive numerical simulations the dynamics of a TP which evolves on a square random Manhattan lattice of frozen (i.e. not varying in time) convection flows in presence of a lattice gas (LG) of mobile hard-core particles. The latter are either insensitive to convection flows, performing standard random walks among the nearest-neighbouring sites of a lattice with the probability 1/4 to go in any direction (Model A), or follow the convection flows (similarly to the TP) by choosing randomly between the two directions prescribed by a local directionality of bonds of the random Manhattan lattice (Model B). In the latter case the backward jumps of any LG particle are completely suppressed. The backward jumps of the TP are forbidden in both models. For both models, the TP and the LG particles obey a simple exclusion constraint, which effectively correlates the TP random motion and the evolution of LG particles. We focus on such characteristics of the TP dynamics as its DA MSD, and generally, the moments of arbitrary order, the distribution of its position at time moment t averaged over disorder, as well as the time evolution of the kurtosis of this distribution. We also address a question of the sample-to-sample fluctuations and analyse the MSD of the TP and the probability distribution of its position for several fixed realisations of disorder. The paper is outlined as follows: In Sec. 2 we define the model under study and introduce basic notations. In Sec. 3 we discuss dynamics of a single TP in absence of the LG particles, appropriately revisiting the arguments presented in Refs. [27,28]. We also present here results of numerical simulations for the DA MSD and for higher moments of the TP displacement, as well for the disorder-averaged probability distribution of the TP position along the x-axis. This sets an instructive framework for the analysis of the TP dynamics in presence of LG particle. We close Sec. 3 addressing the issue of sample-to-sample fluctuations and also examine the spectral properties of the TP trajectories, which reveal several interesting features. In Sec. 4 we consider the TP dynamics in presence of LG particles for both Model A and Model B. Finally, in Sec. 5 we conclude with a brief recapitulation of our results. Model Consider a two-dimensional random Manhattan lattice (see Fig. 1), i.e., an infinite in both directions square lattice with unit spacing, decorated with arrows in such a way that directionality of each of them is fixed along each street (East-West) and an avenue (North-South) for their entire length, but whose orientation varies randomly from a street (an avenue) to a street (an avenue). Let an integer n, n ∈ (−∞, ∞), numerate the columns (avenues) of the lattice, and an integer m, m ∈ (−∞, ∞), -the rows (streets), respectively. Then, the pattern of arrows in a given frozen realisation of convection flows is specified by assigning to each lattice site (with integer coordinates (n, m)) a pair of quenched random, mutually uncorrelated "bias" variables η n and ζ m . We use a convention that η n = +1 if an arrow points to the North, and η n = −1, otherwise; and ζ m = +1 if an arrow points to the East, and ζ m = −1, otherwise. We focus solely on the case when there is no global bias; that being, η n and ζ m assume the values ±1 with equal probabilities, which implies that η n = ζ m = 0. Furthermore, we stipulate that there are no correlations between the directions of arrows at n and n ′ , and at m and m ′ , i.e., η n η n ′ = δ n,n ′ , ζ m ζ m ′ = δ m,m ′ ,(1) where δ a,b is the Kronecker-delta, such that δ a,b = 1 for a = b, and equals zero otherwise. A single tracer particle At time moment t = 0 (t is a discrete time variable, t = 0, 1, 2, . . .), we introduce the TP at the origin of the lattice and let it move, at each tick of the clock, according to the following rules: -at each discrete time instant t, we toss a two-sided "coin" ξ t which can assume, (with equal probabilities = 1/2), the values +1 and −1. -being at position R t = (X t , Y t ), (where X t and Y t are the projections of R t on the x-and y-axes), the TP is moved, after choosing the value of ξ t , to a new position R t+1 = R t + δ t ,(2) where the vectorial increment δ t is defined as δ t = (1 + ξ t ) 2 ζ Yt e x + (1 − ξ t ) 2 η Xt e y ,(3) with e x and e x being the unit vectors in the x-and y-directions, respectively. The expression (2) can also be conveniently rewritten in form of two coupled, non-linear recursion relations for the integer-valued components X t and Y t : X t+1 = X t + (1 + ξ t ) 2 ζ Yt , Y t+1 = Y t + (1 − ξ t ) 2 η Xt .(4) Therefore, once (with probability 1/2) ξ t = 1, the TP is moved onto the neighbouring site along the x-axis in the direction prescribed by ζ Yt , and does not change its position along the y-axis. Conversely, if ξ t = −1, the TP is moved on a unit distance along the y-axis in the direction prescribed by η Xt , and does not change its position along the x-axis. We recall that the ensuing motion of the TP as defined by the recursion relations (4) is super-diffusive, with the dynamical exponent γ = 4/3 [27,28]. We note parenthetically that it may apparently be possible to find an equivalent two-dimensional model in the continuum space and time limit, write down coupled Langevin equations for the time evolution of the components and, eventually, define the associated Fokker-Planck equation obeyed by the probability Π t (X, Y ) of finding the TP at position (X, Y ) at time moment t for a given realisation of disorder. We will address this question in our following work. Second, it was claimed in Refs. [27,28] that at a coarse-grained level the TP dynamics on a random Manhattan lattice becomes equivalent to a Brownian motion in continuum, in a divergenceless random velocity field with power-law decay of the velocity correlation function. We however remark that going to a continuum limit necessitates a generalisation of the model studied here; in our settings, the jumps against an arrow are not permitted which tacitly presumes that the force acting on the particle along a given bond is infinitely large. Therefore, one has to allow for the jumps against an arrow and let them occur with a smaller (but finite) probability, than the probability of the jumps along an arrow. This is tantamount to considering finite forces. We, however, do not expect any substantial change in the dynamics in the finite force case, as compared to our model. The algorithm of our numerical simulations of the TP dynamics on a random Manhattan lattice follows the relations (4). We generate trajectories along the xabd y-axes of a given length t, for a given set of thermal variables {ξ t } and a given realisation of "bias" variables η n and ζ m . The obtained individual trajectories are stored and the characteristic properties of interest -the moments of the TP displacement and the distribution function of the TP position -are evaluated by averaging over different realisations of trajectories. Averaging is first performed over 10 4 trajectories generated for a fixed realisation of a random Manhattan lattice, and then the procedure is repeated for 2 × 10 5 realisations of disorder. Simulations are performed for lattices containing L×L sites with L = 2×10 6 . Care is taken that neither of the TP trajectories reaches the boundaries of the lattice within the observation time, such that the finite-size effects do not matter. For the lattice size used in our numerical modelling, this permits us to safely explore the TP dynamics for times up to t = 10 6 . Lastly, we also analyse the sample-to-sample fluctuations and, in particular, address a question of the TP dynamics in presence of a single fixed realisation of disorder. In this case, for a given random realisation of disorder we run 2 × 10 7 trajectories. The TP dynamics on a crowded random Manhattan lattice The TP dynamics on a random Manhattan lattice populated with N − 1 lattice gas particles is analysed numerically. Due to a significant number of the particles involved, we are only able to consider square lattices with the maximal linear extent L = 2×10 3 . This means that the maximal time t, until which the finite-size effects can be discarded, is of order of 4 × 10 3 . Moreover, due to computational limitations, we record only 50 TP trajectories for each given realisation of disorder, and average over 10 3 realisations of disorder. Such a statistical sample appears to be sufficiently large to probe the behaviour of the DA MSD of the TP, but does not permit us to make absolutely conclusive statements about the shape of the distribution function. Nonetheless, our numerical data rather convincingly demonstrate that the overall behaviour of the latter is very similar to the one observed for the TP dynamics in absence of the LG particles, in which case a more ample statistical analysis has been performed. The simulations are performed as follows: We first place the TP at the origin of a lattice and then distribute N − 1 hard-core particles among the remaining sites by placing a LG particle at each lattice site, at random, with probability ρ = N/L 2 . The latter parameter defines the mean density of particles in the system; in our simulations, we study the TP dynamics for nine values of ρ, ρ = 0.1, 0.2, 0.3, . . . , 0.9. After the particles are introduced into the system, they are let to move randomly subject to a single-occupancy constraint. We distinguish between two possible scenarios: 2.2.1. Model A. In model A we suppose that all the LG particles are not sensitive to the frozen pattern of convection flows and perform symmetric random walks, subject to the constraint that there may be at most a single particle (i.e., either the TP or a LG particle) at each lattice site. On contrary, for the TP the choice of the jump direction is dictated by the arrows present at the site it occupies at time moment t. As described above, the TP chooses at random between the two arrows outgoing from the site it occupies. In this case, the TP (which exhibits a super-diffusive motion in absence of the LG particles) is not identical to the LG particles and moves in a quiescent "fluid" of hard-core particles which exerts some frictional force on it. Note that here the backward jumps are forbidden for the TP only. More specifically, at each step we select at random a particle, which can be either a TP or a LG particle, and let it choose the jump direction: if the selected particle is a TP, it chooses at random between the two arrows. Conversely, a LG particle chooses at random one among four neighbouring sites with probability = 1/4. The jump of a TP or a LG particle is fulfilled, once the target site is empty at this time instant; otherwise, the particle remains at its position. The time t is increased by unity after repeating such a procedure N times, such that all N particles present in the system, on average, have a chance to change their positions. We have already mentioned that in this model the dynamics of LG particles is rather non-trivial due to an enhanced probability of backward jumps; it means that a particle which jumps onto an empty target site will most likely return on the next time step to the site it just left vacant, then will keep on going away from it. Even in absence of the TP and random convection flows acting on it, this circumstance results in a non-trivial dependence of the self-diffusion coefficient D tp of any tagged particle on the overall density of the LG particles. This dependence is known only in an approximate form (see, e.g., Refs. [13] and [57][58][59][60][61][62][63]). The available exact results concern the leading, in the dense limit ρ ≃ 1, behaviour of the self-diffusion coefficient D tp ≃ (1 − ρ)/(4(π − 1)) [64] and of the mobility µ tp ≃ β(1 − ρ)/(4(π − 1)) [65] of a tagged particle subject to a vanishingly small external force, with β being the reciprocal temperature. The appearance of the Archimedes' irrational number "π" seems astonishing and points on a non-trivial behaviour. Model B. In this model, we suppose that all the particles in the system are identical. It means that both the TP and the LG particles move on the lattice subject to a single-occupancy constraint and obey the rules of the random Manhattan lattice, by following the jump directions prescribed by the arrows. Note that in this model the backward jump probability is equal to zero for all the particles, both for the LG particles and the TP. As a consequence, we expect that here the environment in which the TP moves is a kind of a "turbulent" fluid, in which all the particles exhibit a super-diffusive motion. Hence, we may expect that the environment becomes perfectly stirred at sufficiently large times, such that the time t gets simply rescaled by the frequency (1 − ρ) of successful jump events, (which is not the case for Model A). We are going to verify if this is the case in what follows. 3. Dynamics of a single tracer particle Disorder-averaged mean-squared displacement In order to calculate the DA MSD of a single TP moving on a random Manhattan lattice in absence of the LG particles, we suitably revisit the arguments presented in Refs. [27,28]. The latter were based on an estimate of typical fluctuations of sums of quenched random variables η n and ζ m , and a plausible closure relation. Here, we pursue a bit different line of thought. First, we "solve" the recursions in eqs. (4) to get, for t ≥ 1, X t = t−1 τ =0 (1 + ξ τ ) 2 ζ Yτ , Y t = t−1 τ =0 (1 − ξ τ ) 2 η Xτ ,(5) with the initial condition X 0 = Y 0 = 0. Expressions (5) define the TP positions X t and Y t for any t, for fixed realisations of thermal noises ξ t and "biases" η n and ζ m . We concentrate on the x-component and write down formally its squared value: X 2 t = t−1 τ =0 (1 + ξ τ ) 2 4 ζ 2 Yτ + 2 t−2 τ =0 t−1 τ ′ =τ +1 (1 + ξ τ ) 2 (1 + ξ τ ′ ) 2 ζ Yτ ζ Y τ ′ .(6) Let the bar denote averaging over ξ τ -s, which amounts to averaging over thermal histories, and the angle brackets -averaging over random variables η n and ζ m , i.e., averaging over quenched disorder. Consider the averaged first sum in the right-handside (rhs) of eq. (6). Noticing that ζ 2 Yτ ≡ 1, i.e., is not fluctuating, we realise that the averaged first sum is simply t−1 τ =0 (1 + ξ τ ) 2 4 ζ 2 Yτ = t−1 τ =0 (1 + ξ τ ) 2 4 = t 2 .(7) Hence, the contribution of the averaged first sum to the DA MSD of the TP along the x-axis is that of a standard, discrete-time random walk (with the diffusion coefficient D = 1/4) on a two-dimensional undecorated square lattice. Focus on the summand in the second term in the rhs of eq. (6) and write down formally its averaged value: 2 (1 + ξ τ ) 2 (1 + ξ τ ′ ) 2 ζ Yτ ζ Y τ ′ .(8) Note that we are allowed to perform averaging over ξ τ ′ directly, due to the fact that both Y τ ′ and Y τ are statistically independent of a random variable ξ τ ′ . Indeed, Y τ ′ depends on ξ τ ′ −1 , ξ τ ′ −2 , . . . , ξ 0 , while Y τ , with τ < τ ′ , depends on ξ τ −1 , ξ τ ′ −2 , . . . , ξ 0 . Note that only the product ζ Yτ ζ Y τ ′ is dependent on random convection flows. Averaging this product over quenched disorder, we find that, in virtue of the definition in eq. (1), expression (8) takes the form δ ξτ ,1 δ Yτ ,Y τ ′ ,(9) i.e., it is an averaged over thermal noises product of the indicator functions of two events: a) Y τ +1 = Y τ and b) Y τ = Y τ +1 = Y τ ′ . As a consequence, the expression 9 is the joint probability P (Y τ \|t = τ ′ ; Y τ \|t = τ + 1; Y τ \|t = τ ) of the events that the TP trajectory Y t , with Y t=0 = 0, a) paused at its (unspecified) position at t = τ and b) returned at time moment t = τ ′ to the position it occupied at t = τ . The probability P (Y τ \|t = τ ′ ; Y τ \|t = τ + 1; Y τ \|t = τ ) decouples into the product of the probability that the trajectory Y t appeared at an unspecified position Y τ at time moment t = τ , which equals unity since averaging over ξ k with k ∈ (0, τ − 1) implies averaging over all possible Y τ ; the probability that Y t paused at t = τ , which equals 1/2; and the probability that Y t returned to Y τ = Y τ +1 within τ ′ − τ − 1 steps. Making a plausible assumption that the dynamics, at least in the asymptotic limit t → ∞, does not depend of the starting point, we thus find that the expression (9) reduces to 1 2 P τ ′ −τ −1 (Y = 0) ,(10) where P τ ′ −τ −1 (Y = 0) is the probability that the y-component of the TP trajectory returns to Y = 0, (not necessarily for the first time), on the (τ ′ − τ − 1)-th step. Here, P t (Y ) (P t (X)) is a marginal distribution obtained from the full probability distribution function P t (X, Y ) of finding the TP at site (X, Y ) at time moment t by summing the latter over all X (Y ), that is P t (Y ) = ∞ X=−∞ P t (X, Y ) , P t (X) = ∞ Y =−∞ P t (X, Y ) .(11) Summing up the presented above reasonings, we arrive at the following representation of the DA MSD: (20) and (21)). A superdiffusive behaviour sets in from rather short times and the diffusive transient (see the first term in the rhs of eq. (12)) is not observed. The inset displays the rate of a convergence of the time-dependent dynamical exponent γt, eq. (18), to its asymptotic value 4/3. Panel (b). Reduced moments < \|Xt\| q > t 2q/3 as functions of time. The dashed lines (from top to bottom) correspond to m 4 = 1.038, m 3 = 0.687, m 1 = 0.590 and m 2 = 0.556 (see eq. (21)). X 2 t ∼ t 2 + 1 2 t−2 τ =0 t−1 τ ′ =τ +1 P τ ′ −τ −1 (Y = 0) .(12) Further on, the probability P τ ′ −τ −1 (Y = 0) is evidently a decreasing function of the difference τ ′ − τ − 1. Very general arguments (see also the numerical results presented in Fig. 3, panel (a)), suggest that P τ ′ −τ −1 (Y = 0) decays as a power-law : P τ ′ −τ −1 (Y = 0) ∼ A (τ ′ − τ − 1) γ/2(13) in the limit (τ ′ − τ − 1) → ∞, where A is the amplitude and γ is the dynamical exponent, both to be defined. Supposing that γ < 2 (γ = 2 corresponds to ballistic motion), we expect that both the inner sum (over τ ′ ) and the outer one (over τ ) in eq. (12) will be dominated by the upper summation limit. As a consequence, in the large-t limit 1 2 t−2 τ =0 t−1 τ ′ =τ +1 P τ ′ −τ −1 (Y = 0) ∼ At 2−γ/2 2(1 − γ/2)(2 − γ/2) ,(14) and hence, in the large-t limit the expression (12) attains the form X 2 t ∼ t 2 + At 2−γ/2 2(1 − γ/2)(2 − γ/2) .(15) In line with the arguments presented in Refs. [27,28], we recall that the dynamical exponent γ defines the characteristic extent of the trajectory Y t ; that being, < Y 2 t >= m 2 t γ , where m 2 is as yet unknown proportionality factor. By symmetry, one expects thus that the DA MSD along the x-axis, i.e., < X 2 t >, obeys exactly the same law, which entails the following closure relation : m 2 t γ ∼ t 2 + At 2−γ/2 2(1 − γ/2)(2 − γ/2) .(16) Inspecting the behaviour of the latter expression in the limit t → ∞, we infer that the contribution of the first term in the rhs of eq. (16) becomes negligible in the limit t → ∞, so that the dominant contribution is provided by the second term. Comparing the power-law on the left-hand-side (lhs) of eq. (16) with the second term on the rhs of this equation, we find that the exponent γ obeys γ = 2 − γ/2 ,(17) which yields γ = 4/3 -the value which has been previously conjectured and verified numerically in Refs. [27,28]. Therefore, our reasonings correctly reproduce the value of the dynamical exponent γ. However, inferring a numerical value of the prefactor m 2 from eq. (16), (which predicts m 2 ∼ 9A/8), should lead to a somewhat higher m 2 than the actual one, because the rhs in eq. (14) evidently overestimates the value of the double sum in the lhs of this equation. The point is that the algebraic form in eq. (13) is only valid for such realisations of the TP trajectories, for which the sum of the number of jumps and of the number of the pausing events is even. Otherwise, P τ ′ −τ −1 (Y = 0) is exactly equal to zero. As a consequence, eq. (16) overestimates m 2 . Lastly, we note that a similar type of arguments was invoked to characterise a decay of the number of tree-like clusters with a growing pattern height in a process of ballistic deposition of sticky particles on a line [66]. Both the decay and the ensuing thinning of the forest of such clusters appear to be controlled by a random wandering of the inter-cluster boundaries with the super-diffusive exponent γ = 4/3. In Fig. 2, panel (a), we present numerical results (open circles) describing the time evolution of the DA MSD of a single TP. The dashed line indicates the superdiffusive power-law behaviour of the form < X 2 t >∼ m 2 t 4/3 , with m 2 = 0.556. This estimate of m 2 is based on the fitting of the full probability distribution, which is discussed below in subsection 3.2. We observe that the super-diffusive behaviour sets in from rather early times and the transient diffusive law, as predicted by the first term in the rhs of eq. (16), is not observed. Next, the inset in the panel (a) illustrates the convergence of the dynamical exponent γ t , defined by γ t = ln X 2 t − ln X 2 t z (1 − z) ln(t) ,(18) to its asymptotic value 4/3. Such a representation of γ t (as compared to the standardly used one, γ t = ln X 2 t / ln(t)) is particularly well-suited for a numerical analysis of the dynamical exponent in an expected power-law dependence on time with an unknown numerical prefactor, since the latter cancels out automatically. In eq. (18) the parameter z is a trial exponent, 0 < z < 1, which rescales time in the second term; in principle, z can be chosen rather arbitrarily; we use z = 0.9. We also observe that γ t converges to its asymptotic value very rapidly, in line with the behaviour of the DA MSD. In panel (b) of Fig. 2 we plot the reduced moments < \|X\| q t > /t 2q/3 for q = 1, 2, 3 and 4 as functions of time. We observe that the reduced moments saturate as some constant values m q as time progresses, indicating that the moments themselves obey < \|X\| q t >= m q t 2q/3 (see eq. (20)). Here, the dashed lines indicate our estimates for the values of the numerical prefactors m q (see eq. (21)). Probability distribution and moments of arbitrary order In Fig. 3 we depict different facets of the numerically evaluated full probability distribution P t (X, Y ) and of the marginal distribution P t (X), (see eq. (11)). Panel (a) presents the time evolution of P t (X = X ⋆ ) for six fixed values of X ⋆ : X ⋆ = 0, 60, 1000, 1400, 1800 and 3000 (curves from top to bottom, with lighter colours corresponding to smaller values of X ⋆ ). Our numerical results show that, unequivocally, P t (0) obeys a power-law of the form P t (0) ≃ A/t 2/3 , which is fully in line with our above analysis. The decay amplitude is defined with a good accuracy by A ≈ 0.568. Moreover, comparing our numerical results with the form P t (0) ≃ A/t 2/3 , we conclude that the latter provides a very accurate estimate for P t (0) starting from rather short times -the dashed line representing A/t 2/3 and the numerical data (light blue curve) are almost indistinguishable. In turn, P t (X = X ⋆ ) for X ⋆ = 60, 1000, 1400, 1800 and 3000 converges ultimately to P t (0) ≃ A/t 2/3 , which is, of course, not an unexpected behaviour. The panel (b) presents the time evolution of P t (0, 0) -the probability of being at the origin at time moment t. We observe that the power-law form P t (0, 0) ≃ A ′ /t 4/3 (with A ′ ≈ 0.555) describes the numerical data fairly well. Note also that this form implies that a random walk on a random Manhattan lattice is not certain to return to the origin. Further on, in Fig. 3, panels (c) and (d), we plot t 2/3 P t (X) and t 4/3 P t (X, Y ) with Y = 0 as functions of the scaled variable u = X/t 2/3 . The data collapse evidenced by our numerical results for both the central part of the distribution and for its tails, suggests, again rather unequivocally, that the marginal distribution P t (X) of the TP position along the x-axis at (sufficiently large) time t has the following form: P t (X) = 1 t 2/3 A exp −au 2 for \|u\| < 1 B exp −b\|u\| 4/3 for \|u\| > 1(19) where B ≈ 1.249, a ≈ 1.049 and b ≈ 1.730. We observe, as well, that the full distribution P t (X, Y ) (with Y = 0) exhibits essentially the same functional behaviour as a function of u, (see Fig. 3, panel (d)), as the marginal distribution P t (X) and only the values of the parameters are slightly different. We therefore conclude that a) the central part of both distributions is a Gaussian, with the variance which grows super-diffusively with t, and b) the tails of both distributions deviate from a Gaussian and have a form ∼ exp(−\|u\| 4/3 ), i.e., are "heavier" than a Gaussian. The presence of such tails also manifests itself in the anomalously high asymptotic value ≈ 3.5 attained by the kurtosis of the marginal distribution P t (X) (see the dashed curve in Fig. 7, panel (d)). Recall that the kurtosis of a Gaussian distribution is equal to 3. We note that the large-u tail of P t (X) and P t (X, Y = 0) has a very different form, as compared to the prediction made in Refs. [27,28]. Assuming the validity of the usual relation between the shape exponent δ and the dynamical exponent γ, δ = 1/(1 − γ), it was conjectured that the shape exponent should be δ = 3. Our data shows that this is not the case and, surprisingly enough, the distribution in the second line in eq. (19) has exactly the same shape exponent δ = 4/3 as the one appearing in the MdM model with random layered convection flows (see Refs. [27][28][29]). Capitalising on the expression in eq. (19), we estimate the behaviour of the moments of P t (X) of arbitrary order q. Multiplying both sides of eq. (19) by \|X\| q , changing the integration variable for u = X/t 2/3 , and integrating the expression in the first line over u ∈ (−1, 1) and in the second line -over u ∈ (−∞, −1) and (1, ∞), we get \|X t \| q = m q t 2q/3 ,(20) with m q ≈ A a (q+1)/2 Γ(q + 1) − Γ(q + 1, a) + 3B 2b 3(q+1)/4 Γ 3(q + 1) 4 , b ,(21) where Γ(a, b) is the incomplete Gamma-function. Note that here we discard the transient region between two asymptotic regimes, supposing that the second regime is valid starting from \|u\| = 1. This is, of course, not true and hence, m q in eq. (21) overestimates the actual value of the numerical prefactor m q in eq. (20). We however believe that such an estimate is quite plausible and would not incur any significant error. The plot of the numerical results for the first four moments together with the estimates for m q presented in Fig. 2, panel (b), shows that it is indeed the case. We close this subsection with two following remarks: a) the value of m 2 deduced from eq. (16), i.e., m 2 ≈ 9A/8 ≈ 0.639, slightly overestimates the value of m 2 obtained from eq. (21), i.e., m 2 = 0.556. This is completely in line with our argument that the second term in the rhs in eq. (16) provides an upper bound on the actual value of m 2 . b) For the kurtosis κ of the marginal distribution P t (X), i.e., κ = X 4 t X 2 t 2 ,(22) we find from eqs. (20) and (21) that κ = m 4 /(m 2 ) 2 ≈ 3.360, which value favourably agrees with κ ≈ 3.5 deduced from our numerical simulations (see Fig. 7, panel (d)). Sample-to-sample fluctuations Up to the present moment we discussed only the averaged behaviour -both over thermal histories and over realisations of random convection flows. However, a legitimate question is how the pertinent parameters themselves vary from a realisation to a realisation of the latter pattern. This question has been first addressed in Refs. [27,28] for the MdM model with random layered flows and significant sample-to-sample fluctuations have been predicted. However, for the dynamics on a random Manhattan lattice this issue has not been analysed and we concentrate on it below, focusing on the mean-squared (averaged over thermal histories only) displacement X 2 t of the TP on a given pattern of arrows as well as on the corresponding probability distribution function Π t (X) of its position X along the x-axis. Recall that P t (X) = Π t (X) . In Fig. 4, panel (a), we depict the corresponding MSD X 2 t (averaging over thermal histories is performed over 2 × 10 7 realisations of trajectories, for a given realisation of convection flows) for 50 realisations of disorder as functions of time. We do indeed observe some scatter in the values of the prefactor m 2 , which here is a random variable dependent on a particular realisation of disorder. On the other hand, the amplitude of fluctuations does not seem to be very significant and all the curves concentrate essentially around the DA MSD X 2 t = 0.556 t 4/3 (see Fig. 2). Moreover, we realise that the MSD averaged over 50 realisations of disorder only, appears to be fairly close to the DA MSD evaluated using an ample statistical sample; recall that in subsection 3.2 we used 2 × 10 5 realisations of disorder in order to perform averaging over random convection flows. On contrary, the sample-to-sample fluctuations do affect in a significant way the shape of the distribution function Π t (X), both in the central part and especially in the region of anomalous tails, for which the numerical data looks quite nebulous. However, it is quite surprising to realise that being averaged over just 50 realisations of disorder, Π t (X) gets rather close to P t (X) depicted in Fig. 3 (see thin solid and dashed curves in Fig. 4), which again was evaluated using a much bigger statistical sample. Here, an agreement between the blue zigzag curve and the thin black solid line looks nearly perfect within the central, Gaussian part of the distribution, while also for the tails it shows a rather convincing agreement. Therefore, we may conclude that sample-to-sample fluctuations are essentially less important for a random motion on a random Manhattan lattice than in the MdM model with layered random flows [27,28]. Spectral analysis of the TP trajectories Complementary information about anomalous diffusion of the TP can be inferred from the so-called single-trajectory power spectral density S(T, f ), where T is the observation time and f is the frequency (see, e.g., Ref. [67] for more details). This property is a random, realisation-dependent variable, parametrised by f and T . Note that in the model at hand, it depends on both a given pattern of arrows in a random Manhattan lattice and on a given realisation of a thermal history. For an integer-valued X t , S(T, f ) is defined as S(T, f ) = 1 T T t=0 e if t X t 2 ,(23) and hence, is a periodic function of f T with the prime period 2πT . In a standard text-book analysis, one considers the ensemble-averaged (and also disorder-averaged for our case) value of the random variable in eq. (23), i.e. its first moment : µ(T, f ) = S(T, f ) = 1 T T t1,t2=0 cos (f (t 1 − t 2 )) X t1 X t2 ,(24) which probes the frequency-dependence of the Fourier-transformed covariance function of X t . Moreover, one also takes formally the limit of an infinitely long observation time, i.e., sets T = ∞. We will demonstrate below that, although one has indeed to consider very large values of T in order to extract a meaningful information about the f -dependence of the power spectral density in eq. (24)), taking the formal limit T = ∞ in our case renders such a standard definition meaningless. In Fig. 5 we present the results of a numerical analysis of the functional form of µ(T, f ), which reveals two rather surprising features. First, it appears that µ(T, f ) is ageing, i.e., its amplitude is dependent on the observation time T , µ(T, f ) ∼ T 1/3 , which is demonstrated in the inset to this figure. As a consequence, setting T = ∞ is meaningless. Second, we observe that f 2 µ(T, f ), (for three fixed values of T ), approaches constant f -independent values for sufficiently large f . This signifies that µ(T, f ) ∼ 1/f 2 , i.e., it exhibits exactly the same f -dependence as the power spectral density of a standard Brownian motion (see, e.g., Ref. [67] and references therein), although the process under study is clearly not a Brownian motion. Therefore, fixing T and focussing only on the f -dependence of µ(T, f ) garnered from numerical simulations, one can be led to an erroneous conclusion that the observed process is a Brownian motion. As an actual fact, this is precisely the T -dependence of µ(T, f ) which helps to realise that this is not the case (see also Ref. [71]). We note that such a "deceptive" f -dependence has been previously reported for the running maximum of Brownian motion [68], diffusion in a periodic Sinai disorder [69], diffusion with stochastic reset [70] and also for a variety of diffusing diffusivity models [43]. Further on, the law µ(T, f ) ∼ T 1/3 /f 2 was observed for other super-diffusive processes, such as a fractional Brownian motion with the Hurst index H = 2/3 (i.e., γ = 4/3) [71] or a super-diffusive scaled Brownian motion Z t described by the Langevin equationŻ t = t 1/6 ζ t [72], with ζ t being a Gaussian white-noise with zero mean. This latter process also produces a super-diffusive motion with γ = 4/3, suggesting that the law µ(T, f ) ∼ T 1/3 /f 2 might be a generic feature of processes with γ = 4/3. We note parenthetically that this questions the robustness of the textbook approach, based solely on the evaluation of µ(T, f ), which cannot distinguish between these three distinctly different random processes. The difference between these processes becomes apparent, however, when one considers higher-order moments of S(T, f ), e.g., its variance. In particular, one may focus on the coefficient of variation C v , which is defined by C v = σ(T, f ) µ(T, f ) = S 2 (T, f ) − S(T, f ) 2 µ(T, f ) ,(25) where σ(T, f ) is the standard deviation of a random variable S(T, f ). This characteristic parameter shows a completely different behaviour as a function of f for a super-diffusive fractional Brownian motion and a super-diffusive scaled Brownian motion. For the former C v approaches for sufficiently large T and f a universal (i.e., regardless of the actual value of H > 1/2), time T -independent constant value √ 2 [71], while for the latter -a universal time T -independent constant value √ 5/2 [72], the same which is observed for a standard Brownian motion [67]. To this end, we have studied via numerical simulations the frequency and the observation time dependence of C v for the TP random motion on a random Manhattan lattice. This dependence is presented in Fig. 5, panel (b), in which we plot the coefficient C v of variation of a single-trajectory power spectral density as a function of f for three values of T . We observe that C v tends to a higher than √ 2 value as frequency increases. Moreover, C v is clearly ageing, i.e., its limiting behaviour is dependent on the observation time. In conclusion, we observe a behaviour of C v which is markedly different from the two above mentioned examples of super-diffusion with γ = 4/3. Tracer particle dynamics on a populated random Manhattan lattice In this last section we discuss the results of the numerical analysis of the TP dynamics on a crowded Manhattan lattice. In Fig. 6 we present the DA MSD of the TP for Model A and Model B, prefactor m 2 (ρ) in the super-diffusive law X 2 t = m 2 (ρ) t 4/3 as a function of the density ρ of the LG particles, and also the kurtosis of the distribution P t (X) for Model A and Model B. First of all, we realise that for both models the DA MSD obeys the same super-diffusive law X 2 t ∼ t 4/3 , for any density of the LG particles. Prefactor m 2 (ρ) depends, of course, on the density of the LG particles and their dynamics; indeed, m 2 (ρ) shows apparently different dependences on ρ for Model A and Model B. For Model A, in which the TP follows random convection flows while the LG particle perform constrained random walks, this dependence is most strong and is rather close to a parabolic law m 2 (ρ) = 0.556 (1 − ρ) 2 (thin solid curve in panel (c)). This parabolic law very accurately describes the actual dependence of m 2 (ρ) on ρ for ρ < 1/2. For higher densities, however, some deviations are clearly seen, although such a discrepancy can be also attributed to the lack of a large enough statistical sample. For Model B, in which all particles are identical and all perform a super-diffusive motion, the ρ-dependence of the prefactor m 2 (ρ) is given by m 2 (ρ) = 0.556 (1 − ρ) 4/3 (thin dashed curve in panel (c)). This law agrees fairly well with the numerical data for any value of the LG particles density. In turn, as we have mentioned above, such a dependence implies that the system is perfectly stirred and the time variable t gets merely rescaled by the fraction of successful jump events of the TP, i.e., by (1 − ρ), as one can expect from simple mean-field-type arguments. Arguably, this expression for m 2 (ρ) is exact. Next, in Fig. 6, panel (d), we depict the time evolution of the kurtosis of the distribution P t (X) in case of a single TP, as well as for Model A and Model B. Thick dashed line corresponds to the kurtosis of P t (X) in case of a single TP. We observe that κ saturates as t evolves at a constant value which is close to 3.5, i.e., it exceeds the value 3 specific to a Gaussian distribution and thus implies that P t (X) is not Gaussian. This happens, of course, due to the presence of anomalous slower-than-Gaussian tails, which we discussed in subsection 3.2. Next, noisy green curves depict our results for κ in Model A, with the LG particles densities ρ = 0.1, 0.2, 0.3 and ρ = 0.4, plotted versus a rescaled time variable t ρ = (1 − ρ) 3/2 t. Although there is a significant scatter of these curves at short times, we notice that for sufficiently large t ρ all these curve collapse on the dashed line representing a single TP case. For Model B we analyse the TP dynamics for a broader range of the LG particles densities; we considered nine values of ρ, ρ = 0.1, 0.2, . . . , 0.9. Here, the curves defining the evolution of κ for different values of ρ, merge altogether even for short times when plotted versus a rescaled time t ρ = (1 − ρ)t, and eventually approach the value of the kurtosis for a single TP case. Such a behaviour signifies that for sufficiently large times P t (X) possesses some universal scaling properties for both Model A and Model B. Lastly, in Fig. 7 we present the numerical data for the distribution function P t (X) for Model A and Model B. We observe that for both models the central Gaussian part of the distribution is described with a very good accuracy by a Gaussian function: P t (X) = A t 2/3 ρ exp −a X 2 t 4/3 ρ ,(26) where the choice of t ρ depends on the model under study. For a single TP case, t ρ = t. The lack of a big statistical sample does not permit to make completely conclusive statements about the tails of the distribution. We notice, however, that such tails are definitely present and a departure of the distribution from a purely Gaussian form is apparent in Fig. 7. We also observe that upon an increase of t the curves get closer for both Model A and Model B. We thus find it absolutely plausible that such a form of distribution is also valid for the dynamics of a TP on a populated random Manhattan lattice. Conclusions To recapitulate, we studied the tracer particle (TP) dynamics in presence of two interspersed and competing types of disorderquenched random convection flows on a random Manhattan lattice, which prompt the TP to move super-diffusively, and a crowded dynamical environment formed by a lattice gas (LG) of hard-core particles, which hinder the TP motion. The random Manhattan lattice is a square lattice decorated with arrows in such a way that directionality of each arrow is fixed along each raw (a street) or a column (an avenue) along their entire length, but whose orientation randomly fluctuates from a street to a street and from an avenue to an avenue. The hard-core LG particles perform a random motion, constrained by the singleoccupancy condition; that being, each lattice site can be occupied by at most a single particle -a LG particle or a TP, or be vacant. We have considered two possible scenarios of the LG particles random motion. In Model A, we supposed that the LG particles are insensitive to the random convection flows and perform symmetric random walks -a simple exclusion process -among the sites of a two-dimensional square lattice. In this case, the TP moves subject to random convection flows and interacts with a fluid-like quiescent environment, which imposes some frictional force on it. In Model B, we supposed that all the particles -the TP and the LG particlesare identical and follow a local directionality of bonds in a random Manhattan lattice. In this case, the system under study is a kind of a "turbulent" fluid in which all the particles perform a super-diffusive motion. We focused on such characteristics of the TP dynamics as its disorder-averaged mean-squared displacement (DA MSD), (and generally, the moments of arbitrary order for a single TP), the distribution of its position at time moment t averaged over disorder, and the time evolution of the kurtosis of this distribution. We have shown that for both Model A and Model B the DA MSD obeys a super-diffusive law X 2 t ∼ m 2 (ρ) t 4/3 , where the functional dependence of the prefactor m 2 (ρ) on the mean density ρ of the LG particles depends on the model under study. For the case of a single TP (i.e. in absence of the LG particles, ρ = 0), we provided some analytical arguments explaining such a super-diffusive behaviour. We showed that the distribution of the TP position has a Gaussian central part and exhibits slower-than-Gaussian tails of the form exp(−(\|X\|/t 2/3 ) 4/3 ) for sufficiently large X and t. Such a form was evidenced in case of a single TP through an analysis of a very big statistical sample, and also shown to hold, although not in a completely conclusive way, for the dynamics on a populated random Manhattan lattice. As a consequence of presence of anomalous tails, the kurtosis of the distribution in all the situations under study, was shown to attain a bigger value (3.5) than the value (3) specific to a Gaussian distribution. Finally, we addressed the question of sample-to-sample fluctuations in the system under study and performed an analysis of spectral properties of the TP trajectories, which revealed some interesting features. Figure 1 . 1Random Manhattan lattice and the TP trajectories. Panel (a). A realisation of a random Manhattan lattice -a square lattice decorated in a random fashion with arrows, indicating the possible jump directions. Jumps against an arrow are not permitted in our model. The directionality of each arrow is fixed along each street (East-West) and an avenue (North-South) along their entire, infinite in both directions length, and fluctuates randomly from a street (an avenue) to a street (an avenue). The pattern of arrows is frozen and does not vary with time. A square (blue) indicates the TP instantaneous position, while the circles (red) denote the instantaneous positions of the LG particles. Panel (b). Five individual TP trajectories on a random Manhattan lattice in absence of the LG particles. Figure 2 . 2Disorder-averaged mean-squared displacement of the TP and higher moments of X(t). Panel (a). Disorder-averaged mean-squared displacement < X 2 t > of the TP in absence of the dynamical environment (LG particles). The open circles depict our numerical results, while the dashed (red) line indicates the prediction < X 2 t >= m 2 t 4/3 with m 2 = 0.556 (see eqs. Figure 3 . 3Probability distribution of the TP position. Panel (a). Temporal evolution of the marginal distribution Pt(X = X ⋆ ), eq. (11), for six fixed values of X ⋆ = 0, 60, 1000, 1400, 1800 and 3000 (solid curves from top to bottom with lighter curves corresponding to smaller values of X ⋆ ). The dashed line denotes the power-law A/t 2/3 with A ≈ 0.568. Note that Pt(0) ≃ A/t 2/3 provides a very accurate estimate for Pt(0) starting from rather short times; Pt(X = X ⋆ ) for X ⋆ > 0 converges ultimately to Pt(0). Panel (b). The probability Pt(0, 0) of being at the origin at time moment t. Thick blue curve presents the numerical data. A dashed line is a power-law A ′ /t 4/3 with A ′ ≈ 0.555. It provides a fairly good estimate for the numerical data starting from t ≥ 10 2 . Panel (c). The marginal distribution Pt(X), multiplied by t 2/3 , is plotted as a function of the scaled variable u = X/t 2/3 . Panel (d). The full distribution Pt(X, Y ) with Y = 0, multiplied by t 4/3 , is plotted as a function of the scaled variable u = X/t 2/3 . In panels (c) and (d) the histograms show the results of numerical simulations: light brown, green and blue colours correspond to the numerical data for t = 10 4 , 4 × 10 4 and t = 10 5 , respectively; thin solid curves are the Gaussian function A exp(−au 2 ), and the dashed curves -a stretched-exponential function of the form B exp(−b\|u\| 4/3 ). Vertical dotted line in panel (c) is a guide to an eye which indicates the crossover value u = 1 between the two asymptotic regimes. Purple circles in the panel (c) depict our data for a shorter time -t = 10 3 . A deviation from the stretched-exponential form signifies that the anomalous tails of Pt(X) appear only for sufficiently large values of t. Figure 4 . 4Sample-to-sample fluctuations of the TP trajectories. Panel (a). MSD X 2 t for fifty (chosen at random) realisations of random convection flows plotted as a function of time. The dashed line represents the DA MSD X 2 t = 0.556 t 4/3 (see Fig. 2), while the green line indicates the MSD averaged over 50 realisations of disorder. Panel (b). Realisation-dependent probability distribution function Πt(X) of the TP position along the x-axis, multiplied by t 2/3 , is plotted versus a scaled variable X/t 2/3 . We present 50 (thin grey) curves corresponding to 50 fixed realisations of frozen convection flows. Thin solid and dashed curves depict the Gaussian function A exp(−au 2 ) and a stretched-exponential function of the form B exp(−b\|u\| 4/3 ), respectively, with the parameters A, a, B and b as defined in Fig. 3. Blue zigzag curve represents Πt(X) averaged over 50 realisations of random convection flows. Figure 5 . 5Spectral properties of the TP trajectories. Panel (a). Ensemble-and disorder-averaged power spectral density µ(T, f ), eq. (24), multiplied by f 2 , as a function of the frequency f for three values of the observation time T (from top to bottom T = 4800, 2400 and 1200). The inset displays the ageing behaviour of µ(T, f = f ⋆ ) as a function of the observation time T for a fixed frequency f ⋆ = 5 × 10 2 , and evidences the dependence µ(T, f = f ⋆ ) ∼ T 1/3 . Panel (b). The coefficient Cv of variation, eq. (25), of the distribution of a random variable S(T, f ), eq. (23), as a function of the frequency f for three values of the observation time T (the same asin panel (a)). The dashed line indicates Cv = √ 2 -a value specific to a super-diffusive fractional Brownian motion (with arbitrary Hurst index in the interval 1/2 < H < 1 and hence, with γ in the interval 1 < γ < 2). Figure 6 . 6DA MSD of the TP, prefactor m 2 (ρ) and the kurtosis for Model A and Model B. Panel (a). The DA MSD for the Model A as a function of time for different densities of the LG particles. Curves from top to bottom correspond to ρ = 0.1, 0.2, . . . , 0.9. Black dashed line depicts the DA MSD X 2 t = 0.556 t 4/3 of a single TP (in absence of the LG particles, i.e., for ρ = 0). Panel (b). The DA MSD as a function of time for Model B. Curves from top to bottom correspond to ρ = 0.1, 0.2, . . . , 0.9 and the black dashed line corresponds to ρ = 0. Panel (c). Numerical prefactor m 2 (ρ) in the law X 2 t = m 2 (ρ) t 4/3 for Model A and Model B. Symbols present the results of numerical simulations: filled circles (Model A) and filled squares (Model B). Thin solid and dashed lines are fits to the numerical data: solid line is m 2 (ρ) = 0.556 (1 − ρ) 2 (Model A) and the dashed line is m 2 (ρ) = 0.556 (1 − ρ) 4/3 (Model B). Panel (d). The kurtosis κ of the distribution Pt(X), (defined in eq. (22)), as a function of a rescaled time tρ. Dashed line depicts the kurtosis of Pt(X) of a single TP, here, tρ = t; noisy green curves present the time evolution of the kurtosis for Model A for ρ = 0.1, 0.2, 0.3 and 0.4, here tρ = (1−ρ) 3/2 t; and eventually the grey curves, which collapse on a single master curve (dashed line), depict the kurtosis of the corresponding distribution Pt(X) for Model B with ρ = 0.1, 0.2, . . . , 0.9, here, tρ = (1 − ρ)t. Figure 7 . 7Probability distribution of the TP position along the x-axis on a populated random Manhattan lattice. Panel (a). Pt(X) multiplied by (1 − ρ)t 2/3 is plotted as a function of a scaled variable u 1 = X/((1 − ρ) 3/2 t) 2/3 . Green zigzag curves correspond to ρ = 0.2, while the blue ones -to ρ = 0.8. Dot-dashed, dashed and thick solid zigzag curves depict the numerical data for t = 10 2 , 2 × 10 3 and 3 × 10 3 , respectively. Thin solid curve is a Gaussian function A exp −au2 1 , while the thin dashed curve is an anomalous tail of the form B exp −b\|u 1 \| 4/3 . Panel (b). 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[ "THE BUTCHER-OEMLER EFFECT IN 295 CLUSTERS: STRONG REDSHIFT EVOLUTION AND CLUSTER RICHNESS DEPENDENCE", "THE BUTCHER-OEMLER EFFECT IN 295 CLUSTERS: STRONG REDSHIFT EVOLUTION AND CLUSTER RICHNESS DEPENDENCE" ]	[ "V E Margoniner ", "R R De Carvalho ", "R R Gal ", "S G Djorgovski " ]	[]	[]	We examine the Butcher-Oemler effect and its cluster richness dependence in the largest sample studied to date: 295 Abell clusters. We find a strong correlation between cluster richness and the fraction of blue galaxies, f B , at every redshift. The slope of the f B (z) relation is similar for all richnesses, but at a given redshift, f B is systematically higher for poor clusters. This is the chief cause of scatter in the f B vs. z diagram: the spread caused by the richness dependence is comparable to the trend in f B over a typical redshift baseline, so that conclusions drawn from smaller samples have varied widely. The two parameters, z, and a consistently defined projected galaxy number density, N , together account for all of the observed variation in f B within the measurement errors. The redshift evolution of f B is real, and occurs at approximately the same rate for clusters of all richness classes.	10.1086/319099	[ "https://export.arxiv.org/pdf/astro-ph/0011210v1.pdf" ]	17,072,353	astro-ph/0011210	dbde454851adab8a59b63d6150f66e99c36dd67d	THE BUTCHER-OEMLER EFFECT IN 295 CLUSTERS: STRONG REDSHIFT EVOLUTION AND CLUSTER RICHNESS DEPENDENCE arXiv:astro-ph/0011210v1 10 Nov 2000 V E Margoniner R R De Carvalho R R Gal S G Djorgovski THE BUTCHER-OEMLER EFFECT IN 295 CLUSTERS: STRONG REDSHIFT EVOLUTION AND CLUSTER RICHNESS DEPENDENCE arXiv:astro-ph/0011210v1 10 Nov 2000accepted for publication in the Astrophysical Journal Lettersaccepted for publication in the Astrophysical Journal Letters Preprint typeset using L A T E X style emulateapjSubject headings: galaxies: clusters -evolution We examine the Butcher-Oemler effect and its cluster richness dependence in the largest sample studied to date: 295 Abell clusters. We find a strong correlation between cluster richness and the fraction of blue galaxies, f B , at every redshift. The slope of the f B (z) relation is similar for all richnesses, but at a given redshift, f B is systematically higher for poor clusters. This is the chief cause of scatter in the f B vs. z diagram: the spread caused by the richness dependence is comparable to the trend in f B over a typical redshift baseline, so that conclusions drawn from smaller samples have varied widely. The two parameters, z, and a consistently defined projected galaxy number density, N , together account for all of the observed variation in f B within the measurement errors. The redshift evolution of f B is real, and occurs at approximately the same rate for clusters of all richness classes. INTRODUCTION The Butcher-Oemler (BO) effect provided some of the first evidence for the evolution of galaxies and clusters. Butcher & Oemler (1978, 1984 found an excess of blue galaxies in high redshift clusters in comparison with the typical early-type population observed in the central region of local clusters (Dressler 1980). The BO effect has also been observed in more recent studies, which indicate an even stronger evolution of the fraction of blue galaxies in clusters (Rakos & Schombert 1995, Margoniner & de Carvalho 2000. The BO effect is an indicator of evolution in the cluster galaxy population, which may result from changes in the morphology and star-formation rates of member galaxies with redshift. The fact that blue galaxies are more commonly found at higher redshifts, and the observation of an apparent excess of S0 galaxies in low redshift clusters, lead to the suggestion by Larson et al. (1980) of an evolutionary connection between S0 and spiral galaxies. This idea can explain the population of blue galaxies observed by Butcher & Oemler as spiral galaxies seen just before running out of gas, and the disappearance of this population in more evolved, low redshift rich clusters , Couch et al. 1998). In the last decade, highresolution Hubble Space Telescope images have allowed the determination of the morphology of these high redshift blue galaxies. Dressler et al. (1994), Couch et al. (1994 found that the Butcher & Oemler galaxies are predominantly normal late-type spirals, and that dynamical interactions and mergers between galaxies may be a secondary process responsible for the enhanced star formation. A recent study by Fasano et al. (2000) indicates that as the redshift decreases, the S0 cluster population increases while the number of spiral galaxies decreases, supporting Larson's original idea of spirals evolving into S0s. A second factor affecting the number of blue galaxies may be their environment. The fraction of blue galaxies seems to be correlated with local galaxy density, and with the degree of substructure in the cluster. Many studies have shown that the blue galaxies lie preferentially in the outer, lower density cluster regions (Butcher & Oemler 1984, Margoniner & de Carvalho 2000. Also, Smail et al. (1998) studied 10 massive clusters and found that the fraction of blue galaxies in these clusters is smaller than observed in Butcher & Oemler (1984) clusters at the same redshift range. However, an opposite trend is found in non-relaxed clusters with a high degree of substructure, where the fraction of blue galaxies is higher than that observed in regular clusters (Caldwell & Rose 1997, Metevier et al. 2000. The evolution of these galaxies is probably correlated with its environment in the sense that spirals might consume and/or lose their gas, evolving to SOs, while falling into the higher density cluster regions. Although the exact mechanism responsible for the BO effect is not completely understood (Kodama & Bower 2000), most authors seem to agree that the effect is real. A different idea is presented by Andreon & Ettori (1999) who show that the mean X-ray luminosity and richness of Butcher & Oemler (1984) clusters increases with redshift, and argue that the increase in the fraction of blue galaxies with redshift may not represent the evolution of a single class of objects. We present observations of the BO effect in a large sample of 295 Abell (1958Abell ( , 1989 clusters of all richnesses, with no further selection on the basis of morphology or X-ray luminosities. This is important because all previous studies were based on samples biased toward richer clusters, with some samples being further selected according to morphology and/or X-ray luminosities, so that a combination of different selection effects could be mimicking the observed f B (z) relation. Although our sample inherits biases existent in the Abell catalog, it should contain a more representative variety of clusters (in terms of degree of substructure, richness, and mass) at each redshift than any previous sample, and because of its large size should allow a better determination of the relation. Any results¸driven by selection effects present in previous samples might become apparent when compared with this one. We describe the data in Section 2, the BO effect analysis in Section 3, and our conclusions in Section 4. SAMPLE SELECTION AND INPUT DATA The data presented in this paper were obtained to calibrate the DPOSS-II (the Digitized Second Palomar Observatory Sky Survey, Djorgovski et al. 1999). It comprises 44 Abell clusters imaged at the 0.9m telescope at the Cerro Tololo Interamerican Observatory (CTIO) (Margoniner & de Carvalho 2000, hereafter MdC00), and 431 clusters observed at the Palomar Observatory 1.5m telescope (Gal et al. 2000, hereafter G00, and in preparation). The CCD images were taken in the g, r and i filters of the Thuan & Gunn (1976) photometric system, with typical 1-σ magnitude errors of 0.12 in g, 0.10 in r, and 0.16 in i at r = 20.0 m . We have also observed 22 control fields in order to assess the background contribution. More details concerning the data reduction, photometry, and catalog construction can be found in MdC00 and G00. From this original sample we excluded 120 clusters observed with a small field of view CCD, and 31 very low redshift (z < 0.05) clusters, for which only the core region can be observed with our 13 ′ × 13 ′ images. Also, 26 cluster fields which exhibited galaxy counts comparable to the mean background, and one cluster with a bright star occupying ∼ 25% of the CCD region, were excluded from the analysis. Since 2 Abell clusters had repeated observations from G00 and MdC00, our final sample comprises 295 clusters. It is important to note that at variance with previous studies, the sample presented in this work is representative of all richness class clusters (21% R = 0, 50% R = 1, 21% R = 2, and 8% R ≥ 3). No further selection regarding richness, morphology or degree of sub-clustering was applied when determining our sample. Spectroscopic redshifts for 77 clusters were obtained from the literature, and for the remainder photometric redshifts were estimated with an rms accuracy of approximately 0.04 using the methodology described in MdC00. ANALYSIS OF THE BUTCHER-OEMLER EFFECT The BO effect can be measured by comparing the fraction of blue galaxies in clusters at different redshifts. The (g−r) vs. r color-magnitude (CM) relation was determined by fitting a linear relation to the red galaxy envelope using the same prescription as in MdC00. Red envelopes were subjectively classified as well-defined or not by visual inspection. Galaxies were defined as blue if they had (g − r) colors 0.2 m below the linear locus in the CM relation. We used the spectral energy distribution of a typical elliptical galaxy (Coleman et al. 1980) to derive k(z) corrections for our sample, and also corrected the data for extinction using the maps of Schlegel et al. (1998). Because the main purpose of this work is to study the evolution of galaxy populations, fainter galaxies play a crucial role. Whereas Butcher & Oemler were limited to brighter galaxies by their photographic data, our CCD sample allows us to probe significantly deeper, selecting galaxies with r magnitude between M * − 1 and M * + 2, inside a region of radius 0.7 Mpc around the cluster. This fixed linear size was chosen because our CCD images cover a field of radius ∼ 0.5 − 4.0 Mpc at z = 0.03 − 0.38, and we want to study the same physical region and the same galaxy luminosity range for the entire sample. We assume a cosmology with H • = 67 Km s −1 Mpc −1 , and q • = 0.1, in which M * r = −20.16 (Lin et al. 1996). Most (61%) of the clusters in our sample are at z = 0.1−0.2, so that the 0.7 Mpc central region, and the entire luminosity range between M * − 1 and M * + 2 can be observed. Clusters at higher redshifts are limited at brighter absolute magnitudes and a correction was applied in order to compare the fraction of blue galaxies in these clusters with the rest of our sample. The lower redshift clusters suffer from the opposite problem, since the brighter galaxies will be saturated, and we are also limited by the field of view. Details about these corrections can be found in MdC00, the only difference being that in the present work we adopt a more conservative brighter limiting magnitude of M * + 2 instead of M * + 3. The blue and total counts contain a mix of cluster members and background galaxies. Background variations place a fundamental limit on the accuracy of f B measurements: no matter how well one determines the mean background from a number of control fields, the background estimate for any particular cluster is no more accurate than the scatter in the control fields. For each cluster, we measured the effect of this scatter by computing f B using background corrections from the 22 individual control fields scaled to the area of the CCD and the CM relation appropriate to that cluster. The final f B is given by the median and the rms is based on the central two quartiles of the distribution. Simple propagation of errors through the equation f B ≡ n blue,cluster −n blue,background n total,cluster −n total,background would give an overestimate because n blue,background and n total,background are correlated. The usual assumption that the error in f B is mainly due to Poisson statistics in the backgroundcorrected cluster counts does not properly take into account the background variations and errors thus derived are about three times smaller than ours when applied to the same data. The final fractions of blue galaxies for the entire sample are shown in the upper panel of Figure 1. For those clusters with multiple observations, we have used the observation with smaller σ fB . Clusters with spectroscopic and photometric redshift measurements are indicated by solid and open circles respectively. The individual error bars are not presented to avoid confusion in the plot, but the median σ fB is 0.071. The lower panel presents only clusters that have: (1) σ fB < 0.071, (2) spectroscopic redshift measurements, and (3) a well-defined CM relation. The 1-σ fB errors are indicated for these clusters. In both panels, the solid lines indicates a linear fit derived from the clusters with z ≤ 0.25 (f B = (1.24 ± 0.07)z − 0.01, with an rms scatter of 0.1 for the entire sample shown in the upper panel, and f B = (1.34 ± 0.11)z − 0.03, with an rms of 0.07 for the sample shown in the lower panel). The fits were done with the GaussFit program (Jefferys et al. 1988) taking into account the measurement errors in f B . The dashed line indicates the rms scatter around the fit, and while the upper panel shows a larger scatter, the two derived relations are the same within the errors. A clear trend of strong evolution with redshift is seen. A χ ν test applied to the fit presented in the upper panel results in 1.36 if only f B errors are considered, and 1.13 when the uncertainties in the photometric redshifts are also taken into account. Such small χ ν is due however to large measurement errors in this sample. A higher χ ν of 1.72 is obtained for the subsample of clusters with smaller σ fB and spectroscopic redshift (lower panel of Figure 1). Richness is a natural second parameter which might cause the range in f B at given redshift. The left upper panel of Figure 2 shows the f B (z) diagram with symbol sizes scaled by N , the number of galaxies between M * − 1 and M * + 2, inside 0.7 Mpc, after background correction. Only clusters from the lower panel in Figure 1 and at redshifts between 0.1 and 0.2, where no corrections needed to be applied to compute f B , are presented in this figure. The solid line is the best-fit linear f B (z) relation for the 26 clusters shown, and the dashed lines indicate the relations obtained when the sample is subdivided in two 13 cluster samples according to richness. The slopes of the three relations are the same within the errors (0.86 ± 0.23 for the entire sample, 0.96 ± 0.26 for the subsample of richest clusters, and 0.86 ± 0.56 for the poorest ones). Although the uncertainties are large, the rate of evolution is approximately constant for all richnesses, and there is clearly an effect, with richer clusters tending to lower f B . The suggestion by Andreon & Ettori (1999) that the observed increase of f B with redshift is caused by missing poor clusters at higher redshift is therefore no longer tenable, because any such selection effects would serve to decrease the redshift evolution of f B . The evolution in f B is real, and takes place in all richness classes. To ensure that richness and redshift are not correlated in this subsample, we plot N vs. z in the right upper panel of Figure 2. The best-linear fit (N = −(38.2 ± 106.2)z + (89.5 ± 14.5)) is indicated by the solid line, and the dashed lines represent adding/subtracting 1-σ uncertainties to its coefficients. Richness and redshift are uncorrelated in this sample. To gauge the richness importance, we investigate f B (z, N ) relations with various richness dependences (N −1 , N −3/2 , N −2 , and N −3 ), and found that N −2 correlates slightly better with f B . The final best-fit relation for the data, when both redshift and richness are considered, and taking into account the errors in f B and N , is f B = (1.03 ± 0.25)z + (388.3 ± 111.4)N −2 − 0.04, with χ ν = 0.99. The F test indicates with > 99% confidence (F χν = 23.8, for 25 to 24 degrees of freedom) that richness is responsible for the improvement observed in χ ν . Richness is therefore extremely important in determining f B and it is in fact enough to account for the scatter observed in a simple f B (z) relation. In the lower left panel of Figure 2 we plot f B vs. z + 378.6N −2 , which represent an edge-on view of the best-fit plane f B (z, N ), and in the lower right panel we present the redshift dependence of a richness-corrected-f B : the redshift evolution is clear. Although all 26 clusters shown in Figure 2 were used to compute the relations indicated by solid lines in the figure, the two clusters represented by open circles (Abell 520, and Abell 1081) seem to deviate from the trend. These clusters also have the largest σ N (derived from the scatter in the 22 control regions), and if excluded from the fitting, the relation changes to f B = 1.13z + 480.56N −2 − 0.06 (χ ν = 0.95), indicating a slightly stronger redshift dependence. Finally, we used 11 clusters with ROSAT X-ray luminosities to check for correlations with f B , but this small sample did not allow any conclusions. There is a possible trend of increasing L X with redshift in the sense found by Andreon & Ettori (1999) in BO's original sample. However, there is no clear trend of f B with L X , arguing against any contamination of the f B (z) relation by selection effects. A comparison of the X-ray luminosities of these clusters with their f B could provide new insight to the BO effect, and we are in the process of obtaining X-ray fluxes and upper limits from the RASS (Rosat All Sky Survey) for all of these clusters, as well as a larger sample of new clusters generated from DPOSS-II. We stress that these results are not to be directly compared with fractions of blue galaxies as originally defined in BO84. The most important reason for differences is the magnitude range used to calculate f B . The number of blue galaxies increases at fainter magnitudes and it is therefore natural that our f B measurements are in general higher. Also, the usage of a rigid physical scale for all clusters, instead of regions determined individually for each cluster according to its density profile, should yield different f B estimates. When f B is computed using the same absolute magnitude range used by BO84, we find signs of evolution that are stronger than originally suggested in their work, and which are consistent with recent works by Rakos & Schombert (1995) and MdC00. Limiting our sample at brighter absolute magnitudes however increases the σ fB (and σ N ) because of the smaller galaxy count statistics, and probes a smaller fraction of the cluster population. SUMMARY AND CONCLUSIONS We compute f B for 295 randomly selected Abell (1989) clusters, including galaxies as faint as M * + 2 to sample a larger range of the luminosity function and provide better statistics in each cluster. The resulting f B shows a stronger trend with redshift than did f B as originally defined by BO84, consistent with the idea that the BO effect is stronger among the late-type spirals and irregulars which dominate the galaxy populations at intermediate and lower luminosities. This is the first sample large enough to allow the study of the f B variations at a given redshift. A χ ν test applied to a simple linear f B (z) relation shows that the scatter in f B at a given redshift is consistently larger than the measurement error, indicating a real cluster-to-cluster variation. We investigate the richness dependence of the Butcher-Oemler effect, and find a strong correlation between f B and galaxy counts in the sense that poorer clusters tend to have larger f B than richer clusters at the same redshift. The inclusion of poor clusters tends therefore to increase the slope of the f B (z) relation, and is another factor (to-gether with the inclusion of fainter galaxies) responsible for the stronger evolution observed in this sample when compared to previous works based mostly on rich clusters. We show that the fraction of blue galaxies in a cluster can be completely determined by its redshift and richness (within the measurement errors). The evolution in f B with redshift is real, and occurs at approximately the same rate for clusters of all richnesses. We thank D. Wittman, J.A. Tyson, A. Dressler, S. An-dreon, and the anonymous referee for very helpful comments and suggestions which helped to improve the paper. We also thank the Palomar TAC and Directors for generous time allocations for the DPOSS calibration effort, and numerous past and present Caltech undergraduates who assisted in the taking of the data utilized in this paper. RRG was supported in part by an NSF Fellowship and NASA GSRP NGT5-50215. The DPOSS cataloging and calibration effort was supported by a grant from the Norris Foundation. Redshift Fig. 1 . 1-Fraction of blue galaxies in the magnitude range between M * − 1 and M * + 2, and within 0.7 Mpc from the center of the cluster. The entire 295-cluster sample is shown in the upper panel, and only clusters with (1) σ fB < 0.071, (2) spectroscopic redshift measurements, and (3) well-defined CM relation are presented in the lower panel. In each panel the solid line indicates a linear f B (z) fit derived from the clusters with z ≤ 0.25, and the dashed line represents the rms scatter of the clusters around the fit. Fig. 2 . 2-Subsample of clusters with 0.1 ≤ z ≤ 0.2 from the lower panel in Figure 1. The solid line in each panel represents the best-fit to the entire sample. Left upper panel: f B (z) diagram with marker sizes scaled by N (galaxy counts). 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[ "Neutrino-Nucleus Reactions and Muon Capture in 12 C", "Neutrino-Nucleus Reactions and Muon Capture in 12 C" ]	[ "F Krmpotić \nInstituto de Física\nUniversidade de São Paulo\n05315-970São Paulo-SPBrazil\n\nInstituto de Física La Plata\nCONICET\n1900La PlataArgentina\n\nFacultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\n1900 La PlataArgentina,\n", "A Samana \nInstituto de Física\nUniversidade de São Paulo\n05315-970São Paulo-SPBrazil\n", "A Mariano \nInstituto de Física La Plata\nCONICET\n1900La PlataArgentina\n\nDepartamento de Física\nUniversidad Nacional de La Plata\n1900La PlataC. C. 67Argentina\n" ]	[ "Instituto de Física\nUniversidade de São Paulo\n05315-970São Paulo-SPBrazil", "Instituto de Física La Plata\nCONICET\n1900La PlataArgentina", "Facultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\n1900 La PlataArgentina,", "Instituto de Física\nUniversidade de São Paulo\n05315-970São Paulo-SPBrazil", "Instituto de Física La Plata\nCONICET\n1900La PlataArgentina", "Departamento de Física\nUniversidad Nacional de La Plata\n1900La PlataC. C. 67Argentina" ]	[]	The neutrino-nucleus cross section and the muon capture rate are discussed within a simple formalism which facilitates the nuclear structure calculations. The corresponding formulae only depend on four types of nuclear matrix elements, which are currently used in the nuclear beta decay. We have also considered the non-locality effects arising from the velocity-dependent terms in the hadronic current. We show that for both observables in 12 C the higher order relativistic corrections are of the order of ∼ 5% only, and therefore do not play a significant role. As nuclear model framework we use the projected QRPA (PQRPA) and show that the number projection plays a crucial role in removing the degeneracy between the proton-neutron two quasiparticle states at the level of the mean field. Comparison is done with both the experimental data and the previous shell model calculations. Possible consequences of the present study on the determination of the ν µ → ν e neutrino oscillation probability are briefly addressed.The weak interaction formalism most frequently employed in the literature is that of Donnelly and Walecka[6,7], where the nuclear form factors are classified as Coulomb (M), longitudinal (L), transverse electric (T el ) and transverse magnetic (T mag ), in close analogy with the electromagnetic transitions. They in turn depend on seven nuclear matrix elements, denoted as: , in studying neutrino induced reactions[8,9]it is sometimes preferred to employ the formulation done by Kuramoto et al.[10], mainly because of its simplicity. The later formalism does not include the velocity dependent terms in the hadronic current, and therefore the reaction cross section only depends on three nuclear form factors, denoted as \| f \|1\|i \| 2 , \| f \|σ\|i \| 2 and Λ. What is more, From the Wigner-Eckart theorem we also get:	10.1103/physrevc.71.044319	[ "https://arxiv.org/pdf/nucl-th/0410086v2.pdf" ]	118,985,081	nucl-th/0410086	146c1577443facf49918874ff836ac3945d0915c	Neutrino-Nucleus Reactions and Muon Capture in 12 C 28 Feb 2005 F Krmpotić Instituto de Física Universidade de São Paulo 05315-970São Paulo-SPBrazil Instituto de Física La Plata CONICET 1900La PlataArgentina Facultad de Ciencias Astronómicas y Geofísicas Universidad Nacional de La Plata 1900 La PlataArgentina, A Samana Instituto de Física Universidade de São Paulo 05315-970São Paulo-SPBrazil A Mariano Instituto de Física La Plata CONICET 1900La PlataArgentina Departamento de Física Universidad Nacional de La Plata 1900La PlataC. C. 67Argentina Neutrino-Nucleus Reactions and Muon Capture in 12 C 28 Feb 2005number: 2340-s2340Bw2340Hc2530Pt2160Jz Keywords: neutrino-nucleus cross sectionmuon captureprojected QRPA The neutrino-nucleus cross section and the muon capture rate are discussed within a simple formalism which facilitates the nuclear structure calculations. The corresponding formulae only depend on four types of nuclear matrix elements, which are currently used in the nuclear beta decay. We have also considered the non-locality effects arising from the velocity-dependent terms in the hadronic current. We show that for both observables in 12 C the higher order relativistic corrections are of the order of ∼ 5% only, and therefore do not play a significant role. As nuclear model framework we use the projected QRPA (PQRPA) and show that the number projection plays a crucial role in removing the degeneracy between the proton-neutron two quasiparticle states at the level of the mean field. Comparison is done with both the experimental data and the previous shell model calculations. Possible consequences of the present study on the determination of the ν µ → ν e neutrino oscillation probability are briefly addressed.The weak interaction formalism most frequently employed in the literature is that of Donnelly and Walecka[6,7], where the nuclear form factors are classified as Coulomb (M), longitudinal (L), transverse electric (T el ) and transverse magnetic (T mag ), in close analogy with the electromagnetic transitions. They in turn depend on seven nuclear matrix elements, denoted as: , in studying neutrino induced reactions[8,9]it is sometimes preferred to employ the formulation done by Kuramoto et al.[10], mainly because of its simplicity. The later formalism does not include the velocity dependent terms in the hadronic current, and therefore the reaction cross section only depends on three nuclear form factors, denoted as \| f \|1\|i \| 2 , \| f \|σ\|i \| 2 and Λ. What is more, From the Wigner-Eckart theorem we also get: I. INTRODUCTION The semileptonic weak interactions with nuclei include a rich variety of processes, such as the neutrino (antineutrino) scattering, charged lepton capture, e ± decays, etc, and we have at our disposal the results of more than a half-century of beautiful experimental and theoretical work. At present their study is mainly aimed to inquire about possible exotic properties of the neutrino associated with its oscillations and massiveness by means of exclusive and inclusive scattering processes on nuclei, which are often used as neutrino detectors. An example is given by the recent experiments performed by both the LSND [1,2] and the KARMEN [3] Collaborations, looking for ν µ → ν e andν µ →ν e oscillations with neutrinos produced by accelerators. When the ν e come from decay-at-rest (DAR) of µ + the flux contains neutrinos with a maximum energy energy of 50 MeV and can be detected through both exclusive and inclusive ν e + 12 C → 12 N +e − reactions [4]. In the case of ν µ coming from the decay-in-flight (DIF) of π + , the neutrino flux extends over the range (0, 300) MeV, and the ν µ → ν e appearance mode is looked for experimentally through the reaction ν µ + 12 C → 12 N + µ − , which also has been measured [5]. On the other hand we don't have at our disposal experimental information on the ν e -reaction in the DIF energy range, which is necessary in the evaluation of the oscillation probabilities. Therefore, it is imperative to develop nuclear models, capable of reproducing the ν e -DAR and ν µ -DIF data, to be used to predict reliable values for the ν e + 12 C → 12 N + e − cross section in the DIF energy range, and to calibrate forthwith the ν µ → ν e appearance probability. Needless to say, in addition and for consistence, the implemented model should also describe properly the well-known 12 N → 12 C + ν e + e + β + -decay and the µ − + 12 C → ν µ + 12 B muon-capture modes. the formalism of Kuramoto et al. [10] does not include the muon capture rates. Therefore, to describe simultaneously the neutrino-nucleus reactions and µ-capture processes it is necessary to resort to additional theoretical developments, such as those of Luyten et al. [11] and Auerbach and Klein [12], where one uses the matrix elements M 2 V , M 2 A and M 2 P that are related to the former ones in a non-trivial way. Quite recently, we have carried out a multipole expansion of the V − A hadronic current, similar to the one used by Barbero et al. [13] for the neutrinoless double beta decay, expressing all above mentioned observables in terms of the vector (V ) and axial vector (A) nuclear form factors M V and M M A , with M = −1, 0, +1 [14]. Such a classification is closely related to the V − A structure of the weak current and to the currently used nuclear β-decay formalisms, where the nuclear moments are expressed in a way that the forbiddeness of the transitions is easily recognized. This in no way means that we have developed a new theoretical framework; the final results can be found in one form or another in the literature. The main difference stems from the fact that we use the Racah algebra from the start, employing the spherical spatial coordinates instead of the cartesian ones. Concomitantly, we also express the lepton trace in spherical coordinates, as done for instance in the book of Supek [15]. Here we present a few more details than in Ref. [14] on the procedure which we have followed and also consider the first order nonlocal corrections, which give rise to the additional matrix elements M M V ′ and M A ′ . In Ref. [14] we have analyzed the inconveniences that appear in applying RPA-like models to describe the nuclear structure of the { 12 B, 12 C, 12 N} triad. We have established that the projected quasiparticle RPA (PQRPA) was the proper approach to treat both the short range pairing and the long range RPA correlations. More details are given in the present work, putting special emphasis on the differences between the projected and the usual QRPA approximations. We also compare our PQRPA cross sections with recent shell model (SM) calculations [16,17], and analyze the reliability of the calculated energy dependence of the ν e + 12 C → 12 N + e − cross-section in the DIF energy region to be used in the evaluation of the oscillation probabilities. The work will be organized as follows. In Section II we explain our multipole decomposition of the weak current and present the corresponding formula for the neutrino-nucleus cross section and the µ-capture. In Section III we briefly overview the projected BCS (PBCS) and the PQRPA equations. The main reason for the last is to emphasize the relationship with the usual QRPA approximation, which is difficult to find in the existing literature. Finally, in Section IV we show the numerical results which come from the PQRPA model and compare them with the most recent shell model calculations. II. FORMALISM FOR THE WEAK INTERACTING PROCESSES The weak Hamiltonian is expressed in the form [6,7,13,18] H W (r) = G √ 2 J α l α e −ir·k ,(2.1) where G = (3.04545 ± 0.00006)×10 −12 is the Fermi coupling constant (in natural units), J α = iγ 4 g V γ α − g M 2M σ αβ k β + g A γ α γ 5 + i g P m ℓ k α γ 5 ≡ (J, iJ ∅ ) (2.2) is the hadronic current operator [40], and l α = −iu s ℓ (p, E ℓ )γ α (1 + γ 5 )u sν (q, E ν ) ≡ (l, il ∅ ) (2.3) is the plane wave approximation for the matrix element of the leptonic current in the case of neutrino reactions. Here α, β = 1, 2, 3, 4, and Walecka's notation [7] with the Euclidean metric for quadrivectors is employed, i.e. x = {x, x 4 = ix ∅ }. The only difference is that we substitute his indices (0, 3) by our indices (∅, 0), which means that we use the index ∅ for the temporal component and the index 0 for the third spherical component. The quantity k = P i − P f ≡ {k, ik ∅ } (2.4) is the momentum transfer (P i and P f are momenta of the initial and final nucleon (nucleus)), M is the nucleon mass, m ℓ is the mass of the charged lepton, and g V , g A , g M and g P are, respectively, the vector, axial-vector, weak-magnetism and pseudoscalar effective dimensionless coupling constants. Their numerical values are g V = 1; g A = 1.26; g M = κ p − κ n = 3.70; g P = g A 2Mm ℓ k 2 + m 2 π . (2.5) These estimates for g M and g P come from the conserved vector current (CVC) hypothesis, and from the partially conserved axial vector current (PCAC) hypothesis, respectively. In the numerical calculation we will use an effective axial-vector coupling g A = 1 [19,20,21]. The finite nuclear size (FNS) effect is incorporated via the dipole form factor with a cutoff Λ = 850 MeV, i.e. as: g → g Λ 2 Λ 2 + k 2 2 . To use (2.1) with the non-relativistic nuclear wave functions, the Foldy-Wouthuysen transformation has to be performed on the hadronic current (2.2). When the velocity dependent terms are included this yields: J ∅ = g V + (g A + g P1 )σ ·k − g A σ · v, J = −g A σ − ig W σ ×k − g Vk + g P2 (σ ·k)k + g V v, (2.6) where v ≡ −i∇/M is the velocity operator, acting on the nuclear wave functions. The following short notation has been introduced: g V = g V κ 2M ; g A = g A κ 2M ; g W = (g V + g M ) κ 2M , g P1 = g P κ 2M k ∅ m ℓ ; g P2 = g P κ 2M κ m ℓ , (2.7) where κ ≡ \|k\|. In performing the multipole expansion of the nuclear operator O α ≡ (O, iO ∅ ) = J α e −ik·r ,(2.8) the momentum k is taken to be along the z axis, i.e. e −ik·r = L i −L 4π(2L + 1)j L (κr)Y L0 (r),(2.9) and one gets O ∅ = J i −J 4π(2J + 1)j J (κr)Y J0 (r)J ∅ , O M = JL i −L F JLM 4π(2J + 1)j L (κr) [Y L (r) ⊗ J] JM . (2.10) The geometrical factors F JLM ≡ (−) J+M (2L + 1)    L 1 J 0 −M M    ,(2.M L F JLM 0 J + 1 − J+1 1 √ 2 J+1 2J+1 which will be seldom used in our work, fulfill the sum rule L F JLM F JLM ′ = δ M M ′ ,(2.12) and their explicit values are listed in Table I. After some Racah algebra we find O α = J √ 2J + 1O α (J), (2.13) with O ∅ (J) = i −J √ 4π [g V Y J0 (κr) − g A Y J0 (κr, σ · v)] + √ 4π (g A + g P1 ) L=J±1 i −L F JL0 S JL0 (κr) (2.14) and Operator Parity O M (J) = √ 4π L i −L F JLM −g A S JLM (κr) + g V P JLM (κr) − i(−) J g V F JL0 Y JM (κr) + I i(−) L g W G JLI + g P2 F JL0 F JI0 S JIM (κr) ,(2.Y JM (κr) = j J (κr)Y JM (r) (−) J S JLM (κr) = j L (κr) [Y L (r) ⊗ σ] JM (−) L P JLM (κr) = j L (κr)[Y L (r) ⊗ v] JM (−) L+1 Y JM (κr, σ · v) = j J (κr)Y JM (r)(σ · v) (−) J+1 where we have introduced the operators listed in Table II, and the coefficients G JLI = (−) J 6(2L + 1)(2I + 1)      1 1 1 I L J         I 1 L 0 0 0    ,(2.O M (J) = √ 4π L i −L F JLM (−g A + Mg W + g P2 δ M 0 )S JLM (κr) + g V P JLM (κr) − √ 4πi −J g V δ M 0 Y J0 (κr). (2.18) For the neutrino-nucleus reaction the momentum transfer is k = p ℓ − q ν , with p ℓ ≡ {p ℓ , iE ℓ } and q ν ≡ {q ν , iE ν }, and the corresponding cross section reads σ(E ℓ , J f ) = \|p ℓ \|E ℓ 2π F (Z + 1, E ℓ ) 1 −1 d(cos θ)T σ (κ, J f ),(2.19) where F (Z + 1, E ℓ ) is the Fermi function, θ ≡q ν ·p ℓ , and T σ (κ, J f ) ≡ 1 2J i + 1 s ℓ ,sν M i ,M f \| J f M f \|H W \|J i M i \| 2 ,(2.20) with \|J i M i and \|J f M f being the nuclear initial and final state vectors. The weak hamiltonian matrix element reads J f M f \|H W \|J i M i = G √ 2 O α l α , (2.21) where O α ≡ J f M f \|O α \|J i M i ,(2.22) and l α is the leptonic current defined in (2.3). Thus T σ (κ, J f ) = G 2 2J i + 1 M i M f O α O * β L αβ ,(2.23) where, the lepton trace L αβ , when expressed in cartesian spatial coordinates, reads L αβ = 1 2 s ℓ sν l α l * β = − 1 E ℓ E ν [p α q β + q α p β − δ αβ (p · q) ± ǫ αβγδ q γ p δ ] ,(2.24) the positive (negative) sign standing for neutrino (antineutrino) scattering. It is convenient to follow Ref. [15] and express the spatial parts of O and L in spherical coordinates (M, M ′ = 0, −1, 1). In this way one might write T σ (κ, J f ) = G 2 2J i + 1 M f M i \|O ∅ \| 2 L ∅∅ + M M ′ O M O * M ′ L M M ′ − 2ℜ O * ∅ M (−) M O −M L ∅M ,(2.25) with [15] L ∅∅ ≡ L 44 = 1 + p · q E ℓ E ν , (2.26) L ∅M ≡ −iL 4M = 1 E ℓ E ν [E ℓ q M + E ν p M ∓ i(q × p) M ] ,(2.27)L M M ′ = δ M M ′ + 1 E ℓ E ν (q * M p M ′ + q M ′ p * M − δ M M ′ p · q) ± √ 6(−) M    1 1 1 −M M ′ M − M ′    q * M −M ′ E ν − p * M −M ′ E ℓ . (2.28) and O M = J (−) J f −M f √ 2J + 1    J f J J i −M f M M i    J f \|\|O M (J)\|\|J i . (2.30) Using now the orthogonality condition M i M f    J f J J i −M f M M i       J f J ′ J i −M f M ′ M i    = 1 (2J + 1) δ JJ ′ δ M M ′ (2.31) one obtains T σ (κ, J f ) = G 2 2J i + 1 J \| J f \|\|O ∅ (J)\|\|J i \| 2 L ∅∅ + M \| J f \|\|O M (J)\|\|J i \| 2 L M M −2ℜ ( J f \|\|O ∅ (J)\|\|J i * J f \|\|O 0 (J)\|\|J i L ∅0 )] ,(2.32) where, from (2.14) and (2.18), J f \|\|O ∅ (J)\|\|J i = 2J i + 1 g V M V (J) − g A M A ′ (J) + (g A + g P1 ) M 0 A (J) J f \|\|O M (J)\|\|J i = 2J i + 1 (−g A + Mg W + g P2 δ M 0 )M M A (J) + g V M M V ′ (J) − g V δ M 0 M V (J) ,(2.33) with the nuclear matrix elements defined as M V (J) = i −J 4π 2J i + 1 J f \|\|Y J (κr)\|\|J i , M M A (J) = 4π 2J i + 1 L i −L F JLM J f \|\|S JL (κr)\|\|J i , (2.34) M A ′ (J) = i −J 4π 2J i + 1 J f \|\|Y J (κr, σ · v)\|\|J i , M M V ′ (J) = 4π 2J i + 1 L i −L F JLM J f \|\|P JL (κr)\|\|J i . The explicit expressions for L ∅M and L M M ′ that appear in (2.32) are [14]: L ∅0 = q 0 E ν + p 0 E ℓ , L 00 = 1 + 2q 0 p 0 − p · q E ℓ E ν , L ±1,±1 = 1 − q 0 p 0 E ℓ E ν ± q 0 E ν − p 0 E ℓ , (2.35) with q 0 =k · q = E ν (\|p\| cos θ − E ν ) κ , p 0 =k · p = \|p\|(\|p\| − E ν cos θ) κ . (2.36) Finally, the transition amplitude can be cast in the form: T σ (κ, J f ) = G 2 J L ∅∅ g 2 V \|M V (J)\| 2 + (g A + g P 1 ) M 0 A (J) − g A M A ′ (J) 2 +L 00 ℜ g V M V (J) − 2g V M 0 V ′ (J) g V M * V (J) + g 2 P2 − 2g A g P2 \|M 0 A (J)\| 2 + M =0,±1 L M M (g A − Mg W ) M M A (J) − g V M M V ′ (J) 2 +2L ∅0 ℜ g V g V M V (J) − g V M 0 V ′ (J) M * V (J) + (g A − g P2 ) (g A + g P1 )M 0 A (J) − g A M A ′ (J) M 0 * A (J) . (2.37) The muon capture transition amplitude T Λ (J f ) can be derived from the result (2.32) for the neutrino-nucleus reaction amplitude, by keeping in mind that: i) the roles of p and q are interchanged within the matrix element of the leptonic current, which brings in a minus sign in the last term of L ±1,±1 , ii) the momentum transfer turns out to be k = q − p, and therefore the signs of the right-hand sides of (2.36) have to be changed, and iii) the threshold values (p → 0 : k → q, k ∅ → E ν − m ℓ ) must be used for the lepton traces. All this yields: L ∅∅ = L 00 = L ∅0 = 1, L ±1,±1 = 1 ∓ 1. (2.38) One should also remember that instead of summing over the initial lepton spins s ℓ , as done in (2.20), one has now to average over the same quantum number, getting Λ(J f ) = E 2 ν 2π \|φ 1S \| 2 T Λ (J f ), (2.39) where φ 1S is the muonic bound state wave function evaluated at the origin, and E ν = m µ − (M n − M p ) − E µ B − E f + E i , where E µ B is the binding energy of the muon in the 1S orbit. In view of (2.39) the transition amplitude reads: T Λ (J f ) = G 2 2J i + 1 J \| J f \|\|O ∅ (J) − O 0 (J)\|\|J i \| 2 + 2\| J f \|\|O −1 (J)\|\|J i \| 2 ,(2.40) which, based on the parity considerations, can be expressed as T Λ (J f ) = G 2 J (g V + g V )M V (J) − g V M 0 V ′ (J) 2 + (g A + g A − g P )M 0 A (J) − g A M A ′ (J) 2 + 2 (g A + g W )M −1 A (J) − g V M −1 V ′ (J) 2 ,(2.41) where g P = g P2 − g P1 . In the case of muon capture, it is convenient to rewrite the effective coupling constants as [11] g V = g V E ν 2M ; g A = g A E ν 2M ; g W = (g V + g M ) E ν 2M ; g P1 = g P E ν 2M . (2.42) Lastly, we mention that the B-values for the Gamow-Teller (GT) beta transitions are defined and related to the ft-values as [22]: \|g A J f \|\|σ\|\|J i \| 2 2J i + 1 ≡ B(GT ) = 6146 f t sec. (2.43) Let us now compare our matrix elements with those currently used in the literature. First, in the Walecka's notation (see Eqs. (45.13) in Ref. [7]) one has O ∅ (J) = M J0 , O M (J) =      L J0 , for M = 0 − 1 √ 2 MT mag JM + T el JM , for M = ±1 , (2.44) where the meaning of M J0 and L J0 is self evident, while T el JM = g W S JJM − i J √ 2 L=J±1 i −L F JLM g A S JLM − g V P JLM , T mag JM = −g A S JJM + g V P JJM + i J √ 2g W L=J±1 i −L F JLM S JLM . (2.45) The matrix elements defined by Donelly [6] are related to ours as: M M J (κr) = Y JM (κr), ∆ M J (κr) = iM κ P JJM (κr), ∆ ′ M J (κr) = i J √ 2 M κ L=J±1 i −L F JLM P LJM (κr), (2.46) Σ M J (κr) = S JJM (κr), Σ ′ M J (κr) = i J−1 √ 2 L=J±1 i −L F JLM S JLM (κr), Σ ′′ M J (κr) = i J−1 L=J±1 i −L F JL0 S JLM (κr) , Ω M J (κr) = iM κ Y JM (κr, σ · v). The relationship between our formalism and those from Refs. [10,11] can be obtained from the formula    L 1 J 0 M −M       L ′ 1 J 0 M −M    = δ LL ′ 3(2L + 1) − M 3 2    L L ′ 1 0 0 0         L 1 J 1 L ′ 1      + (−) J+1 5 6 (3M 2 − 2)    L L ′ 2 0 0 0         L 1 J 1 L ′ 2      . (2.47) For the matrix elements of Kuramoto et al. [10] we get \| f \|1\|i \| 2 = J \|M V (J)\| 2 , f \|σ\|i \| 2 = J M =0,±1 \|M M A (J)\| 2 , Λ = 1 3 J \|M 0 A (J)\| 2 − \|M 1 A (J)\| 2 ; (2.48) and for those of Luyten et al. [11] M 2 V = E ν m µ 2 J \|M V (J)\| 2 , M 2 A = E ν m µ 2 J M =0,±1 \|M M A (J)\| 2 , M 2 P = E ν m µ 2 J \|M 0 A (J)\| 2 . (2.49) III. PROJECTED QRPA FORMALISM We have shown in Ref. [14] that to account for the weak decay observables in a light N = Z nucleus in the framework of the RPA, besides including the BCS correlations, it is imperative to perform the particle number projection. It should be remembered that in heavy nuclei the neutron excess is usually large, which makes the projection procedure less important that in light nuclei [23]. In this section we give a more detailed overview of the projected BCS (PBCS) and projected QRPA (PQRPA) approximations. When the the excited states \|J f in the final (Z ± 1, N ∓ 1) nuclei are described within the PQRPA, the transition amplitudes for the multipole charge-exchange operators Y J , etc, listed in Table II, read J f , Z + µ, N − µ\|\|Y J \|\|0 + = 1 (I Z I N ) 1/2 pn Λ µ (pnJ) (I Z−1+µ (p)I N −1+µ (n)) 1/2 X * µ (pnJ f ) + Λ −µ (pnJ) (I Z−1−µ (p)I N −1−µ (n)) 1/2 Y * µ (pnJ f ) ,(3.1) with the one-body matrix elements given by Λ µ (pnJ) = − p\|\|Y J \|\|n √ 2J + 1      u p v n , for µ = +1 u n v p , for µ = −1 , (3.2) where I K (k 1 k 2 · ·k n ) = 1 2πi dz z K+1 σ k 1 · · · σ kn k (u 2 k + z 2 v 2 k ) j k +1/2 ; σ −1 k = u 2 k + z 2 k v 2 k , (3.3) are the PBCS number projection integrals. The PBCS gap equations are 2ē k u k v k − ∆ k (u 2 k − v 2 k ) = 0, (3.4) where ∆ k = − 1 2 k ′ (2j k ′ + 1) 1/2 (2j k + 1) 1/2 u k ′ v k ′ G(kkk ′ k ′ ; 0) I Z−2 (kk ′ ) I Z (3.5) are the pairing gaps, and e k = e k I Z−2 (k) I Z + k ′ (2j k ′ + 1) 1/2 (2j k + 1) 1/2 v 2 k ′ F(kkk ′ k ′ ; 0) I Z−4 (kk ′ ) I Z + ∆e k ,(3.6) are the dressed single-particle energies. The PBCS correction term ∆e k can be found in Ref. [23], and F and G stand for the usual particle-hole (ph) and particle-particle (pp) matrix elements, respectively. The forward, X µ , and backward, Y µ , PQRPA amplitudes are obtained by solving the RPA equations    A µ B −B * −A * −µ       X µ Y µ    = ω µ    X µ Y µ    ,(3.7) with the PQRPA matrices defined as: A µ (pn, p ′ n ′ ; J) = (ε Z−1+µ p + ε N −1−µ n )δ pn,p ′ n ′ + N µ (pn) −1/2 N µ (p ′ n ′ ) −1/2 × [u p v n u p ′ v n ′ I Z−1+µ (pp ′ )I N −3+µ (nn ′ ) + v p u n v p ′ u n ′ I Z−3+µ (pp ′ )I N −1+µ (nn ′ )]F(pn, pn; J) + [u p u n u p ′ u n ′ I Z−1+µ (pp ′ )I N −1+µ (nn ′ ) + v p v n v p ′ v n ′ I Z−3+µ (pp ′ )I N −3+µ (nn ′ )]G(pn, pn; J) , B(pn, p ′ n ′ ; J) = N µ (pn) −1/2 N −µ (p ′ n ′ ) −1/2 I Z−2 (pp ′ )I N −2 (nn ′ ) × [(v p u n u p ′ v n ′ + u p v n v p ′ u n ′ )F(pn, pn; J) + (u p u n v p ′ v n ′ + v p v n u p ′ u n ′ )G(pn, pn; J)], (3.8) where N µ (pn) = I Z−1+µ (p)I N −1+µ (n), (3.9) are the norms, ε K k = R K 0 (k) + R K 11 (kk) I K (k) − R K 0 I K (3.10) are the projected quasiparticle energies, and the quantities R K are defined as [23] R K 0 (k) = k 1 (2j k 1 + 1)v 2 k 1 e k 1 I K−2 (kk 1 ) + 1 4 k 1 k 2 (2j k 1 + 1) 1/2 (2j k 2 + 1) 1/2 × v 2 k 1 v 2 k 2 F(k 1 k 1 k 2 k 2 ; 0)I K−4 (k 1 k 2 k) + u k 1 v k 1 u k 2 v k 2 G(k 1 k 1 k 2 k 2 ; 0)I K−2 (k 1 k 2 k) , R K 11 (kk) = e k [u 2 k I k 1 (kk) − v 2 k I K−2 (kk)] + k 1 (2j k 1 + 1) 1/2 (2j k + 1) 1/2 × v 2 k 1 F(k 1 k 1 kk; 0)[u 2 k I K−2 (k 1 kk) − v 2 k I K−4 (k 1 kk)] − u k 1 v k 1 u k v k G(k 1 k 1 kk; 0)I K−2 (k 1 kk) . (3.11) It is worth to note that the PQRPA formalism is valid not only for the particle-hole chargeexchange excitations (Z ± 1, N ∓ 1), but also for the charge-exchange pairing-vibrations (Z ± 1, N ± 1). In the later case one has simply to do the replacement µ → −µ in the neutron sector. The usual gap equations are obtained from Eqs. (3.4)-(3.6) by: 1. Making the replacement e k → e k − λ k , with λ k being the chemical potential, and taking the limit I K → 1. That is, the Eq. (3.4) remains as it is, but instead of (3.5) and (3.6) one has now ∆ k = − 1 2 k ′ (2j k ′ + 1) 1/2 (2j k + 1) 1/2 u k ′ v k ′ G(kkk ′ k ′ ; 0),(3.12)andē k = e k − λ k + k ′ (2j k ′ + 1) 1/2 (2j k + 1) 1/2 v 2 k ′ F(kkk ′ k ′ ; 0). (3.13) 2. Imposing the subsidiary conditions Z = jp (2j p + 1) 2 v 2 jp , N = jn (2j n + 1) 2 v 2 jn ,(3.14) as the number of particles is not any more a good quantum number. Finally, the plain QRPA equations are recovered from (3.7) and (3.8) by: i) dropping the index µ and taking the limit I K → 1, and ii) substituting the unperturbed PBCS energies by the BCS energies relative to the Fermi level, i.e. by E (±) k = ±E k + λ k ,(3.15) where E k = (ē 2 k + ∆ 2 k ) 1/2 are the usual BCS quasiparticle energies. In this way the the unperturbed energies in (3.8) are replaced as [41] ε Z−1+µ jp + ε N −1−µ jn → E jp + E jn + µ(λ p − λ n ). (3.16) IV. NUMERICAL RESULTS AND DISCUSSION In this section, our theoretical results within the PQRPA are confronted with the experimental data for the µ − + 12 B → 12 C + ν µ muon capture rates, as well as for the neutrino cross sections involving the DAR reaction ν e + 12 C → 12 N + e − and DIF reaction ν µ + 12 C → 12 N + µ − . We also exhibit our predictions for the ν e + 12 C → 12 N + e − differential cross section for ν e -energies in the DIF energy range. At variance with our previous work, we consider here also the velocity dependent matrix elements M A ′ (J) and M M V ′ (J), defined in (2.34). The calculations shown here were done in the same way as in the previous work [14]. That is, for the residual interaction we adopted the delta force, V = −4π (v s P s + v t P t ) δ(r),(4.1) which has been used extensively in the literature [24,25,26] for describing the single and double beta decays. The configuration space includes the single-particle orbitals with nl = (1s, 1p, 1d, 2s, 1f, 2p) for both protons and neutrons. in the fitting procedure, and e j are the resulting s.p.e. The underlined quasiparticle energies correspond to single-hole excitations (for j h = 1s 1/2 , 1p 3/2 ) and to single-particle excitations (for j p = 1p 1/2 , 1d 5/2 , 2s 1/2 , 1d 3/2 , 1f 7/2 , 2p 3/2 , 2p 1/2,1f 5/2 ). The non-underlined energies are mostly two hole-one par- Most of the bare single-particle energies (s.p.e.) e j , as well as the value of the singlet strength within the pairing channel (v pair s ), were fixed from the experimental energies E exp j of the odd-mass nuclei 11 C, 11 B, 13 C and 13 N. That is, in the BCS case: 1) we assume that the ground states in 11 C and 11 B are pure quasi-hole excitations E (−) j h , with j h = 1p 3/2 , and that the lowest observed 1/2 − , 5/2 + , 1/2 + , 3/2 + , 7/2 − and 3/2 − states in 13 C and 13 N are pure quasi-particle excitations E (+) jp with j p = 1p 1/2 , 1d 5/2 , 2s 1/2 , 1d 3/2 , 1f 7/2 , 2p 3/2 , and 2) the s.p.e. of these 7 states and the pairing strength, that appear in the BCS gap equation (3.4), (3.12)-(3.14), were varied in a χ 2 search in order to account for the experimental spectra E exp j [27]. In the PBCS case we proceed in the same way, i.e. we solve the Eqs. BCS P BCS Shell E exp j E (+) j E (−) j e j ε N j −ε N −2 j e j1s BCS P BCS Shell . We have considered the faraway orbitals 1s 1/2 , 2p 1/2 and 1f 5/2 as well, assuming the first one to be a pure hole state and the other two pure particle states. Their s.p.e. were taken to by that of a harmonic oscillator (HO) with standard parameterization [28]. The single-particle wave functions were also approximated with that of the HO with the length parameter b = 1.67 fm, which corresponds to the estimatehω = 45A −1/3 − 25A −2/3 MeV for the oscillator energy. The final results for neutrons are shown in Table III and those for protons in Table IV. It is worth noting that the PBCS neutron-energy ε N 1p 3/2 = −1.28 MeV nicely agrees with the experimental energy E 3/2 − Before proceeding let us remember an important issue in the description of the N ∼ = Z nuclei within the QRPA, which is more inherent to the model itself that to the occasional parameterization that might be employed. In fact, a few years ago Volpe et al. [9] have called attention to the inconveniences of applying QRPA to 12 N , since the lowest state turned out not to be the most collective one. Later on we have shown [14] that the origin of this difficulty was the degeneracy among the four lowest proton-neutron two-quasiparticle states \|1p 1/2 1p 3/2 , \|1p 3/2 1p 3/2 , \|1p 1/2 1p 1/2 and \|1p 3/2 1p 1/2 . It also has been shown in Ref. [14] that it is imperative to use the projected QRPA for a physically sound description of the weak processes among the ground states of the triad { 12 B, 12 C, 12 N }. In fact, when the Fermi level is fixed at N = Z = 6, their BCS energies are almost degenerate for all four odd-odd (Z ± 1, N ∓ 1) and (Z ± 1, N ± 1) nuclei. As illustrated in Fig. 1, this, in turn, comes from the fact that for N = Z = 6 the quasiparticle energies E 1p 1/2 and E 1p 3/2 are very close to each other. In the upper panel of Fig. 2 are shown the BCS energies (4.2), as a function of N, of the just mentioned four states in the case of nitrogen isotopes. One notices that for N = Z the degeneracy is removed yet only partially. But, as seen from the lower panel in Fig. 2, the degeneracy discussed above is totally removed when the number projection is done. Moreover, within the PBCS in the case of 12 N, for instance, we get from numerical to the shell-model and analyzing the particle-hole (ph) limits of the proton-neutron twoquasiparticle states, which are pictorially shown in Fig. 3. One sees that, while \|1p 1/2 1p 3/2 corresponds to a 1p1h state in 12 N , \|1p 3/2 1p 3/2 and \|1p 1/2 1p 1/2 correspond to 2p2h states, and \|1p 3/2 1p 1/2 to a 3p3h state in the same nucleus. Therefore one can expect that the energy ordering of these states would be given by (4.3), with ∆ = ∆ ls − ∆ pair being the energy difference between the spin-orbit splitting, ∆ ls , and the pairing energy, ∆ pair . A similar discussion is pertinent to the remaining three nuclei 10 B, 14 N, and 12 B. That is, one expects that their lowest states would be, respectively, \|1p 3/2 1p 3/2 , \|1p 1/2 1p 1/2 and calculation of the unperturbed GT strength in 12 N, given by E exp j E (+) j E (−) j e j ε Z j −ε Z−2 j e j1sE jpjn =                    E (+) jp − E (−) jn = E jp + E jn + λ p − λ n ; for 12 N , −E (−) jp + E (+) jn = E jp + E jn − λ p + λ n ; for 12 B, E (+) jp + E (+) jn = E jp + E jn + λ p + λ n ; for 14 N, −E (−) jp − E (−) jn = E jp + E jn − λ p − λ n ; for 10 B,calculations E 1p 3/2 1p 3/2 ∼ = E 1p 1/2 1p 1/2 ∼ = E 1p 1/2 1p 3/2 + ∆, E 1p 3/2 1p 1/2 ∼ = E 1p 1/2 1p 3/2 + 2∆,S GT (E) ≡ 1 π pn \|g A p\|\|σ\|\|n \| 2 η η 2 + (E − E jpjn ) 2 . (4.4) The BCS and PBCS results for 12 N, when folded with η = 1 MeV, are compared in Fig. 4. The PBCS energy ordering, given by (4.3), is accompanied by the partial shifting of the GT strength to higher energies. Therefore it can be said that within the PBCS the GT resonance is quenched even at the level of the mean field. In view of the above mentioned disadvantages of the standard BCS approach, from now on we will mainly discuss the number projection results. In our previous work [14] we have also pointed out that the values of the coupling strengths v s and v t within the particleparticle (pp) and particle-hole (ph) channels which are used in N > Z nuclei (v pp s ≡ v pair s , and v pp t > ∼ v pp s ), might not be suitable for N = Z nuclei. In fact, the best agreement with data in 12 C was obtained when the pp channel is totally switched off, i.e. v pp s ≡ v pp t = 0, and three different set of values for the ph coupling strengths [14], namely: The values of v ph s and v ph t correspond to PII. Parameterization II (PII): v ph s = 27 MeV-fm 3 , and v ph t = 64 MeV-fm 3 . These values were first used in Refs. [20,30] and later on in the QRPA calculations of 48 Ca [13,24,31]. Parameterization III (PIII): v ph s = v ph t = 45 MeV-fm 3 . With these coupling constants it is possible to reproduce fairly well the energies of the J π = 0 + 1 and 1 + 1 states in 12 B and 12 N. The results displayed in Fig. 5 suggest that the choice v pp t = 0 for the pp parameter in the S = 0, T = 1 channel could be appropriate for the description of the N = Z nuclei. They are shown as functions of the parameter t = 2v pp t v pair s (p) + v pair s (n) , together with the experimental data for: 1) the energy difference ω 1 + between the 1 + ground state in 12 N and the 0 + ground state in 12 C, 2) the B-value for the GT beta transition between these two states, and 3) the corresponding exclusive muon capture rate Λ(1 + 1 ) ≡ Λ exc . The values of v ph s and v ph t are that from the PII, but quite similar results are obtained with other two sets of parameters. Note that the ph interaction first shifts the energy ω 1 + upwards by ∼ 1.5 MeV, from its unperturbed value E 1p 3/2 1p 3/2 = 16.8 MeV. Then, when t is increased, we have an opposite attractive effect. That is, the pp interaction diminishes ω 1 + all up to t > ∼ 0.6, where the well known collapse of the QRPA approximation occurs. At variance with what happens in the case of heavy nuclei, here the values of B(GT ) and Λ(1 + 1 ) basically rise with t, and the agreement with the data is achieved only when the pp interaction is totally switched off. Quite generally the nuclear moments (2.34) also depend weakly on the S = 1, T = 0 channel parameter v pp s , for which we adopt as well the null value, just to be consistent with our election of v pp t . Results for the muon capture rates, and the neutrino (ν e , e − ) DAR, and (ν µ , µ − ) DIF reaction flux-averaged cross sections are shown, respectively, in Tables V, Tables VI and Tables VII. The flux-averaged cross section is defined as σ ℓ (J f ) = ∆ J f dE ν σ ℓ (E ν , J f )Φ ℓ (E ν ); ℓ = e, µ,(4.5) where Φ ℓ (E ν ) is the normalized neutrino flux. For electron neutrinos this flux was approximated by the Michel spectrum, and for the muon neutrinos we used that from Ref. [37]. The energy integration is carried out in the DAR interval m e + ω J f ≤ ∆ DAR Contributions of the other two operators, P JL (κr) and Y J (κr, σ · v), to the muon capture rates are small (of the order of 5%) as displayed in Table V. The only exception are the 0 − states where the relativistic operator Y 0 (κr, σ · v) dominates over the non-relativistic one S 01 (κr). We also see from Tables VI and VII that the nonlocality effects on the neutrinonucleus reactions are of the order of 1% and therefore they can be neglected. • (ν µ , µ − )-reaction (Table VII): Here also the exclusive cross section, σ exc µ , is in full agreement with the data, whereas the inclusive one, σ inc µ , is overpredicted by ∼ 20%. In the last three tables we also confront our PQRPA results with the shell model calculations performed by: a) Hayes and Towner [16], within the model spaces called by them as (iii) and (iv), and which are labeled here, respectively, as SM1 and SM2, and b) Auerbach and Brown [17] which we label as SM3. The multipole breakdown in the contributions to the cross sections from each multipole is only given for the SM1 scheme [16]. At first glance our results seem to agree fairly well with the shell model ones, particularly when the amounts of allowed and forbidden transition strengths are confronted. However, this is not true, as can be seen from a more careful analysis of the multipole structure of the transition rates. For instance, the total positive parity capture rates with the 1 + 1 state excluded, i.e. J f =1 + 1 Λ(J + f ), are equal to 5.9 and 3.6 in SM1 and SM3, respectively, while we get 16.4 (in units of 10 3 s −1 ). Note also that in shell model calculations almost all GT strength is exhausted by the ground state transition, while within the PQRPA only 35% of this strength goes into the 1 + 1 state. Another important discrepancy is in the Fermi transitions, for which we get f Λ(0 + f )= 2.86, while they obtain only f Λ(0 + f ) = 0.21 (in units of 10 3 s −1 ). One sees that our σ exc e is consistent with the SM2 and SM3 shell model calculations, while our σ inc e falls in between the calculations SM1 and SM2 and is ∼ 20% larger than the SM3 result. However, same as in the case of muon capture, we find much more σ e (1 + )-strength in the excited states of 12 N nucleus than appears in shell model calculations. Namely, we obtain that J f =1 + 1 σ e (J + f ) = 7.44, while in SM1 and SM3 this quantity is, respectively, 1.77 and 0.40 (in units of 10 −42 cm 2 ). More, in all three shell model calculations more than 90% of the σ e (1 + )-strength is concentrated in the 12 N ground state meanwhile we find only 57%. In the same way as in the shell model calculations, within the PQRPA the total DIF cross section is mainly built up from forbidden excitations. Nevertheless, although the PQRPA results agree with the SM2 calculation, they are quite small when compared with those provided by the SM1 and SM3 models, for both σ exc µ and σ inc µ . The differences in σ inc µ come not only from the positive parity contributions but also from the negative parity ones. As a corollary of the above discussion we would like to stress that it is not easy to asses whether the PQRPA results are better or worse than the shell model ones. We just can say that with the use of only a few essentially phenomenological parameters the PQRPA is able to account for a large number of weak processes in 12 C. (The experimental energies of the 3/2 − 1 state in 13 C and 13 N are also predicted within the PBCS.) That is, the s.p.e., e j , for j = 1p 3/2 , 1p 1/2 , 1d 5/2 , 2s 1/2 , 1d 3/2 , 1f 7/2 , 2p 3/2 , and the pairing strength v pair s have been fixed from of the experimental energies of the odd-mass nuclei 11 C, 11 B, 13 C and 13 N, while the particle-hole coupling strengths v ph s and v ph t were taken from the previous QRPA calculations of 48 Ca [13,24,31]. Only the particle-particle couplings v pp s and v pp t has been treated as free parameters, and we have used here v pp s = v pp t = 0. It could be interesting to inquire whether, this rather extreme parameterization is also appropriate for the description of other light N = Z nuclei, such as 14 N and 16 O, which have been discussed recently within a shell model scheme [17]. Finally, we would like to point out that in the DIF neutrino oscillation search [1] an excess signal of N exp νµ→νe = 18.1 ± 6.6 ± 4.0 events has been observed in the 12 C(ν e , e − ) 12 N reaction, which are evaluated theoretically by the expression N th νµ→νe = ∆ osc dE ν σ e (E ν )P νµ→νe (E ν )Φ µ (E ν ); σ e (E ν ) ≡ J f σ e (E ν , J f ), where 77.3 MeV ≤ ∆ osc ≤ 217.3 MeV stands for the experimental energy window. The oscillation probability reads P νµ→νe (E ν ) = sin 2 (2θ) sin 2 1. 27∆m 2 L E ν , where θ is the mixing angle between the neutrino mass eigenstates, ∆m 2 is the difference in neutrino eigenstate masses squared, in eV 2 , and L is the distance in meters traveled by the neutrino from the source. One sees therefore that the extraction of the permitted values of θ and ∆m 2 from experimental data might depend critically on the theoretical estimate of σ e (E ν ). So far, the electron cross section obtained within the continuum random phase approximation (CRPA) [38] has been used [1]. This σ e (E ν ) is confronted in Figure 6 with our QRPA and PQRPA results calculated with PII. As can be noticed the PQRPA yields a substantially different σ e (E ν ), inside the experimental energy window for the neutrino energy. The consequences of this difference on the confidence regions for sin 2 (2θ) and ∆m 2 will be discussed in a future paper. After finishing this work we have learned that quite recently Nieves et al. [39] were able to describe rather well the inclusive muon capture rate in 12 C, and the inclusive 12 C(ν µ , µ − ) 12 N and 12 C(ν e , e − ) 12 N cross sections, within the framework of a Local Fermi Gas picture which includes the RPA correlations. F JLM G JLI = −MF JIM . (2.17) Using (2.12) and (2.17) we can rewrite the spherical components of O M (J) as ticle and two particle-one hole excitations. The fitted values of the pairing strengths v pair s in units of MeV-fm 3 are also displayed. 1 = 1−1.26 MeV in 13 C. In the same way the PBCS proton-energy ε Z 1p 3/2 = 1.46 MeV agrees with the measured energy E /2 = −2.44 MeV for protons, which clearly shows the necessity for the number projection procedure. At this point it could be useful to remember that, while in 11 C and 11 B the state 3/2 − 1 is dominantly a hole state, in 13 C and 13 N it is basically a two-particle-one-hole state. FIG. 1 : 1Neutron quasiparticle excitation energies for N = 6 and N = 8. The states are ordered as 1s 1/2 , 1p 3/2 , 1p 1/2 , 2s 1/2 , 1d 3/2 , 1f 7/2 , 2p 3/2 , 2p 1/2 , and 1f 7/2 , and the energies are indicated by circles. FIG. 2 : 2Unperturbed two-quasiparticle energies E jpjn for nitrogen isotopes, as a function of N , of the states \|1p 1/2 1p 3/2 , \|1p 3/2 1p 3/2 , \|1p 1/2 1p 1/2 and \|1p 3/2 1p 1/2 . In the upper (lower) panel are shown the BCS (PBCS) results. = 3.4 MeV. The meaning of this results can be easily disentangled by referring FIG. 3 : 3Schematic representation of the particle-hole limits for the seniority-two pn-states. The zeroangular-momentum couplings of two-particles or two-holes are indicated by a horizontal bracket. ε F represents the Fermi energy. The unperturbed energies E jpjn are given in MeV, being the PBCS results given in brackets. \|1p 3/ 2 FIG. 4 : 241p 1/2 , as it happens in the PBCS case but not within the BCS. Finally, it could be worthwhile to indicate that the same pn quasiparticle excitation has quite different ph compositions in different nuclei. This can be observed by scrutinizing the four columns in Fig. 3. The improvement introduced by the PBCS can also be visualized by making a direct The BCS (upper panel) and the PBCS (lower panel) Gamow-Teller strength function in 12 N . Parameterization I (PI): v ph s = v pair s = 24 MeV-fm 3 , and v ph t = v ph s /0.6 = 39.86 MeV-fm 3 . This means that, the singlet ph strength is the same as v pair s obtained from the proton and neutron gap equations, while the triplet ph depth is estimated from the relation used by Goswami and Pal[29] in the RPA calculation of 12 C. FIG. 5 : 5The results of the PQRPA calculations, as functions of the pp parameter t, are confronted with the experimental data taken from Refs.[27,32,33] for: 1) the energy difference ω 1 + between the 1 + ground state in 12 N and the 0 + ground state in 12 C (upper panel), 2) the B-value for the GT beta transition between these two states (middle panel), and 3) the corresponding muon capture rate Λ(1 + 1 ) (lower panel). in the DIF interval m µ + ω J f ≤ ∆ DIF J f ≤ 300 MeV for muons. The full PQRPA calculations, which include the relativistic corrections are listed for all three parameterizations I, II, and III. On the contrary, the theoretical results involving just the velocity-independent operators Y J (κr) and S JL (κr) are displayed only for the case PII. The numerical results are sorted according to the order of forbiddeness of the transition moments, which can be: allowed (A): J π = 0 + , 1 + , first forbidden (F1): J π = 0 − , 1 − , 2 − , second forbidden (F2): J π = 2 + , 3 + , third forbidden (F3): J π = 3 − , 4 − , and so on. The response of the three weak processes to successive multipoles is strongly correlated with the average momentum transfer, κ, involved in each: (ν e , e − )-reaction, κ ∼ 0.2 fm −1 ; µ-capture, online) Calculated cross section σ νe as function of the neutrino energy. The dashed region indicates the experimental energy window. 11) TABLE I : IValues of the geometrical factors F JLM . 15) TABLE II : IIElemental operators and their parities. TABLE III : IIIBCS and PBCS results for neutrons. E expj stand for the experimental energies used TABLE IV : IVSame as Table III but for protons. TABLE V : VExperimental and calculated muon capture rate in units of 10 3 s −1 . The full PQRPA calculations, which include the relativistic corrections are listed for all three parameterizations . Theoretical results that involve only the velocity-independent matrix elements are displayed in parentheses in the third column for the case PII. The rates are grouped by their degrees of forbiddeness. We show: i) the exclusive rates κ ∼ 0.5 fm −1 ; and (ν µ , µ − )-reaction, κ ∼ 1 fm −1 . As a consequence, in the first case the A-moments are by far the dominant ones, contributing with ∼ 83.0%, with the remaining part of the reaction strength carried almost entirely (∼ 16.6%) by the F1-moments. ForΛ(J π ), for J π f = 1 + 1 , 1 − 1 , 2 − 1 , 2 + 1 , ii) the multipole decomposition of the rates f Λ(J π f ) for each final state with spin and parity J π f , and iii) the inclusive decay rate Λ inc ≡ J π f Λ(J π f ). In column four are listed the results of recent shell-model calculations, which are explained in the text. The measured capture rates are given in the last column. µ capture PQRPA Shell Model Experiment rate (I) (II) (III) SM1[16] SM2[16] SM3[17] allowed 49.3 % 39.3 % Λ(1 + 1 ) 7.52 6.27(6.50) 6.27 11.56 6.3 6.0 6.2 ± 0.3 [33] f Λ(0 + f ) 3.68 2.86 (3.15) 3.77 0.21 f Λ(1 + f ) 20.28 18.14 (18.63) 18.22 15.43 1st forbidden 46.6 % 55.7 % Λ(1 − 1 ) 1.06 0.49 (0.51) 0.98 1.86 0.62 ± 0.20 [34, 35] Λ(2 − 1 ) 0.31 0.18 (0.18) 0.16 0.22 0.18 ± 0.10 [34, 35] f Λ(0 − f ) 2.62 2.35 (0.72) 2.35 2.12 f Λ(1 − f ) 11.84 10.37 (9.51) 11.37 12.25 f Λ(2 − f ) 7.78 7.12 (6.90) 7.15 7.79 2nd forbidden 3.9 % 4.6 % Λ(2 + 1 ) 0.19 0.14 (0.16) 0.15 0.25 0.21 ± 0.10 [34, 35] f Λ(2 + f ) 1.26 1.09 (0.89) 1.17 1.36 f Λ(3 + f ) 0.63 0.57 (0.57) 0.58 0.46 Λ inc 48.16 42.56 (40.7) 44.67 39.82 41.9 33.5 38 ± 1 [36] TABLE VI : VIExperimental and calculated flux-averaged cross section for the 12 C(ν e , e − ) 12 N DAR reaction in units of 10 −42 cm 2 . The full PQRPA calculations, which include the relativistic corrections, are listed for all three parameterizations. Theoretical results that involve only the velocity-independent matrix elements are displayed in parentheses for the case PII in the third column. The multipole decomposition f σ e (J π f )for each final state with spin and parity J π f , as well as the exclusive, σ exc TABLE VII : VIIIdem Table VI but for the averaged exclusive, σ excreproduced within the PQRPA, the inclusive one, σ inc e , is ∼ 40% above the data. A plausible explanation for this difference could be the fact that we find a very significant amount of the GT strength (32%) and the Fermi strength (7%) within the DAR energy interval, 18 MeV < ∼ E ν < ∼ 50 MeV, where the electron-neutrino flux, Φ e (E νe ), changes very abruptly, making the inclusive cross section to be very sensitive to the strength distribution of low-lying excited states. ACKNOWLEDGEMENTSThe authors wish to express his sincere thanks to Tatiana Tarutina for the meticulous reading of the manuscript. One of us (A.S.) acknowledges support received from the CLAF-CNPq Brazil.µ-capture, and (ν µ , µ − )-reaction, the exclusive contributions are, respectively, 43%, 15%, and 5%.Let us say a few words on the comparison of our results with the experimental data:• µ capture(Table V): All exclusive rates Λ(J π ) with J π f = 1 + 1 , 1 − 1 , 2 − 1 , 2 + 1 , are fairly well accounted for by the theory, while the inclusive rate, Λ inc , is overpredicted by ∼ 10%.• (ν e , e − )-reaction(Table VI): Even though the exclusive cross section, σ exc e , is well . C Athanassopoulos, LSND CollaborationPhys. Lett. 811774LSND Collaboration, C. Athanassopoulos et al., Phys. Lett. 81, 1774 (1998); . Phys. Rev. C. 582489Phys. Rev. C 58, 2489 (1998). . A Aguilar, LSND CollaborationPhys. Rev. D. 64112007LSND Collaboration, A. Aguilar et al., Phys. Rev. D 64, 112007 (2001). . 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C, in press, nucl-th/0408005, ibidem nucl-th/0408008. J Nieves, J E Amaro, M ; J E Valverde, C Amaro, J Maieron, M Nieves, Valverde, nucl-th/0409017J. Nieves, J.E. Amaro, and M. Valverde, Phys. Rev. C, in press, nucl-th/0408005, ibidem nucl-th/0408008; J.E. Amaro, C. Maieron, J. Nieves, and M. Valverde, nucl-th/0409017. To avoid confusion, we will be using roman fonts (M,m) for masses and math italic fonts (M ,m) for azimuthal quantum numbers. To avoid confusion, we will be using roman fonts (M,m) for masses and math italic fonts (M ,m) for azimuthal quantum numbers.	[]
[ "Structure of the photon and magnetic field induced birefringence and dichroism", "Structure of the photon and magnetic field induced birefringence and dichroism" ]	[ "J A Beswick ", "C Rizzo ", "\nLaboratoire Collisions\nCNRS\n5589AgrégatsRéactivité (UMR\n", "\nUniversité Paul Sabatier Toulouse\n\n", "\nIRSAMC\n31062, Cedex 9ToulouseFrance\n" ]	[ "Laboratoire Collisions\nCNRS\n5589AgrégatsRéactivité (UMR", "Université Paul Sabatier Toulouse\n", "IRSAMC\n31062, Cedex 9ToulouseFrance" ]	[]	In this letter we show that the dichroism and ellipticity induced on a linear polarized light beam by the presence of a magnetic field in vacuum can be explained in the framework of the de Broglie's fusion model of a photon. In this model it is assumed that the usual photon is the spin 1 state of a particle-antiparticle bound state of two spin 1/2 fermions. The other S = 0 state is referred to as the second photon. On the other hand, since no charged particle neither particles having an electric dipole are considered, no effect is predicted in the presence of electric fields and this model is not in contradiction with star cooling data or solar axion search.	null	[ "https://arxiv.org/pdf/quant-ph/0702128v1.pdf" ]	118,989,623	quant-ph/0702128	d306f72fccdffa23d57e840350006ef12618a745	Structure of the photon and magnetic field induced birefringence and dichroism arXiv:quant-ph/0702128v1 13 Feb 2007 J A Beswick C Rizzo Laboratoire Collisions CNRS 5589AgrégatsRéactivité (UMR Université Paul Sabatier Toulouse IRSAMC 31062, Cedex 9ToulouseFrance Structure of the photon and magnetic field induced birefringence and dichroism arXiv:quant-ph/0702128v1 13 Feb 2007(Dated: November 3, 2018) In this letter we show that the dichroism and ellipticity induced on a linear polarized light beam by the presence of a magnetic field in vacuum can be explained in the framework of the de Broglie's fusion model of a photon. In this model it is assumed that the usual photon is the spin 1 state of a particle-antiparticle bound state of two spin 1/2 fermions. The other S = 0 state is referred to as the second photon. On the other hand, since no charged particle neither particles having an electric dipole are considered, no effect is predicted in the presence of electric fields and this model is not in contradiction with star cooling data or solar axion search. Very recently an experimental observation of optical activity of vacuum in the presence of a magnetic field has been reported [1] by the PVLAS collaboration. The observed results could not be explained in the framework of the standard Quantum ElectroDynamics (QED), and the existence of light pseudoscalar spinless bosons of the same nature of the Peccei and Quinn axion [2] has been suggested (see e.g. [3]). This explanation however is in contradiction with other existing experimental data. In particular, the particle needed to justify the PVLAS results should be largely produced in the star core by interaction of photons with plasma electric fields. Such a particle should escape because of its very low coupling with matter, and induce a fast cooling of stars at a level already excluded by astrophysical observations [4]. Moreover, CAST experiment [5] devoted to detect solar axions by conversion in a magnetic field, has already excluded the existence of such a particle in the range of mass and coupling constant necessary to give the PVLAS effect. To get rid of this contradiction more exotic solutions have been proposed. In particular, the existence of a massive paraphoton which would couple with the standard photon [6] and with the axionlike particle (ALP), the photon-initiated real or virtual production of pair of low mass millicharged particles [7], and the existence of an ultralight pseudo-scalar particle interacting with two photons and a scalar boson and the existence of a low scale phase transition in the theory [8]. In this letter we show that dichroism and ellipticity induced on a linear polarized light beam by the presence of a magnetic field in vacuum can be predicted in the framework of the de Broglie's fusion model of a photon [9]. In this model it is assumed that the usual photon is the spin 1 state of a particle-antiparticle bound state of two spin 1/2 fermions. The other S = 0 state is referred to as the second photon. The mass of the usual photon is supposed to be zero or negligible. In particular, we show that taken the spin-spin coupling and the interaction with an external magnetic field proportional to s 1 .s 2 and B.(s 1 − s 2 ), respectively, magnetic induced birefringence and dichroism are obtained with a (ǫ.B) 2 pseudo-scalar symmetry where ǫ is the polarization of the photon (as usual ǫ is defined by the direction of the electric field). Thus both dephasing and absorption appear for linearly polarized light parallel to the external applied magnetic field. On the other hand, since no charged particle neither particles having an electric dipole are considered, no effect is predicted in the presence of electric fields and this model is not in contradiction with star cooling data or solar axion search. We consider the photon as composed of a spin 1/2 particle and its antiparticle. The spin Hamiltonian is assumed to be approximate by H 0 = − ∆ 2 s 1 .s 2(1) with ∆ > 0. The ground state eigenstates are then given by: \|S=1, M z =1 = \|↑, ↑ ; \|S=1, M z = − 1 = \|↓, ↓ \|S=1, M z =0 = 1 √ 2 \|↑, ↓ + \|↓, ↑(2) with energy E 1 = −∆/4, corresponding to the ordinary photon γ 1 . The second photon γ 0 is then given by the excited singlet state \|S=0, M z =0 = 1 √ 2 \|↑, ↓ − \|↓, ↑(3) with energy E 0 = (3/4) ∆. The energy difference between the two photons γ 1 and γ 0 is then given by ∆. We assume the particle/antiparticle have magnetic moments m 1 = (β µ B / ) s 1 and m 2 = −µ 1 . Thus the total magnetic moment m = (β µ B / ) (s 1 − s 2 ) has zero average value for the γ 1 photon and m z = β µ B for the γ 0 photon. In the presence of a magnetic field B along Oz we shall have V = (β µ B B/ ) (s 1z − s 2z )(4) The only non-zero matrix element of V is S=1, M z =0\|V \|S=0, M z =0 = β µ B B(5) After diagonalisation \|Ψ 1,0 = cosθ \|S=1, M z =0 + sinθ \|S=0, M z =0 \|Ψ 0,0 = −sinθ \|S=1, M z =0 + cosθ \|S=0, M z =0 (6) with tan(2 θ) = 2β µ B B/∆(7) and eigenvalues E 1 = E 1 + E 0 2 − ∆ 2 1 + tan 2 (2 θ) E 0 = E 1 + E 0 2 + ∆ 2 1 + tan 2 (2 θ)(8) with the new energy difference ∆ = ∆ 1 + tan 2 (2 θ)(9) The ordinary photon γ 1 can be described by a linear combination of the two helicity states \|S=1, M k = ± 1 where M k is the projection of the spin angular momentum in the direction of propagation of the photon. Let first note that if the γ 1 is propagating along the direction of the magnetic field, only \|S=1, M z = ± 1 will be involved and no effect is expected. Consider now a γ 1 propagating along Oy and linearly polarized in the direction of Oz. We shall have \|ǫ z = − 1 √ 2 \|S=1, M y =1 − \|S=1, M y = − 1 (10) but \|S, M y = Mz=0,±1 \|S, M z d S Mz ,My (π/2)(11) where the d S Mz ,My are the Wigner d-functions. Using d 1 1,±1 (Θ) = 1 2 (1 ± cos Θ) d 1 0,±1 (Θ) = ± 1 2 √ 2 sin Θ d 1 −1,±1 (Θ) = 1 2 (1 ∓ cos Θ)(12) we get from (10) and (11) \|ǫ z = −\|S=1, M z =0(13) and this state will be affected by the magnetic field through its coupling to the \|S=0, M z =0 state. We note in passing that in the case of a linear polarization along the Ox axis we have (14) and this state will not be affected by the magnetic field. \|ǫ y = 1 √ 2 \|S=1, M y =1 + i \|S=1, M y = − 1 = 1 √ 2 \|S=1, M z =1 + i \|S=1, M z = − 1 We assume that the magnetic field is switch-on between t = 0 and t = τ = L/c, where L is the field length (of the order of 1 m in PVLAS experiment). We shall have \|ψ(0) = −\|1, 0 = − cos θ \|Ψ 1,0 + sin θ \|Ψ 0,0 where from now on the kets correspond to \|S, M z . At time τ \|ψ(τ ) = − cos θ e −i E1 τ / \|Ψ 1,0(15)+ sin θ e −i E0 τ / \|Ψ 0,0(16) which in terms of the non-perturbed kets \|1, 0 and \|0, 0 , will be given by \|ψ(τ ) = − cos 2 θ e −i E1 τ / + sin 2 θ e −i E0 τ / \|1, 0 − cos θ sin θ e −i E1 t/ − e −i E0 τ / \|0, 0 (17) This can be written as \|ψ(τ ) = −e −i (E1+E0) τ /2 cos(∆ τ /2 ) + i cos(2θ) sin(∆ τ /2 ) \|1, 0 + i sin(2θ) sin(∆ τ /2 ) \|0, 0(18) From (18) the probability to produce \|0, 0 is P γ1→γ0 = \| 0, 0\|ψ(τ ) \| 2 which gives P γ1→γ0 = tan 2 (2θ) 1 + tan 2 (2θ) sin 2 ∆ 1 + tan 2 (2θ) τ /2 (19) with tan 2 (2θ) given by (7). In the limit where 2β µ B B ≪ ∆, tan 2 (2θ) ≪ 1, and P γ1→γ0 ≃ β µ B B τ 2   sin ∆ τ /2 ∆ τ /2   2(20) Thus, when ∆ τ /2 ≪ 1, P γ1→γ0 does not depend on ∆. In an apparatus like the PVLAS one where a linearly polarized laser passes through a region where a magnetic field pointing at 45 degrees with respect to light polarization plane is present, such a conversion probability will show as a linear dichroism giving an apparent rotation of the polarization plane ρ = 1 2 P γ1→γ0 . It is worth to stress that standard QED [10] does not predict any dichroism for light propagating in vacuum in the presence of a magnetic field. As for the phase of the \|1, 0 , this is given by φ 1 = −(E 1 + E 0 ) τ /2 + arctan cos(2θ) tan(∆ τ /2 ) (21) On the other hand, for the Ox polarization the phase is φ 1 = E 1 τ / . The phase difference between γ 1 states for polarization along and perpendicular to the magnetic field is then given by δφ ≡ φ 1 − φ 1 = arctan cos(2θ) tan(∆ τ /2 ) − ∆ τ /2 (22) Expanding this function in powers of θ around zero, we found δφ = β µ B B ∆ 2 ∆ τ − sin(∆ τ / )(23) Again, in the case of an apparatus like the PVLAS one, this dephasing will show as an ellipticity ǫ = δφ/2 acquired by the polarized beam passing through the magnetic field region. Ellipticity is associated to the existence of a birefringence by the formula ǫ = π L λ (n − n ⊥ )(24) where λ is the light wavelength, and n and n ⊥ are the indexes of refraction for light polarized parallel and perpendicular with respect to the magnetic field, respectively. Thus, in the framework of our model a vacuum will show an apparent magnetic birefringence (n − n ⊥ ) = λ 2π c τ β µ B B ∆ 2 ∆ τ − sin(∆ τ / ) (25) that depends on the time the photon stays in the magnetic field region. Standard QED predicts that a vacuum is a magnetic birefringent medium showing a (n − n ⊥ ) ≃ 4 × 10 −24 B 2 where B is given in Tesla. That only depends on the value of fundamental constants and the square of the magnetic field intensity [10]. QED also predicts that a corresponding effect exists in the presence of an electric field, such an effect is absent in the framework of our model. We note that our formulas for the conversion probability and the dephasing are equivalent to the ones obtained in the axion case [11] since axion-photon coupling can be also treated as a two level system [12]. Our ∆ corresponds to the ratio m 2 a /ω and β to g aγγ , where m a is the axion mass, ω the photon energy, and g aγγ the axionphoton coupling constant. The mass and coupling constant associated to the ALP needed to explain PVLAS results, m a ≈ 10 −3 eV and g aγγ ≈ 3 × 10 −6 GeV −1 [1], have been chosen by comparing the dichroism signal of PVLAS with limits published by the BRFT collaboration in 1993 [13]. In fig. 1 we show the corresponding graph following equation (20). We have assumed as usual that the measured effect is simply the effect predicted by this formula multiplied by the number of passages in the magnetic field due of the presence of optical cavities. Dotted line represents the lower border of the parameters plane forbidden by BRFT results at a 2σ level, while full line represents the PVLAS signal. The main difference between our model and the axion model is that in our case the optical effects do not depend on the photon energy. Thus, in our case, the oscillations between the two states of the hamiltonian only depend on the time the γ 1 stay in the magnetic field i.e. the length of the magnetic field region. In the axion case the oscillations depend on the length divided by the photon energy ω. Oscillations can therefore be avoided by choosing higher energy photons for longer magnets, and that was the case of BRFT collaboration with respect to PVLAS collaboration. Eventually, this explains why in our case the allowed window for the parameters that could explain the PVLAS dichroism is larger that in the axion case treated in ref. [1]. In our model the mixing between the ordinary photon γ 1 and the second photon γ 0 only appears in a magnetic field. This will not affect the energy balance and star evolution, but should be important in the case of photon emission from neutron stars which show magnetic fields as high as 10 9 T. This is anyway an important issue also for ALP (see e.g. ref. [14], and [15]). Since the fusion model assumes structure of the photon, we should expect excited states associated to internal motion and presumably non zero mass of the constituents. However, if the masses and the spatial dimension of the photon are very small, the first excited state can be very high up in energy. In conclusion, once the PVLAS signal will be confirmed, the exotic but simple de Broglie's fusion model for the photon can provide an explanation for this signal that is not in contradiction with star observation or solar axion search. On the other hand, experiments testing the propagation of light in the presence of a magnetic field in terrestrial laboratories or by astrophysical observations can put more and more stringent limits on its free parameters. FIG. 1 : 1Comparison between PVLAS signal and BRFT limits ACKNOWLEDGEMENTSWe thanks E. Massó for very helpful discussion and carefully reading our manuscript. . E Zavattini, Phys. Rev. Lett. 96110406E. Zavattini et al., Phys. Rev. Lett. 96, 110406 (2006). . R D Peccei, H R Quinn, Phys. Rev. Lett. 381440R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1997). . S Lamoreaux, Nature. 44131S. 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B 175, 359 (1986). . G Raffelt, L Stodosky, Phys. Rev. D. 371237G. Raffelt and L. Stodosky, Phys. Rev. D 37, 1237 (1988). . R Cameron, Phys. Rev. D. 473707R. Cameron et al., Phys. Rev. D 47, 3707 (1993). . A Dupays, C Rizzo, M Rocandelli, G F Bignani, Phys. Rev. Lett. 95211302A. Dupays, C. Rizzo, M. Rocandelli, and G.F. Bignani Phys. Rev. Lett. 95, 211302 (2005). . D Lai, Phys. Rev. D. 74123003D. Lai and J Heyl, Phys. Rev. D 74, 123003 (2006).	[]
[ "Fermionic fields in the pseudoparticle approach Fermionic fields in the pseudoparticle approach", "Fermionic fields in the pseudoparticle approach Fermionic fields in the pseudoparticle approach" ]	[ "Marc Wagner [email protected] \nInstitut für Physik\nHumboldt-Universität zu Berlin\nNewtonstraße 15D-12489Berlin, RegensburgGermany, Germany\n", "Marc Wagner \nInstitut für Physik\nHumboldt-Universität zu Berlin\nNewtonstraße 15D-12489Berlin, RegensburgGermany, Germany\n" ]	[ "Institut für Physik\nHumboldt-Universität zu Berlin\nNewtonstraße 15D-12489Berlin, RegensburgGermany, Germany", "Institut für Physik\nHumboldt-Universität zu Berlin\nNewtonstraße 15D-12489Berlin, RegensburgGermany, Germany" ]	[ "The XXV International Symposium on Lattice Field Theory" ]	The pseudoparticle approach is a numerical method to compute path integrals without discretizing spacetime. The basic idea is to consider only those field configurations, which can be represented as a linear superposition of a small number of localized building blocks (pseudoparticles), and to replace the functional integration by an integration over the pseudoparticle degrees of freedom. In previous papers we have successfully applied the pseudoparticle approach to SU(2) Yang-Mills theory. In this work we discuss the inclusion of fermionic fields in the pseudoparticle approach. To test our method, we compute the phase diagram of the 1+1-dimensional Gross-Neveu model in the large-N limit as well as the chiral condensate in the crystal phase.	10.22323/1.042.0339	[ "https://arxiv.org/pdf/0708.2359v1.pdf" ]	16,569,966	0708.2359	6f1ea5c30d8c6c9775c6b8fb8f7bb96aa017367e	Fermionic fields in the pseudoparticle approach Fermionic fields in the pseudoparticle approach 17 Aug 2007 July 30 -August 4 2007 Marc Wagner [email protected] Institut für Physik Humboldt-Universität zu Berlin Newtonstraße 15D-12489Berlin, RegensburgGermany, Germany Marc Wagner Institut für Physik Humboldt-Universität zu Berlin Newtonstraße 15D-12489Berlin, RegensburgGermany, Germany Fermionic fields in the pseudoparticle approach Fermionic fields in the pseudoparticle approach The XXV International Symposium on Lattice Field Theory 17 Aug 2007 July 30 -August 4 2007* Speaker. The pseudoparticle approach is a numerical method to compute path integrals without discretizing spacetime. The basic idea is to consider only those field configurations, which can be represented as a linear superposition of a small number of localized building blocks (pseudoparticles), and to replace the functional integration by an integration over the pseudoparticle degrees of freedom. In previous papers we have successfully applied the pseudoparticle approach to SU(2) Yang-Mills theory. In this work we discuss the inclusion of fermionic fields in the pseudoparticle approach. To test our method, we compute the phase diagram of the 1+1-dimensional Gross-Neveu model in the large-N limit as well as the chiral condensate in the crystal phase. Introduction Recently there have been several papers proposing models for SU (2) Yang-Mills theory with a small number of physically relevant degrees of freedom. These models include ensembles of regular gauge instantons and merons [1,2], the pseudoparticle approach [3,4,5], superpositions of calorons with non-trivial holonomy [6,7] and an ensemble of dyons [8]. The common basic principle is to restrict the Yang-Mills path integral to those gauge field configurations, which can be represented as a linear superposition of a small number of localized building blocks (pseudoparticles), e.g. instantons, merons, akyrons, calorons or dyons. These models have been quite successful, when dealing with problems related to confinement. First of all, the potential of two static charges is essentially linear within phenomenologically relevant distances. Moreover, a confinement-deconfinement phase transition can be modeled, and numerical results for various quantities, e.g. the string tension, the topological susceptibility, the critical temperature or the low lying glueball spectrum, are in qualitative agreement with results from lattice calculations. However, all these models exclusively consider pure Yang-Mills theory. Therefore, incorporating fermions is an interesting issue. In this paper we present first steps in this direction: we propose a method how to deal with fermionic fields in the pseudoparticle approach, and we test this method by applying it to a simple interacting fermionic theory, the 1+1-dimensional Gross-Neveu model in the large-N-limit. Fermionic fields in the pseudoparticle approach Basic principle The starting point is action and partition function of any theory with quadratic fermion interaction: S[ψ,ψ, φ ] = d d+1 x ψQ(φ )ψ + L (φ ) (2.1) Z = Dψ Dψ Dφ e −S[ψ,ψ,φ ] ,(2.2) where φ denotes any type and number of bosonic fields, e.g. the non-Abelian gauge field in QCD, and Q is the Dirac operator, which, of course, depends on these bosonic fields. To stay close to the spirit of the pseudoparticle approach, we consider fermionic field configurations ψ, which can be represented as a linear superposition of a fixed number of pseudoparticles: ψ(x) = ∑ j η j G j (x) j-th pseudoparticle . (2.3) Each pseudoparticle is a product of a Grassmann valued spinor η j and a function G j , which is localized in space as well as in time (the term pseudoparticle refers to this localization). The integration over all fermionic field configurations is defined as the integration over the Grassmann valued spinors η j : Dψ Dψ . . . = ∏ j dη j dη j . . . (2.4) Integrating out the fermions yields S effective [φ ] = d d+1 x L (φ ) − ln det G j \|Q\|G j ′ (2.5) Z ∝ Dφ e −S effective [φ ] ,(2.6) where the "fermionic matrix" G j \|Q\|G j ′ is the Dirac operator represented in the pseudoparticle basis. We will refer to this pseudoparticle regularization as Q-regularization, and we will shortly point out that this Q-regularization is not suited to produce physically meaningful results. In the case that det(Q) is real and positive, det(Q) = det(Q † Q). This suggests another pseudoparticle regularization: S effective [φ ] = d d+1 x L (φ ) − 1 2 ln det G j \|Q † Q\|G j ′ . (2.7) In the following section we will argue that this Q † Q-regularization has significant advantages over the Q-regularization (2.5). Note that using eigenfunctions of the Dirac operator as "pseudoparticles" yields the well known finite mode regularization [9,10]. The Q-regularization versus the Q † Q-regularization The problem of the Q-regularization (2.5) is that applying the Dirac operator Q to one of the pseudoparticles G j ′ in general yields a function, which is partially outside the pseudoparticle function space span{G n }: QG j ′ (x) = ∑ k a j ′ k G k (x) + h j ′ H j ′ (x) (2.8) with H j ′ normalized and H j ′ ⊥ span{G n }. If \| ∑ k a j ′ k G k \| ≫ \|h j ′ \|, the situation is uncritical. However, as soon as \| ∑ k a j ′ k G k \| < ∼ \|h j ′ \|, serious problems arise: when computing the fermionic matrix elements G j \|Q\|G j ′ , a significant part of QG j ′ is simply ignored, namely h j ′ H j ′ , because it is perpendicular to the pseudoparticle function space span{G n }. On the other hand, the Q † Q-regularization (2.7) has the following advantage: both the left hand sides G j \|Q † and the right hand sides Q\|G j ′ of the fermionic matrix elements G j \|Q † Q\|G j ′ might be (partially) outside to the pseudoparticle function space span{G n }, but they form the same function space span{QG n }, in which their overlap is computed. Of course, the above problem of partially perpendicular left and right hand side function spaces does not exist anymore. For more elaborate arguments, especially why one can expect to obtain correct results from the Q † Q-regularization, we refer to [11]. Testing the method: the Gross-Neveu model in the pseudoparticle approach The 1+1-dimensional Gross-Neveu model in the large N-limit As a testbed for our pseudoparticle method we use the Gross-Neveu model [12], which is a four fermion interacting theory with N identical flavors. Action and partition function of the 1+1-dimensional Gross-Neveu model are given by S = d 2 x N ∑ n=1ψ (n) γ 0 (∂ 0 + µ) + γ 1 ∂ 1 ψ (n) − g 2 2 N ∑ n=1ψ (n) ψ (n) 2 (3.1) Z = N ∏ n=1 Dψ (n) Dψ (n) e −S ,(3.2) where µ is the chemical potential and g the dimensionless coupling constant. To get rid of the four fermion term, one usually introduces a scalar field σ . Integrating out the fermions yields S effective = N 1 2λ d 2 x σ 2 − ln det γ 0 (∂ 0 + µ) + γ 1 ∂ 1 + σ (3.3) Z ∝ Dσ e −S effective (3.4) with λ = Ng 2 . In the following we consider the large-N limit, in which the model can be solved analytically [13,14,15]. This amounts to using an infinite number of flavors N, while λ = Ng 2 is kept constant. Note that in the N → ∞ limit only a single σ -field configuration contributes to the partition function (3.4) minimizing the effective action. Note also that in the large-N limit σ is proportional to the chiral condensate, i.e. σ = −g 2 ∑ N n=1ψ (n) ψ (n) . Numerical results: the phase diagram and the chiral condensate From a technical point of view computations in the pseudoparticle approach are quite similar to those in lattice field theory. The number of pseudoparticles corresponds to the number of lattice sites, while the distance between neighboring pseudoparticles plays a role similar to the lattice spacing. The scale can be set by any dimensionful quantity and it can be changed by choosing a different value for the dimensionless coupling constant. For a recent lattice study of the Gross-Neveu model we refer to [16]. For the following computations we apply the Q † Q-regularization (2.7). As pseudoparticles we use a large number of uniformly distributed hat functions, more precisely B-spline basis functions of degree 2 (cf. e.g. [17]), which are shown in Figure 1. There is one fermionic pseudoparticle per unit volume, the spatial extension of the periodic spacetime region is L 1 = 144 and the temporal extension L 0 varies, corresponding to different temperatures T = 1/L 0 . The main reason for considering such pseudoparticles is that they yield a sensible set of field configurations: they form a piecewise polynomial basis of degree 2, i.e. any not too heavily oscillating field configuration can be approximated. Therefore, if the pseudoparticle method we have presented in Section 2 is a useful numerical technique, we can expect to reproduce correct Gross-Neveu results. In other words, B-spline basis functions are suitable pseudoparticles for testing our approach. At first we perform computations of the chiral condensate σ at chemical potential µ = 0 and temporal extension L 0 = 8 for various values of the coupling constant λ . As it is in lattice calculations different values of λ correspond to different physical extensions of the spacetime region and, therefore, to different values of the temperature. From these computations we determine that value of λ , where σ just vanishes: λ critical = 1.153. For all further computations we use λ = λ critical . By doing this we have set the scale, since from now on L 0 plays the role of inverse temperature such that L 0 = 8 corresponds to the critical temperature of chiral symmetry breaking. After that, we perform a low temperature computation at L 0 = 48 or equivalently T = T critical /6, to obtain an approximation of the zero temperature value of the chiral condensate: σ 0 = 0.221. This allows us to express all dimensionful quantities in terms of σ 0 . Now we are in a position to compute the chiral condensate at arbitrary temperature T /σ 0 and chemical potential µ/σ 0 . Results for homogeneous chiral condensate are shown in Figure 2a together with the analytically obtained phase boundary [13,14] and the tricritical point separating first and second order phase transitions. Pseudoparticle and analytical results are in excellent agreement both for the phase boundary (cf. also Figure 2b) and for the order of the phase transition (cf. also Figure 2c, where we have plotted σ /σ 0 as a function of µ/σ 0 for two different values of T /σ 0 , one in the first order region and the other in the second order region). For inhomogeneous chiral condensate a third so called crystal phase appears [15], where the minimum of the effective action (3.3) is not anymore given by a homogeneous chiral condensate σ . In addition to the fermionic fields we also represent σ in terms of B-spline pseudoparticles (for details cf. [11]). As before, the pseudoparticle phase diagram and the analytically obtained phase diagram are essentially indistinguishable (cf. Figure 3a). We have also compared the pseudoparticle chiral condensate and the analytically obtained chiral condensate at various points (µ/σ 0 , T /σ 0 ) inside the crystal phase; again, there is excellent agreement. Figure 3b shows the emergence of a crystalline structure: the kink-antikink structure close to the left phase boundary changes to a sin-like behavior, when approaching the center of the crystal phase. Note that we have performed the same computations also with the naive Q-regularization. As expected the results are completely wrong, e.g. there is no chirally symmetric phase even in the simple case of homogeneous chiral condensate. One can easily show that this is inherent to the Q-regularization and not a problem of the number or the type of pseudoparticles applied [11]. Summary and outlook We have proposed a method to incorporate fermionic fields in the pseudoparticle approach. While the naive Q-regularization is not suited to produce any useful results, the Q † Q-regularization has the potential to yield correct and physically meaningful results. The computation of the phase diagram of the Gross-Neveu model with the Q † Q-regularization both for homogeneous and for inhomogeneous chiral condensate has been a first successful test of the pseudoparticle approach applied to fermionic theories. The next step is to apply the pseudoparticle approach to QCD and to identify a small number of physically relevant degrees of freedom, probably fermionic pseudoparticles, which are able to approximate typical low lying eigenmodes of the Dirac operator. The goal is to obtain a model with a small number of degrees of freedom, which exhibits both chiral symmetry breaking and a confinement deconfinement phase transition at the same time. Figure 1 : 1B-spline basis functions in one and two dimensions. Figure 2 2: a) σ /σ 0 as a function of µ/σ 0 and T /σ 0 (red dots) together with the analytically obtained phase boundary (blue line) and the tricritical point (µ/σ 0 , T /σ 0 ) = (0.608, 0.318) separating first and second order phase transitions (black dot). b) Phase diagram for homogeneous chiral condensate (red dots: pseudoparticle results; green line: analytical result). c) Two sections trough the phase diagram showing σ /σ 0 as a function of µ/σ 0 at T /σ 0 = 0.283 (first order phase transition) and T /σ 0 = 0.377 (second order phase transition). Figure 3 3: a) Phase diagram for inhomogeneous chiral condensate (red dots: pseudoparticle results; green line: analytical result). b) The pseudoparticle chiral condensate for T /σ 0 = 0.141 and different values of µ/σ 0 . Acknowledgments . F Lenz, J W Negele, M Thies, arXiv:hep-th/0306105Phys. Rev. D. 6974009F. Lenz, J. W. Negele and M. Thies, Phys. Rev. D 69, 074009 (2004) [arXiv:hep-th/0306105]. . 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[ "DRBM-ClustNet: A Deep Restricted Boltzmann-Kohonen Architecture for Data Clustering", "DRBM-ClustNet: A Deep Restricted Boltzmann-Kohonen Architecture for Data Clustering" ]	[ "Senior Member, IEEEJ Senthilnath ", "Nagaraj G ", "Sumanth Simha ", "C ", "Sushant Kulkarni ", "Meenakumari Thapa ", "Indiramma M ", "Fellow, IEEEJón Atli Benediktsson " ]	[]	[]	A Bayesian Deep Restricted Boltzmann-Kohonen architecture for data clustering termed as DRBM-ClustNet is proposed. This core-clustering engine consists of a Deep Restricted Boltzmann Machine (DRBM) for processing unlabeled data by creating new features that are uncorrelated and have large variance with each other. Next, the number of clusters are predicted using the Bayesian Information Criterion (BIC), followed by a Kohonen Network-based clustering layer. The processing of unlabeled data is done in three stages for efficient clustering of the non-linearly separable datasets. In the first stage, DRBM performs non-linear feature extraction by capturing the highly complex data representation by projecting the feature vectors of d dimensions into n dimensions. Most clustering algorithms require the number of clusters to be decided a priori, hence here to automate the number of clusters in the second stage we use BIC. In the third stage, the number of clusters derived from BIC forms the input for the Kohonen network, which performs clustering of the feature-extracted data obtained from the DRBM. This method overcomes the general disadvantages of clustering algorithms like the prior specification of the number of clusters, convergence to local optima and poor clustering accuracy on non-linear datasets. In this research we use two synthetic datasets, fifteen benchmark datasets from the UCI Machine Learning repository, and four image datasets to analyze the DRBM-ClustNet. The proposed framework is evaluated based on clustering accuracy and ranked against other state-of-the-art clustering methods. The obtained results demonstrate that the DRBM-ClustNet outperforms stateof-the-art clustering algorithms.	10.1109/tnnls.2022.3190439	[ "https://arxiv.org/pdf/2205.06697v1.pdf" ]	248,798,572	2205.06697	bbbcb8c6b86fc8bb6d69b0b79b3e6f1208e2f014	DRBM-ClustNet: A Deep Restricted Boltzmann-Kohonen Architecture for Data Clustering Senior Member, IEEEJ Senthilnath Nagaraj G Sumanth Simha C Sushant Kulkarni Meenakumari Thapa Indiramma M Fellow, IEEEJón Atli Benediktsson DRBM-ClustNet: A Deep Restricted Boltzmann-Kohonen Architecture for Data Clustering PREPRINT VERSION 1Index Terms-Data ClusteringBayesian Information Crite- rionDeep Restricted Boltzmann MachineKohonen Network A Bayesian Deep Restricted Boltzmann-Kohonen architecture for data clustering termed as DRBM-ClustNet is proposed. This core-clustering engine consists of a Deep Restricted Boltzmann Machine (DRBM) for processing unlabeled data by creating new features that are uncorrelated and have large variance with each other. Next, the number of clusters are predicted using the Bayesian Information Criterion (BIC), followed by a Kohonen Network-based clustering layer. The processing of unlabeled data is done in three stages for efficient clustering of the non-linearly separable datasets. In the first stage, DRBM performs non-linear feature extraction by capturing the highly complex data representation by projecting the feature vectors of d dimensions into n dimensions. Most clustering algorithms require the number of clusters to be decided a priori, hence here to automate the number of clusters in the second stage we use BIC. In the third stage, the number of clusters derived from BIC forms the input for the Kohonen network, which performs clustering of the feature-extracted data obtained from the DRBM. This method overcomes the general disadvantages of clustering algorithms like the prior specification of the number of clusters, convergence to local optima and poor clustering accuracy on non-linear datasets. In this research we use two synthetic datasets, fifteen benchmark datasets from the UCI Machine Learning repository, and four image datasets to analyze the DRBM-ClustNet. The proposed framework is evaluated based on clustering accuracy and ranked against other state-of-the-art clustering methods. The obtained results demonstrate that the DRBM-ClustNet outperforms stateof-the-art clustering algorithms. samples and group the related samples [2]. Clustering is very useful in organizing data and understanding the hidden structure of the data. Clustering finds its plethora of applications in text analytics [3] [4], anomaly detection [5], hand-written digit recognition [6], social network analysis [7], customer grouping based on preferences in marketing sites [8], dynamic clustering [9], face recognition [10], model order detection [11] and heterogeneous system placement problems [12]. There are various types of clustering algorithms such as partitional clustering, hierarchical clustering, model-based clustering, and density-based clustering [13]. Most popular partitional clustering algorithms are K-means and the Kohonen Network. In the case of model-based clustering, the expectation maximization (EM) Algorithm is most popular [14] [15] [16]. For K-means, the cluster centers are initialized randomly, and the samples are assigned to any one of the clusters using a similarity measure [14]. In every iteration, these cluster centers are updated until convergence is reached. In the Kohonen Network, the attributes that are assigned weights are updated iteratively using a competitive learning mechanism [15]. The EM algorithm groups the data based on the maximum likelihood approach using the Gaussian mixture model [16]. The main drawback of the aforementioned clustering algorithms is that their performance mainly depends on the dimensionality of the feature vector and the prior assignment of the number of clusters [17] [18]. By using the Bayesian Information Criterion (BIC), we can remove the problem of specifying the number of clusters a priori [19]. A common misconception in clustering is that having a larger number of samples with cluster information leads to a better discriminative cluster since the features contain information about the target (class/cluster). However, there are many reasons like the presence of irrelevant, redundant or noisy features, which thwarts the clustering accuracy. Thus, it is essential to project features into a higher or lower order (and vice versa) by transforming non-linearly separable data into linearly separable data. In the literature, the Restricted Boltzmann Machine (RBM) has overcome such issues [20] [21] [22]. The RBM is a non-linear feature extraction technique that is superior in comparison to linear feature extraction techniques such as Principal Component Analysis [23]. The first Boltzmann Machine (BM) model was developed in the 1980s and was inspired by statistical mechanics [24]. Smolensky in 1986 [25] implemented the Harmonium, which is a variant of the BM. The RBM is often described as the Monte Carlo version of Hopfield networks [26]. Even though the idea of BM has existed since the 1980s, it was more widely used from 2006 when Hinton implemented a modified version of BM, termed as RBM, for a fast and efficient way of learning [27]. RBM overcomes problems with the time required for the network to converge, which grows exponentially with the size of the samples [28]. In BM, it is difficult to obtain an unbiased sample from the posterior distribution given the data vector; the complexity is due to the connectivity within and between the layers. In RBM, the restrictedness arises due to the absence of visible to visible and hidden to hidden interactions in the network. The learning algorithm used to train the RBM is based on a k-step contrastive divergence (CD) technique, which approximates the Markov Chain Monte Carlo (MCMC) technique by sampling all the units in a layer at once [29]. After the emergence of this technique, and with advances in the availability of computational power, many researchers realized that shallow neural networks were not sufficient. Thus, there was a need for networks to learn deeper representations of the data [30] which is essential for computer vision [31], timeseries forecasting and other applications [32]. Later, Hinton et al., [33] developed Deep Boltzmann Machines by stacking many layers of RBM and forming an undirected graphical model, which is unlike Deep Belief Networks, and uses directed graphical models [27] [34]. The Deep Restricted Boltzmann Machine (DRBM) has been used as a generative model to learn the non-linear distribution of a dataset [32]. The development of better computational hardware and highperformance processors has recently led to applications of the approach with larger datasets [35] [36]. Sankaran et al., [37] proposed an RBM-based semi-supervised class sparsity signature by combining the unsupervised generative training with a supervised sparsity regularizer to learn better generative features. The RBM has also been applied for cancer detection [38], time-series prediction [40], detecting event-related potential [39], and estimation of 3D trajectories from 2D trajectories [41]. In this paper, we propose the DRBM-ClustNet framework, which performs feature extraction using DRBM. The output of the first stage is used to predict the number of clusters with BIC. The feature-extracted data obtained with DRBM and the predicted number of clusters are used as an input for the clustering algorithm using the Kohonen Network. The proposed framework's performance is compared with six state-of-the-art clustering methods, namely, K-means [42], Self-Organizing Maps (SOM) [43], Expectation Maximization (EM) [44], Density-Based Spatial Clustering of Applications with Noise (DBSCAN) [45], Unsupervised Extreme Learning Machine (US-ELM) [46], and Bayesian Extreme Learning Machines Kohonen Network (BELMKN) [47]. The proposed DRBM-ClustNet and the state-of-the-art clustering algorithms are applied on 2 synthetic datasets, 15 benchmark datasets from the UCI Machine Learning Repository [48], and 4 image datasets. The performance evaluation is carried out using various statistical approaches. The remaining content of the paper is organized as follows: The architecture diagram of the DRBM-ClustNet framework with an abstract code is discussed in Section II. The clustering performance for 2 synthetic datasets with the DRBM-ClustNet framework and other available clustering algorithms is narrated in Section III. The results for various clustering methods when applied on 15 benchmark tabular datasets and 4 image datasets compared to the proposed approach are discussed in Section IV. Conclusions are drawn in Section V. II. DRBM-CLUSTNET ARCHITECTURE The proposed DRBM-ClustNet architecture for data clustering is discussed in this section. The DRBM is used for feature extraction, followed by BIC which is applied to predict the number of cluster centers for the feature extracted data. Finally, the above two levels are used as input for data clustering using the Kohonen Network. The detailed flow of the DRBM-ClustNet architecture is shown in Fig. 1. A. Feature extraction using DRBM The RBM is an energy-based undirected bipartite graphical model as shown in layer 1 (L 1 ) of Fig. 1. For a given dataset, initially, we use the min-max normalization on a realvalued dataset for feature extraction, whereas the conventional RBM uses binary-valued data. The normalization of the dataset performed is the same as in [49], [50] F (x s ) = 0.9 − 0.1 max(x i ) − min(x i ) (x s − min(x i )) + 0.1,(1) where x s is a dataset with s = {1, 2, ..., N }, min(x i ) and max(x i ) corresponds to the minimum and maximum value for a particular feature with i = {1, 2, ..., d}. The normalized dataset is passed to L 1 of the DRBM network but L 1 consists of a visible layer and a hidden layer. The visible units v i , i ∈ 1 to d are real-valued and the hidden units are given by h j , j ∈ 1 to p . Let W 1 ∈ IR d×p denote the weights between the hidden layer and visible layer units in L 1 and w ij indicate the weights between the i th visible unit and the j th hidden unit. As RBM is an energy-based model, the probability distribution of the visible layer and hidden layer units are stated in the form of an energy equation. The joint probability distribution of the visible layer and the hidden layer units P (v, h) is given by P (v, h) = 1 Z e −E(v,h) ,(2) where E(v, h) is the energy of the joint configuration between visible layer and hidden layer units and is given by E(v, h) = − d i=1 p j=1 w ij v i h j − d i=1 b i v i − p j=1 c j h j ,(3) where b i is the visible layer bias and c j is the hidden layer bias. Z is the partition function defined as The partition function (Z) is intractable. As a result, the joint probability distribution of the visible and the hidden units P (v, h) which depends on Z is also intractable. Therefore, we express the joint probability distribution P (v, h) in (2) in terms of the conditional probability distribution P (v\|h) and P (h\|v). This is much easier to compute and is defined as Z = d i=1 p j=1 e −E(vi,hj ) .(4)P (v\|h) = d i=1 P (v i \|h),(5)P (h\|v) = p j=1 P (h j \|v).(6) From (5) and (6) we can observe that the conditional probabilities are all independent. This indicates that the state of the hidden layer units is independent given the states of the visible layer units (without interconnecting the neurons in the same layer either in visible or hidden layer). Therefore, this type of Boltzmann Machine is defined as a Restricted Boltzmann Machine (RBM). The conditional probabilities from (5) and (6) are derived as a sigmoidal function given by P (h j \|v) = s( d i=1 v i w ij + c j ),(7)P (v i \|h) = s( p j=1 h j w ij + b i ),(8) where s(.) denotes the sigmoidal function given by s( x) = 1 1+e −x . RBM is a generative model from which we predict P (h\|v) by clamping the visible units and P (v\|h) by using the hidden activation obtained from P (h\|v). Hinton proposed the k-step Contrastive Divergence algorithm (CD) [29] for training the RBM, which jointly performs the Gibbs Sampling for all variables in one layer instead of sampling the new values of all variables one-by-one. The one-step CD is divided into two phases namely, positive contrastive divergence and negative contrastive divergence followed by weight updating. 1) Positive contrastive divergence: The probability that a hidden state is evaluated by clamping all the visible units: P (h\|v) = s(v <0> W + c) = h <0> .(9) Then, we calculate the interactions between the visible layer and hidden layer units as a + = v <0> h <0> = (vh) <0> ,(10) where a + denotes the positive associations. 2) Negative contrastive divergence: The probability of a visible state given the hidden activations from (9) to get v <1> . Then we use these visible units to find h <1> : P (v\|h) = s(W h <0> + b) = v <1> ,(11)P (h\|v) = s(W v <1> + c) = h <1> .(12) Then, we calculate the negative associations between the visible layer and hidden layer units a − = v <1> h <1> = (vh) <1> .(13) 3) Weight update step: Update the weights by considering the difference between the positive associations from (10) and negative associations from (13) using adaptively decreasing learning rate ( ) ∆w = (a + − a − ).(14) The above one-step CD algorithm is applied and the states of visible and hidden units are sampled alternately. In many cases, a single layer (L 1 ) RBM will not be able to capture the complete non-linearity in the dataset. As a result, we need to stack multiple RBM layers (L 1 , L 2 ,..., L n ) to produce a DRBM. In DRBM, the activation of one hidden layer forms the training samples for the next hidden layer. This technique is efficient in learning the complex data representations. DRBM can efficiently learn a generative model of high dimensional input and large-scale dataset [20]. Consider the DRBM network in the proposed architecture shown in Fig. 1, where feature extraction is carried out using the layers (L 1 , L 2 ,...,L n ). Let h l (l ∈ 1 to n) denote the l th hidden layer, W l denote the weights between the layers. The energy equation for the DRBM is given by E(v, h 1 , .., h n ; θ) = −vW 1 h 1 − h 1 W 2 h 2 − .. − h n−1 W n h n ,(15) where θ = {W 1 , W 2 , ..., W n } are the model parameters which have to be trained and n is the number of hidden layers in the DRBM network [51]. In DRBM, the first layer RBM is trained using one-step contrastive divergence to obtain the reconstructions of the visible vectors. Then, the next layer RBM is trained using the first layer sampled hidden layer activations h l obtained from P (h l \|v; W l ) as the training data for the second layer RBM and obtain the reconstructions of the visible units. Continue this process recursively until layers n−1. Using all the trained weights θ = {W 1 , W 2 , ..., W n } and the input data features as the visible units, a feedforward pass is performed until the last hidden layer n. The final activations of the hidden layer n are the feature extracted data (x l s ). B. BIC for Cluster Prediction To automate the process of cluster prediction, DRBM-ClustNet uses the BIC. This is applied to the feature extracted data obtained from DRBM instead of the actual dataset as shown in Fig. 1. Most of the clustering algorithms require a parameter called the number of clusters to be inputted manually. As in real-world problems, datasets may be with or without labels, hence BIC is applied to statistically predict the number of clusters. BIC uses multivariate Gaussian distribution and parameters, which are the mean and covariance matrices. These parameters are estimated using the EM algorithm. BIC for cluster prediction is defined as BIC = ln(N )k − 2ln(L),(16) where N is the total number of samples,L is the maximized value of the likelihood function and k is the total number of free parameters to be estimated. The BIC is computed for c = 1, 2, ..., N and the model value with minimum BIC value is selected and its corresponding number of clusters n c are used for clustering the data. C. Kohonen Network for data clustering The DRBM-ClustNet uses the feature extracted data from DRBM and the BIC selected number of clusters (n c ). The data clustering is performed using the Kohonen Network. The Kohonen Network (KN) is made up of input and output layer. The number of output layer neurons (n o ) depends on the total number of clusters (n c ) predicted using BIC. The weights between the input and the output layer are W k ∈ R ni×nc , where n i is the number of input neurons and the weight matrix is randomly initialized. The weight is updated iteratively using the discriminant function considering Euclidean distance given by d(j) = ni i=1 (x i − w ij ) j ∈ 1 to n o .(17) The winning neuron is the one, which has minimum separation with the input sample. The weight is updated iteratively considering the neighborhood function (i.e., the neurons that are within its vicinity) using h ci (t) = α(t)exp −d 2 ij 2σ(t) ,(18) where t is the iteration, α(t) is the learning rate for given iteration t and σ(t) is the spread of the data points in consideration and is given by α(t) = α 0 exp −t T 1 ,(19)σ(t) = σ 0 exp −t T 2 ,(20) where T 1 and T 2 are time constants. The updating of the weights for the winning neuron and its neighboring neurons within its vicinity are computed as follows: δw ij = h ci (t)(y i − w ij ).(21) DRBM-ClustNet uses this as a final layer for data clustering. To know the grouping of the data points to individual class label is evaluated using a performance measure. The clustering efficiency can be evaluated using the clustering accuracy (η) η = nc i=1 a l N ,(22) where a l is the correctly clustered samples and N is the total number of samples in the data. The proposed DRBM-ClustNet is summarized in Algorithm 1. Algorithm 1 DRBM-ClustNet Algorithm Input: clustering dataset X={x s ∈ IR d } N s=1 , DRBM-ClustNet (v 1 , ..., v d , h 1 , ..., h p ), θ 1 ={W l } n l=1 , θ 2 ={W K }, learning rate ( , α). Output: cluster prediction (n c ), clustering accuracy (η). for x s in X do v s i <-x s for l = 1 to n do Compute CD using (9)- (14) to obtain W l Perform feedforward pass using W l Extract featuresx l s from layer h l end Using extracted featurex l s predict n c (16) Perform clustering with n c onx l s using (17)-(21) Evaluate η using (22) end return: n c , η. III. DRBM-CLUSTNET APPLIED ON SYNTHETIC DATASET In this section, we discuss the results of the proposed framework and other clustering algorithms for two synthetic datasets as shown in Figs. 2 and 3. The results obtained are evaluated and benchmarked against the state-of-the-art clustering algorithms such as K-means [42], SOM [43], EM [44] and DBSCAN [45]. Similarly, clustering is compared with feature learning algorithms like US-ELM [46], BELMKN [47], and Single Layer RBM [18]. The first dataset is of the flame distribution, which consists of 600 instances. It has two classes with 300 instances each. The second dataset is of the moon distribution, which consists of two classes (two half-moons) each of which has 150 samples. The flame distribution and the moon distribution datasets are non-linearly distributed and clustering them accurately is a challenge. For each of the two datasets, DRBM-ClustNet is applied and compared with the aforementioned clustering techniques and their performance is analyzed. DRBM-ClustNet uses DRBM for feature extraction by setting the parameters such as the number of hidden layers and the number of hidden neurons in each layer. We can observe from Fig. 2 that K-means, SOM, EM, DBSCAN and US-ELM have failed to cluster the dataset efficiently. However, BELMKN clusters the dataset efficiently as it uses ELM for feature learning and the Kohonen Network for clustering. The clustering accuracy using BELMKN for the flame distribution dataset is 98.5%. The single layer RBM has few misclassification samples to capture the non-linearity in the dataset. It is overcome by stacking the multiple layers of RBM to form a DRBM, which is used in the proposed DRBM-ClustNet. The DRBM-ClustNet architecture for the flame distribution dataset consists of five hidden layers. For the proposed DRBM-ClustNet, we can observe that there is no misclassification for the flame distribution dataset, where the performance is better compared to the other clustering algorithms used in this paper. Similarly, we can observe from Figs. 3(a)-(g) that for the moon distribution dataset, all the clustering algorithms fail to cluster the corners of the moon-shaped distribution. Fig. 3(h) shows the proposed method DRBM-ClustNet can cluster even the corners without any samples being misclassified. By observing the two synthetic datasets used here, we can infer that the proposed DRBM-ClustNet which uses DRBM for feature extraction gives a better clustering accuracy in comparison to state-of-the-art clustering algorithms. Hence, we conclude that the DRBM-ClustNet can be used for efficient clustering of non-linearly distributed datasets. IV. RESULTS AND DISCUSSION This section discusses the results for the DRBM-ClustNet framework on fifteen different benchmark datasets from the UCI repository. Initially, we discuss the characteristics of the datasets shown in Table I. Next, the number of clusters are predicted using BIC with the feature extracted data obtained using three feature learning techniques, namely, ELM, Single Layer RBM and DRBM-ClustNet. The cluster prediction of these feature learning techniques is used for comparison against the BIC evaluated on the actual dataset. Finally, clustering performance of the DRBM-ClustNet algorithm is compared with other prominent clustering algorithms such as K-means [42], SOM [43], EM [44], DBSCAN [45], USELM [46] and BELMKN [47]. The algorithms were executed on a Core i3 processor, 4 GB RAM, Python 2.7 and Windows 10 OS. A. Dataset Description In this research, we have applied the proposed DRBM-ClustNet on 15 benchmark datasets as shown in Table I (https://archive.ics.uci.edu/ml/index.php). We have adopted 15 widely used datasets of the UCI repository to compare with other prominent clustering methods. The number of samples/observations, the number of features/attributes and the actual number of clusters for every dataset used in this work is shown in Table I. The detailed description for the datasets are as follows: Dataset 1: The Balance data classifies a sample into one of the three classes, namely, balance scale tip to the right, tip to the left or to be balanced. It consists of four attributes, which includes the left weight, the left distance, the right weight, the right distance, and 625 samples. Dataset 2: The Cancer data categorizes a tumor either as benign or malignant. It has 30 attributes and 569 observations in total. Dataset 3: The Cancer-Int data classifies a breast tumor either as malignant or benign. It has 699 observations and 9 attributes. Dataset 4: The Credit data has two classes, namely, to grant a credit approval or not. It has 14 attributes with 690 observations. Dataset 5: The Dermatology data has different diagnosis details for the erythemato-squamous diseases in dermatology. It has 6 classes, 34 attributes and 366 observations. Dataset 6: The Diabetes data is to diagnostically predict whether a patient has diabetes based on certain diagnostic measurements. It has 768 observations and 8 attributes. Dataset 7: The E.Coli data has details of the proteins cellular localization sites. It contains 5 classes, 327 observations and 7 attributes. B. Parameters for Algorithm All the datasets used in the paper are pre-processed which includes imputation, encoding and scaling. In the imputation process, all the missing values were imputed with a zero. All the categorical features were label encoded or one-hot encoded based on feature description. All distance calculation is based on the Euclidean distance. Distance-based techniques are sensitive to the scale of features. Therefore, to avoid bias towards high-value features, all the features were normalized between 0.1 to 0.9 using min-max scaling. This also ensures a faster convergence for neural network-based approaches. It is necessary to tune the hyper-parameters for effective learning. Since our hyper-parameters are in a well-defined range, the standard grid search approach [52] has been used to find an optimal set of hyper-parameters for the data clustering. A single RBM had one hidden layer with 50 hidden units. The deep RBM has five hidden layers with 50 hidden units in each of the first four layers and 10 hidden units in the final layer. The maximum epochs and the learning rate are set to 50000 and 0.1, respectively. Furthermore, as all these algorithms are dependent on initial random points or weights, ten runs were carried out, and the average and the standard deviation were recorded. C. Assessment of the cluster prediction by DRBM-ClustNet The BIC is used initially to evaluate the number of clusters for a given dataset. The proposed DRBM-ClustNet uses DRBM for feature extraction on the above-mentioned dataset. The feature extracted dataset is used to predict the number of clusters with BIC. Furthermore, the obtained results is compared with the feature extracted data based on ELM, the single layer RBM for the number of clusters predicted by BIC, along with the BIC applied on the actual data as shown in Table II. In Table II, we observe a slight change in the actual number of clusters in the database when compared with the BIC predicted number of clusters for various feature learning techniques. From Table II, we can observe that for the Cancer and Cancer-Int datasets, none of the feature extracted data with BIC were able to predict the actual number of clusters. Here, instead of two clusters, all the feature learning techniques with BIC predict three clusters. This kind of behaviour is observed in Cancer and Cancer-Int datasets as they are linearly separable. By applying feature learning on these datasets, we are introducing additional non-linearity. As a result, feature learning techniques like ELM, Single Layer RBM and DRBM fail for these two datasets. In contrast, for the Dermatology, Iris, Thyroid and Wine datasets, the methods predicted the exact number of clusters. Furthermore, the Glass and SECOM datasets are highly non-linear, hence, we get the actual number of clusters by applying the feature learning techniques on these two datasets. For the Balance, Horse and Vehicle datasets, the USELM, BELMKN and single layer RBM feature learning data on BIC fail to accurately predict the number of clusters. Only the DRBM feature learning method accurately predicts the exact number of clusters as three, three and four for Balance, Horse and Vehicle datasets, respectively. This is observed, as these three datasets are highly non-linear and the non-linearity in the datasets can be captured by stacking many layers of RBM. As a result, we can conclude that the DRBM-BIC is a good approach for the prediction of the number of clusters for non-linearly distributed data. D. Data visualization for DRBM To analyze the behaviour of data distribution in some of the scenarios where DRBM can pick the exact clusters, it is possible to visualize the data from high-dimension to lowdimension and vice versa. This is analyzed with and without applying DRBM on the dataset using a parallel coordinates plot, a scatter plot and a correlation matrix [53]. The architecture used for DRBM feature learning has 10 hidden neurons in the last layer. The Iris dataset has four features and the DRBM feature learning, obtained in the last layer, contains 10 dimensions. Similarly, the Wine dataset has 13 features reduced to 10 features using 10 hidden neurons in the last layer of the DRBM architecture. The parallel coordinates and scatter plots for the Iris and Wine datasets without DRBM (actual data) and with DRBM feature extracted data are shown in Figs. 4 and 5. From Fig. 4(a), for the Iris dataset, we can observe that there is an overlap between the two classes (class 2 and class 3). However, after applying DRBM feature learning, in Fig. 4(b), we can observe that the overlap is reduced with a clear separation between these classes. For the Wine dataset, we can observe from Figs. 5(a) and 5(b) that there is a significant difference between the two scatter plots. Because of the non-linear dimensionality reduction from 13 to 10 dimensions in the DRBM feature extraction, linear separability is introduced to the dataset. Hence, by applying DRBM feature extraction, the number of clusters is well-predicted using BIC as shown in Table II. Most machine learning algorithms involving sigmoidal activation functions or logistic regression functions, often exhibit poor performance if they have highly correlated input variables or feature vectors in the data. Fig. 6(a) shows the correlation matrix for the actual Iris dataset. From this figure, we can observe that the percentage of feature vector pairs having a high correlation (in the range 0.75-1.0) is 6 out of 12 i.e. 50%. In the case of DRBM feature leaning, as shown in Fig. 6(b), we can observe that the percentage of attribute pairs having a high correlation (0.75-1.0) is 2 out of 12, i.e., 16.67%. Hence, by applying DRBM feature extraction to the Iris dataset, we can significantly reduce the correlation between the feature vectors, which in turn results in high clustering accuracy. Overall, from the above analysis, we can conclude that by decreasing the dimension, DRBM performs well for five datasets (Dermatology, Horse, SECOM, Vehicle and Wine). Hence, DRBM with BIC performs well for 10 out of the 15 datasets, i.e., in 66.6% of the cases, whereas BIC applied on the actual dataset performs well for 7 out of 15 datasets, i.e., for 46.6% of the datasets. Both ELM and the single layer RBM perform well for 7 out of 15 datasets, i.e., in 46.6% of the datasets. Another observation from Table II is that DRBM with BIC predicts cluster consistently as the standard deviation is almost zero for most of the datasets whereas a large variation is present for other techniques. Hence, DRBM works effectively and efficiently for the non-linearly distributed datasets. Iris 3 3 ± 0.31 3 ± 0.02 3 ± 0 3 ± 0 SECOM 2 3 ± 0.15 2 ± 0.54 2 ± 0.44 2 ± 0.22 Thyroid 3 3 ± 0.39 3 ± 0.31 3 ± 0 3 ± 0 Vehicle 4 3 ± 1.25 3 ± 1.10 3 ± 0.98 4 ± 0.65 Wine 3 3 ± 0.49 3 ± 0.23 3 ± 0 3 ± 0 E. Assessment of the cluster accuracy of DRBM-ClustNet The proposed DRBM-ClustNet is applied to the abovementioned 15 benchmark UCI machine learning repository datasets [48]. In the DRBM-ClustNet, the feature extracted data from the DRBM network and the cluster prediction output from BIC are given for clustering using the Kohonen Network. This approach shows the best performance when compared to the traditional clustering algorithms such as K-means, EM, SOM, DBSCAN and feature learning based clustering techniques such as USELM, BELMKN, and single layer RBM. The K-means algorithm involves random initialization of centroids and the cluster centers are iteratively updated until it converges. The main drawback of the k-means is the random choice of initial clusters. In K-means, we assume that each attribute has the same weight, i.e., all attributes are assumed to have the same contribution towards clustering. In addition, it gets stuck in local optima, and we can observe from Table III that the performance of the K-means algorithm is the worst of all the approaches. Unlike K-means, SOM gives weights for each attribute by using the neighbourhood concept and iteratively computes the cluster centers. The main drawback of SOM is ineffective in clustering non-linearly separable datasets as the hidden layers are absent. The EM algorithm iteratively computes the cluster centers using the maximum likelihood approach. The performance of EM is better than SOM and K-means as it is a soft clustering algorithm. The DBSCAN finds core samples of high density and expands clusters from them. However, as the quality of DBSCAN also depends on the distance measure used in the function region, it becomes challenging at higher dimensions to find appropriate separation or clusters. The USELM and BELMKN perform better in comparison to K-means, DBSCAN, SOM and EM. USELM and BELMKN use ELM with K-means and Kohonen Network as the clustering algorithms, respectively. As BELMKN uses the Kohonen Network, this clustering algorithm overcomes the stated drawbacks of K-means. As both USELM and BELMKN use ELM for feature learning, they suffer from the drawbacks of ELM. ELM is a single hidden layer neural network where the weights between the input layer and hidden layer are randomly initialized and the weights between the hidden layer and the output layer are computed using a closed-form solution. This sometimes turns out to be a major drawback as it increases the amount of randomness in the network and hence the clustering results obtained through ELM feature learning are not consistent which may result in either underfitting or overfitting. From Table III, we can observe that the standard deviation for USELM and BELMKN is greater than 1 for all the datasets. For the datasets that are highly non-linear like Balance, Credit, Dermatology, Glass and Heart, the standard deviation is very high compared to the other datasets which indicate that the clustering results are not consistent for ELM. Hence, for USELM and BELMKN we need to run the program several trials to obtain better accuracy. Another disadvantage of ELM is that it sometimes overfits the data. As a result, more samples are drawn into one of the clusters that dominate the data and the remaining clusters suffer from the sparsity of samples. Hence, we can observe from Table III that for datasets like Balance, Dermatology and Thyroid, the clustering accuracies of the ELM based algorithms USELM and BELMKN are lower than the ones obtained by the proposed DRBM-ClustNet. DRBM-ClustNet is modelled as bipartite graphical model to exploit the generative nature of data and to estimate better probability distributions. Each hidden neuron in a DRBM-ClustNet is a stochastic processing unit, which learns a probability distribution over the inputs. The hidden units in a DRBM-ClustNet capture higher-order correlations between the inputs. In DRBM-ClustNet, the convergence to the minima is achieved in two steps, namely based on the positive contrastive divergence and the negative contrastive divergence, which is accomplished by Gibbs sampling [20]. Hence, the convergence to a global minimum is better guaranteed when compared to ELM-based feature learning. As a result, the clustering accuracy for the datasets using DRBM-ClustNet is higher when compared to USELM and BELMKN. In addition, there are higher accuracies obtained by DRBM-ClustNet for the non-linear datasets like Balance, Credit, Dermatology, Glass and Heart. As DRBM-ClustNet is less immune to the nonlinearity in the dataset, it does not overfit the data. The RBM network with one hidden layer is not sufficient to capture the non-linearity of a dataset even with an increase in the number of hidden neurons in that layer. This is evident for almost all the datasets from Table III. The proposed DRBM-ClustNet overcomes this problem during the processing of input data. Initially it has one hidden layer but the output of this layer is taken to the next hidden layer and a stack of RBM layers are built up producing a better generative model. The DRBM-ClustNet has the potential to learn internal representations of the data that are increasingly complex. As a result, DRBM-ClustNet performs better in terms of accuracy when compared to the RBM with one hidden layer and also in comparison with the other clustering algorithms used in this study. Furthermore, Table III shows that the clustering results using single layer RBM and DRBM-ClustNet are consistent for every dataset. The standard deviation is less than about 0.7 and 0.41 in all cases for the single layer RBM and DRBM-ClustNet, respectively. The proposed approach also works for clustering of high dimensional and highly imbalanced data. This can be observed by looking at the results in the Table for the SECOM data which has 590 attributes and a minority class of 6%. Here, it can be stated that the DRBM-ClustNet not only outperforms the other algorithms in terms of accuracy but also does well when a minority class is in the data. From Table III, for the Thyroid dataset, average clustering accuracies with RBM with one hidden layer and DRBM-ClustNet is 94.1% and 92.0%, respectively. The same kind of results are also observed for the Cancer dataset. There is an exception here in that the feature extraction with RBM with one hidden layer produces a higher clustering accuracy when compared to the DRBM-ClustNet. This occurs because the Thyroid dataset is a more linearly separable dataset. Increasing the number of hidden layers for a simple dataset may result in overfitting. As a result, a drop in the accuracy of the DRBM-ClustNet is observed. Overall, from Table III it is evident that for 13 out of 15 datasets (i.e. 86.6%) the DRBM-ClustNet performs best in terms of accuracy when compared to all other clustering approaches. Table IV shows the average clustering accuracy for each of the clustering approaches applied on the 15 benchmark datasets. Here, we can infer that the DRBM-ClustNet has the maximum average clustering accuracy. The single-layer RBM ranks second, then BELMKN, followed by USELM. Amongst the traditional clustering approaches, EM fares better than DBSCAN, SOM and K-means. Table V represents the sum of the ranks for all the datasets and clustering approaches shown in Table III. The ranking based on the sum of ranks indicates that the proposed DRBM-ClustNet outperforms the other clustering approaches. The Single RBM, BELMKN, EM, USELM, SOM, K-means and DBSCAN follow DRBM-ClustNet in ranking order. F. Assessment of the clustering on image datasets In this section, the proposed algorithm is compared with state-of-the-art algorithms based on RBM, namely Graph RBM, mixed GraphDBN (mGraphDBN) and full GraphRBMbased DBN (fGraphDBN) [54]. The performance of the DRBM-ClustNet algorithm is assessed on three publicly available image datasets, namely, COIL-20, the Extended Yale database B (YaleB) and MNIST. All the observations in COIL-20 and YaleB are used for experimentation. In the case of MNIST data, 60000 training dataset observations are used in the experimental study as in [54]. The Whitening Principal Component Analysis is applied for reducing the dimension of all the three image datasets to 400 as in [54]. The setting of DRBM-ClustNet remains the same as in the previous section. The obtained results are also compared with the normalized mutual information (NMI) as performance metric [54] [55] and tabulated in Table VI. The populated values are from ten runs and indicate the mean with its standard deviation. The performance of other algorithms in [54] is listed for reference in the Table. From Table VI it can be observed that DRBM-ClustNet performs best for the MNIST and YaleB datasets. In the case of COIL-20, DRBM-ClustNet performs better than GraphRBM and mGraphDBN but shows a fractionally lower performance than fgraphDBN. This performance could be attributed to better subspace clustering by DRBM-ClustNet in the final stage, which uses the Kohonen network. Furthermore, it is also observed that the average number of clusters predicted by the second stage of DRBM-ClustNet for COIL-20, YaleB, and MNIST datasets are 19 ± 0.87, 38 ± 0.61, and 10 ± 0.21, respectively. The actual number of clusters for COIL-20, [54] YaleB, and MNIST dataset are 20, 38, and 10, respectively. This indicates the ability of DRBM-ClustNet to accurately pick the number of clusters without having to define them explicitly like other methods. Overall, these results demonstrate the superior performance of the DRBM-ClustNet framework in comparison to other state-of-the-art RBM algorithms for various non-linear image datasets. Assessment of the flooded region using satellite imagery: The DRBM-ClustNet is further evaluated for clustering actual satellite images of the flood-prone region. This application would aid in the quick identification of the extent of floodprone regions. Here, three MODIS satellite images of various stages, namely, before the flood, during the flood and after the flood are used as shown in Fig. 7. The dimension of each of the three images is 482 x 627 pixels. Details of the MODIS images used in this study are discussed in [56]. All three images were processed using DRBM-ClustNet for spectral clustering to assess the extent of water (flood) cover. The extracted flood regions across all three images are highlighted in the figure with blue (actual river) and red (flooded region) colors. MODIS satellite images are of low spatial resolution (250 m), hence the study prominently focuses on city granularity for the current application. Fig. 7 (b) shows results of the proposed approach during a flood timeline wherein white dots signify cities that were not flooded while the flooded cities are indicated with white dots with black dots inside. Here, it can be observed that the DRBM-ClustNet is able to correctly identify flooded regions for 12 out of 15 cities with just one false positive. Other clustering approaches such as K-means and SOM are able to identify only 6 and 8 cities with false positives being 4 and 5 respectively. These numbers reflect the inability to identify the majority of flooded regions accurately. In the case of before the flood and after the flood, K-means clustering accuracies were 83.87% and 85.21% and SOM clustering accuracies were 87.54% and 89.05%, respectively. However, the clustering accuracy of V. CONCLUSION In this paper, DRBM-ClustNet is proposed, a Bayesian Deep Restricted Boltzmann-Kohonen Network based on clustering. DRBM-ClustNet uses three stages to maximize the clustering efficiency, particularly for non-linearly separable datasets. The first stage uses the DRBM, a generative model built by stacking multiple layers of RBM for feature learning. The second stage uses BIC to calculate the number of actual clusters in the dataset. The final stage uses the Kohonen Network to cluster the feature extracted data from DRBM using BIC predicted clusters. The parameters of the DRBM-ClustNet are set by the grid search approach, i.e., to determine the number of hidden layers and the number of hidden neurons to obtain a better clustering accuracy. The data clustering task was successfully accomplished with DRBM-ClustNet on two synthetic datasets, fifteen UCI repository benchmark data and four image datasets.The results demonstrate that the proposed DRBM-ClustNet approach outperforms the other clustering techniques used in experiments. DRBM-ClustNet is also able to accurately predict the number of clusters for various datasets, giving it a further edge over other techniques. The obtained results show that DRBM-ClustNet is a very consistent, efficient and reliable algorithm for clustering of non-linearly separable datasets. APPENDIX Recall, in DRBM-ClustNet framework the stage 1 consists of DRBM, here the joint probability of the visible and hidden units is given by P (v, h) = e −E(v,h) v h e −E(v,h) ,(23) where E(v, h) = − i,j v i h j W ij − i v i b i − j h j c j ,(24)P (v, h) = 1 Z e −E(v,h) ,(25) where Z is called the partition function which is given by Z = v h e −E(v,h) .(26) The partition function Z is intractable and the joint probability P (v, h) is also intractable. Hence, we derive the conditional probability distribution P (h\|v) and P (v\|h) from the joint probability distribution P (v, h) which is easy to compute and to sample from. Equation (24) can be expressed in matrix form as E(v, h) = −b T v − c T h − v T W h,(27)P (v, h) = 1 Z e (b T v+c T h+v T W h) ,(28)P (h\|v) = P (h, v) P (v) ,(29)P (h\|v) = 1 P (v) 1 Z e (b T v+c T h+v T W h) .(30) The derivative Z using (28) can be expressed in terms of Z as Z = e (b T v) P (v)Z .(31) Substituting (31) in (30), we get P (h\|v) = 1 Z p j=1 e (cj hj +v T Wij hj ) . Using Bayes theorem P (h j = 1\|v) = P (h j = 1, v) P (h j = 0, v) + P (h j = 1, v) . Substitute h j = 1, in (32) to get P (h j = 1\|v) = sigmoid(c j + v T W ij ).(34) Similarly for P (v\|h) we obtain P (v i = 1\|h) = sigmoid(b i + W ij h).(35) Therefore P (h\|v) = p j=1 sigmoid(c j + v T W ij ),(36) and P (v\|h) = d i=1 sigmoid(b i + W ij h).(37) Figs. 5(a) and 5(b) show the cluster separation for the Wine dataset before applying DRBM and after applying DRBM respectively, using the conditional probability distributions as mentioned above. We can observe that in Fig. 5(a), the clusters are overlapping before applying DRBM and the clusters are not so obvious, making it difficult for BIC to predict the number of clusters and also for SOM to calculate the cluster centroids. In Fig. 5(b), the clusters are well separated which makes it easy for BIC and SOM to predict the number of clusters and calculate the cluster centroids respectively. The same behavior is also observed for the other datasets used for comparing the clustering accuracy. BIC uses the maximum likelihood parameter to determine the number of cluster which can also be expressed in terms of Mean Squared Error (MSS) as follows BIC = ln(N )k − 2ln(L), (38) whereL is the maximum likelihood function Assuming model errors are independent and identically distributed, according to the normal distribution theory h M L = p i=1 P ( d i p ),(39) where h M L is the maximum likelihood hypothesis, p is the number of hidden layers and d i is the output from DRBM h M L = p i=1 e − (d i −µ) 2 2σ 2 √ 2πσ 2 ,(40) where d i = f (x i ) + e i , µ and σ are the mean and variance of normal distribution, f (x i ) is a function of x i and e i is random variable representing noise. Hence, h M L = p i=1 e − (d i −f (x i )) 2 2σ 2 √ 2πσ 2 ,(41)log(h M L ) = p i=1 ln( 1 √ 2πσ 2 ) − (d i − f (x i )) 2 2σ 2 .(42) The first term in the above equation is independent of the function f and it can be discarded, therefore log(h M L ) = p i=1 − (d i − f (x i )) 2 2σ 2 ,(43)− log(h M L ) = − p i=1 − (d i − f (x i )) 2 2σ 2 = RSS n = M SE. (44) As BIC is expressed in terms of MSE as described above and as DRBM makes the cluster separation linearly separable as observed in Fig. 5(b), it becomes easy for BIC to predict the number of clusters. SOM uses the Gaussian neighborhood function to find the nearest neighbors and uses Euclidean distance to find the winning neuron. Hence, SOM works well for datasets which are more linearly separable and for datasets where the clusters are non-overlapping. Hence, applying a DRBM transformation on the existing dataset facilitates SOM to calculate the cluster centroids more accurately which in turn increases SOM clustering efficiency. VI. ACKNOWLEDGEMENT Fig. 1 : 1DRBM-ClustNet architecture Fig. 2 :Fig. 3 : 23Clustering of flame pattern distribution using (a) k-means; (b) SOM; (c) EM; (d) DBSCAN; (e) US-ELM; (f) BELMKN; (g) Single Layer RBM; and (h) DRBM-ClustNet Clustering of moon pattern distribution using (a) k-means; (b) SOM; (c) EM; (d) DBSCAN; (e) US-ELM; (f) BELMKN; (g) Single Layer RBM; and (h) DRBM-ClustNet Dataset 8: The Glass data has details for the oxide content and refractive index for different glass types. It consists of 6 classes, 214 observations and 9 attributes. Dataset 9: The Heart data consists of details for the diagnosis of heart diseases. It consists of 2 classes, 270 observations and 13 attributes. Dataset 10: The Horse data has the details for the different types of horses. It consists of 3 classes, 364 observations and 26 attributes. Dataset 11: The Iris data classifies a sample into one of the three types of flower classes, namely, setosa or virginica or versicolor. It has 150 observations and 4 attributes, namely, width of petals, length of petals, the width of sepals and length of the sepals. Dataset 12: The SECOM data has signals from various sensors during the process of manufacturing semiconductors. There are 1567 observations and 590 features. The dataset consists of two classes, non-fail and fail, and is highly imbalanced. The missing values in attributes were imputed with zeroes. Dataset 13: The Thyroid data consists of three classes, namely, normal function, under functional and over-functional. There are 215 observations and 5 attributes. Dataset 14: The Vehicle data consists of different types of vehicles, namely, bus, Opel, van and Saab. There is a total of 846 observations and 8 attributes. Dataset 15: The Wine data lists different types of chemical analysis of wine. It consists of 3 classes, 178 observations and 13 attributes. Fig. 4 : 4Parallel coordinates plot for (a) Actual Iris dataset, (b) Iris dataset after DRBM feature extraction Fig. 5 : 5Scatter plot for (a) Actual Wine dataset, (b) Wine dataset after DRBM feature extraction Fig. 6: Correlation matrix plot for (a) Original Iris dataset, (b) Iris dataset after RBM feature extraction Fig. 7 : 7Assessment flooded region: (a) before flood, (b) during flood, (c) after flood DRBM-ClustNet for before the flood and after the flood were 97.3% and 98%, respectively. Overall, the DRBM-ClustNet extracted images show a better performance than the other considered approaches. Therefore, DRBM-ClustNet can be further explored for other practical image clustering tasks. TABLE I : IProperties of the DatasetsSl. No Dataset Number of Input Number of name samples Dimensions Cluster 1 Balance 625 4 3 2 Cancer 569 30 2 3 Cancer Int 699 9 2 4 Credit 690 14 2 5 Dermatology 366 34 6 6 Diabetes 768 8 2 7 E.Coli 327 7 5 8 Glass 214 9 6 9 Heart 270 13 2 10 Horse 364 26 3 11 Iris 150 4 3 12 SECOM 1567 590 2 13 Thyroid 215 5 3 14 Vehicle 846 18 4 15 Wine 178 13 3 TABLE II : IICluster Prediction using BIC for 15 datasetsDataset Actual BIC on ELM with Single Layer DRBM Clusters actual data BIC RBM BIC with BIC Balance 3 5 ± 0.04 7 ± 0.31 5 ± 0 3 ± 0 Cancer 2 3 ± 0.47 3 ± 0.26 3 ± 0 3 ± 0 Cancerint 2 2 ± 0.63 3 ± 0.37 3 ± 0 3 ± 0 Credit 2 2 ± 0.39 5 ± 0.31 4 ± 0 6 ± 0 Dermatology 6 5 ± 0.99 6 ± 0.87 6 ± 0.44 6 ± 0.39 Diabetes 2 2 ± 0.93 3 ± 0.79 5 ± 0.31 5 ± 0.22 E.Coli 5 4 ± 0.67 5 ± 0.61 5 ± 0 5 ± 0 Glass 6 3 ± 0.81 6 ± 0.73 6 ± 0 6 ± 0 Heart 2 2 ± 1.07 3 ± 1.18 4 ± 0.94 4 ± 0.81 Horse 3 2 ± 0.75 2 ± 0.88 2 ± 0.18 3 ± 0.18 TABLE III : IIIClustering accuracy percentage with standard deviation and ranking (in parenthesis below) of various techniques on each dataset Dataset K-means SOM EM USELM BELMKN DBSCAN Single RBM DRBM-ClustNet Balance 51.4 ± 0.9 52.5 ± 0.2 53.0 ± 0.6 45.9 ± 3.3 47.1 ± 3.4 52.5 ± 0.5 68.1 ± 0.16 70.8 ± 0.06 (6) (5) (3) (8) (7) (4) (2) (1) Cancer 83.2 ± 1.9 86.0 ± 1.0 91.2 ± 1.6 83.8 ± 1.6 86.7 ± 2.0 88.8 ± 0.8 95.0 ± 0.2 93.4 ± 0.2 (8) (6) (3) (7) (5) (4) (1) (2) Cancerint 95.8 ± 1.1 94.2 ± 0.4 93.6 ± 1.6 92.0 ± 1.5 92.5 ± 1.8 91.9 ± 0.9 96.1 ± 0.7 97.0 ± 0.41 (3) (4) (5) (7) (6) (8) (2) (1) Credit 52.2 ± 2.8 54.8 ± 1.2 51.4 ± 1.8 58.5 ± 4.1 61.5 ± 3.8 67.1 ± 1.3 85.3 ± 0.13 86.5 ± 0.26 (7) (6) (8) (5) (4) (3) (2) (1) Dermatology 24.2 ± 3.9 29.8 ± 1.5 67.7 ± 0.4 71.0 ± 3.9 76.6 ± 3.6 63.2 ± 0.9 92.8 ± 0.4 95.5 ± 0.28 (8) (7) (5) (4) (3) (6) (2) (1) Diabetes 63.6 ± 1.5 64.8 ± 0.8 52.4 ± 0.5 65.0 ± 1.9 67.9 ± 2.4 61.7 ± 0.8 69.2 ± 0.23 71.0 ± 0.25 (6) (5) (8) (4) (3) (7) (2) (1) E.Coli 53.8 ± 5.7 61.0 ± 3.8 82.0 ± 0.5 80.1 ± 2.7 80.7 ± 1.8 52.8 ± 4.2 80.9 ± 1.2 83.1 ± 0.19 (7) (6) (2) (5) (4) (8) (3) (1) Glass 52.1 ± 3.8 53.8 ± 2.9 47.7 ± 0.4 40.2 ± 4.1 41.5 ± 3.5 39.8 ± 3.7 54.4 ± 0.21 57.6 ± 0.26 (4) (3) (5) (7) (6) (8) (2) (1) Heart 58.2 ± 3.4 58.8 ± 2.2 52.6 ± 1.6 66.6 ± 2.2 71.4 ± 2.9 58.3 ± 2.0 79.2 ± 0.27 82.4 ± 0.24 (7) (5) (8) (4) (3) (6) (2) (1) Horse 49.7 ± 4.4 48.3 ± 2.3 43.4 ± 1.9 58.6 ± 4.8 60.9 ± 4.1 47.6 ± 3.1 62.0 ± 0.56 65.3 ± 0.26 (5) (6) (8) (4) (3) (7) (2) (1) Iris 88.6 ± 0 81.3 ± 0.8 90.0 ± 0.3 91.1 ± 5.3 92.3 ± 4.8 77.2 ± 1.8 91.5 ± 0.33 93.8 ± 0.33 (6) (7) (5) (4) (2) (8) (3) (1) SECOM 62.2 ± 2.2 60.6 ± 0.3 66.1 ± 0.7 65.9 ± 1.2 69.4 ± 2.3 61.6 ± 0.9 73.1 ± 0.40 73.5 ± 0.28 (6) (8) (4) (5) (3) (7) (2) (1) Thyroid 85.8 ± 2.2 86.2 ± 0.3 90.1 ± 0.7 86.5 ± 1.2 87.6 ± 2.3 85.6 ± 0.9 94.1 ± 0.09 92.0 ± 0.09 (7) (6) (3) (5) (4) (8) (1) (2) Vehicle 44.0 ± 2.2 44.0 ± 1.5 45.0 ± 1.8 40.5 ± 3.8 42.0 ± 2.2 43.3 ± 1.5 46.0 ± 0.62 48.4 ± 0.22 (4) (4) (3) (8) (7) (6) (2) (1) Wine 70.0 ± 3.4 75.0 ± 0.0 90.4 ± 0.3 91.9 ± 1.1 95.5 ± 1.7 74.8 ± 0.4 95.8 ± 0.08 97.2 ± 0.08 (6) (8) (4) (5) (3) (7) (2) (1) TABLE IV : IVAverage clustering accuracy and general ranking of the techniques for all datasetsDataset K-means SOM EM USELM BELMKN DBSCAN Single RBM DRBM-ClustNet Average 62.32 63.41 67.77 69.16 71.55 64.41 78.92 80.59 Rank 8 7 5 4 3 6 2 1 TABLE V : VThe sum of ranking of the techniques and general ranking based on total rankingDataset K-means SOM EM USELM BELMKN DBSCAN Single RBM DRBM-ClustNet Average 92 84 75 81 63 97 30 17 Rank 7 6 4 5 3 8 2 1 TABLE VI : VIClustering performance (NMI value) on the COIL-20, YaleB and MNIST datasets Algorithms COIL-20 YaleB MNIST SSC 1 80.72 ± 0.88 63.42 ± 0.79 68.35 ± 0.02 LSC 1 65.25 ± 2.66 41.96 ± 0.98 74.06 ± 2.75 SCC 1 88.23 ± 1.37 34.39 ± 0.61 57.00 ± 0.04 LRR 1 65.60 ± 1.02 10.19 ± 0.18 46.10 ± 0.01 LRSC 1 67.55 ± 1.52 14.94 ± 0.39 50.96 ± 0.00 LSR1 1 72.33 ± 0.88 66.93 ± 1.61 45.73 ± 0.01 LSR2 1 72.18 ± 1.08 67.04 ± 1.62 45.72 ± 0.02 GGMM 1 79.53 ± 1.26 12.37 ± 0.31 62.34 ± 1.69 GSC 1 78.02 ± 3.58 26.55 ± 1.00 70.57 ± 2.95 GNMF 1 72.00 ± 0.08 49.10 ± 1.37 72.00 ± 1.01 gSAE 1 57.10 ± 2.88 16.53 ± 1.40 17.12 ± 0.54 RBM 1 73.81 ± 1.79 18.53 ± 1.64 53.50 ± 1.61 DBN 1 76.32 ± 2.63 21.77 ± 1.57 52.79 ± 2.44 DAE 1 75.29 ± 1.73 18.37 ± 2.00 50.57 ± 1.31 SDAE 1 77.72 ± 0.51 20.99 ± 1.84 52.81 ± 1.61 GraphRBM 1 94.46 ± 2.14 73.03 ± 1.32 86.66 ± 1.23 mGraphRBM 1 95.19 ± 1.98 73.36 ± 0.93 88.73 ± 1.49 fGraphRBM 1 95.54 ± 1.43 75.18 ± 1.29 89.97 ± 0.98 DRBM-ClustNet 95.06 ± 1.56 76.06 ± 1.11 90.11 ± 1.03 1 Reproduced from A Comprehensive Survey of Clustering Algorithms. 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[ "A spectroscopy study of nearby late-type stars, possible members of stellar kinematic groups ⋆ ⋆⋆", "A spectroscopy study of nearby late-type stars, possible members of stellar kinematic groups ⋆ ⋆⋆" ]	[ "J Maldonado \nDpto. Física Teórica\nFacultad de Ciencias\nUniversidad Autónoma de Madrid\nMódulo 15, Campus de CantoblancoE-28049MadridSpain\n", "R M Martínez-Arnáiz \nDpto. Astrofísica\nUniversidad Complutense de Madrid\nFacultad Ciencias FísicasE-28040MadridSpain\n", "C Eiroa \nDpto. Física Teórica\nFacultad de Ciencias\nUniversidad Autónoma de Madrid\nMódulo 15, Campus de CantoblancoE-28049MadridSpain\n", "D Montes \nDpto. Astrofísica\nUniversidad Complutense de Madrid\nFacultad Ciencias FísicasE-28040MadridSpain\n", "B Montesinos \nCentro de Astrobiología\nLaboratorio de Astrofísica Estelar y Exoplanetas\nLAEX-CAB (CSIC-INTA)\nESAC Campus\nVillanueva de la CañadaP.O. BOX 78E-28691MadridSpain\n" ]	[ "Dpto. Física Teórica\nFacultad de Ciencias\nUniversidad Autónoma de Madrid\nMódulo 15, Campus de CantoblancoE-28049MadridSpain", "Dpto. Astrofísica\nUniversidad Complutense de Madrid\nFacultad Ciencias FísicasE-28040MadridSpain", "Dpto. Física Teórica\nFacultad de Ciencias\nUniversidad Autónoma de Madrid\nMódulo 15, Campus de CantoblancoE-28049MadridSpain", "Dpto. Astrofísica\nUniversidad Complutense de Madrid\nFacultad Ciencias FísicasE-28040MadridSpain", "Centro de Astrobiología\nLaboratorio de Astrofísica Estelar y Exoplanetas\nLAEX-CAB (CSIC-INTA)\nESAC Campus\nVillanueva de la CañadaP.O. BOX 78E-28691MadridSpain" ]	[]	Context. Nearby late-type stars are excellent targets for seeking young objects in stellar associations and moving groups. The origin of these structures is still misunderstood, and lists of moving group members often change with time and also from author to author. Most members of these groups have been identified by means of kinematic criteria, leading to an important contamination of previous lists by old field stars. Aims. We attempt to identify unambiguous moving group members among a sample of nearby-late type stars by studying their kinematics, lithium abundance, chromospheric activity, and other age-related properties. Methods. High-resolution echelle spectra (R ∼ 57000) of a sample of nearby late-type stars are used to derive accurate radial velocities that are combined with the precise Hipparcos parallaxes and proper motions to compute galactic-spatial velocity components. Stars are classified as possible members of the classical moving groups according to their kinematics. The spectra are also used to study several age-related properties for young late-type stars, i.e., the equivalent width of the lithium Li i 6707.8 Å line or the R ′ HK index. Additional information like X-ray fluxes from the ROSAT All-Sky Survey or the presence of debris discs is also taken into account. The different age estimators are compared and the moving group membership of the kinematically selected candidates are discussed. Results. From a total list of 405 nearby stars, 102 have been classified as moving group candidates according to their kinematics. i.e., only ∼ 25.2 % of the sample. The number reduces when age estimates are considered, and only 26 moving group candidates (25.5% of the 102 candidates) have ages in agreement with the star having the same age as an MG member.	10.1051/0004-6361/201014948	[ "https://arxiv.org/pdf/1007.1132v1.pdf" ]	119,209,183	1007.1132	944f5fc40a4759106bbed01a97679327dbc4faa3	A spectroscopy study of nearby late-type stars, possible members of stellar kinematic groups ⋆ ⋆⋆ 7 Jul 2010 July 8, 2010 J Maldonado Dpto. Física Teórica Facultad de Ciencias Universidad Autónoma de Madrid Módulo 15, Campus de CantoblancoE-28049MadridSpain R M Martínez-Arnáiz Dpto. Astrofísica Universidad Complutense de Madrid Facultad Ciencias FísicasE-28040MadridSpain C Eiroa Dpto. Física Teórica Facultad de Ciencias Universidad Autónoma de Madrid Módulo 15, Campus de CantoblancoE-28049MadridSpain D Montes Dpto. Astrofísica Universidad Complutense de Madrid Facultad Ciencias FísicasE-28040MadridSpain B Montesinos Centro de Astrobiología Laboratorio de Astrofísica Estelar y Exoplanetas LAEX-CAB (CSIC-INTA) ESAC Campus Villanueva de la CañadaP.O. BOX 78E-28691MadridSpain A spectroscopy study of nearby late-type stars, possible members of stellar kinematic groups ⋆ ⋆⋆ 7 Jul 2010 July 8, 2010Received ; AcceptedAstronomy & Astrophysics manuscript no. 14948stars: activity -stars: ages -stars:late-type -stars: kinematics -open clusters and associations: general Context. Nearby late-type stars are excellent targets for seeking young objects in stellar associations and moving groups. The origin of these structures is still misunderstood, and lists of moving group members often change with time and also from author to author. Most members of these groups have been identified by means of kinematic criteria, leading to an important contamination of previous lists by old field stars. Aims. We attempt to identify unambiguous moving group members among a sample of nearby-late type stars by studying their kinematics, lithium abundance, chromospheric activity, and other age-related properties. Methods. High-resolution echelle spectra (R ∼ 57000) of a sample of nearby late-type stars are used to derive accurate radial velocities that are combined with the precise Hipparcos parallaxes and proper motions to compute galactic-spatial velocity components. Stars are classified as possible members of the classical moving groups according to their kinematics. The spectra are also used to study several age-related properties for young late-type stars, i.e., the equivalent width of the lithium Li i 6707.8 Å line or the R ′ HK index. Additional information like X-ray fluxes from the ROSAT All-Sky Survey or the presence of debris discs is also taken into account. The different age estimators are compared and the moving group membership of the kinematically selected candidates are discussed. Results. From a total list of 405 nearby stars, 102 have been classified as moving group candidates according to their kinematics. i.e., only ∼ 25.2 % of the sample. The number reduces when age estimates are considered, and only 26 moving group candidates (25.5% of the 102 candidates) have ages in agreement with the star having the same age as an MG member. Introduction Last years have been very productive in identifying small associations and kinematic groups of young late-type stars in the solar vicinity. Although the study of moving groups (MGs) goes back more than one century, their origin and evolution remain still unclear, and this term is commonly used in the literature to indicate any system of stars sharing a common spatial motion. The best-studied MGs are the so-called classical MGs. Examples are Castor, IC2391, Ursa Major, the Local Association and the Hyades (e.g. Montes et al. 2001b;López-Santiago et al. 2006, 2009, 2010, and references therein). Send offprint requests to: J. Maldonado, e-mail: [email protected] ⋆ Based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC) and observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias ⋆⋆ Appendices and Tables 5-15 are only available in the electronic version of the paper. Table 1 is also available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ In the classical theory of MGs developed by O. Eggen (Eggen 1994), moving groups are the missing link between stars in open clusters and associations on one hand and field stars on the other. Open clusters are disrupted by the gravitational interaction with massive objects in the Galaxy (like giant molecular clouds), and as a result, the open cluster members are stretched out into a "tube-like" structure and dissolve after several galactic orbits. The result of the stretching is that the stars appear, if the Sun happens to be inside the "tube", all over the sky, but they may be identified as a group through their common space velocity. Clusters disperse on time scales of a few hundred years (Wielen 1971); therefore, most of these groups should be moderately young (∼ 50 -650 Myr). However, Eggen's hypothesis is controversial and some of the MGs may also be the result of resonant dynamical structures. For instance, Famaey et al. (2007) studied a large sample of stars in the Hyades MG, and determined that it is a mixture of stars evaporated from the Hyades cluster and a group of older stars trapped at a resonance. MGs may also be produced by the dissolution of larger stellar aggregates, such as stellar complexes or fragments of old spiral arms. The young MGs (8 -50 Myr) are probably the most immediate dissipation products of the youngest associations. Examples of such associations are TW Hya, β Pic, AB Dor, η Cha, ǫ Cha, Octans, Argus, the Great Austral complex (GAYA), and the Hercules-Lyra association Torres et al. 2008;Fuhrmann 2004;López-Santiago et al. 2006;Montes 2010). Some of the young MGs are in fact related to star-forming regions like the Scorpius-Centaurs-Lupus complex , Ophiuchus or Corona Australis (Makarov 2007). The availability of accurate parallaxes provided by the Hipparcos satellite became a milestone in the study of MGs. Statistical, unbiased studies of large samples of stars have confirmed the existence of the classical MGs and have given rise to new clues and theories about the origin of such structures. Examples of these studies are those by Chereul et al. (1999), Asiain et al. (1999), Skuljan et al. (1999), and Antoja et al. (2008). Identifying a star or group of stars as members of an MG is not a trivial task, and in fact, lists of members change among different works. Most members of MGs have been identified by means of kinematic criteria; however, this is not sufficient since many old stars can share the same spatial motion of those stars in MGs. For example, López-Santiago et al. (2009) show that among previous lists of Local Association members, roughly 30% are old field stars. The membership issue can be partially solved if high-resolution spectroscopy is used. Recent studies have shown that stars belonging to a given MG share similar spectroscopic properties (e.g. Montes et al. 2001a;López-Santiago et al. 2009,2010. These studies exploit the many advantages of the nearby late-type stars. First, spectra of late-type stars are full of narrow absorption lines, allowing determination of accurate radial velocities. In addition, it is unlikely that an old star by chance shares chromospheric indices or a lithium abundance similar to those of young solar-like stars, which provides means for assessing the likelihood of membership of a given star that are independent of its kinematics (e.g. Soderblom & Mayor 1993b). In this paper we present a search for classical MG members by analysing the kinematic and spectroscopic properties of a sample of nearby late-type stars. Section 2 describes the stellar sample and the observations and data reduction are described in Section 3. A detailed analysis of the kinematic properties of the stars is given in Section 4. Age indicators for solar-like stars are analysed in Section 5. A combination of the results from Sections 4 and 5 is used in Section 6 to analyse the MG membership of the stars. Section 7 summarizes our results. The stellar sample, observations, and data reduction Our reference stellar sample consists of main-sequence (luminosity classes V/IV-V) FGK stars located at distances less than 25 pc. The stars have been selected from the the Hipparcos catalogue (ESA 1997), since it constitutes a homogeneous database especially for distance estimates -parallax errors are typically about 1 milliarcsec. In this work we have taken the revised parallaxes computed by van Leeuwen (2007) from Hipparcos' raw data. No other selection criteria have been applied to the sample. The sample is most likely complete for FG-type stars; i.e., it constitutes a volume-limited sample since the Hipparcos catalogue is complete for these spectral types. In the case of K-type stars, Hipparcos is incomplete beyond ∼ 15 pc; however, the number of the K-type stars is high enough for our purposes. The final selection contains 126, 220, and 477 stars of spectral types F,G and K respectively. In this contribution we present our first results for an observed subsample of 405 stars. The completeness of the observed sample can be seen in Figure 1 where the number of objects is plotted as a function of distance, and the distribution fits well a cubic law, which indicates that they are homogeneously distributed. M-type stars have in principle been excluded from this study; nevertheless, six M-type stars, members or candidate member of MGs, which exhibit high levels of chromospheric activity and are suspected to be young, have been included in order to better understand the properties of such stellar groups. The observed stars are listed in Table 1, and Figure 2 shows the HR diagram of the sample. Several stars are clearly under the main sequence: HIP 4845, HIP 42525, HIP 49986, HIP 57939, HIP 72981, and HIP 96285. Hipparcos' spectral types for these stars are quite similar to those reported in other catalogues such as Wright et al. (2003), Skiff (2009), or SIMBAD. Only for HIP 72981 is incomplete, giving simply 'K:', whereas SIMBAD gives M1 and the most updated reference in Skiff (2009) gives M2. However, the colour index B − V = 1.17 suggests an early type, around K5. HIP 42525 is a star in a double system and has a large uncertainty in the parallax (σ π = ±15.51 mas). Stars with uncertainties over 10 milliarcsec are identified with a symbol † in Table 1. The original selection (and therefore the observations) of the sample was made before the release of the revised Hipparcos parallaxes (van Leeuwen 2007), and some of our stars are now out of the 25 pc distance because their revised parallaxes are slightly smaller. These stars are identified with a symbol ‡ in Table 1. The most "extreme" case is HIP 1692 whose parallax has changed from 43.42 ± 1.88 mas to 3.23 ± 1.43 mas. This new parallax places the star in the giant branch as is clearly shown in Figure 2 (square in the upper right corner). Fig. 2. HR Diagram for our sample of nearby late-type stars. Ftype stars are plotted with circles; G-type stars with triangles; K-type stars with squares and M-type stars with stars. Observations and data reduction High-resolution spectra of 315 stars were obtained at the Calar Alto (Almería, Spain) and La Palma (Canary Islands, Spain) observatories during eight observing runs. Some stars (the most active ones) were observed more than once. The Calar Alto observations were taken with the fiber optics echelle spectrograph FOCES (Pfeiffer et al. 1998) attached at the Cassegrian focus of the 2.2 meter telescope. FOCES spectra have a resolution of ∼ 57000 and cover a spectral range λλ 3800 -10000 Å. La Palma observations were done at the 3.56 m Telescopio Nazionale Galileo (TNG) using the cross dispersed echelle spectrograph SARG (Gratton et al. 2001). In this case the resolution and spectral range are ∼ 57000 and λλ 4960-10110 Å, respectively. Further details are given in Table 2. The spectra were reduced using the standard procedures in the IRAF 1 packages imred, ccdred, and echelle, i.e. overscan, scattered light correction, and flat-fielding. Spectral orders were extracted with the routine apall and were normalized using the IRAF task continuum in order to compare the intensity of the lines and to measure equivalent widths. Thorium-Argon spectra were used for wavelength calibration. Figure 3 shows examples of representative stars in different spectral regions. Since observations were done from northern observatories, most targets have δ > −25 o . Therefore additional spectra from public libraries have also been analysed. Specifically, 90 spectra were taken from the public library "S 4 N" (Allende Prieto et al. 2004), which contains spectra taken with the 2dcoudé spectrograph at Mc Donald Observatory and the FEROS instrument at the ESO 1.52 m telescope in La Silla. Both the resolution and the spectral range are similar to those of our own observations (R ∼ 57000, λλ 3620-9210 Å). FEROS spectra contribute partially to covering for the lack of southern targets. 1 IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation. Kinematic analysis Radial velocities Radial velocities were measured by cross-correlating order by order, using the IRAF routine fxcor, the spectra of our programm stars with spectra of radial velocity standard stars of similar spectral types (Table 3), taken from Barnes et al. (1986), Beavers et al. (1979), and Udry et al. (1999a,b). Spectral orders with chromospheric features and prominent telluric lines were excluded when determining the mean radial velocity. Typical uncertainties are between 0.15 and 0.25 km/s, while maximum uncertainties are around 1-2 km/s. Column 9 of Table 1, gives our results for the radial velocities. A large number of stars in our sample (51) are known spectroscopic binaries and are listed in The 9th catalogue of spectroscopic binaries (Pourbaix et al. 2004, hereafter SB9) and The 3rd Catalogue of Chromospherically Active Binary Stars (Eker et al. 2008). They are identified in Table 1 with the label 'Spec. Binary'. For those stars we have considered the radial velocity of the centre of mass of the system. We have compared our results with radial velocity estimates by Kharchenko et al. (2007, hereafter KH07), Nordström et al. (2004, hereafter NO04), Valenti & Fischer (2005, hereafter VF05), and Nidever et al. (2002, hereafter NI02). These values are also given columns 10 to 13 of Table 1. Of the 405 stars in our sample, 366 are found in KH07, and the differences among the radial velocity values in that work and our results are less than 2 km/s for 290 stars, i.e., 79.2% of the common stars. A comparison with the NO04 data shows that, for 215 out of 251 common stars (i.e. 85.6%), the corresponding differences between the radial velocities are less than 2 km/s. A similar result, 85.3% (177 out of the 151 common stars), is found when considering VF05 data. The comparison with NI02 is even better, because 179 out of 190 stars (i.e. 94.2 %) show differences lower than 2 km/s. Figure 4 illustrates these comparisons. One can see that the differences are slightly greater with KH07, likely because of the non-homogeneous origin of their radial velocities values, mainly taken from The general catalogue or radial velocities (Barbier-Brossat & Figon 2000). Galactic spatial-velocity components (U, V, W) were computed using our radial velocity results listed in Table 1, together with Hipparcos parallaxes (van Leeuwen 2007) and Tycho-2 proper motions (Høg et al. 2000). To compute (U, V, W) we followed the procedure of Montes et al. (2001b) who updated the original algorithm of Johnson & Soderblom (1987) to epoch J2000 in the International Celestial Reference System (ICRS) as described in section 1.5 of The Hipparcos and Tycho Catalogues' (ESA 1997). To take the possible correlation be-tween the astrometric parameters into account, the full covariance matrix was used in computing the uncertainties. To identify possible members of MGs we proceeded in two steps: -i) Selection of young stars. Young stars are assembled in a specific region of the (U, V) plane with (−50 km/s < U < 120 km/s; −30 km/s < V < 0 km/s), although the shape is not a square, see Figure 5. -ii) Selection of possible members of MGs with small V dispersion. Considering previous results (Skuljan et al. 1997(Skuljan et al. , 1999Montes et al. 2001b), a dispersion of 8 km/s in the U, V components with respect to the central position of the MG in the (U, V) plane is allowed. The same dispersion is considered when taking the W component into account. One hundred two stars of the sample have been classified as possible members of the different MGs: 29 for the Local Udry et al. (1999b) b Barnes et al. (1986) c Udry et al. (1999a) d Beavers et al. (1979) Association, 29 for the Hyades, 18 for the Ursa Major, 19 for IC2391, and 7 for Castor. Column 2 of Table 4 lists these numbers, while the specific stars are listed in Tables 9 to 13. Their contents are described in Appendix A. Another 78 stars have been selected as young disc stars. These stars are inside or in the boundaries that determine the young disc population, but their possible inclusion in one of the stellar kinematic groups is not clear. The identified young disc stars are given in Table 14. Figure 5 shows the (U, V) and (W, V) planes, usually known as Bottlinger's diagrams, for these stars. Eggen's astrometric criteria To test whether a star "belongs" to a kinematic group, Eggen tried to establish "strict" criteria for MG-membership (Eggen 1958(Eggen , 1995. Eggen's criteria basically treat MGs, whose stars are extended in space, like open clusters whose stars are concentrated in space. Therefore, it is assumed that the total space velocities of the stars in the MG are parallel and move towards a common convergent point. The same relations of the moving-cluster method for the total and tangential velocities are applied, but taking into account only the components of the proper motion (µ) oriented towards the convergence point (ν) and the component of the proper motion oriented perpendicularly to the great circle between the star and the convergence point (τ). The total (V T ) and tangential velocity (denoted as Peculiar Velocity, PV, by Eggen) can be combined to define a predicted radial velocity (ρ c ). The first membership criterion, namely the peculiar velocity criterion, is to compare the proper motion of the candidate to the proper motion expected if the star were a member of the Kharchenko et al. (2007); top right panel: Nordström et al. (2004); bottom left panel: Nidever et al. (2002); bottom right panel: Valenti & Fischer (2005) MG; i.e., the candidate is accepted as an MG member if the ratio τ/ν or PV/V Total is "sufficiently small". Eggen (1995) considered a candidate to be a member if its peculiar velocity is less than 10% of the total space velocity. The second membership criterion, the radial velocity criterion, compares the observed and the predicted radial velocities. Eggen (1958) considered a star to be a member if the difference between both radial velocities is less than 4-8 km/s. A more detailed discussion of these criteria can be found in Montes et al. (2001b). Table 4 gives the number of stars in each MG that satisfy both criteria (column 3), only the peculiar velocity criterion (column 4), and only the radial velocity criterion (column 5). Only a low percentage of the MGs members selected in the previous section satisfies both criteria (from ≈ 49% in the Local Association to roughly 16% in the IC2391 MG). The results for individual stars are given in columns 8 and 9 of Tables 9 to 13. For both PV (column 8) and ρ c (column 9) criteria, there is a label, 'Y' or 'N', which indicates if the star satisfies the criteria. Local Association 29 14 9 1 IC2391 19 3 7 5 Castor 7 2 0 4 Ursa Major 18 6 1 7 Hyades SC 29 9 5 9 Eggen's criteria are not conclusive since they assumed a constant V within the stars of a given MG. Anticipating some of the results in Section 6, some stars for which both age estimates and (U, V, W) components indicate that they are probable MG members do not satisfy these criteria. Age estimates Members of a given MG should be coeval and moderately young (only several Myr old, see Section 1) therefore it is expected that MGs members share age related-properties, such as similar chromospheric emission or lithium abundance. This provides the means of assessing the likelihood of membership for a given star that is independent of its kinematics. Lithium abundance Lithium abundance in late-type stars is a well-known age indicator since this element is destroyed as the convective motions gradually mix the stellar envelope with the hotter (T ∼ 2.5 × 10 6 K) inner regions. However, it should only be regarded as an additional age indicator when compared with others since Li i equivalent width has a wide spread at a given age and mass, and consequently, the relation lithium-age is poorly constrained. Furthermore, for late K, M-type stars, lithium is burned so rapidly that it is only detectable for extremely young stars. Thus, the use of Li i as an age tracer is biased toward young stars, and it only provides low limits for stars of the age of the Hyades or older. An age estimate of the stars in our sample can be carried out by comparing their Li i equivalent width, with those of stars in well known young open clusters of different ages (e.g. Montes et al. 2001a;López-Santiago et al. 2006). Lithium EWs have been obtained using the IRAF task sbands, performing an integration within a band of 1.6 Å centred in the lithium line. At the spectral resolution of our observations, the Li i 6707.8Å line is blended with the Fe i 6707.41 Å line. To correct for a possible contamination by Fe i, Soderblom et al. (1990) obtained an empirical relationship between the colour index (B − V) and the Fe i equivalent width, measured in stars that showed only the Fe i feature and no Li i. Soderblom's equation was obtained by using main-sequence and subgiant stars, so it does not account for possible luminosity-class effects. Therefore we have built a new relationship, using only main-sequence stars without lithium detected in the spectrum: Fe i (EW) = (0.020 ± 0.005)(B − V) − (0.003 ± 0.0015)(Å). (1) Which is fairly similar to the one obtained by Soderblom: Fe i (EW) = 0.040(B − V) − 0.015(Å).(2) The EWs obtained are shown in column 10 in Tables 9 to 13, for each individual MG and in column 6 in Tables 14 to 15 for the stars classified as Other young disc stars and for the stars not selected as possible MGs, respectively. Figure 6 shows the EW Li i versus colour index (B − V) diagram. We have overplotted the upper envelope of the Li i EW of IC2602 (10-35 Myr) given by Montes et al. (2001a), the Pleiades cluster (78-125 Myr) upper envelope determined by Neuhaeuser et al. (1997), and the lower envelope adopted by Soderblom & Mayor (1993a), as well as the Hyades open cluster (600 Myr) envelope adopted by Soderblom et al. (1990). These clusters cover the range of ages of the MGs studied here (35-600 Myr). Nearly 4 % of the stars are between the Pleiades envelopes, consistent with an age of ∼ 80 Myr. Roughly 8% of the stars are between the Hyades and the Pleiades lower envelope with an age similar to those stars in the Ursa Major ∼ 300 Myr. Stars with lithium EW below the Hyades envelope are likely to be older than 600 Myr. They are around the 23% of the sample. Thus, approximately 35% of the stars are moderately young (younger than 1 Gyr). Roughly 50% of the stars lie below the Pleaides lower envelope (but not below the Hyades' one). For these stars we can only state that they should be older than the Pleiades. Finally, stars with no photospheric Li i detected are expected to be older than 1 Gyr (around 15% of the whole sample). Concerning spectral types, the majority of the F stars are in the Hyades-like region of the diagram, with only five out of 61 stars in the Pleiades-like region. For G-type stars, 71 out of 129 are in the Hyades-like region, 34 in the Ursa Major-like region and only HIP 63742 shows an EW comparable to those stars in the Pleiades. Finally, six out of 209 K-type stars are in the Pleiades-like region and 15 in the Ursa Major-like. The stars with the largest Li i EW are HIP 46816, HIP 46843, HIP 13402, HIP 63742 HIP 75809, HIP 75829 (in this order). According to their kinematics, HIP46816 has been classified in the young discs stars category, whereas the three other stars have velocity-components (U, V, W) in the boundaries of the Local Association (discussed in some detail in Section 6.1). Stellar activity indicators It is well known that for cool stars with convective outer-layers, chromospheric activity and rotation are linked by the stellar dynamo (e.g. Kraft 1967;Noyes et al. 1984;Montesinos et al. 2001) and both (activity and rotation) diminish as the stars evolve. Thus, activity/rotation tracers, such as R ′ HK , L X or rotational periods are often used to estimate stellar ages (for a recent detailed work on this subject see Mamajek & Hillenbrand 2008). Chromospheric emission: Ca ii H & K lines The stellar chromospheric activity is usually quantified by the R ′ HK index, defined as the ratio of the chromospheric emission in the cores of the broad Ca ii H & K absorption lines to the total bolometric emission of the star (e.g. Noyes et al. 1984 The dotted line represents the boundary of the young disc population as defined by Eggen (Eggen 1984(Eggen , 1989. Stars that satisfy both Eggen's criteria are shown with filled symbols, while open symbols indicate stars that do not satisfy at least one of the Eggen's criteria. Several relations between log R ′ HK and stellar chromospheric age are available in the literature (e.g. Soderblom et al. 1991). In this paper we take those given by Mamajek & Hillenbrand (2008, Eq.3): log(τ/yr) = −38.053 − 17.912 log R ′ HK − 1.6675 log R ′ 2 HK ,(3) which is valid between log R ′ HK values of -4.0 and -5.1 (i.e. log τ of 6.7 and 9.9). Although the stars used in the calibration of Eq. 3 are all stars with (B − V) < 0.9, we assume it holds for the entire (B-V) range of our stars. As in the case of the lithium abundance, activity indicators are also biased towards younger stars. The accuracy of Mamajek's relation is 15-20% for young stars (younger than 0.5 Gyr), but beyond this age, uncertainties can grow up to more than 60%. The log R ′ HK values and derived ages are shown in columns 11 and 12 in Tables 9 to 13, and columns 7 and 8 in Tables 14 to 15. Figure 7 shows the log R ′ HK versus (B − V) diagram of stars in clusters of known ages. Following Henry et al. (1996), we used log R ′ HK to classify stars into "very inactive" (log R ′ HK < −5.1), "inactive" (−5.1 < log R ′ HK < −4.75), "active" (−4.75 < log R ′ HK < −4.2), and "very active" if log R ′ HK > −4.2. The percentages of stars in each region are 8%, 47%, 41%, and 4%, respectively. Mean log R ′ HK value for inactive stars is -4.93 with a standard deviation of 0.09, and < log R ′ HK >= −4.53 with a standard deviation of 0.13 for active stars. These numbers are quite similar to those found by Henry et al. (1996) and Gray et al. (2003). Most of the stars in the "very active" category are, according to their kinematics, candidate members to MGs. HIP 46843 and HIP 86346 (Local Association), HIP 21482, and HIP 25220 (Hyades), HIP 8486 (Ursa Major), HIP 66252 (IC2391) HIP 33560 and HIP 46816 (young disc population). Three of the stars in this "very active" region, namely HIP 45963, HIP 21482 and HIP 91009 are well-known variable chromospherically active binaries (included in The 3rd Catalogue of chromospherically active binary stars (Eker et al. 2008)). In those systems, stellar activity/rotation are enhanced by tidal interaction with the companion star, leading to high levels of chromospheric and coronal emission, up to two orders of magnitude higher than the level expected for a single star with the same rotation period (Basri et al. 1985;Simon & Fekel 1987;Montes et al. 1996). Therefore their log R ′ HK values cannot provide any information on their age or membership to MGs. Lithium abundance is also affected in this kind of systems, showing overabundances with respect to the typical values for single stars of the same mass and evolutionary stage (Barrado y Navascués et al. 1997). Henry et al. (1996). Coronal emission: ROSAT data In addition to their chromospheric activity, the rapid rotation of young stars drives a vigorous stellar dynamo, producing a strong, coronal X-ray emission. Even though there are L X values already published in several catalogues (e.g. Hünsch et al. 1999), in order to be self-consistent we have re-computed them with the revised Hipparcos parallaxes (van Leeuwen 2007) used in this work. To compute L X , we searched for X-ray counterparts in the ROSAT All-Sky Survey Bright Source Catalogue ) and the Faint Source Catalogue (Voges et al. 2000). To determine the X-ray fluxes we used the count rate-to-energy flux conversion factor (C X ) relation given by Fleming et al. (1995): C X = (8.31 + 5.30 HR1)10 −12 erg cm −2 counts −1 . (4) Where HR1 is the hardness ratio of the star in the ROSAT energy band 0.1-2.4 KeV. Combining the X-ray count rate, f X (counts s −1 ), and the conversion factor C X with the distance D, the stellar X-ray luminosity L X (erg s −1 ) can be estimated: Figure 8 shows the fractional X-ray luminosity L X /L Bol versus the colour index (B − V). Bolometric corrections were derived from the (B − V) colour by interpolating in Flower (1996 , Table 3) Data for the Pleaides (Stauffer et al. 1994) and Hyades (Stern et al. 1995) clusters have been overplotted for a comparison. Approximately 23% of the stars are in the Pleiades region of the diagram, 51% of the stars are in the Hyades region, and ∼ 26% of the stars are below the Hyades' sequence. L X = 4πD 2 C X f x .(5) To compute the stellar age from the X-ray luminosity, we followed the work by Garcés et al. (2010, in prep): L X = 6.3 × 10 −4 L Bol (τ < τ i ) L X = 1.89 × 10 28 τ −1.55 (τ > τ i ). (6) Fig. 8. Fractional X-ray luminosity log(L X /L Bol ) vs colour index B − V. Stars classified as "very active" and "very inactive" according to their log R ′ HK value are plotted in red and green colours, respectively. With τ i = 2 × 10 20 L −0.65 Bol , and both L X and L Bol are expressed in erg/s and τ is given in Gyr. Columns 13 and 14 in Tables 9 to 13 show the L X /L Bol values and derived ages, while in Tables 14 to 15 these data are in columns 9 and 10. The critical parameter τ i marks the change from a nonsaturated regime in which there is an inverse relation between the stellar rotation and L X and the saturated regime in which the star reaches a maximum L X such that L X /L Bol ≈ 10 −3 (e.g. Pizzolato et al. 2003, and references therein). Only one star, HIP 86346, is in the saturated regime. For this star, Eq. 6 only provides an upper limit to the age, close to the "real" age of the star. This star is discussed in some detail in Section 6.1. Most of the stars included in the "very inactive" category defined before do not have ROSAT data, and for the few of them that do, X-ray data place them below the Hyades' sequence (Fig 8). Lithium abundance shows a similar behaviour, and these stars are below the Hyades' envelope or do not show lithium at all. Although some of them have been identified by means of their kinematics as young disc stars or MG members, age diagnostics show that they are, however, old stars. For the stars in the "very active" category, the situation is the opposite one. All of them have ROSAT data, and most of them have fractional X-ray luminosities similar to those of the Pleiades (Fig 8). They also show higher lithium abundances than the "very inactive" or the "inactive" stars. Age from stellar rotation: Gyrochronology Stars are born with relatively high rotational velocities. In the course of their evolution, rotation decreases due to the loss of angular momentum with stellar winds and magnetic breaking (Weber & Davis 1967;Jianke & Collier Cameron 1993;Aibéo et al. 2007). Thus stellar rotation can be used to estimate stellar ages, and it is well known that solar-type stars follow Prosser et al. (1995), whereas data from the Hyades are from Radick et al. (1987). Three gyrochrones (at the ages of the Pleiades, Hyades, and the Sun) have been overplotted for a comparison. a law of the form P Rot ∝ t 1/2 (Skumanich 1972). Subsequent works have refined this relationship, e.g., by establishing a mass dependence in the evolution of rotational periods (e.g. Kawaler 1989) or deriving a rotation-age relationship as a function of the stellar colour (Barnes 2007;Mamajek & Hillenbrand 2008). To compute ages, we follow the relationship given by Mamajek & Hillenbrand (2008): P Rot = 0.407((B − V) − 0.495) 0.325 × t 0.566 .(7) With the age of the star, t, given in Myr and the period in days. Rotational periods have been taken from Noyes et al. (1984), Baliunas et al. (1996), Saar & Osten (1997), and Messina et al. (2001). Unfortunately, only 17.3% of the stars have measured rotational periods. Rotational periods and derived ages are given in columns 15 and 16 in Tables 9 to 13 and columns 11 and 12 in Tables 14 to 15. Figure 9 shows the rotation period for the stars of our sample as a function of the colour index (B − V). Percentages of Pleiades-like, Hyades-like, and older stars are 22%, 28%, and 50% respectively. All stars with rotation periods lower than seven days are MGs candidates. The fastest rotator is HIP 86346 with a period of only 1.8 days, while the slowest ones are HIP 3093 and HIP 104217 with 48 days. Figure 10 shows the age distribution for the different activity indicators. The results can be compared with those of Mamajek & Hillenbrand (2008, Figure 14). Chromospheric age shows an enhancement of the star formation rate in the last 2 Gyr, then the distribution becomes more or less flat. We do not find a clear minimum at 2 Gyr, the so-called Vaughan-Preston gap (Vaughan & Preston 1980). ROSAT ages are biased towards stars younger than 3-4 Gyr; i.e., older stars have negligible (or undetectable) X-ray emission, and therefore their distribution does not offer information on the stellar formation history. As far as rotational ages are concerned, there are not enough stars with measured rotational periods to draw robust conclusions. Discussion Although the agreement between the ROSAT and the chromospheric distribution is overall good, when considering individual stars there can be discrepancies, which can for to different reasons. For example, some stars present variability in their levels of activity, which leads to very different age estimates if the activity indicators are taken in different epochs of the activity cycle. For example, HIP 37349 is a known variable observed three times: log R ′ HK values are -4.54, -4.57, and -4.28 , which lead to ages 800, 950, and 115 Myr, respectively, while the ROSAT-derived age is 1.17 Gyr, which is compatible with 800-950 Myr but not with 115 Myr. In addition, stellar rotation can be influenced by tidal interaction in binary systems, leading to completely different ages. Finally there could be other aspects like possible mismatches of X-ray sources with their optical counterparts. Additional criteria Presence of debris discs It is now well established that debris discs are more common around young stars (e.g. Habing et al. 2001;Siegler et al. 2007). As stars age they are on average orbited by increasingly fewer dust particles so a high value of the fractional dust luminosity, f d , can be used as an additional indicator of youth. There is evidence that debris systems of high infrared luminosity are more intimately linked to young stellar kinematic groups than the majority of normal stars (e.g Moór et al. 2006); indeed, several of the stars with the strongest infrared excesses are members of MGs (e.g., β Pic, Barrado y Navascués et al. 1999). However, f d is a rather inaccurate age diagnostic. First, the amount of excess emission shows large differences among stars within the same age range (e.g Siegler et al. 2007, Fig. 7). Even though stars with significant excess emissions should in principle be young, no further information can be given without additional age estimates. Moreover, there are relatively old systems (age 500 Myr) with high f d values ( f d ≃ 10 −3 ) possibly associated with stochastic collisional events. Stars with known infrared excesses and their inclusion to MGs are given Table 5. As shown in that table, the IR excesses of those stars are relatively moderate ( f d ≃ 10 −5 ) and most stars with excess are not related to MGs. For example, the three stars with the largest IR-excess (HIP 76375, HIP 40693, and HIP 32480) are old field stars. Both kinematics and activity-derived age confirm this. Metallicity Moving group members are supposed to have formed in the same molecular cloud, so, they should have similar metallicity. Local Association and Ursa Major members are expected to have metallicities compatible with the solar value. Recently, Soderblom et al. (2009) have obtained [Fe/H] = +0.03 for a sample of 20 Pleaides' stars with statistical and systematic uncertainties of +0.002 and +0.05, respectively. For members of the Ursa Major Group, Boesgaard & Friel (1990) found [Fe/H] = −0.085, σ = 0.021. Hyades' members should be slightly metal rich [Fe/H] = +0.14, σ = 0.05. Therefore we discard very metal-poor stars as "good" MG candidates. Considering that the "old" (2 Gyr) and metal-rich MG HR1614 has a mean metallicity of [Fe/H] = +0.19 ± 0.06 (Feltzing & Holmberg 2000), stars with metal overabundances over ≈ +0.20 should also be discarded. Reliable spectroscopic determinations of the metallicity for our stars were taken from the literature (Fuhrmann 2008(Fuhrmann , 2004Santos et al. 2004;Sousa et al. 2008;Takeda et al. 2005;Valenti & Fischer 2005). When no spectroscopic metallicities were found there, they were computed from Strömgren indices (Hauck & Mermilliod 1997) by using the calibrations given by Schuster & Nissen (1989). These values are given in columns 17 (Tables 9 to 13) and 13 (Tables 14 to 15). An inspection of the metallicities obtained reveals that there are no MGs candidates among the most metal-poor stars. However, some MGs candidates have positive metallicities. This is especially evident in the Hyades MG where the 45% of the candidates are more metal-rich than the Sun. Between them we find HIP 43587 and HIP 67275, which are among the most metal rich in our sample, [Fe/H] = +0.35, [Fe/H] > +0.30, respectively, and they both are known to have planets. As we will see in Section 6.2, their age estimates confirm that they are old stars and not "good" Hyades' members. Comparison between kinematic and age estimates. Final membership. Tables 9 to 13 show a summary of the kinematic and spectroscopic properties, as well as age estimates, of the stars which are candidate members to the different MG, according to their (U, V, W) velocity components. Each table refers to one specific MG. This summary classifies the MGs candidates into three different categories, which are similar to the ones by Soderblom & Mayor (1993a): -Probable non-member: If the derived ages from the different indicators agree, but they are in conflict with the object having an age as an MG member. -Doubtful member: If there is important disagreement among the different age indicators, including here the assigned age of the corresponding MG, or there is lack of information (i.e., some age indicators are not available) -Probable member: If age indicators agree and also do with the position of the star in the (U, V) plane. The following subsections describe the membership of the stars studied in this work and the properties of each MG individually Membership and properties of the Local Association candidates The concept of a Local Association of stars was introduced by Eggen (e.g. Eggen 1975). This association, also known as Pleiades MG or Pleiades stream, includes stars in the Pleiades, α Persei, and IC2602 clusters, as well as stars in the Scorpius-Centaurus star-forming region. In the past years, small associations or groups of very young stars have been detected among Local Association members (AB Dor, TW Hydrae, β Pic, and others). The spatial motions of these new associations are quite similar, but they present a wide range in ages and distributions around the Sun (e.g. , which leads to the question of whether it is reasonable to consider the Local Association as a single entity. Addressing this problem is beyond the scope of this paper, so we consider the Local Association as a single MG. There are 29 stars ( Table 9) that have velocity components (U, V, W) consistent with the stars being candidates to the Local Association. Eight out of the 29 stars do not satisfy other criteria; i.e., their age estimates suggest that they are older than 20-150 Myr commonly adopted for the Local Association, and we consider them to be non-members; Seven out of the 29 stars are considered as doubtful members while, 14 are good candidates, i.e., probable members. Although the number of candidates is too small to draw robust conclusions we can infer that the contamination by old main-sequence stars vary from roughly 25% to 50%. Table 6 lists the candidates and our final classification (column 3) for the Local Association stellar membership. Some of our candidates have already been classified as members of the young association around the star AB Dor or members of the so-called Hercules-Lyra association introduced by Fuhrmann (2004), which are listed in columns 4 to 6 of Table 6. All the previously proposed members of AB Dor or Hercules-Lyra fall into our classification of probable Local Association members, with the only exception of HIP 62523, which we have classified as a "doubtful member". Thirteen out of our 29 candidates have not been included in these previous studies. Among the Local Association members there are some interesting stars: -HIP 13402 (HD 17925): Although it is classified as an RS CVn variable (Eker et al. 2008), Cutispoto et al. (2001) shows that the binary hypothesis does not seem to be consistent with the Hipparcos photometric data. The estimated EW Li i = 182.52 ± 4.63 mÅ agrees with the 208 mÅ given by Montes et al. (2001a) and the 197 mÅ given by Favata et al. (1995), which suggests an age similar to the Pleaides (≈ 80 Myr). In addition, the different age estimates agree very well and confirm that it is a young star; furthermore, the star is known to have IR-excess at 70 µm, see Table 5 ( Trilling et al. 2008). Thus, we consider that it is a reliable member of the Local Association. -HIP 18859 (HD 25457): This star is classified as a weak-line T Tauri (e.g. Li et al. 2000), and has a remarkable infrared excess of f d = 1.0 ± 0.2 × 10 −4 , (Reid et al. 1995;). This object is a flare star identified as a spectroscopic binary by Gálvez et al. (2006). A close companion was detected using lucky imaging techniques by Hormuth et al. (2007). Our radial velocities vary between -25.37 and -28.39 km/s. In all epochs the main optical activity tracers (Ca ii IRT, H α , Na i D 1 , D 2 , Ca ii H & K) are in emission. This star is a very rapid rotator (for an M-type star) with v sin i between 21 and 23 km/s. Our EWs Li i measurements vary from 11.1 to 55.7 mÅ and are agree with the 40 mÅ reported by . ROSAT-age indicates a star younger than 100 Myr (it is in the "saturated" regime in the log L X /L Bol vs age diagram), the position of the star in a colour-magnitude diagram (M V = 7.55 ± 0.15; (V − I) = 2.58 ± 0.91) suggests it is a pre-main sequence star. Membership and properties of the Hyades candidates The Hyades MG group or Hyades Supercluster 2 has a venerable history in the study of MGs since references to the Hyades MG group go back in time to the first works in this area (Proctor 1869). It is commonly related with the Hyades and Praesepe clusters, both of them with ages around 600 Myr. Recently, Famaey et al. (2007) has found that the MG is in reality a mixture of two different populations: a group of coeval stars related to the Hyades cluster (the evaporating halo of the cluster) and a second group of old stars with similar space motions. Age diagnostics analysed in Section 5 allow us, in principle, to distinguish between the two populations. There are 29 stars in the region of the (U, V, W) planes occupied by the Hyades MG. Eleven out of these 29 candidates have been classified as probable members 3 , nine as probable non-members, whereas the classification of the other nine stars remains unclear (Table 10). Our selection contains 14 stars in common with López-Santiago et al. (2010). A comparison between our final classification and those given by López-Santiago et al. (2010) is shown in Table 7. There is good agreement with two exceptions, HIP 17420 (for which our age estimates suggest an old star) and HIP 19335 (discussed below). We briefly describe some interesting stars concerning this MG: -HIP 19335 (HD 25998): This F7V star has been identified as a T-Tauri star in the surroundings of the Taurus-Auriga star formation region (Li & Hu 1998), although it is located at a significantly shorter distance, 21 pc, than the commonly accepted distance of ∼ 140 pc to that star forming region. The Li i EW = 93.1 ± 3.0 mÅ confirms its youth and agrees well with the rest of age indicators, between 96 and 300 Myr. The star has infrared-excesses at both Spitzer 24 µm and 70 µm MIPS bands . All this information also confirms the youth of this star, but it is likely too young to be a member of the Hyades MG. -HIP 43726 (HD 76151): The Li i EW of 31.42 ± 3.66 mÅ of this star, as well as the estimated chromospheric and rotational ages of ∼ 1.0 Gyr, suggests that it is not a member of the Hyades MG. Interestingly, this is a relatively old star with a debris disc (Trilling et al. 2008;Beichman et al. 2006). -HIP 67275 (HD 120136, τ Boo): τ Boo is one of the first cases where an exoplanet was found (Butler et al. 1997). There is a strong disagreement between the X-ray age estimate, 0.36 Gyr, and the chromospheric age, 4.78 Gyr. Li i EW also suggests an old star. This agrees with other published ages, 1.3 Gyr (Valenti & Fischer 2005), 2.1 Gyr (Nordström et al. 2004), and 2.52 Gyr (Saffe et al. 2005). It is therefore unlikely that HIP 67275 is a member of the Hyades MG. Membership and properties of the Ursa Major moving group The concept of a group of stars sharing the same kinematic as Sirius goes back more than one century ago. Nowadays the group includes more than 100 stars (Eggen 1992;Soderblom & Mayor 1993b;King et al. 2003;Fuhrmann 2004;Ammler-von Eiff & Guenther 2009). Eighteen stars have velocity components (U, V, W) consistent with the star being a candidate for Ursa Major (Table 11). Four out of the 18 stars do not satisfy other criteria, and their age estimates indicate that they are older than the 300 Myr commonly adopted for the Ursa Major MG. Another eight out of the 18 stars stars are considered as doubtful members, while six stars are probable members. Table 8 shows a comparison between our classification and those reported in the literature. There is good agreement specially in the stars classified as good members. Three candidates of this MG are of special interest: -HIP 42438 (HD 72905): This star is known to have infrared excesses at 60 and 70 µm (Spangler et al. 2001;Bryden et al. 2006). All age estimates agree with an age of ≈ 300 Myr, which indicate that it is a probable member of the group. -HIP 71395 (HD 128311): This object is an example of a star with a planetary system (Butler et al. 2003;Vogt et al. 2005) in a debris disc (excess at 70 µm found by Trilling et al. 2008). Our chromospheric-derived age of 430 Myr agrees with the 390 Myr given by Saffe et al. (2005) and confirms that this star is a probable member of the Ursa Major group. -HIP 80337 (HD 147513): This star is also known to have a planet ). Due to a problem with the header's spectra, no radial velocity could be obtained so we have adopted the value given by NO04. Measured Li i EW = 35.51 ± 3.5 mÅ suggests that it is older than the Hyades, which agrees with both chromospheric and rotational ages, around 700 Myr. However the ROSAT age is much shorter, only 370 Myr. Therefore, we have classified this star as a "doubtful" member. Membership and properties of the IC2391 moving group The identification of an MG related to the IC2391 cluster is from Eggen (1991Eggen ( , 1995. Most of the stars listed as members of this MG are in fact early-type star members of the cluster. By using the member's position in colour-magnitude diagrams Eggen obtained an age of ∼ 100 Myr, within an interval spreading from 80 to 250 Myr. Recently, López-Santiago et al. (2010) has suggested the presence of two subgroups mixed in the (U, V) plane with ages of 200-300 and 700 Myr. Table 12 summarizes our membership criteria for the IC2391 MG. Five out of 19 candidate stars have been classified as probable members, 10 as doubtful, and four as probable nonmembers. Our sample contains three stars in common with López-Santiago et al. (2010). We confirm that HIP 11072 is a doubtful member and HIP 59280 is a member of the old subgroup, but our age estimates for HIP 25119 disagree with López-Santiago et al. (2010). We therefore consider this star as a "non" member instead of a member of the old subgroup since both chromospheric and ROSAT ages are around 3 -5 Gyr. We have identified four new stars as probable-members of the young subgroup: HIP 19076, HIP 22263, HIP 29568, and HIP 66252. In addition, HIP 71743 has been classified as a probable member of the old subgroup. HIP 66252 (HD 118100, EQ Vir) is known to have flares, and therefore chromospheric and ROSAT ages can be greater than our estimates, although the lithium abundance confirms that it is a young star. HIP 34567 (ages between 320 and 470 Myr) should remain in the "doubtful category" since it is a known chromospherically active binary (Eker et al. 2008). Membership and properties of the Castor moving group The Castor MG was originally suggested by Anosova & Orlov (1991). This group includes, among other stars, three spectroscopic binaries (Castor A, Castor B, and YY Gem) and two prototypes of the β Pic stars (Vega and Fomalhaut). Barrado y Navascués (1998) estimated an evolutionary age for this association of 200 ± 100 Myr. Only seven stars have been identified on the basis of their kinematics as candidate members of this group (Table 13). Four were classified as probable members and three as doubtful members. HIP 29067 and HIP 109176 have been previously studied in detail by Barrado y Navascués (1998), and since we obtain similar results, we concentrate on the rest of candidates: -HIP 12110 (HD 16270): There is a strong discrepancy between ROSAT and chromospheric ages, therefore the star remains as a doubtful member. -HIP 45383 (HD 79555): This star is a long-period astrometric binary (Mason et al. 2001). Both ROSAT and chromospheric ages agree with the star being coeval with Castor MG members. As an additional test of youth, we plotted the star in a M V vs (V − I) diagram ( Figure 11). The position of the star in this diagram suggests an age around 35 Myr. Therefore we conclude that HIP 45383 is a young star and a probable Castor member. -HIP 67105 (HD 119802): There is a strong discrepancy between ROSAT and chromospheric ages, therefore the star remains as doubtful member. -HIP 110778 (HD 212697): Both ROSAT and calcium ages agree with the star being coeval with the Castor MG. Since this is a star in a multiple system, we have confirmed its youth nature by using colour-magnitude diagrams. -HIP 117712 (HD 22378): This star is a known spectroscopic binary. Chromospheric ages suggest a moderately young star, between 600 and 860 Myr. The position of the star in colour magnitude diagrams suggests also it is a young star, and hence a probable Castor MG member. Summary In this paper we have addressed the problem of identifying unambiguous MG members. Making use of a large quantity of data from the literature and data from our own spectroscopic observations, we were able to study the kinematics and age of the nearby late-type population, identifying a considerable group of stars that are members of moderately young (35-600 Myr) kinematic groups. Based on both the kinematics and different age estimates, our results allow us to identify new members, confirm previously suggested members of MGs, and discard previously claimed members. We find that approximately ∼ 25% of the nearby stars can be classified as members of MGs according to their kinematics, but that only 10% have ages that agree with the accepted ages of the corresponding MG members. Specifically, we find that among the stars studied in this work, the bona fide members for each MG are 14 stars (out of 29 kinematic candidates) for the Local Association, 11 (29 of kinematic candidates) for the Hyades MG, six (out of 18 kinematic candidates) for the Ursa Major MG, six (out of 19 kinematic candidates) for IC2391, and four (out of seven kinematic candidates) for the Castor MG. Some of the bona fide members identified here have not been reported before (at least to our knowledge), especially when considering the less-studied groups: Hyades (four new probable members), IC2391 (five new probable members), and Castor (three new probable members). We find discrepancies with previously reported lists in eight stars. Additional observations are required to identify new bona fide members in each group and to address further investigations as suggested in Appendix B. manuscript. J.M., C.E., and B.M. acknowledge support from the Spanish Ministerio de Ciencia e Innovación (MICINN), Plan Nacional de Astronomía y Astrofísica, under grant AYA2008-01727, and the Comunidad de Madrid project ASTRID S-0505/ESP/00361. R.MA., and D.M. acknowledges support from the Spanish Ministerio de Ciencia e Innovación (MICINN), Plan Nacional de Astronomía y Astrofísica, under grant AYA2008-00695, and the Comunidad de Madrid project AstroMadrid S2009/ESP-1496. We would like to thank the staff at Calar Alto and Telescopio Nazionale Galileo for their assistance and help during the observing runs. This research has made use of the VizieR catalogue access tool and the SIMBAD database, both operated at the CDS, Strasbourg, France. We also thank the anonymous referee for his/her valuable suggestions on how to improve the manuscript. Appendix A: Tables Results produced in the framework of this project are published in electronic format only. Table 1 is also available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ Table 1 contains the following information: HIP number (column 1), HD number (column 2), right ascension and declination (ICRSJ2000) (columns 3 and 4), parallax and its uncertainty (column 5), proper motions in right ascension and declination with their uncertainties (columns 6 and 7), observing run identifier (column 8), radial velocity used in this work and its uncertainty (column 9), and radial velocities reported in KH07, NO04, NI02, and VF05 works (columns 10 to 13) with their uncertainties, if available. Column 14 contains important notes: spectroscopic binaries radial velocities standards, and stars in chromospherically active binary systems are identified in this column. Tables 9 to 13 contain the properties of the potential candidates to MG members for the different MGs studied in this work. These tables give: HIP number (column 1), (B-V) colour (column 2), spatial-velocity components (U, V, W) with their uncertainties (columns 3, 4 and 5), V Total , V T , PV and ρ c as defined by Eggen (columns 6, 7, 8 and 9), measured Li i EW (column 10), R ′ H,K value and derived age (columns 11 and 12), log(L X /L Bol ) and derived age (columns 13 and 14), rotational period and derived age (columns 15 and 16) and metallicity (column 17). For each Eggen's criteria, PV and ρ c (columns 8 and 9), there is label indicating if the star satisfies the criteria (label 'Y') or not (label 'N'). Tables 14 to 15 are similar to the previous ones, but they show the properties of the stars classified as Other young disc stars and stars not selected as possible MG members, respectively. Tables 9 to 15 References in Appendix B: Further applications of MGs members. Lists of nearby MGs members constitute promising targets for a wide variety of further investigations. We briefly summarized some of them: First, we investigated whether there is a connection between the so-called "solar-analogues" (e.g. Porto de Mello & da Silva 1997; Meléndez et al. 2009;Ramírez et al. 2009) and MGs members. Taking as a reference the list of analogues published by Gaidos et al. (2000), we have found 25 matches between their list and our sample, where 22 out of these 25 stars, have been classified as bona fide MG members. Another three stars, HIP 29525, HIP 80337, and HIP 116613, also candidates for MGs, satisfy Gaidos' criteria for being considered as solar analogues. These stars are listed in Table B.1. These "young-suns" are Ribas et al. 2005). As we have shown in Section 5.3.1, debris discs are linked to stars in MGs. It is therefore natural to check if there is a similar relation between stars with known planets and MGs. Nineteen stars of our sample have detected planets 4 , and seven of them are MGs candidates: HIP 21482 (Hyades MG, but it is doubtful member and in addition the planet is not confirmed); HIP 43587 (Hyades MG, but it is a doubtful member), HIP 71395 (Ursa Major MG, probable member); HIP 80337 (Ursa Major, doubtful member); HIP 95319 (IC2391 MG, but doubtful member and the planet is not confirmed); HIP 49669 and HIP 53721 (young disc stars according to their kinematics, but their calcium ages suggest that they are old stars). Other applications are related to activity studies, i.e., flux-flux and rotation-activity-age relationships (e.g Martínez-Arnáiz et al. 2010), or search programmes to detect stellar and substellar companions (e.g Hormuth et al. 2007). Finally, we point out that an important fraction of the stars analysed in this paper will be observed in the framework of the DUNES (DUst around NEarby Stars) programme, an approved Herschel Open Time Key Project with the aim of detecting cool faint dusty discs, at flux levels as low as the Solar EKB Eiroa et al. 2010). Trilling et al. (2008) -35.15 degrees; U =-11.6 km/s, V = -21.0 km/s, W = -11.4 km/s; Age: 20-150 Myr) Point: 5.82 hours, degrees; U =-20.6 km/s, V = -15.7 km/s, W = -9.1 km/s; Age: 35-55 Myr) Kinematics Lithium CaII H,K ROSAT-data Rotation HIP (B − V) U V W V T otal V T PV ⋆ ρ ⋆ c EW(LiI) R ′ HK Age L X /LHIP (B − V) U V W V T otal V T PV ⋆ ρ ⋆ c EW(LiI) R ′ HK Age L X /L Kinematics Lithium CaII H,K ROSAT-data Rotation HIP (B − V) U V W V T otal V ⋆ T PV ⋆ ρ c EW(LiI) R ′ HK Age L X /L Fig. 1 . 1Number of stars versus distance (normalized to 25 pc) for the F stars (green), G stars (orange), K stars (red) and for the observed 405 stars. Fits to cubic laws are plotted in blue. Fig. 4 . 4Comparison of radial velocities taken from the literature and obtained in this work. Top left panel: Fig. 5 . 5(U, V) and (W, V) planes for the observed stars. Different colours and symbols indicate membership to different MGs. Large crosses represent the convergence point of the young MGs shown in the figure. Fig. 6 . 6Li i vs (B − V) diagram. Lines indicate the envelopes for the IC2602 (green), Pleiades (red), and Hyades (dashed blue). Fig. 7 . 7log R ′ HK vs (B − V) colour. The position of the Pleiades (∼ 120 Myr), Hyades (600 Myr), and M67 (4 Gyr) stars are indicated with dotted lines (Mamajek & Hillenbrand 2008). The position of the Sun is also shown with a dotted circle. Dashed lines are the limits for very active, active, inactive, and very inactive stars, according to Fig. 9 . 9Rotation periods vs (B−V) colour. Data from the Pleiades were taken from Fig. 10 . 10Age distribution for chromospheric-derived ages (black solid line), ROSAT ages (blue line), and rotational ages (red line). Fig. 11 . 11Colour-magnitude diagram for the Castor MG candidates. Pre-main sequence isochrones fromSiess et al. (2000) are plotted at 10, 20, 30 and 50 Myr. are indicated in parenthesis: (1) Martínez-Arnáiz et al. (2010) (2) Baliunas et al. (1996); (3) Duncan et al. (1991) calculated using equations in Noyes et al. (1984); (4) Gray et al. (2003); (5) Gray et al. (2006); (6) Hall et al. (2007); (7) Henry et al. (1996); (8) Jenkins et al. (2006); (9) Saffe et al. (2005); (10) Wright et al. (2004); (11) estimated form ROSAT-data using equation A1 in Mamajek & Hillenbrand (2008); (12)Noyes et al. (1984); (13)Saar & Osten (1997);(14) Messina et al. (2001). Table 2 . 2Observing runs between 2005 and 2008.Date Instrument Spectral Range orders dispersion FWHM (Å) (Å/pixel) (Å) July 05 FOCES 3470-10700 111 0.04-0.13 0.07-0.42 Jan 06 FOCES 3470-10700 111 0.05-0.13 0.09-0.39 Feb 06 SARG 5600-10000 50 0.02-0.04 0.09-0.14 Dec 06 FOCES 3640-10700 106 0.05-0.13 0.09-0.22 Jan 07 SARG 5600-10000 50 0.02-0.04 0.09-0.14 Apr 07 SARG 5600-10000 50 0.02-0.04 0.10-0.13 07A ‡ FOCES 3470-10700 106 0.04-0.13 0.09-0.25 Nov 08 SARG 5600-10000 50 0.02-0.04 0.09-0.14 ‡ First semester 07. Service Mode. Table 3 . 3Radial velocity standard starsStar SpT V r ± σ Vr Reference (km/s) HD 102870 F8V 4.30 a HD 50692 G0V −15.05 a HD 84737 G0.5 6.0 ± 1.1 b HD 20630 G5V 18.0 ± 1.0 b HD 159222 G5V −51.60 a HD 82885 G8III 14.40 a HD 65583 G8V 14.80 a HD 144579 G8V −59.45 a HD 182488 G8V −21.55 a HD 102494 G9IV −22.1 ± 0.3 c HD 62509 K0III 3.2 ± 0.3 c HD 100696 K0III 0.2 ± 0.5 b HD 3651 K0V −32.96 ± 0.8 b HD 38230 K0V −29.25 a HD 136442 K0V −45.6 ± 0.8 b HD 92588 K1IV 43.5 ± 0.3 d HD 10476 K1V −33.9 ± 0.9 b HD 73667 K1V −12.10 a HD 124897 K2III −5.3 ± 0.3 c HD 4628 K2V −10.1 ± 0.4 d HD 82106 K3V 29.75 a HD 139323 K3V −67.20 a HD 29139 K5III 54.29 ± 0.2 c a Table 4 . 4Number of MGs candidates according to Eggen's criteria.Group Total stars Both criteria Only PV Only ρ c ). The R ′ HK values used in this work were taken from Martínez-Arnáiz et al. (2010) since they were obtained from the spectra in this paper. For those stars with no R ′ HK value in Martínez-Arnáiz et al. (2010), the R ′ HK values have been taken from the literature (see references in Appendix A). Table 5 ( 5Moór et al. 2006). The EW Li i and age estimates confirm its young evolution- ary state. -HIP 86346 (HD 160934): This star is one of the few Hipparcos' M-type stars we have observed in this project. Available spectral types in the literature vary between K7 to M0 Table B B.1. Solar analogues and their ascription to MGs.Label 'Y' indicates probable members, '?' doubtful members, and 'N' probable non-members, respectively. Local Association; HS: Hyades; UMa: Ursa Major IC2: IC2391; Cas: Castor essential to study the history and formation of our own Solar System, indeed three of them, namely HIP 15457, HIP 42438, and HIP 64394, are included in the ambitious project The Sun in Time aimed at reconstructing the spectral irradiance evolution of the Sun (e.gHIP MG Membership HIP MG Membership 544 LA Y 46843 LA Y 1803 HS Y 54745 LA Y 5944 UMa Y 63742 LA Y 7576 LA Y 65515 LA Y 8362 IC2 N 71743 IC2 Y 8486 UMa Y 72567 15457 YD Y 74702 UMa N 22263 IC2 Y 77408 LA Y 26779 LA Y 80337 UMa ? 29525 YD Y 82588 29568 IC2 Y 94346 HS Y 42074 HS Y 107350 LA Y 42333 HS Y 115331 42438 UMa Y 116613 HS Y LA: Table 1 . 1Positions, proper motions and radial velocities for the observed stars.Radial Velocity Table 5 . 5Stars with known debris discsHIP HD f d Reference MG Age (10 −5 ) (Myr) 544 166 5.9 [4] LA 20-150 1599 1581 0.2-1.6 [4] 13402 17925 2.2-4.4 [4] LA 20-150 15371 20807 0.4-1.5 [4] 16537 22049 8.3 [3] 16852 22484 1.2-4.3 [4] YD 18859 25457 10±2 [2] LA 20-150 19335 25998 2.7 [1] HS 600 22263 30495 2.0-3.0 [4] IC2391 35-55 23693 33262 0.2-1.1 [4] 27072 38393 0.77 [3] UMa 300 27435 38858 10 [1] YD 28103 40136 2.04 [3] 32480 48682 11 [1] 40693 69830 20 [2] 42430 73752 3.21 [3] 42438 72905 0.6-1.5 [4] UMa 300 43726 76151 0.4-1.0 [4] HS 600 51502 90089 0.85 [1] 62207 110897 1.4-2.3 [4] 64924 115617 1.9-3.3 [4] 65721 117176 1.8-7.7 [4] 71284 128167 0.49 [3] 71395 128311 1.3-2.7 [4] UMa 300 76375 139323 78.6 [3] 85235 158633 4.1 [1] 107350 206860 0.6-1.5 [4] LA 20-150 [1] Beichman et al. (2006); [2] Moór et al. (2006) [3] Rhee et al. (2007); [4] Table 6 . 6Comparison between our final membership for the Local Association and previous studies. †Hercules-Lyra AB Dor Table 7 . 7Comparison between our final memberships for the Hyades MG and those given byLópez-Santiago et al. (2010). † Label 'Y' indicates probable members, '?' doubtful members and 'N' probable non-members, respectively.HIP HD This work LS10 1803 1835 Y Y 4148 5133 N 12709 16909 Y 13976 18632 Y ? 16134 21531 Y ? 17420 23356 N Y 18774 24451 ? 19335 25998 N Y 21482 283750 ? 25220 35171 ? Y 40035 68146 N 42074 72760 Y Y 42333 73350 Y 43587 75732 ? ? 43726 76151 N 44248 76943 ? 46580 82106 Y 47592 84117 N 48411 85488 ? ? 63257 112575 ? 66147 117936 N 67275 120136 N ? 69526 124642 ? 72848 131511 Y 90790 170657 N 94346 180161 Y ? 96085 183870 ? ? 104239 200968 Y ? 116613 222143 Y ? † Table 8 . 8Comparison between our final memberships for the Ursa Major MG and those previously reported in the literature. †HIP HD This work SO93 KI03 FU04 LS10 5944 7590 Y Y 8486 11131 Y Y Y? Y Y 27072 38393 N ? Y? Y 27913 39587 ? Y Y Y 33277 50692 N N ? 36827 60491 ? N?/? ? 37349 61606A ? N? Y 42438 72905 Y Y Y? Y ? 60866 108581 ? 71395 128311 Y ? Y 72659 131156A Y Y ? Y ? 73996 134083 N N? Y 74702 135599 N ? Y 80337 147513A ? Y N?/? Y 80686 147584 ? Y 96183 184385 ? 102485 197692 Y 108028 208038 ? S093: Soderblom & Mayor (1993b); KI03: King et al. (2003) FU04: Fuhrmann (2004); LS10: López-Santiago et al. (2010) Table 9 . 9Membership criteria for the Local Association candidate stars. (Convergence Point: 5.98 hours, Upper limit ⋆ 'Y', 'N' labels indicate if the star satisfies or not the criteria.Bol Age Prot Age [Fe/H] (km/s) (km/s) (km/s) (km/s) (km/s) (km/s) (km/s) (mÅ) (log) (Gyr) (log) (Gyr) (days) (Gyr) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) Probable Non-members 3979 0.68 -15.58 ± 0.32 -25.43 ± 0.42 -0.53 ± 0.18 29.83 29.83 5.48 Y -7.01 Y 35.84 ± 3.72 -4.92 [4] 5.04 -0.19 7751 0.88 -4.19 ± 0.16 -18.73 ± 0.14 -18.60 ± 0.07 26.72 15.16 -0.06 Y 10.19 N 2.14 ± 4.65 -4.94 [1] 5.49 30 [13] 3.46 -0.23 12929 1.18 -14.65 ± 0.21 -22.21 ± 0.58 -8.39 ± 0.12 27.90 27.82 4.30 Y 11.59 Y 34.29 ± 6.13 -4.57 [3] 0.97 26779 0.84 -13.82 ± 0.10 -23.07 ± 0.18 -14.40 ± 0.13 30.50 30.44 -1.69 Y 0.68 Y 13.54 ± 4.49 -4.55 [1] 0.84 -4.72 0.66 11 [2] 0.62 0.16 69357 0.86 -10.90 ± 0.42 -33.09 ± 0.98 -1.65 ± 0.19 34.88 36.13 5.33 Y 11.26 N 21.43 ± 4.56 -4.60 [1] 1.14 -5.34 2.02 -0.15 79755 1.32 -9.71 ± 0.15 -29.99 ± 0.22 3.82 ± 0.22 31.75 32.24 -17.23 N -26.32 N -4.58 [3] 1.03 -4.72 2.45 108156 0.93 -2.70 ± 0.16 -21.94 ± 0.18 -24.19 ± 0.49 32.77 35.10 -13.76 N -23.30 N 18.73 ± 4.88 -4.68 [1] 1.79 -5.33 2.04 -0.08 115341 1.06 -9.89 ± 0.30 -20.49 ± 0.19 -9.14 ± 0.28 24.52 23.37 0.44 Y -11.71 Y 18.15 ± 5.52 -4.73 [1] 2.31 -4.75 0.99 Doubtful Members 19422 0.96 -7.92 ± 0.26 -23.86 ± 0.23 -17.19 ± 0.22 30.46 28.17 -4.37 Y -8.04 Y 4.54 ± 5.02 -4.86 [3] 4.08 -0.06 37288 1.40 -12.25 ± 0.17 -22.15 ± 0.27 -12.50 ± 0.39 28.23 28.58 0.31 Y 20.51 Y 43.42 ± 7.27 -4.72 [3] 2.2 57494 1.17 -14.43 ± 0.52 -26.95 ± 0.44 3.59 ± 0.33 30.78 23.66 6.46 N 3.76 N 14.26 ± 6.09 -4.80 [1] 3.2 62523 0.71 -18.16 ± 0.20 -21.36 ± 0.23 -8.35 ± 0.05 29.25 30.17 5.74 Y -12.14 Y 22.48 ± 3.85 -4.43 [1] 0.37 -5.08 0.83 0.08 72146 0.93 -6.08 ± 0.09 -30.02 ± 0.48 -6.84 ± 0.10 31.38 38.31 -4.74 Y -25.49 N 8.69 ± 4.89 -4.84 [1] 3.72 73695 0.59 -17.06 ± 0.32 -26.77 ± 0.41 -2.37 ± 0.42 31.83 46.30 -1.67 Y -38.12 N 29.62 ± 3.30 -4.60 [3] 1.16 -3.96 0.12 -0.19 105038 1.04 -10.74 ± 0.16 -20.02 ± 0.16 -4.74 ± 0.25 23.21 18.27 -0.34 Y -11.30 N 12.14 ± 5.40 -4.69 [1] 1.89 -0.10 Probable Members 544 0.75 -15.02 ± 0.15 -21.38 ± 0.14 -10.12 ± 0.12 28.02 27.92 3.44 Y -7.12 Y 79.89 ± 4.05 -4.320 [3] 0.16 -4.35 0.32 0.12 7576 0.79 -13.31 ± 0.25 -18.41 ± 0.42 -11.14 ± 0.12 25.31 24.45 2.34 Y 9.95 Y 117.41 ± 4.25 -4.29 [1] 0.13 -4.43 0.44 -0.03 13402 0.88 -15.30 ± 0.07 -21.81 ± 0.11 -9.21 ± 0.06 28.19 28.78 4.06 Y 19.44 Y 182.52 ± 4.63 -4.33 [1] 0.17 -4.12 0.30 6.6 [12] 0.24 0.10 18859 0.51 -7.98 ± 0.18 -28.15 ± 0.21 -11.89 ± 0.15 31.58 37.55 -1.77 Y 26.94 N 96.79 ± 2.98 -4.291 [4] 0.13 -4.16 0.12 -0.01 46843 0.79 -9.93 ± 0.13 -22.89 ± 0.26 -5.64 ± 0.18 25.58 24.18 -4.59 Y 4.17 Y 192.41 ± 4.24 -4.08 [1] 0.02 -3.93 0.22 6 [2] 0.23 54155 0.79 -15.14 ± 0.38 -28.39 ± 0.44 -0.70 ± 0.42 32.18 26.27 2.33 Y 6.12 N 103.69 ± 4.22 -4.34 [1] 0.18 -3.65 0.12 -0.09 54745 0.62 -16.07 ± 0.23 -23.41 ± 0.29 -11.04 ± 0.12 30.47 30.87 1.94 Y -6.30 Y 88.31 ± 3.44 -4.27 [1] 0.11 -4.52 0.29 7.6 [12] 0.59 0.08 63742 0.86 -8.87 ± 0.15 -23.28 ± 0.51 -19.50 ± 0.22 31.64 29.87 -6.54 N -5.26 Y 140.60 ± 4.56 -4.34 [4] 0.19 -4.25 0.38 65515 0.81 -15.76 ± 0.24 -18.47 ± 0.18 -8.52 ± 0.08 25.73 29.12 2.85 Y -18.87 N 39.95 ± 4.33 -4.22 [1] 0.07 -4.36 0.36 -0.10 75809 0.65 -14.73 ± 0.22 -25.95 ± 0.17 -2.33 ± 0.14 29.93 32.31 -9.56 N -21.82 N 141.10 ± 3.59 -4.41 [1] 0.32 -4.01 0.15 -0.29 75829 0.78 -14.16 ± 0.26 -25.03 ± 0.18 -2.30 ± 0.14 28.85 31.11 -9.19 N -21.02 N 127.81 ± 4.19 -4.41 [4] 0.31 -3.79 0.15 0.07 77408 0.80 -17.81 ± 0.10 -27.25 ± 0.44 -12.44 ± 0.13 34.85 34.04 3.56 Y -23.96 Y 10.37 ± 4.30 -4.35 [1] 0.2 -4.40 0.43 -0.05 86346 1.18 -6.86 ± 0.68 -24.72 ± 0.30 -11.76 ± 0.46 28.22 16.54 -2.27 Y -14.83 N 28.19 ± 6.25 -3.96 [11] -2.90 0.10 † 1.8 [14] 107350 0.59 -13.92 ± 0.13 -20.05 ± 0.10 -11.10 ± 0.18 26.81 26.73 2.55 Y -15.61 Y 91.91 ± 3.30 -4.48 [1] 0.53 -4.40 0.24 4.7 [12] 0.3 -0.01 † Table 10 . 10Membership criteria for the Hyades candidate stars. 'Y', 'N' labels indicate if the star satisfies or not the criteria.(Convergence Point: 6.40 hours, 6.50 degrees; U =-39.7 km/s, V = -17.7 km/s, W = -2.4 km/s; Age: 600 Myr) Kinematics Lithium CaII H,K ROSAT-data Rotation Table 11 . 11Membership criteria for the Ursa Major MG candidate stars.(Convergence Point: 20.55 hours, -38.10 degrees; U =14.9 km/s, V = 1.0 km/s, W = -10.7 km/s; Age: 300 'Y', 'N' labels indicate if the star satisfies or not the criteria.Bol Age Prot Age [Fe/H] (km/s) (km/s) (km/s) (km/s) (km/s) (km/s) (km/s) (mÅ) (log) (Gyr) (log) (Gyr) (days) (Gyr) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) Probable Non-members 27072 0.49 18.18 ± 0.13 4.44 ± 0.13 -11.67 ± 0.09 22.06 20.82 1.62 Y -6.32 Y 68.82 ± 2.90 -4.78 [7] 2.93 -0.09 33277 0.60 13.53 5.66 -4.98 15.48 2.72 3.47 N -2.48 N 15.28 ± 3.35 -4.94 [3] 5.45 -0.13 73996 0.45 16.71 ± 0.39 -1.81 ± 0.27 -18.15 ± 0.75 24.74 23.13 -1.39 Y -3.54 Y -4.85 [3] 3.95 -5.34 0.50 0.05 74702 0.82 9.28 ± 0.17 1.13 ± 0.09 -14.68 ± 0.16 17.40 17.04 0.34 Y -2.35 Y -4.35 [1] 0.20 -5.11 1.34 -0.09 Doubtful Members 27913 0.59 13.63 ± 0.30 2.46 ± 0.06 -7.40 ± 0.05 15.70 12.78 1.27 Y -10.00 Y 102.91 ± 3.30 -4.49 [1] 0.57 -4.54 0.30 5.2 [12] 0.36 -0.01 36827 0.87 6.34 ± 0.13 6.30 ± 0.13 -11.69 ± 0.32 14.72 10.01 7.72 N -6.82 Y 53.47 ± 4.59 -4.24 [1] 0.08 -4.49 0.61 -0.26 37349 0.96 25.33 ± 0.14 -2.41 ± 0.17 -7.36 ± 0.11 26.49 24.24 -9.54 N -17.59 Y 8.58 ± 5.02 -4.46 [1] 0.47 -4.87 1.17 -0.05 80337 0.63 13.23 ± 0.10 -1.03 ± 0.06 -1.40 ± 0.06 13.34 5.50 1.65 N 3.70 N 35.52 ± 3.49 -4.53 [7] 0.69 -4.62 0.37 8.5 [12] 0.68 0.05 60866 1.16 23.76 ± 0.72 1.69 ± 0.19 -16.49 ± 0.39 28.97 29.02 0.36 Y -5.58 Y 26.68 ± 6.05 -5.14 [1] 80686 0.55 12.80 5.30 -5.99 15.09 17.66 3.89 N 12.53 Y 116.12 ± 3.15 -4.65 [1] 1.52 -4.59 0.29 13 [13] 0.24 -0.08 96183 0.75 21.40 ± 0.49 -0.95 ± 0.69 -7.63 ± 0.16 22.74 21.54 -6.61 N 10.26 Y -4.75 [4] 2.56 0.11 108028 0.92 7.78 ± 0.25 -1.68 ± 0.71 -8.53 ± 0.38 11.67 12.04 -2.91 N 5.66 Y 27.80 ± 4.85 -4.40 [1] 0.30 -4.33 0.60 Probable Members 5944 0.58 17.87 ± 0.17 -3.60 ± 0.14 0.16 ± 0.14 18.23 11.82 5.46 N -2.74 N 95.70 ± 3.29 -4.47 [1] 0.50 -4.68 0.39 -0.07 8486 0.62 18.93 ± 1.21 2.03 ± 0.22 -2.55 ± 0.45 19.21 19.35 1.82 Y -5.05 Y 91.56 ± 3.46 -4.19 [1] 0.05 -4.35 0.27 -0.11 42438 0.61 10.69 ± 0.06 0.24 ± 0.07 -10.55 ± 0.07 15.02 13.07 2.25 N -11.65 Y 106.36 ± 3.41 -4.38 [1] 0.26 -4.45 0.29 5 [2] 0.29 -0.20 71395 0.98 16.90 ± 0.32 -4.47 ± 0.12 -20.53 ± 0.19 26.96 24.75 -5.66 N -2.33 N 12.30 ± 5.12 -4.45 [1] 0.43 -4.54 0.70 0.02 72659 0.72 4.99 ± 0.04 1.99 ± 0.02 -1.35 ± 0.07 5.54 5.25 1.33 N 0.75 Y 99.03 ± 3.92 -4.36 [2] 0.23 -4.37 0.37 6.2 [12] 0.29 -0.10 102485 0.43 19.67 ± 2.91 -5.05 ± 1.06 -11.47 ± 2.21 23.32 50.17 1.39 Y 48.86 N -8.08 ± 2.66 -4.40 [5] 0.31 -5.06 0.30 0.00 ⋆ Table 12 . 12Membership criteria for the IC2391 MG candidate stars. (Convergence 'Y', 'N' labels indicate if the star satisfies or not the criteria.Bol Age Prot Age [Fe/H] (km/s) (km/s) (km/s) (km/s) (km/s) (km/s) (km/s) (mÅ) (log) (Gyr) (log) (Gyr) (days) (Gyr) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) Probable Non-members 8362 0.79 -24.86 ± 0.13 -16.48 ± 0.11 -5.36 ± 0.07 30.30 29.81 4.72 N 0.58 Y 7.98 ± 4.22 -5.05 [1] 7.54 -4.96 0.89 23 [2] 2.53 0.02 18267 0.73 -21.33 ± 0.62 -24.22 ± 0.41 -5.06 ± 0.34 32.66 40.76 3.16 Y 30.95 N -4.89 [4] 4.67 -5.17 1.03 -0.01 25119 0.94 -26.59 ± 0.17 -26.68 ± 0.40 -12.84 ± 0.14 39.80 56.71 0.30 Y 54.40 N 10.76 ± 4.93 -4.88 [4] 4.42 -5.65 3.46 -0.23 54810 1.23 -24.97 ± 0.54 -20.40 ± 0.31 3.09 ± 0.32 32.39 27.57 3.24 N 3.73 N -4.50 [1] 0.61 -5.07 2.34 Doubtful Members 11072 0.61 -17.46 ± 0.27 -15.84 ± 0.27 -4.79 ± 0.15 24.06 26.58 -1.79 Y 17.04 Y 35.24 ± 3.39 -4.99 [5] 6.32 -4.51 0.14 19.3 [13] 3.25 -0.06 19832 1.18 -16.71 ± 0.23 -17.91 ± 0.93 -12.15 ± 0.25 27.34 27.13 -5.37 N 24.67 Y 25.86 ± 6.13 -4.66 [1] 1.61 22449 0.48 -25.65 ± 0.07 -14.77 ± 0.03 4.37 ± 0.04 29.92 25.13 14.35 N 22.89 Y 12.88 ± 2.82 -4.79 [3] 3.04 -4.99 0.32 -0.03 34567 0.70 -21.68 -18.03 -15.92 32.39 37.46 2.59 Y 27.44 N 54.66 ± 3.82 -4.46 [4] 0.47 -4.45 0.32 0.00 40170 1.21 -17.68 ± 0.78 -20.66 ± 1.22 -10.07 ± 0.74 29.00 29.90 -3.70 N 9.55 Y 23.26 ± 6.43 -5.15 [1] 53486 0.93 -16.63 ± 0.29 -14.39 ± 0.22 -6.15 ± 0.21 22.83 22.40 -0.94 Y 4.32 Y -4.59 [1] 1.08 -4.59 0.67 -0.03 75201 1.30 -28.10 ± 0.47 -17.90 ± 0.61 0.50 ± 0.48 33.32 37.79 10.37 N -28.78 N 16.31 ± 6.77 -4.54 [1] 0.79 76051 0.77 -27.90 ± 4.87 -16.55 ± 8.59 -18.44 ± 6.25 37.31 25.02 0.45 Y -20.80 N -4.67 [4] 1.69 95319 0.81 -21.00 -14.04 -2.91 25.43 24.42 5.82 N -21.16 Y 22.77 ± 4.31 -5.08 [3] 0.16 Probable Members 19076 0.64 -25.68 ± 0.09 -13.46 ± 0.12 -6.61 ± 0.07 29.74 24.74 4.18 N 18.14 N 44.90 ± 3.53 -4.30 [1] 0.14 -4.61 0.34 0.04 22263 0.62 -24.11 ± 0.08 -8.65 ± 0.07 -3.06 ± 0.08 25.80 39.73 8.25 N 38.27 N 55.79 ± 3.47 -4.511 [2] 0.65 -4.73 0.44 7.6 [12] 0.57 0.00 29568 0.70 -20.82 ± 0.12 -10.26 ± 0.11 -7.01 ± 0.10 24.24 43.31 1.05 Y 42.23 N 51.29 ± 3.79 -4.39 [1] 0.28 -4.29 0.28 -0.01 59280 0.80 -28.62 ± 0.50 -22.55 ± 0.40 -7.13 ± 0.10 37.13 37.79 -1.89 Y -7.77 Y 26.16 ± 4.26 -4.56 [1] 0.90 -5.24 1.26 0.14 66252 1.16 -27.83 ± 0.34 -14.63 ± 0.39 -17.03 ± 0.22 35.76 31.50 0.34 Y -12.54 N 65.40 ± 6.06 -3.89 [1] -3.05 0.15 3.9 [13] 71743 0.73 -22.30 ± 0.18 -6.93 ± 0.27 -13.83 ± 0.15 27.14 18.29 -0.15 Y -10.56 N 34.70 ± 3.96 -4.51 [1] 0.65 -4.86 0.68 0.04 ⋆ Table 13 . 13Membership criteria for the Castor MG candidate stars. labels indicate if the star satisfies or not the criteria.(Convergence Point: 4.57 hours, -18.44 degrees; U =-10.7 km/s, V = -8.0 km/s, W = -9.7 km/s; Age: 200 Myr) Kinematics Lithium CaII H,K ROSAT-data Rotation Table 14 . 14Properties of the stars classified as Other young discs starsKinematics Lithium Ca ii H,K X-ray Rotation HIP (B − V) U V W EW Li i R ′ HK Age L X /L Bol Age Prot Age [Fe/H] (km/s) (km/s) (km/s) (mÅ) (log) (Gyr) (log) (Gyr) (Days) (Gyr) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) 1692 1.31 -18.78 ± 8.10 -1.08 ± 3.74 -18.51 ± 1.00 3810 0.52 7.43 -10.88 -18.45 -4.92 [4] 5.07 -6.35 1.88 -0.07 3998 1.22 3.40 ± 0.27 -14.96 ± 0.68 -23.70 ± 0.68 21.94 ± 6.34 -4.96 [3] 5.80 -4.53 1.18 4845 1.10 5.05 ± 0.32 -13.10 ± 0.62 -13.53 ± 0.26 4.35 ± 5.90 -4.54 [1] 0.79 7235 0.75 -10.89 ± 0.18 -26.44 ± 0.29 0.94 ± 0.08 -4.72 [1] 2.20 -5.29 1.45 -0.04 7918 0.61 -38.63 ± 0.28 -31.20 ± 0.26 -0.07 ± 0.07 7.60 ± 3.42 -4.99 [3] 6.36 0.06 8275 1.02 -11.10 ± 0.15 0.75 ± 0.30 -20.62 ± 0.19 4.53 ± 5.34 -4.51 [1] 0.65 8768 1.40 -35.88 ± 0.45 -28.55 ± 0.38 -1.87 ± 0.22 26.69 ± 7.20 -4.59 [1] 1.08 -4.52 1.92 9829 0.65 6.34 ± 0.41 -12.38 ± 0.35 -9.20 ± 0.41 6.47 ± 3.57 -4.98 [4] 6.13 -0.20 10337 1.37 -32.70 ± 1.18 -21.64 ± 0.83 10.03 ± 0.50 13.06 ± 7.18 -4.68 [1] 1.79 -4.49 1.27 13081 0.83 -24.79 ± 0.60 -26.53 ± 0.99 -1.49 ± 0.19 -4.58 [4] 1.00 -4.56 0.48 -0.08 13642 0.90 -40.49 ± 0.35 -21.47 ± 0.71 -16.62 ± 0.17 -5.11 [3] 0.21 14150 0.70 -18.54 ± 0.13 -22.45 ± 0.28 -5.69 ± 0.10 -4.85 [4] 4.01 0.12 15442 0.64 -25.57 ± 0.22 -30.64 ± 0.57 -6.09 ± 0.24 5.90 ± 3.56 -4.75 [1] 2.55 -0.20 15457 0.67 -21.03 ± 0.73 -4.33 ± 0.04 -4.02 ± 0.68 38.88 ± 3.68 -4.42 [2] 0.35 -4.62 0.40 9.4 [12] 0.70 0.11 16852 0.57 1.66 ± 0.18 -15.51 ± 0.12 -42.40 ± 0.19 46.48 ± 3.22 -5.05 [3] 7.59 17.6 [13] 3.50 -0.04 23835 0.64 -26.31 ± 0.08 -23.11 ± 0.13 27.54 ± 0.19 13.06 ± 3.55 -5.13 [4] -0.16 27435 0.62 -18.69 ± 0.10 -29.66 ± 0.12 -12.38 ± 0.07 33.59 ± 3.46 -4.97 [4] 6.00 -0.22 29525 0.66 3.04 ± 0.76 -25.79 ± 0.37 -6.62 ± 0.16 39.19 ± 3.65 -4.33 [1] 0.17 -4.78 0.55 7.8 [14] -0.10 29650 0.45 -33.85 ± 0.13 -18.52 ± 0.12 -16.37 ± 0.16 4.90 ± 2.72 -4.71 [4] 2.13 -5.30 0.51 0.07 30422 0.45 -15.95 -4.07 -2.82 4.66 ± 2.72 -4.58 [4] 0.99 -5.05 0.40 33560 1.07 14.67 ± 0.37 -8.13 ± 0.36 -1.26 ± 0.19 58.70 ± 5.62 -4.14 [1] 0.03 -3.37 0.22 37279 0.43 5.48 -8.48 -18.84 2.10 ± 2.65 -4.78 [2] 2.89 -6.12 0.99 3.0 [2] 0.00 42808 0.93 -25.93 ± 0.12 -7.93 ± 0.10 -0.82 ± 0.07 69.52 ± 4.89 -4.48 [1] 0.53 -4.41 0.57 -0.03 44897 0.58 -30.64 ± 0.14 -14.03 ± 0.16 4.33 ± 0.16 28.44 ± 3.29 -4.47 [1] 0.50 -4.74 0.36 10.0 [2] 1.15 0.03 45038 0.49 -1.13 ± 0.08 -9.16 ± 0.11 1.93 ± 0.09 -5.31 [3] -6.09 1.32 -0.05 45333 0.59 8.64 -7.61 -9.46 60.35 ± 3.32 -5.07 [4] -0.13 45343 1.36 -39.13 ± 1.18 -13.70 ± 0.58 -21.71 ± 1.08 -4.56 [3] 0.91 -4.59 2.37 46509 0.46 0.94 ± 0.36 -7.33 ± 0.22 11.37 ± 0.35 3.36 ± 2.77 -4.83 [3] 3.61 -4.96 0.27 -0.01 46816 0.89 -20.11 ± 0.33 -4.35 ± 0.25 -10.13 ± 0.28 202.25 ± 4.72 -3.66 [1] -3.03 0.08 -0.21 47080 0.76 -36.76 -17.35 -15.85 10.43 ± 4.11 -4.57 [1] 0.95 -5.11 0.83 18.0 [2] 1.72 0.33 48113 0.61 11.12 ± 0.72 -5.54 ± 0.13 17.93 ± 0.84 28.30 ± 3.41 -5.04 [3] 7.39 0.13 49699 0.97 -8.69 ± 0.11 -7.09 ± 0.13 4.27 ± 0.09 11.26 ± 5.09 -5.00 [4] 6.58 0.04 49908 1.33 -8.81 ± 0.10 -20.71 ± 0.05 -36.38 ± 0.11 37.55 ± 6.82 -5.00 [1] 6.60 -5.09 3.46 6.0 [14] 49986 1.29 -2.20 ± 0.07 -12.46 ± 0.15 -2.99 ± 0.15 14.13 ± 6.75 -4.41 [11] 0.32 -4.47 3.86 51459 0.52 -13.58 ± 0.05 -1.99 ± 0.05 1.82 ± 0.05 56.71 ± 3.02 -4.78 [3] 2.97 -5.51 1.02 -0.05 53721 0.61 -24.18 ± 0.09 -2.38 ± 0.05 0.74 ± 0.08 16.15 ± 3.42 -4.91 [3] 4.93 0.03 54426 0.98 -6.42 ± 0.48 -3.54 ± 0.21 -41.94 ± 0.22 2.23 ± 5.12 -4.77 [1] 2.80 54906 0.83 -10.70 ± 0.25 1.19 ± 0.15 -4.76 ± 0.11 5.05 ± 4.42 -5.06 [4] 7.72 -0.27 56242 0.58 -21.34 ± 0.67 -30.27 ± 0.94 -20.26 ± 0.47 58.86 ± 3.26 -4.92 [2] 5.16 -5.95 2.09 14.0 [12] 2.21 -0.02 56997 0.73 7.85 ± 0.04 -16.01 ± 0.06 -3.74 ± 0.06 7.26 ± 3.95 -4.55 [2] 0.82 -5.16 1.08 17.1 [12] 1.70 -0.03 61941 0.38 -35.51 ± 0.27 -6.47 ± 0.31 -18.47 ± 0.58 43.96 ± 2.44 -4.50 [11] 0.61 -4.79 0.18 62505 0.93 4.37 ± 0.36 5.64 ± 0.50 4.00 ± 0.28 12.13 ± 4.92 -4.57 [5] 0.96 -4.68 0.68 64241 0.47 -39.00 ± 0.61 -9.53 ± 0.22 -13.70 ± 0.24 12.33 ± 2.81 -4.53 [2] 0.74 -4.91 0.21 3.0 [12] -0.21 Table 14 Continued 14Kinematics Lithium Ca ii H,K X-ray Rotation HIP (B − V) U V W EW Li i R ′ HK Age L X /L Bol Age Prot Age [Fe/H] (km/s) (km/s) (km/s) (mÅ) (log) (Gyr) (log) (Gyr) (Days) (Gyr) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) 65343 1.40 -49.45 ± 0.89 -11.19 ± 0.22 -34.57 ± 0.20 46.18 ± 7.24 -4.59 [11] 1.08 -5.10 3.06 65352 0.80 3.20 ± 0.18 4.04 ± 0.28 31.80 ± 0.15 -5.07 [4] 7.81 -0.30 66886 1.25 -35.59 ± 0.52 -21.62 ± 0.99 -38.36 ± 0.13 -4.67 [4] 1.67 67422 1.22 -20.07 ± 0.21 -25.35 ± 0.27 -13.95 ± 0.09 6.49 ± 6.30 -4.68 [1] 1.79 -4.57 1.09 -0.35 68030 0.51 -29.60 ± 0.29 -20.32 ± 0.25 -1.46 ± 0.20 32.74 ± 2.95 -4.92 [10] 5.13 -0.41 68184 1.03 -2.56 ± 0.07 -10.09 ± 0.07 -26.64 ± 0.08 19.60 ± 5.39 0.10 68337 1.16 -29.65 ± 0.39 -16.50 ± 0.30 -50.29 ± 0.21 6.46 ± 6.02 -4.72 [1] 2.20 68682 0.75 14.92 ± 0.26 -13.83 ± 0.20 -19.46 ± 0.30 5.62 ± 4.04 -4.87 [3] 4.25 -0.02 69972 1.03 -43.04 ± 0.24 -11.54 ± 0.24 -34.89 ± 0.29 25.50 ± 5.35 -5.01 [5] 6.79 0.22 72603 0.41 -24.54 ± 0.14 -8.16 ± 0.15 -12.07 ± 0.10 71.44 ± 2.54 -4.58 [11] 1.01 -5.05 0.30 -0.09 75253 0.97 -0.44 ± 0.16 -19.12 ± 0.55 -12.74 ± 0.28 30.26 ± 5.10 -4.83 [1] 3.64 0.11 75277 0.79 -19.79 ± 0.52 -11.30 ± 0.30 7.04 ± 0.75 11.75 ± 4.25 -4.90 [4] 4.81 -0.07 75722 0.91 2.08 ± 0.33 -19.22 ± 0.42 -34.75 ± 0.47 15.80 ± 4.81 -5.18 [3] -4.53 0.57 0.18 76602 0.51 -1.70 ± 0.30 -0.50 ± 0.35 -5.43 ± 0.40 65.11 ± 2.97 -4.79 [4] 3.10 0.11 76603 0.49 1.72 ± 0.40 -1.48 ± 0.41 -4.63 ± 0.56 73.03 ± 2.88 -4.86 [4] 4.12 0.00 77052 0.67 18.73 ± 0.10 -7.24 ± 0.11 11.28 ± 0.09 26.13 ± 3.69 -4.89 [7] 4.61 -5.29 1.10 0.06 81300 0.83 1.00 ± 0.07 -0.38 ± 0.04 -28.39 ± 0.10 15.43 ± 4.43 -4.73 [1] 2.31 -5.08 1.19 21.3 [12] 2.00 0.05 85561 1.24 -33.73 ± 0.18 -21.14 ± 0.72 14.61 ± 0.66 2.79 ± 6.51 -4.40 [1] 0.30 86400 0.95 20.44 ± 0.07 0.70 ± 0.09 12.03 ± 0.07 24.20 ± 5.01 -4.76 [1] 2.67 -5.58 3.15 33.5 [12] -0.16 87579 0.96 -13.61 ± 0.17 -9.89 ± 0.11 4.23 ± 0.27 3.65 ± 5.06 -4.43 [1] 0.37 -4.74 1.04 88601 0.83 7.29 ± 0.11 -19.30 ± 0.10 -18.97 ± 0.10 7.40 ± 4.44 -4.86 [1] 4.11 -4.88 0.76 19.7 [12] 1.76 0.04 90656 1.10 -42.67 ± 0.15 -32.80 ± 0.15 -25.17 ± 0.38 39.46 ± 5.71 0.19 98698 1.11 -22.88 ± 0.09 -16.92 ± 0.10 15.71 ± 0.10 29.15 ± 5.74 -4.69 [2] 1.91 -5.11 2.03 29.3 [12] 98828 0.93 3.69 ± 0.17 -9.96 ± 0.12 4.92 ± 0.15 16.57 ± 4.89 -4.81 [1] 3.34 99240 0.75 -48.79 ± 0.08 -13.38 ± 0.06 -14.92 ± 0.06 24.75 ± 4.07 -5.00 [7] 6.60 0.30 99316 0.80 -29.52 ± 0.36 -15.73 ± 0.28 14.22 ± 0.30 -5.08 [4] 99452 0.82 -31.69 ± 0.45 -27.29 ± 0.69 61.66 ± 0.73 13.97 ± 4.40 -5.09 [4] -0.09 99764 1.37 14.87 ± 0.38 -17.35 ± 0.67 -14.35 ± 0.45 10.96 ± 7.36 -4.71 [1] 2.09 105152 1.03 -36.58 ± 0.50 -32.18 ± 0.32 -24.89 ± 0.85 8.82 ± 5.39 -5.11 [3] 8.53 -0.09 106400 1.16 -41.62 ± 1.05 -18.29 ± 0.21 -13.82 ± 0.40 22.26 ± 6.02 -3.98 [1] 0.01 -3.68 111449 0.45 -16.42 ± 0.40 -21.61 ± 0.32 -10.08 ± 0.77 -4.55 [5] 0.83 -5.05 0.31 0.00 113357 0.65 -15.42 ± 0.12 -28.13 ± 0.05 14.43 ± 0.06 12.20 ± 3.58 -5.07 [2] 37.0 [2] 8.49 0.18 116771 0.51 -7.82 ± 0.03 -26.07 ± 0.07 -26.35 ± 0.06 24.82 ± 2.96 -5.11 [3] -6.15 1.59 -0.05 120005 1.40 -44.78 ± 1.97 -17.49 ± 1.11 -22.54 ± 1.75 3.47 ± 7.15 -4.39 [1] 0.28 -4.59 2.09 Table 15 . 15Properties of the stars non-members of moving groupsKinematics Lithium Ca ii H,K X-ray Rotation HIP (B − V) U V W EW Li i R ′ HK Age L X /L Bol Age Prot Age [Fe/H] (km/s) (km/s) (km/s) (mÅ) (log) (Gyr) (log) (Gyr) (Days) (Gyr) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) 171 0.65 -8.36 -76.17 -33.38 -4.85 [3] 3.96 -5.89 3.23 -0.52 910 0.48 18.82 ± 0.26 -12.66 ± 0.23 -19.95 ± 0.09 28.53 ± 2.83 -4.79 [5] 3.04 -0.34 1499 0.69 -31.47 ± 0.42 -38.69 ± 0.45 -1.61 ± 0.17 9.01 ± 3.75 -5.06 [4] 7.70 0.17 1532 1.23 18.83 ± 0.70 -26.56 ± 0.85 1.71 ± 0.38 42.26 ± 6.47 1598 0.62 18.02 ± 0.47 10.64 ± 0.58 -33.64 ± 0.52 32.07 ± 3.46 -4.98 [4] 6.21 -0.41 1599 0.56 -71.53 ± 0.12 -4.39 ± 0.04 -45.35 ± 0.08 39.59 ± 3.20 -4.84 [7] 3.79 -0.17 2941 0.72 -85.72 -52.37 -23.99 -4.90 [2] 4.83 3093 0.85 39.89 ± 0.31 -19.53 ± 0.52 8.50 ± 0.53 10.50 ± 4.53 -4.99 [2] 6.43 48.0 [12] 8.25 0.14 3206 0.96 17.95 ± 0.22 -82.78 ± 0.50 -27.33 ± 0.76 -5.02 [3] 6.96 -5.41 2.20 0.12 3418 1.10 43.02 ± 0.53 -30.44 ± 0.18 -12.44 ± 0.60 6.29 ± 5.70 -4.71 [3] 2.04 3535 1.02 29.79 ± 0.50 -36.44 ± 0.66 -36.23 ± 0.80 32.96 ± 5.32 -5.05 [3] 7.59 0.22 3765 0.88 -1.26 ± 0.16 -47.44 ± 0.28 -13.30 ± 0.35 -5.41 [1] 38.0 [12] 5.21 -0.22 3821 0.57 27.12 ± 3.24 -4.96 [3] 5.83 -6.24 3.56 -0.19 5286 1.11 4.43 ± 0.47 -28.76 ± 0.87 -41.33 ± 0.94 12.06 ± 5.79 -4.59 [1] 1.08 5336 0.68 -42.40 -157.40 -35.23 -4.91 [3] 4.90 -0.81 5799 0.45 -33.17 ± 0.25 21.12 ± 0.18 -9.34 ± 0.16 9.47 ± 2.70 -5.46 [3] -5.32 0.44 -0.33 5957 1.35 -23.43 ± 1.53 -48.75 ± 1.48 10.24 ± 0.29 32.64 ± 7.15 -4.78 [3] 2.89 6290 1.34 15.32 ± 1.05 33.23 ± 0.98 36.63 ± 1.70 -4.40 [1] 0.30 6917 0.98 -45.69 -31.09 -22.23 -4.56 [4] 0.88 -4.63 0.58 -0.32 7339 0.68 50.88 ± 0.30 -4.15 ± 0.22 1.12 ± 0.14 -5.02 [4] 6.99 7513 0.54 28.77 ± 0.04 -22.50 ± 0.02 -14.28 ± 0.07 31.01 ± 3.10 -5.04 [3] 7.26 -5.98 1.24 0.11 7981 0.83 34.82 ± 0.52 -24.97 ± 0.45 2.61 ± 0.60 3.69 ± 4.44 -5.19 [1] 35.0 [2] 4.87 -0.04 8102 0.71 18.71 ± 0.01 29.42 ± 0.02 12.80 ± 0.04 -4.96 [2] 5.82 -6.86 16.30 34.0 [2] 5.95 -0.46 9269 0.79 8.32 ± 0.25 -60.07 ± 0.61 -11.59 ± 0.46 -5.13 [4] 0.15 10138 0.82 -99.05 ± 0.38 -77.08 ± 0.22 -31.44 ± 0.12 2.31 ± 4.35 -4.68 [1] 1.79 30.0 [13] 3.83 -0.22 10416 1.05 17.30 ± 0.35 -17.85 ± 0.44 -2.60 ± 0.26 -4.30 [1] 0.14 -4.66 0.99 -0.06 10644 0.60 -35.30 -47.96 11.12 38.93 ± 3.37 -4.64 [3] 1.47 -5.05 0.65 10798 0.73 -11.03 ± 0.08 26.81 ± 0.15 -9.94 ± 0.04 0.76 ± 3.95 -4.86 [7] 4.11 -0.46 11452 1.41 -16.43 ± 0.30 10.70 ± 0.22 8.56 ± 0.30 61.17 ± 7.23 11565 1.22 -59.01 ± 1.29 -27.82 ± 0.65 -0.90 ± 0.63 -4.58 [1] 1.01 12114 0.97 -76.46 ± 0.20 0.41 ± 0.03 32.01 ± 0.17 1.15 ± 5.07 -4.96 [2] 5.82 45.0 [12] 12777 0.50 -30.27 ± 0.05 1.28 ± 0.05 -0.85 ± 0.02 69.56 ± 2.92 -5.07 [3] 7.92 -5.92 1.46 0.04 12843 0.48 14.25 ± 2.85 -4.52 [5] 0.71 -4.80 0.25 -0.07 13258 1.23 11.52 ± 0.26 -62.02 ± 1.17 6.66 ± 0.28 33.57 ± 6.38 -4.76 [1] 2.67 14286 0.67 -71.34 ± 1.28 -85.48 ± 1.32 -29.36 ± 0.47 3.47 ± 3.68 -4.99 [4] 6.36 -0.26 14632 0.59 -75.90 ± 0.10 -15.61 ± 0.12 21.45 ± 0.08 54.94 ± 3.33 -5.00 [3] 6.58 0.13 14879 0.50 -38.59 ± 0.29 16.45 ± 0.07 29.26 ± 0.10 12.88 ± 2.93 -4.90 [5] 4.79 -4.50 0.12 -0.19 15099 0.89 5.75 ± 0.56 -62.45 ± 1.20 8.88 ± 0.47 5.34 ± 4.71 -4.84 [3] 3.82 -0.04 15330 0.64 -70.01 ± 0.20 -46.56 ± 0.13 15.82 ± 0.11 2.83 ± 3.53 -4.86 [1] 4.11 -5.71 2.15 -0.22 15371 0.59 -69.72 ± 0.17 -46.46 ± 0.09 15.85 ± 0.06 4.36 ± 3.31 -4.79 [7] 3.06 -0.23 15510 0.70 -79.86 ± 0.08 -96.10 ± 0.07 -34.11 ± 0.07 -4.98 [7] 6.22 -6.70 10.59 -0.34 15673 1.01 42.65 ± 0.26 -14.07 ± 0.25 -40.49 ± 1.05 22.95 ± 5.31 -4.54 [1] 0.79 15919 1.15 48.04 ± 0.69 -29.02 ± 0.36 -21.33 ± 0.69 20.02 ± 5.95 -4.60 [1] 1.14 -5.12 1.99 16537 0.89 -3.60 ± 0.03 7.06 ± 0.01 -20.70 ± 0.03 7.30 ± 4.69 -4.62 [1] 1.28 -4.80 0.94 11.3 [12] 0.61 -0.04 Table 15 Continued 15Kinematics Lithium Ca ii H,K X-ray Rotation HIP (B − V) U V W EW Li i R ′ HK Age L X /L Bol Age Prot Age [Fe/H] (km/s) (km/s) (km/s) (mÅ) (log) (Gyr) (log) (Gyr) (Days) (Gyr) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) 17147 0.54 -111.06 ± 0.39 -89.27 ± 1.11 -42.84 ± 0.60 8.46 ± 3.11 -4.92 [10] 5.13 -0.79 17496 1.18 -89.58 ± 0.51 -6.83 ± 0.35 -16.85 ± 0.74 16.62 ± 6.15 -4.71 [3] 2.06 17651 0.43 35.18 ± 0.20 -22.45 ± 0.14 -20.85 ± 0.23 0.51 ± 2.66 -4.68 [5] 1.83 -6.88 3.70 0.04 18324 0.85 -59.46 ± 0.54 -16.91 ± 0.75 12.98 ± 0.25 -5.03 [3] 7.11 -0.32 18413 0.67 33.35 ± 0.27 16.10 ± 0.38 -5.75 ± 0.12 12.72 ± 3.67 -4.93 [4] 5.25 -0.10 19849 0.81 96.26 ± 0.08 -12.27 ± 0.03 -41.12 ± 0.08 -5.38 [1] 43.0 [2] 7.30 -0.23 20917 1.34 33.30 ± 0.14 6.86 ± 0.13 14.26 ± 0.08 55.68 ± 6.90 -4.62 [1] 1.28 -4.45 1.25 22498 1.34 -69.12 ± 1.06 -1.81 ± 1.71 19.17 ± 0.58 27.95 ± 6.93 -4.55 [11] 0.84 -4.96 1.88 23311 1.07 4.64 ± 0.12 -54.29 ± 0.21 -10.90 ± 0.05 29.34 ± 5.56 -5.29 [1] -5.90 5.25 47.0 [2] 0.30 23693 0.52 -6.12 ± 0.05 2.55 ± 0.24 -1.51 ± 0.18 75.41 ± 3.00 -4.49 [1] 0.57 -5.02 0.52 -0.12 23786 0.79 -28.29 ± 0.86 -44.02 ± 0.94 2.34 ± 0.40 5.04 ± 4.26 -4.49 [1] 0.57 -5.07 1.22 -0.17 24786 0.56 -37.98 ± 0.14 -44.72 ± 0.25 20.22 ± 0.42 27.63 ± 3.20 -4.98 [5] 6.17 -0.08 24813 0.61 -76.26 ± 0.05 -34.97 ± 0.18 4.41 ± 0.02 29.94 ± 3.42 -5.01 [3] 6.81 0.10 24819 1.01 -84.37 ± 0.21 -56.47 ± 0.43 13.01 ± 0.82 -4.53 [1] 0.74 24874 1.03 -8.38 ± 0.55 10.71 ± 0.18 30.86 ± 0.71 22.59 ± 5.39 -4.14 [1] 0.03 -4.91 1.46 0.09 25623 1.14 74.23 ± 0.30 -2.04 ± 0.28 -18.18 ± 0.39 30.00 ± 5.89 -4.60 [1] 1.14 -0.20 26505 0.82 47.65 ± 0.74 20.43 ± 0.60 -47.67 ± 0.61 4.71 ± 4.39 -5.16 [4] -0.31 27207 0.84 21.36 ± 1.10 -74.47 ± 1.20 13.76 ± 0.31 6.32 ± 4.49 -5.04 [3] 7.39 -0.02 28103 0.35 -6.01 ± 0.41 8.23 ± 0.34 1.62 ± 0.18 22.18 ± 2.33 -4.78 [11] 2.93 -5.75 0.64 28267 0.71 -106.69 ± 0.22 -92.53 ± 0.34 -37.64 ± 0.13 5.55 ± 3.87 -5.06 [4] 7.70 -0.09 29271 0.72 18.97 ± 0.06 -30.21 ± 0.17 -11.64 ± 0.10 11.36 ± 3.91 -4.94 [7] 5.49 -6.72 8.88 0.08 29432 0.64 61.84 ± 0.20 -12.69 ± 0.50 11.15 ± 0.20 13.08 ± 3.53 -4.97 [4] 6.11 -0.09 29800 0.44 -13.07 ± 0.57 7.80 ± 0.21 14.32 ± 0.11 13.04 ± 2.67 -4.51 [1] 0.65 -4.98 0.32 -0.14 29860 0.60 -15.87 ± 0.09 16.53 ± 0.19 -9.46 ± 0.12 43.55 ± 3.35 -5.00 [2] 6.62 20.0 [2] 3.63 -0.08 32010 1.02 49.16 ± 0.14 -19.57 ± 0.59 -1.36 ± 0.16 4.00 ± 5.35 -4.38 [1] 0.26 -5.04 1.91 -0.18 32423 0.97 18.90 ± 0.34 35.10 ± 0.91 -52.65 ± 1.21 11.43 ± 5.09 -4.64 [1] 1.44 -0.18 32439 0.52 44.74 ± 3.01 -5.02 [3] 7.05 -6.53 4.34 -0.16 32480 0.56 25.61 ± 0.07 8.92 ± 0.09 -2.76 ± 0.06 70.00 ± 3.17 -4.99 [3] 6.32 0.09 32919 1.22 -40.58 ± 0.44 -43.65 ± 0.88 -31.68 ± 0.71 30.23 ± 6.33 -5.06 [1] 7.72 32984 1.07 0.47 ± 0.08 13.15 ± 0.08 -19.76 ± 0.10 15.91 ± 5.55 -4.36 [1] 0.22 -4.84 1.36 -0.02 33373 1.10 -53.29 ± 0.19 -49.11 ± 1.73 10.55 ± 0.23 25.23 ± 5.73 -4.51 [1] 0.65 33537 0.64 25.58 ± 0.05 14.29 ± 0.14 4.37 ± 0.16 2.24 ± 3.52 -4.94 [4] 5.40 -0.33 33955 1.09 34.33 ± 0.23 7.57 ± 0.25 -27.68 ± 0.50 -4.74 [4] 2.43 -5.15 2.35 -0.17 34017 0.59 -18.07 ± 0.09 -77.69 ± 0.59 -9.00 ± 0.15 24.82 ± 3.32 -4.94 [3] 5.54 -0.11 35136 0.57 -79.85 ± 0.06 -1.63 ± 0.12 32.37 ± 0.07 17.60 ± 3.24 -4.93 [4] 5.25 -0.33 36357 0.93 9.05 ± 0.13 10.60 ± 0.20 14.78 ± 0.29 6.40 ± 4.88 -4.38 [1] 0.25 -5.00 1.46 36366 0.33 22.75 ± 8.04 13.51 ± 1.01 9.86 ± 3.17 31.29 ± 2.26 -4.66 [11] 1.61 -5.35 0.36 36439 0.46 25.58 ± 0.11 -15.42 ± 0.16 -4.26 ± 0.13 31.70 ± 2.77 -5.31 [3] -0.35 36551 1.14 -47.20 ± 0.28 -52.83 ± 0.66 12.47 ± 0.18 -4.44 [1] 0.40 38382 0.60 27.94 2.47 -19.69 22.01 ± 3.36 -4.88 [5] 4.49 9.7 [14] 0.99 -0.02 38657 0.97 30.56 ± 0.27 -36.15 ± 0.79 -15.14 ± 0.18 24.29 ± 5.10 -5.15 [4] 0.09 38784 0.72 -8.99 ± 0.12 8.77 ± 0.12 -38.09 ± 0.23 8.08 ± 3.92 -4.84 [3] 3.75 -5.84 2.95 -0.14 38931 1.01 -3.54 ± 0.18 5.97 ± 0.11 -12.82 ± 0.28 -5.05 [4] 7.52 -5.23 2.43 -0.23 39064 0.83 45.98 ± 0.46 -52.11 ± 1.06 -14.95 ± 0.19 9.21 ± 4.41 -5.11 [4] -0.04 39157 0.72 -13.31 -88.96 -29.79 6.29 ± 3.91 -4.92 [3] 5.04 -0.48 Table 15 Continued 15Kinematics Lithium Ca ii H,K X-ray Rotation HIP (B − V) U V W EW Li i R ′ HK Age L X /L Bol Age Prot Age [Fe/H] (km/s) (km/s) (km/s) (mÅ) (log) (Gyr) (log) (Gyr) (Days) (Gyr) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) 40118 0.66 -48.25 ± 0.29 -60.95 ± 0.69 -39.98 ± 0.68 11.67 ± 3.64 -4.85 [3] 3.92 -0.30 40375 1.19 -39.30 ± 0.51 3.93 ± 0.32 -19.41 ± 0.63 15.10 ± 6.16 -4.56 [4] 0.87 40671 1.09 -21.87 ± 0.32 -83.58 ± 2.26 -38.22 ± 1.26 24.01 ± 5.66 -4.64 [1] 1.44 40693 0.77 28.41 ± 0.21 -60.96 ± 0.17 -9.88 ± 0.10 8.03 ± 4.12 -4.99 [5] 6.43 -5.90 3.32 -0.04 40843 0.48 -24.26 ± 0.08 -38.80 ± 0.21 7.19 ± 0.09 62.43 ± 2.82 -5.19 [3] 9.83 -0.22 41484 0.62 20.11 ± 0.13 -38.97 ± 0.39 -22.89 ± 0.12 30.59 ± 3.47 -4.91 [4] 5.00 -0.01 41926 0.78 -75.43 ± 0.41 6.06 ± 0.13 -24.57 ± 0.17 -4.95 [7] 5.67 -0.37 42173 0.69 -33.42 ± 0.86 -36.33 ± 0.82 -0.79 ± 0.96 98.98 ± 3.77 -4.48 [3] 0.53 -4.80 0.51 0.08 42430 0.87 -62.62 ± 0.64 -27.29 ± 0.31 13.98 ± 0.36 24.67 ± 4.63 -5.03 [5] 7.18 -5.52 1.60 0.30 42499 0.82 18.00 ± 1.09 -30.28 ± 0.89 -29.05 ± 0.80 4.59 ± 4.38 -4.99 [3] 6.39 -0.36 42525 0.58 -49.27 ± 0.40 -2.01 ± 0.44 34.20 ± 0.50 70.66 ± 3.45 43557 0.63 19.51 -25.38 4.81 3.19 ± 3.51 -4.67 [1] 1.70 -4.93 0.52 -0.14 44075 0.52 -48.67 ± 0.14 -91.58 ± 0.18 69.61 ± 0.25 ± -4.78 [5] 2.94 -0.90 45170 0.74 -75.01 -1.37 1.41 ± -4.85 [4] 3.89 -5.60 1.59 -0.30 45617 0.99 24.59 ± 0.65 -35.93 ± 0.60 -17.17 ± 0.62 3.65 ± 5.18 -4.51 [1] 0.65 0.07 45839 1.15 -39.21 ± 0.79 -36.68 ± 0.36 -16.62 ± 1.22 18.50 ± 6.01 -4.78 [4] 2.90 45963 1.04 -24.69 -43.60 -30.49 ± -4.10 [4] 0.02 -3.11 0.05 49081 0.67 -56.14 ± 0.13 -44.12 ± 0.19 20.90 ± 0.13 17.99 ± 3.66 -4.97 [3] 6.02 0.19 49366 0.91 -12.45 ± 0.41 4.06 ± 0.16 -20.89 ± 0.38 -4.54 [1] 0.79 -5.05 1.41 -0.14 50125 1.11 22.51 ± 0.35 -82.97 ± 2.48 10.69 ± 0.90 5.62 ± 5.77 -4.81 [4] 3.30 50384 0.50 -51.29 ± 0.29 -29.25 ± 0.18 4.33 ± 0.23 53.08 ± 2.92 -5.02 [3] 6.96 -0.39 50505 0.68 12.09 ± 0.13 -27.44 ± 0.31 2.47 ± 0.12 5.29 ± 3.71 -5.00 [4] 6.60 -0.18 51248 0.59 20.38 ± 0.25 -91.34 ± 0.91 22.88 ± 0.34 14.63 ± 3.33 -4.91 [3] 4.88 -0.40 51502 0.39 -10.86 ± 1.92 5.56 ± 2.44 -1.22 ± 2.03 -4.55 [11] 0.84 -4.94 0.30 -0.30 51525 1.35 -40.54 ± 0.52 -50.66 ± 0.92 5.20 ± 0.27 27.67 ± 6.93 -4.88 [1] 4.44 -4.59 1.38 51933 0.53 69.87 ± 0.66 -35.35 ± 0.39 -36.38 ± 0.32 49.82 ± 3.05 -4.85 [5] 3.93 -0.21 52369 0.63 34.72 ± 0.42 9.05 ± 0.12 -10.15 ± 0.11 27.56 ± 3.50 -4.83 [5] 3.58 -0.10 54646 1.33 38.38 ± 0.35 2.45 ± 0.06 -2.01 ± 0.16 54.96 ± 6.84 -4.86 [1] 4.11 54651 1.08 -107.88 ± 2.12 -20.98 ± 0.19 26.81 ± 0.13 -4.94 [4] 5.47 54677 1.13 94.05 ± 2.75 -2.09 ± 0.38 -19.65 ± 0.42 21.42 ± 5.90 -4.73 [5] 2.35 54966 1.34 43.38 ± 18.37 -22.70 ± 7.29 -18.94 ± 10.14 26.16 ± 6.87 55210 0.74 77.36 ± 1.12 8.89 ± 0.28 31.03 ± 0.31 2.29 ± 4.01 -4.94 [10] 5.49 -0.22 55848 1.04 -62.03 ± 1.62 -12.75 ± 0.31 -9.85 ± 0.37 4.09 ± 5.41 0.24 56452 0.80 -47.58 ± 0.17 19.65 ± 0.05 12.37 ± 0.10 18.93 ± 4.26 -4.86 [7] 4.11 -0.32 56809 0.58 -50.90 ± 1.01 -27.36 ± 0.46 -36.02 ± 0.40 34.53 ± 3.27 -5.07 [3] 7.83 -0.17 56829 0.98 -29.95 34.34 1.33 -4.34 [11] 0.19 -4.22 0.53 57443 0.66 -59.64 ± 0.15 -38.48 ± 0.10 5.17 ± 0.05 0.95 ± 3.63 -4.95 [7] 5.67 24.0 [13] 3.81 -0.27 57757 0.56 40.39 3.45 6.65 18.12 ± 3.18 -4.99 [6] 6.41 -5.75 0.84 0.14 57939 0.74 277.96 ± 0.94 -157.18 ± 0.57 -13.57 ± 0.33 -4.90 [2] 4.71 31.0 [2] 4.71 -1.22 59000 1.37 20.65 ± 0.47 -42.11 ± 0.62 -0.88 ± 0.77 45.91 ± 7.16 -4.12 [1] 0.03 -3.68 0.44 59750 0.46 53.11 ± 0.98 -71.06 ± 1.28 -60.92 ± 1.20 10.35 ± 2.78 -4.65 [2] 1.53 -5.81 1.69 7.0 [2] -0.76 61317 0.59 8.51 ± 3.31 -4.85 [3] 3.96 -0.16 61901 1.09 22.47 ± 0.24 -22.12 ± 0.21 18.62 ± 0.10 -5.02 [4] 6.98 62207 0.55 -41.67 ± 0.19 6.94 ± 0.08 75.31 ± 0.09 38.10 ± 3.16 -4.98 [3] 6.24 -0.50 63366 0.81 -74.85 ± 1.47 -34.15 ± 0.64 17.35 ± 0.30 8.40 ± 4.32 -4.88 [3] 4.37 -0.30 Table 15 Continued 15Kinematics Lithium Ca ii H,K X-ray Rotation HIP (B − V) U V W EW Li i R ′ HK Age L X /L Bol Age Prot Age [Fe/H] (km/s) (km/s) (km/s) (mÅ) (log) (Gyr) (log) (Gyr) (Days) (Gyr) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) 85235 0.76 1.59 ± 0.08 -49.84 ± 0.09 5.49 ± 0.12 -4.93 [10] 5.31 -0.33 85295 1.35 0.01 ± 0.15 -52.91 ± 0.26 -9.92 ± 0.05 28.77 ± 6.92 -4.72 [1] 2.20 -4.95 2.68 85810 0.65 -31.01 -49.90 -1.88 33.56 ± 3.59 -4.46 [1] 0.46 -5.83 1.92 0.09 86036 0.59 35.97 -4.43 -22.33 61.79 ± 3.31 -4.76 [1] 2.67 -5.04 0.55 -0.17 86722 0.78 67.13 ± 1.05 -28.04 ± 0.90 -1.06 ± 0.27 7.55 ± 4.20 -5.09 [4] 8.32 -0.34 88622 0.60 -78.72 ± 0.32 -88.27 ± 0.43 -38.62 ± 0.26 -4.61 [3] 1.23 -5.31 1.07 -0.49 88972 0.88 16.61 ± 0.11 -31.28 ± 0.07 0.33 ± 0.08 12.59 ± 4.66 -4.96 [2] 5.76 42.0 [12] 6.24 -0.05 89937 0.50 3.64 ± 0.10 39.90 ± 0.24 -3.15 ± 0.18 29.09 ± 2.94 -4.90 [11] 4.78 -6.16 2.29 -0.40 91009 1.17 18.22 -17.57 -28.39 4.29 ± 6.06 -3.66 [1] -3.06 0.10 91438 0.65 38.24 ± 0.10 -2.15 ± 0.08 -4.50 ± 0.05 24.36 ± 3.61 -4.89 [7] 4.61 -5.76 2.55 -0.24 92043 0.47 37.18 ± 0.20 1.21 ± 0.22 -8.01 ± 0.08 -4.90 [3] 4.74 -5.27 0.29 0.04 92200 1.23 23.39 ± 0.25 -10.67 ± 0.34 -0.75 ± 0.10 6.96 ± 6.37 -4.65 [1] 1.52 92283 1.06 14.11 ± 0.30 -23.24 ± 0.29 -25.33 ± 0.38 20.56 ± 5.52 -4.68 [1] 1.79 0.06 93017 0.56 -15.02 ± 0.30 -37.95 ± 0.60 -27.23 ± 0.19 42.51 ± 3.18 -4.87 [2] 4.34 -5.67 1.39 16.0 [2] 3.21 -0.15 93871 1.06 70.55 ± 2.26 -66.91 ± 2.56 -7.61 ± 0.33 1.12 ± 5.55 95995 0.85 45.27 3.20 27.64 -5.05 [4] 7.48 -0.26 96100 0.79 31.40 ± 0.07 43.23 ± 0.05 -18.93 ± 0.06 0.78 ± 4.21 -4.83 [2] 3.67 -5.59 2.66 27.0 [2] 3.37 -0.16 96285 1.19 -69.51 ± 0.63 -14.56 ± 0.60 -13.72 ± 0.51 29.42 ± 6.18 -5.08 [1] 97944 1.03 3.24 -28.81 0.37 ± -4.58 [5] 1.03 -4.70 0.49 98677 0.73 52.84 ± 0.70 -15.60 ± 0.77 -17.19 ± 0.26 13.84 ± 3.97 -4.83 [3] 3.61 -0.30 98819 0.60 42.36 ± 0.27 -20.31 ± 0.17 9.52 ± 0.12 53.60 ± 3.37 -4.80 [2] 3.16 -5.88 2.01 13.5 [12] 1.75 0.04 99461 0.85 -118.41 ± 0.09 -51.77 ± 0.08 47.08 ± 0.07 -5.39 [1] -0.44 99711 0.94 -22.84 ± 0.59 4.19 ± 0.58 22.21 ± 0.38 8.15 ± 4.95 -4.60 [1] 1.14 -5.07 1.49 -0.05 99825 0.89 -73.00 ± 0.19 -11.63 ± 0.06 -18.69 ± 0.17 14.56 ± 4.70 -5.05 [5] 7.50 -0.02 101345 0.69 -18.58 ± 0.24 18.86 ± 0.15 -28.04 ± 0.23 48.37 ± 3.76 -5.15 [3] 0.04 101955 1.34 -68.74 -16.82 -37.14 0.87 ± 6.94 -4.97 [3] 6.08 -4.36 1.31 101997 0.72 -58.88 ± 0.18 20.34 ± 0.21 4.22 ± 0.14 -4.93 [7] 5.31 -0.28 103256 1.01 -81.52 ± 1.33 -8.01 ± 0.56 -8.59 ± 0.52 18.34 ± 5.29 -4.50 [1] 0.61 104092 1.15 -15.09 ± 0.31 -76.68 ± 0.42 3.91 ± 0.40 23.52 ± 5.95 -5.23 [1] 104214 1.15 -94.40 ± 2.10 -54.60 ± 0.26 -8.28 ± 0.45 -4.76 [2] 2.72 37.9 [12] 104217 1.29 -92.19 ± 0.18 -53.34 ± 0.11 -9.41 ± 0.06 -4.89 [2] 4.62 48.0 [12] 104858 0.51 5.77 -28.95 -10.05 44.03 ± 2.96 -4.91 [4] 4.86 -0.15 105858 0.48 -14.30 ± 0.05 44.80 ± 0.06 7.01 ± 0.05 45.15 ± 2.84 -4.49 [5] 0.57 -0.64 109378 0.75 5.01 ± 0.17 -49.87 ± 0.42 -7.53 ± 0.25 4.65 ± 4.06 -5.10 [3] 0.20 109527 0.82 14.04 ± 0.27 -25.35 ± 0.14 -17.31 ± 0.37 12.56 ± 4.36 -4.41 [1] 0.32 -4.88 0.70 0.16 110109 0.59 -29.52 ± 0.13 -42.13 ± 0.24 6.19 ± 0.06 22.21 ± 3.32 -4.86 [7] 4.11 -6.18 3.45 -0.19 111888 0.89 -20.60 ± 0.50 13.18 ± 0.16 -13.63 ± 0.16 29.77 ± 4.69 -4.59 [1] 1.08 -4.97 1.52 112190 0.97 22.56 ± 0.60 -27.20 ± 0.58 7.55 ± 0.57 17.36 ± 5.11 -4.93 [4] 5.22 -0.10 112447 0.50 3.87 ± 0.08 -31.77 ± 0.14 -27.86 ± 0.14 43.13 ± 2.91 -5.28 [3] -0.18 112870 0.85 -45.31 ± 1.01 1.20 ± 0.12 -5.99 ± 0.17 3.65 ± 4.54 -4.97 [3] 5.96 -0.47 113576 1.32 34.84 ± 0.19 16.95 ± 0.11 0.11 ± 0.19 26.85 ± 6.80 -4.63 [1] 1.36 113718 0.94 -28.52 ± 0.37 -47.12 ± 0.42 16.73 ± 0.35 15.17 ± 4.94 -5.09 [4] 114622 1.01 -52.91 ± 0.12 -39.48 ± 0.05 -14.27 ± 0.04 -5.10 [3] -6.46 12.36 0.09 114886 0.91 -41.75 ± 0.74 -1.78 ± 0.24 -10.97 ± 0.19 8.63 ± 4.79 -5.11 [1] -0.05 115331 0.81 -66.57 ± 0.77 -16.76 ± 0.25 -2.91 ± 0.10 49.87 ± 4.31 -4.15 [1] 0.04 -4.61 0.55 -0.01 The terms moving group and supercluster are used here without distinction.3 To avoid confusion we recall that by "probable member" of the Hyades supercluster we mean member of the group of coeval stars evaporated from the primordial Hyades cluster The Extrasolar Planets Encyclopedia, http://www.obspm.fr/encycl/es-encycl.html † Label 'Y' indicates probable members, '?' doubtful members and 'N' probable non-members, respectively. 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[ "Constraints and Soliton Solutions for the KdV Hierarchy and AKNS Hierarchy", "Constraints and Soliton Solutions for the KdV Hierarchy and AKNS Hierarchy" ]	[ "Nianhua Li \nCenter for Nonlinear Science Center\nNingbo University\n315211NingboChina\n", "Yuqi Li \nCenter for Nonlinear Science Center\nNingbo University\n315211NingboChina\n" ]	[ "Center for Nonlinear Science Center\nNingbo University\n315211NingboChina", "Center for Nonlinear Science Center\nNingbo University\n315211NingboChina" ]	[]	It is well-known that the finite-gap solutions of the KdV equation can be generated by its recursion operator. We generalize the result to a special form of Lax pair, from which a method to constrain the integrable system to a lower-dimensional or fewer variable integrable system is proposed. A direct result is that the n-soliton solutions of the KdV hierarchy can be completely depicted by a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable. *	10.1088/0253-6102/56/4/01	[ "https://arxiv.org/pdf/1011.5752v1.pdf" ]	119,680,429	1011.5752	b1f563aee311a26e50020eb7c9f5970d0f37e546	Constraints and Soliton Solutions for the KdV Hierarchy and AKNS Hierarchy 26 Nov 2010 November 29, 2010 Nianhua Li Center for Nonlinear Science Center Ningbo University 315211NingboChina Yuqi Li Center for Nonlinear Science Center Ningbo University 315211NingboChina Constraints and Soliton Solutions for the KdV Hierarchy and AKNS Hierarchy 26 Nov 2010 November 29, 2010 It is well-known that the finite-gap solutions of the KdV equation can be generated by its recursion operator. We generalize the result to a special form of Lax pair, from which a method to constrain the integrable system to a lower-dimensional or fewer variable integrable system is proposed. A direct result is that the n-soliton solutions of the KdV hierarchy can be completely depicted by a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable. * Introduction Much effort has been devoted to finding the exact solutions to integrable systems since Gardner, Greene, Kruskal and Miura found the inverse scattering (IST) transform method to solve the KdV. In general on the full line the reflectionless potential is solved by the IST as soliton or multi-soliton solutions. It is widely believed that the IST is inefficient to obtain the other kinds of solutions. Therefore, from this point of view it is amazing that the periodic KdV is completely solved by the algebraic-geometric solutions. The crucial fact in obtaining the algebraic-geometric solutions is that the stationary solutions of any higher-order KdV is invariant to the usual KdV. The idea that solving PDEs in its finite-dimensional invariant subspaces somehow has been developed to the method of nonlinearization of Lax pair [2] or symmetry constraint [3]. In most cases, the solutions obtained by nonlinearization of Lax pair are algebraic-geometric solutions, from which useful information is hard to get because of the complex expressions of the solutions. Furthermore, returning to get the soliton solutions some proper constraints has to be imposed for the algebraic-geometric solutions. This seems to be not straightforward. It is even more complicated to characterize the other kinds of solutions such as the elliptic solutions of the KdV. What is even worse is that there is no a rigid classification for the kinds of solutions for integrable partial differential equations (IPDEs). In this paper we will neatly characterize the soliton solutions by a less popular Lax pair for the KdV hierar-chy without any knowledge of the algebraic-geometric solutions or any other old methods for obtaining the soliton solutions such as the Bäcklund or Darboux transformations. For an IPDE the kind of solutions that can be gotten by only solving some linear PDE are of special interest. At first glance there seems to be little chance to realize this for an S-integrable system such as KdV or AKNS. The most desirable situation for solving a nonlinear partial differential equation is that it can be linearized by an appropriate change of variables, which is called C-integrable. The famous example of such kind is the Burgers equation. But most researchers firmly believe that a true S-integrable system such as KdV or AKNS will never be linearized by a common change of variables. So it will be very interesting to know to what an extent the S-integrable system is solvable by the change of variables. In this paper we will give a series of constraints on the AKNS hierarchy. The final result is that under the constraints the resulting equations are all linearizable by some proper transformations. The paper is organized as follows. Section 2 introduces a special form of Lax pair, about which a basic theorem is given. The theorem states how to construct invariant manifolds corresponding to the special form of Lax pair. Section 3 introduces a special form of Lax pair generating the KdV hierarchy. By the theorem introduced in Section 2, the invariant manifolds corresponding to the Lax pair are constructed. It turns out that the invariant manifolds are just the n-soliton solutions of the KdV hierarchy. Section 4 deals with a special form of Lax pair of the AKNS hierarchy. We will first obtain the special kind of constraints. Then we will solve the first few invariant manifolds in detail. At last we will prove the main theorem for the constraints of the AKNS hierarchy. Constraint for evolution equations with a special form of Lax pair Integrable equations are consistency conditions of the Lax pair Lϕ = λϕ,(1)ϕ t =Pϕ,(2) where the eigenfunction ϕ = ϕ(x, t) is n-dimensional vector and linear operatorsL andP are differential polynomials of potential u = u(x, t). The method of nonlinearization of Lax pair or symmetry constraint method set up additional constraints between the potential u and the eigenfunction ϕ. With the additional constraints Equations (1) and (2) will become ODEs in most cases. Let us still take the KdV as an example. The KdV u t = 6uu x + u xxx has a wellknown Lax pair (∂ 2 + u)ϕ i = λ i ϕ i ,(3)∂ t ϕ i = (4∂ 3 + 3(u∂ + ∂u))ϕ i(4) and the well-known constraint for the KdV is u = c 0 + n i=1 c i ϕ 2 i .(5) With the constraint (5), Equation (3) and Equation(4) become two sets of ODEs. In most cases the efficient way to find a constraint for Lax equations is the symmetry constraint method. But symmetry constraints are not all constraints. It is observed in [7] that systems with the following special form of Lax pair have natural constraints. • OperatorL has formL =L + + n i=1 h i ∂ −1 g i , whereL + is a differential operator and h i and g i are differential polynomials of potential u. • OperatorP is a differential operator. The following theorem guarantees a natural constraint. Theorem 2.1 For systems with Lax pairs in the above form, there is a function It is also well-known that Lax pair has a less popular form L F = n i=1 a i h i such that a constraint L F = m j=1 b j ϕ j exists, where ϕ j is the eigenfunctionLϕ j = λ j ϕ j ,f j+1 =L f j ,(6) f jt =P f j . Correspondingly Theorem 2.1 has a variant form: Theorem 2.2 There is a function L F = n i=1 a i h i such that a constraint m j=1 b j f j = 0 exists, where f 1 = L F and the requirements forL,P, a i s, b j s are the same as in Theorem 2.1. In the following paper Theorem 2.2 will be applied more frequently. The soliton constraint for the KdV hierarchy The KdV hierarchy is defined by its recursion operatorφ 1 = ∂ 2 + 4u + 2u x ∂ −1 u t =φ n 1 u x ,(8) where n is an arbitrary positive integer. The first nontrivial equation of the hierarchy is the KdV equation u t = 6uu x + u xxx .(9) Theorem 3. 1 The following Lax pair [6], Lϕ = (∂ + u∂ −1 )ϕ = λϕ, (10) ϕ t =P m ϕ = (L m ) + ϕ(11) generates the KdV hierarchy, where m is an odd integer andL m + is the differential part [5] of pseudo-differential operatorL m . Proof: First we proveL n + = ∂h,(12) where n is an odd positive integer and h is a differential operator. In fact for odd n we will prove (∂ + u∂ −1 ) n + ∂ = ∂(∂ + ∂ −1 u) n + .(13)By ∂ −1 (∂ + u∂ −1 ) n ∂ = (∂ + ∂ −1 u) n , we obtain (∂ + u∂ −1 ) n ∂ = ∂(∂ + ∂ −1 u) n . Then we immediately get [(∂ + u∂ −1 ) n ∂] + = [∂(∂ + ∂ −1 u) n ] + .(14) Equation (14) is equivalent to (∂ + u∂ −1 ) n + ∂ + res(∂ + u∂ −1 ) n = ∂(∂ + ∂ −1 u) n + + res(∂ + ∂ −1 u) n .(15) In fact for odd n we have res (∂ + u∂ −1 ) n = res (∂ + ∂ −1 u) n ,(16) because res (∂ + u∂ −1 ) n = res (−1) n [(∂ + ∂ −1 u) n ] * = (−1) n+1 res (∂ + u∂ −1 ) n . Formula (13) is just a direct result of Equation (15) and Equation (16). Secondly we prove that d dtL = [P m ,L] is only one PDE for u, or in other words we will prove [P m ,L] = f [u]∂ −1 ,(17) where f [u] denotes a differential polynomial of u. By (12) [P m ,L] =Ĝ + f [u]∂ −1 ,(18) whereĜ is a differential operator. But we also have [5] [ P m ,L] = [L m −L m − ,L] = −[L m − ,L]. So the order of [P m ,L] is less than 0. This fact and Equation (18) = [P m ,L] is only a PDE u t = f [u]. At last we prove d dtL = [P m ,L], m = 1, 2, 3, · · ·, is the KdV hierarchy. This can be verified by its recursion operatorφ = ∂ 2 + 4u + 2u x ∂ −1 , which may be easily carried out by the method established by [6]. By Theorem 3.1 and Theorem 2.2 we immediately know m j=1 b j (∂ + u∂ −1 ) j−1 u = 0 is a proper constraint, which has been proved [7] to be all the soliton solutions of the KdV equation. Special constraints for the AKNS hierarchy The special type of Lax pair for the AKNS hierarchy The AKNS hierarchy [1,8] is q r t =φ n −iq ir , whereφ is the recursion operator ϕ = 1 i −∂ + 2q∂ −1 r 2q∂ −1 q −2r∂ −1 r ∂ − 2r∂ −1 q and n is an arbitrary positive integer. The first equation of the hierarchy is q r t = 1 i −q xx + 2q 2 r r xx − 2qr 2 .(19) The natural Lax pairφ t = [P m ,φ] generates the AKNS hierarchy. Proof: First we must prove the Lax equation (20) is just two PDEs for q and r respectively. In fact we will prove q r t = P m · q −P * m · r ,(21) whereP m · q denotes the differential polynomial gotten by acting the operatorP m on q andP * is the conjugate operator ofP. Because [P m ,L] = 1 i [L m −L m − ,L] = i[L m − ,L], we know [P m ,L] + = 0. So [P m ,L] = (P m · q)∂ −1 r − q∂ −1 (P * m · r).(22) With Equation (22) and Equation (20) we immediately get Equation (21). Secondly we will prove the recursion operator of the hierarchy (22) iŝ ϕ = 1 i −∂ + 2q∂ −1 r 2q∂ −1 q −2r∂ −1 r ∂ − 2r∂ −1 q .(23)t n+1 =L t nL + [R n ,L],(24) whereR n = 1 i (a n + b n ∂ −1 r). Equation (24) is just the recursion equation q t n+1 r t n+1 =φ q t n r t n ,(25) whereφ is the recursion operator (23). Proof: We will first prove Equation (24) . SinceL n+1 =LL n , we havê P n+1 = 1 i (L n+1 ) + = 1 i (L nL ) + = 1 i ((L n ) +L + (L n ) −L ) + = 1 i L n +L − (L n +L ) − + (L n −L ) + , which leads directly toL t n+1 = [P n+1 ,L] =L t nL + [R n ,L], whereR n has the expressionR n = 1 i −(L n +L ) − + (L n −L ) + . SoR n can be expressed asR n = 1 i (a n + b n ∂ −1 r).(26) Then we will prove (25). Substituting (26) to (24) we get i × q t n+1 ∂ −1 r + q∂ −1 r t n+1 = (q t n ∂ −1 r + q∂ −1 r t n )(−∂ + q∂ −1 r) + [a n + b n ∂ −1 r, −∂ + q∂ −1 r]. (27) The positive part of Equation (27) gives a n = (q t n r + qr t n )dx. Rearranging the negative part of Equation (27) we get (iq t n+1 − a n q − b ′ n )∂ −1 r + q∂ −1 (ir t n+1 − r ′ t n + ra n ) = (q t n + b n )∂ −1 (r ′ + rq∂ −1 r) + q∂ −1 (r t n q − rb n )∂ −1 r. Left multiplying (29) with 1 r ∂ and right multiplying (29) with ∂ 1 q simultaneously, considering its negative part we get b n = −q t n . (30) Substituting b n = −q t n to (29) and considering q t n r + qr t n = a ′ n , we get (iq t n+1 − 2a n q − b ′ n )∂ −1 r + q∂ −1 (ir t n+1 − r ′ t n + 2ra n ) = 0. Note we have applied the well-known formula ∂ −1 f ′ ∂ −1 = f ∂ −1 − ∂ −1 f . Now it is clear q t n+1 = 1 i (2a n q + b ′ n ), r t n+1 = 1 i (r ′ t n − 2ra n ). By substituting (28) and (30) to the above equations we immediately get the recursion equation (25). Specially constraints for AKNS hierarchy Applying Theorem 2.2 to the Lax pair in Theorem 4.1 does not generate two systems of ODEs, because the constraintL n (0) = 0 only offers one constraint between q and r and still another constraint between q and r must be given for q and r being ODEs of the independent variable x. The situation is best explained by the case n = 1. When n = 1,byL(0) = 0 we immediately get q = 0. So for the usual AKNS (19) only one PDE is left r t = −ir xx . The second constraintL 2 (0) = 0 can be simplified to qr = ( q x q ) x .(31) It has been noticed [4] that with the constraint (31) the AKNS hierarchy is constrained to the Burgers hierarchy of w by the transformation w x = qr = ( q x q ) x . And it is also well-known that the Burgers hierarchy can be linearized by the famous Cole-Hopf transformation. So with the constraint (31) the AKNS hierarchy may be linearized, which is best explained by the following theorem: Theorem 4.3 With the constraint (31) the AKNS hierarchy can be constrained to linear equations u t = (−i) m−1 u (m) , where u = 1 q . Proof: Let's prove this theorem by mathematical induction. When k = 2 the AKNS, which is just the NLS equation in this case, is constrained to iq t + q xx − 2q( q x q ) x = 0. Substituting q with 1 u , we get the linear equation u t 2 = −iu xx . When k = n, we assume the equation has been transformed to u t n = (−i) n−1 u (n) , i.e., q t n r t n =φ n −iq ir = −(−i) n−1 q 2 ( 1 q ) (n) ((ln q) xx /q) t n . When k = n + 1, by qr = ( q x q ) x we get ∂ −1 (qr) t n = ( q x q ) t n , then by the recursion operator we get q t n+1 = 1 i −∂ + 2q∂ −1 r 2q∂ −1 q φ n −iq ir . Substituting q = 1 u , we finally we get u t n+1 = (−i) n u (n+1) . This finishes the proof. Then we consider the constraintL 3 (0) = 0, which is equivalent to qr = ( (q 3 r) x − qq xxx + q x q xx −qq xx + q 2 x + q 3 r ) x = ln(q 3 r − qq xx + q 2 x ) xx .(32) Theorem 4.4 With (32) the standard AKNS is constrained to g t = −ig xx by the transformation g = q q 3 r − qq xx + q 2 x . (33) Proof: Direct calculation shows g t + ig xx = 2ig qr − ln(q 3 r − qq xx + q 2 x ) xx . By (33) and (32) q can be easily obtained q = − 1 g(ln g) xx .(34) Let us explain how (33) is obtained. Equation iq t + q xx − 2q 2 r = 0 is transformed to −4u 2 x u 2 t + 7u 2 t uu xx − 2u t u 2 u xxt + 2u 2 xt u 2 − 2u 2 xxx u 2 + 11iu t uu 2 xx − 2iu t u 2 u xxxx − iu 3 t u −5u 3 xx u + 4iu 2 u xt u xxx − 2iu 2 u xxt u xx − 8iu 2 x u t u xx + 4u 2 x u 2 xx + 2u xxxx u 2 u xx = 0(35) by (32) and q = 1 u . Here u t = −iu xx is still a solutions of Equation (35). Then Equation (35) is transformed to iv xxt − iv xx v t − v xxxx + 2v xxx v x − v xx v 2 x + 3v 2 xx = 0(36) by e v = u t +iu xx u 2 . Now the solution u t = −iu xx has been ruled out. Equation (36) can be written into a more compact form i(∂ 2 − v xx )(v t + iv xx − iv 2 x ) = 0.(37) Equation (37) suggests us to calculate v t + iv xx − iv 2 x , which turned out to be 0. But v t + iv xx − iv 2 x = 0 is linearized by transformation v = −ln(g). Therefore we get the final transformation (33). Note that v t + iv xx − iv 2 x = 0 can also be gotten by considering the compatibility condition between equation ir t − r xx + 2qr 2 = 0, Equation (32) and (36). The third equation of the AKNS hierarchy is the coupled KdV q r t = 6qrq x − q xxx 6qrr x − r xxx .(38) It is easy to verify that under constraint (32) Equation (38) is simplified to g t = −g xxx , where g is also defined by (33). Then we consider the caseL 4 (0) = 0, which is equivalent to qr = (ln p 3 ) xx ,(39) where p 3 is defined as p 3 = 5q 2 x r 2 q 3 − q xxxx qq xx + q xxxx q 3 r − 2q xxx q 3 r x − 6q xxx q x q 2 r − 2q xxx q x q xx +7q xx q 2 x qr + 8q x r x q 2 q xx + q xxxx q 2 x + 3q 2 xx q 2 r − 5r 2 q xx q 4 − 6q 3 x qr x − q 5 r xx r +q 3 r xx q xx − q 2 q 2 x r xx − 5q 4 x r + q 2 xxx q + q 5 r 2 x + q 6 r 3 + q 3 xx .(40) Let us define p 2 as p 2 = q 3 r − qq xx + q 2 x .(41) It can be direct verified that under constraint (39) the standard AKNS is simplified to h = −ih xx , where h = p2 p3 . Then q can be obtained by 1 q = h(ln h) xx (ln(h h(ln h) xx )) xx .(42) Before summarizing all results above, let us first define p k and θ k (g) recursively. Function p k is completely determined by k: p −1 = r, p 0 = 1, p 1 = q, p k+1 p k = qr − (ln p k ) xx p k p k−1 .(43) Function θ k (g) is determined by both k and g: θ 1 (g) = g, θ k+1 (g) = (ln(θ 1 (g)θ 2 (g) · · · θ k (g))) xx θ k (g). Now we summarize our main result as following: Theorem 4.5 The constraintL n q = 0 is equivalent to qr = (ln p n ) xx , whereL = 1 i (−∂ + q∂ −1 r). With the constraint the m-th AKNS equation is equivalent to ∂u n ∂t = (−i) m−1 u (m) n , where u n = p n−1 p n and p is defined by (43). Given u n , q and r are determined by q −1 = (−1) n−1 θ n (u n ) and r = (−1) n θ n+1 (u n ) respectively, where θ is defined by (44). The proof of Theorem 4.5 consists of several parts. The following theorems greatly reduce the complexity of the proof of Theorem 4.5. So we will first prove the following Lemma 4.6, Theorem 4.7 and Proposition 4.8, Theorem 4.9. At last we will prove Theorem 4.5. Let us define (iL) k q = f k , and define α j by α 0 = q for j = 0 and α j = qr − (ln p j ) xx for j 0. Lemma 4.6 f k = α 0 ∂ −1 α 1 ∂ −1 α 2 · · · α k−1 ∂ −1 α k . Proof: Obviously the theorem is true for k = 1. Suppose the theorem is true for k = s. Then we must compute f s+1 = (−∂ + q∂ −1 r) f s . Let us defineĈ j byĈ j = α j ∂ −1 . Then we have f k =Ĉ 0Ĉ1 · · ·Ĉ k 0, where we have set ∂ −1 0 = 1. It can be verifiedB kĈk =Ĉ k B k+1 , whereB k is defined asB 0 = iL for k = 0 andB k = −∂ − (ln p k p k−1 ) x + α k ∂ −1 for k 0. Therefore, the theorem must be true for k = s + 1, becauseB s+1 0 = α s+1 . By Lemma 4.6, we immediately get: So Equation (48) is proved. Let us give a simple example to illustrate Theorem 4.5. Choose n = 4 and m = 3 in Theorem 4.5. Then u 4 ∂t = −(u 4 ) xxx . Choose a solution of it such as u 4 = 1 + e 8t−2x + e t−x + e x−t + e 2x−8t . Then we obtain q = −e 18t − e 6x − 9e 17t+x − 36e 10t+2x − 9e 16t+2x − 65e 9t+3x − 9e 2t+4x − 36e 8t+4x − 9e t+5x 36e 17t+x + 576e 10t+2x + 1296e 9t+3x + 576e 8t+4x + 36e t+5x , r = 576e 8t+2x e 16t + e 4x + 16e 9t+x + 36e 8t+2x + 16e 7t+3x . It is easy to check that the above solution is indeed a solution of (38), the m-th equation of AKNS hierarchy. Conclusions In this work, we apply a special form of Lax pair to analyze the solutions of KdV hierarchy and AKNS hierarchy. For the KdV hierarchy, the soliton solutions are completely depicted by a new, simple and direct way. For the AKNS Hierarchy a special form of Lax pair is analyzed. With the special kind of Lax pair a wide class of solutions of the AKNS hierarchy have been obtained. At last we give the main theorem which states how to linearize all equations of the AKNS hierarchy with the special kind of constraints mentioned in this paper. a i s are some proper constants, b j s are arbitrary constants and m is an arbitrary positive integer. of the AKNS hierarchy fulfills Theorem 2.2, whereP m is the linearization operator of the AKNS. So the constraint proper constraint. But here we will not discuss this useful symmetry constraint. We will investigate the constraints induced by the following Lax pair of the AKNS.Theorem 4.1 [9] LetL = 1 i (−∂ + q∂ −1 r),P m = 1 iL m + . Then Lax equation L t = [P m ,L] imply Equation (17). So we have proved that d dtL AcknowledgmentsThe authors would like to thank Prof. Lou S. Y., who had read the draft and given a lot of instructions. The work is partly supported by NSFC (No.10735030), NSF of Zhejiang Province (R609077, Y6090592), NSF of Ningbo City (2009B21003, 2010A610103, 2010A610095).Theorem 4.7L n q = 0 implies qr = (ln p n ) xx . The following proposition is also crucial for the proof of our final result. Proposition 4.8 For the AKNS hierarchyandSketch of the proof: According to the recursion equation (25) the proposition is clearly true for k = 1. So let us suppose the proposition is true for for k = j. Then we must prove it is also true for k = j + 1. First we can solve (p j−1 ) t n+1 and (p j ) t n xx by(45)and(46)when k = j. When k = j + 1, we can verify, by a lengthy but straightforward calculations, Equations(45)and(46)are simply two identities after considering (43). Theorem 4.9 If q and θ are defined by(43)and(44)respectively. Given arbitrary function g, set θ 1 = g, q −1 = (−1) n−1 θ n (g) and r = (−1) n θ n+1 (g). Then θ n−k+1 = (−1) n−k p k−1 p k . As a result, g = p n−1 p n , qr = (ln p n ) xx . Conversely if p is defined by (43), θ is defined by θ n−k+1 = (−1) n−k p k−1 p k and there is a constraint qr = (ln p n ) xx between q and r, then θ satisfies (44), q −1 = (−1) n−1 θ n (g) and r = (−1) n θ n+1 (g), where g = p n−1 p n . Proof: The proof is obvious by mathematical induction. Now we will give the proof of Theorem 4.5. Proof of Theorem 4.5: We only need to proveIn fact (47) is a direct result of the following equationbecause(47)is obviously true for m = 1. Suppose (48) has been true for m = k.Then u t k+1 can be directly calculated as(p n ) t k p n = i p n−1 p n 2(ln p n ) t k x − (p n−1 ) t k x p n−1 + (p n ) x p n (p n−1 ) t k p n−1 − (p n ) t k x p n + (p n−1 ) x p n−1 (p n ) t k p n = −i − p n−1 (p n ) t k p 2 n + (p n−1 ) t k p n x = −i u t k x . Nonlinear evolution equations of physical significance. M J Ablowitz, D J Kaup, A C Newell, H Segur, Phys. Rev. Lett. 31Ablowitz M. J., Kaup D. J., Newell A. C. and Segur H. Nonlinear evolution equations of physical significance, Phys. Rev. Lett. 31(1973), 125-127. Nonlinearization of the Lax system for AKNS hierarchy. C W Cao, Science in China A. 33Cao C. W., Nonlinearization of the Lax system for AKNS hierarchy, Science in China A, 33(1990), 528-536. The constraint of the Kadomtsev-Petviashvili equation and its special solutions. Y Cheng, Y S Li, Phys.Left.A. 157Cheng Y. and Li Y. S., The constraint of the Kadomtsev-Petviashvili equation and its special solutions, Phys.Left.A, 157(1991), 22-26. Transformation from AKNS hierarchy to Burgers hierarchy. Y Cheng, Journal of henan normal university. 37Cheng Y., Transformation from AKNS hierarchy to Burgers hierarchy, Journal of henan normal university, 37(2009), 6-10. Soliton equations and Hamiltonian systems. L A Dickey, World Scientific. Dickey L. A., Soliton equations and Hamiltonian systems, World Scientific, Singapore, 1991. On construction of recursion operators from Lax representation. M Gürses, A Karasu, V V Sokolov, J. Math. Phys. 40Gürses M., Karasu A. and Sokolov V. V., On construction of recursion operators from Lax representation, J. Math. Phys., 40(1999), 6473-6490. . Y Q Li, B Li, S Y Lou, arXiv:1008.1375Li Y. Q., Li B. and Lou S. Y., arXiv:1008.1375. Soliton and Integrable System, Shanghai Scientific and Technological Education Publishing House. Y S Li, ShanghaiLi Y. S., Soliton and Integrable System, Shanghai Scientific and Technological Education Publishing House, Shanghai, 1999. Constrained KP hierarchy and bi-Hamiltonian structures. W Oevel, W Strampp, Commun. Math. Phys. 157Oevel W. and Strampp W., Constrained KP hierarchy and bi-Hamiltonian structures, Commun. Math. Phys. 157(1993), 51-81.	[]
[ "Galactic kinematics and dynamics from RAVE stars", "Galactic kinematics and dynamics from RAVE stars" ]	[ "J Binney 1⋆ \nRudolf Peierls Centre for Theoretical Physics\nKeble RoadOX1 3NPOxfordUK\n", "B Burnett \nRudolf Peierls Centre for Theoretical Physics\nKeble RoadOX1 3NPOxfordUK\n", "G Kordopatis \nInstitute of Astronomy\nMadingley RoadCB3 0HACambridgeUK\n", "M Steinmetz \nLeibniz-Institut fr Astrophysik Potsdam (AIP)\nAn der Sternwarte 1614482PotsdamGermany\n", "G Gilmore \nInstitute of Astronomy\nMadingley RoadCB3 0HACambridgeUK\n", "O Bienaymé \nObservatoire Astronomique de Strasbourg\n11 rue de l'UniversitéStrasbourgFrance\n", "J Bland-Hawthorn \nSydney Institute for Astronomy\nSchool of Physics A28\nUniversity of Sydney\n2006NSWAustralia\n", "B Famaey \nObservatoire Astronomique de Strasbourg\n11 rue de l'UniversitéStrasbourgFrance\n", "E K Grebel \nZentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMönchhofstr 12-14D-69120HeidelbergGermany\n", "A Helmi \nKapteyn Astronomical Institut\nUniversity of Groningen\nLandleven 129747 ADGroningenThe Netherlands\n", "J Navarro \nSenior ClfAR Fellow\nUniversity of Victoria\nV8P 5C2BCCanada\n", "Q Parker \nMacquarie University\nBalaclava Road2109NSWAustralia\n\nAustralian Astronomical Observatory\nNorth Ryde NSW 1670PO Box 915Australia\n", "W A Reid \nMacquarie University\nBalaclava Road2109NSWAustralia\n", "G Seabroke \nMullard Space Science Laboratory\nUniversity College London\nHolmbury St Mary\nRH5 6NTDorkingUK\n", "A Siebert \nObservatoire Astronomique de Strasbourg\n11 rue de l'UniversitéStrasbourgFrance\n", "F Watson \nAustralian Astronomical Observatory\nNorth Ryde NSW 1670PO Box 915Australia\n", "M E K Williams \nLeibniz-Institut fr Astrophysik Potsdam (AIP)\nAn der Sternwarte 1614482PotsdamGermany\n", "R F G Wyse \nDepartement of Physics and Astronomy\nJohns Hopkins University\n366 Bloomberg center, 3400 N. Charles St21218BaltimoreMDUSA\n", "T Zwitter \nFaculty of Mathematics and Physics\nCenter of Excellence SPACE-SI\nUniversity of Ljubljana\nJadranska 19, Aškerčeva cesta 121000, 1000Ljubljana, LjubljanaSlovenia, Slovenia\n" ]	[ "Rudolf Peierls Centre for Theoretical Physics\nKeble RoadOX1 3NPOxfordUK", "Rudolf Peierls Centre for Theoretical Physics\nKeble RoadOX1 3NPOxfordUK", "Institute of Astronomy\nMadingley RoadCB3 0HACambridgeUK", "Leibniz-Institut fr Astrophysik Potsdam (AIP)\nAn der Sternwarte 1614482PotsdamGermany", "Institute of Astronomy\nMadingley RoadCB3 0HACambridgeUK", "Observatoire Astronomique de Strasbourg\n11 rue de l'UniversitéStrasbourgFrance", "Sydney Institute for Astronomy\nSchool of Physics A28\nUniversity of Sydney\n2006NSWAustralia", "Observatoire Astronomique de Strasbourg\n11 rue de l'UniversitéStrasbourgFrance", "Zentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMönchhofstr 12-14D-69120HeidelbergGermany", "Kapteyn Astronomical Institut\nUniversity of Groningen\nLandleven 129747 ADGroningenThe Netherlands", "Senior ClfAR Fellow\nUniversity of Victoria\nV8P 5C2BCCanada", "Macquarie University\nBalaclava Road2109NSWAustralia", "Australian Astronomical Observatory\nNorth Ryde NSW 1670PO Box 915Australia", "Macquarie University\nBalaclava Road2109NSWAustralia", "Mullard Space Science Laboratory\nUniversity College London\nHolmbury St Mary\nRH5 6NTDorkingUK", "Observatoire Astronomique de Strasbourg\n11 rue de l'UniversitéStrasbourgFrance", "Australian Astronomical Observatory\nNorth Ryde NSW 1670PO Box 915Australia", "Leibniz-Institut fr Astrophysik Potsdam (AIP)\nAn der Sternwarte 1614482PotsdamGermany", "Departement of Physics and Astronomy\nJohns Hopkins University\n366 Bloomberg center, 3400 N. Charles St21218BaltimoreMDUSA", "Faculty of Mathematics and Physics\nCenter of Excellence SPACE-SI\nUniversity of Ljubljana\nJadranska 19, Aškerčeva cesta 121000, 1000Ljubljana, LjubljanaSlovenia, Slovenia" ]	[ "Mon. Not. R. Astron. Soc" ]	We analyse the kinematics of ∼ 400 000 stars that lie within ∼ 2 kpc of the Sun and have spectra measured in the RAdial Velocity Experiment (RAVE). We decompose the sample into hot and cold dwarfs, red-clump and non-clump giants. The kinematics of the clump giants are consistent with being identical with those of the giants as a whole. Without binning the data we fit Gaussian velocity ellipsoids to the meridionalplane components of velocity of each star class and give formulae from which the shape and orientation of the velocity ellipsoid can be determined at any location. The data are consistent with the giants and the cool dwarfs sharing the same velocity ellipsoids, which have vertical velocity dispersion rising from 21 km s −1 in the plane to ∼ 55 km s −1 at \|z\| = 2 kpc and radial velocity dispersion rising from 37 km s −1 to 82 km s −1 in the same interval. At (R, z) the longest axis of one of these velocity ellipsoids is inclined to the Galactic plane by an angle ∼ 0.8 arctan(z/R). We use a novel formula to obtain precise fits to the highly non-Gaussian distributions of v φ components in eight bins in the (R, z) plane.We compare the observed velocity distributions with the predictions of a published dynamical model fitted to the velocities of stars that lie within ∼ 150 pc of the Sun and star counts towards the Galactic pole. The predictions for the v z distributions are exceptionally successful. The model's predictions for v φ are successful except for the hot dwarfs, and its predictions for v r fail significantly only for giants that lie far from the plane. If distances to the model's stars are over-estimated by 20 per cent, the predicted distributions of v r and v z components become skew, and far from the plane broader. The broadening significantly improves the fits to the data.The ability of the dynamical model to give such a good account of a large body of data to which it was not fitted inspires confidence in the fundamental correctness of the assumed, disc-dominated, gravitational potential.	10.1093/mnras/stt2367	[ "https://arxiv.org/pdf/1309.4285v2.pdf" ]	118,471,110	1309.4285	8f8dd9fbededcd1b3497babecf19906e74ffb380	Galactic kinematics and dynamics from RAVE stars 2012 J Binney 1⋆ Rudolf Peierls Centre for Theoretical Physics Keble RoadOX1 3NPOxfordUK B Burnett Rudolf Peierls Centre for Theoretical Physics Keble RoadOX1 3NPOxfordUK G Kordopatis Institute of Astronomy Madingley RoadCB3 0HACambridgeUK M Steinmetz Leibniz-Institut fr Astrophysik Potsdam (AIP) An der Sternwarte 1614482PotsdamGermany G Gilmore Institute of Astronomy Madingley RoadCB3 0HACambridgeUK O Bienaymé Observatoire Astronomique de Strasbourg 11 rue de l'UniversitéStrasbourgFrance J Bland-Hawthorn Sydney Institute for Astronomy School of Physics A28 University of Sydney 2006NSWAustralia B Famaey Observatoire Astronomique de Strasbourg 11 rue de l'UniversitéStrasbourgFrance E K Grebel Zentrum für Astronomie Astronomisches Rechen-Institut Universität Heidelberg Mönchhofstr 12-14D-69120HeidelbergGermany A Helmi Kapteyn Astronomical Institut University of Groningen Landleven 129747 ADGroningenThe Netherlands J Navarro Senior ClfAR Fellow University of Victoria V8P 5C2BCCanada Q Parker Macquarie University Balaclava Road2109NSWAustralia Australian Astronomical Observatory North Ryde NSW 1670PO Box 915Australia W A Reid Macquarie University Balaclava Road2109NSWAustralia G Seabroke Mullard Space Science Laboratory University College London Holmbury St Mary RH5 6NTDorkingUK A Siebert Observatoire Astronomique de Strasbourg 11 rue de l'UniversitéStrasbourgFrance F Watson Australian Astronomical Observatory North Ryde NSW 1670PO Box 915Australia M E K Williams Leibniz-Institut fr Astrophysik Potsdam (AIP) An der Sternwarte 1614482PotsdamGermany R F G Wyse Departement of Physics and Astronomy Johns Hopkins University 366 Bloomberg center, 3400 N. Charles St21218BaltimoreMDUSA T Zwitter Faculty of Mathematics and Physics Center of Excellence SPACE-SI University of Ljubljana Jadranska 19, Aškerčeva cesta 121000, 1000Ljubljana, LjubljanaSlovenia, Slovenia Galactic kinematics and dynamics from RAVE stars Mon. Not. R. Astron. Soc 0002012Draft, November 14 2013arXiv:1309.4285v2 [astro-ph.GA] Printed 11 (MN L A T E X style file v2.2) 2 J. Binney et al.Galaxy: disc -kinematics and dynamics solar neighbourhood -galaxies: kinematics and dynamics - We analyse the kinematics of ∼ 400 000 stars that lie within ∼ 2 kpc of the Sun and have spectra measured in the RAdial Velocity Experiment (RAVE). We decompose the sample into hot and cold dwarfs, red-clump and non-clump giants. The kinematics of the clump giants are consistent with being identical with those of the giants as a whole. Without binning the data we fit Gaussian velocity ellipsoids to the meridionalplane components of velocity of each star class and give formulae from which the shape and orientation of the velocity ellipsoid can be determined at any location. The data are consistent with the giants and the cool dwarfs sharing the same velocity ellipsoids, which have vertical velocity dispersion rising from 21 km s −1 in the plane to ∼ 55 km s −1 at \|z\| = 2 kpc and radial velocity dispersion rising from 37 km s −1 to 82 km s −1 in the same interval. At (R, z) the longest axis of one of these velocity ellipsoids is inclined to the Galactic plane by an angle ∼ 0.8 arctan(z/R). We use a novel formula to obtain precise fits to the highly non-Gaussian distributions of v φ components in eight bins in the (R, z) plane.We compare the observed velocity distributions with the predictions of a published dynamical model fitted to the velocities of stars that lie within ∼ 150 pc of the Sun and star counts towards the Galactic pole. The predictions for the v z distributions are exceptionally successful. The model's predictions for v φ are successful except for the hot dwarfs, and its predictions for v r fail significantly only for giants that lie far from the plane. If distances to the model's stars are over-estimated by 20 per cent, the predicted distributions of v r and v z components become skew, and far from the plane broader. The broadening significantly improves the fits to the data.The ability of the dynamical model to give such a good account of a large body of data to which it was not fitted inspires confidence in the fundamental correctness of the assumed, disc-dominated, gravitational potential. INTRODUCTION A major strand of contemporary astronomy is the quest for an understanding of how galaxies formed and evolved within the context of the concordance cosmological model, in which the cosmic energy density is dominated by vacuum energy and the matter density is dominated by some initially cold matter that does not interact electromagnetically. This quest is being pursued on three fronts: observations of objects seen at high redshifts and early times, simulations of clustering matter and star formation, and by detailed observation of the interplay between the chemistry and dynamics of stars in our own Galaxy. As a contribution to this last "Galactic archaeology" strand of the quest for cosmic understanding, the RAdial Velocity Experiment (Steinmetz et al. 2006) has since 2003 gathered spectra at resolution ∼ 7500 around the CaII near-IR triplet of ∼ 400 000 stars. The catalogued stars are roughly half giants and half dwarfs, and mostly lie within 2.5 kpc of the Sun (Burnett et al. 2011;. The RAVE survey is complementary to the Sloan Digital Sky Survey (SDSS; York et al. 2000) and the latter's continuations (Yanny et al. 2009;Eisenstein et al. 2011) in that it observes stars at least as bright as I = 9 − 13, whereas the SDSS observes stars fainter than g = 14. On account of the faint magnitudes of the SDSS stars, they are overwhelmingly at distances greater than 0.5 kpc so the Galaxy's thin disc, which has a scale height ∼ 0.3 kpc and is by far the dominant stellar component of the Galaxy, contributes a small proportion of the stars in the SDSS data releases. The thin and thick discs, by contrast, completely dominate the RAVE catalogue. Recently derived distances to ∼ 400 000 stars from 2MASS photometry and the stellar parameters produced by the VDR4 spectral-analysis pipeline described by Kordopatis et al. (2013). We use these distances to discuss the kinematics of the Galaxy in the extended solar neighbourhood, that is, in the region within ∼ 2 kpc of the Sun. Since the selection criteria of the RAVE survey are entirely photometric, we can determine the distribution of the velocities of survey stars within the surveyed region without determining the survey's complete selection function, which is difficult (see Piffl & Steinmetz in preparation, Sharma et al in preparation). We characterise the kinematics in several distinct ways. In Section 3 we obtain analytic fits to the variation within the (R, z) plane of the velocity ellipsoid by a technique that avoids binning stars (Burnett 2010). In Section 4 we bin stars to obtain histograms of the distribution of three orthogonal components of velocity. We use a novel formalism to obtain analytic fits to the distributions of the azimuthal component of velocity. We examine the first and second moments of the distributions of the velocity components parallel to the principal axes of the local velocity ellipsoid. The second moments are consistent with our previously derived values, but some first moments are non-zero: values ∼ 1.5 km s −1 are common and values as large as 5 km s −1 occur. In Section 5 we compare our results with the predictions of a dynamical model Galaxy that is based on Jeans' theorem. Although this model, which was described by Binney (2012;hereafter B12), was not fitted to any RAVE data, we find that its predictions for the distributions of vertical com-ponents are extremely successful, while those for the radial components are successful at \|z\| < 0.5 kpc but become less successful further from the plane, where they produce velocity distributions that are too narrow and sharply peaked. In Section 5.3 we investigate the impact of systematically overestimating distances to stars. When distances to the model's stars are over-estimated by 20%, the predicted distributions of vr and vz acquire asymmetries that are similar to those sometimes seen in the data. Systematic over-estimation of distances brings the model into better agreement with data far from the plane by broadening its vr distributions. INPUT PARAMETERS AND DATA Throughout the paper we adopt R0 = 8 kpc as the distance of the Sun from the Galactic centre, Θ0 = 220 km s −1 for the local circular speed and from Schönrich et al. (2010) (U0, V0, W0) = (11.1, 12.24, 7.25) km s −1 as the velocity of the Sun with respect to the Local Standard of Rest. While our values of R0 and Θ0 may be smaller than they should be (e.g. McMillan 2011), we adopt these values in order to be consistent with the assumptions inherent in the B12 model. Proper motions for RAVE stars can be drawn from several catalogues. Williams et al. (2013) compares results obtained with different proper-motion catalogues, and on the basis of this discussion we originally decided to work with the PPMX proper motions (Röser et al. 2008) because these are available for all our stars and they tend to minimise anomalous streaming motions. However, when stars are binned spatially and one computes the dispersions in each bin of the raw velocities 4.73µ(s/kpc) + v los from the PPMX proper motions, the resulting dispersions for bins at distances > ∼ 0.5 kpc are often smaller than the contributions to these from proper-motion errors alone. It follows that either our distances are much too large, or the quoted proper-motion errors are seriously over-estimating the true random errors. The problem can be ameliorated by cutting the sample to exclude stars with large proper-motion errors, but there are still signs that the velocity dispersions in distant bins are coming out too small on account of an excessive allowance for the errors in the proper motions of stars that have small proper motions. The errors in the UCAC4 catalogue (Zacharias et al. 2013) are ∼ 60 percent of those in the PPMX catalogue and the problem just described does not arise with these proper motions, so we have used them. We do, however, exclude stars with an error in µ b greater than 8 mas yr −1 . In addition to this cut on proper-motion error, the sample is restricted to stars for which determined a probability density function (pdf) in distance modulus. To belong to this group a star has to have a spectrum that passed the Kordopatis et al. (2013) pipeline with S/N ratio of 10 or more. FITTING MERIDIONAL COMPONENTS WITHOUT BINNING THE DATA At each point in the Galaxy a stellar population that is in statistical equilibrium in an axisymmetric gravitational potential Φ(R, z) should define a velocity ellipsoid. Two of the principal axes of this ellipsoid should lie within the (R, z) plane, with the third axis in the azimuthal direction e φ . Near the plane the ellipsoid's longest axis is expected to point roughly radially and the shortest axis vertically. Let e1 be the unit vector along the longest axis, and e3 be the complementary unit vector, and let θ(R, z) denote the angle between e1 and the Galactic plane. The lengths of the principal semi-axes of the velocity ellipsoid are of course the principal velocity dispersions σ1(R, z) = (v · e1) 2 1/2 σ φ (R, z) = (v · e φ ) 2 − v · e φ 2 1/2 (1) σ3(R, z) = (v · e3) 2 1/2 . In the following we shall use the notation V1 ≡ v · e1 and V3 ≡ v · e3. (2) We estimate the functional forms of σ1 and σ3 as follows. We let θ(R, z) be determined by a single parameter a0 through θ = a0 arctan (z/R) . ( We use four further parameters ai to constrain the behaviour of σ1, and similarly for σ3, by writing σ1(R, z) = σ0a1 exp[−a2(R/R0 − 1)][1 + (a3z/R) 2 ] a 4 σ3(R, z) = σ0a5 exp[−a6(R/R0 − 1)][1 + (a7z/R) 2 ] a 8 ,(4) where σ0 ≡ 30 km s −1 ensures that all the ai are dimensionless and of order unity. These forms are the fruit of a combination of physical intuition and some experimentation. In particular, by symmetry we require even functions of z that have vanishing vertical gradients in the plane, and experimentation shows that power series in z 2 do not work well. Second, it has been conventional to assume exponential dependence of velocity dispersion on R since the scale heights of discs were found to be roughly constant (van der Kruit & Searle 1981). Moreover, the data cover a significant range in R only at large \|z\|, so we are not in a position to consider elaborate dependence on R. The parameters a1 and a5 set the overall velocity scale of σ1 and σ3, respectively, while a2 and a6 determine how fast these dispersions decrease with increasing radius. The parameter pairs (a3, a4) and (a7, a8) determine how the dispersions vary with distance from the plane. From equations (4) it is straightforward to calculate the derivatives with respect to the nine parameters ai of the components V1, V3 of a star's velocity and of the dispersions σi, so we use a conjugate-gradient method to extremise the log-likelihood stars i=1,3 ln[σ 2 i + e 2 (Vi)] + V 2 i σ 2 i + e 2 (Vi) ,(5) associated with a correctly normalised biaxial Gaussian pdf in (V1, V2) space. Here e(Vi) is the formal error in Vi for a given star. This is computed from the quoted errors on the proper motions and the line-of-sight velocity assuming the distance to be inverse of the expectation of the parallax given by , who found this to be the most reliable distance estimator. With the present method it is exceedingly hard to allow for distance errors, and we do not do this. The code for extracting the values of the ai from a catalogue of stellar phase-space coordinates was tested as follows. The velocity of each RAVE star was replaced by a velocity chosen at random from a triaxial Gaussian velocity distribution with variances σ 2 i (R, z) + e 2 (Vi), where the σi were derived from plausible values of the ai and the errors e(Vi) are the actual errors on that star's velocity compo- Table 1. Test of the fitting procedure. The bottom row gives the parameters used to choose the velocities, while top row gives the values of the parameters in equation (4) from which frprmn started. The second row shows the values of the parameters on which it converged given data at the locations of the 40 175 clump giants. The third, fourth and fifth rows give the parameters values similarly obtained using data at the locations of 181 725 non-clump giants, 55 398 hot dwarfs and 95 470 cool dwarfs, respectively. nents. Then the routine frprmn of Press et al. (1994) was used to maximise the function (5) starting from another set of values of the ai. The conventional χ 2 is χ 2 = stars i=1,3 V 2 i σ 2 i + e 2 (Vi) .(6) In all tests the chosen model yielded a value of χ 2 per degree of freedom that differed from unity by less than 3 × 10 −4 . We have analysed separately four classes of stars: clump giants (0.55 ≤ J − K ≤ 0.8 and 1.7 ≤ log g < 2.4), nonclump giants (log g < 3.5), hot (T eff > 6000 K) dwarfs and cool dwarfs. The first row of Table 1 shows the parameters from which fitting started, while the bottom row gives the values of the parameters that were used to assign velocities to the stars. The second row shows the parameter values upon which frprmn converged with data at the locations of 40 175 red-clump stars in the RAVE sample. The third row gives the results obtained using the sample's 181 726 non-clump giants. The fourth and fifth rows give, respectively, results obtained using the 55 398 hot dwarfs and 95 469 cool dwarfs. Naturally the precision with which the parameters can be recovered from the data increases with the size and spatial coverage of the sample. Hence the cold dwarfs deliver the least, and the giants the most, accurate results. The parameters that are most accurately recovered are a1 and a5, which control the magnitudes of dispersions, and a0, which controls the tilt of the velocity ellipsoid. The parameters a3 and a4, which control the vertical variation of the radial dispersion, and a7 and a8, which control the vertical variation of the vertical dispersion, are recovered quite well from the giants but rather poorly from the dwarfs. However, even the dwarfs yield quite accurate values for the products a 2 3 a4 and a 2 7 a8 that occur in the first non-trivial term in the Maclaurin series of the final brackets of equations (4). The parameters a2 and a6, which control radial gradients are recovered only moderately well by all star classes. When fitting the measured velocities of RAVE stars, the difference between unity and χ 2 per degree of freedom for the chosen model ranged from 3.5×10 −3 for cold dwarfs to 1.7× 10 −2 for non-clump giants. Table 2 shows the parameters of the chosen models. Both classes of giants and the cool dwarfs yield similar values a0 ≃ 0.8 of the parameter that controls the orientation of the velocity ellipsoid. Since this value lies close to unity, the long axis of the velocity ellipsoid points almost to the Galactic centre ( Fig. 1) consistent with the findings of Siebert et al. (2008). The hot dwarfs yield a much smaller value, a0 ≃ 0.2, so the long axis of their velocity ellipsoid does not tip strongly as one moves up. The velocity dispersions in the plane are σR = 30a1 km s −1 and σz = 30a5 km s −1 . The smallest dispersions, (σR, σz) = (29.3, 14.0) are for the hot dwarfs and the largest, (37.3, 21.4) are for the giants. For the giants and cool dwarfs we have σz/σR = a5/a1 ≃ 0.6, while for the hot dwarfs we have σz/σR ≃ 0.48, significantly smaller. The scale lengths on which the dispersions vary are Rσ = R0/a2 for σr and Rσ = R0/a6 for σz. For the giants these are ∼ 2.5R0, which is surprisingly large: one anticipates Rσ < ∼ 3R d ≃ R0. The cool dwarfs, by contrast yield Rσ < R0. For σr the hot dwarfs yield Rσ ≃ 1.4R0, but for σz they yield a negative value of Rσ, implying that σz increases with radius. Given that the survey volume is a cone that excludes the plane, not only is it hard to disentangle radial and vertical gradients, but stars such as hot dwarfs that are strongly concentrated to the plane do not probe a large volume and consequently are not suited to measuring gradients. Moreover, the longest axis of the velocity ellipsoids of populations of young stars are known not to lie within the (R, z) plane -the "vertex deviation" (e.g. Dehnen & Binney 1998). This phenomenon is evidence that these populations are not in dynamical equilibrium as our methodology assumes, either because they are too young, or because they are strongly disturbed by spiral structure. The upper panel of Fig. 2 shows the dependencies on z at R = 8 kpc of σ1 (dashed line) and σ3 (full line) that are implied by Table 2 for non-clump giants. The squares and triangles show velocity dispersions estimated by binning the data as described in Section 4 below. The lower panel Table 2. Velocity ellipsoids from measured velocities. When the values given here are inserted into equations (3) and (4) one obtains expressions for the semi-axis lengths and orientation of the velocity ellipsoids at a general point (R, z). From top to bottom the rows give results for clump giants, non-clump giants, and hot and cool dwarfs. shows the corresponding radial dependencies at z = 0.22 and z = 0.86 kpc. In Fig. 3 the full black curves show the runs with z at R = R0 of σ1 and σ3 for non-clump giants, while the dashed red curves show the same quantities for the cool dwarfs. From these plots we infer that the dispersions of the cool dwarfs are probably consistent with those for non-clump giants except very near the plane where σ1 may be lower for the dwarfs. The blue dotted curves show the distinctly lower velocity dispersions of the hot dwarfs: lower dispersions are to be expected of such relatively young stars since they have experienced less stochastic acceleration than older stars. USING BINNED DATA Azimuthal velocities In a disc galaxy, the distribution of v φ components is inherently skew and the skewness of the distribution contains essential information about the system's history and dynamics. Consequently, it is not appropriate to use the machinery described in the last section to fit observed v φ distributions. The v φ distributions of the dynamical models described by B12, which will be discussed in Section 5 below, can be fitted extremely well by the following analytic distribution P (v φ ) = constant × e −(v φ −b 0 ) 2 /2σ 2 φ ,(7) where σ φ is a cubic in v φ : σ φ (v φ ) = b1 + b2v φ100 + b3v 2 φ100 + b4v 3 φ100 ,(8) with v φ100 ≡ v φ /100 km s −1 . The general idea here is that b0 defines a characteristic streaming velocity, while b1 is a basic azimuthal velocity dispersion. The parameters b2 to b4 cause the velocity dispersion σ φ to increase/decrease as v φ moves below/above the circular speed, thus making the v φ distribution skew. In principle functional forms could be adopted for the dependence on (R, z) of the parameters bi appearing in equations (7) and (8), and then, in strict analogy to the work of the previous section, the values of the parameters appearing in these functional forms could be determined by maximising the likelihood of the data given the distribution (7). Unfortunately, for this scheme to be viable we require an expression for the value of the normalising constant as a function of the parameters, and no such formula is available. Therefore we have determined the bi by binning the data and doing a least-squares fit of equation (7) convolved with the observational errors to the histogram of the binned data. (7) and (8), while the full curve shows the model convolved with the mean errors in v φ . The red points show the predictions of the B12 dynamical model. The mean coordinates of the stars in each bin are given at top left, followed by the rms velocity error (eV) and the sample mean of v φ (vbar). In this and all subsequent histograms, the horizontal bars span the width of the bins and the vertical bars indicate Poisson errors. The stars were divided into 8 spatial bins according to whether R < R0 or R > R0 and \|z\| lay in intervals bounded by (0, 0.3, 0.6, 1, 1.5) kpc for giants or (0, 0.15, 0.3, 0.45, 0.6) kpc for dwarfs. Table 3 gives the parameters that fit the v φ distributions of the clump stars (upper block) and non-clump giants (lower block). Table 4 gives values of the parameters for the hot (upper block) and cool dwarfs. The black points in Figs. 4 to 7 show the observational histograms. At the top left of each panel we give the mean values of (R, \|z\|) and e(v φ ) for stars in the bin, where the latter is the r.m.s. error for the stars in the given bin. Also given at the top of each panel is the mean velocity, v φ , which of course is sensitive to our adopted values Θ0 = 220 km s −1 and v φ⊙ = Θ0 + 12.24 km s −1 . The values of v φ are also given in Tables 3 and 4, where we see that on account of the skewness of the v φ distributions, v φ is sys- Table 3. Values of the mean streaming velocity and the parameters defined by equations (7) and (8) required to fit the v φ distributions of RAVE stars. The upper block refers to red clump stars and the lower one to non-clump giants. tematically smaller than the fit parameter b0, which would be the mean velocity if σ φ were not a function of v φ . In Figs. 4 to 7 bins with R < R0 are shown in the left column, bins with R > R0 are shown in the right column, and \|z\| increases downwards. The dotted curves show the functions defined by the bi in Tables 3 and 4 while the full curves show the results of convolving these curves with the Gaussian of dispersion e(v φ ). The dotted curves are mostly obscured by the full curves because observational errors do not have a big impact on these data. All histograms are fitted to great precision by the full curves. Figs 8 and 9 show, respectively, the mean rotation ve- locity of the giants and dwarfs as functions of distance from the plane. The data points were obtained by fitting the analytic model convolved with the measurement errors to histograms of v φ components with the stars placed in seven bins at each of R < R0 and R > R0, and then calculating for each bin the mean velocity of the model distribution before convolution by error. We do not show error bars, but the statistical errors on these points are very small. All these points would move upwards by 20 km s −1 if we increased our estimate of the local circular speed from Θ0 = 220 km s −1 to Θ0 = 240 km s −1 , and they would move down by 5 km s −1 if we decreased our estimate of v φ⊙ − Θ0 from 12.24 km s −1 to 7.24 km s −1 . In Fig. 8 the points for giants show a clear trend for v φ to decline with distance from the plane, as we expect given that along this sequence σ1 rises and increases the asymmetric drift va ∼ σ 2 1 /vc. In Fig. 9 the point for hot dwarfs at z < ∼ 50 pc and R < R0 is ∼ 25 km s −1 larger than the corresponding point at R > R0, so both points are highly anomalous. However, the histograms for the associated bins (which we do not show) indicate that the anomaly is not caused by small-number statistics. The points for larger distances from the plane lie close to the circular speed at R < R0 and fall about 4 km s −1 lower at R > R0. These differences could well reflect spiral structure. The points for cool dwarfs show a slight fall with increasing distance from the plane and a tendency to be up to 2 km s −1 lower at R > R0 than at R < R0. The fall in v φ between the plane and 0.5 kpc is consistent with that of the giants. (R, \|z\|) v φ b 0 b 1 b 2 b 3 b 4((R, \|z\|) v φ b 0 b 1 b 2 b 3 b 4( Moments of the V1 and V3 distributions The black points in Figs. 11 to 14 show, for hot dwarfs, cool dwarfs, clump and non-clump giants respectively, the distributions of the meridional-plane components V1 and V3 defined by equations (2). At the bottom-centre of each panel the numbers in brackets give the mean values of R and \|z\| for the stars in each bin, the standard deviation of the data (sD), the value at this location of the relevant velocity dispersion from the Gaussian model of Section 3 (sM), the mean velocity of the stars in the bin (mV) and the rms measurement error for those stars (eV). The agreement between the standard deviations of the data and the model dispersion at the bin's barycentre is typically excellent. If the Galaxy were in an axisymmetric equilibrium and we were using the correct value for the Sun's peculiar velocity, the mean velocities would all vanish to within the discreteness noise, but they do not. All the three older populations show similar trends in mean velocities: the means of V3 tend to be negative at R > R0 and increase in absolute value away from the plane, while the mean values of V1 fall from positive to negative as one moves away from the plane with the largest absolute values occurring for giants near the plane. Siebert et al. (2011) and Williams et al. (2013) have analysed similar statistically significant mean velocities in velocities of RAVE stars drawn from an earlier spectralanalysis pipeline than that used here. We defer discussion of this phenomenon until Section 5.3. COMPARISONS WITH DYNAMICAL MODELS It is interesting to compare the observed distributions with ones predicted by the favoured equilibrium dynamical model of B12. This model is defined by a gravitational potential and a distribution function. The potential is generated by thin and thick exponential stellar discs, a gas layer, a flattened bulge and a dark halo. Fig. 10 shows the contributions to the circular speed from the baryons (dotted curve) and from the dark halo (dashed curve). One sees that this is a maximum-disc model. In fact, 65% of the gravitational force on the Sun is produced by baryons rather than dark matter. The distribution function (df) is an analytic function f (J) of the three action integrals Ji. The function, which specifies the density of stars in three-dimensional action space, has nine parameters. Four parameters specify each of the thin and thick discs and one parameter specifies the relative weight of the thick disc. Their values are given in column (b) of Table 2 in B12. They were chosen by fitting the model's predictions for the velocity distribution of solar-neighbourhood stars to that measured by the Geneva-Copenhagen survey (GCS) of F and G stars (Holmberg et al. 2007), and to the vertical density profile of the disc determined by Gilmore & Reid (1983). Hence the data to which this df was fitted do not include velocities in the region distance s > ∼ 150 pc within which most RAVE stars lie, and whatever success the df has in predicting the velocities of RAVE stars must be considered a non-trivial support for the assumptions that went into the model, which include the use of a particular, disc-dominated, gravitational potential and the functional form of the df. We have used the B12 df to generate pseudo-data for each star in the RAVE sample from the model's velocity distribution as follows. We start by choosing a possible true location x ′ by picking a distance s ′ from the multi-Gaussian model of the star pdf in distance s that produced. We then sample the velocity distribution of the dynamical model for that class of star at x ′ and compute the corresponding proper motions and line-of-sight velocity v los . To these observables we add random errors drawn from the star's catalogued error distributions, and from the modified observables compute the space velocity using the catalogued distance s rather than the hypothesised true distance s ′ . This procedure comes very close to reproducing the data that would arise if the Galaxy were correctly described by the model, each star's distance pdf were sound and the errors on the velocities had been correctly assessed: it does not quite achieve this goal on account of a subtle effect, which is costly to allow for. This effect causes the procedure to overweight slightly the possibility that stars lie at the far ends of their distance pdfs (Sanders & Binney in preparation). We believe the impact of this effect to be small, so our model histograms correctly represent the model's predictions for a survey with the selection function and errors of RAVE. We assume that the hot dwarfs are all younger than 5 Gyr (e.g., Fig. 2 of Zwitter et al. 2010) and correspondingly restrict the B12 df of these objects to the portion of the thin disc that is younger than 5 Gyr. The distributions of clump and non-clump giants and cool dwarfs are (rather arbitrarily) assumed to sample the whole df. Azimuthal velocities distributions The red points in Figs 4 to 7 show the model's predictions for the v φ components. Figs 4 and 5 show that the velocities of the clump giants are very similar to those of the nonclump giants. This result is in line with expectations, but serves to increase our confidence in our distance estimates for, as we shall see in Section 5.3, systematic errors in the distances of whole groups of stars distort the derived velocity distributions. Hence consistency between the histograms for clump and non-clump giants suggests that our distances to non-clump giants, which are the hardest to determine, are no more in error than are the distances to clump giants. In Figs 4 and 5 the models definitely under-populate the wing at v φ > Θ0, especially away from the plane. This is likely to reflect the model's thick disc being radially too cool, as discussed below. A notable difference between the observed and predicted distributions for both the giants and the hot dwarfs (Figs 4 to 6) is that at R < R0 and \|z\| ∼ 0.5 kpc the black, measured, distribution is shifted to larger values of v φ than the red predicted one. In the case of the hot dwarfs, a similar but distinctly smaller shift is seen at R > R0. The smaller shift at R > R0 is clearly connected to the fact that in Fig. 9 the v φ points for R > R0 lie below those for R < R0. At z < 0.5 kpc the same phenomenon is evident for giants in Fig. 8. One possible explanation is that the Galaxy's circular-speed curve is falling with R relative to that of the model. While the theoretical distribution depends only on the model's value 220 km s −1 for the local circular speed Θ0, the observed velocities have been derived using both Θ0 and a value V0 = 12.24 km s −1 from Schönrich et al. (2010) for the amount by which the Sun's v φ exceeds Θ0. Hence an offset between the red and black curves in Figs 4 to 7 can be changed by changing the assumed value of V0: reducing V0 shifts the black distribution to the left. However, the case for such a change is less than unconvincing because the shift is clear only at R < R0 and \|z\| < ∼ 0.5 kpc. Moreover in Fig. 7 for the cool dwarfs the model histograms provide excellent fits to the data. In Fig. 6 for the hot dwarfs the offset between the red and black histograms vanishes at R > R0 near the plane but grows with \|z\|. A more convincing case can be made for an increase in the width of the theoretical distributions of giants away from the plane. In addition to a possibly incorrect value of V0, there are four other obvious sources of offsets between the observational and theoretical distributions of v φ : • Spiral arms must generate fluctuations in the mean azimuthal velocity of stars. Judging by oscillations with Galactic longitude in the observed terminal velocity of interstellar gas (e.g. Malhotra 1995), the magnitude of this effect is probably at least as great as 7 km s −1 in a population such as hot dwarfs that has a low velocity dispersion. Moreover, it is now widely accepted that the irregular distribution of Hipparcos stars in the (U, V ) plane of velocities (Dehnen 1998) is in large part caused by spiral arms perturbing the orbits of stars (De Simone et al. 2004;Antoja et al. 2011;Siebert et al. 2012;McMillan 2013). The large (up to 20 • ) value of the vertex deviation for hot dwarfs is surely also due to spiral structure. Spiral-induced modulations in v φ will Figure 11. Distributions of V 1 ≃ −vr and V 3 ≃ vz for hot dwarfs. Black points show the RAVE data, red points the predictions of the B12 model when it is assumed that all hot dwarfs are younger than 5 Gyr and as such belong to the thin disc. At the lower middle of each panel are given: the mean (R, z) coordinates of the bin; the standard deviation of the data after correction for error and the velocity dispersion at the mean coordinates of the Gaussian-model described in Section 3; the mean of the data and the rms error of the velocities. Figure 12. As Fig. 11 but for cool dwarfs. The red points now show the predictions of the B12 model when cool dwarfs are assumed to sample the entire df. In the last two panels of the top row we show the Gaussian distributions that were fitted in Section 3 to illustrate how well the dynamical model captures the deviations of the observed distribution from Gaussianity. vary quite rapidly with radius and thus could make significantly different contributions to v φ in our bins at R < R0 and R > R0. • The mean age of the stellar population is expected to decrease with increasing Galactocentric distance. Such a decrease would introduce a bias into a sample selected to be young such that there were more stars seen near pericentre than near apocentre than in a sample of older stars, so stars in the younger sample would tend to have larger values of v φ than stars in the older sample. This effect could explain why the histograms for hot dwarfs show larger offsets than do those for cool dwarfs. • We are probably using a value of R0 that is too small by ∼ 3%. Changing the adopted value of R0 changes the supposed direction of the tangential vector e φ (⋆) at the location of a star and thus changes the component of a star's Galactocentric velocity v that we deem to be v φ . The velocity v is made up of the star's heliocentric velocity v h and the Sun's largely tangential velocity v⊙ = Θ0e φ (⊙)+(U0, V0, W0). For a star at a given distance, increasing R0 diminishes the angle between e(⋆) and e(⊙), and thus, by diminishing the angle between e φ (⋆) and v⊙, tends to increase v φ . Consequently, in Figs 4 to 7 increasing R0 moves the black points to the right, away from the model's predictions. • We are probably using a value of Θ0 that is too small by ∼ 9%. Increasing Θ0 by δΘ simply moves the observational histogram to the right by δΘ. However, since the asymmetric drift va of a population that has radial velocity dispersion σr scales as σ 2 r /Θ0, increasing Θ0 moves the theoretical histogram to the right by δΘ − δva = 1 + σ 2 r Θ 2 0 δΘ,(9) so this upward revision will reduce by (σr/Θ0) 2 δΘ0 ∼ 0.04δΘ0 the offsets we obtained with our traditional choices of R0 and Θ0. Velocities in the meridional plane Figs. 11 to 14 are the analogues of Figs 4 to 7 for components of velocity V1 and V3 (equation 2) in the meridional plane: black points show observational histograms and red ones the predictions of the B12 model. V1 is the component of velocity along the longest principal axis of the velocity ellipsoid at the star's location according to the Gaussian model fitted in Section 3. The sign convention is such that at the Sun V1 ≃ U = −vR. V3 ≃ W = vz is the perpendicular velocity component. The left two columns are for bins with R < R0 while the right two columns are for bins with R > R0. At the lower middle of each panel are given: the mean (R, z) coordinates of stars in the bin; the standard deviation of the data Figure 14. As Fig. 12 but for non-clump giants. Figure 16. As Fig. 5 but when the adopted distances to these (giant) stars are 20% larger than they should be. after correction for error (sD) and the velocity dispersion at the mean coordinates of the Gaussian-model described in Section 3 (sM); the mean of the data (mV) and the rms error of the velocities (eV). All distributions are significantly non-Gaussian (i.e. the distributions are far from parabolic) and the B12 model captures this aspect of the data beautifully. The last two panels in the top row of Fig. 12 illustrate this phenomenon by showing the parabolas of the Gaussian distributions fitted in Section 3. Notwithstanding the non-Gaussian nature of the velocity distributions, in every bin there is good agreement between the standard deviation of the data sD and the dispersion at of the Gaussian model sM at the barycentre of the bin. This result implies that equations (4) can be safely used to recover the principal velocity dispersions throughout the studied region. The model is particularly successful in predicting the V3 distributions of both dwarfs and giants. In the case of the dwarfs, the only blemish on its V3 distributions is a marginal tendency for the distribution of hot dwarfs to be too narrow at high \|z\|. The principal differences between the model and observed V1 distributions of dwarfs arise from left-right asymmetries in the data. For example, in the third panels from the left in the first and second rows of Fig. 11 for hot dwarfs, the black points lie systematically above the red points for Figure 15. The black points and curves are identical to those plotted in Fig. 14. The red model histograms have been modified by supposing that the catalogued distance to each (giant) star is 20% larger than it should be. The values sD and mV given at the bottom are now the standard deviation and mean of the red histogram. V1 > 0 (inward motion), a phenomenon also evident in the top left panel of that figure. In the first and third panels in the second row of Fig. 12 for cool dwarfs, a similar phenomenon is evident in that the red points lie above the black points at V1 < 0. A contribution to these divergences must come from star streams, which Dehnen (1998) showed to be prominent in the local U V plane. Figs 13 and 14 for clump and non-clump giants show V1 and V3 distributions in bins that extend to much further from the plane. In both cases the model and observed V3 distributions agree to within the errors. Given the smallness of the error bars in the case of the giants and the fact that the data extend to a distance from the plane that is more than ten times the extent of the GCS data to which the B12 model was fitted, the agreement between the observed and theoretical V3 histograms in Fig. 14 amounts to a very strong endorsement of the B12 model. The observed V1 distributions for clump and non-clump giants are consistent with one another, and the superior statistics of non-clump giants highlight the deviations from the model predictions. Near the plane the model fits the data well, but the further one moves from the plane, the more clear it becomes that the model distribution of V1 is too narrow. This phenomenon arises because in B12, contrary to expectation, the thick disc needed to be radially cooler than the thin disc. The RAVE data are indicating that this was a mistake. In B12 two factors shared responsibility for the radial coolness of the thick disc. One was the ability of the thin-disc df to fit the wings of the U and V distributions in the GCS, leaving little room for the thick disc's contribution there. The other factor was an indication from SDSS that v φ does not fall rapidly with distance from the plane. Fig. 5 relates to this second point, and indeed the RAVE data show more stars with large v φ than the model, especially at large \|z\|. In B12 it was demonstrated that there is a clean dynamical trade-off between v φ and σ φ in the sense that an increase in the former has to be compensated by a decrease in the latter. Moreover, σ φ is dynamically coupled to V 2 1 1/2 , so if one is reduced the other must be reduced as well. Hence large v φ implies small V 2 1 1/2 . There is a puzzle here that requires further work. Effect of distance errors Our model predictions already include the effects of random distance (and velocity) errors. Now we investigate how systematic errors in our spectrophotometric distances affect the derived kinematics. This investigation is motivated in part by the indication in from the kinematic test of Schönrich et al. (2012) that distances to giants might be over-estimated by as much as 20%, and distances to the hottest dwarfs under-estimated by a similar amount. The black points in Fig. 15 are identical to those in the corresponding panels of Fig. 14 but the red model points have been modified by adding −5 log 10 (e) × 0.2 to the randomly chosen distance modulus of each star before evaluating the df. This modification enables us to model the impact on the survey of catalogued distances being on average 20 per cent too large. The figure shows that such distance errors introduce left-right asymmetry into the model distributions of both V1 and V3 similar to that evident in the V1 distribution of hot dwarfs. The red values of mV at the bottom middle of each panel, show the mean values of V1 and V3 for the model histograms. We see that these values are non-zero and of comparable magnitude to the mean values of the observed histograms given in Fig. 14. Thus non-zero mean values of V1 and V3 may arise from distance errors rather than from real streaming motion. However, near the plane our distance errors induce negative mean values of V1 (net outward motion) whereas the data histogram shows a smaller positive mean value of V1. Physically, over-estimating distances makes the V1 distribution skew to positive V1 because the survey volume is not symmetric in Galactic longitude, and at certain Galactic longitudes proper motion generated by the disc's differential rotational is wrongly interpreted to be proper motion associated with motion towards the Galactic centre. The assumption that distances are over-estimated also broadens the model distribution of V1 far from the plane, with the result that, for example, in the third row of Fig. 15 the red and black points for V1 lie significantly closer than in the corresponding panels of Fig. 14. Fig. 16 is the analogue of Fig. 5 for the case in which the distances to giants have been over-estimated by 20%. In the top left panel for small \|z\| and R < R0 the agreement between model and data is now less good than it is in Fig. 5, but in every other panel the agreement is at least as good in Fig. 5 and for R > R0 it is distinctly improved. Thus the v φ distributions by no means speak against the suggestion that many distances have been over-estimated by ∼ 20%. While in Fig. 15 distance errors have improved the fit to the data only at \|z\| > 0.5 kpc and weakened the fit closer to the plane, it is perfectly possible that systematic errors are largely confined to more distant stars and/or ones further from the plane. In fact, such an effect is inevitable even if the errors in distances of individual stars were inherently unbiased because stars that happen to pick up a positive distance error will tend to accumulate in the distant bins, and conversely for stars that happen to pick up a negative distance error. When we modified the model's predictions to allow for random distance errors, we did not capture this effect because the spatial bin to which a star is then assigned is not affected by whether it is supposed to have had its distance over-or under-estimated. Siebert et al. (2011) reported a significant radial gradient in the mean vR of velocities of stars reduced by the RAVE VDR2 pipeline. Williams et al. (2013;hereafter W13) used data from the VDR3 pipeline to analyse the mean velocity field v of clump stars. In a steady-state, axisymmetric Galaxy the only non-vanishing component of this field would be v φ and it would have a maximum in the plane, falling away with \|z\| symmetrically on each side. Instead Fig. 11 of W13 indicates that the velocity field of the clump stars has both vR and vz components non-zero and with gradients in both the R and z directions, and there is a lack of symmetry about the plane. W13 strike a cautionary note by showing that the vR and vz components are sensitive to which proper motions one adopts, but they demonstrate that v is insensitive to the adopted absolute magnitude of clump stars. DISCUSSION As W13 show, probing the observed velocity field is made difficult by the complexity of the three-dimensional volume surveyed by RAVE: samples assembled to have a progression of values of one coordinate inevitably differ systematically in another coordinate as well. For this reason it is crucial to compare observational results with the predictions of a model that suffers the same selection effects. W13 compare the observations to mock catalogues selected by the code Galaxia (Sharma et al. 2010) from the Besançon model (Robin et al. 2003). Our comparisons differ in that (i) we have used a fully dynamical model, based on Jeans' theorem, rather than the essentially kinematic Besançon model, and (ii) we assign new velocities to existing stars rather than drawing an entirely new sample from the model -this procedure has the great advantage that we do not have to engage with the survey's complex photometric selection function. Our emphasis has been different in that we have focused on entire velocity distributions rather than just the distributions' means. This has been possible because we have a more prescriptive dynamical model, but it has resulted in our using much bigger bins than W13. In particular, we have grouped together stars above and below the plane, which will inevitably wash out some of the structure in the (R, z) plane seen by W13. Our demonstration that introducing plausible systematic errors in the assumed distances to stars causes the model histograms to acquire mean velocities that are similar in magnitude to those found by Williams et al. (2013) must be a concern even though the particular systematic in distance error that we have considered does not generate the observed pattern of mean velocities. The extent to which distance errors broaden the distributions of V1 is surprising and interesting given the difficulties one encounters finding a dynamical model that is consistent with all the data for v φ and V 2 1 1/2 in the absence of systematic distance errors. CONCLUSIONS We have analysed the kinematics of ∼ 400 000 RAVE stars for which have deduced pdfs in distance modulus. The sample divides naturally into clump and non-clump giants, hot and cool dwarfs. For each of these classes, and without binning the data, we have obtained analytic formulae for the structure of the velocity ellipsoid at each point in the (R, z) plane. We are able to map the velocity ellipsoid of the giants to distances ∼ 2 kpc from the Sun and find that at (R, z) the direction of the longest axis is inclined to the Galactic plane by an angle ∼ 0.8 arctan(z/R). The lengths of the (R, z) semi-axes are in the ratio σ3/σ1 ≃ 0.6. The velocity dispersions rise with distance from the plane, from σr ≃ 37 km s −1 , σz ≃ 21 km s −1 at (R0, 0) to σr ≃ 82 km s −1 , σz ≃ 54 km s −1 at (R0, 2 kpc). The velocity ellipsoid of the cool dwarfs cannot be traced to great distances, but it is consistent with being the same as that of the giants. In the plane the velocity dispersions of the hot dwarfs are σr ≃ 29 km s −1 and σz ≃ 14 km s −1 and they increase rather slowly with distance from the plane. From equations (3) and (4) and Table 2 one can compute for any of our four classes of star the structure of the velocity ellipsoid at a general point in the (R, z) plane. We have used a novel formula to obtain remarkably precise analytic fits to the distinctly non-Gaussian v φ distributions for eight bins in the (R, z) plane. The complete v φ distributions at these points can be recovered for any of the four classes of stars by inserting values from either Table 3 or Table 4 into equations equations (7) and (8). We have compared our observational velocity histograms with the predictions of a dynamical model that was fitted to the local velocity distribution and the Gilmore & Reid (1983) vertical density profile. When making this comparison we assume only that the survey's selection function is velocity-blind (which it certainly is) and we are able to model the effects of errors in both distances and velocities with considerable completeness. Overall the agreement between the model's predictions and the data is remarkably good and offers strong support for the assumptions on which the dynamical model rests, including its gravitational potential. There is, however, a tendency for the distribution of observed v φ components to be shifted to larger values than the model predicts. A possible contributory factor to this offset may be over-estimation of the Sun's peculiar V velocity, but the offset can be generated in several ways, including spiral arms, the age gradient within the disc, and use of incorrect values of R0 and Θ0. The dynamical model performs outstandingly well in predicting the distributions of vertical velocity components V3 of all star classes. These distributions are considerably more sharply peaked than Gaussians and the model captures this phenomenon beautifully. At \|z\| < 0.5 kpc the model predicts the distributions of radial components V1 nearly as successfully, but at greater distances from the plane the model predicts distributions of V1 that are too narrow. This problem is undoubtedly connected to the surprising conclusion of B12 that the thick disc is radially cooler than the thin disc, a conclusion driven by both the structure of the GCS histograms for U and the strong mean rotation of SDSS stars far from the plane. The RAVE data also require that at \|z\| > 1 kpc there are unexpectedly many stars at large v φ , and this fact constraints our ability to make the thick disc radially hotter as the V1 histograms imply. One way to resolve, or at least ameliorate, the problem is to suppose that stars in the most distant bins have had their distances over-estimated by ∼ 20%. Similar distance over-estimates in the nearer bins would impair the nice agreement between theory and observation. However, it is inevitable that stars placed in the most distant bins have, on average, over-estimated distances, so it is plausible that distance over-estimates contribute significantly to the anomalies in the high-\|z\| bins. This study clearly indicates that the approach to Galaxy modelling developed in B12 is well worth developing. There are several directions in which to go. First a new df of the current type should be fitted to the richer body of observational data that is now available using an updated Galactic potential Φ. Next this df and these data should be used as a starting point for a re-determination of Φ along the lines outlined by McMillan & Binney (2013). Currently the df is being extended to include chemistry alongside age (Binney & Sanders 2013): this extension should markedly increase our ability to diagnose Φ because the requirement that several stellar populations that differ in both their chemistry and their kinematics exist harmoniously in a common potential will strongly constrain Φ. Figure 1 . 1Representation of the velocity ellipsoids of giant stars; the lengths of the principal axes of each ellipse are proportional to the corresponding velocity dispersion at the centre of the ellipse. Figure 2 . 2The curves show the spatial variation of the values of σ 3 and σ 1 at fixed R (top) or z (below) that are extracted from the raw data for non-clump giant stars by a maximum-likelihood technique that takes into account random measuring errors. The black dots show the result of correcting the dispersions of binned data for measurement errors by simple quadrature subtraction. In the upper panel the upper point of each pair refers to a bin that lies inside R 0 and the lower point refers to a bin at R > R 0 . In the lower panel results are shown for z = 0.22 and 0.86 kpc. Figure 3 . 3The dependence on \|z\| of the velocity dispersions at R 0 . Full black curves are for non-clump giants, red dashed curves are for cool dwarfs and dotted blue curves are for hot dwarfs. Figure 4 . 4The distributions of v φ for red-clump giants (black data points) and fits to them -in each panel the dashed curve shows the kinematic model specified by equations Figure 5 . 5As Fig. 4 but for non-clump giant stars. Figure 6 . 6As Fig. 4 but for hot dwarfs. Figure 7 . 7As Fig. 4 but for cool dwarfs. Figure 8 . 8The mean rotation velocity of the giants as a function of distance from the plane. The full curve is for bins at R < R 0 . The data points are the means of model distributions like those plotted as dotted curves inFig. 5. The statistical errors on these points are very small. Figure 9 . 9As Fig. 8 but for the dwarfs: hot (top) and cool (below). Figure 10 . 10Dotted curve: the contribution to the circular speed from the disc and bulge components; dashed curve: the contribution of the dark halo. Figure 13 . 13As Fig. 12 but for clump giants. Table 4 . 4Thesame as Table 3 but for hot (upper block) and cool (lower block) dwarfs. c 2012 RAS, MNRAS 000, 1-15 . 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[ "Teach Network Science to Teenagers", "Teach Network Science to Teenagers" ]	[ "Heather A Harrington \nDivision of Molecular Biosciences\nImperial College London\nSW7 2AZLondonUK\n", "Mariano Beguerisse Díaz \nDepartment of Mathematics\nImperial College London\nSW7 2AZLondonUK\n", "M Puck Rombach \nOxford Centre for Industrial and Applied Mathematics\nMathematical Institute\nUniversity of Oxford\nOX1 3LBUK\n", "Laura M Keating \nOxford Centre for Industrial and Applied Mathematics\nMathematical Institute\nUniversity of Oxford\nOX1 3LBUK\n", "Mason A Porter \nOxford Centre for Industrial and Applied Mathematics\nMathematical Institute\nUniversity of Oxford\nOX1 3LBUK\n\nCABDyN Complexity Centre\nUniversity of Oxford\nOX1 1HPOxfordUK\n" ]	[ "Division of Molecular Biosciences\nImperial College London\nSW7 2AZLondonUK", "Department of Mathematics\nImperial College London\nSW7 2AZLondonUK", "Oxford Centre for Industrial and Applied Mathematics\nMathematical Institute\nUniversity of Oxford\nOX1 3LBUK", "Oxford Centre for Industrial and Applied Mathematics\nMathematical Institute\nUniversity of Oxford\nOX1 3LBUK", "Oxford Centre for Industrial and Applied Mathematics\nMathematical Institute\nUniversity of Oxford\nOX1 3LBUK", "CABDyN Complexity Centre\nUniversity of Oxford\nOX1 1HPOxfordUK" ]	[]	We discuss our outreach efforts to introduce school students to network science and explain why networks researchers should be involved in such outreach activities. We provide overviews of modules that we have designed for these efforts, comment on our successes and failures, and illustrate the potentially enormous impact of such outreach efforts.	null	[ "https://arxiv.org/pdf/1302.6567v1.pdf" ]	10,387,565	1302.6567	be8ed4982d6b785dc1ffa3d8a3b1ea4f9114694b	Teach Network Science to Teenagers 26 Feb 2013 Heather A Harrington Division of Molecular Biosciences Imperial College London SW7 2AZLondonUK Mariano Beguerisse Díaz Department of Mathematics Imperial College London SW7 2AZLondonUK M Puck Rombach Oxford Centre for Industrial and Applied Mathematics Mathematical Institute University of Oxford OX1 3LBUK Laura M Keating Oxford Centre for Industrial and Applied Mathematics Mathematical Institute University of Oxford OX1 3LBUK Mason A Porter Oxford Centre for Industrial and Applied Mathematics Mathematical Institute University of Oxford OX1 3LBUK CABDyN Complexity Centre University of Oxford OX1 1HPOxfordUK Teach Network Science to Teenagers 26 Feb 2013arXiv:1302.6567v1 [physics.ed-ph] We discuss our outreach efforts to introduce school students to network science and explain why networks researchers should be involved in such outreach activities. We provide overviews of modules that we have designed for these efforts, comment on our successes and failures, and illustrate the potentially enormous impact of such outreach efforts. V. In Section VI, we present an outlook and encourage others to participate in and develop similar activities. In Supplementary Online Material (SOM), we include lesson materials for five of the modules, and we encourage you to use and adapt them for your own activities. II. OUR OUTREACH ACTIVITIES We ran events in which students visited us at Somerville College in Oxford and others in which we visited them. The number of students varied significantly (from about 5 to about 50) from one event to another, and their ages ranged from 13 to 16. After introducing ourselves to the students and teachers, we gave a short introductory talk [27] in which we defined "network" and some other relevant terms. We also introduced "network science" as the science of connectivity, showed several diverse examples of networks, and gave tantalizing hints as to how investigating network structure can provide information about dynamics or function. We then split the students into smaller groups (e.g., 50 students into 3 groups) in breakout sessions so that they could delve deeper into a specific topic. An example of an introductory talk on networks is available at [20]. This introductory talk takes 20-30 minutes. (We varied the amount of time that was spent discussing the various examples.) The presentation gives the definition of a network and shows how to represent a network as an adjacency matrix [28] using a small example. The speaker indicates different types of networks (e.g., unweighted, weighted, directed, etc.) and asks the students to think about how the matrix representation can be generalized for the different cases. The presentation then includes numerous examples, which are introduced via pretty pictures (just like in other talks for general audiences), and comments on how it relates to the students' experiences. Examples that we discussed include London's metropolitan transportation network ("The Tube"), Facebook friendships, food webs, networks in online role-playing games, and Web pages connected by hyperlinks. The talk also purposely introduces a small amount of jargon-such as "node", "edge", "small world", and "degree"-to provide some terminology to facilitate discussions in the breakout sessions. The introductory talk also includes hints of the mathematics under the hood, but it focuses predominantly on using broad brush strokes to introduce a few important ideas. After the introductory talk, we break out into sessions for students to explore in detail in smaller groups. (See Section III for the topics and the appendices and SOM for lots of detail.) Each of these sessions, which had 3-5 students per volunteer, was led by 1-2 people with 1-2 others helping. In many cases, the school teachers participated actively and were extremely helpful. Depending on the event, our sessions lasted 30-45 minutes, and each student participated in 1-2 such sessions [29]. On many occasions, we also asked the students to present their findings to the other groups. Naturally, our sessions had broader aims beyond discussions of specific ideas from network science: we wanted the students to think independently (and in small groups) and to investigate difficult, open-ended problems instead of problems with neat answers (to which they were more accustomed). We wanted to give the students a sense of how a scientist might tackle a problem or at least to give them something closer to what a university student might experience. We had two types of outreach events: (1) ones in which students and teachers visited us at Somerville College in Oxford, and (2) ones in which we visited the schools. For the events at Oxford, the logistics (including essential items like food, travel, and coffee) were arranged and run by Amy Crosweller, Somerville's Access and Communications Officer. Amy also helped recruit Somerville undergraduates (usually people studying mathematics) to assist us by giving tours of the College, answering questions about what it is like to be an undergraduate at University of Oxford and in Somerville College, and occasionally even acting as helpers for the breakout sessions. In one of the events in Oxford, we had a panel discussion about careers that included a volunteer from Google. Hosting the students in Somerville College had several advantages-it made it possible to draw students from different schools to the same event, it allowed more control and knowledge of the local facilities (flip charts, markers, etc.), and it gave us access to friendly and helpful undergraduates. However, we ultimately decided that the "travelling road show" format was more effective. This format allows us to work with students who are farther away geographically, as only nearby schools came to the Oxford events in practice even though we had money to offer them accommodation. Traveling to the schools is also important for attempting to work with students from schools who do not typically send students to University of Oxford (or, in some cases, even consider that as a possibility that's actually on the radar). It also makes it easier to have an event on a weekday rather than a weekend [30], as the students can attend this special event rather than a normal class for an hour or two. We varied details of the event components to accommodate school schedules, road-show versus Oxford events, number of volunteers, number and type (e.g., age and quality) of students who would be working with us, and lessons we learned regarding what seemed to work and what didn't. For example, in one event held in Somerville College, we worked with the same set of students for 6 hours [31]: we held two 45-minute sessions (with parallel breakout modules during each session), discussions over lunch, and more. Our other events were shorter. In our visits to schools, we experienced a wide range of abilities and behavior among the students. Sometimes we had students from the same grade level and other times there was a mixture. The latter was particularly enjoyable, as it appears to be unusual for students of different ages to work with each other on equal footing. Schools varied on whether they asked us to work with their top students, standard students, or underachievers. Our outreach events benefited tremendously from numerous volunteers-including professors, postdoctoral scholars from University of Oxford and Imperial College London, doctoral students from several different disciplines (predominantly mathematical scientists but also several biologists), visiting researchers, undergraduates, staff members, and the students' own teachers. The Web pages http://www.some.ox.ac.uk/191-5717/all/1/Somerville_tutor_helps_students_harness_the_science_behind_social_n and http://blogs.some.ox.ac.uk/access/2012/04/30/motivating-maths-pupils-and-reflecting-on-the-latest-sutto contain descriptions of our outreach efforts. III. LESSON MATERIALS Our session plans are designed for modules that last about 30-45 minutes, but they can be adapted readily for other formats. For example, we have occasionally merged multiple modules into a single module that covers a broader set of topics. The modules are interactive, and each one allows participating students to learn about one or more areas (or applications) of network science. In the appendices, we describe the following modules: 1. Appendix A: climate change and food webs; 2. Appendix B: small worlds and social networks; 3. Appendix C: disease spread and vaccination strategies; 4. Appendix D: Google's PageRank algorithm; 5. Appendix E: coloring maps and other puzzles as an introduction to proving theorems in graph theory; 6. Appendix F: why your friends have more friends than you do; 7. Appendix G: structural balance in networks. Modules 1-5 were our most successful ones, though we're sure you'll have ideas for how to improve them. We discuss all seven of the above modules in the appendices and go into greater depth for Modules 1-5 in the SOM. IV. SUCCESSES AND CHALLENGES Our outreach events have been largely successful, and (unsurprisingly) more iterations have led to greater polish and greater success. In this section, we'll highlight a few points that we hope will be helpful for your outreach events. The introductory talk [20] was extremely challenging, but it interested the students immensely and its introduction to network theory, its applications, and some of its jargon helped to provide a solid foundation for the interactive modules to follow. Put another way, the initial challenge helped make life easier for the rest of the event. We worked with students from very different backgrounds, but (for the most part) they were genuinely interested in working with us. We learned from our first outreach event that many students learned the key material in our lesson plans extremely quickly (despite the fact that we discussed topics on which there is active research). To address this, we sometimes needed to come up with impromptu activities to fill the allotted time. In Module B (Small Worlds), for example, we sometimes discussed material from a different module (such as Networks and Disease) that we were not running that day. When we were able to work with the same students on more than one module, students usually picked up the key ideas faster in the second module. Our most successful modules were the most interactive ones, and we tried to increase the interactivity in the modules as we refined them. The sessions instigated many interesting and thought-provoking conversations, and sometimes the best thing we could do was keep our mouths shut for a while. In Module A, for example, students used a networkscience perspective to think about the importance of grass at the bottom for survival in an entire food web. It was inspiring to see individual thinking, group discussions, and "ah-ha" moments without our intervention. Despite the overall success of our efforts, we faced many challenges, and we haven't yet figured out how to overcome all of them. We have gained much more of an appreciation for the difficulties faced by primary and secondary school teachers than we had before [32]. For example, sometimes it was difficult to get students to choose a module in the first place. (We preferred, when possible, not to choose modules for them.) Sometimes the majority of students picked the same topic, and we had to balance group sizes with the students' interests. Many students were hesitant to participate, and we sometimes had to tailor the module format to encourage more group-based answers rather than individuals ones. Some students were disruptive, though teachers from most of the schools were very helpful for managing these situations and otherwise maintaining order. (This became a serious problem at only one school.) It is worth remarking that we often noticed a difference in student comfort level when they visited us versus when we visited them. It is also important to note that the smoothest outreach days are not necessarily the most important ones to undertake. For example, our outreach activities were more difficult when we worked with weaker students, but this type of struggle is a necessary one. We welcome ideas for how to improve our outreach activities to make them more accessible for a wide variety of students and schools. V. WHY BOTHER? Network science offers an exciting supplement to standard mathematics curricula in schools. Spending time on it does take time away from the existing school curriculum, but we think that it is time well spent. Examining problems in networks demonstrates that mathematics is about ideas and abstractions rather than just calculations, and it does so in a way that is highly visual and closely connected to students' everyday experiences. Working through problems like those in our modules helps students (and teachers) to see a side of mathematics that is different from most of what they have experienced. We emphasize intuition, modeling, and problem-solving, and this provides a nice complement to the calculations to which students are accustomed. Because of the ubiquity of networks in everyday life, network science is an ideal subject for these kinds of outreach efforts. Once we explain to students what a network is, theyjust like professional scientists-see them everywhere they look, and hopefully they will also start to see mathematics everywhere. Based on our discussions with the students (and their teachers) and the comments in the survey forms they filled out for the events in Oxford, the students with whom we have worked certainly seem to have gained a significant interest in networks and mathematics. They viewed the problems that we presented as puzzles to ponder rather than rote calculations to get out of the way. It is also a positive experience for students and teachers to have direct contact with professional mathematicians and scientists to get a better idea of what it is that we do (which is to try to solve problems, just like the students were doing). It is also good to promote a view of scientists as regular human beings, which contrasts sharply with what often seems to be the case. Meanwhile, the outreach activities help improve the communication skills of the instructors, and that too is a significant boon. We target our outreach activities for students of age 13-16, as the younger students in this range have still not decided what subjects to study at university. (Students specialize very early in the United Kingdom.) We hope to persuade more students to pursue mathematics and science, and we hope that those who don't pursue those subjects will at a minimum gain a greater appreciation of mathematics as well as its importance. Network science is an eminently accessible subject-that is why many professionals enjoy it, after all-which makes it perfect for these kinds of outreach efforts. VI. CONCLUSIONS AND OUTLOOK We are continuing to conduct outreach activities across England, but the only way to make a really big impact is if these activities spread far and wide. (Let's turn this into a social contagion, okay?) We hope that we have wet your appetite to conduct outreach activities in schools in your area, and we encourage you to use, steal, adapt, and improve any material in this article or in the SOM. We recognize that many of you will have different opinions as to what should constitute the contents of modules, and we have already seen that the outreach activities and modules need to be adaptable from one school to another. Please send us your new modules and your improvements to our modules so that we can develop a large repository for networks-related school activities. We are also happy to provide advice or send you less formal notes if you think that they can be helpful. We also encourage school teachers to include network science in their lessons and math/science-club projects (and, of course, to use our materials and to contact us for any desired discussions or feedback). We have written only a few modules, but we hope that they convey the hugely important role that networks have the potential to play in getting school students interested in mathematics and science. This is a big challenge, but it is also an exciting opportunity. We hope that we have convinced you to engage in outreach efforts in your neighborhoods. It is a valuable use of your time, and it can have a very large impact. Please don't hesitate to contact us if you would like to discuss this further. Council of Canada (CGSM #403601-2011) and TERA Environmental Consultants. We thank Daniel Kim for assistance in developing some materials and Amy Crosweller, Karen Daniels, Lucas Jeub, Sang Hoon Lee, Jesús San Martin, and Stan Wasserman for helpful comments on this manuscript. We thank Somerville College for supplying room facilities free of charge on several occasions and Amy Crosweller of Somerville College for helping to arrange the events. We our particularly indebted to the numerous volunteers who helped design and/or run modules. In this module, which was designed by Laura Keating and Puck Rombach, the students examine how climate change can affect animal species. The students construct and analyze food webs, and they investigate how the extinction of one species can affect other species. We start the session by informing the students that they have been hired by the government as applied mathematicians to advise how climate change might affect species survival in the Arctic. Many of the students already possess some knowledge of food chains from their biology lessons, and we build on this to present food webs as networks rather than chains. To help introduce background material, the students first fill out a worksheet that includes questions about what a food web is and how one might construct a food web as a network. These initial questions also introduce relevant network properties, such as directedness of edges to describe unidirectional energy flow between species. To further grasp the concept of a food web as a network, we construct a network using a small set of familiar species (e.g., grass, a mouse, an insect, a small bird, and a large bird). We then explore what happens if one or more of those species becomes endangered. We begin to discuss the potential for extinction cascades and the implications that network structure (e.g., high in-degree or out-degree) can have for an entire ecosystem. The final part of the module is spent on a 'game' that explores food webs in Arctic ecosystems. We split the students into two groups and give each group a set of Arctic species that are either aquatic or terrestrial. Both sets of animals include the polar bear, which connects the aquatic and terrestrial food webs. See the Supplementary Online Material (SOM) for further details and the handouts that we give to the students. Each group of students constructs a network from their respective species, and they write down the adjacency matrix representation of it. We then ask the students what happens if one of the nodes is removed (i.e., if a species becomes extinct). Even these small networks are able to help illustrate the complicated and far-reaching effects of species extinction. At the end of the module, the students present their findings-including which species are most vulnerable, how climate change can affect food webs, and potential ways to reduce the negative effects of extinction. This module, which was designed by Mariano Beguerisse Díaz and Pau Erola, illustrates that social networks are everywhere and introduces some of their features. We examine the notion of a small world-the introductory talk included the jargon and a picture of a Watts-Strogatz network-using several examples that we hope are poignant for the students. At the beginning, we ask the students whether they have heard of the idea of "six degrees of separation". The term itself tends to be unfamiliar to most students, but many of them recognize the idea as a familiar one once we explain the jargon. We discuss Stanley Milgram's package-passing experiments and how it reached a more mainstream audience through venues such as The Oracle of Kevin Bacon [22]. We play the Kevin Bacon game with the students to try to find paths between movie actors so that they can find their own "surprisingly" short paths (e.g., a short path between an actor in a horror film and one in a children's movie), and we also think about trying to navigate networks. See the SOM for an example of a handout used for this activity. We ask the students to explain what is meant by a "social network" and to give examples (and to indicate the nodes and edges). We expand on the Kevin Bacon game by examining short paths and navigation in social networks. For example, we might ask the students "How many degrees of separation are there between you and the Queen of England?" We also tried other famous people, and students on two occasions had very short paths to Nelson Mandela (length 1 in one case and length 2 in another). We again find short paths and try to reconcile these close connections with the fact that famous people are supposedly distant and unreachable. We discuss who in a social network might help lead to the presence of many short paths, introduce the notion of hubs, and ask the students to consider this in the context of their Facebook friendships [4,26]. We stress that the notion of a small world applies throughout a network and that many short paths go through hubs. We also discuss different "communities" of people and which types of connections might serve as "shortcuts" between different communities rather than "local" connections within a community. Typical examples of the former arose via friendships from vacations or a summer spent abroad, and naturally the latter tended to be people-classmates, neighbors, etc.-in close geographical proximity. As we developed this module, we added more exercises for the students, as early versions of the module were sometimes less interactive than we would have preferred. One exercise, which did not include a worksheet, concerns how Facebook chooses which friendships it recommends to its users. We encouraged the students to develop their own recommendation algorithms, and we discussed their ideas on a whiteboard or flip chart. In another (very successful) exercise, we returned to movies and considered the social networks of characters in movies such as Toy Story (see the SOM and [16]). We gave the students a handout with an unlabeled version of the Toy Story network, and we asked them to identify which nodes represent the protagonists and other characters. We also asked the students to identify some communities of toys that had things in common. Appendix C: Module 3: Networks and Disease This module, which was designed by Heather Harrington and Mariano Beguerisse Díaz, illustrates how thinking about networks arises naturally when one tries to understand how diseases spread and how to develop good strategies to contain them. We start this module by asking students to discuss the main characteristics of an infectious disease and to think about how it might spread (e.g., what is the main difference between a non-infectious disease and an infectious one). We encourage the students to think about a fictional disease that can only spread by shaking hands and to discuss how it might spread in their school. This quickly leads to the notion of disease spread along social networks in schools, and we sometimes combined this module with Module 2 (on social networks and small worlds). For this discussion, it can be useful to distinguish online and offline social networks (and also to distinguish between the spread of ideas and rumours versus diseases). We also briefly discuss what other types of networks (e.g., transportation networks or trade networks) might be important for understanding, containing, and preventing diseases. We steer the discussion towards how different types of network topologies can affect disease spread and vaccination strategies. (One can ask similar questions in Module 2.) If it is too expensive to vaccinate everybody, then who should be vaccinated? Some students brought up node labels in this discussion-one rebellious student proposed that the youngest people should be vaccinated because they (supposedly) had the longest left to live-though our primary focus was on network topology and how it affects disease spread. In some sessions, students brought up air travel, which led to a discussion of how such travel has changed the ways diseases spread. Students also pointed out that some diseases could be spread by insects like mosquitos, and we used this opportunity to introduce bipartite networks and to explain how vaccination strategies differ in this situation. For example, fumigation can eliminate many mosquitoes (reducing the number of one type of node) and slow down the spread of a disease. The largest portion of the module is a hands-on activity in which we distribute handouts (see the SOM) with various example networks (see Fig. 1) to the students and ask them to devise possible vaccination strategies in each case if they are only allowed to vaccinate three or fewer nodes. The students realized quickly that this question was much harder to answer for some network topologies than for others. This was an interesting point of discussion, as it allows the students to consider how one might develop a vaccination strategy in real networks, which are much more complicated. Moreover, given that one needs to think about the answer even if one knows network structure exactly, we can discuss how to develop strategies when some (or even a lot) of the network structure is not known. We ask the students what would they do if we only know a network has a particular structure (e.g., suppose that one knows that it was generated using a Barabási-Albert mechanism) but do not know anything else. This question generated a lively discussion. A useful hint for many students is to ask what would happen if we choose a node at random and then ask him/her to choose a friend to vaccinate (rather than vaccinating the original node). We also sometimes discuss the time-ordering of contacts in social networks and how that can influence disease spread. In addition to discussing diseases specifically, it can be useful to encourage the students to think about other contexts in which "vaccination" strategies might be useful. One key question is the difference between the spread of an idea and a disease, and one might also wish to discuss other dynamical processes on networks. This module is particularly nice for illustrating that mathematics shows up in many situations that the students (and their teachers) did not previously consider to be mathematical. This occasionally came up in discussions of viable careers for people who study mathematics at the university level, and we highlighted that nowadays mathematicians work closely alongside health professionals. Appendix D: Module 4: How Google Works This module, which was designed by Mariano Beguerisse-Díaz, Sang Hoon Lee, and Lucas Leub, aims to introduce Google's PageRank algorithm for ranking pages on the World Wide Web [6,19]. (Daniel Kim also assisted in developing materials for this module.) To start the module, we ask the students to imagine a world without Google or other search engines and to develop their own strategies for finding information on the Web. Usually, one of the first ideas is to compile an exhaustive list of every Web page. We use this to introduce the idea of a crawler to navigate Web pages, and this leads naturally to the notion of representing the Web as a network with directed edges (the hyperlinks) between nodes (the Web pages). We ask the students to think about how to figure out whether a Web page is relevant for the information one seeks, and this leads almost immediately to the issue of how one should rank Web pages in order of importance. One possibility that the students quickly bring up is that one can develop rankings based on the textual content of a page. (In one case, we had a good discussion about how we would try to use an automated method to distinguish the Amazon rain forest from Amazon.com.) We let the students know that the first Web search engines used to be FIG. 2: An example of a directed network whose nodes we ask students to rank in order of importance. This network is strongly connected (so any Markov chain on it is ergodic), so we can ignore the problem of dead-end nodes (i.e., nodes with an out-degree of 0). However, we did discuss the notion of dead ends on many occasions that we ran this module. The ranking of the nodes in this graph from largest to smallest PageRank score (in parenthesis) is as follows: D (0. "curated" by hand, which limited how much of the Web could be explored. This limitation was an incentive to people to seek algorithmic methods to rank pages, which is what we want the students to explore. We ask the students to develop ideas for how to use the network structure of the Web to rank pages. (The difficulty of this transition in the discussion varied strongly from one to school to another.) Most of the time, the first structure-based ranking that the students propose is to rank Web pages according to the number of incoming hyperlinks (i.e., according to in-degree). We discuss whether someone can cheat this system (as well as simple text-based systems) to improve the ranking of a page and whether better methods are available. In Fig. 2, we show an example network that we use to help guide our discussion of how to rank Web pages. This example is particularly useful for moving beyond ideas for ranking based on the text on a page, as we can pretend that no such text exists (or that it is otherwise impossible to distinguish Web pages based on their text). We give the students a handout with this network (see the SOM), and we ask them to rank the nodes in order of importance (and also to indicate how they have defined "important"). Motivated by the fact that people often seem to explore the Web by "randomly" following hyperlinks-who hasn't done this on Wikipedia?-we ask the students whether they can develop a ranking method based on this idea. We use phrasing along the following lines: If we have a large number of monkeys-it is very compelling to refer to random walkers as "monkeys" [15]-who are clicking on hyperlinks randomly, what ranking would we obtain based on the number of times each page is visited. It can also be useful to discuss why Wikipedia is a "monkey trap", in the sense that many Wikipedia pages have high rankings in Google searches. (We occasionally discussed having one random walker versus having a large number of random walkers.) Using these questions, we introduce the rationale behind PageRank. Crucially, we try to avoid words like "eigenvalue" and "eigenvector" (and "ergodic", "Markov", etc.), though we do attempt to get the students to compute (by hand) the PageRank eigenvector for a network like the one in Fig. 2. They just don't know that what they are computing is called an eigenvector. The example network in Fig. 2 is very instructive. Different choices for how to measure node importance (e.g., in-degree versus out-degree) lead to different rankings, and we have interesting discussions regarding which ranking is "correct". (These ideas could also be used to develop a module that focuses on centralities more generally-e.g., intuition related to the notion of "betweenness" sometimes comes up in Modules 2 and 3-as well as how one might change the notion of importance depending on the question one wants to answer.) An interesting feature of the network in Fig. 2, which is worth asking the students to try to prove, is that nodes B and C have the same PageRank score. To compute the rankings, the students count the number of times the nodes are visited on different walks through the network. We and the students use two primary techniques for this calculation: (1) start from a uniform distribution and iteratively count the number of walkers on each node, or (2) try to identify relative orderings for the ranking of different nodes without calculating individual probabilities. The maximum out-degree in the example network is 2, which makes it easy to simulate a random walk by flipping a coin to decide which edge to follow. The students soon realize that one can get to node D from almost everywhere, and that it is indeed the most visited node in a (conventional) random walk. This then makes it the most important node in this ranking scheme. The students then realize that the nodes receiving edges from it (C and F) must come next in the rankings, and they discuss how to break the tie. Students also note that C and B are always visited the exact same number of times (as long as we don't stop the walk before a monkey has had the chance to leave C). In some cases, we were able to get the students to calculate the actual percentage of visits for each node rather than only determining the rank order. When there is enough time, we discuss that a monkey gets "trapped" on a Web page with an out-degree of 0. This problem can be illustrated by adding a dead-end node G to the network in Fig. 2. We ask the students to think about how they can change their ranking methodology to be able to deal with this situation. This gives the opportunity to discuss the idea of a random walk with "teleportation" (e.g., at each node, one follows an edge with a probability p or otherwise chooses some other node in the network via a random process). Once the dead-end node has been added, the graph is no longer "strongly connected", which we can use to illustrate that even a small perturbation of a network can change its properties in a fundamental manner. As part of this module, we sometimes discuss clever scientific ways to use Google-such as trying to measure the similarity between two football players by examining how often they show up together in Google searches [17]. Appendix E: Module 5: Introduction to Graph Theory This module, which was designed by Puck Rombach, provides an introduction to graph theory through the discussion of some famous mathematical "puzzles". In contrast to the other modules, it focuses on the theoretical (or "pure") side of mathematics. It is also arguably our most popular and successful module. Graph theory's deep connection with networks makes it an area of pure mathematics that can resonate with students. The problems that we discussed with the students happen to also be relevant for applications (and we mention them when we are asked about them), but that is not our focus. We want to show the students how to formulate and prove theorems, and we want to convey our excitement for and the value of abstract mathematical ideas for their own sake. This module also illustrates that university mathematics need not require any numbers or calculations, which is an important difference from the kind of mathematics with which students are familiar from schools. Graph theory is wonderful because its basic ideas and many of its interesting problems can be explained in a few minutes in a way that allows people with little or no mathematical background (such as young children) to understand them. Two examples that we discuss with the students are the Bridges of Königsberg (see Fig. 3) and the Water, Gas, and Electricity puzzle (see Fig. 4). Both of these puzzles are unsolvable, which one can demonstrate by systematically exhausting all possible solutions. We ask the students to try to find solutions to these puzzles and then, when they can't, to try to prove that no solutions exist. Excitingly, both puzzles hint at deep and general theorems. The Bridges of Königsberg [11] is unsolvable because it is both necessary and sufficient that every node have an even degree for a graph to have an Euler cycle (i.e., a circuit that runs along every edge exactly once and returns to its starting point). In this case, no nodes have an even degree. Proving the necessity is easy-every time that an island (i.e., node) is "entered", it must also be exited-but proving sufficiency is more difficult. This problem also allows us to discuss the notion of necessary and sufficient conditions for results to be true. The Water, Gas, and Electricity puzzle is unsolvable because K 3,3 (the complete bipartite graph on two sets of three nodes) is not planar [33]. Another family of problems that we discuss at length with the students concerns graph coloring. Graph-coloring problems are easy to explain, but they can be excruciatingly difficult to solve. The most famous graph-coloring result is the Four-Color Theorem [8], which we explore in detail with the students. We ask them the following question: How many colors do we need to color a map such that two countries that share a border are not colored with the same color? (Maps can be represented as planar graphs and vice versa, so map coloring is the same as planar graph coloring.) We provide paper and markers and examine the colorings that the students draw. This allows us to determine quickly if they understand the definition of "coloring". We keep challenging them to try to construct maps that require more colors (and we typically need to make an explicit statement that we are disallowing islands). The students find through empirical observation and discussion that none of them ever need more than four colors, and we eventually let them know that it impossible to construct a planar map of contiguous countries that isn't four-colorable. [34] In one memorable incident, a student insisted (despite our statement that it was impossible) that he was going to construct a map that couldn't be four-colored, he kept working on that for the rest of the day, and then he insisted as the students were leaving that he was going to continue working at it and get it to work. He'll probably be a famous mathematician someday, as he's already got the right persistent attitude for it. To get the students to get their hands dirty with mathematical proofs, we then discuss a much simpler result that we call a Two-Color Theorem. We consider maps with the special rule that borders cannot end. That is, every border must either go off of the page in all directions (imagining that it goes to infinity) or it has to connect to itself in a cycle. To make things simpler (although the result holds without this simplification), we do not allow borders to cross themselves. In Fig. 5, we show an example of this special "infinite-border" map (IB-map). We teach the students how to prove this Two-Color Theorem using induction [14], which we explain is a common approach in mathematics. (The proof is in the SOM.) If there is extra time or if it is useful to discuss something different, we also go through the proof of Hall's Marriage Theorem. (This problem can also be discussed without bringing up marriage, though that is the traditional setting of the problem.) This theorem gives a necessary and sufficient condition for marrying each member of one group of people to a member of another group of people, given that some of the latter have conditions that limit who they are willing to marry. [35] The setting for HMT is enjoyable to explain, and it usually elicits a few laughs from the students. Student reaction to this module has been overwhelmingly positive. To our great pleasure, most students have enjoyed solving problems for their own sake without having to be encumbered by any external importance. Most of the questions are very challenging and difficult to answer straight away. When running this module, we encourage the students to think like mathematicians. If they find a problem to be too hard, then we encourage them to consider a simpler but related question that they can try to answer first. If students doesn't know an answer, it is good to ask them to indicate what they do know. For particularly keen students, of course, this module can be scaled up to make it more challenging. Two decades ago, Scott Feld wrote a well-known paper that discusses why your friends have more friends than you do [12]. Because this can be explained using simple mathematical arguments on some networks (such as the configuration model [5,19]), we decided that this idea would make a very cool module. This module, which was designed by Lucas Jeub, starts by asking the students whether or not they have heard about this result. In the one time we ran the module, none of the students had heard about anything like this (which isn't terribly shocking), and we actually want students to express skepticism about this kind of result. Our plan in such a situation is ask the students to come up with an argument of why such a result can't be true followed by guiding them through the intuition for the correct result that the mean number of friends of a friend is larger than one's mean number of friends. To illustrate this result, we ask the students to think about their Facebook friends and the number of friends of their most popular Facebook friend. We also thought about getting the students to write down the number of people in the room that they consider to be friends, as this can of course help illustrate the same phenomenon. However, you can already probably see a major flaw in this attempt to illustrate the key idea: the students were rather shy about saying how many friends they had. We used the configuration model as a toy situation to illustrate the key phenomenon. We asked the students to write down their number of friends on a piece of paper. [36] We tabulated the numbers and asked the students to try and construct a network with the given degree sequence, and the idea of the configuration model than comes naturally by considering the set of all possible networks that one can construct from this information. We were then hoping to get into a discussion on an intuitive level of whether the fact that social networks are rather different from the configuration model (e.g., because of triadic closure) changes the result, and we thereby wanted to get into some issues regarding the structure of social networks. However, although we were able to get the students to understand the arguments with the configuration model, our attempts to scale up from this point were unsuccessful. An even bigger issue is that the calculations necessary to verify the key result are tedious even for small networks. We ended up finding something else to discuss with the remaining time and decided that this module needed to go back to the drawing board. A while after we ran this module, Steve Strogatz presented an excellent explanation of Scott Feld's result in a New York Times article [24]. We think that his explanation can serve as a useful springboard for a good module, and we suggest starting from there if you like the idea of a module with this theme. We believe that this theme has the potential to make an excellent module. This module wasn't "designed" by anybody, though we ran it once on an ad hoc basis instead of Module 6 when the latter hadn't worked out as well as we would have liked earlier in the day. We needed something at the last minute, as trying Module 6 without going back to the drawing board first was not an option. Motivated by a brilliant colloquium by Steve Strogatz the previous day, we decided that structural balance would make an excellent topic. We covered the basic ideas behind structural balance, including whether or not three mutually antagonistic connections in a triad should be considered as balanced. We also guided the students through work by Cartwright and Harary [7] and asked them to consider this in real life as well as on Massively Multiplayer Online Role-Playing Games (MMORPGs) [25]. (For some students, it might not be entirely clear which of these is more real.) In our original brainstorming for sessions to design, we had actually considered developing a module on structural balance. The ad hoc module based on that idea was surprisingly successful given the lack of formal design and preparation, so we feel that structural balance would make a good module to develop more carefully. One can of course bring up alliances and conflicts in war and among schoolmates, and one can also discuss games like Risk in FIG. 1 : 1Example networks for which one can apply different vaccination strategies (see the SOM). (A) star network, (B) circular lattice, (C) regular circular lattice with four neighbors (D) Erdős-Rényi random graph, and (E): Barabási-Albert network. FIG. 2: An example of a directed network whose nodes we ask students to rank in order of importance. This network is strongly connected (so any Markov chain on it is ergodic), so we can ignore the problem of dead-end nodes (i.e., nodes with an out-degree of 0). However, we did discuss the notion of dead ends on many occasions that we ran this module. The ranking of the nodes in this graph from largest to smallest PageRank score (in parenthesis) is as follows: D (0.3077), F (0.2051), B (0.1538), C (0.1538), E (0.1026), and A (0.0769). FIG. 3 : 3The Bridges of Königsberg. Is it possible to create a walk that crosses each bridge exactly once and also returns to its starting point? FIG. 4 : 4The Water, Gas, and Electricity Puzzle. Is it possible to connect every house to all of water, gas, and electricity such that no lines cross? FIG. 5 : 5An infinite-border map, for which one can prove a Two-Color Theorem. These include Ross Atkins, Senja Barthel, Andrew Elliott, Pau Erola, Virginia Faircloud, Martin Gould, Lucas Jeub, Sang Hoon Lee, Peter Neumann, Sofia Piltz, Jesús San Martín, Andrew McDowell, Atieh Mirshahvalad, Ioannis Psorakis, Marta Sartzynska, Karin Valencia, and Kerstin Weller. (We apologize to anybody who we forgot to mention.) During events at Somerville College, we also received help from Zoe Fannon, Martin Griffiths, Stanislav Kasjalov, Jess King, Jen Kitson, Josie Messa, Steve Strogatz, and Almat Zhantikin. Wiesner Vos and Philip Clarkson of Google helped during a panel discussion about mathematics careers. MAP thanks Chris Budd, Amy Crosweller, Martin Griffiths, and Siân Owen for helpful advice in designing this outreach program. Appendix A: Module 1: Networks and the Environment Appendix F: Module 6: Do Your Friends Have More Friends Than You Do? AcknowledgementsMAP acknowledges University of Oxford for twice funding these outreach projects as a component of their Pathways to Impact grant from the EPSRC. MAP was also supported by the FET-Proactive project PLEXMATH(FP7-ICT The Prize in Economic Sciences. (2012). The Prize in Economic Sciences 2012. available at http://www.nobelprize.org/nobel_prizes/economics/laureates/2012/. Improbable Research. (2013). Improbable Research. available at http://www.improbable.com/. Every Planar Map is Four Colorable. K Appel, W Haken, Bulletin of the american mathematical society. 825Appel, K., & Haken, W. (1976). Every Planar Map is Four Colorable. Bulletin of the american mathematical society, 82(5), 711-712. L Backstrom, P Boldi, M Rosa, J Ugander, S Vigna, arXiv:1111.4570Four Degrees of Separation. Backstrom, L., Boldi, P., Rosa, M., Ugander, J., & Vigna, S. (2011). Four Degrees of Separation. arXiv:1111.4570, Nov. Random graphs. B Bollobás, Cambridge University Press2nd ednBollobás, B. (2001). Random graphs. 2nd edn. Cambridge University Press. The anatomy of a large-scale hypertextual web search engine. S Brin, L Page, Proceedings of the seventh international world-wide web conference. the seventh international world-wide web conferenceBrin, S., & Page, L. (1998). The anatomy of a large-scale hypertextual web search engine. Proceedings of the seventh international world-wide web conference (www 1998). Structural Balance: A Generalization of Heider's Theory. Psychological review. D Cartwright, F Harary, 63Cartwright, D., & Harary, F. (1956). Structural Balance: A Generalization of Heider's Theory. Psychological review, 63, 277-293. Graph theory. R Diestel, Springer3rd ednDiestel, R. (2005). Graph theory. 3rd edn. Springer. Presidential Views: Interview with Hyman Bass. Notices of the american mathematical society. J F Dwyer, 48Dwyer, J. F. (2001a). Presidential Views: Interview with Hyman Bass. Notices of the american mathematical society, 48(3), 312-315. Reflections of an Outreach Mathematician. Notices of the american mathematical society. J F Dwyer, 48Dwyer, J. F. (2001b). Reflections of an Outreach Mathematician. Notices of the american mathematical society, 48(10), 1173-1175. Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum petropolitanae. L Euler, 8Euler, L. (1741). Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum petropolitanae, 8, 128-140. Why Your Friends Have More Friends Than You Do. S L Feld, American journal of sociology. 966Feld, S. L. (1991). Why Your Friends Have More Friends Than You Do. American journal of sociology, 96(6), 1464-1477. The pleasure of finding things out: The best short works of richard p. R P Feynman, feynman. Helix BooksFeynman, R. P. (19). The pleasure of finding things out: The best short works of richard p. feynman. Helix Books. New mathematical diversions. M Gardner, University of Chicago PressGardner, M. (1984). New mathematical diversions. University of Chicago Press. Virtual Voters Pick Best Teams. M Hopkin, Hopkin, M. (2003). Virtual Voters Pick Best Teams. available at http://www.nature.com/news/1998/031110/full/news031110-15.htm J Kaminski, M Schober, Moviegalaxies. available at. Kaminski, J., & Schober, M. (2013). Moviegalaxies. available at http://moviegalaxies.com/. Googling Social Interactions: Web Search Engine Based Social Network Construction. S H Lee, P-J Kim, Y-Y Ahn, H Jeong, Plos one. 5711233Lee, S. H., Kim, P-J, Ahn, Y-Y, & Jeong, H. (2010). Googling Social Interactions: Web Search Engine Based Social Network Construction. Plos one, 5(7), e11233. Being a Professional Mathematician. A Mann, C Good, Mann, A., & Good, C. (2013). Being a Professional Mathematician. available at http://www.beingamathematician.org/. Networks: An introduction. M E J Newman, Oxford University PressNewman, M E J. (2010). Networks: An introduction. Oxford University Press. . M A Porter, Porter, M A. (2012). An Introduction to the Mathematics of Networks. An Introduction to the Mathematics of Networks. available at http://people.maths.ox.ac.uk/~porterm/temp/networks-day_intro-fall2012.pdf. . J P Preskill, Preskill, J. P. (2013). The Oracle of Bacon. P Reynolds, Reynolds, P. (2013). The Oracle of Bacon. available at http://oracleofbacon.org/. Engaging the Clutch of the Science Communication Continuum Shifting Science Outreach into High Gear. B J Saab, Hypothesis journal. 81Saab, B. J. (2010). Engaging the Clutch of the Science Communication Continuum Shifting Science Outreach into High Gear. Hypothesis journal, 8(1), 1-7. Friends You Can Count On. S Strogatz, Strogatz, S. (2012). Friends You Can Count On. available at http://opinionator.blogs.nytimes.com/2012/09/17/friends-you-can- Multi-Relational Organization of Large-Scale Social Networks in an Online World. M Szell, R Lambiotte, S Thurner, Proceedings of the national academy of sciences of the united states of america. the national academy of sciences of the united states of america107Szell, M., Lambiotte, R., & Thurner, S. (2010). Multi-Relational Organization of Large-Scale Social Networks in an Online World. Proceedings of the national academy of sciences of the united states of america, 107, 13636-13641. J Ugander, B Karrer, L Backstrom, C Marlow, arXiv:1111.4503The Anatomy of the Facebook Social Graph. Ugander, J., Karrer, B., Backstrom, L., & Marlow, C. (2011). The Anatomy of the Facebook Social Graph. arXiv:1111.4503, Nov. For events at Somerville College, we also included a short introduction to the University of Oxford, studying mathematics and science at a university, and Somerville College before we started discussing network science. For events at Somerville College, we also included a short introduction to the University of Oxford, studying mathematics and science at a university, and Somerville College before we started discussing network science. Some of the students had seen matrices before, but others had not. Some of the students had seen matrices before, but others had not. As we became more experienced with our efforts, we concluded that shorter sessions were a better format than longer sessions for most of the students. As we became more experienced with our efforts, we concluded that shorter sessions were a better format than longer sessions for most of the students. On one occasion, we tried having a weekday event in Oxford, but attendance was sparse. On one occasion, we tried having a weekday event in Oxford, but attendance was sparse. In retrospect, this was probably too long for them. In retrospect, this was probably too long for them. Occasional outreach activities are wonderful, but we wouldn't want to do this every day!. Occasional outreach activities are wonderful, but we wouldn't want to do this every day! . A "planar, graph is a graph that can be drawn on a plane such that no edges cross each otherA "planar" graph is a graph that can be drawn on a plane such that no edges cross each other. This was proved in the 1970s by Kenneth Appel and Wolfgang Haken [3], but it required exhaustive computer searches and hundreds of pages of analysis. There isn't (yet) an elegant proof for of this theorem. This was proved in the 1970s by Kenneth Appel and Wolfgang Haken [3], but it required exhaustive computer searches and hundreds of pages of analysis. There isn't (yet) an elegant proof for of this theorem. We used this as part of our response to a skeptical question about HMT ever being useful-the reaction to our answer was wonderful-though we also stressed that that didn't matter. Conveniently, half of the 2012 Nobel Prize in Economics was awarded for theoretical work on generalizing. as we were only concerned with the fact that proving the theorem is interestingConveniently, half of the 2012 Nobel Prize in Economics was awarded for theoretical work on generalizing HMT [1]. We used this as part of our response to a skeptical question about HMT ever being useful-the reaction to our answer was wonderful-though we also stressed that that didn't matter, as we were only concerned with the fact that proving the theorem is interesting. As indicated above, it would have been better to use a less personal mechanism to illustrate this example. For example, one can ask students to choose an arbitrary positive integer bounded above by the number of people in the room and to. just pretend that that number is their number of friendsAs indicated above, it would have been better to use a less personal mechanism to illustrate this example. For example, one can ask students to choose an arbitrary positive integer bounded above by the number of people in the room and to just pretend that that number is their number of friends.	[]
[]	[ "Ariel R Zhitnitsky \nDepartment of Physics & Astronomy\nUniversity of British Columbia\nV6T 1Z1VancouverB.CCanada\n" ]	[ "Department of Physics & Astronomy\nUniversity of British Columbia\nV6T 1Z1VancouverB.CCanada" ]	[]	The violation of local P and CP invariance in QCD has been a subject of intense discussions for the last couple of years as a result of very interesting ongoing results coming from RHIC. Separately, a new thermalization scenario for heavy ion collisions through the event horizon as a manifestation of the Unruh effect, has been also suggested. In this paper we argue that these two, naively unrelated phenomena, are actually two sides of the same coin as they are deeply rooted into the same fundamental physics related to some very nontrivial topological features of QCD. We formulate the universality conjecture for P and CP odd effects in heavy ion collisions analogous to the universal thermal behaviour observed in all other high energy interactions.	10.1016/j.nuclphysa.2011.01.020	[ "https://arxiv.org/pdf/1008.3598v3.pdf" ]	117,026,633	1008.3598	29fc4a2b953e0dfaf50cebdb68011361a6e3b541	Ariel R Zhitnitsky Department of Physics & Astronomy University of British Columbia V6T 1Z1VancouverB.CCanada Preprint typeset in JHEP style -HYPER VERSION P and CP Violation and New Thermalization Scenario in Heavy Ion Collisions The violation of local P and CP invariance in QCD has been a subject of intense discussions for the last couple of years as a result of very interesting ongoing results coming from RHIC. Separately, a new thermalization scenario for heavy ion collisions through the event horizon as a manifestation of the Unruh effect, has been also suggested. In this paper we argue that these two, naively unrelated phenomena, are actually two sides of the same coin as they are deeply rooted into the same fundamental physics related to some very nontrivial topological features of QCD. We formulate the universality conjecture for P and CP odd effects in heavy ion collisions analogous to the universal thermal behaviour observed in all other high energy interactions. Introduction. Motivation The main goal of this paper is to argue that two, naively unrelated, phenomena: 1. local P and CP violation in QCD as studied at RHIC [1,2,3,4,5]; and 2. universal behaviour of multihadron production described by a universal hadronization temperature T H ∼ (150 − 200) MeV are in fact tightly related, as they describe different sides of the same fundamental physics. Before we present our arguments suggesting the common nature of these two phenomena, we review each effect separately as it is conventionally treated today. Our next step is to take a fresh look at these effects and present some arguments suggesting that both these phenomena are in fact originated from the same fundamental physics and both are related to the very deep and nontrivial topological features of QCD. Local P and CP violation in QCD. Charge separation effect The charge separation effect [6,7] can be explained in the following simple way. Let us assume that an effective θ( x, t) ind = 0 is induced as a result of some non-equlibrium dynamics as suggested in refs. [8,9,10,11,12,13]. The θ( x, t) ind parameter enters the effective lagrangian as follows, L θ = −θq where q ≡ g 2 64π 2 µνρσ G aµν G aρσ such that local P and CP invariance of QCD is broken on the scales where correlated θ( x, t) ind = 0 is induced. As a result of this violation, one should expect a number of P and CP violating effects taking place in the region where θ( x, t) ind = 0. In particular, one should expect the separation of electric charge along the axis of magnetic field B or along the angular momentum l if they are present in the region with θ( x, t) ind = 0. This area of research became a very active field in recent years mainly due to very interesting ongoing experiments [1,2,3,4,5]. There is a number of different manifestations of this local P and CP violation, see [14,15,16,17,18,19,20,21,22] and many additional references therein. In particular, in the presence of an external magnetic field B or in case of the rotating system with angular velocity Ω there will be induced electric current directed along B or Ω correspondingly, resulting in separation of charges along those directions as mentioned above. One can interpret the same effects as a generation of induced electric field E directed along B or Ω resulting in corresponding electric current flowing along J ∼ B or J ∼ Ω directions. All these phenomena are obviously P and CP odd effects. Non-dissipating, induced vector current density has the form: J = (µ L − µ R ) e B 2π 2 ,(1.1) where P odd effect is explicitly present in this expression as the difference of chemical potentials of the right µ R and left µ L handed fermions is assumed to be nonzero, (µ L −µ R ) = 0. The combination (µ L − µ R ) can be thought asθ(t) after a corresponding U (1) A chiral time-dependent rotation is performed, see also [17] for a physical interpretation of the relation (µ L − µ R ) =θ. Originally, formula (1.1) has been derived in [23], though in condensed matter context. In QCD context formula (1.1) has been used in applications to neutron star physics where magnetic field is known to be large, and the corresponding (µ L −µ R ) = 0 can be generated in neutron star environment as a result of continuos P violating processes happening in nuclear matter [24,25]. It has been also applied to heavy ion collisions where an effective (µ L −µ R ) = 0 is locally induced. The effect was estimated using the sphaleron transitions generating the topological charge density in the QCD plasma [14,15]. The effect was coined as "chiral magnetic effect" (CME) [14,15]. Formula (1.1) has been also derived a numerous number of times using numerous variety of techniques such as: effective lagrangian approach developed in [26]; explicit computation approach developed in [27]; direct lattice computations [18,19]. In addition, the effect has been studied in holographic models of QCD [28,29,30]. While there is a number of subtitles in holographic description of the effect [29,30], it is fair to say: on the theoretical side the effect is a well established phenomenon. It remains to be seen if this phenomenon is related in anyway to what has been actually experimentally observed [1,2,3,4,5]. Universal hadronization temperature T H ∼ (150 − 200) MeV Naively unrelated story goes as follows. We start from the following general observation: over the years, hadron production studies in a variety of high energy collision experiments have shown a remarkably universal feature, indicating a universal hadronization temperature T H ∼ (150 − 200) MeV . From e + e − annihilation to pp and pp interactions and further to collisions of heavy nuclei, with energies from a few GeV up to the TeV range, the production pattern always shows striking thermal aspects, connected to an apparently quite universal temperature around T H ∼ (150 − 200) MeV [31]. While experimentally it is well established phenomenon, it is very difficult to understand its nature as number of incident particles in e + e − annihilation as well as in pp and pp interactions is not sufficient even to talk about statistical averages. This observation motivated a number of early attempts [32] to interpret the resulting spectrum of particles as the Unruh radiation [33,34,35] when the event horizon emerges as a result of strong interactions. The modern, QCD based formulation of this idea has been developed quite recently in refs. [36,37,38,39,40,41,42,43]. See also alternative approaches [46,47,48] leading to the same "apparently thermal" aspects of the produced particles. The key ingredient of the approach suggested in refs. [36,37,38,39,40,41,42,43] is as follows: an observer moving with an acceleration a experiences the influence of a thermal bath with an effective temperature T = a 2π (1.2) which is conventional Unruh effect [33]. The corresponding acceleration parameter a in QCD is determined by the so-called "saturation scale" Q s [44,45] as suggested in refs. [36,37,38,39], or by the string tension σ as advocated in refs. [40,41,42,43], a Q s or a √ 2πσ. (1. 3) The problem of calculating of the effective acceleration "a" is obviously very hard problem of strongly interacting QCD. This problem of computation "a" is not addressed in the present paper. We simply assume that such a description exists, in which case the relation (1.3) explains the puzzle on why the temperature T given by eq. (1.2) is so universal, as it it almost independent on type of processes and the energy of colliding particles, as it is entirely determined by the fundamental Λ QCD scale. This "apparent thermalization" originates from the event horizon in an accelerating frame: the incident hadron decelerates in an external colour field, which causes the emergence of the causal horizon. Quantum tunnelling through this event horizon then produces a thermal final state of partons, in complete analogy with the thermal character of quantum Unruh radiation [33,34,35]. One should emphasize that the Planck spectrum in this approach is not resulted from the kinetics when the thermal equilibrium with temperature T given by eq. (1.2) is reached due to the large number of collisions. Rather, the Planck spectrum in high energy collisions is resulted from the stochastic tunnelling processes when no information transfer occurs. In such circumstances the spectrum must be thermal. Such interpretation would naturally explain another puzzle with the thermal spectrum in e + e − , pp and pp high energy collisions when the statistical thermalization could never be reached in those systems. We adopt this viewpoint, and we have nothing new to add to the computations presented in refs. [36,37,38,39,40,41,42,43] in strongly coupled QCD. However, we interpret that the Planck spectrum observed in high energy collisions somewhat differently in comparison with papers mentioned above. Namely, we interpret the observed spectrum as the result of complete reconstruction of the QCD vacuum state in accelerating frame "a", rather than due to any specific properties of its excitations -the partons. Our interpretation does not change any previous results within this framework. However, the new interpretation will play a crucial role when we apply the same logic and the same technique for discussions of local P and CP violation in QCD, which is the main subject of the present paper. • Our original contribution is the study of some specific topological fluctuations, which we believe are responsible for local P and CP violating processes observed at RHIC. Those vacuum topological fluctuations will be changed along with many other vacuum fluctuations in accelerating frame "a". These changes of the ground state due to the acceleration as we shall see are describable in terms of the Veneziano ghost 1 which solves the U (1) A problem in QCD [49,50,51,52]. This key degree of freedom (the Veneziano ghost) has η − quantum numbers, and plays the role similar to θ parameter from section 1.1. Its 0 −+ quantum numbers, as we shall see, play the crucial role in linking two naively unrelated problems outlined in two sections above: local P and CP violation in QCD, section 1.1, and universal hadronization temperature, section 1.2. The paper is organized as follows. In section 2 we discuss the nature of universality of hadronization temperature observed in numerous high energy experiments. We adopt the basic logic and philosophy of refs. [36,37,38,39,40,41,42,43]. We shall explicitly demonstrate the observed thermal spectrum with temperature (1.2) can be interpreted as the direct consequence of the basic features of known Bogolubov's coefficients in the accelerating frame for a system moving with effective acceleration "a". The temperature will be universal for all types of produced particles: massive or massless, charged or neutral, scalars, spinors, or vectors. Such a universality is similar to universal features of the Unruh radiation [33,34,35]. This result will be our basic explanation for the thermal spectrum observed in e + e − , pp and pp high energy collisions when the statistical thermalization (due to the conventional collisions) can not be ever reached in those systems. In section 3 we review the resolution of the U (1) A problem and structure of the θ vacua [49,50,51,52] by constructing the corresponding effective lagrangian. We pay special attention to the structure of the Veneziano ghost, its contribution to the topological susceptibility with a "wrong sign" (which is a key element in resolution of the U (1) A problem), the unitarity, anomalous Ward Identities and other important properties of QCD. We formulate the physics of topological fluctuations (described by the Veneziano ghost in our framework) in such a way that the relevant technique can be easily generalized for the case of accelerating frame. Section 4 is devoted to analysis of the Veneziano ghost in the accelerating frame. We shall explicitly compute the Bogolubov's coefficients to demonstrate that the Veneziano ghost contribution to energy (being identically zero in Minkowski space) does not vanish anymore in accelerating frame, in the Rindler space. Therefore we identify the local P and CP violation in QCD as a result of fluctuations of the vacuum 0 −+ ghost field in the accelerating frame. We conclude with section 5 where we list the possible tests discriminating this framework from the previously suggested mechanisms. The readers interested in applications only may skip sections 2,3,4 and immediately jump to section 5. Universal hadronization temperature as the Unruh effect As we mentioned above, we adopt the basic logic and philosophy of refs. [36,37,38,39,40,41,42,43]. Essentially, the previously developed picture can be explained in a simplified way as follows: the incident parton decelerates in an external colour field. The causal horizon emerges as a result of this strong interaction. Quantum tunnelling through the emergent event horizon then produces a thermal final state of partons. The hard part of the problem, the computation of the acceleration "a" is not addressed in the present work. The acceleration must be an universal number (which we assume to be the case), not sensitive to a colour representation of the incident particles. Once the universal acceleration is reached, the produced particles will automatically have a thermal spectrum. Our interpretation of this physics is somewhat different: instead of tunnelling of real particles we rather speak about changes of the ground state. Initially, the ground state was a pure quantum state in Minkowski space. In accelerating frame the Rindler observers (who do not ever have access to the entire space-time as a result of emerging horizon) would see the same ground state as a mixed state (rather than pure state) filled by particles with Planck spectrum. This new interpretation will be quite important when we apply the same technique in section 4 for discussions of the topological vacuum fluctuations in accelerating frame because there will be no real asymptotic states which would correspond to those topological vacuum fluctuations. Indeed, as we argue below, the Veneziano ghost does not contribute to absorptive parts of any correlation functions, but only to the real parts. According to our logic these topological vacuum fluctuations in accelerating frame will serve as a P and CP odd background for physical fluctuations of quarks and gluons. As we argue below, these topological vacuum fluctuations will be eventually responsible for the P and CP violating correlations observed at RHIC. As we mentioned above, the computation of the acceleration parameter "a" is a hard problem of strongly interacting gauge theory which is not addressed in the present work. Instead, we adopt the entire framework and treat the acceleration "a" as a free parameter of the theory. This parameter, in principle, could have had any value. In nature "a" is not really a free parameter, but expressed in terms of Λ QCD as eq. (1.3) states. However, to simplify things, we will be working in the limit 1 GeV a m q where our arguments can be made precise, though numerically a 1 GeV as estimate (1.3) shows. Also: we have nothing new to add to the previous arguments [36,37,38,39,40,41,42,43] suggesting that the acceleration (1.3) and the corresponding temperature (1.2) are universal for different processes. Rather we take a given accelerating parameter assuming a 1 GeV and study the changes in the ground state (in comparison with Minkowski vacuum state) which occur due to the acceleration. In our analysis in what follows we consider the radiation of a single massless scalar field to demonstrate the most important features of the accelerating frame; generalization to vector and spinor fields in the Rindler space is also known, but shall not be discussed in the present work. Rindler space We follow notations [35] in our analysis and separate the space time into four quadrants F (future), P (past), L (left wedge) and R (right wedge). We will choose the origin such that these regions are defined by t > \|x\|, t < −\|x\|, x < −\|t\| and x > \|t\| respectively. While no single region contains a Cauchy surface, the union of the left and right regions L and R plus the origin does contain many Cauchy surfaces, for example t = 0. We will write the Minkowski metric with the sign convention ds 2 = dt 2 − dx 2 − dy 2 − dz 2 ,(2.1) In the quadrant R, called the right Rindler wedge, one may define the coordinates (ξ R , η R , y, z) via the transformations t = e aξ R a sinh aη R , x = e aξ R a cosh aη R (2.2) where a is a dimensional constant. We may define coordinates (ξ L , η L , y, z) in the left Rindler wedge L in a similar way with the signs of both t and x reversed [35]. In these new coordinates the metric is presented as ds 2 = e 2aξ (dη 2 − dξ 2 ) − dy 2 − dz 2 . (2.3) Without loosing any generalities we ignore in what follows a trivial dependence on y, z coordinates. It is important to emphasize that the coordinates (η R , ξ R ) cover only a quadrant of Minkowski space, namely the wedge x > \|t\|. Lines of constant ξ are hyperbolae Uniformly accelerated observers will be referred to as Rindler observers. It is important to emphasize that L and R regions are separated by the event horizons such that no events in L can be witnessed in R and vice versa. In different words, regions L and R represent two causally disjoint universes. Left wedge L with x < \|t\| is obtained by changing the signs in eq. (2.2). The sign reversals in L mean that increasing t corresponds to decreasing η which implies that time-like Killing vector being +∂ η in R becomes −∂ η in L in contrast with Minkowski space where ∂ t is time-like Killing vector in entire space. This feature plays an important role in selection of positive frequency modes in L and R wedges as discussed below. In terms of these variables the picture of high energy collision (in very simplified way) can be represented as follows (detail derivations and explanations are presented below in next subsection). Two energetic particles approach the interaction region (x = t = 0) along the light cone from x = t = −∞ and x = −t = ∞. When the colliding particles have non-vanishing space-like transverse momenta k 2 = −k 2 ⊥ their world lines are located off the light cone as described in [36]. Due to the interaction the initial particles experience acceleration (2.5) which we assume to be a constant "a" to simplify our computations. In this case one can switch to the Rindler frame where the question is formulated as follows: how is the initial ground state expressed as a Fock state in the Rindler frame? As we shall see below it will not be a ground state any more in the accelerating frame. Rather it will be a superposition of excited states which include both: the L and R components separated by the horizon. We shall see that the corresponding excitations have the thermal spectrum as the direct consequence of the Bogolubov's coefficients' properties. It is a different way to explain the same physics which was described in refs. [36,37,38,39,40,41,42,43] as a tunnelling through the event horizon. Quantum Fields in Rindler space As we mentioned above, the dynamics along y, z directions is trivial, and we ignore it in what follows to simplify the notations. The wave equation of a free massless field φ(t, x) possesses standard orthonormal mode solutions u k = 1 √ 4πω e −iωt+ikx . (2.7) such that we can expand it in terms of complete orthonormal basis u k (t, x) φ(t, x) = k b k u k (t, x) + b † k u * k (t, x) . (2.8) The commutation relations take the form, [b k , b k ] = 0 , [b † k , b † k ] = 0 , [b k , b † k ] = δ kk ,(2.9) while the ground state in Minkowski space \|0 M is defined as usual b k \|0 M = 0 , ∀k . (2.10) The number operator N and the Hamiltonian H for φ field in these notations have the standard form N = k b † k b k , H = k ω k b † k b k ,(2.11) where normal ordering is implied (but not explicitly shown) in all formulae presented below, such that the ground state satisfies the standard conditions 0 M \|H\|0 M = 0 , 0 M \|N\|0 M = 0 . (2.12) We want to describe the ground state defined by eq. (2.10) in terms of the Rindler coordinates. For the metric (2.3) the corresponding modes are: R u k = 1 √ 4πω e ikξ R −iωη R in R, R u k = 0 in L (2.13) L u k = 1 √ 4πω e ikξ L +iωη L in L, L u k = 0 in R (2. 14) The set (2.13) is complete in region R, while (2.14) is complete in L, but neither is complete in on all of Minkowski space. However, both sets together are complete. The sign difference corresponds to the fact that a right moving wave in R moves towards increasing value of ξ, while in L it moves toward decreasing value of ξ. In any case, these modes are positive frequency modes with respect to the time-like Killing vector +∂ η in R and −∂ η in L. The fact that (2.13) and (2.14) have the same functional form as (2.7) is a consequence of the conformal triviality of the system. Thus the Rindler modes (2.13) and (2.14) represent a good basis for quantizing the φ field, as good as the Minkowski basis (2.7). Therefore, one can use modes (2.13) and (2.14) to expand the field φ 16) and similar for L− Rindler wedge operators b L k . The Rindler vacuum state is defined in terms of these operators as follows, φ = k 1 √ 4πω (b L k e ikξ L +iωη L + b L † k e −ikξ L −iωη L + b R k e ikξ R −iωη R + b R † k e −ikξ R +iωη R ),(2.15) where b L k , b R k satisfy the following commutation relations, b R k , b R k = 0 , [b R † k , b R † k ] = 0 , [b R k , b R † k ] = δ kk ,(2.b R k \|0 R = 0 , ∀k . (2.17) We need to compute the corresponding Bogolubov's coefficients which relate two alternative expansions (2.8) and (2.15) in order to answer the question: how is the initial ground state expressed as a Fock state in the Rindler frame? The simplest way to compute these coefficients is to note that although R u k and L u k are not analytic, the two combinations exp ( πω 2a ) R u k + exp (− πω 2a ) L u * −k (2.18) exp (− πω 2a ) R u * −k + exp ( πω 2a ) L u k are analytic and bounded [33]. These modes share the positivity frequency analyticity properties of the Minkowski modes (2.7), therefore, they must also share a common vacuum state, see below precise definition. Therefore, instead of expansion (2.8) with modes (2.7) we can expand φ in terms of (2.18) as φ = k 1 √ 4πω · 1 (e πω/a − e −πω/a ) b 1 k (e πω 2a +ikξ R −iωη R + e −πω 2a +ikξ L −iωη L ) + b 2 k (e πω 2a +ikξ L +iωη L + e −πω 2a +ikξ R +iωη R ) + b 1 † k (e πω 2a −ikξ R +iωη R + e −πω 2a −ikξ L +iωη L ) + b 2 † k (e πω 2a −ikξ L −iωη L + e −πω 2a −ikξ R −iωη R ) , (2.19) where b 1 k , b 2 k satisfy the following commutation relations, b (1,2) k , b (1,2) k = 0 , [b (1,2) † k , b (1,2) † k ] = 0 , [b (1,2) k , b (1,2) † k ] = δ kk . (2.20) The Minkowski vacuum state is determined in terms of these operators as b 1 k \|0 M = 0 , b 2 k \|0 M = 0 , ∀k . (2.21) This equation replaces eq. (2.10) as it defines the same ground state \|0 M because both sets share a common vacuum state as explained above. Matching coefficients in (2.15) with (2.19) one finds the Bogoliubov's coefficients [33,35], b L k = e −πω/2a b 1 † −k + e πω/2a b 2 k e πω/a − e −πω/a b R k = e −πω/2a b 2 † −k + e πω/2a b 1 k e πω/a − e −πω/a . (2.22) Now consider an accelerating Rindler observer at ξ =const. As we mentioned above, such an observer's proper time is proportional to η, see eq. (2.6). The vacuum for this observer is determined by (2.17) as this is the state associated with the positive frequency modes with respect to η. A Rindler observer in R will measure the energy using the Hamiltonian H R and the number operator N R which are given by N R = k b R † k b R k , H R = k ω k b R † k b R k . (2.23) Similar expressions are also valid for a L-Rindler observer. The Hamiltonian H R and the number operator N R in the R-Rindler accelerating frame have the same form as for conventional Minkowski expressions (2.11). However, they are expressed in terms of the different operators which select a different ground state \|0 R as defined by eq. (2.17). It is obvious that the ground state in R-wedge, \|0 R satisfies the standard conditions, 0 R \|H R \|0 R = 0, 0 R \|N R \|0 R = 0, (2.24) as it should. However, if the initial system is prepared as the Minkowski vacuum state \|0 M defined by (2.21) (or what is the same (2.10)) a Rindler observer using the same expression for the number operator (2.23) will observe the following number of particles in mode k, 0 M \|N R \|0 M = 0 M \|b R † k b R k \|0 M = e −πω/a (e πω/a − e −πω/a ) = 1 (e 2πω/a − 1) , (2.25) where we used the Bogolubov's coefficients (2.22) to express b R k in terms of b (1,2) k . This is the central formula of this section and represents nothing but the the conventional Unruh effect [33,35]. In context of high energy collisions the Planck spectrum given by eq. (2.25) was described in refs. [36,37,38,39,40,41,42,43] as a tunnelling through the event horizon. In our description the same physical effect is resulted from restriction of Minkowski vacuum \|0 M to a single Rindler wedge region where it becomes a thermal state (rather than a pure quantum state) with temperature T = a 2π . This structure is precisely resulted from expression of the Minkowski ground state \|0 M in terms of the excited states in L and R regions when the combination b R † k b L † −k which includes the operators from causally disconnected regions L and R, enters formula for \|0 M as can be seen from explicit construction: \|0 M = k 1 (1 − e −2πω/a ) exp e −πω/a b R † k b L † −k 0 R ⊗ 0 L , (2.26) where we take into account that the operators in the L, R basis correspond to the decompositions with support in only one wedge such that the right hand side is represented by the tensor product 0 R ⊗ 0 L . The crucial point in this relation is the fact that the operators from different causally disconnected regions L and R enter the same expression (2.26), and therefore, there is a correlation between causally disconnected regions L and R. However, as discussed in ref. [33], one can not use these correlations to send signals. In context of high energy collisions the picture advocated in refs. [36,37,38,39,40,41,42,43] when two causally disconnected regions are connected as a result of the tunnelling through the event horizon, manifests itself in eq. (2.26) by emergence of the combination b R † k b L † −k when L and R components are simultaneously present in eq. (2.26). The expression (2.26) also shows that the Planck spectrum observed in high energy collisions can be interpreted as a result of preparation of the ground state \|0 M even before collision develops. This interpretation naturally explains the puzzle with rapid "thermalization" observed in all high energy collisions.Many other observed consequences (such as dependence on transverse momenta k 2 ⊥ , saturation scale at low energies and/or peripheral AA collisions, dependence on strange mass quark, and many others) also find their simple explanations in this framework. They have been discussed in those references in great details, and we have nothing new to add to those discussions as the basic logic of our approach and the one advocated in refs. [36,37,38,39,40,41,42,43] is the same as both pictures naturally lead to the Planck spectrum (2.25). Still, we want to add one more comment which my shed some light on mysterious "apparent thermalization" effect resulted from the acceleration. The "apparent thermalization" can be understood as entanglement type behaviour when the Planck spectrum emerges as a result of the description in terms of the density matrix in R region by "tracing out" over the degrees of freedom associated with inaccessible states in L-region. In different words, the Bogolubov transformations (2.22) describe a construction when a total system is divided into two subsystems with the horizon separating them. This is the deep physics reason why the Planck spectrum (2.25) emerges for a subsystem. It is known that a number of nontrivial physics effects (including the entanglement) are described by the common surface separating such two sub-systems, i.e. by the horizon separating L and R Rindler wedges. The particle production in the framework advocated in refs. [36,37,38,39,40,41,42,43] occur exactly from this horizon region, while in our framework it can be interpreted as a result of entanglement. One should also note that typical quark and gluon vacuum fluctuations develop in this accelerating environment characterized by temperature (1.2), not in laboratory frame, and not in center of mass frame. Once acceleration ceases to exist, a detector in laboratory frame will measure a particle distribution according to the temperature (1.2) as the acceleration ends almost instantaneously, and produced particles do not have time to make any adjustments to a new environment. In this respect, it is very similar to measurements of CMB (cosmic micro wave background) radiation as the CMB photons being produced at the last scattering at temperature T 2.7 K nevertheless do not change their properties during the next ∼ 14 billion years. • To conclude this section: we have not derived anything new which was not previously known. However, we presented the "new thermalization" scenario advocated in refs. [36,37,38,39,40,41,42,43] as a result of reconstruction of the vacuum state in accelerating system. This new interpretation will be a crucial element in section 4 when we argue that some very specific topological fluctuations are responsible for violation of local P and CP invariance in QCD observed at RHIC. We would not be able to use the quasiclassical technique developed in refs. [36,37,38,39,40,41,42,43] to discuss corresponding fluctuations as no classical trajectories of real particles propagating in external classical colour fields exist for these type of fluctuations. This is because the relevant topological vacuum fluctuations, as we discuss below, are not related to any absorptive parts of any physical correlation functions. Rather, the topological fluctuations which will be main subject of this paper may contribute only to the real parts of the correlation functions (such as topological susceptibility) see next section. In Minkowski space similar contributions normally treated as the subtraction constants. In the present case of the accelerating frame, the corresponding "subtraction constant" becomes a "subtraction function" which depends on acceleration and which is sensitive to the global properties of the space-time. As we shall see below, the approach developed in the present section is well suited to attack this problem as everything in section 4 will be formulated precisely in appropriate terms of vacuum fluctuations. The θ vacua, U (1) A effective lagrangian and the Veneziano ghost in Minkowski spacetime The main goal of this section is to single out (identify) the fields which describe the topological fluctuations in QCD. It turns out that these relevant fields (in a specific gauge) can be represented as (pseudo)scalar colour-singlet fields such that one can immediately apply the technique developed in previous section 2 to describe the corresponding vacuum fluctuations in accelerating frame. As we shall argue below in section 4 those vacuum topological fluctuations in accelerating frame may be responsible for P and CP violation observed at RHIC. It is quite obvious that those configurations must be related to θ dependence, topological charge density, and other related problems. Therefore, we start this section by reviewing the θ dependence in QCD and the standard resolution of U (1) A problem [49,50,51,52]. We follow [53] to identify the relevant for the present work degrees of freedom (Veneziano ghost) which saturates topological correlation functions. After integrating the ghost out (as was done in the original paper [50]) one reproduces the η mass, which was the main result of [50]. We keep the Veneziano ghost explicitly as it can not be integrated out in accelerating frame. Moreover, it will play a central role in our following discussions when we consider the accelerating frame relevant for description of high energy collisions. As we shall argue in section 4 the corresponding topological fluctuations in accelerating frame might be the pivotal source of P and CP violating fluctuations observed at RHIC. The Lagrangian and the ghost The starting point for our analysis will be the Lagrangian as proposed in [50,52] in large N c 1 limit whose general form reads L = L 0 + 1 2 ∂ µ η ∂ µ η + N c bf 2 η q 2 − θ − η f η q + N f m q \| qq \| cos η f η + g.f. , q = g 2 64π 2 µνρσ G aµν G aρσ ≡ 1 4 µνρσ G µνρσ , (3.1) G µνρσ ≡ ∂ µ A νρσ − ∂ σ A µνρ + ∂ ρ A σµν − ∂ ν A ρσµ , A νρσ ≡ g 2 96π 2 A a ν ↔ ∂ ρ A a σ − A a ρ ↔ ∂ ν A a σ − A a ν ↔ ∂ σ A a ρ + 2gC abc A a ν A b ρ A c σ . where we explicitly keep only relevant for the present work degrees of freedom, such as η and the topological density q, while all others (including π, K, η) are assumed to be in L 0 , and shall not be mentioned in this paper. In this Lagrangian g.f. means gauge fixing term for three-form A µνρ , and the coefficient b ∼ m 2 η is fixed by the Witten-Veneziano relation for the topological susceptibility in pure gluodynamics. The fields A a µ are the usual N 2 c − 1 gauge potentials for chiral QCD and C abc the SU (N c ) structure constants. The constant f η f π is the η decay constant, while m q is the quark mass, and qq is the chiral condensate. The three-form A νρσ is an abelian totally antisymmetric gauge field which, under colour gauge transformations Λ a behaves as A νρσ → A νρσ + ∂ ν Λ ρσ − ∂ ρ Λ νσ − ∂ σ Λ ρν , Λ ρσ ∝ A a ρ ∂ σ Λ a − A a σ ∂ ρ Λ a ,(3.2) such that the four-form G µνρσ is gauge invariant as it should. The term proportional to θ is the usual θ-term of QCD and appears in conjunction with the η field in the correct combination as dictated by the Ward Identities (WI). The constant b is a positive constant which determines the topological susceptibility in pure gauge theory, i d 4 x T {q(x), q(0)} Y M = − f 2 π 8N c b . (3.3) The constant b saturates the topological susceptibility and has a wrong sign (in comparison with contribution from any real physical states), the property which motivated the term "Veneziano ghost". Indeed, a physical state of mass m G , momentum k → 0 and coupling 0\|q\|G = c G contributes to the topological susceptibility with the sign which is opposite to (3.3), i lim k→0 d 4 xe ikx T {q(x), q(0)} ∼ i lim k→0 0\|q\|G i k 2 − m 2 G G\|q\|0 \|c G \| 2 m 2 G ≥ 0. (3.4) However, the positive sign for b (and negative sign for the topological susceptibility (3.3)) is what is required to extract the physical mass for the η meson, m 2 η b/2N c = 0, see the original reference [50] for a thorough discussion. One can interpret the field A µνρ as a collective mode of the original gluon fields, which in the infrared leads to a pole in the unphysical subspace and provides a finite contribution with a wrong sign to the topological susceptibility (3.3). We know about the existence of this very special degree of freedom and its properties from the resolution of the famous U (1) A problem: integrating out the q field (as shown below) provides the mass for the η meson. Of course b = 0 to any order in perturbation theory because q(x) is a total divergence q = ∂ µ K µ . However, as we learnt from [49,51], b = 0 due to the non-perturbative infrared physics; in fact, in the chiral limit m 2 η ∼ b. One can integrate out the scalar field q since there is no kinetic term associated to it. This is indeed the procedure followed by Di Vecchia and Veneziano in their original paper [50], the outcome of which, as follows from (3.1), is L = L 0 + 1 2 ∂ µ η ∂ µ η − bf 2 η 4N c θ − η f η 2 + N f m q \| qq \| cos η f η , (3.5) where all the dependence on the three-form A νρσ has disappeared. This formula explicitly shows that η receives a non-vanishing mass in the chiral limit, m 2 η b/2N c = 0 due to the non-zero magnitude of the coefficient b, which enters (3.1) and (3.3). This formula also reproduces the notorious Witten-Veneziano relation for the topological susceptibility in pure gluodynamics if one substitutes b = 2N c m 2 η into the expression (3.3) for the topological susceptibility. As we mentioned above, we want to keep the ghost hidden in A µνρ explicitly. We shall now choose the Lorenz-like gauge (∂ ρ A µνρ ) (∂ µ K ν − ∂ ν K µ ) = 0 , K µ ≡ µνρσ A νρσ , q = ∂ µ K µ ,(3.6) in which we will carry out our manipulations. It is the same gauge which was discussed in the original paper [50]. We choose to work only with the longitudinal part of the K µ field because only this longitudinal part determines the topological density q = ∂ µ K µ , and eventually leads to a non-vanishing contribution to the topological susceptibility (3.3). Therefore, we write the longitudinal part of K µ as K µ ≡ ∂ µ Φ ,(3.7) such that the expression for the topological density takes the form q = ∂ µ K µ = 2Φ ,(3.8) where Φ is a new scalar field of mass dimension 2. We notice that the gauge condition (3.6) is automatically satisfied with our definition (3.7). Now our Lagrangian (3.1) can be expressed in terms of the Φ field as follows L = L 0 + 1 2 ∂ µ η ∂ µ η + N f m q \| qq \| cos η f η (3.9) + 1 2m 2 η f 2 η Φ22Φ + η f η 2Φ − θ2Φ , where we plugged in the coefficient b → 2N c m 2 η as the Witten-Veneziano relation requires. If we integrate out the 2Φ field we return to the expression (3.5) which describes the physical massive η field alone. As usual, the presence of 4-th order operator Φ22Φ is a signal that the ghost is present in the system and may be quite dangerous. However, we know from the original form (3.1) that the system is unitary, well defined etc, in different words, it does not present any problem associated with the ghost. One can redefine all fields in such a way that the final Lagrangian can be expressed as follows [53]: L = 1 2 ∂ µφ ∂ µφ + 1 2 ∂ µ φ 2 ∂ µ φ 2 − 1 2 ∂ µ φ 1 ∂ µ φ 1 (3.10) − 1 2 m 2 η φ 2 + N f m q \| qq \| cos φ + φ 2 − φ 1 f η , where all fields have now canonical dimension one in four dimensions. We claim that the Lagrangian (3.10) is that part of QCD which describes long distance physics in our context. Notice that (3.10) is exactly identical to that proposed by Kogut and Susskind (KS) in [55] for the 2d Schwinger model, see also [57] in given context. The unitarity and other important properties of QFT are satisfied in our 4d system (3.10) in the same way as they are satisfied in [55] for the 2d Schwinger model as will be reviewed in the next subsection 3.2. The Veneziano ghost in QCD is represented in our notations by the φ 1 field in (3.10) and it is always accompanied by its companion, the massless field φ 2 . These two fields cancel each other in every gauge invariant matrix element once the auxiliary (similar to Gupta-Bleuler [58,59]) conditions on the physical Hilbert space are imposed, see below. Using the explicit expression for the Green's function and the expression for the topological density q = ∂ µ K µ = 2Φ after simple algebraic manipulations one can represent the topological susceptibility for QCD in model (3.1) in the chiral limit m q → 0 in the following way, χ QCD ≡ i d 4 x T {q(x), q(0)} QCD = − f 2 η m 2 η 4 · d 4 x δ 4 (x) − m 2 η D c (m η x) (3.11) where D c (m η x) is the Green's function of a free massive particle with standard normalization d 4 xm 2 η D c (m η x) = 1. In this expression the δ 4 (x) represents the ghost contribution with the required "wrong" sign while the term proportional to D c (m η x) represents the η contribution. The ghost's contribution can be also thought as the Witten's contact Figure 1: The density of the topological susceptibility χ(r) ∼ q(r), q(0) as function of separation r such that χ QCD ≡ drχ(r), adapted from [60]. Plot explicitly shows the presence of the contact term with the "wrong sign" (narrow peak around r 0) represented by the Veneziano ghost in our framework. term not related to any propagating degrees of freedom. The topological susceptibility χ QCD (m q = 0) = 0 vanishes in the chiral limit as a result of exact cancellation of two terms entering (3.11) in complete accordance with WI. When m q = 0 the cancellation is not complete and χ QCD m q qq . One should emphasize that the presence of the contact term (described by the Veneziano ghost in our framework) ∼ δ 4 (x) in eq. (3.11) is well established and well understood phenomenon in QCD. In particular, it has been studied on the lattice, see e.g. [60] and references therein 2 where a narrow peak around r 0 and a smooth behaviour in extended region of r ∼ fm with the opposite signs have been seen as a result of numerical computations. We reproduce Fig.1 for illustration purposes adapted from ref. [60] where these crucial elements are explicitly present on the plot. As we mentioned previously, the main goal of the present paper is to study precisely this (unphysical and non-propagating) effective degree of freedom leading to ∼ δ 4 (x) in eq. (3.11) (and represented by a narrow peak at r ∼ 0 on Fig. 1) when we accelerate our system. As we discuss in great details in [54] the ghost does not contribute to absorptive parts of any correlation functions (after all, it is not an asymptotic degree of freedom). However, it does contribute to the real part as plot on Fig.1 explicitly shows. As we shall argue below, the topological nature of the ghost and its 0 −+ quantum numbers may play a crucial role in understanding of local P and CP violation in QCD observed at RHIC [1,2,3,4,5]. Before we proceed with our description of the Veneziano ghost in accelerating frame we would like to demonstrate that the Veneziano ghost is harmless (e.g. it does not violate unitarity) in spite of its negative sign in the Lagrangian (3.10). Unitarity and the ghost We follow Kogut and Susskind construction [55] in order to demonstrate the unitarity of our system when the Veneziano ghost φ 1 and its partner φ 2 explicitly present in the Lagrangian (3.10). The cosine interaction term (3.10) includes vertices between the ghost and the other two scalar fields, but it can in fact be shown in complete analogy with [55] that, once appropriate auxiliary (similar to Gupta-Bleuler [58,59]) conditions on the physical Hilbert space are imposed, the unphysical degrees of freedom φ 1 and φ 2 drop out of every gauge-invariant matrix element, leaving the theory well defined, i.e., unitary and without negative normed physical states, just as in the Lorentz invariant quantization of electromagnetism. Specifically, this is achieved by demanding that the positive frequency part of the free massless combination (φ 2 − φ 1 ) annihilates the physical Hilbert space: (φ 2 − φ 1 ) (+) \|H phys = 0 . (3.12) With this additional requirement the quantum theory built on the Lagrangian (3.10) is well defined in any respect, and the physical sector of the theory exactly coincides with (3.5) which was obtained by a trivial integrating out procedure [50]. In particular, one can explicitly check that the expectation value for any physical state in fact vanishes as a result of the subsidiary condition (3.12): H phys \|H\|H phys = 0 . (3.13) In different words, all these "dangerous" states which can produce arbitrary negative energy do not belong to the physical subspace defined by eq. (3.12). Therefore, the main conclusion is that the description in terms of the ghost is equivalent to well-known standard procedure of integrating out the ghost field leading to well-known expression (3.5) for the effective low energy lagrangian. In the next section we want to study the dynamics of the ghost fields in the accelerating frame. The corresponding technique for a non-interacting (pseudo)scalar field has been previously developed and presented in section 2. In what follows we consider the chiral limit m q → 0 such that the lagrangian describing the ghost field and its partner (3.10) is precisely represented by the combination of two massless non-interacting fields 3 in which case the corresponding Bogolubov's coefficients have been previously computed (2.22). Numerically, the chiral limit in fact implies that the acceleration parameter a m q . In addition, as we mentioned previously, we want to keep "a" as a free parameter of the theory in spite of the fact that in nature it is fixed by the string tension (1.3). Therefore, in all discussions below we consider the following hierarchy of scales 1 GeV a m q (3.14) in order to separate the effects topological fluctuations due to the acceleration "a" from conventional QCD fluctuations with typical scales ∼ GeV. P and CP violating fluctuations in accelerating frame Our goal here is to understand the behaviour of the system (3.10) in the chiral limit m q = 0 in accelerating frame. These fields have quantum numbers 0 −+ and their long wave fluctuations may produce observable P and CP odd effects as discussed below. There are many other fluctuations, of course, in the system due to conventional quarks and gluons. However, the ghost field φ 1 and its partner φ 2 are unique degrees of freedom in many respects, and their fluctuations, hopefully, can be separated from all other vacuum fluctuations. The main point is as follows. As we discussed above, the Bogolubov's coefficients (2.22) have the property that they are exponentially suppressed for ω a. Therefore, the typical wave lengths of fluctuations related to acceleration are λ ≥ a −1 . If a 1 GeV as estimation (1.3) suggests, it would be very difficult to disentangle these fluctuations from conventional fluctuations of quarks and gluons with the same typical scale λ GeV −1 . However, we work in the limit a 1 GeV where such separation (at least theoretically) is a possibility 4 . In case a 1 GeV all colour fields will still fluctuate with typical λ GeV −1 as a result of confinement which implies that all fluctuations are effectively gapped. It is in a drastic contrast with fluctuations of colourless ghost field φ 1 and its partner φ 2 . The ghost remains massless even when interactions are present, as its pole (in unphysical Hilbert space) is topologically protected as discussed above in section 3. Indeed, non vanishing contribution to the topological susceptibility (3.3) constructed from the operators which are total derivatives q = ∂ µ K µ may only come from (unphysical) massless pole. In different words, the typical wave lengths of φ 1 and φ 2 fields related to acceleration are λ ≥ a −1 for arbitrary small a. Based on this comment, we ignore in this section the conventional fluctuations of quarks and gluons with λ GeV −1 ; we return to them in section 5 when we produce some numerical estimates by comparing the energetics of the ghost fields φ 1 , φ 2 and conventional quarks and gluons. Veneziano ghost in the accelerating frame at 1 GeV a m q As we mentioned above, in the region 1 GeV a m q the problem is reduced to free massless fields φ 1 and φ 2 with GB like constraint. Therefore, one can explicitly use the formalism developed earlier in section 2. In particular, one can compute the Bogolubov's coefficients (2.22), construct the Hamiltonian and the number operator for the ghost field φ 1 and its partner φ 2 in accelerating frame as it was done previously, see eq.(2.23). The next step is to compute the corresponding expectation values when the system is being prepared as Minkowski vacuum \|0 M evolves in the accelerating background. Technically it is exactly the same problem of our previous computations of the Planck spectrum (2.25) detected by a Rindler observer in a model of a single massless particle. We start with expansion of the ghost field φ 1 and its partner φ 2 using the modes (2.13) and (2.14) as we have done previously (2.15), φ 1 = k 1 √ 4πω (a L k e ikξ L +iωη L + a L † k e −ikξ L −iωη L + a R k e ikξ R −iωη R + a R † k e −ikξ R +iωη R ) (4.1) φ 2 = k 1 √ 4πω (b L k e ikξ L +iωη L + b L † k e −ikξ L −iωη L + b R k e ikξ R −iωη R + b R † k e −ikξ R +iωη R ). The Rindler vacuum state is defined as usual, a R k \|0 R = 0 , b R k \|0 R = 0 , ∀k . (4.2) In Minkowski space we can proceed exactly along the same line as we have done in section 2. Namely, instead of expansion with modes (2.7) we can expand φ 1 and φ 2 in terms of (2.18) as follows: φ 1 = k 1 √ 4πω · 1 (e πω/a − e −πω/a ) a 1 k (e πω 2a +ikξ R −iωη R + e −πω 2a +ikξ L −iωη L ) + a 2 k (e πω 2a +ikξ L +iωη L + e −πω 2a +ikξ R +iωη R ) + a 1 † k (e πω 2a −ikξ R +iωη R + e −πω 2a −ikξ L +iωη L ) + a 2 † k (e πω 2a −ikξ L −iωη L + e −πω 2a −ikξ R −iωη R ) , φ 2 = k 1 √ 4πω · 1 (e πω/a − e −πω/a ) b 1 k (e πω 2a +ikξ R −iωη R + e −πω 2a +ikξ L −iωη L ) + b 2 k (e πω 2a +ikξ L +iωη L + e −πω 2a +ikξ R +iωη R ) + b 1 † k (e πω 2a −ikξ R +iωη R + e −πω 2a −ikξ L +iωη L ) + b 2 † k (e πω 2a −ikξ L −iωη L + e −πω 2a −ikξ R −iωη R ) ,(4.3) where b 1 k , b 2 k satisfy the following commutation relations, b (1,2) k , b (1,2) k = 0 , [b (1,2) † k , b (1,2) † k ] = 0 , [b (1,2) k , b (1,2) † k ] = δ kk ,(4.4) whereas a 1 k , a 2 k for the ghost field φ 1 satisfy a (1,2) k , a (1,2) k = 0 , [a (1,2) † k , a (1,2) † k ] = 0 , [a (1,2) k , a (1,2) † k ] = −δ kk (4.5) where again the sign minus appears in these commutation relations. The Minkowski vacuum state is determined as usual a 1 k \|0 = 0 , a 2 k \|0 = 0 , b 1 k \|0 = 0 , b 2 k \|0 = 0 , ∀k . (4.6) The Bogolubov's coefficients for φ 1 and φ 2 fields relating the description in Minkowski and Rindler spaces can be computed exactly in the same way as it was done before, see eq. . Now consider an accelerating Rindler observer at ξ =const. As we discussed previously, such an observer's proper time is proportional to η. The vacuum for this observer is determined by (4.2) as this is the state associated with the positive frequency modes with respect to η. A Rindler observer in R wedge will measure the energy and particle density using the Hamiltonian H R and density operator N R which are given by (a similar formula applies for L wedge as well), H R = k ω k b R † k b R k − a R † k a R k , N R = k b R † k b R k − a R † k a R k . (4.8) The subsidiary condition (3.12) defines the physical subspace for accelerating Rindler observer a R k − b R k H R phys = 0 ,(4.9) such that the exact cancellation between φ 1 and φ 2 fields holds for any physical state defined by eq. (4.9), i.e. H R phys \|H R \|H R phys = 0 (4.10) as it should. However, if the system is prepared as the Minkowski vacuum state \|0 M defined by eq.(4.6) a Rindler observer using the same expressions for the number operator and Hamiltonian (4.8) will observe the following amount of energy in mode k, 0\|ω k b R † k b R k − a R † k a R k \|0 = 2ωe −πω/a (e πω/a − e −πω/a ) = 2ω (e 2πω/a − 1) , (4.11) where we used the Bogolubov's coefficients (4.7) to express a R k , b R k in terms of a (1,2) k , b (1,2) k . This formula (up to a numerical coefficient) has been reproduced in ref. [56] by using a different technique. This is the central result of this section and is a direct analog of Planck spectrum given by eq. (2.25) discussed previously for the conventional massless particle with the only difference of factor 2 in front which is result of extra degeneracy: we have two degrees of freedom φ 1 and φ 2 instead of one scalar field φ from section 2. The crucial point here is as follows. No cancellation between the ghost φ 1 and its partner φ 2 could occur in the expectation value (4.11), in net contrast with eq. (4.10). Technically, a "non-cancellation" of unphysical degrees of freedom (4.11) in accelerating frame is a result of opposite sign in commutator (4.5) along with negative sign in Hamiltonian (4.8). We will discuss this important point in great details in the next subsection in nontechnical, intuitive way. However, we want to emphasize that this result (4.11) should not be interpreted as actual emission of ghost modes, as they are not the asymptotic states in Minkowski spacetime in the remote past and future, and therefore they can not propagate to infinity in contrast with conventional Unruh effect, see appendix A for details. Rather, one should interpret (4.11) as an additional time dependent contribution to the vacuum energy in accelerating background in comparison with Minkowski space-time. This extra energy is entirely ascribable to the presence of the unphysical (in Minkowski space) degrees of freedom. We interpret the extra contribution to the energy observed by the Rindler observer as a result of formation of a specific configuration, the "ghost condensate" [54], rather than a presence of "free particles" prepared in a specific mixed state which can be detected. This extra term should be treated as a result of very unique vacuum fluctuations, not related to any absorptive contributions. The observational effects of this extra vacuum contribution will be discussed in the next section 5. Interpretation • As explained above the nature of the effect (4.11) is the same as the conventional Unruh effect [33] discussed in section 2 when the Minkowski vacuum \|0 M is restricted to the Rindler wedge with no access to the entire space time. A pure quantum state \|0 M becomes a thermo mixed state as a result of this quantum restriction. The result (4.11), by definition, implies that the states which were unphysical (in Minkowski space) lead to physically observable phenomena, though it can not be interpreted in terms of pure states of individual particles, see Appendix A for details. The effect is obviously IR in nature, and it is basically due to the presence of the horizon which itself dynamically emerges as a result of strong interactions as advocated in refs. [36,37,38,39,40,41,42,43]. • One can explicitly see why the cancellation of unphysical degrees of freedom φ 1 and φ 2 in Minkowski space fail to hold for the accelerating Rindler observer (4.11). The selection of the physical Hilbert subspace (3.12) is based on the properties of the operator which selects positive -frequency modes with respect to Minkowski time t. At the same time the Rindler observer selects the physical Hilbert space (4.9) by using positive -frequency modes with respect to observer's proper time η. These two sets are obviously not equivalent, as e.g. they represent a mixture of positive and negative frequencies modes defined in R-and L-Rindler wedges. At the same time, the Rindler observers do not ever have access to the entire space time. Therefore, from the Rindler's view point the cancellation in Minkowski space can be only achieved if one uses both sets (L and R). Of course, using the both sets would contradict to the basic principles as the R-Rindler observer does not have access to the L wedge even for arbitrary small acceleration parameter a. • This is not the first time when unphysical (in Minkowski space) ghost contributes to a physically observable quantity. The first example is the famous resolution of the U (1) A problem in QCD, see section 3. As long as we work in Minkowski spacetime the two constructions (based on the Veneziano ghost [49,50] and on the Witten's subtraction constant [51]) are perfectly equivalent as the subsidiary condition (3.12) ensures that the ghost degrees of freedom are decoupled from the physical Hilbert subspace, leaving both schemes with the identical physical spectrum. In different words, these unphysical degrees of freedom do not contribute to absorptive parts, but only to the real parts of the correlation functions. In Minkowski space such a contribution is normally represented by a "subtraction constant", while in a time dependent background this subtraction constant becomes a "subtraction function" which depends on acceleration. We advocate the ghostbased technique to account for this physics because the corresponding description can be easily generalized for accelerating background, while a similar generalization (without the ghost, but with explicit accounting for the infrared behaviour at the boundaries/horizon) is unknown and likely to be much more technically complicated. In different words, the description in terms of the ghost is a matter of convenience in order to effectively account for the boundary/horizon effects in topologically nontrivial sectors of the theory. • One should emphasize that the Veneziano ghost we are dealing with in this paper is very different from all other ghosts, including the conventional Fadeev Popov ghosts. The Veneziano ghost is not an asymptotic state, it does not propagate, it does not contribute to the absorptive parts of the correlation functions, as explained in Appendix A, though, it does fluctuate and does contribute to the energy through the boundary/horizon conditions (similar to the Casimir effect). The spectrum of these fluctuations is very different from conventional Fadeev Popov ghosts (when momenta could be arbitrary large in order to cancel the corresponding unphysical polarizations of the gauge fields). A typical frequency of the Veneziano ghost is determined by the horizon scale ω ∼ a, while the higher frequency modes ω a are exponentially suppressed. • The unique feature of the Veneziano ghosts is due to its close connection to the topological properties of the theory. Indeed, the topological density operator q is explicitly expressed in terms of the Veneziano ghost φ 1 as follows, q ∼ 2Φ ∼ 2φ − 2φ 1 such that the contact term (representing the real, not absorptive part of the topological susceptibility) ∼ δ 4 (x) in eq. (3.11) is saturated by the ghost. One should also note that the appearance of the ghost degree of freedom in the formalism can be traced from the induced q 2 term (3.1) which contains Φ2 2 Φ operator (3.9). As is known the 2 2 operator can be always re-written in terms of a degree of freedom with a negative kinetic term. This explains the origin and uniqueness of the Veneziano ghost and its relation to topological features of the theory. A number of very nontrivial properties of this ghost which are discussed in this paper are intimately related to its topological nature. Observations of the P and CP odd fluctuations at RHIC The goal of this section is to apply our previous formal analysis to the very concrete subject: we want to interpret the recent RHIC experimental results [1,2,3,4,5] as violation of local P and CP symmetries in QCD. The key point of all our previous discussions can be formulated in one line: QCD supports the topologically nontrivial unique vacuum fluctuations (Veneziano ghost) in the accelerating system. The fluctuations are IR in nature, sensitive to the horizon scale λ ≥ a −1 for arbitrary small a, they do not propagate, do not contribute to the absorptive parts of the correlation functions, but they do contribute to the real portion of the correlation functions. Their IR nature is protected by topology: they remain gapless even in the presence of the strong confined forces. Such a property is in huge contrast with conventional fast quark and gluon fluctuations which have a sharp cutoff at wavelengths λ ∼ Λ −1 QCD . These topological fluctuations have 0 −+ quantum numbers, and in all respects very similar to the induced, slowly fluctuating θ ind discussed in section 1.1 withθ ind ∼ a. We know about the existence of the Veneziano ghost from the resolution of the U (1) A problem when it saturates the contact term in the topological susceptibility. In the accelerating frame this contact term becomes a "subtraction function" and the corresponding topological fluctuations lead yet to another observable phenomena as we shall argue below. The basic picture In what follows we assume that 1 GeV a m q such that we can separate the topological fluctuations with very large wave lengths λ ≥ a −1 which carry 0 −+ quantum numbers from conventional fluctuations of quarks and gluons with typical λ ∼ 1 GeV −1 . We also neglect the interacting term ∼ m q in eq. (3.10) such that our consideration of free fields in accelerating frame leading to (4.11) is justified. Our basic picture in this regime can be formulated as follows. The conventional quark and gluon fluctuations with typical λ ∼ 1 GeV −1 are propagating in the environment of very slow topological fluctuations with wave lengths λ ≥ a −1 . These slow topological fluctuations can be thought as P and CP odd environment for the fast conventional fluctuations with typical λ ∼ 1 GeV −1 . The fast conventional fluctuations are distributed according to the Planck formula as discussed in section 2 and described by eq. (2.25). While this formula was derived for a massless scalar particle for illustration purposes, it is known that a similar thermal distribution is expected to hold for vector and spinor fields as well. This picture for the fast fluctuations is equivalent to the "new thermalization" scenario advocated in refs. [36,37,38,39,40,41,42,43] as it produces hadrons distributed according to the thermal law determined by temperature (1.2). The only new element of this work is the observation that these conventional fast fluctuations with typical λ ∼ 1 GeV −1 are propagating in the P and CP odd environment described by new type of topological fluctuations with very large wave lengths λ ≥ a −1 . We emphasize that the soft topological fluctuations can not propagate to infinity by themselves as they are not asymptotic states; rather they produce the P and CP odd environment for conventional fast fluctuations which eventually will be observed as the hadrons produced in this odd environment. Our main conjecture is that the P and CP odd fluctuations observed at RHIC [1,2,3,4,5] is a direct consequence of this odd environment. In next subsection 5.2 we present a number of qualitative consequences of the entire framework supporting this basic picture. Before we do so, we want to compare the energetics of slow topological fluctuations with conventional fast fluctuations of quarks and gluons. Our starting formula is the Planck spectrum for the Veneziano ghost and its partner (4.11) valid for a m q . Number density of the P odd domains with size λ 2π ω is given by dN ω = d 3 k (2π) 3 2 (e 2πω/a − 1) ,(5.1) while the total contribution to the energy associated with these soft fluctuations is E ghost d 3 k (2π) 3 2ω (e 2πω/a − 1) = π 2 15 aE q+g π 2 15 a 2π 4 (N 2 c − 1) + 7N c N f 4 . (5.3) Therefore, the relative energy associated with slow ghost fluctuations with 0 +− quantum numbers in comparison with conventional fast fluctuations of quarks and gluons is estimated to be κ ≡ E ghost E q+g ∼ 1 (N 2 c − 1) + 7NcN f 4 ,(5.4) which is numerically ∼ 0.05. The effect is parametrically small at large N c and proportional ∼ 1/N 2 c which is a typical suppression for any phenomena related to topological fluctuations. The effects related to the ghost obviously vanish at a = 0 as eq. (5.2) states. This limit corresponds to the transition to Minkowski space when the Veneziano ghost is decoupled from physical Hilbert space (3.12). The factor κ essentially counts number of fluctuating degrees of freedom which lead to the P and CP odd environment. However, these degrees of freedom are not the asymptotic states, and they do not propagate to infinity as explained above, and they do not contribute to the absorptive parts of any correlation functions. The information about the P and CP odd environment must be transferred to the conventional propagating degrees of freedom which can be observed and analysed. In the ideal world with a 1 GeV all strong interactions can be treated as fast fluctuations in slow varying background of the Veneziano ghost φ 1 and its parter φ 2 at nonzero acceleration a 1 GeV. As we mentioned above, such background can be thought as slowly varying effective θ ind = 0. The spectral properties of these fluctuations are determined by eq. (5.1) while its energetics is determined by eq. (5.2). Therefore, in the limit a 1 GeV we can apply our previous knowledge about physical properties of hadrons in θ ind = 0 background. In particular, all previous estimates on P and CP odd effects reviewed in section 1.1 (including the charge separation effect as a result of anomalous interaction of slow varying θ ind = 0 with electromagnetic field) remain valid in this limit when P odd domain is much larger in size than conventional QCD fluctuations. Therefore, we shall not elaborate on this point in the present work. Instead, we concentrate below on immediate qualitative consequences of the picture developed in this work when acceleration "a" being the key parameter of the system is parametrically small. Observational consequences. The universality. •1. First immediate consequence of the developed picture is the presence of P and CP odd fluctuations in any accelerating system with a = 0 including all energetic e + e − , pp and pp interactions when the thermal aspects corresponding to the universal temperature around T H ∼ (150 − 200) MeV have been already observed. According to the entire logic of our framework the presence the thermal aspects in observations is resulted from the acceleration "a" when the QCD vacuum structure is completely reconstructed. The corresponding reconstruction, among many other things, leads to topological fluctuations which play the role of the P and CP odd environment where hadrons are being produced. These topological fluctuations will be developed in all energetic e + e − , pp, pp and heavy ion collisions. However, the heavy ion collisions are unique in comparison with other types of energetic interactions as they allow to study the dependence of these odd effects as function of centrality which, as we argue below, effectively corresponds to variation of the acceleration parameter "a" with centrality. •2. As we reviewed in section 1.2 the acceleration realized in nature for e + e − , pp, pp (not heavy ions) collisions is a universal number given by (1.3) and close to a 1 GeV, rather than a free parameter "a". A typical domain size where P and CP odd fluctuations will be developed is determined by eq. (5.1). The fluctuations will be order of λ 2π/a ∼ fm for a 1 GeV which is about the size of conventional fluctuations of quarks and gluons. In such circumstances the correlations similar the ones studied at RHIC [1,2,3,4,5] are expected to be suppressed for e + e − , pp, pp collisions as the formation of different hadrons most likely will occur in different P odd domains rather than in one large domain. Nevertheless, the effect being suppressed for e + e − , pp, pp collisions, still does not vanish. The intensity of the corresponding correlations for e + e − , pp, pp collisions is predicted to have the same intensity as in heavy ion collisions for the most central events, see below. •3. This conclusion (on suppression of the correlations for e + e − , pp, pp collisions) changes drastically when we consider the heavy ion collisions. In this case it has been argued [40] that the temperature (1.2), and therefore, the acceleration "a" will be reduced for noncentral collisions, which for small angular momentum J can be approximated as follows [40], a(J) a J=0 1 − cJ 2 , c > 0, (5.5) where c is some positive constant. Reduction of the acceleration "a" will increase a typical domain size where P and CP odd fluctuations develop as eq. (5.1) suggests. At the same time, the conventional fluctuations responsible for the formation of hadrons are not much affected by the reduction of "a" as they keep a typical (for Minkowski space) scale ∼ fm determined by confinement forces rather than by acceleration "a". When the size of the P odd domain becomes sufficiently larger than ∼ fm scale the strength of correlations should drastically increase as quite a few particles could be formed in the same large P odd domain. Even a slight reduction of "a" (which corresponds to moving toward the least central collisions), may produce some drastic changes in strength of correlations as dependence on "a" is exponential, see eq. (5.1). Strong dependence on centrality is indeed supported by observations [5], though the acceleration parameter "a" is obviously not identically the same variable as centrality defined in [1,2,3,4,5]. •4. Another immediate consequence of the developed picture is that the correlations should demonstrate the universal behaviour similar to the universality discussed in section 1.2 as the source for the both effects (" new thermalization" scenario and P and CP odd effects in QCD) is the same as argued in this paper. In particular, the effect should not depend on energy of colliding ions. Indeed, the size of the P odd domains as well as the spectrum of the formed particles is determined exclusively by the acceleration "a" and should not depend on energy of colliding ions. Such independence on energy is indeed supported by observations where correlations measured in Au+Au and Cu+Cu collisions at √ s N N = 62 GeV and √ s N N = 200 GeV are almost identical and independent on energy [5]. These similarities in behaviour of the correlations is definitely a strong argument supporting entire framework based on universality and common origin of both effects as formulated in introductory section 1. Based on this universality we predict that the corresponding correlations at the LHC energies would demonstrate a similar strength and a similar features found at RHIC. •5. One should emphasize that the universal features as formulated above are related exclusively to the portion of the "apparent thermalization" of the system as a result of acceleration "a". Those aspects are expected to be universal for all high energy collisions: from e + e − to AA. In case of heavy ion collisions however, in addition to those "apparent thermalization" aspects there are very real thermodynamical features of the system resulting from the conventional collisions which are normally described using the hydrodynamical equations. This conventional "hydro" portion of the dynamics, of course is not universal. This portion, for example, is not present in pp collisions, and must be subtracted from analysis when comparison of AA with pp collisions is made in order to test the universality conjecture. •6. The arguments presented above on universal behaviour do not explicitly depend on the strength of the magnetic field which is a key player in CME, see eq. (1.1). This is a consequence of the same universal behaviour discussed above. The direction of B (or angular momentum L) does play a role of selecting the reaction plane, while the absolute value of \| B\| is less important parameter in our arguments. The corresponding \| B\|dependence is implicitly hidden in the magnitude of acceleration parameter "a" which is a function of many other things, including centrality, \| B\|, etc. Therefore, one should not expect a strong dependence of the effect on charges Z i of colliding ions which would lead to very different magnetic fields \| B\| for Au+Au and Cu+Cu collisions. Observed similarity in behaviour for Au+Au and Cu+Cu is another manifestation of universality discussed in item 4 above. A relatively mild Z i dependence of the effect is indeed consistent with observations [5]. •7. The arguments presented above on universality of the correlation strength do not depend on transverse momenta k 2 ⊥ , even for relatively large k ⊥ > 1 GeV. This is a consequence of the same universal behaviour discussed in items 4 and 5. Indeed, the entire picture described above, assumes that all hadrons are formed with k ⊥ determined by the thermal distribution in the P odd background (5.1). The spectrum of both: slow and fast fluctuations is the result of preparation of the vacuum state \|0 M in the accelerating frame even before the collision develops as explained above and expressed by eq. (2.26). Therefore, one should not expect strong dependence on \|k ⊥,α − k ⊥,β \| in the correlations for particles α and β even for large \|k ⊥,α − k ⊥,β \| ≥ 1 GeV as the corresponding distributions are not much affected by P odd fluctuations (5.1). This consequence of the universality is also consistent with observations [5] where it is found that the correlation depends very weakly on \|k ⊥,α − k ⊥,β \|. •8. A picture outlined above assumes an ideal world with a 1 GeV when a very large P odd domain is formed with size ∼ (2π)/a which is much larger than conventional QCD fluctuations with typical sizes ∼ fm. In reality, one should expect some deviations from this universal behaviour due to a number of complications in the real (rather than ideal) world, e.g. finite size of the system. In particular, the strength of the correlations is expected to increase when \|k ⊥,α + k ⊥,β \| increases for finite (rather than very large) P odd domain. This is due to the fact that the probability to form two fm− size particles within one finite size domain is larger if the particles have smaller sizes, and correspondingly larger \|k ⊥,α + k ⊥,β \|. This deviation from the universality apparently consistent with observations [5] where it is found that the correlation in fact increases with \|k ⊥,α + k ⊥,β \| even for relatively large k ⊥ > 1 GeV. Such a behaviour naively contradicts to a conventional intuition that all nonperturbative effects must be suppressed for large k ⊥ > 1 GeV but in fact it has a simple and natural explanation within our framework as suggested above. Another manifestation of the same finite size effect would be a sharp cutoff in the correlations when the centrality continues to increase. This pure geometrical effect occurs when the available overlapping portion of the colliding nucleus and the size of a P odd domain become approximately equal in sizes. The observation of the corresponding peak in strength of the correlations gives a precise experimental tool to measure the maximal effective size of P odd domains for a given system. This feature, of course, can not have an universal description as it is related to the finite size effects, and therefore, is sensitive to specific properties of colliding nucleus. To conclude: The qualitative consequences which follow from the picture outlined above apparently consistent with all presently available data. In reality "a" is not a small number, and the size of P odd domain is not very large even for non-central collisions. The finite size effects and other non-universal features may lead to some corrections from the universal picture as we mentioned in item 8 above. Moreover, those corrections themselves are not expected to follow the universal behaviour. Much work needs to be done before a qualitative picture sketched above becomes a quantitative description of the correlations observed at RHIC [1, 2, 3, 4, 5]. Conclusion. Future Directions. In this paper we adopt the approach suggested in refs. [36,37,38,39,40,41,42,43] and treat the universality observed in all high energy collisions as a result of the Unruh radiation characterized by a single parameter "a". The problem of computation the acceleration "a" is not addressed in the present paper. It is is obviously very hard problem of strongly interacting QCD. Instead, we study the topological fluctuations (represented in our framework by the Veneziano ghost) in the given accelerating background "a". Such a treatment is a consistent description in large N c limit when the influence of these degrees of freedom on acceleration "a" itself is negligible. There is a number of immediate consequences from this picture relevant for analysis of the correlations observed at RHIC [1,2,3,4,5] and which were outlined in section 5.2. We formulate the universal properties for the P odd effects, similar to well-known universal thermal behaviour of the spectrum studied in all high energy collisions. We observe that our predictions are consistent with all presently available data. We formulate below some possible directions for the future study which may confirm or rule out this entire framework. • It would be very desirable (if not crucial) to understand the connection between the acceleration parameter "a" which is the key element of the present paper and familiar parameters such as centrality, initial energy and the charges of the colliding ions in realistic (rather than ideal) world. Such knowledge would allow us to (quantitatively) test the basic conjecture on universality of the P odd effects formulated in section 5.2. • It would be very interesting to study the P odd correlations for other high energy collisions, beyond the heavy ion systems. As we mentioned in the text the P odd domains are also produced in e + e − , pp and pp systems, though the size of the produced P odd domain would be quite small as it is determined by 2π/a with a given by eq. (1.3). If one uses the same correlation [1] which has been used for analysis of heavy ion system, the universality arguments suggest that the intensity of the correlations in e + e − , pp and pp systems would be exactly the same as measured in heavy ions in most central collisions when the effective acceleration "a" is determined by the same expression for all systems (1.3). • The Veneziano ghost which is the subject of this paper may in fact lead to another IR effect demonstrating the sensitivity to the boundaries (in addition to sensitivity to the horizon scale studied in this work). Specifically, the Veneziano ghost which is protected by topology and which saturating the subtraction term in the topological susceptibility, may lead to the Casimir type effects as argued in [61] though no massless physical degrees of freedom are present in the system. This effect can be exactly computed in 2d QED which is known to be the model with a single massive degree of freedom [55]. Still, the Casimir like effect is present in this system [57]. The Casimir type effects in 4d QCD appear to be present on the lattice where the power like behaviour (1/L) α as a function of the total lattice size L has been observed in measurements of the topological susceptibility [62]. Such a behaviour is in huge contrast with exponential exp(−L) decay law which one normally expects for any theories with massive degrees of freedom 5 . • The obtained results may have some profound consequences for our understanding of physics at the largest possible scales in our universe as a result of dynamics of the same (protected by topology) Veneziano ghost. Namely, the Casimir type effects in 4d QCD may be observable using the CMB analysis as suggested in ref. [63]. • Another manifestation of the same physics is as follows. The dark energy observed in our universe might be a result of mismatch between the vacuum energy computed in slowly expanding universe with the expansion rate H and the one which is computed in flat Minkowski space. If true, the difference between two metrics would lead to an estimate ∆E vac ∼ HΛ 3 QCD ∼ (10 −3 eV ) 4 which is amazingly close to the observed value today [53]. As explained in the text the typical wavelengths λ associated with this energy density are of the order of the inverse Hubble parameter, λ ∼ (2π/H) ∼ 10 Gyr, and therefore, these modes do not clump on distances smaller than H −1 , in contrast with all other types of matter. This makes these fluctuations to be the perfect dark energy candidates [53]. • Also, a nature of the magnetic field in the universe with characteristic intensity of around a few µG correlated on very large scales and observed today is still unknown. One can argue that the very same Veneziano ghost which is the subject of the present work may in fact induce the large scale magnetic field with correlation length ∼ (2π/H) as a result of anomalous interaction similar to the one which leads to the charge separation and chiral magnetic effects (1.1). In the case of heavy ions the correlation length for charge separation effect and CME is determined by the size of the P odd domain ∼ (2π/a), see section 1.1, while in case of expanding universe the correlation length ∼ (2π/H). More than that, the corresponding induced magnetic field in the universe is expected to be helical (i.e. d 3 x A · B = 0) and would naturally have the intensity B α 2π HΛ 3 QCD ∼ nG, which by simple adiabatic compression during the structure formation epoch, could explain the field observed today at all scales, from galaxies to superclusters [64]. • Finally, the cosmological observations on the largest scales exhibit a solid record of unexpected anomalies and alignments, apparently pointing towards a large scale violation of statistical isotropy. These include a variety of CMB measurements, as well as alignments and correlations of quasar polarization vectors over huge distances of order of 1 Gpc. The only comment I would like to make here that such anomalies may in fact be originated from the same fundamental topological P odd fluctuations studied in this work, see [65] for the details. To conclude: the development of the early Universe is a remarkable laboratory for the study of most nontrivial properties of particle physics such as P odd effects on the scale of the entire universe. What is more remarkable is the fact that the very same phenomena can be, in principle, experimentally tested in heavy ion collisions, where a similar environment can be achieved. Acknowledgements I am thankful to Dima Kharzeev, Larry McLerran, Valery Rubakov, Edward Shuryak, and also Berndt Mueller and Andrey Leonidov and other participants of the workshops "P -and CP -odd Effects in Hot and Dense Matter" at Brookhaven, April 2010 and "the first heavy ion collisions at the LHC" at CERN, August 2010, where this work has been presented, for useful and stimulating discussions. I am also thankful to James Bjorken for many hours of discussions during his visit to Vancouver in October 2010. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada. A. The Veneziano ghost is not an asymptotic state. The main goal of this Appendix is to argue that while the Veneziano ghost leads to a number of profound effects in Minkowski space (the resolution of the U (1) A problem) as well as in the accelerating frame (present work), it nevertheless is not a conventional propagating asymptotic state. In different words, it does not contribute to absorptive parts of any correlation functions. However, it does contribute to the real parts of the correlation functions. In Minkowski space such kind of contributions normally treated as the subtraction constants. In the present case of the accelerating frame, the corresponding "subtraction constant" becomes a "subtraction function" which depends on acceleration and which is sensitive to the global properties of the space-time. The crucial observation for future analysis is as follows: the fields φ 1 , φ 2 which are originated from unphysical (in Minkowski space) degrees of freedom can couple to other fields only through a combination (φ 1 −φ 2 ) as a consequence of the original gauge invariance (3.10). Precisely this property along with Gubta-Bleuler auxiliary condition (3.12) provides the decoupling of physical degrees of freedom from unphysical combination (φ 2 − φ 1 ) as discussed in great details in [55]. We follow analysis of ref. [34] to study the propagating features of fields in accelerating frame. To achieve this goal we replace a single physical field Φ from ref. [34] by specific combination (φ 2 − φ 1 ) fields for our system (3.10). It leads to some drastic consequences as instead of conventional expectation values such as < 0\|a k ...a † k \|0 > = 0 from ref. [34] we would get < 0\|(a k − b k )...(a † k − b † k )\|0 >= 0. The relevant matrix elements vanish as a result of the corresponding commutation relation [(a † k −b † k ), (a k −b k )] = 0. Furthermore, as [H, (a k −b k )] = (a k −b k ) the structure (a k −b k ) is preserved such that a k and b k never appear separately. Based on this observation, one can argue that the same property holds for any other operators which constructed from the combination (φ 2 − φ 1 ). In different words, no actual radiation of real particle occurs in our case in contrast with real Unruh radiation [34], i.e the absorptive parts of relevant correlation functions always vanish. Therefore, there are some fluctuating degrees of freedom in the system observed by a Rindler observer without radiation of any real particles. In many respects, this feature is similar to the Casimir effect though spectral density distribution describing the fluctuations of the vacuum energy has a nontrivial ω dependence in contrast with what happens in the Casimir effect. Another way to arrive to the same conclusion is to consider the particle detector moving along the world line described by some function x µ (τ ) where τ is the detector's proper time. In the case for the Rindler space the corresponding τ is identified with η defined by formula (2.2). As is known, the corresponding analysis in the lowest order approximation is reduced to study of the positive frequency Wightman Green function defined as where we use notations from [35]. In case of inertial trajectory for massless scalar field Φ the positive frequency Wightman Green function is given by In our case the detector-field interaction is described by the combination (φ 1 − φ 2 ) rather by a single field Φ discussed above. Therefore, the relevant response function in our case is described by the positive frequency Green's function defined as ∼ 0\| φ 1 (x) − φ 2 (x) , φ 1 (x ) − φ 2 (x ) \|0 , (A.5) which replaces eq. (A.1). One can easily see that this Green's function given by eq. (A.5) identically vanishes as the consequence of the opposite signs in commutation relations describing φ 1 and φ 2 fields, in complete agreement with the arguments presented above. Therefore, the Rindler observer will see an extra energy (4.11) without detecting any physical particles. One can rephrase the same statement by saying that the ghost and its partner do not contribute to the absorptive part of the Green's function, but do contribute to its real part. In Minkowski space the contribution to the real part of the topological susceptibility is nothing but the well known subtraction constant which has been precisely studied and measured on the lattice, see Fig.1. The ghost-based technique we advocate in this paper is well suited to study the corresponding physics in accelerating or time dependent background. x 2 − 2t 2 = a −2 e 2aξ = world lines. (2.4) They represent the world lines of uniformly accelerated observers with proper acceleration given by ae −aξ = proper acceleration. (2.5) Thus, lines of large positive ξ (far from x = t = 0) represent weakly accelerated observers, while large negative ξ correspond a high proper acceleration. The observer's proper time τ is τ = ηe aξ = observer's proper time. (2.6) e −πω/2a b 1 † −k + e πω/2a b 2 k e πω/a − e −πω/a b R k = e −πω/2a b 2 † −k + e πω/2a b 1 k e πω/a − e −πω/a should be compared with standard contribution to the thermal energy associated with the fast fluctuations of N f light quark flavours and N 2 c − 1 gluons at temperature T = (a/2π), D + (x, x ) = 0\|Φ(x), Φ(x )\|0 , (A.1)while the transition probability per unit proper time is proportional to its Fourier trans∆τ )e −iω∆τ D + (∆τ ) (A.2) corresponding Fourier transform (A.2) obviously vanishes. No particles are detected as expected. In case if the detector accelerates uniformly with acceleration a the corresponding Green's function is given by[35] are infinite number of poles in the lower -half plane at ∆τ = −2iπ k a for positive k the corresponding Fourier transform (A.2) leads to the known result ∼ ω[exp(2πω/a)−1] −1 . not to be confused with conventional Fadeev Popov ghosts which appear in covariant quantization of non-abelian gauge theories A warning signal with the signs: the physical degrees of freedom in Euclidean space (where the lattice computations are performed) contribute to topological susceptibility χQCD with the negative sign, while the contact term (the Veneziano ghost) contributes with the positive sign, in contrast with our discussions in Minkowski space. The fluctuations of the physical massive η field can be obviously neglected. It will be ignored in what follows. In fact, we argue in the next section 5 that in heavy ion collisions such conditions indeed could be achieved experimentally. 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[ "Devil's staircases in the IV characteristics of superconductor/ferromagnet/superconductor Josephson junctions", "Devil's staircases in the IV characteristics of superconductor/ferromagnet/superconductor Josephson junctions", "Devil's staircases in the IV characteristics of superconductor/ferromagnet/superconductor Josephson junctions", "Devil's staircases in the IV characteristics of superconductor/ferromagnet/superconductor Josephson junctions" ]	[ "M Nashaat \nDepartment of Physics\nCairo University\n12613CairoEgypt\n\nBLTP\nJINR\nDubna\n\nMoscow Region\n141980Russia\n", "A E Botha \nDepartment of Physics\nScience Campus\nUniversity of South Africa\nPrivate Bag X61710Florida ParkSouth Africa\n", "Yu M Shukrinov \nDepartment of Physics\nScience Campus\nUniversity of South Africa\nPrivate Bag X61710Florida ParkSouth Africa\n\nBLTP\nJINR\nDubna\n\nMoscow Region\n141982Russia\n\nDubna State University\nDubnaRussian Federation\n", "M Nashaat \nDepartment of Physics\nCairo University\n12613CairoEgypt\n\nBLTP\nJINR\nDubna\n\nMoscow Region\n141980Russia\n", "A E Botha \nDepartment of Physics\nScience Campus\nUniversity of South Africa\nPrivate Bag X61710Florida ParkSouth Africa\n", "Yu M Shukrinov \nDepartment of Physics\nScience Campus\nUniversity of South Africa\nPrivate Bag X61710Florida ParkSouth Africa\n\nBLTP\nJINR\nDubna\n\nMoscow Region\n141982Russia\n\nDubna State University\nDubnaRussian Federation\n" ]	[ "Department of Physics\nCairo University\n12613CairoEgypt", "BLTP\nJINR\nDubna", "Moscow Region\n141980Russia", "Department of Physics\nScience Campus\nUniversity of South Africa\nPrivate Bag X61710Florida ParkSouth Africa", "Department of Physics\nScience Campus\nUniversity of South Africa\nPrivate Bag X61710Florida ParkSouth Africa", "BLTP\nJINR\nDubna", "Moscow Region\n141982Russia", "Dubna State University\nDubnaRussian Federation", "Department of Physics\nCairo University\n12613CairoEgypt", "BLTP\nJINR\nDubna", "Moscow Region\n141980Russia", "Department of Physics\nScience Campus\nUniversity of South Africa\nPrivate Bag X61710Florida ParkSouth Africa", "Department of Physics\nScience Campus\nUniversity of South Africa\nPrivate Bag X61710Florida ParkSouth Africa", "BLTP\nJINR\nDubna", "Moscow Region\n141982Russia", "Dubna State University\nDubnaRussian Federation" ]	[]	We study the effect of coupling between the superconducting current and magnetization in the superconductor/ferromagnet/superconductor Josephson junction under an applied circularly polarized magnetic field. Manifestation of ferromagnetic resonance in the frequency dependence of the amplitude of the magnetization and the average critical current density is demonstrated. The IV characteristics show subharmonic steps that form devil's staircases, following a continued fraction algorithm. The origin of the found steps is related to the effect of the magnetization dynamics on the phase difference in the Josephson junction. The dynamics of our system is described by a generalized RCSJ model coupled to the Landau-Lifshitz-Gilbert equation. We justify analytically the appearance of the fractional steps in IV characteristics of the superconductor/ferromagnet/superconductor Josephson junction.	10.1103/physrevb.97.224514	[ "https://arxiv.org/pdf/1802.09212v2.pdf" ]	119,084,663	1802.09212	3c5eca0a95aacdf07a7e63abe7cf54f38f2c5def	Devil's staircases in the IV characteristics of superconductor/ferromagnet/superconductor Josephson junctions 17 Jun 2018 (Dated: June 19, 2018) M Nashaat Department of Physics Cairo University 12613CairoEgypt BLTP JINR Dubna Moscow Region 141980Russia A E Botha Department of Physics Science Campus University of South Africa Private Bag X61710Florida ParkSouth Africa Yu M Shukrinov Department of Physics Science Campus University of South Africa Private Bag X61710Florida ParkSouth Africa BLTP JINR Dubna Moscow Region 141982Russia Dubna State University DubnaRussian Federation Devil's staircases in the IV characteristics of superconductor/ferromagnet/superconductor Josephson junctions 17 Jun 2018 (Dated: June 19, 2018)numbers: 7450+r7445+c7650+g We study the effect of coupling between the superconducting current and magnetization in the superconductor/ferromagnet/superconductor Josephson junction under an applied circularly polarized magnetic field. Manifestation of ferromagnetic resonance in the frequency dependence of the amplitude of the magnetization and the average critical current density is demonstrated. The IV characteristics show subharmonic steps that form devil's staircases, following a continued fraction algorithm. The origin of the found steps is related to the effect of the magnetization dynamics on the phase difference in the Josephson junction. The dynamics of our system is described by a generalized RCSJ model coupled to the Landau-Lifshitz-Gilbert equation. We justify analytically the appearance of the fractional steps in IV characteristics of the superconductor/ferromagnet/superconductor Josephson junction. I. INTRODUCTION An important challenge, in superconducting spintronics dealing with the Josephson junctions coupled to magnetic systems, is the achievement of electric control over the magnetic properties by the Josephson current and its counterpart, i.e. the achievement of magnetic control over the Josephson current. [1][2][3][4] In some systems, spin-orbit coupling plays a major role in the attainment of such control. 5 For example, a recent study showed a full magnetization reversal in a superconductor/ferromagnet/superconductor (S/F/S) structure, with spin-orbit coupling, by adding an electric current pulse. 6 Such a reversal may be important for certain applications. 6 Another approach was followed in Refs. 7 and 8, where the authors demonstrated the interaction of a nanomagnet with a weak superconducting link and the reversal of single domain magnetic particle magnetization by an ac field. The superconducting current of a Josephson junction (JJ) coupled to an external nanomagnet driven by a time-dependent magnetic field both without and in the presence of an external ac drive were studied in Ref. 9. The authors showed the existence of Shapiro-type steps in the IV characteristics of the JJ subjected to a voltage bias for a constant or periodically varying magnetic field and explored the effect of rotation of the magnetic field and the presence of an external ac drive on these steps. Furthermore, a uniform precession mode (spin wave) could be excited by a microwave magnetic field, at ferromagnetic resonance (FMR), when all the elementary spins precess perfectly in phase. 10 Finally, coupling between the Josephson phase and a spin wave was studied in the series of papers. 4,[11][12][13][14][15][16] In Josephson junctions driven by external microwave radiation the Shapiro steps 17 that appear in the IV characteristics can form the so-called devil's staircase (DS) structure as a consequence of the interplay between Josephson plasma and applied frequencies. [18][19][20][21] The DS structure is a universal phenomenon and appears in a wide variety of different systems, including infinite spin chains with long-range interactions, 22 frustrated quasi-two-dimensional spin-dimer systems in magnetic fields, 23 and even in the fractional quantum Hall effect. 24 In Ref. 25 the authors considered symmetric dual-sided adsorption, in which identical species adsorb to opposite surfaces of a thin suspended membrane, such as graphene. Their calculations predicted a devil's staircase of coverage fractions for this widely studied system. 25 In Ref. 26 a series of fractional integer size steps was observed experimentally in the Kondo lattice CeSbSe. In this system the application of a magnetic field resulted in a cascade of magnetically ordered states -a possible devil's staircase. A devil's staircase was also observed in soft-x-ray scattering measurements made on single crystal SrCo 6 O 11 , which constitutes a novel spinvalve system. 27 An extension of the investigation of this problem on the S/F/S Josephson junction might open horizons in this field. The problem of coupling between the superconducting current and magnetization in the S/F/S Josephson junction attracts much attention today (see Ref. 2 and the references therein). An intriguing opportunity is related to the connection between the staircase structure and current-phase relation. 28 Particularly, the manifestation of the staircase structure in the IV characteristics of S/F/S junctions might provide the corresponding information on current-phase relation and, in this case, serve as a novel method for its determination. The ap-pearance of the DS structure and its connection to the current-phase relation in experimental situations has not yet been investigated in detail. It stresses a need for a theoretical model which would fully describe the dynamics of the S/F/S Josephson junction under external fields, features of Shapiro-like steps and their DS staircase structures. In Ref. 13 the Josephson energy in the expression for the effective field was not considered. Consequently, the IV characteristics of the S/F/S junction at FMR only showed current steps at voltages corresponding to even multiples of the applied frequency. The authors related these steps to the interaction of Cooper pairs with an even number of magnons. 13 In this paper we investigate the effect of coupling between the superconducting current and magnetization in the superconductor/ferromagnet/superconductor Josephson junction under an applied circularly polarized magnetic field. Taking into account the Josephson energy in the effective field, we demonstrate an appearance of odd and fractional Shapiro steps in IV characteristics, in addition to the even steps that were reported in Ref. 13. We demonstrate the appearance of devil's staircase structures and show that voltages corresponding to the subharmonic steps under applied circularly polarized magnetic field follow the continued fraction algorithm. [19][20][21] An analytical consideration of the linearized model, based on a generalized RCSJ model and Landau-Lifshitz-Gilbert (LLG) equation, including the Josephson energy in the effective field, justifies the appearance of the fractional steps in IV characteristics, in agreement with our numerical results. We also show the manifestation of ferromagnetic resonance in the frequency dependence of the amplitude of the magnetization and the average critical current density. An estimation of the model parameters shows that there is a possibility for the experimental observation of this phenomenon. The plan of the rest of the paper is as follows. In Sec. II, we describe the model and present an explicit form of the equations. Ferromagnetic resonance is demonstrated in Sec. III, where the effect of Gilbert damping is shown and a comparison with the linearized case is presented. This is followed by a discussion of the IV characteristics and observed staircase structures in Sec. IV. In Sec. V we discuss the additional effect of an oscillating electric field on the Shapiro steps. Demonstration of different possibilities of the frequency locking and discussion of the experimental realization of the found effects is presented in Sec. VI. Finally, we conclude in Sec. VII and specify our calculations for linearized case. 29 II. MODEL AND METHODS The geometry of the S/F/S Josephson junction under an applied circularly polarized magnetic field is shown in Fig. 1. There is a uniform magnetic field of magnitude H 0 applied in the z-direction. Additionally, a circularly polarized magnetic field, of amplitude H ac and fre-quency ω, is applied in xy-plane. The total applied field is thus H(t) = (H ac cos(ωt), H ac sin(ωt), H 0 ). A bias current I flows in the x-direction. Φ z (t) = 4πdL y M z (t)/Φ 0 , Φ y (t) = 4πdL z M y (t)/Φ 0 , where M z and M y are components of magnetization and d is the thickness of ferromagnet. Using the equation ∇θ(y, z, t) = − 2πd Φ0 B(t) × n, n is the unit vector in x direction and the fact that two superconductors are thicker than London's penetration depth, we obtain an expression for the gauge-invariant phase difference, θ(y, z, t) = θ(t) − 8π 2 dMz(t) Φ0 y + 8π 2 dMy(t) Φ0 z, where Φ 0 = h/(2e) is the magnetic flux quantum. Hence, within the framework of the modified RCSJ model, which takes into account the gauge invariance including the magnetization of the ferromagnet, 13 the electric current reads I/I 0 c = sin πΦz (τ ) Φ0 sin πΦy(τ ) Φ0 (πΦ z (τ )/Φ 0 )(πΦ y (τ )/Φ 0 ) sin θ(τ ) + dθ(τ ) dτ + β c d 2 θ(τ ) dτ 2 ,(1) where τ = tω c is the normalized time, ω c = 2πI 0 c R/Φ 0 is the characteristic frequency, R is the junction resistance, β c = RCω c is the McCumber parameter, 3 and C is the junction capacitance. In the present paper we will only consider the overdamped case for which β c = 0. The applied circularly polarized magnetic field in the xy-plane causes precession of the magnetization M in the ferromagnetic (FM) layer. The dynamics of the magnetization is described by the LLG equation 10 (1 + α 2 ) dM dt = −γM × H e − γα \|M\| M × (M × H e ),(2) where α is the Gilbert damping, γ is the gyromagnetic ratio, and H e is an effective field. Taking into account that the phase difference depends on the magnetization components, we write the total energy of our system as E = E s + E M + E ac , where E s = − Φ 0 2π θ(t) − 8π 2 d Φ 0 (M z (t)y − M y (t)z) I + (3) E J 1 − cos θ(t) − 8π 2 d Φ 0 (M z (t)y − M y (t)z) , E M = −vH 0 M z (t), E ac = −vM x (t)H ac cos(ωt) − vM y (t)H ac sin(ωt). Here H 0 = ω 0 /γ, ω 0 is the ferromagnetic resonance frequency, and v is the volume. When we switch on H 0 and H ac , the phase difference starts to depend on M , and so does the Josephson energy. The addition of E s leads to the dependence of the effective field on the ratio E J /E M , and generalizes the considerations made in Ref. 13. The effective field is now given by H e = − 1 v ∇ M E.(4) In dimensionless form, we write m = M/M 0 , M 0 = \|M\|, h e = H e /H 0 , h ac = H ac /H 0 , Ω = ω/ω c , and Ω 0 = ω 0 /ω c , After integrating the total effective field over the junction area, it has the following form h e = (h ac cos Ωτ )ê x + (h ac sin Ωτ + Γ yz ǫ J cos θ)ê y + (1 + Γ zy ǫ J cos θ)ê z ,(5) where ǫ J = E J / (vM 0 H 0 ) and Γ yz = sin (φ sy m z ) m y (φ sy m z ) cos(φ sz m y ) − sin(φ sz m y ) (φ sz m y ) ,(6)Γ zy = sin (φ sz m y ) m z (φ sz m y ) cos(φ sy m z ) − sin(φ sy m z ) (φ sy m z ) , with φ sy =4π 2 L y dM 0 /Φ 0 , and φ sz =4π 2 L z dM 0 /Φ 0 . If we set Γ yz = Γ zy = 0, our system reduces to that of Ref. 13. We note that the first term for E s in Eq. (3), does not contribute to the effective field after integration over the junction area and taking the derivative with respect to the magnetization. The LLG equation in the dimensionless form reads dm dτ = − Ω 0 (1 + α 2 ) m × h e + α [m × (m × h e )] .(7) The magnetization and phase dynamics of the considered S/F/S Josephson junction is determined by Eqs. (1) and (7). To solve this system and calculate the IV characteristics, we assume a constant bias current and calculate the voltage from the Josephson relation V (τ ) = dθ/dτ . We employ a 4th-order Runge-Kutta integration scheme which conserves the magnetization magnitude in time. The dc bias current I is normalized to the critical current I 0 c , and the voltage V (t) to ω c /(2e). As a result, we find the temporal dependence of the voltage in the JJ at a fixed value of bias current I. Then, the current value is increased or decreased by a small amount, δI (the bias current step), to calculate the voltage at the next point of the IV characteristics. We use the final phase and voltage achieved at the previous point of the IV characteristics as the initial condition for the next current point. The average of the voltage V (τ ) is given by V = 1 T f −Ti T f Ti V (τ )dτ , where T i and T f determine the interval for the temporal averaging. Further details of the simulation procedure are described in Ref. 31. The initial conditions for the magnetization components are assumed to be m x = 0, m y = 0.01 and m z = 1 − m 2 x − m 2 y , while for the voltage and phase we take zeros. The numerical parameters (if not mentioned) are α = 0.1, h ac = 1, φ sy = φ sz = 4, ǫ J = 0.2 and Ω = Ω 0 = 0.5. III. FERROMAGNETIC RESONANCE First we show that the system displays ferromagnetic resonance. Its manifestation, in the frequency dependence of the amplitude of the magnetization component m y and the average critical current density, is presented in Fig. 2, where we see that the maximum in both cases occurs at the resonance frequency Ω = Ω 0 = 0.5. Furthermore, the oscillation amplitude is not symmetric relative to Ω 0 , which reflects the influence of H s in the effective field. The behavior of the amplitude of the magnetization component m x is qualitatively the same. In Fig. 2 (b) we show the frequency dependence of the maximum of magnetization component m y at different damping α and amplitude of circularly polarized magnetic field h ac . We see that the resonance line width changes with changing h ac (curves with label 1 and 2) and α (curves with label 2 and 3). For comparison, we also demonstrate the manifestation of the ferromagnetic resonance in the linearized case. 2,29 In the linearized case, the RSJ equation reduced to I/I c = sin (φ s m y ) (φ s m y ) sin θ(t) + dθ(t) dt ,(8) where I c = I 0 c sin(φ sy )/(φ sy ), φ sy = 4π 2 L y dM z /Φ 0 and the expression for y-component of magnetization has a form m y = −2α Ω 2 Ω 2 0 cos(Ωt) + 1 − η 1 Ω 2 Ω 2 0 sin(Ωt) 1 − η 2 Ω 2 Ω 2 0 2 + ∆ J 1 − η 1 Ω 2 Ω 2 0 + 4α 2 Ω 2 Ω 2 0 ,(9) where ∆ J = ǫ J φ 2 sz cos θ(t)/3, η 1 = 1−α 2 and η 2 = 1+α 2 . Results of calculations based of these formulas are presented in Fig. 2(c). We see a qualitative agreement of the ferromagnetic resonance features in both cases. IV. DS STRUCTURE IN THE IV CHARACTERISTICS Let us now discuss the S/F/S junction at FMR, when the coupling between Josephson and magnetic system is strongest. In Fig. 3(a) the IV characteristic demonstrates current steps at V = mΩ 0 , with m integer, and also some fractional steps. In the case of conventional JJs the widths of the first Shapiro step is larger than the second. In the present case, we see that the width of the first step is much narrower than that of the second. So, the width of the harmonics are different for even and odd m: large steps are at even m and small steps at odd m. In Ref. 13, which did not consider the Josephson energy in the expression for the effective field, only the steps with even m were observed. In our case, taking into account the Josephson energy in the effective field, we have obtained additional steps with odd and fractional values of m, as we see in Fig. 3(a). The structure of those fractional steps can be clarified by analysis of their positions on the voltage scale, using an algorithm based on the generalized continued fraction formula: 19 -21 V =   N ± 1 n ± 1 m± 1 p±..   Ω,(10) where N , n, m, p, . . . are positive integers. The locking of the Josephson frequency to the frequency of magnetic precession occurs due to the additional terms (Γ yz ǫ J cos θ, Γ zy ǫ J cos θ) in the effective field, as given by Eq. (5). Fig. 3(b) and (c) demonstrate the enlarged parts of the IV characteristic shown in Fig. 3(a). There are the fractional current steps between V = 0 and V = 0.5 which can be described by the continued fractions of second level 19 (N − 1) + 1/n and N − 1/n with N = 1 in both cases (see Fig. 3(b)). In addition, there is a manifestation of two third-level continued fractions (N − 1) + 1/(n − 1/m) with N = 1, n = 2 (shown in the inset) and n = 3. The steps between V = 0.5 and V = 1 follow the continued fractions of second level (N −1)+1/n and N − 1/n with N = 2 in both cases. In Fig. 3(c) we see clearly the manifestation of second level continued fractions N − 1/n with N = 3 and (N − 1) + 1/n with N = 4 between voltage steps V = 1 and V = 2. V. EFFECT OF OSCILLATING ELECTRIC FIELD The ac field can affect the Josephson junction directly, and not only via the oscillating magnetization. The effect of an oscillating electric field from microwave radiation is usually taken into account by adding the term A sin Ω r t in Eq. (1), where A is the amplitude and Ω r = ω r /ω cthe frequency of the external electromagnetic radiation. Figure 4 shows the IV characteristics without the effect See in (Fig.3-b) See in (Fig.3-c of the oscillating electric field (i.e. for A = 0) and two curves at amplitudes A = 0.3 and A = 1. In comparison to A = 0, where the width of the first step at V = 0.5 is smaller relatively to the step at V = 1 (a signature of the S/F/S IV characteristics), we see that at A = 0.3 the first step has widened in comparison to the second step at A = 1. But, even in this case, the IV characteristics show the unusual behavior of Shapiro step widths for a conventional Josephson junction, specifying width of odd and even steps. Fig. 3. For clarity, the curves at A = 0.3 and A = 1 have been shifted to the right, by ∆I = 0.6 and ∆I = 1.2 respectively, relative to the IV characteristic at A = 0. VI. DISCUSSION We have also found that one can control the structure of the devil's staircase by tuning the frequency of the acmagnetic field out of resonance. Of course, the width of the subharmonic steps is largest at the FMR. The step structure depends on the junction parameters (Gilbert damping, cross-section, etc). The main parameter determining the appearance of the DS structure is the ratio of the Josephson to magnetic energy. If this ratio is close to zero, we observe only even steps. In the present work the appearance of the fractional steps and the formation of the devil's staircases in the IV-curve are consequences of including the Josephson energy in the effective field, i.e. the term ǫ J in (5). We justify this claim by solving the linearized LLG equation analytically using well known mathematical methods. 5? ,6 As demonstrated in Fig. 5, our proposed model shows different possibilities of the frequency locking leading to even, odd and fractional current steps in IV characteristics of S/F/S junction under an external circularly polarized magnetic field. This fact is in an agreement with the results presented in Fig. 3. ¢ ±2(m+n+1) £ ±2(m-n) ¤ ±2(m+n+1) ¥ ±[(m-n)/(k-r)]¦ ±[(m-n)/(k-r+1)] § ±[(m+n+1)/(k-r)]± [(m+n+1)/(k-r+1)]© ±[2(m-n)/(1-2(k-r))] ±[2(m-n)/(2(k-r)+1)] ±[2(m+n+1)/(2k-r+1)] ±[2(m+n+1)/(1-2(k-r))] If H e =H ac +H 0 +H s Let us now discuss the possibility of experimentally observing the effects found in this paper. The main parameter which controls the appearance of the current steps is ǫ J = E J / (vM 0 H 0 ). Using typical junction parameters d = 5 nm, L y = L z = 75 nm, critical current I 0 c ≈ 160 µA, saturation magnetization M 0 ≈ 4 × 10 5 A/m, H 0 ≈ 26 mT and gyromagnetic ration γ = 3π MHz/T, we find the value of φ sy(z) =4π 2 L y(z) dM 0 /Φ 0 = 3.6 and ǫ J = 0.18, which are very close to the values we used in our simulations and justify the choice of parameters: φ sy,sz = 4, ǫ J = 0.2. With the same junction parameters one can control the appearance of the subharmonic steps by tuning the strength of the constant magnetic field H 0 . Estimations show that, for H 0 = 90 mT, the fractional subharmonic steps are disappear at ǫ J = 0.05. For junctions with L y = L z = 50 nm, H 0 = 10 mT, we find φ sy(z) = 2.4 and ǫ J = 1.05, which are rather good for the step manifestation. Of course, in general, the subharmonic steps are sensitive to junction parameters, Gilbert damping and the frequency of the magnetic field. VII. CONCLUSION The S/F/S Josephson junction is of considerable importance for the development of certain spintronic applications/devices. Motivated by physical considerations, our paper has presented a major advance in modeling the S/F/S Josephson junction, by including a previously neglected physical effect, i.e. of the Josephson energy on the effective magnetic field. Our calculations predict that the addition of the Josephson energy should manifest itself (measurably) through the appearance of devil's staircase structures in the IV characteristics, thus providing insight into the precise nature of the current-phase relation and opportunities for potential applications. In our paper we have developed a model which fully describes the dynamics of the S/F/S Josephson junction under an applied circularly polarized magnetic field. Manifestation of ferromagnetic resonance in the frequency dependence of the amplitude of the magnetization and the average critical current density was demonstrated. The IV characteristics showed subharmonic steps which formed devil's staircase structures, following the continued fraction algorithm. 19 The origin of the found steps was related to the effect of the magnetization dynamics on the phase difference in the Josephson junction. Analytical considerations of the steps were in agreement with the numerical results. An interesting question appears about whether the manifestation of the staircase structure in the IV characteristics can provide information on the current-phase relation of the S/F/S Josephson junction and, in some cases, serve as a novel method for its determination. The results on the developed model might serve for better understanding of the coupling between the superconducting current and magnetization in the S/F/S Josephson junction. The appearance of the staircase structure in experimental situations and its connection with the currentphase relation may open horizons in this field. The observed features might also find application in some fields of superconducting spintronics. The Landau-Lifshitz-Gilbert (LLG) equation for the superconductor/ferromagnet/ superconductor (S/F/S) structure describes the behavior of magnetization in the effective magnetic field H e . Being nonlinear, it cannot in general be solved analytically 1 . Here we investigate the system of equations describing the SFS Josephson junction under the application of a circularly polarized magnetic field in xy-plane. We derive a linearized form of the equations by using the method of complex amplitudes 2 and find expressions for the magnetization components. After that, we calculate the effective field components and obtain an expression for m y , which we will subsequently use in the RSJ equation. In its general form, the Landau-Lifshitz-Gilbert equation reads dM dt = −γM × H e + α \|M \| M × dM dt ,(11) where α is the Gilbert damping and γ is the gyromagnetic ratio. We assume that the effective magnetic field and magnetization can be written as sums of constant and alternating parts H e = H 0 +H, M = M 0 +M ,(12)dM dt + γM × H 0 + α \|M \| dM dt × M 0 = −γM 0 ×H.(13) We consider a harmonic time dependence forH. In this case, the time dependence ofM will be also harmonic. Our aim is to find harmonic solutions to the linearized LLG equation, i.e. in the form M =m e iωt ,H =h e iωt .(14) This can be done by inserting Eq. (14) into Eq. (13) to obtain iωm + γm × H 0 + iωα \|M \|m × M 0 = −γM 0 ×h.(15) Projecting Eq. (15) onto the axes of Cartesian coordinate system, we get iωm x + (ω 0 + iαω)m y = γM zhy , −(ω 0 + iαω)m x + iωm y = −γM zhx ,(16) where we use \|M \| ≈ M z , H 0 = ω 0 /γ. The solution to Eq. (16) is m x = (ω 0 + iαω)γM zhx + iωγM zhy ω 2 0 − (1 + α 2 )ω 2 + 2iαωω 0 , m y = −iωγM zhx + (ω 0 + iαω)γM zhy ω 2 0 − (1 + α 2 )ω 2 + 2iαωω 0 ,(17) m x (t) andm y (t) can be written in the following forms m x = χ ′ 1hx + χ ′ 2hy − i(χ ′′ 1hx − χ ′′ 2hy ), m y = −χ ′ 2h x + χ ′ 1h y − i(χ ′′ 2h x + χ ′′ 1h y ),(18) where χ ′ 1 = 1 Γ γM z ω z [ω 2 0 − (1 − α 2 )ω 2 ], χ ′′ 1 = 1 Γ αγM z ω[ω 2 0 + (1 + α 2 )ω 2 ], χ ′ 2 = 1 Γ 2αγM z ω 2 ω 0 , χ ′′ 2 = 1 Γ γM z ω[ω 2 0 − (1 + α 2 )ω 2 ], Γ = [ω 2 0 − (1 + α 2 )ω 2 ] 2 + 4α 2 ω 2 ω 2 0 .(19) Using Eq. (14), the real part for M x (t) and M y (t) can be written as Re{M x (t)} = γM z ω 0    1 − (1 − α 2 ) ω 2 ω 2 0 H x (t) + 2α ω 2 ω 2 0 H y (t) 1 − (1 + α 2 ) ω 2 ω 2 0 2 + 4α 2 ω 2 ω 2 0    ,(20)Re{M y (t)} = γM z ω 0    −2α ω 2 ω 2 0 H x (t) + 1 − (1 − α 2 ) ω 2 ω 2 0 H y (t) 1 − (1 + α 2 ) ω 2 ω 2 0 2 + 4α 2 ω 2 ω 2 0    .(21) B. Expression for Effective Field and Magnetization Component Next, we find the effective field components H x (t) and H y (t) which should have harmonic dependence. The total energy of the S/F/S Josephson junction in the circularly polarized ac field is given by E = E s + E M + E ac , where E s = − Φ 0 2π θ(y, z, t)I + E J [1 − cos(θ(y, z, t))], E M = −vH 0 M z , E ac = −vM x H ac cos ωt − vM y H ac sin ωt,(22) Here E J = I c Φ 0 /2π, I c is the critical current, Φ 0 = h/(2e) is the magnetic flux quantum, θ(y, z, t) is the gauge invariant phase difference between superconducting electrodes, θ(y, z, t) = θ(t) − 8π 2 dM z y/Φ 0 + 8π 2 dM y (t)z/Φ 0 , d is the magnetic thickness, and v is the volume. H ac and ω are the amplitude and frequency of the ac magnetic field. The effective field can be found using H e = −∇ M E/v. In the dimensionless form we use m = M /M z (we omit the time argument for simplicity of notation), h e = H e /H 0 , h ac = H ac /H 0 , Ω = ω/ω c , Ω 0 = ω 0 /ω c , and t −→ tω c . Here ω c = 2πI c R/Φ 0 is the characteristic frequency. After integrate over the junction area, the effective field components in x and y-direction are given in dimensionless form by h x = h ac cos(Ωt),(23)h y = h ac sin(Ωt) + ǫ J cos θ(t) m y cos (φ sz m y ) − sin (φ sz m y ) φ sz m y ,(24) where φ sz = 4π 2 L z dM z /Φ 0 , d is the thickness of ferromagnet, L y , L z are the junction lengths in y and z-direction. Since M z is constant, we define I c = I 0 c sin(φ sy )/(φ sy ), ǫ J = γE J /(vM z Ω 0 ), and φ sy = 4π 2 L y dM z /Φ 0 . Since we assume thatM << M z , we use series expansion for cos(φ sz m y ) and sin(φ sz m y ) in the second term of h y (see Eq. (24)). So, we obtain for first order approximation 1 m y cos (φ sz m y ) − sin (φ sz m y ) φ sz m y ≈ − φ 2 sz m y 3 + ....(25) The effective field reads h y = − ǫ J φ 2 sz m y 3 cos θ(t) + h ac sin Ωt.(26) This term is considered as a modulated harmonic behavior that depends on the value of (ǫ J φ 2 sz m y /3) cos θ(t). So, we can rewrite m y , using Eq. (21) and Eq. (26) in the following form m y = −2α Ω 2 Ω 2 0 cos(Ωt) + 1 − (1 − α 2 ) Ω 2 Ω 2 0 sin(Ωt) 1 − (1 + α 2 ) Ω 2 Ω 2 0 2 + ∆ J 1 − (1 − α 2 ) Ω 2 Ω 2 0 + 4α 2 Ω 2 Ω 2 0 ,(27) where ∆ J = ǫ J φ 2 sz cos θ(t)/3. At α = 0 we have: m y = 3Ω 2 0 sin(Ωt) 3(Ω 2 0 − Ω 2 ) + Ω 2 0 ǫ J φ 2 sz cos(θ(t)) . Now, let us describe the result shown by Eq. (33). If H e = H ac + H 0 , the oscillation of m y is purely harmonic in time. While if we take Josephson energy in the effective field, the oscillation of m y can be considered as harmonic with small changes due to Josephson energy. The validity of the above method require a harmonic dependence of the effective field which can be satisfied in Eq. (26). with Υ 1 = 2α Ω 2 Ω 2 0 , Υ 2 = 1 − (1 − α 2 ) Ω 2 Ω 2 0 , D = 1 − (1 + α 2 ) Ω 2 Ω 2 0 2 + 4α 2 Ω 2 Ω 2 0 , ∆ = 1 + φ 2 sz ǫ J Υ 2 3D cos θ(t).(34) We rewrite m y in Eq. (33) as m y = R ∆ sin(Ωt − ζ),(35) where R = 1 D Υ 2 1 + Υ 2 2 , ζ = arctan(Υ 2 /Υ 1 ), ∆ = 1 + ξ cos(Ω J t + θ 0 ) and ξ = φ 2 sz Υ 2 ǫ J /3D. Substituting this expression to the formula (30), we get I s = sin(φ s R sin(Ωt − ζ)/∆) (φ s R sin(Ωt − ζ)/∆) sin(Ω J t + θ 0 ).(36) Using the following series expansions 5 : sin(φ s R sin(Ωt − ζ)/∆) = 2∆ φ s R ∞ n=0 J 2n+1 φ s R ∆ sin((2n + 1)(Ωt − ζ)), csc(Ωt − ζ) = 2 ∞ m=0 sin((2m + 1)(Ωt − ζ)),(37) where J 2n+1 (φ s R/∆) are Bessel functions of the first kind, J 2n+1 φ s R ∆ = ∞ p=0 (−1) p 2 2p+2n+1 Γ(p + 2n + 2)(2n + 1)! φ s R ∆ 2p+2n+1 .(38) The expression for the supercurrent then becomes I s = 2∆ sin(Ω J t + θ 0 ) φ s R ∞ m=0 ∞ n=0 J 2n+1 φ s R ∆ [cos(2(n − m)(Ωt − ζ)) − cos(2(n + m + 1)(Ωt − ζ))],(39) So, substituting Eq. (38) into Eq. (39), we have I s = 2 sin(Ω J t + θ 0 ) φ s Rˆ m,n,p 1 1 + ξ cos(Ω J t + θ 0 ) 2p+2n cos(2(n − m)(Ωt − ζ)) − cos(2(n + m + 1)(Ωt − ζ)) ,(40) whereˆ m,n,p = ∞ m=0 ∞ n=0 ∞ p=0 (−1) p (φ s R) 2p+2n+1 2 2p+2n+1 Γ(p + 2n + 2)(2n + 1)! .(41) Using the binomial expansion 5 (1 + ξ cos(Ω J t + θ 0 )) −2n−2p = ∞ k=0 (2n + 2p) k k! (ξ cos(Ω J t + θ 0 )) k , \|ζ cos(Ω J t + θ 0 )\| < 1, and trigonometric power formulas 6 cos 2k (Ω J t + θ 0 ) = (2k)! 2 2k (k!) 2 + 1 2 2k−1 k−1 r=0 (2k)! (2k − r)!r! cos((2(k − r)(Ω J t + θ 0 ))), cos 2k+1 (Ω J t + θ 0 ) = 1 4 k k r=0 (2k + 1)! (2k − r + 1)!r! cos((2k − 2r + 1)(Ω J t + θ 0 )). we can rewrite Eq. (42) in terms of even and odd powers as (1 + ξ cos(Ω J t + θ 0 )) −2n−2p = ∞ k=0 (2n + 2p) 2k (2k)! (ξ cos(Ω J t + θ 0 )) 2k + ∞ k=0 (2n + 2p) 2k+1 (2k + 1)! (ξ cos(Ω J t + θ 0 )) 2k+1 ,(44) where the Pochhammer symbol (2n + 2p) k = Γ(2n + 2p + k)/Γ(2n + 2p). By inserting Eq. (2k + 1)! (2k − r + 1)!r! cos((2k − 2r + 1)(Ω J t + θ 0 )) sin(Ω J t + θ 0 ) . FIG. 1 . 1(Color online) Geometry of the S/F/S Josephson junction with cross-sectional area LyLz in uniform magnetic field H0 and circularly polarized magnetic field Hac.the precessional motion of the magnetization in the presence of Gilbert damping. The magnetic fluxes in the zand y-directions are given by FIG. 2 . 2(Color online) (a) Manifestation of the FMR in the frequency dependence of the maximum of magnetization component my and the average critical current density at bias current I = 1.16. Lines added to guide the eye; (b) Frequency dependence of the maximum of magnetization component my at different damping α and amplitude of circularly polarized magnetic field hac. Other parameters are the same as in (a); (c) Comparison with a linearized case at ǫJ = 0.02. FIG . 3. (Color online) (a) IV characteristic of S/F/S junction at ferromagnetic resonance. The case in Ref. 13 is shown by the dashed line for comparison, shifted by 0.8 to the right for clarity; (b) and (c) enlarge the parts of IV characteristic marked by rectangles in (a). FIG. 4 . 4(Color online) IV characteristics of the S/F/S junction at ferromagnetic resonance without oscillating electric field (A = 0) and two characteristics at amplitudes A = 0.3 and A = 1. Here Ωr = ωr/ωc, other parameters are the same as in FIG. 5 . 5(Color online) Different possibilities of the frequency locking excluding (red) and including (blue) the Josephson energy in the effective magnetic field. where the components of H 0 are (0, 0, H 0 ), with H 0 = ω 0 /γ and ω 0 is the ferromagnetic resonance frequency.The components ofH are (H x ,H y , 0). On other hand, the components of M 0 are (0, 0, M z ) and those ofM are (M x ,M y , 0). The magnitude of alternating parts are considered smaller than the steady parts, i.e.H << H 0 , M << M z . The linearization of Eq. (11) can be found by inserting Eq. (12) into Eq. (11) and neglecting the products of the alternating parts. This gives s Rˆ m,n,p [cos(2(n − m)(Ωt − ζ)) − cos(2(n + m + 1)(Ωt − ζ))] J after using the trigonometry relation sin A cos B = (1/2)[sin(A + B) + sin(A − B)], the supercurrent can be [(Ω J t + θ 0 ) + 2(n − m)(Ωt − ζ)] + sin[(Ω J t + θ 0 ) − 2(n − m)(Ωt − ζ)] − sin[(Ω J t + θ 0 ) + 2(n + m + 1)(Ωt − ζ)] − sin[(Ω J t + θ 0 ) − 2(n + m + 1)(Ωt − ζ)] 2k (2n + 2p) 2k 2 2k (2k − r)!r!   sin[(2(k − r) + 1)(Ω J t + θ 0 ) + 2(n − m)(Ωt − ζ)] + sin[(2(k − r) + 1)(Ω J t + θ 0 ) − 2(n − m)(Ωt − ζ)] − sin[(2(k − r) + 1)(Ω J t + θ 0 ) + 2(n + m + 1)(Ωt − ζ))] − sin[(2(k − r) + 1)(Ω J t + θ 0 ) − 2(n + m + 1)(Ωt − ζ)] + sin[(1 − 2(k − r))(Ω J t + θ 0 ) + 2(n − m)(Ωt − ζ)] + sin[(1 − 2(k − r))(Ω J t + θ 0 ) − 2(n − m)(Ωt − ζ)] − sin[(1 − 2(k − r))(Ω J t + θ 0 ) + 2(n + m + 1)(Ωt − ζ)] − sin[(1 − 2(k − r))(Ω J t + θ 0 ) − 2(n + m + 1)(Ωt − ζ)] [2(k − r + 1)(Ω J t + θ 0 ) + 2(n − m)(Ωt − ζ)] + sin[2(k − r + 1)(Ω J t + θ 0 ) − 2(n − m)(Ωt − ζ)] − sin[2(k − r + 1)(Ω J t + θ 0 ) + 2(n + m + 1)(Ωt − ζ)] − sin[2(k − r + 1)(Ω J t + θ 0 ) − 2(n + m + 1)(Ωt − ζ)] − sin[2(k − r)(Ω J t + θ 0 ) − 2(n − m)(Ωt − ζ)] − sin[2(k − r)(Ω J t + θ 0 ) + 2(n − m)(Ωt − ζ)] + sin[2(k − r)(Ω J t + θ 0 ) − 2(n + m + 1)(Ωt − ζ)] + sin[2(k − r)(Ω J t + θ 0 ) + 2(n + m + 1)(Ωt − ζ)]   .(46) A special case occurs if one consider H e = H 0 + H ac , where H 0 = (0, 0, ω 0 /γ), H ac = (H ac cos ωt, H ac sin ωt, 0) represents circularly polarized magnetic field in the xy-plane with amplitude H ac . In this case ∆ = 1 and Eq. 2n+1 (φ s R) [sin[Ω J t + θ 0 + 2(n − m)(Ωt − ζ)] + sin[Ω J t + θ 0 − 2(n − m)(Ωt − ζ)] − sin[Ω J t + θ 0 + 2(n + m + 1)(Ωt − ζ)] − sin[Ω J t + θ 0 − 2(n + m + 1)(Ωt − ζ)]]. D. Zwillinger, Standard Mathematical Tables and Formulae (CRC Press, Boca Raton, 2002), 33rd ed. W.C.Stewart, Appl. Phys. Lett. 12, 277 (1968). 4 D. E. McCumber, J. Appl. Phys. 39, 3113 (1968). 5 K. B. Oldham, J. Myland, and J. Spanier. An Atlas of Functions: with Equator, the Atlas Function Calculator, 2nd edi- tion (Springer, New York, 2009). 6 D. Zwillinger, Standard Mathematical Tables and Formulae, 33rd edition (CRC Press, Boca Raton, 2002). ACKNOWLEDGMENTSThe authors thank A. Buzdin, S. Maekawa, S. Takahashi, S. Hikino, I. Rahmonov, K. Kulikov, I. Bobkova, and A. Bobkov for useful discussions, and D. Kamanin, V. V. Voronov, and H. El Samman for supporting this work. The reported study was partially funded by RFBR according to the research project 18-02-00318, and the SA-JINR and Egypt-JINR collaborations. Y.M.S. and A.E.B. thank the visiting researcher program at the University of South Africa for financial support.C. RSJ EquationAccording to RCSJ model, the current through the junction in the dimensionless form is given by3,4where I 0 c is the critical current, β c = RCω c is the McCumber parameter (Here we consider β c = 0). The Modified RCSJ equation is found by inserting θ(y, z, t) = θ(t) − 8π 2 dM z y/Φ 0 + 8π 2 dM y (t)z/Φ 0 into Eq.(29) then take the integration over junction area (The final RCSJ equation in the dimensionless form is given bywhere I c = I 0 c sin(φ sy )/(φ sy ), φ sy = 4π 2 L y dM z /Φ 0 , and φ s = 4π 2 L z dM z h ac /Φ 0 . In the next section, we will use the expression for m y given by Eq.(27)in the supercurrent term in the RSJ equation to find the conditions for the appearance of current steps.D. 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[ "Holomorphic spinors and the Dirac equation", "Holomorphic spinors and the Dirac equation" ]	[ "K.-D Kirchberg ", "Berlin " ]	[]	[]	A closed spin Kähler manifold of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator is characterized by holomorphic spinors. It is shown that on any spin Kähler-Einstein manifold each holomorphic spinor is a finite sum of eigenspinors of the square of the Dirac operator. Vanishing theorems for holomorphic spinors are proved.	10.1023/a:1006537717534	[ "https://arxiv.org/pdf/math/9802050v1.pdf" ]	18,829,079	math/9802050	b4e5236a4db6faa5ffd0a54e7cf01f68814b4386	Holomorphic spinors and the Dirac equation 10 Feb 1998 October 13, 2018 K.-D Kirchberg Berlin Holomorphic spinors and the Dirac equation 10 Feb 1998 October 13, 2018 A closed spin Kähler manifold of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator is characterized by holomorphic spinors. It is shown that on any spin Kähler-Einstein manifold each holomorphic spinor is a finite sum of eigenspinors of the square of the Dirac operator. Vanishing theorems for holomorphic spinors are proved. Introduction Let M 2m be a spin Kähler manifold of complex dimension m and scalar curvature R. If M 2m is closed and R 0 := min(R) > 0, then it is known from [9] that the first eigenvalue λ 1 of the Dirac operator D satisfies the inequality λ 2 1 ≥            m m − 1 · R 0 4 , m even m + 1 m · R 0 4 , m odd. M 2m is called to be a limiting manifold iff this inequality is an equality. There are good descriptions of limiting manifolds for odd m (see [8], [10], [11], [16]). The most important result in this direction was proved by A. Moroianu which says that each limiting manifold of odd complex dimension m ≥ 3 is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. By Theorem 6 in [11], this implies that limiting manifolds of complex dimension m = 4k + 3 are just the closed Kähler-Einstein manifolds carrying a complex contact structure. Moreover, the complex projective spaces CP m are the only limiting manifolds for m = 4k + 1. In this paper we add a holomorphic characterization of limiting manifolds: If m is odd, then we prove that a closed spin Kähler manifold M 2m of positive scalar curvature is a limiting one iff M 2m is Einstein and the bundle Λ 0, m+1 2 ⊗ √ Λ m,0 admits a holomorphic section, where Λ m,0 is the canonical bundle and √ Λ m,0 the spin structure of M 2m . By Moroianu's theorem, this is an assertion on the twistor spaces of quaternionic Kähler manifolds of positive scalar curvature. In case where m is even we obtain an analogous description of limiting manifolds using a result of A. Moroianu concerning the eigenvalues of the Ricci tensor (see [16]). We recall that the situation is not clear for even m ≥ 4. In 1990 A. Lichnerowicz announced a theorem (see [13]) which asserts that in this case each limiting manifold is the product of the flat torus T 2 by a limiting manifold of complex dimension m − 1. But up to now there is no proof of this theorem. Thus, for even m, the only classification result is that of Th. Friedrich in case m = 2 (see [2]). The holomorphic characterization of limiting manifolds given here is a generalization of the holomorphic description in case m = 2 that was used by Th. Friedrich to obtain his classification. Our paper contains some other results, too. For example, we show that on any spin Kähler-Einstein manifold M 2m each holomorphic spinor is the sum of at most m + 1 eigenspinors of the square of the Dirac operator. Moreover, we obtain vanishing theorems for holomorphic spinors using basic Weitzenböck formulas of spin Kähler geometry. The author would like to thank Th. Friedrich and A. Moroianu for useful hints and discussions. Basic equations satisfied by holomorphic spinors. Let M 2m be a spin Kähler manifold of complex dimension m with complex structure J, Kähler metric g and spinor bundle S. Then the Kähler form Ω defines a canonical splitting S = S 0 ⊕ S 1 ⊕ · · · ⊕ S m (1) into holomorphic subbundles S r ∼ = Λ 0,r ⊗S o (r = 0, 1, . . . , m, ) of rank m r , where S 0 = √ Λ m,0 is the square root of the canonical bundle representing the spin structure of M 2m (see [6] or [3], Section 3.4.). Considering Ω as an endomorphism of S the action of Ω on S r is just the multiplication by i(m − 2r). Let p(X) := 1 2 (X − iJX) andp(X) := 1 2 (X + iJX) for any real vector field X and let ψ ∈ Γ(S r ). Then we have p(X) · ψ ∈ Γ(S r+1 ) andp(X) · ψ ∈ Γ(S r−1 ). Furthermore, let (X 1 , . . . , X 2m ) be any local frame of vector fields and (ξ 1 , . . . , ξ 2m ) the corresponding coframe. Using the notations g kl := g(X n , X l ), (g kl ) := (g kl ) −1 , X k := g kl X l (Einstein' s convention of summation) and g(X) := g(X, ·) the holomorphic and antiholomorphic part of the covariant derivative ∇ψ of any spinor ψ ∈ Γ(S) are locally defined by ∇ 1,0 ψ := ξ k ⊗ ∇ p(X k ) ψ = g(p(X k )) ⊗ ∇ p(X k ) ψ, ∇ 0,1 ψ := ξ k ⊗ ∇p (X k ) ψ = g(p(X k )) ⊗ ∇p (X k ) ψ,(2) respectively. This yields the decomposition ∇ψ = ∇ 1,0 ψ + ∇ 0,1 ψ(3) with ∇ 1,0 ψ ∈ Γ(Λ 1,0 ⊗ S) and ∇ 0,1 ψ ∈ Γ(Λ 0,1 ⊗ S). ψ is called to be holomorphic (antiholomorphic) iff ∇ 0,1 ψ = 0 (∇ 1,0 ψ = 0). The Dirac operator D and its Kähler twistD are locally given by D = X k · ∇ X k ,D = J(X k ) · ∇ X k .(4) In the following the operators D + and D − appear defined by D ± := 1 2 (D ∓ iD). Then it holds D + = p(X k ) · ∇ X k = p(X k ) · ∇p (X k ) = X k · ∇p (X k ) , D − =p(X k ) · ∇ X k =p(X k ) · ∇ p(X k ) = X k · ∇ p(X k )(5) and there are the operator equations D = D + + D − ,(6)D 2 + = D 2 − = 0,(7)D 2 = D + D − + D − D + .(8) Moreover, for ψ ∈ Γ(S r ), we have D ± ψ ∈ Γ(S r±1 ). Proposition 1: Let ψ ∈ Γ(S) be any holomorphic (antiholomorphic) spinor. Then ψ satisfies the equation D + ψ = 0 (D − ψ = 0) (9) and, moreover, the equation ∇p (X) D − ψ + 1 2p (Ric(X)) · ψ = 0 (∇ p(X) D + ψ + 1 2 p(Ric(X)) · ψ = 0)(10) for each real vector field X, where Ric is the Ricci tensor. Proof: By definition, ψ ∈ Γ(S) is holomorphic (antiholomorphic) iff there is the equation ∇p (X) ψ = 0 (∇ p(X) ψ = 0)(11) for each real vector field X. Thus, for example, we have 0 = X k · ∇p (X k ) ψ = D + ψ and hence (9). For any ϕ ∈ Γ(S) and any X ∈ Γ(T M 2m ), there is the well-known relation X k · C(X k , X)ϕ = 1 2 Ric(X) · ϕ,(12) where C is the curvature tensor of S. Hence, since M 2m is Kähler, it holds 1 2p (Ric(X)) · ϕ = 1 2 Ric(p(X)) · ϕ = 1 2 X k · C(X k ,p(X))ϕ = = X k · C(p(X k ),p(X))ϕ =p(X k ) · C(p(X k ),p(X))ϕ = =p(X k ) · C(X k ,p(X))ϕ. Thus, (12) implies in Kähler case the relations 1 2p (Ric(X)) · ϕ =p(X k ) · C(X k ,p(X))ϕ, 1 2 p(Ric(X)) · ϕ = p(X k ) · C(X k , p(X))ϕ. Now, let P ∈ M 2m be any point and let (X 1 , . . . , X 2m ) be a frame on a neighbourhood of P such that (∇X k ) P = 0 (k = 1, · · · , 2m). Using (5), (11), (13) and (14) we have at P ∇p (X) D − ψ + 1 2p (Ric(X)) · ψ = ∇p (X) D − ψ +p(X k ) · C(X k ,p(X))ψ = = ∇p (X) (p(X k )∇ X k ψ) +p(X k )(∇ X k ∇p (X) ψ − ∇p (X) ∇ X k ψ − ∇ [X k ,p(X)] ψ) = = −p(X k )∇ [X k ,p(X)] ψ = −p(X k )∇ ∇ X kp (X) ψ = −p(X k )∇p (∇ X k X) ψ = 0. This yields (10). ✷ Corollary 2: Let ψ ∈ Γ(S) be holomorphic (antiholomorphic). Then there is the equation D + (−) D − (+) ψ = R 4 ψ + (−) i 2 ρ · ψ,(15) where R is the scalar curvature and ρ the Ricci form. Proof: Using (10) and the first one of the well-known relations X k ·p(Ric(X k )) = − R 2 − iρ, X k · p(Ric(X k )) = − R 2 + iρ,(16) we have 0 = X k · ∇p (X k ) D − ψ + 1 2 X k ·p(Ric(X k )) · ψ = = D + D − ψ − R 4 ψ − i 2 ρ · ψ. ✷ Let ∇ * ∇ be the Bochner Laplacian on Γ(S) locally given by ∇ * ∇ = −g kl (∇ X k ∇ X l − ∇ ∇ X k X l ).(17) Then there is the well-known relation (see [12]) ∇ * ∇ = D 2 − R 4 (18) Using (8), (9), (15) and (18) we immediately obtain Corollary 3: If ψ ∈ Γ(S) is holomorphic (antiholomorphic), then ψ satisfies the equivalent equations D 2 ψ = R 4 ψ + (−) i 2 ρ · ψ,(19)∇ * ∇ψ = + (−) i 2 ρ · ψ.(20) 2 Holomorphic spinors and limiting manifolds. Proof: We recall that there are canonical unitary isomorphisms [7], Prop. 4). From Section 4 in [9] we know that limiting manifolds of odd complex dimension m are Einstein. Moreover, there is a holomorphic eigenspinor ψ ∈ Γ(S m+1 2 ) to the first eigenvalue λ 2 α r : Λ 0,r ⊗ S 0 ∼ −→ S r (r = 0, 1, . . . , m) (21) defined by α r (ω ⊗ ψ 0 ) := 2 − r 2 ω · ψ 0 (see1 = m+1 4m R of D 2 . Thus, α m+1 2 −1 • ψ is a holomorphic section of Λ 0, m+1 2 ⊗ S 0 . Conversely, let M 2m be Einstein and let ϕ ≡ 0 be a holomorphic section of Λ 0, m+1 2 ⊗ S 0 . Then ψ := α m+1 2 • ϕ ∈ Γ(S m+1 2 ) is holomorphic. Hence, using Corollary 3, the Einstein condition ρ = R 2m Ω and Ωψ = i(m − 2 · m+1 2 )ψ = −iψ we have D 2 ψ = R 4 ψ + i 2 ρψ = R 4 ψ + R 4m ψ = m + 1 4m Rψ. Thus, M 2m is a limiting manifold. ✷ The corresponding holomorphic characterization of limiting manifolds in case of even complex dimension m is more complicated since such manifolds are not Einstein for m ≥ 4. In case m = 2 the complete list of limiting manifolds is given by S 2 × S 2 and S 2 × T 2 (see [2]). In case m = 2l ≥ 4 it is known that the scalar curvature R is a (positive) constant and that there is a holomorphic eigenspinor ψ ∈ Γ(S m 2 ) to the first eigenvalue λ 2 1 = m 4(m−1) R of D 2 which additionally satisfies the eigenvalue equation ρ · ψ = −i R 2m − 2 ψ(22) (see [9], Section 4). A previous result is Proposition 5: Let M 2m be a closed spin Kähler manifold of even complex dimension m ≥ 4 with positive scalar curvature R. Then M 2m is a limiting manifold iff R is constant and S m 2 admits a holomorphic section satisfying equation (22). Proof: Let 0 ≡ ψ ∈ Γ(S m 2 ) be holomorphic and also a solution of equation (22) and let R > 0 be constant. Then, by Corollary 3, it holds D 2 ψ = R 4 ψ + i 2 ρψ = R 4 ψ + R 4(m − 1) Rψ = m 4(m − 1) Rψ. Hence, M 2m is a limiting manifold. The converse is true by the preceeding remarks. ✷ In the following we replace the condition (22) by a more geometrical one. We use the result of A. Moroianu that the Ricci tensor of a limiting manifold M 2m with m = 2l ≥ 4 has exactly the two constant eigenvalues R 2m−2 and 0 of multiplicity 2m − 2 and 2, respectively (see [16]). This property is called the condition (Ric). S r = S r,0 ⊕ S r,1 (23) into complex subbundles with rank C (S r,ε ) = m − 1 r − ε (ε = 0, 1) such that −iρ acts on S r,ε as multiplication by R 2m−2 · (m − 1 − 2(r − ε)). Proof: Let P ∈ M 2m be any point. The Ricci form ρ defines an endomorphism ρ P : S P → S P being compatible with the splitting S P = (S 0 ) P ⊕ · · · ⊕ (S m ) P . If we look at the spin representation, then there is an identification such that Ω P = m k=1 e 2k−1 · e 2k , ρ P = m k=1 ρ k (P )e 2k−1 · e 2k ,(24) where ρ 1 (P ), . . . , ρ m (P ) are the eigenvalues of the Ricci tensor at P and (e 1 , · · · , e 2m ) is the canonical basis of R 2m corresponding to a suitable orthonormal frame of T P M 2m . Let {u ε 1 ...εm \|ε 1 , · · · , ε m ∈ {1, −1}} be the standard basis of the spin module. Then there are the relations e 2k−1 · e 2k · u ε 1 ...εm = iε k · u ε 1 ...εm (k = 1, · · · , m). This yields ρ P · u ε 1 ...εm = i m k=1 ε k ρ k (P ) · u ε 1 ...εm .(26) In case of condition (Ric) we can assume that ρ 1 (P ) = · · · = ρ m−1 (P ) = R 2m−2 , ρ m (P ) = 0. Then (26) implies ρ P · u ε 1 ...εm = i R 2m − 2 m−1 k=1 ε k · u ε 1 ...εm . Now, (S r ) P corresponds to the vector space over C spanned by all u ε 1 ...εm for which exactly r of the ε k are equal to −1. Hence, corresponding to the two possibilities ε m = 1 and ε m = −1 the endomorphisms ρ P restricted to (S r ) P has exactly the two eigenvalues i R 2m−2 (m−1−2r) and i R 2m−2 (m−1−2(r −1)) of multiplicity m − 1 r and m − 1 r − 1 , respectively. ✷ Lemma 6 shows that in case of condition (Ric) a spinor ψ ∈ Γ(S m 2 ) satisfies the equation (22) iff ψ is a section of the subbundle S m 2 ,0 of S m 2 . Now we will see how this subbundle can be constructed with help of the Ricci tensor. The condition (Ric) provides an orthogonal J-invariant decomposition T M 2m = E ⊕ F,(27) where the fibres of the subbundles E and F at any point P ∈ M 2m are given by E P := ker(Ric P − R 2m−2 ) and F P := ker(Ric P ), respectively. This implies the decomposition T 0,1 * M 2m = E 0,1 * ⊕ F 0,1 * (28) with rank C (E 0,1 * ) = m − 1 and rank C (F 0,1 * ) = 1. Thus, we have Λ 0,r : = Λ r (T 0,1 * M 2m ) = Λ r (E 0,1 * ⊕ F 0,1 * ) = = (Λ r E 0,1 * ) ⊕ ((Λ r−1 E 0,1 * ) ⊗ F 0,1 * ). Hence, using the isomorphisms (21) we obtain (Λ r E 0,1 * ) ⊗ S 0 ⊕ (Λ r−1 E 0,1 * ) ⊗ F 0,1 * ⊗ S 0 ∼ = S r .(29) By construction, it holds rank C ((Λ r E 0,1 * ) ⊗ S 0 ) = m − 1 r , rank C ((Λ r−1 E 0,1 * ) ⊗ F 0,1 * ⊗ S 0 ) = m − 1 r − 1 .(30) Lemma 7: Let r ∈ {1, . . . , m − 1}. Then the isomorphism (29) induces isomorphisms (Λ r E 0,1 * ) ⊗ S 0 ∼ = S r,0 , (Λ r−1 E 0,1 * ) ⊗ F 0,1 * ⊗ S 0 ∼ = S r,1 .(31) Proof: By Lemma 6 and (30), it is sufficient to show that α r ((Λ r E 0,1 * ) ⊗ S 0 ) = S r,0 . Hence, we have to prove that ρ is the multiplication by i R 2m−2 (m − 1 − 2r) on α r ((Λ r E 0,1 * ) ⊗ S 0 ). For any local frame (X 1 , . . . , X n ), n := 2m, the Ricci form ρ acts on S by ρ = 1 2 J(X k ) · Ric(X k ) (32) (see [9], (54)). Now, let (X 1 , . . . , X n ) be orthonormal such that J(X 2k ) = X 2k−1 (k = 1, . . . , m),(33)Ric(X 2k ) = R 2m − 2 X 2k (k = 1, . . . , m − 1), Ric(X n ) = 0.(34) Then we have Ω = 1 2 n k=1 J(X k ) · X k(35) and, moreover, by (32) and (34), ρ = R 2(n − 2) · n−2 k=1 J(X k ) · X k .(36) Let us consider the 2-form η := Ω − n−2 R ρ. Then, by (33), (35) and (36), we obtain η = 1 2 (J(X n−1 ) · X n−1 + J(X n ) · X n ) = 1 2 (−X n · X n−1 + X n−1 · X n ) = X n−1 · X n . Using this we determine the action of η P on the space α r ((Λ r E 0,1 * ) ⊗ S 0 ) P = {ω · ψ 0 \|ω ∈ (Λ r E 0,1 * ) P , ψ 0 ∈ (S 0 ) P } for any P ∈ M 2m . We remark that ω ∈ (Λ r E 0,1 * ) p implies X n−1 ω = X n ω = 0. Hence, using (25) and the properties of Clifford multiplication we calculate η · (ω · ψ 0 ) = X n−1 · X n · (ω · ψ 0 ) = (−1) r X n−1 · (ω · (X n · ψ 0 )) = = (−1) r · (−1) r ω · (X n−1 · X n · ψ 0 ) = iω · ψ 0 . Thus, η is the multiplication by i on α r (Λ r E 0,1 * ⊗ S 0 ). This proves the assertion since ρ = R n−2 (Ω − η) and Ω is the multiplication by i(m − 2r) on S r . We remark that the conditon (Ric) holds obviously if M 2m = N 2m−2 × T 2 , where N 2m−2 is a limiting manifold of odd complex dimension m − 1 (see [9], Section 5). In these cases the Ricci tensor is always parallel. Hence, if the theorem of Lichnerowicz is valid, then the Ricci tensor must be parallel for each limiting manifold of even complex dimension m ≥ 4. But Theorem 8 suggests the conjecture that there are examples with non-parallel Ricci tensor. Holomorphic spinors on Einstein manifolds. Let j : S → S be the j-structure of the spinor bundle S which always exists in even real dimensions. We recall that j is parallel and anti-linear, preserves the length of spinors, commutes with Clifford multiplication by real vectors and has the properties jS r = S m−r (r = 0, 1, . . . , m),(37)j 2 = (−1) m(m+1) 2 .(38) Then we have obviously j ker(∇ 1,0 ) = ker(∇ 0,1 ) , j ker(∇ 0,1 ) = ker(∇ 1,0 ). (39) Furthermore, let λ be any eigenvalue of D 2 and let E λ (D 2 ) denote the corresponding eigenspace. Then the relation jE λ (D 2 ) = Eλ(D 2 ) (40) is also obvious. E 1,0 λ (D 2 ) := E λ (D 2 ) ∩ ker(∇ 0,1 ) E 0,1 λ (D 2 ) := E λ (D 2 ) ∩ ker(∇ 1,0 ) is the corresponding holomorphic (antiholomorphic) eigenspace. Moreover, we say that λ is holomorphic (antiholomorphic) iff E 1,0 λ (D 2 ) = 0 (E 0,1 λ (D 2 ) = 0). From (39) and (40) we see that jE 1,0 λ (D 2 ) = E 0,1 λ (D 2 ) , jE 0,1 λ (D 2 ) = E 1,0 λ (D 2 ).(41) Thus, λ is holomorphic iffλ is antiholomorphic. Proposition 9: Let M 2m be any spin Kähler-Einstein manifold. Then the sets of holomorphic eigenvalues of D 2 coincides with the set of antiholomorphic eigenvalues and is contained in { rR 2m \| r = 0, 1, . . . , m}. Moreover, it holds E 1,0 rR 2m (D 2 ) ⊆ Γ(S r ), E 0,1 rR 2m (D 2 ) ⊆ Γ(S m−r ). (r = 0, 1, . . . , m) (42) Proof: Let λ be any holomorphic eigenvalue of D 2 . Moreover, let 0 ≡ ψ ∈ E 1,0 λ (D 2 ) and ψ = ψ 0 + ψ 1 + · · · + ψ m the decomposition according to (1). Using the equation (19) and the Einstein condition ρ = R 2m Ω (43) we have 0 = λψ − D 2 ψ = λψ − R 4 ψ − i 2 ρψ = λ − R 4 ψ − R 4m iΩψ = = λ − R 4 ψ + R 4m m s=0 (m − 2s)ψ s = m s=0 λ − sR 2m ψ s and hence (λ − sR 2m )ψ s = 0 for s = 0, 1, . . . , m. Since ψ ≡ 0 there is an r with ψ r ≡ 0. This implies λ = rR 2m and ψ s = 0 for s = r. ✷ Proposition 10: Let M 2m be a spin Kähler-Einstein manifold, let r ∈ {0, 1, . . . , m} and let ψ ∈ Γ(S r ) be holomorphic (antiholomorphic). Then ψ satisfies the eigenvalue equation D 2 ψ = rR 2m ψ (D 2 ψ = (m − r)R 2m ψ).(44) Proof: For example, let ψ ∈ Γ(S r ) be holomorphic. Then the equations (19), (43) and Ωψ = i(m − 2r)ψ imply (44). ✷. Theorem 11: If M 2m is a spin Kähler-Einstein manifold of scalar curvature R, then there are the decompositions ker(∇ 0,1 ) = m r=0 E 1,0 rR 2m (D 2 ), ker(∇ 1,0 ) = m r=0 E 0,1 rR 2m (D 2 ).(45) Proof: Let ψ ∈ ker(∇ 0,1 ) and ψ = ψ 0 + ψ 1 + · · · + ψ m the decomposition according to (1). Since the splitting (1) is parallel, ψ is holomorphic iff each of its components ψ r is holomorphic. Thus, Proposition 10 implies ψ r ∈ E 1,0 rR 2m (D 2 ). ✷ From Proposition 10 and the structure (45) of the spaces of holomorphic or antiholomorphic spinors we immediately obtain Theorem 12: Let M 2m be a spin Kähler-Einstein manifold and R its scalar curvature. Then the following holds: (i) If rR 2m is not an eigenvalue of D 2 for an r ∈ {0, 1, . . . , m}, then there is no holomorphic section in the bundle S r and no antiholomorphic section in S m−r . (ii) If rR 2m is not an eigenvalue of D 2 for each r ∈ {0, 1, . . . , m}, then there are no holomorphic and no antiholomorphic spinors on M 2m . Holomorphic spinors on closed manifolds. Let Z, W be any complex vector fields on a spin Kähler manifold M 2m . Then we use the notation ∇ Z,W := ∇ Z ∇ W − ∇ ∇ Z W for the corresponding second order derivative of spinors. We consider the Kähler-Bochner Laplacians on Γ(S) locally defined by ∇ 1,0 * ∇ 1,0 := −g kl ∇p (X k ),p(X l ) , ∇ 0,1 * ∇ 0,1 := −g kl ∇ p(X k ),p(X l ) .(46) By definition, we have the inclusions ker(∇ 1,0 ) ⊆ ker(∇ 1,0 * ∇ 1,0 ) , ker(∇ 0,1 ) ⊆ ker(∇ 0,1 * ∇ 0,1 ).(47) Proposition 13: There are the operator equations 2∇ 1,0 * ∇ 1,0 = D 2 − R 4 + i 2 ρ, 2∇ 0,1 * ∇ 0,1 = D 2 − R 4 − i 2 ρ.(48) Proof: For example, we prove the first one of these equations. Let P ∈ M 2m be any point and (X 1 , . . . , X 2m ) a frame in a neighbourhood of P with property (∇X k ) P = 0 for k = 1, . . . , 2m. Using the formulas (5), (8), (13) and (16) we have at the point P 2∇ 1,0 * ∇ 1,0 = −2g kl ∇p (X k ) ∇ p(X l ) = (X k X l + X l X k )∇p (X k ) ∇ p(X l ) = = D + D − + X l X k (∇ p(X l ) ∇p (X k ) − C(p(X l ),p(X k ))) = = D + D − + D − D + − X l X k C(p(X l ),p(X k )) = = D 2 + X l p(X k )C(X k , p(X l )) = D 2 + 1 2 X l p(Ric(X l )) = = D 2 − R 4 + i 2 ρ. ✷ In 1979 M.L. Michelsohn proved Weitzenböck formulas which are very similar to (48). M.L. Michelsohn also showed that ∇ 0,1 * ∇ 0,1 and ∇ 1,0 * ∇ 1,0 are non-negative, elliptic and formally self-adjoint differential operators (see [14], Prop. 7.2, 7.6). We prefer the equations (48) since the Ricci form ρ enters here explicitely being more convenient for applications considered here. For example, we remark that Corollary 3 also follows from (47) (ii) In case R > 0 the bundles S r with r ≤ m/2 (r ≥ m/2) do not admit any holomorphic (antiholomorphic) section. In case R = 0 each holomorphic or antiholomorphic spinor is parallel. (iii) It holds ker(∇ 0,1 ) = r>m/2 E 1,0 rR 2m (D 2 ) and ker(∇ 1,0 ) = r>m/2 E 0,1 rR 2m (D 2 ) . Proof: In Einstein case the equations (19) and (44) are equaivalent for ψ ∈ Γ(S r ). Thus, assertion (i) follows from Theorem 14 immediately. Moreover, using the equation (20) and the Einstein condition (43) we see that ψ ∈ Γ(S r ) is holomorphic (antiholomorphic) iff ψ satisfies the equation ∇ * ∇ψ = − (+) m − 2r 4m Rψ which implies ∇ * ∇ψ, ψ = − (+) m − 2r 4m R\|ψ\| 2 . Integrating this equation we find \|\|∇ψ\|\| 2 = − (+) m − 2r 4m R\|\|ψ\|\| 2 . Let R > 0 and ψ ≡ 0. Then we obtain a contradiction for m > 2r (m < 2r). The case m = 2r provides ∇ψ = 0. But it is known that the existence of a parallel spinor implies Ric = 0 being a contradiction to our assumtion R > 0. Finally, in case R = 0 we always obtain ∇ψ = 0. This proves (ii). The assertion (iii) follows from (ii) and Theorem 11. ✷ An essential generalization of Theorem 15, (ii) is Theorem 16: Let M 2m be a closed non-Ricci-flat spin Kähler manifold and let ρ 1 (P ) ≥ ρ 2 (P ) ≥ · · · ≥ ρ m (P ) denote the eigenvalues of Ric at P ∈ M 2m . Then the bundle S r (r ∈ {0, 1, . . . , m}) does not admit any holomorphic (antiholomorphic) section if at each point P the condition ρ 1 (P ) + · · · + ρ r (P ) ≤ 1 4 R(P ) (ρ m−r+1 (P ) + · · · + ρ m (P ) ≥ 1 4 R(P )) (49) is satisfied. Proof: Since ρ 1 (P ) + · · · + ρ m (P ) = 1 2 R(P ), we see from (26) that the set of all eigenvalues of the endomorphism −iρ P restricted to (S r ) P is given by 1 2 R(P ) − 2(ρ i 1 (P ) + · · · + ρ ir (P ))\|1 ≤ i 1 < · · · < i r ≤ m . Thus, for any ψ ∈ Γ(S r ), the condition (49) yields −iρψ, ψ ≥ 0 ( −iρψ, ψ ≤ 0). On the other hand, if ψ ∈ Γ(S r ) is holomorphic (antiholomorphic), then (20) implies the equation \|\|∇ψ\|\| 2 = − (+) 1 2 M 2m −iρ · ψ, ψ which provides a contradiction for −iρ·ψ, ψ ≥ 0 ( −iρ·ψ, ψ ≤ 0). ✷ We remark that the assertion of Theorem 16 is also valid if the condition (49) is not satisfied on a subset of M 2m of measure zero. Moreover, in case r = 0 the condition (49) simply reduces to R ≥ 0 (R ≤ 0). Hence, if M 2m is closed and non-Ricci-flat, the bundle S 0 (S m ) does not admit any holomorphic (antiholomorphic) section in case R ≥ 0 (R ≤ 0). Theorem 16 immediately yields Corollary 17: Let M 2m be a closed non-Ricci-flat spin Kähler manifold of complex dimension m ≥ 3 with scalar curvature R ≥ 0 such that at each point P ∈ M 2m the Ricci tensor has the eigenvalues 1 2m−2 R(P ) and 0 of multiplicity 2m − 2 and 0, respectively. Then the bundles S r with r ≤ m−1 2 (r ≥ m+1 2 ) do not admit any holomorphic (antiholomorphic) section. By Theorem 8 and Corollary 17, we obtain Proposition 18: If M 2m is a limiting manifold of even complex dimension m ≥ 4, then the bundles S r with r ≤ m−2 2 (r ≥ m+2 2 ) do not admit any holomorphic (antiholomorphic) section. Theorem 4 : 4Let M 2m be a closed spin Kähler manifold of odd complex dimension m with positive scalar curvature and spin structure S 0 = √ Λ m,0 . Then M 2m is a limiting manifold iff M 2m is Einstein and the bundle Λ 0, m+1 2 ⊗ S 0 admits a holomorphic section. Lemma 6 : 6Let M 2m be a spin Kähler manifold satisfying the condition (Ric) and let r ∈ {1, . . . , m − 1}. Then there is an orthogonal splitting ✷: Let M 2m be a spin Kähler manifold of even complex dimension m satisfying the condition (Ric). Then the subbundle(Λ m 2 E 0,1 * ) ⊗ S 0 ⊂ Λ 0, m 2 ⊗ S 0constructed with help of the Ricci tensor only is not holomorphic in general. (Clearly, it is holomorphic if Ric is parallel). But also in case of a limiting manifold we can not say up to now whether this bundle must be holomorphic or not. Let M 2m be a closed spin Kähler manifold of even complex dimension m ≥ 4 and positive scalar curvature. Then M 2m is a limiting manifold iff M 2m satisfies the condition (Ric) and Λ 0, m 2 ⊗ S 0 admits a holomorphic section which simultaneously is a section of the subbundle (Λ m 2 E 0,1 * ) ⊗ S 0 . and (48) immediately. 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Geom. 11Kirchberg, K.-D.: Killing spinors on Kähler manifolds, Ann. Global Anal. Geom. 11 (1993), 141-164. Complex contact structures and the first eigenvalue of the Dirac operator on Kähler manifolds. K.-D Kirchberg, U Semmelmann, GAFA. 53Kirchberg, K.-D.; Semmelmann, U.: Complex contact structures and the first eigenvalue of the Dirac operator on Kähler manifolds, GAFA 5 No. 3 (1995), 604-618. A Lichnerowicz, Spineurs harmoniques. 257Lichnerowicz, A.: Spineurs harmoniques, C.R. Acad. Sci. Paris, Série A-B 257 (1963), 7-9. La premiére valeur de l'opérateur de Dirac pour une variété kählérienne et son cas lemite. A Lichnerowicz, C.R. Acad. Sci. Paris, Série I. Lichnerowicz, A.: La premiére valeur de l'opérateur de Dirac pour une variété kählérienne et son cas lemite, C.R. Acad. Sci. Paris, Série I 311 (1990), 717-722. Clifford and spinor cohomology of Kähler manifolds. M L Michelsohn, Amer. J. Math. 102Michelsohn, M.L.: Clifford and spinor cohomology of Kähler manifolds, Amer. J. Math. 102 (1980), 1084-1146. La premiére valeur de l'opérateur de Dirac sur les variétés kählerériennes compactes. A Moroianu, Comm. Math. Phys. 169Moroianu, A.: La premiére valeur de l'opérateur de Dirac sur les variétés kählerériennes compactes, Comm. Math. Phys. 169 (1995), 373-384. On Kirchberg's inequality for compact Kähler manifolds of even complex dimension. A Moroianu, Ann. Global Anal. Geometry. 15Moroianu, A.: On Kirchberg's inequality for compact Kähler manifolds of even complex dimension, Ann. Global Anal. Geometry 15 (1997), 235-242.	[]
[ "CCD BVRI and 2MASS photometry of the poorly studied open cluster NGC 6631", "CCD BVRI and 2MASS photometry of the poorly studied open cluster NGC 6631" ]	[ "A L Tadross \nNational Research Institute of Astronomy & Geophysics\nNRIAG\nElmarsad St11421Helwan, CairoEgypt\n", "R Bendary \nNational Research Institute of Astronomy & Geophysics\nNRIAG\nElmarsad St11421Helwan, CairoEgypt\n", "P Hasan \nMuffakham Jah College of Engineering and Technology\nof National Research Institute of Astronomy and Geophysics\nBanjara Hills\n500 034HyderabadIndia\n", "A Essam \nNational Research Institute of Astronomy & Geophysics\nNRIAG\nElmarsad St11421Helwan, CairoEgypt\n", "A Osman \nNational Research Institute of Astronomy & Geophysics\nNRIAG\nElmarsad St11421Helwan, CairoEgypt\n" ]	[ "National Research Institute of Astronomy & Geophysics\nNRIAG\nElmarsad St11421Helwan, CairoEgypt", "National Research Institute of Astronomy & Geophysics\nNRIAG\nElmarsad St11421Helwan, CairoEgypt", "Muffakham Jah College of Engineering and Technology\nof National Research Institute of Astronomy and Geophysics\nBanjara Hills\n500 034HyderabadIndia", "National Research Institute of Astronomy & Geophysics\nNRIAG\nElmarsad St11421Helwan, CairoEgypt", "National Research Institute of Astronomy & Geophysics\nNRIAG\nElmarsad St11421Helwan, CairoEgypt" ]	[]	Here we have obtained the BVRI CCD photometry down to a limiting magnitude of V $ 20 for the southern poorly studied open cluster NGC 6631. It is observed from the 1.88 m Telescope of Kottamia Observatory in Egypt. About 3300 stars have been observed in an area of $ 10 0 Â 10 0 around the cluster center. The main photometric parameters have been estimated and compared with the results that determined for the cluster using JHKs 2MASS photometric database. The cluster's diameter is estimated to be 10 arcmin; the reddening EðB À VÞ ¼ 0:68 ± 0.10 mag, EðJ À HÞ ¼ 0:21 AE 0.10 mag, the true modulus ðm À MÞ o ¼ 12:16 AE 0.10 mag, which corresponds to a distance of 2700 AE 125 pc and age of 500 AE 50 Myr.ª 2014 Production and hosting by Elsevier B.V. on behalf	10.1016/j.nrjag.2014.11.003	null	119,285,802	1403.0546	8cdead15b5d896a8736d946695417d0adeb3a01a	CCD BVRI and 2MASS photometry of the poorly studied open cluster NGC 6631 Available online 22 January 2015 A L Tadross National Research Institute of Astronomy & Geophysics NRIAG Elmarsad St11421Helwan, CairoEgypt R Bendary National Research Institute of Astronomy & Geophysics NRIAG Elmarsad St11421Helwan, CairoEgypt P Hasan Muffakham Jah College of Engineering and Technology of National Research Institute of Astronomy and Geophysics Banjara Hills 500 034HyderabadIndia A Essam National Research Institute of Astronomy & Geophysics NRIAG Elmarsad St11421Helwan, CairoEgypt A Osman National Research Institute of Astronomy & Geophysics NRIAG Elmarsad St11421Helwan, CairoEgypt CCD BVRI and 2MASS photometry of the poorly studied open cluster NGC 6631 Available online 22 January 201510.1016/j.nrjag.2014.11.003Received 24 February 2014; revised 25 October 2014; accepted 26 November 2014Galaxy: open clusters and associationsIndividual: NGC 6631AstrometryStars: luminosity functionMass function Here we have obtained the BVRI CCD photometry down to a limiting magnitude of V $ 20 for the southern poorly studied open cluster NGC 6631. It is observed from the 1.88 m Telescope of Kottamia Observatory in Egypt. About 3300 stars have been observed in an area of $ 10 0 Â 10 0 around the cluster center. The main photometric parameters have been estimated and compared with the results that determined for the cluster using JHKs 2MASS photometric database. The cluster's diameter is estimated to be 10 arcmin; the reddening EðB À VÞ ¼ 0:68 ± 0.10 mag, EðJ À HÞ ¼ 0:21 AE 0.10 mag, the true modulus ðm À MÞ o ¼ 12:16 AE 0.10 mag, which corresponds to a distance of 2700 AE 125 pc and age of 500 AE 50 Myr.ª 2014 Production and hosting by Elsevier B.V. on behalf Introduction Open star clusters (OCs) are ideal objects for studying the main properties of the Milky Way Galaxy, i.e. star formation, stellar evolution, and distance scale of the Galaxy. The fundamental parameters of an open cluster; e.g. distance, age, and interstellar extinction; can be determined by comparing color-magnitude (CM) and color-color (CC) diagrams with the modern theoretical models. As we know, more than half of the currently cataloged open clusters ($2000 OCs) have been poorly studied or even unstudied at all. The current paper is thus a part of our goal to obtain good-quality photometric data for poorly studied or unstudied OCs and to estimate their fundamental parameters more accurately. The open cluster NGC 6631 located in the direction of Galactic center at constellation SCT-Scutum at 2000.0 coordinates a ¼ 18 h 27 m 11 s ; d ¼ À12 01 0 48 00 ; ' ¼ 19:47 ; b ¼ À0:19 . The cluster is classified as II2m by Ruprecht (1966) and as II1m by Lynga˚and Palous (1987). Alter et al. (1970) has presented some rough estimations for angular diameter of this cluster ranging from 4 to 16 arcmin, while the distance varies from 880 to 5000 pc. We can say that no real study has been done for this cluster before Sagar et al. (2001). They presented the first VI CCD photometric study of NGC 6631 and estimated the cluster main parameters by getting the best fit of the theoretical isochrones on V $ ðV À IÞ diagram of the cluster. They obtained distance = 2.6 AE 0.5 kpc, age = 400 AE 100 Myr, and EðV À IÞ ¼ 0:60 AE 0:05 mag. The cluster diameter estimated from the radial density profile was 4.8 AE 0.5 pc, while the mass function of the cluster has a slope of 2.1 AE 0.5. This paper is organized as follows: Observations and data reduction are presented in Section 2. Radial density profile of the cluster is described in Section 3. Analysis of the colormagnitude diagrams is declared in Section 4. The mass function of the cluster is estimated and discussed in Section 5. Finally, the conclusion of our study is devoted to Section 6. Observations and data reductions The BVRI CCD photometric observations of the star cluster NGC 6631 were obtained during June 12-14, 2012, using the Newtonian focus scale 22.53 arcsec per mm of the 1.88 m Reflector Telescope at Kottamia Observatory in Egypt. Fig. 1 represents the image of the cluster in I band. The characteristics of the CCD camera are listed in Table 2, while the observation log is given in Table 3; for more details, see Azzam et al. (2010). The images were bias subtracted and flat-fielded using standard procedures in IRAF and the photometry was done using IRAF/DAOPHOT (Stetson, 1987(Stetson, , 1992. The standard stars field SA 110 230 (Landolt, 1992) was observed for standardization and the APPHOT photometry was used to derive the observed magnitudes. Extinction coefficients and zero points were determined to standardize the data. Fig. 2 shows the magnitude versus error for the BVRI bands from the DAOPHOT photometry. The total number of stars in the B, V, R and I bands are 1259, 1806, 3043 and 3250 respectively. The limits of errors in our final photometry are 0.363, 0.068, 0.034 and 0.094 respectively. The adopted calibration equations using the transformation coefficients between instrumental and standard magnitude are the following equations: b ¼ B þ z b þ k b Â X þ a b Â ðB À VÞ v ¼ V þ z v þ k v Â X þ a v Â ðB À VÞ r ¼ R þ z r þ k r Â X þ a r Â ðV À RÞ i ¼ I þ z i þ k i Â X þ a i Â ðV À IÞ where B, V, R and I are standard magnitudes and b,v,r and i are the instrumental magnitudes respectively, z b , z v , z r and z i are the zero point in b, v, r and i respectively, X is the air mass for each filter and k b , k v , k r and k i are the extinction coefficients in B, V, R and I filters respectively. The values of the zero point, the color coefficients and the extinction coefficients for each filter are listed in Table 1. On the other hand, the near-IR JHK data are taken from the digital Two Micron All Sky Survey (2MASS) of Skrutskie et al. (2006). It is uniformly scanning the entire Sky in three IR bands J (1.25 lm), H (1.65 lm) and Ks (2.17 lm). Data extraction has been performed at a preliminary radius of 10 arcmin using the known tool of VizieR for 2MASS 1 database. A cutoff of photometric completeness limit at J P 16:5 mag is applied to the data to avoid the over-sampling (cf. Bonatto et al., 2004). Also, for photometric quality, stars with errors in J, H and Ks bigger than 0.20 mag have been excluded (cf. Tadross, 2011 and references therein). 3. The radial density profile of the cluster To establish the radial density profile (RDP) of NGC 6631, we counted the stars of the cluster (taken from 2MASS database) within concentric shells in equal incremental steps of 0.1 arcmin from the cluster center. We repeated this process for 0:1 < r 0:2 up to 10 arcmin, i.e. the stellar density is derived out to the preliminary radius of the cluster. The stars of the next steps should be subtracted from the later ones, so that we obtained only the amount of the stars within the relevant shell's area, not a cumulative count. Finally, we divided the star counts in each shell to the area of that shell those stars belong to. The density uncertainties in each shell were calculated using Poisson noise statistics. Fig. 3 shows the RDP for NGC 6631 to the maximum angular separation of 5 arcmin where the decay becomes asymptotically at that point, i.e. reaching the stability with the background density. Applying the empirical King model (1966), it parameterizes the density function qðrÞ as follows: qðrÞ ¼ f bg þ f 0 1 þ ðr=r c Þ 2 where f bg ; f 0 and r c are background, central star density and the core radius of the cluster respectively. In this context, f bg $ 36 stars per arcmin 2 , f 0 ¼ 42 stars per arcmin 2 , and r c ¼ 0:59 arcmin. According to the next section, consequently, the radial diameter of the cluster is determined to be 7.85 pc. Color-magnitude diagrams The Color-Magnitude Diagrams (CMDs) of the observed stars: V $ ðB À VÞ; V $ ðV À IÞ, V $ ðV À RÞ, R $ ðR À IÞ, and of the obtained JHK-2MASS: J $ ðJ À HÞ and K $ ðJ À KÞ are constructed for the cluster. The theoretical isochrones of Padova 2 that computed by Marigo et al. (2008) are used in fitting processes. Several solar isochrones (Z $ 0:02) of different ages have been applied to the CMDs of NGC 6631. The best fittings for BVRI diagrams are obtained at distance modulus of 14.20 AE 0.10 mag, age of 500 AE 50 Myr, and reddening of 0.68, 1.00, 0.54, and 0.47 AE 0.10 mag respectively, from left to right as shown in Fig. 4. On the other hand, the fittings for JHK-2MASS diagrams are obtained at distance modulus of 12.75 AE 0.10 mag, age of 500 AE 50 Myr, and reddening of 0.21 and 0.33 AE 0.10 mag, from left to right as shown in Fig. 5. The resulting total visual absorption is taken from the ratio A v =EðB À VÞ ¼ 3:1, following Garcia et al. (1988). JHK-2MASS data have been corrected for interstellar reddening using the coefficient ratios A J A V ¼ 0:276 and A H A V ¼ 0:176, which were derived from absorption rations in Schlegel et al. (1998), while the ratio A Ks AV ¼ 0:118 was derived from Dutra et al. (2002). Applying the calculations of Fiorucci and Munari (2003) for the color excess of 2MASS photometric system, we ended up with the following results: E JÀH E BÀV ¼ 0:309 AE 0:130, EJÀK s EBÀV ¼ 0:485 AE 0:150, where R V ¼ AV EBÀV ¼ 3:1. Also, we can de-reddened the distance modulus using the following formulae: AJ EBÀV ¼ 0:887, AK s EBÀV ¼ 0:322. Therefore, the true distance modulus is calculated to be ðV À M v Þ o ¼ Fig. 2 The BVRI errors of the observed magnitudes for the stars of NGC 6631. where N is the number of stars used in the density estimation at that point). The background field density f bg $ 36 stars per arcmin 2 . The core radius r c ¼ 0:59 arcmin. 12:16 AE 0:10 mag, corresponding to a distance of 2700 AE 125 pc. After estimating the cluster's distance from the Sun, R , the distance from the galactic center (R g ), the projected distances on the galactic plane from the Sun (X &Y ) and the distance from the galactic plane (Z ) are estimated to be 6000, À2545, 900 and À8.95 pc respectively. The mass function of NGC 6631 It is difficult to determine the membership of the cluster using only the stellar RDP. The stellar membership is found more precisely for those stars are closer to the cluster's center and in the same time very near to the main-sequence (MS) in CMDs. These MS stars are very important in determining the luminosity and mass functions of the investigated cluster. The number of stars per luminosity interval, or in other words, the number of stars in each magnitude bin, gives us what so-called the luminosity function (LF) of the cluster. In order to estimate the LF of NGC 6631, we count the observed Fig. 4 Theoretical BVRI-isochrones fit to the observed CMDs of NGC 6631. The distance modulus is found to be 14.20 mag, and the color excesses are found to be (from left to right) 0.68, 1.00, 0.54 and 0.47 mag respectively. Fig. 5 Theoretical JHK-isochrones fit to the obtained CMDs of NGC 6631. The distance modulus is found to be 12.75 mag, and the color excesses are found to be (from left to right) 0.21 and 0.33 mag respectively. Fig. 6 The BVRI and JHK mass functions of NGC 6631. The slopes of the two panels are close to Salpeter's value, see Section 5. stars in terms of absolute magnitude after applying the distance modulus. The magnitude bin intervals are selected to include a reasonable number of stars in each bin and for the best possible statistics of the luminosity and mass functions. From LF, we can infer that the massive bright stars seem to be centrally concentrated more than the low masses and fainter ones (Montgomery et al., 1993). The LF and the mass function (MF) are correlated to each other according to the known mass-luminosity relation. The accurate determination of both of them (LF and MF) suffers from some problems e.g. the contamination of field stars; the observed incompleteness at low-luminosity (or low-mass) stars; and mass segregation, which may affect even poorly populated, relatively young clusters (Scalo, 1998). On the other hand, the properties and evolution of a star are closely related to its mass, so the determination of the initial mass function (IMF) is needed, that is an important diagnostic tool for studying large quantities of star clusters. IMF is an empirical relation that describes the mass distribution (a histogram of stellar masses) of a population of stars in terms of their theoretical initial mass (the mass they were formed with). The IMF is defined in terms of a power law as follows: dN dM / M Àa where dN dM is the number of stars of mass interval (M:M + dM), and a is a dimensionless exponent. The IMF for massive stars (> 1M ) has been studied and well established by Salpeter (1955), where a ¼ 2:35. This form of Salpeter shows that the number of stars in each mass range decreases rapidly with increasing mass. Fig. 6 shows the BVRI and JHK mass functions of NGC 6631, where the slopes of the two MFs close to Salpeter's value. The right panel of Fig. 6 seems to complete the left panel. The mean slope of the mass function taken to be 2.3 AE 0.05. Conclusion The open star cluster NGC 6631 has been observed using BVRI pass-band of the 1.88 m Kottamia Telescope of Egypt. The main astrophysical properties of the cluster have been estimated and confirmed by the JHK 2MASS bass-band data. It is noted that the determination of the cluster radius made by the uniformity of 2MASS database allows us to obtain reliable data on the projected distribution of stars for large extensions to the clusters' halos. However, a comparison between the results of the present work with those of Sagar et al. (2001) is given in Table 4. Fig. 1 1The CCD I image of open star cluster NGC 6631 as observed by 1.88 m Kottamia Telescope of Egypt. North is up, East on the left. Fig. 3 3The radial density distribution of the stars in NGC 6631. The decay of the density reaches a value of q ¼ 37 stars/arcmin 2 at 5.0 arcmin, where the decay becomes asymptotic. The curved solid line represents the fitting of King (1966) model. Error bars are determined from sampling statistics (1= ffiffiffiffi N p Table 1 1The zero point, color coefficients and extinction coefficients for each filter.Filter k a z b 0.26 0.0043 3.240 v 0.16 0.0021 2.755 r 0.09 0.0010 2.650 i 0.05 À0.003 3.108 Table 2 2The characteristics of CCD camera used in the observations.Type EEV CCD 42-40 Version Back-illuminated with BPBC (Basic Process Broadband Coating) Format 2048 Â 2048 pixel 2 Pixel size 13.5 Â 13.5 lm 2 Grade 0 Dynamic range 30:1 A/D converter 16 bit Imaging area 27.6 Â 27.6 mm 2 Read out noise@20 KHz 3.9 e À = pixel Gain 2.26 e À =ADU (Left amplifier) and 2.24 e À =ADU (right amplifier) Table 3 3The log of optical photometric CCD observations.Filter No. of frames Exposure time in seconds B 8 6 0 3 8 0 4 120 V 4 120 R 8 120 I 4 120 Table 4 4Comparison between the present and previous studies.Parameter The present work Sagar et al. (2001) EðV À IÞ 1.00 AE 0.10 mag 0.60 AE 0.05 mag EðB À VÞ 0.68 AE 0.10 mag - EðV À RÞ 0.54 AE 0.10 mag - EðJ À HÞ 0.21 AE 0.10 mag - EðJ À KÞ 0.33 AE 0.10 mag - ðm À MÞ BVRI 14.20 AE 0.10 mag 13.50 AE 0.30 mag ðm À MÞ JHK 12.75 AE 0.10 mag - Distance (R ) 2700 AE 125 pc 2600 AE 500 pc R g 6000 pc - X -2545 pc - Y 900 pc - Z -8.95 pc -- Age 500 AE 50 Myr 400 AE 100 Myr Metallicity (Z) 0.02 0.05 Radius 5.0 arcmin 3.2 arcmin Linear diameter 7.85 pc 4.80 pc Core radius %0.59 arcmin ($0.5 pc) - Mass function slope 2.3 AE 0.05 2.1 AE 0.50 http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=II/246. http://stev.oapd.inaf.it/cgi-bin/cmd. AcknowledgmentsThis paper is a part of the Project No. STDF-1335; funded by Science & Technology Development Fund (STDF) under the Egyptian Ministry for Scientific Research. The project team expresses their deep appreciation to the administrators of STDF and its organization. Catalogue of Star Clusters and Associations. G Alter, Budapestsecond ed. 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[ "Inflation in models with Conformally Coupled Scalar fields: An application to the Noncommutative Spectral Action", "Inflation in models with Conformally Coupled Scalar fields: An application to the Noncommutative Spectral Action" ]	[ "Michel Buck \nDepartment of Physics\nKing's College London\nWC2R 2LSLondonStrandU.K\n", "Malcolm Fairbairn \nDepartment of Physics\nKing's College London\nWC2R 2LSLondonStrandU.K\n", "Mairi Sakellariadou \nDepartment of Physics\nKing's College London\nWC2R 2LSLondonStrandU.K\n" ]	[ "Department of Physics\nKing's College London\nWC2R 2LSLondonStrandU.K", "Department of Physics\nKing's College London\nWC2R 2LSLondonStrandU.K", "Department of Physics\nKing's College London\nWC2R 2LSLondonStrandU.K" ]	[]	Slow-roll inflation is studied in theories where the inflaton field is conformally coupled to the Ricci scalar. In particular, the case of Higgs field inflation in the context of the noncommutative spectral action is analyzed. It is shown that while the Higgs potential can lead to the slow-roll conditions being satisfied once the running of the self-coupling at two-loops is included, the constraints imposed from the CMB data make the predictions of such a scenario incompatible with the measured value of the top quark mass. We also analyze the rôle of an additional conformally coupled massless scalar field, which arises naturally in the context of noncommutative geometry, for inflationary scenarios.	10.1103/physrevd.82.043509	[ "https://arxiv.org/pdf/1005.1188v2.pdf" ]	119,118,521	1005.1188	b52c9184a553f051e28df1777bc39a48590b3b2f	Inflation in models with Conformally Coupled Scalar fields: An application to the Noncommutative Spectral Action Aug 2010 Michel Buck Department of Physics King's College London WC2R 2LSLondonStrandU.K Malcolm Fairbairn Department of Physics King's College London WC2R 2LSLondonStrandU.K Mairi Sakellariadou Department of Physics King's College London WC2R 2LSLondonStrandU.K Inflation in models with Conformally Coupled Scalar fields: An application to the Noncommutative Spectral Action Aug 2010 Slow-roll inflation is studied in theories where the inflaton field is conformally coupled to the Ricci scalar. In particular, the case of Higgs field inflation in the context of the noncommutative spectral action is analyzed. It is shown that while the Higgs potential can lead to the slow-roll conditions being satisfied once the running of the self-coupling at two-loops is included, the constraints imposed from the CMB data make the predictions of such a scenario incompatible with the measured value of the top quark mass. We also analyze the rôle of an additional conformally coupled massless scalar field, which arises naturally in the context of noncommutative geometry, for inflationary scenarios. I. INTRODUCTION Cosmological inflation is the most widely accepted mechanism to resolve the shortcomings of the standard Hot Big Bang model. This mechanism, leading to a phase of exponential expansion in the very early universe, is deeply rooted in the fundamental principles of General Relativity and Field Theory, and once combined with the principles of Quantum Mechanics, it can account for the origin of the observed large scale structures and the measured temperature anisotropies of the Cosmic Microwave Background (CMB). However, despite its success, cosmological inflation remains a paradigm in search of a model which should be motivated by a fundamental theory. The strength of the inflationary mechanism is based on the assumption that its onset is generically independent of the initial conditions. Nevertheless, even this issue is under debate [1][2][3][4][5][6] given the lack of a complete theory of Quantum Gravity. The inflaton field (usually a scalar field) is assumed to dominate the evolution of the universe at early times, but its origin and the form of its effective potential both remain unknown; for this reason it would be attractive if the one scalar field that is commonly thought to exist, namely the Higgs field, also doubled as the long-searched for inflaton. Unfortunately, it seems that if the Higgs field is minimally coupled to gravity this cannot be achieved, which has led some authors to consider large nonminimal couplings of the Higgs field to gravity where inflation might be achieved [7]. It is commonly assumed/chosen that there is no coupling (i.e., minimal coupling) between the inflaton field and the background geometry (the Ricci curvature). However, this assumption/choice seems to lack a solid justification. A first (and merely aesthetic) motivation comes from the observation that in the early universe (where masses are negligible), the equations of motion * [email protected] † [email protected] ‡ [email protected] for spinors and gauge bosons have a natural conformal invariance in four space-time dimensions, while the same is true for scalar fields only when they couple to the Ricci scalar in a specific way. More compelling is the fact that even if classically the coupling between the scalar field and the Ricci curvature could be set equal to zero, a nonminimal coupling will be induced once quantum corrections in the classical field theory are considered. Moreover, a nonminimal coupling seems to be needed in order to renormalize the scalar field theory in a curved space-time. The precise value of the coupling constant (denoted by ξ) then depends on the choice of the theory of gravity and the scalar field [8]. It has also been argued that in all metric theories of gravity, including General Relativity, in which the scalar field is not part of the gravitational sector (e.g., when the scalar field is the Higgs field), the coupling constant should be conformal in order for the short distance propagators of the theory to match those found in a Minkowski space-time -a requirement of the strong equivalence principle [8,9] (in our notation, conformal coupling means ξ = 1/12). Finally, in the context of finite theories at one-loop level, it was shown [10,11] that the nonminimal coupling ξ tends either to its conformal value or increases exponentially in modulus, depending on the specific structure of the theory. In what follows, we will investigate whether scalar fields, and in particular the Higgs field, could play the rôle of the inflaton in the presence of a small positive nonminimal coupling between the scalar field and the background geometry. The coupling constant ξ is not a free parameter which could be tuned to achieve a successful inflationary scenario avoiding severe fine-tuning of inflationary parameters (e.g., the self-coupling of the inflaton field), ξ should instead be dictated by the underlying theory. For negative values of ξ, exponential expansion is more easily achieved than in the minimal case, and it can in fact lead to inflation consistent with observational data in the strong coupling limit [7,12]. In fact, the slow-roll parameters for large \|ξ\| are independent of ξ and only depend on the number of e-folds. However, exponential expansion is less favoured for positive values (in our conventions) such as conformal coupling [13]. In light of the motivations for a small positive ξ outlined above, we will investigate whether quantum corrections to the Higgs potential can lead to a slow-roll inflationary era and if so, whether the constraints imposed from the CMB temperature anisotropies are satisfied. We will apply this analysis to the Spectral Action of NonCommutative Geometry (NCG). This theory leads naturally to a Lagrangian with a conformal coupling between the Higgs field and the background geometry, in the form of a boundary condition at high energies E ≥ Λ, where Λ is a characteristic scale of the model. NCG provides an elegant way of accounting for the Standard Model (SM) of Particle Physics and its phenomenology [14]. Our motivation is to investigate cosmological consequences of the NCG Spectral Action and, in particular, to test whether slow-roll inflation driven by one of the scalar fields arising naturally within NCG could be realized in agreement with experimental data and astrophysical measurements. In a previous study, we (one of us and a collaborator) have studied [15] the conditions on the couplings so that the Higgs field could play the rôle of the inflaton in the context of the NCG. Since however the running of couplings with the cut-off scale had been only analyzed [14] neglecting the nonminimal coupling between the Higgs field and the curvature, we were not able to reach a definite conclusion. In this respect, the study below is a follow-up of Ref. [15]. Moreover, it has been argued [16] that inflation with a conformally coupled Higgs boson could be realized in the context of NCG due to the running of the effective gravitational constant. In what follows, we will also analyze the validity of this statement. Finally, the NCG Spectral Action provides, in addition to the Higgs field, another conformally coupled (massless) scalar field, which exhibits no coupling to the matter sector [17]. One may a priori wish/expect that this field could be another candidate for the inflaton; we will examine this scenario as well. Concluding, we analyze slow-roll inflation within models that exhibit a conformal coupling between the Higgs field and the Ricci curvature. Our motivation is to investigate whether any of the two scalar fields arising naturally within the NCG Spectral Action could be identified as the inflaton. As we will explicitly show, our analysis leads us to the conclusion that unfortunately such a slow-roll inflationary scenario fails to remain in agreement with current data from high energy physics experiments and astrophysical measurements. This paper is organized as follows: In Section II, we study the issue of the realization of slow-roll inflation within theories with a nonminimal coupling between the scalar field and the Ricci curvature, classically. The analysis is first performed in the Jordan frame and then in the Einstein frame. In Section III, we consider corrections to the Higgs potential through two-loop renormalization group analysis of the minimally coupled Standard Model; we then enlarge this study in the case of a conformal coupling. We focus on the gravitational and Higgs field sector of the Lagrangian density, obtained within the noncommutative spectral action, which has a conformal coupling, in Section IV. We find that even though we can accommodate an era of slow-roll inflation, it seems difficult to reach an agreement with the CMB data. This conclusion holds not only for the Higgs field but also for the other scalar field which appears generically in the theory. We then examine in Section V, whether running of the gravitational constant could modify our conclusions with regards to the realization of a successful inflationary scenario driven through one of the scalar fields in the NCG theory. We round up our conclusions in Section VI. Our signature convention is (− + + +); the Riemann and Ricci tensors are defined as R σ µνρ = Γ σ µρ,ν − Γ σ νρ,µ + Γ τ µρ Γ σ τ ν − Γ τ νρ Γ σ τ µ , R µν = R ρ µρν , respectively. Note that within our definition of ξ, conformal coupling means ξ = 1/12. II. SLOW-ROLL INFLATION WITH NONMINIMALLY COUPLED SCALAR FIELDS In this section, we will study whether slow-roll inflationary scenarios can be realized within models with an implicit nonminimal coupling between the inflaton field and the scalar curvature. We will first work in the Jordan frame and then we will perform the analysis in the Einstein frame. The Jordan frame is natural (physical) and offers some useful insights on the effect of conformal coupling, while the Einstein frame is mathematically more convenient, especially when including more complicated corrections to the potential. A. Analysis in the Jordan frame Let us consider the action of a Higgs boson (or any other scalar field φ) nonminimally coupled to gravity: S = d 4 x √ −g 1 2κ 2 f (φ)R − 1 2 (∇φ) 2 − V (φ) ,(1) where f (φ) = 1 − 2κ 2 ξφ 2 , with κ ≡ √ 8πG = m −1 Pl and g being the determinant of the metric tensor. The scalar potential of φ is: V (φ) = λφ 4 − µ 2 φ 2 .(2) The term −ξφ 2 R in the action encodes the explicit nonminimal coupling of the scalar field φ to the Ricci curvature R. The background geometry during inflation is of the Fridemann-Lemaître-Robertson-Walker (FLRW) form: ds 2 = dt 2 − a 2 (t)dΣ ,(3) where t stands for cosmological time, a(t) is the scale factor and dΣ describes spatial sections of constant curvature. Einstein's equations read R µν − 1 2 g µν R = κ 2 f (φ) −1 T µν (φ) ,(4) where the energy-momentum tensor, obtained by varying the action with respect to the metric, is [18,19] T µν (φ) = (1 − 4ξ) ∇ µ φ∇ ν φ + 4ξφ (g µν − ∇ µ ∇ ν ) φ +g µν − 1 2 − 4ξ ∇ ρ φ∇ ρ φ − V (φ) . (5) Here ≡ g µν ∇ µ ∇ ν is the Laplace-Beltrami operator and Greek and Latin indices take values 0,1,2,3 and 1,2,3, respectively 1 . The equation of motion (Klein-Gordon equation) of the Higgs field reads φ − 2ξRφ − dV dφ = 0 .(6) For vanishing and quartic potentials, Eq. (6) is invariant under conformal transformations g µν → Ω(x) 2 g µν and φ → Ω(x) −1 φ at conformal coupling ξ = 1/12. For a FLRW background and a spatially homogeneous φ, Eqs. (4),(6) combine to H 2 = κ 2 3f (φ) 1 2φ 2 + V (φ) + 12ξHφφ ,(7)0 =φ + 3Hφ − 2ξ(1 − 12ξ)κ 2 φφ 2 1 − 2ξ(1 − 12ξ)κ 2 φ 2 + 8ξκ 2 φV (φ) + f (φ)V ′ (φ) 1 − 2ξ(1 − 12ξ)κ 2 φ 2 ,(8) where overdots denote time derivatives and primes stand for derivatives with respect to the argument (e.g., V ′ (φ) ≡ dV /dφ). Note that 2ξ(1 − 12ξ) is zero at both, minimal (i.e., ξ = 0) and conformal (i.e., ξ = 1/12) couplings. Inflationary models are usually built upon the slow-roll approximation, consisting of neglecting the most slowly varying terms in the equation of motion for the inflaton field. However, in the case of nonminimal coupling (i.e., ξ = 0), it is more difficult to achieve the slow-rolling of the inflaton field. More precisely, the nonminimal coupling term in the action, −ξφ 2 R, plays the rôle of an effective mass term for the scalar field, distorting the flatness of the scalar potential. Thus, in the case of a nonminimal coupling, inflationary requirements such as −Ḣ < H 2 (where H denotes the Hubble parameter) do 1 Note that it is really the tensor T µν (φ) = f (φ) −1 T µν (φ) which is covariantly conserved rather than T µν (φ) [13], but this ambiguity in the choice of the energy-momentum tensor will not be relevant in our analysis. not translate in an equally straight-forward manner to relations on the inflaton fields and their scalar potentials. Indeed, there is no common choice of conditions (see e.g., Refs. [18,20,21]), and no analog of slow-roll parameters in terms of which quantities such as the number of e-folds of expansion or perturbation amplitudes are evaluated. With a tentative choice of conditions [18] \|φ φ \| ≪ H, \|φ φ \| ≪ H and 1 2φ 2 ≪ V (φ) ,(9) and a negligible mass term in the potential at high energies, the energy constraint, Eq. (7), and field equation, Eq. (8), reduce to H 2 ≈ λκ 2 φ 4 3f (φ) 1 − 16ξ 1 − 2ξ(1 − 12ξ)κ 2 φ 2 ,(10)3Hφ ≈ − 4λφ 3 1 − 2ξ(1 − 12ξ)κ 2 φ 2 ,(11) respectively. These equations determine the background solution, given by a(φ) = 1 − 2ξκ 2 φ 2 1 4 exp − 1 − 12ξ 8 κ 2 φ 2 .(12) It is the second factor, in Eq. (12) above, which has the potential to generate sufficient number of e-folds, as the first one will only lead to logarithmic corrections. For ξ = 1/12 (i.e., nonconformal coupling), a large enough change in \|ξ\|κφ can lead to sufficient inflation to resolve the horizon problem. This leaves some room to play with the coupling and the field values, and it has indeed been shown [12,22] in recent literature that inflation can be achieved in a manner consistent with CMB data for large negative \|ξ\| ∼ 10 4 . At conformal coupling (ξ = 1/12) however, the argument in the exponential vanishes identically. For this particular value the smallness of \|ξ\| can thus not be compensated by a larger value of φ during inflation to generate the required expansion. What about quantum corrections to ξ? For values close to conformal coupling, δξ = ξ − 1/12, the number of efolds is approximately N (φ) = 3 2 δξκ 2 φ 2 − φ 2 e ,(13) (φ e denotes the value of φ at the end of the inflationary era) which requires a minimum initial Higgs field of the order of φ ≈ N/\|δξ\|. Renormalization group analysis shows that δξ (as a function of the energy scale) is small in the inflationary region, namely less than O(ξ) [10,11]. The initial Higgs amplitude required for sufficient number of e-folds with such values of δξ generally lies above the Planck scale. Whether this implies energies above the Planck mass relies in turn on the value of the parameter λ. Note however that the same renormalization group analysis of the nonminimally coupled Standard Model suggests that there are no quantum corrections to ξ, if it is exactly conformal at some energy scale [10,11]. This is based on the observation that there are no nonconformal values for the coupling ξ for which there is a renormalization group flow towards the conformal value as one runs the Standard Model parameters up in the energy scale. It thus indicates that if one expects an exactly conformal coupling for the Higgs field at some specific scale, it will be exactly conformal at all scales, hence δξ = 0. The fact that conformal coupling destroys the accelerated expansion has been noted previously [13]. How can conformal invariance be connected to the conditions for inflation? The implications of conformal invariance on the stress-energy tensor are well-known: if the matter sector of the theory is invariant under the conformal transformation g µν → Ω 2 g µν , φ → Ω −1 φ ,(14) then the trace of T µν vanishes covariantly, and hence the scalar curvature R is zero. However, for a FLRW universe the scalar curvature reads R = 6(Ḣ + 2H 2 ) ,(15) and therefore R = 0 implies −Ḣ H 2 = 2 ,(16) which is, for example, satisfied during the radiationdominated period of the evolution of a universe in the context of General Relativity. However, it rules out inflationary solutions which require 2 −Ḣ/H 2 < 1. Indeed, taking T µν (φ) from Eq. (5), its trace evaluates to T µ µ (φ) = − [1 − 12ξ] ∇ ρ φ∇ ρ φ + [12ξφV ′ (φ) − 4V (φ)] + 24ξ 2 Rφ 2 ,(17) having used the equation of motion for the scalar, Eq. (6). However, from Eq. (4), the trace of the energymomentum thensor of φ reads T µ µ (φ) = −κ −2 f (φ)R = −κ −2 (1 − 2ξκ 2 φ 2 )R .(18) Thus, Eqs. (17), (18) imply − [1 − 12ξ] ∇ ρ φ∇ ρ φ + [12ξφV ′ (φ) − 4V (φ)] + 24ξ 2 Rφ 2 = −κ −2 (1 − 2ξκ 2 φ 2 )R .(19) Let us analyze Eq. (19): For vanishing (V = 0) or quartic (V = λφ 4 ) potential, conformal invariance (ξ = 1/12) implies that the terms in square brackets vanish and the last term on the left-hand side cancels with the last term on the right-hand side, leading to zero scalar curvature R and thus zero trace of the energy-momentum tensor. However, when conformal invariance is broken, due for example to a nonzero mass term for the inflaton field (i.e., µ = 0), the induced corrections to the scalar curvature are δR = 2µ 2 κ 2 φ 2 .(20) In this case, the inflationary condition −Ḣ/H 2 < 1 requires that µκφ > √ 3\|H\|, which is not satisfied by a light scalar inflaton. For a V (φ) = λφ 4 potential, classical analysis therefore seems to exclude an inflationary regime. However, it is worth investigating whether quantum corrections to the quartic self-coupling λ can induce potential terms that break conformal invariance, and whether this can have a sufficiently strong effect as to enable inflationary solutions. This can happen if these corrections are drastic enough to generate terms in the effective potential which alter the local profile of the potential, i.e., V (φ) → V eff = V (φ) + αδφ with O ((δφ) ′ ) ∼ O (V ′ ). Then the slow-roll parameters will have a different form and may allow inflation. For slow-roll analysis with more complex potentials, it is convenient to perform a transformation to the Einstein frame, where the action is formulated in terms of a rescaled metric and a new scalar field with a minimal coupling to the curvature scalar of the new metric. Any meaningful conclusions should of course be independent of the choice of conformal frame used during the calculation. B. Analysis in the Einstein Frame Performing a suitable Weyl transformation, the action, Eq. (48), can be recast in terms of a new metriĉ g µν = f (φ)g µν = 1 − 2ξκ 2 φ 2 g µν ,(21) and a canonical scalar field χ(φ) that is minimally coupled and related to the Higgs field by dχ dφ = 1 − 2ξ(1 − 12ξ)κ 2 φ 2 f (φ) .(22) It should be noted that the transformation is singular for φ s = 1/(κ √ 2ξ). In fact, solving for the canonical field χ, one can show that it covers only the range \|φ\| ≤ φ s , implying that the analysis in the Einstein frame is valid only for this restricted domain of the original scalar. The value φ s also has special status in the Jordan frame itself. At ξ = 1/12 in particular, it was shown [23] that although the scalar field evolves smoothly through φ s in isotropic background cosmologies, its anisotropic shear diverges. We will safely stay below this point in the Einstein frame analysis, still having access to Higgs field values all the way up to the Planck scale, as long as ξ ≤ 1. The Weyl transformation is not a diffeomorphism and the space-time coordinates are left unchanged,x µ = x µ . Now a(t) = a(t) is not the FRWL scale factor of the universe described by the Einstein frame variables. However, by defining a new time coordinate dτ = a(t)dt = a(t)dt ,(23) the metric takes the FRWL form in the Einstein frame with a scale factorâ (τ ) = f (φ)a(t) ,(24) and the Hubble parameter can be defined aŝ H = 1 a dâ dτ .(25) This leaves us with an Einstein frame action S E = d 4 x −ĝ 1 2κ 2R − 1 2 (∇χ) 2 −V (χ) ,(26) and a scalar potential V (χ) = V (φ(χ)) [f (φ(χ))] 2 = λ[φ(χ)] 4 − µ 2 [φ(χ)] 2 [f (φ(χ))] 2 .(27) The expression for φ(χ) is obtained from Eq. (22) and can be solved analytically for any ξ [24]. In this study however we shall express any functions (e.g., slow-roll parameters) of the Einstein frame in terms of φ, the physical degree of freedom, so we leave the new potential in terms of φ. Of course, our interpretation of the Einstein frame as unphysical but mathematically convenient presupposes that the "observables" computed therein have no immediate physical meaning. We will come back to this point, and particularly the translation from Einstein frame observables to physical Jordan frame observables, later. It is now possible to look for an inflationary regime within the Einstein frame cosmology. We shall neglect the mass term in the following analysis 3 since we consider energy scales E ≫ µ. In terms of the Higgs field φ, the canonical first and second slow-roll parameters are given 3 It is worth noting that the potential takes a particularly simple form at ξ = 1/12 when the (conformal invariance breaking) mass term is neglected: V (χ) = 36λκ −4 sinh 2 (κχ/ √ 6). by the formulae: ǫ(φ) = 1 2κ 2 1 V 2 dV dφ 2 dχ dφ −2 (28) η(φ) = 1 κ 2 1 V dχ dφ −2 d 2V dφ 2 − d 2 χ dφ 2 dχ dφ −3 dV dφ .(29) The number of Einstein frame e-folds iŝ N = t end tĤ dτ = κ χ χ end 1 2ǫ(χ) dχ = κ φ φ end 1 2ǫ(φ) dχ dφ dφ ,(30) and is related to the true number of e-folds in the Jordan frame by N =N + 1 2 ln f (φ) f (φ end ) .(31)Classically, we haveV (φ) = λφ 4 /[f (φ)] 2 , which giveŝ ǫ(φ) = 8 κ 2 φ 2 [1 − 2ξ(1 − 12ξ)κ 2 φ 2 ] CC − − → 8 κ 2 φ 2 ,(32)η(φ) = 4 3 + 24ξ 2 κ 2 φ 2 − 2ξ(1 − 12ξ)κ 2 4ξκφ 2 + 1 φ 2 κ 2 φ 2 [1 − 2ξ(1 − 12ξ)κ 2 φ 2 ] CC − − → 4 3 + 12 κ 2 φ 2 ,(33) where CC denotes the conformal coupling limit. It thus emerges that the slow-roll parameters admit no slow-roll region at all at conformal coupling. This can also be seen from the total number of e-folds: N (φ) = (1 − 12ξ)κ 2 8 φ 2 − φ 2 end − 3 4 ln f (φ) f (φ end ) ,(34) which lacks the first, exponential expansion generating, term when ξ = 1/12. Comparing the number of e-folds in the Jordan frame, obtained in the Einstein frame analysis, namely from Eq. (31), with the scalar factor a(t) given from Eq. (12), one can confirm that it indeed agrees with the previous result obtained within the Jordan frame. This shows that the canonical slow-roll conditions in the Einstein frame and the ones chosen in the Jordan frame produce agreeing results, at least at the level of the observed expansion. Of course this does not imply the equivalence of other quantities such as the perturbation amplitudes in the two frames. As it has been explicitly shown in Ref. [25], the scalar two-point correlation functions evaluated in the Jordan frame are different than those calculated after the field redefinitions in the Einstein frame. Therefore, one should keep in mind that there is a number of ambiguities when quantum fluctuations of the scalar fields are studied in different frames in the context of generalized Einstein theories. Primordial spectral indices are calculated to second order in slow-roll parameters in Ref. [20] for different inflationary models, in the context of theories with a nonminimal coupling between the inflaton field and the Ricci curvature scalar. It has been shown that there are inflationary models (e.g., new inflation) for which there are discrepancies between the values of the spectral index n s calculated in the Einstein and the Jordan frame, while for others (e.g., chaotic inflation) there are not. Finally, the reader should keep in mind that while realization of slow-roll inflation in the (physical) Jordan frame, in which the inflaton is nonminimally coupled to the Ricci curvature, implies slow-roll inflation in the (unphysical) Einstein frame, the vive versa does not hold [13]. III. FLAT POTENTIAL THROUGH QUANTUM CORRECTIONS The Higgs potential takes the classical form V (φ) = λφ 4 − µ 2 φ 2 ,(35) however both µ and λ are subject to radiative corrections as a function of energy. For very large values of the field φ one therefore needs to calculate the renormalized value of these parameters at the energy scale µ ∼ φ. The running of the top Yukawa coupling and the gauge couplings cannot be neglected and must be evolved simultaneously. We follow the analysis of Ref. [26], which relies upon the β-functions and improved effective potential presented in Refs. [27][28][29]. This involves taking the measured values of the gauge couplings at low energy and evolving them upwards in energy, taking into account the thresholds where quark species come into the running. It is neccesary to simultaenously evolve all three gauge couplings and the top quark Yukawa coupling in order to accurately predict the full effect upon the Higgs self-coupling. Care must be taken to use the correct relationship between the pole masses and the parameters used in the running [26]. At high energies the mass term is sub-dominant and one can write the effective potential as V (φ) = λ(φ)φ 4 .(36) Then for a given mass m t of the top quark, a smaller value of the Higgs mass will result in the quartic coupling being driven down at large values of φ, such that it may develop a metastable or true vacuum at expectation values of φ, far in excess of that observed from Standard Model physics φ = 246GeV. For typical values of m t , if this false vacuum appears at all, it will show up relatively close to the Planck scale. When calculating the running of λ it is in fact necessary to go to two-loop accuracy since at one-loop this second minimum develops at scales typically far in excess of the Planck scale, where we would really expect higher order nonrenormalizable contributions to the potential to become important. For each value of m t there is therefore a value of the Higgs mass, m h , where the effective potential is on the verge of developing a metastable minimum at large values of φ and the Higgs potential is locally flattened. This is illustrated in Fig. 1. Since the region where the potential becomes flat is narrow, slow-roll must be very slow (i.e., the slow-roll parameters very small), in order to provide a sufficiently long period of quasi-exponential expansion. The slow-roll parameters for the top (black curve) potential profile of Fig. 1 are shown in Fig. 2, and one can see that the region where ǫ is extremely small takes the form of a narrow dip. It is there that the integral N ∼ ǫ − 1 2 dφ can generate the required number of e-folds. It was noted in Ref. [30] that in the minimally coupled model, slow-roll through this flat region will not match the observed amplitude of density perturbations ∆ 2 R in the cosmic microwave background. Inflation predicts the latter to be related to the potential and first-slow roll parameter at horizon crossing (labelled by stars). Its value as measured by WMAP7 [31] imposes the constraint V * ǫ * 1 4 = 2 √ 3πm Pl ∆ 1 2 R = (2.75 ± 0.30) × 10 −2 m Pl ,(37) where ǫ * ≤ 1. The mismatch arises because ǫ needs to be extremely small in order to allow for sufficient e-folds and the potential energy is then too large to fit the condition. However, even in the minimally coupled model, there remains the possibility that horizon crossing occurs close to the beginning of inflation, where ǫ is not yet so small, provided the flat region occurs at low enough energy. Since ǫ * ≤ 1, the maximum potential energy at horizon crossing is 5.7 × 10 −7 m 4 Pl . We shall see in the renormalization group analysis that there exist values of the top quark mass for which the flattening does happen at energies below this value. Furthermore, the presence of nonminimal coupling has additional effects since it changes the potential felt by the Higgs field. When the nonminimal coupling ξ of the Higgs boson to gravity is included in the Standard Model, it has a βfunction induced by the coupling between the Higgs field and the matter sector whose behaviour has been analyzed to one-loop [10,11]. As previously stated, we take β ξ = 0, since the presence of a boundary value ξ = 1/12 at some energy scale suggests that ξ = 1/12 at all scales. The β-function of the quartic Higgs self-coupling changes as well due to the −ξRφ 2 term, and this can have significant effects on the remaining Standard Model parameters when ξ is large [32]. We have worked out how large ξ needs to be to impact the normal Standard Model running by considering the two cases ξ = 1 and ξ = −1 at low energies and running these up with the other parameters. The effect that either of these choices has on the potential is very small and looks like a minute change in the Higgs mass, much less than than any possible experimental error. Because we are well within this range we can neglect these corrections. We therefore calculate the renormalization of the Higgs self-coupling in the minimally coupled Standard Model and construct an effective potential which fits the renormalization group improved potential around the flat region. The modifications in that fit are very small when the conformal coupling is included. We first consider the implications for the minimally coupled model (where the Jordan and Einstein frames coincide), which had been mentioned in Ref. [30], and then extend the analysis to a conformally coupled model, which is of particular relevance to the Noncommutative Spectral Action approach to the Standard Model of Particle Physics. There is a very good analytic fit to the Higgs potential in the region around this plateau/minimum, which takes the form V E = λ E (φ)φ 4 = [a ln 2 (bκφ) + c]φ 4 .(38) The parameters are found to relate to the low energy values of m t in the following way: The third parameter, c = c(m t , m φ ), encodes the appearance of an extremum (see, Fig. 1) and depends on the values for m t and m φ . Indeed, V E (φ) exhibits a sub-Planckian flat region (or local minimum) for suitably tuned parameters. An extremum occurs if and only if c/a ≤ 1/16, the saturation of the bound corresponding to a perfectly flat region, i.e., V ′ E (φ 0 ) = V ′′ E (φ 0 ) = 0, where φ 0 = e − 1 4 /b(m t ) and e is Euler's constant. The energy at these points is given by V E (φ 0 ) = a(m t ) 8eb(m t ) 4 κ 4 ,(40) which for 169 ≤ m t ≤ 175 lies within 10 −10 κ −4 ≤ V 0 ≤ κ −4 (note that V E (φ 0 ) increases with m t ). This shows that there are regions where the flattening occurs at scales potentially consistent with perturbation amplitudes, given in Eq. (37). It is convenient to write c = [(1 + δ)/16]a, where δ = 0 saturates the bound below which a local minimum is formed. We restrict ourselves to δ > 0, so that the potential contains no metastable vacua. The slow-roll conditions are met only for a narrow region, but for the points in parameter space which are close to δ = 0, both slow-roll parameters vanish simultaneously and we get slow-roll inflation with extremely small ǫ. From Eq. (37) it follows that for m t > 171.42GeV the two conditions cannot be simultaneously met since the flat region occurs at too high energies. Slow-roll is restricted to the domain where max (ǫ, \|η\|) ≤ 1, and for inflation one should find a point in parameter space which: (i) leads to sufficient e-folds within a region [φ end , φ * ], (ii) has an ǫ * which lies within the bounds imposed by COBE normalization, and (iii) satisfies the observational constraints on n s and r. The measured value of perturbation amplitudes serves as a convenient first test of the model. For the scenario to be viable, ǫ at horizon crossing cannot be too small. Since the requirement on a sufficient number of e-folds relies on a potential that has very small ǫ in a small region, the problem is that the valley in ǫ is far narrower than that in η. As a result, within the region \|η\| ≤ 1, ǫ tends to be very small. The best fit to the observed perturbation amplitude will occur for scenarios in which horizon crossing occurs close to the onset of inflation, i.e., η(φ ⋆ ) ∼ 1, so that ǫ ⋆ takes its largest possible value. The corrections due to conformal coupling to the potential in the Einstein frame are entirely embodied in the function f (φ) ∼ 1 + O(κ 2 φ 2 ), since the canonical field χ feels the potential V E /f 2 . The value of the Higgs field where the plateau occurs in the potential rises with increasing top quark mass, so the greatest effect will be at the highest top quark mass. However, the lower bound on ǫ * then gets more stringent since V * is larger. Due to the change in the potential, flatness does not occur at δ = 0 anymore but for fixed values of δ depending on the value of the top quark mass. Sub-Planckian inflation is again reliant on a relationship between the Higgs field and the top quark masses. The values of δ for which the potential has the right flatness are not anymore centered around δ = 0 due to the altered form of the potential. This has an effect on the Higgs masses where flattening occurs: for any −1 ≤ δ ≤ 0, a given top quark mass fixes the Higgs mass to a value in the range (120 − 130) GeV with an accuracy of ∆m φ /m φ ∼ 10 −6 . This means that for inflation to occur via this mechanism, the top quark mass fixes the Higgs mass extremely accurately. As an example, for m t = 171.70 GeV and δ = −0.2867 (corresponding to m φ = 125.735368 GeV), we obtainN = 62 of e-folds between κφ = 0.9570 and κφ end = 0.9417. Scanning through parameter space it emerges that sufficient e-folds are indeed generated provided a suitably tuned relationship between m t and m φ holds. Numerical integration needs to be performed carefully since the slow-roll approximation implies a strongly peaked integrand in the number of e-folds. Using a Runge-Kutta integrator in FORTRAN we identify the curve in parameter space along which sufficient expansion occurs during almost perfect de Sitter inflation, both for minimal and conformal couplings. The next step is a comparison with astrophysical measurements. To probe the parameter space more finely we use a Monte-Carlo chain. It turns out that in both the minimal and conformal cases, the perturbation amplitudes are too large -the best fit to the ratio (V ⋆ /ǫ ⋆ ) 1 4 is still too large by two orders of magnitude. Small positive nonminimal couplings such as ξ = 1/12 improve the fit only minimally. It should be noted also that when perturbation amplitudes are too large, scenarios where perturbations are generated by a curvaton are in turn ruled out as well, because the quantum fluctuations of the inflaton are already too large. In Fig. 3 we show the best fit, i.e., the scenario with the largest possible values of ǫ * , the first slow-roll parameter at horizon crossing, for a given top quark mass along with the potential energy V * , at horizon crossing. The resulting ratio of perturbation amplitudes is too large for any value of m t . On a side note, let us mention that the renormalization of Standard Model parameters is generally performed in Minkowski space-time, while inflationary perturbations are calculated on a general de Sitter background. The conditions for the geralization of a slow-roll inflationary era should of course be studied in a de Sitter space and the Coleman-Weinberg result should then be recovered as a limit to the flat Minkowski space-time. This analysis has been performed in a recent study [21], where the one-loop improved potential for the nonminimally coupled scalar λφ 4 theory in de Sitter space was calculated. Their analysis poses a stringent constraint on the coupling parameter ξ. The assumption \|Ḣ\| < H 2 along with the requirement that f (φ) in the equations of motion remain nonsingular, f (φ) < ∞, implies 1 16Ñ ≪ \|ξ\| ≪ 1 48 ,(41) whereÑ ≈ N +1−ξ. This rules out most values of ξ used in literature. However, \|Ḣ\| < H 2 is in fact a stronger condition than the condition −Ḣ < H 2 for inflationary expansion. The latter implies the former only whenḢ is negative. The stronger condition \|Ḣ\| < H 2 could be circumvented in an inflationary universe whereḢ is large and positive. In the minimally coupled case this is clearly not possible, sinceḢ = −φ 2 /2. For nonzero ξ we have howeveṙ H = φφ f (φ) − 1 2 (1 − 4ξ)φ φ − 2ξH + 2ξφ φ ,(42) wich for the slow-roll conditions, Eq. (9), reduces tȯ H = − 2ξφφ f (φ) H .(43) Since the field rolls down the potential, sign(φ) = −sign(φ) andḢ is indeed positive when the nonminimal coupling is positive (e.g., conformal) in our notation. This means that the above constraint does not apply in the conformally coupled case. However, it should be mentioned that for negative choices of ξ, popular due to their promise in achieving Higgs driven inflation,Ḣ < 0. The constraint in Eq. (41) is then valid and seems to be in contradiction with large \|ξ\|. IV. NONCOMMUTATIVE SPECTRAL ACTION AND INFLATION Using the language of noncommutative geometry and spectral triples, Connes and collaborators have reformulated the Standard Model in terms of purely geometric data [14]. Based on spectral triples, A. Connes [33] has developed a new calculus that deals not with the underlying spaces, but with the algebra of functions defined upon them instead. This reformulation allows a natural generalization of the differential calculus on Riemannian manifolds to a wider class of geometric structures, i.e., noncommutative spaces. It is the geometry of these spaces that encodes not only space-time and gravity, but also the matter content of the Standard Model. In NCG, the fundamental particles and interactions derive from the spectral data of an action functional defined on noncommutative spaces, the Spectral Action. The Standard Model emerges as the asymptotic expansion of this action at an energy Λ below the Planck scale, at which the fundamental noncommutative space is approximated by an almost-commutative space. This space is assumed to be the simplest noncommutative extension of the smooth four-dimensional space-time manifold, and is obtained by taking its tensor product with a finite noncommutative space. Having recovered low energy physics in the framework of NCG, the next step will be to find the true geometry at Planckian energies, for which this product is a low energy limit. We consider here the effective action functional at the scale Λ. In this section, we will first highlight the main principles of the noncommutative geometry approach and we will then investigate possible inflationary mechanisms driven by one of the available scalar fields. A. Elements of NCG spectral action Within General Relativity, the group of symmetries of gravity is given by diffeomorphism of the underlying differentiable manifold of space-time; a key ingredient that one would like to extend to the theory of elementary particles. To achieve such a geometrization of the Standard Model coupled to gravity, one should turn the SM coupled to gravity into pure gravity on a preferred space, whose group of diffeomorphisms is given by the semidirect product of the group of maps from the background manifold to the gauge group of the SM, with the group of diffeomorphisms of the background manifold. Such preferred space cannot be obtained however within ordinary spaces, while noncommutative spaces can easily lead to the desired answer. This is the main reason for extending the framework of geometry to spaces whose algebra of coordinates is noncommutative. To extend the Riemaniann paradigm of geometry to the notion of metric on a noncommutative space, the latter should contain the Riemaniann manifold with the metric tensor (as a special case), allow for departures from commutativity of coordinates as well as for quantum corrections of geometry, contain spaces of complex dimension, and offer the means of expressing the Standard Model coupled to Einstein gravity as pure gravity on a suitable geometry. A metric NCG is given by a spectral triple (A, H, D), in the sense that we will discuss below. Thus, within NCG, geometric spaces emerge naturally from purely spectral data. The fermions of the Standard Model provide the Hilbert space H of a spectral triple for a suitable algebra A, and the bosons arise naturally as inner fluctuations of the corresponding Dirac operator D. To study the implications of this noncommutative approach coupled to gravity for the cosmological models of the early universe, we will only consider the bosonic part of the action; the fermionic part is however crucial for the particle physics phenomenology of the model. More precisely, let us consider a geometric space defined by the product of a continuum compact Riemaniann manifold, M, and a tiny discrete finite noncommutative space, F , composed of only two points. The product geometry M × F has the same dimension as the ordinary space-time manifold, namely 4. Hence, the noncommutative space F has zero metric dimension. The space F represents the geometric origin of the Standard Model and it is specified in terms of a real spectral triple (A, H, D), where A is a noncommutative ⋆ -algebra, H is a Hilbert space on which A is realized as an algebra of bounded operators, and D is a suitably defined Dirac operator on H. The Dirac operator can be seen as the inverse of the Euclidean propagator of fermions. Since the action functional only depends on the spectrum of the line element, it is a purely gravitational action. In other words, the physical Lagrangian is entirely determined by the geometric input, which implies that the physical implications are closely dependent on the underlying chosen geometry, see, Ref. [14]. By assuming that the algebra constructed in M × F is symplectic-unitary, the algebra A is restricted to be of the form A = M a (H) ⊕ M k (C) ,(44) where k = 2a and H is the algebra of quaternions. The choice k = 4 is the first value that produces the correct number (k 2 = 16) of fermions in each of the three generations [34]. The Dirac operator D connects M and F via the spectral action functional on the spectral triple. It is defined as Tr (f (D/Λ)), where f > 0 is a test func-tion and Λ is the cut-off energy scale. The asymptotic expression for the spectral action, for large energy Λ, is of the form Tr f D Λ ∼ k∈DimSp f k Λ k −\|D\| −k +f (0)ζ D (0)+O(1) , (45) where f k = ∞ 0 f (v)v k−1 dv are the momenta of the function f , the noncommutative integration is defined in terms of residues of zeta functions, and the sum is over points in the dimension spectrum of the spectral triple. The test function enters through its momenta f 0 , f 2 , f 4 ; these three additional real parameters are physically related to the coupling constants at unification, the gravitational constant and the cosmological constant. In the four-dimensional case, the term in Λ 4 in the spectral action, Eq. (45), gives a cosmological term, the term in Λ 2 gives the Einstein-Hilbert action functional with the physical sign for the Euclidean functional integral (provided f 2 > 0), and the Λ-independent term yields the Yang-Mills action for the gauge fields corresponding to the internal degrees of freedom of the metric. The scaleindependent terms in the spectral action have conformal invariance. Note that the arbitrary mass scale Λ can be made dynamical by introducing a scaling dilaton field. Writing the asymptotic expansion of the spectral action, a number of geometric parameters appear; they describe the possible choices of Dirac operators on the finite noncommutative space. These parameters correspond to the Yukawa parameters of the particle physics model and the Majorana terms for the right-handed neutrinos. The Yukawa parameters run with the renormalization group equations of the particle physics model. Since running towards lower energies, implies that nonperturbative effects in the spectral action cannot be any longer safely neglected, any results based on the asymptotic expansion and on renormalization group analysis can only hold for early universe cosmology. For later times, one should instead consider the full spectral action. Applying the asymptotic expansion of Eq. (45) to the spectral action of the product geometry M × F gives a bosonic functional S which includes cosmological terms, Riemannian curvature terms, Higgs minimal coupling, Higgs mass terms, Higgs quartic potential and Yang-Mills terms. Moreover, one can introduce a relation between the parameters of the model, namely a relation between the coupling constants at unification. More precisely, we impose the relation g 2 3 f 0 2π 2 = 1 4 and g 2 3 = g 2 2 = 5 3 g 2 1 ,(46) between the coefficient f 0 and the coupling constants g 1 , g 2 , g 3 , which is dictated by the normalization of the kinetic terms. This condition means that the so-obtained spectral action has to be considered as the bare action at unification scale Λ, where one supposes the merging of the coupling constants to take place. The gravitational terms in the spectral action, in Euclidean signature, are of the form S E grav = 1 2κ 2 R + α 0 C µνρσ C µνρσ + τ 0 R ⋆ R ⋆ − ξ 0 R\|H\| 2 √ gd 4 x . (47) Note that H is a rescaling H = ( √ af 0 /π)φ of the Higgs field φ to normalize the kinetic energy; the momentum f 0 is physically related to the coupling constants at unification and the coefficient a is related to the fermion and lepton masses and lepton mixing. In the above action, Eq. (47), the first two terms only depend upon the Riemann curvature tensor; the first is the Einstein-Hilbert term with the second one being the Weyl curvature term. The third term R ⋆ R ⋆ = 1 4 ǫ µνρσ ǫ αβγδ R αβ µν R γδ ρσ , is the topological term that integrates to the Euler characteristic and hence is nondynamical. Notice the absence of quadratic terms in the curvature; there is only the term quadratic in the Weyl curvature and topological term R ⋆ R ⋆ . In a cosmological setting namely for Friedmann-Lemaître-Robertson-Walker geometries, the Weyl term vanishes. The spectral action contains one more term that couples gravity with the SM, namely the last term in Eq. (47), which should always be present when one considers gravity coupled to scalar fields. B. Higgs field inflation The asymptotic expansion of the Spectral Action, proposed in Ref. [14], gives rise to the following Gravity-Higgs sector L GH ⊂ L NCG : S GH = d 4 x √ −g 1 − 2κ 2 ξH 2 2κ 2 R − 1 2 (∇H) 2 − V (H) ,(48)where V (H) = λH 4 − µ 2 H 2 . In the derivation of the Standard Model from the Spectral Action principle, the metric carries Euclidean signature. The discussion of phenomenological aspects of the theory relies on a Wick rotation to imaginary time, into the standard (Lorentzian) signature. While sensible from the phenomenological point of view, there exists as yet no justification on the level of the underlying theory. To discuss the phenomenology of the aspects of the cut-off scale Λ, the Spectral Action principle leads to a number of boundary conditions on the parameters of the Lagrangian. These conditions encode the geometric origin of the Standard Model parameters. Normalization of the kinetic terms in the action implies the following now accompanied by the new fields χ H and χ σ related to the Jordan frame fields by dχ H dH = 1 − 2ξ H (1 − 12ξ H )H 2 f (H, σ) CC − − → 1 f (H, σ) ,(55)dχ σ dσ = 1 − 2ξ σ (1 − 12ξ σ )σ 2 f (H, σ) CC − − → 1 f (H, σ) . The Einstein frame Lagrangian reads S = d 4 x −ĝ 1 2κ 2R − 1 2 (∇χ H ) 2 − 1 2 (∇χ σ ) 2 − P (χ H , χ σ )∇ µ χ H∇ µ χ σ − V (χ H , χ σ ) , (57) withV (χ H , χ σ ) = V (H, σ) f (H, σ) 2 ,(58) and a novel coupling P (χ H , χ σ ) = 24κ 2 ξ H ξ σ f (H, σ) 2 dH dχ H dσ dχ σ Hσ CC − − → κ 2 6 σH .(59) Note that there exists no conformal transformation which gets rid of both the nonminimal coupling to gravity and the cross-term P (χ H , χ σ ) [35]. However, at conformal coupling P (χ H , χ σ ) can be neglected as long as σH ≪ 6κ 2 . We are then left with a minimally coupled theory of two scalar fields with potentialV (χ H , χ σ ). When the mass term is negligible the theory is symmetric in the two fields. Consider the first slow-roll parameter for the σ-field, defined aŝ ǫ σ = 1 2κ 2 1 V 2 ∂V ∂σ 2 ∂χ σ ∂σ −2(60) For H = 0 this reduces to the earlier case and one gets an insufficient numebr of e-folds below the Planck scale. If we have a nonzero H however, say H ∼ κ −1 close to the Planck mass, then the situation changes somewhat. Due to the additional terms inǫ σ , the coupling constants do not fall out of the expression, and they can therefore influence the magnitude of the integrand in the number of e-folds. For this effect to take place however, it is necessary that the assisting field maintains a relatively large value throughout the inflationary era driven by the inflaton. This in turn requires the curvature of the potential to be much less in the direction of the constant field. Since only quartic terms arise in the model, the quartic self-coupling of the assisting field is then required to be much lower than that of the inflaton. But in that case, the new terms due to the assisting field are not large enough to enable the inflaton to generate sufficient number of e-folds. The situation is of course entirely symmetric in the two fields (except for the nonzero H-mass which is negligible at high energies), so the rôles of the two fields may well be interchanged depending on which constraints lay on the respective coupling constants. V. RUNNING OF THE GRAVITATIONAL CONSTANT region is confined to a small range of energies where the potential is flattened. In conclusion, unless modification of the spectral triple allows for a nonconformal boundary value of ξ, there seems to be no viable slow-roll scenario for any of the two scalars. Furthermore, if one assumes the validity of the suggestion in Ref. [16] in relating the running of c and κ 2 , the only situation in which this could trigger inflation (with conformal coupling) would be one in which the running changes drastically, e.g., through the see-saw mechanism. However, the inevitable lack of differentiability of the renormalized couplings at see-saw scales [16,36] makes such a scenario very unlikely and also inaccessible to slow-roll analysis. VI. CONCLUSIONS In many realistic cosmological models, the nonminimal coupling of the scalar field to the Ricci curvature cannot be avoided. In particular, there are arguments requiring a conformal coupling between the scalar field and the background curvature. The existence of such a term will generically lead to difficulties in achieving a slowroll inflationary era. In this paper, we have investigated whether two-loop corrections to the Higgs potential could lead to a slow-roll inflationary period in agreement with the constraints imposed by the CMB measurements. Our findings do not favor the realization of such an era. More precisely, even though slow-roll inflation can be realized, we cannot satisfy the COBE normalization constraint for any values of the top quark and Higgs masses allowed from current experimental data. We have in particular investigated Higgs inflation in the context of the Noncommutative Geometry Spectral Action, which provides an elegant explanation for the phenomenology of the Standard Model. Within this context, a conformal coupling arises naturally between the Higgs field and the Ricci curvature. It is also important to note that once conformal coupling is set at the preferential (boundary) energy scale of the spectral action model, then it will remain conformal at all scales. Running of the gravitational constant and corrections by considering the more appropriate de Sitter, instead of a Minkowski, background do not favor the realization of a successful inflationary era. The NCG Spectral Action provides in addition to the Higgs field, another (massless) scalar field which exhibits no coupling to the matter sector. Our analysis has shown that neither this field can lead to a successful slow-roll inflationary era if the coupling values are conformal. One may be able to improve upon this (negative) conclusion, if important deviations of ξ from its conformal value can be allowed; the value ξ = 1/12 may turn out that is not a generic feature of NCG models. VII. ACKNOWLEDGMENTS This work is partially supported by the European Union through the Marie Curie Research and Training Network UniverseNet (MRTN-CT-2006-035863). FIG. 1 : 1Sub-Planckian flattening of the Higgs potential due to two-loop corrections in the Standard Model (ξ = 0). We analyze slow-roll for profiles just above the top (black) curve, which feature no metastable vacua. m t ) = exp −0.979261 m t GeV − 172.051 . FIG. 2 : 2Typical profiles of ǫ (blue) and η (red) with a small sub-Planckian region of slow-roll, plotted here for mt = 172GeV and δ = 0. There is a narrow region in which both are very small. FIG. 3 : 3The value of the potential (solid) in units of κ −4 and the maximum value of the first slow-roll parameter (dashed) at horizon crossing for minimal ξ = 0 (black) and conformal ξ = 1/12 (blue). The striped area represents the region of the top mass excluded by Eq. (37) from the height of the plateau in the potential. The inset shows the ratio (V⋆/ǫ⋆)1 4 in both cases and WMAP7 observations (red region). The calculated value of perturbation amplitudes is off by several orders of magnitude and the improvement at conformal coupling minimal. [λ H H 4 + λ σ σ 4 + λ Hσ \|H\| 2 σ 2 ] −2 × σ 2 4λ σ σ 2 + 2λ Hσ H 2 f (H, σ) λ σ σ 4 + λ H H 4 + λ Hσ H 2 σ 2= 1 2κ 2 + 2 3 κ 2 2 . Note that conformal invariance is considered here solely in the matter sector. The Einstein-Hilbert term is not conformally invariant. relations:We emphasize that the action, Eq. (48), has to be taken as the bare action at some cutoff scale Λ. The renormalized action will have the same form but with the bare quantities κ, µ, λ and the three gauge couplings g 1 , g 2 , g 3 replaced with physical quantities. The factor f 0 is fixed by the canonical normalization of the Yang-Mills terms (not included here) in terms of the common value of the gauge coupling constants g at unification, f 0 = π 2 /(2g 2 ). The value of g at the unification scale is determined by standard renormalization group flow, i.e., it is given a value which reproduces the correct observed coupling at low energies. Note that it is not unique since the gauge couplings fail to meet exactly in the nonsupersymmetric Standard Model (or its extension by right-handed neutrinos). The coefficients a, b, c are the Yukawa and Majorana parameters subject to renormalization group flow, see e.g. Ref.[14]. The parameter f 2 is a priori unconstrained in R * + . Assuming the big desert hypothesis, we can connect the physics at low energies with those at E = Λ through the standard renormalization procedure. This was carried out at one loop in Ref.[14], and more recently in Ref.[16]where Majorana mass terms for right-handed neutrinos were included and the see-saw mechanism was taken into account. In our renormalization group analysis of the Higgs potential, following Ref.[26], the choice of boundary conditions is the standard one motivated by particle physics considerations. The focus here has of course been on the different boundary conditions at low energies for which a flat section develops in the Higgs potential.The relations above rely on the validity of the asymptotic expansion at Λ, and are therefore tied intimately to the scale at which the expansion is performed. There is no a priori reason for the constraints to hold at scales below Λ -they represent mere boundary conditions. The constraint ξ(Λ) = 1/12 by itself therefore does not require the coupling to remain conformal all the way down to present energy scales, or even during an inflationary epoch, since it may run with the energy scale. However, we will assume no running in ξ as the arguments laid out (see, discussion in Section IIA) above still apply.As we can see from the results presented above, the conformally coupled Higgs field in the Spectral Action Standard Model is not a viable candidate for inflaton if the coupling remains conformal at all scales. However, at present it is still unclear whether conformal 4 invariance and ξ = 1/12 is a generic feature of models from noncommutative geometry. If it turns out not to be, one can proceed along the line of the analyses presented in Refs.[15,16].C. Inflation through the massless scalar fieldThe spectral action gives rise to an additional massless scalar field 5[17], denoted by σ. Including this field, the cosmologically relevant terms in the Wick rotated action readwhereThe constants are related to the underlying parameters as follows6:This action also admits a rescaling of the metric which transforms it to the Einstein frame. The rescaled metriĉThe coupling term between the Higgs field and the Ricci curvature, appearing in the spectral action functional, is −f 0 /(12π 2 )aR\|φ\| 2 , which after rescaling H = ( √ af 0 /π)φ, leads to the term −R\|H\| 2 /12. This indeed shows the conformal coupling between the background and the Higgs field.5The field σ is unlike all other fields in the theory, such as the Higgs field and gauge fields. Usually one starts with a parameter in the Dirac operator of the discrete space, and then inner fluctuations of the product space would generate the dynamical fields. The only exception being the matrix entry that gives mass to the right-handed neutrinos, where the parameter can either remain as such, or one can use the freedom to make it a dynamical field, which a priori may lead to important cosmological consequences[? ]. Note that the σ field was not considered in the original noncommutative geometry spectral action analysis presented in Ref.[14], where the authors were mainly interested in recovering the Standard Model.6Note that a similar action has been studied in Ref[32], but in their analysis the additional scalar field has a nonzero mass and the nonminimal couplings are studied in the previously mentioned large negative ξ regime, which flattens the classical quartic potential in the Einstein frame.At the scale Λ, the gravitational constant is related to the geometric parameters of the theory byf 0 is fixed by one of the unification conditionsf 2 is an unconstrained parameter in R * + , and c is determined by the renormalization group equations. 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[ "ON SUPERSIMPLICITY AND LOVELY PAIRS OF CATS", "ON SUPERSIMPLICITY AND LOVELY PAIRS OF CATS" ]	[ "Itay Ben-Yaacov " ]	[]	[]	We prove that the definition of supersimplicity in metric structures from [Ben05b] is equivalent to an a priori stronger variant. This stronger variant is then used to prove that if T is a supersimple Hausdorff cat then so is its theory of lovely pairs.	null	[ "https://arxiv.org/pdf/0902.0118v1.pdf" ]	115,171,104	0902.0118	aeae08346f72d749c818ee5f512fbe984a7c21fa	ON SUPERSIMPLICITY AND LOVELY PAIRS OF CATS 1 Feb 2009 Itay Ben-Yaacov ON SUPERSIMPLICITY AND LOVELY PAIRS OF CATS 1 Feb 2009arXiv:0902.0118v1 [math.LO] We prove that the definition of supersimplicity in metric structures from [Ben05b] is equivalent to an a priori stronger variant. This stronger variant is then used to prove that if T is a supersimple Hausdorff cat then so is its theory of lovely pairs. Introduction A superstable first order theory is one which is stable in every large enough cardinality, or equivalently, one which is stable (in some cardinality), and in which the type of every finite tuple over arbitrary sets does not divide over a finite subset. In more modern terms we would say that a first order theory is superstable if and only if it is stable and supersimple. Stability and simplicity were extended to various non-first-order settings by various people. Stability in the setting of large homogeneous structures goes back a long time (see [She75]), and some aspects of simplicity theory were also shown to hold in this setting in [BL03]. The setting of compact abstract theories, or cats, was introduced in [Ben03b] with the intention, among others, to provide a better non-first-order setting for the development of simplicity theory, which was done in [Ben03c], and under the additional assumption of thickness (with better results) in [Ben03d]. Hausdorff cats are ones whose type spaces are Hausdorff. Many classes of metric structures arising in analysis can be viewed as Hausdorff cats (e.g., the class of probability measure algebras [Ben], elementary classes of Banach space structures in the sense of Henson's logic, etc.) Conversely, a Hausdorff cat in a countable language admits a definable metric on its home sort which is unique up to uniform equivalence of metrics [Ben05b] (and even if the language is uncountable this result remains essentially true). Thus Hausdorff cats form a natural setting for the study of metric structures. There is little doubt about the definitions of stability and simplicity in the case of (metric) Hausdorff cats: all the approaches mentioned above, and others, agree and give essentially the same theory as in first order logic. Many natural examples are indeed stable. Unfortunately, no metric structure can be superstable or supersimple according to the classical definition, unless it is essentially discrete: Indeed, in most cases that b n → a in the metric we have a \| ⌣b <n b <ω for all n. A very illustrative example is the following. Let T be a first order theory. For every M T , we can view M ω as a metric structure, with d(a <ω , b <ω ) = inf{2 −n : a <n = b <n }. The class of metric structure {M ω : M T } is the class of complete models (in the sense of [Ben05b]) of a compact abstract theory naturally called T ω . If T is stable (simple), then so is T ω , but T ω is never supersimple: this can be seen using the argument in the preceding paragraph, or directly from the fact that finite tuples in the sense of T ω are in fact infinite tuples in the sense of T (these two arguments eventually boil down to the same thing). In an arbitrary metric cat define that a ε \| ⌣C B if there is a ′ such that d(a, a ′ ) ≤ ε and a ′ \| ⌣C B (later on we will slightly modify this). Then for every simple first order theory T and the corresponding T ω we have: a (2 −n ) <ω \| ⌣ C B (in the sense of T ω ) ⇐⇒ a <n \| ⌣ C B (in the sense of T ). (Knowing a <ω ∈ M ω up to distance 2 −n is the same as knowing a <n .) It follows that T is supersimple if and only if, in T ω , for everyā ∈ M ω , ε > 0, and set B, there is B 0 ⊆ B finite such thatā ε \| ⌣B 0 B. Generalising from this example, we suggested in [Ben05b] that: -Finite tuples in metric structures behave in some sense like infinite tuples in classical first order structures, and the right way to extract a "truly finite" part of them is to consider them only up to some positive distance. -As a consequence, the above characterisation of the supersimplicity of T by properties of T ω should be taken as the definition of the supersimplicity of a metric theory (so T ω would be supersimple if and only if T is). One can now define that a Hausdorff cat is superstable if it is stable and supersimple. An alternative approach to superstability was suggested by Iovino in the case of Banach space structures through the re-definition of λ-stability in a manner that takes the metric into account [Iov99]. The two definitions agree: T is λ-stable for all big enough λ (by Iovino) if and only if it is stable and supersimple by the definition above. (This follows from the metric stability spectrum theorem [Ben05b,Theorem 4.13] as in the classical case.) We find this a fairly reassuring evidence that the definitions are indeed "correct". The present paper attempts to address some questions these definitions raise: First, the definition of supersimplicity above is somewhat disturbing, as it translates the two occurrences of "finite" in the original definition differently. We would prefer something of the form: "T is supersimple if for every a, ε > 0 and B, there is a distance δ > 0 such that a ε \| ⌣B δ B." Of course, in order to do that we would first have to give meaning to a ε \| ⌣B δ B. This is addressed in Section 1. A second issue arises from the theory T P of beautiful (or lovely) pairs of models of a stable (or simple) theory T [Poi83, BPV03,Ben04]. It was shown (by Buechler [Bue91], later extended by Vassiliev [Vas02]) that such theories of pairs of models of T can be used as means for obtaining information on T itself. More precisely, for a rank one theory T , the rank of T P yields information about the geometry of T . While metric structures can never have "rank one", it is natural to seek to compare the rank of T P with that of T , when T is superstable or supersimple. In order for such a course of action to be feasible, one would first have to show that in that case T P is supersimple as well. In [BPV03] and in [Ben04] two distinct proofs are given to the effect that if T is a supersimple first order theory, or more generally, a "supersimple" cat in the wrong sense that does not take into account the metric, then so is the theory of its lovely pairs T P . Due to the nature of independence in T P , both proofs inevitably use the fact that T is supersimple at least twice. These proofs do not extend to the corrected definition of supersimplicity: without entering into details, on the first application of supersimplicity we see that a ε \| ⌣B 0 B for some finite B 0 ⊆ B, but then we cannot apply supersimplicity to the type of B 0 over something else. On the other hand, if we did have a ε \| ⌣B δ B for some δ > 0, we could apply supersimplicity to types of B δ over another set, and the proof may be salvaged. This is addressed in Section 2. Thus the novelty of this paper is a notion of independence over virtual tuples, i.e., over objects of the form B δ . This is a venture into difficult and unsound terrain (for example, the results of [Ben03a], while dealing with ultraimaginary elements rather than virtual ones, suggest that independence over objects which are not "at least" hyperimaginary should be approached with extreme caution and without too many hopes). While one can come up with many definitions for such a notion of independence it is not at all obvious to come up with one which satisfies the usual axioms, or even any "large" subset thereof. Our notion of independence is merely shown to satisfy some partial transitivity properties (Proposition 1.15), and at the same time to yield an equivalent characterisation of supersimplicity (Theorem 1.18). We content ourselves with such a modest achievement as it does suffice to close the gap in the proof that if T is supersimple then so is T P (Theorem 2.4). Other properties, such as symmetry, are lost (when considering independence over virtual tuples). For example, in a Hilbert space, if u and v are unit vectors and ε ≤ √ 2, then one can show that: u \| ⌣ v ε v ⇐⇒ u − v ≥ ε v \| ⌣ v ε u ⇐⇒ u − v = √ 2 (⇐⇒ u \| ⌣ v). The question of finding a notion of independence which has more of the usual properties (e.g., symmetry, full transitivity, extension) without losing those we need for the results presented in this paper, remains an open (and difficult) one. We assume basic familiarity with the setting of compact abstract theory and simplicity theory in this setting (see [Ben05a]). We use a, b, c, . . . to denote possibly infinite tuples of elements in the universal domain of the theory under consideration. When we want them to stand for a single element, we say so explicitly. Similarly, x, y, z, . . . denote possibly infinite tuples of variables. Greek letters ε, δ, . . . denote values in the interval [0, ∞], or possibly infinite tuples thereof. Supersimplicity Convention 1.1. We work in a Hausdorff cat T . We recall from [Ben05b] that every sort admits a definable metric (i.e., a metric d such that for every r ∈ R + , the properties d(x, y) ≤ r and d(x, y) ≥ r are type-definable), or, if not, can be decomposed into uncountably many imaginary sorts each of which does admit such a metric. Therefore, at the price of possibly working with a multi sorted language, we may assume that all sorts admit a definable metric. For convenience we will proceed as if there is a single home sort, but the generalisation to many sorts should be obvious. Let us fix, once and for all, a definable metric on the home sort. By [Ben05b] we know that any two such metrics are uniformly equivalent, so notions such as supersimplicity and superstability are not affected by our choice of metric. If the reader wishes nevertheless to avoid such an arbitrary choice, she or he may use the notion of abstract distances from [Ben05b] instead of real-valued distances whose interpretation depends on a metric function. Distances on tuples will be viewed as tuples of distances of singletons: Definition 1.2. Let I be a set of indices. (i) Ifā andb are I-tuples then we consider d(ā,b) to be the I-tuple (d(a i , b i ) : i ∈ I) ∈ [0, ∞] I (In fact, a definable metric is necessarily bounded, so we can replace ∞ with some real number here, but keeping ∞ as a special distance is convenient). (ii) Ifε,ε ′ ∈ [0, ∞] I , we say thatε ≤ε ′ if ε i ≤ ε ′ i for all i ∈ I. (iii) Ifε,ε ′ ∈ [0, ∞] I , we say thatε <ε ′ if ε i < ε ′ i for all i ∈ I, and ε ′ i = ∞ for all but finitely many i ∈ I. For the purpose of this definition we use the convention that ∞ < ∞. (iv) Givenε ∈ [0, ∞] I and ε ′ ∈ [0, ∞], we understand statements such asε < ε ′ , ε ≤ ε ′ , etc., by replacing ε ′ with the I-tuple all of whose coordinates are ε ′ . From our convention that ∞ < ∞ it follows thatε < ∞ for allε ∈ [0, ∞] I , and ε > 0 =⇒ε > 1 2ε (where 1 2 (ε i ) i∈I = ( ε i 2 ) i∈I , and ∞ 2 = ∞ < ∞). Superstability and supersimplicity in the first order context deal with properties of independence of finite tuples of elements. In the metric setting we replace "finite tuple" with a "virtually finite" one: Definition 1.3. (i) A virtual element is formally a pair (a, ε), where a is a singleton and ε ∈ [0, ∞]. Usually a virtual element (a, ε) will be denoted by a ε , and we think of this conceptually as "the element a up to distance ε". (ii) A virtual tuple is a tuple of virtual elements, i.e., an object of the form (a ε i i : i ∈ I). This can also be denoted byāε, or simply a ε , as single lowercase letters may denote arbitrary tuples. (iii) As in Definition 1.2, ifā is an I-tuple, and ε ∈ [0, ∞] a single distance, we understandā ε asāε, whereε is a tuple consisting of I repetitions of ε. (iv) A virtually finite tuple is a virtual tupleāε such thatε > 0. We remind the reader that according to Definition 1.2, this means that ε i > 0 for all i, and ε i = ∞ for all but finitely many i ∈ I. Thus a ε denotes a virtual tuple, possibly infinite, with the implicit understanding that a and ε are of the same length. If we wish to render explicit the fact that these are tuples we may use notation such asāε etc. We identify a tuple a with the virtual tuple a 0 : knowing a up to distance 0 means knowing a precisely. More generally, if the relation d(x, y) ≤ ε is transitive (e.g., in the rare case where the metric is an ultrametric) then it is an equivalence relation, and we can identify the virtual tuple a ε with the hyperimaginary a/[d(x, y) ≤ ε]. Similarly, we identify a virtual tupleāε with any virtual tuple obtained by omitting or adding virtual elements of the form a ∞ i : knowing a i up to distance ∞ means not knowing a i at all. (The reader will see that these identifications are consistent with the way we use virtual tuples later on.) Note that every virtually finite I-tupleāε can be thus identified with the sub-tuple corresponding to J = {i ∈ I : ε i < ∞}, which is finite asε > 0. This identification allows a convenient re-definition of the notion of a sub-tuple: Definition 1.5. A virtual sub-tuple of a virtual tuple a ε is a virtual tuple (which can be identified with) a ε ′ for some ε ′ ≥ ε. Remark 1.6. The notion of a virtual sub-tuple extends the "ordinary" notion of subtuple. Indeed, letāε be a virtual I-tuple, andbδ a sub-tuple in the ordinary sense, i.e., given by restricting so a subset of indices J ⊆ I. For i ∈ I define ε ′ i = ε i if i ∈ J, and ε ′ i = ∞ otherwise. Thenε ′ ≥ε, andbδ can be identified with the virtual sub-tupleāε ′ . We define types of virtual tuples: Definition 1.7. As we defined in [Ben05b], if p(x) is a partial type in a tuple of variables x, and ε is a tuple of distances of the same length, then p(x ε ) is defined as the partial type ∃y (p(y) ∧ d(x, y) ≤ ε). Since d(x, y) ≤ ε is a type-definable property, this is indeed expressible by a partial type. We define tp(a ε ) as p(x ε ) where p = tp(a). Similarly, if p(x, y) = tp(a, b) then tp(a ε /b δ ) is p(x ε , b δ ). Remark 1.8. If a ′ tp(a ε /b δ ) then we say that a ε and a ′ ε have the same type over b δ , in symbols a ε ≡ b δ a ′ ε . This is a symmetric relation. Proof. Assume that a ′ tp(a ε /b δ ). Then there are a ′′ b ′ ≡ ab such that d(a ′ b, a ′′ b ′ ) ≤ εδ. Let a ′′′ , b ′′ be such that a ′′ b ′ a ′ b ≡ aba ′′′ b ′′ . Then d(ab, a ′′′ b ′′ ) ≤ εδ and a ′′′ b ′′ ≡ a ′ b, whereby a tp(a ′ ε /b δ ). qed 1.8 We recall from [Ben05b]: Definition 1.9. T is supersimple if for every virtually finite tuple a ε and set A, there is a finite subset A 0 ⊆ A such that tp(a ε /A) does not divide over A 0 . If we replace "virtually finite tuple" with "virtually finite singleton" (i.e., a is a singleton) we obtain an equivalent definition. We now turn to the principal new definition in this paper, of independence over virtual tuples. Definition 1.10. We say that an indiscernible sequence (b i : i < ω) could be in tp(b/c ρ ) if there are (c i ) such that: (i) (b i c i : i < ω) is an indiscernible sequence in tp(bc). (ii) For all i, j ≤ ω: d(c i , c j ) ≤ ρ. Remark 1.11. Let E ρ (x, y) be the relation d(x, y) ≤ ρ. Assume that E ρ happens to be transitive, and therefore an equivalence relation (this would happen in the rare case that the metric is an ultrametric, and also if ρ = 0). Then a sequence (b i : i < ω) could be in tp(b/c ρ ) if and only if it has an automorphic image in tp(b/(c/E ρ )). In particular, if ρ = 0 then E is equality, and (b i ) could be in tp(b/c 0 ) if and only if it has an automorphic image in tp(b/c). This justifies the terminology, as well as the identification between c 0 and c. Definition 1.12. We say that a ε \| ⌣c ρ b if every indiscernible sequence that could be in tp(b/c ρ ) could also be in tp(b/a ε c ρ ). As explained in the introduction, this notion of independence has very few "nice" properties, although these suffice for the application we seek. We will not dare to extend it to, say, independence of the form a ε \| ⌣c ρ b δ without being able to show that such extension has useful properties. When restricting to independence over non-virtual (real or even hyperimaginary) tuples, it is not true that a ε \| ⌣b c if and only of tp(a ε /bc) does not divide over c. These notions are close enough to being equivalent, though: Lemma 1.13. Assume that T is simple. For a ε , b and c, the following conditions imply one another from top to bottom: (i) a ε \| ⌣c b (remember to identify c with c 0 ). (ii) There is a Morley sequence for b over c which could be in tp(b/a ε c). (iii) tp(a ε /bc) does not divide over c. (iv) a 2ε \| ⌣c b. Proof. (i) =⇒ (ii). By definition. (ii) =⇒ (iii). We recall the D(−, Ξ) ranks from [Ben03d]: Fixing the tuple x, we define Ξ = Ξ(x) as the set of all pairs (ϕ(x, y), ψ(y <k )) (y and k may vary) such that ϕ and ψ are positive formulae and ψ is a k-inconsistency witness for ϕ, i.e., T ⊢ ¬∃xy <k ψ(y <k ) ∧ i<k ϕ(x, y i ) . If p(x) is a partial type with parameters in A, D(p, Ξ) is a subset of Ξ <\|T \| + such that for ξ <α ∈ Ξ α : • α = 0: ξ <α ∈ D(p, Ξ) if and only if p is consistent. So let (b i : i < ω) be a Morley sequence for b over c which could be in tp(b/a ε c). Then there are (a i : i < ω) and c ′ such that (a i b i c ′ : i < ω) is an indiscernible sequence in tp(abc) and d(a i , a j ) ≤ ε for all i, j < ω. Extend the sequence to length ω + 1. Then by standard arguments we have b ω \| ⌣b <ω a <ω c ′ , so: D(b ω /b <ω , Ξ) = D(b ω /b <ω a <ω c ′ , Ξ) ⊆ D(b ω /a 0 c ′ , Ξ) ⊆ D(b ω /c ′ , Ξ). On the other hand, since (b i ) is a Morley sequence over c: D(b ω /b <ω , Ξ) = D(b/c, Ξ) = D(b ω /c ′ , Ξ) Therefore equality holds all the way and we have b ω \| ⌣c ′ a 0 . Since d(a 0 , a ω ) ≤ ε, it follows that tp(a ε ω /b ω c ′ ) does not divide over c ′ , and by invariance tp(a ε /bc) does not divide over c. (iii) =⇒ (iv). We assume that tp(a ε /bc) does not divide over c. Then there exists a ′ such that a ′ \| ⌣c b and d(a, a ′ ) ≤ ε. Let (b i ) be any indiscernible sequence that could be in tp(b/c). Then we might as well assume that it is in tp(b/c) and since a ′ \| ⌣c b it can even be in tp(b/a ′ c). Find now (a i ) such that a i b i ≡ a ′ c ab. Then we may always choose them such that (a i b i ) is c-indiscernible, and d(a i , a ′ ) ≤ ε for all i yields d(a i , a j ) ≤ 2ε for all i, j, as required. qed 1.13 This means that a \| ⌣c b if and only if tp(a/bc) does not divide over c (since 2 · 0 = 0), so this definition agrees with the usual definition of independence of ordinary (i.e., nonvirtual) elements. We can continue Remark 1.11 to show that if the tuple of distances ρ defines an equivalence relation E ρ then a ε \| ⌣c ρ b if and only if a ε \| ⌣c/E ρ b (here c/E ρ is viewed as hyperimaginary, rather than virtual). If ε also defines an equivalence relation E ε , then a ε \| ⌣c ρ b if and only if a/E ε \| ⌣c/E ρ b. Also, Lemma 1.13 and the fact that ε > 0 =⇒ 1 2 ε > 0 give: Proposition 1.14. T is supersimple if and only if for every virtually finite tuple (singleton) a ε and set A there is a finite subset A 0 ⊆ A such that a ε \| ⌣A 0 A. Proposition 1.15. Independence satisfies right downward transitivity, left upward transitivity, and two-sided monotonicity: (i) If a ε \| ⌣c ρ b and δ is any tuple of distances of the length of b, then a ε \| ⌣b δ c ρ b. (ii) If a ε \| ⌣c ρ b and d υ \| ⌣a ε c ρ b then a ε d υ \| ⌣c ρ b. (iii) If a ε \| ⌣c ρ bd and ε ′ ≥ ε (i.e., if a ε ′ is a virtual sub-tuple of a ε ) then a ε ′ \| ⌣c ρ b. Proof. (i) Let (b i ) be an indiscernible sequence that could be in tp(b/b δ c ρ ). This is the same as saying that (b i ) could be in tp(b/c ρ ) and d(b i , b j ) ≤ δ for all i, j < ω. As we assume that a ε \| ⌣c ρ b, the sequence (b i ) could be in tp(b/a ε c ρ ); since d(b i , b j ) ≤ δ it could also be in tp(b/a ε b δ c ρ ), as required. (ii) If (b i ) is indiscernible and could be in tp(b/c ρ ) then it could also be in tp(b/a ε c ρ ) and therefore in tp(b/a ε c ρ d υ ). (iii) Let (b i ) be an indiscernible sequence that could be in tp(b/c ρ ). By standard arguments we can find (d i ) such that (b i d i ) is indiscernible and could be in tp(bd/c ρ ). As we assume that a ε \| ⌣c ρ bd, it could also be in tp(bd/a ε c ρ ). Therefore (b i ) could be in tp(b/a ε c ρ ) and a fortiori in tp(b/a ε ′ c ρ ). qed 1.15 We obtain a more general form, in this context, of the finite character of independence: Proposition 1.16. For all a ε , b and c ρ : a ε \| ⌣c ρ b if and only if a ε ′ \| ⌣c ρ b ′ for all ε ′ > ε and finite b ′ ⊆ b. (By our approach, a ε ′ should be viewed as a finite sub-tuple of a ε , since ε ′ > ε.) In particular, a \| ⌣c b if and only if a ε \| ⌣c b for all ε > 0. Proof. Left to right is by monotonicity. For right to left, assume that a ε \| ⌣c ρ b. Then there is an indiscernible sequence (b i ) that could be in tp(b/c ρ ) but not in tp(b/a ε c ρ ). Letting p(x, y, z) = tp(a, b, c), the latter means that the following is inconsistent: i<ω p(x i , b i , z i ) ∧ i,j<ω [d(x i , x j ) ≤ ε ∧ d(z i , z j ) ≤ ρ]. Since d(x i , x j ) ≤ ε is logically equivalent to ε ′ >ε d(x i , x j ) ≤ ε ′ , and the family of all ε ′ > ε is closed for finite infima, we obtain by compactness some ε ′ > ε such that the above is still inconsistent with ε ′ instead of ε. Therefore a ε ′ \| ⌣c ρ b. Replacing b with a finite sub-tuple is similar (and fairly standard). qed 1.16 Let us recall from [Ben03c, Lemma 1.2] the following useful fact about "extraction" of indiscernible sequence from long sequences: δ =δ > 0 such that a ε \| ⌣b δ b. Let B 0 = {b i : δ i < ∞}. Then B 0 is finite, and by right downward transitivity: a ε \| ⌣B 0 B. We now prove left to right: aiming for a contradiction, we assume that T is supersimple, and yet there is no δ as in the statement. Let p(x, y) = tp(a, b), q(y) = tp(b). We will construct by induction a sequence of tuples (b n : n < ω) in q, and a sequence of tuples of distances (δ n : n < ω). These will satisfy, among other things, that δ n ≥ 2δ n+1 > 0 and d(b n , b n+1 ) ≤ δ n . For convenience, let δ −1 = ∞. At the nth step, assume we already have b <n satisfying q and δ n−1 > 0. By assumption a ε \| ⌣b δ n−1 b. Therefore there is an indiscernible sequence (b i n : i < ω) such that d(b i n , b j n ) ≤ δ n−1 for all i, j < ω and yet the following is inconsistent: i<ω p(x i , b i n ) ∧ i,j<ω d(x i , x j ) ≤ ε (* n ) By a compactness argument as in the proof of Proposition 1.16, there exists δ n > 0 such that the following weakening of (* n ) is still inconsistent: i<ω p(x i , b i n 2δn ) ∧ i,j<ω d(x i , x j ) ≤ ε ( n ) We may always assume that δ n ≤ 1 2 δ n−1 . If n = 0, the sequence (b i n : i < ω) is b <n -indiscernible, and we skip the following paragraph. If n > 0, note that all that matters for ( n ) is the type of the sequence (b i n : i < ω): we may therefore replace it with another sequence which has the same type, such that in addition (b i n : i < ω) is b <n -indiscernible and satisfy d(b i n , b n−1 ) ≤ δ n−1 . In order to see this, extend this sequence to arbitrary length λ + 1 (b i n : i ≤ λ). Since b λ n ≡ b ≡ b n−1 , we may assume (up to replacing (b i n : i ≤ λ) with an automorphic image) that b λ n = b n−1 . Applying Fact 1.17 to the sequence (b i n : i < λ) over b <n , we can find a sequence (c i n : i < ω) which is b <n -indiscernible, and such that for all m < ω there are i 0 < · · · < i m−1 < λ such that Therefore (c i n : i < ω) ≡ (b i n : i < ω), and d(c i n , b n−1 ) ≤ δ n−1 , so the sequence (c i n : i < ω) has the required properties. Let b n = b 0 n , so in particular d(b n , b n−1 ) ≤ δ n−1 , and continue the construction. Once the construction is done, let δ ω = inf δ n . If b is an I-tuple then so are δ n = δ n,∈I and δ ω = δ ω,∈I . Since δ n ≥ 2δ n+1 for all n, we must have δ ω,i ∈ {0, ∞} for all i ∈ I. But if i ∈ I is such that δ ω,i = ∞, then δ n,i = ∞ for all n, which means that the ith coordinate of b and the b n played absolutely no role throughout the construction, and may be entirely dropped. Therefore, replacing b, b n , δ n , etc., with sub-tuples we may assume that δ ω = 0. The fact that δ n ≥ 2δ n+1 implies that for every n ≤ m: d(b n , b m ) ≤ 2δ n . Thus the partial type q(y) ∧ n d(b n , y) ≤ 2δ n is consistent, and has a realisation b ω . Since inf δ n = 0, b ω is the unique realisation of this type, so b ω ∈ dcl(b <ω ) (we say that b ω is the limit of the Cauchy sequence (b n : n < ω)). Since b ω ≡ b, there is a ω such that p(a ω , b ω ). By supersimplicity there is n such that a ε ω \| ⌣b <n b <ω , whereby a ε ω \| ⌣b <n b ω . Let us go back to our b <n -indiscernible sequence (b i n : i < ω), and we recall that b n = b 0 n . Find (b i ω : i < ω) such that b 0 ω = b ω and (b i n b i ω : i < ω) is b <n -indiscernible. Since a ε ω \| ⌣b <n b ω , there are (a i ω : i < ω) realising i<ω p(a i ω , b i ω ) ∧ i,j<ω d(a i ω , a j ω ) ≤ ε But d(b n , b ω ) ≤ 2δ n =⇒ d(b i n , b i ω ) ≤ 2δ n . This shows that: i<ω p(a i ω , b i n 2δn ) ∧ i,j<ω d(a i ω , a j ω ) ≤ ε, so (** n ) was consistent after all. qed 1.18 Having given meaning to a ε \| ⌣c b, it is natural to define the corresponding SU-ranks: Definition 1.19. SU(a ε /b) is the minimal rank taking ordinal values or ∞ satisfying: SU(a ε /b) ≥ α + 1 if and only if there is c such that a ε \| ⌣b c and SU(a ε /bc) ≥ α. Proposition 1.20. (i) T is supersimple if and only if SU(a ε /b) < ∞ for all b and virtually finite a ε . (ii) Assuming that T is supersimple, a \| ⌣b c ⇐⇒ SU(a ε /bc) = SU(a ε /b) for all ε > 0. Proof. (i) Standard argument, using Proposition 1.14. (ii) If SU(a ε /bc) = SU(a ε /b) for all ε > 0, then a ε \| ⌣b c for all ε > 0, whereby a \| ⌣b c by the finite character (Proposition 1.16). Conversely, assume that a \| ⌣b c. Clearly, SU(a ε /b) ≥ SU(a ε /bc). We prove by induction on α that SU(a ε /b) ≥ α =⇒ SU(a ε /bc) ≥ α. For α = 0 and limit this is clear, so we need to prove for α = β + 1. Since SU(a ε /b) ≥ β + 1, there is d such that SU(a ε /bd) ≥ β and a ε \| ⌣b d. We may assume that d \| ⌣ab c. We assumed that a \| ⌣b c, whereby ad \| ⌣b c and a \| ⌣bd c. Therefore, by the induction hypothesis, SU(a ε /bcd) ≥ β. On the other hand, a ε \| ⌣bc d: otherwise we'd get a ε c \| ⌣b d by Proposition 1.15, contradicting prior assumptions. This shows that SU(a ε /bc) ≥ β + 1 = α. qed 1.20 Question 1.21. It is fairly easy to prove that for all virtually finite a ε and b δ , and all tuples c: SU(a ε /bc) + SU(b δ /c) ≤ SU(a ε b δ /c). Note, however, that we use SU(a ε /bc) rather than SU(a ε /b δ c), to which we haven't given a meaning. This is a serious problem, since the converse inequality may easily be false (for example, if a = b and δ = ∞.): SU(a ε b δ /c) SU(a ε /bc) ⊕ SU(b δ /c). Is there a way to give meaning to SU(a ε /b δ c) such that the standard Lascar inequalities (or reasonable variants thereof) hold? Lovely pairs We assume familiarity at least with the basics of lovely pairs as exposed in [BPV03], where for every simple first order theory T we constructed its theory of lovely pairs T P , and proved that if T has the weak non-finite-cover-property then T P is simple and independence in T P was characterised. In [Ben04] we generalised the latter result to the case where T is any thick simple cat. Namely, for each such T we constructed a cat T P whose \|T \| + -saturated models are precisely the lovely pairs of models of T , and proved it is simple with the same characterisation of independence. If T is a first order theory then T P is first order if and only if T has the weak non-finite-cover-property, in which case T P coincides with T P . Convention 2.1. If (M, P ) is a lovely pair of models of T and a ∈ M then tp(a) denotes the type of a in M (in the sense of T ) while tp P (a) denotes its type in (M, P ) (i.e., in the sense of T P ). We are going to use a few results from [Ben04] which do not appear explicitly in [BPV03]. If (M, P ) is a lovely pair and a ∈ M, then tp P (a) determines the set of all possible types of Morley sequences (both in the sense of T ) for a over P (M). Conversely, any of these types determines tp P (a). Also, the property "the sequence (a i : i < ω) has the type of a Morley sequence for a over P " is definable by a partial type in x <ω , which is denoted by mcl(a) (the Morley class of a). Finally, let a, b, c ∈ M, and (a i b i c i : i < ω) mcl(abc) be a sequence in some model of T . Then a is independent from b over c in the sense of T P , in symbols a \| P ⌣c b, if and only if a <ω \| ⌣c <ω b <ω (here in the sense of T ). Convention 2.2. T is a simple Hausdorff cat, and in particular thick. Therefore T P exists and the properties mentioned above hold. Since T is Hausdorff, so is T P . Also, as T is a reduct of T P , any definable metric we might have fixed for T is also a definable metric in the sense of T P . (Since T P is richer, there may be new definable metrics: however, as all definable metrics are uniformly equivalent, this makes no difference.) As we said earlier, the notation a ε <ω here means the virtual tuple (a ε i : i < ω), and similarly for tuples of other lengths. As ε is a tuple of distances of the length of a, there should be no ambiguity about this. Proof. Assume that tp P (a ε /bc) does not divide over c. Then there is a ′ ∈ M such that d(a, a ′ ) ≤ ε and a ′ \| P ⌣c b. There exist (a ′ i : i < ω) such that (a ′ i a i b i c i : i < ω) mcl(a ′ abc). Then a ′ \| P ⌣c b =⇒ a ′ <ω \| ⌣c <ω b <ω , and d(a, a ′ ) ≤ ε =⇒ d(a i , a ′ i ) ≤ ε for all i < ω, whereby tp(a ε <ω /b <ω c <ω ) does not divide over c <ω . Conversely, assume that tp(a ε <ω /b <ω c <ω ) does not divide over c <ω . Then there exist (a ′ i : i < ω) such that d(a i , a ′ i ) < ε for all i < ω and a ′ <ω \| ⌣c <ω b <ω , but the sequence (a ′ i a i b i c i : i < ω) needs not be indiscernible. Continue the sequence (a i b i c i ) to an indiscernible sequence of length λ > ω, big enough to allow us to apply Fact 1.17 later on. Let e = Cb(b ω c ω /b <ω c <ω ) and f = Cb(c ω /c <ω ) (in the sense of T ), so (b i c i : i < λ) is a Morley sequence over e, and (c i : i < λ) is a Morley sequence over f , and indiscernible over ef , whereby c <λ \| ⌣f e. It follows that for all w ⊆ ω: b ∈w c ∈w \| ⌣ e c ∈ω w =⇒ b ∈w \| ⌣ c∈we c <ω =⇒ b ∈w e \| ⌣ c∈wf c <ω Whereby: b <ω \| ⌣ c<ω a ′ <ω =⇒ b ∈w \| ⌣ c<ω a ′ <ω =⇒ b ∈w \| ⌣ f c∈w a ′ ∈w . For any finite w ⊆ λ, tp(b ∈w c ∈w f ) depends solely on \|w\| (where the tuples b ∈w and c ∈w are enumerated according to the ordering induced on w from λ). Thus by the definability of independence for known complete types, for every n < ω and tuple of variables x there is a partial type ρ n,x (x, y <n , z <n , f ) such that for every w ∈ [λ] n and every g in the sort of x: b ∈w \| ⌣ c∈wf g ⇐⇒ ρ n,x (g, b ∈w , c ∈w , f ). Putting these two facts together and applying compactness we can find a sequence (a ′′ i : i < λ) such that d(a i , a ′′ i ) ≤ ε for all i < λ, and b ∈w \| ⌣c ∈wf a ′′ ∈w for every finite w ⊆ λ. As we could have chosen λ arbitrarily big, by standard extraction arguments (i.e., Fact 1.17) there exists a sequence (ã i : i < ω) such that (ã i a i b i c i : i < ω) is f -indiscernible, and in addition d(a i ,ã i ) ≤ ε for all i < ω, and b ∈w \| ⌣c ∈wfã ∈w for all finite w ⊆ ω. As every formula in tp(ã <ω /b <ω c <ω f ) only involves finitely many variables and parameters, it does not divide over c ∈w f for some finite w ⊆ ω, and a fortiori over c <ω f . Thereforẽ a <ω \| ⌣c <ω f b <ω . As f ∈ dcl(c <ω ), we conclude thatã <ω \| ⌣c <ω b <ω . As the sequence (ã i a i b i c i : i < ω) is indiscernible, we may extend it to length ω + 1. Then one can find a lovely pair (M, P ) such thatã ≤ω a ≤ω b ≤ω c ≤ω ∈ M,ã <ω a <ω b <ω c <ω ∈ P , andã ω a ω b ω c ω \| ⌣ã <ω a<ωb<ωc<ω P . It follows that (ã i a i b i c i : i < ω) mcl (M,P ) (ã ω a ω b ω c ω ). Thus d(a ω ,ã ω ) ≤ ε andã ω \| P ⌣c ω b ω . In particular, tp P (a ω ε /b ω c ω ) does not divide over c ω . As we assumed that (a i b i c i : i < ω) mcl(abc), we have a ω b ω c ω ≡ P abc, so tp P (a ε /bc) does not divide over c. Theorem 2.4. If T is supersimple, then so is T P . Proof. We need to show that for every virtually finite element a ε and every tuple B = b ∈I in a model of T P , there exists a finite sub-tuple B ′ ⊆ B such that tp P (a ε /B) does not divide over B ′ . We follow the path of [Ben04, Corollary 3.6]. Choose (a j B j : j < 2ω) mcl(aB) in some model of T . Let b ∈I,j be the enumeration of each B j corresponding to B = b ∈I . By supersimplicity of T there are tuples of distances υ = υ <ω > 0 and ρ = ρ ∈I,<2ω > 0 such that: a ε ω \| ⌣ (a<ω ) υ (B <2ω ) ρ a <ω B <2ω .(1) Then by definition, there are only finitely many j < ω such that υ j = ∞, and only finitely many pairs (i, j) ∈ I × 2ω such that ρ i,j = ∞, so we can define: n = 1 + max{j < ω : υ j = ∞ or there exists i ∈ I such that ρ i,j = ∞}, δ = min{ε, υ j : j < n}, J 0 = {i ∈ I : ρ i,j = ∞ for some j < 2ω}. In particular, ε ≥ δ > 0 and J 0 ⊆ I is finite. By right downward transitivity, (1) becomes: a ε ω \| ⌣ a δ <n ,b ∈J 0 ,∈[0,n)∪[ω,2ω) a <ω B <2ω . Applying supersimplicity again, there is J 1 ⊆ I finite such that: a δ <n \| ⌣ b ∈J 1 ,<ω B <ω . Let J = J 0 ∪ J 1 ⊆ I, and let B ′ = b ∈J , B ′ j = b ∈J,j . These are finite sub-tuples of B and B j , respectively, and: a ε ω \| ⌣ a δ <n ,B ′ ∈[0,n)∪[ω,2ω) a <ω B <2ω , (2) a δ <n \| ⌣ B ′ <ω B <ω .(3) We now prove by induction on n ≤ m < ω that: a δ <n a ε <m \| ⌣ B ′ <ω B <ω .(4) For m = n, this follows from (3) since ε ≥ δ. Assume now (3) for some n ≤ m < ω. From (2) we obtain by monotonicity and right downward transitivity: Since (a j , b ∈I,j : j < 2ω) is an indiscernible sequence, there is an automorphism sending a ω+j B ω+j to a m+j B m+j for every j < ω, while keeping a <m B <m in place. Applying such an automorphism to (5) we get: a ε m \| ⌣ a δ <n a ε <m ,B ′ <ω B <ω . By left upward transitivity and the induction assumption (4) we obtain: a δ <n a ε <m+1 \| ⌣ B ′ <ω B <ω . This concludes the proof of (4). By the finite character of independence (Proposition 1.16) we conclude that: a δ <n a ε <ω \| ⌣ B ′ <ω B <ω . In particular, tp(a ε <ω /B <ω ) does not divide over B ′ <ω . By Lemma 2.3 tp P (a ε /B) does not divide over B ′ , which is finite, as required. qed 2.4 Corollary 2.5. If T is superstable, then so is T P . Proof. T is superstable if and only if it is stable and supersimple. In that case T P is stable by [Ben04,Theorem 3.10] and supersimple by Theorem 2.4. Therefore T P is superstable. qed 2.5 Question 2.6. Assume that T is ω-stable. Is T P ω-stable as well? Notation 1. 4 . 4Unless explicitly said otherwise, a, b, etc., denote possibly infinite tuples of elements in a model. Similarly, ε, δ, etc., denote possibly infinite tuples in [0, ∞]. • α limit: ξ <α ∈ D(p, Ξ) if and only if ξ <β ∈ D(p, Ξ) for all β < α.• α = β + 1, ξ β = (ϕ(x,y), ψ(y <k )): ξ <α ∈ D(p, Ξ) if and only if there exists an A-indiscernible sequence (b i : i < ω) in the sort of y such that ψ(b <k ) and ξ <β ∈ D(p ∪ {ϕ(x, b 0 )}, Ξ). We recall that this rank characterises independence (for T simple and thick, and thus in particular simple and Hausdorff): If A ⊆ B then p ∈ S(B) does not divide over A if and only if D(p, Ξ) = D(p↾ A , Ξ). Fact 1 . 17 . 117Let A be a set of parameters, and λ ≥ \| Sκ(A)\| + . Then for any sequence(a i : i < λ) of κ-tuples there is an A-indiscernible sequence (b i : i < ω) such that for all n < ω there are i 0 < . . . < i n−1 < λ for which tp(b 0 . . . b n−1 /A) = tp(a i 0 . . . a i n−1 /A).Theorem 1.18. T is supersimple if and only if for every virtually finite tuple (singleton) a ε , and any tuple b, there is a virtually finite sub-tuple b δ of b (i.e., there exists δ > 0 of the appropriate length) such that a ε \| ⌣b δ b. Proof. In order to prove right to left, it would suffice to show that for every virtually finite singleton a ε and set B there is B 0 ⊆ B finite such that a ε \| ⌣B 0 B. Let b =b be an I-tuple enumerating B. Then by assumption there is Lemma 2 . 3 . 23Let a, b and c be tuples in a lovely pair (M, P ), and let (a i b i c i : i < ω) mcl(abc) in a universal domain for T . Let a ε be a virtual sub-tuple of a. Then tp P (a ε /bc) divides over c (in the sense of T P ) if and only if tp(a ε <ω /b <ω c <ω ) divides over c <ω (in the sense of T ). Schrödinger's cat. Itay Ben-Yaacov, Israel Journal of Mathematics. to appearItay Ben-Yaacov, Schrödinger's cat, Israel Journal of Mathematics, to appear. Discouraging results for ultraimaginary independence theory. Journal of Symbolic Logic. 683, Discouraging results for ultraimaginary independence theory, Journal of Symbolic Logic 68 (2003), no. 3, 846-850. Positive model theory and compact abstract theories. Journal of Mathematical Logic. 31, Positive model theory and compact abstract theories, Journal of Mathematical Logic 3 (2003), no. 1, 85-118. Simplicity in compact abstract theories. Journal of Mathematical Logic. 32, Simplicity in compact abstract theories, Journal of Mathematical Logic 3 (2003), no. 2, 163-191. Thickness, and a categoric view of type-space functors. Fundamenta Mathematicae. 179, Thickness, and a categoric view of type-space functors, Fundamenta Mathematicae 179 (2003), 199-224. Lovely pairs of models: the non first order case. Journal of Symbolic Logic. 693, Lovely pairs of models: the non first order case, Journal of Symbolic Logic 69 (2004), no. 3, 641-662. Compactness and independence in non first order frameworks. Bulletin of Symbolic Logic. 111, Compactness and independence in non first order frameworks, Bulletin of Symbolic Logic 11 (2005), no. 1, 28-50. Uncountable dense categoricity in cats. Journal of Symbolic Logic. 703, Uncountable dense categoricity in cats, Journal of Symbolic Logic 70 (2005), no. 3, 829-860. Simple homogeneous models. Steven Buechler, Olivier Lessmann, Journal of the American Mathematical Society. 16Steven Buechler and Olivier Lessmann, Simple homogeneous models, Journal of the American Mathematical Society 16 (2003), 91-121. Lovely pairs of models. Itay Ben-Yaacov, Anand Pillay, Evgueni Vassiliev, Annals of Pure and Aplied Logic. 122Itay Ben-Yaacov, Anand Pillay, and Evgueni Vassiliev, Lovely pairs of models, Annals of Pure and Aplied Logic 122 (2003), 235-261. Pseudoprojective strongly minimal sets are locally projective. Steven Buechler, Journal of Symbolic Logic. 564Steven Buechler, Pseudoprojective strongly minimal sets are locally projective, Journal of Sym- bolic Logic 56 (1991), no. 4, 1184-1194. Stable Banach spaces and Banach space structures, I and II, Models, Algebras, and Proofs. José Iovino, Lectures Notes in Pure and Appl. Math. 203DekkerJosé Iovino, Stable Banach spaces and Banach space structures, I and II, Models, Algebras, and Proofs (Bogotá, 1995), Lectures Notes in Pure and Appl. Math., no. 203, Dekker, New York, 1999, pp. 77-117. Bruno Poizat, Paires de structures stables. 48Bruno Poizat, Paires de structures stables, Journal of Symbolic Logic 48 (1983), no. 2, 239-249. The lazy model-theoretician's guide to stability. Saharon Shelah, Logique et Analyse 71-72Saharon Shelah, The lazy model-theoretician's guide to stability, Logique et Analyse 71-72 (1975), 241-308. Generic pairs of SU-rank 1 structures. Evgueni Vassiliev, Annals of Pure and Applied Logic. 120Evgueni Vassiliev, Generic pairs of SU-rank 1 structures, Annals of Pure and Applied Logic 120 (2002), 103-149. . Itay Ben-Yaacov, University of Wisconsin -Madison, Department of MathematicsItay Ben-Yaacov, University of Wisconsin -Madison, Department of Mathematics, . Lincoln Drive, Madison, Usa Wi 53706, Url, Lincoln Drive, Madison, WI 53706, USA URL: http://www.math.wisc.edu/~pezz	[]
[ "Non-vanishing forms in projective space over finite fields", "Non-vanishing forms in projective space over finite fields" ]	[ "Samuel Lundqvist " ]	[]	[]	We consider a subset of projective space over a finite field and give bounds on the minimal degree of a non-vanishing form with respect to this subset.	10.1216/jca-2010-2-4-437	[ "https://arxiv.org/pdf/0905.0342v2.pdf" ]	115,171,996	0905.0342	65743141fa4513a59b66e4b0ec2fffc7720102a6	Non-vanishing forms in projective space over finite fields 4 May 2009 May 4, 2009 Samuel Lundqvist Non-vanishing forms in projective space over finite fields 4 May 2009 May 4, 2009 We consider a subset of projective space over a finite field and give bounds on the minimal degree of a non-vanishing form with respect to this subset. Introduction Let X = {p 1 , . . . , p m } be a set of points in P n (k), where k is a field. We say that a form f in k[x 0 , . . . , x n ] is non-vanishing with respect to X if f (p i ) = 0 for all i. When k is an infinite field, then there is an infinite number of linear forms which are non-vanishing on X. But consider for instance X = {(1 : 0), (0 : 1), (1 : 1)} in P 1 (F 2 ), where F 2 denotes the finite field with two elements. There are three linear forms in F 2 [x 0 , x 1 ]: x 0 , x 1 and x 0 + x 1 . We have x 0 ((0 : 1)) = x 1 ((1 : 0)) = (x 0 + x 1 )((1 : 1)) = 0. Thus, there is no linear non-vanishing form with respect to X. When X ⊆ P n (F q ), where F q denotes a field with q elements, let DegNz(X) denote the least degree of a non-vanishing form f . The aim of this paper is to give bounds on DegNz(X) and combining Theorem 3.1 and Theorem 4.2 we show Theorem 1.1. d 1 ≤ DegNz(X) ≤ d 2 , where d 1 is the biggest integer such that q n−d1+2 + · · · + q n < \|X\| and where d 2 is the least integer such that \|X\| ≤ q + · · · + q d2 . In the language of commutative algebra, a form f in k[x 0 , . . . , x n ] is nonvanishing with respect to a set of projective points X if and only if [f ] is a nonzero divisor in the quotient ring k[x 0 , . . . , x n ]/I(X), where I(X) is the vanishing ideal with respect to X and [f] denotes the equivalence class in k[x 0 , . . . , x n ]/I(X) containing f . The problem of finding a non-vanishing form came from an algorithm to compute the variety of an ideal of projective dimension zero [3]. A crucial point of the algorithm is to find a minimal degree of a non-zero divisor in the quotient ring k[x 0 , . . . , x n ]/I(X). It should be mentioned that the existence of a non-zero divisor could be rephrased in terms of the union of finite subspaces of k[x 0 , . . . , x n ] as follows. Let X be a set of projective points and let I(p i ) be the set of all polynomials vanishing on p i . Let A = ∪ i I(p i ) and let d be the least positive degree such that k[x 0 , . . . , x n ] d is not contained in A. Then d = DegNz(X). Preliminaries To give the bounds, we use Warning's theorem (Satz 3 in [4]), which states that if the equation f = 0, where f in F q [x 1 , . . . , x n ] is an element of degree d < n, has a solution, then it has at least q n−d solutions. For simplicity, we state the projective version of this result as a lemma. Lemma 2.1. Let f be a form in F q [x 0 , . . . , x n ] of degree d < n + 1. Then there are at least 1 + q + · · · + q n−d solutions to f = 0 in P n (F q ). Proof. Removing the trivial solution, we are left with at least q n−d+1 − 1 zeroes. Thus, the number of projective solutions is at least (q n−d+1 − 1)/(q − 1) = 1 + q + · · · + q n−d . The requirement on d in Warning's theorem is sharp. Indeed, Lang [2] gives a construction of a form of degree n + 1 in F q [x 0 , . . . , x n ] which is non-vanishing with respect to P n (F q ) -it is the norm of the element x 0 e 0 + · · · + x n e n , where e 0 , . . . , e n is a basis for an extension F q n+1 of degree n + 1 of F q . Recall that the norm of an element α in F q n+1 is defined as Nm(α) = α · Fr q (α) · Fr 2 q (α) · · · Fr n q (α), where Fr q is the Frobenius map Fr q : F q n+1 → F q n+1 , α → α q . Example 2.2. Suppose that we want to find a quadratic non-vanishing form with respect to P 1 (F 2 ). Since 1 + y + y 2 is irreducible over F 2 [y], a basis for the extension field F 4 of F 2 is {1, y}. We get Nm(x 0 +x 1 y) = (x 0 +x 1 y)·Fr 2 (x 0 +x 1 y) = (x 0 +x 1 y)·(Fr 2 (x 0 )+Fr 2 (x 1 ) Fr 2 (y)) = (x 0 + x 1 y) · (x 0 + x 1 (1 + y)) = (x 0 + x 1 y)(x 0 + x 1 + x 1 y) = x 2 0 + x 0 x 1 + x 2 1 . The upper bound Theorem 3.1. Let X ⊆ P n (F q ) and let d be the least integer such that \|X\| ≤ q + · · · + q d . Then DegNz(X) ≤ d. This bound is the best possible. Proof. Induction on n. Consider first P 1 (F q ). We have \|X\| ≤ q + 1. When \|X\| = P 1 (F q ), it follows from Lang's construction that there is a quadratic form which is non-vanishing on X. If \|X\| < q + 1, we need to show that there is a linear form which is non-vanishing on the points. Let p a point not in \|X\| and consider coordinates x 0 , x 1 such that p = (0 : 1). It is then clear that x 0 evaluated at any point in X is non-zero. Let now X be a subset of P n (F q ) for n > 1. If X = P n (F q ), then there is a non-vanishing form of degree n + 1 due to Lang's construction. Since \|P n (F q )\| = 1 + q + · · · + q n , it is clear that the least integer d such that q + · · · + q d ≥ \|X\| equals n + 1. If X is a proper subset of P n (F q ), pick a point p in P n (F q ) \ X. It is possible to choose coordinates x 0 , . . . , x n such that p = (0 : · · · : 0 : 1). With respect to these coordinates, the map π : X → P n−1 (F q ), (a 0 : · · · : a n ) → (a 0 : · · · : a n−1 ) is well defined. By the induction assumption, there is a non-vanishing form f of degree at most d, with respect to the set π(X) ⊆ P n−1 (F q ), where d is the least integer such that q d + · · · + q ≥ \|π(X)\|. Butf (x) = f (π(x)) for all points x ∈ X, wheref is the image of f with respect to the natural embedding of F q [x 0 , . . . , x n−1 ] into F q [x 0 , . . . , x n ]. Thus,f is non-vanishing on X. Since \|π(X)\| ≤ \|X\|, the first part of the theorem follows. For the second part, consider a set X ⊂ P n (F q ) with q + · · · + q n elements. Let f be a non-vanishing form of degree d with respect to X. By the first part of the theorem, d ≤ n. By Lemma 2.1, the number of points where f is vanishing is at least 1 + q + · · · + q n−d . Hence, we must have d = n. It follows that the bound is sharp. ∈ X. We project down to P 2 (F 2 ) and get the points π(X) = {(1 : 1 : 1), (0 : 0 : 1), (1 : 0 : 0), (0 : 1 : 0), (1 : 1 : 0)}. Notice that π(0 : 0 : 1 : 1) = π(0 : 0 : 1 : 0). Now we are looking for a non-vanishing form with respect to these five points in P 2 (F 2 ). We notice that the point p ′ = (1 : 0 : 1) is missing, so we consider the linear change of coordinates z 0 = y 0 + y 2 , z 1 = y 1 , z 2 = y 2 for which p ′ = (0 : 0 : 1), and we get π(X) = {(0 : 1 : 1), (1 : 0 : 1), (1 : 0 : 0), (0 : 1 : 0), (1 : 1 : 0)} with respect to these coordinates. We project down to P 1 (F 2 ) to get the points {(0 : 1), (1 : 0), (1 : 1)}. From Example 2.2, we know that z 2 0 + z 2 1 + z 0 z 1 is non-vanishing and hence (y 0 + y 2 ) 2 + (y 0 + y 2 )y 1 + y 2 1 is non-vanishing on P and finally we get (x 0 + x 2 + x 3 ) 2 + (x 0 + x 2 + x 3 )x 1 + x 2 1 , a quadratic form which is non-vanishing on X. Note that to actually construct a non-vanishing form, we have to compute a norm-form. Thus, our method relies on finding an irreducible over F q [y]. We refer the reader to the paper [1], where algorithms to construct irreducibles are discussed. The lower bound Lemma 4.1. There is a point set X, with \|X\| = q n−d+1 + · · · + q n , such that DegNz(X) ≤ d. Proof. Let τ be the embedding of P n−d into P n defined by (a 0 : · · · : a n−d ) → (0 : · · · : 0 d times : a 0 : · · · : a n−d ). Let X = P n (F q ) \ τ (P n−d (F q )). We can now define a map π from X to P d by (a 0 : · · · : a n ) → (a 0 : · · · : a d ). Notice that (0 : · · · : 0 : 1) / ∈ π(X). Hence, \|π(X)\| = \|P d \| − 1 = q + · · · + q d and by Theorem 3.1, there is a form f in F q [x 0 , . . . , x d ] of degree bounded by d which is non-vanishing with respect to π(X). This form is naturally embedded into F q [x 0 , . . . , x n ]. Theorem 4.2. Let X ⊆ P n (F q ) and let d be the biggest integer such that q n−d+2 + · · · + q n < \|X\|. Then DegNz(X) ≥ d. This bound is the best possible. Proof. Let f be a form of degree d − 1. By Lemma 2.1, the number of projective solutions is at least 1 + q + · · · + q n−d+1 . It follows that there are at most P n (F q ) − (1 + q + · · · + q n−d+1 ) = q n−d+2 + · · · + q n points where f is nonvanishing. By assumption, q n−d+2 + · · · + q n < \|X\|. Thus, there is a point p in X such that f (p) = 0. Hence DegNz(X) > d − 1. By Lemma 4.1, there is a point set X with q n−d+2 + · · · + q n < \|X\| = q n−d+1 + · · · + q n for which there is a non-vanishing form of degree at most d. Thus, the degree of the non-vanishing form in Lemma 4.1 must be equal to d. It follows that the bound is the best possible. Example 3 . 2 . 32Consider P 3 (F 2 ) and the point set X = {(1 to some coordinates x 0 , x 1 , x 2 , x 3 . We have 2 2 + 2 = 6 and hence, by Theorem 3.1, there is a form of degree at most two which is non-vanishing on X. To find it, pick the point (0 : 0 : 1 : 1) / ∈ X. With respect to the linear change y 0 = x 0 , y 1 = x 1 , y 2 = x 2 + x 3 and y 3 = x 3 of coordinates, this point reads (0 : 0 : 0 : 1). Thus, with respect to the coordinates y 0 , y 1 , y 2 , y 3 , X = {(1 L Adleman, H LenstraJr, Finding irreducible polynomials over finite fields, STOC 1986. L. Adleman and H. Lenstra Jr, Finding irreducible polynomials over finite fields, STOC 1986, 350-355. On quasi algebraic closure. S Lang, Ann. of Math. 255S. Lang, On quasi algebraic closure, Ann. of Math. 2, no. 55, 1952, 373-390. S Lundqvist, arXiv:0903.2028Multiplication matrices and ideals of projective dimension zero. Submitted to Journal of Symbolic ComputationS. Lundqvist, Multiplication matrices and ideals of projective dimension zero, arXiv:0903.2028, 2009. Submitted to Journal of Symbolic Computa- tion. Bemerkung zur vorstehenden Arbeit von Herr Chevalley. E Warning, Abh. Math. Sem. Univ. Hamburg. 11E. Warning, Bemerkung zur vorstehenden Arbeit von Herr Chevalley, Abh. Math. Sem. Univ. Hamburg 11, 1936, 76-83.	[]
[ "COSMOGRAPHY APPROACH TO DARK ENERGY COSMOLOGIES: NEW CONSTRAINS USING THE HUBBLE DIAGRAMS OF SUPERNOVAE, QUASARS AND GAMMA-RAY BURSTS", "COSMOGRAPHY APPROACH TO DARK ENERGY COSMOLOGIES: NEW CONSTRAINS USING THE HUBBLE DIAGRAMS OF SUPERNOVAE, QUASARS AND GAMMA-RAY BURSTS" ]	[ "Mehdi Rezaei \nDepartment of Physics\nBu-Ali Sina University\n65178HamedanIran\n", "Saeed Pour Ojaghi \nDepartment of Physics\nBu-Ali Sina University\n65178HamedanIran\n", "Mohammad Malekjani \nDepartment of Physics\nBu-Ali Sina University\n65178HamedanIran\n" ]	[ "Department of Physics\nBu-Ali Sina University\n65178HamedanIran", "Department of Physics\nBu-Ali Sina University\n65178HamedanIran", "Department of Physics\nBu-Ali Sina University\n65178HamedanIran" ]	[]	In the context of cosmography approach and using the data of Hubble diagram for supernovae, quasars and gamma-ray bursts, we study some DE parametrizations and also the concordance ΛCDM universe. Using the different combinations of data sample including (i) supernovae (Pantheon), (ii) Pantheon + quasars and (iii) Pantheon + quasars + gamma-ray bursts and applying the minimization of χ 2 function of distance modulus of data samples in the context of Markov Chain Monte Carlo method, we first obtain the constrained values of the cosmographic parameters in model independent cosmography scenario. We then investigate our analysis, for different concordance ΛCDM cosmology, wCDM, CPL and Pade parametrizations. Comparing the numerical values of the cosmographic parameters obtained for DE scenarios with those of the model independent method, we show that the concordance ΛCDM model has a serious tension when we involve the quasars and gamma-ray bursts data in our analysis. While the high redshift quasars and gamma-ray bursts can falsify the concordance model, our results of cosmography approach indicate that the other DE parametrizations are still consistent with these observations.	10.3847/1538-4357/aba517	[ "https://arxiv.org/pdf/2008.03092v1.pdf" ]	221,083,255	2008.03092	b8700f1c6c3b63f79f8b35d878f89734453687e8	COSMOGRAPHY APPROACH TO DARK ENERGY COSMOLOGIES: NEW CONSTRAINS USING THE HUBBLE DIAGRAMS OF SUPERNOVAE, QUASARS AND GAMMA-RAY BURSTS 7 Aug 2020 Mehdi Rezaei Department of Physics Bu-Ali Sina University 65178HamedanIran Saeed Pour Ojaghi Department of Physics Bu-Ali Sina University 65178HamedanIran Mohammad Malekjani Department of Physics Bu-Ali Sina University 65178HamedanIran COSMOGRAPHY APPROACH TO DARK ENERGY COSMOLOGIES: NEW CONSTRAINS USING THE HUBBLE DIAGRAMS OF SUPERNOVAE, QUASARS AND GAMMA-RAY BURSTS 7 Aug 2020Preprint typeset using L A T E X style AASTeX6 v. 1.0 In the context of cosmography approach and using the data of Hubble diagram for supernovae, quasars and gamma-ray bursts, we study some DE parametrizations and also the concordance ΛCDM universe. Using the different combinations of data sample including (i) supernovae (Pantheon), (ii) Pantheon + quasars and (iii) Pantheon + quasars + gamma-ray bursts and applying the minimization of χ 2 function of distance modulus of data samples in the context of Markov Chain Monte Carlo method, we first obtain the constrained values of the cosmographic parameters in model independent cosmography scenario. We then investigate our analysis, for different concordance ΛCDM cosmology, wCDM, CPL and Pade parametrizations. Comparing the numerical values of the cosmographic parameters obtained for DE scenarios with those of the model independent method, we show that the concordance ΛCDM model has a serious tension when we involve the quasars and gamma-ray bursts data in our analysis. While the high redshift quasars and gamma-ray bursts can falsify the concordance model, our results of cosmography approach indicate that the other DE parametrizations are still consistent with these observations. INTRODUCTION Recent advances in observational cosmology have revealed that the current universe has experienced a stage of accelerated expansion. This expansion can be well explained by introducing an exotic component with negative pressure, dubbed dark energy (DE)which violates the strong energy conditions, ρ x + 3p x > 0. This expansion can also be justified by modifying the standard theory of gravity on extragalactic scales (Riess et al. 1998;Perlmutter et al. 1999;Kowalski et al. 2008). In the framework of general relativity (GR), it appears that approximately 70% of the energy budget of the universe is in the form of dark energy (Bennett et al. 2003;Spergel et al. 2003;Peiris et al. 2003). The cosmological constant Λ in which the equation of state (EoS) parameter is equal to −1, is the most likely possibility for dark energy. Although by assuming the cosmological constant and cold dark matter (CDM) in the context of standard ΛCDM cosmology, one can serve the purpose the accelerated expansion of the universe and the model is in good agreement with the cosmological observations, it suffers from the serious problems of cosmic coincidence and the fine tuning (Weinberg 1989;Padmanabhan 2003;Copeland et al. 2006). Also, from the observational point of view, the ΛCDM cosmology plagued with some significant tensions in estimation of some key cosmological parameters. In particular, the first tension concerns the discrepancy between the amplitude of matter fluctuations from large scale structure (LSS) data (Macaulay et al. 2013), and the value predicted by CMB experiments based on the ΛCDM. As the other tension, the Lyman-α forest measurement of the Baryon Acoustic Oscillations (BAO) reported in (Delubac et al. 2015), suggests a smaller value of the matter density parameter (Ω m ) in comparison with the value obtained by CMB data. Furthermore, there is a statistically significant disagreement between the value of Hubble constant measured by the classical distance ladder and that obtained from the Planck CMB data (Freedman 2017). Quantitatively speaking, the ΛCDM cosmology deduced from Planck CMB data predicts H 0 = 67.4 ± 0.5 km/s/Mpc (Aghanim et al. 2018), while from the Cepheid-calibrated SnIa (Riess et al. 2019) we have H 0 = 74.03 ± 1.42 km/s/Mpc. To solve these problems, various kinds of DE models have been proposed in literature (Veneziano 1979;Erickson et al. 2002;Thomas 2002;Armendariz-Picon et al. 2001;Caldwell 2002;Padmanabhan 2002;Gasperini & Veneziano 2002;Elizalde et al. 2004;Gomez-Valent & Sola 2015). Comparing with different observations, some of these models have been ruled out and many of them lead to good consistency with data (see also Malekjani et al. 2017;Rezaei et al. 2017;Malekjani et al. 2018;Lusso et al. 2019;Rezaei 2019a;Lin et al. 2019;Rezaei et al. 2019Rezaei et al. , 2020. The results of these investigations show that by using the current observations, it is difficult to distinguish different DE models. This confusion about different DE models suggests that a more conservative way to justify the cosmic acceleration, relying on as less model dependent quantities as possible, is welcome. As a solution, the well known model independent approach which is commonly used in the literature for testing the fitting capability of models with data, is cosmography (see Sect. 2)). Applying the cosmographic approach to distinguish between different DE models was proposed in Alam et al. 2003). Using cosmographic parameters in (Capozziello & Salzano 2009), authors tried to constraint a cosmological model, f (R)gravity. Because these parameters are model-independent, they lead to natural "priors" to any theory. Using cosmography, the authors of (Capozziello & Salzano 2009) have discussed how f (R)−gravity could be useful to solve the problem of mass profile and dynamics of galaxy clusters. In (Capozziello et al. 2011), they studied the possibility to extract the model independent information about the dynamics of the universe by using cosmography approach. Their results showed that in the context of cosmography approach, our predictions considerably deviate from the ΛCDM cosmology. Based on the cosmography approach, authors of ) constrained the late time evolution of the Universe using the low-redshift observations. Their results confirmed the tensions with ΛCDM model at low redshift universe. The authors of (Lusso et al. 2019) assumed two different cosmographic models consisting of a fourth-order logarithmic polynomial and a fifth-order linear polynomial, and fitted these models with different data sets. Then, by comparing the results with the expectations from concordance ΛCDM model, they found significant tensions between the best-fit cosmographic parameters and the concordance ΛCDM model. The cosmographic approach also is used in (Li et al. 2019) to determine the spatial curvature of the Universe. They showed that by combining the supernovae (Pantheon sample), the latest released cosmic chronometers and the BAO measurements, the most favored cosmography model prefers a non-flat universe with Ω K = 0.21 ± 0.22. Following these works, in this paper we want to study some relevant DE models including the standard ΛCDM, wCDM, Chevallier-Polarski-Linder (CPL) and Pade parameterizations in the context of cosmography approach. By combining different data sets including the distance modulus of quasars, the Pantheon and and publicly available gamma-ray burst (GRB) data, we try to find the best-fit values of cosmographic parameters using the minimization of χ 2 function based on the Markov Chain Monte Carlo (MCMC) method. Notice that we first obtain the best fit values of the cosmographic parameters without considering a cosmological model. We then put constraints on the free parameters of the models under study. Using the constrained values and their confidence regions within 1 − σ uncertainties of cosmological parameters of the models, we compute the best fit values of the cosmographic parameters for each model. By comparing the computed cosmographic parameters of the models and those obtained from model independent approach, one can examine the cosmological models against observation. The layout of our paper is as follows: In Sect. 2, we present the cosmographic approach. Then we introduce the observational data which we have used in our analysis. In Sect.3, we first briefly introduce the DE models and parametrizations in our study and then present the numerical results. In Sect.4 we present discussions based on our numerical results for different models . Finally in Sect.5, the paper is concluded. COSMOGRAPHIC APPROACH Recently, the cosmographic approach to cosmology commonly used in the literature in order to obtain as much information as possible directly from observations. In this approach without addressing issues such as which model of DE is required to satisfy the accelerated expansion of the Universe, and just by assuming the minimal priors of homogeneity and isotropy we can study the evolution of the Universe. Cosmography provides information about cosmic flow and it's evolution derived from measured distances, by using Taylor expansions of the basic observables (Demianski et al. 2017b). The distance -redshift relations obtained from these expansions only rely on the assumption of the Friedman-Lemaitre-Robertson-Walker(FLRW) metric and are therefore fully model independent. Firstly, we introduce the cosmographic functions by the first five derivatives of scale factor a(t) as follows (Visser 2004): Hubblef unction : H(t) = 1 a da dt ,(1)decelerationf unction : q(t) = − 1 aH 2 d 2 a dt 2 ,(2) jerkf unction : j(t) = 1 aH 3 d 3 a dt 3 ,(3) snapf unction : s(t) = 1 aH 4 d 4 a dt 4 ,(4) lerkf unction : l(t) = 1 aH 5 d 5 a dt 5 .(5) The cosmographic parameters (H 0 , q 0 , j 0 , s 0 &l 0 ) are corresponding to the present values of the above functions. Furthermore, it is easy to find the relation between the derivatives of the Hubble parameter and the cosmographic parameters as follows:Ḣ = −H 2 (1 + q) ,(6)H = H 3 (3q + j + 2) ,(7) ... H = H 4 (−3q 2 − 12q − 4j + s − 6) ,(8) .... H = H 5 (30q 2 + 60q + 10qj + 20j − 5s + l + 24) , (9) where each over-dot denotes a derivative with respect to cosmic time t. One can compute the Taylor Series expansion of the Hubble parameter to the forth order in redshift z around it's present value z = 0 H(z) = H\| z=0 + dH dz \| z=0 z 1! + d 2 H dz 2 \| z=0 z 2 2! + d 3 H dz 3 \| z=0 z 3 3! + d 4 H dz 4 \| z=0 z 4 4! ,(10) The above Taylor series expansion is valid for small redshifts z < 1, whereas much of the most interesting recent observational data sets occur at higher redshifts z > 1. In the other word, the radius of convergence of any series expansion in redshift is equal or less than (z 1), and thus any z-based expansion will break down at z > 1. To solve this problem we use an improved redshift definition which commonly used in literature, the y-redshift y = z z+1 (Capozziello et al. 2011). Although changing the z-redshift in to the y-redshift will not change the physics, but it can improve the series of convergence. In terms of the y-redshift, we see that the radius of convergence of a Taylor expansion is ( 1) which correspond to z → ∞. Thus, using y-redshift definition, we can use the Taylor expansion of the Hubble parameter at any higher redshifts as the following form (Capozziello et al. 2011): H(y) = H\| y=0 + dH dy \| y=0 y 1! + d 2 H dy 2 \| y=0 y 2 2! + d 3 H dy 3 \| y=0 y 3 3! + d 4 H dy 4 \| y=0 y 4 4! .(11) We note that there are some other procedures which can solve the convergence problem. In (Capozziello et al. 2020), authors compared some of these procedures to find the best approach to explain low and high redshift data sets. They have expanded the luminosity distance d L , using Taylor series and its alternatives, rational polynomials and auxiliary variables. Their results show that at low redshifts there is no apparent need to adopt the y-variables or rational polynomials instead of Taylor series. But, differences appear at high redshifts, where the results of (Capozziello et al. 2020) indicate that (2,1) polynomial performs better than the yvariables. In this work we use different observations in the redshift range up to z ∼ 7. Thus we can not use the Taylor expansion and we should apply one of its alternatives. Since we have not using the high redshift CMB data, so we can use an alternative approach having good performance at low and intermediate redshifts. Assuming this condition and in order to prevent the complexity arising from inserting more additional degrees of freedom, in this work we select the yredshift procedure. Now by changing the time derivatives of Eqs.(6-9) in to derivatives with respect to y, inserting the results in Eq.(11) and using Eqs.(1-5), we will have: E(y) = H(y) H\| y=0 = 1 + k 1 y + k 2 y 2 2 + k 3 y 3 6 + k 4 y 4 24 .(12) where different k i are: k 1 = 1 + q 0 ,(13)k 2 = 2 − q 2 0 + 2q 0 + j 0 ,(14)k 3 = 6 + 3q 3 0 − 3q 2 0 + 6q 0 − 4q 0 j 0 + 3j 0 − s 0 ,(15)k 4 = −15q 4 0 + 12q 3 0 + 25q 2 0 j 0 + 7q 0 s 0 − 4j 2 0 − 16q 0 j 0 − 12q 2 0 + l 0 − 4s 0 + 12j 0 + 24q 0 + 24 . (16) In the above equations q 0 , j 0 , s 0 and l 0 are the current values of cosmographic parameters. By knowing the evolution of E as a function of redshift, we can investigate the evolution of cosmic fluid. In this paper we want to put constraint on the cosmographic parameters using the Hubble diagrams of low redshift observational data. To do this, we set the current value of cosmographic parameters (q 0 , j 0 , s 0 and l 0 ) as the free parameters in MCMC algorithm. Then, the best fit values for the free parameters are those which can minimize the χ 2 function. Notice that the χ 2 function is defined based on the distance modulus of observational objects. The Hubble diagram of low-redshift observational data used in this work is as follows: • Pantheon sample: This sample as a set of latest data points for the apparent magnitude of type Ia supernovae (SNIa) (Scolnic et al. 2018) in the range 0.01 < z < 2.26, is one of three sample of data points we use in this work. This sample includes 279 spectroscopically confirmed SNIa discovered by the Pan-STARRS1(PS1) Medium Deep Survey (Rest et al. 2014;Scolnic et al. 2018) in the redshift range 0.03 < z < 0.68. The pantheon sample also includes the SNIa data from the Sloan Digital Sky Survey (SDSS) (Frieman et al. 2008;Sako et al. 2018) and the Supernova Legacy Survey (SNLS) (Conley et al. 2011;Sullivan et al. 2011). This sample is the largest combined sample of SNIa data consisting of a total of 1048 data points up to redshift ∼ 2.3. • Gamma-ray bursts (GRBs): The GRBs are the most energetic and powerful explosions in the universe and can be detectable up to very high redshifts. GRBs are the mysterious objects in the universe. A mechanism which indicates the high amounts of releasing energy from a typical GRB emits is not yet completely known. Some investigations show that the GRBs are produced by core-collapse events (Meszaros 2006). Despite these difficulties, the GRBs are astrophysical objects for studying the expansion scenario of the universe at high redshifts. In fact, using the Hubble diagrams of GRBs, one can study the expansion rate of the universe and investigate the observational properties of DE up to higher redshifts. One of the most important aspects of the observational property of GRBs is that they show several correlations between spectral and intensity properties (luminosity, radiated energy). (Demianski et al. 2017a) proposed an empirical correlation between the observed photon energy of the peak spectral flux, E p,i , and the isotropic equivalent radiated energy, E iso . This correlation not only provides constraints on the model of the prompt emission, but also naturally suggests that the GRBs can be used as distance indicators. In fact to use the GRBs as distance indicators, it is necessary to consistently calibrate this correlation. Unfortunately, due to the lack of GRBs at very low redshifts, the calibration of GRBs is more difficult than that of SNIa. In this regard, several calibration procedures have been suggested so far (Dainotti et al. 2008;Demianski & Piedipalumbo 2011;Demianski et al. 2012;Postnikov et al. 2014). Recently, (Demianski et al. 2017a) by applying a local regression technique and using the SNIa sample, have constructed a new calibration for the GRB Hubble diagram that can be used for cosmological investigations. They showed that how the E p,i − E iso correlation can be calibrated to standardize the long GRBs and to build a GRB Hubble diagram, which we use to investigate the cosmology at very high redshifts (Demianski et al. 2017a). Notice that their E p,i − E iso correlation has no significant redshift dependence. In this work, we use the 162 data points for distance modulus of GRBs derived and reported in (Demianski et al. 2017a). This sample contains the low and high redshifts GRBs in the range of 0.03 < z < 6.67. More details and discussions about the calibration method and construction of the Hubble diagram of GRBs can be found in (Demianski et al. 2017a,b;Amati & Della Valle 2013). • Quasars: quasars are extremely luminous active galactic nucleus (AGN), in which a supermassive black hole (SMBH) is surrounded by a gaseous accretion disk. As gas in the disk falls towards the SMBH, energy is released, which can be observed across the electromagnetic spectrum. The observed properties of a quasars depend on factors such as the mass of the SMBH and the rate of gas accretion. The spectral energy distribution (SED) of quasars shows the significant emis-sion in the optical-UV band L UV , the so-called big blue bump (BBB), with a softening at higher energies (Sanders et al. 1989;Elvis et al. 1994;Trammell et al. 2007;Shang et al. 2011) . This emission is thought to origin from an optically thick disc surrounding the SMBH. Also, the X-ray photons, L X , are generated by inverse Compton scattering of disc UV photons by a hot electron plasma, the so-called X-ray corona. Notice that the energy loss through X-ray emission may cool down the electron plasma, if there is no efficient energy transfer mechanism from the disc to the corona. However, the physical nature of such a process is still poorly understood. An important observational feature concerning the connection between the UV disc and Xray corona is provided by the non-linear correlation between the L UV from the disc and L X from the corona. The non-linear relationship between L X and L UV as log L X = γL UV + β, has been obtained in both optically and X -ray AGN samples with slope parameter γ around 0.5 − 0.7 (Vignali et al. 2003;Strateva et al. 2005;Steffen et al. 2006;Just et al. 2007;Green et al. 2009;Lusso & Risaliti 2016a;Young et al. 2009Young et al. , 2010Jin et al. 2012) representing that optically bright AGN emits relatively less X-rays than optically faint AGN. It has been shown that such relation is independent of redshift and it is very tight (Lusso & Risaliti 2016b). This relation has also been used as a distance indicator to estimate cosmological parameters. Using the L X − L UV relationship, Risaliti & Lusso (2015) have constructed the complete sample of quasar Hubble diagram up to z ∼ 6, which is in excellent agreement with the analogous Hubble diagram for SNIa in the common redshift range (i.e., z ∼ 0.01 − 1.4). This capability turns quasars into a new class of standard candles (Lusso & Risaliti 2017). The main quasars sample is composed of 1598 data points in te range 0.04 < z < 5.1. In this work instead of the main sample, we use a binned catalog including 25 datapoint from (Risaliti & Lusso 2015;Lusso & Risaliti 2016b). All the details on sample selection, X-ray, and UV flux computation and the analysis of the nonlinear relation, calibration, and a discussion on systematic errors are provided in (Risaliti & Lusso 2019). Combining Gamma ray bursts and quasars with Pantheon is motivated, because we can probe a redshift range (0.03 < z < 6.67) better suited for investigating DE than the one covered by Pantheon sample (0.01 < z < 2.26). Hence, by adding these data samples to Hubble diagram, we have more observational data at higher redshifts. Using these datasets we calculate the χ 2 function of the distance modulus based on the MCMC algorithm to find the best fit values of cosmographic parameters in a model independent cosmology. To run the MCMC algorithm, we select two different sets of the initial values for free parameters. This can guaranty that our results are independent from the initial values of free parameters. For all of the free parameters we choose big σ values to ensure that the MCMC can sweep the whole of the parameters space. Using these choices, we have removed the risk of finding a local best fit values in the parameters space. Notice that for both of the initial value sets, we obtained similar posteriors for q 0 and j 0 . But in the case of s 0 and l 0 , the posteriors are slightly different. Thus we have repeated our analysis by setting an initial value for s 0 and l 0 between two best fit values obtained in previous steps (we presented the initial values of free parameters in Table 1). In order to see the influence of each data sample of quasars and GRB in our analysis, we consider different combinations of data samples as Pantheon, Pantheon+GRB, Pan-theon+quasars and Pantheon+GRB+quasars. For all of these combinations, we do our analysis in order to find the best fit values of the free parameters and their 1− and 2 − σ uncertainties. Notice that a procedure to chose the proper initial value for each of the free parameters were described above. The results of our analysis are presented in Tab. 2. For all combinations of data samples, we can see that the deceleration parameter q 0 is tightly constrained. The constraints for jerk parameter j 0 is approximately tight. However, our analysis can not put the tight constraints on the snap s 0 and lerk l 0 parameters. We observe that adding the high redshift observational data of quasars and GRB causes to get higher values of q 0 and j 0 . Due to the large values of uncertainties for s 0 and l 0 , we cannot reach to clear conclusion when we compare the results of different combinations. Notice that the s 0 and l 0 parameters are appearing in the forth and fifth term of the Eq.12 as the coefficient of third and forth order of redshift respectively. In these terms, the big error bar of data points leads to very weak constraints on these two parameters. DE MODELS AND PARAMETERIZATIONS In this section we first briefly introduce some DE models and parameterizations which we want to study in cosmography approach. Notice that we also consider the standard ΛCDM cosmology as a concordance model. Then, by using the data samples presented in previous section and by applying the minimization of χ 2 function based on the MCMC algorithm, we find the best fit values of the cosmological parameters of DE models. Using the chain obtained for cosmological parameters of each model within 1 − σ level, we compute the best fit and 1 − σ uncertainty of cosmographic parameters for each model. Finally, we will compare the best fit cosmographic parameters of each model with those of the model independent approach obtained in Table (2). The DE models that we examine in our analysis are: 1. wCDM : The first model is the DE model with constant equation of state (EoS) parameter w de . The Hubble parameter of the model in a flat FRW universe reads (Mota & Barrow 2004;Barger et al. 2007): E 2 (z) = Ω m,0 (1 + z) 3 + (1 − Ω m0 )(1 + z) 3(1+w de ) ,(17) where Ω m,0 is the energy density of pressure-less matter at the present time. Notice that we study the model in late time cosmology where the energy density of radiation is negligible. Using the above equation and rewriting Eqs. (1-5)in term of redshift, we can obtain cosmographic parameters in the context of wCDM cosmology as follows: q(z) = Ω m,0 (1 + z) 3 + (1 + 3w)Ω d,0 (1 + z) 3(1+w) 2E 2 (z) ,(18)j(z) = 1 + 9w(1 + w)Ω d,0 (1 + z) 3(1+w) 2E 2 (z) ,(19) In order to obtain the best fit values and the confidence regions of the cosmographic parameters, we need to obtain the best fit and also the confidence regions of the cosmological parameters Ω m,0 and w de of the model. Notice that in the flat FRW universe, we have Ω d,0 = 1 − Ω m,0 . So using the different combinations of observational data: Pantheon, Pantheon + GRB, Pantheon + quasaras and finally Pan-theon+GRB+quasars, we obtain the best best fit values of Ω m,0 and w de as well as their confidence regions in 1 − σ uncertainty. Our results are reported in the left part of Table (3). Using Eqs. (18,19) and the data of Ω m,0 and w de in 1σ error, we put constraints on the cosmographic parameters in wCDM cosmology. Results for the best fit values and 1−σ confidence regions are presented in the right part of f (x) = a 0 + a 1 x + a 2 x 2 + ... + a n x n b 0 + b 1 x + b 2 x 2 + ... + b m x m ,(20) where exponents (m, n) are positive and the coefficients (a i , b i ) are constants (Pade 1892). In this work, we consider the Pade expansion of the Eos parameter w de (a) up to the order (1, 1) around the variable (1 − a), where a is scale factor. Previously, this parametrization has been studied in in the light of different observational data. But, here we investigate this parametrization from the cosmography point of view. The EoS parameter for the Pade (1, 1) parametrization can easily be written as follows Rezaei 2019b): w d (z) = w 0 + (w 0 + w 1 )z 1 + z + w 2 z .(21) Following Rezaei 2019b), we can find the evolution of dimensionless Hubble parameter of Pade parametrization, E(z), as E 2 (z) = Ω m,0 (1 + z) 3 + (1 + w 2 − w 2 1 + z ) p1 × (1 − Ω m,0 )(1 + z) p2 ,(22) where p 1 and p 2 are: p 1 = −3( w 1 − w 0 w 2 w 2 (1 + w 2 ) ) , p 2 = 3 1 + w 0 + w 1 + w 2 1 + w 2 .(23) Using Eq. (22) in Eqs. (1-3), we can obtain the cosmographic parameters for Pade the parametrization as follows: q(z) = 3Ω m,0 (1 + z) 3 + (1 + z)(A 1 B 1 + C 1 D 1 ) 2E 2 (z) − 1 ,(24)j(z) = 1 + (1 + z) 2 2A 1 C 1 + B 1 F 1 + G 1 D 1 2E 2 (z) − (1 + z) A 1 B 1 + C 1 D 1 E 2 (z) ,(25) where constants A 1 , B 1 , C 1 , D 1 , F 1 and G 1 are, respectively, given by: A 1 = w 2 p 1 (1 + z) 2 (1 + w 2 − w 2 1 + z ) −1+p1 , B 1 = Ω d,0 (1 + z) p2 , C 1 = p 2 Ω d,0 (1 + z) −1+p2 , D 1 = (1 + w 2 − w 2 1 + z ) p1 , F 1 = p 1 (1 + w 2 − w 2 1 + z ) p1−1 − 2p 1 w 2 (1 + z) 3 + (p 2 1 − p 1 )w 2 2 (1 + z) 4 (1 + w 2 − w 2 1 + z ) p1−2 , G 1 = p 2 (p 2 − 1)Ω d,0 (1 + z) −2+p2 . (26) In a cosmology based on the Pade parametrization for EoS parameter of DE, we have four free parameters including Ω m,0 , w 0 , w 1 and w 2 . We redo our analysis for Pade parametrization like what was done for wCDM and ΛCDM cosmologies. So, we first find the best fit as well as the confidence regions of the parameters within 1 − σ level. Then, we obtain the best fit and the error bar of the cosmographic parameters for Pade approximation for different combinations of data samples. Results are presented in Table (5). 4. CPL parametrization: The other parametrization that we study in this work is the well-known Chevallier-Polarski-Linder (CPL) parametrization of DE in which the EoS parameter is simply expanded around (1 − a) by Taylor approximation up to first order, e.g., w = w 0 + w 1 z/(1 + z) (Chevallier & Polarski 2001;Linder 2003). It is easy to see that for a particular value of w 2 = 0, we can recover the CPL parametrization from Pade formula. In the CPL parametrization, the Hubble parameter is written as (Chevallier & Polarski 2001;Linder 2003): E 2 (z) = Ω m,0 (1 + z) 3 + (1 − Ω m,0 )(1 + z) 3(1+w0+w1) × exp[−3w 1 z 1 + z ] .(27) Hence, inserting Eq. (27) into Eqs. (1-3), the cosmographic parameters in CPL cosmology are obtained as follows: q(z) = A 2 [1 + z + 3(1 + z)w 0 + 3zw 1 ] + (1 + z)B 2 2(1 + z)[A 2 + B 2 ] ,(28)j(z) = A 2 C 2 + 2(1 + z) 2 B 2 2(1 + z) 2 [A 2 + B 2 ] ,(29) where the constants A 2 , B 2 and C 2 are: A 2 = Ω d,0 (1 + z) 3(w0+w1) , B 2 = Ω m,0 exp[3w 1 z 1 + z ] , C 2 = 9z 2 w 2 1 + 3w 1 (1 + z)(6w 0 z + 3z + 1) + (1 + z) 2 (9w 2 0 + 9w 0 + 2) .(30) In this case we have three free parameters including Ω m,0 , w 0 and w 1 . The best fit values, the 1 − σ confidence region of these parameters and also the best fit values of the cosmographic parameters of the model are reported in Table (6). Using these numerical results, in the next section we will compare the cosmographic parameters of the above DE parametrizations with the cosmographic parameters obtained from model independent approach presented in Table (2). In Fig. (1), we plot the 1− and 2 − σ confidence regions of the cosmographic parameters q 0 and j 0 obtained for model independent approach. We can easily observe that the 2 − σ confidence region of the jerk parameter j 0 for different combinations of Pantheon+GRB, Pantheon+quasars and Pantheon+GRB+quasars is above the critical value j 0 = 1. While the confidence region of j 0 for Pantheon sample covers the critical point j 0 = 1. Hence as a quick result, we can see that the ΛCDM cosmology has a significant tension with the high redshift GRB and quasars observations. DISCUSSIONS In this section we will compare the numerical results of our analysis obtained in previous sections. As one can see in Tab.2 our analysis leads to fairly tight constraints on two of the cosmographic parameters, q 0 and j 0 , while the other two parameters of cosmography, s 0 and l 0 , have not been tightly constrained. Furthermore, the best fit values of s 0 and l 0 are significantly varying based on then the initial conditions of MCMC algorithm. Also their related confidence regions are also so big. Therefore, in order to compare DE models, we just focus on our results for q 0 and j 0 and ignore the other ones. Notice that the impact of q 0 and j 0 on the Taylor expansion of the Hubble parameter is much bigger than s 0 and l 0 . Hence in overall, our consideration to compare DE parametrizations based on q 0 and j 0 can not restrict our conclusion. Based on the results of Table (2), when we just use the Pantheon sample, the deceleration parameter q 0 has the largest value,q 0 = −0.702, while adding other data samples to Pantheon leads to smaller value for q 0 . Thus we can say that the larger value of deceleration parameter q 0 is favored by low redshift data points, while using relatively higher redshift data points causes the smaller q 0 . Oppositely, in the case of jerk parameter we obtain smaller value of j 0 when we use the Pantheon sample and larger value of j 0 when we add the other data samples to Pantheon. These results are completely in agreement with those of (Lusso et al. 2019) which have used one of our combination of data samples (Pantheon+GRB+quasars) in a different way to constraint cosmographic parameters. Our results also nearly confirm the results of (Li et al. 2019) which were obtained using different data samples and different approach. Now we compare the best fit values of the cosmographic parameters q 0 and j 0 for each DE parametrization obtained in previous section with those of the model independent way. Our comparison for different combinations of data samples is described as follows. 1. Pantheon sample: Using the Pantheon sample, The best fit values of deceleration and jerk parameters within 1σ uncertainty are q 0 = −0.702 ± 0.104 and j 0 = 1.60 ± 0.71 for model independent approach. Our results for wCDM model (Table3) show that both of q 0 and j 0 are in full agreement (at 1 − σ confidence level) with those we obtained from model independent constrains. In the case of ΛCDM model, the results are almost different from those of model independent case. For ΛCDM we have q 0 = −0.572 ± 0.019 ( j 0 = 1.0) which is in 1.25σ (0.85σ) tension with the result of q 0 (j 0 ) which we have obtained for model independent case (see Tables 2 & 4). Both of Pade and CPL parametrizations have nearly the same re-sults. The results of q 0 in these two parametrizations is in full agreement with the value of q 0 which we have found for model independent case, while the value of j 0 that we obtained for these parameterizations is in more than 1σ tension with the j 0 of model independent case (see Tables 2, 5 and 6). In Summary, our results are presented in the top-left panel of Fig.2 in which the contour plot shows the model independent constraints on j 0 − q 0 plan up to 3 − σ confidence level and the error bars have used to show the computed value of cosmographic parameters for different cosmological models up to 1 − σ. As a result of this part, we can say that using the solely Pantheon sample, the constrained parameters q 0 and j 0 for wCDM, Pade and CPL parametrizations are compatible with model independent constraints in ∼ 1σ error. While the standard ΛCDM model can be falsified by 1σ uncertainty because of its jerk parameter. Notice that the ΛCDM model is still consistent with model independent results in 2σ level. 2. Pantheon + GRB data Using this sample the best fit value of cosmographic parameters and their 1 − σ confidence levels in a model independent approach are q 0 = −0.755 ± 0.048 and j 0 = 2.61 +0.29 −0.19 . We see that adding the GRB to Pantheon data leads to smaller value of deceleration parameter and larger jerk parameter compare to solely Pantheon sample. Same as previous part, the q 0 parameter in wCDM is in full agreement with model independent result. But the j 0 parameter in this model has a ∼ 3σ tension with the that of the model independent scenario. Notice that here 1σ is the average of error bar obtained in model independent j 0 (see Table 2. The results of ΛCDM model are disappointing in this part. The best fit value of q 0 in this model leads to a 3.8σ tension with that of the model independent one and it's j 0 is in more than 5σ tension with the bests of the model independent approach. We emphasize that for all comparisons, we define the agreement or tension between DE models and the model independent approach, based on the average of error bar obtained for cosmographic parameters in model independent way presented in Table (2.) Like previous part, the results of q 0 parameter in the Pade and CPL parametrizations are completely compatible with those of the model independent approach, while the j 0 is in a ∼ 2σ tension with the best value of j 0 in the model independent approach. Therefore we can say that using the combination of GRB and Pantheon data points, Pade and CPL are the best models and ΛCDM is completely dis-favorable. In the upright panel of Fig.2, the contour plot shows the best fit of cosmographic parameters and related confidence levels up to 3 − σ, obtained using Pantheon+GRB data points in the model independent approach. The best fit values and their error bars obtained for different DE models also are plotted for comparison. We can see that in q 0 − j 0 plan, wCDM, CPL and Pade parametrizations are located inside the confidence regions while the standard ΛCDM model is in outside. 3. Pantheon + quasars: Now let see the effect of adding quasars data to Pantheon sample in our analysis. By combination of quasars and Pantheon data sets, the model independent approach leads to q 0 = −0.844 ± 0.048 and j 0 = 2.42 ± 0.25. Now we compare this result with the best fit values of q 0 and j 0 obtained for different DE models. In the case of wCDM (see Ta-ble3), we observe that q 0 (j 0 ) of the model deviates from model independent values as 0.98σ (1σ) region. This result for CPL parametrization (see Table6) is 1σ and 1.2σ deviation, respectively, for q 0 and j 0 . In the case of Pade parametrization (see Table5), we obtain 1.9σ (1.6σ) deviation from model independent constraints of q 0 (j 0 ). Finally in the case of concordance ΛCDM (see Table4), we see the big tension between the values of cosmographic parameters of the model and those of model independent approach. Numerically, this tension is approximately 6σ for both q 0 and j 0 . In the down-right panel of Fig.2, the contour plots show the confidence levels of cosmographic parameters up to 3 − σ, obtained using Pantheon+quasars data points in the model independent approach. The best fit values and their error bars of cosmographic parameters obtained for different DE models also are plotted for comparison. We see that the ΛCDM model is completely outside of the confidence regions, while wCDM, CPL and Pade parametrizations are still inside the regions. 4. Pantheon + quasars + GRB: In the last step, we combine all of our data samples and compare the results of model independent approach with those of DE parametrizations. As one can see in Tab.2, the best fit values of cosmographic parameters in model independent approach are q 0 = −0.819 ± 0.065 and j 0 = 2.21 +0.37 −0.042 . Assuming the results obtained for our models (see Tables 3, 4, 6, 5), we observe that the best fit values of q 0 and j 0 for wCDM model are respectively in 0.3σ and 1σ tension with the results of model independent approach. These tensions in the case of ΛCDM enhance to 3.7σ for q 0 and 4σ for j 0 . For Pade parametrization the differences are smaller. Here we have tensions about 0.6 − σ for q 0 and 0.8σ for j 0 .In the CPL case, q 0 parameter has 0.3σ tension and j 0 has 1.3σ tension with the best fit values in model independent analysis. In the bottom-right panel of Fig.2, the contour plots show the confidence levels of cosmographic parameters obtained using Pan-theon+GRB+quasars data points in a model independent approach. The best fit values and their error bars obtained for different DE parametrizations also plotted for comparison. Same as previous parts, the ΛCDM cosmology are far from the confidence region in q 0 −j 0 space, while other models are still not refuted. Now we examine the DE parametrizations and also concordance ΛCDM universe by reconstructing the Hubble parameter in the context of cosmography approach. In Fig.3, we have reconstructed the redshift evolution of Hubble parameter, H(z), within 1 − σ confidence region, using Eqs. (12 -16). Notice that we consider Eq.12 up to y 2 which involves q 0 and j 0 parameters. So we can use the best fit values of cosmographic parameters q 0 and j 0 for model independent approach in Table (Table2) and also for DE models and parametrizations in Tables (3 -6). Each of the panels of the figure obtained from one of our combinations of data samples. In all cases, we set H 0 = 70 km/s/Mpc (Abbott et al. 2017). The 1 − σ confidence level of H(z) (green band) is calculated by using the upper and lower limits of best fit values of q 0 and j 0 obtained in model independent approach from Tab.2. In the up-left panel we show the reconstructed H(z) obtained from Pantheon sample. The evolution of H(z) for different DE models and parametrizations also plotted for comparison. As one can see in this panel, H(z) curve of ΛCDM deviates from 1 − σ region at redshifts higher than z ∼ 0.8. While H(z) for other DE parametrizations evolve within 1 − σ region even at high redshifts. The results obtained from Pantheon+GRB sample are presented in the upright panel. In this plot same as the previous one, the H(z) curve of ΛCDM has the maximum differences from the best curve among different models. The bottom-left panel which shows the reconstruction of H(z) for Pantheon+quasars sample, represents the ΛCDM cosmology as the most incompatible model again. We see that the deviation from confidence region is so big at higher redshifts. Furthermore in this plot, the Pade parametrization also evolves outside of 1 − σ region. Finally in the bottom-right panel, we present the results obtained using Pantheon+GRB+quasars sample. This plot confirms the results of previous panels again. We see that the reconstructed Hubble parameters of ΛCDM cosmology evolves outside of confidence region at redshifts bigger than z ∼ 0.8. So among cosmological DE scenarios studied in this work, the ΛCDM is the worst one. CONCLUSIONS In this work we first used the data points of low-redshifts Hubble diagrams for Pantheons, quasars and GRB's to put constraints on the present value of cosmographic parameters in a independent cosmography approach. To do this, we used different combinations of data samples including Pantheon, Pantheon + quasars, Pantheon + GRB and finally Pantheon + quasars + GRB. In the context of cosmography approach, we obtained the best fit values of cosmographic parameters as well as their confidence regions up to 3 − σ uncertainties for different combinations of data samples. Our results showed that the best fit value of deceleration parameter q 0 varies in the range of −0.844 to −0.702 and the best fit of jerk parameter j 0 varies in the range of 1.60 to 2.61 for different combinations of data samples. Notice that here we used the Hubble diagrams of quasars and GRB, respectively, derived in (Lusso & Risaliti 2016b) and (Demianski et al. 2017a). In the calibration procedure to form the Hubble diagrams of both quasars and GRBs, they have used the SNIa data at low redshifts. Their results for quasars and GRBs samples are consistent with that of the SNIa samples at low redshift universe. Hence we adopted their calibrations and used their Hubble diagrams for quasaras and GRBs. In the case of concordance ΛCDM cosmology, our results are also compatible with recent work in Lusso et al. (2019). They confirmed the presence of a tension between Λ cosmology and the best-fit cosmographic parameters ∼ 4σ with SnIa+quasars, at ∼ 2σ with SnIa+GRBs, and at 4σ with the whole SnIa+quasars+GRB data set Lusso et al. (2019). Furthermore, we studied some relevant DE parametrizations as well as the concordance ΛCDM cosmology using the Hubble diagrams of Pantheons, quasars and GRB observations in the context of cosmography approach. The DE parametrizations studied in our analysis are wCDM, CPL and Pade parametrizations. Firstly, by using the different combinations of data samples and in the context of MCMC algorithm, we calculate the χ 2 function of the distance modulus to find the best fit values and also the 1 − σ uncertainty of cosmological parameters for each DE parametrization. Using the chain of data obtained for cosmological parameters, we found the best fit values and 1 − σ confidence region of the cosmographic parameters of DE parametrizations. Comparing the results for DE models with those of obtained for model independent approach leads to conclude that which of the model is in better (worse) agreement with Hubble diagrams of Pantheons, quasars and GRB's. In the first stage, using the solely Pantheon sample, we found that the wCDM model is the most compatible model with the result of model independent constraints and on the other hand the concordance ΛCDM model is the worst model. In the second step, by combining the GRB data to the Pantheon sample, we obtained disappointed results for ΛCDM model. In this case q 0 parameter of the ΛCDM has a 3.8 − σ tension with that of the model independent cosmography approach. Moreover, the j 0 parameter of ΛCDM cosmology, has roughly 5 − σ tension with that of the model independent approach. These results will be more frustrated when we see the results of other DE models and parametrizations that we studied in this work. We observed that wCDM, CPL and Pade parametriza-tions are in better agreement with the results of model independent cosmography approach rather than concordance model. In the third and fourth steps, by using the combinations Pantheon+quasars and Pantheon+GRB+quasars data points, we obtained the same results again, supporting our results in previous steps. So we conclude that the concordance ΛCDM cosmology has a big tension with the observations of quasars and GRB at higher redhsift. Notice that the DE parametrizations studied in this work sre in better agreement with Hubble diagrams of high redshift quasars and GRB observations. Finally, we reconstructed the Hubble parameter by using the best fit value of cosmographic parameters for both model independent approach, ΛCDM model and DE parametrizations. We observed that for different data sample combinations, the evolution of reconstructed H(z) in concordance ΛCDM model has the maximum deviation from the confidence region compare to different DE parametrizations. Upon this result, we can conclude that among different cosmological models studied in this work, the ΛCDM has the minimum compatibility with the predictions of model independent approach and thus it is falsified by cosmography approach. The big value of tensions (between 3σ to 6σ for different data combinations) that we observed between the cosmographic parameters of ΛCDM and those we obtained in model independent approach support this claim again that we should explore other alternatives for standard ΛCDM cosmology. We observed that other DE parametrizations in this study can not be refuted in the context of cosmography approach. Our results for ΛCDM cosmology are in agreement with the results of recent work in (Yang et al. 2019;Khadka & Ratra 2019) which was obtained by using a different approach and different data sets. Although, in the literature it has been thoroughly affirmed that the ΛCDM well describes the evolution of the universe until recent times, but our conclusion confirms the result of (Benetti & Capozziello 2019) representing some big tensions emerge at higher redshifts for ΛCDM. Our analysis can be extended by calling the other cosmic observations in the context of cosmography approach. Figure 1 . 1The confidence regions in q0 − j0 plan obtained in model independent approach using different combinations of Pantheon, quasars and GRB data samples. Figure 2 .3 2− σ confidence levels of cosmographic parameters q0 and j0 obtained from model independent approach. Also the best fit values of same parameters with their error bar for different DE scenarios have been shown. The up-left(up-right) panel shows the results obtained using Pantheon (Pantheon+GRB) sample. The bottom-left(bottom-right) panel shows the results obtained using Pantheon+quasars (Pantheon+GRB+quasars) sample. Figure 3 . 3The reconstructed Hubble parameter H(z) based on the best fit values of the cosmographic parameters q0 and j0 for different model independent approach and DE scenarios studied in this work. The green band shows the 1 − σ confidence region of reconstructed Hubble parameter in model independent method. The up-left(up-right) panel shows the results obtained using Pantheon (Pantheon+GRB) sample. The bottom-left(bottom-right) panel shows the results obtained using Pantheon+quasars (Pantheon+GRB+quasars) sample. Table Table 1. Two different sets of initial values for MCMC algorithm (left part) and the the final results (right part).Table 2. The best fit values of cosmography parameters and their 1 − σ uncertainties obtained for different combinations of data samples.(3). 2. Concordance ΛCDM: In fact when we do an analysis on a given DE model, we should redo our analysis for standard ΛCDM model as a concordance model. So in this part we study the standard model from the view- point of cosmography approach. In order to obtain the cosmographic parameters for ΛCDM model, we can easily set w de = −1 in Eqs.(17-19). Then we fol- low the procedure implemented for wCDM model to find the cosmographic parameters in ΛCDM cosmol- ogy. Our results are presented in Table (4). Notice that in the ΛCDM model, the jerk parameter is exactly equal to one independent of the values of cosmological parameters. initial values \| best fit values parameter q0 j0 s0 l0 \| q0 j0 s0 l0 Set (1) −2.0 5.0 −4.0 −5.0 \| −0.838 +0.06 −0.04 2.27 +0.25 −0.33 −3.8 +0.67 −1.0 −5.2 +2.2 −3.0 Set (2) 2.0 −5.0 4.0 5.0 \| −0.811 ± 0.090 2.51 +0.24 −0.31 −0.11 +0.8 −1.7 0.91 +2.10 −3.7 Final Set −0.8 2.5 −2.0 −3.0 \| −0.819 ± 0.065 2.21 +0.37 −0.42 −3.44 +0.46 −1.5 −3.8 +8.2 −6.2 Data sample q0 j0 s0 l0 Pantheon −0.702 ± 0.104 1.60 ± 0.71 −3.54 +0.38 −1.5 −4.9 +6.3 −5.0 Pantheon+GRB −0.775 ± 0.048 2.61 +0.29 −0.19 2.8 ± 1.4 −1.3 +3.0 −3.6 Pantheon+quasars −0.844 ± 0.048 2.42 ± 0.25 −2.5 +1.4 −1.2 −3.2 +2.5 −2.1 Pantheon+GRB+quasars −0.819 ± 0.065 2.21 +0.37 −0.42 −3.44 +0.46 −1.5 −3.8 +8.2 −6.2 Table 3 . 3The best fit values of cosmological parameters in wCDM DE parametrization obtained from the minimization of χ 2 function based on MCMC algorithm (left part) and the best fit values of the cosmographic parameters computed in wCDM model (right part).best fit parameters \| computed values Data Ωm,0 w \| q0 j0 Pantheon 0.349 +0.037 −0.029 −1.25 +0.15 −0.13 \| −0.709 +0.086 −0.075 1.92 +0.45 −0.66 Pantheon+GRB 0.352 +0.036 −0.027 −1.26 ± 0.14 \| −0.714 ± 0.081 1.97 +0.49 −0.61 Pantheon+quasars 0.389 +0.027 −0.023 −1.42 +0.15 −0.13 \| −0.801 +0.088 −0.077 2.69 +0.53 −0.74 Pantheon+GRB+quasars 0.388 +0.028 −0.022 −1.42 +0.15 −0.13 \| −0.798 +0.086 −0.077 2.66 +0.53 −0.72 Table 4 . 4The best fit values of cosmological parameters obtained for ΛCDM universe (left part) and the the best fit of cosmographic parameters of the model (right part).best fit parameters \| computed values Data Ωm,0 \| q0 j0 Pantheon 0.285 ± 0.013 \| −0.572 ± 0.019 1.0 Pantheon+GRB 0.285 ± 0.012 \| −0.572 ± 0.019 1.0 Pantheon+quasars 0.294 ± 0.012 \| −0.559 ± 0.019 1.0 Pantheon+GRB+quasars 0.294 ± 0.012 \| −0.559 ± 0.019 1.0 Table 5 . 5The best fit values of cosmological parameters for Pade parameterization (left part) and the best fit of cosmographic parameters (right part).best fit parameters \| computed values Data Ωm,0 w0 w1 w2 \| q0 j0 Pantheon 0.330 +0.060 −0.045 −1.24 +0.15 −0.13 0.21 +0.72 −0.40 0.16 +0.66 −0.42 \| −0.741 ± 0.097 2.4 ± 1.0 Pantheon+GRB 0.327 +0.063 −0.045 −1.22 +0.15 −0.12 0.08 ± 0.59 0.21 +0.64 −0.40 \| −0.725 ± 0.094 2.22 +0.91 −1.1 Pantheon+quasars 0.402 +0.033 −0.024 −1.40 +0.15 −0.12 −0.098 +0.58 −0.74 −0.36 +0.21 −0.63 \| −0.756 +0.11 −0.098 2.05 +0.94 −1.3 Pantheon+GRB+quasars 0.391 +0.038 −0.026 −1.41 +0.15 −0.13 −0.10 +0.55 −0.67 −0.07 +0.55 −0.61 \| −0.78 ± 0.10 2.5 +1.0 −1.3 Table 6 . 6The best fit values of cosmological parameters for CPL parametrization (left part) and the best fit of cosmographic parameters of the model (right part).best fit parameters \| computed values Data Ωm,0 w0 w1 \| q0 j0 Pantheon 0.281 +0.12 −0.059 −1.17 ± 0.17 0.55 +1.1 −0.52 \| −0.74 ± 0.10 2.33 +1.2 −0.91 Pantheon+GRB 0.326 +0.061 −0.033 −1.22 ± 0.15 0.30 +0.53 −0.41 \| −0.724 +0.086 −0.075 2.17 +0.59 −0.74 Pantheon+quasars 0.382 +0.035 −0.024 −1.41 ± 0.14 0.08 +0.60 −0.51 \| −0.798 ± 0.090 2.70 ± 0.85 Pantheon+GRB+quasars 0.384 +0.033 −0.022 −1.41 ± 0.14 0.05 ± 0.50 \| −0.801 ± 0.090 2.71 +0.82 −0.91 . 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[ "Coherent Baryogenesis and Nonthermal Leptogenesis: A comparison ", "Coherent Baryogenesis and Nonthermal Leptogenesis: A comparison " ]	[ "Björn Garbrecht [email protected] \nInstitut für Theoretische Physik\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany\n", "Tomislav Prokopec [email protected] \nInstitut für Theoretische Physik\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany\n", "Michael G Schmidt [email protected] \nInstitut für Theoretische Physik\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany\n" ]	[ "Institut für Theoretische Physik\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany", "Institut für Theoretische Physik\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany", "Institut für Theoretische Physik\nHeidelberg University\nPhilosophenweg 16D-69120HeidelbergGermany" ]	[]	We present a new mechanism for baryogenesis: at preheating after inflation fermions acquire a varying mass by their coupling to a time dependent field. Their CP-violating mass matrix can generate a charge asymmetry to be transformed into a lepton asymmetry through decay into standard model particles and heavy Majorana neutrinos. In a concrete model of hybrid inflation we compare "coherent baryogenesis" with nonthermal leptogenesis by perturbative decay of the inflation condensates.	10.1142/9789812702159_0052	[ "https://arxiv.org/pdf/hep-ph/0410132v1.pdf" ]	119,082,222	hep-ph/0410132	06df7ccf275640ce196fcba10ce85e5cfd3b21ad	Coherent Baryogenesis and Nonthermal Leptogenesis: A comparison * 8 Oct 2004 Björn Garbrecht [email protected] Institut für Theoretische Physik Heidelberg University Philosophenweg 16D-69120HeidelbergGermany Tomislav Prokopec [email protected] Institut für Theoretische Physik Heidelberg University Philosophenweg 16D-69120HeidelbergGermany Michael G Schmidt [email protected] Institut für Theoretische Physik Heidelberg University Philosophenweg 16D-69120HeidelbergGermany Coherent Baryogenesis and Nonthermal Leptogenesis: A comparison * 8 Oct 2004 We present a new mechanism for baryogenesis: at preheating after inflation fermions acquire a varying mass by their coupling to a time dependent field. Their CP-violating mass matrix can generate a charge asymmetry to be transformed into a lepton asymmetry through decay into standard model particles and heavy Majorana neutrinos. In a concrete model of hybrid inflation we compare "coherent baryogenesis" with nonthermal leptogenesis by perturbative decay of the inflation condensates. I. INTRODUCTION The baryon asymmetry of the universe n B /s ≃ 7 − 8 × 10 −11 (BAU), which is now deduced from early nucleosynthesis data and from WMAP data on microwave background fluctuations, is in surprisingly good agreement. Its theoretical explanation requires detailed knowledge on B, C/CP violation and nonequilibrium and this is an ideal testing ground for elementary particle physics models. Electroweak baryogenesis [1] in a strong first order phase transition requires a rather low Higgs mass. This and other ingredients can be tested in present and near future experiments -a great advantage. But this also led to the conclusion that the Standard Model (SM) is ruled out for such a mechanism and that the MSSM is close to this borderline. Out of equilibrium decay of heavy Majorana neutrinos -CP-violating at one loop level through a matrix of Yukawa couplings -has become very attractive again because such Majorana neutrinos also play an important role in explaining the newly found neutrino mass pattern. If one starts with thermalised Majorana neutrinos one can arrive at a simple and very restrictive picture [2] though in the SUSY case there are some problems with gravitino overproduction. We here present a new 'coherent baryogenesis' mechanism, which also contains lepton number violation by Majorana neutrinos and compare it with nonthermal leptogenesis by perturbative decay of the inflaton field and with nonthermal leptogenesis during preheating. II. COHERENT BARYOGENESIS, FORMALISM, AND A CONCRETE MODEL At the end of inflation, there is an oscillating condensate which couples to fermions and induces a varying mass matrix. This, in turn, yields fermion production, known as preheating. Moreover, the mass matrix induces CP-violating flavour oscillations, which is a tree level effect. Charge is produced and is frozen in when the scalar condensate settles to its minimum. Eventually, this charge gets converted to (B-L). This mechanism we call "coherent baryogenesis" [3]. We use the following system of equations for charge and currents of a fermionic system derived in the Schwinger-Keldish formalism for two point functions after Wigner transformḟ 0h + i [M H , f 1h ] + i [M A , f 2h ] = 0 f 1h + 2h\|k\|f 2h + i [M H , f 0h ] − {M A , f 3h } = 0 f 2h − 2h\|k\|f 1h + {M H , f 3h } + i [M A , f 0h ] = 0 f 3h − {M H , f 2h } + {M A , f 1h } = 0(1) with f 0h : charge density, f 3h : axial charge density, f 1h : scalar density, f 2h : pseudoscalar density, h: helicity, k momentum, and hermitean and antihermitean part of the mass matrix, M H = 1 2 (M + M + ) and M A = 1 2i (M − M + ). For pure quantum states this is equivalent to using the Dirac equation for mixing fermions, but it is also a generalization for mixed states. In order to demonstrate our new proposal we implement it in a realistic model for hybrid inflation [4], the supersymmetric Pati-Salam model with gauge group G P S = SU(4) c × SU(2) L × SU(2) R which after modifications does not suffer from the monopole problem as . ( During inflation SUSY is broken, the sneutrino like scalar Higgses have an expectation value \|ν c H \| = M S κ/2β. This field and the inflaton field S begin to fall rapidly during the waterfall regime at the end of inflation (see figures below) before the SUSY vacuum is approached at S = 0. This induces a time dependent mass matrix for the Dirac fermions χ 1j = −Ψ d c H j ,ΨD j , χ 2 j = Ψ D j ,Ψ d c H j (χ 1j ,χ 2j )       ℜ [ ν c H ξ] 1 2 m d 1 2 m d ℜ [ ν c H ζ]   +iγ 5    −ℑ [ ν c H ξ] − i 2 m d i 2 m d −ℑ [ ν c H ζ]          χ 1j χ 2j    ,(4) where m d = S (κ/2 − β \|ν c H \| 2 /M 2 S ) and we have chosen the parameters κ = 0.007, β = 1, ξ = 0.12, ζ = 0.12i, M s = 100µ, µ = 3.9 × 10 16 GeV, γ = γ 1 = 0.0001, Γ S = Γ ν = 0.1µ (a phenomenological damping term, which models tachyonic preheating). Note that the source of CP violation is here the phase between ξ and ζ, which enters at tree level. III. CONTRIBUTION FROM COHERENT BARYOGENESIS Resonant fermion production occurs whenever the fermion mode frequency changes nonadiabatically, which happens during the waterfall regime. The charges stored finally in the χ fermions are displayed in the figure below. Coupling to quarks and leptons is through the superpotential terms γF cH c F cH c /M s and δF c H c F c H c /M s (5) where F c = (4, 1, 2). We then have the decay reactions χ 1j → d c * + ν c * , χ 2j → d c + u c(6) Because of the lepton number violating terms in (5) the charges hence get transformed to q = 1 3 q 1 − 2 3 q 2 . This results in a baryon to entropy ratio n B s = 3 4 n (0) B T R V 0 ≈ 1 × 10 −10 .(7) where n (0) B ≃ 1.5 × 10 45 GeV 3 is the baryon density produced at preheating (the charge density q obtained from figure below multiplied by 3 colors and 1/3, the sphaleron conversion efficiency), V 0 ≃ 3 × 10 64 GeV 4 is the energy density at the end of inflation, T R = [90/(π 2 g * )] 1/4 √ ΓM P ≃ 2. IV. REHEATING AND NONTHERMAL LEPTOGENESIS In section III we have already mentioned the decay of the ν c H -condensate into the lightest Majorana neutrinos after preheating. The reheating temperature we obtained is T R ≃ 2.7 × 10 9 GeV, which meets the gravitino bound. It is much below the Majorana neutrino mass M 1 ≃ 3.9 × 10 10 GeV. The two other masses M 2,3 we assume to be much heavier. n L s ≤ 3 × 10 −10 T R m ν c H M 1 10 6 GeV m ν 3 0.04eV ≃ 8 × 10 −11 ,(8) where we assumed the last factor to be one, as suggested by atmospheric neutrino oscillations. This gives n B /s ≤ 3 × 10 −11 , which is significantly smaller than the coherent baryogenesis result (7). Increasing the cutoff scale M s lowers the value (7), but reduces (8) at least equally fast. Majorana neutrinos can also be produced nonperturbatively [7], just like the χ-particles in coherent baryogenesis. Nonperturbative production of the lightest Majorana neutrino N 1 is dominated by production of N 2 and N 3 with mass of order the inflaton mass, M 2,3 ≃ m ν c H . Their decay asymmetry can be much larger than the asymmetry for N 1 [8]. We conclude that, like coherent baryogenesis, this production channel should also be regarded as a possible source for generating the BAU. V. CONCLUSIONS We considered two sources for baryogenesis during (p)reheating: (1) Direct charge production through coherent baryogenesis; (2) Perturbative decay of the inflaton into Majorana neutrinos. For the given parameters, we found coherent baryogenesis to be the only viable baryogenesis mechanism. For this choice of parameters the gravitino bound is met. For a different choice of parameters (smaller M s ), coherent baryogenesis is further enhanced, making it 7×10 9 9GeV is the reheat temperature, g * = 221.5 is the number of relativistic degrees of freedom of the MSSM, M P ≃ 2.4×10 18 GeV the reduced Planck mass, Γ ≡ H R ≃ 15 GeV is the perturbative inflaton decay rate, and s = 2π 2 g * T 3 R /45 is the entropy density. This estimate of T R is based on the assumption that tachyonic preheating thermalises the inflaton sector, and all other (light) species thermalise at T ≃ T R . The decay rate of the inflaton Γ is dominated by the perturbative decay of the ν c H -condensate into the lightest Majorana neutrinos with mass M 1 = γ 1 ν c H 2 0 /M s ≈ 4 × 10 10 GeV, such that Γ ≃ (1/8π)m ν c H (γ 1 ν c H 0 /M s ) 2 ≃ 15 GeV, where ν c H 0 ≃ µ and m ν c H ≃ 4 × 10 14 GeV with our choice of parameters. The baryon to entropy ratio (7) is somewhat larger than the observed value. It can be easily reduced by chosing a smaller CP violation (by a different choice of ξ and ζ), or by reducing the reheat temperature. produced Majorana neutrino is certainly nonthermal. For maximal mixing and CP violation via one-loop interference like in thermal leptogenesis, one obtains[6] Since the CP-violating sources of coherent baryogenesis arise already at tree level and not at one-loop, as for leptogenesis, it is a competitive source of baryons over an ample section of phase space. robust baryogenesis mechanismrobust baryogenesis mechanism. Since the CP-violating sources of coherent baryogen- esis arise already at tree level and not at one-loop, as for leptogenesis, it is a competitive source of baryons over an ample section of phase space. T Prokopec, K Kainulainen, M G Schmidt, S Weinstock, hep-ph/0302192SEWM 2002 Proceedings. and refs. thereinT. Prokopec, K. Kainulainen, M. G. Schmidt, S. Weinstock, SEWM 2002 Proceedings, hep-ph/0302192 and refs. therein; . M Carena, M Quiros, C E M Wagner, hep-ph/9710401Nucl. Phys. B. 524M. Carena, M. Quiros, C.E.M. Wagner Nucl. Phys. B 524 (1998) 3, hep-ph/9710401 . W Buchmüller, P Di Bari, M Plümacher, hep-ph/0401240 and refs. thereinW. Buchmüller, P. Di Bari, M. Plümacher, hep-ph/0401240 and refs. therein . B Garbrecht, T Prokopec, M G Schmidt, hep-ph/0304088Phys. Rev. Let. 9261302B. Garbrecht, T. Prokopec, M. G. Schmidt, Phys. Rev. Let. 92, 061302 (2004), hep-ph/0304088; . hep-th/0211219Europhys. J. hep-th/0211219, Europhys. J. . 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[ "RELAXATION OF THE CHIRAL CHEMICAL POTENTIAL IN THE DENSE MATTER OF A NEUTRON STAR", "RELAXATION OF THE CHIRAL CHEMICAL POTENTIAL IN THE DENSE MATTER OF A NEUTRON STAR" ]	[ "Maxim Dvornikov \nPushkov Institute of Terrestrial Magnetism\nIonosphere, and Radio Wave Propagation\nMoscowRussia\n\nNational Research Tomsk State University\nTomskRussia\n" ]	[ "Pushkov Institute of Terrestrial Magnetism\nIonosphere, and Radio Wave Propagation\nMoscowRussia", "National Research Tomsk State University\nTomskRussia" ]	[]	A model of the generation of a magnetic field in a neutron star is developed, based on an instability of the magnetic field caused by the electroweak interaction between electrons and nucleons in nuclear matter. The rate of change of the helicity of the electrons as they scatter on protons in the dense matter of a neutron star is calculated with the help of methods of quantum field theory. The influence of the electroweak interaction between electrons and background nucleons on the process of change of the helicity is examined. A kinetic equation is derived for the evolution of the chiral chemical potential. The results obtained are used to describe the evolution of the magnetic field in magnetars. *	10.1007/s11182-017-0991-0	[ "https://arxiv.org/pdf/1702.05737v1.pdf" ]	119,084,623	1702.05737	9599d494e07a2fdff717c40be2a3e6af3eed6f6a	RELAXATION OF THE CHIRAL CHEMICAL POTENTIAL IN THE DENSE MATTER OF A NEUTRON STAR Maxim Dvornikov Pushkov Institute of Terrestrial Magnetism Ionosphere, and Radio Wave Propagation MoscowRussia National Research Tomsk State University TomskRussia RELAXATION OF THE CHIRAL CHEMICAL POTENTIAL IN THE DENSE MATTER OF A NEUTRON STAR 1 A model of the generation of a magnetic field in a neutron star is developed, based on an instability of the magnetic field caused by the electroweak interaction between electrons and nucleons in nuclear matter. The rate of change of the helicity of the electrons as they scatter on protons in the dense matter of a neutron star is calculated with the help of methods of quantum field theory. The influence of the electroweak interaction between electrons and background nucleons on the process of change of the helicity is examined. A kinetic equation is derived for the evolution of the chiral chemical potential. The results obtained are used to describe the evolution of the magnetic field in magnetars. * INTRODUCTION Some neutron stars can possess extremely powerful magnetic fields 15 10 B ≥ G. Such neutron stars are called magnetars [1]. Despite the long history of observations of magnetars and the existence of numerous theoretical models of the generation of their magnetic fields, at the present time there does not exist a generally accepted mechanism explaining the origin of the magnetic field in these compact stars. Recently, in [2,3], efforts were undertaken to solve the problem of the generation of the magnetic field in magnetars with the help of the chiral magnetic effect. The given effect was used to explain the origin of toroidal magnetic fields in neutron stars in [4]. An alternative mechanism predicting the creation of strong cosmic magnetic fields based on the instability of the magnetic field caused by the interaction violating spatial parity was proposed in [5]. A new model of the generation of magnetic fields in magnetars was proposed in [6][7][8]. The main mechanism underlying the proposed model is the growth of a seed magnetic field due to its instability in nuclear matter caused by the electron-nucleon ( eN ) electroweak interaction. Within the framework of the proposed approach, some observed characteristics of magnetars have been explained, such as the strength and the length scale of the magnetic field, and also the age of these stars. Nevertheless, some aspects of the model require a more thorough grounding on the basis of calculations making use of methods of quantum field theory (QFT). The present paper is dedicated to a further development of the proposed description of the generation of magnetic fields in magnetars. MODEL OF THE GENERATION OF MAGNETIC FIELDS IN MAGNETARS As is well known, the dense matter of a neutron star consists of ultra-relativistic electrons and non-relativistic nucleons: neutrons and protons. It is assumed that the given matter possesses zero macroscopic velocity and zero polarization. Electrons in such matter interact with nucleons through the electroweak forces, which violate parity. In [6,7] it was found that in this case, in an external magnetic field B , an induced anomalous electric current of electrons 5 J arises having the following form: ( ) em 5 5 5 2 = , = V α Π Π µ + π J B ,(1)5 L R F = / 2 / 2 2 n V V V G n − ≈ , L,R V8 () ( , ) 2 = ( ,) ( ,) t V h k t k h k t k t t α µ + ∂ − + ρ ∂ σ π σ ,(2) [ ] 2 em 5 5 2 B B cond cond 2 () ( , ) 2 = ( ,) ( ,) t V k t k k t k h k t t α µ + ∂ρ − ρ + ∂ σ π σ ,(3) [ ] 2 5 em em 5 5 B 5 2 cond d ( ) 4 = d ( ,) () ( ,) () d f e t k k h k t t V k t t t µ πα α ⎧ ⎫ − µ + ρ −Γ µ ⎨ ⎬ π µ σ ⎩ ⎭ ∫ ,(4) where f Γ is the rate of change of the helicity of the electrons in electron-proton ( ep ) collisions (see Section 2), and e µ is the average chemical potential of the electron gas. The functions ( , ) h k t and B ( , ) k t ρ are related to the total magnetic helicity ( ) H t and the magnetic field strength via the relations 2 B 1 ( ) = ( , )d , ( ) = ( , )d 2 H t V h k t k B t k t k ρ ∫ ∫ ,(5) where V is the normalization volume. The integration in formula (5) is performed over the entire range of the variation of the wave number k . It is necessary to point out that formula (5) assumes isotropic spectra. Equations (2) Γ µ , has been taken into account phenomenologically. It is based on the fact that the helicity of an electron changes in ep collisions. As a rule, electrons are ultra-relativistic in neutron stars. However, they have nonzero mass. Thus, the following estimate was used in [6] for f Γ : Γ f ∼ m µ e ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ν coll ∼ m µ e ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ω p 2 σ cond ,(6) where m is the mass of an electron, coll ν is the frequency of the ep collisions, and p ω is the plasma frequency in a degenerate plasma. Equation Thus, in order to complete the theoretical grounding of the main equations of the model in [6][7][8], it is necessary to consider the variation of the helicity of the electrons in ep collisions in the dense matter of a neutron star using methods of QFT. Moreover, it is not without interest to investigate the influence of the electroweak interaction between the electrons and the nucleons on the given process. ELECTRON-PROTON COLLISIONS IN A DENSE PLASMA In a neutron star, the helicity of a massive electron can vary during ep and electron-electron collisions due to the electromagnetic interaction with exchange of a virtual plasmon, and also upon the interaction of an electron with the anomalous magnetic moment of the neutron. In [10] it was found that the frequency of ep collisions in the dense matter of a neutron star is much higher than for other reactions. Thus, only ep collisions need be taken into account. Rate of change of helicity in ep collisions The matrix element of ep -scattering due to the electromagnetic interaction has the form M = ie 2 k 1 − k 2 ( ) 2 e ( p 2 )γ µ e( p 1 ) p(k 2 )γ µ p(k 1 ) ,(7) where > 0 e is the absolute value of the charge of the electron, γ µ = γ 0 ,γ ( ) are the Dirac matrices, ( ) 1,2 1,2 1,2 = , p E µ p and ( ) 1,2 1,2 1,2 = , k E µ k are the 4-momenta of the electrons and protons. The momenta of the particles before and after the scattering have the indices 1 and 2. Besides the exchange of a plasmon with the proton, the electron can interact via the electroweak interaction with the nucleons of the matter of a neutron star. In order to take this interaction into account, in matrix element (7), instead of solutions of the Dirac equation in a vacuum it is necessary to use the spinors corresponding to exact solutions of the wave equation for an electron interacting with the background matter. These exact solutions are found in [11]. We will be interested in reactions in which the helicity of the electron changes sign (flips). Let us first consider R L e e → transitions. According to Eq. (7), it is necessary to calculate the following quantity: ( ) 0 2 1 = , = ( ) ( ) J J u p u p µ µ − + γ J ,(8) where u ± are the basis spinors corresponding to the different polarizations of the electrons. They are normalized by the condition † = 1 u u ± ± . Direct calculation of J µ gives [ ][ ] 0 1 2 1 2 0 0 1 0 2 2 2 R 1 1 L ( ) ( ) 2 = 2 ( ) ( ) ( ) ( ) mP p p E p E p V J E p E p E p p V E p p V + − + − − + ⎡ ⎤ + + + − ⎣ ⎦ − + − + − , [ ] [ ][ ] 1 2 1 2 5 0 1 0 2 2 2 R 1 1 L ( ) ( ) 2 = 2 ( ) ( ) ( ) ( ) m p p E p E p V E p E p E p p V E p p V + − + − − + − + − − − + − + − P J ,(9) where † † 0 2 1 2 1 = ( ) ( ), = ( ) ( ) P w w w w − + − + p p P p p σ .(10) Here σ are the Pauli matrices, E 0± ( p) =\| p ∓ V 5 \| , V = V L + V R ( ) / 2 , and ( ) w ± p are the two-component spinors appearing on p. 86 in [12]. To derive formula (9), we have made use of the Dirac matrices in the chiral representation. As was shown in [13, pp. 205-209], to describe the collisions in relativistic plasma due to the long-range Coulomb forces, it is necessary to use the elastic scattering approximation. Thus, in the treatment of the R L → transitions, it is necessary to assume that 1 2 ( ) = ( ) E p E p + − . Using the expression for the energies of the ultrarelativistic electrons [11] 1,2 1,2 R,L ( )= E p p V ± + , we find that the condition for elasticity of the collisions is written in the form 1 2 5 = 2 p p V − . In this case, we find from relation (9) that = 0 J . The quantities 0 P in formula (10) can also be calculated in explicit form: ( ) 2 0 1 2 \| \| = 1 /2 P ⎡ − ⋅ ⎤ ⎣ ⎦ n n , where 1,2 n are the unit vectors in the direction 1,2 p . Thus, employing formula (9), we find that the square of the matrix element given by formula (7) has the form \| M \| 2 = e 4 m 2 p 1 + p 2 ( ) 2 1− n 1 ⋅ n 2 ( ) ⎡ ⎣ ⎤ ⎦ 8 p 1 − V 5 ( ) 2 p 2 + V 5 ( ) 2 E 1 E 2 + M 2 + k 1 ⋅ k 2 ( ) ⎡ ⎣ ⎤ ⎦ E 1 − E 2 ( ) 2 − k 1 − k 2 ( ) 2 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 ,(11) where the leading order in the mass of the electron is retained. Note that the contribution of the protons to \| M \| 2 , which are assumed to be non-polarized, is found according to the standard scheme (see, for example, [12, pp. 252-256] W = V 2(2π) 8 ∫ d 3 p 1 d 3 p 2 d 3 k 1 d 3 k 2 E 1 E 2 δ 4 p 1 + k 1 − p 2 − k 2 ( ) \| M \| 2 1 R 2 L 1 2 ( ) 1 ( ) ( ) 1 ( ) e e p p p p f E f E f f + − ⎡ ⎤ ⎡ ⎤ × −µ − −µ −µ − −µ ⎣ ⎦ ⎣ ⎦ E E ,(12) where we have carried out the summation over the polarizations of the proton after scattering. Here Direct calculation of the integrals over the momenta of the ultra-relativistic electrons and non-relativistic protons in formula (12) gives W (R → L) = W 0 µ R − µ L ( ) θ µ R − µ L ( ) , W 0 = Ve 4 32π 5 m 2 M µ e T ln 48π α em ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 4 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ,(13) where M is the mass of a proton. Note that in the derivation of expression (13), we took the dependence on the interaction potentials of the electrons with matter L,R V exactly into account. In the treatment of the L R → transitions, the calculation of the total scattering probability is analogous to the R → L case. It can be shown that (L R) W → in this situation coincides completely with formula (13) with the substitution R L µ ↔ µ taken into account. For the sake of brevity, the corresponding calculations are not presented. Kinetics of the chiral chemical potential Proceeding from formula (13) and the analogous relation for the L R → transitions, we obtain the kinetic equations for the total number of right and left electrons R,L N in the form ( ) ( ) R L 0 R L 0 L R d d = (R L) (L R)= , = (L R) (R L)= d d N N W W W W W W t t − → + → − µ −µ − → + → − µ −µ .(14) Introducing the number densities of left and right electrons R,L R,L = / n N V and employing the expression for R,L n in terms of the distribution function ( ) ( ) 3 3 R,L R,L R,L 3 2 R,L R,L d 1 = 2 (2 ) 3 exp 1 V p n p V µ − ≈ ⎡ ⎤ π π β + −µ + ⎣ ⎦ ∫ ,(15) we obtain that d n R − n L ( ) / dt ≈ 2 ! µ 5 µ e 2 / π 2 , where it has been taken into account that ! V 5 = 0 and 5 e µ << µ . Finally, we can derive the kinetic equation for 5 µ : 2 2 5 em 5 em d 4 8 = , = ln 4 d f f e e m M T t ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ µ α π −Γ µ Γ − ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ π α µ µ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ .(16) Here we have made use of formulas (13) and (14). It should be noted that the quantity f Γ in formula (6) differs from that used in [6][7][8]. The reason for the difference in f Γ in formulas (6) and (16) consists in the fact that in [6] we used the results of [10], which considered scattering of non-polarized electrons on protons. However, in the problem considered in the present work, fixed polarizations of electrons, which have opposite values before and after scattering, are important. This fact explains, for example, the result that f Γ in formula (16) is linear in T , whereas in relation (6) the given quantity ∼ T 2 . Note also that f Γ was recently calculated in [14]. The value of f Γ obtained in [14] does not depend on T , since in [14] it was assumed that the protons are non-degenerate. This assumption is valid in the early stages of the evolution of neutron stars. The present work examines the generation of a magnetic field in a neutron star found in a state of thermodynamic equilibrium at Thermodynamic description of relaxation of the chiral chemical potential Recently, Sigl and Leite [2] advanced the hypothesis that the kinetics of the chiral chemical potential in the systems of left and right electrons interacting via the electroweak interaction with matter satisfies the equation 5 5 5 d = ( ) d f V t µ −Γ µ + ,(17) and not Eq. (16), which follows from the results of our calculations. It is possible, nevertheless, to show that Eq. (17) contradicts the laws of thermodynamics. Bringing formula (15) to bear, it is possible to rewrite Eq. (17) in the form ( ) 2 R L 5 5 2 2 d = ( ) d e f n n V t µ − − Γ µ + π . (18) It is clear from formula (18) GENERATION OF MAGNETIC FIELDS IN MAGNETARS As the initial condition for the magnetic energy density spectrum, we choose the Kolmogorov spectrum: ρ B (k,t 0 ) = C k −5/3 , where the constant C is related to the seed field in a young pulsar where B and eq B are assigned in formulas (5) and (21). Substitution (19) also makes it possible to eliminate the excessive growth of the magnetic field at 0 t t >> . A renormalization analogous to that implied by substitution (19) for B << B eq was also implemented in [8]. Let us consider the evolution of the magnetic field in a neutron star found in a state of thermal equilibrium that sets up for The maximum magnetic field strength B max ∼ 10 14 − 10 15 G is determined by the initial thermal energy. After it is reached, the field begins slowly to decrease. This is explained by the continuing energy losses due to the emission of neutrinos. Note that max B in Fig. 1a is less than in Fig. 1b. This follows from the fact that the time scale of the field in Fig. 1a DISCUSSION OF RESULTS The rate of change of the helicity of the electrons during ep collisions was estimated in [6,7] on the basis of qualitative arguments of classical physics [9, pp. 66-67]. As is well known, the spin of a particle is an intrinsically quantum object. It is namely for this reason that its evolution must be studied in a corresponding fashion. In the present work, we have implemented the methods of QFT to calculate the rate of the change of the helicity. This can explain the difference between the results obtained here and those of [6,7] (see formulas (16) and (6)). Another important result obtained in the given work consists in the findings of our study of the influence of the electroweak interaction of electrons with neutrons on the process of the change of their helicity during collisions with protons. Employing the method of exact solutions of the Dirac equation in an external field and assuming that the scattering is elastic and the electrons are ultra-relativistic, we found that the effective potentials L,R V do not enter into the expression for the total probability of the processes L,R R,L e e ↔ in formula (13) in the explicit form. Hence it follows that the kinetic equation for the chiral chemical potential (Eq. (16)) coincides with the kinetic equation used in [6][7][8], in contradiction to the recent assertion otherwise in [2] (see Eq. (17)). Moreover, the kinetic equation derived here (Eq. (16)) is confirmed by the laws of thermodynamics (see Subsection 2.3). Finally, an accurate account of the energy conservation law in the Appendix allowed us to alter the form of the quenching of the parameter Π in formula (19) in comparison with the results of [8]. This led to a more correct description of the evolution of the magnetic field in magnetars in Section 3, especially for B ∼ B eq . Nevertheless, the parameters of the generated field, B max ∼ 10 14 − 10 15 G, and the time required for the field to grow to its maximum value 4 5 10 ≤ ⋅ years, are in agreement with astrophysical predictions for magnetars [1]. Note that it is sufficient to take the given dependence into account in Eqs. (2)-(4) only in the terms containing 5 5 V µ + since only they are responsible for the instability of the magnetic field. If we also allow for cooling of the neutron star due to the emission of neutrinos, then it is necessary to make the substitution 2 2 eq 0 eq ( ) ( ) B T B T → in formula (23). The given modification of Eqs. (2)-(4) is equivalent to formula (19). It is interesting to note that, despite the cooling of a neutron star due to the growth of its magnetic field, the second law of thermodynamics is not violated. This fact can be verified using the equation of the heat transfer in magnetohydrodynamics [17, p. 335], from which it follows that the total entropy of the neutron star always grows: ! S > 0 . and (3) for ( , ) h k t and B ( , ) k t ρ are a direct consequence of the modified Faraday equation, in which the anomalous current in relation (1) is taken into account. The first two terms on the right-hand side of Eq. (4) for 5 ( ) t µ result from the Adler anomaly for ultra-relativistic electrons and from Eq. (2). The last term on the righthand side of Eq. (4), 5 f ( 6 ) 6was derived on the basis of the relation between coll ν and cond σ in a classical Lorentz plasma [9, pp. 66-67]. Fermi-Dirac distributions of the electrons and protons, = 1/T β is the inverse temperature, and p µ is the chemical potential of the protons. In Eq.(12) it is assumed that electrons before and after a collision have different chemical potentials: R µ and L µ . The protons and electrons are assumed to be in thermal equilibrium with the same temperature T . the onset of the collapse of a supernova. At the given stage of the evolution of a neutron star, the proton component of matter is degenerate. It should be noted that Γ f ∼ α em 2 in formula(16), which coincides with the result of[14]. that the equilibrium state in which R,L = const n would be reached at ! µ R = ! µ L , where ! µ L,R = µ L,R − V L,R , and not at R L = µ µ , as is required by the laws of thermodynamics[15, p. 306]. Note that the formally introduced quantities ! µ L,R = ! µ L,R (P,T ) , where P is the pressure in the system, are the chemical potentials in the absence of the background matter. Analysis of the equilibrium state in a system of left and right electrons is a particular case of the description of the equilibrium of a body in an external field V(r) . As was shown in[15, p. 73-74], the equilibrium in the given case is reached when the full chemical potential µ = ! µ(P,T ) + V takes constant values inside the system. The results of[15, p. 73-74] are easily generalized to the case of a system consisting of two types of particles: left and right electrons.In this situation we find that in the equilibrium state the full chemical potentials, which include the interaction potentials with the background matter L,R V , should coincide: L R = µ µ . Thus, Eq.(17), proposed in[2], not only is not confirmed by direct calculation of the probability of the processes L k min = 2 ⋅10 −11 eV = R NS −1 , NS = 10 R km is the radius of the neutron star, and B Λ is the minimal scale of the magnetic field and is a free parameter. The initial magnetic energy density spectrum is chosen in the form is a parameter defining the initial helicity: = 0 q corresponds to zero helicity, and =1 q corresponds to the maximal helicity. We choose the initial value of the chiral chemical potentialMeV. Note that the evolution of the magnetic field has practically no dependence on relativistic electrons. Matter with such parameters can be present in neutron stars. In order to account for the energy balance in the system consisting of the magnetic field and the background matter, it is necessary to renormalize the parameter Π in formula (1) (see the Appendix): T . The solid curves correspond to magnetic fields with zero initial helicity ( = 0 q ), and the dashed curves, to those with maximum initial helicity ( = 1 q ): a) evolution of the magnetic field for K years, where t 0 ∼ 10 2 years. In the given time interval, the neutron star cools down due to the emission of neutrinos in modified Urca processes. This leads to the following time dependence of the temperature[16]: we have employed the cooling law of the neutron star and the chosen electron number density.Fig. 1plots the time dependence of the magnetic field strength based on a numerical solution of Eqs. (2)-(4) with the chosen initial conditions. It is clear fromFig. 1that the magnetic field displays an exponential growth in the initial stage of the evolution. Such a growth of the field is mediated by the electroweak eN -interaction and is governed by the nonzero parameter 5 V . The growth of the field takes place for t − t 0 ∼ (10 − 10 5 ) years, depending on B Λ and 0 T . The most rapid growth is observed for 9 0 =10 T K, which corresponds to the smallest value of cond σ , and also for small-scale fields. Note that the time it takes the field to grow to its maximum value t ∼ (10 3 − 10 5 ) (seeFigs. 1a and b)is close to the observed age of young magnetars [1]. are the effective interaction potentials of the left and right electrons with the nucleons of the medium (mainly with neutrons),5 F 1.17 10 G − ≈ ⋅ GeV -2 is the Fermi constant, and n n is the neutron density. The current in equation (1) was derived in [6, 7] using the exact solution of the Dirac equation for an ultra-relativistic electron interacting with matter under the influence of an external magnetic field. Basing ourselves on expression (1) for the current, in [7] we obtained a system of equations for the evolution of the spectrum of the density of the magnetic helicity ( , ) h k t , the spectrum of the density of the magnetic energy B ( , ) k t ρ , and the chiral chemical potential in the form [ ] 2 em 5 5 B cond cond ). The total scattering probability has the form[12, pp. 248-249] is greater and correspondingly the neutrinos have enough time to carry off more energy from the neutron star. It should be noted that the values of B max ∼ 10 15 G in Figs. 1c and d correspond to the predictions of magnetic fields in magnetars [1]. The author expresses his gratitude to V. G. Bagrov and V. B. Semikoz for fruitful discussions.APPENDIX. ENERGY SOURCE UNDERLYING GROWTH OF THE MAGNETIC FIELDDespite the fact that the fermions in a neutron star are strongly degenerate, they possess a nonzero temperature.For example, t ∼ 10 2 years after the explosion of a supernova the temperature may reach T !10 8 K . In[8]the hypothesis was advanced that the growth of the magnetic field predicted in[6,7], can be supported by the transformation of the thermal energy of the fermions of matter into the energy of the magnetic field. In order to ground the possibility of the given process, it is necessary to consider the equation describing the conservation of energy in magnetohydrodynamics[17, pp. 226 -227]:where ρ is the mass per unit volume of the matter of a neutron star, T ε is the internal energy per unit volume, v is the velocity, and q is the energy flux density.It is possible to represent T ε in the formenergy of the degenerate gas, and T δε is the thermal correction. It was shown in[8]that the magnetic field can acquire energy from δε T . Moreover, it was found in[8]that 1/T σ >>[10], we find that the conductivity now depends on the growing magnetic field: . S Mereghetti, J A Pons, A Melatos, Space Sci. Rev. 191S. Mereghetti, J. A. Pons, and A. Melatos, Space Sci. Rev., 191, 315-338 (2015). . G Sigl, N Leite, J. Cosmol. Astropart. Phys. 0125G. Sigl and N. Leite, J. Cosmol. Astropart. Phys., 01, 025 (2016). . N Yamamoto, Phys. Rev. 9365017N. Yamamoto, Phys. Rev., D93, 065017 (2016). . J Charbonneau, A Zhitnitsky, J. Cosmol. Astropart. Phys. 0810J. Charbonneau and A. Zhitnitsky, J. Cosmol. Astropart. Phys., 08, 010 (2010). . A Boyarsky, O Ruchayskiy, M Shaposhnikov, Phys. Rev. Lett. 109111602A. Boyarsky, O.Ruchayskiy, and M. Shaposhnikov, Phys. Rev. Lett., 109, 111602 (2012). . M Dvornikov, V B Semikoz, Phys. Rev. 9161301M. Dvornikov and V. B. Semikoz, Phys. Rev., D91, 061301 (2015). . M Dvornikov, V B Semikoz, J. Cosmol. Astropart. Phys. 0532M. Dvornikov and V. B. Semikoz, J. Cosmol. Astropart. Phys., 05, 032 (2015). . M Dvornikov, V B Semikoz, Phys. Rev. 9283007M. Dvornikov and V. B. Semikoz, Phys. Rev., D92, 083007 (2015). A F Aleksandrov, L S Bogdankevich, A A Rukhadze, Principles of Plasma Electrodynamics. BerlinSpringer VerlagA. F. Aleksandrov, L. S. Bogdankevich, and A. A. Rukhadze, Principles of Plasma Electrodynamics, Springer Verlag, Berlin (1984). . D C Kelly, Astrophys. J. 179D. C. Kelly, Astrophys. J., 179, 599-606 (1973). . A Grigoriev, S Shinkevich, A Studenikin, Grav. Cosmol. 14A. Grigoriev, S. Shinkevich, A. Studenikin, et al., Grav. Cosmol., 14, 248 -255 (2008). . V B Berestetskii, E M Lifshitz, L P Pitaevskii, Quantum Electrodynamics. 4V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, Vol. 4, Butterworth- Heinemann, London (1982). . E M Lifshitz, L P Pitaevskii, Physical Kinetics. 10Pergamon PressE. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Vol. 10, Pergamon Press, Oxford (1981). . D Grabowska, D B Kaplan, S Reddy, Phys. Rev. 9185035D. Grabowska, D. B. Kaplan, and S. Reddy, Phys. Rev., D91, 085035 (2015). . L D Landau, E M Lifshitz, Statistical Physics. 5Butterworth-HeinemannL. D. Landau and E. M. Lifshitz, Statistical Physics, Vol. 5, Butterworth-Heinemann, London (1980). . C J Pethick, Rev. Mod. Phys. 64C. J. Pethick, Rev. Mod. Phys., 64, 1133-1140 (1992). L D Landau, E M Lifshitz, Electrodynamics of Continuous Media. LondonButterworth-Heinemann8L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Vol. 8, Butterworth-Heinemann, London (1984).	[]
[ "RESPONSE OF THE TWO-DIMENSIONAL ELECTRON GAS OF AlGaAs/GaAs HETEROSTRUCTURES TO PARALLEL MAGNETIC FIELD", "RESPONSE OF THE TWO-DIMENSIONAL ELECTRON GAS OF AlGaAs/GaAs HETEROSTRUCTURES TO PARALLEL MAGNETIC FIELD" ]	[ "V S Khrapai \nInstitute of Solid State Physics\n142432ChernogolovkaMoscow DistrictRussia\n", "E V Deviatov \nInstitute of Solid State Physics\n142432ChernogolovkaMoscow DistrictRussia\n", "A A Shashkin \nInstitute of Solid State Physics\n142432ChernogolovkaMoscow DistrictRussia\n", "V T Dolgopolov \nInstitute of Solid State Physics\n142432ChernogolovkaMoscow DistrictRussia\n" ]	[ "Institute of Solid State Physics\n142432ChernogolovkaMoscow DistrictRussia", "Institute of Solid State Physics\n142432ChernogolovkaMoscow DistrictRussia", "Institute of Solid State Physics\n142432ChernogolovkaMoscow DistrictRussia", "Institute of Solid State Physics\n142432ChernogolovkaMoscow DistrictRussia" ]	[]	We study the transport properties of the two-dimensional electron gas in AlGaAs/GaAs heterostructures in parallel to the interface magnetic fields at low temperatures. The magnetoresistance in the metallic phase is found to be positive and weakly anisotropic with respect to the orientation of the in-plane magnetic field and the current through the sample. At low electron densities (ns < 5 × 10 10 cm −2 ) the experimental data can be described adequately within spin-related approach while at high ns the magnetoresistance mechanism changes as inferred from ns-independence of the normalized magnetoresistance.	null	[ "https://arxiv.org/pdf/cond-mat/0005377v3.pdf" ]	117,270,871	cond-mat/0005377	ceafad69e1c9dd35027ae81e775196062f9ff1b4	RESPONSE OF THE TWO-DIMENSIONAL ELECTRON GAS OF AlGaAs/GaAs HETEROSTRUCTURES TO PARALLEL MAGNETIC FIELD 26 Sep 2000 V S Khrapai Institute of Solid State Physics 142432ChernogolovkaMoscow DistrictRussia E V Deviatov Institute of Solid State Physics 142432ChernogolovkaMoscow DistrictRussia A A Shashkin Institute of Solid State Physics 142432ChernogolovkaMoscow DistrictRussia V T Dolgopolov Institute of Solid State Physics 142432ChernogolovkaMoscow DistrictRussia RESPONSE OF THE TWO-DIMENSIONAL ELECTRON GAS OF AlGaAs/GaAs HETEROSTRUCTURES TO PARALLEL MAGNETIC FIELD 26 Sep 2000 We study the transport properties of the two-dimensional electron gas in AlGaAs/GaAs heterostructures in parallel to the interface magnetic fields at low temperatures. The magnetoresistance in the metallic phase is found to be positive and weakly anisotropic with respect to the orientation of the in-plane magnetic field and the current through the sample. At low electron densities (ns < 5 × 10 10 cm −2 ) the experimental data can be described adequately within spin-related approach while at high ns the magnetoresistance mechanism changes as inferred from ns-independence of the normalized magnetoresistance. Much interest has been aroused recently by the behaviour of two-dimensional (2D) electron systems in a parallel magnetic field. The resistance of a 2D electron gas in Si MOSFETs was found to rise steeply with parallel field B saturating to a constant value above a critical magnetic field B c which depends on electron density [1,2,3,4]. Such a behaviour of the resistance agrees well with data on the metal-insulator phase diagram for the case of parallel fields of Ref. [5] where the suppression of the metallic phase by B was observed. The insensitivity of the effect to the orientation of B with respect to the current through the sample [1] as well as its isotropy in weak, tilted magnetic fields [2,5] hinted at the spin origin of the effect. Recently, an analysis of Shubnikov-de Haas oscillations in tilted magnetic fields has established that the field B c corresponds to the onset of full spin polarization of the electron system [6,7]. The influence of B on the resistance of the 2D hole gas in GaAs heterostructures was found to be basically similar to the case of Si MOSFETs [8] with two noteworthy distinctions: (i) above B c , the resistance keeps on increasing less steeply with no sign of saturation [9]; and (ii) the magnetoresistance is strongly anisotropic depending upon the relative orientation of the in-plane magnetic field and the current [10]. The early version of the theory of the spin origin of parallel field effects exploits scaling arguments for calculating the temperature-dependent magnetoresistance in the metallic phase in the low field limit [11]. An alternative concept has been expressed recently based on the fact that the 2D electron screening of a random potential depends on the relative population of spin-up and spin-down subbands [12]: at zero temperature the magnetoresistance is expected to be positive for relatively low electron densities in the metallic phase and is determined by the spin polarization of 2D electrons which is defined as the ratio of the Zeeman splitting and the Fermi energy ξ = gµ B B/2E F . Above the critical field B c corresponding to the condition ξ = 1, the resistance R(B ) should saturate at the level of the four-fold zero-field resistance. While the spin-related approach [12] allows the interpretation of the resistance rise with B in Si MOSFETs, the strongly anisotropic magnetoresistance observed on the 2D holes in GaAs heterostructures is likely to point to a contribution of the orbital effects of Ref. [13] where it was shown that for a 2D system with finite thickness the form of the Fermi surface changes in a parallel magnetic field. In a number of recent publications, the occurrence of a zero-magnetic-field metal-insulator transition in Si MOS-FETs and for the 2D holes in GaAs as well as the origin of the effects observed in parallel magnetic fields have been attributed to strong particle-particle interaction as characterised by the Wigner-Seitz radius r s (see, e.g., Ref. [6]). Oppositely, for the 2D electrons in GaAs heterostructures the values of r s are almost an order of magnitude lower, particularly, because of the small effective mass, which is traditionally expressed in terms of weak electron-electron interaction in GaAs. Nevertheless, for the 2D electrons in both GaAs [14] and Si MOSFETs [15], similar metal-insulator phase diagrams were obtained in normal magnetic fields including a zero field [16] and so the parameter r s is not crucial in this case. The obvious consequence of the small effective electron mass in GaAs is that much higher values of the critical magnetic field B c are expected [12]. Here, we investigate the influence of parallel magnetic field on the resistance in the metallic phase of the 2D electron system in GaAs heterostructures. We observe a positive magnetoresistance which is weakly anisotropic with respect to the orientation of the in-plane magnetic field and the current through the sample. This finding is similar to results reported for the 2D electrons in Si MOSFETs and the 2D holes in GaAs heterostructures and enables us to split the parallel magnetic field effect from the problem of the interaction-induced metalinsulator transition. The spin mechanism allows the description of the experimental data at low electron densities but fails in the opposite limit in which the normalized magnetoresistance is found to be independent of n s . Our devices are 170 µm wide conventional Hall bars based on an AlGaAs/GaAs heterostructure that is grown on a (100) GaAs substrate and contains a high mobility 2D electron gas. The density n s of the 2D electrons is controlled using a gate on the front surface of the device. The behaviour of the low-temperature mobility µ in the studied range of electron densities is depicted in the top inset to Fig. 1. For our samples the conductivity remains in the metallic regime down to n s ∼ 2 × 10 10 cm −2 . The sample is placed in the mixing chamber of a dilution refrigerator with a base temperature of 30 mK. The measurements are performed using a standard fourterminal lock-in technique at a frequency of 10 Hz in magnetic fields up to 14 T. The ac current I through the device does not exceed 1 nA. Two samples made from the same wafer have been investigated; the results obtained on these are practically identical. To change the sample position in the mixing chamber we warm the sample up, rotate it at room temperature, and cool it down again. The alignment uncertainty of the sample plane with the magnetic field is kept within 0.3 • . We use small misalignments ≤ 2 • to observe quantum oscillations caused by a perpendicular component of the magnetic field and evaluate the g factor from the oscillation beating pattern [18], see the bottom inset to Fig. 2. The electron density as a function of gate voltage is determined from quantum oscillations in normal magnetic fields. We have checked that the gate voltage dependence of the resistance at B = 0 is well reproducible in different coolings of the sample with an accuracy of insignificant threshold shifts. In parallel magnetic fields, this dependence is used for determining the threshold voltage. A typical experimental trace of the resistivity ρ(B ) is shown in Fig. 1 for the parallel and perpendicular orientations of B relative to the current I. The magnetoresistance is close to a parabolic dependence, being smaller in the parallel configuration. As is evident from Fig. 1, the magnetoresistance anisotropy is not strong, approximately a factor of 1.2. The observed resistance rise reaches a factor of three at the lowest n s and the highest magnetic fields, displaying no tendency of saturation. The temperature dependence of the resistance is practically absent in the interval between 30 and 600 mK. We emphasize that in the presence of a small normal component of the magnetic field as caused by misalignment the dependence ρ(B ) is drastically distorted. As seen in the bottom inset to Fig. 1, the effect is dramatic even for small misalignments ∼ 1 • . By scaling the B -axis we make the normalized resistivity traces ρ(B )/ρ(0) at different electron densities collapse onto a single curve simultaneously for each of the two field directions (Fig. 2). At low n s the scaling factor B c is found to enhance approximately linearly with electron density (top inset to Fig. 2). At higher n s > 5 × 10 10 cm −2 the normalized resistivity ρ(B )/ρ(0) becomes independent of n s (Fig. 3) such that the scaling parameter B c saturates. Thus, we observe a strong rise of the resistance with parallel magnetic field in the metallic regime in the 2D electron system with r s spanning between 2 and 3.5, the range in which ρ(0) changes by more than an order of magnitude. The effect is weakly anisotropic relative to the orientation of B and I and is qualitatively similar to that found in 2D systems with strong particle-particle interaction, r s > ∼ 10 [1,2,3,4,8,9]. In contrast to the conclusion of Refs. [1,2,3,4,8,9], we do not suppose that the observed dependence ρ(B ) is due to electron-electron interaction because in our case the r s values are considerably smaller. Particularly, the parallel field effect can be considered regardless of the interaction-induced metal-insulator transition. Two different approaches that predict the change of ρ with B in a 2D system with weak electron-electron interaction have been formulated [12,13]. To compare our data with the theoretical predictions we reason as follows. The absence of strong anisotropy of the observed magnetoresistance allows one to presume the dominance of spin effects as discussed in the theory of Ref. [12]. This theory demands, in particular, identifying B /B c with the spin polarization ξ, i.e., B c = 2E F /gµ B (where E F = πh 2 n s /m and m is the effective mass). The dashed line in Fig. 2 is drawn in accordance with the theory; its best fit to the data as shown in the figure yields the normalizing condition for the parameter B c . Although the consistency between experiment and theory is fairly good, there are problems with such a description of the data. Firstly, the accessible magnetic fields are not high enough to reach the expected saturation of the resistance. Secondly, the scaling parameter B c becomes independent of electron density at n s > 5 × 10 10 cm −2 . Thirdly, the so-defined critical field B c (n s ) corresponds to the g factor g ≈ 2.2 which is much larger than its bulk GaAs value of g = 0.44. In fact, at low electron densities the g factor is expected to be enhanced due to electron-electron interaction as discussed in the Fermi liquid model (see, e.g., Ref. [19]). An independent check of the beating pattern of Shubnikov-de Haas oscillations in slightly nonparallel magnetic fields gives a crude estimate 0.7 < g < 1.4, which is still small compared to the parallel field data [20]. The noticeably smaller values of B c obtained in the experiment as well as the saturation of B c at high electron densities are in contrast to the behaviour of the critical field B c found in Refs. [6,9] and cause us to invoke alternative mechanisms of the parallel field magnetoresistance. The most likely candidate is an orbital effect caused by the finite thickness of the 2D electron system [13]. Its contribution would naturally explain the observed magnetoresistance anisotropy and weaker dependence B c (n s ). Nevertheless, we believe that at low electron densities the spin-related concept [12] describes the magnetoresistance adequately because with decreasing n s the orbital effect (or any other mechanism that yields n s -independent scaling parameter B c ) is overpowered by the spin effect as will be discussed below. In a 2D electron system with finite thickness the parallel magnetic field deforms the Fermi surface so that the effective mass in the normal to B direction increases leading to a positive magnetoresistance, whereas the one in the parallel direction remains unchanged and so the resistance [13]. An example of the deformed Fermi surface as calculated in triangular potential approximation is displayed in the inset to Fig. 4 for different magnitudes of B . With increasing magnetic field the Fermi surface broadens in the k ⊥ direction and narrows in the k direction to keep its area constant, shifting as a whole along k ⊥ . For lower n s , the distortion of the Fermi sur-face is stronger and so the magnetoresistance is larger, see Fig. 4. Although the model [13] yields the correct order of magnitude of the magnetoresistance for the perpendicular field to current orientation (cf. Figs. 3 and 4), it cannot describe the relatively weak magnetoresistance anisotropy as well as ρ(B )/ρ(0) at different n s . At the same time, the theoretical magnetoresistance changes with electron density not so strongly as the one predicted by the spin-related model [12]. Apparently, this is the condition for switching the dominant magnetoresistance mechanism: at low n s the spin mechanism of the magnetoresistance prevails while at high n s the orbital effect is likely to become dominant. In our opinion, the approach of Ref. [13] should be completed by including a change of the relaxation time in a parallel magnetic field. The following aspects seem important: (i) the increase of the Fermi surface perimeter leads to shortening the relaxation time; (ii) the increase of the density of states at the Fermi energy results in a better screening by the 2D system and, hence, increasing the relaxation time; and (iii) the anisotropy of screening. Their account should cause, at least, a reduction of the anisotropy of the theoretical magnetoresistance. In summary, we have investigated the transport properties of the 2D electrons in AlGaAs/GaAs heterostructures in parallel to the interface magnetic fields at low temperatures. It has been found that the magnetoresistance in the metallic phase is positive and weakly anisotropic with respect to the orientation of the in-plane magnetic field and the current through the sample. This is basically similar to data obtained for the 2D electrons in Si MOSFETs and the 2D holes in GaAs heterostructures, although the electron-electron interaction in GaAs is considerably weaker. Therefore, our experiment splits the parallel magnetic field effect from the problem of the interaction-induced metal-insulator transition. At low electron densities the spin-related model is capable of describing the experimental results while at high n s neither approach can explain n s -independence of the normalized magnetoresistance. FIG. 1 . 1Dependence of the sample resistivity on parallel magnetic field for B ⊥ I at n s = 7.4 × 10 10 cm −2 (solid line) and B I at n s = 7.5 × 10 10 cm −2 (dashed line). The top inset shows the mobility as a function of electron density. Bottom inset: behaviour of the normalized resistivity with magnetic field in the perpendicular field to current orientation for the in-plane B (solid line) and the field tilted by 1.2 • relative to the sample plane (dashed line). FIG. 2 . 2Scaling the magnetic field dependence of the normalized resistivity of the sample at low n s . Also shown by a dashed line is a fit using the theory of Ref.[12]. The scaling parameter B c as a function of electron density is displayed in the top inset. The bottom inset shows the resistivity as a function of filling factor ν for two tilt angles of the magnetic field. FIG. 3 . 3Change of the normalized resistivity of the sample with parallel magnetic field at high electron densities. FIG. 4 . 4Magnetoresistance calculated in a similar way to the theory of Ref.[13] for n s = 1 × 10 11 cm −2 (solid lines) and n s = 7 × 10 10 cm −2 (dashed lines). The corresponding transformation of the Fermi surface with parallel magnetic field at n s = 1 × 10 11 cm −2 is depicted in the inset. We are grateful to S.V. Kravchenko . D Simonian, S V Kravchenko, M P Sarachik, V M Pudalov, Phys. Rev. Lett. 792304D. Simonian, S.V. Kravchenko, M.P. Sarachik, and V.M. Pudalov, Phys. Rev. Lett. 79, 2304 (1997). . S V Kravchenko, D Simonian, M P Sarachik, A D Kent, V M Pudalov, Phys. Rev. B. 583553S.V. Kravchenko, D. Simonian, M.P. Sarachik, A.D. Kent, and V.M. 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[]	[ "Pedro Morais [email protected] ", "Rui Pacheco [email protected] ", "\nDepartamento de Matemática\nDepartamento de Matemática\nUniversidade da Beira Interior\nCovilhãPortugal\n", "\nUniversidade da Beira Interior\nCovilhãPortugal\n" ]	[ "Departamento de Matemática\nDepartamento de Matemática\nUniversidade da Beira Interior\nCovilhãPortugal", "Universidade da Beira Interior\nCovilhãPortugal" ]	[ "SURFACES IN R 7 OBTAINED FROM HARMONIC MAPS IN S 6" ]	We will investigate the local geometry of the surfaces in the 7dimensional Euclidean space associated to harmonic maps from a Riemann surface Σ into S 6 . By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions ϕ : Σ → S 6 whose associated immersions F : Σ → R 7 belong to certain remarkable classes of surfaces, namely: minimal surfaces in hyperspheres; surfaces with parallel mean curvature vector field; pseudo-umbilical surfaces; isotropic surfaces.2010 Mathematics Subject Classification. 53C43,53C42, 53A10,53A07.	10.1007/s10711-017-0266-5	[ "https://arxiv.org/pdf/1605.09344v2.pdf" ]	119,150,283	1605.09344	06c3438cb68ea8d9c2742f573fc678c4a01b45c9	30 May 2016 Pedro Morais [email protected] Rui Pacheco [email protected] Departamento de Matemática Departamento de Matemática Universidade da Beira Interior CovilhãPortugal Universidade da Beira Interior CovilhãPortugal SURFACES IN R 7 OBTAINED FROM HARMONIC MAPS IN S 6 30 May 2016and phrases Harmonic mapsminimal surfacesparallel mean curvaturepseudo- umbilical surfacesseven dimensional cross product We will investigate the local geometry of the surfaces in the 7dimensional Euclidean space associated to harmonic maps from a Riemann surface Σ into S 6 . By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions ϕ : Σ → S 6 whose associated immersions F : Σ → R 7 belong to certain remarkable classes of surfaces, namely: minimal surfaces in hyperspheres; surfaces with parallel mean curvature vector field; pseudo-umbilical surfaces; isotropic surfaces.2010 Mathematics Subject Classification. 53C43,53C42, 53A10,53A07. Introduction It is a well-known fact that any non-conformal harmonic map ϕ from a simplyconnected Riemann surface Σ into the round 2-sphere S 2 is the Gauss map of a constant Gauss curvature surface, F : Σ → R 3 , and of two parallel constant mean curvature surfaces, F ± = F ± ϕ : Σ → R 3 ; the surface F integrates the closed 1-form ω = ϕ × * dϕ, where × denotes the standard cross product of R 3 . The immersion F can also be obtained from the associated family of ϕ by applying the famous Sym-Bobenko's formula [18,20]. Eschenburg and Quast [11] replaced S 2 by an arbitrary Kähler symmetric space N = G/K of compact type and applied a natural generalization of Sym-Bobenko's formula [18] to the associated family of an harmonic map ϕ : Σ → N in order to obtain an immersion F of Σ in the Lie algebra g of G. This construction was subsequently generalized to primitive harmonic maps from Σ to generalized flag manifolds [21]. Surfaces associated to harmonic maps into complex projective spaces have also been constructed in [14,15,24]. In the present paper, we will investigate the local geometry of surfaces in R 7 associated to harmonic maps from a Riemann surface Σ into the nearly Kähler 6sphere S 6 . In this setting, the harmonicity of a smooth map ϕ : Σ → S 6 amounts to the closeness of the differential 1-form ω = ϕ × * dϕ, where × stands now for the 7dimensional cross product. This means that we can integrate on simply-connected domains in order to obtain a map F : Σ → R 7 . When ϕ is a conformal harmonic immersion, F is a conformal immersion; and, in contrast with the 3-dimensional case, where F is necessarily a totally umbilical surface, F can exhibit a wide variety of geometrical behaviors in the 7-dimensional case. By applying methods based on the use of harmonic sequences [2,4,8,10,12,22], we will characterize the conformal harmonic immersions ϕ : Σ → S 6 whose associated immersions F : Σ → R 7 belong to certain remarkable classes of surfaces, namely: minimal surfaces in hyperspheres (Theorem 2); surfaces with parallel mean curvature vector field (Theorem 3); pseudo-umbilical surfaces (Theorem 4); isotropic surfaces (Theorem 5). From our results, it is interesting to observe that SO(7)-congruent harmonic maps into S 6 may produce surfaces in different classes of surfaces. For instance, if ϕ is superconformal in S 6 ∩ W for some 4-plane W , then, by Theorem 3, up to translation, F will be a constant mean curvature surface in W ⊥ if W is coassociative; but, if the 4-plane W admits a ×-compatible decomposition, then, by Theorem 4, F will be pseudo-umbilical with non-parallel mean curvature vector. Theorem 2 says, in particular, that, if F is minimal in a hypersphere of R 7 , then, up to change of orientation of Σ, ϕ is SO(7)-congruent to some almost complex curve. Almost complex curves can be characterized as those weakly conformal harmonic maps from Σ in S 6 with Kähler angle θ = 0 and were classified in [2]. If ϕ is an almost complex curve, F + = F + ϕ is a constant map and F − = F − ϕ is just a dilation of ϕ. Apart from almost complex curves, totally real minimal immersions are the most investigated minimal immersions with constant Kähler angle [3]. In Theorem 6 we will identify those totally real harmonic immersions ϕ : Σ → S 6 with respect to which both F + and F − are immersions with parallel mean curvature vector. Preliminaries 2.1. The seven dimensional cross product. Let · be the standard inner product on R 7 and e 1 , . . . , e 7 be the canonical basis of R 7 . Fix the 7-dimensional cross product × defined by the multiplication table The cross product × satisfies the following identities, for all x, y ∈ R 7 : (P1) x · (x × y) = (x × y) · y = 0; (P2) (x × y) · (x × y) = (x · x)(y · y) − (x · y) 2 ; (P3) x × y = −y × x; (P4) x · (y × z) = y · (z × x) = z · (x × y); (P5) (x × y) × (x × z) = ((x × y) × z) × x + ((y × z) × x) × x + ((z × x) × x) × y; (P6) x × (x × y) = −(x · x)y + (x · y)x; (P7) x × (y × z) + (x × y) × z = 2(x · z)y − (x · y)z − (y · z)x. Extend the inner product · and the cross product × by complex bilinearity to C 7 = R 7 ⊗ C. We also denote these complex bilinear extensions by · and ×, respectively. Later on we will need the following lemma. Lemma 1. Let x and y be two vectors in C 7 and suppose that y = 0 is isotropic. If x × y = 0 then x is isotropic and x · y = 0. Proof. If x and y are collinear, it is clear that x is also isotropic and x · y = 0. Otherwise, if x × y = 0, by (P6) we have 0 = x × (x × y) = −(x · x)y + (x · y)x, which implies that x · x = 0 (that is, x is isotropic) and x · y = 0. Let V ⊂ R 7 be a 3-dimensional subspace and set W := V ⊥ . The subspace V is an associative 3-plane if it is closed under the cross product, and W is a coassociative 4-plane if V is an associative 3-plane. The exceptional Lie group G 2 , which is precisely the group of isometries in SO(7) preserving the vector cross product, acts transitively on the Grassmannian G a 3 (R 7 ) of associative 3-dimensional subspaces of R 7 , with isotropy group isomorphic to SO(4) (see [16] for details). Lemma 2. Let V be a 3-dimensional subspace of R 7 and set W := V ⊥ . Then: 1) if V is an associative 3-plane, V × W = W and W × W = V ; 2) if V is an associative 3-plane, for any orthogonal direct sum decomposition W = W 1 ⊕ W 2 , with dim W 1 = dim W 2 = 2,(2)we have W 1 × W 1 = W 2 × W 2 ; 3) conversely, if W admits an orthogonal direct sum decomposition (2) satis- fying W 1 × W 1 = W 2 × W 2 , then V is an associative 3-plane; Proof. The first two statements can be proved by direct application of the properties of ×. Alternatively, one can use the multiplication table (1) in order to check the statements for a suitable associative 3-plane, and then apply the transitivity of the G 2 -action. Given a 4-plane W in the conditions of the third statement, fix an orthonormal basis v 1 , v 2 , v 3 , v 4 , with W 1 = span{v 1 , v 2 } and W 2 = span{v 3 , v 4 }, such that v 5 := v 1 ×v 2 = v 3 ×v 4 . By (P1), it is clear that v 5 ∈ W ⊥ . Define also v 6 := v 1 ×v 3 and v 7 := v 1 × v 4 . One can now apply the properties of × in order to show that V = span{v 5 , v 6 , v 7 } is an associative 3-plane. Given a 4-dimensional subspace of R 7 , an orthogonal direct sum decomposition of the form (2) is said to be ×-compatible if W 1 × W 1 ⊥ W 2 × W 2 . Lemma 3. If the 4-dimensional subspace W admits one ×-compatible decomposition, then any decomposition of W of the form (2) is ×-compatible. Proof. Fix an orthonormal basis v 1 , v 2 , v 3 , v 4 of W such that the decomposition W = W 1 ⊕ W 2 is ×-compatible, where W 1 = span{v 1 , v 2 } and W 2 = span{v 3 , v 4 }, which means that (v 1 × v 2 ) · (v 3 × v 4 ) = 0. By applying the properties of the cross product, this implies that (v σ(1) × v σ(2) ) · (v σ(3) × v σ(4) ) = 0 for any permutation σ of {1, 2, 3, 4}. The result follows now by linearity. For example, any decomposition of W = span{e 1 , e 2 , e 3 , e 4 } of the form (2) is ×-compatible. On the contrary, coassociative 4-planes do not admit ×-compatible decompositions. 2.2. Harmonic sequences. Let Σ be a Riemann surface with local conformal coordinate z. We will view any smooth map into the Grassmannian G k (C n ) of k-planes in C n as a vector subbundle of the trivial bundle C n = Σ × C n . Given a harmonic map ϕ : Σ → G k (C n ), let {ϕ i } i∈Z , with ϕ 0 = ϕ, be the correspondent harmonic sequence [2,4,8,10,12,22] of ϕ. Any two consecutive elements in this sequence are orthogonal with respect to the standard Hermitian inner product h(x, y) = x ·ȳ on C n = R n ⊗ C, where · is the standard inner product of R n . The harmonic map ϕ has isotropy order r > 0 if ϕ ⊥ h ϕ i for all 0 < i ≤ r but ϕ ⊥ h ϕ r+1 . In the harmonic sequence of ϕ, the harmonic maps ϕ 1 and ϕ −1 are precisely the images of the second fundamental forms A ′ ϕ and A ′′ ϕ , respectively. These are defined by A ′ ϕ (v) = P ⊥ h ϕ ( ∂v ∂z ) and A ′′ ϕ (v) = P ⊥ h ϕ ( ∂v ∂z ), where P ⊥ h ϕ is the Hermitian projection onto the orthogonal bundle of ϕ, for any smooth section v of the vector bundle associated to ϕ. The harmonicity of ϕ can be reinterpreted in terms of the holomorphicity of its second fundamental forms with respect to the Koszul-Malgrange holomorphic structures on ϕ and on the orthogonal bundle ϕ ⊥ h . This allows one to remove singularities and define ϕ 1 and ϕ −1 globally on Σ. The remaining elements of the harmonic sequence are obtained by iterating this construction. Given a harmonic map ϕ : Σ → CP n into the complex projective space, any harmonic map ϕ i in the harmonic sequence of ϕ has the same isotropy order of ϕ. Moreover, ϕ is non-constant weakly conformal if, and only if, it has isotropy order r ≥ 2. If ϕ has finite isotropy order r, then it is clear that r ≤ n. Following [1], we say that ϕ is superconformal if r = n. If ϕ j = {0} for some j > 0 (which implies that ϕ −j ′ = {0} for some j ′ > 0), then ϕ has infinite isotropy order. Such harmonic maps are said to be isotropic [12] or superminimal [6]. An isotropic harmonic map ϕ : Σ → CP n has length l > 0 if its harmonic sequence has length l. Now, consider the totally geodesic immersion of the unit round sphere S n in RP n ⊂ CP n . Let ϕ : Σ → S n be a harmonic map, which can also be seen either as a harmonic map ϕ : Σ → CP n satisfying ϕ = ϕ or as a parallel section of the corresponding line subbundle. We also have ϕ j = ϕ −j . Moreover, for each j, there exists a local meromorphic section f j of ϕ j such that [2,4]: ∂f j ∂z = f j+1 + ∂ ∂z log \|f j \| 2 f j ; ∂f j+1 ∂z = − \|f j+1 \| 2 \|f j \| 2 f j ; \|f j \|\|f −j \| = 1 (if f j = 0),(3)with f 0 = ϕ. Observe that, since \|f 1 \| 2 = f 1 ·f 1 = ∂f0 ∂z · ∂f0 ∂z = − 1 \|f−1\| 2 f 1 · f −1 , we have f 1 · f −1 = −1.(4) If ϕ is non-constant weakly conformal, then ϕ 1 ⊥ h ϕ 1 (ϕ has isotropy order r ≥ 2), which means that f 1 · f 1 = 0. Differentiating we obtain f 2 · f 1 = 0, that is, ϕ 2 ⊥ h ϕ −1 . Hence ϕ has isotropy order r ≥ 3. This argument can be extended in order to prove the following. Lemma 4. If the harmonic map ϕ : Σ → S n has finite isotropy order r, then r is odd. On the other hand, since ϕ −j = ϕ j , it is clear that if ϕ : Σ → S n is an isotropic harmonic map then its length l must also be odd. Lemma 5. If ϕ : Σ → S n has isotropy order r ≥ m, m + 1 is even and ϕ k = {0} is real, with k = m+1 2 , then the harmonic map ϕ is superconformal in S m = S n ∩ W , where the constant m + 1-dimensional subspace is given by W ⊗ C = ϕ k ⊕ k−1 i=−k+1 ϕ i . Proof. Since ϕ k = {0} is real, we have ϕ k+1 = ϕ k−1 = ϕ −k+1 . Consequently, ϕ has finite isotropy order r = m and W ⊗ C = ϕ k ⊕ k−1 i=−k+1 ϕ i is a constant bundle, that is, ϕ is superconformal in S m = S n ∩ W . 2.3. Almost complex curves in the nearly Kähler 6-sphere. The standard nearly Kähler structure J on the 6-dimensional unit sphere S 6 can be written in terms of the cross product × as follows: Ju = x × u, for each u ∈ T x S 6 and x ∈ S 6 . An almost complex curve in S 6 is a non-constant smooth map ϕ : Σ → S 6 whose differential is complex linear with respect to the nearly Kähler structure J. Any such map ϕ is a weakly conformal harmonic map for which ϕ 1 ⊕ ϕ ⊕ ϕ 1 is a bundle of associative 3-planes. In [2], the authors showed that there are four basic types of almost complex curves: (I) linearly full in S 6 and superminimal; (II) linearly full in S 6 but not superminimal; (III) linearly full in some totally geodesic S 5 in S 6 ; (IV) totally geodesic. Twistorial constructions of almost complex curves of type (I) can be found in [5]; almost complex-curves of type (II) and (III) from the 2-tori are obtained by integration of commuting Hamiltonian ODE's [1,17]. The Kähler angle θ : Σ → [0, π] of a weakly conformal harmonic map ϕ : Σ → S 6 is given by [2] ( f 1 × f −1 ) · f 0 = −i cos θ.(5) Almost complex curves are precisely those weakly conformal harmonic maps with θ = 0. In [2], the authors also proved the following. Theorem 1. [2] Let ϕ : Σ → S 6 be a weakly conformal harmonic map with constant Kähler angle θ = 0, π and assume that the ellipse of curvature is a circle. Then there is a unit vector v such that ϕ is linearly full in the totally geodesic S 5 = S 6 ∩ v ⊥ . Moreover, there exist an angle β and an almost complex curveφ of (7) is the rotation defined as follows: type (III) such that ϕ = R v (β)φ, where R v (β) ∈ SOR v (β)x = v, if x = v; cos β x + sin β v × x, if x ∈ S 6 ∩ v ⊥ .(6) These conditions on ϕ can be expressed in terms of the harmonic sequence as follows. Lemma 6. [2] Let ϕ : Σ → S 6 be a weakly conformal harmonic map. Then, we have: 1) The ellipse of curvature of ϕ is a circle if, and only if, ϕ 2 is an isotropic line bundle. 2) The ellipse of curvature of ϕ is a point if, and only if, ϕ 2 = {0}. 3) ϕ has constant Kähler angle if, and only if, ϕ 0 × ϕ 1 ⊥ h ϕ 2 . 3. Surfaces obtained from Harmonic maps into the 6-sphere Let Σ be a Riemann surface with local conformal coordinate z = x + iy, and let ϕ : Σ → S n be a harmonic map, that is, △ϕ ⊥ T ϕ S n = {u ∈ R n \| ϕ · u = 0}. For n = 7, this means that ϕ × △ϕ = 0, which is equivalent to the closeness of the one form ω = ϕ × * dϕ. Hence, if ϕ : Σ → S 6 is a harmonic immersion and Σ is simply-connected, we can integrate to obtain an immersion F : Σ → R 7 such that dF = ϕ × * dϕ. Lemma 7. At corresponding points, the tangent spaces T ϕ and T F satisfy T ϕ × T ϕ ⊥ T F . Proof. The tangent space T F at z = x + iy is generated by F x = ϕ × ϕ y and F y = −ϕ × ϕ x ; and T ϕ × T ϕ at z = x + iy is generated by ϕ x × ϕ y . Taking into account the properties for the cross product, we have (ϕ x × ϕ y ) · (ϕ × ϕ y ) = −ϕ x · ϕ y × (ϕ y × ϕ) = ϕ x · (\|ϕ y \| 2 ϕ − (ϕ · ϕ y )ϕ y ) = \|ϕ y \| 2 (ϕ x · ϕ) − (ϕ y · ϕ)(ϕ x · ϕ y ) = 0, since ϕ x · ϕ = ϕ y · ϕ = 0. Hence F x · (ϕ x × ϕ y ) = 0; and, similarly, F y · (ϕ x × ϕ y ) = 0. Making use of the properties for the cross product, we also obtain the following formulae for the first and second fundamental forms of the immersion F in terms of ϕ and its derivatives. Proposition 1. Let I F and II F be the first and the second fundamental forms of F : Σ → R 7 , respectively. Let N be a vector field of the normal bundle T F ⊥ . Then, with respect to the local conformal coordinate z = x + iy of Σ, we have I F = \|ϕ y \| 2 −ϕ x · ϕ y −ϕ x · ϕ y \|ϕ x \| 2 (7) II N F := II F · N = (ϕ x × ϕ y ) · N + (ϕ × ϕ xy ) · N (ϕ × ϕ yy ) · N (ϕ × ϕ yy ) · N (ϕ x × ϕ y ) · N − (ϕ × ϕ xy ) · N .(8) If ϕ is conformal, F is also conformal. Let e 2α be the common conformal factor of ϕ and F . Taking Lemma 7 into account, one can check that the mean curvature vector of F is given by h F := 1 2 tr I −1 F II F = e −2α 2 II F (F x , F x ) + II F (F y , F y ) = e −2α ϕ x × ϕ y . Consider the harmonic sequence {ϕ j } of ϕ : Σ → S 6 , viewed as map into RP 6 ⊂ CP 6 . Since ϕ is conformal, we already know that it has isotopy order r ≥ 3. Let {f j } be a sequence of meromorphic sections satisfying (3). We have ϕ = f 0 , ϕ z = f 1 , ϕz = − f−1 \|f−1\| 2 , and the conformal factor e 2α of ϕ satisfies e 2α 2 = \|f 1 \| 2 = 1 \|f−1\| 2 . Hence we can also write h F = if 1 × f −1 . (9) Observe that \|h F \| = 1. In the conformal case, u = e −α (cos θ F x + sin θ F y ) is a unit tangent vector for each θ ∈ R and the ellipse of curvature of F is given by E F = h F + e −2α cos 2θ P ⊥ T F (ϕ × ϕ xy ) + e −2α sin 2θ P ⊥ T F (ϕ × ϕ xx )\| θ ∈ R . (10) Next we will characterize those conformal harmonic maps ϕ : Σ → S 6 for which the corresponding immersions F : Σ → R 7 belong to certain remarkable classes of surfaces, namely: minimal surfaces; surfaces with parallel mean curvature vector field; pseudo-umbilical surfaces; isotropic surfaces. Remark 1. When ϕ is a conformal, we have I ϕ = I F . Hence the Gauss curvatures K ϕ of ϕ and K F of F coincide on Σ; and those conformal harmonic maps ϕ producing a surface F with constant Gauss curvature are precisely those minimal surfaces ϕ of constant curvature in S 6 . For a general n, minimal surfaces of constant Gauss curvature in S n were classified by R. Bryant [7]. As a consequence of Bryant's results, if F has constant Gauss curvature K F , then there are three possibilities: K F = 1 and ϕ is totally geodesic; K F = 1/6 and ϕ is an open subset of a Boruvka sphere; K F = 0 and ϕ is a flat minimal surface which can be written as a sum of certain exponentials. More precisely, in the flat case, ϕ takes values in S 5 = S 6 ∩ V , for some 6-dimensional subspace of R 7 , and can be written in the form [7] ϕ(z) = 3 k=1 v k e µ k z−µ kz + v k e −µ k z+µ kz ,(11) where ±µ 1 , ±µ 2 , ±µ 3 are six distinct complex numbers in S 1 and v k ∈ V ⊗ C are nonzero vectors satisfying v k · v j = 0, v k · v j = 0 if j = k, 3 k=1 v k · v k = 1 2 , 3 k=1 µ 2 k v k · v k = 0.(12) 3.1. F is minimal in some hypersphere of R 7 . In such cases, the immersions F have parallel mean curvature vector h F in the normal bundle of F in R 7 and, simultaneously, are pseudo-umbilical, that is, II hF F = λI F for some smooth function λ on Σ. As a matter of fact, for a general dimension, we have: iϕ z = ϕ × ϕ z . If F is such that dF = ϕ × * dϕ, we have F z = iϕ × ϕ z . Hence F z = −ϕ z , which means that, up to an additive constant, F = −ϕ. In this case, it is clear that F is minimal in S 6 . Theorem 2. Let ϕ : Σ → S 6 be a conformal harmonic immersion and F : Σ → R 7 such that dF = ϕ × * dϕ. F is minimal in some hypersphere of R 7 if, and only if, up to change of orientation of Σ, one of the following statements holds: 1) ϕ is an almost complex curve; 2) there exists a unit vector v, an angle β and an almost complex curveφ of type (III), withφ(Σ) ⊂ S 5 = S 6 ∩ v ⊥ , such that ϕ = R v (β)φ, where R v (β) is the rotation given by (6); 3) ϕ is totally geodesic. Proof. Since h F = 0, F can not be minimal in R 7 . We start the proof with the following lemmas. (f 2 × f −1 ) × f 1 · f 0 = 0.(13) Proof. In view of (7) and (8), F is pseudo-umbilical if, and only if, (ϕ × ϕ yy ) · h F = 0, (ϕ × ϕ xy ) · h F = 0.(14) Since h F is a vector field of T F ⊥ and T F ⊗ C = (ϕ × ϕ 1 ) ⊕ (ϕ × ϕ −1 ), we have (f 0 × f 2 ) · h F = (f 0 × P ⊥ ϕ1 f 0zz ) · h F = (f 0 × f 0zz ) · h F . Equating the real and the imaginary parts, and taking into account that △ϕ is collinear with ϕ, we conclude that (f 0 × f 2 ) · h F = 0 if, and only if, equations (14) hold. Making use of properties (P3), (P4) and (P7), we see that, since h F is given by (9), (f 0 × f 2 ) · h F = i (f 2 × f −1 ) × f 1 − 2(f 2 · f 1 )f −1 + (f 2 · f −1 )f 1 + (f −1 · f 1 )f 2 · f 0 . Now, since ϕ is conformal, by Lemma 4 it has isotropy order r ≥ 3, which means, in particular, that f 0 ·f 2 = 0. By definition of harmonic sequence, we also have f 0 ·f 1 = f 0 ·f −1 = 0. Hence, F is pseudo-umbilical if, and only if, (f 2 ×f −1 )×f 1 ·f 0 = 0. Proof. The mean curvature vector field h F is parallel in the normal bundle T F ⊥ if, and only if, ∂hF ∂z is a vector field of T F ⊗ C = (ϕ × ϕ 1 ) ⊕ (ϕ × ϕ −1 ). Taking the z-derivative of h F = if 1 × f −1 , we obtain, in view of (3), ∂h F ∂z = i(f 2 × f −1 + f 1 × f 0 ). Hence ∂hF ∂z is a vector field of T F ⊗ C if, and only if, f 2 × f −1 = af 0 × f −1 + bf 0 × f 1 for some complex functions a and b on Σ. But (f 2 ×f −1 )·(f 0 ×f −1 ) = −f 0 ·(f −1 ×(f −1 ×f 2 )) = f 0 · (f −1 ·f −1 )f 2 −(f −1 ·f 2 )f −1 = 0; hence the component b of f 2 × f −1 along the isotropic section f 0 × f 1 vanishes. Lemma 10. The following statements are equivalent: F is minimal in some hypersphere of R 7 , that is, F is pseudo-umbilical with parallel mean curvature vector; f 2 × f −1 = 0; ϕ × h F is a constant vector v. Proof. Assume that F is pseudo-umbilical with parallel mean curvature vector, that is, condition (13) holds and f 2 × f −1 = af 0 × f −1 , for some complex function a. Then, since (f 2 × f −1 ) · (f 0 × f 1 ) = − (f 2 × f −1 ) × f 1 · f 0 = 0, we also have a = 0. Hence, f 2 × f −1 = 0. Conversely, if f 2 × f −1 = 0, F is pseudo-umbilical by Lemma 8, and h F is parallel by Lemma 9. Taking into account (3) and (4), we have (ϕ × h F ) z = i f 0 × (f 1 × f −1 ) z = if 0 × (f 2 × f −1 ).(15) Assume that ϕ × h F is a constant vector v, that is, (ϕ × h F ) z = 0. Since f 0 is real, this implies that f 2 × f −1 = af 0 for some complex function a on Σ. However, since f −1 is isotropic and f 2 · f −1 = 0, because f 1 and f 2 are consecutive in the harmonic sequence of f 0 , we have (f 2 × f −1 ) · (f 2 × f −1 ) = (f 2 · f 2 )(f −1 · f −1 ) − (f 2 · f −1 ) 2 = 0, which means that f 2 × f −1 is isotropic; hence a = 0 and f 2 × f −1 = 0. Conversely, if f 2 × f −1 = 0, from (15) we conclude that ϕ × h F is constant. Observe that ϕ × h F being a constant v implies that ϕ · h F = cos θ is constant, that is, taking account of (5) and (9), ϕ has constant Kähler angle θ; on the other hand, the condition f 2 × f −1 = 0 implies, by Lemma 1, that f 2 · f 2 = 0, meaning, by Lemma 6, that the ellipse of curvature E ϕ of ϕ is a point or a circle. Now, assume that F is minimal in some hypersphere of R 7 and consider the constant vector v = ϕ × h F . If v = 0, then ϕ = ±h F . Consequently, f 0 × f 1 = ±i(f 1 × f −1 ) × f 1 = ∓if 1 × (f 1 × f −1 ) = ∓i(f 1 · f −1 )f 1 = ±if 1 , that is, up to change of orientation, ϕ is an almost complex curve. Next we take v = 0, which corresponds to a constant kähler angle θ = 0, π. In this case, ϕ(Σ) ⊂ S 6 ∩ v ⊥ . We have two possibilities: either E ϕ is a point or E ϕ is a circle. If E ϕ is a point, then ϕ 2 = {0}, which means that ϕ is superminimal in S 2 = S 6 ∩ E, where E ⊗ C := ϕ ⊕ ϕ −1 ⊕ ϕ 1 is a (constant) non-associative 3-plane. If E ϕ is a circle, then, by Theorem 1, ϕ = R v (β)φ for some almost complex curveφ of type (III), withφ(Σ) ⊂ S 6 ∩ v ⊥ . Reciprocally, if ϕ is an almost complex curve, we have, up to translation, F = −ϕ, as seen in Example 1, and, consequently, F is minimal in S 6 . If the conformal harmonic map ϕ is totally geodesic, we have f 2 = 0; then f −1 × f 2 = 0, and, in view of Lemma 10, F is minimal in some hypersphere. Finally, assume that ϕ = R v (β)φ for some almost complex curveφ of type (III), withφ(Σ) ⊂ S 6 ∩ v ⊥ . Since R v (β) ∈ SO(7), we also have ϕ i = R v (β)φ i . Being an almost complex curve,φ certainly satisfiesφ −1 ×φ 2 = 0. By applying the properties of × and the definition of R v (β), we see now that ϕ −1 × ϕ 2 = R v (β)φ −1 × R v (β)φ 2 = 0, which means, by Lemma 10, that F is minimal in some hypersphere of R 7 . Example 2. Consider the conformal harmonic immersion ϕ : Σ → S 6 ∩ e ⊥ 4 of the form (11) with v k = 1 2 √ 3 exp(πi/6)(e k + ie k+4 ) for k = 1, 2, 3, µ 1 = 1, µ 2 = exp(2πi/3), and µ 3 = exp(4πi/3). The numbers µ i and the vectors v j satisfy (12). The harmonic map ϕ has constant Kähler angle θ = π/2 and ϕ 2 is an isotropic line bundle. Then, by Theorem 1, ϕ is SO(7)-congruent with some almost complex curve. The associated surface F is minimal in a hypersphere of R 7 , by Theorem 2, and F has constant Gauss curvature K F = 0, by Remark 1. Up to translation, F is given by F (z) = 3 k=1ṽ k e µ k z−µ kz +ṽ k e −µ k z+µ kz ,(16) withṽ k = 1 2 √ 3 exp(2πi/3)(e k + ie k+4 ) for k = 1, 2, 3. 3.2. F has parallel mean curvature. Theorem 3. Let ϕ : Σ → S 6 be a conformal harmonic immersion. F : Σ → R 7 has parallel mean curvature vector field and it is not pseudo-umbilical if, and only if, ϕ is superconformal in some 3-dimensional sphere S 3 = S 6 ∩ W , where W is a coassociative 4-space. Proof. If F has parallel mean curvature and it is not pseudo-umbilical, F is not minimal neither in R 7 nor in a hypersphere of R 7 . Since F is not minimal, the normal curvature of F vanishes, by Lemma 4 of [9]. This is known [13] to be equivalent to the fact that the ellipse of curvature E F of F degenerates. Hence, either E F is a point or E F is a line segment. But E F can not be a point because, in that case, in view of (10), we would have f 0 × f 2 = 0, which means that f 2 = 0, and therefore ϕ would be totally geodesic, that is, F would be minimal, by Theorem 2. Then E F must be a line segment, which implies, in view of (10), that ϕ 2 is real. Consequently, by Lemma 5, ϕ is superconformal in S 3 = S 6 ∩ W , with W ⊗ C = ϕ ⊕ ϕ −1 ⊕ ϕ 1 ⊕ ϕ 2 . We also have, by Lemma 9, f 0 × f −1 = a −1 f 2 × f −1 (with a = 0); equivalently, (f 0 − a −1 f 2 ) × f −1 = 0. Hence, by Lemma 1, f 0 − a −1 f 2 is isotropic, which implies that 1 + a −2 f 2 · f 2 = 0. Then we can define orthonormal real vector fields v 1 , v 2 , v 3 , v 4 by v 1 = f 0 , v 2 + iv 3 √ 2 = f −1 \|f −1 \| , iv 4 = a −1 f 2 . Set W 1 = span{v 1 , v 2 } and W 2 = span{v 3 , v 4 }. Taking the real part of f 0 × f −1 = a −1 f 2 × f −1 , we see that v 1 × v 2 = v 3 × v 4 . Hence, Lemma 2 implies that W ⊥ is an associative 3-plane. Reciprocally, again by Lemma 2, if ϕ : Σ → S 3 = S 6 ∩ W is a superconformal harmonic map, with W ⊥ an associative 3-plane, f 0 × f −1 is complex collinear with f 2 × f −1 . Consequently, F : Σ → R 7 has parallel mean curvature and it is clear that F can not be pseudo-umbilical (since ϕ 2 is real, ϕ 2 × ϕ −1 = 0). The parallel mean curvature surfaces F arising from a superconformal harmonic map ϕ in some 3-dimensional sphere S 3 = S 6 ∩ W , with V = W ⊥ an associative 3-plane, are constant mean curvature surfaces in V , up to translation, because F satisfies dF = ϕ × * dϕ and, by Lemma 2, W × W = V . Example 3. Let W = span{e 4 , e 5 , e 6 , e 7 } and ϕ : C → S 3 = S 6 ∩ W be defined by ϕ(x, y) = 1 √ 2 cos x e 4 + sin x e 5 + cos y e 6 + sin y e 7 , which is a superconformal harmonic map and parameterizes a Clifford torus. Taking into account the multiplication table (1), one can check that the associated surface F : C → V ⊂ R 7 is the cylinder given by F (x, y) = 1 2 − (x + y) e 1 − cos(x − y) e 2 + sin(x − y) e 3 . 3.3. F is pseudo-umbilical. Theorem 4. Let ϕ : Σ → S 6 be a conformal harmonic immersion. If F is pseudoumbilical with non-parallel mean curvature vector field, then ϕ belongs to one of the following classes of harmonic maps: 1) ϕ is full and superminimal in S 4 = S 6 ∩ W , for some 5-dimensional vector subspace W of R 7 ; 2) ϕ has finite isotropy order r = 3 and is full in S 4 = S 6 ∩ W , for some 5-dimensional vector subspace W of R 7 ; 3) ϕ is superconformal in S 3 = S 6 ∩ W , for some 4-plane W admitting a ×-compatible decomposition. Conversely, if ϕ is a superconformal harmonic map in S 3 = S 6 ∩ W , for some 4plane W admitting a ×-compatible decomposition, then F is pseudo-umbilical with non-parallel mean curvature vector field. Proof. The harmonic map ϕ takes values in S 6 ∩ W for some W ⊆ R 7 and has isotropy order r ≥ 3. Assume that F is pseudo-umbilical, that is, (f 0 × f 2 ) · h F = 0. Differentiating this equation and using the properties of cross product, we obtain (f 0 × f 3 ) · h F = 0. Since h F is in the normal bundle of F and f 0 × f 1 is a section of T F ⊗ C, we also have (f 0 × f 1 ) · h F = 0. Hence, the sections f 0 × f i and their conjugates f 0 × f −i , with i = 1, 2, 3, are all orthogonal to h F . This is equivalent to say that v = ϕ× h F is orthogonal to the vector subspace W ⊗ C, which is generated by the meromorphic sections f 0 , f i and f −i , with i = 1, 2, 3. Clearly, ϕ is full in S 6 ∩ W . If dim W = 7 (that is, ϕ is full in S 6 ), then v = ϕ × h F vanishes, which implies that ϕ = ±h F ; and, consequently, as we have seen in the proof of Theorem 2, ϕ is, up to orientation, an almost complex curve, and F has parallel mean curvature vector. If dim W = 6, then v = f 0 × h F generates a constant space. Differentiating, we see that f 0 × (f 2 × f −1 ) and v are complex linearly dependent. This occurs if, and only if, f 2 × f −1 = a(f 1 × f −1 ) + bf 0 . However (f 2 × f −1 ) · (f 1 × f −1 ) = 0, hence a = 0 and f 2 × f −1 = bf 0 . Since f 2 × f −1 is isotropic and f 0 is real, we have b = 0, that is, f 2 × f −1 = 0, and F has parallel mean curvature vector, by Lemma 10. When dim W = 3, ϕ is totally geodesic in S 6 , and F is minimal in S 6 , by Theorem 2. So it remains to investigate the cases dim W = 4 and dim W = 5. In the second case, if ϕ has finite isotropy order r, then, by Lemma 4, we must have r = 3; otherwise, ϕ is full and superminimal in S 4 = S 6 ∩ W . Assume now that dim W = 4. In this case, ϕ 2 = {0} is real and ϕ has isotropy order r = 3; hence, by Lemma 5, ϕ is superconformal in S 3 = S 6 ∩W . Consider the real subspaces W 1 and W 2 defined by W 1 ⊗C = ϕ −1 ⊕ϕ 1 and W 2 ⊗C = ϕ⊕ϕ 2 . Since F is pseudo-umbilical, we have (f 0 ×f 2 )·h F = 0; consequently, W 1 ×W 1 ⊥ W 2 ×W 2 , that is, the decomposition W = W 1 ⊕W 2 is ×-compatible. Conversely, suppose that ϕ is a superconformal harmonic map in S 3 = S 6 ∩W and that W = ϕ⊕ϕ −1 ⊕ϕ 1 ⊕ϕ 2 admits a ×-compatible decomposition. Then, by Lemma 3, the decomposition W = W 1 ⊕ W 2 , with W 1 ⊗ C = ϕ −1 ⊕ ϕ 1 and W 2 ⊗ C = ϕ ⊕ ϕ 2 , is ×-compatible at each point of Σ, which implies that F is pseudo-umbilical: (f 0 ×f 2 )·h F = 0. Finally, observe that W can not be a coassociative 4-plane because it admits ×-compatible decompositions. Hence, by Theorem 3, F has non-parallel mean curvature vector field. Since W admits ×-compatible decompositions and ϕ is superconformal in S 3 = S 6 ∩ W , the immersion F is pseudo-umbilical with non-parallel mean curvature vector field, by Theorem 4. Examples and the existence of pseudo-umbilical surfaces F arising from full harmonic maps ϕ in S 4 = S 6 ∩ W , with W a 5-dimensional subspace of R 7 , seem not so easy to establish. 3.4. F is isotropic. An isometric immersion is isotropic provided that, at each point, all its normal curvature vectors have the same length [19]. Theorem 5. Let ϕ : Σ → S 6 be a conformal harmonic immersion. F is isotropic if, and only if, F is pseudo-umbilical and ϕ 2 is isotropic (either ϕ 2 = {0} or ϕ 2 is an isotropic line bundle). In particular, if F is isotropic, then, up to change of orientation of Σ, one of the following statements holds: 1) ϕ is an almost complex curve; 2) ϕ = R v (β)φ for some almost complex curveφ of type (III) and some rotation R v (β) of the form (6); 3) ϕ is totally geodesic; 4) ϕ is superminimal in S 4 = S 6 ∩ W , for some 5-dimensional vector subspace W of R 7 . Proof. From (10) we see that F is isotropic if, and only if, the following holds at each point: h F · P ⊥ T F (ϕ × ϕ xy ) =h F · P ⊥ T F (ϕ × ϕ xx ) = 0;(17)\|P ⊥ T F (ϕ × ϕ xy )\| = \|P ⊥ T F (ϕ × ϕ xx )\|; P ⊥ T F (ϕ × ϕ xy ) · P ⊥ T F (ϕ × ϕ xx ) = 0.(18) Observe that (17) means that F is pseudo-umbilical. On the other hand, (18) means that P ⊥ T F (ϕ × ϕ zz ) = ϕ × P ⊥ T ϕ ϕ zz is isotropic, that is, 0 = ϕ × P ⊥ T ϕ ϕ zz · ϕ × P ⊥ T ϕ ϕ zz = P ⊥ T ϕ ϕ zz · P ⊥ T ϕ ϕ zz − ϕ · P ⊥ T ϕ ϕ zz 2 .(19) Since P ⊥ T ϕ ϕ zz is a meromorphic section of ϕ 2 and ϕ is conformal (in particular, ϕ · ϕ 2 = 0), we see from (19) that (18) is equivalent to the isotropy of ϕ 2 . We know [2] that the ellipse of curvature of an almost complex curve is either a point or a circle, which implies, by Lemma 6, that either ϕ 2 = {0} or ϕ 2 is an isotropic line bundle. Observe also that if the isotropy order of ϕ is r = 3, then ϕ 2 can not be isotropic. Hence, the statement follows now from Theorem 2 and Theorem 4. The parallel surfaces Given any harmonic map ϕ : Σ → S 6 , we define also the maps F + , F − : Σ → R 7 by F ± = F ± ϕ. We have \|F ± x \| 2 = \|F ± y \| 2 = \|ϕ x \| 2 + \|ϕ y \| 2 ∓ 2ϕ · (ϕ x × ϕ y ), F ± x · F ± y = 0.(20) 4.1. The conformal case. If ϕ : Σ → S 6 is an almost complex curve, then F = −ϕ up to translation, and F + is just a constant map. In such case, V ϕ := ϕ ⊕ T ϕ is a bundle of associative 3-planes. Otherwise, we have the following. Lemma 11. Let ϕ : Σ → S 6 be a conformal harmonic map such that V ϕ is everywhere non-associative. Then F ± is a conformal immersion with mean curvature vector field given by h ± = e −2ω ± ϕ x × ϕ y ± △ϕ 2 ,(21) where e 2ω ± = 2(\|ϕ x \| 2 ∓ ϕ · ϕ x × ϕ y ) is the conformal factor of F ± . Proof. If ϕ is conformal, we have \|ϕ · (ϕ x × ϕ y )\| ≤ \|ϕ x \| 2 , where the equality holds if, and only if, ϕ is collinear with ϕ x × ϕ y , that is, if, and only if, V ϕ is associative at that point. Hence, assuming that ϕ is everywhere non-associative, we see from (20) that (21) can be deduced by straightforward computations. \|F ± x \| 2 = \|F ± y \| 2 ≥ 2(\|ϕ x \| 2 − \|ϕ · (ϕ x × ϕ y )\|) > 0, which means that F ± is a conformal immersion. Formula In terms of the meromorphic sections {f j } satisfying (3), the mean curvature vector field h ± is given by h ± = if 1 × f −1 ∓ f 0 2 ∓ 2if 0 · (f 1 × f −1 ) ,(22) assuming that ϕ is conformal. From this we see that \|h ± \| is constant if f 0 ·(f 1 ×f −1 ) is constant, that is, if ϕ has constant Kähler angle. Apart from almost complex curves, totally real minimal surfaces are the most investigated minimal surfaces type (III) and some rotation R v (β) of the form (6), or ϕ takes values in S 2 = S 6 ∩V for some non-associative 3-space V satisfying V × V ⊂ V ⊥ . Conversely, assume that the totally real minimal immersion ϕ belong to one of these two classes of conformal immersions. Then, by Theorem 2 and Lemma 10, we certainly have f 2 × f −1 = 0; and, from (25), we conclude that both F + and F − are immersions with parallel mean curvature vector. Remark 2. Let ϕ : Σ → S 6 be a totally real conformal harmonic map such that F ± have parallel mean curvature. Then f 0 · (f −1 × f 1 ) = 0 and, as we have seen in the proof of the previous theorem, we also have f −1 × f 2 = 0. By using these equations, one can check that the parallel surfaces F + and F − are given, up to translation, by F ± = if −1 × f 1 ± f 0 and that both surfaces are minimal in the sphere S 6 ( √ 2) of R 7 of radius √ 2 with center at the origin. Let {F ± i } be the harmonic sequence of the minimal surface F ± , with F ± 0 = F ± . If ϕ is totally geodesic, we have ϕ 2 = {0} and F ± 2 = {0}. This means that F + and F − are totally geodesic. If ϕ is SO(7)-congruent with some almost complex curve of type (III), then ϕ 2 is an isotropic line bundle and we have F ± 1 = span{if 0 ×f 1 ±f 1 }, F ± 2 = span{if 0 × f 2 ± f 2 }, and F ± 3 = span{if 1 × f 2 + if 0 × f 3 ± f 3 }. F ± 1 and F ± 2 are isotropic line bundles. On the other hand, differentiating (28) we see that f 3 is collinear with f −1 ×f 1 , and from this it follows, taking the properties of × into account, that the line bundle F ± 3 is not isotropic: F ± 3 · F ± 3 = −2a 2 2 \|f 1 \| 2 = 0. Hence F ± has finite isotropy order r = 5. Example 5. The conformal immersion ϕ : Σ → S 6 ∩ e ⊥ 4 of Example 2 is a totally real minimal immersion and is SO(7)-congruent with some almost complex curve. Then F ± = F ± ϕ, with F given by (16), has parallel mean curvature vector. Moreover, by Remark 2, F ± is minimal in S 6 ( √ 2) ∩ e ⊥ 4 , with finite isotropy order r = 5. 4.2. The non-conformal case. When ϕ : Σ → S 2 is non-conformal, it is well known that F ± = F ± ϕ are conformal immersions with constant mean curvature ∓ 1 2 . For non-conformal harmonic maps into S 6 we have the following. Proposition 3. Assume that the harmonic map ϕ : Σ → S 6 is everywhere nonconformal and has no branch-points (dϕ p = 0 for all p ∈ Σ). Then: 1) F + , F − : Σ → R 7 are conformal immersions; 2) the mean curvature vectors h + and h − of F + and F − , respectively, are given by h ± = e −2ω ± ϕ x × ϕ y ± △ϕ 2 ,(29) where e 2ω + and e 2ω − are the conformal factors of F + and F − , respectively. 3) the conformal immersions F + and F − have constant mean curvature along ϕ, with h ± · ϕ = ∓ 1 2 . Proof. From (20), we see that \|F ± x \| 2 = \|F ± y \| 2 ≥ (\|ϕ x \| − \|ϕ y \|) 2 . If ϕ is everywhere non-conformal and has no branch-points, then (\|ϕ x \| − \|ϕ y \|) 2 > 0, and we conclude that F + and F − are conformal immersions. Formula (29) follows directly from the definitions. Finally, differentiating twice the equality ϕ · ϕ = 1 one obtains △ϕ · ϕ = −\|ϕ x \| 2 − \|ϕ y \| 2 , and it follows that ϕ x × ϕ y ± △ϕ 2 · ϕ = ∓ 1 2 \|ϕ x \| 2 + \|ϕ y \| 2 ∓ 2ϕ · (ϕ x × ϕ y ) = ∓ e 2ω± 2 , and consequently h ± · ϕ = ∓ 1 2 . Up to change of orientation, almost complex curves are precisely those weakly conformal harmonic maps ϕ : Σ → S 6 satisfying ϕ 1 × ϕ −1 = ϕ. We finalize this paper by observing that their non-conformal analogous take values in a 2dimensional sphere. Theorem 7. Let ϕ : Σ → S 6 be a non-conformal harmonic map such that ϕ 1 × ϕ −1 ⊆ ϕ. Then ϕ takes values in S 6 ∩ V , for some associative 3-plane V . Consequently, under the additional assumptions of Proposition 3, the conformal immersions F + and F − are constant mean curvature immersions in V . Proof. If ϕ 1 × ϕ −1 ⊆ ϕ, we have f 1 × f −1 = af 0 for some complex function a on Σ and V ϕ := ϕ ⊕ ϕ 1 ⊕ ϕ −1 is a bundle of associative 3-spaces. Differentiating, we get f 2 × f −1 + f 1 × f 0 = af 1 + a z f 0 . Since the fibres of V ϕ are associative, f 0 × f 1 is a section of V ϕ . Hence f 2 × f −1 is a section of V ϕ , and, by associativity, f −1 × (f 2 × f −1 ) is also a section of V ϕ . But f −1 × (f 2 × f −1 ) = (f −1 · f −1 )f 2 − (f −1 · f 2 )f −1 . This implies that P ⊥ Vϕ (f 2 ) = 0, that is, f 2 ∈ V ϕ . Hence V := V ϕ is a constant associative 3-space. Proposition 2 . 2[23] M n is a pseudo-umbilical submanifold of R m such that the mean curvature vector h is parallel in the normal bundle if, and only if, M n is either a minimal submanifold of R m or a minimal submanifold of a hypersphere of R m . Example 1. Let ϕ : Σ → S 6 be an almost complex curve, meaning that Lemma 8 . 8F is pseudo-umbilical if, and only if, Lemma 9 . 9F has parallel mean curvature vector in T F ⊥ if, and only if, f 2 × f −1 = af 0 × f −1 for some complex function a on Σ. Example 4 . 4Let W = span{e 1 , e 2 , e 3 , e 4 } and ϕ : C → S 3 = S 6 ∩ W be the the Clifford torus ϕ(x, y) = 1 √ 2 cos x e 1 + sin x e 2 + cos y e 3 + sin y e 4 . The corresponding immersion F : C → R 7 is given by F (x, y) = 1 2 (cos x sin y e 1 + sin x sin y e 2 − y e 3 + sin x cos y e 5 − cos x cos y e 6 + x e 7 ). with constant Kähler angle[3]. Next we will identify those totally real minimal surfaces ϕ in S 6 producing immersions F ± with parallel mean curvature. Theorem 6. Let ϕ : Σ → S 6 be a totally real minimal immersion. Then the conformal immersions F + = F + ϕ and F − = F − ϕ have parallel mean curvature vector field if, and only if, up to change of orientation of Σ, either ϕ = R v (β)φ, for some almost complex curveφ of type (III) and some rotation R v (β) of the form (6), or ϕ is totally geodesic.Proof. Recall that ϕ is totally real if, and only if,(the Kähler angle is π 2 ). Consequently, in view of (22), we have \|h ± \| = √ 22 andDifferentiating(24), we obtainSince the tangent bundle T C F ± is spanned by the isotropic sectionsFrom the properties of × and the properties of the harmonic sequence, we obtain (f 2 × f −1 ) ·s ± = 0. Hence a = 0 andOn the other hand, since ϕ is totally real, the sectionsform a moving unitary frame along ϕ (see[3], pg. 629). Using this, we can writewhere a 1 , a 2 , a 3 , a 4 are complex functions on Σ. By Theorem 4.2 of[3], these complex functions satisfyā 3 a 4 − a 3ā4 = 0. Differentiating (23) and applying property (P4), we see that f 2 · (f 0 × f −1 ) = 0, hence a 1 = 0. We also haveThenNow, assume that h ± is parallel. Equating(26)and(27),we see that b = ∓a 4 = −a 3 i. In particular, we can not haveā 3 a 4 − a 3ā4 = 0, unless a 3 = a 4 = 0. In this case,Hence f 2 × f −1 = 0, which means, by Lemma 10, that F is minimal in a hypersphere. 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[ "Unifying several separability conditions using the covariance matrix criterion", "Unifying several separability conditions using the covariance matrix criterion" ]	[ "O Gittsovich \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 256020InnsbruckAustria\n\nInstitut für Quantenoptik und Quanteninformation\nOsterreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 16020InnsbruckAustria\n", "O Gühne \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 256020InnsbruckAustria\n\nInstitut für Quantenoptik und Quanteninformation\nOsterreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 16020InnsbruckAustria\n", "P Hyllus \nInstitut für Theoretische Physik\nUniversität Hannover\nAppelstraße 230167HannoverGermany\n", "J Eisert \nBlackett Laboratory\nQOLS\nImperial College London\nPrince Consort RoadSW7 2BWLondonUK\n\nInstitute for Mathematical Sciences\nPrince's Gate\nImperial College London\nSW7 2PELondonUK\n" ]	[ "Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 256020InnsbruckAustria", "Institut für Quantenoptik und Quanteninformation\nOsterreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 16020InnsbruckAustria", "Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 256020InnsbruckAustria", "Institut für Quantenoptik und Quanteninformation\nOsterreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 16020InnsbruckAustria", "Institut für Theoretische Physik\nUniversität Hannover\nAppelstraße 230167HannoverGermany", "Blackett Laboratory\nQOLS\nImperial College London\nPrince Consort RoadSW7 2BWLondonUK", "Institute for Mathematical Sciences\nPrince's Gate\nImperial College London\nSW7 2PELondonUK" ]	[]	We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria which have been proposed to complement the positive partial transpose criterion -the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others -are in fact a corollary of the covariance matrix criterion.	10.1103/physreva.78.052319	[ "https://arxiv.org/pdf/0803.0757v3.pdf" ]	119,205,091	0803.0757	0e8df2e5ed6f3787f205bf7f67caade3f8f7f40c	Unifying several separability conditions using the covariance matrix criterion 22 Aug 2008 (Dated: August 22, 2008) O Gittsovich Institut für Theoretische Physik Universität Innsbruck Technikerstraße 256020InnsbruckAustria Institut für Quantenoptik und Quanteninformation Osterreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 16020InnsbruckAustria O Gühne Institut für Theoretische Physik Universität Innsbruck Technikerstraße 256020InnsbruckAustria Institut für Quantenoptik und Quanteninformation Osterreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 16020InnsbruckAustria P Hyllus Institut für Theoretische Physik Universität Hannover Appelstraße 230167HannoverGermany J Eisert Blackett Laboratory QOLS Imperial College London Prince Consort RoadSW7 2BWLondonUK Institute for Mathematical Sciences Prince's Gate Imperial College London SW7 2PELondonUK Unifying several separability conditions using the covariance matrix criterion 22 Aug 2008 (Dated: August 22, 2008)PACS numbers: 0367-a, 0365Ud We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria which have been proposed to complement the positive partial transpose criterion -the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others -are in fact a corollary of the covariance matrix criterion. We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria which have been proposed to complement the positive partial transpose criterion -the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others -are in fact a corollary of the covariance matrix criterion. I. INTRODUCTION Entanglement is the feature of quantum theory that renders it crucially different from a classical statistical theory. It also plays the central role in quantum information science, as a resource for information processing tasks. Consequently, a lot of effort has been made in the last decade to understand the meaning and the structure of entangled states [1,2]. One of the most elementary yet notorious questions is how to find good criteria to decide whether a state is entangled or classically correlated in the first place. Formally, one asks whether a given state ρ in a bipartite system is contained in the convex hull of product states and can hence be written as ρ = k p k \|a k a k \| ⊗ \|b k b k \|,(1) where the coefficients p k form a probability distribution. If so, all correlations can come from classical shared randomness, and a state is called classically correlated or separable. Otherwise, ρ is an entangled state. The decision problem of deciding whether a state is entangled or separable is known to be a computationally hard problem in the physical dimension, for a certain scaling of the error in the weak membership problem [3]. Yet, the problem one typically faces is the one where one has just a physical state given -having some fixed dimension -and one aims at finding criteria to make a judgment based on these criteria. Not surprisingly, given the central status of entanglement in quantum information theory, a lot of effort has been devoted to identifying such good and practical and computable criteria for separability in composite quantum states. Historically, the first criterion of this sort was derived from the observation that every separable state will have a positive partial transpose, and that the positivity of the latter can hence be used as an entanglement crite-rion (PPT criterion) [4,5]. This criterion later turned out to be necessary and sufficient for separability for low dimensional systems (2 × 2 and 2 × 3), whereas in higher dimensions this is no longer the case [6]. The PPT criterion is an example of a criterion based on positive maps: In fact, it has been proven that a state ρ is separable if and only if for any positive map Λ the operator (½⊗Λ)(ρ) is also positive [5]. Consequently, the systematic investigation of positive maps has led to a number of new separability criteria [7]. A quite remarkable criterion of this type is the reduction criterion, which is equivalent to the PPT criterion for 2× 2 and 2 × 3 cases and weaker for higher dimensions [8]. There are also other criteria that turned out to be directly related to the PPT criterion: The majorization criterion [9] and also entropic criteria [10] have been shown to be weaker than the PPT criterion [11,12]. Moreover, one can extend the PPT condition to a test based on a complete hierarchy of symmetric extensions, where each step constitutes a semidefinite program [13] (another complete family of semidefinite tests has been described in [14]). In such a hierarchy, every entangled state is necessarily detected as such in some step of the hierarchy. However, the steps in the hierarchy require more and more computational effort, which makes this approach difficult already for modest system sizes. Apart from these criteria that are directly related to the PPT criterion, a number of other separability criteria have been suggested where such a connection seemingly does not exist. The most prominent criterion of this type is the computable cross-norm or realignment criterion (CCNR) [15]. Other criteria in this category use the Bloch representation of density matrices [16,17], local uncertainty relations (LURs) [18], local orthogonal observables [19], or extensions of the realignment map [20,21]. These criteria are complementing the PPT criterion in an interesting way: In fact they detect some states as being entangled where the PPT criterion fails. They also do not rely on positive maps. At first sight, one might think that these criteria form a collection of quite beautiful, but strangely disconnected results. They have been derived using a variety of unrelated methods, and their connection often seems quite unclear. It is the main purpose of this work to develop a framework for the systematic understanding of all these latter approaches. In Ref. [22] we have proposed to investigate the separability problem using covariance matrices (CMs) of certain observables. In this context we have developed a separability criterion in terms of covariance matrices (covariance matrix criterion or CMC). We have shown that the CMC is, when augmented with appropriate local filtering, despite its simplicity a surprisingly strong entanglement criterion, which can detect states where the PPT criterion fails and which is at the same time necessary and sufficient for two qubits. Here we complete this approach and present new results in various directions. Specifically, we show that a number of criteria which have been proposed to improve the PPT criterion -namely the CCNR criterion [15], a criterion using the Bloch representation [16,17], the LURS [18], and recent criteria from Refs. [20,21] -follow directly from the CMC. In this way the CMC can be seen as complementary to the PPT criterion. We also tighten previous formulations of the CMC. We discuss several examples, and compare the performance of the criteria to instances of random states from a families of bound entangled states. This manuscript is organized as follows: In the second section, we introduce CMs and discuss their mathematical properties. Those readers who are mainly interested in separability criteria may only consume Definitions II.1 and II.2 and Propositions II.7, II.8 and II.12 and may then directly jump to Section III. In that Section, we introduce the CMC and evaluate it in several different ways. By doing this we establish the mentioned connection to the other separability criteria which will turn out to be corollaries of the former. In Section IV of the paper we will consider the connection between the CMC and the LURs. In the fifth section we will scrutinize the CMC for the two qubit case. In Section VI we will assess the strength of the mentioned criteria by considering a family of bound entangled states. We will then conclude and elaborate on possible extensions of the work presented here. Some more technical proofs of our theorems will finally be presented in the Appendix. II. COVARIANCE MATRICES In this section we will investigate covariance matrices as our main tool. In the first subsection we will introduce the different definitions of CMs [21,22,23] and fix our notation. In the second subsection we will address the question to which extent CMs can be used as a unique description of quantum states besides density matrices. Finally, in the third and fourth subsection we will men-tion and prove some useful properties of CMs, which will be used later in our study of entanglement. A. Definition of covariance matrices In what follows let ρ be a pure or mixed quantum state, described by a (positive) density operator in a ddimensional Hilbert space H and let {M k : k = 1, . . . , N } a suitable set of observables. Unless stated otherwise, we will always assume that these observables are orthonormal observables with respect to the Hilbert-Schmidt scalar product between observables, i.e., they fulfill tr(M i M j ) = δ i,j .(2) Furthermore, we will typically assume that the M i form a complete basis and span the whole observable algebra. This implies that there are N = d 2 different M i , and that any other observable can be expressed as a linear combination of the M i . As an example for such a set of observables for the case of a single qubit, one can consider the (appropriately normalized) Pauli matrices, M 1 = ½ √ 2 , M 2 = σ x √ 2 , M 3 = σ y √ 2 , M 4 = σ z √ 2 .(3) We can now formulate the main definitions for this work. Definition II.1 (Covariance matrix). The d 2 × d 2 co- variance matrix γ = γ(ρ, {M k }) and the d 2 × d 2 sym- metrized covariance matrix γ S = γ S (ρ, {M k }) are defined by their matrix entries as γ i,j = M i M j − M i M j ,(4)γ S i,j = M i M j + M j M i 2 − M i M j .(5) Sometimes, the difference between the linear part of a CM and the nonlinear part becomes relevant. Therefore, we define the linear part of γ as g i,j = M i M j and the linear part of the symmetric CM as g S i,j = M i M j + M j M i /2. We will often for simplicity of notation also write γ(ρ) or γ({M k }) instead of γ(ρ, {M k }), or simply γ. We will also sometimes indicate with respect to what state an expectation value is taken, so M i = M i ρ . It is straightforward to see that γ is a complex Hermitian matrix. The matrix γ S in turn is real and symmetric. Both γ and γ S are positive semidefinite, γ, γ S ≥ 0 [24]. Note finally that for odd d, there is another basis of orthonormal observables that can equally be used and that is commonly employed in the mathematical physics literature in the context of discrete Weyl systems [25]. Let A(0, 0) be the parity operator that maps P (0, 0) : \|x → \|−x , where \|x ∈ {\|0 , . . . , \|d−1 }, meant modulo d. Then, for (q, p) ∈ 2 d let P (q, p) = W (q, p)P (0, 0)W (q, p) † the translated versions of P (0, 0) in discrete phase space, where W (q, p) are the discrete Weyl operators [26]. The operators {M (q,p) } = {P (q, p) √ d} then form a set of Hilbert-Schmidt orthonormal Hermitian matrices. This is the standard set of observables when phase-space methods are made use of. B. Covariance matrices for bipartite systems In the focus of this work is the situation where the Hilbert space is a tensor product H = H A ⊗H B of Hilbert spaces of two subsystems A and B. We consider finitedimensional systems, and denote the dimension of H A (H B ) with d A (d B ),{M k } = {A k ⊗ ½, ½ ⊗ B k }.(7) Note that this set is not tomographically complete, since observables like A k ⊗ B l are missing. However, this set can be employed to define a very useful form of CMs. γ = A C C T B ,(8) where A = γ(ρ A , {A k }) and B = γ(ρ B , {B k }) are CMs of the reduced states of systems A and B, and C i,j = A i ⊗ B j − A i B j .(9) Similarly, we can define a symmetric block covariance matrix γ S ({M k }), for which A and B are the corresponding symmetric CMs, while C remains unchanged. C. Covariance matrices as description of quantum states Is it possible to completely reconstruct the state from a given CM? As our separability criterion uses the CM to decide separability, this question is important in order to understand, whether all states can be detected. We will discuss it in this subsection. Let i = N i=1 O i,j M j with some ma- trix O then γ({K k }) is given by γ({K k }) = Oγ({M k })O T .(10) Note that O is an orthogonal matrix if K i and M i are orthonormal bases. Proof: A direct calculation gives γ({K k }) i,j = l,m O i,l M l O j,m M m − O i,l M l O j,m M m = l,m O i,l γ({M k }) l,m O T m,j ,(11) which proves the claim. The main point is that the previous proposition allows us to choose the basis which we want to express our CM in arbitrarily, since we know how the CM will be transformed under a basis transformation in the space of observables. We can now come back to the initial question: Suppose we are given some CM with a fixed basis of observables. Are we able to reconstruct the physical state from this CM uniquely? We will start answering this question by considering a single system. Proposition II.4 (Characterization of states via non-symmetric covariance matrices). Given a nonsymmetric CM with tomographically complete set of observables, we can reconstruct the corresponding physical state unambiguously. Proof: We choose the following basis of the observables: D i = \|i i\|, i = 1, . . . , d,(12)X i,j = 1 √ 2 (\|i j\| + \|j i\|), 1 ≤ i < j ≤ d, (13) Y k,l = i √ 2 (\|k l\| − \|l k\|), 1 ≤ k < l ≤ d.(14) These observables form an orthonormal basis, and we will refer to this basis as to the standard basis later on. As in any basis M k , we can write the state as ρ = k M k M k , it suffices to know the first moments M k . From Eq. (4) one can see that γ i,j − γ j,i = [M i , M j ] . In the following we will show that in the chosen basis, all first moments can be obtained from expectation values of commutators. For the chosen standard basis we can explicitly calculate all commutators [D k , X k,l ] = i √ 2 Y k,l , [D k , Y k,l ] = − i √ 2 X k,l ,(15) [X k,l , Y k,l ] = i(\|k k\| − \|l l\|). Hence, all expectation values of the X i,j and Y k,l can be calculated. The same is true for the diagonal elements: Using the fact that the trace of the density matrix is equal to one, we can calculate all the diagonal elements from the mean values of [X k,l , Y k,l ]. Clearly, the same approach can be used for bipartite systems, if we use the CM in the full (and not in a block) form. In this case we can use a product basis {\|i 1 , i 2 }. Identifying (i 1 , i 2 ) =: i we can define the standard basis as above and find all first moments from the covariance matrix. As we have seen, the non-symmetric CM defined in Eq. (4) describes the physical state completely. The knowledge of the symmetric CM in Eq. (5) is, however, not enough: Proposition II.5 (Inequivalence of states and symmetric covariance matrices). The knowledge of the symmetric CM γ S does, in general, not determine the state ρ completely. Proof: We prove the claim by providing a counterexample. Let us take a single qubit. As observables we take the appropriate normalized Pauli matrices. The symmetric CM has the following entries γ S 0,j = ½σ j + σ j ½ 4 − ½ σ j 2 = 0 = γ S i,0 ,(17)γ S i,j = {σ i , σ j } 4 − σ i σ j 2 = δ i,j − σ i σ j 2 .(18) From this we can determine the norm of the mean value of the spin component in a certain direction, but not its sign. Hence we know the length of the Bloch vector of the system, up to some reflection to the origin, which corresponds to simultaneous change of signs of all σ i 's. One might think that the case of one qubit constitutes a special case. However, the same ambiguity will arise if one embedded a qubit in a higher dimensional, say, three level system. As it can be checked, the additional observables in the basis of observables {M k } will not provide any further information. To summarize: The knowledge of the symmetric CM of a qubit alone is not sufficient to decide between two alternatives of states which have opposite (symmetric to the origin) Bloch vectors. Also, merely the additional knowledge of a single bit (the sign) is needed to make this correspondence unambiguous. This, however, is specific to the qubit case. We will now turn to investigating the same question for the block CM defined in Eq. (8): Proposition II.6 (Relationship between bipartite states and block covariance matrices). For block CMs γ and γ S on a bipartite system, the following statements hold: (i) The (non-symmetric) block CM γ determines the bipartite state ρ AB completely. (ii) The symmetric block γ S does not determine ρ AB completely. Proof: Obviously, given a non-symmetric block CM for the set of variables A k ⊗ ½ and ½ ⊗ B l we can determine first all A k and B l for the reduced state ρ A in the same way as in Proposition II.4 from the blocks A and B of γ. Then, knowing the block C we can fix the rest A k ⊗ B l as A k ⊗ B l = C k,l + A k B l(19) and hence (i) is proved. The validity of (ii) is straightforward to see for two qubit states, as there will be the same lack of information on the mean values of observables as in Proposition II.5 and hence γ S AB does not provide the whole information about the state. The fact that the symmetric block CM γ S does not determine the state completely will later be important for the discussion of our entanglement criteria. Therefore, let us investigate this correspondence for the case of two qubits in some more detail. For that, let A i and B j be Pauli matrices. We may write the state in the form ρ AB = 1 4 i,j λ i,j σ A i ⊗ σ B j ,(20) where λ i,j = tr(ρσ A i ⊗ σ B j ). As one can see from Eq. (19) we have two possibilities of changing the λ i,j while keeping the C i,j invariant: We can (i) flip the signs of both of the Bloch vectors of the reduced density matrices, (λ 0,j and λ i,0 for i, j = 1, 2, 3), while keeping the left hand side of Eq. (19) invariant. Alternatively, we can (ii) flip the sign of only one of them, which implies that we also have to change the left hand side of Eq. (19). Concerning (i), one can directly calculate the transformed state ρ inv . It turns out that the eigenvalues of ρ and ρ inv are the same, suggesting that they are connected by a unitary transformation maybe in addition with a global transposition which transforms one state to the other. Actually the unitary transformation is a local unitary one, and one has the following transformation: ρ inv T = U † ρ AB U, U = 0 −1 1 0 ⊗ 0 1 −1 0 .(21) Since there is no physical process which corresponds to a transposition of a state, there are two physically different states ρ and ρ inv which give rise to the same covariance matrix and which are connected by the simultaneous flip of the Bloch vectors of their subsystems. Nevertheless we can see from Eq. whole covariance matrix will remain unaltered. This kind of transformation is done by σ A i → − σ A i , σ A i ⊗ σ B j → σ A i ⊗ σ B j − 2 σ A i σ B j ,(22) resulting in a transformation of ρ to a different ρ inv . Such a change of the state is nontrivial and can give rise to a matrix ρ inv with negative eigenvalues, which clearly does not correspond to any state. Two more cases that should be discussed. As one can see from a numerical search, there are some states ρ, for which ρ inv is still a state and ρ and ρ inv are either both separable or both entangled. But there exist also states which alter their separability properties after a Bloch vector inversion. As an example of states where ρ and ρ inv have different separability properties, consider the states of the form ρ ε = ε 2    1 + r 0 0 t 0 0 0 0 0 0 s − r 0 t 0 0 1 − s    + (1 − ε)    0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0    . (23) ρ ε is a slight modification of the family of the states introduced in Ref. [27] and which are known to be detected by PPT but not by CCNR criterion for certain parameters. The inverted form ρ inv ε of this states can be calculated analytically. The states ρ ε are known to be PPT for (t = 0, ε = 1). Going away from ε = 1 and changing other parameters one can find regions where ρ ε and ρ inv ε have different entanglement properties. As we can see from Fig. 1 there are three different regions corresponding to the different physical situations. The most interesting region is the "Different" region where entanglement properties of the inverted state are different from that of the initial one. This means that any separability criterion which uses only the symmetric CM will not detect these states, as the symmetric CM is compatible with a separable as well as with an entangled state. These states will not be detected by the CMC, and also not by a variety of other criteria, as we will see later. D. Properties of covariance matrices In this section we will prove several properties of CMs which are important for our later discussion. This concerns mainly properties of CMs for pure states and the behavior of CMs under the mixing of states. We will first show in the subsequent proposition that a suitable choice of observables can dramatically simplify the form of CM γ for pure states. Proposition II.7 (Covariance matrices of pure states). Let G i be a tomographically complete set of observables of a d-dimensional system. If ρ is a pure state then γ (as a d 2 × d 2 matrix) fulfills: (i) The rank is given by Rank(γ) = d − 1. (ii) The nonzero eigenvalues of γ are equal to 1, hence tr(γ) = d − 1. (iii) Consequently, we have γ 2 = γ. Proof: Without any loss of generality we assume ρ = \|1 1\| and take as observables the ones of the standard basis (14). Calculating directly and reordering of the matrix elements afterwards gives a block structure [28] γ = d−1 k=1 B k Ç d 2 −2d+2 with B k = 1/2 i/2 −i/2 1/2 ,(24) where Ç k denotes a k × k matrix of zeros. This matrix has the desired properties. From this we can directly read off the properties of the symmetric form of the covariance matrix: Corollary II.8 (Properties of symmetric CMs for pure states). Let {G i } be a tomographic complete set of observables of a d-dimensional quantum system. If ρ is a pure state, then γ S (as d 2 × d 2 symmetric matrix) fulfills: (i) The rank is given by Rank(γ S ) = 2(d − 1). (ii) The nonzero eigenvalues of γ S are equal to 1/2, hence tr(γ) = d − 1. We now turn to a proposition concerning the trace of a CM for mixed states. Proposition II.9 (Trace of CMs). Let ρ be a mixed state. Then tr[γ(ρ)] = d − tr(ρ 2 ) (25) which implies that d − 1/d ≥ tr(γ(ρ)) ≥ d − 1. This holds also for γ S . Proof: By definition tr(γ) = i γ i,i = i δ 2 (M i ) = i ( M 2 i − M i 2 ) . The first summation is trivial, since we have k M 2 k = d½ [37]. Furthermore we can write ρ = k M k M k which implies that k M k 2 = tr(ρ 2 ), and further 1/d ≤ tr(ρ 2 ) ≤ 1. The statement for γ S follows directly from the fact that tr(γ) = tr(γ S ). We can also estimate the operator norm (i.e., the maximal eigenvalue) of CMs. Proposition II.10 (Operator norm of CMs). For the CM γ(ρ) and its linear part g(ρ) the operator norm is bounded by g(ρ) ≤ ρ and γ(ρ) ≤ ρ .(26) The same bounds hold for symmetric CMs. Proof: Let us first consider g(ρ). We have g(ρ) = max \|x x\|g(ρ)\|x = x 0 \|g(ρ)\|x 0 = tr(ρAA † ) with tr(AA † ) = 1. This is clearly smaller than ρ . For γ(̺) this follows then from AA † − A A † ≤ AA † . Finally, CMs also satisfy an interesting majorization relation. This has its root in the way how one can relate CMs to the rotated CMs of the pure states occurring in their convex decompositions in terms of pure states. Proposition II.11 (Majorization relation for CMs). For any (mixed) state ρ, both the linear part g(ρ) with entries g i,j = M i M j as well as the CM γ(ρ) satisfy k j=1 λ j [g(ρ)], k j=1 λ j [γ(ρ)] ≤ min(k, d − δ γ 1 d ),(27) for the non-increasingly ordered eigenvalues, where δ γ = 1 for γ(ρ) δ γ = 0 if g(ρ) is considered. Proof: This is a consequence of γ(ρ) , g(ρ) ≤ 1 as well as of tr[γ(ρ)] ≤ d − 1 d and tr(g(ρ)) ≤ d. E. Mixing property of covariance matrices Separable states are those states that can be written as convex combinations of product states. Therefore we have to understand the behavior of CMs under mixing of states for the derivation of separability criteria. An important property of covariance matrices which we refer to as concavity property is the following: Proposition II.12 (Concavity property). Let ρ = k p k ρ k be a convex combination of states ρ k , then γ(ρ) ≥ k p k γ(ρ k ).(28) Clearly, this implies the same relation for the symmetrized CM γ S . Proof: As shown in Ref. [29] this inequality holds for an arbitrary symmetric CM γ S . Moreover, since M i M j ρ = k p k M i M j ρ k for all i, j, we have for the non-linear part that − M i ρ M j ρ ≥ − k p k M i ρ k M j ρ k(29) as a matrix inequality for the matrices X i,j = − M i ρ M j ρ and Y i,j = − k p k M i ρ k M j ρ k . From this, the above inequality follows for the non-symmetric CM γ. This property will later be used to derive the separability criterion. F. Transformations of observables and validity of covariance matrices Transformations as generated by a general orthogonal matrix O used in Proposition (II.3) do in general not preserve the positivity of the state ρ (see Ref. [19] for discussion). Only a subgroup will correspond to unitary transformations on the level of states. Here, we will clarify how unitary transformations of the state are reflected by orthogonal transformations on the level of CMs. For this aim, let us consider the case that ρ is transformed by some unitary transformation ρ → U † ρU . Equivalently, we can transform the operator basis, denoted as {G i }, as G i → H i = U G i U † = j O i,j G j .(30) It is then easy to see that the transformation of the CM is γ(ρ) → Oγ(ρ)O T = γ(U † ρU ),(31) We can now ask in what way O depends on U , and which orthogonal O ∈ O(d 2 ) correspond to a unitary U ∈ U (d) acting in state space as described above. That is, we look for the group representation of U (d) in the space of CMs (compare also the metaplectic representation of symplectic transformations in discrete Weyl systems, see Ref. [25]). The following theorem gives an answer to this question. Proposition II.13 (Transformation laws for CMs). Let U ∈ U (d). Then the O ∈ O(d 2 ) representing U as described above (30) is given by O = Γ T (U T ⊗ U † )Γ * ,(32) where Γ is a d 2 ×d 2 square matrix constructed as Γ α,β\|i = G α,β i = (\|G 1 , \|G 2 , . . . ), where we understand α, β as a row index, G α,β i as vectors and construct Γ α,β\|i from them. Proof: The proof is given in the Appendix. It is an interesting open question to see how CMs are transformed under general completely positive maps, ρ → i A i ρA † i , where {A i } are Kraus operators, directly expressed in terms of the Kraus operators. At the very beginning we have given two definitions of covariance matrices for the symmetric and nonsymmetric case (4,5). We will discuss this difference also later in the paper. However at this stage we mention a single connection between these two definitions for block CMs: Proof: The proof is given in the Appendix. III. THE COVARIANCE MATRIX CRITERION FOR SEPARABILITY In this section, we introduce the covariance matrix criterion (CMC) for separability as our main topic of this paper. This is derived from the concavity property for CMs and the fact that the block CM for product states is block diagonal. spaces are d A (d B , respectively). Define M i = A i ⊗ ½ for i = 1, . . . , d 2 A and M i = ½⊗B i for i = d 2 A +1, . . . , d 2 B +d 2 A . Then there exist pure states \|ψ k ψ k \| for A and \|φ k φ k \| for B and convex weights p k such that if we define κ A = k p k γ(\|ψ k ψ k \|) and κ B = k p k γ(\|φ k φ k \|) the inequality γ S (ρ, {M i }) ≥ κ A ⊕ κ B ⇔ A C C T B ≥ κ A 0 0 κ B(33) holds. This means that the difference between left and right hand side must be positive-semidefinite. If there are no such κ A,B then the state ρ must be entangled. Proof: First note that for this special choice of M i , for any product state γ(ρ A ⊗ ρ B , {M i }) = γ(ρ A , {A i }) ⊕ γ(ρ B , {B i }) (34) holds. Now, since any separable state can be written as ρ = k p k \|ψ k ψ k \| ⊗ \|φ k φ k \|, we can apply Prop. (II.12) and arrive at the conclusion. Note that the CMC is manifestly invariant under a change of the observables {A k } and {B k }, as we know from Proposition II.3 [see also Eq. (97)]; however, a suitable choice of them may simplify the evaluation a lot. Also, note that we have formulated the CMC for symmetric CMs, we will later discuss the case of non-symmetric CMs. Obviously, as such, as formulated as in Prop. III.1, it is not clear that the CMC leads to an efficient and physically plausible test for separability: The main problem is to characterize the possible κ A and κ B . As such, the formulation still contains an optimization over all pure product states. We will refer to an "evaluation of the CMC" hence whenever we can identify a property of κ A,B that will render the above criterion an efficient test. Some properties of we have derived above, notably tr(κ A ) = d A − 1(35) (see Proposition II.8), which we will use subsequently. We will now turn to feasible ways to evaluate the CMC. As a first step, we have to derive conditions on the blocks of a block matrix as in Eq. (33), which follow from the positivity condition in Eq. (33). Then, we ask how the observables {A k } and {B k } must be chosen in order to make a violation of Eq. (33) manifest. Note the formal similarity of the condition γ ≥ κ A ⊕κ B to tests for separability for Gaussian states for systems with canonical coordinates. A CM in that context [30,31] is any symmetric matrix satisfying γ + iσ ≥ 0, where σ = n k=1 0 1 −1 0 .(36) A Gaussian state ρ with CM γ is now separable if and only if [32] there exist covariance matrices γ A , γ B , each satisfying γ A , γ B + iσ such that γ ≥ γ A ⊕ γ B .(37) For non-Gaussian states, a violation of this condition is still sufficient to detect entanglement. Eq. (37), in turn, is simply a semi-definite program (SDP), so it can be efficiently decided [33]. The characterization of the right hand side is here an easy task, as these matrices are again constrained by a semi-definite constraint. We will see in section VI that for two qubits the CMC can be solved in a similar way as in the continuous variable case [34]. IV. EVALUATION OF THE CMC We would like to follow two strategies to evaluate the CMC as presented in the previous section. Both strategies are based on matrix invariants such as eigenvalues or singular values. Let us first characterize positive semidefiniteness of a block matrix of the type as in (33) in terms of singular values of its submatrices. A. Evaluation of the CMC via singular values of submatrices As a start, we state the following lemma: Lemma IV.1 (Block covariance matrices and unitarily invariant norms). If a positive matrix partitioned in block form is positive semi-definite, A C C T B ≥ 0,(38) then Proof: The proof of this statement is actually a corollary of Theorem 3.5.15 of Ref. [35]. It is shown that \|\|\|A\|\|\| \|\|\|B\|\|\| ≥ \|\|\| \|C\| \|\|\| 2 ,(39)\|\|\|A p \|\|\| \|\|\|B p \|\|\| ≥ \|\|\|(C † C) p/2 \|\|\| 2 ,(40) for any p > 0 and any unitarily invariant norm. For p = 1 this is the result we are interested in. We will nevertheless present an alternative proof of this statement for Ky-Fan norms A KF B KF ≥ C 2 KF , in the Appendix, as the proof of Proposition IV.9 will make use of this proof. Using the last Lemma we have: Proposition IV.2 (CMC evaluated using singular values). Let γ = A C C T B(41) be a CM. Then, if ρ is separable, we have C 2 T r ≤ 1 − tr ρ 2 A 1 − tr ρ 2 B .(42) If this inequality is violated, then ρ must be entangled. Proof: We prove the claim applying the formula (39) directly, yielding C 2 T r ≤ A − κ A T r B − κ B T r(43) Since A − κ A as well as B − κ B are Hermitian positive semi-definite matrices (due to concavity property of CMs) their trace norm will coincide with their trace. Hence A − κ A T r = tr(A) − tr(κ A ) = 1 − tr(ρ 2 A ), where we have used Corollary II.9 and the fact that i A i,i = i ( A 2 i − A i 2 ) = d A ½ − tr(ρ 2 A ), since tr(A i A j ) = δ i,j and ρ 2 A = i,j A i A j A i A j . Interestingly, this criterion has been proven already in a different context: Remark IV.3 (CMC and the criterion of Ref. [21]). The separability criterion in Proposition IV.2 is nothing but the separability criterion proposed in Theorem 1 of Ref. [21], hence the criterion of Ref. [21] is a corollary of the CMC. Let us now connect the CMC to another type of entanglement criteria: There are several separability criteria in the literature which are based on the Bloch representation of density matrices. This representation in our case is just some particular choice of observables, namely one has to detach the identity from all others generators, which then have to be traceless. The fact that one of the observables is the identity, can simplify the CMC sometimes. By definition the entries of the matrix C are given by C i,j = A i ⊗ B j − A i B j ,(44) which consists of a linear (in the sense of mean values) and quadratic part. We define C as the linear part of C, i.e., C i,j = A i ⊗ B j . Let us further consider C red as the submatrix of C, where the entries ½ A ⊗B j and A i ⊗½ B are omitted, i.e., the first row and the first column are removed. Similarly, we can define matrices like A and B red from A and B. In the same spirit, we can define a submatrix of κ as κ red . Note that tr(κ) = tr(κ red ), as the missing diagonal entry is the variance of 1 1, which is vanishing. We are now able to establish a connection between the CMC and criteria based on the Bloch representation of density matrices: Proposition IV.4 (Relationship between CMC and criteria based on Bloch representations). Let γ = A C C T B(45) be a CM. Then if ρ is separable, we have C red 2 T r ≤ 1 − 1 d A 1 − 1 d B .(46) If this inequality is violated, then ρ must be entangled. Proof: First, we can define two vectors \|ψ A/B with entries \|ψ A i = A i \|ψ B i = B i(47) resulting in C = C − \|ψ A ψ B \|. Similar relations hold for A and B, so we can write the condition in CMC (33) in the form A − κ A C C T B − κ B X − \|ψ A \|ψ B ψ A \| ψ B \| T ≥ 0.(48) Positivity of the left hand side implies positivity of the first term X alone, since we subtract only one projector which is itself positive. Concerning positivity of X we can take A red , B red , C red , and κ red instead, since positivity of a matrix implies positivity of all its main minors. Using Eq. (39), we get C red T r ≤ A red − κ red A T r B red − κ red B T r .(49) Using that tr(Â red ) = i≥2 A 2 i = d A ½ A − ½ A /d A and tr(κ red A ) = d A − 1 proves the claim. Interestingly, this separability criterion has also been proven before: Remark IV.5 (CMC and the criterion of Ref. [16]). The separability criterion in Proposition IV.4 is nothing but the separability criterion for Bloch representations proposed in Ref. [16], hence the criterion of Ref. [16] is a corollary of the CMC. Note that in Ref. [16] the observables have been normalized in a different way, leading to a slightly different formula. Remark IV.6 (Connection between Propositions IV.4 and IV.2). Proposition IV.2 is strictly stronger than Proposition IV. 4. This fact was proven in version 5 of [21], which came out after submission of our paper to the arXiv. λ j (κ A ) ≤ min(k, d A − 1),(50) for the non-increasingly ordered eigenvalues of κ A (and κ B ). Proof: One can argue as in Proposition II.11, using the fact that a convex combination of matrices leads to more mixed matrices in the sense of majorization [35]. This property can immediately be applied to evaluate the CMC, making use of proposition IV.1 and Weyl's inequalities [36]. For example, if we consider d A = d B = d and the k-Ky-Fan norm . KF (k) for k = (d 2 −d+1+s), we can apply the first of Weyl's inequalities with i = 1, and s = 1, . . . , d − 1, to conclude that A − κ A KF (k) = k j=1 λ j (A − κ A ) ≤ k j=1 λ 1 (A) + k j=1 λ j (−κ A ) = (d 2 − d + 1 + s) A − d 2 j=d 2 −k+1 λ j (κ A ),(51) where A denotes the spectral norm of A. Using that κ A will be more mixed in the sense of majorization than diag(1, . . . , 1, 0, . . . , 0) of rank d − 1 and Proposition II.9, one arrives at A − κ A KF (k) ≤ (d 2 − d + 1 + s) A − s,(52) and a corresponding statement for κ B . Using Proposition IV.1, one hence arrives at the observation that any separable state ρ on a bipartite Hilbert space satisfies (d 2 − d + 1 + s) A − s (d 2 − d + 1 + s) B − s − C 2 KF (k) ≥ 0.(53) It is an interesting open question whether more sophisticated uses of the knowledge of spectral properties of κ A and κ B can be employed the further sharpen the evaluation of the CMC. B. Evaluation of the CMC via traces of submatrices Let us first prove a simple condition on the traces of A, B and C, which follows from the CMC. In the following, we always assume that d A ≤ d B . Sometimes we assume that the dimensions are the same, meaning that d = d A = d B . Proposition IV.8 (CMC evaluated from traces). Let γ = A C C T B(54) be the symmetric CM of a state ρ and let J = {j 1 , . . . , j d 2 A } ⊂ {1, . . . , d 2 B } be a subset of d 2 A pairwise different indices. Then if ρ is separable, we have 2· d 2 A i=1 j∈J \|C i,j \| ≤ d 2 A i=1 A i,i − d A + 1 + (55) + d 2 B i=1 B i,i − d B + 1 = 1 − tr(ρ 2 A ) + 1 − tr(ρ 2 B ) , If this inequality is violated, then ρ must be entangled. Proof: First, note that a necessary condition for a 2×2 matrix X = a c c b(56) to be positive semidefinite is that 2\|c\| ≤ a + b. If ρ is separable, then by the CMC we have Y = γ−κ a ⊕κ B ≥ 0. This implies that all 2 × 2 minor submatrices of Y have to be positive semidefinite as well. Hence for all i, j we have 2\|C i,j \| ≤ A i,i + B j,j − (κ A ) i,i − (κ B ) j,j .(57) Summing over i, j and using Corollary II.9 proves the claim. We will use this Proposition mainly for the case that d A = d B and where we sum over the diagonal entries of C. In this case, it just gives the condition that for separable states, 2tr(C) ≤ 2 − tr(ρ 2 A ) − tr(ρ 2 B ). This is a quadratic polynomial in the entries of the state, and may be viewed as a suitable entanglement witness on two specimens on a state. In the light of this fact, the criterion evaluated in this fashion is surprisingly strong. It is also worth mentioning here that one can improve Proposition IV.8 by taking 4 × 4 minor submatrices for evaluation. Then, however, also off diagonal terms of κ A/B will occur, for which not many properties are known. This makes the resulting conditions difficult to evaluate. Physically, Proposition IV.8 says that if the correlations C i,j are sufficiently large, then ρ must be entangled. The question arises, how to find the observables, for which the C i,j are large. There are several ways of doing this. A first result is the following: Proposition IV.9 (Criterion in Proposition IV.8 and diagonal block matrices). The criterion in Proposition IV.8 detects most states if the observables are chosen in such a way that C is diagonal. For any state there exist a choice of observables that this can be achieved. However, even with this optimal choice of observables Proposition IV.8 delivers a strictly weaker separability criterion than Proposition IV.2. Proof: It is clear that the criterion is optimal, if the trace of C is maximal, which is the case if it is brought into the singular value diagonal form [21,22]. This can always be achieved [see Eq. (97)]. The fact that Proposition IV.2 is stronger, was in a different language proven in Ref. [20]. Interestingly, the fact that Proposition IV.8 is weaker than Proposition IV.2 can also be seen from Eq. (100) from the alternative proof of Lemma IV.1. If C is chosen to be diagonal, then Proposition IV.8 reduces to this equation with α = β. Clearly, allowing α and β to be different, improves the criterion. In the following, however, we will consider two different strategies: Firstly, we use the Schmidt decomposition in operator space of the density matrix [37]. This will lead to a natural choice of the observables {A k } and {B k }, and will further connect the CMC to the CCNR criterion. Secondly, we will consider appropriate local filterings of the state [38,39,40,41]. These are active transformations of the state, which, however, do not change the entanglement properties. Under this transformations, the state can be transformed into what is called its standard form in Ref. [38]. In this standard form, the CMC becomes very strong and even necessary and sufficient for two qubits. C. Schmidt decomposition and the CMC We will first remind ourselves of what is called the Schmidt decomposition in operator space. It is the same construction as the ordinary Schmidt decomposition in the vector space now equipped with the Hilbert-Schmidt scalar product. A general density matrix of a composite system can be written as ρ = d 2 A k=1 d 2 B l=1 ξ k,lG A k ⊗G B l ,(58) with real ξ k,l and the {G A l } (respectively, {G B l }) form an orthonormal Hermitian basis of observables. The Schmidt decomposition can now be achieved by diagonalizing the above expression using the singular value decomposition of the matrix ξ, ρ = d 2 A k=1 λ k G A k ⊗ G B k ,(59) where we made the assumption that d A ≤ d B . Clearly, the Schmidt coefficients λ k are real and non-negative. Using the new basis observables {G A k } and {G B k } as observables for the construction of the symmetric block CM, we have a normal form of the CMC, which we will call the Schmidt CMC. Proposition IV.10 (Schmidt CMC). If ρ is separable, then 2 i \|λ i − λ 2 i g A i g B i \| ≤ 2 − i λ 2 i (g A i ) 2 + (g B i ) 2 ,(60) where we defined g A i = tr(G A i ) and g B i = tr(G B i ). If this condition is violated, the state must be entangled. Proof: Using the orthonormality of the {G A/B i }, it is not difficult to see that with the observables from the Schmidt decomposition C i,j = λ i δ i,j − λ i λ j g A j g B i holds. In addition, we have tr(ρ 2 A ) = i λ 2 i (g B i ) 2 . Together with Proposition IV.8 this proves the claim. Corollary IV.11 (CMC and CCNR). If a state ρ is separable, then in the Schmidt decomposition k λ k ≤ 1 (61) has to hold. This condition is just the CCNR criterion, hence the CCNR criterion is a corollary of the CMC. Proof: Using the relations \|a − b\| ≥ \|a\| − \|b\| and a 2 + b 2 ≥ 2\|ab\| we have 2 i \|λ i − λ 2 i g A i g B i \| ≥ 2 i λ i − 2 i λ 2 i \|g A i g B i \| and 2 − i λ 2 i [(g A i ) 2 + (g B i ) 2 ] ≤ 2(1 − i λ 2 i \|g A i g B i \|) , which, due to the Proposition IV.10, proves the claim. D. Filtering and the CMC Let us now consider local filtering operations or SLOCC (stochastic local operations assisted by classical communication) [41] of the form ρ →ρ = (F A ⊗ F B )ρ(F A ⊗ F B ) † ,(62) where and F A ∈ SL(d A , ) and F B ∈ SL(d B , ) are invertible matrices on the respective Hilbert spaces. Clearly, such operations cannot map a separable state into an entangled one (although they might increase entanglement measures). Also, since F A and F B are invertible, they will also not destroy any entanglement that may be present in the state. In other words, ρ is entangled if and only ifρ is entangled. As has been shown in Refs. [38,41] we can bring any state of full rank (i.e., ρ > 0) by such filtering operations in its standard form which is given bỹ ρ = 1 d A d B ½ + d 2 A −1 k=1 ξ kĜ A k ⊗Ĝ B k (63) where the {Ĝ A k }, {Ĝ B k } are traceless orthogonal observ- ables. Here, we again assumed that d A ≤ d B . The idea now is to first apply a filtering operation and bring the state into its normal form. Then, the new separability criteria are applied afterwards. Note that the reduction to the normal form is always possible. The merits of this approach are twofold: Firstly, the normal form reduces the number of relevant parameters, while still encoding all information about entanglement and separability. Secondly, the normal form is in a certain sense "more entangled" than the original state, as it was shown in Ref. [40]: Remark IV.12 (Extremality of states in normal form). The local filtering operations bringing a mixed state into its normal form are those operations which maximize all entanglement monotones that remain invariant under determinant 1 SLOCC operations. Therefore, it may be expected that many separability criteria become stronger if we first bring the state into its normal form. Note, however, that this does not hold for the PPT criterion, as local filtering leaves this criterion invariant. Following Ref. [38], let us explain briefly an algorithm for transforming a state of a form as in Eq. (58) to its normal form in Eq. (63). As a starting point, one considers the compact space D A ⊗D B of all normalized product density matrices ρ A ⊗ ρ B . For any given density matrix ρ one can define a function f of ρ A and ρ B via Minimization of the function f ρ will, as proven in Ref. [38], yield the filtering operations needed. Suppose the minimum value for f ρ attained for some product density matrix τ A ⊗ τ B with det τ A > 0, det τ B > 0. Each of them can be decomposed as (see Eq. (66) in Ref. [38]) f ρ (ρ A , ρ B ) = tr [ρ(ρ A ⊗ ρ B )] (det ρ A ) 1/dA (det ρ B ) 1/dB . (64) f ρ (ρ A , ρ B )τ A = T † A T A , τ B = T † B T B , T A/B ∈ SL(d A/B , C) (65) where the T A and T B are desired local filtering operations. Normalization factors have been ignored. Using this filtering operations one obtains the new stateρ which has a form ρ = 1 d A d B ½ + d 2 A −1 i=1 d 2 B −1 k=1 ξ ikĜ A i ⊗Ĝ B k (66) The final step involves a standard singular value decomposition of ξ ik , which leads to Eq. (63). A priori, it is not clear whether the normal form is in some sense unique or not. However, it is easy to see that if we start from a given state and convert it into two different states in a normal form, then these two normal forms have to be connected by a local filtering operation. Using the fact that the reduced states of a state in the normal form are maximally mixed, one can further conclude that two different normal forms can only differ by a local unitary transformation. In practice, the minimization of f ρ (ρ A , ρ B ) in Eq. (64) can be performed by an iteration as follows: let us fix ρ B and consider only the minimization over ρ A . This minimization can further be split into a minimization over the spectrum of ρ A and a local unitary transformation. If the spectrum is fixed, the optimal unitary is constructed such that ρ A and X = tr B [ρ(1 1 ⊗ ρ B )] are diagonal in the same basis where the maximal eigenvalue of X has the same eigenvector as the minimal eigenvalue of ρ A and the second largest eigenvalue of X has the same eigenvector as the second smallest eigenvalue of ρ A etc. If the basis is fixed, and λ k (µ k ) are the eigenvalues of ρ A (X) then a simple calculation using Lagrange multipliers shows that the optimal λ k fulfill λ k ∼ ( i =k µ i λ i )/( i =k λ i ) 1/2 ,(67) which can be used for an iterative determination of the optimal λ k . In this way, the optimization can be iterated, converging to a minimum. Note while it is known that every state can be brought into this normal form, the above procedure of Ref. [38] is not known to be strictly efficient in the physical dimension d. Yet, for "reasonable physical dimensions", the method in practice converges quickly. Moreover, and importantly, at the end of the procedure, one can easily (and efficiently) check via direct inspection whether the obtained filters map the state onto the normal form or not. Global optimality can hence be easily certified. As one can directly calculate, for a state in the normal form the CM takes a really simple form, namely γ = 1 d A d B diag(0, d B , d B , . . . ) diag(0, ξ 1 , ξ 2 , . . . ) diag(0, ξ 1 , ξ 2 , . . . ) diag(0, d A , d A , . . . ) . (68) Using this form we obtain the following strong separability criterion, which we call the filter CMC. Interestingly, for two qubits we have: Remark IV.14 (Filter CMC for two qubits). For two qubits, the filter CMC in Proposition IV.13 is a necessary and sufficient criterion for separability. Proof: If a two-qubit state is of full rank, the normal form readsρ = 1 4 ½ + 3 k=1 ξ k σ A k ⊗ σ B k ,(70) where {σ A/B k } are the Pauli matrices [38]. Such states are diagonal in the Bell basis, and it is known that for these states 3 k=1 ξ k ≤ 2 is necessary and sufficient for separability [38,46]. Note also that the filter normal form can be explicitly stated for two-qubit systems. If an entangled (or separable) state is not of full rank, it can, as explicitly shown in Ref. [39], be brought by filtering operations arbitrarily close to a Bell diagonal state with finite (or vanishing) concurrence. Such a state will also be detected by the CMC (or not). Direct comparison of this result with the discussion in Section II and Fig. 1 (and later the result of Proposition V.2) might be confusing at this point, since we know already that the CMC itself cannot be necessary and sufficient for two qubits. This can be resolved in the following way: We have already learned that filtering brings the state in the form which in a certain sense contains the maximum amount of entanglement (it maximizes all monotones). This indeed shows that the filter CMC is sometimes a real improvement of the "bare" CMC, and filtering is more than just an appropriate choice of the observables. Let us now consider the asymmetric case, when d A < d B . We can formulate for this case following statement: Proposition IV.15 (Separability criterion for uneven local dimension). If ρ is separable, then the following inequalities hold i ξ i ≤ d A d B 2 1 − 1 d A + (d 2 A − 1) 1 d B + min(0, −(d B − 1) + (d 2 B − d 2 A ) 1 d B ) (71) and i ξ i ≤ [d A d B (d A − 1)(d B − 1)] 1/2 .(72) holds. If one of these inequalities is violated, the state must be entangled. Proof: Eq. (72) is nothing but an application of Proposition IV.4 (or IV.2), it has already been derived in Ref. [16]. Concerning Eq. (71), we will again apply Proposition IV.8, but with two modifications. First, when carrying out the sum over 2\|C i,j \| ≤ A i,i + B j,j − κ A,i,i − κ B,j,j [see Eq. (57) in Proposition IV.8] we do not sum over all B i,i . But then, we cannot subtract all of the κ B,i,i anymore, since d 2 B − d 2 A diagonal elements of κ B do not occur in the sum. As a first approach, we can drop completely the summation over all κ B,j,j , since they are positive anyway. This gives 2 d A d B d 2 A i=1 ξ i ≤ 1 − 1 d A + d 2 A − 1 d B ,(73) justifying one part of Eq. (71). In a second approach, we estimate d 2 A i=1 κ B,i,i . As one can see by direct inspection, the non-vanishing elements of γ in Eq. (68) origin only from the linear part of CM (in the spirit of Proposition IV.4 this linear part is denoted by g). But as we have seen in the proof of Proposition II.12 that this linear part g is just the same as the linear part of the direct sum of κ A ⊕ κ B (denoted by k A ⊕ k B ) for separable states, i.e. g = k A ⊕ k B , hence B = B = k B . This implies that for the diagonal elements of κ B the relation κ B,i,i ≤ B i,i = B i,i = 1/d B holds, leading to d 2 A i=1 κ B,i,i = d B − 1 − d 2 B i=d 2 A +1 κ B,i,i ≥ d B − 1 − d 2 B − d 2 A 1 d B .(74) This proves the second part of Eq. (71). V. CONNECTION TO LOCAL UNCERTAINTY RELATIONS In this section we will further analyze the connection of CMC with the separability criterion based on local uncertainty relations (LURs) [18]. To start with, we again state the LUR criterion as a reminder: Proposition V.1 (Criterion based on local uncertainty relations). Let beÂ k andB k observables in system A and B, respectively, for which some of the variances on single systems is bounded by constants U A , U B such that k δ 2 (Â k ) ≥ U A and k δ 2 (B k ) ≥ U B .(75) Then, we have for separable states k δ 2 (Â k ⊗ ½ + ½ ⊗B k ) ≥ U A + U B(76) and violation implies the presence of entanglement. Physically, the LURs state that separable states inherit the uncertainty relations from their reduced states, which is not the case for entangled states. Due to this observation the LURs have attracted a considerable interest, and a number of interesting properties have been discovered: LURs can detect bound entangled states [42] and can be used to estimate the concurrence [43]. They can be extended to other formulations of the uncertainty principle [44,45] and they can be generalized to nonlocal observables [29]. Finally, they can be viewed as nonlinear entanglement witnesses, which improve the CCNR criterion [37]. For the connection to the CMC we have the following: Proposition V.2 (Connection to local uncertainty relations). A state ρ violates the CMC for symmetric CMs iff it can be detected by a LUR. Proof: The proof is given in the Appendix. This result show that the LURs for appropriate observables and the CMC are equivalent, however, the CMC has the major advantage that it can be directly evaluated, while for the LURs the appropriate observables have to be identified. Moreover, we can state: Corollary V.3 (Insufficiency of LUR to detect all entangled states). There exist entangled two qubit states which can not be detected by a LUR, hence LURs are not a necessary and sufficient criterion for separability. Proof: In the Section II C we have already constructed a family of states ρ ε which cannot be detected by the CMC, as their symmetric block CM is compatible with a separable as well as an entangled state. This proves the claim. VI. THE CMC FOR TWO QUBITS After the previous discussion of the situation of Hilbert spaces of arbitrary finite dimension, we now turn to the important simple case of a 2 × 2-system -two qubits -in some more detail. We take as observables the set {A k } = {B k } = {½/ √ 2, σ x / √ 2, σ y / √ 2, σ z / √ 2} as in Eq. (3) . Since these observables contain the identity, one can easily check that many terms in the symmetric block CM vanish. Effectively, γ is actually a 6 × 6 matrix (denoted by γ eff ) originating only from the {A k } and {B k } with k = 1, 2, 3 which are not proportional to the identity, and not by an 8 × 8 as one could guess from the general theory. To characterize the κ A in the CMC, note that for a pure state \|a on system A we find, according to Proposition II.8, the following properties of the 4 × 4 matrix γ(\|a a\|): (i) Rank(γ) = 2. (ii) The nonzero eigenvalues of γ are equal to 1/2 in a suitable basis. We also know that in the chosen basis, the first row as well as the first column of γ(\|a a\|) vanish, and we have γ(\|a a\|) = Ç 1 ⊕ γ(\|a a\|) eff(77) where γ eff is the effective 3 × 3 CM as above. This has to be of rank two with eigenvalues 1/2. This implies that γ(\|a a\|) eff can be written as γ(\|a a\|) eff = 1 2 (½ 3 − \|φ a φ a \|),(78) where ½ 3 denotes a 3 × 3 identity matrix, and \|φ a ∈ Ê 3 . In fact, any matrix of this form is a valid CM: Lemma VI.1. For any vector \|φ ∈ Ê 3 a matrix of the form (½ 3 − \|φ a φ a \|)/2, is a valid CM of some two qubit state. Consequently, the set of valid κ A is given by all matrices of the form κ A = 1 2 (½ 3 − ρ A )(79) where ρ A is a real 3×3 matrix with trace one and positive eigenvalues. Proof: We have already shown that the CMs are of the required form, and only have to argue that any matrix of the form X = (½ 3 − \|φ x φ x \|)/2 is a valid CM. To see this, note that unitary transformations of the \|a result in orthogonal transformation on γ(\|a a\|) eff . Moreover, for the special case of a single qubit any orthogonal transformation on γ eff can be generated by a unitary transformation on state space [46], expressing the isomorphism between the Lie-algebras su(2) and so (3). Therefore, we can transform X into γ(\|a a\|) eff and construct the corresponding state vector \|x . To finish the argument, note that the set of all κ A is by definition the set of all convex combinations of pure state CMs. After having proven this Lemma we can formulate the two-qubit version of the CMC: Proposition VI.2 (CMC for two qubits). Let ρ be a state of two qubits and let {A k } = {B k } = {½/ √ 2, σ x / √ 2, σ y / √ 2, σ z / √ 2} (80) be the chosen set of observables. Let γ eff be the 6 × 6 CM as mentioned before. Then the state ρ fulfills the CMC iff there exist 3 × 3 density matrices ρ A and ρ B such that γ eff − 1 2 ½ 6 + 1 2 (ρ A ⊕ ρ B ) ≥ 0.(81) Proof: The claim follows if we insert the κ's from Eq. (79) into Proposition III.1. Note that it suffices to find complex ρ A and ρ B . If we can identify such matrices, their real part will saturate Eq. (81) as well. In this form, the problem is a special instance of an efficiently solvable semidefinite program (SDP) [33] in primal form, a feasibility problem . In general, a SDP consists of a linear function c T x which is minimized subject to a semi-definite constraint F (x) = F 0 + i x i F i ≥ 0,(82) which is linear in the problem variables x i . Hence the problem is defined by the real vector c and by the hermitian or symmetric matrices F i . If c = 0, then the problem is referred to as a feasibility problem. Via Lagrangeduality, a dual problem can be formulated in which the expression −tr(F 0 Z) is maximized over a positive semidefinite (hermitian or symmetric) matrix Z, with the constraints that tr(F i Z) = c i . Since c T x + tr(F 0 Z) = tr(F (x)Z) ≥ 0(83) holds true due to the positive semi-definiteness of F (x) and Z, solutions of the dual problem deliver a bound on the solutions of the primal problem and vice versa, which is referred to as weak duality. Finally, if there is a solution to the primal problem with F (x) > 0 or a solution to the dual problem with Z > 0, then strong duality holds, meaning that a pair (x * , Z * ) exists such that c T x * + tr(F 0 Z * ) = 0 holds. See also Ref. [47] for an extensive treatment of the subject. For the evaluation of the CMC, we can formulate the problem differently, such that if the primal problem detects the state as entangled, then from the solution of the dual problem local operators can be extracted which allow for the detection of the state with LURs. This is similar in spirit as the solution in the continuous variable case [34]. Explicitly, we formulate the primal problem as min −λ (84) subject to γ eff − κ A ⊕ κ B ≥ 0 κ A,B = 1 2 (1 + λ)½ 3 − ρ A,B ≥ 0 tr(ρ A,B ) = 1 + λ. In this formulation, the matrices κ A,B are positive and have trace 1 + λ. If the constraints can be fulfilled for λ < 0 only, then the state corresponding to γ eff is entangled. The SDP can be formulated with block-diagonal matrices {F i } collecting all the constraints. For instance, by inserting the definition of κ A,B into the first constraint and expressing the equality constraints by a '≥' and a '≤' constraint, we can write F 0 as F 0 = (γ eff − 1 2 ½ 6 )⊕ 1 2 ½ 3 ⊕ 1 2 ½ 3 ⊕(−1)⊕1⊕(−1)⊕1,(85) and the matrices F i accordingly by choosing a basis for real, symmetric matrices for the blocks. Without loss of generality, the matrix Z can be chosen block-diagonal accordingly. In the order from above we have Z = Z 1 ⊕ Z A 2 ⊕ Z B 2 ⊕ Z A1 3 ⊕ Z A2 3 ⊕ Z B1 3 ⊕ Z B2 3 , where Z 1 is a 6 × 6 matrix, Z A,Bsubject to − 1 2 [tr(Z 1 ) − tr(Z A 2 ) − tr(Z B 2 )] = Z A1 3 − Z A2 3 + Z B1 3 − Z B2 3 − 1 (Z A,B 1 ) i,i − (Z A,B 2 ) i,i = −2(Z A,B;1 3 − Z A,B;2 3 ) (Z A,B 1 ) i<j = (Z A,B 2 ) i<j ,(86) where Z A,B 1 are the single-particle subblocks of system A and B, respectively, and i and j run from 1 to 3. It turns out that Z 1 has the properties of an entanglement witness in the space of covariance matrices (CM-witness) as in the continuous-variables case [34]: Proposition VI.3 (CM-Witness from dual program). For every feasible solution Z to the dual problem formulated above, the matrix Z 1 is a CM-witness in the sense that it fulfills tr(γ eff S Z 1 ) ≥ 1 for all CMs γ eff S from separable states. Hence if tr(γ eff Z 1 ) < 1 then the corresponding state is entangled. Further, it is optimal in the sense that tr(γ eff Z 1 ) is the minimal value of tr(γ eff X) for any X ≥ 0 of the same dimensions. Proof: It follows from weak duality that tr(γ eff Z 1 ) ≥ 1+λ, hence tr(γ eff S Z 1 ) ≥ 1 holds for all γ eff S from separable states. In this case, strong duality holds, which we prove by providing an example: Z = 3 2 ½ 6 ⊕ ½ 3 ⊕ ½ 3 ⊕ 3 4 ⊕ 1 ⊕ 3 4 ⊕ 1 > 0 (87) fulfills all constraints. Hence there exist (λ * , Z * ) such that tr(γ eff Z * 1 ) = 1 + λ * holds, and the dual program reaches the minimal value of tr(γ eff Z 1 ). If the entanglement of a state is detected by a CWwitness Z 1 , then it is possible to write down a LUR detecting the state as well. This is remarkable because it is in general very difficult to find a LUR detecting the entanglement of a given state. Proposition VI.4 (LUR observables from witness). Given a CM-witness Z 1 , it is possible to define LUR matrices {Â k } and {B k } from Z 1 such that tr(γ eff Z 1 ) = k δ 2 (Â k ⊗ ½ + ½ ⊗B k ) (88) holds. Proof: The LUR corresponding to Z 1 can be extracted as shown in the proof of Proposition V.2 in the Appendix: we can spectrally decompose Z 1 = k λ k \|ψ k ψ k \| =: k λ k \|α (k) ⊕ β (k) α (k) ⊕ β (k) \|. Defining the local LUR variablesÂ k = √ λ k l α (k) l A l andB k = √ λ k l β (k) l B l we have for ρ that tr(Z 1 γ eff ) = k δ 2 (Â k ⊗ ½ + ½ ⊗B k ), where {A k } and {B k } are defined in Eq. (80). VII. EXAMPLES In this section, we consider bound entangled states of two different types, and investigate the strength of the different criteria discussed in this paper. In the first example, we take the 3 × 3 bound entangled states, called chessboard states, introduced by D. Bruß and A. Peres [48]. They are defined as ρ = N 4 j=1 \|V j V j \|,(89) where N denotes the normalization, and we used the unnormalized vectors Characterization of the family is done by six real parameters. We tested all criteria, presented in this paper on randomly generated chessboard states, where parameters have been drawn from the normal distribution with zero mean value and standard deviation of two. The results of this test are presented on the Fig. 2. As one can see from Fig. 2 the most of the states are detected by bringing first the state in its normal form (Proposition IV.13) -98.86% of all states. The criterion, which uses an estimation of singular values of the off diagonal block of CM (Proposition IV.2), which was also proposed earlier in [20] detects 22.57%, whereas another criterion proposed in this paper (Proposition IV.9) detects 22.00%. Moreover the criterion, which uses Schmidt decomposition (Proposition IV.10) detects 20.00% which is more or less the same amount as is detected by CCNR criterion -19.52%. Finally the criterion presented in Proposition IV.4, which was first proposed by de Vicente [16] detects only 8.57% of randomly generated chessboard states. As the second example, we consider 3 × 3 bound entangled states arising from an unextendible product basis [49], mixed with white noise: \|ψ 0 = 1 √ 2 \|0 (\|0 − \|1 ), \|ψ 1 = 1 √ 2 (\|0 − \|1 )\|2 , \|ψ 2 = 1 √ 2 \|2 (\|1 − \|2 ), \|ψ 3 = 1 √ 2 (\|1 − \|2 )\|0 , \|ψ 4 = 1 3 (\|0 + \|1 + \|2 )(\|0 + \|1 + \|2 ), ρ BE = 1 4 ½ − 4 i=0 \|ψ i ψ i \| , ρ UP (p) = pρ BE + (1 − p) ½ 9(91) These states are detected by Proposition IV.13 for p ≥ 0.8723 while the best known positive map detects them only for p ≥ 0.8744 (see [37] and references therein). Besides this we have also tested all other criteria presented in this paper. Criteria of Propositions IV.2, IV.9 both detect these states for p ≥ 0.8822. The criterion derived for Schmidt decomposed states (Proposition IV.10) detects the states for p ≥ 0.8834, whereas the CCNR criterion detects them for p ≥ 0.8897. Finally Proposition IV.4 detects the states for p ≥ 0.9493. Finally, let us shortly comment on the efficiency of the implementation of all these criteria. The filtering operation can be implemented quite fast, using the simple algorithm outlined above takes a few seconds on a desktop computer (5 × 5 system: ca. 6 sec., 10 × 10 system: ca. 24 sec., 15 × 15 system: ca. 72 sec.). Then, the trace norm of C can be quickly computed as the trace norm of the realignment of the matrix ρ − ρ A ⊗ ρ B [21]. For comparison, only the first step of the semidefinite program of Ref. [13] requires already ca. 10 min. for a 4 × 4 system, becoming practically unfeasible for higher dimensions. VIII. CONCLUSION AND OUTLOOK In this work, we have further developed the ideas of Ref. [22] and investigated the covariance matrix criterion (CMC). We have shown that this is a strong separability criterion, which can be simply evaluated. Combined with filtering it is necessary and sufficient for two qubits and in higher dimensions it detects states where the PPT criterion fails. Moreover, it contains many other separability criteria, which have been proposed to complement the PPT criterion as corollaries. There are several open problems which deserve a further investigation: • First, one might study the exact relation between the CMC for symmetric CMs and non-symmetric CMs. In the present paper, we have used only the trace of κ A/B for the evaluation, hence this difference did not become apparent. However, as the non-symmetric CM describes the state completely and encodes therefore all information about the separability properties, this difference might be a way to improve the CMC. In addition, one could investigate the relation between the linear part and the nonlinear part of the CM in some more detail. In the proof of Prop. IV.15 we have seen already that such an investigation may indeed improve the CMC. • Another interesting open question is to relate the CMC to quantitative statements, such as to the estimation of entanglement measures. One should expect that the program of Ref. [51] linking violations to quantitative statements on the entanglement content is applicable to the discussed criteria. Also, similar relations hold for Gaussian states [52]. First steps in this direction have already been taken in Refs. [17,20,43]. • Finally, it would be very interesting to develop a theory similar to ours for entanglement of multiparticle systems. Here, however, a significant amount of work has yet to be done, as it is not even obvious how to identify the object corresponding to the block CM for multipartite systems. IX. ACKNOWLEDGMENTS We thank H.J. Briegel, R. Horodecki and M. van den Nest for discussions. This work has been supported by the EU (OLAQUI, QAP, QICS, SCALA), the FWF, Microsoft Research, the EPSRC, and the EURYI. X. APPENDIX In this appendix, we present the more technical proofs of our previous statements. Proof of Proposition II.13: Let us first explain some properties of the matrix Γ. This matrix has entries which are just the basis vectors G i written as columns. Moreover, ΓΓ † = ½ = Γ † Γ, i.e., Γ is a unitary, since (Γ † Γ) i,j = k Γ † ik Γ kj = α,β (G α,β i ) * G α,β j = α,β (G β,α i )G α,β j = tr(G i G j ) = δ i,j ,(92) where we have used the orthogonality and hermiticity of G i . However Γ is a special unitary, since the columns correspond to orthonormal Hermitian observables. Now we have in Eq. (30) j O i,j G α,β j = j G α,β j O T j,i = ΓO T α,β\|i ,(93) where we have used the definition of Γ and the fact that the expression in the middle of Eq. (93) is nothing but i-th column of ΓO T . Conversely, (U G i U † ) α,β = U α,δ G δ,γ i U † γ,β = U α,δ U * β,γ Γ δ,γ\|i = (U ⊗ U * ) α,β,δ,γ Γ δ,γ\|i = (U ⊗ U * Γ) α,β\|i , where we have used the definition of Γ and that A i,k ⊗ B l,m ≡ (A ⊗ B) i,l,k,m . Therefore we can write O T = Γ † (U ⊗ U * )Γ = Γ T (U * ⊗ U )Γ * , O = Γ † (U † ⊗ U T )Γ = Γ T (U T ⊗ U † )Γ * ,(95) where we used that O is real. With these representations, Proof of Proposition II.14: First note that if we write γ as in Eq. (8) in the blockwise form, A, B correspond to CMs of the subsystems A, B and C has entries of the form A i ⊗ B j − A i B j , where A i , B j are observables taken for subsystems A, B. The condition γ = γ S is equivalent to the condition γ = γ T , in particular A = A T and B = B T . If we change the local bases on A and B via O = O A ⊕ O B the CM gets to γ ′ = (O A ⊕ O B )γ(O A ⊕ O B ) T .(97) As we can immediately see γ ′T = (O A ⊕ O B )γ T (O A ⊕ O B ) T = γ ′ if and only if γ T = γ, so the symmetry of CM does not depend on the particular choice of basis in observable space. Therefore we are able to choose the standard basis. Let us consider only subsystem A, i.e., left upper block of matrix γ, and let us assume that A = A T holds. As we have showed already, we can obtain ρ A from the matrix A by use of the commutators A i,j − A j,i = [M A i , M A j ] . However, all those commutators vanish for the case A = A T . Since then X k,l = Y k,l = 0 for all k, l, it follows that ρ A is diagonal. The diagonal elements can be also determined as in Prop. II.4 and since also Z k,l = 0 for all k, l, it follows that ρ A = ½/d A , which completes the first part of the proof. Finally, local unitary transformations are only a subclass of the orthogonal transformations considered before, hence γ = γ T cannot be achieved by a local unitary transformation of ρ neither. Alternative proof of Proposition IV.1 for Ky-Fan norms: For a matrix as in Eq. (38) the following condition has to be fulfilled: α\| β\| A C C T B \|α \|β ≥ 0,(98) for all vectors \|α , \|β , which implies that α\|A\|α + β\|B\|β ≥ 2 α\|C\|β , where we took −\|β instead of \|β for convenience. Especially, we can take \|α = α\|ψ k and \|β = β\|φ k , where the vectors \|ψ k and \|φ k are singular vectors from the singular value decomposition of C and ψ k \|C\|φ k = σ k (C) is the k-th singular value. Hence α 2 ψ k \|A\|ψ k + β 2 φ k \|B\|φ k ≥ 2αβ ψ k \|C\|φ k .(99) Note that ψ k \|A\|ψ k and ψ k \|A\|ψ k are greater than zero, because A and B are positive semi-definite matrices. Taking the sum over k and noting that for A and B expressions like K k=1 ψ k \|A\|ψ k are a lower bound on the K-th Ky-Fan norm [35] we get α 2 A KF + β 2 B KF ≥ 2αβ C KF .(100) The last formula is necessary and sufficient condition for the 2 × 2 matrix A KF C KF C KF B KF ≥ 0(101) to be positive semi-definite and having a non-negative determinant, from which the claim follows. Proof of Proposition V.2: The proof is an adaption of a similar proof given in Ref. [29]. We will often use the property that CMs can be used to compute variances. Imagine N = k ν k M k is a linear combination of the M k with ν k ∈ Ê, then δ 2 (N ) = k,l ν k ν l ( M k M l − M k M l ) = ν\|γ({M })\|ν . (102) Let us now assume that ρ violates the LURs and we can findÂ k ,B k , U A and U B as in Proposition V.1. We assume that the CMC is fulfilled, i.e., there exist κ A and κ B such that for the CM γ we have γ ≥ κ A ⊕ κ B . We can writê A k = l α (k) l A l andB k = l β (k) l B l(103) where the {A k } and {B k } are the observables chosen in the definition of γ. This leads to δ 2 (Â k ⊗ ½ + ½ ⊗B k ) = α (k) ⊕ β (k) \|γ\|α (k) ⊕ β (k) . Also, by definition κ A ⊕ κ B = l p l γ(\|a l a l \|) ⊕ γ(\|b l b l \|)(104) and hence α (k) ⊕ β (k) \|κ A ⊕ κ B \|α (k) ⊕ β (k) = l p l [δ 2 (Â k ) \|a l a l \| + δ 2 (B k ) \|b l b l \| ]. But then summing over k yields k δ 2 (Â k ⊗ ½ + ½ ⊗B k ) (105) ≥ k,l p l δ 2 (Â k ) \|a l a l \| + δ 2 (B k ) \|b l b l \| ≥ min \|a a\| k δ 2 (Â k ) \|a a\| + min \|b b\| k δ 2 (B k ) \|b b\| ≥ U A + U B , which is a contradiction to our assumption that ρ violates the LURs. To show the converse direction, let us assume that ̺ violates the CMC. Let us define a set of matrices as X = {x\|x = κ A ⊕ κ B + P with P ≥ 0}, which geometrically is a closed convex cone. Using this definition, we can formulate the CMC differently, by saying that if ρ is separable, then γ ∈ X. As our ρ violates the CMC, we have γ / ∈ X. According to a corollary to the Hahn-Banach theorem [50] for each γ / ∈ X there exist a symmetric matrix W and a number C such that tr(W γ) < C while tr(W x) > C ∀x ∈ X. Since X is a non-compact cone, and we can add arbitrary positive operators to the elements of X, we can conclude that tr(W P ) ≥ 0 has to hold for all P ≥ 0, and consequently we have W ≥ 0. Now let us make use of spectral decomposition of W and write W = k λ k \|ψ k ψ k \| =: k λ k \|α (k) ⊕ β (k) α (k) ⊕ β (k) \|. Defin-ingÂ k = √ λ k l α (k) l A l andB k = √ λ k l β (k) l B l we have for ρ that tr(W γ) = k δ 2 (Â k ⊗ ½ + ½ ⊗B k ). (107) Furthermore, by definition we have for all κ A ⊕ κ B ∈ X and from Proposition II.12 it follows that all γ A ⊕ γ B ∈ X. Hence for each product state ρ = ρ A ⊗ ρ B we have C < tr(W γ A ⊕ γ B ) = k [δ 2 (Â k ) ρA + δ 2 (B k ) ρB ]. This implies that C < min ρA,ρB k (δ 2 (Â k ) ρA + δ 2 (B k ) ρB ) < min ρA k δ 2 (Â k ) ρA + min ρB k δ 2 (B k ) ρB =: U A + U B(108) Finally, since the CMC is violated, γ / ∈ X and k δ 2 (Â k ⊗ ½ + ½ ⊗B k ) = tr(W γ) < C < U A + U B leading to a violation of the LURs criterion. Note that in principle this proof also applies to the CMC for non-symmetric CMs. Then, however, the "observables" in the LURs will be non-hermitian, their variance has to be defined as δ 2 (X) = XX † − X X † and their physical interpretation is not so clear. PACS numbers: 03.67.-a, 03.65.Ud FIG. 1 : 1(21) that these states have the same entanglement properties, because there is a local unitary operation in addition to a global transposition connecting them. These transformations do not change the outcome of the PPT criterion, and in fact do not change the entanglement properties of any two-qubit quantum state.Concerning (ii), it also possible to flip the Bloch vector of only one of the subsystems in a such a way that the (Color online) Entanglement properties of ρε and ρ inv ε are revealed by the PPT criterion for s = 0.45 and t = 116 . ε and r are varied. Three regions corresponds to three different cases. The region "Same" corresponds to the case where ρε and ρ inv ε are either both separable or both entangled. The region "Different" corresponds to the situation where ρε is separable but ρ inv ε is entangled or vice versa. The last region consists of states ρε for which the inversion of the Bloch vector of one of the subsystems leads to ρ inv ε which is not positive semi-definite anymore. Proposition II.14 (Block forms of CMs under local basis transformations). It is not possible to achieve for the block CMs γ = γ S via local basis transformations of the operator basis. The only states for which this relation holds have the reduced states ρ A = tr B (ρ) = ½/d A and ρ B = tr A (ρ) = ½/d B , where d A,B are the dimensions of ρ A,B . It follows that γ = γ S cannot be achieved by local unitary operations either. Let us finish this subsection with a remark on the possible use of other Ky-Fan norms in the above argument. In fact, we do know more about the singular values (here eigenvalues) of κ A and κ B than their sum: Lemma IV.7 (Ky-Fan norms of matrices in the CMC). The matrices κ A (and similarly κ B ) in Proposition III.1 satisfy k j=1 is a family of positive well defined functions on the interior of D A ⊗ D B , where the reduced density matrices both have full rank. Since ρ has also full rank, we have tr [ρ(ρ A ⊗ ρ B )] > 0 and because of compactness of D A ⊗ D B one has even stronger tr [ρ(ρ A ⊗ ρ B )] ≥ c ρ > 0. Divergence of f ρ (ρ A , ρ B ) on the boundary implies that it has a positive minimum on the interior of D A ⊗ D B . Proposition IV.13 (Filter CMC). If d = d A = d B and ρ is separable, then the coefficients in the filter normal form fulfilli ξ i ≤ d 2 − d.(69)Proof: The claim obviously follows from Proposition IV.8 and the form of the CM for the normal form of the state. . The dual problem can then be formulated asmax −[tr(γ eff Z 1 ) − 1] \|V 1 = 1\|m, 0, ac/n; 0, n, 0; 0, 0, 0 , \|V 2 = \|0, a, 0; b, 0, c; 0, 0, 0 , \|V 3 = \|n, 0, 0; 0, −m, 0; ad/m, 0, 0 , \|V 4 = \|0, b, 0; −a, 0, 0; 0, d, 0 . we can finally check the orthogonality of the O asO T O = Γ T (U * ⊗ U )Γ * Γ T (U T ⊗ U † )Γ * = Γ T (U * ⊗ U )½(U T ⊗ U † )Γ * = Γ T (U * U T ⊗ U U † )Γ * = Γ T Γ * = ½. (96) respectively, such that the dimension of the tensor product Hilbert space isd = d A × d B .We can choose a basis of the observable algebra in A as {A k : k = 1, . . . , d 2A } and in B as {B k : k = 1, . . ., d 2 B }, and consider the set of d 2 A + d 2 B observables Definition II.2 (Block covariance matrices). Let ρ be a state of a bi-partite system, and let M k = {A k ⊗ ½, ½ ⊗ B k } be a set of observables as outlined above. Then, the block covariance matrix γ(ρ, {M k }) has the entries γ i,j = M i M j − M i M j and consequently a block structure: us first show how CMs depend on the set of observables {M k : k = 1, . . . , N }: Proposition II.3 (Transformation of covariance matrices). Let γ({M k }) be a CM as defined in (4). If {K k } is another set of observables, connected to the {M k } by a basis transformation K Proposition III.1 (Covariance matrix criterion). Let ρ be a separable state and A i (B i ) be orthogonal observables on H A (H B ), where the dimensions of the Hilbert PPT CCNR Prop. IV.10 Prop. IV.13 Prop. IV.FIG. 2: (Color online) Detection of 3 × 3 chessboard states. For the different criteria the fraction of states which are detected is shown. See text for further details.2 Prop. IV.9 Prop. IV.4 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% PPT CCNR Prop. IV.10 Prop. IV.13 Prop. IV.2 Prop. IV.9 Prop. IV.4 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Interestingly, this Proposition includes the CCNR criterion as a corollary. This shows that the CMC, even without filtering, and evaluated merely via the trace of the blocks, once the matrix is brought to Schmidt form, is stronger than the CCNR, which it implies as a corollary. . R Horodecki, P Horodecki, M Horodecki, K Horodecki, quant-ph/0702225R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, quant-ph/0702225. . 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[ "Lowering the low-energy threshold of xenon detectors Lowering the low-energy threshold of xenon detectors", "Lowering the low-energy threshold of xenon detectors Lowering the low-energy threshold of xenon detectors" ]	[ "P Sorensen \nLawrence Livermore National Laboratory\n7000 East Ave94550LivermoreCAUSA\n", "J Angle \nDepartment of Physics\nUniversity of Florida\n32611GainesvilleFLUSA\n\nPhysics Institute\nUniversity of Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland\n", "E Aprile \nDepartment of Physics\nColumbia University\n10027New YorkNYUSA\n", "F Arneodo \nGran Sasso National Laboratory\n67010AssergiL'AquilaItaly\n", "L Baudis \nDepartment of Physics\nUniversity of Florida\n32611GainesvilleFLUSA\n\nPhysics Institute\nUniversity of Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland\n", "A Bernstein \nLawrence Livermore National Laboratory\n7000 East Ave94550LivermoreCAUSA\n", "A Bolozdynya \nDepartment of Physics\nCase Western Reserve University\n44106ClevelandOHUSA\n", "L C C Coelho \nDepartment of Physics\nUniversity of Coimbra\nR. Larga3004-516CoimbraPortugal\n", "C E Dahl \nDepartment of Physics\nCase Western Reserve University\n44106ClevelandOHUSA\n\nDepartment of Physics\nPrinceton University\n08540PrincetonNJUSA\n", "L Deviveiros \nDepartment of Physics\nBrown University\n02912ProvidenceRIUSA\n", "A D Ferella \nPhysics Institute\nUniversity of Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland\n\nGran Sasso National Laboratory\n67010AssergiL'AquilaItaly\n", "L M P Fernandes \nDepartment of Physics\nUniversity of Coimbra\nR. Larga3004-516CoimbraPortugal\n", "S Fiorucci \nDepartment of Physics\nBrown University\n02912ProvidenceRIUSA\n", "R J Gaitskell \nDepartment of Physics\nBrown University\n02912ProvidenceRIUSA\n", "K L Giboni \nDepartment of Physics\nColumbia University\n10027New YorkNYUSA\n", "R Gomez \nDepartment of Physics and Astronomy\nRice University\n77251HoustonTXUSA\n", "R Hasty \nDepartment of Physics\nYale University\n06511New HavenCTUSA\n", "L Kastens \nDepartment of Physics\nYale University\n06511New HavenCTUSA\n", "J Kwong \nDepartment of Physics\nCase Western Reserve University\n44106ClevelandOHUSA\n\nDepartment of Physics\nPrinceton University\n08540PrincetonNJUSA\n", "J A M Lopes \nDepartment of Physics\nUniversity of Coimbra\nR. Larga3004-516CoimbraPortugal\n", "N Madden \nLawrence Livermore National Laboratory\n7000 East Ave94550LivermoreCAUSA\n", "A Manalaysay \nDepartment of Physics\nUniversity of Florida\n32611GainesvilleFLUSA\n\nPhysics Institute\nUniversity of Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland\n", "A Manzur \nDepartment of Physics\nYale University\n06511New HavenCTUSA\n", "D N Mckinsey \nDepartment of Physics\nYale University\n06511New HavenCTUSA\n", "M E Monzani \nDepartment of Physics\nColumbia University\n10027New YorkNYUSA\n\nDepartment of Physics\nYale University\n06511New HavenCTUSA\n", "K Ni ", "U Oberlack \nDepartment of Physics and Astronomy\nRice University\n77251HoustonTXUSA\n", "J Orboeck \nDepartment of Physics\nRWTH Aachen University\n52074AachenGermany\n", "G Plante \nDepartment of Physics\nColumbia University\n10027New YorkNYUSA\n", "R Santorelli \nDepartment of Physics\nColumbia University\n10027New YorkNYUSA\n", "J M F Dos Santos \nDepartment of Physics\nUniversity of Coimbra\nR. Larga3004-516CoimbraPortugal\n", "S Schulte \nDepartment of Physics\nRWTH Aachen University\n52074AachenGermany\n", "P Shagin \nDepartment of Physics and Astronomy\nRice University\n77251HoustonTXUSA\n", "T Shutt \nDepartment of Physics\nCase Western Reserve University\n44106ClevelandOHUSA\n", "C Winant \nLawrence Livermore National Laboratory\n7000 East Ave94550LivermoreCAUSA\n", "M Yamashita \nDepartment of Physics\nColumbia University\n10027New YorkNYUSA\n", "Montpellier France \nPhysics Institute\nUniversity of Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland\n", "P Sorensen " ]	[ "Lawrence Livermore National Laboratory\n7000 East Ave94550LivermoreCAUSA", "Department of Physics\nUniversity of Florida\n32611GainesvilleFLUSA", "Physics Institute\nUniversity of Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland", "Department of Physics\nColumbia University\n10027New YorkNYUSA", "Gran Sasso National Laboratory\n67010AssergiL'AquilaItaly", "Department of Physics\nUniversity of Florida\n32611GainesvilleFLUSA", "Physics Institute\nUniversity of Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland", "Lawrence Livermore National Laboratory\n7000 East Ave94550LivermoreCAUSA", "Department of Physics\nCase Western Reserve University\n44106ClevelandOHUSA", "Department of Physics\nUniversity of Coimbra\nR. Larga3004-516CoimbraPortugal", "Department of Physics\nCase Western Reserve University\n44106ClevelandOHUSA", "Department of Physics\nPrinceton University\n08540PrincetonNJUSA", "Department of Physics\nBrown University\n02912ProvidenceRIUSA", "Physics Institute\nUniversity of Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland", "Gran Sasso National Laboratory\n67010AssergiL'AquilaItaly", "Department of Physics\nUniversity of Coimbra\nR. Larga3004-516CoimbraPortugal", "Department of Physics\nBrown University\n02912ProvidenceRIUSA", "Department of Physics\nBrown University\n02912ProvidenceRIUSA", "Department of Physics\nColumbia University\n10027New YorkNYUSA", "Department of Physics and Astronomy\nRice University\n77251HoustonTXUSA", "Department of Physics\nYale University\n06511New HavenCTUSA", "Department of Physics\nYale University\n06511New HavenCTUSA", "Department of Physics\nCase Western Reserve University\n44106ClevelandOHUSA", "Department of Physics\nPrinceton University\n08540PrincetonNJUSA", "Department of Physics\nUniversity of Coimbra\nR. Larga3004-516CoimbraPortugal", "Lawrence Livermore National Laboratory\n7000 East Ave94550LivermoreCAUSA", "Department of Physics\nUniversity of Florida\n32611GainesvilleFLUSA", "Physics Institute\nUniversity of Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland", "Department of Physics\nYale University\n06511New HavenCTUSA", "Department of Physics\nYale University\n06511New HavenCTUSA", "Department of Physics\nColumbia University\n10027New YorkNYUSA", "Department of Physics\nYale University\n06511New HavenCTUSA", "Department of Physics and Astronomy\nRice University\n77251HoustonTXUSA", "Department of Physics\nRWTH Aachen University\n52074AachenGermany", "Department of Physics\nColumbia University\n10027New YorkNYUSA", "Department of Physics\nColumbia University\n10027New YorkNYUSA", "Department of Physics\nUniversity of Coimbra\nR. Larga3004-516CoimbraPortugal", "Department of Physics\nRWTH Aachen University\n52074AachenGermany", "Department of Physics and Astronomy\nRice University\n77251HoustonTXUSA", "Department of Physics\nCase Western Reserve University\n44106ClevelandOHUSA", "Lawrence Livermore National Laboratory\n7000 East Ave94550LivermoreCAUSA", "Department of Physics\nColumbia University\n10027New YorkNYUSA", "Physics Institute\nUniversity of Zürich\nWinterthurerstrasse 190CH-8057ZürichSwitzerland" ]	[]	We show that the energy threshold for nuclear recoils in the XENON10 dark matter search data can be lowered to ∼ 1 keV, by using only the ionization signal. In other words, we make no requirement that a valid event contain a primary scintillation signal. We therefore relinquish incident particle type discrimination, which is based on the ratio of ionization to scintillation in liquid xenon. This method compromises the detector's ability to precisely determine the z coordinate of a particle interaction. However, we show for the first time that it is possible to discriminate bulk events from surface events based solely on the ionization signal.	10.5167/uzh-46842	[ "https://arxiv.org/pdf/1011.6439v1.pdf" ]	7,154,986	1011.6439	f84330fd93624bc9f0956b6affe56dd9db38d67a	Lowering the low-energy threshold of xenon detectors Lowering the low-energy threshold of xenon detectors July 26-30, 2010 30 Nov 2010 P Sorensen Lawrence Livermore National Laboratory 7000 East Ave94550LivermoreCAUSA J Angle Department of Physics University of Florida 32611GainesvilleFLUSA Physics Institute University of Zürich Winterthurerstrasse 190CH-8057ZürichSwitzerland E Aprile Department of Physics Columbia University 10027New YorkNYUSA F Arneodo Gran Sasso National Laboratory 67010AssergiL'AquilaItaly L Baudis Department of Physics University of Florida 32611GainesvilleFLUSA Physics Institute University of Zürich Winterthurerstrasse 190CH-8057ZürichSwitzerland A Bernstein Lawrence Livermore National Laboratory 7000 East Ave94550LivermoreCAUSA A Bolozdynya Department of Physics Case Western Reserve University 44106ClevelandOHUSA L C C Coelho Department of Physics University of Coimbra R. Larga3004-516CoimbraPortugal C E Dahl Department of Physics Case Western Reserve University 44106ClevelandOHUSA Department of Physics Princeton University 08540PrincetonNJUSA L Deviveiros Department of Physics Brown University 02912ProvidenceRIUSA A D Ferella Physics Institute University of Zürich Winterthurerstrasse 190CH-8057ZürichSwitzerland Gran Sasso National Laboratory 67010AssergiL'AquilaItaly L M P Fernandes Department of Physics University of Coimbra R. Larga3004-516CoimbraPortugal S Fiorucci Department of Physics Brown University 02912ProvidenceRIUSA R J Gaitskell Department of Physics Brown University 02912ProvidenceRIUSA K L Giboni Department of Physics Columbia University 10027New YorkNYUSA R Gomez Department of Physics and Astronomy Rice University 77251HoustonTXUSA R Hasty Department of Physics Yale University 06511New HavenCTUSA L Kastens Department of Physics Yale University 06511New HavenCTUSA J Kwong Department of Physics Case Western Reserve University 44106ClevelandOHUSA Department of Physics Princeton University 08540PrincetonNJUSA J A M Lopes Department of Physics University of Coimbra R. Larga3004-516CoimbraPortugal N Madden Lawrence Livermore National Laboratory 7000 East Ave94550LivermoreCAUSA A Manalaysay Department of Physics University of Florida 32611GainesvilleFLUSA Physics Institute University of Zürich Winterthurerstrasse 190CH-8057ZürichSwitzerland A Manzur Department of Physics Yale University 06511New HavenCTUSA D N Mckinsey Department of Physics Yale University 06511New HavenCTUSA M E Monzani Department of Physics Columbia University 10027New YorkNYUSA Department of Physics Yale University 06511New HavenCTUSA K Ni U Oberlack Department of Physics and Astronomy Rice University 77251HoustonTXUSA J Orboeck Department of Physics RWTH Aachen University 52074AachenGermany G Plante Department of Physics Columbia University 10027New YorkNYUSA R Santorelli Department of Physics Columbia University 10027New YorkNYUSA J M F Dos Santos Department of Physics University of Coimbra R. Larga3004-516CoimbraPortugal S Schulte Department of Physics RWTH Aachen University 52074AachenGermany P Shagin Department of Physics and Astronomy Rice University 77251HoustonTXUSA T Shutt Department of Physics Case Western Reserve University 44106ClevelandOHUSA C Winant Lawrence Livermore National Laboratory 7000 East Ave94550LivermoreCAUSA M Yamashita Department of Physics Columbia University 10027New YorkNYUSA Montpellier France Physics Institute University of Zürich Winterthurerstrasse 190CH-8057ZürichSwitzerland P Sorensen Lowering the low-energy threshold of xenon detectors Lowering the low-energy threshold of xenon detectors July 26-30, 2010 30 Nov 2010Identification of Dark Matter 2010-IDM2010 We show that the energy threshold for nuclear recoils in the XENON10 dark matter search data can be lowered to ∼ 1 keV, by using only the ionization signal. In other words, we make no requirement that a valid event contain a primary scintillation signal. We therefore relinquish incident particle type discrimination, which is based on the ratio of ionization to scintillation in liquid xenon. This method compromises the detector's ability to precisely determine the z coordinate of a particle interaction. However, we show for the first time that it is possible to discriminate bulk events from surface events based solely on the ionization signal. Introduction The XENON10 detector [1] is a liquid xenon time-projection chamber with a an active target mass of 13.7 kg (15 cm height and 10 cm radius). It was designed to directly detect galactic dark matter particles which scatter off xenon nuclei. Typical velocities of halo-bound dark matter particles are of order 10 −3 c. This leads to the prediction of featureless exponential recoil energy spectra for spin-independent elastic scattering of dark matter particles on a xenon target [2]. Typical nuclear recoil energies are few keV. A particle interaction in liquid xenon results in a prompt scintillation signal (S1) and an ionization signal (S2). The generation and collection of these signals are discussed in detail in [1]. A key feature of the XENON10 detector is the event-by-event discrimination between incident particle type, based on the ratio S2/S1. The energy threshold of previously reported XENON10 data [3 -5] depends on the primary scintillation efficiency of liquid xenon for nuclear recoils L e f f [6,7], and for a conservative assumption of the energy dependence of L e f f [6], is about 5 keV. This energy threshold is dictated by the collection of primary scintillation photons following a nuclear recoil in the xenon target. But it is possible to obtain a lower energy threshold from the existing XENON10 data, using only the ionization signal. We discuss an analysis of XENON10 dark matter search data, with the energy scale set by the detected ionization signal and no requirement that valid events contain both an S1 and an S2 signal. This requires a compromise on two important aspects of the detector performance: the ability to precisely reconstruct the z coordinate of a particle interaction (which is normally obtained by the time delay between S1 and S2 signals), and the discrimination between incident particle types. Calibration of the ionization energy scale for nuclear recoils in liquid xenon is a pre-requisite. Direct measurements exist in the literature [6,8] (and are shown in Fig. 2b), but do not extend to the smallest signals observed in our detector. We describe a new measurement of the ionization yield of liquid xenon for nuclear recoils, down to a nuclear recoil energy of about 1 keVr. We then describe a method for making an approximate determination of the z coordinate of the interaction. The method exploits the measured width of the S2 signal, which is broadened by electron diffusion as a cloud of electrons drifts across the liquid xenon target. The precision is low (∼cm), but is sufficient to discriminate bulk events from surface events. This is important because the latter are more likely to result from radioactive background. Ionization yield of liquid xenon for nuclear recoils The ionization yield Q y was obtained directly from in-situ neutron calibration data. The neutron calibration experiment and general analysis technique are both described in a previous study by the XENON10 collaboration [9]. Note that here we circumvent the false single scatter pathology described in [3], since our calibration technique relies only on the S2 signal. The S2 spectrum for single elastic neutron scatters is shown in Fig. 1a, with 1σ uncertainty. The S2 signal was scaled to an absolute number of electrons via the measured 24 photoelectrons per liquid electron [1]. Only the most basic data quality cuts were applied, along with a fiducial cut r < 8 cm. No z fiducial cut was applied because a substantial fraction of events at low energy have no S1 (and hence have indeterminate z). The fiducial target volume used in this calibration thus differs from Shown in light gray is the spectrum of events in which an S1 signal was found in the 80 µs before the S2. Approximate keVr-equivalent is indicated along the top. (b) The ionization yield from nuclear recoils Q y from the present work ( ), with 1σ statistical uncertainty. Systematic uncertainty is shown along the axis. Also shown are data from [6] ( ), [9] ( ) and [8] ( and ). previous analyses [4,5,3,9] in that it considers the full ∆z = 15 cm active target (8.6 kg target mass), rather than just the central ∆z = 9.3 cm (5.4 kg target mass). With no self-shielding liquid xenon above or below the chosen xenon target volume, it is important to take account of possible gamma and beta background contamination in the nuclear recoil data. This background is assessed by counting the number of events within ±2σ of the electron recoil centroid, in the region around 30 − 35 keVee. The chosen energy window is above the tail of the elastic nuclear recoil distribution, and below the 40 keVee inelastic scatters [9]. The electromagnetic background spectrum below about 50 keVee is predicted and observed to be flat with energy. Therefore, we were able to subtract the resulting background prediction of about 0.75 counts/electron. Considering the number of nuclear recoil events in the data sample shown in Fig. 1a, this is a very small correction. In comparing the measured energy spectrum with the Monte Carlo predicted spectrum, the energy resolution in the electron signal was assumed to be Poisson in the number of detected electrons; account was also taken of the 20% 1σ width of the single electron distribution [10]. The S2-sensitive trigger threshold for the neutron calibration data had full efficiency for events with at least 182 photoelectrons [1], or 7.6 S2 electrons. This is valid for r ≤ 3 cm, and there is a slight radial dependence to the trigger such that by r = 8 cm the efficiency is unity for events with greater than 11 S2 electrons. As a result, events with S2 < 12 electrons were not used in the calibration analysis. No attempt was made to model the S2 trigger efficiency, as is apparent in Fig. 1a from the discrepancy between data and simulation below ∼ 8 electrons. The ionization yield Q y was modeled as a continuous cubic Hermite spline interpolation as in [9], and a maximum likelihood comparison was applied to find Q y (E nr ) as shown in Fig. 1b ( , with 1σ statistical uncertainty). No constraints were applied to the spline other than the re-quirement that Q y = 0 at E nr = 0.1 keVr. The exact minimum recoil energy that can result in a single detectable electron is not known, but is limited by the ionization potential of xenon and by nuclear recoil quenching [11]. Our chosen boundary condition is conservative, and our results are insensitive to a ×5 shift in this value. The spline points were chosen at fixed values of keV nuclear recoil energy (keVr), and the results do not depend on the location of the spline points (as long as there are ∼ 10 points spanning the full energy range). The best fit shown in Fig. 1a has χ 2 /d.o. f . = 414/425 . For reference, the ionization yield of 122 keV gammas from a 57 Co source were found to have an ionization yield of 46.5 ± 7.0 (stat.) electrons/keV, for events with radial positions in the range 8.5 < r < 9.0 cm. Two sources of systematic uncertainty are indicated as vertical bars for each spline point, along the x-axis. These arise from an assumed ±10% uncertainty in the single electron calibration (left bar), and from uncertainty below 2 MeV in the spectrum of initial neutron energies E n from the AmBe neutron source [12] (right bar). The neutron energy spectrum was taken from Fig. 5 of [12], and the uncertainty was parameterized as 1 ± exp(−E n − 1 2 ) for E n < 2 MeV. The effect of this conservative assumption on the Monte Carlo nuclear recoil energy spectrum was a change of about ±15% in bin counts at 1 keVr. A third source of uncertainty, arising from the Xe(n,n)Xe elastic cross-section data [13], would appear almost point-like in Fig. 1b and is not shown. Our Q y results were found to be essentially unchanged if the assumed energy resolution was varied by ±25%, and remained very similar if the fit range was instead truncated at either 10 or 15 electrons. The agreement between this measurement and that of [6] (Fig. 1b, ) is quite good above 6 keVr. A possible reason for the rapid rise in Q y below 6 keVr obtained in [6] is discussed in [14]. The previous work [9] (Fig. 1b, ) shows systematically higher values of Q y below about 25 keVr, possibly due to the systematic effect of false single scatters mentioned at the beginning of Sec. 2. It is also worth emphasizing that the previous work [9] relied on the S1 signal, and therefore had very limited sensitivity to recoil energies smaller than about 5 keVr. We have argued that our nuclear recoil data sample does not contain a significant electromagnetic background contamination, and that what little contamination exists can be safely subtracted. This argument is weakened for events in which no S1 signal was detected, since they have an indeterminate z coordinate for the scatter vertex, and also an indeterminate discrimination parameter (S2/S1). It is clear from Fig. 1a that most events with no S1 have S2 40 electrons. A compelling piece of evidence that such events are in fact elastic nuclear recoils is that their spectral shape is similar to that obtained for elastic nuclear recoils with known S1. In contrast, the population of events with no S1 in the gamma calibration data shows the expected flat S2 spectral shape in the range S2 40 electrons. However, this does not fully exclude the possibility of an additional lowenergy background. The most likely origin of such a background, if it exists, would be low-energy gamma or beta scattering at the liquid surface. Bulk events would be excluded by virtue of the ∼µm range of ∼keV particles in liquid xenon. About 95% of events at the liquid surface are measured to have 0.13 σ S2 0.23 µs, as can be seen in Fig. 2a. The distribution of σ S2 for bulk events, and events with indeterminate drift time are both roughly Gaussian. The former shows a 14.7% ± 2.2% excess of events in the region 0.13 σ S2 0.23 µs. This places an upper limit on the fraction of events with no S1 which might be unaccounted surface background events, rather than bulk elastic nuclear recoils. The effect of such a background (assumed flat in the range 7 − 40 electrons) on our energy calibration would be Figure 2: The S2 pulse width σ S2 obtained by Gaussian fit, for single scatter events with 7 ≤ S2 < 100 electrons in the region r < 3 cm. The liquid-gas interface is at z = 0 cm. Events with indeterminate z coordinate due to absent S1 are shown at z=15 cm. The band mean is indicated by , with 1σ width; horizontal bars indicate bin width. A 0.4 µs section of two typical S2 12 electron events are shown inset (top). In the event at z = 0.3 cm, the S1 is visible 1.6 µs before the S2; in the event at z = 13.7 cm, the S1 is far off scale. to increase Q y at 8 keVr and 16 keVr by about 0.2 electrons/keVr, and by half that amount at other energies E nr < 32 keVr. This is a very small effect and is not indicated with the other uncertainties in Fig. 1b. 3. Obtaining the z coordinate of a particle interaction from the S2 width A cloud of ionized electrons resulting from a particle interaction drifts through the liquid xenon target at about 0.20 cm µs −1 [15]. As it drifts, its spatial extent broadens due to diffusion. The amount of diffusion broadening is reflected in the width of the S2 pulse. In this way, the width of S2 signals depends weakly on the z coordinate of the scatter vertex. The width σ S2 was obtained by Gaussian fit to each S2 pulse, and is shown in Fig. 2a for events with r < 3 cm. Events with no S1 pulse do not have a precisely defined drift time, and are shown (arbitrarily) at z = 15 cm. A 0.4 µs section of two typical events with S2 12 electrons are shown inset in Fig. 2a, corresponding to nuclear recoils at the extrema (in z) of the active xenon target. The height of the S2 pulse in the z = 0.3 cm event is about 10 mV referred to the amplifier input. The z dependence of σ S2 is clearly weak, however it is sufficient for us to discriminate events which occur at the surface (z = 0 cm and z = 15 cm) from those which occur in the bulk of the liquid xenon target. Figure 2b shows the differential event rate from a 12.5 live day exposure of the XENON10 detector, obtained between August 23 and September 14,2006. This data set is distinct from the previously reported [3,4] dark matter search data. The most notable difference is that during this time, the detector was operated with an S2-sensitive trigger threshold at the level of a single ionized electron. Event selection criteria were applied as follows. Valid single scatter events were required to have only a single S2 pulse in the event record. The efficiency for making this selection is very high, considering the robustness of the S2 signal (as shown in Fig. 2a). A fiducial cut defined by r < 3 cm was imposed, giving a target mass of 1.2 kg. This central region features optimal self-shielding by the surrounding xenon target. No explicit z cut was made since events were not required to contain an S1 signal. Events in which an S1 signal was found were required to have log 10 (S2/S1) in the ±3σ band for elastic single scatter nuclear recoils [3]. Events in which no S1 signal was found were assumed to be low-energy nuclear recoils and were retained. The fraction of events with no S1 is given by the ratio of the histograms in Fig. 1 for the nuclear recoil calibration data. For the dark matter search data, the fraction is approximately ×3 lower. This behavior is expected for electron recoils, based on the observed ratio of log 10 (S2/S1) for electron and nuclear recoils [3]. The combined acceptance of the two cuts is ε 0.99, for events in the 1.2 kg central region. Dark matter search data We then used the S2 width σ S2 to discriminate bulk events from edge events, as described in Sec. 3. In order to ensure complete rejection of events at the liquid xenon surface (at z = 0 cm), we set the lower bound of the cut at σ S2 ≥ 0.23 µs. The upper bound of the cut was set at σ S2 ≤ 0.30 µs. Considering the total (∼Gaussian) distribution of recorded σ S2 values, these bounds correspond approximately to the region between µ and µ + 2σ . It is clear from Fig. 2a that this only partially targets events from the bottom of the active region of the detector (near z = 15 cm). This choice does not compromise surface event rejection, considering that below the cathode grid which defines z = 15 cm are an additional 1.3 cm of self-shielding liquid xenon. The acceptance of the S2 width cut for single scatter nuclear recoils is mildly energy-dependent, rising monotonically from ε = 0.39 for events with S2 = 7 electrons, to ε = 0.44 for events with S2 ≥ 40 electrons. The S2 spectrum of all single-scatter events that passed these cuts is shown in Fig. 2b ( ). Vertical bars indicate statistical uncertainty, and horizontal bars indicate bin width. The count rate is adjusted for the total acceptance fraction ε. The lowest energy events remaining above an analysis threshold of 7 electrons have S2 signals of 8, 17, 18 and 29 electrons. An additional 4 events were found with 30 < S2 < 44 electrons. For a higher statistics comparison, the spectrum of all single-scatter events within an 8 cm radius (target mass 8.6 kg) is also shown ( ). No S2 width cut was applied to the data in that case, and the statistical uncertainty is smaller than the data points. Both spectra are essentially flat above S2 = 7 electrons, as would be expected from a Compton scatter background. Note that in the calibration analysis of Sec. 2, a lower bound of 7 electrons in the S2 signal was motivated by the trigger efficiency during the neutron calibration. In the dark matter search data the trigger threshold is at the level of a single electron in the S2 signal. However, we retain the 7 electron lower bound in analyzing this data because (i) we can most accurately determine acceptance for nuclear recoils above this value, and (ii) the electronic noise increases significantly below S2 6 electrons, as shown in Fig. 2b. Summary The measured number of S2 electrons can be scaled to nuclear recoil equivalent energy via the Q y curve shown in Fig. 1b. A conservative choice would be the −1σ contour, which is approximately Q y ≈ 4 electrons/keVr below E nr = 20 keVr. As mentioned in Sec. 2, the detector resolution for S2 signals depends primarily on Poisson fluctuations in the number of detected elec-trons, with an additional component due to instrumental fluctuations. This is discussed in detail in [14], and for higher energy signals in [1]. So as not to overstate the energy resolution, we suggest a parameterization R(keV) = 1/2 √ keV which follows the Poisson component only. This is valid for the −1σ contour of Q y . We have shown that it is possible to give up the usual incident particle type discrimination based on log 10 (S2/S1), and analyze the dark matter sensitivity of XENON10 using only the S2 signal. The advantage of this analysis appears to be an increased sensitivity to light ( 10 GeV) dark matter particles, due to the significantly lower energy threshold. For larger particle masses, the usual analyses [3,4] offer superior sensitivity. Dark matter exclusion limits obtained from the present work should offer substaintial constraints on recent interpretations [16 -21] of the excess low-energy events observed by CoGeNT [22] and CRESST-II [23], as well as the DAMA modulation signal [24]. Figure 1 : 1(a) Spectrum of the number of electrons extracted from single scatter elastic nuclear recoil interactions in the fiducial target defined by r < 8 cm, with 1σ statistical uncertainty. The best-fit Monte Carlo is shown as a continuous stair-step. The energy values of the 9 spline points are identical in both plots. . E Aprile, XENON10 Collaborationarxiv:1001.2834v1 [astro-ph.IME. Aprile et al. (XENON10 Collaboration), arxiv:1001.2834v1 [astro-ph.IM] (2010). . G Jungman, M Kamionkowski, K Griest, Phys. Rept. 267G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267 (1996). . J Angle, XENON10 CollaborationPhys. Rev. D. 80115005J. Angle et al. 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[ "STABILITY OF SHARP FOURIER RESTRICTION TO SPHERES", "STABILITY OF SHARP FOURIER RESTRICTION TO SPHERES" ]	[ "Emanuel Carneiro ", "Giuseppe Negro ", "Diogo Oliveira ", "E Silva " ]	[]	[]	In dimensions d P t3, 4, 5, 6, 7u, we prove that the constant functions on the unit sphere S d´1 Ă R d maximize the weighted adjoint Fourier restriction inequalityˇˇˇˆRwhere σ is the surface measure on S d´1 , for a suitable class of bounded perturbations g : R d Ñ C. In such cases we also fully classify the complex-valued maximizers of the inequality. In the unperturbed setting (g " 0), this was established by Foschi (d " 3) and by the first and third authors (d P t4, 5, 6, 7u) in 2015.2010 Mathematics Subject Classification. 42B10, 42B37, 33C55.	null	[ "https://arxiv.org/pdf/2108.03412v2.pdf" ]	236,957,131	2108.03412	44705a148abeb88150afec3e16c616439541a196	STABILITY OF SHARP FOURIER RESTRICTION TO SPHERES Emanuel Carneiro Giuseppe Negro Diogo Oliveira E Silva STABILITY OF SHARP FOURIER RESTRICTION TO SPHERES In dimensions d P t3, 4, 5, 6, 7u, we prove that the constant functions on the unit sphere S d´1 Ă R d maximize the weighted adjoint Fourier restriction inequalityˇˇˇˆRwhere σ is the surface measure on S d´1 , for a suitable class of bounded perturbations g : R d Ñ C. In such cases we also fully classify the complex-valued maximizers of the inequality. In the unperturbed setting (g " 0), this was established by Foschi (d " 3) and by the first and third authors (d P t4, 5, 6, 7u) in 2015.2010 Mathematics Subject Classification. 42B10, 42B37, 33C55. Introduction Let S d´1 Ă R d be the pd´1q-dimensional unit sphere, d ě 2, equipped with the standard surface measure σ " σ d´1 that verifies σ`S d´1˘" 2π d{2 { Γpd{2q. For f P L 1 pS d´1 q, we define the Fourier transform of the measure f σ by x f σpxq "ˆS d´1 e ix¨ω f pωq dσpωq ; px P R d q. (1.1) It is conjectured that the constant functions maximize the adjoint Fourier restriction inequality › › x f σ › › L q pR d q ď C }f } L 2 pS d´1 q (1.2) for all q ě 2pd`1q{pd´1q. Up to date, this claim has only been established in a few cases, all in low dimensions: in the Stein-Tomas endpoint case q " 4 and d " 3 by Foschi [23]; in the cases q " 4 and d P t4, 5, 6, 7u by Carneiro and Oliveira e Silva [13]; and, more recently, in the cases q " 2k with k ě 3 an integer, and d P t3, 4, 5, 6, 7u, by Oliveira e Silva and Quilodrán [37]. Other works related to the sharp adjoint Fourier restriction to the sphere include [2,11,15,18,19,25,28,38,39,40,47]. The general theme of sharp Fourier restriction has flourished over the last two decades with many interesting works for other quadratic surfaces and its relations to partial differential equations. This was inspired by the classical work of Strichartz [49] in 1977, which in turn appeared shortly after Beckner's celebrated sharpening of the Hausdorff-Young inequality [3]. A non-exhaustive list of works in sharp Fourier restriction theory includes [5,10,17,22,26,27,30,32,45] for the paraboloid (Schrödinger equation), [7,8,22,33,42,44] for the cone (wave equation), [14,16,31,41,43] for the hyperboloid (Klein-Gordon equation), and [4,6,9,12,20,21,29,34,35,36,46] in other related settings. We refer the reader to the survey [24] for a more detailed account on the latest developments. Apart from the base case h " 1, for which little is known, this is completely uncharted territory. The purpose of this paper is to provide the first non-trivial results in this direction. Our terminology here is the usual one: the value of the optimal constant in the inequality (1.3) is C " sup 0‰f PL 2 pS d´1 qˇ´R dˇx f σpxqˇˇq hpxq dxˇˇ1 {q }f } L 2 pS d´1 q ,(1.4) and a maximizer is a function f P L 2 pS d´1 qzt0u that realizes the supremum on the right-hand side of (1.4). Throughout the paper, we work with the exponent q " 4 in dimensions d P t3, 4, 5, 6, 7u, in the regime h " 1`g, with g P L 8 pR d q. We are then interested in the sharp form of the inequalityˇˇˇˆR dˇx f σpxqˇˇ4 hpxq dxˇˇˇˇ1 {4 ď C }f } L 2 pS d´1 q . (1.5) Recall that when g " 0 and d P t3, 4, 5, 6, 7u, the constant functions maximize (1.5). Heuristically it is expected that, if g is sufficiently small, then the constant functions should come close to realizing equality in (1.5) in dimensions d P t3, 4, 5, 6, 7u. We refine this stability statement and prove that, if the perturbation g is sufficiently regular and small, as properly described below, then the constant functions continue to be maximizers of (1.5) (and, generically, they are the unique maximizers). This is the content of Theorems 1, 2 and 3 below. We note that, although it may seem more natural to consider non-negative weights h in (1.5), our methods do not require this assumption, and the weight h is free to exhibit sign oscillations. Our notation for a multi-index is standard, letting α " pα 1 , α 2 , . . . , α d q with each α i P Z ě0 (1 ď i ď d), and writing \|α\| :" α 1`. . .`α d and B α g :" B α1 x1 . . . B α d x d g. Throughout the paper we assume the following regularity condition on the perturbation g : R d Ñ C: (R1) g P L 2 pR d q X L 8 pR d q and its Fourier transform p g is radial and non-negative on the closed ball B 4 " tξ P R d ; \|ξ\| ď 4u. There is a second regularity condition in our study, which is related to the smoothness of p gˇˇB 4 . Here we consider two cases of interest: (R2.A) (Analytic version) p gˇˇB 4 : B 4 Ñ R admits an analytic continuation, which we call G, to an open disk in C d containing the closed disk D R " tz P C d ; \|z\| ď Ru for some radius R ą 4. (R2.C) (C k -version) p gˇˇB 4 : B 4 Ñ R belongs to C k pB 4 q X C 0 pB 4 q, with bounded partial derivatives of order up to k in B 4 , where 1 k " kpdq " tpd`3q{2u. Remark: In sympathy with (1.1), our normalization for the Fourier transform in R d is p gpξq "ˆR d e iξ¨x gpxq dx. (1.6) Remark: In this paper we shall always work under conditions pR1q and pR2.Aq, or under conditions pR1q and pR2.Cq. The portion of the distributional Fourier transform p g outside B 4 has no effect on the integral on the left-hand side of (1.5) (see (2.5) below) and we can assume without loss of generality that supppp gq Ă B 4 . Hence, by Fourier inversion, we may assume when convenient that g itself is radial, real-valued, smooth, and that g and all of its partial derivatives belong to L 2 pR d q X L 8 pR d q. Our main results are the following. Theorem 1 (Sharp weighted adjoint Fourier restriction: analytic version). Let g : R d Ñ C be a function verifying the regularity conditions pR1q and pR2.Aq above, and set h " 1`g. Let R ą 4 and G : for any multi-index α P Z d ě0 of the form α " pα 1 , 0, 0, . . . , 0q, with 0 ď α 1 ď kpdq, then the constant functions are maximizers of the weighted adjoint Fourier restriction inequality (1.5). Our constant C d is effective, given by (7.36), corresponding to C 3 " 0.157 . . . ; C 4 " 0.918 . . . ; C 5 " 0.908 . . . ; C 6 " 1.099 . . . ; C 7 " 0.534 . . . . D R Ñ C One readily notices that we put some effort into making the main results not only qualitative but also quantitative. In general, we shall see that the C k -condition (R2.C) corresponds to the minimal regularity required on p g in order to achieve our goal. Nevertheless, we decided to state the slightly more restrictive analytic version (Theorem 1) separately since it is already a fruitful source of examples, with a touch of simplicity in its statement and different insights within the course of its independent proof. Moreover, there are situations, when both Theorems 1 and 2 are available, in which the bounds coming from Theorem 1 are strictly superior (for instance, as in the remark after Theorem 1, where p gˇˇB 4 is a polynomial of low degree in the variable \|ξ\| 2 ). In particular, it is not the case that Theorem 1 follows from Theorem 2. Theorems 1 and 2 are complemented by the following classification result. Theorem 3 (Full classification of maximizers). Under the hypotheses of Theorem 1 or Theorem 2: (i) If p g " 0 on the ball B 4 , then the complex-valued maximizers f P L 2 pS d´1 q of (1.5) are given by f pωq " c e iy¨ω , where y P R d and c P Czt0u. (ii) If p g ‰ 0 on the ball B 4 , then the constant functions are the unique complex-valued maximizers of (1.5). In the unperturbed setting (i.e. g " 0), the conclusion of Theorem 3 (i) was established in [13,23], and we just record here for the convenience of the reader that this continues to hold when p g " 0 on the ball B 4 , by a simple orthogonality argument. The novelty in the classification above occurs in the broad situation of Theorem 3 (ii), where general complex characters e iy¨ω , y P R d zt0u, do not maximize (1.5), in contrast to the previous case. This is ultimately due to the modulation/translation symmetry of the extension operator, { pe iy¨f qσ " x f σp¨`yq, which is naturally incompatible with the assumed radiality of p gˇˇB 4 . There is a myriad of examples of perturbations g that would fit into our framework. The most naive one is perhaps the Gaussian p gpξq " c e´\| ξ\| 2 {2 (so that, in our normalization, gpxq " c p2πq´d {2 e´\| x\| 2 {2 ), provided c ą 0 is sufficiently small. In this situation, Theorem 2 typically provides a better bound than Theorem 1 for the admissible range of the parameter c, due to the growth of Gaussians along the imaginary axis. A concrete choice which falls within the scope of Theorem 2 in every dimension d P t3, 4, 5, 6, 7u is c " 1 11 . More generally, a particularly simple family for the analytic setting of Theorem 1 is given by p gpξq " or ξ " pξ 1 , ξ 2 , . . . , ξ d q P B 4 Ă R d , where G : D 4 Ă C Ñ C is an even analytic function of one complex variable. In fact, given such conditions on p g, then Gpzq is simply Gpz, 0, . . . , 0q for z P D 4 . Conversely, given G : D 4 Ă C Ñ C even and analytic, we can write Gpzq " ř 8 "0 c z 2 (with this series being absolutely convergent for any z with \|z\| ă 4) and then p gpξ 1 , ξ 2 , . . . , ξ d q :" ř 8 "0 c pξ 2 1`ξ 2 2`. . .`ξ 2 d q defines a radial function on B 4 that admits an analytic continuation to D 4 Ă C d . P p\|ξ\| 2 q´8 0 e´λ \|ξ\| 2 {2 dµpλq, Let us briefly comment on the motivation behind the regularity conditions. In Section 9 we discuss how the radiality condition in (R1), the non-negativity condition in (R1), and the smallness condition in (1.7) and (1.9) (associated to (R2.A) and (R2.C), respectively) are reasonable assumptions, in the sense that if one of them is removed, then it is possible to construct explicit examples of perturbations g for which the constant functions do not maximize (1.5). As far as the proof strategy for Theorems 1 and 2 is concerned, and how the assumptions play a role, we highlight the following aspects. The radiality condition in (R1) is present in order to preserve the natural radial symmetry of the problem. As in the predecessors [13,23], which treat the case g " 0, the strategy can be divided into three main steps: I. Symmetrization; II. Magical identity and an application of the Cauchy-Schwarz inequality; III. Spectral analysis of a quadratic form. Step I, in which we reduce the search of maximizers to non-negative and even functions, is similar to the one in [13,23], using the non-negativity condition in (R1). On the other hand, Steps II and III bring new insights. In [23], Foschi had the elegant idea of introducing what we call a magical geometric identity to deal with the singularity of the two-fold convolution of the surface measure of the sphere at the origin: if pω 1 , ω 2 , ω 3 , ω 4 q P pS d´1 q 4 are such that ω 1`ω2`ω3`ω4 " 0, then \|ω 1`ω2 \| \|ω 3`ω4 \|`\|ω 1`ω3 \| \|ω 2`ω4 \|`\|ω 1`ω4 \| \|ω 2`ω3 \| " 4. (1.10) In the presence of a perturbation g, one must find the "correct" magical identity which needs to be applied, a task that in principle is not obvious. We present a new point of view to generate such magical identities, via the underlying partial differential equation (Helmholtz equation) and opportune applications of integration by parts. This general perspective turns out to be amenable to perturbations, and this ultimately enables our progress in Step II (and, as a by-product, we recover (1.10) in the case g " 0). Finally, in Step III we arrive at the analysis of a suitable quadratic form. Conceptually behind our proof lurks the fact that, in the corresponding step in [13,23] for g " 0, there was "some room to spare", in the sense that certain Gegenbauer coefficients which appeared in connection to the problem were not only less than or equal to zero (which would suffice for the argument that constants are maximizers) but, in fact, strictly less than zero. In order to properly understand, quantify and take advantage of such heuristics, we bring in the final regularity assumption (R2.A) in case of Theorem 1, and (R2.C) in case of Theorem 2, since, in essence, smoothness of p g will ultimately yield the required decay of the corresponding Gegenbauer coefficients. By Hölder's inequality, one has }f } L 2 pS d´1 q ď σpS d´1 q 1´2 p }f } L p pS d´1 q for any p ě 2, with equality if f is constant. Hence, under the assumptions of Theorem 1 or Theorem 2, we see that the constant functions are also maximizers of the family of inequalitiešˇˇˇˆR dˇx f σpxqˇˇ4 hpxq dxˇˇˇˇ1 {4 ď C σpS d´1 q 1´2 p }f } L p pS d´1 q , for any p ě 2, with the same optimal C as in (1.5). A related weighted inequality in the regime p " 4 and d " 3, with a simpler setup, has been previously suggested by Christ and Shao in [18,Remark 16.3]. A word on notation. Throughout the text we denote by 1 (resp. 0) the constant function equal to 1 (resp. 0), which may be on R d or S d´1 depending on the context. The indicator function of a set X is denoted by 1 X . Given a radius R ą 0, we let B R " tx P R d ; \|x\| ă Ru be the open ball centered at the origin in R d , and D R " tz P C d ; \|z\| ă Ru be the open disk (we shall use here the term "disk" instead of "ball" just to emphasize the different environment) centered at the origin in C d . Their respective topological closures are denoted by B R and D R . We denote by p gˇˇB 4 the restriction of p g to the ball B 4 . We write A À B if A ď CB for a certain constant C ą 0, and we write A » B if A À B and B À A (parameters of dependence of such a constant C ą 0 might appear as a subscript in the inequality sign). Symmetrization Throughout the paper we keep the notation h " 1`g. Then p h " p2πq d δ`p g , (2.1) where δ is the d-dimensional Dirac delta distribution. For functions f i : S d´1 Ñ C (1 ď i ď 4), define the quadrilinear form Q h pf 1 , f 2 , f 3 , f 4 q :"ˆp S d´1 q 4 p hpω 1`ω2´ω3´ω4 q 4 ź j"1 f j pω j q dσpω j q. (2.2) Further define the quadrilinear forms Q 1 and Q g as in (2.2), with 1 and g replacing h, respectively. Let f P L 2 pS d´1 q. Plancherel's identity leads us tô R dˇx f σpxqˇˇ4 dx " p2πq d › › f σ˚f σ › › 2 L 2 pR d q " Q 1 pf, f, f , f q (2.3) (note that this quantity is always non-negative), and R dˇx f σpxqˇˇ4 hpxq dx " Q 1 pf, f, f , f q`Q g pf, f, f , f q " Q h pf, f, f , f q. (2.5) We now show how to exploit the symmetries of the problem, thus obtaining some estimates which simplify the search for the maximizers. The discussion of the cases of equality will be applied in Section 8. Reduction to non-negative functions. Our first auxiliary result is the following. Lemma 4. Let f P L 2 pS d´1 q. We havěˇQ h pf, f, f , f qˇˇď Q h p\|f \|, \|f \|, \|f \|, \|f \|q. (2.6) Equality holds if f is non-negative pin particular, if f " 1q. Furthermore, if there is equality, then necessarily f σ˚f σ L 2 pR d q " \|f \|σ˚\|f \|σ L 2 pR d q . (2.7) Proof. Inequality (2.6) follows immediately from the definition (2.2) of Q h since, by condition (R1) and (2.1), we have that the measure p h is non-negative on B 4 . Similarly, note that Q 1 pf, f, f , f q ď Q 1 p\|f \|, \|f \|, \|f \|, \|f \|q andˇˇQ g pf, f, f , f qˇˇď Q g p\|f \|, \|f \|, \|f \|, \|f \|q. (2.8) From (2.5), the triangle inequality and (2.8) we actually have the intermediate inequalitiešˇQ h pf, f, f , f qˇˇď Q 1 pf, f, f , f q`ˇˇQ g pf, f, f , f qˇď Q 1 p\|f \|, \|f \|, \|f \|, \|f \|q`Q g p\|f \|, \|f \|, \|f \|, \|f \|q " Q h p\|f \|, \|f \|, \|f \|, \|f \|q. (2.9) In order to have equality in (2.9), we must have equality in both inequalities of (2.8), and the first of these is equivalent to (2.7). From now on, unless otherwise stated, we will assume without loss of generality that f is a non-negative function. In particular,´R dˇx f σpxqˇˇ4 hpxq dx " Q h pf, f, f, f q is also non-negative. Reduction to even functions. Given a function f : S d´1 Ñ R ě0 we define its antipodally symmetric rearrangement f 7 by f 7 pωq :"ˆf pωq 2`f p´ωq 2 2˙1 2 . Observe that }f 7 } L 2 pS d´1 q " }f } L 2 pS d´1 q . Lemma 5. If f : S d´1 Ñ R ě0 belongs to L 2 pS d´1 q then Q h pf, f, f, f q ď Q h pf 7 , f 7 , f 7 , f 7 q. (2.10) There is equality if and only if f " f 7 pin particular, if f " 1q. Proof. Let us abbreviate the notation by writing dσp ωq :" dσpω 1 q dσpω 2 q dσpω 3 q dσpω 4 q, with the vector ω " pω 1 , ω 2 , ω 3 , ω 4 q P pS d´1 q 4 . By changing variables and reordering we observe that Q h pf, f, f, f q "ˆp S d´1 q 4 p hpω 1`ω2´ω3´ω4 qˆf pω 1 qf pω 3 q`f p´ω 1 qf p´ω 3 q 2˙f pω 2 q f pω 4 q dσp ωq ďˆp S d´1 q 4 p hpω 1`ω2´ω3´ω4 qˆf pω 1 q 2`f p´ω 1 q 2 2˙1 2ˆf pω 3 q 2`f p´ω 3 q 2 2˙1 2 f pω 2 q f pω 4 q dσp ωq " Q h pf 7 , f, f 7 , f q , where we used the Cauchy-Schwarz inequality in its simplest form, AB`CD ď ? A 2`C 2 ? B 2`D2 . Recall that both the measure 1 B4 p h and the function f are non-negative. Repeating the argument with the variables pω 2 , ω 4 q instead of pω 1 , ω 3 q finishes the proof of (2.10). To establish the case of equality, we note that, by the same argument as in the proof of Lemma 4, if there is equality in (2.10) then f σ˚f σ L 2 pR d q " f 7 σ˚f 7 σ L 2 pR d q . This implies f " f 7 ; see [13, Lemma 9]. Hence, on top of being non-negative, we may further assume that f is even (i.e. f pωq " f p´ωq for all ω P S d´1 ) in our search for maximizers. In this case we note that x f σ is real-valued, and that R d`x f σpxq˘4 hpxq dx " Q h pf, f, f, f q "ˆp S d´1 q 4 p h˜4 ÿ j"1 ω j¸4 ź j"1 f pω j q dσpω j q. (2.11) Magical identities via partial differential equations Since p h ě 0 on B 4 , one could think of applying the Cauchy-Schwarz inequality to the right-hand side of (2.11). This turns out to be an inappropriate move, which leads to an unbounded quadratic form because of the singularity of the two-fold convolution of the surface measure of the sphere at the origin (one has pσ˚σqpxq » 1{\|x\| near x " 0; see (6.15) below). In order to overcome this obstacle, Foschi [23] had the remarkable idea of introducing a suitable term on the right-hand side of (2.11) (with g " 0) in order to control this singularity. Such a move is only admissible because of the insightful geometric identity (1.10). Our goal in this section is to find a proper replacement in the general weighted situation. We do so by presenting a different perspective on how to look for such magical identities, via the connection with the underlying Helmholtz equation. 3.1. Helmholtz equation and integration by parts. The next result lies at the genesis of our magical identity. Recall from the remark after (1.6) that we may assume that supppp gq Ă B 4 , which implies that g is radial, real-valued, smooth, and that g and all of its derivatives belong to L 2 pR d q X L 8 pR d q. For simplicity, let us assume this is the case throughout §3.1 and §3.2. Proposition 6. Let f P L 2 pS d´1 q be a non-negative and even function. Then R d`x f σpxq˘4 hpxq dx " 3 4ˆRdˇˇ∇`p x f σq 2˘p xqˇˇ2 hpxq dx´1 4ˆRd`x f σpxq˘4∆hpxq dx. (3.1) Proof. Set u :" x f σ and observe that, by dominated convergence, u P C 8 pR d q. The function u is a classical solution to the Helmholtz equation ∆upxq`upxq " 0 for all x P R d . Also, by the assumptions on f , the function u is real-valued. Assume for a moment that f P C 8 pS d´1 q; this extra hypothesis will be removed at the end of the proof. The Helmholtz equation and integration by parts yield R d u 4 h "ˆR d p´∆uq u 3 h "ˆR d ∇u¨∇pu 3 hq. (3.2) Note that there are no boundary terms, since lim RÑ8ˆSd´1 R´∇ u¨ν R¯u 3 h dσ d´1,R pνq " 0, (3.3) where S d´1 R Ă R d denotes the sphere of radius R centered at the origin, and σ d´1,R is its surface measure. Identity (3.3) follows from the fact that f P C 8 pS d´1 q, since a well-known stationary phase argument [48, Chapter VIII, §3, Theorem 1] yields the decay estimate \|∇upxq\|`\|upxq\| À p1`\|x\|q 1´d 2 (3.4) for every x P R d . Estimate (3.4) and the fact that h P L 8 pR d q plainly imply (3.3). Since ∇pu 3 hq " p3u 2 ∇uqh`u 3 ∇h and u 2 \|∇u\| 2 " 1 4 \|∇pu 2 q\| 2 , further partial integrations from (3.2) yield R d u 4 h " 3 4ˆRd \|∇pu 2 q\| 2 h´ˆR d u ∇¨pu 3 ∇hq " 3 4ˆRd \|∇pu 2 q\| 2 h´1 4ˆRd u 4 ∆h, (3.5) which is the desired identity (3.1). The last identity in (3.5) amounts to realizing that ∇¨pu 3 ∇hq " ∇pu 3 q¨∇h`u 3 ∇¨p∇hq " p3u 2 ∇uq¨∇h`u 3 ∆h and thatˆR d u 3 ∇u¨∇h " 1 4ˆRd ∇pu 4 q¨∇h "´1 4ˆRd u 4 ∆h. The boundary terms in the preceding partial integrations vanish, for similar reasons to the ones mentioned in (3.3)-(3.4); here we are using that h P C 2 pR d q with h, ∇h, ∆h P L 8 pR d q. It remains to prove that the smoothness assumption on f can be dropped. For any M ą 0 and any f 1 , f 2 such that f 1 L 2 pS d´1 q` f 2 L 2 pS d´1 q ď M , since h P L 8 pR d q, by the Cauchy-Schwarz inequality and Plancherel's identity we havěˇˇˇˆR d ∇p y f 1 σq 2 2 h´ˆR d ∇p y f 2 σq 2 2 hˇˇˇˇ2 À hˆR d ∇`p y f 1 σq 2´p y f 2 σq 2˘ 2ˆR d ∇`p y f 1 σq 2`p y f 2 σq 2˘ 2 " p2πq 2dˆˆR d \|y\| 2 \|pf 1 σ˚f 1 σqpyq´pf 2 σ˚f 2 σqpyq\| 2 dy˙ˆˆR d \|y\| 2 \|pf 1 σ˚f 1 σqpyq`pf 2 σ˚f 2 σqpyq\| 2 dyÀ dˆR d p y f 1 σq 2´p y f 2 σq 2 2ˆR d p y f 1 σq 2`p y f 2 σq 2 2 À d,M f 1´f2 2 L 2 pS d´1 q . In the last line, we have used the fact that both f 1 σ˚f 1 σ and f 2 σ˚f 2 σ are supported on B 2 , as well as the Stein-Tomas estimate. This proves that the first term on the right-hand side of (3.1) is a continuous function of f P L 2 pS d´1 q. Since ∆h is also in L 8 pR d q, the same argument proves that all the terms in (3.1) are continuous in L 2 pS d´1 q, which concludes the proof by density. 3.2. Magical identity. Taking the gradient (in the variable x) in (1.1) yields ∇`p x f σq 2˘p xq " iˆp S d´1 q 2 e ix¨pω1`ω2q pω 1`ω2 qf pω 1 qf pω 2 q dσpω 1 q dσpω 2 q. Henceˇˇ∇`p x f σq 2˘p xqˇˇ2 "ˆp S d´1 q 4 e ix¨pω1`ω2´ω3´ω4q pω 1`ω2 q¨pω 3`ω4 q 4 ź j"1 f pω j q dσpω j q "´ˆp S d´1 q 4 e ix¨p ř 4 j"1 ωj q pω 1`ω2 q¨pω 3`ω4 q 4 ź j"1 f pω j q dσpω j q , (3.6) where we have used the fact that f is non-negative and even. It follows that ( 4 3 times) the first term on the right-hand side of identity (3.1) is given bŷ R dˇ∇`p x f σq 2˘p xqˇˇ2 hpxq dx "´ˆp S d´1 q 4 p h˜4 ÿ j"1 ω j¸p ω 1`ω2 q¨pω 3`ω4 q 4 ź j"1 f pω j q dσpω j q. (3.7) Similarly, for the second term on the right-hand side of (3.1), we havê R d`x f σpxq˘4∆hpxq dx "ˆp S d´1 q 4 x ∆h˜4 ÿ j"1 ω j¸4 ź j"1 f pω j q dσpω j q "´ˆp S d´1 q 4 p h˜4 ÿ j"1 ω j¸ˇ4 ÿ j"1 ω jˇ2 4 ź j"1 f pω j q dσpω j q. (3.8) At this point, we note that 3pω 1`ω2 q¨pω 3`ω4 q`ˇˇˇˇ4 ÿ j"1 ω jˇ2 " \|ω 1`ω2 \| 2`\| ω 3`ω4 \| 2´p ω 1`ω2 q¨pω 3`ω4 q ě 0 ,(3.9) in light of the Cauchy-Schwarz and the AM-GM inequalities. In fact, the left-hand side of (3.9) is zero if and only if ω 1`ω2 " ω 3`ω4 " 0. Plugging (3.7), (3.8) and (3.9) into (3.1) we arrive at R d`x f σpxq˘4 hpxq dx " 1 4ˆp S d´1 q 4 p h˜4 ÿ j"1 ω j¸`\| ω 1`ω2 \| 2`\| ω 3`ω4 \| 2´p ω 1`ω2 q¨pω 3`ω4 q˘4 ź j"1 f pω j q dσpω j q. (3.10) This is our magical identity. Note that when g " 0 we have p h˜4 ÿ j"1 ω j¸4 ź j"1 dσpω j q " p2πq d δ˜4 ÿ j"1 ω j¸4 ź j"1 dσpω j q, which is a measure supported in the submanifold of pS d´1 q 4 defined by the equation ř 4 j"1 ω j " 0. With respect to this measure, the newly introduced term (3.9) is almost everywhere equal to a multiple of \|ω 1ὼ 2 \| 2 " \|ω 3`ω4 \| 2 " \|ω 1`ω2 \| \|ω 3`ω4 \|, and we thus recover Foschi's identity [23,Eq. (9)]. 3.3. Cauchy-Schwarz. Recall that we are assuming f to be non-negative and even. In light of (3.9) and the fact that 1 B4 p h ě 0, we are now in position to move on by applying the Cauchy-Schwarz inequality on the right-hand side of (3.10). This leads tô R d`x f σpxq˘4 hpxq dx ď 1 4ˆp S d´1 q 2 f pω 1 q 2 f pω 2 q 2 r K h pω 1 , ω 2 q dσpω 1 q dσpω 2 q, (3.11) where r K h pω 1 , ω 2 q :"ˆp S d´1 q 2 p h˜4 ÿ j"1 ω j¸`\| ω 1`ω2 \| 2`\| ω 3`ω4 \| 2´p ω 1`ω2 q¨pω 3`ω4 q˘dσpω 3 q dσpω 4 q. (3.12) Since p h is radial on B 4 , r K h pω 1 , ω 2 q " r K h pρω 1 , ρω 2 q for every rotation ρ P SOpdq and, therefore, r K h depends only on the inner product ω 1¨ω2 . Thus we define K h pω 1¨ω2 q :" r K h pω 1 , ω 2 q. (3.13) Further define the functions K 1 and K g as in (3.12)-(3.13), with 1 and g replacing h, respectively. Remark: Assuming f non-negative and even, equality happens in (3.11) if and only if f is constant. To see this, just split h " 1`g and argue like in the proof of Lemma 5, using that the cases of equality for the analogous of (3.11) with h " 1 have already been completely characterized in [13,Lemma 11], and are only the constant functions. Bilinear analysis The task ahead of us now consists of analyzing the right-hand side of (3.11). A quadratic form. Consider the quadratic form H d,h pϕq :"ˆp S d´1 q 2 ϕpω 1 q ϕpω 2 q K h pω 1¨ω2 q dσpω 1 q dσpω 2 q. (4.1) This defines a real-valued and continuous functional on L 1 pS d´1 q. Indeed, K h " K 1`Kg , and so H d,h " H d,1`Hd,g . From [13,Lemma 5] it follows that (recall that p 1 " p2πq d δ in our setup) K 1 ptq " " p2πq d¨3¨22´d 2 σ d´2`S d´2˘ı p1`tq 1 2 p1´tq d´3 2 ,(4.2) where t :" ω 1¨ω2 . The continuity of H d,1 on L 1 pS d´1 q, as noted in [13,Eq. (5.19)], is a simple consequence of the fact that K 1 P L 8 pr´1, 1sq since one can prove directly from (4.1) thaťˇH d,1 pϕ 1 q´H d,1 pϕ 2 qˇˇď }K 1 } L 8 pr´1,1sq`} ϕ 1 } L 1 pS d´1 q`} ϕ 2 } L 1 pS d´1 q˘} ϕ 1´ϕ2 } L 1 pS d´1 q . The continuity of H d,g follows similarly since p g is bounded on B 4 , whence K g P L 8 pr´1, 1sq and an analogous argument applies. The following proposition is the final piece in our puzzle. Proof of part of Theorems 1 and 2: Assuming the validity of Proposition 7, we apply it with ϕ " f 2 (in which f is non-negative and even), coming from (3.11), to obtain where Y n is a spherical harmonic of degree n. Since ϕ is an even function, in the representation (4.4) we must have Y 2 `1 " 0 for all P Z ě0 . Note also that Y 0 " µ1. The partial sums ř N n"0 Y n converge to ϕ in L 2 pS d´1 q, as N Ñ 8, and hence also in L 1 pS d´1 q. Therefore, from (4.1) and (4.4), we are led to R dˇx f σpxqˇˇ4 hpxq dx ď H d,h p1q 4 σ`S d´1˘2 }f } 4 L 2 pS d´1 q .H d,h pϕq " lim N Ñ8 N ÿ m,n"0ˆS d´1 Y m pω 2 qˆˆS d´1 Y n pω 1 q K h pω 1¨ω2 q dσpω 1 q˙dσpω 2 q. (4.5) The tool to evaluate the latter inner integral is the Funk-Hecke formula: S d´1 Y n pω 1 q K h pω 1¨ω2 q dσpω 1 q " λ d,h pnq Y n pω 2 q ,(4.6) with the constant λ d,h pnq given by λ d,h pnq " σ d´2`S d´2˘ˆ1 1 C d´2 2 n ptq C d´2 2 n p1q K h ptq p1´t 2 q d´3 2 dt. (4.7) Here, C d´2 2 n denotes the Gegenbauer polynomial (or ultraspherical polynomial) of degree n and order d´2 2 . In general, for α ą 0, the Gegenbauer polynomials t Þ Ñ C α n ptq are defined via the generating function p1´2rt`r 2 q´α " 8 ÿ n"0 C α n ptq r n . (4.8) Note that, if t P r´1, 1s, the left-hand side of (4.8) defines an analytic function of r (for small r) and the righthand side of (4.8) is the corresponding power series expansion. We further remark that C α n ptq has degree n, and that the Gegenbauer polynomials C α n ptq ( 8 n"0 are orthogonal in the interval r´1, 1s with respect to the measure p1´t 2 q α´1 2 dt. Differentiating (4.8) with respect to the variable r and comparing coefficients, we obtain the following three-term recursion relation, valid for any n ě 1: 2tpn`αqC α n ptq " pn`1qC α n`1 ptq`pn`2α´1qC α n´1 ptq , which coincides with [51, Eq. (2.1)]. Since, additionally, C α 0 ptq " 1 and C α 1 ptq " 2αt, our normalization agrees with that from [51], which is going to be used later in some of our effective estimates. In this normalization, C α n p1q " Γpn`2αq n! Γp2αq andˆ1 1 C α n ptq 2 p1´t 2 q α´1 2 dt " 2 1´2α π Γpαq 2 Γpn`2αq n! pn`αq ": ph α n q 2 . (4.9) We further note that C α n p´tq " p´1q n C α n ptq and that max tPr´1,1sˇC α n ptqˇˇ" C α n p1q; see [50, Theorem 7.33.1]. Returning to our discussion, since spherical harmonics of different degrees are pairwise orthogonal, we plainly get from (4.5), (4.6), and the fact that Y n " 0 if n is odd, that H d,h pϕq " 8 ÿ "0 λ d,h p2 q }Y 2 } 2 L 2 pS d´1 q . The crux of the matter lies in the following result. Assuming the validity of Lemma 8, the proof of Proposition 7 follows at once since H d,h pϕq " 8 ÿ "0 λ d,h p2 q }Y 2 } 2 L 2 pS d´1 q ď λ d,h p0q}Y 0 } 2 L 2 pS d´1 q " H d,h pµ1q " \|µ\| 2 H d,h p1q, with equality if and only if Y 2 " 0 for all ě 1, which means that ϕ " Y 0 is a constant function. We address the proof of the key Lemma 8 in the next three sections. The spectral gap In this section, we briefly discuss the common strategy for the proof of Lemma 8, both in the analytic and C k -versions, and quantify the available gap. Throughout the rest of the paper we let ν :" d´2 2 . For n P Z ě0 , define the coefficients λ d,1 pnq and λ d,g pnq as in (4.7), with K 1 and K g replacing K h , respectively. From condition (R1), observe that K h ě K 1 ą 0 in p´1, 1q. Since C ν 0 ptq " 1 we plainly get that λ d,h p0q ě λ d,1 p0q ą 0. 5.1. The strategy. For ě 1 we proceed as follows. First we write λ d,h p2 q " λ d,1 p2 q`λ d,g p2 q. (5.1) The following observation from the proof of [13, Lemma 13] is a key ingredient in our argument: for each d P t3, 4, 5, 6, 7u there exists a constant c d ą 0 such that, for every ě 1, λ d,1 p2 q ď´c d ´d ă 0 ; (5.2) see Lemma 9 below for a precise quantitative statement. 2 In order to argue that (5.1) is negative for all ě 1, in light of (5.2) it suffices to show that, for all ě 1, we have \|λ d,g p2 q\| ă c d ´d . (5.3) If we consider the Gegenbauer expansion of K g , namely, K g ptq " 8 ÿ n"0 a ν n C ν n ptq pt P r´1, 1sq, (5.4) we find directly from (4.7) and (4.9) that λ d,g pnq " 2 π ν`1 pn`νq Γpνq a ν n . (5.5) Here we used the fact that σ d´2`S d´2˘" 2 π ν`1 2 { Γpν`1 2 q together with the duplicating formula for the Gamma function, ΓpνqΓpν`1 2 q " 2 1´2ν π 1 2 Γp2νq. Looking back at (5.3)-(5.5), we see that we need good estimates for the decay of the Gegenbauer coefficients a ν 2 in terms of the function K g , and this is ultimately where the smoothness of p gˇˇB 4 will play a role. 5.2. Quantifying the gap. We now provide an effective form of the gap inequality (5.2). Most of the work towards this goal was essentially accomplished in [13,23], and here we just revisit it in a format that is appropriate for our purposes. For convenience, let us recall (4.2) and (4.7): K 1 ptq " " p2πq d¨3¨22´d 2 σ d´2`S d´2˘ı p1`tq 1 2 p1´tq d´3 2 pt P r´1, 1sq; λ d,1 pnq " σ d´2`S d´2˘ˆ1 1 C d´2 2 n ptq C d´2 2 n p1q K 1 ptq p1´t 2 q d´3 2 dt pn P Z ě0 q. Lemma 9 (cf. [13,23]). For d P t3, 4, 5, 6, 7u let κ d :" p2πq d¨3¨23´d " σ d´2`S d´2˘‰ 2 . Then, for each ě 1, λ 3,1 p2 q " κ 3ˆ´8 p4 ´1qp4 `1qp4 `3q˙ď´ˆ2 8 π 5 35˙ ´3 ; (5.6) λ 4,1 p2 q " κ 4ˆ´8 p2 ´1q p2 `1q 2 p2 `3q˙ď´ˆ2 10 π 6 15˙ ´4 ; (5.7) λ 5,1 p2 q " κ 5˜´1 536p2 `1qp2 `2qp4 2`6 ´3q 2 `2 2˘p 4 ´3qp4 ´1qp4 `1qp4 `3qp4 `5qp4 `7qp4 `9q¸ď´ˆ2 15 π 9 2145˙ ´5 ; (5.8) λ 6,1 p2 q " κ 6˜´3 2p2 `2q 2 `3 3˘p 2 ´1qp2 `1qp2 `3qp2 `5q¸ď´ˆ2 15 π 10 1575˙ ´6 ; (5.9) λ 7,1 p2 q " κ 7˜´1 63840p2 `1qp2 `2qp2 `3qp2 `4qp4 2`1 0 ´15qp4 2`1 0 ´3q 2 `4 4˘p 4 ´5qp4 ´3qp4 ´1qp4 `1qp4 `3qp4 `5qp4 `7qp4 `9qp4 `11qp4 `13qp4 `15qḑ´ˆ2 21 π 13 1322685˙ ´7 . (5.10) Proof. The identity in (5.6) follows from [23, Proof of Lemma 5.4] (in the notation of that proof, one has C 1{2 2 " P 2 , which is an even function). The identities in (5.7)-(5.10) follow from [13, Proof of Lemma 13, Steps 2 to 5] (in the notation of that proof, there is a quantity Λ 2 pφ d q which is computed, satisfying λ d,1 p2 q "`κ d {σ d´2`S d´2˘˘Λ 2 pφ d q, with λ d,1 p2 q and κ d as defined above; recall that C d´2 2 2 is even). The upper bounds in (5.6)-(5.10) work as follows. For (5.6), one multiplies the left hand-side by the appropriate power of (in this case, 3 ) and observe that Þ Ñ´8 3 p4 ´1qp4 `1qp4 `3q defines a decreasing function of ě 1. This is a routine verification (e.g. with basic computer aid). The upper bound then comes from evaluating it at " 1. The other cases follow the same reasoning. Proof of Lemma 8: analytic version In this section we work under the hypotheses of Theorem 1; in particular, (R2.A) holds. Recall ν :" d´2 2 . 6.1. Bounds for the Gegenbauer coefficients (analytic version). As previously observed, we need decay estimates for the Gegenbauer coefficients a ν n in terms of the function K g in (5.4). The analogous situation for Fourier series is very classical, via the paradigm that regularity of the function implies decay of the Fourier coefficients. Here we face a similar situation, where the orthogonal basis is the one of Gegenbauer polynomials, and we want to deploy the same philosophy that regularity on one side implies decay on the other side. If our function, initially defined on the interval r´1, 1s, admits an analytic continuation, then we will be able to invoke careful quantitative estimates from the recent work of Wang [51]. In order to state the relevant result, given ρ ą 1, define the so-called Bernstein ellipse E ρ Ă C as E ρ :" z P C ; z " 1 2`ρ e iθ`ρ´1 e´i θ˘, 0 ď θ ď 2π ( , with foci at˘1 and major and minor semiaxes of lengths 1 2 pρ`ρ´1q and 1 2 pρ´ρ´1q, respectively; see Figure 1. The following result from [51] will be convenient for our purposes. Then, for any n ě 1, we have the following explicit estimates: \|a α n \| ď $ ' ' ' ' & ' ' ' ' % Λpn, ρ, αqˆ1´1 ρ 2˙α´1 n 1´α ρ n`1 , if 0 ă α ď 1; Λpn, ρ, αqˆ1`1 ρ 2˙α´1 n 1´α ρ n`1 , if α ą 1, (6.1) where Λpn, ρ, αq :" Γpαq M Υ 1,α n πˆ2ˆρ`1 ρ˙`2´π 2´1¯ˆρ´1 ρ˙˙, (6.2) and Υ 1,α n :" expˆ1´α 2pn`α´1q`1 12n˙. (6.3) u, and the corresponding enveloping disks D r Ă C, r P t 5 4 , 17 8 , 7 2 u. If we succeed in proving that our function K g admits an analytic continuation past a Bernstein ellipse E ρ for some ρ ą 1, then Lemma 10 will be an available tool with K " K g , α " ν and n " 2 . One readily checks that the bounds provided by (6.1) decay exponentially in n " 2 , and from (5.2) and (5.5) we see that it should be possible to achieve (5.3) as long as M " max zPEρ \|K g pzq\| is sufficiently small, which ultimately will be verified provided that the analytic continuation of p gˇˇB 4 is sufficiently small in a certain disk. 6.2. Analytic continuation of K g . Condition pR2.Aq states that p gˇˇB 4 admits an analytic extension to a disk D R 1 Ă C d , with D R Ă D R 1 and R ą 4. Recall that, for t " ω 1¨ω2 , we have K g ptq "ˆp S d´1 q 2 p g˜4 ÿ j"1 ω j¸`\| ω 1`ω2 \| 2`\| ω 3`ω4 \| 2´p ω 1`ω2 q¨pω 3`ω4 q˘dσpω 3 q dσpω 4 q. Set s :" \|ω 1`ω2 \| " p2`2tq 1 2 and K ‹ g psq :" K g ptq. Note that s P r0, 2s. We show that K ‹ g can be extended to an even analytic function on an open disk D R 2 Ă C of radius R 2 :" R 1´2 ą R´2 ą 2, and hence it admits a power series representation of the form K ‹ g psq " ř 8 "0 c 2 s 2 , which is absolutely convergent if \|s\| ă R 2 . This plainly implies that K g can be extended to an analytic function on the open disk D R 3 Ă C with R 3 :"`pR 2 q 2´2˘{ 2 ą 1 since K g ptq " K ‹ g psq " 8 ÿ "0 c 2 ps 2 q " 8 ÿ "0 c 2 p2`2tq " 8 ÿ "0 c 1 t . (6.4) This is going to be sufficient for our purposes since the Bernstein ellipse E ρ is contained in the closed disk of radius 1 2 pρ`ρ´1q; see Figure 1. We are then able to choose ρ ą 1 such that E ρ Ă D R 3 . In particular, we can choose ρ ą 1 such that 1 2 pρ`ρ´1q " 1 2`p R´2q 2´2˘. (6.5) Let us write K ‹ g psq " Ipsq`IIpsq´IIIpsq , (6.6) where the three summands are defined as follows: Ipsq :" s 2ˆp S d´1 q 2 p g˜4 ÿ j"1 ω j¸d σpω 3 q dσpω 4 q ; (6.7) IIpsq :"ˆp S d´1 q 2 p g˜4 ÿ j"1 ω j¸\| ω 3`ω4 \| 2 dσpω 3 q dσpω 4 q ; (6.8) IIIpsq :"ˆp S d´1 q 2 p g˜4 ÿ j"1 ω j¸p ω 1`ω2 q¨pω 3`ω4 q dσpω 3 q dσpω 4 q. (6.9) We show that each of these functions can be extended to an even analytic function on the open disk D R 2 Ă C. The reasoning for I and II is similar to that of III, but simpler. So we focus on III only. 6.2.1. The function III. Recall that coordinates for ω P S d´1 can be defined recursively: ω " pζ sin θ d´1 , cos θ d´1 q, ζ P S d´2 , with dσ d´1 pωq " psin θ d´1 q d´2 dθ d´1 dσ d´2 pζq , where we denote by σ d´j the surface measure on the unit sphere S d´j . Since the arc length measure on S 1 is simply dθ 1 , it follows by induction that dσ d´1 " d´2 ź j"1 psin θ d´j q d´j´1 dθ d´j dθ 1 , where 0 ď θ 1 ď 2π and 0 ď θ j ď π for j P t2, 3, . . . , d´1u. Going back to (6.9), by the radiality of p g\| B4 , no generality is lost in assuming that ω 1`ω2 " s e 1 for some s P r0, 2s, where e 1 " p1, 0, . . . , 0q P R d denotes the first coordinate vector. Writing x " px 1 , x 1 q P RˆR d´1 , we have that IIIpsq "ˆr 0,2πs 2ˆr 0,πs 2d´4 p g`s`pω 3`ω4 q 1 , pω 3`ω4 q 1˘s pω 3`ω4 q 1 d´2 ź j"1 psin θ d´j q d´j´1 dθ d´j d´2 ź j"1 psinθ d´j q d´j´1 dθ d´j dθ 1 dθ 1 , (6.10) where the variables of integration ω 3 and ω 4 are coordinate-wise given as follows: ω 3,1 " d´1 ź j"1 sin θ j , ω 4,1 " d´1 ź j"1 sinθ j ; ω 3,2 " cos θ 1 d´1 ź j"2 sin θ j , ω 4,2 " cosθ 1 d´1 ź j"2 sinθ j ; . . . . . . ω 3,d´1 " cos θ d´2 sin θ d´1 , ω 4,d´1 " cosθ d´2 sinθ d´1 ; ω 3,d " cos θ d´1 , ω 4,d " cosθ d´1 . (6.11) Note that (6.10) can be used to extend the domain of definition of the function III to D R 2 Ă C, by replacing p g\| B4 by its analytic continuation G. Such an extended function s Þ Ñ IIIpsq is clearly continuous. Let γ be an arbitrary 3 closed piecewise C 1 -curve in D R 2 Ă C. Then from (6.10) and Fubini's Theorem it follows that γ IIIpsq ds "ˆr 0,2πs 2ˆr 0,πs 2d´4 pω 3`ω4 q 1ˆˆγ G`s`pω 3`ω4 q 1 , pω 3`ω4 q 1˘s dsḋ´2 ź j"1 psin θ d´j q d´j´1 dθ d´j d´2 ź j"1 psinθ d´j q d´j´1 dθ d´j dθ 1 dθ 1 " 0. (6.12) Indeed, the innermost integral on the right-hand side of (6.12) vanishes by Cauchy's Theorem, since the function G is analytic on D R 1 Ă C d (in particular, in its first coordinate). By Morera's Theorem, it then follows that III defines an analytic function on D R 2 Ă C. Finally, observe in (6.10) that the change of variables pθ 1 ,θ 1 q Þ Ñ p´θ 1 ,´θ 1 q mod 2π, corresponding to a reflection across the hyperplane xe 1 y K , and the radiality of p g\| B4 together reveal that IIIp´sq " IIIpsq, first for every s P p´2, 2q, and hence for every s P D R 2 . This yields the qualitative proof of Lemma 8, and we now proceed to the effective implementation. 6.3. Auxiliary integrals. Let us record two integrals that shall be relevant for the upcoming discussion. Lemma 11. Let ζ P S d´1 be given. We have: pS d´1 q 2 \|ω 3`ω4 \| 2 dσpω 3 q dσpω 4 q " 2 σpS d´1 q 2 ; (6.13) pS d´1 q 2ˇp ω 3`ω4 q¨ζˇˇdσpω 3 q dσpω 4 q "˜2 d´1 Γ`d 2˘3 π Γ`d´1 2 q Γ`d`1 2˘¸σ pS d´1 q 2 ": r d σpS d´1 q 2 . (6.14) Remark: For our purposes, the pertinent values of the constants r d are: r 3 " 2 3 ; r 4 " 2 8 45 π 2 ; r 5 " 18 35 ; r 6 " 2 16 14175 π 2 ; r 7 " 100 231 . Proof. Identity (6.13) follows simply from the relation \|ω 3`ω4 \| 2 " 2`2ω 3¨ω4 , together with the fact that´p S d´1 q 2 pω 3¨ω4 q dσpω 3 q dσpω 4 q " 0. The proof of (6.14) is more interesting. First notice that the left-hand side of (6.14) is independent of ζ P S d´1 and hence we may assume without loss of generality that ζ " e d " p0, 0, . . . , 1q. Recall from [13, Lemma 5] the exact expression for the two-fold convolution of the surface measure σ: pσ˚σqpxq " 2 3´d σ d´2 pS d´2 q p4´\|x\| 2 q d´3 2 \|x\| 1 B2 pxq ; px P R d q. (6.15) The left-hand side of (6.14) is equal tô pS d´1 q 2ˆRd δpx´ω 3´ω4 q \|x¨e d \| dx dσpω 3 q dσpω 4 q "ˆR d pσ˚σqpxq \|x¨e d \| dx " 2 3´d σ d´2 pS d´2 qˆ2 0ˆˆS d´1 \|ω¨e d \| dσpωq˙p4´r 2 q d´3 2 r d´1 dr " 2 3´d`σ d´2 pS d´2 q˘2ˆˆπ 0 \| cos θ\| psin θq d´2 dθ˙ˆˆ2 0 p4´r 2 q d´3 2 r d´1 dr" 2 3´d`σ d´2 pS d´2 q˘2ˆ2 d´1˙ˆ2 2d´4ˆ1 0 p1´sq d´3 2 s d´2 2 ds" 2 d`σ d´2 pS d´2 q˘2 d´1 Γ`d´1 2˘Γ`d 2Γ`d´1 2˘, and the latter is equal to the right-hand side of (6.14). Here, in the second identity we changed variables to polar coordinates x " rω, in the third identity we changed the variable ω as described in (6.11), in the fourth identity we evaluated the trigonometric integral and changed variables r 2 " 4s in the other integral, and in the fifth identity we used the Beta function evaluation´1 0 p1´sq a´1 s b´1 ds " Γpaq Γpbq Γpa`bq for a, b ą 0. 6.4. Quantifying the perturbation. In our setup, recall that ν :" d´2 2 P 1 2 , 1, 3 2 , 2, 5 2 ( . Our objective now is to bound (5.5) using Lemma 10. We choose the particular ρ ą 1 given by (6.5), verifying ρ`ρ´1 " pR´2q 2´2 . (6.16) From (6.2) and (6.3) we obtain Λp2 , ρ, νq " Γpνq π expˆ1´ν 2p2 `ν´1q`1 24 ˙ˆ2ˆρ`1 ρ˙`2´π 2´1¯ˆρ´1 ρ˙˙m ax wPEρ \|K g pwq\| . (6.17) Let M d,R :" max zPD R \|Gpzq\|. At this point, we want to bound max wPEρ \|K g pwq\| in terms of M d,R . We have seen in (6.4) that, via the change of variables w " ps 2´2 q{2, we have K g pwq " K ‹ g psq, and whenever w P E ρ we have s P D R´2 . Using (6.6)-(6.9), it follows that \|K g pwq\| " \|K ‹ g psq\| ď \|Ipsq\|`\|IIpsq\|`\|IIIpsq\|. (6.18) Regarding IIIpsq, we look at it via (6.10), yielding the analytic continuation. Using the elementary estimates \|s\| ď R´2 and \|s e 1`ω3`ω4 \| ď \|s\|`\|ω 3`ω4 \| ď R , (6.19) together with (6.14), we plainly get from definition (6.9) \|IIIpsq\| ď pR´2q M d,Rˆp S d´1 q 2ˇp ω 3`ω4 q 1ˇd σpω 3 q dσpω 4 q ď pR´2q r d σpS d´1 q 2 M d,R . (6.20) Similarly, using the analogous expressions for the analytic continuations of Ipsq and IIpsq, the elementary inequalities (6.19), and identity (6.13) for IIpsq, one finds \|Ipsq\| ď pR´2q 2 σpS d´1 q 2 M d,R and \|IIpsq\| ď 2 σpS d´1 q 2 M d,R . (6.21) Putting together (6.18), (6.20) and (6.21) we find that max wPEρ \|K g pwq\| ď`pR´2q 2`p R´2qr d`2˘σ pS d´1 q 2 M d,R . (6.22) From (5.5), (6.1), (6.17) and (6.22), we obtain \|λ d,g p2 q\| ď β d,R G d,R p q M d,R ,(6.23) with the constant β d,R given by β d,R :" 2 2´ν π ν " 2ˆρ`1 ρ˙`2´π 2´1¯ˆρ´1 ρ˙ˆ1˘1 ρ 2˙ν´1`p R´2q 2`p R´2qr d`2˘σ pS d´1 q 2 (the minus sign above is used for ν P 1 2 , 1 ( and the plus sign for ν P 3 2 , 2, 5 2 ( ) and G d,R p q :" 1 p2 `νq expˆ1´ν 2p2 `ν´1q`1 24 ˙ 1´ν ρ 2 `1 . Final comparison. Let us write the bounds on the right-hand sides of (5.6)-(5.10) as´c d ´d (i.e. β d,R G d,R p q M d,R ă c d ´d . Since this must hold for every P N :" t1, 2, 3, . . .u, equivalently we have to ensure that M d,R ă c d β d,Rˆm ax PN G d,R p q d˙" : A d,R . (6.24) Note that the non-negative function F d,R p q :" G d,R p q d is exponentially decaying on (since ρ ą 1), and hence it must attain its maximum value (over N) at a certain d ,R . For instance, a routine verification yields the following values: This concludes the proof of Lemma 8 in the analytic case. 3 ,R 4 ,R 5 ,R 6 ,R 7 ,R A 3,R A 4,R A 5,R A 6,R A 7,R R " 3 ? 2 2`2 ; (ρ " 2 6.6. Limiting behavior. When R is sufficiently large, one can easily verify that d ,R " 1. Hence, A d,R " c d β d,R G d,R p1q . Recalling (6.16), this plainly implies that L d :" lim RÑ8 A d,R R 2 " c d 2 2´ν π ν`1 σpS d´1 q 2 1 p2`νq expˆ1´ν 2p1`νq`1 24˙. (6.25) The corresponding evaluation yields Proof of Lemma 8: C k -version In this section we work under the hypotheses of Theorem 2; in particular, (R2.C) holds. Recall ν :" d´2 2 . 7.1. Bounds for the Gegenbauer coefficients (C k -version). Not having found in the literature a result that would exactly fit our purposes (like Lemma 10 in the analytic case), we briefly work our way up from first principles. Recall the value of C α n p1q and the definition of the constant h α n in (4.9). We start with the following lemma. Lemma 12. Let K P C 0 r´1, 1s X C k p´1, 1q for some k ě 1. Let α ą 0 and assume further that t Þ Ñ K pjq ptq p1´t 2 q α`j´1 2 is bounded in p´1, 1q for 1 ď j ď k. (7.1) Consider the Gegenbauer expansion Kptq " 8 ÿ n"0 a α n C α n ptq. (7.2) Let 2 ď p ď 8. Then, for any n ě k, we have the following estimate: \|a α n \| ď D α n,k`C α`k n´k p1q˘1´2 p`hα`k n´k˘2 p ph α n q 2ˆˆ1 1ˇK pkq ptqˇˇp 1 p1´t 2 q pα`k´1 2 q dt˙1 p 1 , (7.3) where 1 p`1 p 1 " 1, and D α n,k :" k´1 ź j"0 2pα`jq pn´jqpn`2α`jq . (7.4) Proof. Recall the indefinite integral [1,Eq. 22.13.2], C α n ptqp1´t 2 q α´1 2 dt "´2 α npn`2αq C α`1 n´1 ptqp1´t 2 q α`1 2 . (7.5) From (7. 2), we may apply integration by parts k times, using (7.5) and (7.1) (to eliminate the boundary terms at each iteration 4 ), to get a α n ph α n q 2 "ˆ1 1 Kptq C α n ptq p1´t 2 q α´1 2 dt " D α n,kˆ1 1 K pkq ptq C α`k n´k ptq p1´t 2 q α`k´1 2 dt. (7.6) Let δ " 1´2 p ě 0. 5 Using thatˇˇC α`k n´k ptqˇˇď`C α`k n´k p1q˘δˇˇC α`k n´k ptqˇˇp 1´δq ,(7.7) and applying Hölder's inequality with exponents p and p 1 below, we observe that the right-hand side of (7.6) is, in absolute value, dominated by ď D α n,k`C α`k n´k p1q˘δˆ1 1´ˇK pkq ptqˇˇp1´t 2 q pα`k´1 2 qp 1`δ 2 q¯´ˇC α`k n´k ptqˇˇp 1´δq p1´t 2 q pα`k´1 2 qp 1´δ 2 q¯d t ď D α n,k`C α`k n´k p1q˘δˆˆ1 1ˇK pkq ptqˇˇp 1 p1´t 2 q pα`k´1 2 q dt˙1 p 1ˆˆ1 1 C α`k n´k ptq 2 p1´t 2 q pα`k´1 2 q dt˙1 p " D α n,k`C α`k n´k p1q˘δ`h α`k n´k˘2 pˆˆ1 1ˇK pkq ptqˇˇp 1 p1´t 2 q pα`k´1 2 q dt˙1 p 1 . This yields the proposed estimate. Remark: Observe that in Lemma 12 we are not specializing to k " kpdq " tpd`3q{2u; rather, it holds for any k ě 1. Further observe that, for fixed k, as n Ñ 8, we have D α n,k » n´2 k ; h α n » n α´1 ; h α`k n´k » n α`k´1 ; C α`k n´k p1q » n 2α`2k´1 . Hence, assuming that the integral on the right-hand side of (7.3) is finite, the dependence on n of (7.3) is given by D α n,k`C α`k n´k p1q˘1´2 p`hα`k n´k˘2 p ph α n q 2 » n´2 k`p1´2 p qp2α`2k´1q`pα`k´1qp 2 p q´2α`2 . From (5.3) and (5.5), this decay in n (with α " νq will suffice for our purposes, provided that 2k``1´2 p˘p 2ν`2k´1q`pν`k´1qp 2 p q´2ν`2 ď´d`1. This leads us to k ě dpp´1q`2 2 . (7.8) Since p ě 2, inequality (7.8) plainly implies that k ě pd`2q{2 and, since k is an integer, we end up with k ě tpd`3q{2u as our minimal regularity assumption in this setup. From now on we specialize matters to our particular situation by letting, in the notation of Lemma 12, K " K g ; α " ν " d´2 2 ; k " kpdq " Z d`3 2^; and p " ppdq " 2`ˆ1´p´1 q d 2˙1 d . (7.9) We postpone the discussion of why the function K g verifies condition (7.1) until the next subsection, and for now follow up with a suitable upper bound for the integral appearing on the right-hand side of (7.3) . Lemma 13. Let d P t3, 4, 5, 6, 7u. In the notation of Lemma 12, with the specialization (7.9), we havêˆ1 1ˇK pkq g ptqˇˇp 1 p1´t 2 q pν`k´1 2 q dt˙1 p 1 ď 2 1 p 1 sup tPp´1,1q´ˇK pkq g ptqˇˇp2`2tq k´1¯. Proof. Observe that, for t P p´1, 1q,ˇˇK pkq g ptqˇˇp1´t 2 q 1 p 1 pν`k´1 2 q "ˇˇK pkq g ptqˇˇp1`tq 1 p 1 pν`k´1 2 q p1´tq 1 p 1 pν`k´1 2 q ď 2 1 p 1 pν`k´1 2 qˇK pkq g ptqˇˇp1`tq 1 p 1 pν`k´1 2 q "ˇˇK pkq g ptqˇˇp2`2tq 1 p 1 pν`k´1 2 q "´ˇˇK pkq g ptqˇˇp2`2tq k´1¯p 2`2tq 1 p 1 pν`k´1 2 q´pk´1q ď « sup tPp´1,1q´ˇK pkq g ptqˇˇp2`2tq k´1¯ff p2`2tq 1 p 1 pν`k´1 2 q´pk´1q . (7.10) Note that with the specialization (7.9) we havê ν`k´1 2˙´p 1 pk´1q "´1 2 . (7.11) Hence, from (7.10) and (7.11) we plainly getˆ1 (7.12) which leads us to the desired conclusion. 1ˇK pkq g ptqˇˇp 1 p1´t 2 q pν`k´1 2 q dt˙1 p 1 ď « sup tPp´1,1q´ˇK pkq g ptqˇˇp2`2tq k´1¯ffˆˆ1 1 p2`2tq´1 2 dt˙1 p 1 , Remark: There is a subtle reason for the particular choice of ppdq in (7.9). The reader may wonder why we are not simply choosing p " 2 in all cases. The reason is as follows. There are two competing forces for the value of p in our argument. On the one hand, from (7.8) one sees that, the smaller the value of p, the smaller the number of derivatives we have to require from our function (which we intend to keep to a minimum). On the other hand, larger values of p place us in a better position to control potential singularities arising in the proof of Lemma 13, a crucial intermediate step in our proof. When the dimension d is even, the choice p " 2 yields an integer number on the right-hand side of (7.8) and we proceed with this choice. When the dimension d is odd, the choice p " 2 yields an integer plus a half on the right-hand side of (7.8). Since we are not entering the realm of fractional derivatives in this paper, this would force us to move k to the next integer (as such, in some vague sense, we would have half a derivative to spare). Moreover, for odd dimensions d, such a choice p " 2 and k " rpd`2q{2s " pd`3q{2 would yield exactly´1 in place of´1 2 on the right-hand side of (7.11), which in turn would make the corresponding integral on the right-hand side of (7.12) diverge. The natural solution is then to use this spare half derivative to increase the value of p slightly, making the right-hand side of (7.9) coincide with the integer pd`3q{2. This leads us to the choice ppdq. 7.2. Relating the derivatives of K g and K ‹ g . As in §6.2, set s :" p2`2tq 1 2 and K ‹ g psq :" K g ptq, with t P r´1, 1s and s P r0, 2s. The next task is to express the derivatives of K g ptq in terms of derivatives of K ‹ g psq. We collect the relevant information in the next lemma. Lemma 14. Assume that K ‹ g : p0, 2q Ñ R is sufficiently smooth. For t P p´1, 1q we have K p1q g ptq " pK ‹ g q p1q psq s´1 ; K p2q g ptq p2`2tq " pK ‹ g q p2q psq´pK ‹ g q p1q psq s´1 ; K p3q g ptq p2`2tq 2 " pK ‹ g q p3q psq s´3pK ‹ g q p2q psq`3pK ‹ g q p1q psq s´1 ; K p4q g ptq p2`2tq 3 " pK ‹ g q p4q psq s 2´6 pK ‹ g q p3q psq s`15pK ‹ g q p2q psq´15pK ‹ g q p1q psq s´1 ; K p5q g ptq p2`2tq 4 " pK ‹ g q p5q psq s 3´1 0pK ‹ g q p4q psq s 2`4 5pK ‹ g q p3q psq s´105pK ‹ g q p2q psq`105pK ‹ g q p1q psq s´1. Proof. Note that, for t P p´1, 1q, we have s P p0, 2q and Bs Bt " 1 s . Hence B Bt K g ptq "ˆ1 s B Bs˙K ‹ g psq. The lemma follows by applying the operator`1 s B Bs˘t o K ‹ g psq a total of j times (1 ď j ď 5), and then multiplying by p2`2tq j´1 " s 2j´2 . Recall the representation (6.6)-(6.9) for K ‹ g psq, after the change of variables proposed in (6.10) (which is performed on IIIpsq but applies to Ipsq and IIpsq as well). Note that the function p gˇˇB 4 appears in this expression and its regularity now enters into play. Since p g P C k pB 4 q X C 0 pB 4 q, with bounded partial derivatives of order up to k in B 4 (condition (R2.C)), expression (6.10) and its analogues for Ipsq and IIpsq define K ‹ g psq as an even function that belongs C k pp´2, 2qq X C 0 pr´2, 2sq, with bounded partial derivatives of order up to k in p´2, 2q (for the claim that it is even, the argument is as in §6.2.1). In particular pK ‹ g q p1q p0q " 0 and the mean value theorem yields, for any s P p0, 2q,ˇp K ‹ g q p1q psqˇˇď \|s\| max uPr0,ssˇp K ‹ g q p2q puqˇˇ. (7.13) As a by-product of Lemma 14 observe that all the functions K pjq g ptq p2`2tq j´1 (1 ď j ď 5) are bounded in p´1, 1q and hence condition (7.1) clearly holds. Our next result bounds the expressions pK ‹ g q pjq psq s j´2 (1 ď j ď 5), appearing in Lemma 14, in terms of the supremum of the partial derivatives of p g. Lemma 15. Let d P t3, 4, 5, 6, 7u and r d as in (6.14). Let M d " max α sup ξPB4ˇB α p gpξqˇˇ, where the first maximum is taken over all multi-indexes α P Z d ě0 of the form α " pα 1 , 0, 0, . . . , 0q, with 0 ď α 1 ď kpdq. Then, for s P p0, 2q,ˇˇp K ‹ g q pjq psq s j´2ˇď 2 j´2`j2`3 j`6`pj`2qr d˘σ pS d´1 q 2 M d for 2 ď j ď kpdq. (7.14) andˇˇˇp Proof. The idea is relatively simple, and matters boil down to certain standard computations, so we are brief with the details. We will take j derivatives (0 ď j ď kpdq) of the expressions Ipsq, IIpsq and IIIpsq in the form (6.10) (note that derivatives with respect to s on p g will be associated to a multi-index pα 1 , 0, . . . , 0q in the way we set up things), use the triangle inequality and the elementary estimates K ‹ g q p1q psq s´1ˇˇď s d σpS d´1 q 2 M d ;(7.\|s e 1`ω3`ω4 \| ď 4; sup ξPB4ˇB α p gpξqˇˇď M d . (7.18) Following this procedure, and using (6.13) when bounding the derivatives of IIpsq, we find, 7 for 0 ď j ď kpdq,ˇI In the analysis of IIIpsq we can further take advantage of the fact that p g is non-negative on B 4 as follows. pjq psqˇˇď`jpj´1q`2js`s 2˘σ pS d´1 q 2 M d ,(7. Split the integral on the right-hand side of (6.10) into two integrals IIIpsq " III`psq`III´psq, (7.21) where III`psq (resp. III´psq) is the integral over the region where pω 3`ω4 q 1 ě 0 (resp. pω 3`ω4 q 1 ă 0). Then, for 0 ă s ă 2, we have III`psq ě 0 and III´psq ď 0. Moreover, using (7.18) and (6.14) we havěˇI Arguing similarly for the first derivative III p1q psq (for the part that retains p g we proceed as above, and for the part with B α p g we use (7.18) and (6.14)) we finďˇI II`psqˇˇď s M dˆp S d´1 q 2 pω3`ω4q1ě0 pω 3`ω4 q 1 dσpω 3 q dσpω 4 q " s 2 M dˆp S d´1 q 2ˇp ω 3`ω4 q 1ˇd σpω 3 q dσpω 4 q " s 2 r d σpS d´1 q 2 M d ,II p1q psqˇˇď`s`1 2˘r d σpS d´1 q 2 M d .(7.23) For 2 ď j ď kpdq, only partial derivatives B α p g with \|α\| P tj´1, ju appear in III pjq psq. One proceeds by applying the triangle inequality, (7.18) and (6.14) to geťˇI II pjq psqˇˇď pj`sq r d σpS d´1 q 2 M d . (7.24) 6 In principle, the values of s d can be computed with arbitrary precision, since this amounts to solving a cubic equation on the variable s in (7.16). For simplicity, we shall use the stated bounds in the final computation. 7 The case j " 0 will be used later on in the argument; see §7. 3.2. and, provided 2 ě k, we have that G d p q :" p2 q dˆ1 p2 `νq˙˜k´1 ź j"0 1 p2 ´jqp2 `2ν`jq¸ˆΓ p2 `2ν`kq p2 ´kq!˙1´2 pˆΓ p2 `2ν`kq p2 ´kq! p2 `νq˙1 pˆp 2 q! p2 `νq Γp2 `2νq˙( 7.31) " p2 q d " p2 `νq´ś k´1 j"0 p2 ´jqp2 `2ν`jq¯´ś for any with 2 ě k. In order to establish (7.33), we note that the terms in the first product in the denominator of (7.32) verify p2 ´1qp2 `2ν`1q ě p2 ´2qp2 `2ν`2q ě . . . ě p2 ´k`1qp2 `2ν`k´1q , (7.34) and the latter verifies p2 ´k`1qp2 `2ν`k´1q ě p2 q 2 (7.35) provided 2 ě pk´1qp2ν`k´1q{p2νq. This happens almost always, in which case (7.34) and (7.35) easily lead to (7.33). The only cases left open (recall that we are assuming that 2 ě k) are " 2 in dimensions d P t3, 5, 6u and " 3 in dimension d " 7. In these cases, one simply checks directly that (7.33) holds. 7.3.2. Conclusion. In light of (7.29) and (7.33) it suffices to have, for 2 ě k, M d ă 1 β d ": C d . (7.36) Now it is matter of carefully evaluating (7.30). The bounds in (7.17) for s d , applied in the definition of b k,d in (7.26), suffice to give us a 3-digit precision: There is a final minor point left, which is to ensure the validity of (5.3) for 2 ď 2 ă k (that is, " 1 in dimensions d P t3, 4, 5, 6u, and P t1, 2u in dimension d " 7). To see this, we start with the definition of the Gegenbauer coefficient a ν 2 in (5.4) (recall also (4.9)), and apply the Cauchy-Schwarz inequality to obtaiňˇa ν 2 ph ν 2 q 2ˇ"ˇˆ1 1 K g ptq C ν 2 ptq p1´t 2 q ν´1 2 dtˇˇˇˇď h ν 2 ˆˆ1 1 K g ptq 2 p1´t 2 q ν´1 2 dt˙1 Here h " 1`g, and we assume for the next proposition that g satisfies (R1), with "radial" replaced merely by "even", and (R4.A) or (R4.C). The constant function 1 is a maximizer of (1.4) if and only if Φpf q ď Φp1q for all f P L 2 pS d´1 q. A necessary condition for this is that 1 be a critical point of the functional Φ, and we provide a characterization of this condition in the following result. Note that, by the non-negativity of p gˇˇB 4 , the numerator of (9.1) is strictly positive when f " 1 (recall (2.2)), hence Φ is differentiable at 1. As an immediate consequence, there exist non-radial even functions p g : B 4 Ñ R ě0 for which 1 fails to maximize the corresponding inequality (1.5) (but see the remark after the proof of Proposition 17). A simple example is obtained by setting ξ˘:" p˘4, 0, . . . , 0q P R d , and considering a non-zero, even function p g : R d Ñ R ě0 which is supported on the closure of Bpξ´, 1q Y Bpξ`, 1q and strictly positive on Bpξ´, 1{2q Y Bpξ`, 1{2q. In this case, pp gˇˇB 4˚σ˚3 qp˘1, 0, . . . , 0q ą 0, but since the support of the latter function does not contain S d´1 , it cannot be constant there, and by Proposition 17 the function 1 is not a critical point of Φ. Proof of Proposition 17. As in [18, §16], we may restrict attention to functions of the form f " 1`λϕ, where ϕ K 1, ϕ is real-valued and even, λ ą 0 is small enough, and }ϕ} L 2 pS d´1 q " where γpωq :"´p S d´1 q 3 p g´ω`ř 4 j"2 ω j¯d σpω 2 qdσpω 3 qdσpω 4 q. One easily checks that γ " pp gˇˇB 4˚σ˚3 qˇˇS d´1 . From (9.2), it then follows that 1 is a critical point of Φ if γ is constant, which establishes the first assertion of the proposition. For the converse direction, start by noting that γ defines a continuous, even, non-negative function on S d´1 under our assumptions. If γ is not constant, then there exist ω 1 ‰˘ω 2 P S d´1 such that γpω 1 q ą γpω 2 q ě 0. Let δ ą 0 be small enough, such that inf ωPCpω1,δq γpωq ą sup ωPCpω2,δq γpωq, and so Cpω 1 , δq X Cpω 2 , δq " H, (9.3) Figure 2. Graphs of p H 2 " p H 2 pρq for ρ P r0, 5s and ρ P r 7 2 , 4s, respectively. One can place inequality (1.2) within a larger program, via the following weighted setup. Given a bounded function h : R d Ñ C, which functions maximize the weighted adjoint Fourier restriction inequalityˇˇˇˆR dˇx f σpxqˇˇq hpxq dxˇˇˇˇ1 {q ď C }f } L 2 pS d´1 q ? (1.3) the left-hand side is non-negative; recall(2.11). The fact that(4.3) holds for all f P L 2 pS d´1 q (with the absolute value of the integral on the left-hand side) follows from Lemmas 4 and 5. Note that equality holds in (4.3) if f " 1. This establishes the claim of Theorems 1 and 2 that constant functions maximize the weighted adjoint restriction inequality (1.5).We shall first discuss the quantitative part of Theorems 1 and 2, and then return to the full characterization of maximizers in Section 8 below.4.2.Proof of Proposition 7. In order to prove Proposition 7, we may work without loss of generality with ϕ P L 2 pS d´1 q. The general case, including the characterization of the cases of equality, follows by a density argument as outlined in our precursor [13, Proof ofLemma 12], using the continuity of H d,h in L 1 pS d´1 q.4.2.1.Funk-Hecke formula and Gegenbauer polynomials. If ϕ P L 2 pS d´1 q we write Lemma 8 ( 8Signed coefficients). Let d P t3, 4, 5, 6, 7u. Under the hypotheses of Theorem 1 presp. Theorem 2 q there exists a positive constant A d,R presp. C d q such that, if (1.7) presp. (1.9)q holds, then λ d,h p0q ą 0 and λ d,h p2 q ă 0 for every ě 1. Lemma 10 . 10(Wang [51, Theorem 4.3]) Let K be a function that is analytic inside and on the Bernstein ellipse E ρ for some ρ ą 1. Let M :" max zPEρ \|Kpzq\|. Let α ą 0 and consider the Gegenbauer expansion pt P r´1, 1sq. Figure 1 . 1Bernstein ellipses E ρ in the complex plane, ρ P t2, 4, c 3 " 32 8 π 5 35 , and so on). Hence, from (5.3), (5.6)-(5.10), and (6.23), it suffices to have that L 3 " 31.826 . . . ; L 4 " 24.555 . . . ; L 5 " 98.593 . . . ; L 6 " 296.255 . . . ; L 7 " 579.209 . . . and this concludes the proof of Theorem 1. we left clear where each term is coming from in (7.28), and in (7.32) we proceeded with the full simplification, taking advantage of the fact that 2ν P N and the Gamma functions are all classical factorials. 7.3.1. The upper bound for G d . With our specialization (7.9) one can check directly from (7.32) that lim Ñ8 G d p q " 1. Moreover, we actually have G d p q ď 1(7.33) C 3 " 30.157 . . . ; C 4 " 0.918 . . . ; C 5 " 0.908 . . . ; C 6 " 1.099 . . . ; C 7 " 0.534 . . . . (7.37) \|K g ptq\|. Proposition 17 ( 17Motivation for the radiality assumption). The function 1 is a critical point of Φ if and only if p gˇˇB 4˚σ˚3 is constant on S d´1 . be as in condition pR2.Aq. Then, for each d P t3, 4, 5, 6, 7u, there exists a positive constant A d,R such that the following holds: if Moreover, the limit L d :" lim RÑ8 A d,R R 2 exists and is given by (6.25), corresponding to L 3 " 1.82 . . . ; L 4 " 24.55 . . . ; L 5 " 98.59 . . . ; L 6 " 296.25 . . . ; L 7 " 579.20 . . . . with z " pz 1 , z 2 , . . . , z d q P C d . We emphasize the fact that G only needs to be an analytic continuation of p gˇˇB 4 , and not of p g itself. It may be the case that G ‰ p g in B R zB 4 .Theorem 2 (Sharp weighted adjoint Fourier restriction: C k -version). Let g : R d Ñ C be a function verifying the regularity conditions pR1q and pR2.Cq above, and set h " 1`g. Then, for each d P t3, 4, 5, 6, 7u, there exists a positive constant C d such that the following holds: ifmax zPD RˇG pzqˇˇă A d,R , (1.7) then the constant functions are maximizers of the weighted adjoint Fourier restriction inequality (1.5). Our constant A d,R is effective, given by (6.24). For instance, when R " 5, we have A 3,5 " 6.49 . . . ; A 4,5 " 90.10 . . . ; A 5,5 " 363.29 . . . ; A 6,5 " 1092.17 . . . ; A 7,5 " 2131.26 . . . . (1.8) Remark: As an example, Theorem 1 can be applied to the situation when g verifies pR1q and p gpξq " a`b\|ξ\| 2 for ξ P B 4 , with \|b\| ă L d . In fact, given (1.8), one can choose R large enough so that (1.7) holds, where the analytic continuation is Gpzq " a`bpz 2 1`. . .`z 2 d q sup ξPB4ˇB α p gpξqˇˇă C d , (1.9) where P is an appropriate meromorphic function (e.g. a polynomial) and µ is a suitable non-negative measure on p0, 8q.For instance, one could take p gpξq " c e´\| ξ\| 2 {2 {p36`\|ξ\| 2 q, for some sufficiently small constant c ą 0. Observe that this particular p g admits an analytic continuation to the open disk D 6 but not beyond that. Such examples are prototypical of the analytic case in the following sense: the fact that p g is a radial function on B 4 Ă R d which admits an analytic continuation G to the disk D 4 Ă C d is equivalent to the existence of a representation of the formp gpξ 1 , ξ 2 , . . . , ξ d q " G`pξ 2 1`ξ 2 2`. . .`ξ 2 d q 1{2f Proposition 7. Let d P t3, 4, 5, 6, 7u. Let ϕ P L 1 pS d´1 q be an even function and writeµ " 1 σ`S d´1˘ˆS d´1 ϕpωq dσpωq for the average of ϕ over S d´1 . Under the hypotheses of Theorem 1 presp. Theorem 2 q there exists a positive constant A d,R presp. C d q such that, if (1.7) presp. (1.9)q holds, then H d,h pϕq ď H d,h pµ1q " \|µ\| 2 H d,h p1q , with equality if and only if ϕ is a constant function. Remark: Note that6 7.88 ă s 3 ă 7.89 ; 7.67 ă s 4 ă 7.68 ; 7.53 ă s 5 ă 7.54 ; 7.42 ă s 6 ă 7.43 ; 7.34 ă s 7 ă 7.35.(7.17) 15) with s d " max 0ďsď2ˆm in " s 2`2 s`2`ps`1 2 qr d s , s 2`4 s`4`ps`2qr d ˙. (7.16) 19)ˇI I pjq psqˇˇď 2 σpS d´1 q 2 M d .(7.20) and the exact same bound holds forˇˇIII´psqˇˇ. Going back to(7.21) and recalling that III`psq and III´psq have opposite signs, it follows thatˇˇIIIpsqˇˇď s 2 r d σpS d´1 q 2 M d . (7.22) 1. A straightforward calculation Φp1`λϕq " 4}1}´4 L 2 pS d´1 qˆR pxq hpxq dx » dˆR pϕσ˚σ˚3qpξq p hpξq dξ,where the last identity follows from Plancherel. That p h can be replaced by p g in the latter integral follows from (2.1) together with the observation that 8 ϕpω 1 q δˆξ´ř 4 j"1 ω j˙d σp ωq p gpξq dξ ω j¸d σp ωq "ˆSyields d dλ λ"0 d x ϕσpxq p σ 3 d R d pϕσ˚σ˚3qpξq δpξq dξ " pϕσ˚σ˚3qp0q " σ˚3p1qˆS d´1 ϕ dσ " 0. Consequently, d dλ λ"0 Φp1`λϕq » dˆR d pϕσ˚σ˚3qpξq p gpξq dξ "ˆB 4ˆpS d´1 q 4 "ˆp S d´1 q 4 ϕpω 1 q p g˜4 ÿ j"1 d´1 ϕγ dσ, (9.2) Recall that txu denotes the greatest integer that is less than or equal to x. If d ě 8, then we start to observe that λ d,1 p2q ą 0 and this step of the proof breaks down. "Triangle" would suffice. Condition (7.1) can be weakened, but its present form suffices for our purposes. 5 This is where the hypothesis p ě 2 is needed: in order to have δ ě 0 and, consequently, the valid inequality (7.7). By a slight abuse of notation, we are using the fact that the spherical convolution defines a radial function on R d and, given r ě 0, denote by σ˚3prq the value attained on the sphere \|ξ\| " r. Adding up(7.19),(7.20)and(7.24)we find, for 2 ď j ď kpdq:ˇˇp K ‹ g q pjq psqˇˇďˇˇI pjq psqˇˇ`ˇˇII pjq psqˇˇ`ˇˇIII pjq psqˇď``j pj´1q`2js`s 2˘`2`p j`sq r d˘σ pS d´1 q 2 M d .(7.25) Multiplying (7.25) by s j´2 and using that s ď 2 we arrive at(7.14).Note that we already have two upper bounds forˇˇpK ‹ g q p1q psq s´1ˇˇ. One comes from adding(7.19), (7.20) (with j " 1q and(7.23), and dividing by s, and the other one comes from(7.13), in which we can use(7.25)(with j " 2). We can take the minimum of these two upper bounds, i.e.ˇˇp K ‹ g q p1q psq s´1ˇˇď min " s 2`2 s`2`ps`1 2 qr d s , s 2`4 s`4`ps`2qr d σpS d´1 q 2 M d , which leads to(7.15).By the triangle inequality, Lemmas 14 and 15 together yield the following bounds.Lemma 16. Let d P t3, 4, 5, 6, 7u, r d as in (6.14) and s d as in(7.16).where the first maximum is taken over all multi-indexes α P Z d ě0 of the form α " pα 1 , 0, 0, . . . , 0q, with 0 ď α 1 ď kpdq. Then, for t P p´1, 1q,ˇK pjq g ptq p2`2tq j´1ˇď b d,j σpS d´1 q 2 M d p1 ď j ď kpdqq ,Recall that we are working under the specialization (7.9), and the Gegenbauer expansion of K g is given by(5.4). For 2 ě k, in light of (5.5) and Lemmas 12, 13, 16, we havěˇλ d,g p2 qˇˇ"Writing the bounds on the right-hand sides of (5.6)-(5.10) as´c d ´d , from (5.3) and (7.27) we seekWe now multiply both sides by pc´1 d 2´dqp2 q d and plug in the definitions of D ν 2 ,k in (7.4), and C ν`k 2 ´k p1q, h ν`k 2 ´k , h ν 2 in (4.9). By isolating the terms that depend on , we arrive at the following reformulation of (7.28):One can then directly check that condition (7.36)-(7.37) also implies that the right-hand side of (7.40) is strictly less than c d ´d in the remaining cases 2 ď 2 ă k.This concludes the proof of Lemma 8 in the C k -case, and hence the proof of Theorem 2.8. Full classification of maximizers: proof of Theorem 3Consider the subclass X Ă L 2 pS d´1 q of characters given by X " f P L 2 pS d´1 q ; f pωq " c e iy¨ω , for some y P R d and c P Czt0u ( .Part (i).If p g " 0 on the ball B 4 , for any f P L 2 pS d´1 q we note thatand the complex-valued maximizers of (1.5) coincide with the complex-valued maximizers of the unperturbed adjoint Fourier restriction inequality, which have been classified in[13,23]. This is exactly the subclass X .Part (ii).The following chain of inequalities contains all the steps we followed in the previous sections in order to prove our sharp inequality:ˇˇˇˆRThis chain is sharp because all the inequalities above are equalities if f is constant. So, if at least one of these inequalities is strict, then f is not a maximizer.We claim that, if f P L 2 pS d´1 q is a maximizer, then f P X . In fact, by the conditions for equality in Lemma 4, Lemma 5 and Proposition 7 (recall that in Lemma 4 we only stated a necessary condition) any maximizer f P L 2 pS d´1 q must verifywhere γ ą 0 is a constant. These are exactly the same conditions that were used in[13]in order to show that f P X . We briefly recall the argument. The first condition in(8.2)implies that there exists a measurable function Φ :for σ 2 -a.e. pω 1 , ω 2 q P pS d´1 q 2 ; see[13,Lemma 8]. By the second and third conditions in (8.2), relation(8.3)The only solutions to this functional equation are of the form f pωq " c e ζ¨ω , for ζ P C d and c P Czt0u; see[13,Theorem 4]. Finally, since \|f \| is constant, we must have ζ " iy for some y P R d , and f P X as claimed.This does not mean that any f P X is a maximizer. In fact, we now verify that only the constant functions are maximizers in the general case. This is a distinct feature of our weighted setup.Let f pωq " c e iy¨ω for y P R d and c P Czt0u. By our assumptions, since p g is continuous, non-negative, and not identically zero on B 4 , there is a subset A Ă B 4 of positive measure such that p gpξq ą 0 for every ξ P A.We claim thatfor any y P R d zt0u. Using the triangle inequality in (8.4), followed by an application of (8.5), we then conclude that the only maximizers to (1.5) are the constant functions. To prove our claim, we start by observing that G P L 2 pR d q, and that p Gpzq " p2πq d p gpzq pσ˚σ˚σ˚σqpzq. where Cpω, rq Ă S d´1 denotes the open cap of radius r centered at ω. Let ϕ " 1 Cpω1,δq`1Cp´ω1,δq´1Cpω2,δq1Cp´ω2,δq . Then ϕ P L 8 pS d´1 q is non-zero, real-valued and even, and such that ϕ K 1. Moreover, (9.3) forces 1 2ˆSd´1 ϕγ dσ "ˆC pω1,δq γ dσ´ˆC pω2,δq γ dσ ą 0, which in light of (9.2) implies that 1 is not a critical point of Φ. This concludes the proof.Remark: As noted after the statement of Proposition 17, a non-radial function G : B 4 Ñ R will in general fail to satisfy the property that G˚σ˚3 is constant on S d´1 . However, and perhaps surprisingly, there exist non-radial functions G for which this property does hold. A simple example in dimension d " 3 is given byGpyq " up\|y\|q y \|y\|¨e 3 , where uprq " rp2´rqˆr´3 5 26˙1 r0,2s prq, y P R 3 , r ą 0;(9.4)here, e 3 " p0, 0, 1q. To prove this, recall that σ˚3 is supported on B 3 Ă R 3 and satisfies σ˚3p\|y\|q " 8π 2 for \|y\| ď 1 and σ˚3p\|y\|q " 4π 2 p3{\|y\|´1q for 1 ď \|y\| ď 3; see[37,Eq. (3.11)]. Given ω P S 2 , let R ω P SOp3q be a rotation such that R ω ω " e 3 . Hence,Lettingω " R ω e 3 and r " \|y\|, we have that GpR´1 ω yq " uprq r y 3ω3`u prq r py 1 , y 2 , 0q¨ω. Observe that the second summand does not contribute to any of the integrals in (9.5), as the change of variables py 1 , y 2 , y 3 q Þ Ñ p´y 1 ,´y 2 , y 3 q reveals. Letting t " y 3 {r and invoking the fact that u is supported on B 2 Ă R 3 , the right-hand side of (9.5) then reads, up to an irrelevant factor of 6πω 3 , 9and the latter integral vanishes, since the function u was defined precisely to ensure this (as a side remark, note that any non-zero function u " uprq supported on r0, 2s and orthogonal to the quartic polynomial r 3 p8´3rq on that interval would work). We conclude that G˚σ˚3 " 0 on S 2 , even though G is plainly non-radial on B 2 Ă R 3 .9.2.Non-negativity and smallness. In this section, we construct examples revealing that the nonnegativity condition in (R1) and the smallness condition in (1.7) and (1.9) (associated to (R2.A) and (R2.C), respectively) cannot be dropped entirely from the set of running assumptions. As usual, J α will denote the Bessel function of the first kind of order α. Considering a radial weight h, we will require the following computations, for ϕ P L 2 pS d´1 q satisfying ϕ K 1:9Here we are using the facts that \|y´e 3 \| ď 1 if and only if t ě r 2 , and that \|y´e 3 \| ď 3 if and only if t ě r 2´8 2r .By the previous subsection, 1 is a critical point of the functional Φ. We compute its second variation:A necessary condition for constant functions to maximize (1.5) is that such second variation be non-positive for all test functions ϕ. We will consider ϕ P tY 2 , iY 1 u, where Y k denotes a real spherical harmonic on S d´1 of degree k. We recall that ν " d{2´1 and the formula from[15,Eq. (2.3)]:Proposition 18 (Motivation for the smallness assumption). Assume d " 3. There exist c 0 ą 0 and a radial Schwartz function g : R 3 Ñ R with p g ě 0, such that, letting h " 1`cg for c ą 0, the following holds: Proof. Specializing (9.6) to ϕ " Y 2 and invoking (9.7), we obtain via Plancherel's identity thatwhere H 2 prq :" J 2 ν prqr3J 2 ν`2 prq´J 2 ν prqsr 4´2d . On the right-hand side of (9.9), the hats refer to the Fourier transform of h and H 2 seen as functions of the radial variable of R d , i.e., Next we observe that the polynomial 3ρ 4´1 2ρ 3`6 ρ 2´1 6 has exactly one positive root ρ 0 " 3.556 . . ., and that p H 2 pρq ă 0 for ρ P r0, ρ 0 q, whereas p H 2 pρq ą 0 for ρ P pρ 0 , 4q; seeFigure 2. Consider a non-zero, smooth, compactly supported bump function ψ P C 8 0 pRq such that ψ ě 0 and supppψq Ď rρ 0 , 4s, and define the radial function g : R 3 Ñ R via p g " p gp\|¨\|q " ψ (hence g is Schwartz). Recalling that hp\|x\|q " 1`c gp\|x\|q, hence p hp\|ξ\|q " p2πq 3 δpξq`c p gp\|ξ\|q, we infer from (9.9) thatfor which (9.8) holds with c 0 " 8 5 p2πq 3´´4 ρ0 p H 2 pρqψpρqρ 2 dρ¯´1 ą 0. This concludes the proof.Proposition 19 (Motivation for the non-negativity assumption). Assume d " 3. There exist c 0 ą 0 and a radial and smooth g P L 1 pR 3 q X L 8 pR 3 q satisfying p gp\|ξ\|q ď 0 for all ξ P B 4 , with the following property. If c P p0, c 0 q and h " 1`cg, thenΦp1`λiY 1 q ą 0.(9.11)In particular, the function 1 fails to maximize (1.5) if c ă c 0 .Proof. In a similar way to the proof of Proposition 18, identities (9.6) and (9.7) yieldΦp1`λiY 1 q » dˆ8 0 H 1 prqr d´1 dr`cˆ8 0 H 1 prqgprqr d´1 dr,(9.12)provided c P p0, c 0 q and c 0 ą 0 is sufficiently small (to be chosen below), where H 1 prq " J 2 ν prqr3J 2 ν`1 prqJ 2 ν prqsr 4´2d . The first integral on the right-hand side of (9.12) vanishes (see [28, §5.1]); we remark that this is a direct consequence of the modulation invariance of Φ in the unweighted case h " 1. For d " 3, we compute p H 1 explicitly; it is supported on the interval r0, 4s and given by p H 1 pρq "$ & % 1 8 ρ 2 p3ρ´8q, ρ P r0, 2s, 1 8 ρpρ´4q 2 , ρ P r2, 4s.(9.13) Let g " gp\|¨\|q :" H 1 , which is smooth and belongs to L 1 pR 3 q X L 8 pR 3 q, and satisfies p g ď 0 everywhere, as can easily be checked from (9.13). Next we choose c 0 ą 0 small enough so that the numerator in (9.1) is nonnegative, thus ensuring differentiability of Φ. 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Email address: [email protected] Email address: [email protected]. Matemática Departamento De, Lisboa, PortugalInstituto Superior Técnico, Av. Rovisco Paispt Email address: [email protected] de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. Email address: [email protected] Email address: [email protected]	[]
[ "Fermionic Dark Matter in Radiative Inverse Seesaw Model with U (1) B−L", "Fermionic Dark Matter in Radiative Inverse Seesaw Model with U (1) B−L" ]	[ "Hiroshi Okada \nSchool of Physics\nKIAS\n130-722SeoulKorea\n", "Takashi Toma \nInstitute for Theoretical Physics\nKanazawa University\n920-1192KanazawaJapan\n" ]	[ "School of Physics\nKIAS\n130-722SeoulKorea", "Institute for Theoretical Physics\nKanazawa University\n920-1192KanazawaJapan" ]	[]	We construct a radiative inverse seesaw model with local B − L symmetry, and investigate the flavor structure of the lepton sector and the fermionic Dark Matter. Neutrino masses are radiatively generated through a kind of inverse seesaw framework. The PMNS matrix is derived from each mixing matrix of the neutrino and charged lepton sector with large Dirac CP phase. We show that the annihilation processes via the interactions with Higgses which are independent on the lepton flavor violation, have to be dominant in order to satisfy the observed relic abundance by WMAP. The new interactions with Higgses allow us to be consistent with the direct detection result reported by XENON100, and it is possible to verify the model by the exposure of XENON100 (2012).	10.1103/physrevd.86.033011	[ "https://arxiv.org/pdf/1207.0864v2.pdf" ]	119,205,617	1207.0864	ae375bcfbb78b92027f98857f863ecdc638e3732	Fermionic Dark Matter in Radiative Inverse Seesaw Model with U (1) B−L 9 Aug 2012 Hiroshi Okada School of Physics KIAS 130-722SeoulKorea Takashi Toma Institute for Theoretical Physics Kanazawa University 920-1192KanazawaJapan Fermionic Dark Matter in Radiative Inverse Seesaw Model with U (1) B−L 9 Aug 2012 We construct a radiative inverse seesaw model with local B − L symmetry, and investigate the flavor structure of the lepton sector and the fermionic Dark Matter. Neutrino masses are radiatively generated through a kind of inverse seesaw framework. The PMNS matrix is derived from each mixing matrix of the neutrino and charged lepton sector with large Dirac CP phase. We show that the annihilation processes via the interactions with Higgses which are independent on the lepton flavor violation, have to be dominant in order to satisfy the observed relic abundance by WMAP. The new interactions with Higgses allow us to be consistent with the direct detection result reported by XENON100, and it is possible to verify the model by the exposure of XENON100 (2012). Introduction Inverse seesaw mechanism which generates neutrino masses due to small lepton number violation is one of the intriguing way to describe tiny neutrino masses [1,2]. Thus interesting phenomenological implications have been accommodated [3][4][5]. However Dark Matter (DM) candidate has to be introduced independently if we discuss DM phenomenology in this kind of models. On the other hand, radiative seesaw mechanism is possible to relate neutrino mass generation with the existence of DM [6][7][8]. In particular, the radiative seesaw model which is proposed by Ma [6] is the simplest model with DM candidates. Subsequently there are a lot of recent works of the model [9][10][11] and the extended models [12][13][14][15][16][17][18][19]. The other radiative neutrino mass models are studied in Refs. [20][21][22][23]. The Z 2 parity imposed to the model forbids to have the Dirac neutrino masses, produces neutrino masses at one loop level, and stabilizes DM candidates. Therefore the feature of the radiative seesaw model motivates to connect the existence of DM with the neutrino mass generation due to inverse seesaw mechanism. In this paper, we construct a radiative inverse seesaw model with U(1) B−L as a concrete example and analyze the neutrino masses, mixing and the feature of DM. We add three pairs of B −L charged fermions and a scalar to Ma model [6]. The scalar particle breaks the B −L symmetry spontaneously. In the model, the small Majorana mass terms which violate U(1) B−L weakly and explain the tiny neutrino masses are generated from a higher operator when the U(1) B−L symmetry is spontaneously broken. We assume the structure of neutrino mass matrix that induces the best fit value of θ 12 of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix [24]: sin 2 θ 12 =0.311, which is within 1σ confidence level in the global analysis [25]. Moreover, we introduce the charged-lepton mixing (λ) with nonzero Dirac CP phase to induce non-zero θ 13 recently reported by T2K [26], Double Chooz [27], Daya-Bay [28], and RENO [29]. This method was firstly proposed by S. King [30,31]. Due to the mixing λ, neutrino Dirac Yukawa couplings are strongly constrained by the lepton flavor violation (LFV), especially, µ → eγ process. Hence in the original radiative seesaw model [6], it is hard to derive the observed relic density of DM [32] associated to the annihilation channel if we assume that the lightest right-handed neutrino is DM. However since the radiative inverse seesaw model has the other channels with different coupling due to the additional Higgs boson coming from the B − L symmetry, the observed relic abundance can be naturally obtained via the channel. Since the additional Higgs boson mixes with the SM like Higgs, the direct detection with Higgs portal also comes into the target of a discussion whether the result can be consistent with the experimental limit reported by XENON100 [33]. It implies that the model can be investigated in the direct search of DM near future. Therefore it is favored to be Higgs portal DM in the radiative inverse seesaw model from the consistency with neutrino masses, mixing, LFV and the DM property. This paper is organized as follows. In Section 2, we construct the radiative inverse seesaw model and its Higgs sector. In Section 3, we discuss the constraints from LFV, especially, µ → eγ process and the DM relic abundance. In Section 4, we analyze the direct detection of our DM including all the other constraint. We summarize and conclude the paper in Section 5. Table 1: The field contents and the charges. Three generations of right handed neutrinos N c i and additional fermions S i , S ′ i are introduced. A pair of fermions S and S ′ is required to cancel the anomaly. The SM, inert and B − L Higgs bosons are denoted by Φ, η, and χ, respectively. The Radiative Inverse Seesaw Model Neutrino Mass and Mixing Particle Q u c , d c L e c N c S S ′ Φ η χ SU (2) L 2 1 2 1 1 1 1 2 2 1 Y B−L 1/3 -1/3 -1 1 1 -1/2 1/2 0 0 -1/2 Z 2 + + + + − − + + − + The radiative inverse seesaw mechanism can be realized by introducing the B − L symmetry which is spontaneously broken at TeV scale [4]. The field contents of our model is shown in Table 1 and the Lagrangian in the lepton sector is L = y ℓ LΦe c + y ν LηN c + y S N c χS + λ S Λ χ †2 S 2 + λ S ′ Λ χ 2 S ′ 2 + h.c., (2.1) where Λ is a cut-off scale and the generation indices are abbreviated. Note that the mass term SS ′ can be forbidden by the Z 2 symmetry 1 . We have another dimension 5 operator such as LΦS ′ χ † , which affects on the neutrino mass, and it would be difficult to forbid it. Thus we need to assume the coupling of this operator is small enough. After the symmetry breaking, that is χ = (χ 0 + v ′ )/ √ 2 2 , φ = (φ 0 + v)/ √ 2 with Φ = (φ + , φ) T 1 Another way to induce the mass term of S is proposed by E. Ma [34], in which the term is done at loop level. 2 The value of v ′ can be constrained by the LEP experiment that tells us m Z ′ /g ′ = \|Y χ B−L \|v ′ > 6 TeV [35], where m Z ′ and g ′ are the B − L gauge boson and the B − L guage coupling, respectively. Since Y χ B−L =-1/2 is taken in our case, v ′ > 12 TeV is obtained. and v = 246 GeV, the neutrino sector in the flavor basis can be written as L ν m = y ν LηN c + M T N c S + µ 2 S 2 + h.c., (2.2) where M = y S v ′ / √ 2 and µ = λ S v ′2 /Λ. The scale of µ ∼ 1 keV which corresponds to Λ ∼ 10 14 GeV is required as we will see in the following section. The inert doublet η ≡ (η + , (η R + iη I )/ √ 2) T does not have any vacuum expectation values (VEV). As a result, the 6 × 6 neutrino mass matrix in the basis (N c , S) takes the form: M ν = 0 M T M µ . (2. 3) The mass matrix M and µ cannot be diagonalized simultaneously in general. We assume that the m i± = M 2 i + µ 2 i 4 ± µ i 2 . (2.4) When µ i ≪ M i , the neutrino masses are degenerated. The unitary matrix U i can be expressed as U i =   M i √ M 2 i +m 2 i+ iM i √ M 2 i +m 2 i− m i+ √ M 2 i +m 2 i+ − im i− √ M 2 i +m 2 i−   , (2.5) where (m i+ , m i− ) = U T i M i ν U i and M i ν implies the 2 × 2 mass matrix for i-th family. The flavor eigenstates are rewritten by the mass eigenstates ν i± as follows: N c i = M i M 2 i + m 2 i+ ν i+ + iM i M 2 i + m 2 i− ν i− , (2.6) S i = m i+ M 2 i + m 2 i+ ν i+ − im i− M 2 i + m 2 i− ν i− . (2.7) The light neutrino mass matrix seen in Fig. 1 is given as Ref. [6] by 1-loop radiative correction: where m R and m I imply masses of η R and η I . When µ i ≪ M i , we can obtain the approximate light neutrino mass matrix by expanding m i± up to the leading order, (m ν ) αβ = − 3 i=1 (y ν ) αi (y ν ) βi (4π) 2 M 2 i m i− M 2 i + m 2 i− m 2 R m 2 R − m 2 i− log m 2 R m 2 i− − m 2 I m 2 I − m 2 i− log m 2 I m 2 i− + 3 i=1 (y ν ) αi (y ν ) βi (4π) 2 M 2 i m i+ M 2 + m 2 i+ m 2 R m 2 R − m 2 i+ log m 2 R m 2 i+ − m 2 I m 2 I − m 2 i+ log m 2 I m 2 i+ , (2.8)(m ν ) αβ ≃ − 3 i=1 (y ν ) αi (y ν ) βi µ i 2(4π) 2 m 2 R M 2 i I M 2 i m 2 R − m 2 I M 2 i I M 2 i m 2 I + O µ 2 i , (2.9) where we will define the function I(x) and the parameter Λ i as I(x) = x 1 − x 1 + x log x 1 − x , Λ i = µ i 2(4π) 2 m 2 R M 2 i I M 2 i m 2 R − m 2 I M 2 i I M 2 i m 2 I . (2.10) We can see from Eq. (2.9) that the mixing matrix of the light neutrino mass matrix is determined by the structure of the neutrino Yukawa matrix y ν since the majorana mass matrix µ is assumed to be diagonal in this case. In order to identify the structure of y ν , here we set a specific texture of the neutrino mass matrix, which induces the best fit values of θ 12 , as m ν =     A B −B B (3A + √ 3B)/6 −(3A + √ 3B)/6 −B −(3A + √ 3B)/6 (3A + √ 3B)/6     . (2.11) Then the mass matrix can be diagonalized by the following mixing matrix and the eigenvalues are written as O νL =      (1 + 1/ √ 7)/2 − (1 − 1/ √ 7)/2 0 − 1 − 1/ √ 7/2 − 1 + 1/ √ 7/2 1/ √ 2 1 − 1/ √ 7/2 1 + 1/ √ 7/2 1/ √ 2      , (2.12) m 1 = A + 2B √ 3 1 − √ 7 , m 2 = A + B √ 3 1 + √ 7 , m 3 = 0. (2.13) Thus the squared mass differences are ∆m 2 sol ≡ m 2 2 − m 2 1 and ∆m 2 atm ≡ \|m 2 1 − m 2 3 \|, and the neutrino mass hierarchy is predicted to be inverted. In order to get ∆m 2 sol = 7.62 × 10 −5 eV 2 , ∆m 2 atm = 2.40 × 10 −3 eV 2 , which are the best fit values [25], we find the following solutions: (A, B) = (±4.92 × 10 −2 , ±2.53 × 10 −4 ), (±1.83 × 10 −2 , ∓3.23 × 10 −2 ) eV. (2.14) Notice that the other solutions do not exist any more. We cannot obtain non-zero sin θ 13 from the light neutrino mass matrix Eq. (2.11). Non-zero sin θ 13 is derived from the charged lepton mixing as we will discuss below. We find i m i ≃ 7.7×10 −4 eV, and the effective mass m ee ≡ \| i (O νL ) 2 1i m i \| ≃ 0.026 eV and tan θ 12 = 1 − √ 7 / √ 6 which gives sin 2 θ 12 = 0.311 which is the best fit 1σ value [25]. The recent experimental value for m ee are referred in Ref. [36]. We can choose the following texture which leads the above neutrino mass matrix 3 : y ν =     0 0 b a 0 c −a 0 −c     , (2.15) where the parameters a, b, c are expressed by A and B as follows: a = 1 √ 2AΛ 1 A + 1 + √ 7 √ 3 B A + 1 − √ 7 √ 3 B , b = A √ AΛ 3 , c = B √ AΛ 3 . (2.16) To induce non-zero θ 13 , we consider the charged lepton mixing [30,31]. If we set the Dirac mass matrix of charged leptons m e and the mixing matrix U eL as m e = v √ 2     y ℓ 1 y ℓ 2 0 y ℓ 2 y ℓ 3 0 0 0 y ℓ 4     , U eL ∼     1 λe iδ 0 −λe −iδ 1 0 0 0 1     ,(2.17) where we define (\|m e \| 2 , \|m µ \| 2 , \|m τ \| 2 ) = U † eL m e m † e U eL and δ is the Dirac CP phase. From the mixing matrix O νL and U eL , we can obtain each of the element of the PMNS matrix, which is defined as Here we restrict sin θ 12 within the range of the best fit with 1σ that is 0.303 ≤ sin 2 θ 12 ≤ 0.335 [25]. U P M N S = U † eL O ν P , is found by sin θ 13 ≃ − λ √ 2 , sin θ 12 ≃ − (1 − 1/ √ 7)/2 + λ √ 2 (1 + 1/ √ 7)/2 cos δ, sin θ 23 ≃ 1 √ 2 , where P contains two majorana phases. The allowed value of sin θ 13 is shown in Fig. 2 as the function of δ, within the range of the best fit with 1σ that is 0.303 ≤ sin 2 θ 12 ≤ 0.335 [25]. The light red region is in 1σ range of sin θ 12 . It suggests that large CP Dirac phase is required to satisfy the current global experimental limit of sin θ 13 for inverted hierarchy such as 0.023 ≤ sin 2 θ 13 ≤ 0.030 [25] which is shown as the two green sandwiched regions. Higgs Sector The Higgses φ 0 and χ 0 mix after the symmetry breaking and these are not mass eigenstates. Interactions should be written by mass eigenstates in order to analyze the DM relic density and direct detection in the next section. The Higgs potential of this model is given by V = m 2 1 Φ † Φ + m 2 2 η † η + m 2 3 χ † χ + λ 1 (Φ † Φ) 2 + λ 2 (η † η) 2 + λ 3 (Φ † Φ)(η † η) + λ 4 (Φ † η)(η † Φ) +λ 5 [(Φ † η) 2 + h.c.] + λ 6 (χ † χ) 2 + λ 7 (χ † χ)(Φ † Φ) + λ 8 (χ † χ)(η † η),(2.19) where λ 5 has been chosen real without any loss of generality. The couplings λ 1 , λ 2 and λ 6 have to be positive to stabilize the potential. Inserting the tadpole conditions; m 2 1 = −λ 1 v 2 − λ 7 v ′2 /2 and m 2 3 = −λ 6 v ′2 − λ 7 v 2 /2, the resulting mass matrices are given by m 2 (φ 0 , χ 0 ) = 2λ 1 v 2 λ 7 vv ′ λ 7 vv ′ 2λ 6 v ′2 = cos α sin α − sin α cos α m 2 h 0 0 m 2 H cos α − sin α sin α cos α , (2.20) m 2 (η ± ) = m 2 2 + 1 2 λ 3 v 2 + 1 2 λ 8 v ′2 , (2.21) m 2 R ≡ m 2 (Reη 0 ) = m 2 2 + 1 2 λ 8 v ′2 + 1 2 (λ 3 + λ 4 + 2λ 5 )v 2 , (2.22) m 2 I ≡ m 2 (Imη 0 ) = m 2 2 + 1 2 λ 8 v ′2 + 1 2 (λ 3 + λ 4 − 2λ 5 )v 2 ,(2.23) where h implies SM-like Higgs and H is an additional Higgs mass eigenstate. The tadpole condition for η, which is given by ∂V ∂η VEV = 0, tells us that [38]. The branching ratios of the processes ℓ α → ℓ β γ (ℓ α , ℓ β = e, µ, τ ) are calculated as m 2 2 > 0, λ 2 > 0, λ 3 + λ 4 + 2λ 5 > 0, λ 8 > 0,(2.Br(ℓ α → ℓ β γ) = 3α em 64π(G F M 2 η ) 2 3 i=1 U † eL y † ν αi y ν U eL iβ F 2 M 2 i M 2 η 2 Br(ℓ α → ℓ β ν β ν α ), (3.1) where α em = 1/137, Br (µ → eν e ν µ ) = 1.0, Br (τ → eν e ν τ ) = 0.178, Br (τ → µν µ ν τ ) = 0.174, M η is η + mass, G F is the Fermi constant and the loop function F 2 (x) is given by F 2 (x) = 1 − 6x + 3x 2 + 2x 3 − 6x 2 ln x 6(1 − x) 4 . (3.2) The µ → eγ process gives the most stringent constraint. If the mixing matrix of the charged leptons Eq. (2.17) is diagonal, i.e. λ = 0, the µ → eγ process does not constrain the model since the branching ratio Eq. (3.1) can be zero. Although τ → µγ and τ → eγ processes remain as LFV constraint, these are much weaker than µ → eγ. Instead of that, non-zero θ 13 is not derived. Thus we can insist that µ → eγ constraint is closely correlated with non-zero θ 13 . In order to be consistent with LFV and obtaining non-zero θ 13 , the neutrino Yukawa couplings must be small enough to escape the LFV constraint. DM Relic Density If we assume that DM is fermionic and mass hierarchy M 1 < M 2 < M 3 for the right-handed neutrinos N i , the mass eigenstates ν 1± can be highly degenerated DMs due to the weak lepton number violation term µ i which induces small neutrino masses. Thus we have to take into account coannihilation of ν 1− and ν 1+ . The typical interacting terms are found as L int ≃ (U eL y ν ) α1 ℓ α η + − ν α η 0 − i √ 2 ν 1− + 1 √ 2 ν 1+ + h.c. + (y S ) 11 sin α 2 h ν 2 1− + ν 2 1+ − (y S ) 11 cos α 2 H ν 2 1− + ν 2 1+ + y f cos α √ 2 hf f + y f sin α √ 2 Hf f, (3.3) where the masses of η 0(±) assumed to be always heavier than ν ± to avoid the too short lifetime of DMs through our analysis. Three types of the coannihilation processes exist and these are shown in σ ℓ eff v rel ≃ 1 96π τ α=e \|(U eL y ν ) α1 \| 2 2 M 2 1 M 4 1 + M 4 η M 2 1 + M 2 η 4 v 2 rel , (3.4) σ t eff v rel ≃ 3 (y S ) 2 11 y 2 t M 2 1 64π sin α cos α 4M 2 1 − m 2 h + im h Γ h − sin α cos α 4M 2 1 − m 2 H + im H Γ H 2 1 − m 2 t M 2 1 3/2 v 2 rel , (3.5) σ h eff v rel ≃ (y S ) 4 11 sin 4 αM 2 1 32π (m 2 h − 2M 2 1 ) 2 1 − 1 3 m 2 h − M 2 1 m 2 h − 2M 2 1 + 1 12 m 2 h − M 2 1 m 2 h − 2M 2 1 2 1 − m 2 h M 2 1 v 2 rel (3.6)Γ H = y 2 t sin 2 α 16π m H 1 − 4m 2 t m 2 H 3/2 . (3.7) The contribution of the process H → ν 1± ν 1± is also added to the decay width when the relation m H > 2M 1 is satisfied. Note that the mixing matrix of charged leptons U eL is introduced in the effective cross section σ ℓ eff v rel since the initial Lagrangian (2.1) is not assumed as diagonal base of charged leptons. The s-wave vanishes and p-wave only remains in the above annihilation cross section due to the helicity suppression. Since the neutrino Yukawa couplings y ν are severely restricted by LFV, the annihilation cross section of ℓℓ-channel is too small to obtain the proper DM relic abundance. Thus large contributions from tt and hh channels are required to have sizable effective annihilation cross section. These processes are possible because of the mixing of Higgses φ 0 and χ 0 , namely the DMs ν 1± can be Higgs portal DMs. This is a different aspect from the loop induced neutrino mass model [6] and the various analysis of the right-handed neutrino DM in the original model [9,[39][40][41][42]. The independent parameters which appear in the analysis are Λ 1 , Λ 3 , M 1 , M 3 , M η , m H , (y S ) 11 and sin α. The parameters A and B of the neutrino mass matrix Eq. (2.11) are determined by neutrino mass eigenvalues as Eq. (2.14), and we take the fourth solution of Eq. (2.14) as example. that µ i ∼ 1 keV if I (M 2 i /m 2 R ) ≃ I (M 2 i /m 2 I ) ∼ 0. 1 is assumed as can be seen from Eq. (2.10). The result does not practically depend on Λ i , M 2 and M 3 since the dependence of Λ i appears in only σ ℓ eff v rel which has small annihilation cross section. Thus we can see that the appropriate contribution comes from the tt-channel and hh-channel. The red, green and blue points correspond to each range of sin α as written in Fig. 4. In the case of (y S ) 11 = 0.5, the mass relation 2M 1 ≈ m H is required since the coupling (y S ) 11 is not so large and the annihilation cross section σ t eff v rel has to be enhanced due to a resonance. The resonance relation is not required for the right side one which corresponds to (y S ) 11 = 1.0 if sin α is large since the hh-channel is effective in this case. There is no allowed parameter space in M 1 < m h region because the main channel is ν 1± ν 1± → bb and the contribution to the annihilation cross section is too small. Direct Detection The DM candidates ν 1± interact with quarks via Higgs exchange. Thus it is possible to explore the DM in direct detection experiments like XENON100 [33]. The Spin Independent (SI) elastic cross section σ SI with nucleon N is given by (y S ) 11 = 1.0 2/3 < sin α < 1 1/3 < sin α < 2/3 0 < sin α < 1/3 XENON100 (2011) bound XENON100 (2012) expected for the heavy quarks Q where q ≤ 3 implies the summation of the light quarks. The recent another calculation is performed in Ref. [45]. σ N SI ≃ µ 2 DM π 1 m 2 h − 1 m 2 H 2 (y S ) 11 m N sin α cos α √ 2v q f N q 2 ,(4. The comparison with XENON100 upper bound is shown in Fig. 5 where the other parameters are fixed as same as Fig. 4 and these correspond to red, green and blue points. The violet dotted line is XENON100 (2011) upper bound and the light blue dashed line is XENON100 expected one in 2012. We can see that from the figure, the XENON100 (2011) limit excludes M 1 800 GeV in the large sin α region for (y S ) 11 = 1.0. The almost excluded region of rather small M 1 implies that the parameter region of the hh-channel is the most effective for the DM annihilation (Fig. 4). The other certain region will be verified by the future XENON experiment. In the case of taking into account the lightest right-handed neutrino as DM in the original radiative neutrino mass model [6], the elastic cross section with nuclei is not obtained at tree level because of the leptophilic feature of the DM. However interactions with quarks via Higgses are obtained in the radiative inverse seesaw model and hence verification by direct detection of DM is possible. Conclusions We constructed a radiative inverse seesaw model which generates neutrino masses and includes DMs simultaneously, and studied the mixing of the lepton sector and the DM features. The neutrino mass matrix was radiatively generated via the inverse seesaw framework. Considering the latest data of non-zero θ 13 , we applied the charged lepton mixing effect with almost Maximal Dirac CP phase suggested by S. King. We can obtain the neutrino Dirac Yukawa matrix which produces the best fit value of θ 12 on the diagonal basis of the right-handed neutrinos and additional fermions and non-zero mass matrix M and µ are diagonal as M = diag (M 1 , M 2 , M 3 ) and µ = diag (µ 1 , µ 2 , µ 3 ) for simplicity. Then the 6×6 mass matrix M ν can be diagonalized for every family as diag(m i+ , m i− ) by the unitary matrix U i for i-th family. The mass eigenvalues are expressed as Figure 1 : 1Neutrino mass generation via radiative inverse seesaw. Figure 2 : 2sin θ 13 versus CP Dirac phase δ. to satisfy the condition v η = 0 at tree level. The masses of φ 0 and χ 0 are rewritten in terms of the mass eigenstates of h and H as φ 0 = h cos α + H sin α, χ 0 = −h sin α + H cos α. Lepton Flavor Violation (LFV) under the flavor structure Eq. (2.15). We put a reasonable approximation m i± ≃ M i hereafter, since the scale of µ that is keV scale is negligible compared to the scale of M i that is expected O(10 2∼3 ) GeV. The experimental upper bounds of the branching ratios are Br (µ → eγ) ≤ 2.4 × 10 −12 [37], Br (τ → µγ) ≤ 4.4 × 10 −8 and Br (τ → eγ) ≤ 3.3 × 10 −8 Fig 3 . 3The effective cross section to ℓℓ, f f and hh are given as Figure 3 : 3t, u and s-channel of coannihilation processes of DM ν 1± . where η R , η I and η + masses are regarded as same parameter M η for simplicity and only top pair is taken into account in fermion pair f f because of the largeness of the Yukawa coupling. The SM-like Higgs mass and decay width are fixed to m h = 125 GeV and Γ h = 10 −2 GeV, and the heavy Higgs mass is assumed to be m H < 200 GeV. The decay width of the heavy Higgs Γ H is expressed as Figure 4 : 4The parameter λ of the charged lepton mixing matrix is fixed by the experimental value of sin θ 13 as Eq.(2.18), and Λ 2 and M 2 are not relative with the analysis since the second column of the neutrino Yukawa matrix Eq. (2.15) is zero. The allowed parameter spaces from LFV and the DM relic abundance for (y S ) 11 = 0.5 and 1.0 in M 1 -m H plane are shown in Fig. 4 where the parameters are fixed to Λ i = 1 eV, M 2 = 1.5 TeV and M 3 = 2.0 TeV. The parameter choice Λ i ∼ 1 eV means The allowed parameter spaces for (y S ) 11 = 0.5 and 1.0 in M 1 -m H plane. The other parameters are fixed to Λ i = 1.0 eV, M 2 = 1.5 TeV, M 3 = 2.0 TeV. The unfixed parameters are M 1 , M η , m H and sin α. Figure 5 : 5The comparison with XENON100 experiment. The left figure is for (y S ) 11 = 0.5 and the right one is for (y S ) 11 = 1.0. The parameter choice is same asFig. 4.where µ DM = M −1 1 + m −1N −1 is the DM-nucleon reduced mass and the heavy Higgs contribution is neglected. The parameters f N q which imply the contribution of each quark to nucleon mass are calculated by the lattice simulation[43,44 θ 13 comes from the charged lepton mixing matrix. The size of the neutrino Yukawa couplings is severely constrained by LFV at the same time. As a result, the annihilation cross section which comes from the Yukawa interaction becomes ineffective, however we found that new interactions via Higgs bosons which are independent on LFV. Thus the DM can have the certain annihilation cross section due to the interactions with Higgses. Verification of the model is also possible by direct detection of DM through the interaction with Higgses. 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[ "Symmetry Protected Quantization and Bulk-Edge Correspondence of Massless Dirac Fermions: Application to Fermionic Shastry-Sutherland Model", "Symmetry Protected Quantization and Bulk-Edge Correspondence of Massless Dirac Fermions: Application to Fermionic Shastry-Sutherland Model" ]	[ "Toshikaze Kariyado \nDivision of Physics\nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaIbarakiJapan\n", "Yasuhiro Hatsugai \nDivision of Physics\nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaIbarakiJapan\n" ]	[ "Division of Physics\nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaIbarakiJapan", "Division of Physics\nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaIbarakiJapan" ]	[]	The fermionic Shastry-Sutherland model has a rich phase diagram, including phases with massless Dirac fermions, a quadratic band crossing point, and a pseudospin-1 Weyl fermion. Berry phases defined by the onedimensional momentum as a parameter are quantized into 0 or π due to the inversion symmetry combined with the time reversal, or existence of the glide plane, which also protects the massless Dirac cones with continuous parameters. This is the symmetry protected Z 2 quantization. We have further demonstrated the Z 2 Berry phases generically determine the existence of edge states in various phases and with different types of the boundaries as the bulk-edge correspondence of the massless Dirac fermion systems.PACS numbers: 73.20.-r, 03.65.Vf	10.1103/physrevb.88.245126	[ "https://arxiv.org/pdf/1307.7926v1.pdf" ]	119,206,187	1307.7926	318ee82754c4d0bc281e726ad40fced5661124fe	Symmetry Protected Quantization and Bulk-Edge Correspondence of Massless Dirac Fermions: Application to Fermionic Shastry-Sutherland Model (Dated: May 11, 2014) 30 Jul 2013 Toshikaze Kariyado Division of Physics Faculty of Pure and Applied Sciences University of Tsukuba 305-8571TsukubaIbarakiJapan Yasuhiro Hatsugai Division of Physics Faculty of Pure and Applied Sciences University of Tsukuba 305-8571TsukubaIbarakiJapan Symmetry Protected Quantization and Bulk-Edge Correspondence of Massless Dirac Fermions: Application to Fermionic Shastry-Sutherland Model (Dated: May 11, 2014) 30 Jul 2013 The fermionic Shastry-Sutherland model has a rich phase diagram, including phases with massless Dirac fermions, a quadratic band crossing point, and a pseudospin-1 Weyl fermion. Berry phases defined by the onedimensional momentum as a parameter are quantized into 0 or π due to the inversion symmetry combined with the time reversal, or existence of the glide plane, which also protects the massless Dirac cones with continuous parameters. This is the symmetry protected Z 2 quantization. We have further demonstrated the Z 2 Berry phases generically determine the existence of edge states in various phases and with different types of the boundaries as the bulk-edge correspondence of the massless Dirac fermion systems.PACS numbers: 73.20.-r, 03.65.Vf Massless Dirac fermion systems, which are zero gap semiconductors found in various situations [1][2][3][4][5] and characterized by a linear dispersion, are novel semimetallic materials exhibiting many intriguing phenomena. A typical realization of massless Dirac fermion system is celebrated graphene [1], which has been attracted much attention since its discovery. Not only in conventional solid state materials, but also in optical lattice systems, the fabrication of massless Dirac fermions becomes a hot topic very recently [6,7]. Among the many unusual properties of the massless Dirac fermions, appearance of characteristic edge states [8][9][10] is important in the view of the bulk-edge correspondence, which implies that topologically nontrivial bulk states and appearance of the edge states, i.e., localized modes near the boundaries, are closely related and reflect each others. The concept "bulk-edge correspondence" is established for a topologically nontrivial gapped state [11]. There, a bulk topological number and number of edge states are connected. Actually, although the massless Dirac fermion system is gapless at the Fermi energy and a topological number cannot be well defined, the bulk-edge correspondence is still at work [9,12,13]. Instead of the bulk topological number such as the Chern number, the Berry phase θ(k ) plays a central role in the massless Dirac fermion systems [9,12,13]. Here, θ(k ) is a bulk quantity parameterized by a momentum k , which is parallel to the "edge". Generically the Berry phase θ(k ) is gauge dependent and takes any real number in modulo 2π. It is in contrast to the Chern number that is gauge invariant and intrinsically integer [14]. However, with the help of a supplemental symmetry, the Berry phase is quantized and becomes topological, that is, adiabatic invariant [12,[15][16][17][18]. This is the symmetry protected quantization, which is useful in odd dimension. Note that the Chern number and its generalizations are only well defined in even dimensions. The symmetry further plays a crucial role for the topological stability of the massless Dirac fermions. Since the gap closing point has co-dimension 3 [17,19], the symmetry discussed above is crucial to have a massless Dirac fermions in two-dimensions in a generic situation. As for the bulk-edge correspondence of the massless Dirac fermions and the stability of the doubled Dirac cones, the chiral symmetry is often employed [9,12]. In this paper, with general discrete symmetries, the idea on the bulk-edge correspondence of the massless Dirac fermions and its stability are discussed and demonstrated using the fermionic Shastry-Sutherland (SS) model. This model has not been studied well, while a spin model on the SS lattice, which is known as the orthogonal dimer model, has been extensively studied following the discovery of the exact ground state wave function [20][21][22][23], and has been realized experimentally [24]. In the following, we first show that the fermionic SS model has a rich electronic phase diagram. Interestingly, the phases with massless Dirac fermions, a quadratic band crossing point [25], or a pseudospin-1 Weyl fermion [26,27] at the Fermi energy are accessible by controlling only a few parameters. Then the bulk-edge correspondence in the fermionic SS model is discussed, focusing on the phase with massless Dirac fermions. Although the fermionic SS model does not respect the chiral symmetry, existence of the inversion center or the glide plane play crucial role in quantization of the Berry phase and the stability of the massless Dirac fermions. A possible physical realization of Shastry-Sutherland lattice is visualized as Fig. 1(a). This lattice possesses many symmetries among which the four-fold rotational symmetry, glide plane symmetry, and inversion symmetry play particularly important roles in the following arguments. A simplified picture of the model is shown in Fig. 1(b). A shaded region represents a unit cell, which contains four lattice sites named site 1-4, implying that our model has four bands. Our Hamiltonian is H = ab rr t ab (r − r )c † ra c r b = abk (Ĥ k ) ab c † ka c kb ,(1) with c ka = 1 √ N r e ik·r c ra . Here, indices a and b run from 1 thorough 4, representing four sublattices, while r and r represents lattice vectors on square lattice. We employ four parameters t + , t − , t x , and t y that correspond to transfer integrals between the sites connected by bonds indicated as +, −, Schematic picture of our model. Bonds named as +, −, x, and y are associated with the transfer integrals t + , t − , t x , and t y , respectively. Shaded region denotes the unit cell. We give numbers one through four to four sites in a unit cell in order to distinguish them. (c) Phase diagram for the case of t + = t − = t 0 . x, and y in Fig. 1(b), respectively. The glide plane symmetry is broken when t + t − , while the four-fold rotational symmetry is broken when t + t − or t x t y . In contrast, the inversion symmetry is always kept with this parameterization. For convenience, we also use parameters t 0 , t 1 , ∆ 0 , and ∆ 1 defined as t ± = t 0 ± ∆ 0 , t x = t 1 + ∆ 1 , and t y = t 1 − ∆ 1 . In this study, we neglect spin degrees of freedom, and concentrate on the half filled case. Namely, "Fermi energy" appearing in the following refers to the chemical potential achieving half filling, and "gapped state" means that the system has a gap between the second and third lowest bands. The phase diagram obtained for ∆ 0 = 0 (t + = t − ), which is the case that the two diagonal bonds orthogonal with each other are equivalent, is shown in Fig. 1(c). For t x = t y < 0.5t 0 , the system is in a (trivial) gapped phase. On the other hand, for t x = t y > 0.5t 0 , we find a quadratic band crossing point (QBCP) [25], at which two parabolic bands, one is hole-like and the other is electron-like, touch with each other, at the Γpoint [21,28]. [ Figs. 2(a) and 2(d).] Note that the hole-like band is not parabolic in a strict sense in this case, since it is dispersionless in the Γ-M direction. QBCP is allowed to exist if the system has a four-fold rotational symmetry [29], and has interesting properties. For instance, the four-fold symmetry can be broken by electron-electron interaction effects, leading to emergent nematic phases [25]. For t x = t y = 0.5t 0 , at which the transition between the trivial gapped phase and the phase with QBCP takes place, there exists "pseudospin-1 Weyl fermion" [27], which is characterized by a linear dispersion and a three-fold degeneracy [26], at the Γ-point. A finite ∆ 1 (t x t y ) imposed in the QBCP phase immediately leads to a phase with Dirac cones at the Fermi energy. [Figs. 2(c) and 2(f).] Namely, two Dirac cones (and two Dirac points associated with them) are generated as a pair from the QBCP by a finite ∆ 1 . The Dirac points are located on the k x -axis for ∆ 1 > 0, while they are on the k y -axis for ∆ 1 < 0. Then, if ∆ 1 is continuously modified from positive to negative, the Dirac points first move on the k x -axis towards the Γ-point until they merge, and they next depart from the Γ-point in the direction of the k y -axis. Note that the second lowest band is no longer dispersionless on the Γ-M direction [ Fig. 2(c)], which is important for letting the Dirac fermions being the only feature appearing at the Fermi energy. If t 1 is made smaller and smaller with finite ∆ 1 , the system experiences a transition from the phase with Dirac cone to the trivial gapped phase. The transition between two phases is characterized by an appearance of a semi-Dirac fermion, whose dispersion is linear in one direction and parabolic in the other direction. Actually, this type of disappearance of the Dirac cones is rather general and found in many other models for Dirac fermions [30]. When ∆ 0 0 simultaneously with ∆ 1 0, the Dirac points go into the general points in the Brillouin zone apart from the high-symmetry lines, i.e., the k x -and k y -axes. In this case, the symmetry of the system is much lowered, but the inversion (and time reversal) symmetry is still kept. We will see later that this is sufficient for stabilizing the massless Dirac fermions by means of the quantized Berry phase. In Fig. 3, the trajectories of the Dirac points when (∆ 0 , ∆ 1 ) is changed according to (∆ 0 , ∆ 1 ) = (δ 0 sin φ, δ 1 cos φ) (0 ≤ φ ≤ 2π) are illustrated for t 0 = t 1 = 1.0 and (δ 0 , δ 1 ) = (0.1, 0.1) or (0.2, 0.1). We find that the Dirac points wind around the Γ-point as φ grows from 0 to 2π. Note that the physical state gets back to the original state after 2π change in φ, but each Dirac point does not get back to the original position: two Dirac points interchange their position after 2π change in φ. It is also worth noting that once the fermionic Shastry-Sutherland model is realized in some materials, perturbations leading to ∆ 0 0 and ∆ 1 0 can be induced by applying uniaxial pressure in diagonal or rectangular direction. Now, let us discuss the bulk-edge correspondence of the massless Dirac fermions. For this purpose, we calculate edge spectra and Berry phase for fermionic Shastry-Sutherland model. For simplicity, we discuss the edge parallel to the xaxis, but it is possible to extend the following methods to more general cases [13]. Edge spectra are calculated by making the system with strip (or ribbon) geometries. Here, in order to make a direct connection to the Berry phase arguments, we set a rule to make strips for calculation: edges are given by cutting a periodic system in between the unit cells. With this construction, the edge shapes, or how the system is terminated at the edge, crucially depend on the convention of the unit cell since the position of the cut is fixed in between the unit cells. In this letter, we treat two kinds of unit cell conventions that lead to two kinds of edge terminations, illustrated as Fig. 4(a) and Fig. 4(b), respectively. Hereafter, we call the convention in Figs. 4(a) and 4(b) "type 1" and "type 2", respectively. As we limit our attention to the edge parallel to the x-axis, Berry phase [12,13,15] is defined as k x k y k x k y k x k y (a) (b) (c) (d) (e) (f)iθ(k ) = n∈filled π −π dk ⊥ u nk k ⊥ \|∇ k ⊥ \|u nk k ⊥ ,(2) where k and k ⊥ are essentially k x and k y , and \|u nk k ⊥ is a Bloch wave function that is a four-component vector for our four-band tight-binding model. Although we handle a twodimensional model here, the extension to d dimensional cases is straight forward. Namely, we simply regard k as a d − 1 dimensional vector rather than a number. Actual evaluation of Eq. (2) We explain these in turn in the following. Since we chose the parameters so as to have bulk massless Dirac fermions, bulk continuum, which is the filled region in Figs. 4(c) and 4(d) and contributed from the bulk states, becomes gapless at the projected Dirac points. We find the edge states apart from the bulk continuum connecting the projected Dirac points for both of the type 1 and 2 cases. The edge states for the type 1 and 2 cases are different due to the different edge termination. For the type 1 case, the edge state appears near k (= k x )= 0, while for the type 2 case, it appears near k = π. For both cases, the edge states are dispersive since there is no chiral symmetry that limits the energy of the edge states, but the edge state shows almost flat dispersion for the type 1 case. In general, the quantization of the Berry phase is caused by some symmetry. In the case of Eq. (2), it is proven that the combination of the time reversal and inversion symmetries is important. These symmetries force θ(k ) to obey θ(k ) = −θ(k ) + 2πl − 2π∆ I (k ),(3) where ∆ I (k ) = n∈filled 1 2π π −π dk ⊥ i u nk k ⊥ \|P −1 k (∂ k ⊥P k )\|u nk k ⊥(4) with l being an integer andP k being the inversion symmetry operator satisfyingĤ −k =P kĤkP −1 k . Then, if ∆ I (k ) is zero (becauseP k has no k dependence, for instance) or some integer (by some symmetrical reason), θ(k ) becomes quantized to 0 or π, i.e., Z 2 . Note that the inversion symmetry alone is sufficient for one-dimensional models [15], but it must be combined with the time reversal symmetry for higher dimensional cases. Note also that the reflection symmetry whose reflection plane is parallel to the edge alone can quantize θ(k ). In the case of fermionic SS model, the glide plane symmetry existing if t + = t − , plays a role of the reflection plane symmetry. A physical meaning of the Z 2 quantization can be understood from the fact that θ(k ) has close relation to the electronic polarization [31]. The inversion or reflection symmetry gives restrictions for possible values of the electronic polarization, and these restrictions appear as the Z 2 quantization. However, a special attention is required in the case that the bulk symmetries are broken after introducing edges to the system. In our edge construction, edge shapes depend on the unit cell convention. Then, if we calculate θ(k ) using a unit cell convention that leads to an edge breaking bulk inversion and glide plane symmetries, θ(k ) is not necessarily quantized even FIG. 5. Schematic description of the relation between π-jumps in θ(k ) and Dirac points. if bulk system without edges has inversion and glide plane symmetries. This corresponds to the case that ∆ I (k ) is noninteger. The stability of massless Dirac fermions in twodimensional systems can be clearly addressed using the quantized θ(k ), which we call Z 2 Berry phase. In order to see this, we must realize that π-jump in θ(k ) is directly related to a bulk Dirac fermion. If an infinitesimal change in k , k → k + δk gives a finite change between θ(k ) and θ(k + δk), the electronic dispersion should have a singularity in the area enclosed by the two integration paths for θ(k ) and θ(k + δk), but, a massless Dirac fermion is nothing more than a singularity in the electronic dispersion. Furthermore, the value π is exactly Berry phase acquired when the integration path encloses a Dirac point. The idea is described in Fig. 5 as a deformation of the integration path. Then, as far as the symmetries quantizing θ(k ) are preserved, massless Dirac fermions are topologically stable, since π-jump cannot be suddenly removed by a small change in parameters when θ(k ) is quantized to 0 or π: π-jump only disappears when two jumps are merged, or parameters themselves are discontinuously changed. Inversely speaking, if symmetries preserving θ(k )-quantization is broken, massless Dirac fermion will be no longer stable. In fact, we have checked that when extra terms breaking the inversion and glide plane symmetries are added to the fermionic SS model, θ(k ) deviates from 0 or π, and a gap is induced at the Dirac point. Z 2 Berry phase is also useful in making a criterion for the existence of massless Dirac fermions in a given model [33,34]. As discussed in Refs. 35 and 36, there is no need to explore the entire Brillouin zone to find out Dirac points, thanks to the Z 2 qunatization. Instead, it is sufficient to check the values of θ(k ) at two k s, typically at k = 0 and π. If two θ(k ) take different values, there must be at least one jump, or equivalently, Dirac point, as far as the quantization is retained. The close relation between the appearance of edge states and θ(k ) can be seen in Figs. 4(c) and 4(d). Namely, we find edge states for k with θ(k ) = π mod 2π, while no edge states for k with θ(k ) = 0. Since the π-jumps are related to the bulk Dirac points, existence and nonexistence of the edge states is switched at the Dirac points projected to the edge. Here, we want to emphasize that, although θ(k ) can be calculated only with bulk information, θ(k ) apparently has an ability to capture the difference in edge terminations, i.e., difference between type 1 and type 2 edges. This is because θ(k ) does depend on the choice of the basis set since its definition involves the Bloch wave functions, and different unit cell conventions are actually connected by a unitary transformation, i.e., a transformation of the basis set. In our specific case, θ(k ) in type 1 and type 2 conventions are connected as θ type 2 (k ) = θ type 1 (k ) − 2πρ 1 (k ),(5) where ρ 1 (k ) is k resolved filling of site 1, which is explicitly calculated as ρ 1 (k x ) = n∈filled 1 2π π −π dk y u nk x k y \|P 1 \|u nk x k y . Here, P 1 is a projection operator projecting on the site 1 component. As far as t + = t − , ρ 1 (k ) = 0.5 holds in our model by a symmetrical reason. Consequently, θ type 1 (k ) and θ type 2 (k ) differ by π. Intuitive understanding of this bulk-edge correspondence is possible with the help of adiabatic continuation when θ(k ) is quantized. We briefly explain this for the type 2 edge with parameters used in Fig. 4(d). Recall that the type 2 edge shows the edge states for k = π and no edge state for k = 0. If k is fixed to π, k resolved HamiltonianĤ k can be adiabatically deformed without gap closing and keeping θ(k ) value to the Hamiltonian corresponding to t x = t y = 0. Then, edge states are readily understood as dangling states appearing as a result of cutting remained diagonal bonds for type 2 edge. Importantly, the same adiabatic continuation cannot be applied to k = 0 case since it leads to the gap closing, which allows change in quantized θ(k ) and leads to qualitative changes of the system properties. We have to use different adiabatic continuation, and that continuation should give Hamiltonian without dangling states for the type 2 edge. In summary, we have shown that the fermionic SS model is a quite important model which hosts many peculiar phases including the phase with Dirac cones. Since the spin SS model has been materialized, we believe that it is possible to realize a fermionic counterpart. Alternatively, the model may be realized in some optical lattices. Using the SS model, we have also demonstrated roles of the symmetry for the massless Dirac fermions and the Z 2 quantization of the Berry phase, which also provides the bulk-edge correspondence. This work is partly supported by Grants-in-Aid for Scientific Research, No.23340112, No.25610101, and No.23540460 from JSPS. FIG. 1 . 1(a) The most "physical" Shastry-Sutherland lattice. (b) FIG. 2 . 2Band structures and dispersion relations. (t 0 , ∆ 0 ) = (1.0, 0.0) for the all three cases, while (t 1 , f). In (e), a part of the dispersion is eliminated so as to make the inside visible. FIG. 3 . 3Trajectories of Dirac points for (∆ 0 , ∆ 1 ) = (δ 0 sin φ, δ 1 cos φ) with t 0 = t 1 = 1.0. Solid line is for (δ 0 , δ 1 ) = (0.1, 0.1) while dashed line is for (δ 0 , δ 1 ) = (0.2, 0.1). is performed using a technique in Refs. 31 and 32. Calculated edge spectra and θ(k )/2π for type 1 and type 2 conventions with (t 0 , t 1 , ∆ 0 , ∆ 1 ) = (1.0, 1.0, 0.0, 0.1) are plotted as functions of k in Figs. 4(c) and 4(d). From these figures, we can extract three important points: (1) appearance of edge states, (2) quantization of the Berry phase, and (3) an intimate relation between the edge states and the Berry phase. FIG. 4 . 4(a,b) Unit cell conventions and edge shapes. (c,d) Edge spectra and Berry phase θ(k ) divided by 2π as functions of k for (t 0 , t 1 , ∆ 0 , ∆ 1 ) = (1.0, 1.0, 0.0, 0.1). (c) is for type 1 edge (a), while (d) is for type 2 edge (b). . 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[ "Spatial Patterns of Wind Speed Distributions in Switzerland", "Spatial Patterns of Wind Speed Distributions in Switzerland" ]	[ "Mohamed Laib [email protected] \nInstitute of Earth Surface Dynamics\nFaculty of Geosciences and Environment\nUniversity of Lausanne\nCH1015LausanneSwitzerland\n", "Mikhail Kanevski \nInstitute of Earth Surface Dynamics\nFaculty of Geosciences and Environment\nUniversity of Lausanne\nCH1015LausanneSwitzerland\n" ]	[ "Institute of Earth Surface Dynamics\nFaculty of Geosciences and Environment\nUniversity of Lausanne\nCH1015LausanneSwitzerland", "Institute of Earth Surface Dynamics\nFaculty of Geosciences and Environment\nUniversity of Lausanne\nCH1015LausanneSwitzerland" ]	[]	This paper presents an initial exploration of high frequency records of extreme wind speed in two steps. The first consists in finding the suitable extreme distribution for 120 measuring stations in Switzerland, by comparing three known distributions: Weibull, Gamma, and Generalized extreme value. This comparison serves as a basis for the second step which applies a spatial modelling by using Extreme Learning Machine. The aim is to model distribution parameters by employing a high dimensional input space of topographical information. The knowledge of probability distribution gives a comprehensive information and a global overview of wind phenomena. Through this study, a flexible and a simple modelling approach is presented, which can be generalized to almost extreme environmental data for risk assessment and to model renewable energy.	null	[ "https://arxiv.org/pdf/1609.05012v1.pdf" ]	118,846,842	1609.05012	8bd160b2495aff391aa637255adda3f3c8a38bcb	Spatial Patterns of Wind Speed Distributions in Switzerland Mohamed Laib [email protected] Institute of Earth Surface Dynamics Faculty of Geosciences and Environment University of Lausanne CH1015LausanneSwitzerland Mikhail Kanevski Institute of Earth Surface Dynamics Faculty of Geosciences and Environment University of Lausanne CH1015LausanneSwitzerland Spatial Patterns of Wind Speed Distributions in Switzerland Wind speedExtreme valuesMachine learning algorithmsSpatial modellingSwitzerland This paper presents an initial exploration of high frequency records of extreme wind speed in two steps. The first consists in finding the suitable extreme distribution for 120 measuring stations in Switzerland, by comparing three known distributions: Weibull, Gamma, and Generalized extreme value. This comparison serves as a basis for the second step which applies a spatial modelling by using Extreme Learning Machine. The aim is to model distribution parameters by employing a high dimensional input space of topographical information. The knowledge of probability distribution gives a comprehensive information and a global overview of wind phenomena. Through this study, a flexible and a simple modelling approach is presented, which can be generalized to almost extreme environmental data for risk assessment and to model renewable energy. Introduction Wind can be regarded as either a positive or negative phenomenon. The positive aspect is the renewable energy it produces which has encouraged the Swiss Federation to expand the proportion of power produced by wind speed [1]. On the other hand, enormous losses have been caused by extremely violent wind-storms in the country [2,3], an excellent catalogue of which has been produced by Stucki et al. 2014 [4], and Usbeck et al. [5]. The first wind energy facility in Switzerland was started in 1986 with an energy output of 28 kilowatts. According to the Swiss Federation reports in 2015, there are 34 wind power plants which produce around 110 gigawatts of electricity. The largest wind park is on Mont Crosin in the Bernese Jura. This facility comprises 16 wind turbines with a total output of 29.2 megawatts [6]. To improve the use of this environmental source, a well-developed statistical field for this type of analysis has been proposed. Most methods used to analyse wind data deal with semi-parametric approaches begin by finding the best probability distribution and then confirm results with parametric and nonparametric tools. Since wind data is known for the presence of extremes, the modelling of these data requires extreme probability distribution in order to study the behaviour of tail in data. Extreme value theory (EVT) has been the most frequently applied modelling approach. The main purpose is to find estimators of the suitable distribution for the studied data [7][8][9]. There are many areas where EVT plays an indispensable role for modelling rare events, such as environmental risk (wind, temperature, rainfall, etc.) [10][11][12][13]. Numerous existing parametric and non-parametric estimation methods are used to find estimators. This work uses the maximum likelihood as estimation method. Besides the method mentioned above, machine learning algorithms (MLA) [14,15] are rapidly gaining popularity in modelling environmental phenomena [16][17][18]. Machine learning is a part of artificial intelligence. Whose objective is to find non-linear dependencies observed in data, and to understand better the structure between the input and the output. Several papers propose different approaches to make use of the performance of machine learning in wind speed modelling. One use more information such as environmental data [19], therefore the parameters of the proposed extreme probability distribution are modelled by using other environmental variables as input data. Many algorithms are used for this purpose notably random forest [20]. A comparison between extreme learning machine, support vector machine, and artificial neural networks has been carried out [21,22]. This comparison favours Extreme Learning Machine for its quality of modelling, its rapidity and simplicity. This study uses the Extreme Learning Machine (ELM) proposed by Huang et al., 2006. Its structure is similar to that of a classical multilayer perceptron (MLP). Moreover, ELM has one parameter to optimise which is the number of hidden nodes. This parameter makes ELM easy to apply and to control the complexity of the phenomenon under study. Its main advantage is the speed of the training step, and the capacity to learn complex data. The techniques of cross-validation and data splitting help to avoid overfitting, and also to test the accuracy of the model [15]. Furthermore, the use of ELM requires a consistent methodology to take into account the randomness in generating the weights. The aim of this study is to find a flexible approach to understand the behaviour of extreme wind speed in Switzerland. The proposed approach combines two steps: The first uses a parametric method called maximum likelihood, to estimate probability distribution parameters. The selection of the best extreme distribution is based on the Kolmogorov-Smirnov test and the quantile-quantile plot. These two tools are commonly used to compare if the data are well modelled by the proposed probability distribution. The second step of this work applies the Extreme Learning Machine to model the estimated parameters of the first step. The main results are presented as probability maps, and parameters are also mapped to visualise them. In addition, the ELM results are quantified to show their efficiency to model different distribution parameters. All of these steps are carried out by using extRemes and elmN N packages of the R language [23]. This paper is organized as follows: Section 2 presents an exploratory analysis of the used data. Section 3 explains the first step which consists in finding the suitable distribution. Section 4 gives a brief introduction to ELM, and the proposed spatial modelling. In section 5, the main results are discussed, and in the last section the conclusions are given with suggestions for future research. Study Area and Dataset Study Area This study was performed in Switzerland, which has a total area of 41, 285 km 2 with three basic topographical area: the Jura mountain on the west, the central plateau, and the Swiss Alps to the south which comprise almost all the highest mountains of the Alps. The altitudes varie between: 198 m in canton Ticino, to 4634 m in canton Valais [24]. All this information is summarized in an input space of thirteen variables, including coordinates (X, Y ) at 250 m resolution (Table 1, see details and descriptions in [25]). These variables are used to model parameters provided from each measuring stations. Fig.1 shows some of variables used as input space. Wind Data Wind data used in this work were collected from the website of the Federal Office of Meteorology and Climatology of Switzerland (IDAWEB, MeteoSwiss). They present Fig. 4 shows the presence of extreme wind speed, which leads to propose extreme distributions to model the data. Wind Speed Distribution As mentioned above, this work leads off with a parametric estimation using maximum likelihood. Maximum likelihood is a very simple tool to find estimators. It chooses the value of the parameter which maximizes the following likelihood function [26]: L(θ) = n i=1 f (x i ; θ)(1) where x i are independent realizations of a random variable with a probability density function f (x i ; θ). As is known, it is more convenient to work with the log-likelihood function: The log-likelihood takes its maximum at the same point as the likelihood function, and it is found by differentiating the log-likelihood and equating to zero. The parameters of each proposed distribution are found in order to compare. Several papers link wind data with the following extreme distributions: log L(θ) = n i=1 logf (x i ; θ)(2) Weibull Distribution Proposed as a wind speed distribution [27], Weibull distribution is a two parameters distribution, with the following density function [28]: f (x; λ, k) =          k λ ( x λ ) k−1 e −( x λ ) k x ≥ 0 0 x < 0(3) where k is the shape parameter and λ is the scale. This function is a continuous probability distribution and mostly used to describe wind data. Gamma distribution Gamma distribution is also a two parameter distribution. It used to present several phenomena especially those that varies over time, and it is defined by the following density function: where α is the shape and β is the rate [29]. f (x; α, β) = β α x α−1 e −xβ Γ(α) f or x ≥ 0 and α, β > 0(4) Generalized Extreme Values It combines three families of distributions: Gumbel, Fréchet and Weibull. The GEV distribution is used to model the maxima of long sequences of random variables, and the treatment of risk [26]. It is defined as follow: F (x; µ, σ, ξ) = exp{−[1 + ξ( x − µ σ )] −1 ξ }(5) for 1 + ξ(x − µ)/σ > 0, where µ is the location parameter, σ is the scale parameter and ξ the shape. The latter indicates the tail behaviour of the distribution. Fig 5 shows the different forms of the probability density for each subfamilies according to the value of the shape parameter ξ, and the subfamilies are defined as following: -Gumbel distribution or type I when ξ = 0. -Fréchet or type II when ξ > 0. -Weibull distribution or type III when ξ < 0. The comparison between these distributions is based on the Kolmogorov-Smirnov test, and a graphical method called quantile-quantile plot with a visual statistical protocol, as it is proposed in [30]. Comparison tools The Kolmogorov-Smirnov goodness of fit test There are a number of tests to check the goodness of fit for a probability distribution. Among the most used is the Kolmogorov-Smirnov test, which can be applied on continuous distribution. This test is based on the maximum difference of the empirical and the proposed theoretical distribution [31]. The Kolmogorov-Smirnov goodness of test statistic is defined as follows: D = M ax \| F (X i ) − i N \| f or 1 ≤ i ≤ N(6) The smaller the value of Kolmogorov-Smirnov statistic is, the better the goodness-offits is. In order to confirm the results given by the early goodness of fit, the quantilequantile plot is proposed. Quantile-quantile plot Shortly QQ-plot, which is a graphical method to compare distributions. based on the observation of the quantiles of each distribution. The linearity in the graph is easily verified, or furthermore, it can be quantified by the correlation coefficient [32]. In this case of study, the data are well-modelled by the Generalized Extreme Value. As described by the QQplot (Figs. 6-7 as examples), extremes are well fitted by the GEV. Furthermore, the Kolmogorov-Smirnov test confirms that the best probability distribution, for the used data, is the GEV (Fig. 8). According to this comparison, the remain work is based on the GEV. Therefore these parameters (µ, σ, ξ) are modelled by using ELM. Spatial Modelling The second step of this study deals with a spatial modelling using Extreme Learning Machine [33]. ELM is inspired by the multi-layer perceptron (MLP) with one hidden layer. For a fixed number of hidden nodes N , ELM generates randomly the weights and the biases of each node. Then the result passes through a differentiable activation function g which gives the matrix H where each row corresponds to the output of hidden layer for one input data vector: H ij = g(x i .w j + b j )(7) where x i = x 1 i , x 2 i , . . . , x d i (i = 1, . . . , n) are the input data, w j (j = 1, . . . , N ) are the vectors of weights, and b j are the biases of each node. To get the connection vector β between the hidden layer and the output layer, ELM uses the Moore-Penrose generalized inverse of the matrix H: β = H † y These operations give at the end new predicted data points as well as the validation and the testing errors. And for more efficiency and clarity, the following methodology is used to validate the given model: -The optimal number of nodes N is selected by using k-fold cross-validation with respect to the mean square error (k = 6). -Then the optimal model is generated and evaluated according to the mean square error for the testing set. -This process of learning is repeated 20 times, with random splitting of the data, and at the end the mean of repetition is taken. The same process is carried out for all parameters of the GEV. Fig.9 shows that the estimated parameters by maximum likelihood are adequate with the predicted by ELM models. Results and discussions The presented work allows us to visualize better each parameter of the GEV distribution. Moreover, the produced maps are coherent with the topographical information in Switzerland. The very important parameter of the GEV is the shape ξ, it defines which subfamily fits the used data. A visualization of the shape on the map of Switzerland can help to answer to the question ( fig. 10): which subfamily is used in a random place? Such information is important for making useful analysis regarding risk assessment, and renewable energy produced by wind speed. Furthermore, the advantage of knowing the extreme distribution allows us to predict the probability for a given wind speed and vice versa, fig. 11 presents probability map that a wind speed exceeds 15 m/s. Finally, this work used ELM to model each parameter of the GEV. This modelling approach deals with the repetition of prediction 20 times to consider the randomness of ELM when it generates the weights. In order to compare the values given by ELM, fig. 12 shows different densities generated by the minimum values, the maximum, and the mean of the predicted parameters after the 20 repetition. This difference is checked by using QQ-plot between the predicted values and the testing data. As expected, ELM shows its efficiency to model, by the insignificant different between results. However, repeat ELM several times improves the quality of results, and helps to obtain optimal model. One of the most important result in this work is the possibility to predict extreme wind speed in new places without measuring stations. Furthermore, the comparison between the three proposed distributions offers more precision to the study. Conclusion Wind energy remains the most attractive resource for providing sustainable power. The research presented here gives an initial overview of wind speed data, and a global idea about extremes of this phenomenon in Switzerland. Such modelling can be useful in developing intelligent decisions for wind-powered electrical generators. Spatial modelling of distributions can be used to optimize existing network and to propose new places for the aeolian energy production. The developed methodology provides more efficiency. The combination of parametric estimation and extreme learning machine offers more rapidity to obtain good results. Furthermore, the results are focused on extremes which is important for natural hazards and risk assessments. This methodology could be applied to other extreme environmental phenomena, e.g. precipitation. Further developments could be in application of this methodology for spatio-temporal environmental data and quantification of the uncertainties. Fig. 1 . 1Scatter-plot of some variables from the input space used for training ELM scaled in the [0,1] interval. wind speed measurements at weather stations distributed in all Switzerland (fig.2), at different elevations, from 203 m to 3580 m. In total there are more than 148 stations. However, some stations were eliminated because they contain an important number of missing values. The final dataset contains measurements of 120 stations for two years (2012 and 2013), taken at 10 minutes intervals. This important high frequency allows us to obtain good approximation of wind speed distribution, even the behaviour of extremes. Fig. 3,4 show examples of some measuring stations. The time series plots do not indicate any significant increasing or decreasing trends. Fig. 2 . 2Locations of MeteoSwiss stations. Fig. 3 . 3Observations of some measuring stations. Fig. 4 . 4Boxplot of some measuring stations. Fig. 5 . 5The three subfamilies of the GEV with µ = 0, σ = 1. And ξ = −0.5, 0, 0.5 for Weibull, Gumbel, Fréchet respectively. Fig. 6 . 6Wind Speed at M atro station (2171 meters). The Kolmogorov-Smirnov test gives the following values: 0.1229, 0.1655, 0.1102 for Weibull, Gamma, and Generalized Extreme values respectively. Fig. 7 . 7Wind Speed at Gornergrat station (3129 meters). The Kolmogorov-Smirnov test gives the following values: 0.1984, 0.2023, 0.1026 for Weibull, Gamma, and Generalized Extreme values respectively. Fig. 8 . 8-Data are projected into the interval [0,1], and then are split into training and testing set, in total 30 measuring stations are assigned as testing set and 90 as training set. -The remain data are used to train ELM with N number of hidden nodes where N ∈ {1, . . . , Values of the Kolmogorov-Smirnov test for each probability distribution. Fig. 9 . 9Comparison between predicted and estimated distribution for Jungf raujoch (3580 meters) taken from the testing set. Fig. 10 . 10On the left: Visualisation of the location parameter of GEV distribution. On the right: The shape parameter, the dominated subfamilies in this case of study are more or less the Gumbel and the Fréchet subfamilies. Fig. 11 . 11Probability map for wind speed greater than or equal to 15 m/s. Fig. 12 . 12Comparison between different results given by ELM. Table 1 1Input space variables generated from digital elevation model.ID Name of the variable Scale X X coordinate Y Y coordinate Z Z (elevation) dogs Diff. of Gauss. at small scale σ 1 = 0.25 km σ 2 = 0.5 km dogm Diff. of Gauss. at medium scale σ 1 = 1.75 km σ 2 = 2.25 km dog l Diff. of Gauss. at large scale σ 1 = 3.75 km σ 2 = 5 km Ss Slopes at small scale σ = 0.2 km Sm Slopes at medium scale σ = 1.75 km Sl Slopes at large scale σ = 3.75 km dns Dir. deriv. in South-North dir. at small scale σ = 0.25 km dws Dir. deriv. in East-West dir. at small scale σ = 0.25 km dnm Dir. deriv. in South-North dir. at medium scale σ = 1.75 km dwm Dir. deriv. in East-West dir. at medium scale σ = 1.75 km AcknowledgementsThe authors are grateful to Jean Golay and Michael Leuenberger for many fruitful discussions about machine learning and extreme values. The authors thank MeteoSuisse for giving access to the data via IDAWEB server. 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[ "Training and recovery behaviours of exchange bias in FeNi/Cu/Co/FeMn spin valves at high field sweep rates", "Training and recovery behaviours of exchange bias in FeNi/Cu/Co/FeMn spin valves at high field sweep rates" ]	[ "D Z Yang \nInstitutt for fysikk\nNTNU\nNO-7491TrondheimNorway\n\nThe Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education\nLanzhou University\n730000LanzhouChina\n", "A Kapelrud \nInstitutt for fysikk\nNTNU\nNO-7491TrondheimNorway\n", "M Saxegaard \nInstitutt for fysikk\nNTNU\nNO-7491TrondheimNorway\n", "E Wahlström \nInstitutt for fysikk\nNTNU\nNO-7491TrondheimNorway\n" ]	[ "Institutt for fysikk\nNTNU\nNO-7491TrondheimNorway", "The Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education\nLanzhou University\n730000LanzhouChina", "Institutt for fysikk\nNTNU\nNO-7491TrondheimNorway", "Institutt for fysikk\nNTNU\nNO-7491TrondheimNorway", "Institutt for fysikk\nNTNU\nNO-7491TrondheimNorway" ]	[]	Training and recovery of exchange bias in FeNi/Cu/Co/FeMn spin valves have been studied by magnetoresistance curves with field sweep rates from 1000 to 4800 Oe/s. It is found that training and recovery of exchange field are proportional to the logarithm of the training cycles and recovery time, respectively. These behaviours are explained within the model based on thermal activation. For the field sweep rates of 1000, 2000 and 4000 Oe/s, the relaxation time of antiferromagnet spins are 61.4, 27.6, and 11.5 in the unit of ms respectively, much shorter than the long relaxation time (∼ 10 2 s) in conventional magnetometry measurements.	10.1016/j.jmmm.2012.05.002	[ "https://arxiv.org/pdf/1201.0358v1.pdf" ]	55,702,603	1201.0358	b1674c4bc8b5c65462fb6405a9d5bbe70bac6cf6	Training and recovery behaviours of exchange bias in FeNi/Cu/Co/FeMn spin valves at high field sweep rates 1 Jan 2012 D Z Yang Institutt for fysikk NTNU NO-7491TrondheimNorway The Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education Lanzhou University 730000LanzhouChina A Kapelrud Institutt for fysikk NTNU NO-7491TrondheimNorway M Saxegaard Institutt for fysikk NTNU NO-7491TrondheimNorway E Wahlström Institutt for fysikk NTNU NO-7491TrondheimNorway Training and recovery behaviours of exchange bias in FeNi/Cu/Co/FeMn spin valves at high field sweep rates 1 Jan 2012Antiferromagnetexchange biastraining effect Training and recovery of exchange bias in FeNi/Cu/Co/FeMn spin valves have been studied by magnetoresistance curves with field sweep rates from 1000 to 4800 Oe/s. It is found that training and recovery of exchange field are proportional to the logarithm of the training cycles and recovery time, respectively. These behaviours are explained within the model based on thermal activation. For the field sweep rates of 1000, 2000 and 4000 Oe/s, the relaxation time of antiferromagnet spins are 61.4, 27.6, and 11.5 in the unit of ms respectively, much shorter than the long relaxation time (∼ 10 2 s) in conventional magnetometry measurements. Introduction The exchange bias (EB) effect in ferromagnetic/ antiferromagnetic systems have been intensely studied in the last decade because of their physical complexity and important applications [1,2]. The technological importance lies in the pinning effect of the antiferromagnet (AFM) layers in which the hysteresis loop of the ferromagnet (FM) can be shifted away from the origin point by the amount of the exchange field (H E ), and is usually accompanied with an enhanced coercivity (H C ). Changes of H E and H C are accordingly directly related to the spin configuration of the AFM layer through the exchange coupling [3]. Among the variety of effects related to the EB phe-nomenon, the training effect is an important effect that reflects the AFM spin dynamic process during repeated hysteresis loops. It is ascribed to that the spin structure of the AFM layer deviates from its equilibrium configuration and approaches another equilibrium triggered by subsequent reversals of the FM magnetization. Nowadays, studies of AFM spin dynamic behaviours with training effect in both experiments and theories have been widely reported [4][5][6][7][8][9][10][11][12][13]. Because most of studies are limited to long timescales (> 1s), by the usually quite long measurement time in magnetometry approaches, the relaxation time of AFM spin are usually reported in second timescale (∼ 10 2 − 10 4 s) [4][5][6]. In contrast, at shorter measurement timescales the relaxation time of exchange bias system was demonstrated to cover a wide range ( ∼ 10 −8 − 10 11 s) [15][16][17][18], which has been ascribed to the magnetization reversal mechanism of FM layer [14]. Hence, the report only on AFM spin dynamic behaviour in the millisecond timescale is still sparse. In addition, recently attempt frequencies up to 10 12 Hz in AFM layer have been reported [19], which indicated the much shorter relaxation timescale of AFM spin than earlier anticipated. Therefore, it is necessary and interesting to study the AFM spin dynamic process at short timescale (technologic importance < 1s). In this work, we have studied the EB training and recovery behaviours at the millisecond timescale based on the electrical transport measurements in FeNi/Cu/Co/FeMn spin valves. The experiments show that at high field sweep rates recovery time of exchange field after training procedures is three orders of magnitude shorter than the values observed by usual magnetometry techniques, and the relaxation of magnetoresistance (MR) is demonstrated in the millisecond timescale. These clearly indicate that AFM spin dynamic behaviours can be studied and resolved down to the millisecond timescale utilizing the ordinary resistance measurements. Experiment and Results The spin valves of Si (001)/Cu (10 nm)/Fe 20 Ni 80 (3 nm)/Cu (3 nm)/Co (3 nm)/FeMn (8 nm)/Ta (3 nm) were prepared by a magnetron sputtering system. The base pressure was 2×10 −5 Pa and the Ar pressure was 0.3 Pa during the deposition. The 10 nm Cu buffer layer was used to stimulate the fcc (111) preferred growth of the FeMn layer in order to enhance the EB. A magnetic field of 130 Oe was applied in the film plane during depo-sition to induce the uniaxial anisotropy and thus the EB. Magnetoresistance (MR) measurements were performed to probe the switching behaviours of the pinned layer for different subsequent hysteresis loops. The magnetic field was provided by home-built Helmholtz coils, and MR was measured in realtime system with 2 M/s sampling rate. To study training and recovery of the EB, we first performed forty consecutive MR measurements with a fixed field sweep rate to characterize the training procedures. Then we stopped the magnetic field sweep with an waiting time t. Finally ten consecutive MR measurements with the same field sweep rate were measured in order to observe and confirm the EB recovery. For each sweep rate, t varied from 0.1 to 10 s. The spin valves MR curves of the training and recovery effects at the 1 st , 40 th and 41 st cycles with the field sweep rate of 4000 Oe/s are displayed in Fig. 1 (a). At large negative field the Co and FeNi magnetizations are parallel and pointing down. When the field is increased above the switch field of the Co layer, about -110 Oe, the Co magnetization reverses and resistance switches from low value (-1) to high value (+1). When the field is further increased above the switch field of the FeNi layer, about -15 Oe, its magnetization reverses, the two magnetizations become parallel once more but this time pointing up, and resistance switches to low value (-1). If the field is then decreased, the two magnetizations will remain parallel until the negative switch field of the FeNi layer is reached at -25 Oe, when its magnetization reverses and resistance switches to high value. When the field is further reduced and reverses the Co magnetization, the two magnetizations align in parallel, and resistance changes to its low value. For all MR curves the hysteresis loops of the Co layer are shifted and fully separated from the hysteresis loops of FeNi layer due to the FeMn pinning effect, therefore the MR curves directly reflect the switching behaviours of the Co and the FeNi layers in detail [20]. Comparing the hysteresis loops of the Co layer in the 1 st , and 40 th MR curves, the switching field of the descent branch shifts more sharply than that of the ascent one, demonstrating the asymmetric magnetization reversal. However, after the magnetic field sweep is stopped for 1 s, a recovery is observed in the 41 st MR curve. It contrasts to the behaviour in the case of normally low field sweep rate, in which substantial recovery was only observed after several hours of waiting time [4]. The H E is plotted as function of cycles n in Fig. 1(b). The H E gradually decreases with the cycle n, has an obvious resilience after 1 s waiting time and finally decreases. For the training procedure, the H E versus n is fitted by a linear functions of 1/ √ n , e −0.06n and ln(n). It is found that the logarithm function yields the best fit, except for initial point n=1 [8,10]. To further study the recovery of the trained EB, we measured the recovery rate R as a function of t at different field sweep rates, Figure 2(a) shows the dependence of R on t at different field sweep rates. The R increases with the increasing t as a linear function of log(t). More remarkably, for a fixed waiting time t, R correspondingly increases with the increasing field sweep rate. This logarithm behaviour is in a good agreement with the previous experiments in NiFe/FeMn system, while the recovery rate is several orders of magnitude faster than the value in the low field sweep rate [4]. The slope and the intercept as a function of the field sweep rate are shown in Fig. 2(b). The slope displays little change with different field sweep rates whereas the intercept increases greatly as the field sweep rate increases in approximate linear function of the logarithm of the field sweep rate. where R = [H E (41) − H E (40)]/[H E (1) − H E (40)] × 100(%). To investigate the dynamic behaviour of the EB with high resolution, we observed the evolution of MR after setting the magnetic field from the positive saturation field to -210 Oe (the point A in Fig. 1(a)) near the switch field. As shown in Fig. 3, MR initially decreases sharply and then gradually reaches a constant. The small fluctuations in the curves are caused by 50 Hz AC noise in the amplifying circuit. Remarkably, a crossover of the normalized MR from positive to negative has been observed, demonstrating the reversal of the magnetization of the Co layer. It is possible to link the time dependence of MR with the magnetic viscosity in the Co/FeMn bilayers [7], in which the magnetization of the pinned layer gradually reverses due to the thermally activated process in Co/FeMn bilayers. Because the reversal process in the EB at the first cycle consists in the single domain wall motion [6], the change of MR here is proportional to the amounts of the reversal magnetization in the pinned layer. Shown as the solid line in Fig. 3, the evolutions of MR are described well by a first order exponential decay. From fitting the data, we extracted the relaxation times τ are 11.5, 27.6, and 61.4 ms for the field sweep rate at 4000, 2000 and 1000 Oe/s respectively. This is again in contrast to the long relaxation time (∼800 s) in the conventional approaches [6]. One can also note the relaxation time decreases with the increasing field sweep rate. Discussion The above results show that the recovery and relaxation of the EB at high field sweep rates are faster than that earlier observed [4][5][6][7]. Below we will interpret the experimental results in conventional models for AFM and training effects. Firstly we consider the change and magnitude of the relaxation time constants at different field sweep rates shown in Fig. 3. The time constant for the relaxation can be described by an ordinary Arrhenius law τ = v −1 σ exp(E σ /k B T ), where v σ is the attempting frequency and E σ = KV represents the AFM energy barrier, K is the AFM anisotropy and V is the AFM grain volume. According to the AFM grain volumes distribution, we can divide the E σ into three different categories [10]: i) small E σ (small grain size), which follows the FM magnetization at the timescale of the experiment. ii) medium energy E σ (medium grain size) which will determine the EB dynamics at the timescale we investigate. iii) large E σ (large grain size), which is a stable configuration over the timescale of the experiment. Assuming a typical uniaxial anisotropy constant of 1 × 10 6 erg/cm 3 and v σ to be 1 × 10 9 Hz, then the average grain size of category (ii) is correspondingly about 9 nm extension, based on the relaxation time in Fig. 3. Accordingly, that relaxation time decreasing with the increasing field sweep rate, demonstrates an apparent increase in attempt frequency v σ . The EB recovery and relaxation at high field sweep rates can still be explained well with the model based on thermal activation [4,23]. As shown in Fig. 2(a), the logarithm time recovery relationship indicates a thermally activated reversal process involving the AFM spin configuration. To explain our data, the activation energy spectrum model simply based on a two-level system is adopted [23,24]. In our case the two level system represents an individual AFM grain or domain switching from a positive to a negative exchange energy with respect to the FM layer. For the system with a wide energy barrier distribution, ∆H E can be expressed in terms of the AFM activation energy spectrum q(E): ∆H E = q(E)k B T ln(v σ t), which is taken from equation (1) in ref [4]. According to the equation, the slope observed for all field rates in Fig. 2(b) is a constant due to the same q(E), while the intercept variation is mainly due to the different activated AFM energy ranges and the time delay at the different field sweep rates. Finally, for the training process H E is proportional to ln(n) at high field sweep rates in Fig. 1(b), which can be compared to the usual power law 5 (1/ √ n) and the exponential (e −αn ) relationships. We model this through following Binek et al [21,22]. At beginning, the equilibrium AFM interface magnetization is defined S e AF M = lim n→∞ S AF M (n). Each positive and negative deviation δS n = S AF M (n) − S e AF M of the AFM interface magnetization from its equilibrium value will increase the total free energy F of the system by ∆F . The relaxation of the system towards equilibrium is determined by the Landau-Khalatnikov (LK) equation [25]: However, our understanding of the system is that we have a nonvanishing odd order term. This is an effect of working at a time scale where we also have substantial coupling at the FM/AFM interface due to large grains that are too large to follow the oscillating exchange coupling of the FM. Instead that portion of the ensemble of grain will orient itself gradually according to the mean coupling induced by the FM in a monotonic fashion. Accordingly we also have to considered the expansion of ∆F also from first order of δS. We then assume δS: ∆F = f (n)(δS) 1 + O(δS) 2 , where the f (n) indicates that the change in the AFM interface magnetization and δS n is dependent on the training procedures n. By replacingṠ with [S(n + 1) − S(n)]/τ , with τ being the relevant experimental time constant and the free energy expression of the first order into the LK equation, we obtain an implicit sequence equation: ξ ′ (S(n + 1) − S(n)) = −f (n), where ξ ′ = ξ/τ . The sum of this equation yields H E (n + 1) ∝ S(n + 1) = − f (n)/ξ ′ . Since we do not know the exact energy distribution of our system, we make a first order approximation assuming a constant distribution of areas, which leads to a linearly increasing activation energy with time, then the approximate development f (n) will follow a 1/n dependance. Using this approximation, we obtain a logarithmic relaxation of the system that fits well with the observed dependence. Note, that already at n=6 the approximation holds to within an error of less than 4%. We note that the training process H E is proportional to ln(n) has also been reported at low field sweep rates [8], where the training speed is several orders of magnitude slower than the values reported here. In summary, for the AFM spins the relaxation time in the millisecond timescale is demonstrated when the bilayers are exposed to high field sweep rates. This behaviour can be well explained in terms of a time constrained thermal activation. Our finding gives a new insight into the dynamic behaviour of the AFM spins. ¡ ¢ £ ¤ ¡ ¢ ¤ ¤ ¡ ¥ £ ¤ ¡ ¥ ¤ ¤ ¡ £ ¤ ¤ ¡ ¥ ¦ ¤ ¡ ¤ ¦ £ ¤ ¦ ¤ ¤ ¦ £ ¥ ¦ ¤ ¥ ¦ £ ¤ ¥ ¤ ¢ ¤ § ¤ ¨ ¤ £ ¤ ¡ ¥ © £ ¡ ¥ © ¤ ¡ ¥ £ £ ¡ ¥ £ ¤ ¡ ¥ ¨ £ ! " # $ % & $ '" ( ) 0 1 0 2 3 4 5 ) 6 7 8 9 4 3 Figure 3: The time dependence of the resistance after the external magnetic field is swept to -210 Oe (point A in Fig. 1(a)) from positive saturation field with different field sweep rates (a) 4000 Oe/s (b) 2000 Oe/s (c) 1000 Oe/s. The solid lines are fits to the first order exponential decay. @ 1 A ) B C D E F G H I P Q R S R T U V W X Y R ` a b c d U V U e f g h 6 i ) C p q G r s r t u ¥ ¦ ¤ v F w x t y ξṠ AF M = −∂∆F /∂S AF M , where ξ is a phenomenological damping constant and ∆F is the function of δS. In Binek's model under the assumption ∆F (δS) = ∆F (−δS) , a series expansion of ∆F up to the fourth order in δS yields ∆F = 1 2 a(δS) 2 + 1 4 b(δS) 4 + O(δS) 6 . Evaluating the free energy expression with a leading term of 2 nd and 4 th order in δS will result in the e −αn [22] and 1/ √ n [21] evolution, respectively. Figure 1 :Figure 2 : 12(a) The magnetoresistance curves used to map the training effect for FeNi/Cu/Co/FeMn spin valve at the 1 st , 40 th and 41 st (after 1 s waiting time) cycles with the field sweep rate of 4000 Oe/s. The resistance is dependent on corresponding magnetization configurations of FeNi (black arrow) and Co (red arrow). (b) The exchange field H E as a function of the number of cycles n. The blue dot, green dash dot and black dash lines are the fitted data with the 1/ √ n , e −0.06n and ln(n), respectively. (a) The recovery of H E as a function of the waiting time t with different sweep rates. The solid lines display the linear fits of the ln(t). (b) The slope and the offset values as a function of the field sweep rate. The solid lines are the linear fits of the logarithm of the field sweep rate. Acknowledgments . 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[ "Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots", "Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots" ]	[ "Mina Aganagic \nCenter for Theoretical Physics\nUniversity of California\n94720BerkeleyCAUSA\n\nDepartment of Mathematics\nUniversity of California\n94720BerkeleyCAUSA\n", "Cumrun Vafa \nJefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA\n" ]	[ "Center for Theoretical Physics\nUniversity of California\n94720BerkeleyCAUSA", "Department of Mathematics\nUniversity of California\n94720BerkeleyCAUSA", "Jefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA" ]	[]	We reconsider topological string realization of SU(N) Chern-Simons theory on S 3 . At large N, for every knot K in S 3 , we obtain a polynomial A K (x, p; Q) in two variables x, p depending on the t'Hooft coupling parameter Q = e N gs . Its vanishing locus is the quantum corrected moduli space of a special Lagrangian brane L K , associated to K, probing the large N dual geometry, the resolved conifold. Using a generalized SYZ conjecture this leads to the statement that for every such Lagrangian brane L K we get a distinct mirror of the resolved conifold given by uv = A K (x, p; Q). Perturbative corrections of the refined B-model for the open string sector on the mirror geometry capture BPS degeneracies and thus the knot homology invariants. Thus, in terms of its ability to distinguish knots, the classical function A K (x, p; Q) contains at least as much information as knot homologies. In the special case when N = 2, our observations lead to a physical explanation of the generalized (quantum) volume conjecture. Moreover, the specialization to Q = 1 of A K contains the classical A-polynomial of the knot as a factor.	null	[ "https://arxiv.org/pdf/1204.4709v4.pdf" ]	16,514,501	1204.4709	b7ea05bfe3940df447f21ff4bb737b333e227fc4	Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots 18 Jul 2012 Mina Aganagic Center for Theoretical Physics University of California 94720BerkeleyCAUSA Department of Mathematics University of California 94720BerkeleyCAUSA Cumrun Vafa Jefferson Physical Laboratory Harvard University 02138CambridgeMAUSA Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots 18 Jul 2012 We reconsider topological string realization of SU(N) Chern-Simons theory on S 3 . At large N, for every knot K in S 3 , we obtain a polynomial A K (x, p; Q) in two variables x, p depending on the t'Hooft coupling parameter Q = e N gs . Its vanishing locus is the quantum corrected moduli space of a special Lagrangian brane L K , associated to K, probing the large N dual geometry, the resolved conifold. Using a generalized SYZ conjecture this leads to the statement that for every such Lagrangian brane L K we get a distinct mirror of the resolved conifold given by uv = A K (x, p; Q). Perturbative corrections of the refined B-model for the open string sector on the mirror geometry capture BPS degeneracies and thus the knot homology invariants. Thus, in terms of its ability to distinguish knots, the classical function A K (x, p; Q) contains at least as much information as knot homologies. In the special case when N = 2, our observations lead to a physical explanation of the generalized (quantum) volume conjecture. Moreover, the specialization to Q = 1 of A K contains the classical A-polynomial of the knot as a factor. Introduction The study of knot invariants has a long history in mathematics. With the introduction of Jones polynomials [1] and their physical interpretation in terms of Chern-Simons theories [2] and their embedding in topological strings [3] this has led to an interesting area of interaction between mathematics and physics in modern times. With the advent of string dualities new perspective was gained on this relation. Large N dualities for topological strings [4] lead to new predictions for HOMFLY polynomials. In particular the interpretation of topological strings as computing BPS degeneracies [5] has led to integrality predictions for colored HOMFLY polynomials. Furthermore, consideration of symmetries of the problem, leads to an interpretation of the Khovanov invariants [6] and knot homologies [7,8], in terms of further decomposition of this BPS Hilbert space with respect to an additional U(1) symmetry [9] (see also the recent work [10]). This refinement in turn relates to Nekrasov deformation of topological strings [11]. The deformation of the topological string also leads to a refinement of Chern-Simons theory recently discovered in [12]. In order to compute knot invariants using topological strings, one needs to get a good handle on the Lagrangian brane L K associated to a knot, after the large N transition. Before the large N transition, the geometry of the Calabi-Yau is T * S 3 and L K is obtained from the knot K by the conormal bundle construction [5]. However, for general knots it has been difficult to use the large N duality to make concrete predictions about the knot invariants, as one needs a simple description of the L K after the large N transition to the resolved conifold. For unknot this can be done explicitly, because it turns out that there is a mirror symmetry [13] which is compatible with the Lagrangian associated with the unknot and which maps the computation of the knot invariants to easy computations on the mirror. Until now it has been difficult to apply this idea to general knots and use topological strings as a practical method for computations of all knot invariants. One aim of this paper is to remedy this gap. We formulate a remarkable generalization of mirror symmetry in the spirit of SYZ [14], which allows us to compute, for the resolved conifold, a distinct mirror for each choice of knot K. We find that for every knot K in S 3 , the leading large N limit of the knot invariants, leads to a polynomial A K (e x , e p , Q) = 0, which captures the moduli of L K corrected by disc instantons. The polynomial A K (e x , e p , Q) captures the mirror geometry in the spirit of SYZ. It determines the mirror Calabi-Yau Y K , as a hypersurface A K (e x , e p , Q) = uv. The polynomial A K (e x , e p , Q) can be viewed as an invariant of the knot. Mirror symmetry and large N-duality relate the computation of SL(N) knot invariants to computation of open topological string amplitudes on the mirror Calabi-Yau Y K , with a brane mirror to L K . Nekrasov deformation of the topological string on Y K is relevant for categorified knot invariants [9]. But, topological string amplitudes carry no more information about distinguishing Calabi-Yau than the classical geometry. This leads us to conclude that categorification carries no additional information for distinguishing knots beyond that captured in the polynomial A K (e x , e p , Q). In a seemingly unrelated development, another polynomial for knots was introduced to physics in [15]. This is the A-polynomial of the knot, which characterizes flat SL(2) holonomies on the knot complement (mathematical work on the A-polynomial originated in [16]). In [15,17,18] the generating polynomial for colored Jones was studied and conjectured to satisfy a difference equation (see also [19,20]). These difference equations can be viewed as a quantum operator associated to the A-polynomial. This structure is reminiscent of difference equations satisfied by open topological string amplitudes [21] and in fact in some cases it was checked that the corresponding objects are canonically isomorphic, namely that a quantum deformation of the A-polynomial maps to a local geometry which in turn captures the topological string amplitudes [22,23]. The secondary aim of this paper is to explain these observations in the context of the broader picture of the relation between knots and topological strings and to extend these observations to higher rank N. We find that the A K (e x , e p , Q) = 0 geometry naturally computes the generating function relevant for the quantum volume conjecture. Moreover, the polynomial that captures the mirror associated to each knot turns out to be a Q-deformed version of the classical A-polynomial of the knot, where Q is related to the large N 't Hooft parameter. More precisely as Q → 1 the mirror geometry contains the A-polynomial of the knot as a factor. In this context, the fact that A K is related to an operator annihilating the partition function of the knot, follows from general properties of open topological strings [21]. The plan for this paper is as follows: In section 2 we recall some basic aspects of mirror symmetry picture according to SYZ [14] and introduce an extension of it which is relevant for local non-compact Calabi-Yau. In section 3 we review the relation of Chern-Simons theory to topological strings on T * S 3 and the Lagrangian L K associated to a knot K. We also recall aspects of large N duality. In section 4, using large N duality, we sum up disk instanton corrections to the moduli space of the brane on the resolved conifold, and discover a family of mirrors, one for each knot K. In section 5 we review the A-polynomial for knots and relations to volume and AJ conjectures and derive these conjecture in the setup of topological string. Furthermore we interpret our Q-deformed A-polynomial as a generalization of the volume conjecture. In section 6 we give examples. In section 7, we discuss the relation between our polynomial and a similar one which has been proposed in the context of knot contact homology. In section 8 we make some concluding remarks and suggestions for future direction of research. Mirror Symmetry in the Local Case In this section we review certain features of mirror symmetry for non-compact Calabi-Yau manifolds. We will focus on the local toric Calabi-Yau three-folds, for which mirror symmetry is understood the best. This is also the case that naturally fits with knot theory, as we will see. It is known [13,24] that for the case of non-compact toric Calabi-Yau the mirror Calabi-Yau 3-fold takes the form F (e x , e p ) = uv (2.1) where (e x , e p , u, v) ∈ C * × C * × C × C. In particular the main data of the Calabi-Yau is captured by the curve F (e x , e p ) = 0. (2.2) As discussed in [25] this curve can be viewed as the moduli space of a canonical special Lagrangian brane. These special Lagrangian submanifolds have the topology of S 1 × R 2 and generalize the original constructions of Harvey and Lawson, and Joyce [26,27]. By a theorem of McLean [28], a special Lagrangian with a b 1 = 1 has a one real dimensional moduli space coming for the deformations, and this is accompanied by the moduli of a flat bundle. If we parameterize the moduli space by x = r + iθ, r comes from geometric deformations of the Lagrangian, and θ = S 1 A from the choice of flat connection on the S 1 . The moduli space receives corrections from holomorphic disks ending on the brane. The disk amplitude W (x) depends holomorphicaly on e −x . The area of the disk is determined by the position r of the Lagrangian. Mirror symmetry relates the disk instanton corrections on the A-model side to classical periods of the mirror. As explained in [25], in the mirror, the disk amplitude W (x) is the integral of the holomorphic threeform Ω on a chain whose boundary is the Lagrangian manifold, which for Calabi-Yau manifolds of the form (2.1) descends to a one form pdx on the curve (2.2).: W (x) = x p(x)dx. The disk instanton corrected moduli space of these Lagrangian submanifolds is captured by a choice of a point on the mirror curve F (e x , e p ) = 0. In other words we can parameterize the moduli space of the brane by the choice of x and find p = p(x), such that F (e x , e p ) = 0. A simple example of this is the resolved conifold geometry given by X = O(−1) ⊕ O(−1) → P 1 . Let Q = e −t where t denotes the complexified Kahler class of P 1 . Then the mirror geometry is captured by the curve [13] F (e x , e p ) = 1 + e x + e p + Qe x+p = 0. This curve can, in turn, be viewed as the moduli space of the special Lagragian branes on the resolved conifold. At one point along their moduli the S 1 of the Lagrangian is identified with the equator of P 1 , while the R 2 is a suitable subspace of the O(−1) ⊕ O(−1) vector bundle over it. Note that the classical moduli space of this Lagrangian brane has singularities, where the S 1 shrinks to a point. The singularities are smoothed out in the quantum theory by the disc instantons. In particular the geometry F (e x , e p ) = 0 has no singular points, and this is a reflection of the fact that the classical geometry of the mirror manifold already captures the disk instanton sum on the A-model side. In the SYZ formulation for mirror symmetry [14], in the compact case, one considers on the A-side a special Lagrangian with the topology of T 3 . This has a 3-complex dimensional moduli space. The mirror is identified as the quantum corrected moduli space, where the quantum corrections include the disc instantons. In the non-compact case it is natural to consider extensions of this where the special Lagrangian has a different topology. In particular from the discussion we just had, it is clear that a natural topology for the special Lagrangian to consider on the A-side is that of R 2 × S 1 and in this context the curve F (e x , e p ) = 0 is naturally identified as the disk instanton corrected moduli space of the Lagrangian brane. 1 However the SYZ construction of mirror geometry admits a more general interpretation. Namely, we can consider an arbitrary special Lagrangian manifold L with the topology of R 2 × S 1 . For each such choice, by summing up disk instanton corrections, we obtain a mirror geometry. The disc corrected moduli space is given by the curve In the next section, we will explain how, using the relation of topological string on X = O(−1) ⊕ O(−1) → P 1 to Chern-Simons theory on S 3 , and large N dualities, we 1 The Lagrangian branes with the topology of R 1 × T 2 have also been considered in this context in [29,30] for toric Calabi-Yau manifolds and relating the SYZ mirror picture with the mirror picture of [13]. can derive, for each knot K in the S 3 a distinct mirror Y K mirror to X depending on the knot. The topological string on each such Y K should be the same. Topological String and Chern-Simons Theory Consider topological A-model on X v = T * S 3 with N Lagrangian D-Z top (X, g s ) = Z CS (S 3 , k) where the topological string coupling g s coincides with the Chern-Simons coupling constant, g s = 2πi k + N . A nice way to introduce knot invariants in this setup is to associate a particular Lagrangian L K to the knot [5], so that L K ∩ S 3 = K. This is the conormal bundle construction, which basically means choosing the R 2 planes of L K to be in the cotangent fiber direction, and be orthogonal (relative to symplectic pairing) to the tangent to the knot. Adding a D-brane on L K in the topological string corresponds to inserting Z top (X, L K , x) = det(1 ⊗ 1 − e −x ⊗ U) (−1) S 3 in Chern-Simons theory, where U is the holonomy along the knot, U = P e i K A . Since the Lagrangian L K has topology R 2 × S 1 , the theory on the brane has one modulus, which we will denote by x. The determinant captures the effect of integrating out the bifundamental scalar field of mass x, corresponding to the string stretching between the S 3 and L K [5]. We can rewrite above as 2 Z top (X, L K , x) = ∞ n=0 T r Rn U S 3 e −nx (3.1) On the right hand side, the sum runs over R n , the totally symmetric representations of SU(N) with n boxes. Let T r Rn U S 3 = H n (K), denote the expectation value of the Wilson loop along the knot K, colored by representation R n in the SU(N) Chern-Simons theory. For general G = SU(N), the topological string partition function defines a wave function Z top (X, L K , x) = n H n (K) e −nx = Ψ K (x) (3.2) which is the exact partition function of the D-brane on L K , Ψ K (x) = Ψ K (x, g s , N) depending on g s and N. It is a wave function because L K is a non-compact three manifold, with a T 2 boundary; x here corresponds to the holonomy around the S 1 in T 2 that remains finite in the interior of L K = R 2 × S 1 . Large N duality It is conjectured in [4] that, if we have N branes wrapped on S 3 in X v = T * S 3 , at large N, the geometry undergoes a transition where S 3 shrinks and the N Lagrangian branes disappear, leaving behind a blown up P 1 . In other words it is conjectured that the SU(N) Chern-Simons theory is equivalent at large N to the topological string on X = O(−1) ⊕ O(−1) → P 1 . where the t'Hooft coupling of Chern-Simons theory becomes the size of the P 1 in X, Q = exp(−t), t = Ng s . The geometry of both X and X v are cones over S 2 × S 3 . The geometric transition shrinks the S 3 at the apex of the cone in T * S 3 and replaces it by the P 1 , leaving the geometry far from the tip untouched. Now consider the noncompact branes L K we constructed in the previous section, one for each knot K in the S 3 . This Lagrangian gets pushed through the transition to a Lagrangian L K on X = O(−1) ⊕ O(−1) → P 1 . We will abuse notation, and denote the Lagrangians on both X and X v by the same letter, L K . The topology of the Lagrangian is unchanged by the transition -L K is still R 2 × S 1 . To see this, note that we could have used the real modulus on the Lagrangian to push L K away from the S 3 before the transition -this way, the Lagranian can pass through the transition while remaining far from the apex of the cone where the geometric transition takes place. Thus, large N duality and the geometric transition gives us a way to construct a family of Lagrangians L K of the topology of R 2 × S 1 in O(−1) ⊕ O(−1) → P 1 , one for each knot K in the S 3 . The large N duality also gives a way to compute topological string amplitudes on X with and without branes. The partition function of Chern-Simons theory on the S 3 , as shown in [4], is the same as the closed topological string partition function of this way derive the geometry of the mirror, as will be discussed in the next section. Mirror Symmetry and Knot Invariants We will now combine the various ideas discussed in the previous sections. We start with the T * S 3 geometry and wrap N Lagrangian branes on the S 3 . This leads to SU(N) Chern-Simons gauge theory on S 3 . We consider a knot K on S 3 and let L K be the associated special Lagrangian brane which intersects the S 3 on K. As reviewed in section 3 at large N the T * S 3 undergoes the geometric transition to the resolved conifold geometry with the modulus of the blown up P 1 given by t where Q = e t . Moreover the L K will be represented by a special Lagrangian submanifold, which by abuse of notation we still denote by L K . Note that L K has the topology of R 2 × S 1 and so the discussion of the previous section applies. The theory has a mirror geometry which is captured by a curve which we denote by F K (e x , e p ; Q) = 0. (4.1) Moreover, as discussed in the previous sections, this classical geometry of the mirror captures all the disc corrections involving worldsheets ending on this special Lagrangian. On the other hand, using geometric transition of the large N Chern-Simons theory, the exact open topological string partition function of the brane on L K , in the resolved conifold, is equivalent to the computation of HOMFLY invariants (or knot homologies which arises in the Nekrasov deformation of topological strings and refined Cherns-Simons theory [12,32]). The exact wave function of the brane, computed by Chern-Simons theory and the topological string, is Ψ K (x, g s , N) = n H n (K) e −nx . (4.2) The large N limit of the amplitude is equivalent to summing over all planar diagrams of the Chern-Simons theory. This is dominated by the disk instanton contribution that sums up all the planar graphs in Chern-Simons theory with one boundary on the D-brane. This can be written as Ψ K (x, g s , Q) ∼ exp( 1 g s p K (x, Q)dx). (4.3) Mirror symmetry and large N duality now imply that [5,33] p = p K (x, Q) = lim gs→0 ∞ n=0 g s tr K U n e −nx lives on the mirror curve (4.1). Conversely, from HOMFLY polynomial H Rn (q, N) colored by symmetric representations, and extracting the classical piece we can read off the curve of the mirror Calabi-Yau manifold. We will give examples of this in section 6. Chern-Simons and The Quantum Mirror Before we go on, note that instead of simply studying the classical equation of the mirror, Chern-Simons theory also gives us the exact quantum mirror geometry. In [21] it was found that the branes on the mirror Riemann surface are annihilated by a quantum operator that quantizes the classical geometry. However, except when the underlying Riemann surface has genus zero, finding the exact brane wave functions is hard by direct methods, and correspondingly the quantum mirrors are not known in general. Large N duality and Chern-Simons theory gives us a way to find the exact brane wave function of the mirror. Thus, from the colored HOMFLY polynomial, without taking any limits, we can derive the quantum mirror Riemann surface. The classical approximation to the wave function (4.3) determines the classical mirror geometry F K (e x , e p , Q) = 0. As discussed in [21] this should be viewed as the classical g s → 0 limit of the operator equation F K (ex, ep, Q)Ψ K (x) = 0 (4.4) where we viewx as a multiplication by x, andp = g s ∂ x , so that [p,x] = g s . In this way Chern-Simons theory on the S 3 gives us a way to solve the topological string on the mirror of the brane and get the quantum Riemann surface associated to the knot. This is an example of quantum Riemann surfaces that emerge from studying D-branes in the topological string [21]. Before we go on, note that we can view Ψ K (x, Q) as a discrete Fourier transform of H Rn (K, q, Q). Correspondingly, an equivalent way to represent the quantum mirror curve is via its action on H Rn (K). This giveŝ F K (ex, ep, Q)H Rn (K) = 0 (4.5) where now epH Rn (K) = q n H Rn (K), exH Rn (K) = H R n+1 (K). Relation to the A-polynomial and the Volume Conjecture We have seen in the previous sections that mirror symmetry and large N duality associate a quantum Riemann surface of the form In the classical → 0 limit A K becomes a plane holomorphic curve, A K (x, p) = 0. The curve that arises in this way is called the characteristic variety of the knot. Moreover, they are closely related to another Riemann surface from classical mathematics, the A-polynomial is defined by studying the moduli space of flat SL(2, C) connections on the complement of the knot K in the S 3 . It was studied in [16]. Given a knot K in S 3 one considers its complement, M K = S 3 /K Cutting out the neighborhood of the knot form an S 3 , one obtains a three manifold with a T 2 boundary. The the coordinates p and x parameterize the flat SL(2, C) connections, around the cycles µ and λ that generate π 1 (T 2 ). Here µ is the meridian that links the knot, and the λ is the longitudinal cycle that runs parallel to the knot. The condition that these extend smoothly from the boundary T 2 to the interior of M K imposes a relation between x and p that is the A-polynomial A K (x, p) = 0. We now explain the relation to the Riemann surfaces H K (x, p, Q) that arise in the context of topological strings. The colored Jones polynomial is a special case of the HOMFLY polynomial colored by symmetric representations, where we restrict to N = 2 and set Q = q N = q 2 , J n (K, q) = H Rn (K, q 2 , q). This immediately implies that the quantum A-polynomial A K (x,p) arises as a special case of the quantum mirror curve where we would replace Q = q 2 . This identifies the corresponding quantum Riemann surfaces. The main difference between the descriptions of the wave functions is that in the context of the topological string the x coordinate, which is holonomy along longitudinal cycle of the knot is the natural variable, whereas in the context of the volume conjecture the p coordinate, which is the holonomy along the meridian is more natural. This in particular means that if we consider the Fourier transform of the wave function we have studied in the context of topological strings we would get the wave function relevant for the quantum volume or AJ conjecture:Ψ K (p) = dx exp( px g s ) Ψ K (x) = dx exp( px g s ) n exp(−nx)J n (K) = J p/gs (K) Moreover we expect that as we take q and Q to 1, we obtain the classical A-polynomial. Furthermore, the explanation of why the classical A-polynomial is related to Jones polynomial and how it relates to the volume of the hyperbolic knot complement now follows because all are related to SL(2) Chern-Simons as was already noted in [15]. In particular in the classical limit fixing the x, p holonomies and solving the SL(2) Chern-Simons equations, lead to flat solutions and thus they should lead to the constraint captured by the A-polynomial. There is a subtlety here, as it turns out that the limit q → 1 leads to a polynomial which is not exactly the A-polynomial but has the Apolynomial as a factor. We can offer an explanation of this, namely these could be interpreted as different branches of the brane geometry where the Lagrangian brane L K has different reconnections with branes on S 3 as well as possibly moved off of S 3 . For example, for the unknot we get in this limit (1 − e x )(1 − e p ) = 0. One solution e x = 1, where x corresponds to longitudinal direction along K, is part of the classical A-polynomial and signifies the branch where the brane L K has reconnected with one of the branes on S 3 and has trivialized the holonomy around the non-trivial circle in L K . The other branch e p = 1 could signify the geometry where the L K is lifted off of S 3 , in which case e p is trival because it corresponds to the contractible circle in T 2 ⊂ L K . It would be interesting to study this multi-branch structure for more general knots, and predict the additional factors that multiply the classical A-polynomial. Considering the limit of q → 1 keeping Q fixed, leads to a deformation of the classical A-polynomial to a Q-deformed version. More precisely, we have the following sequence of limits:F K (ex, ep; Q)→ q→1 F K (e x , e p , Q) → Q→1 A K (e x , e p ) andF K (ex, ep; Q) → Q→q 2Â K (ex, ep) → q→1 A K (e x , e p ) More precisely in the last identity we expect that A K (e x , e p ) is a factor of F K (e x , e p ; 1). Thus, it simply follows from large N duality arguments and mirror symmetry that the mirror curve associated to the knot K, F K (e x , e p , Q) is a Q-deformation of a polynomial which has the classical A-polynomial of the knot A K (e x , e p ) as a factor. This is the same as the classical mirror geometry associated to K. In other words, it simply follows from large N duality arguments and mirror symmetry that there exists a Q-deformation of a polynomial A K (e x , e p ) which has the classical A-polynomial as a factor, which in turn is the same as the classical mirror geometry associated to K. We can already make a few checks about this statement. In particular let us consider the limit Q → 1. In this case we should obtain the mirror to the conifold at the singular point, where the size of the P 1 is zero. If A K (e x , e p ) is in fact the mirror geometry for the conifold, then it better be true that the periods we compute for A using the mirror 1-form pdx are zero. More precisely the period should be zero, up to the ambiguity of adding an integer multiple of 2πi, because Q = 1 only leads to t = 2πin for some n. Indeed this is one of the deep properties 4 of the classical A-polynomial for knots [34]! This is already a first indication of the power of our conjecture, which automatically provides an alternative explanation of this fact. For general Q, the mirror curve A K (Q) = 0 should have the same log(Q)-periods as the canonical mirror of the conifold. We will show this in the example of (m, n) torus knot curves, for arbitrary (m, n). Unlike the A-polynomials, the A K (Q) polynomials of these curves are highly non-trivial, as we will see. Hence this is a strong test of our conjecture. Examples As we discussed above, the canonical of mirror of X v = O(−1) ⊕ O(−1) → P 1 is uv = (1 − e p ) − e x (1 − Qe p ). In this section, we will explain that the canonical mirror is associated to the simplest knot in the S 3 , the unknot. Then, by changing the knot type, we will give examples of different mirrors that can arise. For each of these, we conjecture that topological strings on them are indistinguishable. The Unknot The canonnical mirror Riemann surface, which knot in the S 3 is this associated to. As explained in [5], the Lagrangian brane L O in X v corresponds to the unknot in the S 3 . The mirror geometry could have been derived from G = SU(N) Chern-Simons theory along the lines we described above. Recall that for the unknot, the HOMFLY polynomial colored by the n'th symmetric representation R n equals H n ( ) = S 0Rn = [N][N + 1] . . . [N + n − 1] [1][2] . . . [n] where S RQ is the S matrix of SU(N) k Chern-Simons theory, and we need its element with Q = 0, R = R n . We can rewrite this, in terms of Q = q N as follows: H n ( ) = Q − n 2 q n 2 (1 − Q)(1 − Qq) . . . (1 − Qq n−1 ) (1 − q)(1 − q 2 ) . . . (1 − q n ) (6.1) where [n] = q n/2 − q −n/2 . From the recurrence relation satisfied by H n (1 − q n )H n − Q − 1 2 q 1 2 (1 − Qq n−1 )H n−1 = 0. we can read off the equation of the quantum mirror. Up to a redefinition of the holonomy variable x by a constant shift, we get F (ex, ep, Q) = 1 − ep − ex(1 − Qep) = 0 acting on H n , or equivalently, on the wave function Ψ O (x) = n=0 H n e −nx . In particular, in the classical limit,F (e x , e p , Q) gives the Riemann surface of the cannonical mirror Calabi-Yau Y . Torus knots For any knot K we expect, via large N duality and mirror symmetry, to obtain a mirror Calabi-Yau to X. A large family of examples is provided by torus knots. The colored SU(N) k polynomial for an (m, n) torus knot is given by Rosso-Jones formula [1]. It will be useful for us to recall a derivation of it, as understanding it will allow us to make some shortcuts later on. If we view the S 3 as a T 2 fibration over an interval, we can take an (m, n) torus knot to wind around the (m, n) cycle of the T 2 . This provides a way to compute the corresponding knot invariants: we think about the holonomy Tr R U (m,n) = Tr R e mx+np as the holonomy of an unknot knot winding m times around the (1, 0) cycle, but with fractional framing f = m/n, corresponding to the shift of x to x + m/np. Tr R e mx+np = Tr R e n(x+m/np) Using the fact that an unknot in framing f , has expectation value T r Q e x+f p = S 0Q T f Q , where S and T are the S and the T matrices of SU(N) k Chern-Simons theory, and expanding Tr R U n in terms of Tr Q U Tr R U n = Q C (n) RQ Tr Q U. we get the Rosso-Jones formula. H (m,n) R = Tr R U (m,n) = P C (n) RP T m/n P S P 0 . (6.2) To find the D-brane moduli space, we can proceed in two equivalent ways. We can use the exact HOMFLY to find the quantum operatorF (m,n) (ex, ep, Q) annihilating the brane wave functionF It would be very interesting to obtain such a formula, analogously to what was done for the figure 8 knot in [35] and recently in [36] 5 . Instead, one can simply use the leading large N limit of the HOMFLY amplitude, which was obtained in [38]. 6 For the (m, n) torus knot, one finds 5 As we prepared this paper for publication, [37] appeared, where just such a formula was found, and in fact extended to the refined setting. Using the formulas in [37], one could in fact obtain also the quantum mirror curves. 6 In [38] the authors studied mirrors of torus knots. The curves obtained there are different than the ones we find. The difference, roughly, is that in their case, the mirror brane corresponds not to a single point on the Riemann surface as is the case for us, but to n points moving together where n is determined by the (m, n) type of the torus knot. The authors showed that one can reproduce the g s corrections to some closed and open string amplitudes of the conifold with the branes corresponding to this knot. It would be nice to elucidate the physical interpretation of the curves obtained in that paper, as one is sure to exist. From (6.3) and (6.4) for specific m, n we can write down the mirror curves this gives rise to, by simply asking for (x, p(x)) to lie on the curve. We will explain later the method we used to obtain the curves, but for now it suffices to give examples of the results one gets. Ψ(x) ∼ exp( 1 g s p(x)dx) The (2, 3) knot The simplest non-trivial torus knot is the (2, 3) knot, the trefoil. For the trefoil, we find that (x, p(x)) lie on the following curve: F (2,3) (α, β, Q) = (1 − Qβ) + β 3 − β 4 + 2β 5 − 2Qβ 5 − Qβ 6 + Q 2 β 7 α + −β 9 + β 10 α 2 For legibility of the formulas, in this and the following examples, we will denote e x = α, e p = β. To be precise, to reproduce the leading large N limit of the colored HOMFLY polynomial, we need to replace α by αQ 5/2 . While fractional powers of Q are natural in knot theory, they are not natural in the topological string, so we prefer to present the curve in this way. At Q = 1 limit, this becomes F (2,3) (α, β, 1) = (−1 + β) 1 + αβ 3 −1 + αβ 6 The classical A polynomial of the trefoil knot, A (2,3) (α, β) = 1 + α −1 + αβ 3 corresponds to the last two factors in F (2,3) (α, β, 1), after the change of framing that takes α to αβ −3 . In section 5 we suggested the interpretation of the remaining factor, (−1 + β), as corresponding to the branch of the moduli space where the branes on L K move off the S 3 . On this branch, the corresponding cycle becomes contractible in L K , so the holonomy around it vanishes, setting β to 1. The (2, 5) knot For the (2, 5) knot, we get the following curve. F (2,5) (α, β, Q) = (1 − Qβ) + (2β 5 − β 6 − Qβ 6 + 3β 7 − 4Qβ 7 + Q 2 β 7 − 2Qβ 8 + 2Q 2 β 8 + 2Q 2 β 9 − 2Q 3 β 9 − Q 3 β 10 + Q 4 β 11 )α + (β 10 − β 11 + 2β 12 − 2Qβ 12 − 2β 13 + 2Qβ 13 + 3β 14 − 4Qβ 14 + Q 2 β 14 − Qβ 15 − Q 2 β 15 + 2Q 2 β 16 )α 2 + (−β 20 + β 21 )α 3 This reproduces the HOMFLY, up to a shift of α by Q 7/2 . In the Q = 1 limit, the curve degenerates to F (2,5) (α, β, 1) = (−1 + β) 1 + αβ 5 2 −1 + αβ 10 The classical A-polynomial is a factor in F (2,5) (α, β, 1), after one changes the framing by replacing α → αβ −5 . The (2, 7) knot For the (2, 7) knot, we get the following curve. F (2,7) (α, β, Q) = (1 − Qβ) + (3β 7 − β 8 − 2Qβ 8 + 4β 9 − 6Qβ 9 + 2Q 2 β 9 − 3Qβ 10 + 4Q 2 β 10 − Q 3 β 10 + 3Q 2 β 11 − 4Q 3 β 11 + Q 4 β 11 − 2Q 3 β 12 + 2Q 4 β 12 + 2Q 4 β 13 − 2Q 5 β 13 − Q 5 β 14 + Q 6 β 15 )α + (3β 14 − 2β 15 − Qβ 15 + 6β 16 − 8Qβ 16 + 2Q 2 β 16 − 3β 17 + 2Qβ 17 + Q 2 β 17 + 6β 18 − 12Qβ 18 + 10Q 2 β 18 − 4Q 3 β 18 − 3Qβ 19 + 2Q 2 β 19 + Q 3 β 19 + 6Q 2 β 20 − 8Q 3 β 20 + 2Q 4 β 20 − 2Q 3 β 21 − Q 4 β 21 + 3Q 4 β 22 )α 2 + (β 21 − β 22 + 2β 23 − 2Qβ 23 − 2β 24 + 2Qβ 24 + 3β 25 − 4Qβ 25 + Q 2 β 25 − 3β 26 + 4Qβ 26 − Q 2 β 26 + 4β 27 − 6Qβ 27 + 2Q 2 β 27 − Qβ 28 − 2Q 2 β 28 + 3Q 2 β 29 )α 3 + (−β 35 + β 36 )α 4 In the Q = 1 limit, we get F (2,7) (α, β, 1) = (−1 + β) 1 + αβ 7 3 −1 + αβ 14 The classical A-polynomial is a factor in F (2,7) (α, β, 1), after one changes the framing by replacing α → αβ −7 . Periods of the Torus Knot Curves Our claim is that for every knot, in particular for every (m, n) torus knot, the curve gives rise to a mirror Calabi-Yau uv = F (m,n) (e x , e p , Q) of the resolved conifold. From the SYZ perspective, all the mirrors are on the same footing, as there is no a-priori reason to favor one curve over the other. Note that the curves one gets are very high genus. The F (2,3) is nominally a genus three curve, while F (2,5) is a genus ten curve. Yet, we expect the topological strings on them to be equivalent. In particular, the classical periods of these cures should be equivalent. We will now prove that in fact for all torus knots, the latter statement holds. Moreover, the methods we will use to prove this will also be closely related to how we found the curves above, so we will have an opportunity to explain this. To show that the periods of p(x)dx on a curve F (m,n) (e x , e p , Q) = 0 are the same as the periods of the conifold, we will find suitable variables in terms of which we can rewrite as p(x)dx = 1 n n−1 k=0 p k (x k )dx k where each of the one forms p k (x k )dx k in fact has the same periods, and indeed the same periods as on the mirror of the conifold. This, together with the 1/n factor out front, will imply that p(x)dx itself has the same periods, as the one form on the canonical mirror, thus proving our claim. To see how such variables may arise we need to back up, all the way to the beginning and the derivation of the Rosso-Jones formula. In the presence of a Lagrangian brane corresponding to an (m, n) torus knot, the topological string computes the expectation value of Ψ(e x ) = det −1 (1 ⊗ 1 − U n ⊗ e −x ) where U = e x+m/np corresponds to an unknot in the fractional framing. We can write this as Ψ(e x ) = n ℓ=1 det −1 (1 ⊗ 1 − ω ℓ U ⊗ e −x/n ) where ω is a primitive, n-th root of unity. As long as we are interested in the semi classical limit (or more precisely, as long as we are interested in single trace contributions to the amplitude where the multi-trace contributions are suppressed relative to this by powers of g s ), we can think of this as a wave function of n particles, evenly distributed Ψ(e x ) ∼ n ℓ=1 det −1 (1 ⊗ 1 − ω ℓ U ⊗ e −x/n ) = n ℓ=1 Ψ * (e −x/n ω ℓ ) Each of the n particles simply corresponds to an unknot in fractional framing f = m/n. If we denote their coordinates and momenta as (x ℓ , p ℓ ), ℓ = 1, . . . n each of these lives on the Riemann surface which is a simple mirror of the conifold, but with x shifted by (m/n)p, corresponding to the change of framing e mp ℓ /n (1 − e p ℓ ) − e −x ℓ (1 − e p ℓ Q) = 0. The closed periods are independent of the change of framing [40] so in fact each of the one forms p(x ℓ )dx ℓ has the same periods as the conifold. Now in order to relate this to Ψ(x) we need to identify e x ℓ = e x/n ω ℓ . This identification is not in SL(2, Z)it rescales pdx by 1/n, so in fact each particle has the same periods as the canonical mirror of the conifold, albeit rescaled by 1/n. But, this is exactly what we need to recover the periods of the mirror. In summary, while (p(x), x), for a generic (m, n) torus knot live on a complicated curve, there is a way to introduce n sets of variables (p ℓ , x ℓ ) in terms of which it becomes apparent that one is in fact studying the conifold. This idea also helps in finding the explicit curve on which p and x live, F (m,n) (p, x). Namely, as long as we are interested in obtaining the classical curve only, we are allowed to replace Ψ(e x ) by n ℓ=1 Ψ ℓ (e x ℓ ). The latter has the form of a q-hypergeometric multisum. There are by now very powerful computer programs that can be used to find the corresponding q− recurrence relations. Taking the classical limit of this, gives us the curves we need. Explicitly, we have the brane amplitude of the framed unknot: Ψ (f ) (x) = R k T f R k S R k 0 e −kx = k=0 H k ( )q −f k 2 /2 q f k(N −1)/2 e −kx where H k ( ) is the HOMFLY polynomial of the unknot (6.1). Taking f = m/n gives Ψ * (x) we used, for a 1-particle wave function in our n-particle system. Figure Eight Knot The figure eight knot HOMFLY polynomial, colored by symmetric representation with n boxes is given by H 4 1 n = n i=0 i j=0 (−1) i Q 3+2i−3n 2 q −2n 2 +i 2 +4in+2j+7n−5i−5 2 × [(q −i , q) j ] 2 [(q 1−n , q) i ] 2 (Qq −1−i+j+n , q) i−j (Q, q) n−1 (q, q) i (q, q) j (q, q) n−1 (q n−i+j , q) i−j This result was provided to us by K. Kawagoe [35], based on unpublished work. (see also the related work [41]). Recently, another expression for the colored HOMFLY in these representations was obtained in [36]. The formulas should be equivalent, although they are not manifestly so. In the above, (a, q) n = n−1 k=0 (1 − aq k ) is a version of q-factorial. The knot invariant is in the form of q−hypergeometric multi-sum. Recurrences of such sums can be found by a powerful program for finding recurrences of such sums, "HolonomicFunctions.m" 7 . From this, we can find the mirror geometry corresponding to the figure eight knot. F 4 1 (α, β, Q) = (β 2 − Qβ 3 ) + (−1 + 2β − 2Q 2 β 4 + Q 2 β 5 )α + (1 − 2Qβ + 2Q 2 β 4 − Q 3 β 5 )α 2 + (−Q 2 β 2 + Q 2 β 3 )α 3 This is obtained as the q → 1 limit of the quantumF 4 1 polynomial. Viewed as a limit of the quantum curve, one finds additional trivial pieces, (−1 + β 2 ) 2 (α − Q) 2 terms. At Q = 1, our curve becomes F 4 1 (α, β, 1) = (−1 + α)(−1 + β) α − αβ − β 2 − 2αβ 2 − α 2 β 2 − αβ 3 + αβ 4 This agrees with the A-polynomial of the figure eight knot, up to a factor (−1 + β). Relation with Knot Contact Homology After the completion of this work, and during preparation of this manuscript, we became aware 8 of the work by Ng [42] and collaborators [43], which seems to be related with the present work. For a readable review of the basic ideas see [44] and references therein. Their work uses symplectic field theory and knot contact homology. The relation of these with topological strings, Gromov-Witten theory and Chern-Simons theory was not observed previously. Below we propose an explanation of such a relation. Both this paper and the ones by Ng and collaborators use the construction in [5] associating to each knot a Lagrangian brane. In the work [42] one starts with the Lagrangian brane in the T * S 3 but deletes the zero section of the S 3 . In other, words, one is focuing on the geometry far from the apex of the cone where the geometry is 7 The recurrence corresponding to this sum was computed by Christoph Koutschan, who also wrote the program. We are indebted to him for patiently explaining how to use his program. The program is publicly available from http://www.risc.jku.at/research/combinat/software/HolonomicFunctions/index.php 8 We thank Marc Culler, Craig Hodgson and Henry Segerman for pointing this out. Q B · a i = disc D Q n D e n D i u+m D i v a i 1 ...a i k D . Thus the open string states correspond to non-trivial elements of the deformed Q B cohomology. The existence of a 1-dimensional representations of this cohomology leads to a constraint A(e u , e v , Q) = 0. Such one dimensional representations can be interpreted as the disc corrected moduli space of one such Lagrangian brane after the large N transition. Using the large N duality this should be the same as the Q-deformed A-polynomial we defined, which was the quantum corrected moduli space of the single brane L K . We computed this before the transition using the color HOMFLY polynomial at leading order in large N. We have checked that for the examples (2, 3), (2, 5) torus knots and the figure-8 knot, for which the knot contact homology has been computed are in accord with computations done here 9 . This agreement is thus a consequence of the large N duality. Concluding Remarks The embedding of Chern-Simons theory in string theory and its large N duality lead to the statement that the knot invariants of the HOMFLY type as well as their Khovanov-type refinements can be deduced in principle once we know the Lagrangian brane associated with the knot on the large N dual geometry. In this paper, we have shown that, using a generalized SYZ conjecture adapted to the local Calabi-Yau geometries, to every knot we can associate a mirror Calabi-Yau Y K . This mirror is captured by a classical polynomial A K (x, p; Q) which as Q → 1 has the classical A-polynomial of the knot as a factor. Since HOMFLY and their refinement in the form of Khovanov invariants and knot homologies can be computed perturbatively starting with the classical B-model geometry of Y K , it implies that the classical polynomial A K (x, p; Q) is at least as refined as the knot homologies. Namely, having related a knot K to the Calabi-Yau Y K , we relate quantum and homological invariants of a knot K to (open) topological string amplitudes on Y K and their refinement. This implies that, to gain any new information about distinguishing knots by going to SL(N) or Khovanov type invariants, we would need to get new information about distinguishing Calabi-Yau manifolds, beyond that captured by classical geometry, by considering Gromov-Witten or topological string amplitudes on Y K instead. But, we do not: at least in principle, Gromov-Witten theory is a machine that uniquely assigns to a classical geometry, quantum amplitudes. We thus expect A K (x, y; Q) to completely capture all HOMFLY knot invariants, including its refinements. The one possible pitfall in this argument is the fact that the Calabi-Yau manifolds one obtains in this way are singular. Since knot invariants get related to open topological string amplitudes on Y K and we already know that the disc amplitudes are not singular, and 9 We thank Lenhard Ng for communicating his results. moreover the closed string amplitudes are expected to be the same as those for the non-singular conifold, we do not expect the singularities to affect them. This point, given its importance, certainly deserves further study. The fact that A K (x, p; Q) is a mirror of resolved conifold is striking. In particular A K embeds as a 1-dimensional subspace of the mirror of a family of more complicated toric geometries. It would be interesting to understand how the amplitudes of closed topological strings of this more complicated geometry reduces to that of the conifold on this locus. In this paper we have checked some aspects of this relation for some knots, which would be nice to generalize to other knots. The connection with knot contact homology seems interesting to explore further. The fact that knot contact homology and open Gromov-Witten theory are computing the same amplitudes was not known previously. The relation between knot contact homology and HOMFLY we uncovered is also new. In particular it would be nice to explicitly check that the disc corrected Q B formulated in the context of A-model open topological strings agrees with what has been considered in the context of knot contact homology. Moreover the results in [42,43] suggest that there is another algorithm to compute the Q-deformed A-polynomial using the braid representation of the knot. Since the HOMFLY polynomials can also in principle be computed using the braid representation, it would be nice to show that the large N limit of HOMFLY leads to the same result for the A Q K as that of the knot contact homology, thus verifying this aspect of large N duality prediction. Taubes for valuable suggestions and discussions. F L (e x , e p ) = 0, which now depends on the Lagrangian L. Thus we end up with infinitely many mirrors for each toric manifold. In particular O(−1) ⊕ O(−1) → P 1 can have infinitely many mirrors in the SYZ sense. The SYZ conjecture in this case becomes the statement that no matter which one of these choices of the B-model geometry we pick, we should nevertheless get identical topological string amplitudes. Moreover the open string amplitudes involving the Lagrangian L on the A-model side can be captured as the canonical brane on the B-model side in this mirror geometry -this brane is the brane that corresponds to a choice of a point on the curve, and the subspace v = 0 of the mirror Calabi-Yau F L (e x , e p ) = uv. branes on the S 3 . The partition function of the open A-model topological string is the G = SU(N) Chern-Simons partition function on the S 3 [3], X. Similarly, the partition function of the theory on T * S 3 with a D-branes on L K gets related by duality to topological string on X with an A-brane on L K . After the transition, one is computing open topological string amplitudes with contributions from holomorphic maps with boundaries on L K . The N of Chern-Simons theory is captured by Q = e −t = q N . Q keeps track of the degrees of the holomorphic maps -their winding number around the P 1 . In particular, we can use Chern-Simons theory to sum up the disk instanton corrections to the moduli space of the Lagrangian brane on L K and, in F K (ex, ep; Q) = 0. (5.1) to a knot in S 3 . Quantum Riemann surfaces of this type have appeared in the literature in the context of the quantum volume and AJ conjectures. We will now explain what the connection is. Consider Chern-Simons theory with gauge group G = SU(2) and Chern-Simons level k on S 3 with a knot K inside it. The colored Jones polynomial J n (K) computes the expectation value of a Wilson loop along the knot, in the spin n/2 representation of G. It was noticed [15, 17] that the colored Jones polynomial J n (K) are defined by exJ n = J n+1 , epJ n = q n J n , where q = e 2πi k+2 = e , where is the effective coupling constant of Chern-Simons theory. 3 In particular, [x,p] = . (5.3)Notice that epJ n = q n J n implies that p can be identified with ng s . In fact this is most often how the wave functionψ(p) of the N = 2 theory has been defined in the context of the volume conjecture. Namely by the analytic continuation of Ψ(p) = J p/gs (K) F (e x , e p , Q) = (1 − e p ) − e x (1 − Qe p ) = 0 is a quantum corrected moduli space of a simple Lagrangian brane L O = R 2 × S 1 in X = O(−1) ⊕ O(−1) → P 1 , where the S 1 can be roughly thought of as wrapping around the equator of the P 1 , and where the R 2 is the Lagrangian subspace of the O(−1) ⊕ O(−1) fiber to the P 1 . From our previous discussion it is natural to ask m,n) (ex, ep, Q)Ψ(x) = 0 and then take the classical g s to zero limit to find the equation of the mirror. However, this is an overkill, if all we want is the classical geometry of the mirror. Moreover, should one want to find the exact recurrence, one would have to rewrite the colored HOMFLY in some way, to have a more compact formula for the n-th term of the sum. expressions do not have a manifest symmetry corresponding to exchanging (m, n), once the sum in (6.4) is performed, the symmetry is restored. As we explained in section 4, the coefficients V (m,n) k have a direct Chern-Simons interpretation, as a limit of the expectation value of the Wilson loop along the knot V (m,n) k = lim gs→0 trU k (m,n) g s . S 3 × 3S 2 × R which shares the basic feature of the geometry after the transition, where S 2 is no longer contractible. Furthermore one considers the L K , i.e. the Lagrangian L K associated with the knot, which has the topology of T 2 × R. It is natural to view R as the 'time' direction. Here T 2 can be viewed as the geometry of the Lagrangian brane at infinity. The knot leaves its imprint on how T 2 is embedded in S 3 × S 2 . Then oneconsiders the physical open string states ending on T 2 . These are in 1-1 correspondence with 'Reeb chords' a i which are open string trajectories beginning and ending on T 2 and which extend along Reeb flows (which correspond to complex pairing of R with a direction on S 3 × S 2 , and so the holomorphic maps can end on them). In the physical setup we would say that the a i are classically annihilated by the BRST symmetry Q B : Q B · a i = 0 However disk instantons could modify the operation of Q B . This is similar to Witten's formulation of Morse theory [45] where the critical points of the Morse function are in 1-1 correspondence with vacua, but instanton corrections, which correspond to gradient flows, modify the supersymmetry algebra. In the case at hand the role of the gradient flows are played by disc instantons. Consider one such disc D which at t → +∞ starts with the Reeb chord a i and at t → −∞ approaches the Reeb chords a ij as ordered multi-pronged strips with j = 1, ..., k D . Let the boundary of the disc map to H 1 (T 2 ) (upon choosing some fixed intervals capping the ends of the Reeb chords), which picks up the holonomy factor exp(n D i u+m D i v) from the Wilson lines on the Lagrangian brane. Let n D denote the intersection of this disc with the 4-cycle dual to S 2 , i.e. R×S 3 . Then Ng et.al. find a deformed Q B operator (which we interpret as the quantum corrected Q B ) given by This follows from the Cauchy formula in the theory of symmetric functions[31]. In[17], one used the normalization in which the Jones polynomial of the unknot J n ( ) = 1 for all n. We will not do that. More precisely, they are known to have rational periods, which we interpret as defining the correct cover of the variables to lead to integral periods. AcknowledgmentsWe would like to thank the Simons Center for Geometry and Physics where this work was initiated, during the ninth Simons Workshop on math and physics. We are greatly indebted to Robbert Dijkgraaf for many discussions during various stages of this work.We are grateful to K. Kawagoe for computing the color HOMFLY polynomial for the figure eight knot and sharing his unpublished work. We are also thankful to C.Koutschan for sharing his mathematica program for computing recursion relations. We On the invariants of torus knots derived from quantum groups. M Rosso, V F R Jones, J. Knot Theory Ramifications. 2M. Rosso and V. F. R. Jones, "On the invariants of torus knots derived from quantum groups," J. Knot Theory Ramifications 2 (1993) 97-112. Quantum field theory and the Jones polynomial. E Witten, 10.1007/BF01217730Commun. Math. Phys. 121351E. Witten, "Quantum field theory and the Jones polynomial," Commun. Math. Phys. 121 (1989) 351. 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[ "LINEAR COMBINATIONS PRESERVING GENERATORS IN MULTIPLICATIVELY INVARIANT SPACES AND APPLICATIONS TO SYSTEMS OF TRANSLATES", "LINEAR COMBINATIONS PRESERVING GENERATORS IN MULTIPLICATIVELY INVARIANT SPACES AND APPLICATIONS TO SYSTEMS OF TRANSLATES" ]	[ "V Paternostro " ]	[]	[]	Multiplicatively invariant (MI) spaces are closed subspaces of L 2 (Ω, H) that are invariant under multiplications of (some) functions in L ∞ (Ω). In this paper we work with MI spaces that are finitely generated. We prove that almost every linear combination of the generators of a finitely generated MI space produces a new set on generators for the same space and we give necessary and sufficient conditions on the linear combinations to preserve frame properties. We then apply what we prove for MI spaces to system of translates in the context of locally compact abelian groups and we obtain results that extend those previously proven for systems of integer translates in L 2 (R d ).	10.4064/sm226-1-1	[ "https://arxiv.org/pdf/1404.1303v3.pdf" ]	119,677,467	1404.1303	1091a5eae244be2ea2f32123358b8e9520568f09	LINEAR COMBINATIONS PRESERVING GENERATORS IN MULTIPLICATIVELY INVARIANT SPACES AND APPLICATIONS TO SYSTEMS OF TRANSLATES 11 Apr 2014 V Paternostro LINEAR COMBINATIONS PRESERVING GENERATORS IN MULTIPLICATIVELY INVARIANT SPACES AND APPLICATIONS TO SYSTEMS OF TRANSLATES 11 Apr 2014arXiv:1404.1303v2 [math.FA] Multiplicatively invariant (MI) spaces are closed subspaces of L 2 (Ω, H) that are invariant under multiplications of (some) functions in L ∞ (Ω). In this paper we work with MI spaces that are finitely generated. We prove that almost every linear combination of the generators of a finitely generated MI space produces a new set on generators for the same space and we give necessary and sufficient conditions on the linear combinations to preserve frame properties. We then apply what we prove for MI spaces to system of translates in the context of locally compact abelian groups and we obtain results that extend those previously proven for systems of integer translates in L 2 (R d ). Introduction Given a vector valued space L 2 (Ω, H) where Ω is a σ-finite measure space and H is a separable Hilbert space and given D a determining set for L 1 (Ω) (see Section 3.1 for a precise definition) a multiplicatively invariant (MI) space is a closed subspace of L 2 (Ω, H) that is invariant under multiplications of the functions in D. A particular case of MI spaces are the well-known doubly invariant spaces introduced by Helson [16] and Srinivasan [25]. Recently, MI spaces were characterized in terms of range functions in [7]. The reason why MI spaces appear on the scene is because they are strongly connected to shift invariant (SI) spaces. In the classical euclidean case, a SI space is a closed subspace in L 2 (R d ) that is invariant under translations by integers. These type of spaces are typically considered in sampling theory, [1,26,27,28] and they also play a fundamental role in approximation theory as well as in frame and wavelet theory [14,18,22]. Shift invariant spaces have proven to be very useful models in many problems in signal and image processing. Due to their importance in theory and applications, their structure has been deeply analyzed during the last twenty five years [5,11,12,16,23]. Every SI space can be generated by a set Φ of functions in L 2 (R d ) in the sense that it is the closure of the space spanned by the integer translations of the functions in Φ. When Φ is a finite set, we say that the SI space is finitely generated. Concerning finitely generated SI spaces, a particular problem of interest for us, is the following: suppose that Φ = {φ 1 , . . . , φ m } generates the SI space V , this is, V = span{T k φ j : k ∈ Z, j = 1, . . . , m}. For ℓ ≤ m, let Ψ = {ψ 1 , . . . , ψ m } be a set of function constructed by taking linear combinations of the functions in Φ, i.e. ψ i = m j=1 a ij φ j for 1 ≤ i ≤ ℓ. The question is which are the linear combinations that produce new sets of generators for V ? and if in addition we know that {T k φ j } k∈Z,j=1,...,m is a frame for V , when is also {T k ψ i } k∈Z,i=1,...,ℓ a frame for V ? These two questions were completely answered in [6] and [8]. The problem of plain generators was addressed in [6] where the authors proved that almost every linear combination of the original generators of V generates V . Regarding the second question, in [8], the authors exactly characterized those linear combinations that transfer the frame property from {T k φ j } k∈Z,j=1,...,m to {T k ψ i } k∈Z,i=1,...,ℓ . In the present work we study the questions formulated above but for MI spaces. In our main result we show that almost every linear combination of generators of a MI space produces a new set of generators for the same space. We also characterize those linear combination that preserve uniform frames (see Definition 3.5). Our results are then in the spirit of those in [6,8]. As a first step, we work with finite dimensional subspaces. We prove that given a finite set of vectors V in a Hilbert space, almost every finite set of vectors constructed by taking linear combinations of the vectors in V spans the same subspace that V spans. This result will be the core of what we then prove for MI spaces and we also believe it is of interest by itself. As a consequence, we obtain similar results to [6,8] but for SI spaces considered in more general contexts than L 2 (R d ). The theory of shift invariant spaces has been extended to the setting of locally compact abelian (LCA) groups, mainly in two different directions. First, in [7,9,19] SI spaces are subspaces of L 2 (G) where G is an LCA group and the translations are taken along a subgroup H of G such that G/H is compact. The case when H is discrete was addressed in [9,19] and in the recent paper [7] the authors worked with the non-discrete case. Second, one can consider SI spaces in L 2 (X ) where X is a measure space and the translations are defined by the action of a discrete LCA group on X , [4]. In both cases, SI spaces were characterized in terms of range functions using fiberizations techniques obtaining results that extend those proven for SI spaces in L 2 (R d ) in [5]. This last fact is what connects SI spaces with MI spaces. Then, our results in MI spaces allow us to provide a unified treatment to the problem of when linear combinations of generators in system of translates preserve generators and frame generators in both contexts described above. The paper is organized as follows. In Section 2 we show that generators of finite dimensional subspaces are generally preserved by the action of taking linear combinations. Section 3 is devoted to MI spaces. We first summarize in Section 3.1 the basic properties of MI spaces. We prove in Section 3.2 that almost every linear combination of generators of a MI space yields to a new set of generators for the same space (Theorem 3.3). In Section 3.3 we address the problem of preserving uniform frames. Finally in Section 4 we apply the result we obtain for MI spaces to systems of translates. We finish this introduction by stating the notation we use. 1.1. Notation and Definitions. Here we set the notation we will use in the next sections, we recall the definition of frames and some basic results about linear algebra that will be important in what follows. For a set of vectors X = {x 1 , . . . , x n } ⊆ H we denote by S(X ) the subspace spanned by X , i.e. S(X ) = span{x 1 , . . . , x n }. The Gramian associated to X is the matrix G X in C n×n whose entries are (G X ) ij = x i , x j . The Gramian is a positive-semidefinite matrix satisfying G * X = G X . Denote by K X : C n → H the synthesis operator associated to X given by K X c = n j=1 c j x j and by K * X : H → C n its adjoint, the so-called analysis operator, given by K * X h = { h, x j } n j=1 . Note that the matrix representation of the operator K * X K X in the canonical basis on C n is the transpose of G X , G t X . It follows then that, rk(G X ) = rk(G t X ) = dim(Im(K * X K X )) = dim(Im(K X K * X )) = dim(Im(K X )) (1) = dim(S(X )). The set X is always a frame for S(X ) and its frame bounds are related to the Gramian in the following way: 0 < α ≤ β are frame bounds of X if and only if Σ(G X ) ⊆ {0} ∪ [α, β], where Σ(G X ) is the set of eigenvalues of G X . If E ∈ C d we indicate by \|E\| its Lebesgue measure. Linear combinations preserving generators of subspaces In this section, we are interested in studying which are the linear combinations of the generators of a finite dimensional subspace that preserve the property of generating the same subspace. Let us explain the problem in details. Let H be a separable Hilbert space and consider a finite set of elements in H, V = {v 1 , . . . , v m }. Denote by V the vector whose entries are the elements of V, i.e. V = (v 1 , . . . , v m ). For any ℓ such that r ≤ ℓ ≤ m, where r = dim(S(V)), let W = {w 1 , . . . , w ℓ } be a set constructed by taking linear combinations of elements of V. This is, for i = 1, . . . , ℓ, w i = m j=1 a ij v j for some complex scalars a ij . Collecting the coefficients of the linear combinations in a matrix A = {a ij } i,j ∈ C ℓ×m , we can write in matrix notation (2) W = AV t , where W = (w 1 , . . . , w ℓ ). Therefore, the question is which are the matrices A that transfer the property of being a set of generators for S(V) from V to W. We shall answer this by showing that for almost every matrix A ∈ C ℓ×m , the set W spans S(V). In order to prove Theorem 2.1 we first give a description of the set R in terms of the Gramians associated to V and W when V and W are linked by (2). The connection between the Gramians G V and G W is provided in the upcoming lemma (see also [8,Proposition 2.5]). Lemma 2.2. Let V = {v 1 , . . . , v m } ⊆ H. If W = {w 1 , . . . , w ℓ } is constructed from V by taking linear combinations of its elements as in (2) then, the Gramians associated to V and W satisfy G W = AG V A * . Proof. Since W = AV t , we obtain (G W ) ij = m k=1 a ik v k , m r=1 a jr v r = m r,k=1 a ik a jr v k , v r (GV ) kr = (AG V A * ) ij . For V and W as in Theorem 2.1, i.e linked by (2) we have that S(W) ⊆ S(V). Hence, S(W) = S(V) if and only if dim(S(W)) = dim(S(V)). Now, by (1) and Lemma 2.2, dim(S(W)) = dim(S(V)) is true if and only if rk(AG V A * ) = rk(G V ). As a consequence, the set R in Theorem 2.1 can be described as the set of matrices preserving the rank of G V under the action AG V A * (3) R = {A ∈ C ℓ×m : rk(AG V A * ) = rk(G V )}. Having the description of R in terms of the Gramian of V, the proof of Theorem 2.1 follows from the next rank-preserving result: Proposition 2.3. Let G be a positive-semidefinite matrix in C m×m such that G = G * and let r = rk(G). For any r ≤ ℓ ≤ m define the set R(G) = {A ∈ C ℓ×m : rk(G) = rk(AGA * )}. Then, N (G) := C ℓ×m \ R(G) has zero Lebesgue measure. Proof. Since G is a self-adjoint positive-semidefinite matrix, there exists a unitary matrix U ∈ C m×m and positive scalars λ 1 ≥ . . . ≥ λ r > 0 such that U * GU = D where D is the diagonal matrix in C m×m , D = diag(λ 1 , . . . , λ r , 0, . . . , 0). In particular, r = rk(G) = rk(D). Note that for any A ∈ C ℓ×m , A preserves the rank of G under the action AGA * if and only if AU preserves the rank of D under the action AU D(AU ) * . Therefore, N (G)U = {AU : A ∈ C ℓ×m , rk(G) = rk(AGA * )} = {B ∈ C ℓ×m , rk(D) = rk(BDB * )} = N (D). Since U is a unitary matrix, the mapping A → AU from C ℓ×m in itself preserves Lebesgue measure, implying \|N (G)\| = \|N (D)\|. Thus, it is enough to show that \|N (D)\| = 0. Let B ∈ C ℓ×m be written by column-blocks as B = (B 1 \|B 2 ) where the columns of B 1 are the first r columns of B and the columns of B 2 are the last m − r columns of B. Then, rk(BDB * ) = rk(BD 1/2 (BD 1/2 ) * ) = rk(BD 1/2 ) = rk(B 1 ). Thus, N (D) = {B = (B 1 \|B 2 ) ∈ C ℓ×m : B 1 ∈ C ℓ×r , B 2 ∈ C ℓ×m−r , rk(B 1 ) < r}. Since the set of matrices in C ℓ×r which are not full rank has zero Lebesgue measure, the result follows. Linear combination of generators of multilicatively invariant spaces In the previous section we showed that almost linear combination of generators of a finite dimensional subspace produces a new set of vectors spanning the same subspace. We want to study now a similar problem but in the context of multiplicatively invariant (MI) spaces of L 2 (Ω, H). The concept of MI spaces was recently introduce in the general setting of L 2 (Ω, H) in [7] as a generalization of the very well-known doubly invariant spaces proposed by Helson in [16] and Srinivasan in [25] for Ω = T. We shall prove that an analogous result to Theorem 2.1 can be obtained for MI spaces in L 2 (Ω, H). The main difference here lies in the sense of the word "generator" which, for MI spaces, differs from the notion of generator for a subspace. To properly describe and state the result we shall prove in this case, we first summarize the basic properties of MI spaces in Section 3.1. 3.1. Multiplicatively invariant spaces in L 2 (Ω, H). The material we collect here is a summary of the content of [7,Section 2]. See [7] for details and proofs. Let (Ω, µ) be a σ-finite measure space and let H be a separable Hilbert space. The vector valued space L 2 (Ω, H) is the space of measurable functions Φ : Ω → H such that Φ 2 = Ω Φ(ω) 2 H dµ(ω) < +∞. The inner product in L 2 (Ω, H) is given by Φ, Ψ = Ω Φ(ω), Ψ(ω) H dµ(ω). For defining MI spaces in L 2 (Ω, H) it is required the concept of determining set for L 1 (Ω). A set D ⊆ L ∞ (Ω) is said to be a determining set for L 1 (Ω) if for every f ∈ L 1 (Ω) such that Ω f (ω)g(ω) dµ(ω) = 0 ∀g ∈ D, one has f = 0. In the setting of Helson [16], a determining set is the set of exponentials with integer parameter, D = {e 2πik· } k∈Z ⊆ L ∞ (T). Definition 3.1. A closed subspace M ⊆ L 2 (Ω, H) is multiplicatively invariant with respect to the determining set D for L 1 (Ω) (MI space for short) if Φ ∈ M =⇒ gΦ ∈ M, for any g ∈ D. For an at most countable (meaning finite or countable) subset Φ ⊆ L 2 (Ω, H) define M D (Φ) = span{gΦ : Φ ∈ Φ, g ∈ D}. The subspace M D (Φ) is called the multiplicatively invariant space generated by Φ, and we say that Φ is a set of generator for M D (Φ). When Φ is finite, M = M D (Φ) is said to be finitely generated by Φ. In that case, we define the length of M as ℓ(M ) = min{n ∈ N : ∃ Φ 1 , · · · , Φ n ∈ M with M = M D (Φ 1 , . . . , Φ n )}. One of the most important properties of MI spaces is their characterization in terms of measurable range functions. A range function is a mapping J : Ω → {closed subspaces of H} equipped with the orthogonal projections P J (ω) of H onto J(ω). A range functions is said to be measurable if for every a, b ∈ H, ω → P J (ω)a, b is measurable as a function from Ω to C. M = {Φ ∈ L 2 (Ω, H) : Φ(ω) ∈ J(ω) a.e. ω ∈ Ω}. Identifying range functions that are equal almost everywhere, the correspondence between MI spaces and measurable range functions is one-to-one and onto. Moreover , when M = M D (Φ) for some at most countable set Φ ⊆ L 2 (Ω, H) the range function associated to M is J(ω) = span{Φ(ω) : Φ ∈ Φ}, a.e. ω ∈ Ω. 3.2. Linear combinations of MI-generators. We can now properly state what we want to prove. Fix D ⊆ L ∞ (Ω) a determining set for L 1 (Ω). Let us suppose that M is a finitely generated MI space with respect to D. This is, . . , Φ m } as we did for the case of generators for a finite dimensional Hilbert spaces in Section 2. More precisely, for each 1 ≤ i ≤ ℓ, Ψ i = m j=1 a ij Φ j , and collecting the coefficients in a matrix A ∈ C ℓ×m , we write Ψ = AΦ t where Φ = (Φ 1 , . . . , Φ m ) and Ψ = (Ψ 1 , . . . , Ψ ℓ ). The question is now, which are the matrices A ∈ C ℓ×m that transfer the property of being a generator set for M from Φ to Ψ . M = M D (Φ) where Φ = {Φ 1 , . . . , Φ m } ⊆ L 2 (Ω, H). Theorem 3.3. Let M be a finitely generated MI space and Φ = {Φ 1 , · · · , Φ m } ⊆ L 2 (Ω, H) be such that M = M D (Φ) where ℓ(M ) ≤ m. For ℓ(M ) ≤ ℓ ≤ m, consider the set of matrices R = {A ∈ C ℓ×m : M = M D (Ψ ), Ψ = AΦ t }. Then, C ℓ×m \ R has zero Lebesgue measure. Observe that this result is analogous to the one we proved for the case of generators for subspaces, Theorem 2.1. As we mentioned before, the generator set Φ generates M D (Φ) as a MI space. This fact changes the nature of the problem and as a consequence, the proof of Theorem 3.3 requires more subtle techniques than those used for proving Theorem 2.1. For the proof of the above theorem we need the following known result. Proof of Theorem 3.3. Along all this proof the relationship between Φ and Ψ will be always Ψ = AΦ t for some matrix A ∈ C ℓ×m so, we will not repeat this again. We denote by J Φ and J Ψ the measurable range functions associated to M D (Φ) and M D (Φ) respectively and for each ω ∈ Ω, Φ(ω) = {Φ 1 (ω), . . . , Φ m (ω)} and Ψ (ω) = {Ψ 1 (ω), . . . , Ψ ℓ (ω)}. Note that since Ψ = AΦ t , Ψ (ω) = AΦ(ω) t , where Φ(ω) = (Φ 1 (ω), . . . , Φ m (ω)) and Ψ (ω) = (Ψ 1 (ω), . . . , Ψ ℓ (ω)) . We now proceed as in [6]. By Theorem 3.2 and and the reasoning we used to obtain (3), we deduce that where in the last equality G Φ(ω) is the Gramian associated to Φ(ω). Since for a.e. ω ∈ Ω, rk(G Φ(ω) ) ≥ rk(AG Φ(ω) A * ), we then want to prove that the set (4) {A ∈ C ℓ×m : rk(G Φ(ω) ) > rk(AG Φ(ω) A * ) for ω belonging to a set of positive measure} has zero Lebesgue measure. Let F = {(ω, A) ∈ Ω × C ℓ×m : rk(G Φ(ω) ) > rk(AG Φ(ω) A * )}. Since for each 1 ≤ i ≤ m, Φ i is a measurable function, so are the entries of G Φ(ω) . On the other hand, the rank of any matrix is the largest of the absolute values of its minors. Thus, since the determinant is a polynomial on the entries of the matrix, it follows that the rank of a matrix with measurable entries is a measurable function. Now, R = {A ∈ C ℓ×m : M D (Φ) = M D (Ψ )} = {A ∈ C ℓ×m : J Φ (ω) = J Ψ (ω),since F = f −1 ((0, +∞)) where f is the measurable function f (ω, A) = rk(G Φ(ω) ) − rk(AG Φ(ω) A * ), it turns out that F is a measurable set of Ω × C ℓ×m . The sections of the F are denoted by F ω and F A . By Proposition 2.3 we know that \|F ω \| = 0 for a.e ω ∈ Ω and hence, by Lemma 3.4 µ(F A ) = 0 for a.e. A ∈ C ℓ×m . Note that the set given in (4) is exactly {A ∈ C ℓ×m : µ(F A ) > 0}. Therefore, it has zero Lebesgue measure. 3.3. Linear combinations preserving uniform frames. As we mentioned in the introduction, we want to give a unified treatment for the problem of when linear combinations preserve generators and frame generators in systems of translates, where the "systems of translates" are considered in different contexts. This is why we work at the level of the vector valued functions. For addressing the frame case, we need to introduce the following definition which, at this point, may seem a bit artificial. However, we shall see that it has complete sense in each of the different contexts we want to consider. Definition 3.5. Let Φ ⊆ L 2 (Ω, H) be a at most countable set and let J be the measurable range function defined as J(ω) = span{Φ(ω) : Φ ∈ Φ}, a.e. ω ∈ Ω. We say that Φ is a uniform frame for J if there exist constant 0 < α ≤ β such that, for a.e. ω ∈ Ω, the set {Φ(ω) : Φ ∈ Φ} is a frame for J(ω) with frame bounds α and β. Fix D ∈ L ∞ (Ω) a determining set for L 1 (Ω) and suppose that Φ is a finite set of functions in L 2 (Ω, H) such that it is a uniform frame for J where J is the measurable range function associated to M = M D (Φ). Then, Theorem 3.3 tells us that almost every linear combination of the functions in Φ produces a new set of generator Ψ of M . In particular, this is saying us that for a.e. ω ∈ Ω, Ψ (ω) is a new set of generators for J(ω). Thus, we are interested in knowing which are the linear combinations that also preserve uniform frames. This is, if A is such that Ψ = AΦ t , what is the property A must satisfy so that Ψ is a uniform frame for J? We shall answer this question by completely characterizing matrices A that preserve uniform frames it terms of angles between subspaces. To this end, we first recall the notion of Friedrichs angle, [13,15,20]. Let S, T = {0} be subspaces of C n . The Friedrichs angle between S and T is the angle in [0, π 2 ] whose cosine is defined by G[S, T ] = sup{\| x, y \| : x ∈ S ∩ (S ∩ T ) ⊥ , x = 1, y ∈ T ∩ (S ∩ T ) ⊥ , y = 1}. We define G[S, T ] = 0 if S = {0}, T = {0}, S ⊆ T or T ⊆ S. As usual, the sine of the Friedrichs angle is defined as F [S, T ] = 1 − G[S, T ] 2 . We can now state the characterization of matrices that preserve uniform frames. (1) A ∈ R where R is as in Theorem 3.3. (2) There exists δ > 0 such that F [Ker(A), Im(G Φ(ω) )] ≥ δ for a.e. ω ∈ Ω. The proof of Theorem 3.6 is based in the fact that Φ is a uniform frame with frame bounds α and β for J if and only if Σ(G Φ(ω) ) ⊆ [α, β] ∪ {0} for a.e. ω ∈ Ω. Therefore, the task is to prove that conditions (1) and (2) of Theorem 3.6 guarantee that the positive eigenvalues of AG Φ(ω) A * are uniformly bounded. This can be done using [8,Proposition 3.3], which is an adaptation of a result on singular values of composition of operator of Antezana et al. [2]. Having at hand these results, the complete proof of Theorem 3.6 is a readily adaptation of the proof of Theorem 4.4 in [8]. For the convenience of the reader we provide it here. Proof of Theorem 3.6. For a matrix G such that it is positive-semidefinite and G = G * we denote by λ − (G) its smallest non-zero eigenvalue. For any matrix B we denote by σ(B) the smallest non-zero singular value of B. Let 0 < α ≤ β be the frame bounds of Φ. Then, since Σ(G Φ(ω) ) ⊆ [α, β] ∪ {0} for a.e. ω ∈ Ω, we have that α ≤ λ − (G Φ(ω) ) and G Φ(ω) ≤ β for a.e. ω ∈ Ω. Suppose first that Ψ is a uniform frame for J with frame bounds 0 < α ′ ≤ β ′ . In particular, since correspondence between MI spaces and range functions is oneto-one and onto, Ψ is a generator set for M = M D (Φ) and then A ∈ R. Thus, we are under hypotheses of [8,Proposition 3.3 ] and so λ − (G Ψ (ω) ) = λ − (AG Φ(ω) A * ) ≤ A 2 G Φ(ω) F[Ker(A), Im(G Φ(ω) )] ≤ A 2 β F [Ker(A), Im(G Φ(ω) )]. Thus, α ′ ≤ A 2 β F [Ker(A), Im(G Φ(ω) )] and condition (2) follows with δ = α ′ A 2 β . Suppose now that (1) and (2) are satisfied for some matrix A. Then, we apply again [8,Proposition 3.3] to get (5) λ − (AG Φ(ω) A * ) ≥ σ(A) 2 λ − (G Φ(ω) ) F [Ker(A), Im(G Φ(ω) )] 2 ≥ σ(A) 2 αδ 2 , for a.e. ω ∈ Ω. Note that, G Ψ (ω) = AG Φ(ω) A * ≤ A 2 G Φ(ω) ≤ A 2 β and hence the eigenvalues of G Ψ (ω) are bounded above by A 2 β. Combining this fact together with (5) we obtain that Σ(G Ψ (ω) ) ⊆ σ(A) 2 αδ 2 , A 2 β ∪ {0} for a.e. ω ∈ Ω and then Ψ is a uniform frame for J. When the new set of generators has exactly ℓ(M ) elements the following theorem can be shown. For its proof see [8,Theorem 4.7]. (I m − A * (AA * ) −1 A)G Φ(ω) G † Φ(ω) < 1. Here, I m is the identity in C m×m and G † Φ(ω) is the Moore-Penrose pseudoinverse of G Φ(ω) . Remark 3.8. It might be the case that condition (2) in Theorem 3.6 is not satisfied for any matrix A. An example of this situation is given in [8,Example 4.12] for the case of system of translates in R d but it can be easily adapted to the setting of MI spaces. Indeed. In L 2 ((−1/2, 1/2] 2 , ℓ 2 (Z 2 )) consider Φ 1 and Φ 2 the vector valued functions given by Φ 1 (ω 1 , ω 2 ) = − sin(2πω 1 )e o and Φ 1 (ω 1 , ω 2 ) = e 2πiω2 cos(2πω 1 )e o where e 0 is the sequence in ℓ 2 (Z 2 ) that takes the value 1 at (0, 0) and 0 otherwise. As a determining set take D = {e 2πi (k,j),· } (k,j)∈Z 2 . Then, for M D (Φ 2 , Φ 2 ) there is no matrix satisfying condition (2) in Theorem 3.6. See [8,Example 4.12] for details. Application to systems of translates In this section we show how the previous results can be applied to systems of translates. As we will see, there exists a connection between systems of translates and vector valued functions which of course depends on the context where the systems of translates are considered. The link is what we call fiberization isometry. 4.1. Systems of translates on LCA groups. Here we work with systems of translates on the context of locally compact abelian groups. Given G a second countable LCA group written additively, we consider translates of function in L 2 (G) along a subgroup H ⊆ G such that G/H is compact. A closed subspace V ⊆ L 2 (G) is said to be H-invariant (or invariant under translations in H) if for every f ∈ V , T h f ∈ V for all h ∈ H where T h denotes the translation by h, i.e T h f (x) = f (x−h). Subspaces that are H-invariant were characterized using range functions and fiberization techniques in [9,19] when H is discrete. Recently in [7], a similar characterization was obtained only assuming that G/H is compact (i.e H not necessarily discrete). An important point to get these characterizations is to see the space L 2 (G) as a vector valued space of the form L 2 (Ω, H) for some particular choices of Ω and H. Now we briefly describe how to do this. Let G be the dual group of G, that is, the set of continuous characters on G. For x ∈ G and γ ∈ G we use the notation (x, γ) for the complex value that γ takes at x. For any subgroup H ⊆ G, H * the annihilator of H is the subgroup of G, H * = {γ ∈ G : (h, γ) = 1, ∀ h ∈ H}. Let us assume from now on that H is a co-compact subgroup of G, this is, G/H is compact. Then, by the duality theorem [24, Lemma 2.1.3], it follows that H * is discrete. Now fix Ω ⊆ G a measurable section of the quotient G/H * whose existence is a consequence of [21, Lemma 1.1]. When the Haar measures of the groups involved here are appropriately chosen, the following result shows that L 2 (G) is isometrically isomorphic to the vector valued space L 2 (Ω, ℓ 2 (H * )). For its proof see [9, Once one identifies range functions that are equal almost everywhere, the correspondence between measurable range functions and H-invariant spaces is one-toone and onto. When V = span{T h ϕ : h ∈ H, ϕ ∈ A} for an at most countable set A ⊆ L 2 (G), the measurable range function associated to V is given by J(ω) = span{T ϕ(ω) : ϕ ∈ A}, a.e. ω ∈ Ω. Frames of translates can be also characterized using range functions and the fiberization isometry. Indeed, when A ⊆ L 2 (G) is a countable set, frames of the form {T h ϕ : h ∈ H, ϕ ∈ A} for V = span{T h ϕ : h ∈ H, ϕ ∈ A} correspond to uniform frames for J where J is the measurable range function associated with V . For the case when H is discrete, this fact was proven in [9,Theorem 4.1]. When H is not discrete but co-compact, the set {T h ϕ : h ∈ H, ϕ ∈ A} is not indexed by a discrete set and then one needs to work with the notions of continuous frame (see Definition 5.1 in [7] for details). The characterization of continuous frames in terms of range functions was given in [7,Theorem 51]. In the upcoming theorem, we state the characterization of frames of translates using range functions without distinguishing between the discrete and the continuous case. The reader must have in mind that when H is not discrete the word "frame" refers to the notion of continuous frame as [7,Definition 5.1]. Theorem 4.3. Let A ⊆ L 2 (G) be a countable set, let J be the measurable range function associated to V = span{T h ϕ : h ∈ H, ϕ ∈ A} and let T be the mapping of Proposition 4.1. Then, the following conditions are equivalent: (1) {T h ϕ : h ∈ H, ϕ ∈ A} is a frame for V with frame bounds 0 < α ≤ β. (2) {T ϕ : ϕ ∈ A} is a uniform frame for J with frame bounds 0 < α ≤ β. This is, for a.e. ω ∈ Ω, {T ϕ(ω) : ϕ ∈ A} is a frame for J(ω) with (uniform) frame bounds 0 < α ≤ β. We already have all the ingredients we need to see how the results of Section 3 can be applied to this setting. Fix {φ 1 , . . . , φ m } ⊆ L 2 (G) and consider the Hinvariant space generated by {φ 1 , . . . , φ m }, V = span{T h φ j : h ∈ H, 1 ≤ j ≤ m}. By taking linear combinations of {φ 1 , . . . , φ m } we want to construct new sets of generators for V . As we did before for the case of generator for subspaces and for MI spaces, we consider sets of functions in L 2 (G), {ψ 1 , . . . , ψ ℓ } where for every 1 ≤ j ≤ m, ψ j = m i=1 a ij φ j and ℓ is a number between the length of the MI space T V and m. Collecting the coefficients of the linear combinations in a matrix A ∈ C ℓ×m and letting Φ and Ψ be the vectors of functions Φ = (φ 1 , . . . , φ m ) and Ψ = (ψ 1 , . . . , ψ ℓ ) we can write Ψ = AΦ t . In the next theorem we prove that for almost every matrix A ∈ C ℓ×m , the functions {ψ 1 , . . . , ψ ℓ } generate V . This result extends [6,Theorem 1] to the context of LCA groups, and moreover, since H is allowed to be non discrete, it is new even in the case when G = R d . The next theorem is an extension to LCA groups of [8,Theorem 4.4]. Theorem 4.4. Given {φ 1 , . . . , φ m } ⊆ L 2 (G) let V = span{T h φ j : h ∈ H, 1 ≤ j ≤ m} and let ℓ(M ) be the length of M = T V where T is the fiberization isometry of Proposition 4.1. For ℓ(M ) ≤ ℓ ≤ m, let R be the set of matrices A = {a ij } ij ∈ C ℓ×m such that the linear combinations ψ j = m i=1 a ij φ j generate V , i.e. V = span{T h ψ i : h ∈ H, 1 ≤ i ≤ ℓ}.Theorem 4.5. Let {φ 1 , · · · , φ m } ⊆ L 2 (G) such that {T h φ j : h ∈ H, 1 ≤ j ≤ m} is a frame for V = span{T h ϕ j : h ∈ H, 1 ≤ j ≤ m} and suppose that ℓ(M ) ≤ ℓ ≤ m, where M = T V and T is as in Proposition 4.1. Let A ∈ C ℓ×m be a matrix and consider {ψ 1 , · · · , ψ ℓ } where Ψ = AΦ t . Then, {T h ψ i : h ∈ H, 1 ≤ i ≤ ℓ} is a frame for V if and only if A satisfies the following two conditions (1) A ∈ R, where R is as in Theorem 4.4. (2) There exists δ > 0 such that F [Ker(A), Im(G Φ(ω) )] ≥ δ for a.e. ω ∈ Ω, where G Φ(ω) is the Gramian associated to {T φ 1 (ω), . . . , T φ m (ω)}. Remark 4.6. Given ∆ ⊆ G a co-compact subgroup of the dual group of G and A ⊆ L 2 (G), let us consider the system {M δ φ : φ ∈ A, δ ∈ ∆} where M δ is the modulation operator given by M δ φ(x) = (x, δ)φ(x). Since under the Fourier transform modulations become translations, all the results we have proven for systems of translates can be reformulated for systems of modulations. Furthermore, one may also consider systems of time-frequency translates {M δ T h φ : φ ∈ A, δ ∈ ∆, h ∈ H} where H ⊆ G and ∆ ⊆ G are discrete subgroups and A ⊆ L 2 (G). Spaces that are the closure of the span of systems of time-frequency translates are called shiftmodulation invariant spaces or Gabor spaces. Using fiberization techniques and range functions, a characterization of theses spaces was given in [10]. Therefore, this setting is one more example where the results of Section 3 can be applied. 4.2. Discrete LCA groups acting on σ-finite measure spaces. We are interested now is systems of functions constructed by the action of a discrete LCA group Γ on L 2 (X ) where (X , µ) is a σ-finite measure space. We will work with quasi-Γ-invariant actions. This notion was introduced in [17] and then extended to the non abelian case in [3]. Fix Γ a discrete countable LCA group. Let (X , µ) is a σ-finite measurable space and σ : Γ × X → X a measurable action satisfying the following conditions: (i) for each γ ∈ Γ the map σ γ : X → X given by σ γ (x) := σ(γ, x) is µmeasurable; (ii) σ γ (σ γ ′ (x)) = σ γγ ′ (x), for all γ, γ ′ ∈ Γ and for all x ∈ X ; (iii) σ e (x) = x for all x ∈ X , where e is the identity of Γ. The action σ is said to be quasi-Γ-invariant if there exists a measurable function J σ : Γ × X → R + , called Jacobian of σ, such that dµ(σ γ (x)) = J σ (γ, x)dµ. To each quasi-Γ-invariant action σ we can associate a unitary representation T σ of Γ on L 2 (X ) given by T σ (γ)f (x) = J σ (−γ, x) 1 2 f (σ −γ (x)). Given a quasi-Γ-invariant action σ, we say that a closed subspace V of L 2 (X ) is Γ-invariant if f ∈ V =⇒ T σ (γ)f ∈ V, for any γ ∈ Γ. When L 2 (X ) is separable, each Γ-invariant spaces is of the form V = span{T σ (γ)ϕ : γ ∈ Γ, ϕ ∈ A} for some at most countable set A ⊆ L 2 (X ). In order to obtain the analogous results to Theorems 4.4 and 4.5 for systems of the form {T σ (γ)φ j } m j=1 using the machinery of MI spaces of Section 3 we first need to establish a connection between L 2 (X ) and a vector valued space of the type L 2 (Ω, H). We can do this assuming that the quasi-Γ-invariant action σ satisfies the tiling property. This is, there exists a measurable subset C ⊆ X such that µ(X \ γ∈Γ σ γ (C)) = 0 and µ(σ γ (C) ∩ σ γ ′ (C)) = 0 whenever γ = γ ′ . In this case it can be shown (see [3] and [4]) that there exists a isometric isomorphism between L 2 (X ) and the vector valued space L 2 ( Γ, L 2 (C)). is an isometric isomorphism and it satisfies T σ [T σ (γ)ψ] = (γ, α)T σ [ψ]. As for the case of ordinary translates of the previous section, the isomorphism T σ of Proposition 4.7 connects Γ-invariant spaces of L 2 (X ) with MI spaces in L 2 ( Γ, L 2 (C)). Here the determining set D is the set of characters of Γ. More precisely, for every γ ∈ Γ, let X γ : Γ → C be the homomorphism defined as X γ (α) = (γ, α). Then, by the Pontrjagin Duality [24, Theorem 1.7.2], {X γ } γ∈Γ is the set of characters of Γ and thus, as a consequence of the uniqueness of the Fourier transform, D = {X γ } γ∈Γ is a determining set for L 1 ( Γ). Therefore, it is possible to characterize Γ-invariant spaces using range functions obtaining a similar result to Theorem 4.2. Furthermore, it can be also proven characterizations of frames of the form {T σ (γ)ϕ : γ ∈ Γ, ϕ ∈ A} for V = span{T σ (γ)ϕ : γ ∈ Γ, ϕ ∈ A} in the same spirit of Theorem 4.3. We do not include here the complete statements of these results because we consider it is clear for the reader how they must be (see [4] for details and proofs). In a similar way as we proved Theorem 4.4, the following result can be shown: Theorem 4.8. Given {φ 1 , . . . , φ m } ⊆ L 2 (X ) let V = span{T σ (γ)φ j : γ ∈ Γ, 1 ≤ j ≤ m} and let ℓ(M ) be the length of M = T σ [V ] where T σ is as in Proposition 4.7. For ℓ(M ) ≤ ℓ ≤ m, let R be the set of matrices A = {a ij } ij ∈ C ℓ×m such that the linear combinations ψ j = m i=1 a ij φ j generate V , i.e. V = span{T σ (γ)ψ i : γ ∈ Γ, 1 ≤ i ≤ ℓ}. Then, C ℓ×m \ R has Lebesgue zero measure. If in addition {T σ (γ)φ j : γ ∈ Γ, 1 ≤ j ≤ m} is a frame for V , it holds that {T σ (γ)ψ i : γ ∈ Γ, 1 ≤ i ≤ ℓ} is also a frame for V in and only if A ∈ R and there exists δ > 0 such that F [Ker(A), Im(G Φ(α) )] ≥ δ for a.e. α ∈ Γ, where G Φ(α) is the Gramian associated to {T σ [φ 1 ](α), . . . , T σ [φ m ](α)}. Theorem 2. 1 . 1Let V = {v 1 , . . . , v m } ⊆ H and let r = dim(S(V)). For any ℓ such that r ≤ ℓ ≤ m, consider the set of matrices R = {A ∈ C ℓ×m : S(V) = S(W)} where W is obtained from V by the relationship W = AV t . Then, C ℓ×m \ R has zero Lebesgue measure. Suppose that L 2 (Ω) is separable, so that L 2 (Ω, H) is also separable. Let M be a closed subspace of L 2 (Ω, H) and D a determining set for L 1 (Ω). Then, M is an MI space with respect to D if and only if there exists a measurable range function J such that For a number ℓ such that ℓ(M ) ≤ ℓ ≤ m we construct a new set of functions of M , Ψ = {Ψ 1 , . . . , Ψ ℓ } say, by taking linear combinations of {Φ 1 , . Lemma 3. 4 . 4Let (X, µ) and (Y, ν) be measure spaces and F ⊆ X × Y be a measurable set. The sections of F are F x = {y ∈ Y : (x, y) ∈ F } and F y = {x ∈ X : (x, y) ∈ F }. Then, (µ × ν)(F ) = 0 if and only if ν(F x ) = 0 for µ-a.e. x ∈ X if and only if µ(F y ) = 0 for ν-a.e. y ∈ Y . a.e. ω ∈ Ω} = {A ∈ C ℓ×m : S(Φ(ω)) = S(Ψ (ω)) a.e. ω ∈ Ω} = {A ∈ C ℓ×m : rk(G Φ(ω) ) = rk(AG Φ(ω) A * ) a.e. ω ∈ Ω}, Theorem 3. 6 . 6Let Φ = {Φ 1 , . . . , Φ m } ⊆ L 2 (Ω, H) be a uniform frame for J where J is the measurable range function associated to M = M D (Φ) and suppose that ℓ(M ) ≤ ℓ ≤ m. Let A ∈ C ℓ×m be a matrix and consider Ψ = {Ψ 1 , . . . , Ψ ℓ } where Ψ = AΦ t . Then, Ψ is a uniform frame for J if and only if A satisfies the following two conditions Theorem 3 . 7 . 37Let Φ = {Φ 1 , . . . , Φ m } ⊆ L 2 (Ω, H) be a uniform frame for J where J is the measurable range function associated to M = M D (Φ) and call ℓ(M ) = ℓ ≤ m. Let A ∈ C ℓ×m be a matrix and consider Ψ = {Ψ 1 , . . . , Ψ ℓ } where Ψ = AΦ t . Then, Ψ is a uniform frame for J if and only if AA * is invertible and ess sup ω∈Ω Proposition 4 . 1 . 41The fiberization mapping T : L 2 (G) → L 2 (Ω, ℓ 2 (H * )) defined byT f (ω) = { f (ω + δ)} δ∈H * is a isometric isomorphism and it satisfies T T h f (ω) = (−h, w)T f (ω) for all f ∈ L 2 (G)and all h ∈ H. Here, f denotes the Fourier transform of f . The fiberization isometry of Proposition 4.1 allows us to see L 2 (G) as the vector valued space L 2 (Ω, ℓ 2 (H * )). Under this isometry, H-invariant spaces of L 2 (G) exactly correspond with MI spaces of L 2 (Ω, ℓ 2 (H * )). Let us explain this correspondence in details. The determining set behind the notion of MI spaces in L 2 (Ω, ℓ 2 (H * )) is the set of functions D = {(h, ·)χ Ω (·)} h∈H , [7, Corollary 3.6]. Thus, by Proposition 4.1, one has that V ⊆ L 2 (G) is a H-invariant space if and only if M = T V is a MI space with respect to D. Therefore, we can also identify Hinvariant spaces with measurable range functions as it was shown in [9, Theorem 3.10] and [7, Theorem 3.8]: Theorem 4.2. Let V ⊆ L 2 (G) be a closed subspace and T the mapping defined in Proposition 4.1. Then, V is H-invariant if and only if there exists a measurable range function J such that V = {f ∈ L 2 (G) : T f (ω) ∈ J(ω) for a.e. ω ∈ Ω}. Then, C ℓ×m \ R has Lebesgue zero measure. Proof. Let A ∈ C ℓ×m and consider the functions {ψ 1 , . . . , ψ ℓ } where Ψ = AΦ t . Then, {ψ 1 , . . . , ψ ℓ } is a set of generators for V if and only if {T ψ 1 , . . . , T ψ ℓ } generates M as a MI space with respect to D = {(h, ·)χ Ω (·)} h∈H . Denoting Φ = {T φ 1 , . . . , T φ m }, Ψ = {T ψ 1 , . . . , T ψ ℓ }, Ψ = (T ψ 1 , . . . , T ψ ℓ ) and Φ = (T φ 1 , . . . , T φ m ), we that have that R = {A ∈ C ℓ×m : M = M D (Ψ ), Ψ = AΦ t }. Thus, by Theorem 3.3, C ℓ×m \ R has Lebesgue zero measure. Proposition 4 . 7 . 47The mapping T σ : L 2 (X ) −→ L 2 ( Γ, L 2 (C)) defined by T σ [ψ](α)(x) := γ∈Γ [(T σ (γ)ψ)(x)] (−γ, α) Definition 1.1. Let H be a separable Hilbert space and {f k } k∈Z be a sequence in H. The sequence {f k } k∈Z is said to be a frame for H if there exist 0 < α ≤ β such that for all f ∈ H. The constants α and β are called frame bounds.α f 2 ≤ k∈Z \| f, f k \| 2 ≤ β f 2 Nonuniform sampling and reconstruction in shift-invariant spaces. A Aldroubi, K Gröchenig, SIAM Rev. 434A. Aldroubi and K. Gröchenig. Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev., 43(4):585-620 (electronic), 2001. Nullspaces and frames. J Antezana, G Corach, M Ruiz, D Stojanoff, J. Math. Anal. Appl. 3092J. Antezana, G. Corach, M. Ruiz, and D. Stojanoff. Nullspaces and frames. J. Math. Anal. Appl., 309(2):709-723, 2005. Riesz and frame systems generated by unitary actions of discrete groups. D Barbieri, E Hernández, J Parcet, PreprintD. Barbieri, E. Hernández, and J. 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Theory Signal Image Process. W Sun, 4W. Sun. Sampling theorems for multivariate shift invariant subspaces. Sampl. Theory Signal Image Process., 4(1):73-98, 2005. Perturbation of regular sampling in shift-invariant spaces for frames. P Zhao, C Zhao, P G Casazza, IEEE Trans. Inform. Theory. 5210P. Zhao, C. Zhao, and P. G. Casazza. Perturbation of regular sampling in shift-invariant spaces for frames. IEEE Trans. Inform. Theory, 52(10):4643-4648, 2006. . V Paternostro, 10623Institut für Mathematik, Technische Universität BerlinV. Paternostro) Institut für Mathematik, Technische Universität Berlin, 10623 . Germany Berlin, [email protected], Germany E-mail address: [email protected]	[]
[ "Heat transport in nonuniform superconductors", "Heat transport in nonuniform superconductors" ]	[ "Caroline Richard \nDepartment of Physics\nMontana State University\n59717BozemanMontanaUSA\n", "Anton B Vorontsov \nDepartment of Physics\nMontana State University\n59717BozemanMontanaUSA\n" ]	[ "Department of Physics\nMontana State University\n59717BozemanMontanaUSA", "Department of Physics\nMontana State University\n59717BozemanMontanaUSA" ]	[]	We calculate electronic energy transport in inhomogeneous superconductors using a fully selfconsistent non-equilibrium quasiclassical Keldysh approach. We develop a general theory and apply it a superconductor with an order parameter that forms domain walls, of the type encountered in Fulde-Ferrell-Larkin-Ovchinnikov state. The heat transport in the presence of a domain wall is inherently anisotropic and non-local. Bound states in the nonuniform region play a crucial role and control heat transport in several ways: (i) they modify the spectrum of quasiparticle states and result in Andreev reflection processes, and (ii) they hybridize with impurity band and produce local transport environment with properties very different from those in uniform superconductor. As a result of this interplay, heat transport becomes highly sensitive to temperature, magnetic field and disorder. For strongly scattering impurities we find that the transport across domain walls at low temperatures is considerably more efficient than in the uniform superconducting state. arXiv:1605.01634v2 [cond-mat.supr-con]	10.1103/physrevb.94.064502	[ "https://arxiv.org/pdf/1605.01634v2.pdf" ]	118,588,311	1605.01634	9d4c550d5d2181ca4dbfc7aca2cff03017a4ca9e	Heat transport in nonuniform superconductors Caroline Richard Department of Physics Montana State University 59717BozemanMontanaUSA Anton B Vorontsov Department of Physics Montana State University 59717BozemanMontanaUSA Heat transport in nonuniform superconductors (Dated: May 12, 2016) We calculate electronic energy transport in inhomogeneous superconductors using a fully selfconsistent non-equilibrium quasiclassical Keldysh approach. We develop a general theory and apply it a superconductor with an order parameter that forms domain walls, of the type encountered in Fulde-Ferrell-Larkin-Ovchinnikov state. The heat transport in the presence of a domain wall is inherently anisotropic and non-local. Bound states in the nonuniform region play a crucial role and control heat transport in several ways: (i) they modify the spectrum of quasiparticle states and result in Andreev reflection processes, and (ii) they hybridize with impurity band and produce local transport environment with properties very different from those in uniform superconductor. As a result of this interplay, heat transport becomes highly sensitive to temperature, magnetic field and disorder. For strongly scattering impurities we find that the transport across domain walls at low temperatures is considerably more efficient than in the uniform superconducting state. arXiv:1605.01634v2 [cond-mat.supr-con] I. INTRODUCTION Electronic heat transport is a powerful tool to explore properties of the superconducting state. It is a bulk probe, that encodes information about both density of electronic states and quasiparticle relaxation times. Heat conductivity experiments have been used extensively to study structure and symmetries of the superconducting order parameter in many different compounds. 1 The low-temperature behavior of thermal conductivity is a signature of either the absence or presence of lowenergy excitations. 2 It can also be used as a directional probe of the gap structure, since it depends on the velocity of the low-energy excitations. One can measure the anisotropy of thermal conductivity along different directions and identify the Fermi velocity vectors of nodal quasiparticles. [3][4][5] Another way to study the nodal structure is to observe the response of nodal excitations to a rotated magnetic field. 1 The external magnetic field modifies the density of states 6,7 and the quasiparticles scattering times. [8][9][10] The magnitude of these effects depends on the orientation of the magnetic field relative to the nodes of the order parameter and the direction of the heat flow. 1 The power of this technique, however, is also the reason why the interpretation of thermal transport measurements is a difficult task, since density of states and transport time of quasiparticles may not be independently available. In this respect, experiment and theory must be employed together in the analysis of data for reaching definite conclusions. In uniform superconductors heat conductivity has been investigated in great details, using several approaches: Boltzmann transport theory, 11 Green's functions technique, 12 and quasiclassical methods, 2 that prompted rapid development on the experimental side. There is growing interest in using thermal transport to study nonuniform superconductors 13 and topological surface states. [14][15][16] However, from the theory side, little is known about heat flow in the presence of a spatiallyvarying order parameter. The challenge here is to understand how quasiparticles transport energy from one point to another when both the quasiparticle density of states and scattering mean free path depend on both energy and position. Under these conditions it is important to treat on the same footing Andreev particle-hole conversion processes in inhomogeneous regions 17 and scattering processes on impurities. As a result, in nonuniform superconductors calculation of heat transport is difficult and so far has been carried out only in two different approximations. In the strongly inhomogeneous situation, as in the case of periodic and moderately dense Abrikosov vortex lattice near H c2 , one can average over vortex lattice unit cell, 18,19 assuming local formula relating heat current to the temperature gradient, j h (R) = −κ ∇T (R), to hold everywhere. In this approach one can analyze the effects of disorder and magnetic field on density of states, lifetime and heat transport of spatially extended quasiparticles outside vortex cores. 9,10 In a very different setting, the heat transport through a pinhole supporting Andreev bound states (ABS) was investigated 20 . When a phase bias ϕ applied across the pinhole, highly degenerate Andreev bound states 17 appear at subgap energies controlled by both ϕ and the transparency of the pinhole. The sudden temperature drop across the pinhole produces local heat current that depends on the phase bias, j h = −κ(ϕ)δT . The bound states lying at subgap energy do not directly couple to the continuum of quasiparticles to transport heat. Nevertheless, their presence modifies the effective transparency of the pinhole for quasiparticles above the gap. In particular, for a pinhole with perfect transparency, the subgap bound states reduce locally the spectral weight of continuum states which suppresses the heat flow, κ(ϕ) < κ(0). By contrast, at low transparency, the ABS lie just below the gap edge and enhance heat conductivity, κ(ϕ) > κ(0), due to a resonance with the continuum. 20 However, in topological insulator junctions, the zero energy ABS are topologically protected, preventing such resonance. 16 Both of these approaches have limited applicability. In the pinhole calculation the sudden drop approximation means point-localized ABS and lack of impurity scatter-ing effects. The averaging procedure, on the other hand, works well for high temperature and fields where vortices are dense, but less well at low temperatures and fields, and even worse with fully gapped superconductors. It relies on a presence of significant number of spatially extended low-energy quasiparticles, but has no way of including the contribution of localized vortex core states. An exact theoretical treatment of the thermal transport that simultaneously takes into account impurity scattering in spatially varying order parameter landscape, effects of spatially localized Andreev bound states, and position-dependent density of states, is lacking. In this direction, we provide, in this paper, a basis for future explorations of general nonuniform superconducting configurations. There are several important details that we include in complete treatment of the problem. First, effects of the Andreev states localized in the inhomogeneous region is taken into account on the same footing with the effects of the impurities that also produce midgap Andreev states distributed throughout the sample. Both kind of bound states result in modifications in the quasiparticle spectrum and scattering time, and their interaction is important. Second, the broken translation and rotation symmetry that appear in systems with a spatially modulated order parameter in general result in additional 'vertex corrections' to the transport lifetime. 21,22 Third, the mean free paths of the quasiparticles can be longer than the coherence-length scale of the order parameter variations, thus invalidating the picture of a local equilibrium and local response even for small temperature gradients. As a particular model for the inhomogeneous phase we consider a domain wall between two degenerate configuration of the order parameter that changes sign across the wall, ∆(−∞) = −∆(+∞). We also consider a more complicated configurations with periodic collection of multiple domain walls. We enforce the order parameter modulation through boundary conditions on the edges of the sample, and self-consistently compute the profile of the domain walls and spatially-dependent impurity selfenergies. The domain walls have width of several coherence lengths, and host highly degenerate zero-energy Andreev bound states. Such profile of the order parameter is a realization of Larkin-Ovchinnikov configuration of the speculative Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. [23][24][25] Building a theoretical understanding of the thermal conductivity in this phase is important for experimental attempts to detect spatial modulations of the order parameter using heat transport. 13 At this point it is not known how anisotropic is the heat flow in the presence of the hypothetical FFLO domain wall structures in the order parameter. For example, the typical assumption is that the heat flow across the domains is strongly constricted due to the presence of the ABS that do not carry heat. We find that this is not, in fact, true, and the combination of impurity scattering effects with spatially-extended range of the bound states can produce both suppression and enhancement of thermal transport I I I x=0 L -L T 1 I h T 2 T H #1 T H #2 y FIG. 1. A typical experimental setup for heat conductivity measurement involves establishing a steady state current in the sample and measuring the effective local temperature T1 and T2. Heat conductivity is defined κ = I h /(T1 − T2). We set a two-dimensional superconductor in the xy plane and for a given heat current I h along the x-axis we calculate temperature difference between points TH #1 and #2 at x = ±L. The sin line is a schematic representation of an FFLO modulation of the order parameter with shaded regions representing domain walls. compared to the uniform state. The organization of the paper is as follows. In section II we describe our model and relate it to the experimental measurement technique. In section III we develop the formalism to compute thermal transport in nonuniform superconductor using Keldysh quasiclassical approach and t-matrix treatment of disorder. The linear response is discussed in section III A; in III B we formulate the boundary conditions for the quasiclassical propagator amplitudes. We apply this technique to compute heat flow across a single domain wall in section IV A, and in IV B we generalize it to the case of multiple domain walls and investigate heat transport dependence on number of domains, temperature, disorder and spacing of domain walls. Finally, in section IV C, we address the modifications arising from the Zeeman shifts of quasiparticle energies. II. MODEL We consider a spin-singlet superconductor with quasi two-dimensional cylindrical Fermi surface. We will focus on the case of a d-wave superconductor. The dwave order parameter has nodes in momentum space, ∆(p) ∝ cos 2(φp − α). Impurities are randomly distributed throughout the sample with concentration c imp . The impurity scattering potential is assumed point-like and isotropic with amplitude u. System is set out of equilibrium by introducing a thermal current flowing along the x-axis. As the stationary state is reached a temperature gradient builds up. We define the heat conductivity between two points as κ = I h × 2L/(T 1 − T 2 ) where I h is the stationary-state heat flow, and T 1 and T 2 are the local temperatures as if measured by two distant thermometers positioned at x = ±L. Our goal is to compute the effective temperature bias dT = T 1 − T 2 for a given I h , in the presence of spatially modulated order parameter as shown in Fig. 1. III. THEORY A convenient approach to study nonuniform superconductors out of equilibrium is the quasiclassical formulation of the Keldysh technique. 26 It is formulated in terms of the Green functionĝ(R,p, ), which for stationary states depends on the center of mass coordinate R, direction of the relative momentum on the Fermi surfacê p, and the energy . It is a 8 × 8 matrix in particle-hole (Nambu), spin, and Keldysh space. In the Keldysh space, it is given byĝ = g R g K 0 g A ,(1) where the superscripts R/A/K stand for Retarded, Advanced and Keldysh. The Retarded and Advanced components g R/A carry information about density of states and correlations, while g K encodes the quasiparticles' dynamics and distribution function. Each of the three components are 4 × 4 matrices, parametrized by outer products of 2 × 2 Pauli matrices in spin and particle-hole spaces σ i ⊗ τ j (i, j = x, y, z). We use the quasi-classical propagator to compute the density of electronic states (DoS) N (R,p, ) N (R,p, ) N F = − 1 π Im 1 4 Tr τ z g R (R,p, ) ,(2) and local heat current j h (R), and its spectral density j h (R, ), j h (R) = 2N F v F +∞ −∞ d 4πi dp [ p] 1 4 Tr g K (R,p, ) ≡ +∞ −∞ d j h (R, ) .(3) Here N F is density of state at Fermi energy per spin in the normal metallic state, v F is the Fermi velocity, and Tr = Tr 4 is the trace operator over spin and Nambu space. dp · · · = . . . p = dφp 2π . . . is the normalized Fermi surface integral. We note that to write the heat current as 4-trace over spin and particle-hole space instead of usual spin-trace over just upper left component of g K we used symmetry of the Keldysh component g K (R,p, ) tr = τ y g K (R, −p, − )τ y . 26 The quasiclassical Green functionĝ is normalized g 2 (R,p, ) = −π 2 , Tr[g R,A ] = 0 ,(4) and satisfies the Eilenberger equation [ τ z −σ,ĝ] + iv Fp · ∇ĝ = 0 ,(5) where 8 × 8 self-energyσ(R,p, ) has the same structure in Keldysh space as Eq. (1) andτ z = diag(τ z , τ z ). The retarded and advanced components for singlet superconductor are parametrized as follows (x = R, A): σ x = Σ x ∆ x (iσ y ) (iσ y )∆ xΣx ,(6) and the Keldysh part is σ K = Σ K ∆ K (iσ y ) −(iσ y )∆ K −Σ K .(7) Components of these matrices are related to each other through symmetries 26 defined by the˜-operation that reverses momentum and energy of complex-conjugated quantities, e.g.∆ x (R,p, ) = ∆ x (R, −p, − ) * . Diagonal self-energy terms Σ,Σ are due to impurity scattering effects. The off-diagonal terms contain the mean field order parameter and impurity contributions: ∆ R/A (R,p, ) = ∆(R,p) + ∆ R/A imp (R, ) while the Keldysh mean fields identically zeros, leaving only impurity contributions ∆ K (R, ) = ∆ K imp (R, ) . The mean-field order parameter is computed selfconsistently from ∆(R,p) = + c − c d 4πi dp V (p,p ) f K (R,p, ) .(8) We consider pair potential V (p,p ) = V 0 Y(p)Y * (p ) and f K = 1 4 Tr[ τx+iτy 2 (−iσ y )g K ] is the upper-right singlet component of the Keldysh Green's function. The momentum space basis functions are Y(p) = 1 for s-wave, and Y(p) = cos 2(φp − α) for d-wave. The cut-off energy c and the interaction amplitude V 0 are eliminated, in the usual manner, in favor of clean-case transition temperature T c . The impurity self-energy part is self-consistently determined within thet-matrix approximation. For randomly distributed isotropic scattering centers, the 8×8t-matrix equation ist = u1+u ĝ pt gives self-energyσ imp = c impt . Using traditional definitions of impurity scattering rate Γ = c imp /πN F and phase shift tan δ = uπN F in terms of impurity concentration c imp and amplitude u of the point-like scattering potential, the 4 × 4 self-energy matrices are determined from (x = R, A) σ x imp (R, ) = Γ tan δ + tan δ g x (R,p, ) π p σ x imp (R, ) , σ K imp (R, ) = 1 Γ σ R imp (R, ) g K (R,p, ) π p σ A imp (R, ) .(9) In the following, we will mainly make comparison between the Born (δ → 0) and Unitary (δ = π/2) limits. Note that in the absence of inelastic scattering, the self-consistent calculation of impurity σ imp and order parameter ∆ that includes non-equilibrium effects, guarantees conservation of energy and charge. In particular, self-consistent calculation of self-energies including corrections due to the heat flow will automatically satisfy divj h (R, ) = 0 -condition of no energy accumulation, see Appendix A. The solution of the self-consistent system of coupled equations for nonuniform states is most conveniently obtained using Riccati parametrization 27 . The retarded (− sign) and advanced (+ sign) Green's functions are given by g x = ∓iπ 1 + γ xγx 1 − γ xγx 2γ x (iσ y ) −2γ x (iσ y ) −(1 −γ x γ x ) ,(10) where γ x (R,p, ) are retarded/advanced scalar coherence functions that are zeros in bulk normal state. The Keldysh component takes the form g K = −2πi (1 + γ RγR )(1 + γ AγA ) × x K + γ RxKγA −(γ RxK − x K γ A )iσ y −iσ y (γ R x K −x KγA )x K +γ R x K γ A ,(11) where x K (R,p, ) is the (scalar) distribution function. We explicitly took out the singlet spin dependence of coherence amplitudes, compared with Ref. 27, where the coherence functions are spin matrices. Coherence and distribution amplitudes are related to each other through -relation, as well as 27 γ A (R,p, ) = −γ R (R,p, ) * , x K (R,p, ) = x K (R,p, ) * .(12) The distribution function is not unique, and the usual choice in equilibrium is x K 0 = Φ 0 ( /T )(1 + γ RγA ) ,(13) where Φ 0 ( /T ) = tanh( /2T ) = 1 − 2f ( /T ) and f ( /T ) = [exp( /T ) + 1] −1 is the Fermi distribution at temperature T . Transport-like equations for coherence and distribution functions follow from Eq. (5) iv Fp · ∇γ x + (2 − Σ x +Σ x )γ x +∆ x (γ x ) 2 + ∆ x = 0 , iv Fp · ∇x K + (γ R∆R − Σ R + ∆ AγA + Σ A )x K = γ RΣKγA − ∆ KγA − γ R∆K − Σ K .(14) A. Linear response In the absence of heat current, j h = 0, the system is assumed in global equilibrium at temperature T , with x K = x K 0 (R,p, ) given by Eq. (13), and equilibrium co- herence functions γ x = γ x 0 (R,p, ) found from iv Fp · ∇γ x 0 + (2 − Σ x 0 +Σ x 0 )γ x 0 +∆ x 0 (γ x 0 ) 2 + ∆ x 0 = 0 . (15) In uniform superconductors, ∇γ x 0 (R,p, ) = 0 and the solution of Eq. (15) for the retarded coherence function is γ R u (p, ) = − ∆ R ū R + i ∆ R u∆ R u − (¯ R ) 2 ,(16)with¯ = − (Σ R u −Σ R u )/2. In the following, the subscript "u" stands for 'uniform', and subscript "0" will refer to the equilibrium solution. In the presence of a small heat current j h = 0 that is assumed to be time-independent (stationary state), the system is out-of-equilibrium. In linear response, we expand coherence and distribution functions around their equilibrium values γ x (R,p, ) =γ x 0 (R,p, ) + γ x 1 (R,p, ) , x K (R,p, ) =Φ 0 ( )(1 + γ R 0γ A 0 ) + Φ 0 ( ) (γ R 0γ A 1 + γ R 0γ A 1 ) + x a .(17) The deviation of the distribution function from equilibrium, x K − x K 0 , is described by two terms. The first accounts for change in the density of states through corrections to the retarded and advanced functions, it is weighted by the equilibrium Fermi distribution Φ 0 ( ). The second term, x a (R,p, ), is the anomalous, or dynamical distribution function. It determines the dynamical part of the Keldysh Green's functionĝ a =ĝ K −ĝ K 0 − Φ 0 (ĝ R 1 −ĝ A 1 ), g a = −2πi (1 + γ R 0γ R 0 )(1 + γ A 0γ A 0 ) x a +x a γ R 0γ A 0 (x a γ A 0 −x a γ R 0 )iσ y −iσ y (x aγR 0 −x aγA 0 )x a + x aγR 0 γ A 0 .(18) In linear response the heat current depends only on the equilibrium spectral properties through coherence amplitudes γ R,A 0 , and the dynamical part of distribution function, x a : j h = −2N F v F +∞ −∞ d p x a (1 +γ R 0 γ A 0 ) +x a (1 + γ R 0γ A 0 ) 4(1 + γ R 0γ R 0 )(1 + γ A 0γ A 0 ) p .(19) To obtain equation for x a we linearize Eq. (14). We linearize with respect to the global equilibrium at temperature T , where Φ 0 ( /2T ) in Eq. (17) is position-independent -in this case the linearized equation reads 27 iv Fp · ∇x a + iv F (R,p, ) x a = γ R 0γ A 0Σ a − (∆ aγA 0 +∆ a γ R 0 + Σ a ) ,(20) with equation for thex a -function obtained from this one by employing the definition of˜-operation. In the above equations we introduced parameter (R,p, ) = iv F /(γ R 0∆ R 0 − Σ R 0 +γ A 0 ∆ A 0 + Σ A 0 ) ,(21) that is purely real, as follows from the symmetries of the coherence functions and self-energies, and has dimension of length. In the normal metallic phase = v F /2Γ sin 2 δ = v F τ N = N matches with the elastic mean free path. The dynamical self-energy entering Eq. (20) as the source term is self-consistently computed from x a and γ x 0 : σ a (R, ) ≡ σ a imp ≡ Σ a (iσ y )∆ a −(iσ y )∆ a −Σ a = 1 Γ σ R 0,imp (R, ) g a (R,p, ) π p σ A 0,imp (R, ) .(22) Note that the linearization scheme in Eq. (17) is very convenient because the calculation of j h only requires the knowledge of the anomalous x a , which itself does not depend on spectrum corrections, γ x 1 . B. Boundary conditions To solve the transport equation Eq. (20) for the distribution function, one needs to provide suitable boundary conditions for initial values of x a at the beginning of a trajectory v Fp , and for final value ofx a at the end of this trajectory. For weak links, as in reference 20, one can assume that the system of interest is connected to large reservoirs in equilibrium at temperatures T 1,2 . Then, for a given quasiparticle trajectory, one can take equilibrium values of the coherence and dynamical amplitudes in those reservoirs as initial values. Such assumption seems inadequate to compute heat transport in bulk samples. Instead, given a stationary conserved heat flow in the entire sample, we will construct the Riccati amplitudes at x = ±L in a way that is consistent with Eq. (20), and would give a fixed thermal current j h (±L) = j h = j BC h (23) in a uniform superconducting state, away from the inhomogeneous region. To write such boundary condition we start by making a trivial observation that the linearization procedure that we followed, can be used to find equilibrium functions at +L −L T(x) local +dT −dT T global j h FIG. 2. Local versus global equilibrium picture. The system is driven out of equilibrium by a steady uniform heat current j h . The local equilibrium picture assumes that, when heat flows, a local temperature, T (x), can be defined, and its gradient determines the magnitude of j h . It is typically used in uniform-state problems, and we use it here to define boundary conditions for distribution functions away from the nonuniform (shaded) region of the order parameter. Our main approach, however, is to expand the propagators around some global equilibrium value of the temperature T : g(x) = geq(T ) + gc(x) where gc(x) determines current j h . slightly different temperature T + δT . For example, the distribution function can be written as x K eq (T +δT ) = [1+γ R 0 (T +δT )γ A 0 (T +δT )]Φ 0 2(T + δT ) where we show only the temperature argument explicitly. Decomposition Eq. (17) in this case gives the anomalous contribution as x a eq (T + δT ) = [1 + γ R 0 (T )γ A 0 (T )] ∂Φ 0 ∂T δT ,(24)with ∂Φ 0 /∂T = − /[2T 2 cosh 2 ( /2T )] . This x a eq distribution function, on the other hand, also satisfies Eq. (20) with appropriately determined self-energy through Eq. (22) that can be brought to the form σ a (T ) = (σ R 0,imp − σ A 0,imp )(∂Φ 0 /∂T )δT . Far from the region of spatially varying order parameter, the equilibrium functions, γ R/A 0 (p, ) and x K 0 (p, ) take their uniform values γ x 0 = γ x u , that determine the scattering length u through Eq. (21), and in equilibrium ∇x a eq ∝ ∇T = 0. When a stationary thermal current flows, a local temperature gradient builds up and δT (R) = T (R) − T is a function of position, see Fig. 2. As a result, Eq. (24) with local δT (R) is no longer a solution to Eq. (20). However, one can modify x a eq to include the temperature gradient: x a u (R,p, ) = 1 + γ R u (p, )γ A u (p, ) ∂Φ 0 ∂T × × [δT (R) − u (p, )p · ∇T ] .(25) This expression with uniform gradient ∇T = const satisfies Eq. (20). The ∇T term in Eq. (25) is odd in momentum, and after angular integration in Eq. (22) it does not contribute to self-energy σ a in even-p superconductor. Consequently, only the first term, ∝ δT (R), determines the dynamical self-energy. In entirely uniform superconductor, x a andx a would be trivially related and result in local equilibrium self-energy σ a u = (σ R 0,imp − σ A 0,imp )(∂Φ 0 /∂T ) δT (R) . This is important since after substitution of this expression together with Eq. (25) into Eq. (20) the arbitrary δT (R) drops out. Non-uniform order parameter means different history for x a (p) andx a (p) along a trajectory and the self-energy σ a even in uniform part does not fully recover the local equilibrium dependence on δT . On the other hand, by similar symmetry arguments, the heat current in the uniform part of the superconductor is independent of δT and is completely determined by the ∇T term of x a u . We use it to set the value of the temperature gradient from fixed j BC h . Inserting Eq. (25) into Eq. (19), and using γ-symmetries Eq. (12), we obtain uniform-state current j BC h = −κ u ∇T , κ u = +∞ −∞ d κ u ( ) ,(26) where the thermal conductivity has this Boltzmann-like representation κ u ( ) = v F 2 2T 2 cosh 2 ( /2T ) p 2 x N (p, )v(p, )τ (p, ) p .(27)v( ,p) = v F 1 − \|γ R u (p, )\| 2 1 + \|γ R u (p, )\| 2 .(28). L -L = Tr [⇥ ⌅ , g K ] p + Tr (⌅ R imp g K p g K p ⌅ A imp + ⌅ K imp g A p g R p ⌅ K imp )] = 0 + 0, ( where the first term is null due to the traceless property of a commutator while the second term is the result of self-consistent equation for impurity scattering in Eq. (9). x a u ( L,p x > 0, ⇥) R u,+ 0 (p x > 0, ⇥) ⌅ (47) ⇤ x a u (L,p x > 0, ⇥) R u, 0 (p x < 0, ⇥) (48) T + dT ( T dT ( p imp p p imp imp p p imp = 0 + 0,(46) where the first term is null due to the traceless property of a commutator while the second term is the result of the self-consistent equation for impurity scattering in Eq. (9). x a u ( L,p x > 0, ⇥) R u,+ 0 (p x > 0, ⇥) ⌅ (47) ⇤ x a u (L,p x > 0, ⇥) R u, 0 (p x < 0, ⇥) (48) T + dT (49) T dT (50) = 0 + 0,(46) where the first term is null due to the traceless property of a commutator while the second term is the result of the self-consistent equation for impurity scattering in Eq. (9). x a u ( L,p x > 0, ⇥) R u,+ 0 (p x > 0, ⇥) ⌅ (47) ⇤ x a u (L,p x > 0, ⇥) R u, 0 (p x < 0, ⇥) (48) T + dT (49) T dT (50) \|x a (±L,p, ✏)\|, energy is perfectly transmitted. However, in the presence of domain walls, it yields \|x a (⌥L,p x ? 0, ✏)\| > \|x a (±L,p x ? 0, ✏)\|, meaning that the energy is not perfectly transmitted. We interpret the latter result as a partial Andreev reflection of quasiparticle due to both a suppression of spectral weight and a change of superconducting phase. Classically, we can illustrate it as the result of a tra c jam. Using Eq. (??), we maybe expressed the heat conductivity in terms of a kernel k(✏,p) aŝ  = Z d✏ ✏ @ T 0 hp k(✏,p)ip .(52) As shown in Fig. 6, the amplitude of k(✏,p) is suppressed in the presence of inhomogeneities. In the clean limit, the heat conductivity is reduced in the presence of a single domain wall, alike the pinhole of perfect transparency ? . We now turn to the transport across a comb of N DW domain walls. Each domain wall hosts degenerate ABS, that can overlap if the domain wall interspacing, X FFLO , gets smaller than the ABS spatial extent X ABS . Since X ABS (p) / v F / (p), the bound states with momentum direction around nodal direction will overlap first. Our goal is to provide a numerical fit for the conductivity based on our understanding of the physics in the vary roughly as d✏ 1/N (✏), it is inversely proportional to the number of conducting channels. downsizing X FFLO enlarges the overlaping group x a u ( L,p x > 0, ✏) R u,+ 0 (p x > 0, ✏) (53) ⇢ x a u (L,p x < 0, ✏) R u, 0 (p x < 0, ✏) (54) T + dT (55) T dT (56) R N (57) R L u (58) R A u (59) R L DW (60) R A DW (61) ✏ (62) (63) W imp(64) FIG. 3. Numerical integrations of Eqs. (15) and (20), in the shaded region, performed from x = ∓L to x = ±L for right/left going (px ≶ 0) trajectories alongp. We start the numerical integration at the white/black circles with the uniform Riccati amplitude given by Eqs. (30) and (31), see main text. The (half-) temperature bias dT is the unknown that we numerically determine to satisfy Eq. (23). From this, the velocity of QPs in superconductor is always smaller than v F ; also in the clean limit one recovers the well-known result v(p, ) = v F 2 − ∆ 2 (p)/\| \|. Typically, heat transport in uniform superconductors is analyzed as interplay between the density of states N (p, ) and effective elastic mean free path e ≡τ (p, )v(p, ) ,(29) where the latter plays a more dominant role. This discussion is moved to Appendix B, with the main result presented in Figs. 11 and 12 there. We are now ready to write the initial values of coherence and distribution functions for numerical integration Eqs. (15) and (20) along quasiclassical trajectories. We start the integration well away from any domain walls, inside the uniform part of superconductor, at x = ±L, Fig. 2. The distance between the initial point on the trajectory and the first domain wall should be much greater than (p, ), which might be difficult to satisfy for all energies and momenta, especially in clean superconductor. The equilibrium coherence functions at x = ±L arrive from infinity with the uniform bulk values, Fig. 3: γ R 0 (−L,p x > 0, ) = γ R u,0 (∆(x = −∞,p),p, ) , γ R 0 (+L,p x < 0, ) = γ R u,0 (∆(x = +∞,p),p, ) .(30) We will position the center of domain walls symmetrically around x = 0, ensuring equivalent temperature drops dT on the left and right, to accelerate numerical integration. The initial values for the anomalous distribution x a (±L) is given by Eq. (25) with temperature gradient fixed by the heat current in uniform region: ∇T = − j BC h κ ux , x a (−L,p x > 0, ) = (1 + γ R uγ A u ) ∂Φ 0 ∂T [dT + p · ∇T ] , x a (L,p x < 0, ) = (1 + γ R uγ A u ) ∂Φ 0 ∂T [−dT + p · ∇T ] .(31) The unknown temperature drop dT is determined, for a given j h , through self-consistent calculation of anomalous self-energies σ a (x, ) at each . Starting with some guess for σ a (x, ) we solve Eq. (20) for x a (x,p, ) with boundary conditions (31). From distribution function we find g a (x,p, ), Eq. (18), and then obtain new values for σ a (x, ), Eq. (22). This process is repeated until self-energy has converged. The linearity of all equations assures that all the parameters are linear combinations of dT and ∇T terms: x a (x,p, ) = x a 1 (x,p, )dT + x a 2 (x,p, )∇T , g a (x,p, ) = g a 1 (x,p, )dT + g a 2 (x,p, )∇T , σ a (x, ) = σ a 1 (x, )dT + σ a 2 (x, )∇T , and similarly the current, j h (x) = κ 1 dT + κ 2 ∇T = const = κ u ∇T , that is equal to the input current at the boundary. After self-consistent determination of the coefficients κ 1,2 through the above procedure we determine the temperature drop dT = κ u − κ 2 κ 1 ∇T . In the uniform case one has dT = Lj BC h /κ u . Here we also want to note that the presence of topological domain walls in the order parameter is reflected in features of the heat current arbitrarily far from the nonuniform region, and is indirectly encoded in the choice (31) for x a at the integration boundaries. For example, in a uniform superconductor the spectral current is given by κ u ( )\|∇T \|, whereas in nonuniform superconductor we have j h ( ) = κ u ( )\|∇T \|, which is obvious if we are right at the domain wall, and thus everywhere else due to the conservation of the spectral current, as shown in Appendix A. To recover the heat current spectrum of the uniform state far from the nonuniform region, one requires presence of nonelastic collisions that are not included in the theory. By contrast, the local equilibrium picture includes the nonelastic collisions implicitly in the definition of the local temperature T (x) in Fermi distribution. IV. HEAT FLOW ACROSS DOMAIN WALLS We now apply the developed formalism to nonuniform d-wave superconductor, and investigate heat transport across an array of N DW domains walls equally spaced with a period X FFLO along thex-axis. Each domain wall has a width of several coherence lengths that we define as ξ = v F 2πk B T c (T c is the transition temperature of clean supercnductor). The uniform heat current flows from left to right j h = j hx , and we consider translationally invariant system along theŷ-direction, so that all functions depend only on x-coordinate. For convenience, we now set a unit gradient at the boundaries ∇T = −x in Eq. (31), giving j h = κ u × 1. Due to the factor \| ∂ T Φ 0 \| = 2 /[2T 2 cosh 2 ( /2T )] the heat current is mainly determined by quasiparticles with energies in the window [T, 5T ]. We introduce T = 2.5 T as a characteristic quasiparticle energy at a given temperature T . A. Single domain wall We first look at the heat transport across a single domain wall (DW) centered at x = 0 (N DW = 1). The domain wall is enforced through the boundary condition ∆ 0 (±L) = ±∆ u . It is self-consistently computed together with the local impurity self-energy σ imp (x, ) via Eqs. (8) and (9). With the domain wall centered at x = 0, we use symmetry ∆(−x) = −∆(x) to speed up numerical calculations through relations: γ R 0 (x,p, ) = −γ R 0 (−x, −p, ), γ R 0 (x,p, ) = −γ R 0 (x, −p, ),(33) and similar ones for self-energies, σ R (x,p, ) = τ z σ R (−x,p, )τ z , σ R (x,p, ) = [σ R (x,p, )] tr , σ a (x, ) = −τ z σ a (−x, )τ z .(34) Technically we proceed as follows. First, we obtain the order parameter profile ∆ 0 (x), shown in Fig. 4(a) we integrate Eq. (15) for real energies to determine the equilibrium values of γ R 0 (x,p, ) and impurity σ R/A 0,imp (x, ). They are then used as input parameters in equation (20) for the anomalous amplitude x a . The last step is the self-consistent calculation of the temperature drop dT together with anomalous self-energy σ a . We compare the temperature drop dT with the drop dT u = (j h /κ u )L = \|∇T \| u L that would appear if the superconductor was uniform. Then dT > dT u corresponds to a suppression of ability to transport heat across domain walls, while dT − dT u < 0 represents an enhancement of heat conductivity. The numerical results for transport across the domain wall are presented in Fig. 5, where we plot the temperature drop across a domain wall for a given heat current, relative to the uniform configuration. We define the parameter dL with dimension of length dL = dT − dT u j h /κ u , that can be interpreted as effective "thermal length" of the domain wall, in units of coherence length ξ. At high temperatures the behavior of the thermal transport is the same for all impurities, with a loss of effectiveness in energy transport. At low temperatures, however, the behavior is remarkably different in Born and Unitary limits. For weak impurity scattering potential the domain wall presents a barrier for heat transport resulting in a larger temperature drop required to maintain current j h . The strong scatterers have the opposite effect -the heat current flows through a domain wall more efficiently than in the uniform case. The origin of such peculiar behavior is in the interplay between two effects of the Andreev bound states at the FIG. 6. Energy dependence of the heat current kernel at T = 0.3Tc for transport across a domain wall (solid lines) and the uniform superconductor (dashed lines) for N = πξ/0.3. In Born or clean limit (orange lines) the ability to transport heat at low energy is suppressed by the presence of a domain: at low energy KDW( ) < Ku( ). By contrast, in unitary limit (blue lines) the coupling between impurity band and Andreev bound states enhances energy transport: at low energy KDW( ) > Ku( ). domain wall: the change of spectrum and the hybridization of the bound states with the impurity band states. The spectral effect is a result of Andreev bound states 'stealing' spectral weight from continuum quasiparticles states above the energy gap. In bulk, the only available quasiparticles with > \|∆(p)\| participate in the energy transport. As these quasiparticles enter the domain wall region with fewer available states they experience Andreev reflection that leads to suppression of the heat conductivity. This effect can be quantified by looking at a clean superconductor. In this case equation for the distribution function (20) has no impurity-generated right-hand-side, and the relaxation length (mean free path) 1/ ∆ = 2 Im[γ R∆ ]/v F is determined purely by the density-of-states effects. Details of this analysis are presented in Appendix C. Effects of the spectral weight reduction and Andreev reflection processes appear in the heat current kernel K( ,p) = j h (p, )/( ∂ T Φ 0 ), shown in Fig. 6, at energies ∼ ∆ and play the most important role at higher temperatures. At very low temperatures T T c , the interaction of low-energy bound states with impurities comes out to the front stage, while we find that ∆ is only slightly modified by impurities. The impurity scattering effects appear in Eq. (20) through anomalous self-energy and local scatter- ing length 1/ imp = 2 Im[γ R 0∆ R imp − Σ R ]/v F . This length is positive and finite, depends on directions very weakly and can be approximated by imp (x, ) ≈ imp (x,p, ) p . The impurity scattering creates a band of mid-gap states, which hybridize with Andreev bound states. Such hybridization depends strongly on the strength of the impurities and may lead to a significant 'renormalization' of scattering features in the vicinity of the domain wall, as shown in Fig. 7. In the unitary limit, Andreev states' interaction with impurity band leads to suppression of scattering and long lifetime of close-to-zero-energy quasiparticles. This results in an effective 'wormhole' across the domain wall region for these quasiparticles, and an Inverse local impurity scattering length N / imp(x,p, ) ≈ N / imp( , x) (weakly dependent on momentum directions), as a function of energy, for N = πξ/0.3. In Born limit (orange), the mean free path is large in the bulk (dashed lines) and becomes small at the domain wall (solid lines). For unitary scattering (blue), on the right, this behavior is reversed: the zero-energy peak in the DOS results in suppression of scattering rate at the domain wall and longer mean free path. enhancement of heat conductivity at low temperature, see Fig. 6. In the Born scattering limit, on the other hand, the impurity band is weak, and its presence cannot compensate Andreev reflections. In this case, for all temperatures, the heat transport is suppressed across the domain wall. B. Multiple domain walls To model the periodic structures of FFLO states we investigate transport across a set of domain walls. Since the main effects come from the density of states and scattering, we omit the self-consistent calculation of the order parameter, and simpy 'build' a lattice of N DW equally spaced domains with an arbitrary period X FFLO , taking the single domain profile as a unit cell, as shown in Fig. 4 for N DW = 4. We place the domains symmetrically around x = 0 and use this symmetry to reduce computation time. There are several effects that influence the transport across multiple domain walls. First one is the trivial (incoherent) accumulation of effects from all domains that are independent in this case. This happens when the mean free path, Eq. (29), is shorter than the spacing X FFLO between domain walls, and the spatial extent of the bound states also exceeds this length, X ABS [p] ≈ v F / ∆ 2 (p) + W 2 imp X FFLO , where W imp is the impurity bandwidth. Independent domain walls lead to linear dependence of the heat conductivity on the number of domain walls N DW , based on the temperature regime and single-domain result as in figure 5. Such behavior is expected for reasonably dirty superconductors. Full numerical results for domain wall spacing X FFLO ≈ 18 ξ are shown in Fig. 8 and in independent-domain regime are fitted with straight lines. When the superconductor is in the clean limit, and the figure 5, coming from low-energy states' transport. At intermediate temperature we have a suppression of heat flow due to independent Andreev reflection processes, with positive slope and linear increase in the thermal length dL with NDW. In Born limit at low temperature the dependence is more complicated due to large extent of bound states and more intricate impurity band energy dependence for T ∼ Wimp. domain walls are tightly spaced with X FFLO < X ABS , the bound states belonging to neighboring domains can overlap, hybridize, and build up a conduction band (hybrid transport). This is expected in FFLO phase when the order parameter is small and harmonic-like, with periods ∼ 5 − 10ξ rather than a combination of fully formed domain walls, or when the transport is dominated by the nodal quasiparticles since ABS states can extend far beyond the DW region, especially in Born limit with tiny W imp . If the spacing between the domain walls is somewhat longer, then the hybridization of bound states from different domains depends on their quasiclassical trajectory. In the anti-nodal direction, the ABS spatial extent is smaller than X FFLO and ABS are spatially separated. Each domain is the center of an Andreev reflection process. Consecutive reflections add up and yield a power law reduction of the transmission of anti-nodal quasiparticles. By contrast, in the nodal direction, the ABS extent is large. ABS at consecutive domains overlap and the transmission is rather insensitive to the number of domains. Together, they result in N DW -dependence seen in Fig. 9. The heat conductance can be roughly fitted by a sum of nodal and anti-nodal contribution: g n. +g a.n. t NDW , where conductance contribution from nodal quasiparticles g n. grows with temperature, and transmission coefficient t is only weakly temperature-independent. Effect on thermal conductance of Andreev reflections from a set of NDW domain walls (clean limit). Heat transport through more than ten consequitive domains is dominated by extended bound states along nodal directions on the Fermi surface. Phase space of those states and their contribution to the heat transport grow with temperature. The fitting line through numerical points is explained in the text. C. Zeeman field In this section we present the effects of a Zeeman field on heat transport across the nonuniform state, since the FFLO state is a result of competition between magnetization and condensation energies. Again, the main effect, we assume, is coming from the modification of the density of quasiparticle states that are shifted in energy by ±µH for up/down spins. We neglect the order parameter suppression due to magnetic field, which is relatively small at low temperature. 25 Then spin up and spin down QPs are independent, and their contributions to thermal transport add up. The dependence on spin enters equations (19), (20), and boundary conditions (25) and (31) through energy shift in coherence functions γ R/A 0 ( ± µH). The quasiparticle distribution function prefactor ∂ T Φ 0 ( , T ) is not changed. We can use it to write the heat current as some spin-dependent kernel times the distribution function, j h = s=±1 d K s ( ) ∂ T Φ 0 ( , T ) .(35) We then can re-use the zero-field results to compute the thermal current including the Zeeman splitting. In the Zeeman field the spin dependent kernel is simply the spinindependent kernel shifted energy: K s ( ) = K( − sµH). We then can transfer the dependence on spins into the distribution function, without recalculating the kernel: The effect of Zeeman splitting of the states on thermal conductivity across a single domain wall is shown in Fig. 10 for strong impurities. The bound states contribute most to the low-energy heat current and lead to increase in conductivity at low temperatures T ∆ BS /2.5. From the h = 0 curve the half-width of the bound states can be estimated as ∆ BS ∼ 0.4T c . When the Zeeman field shifts the bound states by h = µH/T c = 0.5 they dominate the heat transport in a wide range of temperatures leading to negative dT − dT u . For even higher fields h = 1, close to the critical field, the contributions of bound states with one spin projection mix with the continuum contribution with the other spin projection, leading to a non-monotonic temperature dependence of the heat conductivity. j H h = 1 2 ± d j 0 h ( ) ( ± µH)∂ T Φ 0 ( ± µH, T ) ∂ T Φ 0 ( , T ) ,(36) V. CONCLUSIONS In this paper we have developed theoretical framework to investigate thermal transport in nonuniform superconductors. Our approach is based on fully self-consistent non-equilibrium quasiclassical Eilenberger-Keldysh technique, that takes into account, on the same footing, combined effects of impurity scattering, spatial variations of the order parameter and density of states, and the presence of Andreev bound states in strongly inhomogeneous environments. We applied this theory to compute the thermal current across a periodic modulations of the order parameter, and domain walls, in a superconductor with d-wave pairing. Here we outline the key effects that govern transport in such systems compared with the uniform superconductors. First, Andreev bound states 'trap' quasiparticles and cause a depletion of the continuum ( > ∆) states near the domain wall, leading to Andreev reflec-tion processes with particle-hole conversions. This results in a reduction of heat transport across the domain wall, and this mechanism is dominant at intermediate temperatures and in clean superconductors. Another effect becomes relevant at low temperatures when disorder is present. Then the bound states at the domain wall interact with the low-energy impurity band. The coupling of the impurity band to localized Andreev states strongly depends on the type of impurity scattering. In Born limit this coupling increases scattering rate, while in unitary limit the scattering of low-energy quasiparticles is suppressed. These states have longer mean free path in the domain wall region resulting in an effective 'wormhole' through the domain wall. At low temperature, below the width of the impurity band, transport is dominated by these states and with unitary impurities heat conductivity across the domain wall is higher than conductivity in the uniform state. This results in a very distinct nonmonotonic feature of heat conductivity as a function of temperature, as one crosses from high-into low-energy regime. In a Zeeman field the difference between thermal transport in uniform and nonuniform phases is softened, but due to the opposite shifts of the up/down spin states, one can observe additional features in T -dependence of the heat conductivity, and non-monotonic T -dependence appears even in the Born limit. A grid of multiple domain walls generally amplifies transport properties of a single domain, but in the clean limit one has to consider multiple-wall Andreev backscattering processes. These results show that thermal transport can be a useful probe to detect and study nonuniform states, such as Fulde-Ferrell-Larkin-Ovchinnikov phase that so far has been only identified using NMR technique. 28 The approach that we developed will pave the way for future theoretical studies of heat transport near surfaces of superconductors with non-trivial surface states, in vortex lattices including vortex core states or for complete analysis of FFLO-type order parameter periodic structures. VI. ACKNOWLEDGEMENTS This work has been done with NSF support through grant DMR-0954342. . Different curves represent different momentum directions spanning the d-wave clover from a node to antinode (solid blue to red dashed lines), as shown in inset. In unitary limit, the low energy impurity band in DoS is large, and the mean free path is reduced by enhanced impurity scattering. By contrast, in Born limit, the impurity band is exponentially small and the mean free path of nodal quasiparticles is longer. The low-energy spectrum of a d-wave superconductor is strongly modified by the scattering of quasiparticles on impurities due to the anisotropy of the order parameter structure. Scattering on impurities results in formation of midgap states. 29 These impurity-bound states are extended in space and form a conduction 'impurity' band with energy width W imp . 2,30 This bandwidth is tiny in the Born limit, W B imp ≈ 4∆ 0 exp(− π∆0 Γ ), but can be large in the Unitary limit where W U imp ≈ π∆ 0 Γ/2. The mean free path reflects the effectiveness of the scattering of quasiparticles by impurities. It depends on the concentration Γ and strength δ of impurities, as well as on the available phase space for scattering, given by the properties of the order parameter ∆. At low energy < W B imp < ∆, in the Born limit, impurity scat- (2Γ sin 2 δ), and it allows quasiparticle to travel long distance between scatterings producing large heat transport. By contrast, in the Unitary limit, scattering is enhanced v F /(2 Im[Σ R imp ]) < N , i.e. low energy QPs bind to impurities forming a wide impurity band. (Color online) Uniform thermal conductivity as a function of temperature. At low temperature T 0.3Tc ( 0.6Tc) thermal conductivity in Born limit (green) is higher than that in Unitary limit (blue), indicating that it is dominated by large mean free path of quasiparticles. Solid and dashed lines correspond to mean free paths N = πξ/0.3 and N = πξ/0.2 respectively. tering is ineffective, v F /2 Im[Σ R imp ] > N = v F τ N = v F / Numerically, we find that thermal transport properties are mainly influenced by the behavior of scattering length e (p, ) rather than that of density of states. In Fig. 12, we plot the temperature dependence of κ u (T ), which we analyze using data from Fig. 11. At low-intermediate temperature 0.05 < T /T c < 0.3, corresponding to energies W imp < 0.6T c the DoS in Born limit is small N B ( ) < N U ( ), while B e U e ,producing κ B u (T ) > κ U u (T ). At higher energy and temperature 0.4 < T /T c , > 0.8T c the result is reversed κ B u (T ) < κ U u (T ), again in agreement with the increase of U e > B e while having about the same values for the DoS in this energy interval. In the very low temperature limit, T W imp , DoS and scattering effects exactly cancel each other, producing the universal limit for heat conductivity, where it does not depend on the disorder properties. 2,31,32 Appendix C: Heat conductivity of a clean constriction In this appendix, we evaluate heat transport properties of a spin-singlet superconducting constriction without impurities and discuss the role of Andreev reflection processes. The constriction can be thought of as a narrow bridge connecting two large reservoirs, that are assumed to be in equilibrium at temperature T ±dT (dT T, T c ). We define the conductance of the clean constriction as G = I h /(2dT ). The global phases of the superconducting order parameter in the reservoirs ∆(p) exp(iϕ L,R ) is set to ϕ L,R = 0, π. The constriction is assumed to be long and narrow, so we neglect the edge effects. In linear response, the energy transport is governed by Eqs. (15) and (20), with σ imp = 0. At boundaries, γ(±L,p x ≶ 0, ) is given by Eq. (30) and we take x a R/L = x a (±L,p x ≶ 0, ) = ∂ T Φ 0 (∓dT )(1 + γ R uγ A u ), (C1) which conveniently describes junctions between reservoirs that have negligible heat currents inside. This is dif- In uniform superconductor it is infinite for above-gap energies 1/ ∆ (\| \| > \|∆(p)\|) = 0, while at the domain wall it is finite for all energies and even changes sign. (c) Kernel of the heat current K(p, ) for four momentum directions and integrated over the Fermi surface. With the domain wall the kernel K( ,p) < 1 is suppressed due to Andreev reflection. ferent from the boundary condition (31) that was aimed at describing a continuous flow of heat. The order parameter ∆(x) and γ 0 (x,p, ) are selfconsistently determined throughout the constriction. From equilibrium γ 0 (x,p, ), using Eq. (15), one can find analytic solution for the distribution function along the constriction: x a (x,p x > 0, ) = t(x,p, ) x a L , x a (x,p x < 0, ) = t(x,p, ) x a R , where t(x,p, ) = 1 − \|γ R 0 (x,p, )\| 2 1 − \|γ R u (p, )\| 2 , plays the role of a transmission coefficient (\|t\| < 1). In a uniform superconductor energy is perfectly transmitted \|t( ,p)\| = 1. However, with a domain wall, one has \|t\| ≤ 1, i.e. energy is not fully transmitted. This is interpreted as a partial Andreev reflection of incident quasiparticles from the spatially varying profile of the order parameter. Inserting Eq. (C2) into heat current expression Eq. (19), we can express the conductance as G = d \|p x \| K( ,p) p ∂Φ 0 ∂T ,(C4) where the kernel K( ,p) is K( ,p) = N F v F (1 − \|γ R 0 ( ,p, x)\| 2 )(1 − \|γ R 0 ( ,p, x)\| 2 ) \|1 + γ R 0 ( ,p, x)γ R 0 ( ,p, x)\| 2 . (C5) Again, because the energy flow is uniform, K( ,p) does not depend on position x, even though γ R 0 does. In Fig. 13 where we plot the heat current kernel together with the density of states and the Andreev reflection length 1/ ∆ = 2 Im[γ R∆ ]/v F appearing in Eq. (20). For uniform order parameter (dashed lines) K( ,p) = 1 for > \|∆(p)\|, and is zero for energies below the gap where there are no quasiparticle states. ∆ ( ,p) is finite for subgap states < ∆(p), and infinite otherwise. At the center of domain wall ∆ ( > ∆(p),p, x) is finite (and can even be negative!) for the above-gap states, their spectral weight is moved into the ABS, and the amplitude of K( ,p) is reduced, as shown by solid lines in Fig. 13. In the clean limit, the conductance is reduced in the presence of a single domain wall, alike the pinhole of perfect transparency. 20 Here N (p, ) is the density of states,τ (p, ) = [ (p, ) + (p, )]/2v F is a scattering time defined using relaxation length (21) (in Unitary or Born limits (p, ) =˜ (p, ) ≡ (−p, − )). The group velocity for quasiparticles (QPs) with momentump and energy is given by FIG. 4 . 4Upper panel: self-consistent OP profile ∆(x) for a single DW (solid line). This solution is used to construct a non-self-consistent profile with NDW , case of 4 DWs with separation XFFLO ≈ 20ξ is shown by the dashed line. Lower panel: local density of states (DoS) in Born (orange) and unitary (blue) limits for N = πξ/0.3. At the domain (solid lines), the peak at zero energy indicates the Andreev bound states (ABS). FIG. 7 . 7FIG. 7. Inverse local impurity scattering length FIG. 8 . 8Effective thermal length dL (normalized by ξ ) across NDW domain walls, for low temperature T /Tc = 0.05 (large symbols, solid lines) and intermediate temperature T /Tc = 0.5 (small symbols, dashed lines). The scattering rate 1/τN = 0.3Tc is used for various impurity strengths: Born (B), Unitary (U) and intermediate δ = π/4 (I). This is an 'independent domain walls' regime where the heat conductivity contributions from each domain add up, as is clear from linear dependency c1NDW + c2 shown by lines. At low temperature the unitary and intermediate strength disorder has negative slope consistent with single-domain result in • T/T c =0.15 ■ T/T c =0.35 FIG. 9. FIG. 10 . 10where j 0 h ( ) is the spectral heat current in the absence of Zeeman field. As a reminder,Fig. 6highlights the effect of impurity on the kernel of heat current in absence of Zeeman field. Effect of the Zeeman field splitting h = µH/Tc on thermal transport across a single domain wall. Unitary limit with scattering rate 1/τN = 0.3 Tc. The bound states, shifted by h = 0.5 contribute to a reduction of the thermal length, dL , in a wide range of temperatures. When the Zeeman shift is very large h = 1 the contributions from bound and continuum states mix up leading to very non-monotonic temperature dependence. FIG. 12. (Color online) Uniform thermal conductivity as a function of temperature. At low temperature T 0.3Tc ( 0.6Tc) thermal conductivity in Born limit (green) is higher than that in Unitary limit (blue), indicating that it is dominated by large mean free path of quasiparticles. Solid and dashed lines correspond to mean free paths N = πξ/0.3 and N = πξ/0.2 respectively. properties of a clean superconductor across a single domain wall (solid lines) compared agains uniform superconductor (dashed lines). (a) Local DoS for momenta directions shown in (d) at the domain wall N (p, , x) with part of the spectral weight (shaded area) moved from continuum states into zero-energy bound states, that form a very sharp peak not resolved on this scale. (b) The Andreev reflection length scale ∆( ,p, x). , using Matsubara technique. With the known mean field profile,�� -�� -� � � �� U B I FIG. 5. Effective change in thermal length dL (in units of ξ) across a single domain wall relative to the uniform case, as function of temperature. Numerical system size is 2L = 16πξ. Different colors correspond to unitary limit (U, blue), Born (B, orange) and intermediate phase shift δ = π/4 (I, green). Solid lines are for scattering rate 2Γ sin 2 δ = 1/τN = 0.6Tc, dashed lines are for a cleaner case 1/τN = 0.2Tc. In the unitary limit, dL is non-monotonous, and at low temperature T < Wimp, the heat conductance through a domain wall is larger than in uniform case. �� ϵ/� � �� FIG. 11. Spectral and transport properties of a uniform dwave superconductor. Angle resolved DoS, N (p, )/NF (top row), mean free path e(p, )/ N (middle row) and impurity scattering length imp( )/ N (bottom row) are plotted in Born and Unitary limits, for the normal state mean free path N ≈ 10 ξ, where imp = vF /2Im[Σimp]�� /� � �� ϵ/� � �� /� � �� ϵ/� � �� ℓ � /ℓ � �� ϵ/� � �� ℓ � /ℓ � �� ϵ/� � �� ℓ �� /ℓ � �� ϵ/� � �� ℓ �� /ℓ � Appendix A: Uniformity of currentsIn the absence of inelastic scattering processes, the selfconsistent solution of the Elenberger transport Eq.(5)together with impurity self-energies (9) guarantees uniform heat flow, and non-accumulation of heat, ∇ · j h = −∂ t Q = 0, even in the presence of spatially-varying order parameter. The heat current is given by Eq. (3) which we repeat here:Tr g K (R,p, ) .(A1) With only energy-conserving impurity collisions, all are independent, and we can consider divergence of the heat current kernel for single energy,(A2) we can split off the mean field self-energy ∆(R,p), common for both retarded and advanced functions and zero for Keldysh component, from the impurity self-energy. This allows us to writewhere the first term is zero due to the traceless property of a commutator and the second zero follows from the self-consistent relations between impurity self-energies and the Fermi-surface averaged propagators, Eq. (9). Note that the order parameter self-consistency was not used in the above argument. It is however needed to conserve the charge/particle number. 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[ "A Simple Adaptive Step-size Choice for Iterative Optimization Methods", "A Simple Adaptive Step-size Choice for Iterative Optimization Methods" ]	[ "I V Konnov 1e-mail:[email protected] \nDepartment of System Analysis and Information Technologies\nKazan Federal University\nul. Kremlevskaya, 18420008KazanRussia\n" ]	[ "Department of System Analysis and Information Technologies\nKazan Federal University\nul. Kremlevskaya, 18420008KazanRussia" ]	[]	We suggest a simple adaptive step-size procedure, which does not require any line-search, for a general class of nonlinear optimization methods and prove convergence of a general method under mild assumptions. In particular, the goal function may be non-smooth and non-convex. Unlike the descent line-search methods, it does not require monotone decrease of the goal function values along the iteration points and reduces the implementation cost of each iteration essentially. The key element of this procedure consists in inserting a majorant step-size sequence such that the next element is taken only if the current iterate does not give a sufficient descent. Its applications yield in particular a new gradient projection method for smooth constrained optimization problems and a new projection type method for minimization of the gap function of a general variational inequality. Preliminary results of computational experiments confirm efficiency of the proposed modification.	null	[ "https://arxiv.org/pdf/1802.00339v2.pdf" ]	119,678,888	1802.00339	36370f843310ce1d313e1c59358e409e351930b5	A Simple Adaptive Step-size Choice for Iterative Optimization Methods 2 Mar 2018 I V Konnov 1e-mail:[email protected] Department of System Analysis and Information Technologies Kazan Federal University ul. Kremlevskaya, 18420008KazanRussia A Simple Adaptive Step-size Choice for Iterative Optimization Methods 2 Mar 2018MSC codes: 90C30, 90C33, 65K05 1Optimization problemsprojection methodsadaptive step- size choicenon-monotone function valuesvariational inequalitiesgap functionconvergence properties We suggest a simple adaptive step-size procedure, which does not require any line-search, for a general class of nonlinear optimization methods and prove convergence of a general method under mild assumptions. In particular, the goal function may be non-smooth and non-convex. Unlike the descent line-search methods, it does not require monotone decrease of the goal function values along the iteration points and reduces the implementation cost of each iteration essentially. The key element of this procedure consists in inserting a majorant step-size sequence such that the next element is taken only if the current iterate does not give a sufficient descent. Its applications yield in particular a new gradient projection method for smooth constrained optimization problems and a new projection type method for minimization of the gap function of a general variational inequality. Preliminary results of computational experiments confirm efficiency of the proposed modification. Introduction Iterative methods are utilized for various difficult problems whose solution in the closed form either is not known or has significant computational drawbacks. For instance, if a system of linear equations has large dimensionality and inexact data, it is better to apply one of well known iterative methods. These methods are a standard tool for nonlinear constrained optimization problems; see e.g. [1,2]. One of the most popular approaches to generation of an iterative sequence consists of solution of direction finding and step-size choice subproblems at each iteration. During rather long time, most efforts were concentrated on developing more powerful and rapidly convergent methods, such as Newton and interior point type ones, which admit complex transformations at each iteration and attain high accuracy of approximations. That is, the direction finding subproblem was considered as the main one, whereas the step-size was chosen by one of the few well known procedures; see e.g. [2,3,4]. However, new significant areas of applications related to data processing in information and communication systems, having large dimensionality and inexact data together with scattered necessary information force one to avoid complex transformations and apply mostly simple methods such as the projection and linearization type methods, whose iteration computation expenses and accuracy requirements are rather low; see e.g. [5,6,7,8]. Therefore, one is also interested in suggesting new step-size choice rules that reduce the total computational expenses of the method. In fact, the existing rules are not completely satisfactory. The exact or approximate one-dimensional minimization line-search requires significant computational expenses per iteration especially in the case where calculation of the function value is almost similar to calculation of its derivative (gradient) and needs solution of complex auxiliary problems; see e.g. [1,3]. In order to remove the line-search, one can calculate the stepsize value via utilization of a priori information such as Lipschitz type constants for the gradient, but then one must take only some their inexact estimates, which leads to slow convergence. This is also the case for the known divergent series rule; see e.g. [1,6]. In this paper, we suggest a new simple adaptive step-size procedure for a general class of iterative optimization methods, which does not require any line-search. In particular, this procedure can be applied to the known projective optimization methods. In creation of the adaptive step-size rule we follow the approach from [9] where a step-size procedure for the conditional gradient method was proposed. However, the procedure in [9] admits only decrease of the step-size and can not be extended to the other optimization methods since it requires the boundedness of the feasible set and is adjusted to the set-valued solution mapping of the direction finding subproblem. Our new step-size procedure admits different changes of the step-size and wide variety of implementation rules. The key element of this procedure consists in inserting a majorant step-size sequence tending to zero. However, we do not change these majorant values continuously such that we take the next element only if the current iterate has not given a sufficient descent. In such a way, the procedure takes into account behavior of the iteration sequence. We show that this strategy can be implemented within a rather general framework of iterative solution methods applied both for smooth and non-smooth optimization problems and variational inequalities. It does not utilize a priori information such as Lipschitz constants of the gradient, besides, the Lipschitz continuity of the gradient of the goal function is not necessary for convergence of the method. Preliminary results of computational experiments confirm efficiency of the proposed modification. The remainder of the paper is organized as follows. Section 2 contains necessary definitions and properties from the theory of non-smooth optimization and set-valued analysis. In Section 3, we describe a general framework of iterative methods with the new step-size procedure applied to constrained non-smooth and non-convex optimization problems and prove its convergence. Afterwards, we describe its specializations as projection type methods. Namely, a new gradient projection method for smooth constrained optimization problems is given in Section 4, whereas a new projection type method for a variational inequality with a general non-integrable mapping, which is re-formulated as a constrained gap function minimization problem, is given in Section 5. We show that they both fall into the general framework of Section 3 and obtain their convergence directly from the basic convergence theorem of Section 3. Section 6 contains results of preliminary computational experiments. Basic Preliminaries We intend to develop the new method for a wide class of optimization problems whose goal functions can be non-smooth and non-convex. For this reason, we first recall some concepts and properties from Non-smooth Analysis; see [10] for more details. If some function f : R n → R is Lipschitz continuous in a neighborhood of any point x of a set X, it is called locally Lipschitz on X. Then we can define its generalized gradient set at x: ∂ ↑ f (x) = {g ∈ R n : g, p ≤ f ↑ (x; p) ∀p ∈ R n }, which must be non-empty, convex and closed. Here f ↑ (x; p) denotes the upper Clarke derivative: f ↑ (x; p) = lim sup y→x,αց0 ((f (y + αp) − f (y))/α). It follows that f ↑ (x, p) = sup g∈∂ ↑ f (x) g, p . In general, the locally Lipschitz function f need not be differentiable. At the same time, it has the gradient ∇f (x) a.e. in X, furthermore, it holds that ∂ ↑ f (x) = conv lim y→x ∇f (y) : y ∈ X f , y / ∈ S ,(1) where X f denotes the set of points where f is differentiable, and S denotes an arbitrary subset of measure zero. If f is convex, then ∂ ↑ f (x) coincides with the subdifferential ∂f (x) in the sense of Convex Analysis, i.e., ∂f (x) = {g ∈ R n : f (y) − f (x) ≥ g, y − x ∀y ∈ R n }. In this case the upper derivative coincides with the usual directional derivative: f ↑ (x; p) = f ′ (x; p).(2) Also, if f is differentiable at x, (2) obviously holds and we have f ′ (x; p) = ∇f (x), p and ∂ ↑ f (x) = {∇f (x)}; cf. (1). The general optimization problem consists in finding the minimal value of some goal function f : R n → R on a feasible set D ⊆ R n . For brevity, we write this problem as min x∈D → f (x),(3) its solution set is denoted by D * and the optimal value of the function by f * , i.e. f * = inf x∈D f (x). We will use the following first set of basic assumptions for problem (3). (A1) The set D is nonempty, convex, and closed, the function f : R n → R is locally Lipschitz on D. (A2) There exists a number γ > f * such that the set D γ = {x ∈ D : f (x) ≤ γ} is bounded. Clearly, (A2) is a general coercivity condition that is necessary in the case where the set D is unbounded. If (A1) and (A2) hold, problem (3) has a solution. This means that we intend to present a method for non-smooth and non-convex optimization problems. Together with problem (3) we will consider the following set-valued variational inequality (VI for short): Find a point x * ∈ D such that ∃g * ∈ ∂ ↑ f (x * ), g * , x − x * ≥ 0 ∀x ∈ D.(4) We denote by D 0 the solution set of VI (4). Solutions of VI (4) are called stationary points of (3) due to the known necessary optimality condition; see e.g. [10,11]. Proposition 1 Let (A1) hold. Then each solution of problem (3) is a solution of VI (4). The reverse implication needs additional conditions. We recall that a function ϕ : R n → R is called (a) pseudo-convex on a set X, if for each pair of points x, y ∈ X, we have ϕ ′ (x; y − x) ≥ 0 =⇒ ϕ(y) ≥ ϕ(x); (b) semi-convex (or upper pseudo-convex) if for each pair of points x, y ∈ X, we have ϕ ↑ (x; y − x) ≥ 0 =⇒ ϕ(y) ≥ ϕ(x); see [12] and also [11]. In case (2), these concepts coincide, but in general (b) implies (a). Besides, the class of convex functions is strictly contained in that of pseudo-convex functions. If (A1) holds and f is semi-convex, then each solution of VI (4) clearly solves problem (3), i.e. D * = D 0 . We need several continuity properties of set-valued mappings; see e.g. [13,11]. Here and below Π(A) denotes the family of all nonempty subsets of a set A. Let X be a convex set in R n . A set-valued mapping Q : X → Π(R n ) is said to be (a) upper semicontinuous (u.s.c.), if for each point y ∈ X and for each open set U such that Q(y) ⊂ U, there is a neighborhood Y of y such that Q(z) ⊂ U whenever z ∈ X ∩ Y ; (b) closed, if for each pair of sequences {x k } → x, {q k } → q such that x k ∈ X and q k ∈ Q(x k ), we have q ∈ Q(x); (c) a K-mapping (Kakutani-mapping), if it is u.s.c. and has nonempty, convex, and compact values. It is known (see e.g. [13, Chapter 1, Lemma 4.4]), that each u.s.c. mapping with closed values is closed and that each closed mapping which maps any compact set into a compact set is u.s.c. Also, if a function f : Y → R is locally Lipschitz on an open convex set Y , then ∂ ↑ f is a K-mapping on Y ; see [10, Section 2.1]. We shall also use the mean value theorem by G. Lebourg for locally Lipschitz functions. f (y) − f (x) ∈ ∂ ↑ f (z), y − x . The Basic Method and Its Convergence We first describe conditions for the solution mapping of the direction finding subproblem. (A3) There exists a single-valued mapping x → y(x), which maps the set D into D such that (i) it is continuous; (ii)x = y(x) if and only ifx is a solution of VI (4); (iii) for each x ∈ D and for all g ∈ ∂ ↑ f (x) it holds that g, y(x) − x ≤ −τ y(x) − x 2 (5) for some τ > 0. We now describe the basic method for problem (3), which involves a simple adaptive step-size procedure without line-search. Method (SBM). Step 0: Choose a point x 0 ∈ D γ , a number β ∈ (0, 1) and a sequence {τ l } → 0, τ l ∈ (0, 1). Set k = 0, l = 0, u 0 = x 0 , choose a number λ 0 ∈ (0, τ 0 ]. Step 1: Take a point y k = y(x k ). If y k = x k , stop. Otherwise set d k = y k − x k and z k+1 = x k + λ k d k . Step 2: If f (z k+1 ) ≤ f (x k ) − βλ k d k 2 ,(6) take λ k+1 ∈ [λ k , τ l ], set x k+1 = z k+1 and go to Step 4. Step 3: Set λ ′ k+1 = min{λ k , τ l+1 }, l = l+1 and take λ k+1 ∈ (0, λ ′ k+1 ]. If f (z k+1 ) ≤ γ, set x k+1 = z k+1 and go to Step 4. Otherwise set x k+1 = u k , u k+1 = u k , k = k + 1 and go to Step 1. Step 4: If f (x k+1 ) < f (u k ), set u k+1 = x k+1 , k = k + 1 and go to Step 1. Therefore, (SBM) in fact represents a general framework of iterative methods for nonlinear optimization problems. Observe that the sequence {u k } simply contains the best current points of the sequence {x k }, i.e. f (u k ) = min 0≤i≤k f (x i ). Due to (A3), termination of (SBM) yields a point of D 0 . Hence, we will consider only the case where the sequence {x k } is infinite. Theorem 1 Let the assumptions (A1)-(A3) be fulfilled and β < τ . Then: (i) The sequence {x k } has a limit point, which belongs to the set D 0 . (ii) If D * = D 0 , then all the limit points of the sequence {x k } belong to the set D * , besides, we have lim k→∞ f (x k ) = f * .(7) Proof. First we observe that the sequence {x k } belongs to the bounded set D γ and must have limit points. By (A3), so is the sequence {y k }, hence {d k }. Next, we take the subsequence of indices {i s } such that f (z is+1 ) > γ, f (x is ) ≤ γ, (8) f (z is+1 ) > f (x is ) − βλ is d is 2 , z is+1 = x is + λ is d is .(9) Let us consider several possible cases. Case 1: The subsequence {x is } is infinite. Take an arbitrary limit point x ′ of the subsequence {x is }. Without loss of generality we can suppose that lim s→∞ x is = x ′ and lim s→∞ y is = y ′ , where y ′ = y(x ′ ) by (A3). Note that λ is ∈ (0, τ ls ], λ is+1 ∈ (0, τ ls+1 ], for some infinite subsequence of indices {l s } where lim s→∞ τ ls = 0. Since the sequence {d is } is bounded, the limit points of the subsequences {x is } and {z is+1 } coincide due to (9). From (8) we now obtain f (x ′ ) = γ > f * .(10) Applying Proposition 2 in (9), we have g is , d is ≥ −β d is 2 for some g is ∈ ∂ ↑ f (x is + θ is λ is d is ) , and θ is ∈ (0, 1). Taking the limit s → ∞ gives g ′ , y ′ − x ′ ≥ −β y ′ − x ′ 2 for some g ′ ∈ ∂ ↑ f (x ′ ). Using (5), we obtain β y ′ − x ′ 2 ≥ τ y ′ − x ′ 2 , i.e. y(x ′ ) = x ′ , hence x ′ ∈ D 0 .(11) Therefore, assertion (i) is true in this case. Case 2: The subsequence {x is } is finite. Without loss of generality we can then suppose that z k = x k for each number k. The further proof depends on the properties of the sequence {λ k }. Case 2a: The number of changes of the index l is finite. Then we have λ k ≥λ > 0 for k large enough, hence (6) gives f (x k+1 ) ≤ f (x k ) − βλ k d k 2 ≤ f (x k ) − βλ d k 2 for k large enough. Since f (x k ) ≥ f * > −∞, we must have lim k→∞ f (x k ) = µ(12) and lim k→∞ y k − x k = 0. Let x ′′ be an arbitrary limit point of the sequence {x k }. From (13) and (A3) we now have y(x ′′ ) = x ′′ , which gives x ′′ ∈ D 0 . Hence in this case all the limit points of the sequence {x k } belong to the set D 0 . Therefore, assertion (i) is true in this case. Case 2b: The number of changes of the index l is infinite. Then there exists an infinite subsequence of indices {k l } such that f (x k l + λ k l d k l ) − f (x k l ) = f (x k l +1 ) − f (x k l ) > −βλ k l d k l 2 .(14) besides, λ k l ∈ (0, τ l ], λ k l +1 ∈ (0, τ l+1 ], and lim l→∞ τ l = 0. Letx be an arbitrary limit point of this subsequence {x k l }. Without loss of generality we can suppose that lim l→∞ x k l =x and lim l→∞ y k l =ȳ. whereȳ = y(x). Applying Proposition 2 in (14), we have g k l , d k l ≥ −β d k l 2 for some g k l ∈ ∂ ↑ f (x k l + θ k l λ k l d k l ), and θ k l ∈ (0, 1). Since λ k l → 0 as l → ∞, taking the limit l → +∞ gives ḡ,ȳ −x ≥ −β ȳ −x 2 for someḡ ∈ ∂ ↑ f (x). Using (5), we obtain β ȳ −x 2 ≥ τ ȳ −x 2 , i.e.x = y(x), hencex ∈ D 0 . Therefore, all the limit points of the subsequence {x k l } belong to the set D 0 . Since x k l +1 = x k l + λ k l d k l , λ k l → 0, and the sequence {d k l } is bounded, the limit points of the subsequences {x k l } and {x k l +1 } coincide and all they belong to the set D 0 . We conclude that assertion (i) is also true in this case. We now suppose in addition that D 0 = D * . Then relations (10) and (11) become inconsistent, hence Case 1 is impossible. This means that the subsequence {x is } is always finite. In Case 2a we now have µ = f * in (12), which gives (7). We conclude that assertion (ii) holds true in this case. In Case 2b, the limit points of the subsequences {x k l } and {x k l +1 } coincide and all they now belong to the set D * . For any index k we define the index m(k) as follows: m(k) = max{j : j ≤ k, f (x j ) − f (x j−1 ) > −βλ j−1 d j−1 2 }, i. e. m(k) is the closest to k but not greater index from the subsequence {x k l +1 }. This means that m (k) = k if f (x k ) − f (x k−1 ) > −βλ k−1 d k−1 2 . By definition, we have f (x k ) ≤ f (x m(k) ).(15) Let now x * be an arbitrary limit point of the sequence {x k }, i.e. lim f * ≤ f (x * ) ≤ f (x) = f * . therefore x * ∈ D * . This means that all the limit points of the sequence {x k } belong to the set D * and that (7) holds true. We conclude that assertion (ii) is also true. The method can be simplified in the case where the set D is bounded. Then we can set γ = +∞ and remove all the calculations of the sequence {u k }. It is easy to verify that all the assertions of Theorem 1 remain true. Application to Smooth Optimization Problems From the results of Section 3 it follows that we can create a number of new solution methods for optimization problems. It suffices to take an optimization problem that satisfies conditions (A1) and (A2) and a method whose solution mapping of the direction finding subproblem satisfies condition (A3). Then we place this method in the framework of (SBM) and obtain its convergence properties directly from Theorem 1. We illustrate diversity of possible specializations of (SBM) by only two basic examples. In this section, we take the well known class of smooth constrained optimization problems. (A1 ′ ) The set D is nonempty, convex, and closed, the function f : R n → R is continuously differentiable on D. Clearly, (A1 ′ ) implies (A1). Let π X (x) denotes the projection of x onto a set X. Fix a number α > 0 and define the mapping y α (x) = π D [x − α −1 ∇f (x)] on the set D. Then setting y(x) = y α (x) in (SBM), we obtain a new version of the gradient projection method for problem (3). We call it (GPMS) for brevity. We now utilize the well known properties of mapping x → y α (x); see e.g. [14] and [11,Lemma 9.5]. (c) For any point x ∈ D it holds that ∇f (x), y α (x) − x ≤ −α y α (x) − x 2 . Therefore, the assumptions in (A3) are fulfilled with τ = α and we can obtain the convergence result for (GPMS) directly from Theorem 1. (7) holds. Observe that the choice α ≥ 1 allows us to take an arbitrary value β ∈ (0, 1) for convergence. The above method can be extended to the case where f (x) = µ(x) + η(x), where µ is continuously differentiable and η is convex, but non-differentiable. We have to replace the projection mapping with the proximal mapping with respect to η and apply the so-called splitting or proximal gradient iteration. A number of these splitting based descent methods were proposed for such composite non-smooth optimization problems; see e.g. [14,4,6,11,7]. Similarly, we can create new versions of splitting methods if we place the corresponding splitting direction finding mapping in the (SBM) framework. Application to Non-smooth Variational Inequality Problems In this section, we take a variational inequality whose underlying mapping is (strongly) monotone, but non-integrable and non-smooth in general. Given a convex set D in R n and a single-valued mapping G : D → R n , one can define the custom variational inequality (VI for short): Find x * ∈ D such that G(x * ), x − x * ≥ 0 ∀x ∈ D.(16) This problem has a great number of applications in different fields, the theory and methods for VIs are investigated very extensively and great advances were made in this field; e.g. see [4,15,16] and the references therein. We denote by D e the solution set of VI (16) and consider this problem under the following basic assumptions. (A1 ′′ ) D is a nonempty, closed, and convex set in R n , the mapping G : Y → R n satisfies the Lipschitz condition in a neighborhood of each point of an open convex set Y such that D ⊂ Y . Fix a number α > 0 and define the usual gap function ϕ α (x) = max y∈D { G(x), x − y − 0.5α x − y 2 };(17) e.g. see [4]. Then there exists a unique element y α (x) ∈ D such that ϕ α (x) = G(x), x − y α (x) − 0.5α x − y α (x) 2 , moreover, y α (x) = π D [x − α −1 G(x)] . Thus, we again have a single-valued mapping x → y α (x) on D. Then, setting y(x) = y α (x) and f = ϕ α in (SBM), we obtain a new version of the projective gap function method for problem (16). We call it (GFPMS) for brevity. We observe that the descent method with Armijo line-search was proposed for this non-smooth VI in [17]. First of all we replace VI (16) with the optimization problem min x∈D → ϕ α (x).(18) From the definition of the function ϕ α in (17) we can easily deduce that it is always nonnegative and that the optimal value in (18) is zero. We shall utilize the other known properties of the mapping x → y α (x); see [17,18]. Therefore, the optimization problem (18) is an equivalent re-formulation of VI (16). Although the gap function ϕ α is non-smooth and non-convex, it is locally Lipschitz on D and we can calculate its generalized gradient set; see [17,Lemma 4]. Lemma 3 Let the assumptions in (A1 ′′ ) be fulfilled. Then, at any point x ∈ D, there exists the generalized gradient set ∂ ↑ ϕ α (x) = G(x) − ∂ ↑ G(x) ⊤ − αI (y α (x) − x), where ∂ ↑ G(x) denotes the generalized Jacobian of G at x. We recall that a mapping G : X → R n is said to be (a) monotone if, for each pair of points x, y ∈ X, it holds that G(x) − G(y), x − y ≥ 0; (b) strongly monotone with constant τ > 0 if, for each pair of points x, y ∈ X, it holds that G(x) − G(y), x − y ≥ τ x − y 2 . Let us take the strong monotonicity assumption on the mapping G. (A4) The mapping G : D → R n is strongly monotone with constant τ > 0. The strong monotonicity enables us to obtain the desired coercivity and stationarity properties; see [17,Lemmas 5 and 6]. ϕ α (x) ≥ σ x − x * 2 ∀x ∈ D, where x * is a unique solution to VI (16); (c) For each point x ∈ D and for all elements V ∈ ∂ ↑ G(x) it holds that G(x) − (V ⊤ − αI)(y α (x) − x), y α (x) − x ≤ −τ y α (x) − x 2 . Therefore, the assumptions in (A2) and (A3) are fulfilled with f = ϕ α , besides, D * = D 0 = D e , and we can obtain the convergence result for (GFPMS) directly from Theorem 1. Theorem 3 Let assumptions (A1 ′′ ) and (A4) be fulfilled. If we apply (GFPMS) with β < τ , then the following assertions are true: (i) The termination of (GFPMS) yields a unique solution to VI (16). (ii) The sequence {x k } converges to a unique solution to VI (16). There exist various gap function based methods with line-search procedures for different classes of VIs; see e.g. [4,16,18,11]. Again, we can create new versions of these methods after the proper mapping substitution in (SBM). Computational Experiments In order to check the performance of the proposed methods we carried out computational experiments. The main goal was to compare them with the methods having the same direction mapping but utilizing the Armijo line-search. For more clarity, we describe now the corresponding modification of (SBM). Method (ABM). Step 0: Choose a point x 0 ∈ D γ , numbers β ∈ (0, 1) and θ ∈ (0, 1). Set k = 0. Step 1: Take a point y k = y(x k ). If y k = x k , stop. Otherwise set d k = y k − x k . Step 2: Determine m as the smallest nonnegative integer such that f (x k + θ m d k ) ≤ f (x k ) − βθ m d k 2 , set λ k = θ m , x k+1 = x k + λ k d k , k = k + 1 and go to Step 1. Various implementations of this method were investigated in many works; e.g. see [14,4,18,11] and the references therein. Its convergence properties are similar to those of the other known descent methods with line-search and it does not require a priori information. The presence of this line-search at each iteration is the main difference from (SBM). Also, we took for comparison the known non-monotone method with the divergent series step-size rule, which does not require line-search or a priori information; see e.g. [1,Chapters V and VII]. We compared all the methods for different dimensionality. They were implemented in Delphi with double precision arithmetic. Namely, we indicate the number of iterations (it) and the total number of calculations of the goal function value (kf) for attaining the same accuracy ε = 0.01 with respect to the error function ∆(x) = x − π D [x − y(x)] . We took θ = 0.5 for (ABM). For (SBM), we simply set λ k+1 = λ k if (6) holds, and λ k+1 = σλ k with σ = 0.9 otherwise. First we applied the methods to smooth convex optimization problems of form (3). More precisely, we chose f (x) = 0.5 P x − q 2 ,(19) the elements of the m × n matrix P were defined by p ij = sin(i) cos(j) if i = j, sin(i) cos(j) + 2 if i = j; and q i = n j=1 p ij , i = 1, . . . , m. We utilized the mapping y α (x) = π D [x − α −1 ∇f (x)] with α = 1 and (GPMS). Analogously, setting y(x) = y 1 (x) in (ABM), we obtain the well known gradient projection method with Armijo line-search. We call it (GPMA) for brevity. We set β = 0.5 for both the methods. We also implemented the gradient projection method with the divergent series step-size rule: x k+1 = π D [x k − λ k ∇f (x k )], λ k = 1/(k + 1), k = 0, 1, . . . ; In the first series, we took the feasible set D = R n + where R n + = {x ∈ R n : x j ≥ 0, j = 1, . . . , n} and the starting point x 0 j = n/2 + sin(j) for j = 1, . . . , n. The results are given in Table 1. (GPMD) showed very slow convergence when n > 5. For the case where m = 5 and n = 10, it attained only the accuracy 0.108 in 5000 iterations. For this reason, we made the further comparison only for (GPMA) and (GPMS). In the second series, we took the same cost function from (19), the feasible set D = {x ∈ R n : −5 ≤ x j ≤ 5, j = 1, . . . , n}, and the starting point x 0 j = −5 for j = 1, . . . , n. The results are given in Table 2. Next we applied the methods to variational inequality problems. We chose the nonlinear strongly monotone mapping in VI (16) as follows: G(x) = Ax + b + µC(x), A = A ′ + A ′′ , the elements of the n × n matrix A ′ were defined by a ′ ij =      sin(i) cos(j)/(i + j) if i < j, sin(j) cos(i)/(i + j) if i > j, s =i \|a ′ is \| + 2 if i = j; the elements of the n × n matrix A ′′ were defined by This means that the matrix A is positive definite and asymmetric. The parameter µ was set to be 10, the mapping C(x) was chosen to be diagonal with the elements C i (x) = arctan(x i − 2), i = 1, . . . , n. We utilized the mapping y α (x) = π D [x−α −1 G(x)] with α = 1 and (GFPMS). Analogously, setting y(x) = y 1 (x) in (ABM), we obtain the well known descent projection method with Armijo line-search. We call it (GFPMA) for brevity. We set β = 0.4 for both the methods. In the third series, we took the feasible set D = {x ∈ R n : 1 ≤ x j ≤ 6, j = 1, . . . , n}, and starting point x 0 j = 6 for j = 1, . . . , n. The results are given in Table 3. In almost all the cases, the implementations of (SBM), which do not use line-search, showed rather rapid convergence, they outperformed the implementations of (ABM) in the total number of goal function calculations. Conclusions We suggested a new simple adaptive step-size procedure in a general class of solution methods for optimization problems, whose goal function may be non-smooth and non-convex. This procedure does not require any line-search or a priori information, but takes into account behavior of the iteration sequence. Therefore, it reduces the implementation cost of each iteration essentially in comparison with the descent linesearch methods. We established convergence of the method under mild assumptions involving the usual coercivity condition. We showed that this new procedure yields in fact a general framework for optimization methods. In particular, a new gradient projection method for smooth constrained optimization problems and a new projection type method for minimization of the gap function of a general variational inequality can be obtained within this framework. The preliminary results of computational tests showed efficiency of the new procedure. Proposition 2 [ 10 , 210Theorem 2.3.7] Let x and y be given points in R n and let f : R n → R be a Lipschitz continuous function on an open set containing the segment [x, y]. Then there exists a point z ∈ (x, y) such that x ts = x * . Create the corresponding infinite subsequence {x m(ts) }. From (15) we have f * ≤ f (x ts ) ≤ f (x m(ts)), but all the limit points of the sequence {x m(ts) } belong to the set D * since it is contained in the sequence {x k l +1 }. Choose any limit pointx of {x m(ts) }. Then, taking a subsequence if necessary we obtain Lemma 1 1Let the assumptions in (A1 ′ ) be fulfilled. Then: (a)x = y α (x) if and only ifx ∈ D 0 ; (b) The mapping x → y α (x) is continuous on D; Theorem 2 2Let assumptions (A1 ′ ) and (A2) be fulfilled. If we apply (GPMS) with β < α, then the following assertions are true:(i) The termination of (GPMS) yields a point of D 0 .(ii) The sequence {x k } has a limit point, which belongs to the set D 0 .(iii) If D * = D 0 , then all the limit points of the sequence {x k } belong to the set D * and Lemma 2 2Let the assumptions in (A1 ′′ ) be fulfilled. Then: (a) VI (16) is equivalent to problem (18); (b)x = y α (x) if and only ifx ∈ D e ; (c) The mapping x → y α (x) is continuous on D. Lemma 4 4Let the assumptions in (A1 ′′ ) and (A4) be fulfilled. Then: (a) VI (16) has a unique solution; (b) There exists a number σ > 0 such that ij , i = 1, . . . , n. Table 1 : 1Convex optimization: Test 1 (it is the number of iterations, kf is the number of function calculations)(GPMA) (GPMS) (GPMD) m n it kf it kf it kf 2 5 4 14 12 21 17 18 4 5 15 57 30 35 40 41 5 10 18 76 28 47 - - 25 50 344 2683 637 679 - - 50 100 1229 12025 2633 2689 - - Table 2 : 2Convex optimization: Test 2 (it is the number of iterations, kf is the number of function calculations) see e.g. [1, Section 7.2.2]. We call it (GPMD) for brevity.(GPMA) (GPMS) m n it kf it kf 2 5 4 24 14 21 4 5 17 65 35 38 5 10 19 80 60 66 25 50 225 1778 440 463 50 100 748 7445 1624 1660 Table 3 : 3Variational inequality: Test 3 (it is the number of iterations, kf is the number of function calculations)(GFPMA) (GFPMS) n it kf it kf 5 4 14 20 26 10 8 23 21 27 20 14 48 40 45 50 47 161 48 53 100 85 320 92 97 200 148 660 145 150 500 375 2143 345 351 1000 761 5076 708 716 AcknowledgementThe results of this work were obtained within the state assignment of the Ministry of Science and Education of Russia, project No. 1.460.2016/1.4. In this work, the author was also supported by Russian Foundation for Basic Research, project No. 16-01-00109a. Introduction to Optimization. B T Polyak, Engl. transl. in Optimization Software. Moscow; New YorkNaukaPolyak, B.T.: Introduction to Optimization. Nauka, Moscow (1983) [Engl. transl. in Optimization Software, New York (1987)] D P Bertsekas, Nonlinear Programming. 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Sci. 2, 183-202 (2009) Convex optimization for big data. V Cevher, S Becker, M Schmidt, Signal Process. Magaz. 31Cevher, V., Becker, S., Schmidt, M.: Convex optimization for big data. Signal Process. Magaz. 31, 32-43 (2014) Decomposition by partial linearization: parallel optimization of multi-agent systems. G Scutari, F Facchinei, P Song, D P Palomar, J.-S Pang, IEEE Trans. Signal Process. 62Scutari, G., Facchinei, F., Song, P., Palomar, D.P., Pang, J.-S.: Decomposition by partial linearization: parallel optimization of multi-agent systems. IEEE Trans. Signal Process. 62, 641-656 (2014) Conditional gradient method without line-search. I V Konnov, Russ. Mathem. (Izv. VUZ). 62Konnov, I.V.: Conditional gradient method without line-search. Russ. Mathem. (Izv. VUZ). 62, 82-85 (2018) Optimization and Nonsmooth Analysis. F H Clarke, John Wiley and SonsNew YorkClarke, F.H.: Optimization and Nonsmooth Analysis. John Wiley and Sons, New York (1983) Nonlinear Optimization and Variational Inequalities. I V Konnov, Kazan Univ. PressKazanIn RussianKonnov, I.V.: Nonlinear Optimization and Variational Inequalities. Kazan Univ. Press, Kazan (2013) [In Russian] Semismooth and semiconvex functions in constrained optimization. R Mifflin, SIAM J. Contr. Optim. 15Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Contr. Optim. 15, 959-972 (1977) H Nikaido, Convex Structures and Economic Theory. New YorkAcademic PressNikaido, H.: Convex Structures and Economic Theory. Academic Press, New York (1968) Cost approximation: a unified framework of descent algorithms for nonlinear programs. M Patriksson, SIAM J. Optim. 8Patriksson, M.: Cost approximation: a unified framework of descent algorithms for nonlinear programs. SIAM J. Optim. 8, 561-582 (1998) Combined Relaxation Methods for Variational Inequalities. I V Konnov, Springer-VerlagBerlinKonnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer-Verlag, Berlin (2001) F Facchinei, J.-S Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. BerlinSpringer-VerlagFacchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Com- plementarity Problems. Springer-Verlag, Berlin (2003) Descent method for nonsmooth variational inequalities. I V Konnov, Comp. Maths and Math. Phys. 46Konnov, I.V.: Descent method for nonsmooth variational inequalities. Comp. Maths and Math. Phys. 46, 1286-1292 (2006) Descent methods for mixed variational inequalities with non-smooth mappings. I V Konnov, Optimization Theory and Related Topics. Contemporary Mathematics. Reich, S. and Zaslavski, A.J.ProvidenceAmer. Math. Soc568Konnov, I.V.: Descent methods for mixed variational inequalities with non-smooth mappings. In: Reich, S. and Zaslavski, A.J. (eds.) Optimization Theory and Re- lated Topics. 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[ "ISM PROCESSING IN THE INNER 20 PC IN GALACTIC CENTER", "ISM PROCESSING IN THE INNER 20 PC IN GALACTIC CENTER" ]	[ "Hauyu Baobab Liu \nAcademia Sinica Institute of Astronomy and Astrophysics\nP.O. Box 23-141106TaipeiTaiwan\n", "Paul T P Ho \nAcademia Sinica Institute of Astronomy and Astrophysics\nP.O. Box 23-141106TaipeiTaiwan\n\nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street02138CambridgeMA\n", "Melvyn C H Wright \nRadio Astronomy Laboratory\nUniversity of California\n601, 94720Berkeley, Campbell Hall, BerkeleyCAUSA\n", "Yu-Nung Su \nAcademia Sinica Institute of Astronomy and Astrophysics\nP.O. Box 23-141106TaipeiTaiwan\n", "Pei-Ying Hsieh \nAcademia Sinica Institute of Astronomy and Astrophysics\nP.O. Box 23-141106TaipeiTaiwan\n\nGraduate Institute of Astronomy\nNational Central University\nNo. 300, Jhongda RdJhongli City\n\nTaoyuan County\n32001Taiwan (R.O.C.\n", "Ai-Lei Sun \nDepartment of Astrophysical Sciences\nPrinceton University\nPeyton Hall08544PrincetonNJUSA\n", "Sungsoo S Kim \nDept. of Astronomy and Space Science\nKyung Hee University\nYongin-shi, Kyungki-do 446-701Korea\n", "Young Chol Minh \nKorea Astronomy and Space Science Institute (KASI)\n776 Daeduk-daero305-348Yuseong, DaejeonKorea\n" ]	[ "Academia Sinica Institute of Astronomy and Astrophysics\nP.O. Box 23-141106TaipeiTaiwan", "Academia Sinica Institute of Astronomy and Astrophysics\nP.O. Box 23-141106TaipeiTaiwan", "Harvard-Smithsonian Center for Astrophysics\n60 Garden Street02138CambridgeMA", "Radio Astronomy Laboratory\nUniversity of California\n601, 94720Berkeley, Campbell Hall, BerkeleyCAUSA", "Academia Sinica Institute of Astronomy and Astrophysics\nP.O. Box 23-141106TaipeiTaiwan", "Academia Sinica Institute of Astronomy and Astrophysics\nP.O. Box 23-141106TaipeiTaiwan", "Graduate Institute of Astronomy\nNational Central University\nNo. 300, Jhongda RdJhongli City", "Taoyuan County\n32001Taiwan (R.O.C.", "Department of Astrophysical Sciences\nPrinceton University\nPeyton Hall08544PrincetonNJUSA", "Dept. of Astronomy and Space Science\nKyung Hee University\nYongin-shi, Kyungki-do 446-701Korea", "Korea Astronomy and Space Science Institute (KASI)\n776 Daeduk-daero305-348Yuseong, DaejeonKorea" ]	[]	We present the Submillimeter Array (SMA) 157-pointing mosaic in 0.86 mm dust continuum emission with 5 ′′ .1×4 ′′ .2 angular resolution, and the NRAO Green Bank 100m Telescope (GBT) observations of the CS/C 34 S/ 13 CS 1-0 and SiO 1-0 emission with ≤20 ′′ ×18 ′′ angular resolution. The dust continuum image marginally resolves at least several tens of 10-10 2 M ⊙ dense clumps in the 5 ′ field including the circumnuclear disk (CND) and the exterior gas streamers. There is very good agreement between the high resolution dust continuum map of the CND and all previous molecular line observations. As the dust emission is the most reliable optically thin tracer of the mass, free from most chemical and excitation effects, we demonstrate the reality of the abundant localized structures within the CND, and their connection to external gas structures. From the spectral line data, the velocity dispersions of the dense clumps and their parent molecular clouds are ∼10-20 times higher than their virial velocity dispersions. This supports the idea that the CND and its immediate environment may not be stationary or stable structures. Some of the dense gas clumps are associated with 22 GHz water masers and 36.2 GHz and 44.1 GHz CH 3 OH masers. However, we do not find clumps which are bound by the gravity of the enclosed molecular gas. Hence, the CH 3 OH or H 2 O maser emission may be due to strong (proto)stellar feedback, which may be dispersing some of the gas clumps.	10.1088/0004-637x/770/1/44	[ "https://arxiv.org/pdf/1304.7573v1.pdf" ]	119,108,060	1304.7573	476c1982eed653a1114760dd9083beb7187d3de0	ISM PROCESSING IN THE INNER 20 PC IN GALACTIC CENTER 29 Apr 2013 Hauyu Baobab Liu Academia Sinica Institute of Astronomy and Astrophysics P.O. Box 23-141106TaipeiTaiwan Paul T P Ho Academia Sinica Institute of Astronomy and Astrophysics P.O. Box 23-141106TaipeiTaiwan Harvard-Smithsonian Center for Astrophysics 60 Garden Street02138CambridgeMA Melvyn C H Wright Radio Astronomy Laboratory University of California 601, 94720Berkeley, Campbell Hall, BerkeleyCAUSA Yu-Nung Su Academia Sinica Institute of Astronomy and Astrophysics P.O. Box 23-141106TaipeiTaiwan Pei-Ying Hsieh Academia Sinica Institute of Astronomy and Astrophysics P.O. Box 23-141106TaipeiTaiwan Graduate Institute of Astronomy National Central University No. 300, Jhongda RdJhongli City Taoyuan County 32001Taiwan (R.O.C. Ai-Lei Sun Department of Astrophysical Sciences Princeton University Peyton Hall08544PrincetonNJUSA Sungsoo S Kim Dept. of Astronomy and Space Science Kyung Hee University Yongin-shi, Kyungki-do 446-701Korea Young Chol Minh Korea Astronomy and Space Science Institute (KASI) 776 Daeduk-daero305-348Yuseong, DaejeonKorea ISM PROCESSING IN THE INNER 20 PC IN GALACTIC CENTER 29 Apr 2013ApJ Accepted; 2013 April 26 ApJ Accepted; 2013 April 26arXiv:1304.7573v1 [astro-ph.GA] Preprint typeset using L A T E X style emulateapj v. 5/2/11Subject headings: Galaxy: center -Galaxy: structure -Galaxy: kinematics and dynamics -ISM: clouds We present the Submillimeter Array (SMA) 157-pointing mosaic in 0.86 mm dust continuum emission with 5 ′′ .1×4 ′′ .2 angular resolution, and the NRAO Green Bank 100m Telescope (GBT) observations of the CS/C 34 S/ 13 CS 1-0 and SiO 1-0 emission with ≤20 ′′ ×18 ′′ angular resolution. The dust continuum image marginally resolves at least several tens of 10-10 2 M ⊙ dense clumps in the 5 ′ field including the circumnuclear disk (CND) and the exterior gas streamers. There is very good agreement between the high resolution dust continuum map of the CND and all previous molecular line observations. As the dust emission is the most reliable optically thin tracer of the mass, free from most chemical and excitation effects, we demonstrate the reality of the abundant localized structures within the CND, and their connection to external gas structures. From the spectral line data, the velocity dispersions of the dense clumps and their parent molecular clouds are ∼10-20 times higher than their virial velocity dispersions. This supports the idea that the CND and its immediate environment may not be stationary or stable structures. Some of the dense gas clumps are associated with 22 GHz water masers and 36.2 GHz and 44.1 GHz CH 3 OH masers. However, we do not find clumps which are bound by the gravity of the enclosed molecular gas. Hence, the CH 3 OH or H 2 O maser emission may be due to strong (proto)stellar feedback, which may be dispersing some of the gas clumps. INTRODUCTION The Galactic center (see Morris &Serabyn 1996 andMezger et al. 1996 for reviews) is a fertile ground for studying the interplay between the supermassive black hole (SMBH; see Genzel et al. 1997 andGhez et al. 2005) and the surrounding interstellar medium (ISM). The overall dynamics, the clumpiness, and the local kinematics may provide clues as to how the SMBH is fed by the ISM, how the circumnuclear gas streams evolve, and how the molecular cores and the young massive-stellar objects (Krabbe et al. 1991;Launhardt et al. 2002;Pfuhl et al. 2011 form and migrate in the Galactic center. Previous molecular line observations indicated that the warm and turbulent gas clouds or streamers in the central ∼20 pc in Galactic center (Ho et al. 1991;Okumura 1991;Coil & Ho 1999McGary et al. 2001;Herrnstein & Ho 2002Oka et al. 2011) are well connected with the 2-4 pc circumnuclear disk (CND; Güsten et al. 1987a;Jackson et al. 1993;Marshall et al. 1995;Christopher et al. 2005;Montero-Castaño et al. 2009;Liu et al. 2012;Martín et al. 2012) surrounding the central black hole. These gas structures are the most extreme environments for highmass star formation in the Milky Way (Morris 1993 and references therein). The thermal dust continuum emission is the least [email protected] ased by the chemical abundances and the excitation conditions. Hence, it is the most reliable tracer for demonstrating the reality and significance of the individual gas clouds or streamers, as well as their internal structures. The IRAM-30m and the JCMT single dish telescope observations of the millimeter and the submillimeter continuum emission have been presented by Zylka & Mezger (1988), Mezger et al. (1989), Dent et al. (1993), Lis & Carlstrom (1994), Pierce-Price et al. (2000), and García-Marín et al. (2011). In this work, we report the first wide-field interferometric mosaic observations (157 pointings, ∼5 ′ ×5 ′ field of view) of the 0.86 mm dust continuum emission using the Submillimeter Array (SMA 1 ; Ho et al. 2004). The improved angular resolution of ∼5 ′′ permits the detection of 0.1-0.2 parsec scale gas clumps, which can be the candidates of high-mass star-forming cores. In addition, we compare the dust continuum image with the HCO + 4-3 line image simultaneously obtained with the SMA. We also compare these maps with the National Radio Astronomy Observatory (NRAO 2 ) Robert C. Byrd Green Bank Telescope (GBT) observa-tions of the CS/C 34 S/ 13 CS 1-0 lines and the SiO 1-0 lines. This allows us to examine the gravitational stabilities of the dense structures and the possible existence of shock fronts (e.g. Kauffmann et al. 2013). The new observations are described in Section 2. The results are presented in Section 3. We compare our observations with the previous observations of OH, CH 3 OH, and H 2 O masers in Section 4. A discussion of the non uniformity of the CND based on the 0.86 mm dust continuum image is given in Section 4.4. A brief summary of our results is given in Section 5. We made mosaic observations toward the Galactic center using the SMA, in its compact and subcompact array configurations. These observations covered the frequency range of 354.1-358.1 GHz in the upper sideband, and 342.1-346.1 GHz in the lower sideband. Details of the subcompact array observations, and the pointing centers of the observations, can be found in Liu et al. (2012). The compact array observations were made in two observing runs, on 2012 May 07 and 2012 May 20, with 6 and 7 available antennas, respectively. The system temperatures T sys during these two runs were ∼180-400 K. We observed two phase calibrators, 1733-130 and 1924-292, every ∼15 minutes during all observations. The amplitude and passband calibrators were Titan and 3C 279 on May 07, Neptune and 3C 279 on May 20. The target loops iterated over 157 pointing centers, with 5 5-second integrations at each pointing center. Each of the 157 pointings was visited more than three times in each observing run (i.e. on source time >3 hours in total in each run). The minimum and maximum projected baselines in our SMA observations are ∼7.0 kλ and ∼82 kλ. All SMA data were calibrated using the MIR IDL software package (Qi 2003). We used the MIRIAD (Sault 1995) task UVAVER to average all line-free channel data and reconstruct the 0.86 mm continuum band data. The short spacing data were complemented by combining the archival JCMT SCUBA image (Appendix A, B). The zeroth order free-free continuum emission model was constructed based on the archival VLA 7 mm observation data, and was subtracted from the combined SMA+JCMT 0.86 mm continuum image (Appendix C). The simultaneously observed HCO + 4-3 line was regrided to 2.1 km s −1 velocity channels for an adequate sensitivity, and will be presented for the purpose of discussing the virial condition. The detailed studies of the submillimeter line data are deferred. GBT Observations We observed the CS 1-0 (48.99095 GHz) and the C 34 S 1-0 (48.20692 GHz) transitions using the NRAO GBT on 2011 November 04 and 07. We observed the 13 CS 1-0 (46.24754 GHz) and the SiO 1-0 (43.42376 GHz) transitions on 2011 November 07 and 09. The field of view of the SiO and 13 CS observations is slightly offset toward the west to better recover the western gas streamers. The angular resolution of the GBT is 763.8 ′′ /ν, where ν is the observing frequency in GHz. The bright point source 1733-130 was observed in the beginning of each session for antenna surface (i.e. by Out of Focus Holography) and pointing calibrations. A line-free reference position RA: 17 h 43 m 43 s .344, Decl.: -29 • 59 ′ 32 ′′ .27 was integrated for 30 seconds before and after the target observations in each block for off-source calibration data. We used the GBTIDL software package (Marganian et al. 2006) to calibrate the GBT data. We note that the CS isotopologue lines are close to the band edge of the GBT, which are subjected to the higher system temperature. We used the AIPS software package to perform imaging. We smoothed the final CS and C 34 S image to an optimized θ maj ×θ min = 20 ′′ ×18 ′′ , and BPA. = 0 • to suppress the striping defects due to the sampling rates, and removed some obvious stripes by fitting zeroth order polynomial. The achieved RMS noise levels in each 24 kHz (∼0.15 km s −1 at 48.99 GHz) spectral channel are ∼0.5 K for the CS and C 34 S observations, and are ∼0.14 K for the 13 CS and SiO observations. RESULTS We present the observing results in this section. Our nomenclature follows Christopher et al. (2005), Amo-Baladrón et al. (2011), andLiu et al. (2012). We also compare our results with the VLA observations of the 1612 MHz OH masers see also Sjouwerman 1998), the VLA observations of the 1720 MHz OH masers , the GBT and VLA observations of the 44.1 GHz Class I CH 3 OH masers (Yusef-Zadeh 2008), the JVLA observations of the 44.1 GHz Class I CH 3 OH masers (Pihlström et al. 2011), the JVLA observations of the 36.2 GHz Class I CH 3 OH masers (Sjouwerman 2010), and the GBT observations of the 22 GHz H 2 O masers (Yusef-Zadeh 2008). Figure 1 shows the high angular resolution SMA+JCMT 0.86 mm continuum image. A model of the free-free emission has been subtracted (see Appendix C). The 0.86 mm continuum image recovers the detailed gas structures in the CND including the Northeast Lobe, the Northeast Extension, the Southwest Lobe, the Southern Extension, and the W-2,3,4 Streamers which connect to the CND from the west. It addition, it presents the detailed structures embedded in the Northern Ridge, the Southern Ridge, the 50 km s −1 cloud, the Molecular Ridge, the 20 km s −1 cloud. A southern dust ridge which appears to connect JCMT-1 to JCMT-2 (see Figure 1 ) in the archival JCMT SCUBA 0.44 mm (678 GHz) continuum image published by Pierce-Price et al. (2000;ProjectID: M98AU64), is also resolved in our SMA+JCMT image (see Figure 2). The spatial location of this southern dust ridge coincides with the Southern Arc reported by the previous observations of the CS 1-0 line emission (Liu et al. 2012). Continuum Emission Excluding the central point source, the 0.86 mm flux of the CND is approximately 68 Jy in a distribution that closely follows that of the inner 5 pc of the CND. At least few tens of dense molecular gas clumps are marginally resolved. Higher angular resolution observations may resolve more blended dense clumps in this field. The precise number of clumps is not important for this discussion. The main conclusion is the impression of a very The angular resolution of this image is θ maj × θ min =5 ′′ .1×4 ′′ .2. Contour spacings are 3σ starting at 3σ (σ=24 mJy beam −1 ). Yellow crosses are the 1720 MHz OH masers taken from . Yellow triangles are the compact, either thermal or low-gain masing 1612 MHz OH line sources discussed in Pihlström et al. (2008) (see also Sjouwerman 1998). Pink diamonds are the 36.2 GHz Class I CH 3 OH masers reported in Sjouwerman (2010). Yellow diamonds are the 44.1 GHz Class I CH 3 OH masers reported in Pihlström et al. (2011). White Crosses and diamonds are the 22 GHz water masers and the 44.1 GHz CH 3 OH masers reported in Yusef-Zadeh (2008). Note the symbol size is not representative to the spatial uncertainty. For our convenience in discussion, we mark the peaks above the 45σ significance level besides the Northeast Lobe and Southwest Lobe as JCMT-1,2,3 (see Appendix A). The regions associated with CH 3 OH masers are labeled by A-H. clumpy structure. The most significant dense clumps are found in the protrusion connecting to the Southwest Lobe. Clusters of very dense clumps are also resolved towards the peaks of the JCMT SCUBA 0.86 mm image (e.g. JCMT-1,2,3 and the peaks in the Northern Ridge; see Appendix A). In the present work, only the dense clumps located near the CH 3 OH masers, and additionally the dense clumps located in JCMT-1 and JCMT-2 are discussed ( Figure 3). We do not attempt to systematically search for and analyze all dense clumps in the entire field because of the limited image quality, and insufficient angular resolution for the CND (see Montero-Castaño et al. 2009 andMartín et al. 2012 for higher angular resolution spectral line images). To estimate the 0.86 mm fluxes of the selected clumps, we first fit ∼1 pc scale 2D Gaussian components to suppress the contribution of the ambient gas, and then fit 2D Gaussian components to the localized peaks. The 0.86 mm fluxes of the individual Gaussian components are summarized in Table 1, which provides a quantitative measure of how massive the subparsec scale gas over-densities may be. The uncertainties in background subtraction can give a few tens of percent of errors in fluxes. In addition, the overestimates of the background emission can lead to underes-timates of the clump sizescale. The main purpose of this analysis is to estimate the virial velocity dispersions of these dense clumps (Section 4.3). In this sense, the effects of overestimates of the background and underestimates of the clumps sizescale are competing. Gaussian components with minor axis FWHM smaller than one standard deviation of the SMA+JCMT Gaussian beam width, 5 ′′ .2/2.355 = 2.2 ′′ are not considered because we cannot estimate the physical deconvolved size scales. The Gaussian components which are located near the edge of the SMA+JCMT 0.86 mm continuum image are also not considered. Molecular Emission The SMA image of HCO + 4-3 is shown in Figure 4, over-plotted with the fitted 0.86 mm dense clumps (Section 3.1; Figure 3). The HCO + 4-3 line is a good tracer of the ring-like CND, and also the gas streamers connecting to the CND. The eastern part of the CND is relatively narrow and smooth, compared to the western part. The CND may be undergoing dynamical evolution, or is composed of distinct streams of molecular gas (Jackson et al. 1993;Wright et al. 2001;see Section 4.4). Towards the more extended 20 km s −1 cloud, the Southern Arc, the Molecular Ridge and the 50 km s −1 cloud, the SMA observations are detecting strong missing flux (see Liu et al. 2012). In addition, the lower gas temperature in the extended cloud than in the CND also causes the weak or non-detection of the warm gas tracer HCO + 4-3 (E up = 43 K). We therefore can only robustly image the HCO + 4-3 line towards a few fitted dense clumps. The detected HCO + 4-3 spectra are presented in Figure 5. The velocity dispersions derived from the single component Gaussian fittings are summarized in Table 1. From Figure 3 and 5 we see that the brightest clump over the selected samples (Table 1), the JCMT-1n, has a centroid velocity of -95 km s −1 . The JCMT-1n clump is likely to be embedded in the very blueshifted Southern Ridge, which shows strong SiO 2-1 emission in the earlier Nobeyama Millimetre Array observations (Sato & Tsuboi 2008, and also see Amo-Baladrón et al. 2011). However, the CS 1-0 line spectrum taken at JCMT-1n indicates that a v lsr ∼7 km s −1 gas component is completely missed from the SMA observations of HCO + 4-3 (ref. spectrum 10 in Liu et al. 2012). The virial velocity of JCMT-1n derived from the 0.86 mm dust continuum emission (Table 1) then should be considered as an upper limit when comparing with the HCO + 4-3 velocity dispersion. The more extended gas streamers are recovered by the GBT observations of the CS/C 34 S/ 13 CS 1-0 lines and the SiO 1-0 line ( Figure 6), which trace cooler gas (E up = 2.1-2.3 K). The SiO 1-0 emission appears to be correlated with the CS 1-0 emission. The flux ratio of these lines will be discussed in Section 4.1. DISCUSSION We discuss the inferred ISM properties based on the presented observations. The Spectral Line Ratio While the abundance of the CS molecule is only mildly enhanced in UV and shocked environments (Amo- Baladrón et al. 2011 and references therein), the abundance of the SiO molecule in shocked environments can be enhanced by up to a factor of 10 6 with respect to the value in quiescent gas (Martin-Pintado et al. 1992). Since SiO and the CS isotopologues have similar dipole moments and energy level distributions, the derived flux ratio between these molecules can trace shocks without being sensitive to the assumed physical conditions. Figure 7 shows the velocity integrated CS/C 34 S 1-0 line ratio. Above ∼40 GHz, difficulties in calibrating the several arcsecond GBT pointing offsets can potentially lead to the ∼30% uncertainty in absolute flux level. This uncertainty does not affect the derived CS/C 34 S 1-0 ratio since these two lines are simultaneously observed. We found that the CS/C 34 S 1-0 ratio is ∼6-12 towards the 50 km s −1 cloud and the 20 km s −1 cloud. The earlier NRO 45m Telescope observations of CS 1-0 and C 34 S 1-0 only robustly detected the C 34 S 1-0 emission at the reference point located 3 ′ north and 3 ′ east of the Sgr A, and showed a CS/C 34 S 1-0 ratio of 8 (Tsuboi et al. 1999). Considering that the NRO 45m Telescope observations were more beam smoothed, our results are consistent with the previous observations. Assuming the abundance ratio of [X(CS)/X(C 34 S)] = 22.6 (Frerking 1980;Tsuboi et al. 1999), the optical depth τ of CS 1-0 can be estimated based on the following relation Table 1). These spectra are regridded to 2.1 km s −1 velocity channels. The rms noise level is 0.6 Jy beam −1 (1 Jy beam −1 ∼0.4 K). Blue lines present the fitted Gaussian components. S CS S C 34 S = 1 − exp(−τ ) 1 − exp(−τ /22.6) ,(1) where S CS and S C 34 S are the fluxes of the CS and the C 34 S lines. The line ratio S CS /S C 34 S =14.2 when τ =1. From Figure 7, the majority of the optically thick CS 1-0 line (τ =1.5-4, i.e. CS/C 34 S 1-0 ratio ∼6-12) is seen towards the 50 km s −1 cloud and the 20 km s −1 cloud. The C 34 S 1-0 line is optically thin over the observed field. The SiO 1-0 line is ∼1.5-2 times fainter than the CS 1-0 line ( Figure 6). Assuming the optically thin limit, the velocity integrated SiO/C 34 S 1-0 line ratio ( Figure 8) traces the abundance ratio. We present the velocity channel maps of the SiO/C 34 S 1-0 line ratio in Figure 9 to trace the velocity structures of the shocked gas. While the SiO/C 34 S 1-0 ratio is already high in the entire map area, it appears to be enhanced in the 20 km s −1 cloud and the Southern Arc. Based on the correlation between NH 3 and 1.2 mm dust emission and the locations of OH masers, Wright et al. (2001) also suggested that these regions could be shock heated. Our observations do not have sufficient angular resolution to resolve the shock fronts. Whether the shocks are created by the supernova shell to the south of the Sgr A East, or are created due to the non-circular orbits of the clouds, are not yet distinguished. The earlier IRAM-30 m Telescope observations additionally showed the enhanced SiO emission toward the 50 km s −1 cloud and around the Southern Ridge and the W-4 Streamer. Our GBT observations of SiO 1-0 only covered a small part of the 50 km s −1 cloud, and did not significantly detect the Southern Ridge. The W-4 Streamer is only marginally detected in our C 34 S 1-0 observations. The Molecular Gas Environment Around the Masers The 1720 MHz OH masers and the 1612 MHz OH masers trace the n H 2 ∼10 5 cm −3 and n H 2 10 7 cm −3 postshock gas in supernova remnants (assuming T∼75 K; Pihlström et al. 2008;Pavlakis & Kylafis 1996). The 22 GHz H 2 O masers are collisionally excited at higher densities (n H 2 ∼10 7−9 cm −3 ; Elitzur et al. 1992), and are often found in star-forming regions. The collisionally excited Class I CH 3 OH masers are regarded as the unambiguous signposts of ongoing high-mass star formation in typical molecular clouds (Menten 1992;Yusef-Zadeh et al. 2008). However, the Galactic center (e.g. inner 2-20 pc) gas streamers are warmer and 10 times more turbulent than the other star-forming molecular clouds. In such extreme environments, the Class I CH 3 OH masers may also be excited in cloud interactions although not yet observationally confirmed (Sjouwerman et al. 2010). From Figure 13, we can see that on ∼10 pc scales, while the distribution of the OH masers follows a shell like geometry, the CH 3 OH and H 2 O masers are more scattered over the field. In the higher angular resolution dust continuum image (Figure 1), we see that the CH 3 OH and H 2 O masers are preferentially detected near the 0.5 pc scale localized over-densities. The only exceptions are the CH 3 OH masers associated with the region H which is located near the central bright free-free continuum sources. The flux at region H may be biased by the uncertainties in interferometric imaging and the free-free model subtraction (Appendix C). In addition, the molecular gas in region H may be photo-ionized by The velocity integrated C 34 S 1-0 emission (grayscale and black contour). Contour spacings are 2σ starting at 2σ (σ ∼0.7 K km s −1 ). (C) The velocity integrated SiO 1-0 emission (grayscale and black contour). Contour spacings are 10σ starting at 10σ (σ ∼0.15 K km s −1 ). (D) The velocity integrated 13 CS 1-0 emission (grayscale and black contour). Contour spacings are 4σ starting at 4σ (σ ∼0.15 K km s −1 ). The few 13 CS peaks at R.A.=17 h 45 m 34-35 s may be caused by ambiguities in baseline subtraction. The rms noise level of the velocity integrated images are estimated based on the integration of the signal over a 13 km s −1 velocity range, which is the median of the fitted velocity dispersion (see Table 1). Green contours in panels (B) and (D) show the free-free model subtracted SMA+JCMT 0.86 mm continuum image. The 0.86 mm continuum image contour spacings are 7.5σ starting at 7.5σ (σ=24 mJy beam −1 ). Blue star labels the location of Sgr A. We summarize the observed intensity of the dust continuum emission and the velocity integrated SiO/C 34 S 1-0 ratio in Figure 10. Except for the 44.1 GHz CH 3 OH masers associated with region H, the other CH 3 OH and H 2 O masers are detected at intensities 0.15 Jy beam −1 . The OH masers do not show a clear correlation with the flux of the dust continuum emission. The clusters of OH maser may only sparsely sample the geometrically thin expanding shell (Coil & Ho 1999) in projection of the bulk of the dense gas. Yusef-Zadeh et al. (2008) and Pihlström et al.(2011) only observed H 2 O and CH 3 OH masers towards selected high density regions, which potentially bias the corresponding maser data in Figure 10 toward higher 'averaged' 0.86 mm intensity. Nevertheless, the size scales of the dense clumps in Figure 1 are generally smaller than what can be discerned by previous observations of molecular gas. We therefore do not think the spatial correlation between the clumps and the maser spots is purely an artifact due to the observational selection. As an example, the good correlation between the 0.86 mm emission clumps in region G (Figure 1, i.e. pointing E in Pihlström et al. 2011) and the JVLA detections of 44.1 GHz CH 3 OH maser spots does not seem Fig. 8.-Velocity integrated SiO/C 34 S 1-0 ratio (color contours). The yellow, orange, and red contour levels are 2, 4, and 6, respectively. Gray contours show the free-free model subtracted SMA+JCMT 0.86 mm continuum image. The velocity integrated SiO 1-0 emission is shown in grayscale. The 0.86 mm continuum image contour spacings are 7.5σ starting at 7.5σ (σ=24 mJy beam −1 ). The GBT images are smoothed to the angular resolution of θ maj × θ min =20 ′′ ×18 ′′ before taking the ratio. Blue star labels the location of Sgr A. to be a coincidence, but should be interpreted with the local physical conditions. For both the maser sources and the ambient gas, our observations do not resolve a clear correlation between the [X(SiO)/X(C 34 S)] abundance ratio and the 0.86 mm intensity ( Figure 10). On the large (e.g. >1 pc) scale, the shocked gas may be well mixed with the ambient material. The enhanced abundance of SiO in shocks last for a few 10 3 years. With the ∼50-100 km s −1 relative motion of gas clouds around the CND, the smeared local SiO shock fronts may have widths of 0.15-0.3 pc (i.e. 3 ′′ .8-7 ′′ .5), which have to be examined with sensitive higher angular resolution observations. Here we refer to Martín et al. (2012) for higher angular resolution observations of SiO emission in the 2 ′ field around the Sgr A. The velocity information is crucial for understanding whether and how the shock can induce the formation of sub-parsec scale dense gas structures. The Dense Molecular Clumps The brightness temperature of the dust emission at the 0.86 mm wavelength (Appendix B) is much lower than the gas temperature (see Herrnstein & Ho 2005). The molecular gas mass can therefore be calculated from the 0.86 mm flux based on the optically thin formula M H2 = 2λ 3 RaρD 2 3hcQ(λ)J(λ, T d ) S(λ),(2) where R is the gas-to-dust mass ratio, a is the mean grain radius, ρ is the mean grain density, D is the distance of the target, Q(λ)∝λ −β is the grain emissivity, T d is the dust temperature, S(λ) is the flux of the dust emission at the given wavelength, J(λ, (Hildebrand 1983;Lis et al. 1998). The c, h, and k B are the light speed, the Planck constant, and the Boltzmann constant, respectively. Following Lis at al. (1998), we adopt R=100, a=0.1 µm, ρ=3 g cm −3 , Q(λ=350 µm)=1×10 −4 . We adopt D=8.33 kpc based on the measurements of Gillessen et al. (2009). The gas temperature measurements which have a comparable angular resolution with our dust continuum map are not yet available. By measuring the rotational temperature of the NH 3 molecule, Herrnstein & Ho (2005) suggested that roughly one quarter of the molecular gas comprises a hot (∼200 K) component, and the remaining gas is cool (∼25 K). We tentatively assumed an averaged gas temperature T gas =70 K in the dense clump. We also assumed the dust temperature T d =T gas (see Chan et al. 1997 for dust temperature in the CND; see also Minh et al.1992, Ao et al. 2012). The ranges of gas mass in the dense clump estimated based on the assumption of β=1-2 is given in Table 1. Based on the aforementioned assumptions, without considering the foreground/background subtractions, the detected 0.86 mm flux in the inner 5 pc CND (68 Jy, see Section 3.1) corresponds to 0.98-2.3×10 4 M ⊙ of gas mass. Our estimates of the CND mass agree reasonably well with those from earlier observations of millimeter and submillimeter dust continuum emission (e.g. Mezger et al. 1989; see also Christopher et al. 2005 for relevant debates). T d ) = 1/[exp(hc/λk B T d ) − 1] If the identified gas dense clumps are virialized, the expected one-dimensional velocity dispersion δv viriral is given by δv viriral = αM G 5δ r ,(3) where α is the geometric factor which equals to unity for a uniform density profile and 5/3 for an inverse square profile (Williams, de Geus & Blitz 1994;Walsh et al. 2007), M is the gas mass, G is the gravitational constant, and δ r is the effective radius of the dense clump. The effective radius δ r can be calculated based on the FWHM of the fitted 2D Gaussian (i.e. δθ maj and δθ min in Table 1), however, is necessary to be corrected for the beam FWHM δθ beam because the identified clumps are only marginally resolved. For simplicity, we adopt the corrected effective circular angular diameter of the dense clump to estimate their linear diameter (c.f. Williams, Note.-The δθ maj and δθ min are the FWHM of the fitted 2D Gaussians. The 1σ velocity dispersion δv CS is measured in a smoothed 20 ′′ ×18 ′′ GBT beam area centered at the peak of the fitted 2D Gaussian. The ranges of gas mass in the dense clump are estimated based on the assumption of β=1-2 (see Section 4.3). de Geus & Blitz 1994) δθ eff = (δθ maj × δθ min ) − (2δθ beam /2.355) 2 (4) and consider the gas mass enclosed in one FWHM of the fitted 2D Gaussian when calculating the virial onedimensional velocity dispersion. For those identified dense clumps with δθ eff > 0, we compare the results of the calculation with the one-dimensional velocity dispersion measured from the CS 1-0 line and the HCO + 4-3 line (when significantly detected) in Table 1. The mean gas number densities are calculated based on the aforementioned estimates of the clump masses and radii, and the assumptions of spherical geometry and the mean molecular weight of 2.33 (Shull & Beckwith 1982). We note that the JCMT-2 region shows double peak profiles in the CS 1-0 spectra. However, the broader line component is very faint and therefore is less likely to be associated with the dense gas clumps. We found that the dense clumps listed in Table 1 have ∼10-10 3 M ⊙ of molecular gas, which can be adequate gas reservoirs to form high-mass stars. However, the mean gas number density in these dense clumps are generally in between 10 5 -10 6 cm −3 , which are marginally unstable against the tidal force (Morris 1993;Liu et al. 2012). The GBT observations of CS 1-0 suggest that the identified dense clumps are all embedded in very turbulent environment, which have one-dimensional velocity dispersion of 10-30 km s −1 . The kinetic energy of the gas may be dissipated on smaller scales. The higher angular reso-lution SMA observations of HCO + 4-3 towards the enclosed Region A, Bn, B, Bs, and JCMT-1n detect the 2-3 times smaller velocity dispersion in the localized dense clumps. Nevertheless, the observed HCO + 4-3 velocity dispersions are still ∼10 times higher than the virial velocity dispersions of the dense clumps. Our preliminary analyses do not yet find any (self-)gravitationally bound gas structures on 0.5 pc scale. Although our HCO + 4-3 image only traces dense clumps in the warmer environments, HCN 1-0 and HCO + 1-0 spectra from previous 13 ′′ ×4 ′′ resolution BIMA array observations also showed broad line profiles in the central 5 ′ ×5 ′ field (Wright et al. 2001). Observations of the lower excitation gas tracers HCN 1-0 and HCO + 1-0 trace the velocity dispersion of the dense clumps in the central 5 ′ ×5 ′ field (Wright et al. 2001). With 13 ′′ ×4 ′′ angular resolution, the BIMA spectra are contaminated by emission from the diffuse and more turbulent ambient gas as traced by CS 1-0, however a deconvolved HCN 1-0 image at ∼ 2 ′′ resolution (Güsten et al. 1987b shows only the integrated image) also shows line widths which are much larger than the virial velocity dispersions of the dense clumps. Unless these dense clumps are composed of dense, bound cores, or the gas kinetic energy can be dissipated efficiently, they may be dispersed in one dynamical timescale (i.e. δ r /δv∼5.3·10 3 -2.6·10 4 years for the HCO + 4-3 emission clumps in Table 1). Higher angular and velocity resolution spectral line observations are required to see whether (self-)gravitationally bound gas structures exist on a smaller scale. Alternatively, these dense clumps embedded in the extremely turbulent molecular clouds may be confined by the high external pressure. For example, the previous Xray studies (e.g. Koyama et al. 1996, Muno et al. 2004 have demonstrated that a high-pressure medium with sound speed >1000 km s −1 pervades the Galactic center region. If small scale virialized gas structures do exist inside the pressurized dense clumps, then hydrostatic cores and stars may still form. Assuming approximate hydrostatic equilibrium of the embedded star-forming cores with masses M core , the radius r s , the mean gas number densityn H , and the r.m.s velocity dispersion v rms of the cores, as well as the final stellar mass m * f =ǫ core M core , can be estimated based on the following formulae (details see McKee & Tan 2002): r s = 0.074(m * f /30M ⊙ ) 1/2 Σ −1/2 pc,(5)n H = 1.0 × 10 6 (m * f /30M ⊙ ) −1/2 Σ 3/2 cm −3 , (6) v rms = 1.65(m * f /30M ⊙ ) 1/4 Σ 1/4 km s −1 ,(7) where Σ is the mean clump surface mass density in units of g cm −2 , and ǫ core (assumed to be 0.5 here) is the fraction of the core mass which is eventually accreted onto the central star. From Figure 1, the majority of the densest clumps are embedded in the region above the 12σ contour level (i.e. 288 mJy beam −1 ), which corresponds to the Σ of 0.31-0.76 g cm −2 . Based on the values in Table 1, we adopt a fiducial valuen H ∼10 6 cm −3 for an example. We estimate the core radius r s ∼0.023-0.056 pc (0.58 ′′ -1.4 ′′ ), the r.m.s. velocity dispersion 0.51-1.3 km s −1 , and the final stellar mass m * f ∼0.89-13 M ⊙ . The former two quantities can be examined with <0.5 ′′ resolution Atacama Large Millimeter/submillimeter Array (ALMA) observations in the near future. The corresponding core mass M core =2×m * f in our estimates is several times smaller than the masses of the parent gas clumps listed in Table 1 so may be reasonable. As discussed in the previous section, the dense clumps embedded in some regions (e.g. A, B, D, E, F, G) are associated with the 22 GHz water maser and the 36.2 GHz and 44.1 GHz Class I CH 3 OH masers, which are often seen in the early phase of high-mass star-formation. In addition, the earlier VLA observations of the 2 cm and the 6 cm continuum emission have found ultracompact Hii regions embedded with several OB stars in the east of the 50 km s −1 cloud (e.g. Ho et al. 1985). The observed high HCO + 4-3 velocity dispersion may be interpreted by (proto)stellar feedback. However, Sjouwerman et al. (2010) also suggested that the 36.2 GHz Class I CH 3 OH masers can be excited at the post shock regions created by cloud-cloud collisions. High angular resolution molecular line observations and studies of the maser proper motion may distinguish these two cases, although they are not mutually exclusive. In fact, it has been argued that the predominant mode of star formation is via external compression of molecular clouds by cloud collisions and supernova (Morris 1993). The high-mass star formation can also be induced by AGN activities (Silk et al. 2012). In these cases, the physical properties of the molecular gas reservoir feeding the high-mass star formation may be very different from the typical OB star-forming cores. The velocity integrated SiO/C 34 S 1-0 ratio and the 0.86 mm intensity at pixels where the velocity integrated C 34 S 1-0 intensities are higher than 0.7 K km s −1 (gray dots). The values at the locations of the 1720 MHz OH masers (green), the 1612 MHz OH masers (light blue), the 22 GHz water masers (black), the 36.2 GHz CH 3 OH masers (magenta), and the 44.1 GHz CH 3 OH masers (blue) are presented by colored symbols. For maser, the lower limit of the velocity integrated SiO/C 34 S 1-0 ratio are given if the C 34 S 1-0 emission is not robustly detected. The data points located outside of the field of view of either the 0.86 mm image or the SiO image are not presented. The Non-Uniform CND Previous interferometric spectral line observations have shown abundant localized structures in the central 2-4 pc CND (see Christopher et al. 2005;Montero-Castaño et al. 2009;Martín et al. 2012, and references therein). The distributions of the spectral line emission, however, are sensitive to the local temperature, volume density, chemical abundances, and other radiation transfer effects such as the foreground absorption or self-absorption. The optically thin 0.86 mm dust thermal continuum emission is a more robust tracer of the gas mass (e.g. Hildebrand 1983). Without being subjected to missing flux, our JCMT+SMA 0.86 mm dust continuum image (Figure 1) successfully reproduces the clumpy CND structures seen in the spectral line observations (e.g. Figure 4). It appears that some previously known clumps in the CND are not merely due to excitation or radiation transfer effects. For example, the 0.86 mm emission clump A and Bn (Table 1; Figure 3) coincide with the HCN 4-3 emission clump CC and BB reported by Montero-Castaño et al.(2009). As can be expected, we found that around the CND, the velocity integrated HCO + 4-3 intensity is correlated with the intensity of the 0.86 mm continuum emission (Figure 11). Fig. 11.-The velocity integrated HCO + 4-3 intensity and the 0.86 mm intensity of pixels within the 60 ′′ radius around the Sgr A. We color code the pixels in the four selected zones in Figure 12. We note that our HCO + 4-3 image is subjected to missing flux, however, should not significantly bias the analysis of the emission from the localized clumps. However, in Figure 11, we see that the correlation between the HCO + 4-3 emission and the 0.86 mm emission is dominated by at least two distinct populations which can be discerned with our current angular resolution and sensitivity. We selected the high HCO + 4-3 intensity parts of the two dominant populations as Zone 1 and Zone 2 in Figure 11, and additionally one high HCO + 4-3 intensity and high 0.86 mm intensity population as Zone 3, and one high 0.86 mm intensity population as Zone 4. We argue that these high intensity pixels are more likely to be associated with compact bright clumps, for which the HCO + 4-3 intensity is less biased by missing flux. The spatial distributions of the pixels in these four zones are not random. For example, the pixels in Zone 1 seems to trace the two arc-shaped clumps west of the CND, the eastern edge of the CND, and the clumps located northwest of the Sgr A. The pixels in Zone 2 are associated with the Northeast Lobe, the Southern Extension, and northern part of the Southwest Lobe. The southern part of the Southwest Lobe is associated with the pixels in Zone 4. Comparing Figure 12 with the ratio of the HCN 4-3 to HCN 1-0 integrated intensity reported by Montero-Castaño et al. (2009), we deduce that the very dense clump in the southern part of the Southwest Lobe has a low gas excitation temperature. This provide hints for the the cooler exterior gas clumps raining down on the warmer CND. For a dynamically evolving CND which is connected with several exterior gas streamers (see Liu et al. 2012), it is not surprising that the excitation conditions, the chemically abundances, or the dust properties are not yet homogenized. SUMMARY We present a wide-field SMA mosaic toward the Galactic center. We also observed the CS/C 34 S/ 13 CS 1-0 and the SiO 1-0 lines using the GBT. The optically thin 0.86 mm dust thermal continuum image with ∼5 ′′ angular resolution confirms the 2-10 pc scale gas streamers and the detailed structures of the CND, which were seen in previous molecular line observations. We marginally resolve more than 22 massive (10 1 -10 3 M ⊙ ) gas clumps, which are embedded in very turbulent molecular gas clouds (δv ∼10-30 km s −1 ). Examination of the brightness ratio of the HCO + 4-3 line and the 0.86 mm continuum emission shows that the non-homogenized central 2-4 pc CND has dense clumps with distinct excitation conditions or chemical abundances. While the distributions of the 1720 MHz and 1612 MHz OH maser clusters do not show obvious correlations with the dust emission, the 22 GHz water masers and especially the 36.2 and 44.1 GHz Class I CH 3 OH masers are seen preferentially near the dense clumps. Even the most significant dense clumps in our selected samples are marginally unstable against the tidal force, and we do not find any self-gravitationally bound gas clump associated with the Class I CH 3 OH masers or the 22 GHz H 2 O masers. How the OB stars form in such environment remains puzzling. The mechanisms to form the dense clumps and the high-mass stars in the Galactic center might be very different from the mechanisms in typical giant molecular clouds. Our simple estimates suggest that if the detected dense clumps are confined by the high external pressure, the presumably existing embedded virialized gas cores can form high-mass stars. Deep JVLA observations to search for the signature of the stellar photoionization, or ALMA observations of the high excitation hot core tracers to look for the gravitationally accelerated rotation, may diagnose the OB star formation in the dense clumps. Our GBT data suggest mildly enhanced SiO/C 34 S line ratio towards the Southern Arc and the 20 km s −1 cloud, relative to the SiO/C 34 S line ratios in the other gas streamers in the observed field. Our observations do not yet resolve the recognizable correlation between the SiO/C 34 S line ratio and the intensity of the submillimeter dust emission. Higher angular resolution and more sensitive observations are required to examine whether the formation of the dense gas structures are predominantly induced by shocks. The GBT and SMA data are from projects GBT11B050, SMA2011AS085 and SMA2011BS040, which are parts of the integrated state-of-art imaging project KISS: Kinematic Processes of the Extremely Turbulent ISM around the Supermassive Black Holes. We acknowledge financial support from ASIAA. S.S.K. was supported by Mid-career Research Program (No. 2011-0016898) through the National Research Foundation (NRF) grant funded by the Ministry of Education, Science and Technology (MEST) of Korea. We thank Dr. Eric Feigelson for the very useful suggestions. We thank Zhao Jun-Hui very much for his efforts in optimizing and upgrading MIRIAD for SMA, which made this project possible. We thank Toney Minter, Glen Langston, and David T. Frayer for assisting the GBT observations. We thank Glen Petitpas and Nimesh Patel for supporting the SMA observations. This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. Facilities: SMA, GBT APPENDIX A. SHORT SPACING INFORMATION Short spacing information, below ∼7.0 kλ, was obtained from single dish telescope observations. We retrieved the archival public processed JCMT 3 SCUBA (ProjectID: M01BU26, observed on 2001 August 05) 0.86 mm continuum image ( Figure 13; rms∼50 mJy beam −1 ). We used the MIRIAD task DEMOS to generate SMA primary beam weighted models for each SMA pointing. For each SMA field, we then resampled the primary beam weighted models in the uv domain with 185 Gaussian randomly distributed visibilities using the MIRIAD tasks UVRANDOM and UVMODEL. We manually assigned a system temperature of 350 K to the visibility model to adjust the weighting. The typical pointing accuracy for JCMT SCUBA is about 3 ′′ and the tracking accuracy ∼1.5 ′′ (Di Francesco et al. 2008). From their JCMT SCUBA legacy survey, Di Francesco et al. (2008) also mentioned that larger pointing offsets (e.g. ∼6 ′′ ) occurred occasionally. When combining with the interferometric observations, pointing offsets during the single dish observations can cause defects especially near bright compact sources. We used a limited uv range of 0-3.8 kλ for the single dish uv model to suppress the potential effect of the single dish pointing offsets, as well as the effect of the single dish primary beam and deconvolution errors. Although the effect of the single dish primary beam is not fully eliminated, it is smaller than the ∼20% absolute flux error in typical SMA observations (Appendix B). Limiting the uv range implies that the single dish primary beam causes defects primarily for structures >33 ′′ , and therefore should not confuse the identification of local gas clumps in the streamers. 3). Contour spacings are 5σ starting at 5σ (σ=50 mJy beam −1 ). For our convenience in discussion, we mark the peaks above the 45σ significance level besides the Northeast Lobe and Southwest Lobe as JCMT-1,2,3. The symbols are described in Figure 1. We jointly deconvolved and imaged the concatenated single dish uv model and the SMA data using the MIRIAD software package (using tasks: INVERT, MOSSDI, RESTOR). The gap (3.8-7 kλ) in the uv sampling may yield uncertainties in reconstructing the 14 ′′ -33 ′′ angular scale structures. However, structures with these angular scales can be directly inspected from the single dish image. We generated the high angular resolution continuum image with the parameters robust=0 fwhm=4,4 in INVERT, which yield a synthesized beamwidth θ maj × θ min =5 ′′ .1×4 ′′ .2, and BPA. ∼28 • . To compare with the earlier observations, we also generated one lower angular resolution (θ maj × θ min =7 ′′ .4×6 ′′ .2, BPA. ∼44 • ) continuum image with the weighting parameters robust=0 fwhm=8,8 to enhance the sensitivity to the extended emission. 2,4,8,16]. By comparing with the GBT CS 1-0 spectra, Liu et al. (2012) suggested that this HCN 4-3 image is subjected to the significant missing flux issue around the 50 km −1 cloud, the Molecular Ridge, the 20 km s −1 , and the Southern Arc. Right: Color image shows the high angular resolution SMA+JCMT 0.86 mm continuum image (θ maj × θ min =5 ′′ .1×4 ′′ .2). Contours show the free-free continuum emission model (see Appendix C), which is smoothed to the same angular resolution with the SMA+JCMT image. Contours are 10 mJy beam −1 × [1,2,4,8,16,32,64]. measure the rms noise level of the high angular resolution SMA+JCMT image by (rms of the residual image) 2 − (rms of the JCMT image) 2 (JCMT beam area)/(SMA+JCMT synthesized beam area) . (B1) This yields an rms noise level of 24 mJy beam −1 (0.011 K) in the high angular resolution SMA+JCMT image, which is ∼1.7 times higher than the theoretical noise level. Generally, the SMA+JCMT image is still dynamic-range limited, especially near the bright compact objects. This effect hampers the systematical search and statistical study of the clumpy structures. C. FREE-FREE MODEL SUBTRACTION The emission from the ionized mini-spiral arms can be recognized from the high angular resolution SMA+JCMT image ( Figure 15). We retrieved the archival VLA Q band observations taken on 2003 February 14, March 17, April 16, and 2004 March 14, April 21, May 16 to generate a model of 0.86 mm free-free emission. These VLA observations were taken with dual 50 MHz IFs centered at 43.314 GHz and 43.364 GHz, with full polarization. The sampling range of these VLA data, is 3.7-490 kλ. However, we assume that only the localized bright components will significantly contribute to the 0.86 mm free-free continuum emission. The basic calibrations and self-calibrations were performed using the Astronomical Image Processing System (AIPS) software package of NRAO. We scaled the VLA image such that the central point source Sgr A* has the flux of 1 Jy (Bower & Backer 1998;Yusef-Zadeh et al. 2011), and then smoothed the VLA image to the angular resolution of the SMA+JCMT image. The smoothed and rescaled VLA image, as the zeroth order model of the 0.86 mm free-free continuum emission, is presented in the right panel of Figure 15. The brightness of the 3 mm and the 1.3 mm emission is comparable, which can be checked in Kunneriath et al. (2012). We then subtracted the free-free emission model from the SMA+JCMT image ( Figure 1). As can be seen from Figure 15 and 1, this free-free model subtraction can only manifestly change the geometry in the Northern and the Eastern ionized mini-spiral arm regions and around Sgr A* (see Zhao et al. 2009 and references therein for the ionized mini-spiral arm). We note that the scaling of the VLA image in this process cannot take into the consideration of the spatial variation of the spectral index (Kunneriath et al. 2012). We have checked that if we scaled the VLA image to be ≥20% brighter, this free-free model subtraction will induce noticeable over-subtracted features in the Northern and the Eastern ionized mini-spiral arms. Except for the location of Sgr A, in the central 1 ′ area, the free-free model subtracted SMA+JCMT image provides a lower limit for the 0.86 mm dust thermal emission. Fig. 1 . 1-The SMA+JCMT 0.86 mm continuum image with a free-free model subtracted. (color and contour). Fig. 2 . 2-The JCMT SCUBA 0.44 mm (678 GHz) continuum image (color; θ maj × θ min =8 ′′ ×8 ′′ ), overlaid with the high angular resolution SMA+JCMT 0.86 mm continuum image (contour; θ maj × θ min =5 ′′ .1×4 ′′ .2). Contour spacings are 3σ starting at 3σ (σ=72 mJy beam −1 ). The public processed JCMT SCUBA 0.44 mm was retrieved from the online data archive (ProjectID: M98AU64), and was published byPierce-Price et al. (2000). The beam of the JCMT SCUBA 0.44 mm observation is shown in the lower left. Fig. 3 . 3-Blowups of the free-free model subtracted SMA+JCMT 0.86 mm continuum image (color and contour) in parsec scale area around regions A, B, D, E, F, G, JCMT-1, and JCMT-2. Contour spacings are 3σ starting at 3σ (σ=24 mJy beam −1 ). Red ellipses show the fitted 2D Gaussian clumps. The major and minor axes of the red ellipses are the FWHM of the fitted 2D Gaussian (not deconvolved). Fig. 4 . 4-The velocity integrated HCO + 4-3 image (θ maj × θ min =5 ′′ .8×4 ′′ .1). Red ellipses show the fitted 0.86 mm clumps (seeFigure 3). Blue star labels the location of Sgr A. Fig. 5 . 5-The HCO + 4-3 line spectra toward the selected regions (black; see Fig. 6 . 6-Molecular gas in the central 20 pc region in the Galactic center. (A) The velocity integrated CS 1-0 emission. (σ ∼0.7 K km s −1 ). The symbols are described in Figure 1. Labels of the large scale gas streamers are consistent with those in Liu et al. (2012). (B) Fig. 7 . 7-Velocity integrated CS/C 34 S 1-0 ratio (color and contour). Contour levels are[6, 12, 18, 24]. Blue star labels the location of Sgr A. the central engine Sgr A. Fig. 9 . 9-Velocity channel maps of the SiO/C 34 S 1-0 ratio. Plus signs label the location of Sgr A. The dash-dotted arcs are drawn to indicate the Southern Arc and the streamers in the east (consist of 50 km s −1 cloud, the Molecular Ridge, and the 20 km s −1 cloud). The GBT images are smoothed to the angular resolution of θ maj × θ min =20 ′′ ×18 ′′ before taking the ratio. Fig. 10 . 10- Fig. 12 . 12-The color coded pixels which reside at Zone 1 (red), 2 (yellow), 3 (green), and 4 (purple) inFigure 11respectively. The color image is overlaid with the velocity integrated HCO + 4-3 image (contours; θ maj × θ min =5 ′′ .8×4 ′′ .1; seeFigure 4). Contours are 3σ×[1, 2, 4, 8] (σ=15.5 mJy beam −1 km s −1 ). Fig. 13 . 13-The JCMT SCUBA 0.86 mm continuum image (θ maj × θ min =14 ′′ .3×14 ′′ . Fig. 15 . 15-The combined SMA+JCMT 0.86 mm continuum image (color). Left: Color image shows the tappered SMA+JCMT 0.86 mm continuum image (θ maj × θ min =7 ′′ .4×6 ′′ .2). The synthesized beam of the SMA+JCMT image is shown in the lower left. Contours show the velocity integrated SMA HCN 4-3 image taken from Liu et al. (2012) (θ maj × θ min =5 ′′ .9×4 ′′ .4). Contours are 75 Jy beam −1 km s −1 ×[1, TABLE 1 1Properties of the fitted 22 0.86 mm continuum clumps.Components R.A. Decl. δθ maj δθ min 0.86mm Flux Gas Mass Number Density δv viriral δv HCO + δv CS (J2000) (J2000) (arcsec) (arcsec) (mJy) (M⊙) (10 5 cm −3 ) (km s −1 ) (km s −1 ) (km s −1 ) Region A 17 h 45 m 43.74 s -29 • 00 ′ 03 ′′ 7.0 3.2 90 13-31 2.5-5.9 0.4-0.8 9.7 19 Region Bn 17 h 45 m 43.72 s -29 • 00 ′ 18 ′′ 5.6 5.3 172 24-60 1.8-4.4 0.45-0.96 4.5 15 Region B 17 h 45 m 43.80 s -29 • 00 ′ 27 ′′ 6.0 4.7 238 34-83 2.9-7.2 0.56-1.1 7.0 14 Region Bs 17 h 45 m 43.65 s -29 • 00 ′ 32 ′′ 5.0 4.3 114 16-40 4.4-11 0.46-0.96 9.4 14 Region Bwn 17 h 45 m 42.89 s -29 • 00 ′ 22 ′′ 4.1 3.1 54 8-19 · · · · · · · · · 21 Region Bws 17 h 45 m 42.81 s -29 • 00 ′ 28 ′′ 6.9 5.7 317 45-110 1.5-3.6 0.55-1.1 · · · 19 Region Dn 17 h 45 m 42.51 s -29 • 01 ′ 53 ′′ 9.6 4.8 271 38-94 0.88-2.1 0.47-0.96 · · · 14 Region Ds 17 h 45 m 42.20 s -29 • 02 ′ 00 ′′ 8.2 4.6 195 28-68 1.0-2.6 0.43-0.87 · · · 12 Region Ew 17 h 45 m 50.28 s -29 • 00 ′ 06 ′′ 4.0 3.0 121 17-42 · · · · · · · · · 13 Region Ee 17 h 45 m 50.51 s -29 • 00 ′ 07 ′′ 10.0 4.9 474 67-160 1.3-3.2 0.61-1.2 · · · 13 Region Fw 17 h 45 m 49.90 s -28 • 59 ′ 41 ′′ 7.1 3.5 134 19-46 2.6-6.4 0.45-0.87 · · · 12 Region Fe 17 h 45 m 50.16 s -28 • 59 ′ 44 ′′ 4.7 3.5 83 12-29 59-140 0.64-1.3 · · · 12 Region Gn 17 h 45 m 49.41 s -28 • 58 ′ 54 ′′ 6.5 6.5 364 51-130 1.5-3.6 0.57-1.1 · · · 12 Region Gs 17 h 45 m 49.63 s -28 • 59 ′ 00 ′′ 4.0 4.0 147 21-51 320-760 1.0-2.1 · · · 13 JCMT-1n 17 h 45 m 36.56 s -29 • 01 ′ 58 ′′ 11.9 8.0 1810 260-630 1.4-3.4 0.96-1.9 6.8 24 JCMT-1s 17 h 45 m 36.10 s -29 • 02 ′ 07 ′′ 6.5 4.9 527 74-180 4.4-11 0.77-1.6 · · · 29 JCMT-2a 17 h 45 m 41.06 s -29 • 02 ′ 20 ′′ 5.5 3.3 119 17-41 16-39 0.58-1.1 · · · 10 (or 19) JCMT-2b 17 h 45 m 40.22 s -29 • 02 ′ 24 ′′ 5.8 4.2 169 24-59 3.6-8.8 0.51-1.0 · · · 11 (or 20) JCMT-2c 17 h 45 m 40.83 s -29 • 02 ′ 26 ′′ 3.8 2.5 48 6.6-16 · · · · · · · · · 11 (or 18) JCMT-2d 17 h 45 m 40.60 s -29 • 02 ′ 39 ′′ 8.8 4.7 412 58-140 1.8-4.1 0.61-1.2 · · · 11 (or 17) JCMT-2e 17 h 45 m 41.90 s -29 • 02 ′ 29 ′′ 6.8 5.4 398 56-140 2.2-5.6 0.63-1.3 · · · 10 (or 23) JCMT-2f 17 h 45 m 39.46 s -29 • 02 ′ 41 ′′ 9.4 5.7 614 87-210 1.4-3.5 0.68-1.4 · · · 11 (or 26) The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica.2 The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. 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In our zero spacing model (see Section A), we did not manually correct for this 4 ′′ offset for the sake of objectiveness. We do not think this potential JCMT 4 ′′ pointing offset can cause very obvious imaging defects since only the ≥33 ′′ scale structures are taken for modeling the zero spacing. However, it can lead to the mismatch between the combined SMA+JCMT image and the JCMT image, which will result in some residual in Figure 14. Beyond the aforementioned regions, the extended structures appear to be reasonably reconstructed in the SMA+JCMT image, and the residual image has an rms noise level of ∼89 mJy beam −1 . The Northern/Southern Ridges, Northeast/Southwest Lobes, and the W-2,3,4 Streamers are detected in the lower angular resolution SMA+JCMT image (Figure 15 left). the SMA+JCMT image. The theoretical rms noise level of the SMA+JCMT image is ∼14 mJy beam −1 . Because of the absence of emissionfree areas in our map. Fig, Left: the residual image constructed by subtracting the JCMT SCUBA image from the smoothed high angular resolution SMA+JCMT 0.86 mm continuum image (see Appendix B). The JCMT beam is shown in the lower left. Contours are 3σ×. 14We note that the averaged 3σ flux density 267 mJy beam −1 in this image corresponds to the averaged flux of ∼28 mJy in each 5 ′′ .1×4 ′′ .2 SMA synthesized beam. The symbols are described in Figure 1. Right: similar to the left panel. However, the original JCMT SCUBA image was shifted toward the south by 4 ′′ before being subtracted from the SMA+JCMT image. B. CONSISTENCY CHECK AND NOISE STATISTICS We smoothed the high angular resolution SMA+JCMT image to the angular resolution of JCMT SCUBA, and then subtracted the JCMT SCUBA image from the smoothed SMA+JCMT image. The residual image after the subtraction is shown in Figure 14. The lower angular resolution SMA+JCMT image, and the previously published SMA HCN 4-3 image (Liu et al. 2012) are presented in Figure 15. it is difficult to directly measure the rms noise level we actually achieved. Since only the ≥33 ′′ structures in the JCMT image. which have high signal to noise ratios, are combined with the SMA data. We assumed the noise in the SMA+JCMT image only has a weak dependence on the noise in the JCMT image. We therefore canFig. 14.-Left: the residual image constructed by subtracting the JCMT SCUBA image from the smoothed high angular resolution SMA+JCMT 0.86 mm continuum image (see Appendix B). The JCMT beam is shown in the lower left. Contours are 3σ×[-2, -1, 1, 2, 3, 4, 5] (σ=89 mJy beam −1 ). We note that the averaged 3σ flux density 267 mJy beam −1 in this image corresponds to the averaged flux of ∼28 mJy in each 5 ′′ .1×4 ′′ .2 SMA synthesized beam. The symbols are described in Figure 1. Right: similar to the left panel. However, the original JCMT SCUBA image was shifted toward the south by 4 ′′ before being subtracted from the SMA+JCMT image. B. CONSISTENCY CHECK AND NOISE STATISTICS We smoothed the high angular resolution SMA+JCMT image to the angular resolution of JCMT SCUBA, and then subtracted the JCMT SCUBA image from the smoothed SMA+JCMT image. The residual image after the subtraction is shown in Figure 14. The lower angular resolution SMA+JCMT image, and the previously published SMA HCN 4-3 image (Liu et al. 2012) are presented in Figure 15. On average the contour intervals in Figure impression by comparing Figure 14 and 15 is that the limited uv sampling rate of our SMA observations cause imaging defects near the bright compact sources. This is most obviously seen around the Sgr A, the Southwest Lobe, and the bright clump in the Southern Arc (coincide with JCMT-1). However, we found that manually shifting the JCMT image towards the south by 4 ′′ can suppress the residual (e.g. Figure 14) around the Sgr A*. In our zero spacing model (see Section A), we did not manually correct for this 4 ′′ offset for the sake of objectiveness. We do not think this potential JCMT 4 ′′ pointing offset can cause very obvious imaging defects since only the ≥33 ′′ scale structures are taken for modeling the zero spacing. However, it can lead to the mismatch between the combined SMA+JCMT image and the JCMT image, which will result in some residual in Figure 14. Beyond the aforementioned regions, the extended structures appear to be reasonably reconstructed in the SMA+JCMT image, and the residual image has an rms noise level of ∼89 mJy beam −1 . The Northern/Southern Ridges, Northeast/Southwest Lobes, and the W-2,3,4 Streamers are detected in the lower angular resolution SMA+JCMT image (Figure 15 left). the SMA+JCMT image. The theoretical rms noise level of the SMA+JCMT image is ∼14 mJy beam −1 . Because of the absence of emission- free areas in our map, it is difficult to directly measure the rms noise level we actually achieved. Since only the ≥33 ′′ structures in the JCMT image, which have high signal to noise ratios, are combined with the SMA data. We assumed the noise in the SMA+JCMT image only has a weak dependence on the noise in the JCMT image. We therefore can	[]
[ "An Efficient Quantum Algorithm for the Hidden Subgroup Problem over Weyl-Heisenberg Groups", "An Efficient Quantum Algorithm for the Hidden Subgroup Problem over Weyl-Heisenberg Groups" ]	[ "Hari Krovi [email protected] \nNEC Laboratories\nAmerica 4 Independence Way, Suite 20008540PrincetonNJU.S.A\n", "Martin Rötteler [email protected] \nNEC Laboratories\nAmerica 4 Independence Way, Suite 20008540PrincetonNJU.S.A\n" ]	[ "NEC Laboratories\nAmerica 4 Independence Way, Suite 20008540PrincetonNJU.S.A", "NEC Laboratories\nAmerica 4 Independence Way, Suite 20008540PrincetonNJU.S.A" ]	[]	Many exponential speedups that have been achieved in quantum computing are obtained via hidden subgroup problems (HSPs). We show that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer. These groups are well-known in physics and play an important role in the theory of quantum error-correcting codes. Our algorithm is based on noncommutative Fourier analysis of coset states which are quantum states that arise from a given black-box function. We use Clebsch-Gordan decompositions to combine and reduce tensor products of irreducible representations. Furthermore, we use a new technique of changing labels of irreducible representations to obtain low-dimensional irreducible representations in the decomposition process. A feature of the presented algorithm is that in each iteration of the algorithm the quantum computer operates on two coset states simultaneously. This is an improvement over the previously best known quantum algorithm for these groups which required four coset states.	10.1007/978-3-540-89994-5_7	[ "https://arxiv.org/pdf/0810.3695v1.pdf" ]	2,067,479	0810.3695	d2c6e161cbee016122f75ae1ced9bc7d830bd163	An Efficient Quantum Algorithm for the Hidden Subgroup Problem over Weyl-Heisenberg Groups 20 Oct 2008 Hari Krovi [email protected] NEC Laboratories America 4 Independence Way, Suite 20008540PrincetonNJU.S.A Martin Rötteler [email protected] NEC Laboratories America 4 Independence Way, Suite 20008540PrincetonNJU.S.A An Efficient Quantum Algorithm for the Hidden Subgroup Problem over Weyl-Heisenberg Groups 20 Oct 2008quantum algorithmshidden subgroup problemcoset states Many exponential speedups that have been achieved in quantum computing are obtained via hidden subgroup problems (HSPs). We show that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer. These groups are well-known in physics and play an important role in the theory of quantum error-correcting codes. Our algorithm is based on noncommutative Fourier analysis of coset states which are quantum states that arise from a given black-box function. We use Clebsch-Gordan decompositions to combine and reduce tensor products of irreducible representations. Furthermore, we use a new technique of changing labels of irreducible representations to obtain low-dimensional irreducible representations in the decomposition process. A feature of the presented algorithm is that in each iteration of the algorithm the quantum computer operates on two coset states simultaneously. This is an improvement over the previously best known quantum algorithm for these groups which required four coset states. Introduction Exponential speedups in quantum computing have hitherto been shown for only a few classes of problems, most notably for problems that ask to extract hidden features of certain algebraic structures. Examples for this are hidden shift problems [DHI03], hidden non-linear structures [CSV07], and hidden subgroup problems (HSPs). The latter class of hidden subgroup problems has been studied quite extensively over the past decade. There are some successes such as the efficient solution of the HSP for any abelian group [Sho97,Kit97,BH97,ME98], including factoring and discrete log as well as Pell's equation [Hal02], and efficient solutions for some non-abelian groups [FIM + 03,BCD05]. Furthermore, there are some partial successes for some non-abelian groups such as the dihedral groups [Reg04,Kup05] and the affine groups [MRRS04]. Finally, it has been established that for some groups, including the symmetric group which is connected to the graph isomorphism problem, a straightforward approach requires a rather expensive quantum processing in the sense that entangling operations on a large number of quantum systems would be required [HMR + 06]. What makes matters worse, there are currently no techniques, or even promising candidates for techniques, to implement these highly entangling operations. The present paper deals with the hidden subgroup problem for a class of non-abelian groups that-in a precise mathematical sense that will be explained below-is not too far away from the abelian case, but at the same time has some distinct non-abelian features that make the HSP over these groups challenging and interesting. The hidden subgroup problem is defined as follows: we are given a function f : G → S from a group G to a set S, with the additional promise that f takes constant and distinct values on the left cosets gH, where g ∈ G, of a subgroup H ≤ G. The task is to find a generating system of H. The function f is given as a black-box, i. e., it can only be accessed through queries and in particular whose structure cannot be further studied. The input size to the problem is log \|G\| and for a quantum algorithm solving the HSP to be efficient means to have a running time that is poly(log \|G\|) in the number of quantum operations as well as in the number of classical operations. We will focus on a particular approach to the HSP which proved to be successful in the past, namely the so-called standard method, see [GSVV04]. Here the function f is used in a special way, namely it is used to generate coset states which are states of the form 1/ \|H\| h∈H \|gh for random g ∈ G. The task then becomes to extract a generating system of H from a polynomial number of coset states (for random values of g). A basic question about coset states is how much information about H they indeed convey and how this information can be extracted from suitable measurements. 1 A fixed POVM M operates on a fixed number k of coset states at once and if k ≥ 2 and M does not decompose into measurements of single copies, we say that the POVM is an entangled measurement. As in [HMR + 06], we call the parameter k the "jointness" of the measurement. It is known that information-theoretically for any group G jointness k = O(log \|G\|) is sufficient [EHK04]. While the true magnitude of the required k can be significantly smaller (abelian groups serve as examples for which k = 1), there are cases for which indeed a high order of k = Θ(log \|G\|) is sufficient and necessary. Examples for such groups are the symmetric groups [HMR + 06]. However, on the more positive side, it is known that some groups require only a small, sometimes even only constant, amount of jointness. Examples are the Heisenberg groups of order p 3 for a prime p for which k = 2 is sufficient [BCD05,Bac08a]. In earlier work [ISS07], it has been shown that for the Weyl-Heisenberg groups order p 2n+1 , k = 4 is sufficient [ISS07]. The goal of this paper is to show that in the latter case the jointness can be improved. We give a quantum algorithm which is efficient in the input size (given by log p and n) and which only requires a jointness of k = 2. Our results and related work: The family of groups we consider in the present paper are well-known in quantum information processing under the name of generalized Pauli groups or Weyl-Heisenberg groups [NC00]. Their importance in quantum computing stems from the fact that they are used to define stabilizer codes, the class of codes most widely used for the construction of quantum error-correcting codes [CRSS97,Got96,CRSS98]. In a more group-theoretical context, the Weyl-Heisenberg groups are known as extraspecial p-groups (actually, they constitute one of the two families of extraspecial p-groups [Hup83]). A polynomial-time algorithm for the HSP for the extraspecial pgroups was already given by Ivanyos, Sanselme, and Santha, [ISS07]. Our approach differs to this approach in two aspects: first, our approach is based on Fourier sampling for the non-abelian group G. Second, and more importantly, we show that the jointness k, i. e., the number of coset states that the algorithm has to operate jointly on, can be reduced from k = 4 to k = 2. Crucial for our approach is the fact that in the Weyl-Heisenberg group the labels of irreducible representations can be changed. This is turn can be used to "drive" Clebsch-Gordan decompositions in such a way that low-dimensional irreducible representations occur in the decomposition. It is perhaps interesting to note that for the Weyl-Heisenberg groups the states that arise after the measurement in the Fourier sampling approach (also called Fourier coefficients) are typically of a very large rank (i. e., exponential in the input size). Generally, large rank usually is a good indicator of the intractability of the HSP, such as in case of the symmetric group when H is a full support involution. Perhaps surprisingly, in the case of the Weyl-Heisenberg group it still is possible to extract H efficiently even though the Fourier coefficients have large rank. We achieve this at the price of operating on two coset states at the same time. This leaves open the question whether k = 1 is possible, i. e., if the hidden subgroup H can be identified from measurements on single coset states. We cannot resolve this question but believe that this will be hard. Our reasoning is as follows. Having Fourier coefficients of large rank implies that the random basis method [RRS05,Sen06] cannot be applied. The random basis method is a method to derive algorithms with k = 1 whose quantum part can be shown to be polynomial, provided that the rank of the Fourier coefficients is constant. 2 Based on this we therefore conjecture that any efficient quantum algorithm for the extraspecial groups will require jointness of k ≥ 2. Finally, we mention that a similar method to combine the two registers in each run of the algorithm has been used by Bacon [Bac08a] to solve the HSP in the Heisenberg groups of order p 3 . The method uses a Clebsch-Gordan transform which is a unitary transform that decomposes the tensor product of two irreducible representations [Ser77] into its constituents. The main difference between the Heisenberg group and the Weyl-Heisenberg groups is that the Fourier coefficients are no longer pure states and are of possibly high rank. Organization of the paper: In Section 2 we review the Weyl-Heisenberg group and its subgroup structure. The Fourier sampling approach and the so-called standard algorithm are reviewed in Section 3. In Section 4 we provide necessary facts about the representation theory that will be required in the subsequent parts. The main result of this paper is the quantum algorithm for the efficient solution of the HSP in the Weyl-Heisenberg groups presented in Section 5. Finally, we offer conclusions in Section 6. The Weyl-Heisenberg groups We begin by recalling some basic group-theoretic notions. Recall that the center Z(G) of a group G is defined as the set of elements which commute with every element of the group i.e., Z(G) = {c : [c, g] = cgc −1 g −1 = e for all g ∈ G}, where e is the identity element of G. The derived (or commutator) subgroup G ′ is generated by elements of the type [a, b] = aba −1 b −1 , where a, b ∈ G. The reader is invited to recall the definition of semidirect products G = N ⋊ H, see for instance [Hup83,Ser77]. In the following we give a definition of the Weyl-Heisenberg groups as a semidirect product and give two alternative ways of working with these groups. Definition 1. Let p be a prime and let n be an integer. The Weyl-Heisenberg group of order p 2n+1 is defined as the semidirect product Z n+1 p ⋊ φ Z n p , where the action φ in the semidirect product is defined on x = (x 1 , . . . , x n ) ∈ Z n p as the (n + 1) × (n + 1) matrix given by φ(x) =        1 . . . 0 0 0 1 . . . 0 . . . . . . 0 . . . 1 0 x 1 x 2 . . . x n 1        . (1) Any group element of Z n+1 p ⋊ φ Z n p can be written as a triple (x, y, z) where x and y are vectors of length n whose entries are elements of Z p and z is in Z p . To relate this triple to the semidirect product, one can think of (y, z) ∈ Z n+1 p and x ∈ Z n p . Then, the product of two elements in this group can be written as (x, y, z) · (x ′ , y ′ , z ′ ) = (x + x ′ , y + y ′ , z + z ′ + x ′ · y),(2) where x · y = i x i y i is the dot product of two vectors (denoted as xy in the rest of the paper). Fact 1 [Hup83] For any p prime, and n ≥ 1, the Weyl-Heisenberg group is an extraspecial p group. Recall that a group G is extraspecial if Z(G) = G ′ , the center is isomorphic to Z p , and G/G ′ is a vector space. Up to isomorphism, extraspecial p-groups are of two types: groups of exponent p and groups of exponent p 2 . The Weyl-Heisenberg groups are the extraspecial p-groups of exponent p. It was shown in [ISS07] that an algorithm to find hidden subgroups in the groups of exponent p can be used to find hidden subgroups in groups of exponent p 2 . Therefore, it is enough to solve the HSP in groups of exponent p. In this paper, we present an efficient algorithm for the HSP over groups of exponent p. Realization via matrices over Z p : First, we recall that the Heisenberg group of order p 3 (which is the group of 3 × 3 upper triangular matrices with ones on the main diagonal and other entries in Z p ) is a Weyl-Heisenberg group and can be regarded as the semidirect product Z 2 p ⋊ Z p . An efficient algorithm for the HSP over this group is given in [BCD05]. Elements of this group are of the type   1 y z 0 1 x 0 0 1   . (3) The product of two such elements is   1 y z 0 1 x 0 0 1     1 y ′ z ′ 0 1 x ′ 0 0 1   =   1 y + y ′ z + z ′ + x ′ y 0 1 x + x ′ 0 0 1   (4) Thus, such a matrix can be identified with a triple (x, y, z) in Z 2 p ⋊ Z p . This matrix representation of the Heisenberg group can be generalized for any n. We can associate a triple (x, y, z) where x, y ∈ Z n p and z ∈ Z p with the (n + 2) × (n + 2) matrix        1 y 1 . . . y n z 0 1 . . . 0 x 1 . . . . . . . . . . . . . . . 0 0 . . . 1 x n 0 0 . . . 0 1        .(5) Realization via unitary representation: Finally, there is another useful way to represent the Weyl-Heisenberg group. The n qupit Pauli matrices form a faithful (irreducible) representation of the Weyl-Heisenberg p-group. For any k = 0, we can associate with any triple (x, y, z) in Z n+1 p ⋊ Z n p , the following matrix: ρ k (x, y, z) = ω kz p X x Z y k ,(6) where the matrix X = u∈Z n p \|u + 1 u\| is the generalized X operator and the matrix Z k = u∈Z n p ω k p \|u u\| is the generalized Z operator, see e. g. [NC00]. Subgroup structure: In the following we will write G in short for Weyl-Heisenberg groups. Using the notation introduced above the center Z(G) (or G ′ ) is the group Z(G) = {(0, 0, z)\|z ∈ Z p } and is isomorphic to Z p . As mentioned above, the quotient group G/G ′ is a vector space isomorphic to Z 2n p . This space can be regarded as a symplectic space with the following inner product: (x, y) · (x ′ , y ′ ) = (x · y ′ − y · x ′ ), where x, y, x ′ , y ′ ∈ Z n p . The quotient map is just the restriction of the triple (x, y, z) ∈ G to the pair (x, y) ∈ Z 2n p . From Eq. (2), it follows that two elements commute if and only if xy ′ − yx ′ = 0. Denote the set of (x, y) pairs occurring in H as S H i.e., for each triple (x, y, z) ∈ H, we have that (x, y) ∈ S H and so \|S H \| ≤ \|H\|. It can be easily verified that S H is a vector space and is in fact, a subspace of Z 2n p . Indeed, for two elements (x, y), (x ′ , y ′ ) ∈ S H , pick two elements (x, y, z), (x ′ , y ′ , z ′ ) ∈ H and so (x + x ′ , y + y ′ , z + z ′ + x ′ y) ∈ H. Therefore, (x + x ′ , y + y ′ ) ∈ S H . To show that if (x, y) ∈ S H , then (ax, ay) ∈ S H for any a ∈ Z p , observe that if (x, y, z) ∈ H, then (x, y, z) a = (ax, ay, az + a(a−1) 2 xy) ∈ H. Therefore, (ax, ay) ∈ S H (in fact, it can be shown that S H ≃ HG ′ /G ′ , but we do not need this result.) Therefore, H ≤ G is abelian if and only if ∀(x, y), (x ′ , y ′ ) ∈ S H , we have that xy ′ − x ′ y = 0. Such a space where all the elements are orthogonal to each other is called isotropic. Now, we make a few remarks about the conjugacy class of some subgroup H. Consider conjugating H by some element of G, say g = (x ′ , y ′ , z ′ ). For any h = (x, y, z) ∈ H, we obtain g −1 hg = (−x ′ , −y ′ , −z ′ + x ′ y ′ )(x, y, z)(x ′ , y ′ , z ′ ) = (−x ′ , −y ′ , −z ′ + x ′ y ′ )(x + x ′ , y + y ′ , z + z ′ + x ′ y) = (x, y, z + x ′ y − xy ′ ) ∈ H g .(7) From this we see that S H g = S H . We show next that S H actually characterizes the conjugacy class of H. Before proving this result we need to determine the stabilizer of H. The stabilizer H S of H is defined as the set of elements of G which preserve H under conjugation i.e., H S = {g ∈ G\|H g = H}. From Eq. (7), we can see that Proof. We have already seen that if H 1 and H 2 are conjugates, then S H1 = S H2 . To show the other direction, we use a counting argument ie., we show that the number of subgroups H ′ of G such that S H ′ = S H is equal to the number of conjugates of H. First, assume that the dimension of the vector space S H1 is k. Now, the number of conjugates of H 1 is the index of the stabilizer of H 1 . From the above result, the stabilizer has a size \|G ′ \|\|S ⊥ H1 \| = p · p 2n−k . Therefore, the index or the number of conjugates of H 1 are p 2n+1 /p 2n−k−1 = p k . Now, the number of different possible subgroups H such that S H = S H1 is p k since each of the k basis vectors of S H1 are generators of the subgroup and they can have any z component independent of each other i.e., there are p possible choices of z for each of the k generators. g = (x ′ , y ′ , z ′ ) ∈ H S if and only if x ′ y − xy ′ = 0 for all (x, y, z) ∈ H. Thus, the stabilizer is a group such that S HS = S ⊥ H , where S ⊥ H is the orthogonal space under the symplectic inner product defined above, i.e., H S = {(x, y, z) ∈ G\|(x, y) ∈ S ⊥ H , z ∈ Z p }. In other words, it is obtained by appending the pairs (x, y) ∈ S ⊥ H with every possible z ∈ Z p . Therefore, \|H S \| = \|G ′ \| · \|S ⊥ H \|. Now, The property G ′ = Z(G) will be useful in that it will allow us to consider only a certain class of hidden subgroups. We show next that it is enough to consider hidden subgroups which are abelian and do not contain G ′ . Recall that that H is normal in G (denoted H G) if g −1 hg ∈ H for all g ∈ G and h ∈ H. Lemma 2. If G ′ ≤ H, then H G. Proof. Since G ′ is the commutator subgroup, for any g 1 , g 2 ∈ G, there exists g ′ ∈ G ′ such that g 1 g 2 = g 2 g 1 g ′ . Now, let h ∈ H and g ∈ G. We have g −1 hg = hg ′ for some g ′ ∈ G ′ . But since G ′ ≤ H, hg ′ = h ′ , for some h ′ ∈ H. Therefore, g −1 hg = h ′ and hence H G. Lemma 3. If H is non-abelian, then H G. Proof. Let h 1 , h 2 ∈ H such that h 1 h 2 = h 2 h 1 . Then h 1 h 2 = h 2 h 1 g ′ for some g ′ ∈ G ′ such that g ′ = e, where e is the identity element of G. This means that g ′ ∈ H. Since G ′ is cyclic of prime order, it can be generated by any g ′ = e and hence, we have G ′ ≤ H. Now, Lemma 2 implies that H G. From these two lemmas, we have only two cases to consider for the hidden subgroup H: (a) H is abelian and does not contain G ′ and (b) H is normal in G. It is possible to tell the cases apart by querying the hiding function f twice and checking whether f (e) and f (g ′ ) are equal for some g ′ = e and g ′ ∈ G ′ . If they are equal then G ′ ≤ H and H G, otherwise H is abelian. If H is normal, then one can use the algorithm of [HRT03], which is efficient if one can intersect kernels of the irreducible representations (irreps) efficiently. For the Weyl-Heisenberg group, the higher dimensional irreps form a faithful representation and hence do not have a kernel. Thus, when the hidden subgroup is normal, only one dimensional irreps occur and their kernels can be intersected efficiently and the hidden subgroup can be found using the algorithm of [HRT03]. Therefore, we can consider only those hidden subgroups which are abelian and moreover do not contain G ′ . Now, we restrict our attention to the case of abelian H. Finally, we need the following two results. Proof. Suppose that for some (x, y) ∈ S H there exist two different elements (x, y, z 1 ) and (x, y, z 2 ) in H, then by multiplying one with the inverse of the other we get (0, 0, z 1 − z 2 ). Since z 1 − z 2 = 0, this generates G ′ , but by our assumption on H, G ′ H. Therefore, \|S H \| = \|H\|. The following theorem applies to the case when p > 2. Proof. We can verify that H 0 is a subgroup by considering elements (x, y, xy/2) and (x ′ , y ′ , x ′ y ′ /2) in H 0 . Their product is (x, y, xy/2) · (x ′ , y ′ , x ′ y ′ /2) = (x + x ′ , y + y ′ , xy/2 + x ′ y ′ /2 + x ′ y) = (x + x ′ , y + y ′ , xy/2 + x ′ y ′ /2 + (x ′ y + xy ′ )/2) = (x + x ′ , y + y ′ , (x + x ′ )(y + y ′ )/2),(8) which is an element of H 0 . Here, we have used the fact that H is abelian i.e., xy ′ −x ′ y = 0, ∀(x, y), (x ′ , y ′ ) ∈ S H . Now for H 0 , since S H0 = S H , H 0 is conjugate to H using Lemma 1. Note that H 0 can be thought of as a representative of the conjugacy class of H since it can be uniquely determined from S H . The above lemma does not apply for the case p = 2. When p = 2, we have that (x, y, z) 2 = (2x, 2y, 2z + xy) = (0, 0, xy). But since we assume that G ′ H, when p = 2 we must have that xy = 0, ∀(x, y, z) ∈ H. Fourier sampling approach to HSP We recall some basic facts about the Fourier sampling approach to the HSP, see also [GSVV04,HMR + 06]. First, we recall some basic notions of representation theory of finite groups [Ser77] that are required for this approach. Let G be a finite group, let C[G] to denote its group algebra, and letĜ be the set of irreducible representations (irreps) of G. We will consider two distinguished orthonormal vector space bases for C[G], namely, the basis given by the group elements on the one hand (denoted by \|g , where g ∈ G) and the basis given by normalized matrix coefficients of the irreducible representations of G on the other hand (denoted by \|ρ, i, j , where ρ ∈Ĝ, and i, j = 1, . . . , d ρ for d ρ , where d ρ denotes the dimension of ρ). Now, the quantum Fourier transform over G, QFT G is the following linear transformation [Bet87,GSVV04]: \|g → ρ∈Ĝ d ρ \|G\| dρ i,j=1 ρ ij (g)\|ρ, i, j .(9) An easy consequence of Schur's Lemma is that QFT G is a unitary transformation in C \|G\| , mapping from the basis of \|g to the basis of \|ρ, i, j . For a subgroup H ≤ G and irrep ρ ∈Ĝ, define ρ(H) := 1 \|H\| h∈H ρ(h). Again from Schur's Lemma we obtain that ρ(H) is an orthogonal projection to the space of vectors that are point-wise fixed by every ρ(h), h ∈ H. Define r ρ (H) := rank(ρ(H)); then r ρ (H) = 1/\|H\| h∈H χ ρ (h), where χ ρ denotes the character of ρ. For any subset S ≤ G define \|S := 1/ \|S\| s∈S \|s to be the uniform superposition over the elements of S. The standard method [GSVV04] starts from 1/ \|G\| g∈G \|g \|0 . It then queries f to get the superposition 1/ \|G\| g∈G \|g \|f (g) . The state becomes a mixed state given by the density matrix σ G H = 1 \|G\| g∈G \|gH gH\| if the second register is ignored. Applying QFT G to σ G H gives the density matrix \|H\| \|G\| ρ∈Ĝ dρ i=1 \|ρ, i ρ, i\| ⊗ ρ * (H), where ρ * (H) operates on the space of column indices of ρ. The probability distribution induced by this base change is given by P (observe ρ) = dρ\|H\|rρ(H) \|G\| . It is easy to see that measuring the rows does not furnish any new information: indeed, the distribution on the row indices is a uniform distribution 1/d ρ . The reduced state on the space of column indices on the other hand can contain information about H: after having observed an irrep ρ and a row index i, the state is now collapsed to ρ * (H)/r ρ (H). From this state we can try to obtain further information about H via subsequent measurements. Finally, we mention that Fourier sampling on k ≥ 2 registers can be defined in a similar way. Here one starts off with k independent copies of the coset state and applies QFT ⊗k G to it. In the next section, we describe the representation theory of the Weyl-Heisenberg groups. An efficient implementation of QFT G is shown in Appendix A. The irreducible representations In this section, we discuss the representation theory of G, where G ∼ = Z n+1 p ⋊ Z n p is a Weyl-Heisenberg group. From the properties of being an extraspecial group, it is easy to see that G has p 2n one dimensional irreps and p − 1 irreps of dimension p n . The one dimensional irreps are given by χ a,b (x, y, z) = ω (ax+by) p ,(10) where ω p = e 2πi/p and a, b ∈ Z n p . Note that χ a,b (H) = 1 \|H\| (x,y,z)∈H ω ax+by p = 1 \|S H \| (x,y)∈SH ω ax+by p .(11) Since S H is a linear space, this expression is non-zero if and only if a, b ∈ S ⊥ H . Suppose we perform a QFT on a coset state and measure an irrep label. Furthermore, suppose that we obtain a one dimensional irrep (although the probability of this is exponentially small as we show in the next section). Then this would enable us to sample from S ⊥ H . If this event of sampling one dimensional irreps would occur some O(n) times, we would be able to compute a generating set of S ⊥ H with constant probability. This gives us information about the conjugacy class of H and from knowing this, it is easy to see that generators for H itself can be inferred by means of solving a suitable abelian HSP. Thus, obtaining one dimensional irreps would be useful. Of course we cannot assume to sample from one dimensional irreps as they have low probability of occurring. Our strategy will be to "manufacture" one dimensional irreps from combining higherdimensional irreps. First, recall that the p n dimensional irreps are given by ρ k (x, y, z) = u∈Z n p ω k(z+yu) p \|u + x u\|,(12) where k ∈ Z p and k = 0. This representation is a faithful irrep and its character is given by χ k (g) = 0 for g = e and χ k (e) = p n . In particular, χ k (H) = p n /\|H\|. The probability of a high dimensional irrep occurring in Fourier sampling is very high (we compute this in Section 5). We consider the tensor product of two such high dimensional irreps. This tensor product can be decomposed into a direct sum of irreps of the group. A unitary base change which decomposes such a tensor product into a direct sum of irreps is called a Clebsch-Gordan transform, denoted by U CG . Clebsch-Gordan transforms have been used implicitly to bound higher moments of a random variable that describes the probability distribution of a POVM on measuring a Fourier coefficient. They have also been used in [Bac08a] to obtain a quantum algorithm for the HSP over Heisenberg groups of order p 3 , and in [Bac08b] for the HSP in the groups D n 4 as well as for Simon's problem. Our use of Clebsch-Gordan transforms will be somewhat similar. For the Weyl-Heisenberg group G, the irreps that occur in the Clebsch-Gordan decomposition of the tensor product of high dimensional irreps ρ k (g) ⊗ ρ l (g) depend on k and l. The Clebsch-Gordan transform for G is given by U CG : \|u, v → w∈Z n p ω l 2 (u+v)w p \|u − v, w for k + l = 0 \|u − v, ku+lv k+l for k + l = 0(13) If k + l = 0, then only one irrep of G occurs with multiplicity p n , namely ρ k (g) ⊗ ρ l (g) UCG → I p n ⊗ ρ k+l (g).(14) If k + l = 0, then all the one dimensional irreps occur with multiplicity one i.e., ρ k (g) ⊗ ρ l (g) UCG → ⊕ a,b∈Zp χ a,b (g).(15) Note, however, that the state obtained after Fourier sampling is not 1 \|H\| g∈H ρ k (g) ⊗ ρ l (g), but rather ρ k (H) ⊗ ρ l (H). When we apply the Clebsch-Gordan transform to this state, we obtain one dimensional irreps χ a,b (H) on the diagonal. Applying this to ρ −l (H) ⊗ ρ l (H) gives us (x,y,z),(x ′ ,y ′ ,z ′ )∈H u,v,w1,w2∈Z n p ω −l(yu+z)+l(y ′ v+z ′ )+ l 2 ((u+v)(w1−w2)+w1(x+x ′ ))× p \|u − v + x − x ′ , w 1 u − v, w 2 \| = (x,y,z),(x ′ ,y ′ ,z ′ )∈H u ′ ,w1,w2∈Z n p ω l 2 (−(y+y ′ )u ′ +2(z ′ −z)+w1(x+x ′ ))× p v ′ ω l 2 (v ′ (w1−w2+y ′ −y)) p \|u ′ + x − x ′ , w 1 u ′ , w 2 \|, where u ′ = u − v and v ′ = u + v. Since v ′ does not occur in the quantum state, the sum over v ′ vanishes unless w 2 = w 1 + y ′ − y. Therefore, the state is (x,y,z),(x ′ ,y ′ ,z ′ )∈H u ′ ,w1∈Z n p ω l 2 (−(y+y ′ )u ′ +2(z ′ −z)+w1(x+x ′ )) p \|u ′ +x−x ′ , w 1 u ′ , w 1 +y ′ −y\|. (16) The diagonal entries are obtained by putting x = x ′ and y = y ′ and since \|H\| = \|S H \|, we get z = z ′ . The diagonal entry is then proportional to (x,y,z)∈H u ′ ,w1∈Z n p ω l(−yu ′ +w1x) p . (17) Up to proportionality, this can be seen to be χ w1,−u ′ (H), a one dimensional irrep. The bottom line is that, although not diagonal in the Clebsch-Gordan basis, the resulting state's diagonal entries correspond to one dimensional irreps we are interested in. The quantum algorithm In this section, we present a quantum algorithm that operates on two copies of coset states at a time and show that it efficiently solves the HSP over G = Z n+1 p ⋊ Z n p , where the input is n and log p. The algorithm is as follows: 1. Obtain two copies of coset states for G. 2. Perform a quantum Fourier transform on each of the coset states and measure the irrep label and row index for each state. Assume that the measurement outcomes are high-dimensional irreps with labels k and l. With high probability the irreps are indeed both high dimensional and k + l = 0, when p > 2 (see the analysis below). When p = 2, there is only one high dimensional irrep which occurs with probability 1/2 and k + l = 0 always, since k = l = 1. We deal with this case at the end of this section. For now assume that p > 2 and k + l = 0. 3. If −k/l is not a square in Z p , then we discard the pair (k, l) and obtain a new sample. Otherwise, perform a unitary U α ⊗ I : \|u, v → \|αu, v , where α is determined by the two irrep labels as α = −k/l. This leads to a "change" in the irrep label 3 of the first state from k to −l. We can then apply the Clebsch-Gordan transform and obtain one dimensional irreps. 4. Apply a Clebsch-Gordan transform defined as U CG : \|u, v → w∈Z n p ω l 2 (u+v)w p \|u − v, w(18) to these states. 5. Measure the two registers in the standard basis. With the measurement outcomes, we have to perform some classical post-processing which involves finding the orthogonal space of a vector space. Now, we present the analysis of the algorithm. 1. In step 1, we prepared the state 1 \|G\| g \|g \|0 and apply the black box U f to obtain the state 1 \|G\| g \|g \|f (g) . After discarding the second register, the resulting state is \|H\| \|G\| \|gH gH\|. We have two such copies. 2. After performing a QFT over G on two such copies, we measure the irrep label and a row index. The probability of measuring an irrep label µ is given by p(µ) = d µ χ µ (H)\|H\|/\|G\|, where χ µ is the character of the irrep. If µ is a one-dimensional irrep, then the character is either 0 or 1 and so the probability becomes 0 or \|H\|/\|G\| accordingly. The character χ µ (H) = 0 if and only if µ = (a, b) ∈ S ⊥ H . Therefore, the total probability of obtaining a one dimensional irrep is \|H\|\|S ⊥ H \|/\|G\|. Now, we have that \|H\| = \|S H \| and so \|H\|\|S ⊥ H \| = p 2n since S ⊥ H is the orthogonal space in Z 2n p . Therefore, the total probability of obtaining a one dimensional irrep in the measurement is p 2n /p 2n+1 = 1/p. This is exponentially small in the input size (log p). Therefore, the higher dimensional irreps occur with total probability of 1 − 1/p. Since all of them have the same χ µ (H) = p n /\|H\|, each of them occurs with the same probability of 1/p. Take two copies of coset states and perform weak Fourier sampling and obtain two high dimensional irreps k and l. The state is then \|H\| 2 p 2n ρ k (H) ⊗ ρ l (H). In the rest, we omit the normalization \|H\| p n of each register. Therefore, the state is proportional to ρ k (H) ⊗ ρ l (H) = (x,y,z),(x ′ ,y ′ ,z ′ )∈H ω k(z+yu)+l(z ′ +y ′ v) p \|u + x, v + y u, v\|. (19) 3. We can assume that k and l are such that k + l = 0 since this happens with probability (p − 1)/p 2 . Now, choose α = −k l . Since the equation lx 2 + k = 0 has at most two solutions for any k, l ∈ Z p , for any given k, l chosen uniformly there exist solutions of the equation lx 2 + k = 0 with probability 1/2. Perform a unitary U α : \|u → \|αu on the first copy. The first register becomes proportional to U α ρ k (H)U † α = (x,y,z)∈H ω k(z+yu) p \|α(u + x) αu\| = (x,y,z)∈H,u1∈Z n p ω k α 2 (z1+y1u1) p \|u 1 + x 1 u\| = ρ k α 2 (φ α (H)),(20) where (x 1 , y 1 , z 1 ) = φ α (x, y, z) = (αx, αy, α 2 z) and u 1 = αu. It can be seen easily that φ α is an isomorphism of G for α = 0 and hence φ α (H) is subgroup of G. In fact, φ α (H) is a conjugate of H since S φα(H) = S H (since if (x, y) ∈ S H , then so is every multiple of it i.e., (αx, αy) ∈ S H ). Thus, we have obtained an irrep state with a new irrep label over a different subgroup. But this new subgroup is related to the old one by a known transformation. In choosing the value of α as above, we ensure that k/α 2 = −l and hence obtain one dimensional irreps in the Clebsch-Gordan decomposition. 4. We now compute the state after performing a Clebsch-Gordan transform U CG on the two copies of the coset states, i.e., perform the unitary given by the action U CG : \|u, v −→ w∈Z n p ω l 2 (u+v)w p \|u − v, w .(21) The initial state of the two copies is ρ −l (φ α (H)) ⊗ ρ l (H) = (x 1 ,y 1 ,z 1 )∈φα(H),(x ′ ,y ′ ,z ′ )∈H u,v∈Z n p ω −l(z1+y1u)+l(z ′ +y ′ v) p \|u + x 1 , v + x ′ u, v\|. The resulting state after the transform is (x 1 ,y 1 ,z 1 )∈φα(H),(x ′ ,y ′ ,z ′ )∈H u,v,w1,w2∈Z n p ω −l(z1+y1u)+l(z ′ +y ′ v)+ l 2 (u+v)(w1−w2)+(x1+x ′ )w1 p × \|u − v + x 1 − x ′ , w 1 u − v, w 2 \| = (x 1 ,y 1 ,z 1 )∈φα(H),(x ′ ,y ′ ,z ′ )∈H u ′ ,v ′ ,w1,w2∈Z n p ω −l(z1+y1 u ′ +v ′ 2 )+l(z ′ +y ′ v ′ −u ′ 2 )+ l 2 (v ′ )(w1−w2)+(x1+x ′ )w1 p × \|u ′ + x 1 − x ′ , w 1 u ′ , w 2 \|, Note that the term in the squared brackets is non-zero only when (v + (1 − α)ŷ, u + (1 − α)x) lies in S ⊥ H . This means that if we measure the above state we obtain pairs (u, v) such that (u + (1 − α)x, v + (1 − α)ŷ) ∈ S ⊥ H . This can be used to determine both S ⊥ H (and hence S H ) and (x,ŷ). Repeat this O(n) times and obtain values for u and v by measurement. 5. From the above, say we obtain n + 1 values (u 1 , v 1 ), . . . , (u n+1 , v n+1 ). Therefore, we have the following vectors in S ⊥ H . (u 1 + (1 − α 1 )x, v 1 + (1 − α 1 )ŷ), (u 2 + (1 − α 2 )x, v 2 + (1 − α 2 )ŷ), . . . . . . (u n+1 + (1 − α n+1 )x, v n+1 + (1 − α n+1 )ŷ). The affine translation can be removed by first dividing by (1 − α i ) and then taking the differences since S ⊥ H is a linear space. Therefore, the following vectors lie in S ⊥ H : (u ′ 1 , v ′ 1 ) = ( u 1 (1 − α 1 ) − u n+1 (1 − α n+1 ) , v 1 (1 − α 1 ) − v n+1 (1 − α n+1 ) ), (u ′ 2 , v ′ 2 ) = ( u 2 (1 − α 2 ) − u n+1 (1 − α n+1 ) , v 2 (1 − α 2 ) − v n+1 (1 − α n+1 ) ), . . . . . . (u ′ n , v ′ n ) = ( u n (1 − α n ) − u n+1 (1 − α n+1 ) , v n (1 − α n ) − v n+1 (1 − α n+1 ) ). With high probability, these vectors form a basis for S ⊥ H and hence we can determine S H efficiently. This implies that the conjugacy class and hence the subgroup H 0 is known. It remains only to determine (x,ŷ). We can set (x,ŷ) = (1 − α 1 ) −1 (u 1 − u ′ 1 , v 1 − v ′ 1 ) since the conjugating element can be determined up to addition by an element of S ⊥ H . H can be obtained with the knowledge of H 0 and (x,ŷ). Finally, for completeness we consider the case p = 2. Assume that after Fourier sampling we have two high dimensional irreps with states given by ρ 1 (H) ⊗ ρ 1 (H) = (x,y,z),(x ′ ,y ′ ,z ′ )∈H,u,v∈Z n 2 (−1) z+z ′ +yu+y ′ v \|u + x, v + x ′ u, v\|. (23) The Clebsch-Gordan transform is given by the base change: \|u, v → w∈Z n 2 (−1) wv \|u + v, w .(24) Applying this to the two states, we obtain (in a similar manner as above) (x,y,z)∈H,u,v∈Z n 2 (−1) z+vx   (x ′ ,y ′ ,z ′ )∈H (−1) uy ′ +vx ′   \|u + x, v + y u, v\|. (25) The inner sum is non-zero if and only if (u, v) ∈ S ⊥ H . Thus, measuring this state gives us S ⊥ H from which we can find S H . We cannot determine H directly from here as in the case p > 2. But since we know S H , we know the conjugacy class of H and we can determine the abelian group HG ′ which contains H. This group is obtained by appending the elements of S H with every element of G ′ = Z 2 i.e., for (x, y) ∈ S H we can say that (x, y, 0) and (x, y, 1) are in HG ′ . Once we know HG ′ , we now restrict the hiding function f to the abelian subgroup HG ′ of G and run the abelian version of the standard algorithm to find H. In summary, we have shown the following result: Sketch of proof. From the above discussion follows that O(n) iterations of Steps 1.-4. in the algorithm will lead to system of equations in Step 5. that with constant probability has a unique solution. The number of queries in each iteration is constant and the computational complexity of each of these steps can be upper bounded as follows: O(n log p log log p) operations for each computation of QFT over G as described in Appendix A. The transform U α and the Clebsch-Gordan transform U CG can easily be implemented using arithmetic modulo p and QFTs over Z p , both of which can be done in O(log p log log p) elementary quantum operations. Hence the running time of the quantum part of the algorithm can be upper bounded by O(n 2 log p log log p) operations and the number of queries by O(n). The overall running time is dominated by the cost for classical post-processing which consists in computing the kernel of an n × n matrix over Z p . This can be upper bounded by O(n 3 ) arithmetic operations over Z p for the Gaussian elimination, leading to a total bit complexity of O(n 3 log p log log p 2 O(log * log p) ) operations when using the currently fastest known algorithm for integer multiplication [Für07]. Conclusions Using the framework of coset states and non-abelian Fourier sampling we showed that the hidden subgroup problem for the Weyl-Heisenberg groups can be solved efficiently. In each iteration of the algorithm the quantum computer operates on k = 2 coset states simultaneously which is an improvement over the previously best known quantum algorithm which required k = 4 coset states. We believe that the method of changing irrep labels and the technique of using Clebsch-Gordan transforms to devise multiregister experiments has some more potential for the solution of HSP over other groups. Finally, this group has importance in error correction. In fact, the state we obtain after Fourier sampling and measurement of an irrep is a projector onto the code space whose stabilizer generators are given by the generators of H. In view of this fact, it will be interesting to study the implications of the quantum algorithm derived in this paper to the design or decoding of quantum error-correcting codes. · · · · · · · · · · · · . . . g s s Fig. 1. QFT for the Weyl-Heisenberg group. The QFT gates shown in the circuit are QFTs for the cyclic groups Zp. Each of these QFTs can be implemented approximately [Kit97,HH00] or exactly [MZ04], in both cases with a complexity bounded by O(log p log log p). It should be noted that the wires in this circuit are actually p-dimensional systems. The meaning of the controlled gates where the control wire is an open circle is that the operation is applied to the target wire if and only if the control wire is in the state \|0 . The meaning of the controlled P gates where the control wire is a closed circle here means that the gate P k is applied in case the control wire is in state \|k with k = 0, and P0 = Ip. Here P k is the permutation matrix for which QFT (k) = P k QFT holds. The complexity of this circuit can be bounded by O(n log p log log p). B Changing labels of irreducible representations In this section, we describe the technique of changing labels of irreducible representations (irreps) in a more abstract, representation theoretic, fashion. We consider a situation slightly more general than the Weyl-Heisenberg groups considered in the paper, namely for semidirect products of the form G = A ⋊ φ B, where A is an Abelian group, B is an arbitrary finite group, and φ : B → Aut(A). We make some further assumptions regarding the irreps of G that arise during Fourier sampling. First, note that in general there might be some irreps of G that arise as inductions [Ser77,Hup83] of irreps of A to G. Suppose that, with high probability, we sample only such irreps, so that we can restrict our attention to this case. This happens for the Weyl-Heisenberg groups discussed in this paper. Other examples are the groups isomorphic to Z n p ⋊ Z p studied in [BCD05] and the affine groups [MRRS04] which are isomorphic to Z p ⋊ Z p−1 . After Fourier sampling and measurement of an irrep label we have the state ρ k (H), where ρ k is an irrep of G and k is its label. We want to apply an operator U B to this state in order to change it to a state ρ k ′ (H ′ ) corresponding to an irrep with label k ′ , possibly with respect to a different subgroup H ′ . In the following we show how this can be done if ρ k (H) = (χ k ↑ G)(H), i. e., if ρ k is an induction of an irrep χ k of A to G. The possible labels k ′ that can be obtained depend on the automorphism group of B, namely on those automorphisms of B that can be extended to automorphisms of G. we can prove the following lemma. Lemma 1. Two subgroups H 1 and H 1 are conjugate if and only if S H1 = S H2 . Lemma 4 . 4If H is an abelian subgroup which does not contain G ′ , then \|S H \| = \|H\|. Lemma 5 . 5Let H be an abelian subgroup which does not contain G ′ . There exists a subgroup H 0 conjugate to H, where H 0 = {(x, y, xy/2)\|(x, y) ∈ S H }. Theorem 1 . 1For n≥1, and p≥2 prime, the hidden subgroup problem for the Weyl-Heisenberg group G of order p 2n+1 can be solved on a quantum computer with O(n) queries. The time complexity of the quantum algorithm can be bounded by O(n 3 log p) operations 4 and the algorithm uses at most k = 2 coset states at the same time. Recall that the most general way to extract classical information from quantum states is given by means of positive operator valued measures (POVMs)[NC00]. This can be obtained by combining the random basis method[Sen06] with the derandomization results of[AE07]. We refer to Appendix B for a description of a technique that allows to change the labels of irreps of semidirect products that are more general than the Weyl-Heisenberg group. [x(v−2αŷ)−y(u−2αx)] p\|u + x, v + y u, v\|. Ignoring factors growing as log log p or weaker. AcknowledgmentsWe thank Sean Hallgren and Pranab Sen for useful comments and discussions.where u ′ = u − v and v ′ = u + v. Notice that v ′ occurs only in the phase and not in the quantum states. Therefore, collecting the terms with v ′ we get v ′ ω l 2 (y ′ −y1+w1−w2) p .(This term is non-zero only when y ′ −y 1 +w 1 −w 2 = 0. Hence w 2 = w 1 −(y 1 −y ′ ). Substituting this back in the equation, we get (x 1 ,y 1 ,z 1 )∈φα (H),(x ′ ,y ′ ,z ′ )∈HReusing the labels u and v by putting u = u ′ and v = w 1 − (y 1 − y ′ ), we obtainThis can be written asSince H is abelian, x 1 y ′ − x ′ y 1 = 0. Now consider the subgroup H 0 defined in the previous section. Let g = (x,ŷ,ẑ) be an element such that H g = H 0 . As discussed in Sec. 2, (x,ŷ) are unique up to an element of S ⊥ H andẑ is any element in Z p . Now, when (x ′ , y ′ , z ′ ) ∈ H is conjugated with g, it gives (x ′ , y ′ , z ′ +xy ′ −ŷx ′ ) = (x ′ , y ′ , x ′ y ′ /2) ∈ H 0 . Therefore, z ′ − x ′ y ′ /2 = x ′ŷ −xy ′ . In order to obtain H 0 from φ α (H) we need to conjugate by φ α (x,ŷ,ẑ). Therefore, z 1 − x1y1 2 = α(ŷx 1 −xy 1 ). Incorporating this into the above expression, we getSeparating the sums over (x, y) and (x ′ y ′ ) we getA QFT for the Weyl-Heisenberg groupsWe briefly sketch how the quantum Fourier transform (QFT) can be computed for the Weyl-Heisenberg groups G n = Z n+1 p ⋊ Z n p . An implementation of the QFT for the case where p = 2 was given in[Høy97]. This can be extended straightforwardly to p > 2 as follows. Using Eq. (9), we obtain that the QFT for G n is given by the unitary operatoran,bn,xn,yn,z∈ZpThe matrix U is given bywhere I p is the p dimensional identity matrix,and QFT First, recall that for χ k ∈Â, the image of an element (a, b) ∈ G under the induction of χ k to G is given byIn order to further simplify this expression, we now suppose that we can extend the automorphism β to an automorphism of the whole group in the form γ = (α, β) ∈ Aut(G), where α ∈ Aut(A). We derive some conditions that α has to satisfy in order for this extension to be possible. First, we have thatThis condition becomesNote that in the above equation, since α and φ t are elements of Aut(A) for all t, we write their product acting on a ∈ A as (αφ t )(a). From Eq. (33) we obtain thatfor all b ∈ B. This means that α ∈ N Aut(A) (Im(φ)) i.e., α lies in the normalizer of Im(φ), the image of φ in Aut(A). Therefore, we need to pick the pair (α, β) such that the condition in Eq. (34) holds. 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[ "THE RIEMANNIAN HEBBARKEITSSÄTZE FOR PSEUDORIGID SPACES", "THE RIEMANNIAN HEBBARKEITSSÄTZE FOR PSEUDORIGID SPACES" ]	[ "João N P Lourenç " ]	[]	[]	We prove Riemann's theorems on extensions of functions over certain mixed characteristic analytic adic spaces, first introduced by Johansson and Newton. We use these results to reprove a theorem of de Jong identifying global sections of an O K -flat normal formal scheme, locally formally of finite type over O K , with locally powerbounded sections over the generic fibre.	null	[ "https://arxiv.org/pdf/1711.06903v1.pdf" ]	119,694,533	1711.06903	477dbc236bba798f54cfbeb1476d35d4b683f162	THE RIEMANNIAN HEBBARKEITSSÄTZE FOR PSEUDORIGID SPACES 18 Nov 2017 João N P Lourenç THE RIEMANNIAN HEBBARKEITSSÄTZE FOR PSEUDORIGID SPACES 18 Nov 2017arXiv:1711.06903v1 [math.AG] We prove Riemann's theorems on extensions of functions over certain mixed characteristic analytic adic spaces, first introduced by Johansson and Newton. We use these results to reprove a theorem of de Jong identifying global sections of an O K -flat normal formal scheme, locally formally of finite type over O K , with locally powerbounded sections over the generic fibre. Introduction In complex analysis, the Riemannian Hebbarkeitssätze 1 concern the possibility of globally extending certain meromorphic functions on a complex manifold X with singularities along an analytic subset Z. Namely, the first (resp. second) Hebbarkeitssatz states that a locally bounded (resp. holomorphic) function f on X \ Z has a unique extension to X, if Z has codimension at least 1 (resp. at least 2). The analogous statements hold true for normal rigid spaces over a complete nonarchimedean field, by the work of Bartenwerfer (cf. [3] for a proof of the first Hebbarkeitssatz) and Lütkebohmert (cf. [29,Satz 1.6] for a proof of the second Hebbarkeitssatz and the first Hebbarkeitssatz in the case of absolute normality). If one works instead with adic spaces, one is then able to produce interesting examples of mixed characteristic analytic adic spaces which do not live over any field. One such example is provided by the class of pseudorigid spaces over a complete discrete valuation ring O K of mixed characteristic. 1 In the English literature, these results usually go by the name of Riemann's removable singularities lemma and Hartogs' theorem, whereas in the German literature these are called Erster Riemmanscher Hebbarkeitssatz and Zweiter Riemannscher Hebbarkeitssatz, respectively. We will adopt the latter terminology, not simply for its greater homogeneity, but also because the ordinals serve as a mnemonic for the lower bounds on the codimension of the appearing singularity loci. These spaces are locally isomorphic over Spa O K = Spa (O K , O K ) to analytic open subsets of a formal scheme X fft over O K , here regarded as an adic space (for a slightly different formulation, cf. Definition 4.3). Although the author was already working with this category during the fall of 2016, it was first introduced in the literature by Johansson and Newton in [22], to whom we essentially owe the designation of these spaces and calling our attention to some results in [1]. We can now state our main result: Theorem 1. Let X be a normal pseudorigid space over O K and Z be a Zariski closed subset of X. The following statements hold: First Hebbarkeitssatz: If codim X Z ≥ 1, then the restriction map O + X (X) → O + X (X \ Z) is an isomorphism of rings, i.e. every locally powerbounded function f defined on the complement of Z admits a unique locally powerbounded extension to X. Second Hebbarkeitssatz: If codim X Z ≥ 2, then the restriction map O X (X) → O X (X \ Z) is an isomorphism of rings, i.e. every function f defined on the complement of Z admits a unique extension to X. We should remark that some notions in the above statement such as normality, codimension and Zariski closed subsets are not defined for general adic spaces. We will define all these notions for sufficiently well behaved analytic adic spaces, called Jacobson adic spaces, in the course of Section 2. Let us briefly describe our proof of the Hebbarkeitssätze. Due to the rigid case, one is reduced to working in the case where X = Spa A is the adic spectrum of an O K -flat normal pseudoaffinoid algebra (cf. Definition 4.1) and Z is contained in the special fibre. The general strategy follows then Lütkebohmert's article. As a first step, we verify the statements for elementary pseudoaffinoid algebras D n X 1 , . . . , X r , where D n = O K [[T ]] π/T n [1/T ], via direct computations with norms. The second ingredient is an adaptation of Noether's normalisation lemma: Proposition 2. Let A be an O K -flat pseudoaffinoid algebra, such that π / ∈ A × , and let T be a topologically nilpotent unit of A. If φ : k((T )) X 1 , . . . , X r → A/πA is a finite injection of k((T ))affinoid algebras, then it can be lifted to a finite injection Φ : D n X 1 , . . . , X r → A π/T n of pseudoaffinoid algebras, for a sufficiently large natural number n. The remaining tool consists of holomorphic, resp. bounded transfer lemmata (cf. Lemma 5.2 and Lemma 5.3) which work as analogues of [29,Hilfssatz 1.7]. For the second Hebbarkeitssatz, the aforementioned steps immediately yield the result. As for the first Hebbarkeitssatz, in virtue of the slight weaknesses of Lemma 5.3, we are forced to develop additional arguments, such as finding Noether normalisations which are unramified at certain given regular primes. This was not the route taken by Lütkebohmert in [29], as he uses a result of Kiehl about rigid spaces for which we could not find a pseudorigid analogue. Next, we provide some applications of these results to formal schemes. For sufficiently well behaved formal schemes, one can show that a locally powerbounded function on the set of analytic points extends to the total space. Proposition 3. Let X be a normal excellent formal scheme, which is nowhere discrete. Then, the restriction map O X (X) → O + Xa (X a ) is an isomorphism of topological rings. In the above proposition, we regard a formal scheme as an adic space and denote by X a its set of analytic points. The claim is clear for principal ideals of definition and can be reduced to that case combining Zariski's Main Theorem with the theorem on formal functions. As a corollary of this and the first Hebbarkeitssatz, we obtain a new proof of a theorem of de Jong (cf. [11,Theorem 7.4.1]). Showing this exact statement was actually our original motivation, before discovering it had already been proven by de Jong. One also has the following corollary: Corollary 4. (Scholze) Let O K be a complete discrete valuation ring with an algebraically closed residue field k. Let C be the category of O K -flat normal formal schemes, locally formally of finite type over O K . Let D be the category of triples (X, Y, p), where X is a K-rigid space, Y is a perfect k-scheme and p : \|X\| → \|Y \| is a map of topological spaces (here, \|X\| means just the classical rigid points). Then the functor F : C → D given by X → (X η , (X red ) perf , sp X ) is fully faithful. In the above corollary X η denotes the generic fibre of X, (X red ) perf the perfection of its reduction and sp X the classical specialisation mapping. This is relevant to the study of integral models of local Shimura varieties, as one may then recover it just by knowing the perfection of its special fibre and the specialisation mapping. The reason why we consider the perfection here is that, if M denotes a formal scheme representing a moduli problem of p-divisible groups as in [30,Chapter 3], then the perfection of its reduction is an affine Deligne-Lusztig variety, by [38,Proposition 3.11]. The ideas behind this paragraph were explained in more detail in Scholze's talk at the Super Arbeitsgemeinschaft held in Bonn in honor of Michael Rapoport. Outline. In Section 2, we review some facts on coherent sheaves over locally noetherian analytic adic spaces and apply these to define Zariski closed subsets globally in terms of coherent sheaves of ideals. In Section 3, we introduce the class of Jacobson adic spaces, whose behaviour resembles that of rigid spaces. We give a sufficient criterion based on work of [1] to determine whether an analytic adic space is Jacobson. Then we define absolute properties such as reducedness, normality and regularity for Jacobson adic spaces, and also define the codimension of a Zariski closed subset. In Section 4, we introduce pseudorigid spaces over a complete DVR and establish some of their properties, including our version of the Noether normalisation lemma. In Section 5, we give our proofs of the Hebbarkeitssätze and the final section is concerned with the applications to formal schemes alluded to above. Notation. We fix the following notation for the rest of the paper: K denotes a discretely valued complete nonarchimedean field, O K its ring of integers, π K a uniformiser of O K and k the residue field of O K . We will always use the letters A and B (resp. X, Y and Z) to denote strongly noetherian complete Tate rings (resp. locally noetherian analytic adic spaces). Acknowledgements. I want to heartily thank my advisor Peter Scholze for his guidance and enthusiasm during this work, and for inviting me to give two talks on the subject at the Arbeitsgemeinschaft Arithmetische Geometrie in Bonn. Evidently, my gratitude extends to everyone present at the talks for their attention and pertinent questions. I am also indebted to Mafalda Santos, Johannes Anschütz and Yichao Tian for helpful conversations regarding this work. Coherent sheaves of O X -modules In this section, we recall the notion of coherent sheaves of O X -modules and review their main properties, following [18]. This will be put to use to obtain a better understanding of the notions of closed immersions and finite morphisms. During the whole section, we will use the notation established during the introduction. Definition 2.2. We say that a sheaf of O X -modules F is coherent if there is an affinoid cover X = i U i such that, for all i, F Ui ∼ = M i for some finite O X (U i )-module M i . Over affinoids, we do not actually encounter any new coherent sheaves: Now we turn to our application of this theorem, namely to a better comprehension of closed immersions and finite morphisms, defined in [20,Section 1.4]. Let us first quickly observe that, given a coherent sheaf of O X -algebras A, we may construct its relative adic spectrum Spa A as the adic space representing the functor Hom(A, −) : Ad X → Set which maps an adic space Y over X to the set Hom OX −Alg (A, f * O Y ). Hence it becomes clear that every finite adic space over X is the relative adic spectrum of a coherent sheaf of O X -algebras, unique up to isomorphism. As a corollary, one obtains Corollary 2.4. Under the hypothesis of Theorem 2.3, every adic space Y finite over X is of the form Spa (B, B + ) for some finite A-algebra B. In particular, closed immersions into X correspond bijectively to ideals of A. We may also use coherent ideal sheaves to define Zariski closed subsets: Definition 2.5. Let X be a locally noetherian analytic adic space. A subset Z ⊆ X is Zariski closed if it equals the support V (I) = {x ∈ X : I x = O X,x } of some coherent sheaf of ideals I on X. Obviously, there are several coherent ideals with the same support. The usual way around this in algebraic geometry is to focus on radical ideals, but in the adic realm we do not know if reducedness is stable under rational localisations. We do have the following safeguard, whose proof we leave to the reader. Proposition 2.6. Keep the notation of Theorem 2.3 and assume that A is a Jacobson ring. Then two ideals I i , i = 1, 2 define the same Zariski closed subset on X if and only if they have the same radical. Jacobson adic spaces The aim of this section is to define a new class of analytic adic space containing a subset of distinguished points controlling several of its properties, very much like classical points of rigid spaces. Definition 3.1. We say that a strongly noetherian complete Tate ring A is a Jacobson-Tate ring if it satisfies the following properties: (1) Every residue field of A is a complete non-archimedean field. (2) For every topologically of finite type A-algebra B, the induced map Spec B → Spec A respects maximal ideals. The reason behind the naming of these rings resides in [9, Chapitre V §3.4, Théorème 3 (ii)], which is actually an equivalence. Using these rings, we may now define the class of Jacobson adic spaces. Definition 3.2. We say that X is a Jacobson adic space if it is locally of the form Spa (A, A + ), where A is a Jacobson-Tate ring. We define its Jacobson-Gelfand spectrum JG(X) as the subset of rank 1 points x ∈ X for which there is an open affinoid Spa (A, A + ) ⊆ X with A Jacobson-Tate such that supp x ⊆ A is a maximal ideal. Proposition 3.3. Let A be a Jacobson-Tate ring. Then the following hold: (1) Every topologically of finite type A-algebra B is a Jacobson-Tate ring. (2) Let B be a rational localisation of A. Then every maximal ideal n ⊆ B is the extension of a unique maximal ideal m ⊆ A and the natural map A m → B mB is an isomorphism. In particular, if x ∈ X = Spa (A, A + ) is the unique rank 1 point supported at m, then the canonical map A m → O X,x is an isomorphism. (3) For every ring of integral elements A + ⊆ A, the support map JG(Spa (A, A + )) → Spec A is an embedding onto Max A and the inclusion JG(Spa (A, A + )) → Spa (A, A + ) is dense. In particular, A is a Jacobson ring. Proof. For the first claim, let n ⊆ B be a maximal ideal. This pulls back to a maximal ideal m of A, whose quotient is a complete non-archimedean field. Thus, we may assume that A is a complete non-archimedean field and B/n must be a field, by the Noether normalisation lemma for affinoid algebras. One checks similarly the second condition for B being a Jacobson-Tate ring. Since rational localisations are topologically of finite type homomorphisms, it is clear that n lies over a maximal ideal m ⊆ A. But if A = K is a complete non-archimedean field, any nonzero rational localisation is isomorphic to K. This shows that n = mB. As for the isomorphy claim, one shows that A/m n → B/m n B is an isomorphism by induction, using flatness of A → B (cf. [18, Proposition 3.3.16]). The last claim follows immediately from the first and the second claims. As for A being Jacobson, it suffices to show density of the maximal spectrum inside the spectrum of every homomorphic image of A, but the adic topology is finer than the Zariski topology. Before developing some more theory around Jacobson adic spaces, let us give a sufficient criterion for an analytic adic space to be Jacobson. We need to introduce some material from [1]. Definition 3.4. A complete noetherian adic ring R is called a valuative order if it is local, integral and one-dimensional. The importance of these rings is revealed by the following result. (1) R/p is a valuative order. (2) p R •• and dim(R/p) = 1. (3) p determines a closed point of X. Proposition 3.6. Assume A has a noetherian ring of definition A 0 ⊆ A whose reduction A 0 /A •• is a Jacobson ring. Then, A is a Jacobson-Tate ring. Proof. The fact that every homomorphic field of A is non-archimedean can be found in [18, Proposition 2.2.10] and it only uses noetherianness of A 0 . For the second property, we let B be a topologically of finite type A-algebra and choose a noetherian ring of definition B 0 of B which is topologically finitely generated over A 0 . Appealing to Proposition 3.5, we simply have to show that the map Spec B 0 → Spec A 0 respect prime ideals whose residue ring is a valuative order. Without loss of generality, assume that B 0 is a valuative order and A 0 injects in the former ring. Now [1, Proposition 1.11.2] yields finiteness of A 0 → B 0 , so A 0 must be local, integral and 1-dimensional. To conclude this section, we will now transfer some ring-theoretic quantities and absolute properties to the world of Jacobson adic spaces. Definition 3.7. Let X be a Jacobson adic space. Its Krull dimension dim.Kr X is the quantity sup x∈JG(X) dim O X,x . Proposition 3.8. Let (A, A + ) be a Jacobson-Tate pair and let X be the corresponding adic space. Then, the equality dim.KrX = dimA holds. Proof. Use Proposition 3.3, part 2. Remark 3.9. When A + = A • , the Krull dimension may in general differ from the spectral dimension of Spa (A, A + ) (e.g., take A to be a complete non-archimedean field). If A + = A • , equality holds for affinoid algebras by [20, Lemma 1.8.6 (ii) and Corollary 1.8.8] and for pseudoaffinoid algebras, as we will see later on, but we do not know whether this is true in general. Definition 3.10. Let I be a coherent sheaf of ideals on a Jacobson adic space X. We say that I has height d at Proof. First we observe that the height of I can be computed by taking the supremum of ht Am I m over all maximal ideals m ⊆ A. Then use the isomorphism A m ∼ = O X,x , where x ∈ JG(X) corresponds to m, and notice that it identifies I m with I x , where I = I. x ∈ JG(X) if ht I x = d, where I x is the maximal-adic completion of I x ⊆ O X,x . The height of I is the quantity ht I = sup x∈JG(X) ht I x ∈ Z ≥0 ∪ {−∞}. Appealing to Proposition 2.6, we may now extend these notions to Zariski closed subsets. Definition 3.12. Let X be a Jacobson adic space and Z ⊆ X a Zariski closed subset defined by a sheaf of coherent ideals I. We say that Z has codimension d at z ∈ JG(Z) if I has height d at z and we define its codimension as codim X Z = ht I. Next we show how to extend an absolute property of rings to the class of Jacobson adic spaces. Definition 3.13. Let P be a property of rings. We say that a Jacobson adic space X has property P if, for all x ∈ JG(X), O X,x satisfies property P . A priori, this may not be such a useful notion, for we lack an understanding of whether affinoid sections satisfy P or not. In our case, this will not be a concern. Proposition 3.14. Let (A, A + ) be a Jacobson-Tate pair, X denote the corresponding adic space and P ∈ {reduced, normal, regular} be a property of rings. If X satisfies P , then so does A. If A is an excellent ring, the converse holds. Proof. This follows from a direct application of [ Remark 3.15. The statement holds more generally if one assumes that the formal fibres of A are regular and P is a local property for regular maps. Other examples of such properties can be found in [15]. Pseudorigid spaces In this section, we introduce the notion of pseudorigid spaces, which first appeared in [22] under the same name, but was also developed independently by the author during this work. Definition 4.1. A pseudoaffinoid 2 O K -algebra is a complete Tate O K -algebra A having a noetherian ring of definition A 0 , which is formally of finite type over O K . A homomorphism of pseudoaffinoid O K -algebras A and B is a continuous O K -homomorphism φ : A → B. We will need the following results on homomorphisms of pseudoaffinoid O K -algebras. Lemma 4.2. Let A be a pseudoaffinoid O K -algebra. (1) Every topologically of finite type A-algebra B is a pseudoaffinoid O K -algebra and if φ : Hence, every rational subset of Spa (A, (A, A • ) → (B, B + )A • ), where A is a pseudoaffinoid O K -algebra, is of the form Spa (B, B • ), where B is a pseudoaffinoid O K -algebra. We now define the following subcategory of analytic adic spaces. Definition 4.3. A pseudorigid space over O K is a Spa (O K )-analytic adic space X, which is locally Spa (O K )-isomorphic to Spa A, where A is a pseudoaffinoid O K -algebra. The category of pseudorigid spaces over O K is the full subcategory of Spa (O K )-adic spaces whose objects are pseudorigid spaces. Now Lemma 4.2 tells us that this category is stable under open immersions and the functor A → Spa (A) from the category of pseudoaffinoid algebras to the category of pseudorigid spaces, is fully faithful. Example 4.4. Let X be a formal scheme, formally of finite type over O K , which we regard here as an adic space in the sense of [19] or [20]. Then the set of analytic points X = X a is a pseudorigid space over O K . For instance, if X = Spf O K [[T ]] , then X is a quasicompact analytic adic space, covered by two affinoids, namely Spa K T /p and Spa O K [[T ]] p/T [1/T ]. If O K is a mixed characteristic DVR, then the latter ring does not live over a field. Further discussion regarding this example can be found in [33]. Before moving on, let us recall that Spa O K consists of two points: the closed point s = Spa k and the generic point η = Spa K. By [20, Proposition 1.2.2], the fibre products X s = X × Spa OK Spa k and X η = X × Spa OK Spa K exist and we call these the special, resp. generic fibre of a pseudorigid space X. It is not very difficult to see that X s is the Zariski closed adic subspace defined by the ideal sheaf πO X and X η its open complement. Proposition 4.5. The generic fibre X η of a pseudorigid space X over O K is a K-rigid space. If X = Spa A with A pseudoaffinoid, then X s is a rigid space over a Laurent series field k((T )). Proof. We may assume that X = Spa A with A being a pseudoaffinoid algebra over O K and π ∈ A × . Then the canonical map K → A is topologically of finite type by Proof. Let R be a formally of finite type O K -algebra. Its reduction R/R •• is a finite type k-algebra, so the first assertion is a consequence of 3.6. As for excellency of R, this is a consequence of [35,Proposition 7] and [36,Theorem 9]. In order to understand pseudoaffinoid algebras better, we will need to work with some very concrete examples. For legibility purposes, we will often shorten D λ X 1 , . . . , X r to D λ,r . Lemma 4.8. For any pseudoaffinoid O K -algebra A, there is a sufficiently small λ ∈ Q >0 such that A is a topologically of finite type D λ -algebra. Proof. Let s ∈ A be a topologically nilpotent unit and choose a sufficiently large n such that π n s −1 ∈ A • . This allows us to define a continuous O K -homomorphism D 1 n → A by mapping T to s, and we are now done by Lemma 4.2. Remark 4.9. This implies that pseudoaffinoid algebras over an equicharacteristic DVR are just affinoid algebras over some field k((T )). Indeed, writing O K = k[[π]], which is always possible in this case, we obtain D λ = k((T )) π, π m /T n . Repeating the same argument, we even conclude that O K → D λ,r is a regular map, so D λ,r is a regular ring by [15, Corollaire 6.5.2 (ii)]. Let now m be a maximal ideal of D λ,r . If m defines a point in the generic fibre, its height may be computed after rational localisation. But the generic fibre of Spa D λ,r is simply an r + 1-dimensional K-rigid semiopen annulus given by the inequalities \|π m \| ≤ \|T \| < 1. On the other hand, if m defines a point in the special fibre, then O K -flatness of D λ,r and Krull's Hauptidealsatz tell us that the ht m = ht m + 1, where m is the reduction of m modulo π. But D λ,r /π ∼ = k((T )) X 1 , . . . , X r is r-equidimensional. As for connectedness, this follows easily from O K -flatness of D λ,r and both its generic and special fibres being connected. Remark 4.11. We should also note that, if one follows step by step the proof of [22, Theorem 2.5.3], then one is capable of showing that D λ,r is a UFD whenever λ ∈ N. Since pseudoaffinoid algebras are catenary, the equidimensionality also holds for integral domains. Proof. Since all generalisations are vertical, we assume without loss of generality that X = Spa A, where A is a pseudoaffinoid domain. Due to Corollary 4.12, we know that dim A = dim B for every rational localisation of A. Notice that the special and generic fibres are specialising subsets which partition X, so we have the identity dim X = max{dim X s , dim X η }. By [20, Lemma 1.8.6], we deduce that dim X s = dim A/πA and dim X η = dim A, as long as the generic fibre is nonempty. But if this were the case, then πA = 0. In any case, the desired equality dim X = dim A = dim.Kr X holds. The next result is a version of Noether's normalisation lemma, which will be extremely useful for proving the Hebbarkeitssätze in the subsequent section. Proposition 4.14. (Noether levantada do chão) 3 Let A be an O K -flat pseudoaffinoid algebra, such that π / ∈ A × , and let T be a topologically nilpotent unit of A. If φ : k((T )) X 1 , . . . , X r → A/πA is a finite injection of k((T ))-affinoid algebras, then it can be lifted to a finite injection Φ : D n,r → A π/T n of pseudoaffinoid algebras, for a sufficiently large natural number n. Proof. Choose a ring of definition A 0 ⊆ A. Since its reduction modulo π is still a ring of definition, we deduce that (A/πA) • is integral over A 0 /(πA ∩ A 0 ). Let f i ∈ A, i = 1, . . . , r be lifts of φ(X i ), respectively. The previous paragraph gives us a monic polynomial p i ∈ A 0 [S] such that p i (f i ) ∈ π/T ki A 0 . By choosing a sufficiently large n and replacing A with the rational localisation A π/T n , we may assume that p i (f i ) ∈ A 0 . Therefore, we may define the map of pseudoaffinoid O K -algebras Φ : D n X 1 , . . . , X r → A by mapping X i to f i for all i = 1, . . . , r, and we notice that this obviously lifts the map φ. Let y j ∈ A, j = 1, . . . , k be topological generators of A as a D n -algebra. Since φ is a finite map of Huber pairs, we may find monic polynomials q j with coefficients in D • N X 1 , . . . , X r such that q j (y j ) ∈ π/T lj A 0 . Repeating the previous trick, we may also assume that q j (y j ) ∈ A •• . Now [20, Lemma 1.4.3] yields finiteness of Φ. To show injectivity, assume on the contrary that there is a nonzero x ∈ ker Φ. As φ is injective and A is π-torsion free, one duduces that x ∈ π k D N X 1 , . . . , X r for all k ≥ 1. But Krull's intersection theorem says that such an element is a zero divisor, which contradicts integrality of D n,r . Later we will also need the following: N ⊗ A A L → M ⊗ A A L . 3 This is a reference to the novel Levantado do Chão (eng. Raised from the Ground) from the Nobel Prize winning author José Saramago. This is motivated by the fact that we always imagine the special fibre depicted as the horizontal axis of the picture in [33, Figure 1 Hebbarkeitssätze In this section, we are going to prove the Riemannian Hebbarkeitssätze for pseudorigid spaces over O K . Let us begin by stating these theorems: Theorem 5.1. Let X be a normal pseudorigid space over O K and Z be a Zariski closed subset of X. The following statements hold: First Hebbarkeitssatz: If codim X Z ≥ 1, then the restriction map O + X (X) → O + X (X \ Z) is an isomorphism of rings, i.e. every locally powerbounded function f defined on the complement of Z admits a unique locally powerbounded extension to X. Second Hebbarkeitssatz: If codim X Z ≥ 2, then the restriction map O X (X) → O X (X \ Z) is an isomorphism of rings, i.e. every function f defined on the complement of Z admits a unique extension to X. For rigid spaces, these were proven in the seventies by Bartenwerfer (cf. [3, Riemann I]) and Lütkebohmert (cf. [29,Satz 1.6]). Hence, in the following pages, we may safely assume that O K is a mixed characteristic DVR, X = Spa A is the adic spectrum of an O K -flat normal pseudoaffinoid domain A with a connected special fibre, and Z is contained in the special fibre. 4 The following is a generalisation of [29, Hilfssatz 1.7]. Let T ⊆ Y be a Zariski closed subset such that codim Y (T ) ≥ 2. If the natural restriction map O X (X) → O X (X \ ϕ(T )) is an isomorphism, then so is O Y (Y ) → O Y (Y \ T ). Proof. Injectivity of the map is immediate, due to connectivity of Y and flatness of rational localisations. Now let b 1 , . . . , b n ∈ B span a basis of Q(B)/Q(A) and consider the free A-submodule M = Ab 1 + · · · + Ab n ⊆ B. By construction, B/M is an A-torsion module, so one can find a nonzero g ∈ A such that gB ⊆ M . This yields a factorisation of the multiplication-by-g map on the coherent sheaf φ * O Y : φ * O Y ·g − → O ⊕n X ֒→ φ * O Y 4 Here we are implicitly using the fact that the extension of a locally powerbounded function f remains locally powerbounded. One argument for this is to note that the constructible set {x ∈ X : \|f (x)\| > 1} is contained in Z, but also contains a nonempty open subset, by [18,Korollar 3.5.7], which is impossible. We now pick some function f ∈ O Y (Y \ T ) and write gf = n i=1 f i b i , where f i ∈ O X (X \ φ(T )) , which holds over Y \ φ −1 (φ(T )). By assumption, every coordinate function f i extends to X, so gf has an extension of the form n i=1 f i b i . Let J be an ideal whose vanishing set equals T . Since ht J ≥ 2, we have that J q for all prime ideals q ⊆ B of height 1. As B is a Jacobson ring, there is a maximal ideal p ⊆ n ⊆ B not containing I. Let y n ∈ JG(Y ) be the point corresponding to n. Then ( n i=1 f i b i )O Y,yn ⊆ gO Y,yn , because f is defined around y n . Now faithful flatness of B n → O Y,yn yields ( n i=1 f i b i )B n ⊆ gB n and thus f ∈ B n ⊆ B q . Applying the identity B = ∩ ht q=1 B q , we conclude that f ∈ B. Proof of the second Hebbarkeitssatz. We start by treating the case A = D n X 1 , . . . , X m and I = (π, X 1 ). Let g be a function defined over X \ V (I) and write it as a series +∞ i=0 a i X i 1 where a i ∈ l≥0 K T, π n /T, T l /π, X 2 , . . . , X m . On the other hand, g ∈ D n X ±1 1 , X 2 , . . . , X m , so it can also be written as i=+∞ i=−∞ b i X i 1 , where b i ∈ D n X 2 , . . . , X m . By uniqueness of the coefficients, we have a i = b i for all i ∈ Z, so b i = 0 when i < 0 and g = +∞ i=0 b i X i 1 ∈ A. In the general case, we apply Noether's normalisation lemma to find a finite injection φ : k((T )) X 1 , . . . , X r → A/πA of k((T ))-affinoid algebras such that (X 1 ) ⊆ φ −1 (I/πA). By Proposition 4.14, we can lift this to a finite injection Φ : D n,r → A π/T n such that (π, X 1 ) ⊆ Φ −1 (I). Since A/πA is connected by hypothesis, there is a unique connected component B of A π/T n with nonempty special fibre. This yields a finite injection D n,r → B of normal pseudoaffinoid domains, to which we may apply the Holomorphic Transfer Lemma. In order to show the first Hebbarkeitssatz, we want to find a powerbounded analogue of the Proof. Without loss of generality, we may assume that φ is genericallyétale. Let g = 0 ∈ A be an element such that B[g −1 ] is a finite free A[g −1 ]-module with basis b 1 , . . . , b n ∈ B + . Over the fraction field K = Q(A), this gives us a dual basis b ∨ 1 , . . . , b ∨ n ∈ L = K ⊗ A B. Consider now the trace elements f k = tr(f b k ) which are defined over the open set X \ (φ(T ) ∪ Z)), where Z is the complement of the locally free locus. Since A and B are normal, Z has codimension at least 2, so f k extends to the complement of φ(T ). Moreover, for genericallyétale extensions the trace is given by a sum of Galois conjugates, so it preserves integrality, thus the functions f k are locally powerbounded. By assumption, we get f k ∈ A • . Note that for all rational subset V ⊆ D X (g), we have an identity f = Proof of the first Hebbarkeitssatz. The proof is going to be divided into several steps. n k=1 f k b ∨ k inside the ring O Y (φ −1 (V )) ⊗ A K, Step 1: Assume A = D n X 1 , . . . , X r . Let f ∈ O + X (X \ Z) be a locally powerbounded function on the generic fibre. The latter space is a union of increasingly large semiopen rigid annuli, for which we have a canonical choice of norm, as in [7, 9.7.1, page 400]. This norm is power multiplicative, which means that powerbounded elements are those with norm ≤ 1. Thus f ∈ n<k∈N O K T, π/T n , T k /π, X 1 , . . . , X r , and, if one looks at the coefficients of its representation as a Laurent series in T , one concludes that f ∈ D • n,r . Step 2: Until Step 6, assume that the residue field k is perfect. Here, we fix a topologically nilpotent unit T of A and assume that A/πA is integral and geometrically regular over k((T )). Then [4,Satz 4.1.12] provides us a finite injection φ : k((S)) X 1 , . . . , X r → A, which is genericallyétale. We can lift this to a finite injection Φ : D n,r → A π/T n , and we note that its ramification locus does not contain the special fibre. Replacing A π/T n by the adequate connected component as in the proof of the second Hebbarkeitssatz, Lemma 5.3 shows now that f is defined everywhere up to a Zariski closed subset of codimension at least 2. The result follows now by the second Hebbarkeitssatz. Step 3: Assume that A/πA is a regular domain. We note that A/πA is geometrically reduced over k((T )) if and only if Q(A/πA) ⊗ k((T )) k((T 1/p )) = Q(A/πA)[X]/(X p − T ) is reduced, i.e, if and only if T / ∈ Q(A/πA) p . At this point, we need to know that for an integral k((T ))-affinoid algebra B, the field Q(B) p ∞ is a finite extension of k. We will give a brief sketch of the argument. First, one shows that if L/K is a finite field extension, then so is L p ∞ /K p ∞ . By the Noether normalisation lemma, one is then reduced to the case where B = k((T )) X 1 , . . . , X r . Using the unique factorisation property of B and looking at the Gauß norm, we deduce that any nonzero b ∈ Q(B) p ∞ must lie in (B • ) × , whence its reduction b ∈ B = B • /B •• = k[X 1 , . . . , X r ] is also a p ∞ -power and thus equal to some constant c ∈ k. Running the same argument with b − c instead, we conclude that it must be zero. Therefore, there is a maximal n ∈ N such that T ∈ Q(A/πA) p n and, using regularity of A/πA, we find an element S ∈ A such that S p n ≡ T mod π. After shrinking X, we may assume that S is a topologically nilpotent unit of A, thus turning A/πA into a geometrically reduced k((S))-affinoid algebra and we return to Step 2. Step 4: Assume that A/πA is generically reduced. By excellency of affinoid algebras, the singular locus of Spec A/πA has codimension 2 in X, so this case follows from the second Hebbarkeitssatz and the Step 3. Step 5: Here we only assume that k is perfect. We claim that there is a finite extension L/K and a finite injection A → B, where B is a normal pseudoaffinoid O L -domain with generically reduced special fibre. This is just some variant of the Reduced Fibre Theorem as in [12, Tag 09IL] and one may basically run the same proof. Indeed, let p i ⊆ A, i = 1, . . . , n be the minimal primes of the special fibre. The statement on p ∞ -powers from Step 3 allows us to apply [12,Tag09F9] Step 6: Here, we drop the assumption that k is perfect. We assume without loss of generality that we have a finite injection D n (O K ) X 1 , . . . , X r → A. Let O K → O L be a local homomorphism of complete DVR's which is weakly unramified and lifts the residue field extension k → k perf . There is always one such choice, by the theory of Cohen rings. We now look at the following pushout diagram in the category of rings: D n,r (O K ) A D n,r (O L ) A L By Lemma 4.15, the vertical arrows are faithfully flat, so the bottom arrow is also injective. Let B be the normalisation of A L in the product of fields Q(A) ⊗ Q(Dn,r(OK )) Q(D n,r (O L )). By excellency, B is finite over A L and the composition A → A L → B is injective. We now fix a basis a 1 , . . . , a k of Q(A)/Q(D n,r (O K )). Arguing as in the proof of the Holomorphic Transfer Lemma, we find nonzero elements g ∈ D n,r (O K ) and h ∈ D n,r (O L ), the latter of which not divisible by π, such that gA ⊆ D n,r (O K )a 1 ⊕ · · · ⊕ D n,r (O K )a k , h / ∈ πD n,r (O L ) and ghB ⊆ D n,r (O L )a 1 + · · · + Remark 5.4. It is also possible to prove the first Hebbarkeitssatz by adapting the formal methods of [11,Section 7.3] to pseudoaffinoid algebras and use our version of Noether's normalisation. Applications to formal schemes In this section, we will give some applications of the previous results to formal schemes. These will be recurrently and without further distinction regarded as adic spaces, following [19]. Recall that, according to [14], a formal scheme is reduced (resp. normal, resp. regular) if their local rings are. If X = Spf A with A an excellent ring, then X is reduced (resp. normal, resp. regular) if and only if A is. Lemma 6.1. Let X be a normal excellent formal scheme, which is locally monogeneous. Then, the natural map O X (X) → O + Xa (X a ) is an isomorphism of topological rings. Proof. We may assume that X = Spf A, where A is a normal domain with ideal of definition sA. Then (Spf A) a = Spa (A[1/s], A), and the map is clearly an isomorphism. Our first goal is to extend this result to all normal excellent formal schemes, which are nowhere discrete. For that, we will need to use normalised admissible blow-ups and Zariski's Main Theorem. Proposition 6.2. Let X be a normal excellent formal scheme, which is nowhere discrete. Then, the restriction map O X (X) → O + Xa (X a ) is an isomorphism of topological rings. Proof. Without loss of generality, assume that X = Spf A, where A is an excellent normal domain. Let Consider furthermore its normalisation Z =Ỹ , which is covered by the affine opens Z i = Spec B i , where B i is the normalisation of A[f j /f i ]/(f i − torsion) in Q(A). In virtue of A being excellent, Z → Y is a finite morphism. As Z → X is birational, Zariski's Main Theorem yields an isomorphism O X (X) → O Z (Z). In the following, we denote by Y, Y i , Z and Z i the I-adic completions of Y , Y i , Z and Z i , respectively. By excellency, Z is still normal. We now look at the following diagram: O X (X) O Z (Z) O + Xa (X a ) O + Za (Z a ) Let us now compute the ring on the upper right corner. The theorem on formal functions gives lim ← −n H 0 (Z, O Z /I n O Z ) ∼ = lim ← −n H 0 (Z, O Z )/I n H 0 (Z, O Z ). Since O Z (Z) = A is already I-adically complete, we get O Z (Z) = A = O X (X) . By the previous lemma, the right arrow is also an isomorphism. Finally, admissible blow-ups induce an isomorphism of the corresponding analytic subsets (cf. [20, (1.1.12)]), so Y a ∼ = X a is already normal and thus Z a ∼ = X a , which gives isomorphy of the bottom map. As for the topological claim, we need to verify that the topology induced on A by the ideals As a corollary of the previous proposition and the first Hebbarkeitssatz, we immediately retrieve a theorem of de Jong: Corollary 6.3. [11, Theorem 7.4.1] Let X be an O K -flat normal formal scheme, locally formally of finite type over O K and let X η be its generic fibre. Then, the restriction map O X (X) → O + X (X η ) is an isomorphism of rings. H 0 (Y, I m O Y ) = lim ← −n H 0 (Y, I m O Y )/I n H 0 (Y, I m O Y ), m ∈ N, Proof. Given a function f ∈ O + X (X η ), we first extend it to X a using the first Hebbarkeitssatz and then to X using Proposition 6.2. Before deriving another corollary of Proposition 6.2, we need to discuss the specialisation mapping. Definition 6.4. We define the specialisation mapping as the continuous function sp : X → X red which sends x to the unique generic point of {x} ∩ X red . If X = Spa A, the trivial valuation y given by \|a(y)\| = 1 if and only if \|a(x)\| = 1 is the unique generic point among all trivial specialisations of x, so the specialisation mapping is well defined and it is functorial in X. We should also remark that, when x ∈ JG((Spa A) a ), its support on A defines a valuative order by [1, Proposition 1.11.2]. The maximal ideal of this order defines a closed point in A red and, by the concrete description given above, coincides with sp(x). Hence this generalises the definition of the specialisation mapping given in [7, 7.1.5]. Corollary 6.5. Let C be the category of normal formal schemes, locally formally of finite type over O K , which are nowhere discrete with adic morphisms over O K as morphisms of C. Let D be the category of triples (X, Y, p), where X is a pseudorigid space over O K , Y is a k-scheme and p : \|X\| → \|Y \| is a map of topological spaces. If the residue field k is algebraically closed, then the functor F : C → D given by X → (X a , X red , sp X ) is fully faithful. Proof. Let X, Y ∈ ObjX and f, g : Y → X be adic morphisms such that F (f ) = F (g). In particular, the topological maps \|f \| and \|g\| coincide. Hence, we may assume that Y = Spf B, X = Spf A and denote by φ f , φ g : A → B the homomorphisms inducing f and g, respectively. By Proposition 6.2, we deduce that φ f = φ g and thus f = g. Consider now a morphism of triples (ϕ, ψ) : (Y a , Y red , sp Y ) → (X a , X red , sp X ). Let Spf A ⊆ X be an affine open subset and cover ψ −1 (Spec A red ) = n i=1 Spec (B i ) red , where Spf B i ⊆ Y are affine open subsets. From the definition of the specialisation map, it follows that ϕ −1 ((Spf A) a ) = n i=1 (Spf B i ) a . Since F is faithful, we are reduced to the case where Y = Spf B and X = Spf A. Now Proposition 6.2 gives us a continuous homomorphism φ : A ∼ = O + X (X a ) → O + Y (Y a ) ∼ = B, inducing ϕ in the analytic loci, thus, in particular, ϕ is adic (cf. [37,Lemma 7.46 (2)]). Now, it suffices to show that ψ is equal to φ red . As the specialisation mapping surjects onto closed points (cf. [1, Proposition 1.11.10]), we get an equality of topological maps \|φ red \| = \|ψ\|. Noticing that A red and B red are reduced and finitely generated over an algebraically closed field k, we deduce the identity φ red = ψ. If we apply instead Corollary 6.3, we can obtain a similar result with X a replaced by the generic fibre X η . Since the former is a rigid space, we may restrict the specialisation map to classical points and furthermore, we can replace the reduced scheme X red by its perfection, as these are canonically homeomorphic. Corollary 6.6. (Scholze) Let C be the category of O K -flat normal formal schemes, locally formally of finite type over O K . Let D be the category of triples (X, Y, p), where X is a K-rigid space (here regarded as in the classical sense), Y is a perfect k-scheme and p : \|X\| → \|Y \| is a map of topological spaces. If the residue field k is algebraically closed, then the functor F : C → D given by X → (X η , (X red ) perf , sp X ) is fully faithful. Proof. Just as in the previous corollary, we reduce to the affine case X = Spa A and Y = Spa B. Then full faithfulness is a consequence of de Jong's theorem as A ∼ = H 0 (X η , O X ) → H 0 (Y η , O Y ) ∼ = B allows us to either recover or construct the map. In order to check the agreement over the reduced locus, we argue as in the previous corollary, noting that X η is a Zariski open subset of X a . Definition 2 . 1 . 21The category of sheaves of O X -modules, denoted by Mod OX , is the category whose objects are sheaves F on X with values in the category of complete topological groups together with a continuous O X -action such that, for all open subsets U ⊆ X, F (U ) becomes an O X (U )-module; and whose morphisms are maps of sheaves respecting the O X -action. Assume now we are given an ring of integral elements A + ⊆ A, and a finite A-module M . Then one can form a presheaf M of complete topological groups on X = Spa (A, A + ) such that M (U ) = M ⊗ A O X (U ) for all rational subsets U ⊆ X, where the right hand side is endowed with the natural topology, and the restriction map M (U ) → M (V ) is the M -tensoring over A of the rational localisation O X (U ) → O X (V ), for any pair of rational subsets V ⊆ U . This presheaf carries an obvious action of O X and it turns out to be an acyclic sheaf of abelian groups (cf. [19, Theorem 2.5]). We are naturally led to the definition of a coherent sheaf: Theorem 2. 3 . 3[18, Satz 3.3.12] Let A be a strongly noetherian complete Tate ring, A + a ring of integral elements of A and X = Spa (A, A + ) the associated adic space. Then, the assignment M → M defines an equivalence between the category of finite A-modules and the category of coherent sheaves on X. Proposition 3.5. [1, Proposition 1.11.8] Let R be a complete noetherian adic ring, X = Spec R − V (R •• ) and p ⊆ R a prime ideal. The following are equivalent: Proposition 3 . 11 . 311Let (A, A + ) be a Jacobson-Tate pair and X = Spa (A, A + ) be the corresponding adic space. Given an ideal I ⊆ A, the equality ht I = ht I holds. denotes the corresponding topologically of finite type morphism, then B + = B • . (2) [21, Corollary A.14] Every homomorphism of pseudoaffinoid O K -algebras A → B is topologically of finite type. Proof. It is clear that B is a pseudoaffinoid O K -algebra. As for B + = B • , this is due to B + being the integral closure of a noetherian ring of definition B 0 ⊆ B, and thus equal to B • (cf. [21, Lemma A.2]). Lemma 4.2. On the other hand, if πA = 0, we choose a topologically nilpotent unit T ∈ A and look at the continuous O K -homomorphism k((T )) → A. Once again, this must be topologically of finite type, which yields the claim. The next proposition allows us to bring all the notions developed in the previous section to the pseudorigid world. Proposition 4. 6 . 6Pseudoaffinoid O K -algebras are Jacobson-Tate rings and they admit excellent rings of definition. In particular, pseudorigid O K -spaces are excellent Jacobson adic spaces. Definition 4. 7 . 7Given a positive rational number λ = n m , (n, m) = 1, we define the λ-elementarypseudoaffinoid O K -algebra as D λ := O K [[T ]] π mT n [1/T ] and we name the topologically freely generated D λ -algebras D λ X 1 , . . . , X r by λ-canonical pseudoaffinoid O K -algebras. Proposition 4 . 10 . 410The pseudoaffinoid algebra D λ,r is a regular domain, whose maximal ideals have constant height equal to r + 1.Proof. First, we observe that the completion O K [[T ]] X 1 , . . . , X r [1/T ] → D λ,r is a regular map (cf. [12, Tag 07BZ]). Indeed, this follows from [15, Scholie 7.8.3 (v)], which is applicable due to excellency of formally of finite type O K -algebras, and stability of regular maps under localisation. Corollary 4 . 12 . 412Let A be an integral pseudoaffinoid O K -algebra. Then A is an equidimensional ring and the irreducible components of A/πA have the same dimension.The next corollary provides an answer to the question[22, page 9]. Corollary 4. 13 . 13Let X be a pseudorigid space over O K . Then, we have an equality dim.Kr X = dim X. Lemma 4 . 15 . 415Let A be a pseudoaffinoid O K -algebra and K ⊆ L the completion of some weakly unramified algebraic extension of K. Then the continuous homomorphism A → A L := A ⊗ OK O L is faithfully flat. Proof. Consider an injection of finite A-modules N → M endowed with the natural topology. First, we notice that the map of topological A ⊗ OK O L modules N ⊗ OK O L → M ⊗ OK O L is a strict injection. Indeed, if M 0 is a A 0 -lattice generating the topology on M , then the subspace topology on N ⊗ OK O L is generated by the sets s n M 0 ⊗ OK O L ∩N ⊗ OK O L for some topologically nilpotent unit s. By flatness of O K → O L , we deduce that this intersection is just (s n M 0 ∩ N ) ⊗ OK O L , and these sets must generate the natural topology, by the Banach open mapping theorem over A. Hence, if we complete these two modules, we still obtain an injection and, arguing as in [2, Proposition 10.13],we conclude that this is the desired injection , page 23]. Now it is enough to show that the canonical map on adic spectra is surjective. Observe that we have an isomorphism of topological spaces Spa (A ⊗ OK O L ) ∼ = lim ← −M Spa (A ⊗ OK O M ), where M/K runs over all finite subextensions of L/K (cf. [23, Remark 2.6.3]). Since the fibres of Spa (A ⊗ OK O M ) → Spa (A) are finite nonempty discrete topological spaces, we may apply [8, Chapitre 1, §9 6 Corollaire 1] to deduce surjectivity of the map. Lemma 5.2. (Holomorphic Transfer) Let φ : A → B be a finite injection of pseudoaffinoid domains and let X = Spa A, Y = Spa B and ϕ = Spa φ be the corresponding adic spaces and morphisms. Holomorphic Transfer Lemma. This requires recalling some properties of the trace map tr B/A of a finite flat homomorphism of noetherian rings. If A is a normal domain, and A → B is genericallý etale, we can define the Dedekind different D B/A in the usual way (cf. [12, Tag 0BW0]). This coincides with a more general definition of the different (cf. [12, Tag 0BW4] and [12, Tag 0BW5]) and its importance lies in the fact that its vanishing set coincides with the ramification locus of B over A, by [12, Tag 0BTC]. Lemma 5.3. (Powerbounded Transfer) Let φ : A → B be a finite injection of normal pseudoaffinoid domains and let X = Spa A, Y = Spa B and ϕ = Spa (φ) be the corresponding adic spaces and morphism. Let T ⊆ Y be a Zariski closed subset of positive codimension. If the restriction map A • → O + X (X \ ϕ(T )) is an isomorphism, then every locally powerbounded function f ∈ O + Y (Y \ T ) extends uniquely to the intersection of the Zariski unramified locus of φ and the preimage of the Zariski locally free locus of φ. by definition of the dual basis. Let d ∈ B be an element in the differentD B[g −1 ]/A[g −1 ] , which implies b ∨ i ∈ B[d −1 ] for all i = 1, . . . , n. Therefore, the identity above yields an extension of f to the Zariski open subset where A[g −1 ] → B[g −1 ] is unramified. to the extensions O K → A pi and obtain a common solution L in the sense of [12, Tag 09EN]. Define now B as a connected component of (A ⊗ OK O L ) norm , on which A injects. Then A → B is a finite injection and B is a pseudoaffinoid domain over O L . We now observe that the minimal primes q j of Spec B/π L B lie over the p i , due to Corollary 4.12 and dimension reasons. By definition of a solution, O L → B qj is formally smooth, which implies that the special fibre of B is generically reduced. Let now f be a locally powerbounded function on X. Then Step 4 yields f ∈ B and, if one looks at the integral closure C of B in the Galois hull of Q(B)/Q(A), one deduces that f is invariant by the Galois action (since this is true in the generic fibre), so f ∈ A. D n,r (O L )a k . Let f be a locally powerbounded function on X \Z. We have an identity gf = f i a i , where the f i are functions on Spa D n,r (O K ) \ V (π). On the other hand, f extends to Spa B by Step 5, so we have another identity hgf = f i a i , where thef i belong to D n,r (O L ). By uniqueness of coefficients, one getsf i = hf i on Spa D n,r (O L ) \ V (π), so f i ∈ D n,r (O L ), by the second Hebbarkeitssatz. This shows that the f i are locally bounded, so by Step 1, we also deduce that f i ∈ D n,r (O K ). In conclusion, f ∈ Q(A) ∩ B. Looking at the characteristic polynomial p of f over Q(D n,r (O K )), we see that it has coefficients in D n,r (O L ). But faithful flatness of D n,r (O K ) → D n,r (O L ) tells us that Q(D n,r (O K ))∩D n,r (O L ) = D n,r (O K ) so f ∈ A. I = (f 1 , . . . , f n ) be an ideal of definition of A and consider the blow-up Y of X = Spec A centred at I. This can be covered by the affine open subsets Y i = Spec A[f j /f i ]/(f i − torsion). is just the usual I-adic topology. Indeed, I m O Y = O Y (m) is induced by the graded module k≥m I m , so it follows that H 0 (Y, O Y (m)) = I m for all sufficiently large m, arguing as in [16, Exercise 5.9(b)] (which works more generally for Nagata rings, cf. 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[ "EXISTENCE OF MINIMAL HYPERSURFACES IN COMPLETE MANIFOLDS OF FINITE VOLUME", "EXISTENCE OF MINIMAL HYPERSURFACES IN COMPLETE MANIFOLDS OF FINITE VOLUME" ]	[ "Gregory R Chambers ", "Yevgeny Liokumovich " ]	[]	[]	We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume.	10.1007/s00222-019-00903-3	[ "https://arxiv.org/pdf/1609.04058v2.pdf" ]	119,702,747	1609.04058	c0f039cccc7324d3175dca20b2400c439de9f468	EXISTENCE OF MINIMAL HYPERSURFACES IN COMPLETE MANIFOLDS OF FINITE VOLUME Gregory R Chambers Yevgeny Liokumovich EXISTENCE OF MINIMAL HYPERSURFACES IN COMPLETE MANIFOLDS OF FINITE VOLUME We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume. Introduction By a result of Bangert and Thorbergsson (see [Th] and [Ba]) every complete surface of finite area contains a closed geodesic of finite length. In this article we generalize this result to higher dimensions. Let M n+1 be a complete Riemannian manifold of dimension n + 1. For an open set U ⊂ M define the relative width of U , denoted by W ∂ (U ), to be the supremum over all real numbers ω such that every Morse function f : U → [0, 1] has a fiber of volume at least ω. Remark 1.2. We make some remarks about Theorem 1.1: 1. The hypersurface Γ intersects a small neighbourhood of U . In fact, for any δ > 0 there exists a finite volume minimal hypersurface that intersects the δ-neighbourhood of U (see Theorem 8.2 and Question 3 in Section 2.5). 2. If M is compact then Γ is compact. If M is not compact then Γ may or may not be compact. In Remark 8.3 we give an example, showing that one can not always expect to obtain a compact minimal hypersurface in a complete manifold of finite volume using a min-max argument. 3. We also obtain upper and lower bounds for the volume of Γ that depend on U (see Theorem 8.2). The condition that there exists a subset U with H n (∂U ) ≤ W ∂ (U ) 10 is satisfied if tM has sublinear volume growth, that is, for some x ∈ M we have lim inf r→∞ V ol(Br(x)) r = 0 . In particular, we have the following corollary. The proof is based on Almgren-Pitts min-max theory [Pi]. We use the version of the theory developed by De Lellis and Tasnady in [DT]. Instead of general sweepouts by integral flat cycles, the argument of [DT] allows one to consider sweepouts by hypersurfaces which are boundaries of open sets. This simplification is used in a crucial way in this paper. We consider a sequence of sweepouts of U and extract a sequence of hypersurfaces of almost maximal volume that converges to a minimal hypersurface. The main difficulty is to rule out the possibility that the sequence completely escapes into the "ends" of the manifold. Proposition 6.1 is the main tool which allows us to rule out this possibility. This Proposition allows us to replace an arbitrary family of hypersurfaces with a nested family of hypersurfaces which are level sets of a Morse function, increasing the maximal area by at most ε in the process. We use this Proposition together with some hands on geometric constructions to show that there exists a sequence of hypersurfaces that converges to a minimal hypersurface and the volume of their intersection with a small neighbourhood of U is bounded away from 0. A number of results about existence of minimal hypersurfaces in non-compact manifolds have appeared recently. Existence results for minimal hypersurfaces (compact and non-compact) in certain classes of complete non-compact manifolds were proved by Gromov in [Gr]. This work was in part inspired by arguments in [Gr]. In [Gr] mean curvature of boundaries plays an important role. Our results do not depend on the curvature of the manifold or mean curvature of hypersurfaces in M . Existence of a compact embedded minimal surface in a hyperbolic 3-manifolds of finite volume was proved by Collin-Hauswirth-Mazet-Rosenberg in [CHMR]. In [Mo] Montezuma gave a detailed proof of the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with mean-concave boundary, as well as satisfying certain conditions on the geometry at infinity. In particular, these manifolds have infinite volume. In [KZ] Ketover and Zhou proved a conjecture of Colding-Ilmanen-Minicozzi-White about the entropy of closed surfaces in R 3 using a min-max argument for the Gaussian area functional on a non-compact space. for many fruitful discussions there. The authors would also like to thank Fernando Coda Marques and André Neves for making several important suggestions. The authors are grateful to Laurent Hauswirth, Daniel Ketover, Laurent Mazet, Rafael Montezuma, Alexander Nabutovsky, Anton Petrunin, Stephane Sabourau, Emanuele Spodaro, Luca Spolaor and Neshan Wickramasekera for discussing this work with them. The authors are grateful to Fernando Coda Marques for pointing out two errors in an earlier draft of this paper. The first author was partly supported by an NSERC postdoctoral fellowship. This paper was partly written during the second author's visit to Max Planck Institute at Bonn; he is grateful for the Institute's kind hospitality. Structure of proof We describe the idea of the proof. 2.1. Families of hypersurfaces and sweepouts. In this article we will be dealing with families of possibly singular hypersurfaces {Γ t }. For the purposes of the introduction the reader may assume that each Γ t is a boundary of a bounded open set Ω and has only isolated singularities of Morse type. In fact, Γ t may differ from ∂ Ω t by a finite set of points. The precise definition of the hypersurfaces and the sense in which the family {∂ Ω t } is continuous are described in Section 3. To follow the outline of the proof we only need to know that the areas of ∂ Ω t i approach the area of ∂ Ω t and the volumes of (Ω t i \ Ω t ) ∪ (Ω t \ Ω t i ) go to zero as t i → t. (We will use the word "volume" for the (n + 1)−dimensional Hausdorff measure and "area" for the n-dimensional Hausdorff measure.) We will consider four types of special families of hypersurfaces, which we will call "sweepouts". We will study the relationship between these four types of families and that will eventually lead us to the proof of Theorem 1.1. Slightly informally we describe them below. 1. An (ordinary) sweepout of a bounded set U is a family of hypersurfaces {∂ Ω t } t∈ [0,1] with Ω 0 ∩U = ∅ and U ⊂ Ω 1 . 2. A good sweepout of U is a sweepout {Γ t } with areas of Γ 0 and Γ 1 less than 5 H n (∂U ). The motivation for this definition is the following. In a "mountain pass" type argument we would like to apply a "pulling tight" deformation to a family {Γ t } so that hypersurfaces that have maximal area in the family converge (in a certain weak sense) to a stationary point of the area functional. When doing this we would like hypersurfaces at the "endpoints" Γ 0 and Γ 1 to stay fixed. We will consider sweepouts of sets with the property that every sweepout must contain a hypersurface of area much larger than the area of the boundary of U (see definition of a good set below). The condition above guarantees that Γ 0 and Γ 1 do not have areas close to the maximum and so the pulling tight deformation will not affect them. 1] with Ω s ⊂ Ω t for every s ≤ t. Moreover, we have ∂ Ω t = f −1 (t) for some Morse function f . Nested sweepouts are a key technical tool in this paper. 4. A relative sweepout of U is a family of hypersurfaces {Σ t } with boundaries ∂ Σ t ⊂ ∂U obtained from some nested sweepout {Γ t } of U by intersecting Γ t with the closure of U , Σ t = Γ t ∩cl(U ). 3. A nested sweepout of U is a sweepout {∂ Ω t } t∈[0, 2.2. Widths. For each notion of a sweepout we define a corresponding notion of width. If S is a collection of families of hypersurfaces we set W (S) = inf {Γt}∈S sup t H n (Γ t ) Let S(U ), S ∂ (U ), S g (U ) and S n (U ) denote the collection of all sweepouts, relative sweepouts, good sweepouts and nested sweepouts correspondingly. We set W (U ) = W (S(U )) to be the width of U , W ∂ (U ) = W (S ∂ (U )) to be the relative width of U , W g (U ) = W (S g (U )) to be the good width of U and W n (U ) = W (S n (U )) to be the nested width of U . Theorem 1.1 is a statement about a bounded open set U ⊂ M with smooth boundary and the property that H n (∂U ) ≤ 1 10 W ∂ (U ). A set satisfying this property will be called a good set. We will show that for a good set U we have the following relationships between the quantities W (U ), W ∂ (U ), W g (U ) and W n (U ): (1) W ∂ (U ) ≤ W n (U ) ≤ W ∂ (U ) + H n (∂U ) (2) W n (U ) = W (U ) (3) W g (U ) = W (U ) The first inequality in (1) follows directly from the definition. The reason for the second inequality in (1) is also clear: to obtain a nested sweepout {Γ t } from a relative sweepout {Σ t } we can take a union of Σ t = Γ t ∩cl(U ) with a subset of the boundary ∂U (the subset varying based on Σ t ). Certain perturbation arguments will guarantee that a sufficiently regular nested sweepout can be obtained in this way. Note that this is also a good sweepout since it starts on a hypersurface of area 0 and ends on a hypersurface of area H n (∂U ) < 5 H n (∂U ). Equation (2) is proved in Proposition 6.1. In fact, (2) holds not only for good sets U , but for any bounded open set U with smooth boundary. The proof of (2) is the most technical part of this paper. Equation (3) is proved below using methods from Section 7. The importance of these equations is the following: we will use (1) and (2) to prove (3); we will use (3) to prove Theorem 1.1. 2.3. Existence of a large slice intersecting U . Now we can outline the proof of Theorem 1.1. We would like to find a minimal hypersurface in M using a min-max argument, developed by Almgren [Al] and Pitts [Pi] and simplified by De Lellis -Tasnady [DT]. Let U be a good set. We choose a sequence of good sweepouts of U with the property that the area of the largest hypersurface converges to W g (U ). We would like to extract an appropriate sequence of hypersurfaces whose areas converge to W g (U ), and argue that they converge (as varifolds) to a minimal hypersurface. The problem with this argument as it stands is that this sequence of hypersurfaces may drift off to infinity, and so strong convergence may not hold. To handle this issue, we will argue that this sequence of hypersurfaces can be chosen so that the intersection of every hypersurface with U is bounded away from 0. This "localization" statement will allow us to conclude that in the limit we obtain a minimal hypersurface with non-empty support in a small neighbourhood of U . Proposition 2.1. For every good set U there exists a positive constant ε(U ) which depends only on U such that the following holds. For every good sweepout {Γ t } of U with associated family of open sets {Ω t }, there is a surface Γ t in the collection which has area at least W g (U ), and such that H n (Γ t ∩ cl(U )) ≥ ε(U ). Theorem 1.1 will follow from by modifying arguments in [DT] (see Section 8). In the remainder of this section we focus on the proof of Proposition 2.1. We explain how we choose ε(U ). In Section 7 (Lemma 7.1) we will show that for every U there exists ε 0 > 0 with the property that every Ω which intersects U in volume at most ε 0 or contains all of U except for a set of volume at most ε 0 can be deformed so that its boundary does not intersect U and the areas of the boundaries in the deformation process are controlled. Specifically, if H n+1 (Ω ∩U ) ≤ ε 0 then there exists a family {Ω t } t∈ [0,1] , such that Ω 0 ∩U = ∅ and Ω 1 = Ω; if H n+1 (U \ Ω) ≤ ε 0 then there exists a family {Ω t } t∈ [0,1] , such that Ω 1 ∩U = U and Ω 0 = Ω. In both cases the boundaries of Ω t satisfy (4) H n (∂ Ω t ) < H n (∂ Ω) + 5 H n (∂U ) Having fixed ε 0 with this property we define ε(U ) = ε(ε 0 ) > 0 to be such that every Ω with min{H n+1 (Ω ∩U ), H n+1 (U \ Ω)} ≥ ε 0 /2 has H n (∂ Ω ∩U ) > ε(U ). Existence of such ε follows from the properties of the isoperimetric profile of U . Suppose now that Proposition 2.1 fails for this value of ε(U ). Let V (t) = H n+1 (Ω t ∩U ) and A(t) = H n (∂ Ω ∩U ). V is a continuous function of t, t ∈ [0, 1], but A(t) may not be continuous. However, the family {∂ Ω t } can be perturbed to make A(t) continuous. In the proof of Proposition 2.1 in section 7 we prove a weaker assertion that A(t) is "roughly" continuous after a small perturbation, in the sense that the oscillation of A at a point t is at most ε/10; this turns out to be sufficient for what we need. For the purposes of this overview we will assume that A(t) is actually continuous. Continuity of A and V and the fact that {∂ Ω t } is a sweepout imply that there exists an interval [a, b] ⊂ [0, 1] with H n (∂ Ω t ∩U ) ≥ ε for all t ∈ [a, b]; H n (∂ Ω a ∩U ) = ε and H n (∂ Ω b ∩U ) = ε; H n+1 (Ω a ∩U ) < ε 0 /2 and H n+1 (Ω b ∩U ) > H n+1 (U ) − ε 0 /2. By our assumption this implies H n (∂ Ω t ) < W g (U ) for all t ∈ [a, b]. Since H n (∂ Ω t ) is a continuous function of t there exists a real number δ > 0 such that ∂ Ω t has area at most W g (U ) − δ for t ∈ [a, b]. LetŨ = U ∩ (Ω b \cl(Ω a ) ). The last paragraph implies that W (Ũ ) ≤ W g (U ) − δ. The boundary ofŨ satisfies H n (∂Ũ ) ≤ H n (∂U ) + 2ε. Even thoughŨ may not be a good set we will show in Section 7 that (3) still holds forŨ . (H n (∂Ũ ) may be larger than 1/10W ∂Ũ , but it is still sufficiently small compared to W ∂Ũ so that the proof of (3) goes through). By ( 3) applied toŨ W g (Ũ ) = W (Ũ ) ≤ W g (U ) − δ and, hence, there exists a good sweepout {∂Ω t } t∈[0,1] ofŨ with areas of all hypersurfaces at most W g (U ) − δ/2. By the definition of a sweepoutΩ 0 ∩ U ⊂ U \Ũ ⊂ (U ∩ Ω a ) ∪ (U \ Ω b ) and hence H n+1 (Ω 0 ∩ U ) ≤ ε 0 . Also, since {∂Ω t } is a good sweepout, ∂Ω 0 has area at most 5 H n (∂U ) − δ/4. By (4) we can deformΩ 0 to a set that does not intersect U through open sets with boundary area at most W g (U )−δ/4. Similarly, we can deformΩ 1 to an open set that contains U through open sets with boundary area at most W g (U )−δ/4. We conclude that there exists a sweepout of U by hypersurfaces of area at most W g (U ) − δ/4. Hence, W (U ) ≤ W g (U ) − δ/4, which contradicts (3). This finishes the proof of Proposition 2.1. 2.4. The good width equals width. In the rest of this section we describe how (3) follows from (1) and (2). The argument is illustrated in Figure 1. We start with a sweepout {∂ Ω t } of a good set U by hypersurfaces of area at most W (U ) + δ. By (2) we can assume that {∂ Ω t } is a nested sweepout. Next, we argue (cf. Lemma 7.4) that there is a hypersurface ∂ Ω t with t ∈ [0, 1] such that H n (∂ Ω t \U ) has area comparable to that of the boundary of U . Indeed, by (1) there is a hypersurface with a large intersection with U , that is, H n (∂ Ω t ∩cl(U )) ≥ W n (U ) − H n (∂U ). The complement then must satisfy H n (∂ Ω t \cl(U )) ≤ W (U ) − W n (U ) + H n (∂U ) + δ = H n (∂U ) + δ. Now consider Ω t \U . Since {∂ Ω t } is nested this set contains Ω 0 and is contained in Ω 1 . By the argument in the previous paragraph we have H n (∂(Ω t \U )) ≤ 2 H n (∂U )+ δ. Let A denote the infimal value of H n (∂ Ω) over all open sets Ω with Ω 0 ⊂ Ω ⊂ Since Ω t \U is one of such sets we have A ≤ 2 H n (∂U ) + δ LetΩ denote a set as above with H n (∂Ω) ≤ A + δ. We replace sweepout {∂ Ω t } with a new sweepout {∂(Ω∪Ω t )}. Perturbation arguments will guarantee that we can smooth out the corners of these hypersurfaces to obtain a sufficiently regular family. This family starts on a surface ∂Ω of area less than 5 H n (∂U ) and ends on Ω 1 . Moreover, it follows form the fact that ∂Ω is δ-nearly area minimizing hypersurface that the area of ∂(Ω ∪ Ω t ) is bounded by W + 2δ (cf. Lemma 5.1). Similarly, we can replace this sweepout with a new sweepout that end on a hypersurface of area less than 5 H n (∂U ), without increasing the areas of other hypersurfaces by more than δ. We conclude that W g (U ) ≤ W (U ) + 3δ, but since δ > 0 was arbitrary (3) follows. The importance of nested sweepouts comes from the fact that it allows us to choose nearly minimizing hypersurfaces like ∂Ω and perform cut and paste procedures as above without increasing the area significantly. The ideas used in the proof of (2) and (3) go back to [CR] by the first author and Regina Rotman. In that article, the authors were interested in nested homotopies of curves, whereas here we use sufficiently regular cycles. Open questions. We list some open questions related to Theorem 1.1. 1. For a positive real number α we say that U is an α-good set if H n (∂U ) ≤ αW ∂ (U ). Theorem 1.1 asserts that if a complete manifold M contains a 1/10-good set, then there is a minimal hypersurface of finite volume in M which intersects a small neighbourhood of U . Question: What is the maximal value of α for which the conclusion of Theorem 1.1 holds? It is conceivable that it may be true for every positive α < 1. 2. In [MN2] Marques and Neves show that a min-max minimal hypersurface has a connected component of Morse index 1, assuming that the manifold has no one-sided hypersurfaces (see [MR], [So], [Zh1], [Zh2] for previous results in that direction). Is it possible to adapt their arguments to construct a minimal hypersurface of finite volume and Morse index 1 for every complete manifold without one-sided hypersurfaces and satisfying the assumptions of Theorem 1.1? 3. In Theorem 8.2 we show that for an arbitrarily small δ > 0 there exists a minimal hypersurface of finite volume intersecting the δ-neighbourhood of a good set U . Does there exist a minimal hypersurface of finite volume intersecting cl(U )? It is plausible that this result follows from a refinement of some of the arguments in Section 8 or from an appropriate compactness argument. 4. In [Gr] it is shown that if a non-compact manifold M does not admit a proper Morse function f , such that all non-singular fibers of f are mean-convex, then M contains a minimal hypersurface of finite volume. The following question was suggested to us by Misha Gromov: Question: Do there exist manifolds of finite volume that admit a Morse function f , such that all non-singular level sets of f have positive mean curvature? More generally, do there exist good sets U (in the sense defined in this paper) which admit Morse foliations by mean convex hypersurfaces (with boundaries of the hypersurfaces contained in the boundary of U )? Preliminaries We begin with fixing notation and introducing several technical definitions which we will use throughout this article. H k k−dimensional Hausdorff measure cl(U ) closure of the set U B r (x) open ball of radius r centered at x N r (U ) the set {x ∈ M : d(x, U ) < r} An(x, t 1 , t 2 ) the open annulus B t 2 (x) \ cl(B t 2 (x)) Following De Lellis -Tasnady we make the following definitions. Families of hypersurfaces and sweepouts. Definition 3.1. Family of hypersurfaces A family {Γ t }, t ∈ [0, 1], of closed subsets of M with finite Hausdorff measure will be called a family of hypersurfaces if: (s1) For each t there is a finite set P t ⊂ M such that Γ t is a smooth hypersurface in M \ P t ; (s2) H n (Γ t ) depends smoothly on t and t → Γ t is continuous in the Hausdorff sense; (s3) on any U ⊂⊂ M \ P t 0 , Γ t → Γ t 0 smoothly in U as t → t 0 . Definition 3.2. Sweepout Let U be an open subset of M . {Γ t }, t ∈ [0, 1], is a sweepout of M if it satisfies (s1)-(s3) and there exists a family {Ω t }, t ∈ [0, 1], of open sets of finite Hausdorff measure, such that (sw1) (Γ t \ ∂ Ω t ) ⊂ P t for any t; (sw2) Ω 0 ∩U = ∅ and U ⊂ Ω 1 ; (sw3) H n+1 (Ω t \ Ω s ) + H n+1 (Ω s \ Ω t ) → 0 as t → s. For a sweepout {Γ t } we will say that {Ω t } is the corresponding family of open sets if it satisfies (sw1) -(sw3). Definition 3.3. Good sweepouts, nested sweepouts and relative sweepouts A good sweepout {Γ t } is a sweepout of U which in addition satisfies: (sw g ) H n (Γ 0 ) ≤ 5 H n (∂U ) and H n (Γ 1 ) < 5 H n (∂U ). A nested sweepout {Γ t } is a sweepout of U which in addition satisfies: 1]; the corresponding family of open sets is given by Ω (sw n ) there exists a Morse function f : M → [−1, ∞), such that Γ t = f −1 (t), t ∈ [0,t = f −1 ((−∞, t)). Suppose ∂U is a smooth manifold and {Γ t } is a nested sweepout of U with the corresponding family of open sets {Ω t }. Set Σ t = (cl(U ) ∩ Γ t ). We will say that {Σ t } is a relative sweepout of U . Definition 3.4. Widths and good sets As described in Section 2 the widths W (U ), W ∂ (U ), W g (U ) and W n (U ) are defined as the min-max quantities corresponding to sweepouts, relative sweepouts, good sweepouts and nested sweepouts respectively. A good set U ⊂ M is a bounded open set with smooth boundary and H n (∂U ) ≤ Σ 2 are disjoint, Σ 1 ∪ Σ 2 = ∂N and ∂ Σ 1 ∩∂ Σ 2 = C is a compact (n − 2)-dimensional submanifold of M . We say that ∂N is a manifold with corner C if for every sufficiently small neigh- bourhood U of a point x ∈ C there exists a diffeomorphism φ from U to R n+1 with φ(N ) = R + × R + × R n−1 , φ(Σ 1 ) = {x 1 = 0}, φ(Σ 2 ) = {x 2 = 0} and C = {x 1 = x 2 = 0}. There is a standard construction of smoothing (or straightening) the corner C of a manifold with corner (see [Mu,Section 7.5]). We briefly describe it here, because we use it several times in this paper. Fix δ > 0. We construct a smooth hypersurface Σ ⊂ cl(N ), such that Σ coincides with ∂N outside of N δ (C). For each x ∈ C let θ(x) ∈ (0, 2π) denote the angle between hyperplanes T x Σ 1 and T x Σ 2 inside tangent space T x M . Define cylindrical coordinates y = (x, θ, r) on cl(N δ (C)∩N ), where x ∈ C, r denotes the radial distance to C and θ ∈ [0, θ(x)] denotes the angle that a minimizing geodesic from C to y makes with the hyperplane T x Σ 1 . Let γ x (t) be a family of smooth convex functions defined on [0, θ( x)] with min t γ x (t) = δ/4, max t γ x (t) = γ x (0) = γ x (θ(x)) = δ/2 and d dk γ x (t) = ∞ for all k > 0 as t ap- proaches 0 or θ(x). We define Σ in N δ (C) by setting Σ ∩ N δ/2 (C) = {(x, θ, r) : r = γ x (θ)} for θ ∈ [0, θ(x)] and r ≤ δ/2 and Σ ∩ (N δ (C) \ N δ/2 (C)) = (Σ 1 ∪ Σ 2 ) ∩ (N δ (C) \ N δ/2 (C)). We make several observations about this construction. 1. Different smoothings Σ corresponding to different choices of the convex functions γ x (t) are all isotopic. 2. For any ε > 0 functions γ x (t) can be chosen in such a way that H n (Σ) < H n (∂N ) + ε. 3. Smoothing can be done parametrically. Given a foliation of a subset of M by hypersurfaces with corners the above construction can be applied to the whole family in such a way that we obtain a foliation by a family of smooth hypersurfaces. 4. For all δ > 0 sufficiently small there exists a choice of Σ and a constant c that depends on M , N and C, so that H n (N 10δ (C) ∩ ∂N 2δ (Σ)) ≤ cδ. The last observation will be important in the proof of Lemma 4.3. It will be convenient to introduce one more definition. Definition 3.5. Let Ω ⊂ M be a bounded open subset and ∂ Ω is a manifold with corner and δ > 0. We will say that Ω +δ is an outward δ-perturbation of Ω if the following holds: (1) Ω Ω +δ ⊂ N δ (Ω); (2) there exists a nested family of open sets {Ξ t } t∈[0,1] and a smooth isotopy Σ t = ∂Ξ t , such that Σ 0 is a smoothing of ∂ Ω, Ξ 1 = Ω +δ and H n (Σ t ) < H n (∂ Ω) + δ for all t ∈ [0, 1]. We will say that Ω −δ is an inward δ-perturbation of Ω if the following holds: (1)' Ω \N δ (∂ Ω) ⊂ Ω −δ Ω; (2)' there exists a nested family of open sets {Ξ t } t∈[0,1] and a smooth isotopy (2) the restriction of f to ∂N is a Morse function. , such that the following holds: Σ t = ∂Ξ t , such that Σ 1 is a smoothing of ∂ Ω, Ξ 0 = Ω −δ and H n (Σ t ) < H n (∂ Ω) + δ for all t ∈ [0, 1] ( 1) g −1 (b) = Σ; (2) f −1 ([a, t)) ⊂ N ε/2 (g −1 ([a, t))) ⊂ N ε (f −1 ([a, t))); (3) H n (g −1 (t)) ≤ H n (∂f −1 ([a, t])) + ε; (4) If dist(x, f (Σ)) > ε then f −1 (x) = g −1 (x). Proof. The idea of the proof is shown in Figure 2 We will define a singular foliation Σ t , t ∈ [0, 1], of N with only finitely many singular leaves that have non-degenerate singularities and with Σ 1 = Σ. It follows then that there exists a Morse function g(x) with g −1 (t) = Σ t . We will prove that this foliation satisfies the desired upper bound on the area. The surfaces in the foliation will coincide with f −1 (t) whenever f −1 (t) is sufficiently far from Σ and so (4) will also follow. Choose r 0 ∈ (0, ε), be sufficiently small, so that the tubular neighbourhood U = N 2r 0 (Σ)∩N does not intersect critical points of f and there exists a diffeomorphism φ from Σ × [0, 2r 0 ) to U . Let φ(x, r), x ∈ Σ, r ∈ [0, r 0 ) denote the normal coordinates on U . For r 0 sufficiently small we may assume that H n ((Σ, r)) ≤ H n (Σ) + ε 2 for r ∈ [0, r 0 ]. Let U r = {φ(x, r ) : r ≤ r}. Let ε 0 = ε 0 (r 0 ) > 0 be a small constant to be specified later and satisfying ε 0 → 0 for r 0 → 0. Let p 0 < ... < p k be critical values of f \| Σ . First we define a singular foliation Σ t , t / ∈ ∪ i (p i − ε 0 , p i + ε 0 ). LetΣ t = ∂(f −1 ([0, t]) \ U (1−t)r 0 ). If t is a singular value of ff \| Σ we have that f −1 (t) intersects φ(Σ, (1 − t)r 0 ) transversally. Hence,Σ t \ s is a manifold with corners. There exists a smoothing of the corners, so that the new foliation {Σ t } coincides with {Σ t } outside of a small neighbourhood of V t = f −1 (t) ∩ φ(Σ, (1 − t)r 0 ) and is smooth in V t . As discussed in subsection 3.2 we can choose it so that H n (Σ t ) − H n (Σ t ) is arbitrarily small. Now we construct the foliation for t ∈ (p i − ε 0 , p i + ε 0 ). Let x i ∈ Σ be the critical point of f \| Σ with f (x i ) = p i . Outside of a small neighbourhood of x i we can define Σ t in the same way as above, since f −1 (t) intersects φ(Σ, (1 − t)r 0 ) transversally and a smoothing of the corners is well-defined. In the neighbourhood of a critical point x i we define the foliation by considering two cases (see Figure 3). Let n i denote the inward pointing unit normal at x i and set s i = ∇f \|∇f \| (p i ), n i . The two cases will depend on the sign of s i . Let y i = φ(x i , (1 − p i )r 0 ). There exists a choice of coordinates u = (u 1 , ..., u n+1 ) in the neighbourhood of y i so that in these coordinates we have f (u) = u n+1 +f (y i ). Let λ denote the index of x i . Let P λ (u 1 , ..., u n ) = −u 2 1 − ... − u 2 λ + u 2 λ+1 + ... + u 2 n . Up to a bilipschitz diffeomorphism of the neighbourhood of y i , the foliation {φ(Σ, (1−t )r 0 )}, t ∈ (p i − ε 0 , p i + ε 0 ), will coincide with the foliation {u n+1 = P λ (u 1 , ..., u n ) − s i t}, t ∈ (−ε 0 , ε 0 ). Case 1: s i = −1. There exists a smoothing of the corners for Σ t so that as t approaches p i from above and below surface Σ t is a graph over {u n+1 = 0} hyperplane in the neighbourhood of y i . There exists a small δ > 0 and a foliation {Γ t } of the neighbourhood of y i so that Γ t = {u n+1 = P λ (u 1 , ..., u n ) + t} for u 2 1 + ... + u 2 n < δ/3 and Γ t is a graph of u n+1 = t for u 2 1 + ... + u 2 n > 2δ/3. The foliation {Γ t } extends the foliation {Σ t } to the neighbourhood of the critical point x i . Case 2: s i = 1. Let Π t = {u n+1 = t} ∩ {P λ (u 1 , ..., u n ) ≤ 2t} and Q t = {u n+1 = P λ (u 1 , ..., u n ) − t} ∩ {u n+1 ≤ t}. After a bilipschitz diffeomorphism in the neighbourhood of y i we may assume that the foliation {Σ t } is given by the smoothing of the union Π t ∪ Q t . By standard Morse theory arguments (see Section 3 of [Mi1] and Section 3 of [Mi2]) Π δ ∪Q δ is obtained from Π −δ ∪Q −δ by surgery of type (λ, n+1−λ) and there exists an elementary cobordism between them of index λ. This cobordism gives the desired foliation in the neighbourhood of the critical point. Observe that in the above operations we applied bilipschitz diffeomorphisms on some small neighbourhood, possibly increasing the areas of hypersurfaces by some controlled constant factor (independent of the size of the neighbourhood). By choosing the neighbourhood to be sufficiently small we ensure that the areas do not increase by more than ε. We will also need a slightly different version of this lemma for a non-compact submanifold N . For every ε > 0 there exists a Morse function g : N → (−∞, b], such that the following holds: (1) g −1 (b) = Σ; ( 2) f −1 ((−∞, t)) ⊂ N ε/2 (g −1 ((−∞, t))) ⊂ N ε (f −1 ((−∞, t))); (3) H n (g −1 (t)) ≤ H n (∂f −1 ((−∞, t])) + ε; (4) If dist(x, f (Σ)) > ε then f −1 (x) = g −1 (x). Proof. Let a be such that f (N ε Σ) ⊂ [a + ε, b]. Since function f is proper we have that N = f −1 ([a, b]) is compact. We apply Lemma 4.1 to N to obtain function g. We set g(x) = f (x) for x not in N and the lemma follows. 4.2. Gluing Morse foliations on a manifold separated by a hypersurface transverse to the boundary. We will also need the following lemma for gluing two Morse foliations on a manifold with boundary separated by a hypersurface which is transversal to the boundary. (1) Ω 1 is an inward ε-perturbation of V 1 ; Ω 2 is an inward ε-perturbation of V 2 . ( 2) f −1 (0) = ∂ Ω 1 ∪∂ Ω 2 and f −1 (1) = ∂N ; (3) H n (f −1 (t)) ≤ H n (∂N ) + 2 H n (Σ) + ε. Proof. The idea of the proof of this lemma is shown in Figure 4. Fix δ > 0 to be specified later. Note that ∂V i is a manifold with a corner Σ ∩ ∂V i . Let V i ⊂ V i be a submanifold with ∂V i a smoothing of ∂V . We have that V i and V i coincide outside of N δ/2 (∂V i ∩ ∂Σ). Let Ω i = V i \ N δ/2 (∂V i ). Let d : M \ (Ω 1 ∪ Ω 2 ) → [0, ∞) denote the distance from x to Ω 1 ∪ Ω 2 . Function d is 1-Lipschitz, but it may not be smooth. However, it is well-known ( [GW]) that for every ε > 0 function d may be approximated by a Morse function f with 1 − ε < \|∇f \| < 1 + ε. We choose such an approximation and consider level sets f −1 (t), t ∈ [0, 2δ]. Define Ω 3 = Ω 1 ∪ Ω 2 ∪f −1 ([0, 2δ]). By curvature comparison arguments from [HK] applied to function f we know that d dt H n (f −1 (t)) only depends on the Ricci curvature of N 2δ (∂N ) and the mean curvature of ∂(Ω 1 ∪ Ω 2 ). The mean curvature of ∂(Ω 1 ∪ Ω 2 ) in turn depends on the mean curvatures of ∂N , Σ and the choice of smoothing of the corners for V 1 and V 2 . As observed in subsection 3.2 we may assume that the contribution that comes from the smoothing of the corners is negligible for sufficiently small δ. If follows that we can find a δ > 0 so that H n (f −1 (t)) ≤ H n (∂V 1 ) + H n (∂V 2 ) + ε. The above construction does not yet give us what we want because f −1 (1) = ∂ Ω 3 , which sits slightly outside of ∂N . To fix this we construct function f as above not for N , but for N = N \ N 3δ (∂N ), for some suitable sufficiently small choice of δ to ensure that ∂N is smooth and intersects Σ transversally. Then Ω 3 sits inside N and there exists a nested isotopy from ∂ Ω 3 to ∂N . Splitting and extension lemmas In this section we prove two important lemmas for nested sweepouts which we will use in sections "Nested sweepouts" and "No escape to infinite". I. Additionally, suppose that Ω is a bounded open set with boundary Γ a smooth embedded manifold such that (1) Ω ⊂ Ω 1 ; (2) There is an ε > 0 such that for every Ω with Ω ⊂ Ω ⊂ Ω 1 we have H n (Γ) < H n (∂ Ω ) + ε/4. Then we can find a nested familyΓ t and an associated family of open setsΩ t such thatΩ 0 ⊂ Ω 0 ,Γ 1 = Γ, and every hypersurface has area at most A + ε. Furthermore, if Ω 0 ⊂ Ω, thenΓ 0 = Γ 0 . II. Suppose that, instead of properties (1) and (2) above, the following are true: (1)' Ω 0 ⊂ Ω; (2)' There is an ε > 0 such that for every Ω with Ω 0 ⊂ Ω ⊂ Ω we have H n (Γ) < H n (∂ Ω ) + ε/4. Then we can find a nested familyΓ t and an associated family of open setsΩ t such that Ω 1 ⊂Ω 1 ,Γ 0 = Γ, and every hypersurface has area at most A + ε. Furthermore, if Ω ⊂ Ω 1 , thenΓ 1 = Γ 1 . Proof. The argument is demonstrated in Figure 5. We begin with a proof of the first half of this lemma. We consider two cases. Suppose first that Ω ⊂ Ω 0 . For a sufficiently small δ > 0 the function g : cl(N δ (Γ) ∩ Ω) → [0, 1] given by g(x) = 1 δ dist(x, Γ) is a smooth function with no critical points andΓ t = g −1 (t) a hypersurface of area at most H n (Γ) + ε/2. By condition (2) H n (Γ) ≤ H n (∂ Ω 0 ) + ε/4 and so H n (Γ t ) ≤ A + ε. We extend g to a Morse function on M in an arbitrary way. {Γ t } is a nested family satisfying the conclusions of the theorem. Suppose now that Ω \ Ω 0 = ∅. Make a small perturbation to the hypersurface Γ = ∂Ω, so that f \| cl(Ω) is Morse and (1) and (2) are still satisfied, possibly replacing ε/4 in (2) by ε/2. Consider f restricted to Ω. After composing with a diffeomorphism of [−1, ∞) we may assume that f (cl(Ω)) ⊂ [−1, 1] and f (cl(Ω \ Ω 0 )) = [0, 1]. We apply Lemma 4.1 with N = Ω and Σ = Γ to obtain a Morse function g : Ω → [− 1 2 , 1], such that g −1 (−1) is a point in Ω, g −1 (1) = Γ and H n (g −1 (t)) ≤ H n (∂(f −1 ([−1, t])∩Ω)+ε/2. It follows that H n (g −1 (t)) ≤ H n (f −1 (t) ∩ Ω) + H n (f −1 ([−1, t]) ∩ Γ) + ε/2. Furthermore, we have g −1 ([−1, 0)) ⊂ N ε (Ω ∩ Ω 0 ) . After a small perturbation of the function g we may assume that g −1 ([−1, 0)) ⊂ (Ω ∩ Ω 0 ). We extend g to a Morse function on M in an arbitrary way. We claim that Γ t = g −1 (t) for t ∈ [0, 1] is the desired nested family. The only thing left to prove is an upper bound for the areas ofΓ t . For any smooth hypersurface Σ t obtained by a small perturbation of ∂(Ω ∪ Ω t ) we have H n (Γ) ≤ H n (Σ t ) + ε/4 by (2). It follows that H n (Γ) ≤ H n (∂(Ω ∪ Ω t )) + ε/2 Since ∂(Ω ∪ Ω t ) = (Γ t ∩ Ω) ∪ (Γ \ Ω t ) we have H n (Γ ∩ Ω t ) + H n (Γ \ Ω t ) ≤ H n (Γ t \ Ω) + H n (Γ \ Ω t ) + ε/2 H n (Γ ∩ Ω t ) ≤ H n (Γ t \ Ω) + ε/2 By Lemma 4.1 we have H n (Γ) ≤ H n (Γ t ∩Ω) + H n (Γ ∩ Ω t ) + ε/2 ≤ H n (Γ t ∩Ω) + H n (Γ t \ Ω) + ε ≤ H n (Γ t ) + ε ≤ A + ε If Ω 0 ⊂ Ω, then by choosing sufficiently small ε > 0 and applying Lemma 4.1 (4) we haveΓ 0 = f −1 (0) = Γ 0 . The proof of the second half is similar. If Ω 1 ⊂ Ω we define the desired nested family {Γ} in a small tubular neighbourhood of Γ. Otherwise, after composing with a diffeomorphism of [−1, ∞) we may assume that f (cl(Ω \ Ω 0 )) = [0, 1]. Definef (x) = −f (x). We apply Lemma 4.2 to the restrictionf : M \ Ω → (−∞, 0]. It follows that there exists a Morse functiong, such thatg −1 (0) = Γ and H n (g −1 (−t)) ≤ H n (∂(f −1 ([t, ∞)) \ Ω)) + ε/2. We define g(x) = −g(x) for x ∈ M \ Ω and extend it to a Morse function from M to [−1, ∞) in an arbitrary way. By property (2) of Lemma 4.2 we have that (possibly after a small perturbation) Ω 1 = g −1 ([−1, 1)) ⊃ Ω 1 . The bound on the area is similar to the argument in the proof of I. It follows by (2)' that H n (G t ) ≤ H n (Γ t \ Ω) + H n (Γ t ∩ Ω) + ε/2 < A + ε. If Ω ⊂ Ω 1 then by property (4) of Lemma 4.2 we may assume that Ω 1 = g −1 (1) = Ω 1 . The second lemma in this section will deal with extending a Morse foliation. The following result of Falconer ( [Fa], see also [Gu1, Appendix 6]) will be used in the proof. Theorem 5.2. (Falconer) There exists a constant C(n) so that the following is true. Let U ⊂ R n+1 be an open set with smooth boundary. There exists a line l ∈ R n+1 , so that projection p l onto l satisfies V ol n (U ∩ p −1 l (t)) < C(n)V ol n+1 (U ) n n+1 for all t ∈ l. Moreover, we can assume that p l restricted to ∂U is a Morse function. (1) H n (Γ t ) ≤ H n (∂ Ω 0 ) + H n (∂(Ω 1 \ Ω 0 )) + C(n)(1 + L) n H n+1 (Ω 1 \ Ω 0 ) n n+1 + ε; (2) Ω 0 is an inward ε-perturbation of Ω 0 and Ω 1 = Ω 1 ; Alternatively, we can require that instead of (2) the family satisfies (2') Ω 1 is an outward ε-perturbation of Ω 1 and Ω 0 = Ω 0 ; Proof. Let Ω be an inward ε/8-perturbation of Ω (Σ a t ) ≤ H n (∂(Ω 1 \ Ω 0 )) + C(n)(1 + L) n H n+1 (Ω 1 \ Ω 0 ) n n+1 + ε/2. Let {Σ b t } be a nested{Ξ c t }, such that Ξ c 1 = Ω 1 , Ξ c 0 = Ξ 1 Ξ 2 , where Ξ 1 is an inward ε/8perturbation of Ω 0 and Ξ 2 is an inward ε/8-perturbation of Ω 1 \ Ω 0 . It follows from the properties of perturbations that, without any loss of generality, we may assume Ξ 1 = Ξ b 1 and Ξ 2 = Ω . We define Γ t = Σ a 2t ∪ Σ b 2t for t ∈ [0, 1/2) and Γ t = Σ c 2t−1 t ∈ [0, 1/2] with the open sets defined correspondingly. We leave it to the reader to verify that a similar construction yields a family satisfying (2') instead of (2). Nested sweepouts In this section we prove the following proposition. The proof proceeds in three steps. 6.1. Step 1. Preliminary modification of the family. We start by replacing the original family {Γ t } with a new family {Γ t } that possesses the property that every hypersurface in the family nearly coincides in the complement of a small ball with some hypersurface from a finite list {Γ t i }. This construction is inspired by constructions of families, which are continuous in the mass norm in the work of Pitts and Marques-Neves (see [Pi,4.5] and [MN1,Theorem 14.1]). i : cl(Ω t i+1 \ Ω t i ) → [t i , t i+1 ], such that Γ t = g −1 i (t) and Ω t = Ω t i ∪g −1 (−∞, t) for t ∈ [t i , t i+1 ]. B. Ω t i+1 ⊂ Ω t i and there exists a Morse function g i : cl(Ω t i \ Ω t i+1 ) → [t i , t i+1 ], such that Γ t = g −1 i (t) and Ω t = Ω t i \g −1 (−∞, t] for t ∈ [t i , t i+1 ] . Proof. Let M be a compact subset of M that contains the closure of Ω t for all t ∈ [0, 1]. Choose r sufficiently small so that for every ball B of radius less or equal to r in M the following holds: (i) B is (1 + ε 100W ) 1/n −bilipschitz diffeomorphic to the Euclidean ball of the same radius; ( ii) H n (B ∩ Γ t ) < ε 20 Let {B i } be a collection of k balls of radius r covering M , such that balls of half the radius cover M . We choose a partition 0 = s 0 < ... < s N = 1, such that (iii) H n+1 (B i ∩ (Ω s j \ Ω s j+1 )) + H n+1 (B i ∩ (Ω s j+1 \ Ω s j )) < min{ rε 10k , ( ε 10 ) n+1 n } for each j = 0, ..., N and i = 1, ..., k. We define the new family {Γ t } as follows. For t = s j we set Ω t = Ω t and Γ t = ∂ Ω t , unless Γ t is a finite collection of points in which case we set Γ t = Γ t and Ω t = ∅. Define a subdivision of [s j , s j+1 ] into 2k subintervals, s j = s 0 j < ... < s 2k j = s j+1 . Let {B i } be a collection of k balls concentric with B i of radius between r/2 and r and such that ∂B i intersects Γ s j and Γ s j+1 transversally. Set U 1 j = Ω s j \ Ω s j+1 and U 2 j = Ω s j+1 \ Ω s j . By coarea formula and property (iii) for our choice of the subdivision 0 = s 0 < ... < s N = 1 we may assume that B i satisfies H n (∂B i ∩ (U 1 j ∪ U 2 j )) ≤ ε 4k . By our choice of B i we have that the collection of balls {B i } k i=1 still cover M . Inductively we define Ω s 0 j = Ω s j Ω s 2i−1 j = Ω s 2i−2 j \(B i ∩ U 1 j ) Ω s 2i j = Ω s 2i−1 j ∪(B i ∩ U 2 j ) for i = 1, ..., k. Surfaces ∂ Ω s l j may not be smooth, but there exists an arbitrarily small perturbation so that the boundaries are smooth (see Section 3.2). We perform these perturbations in the inward direction for Ω s 2i−1 j and in the outward direction for Ω s 2i j . To simplify notation we do not rename the sets after the perturbations; since the perturbations are arbitrarily small all the estimates for areas and volumes remain valid. The following properties follow from the definition and (i)-(ii): (a) \| H n (∂ Ω s l j ) − H n (Γ s j )\| < ε/2; (b) Ω s 2i−1 j ⊂ Ω s 2i j and Ω s 2i−1 j ⊂ Ω s 2i−2 j . We Consider the set Ω s 2i−2 j \ Ω s 2i−1 j = B i ∩ U 1 j . After smoothing the corner (see Section 3.2) we call this set U . We map B j+1 to R n+1 by a (1 + ε 100W ) 1/n -bilipschitz diffeomorphism. Existence of the desired nested families follows by properties (i)-(iii) and Lemma 5.3. 6.2. Step 2. Local monotonization. Assume that family {Γ t } satisfies conclusions of Lemma 6.2 for the subdivision 0 = t 0 < ... < t N = 1. For every ε > 0 and each i = 0, ..., N − 1 we will define sets Ω i 0 and Ω i 1 , such that the following holds: ( 2.1) Ω i 0 ⊂ Ω i 1 ; (2.2) max{H n (∂ Ω i 0 ), H n (∂ Ω i 1 )} ≤ max{H n (Γ t i ), H n (Γ t i+1 )}; (2.3) Ω t i+1 ⊂ Ω i 1 and Ω i 0 ⊂ Ω t i ;((Γ i t ) ≤ max{H n (Γ t i ), H n (Γ t i+1 )}+ ε. Definition of Ω i 0 and Ω i 1 Assume (2.1) -(2.4) are satisfied for all Ω j 0 and Ω j 1 for j < i. By Lemma 6.2 (1.3) we only need to consider the following two cases: (A) Ω t i ⊂ Ω t i+1 . In this first case we define Ω i 0 = Ω t i and Ω i 0 = Ω t i . Properties (2.1)-(2.3) follow immediately from the definition. Property (2.5) follows by Lemma 6.2 (1.3). (B) Ω t i+1 ⊂ Ω t i . In the second case we consider two subcases: (B1) Suppose H n (∂ Ω t i ) ≥ H n (∂ Ω t i+1 ). We define Ω i 0 = Ω t i+1 \cl(N δ (∂ Ω t i+1 )) , where δ > 0 is chosen sufficiently small so that cl(N δ (∂ Ω t i+1 )) is diffeomorphic to ∂N × [−δ, δ] and hypersurfaces equidistant from ∂N in this neighbourhood all have areas less than H n (∂ Ω t i+1 ) + ε/2. We set Ω i 1 = Ω t i+1 . (B2) Suppose H n (∂ Ω t i ) < H n (∂ Ω t i+1 ). We set Ω i 0 = Ω t i and Ω i 1 = N δ (Ω t i ), where δ > 0 is chosen as in (B1) to guarantee property (2.4). It is straightforward to verify that with these definitions Ω i 0 and Ω i 1 satisfy (2.1)-(2.4). The following important property is an immediate consequence of (2.3): (2.5) Ω i+1 0 ⊂ Ω i 1 . Informally, the reason why (2.5) holds is because to construct Ω i+1 0 we push Ω t i+1 inwards (or not at all) and to construct Ω i 1 we push Ω t i+1 outwards (or not at all). 6.3. Step 3. Gluing two nested families. We prove the following (Γ t ) ≤ W + ε, Ω b 1 ⊂ Ω 1 and Ω 0 ⊂ Ω a 0 . Proof. The idea for the proof is shown in Figure 5. Let S denote the collection of all open sets Ω , such that Ω b 0 ⊂ Ω ⊂ Ω a 1 and ∂ Ω is smooth. Let A = inf Ω ∈S H n (∂ Ω ) and choose Ω ∈ S with and H n (∂ Ω) < A + ε/4. We set α = ∂ Ω. We claim that Ω and α satisfy properties (i) and (ii) from Lemma 5.1(I) for Ω t = Ω a t . Indeed, if Ω satisfies Ω ⊂ Ω ⊂ Ω a 1 then Ω ∈ S and H n (∂ Ω ) < H n (α) + ε/4. By b t }, such that Ω b 1 ⊂Ω b 1 ,Γ b 0 = α and H n (Γ b t ) ≤ W + ε. We define the desired nested family Γ t simply by concatenating these two nested families. Now we are ready to complete the proof of Theorem 6.1. We apply local monotonization to define families {Γ i t } for i = 1, ..., N − 1. By (2.5) we have Ω 2 0 ⊂ Ω 1 1 . Hence, we can apply Proposition 6.3 to the nested families and Ω t 2 ⊂ Ω 2 1 ⊂ Ω 1,2 1 . Using (2.5) again we have Ω 3 0 ⊂ Ω 1,2 1 . Hence, we can apply Proposition 6.3 to {Γ 1,2 t } and {Γ 3 t }. We iterate this procedure. At the i-th step we apply Proposition 6.3 to families {Γ 1,...,i t } and {Γ i+1 t } to construct a new nested family {Γ 1,...,i,i+1 t } with Ω 1,...,i 0 ⊂ Ω 0 and Ω 1 ⊂ Ω 1,...,i 1 . Proposition 6.3 and (2.5) guarantee that Ω i+2 0 ⊂ Ω 1,...,i 1 , so we can go to the next step. After performing this operation N times we obtain the desired nested family. This finishes the proof of Theorem 6.1. No escape to infinity In this section we prove Proposition 2.1, which we recall below. Proposition 2.1 For every good set U there exists a positive constant ε(U ) which depends only on U such that the following holds. For every good sweepout {Γ t } of U with associated family of open sets {Ω t }, there is a surface Γ t in the collection which has area at least W g (U ), and such that H n (Γ t ∩ cl(U )) ≥ ε(U ). The proof is by contradiction. We assume that Proposition 2.1 does not hold and construct a good sweepout with volume of hypersurfaces strictly less than W g (U ). The main tool in the proof is Theorem 6.1. Let U be a good set. Lemma 7.1. There exits ε(U ) > 0, ε 0 (U ) > 0 and ε 1 (U ) > 0 such that for any open set Ω the following holds: (1) max{ε, ε 1 } < H n (∂U )/10. (2) If ε 0 < H n+1 (Ω ∩U ) < H n+1 (U ) − ε 0 then H n (∂ Ω ∩U ) > 2ε. (3) A) If H n+1 (Ω ∩U ) < 2ε 0 then there exists a family of open sets {Ξ t } with Ξ 0 = Ω , Ξ t \ N ε 1 (U ) = Ω \N ε 1 (U ), Ξ 1 ∩ U = ∅ and H n (∂Ξ t ) < H n (∂ Ω ) + ε 1 . B) If H n+1 (Ω ∩U ) > H n+1 (U ) − 2ε 0 then there exists a family of open sets {Ξ t } with Ξ 0 = Ω , Ξ t \ N ε 1 (U ) = Ω \N ε 1 (U ), Ξ 1 ∩ U = U and H n (∂Ξ t ) < H n (∂ Ω ) + ε 1 . Proof. Pick any ε 1 ∈ (0, H n (∂U )/10). We will show that for all sufficiently small ε 0 (with the choice of ε 0 depending on ε 1 ) statement (3) holds; we will show that for all sufficiently small ε (with the choice of ε depending on ε 0 ) statement (2) holds. Statement (2) follows from the properties of the isoperimetric profile of cl(U ). Now we will prove Statement (3) A). Statement (3) B) follows by an analogous argument. The argument is similar to the proof of Proposition 4.3 in [GL] (see also Lemma 7.1 in [Mo]). Let r 0 > 0 be sufficiently small, so that every ball B of radius r ∈ (0, r 0 ] centered at a point in U is 2-bilipschitz diffeomorphic to a ball of the same radius in the Euclidean space. Choose a covering {B i } of U by balls of radius r 0 , so that concentric balls of radius r 0 4 , denoted by 1 4 B i , still cover U . Using coarea formula we may choose a covering {B i } of U by N balls of radius r i ∈ (r 0 /2, r 0 ), so that H n (∂B i ∩ Ω ) ≤ 4ε 0 r 0 . Given an (n − 1)-dimensional compact submanifold γ ⊂ B i we say that an ndimensional manifold (with boundary) Σ ⊂ B i is a δ-minimizing filling of γ if ∂ Σ = γ and for every other submanifold Σ filling γ in cl(B i ) we have H n (Σ) ≤ H n (Σ ) + δ. By Lemma 4.6 in [GL] there exists a constant c 0 (n), so that if A is an open set in ∂B i with H n (A) ≤ c 0 (n)r n 0 then for every δ > 0 there exists a δ-minimizing filling Σ of ∂A in B i , so that Σ does not intersect 1 4 B i . Set δ = ε 1 10N and ε 0 = min{ ε 1 r 0 40 , c 0 r n+1 0 4 , ( ε 1 10 ) n+1 n }. We will inductively remove Ω from each 1 4 B i . Since { 1 4 B i } cover U the desired conclusion follows. Start with B 1 . First we use Lemma 5.3 to construct a nested family that starts on ∂ Ω and ends on the smoothing of ∂ Ω \B i . Let Σ i be a δ-minimizing filling for ∂(Ω ∩∂B i ), which does not intersect 1 4 B 1 . Note that by definition of δ-minimizing we have H n (Σ i ) ≤ H n (∂ Ω ∩B i ) + δ. The second step is to construct a family that starts on a smoothing of ∂ Ω \B i and ends on a smoothing of (∂ Ω \B i ) ∪ Σ i . Note that during these two deformations the areas of hypersurfaces are bounded above by H n (∂ Ω ) + ε 1 and in the end of the second step the area of the hypersurface is bounded above by H n (∂ Ω ) + ε 1 10N . We iterate this procedure for each ball B i . Since at the end of the deformation in each ball we only accumulate an increase in area of at most ε 1 10N the total increase in area will be below ε 1 . Proof of Proposition 2.1. Suppose Proposition 2.1 does not hold. Then there exists a good sweepout {Γ t } t∈[0,1] , such that if H n (Γ t ) ≥ W g (U ) then H n (Γ t ∩U ) < ε(U ). Let {Ω t } denote the corresponding family of open sets. Let f (t) = H n (Γ t ∩U ) . Note that f (t) may not be continuous. However, it is easy to see that one can perturb the family {Γ t } so that it is roughly continuous in the following sense. Definition 7.2. Function f (t) is δ-continuous if the oscillation ω f (t) = lim a→0 [sup s∈[t−a,t+a] f (s) − inf s∈[t−a,t+a] f (s)] satisfies ω f (t) < δ for every t.t } of U , such that f (t) = H n (Γ t ∩U ) is δ-continuous, sup t H n (Γ t ) ≤ sup t H n (Γ t ) + δ and sup t H n (Γ t ∩U ) ≤ sup t H n (Γ t ∩U ) + δ. Proof. This follows from the construction in the proof of Lemma 6.2. Hence, without any loss of generality we may assume that sweepout {Γ t } satisfies the conclusions of Lemma 7.3 for δ < ε/10 and that for all Γ t with H n (Γ t ) ≥ W g (U ) we have H n (Γ t ∩U ) < 1.1ε(U ). Let g : [0, 1] → [0, H n+1 (U )] be defined as g(t) = H n+1 (U ∩ Ω t ). Function g(t) is continuous. By Lemma 7.1 (2) each connected component 1], such that f (t) ≥ 3 2 ε for all t ∈ I. Moreover, by Lemma 7.3 we may assume that ε ≤ f (t i ) ≤ 2ε, i = 0, 1. By continuity of g(t) and since {Γ t } is a sweepout there exists an interval I as above with H n+1 (Ω t 0 ∩U ) ≤ ε 0 and H n+1 (Ω t 1 ∩U ) ≥ H n+1 (U ) − ε 0 . I of g −1 ([ε 0 , H n+1 (U )−ε 0 ]) is contained in some interval I = [t 0 , t 1 ] ⊂ [0, By construction we have that H n (Γ t ) < W g (U ) − δ for some δ > 0 and for all t ∈ I. We would like to turn {Γ t } into a good sweepout of U , while retaining an upper bound on the volume below W g (U ). The family {Γ t } t∈I fails to be a good sweepout of U for two reasons: 1. Ω t 0 ∩U and Ω t 1 \U are not empty; 2. H n (Γ t 0 ) and H n (Γ t 1 ) may be larger than 5 H n (∂U ). In fact, they may be as large as the largest hypersurface in {Γ t } t∈I . To address the first problem we note that Ω t 0 ∩U and Ω t 1 \U have volume at most ε 0 and we may use Lemma 7.1 to homotope Γ t 0 and Γ t 1 outside of U while increasing the H n −measure of the hypersurfaces by a controlled amount. Observe, however, that if δ is much smaller than ε and H n (Γ t i ) is almost equal to W g (U ) − δ then the resulting family will have volume larger than W g (U ). The second problem seems even more substantial. The main tool to resolve these two problems is to replace {Γ t } t∈I with a nested family. This allows us to define certain two nearly volume minimizing hypersurfaces. We then modify the nested family so that it starts and ends on these two hypersurfaces, which have small area and can be "homotoped" away form U to produce a good sweepout. We apply Theorem 6.1 to construct a nested family {Γ t }, t ∈ [0, 1], such that H n (Γ t ) < W g (U ) − δ 2 ,Ω 0 ⊂ Ω t 0 and Ω t 1 ⊂Ω 1 . The situation is depicted on Figure 6. It will be useful to define the set P = (Ω t 0 ∩U ) ∪ (U \ cl(Ω t 1 )). P will play an important role for three reasons: ( 2.1) H n+1 (P ) < 2ε 0 (2.2) H n (∂P ∩ U ) ≤ 4ε (2.3)Ω 0 ∩ U and (M \ cl(Ω 1 )) ∩ U are contained in cl(P ). Lemma 7.4. There exists t ∈ [0, 1], such that H n (Γ t \ U ) ≤ 2 H n (∂U ). Proof. Let L = max t {H n (Γ t ∩ U )}. LetŪ denote an inward δ-perturbation of U \ cl(P ). We have that {Γ t } is a nested sweepout ofŪ . By Lemma 4.1 there exists a nested sweepout ofŪ by hypersurfaces of area at most L + H n (∂Ū ) + δ ≤ L + H n (∂U ) + 4ε + 2δ ≤ L + 2 H n (∂U ) Moreover, this sweepout starts on a hypersurface of area 0 and ends on ∂Ū . By Lemma 7.1 we can deform ∂Ū outside of U through hypersurfaces of controlled area. We have produced a good sweepout of U with maximal volume of the hypersurface at most L + 2 H n (∂U ). By definition of W g t we have L + 2 H n (∂U ) ≤ W g . Hence, H n (Γ t ) < W g (U ) implies that for some t ∈ [0, 1] we have H n (Γ t \ U ) < 2 H n (∂U ). We will construct a good sweepout ofŪ with hypersurfaces of area at most W g (U )− δ, starting and ending on hypersurfaces less than 5 H n (∂U ). By Lemma 7.1 we can deform it into a good sweepout of U by hypersurfaces of area at most W g (U ) − δ/4. This contradicts the definition of W g (U ) and so Proposition 2.1 follows. To construct a good sweepout ofŪ with these properties we proceed as follows. Let t be as in Lemma 7.4, and let U 0 denote a collection of all open sets Ω with smooth boundary, such thatΩ 0 ⊂ Ω ⊂Ω t \Ū , whereŪ denotes an inward δ 100 -perturbation of U \ cl(P ). Let U 1 denote a collection of all open sets Ω with smooth boundary, Figure 6. Replacing family {Γ t } t∈I with a nested family {Γ t } such thatΩ t ∪Ū ⊂ Ω ⊂Ω 1 . Let A i = inf{H n (∂ Ω) : Ω ∈ U i }. Observe that a perturbation ofΩ t \ cl(Ū ) is an element of U 0 and a perturbation ofΩ t ∪Ū is an element of U 1 . By Lemma 7.4 the boundary areas of these hypersurfaces are at most 3 H n (∂U ). Hence, it follows from Lemma 7.1 that A i ≤ 3 H n (∂U ). Let Σ 0 = ∂Ξ 0 and Σ 1 = ∂Ξ 1 be two hypersurfaces with Ξ i ∈ U i and H n (Σ i ) ≤ A i + δ/4. We have that Ξ 0 is contained inΩ t , and thatŪ is contained in its complement, and we also have that Ξ 1 contains bothŪ andΩ t . In particular, the set Ξ 1 \ Ξ 0 containsŪ . We apply Lemma 5.1 I to construct a nested sweepout ofŪ that starts on Σ 0 and ends onΩ 1 and is composed of hypersurfaces of area at most W g (U ) − 3δ/4. Here we are using the fact that Ξ 0 is contained inΩ 1 . We then apply Lemma 5.1 II to this sweepout to produce a nested sweepout ofŪ that starts on Σ 0 and ends on Σ 1 and is composed of hypersurfaces of area at most W g (U ) − δ/4. Here we are using the fact that Xi 0 ⊂ Xi 1 . This finishes the proof of Proposition 2.1. This proof is shown in Figure 7. 8. Convergence of a min-max sequence to a minimal hypersurface 8.1. Manifolds with sublinear volume growth. In this section we prove Theorem 1.1 and Corollary 1.3. Corollary 1.3 follows from the following lemma. We show that if M has sublinear volume growth (in particular, if it has finite volume) then it contains a good set. Lemma 8.1. Let M n+1 be a complete non-compact manifold with sublinear volume growth. There exists a good set U ⊂ M , such that 0 < W g (U ) < ∞. Proof. Let x be such that lim inf r→∞ V ol(Br(x)) r = 0 Fix a small geodesic ball B r (x) and define an isoperimetric constant C I = inf{H n (Σ)}, where the infimum is taken over all hypersurfaces in B r (x), subdividing B r (x) into two subsets of equal volume. By the coarea formula we can find R > r with H n (∂B R (x)) < C I 100 and ∂B R (x) smooth. It follows that B R (x) is a good set. The distance function d x (y) = dist(x, y) may not be smooth, but there exists a smoothing of this functiond x (see [GW]), such thatd x = d x in B R (x) and \|∇d x \| ≤ 1 + ε for all y. Moreover, we may assume that d x is a Morse function. Hence, the set of good sweepouts of B R (x) is non-empty. Every sweepout of B R (x) is also a sweepout of B r (x), so it must contain a hypersurface of area at least C I . 8.2. Proof of Theorem 1.1. Theorem 1.1 follows immediately from the following Theorem. Theorem 8.2. Let M n+1 be a complete Riemannian manifold of dimension n + 1. Suppose M contains a good set U . For every δ > 0 there exists a complete embedded minimal hypersurface Γ, satisfying the following properties: (1) H n (Γ) ≤ W ∂ (U ) + H n (∂U ); (2) H n (Γ ∩N δ (U )) ≥ ε(U ) 2 , where ε(U ) is as in Lemma 7.1. The hypersurface is smooth in the complement of a closed set of dimension n − 7. Remark 8.3. a) The min-max argument applied to families of good sweepout of a good set U may produce a non-compact minimal hypersurface. Consider the following example. Let S r denote spheres of radius r in R 3 . We modify the Euclidean metric on R 3 , so that the new metric is invariant under rotations around 0, and so that the areas of S r and lengths of great circles on S r decay exponentially for r > 1. If the decay is fast enough the min-max argument for good sweepouts of the ball B 2 (0) will produce a hyperplane passing through 0 (of area π + ε). b) If U is conformally equivalent to a metric of non-negative Ricci curvature then from [GL] we obtain an upper bound for the volume of the minimal hypersurface H n (Γ) ≤ C(n) H n+1 (U ) n n+1 . To prove Theorem 8.2 we use Proposition 2.1 and arguments from [DT]. For the most part in this section we closely follow [DT]. However, some modifications are necessary in construction of the pull-tight deformation and construction of a min-max sequence, which is almost minimizing in all sufficiently small annuli. The regularity of a stationary varifold obtained from a min-max sequence is proved using the notion of ε-almost minimizing hypersurfaces introduced in [Pi]. We will use the notion of almost minimality from [DT,2.2]. Definition 8.4. Let ε > 0 and U ⊂ M open. A boundary ∂ Ω is called ε-almost minimizing in U if there is NO 1-parameter family of boundaries {∂ Ω t }, t ∈ [0, 1], satisfying the following properties: • (s1), (s2), (s3), (sw1), and (sw3) of Definition 3.2 hold; • ∂ Ω 0 = Ω and ∂ Ω t \U = ∂ Ω \U for every t; • H n (∂ Ω t ) ≤ H n (∂ Ω) + 1 8 ε; • H n (∂ Ω 1 ) ≤ H n (∂ Ω) − ε A sequence {∂ Ω k } of hypersurfaces is called almost minimizing in U if each ∂ Ω k is ε k -almost minimizing in U for some sequence ε k → 0. Let AN r (x) denote the set of all open annuli An(x, t 1 , t 2 ) = B t 2 (x) \ cl(B t 2 (x)) for t 1 < t 2 < r. We have the following result from [DT]: Proposition 8.5. Let r : M → R + be a function and {Γ k } is a sequence of hypersurfaces, s.t. (A) {Γ k } is a.m. in every An(x) ∈ AN r(x) (x); (B) Γ k converges to a stationary varifold V as k → ∞. Then V is induced by an embedded minimal hypersurface, which is smooth on the complement of a closed set of Hausdorff dimension at most n − 7. Proof. This proposition is contained in Propositions 2.6, 2.7 and 2.8 of [DT]. All arguments there are local and therefore they apply to the non-compact case. Proposition 8.6. Let U ⊂ M be a good set and suppose W g (U ) < ∞. For every δ > 0 there exists a function r : M → R + , ε > 0 and a sequence {Γ k }, such that (A) and (B) of Proposition 8.5 hold and (C) H n (Γ k ∩N δ (U )) > ε/2 for every k. Remark 8.7. The statement of the proposition remains true if we replace 1-neighbourhood of U with N r 0 (U ) for any positive r 0 . The function r : M → R + may change depending on r 0 . Combining Proposition 8.5 and 8.6 we obtain that M contains a stationary varifold V induced by a minimal hypersurface Σ with H n (Σ ∩N δ (U )) > ε/2. In particular, the intersection of Σ with N δ (U ) is non-empty and the minimal hypersurface has volume at least ε/2. This implies Theorem 8.2. The rest of this section will be devoted to the proof of Proposition 8.6. 8.3. Pull-tight. Using terminology from [DT] we say that a sequence {Γ i t } of good sweepouts of U is minimizing if lim i→∞ sup t H n (Γ i t ) = W g (U ) and a sequence of hy- persurfaces {Γ i t i } with lim i→∞ H n (Γ i t i ) → W g (U ) will be called a min-max sequence. Let V denote the space of varifolds in M with mass bounded by 2W g (U ). V is endowed with weak* topology. By the Riesz Representation Theorem and the Banach-Alaoglu Theorem this space is compact and metrizable. Let d denote a metric on V which induces this topology. Another important metric on the space of varifolds is given by (see [Pi,2.1(19) ]) F(V 1 , V 2 ) = sup{V 1 (f ) − V 2 (f )\|f ∈ K(Gr n (M )), \|f \| ≤ 1, Lip(f ) ≤ 1} where K(Gr n (M )) denotes the set of Lipschitz functions compactly supported in Gr n (M ). When manifold M is compact the topology of the F metric and the weak* topology on V coincide. When M is not compact these topologies are different. Moreover, in this case V is not compact in the F metric. The standard pull-tight argument (see [Pi,Theorem 4.3], [CD,Proposition 4.1] and [MN1,Proposition 8.5]) uses compactness with the F metric in an important way, so in our case the argument has to be modified. Let V st ⊂ V denote the closed subset of stationary varifolds in V (see [Si,8.2]). If Γ is a hypersurface we will slightly abuse notation and write Γ to denote the varifold induced by Γ. Lemma 8.8. There exists a minimizing sequence {{Γ i t }} of good sweepouts of U , such that for every min-max sequence {Γ i t i } we have lim i→∞ d(Γ i t i , V s ) = 0. Let Ω ⊂ M be an open subset. Let V Ω denote the space of varifolds in Ω with mass bounded by 2W g (U ). For varifolds in Ω we can define metric F Ω (V 1 , V 2 ) = sup{V 1 (f ) − V 2 (f )\|f ∈ K(Gr n (Ω)), \|f \| ≤ 1, Lipf ≤ 1} It follows from the definition that F Ω 1 (V 1 Gr n (Ω 1 ), V 2 Gr n (Ω 1 )) ≤ F Ω 2 (V 1 Gr n (Ω 2 ), V 2 Gr n (Ω 2 )) whenever Ω 1 ⊂ Ω 2 . When Ω is a bounded subset of M the weak topology on V Ω and the topology induced by the F Ω metric coincide. We will also need the following notation. Let V Ω,st denote the set of all stationary varifolds of mass at most 2W g and supported in Gr n (Ω). Lemma 8.9. Let Ω be a bounded open set. There exists a map Φ Ω : V → V and monotone sequences of positive numbers τ 1 ≥ τ 2 ≥ ...τ k → 0 and ε 1 ≥ ε 2 ≥ ...ε k → 0 with the following properties. ( Proof. Fix integer k > 0. Let V Ω,k be the set of varifolds V ∈ V satisfying the following properties: 1. \|\|V \|\|(M ) ∈ [9 H n (∂U ), 2W g (U )]; 2. F Ω (V Gr n (Ω), V Ω,st ) ∈ [ 1 2 k+1 , 1 2 k ] Let p(V ) = V Gr n (Ω) denote the restriction function and let V Ω,k = p(V Ω,k ). It is straightforward to check that V Ω,k is compact in the topology induced by the F Ω metric. We will say that a smooth vector field χ is admissible if χ is compactly supported in Ω, \|χ\| C 1 ≤ 1 and \|χ(x)\| ≤ dist(x, ∂ Ω). Let X Ω denote the set of all admissible vector fields. We claim that there exists a c k > 0, such that sup V ∈V Ω,k inf χ∈X Ω {δV (χ)} < −c k for otherwise there would exist a sequence of varifolds V i ∈ V Ω,k converging (in F Ω ) to a stationary varifold supported in Gr n (Ω), which contradicts condition 2 above. Here, δV (χ) means the first variation of V with respect to the vector field χ. By compactness (cf. arguments in [Pi,Theorem 4.3], [CD,Proposition 4.1] and [MN1,Proposition 8.5]) we can find a locally finite open covering {U k i } of V Ω,k and a collection of admissible vector fields {χ k i }, such that δV (χ) < − c k 2 and U k 1 i 1 is disjoint from U k 2 i 2 whenever \|k 1 − k 2 \| ≥ 2. LetŪ k i = p −1 (U k i ) ∩ {V : \|\|V \|\|(M ) > 6 H n (∂U )}. We have that the collection {Ū k i } covers V Ω,k and the union Ū k i is disjoint from the set of varifolds V with \|\|V \|\|(M ) ≤ 5 H n (∂U ). Choose a partition of unity {φ k i } subordinate to {Ū k i }. Define a continuous family of vector fields χ V = φ k i (V )χ k i . We have that χ V is admissible for all V ∈ V and δV (χ V ) < − min 1 2 {c k−1 , c k , c k+1 }. Hence, for each χ V we can define a 1-parameter family of diffeomorphisms Ψ V : [0, ∞) × M → M with ∂Ψ V (t,x) ∂t = χ V (Ψ V (t, x). By definition of admissible vector field we have that Ψ V is the identity on Gr n (M \ Ω). It follows that there exists a continuous choice of t = t(V ) and ε k > 0 so that \|\|Ψ V # (t V , V )\|\|(M ) ≤ \|\|V \|\|(M ) − ε k for all V ∈ V Ω,k . Moreover, we may assume that t V ≤ 1/k if V ∈ V Ω,k . We define Φ(V ) = Ψ V # (t V , V ). Properties A and B follow by construction. Recall that N r (U ) = {x ∈ M : d(x, U ) < r} denote the r-neighbourhood of U . Let A(r, U ) denote the set of all open subsets V of M , such that either V ∩ cl(U ) = ∅ or V ⊂ N r (U ). Lemma 8.11. Let {{Γ i t }} be a minimizing sequence of good sweepouts as in Lemma 8.8 and assume furthermore that H n (Γ k t ) < W g (U ) + 1 8k . For every r > 0 and N large enough, there exists t N ∈ [0, 1] such that • Γ N = Γ N t N is 1 N -a.m. in all (U 1 , U 2 ) ∈ CO(A(r, U )) • H n (Γ N ) ≥ W − 1 N • H n (Γ N ∩cl(N r (U ))) ≥ ε(U )/2 Proof. The proof is by contradiction (cf. proofs of [CD,5.3] and [DT,3.4]). Assume N to be sufficiently large so that 1 N < ε/2. Let A N = {t ∈ [0, 1] : H n (Γ N t ) ≥ W g (U ) − 1 N } and B N (U, r) = {t ∈ [0, 1] : H n (Γ N t ∩cl(N r (U ))) ≥ ε(U )/2} Define K N (U, r) = A N ∩ B N (U, r). K N (U, r) is a compact set as A N and B N (U, r) are closed. By Proposition 2.1 K N (U, r) is non-empty. Assume the lemma to be false. Then there is a sequence N k , so that Γ N k (a) either {Γ k } is 1/k-a.m. in B r (y) \ cl(U ) for k > k(y) for all y ∈ M \ cl(U ); (b) or there is a (not relabeled) subsequence {Γ k } and a sequence {x k r }, x k r ∈ M \ cl(U ), such that Γ k is 1/k-a.m. in M \ cl(U ∪ B 9r (x k r )). If (a) holds for some positive radius r 0 then condition (A) is satisfied for all y ∈ M \cl(U ) for r(y) = min{r 0 , d(y, cl(U )}. Otherwise, we obtain a sequence {x j } and a (not relabeled) subsequence {Γ k }, such that that Γ j is 1 j -a.m. in M \ cl(U ∪ B 1/j (x j )) for all large j. If sequence {x j } contains a subsequence that converges to a point x ∈ M \cl(U ) then we verify that for a subsequence of {Γ k } condition (A) is satisfied for x with r(x) = d(x, cl(U ) and for all y ∈ M \cl(U ) with r(y) = min{d(y, cl(U )), d(y, x)}. Otherwise there is a subsequence of {x j }, such that either d(x j , cl(U )) → ∞ or d(x j , cl(U )) → 0. In both cases we have that condition (A) is satisfied for all y ∈ M \ cl(U ) with r(y) = d(y, cl(U )). Theorem 1 . 1 . 11Let M n+1 be a complete Riemannian manifold of dimension n + 1. Suppose M contains a bounded open set U with smooth boundary, such that V ol n (∂U ) ≤ W ∂ (U ) 10 . Then M contains a complete embedded minimal hypersurface Γ of finite volume. The hypersurface is smooth in the complement of a closed set of Hausdorff dimension n − 7. Every complete non-compact Riemannian manifold M n+1 of finite volume contains a (possibly non-compact) embedded minimal hypersurface of finite volume. The hypersurface is smooth in the complement of a closed set of Hausdorff dimension n − 7. Figure 1 . 1Cut and paste argument in the proof of 3 Ω t \U . . Smoothing corners. Let N ⊂ M be an open subset and suppose Σ 1 ⊂ ∂N and Σ 2 ⊂ ∂N are n-dimensional submanifolds of M , such that the interiors of Σ 1 and . 4 . 4Morse foliations with controlled area of fibers. Here we present several results about concatenating different Morse foliations and controlling areas of fibers of Morse functions. For PL Morse functions Sabourau proved similar results in [Sa]. 4.1. Gluing Morse foliations. Let N ⊂ M be a compact submanifold of M with boundary. We will say that a Morse function f : N → R is ∂-transverse if (1) there exists an extensionf of f to an open neighbourhood of N in M , such that all critical points are isolated, non-degenerate and lie in the interior of N ; Lemma 4. 1 . 1Let N ⊂ M be a compact submanifold with non-empty boundary and f : N → [a, b] be a ∂-transverse Morse function. Let Σ be a closed submanifold of ∂N . For every ε > 0 there exists a Morse function g : N → [a, b] Figure 2 . 2Constructing a singular foliation of N . thenΣ t has a Morse type singularity at the singular point s of f in the interior of N . Since t is at least ε 0 away from singular values of Figure 3 . 3Procedure for dealing with singularities on the boundary of N . Lemma 4. 2 . 2Let N ⊂ M be a not necessarily compact submanifold with non-empty boundary and f : N → (−∞, b] be a proper Morse function, which is ∂-transverse. Let Σ be a compact submanifold of ∂N . Lemma 4. 3 . 3Let N be a manifold with compact boundary ∂N and Σ be a hypersurface with ∂Σ ⊂ ∂N and such that Σ intersects ∂N transversally. Suppose N \ Σ = V 1 V 2 . For every ε > 0 there exist open sets with smooth boundary Ω 1 and Ω 2 and a Morse function f : cl(N \ (Ω 1 ∪ Ω 2 )) → [0, 1], such that the following holds: Figure 4 . 4Gluing two submanifolds using a Morse foliation. Lemma 5. 1 . 1Suppose that f : M → [−1, ∞) is a Morse function and {Γ t } = {f −1 (t)} t∈[0,1] is a nested family of hypersurfaces of area ≤ A with associated open sets {Ω t } = {f −1 ((−∞, t))}. Lemma 5. 3 . 3Let ε > 0, L > 0. Suppose Ω 0 ⊂ Ω 1 are bounded open sets with smooth boundary and Ω 1 \ Ω 0 ⊂ U , where U is (1 + L)-bilipschitz diffeomorphic to an open subset of R n+1 . There exists a constant C(n) and a nested family {Γ t } with a family of corresponding open sets {Ω t }, such that family with a corresponding family of open sets {Ξ b t }, such that Ξ b 0 is an inward ε/2-perturbation of Ω 0 , Ξ b 1 is an inward ε/8-perturbation of Ω 0 and the areas of all hypersurfaces are at most H n (∂ Ω) + ε/2. By Lemma 4.3 there exists a nested family {Σ c t } with a corresponding family of open sets Proposition 6 . 1 . 61For every ε > 0, given a family of hypersurfaces {Γ t } with the corresponding family of open sets {Ω t } and H n (Γ t ) ≤ A, there exists a nested family {Γ t } with the corresponding family of open sets {Ω t }, such thatΩ 0 ⊂ Ω 0 , Ω 1 ⊂Ω 1 and H n (Γ t ) ≤ A + ε. In particular, for any bounded open set U ⊂ M with smooth boundary we have W (U ) = W n (U ). Lemma 6 . 2 . 62For any ε > 0 there exists a partition 0 = t 0 < ... < t N = 1 of [0, 1] and a family {Γ t } with the corresponding family of open sets {Ω t }, such that the following holds: (1.1) Ω 0 ⊂ Ω 0 and Ω 1 ⊂ Ω 1 ; (1.2) sup{H n (Γ t )} < sup{H n (Γ t )} + ε; (1.3) For each i = 0, ..., N − 1 we have one of the two possibilities: A. Ω t i ⊂ Ω t i+1 and there exists a Morse function g . our construction we need to show existence of two types of nested families: a nested family that starts on Γ s In both cases we want the homotopies to satisfy the desired upper bound on the areas. 1 = α and H n (Γ a t ) ≤ W + ε. We claim that Ω and α also satisfy properties (i)' and (ii)' from Lemma 5.1(II) for Ω t = Ω b t . Indeed, if there is an open set Ω with Ω b 0 ⊂ Ω ⊂ Ω then again we have Ω ∈ S and inequality H n (∂ Ω ) < H n (α) + ε/4 follows by definition of Ω. By Lemma 5.1(II) there exists a nested family {Γ b t } with the corresponding family of open sets {Ω Figure 5 . 5{Γ 1 t } and {Γ 2 t }.We obtain a new nested family Γ Gluing two nested sweepouts. Lemma 7 . 3 . 73Let U be a bounded open set with smooth boundary and {Γ t } be a good sweepout of U . For every δ > 0 there exists a good sweepout {Γ Figure 7 . 7Constructing a good sweepout in the proof of Proposition 2.1. 1 ) 1\|\|Φ Ω (V )\|\|(M ) ≤ \|\|V \|\|(M ) (2) If \|\|V \|\|(M ) ≤ 5 H n (∂U ) then Φ Ω (V ) = V (3) If \|\|V \|\|(M ) ≥ 9 H n (∂U ) and F Ω (V Gr n (Ω), V Ω,st ) ∈ [ 1 2 k+1 , 1 2 k ] then the following holds: A. \|\|Φ(V )\|\|(M ) ≤ \|\|V \|\|(M ) − ε k B. F(V, Φ Ω (V )) ≤ τ k Moreover, if {support(V t )} is a family of hypersurfaces in a sense of Definition 3.1 then so is {support(Φ Ω (V t ))}. 2.4) There exists a nested family of hypersurfaces {Γ i t }, 0 ≤ t ≤ 1, with the corresponding family of nested open sets Ω i t , such that H n there exists a nested family {Γ t } and a corresponding family of open sets {Ω t }, such that H nProposition 6.3. Suppose {Γ a t } and {Γ b t } are two nested families (with correspond- ing families of open sets {Ω a t } and {Ω b t } respectively) and H n (Γ i t ) ≤ W . Suppose moreover that Ω b 0 ⊂ Ω a 1 . For any ε > 0 t is not 1 N k -a.m. in some pair (U 1 t , U 2 t ) ∈ CO(A(r, U )) for every t ∈ K N k (U, r). To simplify notation we will drop sub-and superscript N k . We will modify family Γ t on some Acknowledgements This paper uses, in a crucial way, ideas from the work of Regina Rotman and the first author. The authors are grateful to Regina Rotman for many helpful discussions.The second author would like to thank Camillo De Lellis and André Neves for organizing the Oberwolfach seminar "Min-Max Constructions of Minimal Surfaces" andopen sets with M = U i . Let τ U i l and ε U i l be sequences of numbers from Lemma 8.9 for Ω = U i .Let {{Γ i t }} be a minimizing sequence of good sweepouts. We will construct a minimizing sequence of good sweepouts {{F i (Γ i t )}} satisfying the conclusions of Lemma 8.8. Maps F i 's are defined as follows. We set F i (Γ i t ) to be the hypersurface with, where Φ U i is given by Lemma 8.9. (Here we use the standard notation that \|Σ\| denotes the varifold induced by hypersuface Σ).We claim that for eachWe have two possibilities. Suppose first that for some l ∈ {0, ..., i} the varifoldGr n (U i )) ≤ i l=0 τ l k < δ/2 by the triangle equality and the fact that Φ Um is the identity outside of U m . From our choice of k we obtain, as a result, thatin at least one of the two open sets. Let CO(A) denote the set of pairs (U 1 , U 2 ) of open sets such that inf x∈U 1 ,y∈U 2 d(x, y) ≥ 4 min{diam(U 1 ), diam(U 2 )} and U i ∈ A for i = 1, 2.We have two possibilities. Suppose first that ∂ Ω t satisfies H n (∂ Ω t ) < W − 1 N . Since t is contained in at most two distinct intervals J i we have that H n (∂ Ω t ) ≤ H n (∂ Ω t ) + 2 1 4N < W . So the claim holds. Suppose now that H n (∂ Ω t ∩N r (U )) < ε/2. We have that t is contained in at most two intervals, say, J i and J i+1 . If U j (for j = i or i + 1) intersects both U and its complement then by definition of A(r, U ) we must have U j ⊂ N r (U ) and so H n (∂ Ω t ∩U j ) < ε/2. In other words, mass can be transferred inside U j from U comp to U , but the transfer can only happen from the part of the hypersurface that lies inNow we can prove Proposition 8.6. Fix δ > 0. Let {Γ N t N } be the min-max sequence from Lemma 8.11. We will show that its subsequence satisfies the requirements of Proposition 8.6. Conditions (B) and (C) are satisfied by construction. We will choose a subsequence that also satisfies (A).Observe that it follows from the definition if U ⊂ V and Γ is ε-a.m. in V then Γ is ε-a.m. in U .Step 1. Almost minimizing annuli around points in cl(U ). We start by finding a subsequence of {Γ N t N } that is a.m. for annuli centered at x ∈ cl(U ). By Lemma 8.11 for each 0 < r < δ 10 and each x ∈ cl(U ) we have that Γ k is 1 k -a.m. either in B r (x) or N 1 (U )\cl(B 9r (x)). For a fixed r as above we have two possibilities.(a) either {Γ k } is 1/k-a.m. in B r (y) for k > k(y) for all y ∈ cl(U ); (b) or there is a (not relabeled) subsequence {Γ k } and a sequence {x k r }, x k r ∈ cl(U ), such that Γ k is 1/k-a.m. in N 1 (U ) \ cl(B 9r (x k r ). Choose a sequence of radii r j → 0. If there exists r j > 0 such that (a) holds then condition (A) is satisfied for all y ∈ cl(U ) for r(y) = min{r j , δ}. Suppose not. By compactness of cl(U ) we can select (not relabeled) subsequences x k r j → x j ∈ cl(U ) and x j → x ∈ cl(U ). After choosing an appropriate diagonal subsequence we obtain that Γ k is 1 k -a.m. in N 1 (U ) \ cl(B1 j (x)) for all k > j. In particular, (A) of Proposition 8.5 holds for all annuli centered at x with r(x) = δ. For y ∈ cl(U ) \ x we obtain that {Γ k } is a.m. for annuli centered at y with r(y) = min{δ, d(y, x)}.Step 2. Almost minimizing annuli around points in M \ cl(U ). Let {Γ n } denote the min-max sequence from Step 1. By Lemma 8.11 for each y ∈ M \ cl(U ) we have that open set containing K = K(U, r) ⊂ [0, 1], so that the new family Γ t has H n (Γ t ) < W for all Γ t with H n (Γ t ∩(U )) > ε(U ). open set containing K = K(U, r) ⊂ [0, 1], so that the new family Γ t has H n (Γ t ) < W for all Γ t with H n (Γ t ∩(U )) > ε(U ). to choose a covering J i = (a i , b i ) of K and a collection of sets U i so that • each point of K is contained in at most two intervals J i • U i ∈ A(r, U ) for all i • if cl(J i ) ∩ cl(J j ) = ∅ then inf x∈U i. By Lemma 3.1 in [DT] and refinement of the covering argument on page 13 in [DT] it is possible. y∈U j d(x, y) > 0By Lemma 3.1 in [DT] and refinement of the covering argument on page 13 in [DT] it is possible to choose a covering J i = (a i , b i ) of K and a collection of sets U i so that • each point of K is contained in at most two intervals J i • U i ∈ A(r, U ) for all i • if cl(J i ) ∩ cl(J j ) = ∅ then inf x∈U i ,y∈U j d(x, y) > 0 • there exists a δ > 0 such that {(a i + δ, b i − δ)} still cover K and a family {Ω i,t }. such that• there exists a δ > 0 such that {(a i + δ, b i − δ)} still cover K and a family {Ω i,t }, such that . Ω I,T = Ω T If T / ∈ J I And Ω I,T \u I = Ω T \u, Ω i,t = Ω t if t / ∈ J i and Ω i,t \U i = Ω t \U i for all t; . • , Ω t \(U i ∪ U i+1 )] ∪ [Ω i,t ∩U i ] sup[Ω i+1,t ∩U i+1 ] if t / ∈ (a i , b i• Ω t = [Ω t \(U i ∪ U i+1 )] ∪ [Ω i,t ∩U i ] sup[Ω i+1,t ∩U i+1 ] if t / ∈ (a i , b i ) F Almgren, The theory of varifolds, Mimeographed notes. PrincetonF. 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[ "A NOTE ON LARGE AUTOMORPHISM GROUPS OF COMPACT RIEMANN SURFACES", "A NOTE ON LARGE AUTOMORPHISM GROUPS OF COMPACT RIEMANN SURFACES" ]	[ "Milagros Izquierdo ", "Sebastián Reyes-Carocca " ]	[]	[]	Belolipetsky and Jones classified those compact Riemann surfaces of genus g admitting a large group of automorphisms of order λ(g − 1), for each λ > 6, under the assumption that g − 1 is a prime number. In this article we study the remaining large cases; namely, we classify Riemann surfaces admitting 5(g − 1) and 6(g − 1) automorphisms, with g − 1 a prime number. As a consequence, we obtain the classification of Riemann surfaces admitting a group of automorphisms of order 3(g − 1), with g − 1 a prime number. We also provide isogeny decompositions of their Jacobian varieties.	10.1016/j.jalgebra.2019.11.012	[ "https://arxiv.org/pdf/1811.08371v1.pdf" ]	119,133,567	1811.08371	6a3653744e5eaff2961ebe1656b1e52dc3bfbbce	A NOTE ON LARGE AUTOMORPHISM GROUPS OF COMPACT RIEMANN SURFACES 20 Nov 2018 Milagros Izquierdo Sebastián Reyes-Carocca A NOTE ON LARGE AUTOMORPHISM GROUPS OF COMPACT RIEMANN SURFACES 20 Nov 2018 Belolipetsky and Jones classified those compact Riemann surfaces of genus g admitting a large group of automorphisms of order λ(g − 1), for each λ > 6, under the assumption that g − 1 is a prime number. In this article we study the remaining large cases; namely, we classify Riemann surfaces admitting 5(g − 1) and 6(g − 1) automorphisms, with g − 1 a prime number. As a consequence, we obtain the classification of Riemann surfaces admitting a group of automorphisms of order 3(g − 1), with g − 1 a prime number. We also provide isogeny decompositions of their Jacobian varieties. Introduction and statement of the results The classification of groups of automorphisms of compact Riemann surfaces is a stimulating subject of study, and has attracted a considerable interest since the nineteen century. Let S be a compact Riemann surface of genus g ≥ 2. It is well-known that the full automorphism group of S is finite, and that its order is bounded by 84(g − 1). A group of automorphisms G of S is said to be large if its order is strictly greater than 4(g − 1); this bound arises naturally in the theory of Hurwitz spaces. In this case, it is known that S is quasiplatonic (i.e. cannot be deformed non-trivially in the moduli space together with its automorphisms) or belong to a complex one-dimensional family. See [7,11,12,24]. Compact Riemann surfaces with large groups of automorphisms have been considered from different points of view. For instance, the cyclic case was considered by Wiman [44], Kulkarni [24] and Singerman [42] (see also [19]), and the abelian case was classified by Lomuto in [26]. Riemann surfaces with 8(g +1) automorphisms were considered by Accola [1] and Maclachlan [27], and by Kulkarni in [23]. More recently, Riemann surfaces with 4g automorphisms were studied in [7] (see also [33]), and with 4(g + 1) automorphisms in [12]. The maximal non-large case is considered in [34]. Belolipetsky and Jones [3] proved that under the assumption that g −1 is a prime number (sufficiently large for avoiding sporadic cases), a compact Riemann surface of genus g admitting a large group of automorphisms of order λ(g − 1), where λ > 6, belongs to one of six infinite well-described sequences of Riemann surfaces. In this article, we study and classify compact Riemann surfaces of genus g ≥ 8 admitting a group of automorphisms of order 5(g − 1) and 6(g − 1), where g − 1 a prime number; these cases were not considered in Belolipetsky-Jones's article [3]. We also determine an isogeny decomposition of the corresponding Jacobian varieties. The results of this paper are given in Theorem 1 and Theorem 2. Theorem 1. Let g ≥ 8 such that g − 1 is prime. There exists a compact Riemann surface S of genus g with a group of automorphisms G of order 5(g − 1) if and only if g ≡ 2 mod 5. Moreover, in this case: (1) the group G is isomorphic to C g−1 ⋊ 5 C 5 = a, b : a g−1 = b 5 = 1, bab −1 = a r , where r is a 5-th primitive root of the unity in F g−1 , and G acts with signature (0; 5, 5, 5), (2) the action of G extends to an action of a group G ′ isomorphic to C g−1 ⋊ 10 C 10 = a, c : a g−1 = c 10 = 1, cac −1 = a −r , with r as before, and G ′ acts with signature (0; 2, 5, 10), (3) there are exactly four pairwise non-isomorphic such Riemann surfaces S, and (4) the Jacobian variety JS of each S decomposes, up to isogeny, as the product JS ∼ J(S/ a ) × (J(S/ c )) 10 . Furthermore, possibly up to finitely many sporadic cases in small genera, the full automorphism group of S is G ′ . Theorem 2. Let g ≥ 8 such that g − 1 is prime. There exists a compact Riemann surface of genus g with a group of automorphisms of order 6(g − 1) if and only if g ≡ 2 mod 3. Moreover, in this case: (1) the Riemann surfaces form a closed one-dimensional equisymmetric familyF g of Riemann surfaces S with a group of automorphisms G isomorphic to C g−1 ⋊ 6 C 6 = a, c : a g−1 = c 6 = 1, cac −1 = a m , where m is a 6-th primitive root of the unity in F g−1 , and G acts with signature (0; 2, 2, 3, 3), (2) the Jacobian variety JS of each S inF g decomposes, up to isogeny, as the product JS ∼ J(S/ a ) × (J(S/ c )) 6 ,(3) F g contains two Riemann surfaces X 1 and X 2 with a group of automorphisms G ′ of order 12(g − 1) isomorphic to (C g−1 ⋊ 6 C 6 ) × C 2 = a, c × z acting with signature (0; 2, 6, 6), and (4) the Jacobian variety JX i of each X i can be decomposed, up to isogeny, as 6 . Furthermore, if F g denotes the interior ofF g then: (5) if S ∈ F g then G is the full automorphism group of S, and (6) the boundaryF g \ F g ofF g is {X 1 , X 2 }, and the full automorphism group of X 1 and X 2 is G ′ . JX i ∼ J(X i / a ) × (J(X i / cz )) As a consequence of the proof of the theorem above, we are able to easily derive a classification for the non-large case λ = 3. Corollary 1. Let g ≥ 8 such that g − 1 is prime. There exists a compact Riemann surface S of genus g with a group of automorphisms of order 3(g − 1) if and only if g ≡ 2 mod 3. Furthermore, in this case S belongs to the familyF g of Theorem 2. As a consequence, there is no compact Riemann surfaces of genus g with full automorphism group of order 3(g − 1). In Section 2 we will briefly review the background. The results will be proved in Sections 3, 4 and 5. Preliminaries Fuchsian groups. Let H denote the upper-half plane, and let Γ be a cocompact Fuchsian group; i.e. a discrete group of automorphisms of H with compact orbit space H/Γ. The algebraic structure of Γ is determined by its signature: s(Γ) = (h; m 1 , . . . , m l ), (2.1) where h denotes the topological genus of the surface H/Γ, and m 1 , . . . , m l the branch indices in the (orbifold) universal covering H → H/Γ. If l = 0, then Γ is called a surface Fuchsian group. Let Γ be a Fuchsian group with signature (2.1). Then Γ has a canonical presentation with generators a 1 , . . . , a h , b 1 , . . . , b h , x 1 , . . . , x l and relations x m1 1 = · · · = x m l l = Π h i=1 a i b i a −1 i b −1 i Π l i=1 x i = 1. (2.2) The hyperbolic area of each fundamental region of Γ is given by µ(Γ) = 2π[2h − 2 + Σ l j=1 (1 − 1 mj )] . Let Γ ′ be a group of automorphisms of H. If a Fuchsian group Γ is a finite index subgroup of Γ ′ then Γ ′ is also a Fuchsian group and their hyperbolic areas are related by the Riemann-Hurwitz formula µ(Γ) = [Γ ′ : Γ] · µ(Γ ′ ). The complex dimension of the Teichmüller space associated to a Fuchsian group of signature (2.1) is 3g − 3 + l. See, for example, [15,30,41]. 2.2. Riemann surfaces and group actions. Let S be a compact Riemann surface. We denote by Aut(S) the full automorphism group of S, and say that a group G acts on S if there is a group monomorphism ψ : G → Aut(S). The space of orbits S/G of the action of G induced by ψ(G) is naturally endowed with a Riemann surface structure such that the projection S → S/G is holomorphic. By the uniformization theorem, a Riemann surface S is conformally equivalent (isomorphic) to the quotient H/Γ, where Γ is a surface Fuchsian group. Lifting G to the universal covering H → H/Γ, the group G acts on S if and only if there is a Fuchsian group Γ ′ containing Γ and a group epimorphism θ : Γ ′ → G such that ker(θ) = Γ (see [5,15,37,41]). Such an epimorphism will be called a surface epimorphism. We say that the action of G on S is given or represented by the surface epimorphism θ. Note that the Riemann surface S/G is isomorphic to H/Γ ′ . We shall also say that G acts on S with signature s(Γ ′ ). Let us assume that G is a subgroup of G 1 . The action of G on S ∼ = H/Γ is said to extend to an action of G 1 if and only if there is a Fuchsian group Γ ′′ containing Γ ′ together with a surface epimorphism Θ : Γ ′′ → G 1 in such a way that Θ\| Γ ′ = θ, ker(Θ) = ker(θ) = Γ, and Γ ′ and Γ ′′ have associated Teichmüller spaces of the same dimension. Singerman in [40] determined all those pairs of signatures (s(Γ ′ ), s(Γ ′′ )) for which it may be possible to extend actions. An action is called maximal if it cannot be extended in the aforementioned sense. 2.3. Topologically equivalent actions. Let S be a compact Riemann surfaces and let Hom + (S) denote the group of orientation preserving homeomorphisms of S. Two actions ψ i : G → Aut(S) are said to be topologically equivalent if there exist ω ∈ Aut(G) and h ∈ Hom + (S) such that ψ 2 (g) = hψ 1 (ω(g))h −1 for all g ∈ G. (2.3) Note that topologically equivalent actions have the same signature. Each orientation preserving homeomorphism h satisfying (2.3) yields a group automorphism h * of Γ ′ where H/Γ ′ ∼ = S/G. We shall denote the subgroup of Aut(Γ ′ ) consisting of the automorphisms h * by B. Two surface epimorphisms θ 1 , θ 2 : Γ ′ → G define topologically equivalent actions if and only there are ω ∈ Aut(G) and h * ∈ B such that θ 2 = ω • θ 1 • h * (see [5,17,28]). We remark that if the genus of S/G is zero, then the group B is generated by the braid transformations Φ i,i+1 ∈ Aut(Γ ′ ) defined by: x i → x i+1 , x i+1 → x i+1 x i x −1 i+1 and x j → x j when j = i, i + 1 for each i ∈ {1, . . . , l − 1}. See, for example, [22, p. 31] and also [6,20]. Equisymmetric stratification. Let M g denote the moduli space of compact Riemann surfaces of genus g ≥ 2. It is well-known that M g is endowed with an orbifold structure and that its locus of orbifold-singular points, the so-called branch locus B g , is formed by Riemann surfaces with non-trivial automorphisms for g ≥ 3. For g = 2 the branch locus B 2 consists of the Riemann surfaces admitting other automorphisms than the hyperelliptic involution. See, for example, [30]. It was proved in [6] that the branch locus B g admits an equisymmetric stratification {M G,θ g }, where each equisymmetric stratum M G,θ g , if non-empty, corresponds to one topological class of maximal actions. More precisely, B g can be written as B g = ∪ G,θM G,θ g (2.4) where the closureM G,θ g of the stratum M G,θ g consists of the Riemann surfaces of genus g admitting an action of the group G with fixed topological class given by θ. We recall thatM G,θ g is a closed irreducible algebraic subvariety of M g . Observe that the union in (2.4) is taken over all possible actions of the non-trivial groups G acting on a compact Riemann surface of genus g. See also [17]. In particular, in this work we shall use the following: Definition. A closed familyF of compact Riemann surfaces of genus g whose members admit an action of a group G will be called equisymmetric if its interior F consists of exactly one stratum. 2.5. Decomposition of Jacobian varieties. Let S be a compact Riemann surface of genus g ≥ 2. We denote by JS the Jacobian variety (or simply the Jacobian) of S, and recall that JS is an irreducible principally polarized abelian variety of dimension g. See [4]. The relevance of the Jacobian variety lies in the well-known Torelli's theorem, which asserts that two compact Riemann surfaces are isomorphic if and only if their Jacobians are isomorphic as principally polarized abelian varieties. If a finite group G acts on S then this action induces an isogeny decomposition JS ∼ J(S/G) × A 2 × . . . × A r (2.5) which is G-equivariant. The factors in (2.5) are in bijective correspondence with the rational irreducible representations of G; the factor A 1 ∼ J(S/G) is associated to the trivial representation (see [8,25]). The decomposition of Jacobians with group actions has been extensively studied; the simplest case of such a decomposition was already noticed by Wirtinger in [45] and used by Schottky and Jung in [39]. For decompositions of Jacobians with respect to special groups, we refer to [9,18,20,31,32,36]. Let G be a finite group. For each complex representation ρ : G → GL(V ) of G we shall denote its degree by d V ; i.e. the dimension of V as a complex vector space. If H is a subgroup of G, then we shall denote the dimension of the vector subspace of V fixed under the action H by d H V . By abuse of notation, we shall write V to refer to the representation ρ. See [38] for more details. Let us assume that G acts on a Riemann surface S with signature (2.1), and that this action is determined by the surface epimorphism θ : Γ → G. Let H 1 , . . . , H t be groups of automorphisms of S such that G contains H i for each i. Following [35], the collection {H 1 , . . . , H t } is called G-admissible if d H1 V + · · · + d Ht V ≤ d V for every complex irreducible representation V of G in J, where the elements of J are characterized (by using [37, Theorem 5.12]) as follows: (1) the trivial representation belongs to J if and only if the genus of S/G is different from zero, and (2) a non-trivial representation V belongs to J if and only if d V (γ − 1) + 1 2 Σ l i=1 (d V − d θ(xi) V ) = 0, where the x ′ i s are canonical generators (2.2) of Γ. The collection is called admissible if it is G-admissible for some group G. The main result of [35] ensures that if {H 1 , . . . , H t } is an admissible collection of groups of automorphism of a Riemann surface S then JS ∼ Π t i=1 J(S/H i ) × P for some abelian subvariety P of JS. See also [21]. Notation. Let n ≥ 2 be an integer and let q be a prime. Throughout this article we denote the cyclic group of order n by C n , the dihedral group of order 2n by D n and the field of q elements by F q . Proof of Theorem 1 Let S be a compact Riemann surface of genus g ≥ 8, where q = g − 1 is prime, and assume that S has a group of automorphisms G of order 5q. By the Riemann-Hurwitz formula the signature of the action of G on S is (0; 5, 5, 5). By the classical Sylow's theorems, if q ≡ 1 mod 5 then G is isomorphic to C 5q , and if q ≡ 1 mod 5 then G is isomorphic to either C 5q or to C q ⋊ 5 C 5 = a, b : a q = b 5 = 1, bab −1 = a r , where r is a 5-th primitive root of the unity in F q . Note that since C 5q cannot be generated by two elements of order five, if q ≡ 1 mod 5, then there are no compact Riemann surfaces of genus g with a group of automorphisms of order 5q. From now on we assume that q ≡ 1 mod 5 and that G ∼ = C q ⋊ 5 C 5 . Let Γ be a Fuchsian group of signature (0; 5, 5, 5) with canonical presentation Γ = x 1 , x 2 , x 3 , : x 5 1 = x 5 2 = x 5 3 = x 1 x 2 x 3 = 1 and let θ : Γ → G be a surface epimorphism representing the action of G on S. We recall that G has exactly four conjugacy classes of elements of order 5; namely {a l b j : 1 ≤ l ≤ q} for j = 1, 2, 3, 4. If the epimorphism θ is defined by θ( x 1 ) = a l1 b i , θ(x 2 ) = a l2 b j and θ(x 3 ) = a l3 b k where l 1 , l 2 , l 3 ∈ {1 , . . . , q} and i, j, k ∈ {1, . . . , 4}, then, after applying a suitable inner automorphism of G, we can assume l 3 ≡ 0 mod q and then l 1 ≡ −r i l 2 mod q. As l 2 ≡ 0 mod q (otherwise θ is not surjective), we can consider the automorphism of G given by a → a t2 and b → b, where l 2 t 2 ≡ 1 mod q, to see that θ is equivalent to the epimorphism θ i,j,k defined by θ i,j,k (x 1 ) = a −r i b i , θ i,j,k (x 2 ) = ab j and θ i,j,k (x 3 ) = b k . Now, as the braid automorphisms act by permuting conjugacy classes of elements of Γ and as i+j+k ≡ 0 mod 5, there are at most four pairwise topologically non-equivalent actions of G on S, represented by θ 1 = θ 1,2,2 , θ 2 = θ 2,4,4 , θ 3 = θ 1,1,3 and θ 4 = θ 3,3,4 . Following [40], the action given by each θ n can be possibly extended to actions of signatures (0; 3, 3, 5) and (0; 2, 5, 10), and these actions, in turn, can be possibly extended to a maximal action of signature (0; 2, 3, 10). Now, if an action of G on S extends to an action of signature (0; 3, 3, 5) then S would have 15q automorphisms; however, as proved in [3], possibly up to finitely many sporadic cases in small genera this situation is not possible. Note that this fact also ensures that, possibly up to finitely many sporadic cases in small genera, none of the actions of G extends to an action of signature (0; 2, 3, 10). Let us now consider a Fuchsian group Γ 1 of signature (0; 2, 5, 10) with canonical presentation Γ 1 = y 1 , y 2 , y 3 : y 2 1 = y 5 2 = y 10 3 = y 1 y 2 y 3 = 1 , and a finite group group G ′ = C q ⋊ 10 C 10 with presentation a, b, s : a q = b 5 = s 2 = 1, bab −1 = a r , sas = a −1 , [s, b] = 1 , where r is a 5-th primitive root of the unity in F q . As proved in [3,Example (ii)] (see also [43,Theorem 3]), the surface epimorphisms Θ n : Γ 1 → G ′ ∼ = C q ⋊ 10 C 10 given by Θ n (y 1 ) = as, Θ n (y 2 ) = ab 2n and Θ n (y 3 ) = b −2n s, 1 ≤ n ≤ 4, define four pairwise non-isomorphic Riemann surfaces X 1 , . . . , X 4 of genus g with full automorphism group C q ⋊ 10 C 10 . Note that the subgroup of Γ 1 generated byx 1 = (y 1 y 3 ) −1 ,x 2 = y 2 andx 3 = y 2 3 is isomorphic to Γ, and that Θ n (x 1 ) = a −r 2n b 2n , Θ n (x 2 ) = ab 2n and Θ n (x 3 ) = b −4n . It follows that Θ n \| Γ = θ n for each n ∈ {1, 2, 3, 4} and therefore each action of G ∼ = C q ⋊ 5 C 5 on S with signature (0; 5, 5, 5) extends to an action of G ′ ∼ = C q ⋊ 10 C 10 with signature (0; 2, 5, 10); thus S isomorphic to X i for some i. In particular, there does not exist a Riemann surface of genus g with full automorphism group of order 5q. Finally, we decompose the Jacobian variety JS of each S. If we set c = bs and m = −r then Aut(S) ∼ = a, c : a q = c 10 = 1, cac −1 = a m = C q ⋊ 10 C 10 . We shall use this presentation in the sequel. Set ω t := exp( 2πi t ). The group C q ⋊ 10 C 10 has, up to equivalence, ten complex irreducible representations of degree 1, given by U i : a → 1, c → ω i 10 for 0 ≤ i ≤ 9. Let α = q−1 10 ∈ N and choose integers k 1 , . . . , k α ∈ {1, . . . , q − 1} in such a way that ⊔ α j=1 {k j , k j m, k j m 2 , . . . , k j m 9 } = {1, . . . , q − 1}, where ⊔ stands for disjoint union. Then, the group C q ⋊ 10 C 10 has, up to equivalence, α complex irreducible representations of degree 10, given by V j : a → diag(ω kj q , ω kj m q , ω kj m 2 q , . . . , ω kj m 9 q ), c →   0 1 0 ··· 0 0 0 1 ··· 0 . . . 0 0 0 ··· 1 1 0 0 ··· 0   for 1 ≤ j ≤ α. Consider H = a and H t = a t c for 1 ≤ t ≤ 10, and notice that d H Ui + Σ 10 t=1 d Ht Ui = 1 = d Ui and d H Vj + Σ 10 t=1 d Ht Vj = 10 = d Vj for each i ∈ {1, . . . , 9} and for each j ∈ {1, . . . , α}. Thereby, as explained in Subsection 2.5, the collection {H, H 1 , . . . , H 10 } is admissible and therefore, by [35], there is an abelian subvariety P of JS such that JS ∼ J(S/H) × Π 10 t=1 J(S/H t ) × P ∼ J(S/ a ) × (J(S/ c )) 10 × P, where the second isogeny follows after noticing that, for each t, the groups H t and c are conjugate. Observe that the q-sheeted regular covering map S → S/ a is unbranched, and that the regular covering map S → S/ c ramifies over exactly three values, marked with 2, 5 and 10. Then, it follows from the Riemann-Hurwitz that the genera of S/ a and S/ c are 2 and α respectively; thus P = 0. This completes the proof of Theorem 1. Proof of Theorem 2 Let S be a compact Riemann surface of genus g ≥ 8, where q = g − 1 is prime, and assume that S has a group of automorphism G of order 6q. By the Riemann-Hurwitz formula the possible signatures of the action of G on S are (0; 2, 2, 3, 3), (0; 2, 2, 2, 6) and (0; 3, 6, 6) for each genus and, in addition, the signature (0; 2, 7, 42) for g = 8. First of all, the signature (0; 2, 7, 42) for g = 8 cannot be realized because there is no surface epimorphism from a Fuchsian group of signature (0; 2, 7, 42) to a (necessarily cyclic) group of order 42. In addition, by the classical Sylow's theorems, G contains exactly one normal subgroup isomorphic to C q and therefore G is isomorphic to a semidirect product C q ⋊ H, where H is a group of order 6. Claim 1. H = C 6 . Let us assume that H = D 3 and therefore G ∼ = C q ⋊ D 3 = a, b, s : a q = b 3 = s 2 = 1, (sb) 2 = 1, bab −1 = a u , sas = a v where u is either 1 or a 3-th primitive root of the unity in F q , and v = ±1. (1) If u = 1 and v = 1 then G is isomorphic to the direct product C q × D 3 . However, as among every collection of generators of C q × D 3 there must be an element of order a multiple of q, we see that there are no compact Riemann surfaces of genus g with a group of automorphisms isomorphic to C q × D 3 since the order of the generators of the Fuchsian groups are 2, 3 and 6. (2) If u = 1 and v = −1 then G is isomorphic to D 3q and therefore G has no elements of order 6. Moreover, the elements of order three are (ab) q and (ab) 2q , and the involutions are of the form s(ab) l for 1 ≤ l ≤ 3q. It can be checked that if the product of two involutions and two elements of order three is 1, then these elements generate D 6 . All the above ensures that there are no Riemann surfaces of genus g with a group of automorphisms isomorphic to D 3q . (3) Finally, if u is a 3-th primitive root of the unity in F q , then the equality (sb)a(sb) −1 = a r 2 v yields that the action of the involution sb on C q has order three for v = 1 and order six for v = −1. This is not possible. This proves Claim 1. Thereby, G ∼ = C q ⋊ C 6 = a, b, s : a q = b 3 = s 2 = 1, [s, b] = 1, bab −1 = a u , sas = a v where u is either 1 or a 3-th primitive root of the unity in F q and v = ±1. Claim 2. u is a 3-th primitive root of the unity in F q . Assume u = 1. (1) If v = 1, then G ∼ = C 6q which is not generated by elements of order two and three. Thus, there are no Riemann surfaces of genus g with a group of automorphisms isomorphic to C 6q . (2) If v = −1 then G ∼ = C q ⋊ 2 C 6 where C 6 acts on C q with order two. The elements of order two are of the form a l s, the elements of order six of the form a l bs and a l b 2 s for 1 ≤ l ≤ q, and the elements of order three are b and b 2 . It can be seen that: (a) G cannot be generated by three elements, being two of them of order two and one of order three, in such a way that their product has order three, (b) the product of three elements of order two must have order two, and (c) G cannot be generated by two elements of order six whose product has order three. All the above ensures that there are no Riemann surfaces of genus g with a group of automorphisms isomorphic to C q ⋊ 2 C 6 . This proves Claim 2. Therefore, G ∼ = C q ⋊ C 6 with a presentation a, b, s : a q = b 3 = s 2 = 1, [s, b] = 1, bab −1 = a r , sas = a v , where v = ±1 and r is a 3-th primitive root of the unity in F q . Consequently g − 1 = q ≡ 1 mod 3. We have two cases for the finite group G: Case 1. If v = 1 then G is isomorphic to C q ⋊ 3 C 6 where C 6 acts on C q with order three The elements of order 3 are of the form a l b and a l b 2 , the elements of order 6 of the form a l bs and a l b 2 s for 1 ≤ l ≤ q, and s is the unique element of order two. Case 2. If v = −1 then G is isomorphic to C q ⋊ 6 C 6 where C 6 acts on C q with order six. The elements of order two are of the form a l s, the elements of order three are of the form a l b and a l b 2 , and the elements of order six of the form a l bs and a l b 2 s for 1 ≤ l ≤ q. We now study each possible signature separately. Signature (0; 2, 2, 2, 6). As in both groups C q ⋊ 3 C 6 and C q ⋊ 6 C 6 the product of three elements of order two has order two, we see that there is no group of order 6q acting on a Riemann surface of genus g with signature (0; 2, 2, 2, 6). See also [12] Signature (0; 2, 2, 3, 3). We note that there are no compact Riemann surfaces of genus g admitting an action of C q ⋊ 3 C 6 with signature (0; 2, 2, 3, 3); this follows from the fact that s (which is the unique involution) and an element of order three generate a group of order six. By contrast, we show that there is a complex one-dimensional equisymmetric familyF g of Riemann surfaces S of genus g with a group of automorphisms isomorphic to C q ⋊ 6 C 6 acting on S with signature (0; 2, 2, 3, 3). Indeed, let Γ 3 be a Fuchsian group of signature (0; 2, 2, 3, 3) with canonical presentation Γ 3 = x 1 , x 2 , x 3 , x 4 = x 2 1 = x 2 2 = x 3 3 = x 3 4 = x 1 x 2 x 3 x 4 = 1 . Then the surface epimorphism θ 3,0 : Γ 3 → C q ⋊ 6 C 6 defined by θ 3,0 (x 1 ) = s, θ 3,0 (x 2 ) = as, θ 3,0 (x 3 ) = ab 2 and θ 3,0 (x 4 ) = b, provides the familyF g of Riemann surfaces admitting an action of C q ⋊ 6 C 6 . To prove thatF g is equisymmetric we notice that, up to a permutation of the generators of Γ 3 , a surface epimorphism θ 3 : Γ 3 → C q ⋊ 6 C 6 is of the form: θ 3 (x 1 ) = a l1 s, θ 3 (x 2 ) = a l2 s, θ 3 (x 3 ) = a l3 b 2 and θ 3 (x 4 ) = a l4 b, for some l 1 , . . . , l 4 ∈ {1, . . . , q}. Moreover, after applying a suitable automorphism of G of the form a → a u , b → a v b we can suppose l 1 ≡ 0 mod q and l 2 ≡ 1 mod q. Now, if we set m = l 4 then an epimorphism θ 3 is equivalent to one epimorphism θ 3,m given by θ 3,m (x 1 ) = s, θ 3,m (x 2 ) = as, θ 3,m (x 3 ) = a 1+(1+r)m b 2 , θ 3,m (x 4 ) = a m b, 1 ≤ m ≤ q As Φ 2 3,4 ·θ 3,m = Θ 3,2r+1+m , after iterating Φ 2 3,4 a suitable number of times, we see that each epimorphism θ 3,m is equivalent to θ 3,0 , as desired. We claim that the full automorphism group of a Riemann surface in the interior F g ofF g is G. Indeed, otherwise by [40] the action would extend to an action of a group of order 12q of signature (0; 2, 2, 2, 3); however, this situation is not possible by [3, Theorem 2(a)] for q ≥ 19 and by [10] for the remaining cases q = 7 and q = 13. Signature (0; 3, 6, 6). Let Γ 1 be a Fuchsian group of signature (0; 3, 6, 6) and consider its canonical presentation Γ 1 = x 1 , x 2 , x 3 = x 3 1 = x 6 2 = x 6 3 = x 1 x 2 x 3 = 1 . Applying automorphisms of the finite group, we have that: (1) A surface epimorphism Γ 1 → C q ⋊ 3 C 6 representing an action of C q ⋊ 3 C 6 on S with signature (0; 3, 6, 6) is equivalent to one defined by θ 1,i (x 1 ) = b i , θ 1,i (x 2 ) = a −r i b i s and θ 1,i (x 3 ) = a i bs for i = 1 or i = 2. (2) A surface epimorphism Γ 1 → C q ⋊ 6 C 6 representing an action of C q ⋊ 6 C 6 on S with signature (0; 3, 6, 6) is equivalent to the one defined by θ 2 (x 1 ) = ab, θ 2 (x 2 ) = bs and θ 2 (x 3 ) = a r bs. Using the results of [40], we can ensure that the action of G on S can be extended possibly only to actions with signatures (0; 2, 6, 6) and (0; 2, 4, 6). Let Γ 2 be a Fuchsian group of signature (0; 2, 6, 6) with canonical presentation Γ 2 = y 1 , y 2 , y 3 : y 2 1 = y 6 2 = y 6 3 = y 1 y 2 y 3 = 1 . Following [3, Example (i)], there exist two non-isomorphic Riemann surfaces X 1 and X 2 of genus g with a group of automorphisms of order 12q acting on it with signature (0; 2, 6, 6). Furthermore, Aut(X i ) ∼ = (C q ⋊ 6 C 6 ) × C 2 for i = 1, 2, with corresponding non-equivalent surface epimorphisms Θ i : Γ 2 → (C q ⋊ 6 C 6 ) × C 2 giving the actions of Aut(X i ) on X i defined by: Θ 1 (y 1 ) = as, Θ 1 (y 2 ) = bsz, Θ 1 (y 3 ) = a −r 2 b 2 z and Θ 2 (y 1 ) = as, Θ 2 (y 2 ) = b 2 sz, Θ 2 (y 3 ) = a −r bz, where z generates the C 2 central factor. Claim 3. If S is a compact Riemann surface with an action of a group of order 6q with signature (0; 3, 6, 6) then S is isomorphic to either X 1 or X 2 . First of all, we have seen above that such an action is given by the surface epimorphisms θ 1,i and θ 2 . We see now that these actions extend. Setting x ′ 1 = y 2 2 , x ′ 2 = y 3 and x ′ 3 = (y 2 2 y 3 ) −1 , the subgroup of Γ 2 generated by x ′ 1 , x ′ 2 , x ′ 3 is isomorphic to Γ 1 . Moreover, Θ 1 (x ′ 1 ) = b 2 , Θ 1 (x ′ 2 ) = a −r 2 b 2 z, Θ 1 (x ′ 3 ) = ab 2 z; Θ 2 (x ′ 1 ) = b, Θ 2 (x ′ 2 ) = a −r bz, Θ 2 (x ′ 3 ) = abz. Note that a, b, z ∼ = C q ⋊ 3 C 6 and that the restrictions Θ 1 \| x ′ 1 ,x ′ 2 ,x ′ 3 , Θ 2 \| x ′ 1 ,x ′ 2 ,x ′ 3 : Γ 1 ∼ = x ′ 1 , x ′ 2 , x ′ 3 → C q ⋊ 3 C 6 are precisely θ 1,2 and θ 1,1 respectively. It follows that the action θ 1,1 and θ 1,2 of C q ⋊ 3 C 6 on compact Riemann surfaces S of genus g with signature (0; 3, 6, 6) extend to the action of (C q ⋊ 6 C 6 ) × C 2 with signature (0; 2, 6, 6) represented by Θ 2 and Θ 1 respectively; thus, S isomorphic to X 2 in the first case, and S isomorphic to X 1 in the second case. Now, setting x ′′ 1 = y 2 3 , x ′′ 2 = y 2 and x ′′ 3 = (y 2 3 y 2 ) −1 , the subgroup of Γ 2 generated by x ′′ 1 , x ′′ 2 , x ′′ 3 is isomorphic to Γ 1 . Moreover, Θ 1 (x ′′ 1 ) = ab, Θ 1 (x ′′ 2 ) = b(sz), Θ 1 (x ′′ 3 ) = a −r b(sz); Θ 2 (x ′′ 1 ) = ab 2 , Θ 2 (x ′′ 2 ) = b 2 (sz), Θ 2 (x ′′ 3 ) = a −r 2 b 2 (sz) . Note that a, b, sz ∼ = C q ⋊ 6 C 6 and that the restrictions Θ 1 \| x ′′ 1 ,x ′′ 2 ,x ′′ 3 , Θ 2 \| x ′′ 1 ,x ′′ 2 ,x ′′ 3 : Γ 1 ∼ = x ′′ 1 , x ′′ 2 , x ′′ 3 → C q ⋊ 6 C 6 are equivalent to θ 2 . It follows that the action θ 2 of C q ⋊ 6 C 6 on a Riemann surface S with signature (0; 3, 6, 6) extends to both actions of (C q ⋊ 6 C 6 ) × C 2 with signature (0; 2, 6, 6) represented by Θ 1 or by Θ 2 ; thus, S is isomorphic to X 1 in the first case, and isomorphic to X 2 in the second case. This proves Claim 3. Note thatx 1 = (y 1 y 2 2 y 2 3 ) −1 ,x 2 = y 1 ,x 3 = y 2 2 andx 4 = y 2 3 generate a subgroupΓ of Γ 2 isomorphic to a Fuchsian group of signature (0; 2, 2, 3, 3). Furthermore, the restrictions Θ 1 \|Γ and Θ 2 \|Γ are epimorphisms equivalent to θ 3,0 . This yields that X 1 and X 2 lie in the boundary ofF g as desired. Finally, as before, applying [3, Theorem 2(a)] for q ≥ 19 and [10] for the remaining cases q = 7 and q = 13, we conclude that: (1) the Riemann surfaces X 1 and X 2 are the unique compact Riemann surfaces with a group of automorphisms of order 12q, (2) there are no compact Riemann surfaces of genus g with 24q automorphisms (in particular, the action of G on S of signature (0; 3, 6, 6) cannot be extended to an action of signature (0; 2, 4, 6)), and therefore {X 1 , X 2 } =F g \ F g . We now decompose the associated Jacobian varieties; to do that we proceed analogously as done in the proof of Theorem 1. Let S ∈F g and set ω t := exp( 2πi t ). Note that the group Aut(S) ∼ = C q ⋊ 6 C 6 = a, c : a q = c 6 = 1, cac −1 = a n where n is a 6-th primitive root of the unity in F q , has, up to equivalence, six complex irreducible representations of degree 1, given by U i : a → 1, c → ω i 6 for 0 ≤ i ≤ 5. In addition, C q ⋊ 6 C 6 has β = q−1 6 ∈ N complex irreducible representations of degree 6, namely The q-sheeted regular covering map S → S/ a is unbranched, and the regular covering map S → S/ c ramifies over exactly four values, two marked with 2 and two with marked 3. Thus, the Riemann-Hurwitz formula implies that the genera of S/ a and S/ c are 2 and β respectively; thus Q = 0. Let S be one of the two non-isomorphic Riemann surfaces with 12q automorphisms. Each complex irreducible representation of Aut(S) ∼ = (C q ⋊ 6 C 6 ) × C 2 = a, c × z coincides with the tensor product of a complex irreducible representation of C q ⋊ 6 C 6 and one of C 2 (see, for example [38, p. 27]) Thus, keeping the same notations as above, we see that the complex irreducible representations of (C q ⋊ 6 C 6 ) × C 2 are U ± i : a → 1, c → ω i 6 , z → ±1 for each 0 ≤ i ≤ 5, and for 1 ≤ j ≤ β, where I 6 denotes the 6 × 6 identity matrix. V ± j : a → diag(ω If we write N = a and N t = a t cz for t ∈ {1, . . . , 6}, then it can be checked that the collection {N, N 1 , . . . , N 6 } is admissible. In addition, as N t and cz are conjugate, we apply the result of [35] to ensure the existence of an abelian subvariety R of JS such that JS ∼ J(S/ a ) × (J(S/ cz )) 6 × R. The q-sheeted regular covering map S → S/ a is unbranched and the regular covering map S → S/ cz ramifies over exactly three values, two marked with 2 and one with marked 3. The Riemann-Hurwitz formula implies that the genera of S/ a and S/ cz are 2 and β respectively; thus R = 0. This finishes the proof of Theorem 2. Proof of Corollary 1 Let S be a compact Riemann surface of genus g ≥ 8, where q = g − 1 is prime, and assume that S has a group of automorphisms G of order 3q. By the Riemann-Hurwitz formula the possible signatures for the action of G on S are (1; 3) and (0; 3, 3, 3, 3) for each g and, in addition, the signature (0; 7, 7, 21) for g = 8. The latter exceptional case for g = 8 can be disregarded because there are no surface epimorphisms from a Fuchsian group of signature (0; 7, 7, 21) to a (necessarily cyclic) group of order 21. By the classical Sylow's theorems if q ≡ 1 mod 3 then G is isomorphic to C 3q , and if q ≡ 1 mod 3 then G is isomorphic to either C 3q or to C q ⋊ 3 C 3 = a, b : a q = b 3 = 1, bab −1 = a r , where r is a 3-th primitive root of the unity in F q . As C 3q is abelian, and as the commutator subgroup of C q ⋊ 3 C 3 does not have elements of order three, we see that there are no compact Riemann surfaces of genus g with a group of automorphisms of order 3q acting with signature (1; 3). Furthermore, as C 3q cannot be generated by elements of order three, we obtain that if q ≡ 1 mod 3 then there are no compact Riemann surfaces of genus g with a group of automorphisms of order 3q acting with signature (0; 3, 3, 3, 3). Thus, from now on we assume that g − 1 = q ≡ 1 mod 3 and that G ∼ = C q ⋊ 3 C 3 . Let Γ ′ be a Fuchsian group of signature (0; 3, 3, 3, 3) with canonical presentation Γ ′ = x 1 , x 2 , x 3 , x 4 : x 3 1 = x 3 2 = x 3 3 = x 3 4 = x 1 x 2 x 3 x 4 = 1 and let θ : Γ ′ → G be a surface epimorphism representing the action of G on S. We recall that G has exactly two conjugacy classes of elements of order 3: C 1 = {a l b : 1 ≤ l ≤ q} and C 2 = {a l b 2 : 1 ≤ l ≤ q}. Note that among the elements θ(x 1 ), . . . , θ(x 4 ) of G exactly two of them must belong to C 1 ; otherwise their product is different from 1. Up to a permutation we can suppose that θ(x i ) = a li b 2 for i = 1, 2 and θ(x i ) = a li b for i = 3, 4, for suitable l 1 , . . . , l 4 . Note that if l 1 ≡ l 2 mod q, then l 3 ≡ l 4 mod q and θ is not surjective; thus, without lost of generality, we can assume l 1 ≡ l 2 mod q. Now, by considering an automorphism of G of the form a → a i , b → a j b, we can assume l 1 ≡ 0 mod q and l 2 ≡ 1 mod q; therefore θ is equivalent to the epimorphism θ n defined by θ n (x 1 ) = b 2 , θ n (x 2 ) = ab 2 , θ n (x 3 ) = a −r(n+1) b, θ n (x 4 ) = a n b, with 1 ≤ n ≤ q. By [40], the action of G on S can possibly be extended to an action of signature (0; 2, 2, 3, 3). We shall prove that each action does extend to an action equivalent to the one given by the surface epimorphism θ 3,0 and therefore the surfaces S belong to the familyF g of Theorem 2. Now, let us consider the Fuchsian group Γ 3 of signature (0; 2, 2, 3, 3) with canonical presentation Γ 3 = y 1 , y 2 , y 3 , y 4 : y 2 1 = y 2 2 = y 3 3 = y 3 1 = y 1 y 2 y 3 y 4 = 1 and let G ′ ∼ = C q ⋊ 6 C 6 be the finite group with presentation a, b, s : a q = b 3 = s 2 = 1, bab −1 = a r , sas = a −1 , [s, b] = 1 , as in Section 4. Following the proof of Theorem 2, each surface epimorphism Θ : Γ 1 → C q ⋊ 6 C 6 representing an action of C q ⋊ 6 C 6 is equivalent to the epimorphism Θ m (= θ 3,m in the notation of proof of Theorem 2) given by Θ m (y 1 ) = s, Θ m (y 2 ) = as, Θ m (y 3 ) = a 1+(1+r)m b 2 , Θ m (y 4 ) = a m b for a suitable 1 ≤ m ≤ q. We notice that the group generated byx 1 = y 3 ,x 2 = y 2 4 ,x 3 = y 4 andx 4 = y 2 3 , is isomorphic to Γ ′ and Θ m (x 1 ) = a 1+(1+r)m b 2 , Θ m (x 2 ) = a (1+r)m b 2 , Θ m (x 3 ) = a m b, Θ m (x 4 ) = a m−r b. Now, after considering the automorphism of G given by a → a −1 , b → ab j where j = 1+m(1+r) 1−r , it follows that Θ m \| Γ = θ 2(mr+r+1) 1−r for each m ∈ {1, . . . , q}. Thereby, each action of C q ⋊ 3 C 3 with signature (0; 3, 3, 3, 3) extends to an action of C q ⋊ 6 C 6 with signature (0; 2, 2, 3, 3), as desired. As a consequence, there does not exist compact Riemann surfaces with full automorphism group of order 3q, and the proof of Corollary 1 is complete. Remark. The Riemann surfaces of genus g = 3 admitting the action of a group of order six or twelve are given and classified in [5]. There is no a Riemann surface of genus three admitting an automorphism of order five. The Riemann surfaces of genus g = 4 with a group of automorphisms of order fifteen or eighteen are given and classify in [2] and [13]. Among them there is the equisymmetric family of cyclic trigonal surfaces with two trigonal morphisms (see [14,16]). Finally, the Riemann surfaces of genus g = 6 with twenty-five or thirty automorphisms are given in [29]. 1 ≤ j ≤ β,where k 1 , . . . , k β ∈ {1, . . . , q − 1} are integers chosen to satisfy that ⊔ β j=1 {k j , k j n, k j n 2 , . . . , k j n 5 } = {1, . . . , q − 1}, where ⊔ denotes the disjoint union.Consider the subgroups H = a and H t = a t c for t ∈ {1, . . . , 6}. Note that d H Ui + Σ 6 t=1 d Ht Ui = 1 = d Ui and d H Vj + Σ 6 t=1 d Ht Vj = 6 = d Vj for each i ∈ {1, . . . , 6} and for each j ∈ {1, . . . , β}. 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[ "Local Diversity and Ultra-Reliable Antenna Arrays", "Local Diversity and Ultra-Reliable Antenna Arrays" ]	[ "Jens Abraham ", "Torbjörn Ekman " ]	[]	[]	Ultra-reliable low-latency communication enables new use cases for mobile radio networks. The ultra-reliability (UR) regime covers outage probabilities between 10 −9 and 10 −5 , obtained under stringent latency requirements. Characterisation of the UR-relevant statistics is difficult due to the rare nature of outage events, but diversity defines the asymptotic behaviour of the small-scale fading distributions' lower tail. The UR-relevant regime in large-scale antenna systems behaves differently from the tail. We present the generalising local diversity at a certain outage probability to show this difference clearly. For more than four independent antenna elements, the classic diversity overestimates and underestimates the slope of the cumulative density function for weak and strong deterministic channel components, respectively.Index Terms-channel hardening, massive MIMO, Rician fading, URLLC.	10.1109/ieeeconf53345.2021.9723123	[ "https://arxiv.org/pdf/2108.00712v2.pdf" ]	236,772,065	2108.00712	a3c825c5f6d6e79930d18e2b13f4bdaf34586414	Local Diversity and Ultra-Reliable Antenna Arrays 2 Dec 2021 Jens Abraham Torbjörn Ekman Local Diversity and Ultra-Reliable Antenna Arrays 2 Dec 2021Index Terms-channel hardeningmassive MIMORician fad- ingURLLC Ultra-reliable low-latency communication enables new use cases for mobile radio networks. The ultra-reliability (UR) regime covers outage probabilities between 10 −9 and 10 −5 , obtained under stringent latency requirements. Characterisation of the UR-relevant statistics is difficult due to the rare nature of outage events, but diversity defines the asymptotic behaviour of the small-scale fading distributions' lower tail. The UR-relevant regime in large-scale antenna systems behaves differently from the tail. We present the generalising local diversity at a certain outage probability to show this difference clearly. For more than four independent antenna elements, the classic diversity overestimates and underestimates the slope of the cumulative density function for weak and strong deterministic channel components, respectively.Index Terms-channel hardening, massive MIMO, Rician fading, URLLC. I. INTRODUCTION One of the reoccurring promises in both fifth generation mobile networks (5G) and sixth generation mobile networks (6G) specifications is ultra-reliable low-latency communication (URLLC). The URLLC requires an outage probability of 10 −5 or better within a 1 ms transmission period in 5G [1]. The authors of [2] introduce the terminology of ultra-reliability (UR)-relevant statistics for outage probabilities below 10 −5 . It can be expected that the requirements for 6G will be even more stringent. Hence, we will consider outage probabilities between 10 −9 and 10 −5 as the UR-relevant regime. Generally, the allowed latency can be used to retransmit a packet, if the original message did not reach its destination. By decreasing the permitted latency, only one-shot transmissions can ultimately fulfil the requirement because a retransmission would take too long. This type of requirement is typical in control-loop or event based applications, where the timing is critical. Alternatively, the age of information [3] can used as a design metric, where the state of a system is observed. Here, a non-successful transmission transmission every now and then might be acceptable, since the system can cope with intermittent link failure. Small-scale fading is one of the main reasons for link-loss in rich scattering environments. It can be counteracted with forward error correction (FEC), relying on the assumption that fading events are short enough with respect to the coded packet length. If the coherence time of the channel is longer than the latency requirement, alternative measures have to be used to overcome small-scale fading. Exploiting spatial diversity through massive multiple input multiple output (MIMO) can improve the link robustness due to channel hardening. This approach reduces the variation of the channel gain around its mean and hereby the outage probability. Recenctly, we have suggested to use a fading margin to characterise channel hardening [4]. It describes the required excess gain to provide a certain outage probability at a chosen rate. Hence, the performance of an UR antenna array with varying number of antenna elements can be quantified clearly. An additional caveat for URLLC is power limitation of users. Especially battery powered sensors in wireless sensor network (WSN) should avoid retransmissions. In those cases, minimising the fading margin improves the energy efficiency and allows to meet UR target outage probabilities. Moreover, smaller fading margins reduce the interference levels for users of the same system and systems that share the same spectrum resource. This outlines why large antenna arrays are a technically viable solution for narrow-band URLLC without retransmission of packets. System level simulations based on a 3rd Generation Partnership Project (3GPP) channel model for a specific cell show promising results for a coherence interval based pilot strategy [5]. A fundamental question remains, how can we infer the system behaviour of events that barely ever happen? A neat approach is the characterisation of the lower tail of the cumulative distribution function (CDF) as an intermediate solution between parametric channel models and non-parametric models [6]. The lower tail of multiple common fading distributions follows a power law [2], which gives the possibility to relax the model assumption from a single distribution to a class of distributions. The power law approximation requires two parameters: an offset and the log-log slope of the CDF. E.g. the classic Rayleigh channel shows a well known slope of 10 dB per decade in the lower tail. Furthermore, the outage probability in detection problems [7] for high signal to noise ratio (SNR) corresponds to the lower tail of the channel gain. Using the SNR emphasises the variation introduced due to the small-scale fading channel and avoids a dependency on a specific modulator and detector. Due to that correspondence, the log-log slope in the asymptotic lower tail reveals the diversity of the radio channel. We propose to evaluate the log-log slope at a specific probability, generalising it to the local diversity. Hence, for outage probabilities converging to zero it attains the classic diversity measure. A dual slope behaviour in single antenna Rician fading channels with larger K-factors has already been shown in [8]. For multi-antenna systems in both Rayleigh and Rician fading, the outage probability slope in the UR-relevant regime deviates from the classic diversity. Therefore, a power law approximation of the lower tail can not provide an accurate description of the CDF for massive MIMO systems. Our main contribution is the local diversity to highlight that lower tail approximations do not cover the actual system behaviour in the UR-relevant regime. We motivate the usage of analytical tools to get insight into UR-relevant statistics in the next section, because the number of of necessary observations for a reliable empirical approach is prohibitive for real world scenarios. An uncorrelated Rician multi-antenna fading environment is introduced in section III. It's local diversity is derived to relate the classical diversity to the URrelevant regime. This measure can be seen as the relative error of a power law approximation based on the asymptotic behaviour of the lower tail. New compact expressions for the CDF, probability density function (PDF) and local diversity in terms of the complementary Marcum-Q function are used to evaluate them for large scale antenna systems. We provide a comparison of multi-antenna systems in different Rician fading environments with respect to the fading margin in section IV, to discuss the scaling behaviour. Furthermore, sampling strategies to analyse the UR-relevant regime are outlined. II. PREDICTING THE UNPREDICTABLE? Let us investigate empirical cumulative distribution functions (ECDFs) as non-parameteric model, to understand the value of parametric analytical models for UR-relevant statistics. Basically, the UR-relevant regime covers the behaviour of rare events that barely ever happen and the fewer assumptions necessary the more general is the solution. How many observations are necessary to reliably estimate the UR-relevant statistics without prior knowledge? The Dvoretzky-Kiefer-Wolfowitz (DKW) inequality [9], [10] can be used to bound an R-sample ECDF with respect to the true underlying CDF leading to the error term ǫ with confidence ξ: ǫ = ln 2 1−ξ 2R .(1) This error term is characterising an error floor for the ECDF at low probabilities. Taking R = 10 6 observations as example and aiming at a confidence of ξ = 99 % gives an error term of 1.6 × 10 −3 . The resulting upper bound of the ECDF for a true single-antenna Rayleigh fading channel is shown in Fig. 1. It can be seen that the ECDF in the UR-relevant regime would be much smaller than the error floor, rendering empirical estimation of outage probabilities below 1.6 × 10 −3 practically useless. The number of antenna elements in massive MIMO ranges from a few ten to a few hundred, that can provide potentially correlated parallel observations of the radio channel. The remaining observations have to be gathered in a stationary timefrequency window to belong to the same underlying CDF. This is very unlikely in realistic scenarios, especially for high (environmental) mobility with limited temporal stationarity. Eventually, the characterisation of UR-relevant statistics in the lower tail is prone to large estimation errors for non-parametric models. Additionally, if energy efficient users are required, less spectrum may be used, reducing the number of samples in the spectral domain. Hence, the spatial domain sampling provided by an antenna array has to provide both the robustness of the system as well as a number of observations to estimate the CDF. The large number of observations an obstacle even for simulations. Assuming that outage probabilities of 10 −6 with confidence of 99.9999 % are of interest, on the order of 10 13 observations have to be collected. Both, runtime and memory requirements of Monte Carlo simulations become cumbersome to get reliable results for the ECDF. Hence, only the analytic study of the UR-relevant regime has the possibility to give insight into trade-offs, as long as the model assumptions are not violated. III. RICIAN FADING CHANNEL REVISITED We will consider Rician fading channels with a Rician Kfactor and a diffuse power (gain) P dif , following the parametrisation in [11]. The K-factor describes the ratio between a deterministic component and the diffuse power of the radio channel. Hence, the mean power gain is (K + 1)P dif . To take M uncorrelated antennas at the base station into account, a complex random vector with mean √ KP dif e jϕ1 , e jϕ2 , · · · , e jϕM T and covariance P dif I is constructed: h ∈ C M ∼ CN KP dif e jϕ1 , e jϕ2 , · · · , e jϕM T , P dif I . (2) The phases ϕ m represent the phase front of the deterministic component with respect to the antennas and I is the M × M identity matrix. For Rayleigh fading (K = 0), h is a circularsymmetric complex normal random vector h ∼ CN (0, P dif I). The effective channel H for a maximal ratio combining (MRC) weight vector w at the receiver results in: The CDF F (Q; P dif , K, M ) of the effective power gain Q = \|H\| 2 of this multi-antenna Rician channel is compactly given by: H = w T h = h H h h 2 2 = M m=1 \|h m \| 2 .(3)F (Q; P dif , K, M ) = P M KM, Q P dif ,(4) where P M (·) is the complementary Marcum Q-function [12] with definition 1 : P µ (x, y) = x 1 2 (1−µ) y 0 t 1 2 (µ−1) e −t−x I µ−1 2 √ xt dt. (5) This power gain CDF is a generalised [13] or non-central gamma distribution [14] arising from a sum over squared perantenna channel coefficients in Eqn. (3). The distribution relates to a κ-µ envelope distribution [15], where the number of independent antenna elements corresponds to µ clusters and the K-factor relates to the ratio κ between dominant and scattered channel components for a mean normalised to unity. The connection between a single antenna Rayleigh channel, the complementary Marcum Qfunction and the effective gain CDF to arrive at a non-central gamma distribution is described in detail in the appendix A 1 Note that this definition is a different variant of the implementation found in major numeric computing environments, but the reference [12] provides a Fortran implementation together with the numerical algorithm description. and the connection to the κ-µ envelope distribution follows directly from comparison of the CDFs. The mean effective power gain is: E{Q} = M (K + 1)P dif ,(6) which follows from adding M independent Rician channels with the same K-factor and power in the diffuse component. Varying K-factors for different antenna elements could be accounted for, by using the mean K-factor in the above formulation. Both the K-factor and the number of antenna elements M , have similar influence on the mean of the distribution. Fig. 2a shows a selection of CDFs that describe the behaviour of a single antenna Rice channel. The channel gain is normalised with its mean to allow easier comparison of the small-scale fading aspects for different K-factors. A stronger deterministic component leads to a dual slope behaviour with a steeper gradient closer to the median of the distribution. Nonetheless, the gradient converges to 10 dB per decade in the lower tail and is independent of the K-factor. The very seldom cases occur only when the diffuse components can cancel the deterministic component almost perfectly. For a K-factor of 10 dB, the gradient is steepest in the region between −15 dB and 0 dB with respect to the mean. This indicates that the lower tail approximation underestimates the channel behaviour for outage probabilities ranging from 10 −4 to 0.5. For sake of completeness is the corresponding PDF f (Q) of the effective power gain given in the following equation. f (Q; P dif , K, M ) =    1 Pdif e − Q P dif −K I 0 2 K Q Pdif M = 1 1 Pdif P M−1 (KM, Q Pdif ) − P M (KM, Q Pdif ) M > 1.(7) Here, I 0 (·) is the zero-order modified Bessel function of the first kind. For the multi-antenna case, we can exploit the relation for derivatives of the complementary Marcum-Q function [12,Sec. 2.3]. A. Local Diversity So far, the local diversity has only been introduced conceptually. Let us recall a common rule of thumb: the outage probability of a single antenna Rayleigh fading channel scales with 10 dB per decade in the lower tail. Furthermore, we have observed that a single antenna in narrowband Rician fading provides a diversity of one, too. Therefore, a slope of 10 dB per decade outage probability is used as reference and we define the local diversity as derivative of the scaled logarithmic CDF of the channel power gain Q in dB: probability, respectively. The classic diversity is attained by evaluating the local diversity for Q → −∞ dB. D(Q) = ∂ ∂10 Q/10 10 log 10 (F (Q)) = Q f (Q) F (Q) .(8) Resolving the differentiation in Eqn. (8) reveals the quotient between PDF f (Q) and CDF F (Q), also known as inverse Mills' ratio, multiplied with Q. To study how well a lower tail approximation represents the behaviour of the radio channel in the UR-relevant region for Rician channels, we use Eqns. (7) and (4) for the PDF and CDF of the effective power gain, respectively. The local diversity for antenna arrays can be expressed in terms of the complementary Marcum-Q function for M > 2: D(Q; P dif , K, M ) = Q P dif P M−1 (KM, Q Pdif ) P M (KM, Q Pdif ) − 1 . (9) Fig. 2b presents the local diversity for a single antenna Rician channel (M = 1). Larger K-factors lead to a superelevated region before before convergence to unity. The local diversity quantifies the increased steepness of the CDFs in Fig. 2a. Fig. 3 plots the local diversity with respect to probability to interpret its behaviour in the UR-relevant regime. The superelevation is pronounced in the region from 10 −6 to 0.5 for a K-factor of 10 dB. All other K-factors have converged to a local diversity of unity for probabilities smaller than 10 −3 . This behaviour changes for larger arrays and is exemplified by the normalised local diversity in Figs. 4a and 4b for a Rayleigh and Rician channel with K-factor 10 dB, respectively. Tab. I summarises the results for a probability of 10 −6 over different K-factors and number of antennas M . The normalisation is achieved by dividing the local diversity with the number of antennas. Hence, once the normalised local diversity attains unity, the classic diversity of M for large SNR is achieved. Therefore, the normalised local diversity can be interpreted as relative error between a lower tail approximation and the actual steepness of the effective gain CDF at the chosen probability. IV. DISCUSSION A. Validity of Lower Tail Approximations The relative error of diversity is provided in Tab. I, revealing three different connected regions. The first region (green) is covering small K-factors for small systems, where the normalised local diversity is close to unity. A lower tail approximation will give reasonable results for UR-relevant statistics. The second region (blue) belongs to Rayleigh fading and smaller K-factors for an increasing number of antennas. In this case, the local diversity has not yet converged to unity and a lower tail approximation will overestimate the performance accordingly. E.g. a 64 antenna element array in Rayleigh fading at a probability of 10 −6 will only provide the performance predicted by the asymptotic regime of a 32 antenna system. For large systems, only significant deterministic components will provide superelevation in the region of interest. The last region (red) belongs to large K-factors, where the local diversity is larger than the diversity, e.g. an environment with a K-factor of 10 dB and 4 antennas presents a local diversity of 4 ·3.07 ≈ 12 in the superelevated probability region. The superelevation moves towards smaller probabilities for an increasing number of antennas. Overall, the deterministic component of a Rician fading environment plays a role for every K-factor for large antenna systems and a growing Kfactor increases the local diversity. Tab. I demonstrates clearly that a low tail approximation is giving misleading results for the effective channel gain of massive MIMO systems in Rayleigh and Rician fading. B. Array Deployment Strategies In the following the impact of some array deployment strategies for URLLC applications is discussed. We relate the local diversity to the fading margin, another tangible figure of merit. The fading margin is describing the gap between the median of the effective channel gain distribution and a target outage probability [4]. It has been evaluated for the same parameters as the normalised local diversity and the result is presented in Tab. II. This complementary perspective highlights the return on investment of extra power or antenna gain, to improve the reliability of a system. Regarding each column in the table shows, that every increase of the deterministic component will reduce the margin, thereby improving the robustness of the system. Hence, it is worthwhile to compare a larger co-located system with a smaller K-factor to smaller spatially distributed deployments. It can be assumed that a distributed deployment will have at least one subarray closer to a user, giving a larger K-factor. As an example: a co-located uncorrelated 64 antenna base station in a Rician fading environment with K = 0 dB = 1 would require a fading margin of 2.5 dB at an outage probability for 10 −6 . The mean of the effective channel gain is 64 * (1 + 1)P dif = 128P dif . Instead, placing two noncooperating uncorrelated 32 antenna base stations (BSs) into that environment, which reduces the length of the deterministic path to a half for a user, could increase the K-factor by 6 dB. The closer base stations would then require a fading margin of 2.5 dB at an outage probability for 10 −6 . For this setting the mean gain would be 32 * (4 + 1)P dif = 160P dif . The CDFs of both deployments are shown in Fig. 5, where both slopes of have not yet converged to the asymptotic behaviour of the lower tail in the UR-relevant regime. In this toy example, distributed base stations requiring the same amount of hardware would give equal fading margins and increase the mean effective gain compared to the co-located case. Hence, not only capacity improvements can be achieved by densification of base stations, but UR-relevant statistics can improve too without increasing the amount of deployed hardware. In a more general situation, for fading environments with deterministic propagation components, the number of antenna elements per base station influences where the normalised local diversity shows superelevation. We notice further, in a pure Rayleigh fading environment, increasing the number of base station antennas gives diminishing returns (see Fig. 4a). C. Inferring UR-relevant Statistics? So, how can we infer the system behaviour of events that barely ever happen? Given a limited number of measurable samples from each antenna element, how could the URrelevant statistics be analysed in real world systems? There are two basic approaches for UR antenna arrays: 1) Element Statistics: The first is based on collection of antenna element observations, estimation of each distribution and careful modeling of correlation properties. Antenna elements that belong to the same local area could be lumped into a single distribution to make more samples available. Post-processing of the resulting distributions with combination strategies like selection combining (SC) or MRC result in a CDF to be evaluated in the UR-relevant regime. In case of SC, it is not necessary to have a reliable estimate of the antenna element CDFs in that regime, but rather in the regime resulting from the M -th root of the target outage probability. This follows from the maximum order statistic [16] for the strongest constituent, being the M -th power of the element CDF. Since MRC will give a better combined gain than SC, using a SC result allows to bound the system behaviour in the UR-relevant regime based on reliable estimates of the element CDFs. 2) Combined Statistics: The second approach implements a specific combination strategy, evaluating the UR-relevant statistics directly. This includes intrinsically antenna correlation, avoiding the necessity of explicit characterisation. Unfortunately, this strategy requires prohibitively many observations. Even for the first approach a lot of samples are necessary, but the antenna element observations do not need to be observed in the UR-relevant regime directly, since this regime only matters for the effective channel gain! Furthermore, the correlation is expected to depend to a lesser extent on the combined channel stationarity, allowing them to be studied in more detail with help of all antenna element observations. D. Correlated Channels Even though this manuscript demonstrated a local diversity based analysis for uncorrelated systems, the same ideas can be transferred to correlated antenna arrays. Analytic results can be derived from the effective channel gain PDF and CDF of the correlated system, to avoid Monte Carlo simulations that depend on a large amount of observations to provide reasonable insight. V. CONCLUSION Acquisition of UR-relevant channel statistics is difficult to achieve in practical situations, because the number of required observations is tremendous. Ultimately, the spatial, spectral and temporal stationarity of the radio channel restricts the collection of a sufficient number of observations. The approach of using the asymptotic lower tail behaviour, to avoid determination of a specific fading distribution, can be used for small arrays up to four antennas in low K-factor Rician fading. Systems that provide large diversity, require consideration of the local diversity in the UR-relevant regime. There, the asymptotic behaviour applies to probabilities beyond the URrelevant regime only. Normalisation of the local diversity with the number of antenna elements in an array gives a relative deviation from the classic diversity. Furthermore, the local diversity opens up for performance evaluation, where measurements of correlated antenna systems can be compared to an uncorrelated optimum. For fast and numerically stable calculations, the distribution functions and local diversity of the effective gain of an uncorrelated antenna array in Rician fading can be formulated on the basis of the complementary Marcum-Q function. Evaluation of the fading margin and distribution mean reinforces that a dense deployment of smaller base stations with the potential for increased deterministic radio channels is preferable over very large co-located systems, not only improving system capacity but also robustness. APPENDIX A. CDF of the Effective Power Gain The non-central gamma distribution has PDF w ρ (x; α, µ) for index ρ, scale α, and non-centrality I ρ−1 2 µx α . (10) The corresponding CDF W ρ (x; α, µ) can be directly related to the definition of the complementary Marcum Q-function in Eqn. (5) by substitution of t = x ′ α in the integral relation between CDF and PDF: W ρ (x; α, µ) = x 0 w ρ (x ′ ; α, µ)dx ′ = P ρ (µ, x α ).(11) The gain PDF f (Q; P dif , K, M = 1) of a single antenna Rician channel is readily available by using the Rician envelope PDF from [11, (5.3.7)] applying the transformation to the power PDF [11, (5.2.1)] and replacing the power term of the deterministic component [11, (5.3.8)], resulting in: f (Q;P dif , K, M = 1) = 1 P dif exp − Q P dif − K I 0 2 K Q P dif = w 1 (Q; P dif , K).(12) This PDF is a special case of the non-central gamma distribution PDF in Eqn. (10) for index one, scale P dif with noncentrality K. For M independent single antenna Rician channels with potentially differing K-factors K m ∀m ∈ [1, · · · , M ] the additive property of non-central gamma distributions can be used to get the PDF of the effective channel gain. The addition property allows to represent the sum of independent random variables with the same scale, potentially varying index and non-centrality as non-central gamma distribution [14, (1. Using Eqn. (11) gives the CDF of the effective channel gain based on the inverse Marcum Q-function: F (Q; P dif , K, M ) = P M M m=1 K m , Q P dif .(14) Fig. 1 . 1The CDF of a Rayleigh fading channel and an upper bound for an ECDF is shown. The error term in Eqn. (1) for a million observations and a confidence interval of 99 % is used as example, showing that the estimation of outage probabilities below 1.6 × 10 −3 is unreliable. Fig. 2 . 2The normalised single antenna Rician channel is displayed for different K-factors. The normalisation enforces unit mean. Larger K-factors lead to a dual slope CDF. The steeper slope corresponds to the superelevation of the local diversity. Fig. 3 . 3This ensures a scaling of 10/D dB per decade outage probability locally at Q. E.g. a local diversity of 10, 33 and 100 describes a slope of 1 dB, 0.3 dB and 0.1 dB per decade outage 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 The local diversity with respect to the probability of a single antenna Rician channel for different K-factors. Fig. 4 . 4Normalised local diversity for an M -antenna array with 1, 4, 16 or 64 elements. A tail approximation would underestimate the outage behaviour of the system for larger arrays in Rayleigh fading and overestimate it in Rician fading. Fig. 5 . 5CDFs of a co-located 64 antenna base station with K-factor 0 dB (blue) and the closest distributed 32 antenna base station (orange) with K-factor 6 dB. The stronger deterministic component of the channel in the distributed base station case compensates for the reduced number of antennas, resulting in a similar local diversity at 10 −6 , giving a slight advantage with respect to the mean of the channel gain. 51)]. The generalisation of Eqn. (12) for an M antenna array follows a non-central gamma distribution of index M and noncentrality M m=1 K m : f (Q; P dif , K, M ) = w M Q; P dif , M m=1 K m . TABLE I NORMALISED ILOCAL DIVERSITY D/M EVALUATED AT 10 −6 PROBABILITY. THE DIFFERENT COLOURED REGIONS SHOW WHERE THE ASYMPTOTIC TAIL APPROXIMATION HOLDS (GREEN), UNDERESTIMATES (RED) OR OVERESTIMATES (BLUE) THE SLOPE IN THE UR-RELEVANT REGIME.K Number of Antennas (M ) [dB] 1 2 4 8 16 32 64 128 −∞ 1.00 1.00 0.99 0.92 0.80 0.65 0.50 0.38 0.0 1.00 1.00 1.00 0.97 0.87 0.72 0.57 0.43 3.0 1.00 1.00 1.07 1.13 1.03 0.86 0.67 0.51 6.0 1.00 1.07 1.48 1.56 1.38 1.12 0.86 0.64 10.0 1.09 2.66 3.07 2.77 2.25 1.74 1.31 0.96 20.0 23.39 19.02 14.68 10.99 8.08 5.86 4.22 3.02 TABLE II ANALYTIC IIFADING MARGINS IN DB AT 10 −6 PROBABILITY.K Number of Antennas (M ) [dB] 1 2 4 8 16 32 64 128 −∞ 58.4 30.7 17.1 10.2 6.5 4.3 2.9 2.0 0.0 57.6 29.7 16.0 9.3 5.7 3.7 2.5 1.7 3.0 55.3 27.3 13.9 7.8 4.9 3.2 2.1 1.5 6.0 49.2 21.3 10.2 6.0 3.8 2.5 1.7 1.2 10.0 27.2 10.4 5.9 3.7 2.5 1.7 1.2 0.8 20.0 3.5 2.3 1.6 1.1 0.8 0.5 0.4 0.3 J.Abraham and T. 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[ "Estimating the circumference of a graph in terms of its leaf number", "Estimating the circumference of a graph in terms of its leaf number" ]	[ "Jingru Yan \nDepartment of Mathematics\nEast China Normal University\n200241ShanghaiChina\n" ]	[ "Department of Mathematics\nEast China Normal University\n200241ShanghaiChina" ]	[]	Let T be the set of spanning trees of G and let L(T ) be the number of leaves in a tree T . The leaf number L(G) of G is defined as L(G) = max{L(T )\|T ∈ T }. Let G be a connected graph of order n and minimum degree δ such that L(G) ≤ 2δ − 1.We show that the circumference of G is at least n − 1, and that if G is regular then G is hamiltonian.	null	[ "https://arxiv.org/pdf/2203.02653v1.pdf" ]	247,292,333	2203.02653	00fa9da507e7290e7f66e0e646846436513bb6d0	Estimating the circumference of a graph in terms of its leaf number Jingru Yan Department of Mathematics East China Normal University 200241ShanghaiChina Estimating the circumference of a graph in terms of its leaf number Leaf numbercircumferencehamiltonian Mathematics Subject Classification 05C3805C45 Let T be the set of spanning trees of G and let L(T ) be the number of leaves in a tree T . The leaf number L(G) of G is defined as L(G) = max{L(T )\|T ∈ T }. Let G be a connected graph of order n and minimum degree δ such that L(G) ≤ 2δ − 1.We show that the circumference of G is at least n − 1, and that if G is regular then G is hamiltonian. Introduction We will deal with only finite nontrivial simple graphs. Let G be a graph with vertex set V (G) and edge set E(G). The order and size of a graph G are its number of vertices and edges, respectively. The notations N G (v) and N G [v] denote the neighborhood and closed neighborhood of v ∈ V (G), respectively. The degree of v is d G (v) = \|N G (v)\|. δ(G) and ∆(G) denote the minimum and maximum degree of a graph G, respectively. If the graph G is clear from the context, we will omit it as subscript. For terminology and notations not explicitly described in this paper, the reader is referred the books [1,19]. Let T be the set of spanning trees of G. L(T ) denotes the number of leaves in a tree T , where a leaf means a vertex of degree 1. Then the leaf number L(G) = max{L(T )\|T ∈ T }. Many researchers have estimated the circumference of graphs by various invariants. The purpose of this paper is to estimate the circumference of a connected graph G by the two invariants δ(G) and L(G). DeLaViña's computer program, Graffiti.pc, posed attractive conjectures [3] and some of the conjectures speculate sufficient conditions for traceability based on the minimum degree and leaf number. In 2013, Mukwembi gave a partial solution to the Graffiti.pc 190a. He [14] showed that if G is a finite connected graph with minimum degree δ(G) ≥ 5, and leaf * E-mail address: [email protected] number L(G) such that δ(G) ≥ L(G) − 1, then G is hamiltonian and thus traceable. In the same year, he [16] relaxed the condition δ(G) ≥ 5 to δ(G) ≥ 3. After that, Mukwembi [15] proved that if G is a connected claw-free graph with δ(G) ≥ (L(G) + 1)/2, then G is hamiltonian. In recent years, several authors reported on sufficient conditions for a graph to be hamiltonian or traceable based on minimum degree and leaf number, see [9][10][11][12][13]. We state the following results, some of which will be used later in this paper. Theorem 1. [12] If G is a connected graph with δ(G) ≥ (L(G)+2)/2, then G is hamiltonian. Theorem 2. [11] If G is a connected graph with δ(G) ≥ (L(G) + 1)/2, then G is traceable. Theorem 3. [13] Let G be a connected triangle-free graph with L(G) ≤ 2δ(G) − 1. Then G is either hamiltonian or G ∈ F 2 , where F 2 is the class of non-hamiltonian graphs with leaf number 2δ(G) − 1. Let p(G) and c(G) be the order of a longest path and a longest cycle in a graph G, respectively. Note that c(G) is equal to the circumference of a graph G. Many researchers have investigated the relation between p(G) and c(G) ( [5], [8], [17], [18]). Motivated by Theorem 2, we obtain the following main result. The bound is sharp and the condition cannot be relaxed. We also consider regular graphs. Main results We start with some lemmas that will be used repeatedly. Lemma 5. [15] Let G be a connected graph of order n. If L(G) ≤ 2δ(G) − 1, then n ≤ max{2δ(G) + 6, 3δ(G)}. Lemma 6. [15] Let G be a connected graph with L(G) ≤ 2δ(G) − 1. Then G is 2-connected. For a graph G, κ(G) and α(G) denote the connectivity and independent number of G, respectively. Let σ k (G) be the minimum degree sum of k independent vertices of G if α(G) ≥ k. K n stands for the complete graph of order n. Lemma 7. [2] Let G be a connected graph. If κ(G) ≥ α(G), then G is hamiltonian except for G = K 2 . Lemma 8. [5] Let G be a 2-connected graph of order n. If σ 3 (G) ≥ n + 2, then c(G) ≥ p(G) − 1. Now we first show that the result of Theorem 4 is true when n ≤ 3δ(G). Lemma 9. [16] Let G be a connected graph of order n. If δ(G) = 2 and L(G) ≤ 3, then c(G) ≥ n − 1. Given graphs G and H, the notation G + H means the disjoint union of G and H. Then tG denotes the disjoint union of t copies of G. The notation G ∨ H means the joint of G and H. For graphs we will use equality up to isomorphism, so G = H means that G and H are isomorphic. F(n) = F 1 (n) ∪ F 2 (n) ∪ F 3 (n) ∪ F 4 (n) ∪ F 5 (n) ∪ F 6 (n). For any graph G ∈ F(n), we have \|V (G)\| = n and σ 3 (G) ≥ n. The subclasses are defined as follows (more details can be found in [5]): F 6 (n): G ∈ F 6 (n) if G is a 2-connected spanning subgraph of K s ∨ (s + 1)K 2 with s ≥ 4 (n = 3s + 2). F 1 (n): G ∈ F 1 (n) if V (G) = A ∪ B with A ∩ B = ∅, Theorem 11. Let G be a connected graph with order n ≤ 3δ(G). If L(G) ≤ 2δ(G) − 1, then c(G) ≥ n − 1. Proof. Let G be a connected graph with order n ≤ 3δ(G) and L(G) ≤ 2δ(G) − 1. By Lemma 6, G is 2-connected. If α(G) ≤ 2, by Lemma 7, then G is hamiltonian and hence c(G) ≥ n − 1. Clearly, δ(G) = 1. By Lemma 9, the result holds true for δ(G) = 2. Now, it suffices to consider the case of α(G) ≥ 3 and δ(G) = δ ≥ 3. Note that G is a connected graph with order n > 3, by Theorem 2 and Lemma 10, c(G) ≥ p(G)−1 = n−1 or G ∈ F(n). Suppose to the contrary that G ∈ F(n). Recall that G is 2-connected. This implies that G / ∈ F 1 (n) ∪ F 2 (n). First suppose G ∈ F 3 (n). For any vertex x of V (K a ), d Ka (x) ≥ δ − 2 in G and hence \|V (K a )\| ≥ δ − 1. Similarly, \|V (K b )\| ≥ δ − 1 and \|V (K c )\| ≥ δ − 1. Then \|V (K a )\| + \|V (K b )\| + \|V (K c )\| + 2 ≥ 3(δ − 1) + 2 = 3δ − 1. It implies that either n = 3δ − 1 or n = 3δ. For the first case, \|V (K a )\| = \|V (K b )\| = \|V (K c )\| = δ − 1 and hence G = K 2 ∨ (K a + K b + K c ). It can easily be shown that G contains a spanning tree with leaf number at least 2δ, a contradiction. For the second case, exactly one of \|V (K a )\|, \|V (K a )\| and \|V (K a )\| is equal to δ, and the rest are equal to δ − 1. Without loss of generality, suppose that \|V (K a )\| = δ and \|V ( K b )\| = \|V (K c )\| = δ − 1. Then G[G−V (K a )] = K 2 ∨(K b +K c ). The subgraph induced by the vertex set of G[G−V (K a )] with one vertex of V (K a ) has a spanning tree with leaf number 2δ, contradicting L(G) ≤ 2δ − 1. Thus G / ∈ F 3 (n). Next assume that G ∈ F 4 (n). For n ≤ 3δ − 2, by Lemmas 6 and 8, c(G) ≥ p(G) − 1 since n ≤ σ 3 (G) − 2. For 3δ − 1 ≤ n ≤ 3δ, δ = 4 or 5 since 11 ≤ n ≤ 15. Note that a + b = 4 and n = a + b + c + 2. If δ = 4, n = 11, a = 4, b = 0 or n = 12, a = 3, b = 1. It is easy to check that L(G) ≥ 8 > 2δ − 1 in both cases. If δ = 5, n = 14, a = 1, b = 3 or n = 15, a = 0, b = 4. Since δ = 5, the first case is not allowed. For n = 15, a = 0, b = 4, we have L(G) ≥ 10 > 2δ − 1. Thus G / ∈ F 4 (n). Now assume that G ∈ F 5 (n). For any vertex x of V (sK 2 ), d Ks (x) ≥ δ − 1 in G and hence \|V (K s )\| ≥ δ − 1. Then n = 3s + 3 ≥ 3(δ − 1) + 3 = 3δ. Since n ≤ 3δ, then s = δ − 1. It implies that G[V (K s ) ∪ V (sK 2 )] contains (δ − 1)K 1 ∨ (δ − 1)K 2 as a subgraph. Note that the subgraph induced by V (K 3 ) contains no isolated vertex in G. Then we can split this into two cases. For G[V (K 3 )] = K 3 , it is easy to check that G contains a spanning tree with leaf number at least 2δ, a contradiction. For G[V (K 3 )] = P 3 , let w 1 , w 2 ∈ V (G[V (K 3 )]) and d G[V (K 3 )] (w 1 ) = d G[V (K 3 )] (w 2 ) = 1. Then d Ks (w 1 ) = d Ks (w 2 ) = δ − 1. We also obtain G has a spanning tree with leaf number at least 2δ, contradicting L(G) ≤ 2δ − 1. Thus G / ∈ F 5 (n). It follows that G ∈ F 6 (n). Since n = 3s + 2 ≤ 3δ, we have s ≤ δ − 1. For any vertex x of V ((s + 1)K 2 ), d Ks (x) ≥ δ − 1 in G since d(x) ≥ δ. Then s = δ − 1 and n = 3s + 2 = 3δ − 1. Further, G contains K 1 ∨ δK 2 as a subgraph. Thus, G has a spanning tree with leaf number at least 2δ, a contradiction. This completes the proof of Theorem 11. Before giving the proof of the main theorem, we prove a conclusion about regular graphs. Lemma 12. [7] Every 2-connected k-regular (k ≥ 3) graph of order at most 3k + 3 is hamiltonian except the Petersen graph P and the graph obtained from P by replacing one vertex of P by a triangle. Denote by P be the graph obtained from P by replacing one vertex of P by a triangle. Theorem 13. Let G be a k-regular connected graph. If L(G) ≤ 2k−1, then G is hamiltonian and the condition cannot be relaxed. Proof. It is easy to verify that L(P ) = 6 and L(P ) = 7 (see Fig 1 and Fig 2). The Petersen graph P is non-hamiltonian but satisfies L(P ) = 6 = 2k, so the condition cannot be relaxed. Let G be a k-regular connected graph of order n with L(G) ≤ 2k − 1. Since L(G) ≥ 2, then k ≥ 2. Clearly, G is hamiltonian when k = 2. Next suppose that k ≥ 3. By Lemma 5, we have n ≤ max{2k + 6, 3k}. Note that 3k + 3 ≥ max{2k + 6, 3k} when k ≥ 3. By Lemmas 6 and 12, G is hamiltonian. This completes the proof of Theorem 13. The following lemmas play the key role in the proof of Theorem 4. Lemma 14. Let G be a connected graph with L(G) ≤ 2δ(G) − 1. For δ(G) ≥ 3, if there is one vertex x ∈ V (G) with degree 2δ(G) − 1, then \|V (G) \ N [x]\| ≤ 2. Proof. Let G be a connected graph with L(G) ≤ 2δ(G) − 1. Since L(G) ≤ 2δ(G) − 1, each vertex of N (x) has at most one neighbour in V (G) \ N [x]. By Lemma 6, G is 2-connected. Then there are two vertices y 1 , (1) The vertices c i and c i+1 have no common neighbor in V (G) \ V (C). y 2 ∈ V (G) \ N [x] have neighbors in N (x). Similarly, each vertex of {y 1 , y 2 } has at most one neighbour in V (G) \ N [x] and hence at least δ(G) − 1 neighbors in N (x). Suppose that \|V (G) \ N [x]\| ≥ 3. There exists one vertex y 3 ∈ V (G) \ (N [x] ∪(2) Let x, y ∈ V (G) \ V (C). If c i , c j ∈ N C (x) , then c i+1 and c j+1 cannot both belong to N (y). Proof. It is easy to show that the results of Lemma 17, so we omit them. Finally, we show that the proof of Theorem 4. Proof. Let G be a connected graph of order n. For n ≤ 3δ(G), by Theorem 11, c(G) ≥ n − 1. For n ≥ 3δ(G) + 1, by Lemma 5, we have δ(G) ≤ 5. Clearly, δ(G) = 1. By Lemma 9, the result is true when δ(G) = 2. Denote by δ(G) = δ. Now, it suffices to consider the case of 3δ + 1 ≤ n ≤ 2δ + 6 and 3 ≤ δ ≤ 5. Suppose to the contrary that 6 ≤ k ≤ 8. For k = 6, by Lemma 17 (3), \|V (P C )\| ≤ 2. Recall that δ = 3 and ∆(G) = 4, by Lemma 17 (1) and (2), we obtain at most two isolated vertices in G − V (C). Hence \|V (P C )\| = 2 and G[V (G) \ V (C)] = 2K 1 + K 2 or 2K 2 . Let x, y ∈ V (G) \ V (C) and x is adjacent to y. Since and \|N (y)\| = 3, by Lemma 17 (1) and (2), which is not allowed. For the second case, the proof method is similar to the first case, and will not be repeated here. N [x]. This implies that e(N (x), N 2 (x)) ≤ 6. Since δ = 4, then \|N 2 (x)\| ≤ 2. By Lemma 6, G is 2-connected and hence \|N 2 (x)\| = 2. Clearly, y 1 is not adjacent to y 2 since n = 13. Let z 1 ∈ N (y 1 ) and z 2 ∈ N (y 2 ). Recall that G is 2-connected. z 1 = z 2 . Hence \|N (z 1 ) \ {y 1 }\| ≥ 3. From Fig.5, we obtain G contains a tree with leaf number 8, a contradiction. For ∆(G) = 4, by Theorem 13, c(G) = n ≥ n − 1. It remains the case of ∆(G) = 5. Let C = c 1 , c 2 , . . . , c k , c 1 be a longest cycle in G and let P C be a longest path in G − V (C). By Lemmas 6 and 15, k ≥ 8. Now we show that k ≥ 12. Suppose to the contrary that 8 ≤ k ≤ 11. For k = 8, by Lemma 17 (3), \|V (P C )\| ≤ 3. Since ∆(G) = 5, by Lemma 17 (1) and (2), there are at most three isolated vertices in G − V (C). Then 2 ≤ \|V (P C )\| ≤ 3. If \|V (P C )\| = 2, then G − V (C) = 3K 1 + K 2 or K 1 + 2K 2 . Let x, y ∈ V (G) \ V (C) and x is adjacent to y. Without loss of generality, suppose that c 1 ∈ N (x). By Lemma 17 (1) and (2) Since δ(G) = 4 and \|V (P C )\| = 3, we have x is adjacent to z. Note that we have a new path P C = y, x, z in G − V (C). Similarly, N C (y) = N C (z) and \|N C (y)\| = \|N C (z)\| = 2. So, N C (x) = N C (y) = N C (z) = {c 1 , c 5 } and d(x) = d(y) = d(z) = 4. Then G[V (G) \ V (C)] = 2K 1 + K 3 . Let {u, v} = V (G) \ (V (C) ∪ {x, y, z}). Clearly, N (u) = N (v) = {c 2 , c 4 , c 6 , c 8 }. The subgraph induced by {c 8 , c 1 , c 2 , c 3 , x, y, z, u, v} contains a tree with leaf number 7 (see Fig.6). By Lemma 14, n ≤ 9 + 2 = 11 < 13, a contradiction. For k = 9, by Lemma 17, \|V (P C )\| ≤ 3 and there are at most two isolated vertices in (1) and (2), \|V (P C )\| > 1 since δ = 4. Hence \|V (P C )\| = 2 G − V (C). Then 2 ≤ \|V (P C )\| ≤ 3. If \|V (P C )\| = 2, then G[V (G) \ V (C)] = 2K 1 + K 2 or 2K 2 . Let x, y ∈ V (G) \ V (C) For the sharpness, consider the following graph. The graph G 1 of order n is formed by taking the cycle C n−1 = v 1 , v 2 , . . . , v n−2 , v n−1 , v 1 and add one vertex v n together with edges v 1 v n , v 3 v n . Note that δ(G 1 ) = 2 and L(G 1 ) = 3. Then G 1 satisfying L(G 1 ) ≤ 2δ(G 1 ) − 1 and c(G 1 ) = n − 1. The condition L(G) ≤ 2δ(G) − 1 cannot be relaxed. The graph G 2 with order n ≥ 8 is formed by taking the cycle C n−2 = v 1 , v 2 , . . . , v n−2 , v 1 and add two vertices v n−1 and v n together with edges v 1 v n−1 , v 3 v n−1 , v n−5 v n , v n−3 v n . Clearly, δ(G 2 ) = 2 and L(G 2 ) = 4. Then Theorem 4 . 4Let G be a connected graph of order n. If L(G) ≤ 2δ(G)−1, then c(G) ≥ n−1. For any graph G, G[S] denotes the subgraph of G induced by S ⊆ V (G). Let A, B ⊆ V (G) and A ∩ B = ∅. Denote by E(A, B) the set of edges of G with one end in A and the other end in B and e(A, B) = \|E(A, B)\|. Lemma 10. [5] Let G be a connected graph with order n ≥ 3. If σ 3 (G) ≥ n, then G satisfies c(G) ≥ p(G) − 1 or G ∈ F(n), where F(n) is the class of graphs defined below. F (n) consists six subclasses: G[A] and G[B] are hamiltonian or isomorphic to K 2 , and e(A, B) = 1. F 2 2(n): G ∈ F 2 (n) if V (G) = A ∪ B with A ∩ B = {x},G[A] and G[B] are both hamiltonian or both isomorphic to K 2 , and e(A \ {x}, B \ {x}) = 0. F 3 3(n): G ∈ F 3 (n) if G is a 2-connected spanning subgraph of K 2 ∨ (K a + K b + K c ) with a, b, c ≥ 2 (n = a + b + c + 2). F 4 4(n): G ∈ F 4 (n) if G is a 2-connected spanning subgraph of K 3 ∨ (aK 2 + bK 3 ) with a, b ≥ 0 and a + b = 4 (n = 2a + 3b + 3, 11 ≤ n ≤ 15). F 5 5(n): G ∈ F 5 (n) if G is a 2-connected spanning subgraph of K s ∨ (sK 2 + K 3 ) with s {y 1 , y 2 }) and y 3 is adjacent to y 1 or y 2 . Clearly, N (y 3 ) ∩ N (x) = ∅. Without loss of generality, assume that y 3 is adjacent toy 1 . G[N [x] ∪ {y 1 } ∪ N (y 3 )] contains a tree with leaf number 3δ(G) − 3. Further, since δ(G) ≥ 3, we have 3δ(G) − 3 > 2δ(G) − 1, a contradiction.This completes the proof of Lemma 14.Lemma 15. [4] Let G be a 2-connected graph of order n and let C be a longest cycle in G. Then \|V (C)\| ≥ min{n, 2δ(G)}. Lemma 16. [6] Let G be a connected graph of order n.(1) If δ(G) ≥ 4, then L(G) ≥ 2n+8 5 . (2) If δ(G) ≥ 5, then L(G) ≥ n 2 + 2. Lemma 17. Let G be a connected graph of order n and let C = c 1 , c 2 , . . . , c k , c 1 be a longest cycle in G. The subscripts of the vertices c t are taken modulo k. ( 3 ) 3Let P C = p 1 , p 2 , . . . , p s be a longest path in G − V (C). If the vertices p 1 and p s have distinct neighbors in V (C), then s ≤ k 2 − 1. 10 ≤ n ≤ 12. Since L(G) ≤ 2δ − 1 = 5, we have 3 ≤ ∆(G) ≤ 5. If ∆(G) = 3, by Theorem 13, G is hamiltonian and hence c(G) ≥ n − 1. If ∆(G) = 5, by Lemma 14, n ≤ 6 + 2 = 8 < 10, a contradiction. Next suppose that ∆(G) = 4. We discuss it in two Subcases according to the order of G. Subcase 1.1. Consider n = 10. Let C = c 1 , c 2 , . . . , c k , c 1 be a longest cycle in G and let P C be a longest path in G − V (C). By Lemmas 6 and 15, k ≥ 6. Now we show that k ≥ 9. k = 6, we have d(x) = d(y) = 3 and N (x) \ {y} = N (y) \ {x}. Clearly, N (x) \ {y} = {c 1 , c 4 } or {c 2 , c 5 } or {c 3 , c 6 }. It is not difficult to see that the proof methods for the above three cases are similar. So let us just consider the first case. Note that \|V (G) \ V (C)\| = 4 and ∆(G) = 4. There is one vertex z ∈ V (G) \ V (C) is adjacent to at least one of {c 2 , c 3 , c 5 , c 6 }.Suppose z is adjacent to c 2 . The subgraph induced by {c 6 , c 1 , x, y, c 2 , c 3 , z} contains a tree with leaf number 5. By Lemma 14, n ≤ 7 + 2 = 9, a contradiction. The remaining cases can be proved in the same way.For k = 7, by Lemma 17 (3), \|V (P C )\| ≤ 2. If \|V (P C )\| = 2, then G[V (G) \ V (C)] = K 1 + K 2 . Let x, y, z ∈ V (G) \ V (C) and x is adjacent to y. It is easy to check thatd(x) = d(y) = 3 and N (x) \ {y} = N (y) \ {x}. Without loss of generality, suppose that N (x) \ {y} = {c 1 , c 4 }.Recall that δ = 3 and ∆(G) = 4. Then z is adjacent to at least three vertices in V (C)\{c 1 , c 4 }. If z is adjacent to c 2 , the subgraph induced by {c 6 , c 1 , x, y, c 2 , c 3 , z} contains a tree with leaf number 5. By Lemma 14, n ≤ 7 + 2 = 9, a contradiction. Using a similar argument as above, we deduce that z is not adjacent to c 3 , c 5 and c 6 , contradictingd(z) ≥ 3. Next suppose \|V (P C )\| = 1. Then G[V (G) \ V (C)] = 3K 1 . Let x, y, z ∈ V (G) \V (C). By Lemma 17 (1) and (2), d(x) = d(y) = d(z) = 3 and N (x) = N (y) = N (z). Then there is one vertex of N (x) has degree at least 5, a contradiction. For k = 8, \|V (P C )\| ≤ 2 since n = 10. If \|V (P C )\| = 2, then G[V (G) \ V (C)] = K 2 . Letx, y ∈ V (G) \ V (C). Without loss of generality, suppose x is adjacent to c 1 . Since C is a longest cycle in G, then y is adjacent to c 4 , c 5 or c 6 . Obviously, we can split this into two cases. The first case is where y is adjacent to c 4 . One can easily show that d(x) = d(y) = 3 and N (x) \ {y} = N (y) \ {x} = {c 1 , c 4 }. Consider the vertex c 2 . If c 2 is not adjacent to c 8 , the subgraph induced by N (c 1 ) ∪ N (c 2 ) contains a tree with leaf number 5 since d(c 2 ) ≥ 3. By Lemma 14, n ≤ 7 + 2 = 9 < 10, a contradiction. If c 2 is adjacent to c 8 , G contains a cycle c 2 , c 8 , c 7 , c 6 , c 5 , c 4 , y, x, c 1 , c 2 with length 9 (see Fig. 3), contradicting to k = 8. The second case is where y is adjacent to c 5 . Similarly, we have d(x) = d(y) = 3 and N (x) \ {y} = N (y) \ {x} = {c 1 , c 5 }. The following results which are derived from the above proof: c 2 is adjacent to c 8 and c 4 is adjacent to c 6 . Then G contains a cycle c 2 , c 8 , c 7 , c 6 , c 4 , c 5 , y, x, c 1 , c 2 with length 9, a contradiction. Next suppose \|V (P C )\| = 1 and hence G[V (G) \ V (C)] = 2K 1 . Let x, y ∈ V (G) \ V (C). Since C is a longest cycle in G and L(G) ≤ 5, then d(x) = d(y) = 3. Without loss of generality, suppose that N (x) = {c 1 , c 3 , c 5 } or {c 1 , c 3 , c 6 }. For the first case, assert that N (y) ∩ N (x) = ∅. Otherwise, G contains a tree with leaf number at least 6 if y is adjacent to c 1 or c 5 (see Fig.4), a contradiction. And if y is adjacent to c 3 , the subgraph induced by the vertex set {c 1 , c 2 , c 3 , c 4 , c 5 , x, y} contains a tree with leaf number 5. Then, by Lemma 14, n ≤ 9, a contradiction. So, N (y) ⊆ V (C) \ N (x) Subcase 1. 2 . 1 21Consider n = 11 or 12. Let x ∈ V (G) with d(x) = 4. Set N (x) = {x 1 , x 2 , x 3 , x 4 }. Assert that any vertex of N (x) has at most one neighbor in V (G) \ N [x]. Since L(G) ≤ 5, we have d G−N [x] (x i ) ≤ 2 for i ∈ {1, 2, 3, 4}. If there is one vertex of N (x) has exactly two neighbors in V (G) \ N [x], by Lemma 14, n ≤ 5 + 2 + 2 = 9 < 11, a contradiction. Hence, e(N (x), V (G) \ N [x]) ≤ 4. Let N 2 (x) ⊆ V (G) \ N [x] and each vertex of N 2 (x) has neighbor in N (x). Similarly, by Lemma 14, we can show that each vertex of N 2 (x) has at most one neighbour in V (G)\N [x]. Then each vertex of N 2 (x) has at least two neighbours in N (x), since δ = 3. By Lemma 6, G is 2-connected and hence \|N 2 (x)\| ≥ 2. Then \|N 2 (x)\| = 2 and e(N (x), V (G) \ N [x]) = 4. Set N 2 (x) = {y 1 , y 2 }. Without loss of generality, suppose that N (y 1 ) ∩ N (x) = {x 1 , x 2 } and N (y 2 ) ∩ N (x) = {x 3 , x 4 }. It is easy to check that y 1 is not adjacent to y 2 , since n ≥ 11. Let z 1 = N (y 1 ) \ {x 1 , x 2 } and z 2 = N (y 2 ) \ {x 3 , x 4 }. Since G is 2-connected, then z 1 = z 2 . Note that G[N (x)] contains 2K 2 . Then G contains a path of length 8 with endpoints z 1 and z 2 . For n = 11, it remains two vertices w 1 and w 2 . Obviously, d(w 1 ) = d(w 2 ) = 3 and N (w 1 ) = {w 2 , z 1 , z 2 }, N (w 2 ) = {w 1 , z 1 , z 2 }. Thus c(G) = n. For n = 12, it remains three vertices w 1 , w 2 and w 3 . One can easy show that c(G) ≥ n − 13 ≤ n ≤ 14. For n = 14, by Lemma 16 (1), L(G) L(G) ≤ 2δ − 1 = 7. Then we only need to consider n = 13. Suppose n = 13. Since L(G) ≤ 7, we have 4 ≤ ∆(G) ≤ 7. By Lemma 14, ∆(G) = 7. For ∆(G) = 6, let x ∈ V (G) with d(x) = 6. Then any vertex of N (x) has at most two neighbors in V (G)\N [x]. Let N 2 (x) ⊆ V (G) \ N [x] and each vertex of N 2 (x) has neighbor in N (x). By Lemma 14, any vertex of N (x) ∪ N 2 (x) has at most one neighbor in V (G) \ , N (y) ⊆ {x, c 1 , c 4 , c 5 , c 6 }. Then N (y) = {x, c 1 , c 4 , c 6 }, since C is a longest cycle and δ = 4. Further, we have N (x) = {c 1 , y}, contradicting d(x) ≥ 4. Next suppose \|V (P C )\| = 3.Let P C = x, y, z. It follows that N C (x) = N C (z) and \|N C (x)\| = \|N C (z)\| = 2. Recall that C is a longest cycle in G. Without loss of generality, suppose that N C (x) = N C (z) = {c 1 , c 5 }. and x is adjacent to y. One can easy show that d(x) = d(y) = 4 and N (x) ∩ N (y) = {c 1 , c 4 , c 7 } or {c 2 , c 5 , c 8 } or {c 3 , c 6 , c 9 }. Without loss of generality, suppose that N (x) ∩ N (y) = {c 1 , c 4 , c 7 }. Since L(G) ≤ 7, any vertex of V (G) \ (V (C) ∪ {x, y}) has no neighbor in {c 1 , c 4 , c 7 }. Further, by Lemma 17 (1) and (2), G[V (G) \ V (C)] = 2K 1 + K 2 and G[V (G) \ V (C)] = 2K 2 , since δ = 4. Next suppose \|V (P C )\| = 3. Let P C = x, y, z. Using the same method as the case of k = 8 and \|V (P C )\| = 3, we obtain G[V (G) \ V (C)] = K 1 + K 3 and N C (x) = N C (y) = N C (z). Without loss of generality, suppose thatN C (x) = N C (y) = N C (z) = {c 1 , c 5 }. Consider the vertex c 2 .If c 2 is adjacent to c 9 , G contains a cycle c 1 , c 2 , c 9 , c 8 , c 7 , c 6 , c 5 , z, y, x, c 1 with length 10, a contradiction. If c 2 is not adjacent to c 9 , the subgraph induced by N (c 1 ) ∪ N (c 2 ) contains a tree with leaf number 7. By Lemma 14, n ≤ 10 + 2 = 12 < 13, a contradiction.For k = 10, \|V (P C )\| ≤ 3. For \|V (P C )\| = 3, let V (G) \ V (C) = {x, y, z}. Similarly, one can easy show that G[V (G) \ V (C)] = K 3 and N C (x) = N C (y) = N C (z) and \|N C (x)\| = \|N C (y)\| = \|N C (z)\| = 2. Without loss of generality, suppose that N C (x) = {c 1 , c 5 } or {c 1 , c 6 }. If N C (x) = {c 1 , c 5 }, we consider the vertex c 3 . Note that d(c 1 ) = d(c 5 ) = 5. Then \|N (c 3 ) ∩ (V (C) \ {c 1 , c 2 , c 4 , c 5 })\| ≥ 2, since δ = 4.If c 3 is adjacent to c 6 (seeFig.7a) orc 10 , then G contains a cycle c 5 , c 4 , c 3 , c 6 , c 7 , c 8 , c 9 , c 10 , c 1 , x, y, z, c 5 with length 12 or a cycle c 1 , c 2 , c 3 , c 10 , c 9 , c 8 , c 7 , c 6 , c 5 , z, y, x, c 1 with length 12, a contradiction. If c 3 is adjacent to c 7 (see Fig.7b) or c 9 , then G contains a cycle c 5 , c 4 , c 3 , c 7 , c 8 , c 9 , c 10 , c 1 , x, y, z, c 5 with length 11 or a cycle c 1 , c 2 , c 3 , c 9 , c 8 , c 7 , c 6 , c 5 , z, y, x, c 1 with length 11, a contradiction. Hence we have N (c 3 ) ⊆ {c 2 , c 4 , c 8 }, contradicting δ = 4. If N C (x) = {c 1 , c 6 }, using the same method, we have c 7 , c 10 / ∈ N (c 3 ) and c 7 , c 10 / ∈ N (c 4 ). Now we show that c 3 and c 4 are both adjacent to exactly one of {c 8 , c 9 }. Suppose to the contrary that c 3 and c 4 are adjacent to c 8 and c 9 or c 9 and c 8 , respectively. Then G contains a cycle c 1 , c 2 , c 3 , c 8 , c 9 , c 4 , c 5 , c 6 , z, y, x, c 1 with length 11 or a cycle c 1 , c 2 , c 3 , c 9 , c 8 , c 4 , c 5 , c 6 , z, y, x, c 1 with length 11, contradicting k = 10. Without loss of generality, suppose c 9 ∈ N (c 3 ) ∩ N (c 4 ). Since δ = 4, c 4 is adjacent to c 2 . Then G contains a cycle c 1 , c 10 , c 9 , c 3 , c 2 , c 4 , c 5 , c 6 , z, y, x, c 1 with length 11, a contradiction. For \|V (P C )\| ≤ 2, it implies that there exists at least one isolated vertex in G − V (C). Assert that any isolated vertex of G − V (C) has degree 4. Otherwise, suppose d G−V (C) (x) = 0 and d(x) = 5. Then N (x) = {c 1 , c 3 , c 5 , c 7 , c 9 } or {c 2 , c 4 , c 6 , c 8 , c 10 }. We show that the first case, the second can be proved by same method. Note the subgraph induced by {c 9 , c 10 , c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , c 7 , x} contains a tree with leaf number 7. By Lemma 14, n ≤ 10 + 2 = 12 < 13, a contradiction. Let V (G) \ V (C) = {x, y, z} and d G−V (C) (x) = 0. Then, by Lemma 14, N (y) ∩ N (x) = ∅ and N (z) ∩ N (x) = ∅. By Lemma 17 G 2 satisfying L(G 2 ) ≤ 2δ(G 2 ) but c(G 2 ) = n − 2.This completes the proof of Theorem 4. Since n = 13, by Lemma 14, d(x) = d(y) = 4. For \|V (P C )\| = 2, G[V (G) \ C are 2,2,7 or 2,3,6 or 2,4,5 or 3,3,5 or 3,4,4. The proof methods for the first three cases are similar. Since C is a longest cycle in G, one can easy show that d(y) ≤ 3 < 4, a contradiction. The proofs for the latter two cases are similar, so we only give the proof for one of them here. Without loss of generality, suppose N (x) = {c 1. 2c 1 , c 2 , c 3 , x, y, c 4 , c 6 , c 7; c 1 , x, y, c; {x, y, c 1 , c 5 , c 74 of length 12, contradicting k = 11. Hence N (c 3 ) ∩ {x, y, c 1 , c 5 , c 7 } = ∅. Note that the subgraph induced by. since d(c 3 ) ≥ 4. Then, by Lemma 14, n ≤ 12 < 13, a contradiction. For \|V (P C )\| = 1, G[V (G) \ V (C)] four cases. The lengths of each parts of C are 2,2,2,5 or 2,2,3,4 or 2,3,2,4 or 2,3,3,3. Similarly, by Lemma 17 (1) and (2), one can easy show that in each case there is ay is adjacent to z. Since d(x) = 4, then the neighbors of x divide C into four parts. The lengths of the four parts of C are 2,2,2,4 or 2,2,3,3 or 2,3,2,3 (see Fig.8). By Lemma 17 (1) and (2), it is easy to show that in each case there is a contradiction, so we omit it. For k = 11, \|V (P C )\| ≤ 2. Let x, y ∈ V (G) \ V (P ). Since n = 13, by Lemma 14, d(x) = d(y) = 4. For \|V (P C )\| = 2, G[V (G) \ C are 2,2,7 or 2,3,6 or 2,4,5 or 3,3,5 or 3,4,4. The proof methods for the first three cases are similar. Since C is a longest cycle in G, one can easy show that d(y) ≤ 3 < 4, a contradiction. The proofs for the latter two cases are similar, so we only give the proof for one of them here. Without loss of generality, suppose N (x) = {c 1 , c 4 , c 7 }. Consider the vertex c 3 . We assert that N (c 3 ) ∩ N (x) = {c 4 }. Otherwise, if c 3 is adjacent to c 1 , the subgraph induced by {c 11 , c 1 , c 2 , c 3 , x, y, c 4 , c 6 , c 7 , c 8 adjacent to c 5 , G contains a cycle c 4 , c 3 , c 5 , c 6 , c 7 , c 8 , c 9 , c 10 , c 11 , c 1 , x, y, c 4 of length 12, contradicting k = 11. Hence N (c 3 ) ∩ {x, y, c 1 , c 5 , c 7 } = ∅. Note that the subgraph induced by N [c 3 ] ∪ {x, y, c 1 , c 5 , c 7 } contains a tree with leaf number 7, since d(c 3 ) ≥ 4. Then, by Lemma 14, n ≤ 12 < 13, a contradiction. For \|V (P C )\| = 1, G[V (G) \ V (C)] four cases. The lengths of each parts of C are 2,2,2,5 or 2,2,3,4 or 2,3,2,4 or 2,3,3,3. 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[ "Unfolding Polyelectrolytes in Trivalent Salt Solutions Using DC Electric Fields: A Study by Langevin Dynamics Simulations", "Unfolding Polyelectrolytes in Trivalent Salt Solutions Using DC Electric Fields: A Study by Langevin Dynamics Simulations" ]	[ "Yu-Fu Wei \nDepartment of Engineering and System Science\nNational Tsing Hua University\n300HsinchuR.O.CTaiwan\n", "Pai-Yi Hsiao [email protected] \nDepartment of Engineering and System Science\nNational Tsing Hua University\n300HsinchuR.O.CTaiwan\n" ]	[ "Department of Engineering and System Science\nNational Tsing Hua University\n300HsinchuR.O.CTaiwan", "Department of Engineering and System Science\nNational Tsing Hua University\n300HsinchuR.O.CTaiwan" ]	[]	We study the behavior of single linear polyelectrolytes condensed by trivalent salt under the action of electric fields through computer simulations. The chain is unfolded when the strength of the electric field is stronger than a critical value. This critical electric field follows a scaling law against chain length and the exponent of the scaling law is −0.77(1), smaller than the theoretical prediction, −3ν/2 [Netz, Phys. Rev. Lett. 90 (2003) 128104], and the one obtained by simulations in tetravalent salt solutions, −0.453(3) [Hsiao and Wu, J. Phys. Chem. B 112 (2008) 13179]. It demonstrates that the scaling exponent depends sensitively on the salt valence. Hence, it is easier to unfold chains condensed by multivalent salt of smaller valence. Moreover, the absolute value of chain electrophoretic mobility increases drastically when the chain is unfolded in an electric field.The dependence of the mobility on electric field and chain length provides a plausible way to impart chain-length dependence in free-solution electrophoresis via chain unfolding transition induced by electric fields. Finally, we show that, in addition to an elongated structure, a condensed chain can be unfolded into an U-shaped structure. The formation of this structure in our study is purely a result of the electric polarization, but not of the elasto-hydrodynamics dominated in sedimentation of polymers.	10.1063/1.3129563	[ "https://arxiv.org/pdf/0904.2433v1.pdf" ]	9,573,997	0904.2433	90e7a25d77589a4e8382699cd56ea9d2257b8544	Unfolding Polyelectrolytes in Trivalent Salt Solutions Using DC Electric Fields: A Study by Langevin Dynamics Simulations 16 Apr 2009 Yu-Fu Wei Department of Engineering and System Science National Tsing Hua University 300HsinchuR.O.CTaiwan Pai-Yi Hsiao [email protected] Department of Engineering and System Science National Tsing Hua University 300HsinchuR.O.CTaiwan Unfolding Polyelectrolytes in Trivalent Salt Solutions Using DC Electric Fields: A Study by Langevin Dynamics Simulations 16 Apr 2009(Dated: April 16, 2009)arXiv:0904.2433v1 [cond-mat.soft]PACS numbers: We study the behavior of single linear polyelectrolytes condensed by trivalent salt under the action of electric fields through computer simulations. The chain is unfolded when the strength of the electric field is stronger than a critical value. This critical electric field follows a scaling law against chain length and the exponent of the scaling law is −0.77(1), smaller than the theoretical prediction, −3ν/2 [Netz, Phys. Rev. Lett. 90 (2003) 128104], and the one obtained by simulations in tetravalent salt solutions, −0.453(3) [Hsiao and Wu, J. Phys. Chem. B 112 (2008) 13179]. It demonstrates that the scaling exponent depends sensitively on the salt valence. Hence, it is easier to unfold chains condensed by multivalent salt of smaller valence. Moreover, the absolute value of chain electrophoretic mobility increases drastically when the chain is unfolded in an electric field.The dependence of the mobility on electric field and chain length provides a plausible way to impart chain-length dependence in free-solution electrophoresis via chain unfolding transition induced by electric fields. Finally, we show that, in addition to an elongated structure, a condensed chain can be unfolded into an U-shaped structure. The formation of this structure in our study is purely a result of the electric polarization, but not of the elasto-hydrodynamics dominated in sedimentation of polymers. I. INTRODUCTION To well understand the properties of charged macromolecules in electric fields, including the conformation and the mobility, is very important in many domains of researches such as in polymer science, in biophysics, and in microfluidics, for the reason of a large variety of applications [1]. Applying electric fields stays at the center of the techniques to manipulate charged macromolecules. It can be also used as a tool to separate molecules by sizes. However, for the latter case, experiments are usually performed in a sieving matrix such as in gel, instead of in a solution [1,2,3]. This is because the free draining effect in an electrolyte solution produces an electrophoretic mobility independent of the chain length of macromolecules under a typical electrophoretic condition [4]. Nevertheless, researchers continue to devote their efforts in finding ways for size separation in free solutions for the reason of its high throughput and applications in microfluidics. In 2003, Netz proposed a new strategy to achieve this goal by unfolding condensed polyelectrolytes (PEs) in electric fields [5,6]. He predicted that the chain mobility increases when a condensed PE chain unfolds in an electric field, and the critical electric field to unfold a chain, E * , depends on the chain length N, following the scaling law E * ∼ N −3ν/2 where ν is the chain swelling exponent. Therefore, longer chains will be unfolded and separated out earlier when the applying electric field slowly increases. His idea has been recently verified by simulations [7] in which PE chains were condensed into globules by tetravalent salts and then stretched in electric fields. A more general form of the scaling law has been proposed in the study, reading as E * ∼ V −1/2 where V is the ellipsoidal volume calculated from the three eigenvalues of the chain gyration tensor. According to the scaling law obtained by the simulations, an electric field of 2kV/cm should be applied to unfold collapsed PEs of chain length of order 10 6 . This electric field is relatively strong. For practical reason, we wish E * to be as small as possible. One way to reduce E * is to make less compact the condensed chain structure. This aim can be achieved, for example, by increasing temperature, by performing experiments in high-dielectric solutions, or by using weak condensing agent to collapse PEs. In this paper, we choose the last method, using trivalent salt as the condensing agent, and study the static and dynamic properties of chains and the unfolding electric field. Since the electrostatic interaction with trivalent salt is 25% weaker than that with tetravalent salt, some unexpected situations may take place. A key question is to know if E * still follow the same scaling law of the strong condensation as shown in Netz's [5,6] and in Hsiao and Wu's study [7]. The rest of this paper is organized as follows. In Section II, we describe our model and simulation setup. In Section III, we present our results. The discussed topics include the degree of unfolding, the critical electric field to unfold a chain, the electrophoretic mobility, the distribution of condensed trivalent counterions on a chain, and the chain conformation after unfolding. We give our conclusions in Section IV. II. MODEL AND SIMULATION SETUP Our simulation system contains a single polyelectrolyte and trivalent salt, placed in a rectangular box with periodic boundary condition. The polyelectrolyte dissociates into a polyion chain and many counterions. The polyion is modeled by a bead-spring chain, consisting of N beads; each bead carries a −e charge where e is the elementary charge unit. The counterions are modeled by spheres; each carries +e charge. The trivalent salt dissociates into trivalent cations (counterions) and monovalent anions (coions); these ions are also modeled by charged spheres. Solvent is treated as a uniform dielectric medium with dielectric constant equal to ǫ r . Three kinds of interaction are considered: the excluded volume interaction, the Coulomb interaction, and the bond connectivity. The excluded volume interaction is modeled by a purely repulsive Lennard-Jones potential U ex (r) =    4ε LJ (σ/r) 12 − (σ/r) 6 + ε LJ for r ≤ 2 1/6 σ 0 for r > 2 1/6 σ(1) where r is the distance between two particles, ε LJ is the interaction strength, and σ denotes the diameter of a particle. We assumed that all the beads and spheres have identical ε LJ and σ. We set ε LJ = k B T /1.2 where k B is the Boltzmann constant and T is the temperature. The Coulomb interaction is U coul (r) = Z i Z j λ B k B T r(2) where Z i and Z j are the valences of the two charges and λ B = e 2 /(4πǫ r ǫ 0 k B T ) is the Bjerrum length, at which two unit charges have the Coulomb interaction tantamount to the thermal energy k B T . We set λ B to be 3σ to simulate highly charged PEs, such as polystyrene sulfonate. U coul was calculated by PPPM Ewald method. Two adjacent beads (monomers) on the chain are connected by the bond connectivity, modeled by a finitely extensible nonlinear elastic potential U bond (b) = − 1 2 k b b 2 max ln 1 − b 2 b 2 max (3) where b is the bond length, b max is the maximum bond extention, and k b is the spring constant. We set b max = 2σ and k b = 5.8333k B T /σ 2 . The average bond length under this setup is about 1.1σ. An external uniform electric field E is applied, toward x direction. The equation of motion of a particle is described by the Langevin equation: m i¨ r i = − ∂U ∂ r i − m i γ i˙ r i + Z i eEx + η i (t)(4) where m i is the mass of the particle i, r i is its position vector, m i γ i is the friction coefficient, and η i simulates the random collision by solvent molecules. η i (t) has zero mean over time and satisfies the fluctuation-dissipation theorem: η i (t) · η j (t ′ ) = 6k B T m i γ i δ ij δ(t − t ′ )(5) where δ ij and δ(t−t ′ ) are the Kronecker and the Dirac delta function, respectively. The temperature control is incorporated according to this theorem. We assumed that the particles have the same mass m and damping constant γ. We set γ = 1τ −1 where τ = σ m/(k B T ) is the time unit. We know that the dynamics of polymers in dilute solutions is described by Zimm model [8]. However, when an electric field is applied in a typical electrophoretic condition, the hydrodynamic interaction is largely canceled out due to the opposite motions of the ions in the electrolyte solution [1,9,10]. Therefore, in this study we neglected the hydrodynamic interaction. Hydrodynamic interaction is important only when the chain length is very short [11,12]. We the chain collapsed into a compact globule structure, in the absence of electric field [13,14], with its effective chain charge almost being neutralized. We performed Langevin dynamics simulations [15] with integrating time step equal to △ t = 0.005τ . We ran firstly 10 6 to 10 7 time steps to bring the system to a steady state and then ran 10 8 time steps to cumulate data for analysis. To simplify the notation, we assign in the following text that σ, m, and k B T are the unit of length, mass, and energy, respectively. Therefore, the concentration will be described in unit of σ −3 , the strength of electric field in unit of k B T /(eσ), and so forth. III. RESULTS AND DISCUSSIONS A. Degree of unfolding We start from studying the chain conformation under the action of an electric field. The degree of unfolding, defined as the ratio of the end-to-end distance R e of chain over the chain contour length L c = (N − 1)b, is used to characterized the conformation. The results are plotted in Fig. 1 as a function of E. Each curve in the plot denotes the variation of R e /L c for a given chain length N. We can see that when the electric field is weak, the ratio is a constant. This indicates an unperturbed conformation of chain and the chain remains in a collapsed structure. An abrupt increase appears when E is increased over some critical value E * . R e can become as large as 90% of L c if the applied field is very strong. This indicates a structural transition from a collapsed structure to an elongated structure. We noticed that the value of E * depends on the chain length. The longer the chain length, the smaller the E * will be. Moreover, this structural transition happens in an interval of E. The size of the interval decreases with increasing chain length. Although the transition becomes sharper when chain length is long, R e increases in a continuous way with E, which suggests a second-order transition. B. Critical electric field E * The dependence of the critical electric field E * on chain length N has been investigated in salt-free [5,6] and in tetravalent salt solutions [7]. Both of these studies showed that E * scales roughly as N −0.5 to unfold a condensed chain. It is now important to know if this scaling law is valid for a PE chain condensed by trivalent counterions. To verify it, we follow firstly the method proposed by Netz [5,6]: E * is calculated by equating the polarization energy U pol = p · E/2 and the thermal fluctuation energy k B T . Here p is the dipole moment of the PE-ion complex induced by the electric field and calculated by p = i Z i e( r i − r cm ) where r i is the position vector, running over all the particles inside the complex, and r cm is the center of mass of the PE. The complex is considered as a set of particles, including monomers and ions, inside the region of a worm-shaped tube which is the union of the jointed spheres of radius r t = 3, centered at each monomer center. The component of p at the field direction, p x , is plotted against the field strength E in Fig. 2. As seen in the log-log plot, p x increases linearly with E with a slope equal to 1, when E is small. This is the well-known linear response of a dielectric object, p x = αE, which has been reported in the previous studies [6,7]. But different to the previous, we found that this linear region terminates before intersecting with the dotted line which denotes the relation p x E/2 = k B T , specially when the chain length is long. This is simply because the binding force to condense the PE chain in the trivalent salt solutions is weaker than in the tetravalent salt [7]. For the system studied by Netz [5,6], the chains were strongly p x E/2 = k B T . condensed because of the un-realistically strong Coulomb coupling chosen by him. Therefore, his method can be used only as a rough estimation of E * for the case of strong condensation but not suitable for the weak condensation. If we continue going with his method and calculate the intersection between the extended linear region and the dotted line, we will find that E * scales as N −0.463 (4) (see open circles in Fig. 3(a)). This scaling law seems to follow the prediction of Netz, N −3ν/2 , because the chain swelling exponent in zero electric field is ν = 0.321(2) for this case (cf. Fig. 3(b)). Nevertheless, E * obtained by this method is actually overestimated, going much over away the linear response region, specially when the chain is long. To give a more accurate estimation of E * , we follow here the definition of the unfolding electric field by taking simply the electric field at the inflection point of the curve R e /L c vs. E. The inflection point on each curve in Fig. 1 is indicated by the symbol 'x'. The scaling law obtained by this method reads as E ∼ N −0.77(1) (see close squares in Fig. 3(a)). The exponent −0.77(1) is significantly smaller than the one obtained by the Netz's method. Therefore, E * is smaller than Netz's prediction for a long chain and it is easier to unfold a according to the scaling law. This E * corresponds to about 185 V/cm, much smaller than 2kV/cm predicted for the chains condensed by tetravalent salt in simulations [7]. Our results show that the valence of the condensing agent plays an important role in determination of the scaling law. There must exists more complicated mechanism to polarize and to unfold a PE chain in an electric field than our thinking. This mechanism will be investigated in detail in the future. C. Electrophoretic mobility and ion condensation We now study the electrophoretic mobility µ pe of PE chain in electric fields of different strength and show how µ pe changes with E when the chain is unfolded to an elongated structure. µ pe was calculated by v pe /E where v pe is the velocity of the center of mass of the chain in the field direction. The results are shown in Fig. 4. In weak electric fields, µ pe is nearly zero, indicating that the PE chain is effectively charge-neutral, as reported in experiments [16]. While E is increased over E * , µ pe turns to be negative and the chain starts to drift opposite the field direction, which suggests a negative effective chain charge. We found that the stronger the field, the faster the chain will drift. For a long chain, µ pe shows furthermore a plateau region when E > E * . The dependence of µ pe on the electric field and the chain length gives a plausible way to electrophoretically separate PE chains by size in free solutions by means of chain unfolding transition [7]. trivalent ions are stripped off the chain even more. The effective chain charge is thus more negative and \|µ pe \| increases. D. Conformation of an unfolded PE chain In our simulations, the PE chains were unfolded, for the most of the time, to an extended structure, similar to a straight line, aligned parallel to the field direction (see in Fig. 6(a)). Nonetheless, we observed sometimes that they were unfolded to a U-shaped structure in the tubules [17] and also been shown in simulations of the elastic uncharged/charged chains in stokes flows or in electric fields [18,19,20,21]. These studies showed that a combination of the elastic and the hydrodynamic effect results in the bending of a rigid chain into a horseshoe shape, oriented perpendicular to the direction of motion [18,21]. If chains are charged and the driving force is an electric field, other effect, the electric polarization of the PE complex, will play a role, which favors parallel orientation to the electric field, and compete with the elasto-hydrodynamic effect [19]. In our simulations, the PE chains are flexible and the hydrodynamic interaction is neglected. Therefore, different mechanism drives the chains to form U-shaped structures where the field-induced dipole moments on the two branches of a U-shaped chain establish an equilibrium. This phenomenon can be seen by plotting in produced. Moreover, we notice that the electrophoretic mobility of an U-shaped chain is approximately equal to that of an elongated chain of half of the chain length. For example, µ pe is −0.209(4) in Fig. 6(b), close to the mobility of the elongated chain of N = 96, −0.225 (3). Furthermore, we verified the stability of these U-shaped chains and found that they can persist through the whole simulation period corresponding to, at least, the order of microsecond. However, by introducing some perturbations such as AC electric fields, the U-shaped structure can be transfered into the elongated chain structure but the inverted direction of transfer cannot be realized. Therefore, the U-shaped structure is probably metastable. We have calculated the total energy of the system for the U-shaped chain structure and also for the extended-chain structure. We found that the previous energy is, at least, 5% higher than the latter. Moreover, the U-shaped chain has a slower electrophoretic mobility than the extended chain, which implies a larger number of counterions condensed on the U-shaped chain to decrease the effective chain charge; consequently, fewer ions are presented in the bulk solution and the entropy of the solution is small, compared to the extended-chain structure. Therefore, the free energy of the system is lower for the extended chain than for the U-shaped chain. This estimation supports that the U-shaped structure is metastable. Since the open side of the U-shaped chain can point to one of the two directions, along or against the field direction, we predict the existence of other metastable states, due to polarization, in which the chain shows many bends, such as S-shaped or W-shaped structures, in electric fields. IV. CONCLUSIONS We have studied the behavior of single polyelectrolytes condensed by trivalent salt under the action of an uniform electric field by means of Langevin dynamics simulations. We found that the chains unfolded while the strength of the electric field is stronger than some critical value E * , similar to the previous study where the chains were condensed by tetravalent salt [7]. E * shows scaling-law dependence on the chain length N, reading as E * ∼ N −0.77 (1) . The exponent in the scaling law is different from the prediction by Netz [5,6] and from the simulations in tetravalent salt solutions [7], which demonstrated the importance of the salt valence on the exponent. Therefore, the weaker the condensing agent, the larger the absolute value of the exponent and the easier the unfolding of a condensed chain will be. We showed that the electrophoretic mobility of chain \|µ pe \| drastically increases while the chain is unfolded. The distribution of the condensed counterions on the chain was studied and related to the change of the mobility in different regions of electric field. The dependence of µ pe on the chain length and the electric field enables us to device a way to impart chainlength dependence in free-solution electrophoresis through chain-unfolding mechanism in electric fields. Finally, we pointed out the possibility to unfold a condensed PE chain into FIG. 1 : 1varied the chain length (or the number of monomer) from 24 to 384 and studied the static and dynamic properties of PE under the action of an electric field, up to a field strength E = 2.0 k B T /(eσ). We set the monomer concentration C m = 0.0001σ −3 . In order to keep C m constant, the size of the simulation box needs to change with N. Instead of using a cubic simulation box, we chose a rectangular parallelepiped of 1.6Nσ × 79.06σ × 79.06σ, where the box size in the field direction is linearly proportional to N to prevent overlap under periodic boundary condition when the chain unfolds. The added salt concentration was fixed at C s = C m /3, the equivalence point C * s . It has been shown that at this salt concentration, R e /L c as a function of E at C s = C * s for different chain length N . The symbol 'x' denotes the inflection point of curve. FIG . 2: p x as a function of E for different chain length N . The dotted line denotes the equation FIG . 3: (a) E * vs. N where the open circles denote the data obtained by Netz's method and the close squares denote the ones obtained from the inflection points. (b) Radius of gyration R g vs. N in zero electric field. PE chain in trivalent salt solutions. For example, for a chain of length 10 6 , E * is 1.76 × 10 −4 FIG . 4: µ pe as a function of E for different chain length N . FIG. 5 : 5The variation of µ pe can be related to the ion condensation on the chain under the action of the electric field. Therefore, we studied here the number of the condensed trivalent ions on the chain by counting the ions inside the worm-shaped tube of radius r t = 3 around the chain. The results for N = 384 in different strength of electric field are plotted inFig. 5against the monomer index ι, rescaled from 0 to 1, where ι = 0 denotes the first monomer heading toward the field direction and ι = 1 denotes the last monomer of the other chain end.We saw that N c (ι) is flat when the applied field is small, E ≤ 0.007, which shows an uniform distribution of the condensed trivalent ions along the chain. There is about 0.33 trivalent counterions condensed on each monomer, which indicates the neutralization N c (ι) obtained in different field strength. The value of E is indicated near each curve.of the negatively-charged chain backbone by these condensed counterions. If we further increases the electric field, these condensed ions distribute non-uniformly on the chain where fewer ions condensed near the heading end (ι = 0) than the tailing end (ι = 1). When 0.07 < E < 0.2, N c (ι) looks similar to an inclined line and the slope increases with E, resulting in a decrease of the total number of the condensed trivalent counterions on the chain. Therefore, \|µ pe \| increases with E due to this partial detachment of the condensed ions by the electric field. At this moment, the PE-ion complex is polarized in a way thatthe condensed trivalent counterions are bound, basically immobile, on the chain. For the higher electric field, 0.2 < E < 1.0, N c (ι) becomes a horizontal sigmoidal curve and the value in the middle chain region is independent of E. The appearance of this horizontal region reflects the fact that the condensed trivalent counterions are now gliding on the chain. These ions can be stripped off the chain by the strong electric field and the other ions in the bulk solution then condense onto it, establishing a steady state. The total number of the condensed trivalent counterions is approximately a constant in this electric field, which results in the plateau region of µ pe against E. For an even stronger electric field, such as E = 2.0, the baseline of the horizontal sigmoidal curve moves downward. The condensed FIG. 6: (Color on line) Snapshots of unfolded PE chains in electric fields. The yellow, the white, the red, and the green spheres represent, respectively, the monomers, the monovalent counterions, the trivalent counterions, and the coins. The chain length, the electric field, the field direction, and the chain drifting direction are indicated in the figure. FIG. 7 : 7electric fields. The open side of the U shape can point opposite or toward the chain drifting direction as shown in Fig. 6, panel (b) and (c), respectively.This U-shaped structure has been observed experimentally in electrophoresis of micro-N c (ι) for the two U-shaped chains inFig. 6(b)and (c), respectively. Fig. 7 7the distribution of the condensed trivalent counterions N c (ι) for the two U-shaped chains from Fig. 6(b) and (c), respectively. The symmetry of N c (ι) with respect to the middle point of the chain (ι = 0.5) shows that an equilibrated polarization was established on the two branches of the U-chain in the electric field. The existence of two pointing directions of the open side of the U chain is a feature specially for the electric polarization. It is distinguishable to the elasto-hydrodynamic effect where only the U-shaped structure with the open side opposite to the moving direction is an U-shaped structure in electric fields, in addition to the elongated structure, with the open side of the U heading or tailing the chain drifting direction. This structure is a result of purely electric polarization, different from the formation of the horseshoe-shaped chains in sedimentation experiments caused by elasto-hydrodynamics. V. ACKNOWLEDGMENTS This material is based upon work supported by the National Science Council, the Republic of China, under the contract No. NSC 97-2112-M-007-007-MY3. The computing resources are supported by the National Center for High-performance Computing. . J.-L Viovy, Rev. Mod. Phys. 72813J.-L. Viovy, Rev. Mod. Phys. 72, 813 (2000). . K Klepárník, P Boček, Chem. Rev. 1075279K. Klepárník and P. Boček, Chem. Rev. 107, 5279 (2007). . H Cottet, P Gareil, J.-L Viovy, Electrophoresis. 192151H. Cottet, P. 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[ "Regularity for multi-phase variational problems", "Regularity for multi-phase variational problems" ]	[ "Cristiana De Filippis ", "Jehan Oh " ]	[]	[]	We prove C 1,ν regularity for local minimizers of the multi-phase energy:under sharp assumptions relating the couples (p, q) and (p, s) to the Hölder exponents of the modulating coefficients a(·) and b(·), respectively.	10.1016/j.jde.2019.02.015	[ "https://arxiv.org/pdf/1807.02880v1.pdf" ]	119,152,229	1807.02880	e2a7cb7eed49d0dc4ff7bbda0957a120ae694c5f	Regularity for multi-phase variational problems Jul 2018 Cristiana De Filippis Jehan Oh Regularity for multi-phase variational problems Jul 2018 We prove C 1,ν regularity for local minimizers of the multi-phase energy:under sharp assumptions relating the couples (p, q) and (p, s) to the Hölder exponents of the modulating coefficients a(·) and b(·), respectively. Introduction and results The aim of this paper is to analyze the regularity properties of non-autonomous variational integrals of the type w → Ω F(x, Dw) dx ,(1.1) where Ω ⊂ R n is a bounded open domain with n ≥ 2, and emphasize a few new phenomena emerging when considering non-uniformly elliptic operators. Let us briefly review the situation. In the case of functionals satisfying standard polynomial growth and ellipticity of the type terminology double phase stems from). In the first case we have, following a terminology introduced in [11], the p-phase, in the other we have the (p, q)-phase. In the case of (1.4), the regularity of minimizers is regulated by a subtle interaction between the pointwise behaviour of the partial function x → F(x, ·) and the growth assumption satisfied by z → F(·, z). For instance, as established in the work of Baroni, Colombo and Mingione [2,3,4,11,12], sufficient and necessary conditions for regularity of minimizers of the functional (1.4) are that a(·) ∈ C 0,α (Ω) and q p ≤ 1 + α n . (1.5) Specifically, if (1.5) holds, then minimizers of the functional (1.4) are locally C 1,β , for some β > 0, otherwise, they can be even discontinuous; see also [13,17,18]. After these contributions, functionals with double phase type have become a topic of intense study, see for instance [7,8,9,21,22,32,33]. The condition in (1.5) plays a role also when considering more general functionals of the type in (1.1),under so called (p, q)-growth conditions, i.e.: \|Dw\| p F(x, Dw) \|Dw\| q and \|Dw\| p−2 Id ∂ zz F(x, Dw) \|Dw\| q−2 Id . For this we refer to [10,16,17]. Moreover, it intervenes in the validity of a corresponding Calderón-Zygmund theory [13]. We refer to the papers of Marcellini [28,29,30] for more on general functionals with (p, q)-growth. The aim of this paper is to study a significant generalization of the functional (1.4), considering a functional that exhibit three phases. We shall indeed consider the following multiphase Multi-Phase variational energy and where the functions a(·) and b(·) satisfy the following assumptions a ∈ C 0,α (Ω), a(·) ≥ 0, α ∈ (0, 1], b ∈ C 0,β (Ω), b(·) ≥ 0, β ∈ (0, 1]. (1.8) As not to trivialize the problem, we specifically focus on the case in which the strict inequality (1.6) 2 holds. The analysis of this functional then opens the way to that of functional exhibiting an arbitrary number of phases, and involves several subtle points. The main one can be described as follows. In the double phase case of the functional (1.4) the main game is to control the interaction between the potentially degenerate parte of the energy a(x)\|Dw\| q (here degenerate means that it can be a(x) ≡ 0) with the non-degenerate one \|Dw\| p , that always provides a solid rate of ellipticity. This is done in [4,11,12] via a careful comparison scheme built in order to distinguish between the two phases. Here the situation changes and the game becomes more delicate. Indeed, the problem is to control the interaction between the two possibly degenerate parts of the energy, that is a(x)\|Dw\| q and b(x)\|Dw\| s . A new aspect in fact emerges here. We see that, in presence of a finer structure, conditions of the type in (1.5) can be in a sense relaxed. In fact, an immediate application of (1.5) would provide us with the conditions a, b ∈ C 0,α (Ω) with s/p ≤ 1 + α/n, by considering the global regularity of x → F(x, ·). Instead, we see that the new condition coming into the play takes into account more precisely the way the presence of x affects the growth with respect to the gradient variable. Specifically, we shall assume that q p ≤ 1 + α n and s p ≤ 1 + β n . (1.9) 2 In other words, less regularity is needed on the coefficient affecting the q-growth, intermediate part of the energy density. Our main results is indeed the following main result of the paper (see the next section for more definitions and notation): Theorem 1 (C 1,ν -local regularity) Let u be a local minimizer of the functional (1.6) under assumptions (1.8) and (1.9). Then there exists ν = ν(data) ∈ (0, 1) such that u ∈ C 1,ν loc (Ω). We remark that the sharpness of both conditions in (1.9) can be obtained by the same counterexamples in [17,18]. Moreover, as it is well-known from the regularity theory for the standard p-Laplacean, the one in Theorem 1 is the maximal regularity obtainable for u. A worth singling-out intermediate result towards the proof of Theorem 1 is the following intrinsic Morrey decay estimate, which reduces to a classical estimate in the case of the p-Laplacean and that extends to the multi phase case the one proved in [4,11,12] for minima of functionals with a double phase. Theorem 2 (Intrinsic Morrey Decay) Let u be a local minimizer of the functional (1.6) under assumptions (1.8) and (1.9). Then, for every ϑ ∈ (0, n), there exists a positive constant c = c(data(Ω 0 ), ϑ) such that the decay estimate B ρ H(x, Du) dx ≤ c ρ r n−ϑ B r H(x, Du) dx (1.10) holds whenever B ρ ⊂ B r ⋐ Ω 0 are concentric balls with 0 < ρ ≤ r ≤ 1. Let us quickly describe the techniques we are employing to obtain the aforementioned theorems. The starting point is the recent proof of regularity of minimizers of double-phase variational problems appeared in [4], and based on a suitable use of harmonic type approximations lemmas (see also [12] for a first version). This is just a general blueprint we move from to treat the the real new difficulty here. Indeed, as we are dealing here with the presence of several phase transitions, and we have to carefully handle the regularity of solutions on the zero sets {a(x) = 0} and {b(x) = 0}, that is, when the functional tends to loose part of its ellipticity properties and switch their kind of ellipticity. Therefore we have to handle the presence of two different transitions. We come up with a delicate scheme of alternatives and of nested exit time arguments, carefully controlling the interaction between the two phase transitions. It is then clear that the techniques introduced in this paper allow to prove regularity results for functionals with an arbitrary large numbers of phases, for instance, w → Ω          \|Dw\| p + m i=1 a i (x)\|Du\| p i          dx , 1 <p<p 1 ≤ · · · ≤ p m and a i (·) ∈ C 0,α i (Ω). Notation and preliminaries In this section we establish some basic notation that we are going to use for the rest of the paper. As in the Introduction, Ω will denote an open subset of R n with n ≥ 2. As usual, we shall denote by c a general constant larger than one, which can vary from line to line. Relevant dependencies from certain parameters will be emphasized using brackets, i.e.: c = c(n, p, q, s) means that c depends on n, p, q, s. We denote with B r (x 0 ) = x ∈ R n : \|x − x 0 \| < r the n-dimensional open ball centered at x 0 and with radius r > 0; when non relevant or clear from the context, we will omit to indicate the centre as follows: B r = B r (x 0 ). When not differently specified, in the same context, balls with different radius will share the same center. If A ⊂ R n is any measurable subset with finite and positive Lebesgue's measure \|A\| > 0 and f : A→R N , N ≥ 1 is a measurable map, we shall denote its integral average over A as ( f ) A = − A f (x) dx = 1 \|A\| A f (x) dx. When A = B r , we shall write ( f ) r := ( f ) B r = − B r f (x) dx = 1 \|B r \| B r f (x) dx. The integrand H(·) has already been defined in (1.7). With abuse of notation we shall denote H(x, z) when z ∈ R n and when z ∈ R, that is when z is a scalar, so that we shall intend both H : Ω × R n → [0, ∞) and H : Ω × R → [0, ∞). The modulating coefficients a(·) and b(·) will always satisfy (1.8). Here we recall that, if f : Ω→R is any γ-Hölder continuous map with γ ∈ (0, 1) and A ⊂ Ω, then its Hölder seminorm is defined as [ f ] 0,γ;A = sup x,y∈A, x y \| f (x) − f (y)\| \|x − y\| γ , [ f ] 0,γ = [ f ] 0,γ;Ω . We are going to use several tools from the Orlicz space setting, therefore we start with the following preliminaries. Definition 1 A function ϕ : [0, ∞) → [0, ∞) is said to be a Young function if it satisfies the following conditions: ϕ(0) = 0 and there exists the derivative ϕ ′ , which is right-continuous, non decreasing and satisfies ϕ ′ (0) = 0, ϕ ′ (t) > 0 for t > 0, and lim t→∞ ϕ ′ (t) = ∞. Remark 1 In order to extrapolate good regularity properties for minimizers of functionals with ϕ-growth, we need to assume something more. Precisely, from now on, in addition to the basic assumptions listed in Definition 1 we will also suppose that ϕ ∈ C 1 [0, ∞) ∩ C 2 (0, ∞) and that i ϕ ≤ tϕ ′′ (t) ϕ ′ (t) ≤ s ϕ unifornly in t . (2.1) This is equivalent to the so-called ∆ 2 condition, since t → ϕ(t) is non decreasing, see [14], Section 2. Definition 2 Let ϕ be a Young function in the sense of Definition 1 and Remark 1. Given Ω ⊂ R n , the Orlicz space L ϕ (Ω) is defined as L ϕ (Ω) = u : Ω → R such that Ω ϕ(\|u\|) dx < ∞ and, consequently, W 1,ϕ (Ω) = u ∈ W 1,1 (Ω) ∩ L ϕ (Ω) such that Du ∈ L ϕ (Ω, R N ) . The definitions of the variants W In connection to H(·), we also consider the following Orlicz-Musielak-Sobolev space (Ω) = W 1,H(·) (Ω) ∩ W 1,p 0 (Ω); we refer to [4,21,22] for more on such spaces. W 1,H(·) (Ω) = u ∈ W 1,1 (Ω) : H(·, Du) ∈ L 1 (Ω) ,(2. For later uses, we introduce also the auxiliary Young functions                      H 0 (z) = \|z\| p + a 0 \|z\| q + b 0 \|z\| s , H s 0 (z) = \|z\| p + b 0 \|z\| s , H q 0 (z) = \|z\| p + a 0 \|z\| q , H p 0 (z) = \|z\| p . (2.3) The values of the constants a 0 , b 0 ≥ 0 will vary according to the necessities, but all the estimates we eventually get are independent on their value. In the following we will often use the vector field V t (z) = \|z\| (t−2)/2 z, t ∈ {p, q, s}. (2.4) We recall from [14], important features of (2.4): there exists c = c(n, t) > 0 such that \|V t (z 1 ) − V t (z 2 )\| 2 ≤ c \|z 1 \| t−2 z 1 − \|z 2 \| t−2 z 2 · (z 1 − z 2 ) , (2.5) \|V t (z 1 ) − V t (z 2 )\| ∼ (\|z 1 \| + \|z 2 \|) t−2 2 \|z 1 − z 2 \|, (2.6) where the constants implicit in (2.6) depend only on n, t and, for all z ∈ R n \|V t (z)\| 2 = \|z\| t . (2.7) For later uses, we introduce the following auxiliary functions                      V 0 (z 1 , z 2 ) 2 = \|V p (z 1 ) − V p (z 2 )\| 2 + a 0 \|V q (z 1 ) − V q (z 2 )\| 2 + b 0 \|V s (z 1 ) − V s (z 2 )\| 2 , V s 0 (z 1 , z 2 ) 2 = \|V p (z 1 ) − V p (z 2 )\| 2 + b 0 \|V s (z 1 ) − V s (z 2 )\| 2 , V q 0 (z 1 , z 2 ) 2 = \|V p (z 1 ) − V p (z 2 )\| 2 + a 0 \|V q (z 1 ) − V q (z 2 )\| 2 , V p 0 (z 1 , z 2 ) 2 = \|V p (z 1 ) − V p (z 2 )\| 2 . (2.8) Let us also recall some important tools in regularity. The first one is an iteration lemma from [19]. Lemma 1 Let h : [ρ, R 0 ] → R be a non-negative bounded function and 0 < θ < 1, 0 ≤ A, 0 < β. Assume that h(r) ≤ A(d − r) −β + θh(d) for ρ ≤ r < d ≤ R 0 . Then h(ρ) ≤ cA/(R 0 − ρ) −β holds, where c = c(θ, β) > 0. Along the proof we shall make an intensive use of the regularity properties of ϕ-harmonic maps, so we recall definition and some reference estimates from Lemma 5.8 and Theorem 6.4 in [14]. u 0 + W 1,ϕ 0 (U, R N ) ∋ w → min U ϕ(\|Dw\|) dx. Proposition 1 Let Ω ⊂ R n be open and ϕ ∈ C 2 (0, ∞) ∩ C 1 [0, ∞) be a Young function satisfying (2.1). If h ∈ W 1,ϕ (Ω, R N ) is ϕ-harmonic on Ω, then for any ball B r with B 2r ⋐ Ω there holds sup B r ϕ(\|Dh\|) ≤ c − B 2r ϕ(\|Dh\|) dx, where c depends only on n, N, i ϕ , s ϕ . We conclude this section by giving the definition of a local minimizer of (1.6). u − v)) ≤ H(v, supp(u − v)) is satisfied whenever v ∈ W 1,1 loc (Ω) and supp(u − v) ⊂ Ω. First regularity results In this section we collect a few basic regularity results which can be proved with minor adjustments to the proofs contained in [4,11,12,32]. Lemma 2 (Sobolev-Poincaré inequality) Let 1 < p<q<s and α, β ∈ (0, 1] verifying (1.8), (1.9). Then there exist a constant c = c(n, p, q, s) and an exponent d = d(n, p, q, s) ∈ (0, 1) such that for any w ∈ W 1,H(·) (B r ) with r ≤ 1, − B r H x, w − (w) r r dx ≤ c 1 + [a] 0,α Dw q−p L p (B r ) + [b] 0,β Dw s−p L p (B r )       − B r H(x, Dw) d dx       1 d . (3.1) Furthermore, the same is still true with w − (w) r replaced by w if we consider w ∈ W 1,H(·) 0 (B r ). Proof. We first consider the case sup x∈B r a(x) ≤ 4[a] 0,α r α and sup x∈B r b(x) ≤ 4[b] 0,β r β . (3.2) Then it follows from the classical Sobolev-Poincaré inequality that − B r a(x) \|w − (w) r \| q r q dx ≤ 4[a] 0,α r α − B r \|w − (w) r \| q r q dx ≤ c[a] 0,α r α       − B r \|Dw\| q * dx       q q * , with c = c(n, q) and − B r b(x) \|w − (w) r \| s r s dx ≤ 4[b] 0,β r β − B r \|w − (w) r \| s r s dx ≤ c[b] 0,β r β       − B r \|Dw\| s * dx       s s * , c = c(n, s), where q * := max nq n + q , 1 , s * := max ns n + s , 1 . We see from the assumption (1.9) that q * < p and s * < p. Therefore, we obtain from Hölder's inequality, (1.9) and the fact r ≤ 1 that − B r a(x) \|w − (w) r \| q r q dx ≤ c[a] 0,α r α       − B r \|Dw\| p dx       q−p p       − B r \|Dw\| q * dx       p q * ≤ c[a] 0,α r α− n(q−p) p Dw q−p L p (B r )       − B r \|Dw\| q * dx       p q * ≤ c[a] 0,α Dw q−p L p (B r )       − B r \|Dw\| q * dx       p q * , (3.3) c = c(n, q) and − B r b(x) \|w − (w) r \| s r s dx ≤ c[b] 0,β r β       − B r \|Dw\| p dx       s−p p       − B r \|Dw\| s * dx       p s * ≤ c[b] 0,β r β− n(s−p) p Dw s−p L p (B r )       − B r \|Dw\| s * dx       p s * ≤ c[b] 0,β Dw s−p L p (B r )       − B r \|Dw\| s * dx       p s * ,(3.4) with c = c(n, s). In addition, it is clear that − B r \|w − (w) r \| p r p dx ≤ c       − B r \|Dw\| p * dx       p p * , where c = c(n, p) and p * := max np n+p , 1 . We remark from (1.6) that p * <q * <s * . Combining these estimates, we get − B r H x, w − (w) r r dx ≤ c 1 + [a] 0,α Dw q−p L p (B r ) + [b] 0,β Dw s−p L p (B r )       − B r \|Dw\| s * dx       p s * ≤ c 1 + [a] 0,α Dw q−p L p (B r ) + [b] 0,β Dw s−p L p (B r )       − B r H(x, Dw) d 0 dx       1 d 0 , (3.5) where d 0 := s * /p ∈ (0, 1) and c = c(n, p, q, s). We now turn to the case sup x∈B r a(x) > 4[a] 0,α r α and sup x∈B r b(x) ≤ 4[b] 0,β r β . (3.6) Then there exists a point y 0 ∈ B r such that a 0 := a(y 0 ) > 4[a] 0,α r α . This gives \|a(x) − a 0 \| ≤ [a] 0,α (2r) α ≤ 2[a] 0,α r α < a 0 2 , ∀x ∈ B r , and hence a(x) ≤ 2a 0 and a 0 ≤ 2a(x). Therefore, we have 1 2 \|t\| p + a 0 \|t\| q ≤ \|t\| p + a(x)\|t\| q ≤ \|t\| p + a 0 \|t\| q , ∀x ∈ B r , t ∈ R. This and (3.4) yield − B r H x, w − (w) r r dx ≤ 2 − B r H q 0 w − (w) r r dx + c[b] 0,β Dw s−p L p (B r )       − B r \|Dw\| s * dx       p s * , with c = c(n, s). Using Sobolev-Poincaré inequality for Young function H q 0 , we have − B r H x, w − (w) r r dx ≤ c       − B r H q 0 (Dw) d q dx       1 dq + c[b] 0,β Dw s−p L p (B r )       − B r \|Dw\| s * dx       p s * ≤ c       − B r H(x, Dw) d q dx       1 dq + c[b] 0,β Dw s−p L p (B r )       − B r H(x, Dw) s * p dx       p s * ≤ c 1 + [b] 0,β Dw s−p L p (B r )       − B r H(x, Dw) d 1 dx       1 d 1 , (3.7) where c = c(n, p, q, s), d q = d q (n, p, q) ∈ (0, 1) and d 1 := max d q , s * p ∈ (0, 1). As in the case (3.6), we can obtain the estimate − B r H x, w − (w) r r dx ≤ c 1 + [a] 0,α Dw q−p L p (B r )       − B r H(x, Dw) d 2 dx       1 d 2 ,(3.8) for c = c(n, p, q, s) and some d 2 = d 2 (n, p, q, s) ∈ (0, 1), for the case sup x∈B r a(x) ≤ 4[a] 0,α r α and sup x∈B r b(x) > 4[b] 0,β r β . (3.9) Finally, let us consider the case sup x∈B r a(x) > 4[a] 0,α r α and sup x∈B r b(x) > 4[b] 0,β r β . (3.10) We see that there exist points y 0 , z 0 ∈ B r such that a 0 := a(y 0 ) > 4[a] 0,α r α and b 0 : = b(z 0 ) > 4[b] 0,β r β . This yields \|a(x) − a 0 \| ≤ [a] 0,α (2r) α ≤ 2[a] 0,α r α < a 0 2 , ∀x ∈ B r , and \|b(x) − b 0 \| ≤ [b] 0,β (2r) β ≤ 2[b] 0,β r β < b 0 2 , ∀x ∈ B r . It follows that 1 2 a 0 ≤ a(x) ≤ 2a 0 and 1 2 b 0 ≤ b(x) ≤ 2b 0 , and hence 1 2 H 0 (t) ≤ H(x, t) ≤ 2H 0 (t), ∀x ∈ B r , t ∈ R. We now use Sobolev-Poincaré inequality for Young function H 0 to obtain − B r H x, w − (w) r r dx ≤ 2 − B r H 0 w − (w) r r dx ≤ c       − B r H 0 (Dw) d 3 dx       1 d 3 ≤ c       − B r H(x, Dw) d 3 dx       1 d 3 (3.11) for c = c(n, p, q, s) and some d 3 = d 3 (n, p, q, s) ∈ (0, 1). Setting d := max{d 0 , d 1 , d 2 , d 3 } ∈ (0, 1), we conclude from (3.5), (3.7), (3.8) and (3.11) that − B r H x, w − (w) r r dx ≤ c 1 + [a] 0,α Dw q−p L p (B r ) + [b] 0,β Dw s−p L p (B r )       − B r H(x, Dw) d dx       1 d , which completes the proof. Remark 2 An inequality of the type of (3.1) holds for general Sobolev maps w ∈ W 1,H(·) such that w ≡ 0 on a set A such that \|A\| ≥ γ\|B r \|. Precisely, we have that − B r H x, w r dx ≤ c       − B r H(x, Dw) d dx       1 d , (3.12) where d < 1 is the same as the one appearing in (3.1) and c = c(γ, n, p, q, s, [a] 0,α , [b] 0,β , α, β, Dw L p (B r ) ). Lemma 3 (Caccioppoli Inequalities) Let u ∈ W 1,H(·) loc (Ω) be a local minimizer of (1.6), with a(·), b(·) and p, q, s satisfy (1.8) and (1.9) respectively. Then there exists a constant c = c(n, p, q, s) > 0 such that − B ρ H(x, Du) dx ≤ c − B r H x, u − (u) r r − ρ dx,(3. 13) and for κ ∈ R, B ρ H(x, D(u − κ) ± ) dx ≤ c B R H x, (u − κ) ± R − ρ dx. (3.14) A direct consequence of (3.13) is the following inner local higher integrability result of Gehring type. Lemma 4 (Gehring's Lemma) There are c = c(n, p, q, s, [a] 0,α , [b] 0,β , Du L p (B r ) ) > 0 and a positive inte- grability exponent δ g = δ g (n, p, q, s, [a] 0,α , [b] 0,β , Du L p (B r ) ) such that if u ∈ W 1,p loc (Ω) is a local minimizer, then H(·, Du) ∈ L 1+δ g loc (Ω) and        − B r/2 H(x, Du) 1+δ g dx        1 1+δg ≤ c − B r H(x, Du) dx, ∀B r ⊂ Ω. (3.15) After a standard covering argument, it follows from Lemma 4 that u ∈ W 1,p(1+δ g ) loc (Ω), so u ∈ W 1,p(1+δ g ) (Ω 0 ) for Ω 0 ⋐ Ω. Moreover, by Hölder inequality, (3.15) is true if δ g is replaced by any σ ∈ (0, δ g ). The next one is an up to the boundary higher integrability result for a solution of Dirichlet problems related to the multi-phase energy H. Clearly, when a(·) ≡ a 0 = const and b(·) ≡ b 0 = const, it extends to the auxiliary Young functions H Lemma 5 (Higher integrability up to the boundary) Let B r ⋐ Ω 0 ⋐ Ω, 1 < p ≤ q ≤ s and v ∈ W 1,H(·) u (B r ) be a solution to the Dirichlet problem v → min w∈W 1,H(·) u (B r ) B r H(x, Dw) dx,(3.16) and δ 0 > 0 be such that u ∈ W 1,H(·) 1+δ 0 (B r ). Then there exists 0 < σ g < δ 0 , so that v ∈ W 1,H(·) 1+σg (B r ) and Proof. With x 0 ∈ B r , let us fix a ball B ρ (x 0 ) ⊂ R n . We start with the case in which it is \|B ρ (x 0 ) \ B r \| > \|B ρ (x 0 )\| 10 . Let us fix ρ/2 < t < s < ρ and take a cut-off function η ∈ C 1 c (B s (x 0 )) such that χ B t (x 0 ) ≤ η ≤ χ B s (x 0 ) and \|Dη\| ≤ 2/(s − t). Since v\| ∂B r = u\| ∂B r and η ∂B s (x 0 ) = 0, the function v − η(v − u) coincides with v on ∂B r and on ∂B s (x 0 ) in the sense of traces and therefore, by the minimality of v and the features of η we obtain − B r H(x, Dv) 1+σ g dx ≤ c                − B r H(x, Dv) dx       1+σ g + − B r H(x, Du) 1+σ g dx          ,(3.B s (x 0 )∩B r H(x, Dv) dx ≤ c        (B s (x 0 )\B t (x 0 ))∩B r H(x, Dv) dx + B s (x 0 )∩B r H(x, Du) + H x, v − u r dx        , with c = c(n, p, q, s). By the classical hole-filling technique and Lemma 1, we can conclude that B ρ/2 (x 0 )∩B r H(x, Dv) dx ≤ c B ρ (x 0 )∩B r H(x, Du) + H x, v − u r dx,(3.18) for c = c(n, p, q, s). Now extend v − u as zero outside B r and recall that \|B ρ ( x 0 )\| ≥ \|B ρ (x 0 ) \ B r \| > \|B ρ (x 0 )\| 10 . Poincaré's inequality (3.12) applies, thus getting − B ρ (x 0 )∩B r H x, v − u r dx ≤ c                   − B ρ ∩B r H(x, Dv) d dx        1 d + − B ρ (x 0 )∩B r H(x, Du) dx            ,(3.− B ρ/2 (x 0 )∩B r H(x, Dv) dx ≤ c                   − B ρ ∩B r H(x, Dv) d dx        1 d + − B ρ (x 0 )∩B r H(x, Du) dx            . We next consider the situation when it is B ρ (x 0 ) ⋐ B r , in which case the proof is analogous to the one for the interior case. As mentioned in Remark 2, we can assume that the exponent d < 1 from (3.1) and (3.12) is the same. The two cases can be combined via a standard covering argument. In fact, let us define V(x) =          H(x, Dv(x)) d in B r 0 in R n \ B r and U(x) =          H(x, Du(x)) in B r 0 in R n \ B r , we get − B ρ/2 (x 0 ) V(x) 1 d dx ≤ c                   − B ρ (x 0 ) V(x) dx        1 d + − B ρ (x 0 ) U(x) dx            , with c = c = c(n, p, q, s, [a] 0,α , [b] 0,β , α, β, H(·, Du) L 1 (B r ) ) and 0 < d < 1. At this point the conclusion follows by a standard variant of Gehring's lemma. Furthermore, u is locally bounded. Lemma 6 Let u ∈ W 1,H M (·) loc (Ω) be a local minimizer of (1.6). Then u is locally bounded in Ω and for any Ω 0 ⋐ Ω there is a positive constant c = c(data(Ω 0 )) such that u L ∞ (Ω 0 ) ≤ c. Proof. This can be obtained as in [11], Section 10 as a consequence of (3.14) or by noticing that the generalized Young function H(x, t) = t p + a(x)t q + b(x)t s under the assumptions (1.8) and (1.9) satisfies hypotheses (A0), (A1), (AInc) and (ADec) of Theorem 1.3 in [21]. In fact, with the notation used in [21], it is easy to see that H + (δ) ≤ 1 ≤ H − (1) for δ = 1 3 1 + max a ∞ , b ∞ −1 1 p ∈ (0, 1). (A1) is true by choosing γ = 1 2 min            1 3 1 p ,         ω 1 p n 3[a] 0,α diam(Ω) α−n q−p p         1 q−p ,         ω 1 p n 3[b] 0,β diam(Ω) β−n s−p p         1 s−p            ∈ (0, 1), Different alternatives For later uses, we also define the quantities a i (B r ) = inf x∈B r a(x) and b i (B r ) = inf x∈B r b(x),(4.1) which will play an important role along the proof. In fact, when dealing with those so called non uniformly elliptic problems, the question of the degeneracy of the coefficients is crucial. Precisely we will look at four different scenarios:                      deg(B r ) : a i (B r ) ≤ 4[a] 0,α r α−γ a and b i (B r ) ≤ 4[b] 0,β r β−γ β deg α (B r ) : a i (B r ) ≤ 4[a] 0,α r α−γ a and b i (B r ) > 4[b] 0,β r β−γ b deg β (B r ) : a i (B r ) > 4[a] 0,α r α−γ a and b i (B r ) ≤ 4[b] 0,β r β−γ b ndeg(B r ) : a i (B r ) ≥ 4[a] 0,α r α−γ a and b i (B r ) > 4[b] 0,β r β−γ b , where γ a =          0 if n ≥ p(1 + δ g ) α − n(q−p) p + nδ g (q−p) 2p(1+δ g ) if n < p(1 + δ g ) (4.2) and γ b =          0 if n ≥ p(1 + δ g ) β − n(s−p) p + nδ g (s−p) 2p(1+δ g ) if n < p(1 + δ g ) ,(4.3) where δ g is the higher integrability exponent given by Gehring Lemma which can be found in Section 3. The above four cases, suitably combined, will render the desired regularity. To shorten the notation, we shall summarize the dependencies from the characteristics of the integrand we are dealing with, as Here, λ g = 1 − n p(1+δ g ) is the Hölder continuity exponent coming from Sobolev-Morrey's embedding theorem when n < p(1 + δ g ) and Ω 0 ⋐ Ω is any open set compactly contained in Ω. This will be helpful, since all the existing results we are going to use are of local nature. data(Ω 0 ) ≡          n, p, q, s, [a] 0,α , [b] 0,β , α, β, u L ∞ (Ω 0 ) , H(·, Du) L 1+δg (Ω 0 ) if n ≥ p(1 + δ g ) Exploiting the different phases (deg)-(ndeg) we obtain various forms of the previous Caccioppoli's inequality. We collect them in the next Corollary. Moreover, the constants a 0 and b 0 appearing in the definition of the auxiliary Young functions H p 0 , H q 0 , H s 0 and H 0 will take the values a 0 = a i (B 2r ) and b 0 = b i (B 2r ). Corollary 3 Let u ∈ W 1,H(·) loc (Ω) be a local minimizer of (1.6) and B r , r ∈ (0, 1) be any ball such that B 2r ⋐ Ω 0 ⋐ Ω. Then the following is verified: Proof. First, notice that, by (1.9), γ a ≥ 0 and γ b ≥ 0. Moreover, if n ≥ p(1 + δ g ) we see that deg(B 2r ) ⇒ − B r H(x, Du) dx ≤ c 1 − B 2r H p 0 u − (u) 2r 2r dx, (4.4) deg α (B 2r ) ⇒ − B r H(x, Du) dx ≤ c 2 − B 2r H s 0 u − (u) 2r 2r dx, (4.5) deg β (B 2r ) ⇒ − B r H(x, Du) dx ≤ c 3 − B 2r H q 0 u − (u) 2r 2r dx, (4.6) ndeg(B 2r ) ⇒ − B r H(x, Du) dx ≤ c 4 − B 2r H 0 u − (u) 2r 2r dx, .α − γ a + p − q ≥ n(q − p) p − (q − p) ≥ δ g (q − p) > 0, (4.8) β − γ b + p − s ≥ n(s − p) p − (s − p) ≥ δ g (s − p) > 0, (4.9) while, if n < p(1 + δ g ), α − γ a + (λ g − 1)(q − p) = nδ g (q − p) 2p(1 + δ g ) > 0, (4.10) β − γ b + (λ g − 1)(s − p) = nδ g (s − p) 2p(1 + δ g ) > 0. (4.11) Assume deg(B 2r ). We observe that for any x ∈ B 2r , a(x) = a(x) − a i (B 2r ) + a i (B 2r ) ≤ [a] 0,α (4r) α + 4[a] 0,α r α−γ a ≤ 8[a] 0,α r α−γ a since γ a ≥ 0 and r ∈ (0, 1). Similarly we have b(x) ≤ 8[b] 0,β r β−γ b , ∀x ∈ B 2r . If n ≥ p(1 + δ g ), from (3.13), Lemma 6, (4.8) and (4.9) we get, − B r H(x, Du) dx ≤c − B 2r H x, u − (u) 2r r dx ≤c − B 2r 1 + 8[a] 0,α r α−γ a +p−q u q−p L ∞ (Ω 0 ) + 8[b] 0,β r β−γ b +p−s u s−p L ∞ (Ω 0 ) u − (u) 2r 2r p dx ≤c 1 − B 2r H p 0 u − (u) 2r 2r dx, where c 1 = c 1 (n, p, q, s, [a] 0,α , [b] 0,β , α, β, u L ∞ (Ω 0 ) ). On the other hand, if n < p(1 + δ g ) proceding as before but using Sobolev-Morrey's theorem and (4.10), (4.11) instead of (4.8), (4.9), we obtain − B r H(x, Du) dx ≤c − B 2r H x, u − (u) 2r r dx ≤c − B 2r 1 + 8[a] 0,α r α−γ a +(λ g −1)(q−p) [u] q−p C 0,λg (Ω 0 ) + 8[b] 0,β r β−γ b +(λ g −1)(s−p) [u] s−p C 0,λg (Ω 0 ) u − (u) 2r 2r p dx ≤c 1 − B 2r H p 0 u − (u) 2r 2r dx, where c 1 = c 1 (n, p, q, s, [a] 0,α , [b] 0,β , α, β, [u] C 0,λg (Ω 0 ) ). Now suppose deg α (B 2r ). If n ≥ p(1 + δ g ) , we see from (3.13), (4.8), (4.9) and Lemma 6 that − B r H(x, Du) dx ≤c − B 2r H x, u − (u) 2r r dx ≤c − B 2r 1 + 8[a] 0,α r α−γ a +p−q u q−p L ∞ (Ω 0 ) u − (u) 2r 2r p dx + c − B 2r b(x) − b i (B 2r ) u − (u) 2r 2r s + b i (B 2r ) u − (u) 2r 2r s dx ≤c − B 2r u − (u) 2r 2r p + [b] 0,β (4r) β u − (u) 2r 2r s + b i (B 2r ) u − (u) 2r 2r s dx ≤c − B 2r u − (u) 2r 2r p + 2b i (B 2r ) u − (u) 2r 2r s dx ≤ c 2 − B 2r H s 0 u − (u) 2r 2r dx, since, being r ∈ (0, 1), r β ≤ r β−γ b . Here, c 2 = c 2 (n, p, q, s, [a] 0,α , α, u L ∞ (Ω 0 ) ). If n < p(1 + δ g ) we have, by exploiting (4.10) and (4.11), − B r H(x, Du) dx ≤c − B 2r H x, u − (u) 2r r dx ≤c − B 2r 1 + 8[a] 0,α r α−γ a +(λ g −1)(q−p) [u] q−p C 0,λg (Ω 0 ) u − (u) 2r 2r p dx + c − B 2r b(x) − b i (B 2r ) u − (u) 2r 2r s + b i (B 2r ) u − (u) 2r 2r s dx ≤c − B 2r u − (u) 2r 2r p + [b] 0,β (4r) β u − (u) 2r 2r s + b i (B 2r ) u − (u) 2r 2r s dx 13 ≤c − B 2r u − (u) 2r 2r p + 2b i (B 2r ) u − (u) 2r 2r s dx ≤ c 2 − B 2r H s 0 u − (u) 2r 2r dx, with c 2 = c 2 (n, p, q, s, [a] 0,α , α, [u] C 0,λg (Ω 0 ) ). If deg β (B 2r ) is in force, then, as before, for n ≥ p(1 + δ g ), we have − B r H(x, Du) dx ≤c − B 2r H x, u − (u) 2r r dx ≤c − B 2r 1 + 8[b] 0,β r β−γ b +p−s u s−p L ∞ (Ω 0 ) u − (u) 2r 2r p dx + c − B 2r a(x) − a i (B 2r ) u − (u) 2r 2r q + a i (B 2r ) u − (u) 2r 2r q dx ≤c − B 2r u − (u) 2r 2r p + 2a i (B 2r ) u − (u) 2r 2r q dx ≤ c 3 − B 2r H q 0 u − (u) 2r 2r dx, where c 3 = c 3 (n, p, q, s, [b] 0,β , β, u L ∞ (Ω 0 ) ). Moreover, if n < p(1 + δ g ) we obtain − B r H(x, Du) dx ≤c − B 2r H x, u − (u) 2r r dx ≤c − B 2r 1 + 8[b] 0,β r β−γ b +(λ g −1)(s−p) [u] s−p C 0,λg (Ω 0 ) u − (u) 2r 2r p dx + c − B 2r a(x) − a i (B 2r ) u − (u) 2r 2r q + a i (B 2r ) u − (u) 2r 2r q dx ≤c − B 2r u − (u) 2r 2r p + 2a i (B 2r ) u − (u) 2r 2r q dx ≤ c 3 − B 2r H q 0 u − (u) 2r 2r dx,H u − (u) 2r r dx ≤c − B 2r u − (u) 2r 2r p + a(x) − a i (B r ) u − (u) 2r 2r (q−p)+p + (b(x) − b i (B r )) u − (u) 2r 2r (s−p)+p + a i (B r ) u − (u) 2r 2r q + b i (B r ) u − (u) 2r 2r s dx ≤c − B 2r u − (u) 2r 2r p + [a] 0,α (4r) α u − (u) 2r 2r q + [b] 0,β (4r) β u − (u) 2r 2r s dx + c − B 2r H 0 u − (u) 2r 2r dx ≤ c 4 − B 2r H 0 u − (u) 2r 2r dx, with c 4 = c 4 (n, p, q, s, [a] 0,α , [b] 0,β , α, β). We conclude this section by recalling a quantitative Harmonic-approximation type result from [4]. We shall report it in the form that better fits our necessities. Lemma 7 Let B r ⊂ R n be a ball, ε ∈ (0, 1),H be one of the Young functions defined in (2.3) and v ∈ W 1,H (B 2r ) be a map satisfying the following estimates: − B 2rH (Dv) dx ≤c 1 , (4.12) and − B rH (Dv) 1+σ 0 dx ≤c 2 , (4.13) wherec 1 ,c 2 ≥ 1 and σ 0 > 0 are fixed constants. Moreover, assume that − B r DH(Dv) · Dϕ dx ≤ ε Dϕ L ∞ (B r ) for all ϕ ∈ C ∞ c (B r ), (4.14) for some ε ∈ (0, 1). Then there exists a functionh ∈ W 1,H v (B r ) such that the following conditions are satisfied: By minimality (4.15) is verified, since it is the Euler-Lagrange equation associated to the above variational problem. Moreover, it follows from (4.12) that − B r DH(Dh) · Dϕ dx = 0 for all ϕ ∈ C ∞ c (B r ), (4.15) − B rH (Dh) 1+σ 1 dx ≤ c(n, p, q, s, σ 0 )c 2 , (4.16) − B r V(Dv, Dh) 2 dx ≤ cε m ,(4.− B r H 0 (Dh 0 ) dx ≤ − B r H 0 (Dv) dx ≤ 2 nc 1 . (4.18) Now, by the previous inequality, Lemma 5 with a(·) ≡ const and b(·) ≡ const, and (4.13), we obtain − B r H 0 (Dh 0 ) 1+σ g dx ≤c                − B r H 0 (Dh 0 ) dx       1+σ g + − B r H 0 (Dv) 1+σ g dx          ≤ c (2 nc 1 ) 1+σ g +c 2 =:c 3 (4.19) for some 0 < σ g < σ 0 , which is (4.16) with σ 1 = σ g . Herec 3 =c 3 (n, p, q, s, σ g ,c 1 ,c 2 ). Set w = h 0 − v ∈ W 1,H 0 0 (B r ) and let λ ≥ 1 to be fixed later and consider w λ ∈ W 1,∞ 0 (B r ), the Lipschitz truncation of w given by the main result in [1] and satisfying (4.20) Using such properties, the fact that t → H 0 (t) is increasing, Markov's inequality, (4.13), (4.19) and the maximal theorem we deduce that \|{w λ w}\| \|B r \| ≤ \|B r ∩ {M(\|Dw\|) ≥ λ}\| \|B r \| ≤ 1 H 0 (λ) 1+σ g − B r H 0 (M(\|Dw\|)) 1+σ g dx ≤ c H 0 (λ) 1+σ g − B r H 0 (Dw) 1+σ g dx ≤ c H 0 (λ) 1+σ g − B r H 0 (Dh 0 ) 1+σ g + H 0 (Dv) 1+σ g dx ≤ c H 0 (λ) 1+σ g            − B r H 0 (Dh 0 ) 1+σ g dx +       − B r H 0 (Dv) 1+σ 0 dx       1+σg 1+σ 0            ≤ c(c 3 +c 1+σg 1+σ 0 2 ) H 0 (λ) 1+σ g ≤ c(c 3 +c 2 ) H 0 (λ) 1+σ g ,(4.21) where c = c(n, p, q, s, σ g , σ 0 ). Now we test (4.15) against w λ , which is admissible by density, to get (I) = − B r (DH 0 (Dh 0 ) − DH 0 (Dv)) · Dw λ χ {w λ =w} dx = − − B r DH 0 (Dv) · Dw λ dx − − B r (DH 0 (Dh 0 ) − DH 0 (Dv)) · Dw λ χ {w w λ } dx = (II) + (III). The properties of H 0 and (2.5) give (I) ≥ c − B r V 0 (Dv, Dh 0 )χ {w λ =w} dx,(4.22) where c = c(n, p, q, s) > 0. Moreover, by (4.14) and (4.20) 1 we see that \|(II)\| ≤ cελ,(4.23) with c = c(n). Before estimating term (III), we recall a standard Young type inequality holding for H 0 , see [4]: for all σ ∈ (0, 1), xy ≤ σ 1−s H 0 (x) + σH * 0 (y),(4.24) where H * 0 (y) = sup x>0 {yx − H 0 (x)} is the convex conjugate of H 0 . Furthermore, there holds: H * 0 H 0 (t) t ≤ H 0 (t), see [5] for more details. Now, using (4.20) 1 , (4.12), (4.18), (4.24) and (4.21) we estimate, for a certain fixed σ ∈ (0, 1), \|(III)\| ≤s Dw λ L ∞ (B r ) − B r H 0 (Dh 0 ) \|Dh 0 \| + H 0 (Dv) \|Dv\| χ {w λ w} dx ≤σ − B r       H * 0 H 0 (Dh 0 ) \|Dh 0 \| + H * 0 H 0 (Dv) \|Dv\|       dx + cH 0 ( Dw λ L ∞ (B r ) ) σ s−1 \|{w λ w}\| \|B r \| ≤σ − B r H 0 (Dh 0 ) + H 0 (Dv) dx + c σ s−1 H 0 (λ) σ g ≤ 2 n+1 σc 1 + c σ s−1 λ pσ g ,(4.25) where we also used the fact that H 0 (λ) ≥ λ p since λ ≥ 1. Here c = c(n, p, q, s, σ g ). Collecting (4.22), (4.23) and (4.25) we obtain − B r V 0 (Dv, Dh 0 ) 2 χ {w λ =w} dx ≤c ελ + σ + σ 1−s λ −pσ g ,(4.26) for c = c(c 1 ,c 2 ,c 3 , n, p, q, s, σ g ) and σ ∈ (0, 1) to be fixed. For θ ∈ (0, 1), by Hölder's inequality and (4.26) we estimate       − B r V(Dv, Dh 0 ) 2θ χ {w λ =w} dx       1 θ ≤ c ελ + σ + σ 1−s λ −pσ g . Again, by Hölder's inequality, (4.21), (4.12) and (4.18) we have       − B r V(Dv, Dh 0 ) 2θ χ {w λ w} dx       1 θ ≤c \|{w λ w}\| \|B r \| 1−θ θ − B r V(Dv, D 0 ) 2 dx ≤cH 0 (λ) − (1+σg)(1−θ) θ − B r H 0 (Dh 0 ) + H 0 (Dv) dx ≤ cλ − p(1−θ) θ ,(4.27) where c = c(c 1 ,c 2 ,c 3 , n, p, q, s, σ g ). Choosing in (4.26) and (4.27) λ = ε −1/2 and σ = ε 3pσg 4(s−1) we obtain that       − B r V(Dv, Dh 0 ) 2θ dx       1 θ ≤ cε 2m ,(4.28) with c = c(c 1 ,c 2 ,c 3 , n, p, q, s, σ g ) and m = 1 2 min 1 2 , pσ g 4 , 3pσ g 4(s−1) , p(1−θ) 2θ . Notice that in the above estimates we still have a degree of freedom in θ. Applying Hölder's inequality with exponents 2(1+σ g ) 1+2σ g and 2(1 + σ g ) we obtain − B r V(Dv, Dh 0 ) 2 dx ≤       − B r V(Dv, Dh 0 ) 2(1+σg) 1+2σg dx       1+2σg 2(1+σg)       − B r V(Dv, Dh 0 ) 2(1+σ g ) dx       1 1+σg ≤cε m       − B r (H 0 (Dh 0 ) + H 0 (Dv)) 1+σ g dx       1 1+σg ≤ cε m , with c = c(c 1 ,c 2 ,c 3 , n, p, q, s, σ g ). Here we used (4.13), (4.19), and (4.28) with θ = 1+σ g 1+2σ g < 1. Recalling (2.7), we can conclude from the previous estimate that − B r V(Dv, Dh 0 ) 2 dx ≤ cε m , which is what we wanted. Morrey decay and Theorem 2 The proof of Theorem 2 goes in two moments: first, we prove that a suitable manipulation of a local minimizer u of (1.6) satisfies the assumptions of Lemma 7, then we exploit this to start an iteration which will eventually render the announced decay. Step 1: Quantitative harmonic approximation. Define the quantities E = E(u, B 2r ) =       − B 2r H(x, Du) dx       1 p and v = u E , where u ∈ W 1,H(·) loc (Ω) is a local minimizer of (1.6) and B 2r ⋐ Ω 0 ⋐ Ω is any ball of radius r ≤ 1 2 . From now on, we will consider the following auxiliary Young functions                      H 0 (z) = \|z\| p + a i (B 2r )\|z\| q + b i (B 2r )\|z\| s ,H 0 (z) = \|z\| p + a i (B 2r )E q−p \|z\| q + b i (B 2r )E s−p \|z\| s , H s 0 (z) = \|z\| p + b i (B 2r )\|z\| s ,H s 0 (z) = \|z\| p + b i (B 2r )E s−p \|z\| s , H q 0 (z) = \|z\| p + a i (B 2r )\|z\| q ,H q 0 (z) = \|z\| p + a i (B 2r )E q−p \|z\| q , H p 0 (z) = \|z\| p , and                      V 0 (z 1 , z 2 ) 2 = \|V p (z 1 ) − V p (z 2 )\| 2 + a i (B 2r )\|V q (z 1 ) − V q (z 2 )\| 2 + b i (B 2r )\|V s (z 1 ) − V s (z 2 )\| 2 , V s 0 (z 1 , z 2 ) 2 = \|V p (z 1 ) − V p (z 2 )\| 2 + b i (B 2r )\|V s (z 1 ) − V s (z 2 )\| 2 , V q 0 (z 1 , z 2 ) 2 = \|V p (z 1 ) − V p (z 2 )\| 2 + a i (B 2r )\|V q (z 1 ) − V q (z 2 )\| 2 , V p 0 (z 1 , z 2 ) 2 = \|V p (z 1 ) − V p (z 2 )\| 2 , where a i (·) and b i (·) are defined as in (4.1). Since u is a local minimizer of (1.6), a straightforward computation shows that v is a local minimizer of the functional H(w, Ω) = Ω \|Dw\| p + a(x)E q−p \|Dw\| q + b(x)E s−p \|Dw\| s dx. Then, by scaling, it is easy to see that Lemma 4 holds true also for v with the same extra integrability exponent δ g = δ g (n, p, q, s, v that − B 2rH (Dv) dx ≤ E −p − B 2r H(x, Du) dx ≤ 1,(5.2) which is (4.12) and, by (5.2) and Lemma 4 we obtain, for someσ g ∈ (0, δ g ), − B rH (Dv) 1+σ g dx ≤ − B r H(x, Dv) 1+σ g dx =       − B 2r H(x, Du) dx       −(1+σ g )       − B r H(x, Du) 1+σ g dx       ≤ c, (5.3) where c = c(n, p, q, s, [a] 0,α , [b] 0,β , Du L p (Ω 0 ) ) is the constant appearing in Lemma 4 and this verifies (4.13). So we see that conditions (4.12)-(4.13) of Lemma 7 are matched with σ 0 =σ g no matter what degeneracy (or non degeneracy) condition holds on B 2r . Here σ g is the exponent given by Lemma 5 depending on whetherH denotesH p 0 ,H q 0 ,H s 0 orH 0 . Clearly we have no problem of integrability, sinceσ g < δ g , which is the corresponding exponent coming from Lemma 4. We now define σ a = α − γ a − n(q − p) p(1 + δ g ) and σ b = β − γ b − n(s − p) p(1 + δ g ) . (5.4) A simple computation shows that σ a and σ b are both positive numbers. We first assume deg(B 2r ). From (5.1) we deduce that − B r DH p 0 (Dv) · Dϕ dx ≤qE q−p − B r a(x)\|Dv\| q−1 \|Dϕ\| dx + sE s−p − B r b(x)\|Dv\| s−1 \|Dϕ\| dx=: (I) deg + (II) deg . From the very definition of condition deg, Lemma 4, (5.2), Hölder's inequality and (5.4) we get (I) deg ≤cE q−p q r α−γa q Dϕ L ∞ (B r ) − B r (E q−p a(x)) q−1 q \|Dv\| q−1 dx ≤c Dϕ L ∞ (B r )       − B 2r H(x, Du) dx       q−p pq r α−γa q       − B r E q−p−q a(x)\|Du\| q dx       q−1 q ≤c Dϕ L ∞ (B r ) H(·, Du) q−p pq L 1+δg (Ω 0 ) r α−γa q − n(q−p) pq(1+δg) ≤ c Dϕ L ∞ (B r ) r σa q (5.5) with c 1 = c 1 (n, p, q, [a] 0,α , α, H(·, Du) L 1+δg (Ω 0 ) ). In a totally similar way we obtain (II) deg ≤cE s−p s r β−γ b s Dϕ L ∞ (B r ) − B r (E s−p b(x)) s−1 s \|Dv\| s−1 dx ≤c Dϕ L ∞ (B r )       − B 2r H(x, Du) dx       s−p ps r β−γ b s       − B r E s−p−s b(x)\|Du\| s dx       s−1 s ≤c Dϕ L ∞ (B r ) H(·, Du) s−p ps L 1+δg (Ω 0 ) r β−γ b s − n(s−p) ps(1+δg) ≤ c Dϕ L ∞ (B r ) r σ b s (5.6) where c 2 = c 2 (n, p, s, [b] 0,β , β, H(·, Du) L 1+δg (Ω 0 ) ). Now we defineσ p := 1 2 min{q −1 σ a , s −1 σ b } > 0 and fix a threshold radiusR 1 * such that max{c 1 , c 2 }(R 1 * )σ p ≤ 1 2 and assume that 0 < r ≤ min{R 1 * , 1}. In correspondence of such a choice, by (5.5) and (5.6) we can conclude that − B r DH p 0 (Dv) · Dϕ dx ≤rσ p Dϕ L ∞ (B r ) ,(5.DH s 0 (Dv) · Dϕ dx ≤qE q−p − B r a(x)\|Dv\| q−1 \|Dϕ\| dx + sE s−p − B r b(x) − b i (B 2r ) \|Dv\| s−1 \|Dϕ\| dx ≤(I) deg α + (II) deg α . As before we estimate (I) deg α ≤c Dϕ L ∞ (B r )       − B 2r H(x, Du) dx       q−p pq r α−γa q       − B r E q−p−q a(x)\|Du\| q dx       q−1 q ≤c Dϕ L ∞ (B r ) H(·, Du) q−p pq L 1+δg (Ω 0 ) r α−γa q − n(q−p) pq(1+δg) ≤ c Dϕ L ∞ (B r ) r σa q (5.9) with c 1 = c 1 (n, p, q, [a] 0,α , α, H(·, Du) L 1+δg (Ω 0 ) ), and (II) deg α ≤c Dϕ L ∞ (B r ) E s−p s r β s + γ b (s−1) s       − B r (E s−p r β−γ b ) s−1 s \|Dv\| s−1 dx       ≤c Dϕ L ∞ (B r ) r γ b (s−1) s r 1 s β− n(s−p) p(1+δg)       − B 2r E −p b i (B 2r )\|Du\| s dx       s−1 s ≤ cr γ b (s−1) s + 1 s β− n(s−p) p(1+δg) Dϕ L ∞ (B r ) ,(5.10) where c 2 = c 2 (n, p, s, [b] 0,β , β, H(·, Du) L 1+δg (Ω 0 ) ). Defineσ s := 1 2 min σ a q , γ b (s−1) s + 1 s β − n(s−p) p(1+δ g ) > 0 and fix a threshold radiusR 2 * such that max{c 1 , c 2 }(R 2 * )σ s ≤ 1 2 and assume that 0 < r ≤ min{R 1 * ,R 2 * , 1}. In correspondence of such a choice, by (5.9) and (5.10) we can conclude that − B r DH s 0 (Dv) · Dϕ dx ≤rσ s Dϕ L ∞ (B r ) ,(5.DH q 0 (Dv) · Dϕ dx ≤sE s−p − B r b(x)\|Dv\| s−1 \|Dϕ\| dx + qE q−p − B r a(x) − a i (B 2r ) \|Dv\| q−1 \|Dϕ\| dx ≤(I) deg β + (II) deg β . As above we estimate where c 2 = c 2 (n, p, q, [a] 0,α , α, H(·, Du) L 1+δg (Ω 0 ) ). Letσ q := 1 2 min σ b s , γ a (q−1) q + 1 q α − n(q−p) p(1+δ g ) > 0 and fix a threshold radiusR 3 * such that max{c 1 , c 2 }(R 3 * )σ q ≤ 1 2 and assume that 0 < r ≤ min{R 1 * ,R 2 * ,R 3 * , 1}. In correspondence of such a choice, by (5.9) and (5.10) we can conclude that (I) deg β ≤c Dϕ L ∞ (B r ) E s−p s r β−γ b s       − B r E −p b(x)\|Du\| s            − B 2r E −p a i (B 2r )\|Du\| q dx       q−1 q ≤ cr γa (q−1) q + 1 q α− n(q−p) p(1+δg) Dϕ L ∞ (B r ) ,(5.18) where c 2 = c 2 (n, p, q, [a] 0,α , α, H(·, Du) L 1+δg (Ω 0 ) ). Letσ 0 := 1 2 min γ a (q−1) q + 1 q α − n(q−p) p(1+δ g ) , γ b (s−1) s + 1 s β − n(s−p) p(1+δ g ) > 0 and fix another threshold radiusR 4 * such that max{c 1 , c 2 }(R 4 * )σ 0 ≤ 1 2 and assume that 0 < r ≤ min{R 1 * ,R 2 * ,R 3 * ,R 4 * , 1}. In correspondence of such a choice, by (5.17) and (5.18) we can conclude that H(x, Du) dx where the above holds for 0 < r ≤R * = min{R 1 * ,R 2 * ,R 3 * ,R 4 * , 1}, and all the quantities involved are as described before. Finally, for the sake of clarity, we let m = min{m p , m q , m s , m 0 }. Now take a ball B r with 0 < r ≤ 1 2R * such that B 2r ⋐ Ω 0 ⋐ Ω. Fix τ p ∈ 0, 1 8 and assume deg(B 2r ) and deg(B 4τ p r ). We fix ϑ ∈ (0, n) and we estimate, by (4.4), Poincaré's inequality, Proposition 1 with ϕ = H p 0 , (2.7) and (5.8), F (x, Dw) ≈ \|Dw\| p and ∂ zz F(x, Dw) ≈ \|Dw\| p−2 Id , x)\|Dw\| p dx , 0 < ν ≤ a(x) ≤ L ,(1.3) W 1 , 1H(·) (Ω) ∋ w → H(w, Ω) = Ω H(x, Dw) dx, 1 < p<q≤s ,(1.6) with H(x, z) := \|z\| p + a(x)\|z\| q + b(x)\|z\| s , (1.7) Ω) come in an obvious way from the one of W 1,ϕ (Ω). Definition 3 3Let U ⋐ Ω be an open set and u 0 ∈ W 1,ϕ loc (Ω, R N ) be any function. With ϕ-harmonic map, we mean a map h ∈ u 0 + W 1,ϕ 0 (U, R N ) solving the Dirichlet problem H 0 . In this case, [a] 0,α = [a 0 ] 0,α = 0 and [b] 0,β = [b 0 ] 0,β = 0, so constants and exponents do not depend either on [a] 0,α , [b] 0,β nor on Dv L p (B r ) . 17) where c = c(n, p, q, s, [a] 0,α , [b] 0,β , H(·, Du) L 1 (B r ) ) and σ g = σ g (n, p, q, s, [a] 0,α , [b] 0,β , H(·, Du) L 1 (B r ) ). where ω n is the volume of the unit ball B 1 ⊂ R n . (AInc) clearly holds with γ − = p > 1 and (ADec) is verified by γ + = s ≥ p > 1. n, p, q, s, [a] 0,α , [b] 0,β , α, β, [u] C 0,λg (Ω 0 ) , H(·, Du) L 1+δg (Ω 0 ) if n < p(1 + δ g ) , and data ≡ n, p, q, s, a L ∞ (Ω) , b L ∞ (Ω) , [a] 0,α , [b] 0,β . ( 4 . 7 ) 47Here, if n ≥ (1+ δ g )p, c 1 = c 1 (n, p, q, s,[a] 0,α , [b] 0,β , α, β, u L ∞ (Ω 0 ) ), c 2 = c 2 (n, p, s, q, [a] 0,α , α, β, u L ∞ (Ω 0 ) ), c 3 = c 3 (n, p, q, s, [b] 0,β , α, β, u L ∞ (Ω0 ) ) and c 4 = c 4 (n, p, q, s, [a] 0,α , [b] 0,β , α, β), while, if n < p(1 + δ g ), c 1 = c 1 (n, p, q, s, [a] 0,α , [b] 0,β , α, β, [u] C 0,λg (Ω 0 ) ), c 2 = c 2 (n, p, s, q, [a] 0,α , α, β, [u] C 0,λg (Ω 0 ) ), c 3 = c 3 (n, p, q, s, [b] 0,β , α, β, [u] C 0,λg (Ω 0 ) ) and c 4 = c 4 (n, p, q, s, [a] 0,α , [b] 0,β , α, β, [u] C 0,λg (Ω 0 ) ). with c 3 3= c 3 (n, p, q, s, [b] 0,β , β, [u] C 0,λg (Ω 0 ) ). Finally, if ndeg(B r ) holds, then by (3.13), (1.8), the fact that either if n ≥ p(1 + δ g ) or if n < p(1 + δ g ), α ≥ α − γ a and β ≥ β − γ b , and the very definition of ndeg(B r ) we have − B r H(x, Du) dx ≤c − B 2r 17) whereṼ is the corresponding auxiliary function defined in(2.8), c = c(n, p, q, s,c 1 ,c 2 ), σ 1 = σ 1 (n, p, q, s, σ 0 ) ∈ (0, σ 0 ), m = m(n, p, q, s, σ 0 ) > 0.Proof. The proof forH = H p 0 , H q 0 , H s 0 is contained in [4, Lemma 1], so we focus onH = H 0 . The proof we provide is in some sense a simplified version of the original one since we do not need a powerful result such as Theorem 5.1 from [13]. In fact we can recover some extra boundary integrability from Lemma 5. Define h 0 to be the solution to the Dirichlet problem h 0 → min w∈W 1,H 0 v (B r ) B r H 0 (Dw) dx. Dw λ L ∞ (B r ) ≤ c(n)λ and {w λ w} ⊂ {M(\|Dw\|) > λ} ∪ negligible set. [a] 0,α , [b] 0,β , Du L p (B r ) ) as u. For any open U ⋐ Ω it satisfies the Euler-Lagrange equation 0 = U p\|Dv\| p−2 + qa(x)E q−p \|Dv\| q−2 + sb(x)E s−p \|Dv\| s−2 Dv · Dϕ dx for all ϕ ∈ C ∞ c (U). 0 , we see from the definition of ≤c Dϕ L ∞ (B r ) H(·, Du) L ∞ (B r ) , (5.13) with c 1 = c 1 (n, p, s, [b] 0,β , β, H(·, Du) L 1+δg (Ω 0 ) ), and (II) deg β ≤c Dϕ L ∞ (B r ) , ) · Dϕ dx ≤rσ q Dϕ L ∞ (B r ) ,(5.15) so the assumptions of Lemma 7 are satisfied and there exists aH q 0 -harmonic maph q satisfying in particular(4.17).Clearly, if h q = Eh q , then h q is H q 0 -harmonic, h q ∂B r = u\| ∂B r and, by (Dh q ) 2 dx ≤ cr m q − B 2r H(x, Du) dx,(5.16) where c = c(n, p, q, s, [a] 0,α , [b] 0,β ) and m q = m q (n, p, q, s, [a] 0,α , [b] 0,β ). Finally, suppose ndeg(B 2r ) holds. Then, by (5.1) we obtain − B rDH 0 (Dv) · Dϕ dx ≤sE s−p − B r b(x) − b i (B 2r ) \|Dv\| s−1 \|Dϕ\| dx + qE q−p − B r a(x) − a i (B 2r ) \|Dv\| q−1 \|Dϕ\| dx ≤(I) ndeg + (II) ndeg .As above we estimate (I) ndeg ≤c Dϕ L ∞ (B r ) 1 = c 1 (n, p, s, [b] 0,β , β, H(·, Du) L 1+δg (Ω 0 ) ), and (II) ndeg ≤c Dϕ L ∞ (B r ) DH 0 (,,,V 0 ( 00Dv) · Dϕ dx ≤rσ 0 Dϕ L ∞ (B r ) ,(5.19) so the assumptions of Lemma 7 are satisfied and there exists aH 0 -harmonic maph 0 satisfying in particular(4.17). Clearly, if h 0 = Eh 0 , then h 0 is H 0 -harmonic, h 0 \| ∂B r = u\| ∂B r and, by(= c(n, p, q, s, [a] 0,α , [b] 0,β ) and m 0 = m 0 (n, p, q, s, [a] 0,α , [b] 0,β ). Summarizing we gotdeg(B 2r ) Dh p ) 2 dx ≤ cr m p − Dh s ) 2 dx ≤ cr m s − Dh q ) 2 dx ≤ cr m q − Du, Dh 0 ) 2 dx ≤ cr m 0 − B 2r Dv L p (B r ) by using the minimality of v and the fact that v\| ∂B r = u\| ∂B r . Merging(3.18) and(3.19) we obtain19) with c = c(n, p, q, s, [a] 0,α , [b] 0,β , α, β, H(·, Du) L 1 (B r ) ). Here we dispensed c from the dependence of 7 ) 7so the assumptions of Lemma 7 are matched and there exists a H p 0 -harmonic maph p satisfying in particular (4.17). It is clear that, if h p = Eh p , then h p is still H p 0 -harmonic, h p ∂B r = u\| ∂B r and, by (4.17), where c = c(n, p, q, s, [a] 0,α , [b] 0,β ) and m p = m p (n, p, q, s, [a] 0,α , [b] 0,β ). Suppose now that deg α (B 2r ) holds.− B r V p 0 (Du, Dh p ) 2 dx ≤ cr m p − B 2r H(x, Du) dx, (5.8) Then, by (5.1) we obtain − B r 11 ) 11so the assumptions of Lemma 7 are matched and there exists aH s 0 -harmonic maph s satisfying in particular (4.17). Clearly, if h s = Eh s , then h s is H s 0 -harmonic, h s \| ∂B r = u\| ∂B r and, by (4.17), Dh s ) 2 dx ≤ cr m s − where c = c(n, p, q, s, [a] 0,α , [b] 0,β ) and m s = m s (n, p, q, s, [a] 0,α , [b] 0,β ). This time assume deg β (B 2r ) holds. Then, by (5.1) we obtain− B r V s 0 (Du, B 2r H(x, Du) dx, (5.12) − B r H(x, Du) dx,(5.21)where c = c(data(Ω 0 ), ϑ). For the ease of exposition we set 2r = ρ and adjusting the constants in (5.21) we getSelecting τ p in such a way that cτ ϑ p ≤ 1 2 and a threshold radius R 1 * ∈ (0,R * ] such that cR m τ ϑ−n p ≤ 1 2 , we can conclude that, for all ρ ∈ (0, R 1 * ) and all ϑ ∈ (0, n),H(x, Du) dx,(5.23)where c = c(data(Ω 0 ), ϑ). Again, we name ρ = 2r thus gettingwhere, as before, ϑ ∈ (0, n) is arbitrary. Choose τ s small enough so that cτ ϑ s < 1 2 and a threshold R 2Consider τ q ∈ 0, 1 8 , assume deg β (B 2r ) and that b i (B 4τ q r ) ≤ 4[b] 0,β (4τ q r) β−γ b , where r < 1 2 R 2 * . For ϑ ∈ (0, n), by (4.6), Poincaré's inequality, Proposition 1 with ϕ = H q 0 and (5.16) we obtainH(x, Du) dx,(5.25)where c = c(data(Ω 0 ), ϑ). Again, we set ρ = 2r thus obtainingwhere, as before, ϑ ∈ (0, n) is arbitrary. Take τ q sufficiently small so that cτ ϑ q < 1 2 and a threshold R 3, Poincaré's inequality, Proposition 1 with ϕ = H 0 , (2.7) and (5.20) we obtainwhere c = c(data(Ω 0 ), ϑ). Again, we set ρ = 2r thus obtainingwhere, as before, ϑ ∈ (0, n) is arbitrary. Take τ 0 sufficiently small so that cτ ϑ 0 < 1 2 and a threshold R 4Hence, for all ρ ∈ (0, R 4 * ] and all ϑ ∈ (0, n) we getStep 2: double nested exit time and iteration Now we are in position to develop the announced double nested exit time. Take B r ⋐ Ω with r ∈ (0, R * ], where R * = min i∈{1,2,3,4} {R i * } and consider 0 < ρ < r. For κ ∈ N ∪ {0}, we consider condition deg(B 2τ κ+1 p r ) and define the exit time indexFor any κ ∈ {1, · · · , t p } we apply repeatedly (5.22) to obtainNow we only need to fillet estimates (5.28)-(5.31). For 0 < ρ < r ≤ R * we consider the following five cases.p r. Then there isκ ∈ {0, · · · , t p } such that τ¯κ +1 p r ≤ ρ < τ¯κ p r. We obtain from (5.29) that,where c = c(data(Ω 0 ), ϑ).p r. We see that, by (5.32),p . We then estimate, using (5.31) and (5.35),where c = c(data(Ω 0 ), ϑ).As mentioned before, the procedure is the same if, after deg occurs deg β instead of deg α and it is actually easier if, from deg we jump directly to ndeg. All in all we can conclude that, for all 0 < ρ < r ≤ R * and all ϑ ∈ (0, n) there holds with c = c(data(Ω 0 ), ϑ). Now, if r > R * and R * ≤ ρ < r ≤ 1 we get with c = c(data(Ω 0 ), ϑ).Gradient continuitywhere c andν depend at the most from n, p, q, s. Moreover, for B r ⋐ Ω 0 with 0 < r ≤ R * , where R * is the threshold radius introduced in the previous section, we obtain from Lemma 7 and (6.1) that for some positive exponent κ 1 = κ 1 (κ 0 , n, p, q, s). In this case, c = c(data(Ω 0 )). Now, for 0 < ρ < r ≤ R * , by (6.3), the minimality of h, (6.1) and (6.2) we see thatwith c = c(data(Ω 0 ), κ). Now, first notice that there is no loss of generality in supposing pν ≤ 1. Setting ρ = r 1+ κ 1 4n and κ = κ 1 pν 8n in (6.4), we easily obtain27 for all ρ ∈ (0, R * ), with c = c(data(Ω 0 )). Now, by the integral characterization of Hölder continuity due to Campanato and Meyers we can conclude that Du ∈ C 0,ν loc (Ω, R n ) for ν = κ 1ν 16n . The full proof of Theorem 1 is still not complete, since ν depends on data(Ω 0 ), while we announced that the Hölder continuity exponent of Du depends only on data. So we will retain that, after a covering argument, Du ∈ L ∞ loc (Ω), therefore the non-uniform ellipticity of (1.6) becomes immaterial. Now, for B r ⋐ Ω 0 ⋐ Ω, no matter what degeneracy condition holds, we compare u to h ∈ W 1,H(·) (B r ) solution to the Dirichlet problemNotice that, for a functional like the one in (6.6), the Bounded Slope Condition holds, see[6], so there exists c = c(n, p, q, s, Du L ∞ (B r ) ) such thatFor simplicity, let us adopt the notation H 0 (z) = \|z\| p + a i (B 2r )\|z\| q + b i (B 2r )\|z\| s . 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[ "Cherednik algebras and Hilbert schemes in characteristic p", "Cherednik algebras and Hilbert schemes in characteristic p" ]	[ "Roman Bezrukavnikov ", "Michael Finkelberg ", "Victor Ginzburg " ]	[]	[]	We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H1,c(An−1) over Fp, an algebraic closure of the finite field. In the most interesting special case where c ∈ Fp, we construct an Azumaya algebra Hc on Hilb n A 2 , the Hilbert scheme of n points in the plane, such that Γ(Hilb n A 2 , Hc) = Hc. Our localization theorem provides an equivalence between the bounded derived categories of Hc-modules and sheaves of coherent Hc-modules on Hilb n A 2 , respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland-King-Reid and Haiman. results included in [GS]. Also, we would like to thank A. Premet for pointing out several inaccuracies involving restricted Lie algebras that have occurred in the original version of the paper.	10.1088/4165(06)00309-8	[ "https://arxiv.org/pdf/math/0312474v6.pdf" ]	9,673,200	math/0312474	2311d516fa630c74c7bebc8b712b6f18e7b7bdbc	Cherednik algebras and Hilbert schemes in characteristic p 24 Nov 2021 Roman Bezrukavnikov Michael Finkelberg Victor Ginzburg Cherednik algebras and Hilbert schemes in characteristic p 24 Nov 2021arXiv:math/0312474v6 [math.RT] (with Appendix by Pavel Etingof) To David Kazhdan with admiration We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H1,c(An−1) over Fp, an algebraic closure of the finite field. In the most interesting special case where c ∈ Fp, we construct an Azumaya algebra Hc on Hilb n A 2 , the Hilbert scheme of n points in the plane, such that Γ(Hilb n A 2 , Hc) = Hc. Our localization theorem provides an equivalence between the bounded derived categories of Hc-modules and sheaves of coherent Hc-modules on Hilb n A 2 , respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland-King-Reid and Haiman. results included in [GS]. Also, we would like to thank A. Premet for pointing out several inaccuracies involving restricted Lie algebras that have occurred in the original version of the paper. Table of Contents 1. Introduction 2. Crystalline differential operators 3. Hamiltonian reduction in characteristic p 4. Azumaya algebras via Hamiltonian reduction 5. The rational Cherednik algebra of type A n−1 6. An Azumaya algebra on the Hilbert scheme 7. Localization functor for Cherednik algebras 8. Induction functor and comparison with [EG] 9. Appendix by Pavel Etingof: The p-center of Symplectic reflection algebras 1 Introduction 1.1 Let c ∈ Q be a rational number, and H 1,c (A n−1 ) the rational Cherednik algebra of type A n−1 with parameters t = 1 and c that has been considered in [EG] (over the ground field of complex numbers). For all primes p ≫ n, we can reduce c modulo p. Thus, c becomes an element of the finite field F p . We let k = k p be an algebraic closure of F p , and let H c := H 1,c (A n−1 , k p ) be the Cherednik algebra, viewed as an algebra over k p . Unlike the case of characteristic zero, the algebra H c has a large center, called the p-center. The spectrum of the p-center is isomorphic to [(A 2 ) n /S n ] (1) , the Frobenius twist of the n-th symmetric power of the plane A 2 . 1.2 We consider Hilb n A 2 , the Hilbert scheme (over k p ) of n points in the plane, see e.g. [Na1]. There is a canonical Hilbert-Chow map Υ : Hilb n A 2 → (A 2 ) n /S n that induces an algebra isomorphism Γ(Hilb n A 2 , O Hilb n A 2 ) ∼ = k (A 2 ) n /S n . (1.2.1) Let Hilb (1) denote the Frobenius twist of Hilb n A 2 , a scheme isomorphic to Hilb n A 2 and equipped with a canonical Frobenius morphism Fr : Hilb n A 2 → Hilb (1) . We introduce an Azumaya algebra H c on Hilb (1) of degree n! · p n (recall that an Azumaya algebra has degree r if each of its geometric fibers is isomorphic to the algebra of r × r-matrices). For all sufficiently large primes p, we construct a natural algebra isomorphism (a version of the Harish-Chandra isomorphism from [EG]) Γ(Hilb (1) , H c ) ∼ −→ H c . (1.2.2) The restriction of this isomorphism to the subalgebra Γ(Hilb (1) , O Hilb (1) ) yields, via (1.2.1), the above mentioned isomorphism between the algebra k (A 2 ) n /S n (1) and the p-center. Remark 1.2.3. More generally, for any c ∈ k, not necessarily an element of F p , there is an Azumaya algebra on the Calogero-Moser space with parameter c p − c such that an analogue of isomorphism (1.2.2) holds for the Calogero-Moser space instead of the Hilbert scheme. This case is somewhat less interesting since the Calogero-Moser space is affine while the Hilbert scheme is not. ♦ The main idea used in the construction of isomorphism (1.2.2) is to compare Nakajima's description of Hilb n A 2 by means of Hamiltonian reduction, see [Na1], with (a refined version, see §5) of the construction introduced in [EG] describing the spherical subalgebra of H c as a quantum Hamiltonian reduction of an algebra of differential operators. 1.3 We introduce the following set of rational numbers Q good = {c ∈ Q c ≥ 0 & c ∈ 1 2 + Z}. 1.4 Now, fix ξ ∈ [(A 2 ) n /S n ] (1) , a point in the Frobenius twist of (A 2 ) n /S n . We write Hilb (1) ξ = Υ −1 (ξ) for the corresponding fiber of the Frobenius twist of the Hilbert-Chow map, and let Hilb (1) ξ = Υ −1 (ξ) denote its formal neighborhood, the completion of Hilb (1) along the subscheme Hilb (1) ξ . The theorem below, based on a similar result in [BK], says that the Azumaya algebra H c splits on the formal neighborhood of each fiber of the Hilbert-Chow map, that is, we have the following result (see Theorem 7.4.1) Theorem 1.4.1. For each ξ ∈ [(A 2 ) n /S n ] (1) , there exists a vector bundle V c,ξ on Hilb (1) 0 is isomorphic to the (Frobenius twist of the) punctual Hilbert scheme. We will show that the algebra H c,0 contains a canonical dense Z 2 -graded subalgebra H • = ⊕ k,l∈Z H k,l , see §7.5. The category of Z 2 -graded H • -modules may be thought of as a 'mixed version' of the category H c,0 -Mod, cf. [BGS,Definition 4.3.1]. Recall that the algebra H c contains a canonical sl 2 -triple, see [BEG]. Let h ∈ H c denote the semisimple element of that triple. We expect that the above mentioned Z 2 -grading has the property that, for any u ∈ H k,l , we have h · u − u · h = (k − l) · u. 1.5 Let k[S n ] denote the group algebra of the Symmetric group on n letters. Write Irr(S n ) for the set of isomorphism classes of simple k[S n ]-modules. This set is labelled by partitions of n, since by our assumptions char k > n. In particular, we have the trivial 1-dimensional representation triv, and the sign representation sign. Let H := k[[A 2n ]]#S n be the cross-product of S n with k[[A 2n ]], the algebra of formal power series in 2n variables acted on by S n in a natural way. We consider D b (H-Mod), the bounded derived category of (finitely-generated) complete topological H-modules. Given a simple k[S n ]-module τ , write τ H for the corresponding H-module obtained by pullback via the natural projection H = k[[A 2n ]]#S n → k[S n ], f ⋊ w → f (0) · w. Similarly, let L τ denote the corresponding simple highest weight H c -module, the unique simple quotient of the standard H c -module associated with τ , see [DO], [BEG]. The results of Bridgeland-King-Reid [BKR] and Haiman [H], see also [BK], provide an equivalence of categories BKR : D b (Coh(Hilb n A 2 )) ∼ −→ D b (k[A 2n ]#S n -Mod), F → RΓ(Hilb n A 2 , P L ⊗ F ), where P denotes the Procesi bundle, the 'unusual' tautological rank n! vector bundle on Hilb n A 2 considered in [H]. Restricting this equivalence to the completion of the zero fiber of the Hilbert-Chow map, and using Corollary 1.4.3, one obtains the following composite equivalence (1.5.1) We recall that the equivalence of Corollary 1.4.3 involves a choice of splitting bundle V c,0 , cf. Remark 1.4.2. This choice may be specified by the following Conjecture 1.5.2. Fix c ∈ Q good . Then, for all p ≫ 0, we have (i) One can choose the splitting bundle V c,0 in such a way that Γ(Hilb 0 , V c,0 (−1)) = L sign . (ii) With this choice of V c,0 , the composite equivalence in (1.5.1) preserves the natural tstructures, in particular, induces an equivalence H c,0 -Mod ∼ −→ H-Mod, of abelian categories, such that L τ goes to τ H , for any simple S n -module τ . For c = 1 n , we expect that Γ(Hilb (1) 0 , V c,0 ⊗ BKR −1 (triv H )) is a 1-dimensional vector space that supports the trivial representation of the group S n ⊂ H c . 1.6 In the special case of rank one, i.e., for n = 2, a complete classification and explicit construction of simple H c -modules (for char k > 0) has been obtained by Latour [La]. It seems certain that our results in characteristic p have their characteristic zero counterparts for the double-affine Hecke algebra of type A n−1 , specialized at a root of unity, cf. [Ch]; in that case one has to replace Hilb n A 2 by Hilb n (C * × C * ), cf. [Ob]. Also, it is likely that the results of the present paper can be generalized to the case of symplectic reflection algebras associated with wreath products Γ n = S n ⋉ Γ n , where Γ is a finite subgroup in SL 2 (k), see [EG,§11]. More generally, we are going to study Azumaya algebras arising via quantum Hamiltonian reduction from the general Nakajima quiver varieties, cf. [Na2], (of which wreath-products are special cases). Our technique is ideally suited for such a generalization, that has been, in effect, suggested earlier by Nakajima and the first author. In another direction, the general results of §4 below apply verbatim to quantizations of Slodowy slices considered in [Pr], see also [GG]. We are going to explore these topics elsewhere. 2 Crystalline differential operators. 2.1 Unless specified otherwise, we will be working over the ground field k, an algebraically closed field of characteristic char k = p > 0. We write Fr : k → k, k → k p for the Frobenius automorphism. Given a k-vector space E, it is convenient to introduce E (1) , a vector space with the same underlying additive group as E, but with a 'twisted' k-linear structure given by k • e := Fr −1 (k) · e, ∀k ∈ k, e ∈ E. Note that if A is a k-algebra, then the map A → A, a → a p is an additive but not k-linear map, that becomes k-linear if considered as a map A (1) → A. Given an additive map f : E → F between two k-vector spaces, we say that f is • p-linear, if f (k · e) = k p · f (e) for any k ∈ k, e ∈ E, i.e., if the corresponding map E (1) → F is k-linear; • p-graded if both vector spaces are equipped with Z-gradings E = E(i), F = F (i), and we have f (E(i)) ⊂ F (p · i) for all i. 2.2 Let X be a smooth algebraic veriety over k with structure sheaf O X . Write k[X] for the corresponding algebra of global sections. We let X (1) denote an algebraic variety with the same structure sheaf as X but with the 'twisted' k-linear structure. Thus, O X (1) := (O X ) (1) , and there is a canonical morphism Fr : X → X (1) called Frobenius morphism, such that the map f → f p on regular functions becomes identified with the natural sheaf imbedding Fr q O X (1) ֒→ O X . We write T X for the tangent sheaf on X, and let T * X denote the total space of the cotangent bundle. There is a canonical isomorphism T * [X (1) ] ∼ = [T * X] (1) , and we will use the notation T * X (1) for these two isomorphic varieties, and π : T * X (1) → X (1) for the natural projection. The space T * X, resp. T * X (1) , has a canonical symplectic structure, which makes k[T * X] a Poisson algebra. Let D X denote the sheaf of crystalline differential operators on X, that is, a sheaf of algebras generated by O X and T X . Let D(X) := Γ(X, D X ) denote the corresponding algebra of global sections. More generally, given a locally-free coherent sheaf (= vector bundle) L on X, let D X (L) := L OX D X OX L ∨ be the sheaf of differential operators on L, and D(X, L) := Γ(X, D X (L)) the algebra of its global sections. 2.3 The sheaf D X is known to have a large center. Specifically, for any vector field ξ ∈ T X , the p-th power of ξ acts as a derivation, hence, gives rise to another vector filed, ξ [p] ∈ T X . The assignment ξ → ξ p − ξ [p] extends to a canonical algebra imbedding z D : Sym T X (1) / / Fr qDX , ξ −→ ξ p − ξ [p] , (2.3.1) (of sheaves on X (1) ) whose image, to be denoted Z X (1) ⊂ Fr qDX , equals the center of Fr qDX . Therefore, the isomorphism π qO T * X (1) ≃ Sym T X (1) makes Fr qDX a sheaf of π qO T * X (1) -algebras. This way, we may (and will) view Fr qDX as a coherent sheaf on T * X (1) , to be denoted D (1) . The sheaf D X comes equipped with a standard increasing filtration D ≤k X , k = 0, 1, . . . , by the order of differential operator. For the associated graded sheaf, one has a graded algebra isomorphism gr D X ∼ = Sym T X = π qOT * X . The filtration on D X induces a filtration on Fr qDX and also the filtration Z ≤i X (1) := Fr qD ≤i X ∩ Z X (1) on the central subalgebra Z X (1) . Observe that the latter algebra already has a natural grading obtained, via the isomorphism Z X (1) = Sym T X (1) , from the standard grading on the Symmetric algebra. With this grading, one has a p-graded algebra isomorphism Sym T X (1) ∼ −→ gr Z X (1) , cf. Sect 2.1, i.e. we have Sym i T X (1) ∼ −→ gr p·i Z X (1) , ∀i. Now, view Fr qDX as a Z X (1) -algebra. Thus, gr(Fr qDX ) becomes a gr Z X (1) -algebra that may be viewed, by the isomorphism Spec(gr Z X (1) ) = T * X (1) , as a G m -equivariant coherent sheaf on T * X (1) . On the other hand, consider the Frobenius morphism Fr T * X : T * X → [T * X] (1) and view Fr T * X q O T * X as a G m -equivariant coherent sheaf of algebras on T * X (1) , a G m -variety. With this understood, there is a natural G m -equivariant algebra isomorphism gr Fr qDX ≃ π qFr T * X q O T * X . (2.3.2) 2.4 The Rees algebra. Let D be an associative algebra, and write D[t] := k[t]⊗D, where t is an indeterminate. We put a grading on D[t] by assigning D grade degree zero, and setting deg t = 1. Recall that, given an increasing filtration 0 = D −1 ⊂ D 0 ⊂ D 1 ⊂ . . . , on D, such that D i ·D j ⊂ D i+j and i≥0 D i = D, one defines the Rees algebra of D as the following graded subalgebra: Rees D := i≥0 t i · D i ⊂ D[t]. There are standard isomorphisms (Rees D)\| {0} ∼ = gr D, and (Rees D) (t) ∼ = k[t, t −1 ] ⊗ D, (2.4.1) where, for any k[t]-algebra R we let R (t) := k[t, t −1 ] k[t] R denote the localization of R, and for any s ∈ k we use the notation R\| {s} := R/(t − s)R. Conversely, given a flat Z ≥0 -graded k[t]-algebra R = i≥0 R(i), set D := R\| {1} = R/(t−1)R. This is a k-algebra equipped with a canonical increasing filtration D i , i = 0, 1, . . . , and with a canonical graded algebra isomorphism gr D ∼ = R\| {0} = R/tR. The filtration on D is defined in the following way. Put D[t, t −1 ] := k[t, t −1 ] k D, and view it as a Z-graded k[t, t −1 ]-algebra. Further, inverting t, we get from R a Z-graded k[t, t −1 ]-algebra R (t) that contains R as a k[t]-subalgebra. The definition of D provides an isomorphism φ : R (t) /(t − 1)R (t) ∼ −→ D that admits a unique lift to the following graded k[t, t −1 ]-algebra isomor- phism φ (t) : R (t) ∼ −→ D[t, t −1 ] = k[t, t −1 ] ⊗ R (t − 1)R , R(i) ∋ u → t i ⊗ u mod(t − 1)R . The above mentioned increasing filtration on D is defined by D i := D ∩ φ (t) (t −i ·R), i = 0, 1, . . . . (2.4.2) Assume next that D = i≥0 D(i) is a graded algebra, and view it as a filtered algebra with filtration being induced by the grading, that is, defined by D i := j≤i D(j). Then, we have Rees D = i≥0 t i · ⊕ j≤i D(j) = i≥0 t i · D(i) [t]. We see that, for a graded algebra D, one has the following graded algebra isomorphism k[t] ⊗ D ∼ −→ Rees D, k[t] ⊕ i D(i) ∋ f ⊗ ( i u i ) −→ f · i t i ·u i . (2.4.3) 2.5 The sheaf RD (1) . We apply the Rees algebra construction to Fr qDX , viewed as a sheaf of filtered algebras. Thus, we get a sheaf Rees Fr qDX of graded O X (1) -algebras. In Rees Fr qDX , we also have a central subalgebra Rees Z X (1) ⊂ Rees Fr qDX . By (2.4.3), the canonical grading on Z X (1) provides a p-graded algebra isomorphism Rees Z X (1) ≃ k[t] ⊗ Z X (1) . Thus, we obtain the following canonical algebra maps Sym T X (1) (2.3.1) ∼ / / Z X (1) z →1⊗z / / k[t] ⊗ Z X (1) (2.4.3) Rees Z X (1) / / Rees Fr qDX . (2.5.1) The composite map in (2.5.1) is a p-graded map to be denoted z RD . This map specializes at t = 0 to the map ξ → ξ p , and at t = 1 to (2.3.1). Further, the algebra isomorphism Rees Z X (1) ≃ k[t] ⊗ Z X (1) yields a direct product decompo- sition Spec Rees Z X (1) ∼ = A 1 × Spec Z X (1) = A 1 × T * X (1) . (2.5.2) We will often identify Rees Fr qDX , a graded Rees Z X (1) -algebra, with a coherent sheaf of algebras on Spec(Rees Z X (1) ), that is, on A 1 × T * X (1) . The resulting sheaf on A 1 × T * X (1) , to be denoted RD (1) , is easily seen to be flat with respect to the first factor A 1 . Moreover, (2.4.1) yields, in view of (2.3.2), the following isomorphisms of sheaves of algebras on T * X (1) , resp., on (A 1 {0})× T * X (1) : (RD (1) ) {0}×T * X (1) ∼ = Fr T * X q O T * X , (RD (1) ) (A 1 {0})×T * X (1) ∼ = pr * 2 (D (1) ), (2.5.3) where pr 2 : (A 1 {0}) × T * X (1) → T * X (1) is the second projection. Further, the grading on the Rees algebra Rees Z X (1) makes Spec(Rees Z X (1) ) a G m -variety. It follows from formula (2.4.3) that the natural G m -action on Spec Rees Z X (1) corresponds, via (2.5.2), to the G m -diagonal action on A 1 × T * X (1) . The sheaf RD (1) on A 1 × T * X (1) comes equipped with a canonical G m -equivariant structure. 3 Hamiltonian reduction in characteristic p 3.1 Lie algebras in characteristic p. Let A be a connected linear algebraic group over k. Write A (1) for the Frobenius twist of A, cf. §2, an algebraic group isomorphic to A and equipped with an algebraic group morphism Fr : A → A (1) , called the Frobenius morphism. The kernel of this morphism is an infinitesimal group scheme A 1 ⊂ A, called Frobenius kernel. By definition, one has an exact sequence: 1 −→ A 1 −→ A Fr −→ A (1) −→ 1. The Lie algebra a := Lie A may be viewed as the vector space of left invariant vector fields on A. This vector space comes equipped with a natural structure of p-Lie algebra, i.e., we have a p-power map a → a, x → x [p] , see [J] or [Ja]. Let Sym a, resp. Ua, be the symmetric, resp. enveloping, algebra of a. The group A acts on Sym a and Ua by algebra automorphisms via the adjoint action. The standard increasing filtration U qa on the enveloping algebra gives rise to a graded algebra Rees Ua = i≥0 t i · U i a. Jacobson's argument [J, ch. V, §7, (60)-(64)] shows that the following assignment z RU : Sym a (1) / / Rees Ua, a (1) ∋ x −→ x p − t p−1 ·x [p] (3.1.1) gives a well-defined Ad A-equivariant injective p-graded algebra homomorphism, cf. also [PS]. The image of the map z RU is an Ad A-stable subalgebra contained in the center of Rees Ua. Specialization of the map z RU at t = 0 reduces to the p-graded algebra map Sym a (1) ֒→ Sym a, x → x p . On the other hand, specializing the map z RU at t = 1, one obtains an algebra imbedding z U := z RU \| t=1 : Sym a (1) ֒→ Ua. The image Z(a) := z U (Sym a (1) ) of this imbedding is a central subalgebra in Ua generated by the elements {x p − x [p] } x∈a , usually referred to as the p-center of Ua, cf. e.g. [Ja]. Thus, the map in (3.1.1) may be identified with the composite of the following chain of algebra homomorphisms, completely analogous to those in (2.5.1): Sym a (1) ∼ z U / / Z(a) z →1⊗z / / k[t] ⊗ Z(a) (2.4.3) Rees Z(a) / / Rees Ua . The adjoint action on Z(a) of the Frobenius kernel A 1 ⊂ A is trivial, hence, the A-action on Z(a) factors through A (1) . The Artin-Schreier map. Let a * denote the k-linear dual of a, and write a 1, * := (a (1) ) * for the k-linear dual of a (1) . For any linear function λ ∈ a * , the assignment x → λ(x) p gives a p-linear map a → k, that is, k-linear function on a (1) , to be denoted λ (1) . This way, one obtains a p-linear map a * → (a (1) ) * , λ → λ (1) . The latter map gives a canonical k-vector space isomorphism (a * ) (1) ∼ −→ (a (1) ) * , λ → λ (1) . Let X * (a) ⊂ a * denote the subspace of fixed points of the coadjoint action of A on a * . Such a fixed point may be viewed as an Ad A-invariant Lie algebra homomorphism a → k (note that a Lie algebra homomorphism a → k need not necessarily be Ad A-invariant). Given χ ∈ X * (a), we write χ [1] for the function a → k, x → χ(x [p] ). Lemma 3.2.1. For any χ ∈ X * (a), the function χ [1] : a → k is a p-linear map, that is, χ [1] ∈ a 1, * . Proof. The above mentioned Jacobson's formula implies that, in Ua, one has an equality, cf. also [Ja,Lemma 2.1]: (x + y) [p] − x [p] − y [p] = (x + y) p − x p − y p . Now, extend χ to an algebra homomorphism Ua → k, and apply the resulting map to the equation above. We find χ (x + y) [p] − x [p] − y [p] = χ (x + y) p − x p − y p = χ(x + y) p − χ(x) p − χ(y) p = χ(x) + χ(y) p − χ(x) p − χ(y) p = 0. The Lemma follows. Lemma 3.2.1 shows that the assignment χ → χ [1] , as well as the following assignment κ : X * (a) −→ a 1, * , κ(χ) = χ (1) − χ [1] : x −→ χ(x) p − χ(x [p] ), (3.2.2) gives a well-defined p-linear map X * (a) −→ a 1, * . The map (3.2.2) will play an important role later in this paper, it may be thought of as a Lie algebra analogue of the Artin-Schreier map. Let I ϕ ⊂ Sym a (1) = k[a 1, * ] denote the maximal ideal corresponding to a point ϕ ∈ a 1, * . Now, given χ ∈ X * (a), extend it to an algebra map χ : Ua → k, as in the proof of Lemma 3.2.1. / / Ua χ / / k = I κ(χ) . Proof. For x ∈ a, we compute χ(z U (x)) = χ(x p − x [p] ) = χ(x) p − χ(x [p] ) = κ(χ)(x). Let X * (A) := Hom(A, G m ) be the character lattice of the algebraic group A. The differential of a character f : A → G m at 1 ∈ A is a linear function x → x(f )(1) on the Lie algebra a, which is clearly an element of X * (a). We denote this linear function by dlog f , so that the assignment f → dlog f yields an additive group homomorphism dlog : X * (A) → X * (a). Fix f ∈ X * (A) and put φ := dlog f ∈ X * (a). Observe that, for any x ∈ a viewed as a left invariant vector field on A, we have x(f ) = φ(x) · f . It follows that the p-th power of x, viewed as a left invariant differential operator on A of order p, acts on a character f ∈ X * (A) as multiplication by the constant φ(x) p ∈ k. On the other hand, this differential operator is a derivation, which corresponds to the left invariant vector field x [p] . Thus, we also have x [p] (f ) = φ(x [p] )·f . Combining together the equations above, we deduce φ(x [p] ) = φ(x) p , ∀x ∈ a, that is, κ(φ) = 0. Thus, we have proved dlog(X * (A)) ⊂ Ker κ : X * (a) −→ a 1, * . (3.2.4) 3.3 Restricted enveloping algebras. Fix χ ∈ X * (a), and let χ : Ua → k be the corresponding algebra homomorphism. Here, Ua · I (1) χ ⊂ I χ , is an Ad A-stable two-sided ideal in Ua, and the quotient u χ (a) is an associative algebra of dimension dim u χ (a) = p dim a , called χ-restricted enveloping algebra. By definition there is an exact sequence 0 −→ Ua·I (1) χ −→ Ua r −→ u χ (a) −→ 0. (3.3.2) Corollary 3.2.3 shows that I (1) χ is a maximal ideal in Z(a) that goes, under the isomorphisms Z(a) ∼ = Sym a (1) ∼ = k[a 1, * ], to the maximal ideal in k[a 1, * ] corresponding to the point κ(χ) ∈ a 1, * , that is, to the ideal I κ(χ) . Thus, we have I (1) χ = z U (I κ(χ) ), hence u χ (a) = Ua/Ua·z U (I κ(χ) ). (3.3.3) Observe further that, the A 1 -action on Z(a) being trivial, it preserves the ideal Ua·I (1) χ , hence induces a well-defined A 1 -action on u χ (a) by algebra automorphisms. We set i χ := r(I χ ) = I χ /Ua·I (1) χ ⊂ u χ (a). (3.3.4) Thus, i χ is an A 1 -stable two-sided ideal in u χ (a) generated by the elements {x − χ(x)} x∈a . In the special case χ = 0, the restricted enveloping algebra u 0 (a) := Ua/Ua · I (1) 0 inherits from Ua the structure of a Hopf algebra. This Hopf algebra is dual to k[A 1 ], the coordinate ring of the Frobenius kernel A 1 . For any χ, the map ad x : u → x·u−u·x, x ∈ a, u ∈ u χ (a), extends to a well-defined u 0 (a)-action on u χ (a), that is, to an algebra map ad : u 0 (a) −→ End k u χ (a) . This u 0 (a)-action corresponds to the adjoint action on u χ (a) of the Frobenius kernel A 1 . Quantum Hamiltonian reduction. Let D be any associative, not necessarily commutative, k-algebra equipped with an algebraic action of the group A by algebra automorphisms and with an A-equivariant algebra map ρ : Ua → D such that the adjoint a-action on D, given by ad x : u → ρ(x) · u − u · ρ(x), x ∈ a, u ∈ D, is equal to the differential of the A-action. Let I ⊂ Ua be an Ad A-stable two-sided ideal. Then, D·ρ(I) is an A-stable left ideal in D. It is easy to verify that multiplication in D descends to a well-defined associative algebra structure on (D/D·ρ(I)) A , the space of A-invariants in D/D·ρ(I). Abusing notation, from now on we will write D·I instead of D·ρ(I). Remark 3.4.1. The algebra (D/D·I) A may be thought of as a 'Hamiltonian reduction' of D with respect to I. ♦ Observe also that, if u ∈ D is such that u mod(D·I) ∈ (D/D·I) A , then the operator of right multiplication by u descends to a well-defined map R u : D/D · I → D/D · I. Moreover, the assignment u → R u induces an algebra isomorphism (D/D·I) A ∼ −→ End D (D/D·I) opp . More generally, let M be a left D-module equipped with an A-equivariant structure (i.e., such that the action map D ⊗ M → M is A-equivariant). The algebra map Ua → D makes M an Ua-module. The space (M/I · M ) A acquires a natural left (D/D · I) A -module structure, to be called a Hamiltonian reduction of M . Similar construction applies to right D-modules. Next, fix χ ∈ X * (a), and let u χ (a) be the corresponding χ-restricted enveloping algebra. Recall that this algebra comes equipped with the adjoint action of A 1 , the Frobenius kernel. Let D be an associative algebra equipped with A 1 -action, and with ρ : u χ (a) → D, an A 1 -equivariant algebra morphism. One shows similarly that, given an A 1 -stable two-sided ideal i ⊂ u χ (a), there is a natural associative algebra structure on (D/D · i) A1 . In the special case i = i χ , see (3.3.4), the algebra (D/D · i χ ) A1 may be thought of as a quantum hamiltonian reduction of D with respect to the action of A 1 , an 'infinitesimal' group-scheme. Later on, we will be interested in the following special case of this construction. Let E be a finite dimensional A-module such that the induced action map ρ : Ua → End E descends to the algebra u χ (a), i.e., vanishes on the ideal I (1) χ ⊂ Z(a). Write E χ := {e ∈ E x(e) = χ(x) · e, ∀x ∈ a} for the χ-weight space of Ua. We put D := End E (= End k E). We claim that the quantum Hamiltonian reduction of the algebra D with respect to the u χ (a)-action is canonically isomorphic to End k E χ . In more detail, we form an associative algebra D χ := (D/D · i χ ) A1 . The natural action of D on E descends to a well-defined D χ -action on the weight space E χ . On the other hand, we have a right D χ -action on D/D · i χ . We are going to establish canonical algebra isomorphisms End D (D/D · i χ ) opp l ∼ = D χ r ∼ = End k E χ . (3.4.2) The isomorphisms above follow from a more general result below that involves two characters χ, ψ ∈ X * (a), such that κ(χ) = κ(ψ). In this case, in Ua we have I (1) χ = I (1) ψ . Hence there is a canonical identification u χ (a) = u ψ (a), and we may view the A-representation E either as an u χ (a)or as an u ψ (a)-module. We set D := End k E, as above, and consider the Hamiltonian reductions D χ and D ψ . We have a D χ -action on the weight space E χ ⊂ E, and a similar D ψ -action on the weight space E ψ . Observe that the natural D-bimodule structure on the algebra D, via left and right multiplication, descends to a D-D χ -bimodule structure on the vector space Hom k (E χ , E). Further, the right D χ -action on D/D · i χ , resp., D ψ -action on D/D · i ψ , gives the following space a natural D χ -D ψ -bimodule structure ψ D χ := Hom D D/D·i χ , D/D·i ψ ψ−χ , (3.4.3) where the superscript 'ψ − χ' denotes the (ψ − χ)-weight component of the natural (adjoint) A-action. Lemma 3.4.4. (i) The restriction map End E → Hom(E χ , E), resp. End E → Hom(E ψ , E), induces a D-D χ -bimodule, resp. D-D ψ -bimodule, isomorphism D D·i χ res ∼ −→ Hom k (E χ , E), resp., D D·i ψ res ∼ −→ Hom k (E ψ , E). (ii) We have the following D χ -D ψ -bimodule isomorphisms ψ D χ res ∼ −→ Hom D Hom k (E χ , E), Hom k (E ψ , E) ψ−χ ∼ ←− Hom k (E χ , E ψ ). (3.4.5) For χ = ψ, it follows that the map l in (3.4.2) is an algebra isomorphism, furthermore, the composite map in (3.4.5) induces an algebra isomorphism D χ = χ D χ ∼ −→ End k (E χ ), which gives the isomorphism r in (3.4.2). Sketch of Proof. Associated to a vector subspace F ⊂ E, one has a left ideal J F ⊂ D, defined by J F := {f ∈ D = End k E f \| F = 0}. Moreover, any left ideal J ⊂ D has the form J = J F where the corresponding subspace F ⊂ E can be recovered from J by the formula F = f ∈J Ker f . Applying this to the left ideal J = D · i χ we get D · i χ = {f ∈ End k E f \| F = 0}, where F = x∈iχ Ker ρ(x). The latter space equals E χ , by definition. Thus, we deduce D · i χ = {f ∈ End k E f \| Eχ = 0} = Hom k (E/E χ , E), hence D/D · i χ ∼ = Hom k (E χ , E). The rest of the proof is an elementary exercise which we leave for the reader. 3.5 Moment maps. Let A be a linear algebraic group as in sect.3.1, and let A × X → X be an algebraic action on X, a smooth k-variety. Any element x ∈ a gives rise to an algebraic vector field ξ x on X. We may view ξ x as a regular function on T * X. This way, the assignment x → ξ x extends uniquely to an A-equivariant Poisson algebra map µ alg : Sym a → k[T * X]. Since Sym a ∼ = k[a * ], this algebra map induces an A-equivariant morphism µ : T * X → a * , called moment map, such that the algebra map µ alg becomes the pull-back via µ. There is also a noncommutative analogue of the Poisson algebra map µ alg . Specifically, the Lie algebra morphism x → ξ x extends uniquely to an A-equivariant associative algebra homomorphism µ U : Ua → D X (more generally, given an A-equivariant vector bundle L, one defines similarly an associative algebra homomorphism µ U : Ua → D(X, L)). The morphism µ U is compatible with natural filtrations, hence, induces a canonical graded algebra homomorphism µ R : Rees Ua −→ Rees D X . We may view the later homomorphism as a map Rees Ua −→ π qRD (1) , cf. Sect. 2.5. We have the following commutative diagram: Sym a (1) _ (3.1.1) z RU : x →x p −t p−1 ·x [p] µ (1) alg / / Z X (1) = Sym T X (1) = π qO T * X (1) _ (2.5.1) z RD : ξ →ξ p −t p−1 ·ξ [p] Rees Ua µ R / / Rees(Fr qDX ) = π qRD (1) . (3.5.1) The map µ R in the bottom row specializes at t = 1 to the map µ R \| t=1 = µ U : Ua → D X considered earlier, and specializes at t = 0 to the map µ R \| t=0 = µ alg . The vertical maps in diagram (3.5.1) are the p-graded algebra morphisms considered earlier. Further, fix χ ∈ X * (a), let I (1) χ ⊂ Z(a) be the corresponding ideal, cf. Definition 3.3.1, and µ U (I (1) χ ) ⊂ Z X (1) its image in Fr qDX = π qD (1) . From commutativity of diagram (3.5.1) for t = 1 and formula (3.3.3), we deduce that the canonical isomorphism Z X (1) ∼ −→ π qO T * X (1) takes the ideal µ U (I (1) χ ) to the ideal µ alg (I κ(χ) ) ⊂ π qO T * X (1) . On the other hand, it follows from Poincaré-Birkhoff-Witt theorem that the associated graded ideal gr I (1) χ equals the the augmentation ideal in gr Z(a) = Sym a (1) . Thus, specializing diagram (3.5.1) at t = 0 and t = 1, respectively, we find µ U (I (1) χ ) = µ alg (I κ(χ) ), resp., gr µ U (I (1) χ ) = µ alg (I 0 ). (3.5.2) We introduce the following subscheme in T * X (1) : T 1, * κ(χ) := zero scheme of µ U (I (1) χ ) = zero scheme of µ alg (I κ(χ) ) = µ (1) −1 (κ(χ)), (3.5.3) the scheme-theoretic fiber of the moment map µ (1) : T * X (1) → a 1, * over the point κ(χ), cf. (3.2.2). For example, if f : A → G m is an algebraic group character and χ := dlog f , then we have κ(χ) = 0, see (3.2.4). Hence, T 1, * κ(χ) = µ −1 (0) (1) . 3.6 Hamiltonian reduction of differential operators. We keep the setup of Sect. 3.5. Thus, we have an Ad A-invariant homomorphism χ : a → k, the corresponding ideal I (1) χ ⊂ Z(a), and its image µ U (I (1) χ ) ⊂ π qD (1) under the map µ U : Ua → π qD (1) , induced by the A-action. This image is a central subalgebra in the Azumaya algebra D (1) on T * X (1) , hence, D (1) · π q µ U (I (1) χ ) ⊂ D (1) is an A-stable two-sided ideal. We put D (1) χ := D (1) /D (1) · π q µ U (I (1) χ ). Since π q µ U (I (1) χ ) = π q µ alg (I κ(χ) ), by formula (3.5.2), we have D (1) χ = D (1) /D (1) · π q µ U (I (1) χ ) = D (1) /D (1) · π q µ alg (I κ(χ) ) = D (1) T 1, * κ(χ) . (3.6.1) We see that D (1) χ is a restriction of the sheaf D (1) to T 1, * κ(χ) , the scheme-theoretic fiber of the moment map, see (3.5.3). Thus, D (1) χ is a coherent sheaf of associative algebras on the subscheme T 1, * κ(χ) . By construction, the map Ua → D(X) = Γ(T * X (1) , D (1) ) descends, in view of exact sequence (3.3.2), to an A 1 -equivariant algebra homomorphism ρ χ : u χ (a) −→ Γ T * X (1) , D (1) χ . (3.6.2) Recall the two-sided ideal i χ ⊂ u χ (a), see (3.3.4), and put E χ := D (1) χ D (1) χ ·i χ A1 . (3.6.3) This is a sheaf on T 1, * κ(χ) that may be thought of as a Hamiltonian reduction of the algebra D (1) χ with respect to the action of A 1 . The construction of Sect. 3.4 applied to D := D (1) χ and to the homomorphism (3.6.2), gives E χ the natural structure of a coherent sheaf of associative algebras on the scheme T 1, * κ(χ) ⊂ T * X (1) . Observe further that the A-action on D (1) factors, when restricted to A 1 -invariants, through the quotient A (1) = A/A 1 . Thus, the sheaf E χ acquires an A (1) -equivariant structure. On the other hand, rather than performing the Hamiltonian reduction of D (1) χ with respect to the A 1 -action, one may perform the Hamiltonian reduction of D (1) , a larger object, with respect to the action of A, a larger group, that is, to consider A-invariants in D (1) D (1) ·I χ (abusing notation, we will write D (1) ·I χ instead of D (1) ·π q µ U (I χ ), and use similar notation in other cases). The elementary result below is a manifestation of the general principle saying that Hamiltonian reduction can be performed in stages: to make a Hamiltonian reduction with respect to A, one can first perform Hamiltonian reduction with respect to A 1 , and then make reduction with respect to A (1) = A/A 1 . Lemma 3.6.4. There is a canonical algebra isomorphism Γ(X, D X /D X ·I χ ) A ∼ = Γ T 1, * κ(χ) , E χ A (1) . Proof. It is clear that in D (1) /D (1) ·I (1) χ , one has an equality D (1) ·I χ D (1) ·I (1) χ = D (1) χ ·i χ . Thus, we obtain D (1) D (1) ·I χ ∼ = D (1) χ D (1) χ ·i χ . Taking A 1 -invariants on both sides, we deduce an isomorphism (of sheaves of associative algebras on T 1, * κ(χ) ): D (1) D (1) ·I χ A1 ∼ = D (1) χ D (1) χ ·i χ A1 = E χ . (3.6.5) Applying the functor Γ(T * X (1) , −) A (1) to A (1) -equivariant sheaves in (3.6.5), we obtain a chain of canonical algebra isomorphisms Γ(X, D X /D X ·I χ ) A ∼ = Γ T * X (1) , D (1) /D (1) ·I χ A ∼ = Γ T * X (1) , (D (1) /D (1) ·I χ ) A1 A (1) (by (3.6.5)) ∼ = Γ T * X (1) , E χ A (1) ∼ = Γ T 1, * κ(χ) , E χ A (1) . The Lemma follows. 3.7 The case of free A-action. Keep the notation of §3.6 and let χ = 0, hence I χ = I + ⊂ Ua is the augmentation ideal. Write D X · I + = D X · a for the left ideal generated by the image of I + under the homomorphism µ U : Ua → D(X). Applying the construction of section 3.4 to the algebra D := D X and the two-ideal I + ⊂ Ua one gets an associative algebra D X /D X · I + A . Assume now that the A-action on X is free and, moreover, there is a smooth variety Y , and a smooth universal geometric quotient morphism pr Y : X ։ Y (whose fibers are exactly the A-orbits), see [GIT,Definition 0.7]. It is well-known that the algebra of differential operators on Y can be expressed in terms of differential operators on X as follows D Y ∼ = (pr Y ) q(DX /D X · I + ) A = (pr Y ) q(DX /D X · µ U (a)) A . (3.7.1) More generally, fix an algebraic homomorphism χ : A → G m . Given a free A-action on X, let O Y (χ) be an invertible sheaf on Y defined as the subsheaf of (pr Y ) qOX formed by the functions f such that a * (f ) = χ(a)·f, ∀a ∈ A. Let D Y (O Y (χ)) be the corresponding sheaf of twisted differential operators, and D(Y, χ) := Γ(Y, D Y (O Y (χ)) the algebra of its global sections. There is a χ-twisted version of formula (3.7.1) that provides a canonical isomorphism (pr Y ) q(DX /D X ·I χ ) A ∼ −→ D Y (χ) (isomorphism of sheaves of algebras on Y ) . Taking global sections on each side of the isomorphism, we get algebra isomorphisms: Γ T 1, * κ(χ) , E χ A (1) Lemma 3.6.4 ∼ / / Γ X, D X /D X ·I χ A ∼ = D(Y, χ). (3.7.2) The isomorphism above makes sense, in effect, not only for χ ∈ X * (A), but also in a slightly more general setting where χ ∈ X * (a) is an Ad A-invariant Lie algebra character that does not necessarily exponentiate to an algebraic group homomorphism A → G m . Although, generally, the sheaf O Y (χ) is not defined in such a case, the corresponding sheaf D Y (χ) of twisted differential operators is always well-defined, cf. [BB], and the isomorphism in (3.7.2) still holds. Remark 3.7.3. The algebra D(Y, χ) may be thought of as a quantization of the commutative algebra k[µ −1 (χ)] A , the coordinate ring of the Hamiltonian reduction of T * X with respect to the 1-point orbit {χ} ⊂ a * and the moment map µ : T * X → a * . ♦ 4 Azumaya algebras via Hamiltonian reduction 4.1 The main result. Let X be a smooth A-variety. Below, we are going to extend considerations of section 3.7 to a more general case where the A-action on X is not necessarily free, but the corresponding Hamiltonian A-action on T * X is free on an open subset of T * X. There is a natural action of the multiplicative group G m on the vector space a * and also on the vector bundle T * X, by dilations. The moment map µ : T * X → a * is clearly compatible with these two actions. It is also compatible with the A-actions, and the latter commute with the G m -actions. Thus, µ is an equivariant morphism between G m × A-varieties. Let µ −1 (0) ⊂ T * X be the scheme-theoretic zero fiber of the moment map. This is clearly a G m × A-stable subscheme in T * X. From now on, we make the following Basic Assumptions 4.1.1. There is a Zariski open G m × A-stable subscheme M ⊂ µ −1 (0) which is a reduced smooth locally-closed connected subvariety in T * X such that • The differential of the moment map µ : T * X → a * is surjective at any point of M; • The A-action on M is free, moreover, there is a smooth variety M and a smooth universal geometric quotient morphism M → M (in particular, it is a principal A-bundle whose fibers are precisely the A-orbits in M), see [GIT]; The assumptions above insure that the standard symplectic structure on T * X induces a symplectic structure on M . Thus, the manifold M may be thought of as a Hamiltonian reduction of T * X at 0. Now, let χ ∈ X * (a) be such that κ(χ) = 0. Recall the notation T 1, * κ(χ) := [µ (1) ] −1 (κ(χ)). The equation κ(χ) = 0 implies that T 1, * κ(χ) = [µ −1 (0)] (1) , which is clearly a G m × A (1) -stable subscheme in T * X (1) . Further, by the Basic Assumptions, the scheme [µ −1 (0)] (1) contains M (1) , the Frobenius twist of M, as an open subscheme. Thus, for any χ ∈ X * (a) such that κ(χ) = 0, the Basic Assumptions yield the following diagram T 1, * κ(χ) = [µ −1 (0)] (1) M (1) ? _ o o ̟ principal A (1) -bundle / / / / M (1) . (4.1.2) We are going to define a coherent sheaf A χ on M (1) that will be an Azumaya O M (1) -algebra of degree p 1/2 dim M , to be called the quantum hamiltonian reduction of D X at κ(χ). To this end, we restrict E χ = (D (1) χ /D (1) χ · i χ ) A1 , an A (1) -equivariant sheaf on T 1, * κ(χ) ⊂ T * X (1) , cf. (3.6.3) , to the open subset M (1) , and consider the push-forward of that restriction under the map ̟ : M (1) −→ M (1) , cf. (4.1.2). Definition 4.1.3. We define the following coherent sheaf of associative algebras on M (1) : A χ := ̟ q E χ M (1) A (1) , and put A χ := Γ(M (1) , A χ ). Assume next that we are given two points χ, ψ ∈ X * (a), such that the character χ − ψ : a → k can be exponentiated to a group homomorphism A → G m , i.e., such that χ − ψ ∈ dlog(X * (A)). Then, formula (3.2.4) yields κ(χ) = κ(ψ). Hence we have T 1, * κ(χ) = T 1, * κ(ψ) , and we may view the set M as a geometric quotient of an open subset of either T 1, * κ(χ) or T 1, * κ(ψ) . The result below that will play a key role in subsequent sections, is a generalization of [BK,Proposition 4.8]. Theorem 4.1.4. Let χ ∈ a * be an A-fixed point such that κ(χ) = 0 and such that the Basic Assumptions 4.1.1 hold. Then we have (i) The sheaf A χ is a sheaf of Azumaya algebras on M (1) equipped with a canonical algebra morphism Ξ χ : Γ(X, D X /D X · I χ ) A −→ Γ(M (1) , A χ ). (ii) For all i > 0, we have H i (M (1) , A χ ) = 0. (iii) If the algebra A χ = Γ(M (1) , A χ ) has finite homological dimension then the functors below give mutually inverse equivalences of bounded derived categories of sheaves of coherent A χ -modules and finitely generated A χ -modules, respectively: D b (A χ -Mod) L −→ RΓ(M (1) ,L ) / / D b (A χ -Mod). Aχ L ⊗ Aχ L ←− L o o (iv) Let ψ ∈ X * (a) be another point satisfying all the assumptions above and such that χ − ψ ∈ dlog(X * (A)). Then the corresponding Azumaya algebras A χ and A ψ are Morita equivalent (but not necessarily isomorphic). The rest of this section is devoted to the proof of the Theorem. As will be explained in Sect. 4.5 below, part (iii) of Theorem 4.1.4 is entirely due to [BK,Proposition 2.2]. A result similar to part (iv) of the theorem is also contained in an updated version of [BMR,§2.3.1]. Remark 4.1.5. (i) The assumption of the Theorem that κ(χ) = 0 may be relaxed, as will be explained elsewhere. (ii) We will show, in the course of the proof of Theorem 4.1.4, that the sheaf E χ M (1) is also an Azumaya algebra, specifically, we have an Azumaya algebra isomorphism: E χ M (1) = ̟ * A χ . (4.1.6) (iii) It will also follow from the proof that the sheaf A χ , viewed as a vector bundle on M (1) , is a deformation of the vector bundle Fr qOM ; in particular, in the Grothendieck group K(Coh(M (1) )) on has [A χ ] = [Fr qOM ], furthermore, R i f (1) q A χ = 0 for any i > 0, where f (1) : M (1) → M (1) aff denotes the affinization morphism. Deformation construction. We are going to apply the Rees algebra formation to all the objects involved in the construction of the algebra A χ . In more detail, the standard filtration on Ua induces a filtration on the ideal I χ ⊂ Ua, and we form a graded ideal Rees I χ ⊂ Rees Ua, which is generated by the set {x−t · χ(x)} x∈a . Further, let RZ(a) be a k[t]-subalgebra in Rees Ua generated by the image of the homomorphism z RU : Sym a (1) → Rees Ua, see (3.1.1). Thus, RZ(a) ⊂ Rees Ua is a graded central subalgebra, and RI (1) χ := RZ(a) ∩ Rees I χ is a graded ideal in RZ(a). For any x ∈ a, we have x p − t p−1 · x [p] −t p · κ(χ)(x) = z RU (x) − t p · κ(χ)(x) ∈ RZ(a). On the other hand, the following equations x p − t p−1 ·x [p] − t p ·κ(χ)(x) = x p − t p ·χ(x) p − t p−1 ·x [p] − t p ·χ [1] (x) = x − t·χ(x) p − t p−1 · x [p] − t·χ(x [p] ) ∈ Rees I χ show that x p − t p−1 · x [p] − t p ·κ(χ)(x) ∈ RI (1) χ . Moreover, it is easy to verify that the elements of this form generate RI (1) χ as an ideal. Next, we apply the Rees algebra construction to the filtered sheaf Fr qDX . Inside Fr qDX , we have two left ideals Fr qDX · µ U (I (1) χ ) ⊂ Fr qDX · µ U (I χ ), generated by the images of the sets I (1) χ ⊂ I χ ⊂ Ua, respectively, under the moment map µ U : Ua → Fr qDX . The filtration on Fr qDX induces by restriction natural filtrations on Fr qDX · µ U (I (1) χ ) and Fr qDX · µ U (I χ ). Thus, we obtain graded ideals Rees(Fr qDX · µ U (I (1) χ )) ⊂ Rees(Fr qDX · µ U (I χ )) ⊂ Rees(Fr qDX ). On the other hand, we have a moment map µ R : Rees Ua → Rees(Fr qDX ), which takes the central subalgebra RZ(a) ⊂ Rees Ua into the central subalgebra Rees Z X (1) ⊂ Rees(Fr qDX ), cf. diagram (3.5.1). Hence, the image of RI (1) χ is a subalgebra µ R (RI (1) χ ) ⊂ Rees Z X (1) , and we have µ R (RI (1) χ )·Rees(Fr qDX ) ⊂ Rees(Fr qDX ·µ U (I (1) χ )), where the inclusion is strict, in general. Further, we have a G m -equivariant sheaf RD (1) on A 1 × T * X (1) corresponding to the graded algebra Rees(Fr qDX ). The above constructed graded ideals in the algebra Rees(Fr qDX ) give rise to the following three G m -equivariant sheaves of left ideals in RD (1) : RD (1) ·{x p − t p−1 · x [p] − t p · κ(χ)(x)} x∈a = RD (1) · RI (1) χ (4.2.1) ⊂ R(D (1) · I (1) χ ) ⊂ R(D (1) · I χ ), where we follow our usual convention to drop the symbols µ U and µ R from the notation. By (2.4.3), we have a graded algebra isomorphism Rees Z X (1) ∼ = O A 1 ×T * X (1) . Thus, µ R (RI (1) χ ) may be viewed as a subset in O A 1 ×T * X (1) , and from commutativity of diagram (3.5.1) we deduce that the set µ (1) alg (x)−t p ·κ(χ)(x) x∈a generates the ideal µ R (RI (1) χ )·O A 1 ×T * X (1) . Thus, in A 1 ×T * X (1) , we have Zero-scheme of µ R (RI (1) χ ) = {(t, ξ) ∈ A 1 × T * X (1) µ (1) (ξ) = t p ·κ(χ)}. (4.2.2) The sheaf RD (1) /RD (1) ·RI (1) χ is clearly supported on the subscheme (4.2.2). We conclude that its quotient RD (1) /R(D (1) ·I χ ), cf. (4.2.1), is supported on the subscheme (4.2.2) as well. Now let χ ∈ X * (a) be such that κ(χ) = 0. Then the set in (4.2.2) reduces to a direct product A 1 × [µ −1 (0)] (1) . One checks further, going through the identifications used above, that the G m × A (1) -action on A 1 × T 1, * κ(χ) = A 1 × [µ −1 (0)] (1) arising from the natural grading and from the A (1)action on RI (1) χ , respectively, is the one where the group G m acts diagonally, and the group A (1) acts only on the factor [µ −1 (0)] (1) . Recall the sheaf RD (1) /R(D (1) · I χ ), which is supported on A 1 × [µ −1 (0)] (1) , since κ(χ) = 0. We restrict this sheaf to the open subscheme A 1 × M (1) ⊂ A 1 × [µ −1 (0)] (1) . We have the principal G m -equivariant A (1) -bundle Id A 1 ⊠ ̟ : A 1 × M (1) −→ A 1 × M (1) . We define RF := (Id A 1 ⊠ ̟) q RD (1) /R(D (1) · I χ ) . (4.2.3) This is a G m -equivariant quasi-coherent sheaf of A-modules on A 1 × M (1) which is flat over the A 1 -factor. Let RF A denote the subsheaf of its A-invariant sections. Lemma 4.2.4. There is a G m -equivariant sheaf isomorphism RF A \| {0}×M (1) ∼ = Fr qOM . In the course of the proof below, we will repeatedly use the following elementary result RF A /t·RF A ∼ = Γ(Y (1) , Fr qOY ) = k[Y] = k[Y ] A . (4.2.6) To prove (4.2.6), let • T * X be the Zariski open (possibly empty) subset in T * X formed by the points ξ ∈ T * X such that the differential of µ : T * X → a * is surjective at ξ. Further, let J ⊂ Sym a = k[a * ] denote the augmentation ideal. It is clear that µ −1 (0) is the zero scheme of the ideal O T * X · µ alg (J ) ⊂ O T * X , and that this ideal is reduced at any point of µ −1 (0) ∩ • T * X. It follows, since gr I χ = J , that on the Frobenius twist of • T * X, one has: gr Fr q(DX ·I χ ) • T * X (1) = (gr Fr qDX )·(gr Fr qIχ) • T * X (1) = Fr q O T * X ·µ alg (J ) • T * X (1) , where we identify gr(Fr qDX ) with the corresponding G m -equivariant sheaf on T * X (1) . We now use our Basic Assumptions saying that the differential of µ is surjective at any RF /t·RF ∼ = Γ Y (1) , RF \| {0}×Y (1) ∼ = Γ(Y (1) , ̟ qFr qOY ) ∼ = Γ(Y (1) , Fr qOY ) = k[Y ]. (4.2.8) We conclude, comparing the above isomorphisms with those in (4.2.6) that proving the Lemma reduces to the following result: The canonical map below gives, for Y sufficiently small, an isomorphism RF A /t·RF A ∼ −→ (RF /t·RF) A . (4.2.9) To prove this, we may assume, shrinking Y if necessary, that there is anétale map θ : Y → Y such that the principal A-bundle Y → Y becomes trivial after pull-back via θ, i.e., that there is an G m × A-equivariant isomorphism Y × Y Y ∼ = A × Y. Usingétale base change for the Cartesian square A × Y Y × Y Y θ / / Y Y θ / / Y we obtain G m × A-equivariant graded algebra isomorphisms k[ Y ] = k[ Y] ⊗ k[Y] k[Y ] ∼ = k[Y × Y Y] ∼ = k[A] ⊗ k[ Y]. (4.2.10) In the leftmost term of this formula, we have used the notation Y := Y × Y Y, and in the rightmost term of the formula, the group A acts trivially on the factor k[ Y]. Next, we set RF := Γ A 1 × Y (1) , (Id A 1 × θ (1) ) * RD (1) /R(D (1) · I χ ) \| A 1 ×Y (1) = k[A 1 × Y (1) ] k[A 1 ×Y (1) ] RF . Since k[ Y (1) ] is flat over k[Y (1) ], from (4.2.8) and (4.2.10) we find RF/t· RF = k[ Y (1) ] ⊗ k[Y (1) ] (RF /t·RF) = k[ Y (1) ] ⊗ k[Y (1) ] k[Y ] = k[ Y ] ∼ = k[A] ⊗ k[ Y].0 −→ RF t −→ RF −→ RF/t· RF −→ 0, we deduce the isomorphism RF A /t· RF A ∼ −→ ( RF/t· RF) A . The latter isomorphism yields (4.2.9) since the morphism Y (1) → Y (1) is faithfully flat. The Lemma is proved. 4.3 Deformation of the algebra A χ . We mimick formulas (3.6.1) and (3.6.3) and put RD (1) χ := RD (1) /R(D (1) · I (1) χ ), and (4.3.1) RE χ := RD (1) /R(D (1) · I χ ) A1 ∼ = RD (1) χ R(D (1) χ ·i χ ) A1 . The sheaves RD (1) χ and RE χ are both supported on A 1 × [µ −1 (0)] (1) , see (4.2.2). Following the same strategy as has been used in the construction of the sheaf A χ , we define RA χ := ̟ q RE χ A 1 ×M (1) A (1) . This is a G m -equivariant sheaf of associative algebras on A 1 × M (1) , viewed as a G m -variety with diagonal action. Write pr 2 : A 1 × M (1) → M (1) for the second projection. Lemma 4.3.2. The sheaf RA χ is flat over the A 1 -factor, and we have RA χ {0}×M (1) ∼ = Fr qOM , and RA χ (A 1 {0})×M (1) ∼ = pr * 2 A χ . Proof. Both the flatness statement and the isomorphism on the right are immediate from the corresponding properties of the Rees algebra. It remains to study the restriction of the sheaf RA χ to the special divisor {0} × M (1) . From the definition, we find RA χ {0}×M (1) ∼ = (Id A 1 ⊠ ̟) qREχ A (1) {0}×M (1) ∼ = (Id A 1 ⊠ ̟) q RD (1) /R(D (1) ·I χ ) A1 A (1) {0}×M (1) ∼ = (Id A 1 ⊠ ̟) q RD (1) /R(D (1) ·I χ ) A {0}×M (1) = RF A {0}×M (1) . But RF A{0}×M (1) ∼ = Fr qOM by Lemma 4.2.4, and we are done. Next, we set R := Γ(A 1 × M (1) , RA χ ), a graded flat k[t]-algebra such that R/(t − 1)R = Γ(M (1) , A χ ). Applying formula (2.4.2) to the algebra R := R and using Lemma 4.3.2, we get a natural increasing filtration on the algebra Γ(M (1) , A χ ) such that for the associated graded algebra, to be denoted gr R Γ(M (1) , A χ ), we have gr R Γ(M (1) , A χ ) = R/tR. On the other hand, the G m -action induces a grading on the algebra Γ(M , O M ) = Γ(M (1) , Fr qOM ). Recall the affinization morphism f : M → M aff . Given ξ ∈ M (1) aff , let M (1) ξ := [f (1) ] −1 (ξ) ⊂ M (1) be the fiber of f (1) : M (1) → M(1) aff over ξ. Proposition 4.3.3. (i) The sheaf A χ is locally free, and H i (M (1) , A χ ) = 0 for all i > 0. (ii) There is a graded algebra isomorphism gr R Γ(M (1) , A χ ) ∼ = Γ(M , O M ). (iii) For any q > 0 we have R q f (1) * A χ = 0 and H q (M (1) ξ , A χ ) = 0, ∀ξ ∈ M(1) aff . Proof. We consider the following diagram M Fr / / f M (1) = {0} × M (1) f (1) ĩ / / A 1 × M (1) f :=Id A 1 ×f (1) M aff Fr / / M (1) aff = {0} × M (1) aff i / / A 1 × M(1) aff . (4.3.4) Lemma (4.3.2) says thatĩ * RA χ ∼ = Fr qOM . Thus, RA χ is a G m -equivariant sheaf on A 1 ×M (1) such that its restriction to the subvariety {0} × M (1) is a locally free sheaf. It follows that the sheaf RA χ must be itself locally free. Indeed, every point in M (1) has a G m -stable affine Zariski open neighborhood. Taking global sections of RA χ over such a neighborhood, we see that our claim reduces to Lemma 4.2.5. Thus, we have proved that the sheaf RA χ , hence its restriction to {1} × M (1) , is a locally free sheaf. But RA χ {1}×M (1) = A χ , thus, the first claim of part (i) of the Lemma is proved. Recall further that the variety M is symplectic, hence it has trivial canonical bundle. Therefore, by the Grauert-Riemenschneider theorem (see [EV] for char k > 0 case), the higher direct image sheaves R q f * O M vanish for all q > 0. Therefore, from the commutative square on the left of (4.3.4) we deduce R q f (1) * (ĩ * RA χ ) = R q f (1) * Fr qOM = 0, for all q > 0. (4.3.5) We are going to use (4.3.5) to prove part ( aff . Let RA χ D×M (1) denote the restriction of RA χ to D × M (1) . We will view RA χ D×M (1) as a sheaf of abelian groups on the closed fiber M (1) = {0} × M (1) . This sheaf is clearly isomorphic to an inverse limit of sheaves which are iterated extensions of the sheafĩ * RA χ = Fr qOM . Hence, formula (4.3.5) implies that R q f * (RA χ D×M (1) ) = 0, for all q > 0. On the other hand, let i : D × M (1) aff ֒→ A 1 × M(1) aff denote the imbedding. For any q = 0, 1, . . . , one has i * R qf * RA χ = R q f * (RA χ D×M (1) ), by the Formal Functions Theorem, cf. [Har,III,11.1] or [EGA III,Sect. 4]. Thus, we have proved i * R qf * RA χ = R q f * (RA χ D×M (1) ) = 0, for all q > 0. (4.3.6) Observe next that, for each q, the sheaf R qf * RA χ is a G m -equivariant coherent sheaf on A 1 × M (1) aff . Therefore, the support of R qf * RA χ is a G m -stable closed subscheme in A 1 × M (1) aff . The assumption that the G m -action on M be attracting implies that any non-empty G m -stable closed subscheme in A 1 × M aff , from the vanishing result of part (iii) and the second isomorphism of Lemma 4.3.2, for any q > 0, we find 0 = Γ (A 1 {0}) × M (1) aff , R qf * RA χ = H q (A 1 {0}) × M (1) , RA χ = H q (A 1 {0}) × M (1) , pr * 2 A χ = k[t, t −1 ] ⊗ H q (M (1) , A χ ), This completes the proof of part (i) of the Proposition. Now, the (ordinary) direct image sheaff * RA χ is by construction flat over A 1 , hence, L q i * (f * RA χ ) = 0 for all q > 0. Therefore, the vanishing of the higher direct images implies that the Proper Base Change theorem for the Cartesian square on the right of diagram (4.3.4) involves no higher derived functors. Thus, using Base Change and the definition of affinization we obtain i * f * RA χ = f (1) * ĩ * RA χ = f (1) * Fr * O M = Fr qf * OM = Fr qOM aff . M denote the restriction of the Azumaya algebra D (1) to the subset M (1) ⊂ T 1, * κ(χ) = T 1, * κ(ψ) . We put χ E ψ := Hom D (1) M (D (1) M /D (1) M · i χ , D (1) M /D (1) M · i ψ ). This is an A (1) -equivariant sheaf on M (1) . Therefore, ̟ q(χEψ) is a sheaf on M (1) with fiberwise A (1) -action. Let χ A ψ be the (ψ − χ)-weight component of ̟ q(χEψ). It is clear that χ A ψ is a coherent sheaf of A χ -A ψ -bimodules. Next, we mimic the argument in Sect. 4.2 and observe that the standard increasing filtration on ̟ qD (1) M induces a natural increasing filtration on ̟ q(χEψ). Therefore, using the Rees algebra construction, we may form a G m -equivariant sheaf R( χ E ψ ) on A 1 × M (1) . This gives an increasing filtration on Γ(M (1) , χ A ψ ), an A χ -A ψ -bimodule, and we write gr R Γ(M (1) , χ A ψ ) for the associated graded (gr A χ )-(gr A ψ )-bimodule. Lemma 4.4.1. (i) The sheaf χ A ψ is locally free, moreover, we have a natural E χ -E ψ -bimodule isomorphism χ E ψ = ̟ * ( χ A ψ ). (ii) Assume that R q f * O M (χ − ψ) = 0 holds for all q > 0. Then, there is a graded (gr A χ )-(gr A ψ )-bimodule isomorphism gr R Γ(M (1) , χ A ψ ) ∼ = Γ(M , O M (χ − ψ)). Moreover, we have R q f (1) * ( χ A ψ ) = 0 and H q (M (1) , χ A ψ ) = 0, ∀q > 0. Proof. We argue as in Proposition 4.3.3, and show first that (Id A 1 × ̟) qR(χEψ) ψ−χ {0}×M (1) ∼ = Fr qOM (ψ − χ). (4.4.2) Now, the sheaf Fr qOM (ψ − χ) is clearly locally free. Hence, arguing as in (4.1.6) we deduce that the sheaf R( χ E ψ ) {0}×M (1) is a locally free sheaf on M (1) which is isomorphic to ̟ * Fr qOM (ψ − χ). Using Lemma 4.2.5 one shows that R( χ E ψ ) is a locally free sheaf on A 1 × M (1) . Thus, χ E ψ = R( χ E ψ ) {1}×M (1) is a locally free sheaf on M (1) and, moreover, we have χ E ψ = ̟ * ( χ A ψ ). Remark 4.4.3. The above argument shows that there is a flat family (over A 1 ) of coherent sheaves such that nonzero members of the family are all isomorphic to χ A ψ and the fiber over 0 ∈ A 1 is isomorphic to Fr qOM (ψ − χ). In particular, in the Grothendieck group of coherent sheaves on M (1) one has an equality [ χ A ψ ] = [Fr qOM (ψ − χ)]. Proof of Theorem 4.1.4. We have A χ = ̟ q E χ M (1) A (1) . The sheaf A χ in the left hand side of this equality is locally free, by Proposition 4.3.3(i). It follows that the sheaf E χ M (1) is also locally free, moreover, we have E χ M (1) ∼ = ̟ * A χ , see (4.1.6). Now, let x ∈ M (1) and writex := ̟(x) ∈ M (1) for its image. Let E x , resp. Ax, denote the geometric fiber at x, resp. atx, of the corresponding locally free sheaf. We deduce from E χ M (1) ∼ = ̟ * A χ that there is an algebra isomorphism E x ∼ = Ax, for all x ∈ M (1) andx := ̟(x) ∈ M (1) . (4.5.1) Thus, to prove that A χ is an Azumaya algebra, it suffices to show that E x is a matrix algebra, for any x ∈ M (1) . By definition, we have E x = (D (1) x /D (1) x · i χ ) A1 , where D (1) x is the geometric fiber at x of the sheaf D (1) . We know that D (1) is an Azumaya algebra on T * X (1) . Hence, there is a vector space E and an algebra isomorphism D (1) x ∼ = End k E. Using this, from the last statement of Lemma 3.4.4 we deduce the following algebra isomorphisms E x = (D (1) x /D (1) x · i χ ) A1 ∼ = End E End E ·i χ A1 = D χ ∼ = End k (E χ ). Thus, E x is a matrix algebra, as claimed. This proves that A χ is an Azumaya algebra. To complete the proof of part (i) of the Theorem, consider the following chain of canonical algebra maps Γ(X, D X /D X ·I χ ) A Lemma 3.6.4 −−− −−→ ∼ Γ T 1, * κ(χ) , E χ A (1) restriction −−− −−→ Γ M (1) , E χ M (1) A (1) (4.5.2) ∼ = Γ M (1) , ̟ q(Eχ M (1) ) A (1) ∼ = Γ M (1) , ̟ q(Eχ M (1) ) A (1) ∼ = Γ(M (1) , A χ ). The composite map provides the algebra map claimed in part (i) of Theorem 4.1.4. Part (ii) of the Theorem follows directly from the cohomology vanishing in Proposition 4.3.3(i); To prove (iv), we fix a point x ∈ M (1) , and write D (1) x ∼ = End k E. We know that for the geometric fibers (at x) of the Azumaya algebras E χ , resp., E ψ , one has the following formulas (E χ ) x = End k E χ , resp. (E ψ ) x = End k E ψ . Further, the sheaf χ E ψ is locally free by Lemma 4.4.1(i). The fiber of that sheaf at x is an (E χ ) x -(E ψ ) x -bimodule, and Lemma 3.4.4(ii) yields the following (E χ ) x -(E ψ ) x -bimodule isomorphisms, cf. (3.4.3): ( χ E ψ ) x ∼ = Hom D (1) x D (1) x /D (1) x ·i χ , D (1) x /D (1) x ·i ψ ψ−χ ∼ = Hom End E End E End E ·i χ , End E End E ·i ψ ψ−χ = ψ D χ ∼ = Hom k (E χ , E ψ ). Thus, the sheaf χ E ψ is a sheaf of locally-projective E χ -E ψ -bimodules. In particular, the Azumaya algebras E χ and E ψ are Morita equivalent. Now, forx := ̟(x) ∈ M (1) , we have ( χ A ψ )x = ( χ E ψ ) x . Therefore, we see from (4.5.1) and Lemma 4.4.1(i) that the sheaf χ A ψ is a sheaf of locally-projective A χ -A ψ -bimodules. Hence, it provides the required Morita equivalence between the Azumaya algebras A χ and A ψ . Part ( Then, the functor below provides an equivalence beteen the bounded derived categories of sheaves of coherent A -modules and finitely-generated A-modules, respectively: D b (A -Mod) −→ D b (A-Mod fin. gen. ), F −→ RHom A -Mod (A , F ). The proof of this Proposition exploits the technique of Serre functors, and is similar in spirit to the proof of [BKR,Theorem 2.4]. Part (iii) of our Theorem follows from the Proposition since for any A -module F , one has RHom A -Mod (A , F ) ∼ = RHom O M -Mod (O M , F ) = RΓ(M , F ). This completes the proof of the Theorem 4.1.4. 5 The rational Cherednik algebra of type A n−1 . Basic definitions. Let W := S n denote the Symmetric group and Z[W ] denote the group algebra of W . Write s ij ∈ W for the transposition i ↔ j. We consider two sets of variables x 1 , . . . , x n , and y 1 , . . . , y n , and let W = S n act on the polynomial algebras Z[x 1 , . . . , x n ] and Z[y 1 , . . . , y n ] by permutation of the variables. Let c be an indeterminate. We define the rational Cherednik algebra of type A n−1 as a Z[c]algebra, H, with generators x 1 , ..., x n , y 1 , ..., y n and Z[W ], and the following defining relations, see [EG]: We keep our standing assumption char k > n, and write e = 1 n! g∈W g ∈ k[W ] ⊂ H c for the symmetrizer idempotent. Let eH c e ⊂ H c be the Spherical subalgebra, see [EG]. s ij · x i = x j · s ij , s ij · y i = y j · s ij , ∀i, j ∈ {1, 2, . . . , n} , i = j [y i , x j ] = c · s ij , [x i , x j ] = 0 = [y i , y j ] , ∀i, j ∈ {1, 2, . . . , n} , i = j [y k , x k ] = 1 − c · i =k s ik . Let h := k n be the tautological permutation representation of W . We identify the variables x 1 , . . . , x n , resp. y 1 , . . . , y n , with coordinates on h, resp. on h * . The algebras H c and eH c e come equipped with compatible increasing filtrations such that all elements of W and x i ∈ h * ⊂ H c have filtration degree zero, and elements y i ∈ h ⊂ H c have filtration degree 1. The Poincaré-Birkhoff-Witt theorem for rational Cherednik algebras, cf. [EG], yields graded algebra isomorphisms gr H c ∼ = k[h * × h]#W, and gr(eH c e) ∼ = k[h * × h] W . (5.1.2) Dunkl representation. Write h reg for an affine Zariski open dense subset of h formed by points with pairwise distinct coordinates. The group W acts naturally on the algebra D(h reg ) of crystalline differential operators on h reg and we let D(h reg ) W ⊂ D(h reg ) be the subalgebra of W -invariant differential operators. The standard increasing filtration on the algebra of differential operators induces an increasing filtration on the subalgebra D(h reg ) W , and we have gr D(h reg ) W ∼ = k[T * h reg ] W = k[h * × h reg ] W . According to Cherednik, see also [EG], [DO], there is an injective algebra homomorphism [EG]. Further, the map Θ c in (5.2.1) is known to be filtration preserving, and it was proved in [EG] that the associated graded map, gr Θ c , induces graded algebra isomorphisms Θ c : eH c e ֒→ D(h reg ) W ,(5.) W = k[h * × h reg ] W , which is known to be equal to k[h * × h] W ⊂ k[h * × h reg ] W , seek[h * × h] W Id * * gr(eH c e) gr Θc ∼ / / gr B c k[h * × h] W ⊂ k[h * × h reg ] W . (5.2.2) 5.3 The 'radial part' construction. In this section, we let k be an arbitrary algebraically closed field, either of characteristic zero or of characteristic p. Let V be an n-dimensional vector space over k. In case the field k has finite characteristic we assume throughout that char k > n ≥ 2. We put G = GL(V ) and let g = Lie G = gl(V ) be the Lie algebra of G. We consider the vector space G := g × V . Definition 5.3.1. Let G • ⊂ G = g × V be a Zariski open dense subset formed by the pairs (x, v) such that v is a cyclic vector for the operator x : V → V . We recall that the endomorphism x ∈ g admits a cyclic vector if and only if x is regular (not necessarily semisimple), i.e., the centralizer of x in g has dimension n. Fix a nonzero volume element vol ∈ ∧ n V * . We introduce the following polynomial function on G: (x, v) −→ s(x, v) := vol, v ∧ x(v) ∧ . . . ∧ x n−1 (v) . (5.3.2) It is clear that we have G • = G s −1 (0), in particular, G • is an affine variety. The group G acts on g via the adjoint action, and acts naturally on V . This gives a G-diagonal action on G such that G • is a G-stable subset of G. We compose the first projection G = g×V → g with the adjoint quotient map g → g/ Ad G = h/W , and restrict the resulting morphism to the subset G • ⊂ G. This way we get a morphism p : G • → h/W . The group G clearly acts along the fibers of p, and we have the following well-known result. Lemma 5.3.3. (i) The G-action on G • is free and each fiber of p is a single G-orbit; (ii) Furthermore, the map p : G • → h/W is a universal geometric quotient morphism. It follows from the Lemma that G • is a principal G-bundle over h/W . Given an integer c ∈ Z, we put O(G • , c) := {f ∈ k[G • ] g * (f ) = (det g) c · f, ∀g ∈ G}. It is clear that pull-back via the bundle projection p makes O(G • , c) a k[h/W ]-module. Also, observe that s ∈ O(G • , 1). Corollary 5.3.4. For any c ∈ Z, the space O(G • , c) is a rank one free k[h/W ]-module with generator s c . Notation 5.3.5. For c ∈ k, we consider a Lie algebra homomorphism χ c : g → k, x −→ c · tr(x). Let I c := I χc ⊂ Ug, denote the two-sided ideal generated by the elements {x − χ c (x)} x∈g , cf. Definition 3.3.1. The action of G on G • induces an algebra map Ug → D(G • ). We fix c ∈ k, and perform the Hamiltonian reduction of the sheaf D G • , of crystalline differential operators on G • , at the point χ c . This way, we get an associative algebra [ D(G • )/D(G • )·I c ] G . From Lemma 5.3.3 and the isomorphism on the right of formula (3.7.2) we deduce Proposition 5.3.6. For any c ∈ k, there is a natural algebra isomorphism [D(G • )/D(G • )·I c ] G ∼ = D(h/W ). More explicitly, if c is an integer, then the isomorphism of the Proposition is obtained by transporting the action of differential operators on G • via the bijection k[h/W ] ∼ −→ O(G • , c), f −→ s c · p * (f ), provided by Corollary 5.3.4. Remark 5.3.7. The explicit construction of the isomorphism shows in particular that the algebra D(h/W, χ c ) of twisted differential operators coming from the right hand side of the general formula (3.7.2) turns out to be canonically isomorphic, in our case, to the algebra D(h/W ) of ordinary differential operators. Thus, we have put D(h/W ) on the right hand side of the isomorphism of Proposition 5.3.6 (although we have only justified this for integral values of c, the same holds for arbitrary values of c as well). The isomorphism of Proposition 5.3.6 may be viewed as a refined version of the 'radial part' construction considered in [EG]. Remark 5.3.8. The action of differential operators on k[h reg ] W gives rise to the following natural algebra inclusions: D(h) W ⊂ D(h/W ) ⊂ D(h reg ) W . We also remark that the space D(h) W has infinite codimension in D(h/W ). A Harish-Chandra homomorphism. Let g rs ⊂ g denote the Zariski open dense subset of semisimple regular elements. Observe that the eigen-spaces of an element x ∈ g rs give a direct sum decomposition V = ℓ 1 . . . ℓ n . Hence, any v ∈ V we can be uniquely written as v = v 1 +. . .+v n where v i ∈ ℓ i , i = 1, . . . , n. Such a vector v is a cyclic vector for x if and only if none of the v i 's vanish. We put U := {(x, v) ∈ G • x ∈ g rs }. Thus, U is an affine G-stable Zariski open dense subset in G • , and the geometric quotient morphism p : G • → h/W restricts to a geometric quotient morphism p : U → h reg /W . Now, the group G acts naturally on the algebra k [G]. We observe that the first projection G = g × V → g induces an isomorphism of G-invariants k[g] G ∼ −→ k[G] G , since the center of G acts trivially on k[g] and nontrivially on any homogeneous polynomial f ∈ k [V ] such that deg f > 0. Let ∆ g denote the second order Laplacian on g associated to a nondegenerate invariant bilinear form. We will identify ∆ g with the operator ∆ g ⊗ 1 ∈ D(g) ⊗ D(V ) = D(G) acting trivially in the V -direction. Restricting the latter differential operator to U , we may view ∆ g as an element of the algebra Γ(U, D U /D U ·I c ) G . Write x 1 , . . . , x n for coordinates in h = k n , Proposition 5.4.1. For any c ∈ k, there is a natural filtration preserving algebra isomorphism Ψ c : Γ(U, D U /D U ·I c ) G ∼ −→ D(h reg ) W , that reduces to the 'Chevalley restriction' map: f −→ f h×{0} , k[g × V ] G −→ k[h reg × {0}] W = k[h reg ] W , on polynomial 1 zero order differential operators, and such that Ψ c (∆ g ) = L c , where L c = j ∂ 2 ∂x 2 j − i =j c(c + 1) (x i − x j ) 2 (5.4.2) is the Calogero-Moser operator with rational potential, corresponding to the parameter c. Sketch of Proof. We restrict the isomorphism of Proposition 5.3.6 to U ⊂ G • , equivalently, we apply formula (3.7.2) to the geometric quotient morphism p : U → h reg /W and to the character χ c : g → k. This way, we deduce an algebra isomorphism Γ(U, D U /D U ·I c ) G ∼ −→ D(h reg /W ) (5.4.3) Now, the natural projection h reg → h reg /W is a Galois covering with Galois group W . Therefore, pull-back via the projection gives rise to a canonical isomorphism D(h reg /W ) ∼ = D(h reg ) W . Thus, composing with (5.4.3) yields an algebra isomorphism Ψ ′ c : Γ(U, D U /D U ·I c ) G ∼ −→ D(h reg ) W . Finally, let R + be the set of positive roots of our root system R ⊂ h * of type A n−1 , and set δ := α∈R+ α. We conjugate the map Ψ ′ c by δ. That is, for any u ∈ Γ(U, D U /D U · I c ) G , let Ψ c (u) be a differential operator on h reg given by Ψ c (u) := M δ • Ψ ′ c (u) • M 1/δ , where M f denotes the operator of multiplication by a function f ∈ k[h reg ]. The map u → Ψ c (u) thus defined gives the isomorphism Ψ c of the Proposition. The equation Ψ c (∆ g ) = L c is verified by a direct computation similar to one in the proof of [EG,Proposition 6.2]. We leave details to the reader. We will need the following analogue of the surjectivity part of [EG,Corollary 7.4]. Proposition 5.4.4. Let k = Q be the field of rational numbers. Then, for all c ∈ k, the algebra B c is contained in the image of the following composite map, cf. Proposition 5.4.1: Ψ c : Γ G, D G /D G ·I c G restriction / / Γ(U, D U /D U ·I c ) G Ψc ∼ / / D(h reg ) W . Proof. We repeat the argument used in [EG], which is quite standard. Specifically, the algebra B c contains a subalgebra C c ⊂ B c formed by so-called Calogero-Moser integrals. The algebra C c is a commutative algebra containing the Calogero-Moser operator L c , and isomorphic to (Sym h) W , due to a result by Opdam. Moreover, the associated graded map corresponding to the imbedding C c ֒→ B c induces an isomorphism, cf. (5.2.2): gr C c ∼ = (Sym h) W ֒→ (Sym(h ⊕ h * )) W = k[h * × h] W . Observe next that the imbedding g = g × {0} ֒→ g × V = G induces an isomorphism (Sym g) G ∼ −→ (Sym G) G , very similar to the isomorphism k[g] G ∼ −→ k[G] G explained in Sect. 5.3. Thus, we identify (Sym g) G with (Sym G) G , and view the latter as a subalgebra in D(G) G formed by constant coefficient differential operators. Clearly, this is a commutative subalgebra that con- tains ∆ g = ∆ g ⊗ 1 ∈ D(G) G . The homomorphism Ψ c of Proposition 5.4.1, hence the composite map Ψ c of Proposition 5.4.4, takes the algebra (Sym g) G , viewed as subalgebra in Γ(G, D G /D G ·I c ) G , to a commutative subalgebra of D(h reg ) W containing L c . Further, one proves by a standard argument that Ψ c (Sym g) G ⊂ C c , cf. e.g. [BEG], moreover, the induced map gr Ψ c : gr(Sym g) G −→ gr C c = (Sym h) W is the Chevalley isomorphism (Sym g) G ∼ −→ (Sym h) W . It follows that the map Ψ c induces an isomorphism (Sym g) G ∼ −→ C c . Now the algebra B c is known to be generated by the two subalgebras k[h * ] W and C c (since gr B c = k[h * × h] W , viewed as a Poisson algebra with respect to the natural Poisson structure on h * × h = T * h, is known to be generated by the two subalgebras k[h * ] W and k[h] W ). By Proposition 5.4.1, we have Ψ c (k[G] G ) = k[h * ] W and as we have explained above, one also has Ψ c (Sym g) G = C c . We conclude that B c is equal to the subalgebra in D(h reg ) W generated by Ψ c (k[G] G ) and C c , hence, is contained in the image of the map Ψ c . 6 An Azumaya algebra on the Hilbert scheme 6.1 Nakajima construction reviewed. We keep the notations of §5.3. In particular, we have a vector space V over k, such that char k > n ≥ 2, where n = dim V. We put G := GL(V ) and g := Lie G = gl(V ). We will freely identify g * with g via the pairing g× g → k, (x, y) −→ 1 n tr(x·y). The group G acts naturally on V and also on g, via the adjoint action. We consider the Gdiagonal action on the vector space G = g × V , and the corresponding Hamiltonian G-action on the cotangent bundle: T * G = G * × G ∼ = g × g × V * × V. The moment map for this action is given by the formula µ : T * G = g × g × V * × V −→ g * ∼ = g, (x, y,v, v) −→ [x, y] +v ⊗ v ∈ g. (6.1.1) Recall the notation introduced in 5.3.5. Observe that the Lie algebra homomorphism χ c = c · tr ∈ g * corresponds, under the identification g * ∼ = g, to the element c · Id V ∈ g. Following Nakajima, we introduce the set M c := µ −1 (χ c ) = (x, y,v, v) ∈ g × g × V * × V [x, y] +v ⊗ v = c · Id V . This is an affine algebraic variety equipped with a natural GL(V )-action. If c = 0, then M c is known, see [Na1], [Wi], to be smooth, moreover, the G-action on M c is free. The quotient M c := M c /G is a well-defined smooth affine algebraic variety of dimension 2 dim V , called Calogero-Moser space. It was first considered in [KKS], and studied in [Wi], cf. also [Na1]. By definition, M c is the Hamiltonian reduction of T * G with respect to the 1-point G-orbit χ c ∈ g * . The standard symplectic structure on the cotangent bundle thus induces a symplectic structure on M c . If c = 0, then the set M 0 is not smooth, and G-action on M 0 is not free. Let M s 0 be the subset of 'stable points' formed by quadruples (x, y,v, v) ∈ g × g × V * × V such that v ∈ V is a cyclic vector for (x, y), i.e., such that there is no nonzero proper subspace V ′ ⊂ V that contains v and that is both x-and y-stable. Then, M s 0 is known to be a smooth Zariski open G-stable subset in M 0 . Moreover, the differential of the moment map µ, see (6.1.1), is known, cf. [Na1], to be surjective at any point of M s 0 , and the G-action on M s 0 is free. The following description of Hilb n A 2 , the Hilbert scheme of zero-dimensional length n subschemes in the affine plane A 2 , is essentially due to Nakajima [Na1]. Proposition 6.1.2. There exists a smooth geometric quotient morphism M s 0 → Hilb n A 2 . Remark 6.1.3. It is known that Hilb n A 2 is a smooth connected (non-affine) algebraic variety of dimension 2 dim V . ♦ Thus, the Hilbert scheme Hilb n A 2 may be viewed as a 'Hamiltonian reduction' of T * G at the 1-point G-orbit {0} ⊂ g * . In particular, Hilb n A 2 has a natural symplectic structure. It is known that for any quadruple (x, y,v, v) ∈ M 0 , the operators x, y can be put simultaneously in the upper-triangular form. Hence, the diagonal components of these two operators give a pair of elements diag x, diag y ∈ h, well defined up to simultaneous action of W = S n . The assignment (x, y,v, v) −→ (diag x, diag y) clearly descends to a morphism Υ : Hilb n A 2 −→ (h ⊕ h)/W , called Hilbert-Chow morphism. It is known that the Hilbert-Chow morphism induces an algebra isomorphism Γ Hilb n A 2 , O Hilb n A 2 Υ * ∼ −→ Γ (h × h)/W, O (h×h)/W = k[h ⊕ h] W . (6.1.4) 6.2 The Azumaya algebra. Let κ : A 1 → A 1 , c → κ(c) = c p − c be the classical Artin-Schreier map. This map is related to the map κ : X * (g) → g 1, * defined in (3.2.2) by the formula κ(χ c ) = χ κ(c) = κ(c) · tr (1) . We introduce the following simplified notation for the scheme-theoretic fiber of the moment map µ (1) : T * X (1) → g 1, * over the point κ(χ c ): T 1, * κ(c) := T 1, * κ(χc) = [µ (1) ] −1 (χ κ(c) ) = M (1) κ(c) . We are going to apply the general Hamiltonian reduction procedure of Sect. 4.1 to the algebraic group A := G, the Lie algebra character χ = χ κ(c) , and the natural G-action on the variety X := G. Assume now that κ(c) = c p − c = 0, that is, the element c ∈ k is contained in the finite subfield F p ⊂ k. Then, we apply the construction of Sect. 4.1 to the open subset M := M s 0 ⊂ µ −1 (0), of stable points. By Proposition 6.1.2, we obtain an Azumaya algebra A c := A χc on the Frobenius twist of Hilb n A 2 , to be denoted Hilb (1) . For any c ∈ k, the Azumaya algebra A c comes equipped with canonical algebra homomorphism Ξ c : Γ G, D G /D G ·I c G (4.5.2) −−− −−→ Γ(M (1) κ(c) , A c ) (6.2.1) 6.3 A Harish-Chandra homomorphism for the Azumaya algebra. We are now going to construct a Harish-Chandra homomorphism for the Azumaya algebra A c . Recall the open subset U ⊂ G formed by the pairs (x, v) ∈ g rs × V such that v is a cyclic vector for x. Proposition 6.3.1. For any c ∈ k, there is an algebra homomorphism Ψ A c making the following diagram commute: Γ G, D G /D G ·I c G (6.2.1) Ξc / / restriction Γ(M (1) κ(c) , A c ) Ψ A c Γ U, D U /D U ·I c Proof. We only consider the most interesting case κ(c) = 0. The action of G on U induces a Hamiltonian G-action on T * U . Let µ U : T * U → g * be the corresponding moment map, and set U := µ −1 U (0). Since U is an open subset of G, the map µ U clearly equals the restriction of the moment map µ : T * G −→ g * to the open subset T * U = G * × U ⊂ G * × G. Thus, we have U = µ −1 U (0) = (G * × U ) µ −1 (0). It is crucial for us that one has an open inclusion U ⊂ M s 0 = M. This trivially follows from definitions since a vector v ∈ V which is cyclic for x ∈ g is necessarily also cyclic for any pair of the form (x, y) ∈ g × g. Thus, we have the diagram U (1) open imbedding / / M (1) = [M s 0 ] (1) ̟ geom. quotient map / / / / M (1) = Hilb (1) . Restricting the sheaf E c := E χc , cf. (3.6.3), from [M s 0 ] (1) to U (1) yields a G (1) -equivariant algebra map Γ([M s 0 ] (1) , E c ) −→ Γ( U (1) , E c ) . Further, applying Lemma 3.6.4 to X := U and A := G, we get an algebra isomorphism Γ(U, D U /D U ·I c ) G ∼ = Γ [µ −1 U (0)] (1) , E c G (1) = Γ U (1) , E c G (1) . (6.3.2) By definition, we have A c := ̟ q E c [M s 0 ] (1) G (1) . We obtain the following chain of algebra homomorphisms Γ(M (1) , A c ) = Γ M (1) , ̟ q E c [M s 0 ] (1) G (1) = Γ M (1) , ̟ q E c [M s 0 ] (1) G (1) = Γ [M s 0 ] (1) , E c G (1) restriction −−− −−→ Γ U (1) , E c G (1) ∼ −→ (6.3.2) Γ(U, D U /D U ·I c ) G Ψc −−− −−→ Prop. 5.4.1 D(h reg ) W . We let Ψ A c be the composite homomorphism. Commutativity of the diagram of the Proposition is immediate from the construction above. 7 Localization functor for Cherednik algebras 7.1 From characteristic zero to characteristic p. We begin by reminding the general technique of transferring various results valid over fields of characteristic zero to similar results in characteristic p, provided p is sufficiently large. We fix c = a/b ∈ Q with b > 0. For any prime p > b, reducing modulo p, we may (and will) treat c = a/b as an element of F p . We let k p ⊃ F p denote an algebraic closure of F p , and consider the corresponding k p -algebras H c and eH c e. We begin with the following characteristic p analogue of Proposition 5.4.4. Γ G, D G /D G ·I c G Ψ Z c / / D(h reg ) W B Z c , ? _ j o o where j denotes the inclusion. Proposition 5.4.4 says that Q Z[ 1 n! ,c] B c ⊆ Q Z[ 1 n! ,c] Im(Ψ Z c ) . Since all the algebras involved are finitely generated, it follows that there exists an integer q ∈ Z such that B Z c ⊂ Im(Ψ Z c )[ 1 q ]. Thus, for all primes p > n which do not divide q, reducing the above inclusion modulo p, we get k Z[ 1 n! ,c] B c ⊆ k Z[ 1 n! ,c] Im(Ψ Z c ) . The Lemma is proved. We now consider the algebras H c and eH c e over the ground field Q of the rational numbers, and let H c eH c be the two-sided ideal in H c generated by the idempotent e. We will use the following result from [GS]. We are going to deduce a similar result in characteristic p, which reads Remark 7.1.4. We emphasize that, in this Corollary and in various other results below, a rational value of the parameter c must be fixed first. The choice of c dictates a lower bound d(c) for allowed primes p, and only after that one considers the corresponding Cherednik algebras over k p . Thus, if c ∈ Q and p have been chosen as above, and c ′ ∈ Q is such that c ′ = c mod p, then H c ∼ = H c ′ as k p -algebras; yet, it is quite possible that we have d(c) < p < d(c ′ ), hence, the results of this section do not apply for H c ′ viewed as a k p -algebra. ♦ The proof of the Corollary will exploit the following standard result of commutative algebra, [Gr], Expose IV, Lemma 6.7. Generic Flatness Lemma. Let A be a commutative noetherian integral domain, B a (commutative) A-algebra of finite type, and M a finitely generated B-module. Then, there is a nonzero element f ∈ A such that M (f ) , the localization of M , is a free A (f ) -module. (H), which is a quotient of the smash-product algebra Z[ 1 n! ][c][h × h * ] ⋉ Z[ 1 n! ][c][S n ] (here we regard h as a free rank n module over Z[ 1 n! ][c]; its dual h * is also free of rank n; by the Poincare-Birkhoff-Witt theorem proved in [EG], the algebra gr(H) and the above smash-product become isomorphic after tensoring with Q). We deduce that gr(H/HeH) is a finitely generated module over the commutative algebra Z[ f · 1 H = m i=1 h ′ i ·e·h ′′ i holds in H, (7.1.6) where 1 H denotes the unit of the Z[ 1 n! ][c]-algebra H. We may specialize this equation at any rational value c = a/b ∈ Q to obtain a similar equation for the corresponding Z[ 1 n! , 1 b ]-algebras. If c ∈ Q good then, according to Claim 7.1.5, we may further assume that f (c) = k/l = 0. Now, let p be a prime such that p > max{n, k, l}. Reducing (the specialization at c of) equation (7.1.6) modulo p, for the corresponding F p -algebras we get f (c) · 1 Hc = m i=1 h ′ i · e · h ′′ i . Thus f (c) = k/l is a nonzero, hence, invertible element in F p and, since F p ⊂ k p , in the k p -algebra H c we obtain 1 Hc = 1 f (c) m i=1 h ′ i · e · h ′′ i . Thus, we have proved that 1 Hc ∈ H c eH c , and the first statement of the Corollary follows. It is well-known that this implies the last statement of the Corollary as well. 7.2 Localization of the Spherical subalgebra. We have the following A c -version of Proposition 5.4.4. Theorem 7.2.1. Fix c = a/b ∈ Q. For all sufficiently large primes p, we have: The image of the Harish-Chandra homomorphism Ψ A c of Proposition 6.3.1 is equal to the subalgebra B c ⊂ D(h reg ) W (algebras over k p ). Moreover, the resulting map gives an algebra iso- momorphism Ψ A c : Γ(Hilb (1) , A c ) ∼ −→ B c . Proof of this Theorem will be given later in this section. Remark 7.2.2. A similar construction also produces an isomorphism Ψ A c : Γ(M (1) κ(c) , A c ) ∼ −→ B c , for all c ∈ k p (not only for c ∈ F p ). The proof of this generalization is similar to the proof of Theorem 7.2.1, but involves twisted differential operators and twisted cotangent bundles. It will be presented elsewhere. ♦ Composing the isomorphism of Theorem 7.2.1 with the inverse of the Dunkl representation (5.2.1), we obtain the following Azumaya version of the Spherical Harish-Chandra isomorphism considered in [EG] Φ A c : Γ(Hilb (1) , A c ) Ψ A c −−− −−→ ∼ B c (Θc) −1 −−− −−→ ∼ eH c e. (7.2.3) Thus, we have proved part (i) of the following theorem, which is one of the main results of the paper Theorem 7.2.4. Fix c = a/b ∈ Q. Then there exists a constant d = d(c) such that for all primes p > d(c), we have: (i) The composite morphism in (7.2.3) yields a k p -algebra isomorphism Φ A c : eH c e ∼ −→ Γ(Hilb (1) , A c ). (ii) H i (Hilb (1) , A c ) = 0 for all i > 0. (iii) If c ∈ Q good , then the (derived) global sections functor RΓ : D b (A c -Mod) → D b (eH c e-Mod) is an equivalence of bounded derived categories. Proof. Part (i) follows from Theorem 7.2.1 (to be proved below), and part (ii) is a consequence of Theorem 4.1.4(ii). We are going to deduce part (iii) of Theorem 7.2.4 from Theorem 4.1.4(iii). To do so, we need to know that the algebra eH c e ∼ = Γ(Hilb (1) , A c ) has finite homological dimension. But this follows from the Morita equivalence of Corollary 7.1.3, since the algebra H c is known to have finite homological dimension, which is equal to 2n, cf. [EG]. Proof of Theorem 7.2.1. Fix c ∈ Q, and let p ≫ 0 be such that Lemma 7.1.1 holds for p. Recall an increasing filtration on the algebra Γ(Hilb (1) , A c ), introduced in section 4.2, such that for the associated graded algebra we have gr R Γ(Hilb (1) , A c ) = Γ(Hilb n A 2 , O Hilb n A 2 ), see Proposition 4.3.3(ii). We consider the commutative diagram of Proposition 6.3.1. All maps in that diagram are filtration preserving, and the corresponding commutative diagram of associated graded maps reads gr Γ G, D G /D G ·I c G (6.2.1) gr Ξc / / restriction res G U gr R Γ(Hilb (1) , A c ) gr Ψ A c gr Γ U, D U /D U ·I c G ∼ gr Ψc / / gr D(h reg ) W = k[h * × h reg ] W .(B c ⊂ Im(Ψ c ) := Ψ c res G U Γ(G, D G /D G ·I c ) G ⊆ Im(Ψ A c ). (7.2.7) Hence, using commutativity of diagram (7.2.5), we obtain the following commutative diagram of graded algebra morphisms k[h * × h] W (7.2.6) gr B c ı / / Im(gr Ψ c ) _  gr R Γ Hilb (1) , A c gr Ψ A c / / / / Im(gr Ψ A c ). (7.2.8) We deduce from commutativity of diagram (7.2.8) that both ı and  must be surjective and, therefore, Im gr Ψ A c = gr B c . Now, the inclusions in (7.2.7) show that we must have Im(Ψ A c ) = B c and, moreover, the map Ψ A c gives an isomorphism Γ Hilb (1) , A c ∼ −→ B c . 7.3 Localization of the algebra H c . We introduce the following localization functor L oc : N → A c L ⊗ eHce N , which is the left adjoint to the functor RΓ : D b (A c -Mod) → D b (eH c e-Mod). Since RΓ(A c ) = eH c e, and the functor RΓ is an equivalence by Theorem 7.2.4, we conclude that L oc(eH c e) = A c is also an equivalence which is a quasi-inverse to RΓ(−). Observe next that eH c is a projective eH c e-module, by Corollary 7.1.3. Hence we conclude that R c := L oc(eH c ) = A c ⊗ eHce eH c is a locally free sheaf of A c -modules. Moreover, it is easy to see by looking at the restrictions of the associated graded modules to the generic locus of (h × h)/W that the rank of eH c viewed as a projective eH c e-module equals n!. Therefore, we deduce that R c is a vector bundle on Hilb (1) of rank n! · p 2n . We put H c := End Ac (R c ). This is clearly an Azumaya algebra on Hilb (1) again, and the degree of this Azumaya algebra is equal to n! · p 2n . Further, the left H c -action on each fiber of the sheaf R c induces a natural algebra map H c −→ Γ(Hilb (1) , H c ). (7.3.1) The second main result of the paper reads , H c ) = 0, ∀i > 0. (ii) The functor RΓ : D b (H c -Mod) → D b (H c -Mod) is a triangulated equivalence. Proof. We have RΓ i (Hilb (1) , H c ) ∼ = RΓ i (Hilb (1) , End A (R c )) ∼ = Ext i D b (Ac-Mod) (R c , R c ) (7.3.3) Theorem 7.2.4 ∼ / / Ext i D b (eHce-Mod) (eH c , eH c ). The Ext-group on the right vanishes for all i > 0 since eH c is a projective eH c e-module. This proves the vanishing statement in part (i). The statement of part (i) for i = 0 follows from the isomorphisms: To prove part (ii) we use a commutative diagram D b (eH c e-Mod) Morita equivalence L oc D b (H c -Mod) L oc D b (A c -Mod) Morita equivalence D b (H c -Mod). Since the left vertical arrow is an equivalence by Theorem 7.2.4, it follows that the right vertical arrow is an equivalence as well. The functor RΓ is a right adjoint of L oc, hence, it must also be an equivalence, which is a quasi-inverse of L oc. 7.4 Splitting on the fibers of the Hilbert-Chow map. We have the Hilbert-Chow map Υ : Hilb n A 2 → (h × h)/W . Given ξ ∈ (h × h)/W , write Hilb ξ := Υ −1 (ξ) for the fiber of Υ over ξ, and let Hilb (1) ξ denote the completion of Hilb (1) along this fiber, a formal scheme. The next result is essentially due to [BK]. H c Hilb (1) ξ ∼ = End O Hilb (1) ξ V c,ξ opp , resp., A c Hilb (1) ξ ∼ = End O Hilb (1) ξ W c,ξ opp . The above vector bundle V c,ξ , resp. W c,ξ , is called a splitting bundle for H c , resp. for A c . Corollary 7.4.2. For any ξ ∈ (h × h)/W and i > 0, we have Ext i (V c,ξ , V c,ξ ) = 0, resp., Ext i (W c,ξ , W c,ξ ) = 0, where the Ext-groups are considered in the category Coh( Hilb (1) ξ ). The rank of a splitting bundle is equal to the degree of the corresponding Azumaya algebra. In particular, we have rk W c,ξ = p n and rk V c,ξ = p n ·n!. This suggests the following Conjecture 7.4.3. For any ξ ∈ [(h × h)/W ] (1) , there is a vector bundle isomorphism Fr * V c,ξ ∼ = (Fr * W c,ξ ) ⊗ (P Hilb n Fr(ξ) ), where P denotes the Procesi bundle on Hilb n A 2 , see [H]. In view of Theorem 4.1.4(iii) it is sufficient to prove the Conjecture for c = 0, that is, for the case where H c = D(h)#W . Proof of Theorem 7.4.1. Clearly, it suffices to prove the Theorem for A c . We repeat the argument in the proof of [BK,Proposition 5.4]. First, recall that Morita equivalence classes of Azumaya algebras on a scheme Y are classified by Br(Y ), the Brauer group of Y . Further, the Brauer group of a local complete k-algebra is known to be trivial. Thus, proving the Theorem amounts to showing that, for any c ∈ F p , the class [A c ] ∈ Br(Hilb (1) ) belongs to the image of the pull-back morphism Υ * : Br [(h × h)/W ] (1) → Br(Hilb (1) ). To prove this, we consider the following diagram (7.4.4) In this diagram, S is a Zariski open dense subset in [(h × h)/W ] (1) such that: S _ q S W -covering / / / / S _ S • _ Υ S ∼ o o [h × h] (1) q finite map / / / / [(h × h)/W ] (1) Hilb (1) . Υ o o • The Hilbert-Chow map restricts to an isomorphism S • := Υ −1 (S) ∼ −→ S, to be denoted Υ S , and • The projection q : (1) is unramified over S. Thus, we have a Galois covering S := q −1 (S) → S, to be denoted q S . Our goal is to construct a class β ∈ Br [(h × h)/W ] (1) such that [A c ] = Υ * (β). We will follow the strategy of [BK]. [h × h] (1) −→ [(h × h)/W ] Let Y be an arbitrary affine scheme, and H 2 et (Y, G m ) torsion be the torsion subgroup of the second etale cohomology of Y with coefficients in the multiplicative group. By a theorem of Gabber [Ga], one has an isomorphism Br(Y ) ∼ = H 2 et (Y, G m ) torsion . Further, it is a simple matter to see that the norm-map associated to the projection q : [h × h] (1) −→ [(h × h)/W ] (1) gives rise to a morphism onétale cohomology, cf. [BK]: q * : H 2 et [h × h] (1) , G m W torsion −→ H 2 et (h × h)/W ), G m torsion . Now let D (1) := Fr qDh be the standard Azumaya algebra on T * h (1) = [h×h] (1) , arising from the sheaf of crystalline differential operators on h, cf. §2. The sheaf D (1) has a natural W -equivariant structure, hence the corresponding class [D (1) ] is a W -invariant class in the Brauer group, that is an element of Br [h × h] (1) W ∼ = H 2 et [h × h] (1) , G m W torsion . We set β := q * ([D (1) ]) ∈ H 2 et [(h × h)/W ] (1) , G m torsion ∼ = Br [(h × h)/W ] (1) . The Theorem would follow provided we show that Υ * β = [A c ]. We first prove a weaker claim To complete the proof of (7.4.5), we use Theorem 4.1.4(iv) and deduce that, for all c ∈ F p , the corresponding Azumaya algebras A c are Morita equivalent, hence represent the same class in Br(Hilb (1) ). Thus, we may assume without loss of generality that c = 0. In that case the corresponding algebra eH c e is isomorphic to D(h) W . Furthermore, going through the Hamiltonian reduction construction of the Azumaya algebra A 0 , it is easy to verify that we have an Azumaya algebra isomorphism (Υ S ) * B ∼ = A 0 S • . This yields an equality of the corresponding classes in Br(S • ), and (7.4.5) follows. (Υ * β) S • = [A c ] S • To complete the proof of the Theorem we recall the well-known result saying that restriction to a Zariski open dense subset induces an injective morphism of the corresponding Brauer groups. Thus, we have an injection Br(Hilb (1) ) ֒→ Br(S • ), α → α S • , and we have shown above that (Υ * β) S • = [A 0 ] S • . Hence, Υ * β = [A 0 ] = [A c ], ∀c ∈ F p , and the Theorem is proved. 7.5 Bigrading on H c,0 . Let ξ = 0 be the origin of (h × h)/W and set H c,0 = Γ( Hilb (1) 0 , H c ), the completion of the Cherednik algebra H c at the zero central character. The isomorphism of Theorem 7.4.1 gives a continuous (right) H c,0 -action on the splitting vector bundle V c,0 by vector bundle endomorphisms. In particular, there is an action of the Symmetric group S n on V c,0 . Thus, for any simple S n -representation τ , we have the corresponding τ -isotypic component Hom Sn (τ, V c,0 ). This isotypic component is again a vector bundle on Hilb (1) 0 , moreover, the natural evaluation map gives an S n -equivariant vector bundle isomorphism τ ∈Irrep(Sn) Hom Sn (τ, V c,0 ) ⊗ τ ∼ −→ V c,0 , (7.5.1) where the direct sum on the left runs over the set of (isomorphism classes of) simple S n -representations τ . The cohomology vanishing in Theorem 1.3.2 implies that the splitting vector bundle V c,0 is rigid, i.e., we have Ext 1 (V c,0 , V c,0 ) = 0. It follows that, for each τ , the isotypic component Hom Sn (τ, V c,0 ) is also a rigid vector bundle. The tautological GL 2 -action on A 2 gives rise to a natural GL 2 -action on the punctual Hilbert scheme Hilb (1) 0 . We restrict attention to the subgroup G m × G m ⊂ GL 2 , of diagonal matrices, and view Hilb (1) 0 as a projective G m × G m -variety. Thus, according to Proposition ??, see §??, for each irredicible S n -module τ , the corresponding isotypic component Hom Sn (τ, V c,0 ) may be equipped with a G m × G m -equivariant structure. This gives, via the isomorphism (7.5.1), a G m × G mequivariant structure on the vector bundle V c,0 . Further, the G m × G m -equivariant structure on V c,0 induces one on the Azumaya algebra H c,0 = End O Hilb (1) ξ V c,ξ opp . Therefore, there is a continuous G m × G m -action on the vector space H c,0 = Γ( Hilb (1) 0 , H c ), of global sections. Thus, we have defined an action of the group G m × G m on the topological algebra H c,0 by continuous algebra automorphisms. Furthermore, the G m × G m -action on H c,0 fixes each element of the group S n ⊂ H c,0 , since the G m × G m -equivariant structure is compatible with the S n -action, by construction. Taking the direct sum of the weight spaces of the G m × G m -action, we get a Z 2graded dense subalgebra H • = k,l∈Z H k,l c ⊂ H c,0 , with S n -stable homogeneous components H k,l c . 8 Induction functor and comparison with [EG] In this subsection, we let k be an arbitrary algebraically closed field, either of characteristic zero or of characteristic p. 8.1 Let P be a linear algebraic group with Lie algebra p, and D an associative algebra equipped with a P -action by algebra automorphisms and with a P -equivariant algebra map ρ : Up → D, as in Sect. 3.4. Recall our convention to write D · J instead of D · ρ(J). Definition 8.1.1. Given a two-sided ideal J ⊂ Up, let Ind(D↑J) ⊂ D denote the annihilator of the left D-module D/D · J. This is a two-sided ideal in D, called the ideal induced from J. It follows from the definition that Ind(D↑J) is the maximal two-sided ideal of D contained in the left ideal D · J. Assume next that P is an algebraic subgroup in another connected linear algebraic group G. Set g := Lie G, and let Ug be the corresponding enveloping algebra. Thus, p ⊂ g and Up ⊂ Ug. Given a two-sided ideal J ⊂ Up, as above, we may form an induced ideal Ind(Ug↑ J) ⊂ Ug. Now, let G act on an associative algebra D, and let Ug → D be a G-equivariant algebra map. We consider the composite map Up ֒→ Ug → D. Let J ⊂ Up be a two-sided ideal and Ind(Ug↑J) ⊂ Ug the corresponding induced ideal. Since J ⊂ Ug · J we have D · Ind(Ug↑J) ⊂ D · J. Hence, the projection D/D·Ind(Ug↑J) ։ D/D·J induces an algebra map (D/D·Ind(Ug↑J)) G −→ (D/D·J) P . (8.1.2) 8.2 We recall the setup of section 6.1, so V is an n-dimensional vector space over k, and we put G = GL(V ) and g = gl(V ). We also fix c ∈ k and let χ c := c · tr : g → k be the corresponding Lie algebra character From now on, we fix a non-zero vector v ∈ V . Let P be a parabolic subgroup of G formed by the maps V → V that preserve the line kv. Thus G/P ∼ = P(V ), the (n − 1)-dimensional projective space associated to V . We put p := Lie P , and write χ p c := χ c p for the character χ c restricted to the subalgebra p ⊂ g. Thus χ p c ∈ p * is a P -fixed point for the coadjoint action of P on p * . We extend χ p c to an associative algebra homomorphism χ p c : Up → k, and let J c := Ker(Up → k), denote the corresponding two-sided ideal generated by the elements {x − c · tr(x)} x∈p , see Definition 3.3.1. Also, write Ind c := Ind(Ug↑J c ) for the two-sided ideal in Ug induced from J c . It is known that Ind c is a primitive ideal in Ug, moreover, it is exactly the primitive ideal considered in [EG]. The adjoint action of G on g gives rise to an associative algebra homomorphism ad : Ug → D(g). Thus, we may consider the homomorphism (8.1.2) in the special case D := D(g), J := J c . As usual, we abuse the notation and write D(g) · Ind c instead of D(g) · ad Ind c for the corresponding left ideal in D(g). We propose the following Let X := G × U v ∼ = G × g • × G m . The assignment u −→ 1 ⊗ u ⊗ 1 gives an algebra map D(g • ) −→ D(G) ⊗ D(g • ) ⊗ D(G m ) ∼ −→ D(G × U v ) = D(X). We compose this map with isomorphism (3.7.2) applied to X, Y := G × P U v , and to the P -bundle map in the top row of diagram (8.4.2). This way, we get a chain of algebra morphisms One can show, using the equality Ind(Ug↑J c ) = ∩ g∈G Ad g(Ug · J c ), that a suitably refined version of the above construction produces a well-defined algebra map F ′′ c : Γ(g • , D g • /D g • ·J c P −→ Γ(U, D U /D U ·Ind c ) G . Furthermore, one verifies that composing the map F ′′ c with the natural restriction map Γ(g rs , D rs g /D rs g · J c P −→ Γ(g • , D g • /D g • · J c P one obtains a homomorphism F c that makes the diagram of the Proposition commute. 9 Appendix: The p-center of symplectic reflection algebras by Pavel Etingof 9.1 Let k be an algebraically closed field of characteristic p > 0, V a finite-dimensional symplectic vector space over k, and Γ ⊂ Sp(V ) a finite subgroup. We assume that p is odd and prime to \|Γ\|. Write S ⊂ Γ for the set of symplectic reflections in Γ. Let H t,c (V, Γ) be the symplectic reflection algebra with parameters t and c ∈ k[S] Γ , as defined in [EG]. For t = c = 0, we have H 0,0 ≃ SV #Γ, a cross-product of SV with the group Γ, to be denoted H. In general, the algebra H t,c comes equipped with an increasing filtration such that gr H t,c ≃ H = SV #Γ. Let Z t,c be the center of H t,c , equipped with the induced filtration. The main result of this appendix is the following theorem. Theorem 9.1.1. There is a graded algebra isomorphism gr(Z 1,c ) = ((SV ) p ) Γ . Corollary 9.1.2 (Satake isomorphism). The map Z 1,c → eH c e, z −→ e · z gives an isomorphism between the centers of the algebras H 1,c and eH 1,c e, respectively. Proof of Corollary. The associated graded map is an isomorphism. Remark 9.1.3. One may consider the field k as the residue class field in W (k), the ring of Witt vectors of k. The algebra H 1,c may thus be regarded as a specialization of a W (k)-algebra. Applying the well-known, see [Ha], Hayashi construction to this situation one obtains a natural Poisson bracket on the p-center Z 1,c ⊂ H 1,c . ♦ 9.2 To prove Theorem 9.1.1 we need the following result. Proposition 9.2.1. The Hochschild cohomology of the algebra H is given by the formula HH m (H) = (⊕ g∈Γ:codimV g ≤m Ω m−codimV g (V g )) Γ . (here Ω j denotes the space of differential forms of rank j) Moreover, elements z ∈ Z 0,0 = HH 0 (H) act on this cohomology by multiplying differential forms on V g by the restriction of z to V g . Proof. The first statement is straightforward by using Koszul resolutions, similarly to [AFLS]. The second statement is easy. . 2 . 2Fix c ∈ Q good . Then, there exists a constant d = d(c) such that for all primes p > d(c), the functor RΓ :D b (H c -Mod) → D b (H c -Mod)is a triangulated equivalence between the bounded derived categories of sheaves of coherent H c -modules and finitely generated H c -modules, respectively, whose inverse is the localisation functor M → H c L ⊗ Hc M . Moreover, we have H i (Hilb (1) , H c ) = 0, ∀i > 0. ∼ = (End V c,ξ ) opp . Remark 1.4.2. The splitting bundle is not unique; it is only determined up to twisting by an invertible sheaf. ♦ Given ξ as above, let m ξ be the corresponding maximal ideal in the p-center of H c . Let H c,ξ , resp. H c,ξ = H c Hilb (1) ξ , be the m ξ -adic completion of H c , resp. of H c . We write D b ( H c,ξ -Mod), resp. D b ( H c,ξ -Mod), for the bounded derived category of finitely-generated complete topological H c,ξ -modules, resp. H c,ξ -modules. On the other hand, let D b (Coh( Hilb ξ )) be the bounded derived category of coherent sheaves on the formal scheme Hilb Fix c ∈ Q good . Then, for all primes p > d(c), Theorems 1.3.2-1.4.1 imply the following The category D b ( H c,ξ -Mod) is equivalent to D b (ξ = 0 be the zero point in [(A 2 ) n /S n ] (1) . The fiber Hilb / / D b (H-Mod). Definition 3 .3. 1 . 31Let I χ := Ker(Ua → k) denote the kernel of χ, the two-sided ideal in Ua generated by the elements {x − χ(x)} x∈a . Also, in Z(a), consider the following ideal I (1) χ := I χ ∩ Z(a) ⊂ Z(a) ⊂ Ua, and set u χ (a) := Ua/Ua·I (1) χ . • The natural G m -action on the algebra k[M ] (arising from the G m -action on M) has no negative weights, more geometrically, the induced G m -action on the scheme M aff := Spec k[M ], the affinization of M , is an attraction. • The canonical projection f : M → M aff is a proper morphism. Lemma 4.2. 5 . 5Let D be a graded algebra, and M a t-torsion free, graded D[t]-module (deg t = 1) such that M/tM is a rank m free, resp. projective, graded D-module. Then, M is a rank m free, resp. projective, D[t]-module.Proof of Lemma 4.2.4. Recall that every point in any G m -variety is known to have a G m -stable affine Zariski open neighborhood. Applying this to the G m -action on M , we may replace M by a G m -stable affine Zariski open subset Y ⊂ M . Let Y be the inverse image of Y under the bundle map M → M . Thus, Y is a G m × A-stable affine Zariski open subset in M, and we put RF := Γ(A 1 × Y (1) , RF ) = Γ A 1 × Y (1) , RD (1) /R(D (1) · I χ ) . point of the open subset M ⊂ µ −1 (0). Hence, M ⊂ • T * X ∩ µ −1 (0) is a non-empty Zariski-open subset in µ −1 (0). Since the sheaf gr(Fr qDX )/ gr(Fr qDX ·I χ ) is supported on [µ −1 (0)] (1) , we obtain gr(Fr qDX )/ gr(Fr qDX ·I χ ) M (1) = gr(Fr qDX )/ gr(Fr qDX ·I χ ) • T * X (1) ∩M (1) qOT * X ) Fr q O T * X ·µ alg (J ) • T * X (1) ∩M (1) = Fr q(OT * X • T * X∩M ) = Fr qOM.The definition of Rees algebra implies readily, see (2.4.1) and (2.5.3), that the restriction to{0} × M (1) of the sheaf RD (1) /RD (1) ·RI (1)χ is isomorphic to gr(Fr qDX )/ gr(Fr qDX ·I χ ) M (1) , the sheaf in the top line of (4.2.7). Hence, equations (4.2.7) and a flat base change yield RF {0}×M (1) ∼ = ̟ qFr qOM. Restricting this sheaf isomorphism to our affine open subset Y (1) and taking global sections, we obtain canonical graded space isomorphisms iii) of the Proposition. To this end, let D := Spec k[[t]] ⊂ A 1 denote the completion of the line A 1 at the origin. Thus, D × M (1) , resp., D×M (1) aff , is the formal completion of the scheme A 1 ×M (1) along the closed subscheme {0}×M (1) , resp., formal completion of A 1 × M (1) aff along the subscheme {0} × M (1) aff . We have D × M (1) = f −1 ( D × M (1) aff ), and the restriction off gives a morphism f : D × M (1) −→ D × M . Hence, if the sheaf R qf * RA χ is non-zero, it must have a non-zero restriction to the subscheme {0} × M this would yield, in particular, that i * R qf * RA χ = 0, contradicting (4.3.6). Part (iii) of the Proposition follows.Taking global sections over the affine open set (A 1 {0}) × M f * RA χ ) = Γ(A 1 × M (1) , RA χ ) = R, the graded k[t]-algebra involved in the definition of filtration on Γ(M (1) , A χ ). Therefore, applying the global sections functor Γ(A 1 × M (1) aff , −) to both sides in (4.3.7) we obtain gr R Γ(M (1) , A χ ) = R/tR ∼ = Γ(M (1) aff , Fr qOM aff ) = Γ(M (1) , Fr qOM ) = Γ(M , O M ). This proves part (ii) of the Proposition. 4.4 Comparison of characters. Fix two characters χ, ψ such that χ − ψ ∈ dlog X * (A). Let O M (ψ − χ) be the (ψ − χ)-weight subsheaf in the push-forward of O M under the bundle map M → M . The (ψ − χ)-isotypic component of the regular representation k[A] being 1-dimensional, we conclude that O M (ψ − χ) is a rank 1 locally free sheaf on M .Let D iii) of Theorem 4.1.4 is a special case of the following more general result due to[BK, Proposition 2.2]. . Let M be a smooth connected variety over k with the trivial canonical class, and such that the morphism M → M aff is proper. Let A be an Azumaya algebra on M such that H i (M , A ) = 0, ∀i > 0 and, moreover, the algebra A := Γ(M , A ) has finite homological dimension. field k and c ∈ k, we let H c := k Z[c] H be the k-algebra obtained from H by extension of scalars via the homomorphism Z[c] → k, f −→ f (c). representation of the algebra eH c e. Let B c := Θ c (eH c e) ⊂ D(h reg ) W be the image of the Dunkl representation. We equip the algebra B c with increasing filtration induced from the standard one on D(h reg ) W . Then, gr B c becomes a graded subalgebra in gr D(h reg ♦ Observe further that the algebras [D(G • )/D(G • )·I c ] G and D(h/W ) both come equipped with natural increasing filtrations and the isomorphism of the Proposition is filtration preserving. c) is smooth, the G-action on this variety is free, and there is a smooth geometric quotient map M κ(c) −→ M κ(c) , where M κ(c) is the Calogero-Moser variety with parameter κ(c) = c p − c. Thus, the construction of Sect. 4.1, applied to M = T 1, * κ(c) , produces an Azumaya algebra A c := A χc on M κ(c) . Lemma 7 .1. 1 . 71Fix c = a/b ∈ Q. For all sufficiently large primes p, we have an inclusion of k p -algebras B c ⊆ Im(Ψ c ).Proof. For c = a/b ∈ Q, we consider the ring Z[ 1 n! , c] = Z[ 1 n! , 1 b ] obtained by inverting n! and b. The algebras Γ(G, D G /D G · I c ) G and D(h reg ) W , are both defined over Z, hence have natural Z[ 1 n! , c]-integral structures. We will denote the corresponding Z[ 1 n! , c]-algebras by the same symbols.Therefore, we may (and will) consider the Z[ 1 n! , c]-integral version ΨZ c : Γ(G, D G /D G · I c ) G −→ D(h reg ) W , of the homomorphism of Proposition 5we may choose a finite set of generators in the Q-algebra B c in such a way that the Z[ 1 n! , c]-subalgebra generated by this set is contained in the Z[ 1 n! , c]-integral form of D(h reg ) W . Denote this Z[ 1 n! , c]-subalgebra by B Z c . Thus, we get a diagram of Z[ 1 n! , c]-algebra maps Proposition 7 .1. 2 . 72For any c ∈ Q good , see (1.3.1), we have H c = H c eH c . Thus, the eH c e-H cbimodule eH c provides a Morita equivalence between the algebras H c and eH c e. Corollary 7 .1. 3 . 73Given c ∈ Q good , there exists a constant d = d(c) such that for all primes p > d(c), the eH c e-H c -bimodule eH c provides a Morita equivalence between the k p -algebras H c and eH c e. // / Hom eHce (eH c , eH c ) / Γ(Hilb (1) , H c ). holds in Br(S • ). (7.4.5) To see this, restrict the Azumaya algebra D (1) to the open subset S ⊂ [h × h] (1) . The map q S , see (7.4.4), is a Galois covering with the Galois group W . It follows that the sheaf B := (q S ) q(D (1) ) S W is an Azumaya algebra on S. Furthermore, it is immediate from the construction that, in Br(S), one has an equality β S = [B]. Pulling back via the isomorphism Υ S , see (7.4.4), we deduce that (Υ * β) S • = (Υ S ) * β S = (Υ S ) * [B]. F ′ c : D(g • ) P −→ D(X) P −→ D(X D X /D X ·J c ) P −→ D(U, χ c ).(8.4.3) Thus, RF is a graded flat k[t]-module, where t stands for the coordinate on A 1 .With these notations, the statement of the Lemma amounts to the claim that, for all sufficiently small G m -stable Zariski open affine subsets Y ⊂ M , there is a natural graded space isomorphism Now, the functor M −→ M A , of A-invariants, takes short exact sequences of A-modules of the form k[A] ⊗ E to short exact sequences. Hence applying this functor to the short exact sequenceThus, we see that RF/t· RF is a rank one free k[A] ⊗ k[ Y]-module. We deduce from Lemma 4.2.5 that RF is isomorphic to a rank one free k[A] ⊗ k[A 1 × Y]-module. Furthermore, it is easy to show that this isomorphism can be chosen to be A-equivariant. Proof of Corollary 7.1.3. Let c denote an independent variable, and let Z[ 1 n! ][c], be the localization of the polynomial ring Z[c] at the number n! ∈ Z. Given a nonzero element f ∈ Z[ 1 To prove the Claim, consider the standard increasing filtration F q on H and the induced filtration F q(H/HeH) on the quotient algebra H/HeH. 2 The associated graded gr F (H/HeH), is a finitelyn! ][c] and a Z[ 1 n! ][c]-module M , we write M (f ) for the localization of M at f , a module over the localized ring Z[ 1 n! ][c] (f ) . Let H and eHe be the universal Cherednik algebras, viewed as algebras over the ground ring Z[ 1 n! ][c]. We first establish the following Claim 7.1.5. There exists a polynomial f ∈ Z[ 1 n! ][c] such that H (f ) = (HeH) (f ) and, moreover, such that f (c) = 0 for any c ∈ Q good . generated Z[ 1 n! ][c]-algebra. Clearly, it suffices to show that this algebra vanishes generically over Spec Z[ 1 n! ][c]. To this end, observe that gr(H/HeH) is a finitely generated module over the graded algebra gr ⊗ H c = Q ⊗ (H c eH c ); that is, the fiber of Q ⊗ gr(H/HeH) over c ∈ Q good vanishes. We conclude that H = HeH holds over a nonempty Zariski open subset of Spec(Z[ 1 n! ][c]) that has a nontrivial intersection with any closed subscheme {c = c}, c ∈ Q good . This proves Claim 7.1.5.1 n! ][c][h × h * ]. The Generic Flatness Lemma implies that there exists a nonzero polynomial f ∈ Z[ 1 n! ][c] such that gr(H/HeH) (f ) is free over Z[ 1 n! ][c] (f ) . On the other hand, for c ∈ Q good , by Proposition 7.1.2 we have Q We complete the proof of Corollary 7.1.3 as follows. By Claim 7.1.5, there exist a polynomial f ∈ Z[ 1 n! ][c] and elements h ′ i , h ′′ i ∈ H, i = 1, . . . , m, such that Further, we have the following graded algebra isomorphisms:gr R Γ Hilb (1) , A c Proposition 4.3.3(ii) / / Γ Hilb n A 2 , O Hilb n A 2 ∼ (6.1.4) / / k[h * × h] W .(7.2.6) Moreover, by going through definitions, it is easy to verify that the composite map in (7.2.6) is equal to the map gr Ψ A c in (7.2.5). It follows, in particular, that Ψ A c is an injective morphism. On the other hand, Lemma 7.1.1 and Proposition 6.3.1 yield7.2.5) ∼ Theorem 7.3.2. Fix c ∈ Q good . Then, there exists a constant d = d(c) such that for all primes p > d(c), we have (i) The map (7.3.1) is an algebra isomorphism, moreover, RΓ i (Hilb(1) Theorem 7.4.1. For any point ξ ∈ (h × h)/W , the restriction of the Azumaya algebra H c , resp. A c , to the formal neighborhood of the fiber Hilb (1) ξ ⊂ Hilb (1) splits, i.e., there is a vector bundle V c,ξ , resp. W c,ξ , on Hilb(1) ξ such that one has notice that although h reg × {0} is not a subset of U the restriction to h reg × {0} is well-defined for a polynomial on G = g × V . The argument below is similar to[Q]. Acknowledgments. The authors are grateful to I. Gordon for informing one of us about his unpublished9.3 Proof of Theorem 9.1.1. Since gr H t,c = H, we have the Brylinski spectral sequence[Br]. This spectral sequence has E r,q 1 = HH r+q(H), and E r,q ∞ = gr(HH r+q(H t,c )). Since the algebra H t,c is Z/2Z-graded, the differentials d j with odd subscripts j in this spectral sequence vanish automatically. Therefore we will abuse notation by writing d i , E r,q i instead of d 2i , E r,q 2i . Let us now consider the differential d 1 . Obviously, we have d 1 = td1 is the differential corresponding to c = 0, t = 1. It was checked by Brylinski that d (1) 1 is the De Rham differential.By a result of[EG](which generalizes in a straightforward manner to positive characteristic), we have gr Z 0,c = Z 0,0 . Thus the differentials d (s) 1 are zero in cohomological degree zero, so they are morphisms of modules over Z 0,0 . Thus in cohomological degree 1, these differentials must land in twisted (torsion) components (as generically onwhere Ω 1 (p) (V ) denotes the first cohomology of the De Rham complex of V . We will now show that all the higher differentials d m , m ≥ 2, vanish in cohomological degree zero, and hence E −q,q ∞ = ((SV ) p ) Γ , as desired. Assume the contrary, and let m ≥ 2 be the smallest number such that d m does not vanish in cohomological degree zero. Then we can view d m as a derivation d m : ((SV ) p ) Γ → Ω 1 (p) (V ) Γ . Recall now ([K], Theorem 7.2) that if X is a smooth affine variety over k then the cohomology of the De Rham complex of X twisted by the Frobenius map can be identified with the module of differential forms on X (as a graded O X -module) via a map C −1 : Ω • (X) → H • (Fr qΩ • (X)), called the Cartier operator. This map is defined by the formulas C −1 (a) = a p , C −1 (da) = a p−1 da for functions a on X.Let d ′ m = Cd m C −1 . Then d ′ m : (SV ) Γ → Ω 1 (V ) Γ is a derivation. Thus, d ′ m gives rise to a Γ-equivariant regular function f on V reg with values in V ⊗ V * , where V reg is the set of points of V which have the trivial stabilizer in Γ. Since the complement of V reg has codimension 2, the function f extends to V and defines an element of SV ⊗ V ⊗ V * . Hence d ′ m is obtained by restricting a map d ′ m : SV → Ω 1 (V ) to Γ-invariants. Therefore, d m is obtained by restricting a map d m : (SV ) p → Ω 1 (p) (V ) to Γ-invariants. Now observe that the map d m must have degree −2m+2 < 0. On the other hand, the generators of (SV ) p sit in degree p, while the lowest degree in Ω 1 (p) (V ) is also equal to p. This means that d m = 0, which is a contradiction. The theorem is proved. Conjecture 8.2.1. For any c ∈ k, the following canonical map is an algebra isomorphism. Conjecture 8.2.1. For any c ∈ k, the following canonical map is an algebra isomorphism: . −−− −−→ D, g) D(g) · J c P . (8.2.2)−−− −−→ D(g) D(g) · J c P . (8.2.2) the authors have constructed, for any c ∈ k, an algebra homomorphism, called the deformed Harish-Chandra homomorphism: Φ c : D(g rs ) D(g rs ) · Ind c G −→ D. h regIn chapter 7 of [EG], the authors have constructed, for any c ∈ k, an algebra homomorphism, called the deformed Harish-Chandra homomorphism: Φ c : D(g rs ) D(g rs ) · Ind c G −→ D(h reg ) W . reg ) W . However, it has been shown in [EG] that the latter map vanishes on the two-sided ideal D(g rs ) · Ind c G = D(g rs ) · Ind c ∩ D(g rs ) G , hence, descends to a well-defined homomorphism D(g rs ) D(g rs ) · Ind c G = D(g rs ) G D(g rs ) · Ind c G −→ D(h reg ) W , (8.3.2) where the equality on the left exploits semisimplicity of the ad G-action on D(g rs ). The construction of [EG] can be also carried out over a field k of characteristic p. In that case, the adjoint G-action on D(g rs ) is not semisimple. = L C , The Calogero-Moser Operator ; G −→ D, The reader should be warned that the map referred to as the deformed Harish-Chandra homomorphism in [EG] was actually a map D(g rs ). h. so the equality on the left of (8.3.2) does not hold, in general. A more careful analysis of the construction of [EG], similar to that of Section 5.3 of the present paper, shows that it actually produces, without any semisimplicity assumption, a homomorphism of the form (8.3.1)= L c , the Calogero-Moser operator. The reader should be warned that the map referred to as the deformed Harish-Chandra ho- momorphism in [EG] was actually a map D(g rs ) G −→ D(h reg ) W . However, it has been shown in [EG] that the latter map vanishes on the two-sided ideal D(g rs ) · Ind c G = D(g rs ) · Ind c ∩ D(g rs ) G , hence, descends to a well-defined homomorphism D(g rs ) D(g rs ) · Ind c G = D(g rs ) G D(g rs ) · Ind c G −→ D(h reg ) W , (8.3.2) where the equality on the left exploits semisimplicity of the ad G-action on D(g rs ). The construction of [EG] can be also carried out over a field k of characteristic p. In that case, the adjoint G-action on D(g rs ) is not semisimple, so the equality on the left of (8.3.2) does not hold, in general. A more careful analysis of the construction of [EG], similar to that of Section 5.3 of the present paper, shows that it actually produces, without any semisimplicity assumption, a homomorphism of the form (8.3.1). Recall the open subset U ⊂ g rs × V • formed by all pairs (x, v) such that v is a cyclic vector for x, see Definition 5.3.1. Let g • ⊂ g rs be the set of all elements x ∈ g = gl(V ) such that our fixed vector v ∈ V is a cyclic vector for x. Clearly, g • is an Ad P -stable Zariski open dense subset of g. Recall also the two-sided ideal I c ⊂ Ug introduced in Sect. 5.3, and observe that Ind c ⊂ I c. The following result provides a relation between the Harish-Chandra homomorphism Ψ c ofRecall the open subset U ⊂ g rs × V • formed by all pairs (x, v) such that v is a cyclic vector for x, see Definition 5.3.1. Let g • ⊂ g rs be the set of all elements x ∈ g = gl(V ) such that our fixed vector v ∈ V is a cyclic vector for x. Clearly, g • is an Ad P -stable Zariski open dense subset of g. Recall also the two-sided ideal I c ⊂ Ug introduced in Sect. 5.3, and observe that Ind c ⊂ I c . The following result provides a relation between the Harish-Chandra homomorphism Ψ c of D(g rs ) D(g rs ) · J c P res / / D(g • ) D(g • ). · J c P Fc / / Γ(U, D U /D U ·I c ) GD(g rs ) D(g rs ) · J c P res / / D(g • ) D(g • ) · J c P Fc / / Γ(U, D U /D U ·I c ) G The fiber of this map over the class of the line kv ∈ P(V ) is the set U v = {(x, t · v) x ∈ g • , t ∈ k × }. Thus, we have a G-equivariant isomorphism U ∼ = G × P U v . Further. Sketch of Proof. The assignment (x, v) −→ kv clearly gives a G-equivariant fibration f : U −→ P(V ). it is clear that the map (x, t) → (x, t · v) gives a P -equivariant isomorphism g • × G m ReferencesSketch of Proof. The assignment (x, v) −→ kv clearly gives a G-equivariant fibration f : U −→ P(V ). The fiber of this map over the class of the line kv ∈ P(V ) is the set U v = {(x, t · v) x ∈ g • , t ∈ k × }. Thus, we have a G-equivariant isomorphism U ∼ = G × P U v . 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Ave, Cambridge, MA 02139, USA; [email protected] Independent University of Moscow, 11 Bolshoy Vlasyevskiy per. M F , V G , Department of Mathematics, MIT, 77 Mass. Ave. Moscow, Russia; Chicago, IL 60637, USA; Cambridge, MA 02139, USA119002Department of Mathematics, University of [email protected] P.E.. [email protected].: Independent University of Moscow, 11 Bolshoy Vlasyevskiy per., 119002 Moscow, Russia; [email protected] V.G.: Department of Mathematics, University of Chicago, Chicago, IL 60637, USA; [email protected] P.E.: Department of Mathematics, MIT, 77 Mass. Ave, Cambridge, MA 02139, USA; [email protected]	[]
[ "Influence of finite Hund rules and charge transfer on properties of Haldane systems", "Influence of finite Hund rules and charge transfer on properties of Haldane systems" ]	[ "A E Feiguin \nInstituto de Física Rosario (CONICET-UNR)\nBv. 27 de febrero 210 bis2000RosarioArgentina\n", "Liliana Arrachea \nDepartamento de Fisica\nPUC-Rio\nCaixa Postal\nCep: 22452-970 -RJ38071Rio de JaneiroBrasil\n", "A A Aligia \nCentro Atómico Bariloche and Instituto Balseiro\nComisión Nacional de Energía Atómica\n8400BarilocheArgentina\n" ]	[ "Instituto de Física Rosario (CONICET-UNR)\nBv. 27 de febrero 210 bis2000RosarioArgentina", "Departamento de Fisica\nPUC-Rio\nCaixa Postal\nCep: 22452-970 -RJ38071Rio de JaneiroBrasil", "Centro Atómico Bariloche and Instituto Balseiro\nComisión Nacional de Energía Atómica\n8400BarilocheArgentina" ]	[]	We consider the Kondo-Hubbard model with ferromagnetic exchange coupling JH , showing that it is an approximate effective model for late transition metal-O linear systems. We study the dependence of the charge and spin gaps ∆C , ∆S, and several spin-spin correlation functions, including the hidden order parameter Z(π), as functions of JH /t and U/t, by numerical diagonalization of finite systems. Except for Z(π), all properties converge slowly to the strong-coupling limit. When JH /t ∼ 2 and U/t ∼ 7 (the effective parameters that we obtain for Y2BaNiO5), ∆S is roughly half of the value expected from a strong-coupling expansion.	10.1103/physrevb.59.9916	[ "https://arxiv.org/pdf/cond-mat/9901220v1.pdf" ]	119,424,943	cond-mat/9901220	9d9dae4146f91e1b9efdf9a64254e766e3ccadbc	Influence of finite Hund rules and charge transfer on properties of Haldane systems 21 Jan 1999 A E Feiguin Instituto de Física Rosario (CONICET-UNR) Bv. 27 de febrero 210 bis2000RosarioArgentina Liliana Arrachea Departamento de Fisica PUC-Rio Caixa Postal Cep: 22452-970 -RJ38071Rio de JaneiroBrasil A A Aligia Centro Atómico Bariloche and Instituto Balseiro Comisión Nacional de Energía Atómica 8400BarilocheArgentina Influence of finite Hund rules and charge transfer on properties of Haldane systems 21 Jan 1999 We consider the Kondo-Hubbard model with ferromagnetic exchange coupling JH , showing that it is an approximate effective model for late transition metal-O linear systems. We study the dependence of the charge and spin gaps ∆C , ∆S, and several spin-spin correlation functions, including the hidden order parameter Z(π), as functions of JH /t and U/t, by numerical diagonalization of finite systems. Except for Z(π), all properties converge slowly to the strong-coupling limit. When JH /t ∼ 2 and U/t ∼ 7 (the effective parameters that we obtain for Y2BaNiO5), ∆S is roughly half of the value expected from a strong-coupling expansion. I. INTRODUCTION After Haldane's conjecture that integer-spin antiferromagnetic Heisenberg chains should exhibit a gap in their excitation spectrum 1 , there has been a considerable amount of research in spin S = 1 systems 2,3,5 . Fascinating aspects of these systems are the presence of free spin-1/2 excitations at the end of sufficiently long finite chains 3 , and the presence of a hidden string-topological order [4][5][6] . The compound Y 2 BaNiO 5 is a candidate to a nearly ideal realization of the spin S = 1 antiferromagnetic Heisenberg chain and stimulated intense research on the system recently. The 1D character is supported by experimental evidence which shows that the exchange couplings transverse to the chains are very small and unable to induce long range magnetic order 7 . The representation of the two S = 1/2 holes per Ni by a S = 1 spin in the effective one-band model is, however, an intuitive but not clearly justified simplification, being the underlying assumption a large Hund rule acting on the two relevant Ni-orbitals. Such a point of view has been adopted in most theoretical works of the system, like an alternative interpretation of specific-heat experiments 8 in Zn-doped Y 2 BaNiO 5 which raised doubts about the presence of free spin-1/2 excitations near the end of long chains 9 . Theoretical work motivated by other experiments 10,11 for Ca-doped Y 2 BaNiO 5 , for which the NiO chains are not broken, but doped with holes also retain only the ground state of the 3d 8 configuration of Ni +212- 16 . On general physical grounds one expects the effective Ni-Ni hopping (via O) to be of the order of the corresponding one in CuO chains (t ∼ 0.85eV 17 ). Furthermore, spectroscopic data in atomic Ni show that the Hund rule leads to a ferromagnetic exchange J ′ H ∼ 1.6eV. Thus, it seems that not only the triplet ground state of the 3d 8 configuration of Ni +2 , but also the excited singlet should be taken into account for a realistic description of the system. Except for a brief discussion on the charge gap and Ni L 3 x-ray absorption spectrum 16 , this issue has been unexplored so far, to our knowledge. Even if the effective intratomic repulsion U is large compared with t, as we shall show, the properties of the system differ from those of the strong-coupling limit. In particular, the expression ∆ S = 0.41049J for the spin gap in terms of the effective Ni-Ni exchange J 18,19 , is no longer valid (at least if J is calculated perturbatively, see section IV). Thus, at least a qualitative study of the effects of a realistic J H (instead of infinite) on the properties of these systems seems necessary. In this work we derive and study the Kondo-Hubbard model with ferromagnetic coupling, as an approximate (in the sense which will be clarified in the next section) effective model for linear transition metal-oxygen systems, in which the relevant transition-metal orbitals are the e g ones. The model retains the effects of charge fluctuations at the transition-metal ions and finite Hund rules. Explicit effective parameters are calculated for Y 2 BaNiO 5 . Its version with antiferromagnetic coupling J H has been extensively studied in the context of heavy fermion systems 21 while the ferromagnetic case also adquired relevance in connection with the physics of the perovskite Mn oxides with giant magnetoresistance 22 , being also closely related to the double-exchange model 23,24 and other models used to study these compounds 25,26 . In section II, we explain the derivation of of the Kondo-Hubbard model as a low-energy effective model, in which some terms of lower magnitude, were neglected for simplicity. Section III contains the results for charge and spin gaps, spin expectation values and several spin-spin correlation functions, obtained by numerical diagonalization. Section IV contains the conclusions. II. THE LOW-ENERGY REDUCTION PROCEDURE In the simplest and most usual perovskite structures, the transition metal atoms are in sites of nearly cubic (O h ) symmetry, surrounded by six O atoms, lying in the directions ±x, ±y, ±z. In the particular case of Y 2 BaNiO 5 , these octahedra are linked by their vertices and form well separated chains, making a nearly ideal one-dimensional compound. Since by far, the largest contribution to crystal-field splitting is due to covalency 29 , near the end of the 3d series, the few holes present in the 3d shell of the transition metal enter the e g (d 3z 2 −r 2 or d x 2 −y 2 ) orbitals. This is due to their larger hybridization with the 2p σ orbitals (those pointing towards the transition metal atom) of the nearest-neighbor O atoms. The starting multiband Hamiltonian for the system should include the above mentioned orbitals and can be divided as follows: (1) are a trivial extension of similar terms extensively described and studied in multiband models for the cuprates 30 and will be not reproduced here. The largest energies in the problem and the ingredients of the new physics when more than one relevant orbital per site is present, are contained in H d . The Coulomb and exchange integrals among the e g spin-orbitals can be parameterized in terms of three Slater parameters F 0 , F 2 and F 4 , using usual methods in atomic physics 31 . Denoting a † iσ (b † iσ ) the creation operator for the a 1g d 3z 2 −r 2 (b 1g d x 2 −y 2 ) orbital at site i with spin σ, the result can be written in the form: H = H d + H p + H pd + H pp ,(1)H d = U d i (a † i↑ a i↑ a † i↓ a i↓ + b † i↑ b i↑ b † i↓ b i↓ ) +(U d − J ′ H ) iσσ ′ a † iσ a iσ b † iσ ′ b iσ ′ + J ′ H 2 i [ σσ ′ a † iσ b † iσ ′ a iσ ′ b iσ +(a † i↑ a † i↓ b i↓ b i↑ + H.c.)],(2) where U d = F 0 + 4F 2 + 36F 4 and J ′ H = 8F 2 + 30F 4 . Because of the neglect of the t 2g (d xy , d yz , d zx ) orbitals, H d lost the invariance under rotations of the atom, but retains cubic (O h ) symmetry. From the two lowest excitation energies of atomic Ni, we obtain F 2 = 0.1600eV and F 4 = 0.0108eV, leading to J ′ H = 1.60eV. According to the theoretical interpretation of optical experiments in NiO, U d + J ′ H /2 = 10eV 32 . The variation of J ′ H along the 3d series is only a few per cent, while the value of U d is very similar to that calculated in the cuprates using constrained density-functional theory 33 . More sensitive to the particular system are the hopping parameters (included in H pp and H pd ) and, particularly, the transitionmetal to O charge transfer energy ∆ (defined as the energy necesary to take a hole from the ground state 3d 8 configuration of the transition metal and put it in the p σ orbital of the 2p shell of a nearest-neighbor oxygen atom 32 ), which increases to the left of the periodic table. To derive the effective Hamiltonian for one-dimensional transition metal-O systems, we employ the cell perturbation method 34 . For simplicity in the explanation below, we choose the particular case of NiO 6 octahedra sharing O atoms along the z direction (present in Y 2 BaNiO 5 ) and assume tetragonal symmetry. The p z orbitals of the O atoms lying between two Ni atoms are expressed in terms of Wannier functions π centered at each Ni atom 35 . Each NiO 5 cell, composed of the 3d orbitals at one Ni site, the π orbital at that site and those of the four nearest O atoms along the ±x and ±y directions 36 , is solved exactly. To construct the low-energy effective Hamiltonian, only eight eigenstates of the cell are retained: for two holes, the B 1g triplet (which is essentially the ground-state of the Ni +2 configuration plus corrections due to hybridization), and the first excited state, the B 1g singlet. For one and two holes in the cell, the lowest B 1g doublets are retained. These eight states are mapped into the corresponding ones of the Kondo-Hubbard model H KH (Eq. 3) at the corresponding site. The matrix elements of H in the restricted basis are calculated and mapped into the corresponding ones of H KH . The effect of the remaining states of H could be included perturbatively but we neglect it for simplicity. To retain a simple and more general form of H KH , we also neglect the dependence of the resulting effective hopping on the occupation and spin of the sites involved. The resulting effective Hamiltonian has the form: H KH = −t i a † iσ a i+1σ + U i a † i↑ a i↑ a † i↓ a i↓ −J H i S ia · S ib ,(3) where S ia , S ib are the spin of the fermions represented by the hole creation operators a † iσ and b † iσ respectively. These are effective operators with the same symmetry as those entering Eq. (2), but which differ from them in the general case. In what follows a † iσ and b † iσ refer to these effective operators. The meaning of H KH is easier to understand in the (3) is given by the second-order process which carries a 3d 3z 2 −r 2 hole to the same orbital of a nearest-neighbor Ni atom: t = t 2 pd /∆. However, the case ∆ ≫ U d is not representative of charge transfer systems like Y 2 BaNiO 5 , for which ∆ < U d . In this case, the states a † i↑ a † i↓ b † iσ \|0 actually represent states with occupation close to one in the O π orbitals. limit ∆ ≫ U d ≫ J ′ H , t pd , As a consequence, U is mainly determined by ∆ instead of U d , and the hopping matrix elements become dependent on the occupation and spin of the two sites involved 37 . As mentioned above, this dependence was neglected to keep a simple and more general form of H KH . To estimate the parameters of the effective model for Y 2 BaNiO 5 , we took the values of J ′ H and U d mentioned above, and the (more uncertain) values of ∆ and the different hopping parameters in H pd and H pp were taken from work on NiO 32 , with the p − d and p − p hopping parameters scaled with distance r as r −7/2 and r −2 respectively. The parameters of H KH which result from the mapping procedure are U = 4.4eV, J H = 1.2eV and t ∼ 0.7eV. It is interesting to note that J H has a very small sensitivity to the parameters of H. Instead, changing ∆ and the hoppings of H within reasonable values affects U by ∼ 20% and t by ∼ 30%. III. RESULTS. In this section, we study the behavior of the charge and spin gap, spin expectation values and spin-spin correlation functions of H KH , using Lanczos diagonalizations in periodic rings of length L = 4, 6 and 8. The rapid increase of the Hilbert space with L prevents us to study longer even chains with the present state of the art, but as we shall show, some trends are already clear. The unit of energy is chosen as t = 1. A. Charge gap In Fig. 1 we represent the charge gap ∆ C = E(1) + E(−1) − 2E(0), where E(n) is the ground-state energy for n added holes to the stoichiometric system (which contains one a 1g and one b 1g hole per site). The result for L = 8 is compared to that of a polynomial extrapolation in 1/L to estimate finite-size effects. These effects are small for U ≥ 4. As expected, the gap increases with U and J H . In the strong-coupling limit t = 0, the gap is ∆ 0 = U + J H /2. As t is turned on, but kept small, the leading correction to E(0) is of order t 2 /∆ 0 , while those of E(1), E(−1) are equal and of order t. Assuming a Neel background (alternating spin projections 1 and -1) the correction for one added or one removed hole can be calculated and is − √ 2t. In both cases, it is more convenient to align ferromagnetically the spin at the site of the added or removed hole with those of its nearest neighbors. Thus, for large U , J H , we estimate: ∆ C = ∆ 0 − 2 √ 2t.(4) In Fig. 2 we represent ∆ C − ∆ 0 as a function of U and J H . The results agree with Eq. (4) in the strong-coupling limit. In the opposite limit, for U = 0, and small values of J H , the results have important size effects, and a large positive value of ∆ C − ∆ 0 is not reasonable. However, the extrapolated results show a reasonable behavior and tend to small values in the limit of J H = 0. In any case, the results for U ≥ 4 seem reliable. From the parameter estimates for Y 2 BaNiO 5 given at the end of the previous section (U/t ∼ = 6.3, J H /t ∼ = 1.7, t ∼ = 0.7eV), we obtain ∆ C ∼ 3eV. This is somewhat larger than the experimental value ∆ C ∼ 2eV 10 . This discrepancy is probably due to the fact that the charge transfer gap and possibly the hopping parameters cannot be transferred directly from Ref. 32 (which is a theoretical interpretation of optical spectra in NiO) to Y 2 BaNiO 5 . B. Spin gap In Fig. 3, we show the spin gap ∆ S as a function of J H for several values of U . For U = 0, the result for ∆ S has already been reported 38 . Here, for the smaller values of U and J H , the finite-size effects are too large, and the extrapolated values to the thermodynamic limit (in some cases negative) are meaningless. However, for more realistic values of U , our results allow us to extract some conclusions. The qualitative behavior of ∆ S as a function of J H was to be expected from the limiting cases: if J H = 0, the model is equivalent to the Hubbard model plus L free spin-1/2 states, and therefore ∆ S = 0. For J H → 0, ∆ S ∼ J 2 H has been obtained using bosonization 28 . In the limit of large J H , the low-energy physics of H KH reduces to a spin-1 Heisenberg chain: H Heis = J i S i · S i+1 , J = t 2 ∆ 0 = t 2 U + J H /2 ,(5) where S i = S ia + S ib . This model has a spin gap ∆ Heis S = c(L)J, where the constant c(L) depends on the size of the system. In particular c(4) = 1, c(6) = 0.72, c(8) = 0.59, c(∞) = 0.41049 19 . The first three values of c(L) coincide within 1% with our results extrapolated to infinite J H . As J H increases, first ∆ S increases from zero, and as the system approaches the strong-coupling limit, ∆ S decreases with the effective spin-1 exchange J. In Fig. 4 we show ∆ S /∆ Heis S as a function of J H . For U > 4, the extrapolated values do not differ very much from those of L = 8. Note that even for large values of U , the ratio ∆ S /∆ Heis S is considerably smaller than 1 if J H ∼ t. In particular, for the parameters estimated for Y 2 BaNiO 5 , ∆ S /∆ Heis S ∼ 0.5. However, this ratio was assumed 1 to estimate the value of J from experimental measurements of ∆ S . Using the same set of parameters, we obtain from the extrapolated values ∆ S ∼ 240K, while the experimental value is ∆ S ∼ 100K 7,10 . In view of the approximations made in deriving H KH , the uncertainties in the parameters of H, and the sensitivity of ∆ S and J to these parameters, the result is satisfactory. Part of the overestimate is due to ferromagnetic corrections to J in second-order in the intercell hopping, which involve virtual quadruplet three-hole states. These states are contained in H, but were projected out of the Hilbert space of H KH . We have calculated J by the cell-perturbation method, including these corrections. The effective J is reduced from 0.098eV to 0.088eV, but the result is quite sensitive to the parameters of H. Comparison with exact diagonalizations of a Ni 2 O 11 cluster 16 , shows that the second-order result of the cell-perturbation method is still an overestimation by a factor near 2, due to higher order corrections. C. Spin expectation values The behavior of the spin gap as a function of J H and U displays a slow change from the weak to the strongcoupling regimes. In Fig. 5(a) we show the ratio of the spin gap to the effective exchange, as a function of U for L = 8 and different values of J H . For any nonzero value of J H , the strong-coupling limit is reached for sufficiently large U and the gap tends to the limit ∆ Heis S = 0.59J. Fig 5(b) shows the corresponding change in the total spin of both itinerant (a 1g ) and localized (b 1g ) holes as U is increased, in the lowest-energy state with total spin and projection S t = S z t = 1. In the limit of small J H , both types of holes are decoupled and it is easier to flip a localized hole rather than an itinerant one from the S t = 0 ground state. As a consequence, S z bt ∼ = 1, S z at = 1 − S z bt ∼ = 0. In the opposite limit of very large J H , the singlet states at each site ( (a † i↑ b † i↓ − a † i↓ b † i↓ )\|0 ) can be projected out of the relevant Hilbert space, and in this case, the following equality among spin operators at a given site can be proved: 2S ia = 2S ib = S i . Summing over all sites: 2S at = 2S bt = S t . Thus, S z at changes from 0 to 1/2 as J H increases and S z bt = 1 − S z at . The effect of increasing U is to localize the itinerant a 1g holes and therefore, to contribute to the effect of J H , reaching faster the strong-coupling limit. However, it is noticeable that the approach to this limit is very slow. Comparison of the quantities represented in Fig. 5 for different sizes suggests that this approach is even slower in the thermodynamic limit. D. Spin-spin correlations In addition to the spin-spin correlation functions S 1 (l) = (S z ia + S z ib )(S z i+la + S z i+lb ) , S 2 (l) = (S z ia − S z ib )(S z i+la − S z i+lb ) ,(6) we also study in this section the string correlation function Z(j − i) = S z i exp(iπ j−1 l=i+1 S z l )S z j .(7) This latter has been propposed as a hidden order parameter for S = 1 chains to describe a hidden Z 2 × Z 2 symmetry breaking corresponding to the appearance of the Haldane gap 4,5 . This symmetry has been first implicitly introduced in an elegant variational approach for the excited states 6 . In Fig. 6, we show the different correlation functions. S 2 (l) has an on-site value S 2 (0) ∼ = 0.3 and for other distances S 2 (l) < 0.02 for the parameters of Fig. 6. The antiferromagnetic correlations are evident. They are larger for the localized holes than for the itinerant ones, as expected. For U = 0, a tenedency to antiferromagnetic order with wave vector q = π is expected from the form cos(2k F r) of the oscilating Ruderman-Kittel-Kasuya-Yosida effective interaction between localized b 1g holes at a distance r mediated by the mobile a 1g ones with Fermi wave vector k F = π/2. In the strong-coupling limit, the effective model H Heis (5) leads to the same type of short-range correlations. The Fourier transform (S 1 (q) = l e −iql S 1 (l), etc.) of some of these correlation functions for wave vector q = π, is represented in Fig. 7. As U increases, S 1 (π) and S 2 (π) approach slowly the asymptotic value in the strong-coupling limit, as it was the case of the spin gap and spin expectation values already discussed. Instead, Z(π) seems to saturate faster to a fixed value as J H and U increase. In the strong-coupling limit, Z(π) is a signature of the Haldane state 4,5 . For our model, Z(π) is a possible generalization of this order parameter, when local singlet states and charge fluctuations are allowed. The results of Fig. 7 are a hint that Z(π) can be used as the corresponding parameter of a hidden order that also exists in the Kondo-Hubbard model H KH . The difference in the behavior of Z(π) as a funcion of U , J H , in comparison with that of the other correlation functions, is an indication that Z(π) is more sensitive to the transition to the spin-gap state. That is precisely what one expects from a quantity playing the role of an order parameter. The other spin-spin correlations should be shortrange-like, due to the opening of the spin gap for finite U, J H 27,28 . Bosonization results indicate that this gap increases quadratically with the Hund coupling J H 28 , and is thus very small for J H → 0. In this regime, the correlation length is larger than the maximum system size that we have studied and we are unable to identify the change in the behavior of the correlation functions. For larger values of J H , the change of regime is captured by our results, and more clearly from the behavior of Z(π). IV. SUMMARY AND DISCUSSION We have studied charge and spin gap, spin expectation values, and several spin-spin correlation functions of a Kondo-Heisenberg model H KH , for two particles per site. We have shown that the model can be considered as an approximate effective model for one-dimensional transition metal-O systems, in which only the e g orbitals of the transition metals are relevant. Without any adjustable parameters (taking the parameters of the original multiband Hamiltonian from NiO scaled appropriately with distance), we obtain from the effective H KH , a charge and a spin gap of the correct order of magnitude for Y 2 BaNiO 5 . The model has also been used as a simplified model for the manganites, and our results should be qualitatively valid in the limit in which all Mn ions are Mn +3 . For sufficiently large U or J H , the charge gap of H KH is approximately given by ∆ C ∼ = U + J H /2 − 2 √ 2t. The effective model H KH contains the spin-1 antiferromagnetic Heisenberg model (also called Haldane chain) in the strong-coupling limit t ≪ J H , U . We obtain however that for realistic parameters for Y 2 BaNiO 5 or when the effective Hund-rule exchange coupling J H is not much larger than the effective hopping t, several properties differ from the Haldane limit. In particular, the different dynamics of the itinerant and mobile e g holes (reflecting that they do not behave as part of the same spin-1 object) are clearly manifested in spin-spin correlation functions. In addition, the spin gap ∆ S is roughly half of that expected from a strong coupling expansions. This fact should be taken into account when the effective spin-1 exchange J is extracted from experimental values of ∆ S and in the consistent interpretation of different thermodynamic experiments together with ∆ S . One possible way to interpret our results for ∆ S when U ≫ J H ∼ t, is that the effective spin-1 Heisenberg Hamiltonian Eq. (5) is still valid, but higher order corrections in t reduce appreciably the second-order result J = t 2 /(U + J H /2) for the effective exchange. Fourth-order corrections which include local singlet states as intermediate states are con-sistent with this reduction. However, when t ∼ J H , perturbation theory ceases to be valid, and it seems more adequate to include the local singlets ((a † i↑ b † i↓ − a † i↓ b † i↓ )\|0 in our notation) in the model Hamiltonian. ACKNOWLEDGMENTS Two of us are supported by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina. A. A. A. is partialy supported by CONICET. We would like to thank to H.A. Ceccatto for useful discussions. metal ion along the ±x and ±y directions are lacking, the low-energy reduction procedure and the form of HKH (Eq. (3)) are not substantially modified. However, if one of the mirror symmetries through the planes x = 0 or y = 0 is severely altered, the hopping of the "localized " biσ holes, can no longer be neglected in HKH . 37 For example, a large contribution to the hopping between two nearest-neighbor cells with 2 and 3 holes respectively, is due to πj,σ −πj+1,σ hopping which is favored if both cells have parallel total spin (antiparallel to that of the πj,σ hole σ). The amount of π holes is much smaller in cells with 1 and 2 holes than in those with 3 holes. 38 H. Tsunetsugu Fig. 1 for the charge gap minus its strong-coupling value ∆ 0 = U + J H /2. Fig. 3. Spin gap as a function of J H for several values of U . Solid symbols correspond to L = 8 and open symbols to the extrapolated value. Fig. 4. Ratio of the spin gap ∆ S over its strongcoupling value (∆ Heis S = c(L)J, where J = t 2 /(U + J H /2), see text) as a function of J H . The meaning of the symbols is the same as before. Fig. 6. Spin-spin correlation functions S z i S z i+l as a function of distance l for a 1g (S a (l)) holes, b 1g (S b (l)) holes, the sum of both spins (S 1 (l)), and that defined by Eq. (1), for J H = 2 and two values of U : U = 2 (circles), and U = 10 (diamonds). Fig. 7. Fourier transform of the correlation functions at momentum π for the sum (S 1 (π)) and difference (S 2 (π)) of the spin of both types of holes at a given site, and the hidden order parameter Z(π). FIGURE CAPTIONS Fig. 1 . 1Charge gap as a function of J H for several values of U indicated inside the figure. Solid symbols are the result for L = 8 sites. Open symbols correspond to extrapolation to the thermodynamic limit from the results of L = 4, 6 and 8, using a quadratic polynomial in 1/L. 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[ "Contagion-Preserving Network Sparsifiers: Exploring Epidemic Edge Importance Utilizing Effective Resistance", "Contagion-Preserving Network Sparsifiers: Exploring Epidemic Edge Importance Utilizing Effective Resistance" ]	[ "Alexander Mercier \nUniversity of South Florida\n\n\nSanta Fe Institute\n\n" ]	[ "University of South Florida\n", "Santa Fe Institute\n" ]	[]	Network epidemiology has become a vital tool in understanding the effects of high-degree vertices, geographic and demographic communities, and other inhomogeneities in social structure on the spread of disease. However, many networks derived from modern datasets are quite dense, such as mobility networks where each location has links to a large number of potential destinations. One way to reduce the computational effort of simulating epidemics on these networks is sparsification, where we select a representative subset of edges based on some measure of their importance. Recently an approach was proposed using an algorithm based on the effective resistance of the edges. We explore how effective resistance is correlated with the probability that an edge transmits disease in the SI model. We find that in some cases these two notions of edge importance are well correlated, making effective resistance a computationally efficient proxy for the importance of an edge to epidemic spread. In other cases, the correlation is weaker, and we discuss situations in which effective resistance is not a good proxy for epidemic importance.	null	[ "https://arxiv.org/pdf/2101.11818v1.pdf" ]	231,719,502	2101.11818	e4c9073f4d15de5c02ebb77daf1eb3df4b757014	Contagion-Preserving Network Sparsifiers: Exploring Epidemic Edge Importance Utilizing Effective Resistance Alexander Mercier University of South Florida Santa Fe Institute Contagion-Preserving Network Sparsifiers: Exploring Epidemic Edge Importance Utilizing Effective Resistance Network epidemiology has become a vital tool in understanding the effects of high-degree vertices, geographic and demographic communities, and other inhomogeneities in social structure on the spread of disease. However, many networks derived from modern datasets are quite dense, such as mobility networks where each location has links to a large number of potential destinations. One way to reduce the computational effort of simulating epidemics on these networks is sparsification, where we select a representative subset of edges based on some measure of their importance. Recently an approach was proposed using an algorithm based on the effective resistance of the edges. We explore how effective resistance is correlated with the probability that an edge transmits disease in the SI model. We find that in some cases these two notions of edge importance are well correlated, making effective resistance a computationally efficient proxy for the importance of an edge to epidemic spread. In other cases, the correlation is weaker, and we discuss situations in which effective resistance is not a good proxy for epidemic importance. Introduction Motivation Networks arise in a variety of contexts, from the study of epidemics, social contagions, terrorism, and biological invasions, to how knowledge itself is organized through epistemological networks. Use of network-based models for simulating epidemics has become particularly popular. However, simulating a stochastic epidemic model on a large network is computationally expensive, especially for dense networks, such as those derived from high-resolution mobility data which have been increasingly used in modeling contagion spread [5,7,14]. In these networks, there is a link between every pair of destinations, with weights corresponding to the flow of people who travel between them [14]. Considering all possible links as potential paths of infection takes significant of computation, and the problem is exacerbated by the need to perform many independent runs to get a sense of the probability distribution of epidemic sizes in stochastic models, as well as to test the effect of various intervention strategies. It is common to apply naive heuristics, like simply removing links whose weights are below some threshold, but it is not clear to what extent this preserves the true behavior of an epidemic, since rare events on low-weight edges can have important downstream consequences. but which is far less costly to study. The aim of sparsification is to approximate a network, ( , , ) by a graph sparsifier,˜( ,˜,˜), on the same set of vertices, , but with a reduced number of edges,˜, and modified edge weights,˜such that˜approximates G in some appropriate metric or metrics. Since graphs arise in the study of complex networks, graph sparsification has become both a topically important area of study and an interesting mathematical challenge. Therefore, network sparsification for networks used in stochastic epidemic simulations are motivated by two primary aspirations. First, to lower the computational cost of simulating epidemics on the network while retaining the same average dynamics. Second, the underlying goal is for the sparsification algorithm to conserve edges important to epidemic spread and remove edges that are not. Likewise, when paired with associated metadata, the removal of "unimportant" edges and the conservation of "important" edges may allow further analytic insight into network topological structure. We explore the notion of a contagion-preserving network sparsifier (CPNS) which seeks to reduce the number of edges in a network while simultaneously approximating average epidemic dynamics. In this way, a CPNS reduces the computational costs incurred in dynamical simulations on , by approximating typical dynamics on˜. Yet, how do we determine which links of the original network are the most important in an epidemic? One possible approach comes from an algorithm created by Daniel Spielman and Shang-Hua Teng, for which they won the Gödel Prize in 2015, and a simplification and improvement by Spielman and Srivastava [9,10]. The idea from the Spielman-Srivastava algorithm is to randomly sample edges with probability proportional to their effective resistance: in physical terms, the potential difference between their endpoints when one unit of current flows between them (and where each edge has resistance equal to the reciprocal of its weight). Effective resistance, also called "current-flow betweenness" or "spanning edge betweenness," has been explored as a measure of edge importance [12,13,1]. This resistance is high if an edge is one of the only ways to quickly get from one part of the network to another or if alternate paths are long or consist of low-weight edges. Choosing edges this way preserves important aspects of the graph spectrumapproximately preserving the graph Laplacian -and makes it possible to solve certain systems of equations in nearly-linear time [10]. However, this notion of "importance" may or may not align with the role edges play in an epidemic, since approximately preserving the graph Laplacian (which governs linear dynamics on the network) might not preserve the highly nonlinear dynamics of an epidemic. Are the edges with higher probability of selection in the Spielman-Srivastava algorithm more likely to spread disease? What is the right way to sparsify a network if our goal is to preserve its epidemic behavior rather than its spectrum? Methodology Towards this end, we develop the novel concept of an infection spanning tree, from which a general notion of epidemic edge importance may be formed and contrasted against effective resistance. Another formulation of effective resistance is the probability that a random spanning tree includes a given edge, where the spanning tree is chosen uniformly or with probability proportional to the product of edge weights. In contrast, the infection spanning tree lets us measures which trees and paths are most likely to spread a contagion. Focusing on SI contagion dynamics with a discrete-time SI model, we use the Spielman-Srivastava algorithm utilizing effective resistance in ( log ) time to produce CPNS, drawing parallels between the linear flow conceptualization of a network and contagion spread on that network. This research takes four approaches not taken in previous literature [11]. Swarup et al. also used effective resistance. However, in comparing with epidemics on the original network they used the metric of minimum Hamming distance [11]. We use a suite of metrics to study the performance of CPNS, namely average Hamming distance, mutual information, and how well the fraction of infected vertices over time matches the epidemic on the original network. Second, we compare effective resistance with the epidemic edge importance using infection spanning trees. Third, while preceding work centered around aggregate SI dynamics on CPNS, this research examines SI contagion dynamics on CPNS primarily through time. Lastly, we compare the Spielman-Srivastava algorithm with a simpler method that samples edges uniformly. In order to explore important edges in contagion spread and CPNS, we conduct a range of experiments on four random networks and a real-world air transport network. The results will show that while the linear flow analog to contagion processes is conceptually fertile, a range of diverse metrics must be implemented in order to view the full picture of the effectiveness of a given CPNS. This spectral sparsification algorithm can be used to create effective CPNS, permitting the removal of 75% of edges in some networks while approximately preserving the same average SI dynamics through time. However, more research must be conducted to understand the importance of any given edge within the context of an epidemic in order to fully grasp the workings of a CPNS. We begin by noting the parallels between linear flow in electrical networks and epidemic processes on social contact networks (Table 1). In contagion processes, we assume that transmission can occur along each edge independently. Therefore, the probability of the contagion spreading along an edge, is treated as analogous to the conductance of that edge. The reciprocal of conductance, known as resistance, is the expected time to infection along that edge or commute time, . The flow of a contagion along the network can then be thought of as current flow on the corresponding electrical network. Effective resistance, , is then defined as the potential difference across an electrical network when one unit of current is injected at a vertex, , and extracted at another vertex, , taking into account all possible paths between and . In order to approximate the effective resistance for all pairs of vertices, we created an implementation in R of the Spielman-Srivastava algorithm. Specifically, we implemented the formulation by Koutis, Levin, and Peng which works in time nearly linear to the number of edges [6]. Methods Linear Flow and Contagion Effective resistance between any two given vertices is given by the graph Laplacian when the resistance of each edge is defined as the inverse of its weight. The effective resistance between and is given by = ( − ) + ( − ) where + is the pseudoinverse of the graph Laplacian and is the column vector where there is a 1 at and zero elsewhere. The algorithm to approximate effective resistance between all vertex pairs inverts the Laplacian approximately using a random projection technique based upon the Johnson-Lindenstrauss lemma [10]. This approximation guarantees the approximate effective resistance given by the algorithm is between (1 − ) and (1 + ) for some constant error parameter which can be made as small as desired. In our implementation we typically have ≤ 0.1. Sparsification via effective resistance approximately preserves the effective resistance between all vertices, ensuring the expected time to infection for any vertex remains approximately the same [11]. This is due to the fact the spectrum of the graph will remain similar to that of the original graph; the spectrum of a graph has been shown to govern many aspects of the diffusion on that graph [3]. The Spielman-Srivastava Algorithm takes the effective resistance of all edges of a graph and uses to sample edges with probability, , equal to where is the weight of that edge. Notably, according to Algorithm 1, edge weights are sampled proportional to [11]. An edge selected by the Spielman-Srivastava algorithm is assigned a new weight of˜equal to / such that is the total number of samples taken. Edges are sampled with replacement, so that they can be selected multiple times. If the same edge is selected more than once, the edge weights are added together. Since the expected number of times is selected is , the expectation of˜is equal to its original edge weight . Thus, the expected adjacency matrix will equal the original adjacency matrix and the expected graph Laplacian will be equal to the original graph Laplacian. In this way, Algorithm 1 modifies edge weights to compensate for reduced edge number and can be thought of as being a part of a more general sampling strategy whereby edges are assigned probabilities by some metric of edge importance. Sparsification using Effective Resistance Broadly, if there is an edge which connects vertices and such that no alternate paths exist between and , or more generally if the alternate paths are long or involve lower-weight edges, then will be equal to the resistance of that edge with equal to 1. Conversely, if more paths are added between and , then the expected time to infection decreases and becomes less than 1. This suggests that as more paths are added between and , there is less incentive for the Spielman-Srivastava algorithm to select for the sparsifier. This notion is similar to the concept of the "embeddedness" of an edge from Schaub et al. which is defined as (1 − ) and conveys how important an edge is in weighted cuts of that graph or how much an edge acts as a "bottleneck" on that network [8]. Contagion-Preserving Network Sparsifiers Methodology and Metrics In order to gauge the success of a CPNS, SI discrete-time process on the original network and CPNS are saved as an indexed list of strings. The string is of length where is the number of vertices within the network. All vertices are indexed 1 through whereby entry of the string denotes vertex within the network and can be either 0 or 1 in the string, representing a susceptible or infected vertex, respectively. The cardinally of the indexed list of strings is where is the number of timesteps designated to run the model. The first string of the indexed list corresponds to the state of the SI model at timestep 1, the second string corresponding to timestep 2, and so on. The probability of an edge with weight transmitting a contagion with probability of transmission is given by = 1 − (1 − ) Because we are utilizing a discrete-time SI model, a contagion might be transmitted to a new vertex by two or more of its edges simultaneously, i.e. on the same time step. In this case, the edge that has the opportunity to transmit first is chosen uniformly at random. We employ the following set of metrics between the original network and CPNS contagion processes: average Hamming distance, mutual information score, and fraction of infected vertices in the network. The SI model is examined through time, where each metric is computed per time step. When contagion processes on the initial network are compared to those on the CPNS, the same patient zero is selected. Because the SI model is stochastic, the purpose of the CPNS is not to precisely mimic the progress of any one run of the epidemic: even independent runs of the SI model on the original network will vary and have some typical, nonzero Hamming distance and mutual information. Then, it is appropriate for the CPNS to be evaluated on its preservation of average metrics over multiple runs. Therefore, we begin by calculating a baseline. This baseline is computed by averaging the respective metric over multiple runs on the original network. Additionally, we pick a small number of CPNS to generate and run multiple runs of the SI model and take the average of the Hamming distance, mutual information, and fraction of infected. To ensure that the CPNS is robust, we pick uniformly at random one patient zero to begin the simulation, keeping the same patient zero for both the runs on the original network and the CPNS. For each average metric, a 95% confidence interval is also calculated. An effective CPNS will remain "close" to the given metric's baseline while matching its confidence interval. To better determine if an improved metric score produced by a CPNS is due to the addition of edges important to contagion spread or is merely due to the addition of more edges to the CPNS, we include a null model with which to compare. The null model uses the same framework as the Spielman-Srivastava algorithm, but samples uniformly from the set of edges with replacement. It will be entitled "uniform sampling." If the Spielman-Srivastava Algorithm is selecting edges of relevance to contagion spread, then the Spielman-Srivastava Algorithm CPNS should perform better than the corresponding uniform sampling CPNS while also selecting less edges. For easy comparison across multiple types of networks with a varying number of edges and to control edge number between the uniform sampling and SS sampling procedures, a CPNS with 25%, 50% and 75% of the total edges will be created for each network using both sampling procedures. Epidemic Edge Importance and Networks To investigate the relationship between effective resistance, the Spielman-Srivastava sparsification algorithm, and the spread of contagion, we introduce the measure of epidemic edge importance and an infection spanning tree. An infection spanning tree is created by simulating an SI contagion process from a patient zero until all vertices of the same component as patient zero are infected, keeping track of edges that transmit the contagion. Those edges which transmit the contagion make up the infection spanning tree. This process is performed over multiple runs, considering all possible patient zeros in the given network; the probability that an edge is found in any given infection spanning tree is termed the epidemic edge importance. The probability that an edge is selected by the Spielman-Srivastava algorithm, , and epidemic edge importance of all edges are normalized and organized in a Q-Q scatter plot, whereby the Pearson correlation coefficient can be used as a quantitative measure of similarity between the two metrics of edge importance. To explore the notion of edge importance as it relates to effective resistance, we test this methodology on four random networks and a real world airline network. The four random networks are a random network drawn from the configuration model with exponential logarithmic degree distribution, stochastic block model, complete network with edge weights drawn from a normal distribution, and a complete network with edge weights drawn from a power law distribution. The configuration network and stochastic block model each have 500 vertices and both complete networks have 100 vertices, denoted 100 . The configuration network and stochastic block model both have all edge weights set to one. The airline network (AirNet) contains 500 vertices corresponding to the 500 airports with the most traffic during the year 2002 in the United States [2]. Edge weights correspond to the total number of seats passing between a pair of airports [2]. Results Contagion-Preserving Network Sparsifiers We wish to check if the Spielman-Srivastava algorithm CPNS is close to the dynamics on the original network for average Hamming distance, mutual information, and fraction of vertices infected by the SI model as a function of time . Because the SI model is stochastic, the objective is not for the Hamming distance to be zero. Rather, it should be comparable to what is outputted from multiple independent runs on the original network. Likewise, for both mutual information and fraction of infected. For comparison, we also measure these quantities for CPNS with same percentage of total edges produced by uniform sampling. On the configuration network with an exponential-logarithmic degree distribution, the 25% and 50% uniform and Spielman-Srivastava sampling CPNS are comparable across all metrics (Figure1). The 75% Spielman-Srivastava sampling CPNS performs better than the uniform sampling CPNS, adhering closer to the baseline for all metrics (Figure 1). It should be noted that the 75% uniform sampling CPNS and 50% Spielman-Srivastava sampling CPNS have lower than baseline Hamming distances (Figure 1). This implies that variation found when the SI model was run on the original network was lowered by the CPNS. This could be disadvantageous for the SI model on the CPNS if it wishes to capture the average dynamics found on the original network. For the stochastic block model in Figure 2, the three CPNS of varying edge number for the uniform sampling and SS sampling are comparable. Each of the CPNS adhere closely to the baseline, with the exception of both the uniform sampling and SS sampling 25% CPNS, for both Hamming distance and fraction of infected. No CPNS correctly captured the average mutual information dynamics through time. (Figure 2). Likewise, for a complete graph with 100 vertices, 100 , with edge weights drawn from a normal distribution, the three CPNS of varying edge number for the uniform sampling and Spielman-Srivastava sampling are similar, staying close to the baseline. The only exception is the 75% SS sampling CPNS, which has a higher Hamming distance than the baseline in the middle timesteps of the SI simulation ( Figure 3). However, this elevated average Hamming distance does not effect the 75% Spielman-Srivastava sampling CPNS's performance with either mutual information or the fraction of infected through time. The 75% Spielman-Srivastava sampling CPNS is closer to the mutual information baseline than the corresponding uniform sampling CPNS. Additionally, the uniform and Spielman-Srivastava sampling CPNS perform similarly on the 100 network with edge weights drawn from a power distribution. The 25% and 75% SS sampling are closer to the baseline than the uniform sampling 25% and 75% CPNS (Figure 4). However, the 50% Spielman-Srivastava sampling CPNS performs worse with mutual information as a metric than all other CPNS. Lastly, when compared to the uniform sampling CPNS, it appears that the Spielman-Srivastava sampling CPNS is closer to the fraction of infected baseline through time. Lastly, the we examine a real-world airline network, AirNet. It is notable that neither the uniform nor Spielman-Srivastava sampling CPNS fully capture the SI dynamics on AirNet 5. As measured by Hamming distance, it appears that uniform sampling produced more effective CPNS than the Spielman-Srivastava sampling CPNS. Additionally, both 50% uniform and Spielman-Srivastava CPNS and the 75% Spielman-Srivastava CPNS have a larger than baseline fraction of infected through time. Only both 25% CPNS correctly captured the fraction of infected through time. However, as both 25% CPNS performed poorly in Hamming distance and mutual information, regardless of adherence to the baseline, both the uniform and Spielman-Srivastava sampling 25% CPNS fail to capture the totality of dynamics from AirNet. Comparing Effective Resistance and Epidemic Edge Importance The Q-Q plots showing the similarity of epidemic edge importance and show relatively good correlation for the configuration network ( = 0.93), the stochastic block model (0.8), and 100 with edge weights selected from a normal distribution ( = 0.96) ( Figure 6). However, AirNet has poor correlation, with = 0.067 ( Figure 6). For AirNet, undervalues the majority of edges deemed important by epidemic edge importance and overvalues certain select edges ( Figure 6). When considering AirNet, this discrepancy between and epidemic edge importance seems to be because the majority of highly important edges do not coincide with the edges epidemic edge importance deems as important ( Figure 6). To further investigate the relationship between epidemic edge importance and , two visualizations of AirNet were generated using the NetworkX in Python [4]: one with edge color dependent on and another with edge color dependent on epidemic edge importance (Figure 8) If an edge has high metric importance, the edge will be colored red. The generation of the two network visualizations suggests a key difference: a subset of edges connecting the core to a singular vertex, which is connected to the periphery of the network, are marked important by epidemic edge importance while does not mark the same subset of edges as important. Instead, the measure of importance more evenly picks edges throughout the network, with only a few edges in the core of the network being marked as especially important. . Edges are colored such that red means an edge is more important and blue means an edge is less important for each respective metric. Lastly, on the 100 network with edge weights drawn from a power law distribution, epidemic edge importance and are poorly correlated ( = 0.51) for a larger probability of transmission, = 3 × 10 −3 , and well correlated ( = 0.99) for a probability of transmission that is sufficiently small, = 3 × 10 −5 (Figure 7). Lowering has two consequences. First, lowering lowers the possibility that a vertex can be infected simultaneously by two or more of its edges within our SI model. Second, the SI discrete-time model moves closer to a continuous-time model as is lowered. This suggests that a continuous-time SI model would potentially produce better correlation between and epidemic edge importance. Discussion Contagion-Preserving Network Sparsifiers and the Spielman-Srivastava Sparsification Algorithm To some extent, the Spielman-Srivastava sparsification algorithm successfully created effective CPNS to preserve average SI dynamics across the three metrics with all four random networks. With respect to the configuration network, the Spielman-Srivastava sampling 75% CPNS best adheres to the baseline, allowing for a removal of 25% of the original edges while maintaining approximately the same average SI dynamics as measured by the average Hamming distance, mutual information, and fraction of infected. One quality worth noting is that both the 75% uniform sampling CPNS and the 50% Spielman-Srivastava sampling CPNS have a smaller Hamming distance than the baseline, suggesting that both CPNS lowered the baseline amount of variance between SI runs on the original network and itself. However, by removing some level of variance, this may cause both CPNS to be less faithful to the original network in a probabilistic sense: generating similar distributions of trajectories. For the stochastic block model, 50% of the edges could be removed with both the uniform and Spielman-Srivastava sampling, with both 50% CPNS staying close to the baseline in each metric except mutual information where it was lower for both sampling procedures. Both 100 networks see effective uniform and Spielman-Srivastava sampling, with the 25% CPNS performing adequately. This allows the removal of 75% of edges in both networks while retaining average SI dynamics. The two aspects of note are the relatively high Hamming distance for the 75% Spielman-Srivastava sampling CPNS on 100 with edge weights from a normal distribution and the SS sampling procedure performing marginally better on 100 with edge weights from a power law distribution when viewed through the fraction of infected metric. However, while moderately successful CPNS were produced for the four random networks, the Spielman-Srivastava algorithm only unambiguously performed better than the uniform sampling on the configuration network. Moreover, the small size of the networks used in this research were limited by the need to run the SI model over multiple runs on the original network. This does not necessarily imply that the Spielman-Srivastava performed poorly at generating CPNS. Rather, we suggest that this in part could be explained as a byproduct of the respective network structures. The stochastic block model and complete networks are both well connected, while the configuration network contains more vertices with lower degree. Because this uniform sampling is similar to the performance of the Spielman-Srivastava sampling CPNS such that both are successful at preserving average SI dynamics, this instead implies that edges within those networks have nearly the same level of importance to the epidemic. Conversely, this could be seen that no specific edge is important to the epidemic. In other words, it does not matter which edges are chosen to create the CPNS for those networks. Rather, what matters is that edges are chosen for those specific networks. Nevertheless, the relative success of the Spielman-Srivastava sampling procedure on the configuration network when compared to the uniform sampling procedure suggests that there are some instances where the Spielman-Srivastava algorithm will succeed and the uniform sampling procedure will fall short. The airline network AirNet was the only network where all CPNS that were ineffective. This may be due to how the Spielman-Srivastava algorithm modifies edge weights to compensate for reduced edge number. The Spielman-Srivastava CPNS may be ensuring certain vertices usually infected on the orignal network are almost always infected on the CPNS, inflating the mutual information score above the baseline by removing variation innate to the SI model on the original network. Similarly, the modified edge weights may account for the 50% uniform and Spielman-Srivastava CPNS and the 75% Spielman-Srivastava CPNS having a higher than baseline fraction of infected, where the Spielman-Srivastava algorithm giving higher edge weights to certain edges on the CPNS than found on the orginal network [11]. In this way, the Spielman-Srivastava algorithm may cause the contagion to spread faster, causing a higher than baseline fraction of infected and causing the CPNS Hamming distance to not adhere to the baseline. The relative success of both uniform sampling and Spielman-Srivastava sampling procedures speaks to the effectiveness of random sampling in preserving certain topological features of a network that a deterministic algorithm would not [11]. Consider the example of a network with groups that contain many intra-group edges and few inter-group edges such that edges between groups have lower weight than edges within groups. In this scenario, one common deterministic strategy would be to simply remove edges below a certain weight threshold; this would cut off the communities from one another, resulting in a poor CPNS. In contrast, a random sampling (either the uniform sampling or the Spielman-Srivastava sampling procedures) procedure would most likely retain a few of the inter-group edges and produce a better performing CPNS. Epidemic Edge Importance and Probability of Selection The relatively high correlation of with epidemic edge importance -probability of that same edge appearing in an infection spanning tree -on the unweighted configuration and stochastic block model networks suggests that the Spielman-Srivastava algorithm is selecting edges with high importance to the SI model. This is especially notable in the configuration network, which is sparser. Similarly, 100 with edge weights from a normal distribution also has high correlation between epidemic edge importance and . Particularly, for the configuration network, edges connecting low degree vertices have epidemic edge importance and probability of selection close to 1. Yet, AirNet has relatively low correlation between epidemic edge importance and probability of selection. For AirNet, probability of selection undervalues many edges with high epidemic edge importance, suggesting that the Spielman-Srivastava algorithm is overvaluing certain edges that are not as important to disease spread. Additionally, the correlation between and epidemic edge importance may be dependent on the probability of transmission , whereby if is small enough the difference in edge weight becomes more pronounced and those bottle necks marked important by effective resistance also have higher epidemic edge importance, as supported by Figure 7. Because the epidemic edge importance relies on the SI model, the metric of edge importance only relies on the infection rate and the topological structure of the network. This is in contrast to something like the SIR model, where it would be dependent on both the infection and recovery rates. One consequence of this is that for vertices on the periphery of the network, the SI epidemic will eventually infect them where an SIR metric of edge importance may or may not. This is important when considering all possible patient zeros; An SIR model measure of epidemic edge importance may bias towards the core of the network undervaluing potentially important edges on the periphery. This is ideal when considering potential intervention strategies that necessitate interdiction of an edge. Note that any edge which exists as the only path from one part of the network to another is assigned epidemic edge importance 1 by our method, as well as a effective resistance of 1. If this edge leads to only a single isolated vertex on the periphery of the network, this may seem counter intuitive, since a typical epidemic might not reach this vertex. However, in the SI model, all vertices in the connected component containing patient zero eventually becomes infected. Moreover, this isolated vertex might itself be patient zero, in which case its single edge is crucial. In general, the idea of epidemic edge importance depends on the details of the epidemic model and parameters used; the probability that any given edge transmit a contagion depends on the specifics of the contagion. For instance, "SIR epidemic edge importance" could also be defined. We chose not to examine SIR epidemic edge importance, as this measure of edge importance depends on two parameters, rate of infection and recovery, instead of one, rate of infection. We focus on the SI model version of epidemic edge importance because it is a simple measure of whether the contagion is likely to spread by an edge if the epidemic reaches (or begins at) either of its endpoints. In this way, the SI version of epidemic edge importance is robust. Even so, AirNet had poor correlation between and epidemic edge importance. This may be because even if there are many alternate paths out of the core of the network, those paths may be long or consist of low-weight edges. Then, especially in a discrete-time model where is fairly large, the epidemic will typically cross to the other part of the network before it has time to traverse the alternate paths. This suggests a tension between vertex centrality and effective resistance of an edge in this particular network, where an edge that is connected to a highly central vertex has high epidemic edge importance but low effective resistance. We see this in the subset of edges with high epidemic edge importance connecting a vertex which links the core of the network with the periphery of the network. The fact that AirNet is the only network to have poor correlation of and epidemic edge importance and is the only network to produce poor forming CPNS implies that correlation of may be a good predictor of CPNS effectiveness. Future Outlook The Spielman-Srivastava algorithm was shown to be reasonably effective at producing reliable and robust contagion-preserving network sparsifiers on an assortment of different networks, allowing for the removal of up to 75% of the edges in certain networks. To some extent, this means that the Spielman-Srivastava algorithm, which approximately preserves the graph Laplacian and therefore linear dynamics, can also approximately preserve the nonlinear behavior of a contagion. However, the fact that even a simple uniform sampling procedure also works fairly well reveals that in some networks there is not a strong enough difference between the importance of different edges to illustrate the specific virtues or faults of the Speilman-Srivastava algorithm within the context of pre-serving average epidemic dynamics. Moreover, both the uniform and Spielman-Srivastava sampling procedures failed to generate effective contagion-preserving network sparsifiers for AirNet, demonstrating potential flaws. Therefore, a greater variety of networks tailored to illustrate the workings of the Spielman-Srivastava sampling procedure in the context of an epidemic should be considered. Additionally, for a majority of the networks considered we found a strong correlation between SI epidemic epidemic importance and the importance, , assigned by the Spielman-Srivastava algorithm, suggesting that effective resistance operates as a good approximation of the importance of any given edge in contagion spread. For AirNet, we found that the network that only produced ineffective Spielman-Srivastava CPNS also was the only network that had poorly correlated and epidemic edge importance, suggesting that the correlation between and epidemic edge importance may act as an indicator of Spielman-Srivastava CPNS performance. We also found that this correlation increases when the parameter in a discrete-time simulation is reduced, or equivalently when we approach a continuous-time version of the SI model, for 100 where the edge weights were drawn from a power law distribution. Still, the fact that the real-world airline network did not have well correlated and epidemic edge importance shows that a greater understanding of the relationship between the two metrics is needed. To better explore the notion of epidemic edge importance, a transition from a discrete time to a continuous time SI model is suggested. More comprehensively, further work is needed to understand the importance of an edge within the context of contagion spread. Intervention strategies acting on vertices, such as vaccinations to protect a vertex, and behavioral interventions, through the interdiction of edges, are critical to controlling and containing disease spread [14]. Much future work remains to address the problem of epidemic sparsifiers. A deeper exploration of network sparsifiers which approximately preserve the simplest case of SI dynamics on larger and more complex real-world networks is needed, as well as an exploration of SIR and more complex epidemic models such as SEIR. Additionally, while there are appealing parallels between important edges in epidemics and edges with high effective resistance, a better notion of the importance of an edge in the context of an epidemic may exist. Likewise, other notions of edge importance using more complex models than SI should be explored. In general, a deeper understanding of the relationship between linear flows and epidemic processes on networks would greatly benefit this course of research. The question of what edge importance means in the context of contagion spread is critical to producing effective contagion-preserving network sparsifiers . Lastly, a broader testing of other sparsification algorithms within the same class of random sampling sparsifiers to approximately preserve average contagion dynamics is recommended. This leaves the issue of producing effective contagion-preserving network sparsifiers an open question. Algorithm 1 : 1Sparsification by effective resistance [9] Input: network ( , , ) Output: network˜( ,˜,˜) Parameters: , the number of samples Procedure: Choose random edge from with probability ∝ e Add edge to˜with weight˜= / Take samples with replacement and sum weights if an edge is chosen more than once Figure 1 . 1Comparison of CPNS performance on a configuration network with degree list generated from an exponential logarithmic distribution. From top to bottom, the plot displays the Hamming distance, mutual information, and fraction of infected vertices. On the left are the uniform sampling CPNS, termed "Uniform Sampling", and the right the Spielman-Srivastava CPNS, called "SS Sampling." The baseline is shown in purple, 25% edge sparsifier in red, 50% in green, and 75% in blue, with shaded region in each color representing the 95% confidence interval. Figure 2 . 2Comparison of CPNS performance on a stochastic block model with four communities of equal size. From top to bottom, the plot displays the Hamming distance, mutual information, and fraction of infected vertices. On the left are the uniform sampling CPNS and the right the Spielman-Srivastava CPNS. The baseline is shown in purple, 25% edge sparsifier in red, 50% in green, and 75% in blue, with shaded region in each color representing the 95% confidence interval. Figure 3 . 3Comparison of CPNS performance on a complete network with 100 vertices and edge weights drawn from a normal distribution. From top to bottom, the plot displays the Hamming distance, mutual information, and fraction of infected vertices. On the left are the uniform sampling CPNS, termed "Uniform Sampling", and the right the Spielman-Srivastava CPNS, called "SS Sampling." The baseline is shown in purple, 25% edge sparsifier in red, 50% in green, and 75% in blue, with shaded region in each color representing the 95% confidence interval. Figure 4 . 4Comparison of CPNS performance on a complete network with 100 vertices and edge weights drawn from a power law distribution. From top to bottom, the plot displays the Hamming distance, mutual information, and fraction of infected vertices. On the left are the uniform sampling CPNS, termed "Uniform Sampling", and the right the Spielman-Srivastava CPNS, called "SS Sampling." The baseline is shown in purple, 25% edge sparsifier in red, 50% in green, and 75% in blue, with shaded region in each color representing the 95% confidence interval. Figure 5 . 5Comparison of CPNS performance on an airline network, AirNet, containing the top 500 airports in the United States in 2002, with edge weights corresponding to the number of seats passing between pairs of airports. From top to bottom, the plot displays the Hamming distance, mutual information, and fraction of infected vertices. On the left are the uniform sampling CPNS, termed "Uniform Sampling", and the right the Spielman-Srivastava CPNS, called "SS Sampling." The baseline is shown in purple, 25% edge sparsifier in red, 50% in green, and 75% in blue, with shaded region in each color representing the 95% confidence interval. Figure 6 . 3 Figure 7 . 637Displayed are the Q-Q plots of the (a) complete network with edge weights from a normal distribution, (b) stochastic block model, (c) AirNet, and (d) configuration network with degree drawn from an exponentiallogarithmic distribution. The x-axis corresponds to a normalized epidemic edge importance of edge and the y-axis to a normalized . The value represents the Pearson correlation coefficient. (a) = 3 × 10 −5 (b) = 3 × 10 −Q-Q Plot of epidemic edge importance and at two different probabilities of transmission, , resulting in two Pearson correlation coefficient values: (a) = 0.99 and (b) = 0.51. Figure 8 . 8A visualization of the air traffic network AirNet showing (a) epidemic edge importance and (b) AcknowledgmentsThis work was carried out as part of an REU (Research Experience for Undergraduates) program at the Santa Fe Institute under the mentorship of Cristopher Moore and Maria Riolo, funded by NSF grants OAC-1757923 and IIS-1838251. We are also grateful to Samuel Scarpino, Sayandeb Basu, and Andrew Kramer for their helpful conversations. Effective and efficient approximations of the generalized inverse of the graph Laplacian matrix with an application to current-flow betweenness centrality. 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[ "An Analytic Study of Strain Engineering the Electronic Bandgap in Single-Layer Black Phosphorus", "An Analytic Study of Strain Engineering the Electronic Bandgap in Single-Layer Black Phosphorus" ]	[ "Jin-Wu Jiang \nShanghai Institute of Applied Mathematics and Mechanics\nShanghai Key Laboratory of Mechanics in Energy Engineering\nShanghai University\n200072ShanghaiPeople's Republic of China\n", "Harold S Park \nDepartment of Mechanical Engineering\nBoston University\n02215BostonMassachusettsUSA\n" ]	[ "Shanghai Institute of Applied Mathematics and Mechanics\nShanghai Key Laboratory of Mechanics in Energy Engineering\nShanghai University\n200072ShanghaiPeople's Republic of China", "Department of Mechanical Engineering\nBoston University\n02215BostonMassachusettsUSA" ]	[]	We present an analytic study, based on the tight-binding approximation, of strain effects on the electronic bandgap in single-layer black phosphorus. We obtain an expression for the variation of the bandgap induced by a general strain type that includes both tension in and out of the plane and shear, and use this to determine the most efficient strain direction for different strain types, along which the strongest bandgap manipulation can be achieved. We find that the strain direction that enables the maximum manipulation of the bandgap is not necessarily in the armchair or zigzag direction. Instead, to achieve the strongest bandgap modulation, the direction of the applied mechanical strain is dependent on the type of applied strain.	10.1103/physrevb.91.235118	[ "https://arxiv.org/pdf/1503.08362v2.pdf" ]	40,884,798	1503.08362	18e7b246ba6e76b48110d0e67dc21bee30abd23b	An Analytic Study of Strain Engineering the Electronic Bandgap in Single-Layer Black Phosphorus 28 Mar 2015 Jin-Wu Jiang Shanghai Institute of Applied Mathematics and Mechanics Shanghai Key Laboratory of Mechanics in Energy Engineering Shanghai University 200072ShanghaiPeople's Republic of China Harold S Park Department of Mechanical Engineering Boston University 02215BostonMassachusettsUSA An Analytic Study of Strain Engineering the Electronic Bandgap in Single-Layer Black Phosphorus 28 Mar 2015(Dated: March 31, 2015)numbers: 6865-k7322-f7780bn Keywords: Black PhosphorusElectronic BandStrain Effect We present an analytic study, based on the tight-binding approximation, of strain effects on the electronic bandgap in single-layer black phosphorus. We obtain an expression for the variation of the bandgap induced by a general strain type that includes both tension in and out of the plane and shear, and use this to determine the most efficient strain direction for different strain types, along which the strongest bandgap manipulation can be achieved. We find that the strain direction that enables the maximum manipulation of the bandgap is not necessarily in the armchair or zigzag direction. Instead, to achieve the strongest bandgap modulation, the direction of the applied mechanical strain is dependent on the type of applied strain. I. INTRODUCTION Strain engineering is an efficient mechanical approach to manipulating the physical properties in quasi twodimensional nanostructures such as graphene, MoS 2 , black phosphorus, and others. A huge number of works have been performed to examine the effectiveness of strain in modulating the physical properties of these 2D materials, with the particularly well-known example of using strain to generate a finite electronic bandgap for graphene (for a review, see eg. Ref. 1). Mechanical strain has also been used to modify the physical properties in single-layer black phosphorus (SLBP). [2][3][4][5][6][7][8] In particular, it has been shown in a number of previous works that mechanical strain is an effective means to tune the electronic bandgap in a wide range for SLBP. A large uniaxial strain in the direction normal to the SLBP plane can even induce a semiconductormetal transition. [9][10][11][12] The in-plane uniaxial strains along the armchair and zigzag directions have also been used to modify the bandgap of SLBP, [13][14][15] while the relative efficacy of uniaxial and biaxial strains have been comparatively studied for their effects on the electronic band structure for SLBP. [16][17][18][19] First-principles calculations have shown that both biaxial and uniaxial strains rotate the preferred electrical conducting direction by 90 degrees. 16 However, in nearly all of the above works, the bandgap changes have been obtained through strains applied either in the armchair or zigzag directions, or in the direction normal to the SLBP plane. This is reasonable, because these three directions are principal directions for the C 2h symmetry of the puckered configuration of the SLBP. However, a very recent study has demonstrated that the maximum in-plane Young's modulus for the SLBP is neither in the armchair direction, nor the zigzag direction. Instead, there exists a third principal direction with direction angle φ = 0.268π, along which the SLBP has the largest Young's modulus value. 20 Similarly, there is no guarantee that the most effective modulation of the bandgap by applying strain occurs in the armchair or zigzag direction. Furthermore, in most existing studies, the mechanical strain that is applied to the SLBP has been limited to either uniaxial or biaxial strain. Hence, a natural question to ask and answer is what the optimal direction and type of mechanical strain is that results in the largest variations in the bandgap. A systematic analysis and understanding for the strain effect on the bandgap for a general strain type will be essential for practical strainbased manipulation of the electronic properties in SLBP. This comprises the focus of the present work. In this paper, using the tight-binding approximation (TBA) model, we derive an analytic formula for the strain dependence of the electronic bandgap in SLBP. We obtain an analytic expression for the direction of the applied strain, along which the strain will induce the strongest modulation in the bandgap of the SLBP. In particular, the effects from different strain types (tension, shear, and coupled tension and shear) are systematically compared. The present paper is organized as follows. In Sec.II, we present details regarding the structure of SLBP. The TBA model for SLBP is introduced in Sec.III. Sec.IV (A) is devoted to the derivation of a general analytic formula for the strain dependent bandgap, and the bandgap variations induced by different strain types are compared in Sec.IV (B). The paper ends with a brief summary in Sec.V. There are two principal directions, i.e., the armchair (blue arrows) and zigzag (red arrows) directions. Color is with respect to the atomic z-coordinate. II. STRUCTURE The atomic configuration of the SLBP is shown in Fig. 1. The structure parameters were measured experimentally. 21 The two in-plane lattice constants are a 1 = r 37 = 4.376Å and a 2 = r 24 = 3.314Å, while the out-of-plane lattice constant is a 3 = 10.478Å. The origin of the Cartesian coordinate system is located in the middle of r 12 . The x-axis is in the horizontal direction and the y-axis is in the vertical direction. The z-axis is in the direction normal to the SLBP plane. There are four inequivalent atoms in the unit cell a 1 × a 2 of SLBP, which will be chosen as atoms 1, 2, 3, and 6 in this work. The coordinates of these atoms are r 1 = (−ua 1 , 0, −va 3 ), r 2 = (ua 1 , 0, va 3 ), r 3 = (0.5a 1 − ua 1 , 0.5a 2 , va 3 ), and r 6 = (−0.5a 1 + ua 1 , 0.5a 2 , −va 3 ). The two dimensionless parameters are u = 0.0806 and v = 0.1017. The bond lengths from the experiment are d 1 = r 23 = r 16 = 2.2449Å and d 2 = r 12 = 2.2340Å, and the two angles are θ 328 = 0.535π and θ 321 = 0.567π. III. TBA MODEL FOR SLBP We describe now the electronic band structure for SLBP obtained using a two orbital TBA model, which is derived from a recently proposed four orbital TBA model. 22 Specifically, it was proposed that the electronic band structure of the SLBP can be treated by a four orbital TBA model, 22 with four hopping parameters between atom pairs (2, 3), (2, 1), (2,6), and (3,6) in Fig. 1. Among these four hopping parameters, it was shown that the electronic band structure in SLBP is determined mainly by the first two nearest-neighbor hopping parameters between atom pairs (2, 3) and (2, 1). As a consequence, we use these two leading hopping parameters to describe the electronic band structure for SLBP in the present work. The two hopping parameters in this two orbital model are t 1 between atoms 2 and 3, and t 2 between atoms 2 and 1. For undeformed SLBP, the hopping parameter between atoms 2 and 8 (t 3 ) is the same as that between atoms 2 and 3. After the SLBP is deformed by the mechanical strain, hopping parameters t 1 and t 3 become different, so generally we have three hopping parameters in the following. Based on the two orbital TBA model, the electronic Hamiltonian for the SLBP can be written as, H =    E 0 t 2 0 t 3 + t 1 δ * 2 t 2 E 0 t 1 + t 3 δ * 2 0 0 t 1 + t 3 δ 2 E 0 t 2 δ 1 t 3 + t 1 δ 2 0 t 2 δ * 1 E 0   (1) where δ 1 = e ik1a1 and δ 2 = e ik2a2 are two phase factors, with k = k 1 b 1 + k 2 b 2 as the wave vector. Here b i are two reciprocal bases defined by b i · a j = 2πδ ij for i, j = 1, 2. The atomic energy level, E 0 , is set to 0 in the following calculation. The eigenvalue solution for the Hamiltonian in Eq. (1) gives four electronic bands for SLBP, C 4 E 4 + C 2 E 2 + C 0 = 0,(2) where the coefficients are, C 4 = 1; C 2 = −2 t 2 1 + t 2 2 + t 2 3 + 2t 1 t 3 cos ∆ 2 ; C 0 = t 4 2 − 2t 2 2 {cos ∆ 1 2t 1 t 3 + t 2 1 + t 2 3 cos ∆ 2 − sin ∆ 1 sin ∆ 2 t 2 1 − t 2 3 } + t 2 1 + t 2 3 + 2t 1 t 3 cos ∆ 2 2 , where ∆ 1 = 2πk 1 a 1 and ∆ 2 = 2πk 2 a 2 . Fig. 2 shows the electronic band structure from Eq. (2) for undeformed SLBP along high symmetric lines in the first Brillouin zone. The two hopping parameters are t 0 1 = −0.797 eV and t 0 2 = 2.393 eV. We have used the subscript 0 to denote hopping parameters in undeformed SLBP. For undeformed SLBP, the hopping parameter between atoms 2 and 8 (t 0 3 ) is the same as that between atoms 2 and 3, i.e., t 0 3 = t 0 1 = −0.797 eV. These parameters are obtained from the corresponding hopping parameters in the four orbital model by scaling them with the same factor, so that the bandgap from the two orbital TBA model agrees with that from the original four orbital model. 22 The band structure in Fig. 2 is very similar as that from the four orbital TBA model, and in particular the conductance band and the valence band are very close to the four orbital model. We focus on the bandgap modulated by mechanical strain in the SLBP. For small strain, the direct bandgap locates at the Γ point. The wave vector k = 0 at Γ point, so the four electronic energy states are E 1 = (t 1 + t 3 ) − t 2 ; E 2 = − (t 1 + t 3 ) − t 2 ; E 3 = (t 1 + t 3 ) + t 2 ; E 4 = − (t 1 + t 3 ) + t 2 . The energy gap is, E gap = E 3 − E 2 = 2 (t 1 + t 2 + t 3 ) .(3) For undeformed SLBP, we find that the bandgap E gap = 1.6 eV, which agrees well with the four orbital TBA model and other first-principles calculations. 22,23 IV. STRAIN EFFECT ON ELECTRONIC BANDGAP A. General formula for strain modulated bandgap We now consider the strain effect on the electronic bandgap of the SLBP. The electronic bands for SLBP are composed of s and p orbitals. 22 Moreover, the hopping parameter (t ) between s and p orbitals depends on the bond length (r ) as 24,25 t ∝ 1 r 2 . Thus, the applied mechanical strain can affect electronic states (including the bandgap) through modifying the hopping parameters in the TBA model. We consider the deformation of SLBP under a general mechanical strain in the direction with angle φ. The direction angle φ is determined starting from the x-axis, and so the armchair direction is for φ = 0, while the zigzag direction is for φ = π 2 . We perform a coordinate transformation, by rotating the x-axis in Fig. 1 to the strain directionê φ =ê x cos φ +ê y sin φ. The coordinates for a vector in this new coordinate system become   x φ y φ z φ   =   cos φ sin φ 0 − sin φ cos φ 0 0 0 1     x y z   ,(4) where (x, y, z) is the original coordinate for the vector, and the subscript φ is to denote quantities in the new coordinate system. In the new coordinate system, the coordinates are deformed by an arbitrary linear mechanical strain as   x ǫ y ǫ z ǫ   =   1 + ǫ x γ 0 γ 1 + ǫ y 0 0 0 1 + ǫ z     x φ y φ z φ   , (5) where γ is the shear component, while ǫ x , ǫ y , and ǫ z are normal strains. The subscript ǫ in the coordinate is to denote quantities after deformation. We have decoupled the z component from the other two in-plane components, considering the quasi-two-dimensional nature of the SLBP structure. In the linear deformation regime, the bond length r can be expanded as a function of all strain components, ǫ x , ǫ y , ǫ z , and γ as r = r 0 + ∂r ∂ǫ x ǫ x + ∂r ∂ǫ y ǫ y + ∂r ∂ǫ z ǫ z + ∂r ∂γ γ ≡ r 0 + α x ǫ x + α y ǫ y + α z ǫ z + α s γ,(6) where we have introduced α as the strain-related geometrical coefficients. Recalling the relationship between the hopping parameter and the bond length, t ∝ 1 r 2 , we get the strain effect on the hopping parameter, t = t 0 1 − 2 r 0 α x ǫ x − 2 r 0 α y ǫ y − 2 r 0 α z ǫ z − 2 r 0 α s γ .(7) According to Eq. (7), the key ingredient is to compute the strain-related geometrical coefficients α for each hopping parameter. For the strain ǫ x , we get the following geometrical coefficients for each hopping parameter t i , α x 1 = ∂r 23 ∂ǫ x \| ǫx=0 = 1 r 23 x 2 23φ = 1 d 1 [(0.5 − 2u) a 1 cos φ + 0.5a 2 sin φ] 2 ; α x 3 = ∂r 28 ∂ǫ x \| ǫx=0 = 1 r 28 x 2 28φ = 1 d 1 [(0.5 − 2u) a 1 cos φ − 0.5a 2 sin φ] 2 ; α x 2 = ∂r 21 ∂ǫ x \| ǫx=0 = 1 r 21 x 2 21φ = 1 d 2 (2ua 1 cos φ) 2 . Here, α x 1 is the coefficient corresponding to the hopping parameter t 1 . For the strain ǫ y , we obtain the following geometrical coefficients, α y 1 = ∂r 23 ∂ǫ y \| ǫy=0 = 1 r 23 y 2 23φ = 1 d 1 [− (0.5 − 2u) a 1 sin φ + 0.5a 2 cos φ] 2 ; α y 3 = ∂r 28 ∂ǫ y \| ǫy=0 = 1 r 28 y 2 28φ = 1 d 1 [(0.5 − 2u) a 1 sin φ + 0.5a 2 cos φ] 2 ; α y 2 = ∂r 21 ∂ǫ y = 1 r 21 y 2 21φ = 1 d 2 (2ua 1 sin φ) 2 . For the ǫ z strain, we get the following geometrical coefficients, α z 1 = ∂r 23 ∂ǫ z \| ǫz=0 = 1 r 23 z 2 23φ = 0; α z 3 = ∂r 28 ∂ǫ z \| ǫz=0 = 1 r 28 z 2 28φ = 0; α z 2 = ∂r 21 ∂ǫ z \| ǫz=0 = 1 r 21 z 2 21φ = 1 d 2 (2va 3 ) 2 . We can derive similar expressions for the geometrical coefficients, α s , corresponding to shear strain, α s 1 = 2 d 1 x 23φ y 23φ = 2 d 1 [(0.5 − 2u) a 1 cos φ + 0.5a 2 sin φ] × [− (0.5 − 2u) a 1 sin φ + 0.5a 2 cos φ] ; α s 3 = ∂r 28 ∂γ \| γ=0 = 2 r 28 x 28φ y 28φ = 2 d 1 [(0.5 − 2u) a 1 cos φ − 0.5a 2 sin φ] × [− (0.5 − 2u) a 1 sin φ − 0.5a 2 cos φ] ; α s 2 = ∂r 21 ∂γ = 2 r 21 x 21φ y 21φ = 2 d 2 (−2ua 1 cos φ) (2ua 1 sin φ) . Inserting these geometrical coefficients into Eq. (7), and using Eq. (3), we obtain the analytic expression for the strain dependence of the electronic bandgap, E gap − E 0 gap = −4ǫ x t 0 1 d 1 (α x 1 + α x 3 ) + t 0 2 α x 2 d 2 − 4ǫ y t 0 1 d 1 (α y 1 + α y 3 ) + t 0 2 α y 2 d 2 −4ǫ z t 0 1 d 1 (α z 1 + α z 3 ) + t 0 2 α z 2 d 2 − 4γ t 0 1 d 1 (α s 1 + α s 3 ) + t 0 2 α s 2 d 2 . After some algebraic manipulation, we get the strain induced modification in the bandgap, ∆E gap = e 0 ǫ z + (e 1 − 2e 2 ) (ǫ x + ǫ y ) − 2e 2 ǫ cos (2φ + ψ) ,(8) where the parameters e 0 , e 1 and e 2 are as follows We have introduced the following two quantities in the above derivation, tan ψ = 2γ ǫ x − ǫ y ; (12) ǫ = (ǫ x − ǫ y ) 2 + (2γ) 2 .(13) Eq. (8) shows the variation in the bandgap induced by a general strain applied in the direction with directional angle φ. As can be seen from Eq. (8), the variation in the bandgap depends on the strain angle φ with period π. For a given strain ratio, tan ψ = 2γ ǫx−ǫy , the maximum (or minimum) strain effect can be achieved, if the strain is applied in the direction with angle φ satisfying cos (2φ + ψ) = ±1, which gives the strain direction, φ = − ψ 2 + j π 2 ,(14) where j is an integer. This means that mechanical strain can introduce the largest (smallest) modulation of the bandgap if the strain is applied in the direction described by Eq. (14). In particular, we note that, to achieve the strongest strain effect on the bandgap, there is no guarantee that the strain should be applied in the armchair or zigzag direction. Instead, the optimal strain direction is generally dependent on the type of the applied strain. B. Comparison between different strain types In the above, we have derived the bandgap variation induced by a general strain in Eq. (8). We have also obtained the direction for a general strain in Eq. (14), where the direction lies in the 2D plane. This direction represents the most efficient strain direction, in that strain applied in this direction will generate the largest modulation of the bandgap. In this section, we will determine the most efficient direction for some common strain types in SLBP. We first note that e 1 − 2e 2 > 0, e 2 > 0, and ǫ = (ǫ x − ǫ y ) 2 + (2γ) 2 > 0 in Eq. (8). It is obvious that strains ǫ x and ǫ y have similar effects on the bandgap, so we will discuss only one of them in some situations in the following. (1) For uniaxial strain in the z-direction, i.e., ǫ x = ǫ y = 0, γ = 0, and ǫ z = 0, we have ∆E gap = e 0 ǫ z .(15) We can see that the change of the bandgap is a linear function of the applied strain. This is consistent with previous first-principles calculations. [9][10][11][12] (2) For in-plane uniaxial strain, i.e., ǫ x = 0, ǫ y = 0, ǫ z = 0 and γ = 0, we have ∆E gap = (e 1 − 2e 2 ) ǫ x − 2e 2 ǫ x cos 2φ.(16) The most effective direction is determined by the condition that both terms on the right side have the same sign, i.e., cos 2φ = −1. This gives φ = π 2 , which is the zigzag direction in SLBP, and means that uniaxial strain can introduce the strongest effect on the bandgap if it is applied in the zigzag direction in SLBP. For this uniaxial strain in the zigzag direction, the bandgap is ∆E gap = e 1 ǫ x . The coefficient e 1 > 0, leading to an increase of the bandgap due to tensile strain, which is consistent with first-principles calculations. 13 As another example, if we assume that the uniaxial strain ǫ x is applied in the armchair direction (φ = 0), then we have ∆E gap = (e 1 − 4e 2 )ǫ x , where the coefficient (e 1 − 4e 2 ) < e 1 . This means that, to induce the same bandgap variation, a larger strain magnitude is needed if the uniaxial strain is applied in the armchair direction. (3) For in-plane biaxial strain, i.e., ǫ x = ǫ y = ǫ, ǫ z = 0 and γ = 0, we find, ∆E gap = (e 1 − 2e 2 ) (ǫ x + ǫ y ) = 2 (e 1 − 2e 2 ) ǫ. (17) There is no preferred strain direction for biaxial strain, which is consistent with the intrinsically isotropic nature of biaxial strain. (4) For a general in-plane strain with ǫ x = ǫ y , ǫ z = 0 and γ = 0, we find ∆E gap = (e 1 − 2e 2 ) (ǫ x + ǫ y ) − 2e 2 (ǫ x − ǫ y ) cos 2φ.(18) The most efficient strain direction depends on the sign of ∆E gap . More specifically, it requires both terms on the right side to have the same sign as ∆E gap . For ∆E gap > 0, an effective strain application should require ǫ x +ǫ y > 0 according to the first term on the right side. From the second term, we have −(ǫ x − ǫ y ) cos 2φ > 0; i.e., we should have ǫ x < ǫ y for φ = 0 or ǫ x > ǫ y for φ = π 2 . This indicates that the tensile strain should be applied in the two principal directions (armchair and zigzag) of SLBP, so that the bandgap can be enlarged most effectively. Furthermore, for maximum bandgap increase, the tensile strain should be larger in the zigzag direction than the armchair direction. For ∆E gap < 0, the most effective strain application for bandgap reduction should require ǫ x + ǫ y < 0 according to the first term on the right side. From the second term, we have −(ǫ x − ǫ y ) cos 2φ < 0; i.e., we should have ǫ x > ǫ y for φ = 0 or ǫ x < ǫ y for φ = π 2 . This indicates that the axial strain should be applied in the two principal directions (armchair and zigzag) of SLBP, so that the bandgap can be reduced most effectively. Furthermore, the compressive strain should be larger in the armchair direction than the zigzag direction to reduce the bandgap. Considering that the strain is compressive in this situation, we have larger strain magnitude in the zigzag direction than the armchair direction. As a result, for both ∆E gap > 0 and ∆E gap < 0, strains should be applied in the two principal directions (armchair and zigzag) of SLBP. This result is consistent with recent first-principles calculations. 14 Furthermore, the strain magnitude in the zigzag direction should be larger than the strain magnitude in the armchair direction to achieve the largest bandgap change. (5) For pure shear strain, i.e., ǫ x = ǫ y = ǫ z = 0 and γ = 0, we find ∆E gap = 4e 2 γ sin 2φ. (19) It is important to point out that the most effective direction for the shear strain is determined by sin 2φ = ±1, which gives φ = ± π 4 , which illustrates that the most effective direction for pure shear is not in either the armchair or zigzag directions of SLBP. Instead, a pure shear strain should be applied in the direction with φ = ± π 4 , so that it can introduce the strongest effect on the bandgap for the SLBP. (6) For strain with ǫ y = ǫ z = 0, ǫ x = 0, and γ = 0, definition ǫx = ǫy = 0 ǫx = 0, ǫy = 0 ǫx = ǫy = ǫ ǫx = ǫy ǫx = ǫy = 0 ǫx = 0, ǫy = 0 ǫz = 0,γ = 0 ǫz = 0,γ = 0 ǫz = 0,γ = 0 ǫz = 0,γ = 0 ǫz = 0,γ = 0 ǫz = 0,γ = 0 ∆Egap e0ǫz (e1 − 2e2) ǫx 2 (e1 − 2e2) ǫ (e1 − 2e2) (ǫx + ǫy) 2e2γ sin 2φ (e1 − 2e2) ǫx −2e2ǫx cos 2φ −2e2\|ǫx − ǫy\| cos 2φ −2e2 ǫ 2 x + γ 2 cos (2φ + ψ) φmax ∆Egap > 0 N.A. zigzag, φ = π 2 arbitrary φ = π 2 , ǫx > ǫy > 0 φ = ± π 4 φ = − ψ 2 + (2j + 1) π 2 ∆Egap < 0 φ = 0, ǫy < ǫx < 0 φ = − ψ 2 + jπ we simultaneously apply the uniaxial strain ǫ x and the shear strain γ to modulate the bandgap of SLBP. In this situation, we have, ∆E gap = (e 1 − 2e 2 ) ǫ x − 2e 2 ǫ cos (2φ + ψ) . (20) To enlarge the bandgap, i.e., ∆E gap > 0, it can be seen from Eq. (20) that the most effective direction for applying strain is to ensure cos (2φ + ψ) = −1. This determines the angle for the strain direction, φ = − ψ 2 + (2j + 1) π 2 ,(21) where j is an integer. Furthermore, ǫ x and γ are related to each other as, Fig. 3 shows this relation between ǫ x and γ for different ∆E gap . Each curve in the figure indicates the most effective way to generate the corresponding change in the bandgap. It is clear that ǫ x < 0 is not a good choice, because it requires larger shear strain γ. Hence, for ∆E gap > 0, the most effective way is to apply a strain with ǫ x > 0, along with an appropriate, non-zero choice of shear strain γ. If larger ǫ x is applied, then the required shear component γ is smaller. We note again that the strain direction (φ) is determined by the actual applied strain ǫ x and γ, because of the relationship between φ and ψ in Eq. (21), and because tan ψ = 2γ ǫx . Similarly, to reduce the bandgap, i.e. ∆E gap < 0, the most effective direction for applying strain is to ensure cos (2φ + ψ) = 1. This determines the angle for the strain direction, ∆E gap = (e 1 − 2e 2 ) ǫ x + 2e 2 ǫ 2 x + (2γ) 2 . (22)φ = − ψ 2 + 2j × π 2 ,(23) where j is an integer. Furthermore, the strains ǫ x and γ are determined by the following relation, ∆E gap = (e 1 − 2e 2 ) ǫ x − 2e 2 ǫ 2 x + (2γ) 2 . (24) Fig. 4 shows this relation between ǫ x and γ for different ∆E gap . The above discussions on different strain types are summarized in Tab. I. From the third line in the table, uniaxial strain in the direction normal to the SLBP plane is the most effective strain type to modify the bandgap. In other words, to generate the same bandgap variation, this strain type requires the smallest strain magnitude among all strain types that have been discussed, because it has the largest pre-coefficient magnitude, \|e 0 \|. However, the ability to apply different strain types, and combinations of strain types, is dependent on the experimental technique that is utilized. Thus, we expect that Tab. I can serve as a guideline for experimentalists to choose the most appropriate strain type to manipulate the bandgap. V. CONCLUSION In conclusion, we have developed an analytic model based on the tight binding approximation to elucidate strain effects on the electronic bandgap in single layer black phosphorus. We have demonstrated that the direction along which the mechanical strain is applied is critical to achieving the maximum modulation of the bandgap. More specifically, we have performed a detailed comparison between the effects from different strain types, and for each strain type, we present predictions for the most efficient direction for the mechanical strain as summarized in Tab. I. FIG. 1 : 1(Color online) SLBP structure. FIG. 2 : 2Electronic band structure for undeformed SLBP using the two orbital TBA model. The bandgap ∆Egap = 1.60 eV is reached at the Γ point. [ (0.5 − 2u) a 1 ] FIG. 3 :FIG. 4 : 34(Color online) The most effective approach to enlarging the bandgap by a combination of normal strain ǫx and shear strain γ. The direction angle for the strain is φ = − ψ 2 + (2j + 1) π 2 , with tan ψ = γ ǫx . (Color online) The most effective approach to decreasing the bandgap by a combination of normal strain ǫx and shear strain γ. TABLE I : ISummary for the strain dependent bandgap variation. The last line lists the most effective direction for each strain type, along which the maximum bandgap variation can be achieved.strain type uniaxial strain uniaxial strain biaxial strain general strain shear uniaxial strain and shear Acknowledgements The authors thank A. Rudenko for helpful communications. The work is supported by the Recruitment Program of Global Youth Experts of China and the start-up funding from Shanghai University. HSP acknowledges the support of the Mechanical Engineering department at Boston University. . A H Castro Neto, F Guinea, N M R Peres, K S Novoselov, A K Geim, Rev. Mod. Phys. 81109A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. 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[ "SOME PROPERTIES OF PRE-TOPOLOGICAL GROUPS", "SOME PROPERTIES OF PRE-TOPOLOGICAL GROUPS" ]	[ "Fucai Lin ", "Ting Wu ", "Yufan Xie ", "Meng Bao " ]	[]	[]	In this paper, we pose the concepts of pre-topological groups and some generalizations of pre-topological groups. First, we systematically investigate some basic properties of pre-topological groups; in particular, we prove that each T 0 pre-topological group is regular and every almost topological group is completely regular which extends A.A. Markov's theorem to the class of almost topological groups. Moreover, it is shown that an almost topological group is τ -narrow if and only if it can be embedded as a subgroup of a pre-topological product of almost topological groups of weight less than or equal to τ . Finally, the cardinal invariant, the precompactness and the resolvability are investigated in the class of pre-topological groups.theory of pre-topology, we can consider the pre-topology on groups and pose the concept of pretopological groups, then we can systematically investigate some basic properties of pre-topological groups.This paper is organized as follows. In Section 2, we introduce the necessary notation and terminology which are used in the paper. In Sections 3 and 4, some basic properties of pre-topological groups and relationships among (quasi, para, almost) pre-topological groups are investigated. In Section 5, we mainly study quotient spaces of pre-topological groups and give the three isomorphisms of quotient spaces. In Section 6, we extend A.A. Markov's theorem to almost topological group and show that every almost topological group is completely regular. In Section 7, some cardinal invariants of pre-topological groups are studied. In particular, the well-known Guran's Theorem is extended, that is, an almost topological group is τ -narrow if and only if it can be embedded as a subgroup of a pre-topological product of almost topological groups of weight less than or equal to τ . In Section 8, some properties about precompactness and resolvability are investigated.Introduction and preliminariesDenote the sets of real number, rational number, positive integers, the closed unit interval and all non-negative integers by R, Q, N, I and ω, respectively. Readers may refer[16,22]for terminology and notations not explicitly given here.Definition 2.1.[10,19,14]A pre-topology on a set Z is a subfamily T of 2 Z such that T = Z and T ′ ∈ T for any T ′ ⊆ T . Each element of T is called an open set of the pre-topology.The name of pre-topology is given in[22]. Next, we list some definitions of pre-topological spaces which are introduced in [22]; of course, these definitions have important roles in our discussion of pre-topological groups.Definition 2.2.[22] Let (G, τ ) be a pre-topological space and B ⊆ τ . If for each U ∈ τ there exists a subfamily B ′ of B such that U = B ′ , then we say that B is a pre-base of (G, τ ).Let Z be a pre-topological space. For each z ∈ Z, the infimum of the set {\|B(z)\| : B(z) is a pre-base at z} is called the character of a point z[22]in Z, which is denoted by χ(z, Z). The supremum of the set {χ(z, Z) : z ∈ Z} is called the character of a pre-topological space (Z, τ ), which is denoted by χ(Z). Each set of cardinal numbers being well-ordered by <. Then the infimum of the set {\|B\| : B is a pre-base for Z} is said to be the weight of Z [22] and is denoted by w(Z).Definition 2.3. [22]Let h : Y → Z be a mapping between two pre-topological spaces (Y, τ ) andDefinition 2.4.[22] Let h : Y → Z be a bijection between two pre-topological spaces Y and Z. If h and h −1 : Z → Y are all pre-continuous, then we say that h is a pre-homeomorphic mapping. We also say that Y and Z are pre-homeomorphic.Definition 2.5.[22] A pre-topological space (Z, τ ) is called a T 0 -space if for any y, z ∈ Z with y = z there exists W ∈ τ such that W ∩ {y, z} is exact one-point set.Definition 2.6.[22] A pre-topological space (Z, τ ) is called a T 1 -space if for any y, z ∈ Z with y = z there are V, W ∈ τ so that V ∩ {y, z} = {y} and W ∩ {y, z} = {z}.Definition 2.7.[22] A pre-topological space (Z, τ ) is called a T 2 -space, or a Hausdorff space, if for any y, z ∈ Z with y = z there are V, W ∈ τ so that y ∈ V , z ∈ W and V ∩ W = ∅.Definition 2.8.[22] Let Z be a T 1 pre-topological space. We say that Z is a T 3 pre-topological space, or a regular space, if for every z ∈ Z and every closed setDefinition 2.9.[22] Let Z be a T 1 pre-topological space. Then Z is a T 3 1 2 pre-topological space, or a completely regular pre-topological space, or a Tychonoff pre-topological space, provide for each z ∈ Z and each closed subset C ⊆ Z with z ∈ C there exists a pre-continuous mapping r : Z → I so that r(z) = 0 and r(x) = 1 for each x ∈ C.Definition 2.10.[33] Let µ be a family of non-empty subsets of X × X such that the following conditions are satisfied (U1) for any U ∈ µ, △ ⊆ U ; (U2) if U ∈ µ, then U −1 ∈ µ; (U3) if U ∈ µ, then there exist V, W ∈ µ such that V • W ⊆ U ; (U4) if U ∈ µ and U ⊂ V ⊆ X × X, then V ∈ µ; (U5) if µ = △. The family µ is a pre-uniform structure of X if µ satisfies (U1)-(U5), the pair (X, µ) is a preuniform space and the members of µ are called entourage.	null	[ "https://arxiv.org/pdf/2203.10724v1.pdf" ]	247,594,948	2203.10724	37f44be77a6f46caf3aa908c2be4efc2a0b50fd0	SOME PROPERTIES OF PRE-TOPOLOGICAL GROUPS 21 Mar 2022 Fucai Lin Ting Wu Yufan Xie Meng Bao SOME PROPERTIES OF PRE-TOPOLOGICAL GROUPS 21 Mar 2022 In this paper, we pose the concepts of pre-topological groups and some generalizations of pre-topological groups. First, we systematically investigate some basic properties of pre-topological groups; in particular, we prove that each T 0 pre-topological group is regular and every almost topological group is completely regular which extends A.A. Markov's theorem to the class of almost topological groups. Moreover, it is shown that an almost topological group is τ -narrow if and only if it can be embedded as a subgroup of a pre-topological product of almost topological groups of weight less than or equal to τ . Finally, the cardinal invariant, the precompactness and the resolvability are investigated in the class of pre-topological groups.theory of pre-topology, we can consider the pre-topology on groups and pose the concept of pretopological groups, then we can systematically investigate some basic properties of pre-topological groups.This paper is organized as follows. In Section 2, we introduce the necessary notation and terminology which are used in the paper. In Sections 3 and 4, some basic properties of pre-topological groups and relationships among (quasi, para, almost) pre-topological groups are investigated. In Section 5, we mainly study quotient spaces of pre-topological groups and give the three isomorphisms of quotient spaces. In Section 6, we extend A.A. Markov's theorem to almost topological group and show that every almost topological group is completely regular. In Section 7, some cardinal invariants of pre-topological groups are studied. In particular, the well-known Guran's Theorem is extended, that is, an almost topological group is τ -narrow if and only if it can be embedded as a subgroup of a pre-topological product of almost topological groups of weight less than or equal to τ . In Section 8, some properties about precompactness and resolvability are investigated.Introduction and preliminariesDenote the sets of real number, rational number, positive integers, the closed unit interval and all non-negative integers by R, Q, N, I and ω, respectively. Readers may refer[16,22]for terminology and notations not explicitly given here.Definition 2.1.[10,19,14]A pre-topology on a set Z is a subfamily T of 2 Z such that T = Z and T ′ ∈ T for any T ′ ⊆ T . Each element of T is called an open set of the pre-topology.The name of pre-topology is given in[22]. Next, we list some definitions of pre-topological spaces which are introduced in [22]; of course, these definitions have important roles in our discussion of pre-topological groups.Definition 2.2.[22] Let (G, τ ) be a pre-topological space and B ⊆ τ . If for each U ∈ τ there exists a subfamily B ′ of B such that U = B ′ , then we say that B is a pre-base of (G, τ ).Let Z be a pre-topological space. For each z ∈ Z, the infimum of the set {\|B(z)\| : B(z) is a pre-base at z} is called the character of a point z[22]in Z, which is denoted by χ(z, Z). The supremum of the set {χ(z, Z) : z ∈ Z} is called the character of a pre-topological space (Z, τ ), which is denoted by χ(Z). Each set of cardinal numbers being well-ordered by <. Then the infimum of the set {\|B\| : B is a pre-base for Z} is said to be the weight of Z [22] and is denoted by w(Z).Definition 2.3. [22]Let h : Y → Z be a mapping between two pre-topological spaces (Y, τ ) andDefinition 2.4.[22] Let h : Y → Z be a bijection between two pre-topological spaces Y and Z. If h and h −1 : Z → Y are all pre-continuous, then we say that h is a pre-homeomorphic mapping. We also say that Y and Z are pre-homeomorphic.Definition 2.5.[22] A pre-topological space (Z, τ ) is called a T 0 -space if for any y, z ∈ Z with y = z there exists W ∈ τ such that W ∩ {y, z} is exact one-point set.Definition 2.6.[22] A pre-topological space (Z, τ ) is called a T 1 -space if for any y, z ∈ Z with y = z there are V, W ∈ τ so that V ∩ {y, z} = {y} and W ∩ {y, z} = {z}.Definition 2.7.[22] A pre-topological space (Z, τ ) is called a T 2 -space, or a Hausdorff space, if for any y, z ∈ Z with y = z there are V, W ∈ τ so that y ∈ V , z ∈ W and V ∩ W = ∅.Definition 2.8.[22] Let Z be a T 1 pre-topological space. We say that Z is a T 3 pre-topological space, or a regular space, if for every z ∈ Z and every closed setDefinition 2.9.[22] Let Z be a T 1 pre-topological space. Then Z is a T 3 1 2 pre-topological space, or a completely regular pre-topological space, or a Tychonoff pre-topological space, provide for each z ∈ Z and each closed subset C ⊆ Z with z ∈ C there exists a pre-continuous mapping r : Z → I so that r(z) = 0 and r(x) = 1 for each x ∈ C.Definition 2.10.[33] Let µ be a family of non-empty subsets of X × X such that the following conditions are satisfied (U1) for any U ∈ µ, △ ⊆ U ; (U2) if U ∈ µ, then U −1 ∈ µ; (U3) if U ∈ µ, then there exist V, W ∈ µ such that V • W ⊆ U ; (U4) if U ∈ µ and U ⊂ V ⊆ X × X, then V ∈ µ; (U5) if µ = △. The family µ is a pre-uniform structure of X if µ satisfies (U1)-(U5), the pair (X, µ) is a preuniform space and the members of µ are called entourage. Introduction Since from 20th century, many topologists and algebraists have contributed to topological algebra. A.D. Alexandroff, N. Bourbaki, E. van Kampen, A.A. Markov and L.S. Pontryagin were among the first contributors to the theory of topological groups. In 2008, the book "Topological Groups and Related Structures" was published, which points out the direction for the research of topological groups. It mainly studied the generic questions in topological algebra is how the relationship between topological properties depend on the underlying algebraic structure. As we all known, a topological group, that is, a group G is endowed with a topology such that the binary operation G × G → G is jointly continuous and the inverse mapping In : G → G, i.e. x → x −1 , is also continuous. The the properties of topological groups have been widely used in the study of topology, analysis and category, see [1,2,4,27,28,29,30,31,32,38,39]. For more details about topological groups, the reader see [3]. Moreover, the topologies on some kinds of weaker algebra structures than groups are posed and investigated, such as (strongly) topological gyrogroups and rectifiable spaces. Hence it is natural to consider extending some well known results of topological groups to these weaker structures, see [5,6,7,8,9,23,24,25,26]. In 1999, Doignon and Falmagne 1999 introduced the theory of knowledge spaces (KST) which is regarded as a mathematical framework for the assessment of knowledge and advices for further learning [15,17]. KST makes a dynamic evaluation process; of course, the accurate dynamic evaluation is based on individuals' responses to items and the quasi-order on domain Q [15]. In 2009, Danilov discussed the knowledge spaces based on the topological point of view. Indeed, the notion of a knowledge space is a generalization of topological spaces [14], that is, a generalized topology on a set Z is a subfamily T of 2 Z such that T is closed under arbitrary unions. Császár (2002) in [10] introduced the notions of generalized topological spaces and then investigated some properties of generalized topological spaces, see [10,11,12,13]. Further, J. Li first discuss the pre-topology (that is, the subbase for the topology) with the applications in the theory of rough sets, see [19,20,21], and then D. Liu in [34,35] discuss some properties of pre-topology. Recently, Lin, Cao and Li [22] systematically investigated some properties of pre-topology. Based on the Definition 3.3. Let ϕ : G → H be a mapping, where G and H are pre-topological groups. We say ϕ is a morphism if ϕ is pre-continuous and group homomorphism. Further, we say that ϕ is a pre-topological isomorphism if ϕ is pre-homeomorphism and group homomorphism. Definition 3.4. A pre-topological space X is said to be pre-homogeneous if for any x, y ∈ X there exists a pre-homeomorphism ϕ : X → X such that ϕ(x) = y. The following two propositions and corollary are obvious. Proposition 3.5. Let G be a pre-topological group and g ∈ G. Then the left (right) translation mapping L g (R g ) : G → G, defined by L g (x) = gx(R g (x) = xg), is a pre-homeomorphism. Corollary 3.6. Each pre-topological group is a pre-homogeneous space. Therefore, we have the following proposition. Proposition 3.7. Let G be a pre-topological group. If B is a local pre-base at e, then for each g ∈ G the local pre-base at g is equal to B g = {gU : U ∈ B} or B g = {U g : U ∈ B}. The following two theorems give the properties of an open base at the neutral element e of G, which play important roles in the study of pre-topological groups. Theorem 3.8. Let G be a pre-topological group and B e be a pre-base at the neutral element e of G. Then the following statements hold: (1) For each U ∈ B e , there exist V, W ∈ B e such that V W ⊆ U . (2) For each U ∈ B e , there exists V ∈ B e such that V −1 ⊆ U . (3) For each U ∈ B e and any g ∈ U , there exists V ∈ B e such that V g ⊆ U . (4) For each U ∈ B e and any g ∈ G, there exists V ∈ B e such that gV g −1 ⊆ U . Proof. Assume G is a pre-topological group. Properties (1) and (2) follow from the pre-continuity of the multiplication mappings (x, y) → xy and x → x −1 at the neutral element e, respectively. Property (3) follows from the pre-continuity of right translation mapping R g : x → xg in G. Property (4) follows from the pre-continuity of right translation mapping R g −1 : x → xg −1 and left translation mapping L g : xg −1 → gxg −1 are pre-homeomorphism of G. Theorem 3.9. Let G be a group, and let U be a family of subsets of G satisfying conditions (1)-(4) of Theorem 3.8. Then the family B U = {U a : a ∈ G, U ∈ U } is a pre-base for a pre-topology τ on G such that (G, τ ) is a pre-topological group. Proof. Let U be a family of subsets of G satisfying conditions (1)-(4) of Theorem 3.8, and let τ = {W ⊂ G : for each x ∈ W there exists U ∈ U such that U x ⊆ W }. In order to prove that (G, τ ) is a pre-topological group, we divide the proof into the following five claims. Claim 1: τ is a pre-topology on G. Clearly, we have ∅, G ∈ τ . Take any non-empty subfamily γ of τ . Pick any x ∈ γ; then there exists W ∈ γ such that x ∈ W , hence we can find U ∈ U so that U x ⊆ W , that is, U x ⊆ W ⊆ γ, then γ ∈ τ . Therefore, τ is a pre-topology on G. Claim 2: The family B U = {U a : a ∈ G, U ∈ U } is a pre-base for the pre-topology τ on G. We first prove that B U is a subfamily of τ . It suffices to verify that for any a ∈ G and U ∈ U we have U a ∈ τ . Take any y ∈ U a; then ya −1 ∈ U . From condition (3), there exists V ∈ U such that V ya −1 ⊆ U , that is, V y ⊆ U a. Hence, U a ∈ τ . Now we prove that B U is a pre-base for τ . Indeed, take any W ∈ τ and a ∈ W . Since W ∈ τ , there exists U ∈ U such that U a ⊆ W . Therefore, B U is a pre-base for τ . Claim 3: The multiplication of G is pre-continuous with respect to the pre-topology τ . Take any x, y ∈ G, and let O be any element of τ such that ab ∈ O. Since O ∈ τ , there exists W ∈ U such that W ab ⊆ O. In order to prove the pre-continuity of multiplication, it suffices to find U, V ∈ U such that U aV b ⊆ W ab, which is equivalent to U aV a −1 ⊆ W . By condition (1), there exist U, U ′ ∈ U such that U U ′ ⊆ W ; then from condition(4) there exists V ∈ U such that aV a −1 ⊆ U ′ , which implies that U aV a −1 ⊆ U U ′ ⊆ W , thus U aV b ⊆ W ab ⊆ O. Hence, the multiplication in G is pre-continuous with respect to the pre-topology τ . Claim 4: For any b ∈ G and V ∈ U , we have bV ∈ τ . Take any y ∈ bV , we have to find U ∈ U such that U y ⊆ bV . Clearly, we have b −1 y ∈ V . By condition (3), there exists W ∈ U such that W b −1 y ⊆ V . Then from condition (4) there is U ∈ U such that b −1 U b ⊆ W . Hence b −1 U bb −1 y ⊆ W b −1 y ⊆ V , that is, b −1 U y ⊆ V , thus U y ⊆ bV . Therefore, bV ∈ τ . Claim 5: The inverse mapping In of G onto G given by In(x) = x −1 is pre-continuous with respect to the pre-topology τ . By Claims 2 and 4, it suffices to prove that U −1 ∈ τ for any U ∈ U . Take any x ∈ U −1 ; then x −1 ∈ U . From condition (3), there exists V ∈ U such that V x −1 ⊆ U . Then from condition (2), there exists W ∈ U such that W −1 ⊆ V , hence we have W −1 x −1 ⊆ V x −1 ⊆ U , that is, xW ⊆ U −1 . By Claim 4 again, xW is an open neighbourhood of x in τ . Hence, U −1 ∈ τ . Therefore, we prove that (G, τ ) is a pre-topological group and the family B U = {U a : a ∈ U, U ∈ U } is a pre-base of the pre-topology τ . For a finite pre-topological group, we have the following proposition. Proposition 3.11. Let G be a finite pre-topological group and B e be a pre-base at the identity e of G. Then the following statements hold: (1) For each U ∈ B e , there exist V ∈ B e such that V 2 ⊆ U . (2) If U ∈ B e is an atom at e, then U −1 is an atom at e and U n = U for each n ∈ N. Proof. (1) For each U ∈ B e , it follows from the finiteness of G that there exists V ∈ B e such that V is an atom at e and V ⊆ U . Since G is a pre-topological group, there exist V 1 , V 2 ∈ B e such that V 1 V 2 ⊆ V ⊆ U . Then, V 1 = V 2 = V since V is an atom at e. Hence, V 2 ⊆ U . (2) Take any U ∈ B e such that U is an atom at e. It follows from (2) of Theorem 3.8 that there exists V ∈ B e such that V −1 ⊆ U . Since V −1 ∈ B e and U ∈ B e is an atom at e, we have V −1 = U , that is, U −1 = V . Moreover, it easily verify that V is an atom at e. Finally, we check that U n = U for each n ∈ N. From (1), it easily see that V 2 ⊆ V ⊆ U , then we have V = U and U 2 = U . By induction, assume that U k = U for 2 ≤ k ≤ n. For k = n + 1, we have U n+1 = U n U = U U = U . Hence, U n = U for each n ∈ N. By Theorem 3.8 and Proposition 3.11, it is natural to consider the following classes of pretopological groups. Definition 3.12. Let G be a pre-topological and B e be a pre-base at the identity e. Then • we say that G is a symmetrically pre-topological group if B e is symmetric; • we say that G is a strongly pre-topological group if B e satisfies (1) in Proposition 3.11; • we say that G is an almost topological group if G is a symmetrically and strongly pre-topological group. Clearly, each topological group is an almost topological group, each almost topological group is both a strongly pre-topological group and a symmetrically pre-topological group, and each strongly pre-topological group or symmetrically pre-topological group is a pre-topological group. The following example shows the inverse relations among them. Example 3.13. (1) There exists a pre-topological group which is not a strongly pre-topological group. (2) There exists a strongly pre-topological group which is not a symmetrically pre-topological group. (3) There exists an almost topological group which is not a topological group. Proof. (1) Let G be the group (R, +) and endowed with a pre-topology which has a pre-base as follows: B = {(−∞, a) : a ∈ Z} ∪ {(b, +∞) : b ∈ Z}. Then (R, τ ) is a pre-topological group; however, it is easily checked that it not a strongly pretopological group. (2) Let R be the real number with usual addition '+'. Let U = {[a, b) : a, b ∈ R, b > a} ∪ {(b, a] : a, b ∈ R, a > b}. Then U satisfies (1)-(4) of Theorem 3.8 and (1) of Proposition 3.11. Hence the pre-topology τ generated by the family U is a strongly pre-topological groups on R. However, G is not a symmetrically pre-topological group. (3) The pre-topological group in (1) of Example 3.2 is just our requirement. However, the following question is still unknown for us. Question 3.14. Is each symmetrically pre-topological group a strongly pre-topological group? The following proposition shows that the properties of an almost topological group is very similar to a topological group. Proof. The necessity is obvious. In order to prove the sufficiency, it suffices to shows that G has a symmetric pre-base. Take any U in B e . Then we can find V ∈ B e such that V V −1 ⊆ U , hence there exists W ∈ B e such that W W −1 ⊆ V . Put O = W W −1 . Then O is an open neighbour of e and O = O −1 . Moreover, O 2 = O −1 O = (W −1 W ) −1 (W −1 W ) ⊆ V V −1 ⊆ U . Hence, G has a symmetric pre-base. The following theorem shows that the separation of T 0 is equivalent to the regularity in a pre-topological group. Theorem 3.16. Each T 0 pre-topological group is regular. Proof. Let G be a T 0 pre-topological group. We first prove that G is T 1 . Indeed, take two distinct points x and y. Since G is T 0 , without loss of generality, we may assume that there exists an open neighborhood U of e such that y ∈ xU , hence x ∈ yU −1 . From (2) of Theorem 3.8, there exists an open neighborhood V of e such that V ⊆ U −1 , then x ∈ yV . Therefore, G is T 1 . Next we prove that it is regular. From the pre-homogeneous, it suffices to prove that for each open neighborhood U of e there exists an open neighborhood V of e such that V ⊆ U . From (1) of Theorem 3.8, there exist V, W ∈ B e such that V W ⊆ U . We claim that V ⊆ U . Indeed, take any z ∈ V . Then zW −1 ∩ V = ∅, which implies that z ∈ V W ⊂ U. Therefore, V ⊆ U . Thus G is regular. In the theory of topological groups, it is well known that any discrete subgroup in a topological group is closed. However, the situation is different in the theory of pre-topological groups. Indeed, the subgroup H = {0, 1, 5} is a discrete subgroup of the pre-topological group G of (1) in Example 3.2. However, H is not closed since G = H = H. Further, G is also an almost topological group. It is natural to pose the following question. In order to give some partial answers to Question 3.17, we introduce the following concepts. Definition 3.18. Let (G, τ ) be a pre-topological group. Then we say that (G, τ ⋆ ) is a co-reflexion group topology of τ if τ ⋆ is the coarsest topology on G such that τ ⊂ τ ⋆ and (G, τ ⋆ ) is a topological group; we say that (G, τ ⋆ ) is a reflexion group topology of τ if τ ⋆ is the strongest topology on G such that τ ⋆ ⊂ τ and (G, τ ⋆ ) is a topological group. Note that the reflexion group topologies of a pre-topological group are not unique. Moreover, we have the following proposition. Proposition 3.19. Let (G, τ ) be a pre-topological group. Then the family B = { F : F ⊆ τ e , \|F \| < ω} is an open neighborhood base at e for the co-reflexion group topology (G, τ ⋆ ). Proposition 3.20. Let G be a pre-topological group. If H is a discrete subgroup in (G, τ ⋆ ), then H is closed in G. Proof. Since (G, τ ⋆ ) is a topological group and H is discrete in (G, τ ⋆ ), it follows from [3, Corollary 1.4.18] that H is closed in (G, τ ⋆ ), hence H is closed in (G, τ ) since τ ⋆ ⊆ τ . The following proposition is obvious. Proposition 3.21. Let G be a pre-topological group. If H is a discrete subgroup in (G, τ ), then H is closed in (G, τ ⋆ ). Assume that U is a neighborhood of the neutral element of a pre-topological group Proof. It suffices to prove that, for every x ∈ G, the open neighbourhood xV of x intersects at most one element of the family {aV : a ∈ B}. Suppose not, then there exist x ∈ G and distinct elements a, b ∈ B such that xV ∩ aV = ∅ and xV ∩ bV = ∅. Then G. A subset B of G is called U -disjoint if b ∈ aU , for any distinct a, b ∈ B.x −1 a ∈ V 2 and b −1 x ∈ V 2 , hence b −1 a = (b −1 x)(x −1 a) ∈ V 4 ⊆ U , thus a ∈ bU , which is a contradiction. Let (G, τ ) be a pre-topological space. If each locally finite family of open subsets is finite, then we say that G is feebly compact. Theorem 3.23. Each discrete subgroup H of a feebly compact almost topological group G is finite. Proof. Since K is discrete in G, there exists an open neighbourhood U of the neutral element in G such that U ∩ H = {e}. Then there is an open neighbourhood V of e such that V 4 ⊆ U because G is an almost topological group. We claim that H is U -disjoint. Indeed, take any distinct elements h 1 , h 2 ∈ H; then h −1 1 h 2 / ∈ {e} = U ∩ H. Since h −1 1 h 2 ∈ H, it follows that h −1 1 h 2 / ∈ U , that is, h 2 / ∈ h 1 U . Therefore, H is U -disjoint. By Lemma 3.22, the family η = {hV : h ∈ H} is discrete in G. Since G is feebly compact, it follows that H is finite. Corollary 3.24. Each infinite feebly compact almost topological group G contains a non-closed countable subset. Proof. Take any infinite countable subset A of G, and let H be the subgroup of G algebraically generated by A. Then H is countable and infinite. Therefore, by Theorem 3.23, H cannot be discrete. Therefore, the subset B = H \ {e} of H is not closed in H, thus B not closed in G. Finally, we discuss the connectedness [22] of pre-topological groups. Let G be pre-topological group with the neutral element e. The connected component of G is the union of all connected subsets of G containing e. It follows from [22,Theorem 24] that the connected component of G can be described as the biggest connected subspace of G containing e. Proof. Since pre-continuity preserves the connectedness, it follows that H 2 and H −1 are connected, hence H is a group. From [22,Theorem 25], H is also closed. Now we only prove that H is invariant. Since the left and right translations are pre-homeomorphism of G onto itself, it follows that aH and Ha are connected for any a ∈ G, hence aHa −1 and a −1 Ha are also connected. For each a ∈ G, since e ∈ aHa −1 ∩ a −1 Ha and H is the biggest connected subset of G containing e, it follows that aHa −1 ⊆ H and a −1 Ha ⊆ H, which implies that H ⊆ aHa −1 , hence H = aHa −1 . Proof. If K = e, then it is obvious. Therefore, suppose that the subgroup K is not trivial. Take any x ∈ K \ {e}. Since the group K is discrete, there is an open neighbourhood U of x such that U ∩ K = {x}. Then it follows from our assumption that there exists an open neighborhood V of e such that V xV ⊆ U . For any y ∈ V , since K is an invariant subgroup of G, we have that yxy −1 ∈ K; it is obvious that yxy −1 ∈ V xV ⊆ U . Thus, yxy −1 ∈ U ∩K = {x}, that is, yxy −1 = x. Thus yx = xy and y −1 x = xy −1 for each y ∈ V . This shows that x commutes with every element of V ∪ V −1 . Next we prove that x commutes with every element of G. Since G is connected, it follows from Proposition 3.26 that G = n∈N (V ∪ V −1 ) n . Then, for every g ∈ G, g can be written in the form g = y 1 y 2 · · · y n , where y 1 y 2 · · · y n ∈ V ∪ V −1 and n ∈ N. Since x commutes with every element of V ∪ V −1 , we conclude that gx = y 1 y 2 · · · y n x = y 1 y 2 · · · y n−1 xy n−1 = · · · = xy 1 y 2 · · · y n = xg. Since x is an arbitrary element of K, we conclude that the center of G contains K. From Theorem 3.27, we have the following question. Generalizations of pre-topological groups In this section, we give some generalizations of pre-topological group, such as, semi-pre-topological group, quasi-pre-topological group and para-pre-topological group. Moreover, we study some basic properties of them. Definition 4.1. Let τ be a pre-topology on a group G. We say that (G, τ ) is a • right (left) pre-topological group if for each a ∈ G, the right (left) action R a (L a ) of a on G is a pre-continuous mapping of the space G to itself. • semi-pre-topological group if (G, τ ) are both right pre-topological group and left pre-topological group. • quasi-pre-topological group if (G, τ ) is a semi-pre-topological group such that the inverse mapping In : G → G is pre-continuous. • para-pre-topological group if the multiplication mapping G × G → G is pre-continuous, where G × G is given the product pre-topology. Definition 4.2. Let G be a semigroup, and let (G, τ ) be a pre-topological space. We say that (G, τ ) is a • right (left) pre-topological semigroup if for each a ∈ G, the right (left) action R a (L a ) of a on G is a pre-continuous mapping of the space G onto itself. • semi-pre-topological semigroup if (G, τ ) both right pre-topological semigroup and left pretopological semigroup. • pre-topological semigroup if the multiplication mapping G × G → G is pre-continuous, where G × G is endowed with the product pre-topology. The following examples shows that the class of pre-topological groups is strictly contained in the class of quasi-pre-topological groups and the class of para-pre-topological groups respectively. Example 4.3. There exists a finite quasi-pre-topological group G such that G is not a pretopological group. Proof. Indeed, let G = {0, 1, 2, 3} be the set of surplus class with respect to module 4. We endow G with a pre-topology as follows: τ = {∅, {0, 1, 3}, {0, 1, 2}, {1, 2, 3}, {0, 2, 3}, G}. It easily check that (G, τ ) is a quasi-pre-topological group. However, (G, τ ) is not a pre-topological group since the multiplication of (G, τ ) is not jointly pre-continuous (Indeed, for any open neighborhoods U, V of e we have 0 ∈ U V , hence U V {0, 1, 3}). Example 4.4. There exists a para-pre-topological group G such that G is not a pre-topological group and para-topological group. Proof. Let R be the real number with usual addition '+'. Let τ 1 be the topology generated by the family {[x, x + y) : x ∈ R, y ∈ R + } (that is, Sorgenfrey line), and let τ 2 be the topology generated by the family {{x} ∪ {m + x : m ≥ n} : n ∈ N}. It easily check that both (G, τ 1 ) and (G, τ 2 ) are para-topological groups on R. Clearly, τ 1 τ 2 and τ 2 τ 1 . Let (G, τ ) be the pre-topology which is generated by the family τ 1 ∪ τ 2 as follows τ = U ⊂ R : there exist U 1 ⊆ τ 1 and U 2 ⊆ τ 2 such that U = ( U 1 ) ∪ (U 2 ) . Since {0} is not open in (G, τ ), it follows that (G, τ ) is not a para-topological group. Moreover, for any open neighborhood W of e in τ , we have −W [0, 1), hence G is not a pre-topological group. Indeed, we have the following proposition which is a machine to generate a para-pre-topological group from a para-topological group. Proposition 4.5. Let (G, τ ) be a para-topological group which is not a topological group. Then the pre-topology σ on G, which has a pre-base τ ∪ τ −1 , is a para-pre-topological group and is not a para-topological group, where τ −1 = {U −1 : U ∈ τ }. Proof. It easily check that (G, σ) is a para-pre-topological group. However, since (G, τ ) is not a topological group, there exists U ∈ τ e such that U ∩ U −1 ∈ τ ∪ τ −1 , hence (G, σ) is not a para-topological group. Hence it is natural to pose the following question. Question 4.6. When is a para-pre-topological group (or quasi-pre-topological group) pre-topological group? The next theorem gives a partial answer to Question 4.6. First, we introduce a concept. Definition 4.7. Let (X, τ ) be a pre-topological space. The space X is locally finite [36] if each point of X has an open neighborhood which is a finite set. Clearly, each finite pre-topological space is locally finite. Theorem 4.8. If (G, τ ) is a locally finite para-pre-topological group, then G is an almost topological group. Proof. Clearly, since G is locally finite, each open set in G must contain a minimally open set. By the proof of (1) in Proposition 3.11, it suffices to prove that U = U −1 for any minimally open set U at e. If G is discrete, then it is obvious. Hence we assume that G is not discrete. Take any minimally open set U at e, and pick any x ∈ U \ {e} = ∅. Clearly, U is finite. Then it follows from the pre-continuous of the multiplication of G that we have U 2 = U , hence xU ⊂ U . Enumerate U as {e, x 1 , x 2 , . . . , x n }. Since \|xU \| = \|U \| and xU ⊂ U , we conclude that xU = U , then there exists i ≤ n such that xx i = e, which implies that x −1 = x i ∈ U . Therefore, U = U −1 . Thus G is an almost topological group. Remark 4.9. From Example 4.3, there exists a finite quasi-pre-topological group which is not a pretopological group. Moreover, R. EIlis proved that each locally compact Hausdorff semi-topological group is a topological group. It natural to pose the following question. A pre-topological space G is said to be compact if each open cover of G has a finite subcover; G is said to be locally compact if for each x ∈ G there exists a compact neighborhood U of x. Question 4.10. If G is a (locally) compact Hausdorff para-pre-topological group, then is G a pre-topological group? Proposition 4.11. Let (G, τ ) be a right pre-topological group and g be any element of G. Then the following statements hold: (1) The right translation r g of G by g is a pre-homeomorphism of the pre-topological space G onto itself. (2) For any pre-base B e of G at e, the family B g = {U g : U ∈ B e } is a pre-base of G at g. Proof. Clearly, (2) follows from (1). Hence we only need to prove (1). For any g ∈ G, r g and r g −1 are all pre-continuous bijection; moreover, since r g r g −1 = e, we have r g −1 = r g −1 . Therefore, r g and (r g ) −1 are all pre-continuous bijection, hence r g is a pre-homeomorphism of the pre-topological space G onto itself. Proof. The family γ = {Ha : a ∈ G} is all the right cosets of H in G. Since each right translation is pre-homeomorphism mapping, it follows that Ha is open in G for each a ∈ G. Then the family γ is a disjoint open cover of G, this implies that H = G \ ( a∈G\{e} Ha). Thus, H is closed in G. U of e G in G such that f (U ) ⊆ V . Since L x is a pre-homeomorphism of G onto itself, the set xU is an open neighbourhood of x in G; therefore, we can conclude that f (x) ∈ f (xU ) = f (x)f (U ) = yf (U ) ⊆ yV ⊆ O. Hence, f is pre-continuous.Corollary 4.16. Each right (left) pre-topological group G is a pre-homogeneous pre-topological space. Proof. Take any elements x and y in G and put z = x −1 y; then R z (x) = xz = xx −1 y = y. Hence each right pre-topological group G is a pre-homogeneous pre-topological space. A similar argument can be applied in the case of left pre-topological group. Proof. By (1) of Proposition 4.11, every right translation of G is pre-homeomorphism of the pretopological space G onto itself, then U A = a∈A U a = a∈A r a (U ). Hence U A is open in G. By a similar method, we can prove that AU is open in G when G is a left pre-topological group. Finally, we mainly discuss the closure of a subset of a left (right) pre-topological group G. The following proposition gives an intimate relationship between the sets AU or U A and the closure operation, where U is open in G and A is a subset of G. V of e such that V −1 ⊆ U . Now take any x ∈ A; since xV is an open neighbourhood of x, we have xV ∩ A = ∅, then there are a ∈ A and b ∈ V such that a = xb. Then x = ab −1 ∈ AV −1 ⊆ AU ; hence, A ⊆ AU .U of e such that U −1 ⊆ W , then (xU −1 ) ∩ A = ∅, that is x ∈ AU ; hence, x ∈ {AU : U ∈ B e }. Therefore, {AU : U ∈ B e } ⊆ A. Then A = {AU : U ∈ B e }. Similarly, the equality A = {U A : U ∈ B e } holds for the right pre-topological groups with pre-continuous inverse. V of e with V −1 ⊂ U . Then G is a quasi-pre-topological group. Proof. Let W be an any open set. We have to show that W −1 is also open in G. Take any x ∈ W −1 . Then x −1 ∈ W . Since G is a left pre-topological group, there exists an open neighbourhood U of e such that x −1 U ⊆ W . By the assumption, there exists an open neighbourhood V of e such that V −1 ⊆ U . Then (V x) −1 = x −1 V −1 ⊆ x −1 U ⊆ W . Hence, the inverse mapping on G is pre-continuous, thus G is a quasi-pre-topological group. V −1 = ∅, that is, V −1 ⊆ G\A = U . Hence, G is a quasi-pre-topological group. Proposition 4.23. Each left pre-topological group with pre-continuous inverse is a right pretopological group and hence, a quasi-pre-topological group. Proof. We can get from x to xa in three steps: from x to x −1 , then from x −1 to a −1 x −1 , and finally from a −1 x −1 to xa. This means that R a = In • L a −1 • In. Since all the mappings on the right side of the above equality are pre-continuous, the right translation is pre-continuous. Proof. First, we prove that H · H ⊆ H. Take any y ∈ H and x ∈ H. Since right translation is pre-continuous ,we have R x (H) ⊆ R x (H) by [22,Theorem 7], that is, Hx ⊆ Hx. It follows from x ∈ H and H is a subsemigroup of G that Hx ⊆ H, then, yx ∈ Hx ⊆ Hx ⊆ H, hence, H · H ⊆ H. And then we can prove that H · H ⊆ H. Since each left translation is pre-continuous, we also have yH ⊆ yH by [22,Theorem 7]. From yH ∈ H · H ⊆ H, it follows that yH ⊆ yH ⊆ H = H by [22,Theorem 7]. Now, take any z ∈ H, we have yz ∈ H. Thus, H is a subsemigroup of G. Proposition 4.25. Let G be an abstract group with a pre-topology τ such that the inverse mapping is pre-continuous. Then, for any symmetric subset A of G, the closure of A in G is also symmetric. Proof. Since the inverse mapping In is pre-continuous and the composition In • In is the identity mapping of G onto itself. Thus, In is a pre-homeomorphism, this implies that In(A) = In(A) by [22,Theorem 7], that is, A −1 = (A) −1 . Since A −1 = A, then A −1 = A = (A) −1 , hence, the closure of A in G is also symmetric. Quotients of pre-topological groups In this section, we mainly discuss the quotient spaces of pre-topological groups and give the three isomorphisms of quotient spaces. Indeed, the authors in [18] has discuss the quotients of generalized topological groups. However, their paper has some gaps in some results, hence we systematically discuss the quotient of pre-topological groups and some results given without proofs. First, we need the following result. Theorem 5.1. Suppose that G is a semi-pre-topological group with the neutral element e and a pre-topology τ , and H is a closed subgroup of G. Denote by G/H the set of all left cosets aH of H in G, and endow it with the quotient pre-topology with respect to the canonical mapping π: G → G/H defined by π(a) = aH, for each a ∈ G. Then the family {π(xU ) : U ∈ τ, e ∈ U } is a local pre-base of the space G/H at the point xH ∈ G/H, the mapping π is open, and G/H is a pre-homogeneous T 1 -space. Proof. Clearly, we have π(xU ) = π(xU H) for any x ∈ G and U ∈ τ with e ∈ U , and the set xU H is the union of a family of left cosets yH, where each y ∈ xU . Hence, π −1 π(xU H) = xU H. Since the set xU H is open in G and the mapping π is quotient, it follows that π(xU H) is open in G/H. Therefore, the mapping π is open. Now we prove that the family {π(xU ) : U ∈ τ, e ∈ U } is a local pre-base of the space G/H at the point xH ∈ G/H. Take any open neighbourhood W of xH in G/H and put O = π −1 (W ); obviously, we have x ∈ O. Since π is pre-continuous, O is open in G. Therefore, there exists an open neighbourhood U of e in G such that xU ⊆ O, then π(xU ) ⊆ W and π −1 π(xU ) ⊆ O. Since xU H = π −1 π(xU ), it follows that π(xU H) = π(xU ) ⊆ W . Finally, we prove G/H is pre-homogeneous. Let a be an arbitrary elements of G; we define a mapping h a of G/H to itself by the rule h a (xH) = axH. Clearly, the mapping is well defined. Since G is a group, it easily check that h a is a bijection of G/H onto G/H. Next we prove h a is pre-homeomorphism. Take any xH ∈ G and any open neighbourhood U of e; then π(xU H) is a neighbourhood of xH in G/H. Similarly, the set π(axU H) is a neighbourhood of axH in G/H. Since h a (π(xU H)) = π(axU H) and h −1 a (π(axU H)) = π(xU H), then the mapping h a is open and pre-continuous. Thus, h a is pre-homeomorphism. For any xH, yH in G/H, take a = yx −1 ; then h a (xH) = yx −1 xH = yH. Therefore, the quotient space G/H is pre-homogeneous. For any xH in G/H, we have π −1 (xH) = xH. Since all left cosets are closed in G and the mapping π is quotient, it follows that xH is closed in G/H. Hence, G/H is T 1 -space. Proposition 5.2. Suppose that G be a pre-topological group, and H is a closed subgroup of G, π is the natural quotient mapping of G onto the left quotient space G/H, a ∈ G, L a is the left translation of G by a (that is, L a (x) = ax, for each x ∈ G), and h a is the left translation of G/H by a (that is, h a (xH) = axH, for each xH ∈ G/H). Then L a and h a are homeomorphisms of G and G/H, respectively, and π • L a = h a • π. If G is a left pre-topological group and H is a closed invariant subgroup of G, then each left coset of H in G is also a right coset of H in G, and a natural multiplication of cosets in G/H is defined by the rule xHyH = xyH, for all x, y ∈ G. This operation turn G/H into a group. Theorem 5.3. Suppose that G is a semi-pre-topological group with the neutral element e and H is a closed subgroup of G. Then G/H with the quotient pre-topology and multiplication is a semi-pre-topological group, and the canonical mapping π : G → G/H is an open pre-continuous homomorphism. If G is a pre-topological group and H is invariant, then G/H is a pre-topological group. Proof. By theorem 5.1, we have that the canonical mapping π is open and pre-continuous. The mapping π is also a homomorphism, since π(ab) = abH = aHbH = π(a)π(b), for any a, b ∈ G. Proof. Suppose that H is open in G, then aH is also open in G. Since π is a quotient mapping and π(aH) = aH, it follows that {aH} is open in G/H, so G/H is discrete. Conversely, suppose that G/H is discrete, then for every x ∈ G, π(x) = xH ∈ G/H, hence {xH} is open in G/H. Since π is a quotient mapping and π −1 (xH) = xH ⊆ G, we conclude that xH is open in G, hence H is also open in G. Lemma 5.5. Suppose that G is a pre-topological group, H is closed subgroup of G and π is the nat- ural quotient mapping of G onto the quotient space G/H. If U , V and W are open neighbourhoods of the neutral element e in G such that W V ⊆ U , then π(V ) ⊆ π(U ). Proof. Take any x in G such that π(x) ∈ π(V ), we only have to prove π(x) ∈ π(U ). Since W −1 x is an open neighbourhood of x and the mapping π is open, we conclude that π(W −1 x) is an open neighbourhood of π(x). Hence, π(W −1 x) ∩ π(V ) = ∅, then there exists a ∈ W −1 and b ∈ V such that π(ax) = π(b), which implies that ax = bh, for some h ∈ H. Therefore, x = (a −1 b)h ∈ (W V )H ⊆ U H, that is, π(x) ∈ π(U H) = π(U ). Theorem 5.6. For any pre-topological group G and any closed subgroup H of G, the quotient space G/H is regular. Proof. Let π be the natural quotient mapping of G onto the quotient space G/H and W be an arbitrary open neighbourhood of π(e) in G/H, where e is the neutral element of G. Since π is pre-continuous, there exists an open neighbourhood U of e in G such that π(U ) ⊆ W . Let V and W are open neighbourhoods of e such that W V ⊆ U . From Lemma 5.5, it follows that π(V ) ⊆ π(U ) ⊆ W . Hence G/H is regular at the point π(e) by Theorem 5.1. Proposition 5.7. Suppose that G, H and K are abstract groups, and suppose that ϕ : G → H and ψ : G → K are homomorphisms such that ψ(G) = K and kerψ ⊆ kerϕ. Then there exists a homomorphism f : K → H such that ϕ = f • ψ. In addition, if G, H, K are pre-topological groups, then ϕ and ψ are pre-continuous, and if for each neighbourhood U of the neutral element e H in H there exists a neighbourhood V of the identity e K in K such that ψ −1 (V ) ⊆ ϕ −1 (U ), then f is pre-continuous. Proof. The algebraic part of the proposition is well known. It only to verify that f is pre-continuous. Let U be a neighbourhood of e H in H. By our assumption there exists a neighbourhood V of the neutral element e K in K such that W = ψ −1 (V ) ⊆ ϕ −1 (U ). Since ϕ = f • ψ, we have f (V ) = f (ψ(ψ −1 (V )) = ϕ(ψ −1 (V )) = ϕ(W ) ⊆ U . Hence, f is pre-continuous at the neutral element of K. Therefore f is pre-continuous. Corollary 5.8. Let ϕ : G → H and ψ : G → K be a pre-continuous homomorphism of semi-pretopological groups G, H and K such that ψ(G) = K and kerψ ⊆ kerϕ. If the homomorphism ψ is open, then there exists a pre-continuous homomorphism f : K → H such that ϕ = f • ψ. Proof. It follows from proposition 5.7 that there exists a homomorphism f : K → H such that ϕ = f • ψ. Then it suffice to prove that f is pre-continuous. Let V be an arbitrary open set in H, then f −1 (V ) = ψ(ϕ −1 (V )). Since ϕ is pre-continuous and ψ is open, we conclude that f −1 (V ) is open in K. Thus f is pre-continuous. Proposition 5.9. Let G and H be pre-topological groups and p be a pre-topological isomorphism of G onto H. If G 0 is a closed invariant subgroup of G and H 0 = p(G 0 ), then the quotient groups G/G 0 and H/H 0 are pre-topologically isomorphic. The corresponding pre-isomorphism φ : G/G 0 → H/H 0 is given by the formula φ(xG 0 ) = yH 0 , where x ∈ G and y = p(x). Proof. Let ϕ : G → G/G 0 and ψ : H → H/H 0 be the pre-quotient homomorphisms. We can easily prove that φ is a homomorphism of G/G 0 onto H/H 0 . From the definition of φ, it follows that ψ • p = φ • ϕ. Since p, ϕ and ψ are open pre-continuous homomorphisms, so is φ. Take an arbitrary element xG 0 of G/G 0 and set y = p(x). If π(xG 0 ) = H 0 , then ψ(y) = H 0 , where y ∈ H 0 and x ∈ G 0 , hence ker of φ is trivial. In other words, φ is an pre-isomorphism. Thus, φ is pre-topological isomorphism. Let G and H be semi-pre-topological groups with neutral elements e G and e H , respectively, and let p be an open pre-continuous homomorphism of G onto H. Then kernel N = p −1 (e H ) of p is a closed invariant subgroup of G, and the fibers p −1 (y) with y ∈ H coincide with the cosets of N in G. The mapping Φ : G/N → H which assigns to a coset xN the element p(x) ∈ H is a pre-topological isomorphism. Theorem 5.11. Let G be an almost topological group, H a closed subgroup of G, and π : G → G/H be the canonical mapping. If K is a dense subgroup of G, then the restriction r = π ↾ K is an open mapping of K onto π(K). Proof. Take any non-empty open set U in K. Then there an open set V in G such that U = K ∩ V . Clearly, r(U ) = π(K ∩ V ) ⊆ π(K) ∩ π(V ) = O. Since the mapping π is open, the set O = π(V ) ∩ π(K) is open in π(K), then we have r(U ) = O. Indeed, for any y ∈ O, there exists x ∈ K such that π(x) = y, which implies that xH ∩ V = π −1 (y) ∩ V = ∅. Since K ∩ H is dense in H, it follows that x(K ∩ H) = K ∩ xH is dense in xH. Thus (K ∩ xH) ∩ V = ∅, so there is a point x ′ ∈ K ∩ xH ∩ V = U ∩ xH. Hence r(x ′ ) = π(x ′ ) = π(x) = y, that is, y ∈ r(U ); then it follows that r(U ) = O. Therefore, the mapping r : K → π(K) is open. Question 5.12. Let G be an almost topological group, H a closed subgroup of G, and π : G → G/H be the canonical mapping. If K is a dense subgroup of G and the restriction r = π ↾ K is an open mapping of K onto π(K), is the intersection K ∩ H dense in H? y ∈ yW 1 ⊆ yp(O) ⊆ p(x)p(O) ⊆ p(xO) ⊆ p(U ). By the arbitrary choice of y, it follows that p(U ) is open in H. The next two results are known as the second isomorphism theorem and the third isomorphism theorem. We give them without any proofs. 6. The complete regularity and the character of pre-topological groups In this section, we mainly discuss the complete regularity of pre-topological groups, and give a characterization of almost topological groups with the character less than or equal to τ . Since each T 0 topological group is completely regular, it is natural to pose the following question. Question 6.1. Is each T 0 pre-topological group completely regular? In this section we shall prove that each almost topological group is completely regular, which gives a partial answer to Question 6.1. First, we give some lemmas. The proofs of these lemmas are similar to the proofs of [3, Lemmas 3.3.7, 3.3.8 and 3.3.10], thus we omit them. Assume N is a prenorm on a group G. Put B N (ε) = {x ∈ G : N (x) < ε} for each ε > 0, which is called the N -ball of radius ε. Obviously, the ball B N (ε) is an open set of G if N is a pre-continuous prenorm. A semi-pre-topological group G is called left pre-uniformly Tychonoff (resp. right pre-uniformly Tychonoff) if for each open neighborhood U of the neutral element e, there exists a left (resp. right) pre-uniformly continuous function f on G such that f (e) = 0 and f (x) ≥ 1 for each x ∈ G \ U . Clearly, each left (or right) pre-uniformly Tychonoff is completely regular. Lemma 6.3. Each pre-continuous prenorm on a pre-topological group G is a pre-uniformly continuous function with respect to left and right group pre-uniformities on G. Lemma 6.4. Let G be an almost topological group, and let {U n : n ∈ ω} be a sequence of symmetric open neighborhoods of the neutral element e such that U n+1 · U n+1 ⊂ U n for each n ∈ ω. Then there exists a prenorm N on G satisfies the following conditions: {x ∈ G : N (x) < 1/2 n } ⊂ U n ⊂ {x ∈ G : N (x) ≤ 2/2 n } for any n ∈ ω. Hence, this prenorm N is pre-continuous. Moreover, if each set U n are invariant, then N on G can be chosen to satisfy N (xyx −1 ) = N (y) for any x, y ∈ G. By Lemmas 6.2 and 6.4, we have the following theorem, which is a generalization of A.A. Markov's theorem. Theorem 6.6. Every almost topological group G is completely regular. Proof. Take any open neighborhood U of the neutral element e in G. From Theorem 6.5, there exists a pre-continuous prenorm N on G such that B N (1) ⊆ U . Therefore, we have N (e) = 0 and N (x) ≥ 1 for each x ∈ G \ U . Since N is pre-continuous, it follows from Lemma 6.3 that G is completely regular. The following theorem gives a characterization of almost topological groups with the character less than or equal to τ . Proof. The sufficiency is obvious. We only need to prove the necessity. Fix a pre-base {U α : α < τ } of G at the neutral element e, where we may assume that each U α is symmetric since G is an almost topological group. Fix any α < τ ; since G is an almost topological group, we can take a subsequence {U αn : n ∈ N} of {U α : α < τ } such that U α1 = U α and U 2 αn+1 ⊆ U αn for each n ∈ N. By Lemma 6.4, there exists a pre-continuous prenorm N α on G such that B Nα ( 1 2 n ) ⊆ U αn for each n ∈ N. Moreover, each B Nα ( 1 2 n ) is open in G. Now, for any x and y in G, put ρ α = N α (xy −1 ). Then it is easy to check that ρ α is a right-invariant pseudometric. For each n ∈ N, since B ρα ( 1 2 n ) = B Nα ( 1 2 n ), it follows that B ρα ( 1 2 n ) is open in G.α < τ such that U α ⊆ O. Hence B ρα ( 1 2 ) ⊆ U α ⊆ O. The proof is completed. One can complement Theorem 6.7 as follows: Proof. As in the proof of Theorem 6.7, for each α < τ take a pre-continuous prenorm N α , and put ρ α = N α (xy −1 ) and σ α = N α (x −1 y) for any x, y ∈ G; then ρ α and σ α are right-invariant and left-invariant pseudometrics on G respectively. As is was shown in Theorem 6.7, both the families {ρ α : α < τ } and {σ α : α < τ } generate the original pre-topology of G respectively. The following corollary generalizes the well known Birkhoff and Kakutani's Theorem. Proof. Necessity. Suppose that G is generated by a family {ρ α : α ∈ I} of invariant pseudometrics. For each α ∈ I and n ∈ N, denote by U α,n the 1 n -ball center at the neutral element e of G with respect to ρ α . Now it suffices to prove that xU α,n x −1 = U α,n for any x ∈ G, α ∈ I and n ∈ N. Fix any x ∈ G, α ∈ I and n ∈ N, since ρ α (e, xyx −1 ) = ρ α (x, xy) = ρ α (e, y) < 1 n , it follows that xU α,n x −1 = U α,n . Hence the pre-topological group G is balanced. Sufficiency. Suppose that G is balanced, and that χ(G) ≤ τ . Then there exists a family N = {U α : α < τ } of open, symmetric, invariant neighborhoods of e in G such that N forms a pre-base for G at e. Hence for each α < τ it follows from Lemma 6.4 that there exists a pre-continuous prenorm N α on G such that {x ∈ G : N α (x) < 1/2} ⊂ U α ⊂ {x ∈ G : N α (x) ≤ 1} and N α (xyx −1 ) = N (y) for any x, y ∈ G; then the pseudometric ρ α defined by ρ α = N α (x −1 y) = N α (xy −1 ) is invariant. It is easily checked that the family {ρ α : α ∈ I} generates the pre-topology of G. Let X be a pre-topological space. If the pre-topology of X is generated by a family {ρ α : α < τ } of pseudometrics, then we say that X is τ -metrizable. Theorem 6.11. Let H be a closed subgroup of a τ -metrizable pre-topological group G. Then the quotient pre-topological space G/H is τ -metrizable. Proof. From Corollary 6.9, there exist a family {ρ α : α < τ } of right-invariant pseudometrics generating the original pre-topology of G. Fix any α < τ . For arbitrary points x, y ∈ G, let d α (xH, yH) be the number d α (xH, yH) = inf{ρ α (xh 1 , yh 2 ) : h 1 , h 2 ∈ H}. By a similar proof of [3, Proposition 3.3.19], it is easily checked that d α is a pseudometric. To finish the proof, we need to prove that the family {d α : α < τ } generates the pre-topology of the quotient pre-topological space G/H. Indeed, denote by π be the quotient mapping of G onto G/H, π(x) = xH for each x ∈ G. For any x ∈ G, α < τ and ε > 0, put O α,ε (x) = {y ∈ G : ρ α (x, y) < ε} and B α,ε (xH) = {yH : y ∈ G, d α (xH, yH) < ε}. For each α < τ , it follows from the definition of the pseudometric ρ α that π(O α,ε (x)) = B α,ε (xH) for each x ∈ G and ε > 0. Since the family {O α,ε (x) : x ∈ G, α < τ, ε > 0} form a pre-base for G and the mapping π is pre-continuous and open by Theorem 5.1, we claim that {B α,ε (xH) : x ∈ G, α < τ, ε > 0} is a pre-base for the original pre-topology of the pre-topological space G/H. Therefore, G/H is τ -metrizable. The following is easily checked, we left the proof to the reader. Theorem 6.13. Assume that G is a pre-topological group and H is a closed subgroup of G. If H and G/H are separable, then G is also separable. Proof. Suppose that π is the natural homomorphism of G onto thequotient pre-topological space G/H. From the separability of G/H, we can fix a dense countable subset A of G/H. Since H is separable and each coset xH is pre-homeomorphic to H, it follows that we can take a dense countable subset D y of π −1 (y) for each y ∈ A. Put D = {D y : y ∈ A}; then D is a countable subset of G and D is dense in π −1 (A). Since π is open, from Lemma 6.12 it follows that π −1 (A) = π −1 (A) = π −1 (G/H) = G. Therefore, G is separable. Question 6.14. Let G be a pre-topological group and H be a closed pre-topological subgroup. If the pre-topological subspace H and G/H are first-countable, is then the pre-topological space G also first-countable? The index τ -narrowness in pre-topological groups In this section, some cardinal invariants of pre-topological groups are studied. In particular, the well-known Guran's Theorem is extended, that is, an almost topological group is τ -narrow if and only if it can be embedded as a subgroup of a pre-topological product of almost topological groups of weight less than or equal to τ . Definition 7.1. A semi-pre-topological group is called left τ -narrow (resp. right τ -narrow) if, for each open neighborhood U of the neutral element in G, there exists a subset F of G such that G = F U (resp. G = U F )) and \|F \| ≤ τ . If G is left τ -narrow and right τ -narrow then G is called τ -narrow. The index of narrowness of a semi-pre-topological group G denoted by ib(G), that is, the minimal cardinal τ ≥ ω such that G is τ -narrow. First, we give some basic properties of τ -narrowness of pre-topological groups. (1) ⇒ (2). Let G be a τ -narrow quasi-pre-topological group. For any open neighbourhood V of e, there is an open neighbourhood U such that U −1 ⊆ V . Therefore, we can find a subset A of G such that G = AU and \|A\| ≤ τ . Put B = A −1 ; then G = G −1 = (AU ) −1 = U −1 A −1 ⊆ V B, that is, G = V B. (2) ⇒ (3). For every V ∈ B e , there exists U ∈ B e such that U −1 ⊆ V . By our assumption, there exist subsets B and A of G such that V A = G ,G = U B, \|A\| ≤ τ and \|B\| ≤ τ , then it The following proposition is obvious, so we leave the proof to the reader. follows from G = G −1 = B −1 U −1 ⊆ B −1 V that G = B −1 V . Put C = A ∪ B −1 ; then \|C\| ≤ τ and CV = G = V C. Proposition 7.4. The pre-topological product of an arbitrary family of τ -narrow pre-topological group is a τ -narrow pre-topological group. It is well-known that each subgroup of a τ -narrow topological group is τ -narrow. Hence we have the following question. Example 7.6. There exists a closed subgroup H of an ω-narrow strongly pre-topological group G such that H is not ω-narrow. Proof. Let G be the group (R 2 , +) with usual addition which is endowed with a pre-topology such that the following family is a pre-basis B e at the neutral element (0, 0): B e = {(− 1 n , 0] × (− 1 n , 0] : n ∈ N} ∪ {[0, 1 n ) × [0, 1 n ) : n ∈ N}. Then G is an ω-narrow strongly pre-topological group. Let H = {(x, y) : x + y = 0}. Then H is a closed subgroup H of G. However, H is a discrete topological group, hence H is not ω-narrow. Example 7.7. There exists an open subgroup H of an ω-narrow pre-topological group G is not ω-narrow. Proof. Let H be the group (R 2 , +) with usual addition and e = (0, 0). Put U = {[0, 1 n ) × {0}, (− 1 n , 0] × {0} : n ∈ N} ∪ {{0} × [0, 1 n ), {0} × (− 1 n , 0] : n ∈ N}. Then U is a family of subsets of H satisfying conditions (1) The next two theorems give partial answers to question 7.5. Then \|C\| ≤ τ and H ⊆ CV . For each c ∈ C, choose an element a c ∈ cV ∩ H, then put A = {a c : a ∈ C}. Since \|C\| ≤ τ , it follows that is a subset of H with \|A\| ≤ τ . We conclude that AW = H. Indeed, since H is a subgroup of G and V 2 ∩ H ⊆ W ⊆ H, we conclude that (AV 2 ) ∩ H ⊆ AW ⊆ H. Obviously, A ⊆ H ⊆ CV . Since V is symmetric, we have C ⊆ AV , hence H ⊆ CV ⊆ AV 2 ⊆ AW . Therefore, H is τ -narrow.V 1 , V 2 of e in G such that V 1 V 2 ∩ H ⊆ W . Since G is τ -narrow, there exists a subset C with \|C\| ≤ τ such that G = CV 1 = CV 2 = CV −1 1 . For each c ∈ C, we have that cV −1 1 ∩ H = ∅, then fix an element a c ∈ cV −1 1 ∩ H. Put A = {a c : a ∈ C}; then \|A\| ≤ τ . We claim that AW = H. Indeed, from our definition of A it follows that C ⊆ AV 1 . Then H ⊆ CV 2 ⊆ AV 1 V 2 , hence H ⊆ A(V 1 V 2 ∩ H) ⊆ AW . Thus AW = H. We say that a pairwise disjoint family consisting of non-empty open subsets of a pre-topological space (Z, τ ) is called a cellular family. The cellularity of Z is defined as follows: c(Z) = sup{\|V \| : V is a cellular family in Z}. Here, the cellularity of a pre-topological space maybe finite. It is well-known that ib(G) ≤ c(G) for any topological group G. Therefore, it is natural to pose the following question. Question 7.10. Let G be a pre-topological group (or strong pre-topology). Does ib(G) ≤ c(G) hold? Next we give some partial answers to Question 7.10. Definition 7.11. Let X be a pre-topological space. The smallest number κ such that each open cover U of X has a subfamily V of U with \|V \| ≤ κ and V = X is called the Lindelöf number of the pre-topological space X and is denoted by l(X). If l(X) = ω, then X is called Lindelöf. x ∈ A} is a cover of G, that is, G = x∈A xU = AU . Hence, G is τ -narrow. Corollary 7.13. If G is a Lindelöf pre-topological group, then G is ω-narrow. By Proposition 4.19 or Corollary 7.13, each separable (left) pre-topological group is ω-narrow. Theorem 7.14. If G is an almost topological group, then ib(G) ≤ c(G). Proof. Let c(G) ≤ τ . We claim that ib(G) ≤ τ . Indeed, pick an arbitrary open neighbourhood U of the neutral element e of G. Then there exists a symmetric and open neighbourhood V of e such that V 2 ⊆ U . Since the family ζ of all V -disjoint subsets of G is (partially) order by inclusion, and the union of any chain of V -disjoint sets is also a V -disjoint set. By the Zorn's Lemma, we can find a maximal element A of the ordered set ζ. Obviously, {aV : a ∈ A} is a disjoint family of non-empty open sets in G. Since c(G) ≤ τ , the set \|A\| ≤ τ . From the maximality of A, it follows that for every x ∈ G \ A there exists a ∈ A such that xV ∩ aV = ∅, then x ∈ aV V −1 = aV 2 ⊆ aU . Hence AU = G. Therefore, ib(G) ≤ τ . Given a pre-topological space X, we denote by e(X) the supremum of cardinalities of closed discrete subsets of X. The cardinal invariant e(X) is called the extent of X. Obviously, each Lindelöf pre-topological space is countable. It is well-known that for any topological group G, we have ib(G) ≤ e(G). However, the following question is still un-known for us in the class of almost topological groups. Question 7.15. If G is an almost topological group, then does ib(G) ≤ e(G) hold? The following theorem gives a complement for Theorem 8.18. V 1 V 2 ⊆ U . Since H is τ -narrow, there is a subset A of H such that A ≤ τ and H ⊆ AV 1 . By Proposition 4.19, G = H ⊆ AV 1 ⊆ AV 1 V 2 ⊆ AU , thus AU = G. Hence G is τ -narrow. The following proposition gives a relation of the weight of an almost topological group G and the narrowness of G. Proof. Clearly, χ(G) ≤ w(G). By Proposition 7.12, we have ib(G) ≤ l(G) ≤ w(G). Hence ib(G)χ(G) ≤ ω(G). We only need to prove ω(G) ≤ ib(G)χ(G). Let ib(G) ≤ τ and χ(G) ≤ κ. Then we assume that {U α : α ∈ κ} is a pre-base at the identity e of G. For every α ∈ κ, there exists a subset C α of G with \|C α \| ≤ τ such that C α U α = G. Then the cardinality of the family B = {xU α : x ∈ C α , α ∈ κ} is at most τ κ. We claim that B is a pre-base of G. Indeed, it suffices to prove that for any neighbourhood O of an arbitrary point a ∈ G there exist α ∈ κ and x ∈ C α such that a ∈ xU α ⊆ O. Take an any neighbourhood O of an arbitrary point a ∈ G. Since G is an almost topological group, it follows that there are α, β ∈ κ such that Remark 7.18. The strongly pre-topological group G in (3) of Example 3.13 is ω-narrow and χ(G) ≤ ω. However, it is easy to see that ω(G) > ω. Moreover, it is natural to consider the following question. aU β ⊆ O and U −1 α U α ⊆ U β . Since C α U α = G, there exists x ∈ C α such that a ∈ xU α , that is, x ∈ aU −1 α , then we have xU α ⊆ (aU −1 α )U α = a(U −1 α U α ) ⊆ aU β ⊆ O. Question 7.19. Let G be a symmetrically pre-topological group. Does ω(G) ≤ ib(G)χ(G) hold? Let G be a pre-topological group. We say that the invariance number of G is less than or equal to τ or in symbols, inv(G) if for any open neighbourhood U of the neutral element e of G, there exists a family γ of open neighbourhoods of e with \|γ\| ≤ τ such that for each x ∈ G there exists V ∈ γ satisfying xV x −1 ⊆ U . Any such family γ will be called subordinated to U . If a pre-topological group G satisfy that inv(G) ≤ τ , then G is called τ -balanced. It is well known that each τ -narrow topological group is τ -balanced. Moreover, it is obvious that each Abelian pre-topological group is τ -balanced. Hence it is natural to pose the following question. Question 7.20. If G is a τ -narrow pre-topological group, then is G τ -balanced? The following proposition gives a partial answer to Question 7.20. Indeed, it is obvious that γ is a family of open neighbourhoods of e and \|γ\| ≤ τ . For any x ∈ G, there exists a ∈ A such that x ∈ V a. Therefore, Proof. Let χ(G) = κ and {U α : α < κ} be a pre-base of G at the neutral element e. Take any open neighborhood V of e. Then for each x ∈ G the set V x is an open neighborhood of x. Since G is a pre-semitopological group, there exists α < κ such that xU α ⊆ V x, hence it follows that xW a x −1 ⊆ V aW a a −1 V −1 ⊆ V V V −1 = V 3 ⊆ U Hence γ is subordinated to U , that is, G is τ -balanced.xU α x −1 ⊆ V . Therefore, inv(G) ≤ κ. We give some properties of the invariance number of pre-topological groups. Proposition 7.23. Each subgroup of a τ -balanced pre-topological group is τ -balanced. Proof. Let G be τ -balanced. Take any subgroup H of G. We claim that H is also τ -balanced. U x , V x and O x of e in G such that U x ∩ H = W x , V 2 x ⊆ U x and O 3 x ⊆ V x . Put ϕ = {O x : x ∈ H}. Clearly, \|ϕ\| ≤ τ . We claim that ϕ is subordinated to U . Indeed, for each y ∈ G there exists x ∈ H such that y ∈ xO x since G = H( W ∈Be W ), where B e is the family of all open neighborhoods of e in G. From the local density of H and Proposition 4.19, it follows that yO x y −1 ⊆ xO x O x O x x −1 ⊆ xV x x −1 ⊆ xV x x −1 = xV x ∩ Hx −1 = x(V x ∩ H)x −1 = xW x x −1 ⊆ V ∩ H = V ⊆ U. However, the following question is still unknown for us. If the following question is positive, then each almost topological group with a dense τ -balanced subgroup is τ -balanced. Question 7.25. Is the closure of a τ -balanced subgroup H of an almost topological group G a τ -balanced subgroup? The following result gives a relation of the invariance number between a pre-topological group and its co-reflexion group topology. i \| ≤ κ such that γ i is subordinated to W i . Put η = { m i=1 W i : W i ∈ γ i , i ≤ m}. From Proposition 3.19, each element of η is open in (G, τ * ). Moreover, \|η\| ≤ κ. We claim that η is subordinated to U in (G, τ * ). Indeed, for any x ∈ G and i ≤ m, then there exists W x,i ∈ γ i such that xW x,i x −1 ⊆ W i . Hence x( m i=1 W x,i )x −1 ⊆ m i=1 W i ⊆ U. Remark 7.27. There exists an Abelian ω-narrow pre-topological group such that the co-reflexion group topology is not ω-narrow, such as (2) in Example 3.13. In order to give a characterization of τ -narrow almost topological groups, we give some lemmas and propositions. 1) γ ⊆ γ * ; 2) for each U ∈ γ * , there exists a symmetric V ∈ γ * such that V 2 ⊆ U ; 3) for each U ∈ γ * and each x ∈ G, there exists V ∈ γ * such that xV x −1 ⊆ U ; 4) \|γ * \| ≤ τ . Proof. For every U ∈ γ, we can find a symmetric open neighborhood V U of e such that V 2 U ⊆ U ; since G is τ -balanced, there exists a family V U of open neighborhoods of e subordinated to U such that \|V U \| ≤ τ . Now put ϕ(γ) = γ ∪ {V U : U ∈ γ} ∪ {V U : U ∈ γ}. Then we put γ 0 = γ, γ 1 = ϕ(γ 0 ), and repeat this operation, which defined by induction families γ 2 , . . . , γ α , and so on, by the rule γ α+1 = ϕ(γ α ) if α is a successor ordinal, and γ α = β<α ϕ(γ β ) for any α < τ . Put γ * = α<τ γ α . Clearly, \|γ α \| ≤ τ for any α < τ and γ α ⊆ γ β for every α < β, then γ * satisfies conditions 1)-4). Lemma 7.29. Let (G, τ ) be an τ -balanced almost topological group, and U an open neighborhood of the neutral element e in G. Then there exists a family {U α : α < τ } of open neighborhoods of e such that, for every α < τ , the following conditions are satisfied: a) U 0 ⊆ U ; b) U α = U −1 α ; c) there exists β < τ such that U 2 β ⊆ U α ; d) for any x ∈ G, there exists δ < τ such that xU δ x −1 ⊆ U α . Proof. By Lemma 7.28, we only put γ = {U }, and then let γ * = {U α : α < τ } such that U 0 ⊆ U , as desired. i ∈ ω} of {U α : α < τ } such that U α0 = U α and {x ∈ G : N α (X) < 1/2 i } ⊆ U αi ⊆ {x ∈ G : N α (X) < 2/2 i }; For any x and y in G, put ρ α (x, y) = N α (x −1 y), where α < τ . Since each N α is pre-continuous, it follows that each ρ α is also pre-continuous. Moreover, it is easily checked that each ρ α is leftinvariant pseudometric on the set G. By a4) and a) of Lemma 7.29 that a1) holds. Moreover, a3) is obviously satisfied. Now, we only need to prove that a2) holds. For each α < τ , put Z α = {x ∈ G : N α (x) = 0}. Then each Z α is closed in G, hence α<τ Z α is closed in G. Put Z = α<τ Z α . Then Z = {x ∈ G : N α (x) = 0, α < τ } is a closed subgroup of G. From the definition of each ρ α , we have Z = {x ∈ G : ρ α (e, x) = 0, α < τ }. We claim that Z is an invariant subgroup of G. Indeed, take any x ∈ G. We have to check that xZx −1 = Z. By a4), it is easily checked that Z = α<τ U α , hence it suffices to show that xZx −1 ⊆ U α for each α < τ . Fix α < τ . It follows from condition d) of Lemma 7.29 that there is β < τ such that xU β x −1 ⊆ U α . Since Z ⊆ U β , we conclude that xZx −1 ⊆ xU β x −1 ⊆ U α . Therefore, Z is invariant. Proof. We continue to use the objects in the proof of Theorem 7.30. In particular, we have a family {ρ α : α < τ } of pseudometrics on G constructed above. Suppose that H = G/Z is the quotient group, and suppose that π is the canonical homomorphism of G onto H. For any A, B ∈ H and α < τ , put d α (A, B) = ρ α (a, b), where a ∈ A, b ∈ B. Indeed, fix any α < τ . For any a ∈ A and b ∈ B, we have A = aZ and B = bZ. It suffices to prove that ρ α (a 1 , b 1 ) = ρ α (a, b) for any a 1 ∈ aZ and b 1 ∈ bZ. We may assume that b = b 1 ; otherwise we simply repeat the argument twice. Then a 1 = az for some z ∈ Z. Hence, N α (z) = N α (z −1 ) = 0, then it follows from [3, Lemma 3.4.16] that α (a, b). ρ α (a 1 , b) = N α (z −1 a −1 b) = N α (z −1 ) + N α (a −1 b) = N α (a −1 b) = ρ We also define a family of functions N α H on H by N α H (A) = N α (a) for each n ∈ ω, A ∈ H and a ∈ A. It is obvious that each N α H is well-defined. From the above definitions, for any α < τ we have that d α (π(a), π(b)) = ρ α (a, b) for any a, b ∈ G, and N α H (π(a) = N α (a) for each a ∈ G. Clearly, each d α is a pseudometric; moreover, each N α H is a prenorm on H satisfying the additional conditions as follows: a5) If N α H (A) = 0 for any α < τ , then A is the neutral element e H of H. For any ε > 0 and α < τ , put B α (ε) = {x ∈ G : N α (x) < ε}, and O α (ε) = {X ∈ H : N α H (X) < ε}. Obviously, we have π(B α (ε)) = O α (ε) for any ε > 0 and α < τ . Note that for each α < τ , it is easily checked that the prenorm N α also satisfies the following conditions: a6) For every x ∈ G and every ε > 0, there exists δ > 0 and β < τ such that xB β (δ)x −1 ⊆ B α (ε). By a6), for each ε > 0 and each X ∈ H, there exists δ > 0 and β < τ such that XO β (δ) X −1 ⊆ O α (ε). a7) For any ε > 0 and α < τ , we have O α (ε) = (O α (ε)) −1 . a8) For any α < τ and δ > 0, there exists β < τ and ε > 0 such that (O β (ε)) 2 ⊆ O α (δ). a9) {e H } = α<τ,m∈ω O α (1/2 m ). Let F H be the pre-topology generated by the family {d α : α < τ } of pseduometrics on H. We will prove that H with this pre-topology is an almost topological group. Indeed, since each pseduometric d α is left-invariant, it suffices to prove that the family {O α (1/2 m ) : α < τ, m ∈ ω} satisfies the axioms in Theorem 3.8 for a pre-base of a group pre-topology at the neutral element. And this is exactly what conditions a6)-a9) guarantee, as is routinely checked. Moreover, by a7), H with the pre-topology F H is an almost topological group. Clearly, χ(H) ≤ τ . Finally, since π(B α (ε)) = O α (ε)) for any α < τ and ε > 0, it follows that π is a pre-continuous at the neutral element, hence π is pre-continuous. Moreover, if x ∈ G, X = π(x), ε > 0 and α < τ , then N α (x) < ε if and only if N α H (X) < ε. Hence, π −1 (O α (ε)) = B α (ε) for any ε > 0. In particular, π −1 (O 0 (1)) = B 0 (1) ⊆ U 0 ⊆ U. H i with χ(H i ) ≤ τ such that π −1 i (V i ) ⊆ U i for some open neighborhood V i of the neutral element in H i . Suppose that = i∈I H i is the pre-topological product of the pretopological groups H i 's, and suppose that ϕ : G → is the diagonal product of the homomorphism π i , where i ∈ I. Obviously, ϕ is the pre-continuous homomorphism of G to . Now it suffices to prove that ϕ is a pre-topological embedding. Let H = ϕ(G). Clearly, it is easily checked that H is an almost topological group. By Theorem 6.6, ϕ : G → H is a bijective mapping. Take π i = p i • ϕ for each i ∈ I, hence ϕ −1 (W ) = π −1 i (V i ) ⊆ U i ⊆ U. Therefore, V = W ∩ H is an open neighborhood of the neutral element in H satisfying ϕ −1 (O) ⊆ U. Therefore, ϕ : G → H is a pre-topological isomorphism. Now we can prove one of the main results in this section. Indeed, the following theorem generalizes the well-known Guran's Theorem. Theorem 7.36. An almost topological group G is τ -narrow if and only if G can be embedded as a subgroup of a pre-topological product of almost topological groups of weight less than or equal to τ . Proof. By Proposition 4.19, each almost topological group of weight ≤ τ is τ -narrow. Then, by Theorem 7.8 and Proposition 7.4, each subgroup of a pre-topological product i∈I H i of almost topological groups is τ -narrow provided that w(H i ) ≤ τ for each i ∈ I. Conversely, suppose that almost topological group G is τ -narrow. By Theorem 7.35, Propositions 7.17 and 7.3, it is easily seen that we can identify G with a subgroup of a pre-topological product = i∈I H i of almost topological groups H i satisfying w(H i ) ≤ τ for each i ∈ I. The following theorem gives another characterization of τ -narrow almost topological groups. First, we give an obviously technical lemma. Proof. By Theorems 7.14 and 7.8, each subgroup of an almost topological group H with c(H) ≤ τ is τ -narrow. Conversely, by Theorem 7.36, a τ -narrow almost topological group G is pre-topologically isomorphic to a subgroup of a pre-topological product H of almost topological groups H i with w(H i ) ≤ τ . By Lemma 7.37, c(H) ≤ τ , as desired. A pre-topological space X is said to be σ-compact if X = n∈ω X n , where each X n is compact. Moreover, a pre-topological group is said to be k-separable if it has a dense σ-compact subgroup. Lemma 7.39. The σ-product of any family of compact T 2 pre-topological spaces is σ-compact. Proof. Let {X α : α ∈ I} be a family of compact T 2 pre-topological spaces and let X = α∈I X α . Suppose that Y be the corresponding σ-product with center at b ∈ X. Then, for each y ∈ Y , only finitely many coordinates y α of y are distinct from the corresponding coordinates b α of b. Denote by r(y) the number of coordinates of a point y ∈ Y distinct from those of b. For each n ∈ ω, put Y n = {y ∈ Y : r(y) ≤ n}. Obviously, we have Y = n∈ω Y n , and each Y n is closed in the product space X since each X α is T 2 . Therefore, Y is σ-compact. By Lemma 7.39, we have the following lemma by a similar proof of [3, Proposition 1.6.41], so we left the proof for the reader. Lemma 7.40. The σ-product of any family of σ-compact T 2 pre-topological spaces is σ-compact. The following lemma is obvious. Lemma 7.41. The σ-product of any family A of pre-topological spaces is dense in the product of the family A . The following theorem gives a generalization of well-known theorem of Pestov's. Theorem 7.42. The class of ω-narrow almost topological groups coincides with the class of subgroups of k-separable almost topological groups. Proof. Let G be a k-separable almost topological group. Then G has a dense σ-compact subgroup. Since each σ-compact almost topological group is Lindelöf, it follows from Corollary 7.13 that H is ω-narrow. From Theorem 7.16, it follows that G is ω-narrow, hence each subgroup of G is also ω-narrow by Theorem 7.8. Conversely, suppose that G is an arbitrary ω-narrow almost topological group. From Theorem 7.36, G can be embedded as a subgroup of a pre-topological product = i∈I H i of secondcountable almost topological groups H i 's. For each i ∈ I, we can fix a countable dense subgroup D i of H i , and then let D be a σ-product of i∈I D i . By Lemmas 7.40 and 7.41, D is σ-compact and dense in i∈I D i . The following proposition shows that, in the class of Abelian almost topological groups G, pre-continuous homomorphic images H with χ(H) ≤ ω of a given almost topological group G determine whether G is τ -narrow or not. Proposition 7.43. Suppose that G is an Abelian almost topological group and suppose that each pre-continuous homomorphic image H of G with χ(H) ≤ ω is τ -narrow. Then the almost topological group G is also τ -narrow. Proof. Take an arbitrary open neighborhood U of the neutral element e in G. Since G is an almost topological group, there exists a sequence {U n : n ∈ ω} of open symmetric neighborhoods of e in G such that U 0 ⊆ U and U 2 n+1 ⊆ U n for every n ∈ ω. Put N = n∈ω U n is a closed subgroup of G. Since G is Abelian, the set of all cosets G/N is a group. Let π : G → G/N be the natural homomorphism. It is easily checked that the family {π(U n ) : n ∈ ω} is a pre-base for a Hausdorff almost topological group pre-topology F on G/N at the neutral element of this group. Let H = (G/N, F ). Then π : G → H is pre-continuous and χ(H) ≤ ω, hence H is τ -narrow. Let V = π(U 1 ). Then there exists a subset K ⊆ H with \|K\| ≤ τ such that KV = H. Let F be any subset of G such that π(F ) = K and \|F \| ≤ τ . We conclude that F U = G. Indeed, take any point x ∈ G. Hence π(x) ∈ bV for some b ∈ K. Hence we can choose an element a ∈ F such that π(a) = b, then π(x) ∈ bV = π(aU 1 ), hence it follows that x ∈ π −1 π(aU 1 ) = aU 1 N ⊆ aU 1 U 1 ⊆ aU 0 ⊆ aU ⊆ F U. Therefore, G is τ -narrow. The precompactness in pre-topological groups In this section, some basic properties about precompactness in pre-topological groups are investigated. Definition 8.1. A semi-pre-topological group is called left precompact (resp. right precompact) if, for each open neighborhood U of the neutral element in G, there exists a finite subset F of G such that G = F U (resp. G = U F )). If G is left precompact and right precompact then G is called precompact. Compare with Proposition 7.2, we also have the following proposition. Proposition 8.2. The following conditions are equivalent for a quasi-pre-topological group G. (1) G is precompact; (2) For every open neighbourhood V of e in G, there exists a finite subset B ⊆ G such that G = V B; (3) For every open neighbourhood V of e in G, there exists a finite subset C ⊆ G such that CV = V C = G. The proof of the following proposition is obvious, thus we omit it. Proposition 8.3. If f is a pre-continuous homomorphism of a precompact pretopological group G onto a pre-topological group H, then H is also precompact. It is obvious that a discrete pre-topological group is precompact if and only if it is finite. Clearly, each precompact pre-topological group is ω-narrow and each compact pre-topological group is precompact. Moreover, we have the following more general fact. x ∈ A} is discrete in G, hence A is finite since G is feebly compact. Then, by the maximality of A, we have G = AV . Hence G is precompact. In order to discuss the relation of the precompactness between pre-topological groups and their subgroups, we introduce the following concept and prove some lemmas. A subset B of a semi-pre-topological group G is called precompact in G if, for each neighborhood U of the neutral element in G, there exists a finite set F ⊆ G such that B ⊆ F U and B ⊆ U F . Then K 1 is finite and is contained in D. We claim that B ⊆ K 1 U . Indeed, take any b ∈ B. Then there is x ∈ F such that b ∈ xV −1 1 . Hence b ∈ B ∩ xV −1 1 = ∅, thus y x ∈ xV −1 1 , then y −1 x x ∈ V 1 . Therefore, it follows that b ∈ xV 2 = y x (y −1 x x)V 2 ⊆ y x V 1 V 2 ⊆ y x U ⊆ K 1 U. Thus B ⊆ K 1 U. Similarly, we can find a finite subset K 2 of S such that B ⊆ U K 2 . Now put K = K 1 ∪ K 2 . Then the finite set K is as required. Proposition 8.6. Each subgroup H of a precompact pre-topological group G is a precompact pretopological group. Proof. Let U be an arbitrary open neighborhood U of the neutral element in H. Then there exists an open neighborhood V of e in G such that V ∩ H = U . Since G is precompact, it follows that H is a precompact subset of G. By Lemma 8.5, there is a finite subset F of H such that H ⊆ F V and H ⊆ V F . We conclude that H ⊆ F U and H ⊆ U F . Indeed, for each h ∈ H, there exist x ∈ F and y ∈ V such that h = xy. Since H is a subgroup, it follows that y = x −1 h ∈ V ∩ H = U , which implies that h ∈ F U . Thus F ⊆ F U . Similarly, we also have H ⊆ U F . Therefore, H is a precompact pre-topological group. Lemma 8.7. Suppose that B is a subset of pre-topological group G such that B contains a dense precompact subset. Then B is also precompact in G. Therefore, the closure of a precompact subset of G is precompact in G. Proof. Let D be a dense precompact subset of B. Take an arbitrary open neighborhood U of the neutral element e in G. Then there exist open neighborhoods V 1 and V 2 of e in G such that V 1 V 2 ⊆ U . Because D is precompact in G, there is a finite subset F such that D ⊆ F V 1 and D ⊆ V 1 F . We conclude that B ⊆ F U ∩ U F . Indeed, take any b ∈ B. Since D is dense in B, then bV −1 2 ∩ D = ∅, hence we can pick a point y b ∈ bV −1 2 ∩ D. Hence y b ∈ xV 1 for some x ∈ F since D ⊆ F V 1 , then b ∈ y b V 2 ⊆ xV 1 V 2 ⊆ xU . Therefore, we have B ⊆ F U . Similarly, we have B ⊆ U F . Corollary 8.8. If a pre-topological group G contains a dense precompact subgroup, then G is also precompact. The following proposition, though quite easy, is nevertheless, rather interesting. Proposition 8.9. The dispersion character of a precompact pre-topological group G is equal to its cardinality. Proof. Let U be an arbitrary open neighborhood of the neutral element e in G. Since G is precompact, there exists a finite subset F such that F U = G, hence \|U \| = \|G\|. We say that a pre-topological group is finite-balanced if for any open neighbourhood U of the neutral element e of G, there exists a finite family γ of open neighbourhoods of e such that for each x ∈ G there exists V ∈ γ satisfying xV x −1 ⊆ U . Any such family γ will be called subordinated to U . By a similar proof of Proposition 7.21, we have the following proposition. Theorem 8.10. Each precompact almost topological group G is finite-balanced. Theorem 8.11. If B i is a precompact subset of a pre-topological group G i for every i ∈ I, then the set B = i∈I B i is precompact in the pre-topological product G = i∈I G i . Proof. Let U be an arbitrary open neighborhood of the neutral element in G = i∈I G i . Then there exist a finite subset C ⊆ I and open neighborhood U i of the neutral element e i of G i for each i ∈ C such that i∈C U i × i∈I\C G i ⊆ U . For each i ∈ C, there exists a finite subset F i such that B i ⊆ F i U i and B i ⊆ U i F i . Put F = i∈C F i × i∈I\C {e i }. Then i∈I B i ⊆ F ( i∈C U i × i∈I\C G i ) ⊆ F U and i∈I B i ⊆ ( i∈C U i × i∈I\C G i )F ⊆ U F . The proof is completed. Corollary 8.12. The product of a family of precompact pre-topological groups is a precompact pre-topological group. By Proposition 8.3 and Theorem 8.11, we have the following corollary. Corollary 8.13. Let A and B be precompact subsets of a pre-topological group G. Then the sets A −1 , B −1 and AB are precompact in G. Next, we discuss the pre-Raǐkov completion of a pre-topological group. Let (G, τ ) be a pretopological group, and let (G, τ * ) be the co-reflexion group topology of (G, τ ). Then (G, τ * ) has a Raǐkov completion (ρG, ρτ * ). We say that the pre-topology σ on ρG is the pre-Raǐkov completion of (G, τ ) if σ has a pre-base {gU : e ∈ U ∩ G ∈ τ, g ∈ ρG, U ∈ ρτ * }, where e is the neutral element of ρG. For convenience, we denote the pre-Raǐkov completion of (G, τ ) by pre-ρG. Theorem 8.14. If G is a pre-topological group, then the pre-Raǐkov completion pre-ρG of G is a homogeneous pre-topological space. Proof. Let G be a pre-topological group. Take any g, h ∈ ρG. Then the left translation mapping l : ρG → ρG, defined by l(x) = hg −1 (x) for each x ∈ ρG, is a pre-homeomorphic mapping. Hence pre-ρG of G is a homogeneous pre-topological space. Theorem 8.15. Let (G, τ ) be a pre-topological group. Then the pre-Raǐkov completion pre-ρG of (G, τ ) satisfies the following conditions: (i) σ is a subbase for ρτ * ; (ii) (G, τ ) is a dense pre-topological subgroup of pre-ρG. (v) If G is an almost topological group, then, for each open neighborhood U of the neutral element in pre-ρG and g ∈ ρG, there exists an open neighborhood O of the neutral element in pre-ρG such that gOg −1 ⊆ U . Proof. (i) It suffices to prove that the family B = {U ∈ ρτ * : e ∈ U ∩ G ∈ τ } is a subbase at the neutral element in (ρG, ρτ * ). Indeed, take any open neighborhood U of the neutral element of ρG in (ρG, ρτ * ). Since U ∩ G is an open neighborhood of e in (G, τ * ), it follows from Proposition 3.19 that there exist finitely many open neighborhoods U 1 , . . . , U n of the neutral element of G in (G, τ ) such that n i=1 U i ⊆ U ∩ G. For each i ≤ n, because U i is open in (G, τ * ), there exists an open neighborhood W i ⊆ U of the neutral element in (ρG, ρτ * ) such that W i ∩ G = U i . Therefore, g ∈ g n i=1 W i ⊆ gU and W i ∈ B for each i ≤ n. (ii) Clearly, G is dense in pre-ρG since (G, τ * ) is dense in (ρG, ρτ * ). From our definition, we have σ\| G = τ . Therefore, (G, τ ) is a dense pre-topological subgroup of pre-ρG. (iii) It is obvious. (iv) Take any open neighborhood U of the neutral element in σ. Then U ∩ G ∈ τ , hence there exist open neighborhoods W 1 and W 2 of the neutral element e in (G, τ ) such that W 1 W 2 ⊆ U . Then W 1 W 2 ⊆ U in (ρG, ρτ * ), hence W 1 W 2 ⊆ U , thus int(W 1 )int(W 2 ) ⊆ int(W 1 W 2 ) ⊆ int(U ) in (ρG, ρτ * ). Put int(W 1 ) = V 1 , int(W 2 ) = V 2 . Clearly, V 1 ∩ G = W 1 , V 2 ∩ G = W 2 and int(U ) = U . Hence both V 1 and V 2 are open neighborhoods of the neutral element in pre-ρG. (v) Take any open neighborhood U of the neutral element in pre-ρG and g ∈ ρG. From the proof of (iv) above, there exist open symmetric neighborhoods W of the neutral element in pre-ρG such that W 3 ⊆ U . By (ii), G is dense in pre-ρG, hence W g ∩ G = ∅ since (G, τ * ) is dense in (ρG, ρτ * ) and W is open in (ρG, ρτ * ), then there exists h ∈ G such that g ∈ W −1 h = W h. Then there exists an open neighborhood V of the neutral element in (G, τ ) such that hV h −1 ⊆ W ∩ G, hence hV h −1 ⊆ W ∩ G in (ρG, ρτ * ), then hint(V )h −1 ⊆ int(W ∩ G) in (ρG, ρτ * ). Put O = int(V ). Since O ∩ G = V and int(W ∩ G) = W in (ρG, ρτ * ), it follows that O ∈ σ and gOg −1 ⊆ W hOh −1 W ⊆ W W W ⊆ U. By Theorems 3.8 and 8.15, it is easily concluded that the following corollary holds. Corollary 8.16. Let G be an almost topological group. If the following condition (⋆) holds, then pre-ρG is an almost topological group. (⋆) For each open neighborhood U of the neutral element in pre-ρG, if g ∈ U , then there exists an open neighborhood V of the neutral element in pre-ρG such that gV ⊆ U Question 8.17. Let G be an almost topological group. Does pre-ρG satisfy the condition (⋆) in Corollary 8.16. The following theorem gives a partial answer to Question 8.17. Theorem 8.18. Let (G, τ ) be a pre-topological group. If (G, τ * ) is locally compact, then pre-ρG = G. Proof. Since (G, τ * ) is locally compact, it follows from [3, Theorem 3.. 6.24] that ρG is just (G, τ * ). Therefore, pre-ρG = G. Corollary 8. 19. Each finite pre-topological group is pre-Raǐkov complete. It is well known that the closure of each precompact subset B in a topological group G is compact in the Raǐkov completion ρG of G. However, in the class of pre-topological groups, we have the following example. Example 8.20. There exists a precompact, non-compact pre-topological group G such that pre-ρG = G. Indeed, let G be the unit circle, that is, G = {\|z\| = 1 : Z ∈ C}. For each 0 < θ < π, let U θ = e iθ and U −θ = e −iθ . Let A = {U θ : 0 < θ < π} ∪ {U −θ : 0 < θ < π}. Then A satisfies the conditions (1)-(4) of Theorem 3.8, hence it follows from Theorem 3.9 that the pre-topology τ , generated by the family A , is a pre-topological group topology on G. Clearly, G is precompact and non-compact. However, (G, τ * ) is discrete, thus it is locally compact, then pre-ρG = G by Theorem 8.18. Remark 8.21. The pre-topological group (G, τ ) in (2) of Example 3.2 is a second-countable and non-precompact pre-topological group. However, (G, τ * ) is locally compact, hence it follows from [3, Theorem 3.. 6.24] that ρG is just (G, τ * ). Therefore, pre-ρG = G by Theorem 8.18. Theorem 8.22. Let G be a pre-topological group and A be a subset of G. Then A is a precompact subset of G if and only if the closure of A in the pre-Raǐkov completion pre-ρG is precompact. Proof. By the proof of Proposition 8.6, the sufficiency is obvious. It suffices to prove the necessity. Let A be a precompact subset of G. Take any open neighborhood U of the neutral element in pre-ρG. By (iV) of Theorem 8. 15, there exist open neighborhoods V 1 and V 2 in pre-ρG such that V 1 V 2 ⊆ U , then V 1 ⊆ U by the proof of Theorem 3.16. Since U ∩ G is open in G, there exist finite subset F ⊆ G such that A ⊆ F (V 1 ∩ G) and A ⊆ (V 1 ∩ G)F . Then A ⊆ F (V 1 ∩ G) = F V 1 and A ⊆ (V 1 ∩ G)F = F V 1 . Since F V 1 ⊆ F U and V 1 F ⊆ U F , it follows that the closure of A in pre-ρG is precompact. Corollary 8.23. A pre-topological group G is precompact if and only if the pre-Raǐkov completion pre-ρG is precompact. Question 8.24. Let G be a pre-topological group. If the pre-Raǐkov completion pre-ρG of G is compact, is pre-ρG = G? Theorem 8.25. Each pre-topological product G = i∈I G i of pre-Raǐkov complete pre-topological spaces is pre-Raǐkov complete. Proof. For each i ∈ I, let τ i be the pre-topology of G i ; since G i is pre-Raǐkov complete, it follows that G i =pre-ρG i , hence (G, τ * ) is Raǐkov complete. Therefore, it follows from [3, Theorem 3.6.22] that the topological product G = i∈I (G i , τ * i ) is Raǐkov complete, then G = i∈I G i is pre-Raǐkov complete. Corollary 8.26. Let G = i∈I G i be a product of pre-topological groups. Then pre-ρG is pretopologically isomorphic to the pre-product pre-topological space i∈I pre-ρG i . Finally, we discuss some applications of the precompactness in pre-topological groups. First, we say that a pre-topological space X is resolvable if there exist dense disjoint subsets A and B. Proposition 8.27. If a subgroup H of a pre-topological group G is resolvable, then so is G. Proof. Let A and B be two dense disjoint subsets in H, and let C = {x α H : x α ∈ G, α ∈ I} be all the cosets of H in G such that x α H ∩ x β = ∅ for any α = β. Put D 1 = {x α A : α ∈ I} and D 2 = {x α B : α ∈ I}. Then it is easily checked that D 1 and D 2 are disjoint dense subsets of G. The following proposition is obvious, so we omit the proof. Proposition 8.28. If a pre-topological group G contains a proper dense subgroup, then G is resolvable. Proposition 8.29. If a pre-topological group G contains a non-closed subgroup, then G is resolvable. Proof. Let H be a non-closed subgroup of G. Then H is a pre-topological subgroup by Proposition 4.26. If H = G, then G is resolvable by Proposition 8.28 since H is non-closed. Now we assume that H = G. Since H = H, it follows that H is resolvable by Proposition 8.28. Then, from Proposition 8.27, it follows that G is resolvable. Lemma 8.30. [37] For any infinite group G, there exists a disjoint family A of cardinality \|G\| of subsets of G such that, for each A ∈ A and any finite subset K of G, AK = G and (G \ A)K = G. Theorem 8.31. For any infinite group G, there exists a disjoint family A of cardinality \|G\| of subsets of G which are dense in any precompact pre-topological group on G. Proof. Let G be a precompact pre-topological group on G. By Lemma 8.30, there exists a disjoint family A of cardinality \|G\| of subsets of G such that, for each A ∈ A and any finite subset K of G, AK = G and (G \ A)K = G. Fix any A ∈ A . We claim that A and G \ A are dense in G. Indeed, if A is not dense in G, then Int(G \ A) = ∅; since G is precompact, there exists a finite subset K of G such that K(G \ A) = G, which is a contradiction. If G \ A is not dense in G, then we can obtain a contradiction by a similar method. Therefore, A and G \ A are dense in G. Corollary 8.32. Each infinite precompact pre-topological group is resolvable; in particular, each infinite precompact topological group is also resolvable. Indeed, by [37, Corollary 10], we have the following more general theorem by a similar proof of Theorem 8.31. Theorem 8.33. Each uncountable ω-narrow pre-topological group is resolvable; in particular, each uncountable ω-narrow topological group is also resolvable. Definition 3 . 10 . 310Let G be a pre-topological space and U be an open neighborhood of a point b of G. We say that U is an atom at b if W = U for any open neighborhood W of b with W ⊆ U . Proposition 3 . 15 . 315Let G be a pre-topological group. Then G is an almost topological group if and only if for each open neighborhood U of e there exists an open neighborhood V of e such that V V −1 ⊆ U . Question 3 . 17 . 317Let (G, τ ) be a pre-topological group. If H is a discrete subgroup G, under what condition is H closed in G? Lemma 3 . 22 . 322Let G be an almost topological group, and let U and V be two open neighborhoods of the neutral element in G such that V 4 ⊂ U and V −1 = V . If a subset B of G is U -disjoint, then the family of open sets {aV : a ∈ B} is discrete in G. Proposition 3 . 25 . 325The connected component H of any pre-topological group G is a closed invariant subgroup of G. Proposition 3 . 26 . 326Let U be an arbitrary open neighborhood of the neutral e of a connected pretopological group G. Then G = ∞ n=1 (U ∪ U −1 ) n . Proof. Let U be an arbitrary open neighborhood of the neutral e. Then H = ∞ n=1 (U ∪ U −1 ) n is an open subgroup of G. According to Proposition 4.15 below, H is closed in G. Since G is connected, we have H = G. Theorem 3 . 27 . 327Let K be a discrete invariant subgroup of a connected pre-topological group G. If, for any x ∈ G and open set U with x ∈ U , there exists an open neighborhood V of e such that V xV ⊂ U , then K is contained in the center of the pre-topological group G. Question 3 . 28 . 328Let G be a strongly pre-topological group (almost topological group). For any x ∈ G and open set U with x ∈ U , does there exist an open neighborhood V of e such that V xV ⊂ U ? Corollary 4 . 12 . 412In each semi-pre-topological group G all, right and left, translations are prehomeomorphism. Corollary 4 . 13 . 413If a subgroup H of a right (or left) pre-topological group G contains a non-empty open subset of G, then H is open in G. Proof. Suppose U is a non-empty open subset of G such that U ⊆ H. By Proposition 4.11, the set R a (U ) = U a is also open in G for each a ∈ H. Hence, the set H = a∈H U a is open in G. Proposition 4. 14 . 14If f : G → H is a homomorphism of left (right) pre-topological groups and f is pre-continuous at the neutral element e of G, then f is pre-continuous. Proof. Take any x ∈ G, and let O be an open neighbourhood of y = f (x) in H. Since the left translation L y is a pre-homeomorphism of H, there exists an open neighbourhood V of the neutral element e H in H such that yV ⊆ O. By the pre-continuity of f at e G , it follows that there exists an open neighbourhood Theorem 4 . 15 . 415Each open subgroup H of a right (left) pre-topological group G is closed in G. Proposition 4 . 17 . 417If G is a left (right) pre-topological group and U is an open subset of G, then the set AU (resp.,U A)is open in G for any subset A of G. Corollary 4 . 18 . 418Let G be a semi-pre-topological group and U an open subset of G. Then the sets U A and U A are open for any subset A of G. Proposition 4 . 19 . 419Let G be a left (right) pre-topological group with pre-continuous inverse. Then, for each subset A of G and each open neighborhood U of e, we have A ⊂ AU (resp., A ⊂ U A) . Proof. Take any subset A of G and any open neighborhood U of e. Since the inverse is precontinuous, there exists an open neighbourhood Proposition 4 . 20 . 420Let G be a left (right) pre-topological group with pre-continuous inverse, and let B e a pre-base of G at the neutral element e. Then, for each subset A of G, we have A = {AU : U ∈ B e } (resp., A = {U A : U ∈ B e } ) . Proof. By Proposition 4.19, we have A ⊆ AU for each U ∈ B e , then A ⊆ {AU : U ∈ B e }. Now assume that x ∈ A. We prove that there exists U ∈ B e such that x ∈ AU . Since x ∈ A, there exists an open neighbourhood W of e such that (xW ) ∩ A = ∅. From the pre-continuity of the inverse, there is an open neighbourhood Proposition 4 . 21 . 421Let G be a semi-pre-topological group such that for every open set U with e ∈ U , there exists an open neighborhood Proposition 4 . 22 . 422Let G be a semi-pre-topological group such that for each closed subset A of G and each point x ∈ A, we have x ∈ AU for some open neighborhood U of e. Then G is a quasi-pre-topological group.Proof. Byproposition 4.21, it suffices to verify that, for each open set U ∈ B e , there exists an open neighbourhood V of e such that V −1 ⊆ U . Take any open neighbourhood U of e, and put A = G\U ; then e ∈ A. By the assumption, there exists an open neighbourhood V of e such that e ∈ AV , then A ∩ Proposition 4 . 24 . 424Let G be a semi-pre-topological semigroup, and H a subsemigroup of G. Then the closure H of H is a subsemigroup of G. Proposition 4 . 26 . 426Let G be a quasi-pre-topological group, and H an algebraic subgroup of G. Then the closure of H is also a subgroup of G.Proof. Take any x, y ∈ H. From proposition 4.24, we have xy ∈ H. By proposition 4.25, we conclude that H −1 = H. Hence, the closure of H is also a subgroup of G. Corollary 4 . 27 . 427Let G be a pre-topological group, and H a subgroup of G. Then the closure of H is also a subgroup of G. Theorem 5 . 4 . 54Let G be a pre-topological group and H is a closed subgroup of G. Then G/H with the quotient pre-topology is discrete if and only if H is open in G. Theorem 5 . 13 . 513Let p : G → H be pre-continuous homomorphism of almost topological groups. Suppose that the image p(U ) contains a non-empty open set in H, for each open neighbourhood U of the neutral element e G in G. Then the homomorphism p is open. Proof. First we claim that for each open neighbourhood U of the neutral element e G in G, there is an open neighbourhood W of neutral element e H of H is contained in p(U ). Indeed, since G is an almost topological group, there exists an open neighbourhood V of the neutral element e G , such that V −1 V ⊆ U . By assumption, p(V ) contains a non-empty open set W in H, then W −1 W is an open neighbourhood of e H and W −1 W ⊆ p(V ) −1 p(V ) = p(V −1 V ) ⊆ p(U ). Now we prove this theorem. Take any open set U in G and any element y ∈ p(U ); there is a point x ∈ U such that y = p(x). So, we can find an open neighbourhood O of the neutral element e G such that xO ⊆ U . According to our claim above, there exists an open neighbourhood W 1 of neutral element e H such that W 1 ⊆ p(O), hence Theorem 5.14. (Second Isomorphism) Let G and H be left pre-topological groups with the neutral elements e G and e H , respectively, and letp : G → H be an open pre-continuous homomorphism of G onto H. Let H 0 be closed invariant subgroup of H, G 0 = p −1 (H 0 ) and N = p −1 (e H ). Then the left pre-topological groups G/G 0 , H/H 0 , and (G/N )/(G 0 /N ) are pre-topologically isomorphic.Theorem 5.15. (Third Isomorphism) Let G be a pre-topological group, H be a closed invariant subgroup of G and M be any pre-topological subgroup of G. Then the quotient group M H/H is pre-topologically isomorphic to the subgroup π(M ) of the pre-topological group G/H, where π : G → G/H is the natural quotient homomorphism. Lemma 6.2. A prenorm N on a pre-topological group G is pre-continuous if and only if for every ε > 0 there is an open neighborhood U of the neutral element e such that U ⊆ B N (ε). Theorem 6. 5 . 5For each open neighborhood U of the neutral element e of an almost topological group G, there is a pre-continuous prenorm N on G such that the unit ball B N (1) ⊆ U .Now we can prove one of main results in this section. Theorem 6. 7 . 7Let G be an almost topological group. Then χ(G) ≤ τ if and only if there exists a family {ρ α : α < τ } of right-invariant pseudometrics such that the family {B ρα ( 1 2 n ) : α < τ, n ∈ N} is a pre-base at the neutral element e. Finally, we prove that the family {B ρα ( 1 2 n ) : α < τ, n ∈ N} is a pre-base at the neutral element e. Indeed, take any open neighborhood O of e; then there exists Corollary 6. 8 . 8Let G be an almost topological group. Then χ(G) ≤ τ if and only if there exist a family {ρ α : α < τ } of right-invariant pseudometrics and a family {σ α : α < τ } of left-invariant pseudometrics, both generating the original pre-topology of G. Corollary 6. 9 . 9An almost topological group G is first-countable if and only if there exists a sequence {ρ n : n ∈ N} of right-invariant pseudometrics such that the family {B ρn ( 1 2 m ) : n, m ∈ N} is a pre-base at the neutral element e. Corollary 6.10. An almost topological group G admits a family of invariant pseudometrics generating its pre-topology if and only if G is balanced. Lemma 6 . 12 . 612Assume that f : X → Y is an open pre-continuous mapping of a pre-topological space X onto a pre-topological space Y . Then f −1 (B) = f −1 (B). Proposition 7 . 2 . 72The following conditions are equivalent for a quasi-pre-topological group G.(1) G is τ -narrow; (2) For every open neighbourhood V of e in G, there exists a subset B ⊆ G with \|B\| ≤ τ such that G = V B; (3) For every open neighbourhood V of e in G, there exists a countable set C ⊆ G with \|C\| ≤ τ such that CV = V C = G. Proof. Clearly,(3) ⇒ (1). Now we only need to prove (1) ⇒ (2) and(2) ⇒ (3). Proposition 7 . 3 . 73If pre-topological group H is a pre-continuous homomorphic image of a τ -narrow pre-topological group G, then H is also τ -narrow.Proof. Let V be an open neighbourhood of neutral element e in H and f : G → H be a precontinuous homomorphic mapping. Since G is τ -narrow, there exists a subset A of G such that Af −1 (V ) = G and A\| ≤ τ . It follows from f is homomorphic that f (G) = f (Af −1 (V )) = f (A)V = H. Clearly, f (A) is a subset of H and \|f (A)\| ≤ τ , hence H is τ -narrow. Question 7 . 5 . 75When is a subgroup H of a τ -narrow pre-topological group G τ -narrow? Indeed, the situation are different in the class of pre-topological groups, see the following two examples. -(4) of Theorem 3.8. Then it follows from Theorem 3.9 that the family B U = {U a : a ∈ H, U ∈ U } is a pre-base for a pre-topology τ on H such that (H, τ ) is a pre-topological group. Clearly, H is ω-narrow and subgroup R × {0} is open in H. However, R × {0} is not ω-narrow since e is open in the uncountable subgroup R × {0}. Theorem 7. 8 . 8Each subgroup H of a τ -narrow almost topological group G is τ -narrow. Proof. Let W be an open neighbourhood of the neutral element e in H. Then there exists an open symmetric neighbourhood V of e in G such that V 2 ∩ H ⊆ W . Since G is τ -narrow, there exists a subset B of G such that BV = G and \|B\| ≤ τ . Let C = {c ∈ B : cV ∩ H = ∅}. Theorem 7. 9 . 9Every dense subgroup H of a τ -narrow pre-topological group G is τ -narrow. Proof. Let W be an open neighbourhood of the identity e in H. Then there exist open neighbourhoods Proposition 7 . 12 . 712If G is a pre-topological group, then ib(G ≤ l(G).Proof. Let l(G) = τ . Take an arbitrary open neighborhood U of the neutral element of e. Then {xU : x ∈ G} is an open cover of G. Since l(G) ≤ τ , there exists a subset A such that \|A\| ≤ τ and {xU : Theorem 7. 16 . 16If a pre-topological group G contains a dense subgroup H such that H is τ -narrow, then G is also τ -narrow.Proof. Let U be any open neighbourhood of the neutral element e in G; then there exist open neighbourhoods V 1 , V 2 of the neutral element e of G such that Proposition 7 . 17 . 717Let G be an almost topological group. Then ω(G) = ib(G)χ(G). Then xU α is an open neighbourhood of point a and xU α ⊆ O. Proposition 7 . 21 . 721If G is a τ -narrow almost topological group, then G is τ -balanced.Proof. Let U be an open neighbourhood of the neutral element e in G; then since G is an almost topological group, there exists an open and symmetric neighbourhood V of the neutral element e such that V 3 ⊆ U . Since G is τ -narrow, there exists a subset A of G with \|A\| ≤ τ such that V A = G. For any a ∈ A, there exists an open neighbourhood W a of the neutral element e such that aW a a −1 ⊆ V . Then we conclude that the family γ = {W a : a ∈ A} subordinated to U . Theorem 7 . 22 . 722Let G be a pre-semitopological group. Then inv(G) ≤ χ(G). Indeed, take any open neighborhood U of e in H. Then there exists an open neighborhood V of e in G such that U = V ∩ H. Since G is τ -balanced, we can find a family η of open neighborhoods of e in G such that \|η\| ≤ τ and for eachx ∈ G there exists W x ∈ η satisfying xW x x −1 ⊆ V . Put η H = {W ∩ H : W ∈ η}. Then, for each x ∈ H, we have x(W x ∩ H)x −1 ⊆ (xW x x −1 ) ∩ H ⊆ V ∩ H = U. Moreover, it is obvious that \|η H \| ≤ τ . Hence H is τ -balanced.A subset D is said to be locally dense in a pre-topological space G if U ∩ D = U for each open set U in G. Clearly, each locally dense subset of a pre-topological space is dense.Proposition 7.24. Let G be an almost topological group. If H is locally dense and τ -balanced subgroup of G, then G is again a τ -balanced subgroup. Proof. Take any open neighborhood U of e in G. Then there exists an open neighborhood V of e in G such that V 2 ⊆ U . Clearly, V ∩ H is an open neighborhood of e in H, hence we can find a family η of open symmetric neighborhoods of e in H such that \|η\| ≤ τ and for each x ∈ H there exists W x ∈ η satisfying xW x x −1 ⊆ V ∩ H. For each x ∈ H, there exist an open symmetric neighborhoods Proposition 7 . 26 . 726Let (G, τ ) be a pre-topological group. If inv(G, τ ) ≤ κ, then inv(G, τ * ) ≤ κ.Proof. Take any open neighborhood U of (G, τ * ). From Proposition 3.19, it follows that there exist open neighborhoods W 1 , . . . , W m in (G, τ ) such that m i=1 W i ⊆ U . For each i ≤ m, since inv(G, τ ) ≤ κ, there exists a family γ i of open neighbourhoods of e in (G, τ ) with \|γ Lemma 7. 28 . 28Let G be an τ -balanced almost topological group, and let γ be a family of open neighborhoods of the neutral element e in G such that \|γ\| ≤ τ . Then there is a family γ * of open neighborhoods of e satisfying the following properties: Theorem 7 . 30 . 730Let G be an τ -balanced almost topological group. Then, for every open neighborhood U of the neutral element e in G, there exists a family {ρ α : α < τ } of pre-continuous left-invariant pseudometrics such that the following conditions are satisfied:(a1) there exists α < τ such that {x ∈ G : ρ α (e, x) < 1} ⊆ U ; (a2) {x ∈ G : ρ α (e, x) = 0, α < τ } is a closed invariant subgroup of G;(a3) for any x and y in G, ρ α (e, xy) ≤ ρ α (e, x) + ρ α (e, y), where α < τ .Proof. From Lemma 7.29, it follows that there exists a family {U α : α < τ } of open neighborhoods of e in G satisfying conditions a)-d) of that lemma. By Lemma 6.4, we can find a family {N α : α < τ } of pre-continuous prenorms on G such that the following conditions are satisfied: a4) for each α < τ , there exists a subsequence {U αi : Theorem 7. 31 . 31Let G be an almost topological group with inv(G) ≤ τ . Then, for each open neighborhood U of the neutral element e in G, there exists a pre-continuous homomorphism π of G onto an almost topological group H with χ(H) ≤ τ such that π −1 (V ) ⊆ U for some open neighborhood V of the neutral element e H of H. Claim 1 : 1The definition of each d α (A, B) does not depend on the choice of a in A and b in B. Corollary 7 . 32 . 732Let G be an ω-narrow almost topological group. Then, for each open neighborhood U of the neutral element e in G, there exists a pre-continuous homomorphism π of G onto a secondcountable almost topological group H such that π −1 (V ) ⊆ U for some open neighborhood V of the neutral element e H of H.Proof. By Theorem 7.33, there exists a pre-continuous homomorphism π of G onto a first-countable almost topological group H and an open neighborhood V of the neutral element in H such that π −1 (V ) ⊆ U. Then it follows from Proposition 7.3 that H is ω-narrow, hence H is second-countable by Proposition 7.17. Corollary 7. 33 . 33Let G be an almost topological group with inv(G) ≤ ω. Then, for each open neighborhood U of the neutral element e in G, there exists a pre-continuous homomorphism π of G onto a first-countable almost topological group H such that π −1 (V ) ⊆ U for some open neighborhood V of the neutral element e H of H. Corollary 7 . 34 . 734Let G be a ω-narrow almost topological group. Then, for each open neighborhood U of the neutral element e in G, there exists a pre-continuous homomorphism π of G onto a second-countable almost topological group H such that π −1 (V ) ⊆ U for some open neighborhood V of the neutral element e H of H. Theorem 7.35. Each almost topological group G with inv(G) ≤ τ can be embedded as a subgroup into a pre-topological product of almost topological groups of character ≤ τ. Proof. Let B = {U i : i ∈ I} of all open neighborhoods of the neutral element e in G. By Theorem 7.33, for each i ∈ I there exists a pre-continuous homomorphism π i of G onto an almost topological group an arbitrary open neighborhood U of e in G. Hence there exists i ∈ I such that U i ⊆ U and then π −1 i (V i ) ⊆ U i by the choice of the open neighborhood V i of the neutral element in H i . Let p i be the projection of onto the factor H i . Then the set W = p −1 i (V i ) is an open neighborhood of the neutral element in . Obviously, we have Lemma 7 . 37 . 737For each α < τ , let H α be a pre-topological space with w(H α ) ≤ τ . Then the pre-topological product H = α<τ H α satisfies c(H) ≤ τ . Theorem 7.38. An almost topological group G is τ -narrow if and only if it can be embedded as a subgroup into an almost topological group H satisfying c(H) ≤ τ . Proposition 8 . 4 . 84Each feebly compact almost topological group is precompact. Proof. Take an arbitrary open symmetric neighborhood V of the neutral element e in G. Then there exists an open symmetric neighborhood U of e in G such that U 4 ⊆ V . From the proof of Theorem 7.14, we can find a maximal V -disjoint set A of G. From Lemma 3.22, it follows that the family of open sets {xU : Lemma 8 . 5 . 1 = 851Let B be a precompact subset of a pre-topological group G and D is dense in B. Then, for each open neighborhood U of the neutral element in G, there exists a finite set K ⊆ D such that B ⊆ KU and B ⊆ U K. Proof. Take an arbitrary open neighborhood U of the neutral element in G. Then there exists open neighborhoods V 1 and V 2 such that V 1 V 2 ⊆ U . Since B is precompact, we can find a finite subset F of G such that B ⊆ F V −1 1 and B ⊆ V −1 1 F . For each x ∈ F , if B ∩ xV −1 1 = ∅, then we can pick a point y x ∈ D ∩ xV −1 1 . Put K 1 = {y x : x ∈ F, B ∩ xV −1 ∅}. (iii) For each open neighborhood U of the neutral element in pre-ρG, we have U −1 ∈ σ. 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Fujian Key Laboratory of Granular Computing and Application, Minnan Normal University. Fucai Lin: 1. School of mathematics and statistics. Zhangzhou; Zhangzhou363000Minnan Normal UniversityChina Email address: [email protected]; [email protected] Lin: 1. School of mathematics and statistics, Minnan Normal University, Zhangzhou 363000, P. R. China; 2. Fujian Key Laboratory of Granular Computing and Application, Minnan Normal Uni- versity, Zhangzhou 363000, China Email address: [email protected]; [email protected] Ting Wu, address: [email protected]. School of mathematics and statistics. Zhangzhou363000Minnan Normal UniversityTing Wu: 1. School of mathematics and statistics, Minnan Normal University, Zhangzhou 363000, P. R. China Email address: [email protected] Yufan Xie: 1. School of mathematics and statistics. address: [email protected]. R. China Email. 363000Minnan Normal UniversityYufan Xie: 1. School of mathematics and statistics, Minnan Normal University, Zhangzhou 363000, P. R. China Email address: [email protected] Chengdu 610064, China Email address: mengbao95213@163. Meng Bao, College of Mathematics, Sichuan UniversityMeng Bao: College of Mathematics, Sichuan University, Chengdu 610064, China Email address: [email protected]	[]
[ "Homogenization and influence of fragmentation in a biological invasion model", "Homogenization and influence of fragmentation in a biological invasion model" ]	[ "Mohammad El Smaily \nLATP\nUniversité Aix-Marseille III\nFaculté des Sciences et Techniques Avenue Escadrille Normandie-NiemenF-13397Marseille Cedex 20France\n", "François Hamel \nLATP\nUniversité Aix-Marseille III\nFaculté des Sciences et Techniques Avenue Escadrille Normandie-NiemenF-13397Marseille Cedex 20France\n", "Lionel Roques \nUnité Biostatistique et Processus Spatiaux\nINRA\n\n" ]	[ "LATP\nUniversité Aix-Marseille III\nFaculté des Sciences et Techniques Avenue Escadrille Normandie-NiemenF-13397Marseille Cedex 20France", "LATP\nUniversité Aix-Marseille III\nFaculté des Sciences et Techniques Avenue Escadrille Normandie-NiemenF-13397Marseille Cedex 20France", "Unité Biostatistique et Processus Spatiaux\nINRA\n" ]	[]	In this paper, some properties of the minimal speeds of pulsating Fisher-KPP fronts in periodic environments are established. The limit of the speeds at the homogenization limit is proved rigorously. Near this limit, generically, the fronts move faster when the spatial period is enlarged, but the speeds vary only at the second order. The dependence of the speeds on habitat fragmentation is also analyzed in the case of the patch model.where the coefficients depend on the space variable x in a L-periodic fashion: 1 arXiv:0907.4951v1 [math.AP]	10.3934/dcds.2009.25.321	[ "https://arxiv.org/pdf/0907.4951v1.pdf" ]	15,043,422	0907.4951	43a37cc97c715b537a251032bb47fbc997994aa6	Homogenization and influence of fragmentation in a biological invasion model Mohammad El Smaily LATP Université Aix-Marseille III Faculté des Sciences et Techniques Avenue Escadrille Normandie-NiemenF-13397Marseille Cedex 20France François Hamel LATP Université Aix-Marseille III Faculté des Sciences et Techniques Avenue Escadrille Normandie-NiemenF-13397Marseille Cedex 20France Lionel Roques Unité Biostatistique et Processus Spatiaux INRA Homogenization and influence of fragmentation in a biological invasion model Domaine St Paul -Site Agroparc 84914 Avignon Cedex 9, France Dedicated to Professor Masayasu Mimura for his 65th birthday In this paper, some properties of the minimal speeds of pulsating Fisher-KPP fronts in periodic environments are established. The limit of the speeds at the homogenization limit is proved rigorously. Near this limit, generically, the fronts move faster when the spatial period is enlarged, but the speeds vary only at the second order. The dependence of the speeds on habitat fragmentation is also analyzed in the case of the patch model.where the coefficients depend on the space variable x in a L-periodic fashion: 1 arXiv:0907.4951v1 [math.AP] Introduction and main hypotheses In homogeneous environments, the probably most used population dynamics reactiondiffusion model is the Fisher-KPP model [13,23]. In a one-dimensional space, it corresponds to the following equation ∂u ∂t = D ∂ 2 u ∂x 2 + u (µ − νu), t > 0, x ∈ R. (1.1) The unknown u = u(t, x) is the population density at time t and position x, and the positive constant coefficients D, µ and ν respectively correspond to the diffusivity (mobility of the individuals), the intrinsic growth rate and the susceptibility to crowding effects. A natural extension of this model to heterogeneous environments is the Shigesada-Kawasaki-Teramoto model [32], ∂u ∂t = ∂ ∂x a L (x) ∂u ∂x + u (µ L (x) − ν L (x)u), t > 0, x ∈ R,(1. 2) Definition 1.1 (L-periodicity) Let L be a positive real number. We say that a function h : R → R is L-periodic if ∀ x ∈ R, h(x + L) = h(x). In this paper, we are concerned with the general equation: ∂u ∂t = ∂ ∂x a L (x) ∂u ∂x + f L (x, u), t ∈ R, x ∈ R. (1.3) The diffusion term a L satisfies a L (x) = a(x/L), where a is a C 2,δ (R) (with δ > 0) 1-periodic function that satisfies ∃ 0 < α 1 < α 2 , ∀ x ∈ R, α 1 ≤ a(x) ≤ α 2 . (1.4) On other hand, the reaction term satisfies f L (x, ·) = f (x/L, ·), where f := f (x, s) : R×R + → R is 1-periodic in x, of class C 1,δ in (x, s) and C 2 in s. In this setting, both a L and f L are L-periodic in the variable x. Furthermore, we assume that: The growth rate µ may be positive in some regions (favorable regions) or negative in others (unfavorable regions). The stationary states p(x) of (1.3) satisfy the equation    ∀ x ∈ R, f (x, 0) = 0, ∃ M ≥ 0, ∀ s ≥ M, ∀ x ∈ R, f (x, s) ≤ 0, ∀ x ∈ R, s → f (x,∂ ∂x a L (x) ∂p ∂x + f L (x, p) = 0, x ∈ R. (1.6) Under general hypotheses including those of this paper, and in any space dimension, it was proved in [4] that a necessary and sufficient condition for the existence of a positive and bounded solution p of (1.6) was the negativity of the principal eigenvalue ρ 1,L of the linear operator L 0 : Φ → −(a L (x)Φ ) − µ L (x)Φ,(1.7) with periodicity conditions. In this case, the solution p was also proved to be unique, and therefore L-periodic. Actually, it is easy to see that the map L → ρ 1,L is nonincreasing in L > 0, and even decreasing as soon as a is not constant (see the proof of Lemma 3.1). Furthermore, ρ 1,L → − 1 0 µ(x)dx as L → 0 + . In this paper, in addition to the abovementioned hypotheses, we make the assumption that 1 0 µ(x)dx > 0. (1.8) This assumption then guarantees that ∀ L > 0, ρ 1,L < 0, whence, for all L > 0, there exists a unique positive periodic and bounded solution p L of (1.6). Notice that assumption (1.8) is immediately fulfilled if µ(x) is positive everywhere. In this work, we are concerned with the propagation of pulsating traveling fronts which are particular solutions of the reaction-diffusion equation (1.3). Before going further on, we recall the definition of such solutions: Definition 1.2 (Pulsating traveling fronts) A function u = u(t, x) is called a pulsating traveling front propagating from right to left with an effective speed c = 0, if u is a classical solution of: where the above limits hold locally in t.                ∂u ∂t = ∂ ∂x a L (x) ∂u ∂x + f L (x, u), t ∈ R, x ∈ R, ∀ k ∈ Z, ∀ (t, x) ∈ R × R, u(t + kL c , x) = u(t, x + kL), 0 ≤ u(t, x) ≤ p L (x), This definition has been introduced in [31,32]. It has also been extended in higher dimensions with p L ≡ 1 in [1] and [35], and with p L ≡ 1 in [5]. Under the above assumptions, it follows from [5] that there exists c * L > 0 such that pulsating traveling fronts satisfying (1.9) with a speed of propagation c exist if and only if c ≥ c * L . Moreover, the pulsating fronts (with speeds c ≥ c * L ) are increasing in time t. Further uniqueness and qualitative properties are proved in [14,15]. The value c * L is called the minimal speed of propagation. We refer to [2,3,11,18,25,27,28,34] for further existence results and properties of the minimal speeds of KPP pulsating fronts. For existence, uniqueness, stability and further qualitative results for combustion or bistable nonlinearities in the periodic framework, we refer to [6,7,12,16,17,19,24,26,35,36,37,38]. In the particular case of the Shigesada et al model (1.2), when a(x) ≡ 1, the effects of the spatial distribution of the function µ L on the existence and global stability of a positive stationary state p L of equation (1.2) have been investigated both numerically [30,31] and theoretically [4,8,29]. In particular, as already noticed, enlarging the scale of fragmentation, i.e. increasing L, was proved to decrease the value of ρ 1,L . Biologically, this result means that larger scales have a positive effect on species persistence, for species whose dynamics is modelled by the Shigesada et al model. The effects of the spatial distribution of the functions a L and µ L on the minimal speed of propagation c * L have not yet been investigated rigorously. This is a difficult problem, since the known variational formula for c * L bears on non-self-adjoint operators, and therefore, the methods used to analyze the dependence of ρ 1,L on fragmentation cannot be used in this situation. However, in the case of model (1.2), when a L ≡ 1, ν L ≡ 1 and µ L (x) = µ(x/L), for a 1-periodic function µ taking only two values, Kinezaki et al [22] numerically observed that c * L was an increasing function of the parameter L. For sinusoidally varying coefficients, the relationships between c * L and L have also been investigated formally by Kinezaki, Kawasaki, Shigesada [21]. The case of a rapidly oscillating coefficient a L (x), corresponding to small L values, and the homogenization limit L → 0, have been discussed in [19] and [38] for combustion and bistable nonlinearities f (u). The first aim of our work is to analyze rigorously the dependence of the speed of propagation c * L with respect to L, under the general setting of equation (1.3), for small L values. We determine the limit of the minimal speeds c * L as L → 0 + (the homogenization limit), and we also prove that near the homogenization limit, the species tends to propagate faster when the spatial period of the environment is enlarged. Next, in the case of an environment composed of patches of "habitat" and "non-habitat", we consider the dependence of the minimal speed with respect to habitat fragmentation. We prove that fragmentation decreases the minimal speed. Main results In this section, we describe the main results of this paper. Unless otherwise mentioned, we make the assumptions of Section 1. The first theorem gives the limit of c * L as L goes to 0. Theorem 2.1 Let c * L be the minimal speed of propagation of pulsating traveling fronts solving (1.9). Then, lim L→0 + c * L = 2 √ < a > H < µ > A , (2.1) where < µ > A = 1 0 µ(x)dx and < a > H = 1 0 (a(x)) −1 dx −1 = < a −1 > −1 A denote the arithmetic mean of µ and the harmonic mean of a over the interval [0, 1]. Formula (2.1) was derived formally in [33] for sinusoidally varying coefficients. Theorem 2.1 then provides a generalization of the formula in [33] and a rigorous analysis of the homogenization limit for general diffusion and growth rate profiles. Remark 2.2 The previous theorem gives the limit of c * L as L → 0 when the space dimension is 1. Theorem 3.3 of El Smaily [11] answered this issue in any dimensions N , but under an additional assumption of free divergence of the diffusion field (in the one-dimensional case considered here, this assumption reduces to da/dx = 0 in R). Lastly, we refer to [6,7,16] for other homogenization limits with combustion-type nonlinearities. Our second result describes the behavior of the function L → c * L , for small L values. Theorem 2.3 Let c * L be the minimal speed of propagation of pulsating traveling fronts solving (1.9). Then, the map L → c * L is of class C ∞ in an interval (0, L 0 ) for some L 0 > 0. Furthermore, lim L→0 + dc * L dL = 0 (2.2) and lim L→0 + d 2 c * L dL 2 = γ ≥ 0. (2.3) Lastly, γ > 0 if and only if the function µ < µ > A + < a > H a is not identically equal to 2. Corollary 2.4 Under the notations of Theorem 2.3, it follows that if a is constant and µ is not constant, or if µ is constant and a is not constant, then γ > 0 and the speeds c * L are increasing with respect to L when L is close to 0. Remark 2.5 The question of the monotonicity of the map L → c * L had also been studied under different assumptions in [11] (see Theorem 5.3). The author answered this question for a reaction-advection-diffusion equation over a periodic domain Ω ⊆ R N , under an additional assumption on the diffusion coefficient (like in Remark 2.2, this assumption would mean again in our present setting that the diffusion coefficient a(x) is constant over R). Our result gives the behavior of the minimal speeds of propagation near the homogenization limit for general diffusion and growth rate coefficients. The condition γ > 0 is generically fulfilled, which means that, roughly speaking, the more oscillating the medium is, the slower the species moves. But the speeds vary only at the second order with respect to the period L. Based on numerical observations which have been carried out in [21] for special types of diffusion and growth rate coefficients, we conjecture that the monotonicity of c * L holds for all L > 0. Lastly, we give a first theoretical evidence that habitat fragmentation, without changing the scale L, can decrease the minimal speed c * . We here fix a period L 0 > 0. We assume that a ≡ 1, and that µ L 0 := µ z takes only the two values 0 and m > 0, and depends on a parameter z. More precisely:    There exist 0 ≤ z and l ∈ (0, L 0 ) such that l + z ≤ L 0 , µ z ≡ m on [0, l/2) ∪ [l/2 + z, l + z), µ z ≡ 0 on [l/2, l/2 + z) ∪ [l + z, L 0 ). (2.4) With this setting, the region where µ z is positive, which can be interpreted as "habitat" in the Shigesada et al model, is of Lebesgue measure l in each period cell [0, L 0 ]. For z = 0, this region is simply an interval. However, whenever z is positive, this region is fragmented into two parts of same length l/2 (see Figure 1). Our next result means that this fragmentation into two parts reduces the speed c * . Remark 2.7 Note that, whenever z > (L 0 − l)/2, the two habitat components in the period cell [l/2 + z, L 0 + l/2 + z] are at a distance smaller than (L 0 − l)/2 from each other. In fact, Theorem 2.6 proves that, when z varies in (0, L 0 − l), c * z is all the larger as the minimal distance separating two habitat components is small, that is as the maximal distance between two consecutive habitat components is large. Remark 2.8 Here, the function µ z does not satisfy the general regularity assumptions of Section 1. However, c * z can still be interpreted as the minimal speed of propagation of weak solutions of (1.9), whose existence can be obtained by approaching µ z with regular functions. The main tool of this paper is a variational formulation for c * L involving elliptic eigenvalue problems which depend strongly on the coefficients a and f. Such a formulation was given in any space dimension in [3] in the case where the bounded stationary state p of the equation (1.3) is constant, and in [5] in the case of a general nonconstant bounded stationary state p(x). The homogenization limit: proof of Theorem 2.1 This proof is divided into three main steps. Step 1: a rough upper bound for c * L . For each L > 0, the minimal speed c * L is positive and, from [5] (see also [3] in the case when p ≡ 1), it is given by the variational formula c * L = min λ>0 k(λ, L) λ = k(λ * L , L) λ * L , (3.1) where λ * L > 0 and, for each λ ∈ R and L > 0, k(λ, L) denotes the principal eigenvalue of the problem a L ψ λ,L + 2λa L ψ λ,L + λa L ψ λ,L + λ 2 a L ψ λ,L + µ L ψ λ,L = k(λ, L)ψ λ,L in R, (3.2) with L-periodicity conditions. In (3.2), ψ λ,L denotes a principal eigenfunction, which is of class C 2,δ (R), positive, unique up to multiplication by a positive constant, and L-periodic. Furthermore, it follows from Section 3 of [5] that the map λ → k(λ, L) is convex and that ∂k ∂λ (0, L) = 0 for each L > 0. Therefore, for each L > 0, the map λ → k(λ, L) is nondecreasing in R + and ∀ λ ≥ 0, ∀ L > 0, k(λ, L) ≥ k(0, L) = −ρ 1,L > 0 (3.3) under the notations of Section 1. Multiplying (3.2) by ψ λ,L and integrating by parts over [0, L], we get, due to the Lperiodicity of a L and ψ λ,L : k(λ, L) L 0 ψ 2 λ,L = − L 0 a L ψ λ,L 2 + λ 2 L 0 a L ψ 2 λ,L + L 0 µ L ψ 2 λ,L , for all λ > 0 and for all L > 0. Consequently, ∀ λ > 0, ∀ L > 0, k(λ, L) ≤ λ 2 a M + µ M , (3.4) where a M = max x∈R a(x) > 0 and µ M = max x∈R µ(x) > 0. Using (3.1), we get that ∀ L > 0, 0 < c * L ≤ 2 √ a M µ M . (3.5) Step 2: the sharp upper bound for c * L . For any λ > 0 and L > 0, consider the functions ϕ λ,L (x) := e λ x ψ λ,L (x), x ∈ R. Since ψ λ,L is unique up to multiplication, we will assume in this step 2 that 2 0 ϕ 2 λ,L (x)dx = 1. (3.6) The above choice ensures that 2 0 ψ 2 λ,L (x)dx ≤ 2 0 e 2λx ψ 2 λ,L (x)dx = 2 0 ϕ 2 λ,L (x)dx = 1. (3.7) We are now going to prove that the families (ψ λ,L ) λ,L and (ϕ λ,L ) λ,L remain bounded in H 1 (0, 1) for L small enough and as soon as λ stays bounded. For each L > 0, we call M L = [1/L] + 1 ∈ N, where [1/L] stands for the integer part of 1/L. Multiplying (3.2) by ψ λ,L and integrating by parts over [0, M L L], we get that − M L L 0 a L ψ λ,L 2 + M L L 0 λ 2 a L ψ 2 λ,L + M L L 0 µ L ψ 2 λ,L = k(λ, L) M L L 0 ψ 2 λ,L . Using (1.4), (3.3) and (3.4), it follows that 0 ≤ M L L 0 ψ λ,L 2 ≤ 1 α 1 × λ 2 a M + µ M × M L L 0 ψ 2 λ,L . Since 1 < M L L ≤ 1 + L for all L > 0, we have that 1 < M L L ≤ 2 for all L ≤ 1. Thus, for all 0 < L ≤ 1, 1 0 ψ λ,L 2 ≤ M L L 0 ψ λ,L 2 and M L L 0 ψ 2 λ,L ≤ 2 0 ψ 2 λ,L ≤ 1 from (3.7) . It follows now that ∀ λ > 0, ∀ 0 < L ≤ 1, 1 0 ψ λ,L 2 ≤ λ 2 a M + µ M α 1 . (3.8) From (3.7) and (3.8), we conclude that, for any given Λ > 0, the family (ψ λ,L ) 0<λ≤Λ, 0<L≤1 is bounded in H 1 (0, 1). On the other hand, ϕ λ,L (x) = λϕ λ,L (x) + e λx ψ λ,L (x). Owing to (3.6) and (3.8), we get: ∀ λ > 0, ∀ L ≤ 1, \|\|ϕ λ,L \|\| L 2 (0,1) ≤ λ \|\|ϕ λ,L \|\| L 2 (0,1) ≤1 +e λ \|\|ψ λ,L \|\| L 2 (0,1) ≤ λ + e λ × λ 2 a M + µ M α 1 . (3.9) From (3.6) and (3.9), we obtain that, for any given Λ > 0, the family (ϕ λ,L ) 0<λ≤Λ, 0<L≤1 is bounded in H 1 (0, 1) and that the family (a L ϕ λ,L ) 0<λ≤Λ, 0<L≤1 is bounded in L 2 (0, 1) (due to (1.4)). Moreover, a L ϕ λ,L = λ 2 a L e λ x ψ λ,L + 2λa L e λ x ψ L + λa L e λ x ψ λ,L + e λ x a L ψ λ,L + e λ x a L ψ λ,L . Multiplying (3.2) by e λ x , we then get a L ϕ λ,L + µ L ϕ λ,L = k(λ, L)ϕ λ,L in R. (3.10) Let v λ,L (x) = a L (x)ϕ λ,L (x) for all λ > 0, L > 0 and x ∈ R. Pick any Λ > 0. One already knows that the family (v λ,L ) 0<λ≤Λ, 0<L≤1 is bounded in L 2 (0, 1). Furthermore, v λ,L + µ L ϕ λ,L = k(λ, L)ϕ λ,L in R. (3.11) Notice that the family (k(λ, L)) 0<λ≤Λ, 0<L≤1 is bounded from (3.3) and (3.4). From (3.6) and (3.11), it follows that the family v λ,L 0<λ≤Λ, 0<L≤1 is bounded in L 2 (0, 1). Eventually, (v λ,L ) 0<λ≤Λ, 0<L≤1 is bounded in H 1 (0, 1). Pick now any sequence (L n ) n∈N such that 0 < L n ≤ 1 for all n ∈ N, and L n → 0 + as n → +∞. Choose any λ > 0 and any sequence (λ n ) n∈N of positive numbers such that λ n → λ as n → +∞. We claim that k(λ n , L n ) → λ 2 < a > H + < µ > A as n → +∞, (3.12) where < a > H = 1 0 (a(x)) −1 dx and < µ > A = 1 0 µ(x)dx. To do so, call ψ n = ψ λn,Ln , ϕ n = ϕ λn,Ln and v n = v λn,Ln . It follows from the above computations that the sequences (ψ n ) and (v n ) are bounded in H 1 (0, 1). Hence, up to extraction of a subsequence, ψ n → ψ and v n → w as n → +∞, strongly in L 2 (0, 1) and weakly in H 1 (0, 1). By Sobolev injections, the sequence (ψ n ) is bounded in C 0,1/2 ([0, 1]). But since each function ψ n is L n -periodic (with L n → 0 + ), it follows from Arzela-Ascoli theorem that ψ has to be constant over [0, 1]. Moreover, the boundedness of the sequence (k(λ n , L n )) n∈N implies that, up to extraction of another subsequence, k(λ n , L n ) → k(λ) ∈ R as n → +∞. We denote this limit by k(λ), we will see later that indeed it depends only on λ. It follows now, from (3.11) after replacing (λ, L) by (λ n , L n ) and passing to the limit as n → +∞, that w + < µ > A e λx ψ = k(λ)ψe λx a.e. in (0, 1). Notice indeed that µ L < µ > A as L → 0 + in L 2 (0, 1) weakly. Meanwhile, ϕ n = λ n e λnx ψ n + e λnx ψ n = v n a Ln < a −1 > A w as n → +∞, weakly in L 2 (0, 1), where < a −1 > A = 1 0 (a(x)) −1 dx. Thus, we obtain w =< a −1 > −1 A λe λx ψ =< a > H λe λx ψ. Consequently, λ 2 < a > H ψ+ < µ > A ψ = k(λ)ψ. Actually, since the functions ψ n are L n -periodic (with L n → 0 + ) and converge to the constant ψ strongly in L 2 (0, 1), they converge to ψ in L 2 loc (R). But 1 = 2 0 ϕ 2 n ≤ e 4λn 2 0 ψ 2 n ≤ e 4M 2 0 ψ 2 n , where M = sup n∈N λ n . Hence, ψ = 0 and λ 2 < a > H + < µ > A = k(λ). (3.13) By uniqueness of the limit, one deduces that the whole sequence (k(λ n , L n )) n∈N converges to this quantity k(λ) as n → +∞, which proves the claim (3.12). Now, take any sequence L n → 0 + such that c * Ln → lim sup L→0 + c * L as n → +∞. For each λ > 0 and for each n ∈ N, one has c * Ln ≤ k(λ, L n ) λ from (3.1), whence lim sup L→0 + c * L = lim n→+∞ c * Ln ≤ k(λ) λ = λ < a > H + < µ > A λ . Since this holds for all λ > 0, one concludes that lim sup L→0 + c * L ≤ 2 √ < a > H < µ > A . (3.14) Step 3: the sharp lower bound for c * L . The aim of this step is to prove that lim inf L→0 + c * L ≥ 2 √ < a > H < µ > A which would complete the proof of Theorem 2.1. For each L > 0, the minimal speed c * L is given by (3.1) and the map (0, +∞) λ → k(λ, L)/λ attains its minimum at λ * L > 0.We will prove that, for L small enough, the family (λ * L ) is bounded from above and from below by λ > 0 and λ > 0 respectively. Namely, one has Lemma 3.1 There exist L 0 and 0 < λ ≤ λ < +∞ such that λ ≤ λ * L ≤ λ for all 0 < L ≤ L 0 . The proof is postponed at the end of this section. Take now any sequence (L n ) n such that 0 < L n ≤ L 0 for all n, and L n → 0 + as n → +∞. From Lemma 3.1, there exists λ * > 0 such that, up to extraction of a subsequence, λ * Ln → λ * as n → +∞. One also has (3.12) and (3.13). c * Ln = k(λ * Ln , L n ) λ * Ln → n→+∞ k(λ * ) λ * = λ * < a > H + < µ > A λ * ≥ 2 √ < a > H < µ > A fromTherefore, lim inf L→0 + c * L ≥ 2 √ < a > H < µ > A . Eventually, lim L→0 + c * L = 2 √ < a > H < µ > A and the proof of Theorem 2.1 is complete. Proof of Lemma 3.1. Observe first that, for λ = 0 and for any L > 0, k(0, L) is the principal eigenvalue of the problem (a L φ L ) + µ L φ L = k(0, L)φ L in R, and we denote φ L = ψ 0,L a principal eigenfunction, which is L-periodic, positive and unique up to multiplication. In other words, k(0, L) = −ρ 1,L under the notations of Section 1. Dividing the above elliptic equation by φ L and integrating by parts over [0, L], one gets k(0, L) = 1 L L 0 a L φ L 2 φ 2 L + 1 0 µ(x)dx ≥ < µ > A > 0. On the other hand, as already recalled, ∂k ∂λ (0, L) = 0 and the map λ → k(λ, L) is convex for all L > 0. Therefore, ∀ λ > 0, ∀ L > 0, k(λ, L) ≥ k(0, L) ≥ < µ > A > 0. Assume here that there exists a sequence (L n ) n∈N of positive numbers such that L n → 0 + and λ * Ln → 0 + as n → +∞. One then gets c * Ln = k(λ * Ln , L n ) λ * Ln ≥ < µ > A λ * Ln → +∞ as n → +∞. This is contradiction with (3.14). Thus, for L > 0 small enough, the family (λ * L ) L is bounded from below by a positive constant λ > 0 (actually, these arguments show that the whole family (λ * L ) L>0 is bounded from below by a positive constant). It remains now to prove that (λ * L ) L is bounded from above when L is small enough. We assume, to the contrary, that there exists a sequence L n → 0 + as n → +∞ such that λ * Ln → +∞ as n → +∞. Call k n = k(λ * Ln , L n ), ψ n (x) = ψ λ * Ln ,Ln (x) and ϕ n (x) = ϕ λ * Ln ,Ln (x) = e λ * Ln x ψ n (x) for all n ∈ N and x ∈ R. Rewriting (3.10) for λ = λ * Ln and for L = L n , one consequently gets ∀ n ∈ N, (a Ln ϕ n ) + µ Ln ϕ n = k n ϕ n in R. But, for each n ∈ N, M Ln ∈ N while a Ln and ψ n are L n -periodic. Hence, a Ln (θ n + M Ln L n ) = a Ln (θ n ), ψ n (θ n + M Ln L n ) = ψ n (θ n ), and ψ n (θ n + M Ln L n ) = ψ n (θ n ) = 0. Then, 0 is given by (1.4)), (3.17) whenever n is large enough so that 2 ≤ e 2λ * Ln M Ln Ln (remember that λ * Ln → +∞ as n → +∞, by assumption). Meanwhile, for all n ∈ N, since ψ n is L n -periodic. One has A(n) = a Ln (θ n )λ * Ln ψ 2 n (θ n ) e 2λ * Ln (θn+M Ln Ln) − e 2λ * Ln θn ≥ α 1 2 × λ * Ln ψ 2 n (θ n )e 2λ * Ln (θn+M Ln Ln) (α 1 >\|C(n)\| ≤ θn+M Ln Ln θn µ( x L n ) e 2λ * Ln x ψ 2 n (x)dx ≤ µ ∞ × ψ 2 n (θ n ) 2λ * Ln × e 2λ * Ln (θn+M Ln Ln) ,(3.Ln 0 ψ n (x) + λ * Ln ψ n (x) 2 dx ≤ ψ 2 n (θ n ) Ln 0 ψ n (x) ψ n (x) + λ * Ln 2 dx. We refer now to equation (3.2). Taking λ = λ * Ln , dividing this equation (3.2) by the L nperiodic function ψ n and then integrating by parts over the interval [0, L n ], we get µ Ln >0 = k n L n ≤ 2 √ a M µ M × λ * Ln L n . Owing to (1.4), it follows that ∀ n ∈ N, Ln 0 ψ n ψ n + λ * Ln 2 ≤ 2 √ a M µ M α 1 × λ * Ln L n . Putting the above result into B(n), we obtain, for all n ∈ N, B(n) ≤ 2α 2 √ a M µ M α 1 × λ * Ln L n ψ 2 n (θ n ) M Ln −1 j=0 e 2λ * Ln (θn+(j+1)Ln) = 2α 2 √ a M µ M α 1 × λ * Ln L n ψ 2 n (θ n )e 2λ * Ln (θn+Ln) × e 2λ * Ln LnM Ln − 1 e 2λ * Ln Ln − 1 ≤ 2α 2 √ a M µ M α 1 × ψ 2 n (θ n ) × λ * Ln L n e 2λ * Ln Ln e 2λ * Ln Ln − 1 × e 2λ * Ln (θn+M Ln Ln) ≤ β × ψ 2 n (θ n )e 2λ * Ln (θn+M Ln Ln) × λ * Ln L n + 1 ,(3.20) where β = 2α 2 √ a M µ M /α 1 × C and C is a positive constant such that ∀x ≥ 0, xe 2x e 2x − 1 ≤ C × (x + 1). Lastly, let us rewrite equation ( Together with (3.17), (3.18), (3.19) and (3.20), one concludes that there exists n 0 ∈ N such that for n ≥ n 0 , α 1 2 × λ * Ln ψ 2 n (θ n )e 2λ * Ln (θn+M Ln Ln) − µ ∞ × ψ 2 n (θ n ) 2λ * Ln × e 2λ * Ln (θn+M Ln Ln) − √ a M µ M × ψ 2 n (θ n )e 2λ * Ln (θn+M Ln Ln) ≤ β × ψ 2 n (θ n )e 2λ * Ln (θn+M Ln Ln) × λ * Ln L n + 1 . (3.21) Divide (3.21) by λ * Ln ψ 2 n (θ n )e 2λ * Ln (θn+M Ln Ln) . Then ∀ n ≥ n 0 , α 1 2 − µ ∞ 2(λ * Ln ) 2 − √ a M µ M λ * Ln ≤ β × L n + 1 λ * Ln . Passing to the limit as n → +∞, one has L n → 0 + and λ * Ln → +∞, whence α 1 ≤ 0, which is impossible. Therefore the assumption that λ * Ln → +∞ as L n → 0 + is false and consequently the family (λ * L ) L is bounded from above by some positive λ > 0 whenever L is small (i.e. 0 < L ≤ L 0 ). This completes the proof of Lemma 3.1. Remark 3.2 From Theorem 2.1, one concludes that the map (0, +∞) L → c * L can be extended by continuity to the right at L = 0 + . Furthermore, for any sequence (L n ) n of positive numbers such that L n → 0 + as n → +∞, one claims that the positive numbers λ * Ln given in (3.1) converge to < a > −1 H < µ > A = < a −1 > A < µ > A as n → +∞. Indeed ∀ n ∈ N, c * Ln = k(λ * Ln , L n ) λ * Ln and Lemma 3.1 implies that, up to extraction of a subsequence, λ * Ln → λ * > 0. Passing to the limit as n → +∞ in the above equation and due (3.13) together with Step 2 of the proof of Theorem 2.1, one gets 2 √ < a > H < µ > A = k(λ * ) λ * = λ * < a > H + < µ > A λ * , whence λ * = < a > −1 H < µ > A . Since the limit does not depend on any subsequence, one concludes that the limit of λ * L , as L → 0 + , exits and lim L→0 + λ * L = < a > −1 H < µ > A = < a −1 > A < µ > A . The sharp lower bound of lim inf L→0 + c * L from the homogenized equation. In the following, we are going to derive the homogenized equation of (1.3), which will lead to the sharp lower bound of lim inf L→0 + c * L . However, to furnish this goal we will only consider for the sake of simplicity a particular type of nonlinearities among those satisfying (1.5). In fact, the following ideas can be generalized to a wider family of nonlinearities which satisfy (1.5), but the proof requires technical extra-arguments which will be the purpose of a forthcoming paper. For each L > 0, let u L be a pulsating travelling front with minimal speed c * L for the reaction-diffusion equation              ∂u L ∂t = ∂ ∂x a L (x) ∂ u L ∂x + µ( x L )g(u L ), t ∈ R, x ∈ R, ∀(t, x) ∈ R × R, 0 < u L (t + L c * L , x) = u L (t, x + L) < 1, lim x→−∞ u L (t, x) = 0 and lim x→+∞ u L (t, x) = 1, (3.22) where a L (x) = a(x/L), a is a C 2,δ (R) 1-periodic function satisfying (1.4), µ is a C 1,δ (R) positive 1-periodic function and g is a C 2 (R + ) function such that g(0) = g(1) = 0 and u → g(u)/u is decreasing in (0, +∞). Up to a shift in time, one can assume that ∀L > 0, (0,1)×(0,1) u L (t, x) dt dx = 1 2 . (3.23) For each L > 0, set f L (x, u) := f (x/L, u) = µ(x/L)g(u). In this setting, there holds p L ≡ 1. From standard parabolic estimates, each function u L is (at least) of class C 2 (R × R). Denote v L (t, x) = a L (x) ∂u L ∂x (t, x) and w L (t, x) = ∂u L ∂t (t, x) in R × R. As already underlined, it follows from [1] that w L = ∂u L ∂t > 0 in R × R for each L > 0. Under the notations of the beginning of this section, it follows from (1.4) and (3.2) that k(λ, L) ≥ λ 2 α 1 + µ m for all L > 0 and λ ∈ R, where µ m = min R µ > 0. Hence, c * L ≥ 2 √ α 1 µ m for each L > 0 and lim inf L→0 + c * L ≥ 2 √ α 1 µ m > 0. We shall now establish some estimates for the functions u L , v L and w L which are independent of L, in order to pass to the limit as L → 0 + . Notice first that standard parabolic estimates and the (t, x)-periodicity satisfied by the functions u L imply that, for each L > 0, u L (−∞, x) = 0 and u L (+∞, x) = 1 in C 2 loc (R), and w L (±∞, x) = 0 in C 1 loc (R). Let k ∈ N\{0} be given. Integrating the first equation of (3.22) by parts over R × (−kL, kL), one obtains ∀L > 0, R×(−kL,kL) f ( x L , u L ) dt dx = 2kL. (3.24) Multiplying the first equation of (3.22) by u L and integrating by parts over R × (−kL, kL), one then gets ∀L > 0, kL = − R×(−kL,kL) a L (x) ∂u L ∂x 2 dt dx + R×(−kL,kL) f ( x L , u L )u L dt dx. (3.25) Notice that the last integral in (3.25) converges because of (3.24) and 0 ≤ f (x/L, u L )u L ≤ f (x/L, u L ). Together with (1.4), one concludes that for each L > 0, the first integral in (3.25) converges and ∀L > 0, R×(−kL,kL) ∂u L ∂x 2 dt dx ≤ kL α 1 . Multiply the first equation of (3.22) by ∂u L ∂t and integrate by parts over R × (−kL, kL). Since R×(−kL,kL) ∂ ∂x a L (x) ∂u L ∂x ∂u L ∂t = − 1 2 R×(−kL,kL) ∂ ∂t a L (x) ∂u L ∂x 2 = 0, one obtains that ∀L > 0, R×(−kL,kL) ∂u L ∂t 2 dt dx = (−kL,kL) F ( x L , 1)dx = 2kL × 1 0 µ × 1 0 g, (3.26) where F (y, s) = s 0 f (y, τ )dτ . It follows from the above estimates that for each compact subset K of R, ∀ 0 < L < 1, R×K ∂u L ∂t 2 + ∂u L ∂x 2 dt dx ≤ C(K), (3.27) where C(K) is a positive constant depending only on K. In particular, for each compact K of R and for each L > 0, \|\|w L \|\| L 2 (R×K) ≤ C(K). Now, differentiate the first equation of (3.22) with respect to t (actually, from the regularity of f , the function w L is of class C 2 with respect to x). There holds ∂w L ∂t = ∂ ∂x a L (x) ∂ w L ∂x + µ( x L )g (u L )w L in R × R. Multiply the above equation by w L and integrate by parts over R × (−kL, kL). From (1.4) and (3.26), it follows that R×(−kL,kL) ∂w L ∂x 2 dtdx ≤ 2kLη α 1 where η is the positive constant defined by η = max x∈R µ(x) max u∈[0,1] \|g (u)\| max x∈R \|F (x, 1)\| ≥ 1 2kL R×(−kL,kL) µ( x L )g (u L )w 2 L dt dx > 0. Then, for each compact K ⊂ R, there exists a constant C (K) > 0 depending only on K such that ∀ 0 < L < 1, R×K ∂w L ∂x 2 dt dx ≤ C (K). (3.28) Let (L n ) n∈N be a sequence of real numbers in (0, 1) such that L n → 0 and c * Ln → lim inf L→0 + c * L > 0 as n → +∞. It follows from (3.27) and the bounds 0 < u Ln < 1 that there exists u 0 in H 1 loc (R×R) such that, up to extraction of a subsequence, u Ln → u 0 strongly in L 2 loc (R × R) and almost everywhere in R × R, and ∂u Ln ∂t , ∂u Ln ∂x ∂u 0 ∂t , ∂u 0 ∂x weakly in L 2 loc (R × R) as n → +∞. Remember that v Ln = a Ln ∂u Ln ∂x and 0 < α 1 ≤ a Ln ≤ α 2 for each n ∈ N. Thus, (3.27) yields that for each compact K of R and for each n ∈ N, \|\|v Ln \|\| L 2 (R×K) ≤ α 2 C(K). Furthermore, (3.22) implies that ∀n ∈ N, ∂v Ln ∂x = ∂u Ln ∂t − f ( x L n , u Ln ) in R × R, while 0 ≤ f (x/L n , u Ln (t, x)) ≤ κ in R × R where κ = max R µ × max [0,1] g > 0 is independent of n.v 0 ∈ H 1 loc (R × R) such that v Ln → v 0 strongly in L 2 loc (R × R) and ∂v Ln ∂t , ∂v Ln ∂x ∂v 0 ∂t , ∂v 0 ∂x weakly in L 2 loc (R × R) as n → +∞. However, a −1 Ln < a −1 > A =< a > −1 H in L ∞ (R) weak- * as n → +∞. Thus, ∂u Ln ∂x = v Ln a Ln v 0 < a > H weakly in L 2 loc (R × R) as n → +∞. By uniqueness of the limit, one gets v 0 =< a > H ∂u 0 ∂x . Passing to the limit as n → +∞ in the first equation of (3.22) with L = L n implies that u 0 is a weak solution of the equation ∂u 0 ∂t = ∂v 0 ∂x + < µ > A g(u 0 ) =< a > H ∂ 2 u 0 ∂x 2 + < µ > A g(u 0 ) in D (R × R). From parabolic regularity, the function u 0 is then a classical solution of the homogenous equation ∂u 0 ∂t =< a > H ∂ 2 u 0 ∂x 2 + < µ > A g(u 0 ) in R × R, such that 0 ≤ u 0 ≤ 1 and ∂u 0 ∂t ≥ 0 in R × R. Lastly, (0,1) 2 u 0 (t, x) dt dx = 1 2 from (3.23) . On the other hand, it follows from the second equation of (3.22) and (3.27) that ∀γ ∈ R, u 0 (t + γ c , x) = u 0 (t, x + γ) in R × R, where c = lim inf L→0 + c * L = lim n→+∞ c * Ln > 0. In other words, u 0 (t, x) = U 0 (x + ct), where U 0 is a classical solution of the equation cU 0 =< a > H U 0 + < µ > A g(U 0 ), 0 ≤ U 0 ≤ 1 in R (3.29) that satisfies U 0 ≥ 0 in R and Standard elliptic estimates imply that U 0 converges as s → ±∞ in C 2 loc (R) to two constants U ± 0 ∈ [0, 1] such that < µ > A g(U ± 0 ) = 0, that is g(U ± 0 ) = 0. The monotonicity of U 0 and the assumption on g imply that U − 0 = 0 and U + 0 = 1. In other words, U 0 is a usual travelling front for the homogenized equation (3.29) with speed c and limiting conditions 0 and 1 at infinity. Since the minimal speed for this problem is equal to 2 √ < a > H < µ > A , one concludes that lim inf L→0 + c * L = c ≥ 2 √ < a > H < µ > A . 4 Monotonicity of the minimal speeds c * L near the homogenization limit This section is devoted to the proof of Theorem 2.3. Before going further in the proof, we recall that for each L > 0, the minimal speed c * L is given by the variational formula c * L = min λ>0 k(λ, L) λ = k(λ * L , L) λ * L , where λ * L > 0 and k(λ, L) is the principal eigenvalue of the elliptic equation (3.2). Notice that k(λ, L) can be defined for all λ ∈ R and L > 0. Step 1: properties of k(λ, L) and definition ofk(λ, L). The principal eigenfunction ψ λ,L of (3.2) is L-periodic, positive and unique up to multiplication. Denote φ λ,L (x) = ψ λ,L (Lx) for all L > 0, λ ∈ R and x ∈ R. Each function φ λ,L is 1-periodic, positive and it is the principal eigenfunction of (aφ λ,L ) + 2Lλaφ λ,L + Lλa φ λ,L + L 2 λ 2 aφ λ,L + L 2 µφ λ,L = L 2 k(λ, L)φ λ,L , associated to the principal eigenvalue L 2 k(λ, L). But the above problem can be defined for all λ ∈ R and L ∈ R. That is, for each (λ, L) ∈ R 2 , there exists a unique principal eigenvaluẽ k(λ, L) and a unique (up to multiplication) principal eigenfunctionφ(λ, L) of (aφ λ,L ) + 2Lλaφ λ,L + Lλa φ λ,L + L 2 λ 2 aφ λ,L + L 2 µφ λ,L =k(λ, L)φ λ,L . (4.1) Furthermore,φ λ,L is 1-periodic, positive and it can be normalized so that 1 0φ 2 λ,L (x)dx = 1 (4.2) for all (λ, L) ∈ R 2 . By uniqueness of the principal eigenelements, it follows that ∀ L > 0, ∀ λ ∈ R,k(λ, L) = L 2 k(λ, L) andφ λ,L and φ λ,L are equal up to multiplication by positive constants for each L > 0 and λ ∈ R. Some useful properties of k(λ, L) as L → 0 + shall now be derived from the study the functionk. Notice first that, since the coefficients of the left-hand side of (4.1) are analytic in (λ, L), the functionk is analytic, and from the normalization (4.2), the functionsφ λ,L also depend analytically in H 2 loc (R) on the parameters λ and L (see [10,20]). In particular, the function k is analytic in R × (0, +∞). Observe also that k(λ, 0) = 0 andφ λ,0 = 1 for all λ ∈ R. Lastly, when λ is changed into −λ or when L is changed into −L, then the operator in (4.1) is changed into its adjoint. But since the principal eigenvalues of the operator and its adjoint are identical, it follows that ∀ (λ, L) ∈ R 2 ,k(λ, L) =k(λ, −L) =k(−λ, L). In particular, it follows that ∀ (i, j) ∈ N 2 , ∂ ik ∂λ i (λ, 0) = ∂ i ∂ 2j+1k ∂λ i ∂L 2j+1 (λ, 0) = 0. (4.3) Therefore, for all λ ∈ R, k(λ, L) =k (λ, L) L 2 → 1 2 × ∂ 2k ∂L 2 (λ, 0) as (λ, L) → (λ, 0 + ). But since this limit is equal to k(λ) = λ 2 < a > H + < µ > A from Step 2 of the proof of Theorem 2.1, one then gets that 1 2 × ∂ 2k ∂L 2 (λ, 0) = λ 2 < a > H + < µ > A for all λ ∈ R. (4.4) It also follows from (4.3) that ∂ 2 k ∂λ 2 (λ, L) = 1 L 2 × ∂ 2k ∂λ 2 (λ, L) → 1 2 × ∂ 4k ∂λ 2 ∂L 2 (λ, 0) as (λ, L) → (λ, 0 + ). (4.5) From (4.4) and (4.5), one deduces that ∂ 2 k ∂λ 2 (λ, L) → 2 < a > H > 0 as (λ, L) → (λ, 0 + ). (4.6) Similarly, as (λ, L) → (λ, 0 + ),                              ∂k ∂L (λ, L) = ∂ ∂L k (λ, L) L 2 → 1 6 × ∂ 3k ∂L 3 (λ, 0) = 0 ∂ 2 k ∂λ∂L (λ, L) = ∂ ∂L 1 L 2 × ∂k ∂λ (λ, L) → 1 6 × ∂ 4k ∂λ∂L 3 (λ, 0) = 0 ∂ 2 k ∂L 2 (λ, L) = ∂ 2 ∂L 2 k (λ, L) L 2 → 1 12 × ∂ 4k ∂L 4 (λ, 0) (4.7) Remark 4.1 As a byproduct of the fact thatk and k are even in λ, it follows that the minimal speed of pulsating fronts propagating from right to left (as in Definition 1.2) is the same as that of fronts propagating from left to right. Step 2: properties of c * L and λ * L in the neighbourhood of L = 0 + . Let us first prove that, for each fixed L > 0, the positive real number λ * L > 0 given in (3.1) is unique. Indeed, if there are 0 < λ 1 < λ 2 such that c * L = k(λ 1 , L) λ 1 = k(λ 2 , L) λ 2 = min λ>0 k(λ, L) λ , then k(λ, L) = c * L λ for all λ ∈ [λ 1 , λ 2 ] since k is convex with respect to λ. Then k(λ, L) = c * L λ for all λ ∈ R by analyticity of the map R λ → k(λ, L). But k(0, L) = −ρ 1,L > 0, which gives a contradiction. Therefore, for each L > 0, λ * L is the unique minimum of the map (0, +∞) λ → k(λ, L)/λ. Furthermore, we claim that L → λ * L and L → c * L are of class C ∞ in a right neighbourhood of L = 0. Indeed, by definition, λ * L satisfies F (λ * L , L) := ∂k ∂λ (λ * L , L) × λ * L − k(λ * L , L) = 0. (4.8) The function (λ, L) → F (λ, L) is of class C ∞ on R × (0, +∞) and ∂F ∂λ (λ, L) = ∂ 2 k ∂λ 2 (λ, L) × λ. But λ * L → λ * = < a > −1 H < µ > A > 0 as L → 0 + from Remark 3.2, and ∂ 2 k ∂λ 2 (λ * L , L) → 2 < a > H > 0 as L → 0 + from (4.6). Therefore, from the implicit function theorem, the map L → λ * L is of class C ∞ in an interval (0, L 0 ) for some L 0 > 0. As a consequence of formula (3.1), the map L → c * L is also of class C ∞ on (0, L 0 ). For each L ∈ (0, L 0 ), one has dc * L dL = 1 λ * L × ∂k ∂λ (λ * L , L) − k(λ * L , L) (λ * L ) 2 × dλ * L dL + 1 λ * L × ∂k ∂L (λ * L , L) = 1 λ * L × ∂k ∂L (λ * L , L) by definition of λ * L and formula (3.1). But λ * L → λ * > 0 and ∂k ∂L (λ * L , L) → 0 as L → 0 + from (4.7). Thus, dc * L dL → 0 as L → 0 + . On the other hand, it follows from (4.6), (4.7) and (4.8) that dλ * L dL = 1 λ * L × ∂ 2 k ∂λ 2 (λ * L , L) × ∂k ∂L (λ * L , L) − λ * L × ∂ 2 k ∂λ∂L (λ * L , L) → 0 as L → 0 + . Therefore, d 2 c * L dL 2 = dλ * L dL × − 1 (λ * L ) 2 × ∂k ∂L (λ * L , L) + 1 λ * L × ∂ 2 k ∂λ∂L (λ * L , L) + 1 λ * L × ∂ 2 k ∂L 2 (λ * L , L) → 1 12λ * × ∂ 4k ∂L 4 (λ * , 0) as L → 0 + ,(4. 9) from (4.7). Step 3: calculation of ∂ 4k ∂L 4 (λ * , 0). In this step, we fix λ * = < a > −1 H < µ > A . Since the functionsφ λ * ,L depend analytically on L ∈ R in H 2 loc (R), the expansioñ φ λ * ,L = 1 + Lφ 1 + L 2 φ 2 + L 3 φ 3 + L 4 φ 4 + . . . is valid in H 2 loc (R) in a neighbourhood of L = 0, where 1 =φ λ * ,0 and φ i = 1 i ! × ∂ iφ λ * ,L ∂L i L=0 for each i ≥ 1. We now put this expansion into (aφ λ * ,L ) + 2Lλ * aφ λ * ,L + Lλ * a φ λ * ,L + L 2 (λ * ) 2 aφ λ * ,L + L 2 µφ λ * ,L =k(λ * , L)φ λ * ,L and remember thatk (4.3) and (4.4). Since bothφ λ * ,L andk(λ * , L) depend analytically on L, it follows in particular that (λ * , 0) = ∂k ∂L (λ * , 0) = ∂ 3k ∂L 3 (λ * , 0) = 0 and ∂ 2k ∂L 2 (λ * , 0) = 2 × (λ * ) 2 < a > H + < µ > A = 4 < µ > A from           (aφ 1 ) + λ * a = 0, (aφ 2 ) + 2λ * aφ 1 + λ * a φ 1 + (λ * ) 2 a + µ = 2 < µ > A , (aφ 3 ) + 2λ * aφ 2 + λ * a φ 2 + (λ * ) 2 aφ 1 + µφ 1 = 2 < µ > A φ 1 , (aφ 4 ) + 2λ * aφ 3 + λ * a φ 3 + (λ * ) 2 aφ 2 + µφ 2 = 2 < µ > A φ 2 + 1 24 × ∂ 4k ∂L 4 (λ * , 0) (4.10) in R. Furthermore, each function φ i is 1-periodic and, by differentiating the normalization condition φ λ * ,L 2 L 2 (0,1) = 1 with respect to L at L = 0, it follows especially that It is then found that, for all x ∈ R, φ 1 (x) = λ * × −x+ < a > H x 0 1 a(y) dy − 1 2 − < a > H 1 0 y a(y) dy and φ 2 (x) = < µ > A × x a(x) − 1 0 y a(y) dy − x 0 1 a(y) dy − < a > H a(x) 1 0 y a(y) dy + 1 a(x) × < a > H 1 0 1 a(y) y 0 µ(z)dz dy − x 0 µ(y)dy + (λ * ) 2 x + 1 2 . Moreover, it follows from the third equation of (4.10) that, for all x ∈ R, a(x)φ 3 (x) = −2λ * x 0 a(y)φ 2 (y)dy − λ * x 0 a (y)φ 2 (y)dy −(λ * ) 2 x 0 a(y)φ 1 (y)dy − x 0 µ(y)φ 1 (y)dy + 2 < µ > A x 0 φ 1 (y)dy+ < a > H c, where c = 1 0 1 a(y) × 2λ * y 0 a(z)φ 2 (z)dz + λ * y 0 a (z)φ 2 (z)dz + (λ * ) 2 y 0 a(z)φ 1 (z)dz + y 0 µ(z)φ 1 (z)dz − 2 < µ > A y 0 φ 1 (z)dz dy On the other hand, by integrating the fourth equation of (4.10) over the interval [0, 1], one gets that 1 24 × ∂ 4k ∂L 4 (λ * , 0) = −2 < µ > A Now, put all the previous calculations into (4.11). After a lengthy sequence of integrations by parts, it is finally found that 1 24 × ∂ 4k ∂L 4 (λ * , 0) = 1 0 A(x) 2 a(x) dx − < a > H 1 0 A(x) a(x) dx 2 , where A(x) = x 0 µ(y)dy + < µ > A < a > H x 0 1 a(y) dy − 2 < µ > A x. From (4.9), it follows that d 2 c * L dL 2 → γ := 2 < a > H < µ > −1 A × 1 0 A(x) 2 a(x) dx − < a > H 1 0 A(x) a(x) dx 2 as L → 0 + . Cauchy-Schwarz inequality yields γ ≥ 0. Furthermore, γ = 0 if and only if A is constant. But since A(0) = 0, the condition γ = 0 is equivalent to A (x) = 0 for all x, which means that µ(x) < µ > A + < a > H a(x) = 2 for all x ∈ R. In particular, if µ is constant and a is not constant (resp. if a is constant and µ is not constant), then this condition is not satisfied, whence lim L→0 + µ(y)dy. Therefore, the speeds c * L are increasing in a right neighbourhood of L = 0 but, in this case, the variation is of the first order. Notice that the formula lim L→0 + dc * L dL = 2 √ β < a > H is coherent with the numerical calculations done by Kinezaki, Kawasaki and Shigesada in [21] (see Figure 3b with < µ > A = 0, that is A = 0 under the notations of [21]). Proof of Theorem 2.6 As in the proofs of the previous theorems, we use the following formula for the minimal speed: c * z = min λ>0 k z (λ) λ = k z (λ * z ) λ * z ,(5.1) where k z (λ) is defined as the unique real number such that there exists a positive L 0 -periodic function ψ satisfying: ψ + 2λ ψ + λ 2 ψ + µ z (x)ψ = k z (λ)ψ in (0, L 0 ). µ(x) := lim s→0 + f (x, s)/s, and µ L (x) := lim s→0 + f L (x, s)/s = µ x L . Figure 1 : 1The L 0 -periodic function x → µ z (x), (a): with z = 0; (b): with z > 0. Theorem 2.6 Let c * z be the minimal speed of propagation of pulsating traveling fronts solving (1.9), with a L 0 ≡ 1 and µ L 0 = µ z defined by (2.4). Assume that l ∈ (3L 0 /4, L 0 ). Then z → c * z is decreasing in [0, (L 0 − l)/2], and increasing in [(L 0 − l)/2, L 0 − l]. the positivity and the L n -periodicity of the C 2 (R) eigenfunction ψ n , it follows that∀ n ∈ N, ∃ θ n ∈ [0, L n ], ψ n (θ n ) = max x∈R ψ n (x) = max x∈[0,Ln] ψ n (x), whence ∀n ∈ N, ψ n (θ n ) = 0.For each n ∈ N, let M Ln = [1/L n ] + 1 ∈ N. Thus, ∀ n ∈ N, ϕ n (θ n + M Ln L n ) = λ * Ln e λ * Ln (θn+M Ln Ln) ψ n (θ n ).Multiplying (3.15) by ϕ n and integrating by parts over the interval [θ n , θ n + M Ln L n ], one then obtains a Ln (θ n + M Ln L n )ϕ n (θ n + M Ln L n )ϕ n (θ n + M Ln L n ) − a Ln (θ n )ϕ n (θ n )ϕ n (θ n ) 18) where µ ∞ = max x∈R \|µ(x)\|. On the other hand, (3.1) and (3.5) yieldk n ≤ 2 √ a M µ M × λ * Ln for all n ∈ N, Ln x ψ 2 n (x)dx ≤ √ a M µ M × ψ 2 n (θ n ) × e 2λ * Ln (θn+M Ln Ln) . a Ln e 2λ * Ln x ψ n (x) + λ * Ln ψ n (x) µ Ln = k n L n for all n ∈ N. 3.16) as ∀ n ∈ N, A(n) + C(n) − k n θn+M Ln Ln θn ϕ 2 n = B(n). Together with(3.27), one concludes that the sequence (∂v Ln ∂x ) n∈N is bounded in L 2 loc (R × R). On the other hand, ∂t ) n∈N is bounded in L 2 loc (R × R).Consequently, up to extraction of another subsequence, there exists 2 > 0 in this case. That completes the proofs of Theorem 2.3 and Corollary 2.4. Remark 4. 2 2In the case when < µ > A = 0 and µ ≡ 0, then ρ 1,L < 0 for each L > 0, and the minimal speed c * L of pulsating traveling fronts is well-defined and it is still positive. From the arguments developed in this section and in the previous one, one can check that, (x) = e λx ψ(x), the above equation and periodicity conditions become equivalent to:   ϕ + µ z (x)ϕ = k z (λ)ϕ in (0, L 0 ), ϕ(L 0 ) = e λL 0 ϕ(0), ϕ (L 0 ) = e λL 0 ϕ (admits, for every positive λ, a unique solution (ϕ, k z (λ)) with ϕ > 0 satisfying the normalisation condition ϕ(0) = 1.Let λ > 0 be fixed. System (5.3), together with the normalization condition ϕ(0) (k z (λ) − m)ϕ on [0, l/2), ϕ = k z (λ)ϕ on [l/2, l/2 + z), ϕ = (k z (λ) − m)ϕ on [l/2 + z, l + z), ϕ = k z (λ)ϕ on [l + z, L 0 ), ϕ(0) = 1, ϕ(L 0 ) = e λL 0 ϕ(0), ϕ (L 0 ) = e λL 0 ϕ (0z ∈ [0, L 0 − l], let λ * z be defined by the formula (5.1). We have the following lemma:Lemma 5.1 Assume that l > 3L 0 /4. Then, for all z ∈ [0, L 0 − l], we have k z (λ * z ) > m.Proof of Lemma 5.1. Let us divide equation (5.2) by ψ and integrate by parts over [0, L 0 ]. Using the L 0 -periodicity of ψ, we obtain: andwith α := L 0 − l and β := L 0 − l − 2z. Each factor in the expression (5.7) is positive, as soon as s > m, for z ∈ [0, L 0 − l]. Thus, whenever k z (λ) > m, system (5.4) is equivalent to the simpler equationFurthermore, from Krein-Rutman theory, since the eigenfunction ψ in (5.2) is positive, k z (λ) is the largest real eigenvalue of the operator ψ → ψ + 2λ ψ + λ 2 ψ + µ z (x)ψ. This result, implies that, for each z ∈ [0, L 0 − l], and each λ > 0, k z (λ) is the largest real root of equation (5.8), as soon as k z (λ) > m.(5.9)for all z ∈ [0, L 0 − l] and λ > 0. Moreover, differentiating (5.6) with respect to z, we obtainThus, for all s > m, and λ > 0,and ∂F ∂z (z, λ, s) < 0 for z ∈ ((L 0 − l)/2, L 0 − l].Now, take z 1 < z 2 in [0, (L 0 − l)/2], and assume that c * z 1 ≤ c * z 2 . It follows from formula (5.1) that k z 2 (λ) ≥ c * z 2 λ, for all λ > 0. In particular,From Lemma 5.1, we know that k z 1 (λ * z 1 ) > m. Thus, (5.11) implies k z 2 (λ * z 1 ) > m. From the above discussion, k z 2 (λ * z 1 ) is therefore the largest real root of the equation F (z 2 , λ * z 1 , k z 2 (λ * z 1 )) = 0, and, similarly, k z 1 (λ * z 1 ) is the largest real root of F (z 1 , λ * z 1 , k z 1 (λ * z 1 )) = 0. Using (5.9) and (5.10), and since 0 ≤ z 1 < z 2 ≤ (L 0 − l)/2, we obtain k z 2 (λ * z 1 ) < k z 1 (λ * z 1 ), which contradicts (5.11). 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[ "Orbital Polarization in Itinerant Magnets", "Orbital Polarization in Itinerant Magnets" ]	[ "I V Solovyev \nComputational Materials Science Center\nNational Institute for Materials Science\n1-2-1 Sengen305-0047TsukubaIbarakiJapan\n" ]	[ "Computational Materials Science Center\nNational Institute for Materials Science\n1-2-1 Sengen305-0047TsukubaIbarakiJapan" ]	[]	We propose a parameter-free scheme of calculation of the orbital polarization (OP) in metals, which starts with the strong-coupling limit for the screened Coulomb interactions in the randomphase approximation (RPA). For itinerant magnets, RPA can be further improved by restoring the spin polarization of the local-spin-density approximation (LSDA) through the local-field corrections. The OP is then computed in the static GW approach, which systematically improves the orbital magnetization and the magnetic anisotropy energies in transition-metal and actinide compounds.	10.1103/physrevlett.95.267205	[ "https://export.arxiv.org/pdf/cond-mat/0510100v1.pdf" ]	16,681,276	cond-mat/0510100	035e8d9adbb6445d9d0eefcfb60f51473b66d474	Orbital Polarization in Itinerant Magnets 5 Oct 2005 I V Solovyev Computational Materials Science Center National Institute for Materials Science 1-2-1 Sengen305-0047TsukubaIbarakiJapan Orbital Polarization in Itinerant Magnets 5 Oct 2005(Dated: December 18, 2021) We propose a parameter-free scheme of calculation of the orbital polarization (OP) in metals, which starts with the strong-coupling limit for the screened Coulomb interactions in the randomphase approximation (RPA). For itinerant magnets, RPA can be further improved by restoring the spin polarization of the local-spin-density approximation (LSDA) through the local-field corrections. The OP is then computed in the static GW approach, which systematically improves the orbital magnetization and the magnetic anisotropy energies in transition-metal and actinide compounds. An electron in solid can carry spin (M S ) and orbital (M L ) magnetic moment. For weakly correlated systems, the problem of spin magnetism alone can be formulated in the fully itinerant fashion, meaning that the effect of other electrons onto a given one can be described by an exchange-correlation field (or spin polarization). The field is typically evaluated in the model of homogeneous electron gas, in the basis of plane waves, which is a limiting case of the extended Bloch waves. This constitutes the ground of the Kohn-Sham (KS) formalism within LSDA [1], which works exceptionally well for the magnetic spin properties of many transition-metal and actinide compounds. They form an extended group of what is currently called the "itinerant electron magnets". The orbital magnetism is an atomic phenomenon. In the majority of cases, it is driven by the spin-orbit interaction (SOI), being proportional to the gradient of the one-electron potential, ∇V , which is large only in a small core region close to the atomic nucleus. Furthermore, the angular momentum operator,L z , does not commute withV . Generally,L z is not an observable quantity, except the same core region, whereV is spherically symmetrical. Therefore, it is more natural to formulate the problem in the basis of Wannier orbitals {φ α } (α being a joint spin-orbital index), localized around each atomic sites [2]. Then, the orbital moment M L =Tr LS {L zn } is specified by the local density matrixn= n αβ (Tr LS being the trace over spin and orbital variables), where n αβ = i n i d αi d † βi , d αi = φ α \|ψ i is the projection of KS eigenstate ψ i onto φ α , n i is the KS occupation number corresponding to the eigenvalue ε i , and the joint index i stands for the spin, band, as well as the position of the k-point in the first Brillouin zone (BZ). In an analogy with the spin polarization for itinerant magnets, one can think of an OP: an exchangecorrelation field in KS equations, which couples with M L . Despite a genuine interest to the problem and wide perspectives of their potential applications, the theories of OP in metals are still in a developing "semi-empirical" stage, as they largely depend on the input parameters, which are typically chosen to fit the experimental data. Although majority of researches agree that OP is controlled by intra-atomic interactions, which are strongly screened in metals, the details of this screening as well as the form of the OP itself remains to be a largely unresolved and disputed problem [3,4,5,6,7,8]. Therefore, there are two important questions, which we would like to address in this work. (i) How the bare on-site interaction u αβγδ = φ α φ γ \|1/r 12 \|φ β φ δ between dor f -electrons is screened in metals? What is the main mechanism of this screening? (ii) Is there any simple and reliable way to evaluate this screening in ab initio calculations of OP? In the atomic limit, the full matrixû= u αβγδ \| is controlled by a small number of Slater integrals {F k }. Then, there is an old empirical rule [9], which states that in metals, the screening affects mainly F 0 , which contribute to the Coulomb matrix elements u ααγγ . Other Slater integrals, which control the exchange and nonsphericity of Coulomb interactions do not change so much. First, we argue that the same type of screening can be naturally obtained in RPA, in the fully deterministic fashion. The screened interaction in RPA [10,11], U (ω) = 1 −ûP (ω) −1û ,(1) depends on the polarizationP = P αβγδ , which is treated in the approximation of noninteracting KS quasiparticles: P RPA αβγδ (ω) = ij (n i − n j )d † αj d βi d † γi d δj ω − ε j + ε i + iδ(n i − n j ) .(2) The ω-dependence ofP contributes mainly to the redistribution of the spectral density, whereas the ωintegrated ground-state properties are controlled byÛ ≡ U (0). Therefore, we consider only the static limit, in which RPA describes the screening ofû caused by the relaxation of {ψ i } upon removal or addition of an electron in terms of the perturbation-theory expansion [12]. The simplest toy model, which illustrates the physics, may consist of two spin-polarized bands, formed by yz (1) and zx (2) orbitals. The model is compatible with the orbital magnetization in the 001 direction. Adopting the following order of orbitals (within one spin channel): αβ (γδ)= 11, 22, 12, and 21, it is easy to show that u =     u u ′ 0 0 u ′ u 0 0 0 0 0 j 0 0 j 0     ,(3) where u=F 0 + 4 49 F 2 + 36 441 F 4 , j= 3 49 F 2 + 20 441 F 4 , and u ′ =u−2j. Due to the orthogonality of the yz and zx orbitals, the Coulomb (αβ= 11, 22) and exchange (αβ= 12, 21) matrix elements are fully decoupled from each other. In order to illustrate the main idea of RPA-screening, P can be taken in the form P αβγδ =P δ αδ δ βγ [13], which yields: U =[u−(u 2 −u ′2 )P ]/[(1−uP ) 2 −(u ′ P ) 2 ], U ′ =u ′ /[(1−uP ) 2 −(u ′ P ) 2 ], and J=j/[1−jP ]. There is certain hierarchy of bare interactions, and for many metals the screening of u and j falls in the strongand weak-coupling regime, respectively, so that u\|P \|≫1 while j\|P \|≪1 [14]. This yields: U ≃−1/(2P )+2J, U ′ ≃−1/(2P ), and J≃j. Thus, this is the inverse polarization, which plays a role of effective Coulomb interaction in metals [12]. U is strongly screened and does not depend on the value of bare interaction. On the other hand, J is insensitive to the screening. The multiplier 1/2 in the expressions for U and U ′ stands for the orbital degeneracy. The result can be easily generalized for an arbitrary number of orbitals M (M = 5 and 7 for d-and f -electrons, respectively), which yields U ′ ≃−1/(M P ) [2]. In this case, in order to justify the strong-coupling regime, it is sufficient to have a milder condition, uM \|P \|≫1, which naturally explains the empirical rule [9]. All these trends are clearly seen in the first-principles calculations for realistic materials shown in Fig. 1 [15], where all Slater integrals except F 0 were calculated inside atomic spheres, and F 0 was treated as a parameter. When F 0 increases, the effective interactions quickly reach the asymptotic limit F 0 →∞, whereÛ is fully de-termined by details of the electronic structure, through the polarizationP , and do not depend on F 0 . This removes the main ambiguity with the choice of interaction parameters for metallic compounds. SinceP depends on the local environment in solid, the screened interactions can be different for different types of Wannier orbitals (e.g., e g and t 2g for d-electrons in the cubic environment). U(t 2u ) U(t 1u ) J(t 2u ) J(t 1u ) Screened Thus, OP in the itinerant magnets can be naturally evaluated in the framework of an universal parameterfree scheme based on the strong coupling limit for the matrix of effective Coulomb interactionsÛ . The selfenergy, incorporating the effects of OP, can be calculated within static approximation in the GW method [10]: Σ αβ = − γδ U αδγβ n γδ .(4) The proper correction to the KS Hamiltonian in LSDA is controlled by ∆n=n− 1 2M 3 r=0 Tr LS {σ rn }σ r . It is obtained after subtracting the charge (r=0) and spin (r= 1, 2, and 3) density elements ofn, which are already taken into account in LSDA (σ 0 being the unity matrix, and σ 1 ,σ 2 , andσ 3 being the Pauli matrices of the dimension 2M ). Therefore, in the actual calculations we uses the change of the self-energy ∆Σ, which was obtained after replacingn by ∆n in Eq. (4). The problem was solved self-consistently with respect to ∆n. The validity of the strong-coupling approach is well justified. So, the effective Coulomb interaction between t 2g electrons in bcc Fe can be estimated in RPA as 1.50, 1.47, and 1.37 eV for F 0 = ∞, 21 eV (the bare Slater integral inside atomic sphere), and 4.5 eV (the value obtained in the constraint-LSDA, which includes the screening by the sp-electrons [12]), respectively. Thus, even if one takes the lowest estimate F 0 =4.5 eV, the additional approximation F 0 →∞ within RPA would overestimate U by less than 10%. For f -electrons, this error is even smaller due to the higher orbital degeneracy. However, this is not the main source of the error. A more fundamental problem is related with the RPA itself, which typically underestimates the spin polarization ∆ RPA =Tr LS {Σσ 3 }, meaning that even for the upper limit in RPA, corresponding to F 0 →∞, the effective Coulomb interaction is overscreened and underestmated. For example, had we replaced the spin part of LSDA by the one of RPA, the spin moment would be underestimated. Obviously, this would destroy the most attractive point of LSDA for itinerant electron magnets. Therefore, there is certain inconsistency in the RPA approach. RPA can be improved by introducing the localfield factor g, which incorporates the effects of exchange-correlation hole for the polarization matrix: (P ) −1 =(P RPA ) −1 −ĝ. Other corrections can be formally reduced toĝ [16]. Our goal is to find such a correction to the matrix of effective Coulomb interactions, which after substitution in Eq. (4) would yield the same spin polarization as LSDA (∆ LSDA ). In order to do so, we search g in the form of local diagonal matrix: g αβγδ =gδ αβ δ γδ . Then, the asymptotic part of the effective Coulomb interaction and the self-energy can be easily recalculated using Eqs. (1) and (4), respectively, and the unknown parameter g is obtained from the condition Tr LS {Σσ 3 }=∆ LSDA , which is solved self-consistently together with the KS equations. In the following, this method will be referred to as corrected-RPA (c-RPA). Let us consider first the canonical example of ferromagnetic transition metals (Fig. 2), where M L is small and typically regarded as a small perturbation to the spindependent properties. M S and M L can be measured sep- arately using the x-ray magnetic circular dichroism combined with the spin and orbital sum rules [17]. Despite an apparent simplicity, LSDA encounters a wide spectrum of problems for bcc Fe, hcp Co, and fcc Ni. We will argue that many of them can be systematically corrected by applying consequently RPA and c-RPA techniques. =0.11µ B ). Fcc Ni is a rare example of ferromagnetic systems for which M L =0.05µ B is well reproduced already in LSDA. Both RPA and c-RPA preserve this good feature of LSDA and do not substantially change M L . However they do change the electronic structure of fcc Ni. Namely, the form of Fermi surface (FS) of fcc Ni has been intensively discussed in the context of the magnetocrystalline anisotropy energy (MAE). It was argued that the reason why LSDA fails to reproduce the correct 111 direction of the magnetization is related with the second pocket of the FS around the X-point of BZ, which is not seen in the experiment [7]. The experimental FS can be reproduced in the LSDA+U approach, by treating U as an adjustable parameter [7]. Therefore, it is important that the same problem can be successfully resolved both in RPA and c-RPA, without any adjustable parameters. The calculated FS, which reveals only one pocket around the X-point, is shown in the inset of Fig. 2. The uranium pnictides (UX, where X= N, P, As, Sb, and Bi) and chalcogenides (X= S, Se, and Te) are ones of the most studied actinide compounds. They crystallize in the rock-salt structure. All chalcogenides are ferromagnets, whereas the pnictides have type-I antiferromagnetic structure, which may also transform into a multi-k structure. The basic difference from the transition metals is that M L in actinides, which can be extracted from the analysis of magnetic form factors [4,18], is very large and typically dominates over M S . According to the third Hund rule, M S and M L in UX are coupled antiferromagnetically. As the U-U distance increases, the U(5f ) states become more localized, and all magnetic moments increase monotonously from UN to UBi and from US to UTe (Fig. 3). UN and US are usually classified as (Color online) Magnetic moments in uranium pnictides (top) and chalcogenides (bottom). The pnictides (chalcogenides) have been computed in the type-I antiferromagnetic (ferromagnetic) structure with 001 ( 111 ) direction of the magnetization. The symbol 'exp' shows the results of neutron diffraction, which were separated into spin and orbital contributions for US (Ref. [4]) and UAs (Ref. [18]). Other notations are the same as in Fig. 2. itinerant magnets. However, the role of intra-atomic correlations is expected to increase for the end-series compounds. Obviously, the real ab initio scheme does not know whether the system is itinerant or not. Therefore, it is important to test both RPA and c-RPA methods for all considered compounds in order to see how they will work for the materials with the different character of the 5f -electrons. The orbital moments are systematically underestimated in LSDA. The error is really large so that the total magnetic moments are typically off the experimental values by 20-50%. RPA systematically improves the LSDA description. However, it is not enough, and for many uranium compounds it is essential to go beyond RPA. For these purposes, c-RPA works exceptionally well and further improves the RPA description. Particularly, we note an excellent agreement with the ex-perimental data for X= S, P, and As. For the end-series compounds (X= Te, Sb, and Bi) the agreement is not so good, signalling at the necessity of more radical improvements, involving both orbital and spin polarization of LSDA. However, even for these complicated systems, c-RPA is a big step forward over conventional LSDA. Finally, let us discuss applications for the MAE. We consider two characteristic examples: CoPt and US. The ordered tetragonal CoPt alloys is a promising candidate for magnetic recording applications. An intriguing point is that although LSDA underestimates M L , MAE is reproduced surprisingly well (Fig. 4) [8]. Therefore, the "correct" OP in CoPt should affect only M L . This requirement is well satisfied both for RPA and c-RPA. The orbital moments systematically increase in the direction LSDA→RPA→c-RPA to reach M Co L =0.14µ B and M Pt L =0.07µ B . The anisotropy of M L also increases (mainly at Co-sites). However, the MAE does not change so much because of large cancellation of on-site interaction energies associated with Co and Pt sites [8]. US has the largest MAE among cubic compounds [19], which is underestimated in LSDA. The situation is corrected in c-RPA, at least qualitatively. It is curious that MAE "anticorrelate" with the anisotropy of orbital magnetization, which decreases in the direction LSDA→RPA→c-RPA. However, this is not surprising, because in cubic compounds, MAE is the forth order effect with respect to SOI. Therefore, there is no simple relation between ∆E and ∆M L and the main correction to MAE in c-RPA comes from the change of the on-site interaction energy. In summary, we have argued that the problem of OP in metals can be naturally formulated "from the first principles", by considering the strong-coupling limit for the screened Coulomb interactions. In the present work, the screenedÛ was computed only once: in LSDA and without SOI. An important extension would be a selfconsistent determination ofÛ , which would incorporate the effects of OP into the screening. (i) It could improve the description of some itinerant actinide compounds (e.g., UN) for which the spin polarization is small, and the screening is strongly influenced by SOI. (ii) Since the OP affects the KS eigenvalues {ε i }, which stand in the denominator of the polarization matrix (2), the screening is expected to decrease. This could extend the applicability of the proposed method for materials with more localized 5f -and 4f -electrons. FIG. 1 : 1Effective Coulomb and exchange interactions in RPA versus bare Slater integral F 0 for 3d-states in bcc Fe and 5fstates in uranium sulfide. The symbols denote the matrix elements corresponding to different representations of the point group O h . The calculations have been performed in the ferromagnetic state without spin-orbit coupling. online) Spin (light blue area), orbital (dark red area), and total (full hatched area) magnetic moments in ferromagnetic transition metals. The experimental data are taken from Ref.[5]. The inset shows the Fermi surface of fcc Ni in the c-RPA approach. LSDA has certain tendency to overestimate M S in bcc Fe and underestimate M L , while RPA and especially c-RPA substantially improve the LSDA description and yield a good agreement with the experimental data. The values of M S (M L ) obtained in LSDA, RPA, and c-RPA are 2.26 (0.04), 2.21 (0.05), and 2.20 (0.06) µ B , respectively, to be compared with the experimental moments of 2.13 (0.08) µ B [5]. Hcp Co has the largest orbital moment among pure transition metals (M L =0.14µ B ), which is strongly underestimated in LSDA (M L =0.08µ B ). The situations is substantially improved in RPA (M L =0.10µ B ) and c-RPA (M L FIG. 3: (Color online) Magnetic moments in uranium pnictides (top) and chalcogenides (bottom). The pnictides (chalcogenides) have been computed in the type-I antiferromagnetic (ferromagnetic) structure with 001 ( 111 ) direction of the magnetization. The symbol 'exp' shows the results of neutron diffraction, which were separated into spin and orbital contributions for US (Ref. [4]) and UAs (Ref. [18]). Other notations are the same as in Fig. 2. online) Magnetocrystalline anisotropy energy (∆E) and the anisotropy of orbital magnetization (∆ML). For each quantity, the anisotropy is defined as the difference between values corresponding to the 100 and 001 (CoPt), and 100 and 111 (US) directions of the magnetization. The experimental values are taken from Ref. [8] (CoPt, at 293 K) and Ref. [19] (US). For CoPt, the values of MS and ML in the 001 direction are shown in the left part of the figure. Other notations are the same as in Fig. 2. * Electronic address: [email protected]. * Electronic address: [email protected] . W Kohn, L J Sham, Phys. Rev. 1401133W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965); . U Bath, L Hedin, J. Phys. C. 51629U. von Bath, and L. Hedin, J. Phys. C 5, 1629 (1972). . I V Solovyev, cond-mat/0506632I. V. Solovyev, cond-mat/0506632. . O Eriksson, Phys. Rev. B. 417311O. Eriksson et al., Phys. Rev. B 41, 7311 (1990); . M R Norman, Phys. Rev. Lett. 641162M. R. Norman, Phys. Rev. Lett. 64, 1162 (1990). . L Severin, Phys. Rev. Lett. 713214L. Severin et al., Phys. Rev. Lett. 71, 3214 (1993). . J Trygg, Phys. Rev. Lett. 752871J. Trygg et al., Phys. Rev. Lett. 75, 2871 (1995). . I V Solovyev, Phys. Rev. Lett. 805758I. V. Solovyev et al., Phys. Rev. Lett. 80, 5758 (1998). . I Yang, Phys. Rev. Lett. 87216405and references thereinI. Yang et al., Phys. Rev. Lett. 87, 216405 (2001), and references therein. . A B Shick, O N Mryasov, Phys. Rev. B. 67172407and references thereinA. B. Shick and O. N. Mryasov, Phys. Rev. B 67, 172407 (2003), and references therein. . D Van Der Marel, G A Sawatzky, Phys. Rev. B. 3710674D. van der Marel and G. A. Sawatzky, Phys. Rev. B 37, 10674 (1988); . M R Norman, 521421M. R. Norman, ibid. 52, 1421 (1995). . L Hedin, Phys. Rev. 139796L. Hedin, Phys. Rev. 139, A796 (1965); . F Aryasetiawan, O Gunnarsson, Rep. Prog. Phys. 61237F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 237 (1998). The matrix multiplication in Eq. 1 implies the convolution over two indices: (ûP ) αβγδ = µν u αβµν P µνγδ. The matrix multiplication in Eq. 1 implies the convolu- tion over two indices: (ûP ) αβγδ = µν u αβµν P µνγδ . . I V Solovyev, M Imada, Phys. Rev. B. 7145103I. V. Solovyev and M. Imada, Phys. Rev. B 71, 045103 (2005). In this approximation, each band is formed by either yz or zx orbitals, which do not mix in KS eigenfunctions. In this approximation, each band is formed by either yz or zx orbitals, which do not mix in KS eigenfunctions. Typical values of parameters for transition metals are u∼u ′ ∼25-30 eV. j∼1 eV, and \|P \|∼0.2-0.3 eV −1Typical values of parameters for transition metals are u∼u ′ ∼25-30 eV, j∼1 eV, and \|P \|∼0.2-0.3 eV −1 . with the experimental lattice parameters. The MAE was calculated in the mesh of 10 5 k-points in the first BZ. The LMTO method was extended beyond the nearly-orthogonal representation [2], which explains a good agreement with more accuarte fullpotentional calculations of MAE [8] and some difference from the previous calculations. O K V Andersen ; I, Solovyev, Phys. Rev. B. 1213419Phys. Rev. BAll calculations have been performed in the linear muffin-tin-orbital (LMTO) method: O. K. Andersen, Phys. Rev. B 12, 3060 (1975), with the experimental lat- tice parameters. The MAE was calculated in the mesh of 10 5 k-points in the first BZ. The LMTO method was ex- tended beyond the nearly-orthogonal representation [2], which explains a good agreement with more accuarte full- potentional calculations of MAE [8] and some difference from the previous calculations: e.g., I. V. Solovyev et al., Phys. Rev. B 52, 13419 (1995). G D Mahan, Many-Particle Physics. New YorkPlenum PressG. D. Mahan, Many-Particle Physics (Plenum Press, New York, 1990). . P Carra, Phys. Rev. Lett. 70694P. Carra et al., Phys. Rev. Lett. 70, 694 (1993). . S Langridge, Phys. Rev. B. 556392S. Langridge et al., Phys. Rev. B 55, 6392 (1997). The experimental situation for US is rather controversial. The measured MAE at 165 K is about 1 meV. Results of extrapolation to 0 K vary from 7.3 meV [D. L. Tillwick and P. de V. du Plessis. G H Lander, J. Magn. Magn. Mater. 3989Appl. Phys. Lett.The experimental situation for US is rather controversial. The measured MAE at 165 K is about 1 meV. Results of extrapolation to 0 K vary from 7.3 meV [D. L. Tillwick and P. de V. du Plessis, J. Magn. Magn. Mater. 3, 329 (1976), shown in Fig. 4] till 86.0 meV [G. H. Lander et al., Appl. Phys. Lett. 57, 989 (1990)].	[]
[ "Horizon supertranslation and degenerate black hole solutions", "Horizon supertranslation and degenerate black hole solutions" ]	[ "Rong-Gen Cai \nInstitute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nCenter for Gravitational Physics\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan\n", "Shan-Ming Ruan [email protected] \nInstitute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n", "Yun-Long Zhang [email protected] \nAsia Pacific Center for Theoretical Physics\n790-784PohangKorea\n" ]	[ "Institute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina", "Center for Gravitational Physics\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan", "Institute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina", "Asia Pacific Center for Theoretical Physics\n790-784PohangKorea" ]	[]	In this note we first review the degenerate vacua arising from the BMS symmetries. According to the discussion in [1] one can define BMS-analogous supertranslation and superrotation for spacetime with black hole in Gaussian null coordinates. In the leading and subleading orders of near horizon approximation, the infinitely degenerate black hole solutions are derived by considering Einstein equations with or without cosmological constant, and they are related to each other by the diffeomorphism generated by horizon supertranslation. Higher order results and degenerate Rindler horizon solutions also are given in appendices.R µν −	10.1007/jhep09(2016)163	[ "https://arxiv.org/pdf/1609.01056v2.pdf" ]	118,641,531	1609.01056	84e674b11e3abee1af843541665a7cd566efd510	Horizon supertranslation and degenerate black hole solutions Rong-Gen Cai Institute of Theoretical Physics CAS Key Laboratory of Theoretical Physics Chinese Academy of Sciences 100190BeijingChina Center for Gravitational Physics Yukawa Institute for Theoretical Physics Kyoto University 606-8502KyotoJapan Shan-Ming Ruan [email protected] Institute of Theoretical Physics CAS Key Laboratory of Theoretical Physics Chinese Academy of Sciences 100190BeijingChina Yun-Long Zhang [email protected] Asia Pacific Center for Theoretical Physics 790-784PohangKorea Horizon supertranslation and degenerate black hole solutions Prepared for submission to JHEP In this note we first review the degenerate vacua arising from the BMS symmetries. According to the discussion in [1] one can define BMS-analogous supertranslation and superrotation for spacetime with black hole in Gaussian null coordinates. In the leading and subleading orders of near horizon approximation, the infinitely degenerate black hole solutions are derived by considering Einstein equations with or without cosmological constant, and they are related to each other by the diffeomorphism generated by horizon supertranslation. Higher order results and degenerate Rindler horizon solutions also are given in appendices.R µν − Introduction Almost half a century ago, Bondi, van der Burg, Metzner and Sachs (BMS) [2,3] independently investigated gravitational waves near the null infinity in asymptotically flat spacetime and showed that the spacetime has an infinitesimal dimensional group associated with the asymptotic symmetries called BMS group now. A few years later, Weinberg found that there is a universal soft theorem [4,5] relating one S-matrix element of n-particles to the other with an additional zero four-momentum photon or graviton that is generally called soft particle, which plays an important role in eliminating the infrared divergence in quantum field theory. In recent years, Strominger got some insights on the infrared structure of quantum gravity [6] and connected these two seemingly different matters mentioned above with his collaborators. It is verified that the soft graviton theorem is exactly equivalent to the Ward identity of the BMS supertranslation [7,8] and there is also equivalence between the subleading soft theorem [9] and the Ward identity of superotation [10][11][12]. Furthermore, the soft theorems and asymptotic symmetries are related to the traditional gravitational memory effect [13] and new spin memory effect [14]. In addition, these elegant connections and equivalences are also found in gauge theories. The large gauge symmetries [15] of gauge theories act as the asymptotic symmetries at the null infinity just like in the BMS group. Of course, BMS transformation can also be considered as large diffeomorphism. Ward identity of large gauge transformation is found to be equivalent to the soft photon [16][17][18][19][20] or gluon theorems [21] which are related to the observable effect, i.e., electromagnetic memory [22]. So all these series of works, starting from fifty years ago, finally illustrate the wonderful triangular connections among soft factors, symmetries and memories for gauge field theory and gravity theory [22,23]. The equivalence is also extended to fermionic symmetry in [24][25][26] where the authors showed that the Ward identity of residual local supersymmetry can be understood as soft gravitino theorem. The asymptotic symmetries or large gauge symmetries not only are related to the BMS group and soft theorems, but also stimulate new insight about black hole physics. Hawking, Perry and Strominger (HPS) [27] recently noticed that there is an infinite family of degenerate vacua, associated with an asymptotically flat spacetime, because of BMS supertranslation which enables black hole to carry soft hair storing information about matter. At the same time, black hole must also carry soft gauge hairs due to the infinite conservation laws coming from large abelian gauge symmetries if Maxwell field is present in the theory under consideration. This claim points out the flaws of Hawking's original argument about the information loss paradox [27,28]. Although the HPS's proposal has not yet solved the information loss paradox, these infinite soft hairs of black hole actually indicate a hopeful direction for information problem of black hole. In addition to the information loss paradox, degenerate black hole states also enable us to count the microstates of black hole [29,30] although it is not exact in the four-dimensional case due to the absence of fully understanding about superrotation. But, some works [31][32][33] give very nice results about the microstate counting in the case of 3-dimensional black hole. Note that most of recent works relevant to the HPS's proposal focus on the symmetries at the null infinity. We here want to directly investigate the asymptotic symmetries near the horizon inspired by the works in [1,34] where the authors showed that there are BMS-analogous supertranslation and superrotation at the horizon. In this note, we first review the BMS supertranslation and degenerate vacuum solutions, then discuss the horizon supertranslation and find the corresponding infinite degenerate black hole solutions of Einstein equations with or without cosmological constant in the near horizon regime. BMS supertranslation and degenerate vacua In this section, we simply review the basics of the supertranslation in BMS group [35][36][37] and the resulting infinitesimal degenerate vacuum states which play a pivot role in understanding the soft hair of black hole. Let us start from the general BMS metric ansatz that can represent asymptotically flat spacetimes 1 ds 2 = e 2β V r du 2 − 2e 2β dudr + g AB (dx A − U A du)(dx B − U B du),(2.1) with four gauge fixing conditions g rr=0 , g rA = 0, det(g AB ) = r 4 det(γ AB ), (2.2) where γ AB is the metric of two-dimensional sphere whose coordinates are described by indices A, B and associated covariant derivative isD. By requiring the Lie derivative L ζ g µν generated by vector ζ defining the asymptotic symmetries to satisfy gauge fixing conditions (2.2) and some fall-off boundary conditions, we can find the solutions of the vector as ζ = (T + u 2D C R C )∂ u − r 2 (D A ζ A − U C ∂ C f )∂ r + (R A − ∂ B T ∞ r e 2β g AB dr )∂ A , (2.3) where T and R are functions of x A and define the supertranslation and conformal transformations, respectively. If one allows R(x A ) to have pole singularities [35,39], the global conformal transformation can be extended to be a local one which is generally called superrotation. But there exist some debates [38,40] on whether the superrotation or extended BMS group is physical or not. We will therefore consider supertranslation only in what follows. The asymptotic Killing vector ζ is derived off shell, which means that it does not rely on equations of motion. Once the Einstein equations are imposed, the on-shell retarded Bondi coordinates can be simply expressed as 2 ds 2 = − du 2 − 2dudr + 2r 2 γ zz dzdz + 2m B r du 2 + rC zz dz 2 + rCzzdz 2 + D z C zz dudz + DzCzzdudz + 1 4r 2 C zz C zz dudr + γ zz C zz C zz dzdz + ... ,(2.4) where γ zz = 2/(1 + zz) 2 is the metric of the two-dimensional sphere defined on complex stereographic coordinate z = e iφ cot(θ/2) and the ellipsis stands for higher order terms which do not affect the following constraint equations. In general, m B (u, z,z) is called Bondi mass aspect constrained by the component of the Einstein equations lim r→∞ r 2 G uu = −2∂ u m B − 1 2 N zz N zz + 1 2 D z D z N zz + 1 2 DzDzNzz = lim r→∞ 8πGr 2 T uu , (2.5) where N zz = ∂ u C zz is the Bondi news tensor, which is relevant to the gravitational radiation and D A stands for the covariant derivative associated with the metric γ zz . More precisely, one can get [36,38] g uz = 1 2 D z C zz + 2 3r N z + 1 6r C zz D z C zz + O( 1 r 2 ), g AB = r 2 γ AB + rC AB + 1 2 γ AB C zz C zz + E AB r + O( 1 r 2 ),(2.6) where N A is angular momentum aspect and C zz = 0, E zz = 0 are determined by determinant condition in (2.2). We only consider the supertranslation generated by the vector [7,37] ζ = T ∂ u − 1 r (D z T ∂ z + DzT ∂z) + (D z D z T )∂ r ,(2.7) whose surface charge can be defined as Q T = 1 4πG I + − d 2 zγ zz T (z,z)m B with {Q T , Q T } = 0,(2.8) where I + − represents the past boundary of future null infinity I + . It can generate infinitesimal supertranslation transformation like L ζ m B = T ∂ u m B , L ζ C zz = T N zz − 2D z D z T, (2.9) which will transform one solution of (2.4) into another one. We can define the vacuum state 3 as N zz = 0 which means there is no gravitational radiation. So the vacuum can be labelled by the u-independent function C zz = −2D z D z C(z,z) that obeys the boundary condition. Amazingly, BMS supertranslation teaches us a lesson that the vacuum is not unique and these infinite degenerate vacua are physically distinct and are related to each other by the BMS supertranslation which leads to a change like C → C + T (z,z) [13]. On the other hand, non-constant BMS supertranslation will be spontaneously broken, which creates soft graviton viewed as Goldstone boson. Furthermore, gravitational memory effect [13,14] makes us able to measure the transition of spacetime metric that is induced by radiation through null infinity. HPS [27] proposed that BMS symmetries and large gauge symmetries in abelian gauge theory enable black hole to carry soft supertranslation hair and soft electric hair, which can carry some information and shed light on resolution of the information loss paradox. But in the Bondi coordinates (2.4) it is not direct to find whether there is a black hole in the bulk because this kind of coordinates is designed to pay attention on the null infinity-the boundary of the spacetime. So it should be more convenient to analyze the analogous asymptotic symmetries of BMS group directly on the horizon. In this aspect, HPS also made some fundamental and important discussions about the horizon supertranslation in their paper. We expect that the horizon supertranslation should be able to repeat similar results mentioned above and the conclusions about BMS supertranslation at the null infinity, inspired by the work in [1] and their recent extended work [34] where the authors showed that event horizon also exhibits the familiar asymptotic symmetries generated by supertranslation and superrotation. With the presence of Maxwell field, the authors in [42] also show that isolated horizon carries a large amount of soft electric hairs which can be considered as the counterpart of soft electric hair discussed in HPS's paper [27]. Horizon supertranslation and degenerate black hole Null infinity I can be considered as the boundary of an asymptotically spacetime. On the other hand, black hole horizon shares some similarities with the null infinity and can be understood as another boundary of the spacetime outside the black hole. So it is reasonable to generalize the discussions near the null infinity to the horizon. In this section we want to study the supertranslation on a black hole horizon. From the previous works, we know the fact that even after one chooses some gauge conditions generally called coordinate conditions in gravity theory, in order to eliminate the extra degrees of freedom, we still have the residual gauge transformations which are called large gauge transformation or large diffeomorphism for gravity. Actually horizon supertranslations have been studied many years ago with other motivations [43,44]. Of course inspired by HPS's work [27], there have been also other works [1,29,30,34,45] concerning supertranslation or superrotation on a black hole horizon recently. Here we want to describe how the degenerate black hole spacetime near the horizon can appear when we have fixed the gauge and we use these degenerate BH solutions to discuss the black hole's ability to store information about the initial state. At the end of this section, we will discuss a little about gravitational memory effect near the horizon which can be considered as the method to measure the information of black hole. Let's start from the simplest Schwarzschild black hole written in the infalling Eddington-Filkenstein coordinates ds 2 = − 1 − 2m r dv 2 + 2dvdr + 2r 2 γ zz dzdz. (3.1) We label this kind of special metric as g 0µν to distinguish it from other degenerate black hole solutions. If one wants to use the near horizon geometry to discuss the infinite dimensional symmetries near the horizon just like the BMS group near the null infinity, the deviation from the metric of black hole (3.1) should be the order of (r − r h ) n with n > 0. It will be convenient to introduce the new radial coordinate ρ = r − r h to describe the region near the horizon located at r h . In this coordinate, the Schwarzschild metric can be expressed as ds 2 = − ρ r h + ρ 2 r 2 h dv 2 + 2dvdρ + 2(ρ + r h ) 2 γ zz dzdz, (3.2) with some neglected higher order terms of ρ. Obviously, it is not the universal near horizon geometry. On the one hand, we need to fix some gauge conditions in order to find the large diffeomorphism. On the other hand, we also need to choose suitable boundary conditions to describe the physical process near the horizon. With the motivation to define the supertranslation on the black hole horizon, we take the four gauges as 4 g ρρ = 0, g ρA = 0, g ρv = 1, (3.3) which are the same as those in [1,30,34] but a little different from the BMS gauges due to the difference in the fourth condition. We will discuss the difference at the end of this section. Actually, the coordinates satisfying these special coordinate conditions (3.3) are commonly known as Gaussian null coordinates (GNC) as the analogues of Gaussian normal coordinates. An arbitrary null surface can be rewritten in GNC [46,47], so the gauge conditions are universal for any isolated horizon. For a simple example, a general black hole solution can be expressed as ds 2 = −f (r)dt 2 + dr 2 f (r) + g AB dx A dx B ,(3.4) and it can be easily rewritten as ds 2 = −f (r)dv 2 + 2drdv + g AB dx A dx B , with v = t + dr f (r) . (3.5) Assuming the event horizon located at r = r h , one can find f (r) ≈ 2κ(r − r h ). From [30] we can find that this kind of gauge conditions play an important role in defining supertranslation at the horizon. For other fall off conditions for other components of metric, we follow Ref. [1] where they showed that the asymptotic symmetries near the horizon of black hole are generated by charges of supertranslation and Virasoro algebra 5 , and they can be expressed explicitly as g vv = −2κρ + ρ 2ã(2) (v, z,z) + O(ρ 2+ ), g vA = ρθ A (v, z,z) + ρ 2 θ (2) A (v, z,z) + O(ρ 2+ ), g AB = Ω(z,z)γ AB + ρλ AB (v, z,z) + ρ 2λ (2) AB (v, z,z) + O(ρ 2+ ), (3.6) where A, B are the complex coordinate z andz indices on the unit 2-dimensional sphere and O(ρ 2+ ) represents the higher order terms which are irrelevant in our following discussions. It can be shown that general stationary black hole can be written in this kind of form with different surface gravity κ and function Ω. For rotating (Kerr) black hole, please see [48] for more detailed coordinate transformation. Of course, the Schwarzschild black hole satisfies the same asymptotic conditions by defining κ = 1/2r h and Ω = r 2 h . So the metric ansatz near the horizon takes the form as ds 2 = −2κρ + ( ρ r h ) 2 dv 2 + 2dvdρ + 2(ρ + r h ) 2 γ zz dzdz + ρ 2 a (2) dv 2 + 2ρθ z dzdv + 2ρθzdzdv + ρλ AB dx A dx B + 2ρ 2 θ (2) z dzdv + 2ρ 2 θ (2) z dzdv + ρ 2 λ (2) AB dx A dx B + O(ρ 2+ ), (3.7) where the first line comes from a general spherically symmetric black hole solution like (3.1), while the second and third lines are higher order terms which also contribute to the Einstein equations at the leading order. In matrix form it can be written as g µν =    −2κρ + ( ρ r h ) 2 + ρ 2 a (2) 1 ρθ A + ρ 2 θ (2) A 1 0 0 ρθ A + ρ 2 θ (2) A 0 (ρ + r h ) 2 γ AB + ρλ AB + ρ 2 λ (2) AB    + O(ρ 2+ ). (3.8) Note that the conventions here are a little different from those in [1] where their λ AB is the same as oursλ AB = λ AB + 2r h γ AB . Supertranslation and charge The horizon supertranslation Killing vector 6 that preserves the asymptotic condition can be derived as [34] ξ = f (z,z)∂ v + D A f ρ 0 dρ g AB g vB ∂ ρ − D B f ρ 0 dρ g AB ∂ A ,(3. 9) 5 Recently this kind of conditions are also extended to a more general case which admits dependence of time [34]. 6 As the above, we here also don't contain the vector associated with superrotation. or asymptotically [1] ξ = f (z,z)∂ v + ρ 2 2r 2 h θ A D A f ∂ ρ + − ρ r 2 h D A f + ρ 2 2r 4 hλ AB D B ∂ A + O(ρ 3 ), (3.10) which can generate the infinitesimal transformation L ξ θ z = −2κD z f (z,z), L ξ θz = −2κDzf (z,z), L ξ λ AB = θ A D B f (z,z) + θ B D A f (z,z) − 2D A D B f (z,z). (3.11) For a general Schwarzschild black hole, one can get a more precise form [30] ξ = f (z,z)∂ v + D z f ( 1 r h + ρ − 1 r h )∂ z + Dzf ( 1 r h + ρ − 1 r h )∂z,(3.12) which can lead to an infinitesimal change given by Lie derivative L ξ g µν = 2∇ (µ ξ ν) =      0 0 −ρ Dzf r −ρ Dzf r 0 0 0 0 −ρ Dzf r 0 −ρ 2rDzDzf r h −ρ 2rDzDzf r h −ρ Dzf r 0 −ρ 2rDzDzf r h −ρ 2rDzDzf r h      ,(3.13) where we have used r = r h +ρ. The result has also been studied in [30,45] where the authors discussed the interesting connection between Goldstone mode and quantum criticality [49]. With the covariant approach developed in [50,51], one can get the charge related to the asymptotic Killing vector (3.10) by calculating the variation of surface charge Q(f ) = 2 8πG H dzdzγ zz Ωκf (z,z) = 1 4πG H dzdzγ zz mf (z,z),(3.14) where H represents horizon and we have used κ = 1/2r h , r h = 2m for a Schwarzschild black hole. In addition, we also added an extra factor 2 in the charge compared with the original definition in [1,34]. Note that they read the charge from the variation of it rather than directly calculating it. The factor 2 can be understood from the difference between the first law of black hole δM = T δS and the Smarr formula M = 2T S in four dimensions. Obviously, the extra factor 2 should be reasonable once the fact is considered that the charge should agree with the result of Komar integral which can be interpreted as the total energy of a stationary spacetime if we set f (z,z) as 1. According to the standard definition, the Komar integral can be written as E t = 1 4πG ∂Σ d 2 x γ (2) n µ σ ν ∇ µ K ν = M,(3.15) where Σ is a spacelike hypersurface and K µ = (1, 0, 0, 0) is Killing vector related to the time translation. Note that the form of charge for horizon supertranslation is also the same as the one for supertranslation in BMS group. This feature should be related to the fact that the ADM and Komar masses agree for stationary solution of general relativity [52,53]. Of course, we can also use the method in [52,54] to calculate the Noether charge (D − 2) form of any infinitesimal diffeomorphism for any covariant gravity theory. In general relativity 7 with or without cosmological constant, one can get the same result as the one in [1]. From the relation between Noether charge and first law of black hole [52,54], one can easily understand why the zero modes of charges defined in [1,34] correspond to entropy and angular momentum of stationary black hole. On the other hand, the supertranslation charges commute with themselves [1]. One can find that the supertranslation (3.10) will not change the energy of black hole because of the commutation {Q(f ), M } = 0. (3.16) It will not leave the black hole invariant but will only produce soft graviton as Goldstone bosons. Degenerate solutions All the results above do not depend on the equations of motion. Now we consider the on-shell case in which the Einstein equations without cosmological constant get satisfied 8 R µν − 1 2 Rg µν = 8πGT µν . (3.17) In the vacuum case without any matter, they can be simplified as R µν = 0. Generally, there are ten components in metric g µν and ten Einstein equations, but only six of them are independent because of the four Bianchi identities. So we have the freedom to choose four coordinate conditions (3.3). This looks like we can solve the whole Einstein equations and get some certain solutions. But as we have said before, there are still residual gauge invariances-supertranslations. We can find how it can happen by solving the Einstein equations. We only consider the leading nontrivial order of the equations in what follows, but of course we can solve them order by order and then get the complete solutions of the Einstein equations. First of all, by direct calculation of Ricci tensor one can find lim ρ→0 R vv = 0, lim ρ→0 R vz = − 1 2 ∂ v θ z = 0, lim ρ→0 R vz = − 1 2 ∂ v θz = 0. (3.18) 7 It is worth checking whether all results about supertranslation and soft graviton still hold for any covariant gravity theory. 8 We put the analysis for the case of Einstein equations with a cosmological constant in Appendix B. So we can set θ z and θz as only functions of (z,z). And one can get the other components of Ricci tensor at the leading order of ρ R ρv = 1 2r 2 h (2 − 4r h κ) + 2r 2 h a (2) − 2θ z θ z − 2κλ z z + D z θ z + Dzθz − 2∂ v λ z z R ρρ = 1 2r 4 h 4r h λ z z + λz z λ zz + λ zz λ zz − 4r 2 h λ (2)z z R ρz = 1 2r 2 h 2r 2 h θ (2) z − θ z λ zz + D z λ zz − Dzλ zz R ρz = 1 2r 2 h 2r 2 h θ (2) z − θzλzz + Dzλzz − D z λ zz R zz = 1 2 (γ zz (2 − 4r h κ) − θ z θz − 2κλ zz + Dzθ z + D z θz − 2∂ v λ zz ) R zz = 1 2 (−θ z θ z − 2κλ zz + D z θ z + D z θ z − 2∂ v λ zz ) Rzz = 1 2 (−θzθz − 2κλzz + Dzθz + Dzθz − 2∂ v λzz) ,(3.19) where we have taken the limit of ρ → 0 and used γ zz to lift the indices z,z. Note that there are nine different functions that appear in the first order of all Ricci tensor components. For the vacuum solution, we have R µν = 0 . From R vρ = 0 = R zz , one can get a (2) = θ z θ z 2r 2 h , λ zz = r h Dzθ z + D z θz − θ z θz + Ae −κv . (3.20) From other components in (3.19), one can directly arrive at λ zz = r h D z θ z + D z θ z − θ z θ z + Be −κv , λzz = r h Dzθz + Dzθz − θzθz + Ce −κv , λ (2) zz = 1 4r 2 h (4r h λ zz + λ z z λ zz + λ z z λ zz ) , θ (2) z = 1 2r 2 h θ z λ zz − D z λ zz + Dzλ zz , θ (2) z = 1 2r 2 h θzλzz − Dzλzz + D z λ zz . (3.21) where we can represent them as the functions of θ z and θz, and A, B, C represent arbitrary functions of (z,z), but are independent of v, which should be determined by the initial conditions. In appendix B, we consider the Einstein equations with cosmological constant and get the solutions with the same forms as those in (3.20) and (3.21). The other components R vv , R vz , R vz begin non-vanishing from the second order but also are not independent on others because of the Bianchi identities. On the contrary, we can use the those components to check the preceding solutions. For example, one can find 22) which is easy to show to be vanishing when λ zz and a with the solution in (3.20) are substituted. Furthermore, we can also calculate R vz at the second order lim ρ→0 R vv = − ρ 2r 4 h 2r 3 h a (2) − 2r h θ z θ z − λ z z + r h (D z θ z + Dzθz − ∂ v λ z z + 2r h ∂ v ∂ v λ z z ) ,(3.lim ρ→0 R vz = − ρ 4r 2 h 8r 2 h κθ (2) z + 4θ z θ z θ z + 2D z D z θ z − 2DzD z θz − 4r 2 h D z a (2) + 4r 2 h ∂ v θ (2) z + θ z (8r h κ + 4κλ z z − 6D z θ z + 2Dzθz + 4∂ v λ z z ) + 2θ z ∂ v λ zz − 2∂ v D z λ zz + 2∂ v Dzλ zz , (3.23) which is also equal to zero when we substitute λ AB and θ (2) z with the solution in (3.21) and notice that [D z , Dz]θ z = −γ zz θ z . So from the all first order Einstein equations, we can not fix the whole components of metric which can influence these first order equations. This feature can be traced back to the fact that there is still residual diffeomorphismsupertranslation corresponding to the asymptotic killing vector (3.10). All these infinitely degenerate black hole solutions can be related with each other by the supertranslation which generates the infinitesimal transformation (3.11). For example, assuming θ A has an infinitesimal transformation δθ A = −2κ∂ A f (z,z), one can get the transformation of λ AB from the solution (3.21) δλ AB = −2D A D B f (z,z) + θ A D B f (z,z) + θ B D A f (z,z), (3.24) which is compatible with Lie derivative of λ AB in (3.13). In addition, there are exponentially decay modes e −κv in these solutions. It can be related to the extension of supertranslation in [34] where they extend the form of function to allow e −κv X(z,z) which can generate another new supertranslation. But all these solutions only satisfy the vacuum Einstein equations, which does not mean they are all physical vacua with absence of all matter and radiation including gravitational radiation. Similar with the case in null infinity where BMS vacuum is defined by the vanishing of Bondi news which means there are no radiative modes, we can define the physical vacuum with black hole as ∂ v λ AB = 0 → A = B = C = 0,(3.25) which can define a stationary spacetime without radiation going in or out. On the other hand, e −κv represents a kind of decay behaviour and must approach zero with the time v increasing. So in the late time v → ∞, we can get a fully static solution and the solution will return back to the physical vacuum state as expected. Obviously all this kind of physical vacua with black hole can be derived by supertranslation from the Schwarzschild black hole (3.1) and written as (3.7) with metric functions where we have used the ellipsis to represent those higher order terms. If we only consider the approximation up to the first order of asymptotic Killing vector, the new metric by supertranslation from the Schwarzschild vacuum can be defined as g µν = g 0µν + L ξ g 0µν , where Lie derivatives is given in (3.13). This has been discussed in Refs [30,55]. Apparently, the new metric g µν will not be able to describe the vacuum solution near the horizon because it ignored all higher order terms of asymptotic Killing vector or function f (z,z). So we use (3.20-3.21) to represent the black hole solutions and label the physical vacuum through function θ(z,z) which can be related to θ A and λ AB by a (2) ≈ 0, θ A ≈ −2κD A f, λ AB ≈ −(D A D B f + D B D A f ), ...a (2) (z,z) = D z θD z θ 2r 2 h , θ A (z,z) = −D A θ(z,z), λ AB (zz) = r h (−2D A D B θ + D A θD B θ) . (3.27) Actually it is also possible for θ A to contain some higher orders of functions θ(z,z) ≈ 2κf (z,z). We need finite transformation to make sure of this point. BMS vacuum is determined by physical argument, see, e.g. (2.35) of [6] . The quantum state with black hole can be expressed as \|M, θ(z,z) or \|M, C ln with − l < n < l, − ∞ < l < ∞ (3.28) where one can use spherically harmonic functions Y ln (θ, φ) as basis to expand function θ(z,z) with expansion coefficients C ln . These infinitely degenerate physical vacua can be distinguished by soft gravitons which play the role as Goldstone bosons of breaking horizon supertranslation symmetry and make black hole possible to storage information about how the black hole was formed or the initial state. Furthermore, the gravitational memory effect near the horizon will make us to detect the variation between two different vacuum states with black hole, whose counterpart generated by supertranslation in BMS group is first pointed out by Strominger and Zhiboedov [13]. It was further illustrated by HPS in [27] that degenerate black hole with infinite soft hairs can open a window for the information loss paradox. Discussion Here we would like to discuss a little about the memory effect near the horizon. Assume a certain process generating a black hole state to another one \|M, C ln radiation − −−−−− → \|M , C ln ,(4.1) where the latter with different mass can be considered as a result of the former black hole absorbing some matter or emitting some radiation. Although thermal Hawking radiation contains no information, but the whole spacetime or black hole horizon actually has the ability to store information about matter. It means that they can have different quantum state or quantum number C l,n , namely the spacetime carries with different information. To be honest, we did not repeat the discussions in [13] to show the exact form of variation between two vacuum metrics induced by radiation. In principle, we can use the Einstein equations with radiation term to relate the degenerate black hole solutions and show that the variation can encode the information about the energy momentum tensor of radiation. We also did not use the charge defined in (3.14) to represent how to create a soft graviton. Technically, we don't have similar constraint equation like (2.5) because of ∂ u m = 0. Obviously, the technical problem derives from our primary ansatz for stationary black hole or constant surface gravity κ. Physically, we don't introduce news tensor like N zz in the Bondi coordinates (2.4), because we pay attention on the on-shell degenerate solutions without radiation in this note. News tensors terms in the BMS supertranslation charges also be viewed as Goldstone of broken BMS supertranslation. For horizon supertranslation, ingoing expansion may have similar role because of similar transformation form. On the hand, Because we want to add radiation to generate the transformation between degenerate black holes, it may make more sense for us to use non-stationary spacetime or apparent horizon to describe the process although the initial and final black holes should be stationary. For example, the Vaidya spacetime ds 2 = − 1 − 2m(v) r dv 2 + 2dvdr + 2r 2 γ zz dzdz, with ∂ v m(v) = 4πr 2 T vv , (4.2) is the simplest one with apparent horizon. But this metric satisfies the BMS gauge fixing conditions in (2.2) rather than the gauges at the horizon (3.3). It is found that Vaidya spacetime admits no BMS supertranslation field [56] due to spherical symmetry. We hope to extend our calculations about horizon supertranslation to apparent horizon in the next work. On the other hand, the discrepancy in these two kinds of gauge fixing conditions also makes us unable to relate the horizon supertranslation with the BMS supertranslation. This point disagrees with the discussion about quotient space BM S H /BM S − in [30] where they only considered the special Schwarzschild metric g 0µν . A Higher order Ricci tensor and solutions In the main content we only consider the nontrivial leading order of the components of all Ricci tensor to get the vacuum degenerate solutions. Here we give these subleading terms of the components of the Ricci tensor, from which we can get the next order components of the metric tensor. The metric conventions with higher order terms take the form g µν =    − ρ r h + ( ρ r h ) 2 + ρ 2 a (2) + ρ 3 a (3) 1 ρθ A + ρ 2 θ (2) A + ρ 3 θ (3) A 1 0 0 ρθ A + ρ 2 θ (2) A + ρ 3 θ (3) A 0 (ρ + r h ) 2 γ AB + ρλ AB + ρ 2 λ (2) AB + ρ 3 λ (3) AB   +O(ρ 3+ ), (A.1) with Ricci tensor defined in the way as R µν = R (0) µν + ρR (1) µν + O(ρ 2 ). (A.2) Besides the leading order terms given above, here we list the other components of Ricci tensor at the subleading order : 1 4r 4 h 8r 3 h γ zz a (2) + 8r h D (z θz ) − 8r 2 h θ (z θ (2) z) − 8r h θ (z θz ) + 4(1 + r 2 h a (2) − θ z θ z )λ zz − 16r 2 h κλ (2) zz + 2κ(λ zz λ zz + λzzλz z ) + 2(λ zz θ z θz + λzzθzθ z ) + 2(λ zz Dzθz + λ zz D z θ z ) − 2(λzzD z θz + λ zz Dzθ z ) + 2(θzDzλ zz + θ z D z λ zz ) − 4(θzD z λzz + θ z Dzλ zz ) + 8r 2 h D (z θ (2) z) + 2(DzD z λ zz − 2D z D z λ zz + D z Dzλzz) + ∂ v (−8r 2 h λ (2) zz + 2λ zz λ zz ) . (A.9) Note that R (1) zz term is equal to zero if we consider the solution of λ (2) zz , so it is a trivial equation that can not help us to get higher order terms of the metric. But all other Ricci tensor components are enough for us to get the full vacuum solutions at the next order, although they can not be presented in a simple form. Note that we want to solve the equations R (1) µν = 0 to get λ (2) AB , a (3) , θ(2) A , and it is easy to find that we firstly need λ (2) AB except for λ (2) zz that we have presented in (3.20). Actually from R (1) zz = 0 or R (1) zz = 0, one can read a special kind of partial differential equations (2κ + ∂ v )λ (2) AB = F AB (v, z,z), (A.10) where F AB (v, z,z) are completely determined by the lower order terms that we have been listed in R where D is an arbitrary function of (z,z). Then we can directly read off other components θ ρρ = 0, respectively. We do not present all these components here, while in principle higher order terms can also be solved order by order. B Degenerate (A)dS black hole In this appendix, we will present the degenerate black hole solution in (A)dS spacetime whose Einstein equations contain a cosmological constant. Let us start with the (A)dS-Schwarzschild black hole ds 2 = −(1 − 2m r − c r 2 l 2 )dv 2 + 2dvdr + 2r 2 γ zz dzdz, (B.1) where l represents the radius of (A)dS spacetime with c = (−)1 . The event horizon is determined by the equation 1 − 2m r h − c r 2 h l 2 = 0, with surface gravity κ = l 2 − 3cr 2 h 2l 2 r h . (B.2) By introducing a new radical coordinate ρ = r − r h , one can arrive at ds 2 = − 2m r 2 h + 2cr h l 2 ρ + c l 2 + 2m r 3 h ρ 2 dv 2 + 2dvdρ + 2(r h + ρ) 2 γ zz dzdz + O(ρ 2+ ) = −2κρ + ρ 2 r 2 h dv 2 + 2dvdρ + 2(ρ + r h ) 2 γ zz dzdz + O(ρ 2+ ), (B.3) near the horizon, whose difference from the leading order of the Schwarzschild black hole solution is just the expression of surface gravity κ. So we can take the degenerate (A)dS-Schwarzschild black hole having the same asymptotic form as that in (3.7), g µν =    −2κρ + ρ 2 r 2 h + ρ 2 a (2) 1 ρθ A + ρ 2 θ (2) A 1 0 0 ρθ A + ρ 2 θ (2) A 0 (ρ + r h ) 2 γ AB + ρλ AB + ρ 2 λ (2) AB    + O(ρ 2+ ), (B.4) where only the surface gravity is different from the one for the Schwarzschild black hole case. As we said before, the definition of supertranslation is off-shell, and determined by the gauge conditions and asymptotic conditions. As a result, one can find the same horizon supertranslation (3.10) for the (A)dS-Schwarzschild black hole. This feature can be considered as an advantage of horizon supertranslation, compared with BMS supertranslation which is based on the null infinity of asymptotically flat spacetime, while the null infinity is absent in asymptotically (A)dS spacetime. C Degenerate Rindler horizon In this appendix, we discuss the Rindler horizon case which is also studied in [29]. Just like the above, we can easily transform traditional Rindler metric into a new set of coordinates which satisfies the horizon gauge fixing conditions (3.3). Starting from the standard Minkowski coordinates, one can arrive at ds 2 = −dT 2 + dX 2 + dY 2 + dZ 2 = e 2κx (−dt 2 + dx 2 ) + dY 2 + dZ 2 , (C.1) by the transformation between the inertial coordinate system and that of uniformly accelerated observer with acceleration κ X = κ −1 e κx cosh κt, T = κ −1 e κx sinh κt. (C.2) Introducing a new frame defined as e 2κx = (1 + κx) 2 = 2κρ, z = Y + iZ √ 2 ,z = Y − iZ √ 2 , (C.3) we can rewrite the Rindler frame in the form ds 2 = −(1 + κx) 2 dt 2 + dx 2 + dY 2 + dZ 2 = −2κρdt 2 + dρ 2 2κρ + 2dzdz, (C.4) in which the horizon is located at ρ = 0. This kind of coordinates can also be obtained from a general black hole solution. Finally, we can rewrite it to the desired form ds 2 = −2κρdv 2 + 2dvdρ + 2dzdz, (C. 5) with the transformation t → v − g(ρ), ρ → e 2κg . (C. 6) Consider degenerate solution in the near horizon region, we take the same asymptotic conditions with (3.6), and write the metric in the matrix form as g µν =    −2κρ + ρ 2 a (2) 1 ρθ A + ρ 2 θ (2) A 1 0 0 ρθ A + ρ 2 θ (2) A 0 δ AB + ρλ AB + ρ 2 λ (2) AB    + O(ρ 2+ ), (C.7) which corresponds to Ω(z,z) = γ zz . From the results in [1], one can see that the horizon supertranslation for the Rindler spacetime reads ξ = f (z,z)∂ v + ρ 2 γ zz 2 θ A D A f ∂ ρ + −ργ zz D A f + ρ 2 γ 2 zz 2 λ AB D B f ∂ A + O(ρ 3 ), (C.8) ( 1 ) 1AB with a little complicate form. So the expected results take the form asλ (2) AB = De −2κv + e −2κv v 1 e 2κv F AB (v , z,z)dv , (A.11) Next we consider the degenerate on-shell solutions which satisfy the Einstein equations with a cosmological constant where we have used the definition of horizon to simplify the expression of λ zz . It is easy to find all results are totally equal to the solutions (3.20) and (3.21) for the Schwarzschild black hole case if one represents κ by 1/2r h . Thus all discussions about infinitesimally degenerate black hole solutions keep valid for the (A)dS black hole as well. Here we follow the notions in[35,37]. The PHD thesis[38] is a simple and good review for BMS gauge in four and three dimensions. Here we adopt the simplified notions in[6,10] which are convenient for discussions on soft theorem and soft hair. See the discussion about the Christodoulou-Klainerman space in[6] and recent study about vacua of gravitational field in[41] In the inverse metric form the coordinate conditions can be written as g vv = 0, g vA = 0, g ρv = 1. AcknowledgmentsThis work was finalized during a visit by R.G. Cai as a visiting professor to the Yukawa Institute for Theoretical Physics, Kyoto University, the warm hospitality extended to him is greatly appreciated. We thank Pujian Mao's suggestions and discussions about the manuscript. This work was supported in part by the National Natural Science Foundation of China under Grants No.11375247 and No.11435006, and in part by a key project of CAS, Grant No.QYZDJ-SSW-SYS006.(2)zz λ zz + λ(2)zz λzz + 2λ(2)zz λ zz ) + 8r 2 h λ z z + 6r h (λ z z λzz + λz z λ zz ) + λ z z λ z z λ z z + 3λ z z λ zz λz z ,Note that in this case, only the equations of R vv , R zz and R vA change at the leading order, compared to the case without the cosmological constant. From the results of (3.19), one can find that the solution readandwith conserved charge at horizonConsidering the vacuum Einstein equations R µν = 0 at the leading order in ρ, one will arrive at the solution as follows,(C.10)Here we did not give explicit expressions for all components of Ricci tensor. 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[ "Strong Structural Controllability of Diffusively Coupled Networks: Comparison of Bounds Based on Distances and Zero Forcing", "Strong Structural Controllability of Diffusively Coupled Networks: Comparison of Bounds Based on Distances and Zero Forcing" ]	[ "Yasin Yazıcıoglu ", "Mudassir Shabbir ", "Waseem Abbas ", "Xenofon Koutsoukos " ]	[]	[]	We study the strong structural controllability (SSC) of diffusively coupled networks, where the external control inputs are injected to only some nodes, namely the leaders. For such systems, one measure of controllability is the dimension of strong structurally controllable subspace, which is equal to the smallest possible rank of controllability matrix under admissible (positive) coupling weights. In this paper, we compare two tight lower bounds on the dimension of strong structurally controllable subspace: one based on the distances of followers to leaders, and the other based on the graph coloring process known as zero forcing. We show that the distance-based lower bound is usually better than the zero-forcing-based bound when the leaders do not constitute a zero-forcing set. On the other hand, we also show that any set of leaders that can be shown to achieve complete SSC via the distance-based bound is necessarily a zero-forcing set. These results indicate that while the zero-forcing based approach may be preferable when the focus is only on verifying complete SSC, the distance-based approach is usually more informative when partial SSC is also of interest. Furthermore, we also present a novel bound based on the combination of these two approaches, which is always at least as good as, and in some cases strictly greater than, the maximum of the two bounds. We support our analysis with numerical results for various graphs and leader sets.	10.1109/cdc42340.2020.9304420	[ "https://arxiv.org/pdf/2008.07495v1.pdf" ]	221,139,608	2008.07495	bc8bd75e0035081602fc5f79f4f5fe18ec0a5f6d	Strong Structural Controllability of Diffusively Coupled Networks: Comparison of Bounds Based on Distances and Zero Forcing Yasin Yazıcıoglu Mudassir Shabbir Waseem Abbas Xenofon Koutsoukos Strong Structural Controllability of Diffusively Coupled Networks: Comparison of Bounds Based on Distances and Zero Forcing We study the strong structural controllability (SSC) of diffusively coupled networks, where the external control inputs are injected to only some nodes, namely the leaders. For such systems, one measure of controllability is the dimension of strong structurally controllable subspace, which is equal to the smallest possible rank of controllability matrix under admissible (positive) coupling weights. In this paper, we compare two tight lower bounds on the dimension of strong structurally controllable subspace: one based on the distances of followers to leaders, and the other based on the graph coloring process known as zero forcing. We show that the distance-based lower bound is usually better than the zero-forcing-based bound when the leaders do not constitute a zero-forcing set. On the other hand, we also show that any set of leaders that can be shown to achieve complete SSC via the distance-based bound is necessarily a zero-forcing set. These results indicate that while the zero-forcing based approach may be preferable when the focus is only on verifying complete SSC, the distance-based approach is usually more informative when partial SSC is also of interest. Furthermore, we also present a novel bound based on the combination of these two approaches, which is always at least as good as, and in some cases strictly greater than, the maximum of the two bounds. We support our analysis with numerical results for various graphs and leader sets. I. INTRODUCTION Networks of diffusively coupled agents, where each node's state is attracted toward the weighted average of its neighbors' states, appear in numerous systems such as sensor networks, distributed robotics, power grids, social networks, and biological systems. Such systems are often modeled by using their interaction graphs where the nodes represent the agents, and the weighted edges denote the couplings among agents. One major research question regarding such systems is whether a desired global behavior can be induced by injecting external inputs to only a subset of agents, so called the leaders. This question has motivated numerous studies on relating network controllability to the structure of the interaction graph. Various graph theoretic tools have been utilized to provide topology-based characterizations of network controllability. Examples include equitable partitions (e.g., [1]), maximum matchings (e.g., [2], [3]), centrality based measures (e.g., [4], [5]), dominating sets (e.g., [6] (e.g., [7], [8], [9], [10]), and zero forcing (e.g., [11], [12], [13], [14], [15]). In this paper, we focus on the strong structural controllability of diffusively coupled networks. More specifically, we consider the dimension of strong structurally controllable subspace (SSCS), i.e., the minimum possible rank of controllability matrix under weighted Laplacian dynamics, as a measure of controllability. Two graph theoretic concepts are known to yield a tight lower bound on this measure: distances and zero forcing. In this paper, we first compare these two approaches. We characterize various cases where the distance-based lower bound is greater than the zeroforcing-based bound. On the other hand, we also show that, for any network of n nodes, any set of leaders that makes the distance-based bound equal to n is necessarily a zero forcing set, i.e., they also make the zero-forcing-based bound equal to n. These results indicate that while the zero-forcing-based approach is better for verifying complete strong structural controllability, the distance-based approach is usually more informative when the leaders do not constitute a zero forcing set. We also propose a novel bound based on the combination of these two methods, which is always at least as good as, and in some cases greater than, the maximum of the two bounds. Finally, we support our analysis with some numerical results. The organization of this paper is as follows: Section II provides some preliminaries. Section III presents our results regarding the comparison of bounds. Section IV provides a novel bound based on the combination of distance-based and zero-forcing-based methods. Some numerical results are given in Section V. Finally, Section VI concludes the paper. II. PRELIMINARIES A. Graph Basics We consider a network represented by a simple directed graph G = (V, E) where the node set V = {v 1 , v 2 , . . . , v n } represent agents, and the edge set E represents interconnections between agents. An edge from a node v i ∈ V to a node v j ∈ V is denoted by e ij . The out-neighborhood of node v i is N i {v j ∈ V : e ij ∈ E}. The in-neighborhood of node v i is N i {v j ∈ V : e ji ∈ E}. The distance d(v i , v j ), is simply the number of edges on the shortest path from v i to v j . Accordingly, d(v i , v i ) = 0 and d(v i , v j ) = ∞ if there is no path from v i to v j . The graph is strongly connected if there is a path from any node to any other node. The weight function w : E → R + assigns a positive weight w(e ij ) to each edge e ij , which will denote how strongly v i is influenced by v j in the dynamical model below. B. System Model While the model can easily be extended to agents with higher-dimensional states, for the sake of simplicity let each agent v i ∈ V have a state x i ∈ R. The overall state of the system is x = x 1 x 2 · · · x n T ∈ R n . The states evolve under the weighted Laplacian dynamics, x = −L w x + Bu,(1) where L w ∈ R n×n is the weighted Laplacian matrix of G and is defined as L w = ∆ − A w . Here, A w ∈ R n×n is the weighted adjacency matrix defined as [A w ] ij = w(e ij ) if e ij ∈ E, 0 otherwise, and ∆ ∈ R n×n is the degree matrix whose entries are [∆] ij = n k=1 A ik if i = j 0 otherwise. The matrix B ∈ R n×m in (1) is an input matrix, where m is the number of leaders (inputs), which are the nodes to which an external control signal is applied. Let V = { 1 , 2 , · · · , m } ⊆ V be the set of leaders, then [B] ij = 1 if v i = j 0 otherwise.(2) C. Strong Structural Controllability A state x f ∈ R n is a reachable state if there exists an input u that can drive the network in (1) from the origin to x f in a finite amount of time. A network G = (V, E) in which edges are assigned weights according to the weight function w, and contains V ⊆ V leaders is called completely controllable if every point in R n is reachable. Complete controllability can be checked via the rank of controllability matrix, i.e., Γ(L w , V ) = [ B (−Lw)B (−Lw) 2 B · · · (−Lw) n−1 B ] , where B is defined as in (2). The network is completely controllable if and only if the rank of Γ(L w , V ) is n, and in such case (L w , B) is called a controllable pair. Note that edges in G define the structure-location of zero and non-zero entries in the Laplacian matrix-of the underlying graph, for instance, see Fig. 1. For any given graph G = (V, E) and V leaders, the rank of controllability matrix depends on the weights assigned to edges. Fig. 1: A graph and its structured Laplacian whose non-zero off-diagonal entries are positive and rows sum to zero. v 1 v 2 v 3 v 4 v 5 v 6           × × 0 0 0 0 0 × 0 × 0 0 × × × 0 0 0 0 0 0 × × 0 0 0 × × × × 0 0 0 0 × ×           A network G = (V, E) with V leaders is strong structurally controllable if (L w , B) is a controllable pair for any choice of weight function w. The dimension of strong structurally controllable subspace (SSCS), denoted by γ(G, V ), is the smallest possible rank of controllability matrix under feasible weights, i.e., γ(G, V ) = min w:E→R + (rank Γ(L w , V )) ,(3) where the minimum is taken over all feasible weight functions w : E → R + . Roughly, γ(G, V ) quantifies how much of the network can be controlled through the leaders V under any feasible choice of edge weights. Remark 2.1: The original notion of strong structural controllability [16] considers the worst-case controllability under any allocation of the non-zero values in a system's structure matrix. Our focus is on the worst-case controllability under weighted Laplacian dynamics, which narrows down the feasible set of system matrices. However, it is worth mentioning that both the distance-based and the zero-forcingbased methods, which will be explained next, are actually applicable to more generalized dynamics (e.g., [10], [14]). D. Distance-based Lower Bound: δ(G, V ) Given a network with m leaders V = { 1 , · · · , m }, we define the distance-to-leaders (DL) vector of each v i ∈ V as D i = d(v i , 1 ) d(v i , 2 ) · · · d(v i , m ) T ∈ Z m . The j th component of D i , denoted by [D i ] j , is equal to the distance of v i to j . Next, we provide the definition of pseudo-monotonically increasing sequences of DL vectors. Definition (Pseudo-monotonically Increasing (PMI) Sequence) A sequence of distance-to-leaders vectors D is PMI if for a vector D i in the sequence, there exists some π(i) ∈ {1, 2, · · · , m} such that [D i ] π(i) < [D j ] π(i) , ∀j > i. We say that D i satisfies the PMI property at coordinate π(i) whenever [D i ] π(i) < [D j ] π(i) , ∀j > i. An example of DL vectors is illustrated in Fig. 2, where a PMI sequence of length six can be constructed as D = 3 0 , 0 4 , 1 4 , 2 1 , 3 2 , 4 3 .(4) Indices of circled values in (4) are the coordinates, π(i), at which the corresponding distance-to-leaders vectors are satisfying the PMI property. The longest PMI sequence of distance-to-leaders vectors is related to the dimension of SSCS as stated in the following result. Theorem 2.2: [9] Consider any network G = (V, E) with the leaders V ⊆ V . Let δ(G, V ) be the length of longest PMI sequence of distance-to-leaders vectors with at least one finite entry. Then, δ(G, V ) ≤ γ(G, V ). (5) Remark 2.3: While the bound in (5) was presented for connected undirected graphs in Theorem 3.2 in [9], it also holds for any choice of leaders on strongly connected graphs as shown in Remark 3.1 in [9]. Such connectivity properties already ensure that all DL vectors have only finite entries. The bound can easily be extended to graphs without strong connectivity by excluding the DL vectors of all ∞, which belong to followers that can not be influenced by any leader. E. Zero-forcing-based Lower Bound: ζ(G, V ) We first give the definitions of zero forcing process and derived set. Definition (Zero Forcing Process) Given a graph G = (V, E) where each node is initially colored either white or black, zero forcing process is defined by the following coloring rule: if v ∈ V is colored black and has exactly one white in-neighbor u, then the color of u is changed to black and u is said to be infected by v. Definition (Derived Set) Given an initial set of black nodes V ⊆ V (called the input set) in a graph G = (V, E), there exists a unique derived set, dset(G, V ) ⊆ V , which is the resulting set of black nodes when no further color changes are possible under the zero forcing process. An input set V is called a zero forcing set (ZFS) if dset(G, V ) = V . Theorem 2.4: [14] For any network G = (V, E) with the leaders V ⊆ V , ζ(G, V ) ≤ γ(G, V ), where ζ(G, V ) = \|dset(G, V )\| is the size of the derived set corresponding to the input set V . Proof: Proof follows from Lemma 4.2 in [14], which shows that for a set of state matrices including weighted Laplacians as a subset, the controllable subspace always contains a \|dset(G, V )\|-dimensional subspace. F. Computation of the Bounds For any given network with n nodes and m leaders, all pair-wise distances can be computed in O(n 3 ) time (e.g., [17]). Given the distances, δ(G, V ) can be computed in O(m(n log n + n m )) time [18]. When the number of leaders makes this computation intractable, an approximation (underestimation), which was shown to be very close to the exact value on various networks, can be obtained in O(mn log n) time [18]. On the other hand, ζ(G, V ) can be computed in O(n 2 ) time by recursively applying the coloring rule to the in-neighbors of infected nodes until no further color change is possible. III. COMPARISON OF BOUNDS In this section, we compare the distance-based bound, δ(G, V ), and the zero-forcing-based bound, ζ(G, V ). It is worth mentioning that both δ(G, V ) and ζ(G, V ) are tight bounds. For instance, in the case of undirected graphs, any path graph in which one of the end nodes is a leader, or any cycle graph in which two adjacent nodes are leaders satisfy ζ(G, V ) = δ(G, V ) = γ(G, V ) = n. Furthermore, neither of these two tight bounds is guaranteed to be at least as good as the other in all possible cases. We provide one example for ζ(G, V ) > δ(G, V ) and one example for Fig. 3. Accordingly, we aim to identify when one bound may be preferable to the other. δ(G, V ) > ζ(G, V ) in(a) v 1 (b) v 2 v 3 v 4 v 1 v 2 v 3 v 4 v 5 v 6 A. Advantages of Using the Distance-based Bound We will present two results, Theorems 3.1 and 3.2, identifying some rich cases where δ(G, V ) > ζ(G, V ). Later in Section V, we will also provide numerical results showing that δ(G, V ) is actually significantly greater than ζ(G, V ) in many cases that are not limited to those captured by Theorems 3.1 and 3.2. Our first result in this section shows that δ(G, V ) is greater than ζ(G, V ) whenever each leader has at least two followers as in-neighbors. Note that this condition is very likely to occur when a small number of leaders are scattered over a large graph where most nodes have an in-degree of two or more (e.g., most regular graphs, random graphs, scale-free networks). Theorem 3.1: Consider any graph G = (V, E) with n nodes and m leaders V ⊆ V . If each leader has at least two followers as in-neighbors, then δ(G, V ) > ζ(G, V ). Proof: If every leader has incoming links from at least two followers, then none of the followers will be forced when only the leaders are the black nodes. Accordingly, the dset(G, V ) = V and ζ(G, V ) = m. On the other hand, we can always find a PMI sequence of DL vectors whose length is greater than m in such a case. As an example, consider the following sequence that has a length of m + 1: 1) start with the DL vectors of leaders in any order, 2) add the DL vector of a follower who has a distance of one to one of the leaders. Since each leader is the only node who has a distance of zero to itself, those self-distance entries can be selected as the entries that satisfy the PMI rule. Hence, the longest possible PMI sequence would have a length of at least m + 1, which implies δ(G, V ) > ζ(G, V ). Our next result shows that for any single-leader network where each follower has a finite distance to the leader, δ(G, V ) < n ensures that δ(G, V ) > ζ(G, V ). Theorem 3.2: For any G = (V, E) with n nodes and a single leader v l ∈ V such that d( v i , v l ) < ∞ for all v i ∈ V , δ(G, V ) < n ⇒ δ(G, V ) > ζ(G, V ). (6) Proof: Since the left side of (6) can never be true for n = 1, we focus on networks with n ≥ 2 and we will prove the claim via contradiction. Suppose that δ(G, V ) < n and ζ(G, V ) ≥ δ(G, V ). Note that if v l has more than one follower as in-neighbor, then the zero forcing process starting with the input set {v l } would not propagate and we would have ζ(G, V ) = 1. Furthermore, for any network with a single leader v l ∈ V such that d(v i , v l ) < ∞ for all v i ∈ V , δ(G, {v l }) = max vi∈V d(v l , v i ) + 1,(7) which is always greater than one. Hence, if ζ(G, V ) ≥ δ(G, V ), then v l must have only one in-neighbor, say v i , who will be infected by v l under the zero forcing process. Now, if n = 2 (there are no other followers), then we end up with δ(G, V ) = ζ(G, V ) = 2, which contradicts with δ(G, V ) < n. On the other hand, if n > 2 then we can repeat the same reasoning by removing v l from the network, since v l has no impact on the infection of nodes at distance of two or more from itself, and treating the remaining network as a system with a single leader v i with d(v j , v i ) < ∞ for every v j = v l (v i being the only in-neighbor of v l implies that the paths from all other nodes to v l goes through v i , hence d(v j , v i ) < ∞). Accordingly, we can show that if ζ(G, V ) ≥ δ(G, V ), then each follower must have a distinct distance from v l , which implies δ(G, V ) = ζ(G, V ) = n and results in a contradiction with δ(G, V ) < n. Remark 3.3: In light of (7), the only connected undirected network with a single-leader that yields δ(G, V ) = n is a path graph with a terminal node being the leader. Hence, Theorem 3.2 implies that for all other connected undirected networks with a single-leader, we have δ(G, V ) > ζ(G, V ). B. Advantages of Using the Zero-forcing-based Bound Here, we show that one major advantage of using the zero-forcing-based approach is that it is better at verifying complete strong structural controllability. More specifically, we show that if δ(G, V ) = n, then V must be a zero forcing set. Note that the converse is not true in general, i.e., it is possible to have a zero forcing set V such that δ(G, V ) < n, as already shown by the example in Fig. 3b. Clearly, such examples do not exist for single-leader networks due to Theorem 3.2. Proof: The claim is trivial for the cases when V = V since δ(G, V ) = ζ(G, V ) = n. Hence we focus on V ⊂ V (n > m) in the proof. Let D = [D 1 D 2 · · · D n ] be a PMI sequence consisting of all the distance-to-leaders (DL) vectors such the first \|V \| vectors belong to the leaders. Note that there is no loss of generality here since for any PMI sequence of DL vectors, the vectors belonging to the leaders can be moved to the beginning of the sequence and the distance of each leader to itself (zero) satisfies the PMI rule. Without any loss of generality, let the nodes be re-labeled based on the order of their DL vectors in the sequence, i.e., D i is the DL vector of v i ∈ V for all i = 1, 2, . . . , n. Furthermore, let π(i) denote the dimension of D i that satisfies the PMI rule, i.e., [D i ] π(i) < [D j ] π(i) , ∀j > i.(8) Due to Lemma 4.1 in [9], if D is the longest possible PMI sequence of DL vectors, then it must satisfy [D i ] π(i) = min j≥i [D j ] π(i) , ∀i ∈ {1, . . . , n − 1}. For each i ∈ {m + 1, . . . , n}, let W i = {v i , . . . , v n } ⊆ V be the owners of the DL vectors in the subsequence of D starting with the i th entry. We will show that ∀i > m, ∃k < i : N k ∩ W i = {v i },(9) where N k is the set of in-neighbors of v k . Note that (9) would imply that if all the nodes {v 1 , . . . , v i−1 } are infected, then v i becomes infected under the zero-forcing process. Accordingly, we can conclude that ζ(G, V ) = n since starting with all the leaders being infected, all the followers would eventually become infected. Note that (9) clearly holds for i = n since W n = {v n } and v n must have at least one out-neighbor in {v 1 , . . . , v n−1 } as otherwise its DL vector would be all ∞ and not included in any PMI sequence, leading to the contradiction δ(G, V ) < n. Now, for the sake of contradiction, suppose that (9) is not true for some i ∈ {m + 1, . . . , n − 1}. Let v k be any out-neighbor of v i such that [D k ] π(i) = [D i ] π(i) − 1. Clearly such a neighbor always exists: v k is either the leader l π(i) or another follower on the shortest path from v i to l π(i) . Furthermore, k < i due to (8). Now suppose that v k has another in-neighbor v j such that j > i. Then, [D j ] π(i) ≤ [D k ] π(i) + 1 = [D i ] π(i) , which contradicts with (8). Hence, (9) must be true, and it implies that ζ(G, V ) = n. IV. COMBINED BOUND: δ(G, DSET(G, V )) Our analysis so far has shown that both the distancebased bound, δ(G, V ), and the zero-forcing-based bound, ζ(G, V ), have their own merits. Given these results, it is only natural to ask if it is possible to find a novel bound that combines the strengths of distance-based and zero-forcingbased methods. In this regard, one trivial approach is taking the maximum of the two bounds. While guaranteed to be at least as good as either of the bounds alone, this approach does not reveal any additional information compared to the two original bounds. In this section, we present a novel bound that fuses the strengths of distance-based and zeroforcing-based approaches. More specifically, we show that the length of the longest PMI sequence of distances to the derived set of leaders, i.e., δ(G, dset(G, V )), provide a tight lower bound on the dimension of SSCS. We show that this novel bound is always at least as good as, and sometimes greater than, either of the bounds alone. To this end, we first provide a result on the invariance of SSCS in diffusively coupled networks to the addition of every node in the derived set, dset(G, V ), as leaders. Theorem 4.1: [14] For any network G = (V, E) with the leaders V ⊆ V , and any weight function w : E → R + , range(Γ(L w , V )) = range(Γ(L w , dset(G, V ))), (10) where range(Γ) is the range space of controllability matrix. Proof: The proof follows from Lemma 4.1 in [14], which shows a stronger condition, i.e., (10) holds for a set of state matrices that contain weighted Laplacians as a subset. δ(G, V ), ζ(G, V ) ≤ δ(G, dset(G, V )) ≤ γ(G, V ). Proof: First, we show that δ(G, dset(G, V )) ≤ γ(G, V ). In light of (3) and (10), γ(G, dset(G, V )) = γ(G, V ).(11) Due to Theorem 2.2, δ(G, dset(G, V )) ≤ γ(G, dset(G, V )).(12) Using (11) and (12), we get δ(G, dset(G, V )) ≤ γ(G, V ). Next, we show that δ(G, dset(G, V )) ≥ ζ(G, V ). Since the DL vectors of leaders can always be included in the beginning of a PMI sequence (self-distances are uniquely zero), δ(G, V ) ≥ \|V \| for any V ⊆ V . Hence, δ(G, dset(G, V )) ≥ \|dset(G, V )\| = ζ(G, V ). Finally, we show that δ(G, dset(G, V )) ≥ δ(G, V ). Since the initial set of infected nodes (input nodes) are always contained in the derived set, we have V ⊆ dset(G, V ). Accordingly, for any PMI sequence D of DL vectors under the leader set V , there is an equally long PMI sequence of DL vectors D under the leader set dset(G, V ), which has the DL vectors of the same nodes in the same order as D. Hence, the longest possible PMI sequence of DL vectors with the additional leaders can not be shorter, i.e., δ(G, dset(G, V )) ≥ δ(G, V ). Remark 4.3: While Theorem 4.2 shows that the combined bound is at least as good as the distance-based and zero-forcing-based bounds, it should also be emphasized that there exist networks G = (V, E) and leader sets V ⊆ V , where the combined bound is strictly better than the two original bounds, i.e., δ(G, dset(G, V )) > δ(G, V ), ζ(G, V ). We provide two such examples in Fig. 4. V. NUMERICAL RESULTS We compare the lower bounds on the dimension of strong structurally controllable subspace on Erdös-Rényi (ER) and Barabási-Albert (BA) graphs. ER graphs are the ones in which any two nodes are adjacent with a probability p. BA graphs are obtained by adding nodes to an existing graph one at a time. Each new node is adjacent to ε existing nodes that are chosen with probabilities proportional to their degrees. In all the simulations, we consider undirected graphs with n = 100 nodes. In Figs. 5 and 6, we plot lower bounds on the dimension of SSCS, including δ(G, V ), ζ(G, V ) and δ(G, dset(V ( )), as a function of number of leaders \|V \| = . We select the leader nodes randomly. Each point on the plots corresponds to the average of 100 randomly generated instances. While we computed the exact value of ζ(G, V ), we used the greedy approximation (underestimation) in [18] for computing δ(G, V ) and δ(G, dset(V ( )) due to the large number of leaders. While this approximation was shown to be very close in [18], the true gap between these two bounds and ζ(G, V ) may be larger than shown in the plots. In all the plots in Figs. 5 and 6, we observe that the distance-based bound δ(G, V ) starts above the ZFS-based bound ζ(G, V ), which is expected due to Theorem 3.2 (or Remark 3.3). Furthermore, δ(G, V ) is usually significantly larger than ζ(G, V ), especially when the number of leaders is small. This can be explained by Theorem 3.1 since most of the nodes in these networks have degrees of two or more. In the ER graphs the expected degree of each node is approximately pn, and each node in the BA graphs has a degree of ε or more. Indeed, all the plots show a linear trend in ζ(G, V ) when the number of leaders is small, indicating ζ(G, V ) ≈ \|V \|. Note that when ζ(G, V ) = \|V \|, trivially δ(V, dset(V )) = δ(G, V ), which explains why the distance-based and combined bounds mostly overlap until the number of leaders is sufficiently large and the zero-forcingbased bound departs from the initial linear regime. While the difference between the combined bound δ(V, dset(V )) and δ(G, V ) was observed to be insignificant in these simulations, it is worth emphasizing that δ(V, dset(V )) is the only bound guaranteed to be at least as good as the other two in all possible cases (Theorem 4.2) and the improvement with respect to δ(G, V ) may be more significant for other families of networks. Finally, we see in all the plots that the three bounds approach each other as they all increase toward n, which is expected due to Theorem 3.4. VI. CONCLUSION In this paper, we focused on the the dimension of strong structurally controllable subspace (SSCS) of networks under weighted Laplacian dynamics. We compared two tight lower bounds on the dimension of SSCS: one based on distances and the other based on zero forcing. We characterized various cases where the distance-based lower bound is guaranteed to be greater than the zero-forcing-based bound. On the other hand, we also show that, for any network of n nodes, any set of leaders that makes the distance-based bound equal to n is necessarily a zero forcing set. These results indicate that while the zero-forcing-based approach may be a better choice for verifying complete strong structural controllability, the distance-based approach is usually more informative when the leaders do not constitute a zero forcing set. We also present a novel bound based on the combination of these two approaches, which is always at least as good as, and in some cases strictly better than, the maximum of the two bounds. Finally, we numerically compared the bounds on various networks. As a future direction, we plan to improve the proposed combined bound, for example by utilizing the invariance of controllable subspace to the addition/removal of links between leaders [19]. Obtaining a formal characterization of cases where the zero-forcing bound is guaranteed to be greater than the distance-based bound is another direction we plan to explore. Furthermore, the distance-based bound was recently utilized for analyzing the robustness-controllability trade-off in networks [20]. We intend to use the combined bound for further exploration of such trade-offs. Fig. 2 : 2A network with two leaders, V = {v 1 , v 6 }, and the corresponding distance-to-leaders (DL) vectors. Fig. 3 : 3Two networks and their leaders show in gray. For the network in (a), δ(G, V ) = 3, ζ(G, V ) = 1. For the network in (b), δ(G, V ) = 5, ζ(G, V ) = 6. Theorem 3 . 4 : 34For any graph G = (V, E) with n nodes and any set of m leaders V ⊆ V , δ(G, V ) = n ⇒ ζ(G, V ) = n. Theorem 4 . 2 : 42Consider any network G = (V, E) with the leaders V ⊆ V . Then, Fig. 4 : 4Two networks and their leaders (gray). In (a): δ(G, dset(G, V )) = 5, δ(G, V ) = 4, ζ(G, V ) = 3. In (b): δ(G, dset(G, V )) = 9, δ(G, V ) = 6, ζ(G, V ) = 5. Fig. 5 :Fig. 6 : 56Comparison of ZFS-based ζ(G, V ), distance-based δ(G, V ) and combined δ(G, dset(V )) bounds on the dimension of SSCS in ER graphs. Comparison of ZFS-based ζ(G, V ), distance-based δ(G, V ) and combined δ(G, dset(V )) bounds on the dimension of SSCS in BA graphs. ), distances Yasin Yazıcıoglu is with the Department of Electrical and Computer Engineering at the University of Minnesota, Minneapolis, MN, USA. Email: [email protected] Mudassir Shabbir is with the Computer Science Department at the Information Technology University, Lahore, Punjab, Pakistan. Email: [email protected] Waseem Abbas and Xenofon Koutsoukos are with the Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, USA (e-mails: [email protected], [email protected]). Controllability of multi-agent systems from a graph-theoretic perspective. 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[]	[ "Jan Harm ", "Van Der ", "Walt " ]	[]	[ "Mathematics Subject Classification. Primary 46A19" ]	It is known that there are complete, Hausdorff and regular convergence vector spaces X and Y such that Lc (X, Y ), the space of continuous linear mappings from X into Y equipped with the continuous convergence structure, is not complete. In this paper, we give sufficient conditions on a convergence vector space Y such that Cc (X, Y ) is complete for any convergence space X. In particular, we show that this is true for every complete and Hausdorff topological vector space Y .	null	[ "https://arxiv.org/pdf/1004.1365v1.pdf" ]	115,175,289	1004.1365	b70c546a44bc2ea02f9f0b3330baaba5b2f970a9	2000 Jan Harm Van Der Walt Mathematics Subject Classification. Primary 46A19 46102000arXiv:1004.1365v1 [math.FA] A NOTE ON THE COMPLETENESS OF C c (X, Y )and phrases Convergence spaceContinuous convergenceCompleteness It is known that there are complete, Hausdorff and regular convergence vector spaces X and Y such that Lc (X, Y ), the space of continuous linear mappings from X into Y equipped with the continuous convergence structure, is not complete. In this paper, we give sufficient conditions on a convergence vector space Y such that Cc (X, Y ) is complete for any convergence space X. In particular, we show that this is true for every complete and Hausdorff topological vector space Y . Introduction It is well known [3] that C c (X), the space of continuous, scalar-valued functions on a convergence space X equipped with the continuous convergence structure, is a complete convergence vector space. An immediate consequence of this fact, see [2], is that the continuous dual L c X of a convergence vector space X is complete. On the other hand, Butzmann [5] gave an example of Hausdorff, regular and complete convergence vector spaces X and Y such that the convergence vector space L c (X, Y ) is not complete. Here, as is standard in the literature, see for instance [2], we denote by L(X, Y ) the vector space of continuous linear mappings from X into Y , and L c (X, Y ) denotes this space equipped with the continuous convergence structure. In this paper, we show that if Y is complete, Hausdorff and topological, then C c (X, Y ) is complete for every convergence space X. An immediate consequence is that L c (X, Y ) is complete whenever X and Y are convergence vector spaces with Y Hausdorff, complete and topological. This is essentially known in the locally convex case [2], [3]. Indeed, if Y is locally convex, Hausdorff and complete, then Y is isomorphic to L c L c Y , which is a closed subspace of C c (L c Y ). Thus L c (X, Y ) is isomorphic to a closed subspace of C c (X, C c (L c Y )). By the Universal Property of the continuous convergence structure, C c (X, C c (L c Y )) is isomorphic to C c (X × L c Y ), which is complete [3]. Hence L c (X, Y ) is a closed subspace of a complete convergence vector space, and is therefore complete. A completeness result We now show that the following more result holds. This result generalizes [2, Theorem 3.1.15]. Theorem 2.1. Let X be a convergence space and Y a Hausdorff, complete topo- logical vector space. Then C c (X, Y ) is complete. Proof. Let Φ be a Cauchy filter on C c (X, Y ) so that ∀ x ∈ X : ∀ F ∈ λ X (x) : ω X,Y (F , Φ − Φ) ∈ λ Y (0) , (2.1) where ω X,Y : X × C(X, Y ) → Y is the evaluation mapping, defined through ω X,Y (x, f ) = f (x) . In particular, upon setting F = [x] in (2.1) we obtain Φ (x) − Φ (x) = Φ ([x]) − Φ ([x]) = ω X,Y ([x], Φ − Φ) ∈ λ Y (0) for every x ∈ X. Therefore Φ(x) is a Cauchy filter in Y for every x ∈ X. Since Y is complete and Hausdorff, it follows that ∀ x ∈ X : ∃! x Φ ∈ Y : Φ(x) ∈ λ Y (x Φ ) . Define the mapping f : X → Y through f : X ∋ x → x Φ ∈ Y. (2.2) We show that f is continuous. Note that, since Y is topological, there is a collection B of closed subsets of Y such that filter G = [B] converges to 0 and ∀ F ∈ λ Y (0) : G ⊆ F . (2.3) Let F converge to x 0 ∈ X. Without loss of generality, we may assume that F ⊆ [x 0 ]. Since the filter ω X,Y (F , Φ − Φ) converges to 0 in Y , it follows by (2.3) that G ⊆ ω X,Y (F , Φ − Φ). We therefore have ∀ B ∈ B : ∃ A B ∈ Φ : ∃ F B ∈ F : ω X,Y (F B , A B − A B ) ⊆ B so that (A B − A B ) (F B ) = g(x) − h(x) g, h ∈ A B x ∈ F B ⊆ B. In particular, ∀ x ∈ F B : ∀ g ∈ A B : A B (x) = {h(x) : h ∈ A B } ⊆ g (x) + B (2.4) Since f (x) is defined as the limit of Φ(x) in Y , it follows that f (x) ∈ a Y (A B (x)), where a Y denotes the adherence operator in Y . Since B is closed in Y , it follows from (2.4) that ∀ g ∈ A B : ∀ x ∈ F B : f (x) ∈ g (x) + B . (2.5) For every B ∈ B and A ∈ Φ, pick some g ∈ A B ∩ A. Since g is continuous, the filter g (F ) converges to g (x 0 ) in Y . It now follows from (2.3 ) that G + g (x 0 ) ⊆ g (F ). Fix B ∈ B. Then ∃ F B,0 ∈ F : g (F B,0 ) ⊆ B + g (x 0 ) so that (2.5) implies f (F B ∩ F B,0 ) ⊆ B + g (x 0 ) ⊆ (A B ∩ A) (x 0 ) + B ⊆ A(x 0 ) + B. Since B ∈ B and A ∈ Φ were arbitrary, it follows that Φ (x 0 ) + G ⊆ f (F ). By definition, Φ (x 0 ) converges to f (x 0 ), and since G converges to 0 it follows that f (F ) converges to f (x 0 ) which shows that f is continuous. Now we show that Φ converges continuously to f . Choose x 0 ∈ X and F ∈ λ X (x 0 ) as above so that we have ∀ B ∈ B : ∃ A B ∈ Φ : ∃ F B ∈ F : g ∈ A B , x ∈ F B ⇒ f (x) ∈ g (x) + B . (2.6) Since f is continuous, we also have ∀ B ∈ B : ∃ F B,0 ∈ F : f (F B,0 ) ⊆ f (x 0 ) + B . (2.7) From (2.6) and (2.7) it follows that ∀ x ∈ F B ∩ F B,0 : A B (x) ⊆ f (x) − B ⊆ f (x 0 ) + B − B. . Therefore ω X,Y (F B ∩ F B,0 , A B ) ⊆ f (x 0 ) + B − B. Consequently, [f (x 0 )] + G − G ⊆ ω X,Y (Φ, F ) so that ω X,Y (Φ, F ) converges to f (x 0 ). Since x 0 ∈ X was chosen arbitrary, it follows that Φ converges continuously to f . This completes the proof. Corollary 2.2. If X and Y are convergence vector spaces, Y Hausdorff, complete and topological, then L c (X, Y ) is complete. Remark 2. 3. It should be noted that the proof of Theorem 2.1 given here cannot be used in the case of a nontopological range space Y . Indeed, the proof depends heavily on the existence of a filter G, with a basis of closed sets, which converges to 0 in Y and satisfies ∀ F ∈ λ Y (0) : G ⊆ F . Clearly the existence of such a filter implies that Y is pretopological, and hence topological. While the techniques used in the proof of Theorem 2.1 does not apply to nontopological spaces, similar arguments suffice if Y is replaced with a complete, Hausdorff and commutative topological group. In particular, the following is true. Theorem 2.4. Let X be a convergence space, and Y a complete, Hausdorff commutative topological group. Then C c (X, Y ) is a complete convergence group. Since the proof is based on almost exactly the same arguments used to verify Theorem 2.1 we do not give it here. Lastly, we mention that the completeness result for L c (X, Y ), with Y a complete locally convex space, or more generally any continuously reflexive convergence vector space [2], mentioned earlier, has been used successfully in infinite dimensional analysis, see for instance [4] and [6]. Our results may therefore have a wide range of applicability in analysis on non locally convex spaces [1]. Foundations of complex analysis in non locally convex spaces: Function theory without convexity conditions. A Bayoumi, ElsevierA. Bayoumi, Foundations of complex analysis in non locally convex spaces: Function theory without convexity conditions, Elsevier, 2003. R Beattie, H.-P Butzmann, Convergence structures and applications to functional analysis. Kluwer Academic PlublishersR. Beattie and H.-P. Butzmann, Convergence structures and applications to functional anal- ysis, Kluwer Academic Plublishers, 2002. E Binz, Continuous convergence in Cc(X). Springer-Verlag469E. Binz, Continuous convergence in Cc(X), Lecture Notes in Mathematics 469, Springer- Verlag, 1975. A general approach to infinite-dimensional holomorphy. S Bjon, M Lindström, Monatshefte für Mathematik. 1011S. Bjon and M. Lindström, A general approach to infinite-dimensional holomorphy, Monat- shefte für Mathematik 101 (1986), no. 1, 11-26. An incomplete function space. H.-P Butzmann, Applied Categorical Structures. 94H.-P. Butzmann, An incomplete function space, Applied Categorical Structures 9 (2001), no. 4, 365-368. Categorical differential calculus for infinite dimensional spaces. L D Nel, Cahiers Topologie Géométrie Différentielle Catégoriques. 294L. D. Nel, Categorical differential calculus for infinite dimensional spaces, Cahiers Topologie Géométrie Différentielle Catégoriques 29 (1988), no. 4, 257-286.	[]
[ "Status of the global fit to electroweak precisions data Global Electroweak Fit and Constraints on the Higgs Mass", "Status of the global fit to electroweak precisions data Global Electroweak Fit and Constraints on the Higgs Mass" ]	[ "Martin Goebel [email protected] \nDESY and Institut für Experimentalphysik\nUniversität Hamburg\n\n", "Paris France \nDESY and Institut für Experimentalphysik\nUniversität Hamburg\n\n", "Martin Goebel \nDESY and Institut für Experimentalphysik\nUniversität Hamburg\n\n" ]	[ "DESY and Institut für Experimentalphysik\nUniversität Hamburg\n", "DESY and Institut für Experimentalphysik\nUniversität Hamburg\n", "DESY and Institut für Experimentalphysik\nUniversität Hamburg\n" ]	[ "35th International Conference of High Energy Physics -ICHEP2010" ]	In this presentation Gfitter results from the global Standard Model (SM) fit to electroweak precision data are discussed. We have used the latest measurements of m top and M W and the most recent results for direct Higgs searches at LEP and Tevatron. We obtain M H = 121 + 17 − 6 GeV and a 95% CL upper limit of 155 GeV for the SM Higgs mass. The forth-order result for the strong coupling constant is given by α S (M 2 Z ) = 0.1193 ± 0.0028(exp)± 0.0001(theo). In addition the electroweak fit has been performed with the top mass determined from the pp → tt + X crosssection as measured at Tevatron.	10.22323/1.120.0570	[ "https://arxiv.org/pdf/1012.1331v1.pdf" ]	119,215,045	1012.1331	fa08e4276aff9295ef5ce921001edff4d5d6d3a3	Status of the global fit to electroweak precisions data Global Electroweak Fit and Constraints on the Higgs Mass July 22-28, 2010 Martin Goebel [email protected] DESY and Institut für Experimentalphysik Universität Hamburg Paris France DESY and Institut für Experimentalphysik Universität Hamburg Martin Goebel DESY and Institut für Experimentalphysik Universität Hamburg Status of the global fit to electroweak precisions data Global Electroweak Fit and Constraints on the Higgs Mass 35th International Conference of High Energy Physics -ICHEP2010 July 22-28, 2010on behalf of the Gfitter Group (www.cern.ch/Gfitter). † Speaker. In this presentation Gfitter results from the global Standard Model (SM) fit to electroweak precision data are discussed. We have used the latest measurements of m top and M W and the most recent results for direct Higgs searches at LEP and Tevatron. We obtain M H = 121 + 17 − 6 GeV and a 95% CL upper limit of 155 GeV for the SM Higgs mass. The forth-order result for the strong coupling constant is given by α S (M 2 Z ) = 0.1193 ± 0.0028(exp)± 0.0001(theo). In addition the electroweak fit has been performed with the top mass determined from the pp → tt + X crosssection as measured at Tevatron. In this presentation Gfitter results from the global Standard Model (SM) fit to electroweak precision data are discussed. We have used the latest measurements of m top and M W and the most recent results for direct Higgs searches at LEP and Tevatron. We obtain M H = 121 + 17 − 6 GeV and a 95% CL upper limit of 155 GeV for the SM Higgs mass. The forth-order result for the strong coupling constant is given by α S (M 2 Z ) = 0.1193 ± 0.0028(exp)± 0.0001(theo). In addition the electroweak fit has been performed with the top mass determined from the pp → tt + X crosssection as measured at Tevatron. Introduction Precision measurements allow us to probe physics at much higher energy scales than the masses of the particles directly involved in experimental reactions by exploiting contributions from quantum loops. Prominent examples are the electroweak precision measurements, which are used in conjunction with the Standard Model (SM) to predict via multidimensional parameter fits unmeasured quantities like the Higgs mass. Such an approach has been used in the Gfitter analysis of the Standard Model (SM) in light of electroweak precision data [1]. In this paper updated results of the global electroweak fit are presented taking into account the latest experimental precision measurements and the results of direct Higgs searches from LEP and Tevatron. Fit Inputs The SM predictions for the electroweak precision observables measured by the LEP, SLC, and Tevatron experiments are fully implemented in Gfitter. State-of-the-art calculations are used, in particular the full two-loop and leading beyond-two-loop corrections for the prediction of the W mass and the effective weak mixing angle [2], which exhibit the strongest constraints on the Higgs mass. The Gfitter library also includes the fourth-order (3NLO) perturbative calculation of the mass-less QCD Adler function [3], allowing the fit to determine the strong coupling constant with negligible theoretical uncertainty. The experimental data used in the fit include the electroweak precision data measured at the Z pole including their experimental correlations [4], the latest W mass world average M W = (80.399± 0.023) GeV [5] and width Γ W = (2.098 ± 0.048) GeV [6], and the newest average of the Tevatron top mass measurements m top = (173.1 ± 1.3) GeV [7]. For the contribution of the five lightest quark flavours to the electromagnetic coupling strength at M Z we use the evaluation from [8]. In addition, for some results we take also into account the information from the direct Higgs searches at LEP [9] and Tevatron [10]. Fit Results The minimum χ 2 value of the fit with (without) using the information of the direct Higgs searches amounts to 17.8 (16.4) which corresponds to a p-value of 0.23 (0.22). Figure 1 and fig. 2 show the corresponding profile curves of the ∆χ 2 estimator. We find for the most probable Higgs mass the value M H = 84 + 30 − 23 GeV (M H = 121 + 17 − 6 GeV). The 95% upper limits are 159 GeV (155 GeV). Figure 3 shows the 68%, 95% and 99% CL contours for the variable pairs of m top vs. M H . Three sets of fits are shown: the largest/blue (narrower/purple) allowed regions are derived from a fit excluding (including) the measured top mass value (indicated by the shaded/light green horizontal band). The fit providing the narrowest constraint (green) uses all available information, ie., including the direct Higgs searches from LEP and Tevatron. The largest/blue contour show nicely the positive correlation factor between the Higgs and top mass, which can be determined to be 0.31. The importance of the top mass for the Higgs mass determination is clearly visible. However, an additional uncertainty for the top mass could arise due to ambiguities in the top mass definition at Tevatron [11]. In an alternative approach the top mass has been determined from the SM pp → tt + X cross-section [12], where the top mass is unambiguous once a renormalisation scheme is defined. The use of this top mass in the electroweak fit leads to a smaller value of M H , but due the larger error of this top mass determination the 95% and 99% CL upper limits does not change significantly (see fig. 1). Figure 4 shows the ∆χ 2 as a function of m top . For a comparison the direct Tevatron measurement and the determination from the tt cross-section are also shown. In fig. 5 only the observable indicated in a given row is included in the fit. The four observables providing the strongest constraint on M H are shown. The compatibility among these measurements can be estimated by repeating the global fit where the least compatible of the measurements (here A 0,b FB ) is removed, and by comparing the χ 2 min estimator obtained in that fit to the one of the full fit. To assign a probability to the observation, the ∆χ 2 min obtained this way must be gauged with toy MC experiments to take into account the "look-elsewhere" effect introduced by the explicit selection of the outlier. We find that in (1.4 ± 0.1)% ("2.5σ ") of the toy experiments, the ∆χ 2 min exceeds the value observed in the current data. From the fit including the direct Higgs searches we find for the strong coupling at the Z-mass scale α s (M 2 Z ) = 0.1193 +0.0028 −0.0027 ± 0.0001, where the first error is experimental and the second due to the truncation of the perturbative QCD series. Figure 6 shows the excellent agreement between our result and 3NLO result from τ decays [13]. 35th International Conference of High Energy Physics -ICHEP2010, July 22-28, 2010 Paris France Fig. 1 : 1∆χ 2 as a function of M H for a fit without the direct Higgs searches from LEP and Tevatron. The red solid line shows the result for a fit using the top mass as determined from the tt cross-section. Fig. 2 : 2∆χ 2 as a function of M H for a fit including the direct Higgs search results from LEP and Tevatron. Fig. 3 : 395%, 99% CL fit contours incl. m 68%, 95%, 99% CL fit contours incl. Contours of 68%, 95% and 99% CL obtained from scans of fits with fixed variable pairs of m top and M H . Fig. 4 : 4∆χ 2 as a function of m top for a fit with and without the direct Higgs searches. Fig. 5 : 5Determination of M H excluding all other sensitive observables from the fit, except for the one given. Fig. 6 : 6Top: Collection of α s (µ) measurements at order 3NLO, 2NLO, and NLO[13]. Bottom: The corresponding α s values evolved to M Z[13]. . H Flächer, arXiv:0811.0009Eur. Phys. J. C. 60hep-phH. Flächer et.al. Eur. Phys. J. C 60 (2009) 543, [arXiv:0811.0009 [hep-ph]]. . M Awramik, hep-ph/0311148Phys. Rev. D. 6953006M. Awramik et al., Phys. Rev. D 69, 053006 (2004), [hep-ph/0311148]; . M Awramik, hep-ph/0608099JHEP. 1148M. Awramik et al., JHEP 11, 048 (2006), [hep-ph/0608099]; . M Awramik, arXiv:0811.1364Nucl. Phys. B. 81348hep-phM. Awramik et al., Nucl. Phys. B 813, 174 (2009), 048 (2006), [arXiv:0811.1364 [hep-ph]]. . P A Baikov, arXiv:0801.1821Phys. Rev. Lett. 10112002hep-phP. A. Baikov et al. Phys. Rev. Lett. 101, 012002 (2008), [arXiv:0801.1821 [hep-ph]] Heavy Flavour Working Groups. Sld Lep, Elektroweak, hep-ex/0509008Phys. Rept. 427LEP and SLD Elektroweak and Heavy Flavour Working Groups, Phys. Rept. 427, 257 (2006), [hep-ex/0509008]. arXiv:1007.3178Elizaveta Shabalina. hep-ex. these proccedingsCDF and D0 Collaboration, arXiv:1007.3178 [hep-ex]], Elizaveta Shabalina, these proccedings. . K Hagiwara, hep-ph/0611102Phys. Lett. B. 649173K. Hagiwara et al., Phys. Lett. B 649, 173 (2007), [hep-ph/0611102]. . hep-ex/0306033Phys. Lett. 565LEP Working Group for Higgs boson searchesLEP Working Group for Higgs boson searches, Phys. Lett. B565 (2003) 61-75., [hep-ex/0306033]. . A H Hoang, I W Stewart, arXiv:0808.0222Nucl. Phys. Proc. Suppl. 185hep-phA. H. Hoang, I. W. Stewart, Nucl. Phys. Proc. Suppl. 185 (2008) 220-226. [arXiv:0808.0222 [hep-ph]]. . U Langenfeld, S Moch, P Uwer, arXiv:0906.5273Phys. Rev. 8054009Peter Uwerhep-ph. these proceedingsU. Langenfeld, S. Moch, P. Uwer, Phys. Rev. D80 (2009) 054009. [arXiv:0906.5273 [hep-ph]], Peter Uwer, these proceedings . M Davier, S Descotes-Genon, A Hocker, B Malaescu, Z Zhang, arXiv:0803.0979Eur. Phys. J. C. 56305hep-phM. Davier, S. Descotes-Genon, A. Hocker, B. Malaescu and Z. Zhang, Eur. Phys. J. C 56 (2008) 305 [arXiv:0803.0979 [hep-ph]].	[]
[ "Successive field-induced transitions in BiFeO 3 around room temperature", "Successive field-induced transitions in BiFeO 3 around room temperature" ]	[ "Shiro Kawachi \nThe Institute for Solid State Physics (ISSP)\nThe University of Tokyo\n277-8581KashiwaChibaJapan\n", "Atsushi Miyake \nThe Institute for Solid State Physics (ISSP)\nThe University of Tokyo\n277-8581KashiwaChibaJapan\n", "Toshimitsu Ito \nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8562TsukubaIbarakiJapan\n", "Sachith E Dissanayake \nQuantum Condensed Matter Division\nOak Ridge National Laboratory (ORNL)\n37831Oak RidgeTennesseeUSA\n", "Masaaki Matsuda \nQuantum Condensed Matter Division\nOak Ridge National Laboratory (ORNL)\n37831Oak RidgeTennesseeUSA\n", "IIWilliam Ratcliff \nNIST center for Neutron Research\nNIST\n20899GaithersburgMarylandUSA\n", "Zhijun Xu \nNIST center for Neutron Research\nNIST\n20899GaithersburgMarylandUSA\n\nDepartment of Materials Science and Engineering\nUniversity of Maryland\n20742College ParkMarylandUSA\n", "Yang Zhao \nNIST center for Neutron Research\nNIST\n20899GaithersburgMarylandUSA\n\nDepartment of Materials Science and Engineering\nUniversity of Maryland\n20742College ParkMarylandUSA\n", "Shin Miyahara \nDepartment of Applied Physics\nFukuoka University\nJonan-ku814-0180FukuokaJapan\n", "Nobuo Furukawa \nDepartment of Physics and Mathematics\nAoyama Gakuin University\n229-8558SagamiharaKanagawaJapan\n", "Masashi Tokunaga \nThe Institute for Solid State Physics (ISSP)\nThe University of Tokyo\n277-8581KashiwaChibaJapan\n" ]	[ "The Institute for Solid State Physics (ISSP)\nThe University of Tokyo\n277-8581KashiwaChibaJapan", "The Institute for Solid State Physics (ISSP)\nThe University of Tokyo\n277-8581KashiwaChibaJapan", "National Institute of Advanced Industrial Science and Technology (AIST)\n305-8562TsukubaIbarakiJapan", "Quantum Condensed Matter Division\nOak Ridge National Laboratory (ORNL)\n37831Oak RidgeTennesseeUSA", "Quantum Condensed Matter Division\nOak Ridge National Laboratory (ORNL)\n37831Oak RidgeTennesseeUSA", "NIST center for Neutron Research\nNIST\n20899GaithersburgMarylandUSA", "NIST center for Neutron Research\nNIST\n20899GaithersburgMarylandUSA", "Department of Materials Science and Engineering\nUniversity of Maryland\n20742College ParkMarylandUSA", "NIST center for Neutron Research\nNIST\n20899GaithersburgMarylandUSA", "Department of Materials Science and Engineering\nUniversity of Maryland\n20742College ParkMarylandUSA", "Department of Applied Physics\nFukuoka University\nJonan-ku814-0180FukuokaJapan", "Department of Physics and Mathematics\nAoyama Gakuin University\n229-8558SagamiharaKanagawaJapan", "The Institute for Solid State Physics (ISSP)\nThe University of Tokyo\n277-8581KashiwaChibaJapan" ]	[]	The effects of high magnetic fields applied perpendicular to the spontaneous ferroelectric polarization on single crystals of BiFeO3 were investigated through magnetization, magnetostriction, and neutron diffraction measurements. The magnetostriction measurements revealed lattice distortion of 2 × 10 −5 , during the reorientation process of the cycloidal spin order by applied magnetic fields. Furthermore, anomalous changes in magnetostriction and electric polarization at a larger field demonstrate an intermediate phase between cycloidal and canted antiferromagnetic states, where a large magnetoelectric effect was observed. Neutron diffraction measurements clarified that incommensurate spin modulation along [110] direction in the cycloidal phase becomes commensurate in the intermediate phase. Theoretical calculations based on the standard spin Hamiltonian of this material suggest an antiferromagnetic cone-type spin order in the intermediate phase.Recently, multiferroic materials have been widely investigated due to their coupling between magnetic and ferroelectric ordering. BiFeO 3 is perhaps the most extensively studied multiferroic material as it possesses robust multiferroicity at room temperature as well as various possible applications [1-10]. The effects of magnetic fields on the coupled multiple degrees of freedom behind these phenomena are not fully understood.BiFeO 3 exhibits a cycloidal magnetic order below 640 K [11]. This state is known to exhibit marginal quadratic magnetoelectric (ME) effect at low magnetic fields[12]. High magnetic fields of ∼20 T stabilize the canted antiferromagnetic (CAFM) phase as opposed to the cycloidal phase[13][14][15][16][17]. Although several groups succeeded in realizing the CAFM phase at zero field[18][19][20][21][22][23], the ME effect in this phase has not been clarified. In this study, several features were observed when a magnetic field was applied normal to the trigonal c-axis, and they indicated that a third magnetic phase emerged in bulk BiFeO 3 between the cycloidal and CAFM phases at approximately room temperature.BiFeO 3 has a crystal structure with the polar space group of R3c. A large switchable spontaneous electric polarization emerges along the c-axis of the trigonal cell[24][25][26]. Degeneracy exists in selecting the polarization direction from eight 111 directions in the cubic unit. Hence, depending on synthesis methods, BiFeO 3 crystals can contain multiple ferroelectric domains[27].Magnetic domains can be present even in single ferroelectric domain crystals. The magnetic propagation vector Q points in one of the 110 directions of the trigonal cell[11]in the cycloidal spin ordered state below ∼640 K. The three-fold rotational symmetry around the c-axis (Z direction in this paper) leads to three equivalent Q i (i = 1, 2, 3), as shown inFig. 1(d). The spins rotated primarily in the Q i -Z plane in a magnetic domain with a given Q i .Recently, Tokunaga et al. indicated the emergence of an electric polarization perpendicular to the Z direction that was controlled by magnetic fields[17]. Theoretical calculations suggested that the cycloidal spin order in BiFeO 3 could involve electric polarization perpendicular to the Q i -Z plane as illustrated as P T inFig. 1(d)[28][29][30]. The existence of P T indicated that three-fold rotational symmetry was broken in the cycloidal state. Therefore, BiFeO 3 had lower symmetry than R3c at room temperature. Sosnowska et al. examined the crystal structure of BiFeO 3 using synchrotron X-ray diffraction and proposed that monoclinic distortion led to the observed broadening of the Bragg peaks below 1038 K[31]. However, relation between the magnetic order and the monoclinic distortion was not clear since this broadening was observed even at temperatures well above 640 K.In this study, the magnetization and magnetostriction of single ferroelectric domain crystals of BiFeO 3 synthesized by the laser-diode heating floating-zone method[32]were measured in pulsed high magnetic fields at ISSP. The magnetization were measured by the induction method. Newly improved capacitance dilatometry enabled the measurement of magnetostriction using the capacitance method[33]. The neutron diffraction experiments were carried out on the BT-7 thermal neutron triple-axis spectrometer at the NIST Center for Neutron Research [34]. The magnetic domains of the crystal were previously aligned by applying magnetic field along the Y direction [seeFig. 1(d)]. The single crystal was oriented	10.1103/physrevmaterials.1.024408	[ "https://arxiv.org/pdf/1703.02306v1.pdf" ]	54,080,755	1703.02306	ba6addc587515dc15406f554b4bfcc803cc149f2	Successive field-induced transitions in BiFeO 3 around room temperature 7 Mar 2017 Shiro Kawachi The Institute for Solid State Physics (ISSP) The University of Tokyo 277-8581KashiwaChibaJapan Atsushi Miyake The Institute for Solid State Physics (ISSP) The University of Tokyo 277-8581KashiwaChibaJapan Toshimitsu Ito National Institute of Advanced Industrial Science and Technology (AIST) 305-8562TsukubaIbarakiJapan Sachith E Dissanayake Quantum Condensed Matter Division Oak Ridge National Laboratory (ORNL) 37831Oak RidgeTennesseeUSA Masaaki Matsuda Quantum Condensed Matter Division Oak Ridge National Laboratory (ORNL) 37831Oak RidgeTennesseeUSA IIWilliam Ratcliff NIST center for Neutron Research NIST 20899GaithersburgMarylandUSA Zhijun Xu NIST center for Neutron Research NIST 20899GaithersburgMarylandUSA Department of Materials Science and Engineering University of Maryland 20742College ParkMarylandUSA Yang Zhao NIST center for Neutron Research NIST 20899GaithersburgMarylandUSA Department of Materials Science and Engineering University of Maryland 20742College ParkMarylandUSA Shin Miyahara Department of Applied Physics Fukuoka University Jonan-ku814-0180FukuokaJapan Nobuo Furukawa Department of Physics and Mathematics Aoyama Gakuin University 229-8558SagamiharaKanagawaJapan Masashi Tokunaga The Institute for Solid State Physics (ISSP) The University of Tokyo 277-8581KashiwaChibaJapan Successive field-induced transitions in BiFeO 3 around room temperature 7 Mar 2017 The effects of high magnetic fields applied perpendicular to the spontaneous ferroelectric polarization on single crystals of BiFeO3 were investigated through magnetization, magnetostriction, and neutron diffraction measurements. The magnetostriction measurements revealed lattice distortion of 2 × 10 −5 , during the reorientation process of the cycloidal spin order by applied magnetic fields. Furthermore, anomalous changes in magnetostriction and electric polarization at a larger field demonstrate an intermediate phase between cycloidal and canted antiferromagnetic states, where a large magnetoelectric effect was observed. Neutron diffraction measurements clarified that incommensurate spin modulation along [110] direction in the cycloidal phase becomes commensurate in the intermediate phase. Theoretical calculations based on the standard spin Hamiltonian of this material suggest an antiferromagnetic cone-type spin order in the intermediate phase.Recently, multiferroic materials have been widely investigated due to their coupling between magnetic and ferroelectric ordering. BiFeO 3 is perhaps the most extensively studied multiferroic material as it possesses robust multiferroicity at room temperature as well as various possible applications [1-10]. The effects of magnetic fields on the coupled multiple degrees of freedom behind these phenomena are not fully understood.BiFeO 3 exhibits a cycloidal magnetic order below 640 K [11]. This state is known to exhibit marginal quadratic magnetoelectric (ME) effect at low magnetic fields[12]. High magnetic fields of ∼20 T stabilize the canted antiferromagnetic (CAFM) phase as opposed to the cycloidal phase[13][14][15][16][17]. Although several groups succeeded in realizing the CAFM phase at zero field[18][19][20][21][22][23], the ME effect in this phase has not been clarified. In this study, several features were observed when a magnetic field was applied normal to the trigonal c-axis, and they indicated that a third magnetic phase emerged in bulk BiFeO 3 between the cycloidal and CAFM phases at approximately room temperature.BiFeO 3 has a crystal structure with the polar space group of R3c. A large switchable spontaneous electric polarization emerges along the c-axis of the trigonal cell[24][25][26]. Degeneracy exists in selecting the polarization direction from eight 111 directions in the cubic unit. Hence, depending on synthesis methods, BiFeO 3 crystals can contain multiple ferroelectric domains[27].Magnetic domains can be present even in single ferroelectric domain crystals. The magnetic propagation vector Q points in one of the 110 directions of the trigonal cell[11]in the cycloidal spin ordered state below ∼640 K. The three-fold rotational symmetry around the c-axis (Z direction in this paper) leads to three equivalent Q i (i = 1, 2, 3), as shown inFig. 1(d). The spins rotated primarily in the Q i -Z plane in a magnetic domain with a given Q i .Recently, Tokunaga et al. indicated the emergence of an electric polarization perpendicular to the Z direction that was controlled by magnetic fields[17]. Theoretical calculations suggested that the cycloidal spin order in BiFeO 3 could involve electric polarization perpendicular to the Q i -Z plane as illustrated as P T inFig. 1(d)[28][29][30]. The existence of P T indicated that three-fold rotational symmetry was broken in the cycloidal state. Therefore, BiFeO 3 had lower symmetry than R3c at room temperature. Sosnowska et al. examined the crystal structure of BiFeO 3 using synchrotron X-ray diffraction and proposed that monoclinic distortion led to the observed broadening of the Bragg peaks below 1038 K[31]. However, relation between the magnetic order and the monoclinic distortion was not clear since this broadening was observed even at temperatures well above 640 K.In this study, the magnetization and magnetostriction of single ferroelectric domain crystals of BiFeO 3 synthesized by the laser-diode heating floating-zone method[32]were measured in pulsed high magnetic fields at ISSP. The magnetization were measured by the induction method. Newly improved capacitance dilatometry enabled the measurement of magnetostriction using the capacitance method[33]. The neutron diffraction experiments were carried out on the BT-7 thermal neutron triple-axis spectrometer at the NIST Center for Neutron Research [34]. The magnetic domains of the crystal were previously aligned by applying magnetic field along the Y direction [seeFig. 1(d)]. The single crystal was oriented The effects of high magnetic fields applied perpendicular to the spontaneous ferroelectric polarization on single crystals of BiFeO3 were investigated through magnetization, magnetostriction, and neutron diffraction measurements. The magnetostriction measurements revealed lattice distortion of 2 × 10 −5 , during the reorientation process of the cycloidal spin order by applied magnetic fields. Furthermore, anomalous changes in magnetostriction and electric polarization at a larger field demonstrate an intermediate phase between cycloidal and canted antiferromagnetic states, where a large magnetoelectric effect was observed. Neutron diffraction measurements clarified that incommensurate spin modulation along [110] direction in the cycloidal phase becomes commensurate in the intermediate phase. Theoretical calculations based on the standard spin Hamiltonian of this material suggest an antiferromagnetic cone-type spin order in the intermediate phase. Recently, multiferroic materials have been widely investigated due to their coupling between magnetic and ferroelectric ordering. BiFeO 3 is perhaps the most extensively studied multiferroic material as it possesses robust multiferroicity at room temperature as well as various possible applications [1][2][3][4][5][6][7][8][9][10]. The effects of magnetic fields on the coupled multiple degrees of freedom behind these phenomena are not fully understood. BiFeO 3 exhibits a cycloidal magnetic order below 640 K [11]. This state is known to exhibit marginal quadratic magnetoelectric (ME) effect at low magnetic fields [12]. High magnetic fields of ∼20 T stabilize the canted antiferromagnetic (CAFM) phase as opposed to the cycloidal phase [13][14][15][16][17]. Although several groups succeeded in realizing the CAFM phase at zero field [18][19][20][21][22][23], the ME effect in this phase has not been clarified. In this study, several features were observed when a magnetic field was applied normal to the trigonal c-axis, and they indicated that a third magnetic phase emerged in bulk BiFeO 3 between the cycloidal and CAFM phases at approximately room temperature. BiFeO 3 has a crystal structure with the polar space group of R3c. A large switchable spontaneous electric polarization emerges along the c-axis of the trigonal cell [24][25][26]. Degeneracy exists in selecting the polarization direction from eight 111 directions in the cubic unit. Hence, depending on synthesis methods, BiFeO 3 crystals can contain multiple ferroelectric domains [27]. Magnetic domains can be present even in single ferroelectric domain crystals. The magnetic propagation vector Q points in one of the 110 directions of the trigonal cell [11] in the cycloidal spin ordered state below ∼640 K. The three-fold rotational symmetry around the c-axis (Z direction in this paper) leads to three equivalent Q i (i = 1, 2, 3), as shown in Fig. 1(d). The spins rotated primarily in the Q i -Z plane in a magnetic domain with a given Q i . Recently, Tokunaga et al. indicated the emergence of an electric polarization perpendicular to the Z direction that was controlled by magnetic fields [17]. Theoretical calculations suggested that the cycloidal spin order in BiFeO 3 could involve electric polarization perpendicular to the Q i -Z plane as illustrated as P T in Fig. 1(d) [28][29][30]. The existence of P T indicated that three-fold rotational symmetry was broken in the cycloidal state. Therefore, BiFeO 3 had lower symmetry than R3c at room temperature. Sosnowska et al. examined the crystal structure of BiFeO 3 using synchrotron X-ray diffraction and proposed that monoclinic distortion led to the observed broadening of the Bragg peaks below 1038 K [31]. However, relation between the magnetic order and the monoclinic distortion was not clear since this broadening was observed even at temperatures well above 640 K. In this study, the magnetization and magnetostriction of single ferroelectric domain crystals of BiFeO 3 synthesized by the laser-diode heating floating-zone method [32] were measured in pulsed high magnetic fields at ISSP. The magnetization were measured by the induction method. Newly improved capacitance dilatometry enabled the measurement of magnetostriction using the capacitance method [33]. The neutron diffraction experiments were carried out on the BT-7 thermal neutron triple-axis spectrometer at the NIST Center for Neutron Research [34]. The magnetic domains of the crystal were previously aligned by applying magnetic field along the Y direction [see Fig. 1(d)]. The single crystal was oriented in the (hhl) [or (X0Z)] scattering plane and was mounted in a 15 T vertical field superconducting magnet. The magnetic field was applied along Y direction. Details of the neutron experiment are described in the supplementary material [35]. Figure 1(a) shows the magnetization (M ) curve at 300 K in the magnetic field (H) applied along the Y direction. The kink at ∼ 15 T indicated the existence of a magnetic transition at this field [17]. Extrapolation of the linear M -H curve at high field to zero field indicated finite offsets in the vertical axis and thus suggested that the high field phase possesses spontaneous magnetization (M S ). Figure 1(b) presents the change in electric polarization along the Y direction (∆P Y ) caused by a magnetic field H Y . The ∆P Y steeply changes below 10 T in the field increasing process of the first field cycle (red). However, this change was suppressed in the second field cycle (blue). This irreversible change (also known as the nonvolatile effect) was ascribed to the reorientation of the magnetoelectric domains [9,17]. It was assumed that the transverse electric polarization P T changed the direction during the reorientation of the cycloidal domains to the Q 2 domain in the initial field scan. µ 0 H ( T ) L \|\| X 1st 2nd (a)! (b)! (c)! (d)! (e)! H! H! H \|\| Y! 210 ps/m! X! Y! a! a! Q 1! Q 3! Q 2! Z! P T! P T! P T! The emergence of P Y indicates that three-fold rotational symmetry was broken at 300 K. We measured the transverse magnetostriction with H Y at 300 K to detect the relevant lattice distortion. The vertical axis in Fig. 1(c) corresponds to the field-induced change in the sample length along X (∆L) normalized by the length at zero field (L). The field dependence of ∆L/L followed that of ∆P Y , in the opposite sign, including the irreversible behavior below 10 T. This measurement demonstrated that the application of a temporal magnetic field slightly compressed the crystal normal to the H-Z plane. Prior to this experiment, we measured the magnetostriction along the X direction for H X while maintaining the capacitance cell set-up [insets of Fig. 1(e)]. This result is indicated by a black solid line in Fig. 1(e). In the case of H X, the crystal length along the X direction increased. The initial position of the ∆L/L for H Y was the same as its final position after the application of H X. Therefore, the data for H Y is plotted with the vertical offset of 2 × 10 −5 . These irreversible behaviors below 10 T indicated that the sample length changed due to the direction of Q. A neutron experiment in a recent study confirmed that application of H Y of 6 T would stabilize the Q 2 domain in Fig. 1(d) [36]. Contraction of the sample along the X direction for H Y indicated that the sample slightly contracted along the Q vector. This result is consistent with the early report of neutron diffraction measurements showing that the population of the magnetic domain with the Q vector normal to the applied pressure direction decreased when uniaxial pressure was applied along the principal axis of the pseudocubic unit cell [37]. A systematic analysis of the data obtained after the domain reorientation in Figs. 1(a)-(c) indicated that all the quantities showed non-monotonic changes between 10 T and 15 T. In order to clarify this anomaly, we measured the magnetization with H X and H Y at various temperatures. The results are shown in Figs. 2(a) and 2(b). As observed in these figures, no intrinsic differences were detected in the M -H curves for H X and H Y . At 150 K, the magnetization showed a step-like change at the transition field. Conversely, M -H curves above 200 K showed the region with a large gradient in the intermediate field region. In the following, we will refer to the magnetic phase in this intermediate region as the IM phase. The transition from the cycloidal phase to the IM phase was discontinuous while that from the IM phase to the CAFM phase was continuous. The IM phase appeared from the lower field as the temperature increased. Figure 2(c) shows the phase diagram in the H-T plane determined by the magnetization curves. Here, characteristic fields were assigned in the magnetization curves as the transition fields as marked by several symbols in the inset of Fig. 2(c). Gareeva et al. theoretically predicted the emergence of the intermediate phase [38]. Their study proposed that an antiferromagnetic cone (AF-cone) phase existed between the cycloidal and CAFM phases in the magnetic fields parallel as well as perpendicular to the trigonal axis. The calculations suggested that the boundary between the cycloidal and AF-cone phases corresponded to a first order phase transition, and this is consistent with the results of the present study. Their calculations focused on a one-dimensional spatially modulated spin structure along a fixed direction. Therefore, it is necessary to consider more general states in order to determine the actual spin order in the IM phase. We performed calculations starting with the following spin Hamiltonian [39][40][41]: H = J 1 n.n. S i · S j + J 2 n.n.n S i · S j + n.n. − 1 2 D S Y i S Z i+x −S Z i S Y i+x + √ 3 2 D S Z i S X i+x − S X i S Z i+x − 1 2 D S Y i S Z i+y − S Z i S Y i+y − √ 3 2 D S Z i S X i+y −S X i S Z i+y + D S Y i S Z i+z − S Z i S Y i+z + n.n (−1) ni D ′ S X i S Y j − S Y i S X j −gµ B B · S i − K S Z i 2 . Here, i, j and x, y, z represent the Fe sites and three adjacent directions in the pseudocubic cell as shown in Fig. 3(c), respectively. J 1 and J 2 represent nearest and next-nearest neighbor interaction. D, D ′ , and K denote DM interactions and single-ion anisotropy. Figure 3(a) shows the energy of several phases as a function of magnetic field applied normal to the Z direction with typical values of parameters [42]. An intermediate (IM) phase is stabilized between the cycloidal and CAFM phases. The IM phase shows an AF cone-type spin structure with the propagation vector pointing to the field direction as shown in Fig. 3(c). In the present Hamiltonian, the D terms stabilize the cycloidal state, while the D ′ term favors the CAFM one. The AF cone state is a kind of the superposition of these two orders caused by the competition of these two DM interactions. Figure 3(b) shows calculated magnetization curves for the three phases. The gradual increase in magnetization for the IM phase reproduces the experimentally observed change in the slope shown in Fig. 2(b). In this AF-cone state, spin susceptibility along the spin modulation vector will be larger than that normal to it. Accordingly, a flop of the modulation vector can be caused by the transition from the cycloidal to the AFcone state. Therefore, we performed neutron diffraction measurements of BiFeO 3 to study the field dependence of the spin modulation through this transition. We performed scans along the X direction at 285 K under applied H Y . The data were taken as we ramped up the magnetic field and these scans are shown in Figure 4(a). As the field increases, the incommensurate peaks at h ∼ ±0.004 are reduced in intensity, while in contrast, the commensurate peak at h = 0 grows. In Fig. 4(b), we show the field dependence of the integrated intensities of the commensurate and incommensurate peaks. At 285 K, application of 14 T was sufficient to reach the IM phase, but insufficient to reach the CAFM phase, at this temperature as denoted by a cross point of vertical and horizontal dashed lines in Fig. 2(c). An increase in magnetic fields from ∼6 T led to a gradual increase and decrease in the commensurate and incommensurate scattering intensities, respectively. The spin structure appeared commensurate along the X direction in the IM phase realized at 14 T. By using the results of the present study, it is not possible to conclude whether the spin structure was commensurate or incommensurate along the Y direction due to the limited resolution out of the scattering plane. A decrease in the magnetic field resulted in the change from the commensurate state to the incommensurate state showing hysteresis. The coexistence of commensurate/incommensurate scattering supports the conclusion that the transition from the cycloidal phase to the IM phase is first order. The disappearance of the incommensurate spin modulation along the X direction in the IM phase did not contradict the picture of the AF-cone phase as shown in Fig. 3(c). We also performed neutron diffraction measurements with H X and found that the transition to the IM phase occurred at ∼14 T [35], consistent with the magnetization results. In the IM phase, magnetization, electric polarization, and magnetostriction changed linearly as a function of magnetic field. The coefficient of the linear ME effect between 12 T and 15 T shown in Fig. 1(b) is approximately 210 ps/m. Although we cannot identify this value determined in a limited field region as the representative one in the IM phase, the observed large ME effect implies the presence of strong ME coupling in this phase. Thus far, we have not succeeded in reproducing the ME effect in the IM phase based on the generalized inverse Dzyaloshinskii-Moriya effect [29]. Phenomenologically, the ME effect might be explained by slight tilting of the trigonal c-axis by the monoclinic distortion. Using the reported lattice parameters [31], we determined that the tilting angle was θ = 0.01 • − 0.04 • [35]. Assuming a spontaneous polarization of 1C/m 2 along the c-axis, the projected component will be between 200 and 800 µC/m 2 , which is the same order of magnitude with the observed transverse component. The important point here is that the tilting is coupled to the spin system, and hence, can be controlled by an external magnetic field. Finally, the study involved examining ways to realize the IM phase at lower magnetic fields than the fields observed in this study. In accordance with the Landau-Ginzburg theory for BiFeO 3 , exchange stiffness, DM interaction, magnetic anisotropy constant (κ c ), differential magnetic susceptibility (χ ⊥ ), and spontaneous magnetization in the CAFM phase (M S ) are fundamental parameters. Among these parameters, χ ⊥ and M S were determined from the experimental results of the M -H curves. Figure 2(d) shows the temperature dependence of these parameters determined from the magnetization curves at various temperatures. These parameters were almost constant at temperatures from 150 K to 320 K. The temperature dependence of the transition field to the IM phase could be a result of the change in κ c because the exchange stiffness and DM interaction are likely insensitive to temperature. Such temperature dependence of κ c may appear as the change in anharmonicity in the cycloidal order, which is observed experimentally [43]. According to a theoretical study [38], when κ c was increased, while holding the other parameters constant, the transition field to the AF-cone phase was reduced. Therefore, future research could examine the AF-cone phase in thin films with large κ c . In summary, magnetization, magnetostriction, and neutron diffraction measurements were performed on sin-gle BiFeO 3 crystals in magnetic fields applied normal to the trigonal axis. The observations indicated that the reorientation of multiferroic domains occurred below 10 T in conjunction with irreversible field-induced change in the lattice distortion. This demonstrated the existence of a monoclinic distortion in the cycloidal phase. The application of an increased magnetic field realized an intermediate phase prior to the transition to the canted antiferromagnetic state. A large ME effect was observed in this intermediate phase. Theoretical calculations indicated the presence of an antiferromagnetic cone-type spin order in this phase, consistent with the neutron diffraction measurements. This work was supported by the MEXT of Japan Grant-in-Aid for Challenging Exploratory Research (16K05413), and Murata Science Foundation. FIG. 1 . 1Magnetic field-induced changes in (a) magnetization, (b) electric polarization along the Y direction, and (c) magnetostriction along the X direction at 300 K for H Y . In (b) and (c), virgin and second traces are presented by red and blue lines, respectively. (d) Schematic drawing of the arrangement of Fe ions, coordinates, magnetic Q−vectors, and transverse electric polarization (PT) in the ab-plane of the trigonal cell. (e) Magnetostriction along X direction in magnetic fields parallel to X and Y at 300 K. FIG. 2 . 2Magnetization curves at various temperatures in magnetic fields applied along (a) X and (b) Y direction. Data obtained at different temperatures are vertically offset for clarity. (c) Temperature dependence of transition fields for H Y . The transition fields are defined in the M -H curves marked by symbols in the inset. Open and closed symbols correspond to the transition fields of the field increasing and decreasing processes, respectively. The blue dashed lines indicate the highest field (14 T) and temperature (285 K) for the neutron experiments in Figs. 4. (d) Temperature dependence of the spontaneous magnetization (MS, closed circles) and differential magnetic susceptibility (χ ⊥ , open squares) in the CAFM phase. FIG. 3 . 3Calculated magnetic field dependences of (a) energy per site and (b) magnetization in the cycloidal, intermediate (IM), and CAFM phases. (c) Schematic drawing of the AFcone spin structure expected in the IM phase. Closed and open circles represent Fe sites which differ by (c/6) along the Z-axis. Red arrows represent the spin moments. x, y, and z denote three adjacent directions in the pseudocubic unit. The propagation vector QAF−cone becomes parallel to the field H ex applied in the Y direction. FIG. 4 . 4(a) Profiles of the neutron diffraction along the X direction in applied magnetic field along the Y direction at 285 K, measured upon increasing the magnetic field. 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We used S = 5/2, J1 = 5.32 meV, J2 = 0.24 meV, D = -0.15 meV, D ′ = 0.1 meV, K = 0.0055 meV, and g =2. which are slightly tuned from the values in [39-41We used S = 5/2, J1 = 5.32 meV, J2 = 0.24 meV, D = -0.15 meV, D ′ = 0.1 meV, K = 0.0055 meV, and g =2, which are slightly tuned from the values in [39-41]. . M Ramazanoglu, W Ratcliff, Y J Ii, Seongsu Choi, S. -W Lee, V Cheong, Kiryukhin, Phys. Rev. B. 83174434M. Ramazanoglu, W. Ratcliff, II, Y. J. Choi, Seongsu Lee, S. -W. Cheong, and V. Kiryukhin, Phys. Rev. B 83, 174434 (2011).	[]
[ "A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems", "A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems" ]	[ "Pinghua Gong \nDepartment of Automation\nZhaosong Lu\nDepartment of Mathematics\nState Key Laboratory on Intelligent Technology and Systems Tsinghua National Laboratory for Information Science and Technology (TNList)\nTsinghua University\n100084BeijingChina\n", "Changshui Zhang [email protected] \nDepartment of Statistics\nSimon Fraser University\nV5A 1S6BurnabyBCCanada\n", "Jianhua Z Huang [email protected] \nComputer Science and Engineering\nTexas A&M University\n77843TXUSA\n", "Jieping Ye [email protected] \nArizona State University\n85287TempeAZUSA\n" ]	[ "Department of Automation\nZhaosong Lu\nDepartment of Mathematics\nState Key Laboratory on Intelligent Technology and Systems Tsinghua National Laboratory for Information Science and Technology (TNList)\nTsinghua University\n100084BeijingChina", "Department of Statistics\nSimon Fraser University\nV5A 1S6BurnabyBCCanada", "Computer Science and Engineering\nTexas A&M University\n77843TXUSA", "Arizona State University\n85287TempeAZUSA" ]	[ "Proceedings of the 30 th International Conference on Ma-chine Learning" ]	Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets.	null	[ "https://arxiv.org/pdf/1303.4434v1.pdf" ]	16,894,434	1303.4434	dd165c31e89fb9dd4c8a7b1d5174a5a2453a7ece	A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems 2013 Pinghua Gong Department of Automation Zhaosong Lu Department of Mathematics State Key Laboratory on Intelligent Technology and Systems Tsinghua National Laboratory for Information Science and Technology (TNList) Tsinghua University 100084BeijingChina Changshui Zhang [email protected] Department of Statistics Simon Fraser University V5A 1S6BurnabyBCCanada Jianhua Z Huang [email protected] Computer Science and Engineering Texas A&M University 77843TXUSA Jieping Ye [email protected] Arizona State University 85287TempeAZUSA A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems Proceedings of the 30 th International Conference on Ma-chine Learning the 30 th International Conference on Ma-chine LearningAtlanta, Georgia, USA282013 Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets. Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-Stage (MS) convex relaxation (or DC programming), which relaxes the original non-convex problem to a sequence of convex problems. This approach is usually not very practical for large-scale problems because its computational cost is a multiple of solving a single convex problem. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm to solve the nonconvex optimization problem for a large class of non-convex penalties. The GIST algorithm iteratively solves a proximal operator problem, which in turn has a closed-form solution for many commonly used penalties. At each outer iteration of the algorithm, we use a line search initialized by the Barzilai- Introduction Learning sparse representations has important applications in many areas of science and engineering. The use of an ℓ 0 -norm regularizer leads to a sparse solution, however the ℓ 0 -norm regularized optimization problem is challenging to solve, due to the discontinuity and non-convexity of the ℓ 0 -norm regularizer. The ℓ 1 -norm regularizer, a continuous and convex surrogate, has been studied extensively in the literature (Tibshirani, 1996;Efron et al., 2004) and has been applied successfully to many applications including signal/image processing, biomedical informatics and computer vision (Shevade & Keerthi, 2003;Wright et al., 2008;Beck & Teboulle, 2009;Wright et al., 2009;Ye & Liu, 2012). Although the ℓ 1norm based sparse learning formulations have achieved great success, they have been shown to be suboptimal in many cases (Candes et al., 2008;Zhang, 2010b;2012), since the ℓ 1 -norm is a loose approximation of the ℓ 0 -norm and often leads to an over-penalized problem. To address this issue, many non-convex regularizers, interpolated between the ℓ 0 -norm and the ℓ 1 -norm, have been proposed to better approximate the ℓ 0 -norm. They include ℓ q -norm (0 < q < 1) (Foucart & Lai, 2009), Smoothly Clipped Absolute Deviation (SCAD) (Fan & Li, 2001), Log-Sum Penalty (LSP) (Candes et al., 2008), Minimax Concave Penalty (MCP) (Zhang, 2010a), Geman Penalty (GP) (Geman & Yang, 1995;Trzasko & Manduca, 2009) and Capped-ℓ 1 penalty (Zhang, 2010b;2012;Gong et al., 2012a). Although the non-convex regularizers (penalties) are appealing in sparse learning, it is challenging to solve the corresponding non-convex optimization problems. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm for a large class of non-convex penalties. The key step of the proposed algorithm is to compute a proximal operator, which has a closed-form solution for many commonly used non-convex penalties. In our algorithm, we adopt the Barzilai-Borwein (BB) rule (Barzilai & Borwein, 1988) to initialize the line search step size at each iteration, which greatly accelerates the convergence speed. We also use a non-monotone line search criterion to further speed up the convergence of the algorithm. In addition, we present a detailed convergence analysis for the proposed algorithm. Extensive experiments on largescale real-world data sets demonstrate the efficiency of the proposed algorithm. The Proposed Algorithm: GIST General Problems We consider solving the following general problem: min w∈R d {f (w) = l(w) + r(w)} . (1) We make the following assumptions on the above formulation throughout the paper: A1 l(w) is continuously differentiable with Lipschitz continuous gradient, that is, there exists a positive constant β(l) such that ∇l(w) − ∇l(u) ≤ β(l) w − u , ∀w, u ∈ R d . A2 r(w) is a continuous function which is possibly non-smooth and non-convex, and can be rewritten as the difference of two convex functions, that is, r(w) = r 1 (w) − r 2 (w), where r 1 (w) and r 2 (w) are convex functions. A3 f (w) is bounded from below. Remark 1 We say that w ⋆ is a critical point of problem (1), if the following holds (Toland, 1979;Wright et al., 2009): 0 ∈ ∇l(w ⋆ ) + ∂r 1 (w ⋆ ) − ∂r 2 (w ⋆ ), where ∂r 1 (w ⋆ ) is the sub-differential of the function r 1 (w) at w = w ⋆ , that is, ∂r 1 (w ⋆ ) = s : r 1 (w) ≥ r 1 (w ⋆ ) + s, w − w ⋆ , ∀w ∈ R d . We should mention that the sub-differential is nonempty on any convex function; this is why we make the assumption that r(w) can be rewritten as the difference of two convex functions. Some Examples Many formulations in machine learning satisfy the assumptions above. The following least square and logistic loss functions are two commonly used ones which satisfy assumption A1: l(w) = 1 2n Xw − y 2 or 1 n n i=1 log 1 + exp(−y i x T i w) , where X = [x T 1 ; · · · ; x T n ] ∈ R n×d is a data matrix and y = [y 1 , · · · , y n ] T ∈ R n is a target vector. The regularizers (penalties) which satisfy the assumption A2 are presented in Table 1. They are non-convex (except the ℓ 1 -norm) and extensively used in sparse learning. The functions l(w) and r(w) mentioned above are nonnegative. Hence, f is bounded from below and satisfies assumption A3. Algorithm Our proposed General Iterative Shrinkage and Thresholding (GIST) algorithm solves problem (1) by generating a sequence {w (k) } via: w (k+1) = arg min w l(w (k) ) + ∇l(w (k) ), w − w (k) + t (k) 2 w − w (k) 2 + r(w),(2) In fact, problem (2) is equivalent to the following proximal operator problem: w (k+1) = arg min w 1 2 w − u (k) 2 + 1 t (k) r(w), where u (k) = w (k) − ∇l(w (k) )/t (k) . Thus, in GIST we first perform a gradient descent along the direction −∇l(w (k) ) with step size 1/t (k) and then solve a proximal operator problem. For all the regularizers listed in Table 1, problem (2) has a closed-form solution (details are provided in the Appendix), although it may Table 1. Examples of regularizers (penalties) r(w) satisfying the assumption A2 and the corresponding convex functions r1(w) and r2(w). λ > 0 is the regularization parameter; r(w) = i ri(wi), r1(w) = i r1,i(wi), r2(w) = i r2,i(wi), [x]+ = max(0, x). Name ri(wi) r1,i(wi) r2,i(wi) ℓ1-norm λ\|wi\| λ\|wi\| 0 LSP λ log(1 + \|wi\|/θ) (θ > 0) λ\|wi\| λ(\|wi\| − log(1 + \|wi\|/θ)) SCAD λ \|w i \| 0 min 1, [θλ−x] + (θ−1)λ dx (θ > 2) λ\|wi\| λ \|w i \| 0 [min(θλ,x)−λ] + (θ−1)λ dx =    λ\|wi\|, if \|wi\| ≤ λ, −w 2 i +2θλ\|w i \|−λ 2 2(θ−1) , if λ < \|wi\| ≤ θλ, (θ + 1)λ 2 /2, if \|wi\| > θλ. =      0, if \|wi\| ≤ λ, w 2 i −2λ\|w i \|+λ 2 2(θ−1) , if λ < \|wi\| ≤ θλ, λ\|wi\| − (θ+1)λ 2 2 , if \|wi\| > θλ. MCP λ \|w i \| 0 1 − x θλ + dx (θ > 0) λ\|wi\| λ \|w i \| 0 min(1, x/(θλ))dx = λ\|wi\| − w 2 i /(2θ), if \|wi\| ≤ θλ, θλ 2 /2, if \|wi\| > θλ. = w 2 i /(2θ), if \|wi\| ≤ θλ, λ\|wi\| − θλ 2 /2, if \|wi\| > θλ. Capped ℓ1 λ min(\|wi\|, θ) (θ > 0) λ\|wi\| λ[\|wi\| − θ]+ Algorithm 1 GIST: General Iterative Shrinkage and Thresholding Algorithm 1: Choose parameters η > 1 and tmin, tmax with 0 < tmin < tmax; 2: Initialize iteration counter k ← 0 and a bounded starting point w (0) ; 3: repeat 4: t (k) ∈ [tmin, tmax]; 5: repeat 6: w (k+1) ← arg min w l(w (k) ) + ∇l(w (k) ), w − w (k) + t (k) 2 w − w (k) 2 + r(w); 7: t (k) ← ηt (k) ; 8: until some line search criterion is satisfied 9: k ← k + 1 10: until some stopping criterion is satisfied be a non-convex problem. For example, for the ℓ 1 and Capped ℓ 1 regularizers, we have closed-form solutions as follows: ℓ 1 : w (k+1) i = sign(u (k) i ) max 0, \|u (k) i \| − λ/t (k) , Capped ℓ 1 : w (k+1) i = x 1 , if h i (x 1 ) ≤ h i (x 2 ), x 2 , otherwise, where x 1 = sign(u (k) i ) max(\|u (k) i \|, θ), x 2 = sign(u (k) i ) min(θ, [\|u (k) i \| − λ/t (k) ] + ) and h i (x) = 0.5(x − u (k) i ) 2 + λ/t (k) min(\|x\|, θ). The detailed procedure of the GIST algorithm is presented in Algorithm 1. There are two issues that remain to be addressed: how to initialize t (k) (in Line 4) and how to select a line search criterion (in Line 8) at each outer iteration. The Step Size Initialization: 1/t (k) Intuitively, a good step size initialization strategy at each outer iteration can greatly reduce the line search cost (Lines 5-8) and hence is critical for the fast con-vergence of the algorithm. In this paper, we propose to initialize the step size by adopting the Barzilai-Borwein (BB) rule (Barzilai & Borwein, 1988), which uses a diagonal matrix t (k) I to approximate the Hessian matrix ∇ 2 l(w) at w = w (k) . Denote x (k) = w (k) − w (k−1) , y (k) = ∇l(w (k) ) − ∇l(w (k−1) ). Then t (k) is initialized at the outer iteration k as t (k) = arg min t tx (k) − y (k) 2 = x (k) , y (k) x (k) , x (k) . Line Search Criterion One natural and commonly used line search criterion is to require that the objective function value is monotonically decreasing. More specifically, we propose to accept the step size 1/t (k) at the outer iteration k if the following monotone line search criterion is satisfied: f (w (k+1) ) ≤ f (w (k) ) − σ 2 t (k) w (k+1) − w (k) 2 ,(3) where σ is a constant in the interval (0, 1). A variant of the monotone criterion in Eq. (3) is a nonmonotone line search criterion (Grippo et al., 1986;Grippo & Sciandrone, 2002;Wright et al., 2009). It possibly accepts the step size 1/t (k) even if w (k+1) yields a larger objective function value than w (k) . Specifically, we propose to accept the step size 1/t (k) , if w (k+1) makes the objective function value smaller than the maximum over previous m (m > 1) iterations, that is, f (w (k+1) ) ≤ max i=max(0,k−m+1),··· ,k f (w (i) ) − σ 2 t (k) w (k+1) − w (k) 2 ,(4) where σ ∈ (0, 1). Convergence Analysis Inspired by Wright et al. (2009); Lu (2012a), we present detailed convergence analysis under both monotone and non-monotone line search criteria. We first present a lemma which guarantees that the monotone line search criterion in Eq. (3) is satisfied. This is a basic support for the convergence of Algorithm 1. Lemma 1 Let the assumptions A1-A3 hold and the constant σ ∈ (0, 1) be given. Then for any integer k ≥ 0, the monotone line search criterion in Eq. (3) is satisfied whenever t (k) ≥ β(l)/(1 − σ). Proof Since w (k+1) is a minimizer of problem (2), we have ∇l(w (k) ), w (k+1) − w (k) + t (k) 2 w (k+1) − w (k) 2 + r(w (k+1) ) ≤ r(w (k) ).(5) It follows from assumption A1 that l(w (k+1) ) ≤l(w (k) ) + ∇l(w (k) ), w (k+1) − w (k) + β(l) 2 w (k+1) − w (k) 2 .(6) Combining Eq. (5) and Eq. (6), we have l(w (k+1) ) + r(w (k+1) ) ≤ l(w (k) ) + r(w (k) ) − t (k) − β(l) 2 w (k+1) − w (k) 2 . It follows that f (w (k+1) ) ≤ f (w (k) ) − t (k) − β(l) 2 w (k+1) − w (k) 2 . Therefore, the line search criterion in Eq. ( 3) is sat- isfied whenever (t (k) − β(l))/2 ≥ σt (k) /2, i.e., t (k) ≥ β(l)/(1 − σ) . This completes the proof the lemma. Next, we summarize the boundedness of t (k) in the following lemma. Lemma 2 For any k ≥ 0, t (k) is bounded under the monotone line search criterion in Eq. (3). Proof It is trivial to show that t (k) is bounded from below, since t (k) ≥ t min (t min is defined in Algorithm 1). Next we prove that t (k) is bounded from above by contradiction. Assume that there exists a k ≥ 0, such that t (k) is unbounded from above. Without loss of generality, we assume that t (k) increases monotonically to +∞ and t (k) ≥ ηβ(l)/(1 − σ). Thus, the value t = t (k) /η ≥ β(l)/(1 − σ) must have been tried at iteration k and does not satisfy the line search criterion in Eq. (3). But Lemma 1 states that t = t (k) /η ≥ β(l)/(1 − σ) is guaranteed to satisfy the line search criterion in Eq. (3). This leads to a contradiction. Thus, t (k) is bounded from above. Remark 2 We note that if Eq. (3) holds, Eq. (4) is guaranteed to be satisfied. Thus, the same conclusions in Lemma 1 and Lemma 2 also hold under the the non-monotone line search criterion in Eq. (4). Based on Lemma 1 and Lemma 2, we present our convergence result in the following theorem. Theorem 1 Let the assumptions A1-A3 hold and the monotone line search criterion in Eq. (3) be satisfied. Then all limit points of the sequence w (k) generated by Algorithm 1 are critical points of problem (1). Proof Based on Lemma 1, the monotone line search criterion in Eq. (3) is satisfied and hence f (w (k+1) ) ≤ f (w (k) ), ∀k ≥ 0, which implies that the sequence f (w (k) ) k=0,1,··· is monotonically decreasing. Let w ⋆ be a limit point of the sequence w (k) , that is, there exists a subsequence K such that lim k∈K→∞ w (k) = w ⋆ . Since f is bounded from below, together with the fact that f (w (k) ) is monotonically decreasing, lim k→∞ f (w (k) ) exists. Observing that f is continuous, we have lim k→∞ f (w (k) ) = lim k∈K→∞ f (w (k) ) = f (w ⋆ ). Taking limits on both sides of Eq. (3) with k ∈ K, we have lim k∈K→∞ w (k+1) − w (k) = 0.(7) Considering that the minimizer w (k+1) is also a critical point of problem (2) and r(w) = r 1 (w) − r 2 (w), we have 0 ∈∇l(w (k) ) + t (k) (w (k+1) − w (k) ) + ∂r 1 (w (k+1) ) − ∂r 2 (w (k+1) ). Taking limits on both sides of the above equation with k ∈ K, by considering the semi-continuity of ∂r 1 (·) and ∂r 2 (·), the boundedness of t (k) (based on Lemma 2) and Eq. (7), we obtain 0 ∈ ∇l(w ⋆ ) + ∂r 1 (w ⋆ ) − ∂r 2 (w ⋆ ), Therefore, w ⋆ is a critical point of problem (1). This completes the proof of Theorem 1. Based on Eq. (7), we know that lim k∈K→∞ w (k+1) − w (k) 2 = 0 is a necessary optimality condition of Algorithm 1. Thus, w (k+1) − w (k) 2 is a quantity to measure the convergence of the sequence {w (k) } to a critical point. We present the convergence rate in terms of w (k+1) − w (k) 2 in the following theorem. Theorem 2 Let {w (k) } be the sequence generated by Algorithm 1 with the monotone line search criterion in Eq. (3) satisfied. Then for every n ≥ 1, we have min 0≤k≤n w (k+1) − w (k) 2 ≤ 2(f (w (0) ) − f (w ⋆ )) nσt min , where w ⋆ is a limit point of the sequence {w (k) }. Proof Based on Eq. (3) with t (k) ≥ t min , we have σt min 2 w (k+1) − w (k) 2 ≤ f (w (k) ) − f (w (k+1) ). Summing the above inequality over k = 0, · · · , n, we obtain σt min 2 n k=0 w (k+1) − w (k) 2 ≤ f (w (0) ) − f (w (n+1) ), which implies that min 0≤k≤n w (k+1) − w (k) 2 ≤ 2(f (w (0) ) − f (w (n+1) )) nσt min ≤ 2(f (w (0) ) − f (w ⋆ )) nσt min . This completes the proof of the theorem. Under the non-monotone line search criterion in Eq. (4), we have a similar convergence result in the following theorem (the proof uses an extension of argument for Theorem 1 and is omitted). Theorem 3 Let the assumptions A1-A3 hold and the non-monotone line search criterion in Eq. (4) be satisfied. Then all limit points of the sequence w (k) generated by Algorithm 1 are critical points of problem (1). Note that Theorem 1/Theorem 3 makes sense only if w (k) has limit points. By considering one more mild assumption: A4 f (w) → +∞ when w → +∞, we summarize the existence of limit points in the following theorem (the proof is omitted): Theorem 4 Let the assumptions A1-A4 hold and the monotone/non-monotone line search criterion in Eq. (3)/Eq. (4) be satisfied. Then the sequence w (k) generated by Algorithm 1 has at least one limit point. Discussions Observe that l(w (k) )+ ∇l(w (k) ), w − w (k) + t (k) 2 w − w (k) 2 can be viewed as an approximation of l(w) at w = w (k) . The GIST algorithm minimizes an approximate surrogate instead of the objective function in problem (1) at each outer iteration. We further observe that if t (k) ≥ β(l)/(1 − σ) > β(l) [the sufficient condition of Eq. (3)], we obtain l(w) ≤l(w (k) ) + ∇l(w (k) ), w − w (k) + t (k) 2 w − w (k) 2 , ∀w ∈ R d . It follows that f (w) = l(w) + r(w) ≤ M (w, w (k) ), ∀w ∈ R d , where M (w, w (k) ) denotes the objective function of problem (2). We can easily show that f (w (k) ) = M (w (k) , w (k) ). Thus, the GIST algorithm is equivalent to solving a sequence of minimization problems: w (k+1) = arg min w M (w, w (k) ), k = 0, 1, 2, · · · and can be interpreted as the well-known Majorization and Minimization (MM) technique (Hunter & Lange, 2000). Note that we focus on the vector case in this paper and the proposed GIST algorithm can be easily extended to the matrix case. Related Work In this section, we discuss some related algorithms. One commonly used approach to solve problem (1) is the Multi-Stage (MS) convex relaxation (or CCCP, or DC programming) (Zhang, 2010b;Yuille & Rangarajan, 2003;Gasso et al., 2009). It equivalently rewrites problem (1) as min w∈R d f 1 (w) − f 2 (w), where f 1 (w) and f 2 (w) are both convex functions. The MS algorithm solves problem (1) by generating a sequence {w (k) } as w (k+1) = arg min w∈R d f 1 (w) − f 2 (w (k) ) − s 2 (w (k) ), w − w (k) ,(8) where s 2 (w (k) ) denotes a sub-gradient of f 2 (w) at w = w (k) . Obviously, the objective function in problem (8) is convex. The MS algorithm involves solving a sequence of convex optimization problems as in problem (8). In general, there is no closed-form solution to problem (8) and the computational cost of the MS algorithm is k times that of solving problem (8), where k is the number of outer iterations. This is computationally expensive especially for large scale problems. A class of related algorithms called iterative shrinkage and thresholding (IST), which are also known as different names such as fixed point iteration and forward-backward splitting (Daubechies et al., 2004;Combettes & Wajs, 2005;Hale et al., 2007;Beck & Teboulle, 2009;Wright et al., 2009;Liu et al., 2009), have been extensively applied to solve problem (1). The key step is by generating a sequence {w (k) } via solving problem (2). However, they require that the regularizer r(w) is convex and some of them even require that both l(w) and r(w) are convex. Our proposed GIST algorithm is a more general framework, which can deal with a wider range of problems including both convex and non-convex cases. Another related algorithm called a Variant of Iterative Reweighted L α (VIRL) is recently proposed to solve the following optimization problem (Lu, 2012a): min w∈R d f (w) = l(w) + λ d i=1 (\|w i \| α + ǫ i ) q/α , where α ≥ 1, 0 < q < 1, ǫ i > 0. VIRL solves the above problem by generating a sequence {w (k) } as w (k+1) = arg min w∈R d l(w (k) ) + ∇l(w (k) ), w − w (k) + t (k) 2 w − w (k) 2 + λq α d i=1 (\|w k i \| α + ǫ i ) q/α−1 \|w i \| α . In VIRL, t (k−1) is chosen as the initialization of t (k) . The line search step in VIRL finds the smallest integer ℓ with t (k) = t (k−1) η ℓ (η > 1) such that f (w (k+1) ) ≤ f (w (k) ) − σ 2 w (k+1) − w (k) 2 (σ > 0). The most related algorithm to our propose GIST is the Sequential Convex Programming (SCP) proposed by Lu (2012b). SCP solves problem (1) by generating a sequence {w (k) } as w (k+1) = arg min w∈R d l(w (k) ) + ∇l(w (k) ), w − w (k) + t (k) 2 w − w (k) 2 + r 1 (w) − r 2 (w (k) ) − s 2 , w − w (k) , where s 2 is a sub-gradient of r 2 (w) at w = w (k) . Our algorithm differs from SCP in that the original regularizer r(w) = r 1 (w) − r 2 (w) is used in the proximal operator in problem (2), while r 1 (w) minus a locally linear approximation for r 2 (w) is adopted in SCP. We will show in the experiments that our proposed GIST algorithm is more efficient than SCP. Experiments Experimental Setup We evaluate our GIST algorithm by considering the Capped ℓ 1 regularized logistic regression problem, that is l(w) = 1 n n i=1 log 1 + exp(−y i x T i w) and r(w) = λ d i=1 min(\|w i \|, θ). We compare our GIST algorithm with the Multi-Stage (MS) algorithm and the SCP algorithm in different settings using twelve data sets summarized in Table 2. These data sets are high dimensional and sparse. Two of them (news20, realsim) 1 have been preprocessed as two-class data sets (Lin et al., 2008). The other ten 2 are multi-class data sets. We transform the multi-class data sets into twoclass by labeling the first half of all classes as positive class, and the remaining classes as the negative class. All algorithms are implemented in Matlab and executed on an Intel(R) Core(TM)2 Quad CPU (Q6600 @2.4GHz) with 8GB memory. We set σ = 10 −5 , m = 5, η = 2, 1/t min = t max = 10 30 and choose the starting points w (0) of all algorithms as zero vectors. We terminate all algorithms if the relative change of the two consecutive objective function values is less than 10 −5 or the number of iterations exceeds 1000. The Matlab codes of the GIST algorithm are available online (Gong et al., 2013). Experimental Evaluation and Analysis We report the objective function value vs. CPU time plots with different parameter settings in Figure 1. From these figures, we have the following observations: (1) Both GISTbb-Monotone and GISTbb-Nonmonotone decrease the objective function value rapidly and they always have the fastest convergence speed, which shows that adopting the BB rule to initialize t (k) indeed greatly accelerates the convergence speed. Moreover, both GISTbb-Monotone and GISTbb-Nonmonotone algorithms achieve the smallest objective function values. (2) GISTbb-Nonmonotone may give rise to an increasing objective function value but finally converges and has a faster overall convergence speed than GISTbb-Monotone in most cases, which indicates that the non-monotone line search criterion can further accelerate the con-vergence speed. (3) SCPbb-Nonmonotone is comparable to GISTbb-Nonmonotone in several cases, however, it converges much slower and achieves much larger objective function values than those of GISTbb-Nonmonotone in the remaining cases. This demonstrates the superiority of using the original regularizer r(w) = r 1 (w) − r 2 (w) in the proximal operator in problem (2). (4) GIST-1 has a faster convergence speed than GIST-t (k−1) in most cases, which demonstrates that it is a bad strategy to use t (k−1) to initialize t (k) . This is because {t (k) } increases monotonically in this way, making the step size 1/t (k) monotonically decreasing when the algorithm proceeds. Conclusions We propose an efficient iterative shrinkage and thresholding algorithm to solve a general class of non-convex optimization problems encountered in sparse learning. A critical step of the proposed algorithm is the computation of a proximal operator, which has a closedform solution for many commonly used formulations. We propose to initialize the step size at each iteration using the BB rule and employ both monotone and non-monotone criteria as line search conditions, which greatly accelerate the convergence speed. Moreover, we provide a detailed convergence analysis of the proposed algorithm, showing that the algorithm converges under both monotone and non-monotone line search criteria. Experiments results on large-scale data sets demonstrate the fast convergence of the proposed algorithm. In our future work, we will focus on analyzing the theoretical performance (e.g., prediction error bound, parameter estimation error bound etc.) of the solution obtained by the GIST algorithm. In addition, we plan to apply the proposed algorithm to solve the multitask feature learning problem (Gong et al., 2012a;b). where C is a set composed of 3 elements or 1 ele- ment. If t 2 (\|u\| − θ) 2 − 4t(λ − t\|u\|θ) ≥ 0, C = {0, t(\|u\| − θ) + t 2 (\|u\| − θ) 2 − 4t(λ − t\|u\|θ) 2t + t(\|u\| − θ) − t 2 (\|u\| − θ) 2 − 4t(λ − t\|u\|θ) 2t + . Otherwise, C = {0}. • SCAD: We can recast problem (9) into the following three problems: x 1 = arg min w 1 2 (w − u) 2 + λ t \|w\| s.t. \|w\| ≤ λ, x 2 = arg min w 1 2 (w − u) 2 + −w 2 + 2θ(λ/t)\|w\| − (λ/t) 2 2(θ − 1) s.t. λ ≤ \|w\| ≤ θλ, x 3 = arg min w 1 2 (w − u) 2 + (θ + 1)λ 2 2t 2 s.t.\|w\| ≥ θλ. We can easily obtain that (x 2 is obtained using the similar idea as LSP by considering that θ > 2): x 1 = sign(u) min(λ, max(0, \|u\| − λ/t)), x 2 = sign(u) min(θλ, max(λ, t\|u\|(θ − 1) − θλ t(θ − 2) )), x 3 = sign(u) max(θλ, \|u\|). The SCP algorithm using the BB rule to initialize t (k) and Eq. (4) as the line search criterion. Note that on data sets 'hitech' and 'real-sim', MS algorithms stop early (the SCP algorithm has similar behaviors on data sets 'hitech' and 'news20'), because they satisfy the termination condition that the relative change of the two consecutive objective function values is less than 10 −5 . However, their objective function values are much larger than those of GISTbb-Monotone and GISTbb-Nonmonotone. Thus, we have w (k+1) = arg min y h i (y) s.t. y ∈ {x 1 , x 2 , x 3 }. • MCP: Similar to SCAD, we can recast problem (9) into the following two problems: x 1 = arg min w 1 2 (w − u) 2 + λ t \|w\| − w 2 2θ s.t. \|w\| ≤ θλ, x 2 = arg min w 1 2 (w − u) 2 + θ(λ/t) 2 2 s.t. \|w\| ≥ θλ. We can easily obtain that x 1 = sign(u)z, x 2 = sign(u) max(θλ, \|u\|), where z = arg min w∈C 1 2 (w − \|u\|) 2 + λ t w− w 2 2θ ; C = 0, θλ, min θλ, max 0, θ(t\|u\|−λ) t(θ−1) , if θ−1 = 0, and C = {0, θλ} otherwise. Thus, we have w (k+1) = x 1 , if h i (x 1 ) ≤ h i (x 2 ) x 2 , otherwise. • Capped ℓ 1 : We can recast problem (9) into the following two problems: x 1 = arg min w 1 2 (w − u) 2 + λ t θ s.t. \|w\| ≥ θ, x 2 = arg min w 1 2 (w − u) 2 + λ t \|w\| s.t. \|w\| ≤ θ. We can easily obtain that x 1 = sign(u) max(θ, \|u\|), x 2 = sign(u) min(θ, max(0, \|u\| − λ/t)). Thus, we have w (k+1) = x 1 , if h i (x 1 ) ≤ h i (x 2 ), x 2 , otherwise. Figure 1 . 1Objective function value vs. CPU time plots. MS-Nesterov/MS-SpaRSA: The Multi-Stage algorithm using the Nesterov/SpaRSA method to solve problem (8); GIST-1/GIST-t (k−1) /GISTbb-Monotone/GISTbb-Nonmonotone: The GIST algorithm using 1/t (k−1) /BB rule/BB rule to initialize t (k) and Eq. (3)/Eq. (3)/Eq. (3)/Eq. (4) as the line search criterion; SCPbb-Nonmonotone: http://www.csie.ntu.edu.tw/cjlin/libsvmtools/datasets/ 2 http://www.shi-zhong.com/software/docdata.zip AcknowledgementsThis work is supported partly by 973 Program (2013CB329503), NSFC (Grant No. 91120301, 61075004, 61021063), NIH (R01 LM010730) and NSF (IIS-0953662, CCF-1025177, DMS1208952).Appendix: Solutions to Problem (2)Observe that r(w) = d i=1 r i (w i ) and problem (2) can be equivalently decomposed into d independent univariate optimization problems:where i = 1, · · · , d and u (k) i is the i-th entry of u (k) = w (k) − ∇l(w (k) )/t(k). 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[ "ABOUT GORDAN'S ALGORITHM FOR BINARY FORMS", "ABOUT GORDAN'S ALGORITHM FOR BINARY FORMS" ]	[ "Marc Olive " ]	[]	[]	In this paper, we present a modern version of Gordan's algorithm on binary forms. Symbolic method is reinterpreted in terms of SL2(C)-equivariant homomorphisms defined upon Cayley operator and polarization operator. A graphical approach is thus developed to obtain Gordan's ideal, a central key to get covariant bases of binary forms. To illustrate the power of the method, we compute a covariant basis of S6 ⊕ S2 and S8.	10.1007/s10208-016-9324-x	[ "https://arxiv.org/pdf/1403.2283v5.pdf" ]	34,532,028	1403.2283	d7ced6e693bc657a5e93b1e9fc0b6a3edeca68f6	ABOUT GORDAN'S ALGORITHM FOR BINARY FORMS Marc Olive ABOUT GORDAN'S ALGORITHM FOR BINARY FORMS Classical invariant theoryCovariantsAlgorithm Contents In this paper, we present a modern version of Gordan's algorithm on binary forms. Symbolic method is reinterpreted in terms of SL2(C)-equivariant homomorphisms defined upon Cayley operator and polarization operator. A graphical approach is thus developed to obtain Gordan's ideal, a central key to get covariant bases of binary forms. To illustrate the power of the method, we compute a covariant basis of S6 ⊕ S2 and S8. Introduction Classical invariant theory was a very active research field throughout the XIX e century. As pointed out by Parshall [43], the birth of this field can be found in the Disquisitiones arithmeticae (1801) of Gauss. He studied in this book linear changes of variables in a quadratic form with integer coefficients. About forty years later, Boole [7] established the main purpose of what will become today classical invariant theory. Cayley [16,17] deeply investigated this field of research and developed important tools still in use nowadays, such as the Cayley Omega operator. During about fifteen years (until 1861 and Cayley's seventh memoir [14]) the English school of invariant theory, mainly led by Cayley and Sylvester, developed important tools to compute explicit invariant generators of binary forms. Thus, the role of calculation deeply influenced this first approach in invariant theory [16]. At that time, a German school mainly conducted by Clebsch, Aronhold and Gordan, developed their own approach, named the symbolic method. In 1868, Gordan, who was called the "King of invariant theory", proved that covariants of any binary forms are always finitely generated [25]. As a great part of the mathematical development of that time, such a result was endowed with a constructive proof: the English and the German school were equally preoccupied by calculation and an exhibition of invariants and covariants. Despite Gordan's constructive proof, Cayley was reluctant to make use of the symbolic method to obtain a new understanding of invariant theory. In the same spirit, Sylvester claimed that Gordan's proof was "so long and complicated and so artificial a structure that it requires a very long study to master and there is not one person in Great Britain who has mastered it" [17]. That's only in 1903, with the work of Grace-Young [27], that the German approach of Gordan and al. became accessible to a wide community of mathematicians. Let also point out that Gordan's constructive approach led to several explicit results: first, and without no difficulty, Gordan [26] gave the quintic and the sextic bases for covariants 1 , then he gave the first part of the septimic and the octic covariant basis. After that, Von Gall finished the computation for the septimic [53] and for the octic [24]. But, in 1890, Hilbert made a critical advance in the field of invariant theory. Using a totally new approach [30], which is the cornerstone of all nowadays abstract algebra, he proved the finiteness theorem for all cases dealing with invariants of a reductive group. But his first proof [30] was criticized for not being constructive. Facing these critics, Hilbert made another contribution [30] which claimed to be more constructive. This effective approach is nowadays widely used to obtain effective results in the field of invariant theory [45,22,11,12]. As pointed out by Hilbert himself in [30], the main scope of this approach can be summarized in three steps. The first step is to compute the Hilbert series of the graded algebra A of invariants 2 . Of course, there exist several methods to compute a priori this Hilbert series [5,37,46] which is always a rational function by the Hilbert-Serre theorem [15]. The second step is to exhibit what is called a system of parameters for the algebra A of invariants 3 . Finally, the Hochster-Roberts theorem [31] ensures us that the algebra A is Cohen-Macaulay 4 . Thanks to that statement, the system of parameters altogether with the Hilbert series give a bound for the degree of invariants still have to be found. We refer the reader to several references [51,11,22,19,20,21] to get a general and modern approach to this subject. But one major lack of this strategy is summed up in the effective computation of a system of parameters. The Noether normalization lemma [35] ensures us that such a system always exists, but as we know, effective algorithms to get such a system [29] are not sufficiently effective because of the extensive use of Grobnër bases. In the case of invariants algebra Inv(S n ) of a single binary form, one has of course the concept of the nullcone and the Mumford-Hilbert criterion [20,9], to check that a finite family of invariants is a system of parameters of Inv(S n ) 5 . But this criterion is not an algorithm to get a system of parameters, and it is no more valid in the case of covariants. Furthermore, in the case of joint invariants, that is invariants algebra of V := S n 1 ⊕ · · · ⊕ S n k , such a system of parameters has, in general, a complex shape. Indeed, Brion [10] showed that only in some very few cases, as for instance in the simple case of joint invariants of S 4 ⊕ S 2 , there exist a system of parameters that respects the multi-graduation of Inv(V ). Let's point out here that an important motivation for this work was to use an effective approach on invariant theory because we had, for example, to compute joint invariants of S 6 ⊕S 2 . In fact, this motivation is directly taken from the field of continuum mechanics, and more precisely from the theory of elasticity in small deformations [1]. As an example, to get one part of the invariant basis of the elasticity tensor [3], Boehler-Kirilov-Onat used a classical isomorphism between SO(3) linear representations over a complex vector space and the one of SL(2, C) linear representations on binary forms [49,6]. Doing so, they directly obtained the part of the invariant bases of the elasticity tensor related to the invariant bases of S 8 , which was first obtained by Von Gall [24] in 1888. Such invariant bases has a direct application to classify orbits space of elasticity tensor, as pointed out by Auffray-Kolev-Petitot [2]. In a forthcoming article, though, we also present a new useful result for continuum mechanics [39], which was a direct consequence of results we obtain in our present paper for the case of joint covariants of S 6 ⊕ S 2 . But we may also observe some other important interests on the subject which come from the field of geometrical arithmetic, illustrated by the work of Lercier-Ritzenthaler [36] on hyperelliptic curves, but also in the field of quantic informatics as illustrated by the work of Luque [38]. Of course, the algebraical geometry approach first developed by Hilbert is not the only constructive one. In the case of a single binary form, Olver [41] exhibits another constructive 1 The case of a binary quintics presented such a level of difficulty for the English school that Cayley conjectured an infinite number of invariant generators for a binary form of order greater than or equal to five [16]. 2 Writing A = Ai we define the Hilbert series to be the formal series HA(z) := dim Aiz i . 3 The set {θ1, · · · , θs} ⊂ A is a system of parameters if A is finitely generated over its subring k[θ1, · · · , θs]. 4 Meaning the algebra A is a finite and free k[θ1, · · · , θs]-module, where {θ1, · · · , θs} is a system of parameters 5 A set {θ1, · · · , θs} ⊂ Inv(Sn) is a system of parameters if θ1(f ) = · · · = θs(f ) = 0 implies that f ∈ Sn has a root which multiplicity is of order strictly greater than n 2 . approach, which was generalized for a single n-ary form and also specified with a "running bound" by Brini-Regonati-Creolis [8]. We also have in Kung-Rota [34] a constructive approach with a combinatorial which became increasingly complex for the cases we had to deal with. Thus, as it appears to us in the case of joint covariants of S 6 ⊕ S 2 , a very simple result stated in Grace-Young (theorem 4.6 of our present paper) gave us a direct algorithm to obtain a covariant basis, although other approaches failed to do so. From this observation, we decided to reformulate Gordan's theorem 6 on binary forms in the modern language of operators. We also decided to represent operators with directed graphs, in the spirit of the graphical approach dealt by Olver-Shakiban [40], and to focus on equivariants morphisms. The paper is organized as follows. In section 2 we recall the mathematical background of classical invariant theory, and we introduce classical operators such as the Omega Cayley operator, polarization operators and the transvectant operator. We then introduce Aronhold molecule and molecular covariants which give graphical representations of equivariant morphisms constructed on the basis of Cayley and polarization operators. We prove Gordan's theorem for joint covariants in section 4 and for simple covariants in section 5. Finally, in Appendix A, we illustrate the method 7 by computing explicitly the basis of joint covariants of a sextic and a quadratic, and of simple covariants of an octic. This result was already obtained by Von Gall [53], Lercier-Ritzenthaler [36], Cröni [18] and Bedratyuk [4], but the computation is summarized and simplified here. Covariants of binary forms Let's take x to be a couple (x, y) ∈ C 2 ; we define: Definition 2.1. The C vector space of nth degree binary forms, noted S n is the space of homogeneous polynomials f (x) = a 0 x n + n 1 a 1 x n−1 y + · · · + n n − 1 a n−1 xy n−1 + a n y n with each a i in C. Now we can take V to be a space of binary forms, that is V := s i=0 S n i There is a natural SL 2 (C) action on C 2 and thus on V , given by (g · f )(x) := f (g −1 · x) for g ∈ Gl 2 (C) or g ∈ SL 2 (C) From this, we naturally define an action 8 on the ring coordinate C[V ⊕ C 2 ]: for p ∈ C[V ⊕ C 2 ] we define the action to be (g · p)(f , x) := p(g −1 · f , g −1 · x) for g ∈ SL 2 (C) Thus, all this lead to the classical definition of the covariant ring of binary forms Definition 2.2. The covariant algebra of a space V of binary forms, noted Cov(V ), is the algebra of SL 2 (C) polynomial invariant: Cov(V ) := C[V ⊕ C 2 ] SL 2 (C) A very important result, first due to Gordan [25] and then generalized by Hilbert [30] is: Theorem 2.3. For every space V of binary forms, the algebra Cov(V ) is finitely generated, meaning there exist a finite set h 1 , · · · , h N in C(V ), called a basis, such that 6 Remark also that Weynman [55] did an algebra formulation of Gordan's theorem. 7 Pasechnik [44] did also an application of this method. 8 for a general and modern approach of invariants and covariants algebra we refer the reader to the online text of Procesi-Kraft [33] We can also attempt to obtain a minimal basis [23]. Let's define the subspace Cov(V ) i ⊂ Cov(V ) of ith degree homogeneous polynomials, and the ideal C + := i>0 Cov(V ) i of the graduated algebra Cov(V ). Then, we can consider for each Cov(V ) i the number δ i to be the cardinal of a supplement to (C + ) 2 i ⊂ C i in Cov(V ) i . Now because of the finiteness, there exist k such that δ i = 0 for i ≥ k ; and we can finally define the invariant number: Cov(C) = C[h 1 , · · · , h N ]n(V ) = i δ i Now: Definition 2.4. A set h 1 , · · · , h N is a minimal basis of Cov(V ) if their image in the vector space C + / (C + ) 2 is a basis. In that case we will have N = n(V ) An important observation is that we have a natural bi-graduation on the covariant algebra Cov(V ): • By the degree, which is the polynomial degree in the coefficients of the space V ; • By the order which is the polynomial degree in the variables x ; If then we put Cov k,r (V ) to be the subspace of kth degree and rth order covariants, we get: Cov(V ) = k≥0,r≥0 Cov k,r (V ) (2.1) A first way to obtain covariant is to make use of Cayley's operator [41], which is a bi-differential operator acting on a tensor product of smooth functions f (x α )g(x β ), given by: Ω αβ (f (x α )g(x β )) := ∂f x α ∂g y β − ∂f y α ∂g x β We will also make use of the polarization operator 9 , defined to be σ α := x ∂ ∂x α + y ∂ ∂y α Cayley's operator and polarization operator commute with SL 2 (C) action [41]. We then naturally get, with these operators, covariants of binary forms. In fact, as we will see further on, these operators suffice to get all covariants (see theorem 2.9). Using Cayley's operator, we can now obtain transvectant operation, defined to be: {f , g} r := Ω r σ n−r α σ p−r β (f α g β ) The classical approach, here, is to give invariant or covariant bases using transvectant operators. For instance, the covariant basis of a cubic f ∈ S 3 is given by table 1. Table 1. Covariant basis of a binary cubic given in terms of transvectant Remark 2.5. The symbolic method developed by XIX e century german school is naturally translated 10 into differential operators, as pointed out by Olver [41]. Order/degree 1 2 3 4 3 f 2 H := {f , f } 2 T := {f , H} 1 0 {H, H} 2 We now define Sym k (V ) to be the space of totally symmetric tensor subspace of ⊗ k V . Here, we have a natural isomorphism between Cov k,r (V ) and the space Hom SL 2 (C) (Sym k (V ), S r ). This isomorphism is a simple trace operation. Indeed, if we take an equivariant morphism ϕ ∈ Hom SL 2 (C) (Sym k (V ), S r ) we just have to take the covariant p(f , x) = ϕ(f (x), · · · , f (x)). Cayley's operator and polarization operator carrying us to a natural way to construct SL 2 (C) equivariant homomorphisms from S n 1 ⊗ S n 2 ⊗ · · · ⊗ S ns to S r . For instance, we can construct the morphism: Ω αβ Ω 2 αγ σ n−3 α σ p−1 β σ q−2 γ : S n ⊗ S p ⊗ S q −→ S r with r = n + p + q − 6 (2.2) Such an equivariant morphism will be represented by a digraph [52,32,42]. We start with atoms α β γ associated to valences val(α) = n, val(β) = p, val(γ) = q. Thus we represent the SL 2 (C) equivariant morphism 2.2 with the digraph 11 α β γ 2 in which val(α) = n − 3, val(β) = p − 1, val(γ) = q − 2 Thus a directed and weighted edge, with weight r, from two given atoms α and β will represent the operator Ω r αβ . Finally, we use polarization operator related to atom's valence to get a morphism ; for instance α β r will represent the SL 2 (C) equivariant morphism Ω r αβ σ n−r α σ q−r β In this example given above, we have val(α) = n − r. Following these ideas, we can now construct a more general object on the space V = s i=1 S n i of binary forms. When given a digraph D, its set of vertices will be denoted by V(D), its set of (oriented) edges by E(D). Given an (oriented) edge e we denote its origin by o(e) and its termination by t(e). Definition 2.6. Let α, β, . . . , be symbols associated to orders n iα , . . . , n i ; an Aronhold molecule D is a digraph constructed on atoms α . . . which represent a SL 2 (C) equivariant morphism τ D := e∈E(D) Ω w(e) o(e) t(e) v∈V(D) σ val(v) v from S n iα ⊗ · · · ⊗ S n i to S r , with r = val(α) + . . . + val( ). The set of all Aronhold molecules will be noted M(V ) and the vector space generated by all Aronhold molecules, will be noted A(V ). Each molecular covariant will aslo be represented using a digraph. For instance, the covariant basis of a binary cubic is given in figure 1. Figure 1. Covariant basis of a binary cubic given in molecular form Taking f (v) ∈ V for each vertex v ∈ V(D), we can thus define a covariant in Cov(V ) taking τ D   v∈V(D) f (v)   This define a map Ψ from A(V ) to Cov(V ). Now:f f f 2 f f f 2 1 f f f f 2 2 In fact, we have a relation between covariants given in transvectant form and the ones given in molecular form (see section 3). When given an Aronhold molecule D ∈ A(V ), we define w(D) to be the weight of the weighted digraph D. We also define the grade gr(D) of D to be the maximal weight of D. Definition 2.8. For a given integer r, we define A r (V ) to be the vector subspace of A(V ) generated by all Aronhold molecules D such that gr(D) ≥ r. Now, if we take a given space V of binary forms, we can define M(V ) to be the algebra generated by all molecular covariants Ψ(M(V )). We then have a very important result, which non trivial proof can be found for example in Olver [41]: Theorem 2.9. Every covariant of a given space of binary forms V is a polynomial in molecular covariants ; that is: Cov(V ) = M(V ) For nineteenth century mathematicians, this result stated that every covariant may be expressible as a polynomial in symbolic forms. Nevertheless, this result doesn't assure us that every covariant of a given space V can be written with transvectant operations. To get this result, one must make use of relations between transvectant covariants and molecular covariants: such a relation is given in Olver [41], but we also give such a result in property 3.5. When we want to express covariants as molecular covariants, we don't have a unique expression. Indeed, (see Olver [41] and Olver-Shakiban [42]) we have fundamental relations, called syzygies, among operators and thus among Aronhold molecules and also among molecular covariants. Take α, β, γ and δ be four symbols associated to valence n 1 , n 2 , n 3 and n 4 . (1) The first syzygie comes from the egality: Ω αβ σ n 1 −1 α σ n 2 −1 β = −Ω βα σ n 1 −1 α σ n 2 −1 β which gives, in graphical form: α β = − α β (2.3) (2) The second one, comes from a determinantal property [41]: Ω αβ σ n 1 −1 α σ n 2 −1 β σ n 3 γ = Ω αγ σ n 1 −1 α σ n 2 −1 β σ n 3 γ + Ω γβ σ n 1 α σ n 2 −1 β σ n 3 −1 γ which gives, in graphical form: α β γ = α β γ + α β γ (2.4)(3) The last one is a peculiar case of the previous one. Ω αβ Ω γδ σ n 1 −1 α σ n 2 −1 β σ n 3 −1 γ σ n 3 −1 δ = Ω αδ Ω βγ σ n 1 −1 α σ n 2 −1 β σ n 3 −1 γ σ n 3 −1 δ +Ω αγ Ω δβ σ n 1 −1 α σ n 2 −1 β σ n 3 −1 γ σ n 3 −1 δ which gives, in graphical form: α β γ δ = α β γ δ + α β γ δ (2.5) One may observe that these syzygies are in fact rewriting rules for molecular covariants. For example, by 2.3 we will have α β 2 = α β 2 = α β 2 thus, for an even number on edges, we will not precise the direction. Another important observation is that the syzygies 2.4 and 2.5 leads to a huge amount of relations among molecular covariants. As an example, let's now 12 Because Cayley's operator and polarization operator commutes, we will have other important relations. One of them is simply an application of the binomial formula: α β γ r = r i=0 r i α β γ i r − i (2.6) Now we can get, with fine enough computations [27] the following relation, obtained by Stroh [50], which can be directly applied to operators Cayley's operators and polarization operators which all commute: Lemma 2.10. Let u 1 , u 2 and u 3 be three commutative variables such that u 1 + u 2 + u 3 = 0 Then we have (−1) k 2 k 1 i=0 g i k 1 + k 3 − i k 3 u g−i 3 u i 1 + (−1) k 3 k 2 i=0 g i k 2 + k 1 − i k 1 u g−i 1 u i 2 + (−1) k 1 k 3 i=0 g i k 3 + k 2 − i k 2 u g−i 2 u i 3 = 0 (2.7) with k 1 + k 2 + k 3 = g − 1. Take here a degree three Aronhold molecule with V = S n so that each atom has the same valence n ; define: D(e 0 , e 1 , e 2 ) := α β γ e 0 e 1 e 2 with weight w = e 0 + e 1 + e 2 (2.8) We then have an important lemma, which proof can be found in Grace-Young [27]: Corollary 2.11. If w ≤ n and m 1 , m 2 , m 3 are integers such that m 1 + m 2 + m 3 = w + 1 then the Aronhold molecule D(e 0 , e 1 , e 2 ) is a linear combination of the Aronhold molecule. α β γ w − i 1 i 1 α β γ w − i 2 i 2 α β γ i 3 w − i 3 with i s = 0 . . . m s . From this we deduce two very important lemmas: One may remark that these relations are upon morphisms, thus these lemmas give new syzygies among molecular covariants. Corollary 2.12. Let D(e 0 , e 1 , e 2 ) be given by 2.8. (1) If w ≤ n then D(e 0 , e 1 , e 2 ) ∈ A r (V ) with r ≥ 2 3 w (2) If w > n then D(e 0 , e 1 , e 2 ) ∈ A r (V ) with r ≥ n − Transvectants and molecular covariants It's important here to understand the way transvectants and molecular covariants are linked. To get molecular covariants when given a transvectant is the easiest way: it is a direct consequence of Leibnitz formula for derivatives. Because molecular covariants come from Aronhold molecules, we will give in fact relations between transvectants and Aronhold molecules. Transvectants can be seen as SL 2 (C) equivariant morphisms ; using composition, we thus can make transvectants of Aronhold molecules. to be a new Aronhold molecule constructed by linking D and E with r edges in a given way ν(r). If we take for example D = β γ α 2 and E = δ we can define D E ν 1 (2) = β γ α δ 2 2 or D E ν 2 (2) = β γ α δ 2 We then get a property which proof can be found in [41]: Proposition 3.2. If D and E are two Aronhold molecules, for every integer r, the r th transvectant {D, E} r can be obtained as a linear combination of Aronhold molecules L ν(r) (D, E), for each possible link ν(r) between D and E: {D, E} r = ν(r) a ν(r) D E ν(r) Because of Aronhold molecule's definition, which differ from Olver-Shakiban's molecular definition, the coefficients are not as simple as the ones given in Olver [41]. In fact, we won't have to use exact expression of these coefficients. As an example, we can take D = β γ α 2 and E = δ We will thus have: {D, E} 2 = a ν(1) β γ α δ 2 2 + a ν(2) β γ α δ 2 2 + a ν(3) β γ α δ 2 2 + a ν(4) β γ α δ 2 + a ν(5) β γ α δ 2 + a ν(6) β γ α δ 2 For the opposite link, that is the link between an Aronhold molecule and transvectants, we will make use of another molecular operation 13 : The proofs of the following two propositions will be omitted. They can be found in Olver [41]: and transvectants {D µ 1 (k 1 ) , E µ 2 (k 2 ) } r with k 1 + k 2 + r = r being constant and r < r. Furthermore we also have: M 1 = λ 1 M 2 + λ 2 α β 3 , γ 1 + λ 3 α β 4 , γ 0 Furthermore property 3.5 assures us that M 1 = µ 1 α β 2 , γ 2 + µ 2 α β 3 , β 1 + µ 3 α β 4 , γ 0 all coefficients depending on the valences degrees of the atoms α, β and γ. Gordan's algorithm for joint covariants Let's take A to be a covariant family taken from a space V of binary forms: A ⊂ Cov(V ) Now, we define Cov(A) to be the covariants algebra taken from A, which can be obtained by doing all possible transvectants 14 from elements of A. First of all it is clear that A ⊂ B ⇒ Cov(A) ⊂ Cov(B) (4.1) Then we have a direct lemma, consequence of theorem 2.9: Furthermore, using (4.1) we get the following lemma: Lemma 4.2. Let A 1 and A 2 be two families of Cov(V ). If A 1 ⊂ A 2 ⊂ Cov(A 1 ) then Cov(A 1 ) = Cov(A 2 ) Now there is an important definition: Definition 4.3. A covariant family A of V is said to be complete if it generates its covariant algebra Cov(A) ; that is C[A] = Cov(A) It is important to notice that the notion of complete family is weaker than the one of a covariant basis 15 . For instance, let us take V = S 3 and f ∈ V to be a cubic. We define We then know that the family A 1 = {f , H, T, ∆} is a covariant basis of Cov(A 1 ) = Cov(S 3 ). Now if we take A 2 = {H, ∆} we will have Cov(A 2 ) Cov(V ) But we also observe that A 2 is exactly the covariant basis [27] of the quadratic form H ∈ S 2 ; thus A 2 is a complete family but is not a covariant basis of Cov(V ). Let now take two finite covariant families A and B: A := {f 1 , · · · , f p } ; B := {g 1 , · · · , g q } We define a i (resp. b j ) to be the order of f i (resp. g j ). If we put U (resp. V ) to be a monomial in CA] (resp. C[B]) we will write U := f α 1 1 · · · f αp p ; V := g β 1 1 · · · g βq q We will also write α := (α 1 , · · · , α p ) ∈ N p and β := (β 1 , · · · , β q ) ∈ N q . To each well defined transvectant {U, V} r we can associate an integer solution κ := (a, b, u, v, r) taken from the system of linear diophantine equations: (S) a 1 α 1 + . . . + a p α p = u + r, b 1 β 1 + . . . + b q β q = v + r,(4.2) Now, it is clear that reciprocally, to each integer solution κ of (S) we can associate a well defined transvectant {U, V} r . For each solution κ, let F(κ) be the finite family of all molecular covariants occurring in the molecular decomposition of the transvectant {U, V} r , directly taken from proposition 3.2. If we take for example the case when A = {f } ,with f ∈ S 5 and B = {g}, with g ∈ S 2 . We then have to consider the system (S) 5α = u + r, 2β = v + r,(4.3) • The solution (a, b, u, v, r) = (1, 2, 3, 2, 2) will correspond to the transvectant {f , g 2 } 2 and the associated family F(1, 2, 3, 2, 2) will contain the molecular covariants Proof. Take the integer solution κ = (a, b, u, v, r) to be reducible, that is κ = κ 1 + κ 2 with κ i = (a i , b i , u i , v i , r i ) solution of (4.2) Thus we will be able to write U = U 1 U 2 and V = V 1 V 2 . Now there exist ν(r), ν 1 (r 1 ) and ν 2 (r 2 ) such that U V ν(r) = U 1 V 1 ν 1 (r 1 ) U 2 V 2 ν 2 (r 2 ) (4.4) which is a non connected covariant molecular occurring in F(κ). Now we know that there exist a finite family of irreducible integer solutions of (4.2) (see [48,47,51] for details). Let then define κ 1 , · · · , κ l to be the irreducible integer solutions of (4.2). We also define τ i to be the transvectant associated to the solution κ i . We thus get a main result [25,27]: Theorem 4.5. Let V 1 and V 2 be two spaces of binary forms. Define A = {f 1 , · · · , f p } ⊂ Cov(V 1 ) and B = {g 1 , · · · , g q } ⊂ Cov(V 2 ) to be two finite and complete families. Then Cov(A ∪ B) is generated by the finite and complete family τ := {τ 1 , · · · , τ l }. Proof. Let first remark that each f i (resp. each g j ) correspond to an irreducible solution of (4.2). Thus we know that A ⊂ τ and B ⊂ τ . From theorem 2.9 we have to prove that each molecular covariant M ∈ Cov(A ∪ B) is in a finite algebra. But, using definition 3.1 we can write the molecular covariant M as M = D E ν(r) with a molecular covariant D ∈ Cov(A) and E ∈ Cov(B) ; r being some integer. Because A is complete, we can suppose D to be a mononomial expression U on the f i 's ; and in the same way we can suppose E to be a mononomial expression V on the g j 's. We then have to consider molecular covariants M = U V ν(r) with U = f a 1 1 · · · f an p and V = g b 1 1 · · · g βp q Now we can make a direct induction on the index r of the transvectant. Put τ 1 , · · · , τ i 1 to be transvectants from the family τ which indexes are lower than r. If we take a transvectant {U, V} r+1 which correspond to a reducible integer solution, then by proposition 3.2, we can extend this transvectant as a linear combination of a non connected molecular covariant T and transvectants {U , V } r of lower index r < r+1. By induction hypothesis, all these transvectants {U , V } r are in k[τ ]. Let suppose without loss of generality that T = T 1 T 2 where each term corresponds to an irreducible integer solution of (4.2). Using proposition 3.5 we can thus write each term as a linear combination of on τ i ∈ τ and transvectants of index r < r + 1. We can thus conclude the first part of the lemma stating that Cov(A ∪ B) is generated by the finite family τ . To conclude, we have to show that τ is a complete family. For that purpose, let just remark that A ∪ B ⊂ τ ⊂ Cov(A ∪ B) and then Cov(τ ) = Cov(A ∪ B) = C[τ ] One direct application of theorem 4.5 is about joint covariants. Indeed, this theorem gives us a constructive approach to get a basis covariant of S n ⊕ S p , once we know a basis covariant of each space S n and S p . Of course, this algorithm depend on the resolution of an integer system. Nevertheless, there is a simple procedure to get a basis covariant of S n ⊕ S 2 , as detailed in theorem 4.6, which proof is given in [27]. From now on, we define u to be a quadratic form. Theorem 4.6. If {h 1 , · · · , h s } is a covariant basis of Cov(S n ), then irreducible covariants of Cov(S n ⊕ S 2 ) are taken from one of this set: • {h i , u r } 2r−1 for i = 1 · · · s ; • {h i , u r } 2r for i = 1 · · · s ; • {h i h j , u r } 2r where h i is of order 2p + 1 and h j is of order 2r − 2p − 1. We also have another important property: Lemma 4.7. Let µ := max(a i ) and ν := max(b j ). If u + v ≥ µ + ν, (4.5) then, the transvectant {U, V } r is reducible. Proof. Condition (4.5) implies that u ≥ µ or v ≥ ν and thus that the transvectant {U, V } r contains a reducible term T (the corresponding integer solution (α, β, u, v, r) is thus not minimal). By virtue of proposition 3.5, the transvectant is a linear combination the term T and transvectants {Ū c(k 1 ) ,V c(k 2 ) } r , where r < r and k 1 + k 2 = r − r . Note that, because both families A and B are supposed to be complete, we haveŪ c(k 1 ) = f α 1 1 . . . f α p p ,V c(k 2 ) = g β 1 1 . . . g β q q , where, moreover, the order of the transvectant {Ū c(k 1 ) ,V c(k 2 ) } r is of order u + v = u + v. Since we have supposed that u + v ≥ µ + ν, we get that u + v ≥ µ + ν and the proof is achieved by a recursive argument on the index of the transvectant r. and then h is not reducible. Note that the lemma 4.7 gives a bound for the order of each element of a minimal basis of joint covariants. More precisely: Corollary 4.9. If V = S n 1 ⊕ · · · ⊕ S ns , and if µ i is the maximal order of a minimal basis for S n i , then, for each element h of a minimal basis for V , we get ord(h) ≤ s i=1 µ i . Gordan's algorithm for simple covariants Now, to get the finiteness result when dealing with a space of binary form V = S n , we will have to introduce a weaker version of the notion of complete family. Note also that we will always consider homogeneous families. As a first observation, it is clear that for V = S n , we have G r (S n ) = {0} as soon as r > n. Furthermore, we will have G i+1 (V ) ⊂ G i (V ) for all i (5.1) We now get the Definition 5.3 (Gordan's ideals). Let r be an integer. We define the Gordan ideal I r (V ) to be the ideal generated by G r (V ) ; we will write I r (V ) := G r (V ) We observe directly that: • I r (S n ) = {0} for all r > n ; • By equation 5.1, we have I r+1 (V ) ⊂ I r (V ) for every integer r. By the property 3.2, we immediately have: Lemma 5.4. If h r ∈ I r (V ), for every covariant h ∈ Cov(V ) and for every integer j, we have {h r , h} j ∈ I r (V ) Let's now take the vector space S n of n th degree binary forms, f ∈ S n . We will write I r to be the associated Gordan's ideal. We also put ∆ to be an invariant. One important result, close to theorem 4.5, is: Theorem 5.5. Let A and B be two families of Cov(S n ). Let's suppose that • f ∈ A ; • A is relatively complete modulo I 2k ; • B is relatively complete modulo I 2k+1 (resp. modulo I 2k+1 + ∆ ). • B contains H 2k = {f , f } 2k Then there exist a finite family C, is relatively complete modulo I 2k+1 (resp. modulo I 2k+1 + ∆ ) such that Cov(C) = Cov(A ∪ B) = Cov(S n ) Proof. Using theorem 2.9 and property 3.5, we can consider transvecants {h A , h B } r avec h A ∈ Cov(A) and h B ∈ Cov(B) (5.2) Now we can write, by hypothesis h A = p(A) + h 2k and h B = q(B) + h 2k+2 (5.3) Thus (5.2) can be decomposed as {p(A), q(B)} r (5.4) {h 2k , q(B)} r (5.5) {p(A), h 2k+2 } r et {h 2k , h 2k+2 } r (5.6) Thus we may directly observe that : • The case (5.4) had been studied in proof of theorem 4.5 ; • all transvectant of (5.6) are in I 2k+2 by lemma 5.4 ; Thus we just have to deal with the case (5.5), when h 2k ∈ I 2k − I 2k+2 . For that purpose, we will make here an induction on : • The order r of the transvectant in (5.5) ; • The degree d in f of the covariant h A ; this degree is the same as the one of h 2k in (5.3). Suppose indeed that for two given integers d and r we have a finite family C 1 , · · · , C l such that, as soon as the degree in f of h A is d 1 < d {h A , q(B)} m = φ 2 (C i ) + h 2k+2 for all m (5.7) and for all r 1 < r {h 2k , q(B)} r 1 = φ 1 (C i ) + h 2k+2 Let's now consider the transvectant {h A , h B } r with h A of degree d in f . If a molecular covariant of this transvectant 3.2 is non connected, then we will have a linear combination of transvectants of order r < r ; either we will only consider transvectants {h 2k , q(B)} r with h 2k of degree d in f . Thus we can write h 2k as M H 2k ν(r) for some integer r and some molecular covariant M ∈ Cov(V ) of degree in f strictly less than d ; and thus {h 2k , q(B)} r will decompose, modulo I 2k+2 , into M H 2k ν(r) q(B) ν(r) thus, modulo I 2k+2 , into M q (B) ν(r ) because H 2k ∈ B and every molecular covariant which come from H 2k and q(B) will be in Cov(B). We can thus make use of (5.7) : we will only have to consider non-connected molecular covariants of {p(A), q(B)} r : we already saw in proof of theorem 4.5 that we only have finite cases. We now give some important lemmas before getting to the proof of theorem 5.5. Let's first define H 2k := {f , f } 2k of order 2n − 4k It is clear that this is the molecular covariant H 2k := f g 2k and thus H 2k ∈ I 2k . Now, using 2.12:, we get: Lemma 5.6. If H 2k is of order strictly greater than n, that is if 2n − 4k > n, then the family B = {H 2k } is relatively complete modulo I 2k+2 Proof. We have to consider Aronhold molecule which contain the Aronhold molecule, all symbol being equivalent: α β γ δ 2k 2k r with 1 ≤ r ≤ 2k When r > k, we can directly use lemma 2.13 with e 0 = 2k and e 1 = r, and conclude that this Aronhold molecule is in A 2k+1 , and thus in A 2k+2 . When r < k, using syzygie (2.6) we may decompose this Aronhold molecule as a linear combination of α β γ δ 2k r 2k − i i with 0 ≤ i ≤ 2k But now; to conclude: • either i ≥ k, and thus 2k + r + i ≥ 3k ; because 2k + r + i ≤ n we may use lemma 2.12 and we will have an Aronhold molecule in A r with r ≥ 2 3 w > 2k ; • or i < k, and thus 2k − i > k: the same argument as above, using lemma 2.13 will be used. And: Lemma 5.7. If H 2k is of order n, that is if si n = 4k, then the family B = {H 2k } is relatively complete modulo I 2k+2 + ∆ where ∆ is an invariant given by: Now, using lemma 4.1, we will have Cov(A 0 ) = Cov(S n ) ; this lemma 5.9 just mean that every covariant h ∈ Cov(S n ) can be written h = p(f ) + h 2 avec h 2 ∈ I 2 where p is a polynomial We then define A k to be a finite family, relatively complete modulo I 2k+2 , and containing f : we will show by induction that such a family exist. Let's first observe that, by lemma 4.1, we will have for every integer k, Cov(A k ) = Cov(S n ). We will also have A k ⊂ A k+1 ; thus, because for some k we will have I 2k+2 = {0}, this induction will give us the desired covariant basis. The main clue is to construct for every integer k an auxiliary familly B k : • If H 2k is of order p > n, we take B k := {H 2k } which, by lemma 5.6, will be relatively complete modulo I 2k+2 ; applying theorem 5.5 leads us to the family A k+1 := C. • If H 2k is of order p = n, we take B k := {H 2k , ∆} which, by lemma 5.7, will be relatively complete modulo I 2k+2 + ∆ ; where ∆ is the invariant ∆ = f f f n 2 n 2 n 2 In that case a direct induction shows that, applying theorem 5.5, we can take A k+1 to be C ∪ {∆}. • If H 2k is of order p < n, we suppose already known a covariant basis of S p ; we then take B k to be this basis, which will be finite and complete, thus finite an relatively complete modulo I 2k+2 ; we directly apply theorem 5.5 to get A k+1 := C. Thus in each case we get the construction of the family A k+1 . Now, depending on n's parity: • If n = 2q is even, we know that the family A q−1 is relatively complete modulo I 2q ; furthermore the family B q−1 only contains the invariant ∆ q := {f , f } 2q ; finally we observe that A p will be given by A p := A p−1 ∪ {∆ q } and it will be relatively complete modulo I 2q+2 = {0} ; this gives us the wanted basis. • If n = 2q + 1 is odd, the family B q−1 will contain the quadratic form H 2q := {f , f } 2q ; we then know that the family B q−1 will be given by the covariant H 2q and the invariant δ q := {H 2q , H 2q } 2 . The family A q obtained using theorem 5.5 will then be relatively complete modulo I 2q+2 = {0} ; which gives us the wanted basis. Recall the covariant algebra Cov(V ) := Cov(S 6 ⊕ S 2 ) is a multi-graded algebra. We can write Cov(V ) = d 1 ≥0,d 2 ≥0,o≥0 Cov(V ) d 1 ,d 2 ,o where d 1 is the degree in the binary form f ∈ S 6 , d 2 is the degree in the binary form u ∈ S 2 and o the degree in the variable x ∈ C 2 . We can define the Hilbert series: H(z 1 , z 2 , t) := d 1 ,d 2 ,o dim(Cov(V ) d 1 ,d 2 ,o )z d 1 1 z d 2 2 t o Hilbert series of the covariant algebra of S 6 ⊕ S 2 has been computed using maple package of Bedratyuk [5]. Thanks to this Hilbert series and theorem 4.6, we finally get a minimal basis of 103 covariants: it's worth noting that, by using this algorithm, we had to check invariant homogeneous space's dimensions up to degree 15. Here we have a direct application of constructive theorem 2.9. • (1) As a first step the family A 0 is simply the binary form f ∈ S 8 ; the set B 0 is simply the form The same kind of argument as above, using lemma such as lemma 2.13 leads to [27,26]: 6,6 } r which is associated to the integer system 8a 1 + 8a 2 + 12a 3 + 12a 4 + 14a 5 + 18a 6 + 18a 7 = u + r 4b 1 + 4b 2 + 6b 3 = v + r (B.1) h 2,12 := {f , f } 2 ∈ S 12(2)Lemma Using Normaliz package [13] of Macaulay2 software [28], we get the integer solutions of B.1. To get a basis reduction, we make use of the fundamental an well known relation between covariants of a binary quartic 12h 2 6,6 + 6h 3 4,4 + 2h 6,0 h 3 2,4 − 3h 2 2,4 h 4,4 h 4,0 = 0 From this, we have a bound on b 3 in the system (B.1), and this remark leads us to important reduction on transvectants. With computations in Macaulay2 [28], we finally get a covariant basis of S 8 given bellow. Definition 2. 7 . 7For every space V of binary forms, we define a molecular covariant M to be a covariant given by M = Ψ(D) where D ∈ M(V ). take the space V = S n . The syzygies 2., in the case we are in Sym 3 (V ), all symbols are equivalent, w 3 3Corollary 2.13. Let D(e 0 , e 1 , e 2 ) be given by 2.8 of grade e 0 and suppose that e 0 ≤ n 2 and e 1 + e 2 > e 0 2 then D(e 0 , e 1 , e 2 ) ∈ A e 0 +1 (V ) unless e 0 = e 1 = e 2 = n 2 . Definition 3. 1 . 1If D and E are two Aronhold molecules, for a given integer r and a given symbol ν(r), we define the Aronhold molecule L ν(r) (D, E), graphically noted D E ν(r) Definition 3. 3 . 3Given an Aronhold molecule D, and an integer k, we define D µ(k) as the Aronhold molecule obtained by adding k edges on D in a certain way µ(k). Proposition 3. 4 . 4Let be given two Aronhold molecules D and E, an integer r and two links ν 1 (r) and ν 2 (r) in the transvectant {D, E} r , then the molecular transvectant Proposition 3. 5 . 5Let be given two Aronhold molecules D and E, an integer r and a link ν(r) in the transvectant {D, E} r , then the Aronhold moleculeD E ν(r)is a linear combination of the transvectants {D, E} r and {D µ 1 (k 1 ) , E µ 2 (k 2 ) } r with k 1 + k 2 + r = r being constant and r < r.If we take for example the Aronhold molecules:D = α β 2and E = γ we can consider the transvectant {D, E} 2 and the two Aronhold molecules: property 3.4 assures us that Lemma 4. 1 . 1Let V = S n and f ∈ V . If any family A ⊂ Cov(V ) contains f then Cov(A) = Cov(V ). H := {f , f } 2 ; T := {f , H} 1 and ∆ := {H, H} 2 first one is a non connected molecular covariant. • The solution (a, b, u, v, r) case there is no non connected molecular covariant. If fact we have: Lemma 4.4. If κ is a reducible integer solution of (S), then F(κ) contains a non connected molecular covariant. Remark 4. 8 . 8The statement u + v ≥ µ + ν can't be replaced by the hypothesis u ≥ µ or v ≥ ν.Indeed, taking f ∈ S 6 and the covariant bases given in A, we can compute the first covarianth 3,8 := {{f , f } 4 , f } 1 from this bases and the second covariant h := {f 2 , f } 5 . For this last covariant we have u = 7 Definition 5 . 1 . 51Let I ⊂ Cov(V ) be an ideal, a family A is said to be relatively complete modulo I if every homogeneous covariant h ∈ Cov(A) can be written h = p(A) + h I with h I ∈ I and p(A) being a polynomial expression in A, all expression having the same degree. Now, related to grade's definition 2.8: Definition 5.2. Let r be an integer ; we define G r (V ) ⊂ M(V ) to be the set of all molecular covariants with grade at least r: G r (V ) := Ψ (A r (V )) For all integer k ≥ 1 we haveI 2k−1 = I 2kAnd a direct lemma:Lemma 5.9. The family A 0 := {f } is relatively complete modulo I 2 Appendix A. Joint covariants of S 6 ⊕ S 2We write h d,o to be a covariant of degree d and order o, taken from the covariant basis of S 6 in table A, issue from Grace-Young[27], and u to be a quadratic form in S 2 . By theorem 4.6 we have to consider covariants given by {h, u r } 2r−1 or {h, u r } 2rD/O 0 2 4 6 1 f 2 {f , f } 6 h 2,4 := {f , f } 4 3 h 3,2 := {h 2,4 , f } 4 h 3,6 := {h 2,4 , f } 2 4 {h 2,4 , h 2,4 } 4 {h 3,2 , f } 2 h 4,6 := {h 3,2 , f } 1 5 {h 2,4 , h 3,2 } 2 {h 2,4 , h 3,2 } 1 6 {h 3,2 , h 3,2 } 2 h 6,6a := {h 3,8 , h 3,2 } 2 h 6,6b := {h 3,6 , h 3,2 } 1 7 {f , h 2 3,2 } 4 {f , h 2 3,2 } 3 8 {h 2,4 , h 2 3,2 } 3 9 {h 3,8 , h 2 3,2 } 4 10 {h 3 3,2 , f } 6 {h 3 3,2 , f } 5 12 {h 3,8 , h 3 3,2 } 6 15 {h 3,8 , h 4 3,2 } 8 D/0 8 10 12 2 h 2,8 := {f , f } 2 3 h 3,8 := {h 2,4 , f } 1 {h 2,8 , f } 1 4 {h 2,8 , h 2,4 } 1 5 h 5,8 := {h 2,8 , h 3,2 } 1 Table 2. Covariant basis of S 6 {f , f } 6 {h 2,4 , h 2,4 } 4 {h 3,2 , h 3,2 } 2 {h 5,4 , u 2 } 4 {h 7,2 , u} 2{f , u 2 } 4 {h 2,4 , u 2 } 3 {h 3,4 , u} 2 {h 4,4 , u 2 } 3 {h 7,2 , u} 1 {h 2,4 , u} 2 {f , u 3 } 5 {h 3,6 , u 2 } 4 {h 5,4 , u} 2 {h 7,4 , u} 2 {h 2,8 , u 3 } 6 {h 3,6 , u 3 } 5 {h 6,6a , u 2 } 4 {h 4,6 , u 2 } 4 {h 6,6b , u 2 } 4 {h 2,8 , u 4 } 7 {h 5,8 , u 3 } 6 {h 3,8 , u 3 } 6 {h 4,10 , u 4 } 8 {h 3,12 , u 5 } 10 , u} 1 {f , u} 2 {f , u 2 } 3 {h 3,6 , u} 2 {h 4,4 , u} 1 {h 6,6b , u} 2 {h 2,8 , u 2 } 4 {h 3,6 , u 2 } 3 {h 5,8 , u 2 } 4 {h 4,6 , u} 2 {h 4,10 , u 3 } 6 {h 2,8 , u 2 } 5 {h 3,12 , u 4 } 8 {h 3,8 , u 2 } 4 {h 6,6a , u} 2 • Order 10: 2 degree 4 covariants h 4,10 and {h 3,12 , u} 2 • Order 12: 1 degree 3 covariant h 3,12Appendix B. Covariant bases of S 8Order 0: 27 invariants Degree 2 Degree 4 Degree 6 Degree 7 Degree 8 Degree 9 Degree 10 {h 8,2 , u} 2 {h 3 3,2 , f } 6 {u, u} 2 {h 3,2 , u} 2 {h 5,2 , u} 2 {h 4,6 , u 3 } 6 {h 7,4 , u 2 } 4 {h 2,4 , u 2 } 4 {h 4,4 , u 2 } 4 {h 3,8 , u 4 } 8 {h 6,6a , u 3 } 6 {f , u 3 } 6 {h 3,6 , u 3 } 6 {h 6,6b , u 3 } 6 {h 2,8 , u 4 } 8 {h 5,8 , u 4 } 8 {h 4,10 , u 5 } 10 {h 3,12 , u 6 } 12 Degree 11 Degree 13 Degree 15 {h 9,4 , u 2 } 4 {h 12,2 , u} 2 {h 3,8 , h 4 3,2 } 2 {h 10,2 , u} 2 • Order 2: 33 covariants Degree 1 Degree 3 Degree 4 Degree 5 Degree 6 Degree 7 Degree 8 u h 3,2 {h 3,2 , u} 1 h 5,2 {h 5,2 , u} 1 h 7,2 h 8,2 Degree 10 Degree 11 Degree 12 Degree 13 h 10,2 {h 10,2 , u} 1 h 12,2 {h 12,2 , u} 1 {h 9,4 , u} 2 {h 9,4 , u 2 } 3 • Order 4: 21 covariants Degree 2 Degree 3 Degree 4 Degree 5 Degree 7 Degree 9 Degree 10 h 2,4 {h 2,4 , u} 1 h 4,4 h 5,4 h 7,4 h 9,4 {h 9,4 • Order 6: 12 covariants Degree 1 Degree 2 Degree 3 Degree 4 Degree 6 f {f , u} 1 h 3,6 h 4,6 h 6,6a {h 2,8 , u} 2 {h 3,8 , u} 2 h 6,6a {h 3,6 , u} 1 {h 5,8 , u} 2 {h 4,10 , u 4 } 4 {h 3,12 , u 3 } 6 • Order 8: 7 covariants Degree 2 Degree 3 Degree 4 Degree 5 h 2,8 h 3,8 {h 2,8 , u 2 } 3 h 5,8 {h 2,8 , u} 1 {h 4,10 , u} 2 {h 3,12 , u 2 } 4 To obtain A 1 we have to consider transvectants {f a , h b 2,12 } r with no reducible molecular covariants modulo I 4 . From lemma 5.8 we deduce that necessarily r ≤ 2. Furthermore, if an Aronhold molecule contain the Aronhold molecule then we can directly use lemma 2.13 with e 0 = 2 and e 1 = 2, and conclude that this Aronhold molecule is in A 3 , and thus in A 4 . We can deduce from all this that A 1 is the family f ; h 2,12 ; h 3,18 := {f , h 2,12 } 1 Now the family B 1 is simply the form h 2,8 := {f , f } 4 ∈ S 8 (3) To get A 2 we have to consider transvectantsα β γ δ 2 2 2 {f a 1 h a 2 2,12 h a 3 3,18 , h b 2,8 } r B.1. The family A 2 is given by the seven covariantsf ; h 2,8 = {f , f } 4 ; h 2,12 = {f , f } 2 ; h 3,12 := {f , h 2,8 } 2 ; h 3,14 := {f , h 2,8 } 1 h 3,18 := {f , h 2,12 } 1 ; h 4,18 := {h 2,12 , h 2,8 } 1We also recall that we have to take into account the invariant{f , h 2,8 } 8The family B 2 is given by the covariant basis ofh 2,4 := {f , f } 6 ∈ S 4As a classical result[27], such a basis is given byh 2,4 ; h 4,4 := {h 2,4 , h 2,4 } 2 ; h 6,6 := {h 2,4 , {h 2,4 , h 2,4 } 2 } 1 and two invariants h 4,0 := {h 2,4 , h 2,4 } 4 ; h 6,0 := {h 2,4 , {h 2,4 , h 2,4 } 2 } 4 (4) To get family B 3 , we have to consider transvectants{f a 1 h a 2 2,8 h a 3 2,12 h a 4 3,12 h a 5 3,14 h a 6 3,18 h a 7 4,18 , h b 1 2,4 h b 2 4,4 h b 3 • Degree 1 : the binary form f of order 8 Covariants h 2,0 := {f , f } 8 h 2,4 := {f , f } 6 h 2,8 h 2,12 • Degree 3 : 8 covariants Covariants {f , h 2,8 } 8 {f , h 2,4 } 4 {f , h 2,4 } 3 {f , h 2,4 } 2 {f , h 2,4 } 1 h 3,12 := {f , h 2,8 } 2 Order 14 18 Covariants h 3,14 := {f , h 2,8 } 1 h 3,18 := {f , h 2,12 } 1 • Degree 4 : 12 covariants Covariants {h 2,4 , h 2,4 } 4 h 4,4 := {h 2,4 , h 2,4 } 2 {h 2,8 , h 2,4 } 3 {h 2,12 , h 2,4 } 4 {h 2,12 , h 2,4 } 3 {h 2,8 , h 2,4 } 4 {h 2,8 , h 2,4 } 1 Covariants {h 2,12 , h 2,4 } 2 {h 2,12 , h 2,4 } 1 {h 2,12 , h 2,8 } 1 • Degree 5 : 11 covariants } 7 {f , h 4,4 } 4 {f , h 4,4 } 3 {f , h 4,4 } 2 {h 3,14 , h 2,4 } 4 {h 3,12 , h 2,4 } 1 Covariants {h 2,12 , h 4,4 } 4 {h 2,12 , h 4,4 } 3 Covariants {h 2,12 , h 3 2,4 } 12 {h 2,12 , h 3 2,4 } 11 {h 2,12 , h 2,4 h 4,4 } 8 {h 2,12 , h 6,6 } 6 {h 2,8 , h 6,6 } 6 Degree 12 : 1 covariants of order 2 {h 4,18 , h 4 2,4 } 16• Degree 2 : 4 covariants Order 0 4 8 12 Order 0 4 6 8 10 12 Order 0 4 6 8 10 Order 12 14 18 Order 0 2 4 6 8 10 14 Covariants {f , h 2 2,4 } 8 {f , h 2 2,4 {f , h 2 2,4 } 6 {f , h 2 2,4 } 5 {h 3,12 , h 2,4 } 3 {f , h 4,4 } 1 • Degree 6 : 9 covariants Order 0 2 4 6 Covariants {h 4,4 , h 2,4 } 4 {h 2,8 , h 2 2,4 } 7 {h 2,12 , h 2 2,4 } 8 h 6,6 := {h 4,4 , h 2,4 } 1 {h 2,8 , h 4,4 } 4 {h 2,12 , h 2 2,4 } 7 {h 2,8 , h 4,4 } 3 Order 8 10 • Degree 7 : 8 covariants Order 0 2 4 6 Covariants {f , h 2,4 h 4,4 } 8 {f , h 6,6 } 6 {h 3,12 , h 2 2,4 } 8 {h 3,14 , h 2 2,4 } 8 {f , h 2,4 h 4,4 } 7 {f , h 6,6 } 5 {h 3,12 , h 2 2,4 } 7 {f , h 6,6 } 4 • Degree 8 : 7 covariants Order 0 2 4 6 {h 2,12 , h 3 2,4 } 10 {h 2,12 , h 2,4 h 4,4 } 7 • Degree 9 : 5 covariants Order 0 2 4 Covariants {h 3,12 , h 3 2,4 } 12 {h 3,14 , h 3 2,4 } 12 {h 3,14 , h 3 2,4 } 11 {h 3,12 , h 3 2,4 } 11 {f , h 2 4,4 } 7 • Degree 10 : 3 covariants Order 0 2 Covariants {h 2,12 , h 2 2,4 h 4,4 } 12 {h 2,12 , h 2,4 h 6,6 } 10 {h 2,12 , h 2 2,4 h 4,4 } 11 • Degree 11 : 2 covariants of order 2 {h 3,18 , h 4 2,4 } 16 ; {h 3,14 , h 2 2,4 h 4,4 } 12 • This operator was named scalling process by Olver[41] 10 A huge amount of work has been done first by Weyl[54] and afterword by Kung-Rota[34] to get a modern version of this symbolic method, which led for example to Umbral calculus. 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[ "FINITE DECOMPOSITION COMPLEXITY AND THE INTEGRAL NOVIKOV CONJECTURE FOR HIGHER ALGEBRAIC K-THEORY", "FINITE DECOMPOSITION COMPLEXITY AND THE INTEGRAL NOVIKOV CONJECTURE FOR HIGHER ALGEBRAIC K-THEORY" ]	[ "Daniel A Ramras ", "Romain Tessera ", "Guoliang Yu " ]	[]	[]	Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proof of the integral K-theoretic Novikov conjecture, regarding split injectivity of the Ktheoretic assembly map, for groups with finite decomposition complexity and finite CW models for their classifying spaces. By work of Guentner, Tessera, and Yu, this includes all (geometrically finite) linear groups.	10.1515/crelle-2012-0112	[ "https://arxiv.org/pdf/1111.7022v6.pdf" ]	55,935,232	1111.7022	474a2250fc836658be59ae20da7c85b07d2a2549	FINITE DECOMPOSITION COMPLEXITY AND THE INTEGRAL NOVIKOV CONJECTURE FOR HIGHER ALGEBRAIC K-THEORY 29 Oct 2012 Daniel A Ramras Romain Tessera Guoliang Yu FINITE DECOMPOSITION COMPLEXITY AND THE INTEGRAL NOVIKOV CONJECTURE FOR HIGHER ALGEBRAIC K-THEORY 29 Oct 2012arXiv:1111.7022v3 [math.KT] Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proof of the integral K-theoretic Novikov conjecture, regarding split injectivity of the Ktheoretic assembly map, for groups with finite decomposition complexity and finite CW models for their classifying spaces. By work of Guentner, Tessera, and Yu, this includes all (geometrically finite) linear groups. Introduction Decomposition complexity for metric spaces, introduced by Guentner, Tessera, and Yu [13,14], is a natural inductive generalization of the muchstudied notion of asymptotic dimension. Roughly speaking, decomposition complexity measures the difficulty of decomposing a metric space into uniformly bounded pieces that are well-separated from one another. The class of metric spaces with finite decomposition complexity (FDC), as defined in Definition 6.1, contains all metric spaces with finite asymptotic dimension [13,Theorem 4.1], as well as all countable linear groups equipped with a proper (left-)invariant metric ([14, Theorem 3.0.1] and [13,Theorems 5.2.2]). In this article, we study the integral Novikov conjecture for the algebraic K-theory of group rings R[Γ], where Γ has FDC. For a discrete group Γ, the classical Novikov conjecture on the homotopy invariance of higher signatures is implied by rational injectivity of the Baum-Connes assembly map [5]. In Yu [31] and Skandalis-Tu-Yu [26], injectivity of the Baum-Connes map was proved for groups coarsely embeddable into Hilbert space. Using this result, Guentner, Higson, and Weinberger [12] proved the Novikov conjecture for linear groups. This inspired the work of Guentner, Tessera, and Yu [14], who proved the integral Novikov conjecture (establishing integral injectivity of the L-theoretic assembly map) The first author was partially supported by NSF grants DMS-0804553/0968766. The second author was partially supported by NSF grant DMS-0706486 and ANR grants AGORA and BLANC. The third author was partially supported by NSF grants DMS-0600216 and DMS-1101195. 1 for geometrically finite FDC groups (i.e. those with a finite CW model for their classifying space), and hence the stable Borel Conjecture for closed aspherical manifolds whose fundamental groups have FDC. The algebraic K-theory Novikov conjecture claims that Loday's assembly map [18] (1) H * (BΓ; K(R)) −→ K * (R[Γ]) is (rationally) injective. Here Γ is a finitely generated group and R is an associative, unital ring (not necessarily commutative). The domain of the assembly map is the homology of Γ with coefficients in the (non-connective) K-theory spectrum of R, and the range is the (non-connective) K-theory of the group ring R [Γ]. For discussions of this conjecture and its relations to geometry, see Hsiang [17] and Farrell-Jones [11]. A great deal is known about the map (1): Bökstedt, Hsiang, and Madsen [6] proved that (1) is rationally injective for R = Z under the assumption that H * (Γ; Z) is finitely generated in each degree. Integral injectivity results were proven for geometrically finite groups with finite asymptotic dimension by Bartels [4] and Carlsson-Goldfarb [9], building on Yu's work [30] (which established injectivity of the Baum-Connes assembly map for groups with finite asymptotic dimension). In Section 7, we prove the following generalization of [4,9]. Theorem 1.1. Let Γ be a group with finite decomposition complexity, and assume there exists a universal principal Γ-bundle EΓ → BΓ with BΓ a finite CW complex. Then for every ring R, the K-theoretic assembly map H * (BΓ; K(R)) −→ K * (R[Γ]) is a split injection for all * ∈ Z. We note that in Theorem 1.1, the ring R may be replaced by any additive category A, as will be clear from our proof. (Then K * (R[Γ]) must be replaced by the K-theory of the category A[Γ], as defined in Bartels [4].) Analogous methods yield an integral injectivity result for the assembly map associated to Ranicki's ultimate lower quadratic L-theory L −∞ . We also obtain a large-scale version of the Borel Conjecture for bounded Ktheory (Theorem 7.1), analogous to [14,Theorems 4.3.1,4.4.1]. Guentner, Tessera, and Yu [14] studied the Ranicki-Yamasaki controlled (lower) algebraic K-and L-groups [23,24] of FDC metric spaces, and established a large-scale vanishing result formulated in terms of Rips complexes. They used this result to study related assembly maps, leading to important geometric rigidity results (in particular, the stable Borel Conjecture). The key technical result in the present paper is a vanishing theorem for continuously controlled K-theory, analogous to [14,Theorem 5.1]. Theorem 1.2. If X is a metric space with bounded geometry and finite decomposition complexity, then colim s K c * (P s X) = 0 for all * ∈ Z. This theorem is proven in Section 6. Here K c * (Z) denotes the continuously controlled K-theory of the metric space Z (see Section 2), and bounded geometry means that for each r > 0, there exists N ∈ N such that each ball of radius r contains at most N elements. Given a bounded geometry metric space X and a positive number s, the Rips complex P s (X) is formed from the vertex set X by laying down a simplex x 0 , . . . , x n whenever the pairwise distances d(x i , x j ) are all at most s. The analogous result from [14] is proven using controlled Mayer-Vietoris sequences for Ranicki and Yamasaki's controlled lower K-and L-groups [23,24]. In that flavor of controlled algebra, one imposes universal bounds on the propagation of morphisms, and the Mayer-Vietoris sequences are only exact in a weak sense involving these bounds. While it may be possible to construct quantitative versions of higher algebraic K-groups (analogous to the Ranicki-Yamasaki controlled lower K-groups, and to recent work of Oyono-Oyono and Yu in operator K-theory [19]) such a theory does not currently exist. Instead, we produce analogous (strictly exact) Mayer-Vietoris sequences in continuously controlled K-theory. Loosely speaking, this corresponds (in low dimensions) to taking colimits over the propagation bounds in the Ranicki-Yamasaki theory. Our Mayer-Vietoris sequences are produced using the machinery of Karoubi filtrations as developed, for instance, in Cárdenas-Pedersen [7]. In broad strokes, the proof of Theorem 1.2 is similar to the arguments in [14,Section 6]. The starting point is that the theorem holds for bounded metric spaces. In [14,Section 6], controlled Mayer-Vietoris sequences were applied to a space X covered by two subspaces, each an r-disjoint union of smaller subspaces. Great care was taken in order to keep r large with respect to the other parameters involved, e.g. the Rips complex parameter and the propagation bound on morphisms. In the present work we consider all at once a sequence of such decompositions of X, whose disjointness tends to infinity. For each continuously controlled K-theory class x ∈ K c * (P s X), we show that at sufficiently high stages in the sequence of decompositions, x can be build from classes supported on the (relative) Rips complexes of the individual factors appearing in the decompositions. An inductive process ensues, in which we further decompose the spaces appearing at each level of the previous sequence of decompositions. Metric spaces with finite decomposition complexity are essentially those for which this process eventually results in (uniformly) bounded pieces. Such considerations lead to the notion of a decomposed sequence, introduced in Section 4. Our approach avoids much of the intricate manipulation of various constants in [14,Section 6], but the price we pay is that we must deal with more complicated objects than simply a metric space decomposed as a union of two subspaces. Our approach to the assembly map makes crucial use of both ordinary Rips complexes P s (X) and the relative Rips complexes introduced in [14]. As the parameter s increases, the simplices in P s (X) wipe out any smallscale features of X and expose the large-scale structure of the space. When X is a torsion-free group Γ equipped with the word metric associated to a finite generating set, the Rips complexes also give a sequence of cocompact Γ-spaces approximating the universal free Γ-space EΓ (if Γ has torsion, they approximate the universal space for proper actions). Theorem 1.1 is deduced from Theorem 1.2 through a comparison between EΓ and the Rips complexes, which shows that when Γ is geometrically finite and has FDC, the controlled K-theory of EΓ vanishes (Theorem 7.8). In earlier work on assembly maps in higher algebraic K-theory, nerves of coverings (as in Bartels [4] or Carlsson-Goldfarb [9]) or compactifications of the universal space EΓ (as in Carlsson-Pedersen [10] or Rosenthal [25]) played roles similar to the Rips complexes used here. Unlike coverings and compactifications, Rips complexes are built in a canonical way from the underlying metric space. Together with their dual relationships to the large-scale geometry of Γ and to the universal space EΓ, this makes Rips complexes ideally suited to the study of assembly maps. Organization: Section 2 reviews notions from geometric algebra. Section 3 establishes algebraic facts about Karoubi filtrations that underly our controlled Mayer-Vietoris sequences. The sequences themselves are constructed in Section 4. This section begins with a general Mayer-Vietoris sequence for proper metric spaces, and then specializes this sequence to Rips complexes and relative Rips complexes. Section 4 also introduces the terminology of decomposed sequences used extensively in Section 6. In Section 5, we review the necessary metric properties of Rips complexes and relative Rips complexes. Section 6 reviews the notion of finite decomposition complexity and establishes our vanishing theorem for continuously controlled K-theory. Assembly maps for K-and L-theory are studied in the final section. To aid readability, we have attempted to make our indexing sets as explicit as possible. In some arguments, the same indexed family occurs several times in one argument, and in such cases we will abbreviate expressions like {Z α } α∈A to {Z α } α after their first appearance. Acknowledgements: We thank Daniel Kasprowski for pointing out an error in a previous version of the paper, and the referee for offering many suggestions that improved the exposition. The first author also thanks Ben Wieland for helpful conversations. Geometric modules Throughout this paper all metrics will be allowed to take on the value ∞, and all categories will be assumed to be small. If X is a metric space and x ∈ X, we set B r (x) = {y ∈ X : d(x, y) < r} and if Z ⊂ X, we set N r (Z) = {y ∈ X : d(y, Z) < r}. We call a metric space proper if the closed ball {y ∈ X : d(x, y) r} is compact for every x ∈ X and every r > 0. Definition 2.1. Let A be an additive category (we think of the objects of A as "modules"). A geometric A-module over a metric space X is a function M : X → Ob(A). We say that M is locally finite if its support supp(M ) = {x ∈ X\|M (x) = 0} is locally finite in X, in the sense that for each compact set K ⊂ X, supp(M ) ∩ K is finite. (If X is proper, this is equivalent to requiring that each x ∈ X has a neighborhood U x such that supp(M ) ∩ U x is finite.) We will usually abbreviate M (x) by M x , and for any subspace Y ⊂ X we define M (Y ) to be the geometric module given by M (Y ) x = M x , x ∈ Y, 0, x / ∈ Y A morphism φ from a geometric module M to a geometric module N is collection of morphisms φ xy : M y → N x for all pairs (x, y) ∈ X × X, subject to the condition that for each x ∈ X, the sets {y ∈ X \| φ xy = 0} and {y ∈ X \| φ yx = 0} are finite. One may think of φ = {φ xy } as a matrix indexed by the points in X, in which each row and each column has only finitely many non-zero entries. We will deal with a fixed additive category A throughout the paper, and we will refer to geometric A-modules simply as geometric modules. The main case of interest is when A is (a skeleton of) the category of finitely generated free R-modules for some associative unital ring R. Geometric modules and their morphisms form an additive category A(X), in which composition of morphisms is simply matrix multiplication (which is well-defined due to the row-and column-finiteness of these matrices) and addition of morphisms is defined via entry-wise sum of matrices (using the additive structure of A). Direct sums of objects in A(X) are formed by taking direct sums pointwise over X. The categories we are interested in will impose important additional support conditions on the morphisms φ. Definition 2.2. We say that a morphism φ : M → N of geometric modules over X has finite propagation (or is bounded) if there exists R > 0 such that φ xy = 0 whenever d(x, y) > R. We may now consider the subcategory of locally finite geometric modules and bounded morphisms A b (X) ⊂ A(X). This is again an additive category, and its K-theory is, by definition, the bounded K-theory of X with coefficients in A. Remark 2.3. Throughout this paper, the K-theory of an additive category C will mean the non-connective K-theory spectrum K(C) as defined, for example, in [7,Section 8]. This means we consider C as a Waldhausen category, in which cofibrations are (up to isomorphism) inclusions of direct summands and weak equivalences are isomorphisms. Since inclusions of direct summands can be characterized in terms of split exact sequences, additive functors C → D always preserve these notions of cofibration and weak equivalence, and hence induce maps K(C) → K(D). We set K * (C) = π * K(C) for * ∈ Z. Next, we will consider the notion of continuously controlled morphisms, which will be the main object of study in this paper. Here and in what follows, we give the half-open interval [0, 1) the usual Euclidean metric d(s, t) = \|s − t\|, and for metric spaces (X, d X ) and (Y, d Y ), we give X × Y the metric d ((x, y), (x ′ , y ′ )) = d X (x, x ′ ) + d Y (y, y ′ ). The following definition appears in Weiss [28], and is a slight variation on the work of Anderson-Connolly-Ferry-Pedersen [1]. Definition 2.4. A morphism φ : M → N of geometric modules over X × [0, 1) is continuously controlled at 1 if for each x ∈ X and each neighborhood U of (x, 1) in X × [0, 1], there exists a (necessarily smaller) neighborhood V of (x, 1) such that φ does not cross U \ V : that is, if v ∈ V and y / ∈ U , then φ yv = φ vy = 0. It is an exercise to check that the collection of continuously controlled morphisms in A b (X) form a subcategory. Since the control condition only depends on the support of the morphism, this collection of morphisms is also closed under addition and negation, and direct sums in this subcategory agree with direct sums in A b (X). Definition 2.5. Let X be a proper metric space. The category of locally finite geometric modules over X × [0, 1) and continuously controlled morphisms, denoted A c (X), is the subcategory of A b (X × [0, 1)) containing all objects, but only those morphisms with continuous control at 1. As explained above, this is an additive subcategory of A(X). For Z ⊂ X, we will write A X c (Z) for the category of controlled modules on Z × [0, 1), where Z has the metric inherited from X. (This will be especially relevant when Z and X are simplicial complexes, since then Z has its own intrinsic simplicial metric, giving rise to a different category of controlled modules.) Given a closed subset Z ⊂ X, we define A X+ c (Z) ⊂ A c (X) to be the full subcategory on those geometric modules M ∈ A c (X) which are supported "near" Z × [0, 1); that is, M ∈ A X+ c (Z) if and only if there exists R > 0 such that M (x,t) = 0 implies d(x, Z) < R. When X is clear from context, we will simply write A + c (Z) rather than A X+ c (Z). Remark 2.6. In Weiss [28, Section 2], a slightly different support condition for modules is used to define an analogue of our category A c (X): namely the support of each module is required to be a discrete, closed subset of X ×[0, 1). This condition is equivalent to our local finiteness condition when X is a proper metric space, so that our category A c (X) is the same as Weiss's A (X × [0, 1], X × [0, 1)). In this paper, we only need to consider A c (X) for proper metric spaces X. The spaces whose controlled K-theory appears in this paper will all be simplicial complexes. We will assume all our simplices have diameter one. More specifically, we identify the simplex with vertices x 1 , . . . , x n with the convex hull of the points √ 2 2 e i ∈ R n , where the e i are the standard basis vectors. Given a simplicial complex K, the simplicial metric d ∆ on P is the unique path-length metric which restricts to the standard Euclidean metric on each simplex. Explicitly, d ∆ (x, y) = inf N −1 i=0 d ∆ (p i , p i+1 ) where the infimum is taken over all sequences x = p 0 , p 1 , . . . , p N = y (with N arbitrary) such that p i and p i+1 lie in the same simplex of K, and d ∆ (p i , p i+1 ) is the Euclidean metric on a simplex containing both points. When x and y lie in different path components of K, we set d ∆ (x, y) = ∞. Note that locally finite simplicial complexes are always proper with respect to their simplicial metrics (this follows, for example, from the argument in Lemma 5.2 below, which can be used to show that each ball contains finitely many vertices). All simplicial complexes in this paper will be equipped with the simplicial metric (possibly restricted from some larger complex). We will need a lemma regarding the functoriality of controlled K-theory for maps between metric spaces. Versions of the following result are stated (without proof) in [3,4,28]; an equivariant version is proven in [2,Lemma 3.3]. For completeness, we sketch the argument. Lemma 2.7. Let f : X → Y be a continuous map of proper metric spaces which is proper (that is, f −1 (C) is compact in X for all compact sets C ⊂ Y ) and metrically coarse (that is, for each R > 0 there exists S > 0 such that d X (x 1 , x 2 ) < R implies d Y (f (x 1 ), f (x 2 )) < S). Then f induces a functor f * : A c (X) → A c (Y ). Moreover, if X ′ ⊂ X and Y ′ ⊂ Y are closed subspaces with f (X ′ ) ⊂ N t (Y ′ ) for some t > 0, then f induces a functor f * : A + c (X ′ ) → A + c (Y ′ ). In particular, given a commutative diagram of simplicial maps between locally finite simplicial complexes (with all but the right-hand vertical map injective) there is an induced functor A P + c (P ′ ) → A Q+ c (Q ′ ), where P ′ and P are given the subspace metrics inherited from the simplicial metric on P ′′ , while Q ′ and Q are given the subspace metrics inherited from the simplicial metric on Q ′′ . Proof. We will construct the functor f * : A + c (X ′ ) → A + c (Y ′ ) ; the other functors are special cases (note that simplicial maps decrease distances). Let M be a geometric module in A + c (X ′ ). If f is injective, we set f * (M ) (y,t) = M f −1 (y,t) . If f is not injective, one needs to redefine the cate- gory A + c (−) so that setting f * (M ) (y,t) = x∈f −1 (y) M (x,t) is well-defined. We will ignore this technicality in what follows; see [28,Section 2] for details. Since M is supported on a neighborhood of X ′ , f is metrically coarse, and f (X ′ ) ⊂ N t (Y ′ ), the module f (M ) will be supported on a neighborhood of Y ′ . The behavior of f on morphisms is defined similarly; since f is metrically coarse, we see that f (φ) has finite propagation for each φ ∈ A + c (X ′ ). Finally, we must check that for each φ ∈ A + c (X ′ ), f (φ) is continuously controlled. Fix y ∈ Y , and consider a neighborhood U of (y, 1) in Y × I. Replacing U with a small ball around (y, 1) if necessary, we may assume that the closure U is compact. Let U ′ = (f × Id I ) −1 (U ). For each x ∈ f −1 (y), U ′ is a neighborhood of (x, 1), so there exists a smaller neighborhood V x of (x, 1) such that φ z ′ ,z = 0 if z ∈ V x , z ′ / ∈ U ′ or z / ∈ U ′ , z ′ ∈ V x . Since f is proper, f −1 (y) is compact, so we may cover (f × Id I ) −1 (y, 1) by finitely many of the sets V x , say to V x 1 , . . . , V xn . Since f is proper and continuous and U is compact, it follows that C := (f × Id I ) (f × Id I ) −1 (U ) \ n i=1 V x i is compact. Now V = U \ C is a neighborhood of (y, 1), and one may now check that φ does not cross U \ V = C ∩ U . Remark 2.8. Most uses of Lemma 2.7 in the sequel will only require the statement regarding simplicial complexes. Note that by setting P ′ = P and Q ′ = Q, we obtain a statement about the categories A c (−). The functors constructed in Lemma 2.7 combine to yield a functor from the category of proper metric spaces and continuous, metrically coarse injections into the category of small categories. This makes the various colimits of categories considered later in the paper well-defined. (For non-injective maps, one needs to be careful in order to make composition strictly associative at the categorical level; this is achieved by Weiss's construction [28,Section 2]. Until Section 7, all the maps we consider are injective.) Karoubi Filtrations We will use the notion of a Karoubi filtration to produce various Mayer-Vietoris sequences in controlled K-theory. Algebraically, a Karoubi filtration is a tool for collapsing a full subcategory of an additive category; geometrically it is a method for producing fibrations of K-theory spectra. By abuse of notation we will write A ∈ A to mean that A is an object in A. Furthermore, we will write A = A 1 ⊕ A 2 to mean that there exist maps i j : A j → A making A the categorical direct sum of A 1 and A 2 . We will always implicitly choose particular maps i j , and we will denote the corresponding projections A → A j by π j . Definition 3.1. Let S ⊂ A be a full additive subcategory of a small additive category A. A Karoubi filtration on the pair (A, S) consists of an index set I and for each A ∈ A and each i ∈ I, a direct sum decomposition A = A i ⊕ A ′ i with A i ∈ S. These data must satisfy the following conditions: (1) For each morphism A f → S (with S ∈ S) there exists i ∈ I such that f factors as A = A i ⊕ A ′ i π 1 −→ A i −→ S (2) For each morphism S g → A (with S ∈ S) there exists i ∈ I such that g factors as S −→ A i i 1 −→ A i ⊕ A ′ i = A (3) The index set I is a directed poset under the relation i j ⇐⇒ for all A ∈ A, A i is a direct summand of A j and A ′ j is a direct summand of A ′ i . (Here directed means that for each i, j ∈ I, there exists k ∈ I such that i, j k.) (4) For each A, B ∈ A and each i ∈ I, we have (A ⊕ B) i = A i ⊕ B i and (A ⊕ B) ′ i = A ′ i ⊕ B ′ i . Remark 3.2. In the literature on Karoubi quotients, the term "filtered" is often used instead of "directed." In category theory, the term "directed" is standard. For any full additive subcategory S ⊂ A, the Karoubi quotient A/S is the category with the same objects as A, but with two morphisms identified if their difference factors through an object of S. The following lemma is surely well-known, but seems not to have been made explicit previously. Lemma 3.3. If S is a full additive subcategory of the additive category A, then A/S is an additive category, and if A 1 i 1 −→ A i 2 ←− A 2 is a direct sum diagram in A, then A 1 [i 1 ] −→ A [i 2 ] ←− A 2 is a direct sum diagram in A/S. Proof. It is elementary to check that A/S is a category. The addition on morphisms is given by [φ] + [ψ] = [φ + ψ]. This is well-defined because if φ: A → B factors through S ∈ S and ψ: A → B factors through S ′ ∈ S, then φ + ψ factors through S ⊕ S ′ ∈ S. Now, say A 1 −→ C makes (2) commute, then f ⊕ g − φ factors through an object of S. i 1 −→ A i 2 ←− A 2 is a direct sum diagram in A. Given a diagram (2) A 1 [i 1 ] / / [f ] ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ A A 2 [i 2 ] o o [g]~⑥⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ C in A/S weWriting φ = (φ • i 1 ) ⊕ (φ • i 2 ), we have (3) f ⊕ g − φ = (f − φ • i 1 ) ⊕ (g − φ • i 2 ). If [φ] makes (2) commute, then f − φ•i 1 and g − φ•i 2 factor through objects S 1 and S 2 in S (respectively), so f − φ • i 1 and g − φ • i 2 are the composites A 1 α 1 −→ S 1 β 1 −→ C and A 2 α 2 −→ S 2 β 2 −→ C, (respectively) for some morphisms α k and β k in A (k = 1, 2). We now see that (f − φ • i 1 ) ⊕ (g − φ • i 2 ) factors through S 1 ⊕ S 2 ∈ S since (letting j 1 and j 2 denote the inclusions of the summands into S 1 ⊕ S 2 ), the diagram A 1 f −φ•i 1 1 1 i 1 / / α 1 ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ A j 1 α 1 ⊕j 2 α 2 A 2 i 2 o o α 2~⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ g−φ•i 2 m m S 1 j 1 / / β 1 $ $ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ S 1 ⊕ S 2 β 1 ⊕β 2 S 2 j 2 o o β 2 z z ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ C commutes in A by construction. The utility of Karoubi filtrations comes from the following result due to Pedersen-Weibel [20]; see also Cárdenas-Pedersen [7, Section 8]. Theorem 3.4. If S ⊂ A is a full additive subcategory of a small additive category A and (A, S) admits a Karoubi filtration, then then there is a long exact sequence in non-connective algebraic K-theory The following lemma is a special case of Bartels and Rosenthal [3, (5.7)]. · · · −→ K * S −→ K * A −→ K * A/S ∂ −→ K * −1 S −→ · · · Lemma 3.6. Let Y and Z be families of subspaces of a proper metric space X, and assume Y and Z are closed under finite unions. Then A c (Y), A c (Z) ⊂ A c (X) are additive subcategories, and if for all Y ∈ Y there exists Z ∈ Z with Y ⊂ Z, then A c (Y) is a full (additive) subcategory of A c (Z). If, in addition, for each Y ∈ Y and each r ∈ N there exists Y ′ ∈ Y such that N r (Y ) ⊂ Y ′ , then the inclusion A c (Y) ⊂ A c (Z) admits a Karoubi filtration. In particular, for any subspace Z ⊂ X, the pair A + c (Z) ⊂ A(X) admits a Karoubi filtration. The direct sum decompositions making up these Karoubi filtration come from the following construction, applied to the subspaces Y ∈ Y. M = M (Z × [0, 1)) ⊕ M ((X \ Z) × [0, 1)) . This follows from the fact that a morphism M → N is defined as a family of morphisms M (x,t) → N (y,s) for all (x, t), (y, s) ∈ X × [0, 1). Definition 3.8. Let A 1 and A 2 be full subcategories of an additive category A. Let A 1 ∩ A 2 (the intersection of A 1 and A 2 ) be the full subcategory generated by those objects lying in both A 1 and A 2 . Remark 3.9. Note that if Y and Z are families of subspaces of a metric space X, then the intersection category A c (Y)∩A c (Z) is simply A c ({Y ∩Z : Y ∈ Y, Z ∈ Z}). In particular, if X 1 , X 2 ⊂ X, then A + c (X 1 ) ∩ A + c (X 2 ) = A c ({N r (X 1 ) ∩ N s (X 2 )} r,s∈N ), and Lemma 3.6 shows that A + c (X 1 ) ∩ A + c (X 2 ) ⊂ A c (X) admits a Karoubi filtration. With the exception of the inclusion A c (X) <1 ⊂ A c (X) discussed in Section 7, all the Karoubi filtrations in this paper follow from Lemma 3.6 by similar arguments. We need another technical condition for some of our arguments. Definition 3.10. Let A 1 and A 2 be full, additive subcategories of the additive category A. We say that (A 1 , A 2 ) is dispersed if every morphism φ : A 1 → A 2 (with A i ∈ A i ) factors through an object in A 1 ∩ A 2 . The dispersion conditions encountered in this paper are all special cases of the following observation. Lemma 3.11. Let X be a metric space, and consider a family of subspaces {X i } i∈I . Assume that for each i ∈ I and each r ∈ N, there exists j ∈ I such that N r (X i ) ⊂ X j . If S ⊂ A c (X) is a full additive subcategory that is closed under restriction of modules (meaning that for all S ∈ Ob(S) and for all Z ⊂ X, S(Z × [0, 1)) ∈ Ob(S)), then the pair (A c ({X i } i ), S) is dispersed. Proof. Consider a morphism φ: M → S in A c (X), with M ∈ Ob(A c ({X i } i )) and S ∈ Ob(S). Then supp(M ) ⊂ X i × [0, 1) for some i ∈ I, and (z, t) : φ (z,t),(z ′ ,t ′ ) = 0 for some (z ′ , t ′ ) ⊂ N r (X i ) × [0, 1) ⊂ X j × [0, 1) for some r > 0 and some j ∈ J. Now φ factors through S(X j × [0, 1)). In Section 4 we will build Mayer-Vietoris sequences in continuously controlled K-theory. These sequences will be applied in Section 6 to spaces of the form ∞ r=1 Z r , covered by subspaces ∞ r=1 U r and ∞ r=1 V r . These decompositions Z r = U r ∪ V r will become finer (in a sense) as r increases, and we will want to ignore the subcategory A c R r=1 Z r R 1 . This will be done through the use of Karoubi quotients, and in the remainder of this section we discuss the necessary categorical set-up. Proof. We begin by examining the full subcategory of A/S on the objects of B. This category is formed by identifying two morphisms φ, ψ : B 1 → B 2 (B i ∈ B) if φ − ψ factors as B 1 α −→ S β −→ B 2 for some S ∈ S. Since (B, S) is dispersed, α factors through an object of B ∩ S, so φ ≡ ψ (modulo B ∩ S). Hence the full subcategory of A/S on the objects of B is precisely B/(B ∩ S). Next, we must show that the inclusion B/(B∩S) ⊂ A/S admits a Karoubi filtration. The filtration on B/(B ∩ S) ⊂ A/S is exactly the same as the filtration on B ⊂ A: for A ∈ A, let I denote the indexing set for the latter filtration. Then for each i ∈ I we have a decomposition A = B i ⊕ B ′ i in A (with B i ∈ B) , and this remains a direct sum decomposition in the category A/S by Lemma 3.3. It now follows from the definitions that these decompositions give a Karoubi filtration on B/(B ∩ S) ⊂ A/S. We record the universal property of Karoubi quotients, which we will use several times. The proof is an elementary exercise. Lemma 3.13. Say G : A → B is a functor between additive categories and S ⊂ A is a full additive subcategory admitting a Karoubi filtration. If G(φ) = 0 whenever φ ≡ 0 (mod S), then there is a unique additive functor G: A/S −→ B such that the composite A → A/S G − → B equals G. Our Mayer-Vietoris sequences will be built using the following version of the Third Isomorphism Theorem from elementary abstract algebra. Proposition 3.14. Let A be a small additive category with full additive subcategories S, A 1 and A 2 , and assume that A 1 and A 2 generate A in the sense that every A ∈ A admits a direct sum decomposition A = A 1 ⊕ A 2 with A i ∈ A i . Set A 12 = A 1 ∩ A 2 , and similarly set S 1 = S ∩ A 1 , S 2 = S ∩ A 2 , and S 12 = S ∩ A 12 . If the triples S, A 1 ⊂ A and S 2 , A 12 ⊂ A 2 satisfy the conditions of and (A 2 , S) are dispersed, then the inclusion A 2 ֒→ A induces an equivalence of categories between Karoubi quotients as follows: A 2 /S 2 A 12 /S 12 ≃ −→ A/S A 1 /S 1 Proof. Lemma 3.12 guarantees that the displayed Karoubi quotients are well-defined. We begin by checking that the composite (4) A 2 ֒→ A −→ A/S −→ A/S A 1 /S 1 factors through the identifications in A 2 /S 2 A 12 /S 12 , so that Lemma 3.13 yields a well-defined functor F : A 2 /S 2 A 12 /S 12 → A/S A 1 /S 1 . If A 2 , A ′ 2 ∈ A 2 and φ : A 2 → A ′ 2 is a morphism in A which is equivalent to zero in A 2 /S 2 A 12 /S 12 , then φ must factor through an object in either S 2 or A 12 . These are subcategories of S and A 1 (respectively), so such morphisms certainly map to zero under the composite (4). Applying Lemma 3.13 twice yields the desired functor F . We must show that, up to isomorphism, every object is in the image of F . Every object A ∈ A can be written in the form A = A 1 ⊕ A 2 with A i ∈ A i , and we claim that in the Karoubi quotient A/S A 1 /S 1 , the objects A and A 2 are isomorphic. Indeed, the inclusion i 2 : A 2 → A and the corresponding projection π 2 : A → A 2 are inverses in this Karoubi quotient (the composite π 2 i 2 is the identity on A 2 by definition, and Id A = i 1 π 1 +i 2 π 2 , so Id A − i 2 π 2 = i 1 π 1 , which factors through A 1 ). This shows that up to isomorphism, every object is in the image of the map F . To complete the proof, we must check that F is full and faithful. Fullness follows from the fact that A 2 is a full subcategory of A. Next, if φ 1 and φ 2 are morphisms in A 2 that are equivalent in (A/S)/(A 1 /S 1 ), then φ 1 − φ 2 factors through an object in either S or A 1 . Dispersion implies that φ 1 − φ 2 actually factors through an object of S 2 or A 12 , so φ 1 and φ 2 are equivalent in the domain of F . Hence F is faithful. Mayer-Vietoris sequences in continuously controlled K-theory In this section we build Mayer-Vietoris sequences in continuously controlled K-theory analogous to the controlled Mayer-Vietoris sequences of Ranicki-Yamasaki [23,24] (see also [14,Appendix B]). First we produce a general Mayer-Vietoris sequence for metric spaces, and then we specialize the construction to the Rips complexes that will be used in later sections. 4.1. A Mayer-Vietoris sequence for the continuously controlled Ktheory of metric spaces. Proposition 4.1. Let X be a proper metric space with subspaces X 1 , X 2 ⊂ X and assume that for some r > 0, N r (X 1 ) ∪ N r (X 2 ) = X. Consider a family {S i } i∈I of subspaces of X such that for each t ∈ N and each i ∈ I, there exists j ∈ I such that N t (S i ) ⊂ S j . Let S = A c ({S i } i∈I ). We denote the intersection of S with A + c (X i ) by S i , and we denote the intersection of S with A + c (X 1 ) ∩ A + c (X 2 ) by S 12 . Then the natural maps from A + c (X 1 )/S 1 and A + c (X 2 )/S 2 to A c (X)/S are isomorphisms onto their images, and the natural map A + c (X 1 ) ∩ A + c (X 2 ) S 12 −→ A c (X)/S is an isomorphism onto the intersection of the images of A + c (X 1 )/S 1 and A + c (X 2 )/S 2 . Moreover, there is a long-exact Mayer-Vietoris sequence in non-connective K-theory · · · −→ K * +1 (A c (X)/S) ∂ −→ K * A + c (X 1 ) ∩ A + c (X 2 ) S 12 ((i 1 ) * ,(i 2 ) * ) − −−−−−− → K * (A + c (X 1 )/S 1 ) ⊕ K * (A + c (X 2 )/S 2 ) (j 1 ) * −(j 2 ) * − −−−−−− → K * (A c (X)/S) ∂ −→ · · · , in which i 1 , i 2 , j 1 , and j 2 are induced by the relevant inclusions of categories. Proof. To construct the Mayer-Vietoris sequence, we will consider the diagram of additive categories (5) A + c (X 1 )∩A + c (X 2 ) S 12 i 1 / / i 2 A + c (X 1 )/S 1 j 1 A + c (X 2 )/S 2 j 2 / / q 2 A c (X)/S q 1 A + c (X 2 )/S 2 (A + c (X 1 )∩A + c (X 2 ))/S 12 F ∼ = / / Ac(X)/S A + c (X 1 )/S 1 , where the q i are the Karoubi projections guaranteed by Lemma 3.12 and (as we will check) the induced map F is an equivalence of categories by Proposition 3.14. Applying K-theory produces two vertical long-exact sequences (Theorem 3.4), which can be weaved together using the isomorphism F * to form the desired Mayer-Vietoris sequence (see, for example, Hatcher [16, Section 2. 2, Exercise 38]). The facts that A + c (X 1 )/S 1 , A + c (X 2 )/S 2 , and A + c (X 1 )∩A + c (X 2 ) S 12 are isomorphic to their images in A + c (X)/S will also follow from Lemma 3.12. The Karoubi filtrations needed to apply Lemma 3.12 come from Lemma 3.6 (see Remark 3.9), while the necessary dispersion conditions can be checked using Lemma 3.11. To complete the proof, we must check that the conditions of Proposition 3.14 are satisfied, so that we obtain a well-defined equivalence of categories A + c (X 2 )/S 2 A + c (X 1 ) ∩ A + c (X 2 ) /S 12 F −→ A c (X)/S A + c (X 1 )/S 1 . The necessary Karoubi filtrations and the dispersion conditions are checked using Lemmas 3.6 and 3.11. To check that A + c (X 1 ) and A + c (X 2 ) generate A + c (X), recall that Construction 3.7 guarantees a direct sum decomposition M = M (N r (X 1 ) × [0, 1)) ⊕ M ((X × [0, 1)) \ (N r (X 1 ) × [0, 1))) . Since N r (X 1 ) ∪ N r (X 2 ) = X, we have (X × [0, 1)) \ (N r (X 1 ) × [0, 1)) ⊂ N r (X 2 ) × [0, 1), and hence M ((X × [0, 1)) \ (N r (X 1 ) × [0, 1))) ∈ A + c (X 2 ). 4.2. Decomposed sequences and Rips complexes. We will apply our general Mayer-Vietoris sequence (Proposition 4.1) to decompositions of Rips complexes arising from decompositions of the underlying metric space. For the proof of our vanishing result for continuously controlled K-theory, it will be necessary to consider an infinite sequence of increasingly refined decompositions of our space. In fact, we will need to consider such sequences all at once by forming an infinite disjoint union of the spaces involved in the decompositions, and we will need to iterate this process (by further decomposing each space in the initial decomposition). Such considerations lead to the notion of decomposed sequence introduced below. We begin by recalling the construction of the Rips complex. Definition 4.2. Given a metric space X and a number s > 0, the Rips complex P s (X) is the simplicial complex with vertex set X and with a simplex x 0 , . . . , x n whenever d(x i , x j ) s for all i, j ∈ {0, . . . , n}. We will often view X as a subset of P s (X) by identifying X with the vertices of P s (X). Note that if X is a metric space with bounded geometry (i.e. if for each r > 0 there exists N > 0 such that for all x ∈ X, the ball B r (x) contains at most N points), then the Rips complex P s (X) is finite dimensional and locally finite. When forming Rips complexes, we will always assume that the underlying metric space has bounded geometry. Note that a finitely generated group, with the word metric arising from a finite generating set, always has bounded geometry. This is our main source of examples. Z = (Z 1 , Z 2 , . . .), Z i ⊂ X, equipped with decomposi- tions (6) Z r = α∈Ar Z r α of each Z r (r = 1, 2, . . .) . We will call this data (the sequence Z together with the families {Z r α } α∈Ar ) a decomposed sequence in X. Note that the Z r α need not be disjoint. Let Seq denote the partially ordered set consisting of all non-decreasing sequences of (strictly) positive real numbers, with the ordering (s 1 , s 2 , . . .) (s ′ 1 , s ′ 2 , . . .) if s i s ′ i for all i. Note that Seq is directed. Given a decomposed sequence Z in X and a sequence s ∈ Seq, the Rips complex P s (Z) is the simplicial complex P s (Z) = ∞ r=1 α∈Ar P sr (Z r α ). Note that Z r α and Z s β may overlap inside of X, so to be precise, points in P s (Z) have the form (x, r, α) where α ∈ A r and x ∈ P sr (Z r α ). Each simplicial complex P sr (Z r α ) is equipped with the metric induced by the simplicial metric on P sr (X) (see Section 2 for a definition of the simplicial metric), and the distance between (x, r, α) and (y, r ′ , β) is set to infinity unless r = r ′ and α = β. In other words, we consider P s (Z) to be a subset of the infinite disjoint union (7) ∞ r=1 α∈Ar P sr (X), with the induced metric. Remark 4.4. Given a decomposed sequence Z = (Z 1 , Z 2 , . . .) with decompositions Z r = α∈Ar Z r α (r 1), we let X Z denote the decomposed sequence X Z = (X, X, . . .) with decompositions X = α∈Ar X for each r 1. Then P s (X Z ) is the metric space (7). We will need to consider coverings of one decomposed sequence by two subsequences. In the applications, the subsequences will have lower "decomposition complexity" than the original sequence, in a sense that will be explained in Section 6. Definition 4.5. Let Z = (Z 1 , Z 2 , . . .) be a decomposed sequence inside the metric space X, with decompositions Z r = α∈Ar Z r α . We write Z = U ∪ V if U and V are decomposed sequences in X whose decompositions are indexed over the same sets A r (r 1) and for each r 1 and each α ∈ A r we have Z r α = U r α ∪ V r α . Similarly, we write U ⊂ Z if U is a decomposed sequence in X with the same indexing sets as Z, and for each r 1 and each α ∈ A r we have U r α ⊂ Z r α . Given a sequence s ∈ Seq, we define A Z+ c (P s (U )) := A Ps(Z)+ c (P s (U )) as in Definition 2.5. Note that both P s (Z) and P s (U ) have the metric induced by the simplicial metric on P s (X Z ). We will sometimes drop Z from the superscript when it is clear from context. In the proof of our vanishing result for continuously controlled K-theory (Theorem 6.4), it will be important to ignore the initial portion of a decomposed sequence. This is done via the following constructions. Definition 4.6. Given proper metric spaces Y 1 , Y 2 , . . . and a subcategory A ⊂ A c ∞ r=1 Y r , we define S = S(A) to be the full subcategory of A consisting of those geometric modules supported on R r=1 Y r × [0, 1) for some R > 0. Note that S = colim R>0 A ∩ A c R r=1 Y r . We then define A = A/S. Given decomposed sequences U ⊂ Z in X, we set A c (Z) = A c (Z) and A c + (U ) = A c Z+ (U ) = A Z+ c (U ). Remark 4.7. The constructions A c and A c + enjoy the same sort of functoriality as A c and A + c . The statements in Lemma 2.7 regarding functoriality of A c and A + c for inclusions of simplicial complexes apply to inclusions of Rips complexes associated to inclusions of decomposed sequences, and Lemma 3.13 yields corresponding statements for A c and A c + . In the sequel, we will simply refer to Lemma 2.7 when constructing functors between categories A c (−) and A c + (−). 4.3. Mayer-Vietoris for Rips complexes. Theorem 4.8. Let Z, U , and V be decomposed sequences in a bounded geometry metric space X, with Z = U ∪ V, and choose s ∈ Seq. Then there is a long exact sequence in non-connective K-theory of the form (8) · · · −→ K * (I s (U , V)) (i 1 ,i 2 ) − −−− → K * A c Z+ (P s (U )) ⊕ K * A c Z+ (P s (V)) (j 1 ) * −(j 2 ) * − −−−−−− → K * (A c (P s (Z))) ∂ −→ K * −1 (I s (U , V)) −→ · · · , where I s (U , V) denotes the intersection in A c (P s (Z)) of A c Z+ (P s (U )) and A c Z+ P s (V). The maps i 1 and i 2 are induced by the relevant inclusions of categories and the maps j 1 and j 2 are the functors associated to the inclusions of simplicial complexes P s (U ) ֒→ P s (Z) and P s (V) ֒→ P s (Z). Proof. By Proposition 4.1 it suffices to check that N 1 (P s (U )) ∪ N 1 (P s (V)) = P s (Z). Given a simplex σ = x 0 , . . . , x n in P s (Z), we either have x 0 ∈ U r α for some r 1 and some α ∈ A r , or we have x 0 ∈ V r α for some r 1 and some α ∈ A r . In the former case, x 0 is a 0-simplex in P s (U ), and σ ⊂ N 1 ( x 0 ) ⊂ N 1 (P s (U )). In the latter case, σ ⊂ N 1 (P s (V)). Mayer-Vietoris for relative Rips complexes. We will need another Mayer-Vietoris sequence for the proof of our vanishing theorem (Theorem 6.4), involving the relative Rips complexes introduced by Guentner-Tessera-Yu [14, Appendix A]. Definition 4.9. Consider a bounded geometry metric space X, along with a subspace Z ⊂ X and a family W of subspaces of X. Given 0 < s < s ′ , the relative Rips complex P s,s ′ (Z, W) is the subcomplex of P s ′ (X) consisting of those simplices x 0 , . . . , x n satisfying at least one of the following conditions: (1) x 0 , . . . , x n ∈ Z and d(x i , x j ) s for all i, j; (2) x 0 , . . . , x n ∈ W for some W ∈ W. Note that in the second case, d(x i , x j ) s ′ for all i, j since we are defining a subcomplex of P s ′ (X). We equip P s,s ′ (Z, W) with the metric induced by the simplicial metric on P s,s ′ (X, W). (It will be crucial for our arguments that we do not use the metric inherited from the simplicial metric on P s ′ (X); see in particular Lemmas 5.3 and 5.4.) Note that in this definition, we do not require that the subspaces W ∈ W satisfy W ⊂ Z. Given a decomposed sequence Z = (Z 1 , Z 2 , . . .) in X with decompositions Z r = α∈Ar Z r α , a set W = {W r α \| r 1, α ∈ A r } of families of subspaces of X, and s, s ′ ∈ Seq satisfying s s ′ , we define the relative Rips complexes P s,s ′ (Z, W) := ∞ r=1 α∈Ar P sr,s ′ r (Z r α , W r α ) ⊂ ∞ r=1 α∈Ar P sr,s ′ r (X, W r α ) =: P s,s ′ (X, W), and we give P s,s ′ (Z, W) the metric induced by the simplicial metric on P s,s ′ (X, W). Given a covering Z = U ∪ V of a decomposed sequence by two subsequences, we will need to consider a relative Rips complex in which the "larger" simplices are constrained to lie near both U and V. Definition 4.10. Consider decomposed sequences Z = (Z 1 , Z 2 , . . .), U = (U 1 , U 2 , . . .), and V = (V 1 , V 2 , . . .) in a metric space X, with decompositions Z r = α∈Ar Z r α , U r = α∈Ar U r α , and V r = α∈Ar V r α . Assume that Z = U ∪ V, and say that we are given additional decompositions (9) U r α = i∈I(r,α) U r αi and V r β = j∈J(r,β) V r βj for each r 1, and each α, β ∈ A r . Given T > 0, r 1, and α ∈ A r , we define W r T,α = {N T (U r αi ) ∩ N T (V r αj ) ∩ Z r α \| i ∈ I(r, α) , j ∈ J(r, β)}, and given T ∈ Seq, we define the set of metric families W T (U , V, Z) to be W T (U , V, Z) = W T = {W r Tr,α \| r 1, α ∈ A r }. (We will suppress the dependence of W T (U , V, Z) on the chosen additional decompositions (9).) For any s, s ′ ∈ Seq, we can now form the relative Rips complexes P s,s ′ (Z, W T ), P s,s ′ (U , W T ) and P s,s ′ (V, W T ) as in Definition 4.9. Following Definition 4.6, we set A c P s,s ′ (Z, W T ) := A c P s,s ′ (Z, W T ) /S. We define A c Z+ P s,s ′ (U , W T ) and A c Z+ P s,s ′ (V, W T ) similarly, by allowing modules supported on neighborhoods of P s,s ′ (U , W T ) (or, respectively, P s,s ′ (V, W T )) inside P s,s ′ (Z, W T ) (recall that these complexes are given the metrics inherited from the simplicial metric on P s,s ′ (X, W T )). Theorem 4.11. Let Z, U , V and X be as in Theorem 4.8, and say that for each r 1 and each α, β ∈ A r we are given additional decompositions U r α = i∈I(r,α) U r αi and U r β = j∈J(r,β) V r βj . Then for any sequences s, s ′ , T ∈ Seq, we can form the sequence W T = W T (U , V, Z) as in Definition 4.10, and there is a long exact Mayer-Vietoris sequence in non-connective K-theory of the form · · · −→ K * +1 A c P s,s ′ (Z, W T ) ∂ −→ K * I ′ s,s ′ ,T (U , V) −→ K * A c Z+ P s,s ′ (U , W T ) ⊕ K * A c Z+ P s,s ′ (V, W T ) −→ K * A c P s,s ′ (Z, W T ) ∂ −→ · · · , where I ′ s,s ′ ,T (U , V) is the intersection of the subcategories A c Z+ P s,s ′ (U ; W T ) and A c Z+ P s,s ′ (V; W T ) inside A c P s,s ′ (Z; W T ). Proof. We apply Proposition 4.1. The conditions are checked just as in the proof of Theorem 4.8: for each r 1 and each α ∈ A r , each point in P sr,s ′ r Z r α , W r Tr,α ) is within distance 1 of P sr,s ′ r U r α , W r Tr,α ∪ P sr,s ′ r V r α , W r Tr,α . 4.5. A comparison of Mayer-Vietoris sequences. For the arguments in Section 6, we will need to compare the absolute and relative Mayer-Vietoris sequences from Sections 4.3 and 4.4. Theorem 4.12. Let Z, U , V, and X be as in Theorem 4.8. Then for any s, s ′ , T ∈ Seq there are functors A c Z+ P s (U ) i U − → A c Z+ P s,s ′ (U , W T ), A c Z+ P s (V) i V − → A c Z+ P s,s ′ (V, W T ), A c P s (Z) γ → A c P s,s ′ (Z, W T ), and I s (U , V) ρ → I ′ s,s ′ ,T (U , V) such that the diagram of Mayer-Vietoris sequences (10) K * (I s (U , V)) ρ * / / K * I ′ s,s ′ ,T (U , V) Proof. The Mayer-Vietoris sequences are produced by Theorem 4.8 and Theorem 4.11. The functors i U , i V , and γ are induced by the relevant inclusions of simplicial complexes, which satisfy the hypotheses of Lemma 2.7. There is then an induced functor I s (U , V) ρ → I ′ s,s ′ ,T between the intersection categories. By Lemma 3.13, these functors produce a commutative diagram D of categories consisting of two diagrams of the form (5), with one mapping to the other. After taking K-theory spectra, we obtain a morphism between two homotopy (co)-cartesian squares of spectra, together with maps between the homotopy cofibers of the vertical maps in these squares. It is a general fact that maps between homotopy (co)-cartesian squares of spectra yield commutative diagrams of Mayer-Vietoris sequences. Metric properties of Rips complexes In this section we record some basic geometric results about Rips complexes, some of which may be found in Guentner-Tessera-Yu [14,Appendix A]. For completeness, we provide detailed proofs. Definition 5.1. If X is a bounded geometry metric space and s is a positive real number, we let C(s, X) = (2 √ 2 + 1) N −1 , where N is the dimension of P s (X) (if P s (X) is zero-dimensional, we set C(s, X) = 1). Lemma 5.2. Let (X, d) be a metric space with bounded geometry and let d ∆ denote the simplicial metric on P s (X). Then for all x, y ∈ X ⊂ P s (X), d(x, y) sC(s, X)d ∆ (x, y). Proof. Given a sequence γ = (p 0 , p 1 , . . . , p k ) of points in P s (X), let l(γ) = k−1 i=0 d ∆ (p i , p i+1 ). By definition of the simplicial metric, we must show that d(x, y) s(2 √ 2 + 1) N −1 l(γ) for all sequences γ = (p 0 , p 1 , . . . , p k ) such that p 0 = x, p k = y, and for i = 1, . . . k, p i and p i−1 lie in a common simplex σ i (which we may assume is the smallest simplex containing p i and p i−1 ). Let dim(γ) = max i dim(σ i ), and note that dim(γ) N . We will show by induction on dim(γ) that d(x, y) s(2 √ 2 + 1) dim(γ)−1 l(γ). Note that if σ i ⊂ σ i+1 or σ i+1 ⊂ σ i , then we may shorten γ by removing p i , so we may assume without loss of generality that σ i ∩ σ i+1 is a proper face of both σ i and σ i+1 . This implies that p i and p i+1 lie in the boundary of σ i+1 for i = 0, . . . , k − 1. If dim(γ) = 1, then p i ∈ X for each i, and we have d(x, y) k−1 i=0 d(p i , p i+1 ) sk = s(2 √ 2 + 1) 0 l(γ). Now assume the result for paths of dimension at most n − 1, and say dim(γ) = n. We will replace γ by a nearby path of lower dimension. By assumption, there exists i ∈ {0, . . . , k − 1} such that σ i+1 = x 0 , . . . , x n for some x 0 , . . . , x n ∈ X. Reordering the x j if necessary, we may further assume that p i ∈ x 0 , . . . , x n−1 and p i+1 ∈ x 1 , . . . , x n . Letting p i and p i+1 denote the orthogonal projections of these points to the affine (n − 2)-plane containing x 1 , . . . , x n−1 (note that these orthogonal projections necessarily lie inside x 1 , . . . , x n−1 ), we will replace γ by the piecewise geodesic path γ ′ = (p 0 , . . . , p i , p i , p i+1 , p i+1 , . . . , p k ). We claim that d ∆ (p i , p i ) and d ∆ (p i+1 , p i+1 ) are at most √ 2d ∆ (p i , p i+1 ). In barycentric coordinates, we may write p i = n i=0 a i x i (with a n = 0) and p i+1 = n i=0 b i x i , (with b 0 = 0). Setting w = (a 0 + a 1 )x 1 + n−1 i=2 a i x i we have d ∆ (p i , w) = √ 2a 0 and d ∆ (p i , p i+1 ) a 0 , so d ∆ (p i , w) √ 2d ∆ (p i , p i+1 ). Hence d ∆ (p i , p i ) d ∆ (p i , w) √ 2d ∆ (p i , p i+1 ), as desired. Similarly, d ∆ (p i+1 , p i+1 ) √ 2d ∆ (p i , p i+1 ). Since orthogonal projections decrease distances, we also have d ∆ (p i , p i+1 ) d ∆ (p i , p i+1 ) 1, and hence l(p i , p i , p i+1 , p i+1 ) (2 √ 2 + 1)d ∆ (p i , p i+1 ). Repeating this procedure for each n-simplex among the σ i , we obtain a new path γ ′ (from x to y) which lies entirely in the (n − 1)-skeleton of P s (X) (meaning that dim(γ ′ ) n − 1) and satisfies l(γ ′ ) (2 √ 2 + 1)l(γ). By induction, we know that d(x, y) (2 √ 2 + 1) n−2 l(γ ′ ), so d(x, y) (2 √ 2 + 1) n−1 l(γ), completing the proof. It is important to note that no bound exists in the opposite direction: if d(x, y) > s, then x and y may lie in different connected components of P s (X), in which case d ∆ (x, y) = ∞. The following result will allow us to compare distances in relative Rips complexes. For this result to hold, it is crucial that we give the relative Rips complex P s,s ′ (Z, W ) the metric inherited from the simplicial metric on P s,s ′ (X, W ) rather than P s ′ (X). Note that each point x in a simplicial complex K can be written uniquely, in barycentric coordinates, in the form x = c v i (x)v i with c v i (x) > 0 for each i. We will refer to the vertices v i as the barycentric vertices of x. Given a vertex v ∈ K, we can extend c v to a continuous function from K to [0, 1] by setting c v (x) = 0 if v is not a barycentric vertex of x. Lemma 5.3. Let W ⊂ X be metric spaces, and assume X has bounded geometry. Given s ′ s > 0, let N t (P s ′ (W )) denote a t-neighborhood of P s ′ (W ) inside P s,s ′ (X, W ). Then for all x ∈ X ∩ N t (P s ′ (W )) (where X is viewed as the 0-skeleton of P s,s ′ (X, W )), we have (11) d(x, W ) (t + 1)C(s, X)s. It follows that inside the simplicial complex P s ′ (X), we have inclusions (12) N t (P s ′ (W )) ⊂ P s,s ′ (N (t+2)C(s,X)s (W ), W ) ⊂ P s ′ (N (t+2)C(s,X)s (W )), where on the left, the neighborhood is still taken with respect to the simplicial metric on P s,s ′ (X, W ). Additionally, for any U ⊂ X, we have inclusions (13) N t (P s,s ′ (U, W )) ⊂ N t (P s ′ (U ∪ W )) ⊂ P s ′ (N (t+2)C(s,X)s (U ∪ W )), where the first neighborhood is taken inside P s,s ′ (X, W ) and the second is taken inside P s,s ′ (X, U ∪ W ). Proof. Say x ∈ X ∩ N t (P s ′ (W )). Then there exists a piecewise geodesic path γ in P s,s ′ (X, W ), starting at x and ending at a point in P s ′ (W ), such that l(γ) < t, where l(γ) is the sum of the lengths of the geodesics making up γ. Since X has bounded geometry, the path γ: [0, 1] → P s,s ′ (X, W ) meets only finitely many (closed) simplices σ 1 , . . . , σ m . Let J ⊂ {1, . . . , m} be the subset of those j such that σ j has a vertex lying in W ; note that J = ∅ since γ ends in P s ′ (W ). Let r ∈ [0, 1] be the minimum element of the compact set j∈J γ −1 (σ j ). If r = 0, then d(x, W ) s and we are done, so we assume r > 0. For r ′ < r, the barycentric vertices of γ(r ′ ) all lie in X \ W , so γ(r ′ ) ∈ P s (X). Continuity of the barycentric coordinate functions implies that the barycentric vertices x 0 , . . . , x n of γ(r) all lie in X \ W as well. By choice of r, we know that γ(r) lies in a simplex σ having a vertex w ∈ W . This simplex must contain x 0 , . . . , x n , and since x i / ∈ W we conclude (from the definition of the relative Rips complex) σ ⊂ P s (X). Concatenating γ\| [0,r] with a geodesic in σ connecting γ(r) and w yields a piecewise geodesic path, inside P s (X), of length at most t + 1. Hence the simplicial distance, in P s (X), from x to w is at most t + 1, and Lemma 5.2 tells us that d(x, w) (t + 1)C(s, X)s. This proves (11). The first containment in (12) follows from the distance estimate (11), since if z ∈ N t (P s ′ (W )) lies in a simplex x 0 , . . . , x n ⊂ P s,s ′ (X, W ), then for each i, the simplicial distance (in P s,s ′ (X, W )) from x i to P s ′ (W ) is at most t + 1, so (11) shows that x i ∈ N (t+2)C(s,X)s (W ). The second containment in (12) is immediate from the definitions. The first containment in (13) follows from the fact that the simplicial metric on P s,s ′ (X, U ∪ W ) is smaller than the simplicial metric on the subcomplex P s,s ′ (X, W ), while the second follows from (12), with U ∪W playing the role of W . The following result, which generalizes (11), will be used in the proof of Lemma 6.18. Lemma 5.4. Let X be a bounded geometry metric space, with subspaces X 1 , X 2 ⊂ X, and let W 1 and W 2 be families of subspaces of X. For i = 1, 2, let W i = W i = {x ∈ X \| x ∈ W for some W ∈ W i } denote the union of the subspaces in W i . Set W = W 1 ∪ W 2 , and let d ∆ denote the simplicial metric on P s,s ′ (X, W) for some fixed s, s ′ > 0. Setting V i = X i ∪ W i and P i = P s,s ′ (X i , W i ) (i = 1, 2), we have (14) d(V 1 , V 2 ) (d ∆ (P 1 , P 2 ) + 2)sC(s, X). Proof. Consider a piece-wise geodesic path γ: [0, 1] → P s,s ′ (X, W) with γ(0) ∈ P 1 and γ(1) ∈ P 2 . It will suffice to show that d(V 1 , V 2 ) (l(γ) + 2)sC(s, X). Arguing as in the proof of Lemma 5.3, let t 1 ∈ [0, 1] denote the maximum time at which γ(t) lies in a simplex with a vertex in V 1 , and let t 2 ∈ [t 1 , 1] denote the minimum time (in the interval [t 1 , 1]) at which γ(t) lies in a simplex with a vertex in V 2 . If t 1 = t 2 , then there is a simplex in P s,s ′ (X, W) containing vertices from both V 1 and V 2 . If this simplex lies outside P s (X), then its vertices must lie entirely inside some set in W = W 1 ∪ W 2 , and we find that V 1 ∩ V 2 = ∅. If this simplex lies in P s (X), then we have d(V 1 , V 2 ) s. In either case, (14) is trivially satisfied. We now assume that t 1 < t 2 . Then for t ∈ (t 1 , t 2 ), if σ ⊂ P s,s ′ (X, W) is a simplex containing γ(t), then σ has no vertex in V 1 ∪ V 2 , and in particular no vertex in W 1 ∪ W 2 . Thus γ(t 1 , t 2 ) ⊂ P s (X). Furthermore, by considering barycentric coordinates as in the proof of Lemma 5.3, one may check that for i = 1, 2, γ(t i ) lies in a simplex σ i ⊂ P s (X) such that at least one vertex v i ∈ σ i satisfies v i ∈ V i . Concatenating γ\| [t 1 ,t 2 ] with geodesic paths inside σ i from γ(t i ) to v i , we obtain a path inside P s (X), of simplicial length at most l(γ) + 2, connecting V 1 and V 2 . The result now follows from Lemma 5.2. Controlled K-theory for spaces of finite decomposition complexity We now apply the results of Sections 4 and 5 to the continuously controlled K-theory of spaces with finite decomposition complexity. We begin by reviewing some definitions from Guentner-Tessera-Yu [13,14], where the notion of decomposition complexity was first introduced. A set of metric spaces will be called a metric family. Let B denote the class of uniformly bounded metric families; that is, a family F lies in B if there exists R > 0 such that diam(F ) < R for all F ∈ F. Given a class D of metric families, we say that a metric family F = {F α } α∈A decomposes over D if for every r > 0 and every α ∈ A there exists a decomposition F α = U r α ∪ V r α and r-disjoint decompositions U r α = r-disjoint i∈I(r,α) U r αi and V r α = r-disjoint j∈J(r,α) V r αj such that the families {U r αi \| α ∈ A, i ∈ I(r, α)} and {V r αj \| α ∈ A, j ∈ J(r, α)} lie in D. 1 Here r-disjoint simply means that if i 1 , i 2 ∈ I(r, α) for some α ∈ A, and i 1 = i 2 , then d(U r αi 1 , U r αi 2 ) > r (and similarly for V in place of U ). We set D 0 = B, and given a successor ordinal γ + 1 we define D γ+1 to be the class of all metric spaces which decompose over D γ . If γ is a limit ordinal, we define D γ = β<γ D β . (This definition will make the limit ordinal cases of all our transfinite induction arguments trivial.) Definition 6.1. We say that a metric space X has finite decomposition complexity if the single-element family {X} lies in D γ for some ordinal γ. (We often write X ∈ D γ rather than {X} ∈ D γ .) Remark 6.2. If X ∈ D γ for some ordinal γ, then in fact there exists a countable ordinal γ ′ such that X ∈ D γ ′ . This is proven in Guentner- Tessera-Yu [13, Theorem 2.2.2]. Given a metric space X, we use the term metric family in X to mean a metric family F such that each F ∈ F is a subspace of X (with the induced metric). Lemma 6.3. Let X be a metric space, and let {Z α } α∈A and {Y β } β∈B be metric families in X. Say {Z α } α∈A ∈ D γ for some ordinal γ. Assume further that there exists t > 0 such that for all β ∈ B, there exists α ∈ A with Y β ⊂ N t (Z α ). Then {Y β } β∈B ∈ D γ as well. (Note here that the parameter t is independent of β ∈ B.) Proof. We use transfinite induction. In the base case, we have a uniform bound D on the diameter of the Z α , and D + t gives a uniform bound on the diameter of the Y β , so {Y β } β ∈ D 0 . Now say γ = δ + 1 is a successor ordinal, and assume the result for D δ . If {Z α } α ∈ D γ , then for each r > 0 and each α ∈ A there exist U r α and V r α such that Z α = U r α ∪ V r α , and there exist decompositions U r α = r-disjoint i∈I(r,α) U r αi , V r α = r-disjoint j∈J(r,α) V r αj such that the families {U r αi \| α ∈ A, i ∈ I(r, α)} and {V r αj \| α ∈ A, j ∈ J(r, α)} 1 In the original definition in [14, Section 2], one assumes instead that there exists a family F ′ ∈ D such that {U r αi \| α ∈ A, i ∈ I(r, α)} ∪ {V r αj \| α ∈ A, j ∈ J(r, α)} ⊂ F ′ . However, since the collections of families Dγ defined here, and the analogous families defined in [14], are closed under forming finite unions of families and under subfamilies, the two definitions of Dγ agree. (With our definition of Dγ , closure under finite unions is checked by transfinite induction; closure under subfamilies follows, for example, from Lemma 6.3.) lie in D δ . For each β ∈ B, we know there exists α = α(β) ∈ A such that Y β ⊂ N t (Z α ). We now have decompositions Y β = (N t (U r α ) ∩ Y β ) ∪ (N t (V r α ) ∩ Y β ) , and (r − 2t)-disjoint decompositions N t (U r α ) ∩ Y β = i∈I(r,β) N t (U r αi ) ∩ Y β and N t (V r α ) ∩ Y β = j∈J(r,β) N t (V r αj ) ∩ Y β . By induction we know that the families {N t (U r βi )∩Y β \| β ∈ B, i ∈ I(r, β)} and {N t (V r βj )∩Y β \| β ∈ B, j ∈ J(r, β)} lie in D δ . Since r − 2t tends to infinity with r, we see that the family {Y β } β decomposes over D δ , as desired. The case of limit ordinals is trivial. We now come to the main result of this section. Theorem 6.4. If X is a bounded geometry metric space with finite decomposition complexity, then for each * ∈ Z we have colim s→∞ K * (A c (P s X)) = 0, where the colimit is taken with respect to the maps K * (A c (P s X)) η s,s ′ − −− → K * (A c (P s ′ X)) induced by applying Lemma 2.7 to the inclusions P s X ֒→ P s ′ X. We will deduce Theorem 6.4 from a closely related vanishing result for the constant and trivially decomposed sequence (15) X = (X, X, X, . . .), where at each level X is decomposed into the one-element family {X}. Definition 6.5. Let Z be a decomposed sequence in X. For each s s ′ ∈ Seq, we define (16) η s,s ′ = η s,s ′ (Z): K * A c (P s (Z)) −→ K * A c (P s ′ (Z)) to be the map induced by the inclusion P s (Z) ⊂ P s ′ (Z). Proposition 6.6. If X is a bounded geometry metric space with finite decomposition complexity, then for each s ∈ Seq and each element x ∈ K * (A c (P s (X ))) there exists s ′ ∈ Seq, with s ′ s, such that η s,s ′ (x) = 0. We will see in the proof that s ′ may depend on x. Remark 6.7. Note that since K-theory commutes with directed colimits of additive categories (see Quillen [22, Section 2]), Proposition 6.6 is equivalent to the statement that colim s∈Seq K * A c (P s X ) = 0. Proof of Theorem 6.4 assuming Proposition 6.6. We apply Proposition 6.6 with s = (s, s, . . .). Given s ′ s and m 1, consider the diagram (17) colim n A c n r=1 P s ′ r (X) colim jn A c (P s X) µ / / A c (P s X ) i / / πs A c (P s ′ X ) qm / / π s ′ A c P s ′ m (X) A c (P s X ) i / / A c (P s ′ X ). Here the maps i,ī, and j n are induced by inclusions of simplicial complexes, π s and π s ′ are the Karoubi projections, the functor µ sends a geometric module M on P s (X) × [0, 1) to the constant sequence (M, M, . . .) (and similarly for morphisms), and q m is the functor which restricts a geometric module to the subspace P s ′ m (X) × [0, 1) ⊂ ∞ r=1 P s ′ r (X) × [0, 1). Let x ∈ K * A c (P s X) be given. In K-theory,ī * is the map (16), so Proposition 6.6 implies that we can choose s ′ s such thatī * (π s • µ(x)) = 0. For m > n the composite q m • j n is the constant functor mapping all objects to 0, so (q m ) * (j n ) * = 0 in K-theory. However, for any m, the composite q m • i • µ is simply the functor induced by the inclusion P s (X) ֒→ P s ′ m (X), so (q m • i • µ) * = η s,s ′ m . Since the third column of Diagram (17) is a Karoubi sequence, chasing the diagram and applying Remark 6.7 shows that for some N 0 and some y ∈ K * A c N r=1 P s ′ r (X) , we have i * µ * (x) = (j N ) * (y), so η s,s ′ (x) = (q N +1 ) * • i * • µ * (x) = (q N +1 ) * (j N ) * (y) = 0. The result now follows, since the colimit in Theorem 6.4 is defined in terms of the maps η s,s ′ . ✷ To prove the desired vanishing result for the map (16), we will proceed through an induction for decomposed sequences inside X. Definition 6.8. Let Z = (Z 1 , Z 1 , . . .) be a decomposed sequence in X with decompositions Z r = α∈Ar Z r α . We say that Z is a vanishing sequence (or more briefly, Z is vanishing) if for each s ∈ Seq and each x ∈ K * (A c (P s Z)), there exists s ′ s such that x maps to zero under K * (A c (P s Z)) η s,s ′ −−→ K * (A c (P s ′ Z)). For each ordinal γ, let D γ (X) denote the set of F ∈ D γ such that F is a metric family in X. By abuse of notation we write Z ∈ D γ (X) if {Z r α } α∈Ar ∈ D γ (X) for each r 1. We say that D γ (X) is vanishing if all decomposed sequences Z ∈ D γ (X) are vanishing. Finally, given a sequence s ∈ Seq, we say that Z is vanishing at s if for each x ∈ K * (A c (P s Z)), there exists s ′ s such that η s,s ′ (x) = 0. Definition 6.9. Given a sequence T ∈ Seq and a decomposed sequence Z = (Z 1 , Z 2 , . . .) in X with decompositions Z r = α∈Ar Z r α , we define N T (Z) to be the decomposed sequence (N T 1 (Z 1 ), N T 2 (Z 2 ), . . .), with decompositions N Tr (Z r ) = α∈Ar N Tr (Z r α ). The next lemma is an immediate consequence of Lemma 6.3. Lemma 6.10. Let Z be a decomposed sequence in X, and say Z ∈ D γ (X) for some ordinal γ. If Y is another decomposed sequence in X, and Y ⊂ N T (Z) for some sequence T of positive real numbers, then Y ∈ D γ (X) as well. Note that a metric space X has finite decomposition complexity if and only if the constant and trivially decomposed sequence X = (X, X, . . .) (see (15)) lies in D γ (X) for some ordinal γ, so Proposition 6.6 is an immediate consequence of the next result. Proposition 6.11. If X is a bounded geometry metric space, then D γ (X) is vanishing for every ordinal γ. The proof of Proposition 6.11 will be by transfinite induction on the ordinal γ, and will fill the remainder of the section. For the rest of the section, we fix a bounded geometry metric space X. We first consider the base case of our induction, Z ∈ D 0 (X). This means Z is a decomposed sequence in X for which each family {Z r α } α∈Ar is uniformly bounded. Hence for each r 1, there exists N (r) such that for all α ∈ A r , the diameter of Z r α is at most N (r). This means that if s ′ N := (N (1), N (2), . . .), the simplicial complex P s ′ (Z) = ∞ r=1 α∈Ar P s ′ r (Z r α ) is a disjoint union of simplices, one for each pair r 1, α ∈ A r . The following lemma will now establish the base case of our induction. Lemma 6.12. Say Z = (Z 1 , Z 2 , . . .) is a decomposed sequence in X with decompositions Z r = α∈Ar Z r α . Assume that there exists a sequence N = (N 1 , N 2 , . . .) ∈ Seq, such that for all r 1 and for all α ∈ A r , the diameter of Z r α is at most N r . Then if s N, we have K * A c (P s (Z)) = 0 for all * ∈ Z. Proof. We have already observed that for s N, P s (Z) is a disjoint union of simplices, all at infinite distance from one another. We claim that the controlled K-theory of such a metric space vanishes. Let W = i∈I W i be such a metric space, meaning that for each i we have W i ∼ = ∆ k i for some k i and the distance between W i and W j is infinite if i = j. Choose inclusions { * } ֒→ W i , where { * } denotes the one-point space, and let j denote the resulting map i∈I { * } ֒→ W . Also, let π denote the natural projection W → i∈I { * }. By Bartels [4,Corollary 3.19], the induced maps π * : K * A c (W ) → K * A c ( i∈I { * }) and j * : K * A c ( i∈I { * }) → K * A c (W ) are inverse isomorphisms: π • j is the identity and j • π is continuously Lipschitz homotopic to the identity (as defined in [4,Definition 3.16]). The category A c ( { * }) has trivial K-theory, because it admits an Eilenberg Swindle (this is analogous to Bartels [4,3.20], which treats the case of a single point). Thus we conclude that A c (P s (Z)) has trivial K-theory. A similar argument shows that the subcategory S = colim n A c n r=1 α∈Ar P sr (Z r α ) ⊂ A c (P s (Z)) has trivial K-theory. We conclude that K * A c P s (Z) = 0 for all * by examining the long exact sequence in K-theory associated to the Karoubi sequence S ֒→ A c P s (Z) → A c P s (Z). If γ is a limit ordinal and Proposition 6.11 holds for all β < γ, it follows immediately from the definitions that Proposition 6.11 also holds for γ. Next, consider a successor ordinal γ = β + 1 and assume that D β (X) is vanishing. For the rest of the section, we fix a decomposed sequence Z = (Z 1 , Z 2 , . . .) ∈ D γ (X), with decompositions Z r = α∈Ar Z r α , and we fix a sequence s ∈ Seq. We will show that Z is vanishing at s. Let C r = (2 √ 2 + 1) dim(Ps r (X))−1 be the sequence of constants from Definition 5.1, and let C = (C 1 , C 2 , . . .). Since Z ∈ D γ (X) and γ = β + 1, for each r 1 and each α ∈ A r we may choose decompositions Z r α = U r α (s) ∪ V r α (s) and (C r s r r)-disjoint decompositions V r αj (s). (18) U r α (s) = Crsrr-disjoint i∈I(r,α) U r αi (s) and V r α = Crsrr-disjoint j∈J(r,α) V r αj (s) such that (19) {U r αi (s) \| α ∈ A r , i ∈ I(r, α)}, {V r αj (s) \| α ∈ A r , j ∈ J(r, α)} ∈ D β . We will denote these more finely decomposed sequences by U ′ s and V ′ s . Note that by (19), we have U ′ s , V ′ s ∈ D β (X). We will use this observation in the proofs of Lemmas 6.13 and 6.14. In the sequel, we will often write U = U s , V = V s , U ′ = U ′ s , and V ′ = V ′ s , suppressing the dependence of these sequences and their underlying data on s (and similarly for U r αi (s) and V r αj (s)). For any s ′ , s ′′ ∈ Seq satisfying s s ′ s ′′ , Theorems 4.8, 4.11, and 4.12 imply that there is a commutative diagram as follows, in which the first column comes from the Mayer-Vietoris sequence in Theorem 4.8 and the second column is the colimit, over t > 0, of the Mayer-Vietoris sequences from Theorem 4.11 (we write t rather than colim t to save space): (21) t K + * P s,s ′ (U , W tCs ) t K + * P s,s ′ (V, W tCs ) i U +i V µ s,s ′ ,s ′′ / / t K + * P s ′′ (N Z tCs U ) t K + * P s ′′ (N Z tCs V) K * (P s (Z)) γ s,s ′ / / ∂ t K * P s,s ′ (Z, W tCs ) ∂ ζ s,s ′ ,s ′′ / / K * (P s ′′ (Z)) K * −1 (I s (U , V)) ρ s,s ′ / / t K * −1 I ′ s,s ′ ,t (U , V) . The importance of Diagram (21) stems from the fact (which follows easily from the definitions below) that the composite ζ s,s ′ ,s ′′ • γ s,s ′ is the natural map η s,s ′ : K * A c (P s (Z)) η s,s ′ −−→ K * A c (P s ′ (Z)) . We now explain the various terms in Diagram (21). • The functor K * is shorthand for K * A c . • tCs is the product sequence with r th term tC r s r , and C r = C r (s r , X) is the constant from Definition 5.1. • The sequence W tCs = W tCs (U , V, Z) was defined in Definition 4.10. • The functor K + * is shorthand for K * A c Z+ (Definitions 4.6 and 4.10). • The maps γ s,s ′ and ρ s,s ′ are simply the compositions of the maps appearing in Theorem 4.12 (for any chosen t > 0) with the natural maps to the colimits. Theorem 4.12 implies that the left-hand square in Diagram (21) commutes. • In the third column, N Z tCs U is the decomposed sequence with rth term (22) α∈Ar Z r α ∩ N tCr sr U r α and with decompositions exactly as shown in (22), and similarly for V in place of U . • The vertical map in the third column arises from the inclusions P s ′′ N Z tCs U ⊂ P s ′′ (Z) and P s ′′ N Z tCs V ⊂ P s ′′ (Z). • To describe the horizontal map ζ = ζ s,s ′ ,s ′′ , note that for each t > 0, the inclusion of simplicial complexes P s,s ′ (Z, W tCs ) ⊂ P s ′′ (Z) induces a functor after applying A c (−) (Lemma 2.7). These maps are compatible as t increases, and ζ is the induced map from the colimit. • The map µ s,s ′ ,s ′′ is the direct sum of maps µ s,s ′ ,s ′′ (U ) and µ s,s ′ ,s ′′ (V) induced by the inclusions P s,s ′ (U , W tCs ) ⊂ P s ′′ (N Z tCs U ) and P s,s ′ (V, W tCs ) ⊂ P s ′′ (N Z tCs V). Note that the term-wise colimit of a (directed) sequence of exact sequences is exact, so the second column of Diagram (21) is exact. Commutativity of the right-hand square in Diagram (21) is immediate from the definitions of the functors inducing the maps. We will prove the following two lemmas, which will allow us to deduce that Z is vanishing by chasing Diagram (21). Lemma 6.13. For each x ∈ K * −1 (I s (U s , V s )), there exists s ′ s such that ρ s,s ′ (x) = 0. Lemma 6.14. For each sequence s ′ s, and for each element x ∈ t K + * P s,s ′ (U s , W tCs ) t K + * P s,s ′ (V s , W tCs ) , there exists s ′′ s ′ such that µ s,s ′ ,s ′′ (x) = 0. Proof of Proposition 6.11 assuming Lemmas 6.13 and 6.14. For simplicity, we drop most subscripts from the maps in Diagram (21). For each element x ∈ K * A c (P s (Z)) , we have ∂(γx) = ρ(∂x). By Lemma 6.13, we can choose s ′ large enough so that ρ(∂x) = 0. Exactness of the second column in Diagram (21) then shows that γ(x) = (i U + i V )(x 1 , x 2 ) for some x 1 , x 2 . Now Lemma 6.14 tells us that for s ′′ large enough, we have µ(x 1 , x 2 ) = 0, and it follows from commutativity of the right-hand square of Diagram (21) that ζ(γx) = ζ((i U + i V )(x 1 , x 2 )) = 0. However, as mentioned above the composite ζ • γ is simply the natural map η s,s ′ : K * A c (P s (Z)) η s,s ′ −−→ K * A c (P s ′ (Z)) . Hence Z is vanishing at s. Since s ∈ Seq was arbitrary, Z is in fact a vanishing sequence, and our induction is complete. ✷ To prove Lemma 6.13, we need to compare two versions of the category of controlled modules on P q (W), where q ∈ Seq and W = (W 1 , W 2 , . . .) is a decomposed sequence in X with decompositions W r = α∈Ar W r α . We may give P q (W) either its intrinsic simplicial metric or the simplicial metric inherited from P q (X W ) (where X W is the decomposed sequence defined in Remark 4.4). The category corresponding to the first metric will be denoted A c W (P q (W)). The latter metric is the one used to define the category A c (P q (W)), and we will sometimes write A c X (P q (W)) = A c (P q (W)) simply to emphasize the chosen metric on P q (W). Lemma 6.15. Let W be a decomposed sequence in X and let q ∈ Seq be any sequence. Then there exist functors (23) Φ q : A c X (P q (W)) −→ colim n∈Seq A c W (P n (W)) that make the diagram (24) A c X (P q W) Φq / / colim n∈Seq A c W (P n W) A c X (P q ′ W) Φ q ′ 7 7 ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ commute whenever q q ′ . Proof. To construct the functors Φ q , first note that objects in A c X (P q W) are also objects in A c W (P q W) because the change of metrics does not affect which sets are compact, and hence the locally finiteness condition is the same in both cases. Letting i q : A c W (P q (W)) −→ colim n∈Seq A c W (P n (W)) denote the structure map for the colimit, we can now define Φ q to be the identity on objects by setting Φ q (M ) = i q (M ). More care is required to define Φ q on morphisms. A morphism in ψ: M → N in A X c (P q W) is a bounded map of geometric modules on P q (W)×[0, 1) (with metric induced from P q (X W )×[0, 1)) which is controlled at 1. Let d < ∞ denote a bound on the propagation of ψ, and let q ′ be the sequence with r th term q ′ r = q r C r (d + 2) (where C r = C r (q r , X) is the constant from Definition 5.1). We claim that ψ ∈ A W c (P q ′ (W)). First we check that ψ is bounded as a morphism on P q ′ (W)×[0, 1), where P q ′ (W) has its intrinsic simplicial metric. Let (a 1 , t 1 ), (a 2 , t 2 ) ∈ P q (W) × [0, 1) be points such that ψ (a 1 ,t 1 ),(a 2 ,t 2 ) = 0. Note that this implies that a 1 , a 2 ∈ P qr (W r ) for some r. Choose barycentric vertices v 1 , v 2 ∈ W r for a 1 and a 2 (respectively). Lemma 5.2 implies that d(v 1 , v 2 ) q r C r (d+2), so v 1 and v 2 lie in a common simplex in P q ′ r (W r ) (by choice of q ′ r ). It follows that a 1 and a 2 are at most distance 3 apart in the simplicial metric on P q ′ r (W r ), so (a 1 , t 1 ) and (a 2 , t 2 ) are at most distance 4 apart in the corresponding metric on P q ′ r (W r ) × [0, 1). Hence ψ has propagation at most 4 in this metric. To check that ψ is controlled as a morphism on P q ′ (W) × [0, 1), note that (continuous) control is a topological condition: it does not refer to the metric on the complex in question. Since ψ is controlled as a morphism on P q (W)×[0, 1), Lemma 2.7 implies that ψ is also controlled on P q ′ (W)×[0, 1) (for the purpose of applying Lemma 2.7, we can give these complexes the metrics inherited from P q (X W ) and P q ′ (X W ), respectively). We can now define Φ q ([ψ]) to be the morphism Φ q (M ) → Φ q (N ) represented by ψ. Since Φ q does not change the underlying data of either geometric modules or morphisms, it follows that Φ q is a functor and that Diagram (24) commutes. The key result behind the proofs of Lemmas 6.13 and 6.14 is a comparison between the categories of controlled modules associated to a decomposed sequence and to a sufficiently good refinement of that sequence. First we record a lemma regarding the construction A c (−). Lemma 6.16. Let K 1 ⊂ K ′ 1 , K 2 ⊂ K ′ 2 , . . . be locally finite simplicial complexes. Then for each R > 0, the inclusion r R K r ֒→ r 1 K r induces an equivalence of categories i R : A c   r R K r   ∼ = −→ A c r 1 K r , where on the left we use the simplicial metric from r R K ′ r and on the right we use the simplicial metric from r 1 K ′ r . In particular, for each decomposed sequence Y = (Y 1 , Y 2 , . . .) in X and each q ∈ Seq, there is an equivalence of categories i R : A c P q(R) (Y(R)) ∼ = −→ A c (P q (Y)) , where q(R) = (q R , q R+1 , . . .) and Y(R) denotes the decomposed sequence (Y R , Y R+1 , . . .), with the same decompositions as in Y. Proof. The functor i R exists by Lemma 2.7, and it follows from the definitions that i R is full and faithful. Each module M ∈ A c r 1 K r is isomorphic, in the Karoubi quotient A c r 1 K r , to its restriction M   r R K r × [0, 1)   . This restriction is in the image of i R , completing the proof. Lemma 6.17. Let Y = (Y 1 , Y 2 , . . .) be a decomposed sequence in X, with decompositions Y r = α∈Ar Y r α . Consider sequences q, f ∈ Seq satisfying lim r→∞ f r /C r q r = ∞, where C r = C(q r , X) is the constant from Definition 5.1. Assume that for each sufficiently large r and each α ∈ A r we have a decomposition Y r α = fr-disjoint i∈I(r,α) Y r αi . Let Y ′ be the decomposed sequence Y ′ = (Y 1 , Y 2 , . . .) with decompositions Y r = α∈Ar i∈I(r,α) Y r αi . Then there are maps Ψ t : K * A c P t (Y ′ ) −→ K * A c (P t (Y)) , natural with respect to t ∈ Seq, and Ψ q is an isomorphism in all dimensions. Consequently, if Y ′ is vanishing at q, then so is Y. Proof. Naturality of the maps Ψ t means that for t ′ t, we will construct a commutative diagram (25) K * A c (P t (Y)) η t,t ′ (Y) / / K * A c (P t ′ (Y)) K * A c (P t (Y ′ )) η t,t ′ (Y ′ ) / / Ψ t O O K * A c (P t ′ (Y ′ )) . Ψ t ′ O O The final statement of the lemma will follow from commutativity of Diagram (25) together with the definition of vanishing, once we establish that Ψ q is an isomorphism. We now define the desired homomorphisms Ψ t for each sequence t ∈ Seq. Our hypotheses imply that there exists R > 0 such that if r R then f r > q r > 0 and d(Y r αi , Y r αj ) f r for all α ∈ A r and all i, j ∈ I(r, α) with i = j. In particular, for such r, α, i and j we have Y r αi ∩ Y r αj = ∅, so there is an injective map of simplicial complexes P t(R) (Y ′ (R)) ֒→ P t(R) (Y(R)). (Note that for r < R, we allow for the possibility that Y r αi ∩ Y r αj = ∅ for some i = j; this will be important when we apply the Lemma 6.17 in the proof of Lemma 6.13.) This map is proper and decreases distances, so by Lemma 2.7 we have an induced functor Φ t : A c P t(R) Y ′ (R) −→ A c P t(R) (Y(R)) . Lemma 6.16 yields a diagram (26) A c (P t (Y ′ )) A c (P t (Y)) A c P t(R) (Y ′ (R)) Φ t / / i ′ ∼ = O O A c P t(R) (Y(R)) , i ∼ = O O and we define K * A c P t (Y ′ ) Ψ t −→ K * A c (P t (Y)) to be the homomorphism obtained from Diagram (26) by inverting (i ′ ) * . Given t ′ t, we obtain a commutative diagram linking the zig-zag (26) to the corresponding zig-zag for t ′ ; this yields commutativity of Diagram (25). We need to check that Ψ q is an isomorphism. We will show that Φ q is an isomorphism of categories (not just an equivalence). For r R, α ∈ A r , and i, j ∈ I(r, α) with i = j, we have d(Y r αi , Y r αj ) > q r , so the simplicial complex P qr (Y r α ) is the disjoint union of the subcomplexes P qr (Y r αi ). This shows that Φ q is bijective on objects: each module on P q (Y r α ) × [0, 1) is the direct sum of its restrictions to the disjoint subspaces P q (Y r αi ) × [0, 1). Next, we check that Φ q is surjective on morphisms. Given [ψ] ∈ A c P q(R) (Y(R)), let T be a bound on the propagation of ψ. Since f r /C r q r → ∞, there exists R T such that f r > (T + 1)q r C r for r R T . Now [ψ] = [ψ(R T )], where ψ(R T ) denotes the morphism ψ(R T ) a,b = ψ a,b , a, b ∈ P qr (Y r α ) × [0, 1) for some r R T , α ∈ A r , 0, else. Lemma 5.2 and our choice of R T imply that ψ(R T ) is a direct sum, over r > R, α ∈ A r and i ∈ I(r, α), of controlled morphisms ψ(R T ) rαi supported on P qr (Y r αi ) × [0, 1). Hence [ψ] = [ψ(R T )] is in the image of Φ q . Note here that both P qr (Y r αi ) and P qr (Y r α ) have the metric inherited from P qr (X), so the propagation of ψ(R T ) = r R T α∈Ar i∈I(r,α) ψ(R T ) rαi is the same whether we consider it as a morphism between modules on P q (Y) or on P q (Y ′ ). Finally, we must check that Φ is faithful. If Φ q ([ψ 1 ]) = Φ q ([ψ 2 ]), then for sufficiently large R 0 , the restrictions of ψ 1 and ψ 2 to Proof of Lemma 6.13. Given t > 0, r 1, and α ∈ A r , we define (27) W r tα = N tCrsr (U r α ) ∩ N tCrsr (V r α ) ∩ Z r α . For each t > 0 we define the decomposed sequence W t = W t (U , V, Z), whose r th term is (28) W r t = α∈Ar W r tα ; the decomposition of W r t is exactly that displayed in (28). We claim that W t is vanishing at s. Consider the decomposed sequence W ′ t , whose r th term is the same as that of W t , but with the finer decomposition (19), the families {U r αi \| α ∈ A r , i ∈ I(r, α)} and {V r αj \| α ∈ A r , j ∈ J(r, α)} lie in D β , so the induction hypothesis and Lemma 6.3 tell us that W ′ t is a vanishing sequence. Our disjointness hypotheses (18) imply that and (C r s r (r − 2t))/C r s r = r − 2t tends to infinity with r. By Lemma 6.17, W t is vanishing at s. For each t > 0, let W t denote the set of metric families W t = {{W r tα } α∈Ar } r 1 . Note that P s ′ (W t ) ⊂ P s,s ′ (X, W t ), where the latter complex was introduced in Definition 4.9. For each s ′ s, we will show that the map ρ s,s ′ factors through a map (30) colim (29) W r t = α∈Ar (i,j)∈I(r,α)×J(r,α) W r tαij , where W r tαij = N tCrsr (U r αi ) ∩ N tCr sr (V r αj ) ∩ Z r α . Byt→∞ K * −1 A c X P s (W t ) (ξ s,s ′ ) * − −−− → colim t→∞ K * −1 A c rel P s ′ (W t ) , where the superscripts indicate that we give these Rips complexes the metrics induced from the simplicial metrics on P s (X Wt ) and P s,s ′ (X, W t ) (respectively). (This choice of metrics will be important in obtaining the desired factorization of ρ s,s ′ ; in particular, if we used the metric on P s ′ (W t ) inherited from the simplicial metric on P s ′ (X) to define the codomain of (ξ s,s ′ ) * , we would not be able to define the map l in (31) below.) Lemma 2.7 shows that the inclusion of simplicial complexes P s (W t ) ⊂ P s ′ (W t ) induces a functor ξ s,s ′ ,t : A c X P s (W t ) −→ A c rel P s ′ (W t ). We set ξ s,s ′ = colim t ξ s,s ′ ,t , and (30) is the induced map on K-theory. We now show that for every class on the left-hand side of (30), there exists s ′ s such that (ξ s,s ′ ) * (x) = 0. (The corresponding result for ρ s,s ′ will follow immediately once we establish the claimed factorization.) It suffices to show that the functor ξ s = colim s ′ ∈Seq ξ s,s ′ induces the zero-map on K-theory. In Lemma 6.15, we constructed functors Φ s,t : A c X (P s (W t )) −→ colim s ′ ∈Seq A c Wt (P s ′ (W t )), where again the superscripts indicate the chosen metrics on the Rips complexes (see the discussion preceding Lemma 6.15). Lemma 2.7 yields functors A c Wt (P s ′ (W t )) Ψ s ′ ,t − −− → A c rel (P s ′ (W t )) for each s ′ ∈ Seq and each t > 0, and now ξ s = colim s ′ ∈Seq ξ s,s ′ factors as colim t→∞ A c X (P s (W t )) colimt Φs,t −−−−−−→ colim t→∞ colim s ′ ∈Seq A c Wt (P s ′ (W t )) colim t,s ′ Ψ s ′ ,t − −−−−−−− → colim t→∞ colim s ′ ∈Seq A c rel P s ′ (W t ) ∼ = colim s ′ ∈Seq colim t→∞ A c rel P s ′ (W t ) . It will suffice to show that the maps Φ s,t induce the zero map on K-theory for all t. Recall (see Diagram (24)) that for each t and each s ′ s, the map Φ s,t factors through the natural map A c X P s (W t ) η s,s ′ ,t −−−→ A c X P s ′ (W t ), and hence colim t Φ s,t factors through colim t,s ′ η s,s ′ ,t . As established above, W t is vanishing at s for every t, so colim s ′ η s,s ′ ,t induces the trivial map on Ktheory for each t. Hence the first map colim t Φ s,t in the above composition induces the trivial map on K-theory, and we conclude that the same is true of ξ s (as desired). The desired factorization of ρ s,s ′ comes from a sequence of functors I s (U , V) i −→ colim t→∞ A c X (P s (Z) ∩ (N t P s U ) ∩ (N t P s V)) j −→ colim t→∞ A c X (P s (W t )) colimt ξ s,s ′ ,t −−−−−−−→ colim t→∞ A c rel (P s ′ (W t )) (31) l −→ colim t→∞ I ′ s,s ′ ,t (U , V). The functors i and l are inclusions of categories that exist by the definitions of the intersection terms in the Mayer-Vietoris sequences. (In the case of l, note that both of these categories are defined using the simplicial metric on P s,s ′ (X, W t ), and use the fact that for metric spaces A ⊂ B, a continuously controlled morphism between geometric modules on A × [0, 1) is also continuously controlled on B × [0, 1). This latter fact was shown in the proof of Lemma 6.15, and also follows from Lemma 2.7.) The functor j exists by Equation (12) in Lemma 5.3 (which may be applied to non-relative Rips complexes simply by setting the two parameters s, s ′ appearing in the Lemma to be equal), and it is immediate from the definitions that the composite of these functors is the functor inducing ρ s,s ′ on K-theory. ✷ For the proof of Lemma 6.14, we need a relative version of (one part of) Lemma 6.17. Lemma 6.18. Let Q = (Q 1 , Q 2 , . . .) and Y = (Y 1 , Y 2 , . . .) be decomposed sequences in X satisfying Q ⊂ Y, and say the decompositions of these sequences are Q r = α∈Ar Q r α and Y r = α∈Ar Y r α . Let q, g ∈ Seq satisfy lim r→∞ g r /C r q r = ∞, where C r = C(q r , X) is the constant from Definition 5.1. Assume that for each r and each α ∈ A r we have decompositions Q r α = i∈I(r,α) Q r αi . Let Q ′ and Y ′ (respectively) be the decomposed sequences Q ′ = (Q 1 , Q 2 , . . .) and Y ′ = (Y 1 , Y 2 , . . .), with decompositions Q r = α∈Ar i∈I(r,α) Q r αi and Y r = α∈Ar i∈I(r,α) Y r αi , where Y r αi = Y r α for each r 1 and each i ∈ I(r, α). Note that Q ′ ⊂ Y ′ . Assume further that we are given a set W = {W r α \| r 1, α ∈ A r } of metric families in X. Let W ′ = {W r αi \| r 1, α ∈ A r , i ∈ I(r, α)} be a refinement of W, in the sense that for each r 1 and each α ∈ A r , we have W r α = {S \| S ∈ W r αi for some i ∈ I(r, α)}. Given r 1, α ∈ A r , and i ∈ I(r, α), let (32) W r αi = W r αi = {x ∈ X \| x ∈ S for some S ∈ W r αi } denote the union of all the sets in the family W r αi . Assume that there exists R 0 > 0 such that for each r R 0 and each α ∈ A r , the family {Q r αi ∪ W r αi } i∈I(r,α) is g r -disjoint, meaning that (33) d(Q r αi ∪ W r αi , Q r αj ∪ W r αj ) > g r for i = j. Then for each q ′ ∈ Seq and each * ∈ Z, there is an isomorphism Ψ q,q ′ : K * A c Y ′ + P q,q ′ (Q ′ , W ′ ) ∼ = −→ K * A c Y+ P q,q ′ (Q, W) . Proof. For each T > 0, define N Y r α T,d ′ ∆ P qr,q ′ r (Q r αi , W r αi ) := P qr,q ′ r (Y r α , W r αi ) ∩ N T,d ′ ∆ P qr,q ′ r (Q r αi , W r αi ) , where on the right, the neighborhood is taken inside the larger complex P qr,q ′ r (X, W r αi ), with its simplicial metric d ′ ∆ . Similarly, let N Y r α T,d ∆ P qr,q ′ r (Q r α , W r α ) := P qr,q ′ r (Y r α , W r α ) ∩ N T,d ∆ P qr,q ′ r (Q r α , W r α ) , where on the right, the neighborhood is taken inside P qr,q ′ r (X, W r α ) with its simplicial metric d ∆ = d ∆ (r, α). Our hypotheses imply that for each T > 0, there exists R T R 0 such that if r R T then (34) g r > (2T + 2)C r q r . Set K ′ T,d ′ ∆ (R T ) := r R T α∈Ar i∈I(r,α) N Y r α T,d ′ ∆ P qr,q ′ r (Q r αi , W r αi ) and K T (R T ) := r R T α∈Ar N Y r α T,d ∆ P qr,q ′ r (Q r α , W r α ) . Let W(R T ) and W ′ (R T ) denote the sets of metric families {W r α \| r R T , α ∈ A r } and {W r αi \| r R T , α ∈ A r , i ∈ I(r, α)}, respectively. We have inclusions of simplicial complexes K ′ T,d ′ ∆ (R T ) ⊂ P q(R T ),q ′ (R T ) X, W ′ (R T ) and K T (R T ) ⊂ P q(R T ),q ′ (R T ) (X, W(R T )) (recall that these relative Rips complexes were introduced in Definition 4.9), and we give K ′ T,d ′ ∆ (R T ) and K T (R T ) the metrics induced from the simplicial metrics on these relative Rips complexes. The inclusion maps N Y r α T,d ′ ∆ P qr,q ′ r (Q r αi , W r αi ) ֒→ N Y r α T,d ∆ P qr,q ′ r (Q r α , W r α ) combine to yield a simplicial map K ′ T,d ′ ∆ (R T ) φ q,q ′ (T ) − −−−− → K T (R T ), which decreases distances (by our choice of metrics). We claim that φ q,q ′ (T ) is injective as well. If not, we would have d ∆ P qr,q ′ r (Q r αi , W r αi ) , P qr,q ′ r Q r αj , W r αj d ′ ∆ P qr,q ′ r (Q r αi , W r αi ) , P qr,q ′ r Q r αj , W r αj < 2T for some r > R T , α ∈ A r , and i, j ∈ I(r, α) with i = j, and then Lemma 5.4 would yield d   Q r αi ∪ W r αi , k∈I(r,α), k =i (Q r αk ∪ W r αk )   (2T + 2)C r q r , contradicting (33) and (34). Since φ q,q ′ (T ) is injective and decreases distances, by Lemma 2.7 it induces a functor A c (K ′ T,d ′ ∆ (R T )) Φ q,q ′ (T ) −−−−− → A c (K T (R T )). Define K ′ T,d ′ ∆ := r 1 α∈Ar i∈I(r,α) N Y r α T,d ′ ∆ P qr,q ′ r (Q r αi , W r αi ) and K T := r 1 α∈Ar N Y r α T,d ∆ P qr,q ′ r (Q r α , W r α ) , and give these complexes the metrics induced by the simplicial metrics on P q,q ′ (X, W ′ ) and P q,q ′ (X, W), respectively. By Lemma 6.16, we have a diagram (35) A c (K ′ T,d ′ ∆ ) A c (K T ) A c (K ′ T,d ′ ∆ (R T )) Φ q,q ′ (T ) / / i ′ ∼ = O O A c (K T (R T )), i ∼ = O O and we define K * A c (K ′ T ) Ψ q,q ′ (T ) −−−−−→ K * A c (K T ) by the equation Ψ q,q ′ (T ) = i * • (Φ q,q ′ (T )) * • (i ′ ) −1 * . By definition, we have A c Y+ P q,q ′ (Q, W) = colim T >0 A c (K T ), and (36) A c Y ′ + P q,q ′ (Q ′ , W ′ ) = colim T >0 A c (K ′ T,d ′ ∆ ). The maps Ψ q,q ′ (T ) are natural with respect to T , so we obtain the desired map Ψ q,q ′ = colim T >0 Ψ q,q ′ (T ): K * A c Y ′ + P q,q ′ (Q ′ , W ′ ) Ψ q,q ′ −−− → K * A c Y+ P q,q ′ (Q, W) . We need to check that Ψ q,q ′ is an isomorphism. We will show that Φ q,q ′ (T ) is an isomorphism of categories for each T > 0. We claim that the maps φ q,q ′ (T ) are actually bijections. We have already shown that φ q,q ′ (T ) is injective, so we need only consider surjectivity. Set N Y r α T,d ∆ P qr,q ′ r (Q r αi , W r αi ) := P qr,q ′ r (Y r α , W r α ) ∩ N T,d ∆ P qr,q ′ r (Q r αi , W r αi ) , where on the right, the neighborhood is taken inside P qr,q ′ r (X, W r α ) with its simplicial metric d ∆ . We claim that for r R T , (37) N Y r α T,d ∆ P qr,q ′ r (Q r αi , W r αi ) = N Y r α T,d ′ ∆ P qr,q ′ r (Q r αi , W r αi ) for all α ∈ A r , i ∈ I(r, α). It follows easily from the definitions that N Y r α T,d ′ ∆ P qr,q ′ r (Q r αi , W r αi ) ⊂ N Y r α T,d ∆ P qr,q ′ r (Q r αi , W r αi ) . Now say x ∈ N Y r α T,d ∆ P qr,q ′ r (Q r αi , W r αi ) . Then there is a piece-wise geodesic path γ in P qr,q ′ r (X, W r α ), of length less than T , from x to P qr,q ′ r (Q r αi , W r αi ). We claim that γ lies inside P qr,q ′ r (X, W r αi ) (which will imply, in particular, that x ∈ N T,d ′ ∆ P qr,q ′ r (Q r α , W r αi ) ). If not, then for some t ∈ [0, 1] and some j ∈ I(r, α) with i = j, we have γ(t) ∈ P q ′ r (W ) for some W ∈ W r αj . Then d ∆ P qr,q ′ r (Q r αi , W r αi ) , P qr,q ′ r Q r αj , W r αj < T. By Lemma 5.4, we have d   Q r αi ∪ W r αi , j∈I(r,α), j =i Q r αj ∪ W r αj   (T + 2)C r q r , contradicting our choice of R T (note that W r αi and W r αj were defined in (32)). A similar argument shows that x ∈ P qr,q ′ r (Y r , W r αi ), establishing (37). From here on we drop the subscripts d ∆ and d ′ ∆ from the sets in (37). By (33) and (34), for r > R T , α ∈ A r , and i, j ∈ I(r, α) with i = j, we have d(Q r αi , Q r αj ) > (2T + 2)C r q r > q r . Hence P qr,q ′ r (Q r α , W r α ) = i∈I(r,α) P qr,q ′ r (Q r αi , W r αi ). Together with (37), this establishes surjectivity of φ q,q ′ (T ). Bijectivity of φ q,q ′ (T ) implies that Φ q,q ′ (T ) is bijective on objects: for all r R T , α ∈ A r , each module on N Y r α T,d ∆ P qr,q ′ r (Q r α , W r α ) × [0, 1) = i∈I(r,α) N Y r α T P qr,q ′ r (Q r αi , W r αi ) × [0, 1) is the direct sum of its restrictions to the disjoint subspaces on the right. Next we check that Φ q,q ′ (T ) is full. Each morphism α in A c Y+ P q,q ′ (Q, W) is represented by a morphism ψ in A c (K T ) for some T > 0. Let D be a bound on the propagation of ψ. Since g r /C r q r → ∞, there exists S = S(ψ) R T such that g r > (D + 2T + 2)C r q r for r S. Setting ψ(S) a,b =    ψ a,b , a, b ∈ N Y r α T P qr,q ′ r (Q r α , W r α ) × [0, 1) for some r S and some α ∈ A r , 0, else, we have [ψ] = [ψ(S)] as morphisms in A c (K T ). As above, Lemma 5.4 and our choice of S imply that for r > S, α ∈ A r , and i, j ∈ I(r, α) with i = j, (38) d ∆ P qr,q ′ r (Q r αi , W r αi ) , P qr,q ′ r Q r αj , W r αj D + 2T. Hence ψ(S) is a direct sum, over r S, α ∈ A r , and i ∈ I(r, α), of morphisms supported on N Y r α T P qr,q ′ r (Q r αi , W r αi ) × [0, 1). When viewed as a morphism between modules on P q(S),q ′ (S) (X, W ′ (S)) × [0, 1), ψ(S) still has propagation at most D: if i ∈ I(r, α) for some r S and some α ∈ A r and there exist points (x, t), (y, s) ∈ N Y r α T P qr,q ′ r (Q r αi , W r αi ) × [0, 1) with ψ(S) (x,t) ,(y,s) = 0, then there exists a simplicial path in P qr,q ′ r (X, W r α ) of length at most D connecting x and y. This path must in fact lie in P qr,q ′ r (X, W r αi ), since otherwise we would have d ∆ P qr,q ′ r (Q r αi , W r αi ) , P qr,q ′ r Q r αj , W r Proof of Lemma 6.14. As in (18) respectively. By (19), we have U ′ , V ′ ∈ D β (X). Given t > 0, let N Z tCs U ′ denote the decomposed sequence with rth term (39) α∈Ar i∈I(r,α) Z r α ∩ N tCrsr U r αi and with decompositions exactly as shown in (39), and similarly for V in place of U . Lemma 6.10 implies that for each t > 0, N Z tCs U ′ and N Z tCs V ′ are in D β (X) as well. By the induction hypothesis, N Z tCs U ′ and N Z tCs V ′ are vanishing sequences. We will show that for any s ′ , s ′′ ∈ Seq with s s ′ s ′′ , the map µ s,s ′ ,s ′′ = µ s,s ′ ,s ′′ (U ) ⊕ µ s,s ′ ,s ′′ (V) factors through the direct sum of the maps (40) colim t→∞ K * A c X (P s ′ (N Z tCs U ′ )) η=colimt η s ′ ,s ′′ (t) − −−−−−−−−−− → colim t→∞ K * A c X (P s ′′ (N Z tCs U ′ )) and (41) colim t→∞ K * A c X (P s ′ (N Z tCs V ′ )) η=colimt η s ′ ,s ′′ (t) − −−−−−−−−−− → colim t→∞ K * A c X (P s ′′ (N Z tCs V ′ )). Since N Z tCs U ′ and N Z tCs V ′ are vanishing sequences, the desired result will follow from this factorization. We will in fact show that µ s,s ′ ,s ′′ (U ) factors through (40) and µ s,s ′ ,s ′′ (V) factors through (41). From here on we deal only with U ; the argument for V is identical. For t > 0, r 1, α ∈ A r , i ∈ I(r, α) and j ∈ J(r, α), let W r tαij = Z r α ∩ N tCr sr (U r αi ) ∩ N tCr sr (V r αj ) and let W r tαi = j∈J(r,α) W r tαij . Furthermore, let W r tαi denote the metric family {W r tαij } j∈J(r,α) . Given T > 0, r 1, α ∈ A r , and i ∈ I(r, α), we have P sr,s ′ r (Z r α , W r tαi ) ∩ N T P sr,s ′ r (U r αi , W r tαi ) (42) ⊆ P sr,s ′ r (Z r α , W r tαi ) ∩ N T P sr,s ′ r (U r αi , W r tαi ) , where the first neighborhood is taken with respect to the simplicial metric on P sr,s ′ r (X, W r tαi ), and the second neighborhood is taken with respect to the (smaller) simplicial metric on P sr,s ′ r (X, W r tαi ). Equation (13) in Lemma 5.3, along with the fact that W r tαi ⊂ Z r α , now shows that P sr,s ′ r (Z r α , W r tαi ) ∩ N T P sr,s ′ r (U r αi , W r tαi ) (43) ⊆ P s ′ r Z r α ∩ N (T +2)Cr sr (U r αi ∪ W r tαi ) ,⊂ P s ′ r Z r α ∩ N (T +t+2)Cr sr (U r αi ) . Let W ′ tCs denote the set of metric families W ′ tCs = {W r tαi \| r 1, α ∈ A r , i ∈ I(r, α)}, and let Z ′ denote the decomposed sequence Z ′ = (V 1 , V 2 , . . .) with decompositions Z r = α∈Ar i∈I(r,α) Z r αi , where Z r αi = Z r α for each r 1, α ∈ A r , and i ∈ I(r, α). For each t, T > 0, we define N Z T P s,s ′ U ′ , W ′ tCs := P s,s ′ (Z ′ , W ′ tCs ) ∩ N T P s,s ′ U ′ , W ′ tCs , where on the right, the neighborhood is taken inside the larger complex P s,s ′ (X, W ′ tCs ) (with respect to the simplicial metric on P s,s ′ (X, W ′ tCs )). We give N Z T P s,s ′ (U ′ , W ′ tCs ) the metric induced by the simplicial metric on P s,s ′ (X, W ′ tCs ). Applying Lemma 2.7 to the inclusions (45) yields functors (46) A c N Z T P s,s ′ U ′ , W ′ tCs j t,T − − → A c X P s ′ N Z (t+T +2)Cs (U ′ ) The colimit, over T > 0, of the categories appearing in the domain of j t,T is precisely A c Z ′ + P s,s ′ (U ′ , W ′ tCs ) . Hence the functors j t,T combine to yield a functor A c Z ′ + P s,s ′ U ′ , W ′ tCs jt=colim T j t,T − −−−−−−−− → colim t→∞ A c X P s ′ N Z tCs (U ′ ) . For each r 1, α ∈ A r , and i ∈ I(r, α) we have U r αi ∪ W r tαi ⊂ N tCr sr (U r αi ), and the families {N tCr sr (U r αi ) \| i ∈ I(r, α)} are (C r s r r − 2tC r s r )-disjoint (by (18)). Since (r − 2t)C r s r /C r s r = r − 2t tends to infinity with r, Lemma 6.18 tells us that for each t > 0 there is an isomorphism Ψ s,s ′ (t): K * A c Z ′ + P s,s ′ U ′ , W ′ tCs ∼ = −→ K * A c Z+ P s,s ′ (U , W tCs ) . The desired factorization of µ s,s ′ ,s ′′ (U ) is obtained by composing the isomorphism colim t→∞ Ψ s,s ′ (t) −1 with the composite colim t→∞ K * A c Z ′ + P s,s ′ U ′ , W ′ tCs colimt(jt) * − −−−−−− → colim t→∞ K * A c X P s ′ N Z tCs (U ′ ) η * −→ colim t→∞ K * A c X P s ′′ (N Z tCs (U ′ )) Ψ s ′′ −→ colim t→∞ K * A c X P s ′′ (N Z tCs (U )) k * −→ colim t→∞ K * A c Z+ P s ′′ (N Z tCs (U )) , where η is the functor from (40), Ψ s ′′ is the colimit (over t) of the isomorphisms from Lemma 6.17, and k * is induced by the colimit of the inclusions A X c P s ′′ (N Z tCs (U )) ⊂ A Z+ c P s ′′ (N Z tCs (U )) . To show that this composite agrees with µ s,s ′ ,s ′′ (U ), we Diagram (47), whose terms are explained below. The dotted arrows in Diagram (47) exist only after passing to K-theory. The maps labelled injective are inclusions of one term into a colimit. To save space, we have written W t 0 and W ′ t 0 rather than W t 0 Cs and W ′ t 0 Cs , and we have written t rather than colim t . In the upper left corner, and δ = colim t→∞ η s ′ ,s ′′ (N Z tCs ). Commutativity of the undotted portion of the diagram follows quickly from the definitions of the functors involved. For instance, the outer square commutes because the maps involved do not change the underlying data of geometric modules or morphisms. Commutativity of the lower right rectangle (after passing to K-theory) now follows from the fact that in the upper left corner of this rectangle, K * A c Z+ P s,s ′ (U ′ , W ′ t 0 ) is (isomorphic to) the colimit over T > 0 of the K-theories of the categories A c N Z T P R s,s ′ (U ′ , W ′ t 0 ) appearing in the upper left-hand corner of the diagram. By definition, the map µ s,s ′ ,s ′′ (U ) is obtained from δ • γ t 0 • β t 0 by passing to the colimit (over t 0 ) in the domain (and then applying K-theory). Commutativity of the lower right rectangle (after passing to K-theory) shows that δ • γ t 0 • β t 0 = (k • Ψ s ′′ • η • j t 0 ) * • Ψ s,s ′ (t 0 ) −1 , and taking colimits over t 0 gives the claimed factorization of µ s,s ′ ,s ′′ (U ). ✷ Assembly for FDC groups In this section, we apply our vanishing result for continuously controlled K-theory (Theorem 6.4) to study assembly maps. We first prove a largescale, bounded version of the Borel Conjecture, analogous to Guentner-Tessera-Yu [14, Theorems 4.3.1, 4.4.1], relating the bounded K-theory of the Rips complexes on an FDC metric space to an associated homology theory. Then we study the classical K-theoretic assembly map, using Carlsson's descent argument [8]. Theorem 7.1. Let X be a bounded geometry metric space with finite decomposition complexity. Then there is an isomorphism colim s→∞ H * (P s (X); K(A)) ∼ = colim s→∞ K * (A b (P s (X))). This result may be thought of as excision statement for bounded Ktheory. Before giving the proof, we need some setup. For a proper metric space X, let A c (X) <1 denote the full additive subcategory of A c (X) on those modules M whose support has no limit points at 1; that is, supp(M ) ∩ (X × 1) = ∅, where the closure supp(M ) is taken in X × [0, 1]. By an argument similar to the proof of Lemma 3.6, the inclusion of categories A c (X) <1 ⊂ A c (X) admits a Karoubi filtration. Definition 7.2. The Karoubi quotient A c (X)/A c (X) <1 is denoted A ∞ (X). Theorem 3.4 yields a long exact sequence in non-connective K-theory · · · ∂ −→ K * A c (X) <1 −→ K * (A c (X)) (48) −→ K * A ∞ (X) ∂ −→ K * −1 A c (X) <1 −→ · · · . As shown by Weiss [28], K * (A ∞ (−)) is the (Steenrod) homology theory associated to the non-connective algebraic K-theory spectrum K(A), with a dimension shift: in particular, if X is a finite CW complex, there are isomorphisms (49) K * (A ∞ (X)) ∼ = H * −1 (X; K(A)) for each * ∈ Z (the result was first proven, in a slightly different form, in Pedersen-Weibel [21]). The two key components of Weiss's proof are the facts that the functor X → K * (A ∞ (X)) is homotopy invariant and satisfies excision. The methods of Weiss and Williams [29] then show that A ∞ (X) ≃ X + ∧ A ∞ ( * ) (at least for X an ENR, and in particular for X a finite CW complex). One then identifies the coefficients A ∞ ( * ) by observing that K * (A ∞ ({ * })) is isomorphic to K * −1 (A c ({ * }) <1 ),A b (X) ∼ = −→ A c (X) <1 . Proof. This equivalence is induced by the inclusion of categories A b (X) = A b (X × {0}) ⊂ A c (X) <1 , which is clearly bijective on Hom sets in the domain. We need to check that every object in A c (X) <1 is isomorphic to an object in A b (X). Given a module M ∈ A c (X) <1 , let M ∈ A b (X) be the module M x = t∈[0,1) M (x,t) . Since objects in A b (X × [0, 1)) <1 stay away from 1, M is finitely generated at each point, and properness of X implies that M is locally finite. We now have an isomorphism M → M sending M (x,t) isomorphically to the corresponding summand of M x . This morphism has propagation less than 1, and is continuously controlled due to the support condition on M . Proof of Theorem 7.1 For each s, the isomorphisms given by (49) and Lemma 7.4 show that the long exact sequence (48) has the form · · · → K * (A c (P s (X))) → H * −1 (P s (X); K(A)) ∂ → K * −1 A b (P s (X)) → · · · . Since directed colimits preserve exact sequences and the K-theory of the category colim s A c (P s X) vanishes (Theorem 6.4), the colimit (over s) of the boundary maps for this sequence yields the desired isomorphism. ✷ We now begin the preparations for the proof of our main result, Theorem 1.1. The proof requires some preliminaries regarding group actions and the "forget-control" description of the assembly map. Let X be a proper metric space with an isometric action of a group Γ. Then Γ acts on A c (X) through additive functors (given by translating modules and morphisms), and this action maps the subcategory A c (X) <1 into itself. It follows from the definitions that the inclusion of fixed point categories A c (X) Γ <1 ⊂ A c (X) Γ admits a Karoubi filtration. We now have a Karoubi sequence (50) A c (X) Γ <1 ⊂ A c (X) Γ −→ A c (X) Γ / A c (X) Γ <1 . When Γ acts freely and cocompactly on X, one may check that there is an equivalence of categories A c (X) Γ <1 ∼ = A[Γ] c (X/Γ) <1 ; note that by compactness, modules in A[Γ] c (X/Γ) <1 have finite support and hence all morphisms in A[Γ] c (X/Γ) <1 lift to bounded morphisms on X × [0, 1). When A is the category of finitely generated free R-modules for some ring R, A[Γ] is the category of finitely generated free R[Γ]-modules. If Γ acts properly discontinuously, there is also an equivalence of categories The following lemma identifies the codomain of this map in the case of interest to us. Lemma 7.5. If K is a compact metric space with diam(K) < ∞ and E is an additive category, then there are equivalences of categories A c (X) Γ / A c (X) Γ <1 ∼ = A ∞ (X/Γ)(52) E c (K) <1 ∼ = −→ E b (K) ∼ = E. Proof. The first equivalence is given by Lemma 7.4. Given x 0 ∈ K, the second equivalence is induced by the inclusion of categories E ∼ = E b ({x 0 }) ⊂ E b (K). This inclusion is an equivalence because compactness implies that any locally finite module M over K is in fact supported on a finite set S ⊂ K, and is isomorphic to the module x∈S M x considered as a module over {x 0 } (this isomorphism has finite propagation because diam(K) < ∞). (For proofs, see [10,15,27,28].) The boundary map for a fibration sequence of spectra can be realized (up to homotopy) as a map of spectra after looping the base spectrum, so we have a map (53) ΩKA ∞ (X) −→ KA c (X) <1 that induces the assembly map after taking fixed-point spectra and then homotopy groups. (We are using the fact that if C is an additive category with an action of a group G by additive functors, then K(C) G ∼ = K(C G ).) Remark 7.6. To be precise, the domain of (53) should be replaced by the homotopy fiber of the map KA c (X) <1 i → KA c (X); then the natural map hofib(i) → KA c (X) <1 is Γ-equivariant and induces the boundary map on homotopy groups. Moreover, since we are dealing with Ω-spectra, the homotopy fiber can be formed level-wise and one finds that hofib(i) Γ = hofib(i Γ ), where i Γ is the restriction of i to the fixed point spectra. The key ingredient in the proof of Theorem 1.1 will be a variation on Theorem 6.4. First, we need a simple lemma about homotopically finite classifying spaces of groups. Note that up to homotopy, there is no difference between assuming that a group admits a finite CW model for BΓ or a finite simplicial complex model, because every finite CW complex is homotopy equivalent to a finite simplicial complex. We have the following lemma. Lemma 7.7. If EΓ → BΓ is a universal principal bundle with BΓ a finite simplicial complex, then the simplicial metric d ∆ on EΓ (corresponding to the simplicial structure lifted from BΓ) is proper and EΓ is uniformly contractible with respect to d ∆ . The statement about uniform contractibility is a special case of Bartels-Rosenthal [3, Lemma 1.5]. Properness follows from the fact that EΓ is a locally finite simplicial complex (this is similar to the proof of Lemma 5.2). Theorem 7.8. Let Γ be a group with finite decomposition complexity, and assume that there exists a universal principal Γ-bundle EΓ → BΓ with BΓ a finite simplicial complex (this implies, in particular, that Γ is finitely generated). Equip EΓ with the simplicial metric corresponding to the simplicial structure lifted from BΓ. Then the category A c (EΓ) has trivial K-theory. Proof. This is similar to the proofs of [14,Lemma 4.3.6] and [3,Lemma 4.4]. We will construct continuous, proper, metrically coarse maps f s : EΓ → P s Γ, g s : P s Γ → EΓ for all sufficiently large s, having the property that each composition EΓ fs −→ P s Γ i ֒→ P s ′ Γ g s ′ −→ EΓ induces the identity on K * A c (EΓ). This suffices, since given any element x ∈ K * A c (EΓ), Theorem 6.4 guarantees that we can choose s ′ large enough that i * (f s ) * (x) = 0 in K * A c (P s ′ Γ); now x = (g s ′ ) * i * (f s ) * (x) = 0. Fix a vertex x 0 ∈ EΓ and consider the embedding Γ ֒→ EΓ, γ → γ · x 0 . The action of Γ on EΓ by deck transformations restricts to left multiplication on Γ, so the simplicial metric d ∆ on EΓ restricts to a proper, left-invariant metric d ∆ on Γ. If we equip Γ with the left-invariant metric d w associated to a finite generating set, then for each R > 0 there exists S > 0 such that d ∆ (γ, γ ′ ) < R implies d w (γ, γ ′ ) < S. In particular, letting D denote the diameter of BΓ = EΓ/Γ, there exists s > 0 such that d ∆ (γ, γ ′ ) < 2(D + 1) implies d w (γ, γ ′ ) < s. By choice of D, the sets U γ = B D+1 (γ · x 0 ) \ {γ ′ · x 0 : γ ′ = γ}. (γ ∈ Γ) form an open cover of EΓ. If {φ γ } γ∈Γ is a partition of unity subordinate to this cover, we can define f s : EΓ → P s Γ by the formula f s (x) = γ∈Γ φ γ (x)γ, Note that f s (x) is a well-defined point in P s Γ, by our choice of s. For each γ ∈ Γ, φ γ (γ · x 0 ) = 1 and hence f s (γ · x 0 ) = x 0 . The maps g s : P s Γ → EΓ (s = 0, 1, . . .) are defined by induction over the simplices in P s Γ. When s = 0, P 0 Γ = Γ and g 0 is just the embedding γ → γ · x 0 . Now assume that g s−1 has been defined (s > 0). Let P (k) s Γ denote the k-skeleton of P s Γ. Viewing P s−1 Γ as a subcomplex of P s Γ, we extend g s−1 inductively over the subcomplexes P (k) s Γ ∪ P s−1 (Γ). Assuming g s has been defined on the P (k−1) s Γ ∪ P s−1 (Γ) for some k 1, we extend over a k-simplex σ / ∈ P s−1 Γ as follows. Let D = diam(g s (∂σ)) and choose x ∈ g s (∂σ). By uniform contractibility of EΓ (Lemma 7.7) there exists D ′ > 0 (depending only on D) and a nullhomotopy of g s \| ∂σ whose image lies inside B D ′ (x). We now extend g s over σ using this nullhomotopy. One may now check that f s and g s are inverse coarse equivalences, hence metrically coarse and proper (since EΓ and P s Γ are proper). To show that g s • f s induces the identity map on continuously controlled K-theory, it suffices to show that this map is Lipschitz homotopic to the identity [4,Proposition 3.17], where a Lipschitz homotopy H: X × I → Y (with X and Y metric spaces) is simply a continuous, metrically coarse map for which {x ∈ X : H(x, t) ∈ C for some t ∈ I} is compact for all compact sets C ⊂ Y . Following Bartels-Rosenthal [3,Lemma 4.4], one constructs a homotopy H: EΓ × I → EΓ connecting g s • f s to Id EΓ by induction over the skeleta of EΓ × I, again using the uniform contractibility of EΓ. (Here it is most convenient to use the cell structure on EΓ × I in which cells are either of the form σ × {0}, σ × {1}, or σ × I, with σ a simplex in EΓ.) To see that H is metrically coarse, note that its restriction to the zero skeleton of EΓ × I is the disjoint union of g s f s and Id EΓ , hence is metrically coarse. Assuming H is metrically coarse on the k-skeleton, one checks metric coarseness on the (k + 1)-skeleton using the fact that there is a uniform bound D(k) on the diameter of H(σ) for σ a k-simplex (note that for 1simplices, this follows from the fact that g s f s is a bounded distance from the identity). For the remaining condition, it suffices to check that {x : d(H(x, t), γ · x 0 ) < R for some t ∈ I} is compact for each γ ∈ Γ, R > 0. This is similar: if x lies in a k-simplex, then d (H(x, t), g s f s (x)) D(k) and d(g s f s (x), x) S (for some constant S independent of x), so if d(H(x, t), γ · x 0 ) < R, we have d(x, γ · x 0 ) < S + D(k) + R, which suffices. Given an Ω-spectrum Y with a level-wise action of a group G, let Y hΓ denote the homotopy fixed point spectrum; that is, the function spectrum F G (EG + , Y ) consisting of (unbased) equivariant maps from EG to Y . The map (54) sits in a commutative diagram (57) (ΩK (A ∞ (EΓ))) Γ ∂ Γ / / i (K (A b (EΓ))) Γ j (ΩK (A ∞ (EΓ))) hΓ ∂ hΓ / / (K (A b (EΓ))) hΓ . The fact that EΓ/Γ = BΓ is a finite CW complex implies that i is a weak equivalence of spectra (see, for example, Carlsson-Pedersen [10, Theorem 2.11]). Theorem 7.8, together with the long exact sequence in homotopy associated to the Karoubi sequence A c (EΓ) ∼ = A c (EΓ) <1 ֒→ A c (EΓ) −→ A ∞ (EΓ), shows that the map (55) is a weak equivalence. It follows that the map ∂ hΓ in Diagram (57) is also a weak equivalence (every G-equivariant map between Ω-spectra with G-actions that is a weak equivalence, in the usual non-equivariant sense, induces a weak equivalence on homotopy fixed point spectra). Commutativity of (57) implies that the assembly map ∂ Γ in (54) is a split injection on homotopy, with splitting given by (i * ) −1 (∂ hΓ * ) −1 j * . ✷ As is usually the case in this area (see Bartels [4,Section 7], for example), Theorem 7.1 has an analogue for Ranicki's ultimate lower quadratic Ltheory spectrum L −∞ (A) of an additive category A with involution. Theorem 7.9. Let Γ be a group with finite decomposition complexity, and assume there exists a universal principal Γ-bundle EΓ → BΓ with BΓ a finite CW complex. Let A be an additive category with involution, and assume that for some r > 0 we have K * (A) = 0 for * < −r. Then the assembly map For our purposes, the key examples of Karoubi filtrations arise from restricting the support of geometric modules. Definition 3 . 5 . 35Given any family of subspaces Y of a proper metric space X we may consider the full subcategory A c (Y) ⊂ A c (X) on those modules supported on Y × [0, 1) for some Y ∈ Y. Note that A X+ c (Z) = A c ({N r (Z) : r ∈ N}) ⊂ A(X). The category A c (Y) is unchanged if we enlarge Y be adding subspaces of elements in Y, so we may always assume that our families are closed under taking subspaces. Construction 3 . 7 . 37Let Z ⊂ X be a subspace of the proper metric space X. For any M ∈ A c (X), the inclusions M (Z × [0, 1)) ֒→ M and M ((X \ Z) × [0, 1)) ֒→ M yield a direct sum decomposition Lemma 3 . 12 . 312Let S, B ⊂ A be full additive subcategories of the additive category A. Assume that: (1) the pairs (B, S ∩ B), (A, S), and (A, B) admit Karoubi filtrations; (2) (B, S) is dispersed. Then the full subcategory of A/S on the objects of B is precisely B/(S ∩ B), and the inclusion B/(S ∩ B) ⊂ A/S admits a Karoubi filtration. Lemma 3.12 (in other words, if (A 1 , S) and (A 12 , S 2 ) are dispersed, and all the relevant inclusions admit Karoubi filtrations), and if the pairs (A 1 , A 2 ) Definition 4 . 3 . 43Let X be a bounded geometry metric space and consider a sequence of subspaces U r (s) = α∈Ar U r α (s) and V r (s) = α∈Ar V r α (s), we have decomposed sequencesU s = (U 1 (s), U 2 (s), . . .) and V s (V 1 (s), V 2 (s), . . .),with decompositions given by(20); note that Z = U (s) ∪ V(s). On the other hand, we can also consider U s and V s as decomposed sequences under the finer decompositions U r (s) = α∈Ar i∈I(r,α) U r αi (s) and V r (s) = α∈Ar j∈J(r,α) r>R 0 α∈Ar P qr (Y r α ) × [0, 1)are identical, and hence [ψ 1 ] = [ψ 2 ]. αj < T + D for some j ∈ I(r, α) with j = i, contradicting (38). This shows that[ψ] = [ψ(S)] is in the image of Φ q,q ′ (T ). Finally, check that Φ q,q ′ (T ) is faithful. If Φ q,q ′ (T )([ψ 1 ]) = Φ q,q ′ (T )([ψ 2 ]), then for sufficiently large R 0 , the restrictions of ψ 1 and ψ 2 tor>R 0 α∈Ar P q,q ′ (Q r α , W r α ) × [0, 1)are identical, and hence [ψ 1 ] = [ψ 2 ]. , let U ′ and V ′ be the decomposed sequences U ′ = (U 1 , U 2 , . . .) and V ′ = (V 1 , V 2 , . . .), with decompositions ( this is essentially Carlsson-Pedersen [10, 2.8]), and (49) yieldsK * A c (X) Γ / A c (X) Γ <1 ∼ = H * −1 (X/Γ; KA).The boundary map for the long exact sequence in K-theory associated to (50) now has the form(51) H * (X/Γ; KA) −→ K * (A[Γ] c (X/Γ) <1 ) . Under the isomorphism induced by (52), the map (51) agrees with the classical assembly mapH * (X/Γ; KA) −→ K * (A[Γ]) . Theorem 1.1 can now be proven exactly as in Bartels' proof for groups with finite asymptotic dimension [4, Theorems 5.3 and 6.5]. For convenience of the reader, we recall the argument.Proof of Theorem 1.1. As explained above, the assembly mapH * (BΓ; K(A[Γ])) −→ K * A[Γ]can be realized (up to homotopy) as the map of fixed-point spectra(54) (ΩK (A ∞ (EΓ))) Γ ∂ Γ −→ (K (A c (EΓ) <1 )) Γassociated to a map of spectra(55) ΩK (A ∞ (EΓ)) ∂ −→ K (A c (EΓ) <1 )that induces, on homotopy groups, the K-theoretic boundary map for H * (BΓ; L −∞ (A)) −→ K * (A[Γ]), is a split injection for all * ∈ Z. The proof is analogous to that of Theorem 7.1. The relevant tools for L-theory are provided in Carlsson-Pedersen [10, Section 4]. The additional condition on K * (A) is needed in order to apply the L-theoretic analogue of Carlsson-Pedersen [10, Theorem 2.11] (see [10, Theorem 5.5]). where the first neighborhood is taken inside P sr,s ′r (X, W r tαi ). From the defi- nitions of W r tαij and W r tαi , we have (44) N (T +2)Cr sr (U r αi ∪ W r tαi ) ⊂ N (T +t+2)Cr sr (U r αi ). Combining (42), (43), and (44) yields P sr,s ′ r (Z r α , W r tαi ) ∩ N T P sr,s ′ r (U r αi , W r tαi ) (45) since the other terms in the long exact sequence (48) vanish when X = * (see, for example Bartels[4, 3.20]), and K * −1 (A c ({ * }) <1 ) ∼ = K * −1 A by Lemma 7.4 below. Details can be found in the above references; see[28, Section 5] in particular. Remark 7.3. Weiss[28] uses a somewhat different description of the category A ∞ (X). He describes the morphisms as "germs" of morphisms in A c (X). It is easy to check, however, that Weiss's germ category is the same as the Karoubi quotient A ∞ (X). Additionally, Weiss works with the idempotent completion of his germ category. This does not affect the results though, since the non-connective K-theory spectrum of an additive category A is weakly equivalent to that for its idempotent completion A ∧ : this follows from Pedersen-Weibel [20, Lemmas 1.4.2 and 2.3].Lemma 7.4. For every proper metric space X there is an equivalence P R s,s ′ (U ′ , W ′ t 0 ) := P s(R),s ′ (R) (U ′ (R), W ′ t 0 (R)), and the other superscripts on the Rips complexes should be interpreted similarly. The isomorphisms labelled i are those from Lemma 6.16. Furthermore, we have set τ = t 0 + T + 2.(47)Diagram (47) exists for each t 0 , T > 0, in the sense that we may choose natural numbers R = R(T, t 0 ) and R t (for each t > 0) such that all the maps exist. Specifically, for each t > 0, choose R t large enough that Φ s ′′ exits (where Φ s ′′ is the colimit over t of the maps constructed in the proof of Lemma 6.17) and then choose R R τ large enough that Φ t 0 s,s ′ (T ) exists, where Φ t 0 s,s ′ (T ) is the map constructed in the proof of Lemma 6.18. After passing to K-theory, the proof of Lemma 6.18 also gives the maps Ψ t 0 s,s ′ (T ) and Ψ t 0 s,s ′ appearing on the right-hand side of the diagram. 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Math. 1391Guoliang Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math., 139(1):201-240, 2000. . P.O. Box. 30001New Mexico State University, Department of Mathematical SciencesMexico 88003-8001 U.S.A. E-mail address: [email protected] Mexico State University, Department of Mathematical Sciences, P.O. Box 30001, Department 3MB, Las Cruces, New Mexico 88003-8001 U.S.A. E-mail address: [email protected]	[]
[ "Birationally rigid Fano fibre spaces. II", "Birationally rigid Fano fibre spaces. II" ]	[ "A V Pukhlikov " ]	[]	[]	In this paper we prove birational rigidity of large classes of Fano-Mori fibre spaces over a base of arbitrary dimension, bounded from above by a constant that depends on the dimension of the fibre only. In order to do that, we first show that if every fibre of a Fano-Mori fibre space satisfies certain natural conditions, then every birational map onto another Fano-Mori fibre space is fibre-wise. After that we construct large classes of fibre spaces (into Fano double spaces of index one and into Fano hypersurfaces of index one) which satisfy those conditions. Bibliography: 35 titles.	10.1070/im2015v079n04abeh002762	[ "https://arxiv.org/pdf/1407.0687v1.pdf" ]	56,133,964	1407.0687	e079df353a6c56dce0fb5196afa7d913d2fa72a4	Birationally rigid Fano fibre spaces. II 2 Jul 2014 A V Pukhlikov Birationally rigid Fano fibre spaces. II 2 Jul 2014 In this paper we prove birational rigidity of large classes of Fano-Mori fibre spaces over a base of arbitrary dimension, bounded from above by a constant that depends on the dimension of the fibre only. In order to do that, we first show that if every fibre of a Fano-Mori fibre space satisfies certain natural conditions, then every birational map onto another Fano-Mori fibre space is fibre-wise. After that we construct large classes of fibre spaces (into Fano double spaces of index one and into Fano hypersurfaces of index one) which satisfy those conditions. Bibliography: 35 titles. Introduction 0.1. Birationally rigid Fano-Mori fibre spaces. In this paper we investigate the problem of birational rigidity of Fano-Mori fibre spaces π: V → S. We assume that the base S is non-singular, the variety V has at most factorial terminal singularities, the anticanonical class (−K V ) is relatively ample and Pic V = ZK V ⊕ π * Pic S. Let π ′ : V ′ → S ′ be an arbitrary rationally connected fibre space, that is, a morphism of projective algebraic varieties, where the base S ′ and the fibre of general position π ′ −1 (s ′ ), s ′ ∈ S ′ , are rationally connected and dim V = dim V ′ . Consider a birational map χ: V V ′ (provided they exist). In order to describe the properties of the map χ, of crucial importance is whether χ is fibre-wise or not, that is, whether this map transforms the fibres of the projection π into the fibres of the projection π ′ . It is expected (and confirmed by all known examples, see subsection 0. 6), that the answer is positive if the fibre space π is "sufficiently twisted over the base". Investigating this problem, one can choose various classes of Fano-Mori fibre space and various interpretations of the property to be "twisted over the base". In the present paper we prove the following fact. Theorem 1. Assume that the Fano-Mori fibre space π: V → S satisfies the conditions (i) every fibre F s = π −1 (s), s ∈ S, is a factorial Fano variety with terminal singularities and the Picard group Pic F s = ZK Fs , (ii) for every effective divisor D ∈ \| − nK Fs \| on an arbitrary fibre F s the pair (F s , 1 n D) is log canonical, and for every mobile linear system Σ s ⊂ \| − nK Fs \| the pair (F s , 1 n D) is canonical for a general divisor D ∈ Σ s , (iii) for every mobile family C of curves on the base S, sweeping out S, and a curve C ∈ C the class of the following algebraic cycle of dimension dim F for any positive N 1 −N(K V • π −1 (C)) − F (where F is the fibre of the projection π) is not effective, that is, it is not rationally equivalent to an effective cycle of dimension dim F . Then every birational map χ: V V ′ onto the total space of a rationally connected fibre space V ′ /S ′ is fibre-wise, that is, there exists a rational dominant map β: S S ′ , such that the following diagram commutes V χ V ′ π ↓ ↓ π ′ S β S ′ . Now we list the standard implications of Theorem 1, after which we discuss the point of how restrictive the conditions (i)-(iii) are. Corollary 1. In the assumptions of Theorem 1 on the variety V there are no structures of a rationally connected fibre space over a base of dimension higher than dim S. In particular, the variety V is non-rational. Every birational self-map of the variety V is fibre-wise and induces a birational self-map of the base S, so that there is a natural homomorphism of groups ρ: Bir V → Bir S, the kernel of which Ker ρ is the group Bir F η = Bir(V /S) of birational self-maps of the generic fibre F η (over the generic non-closed point η of the base S), whereas the group Bir V is an extension of the normal subgroup Bir F η by the group Γ = ρ(Bir V ) ⊂ Bir S: 1 → Bir F η → Bir V → Γ → 1. How restrictive are the conditions (i)-(iii)? The condition (iii) belongs to the same class of conditions as the well known K 2 -condition and the K-condition for fibrations over P 1 (see, for instance, [31,Chapter 4]) and the Sarkisov condition for conic bundles (see [32,33]). This condition measures the "degree of twistedness" of the fibre space V /S over the base S. Below we illustrate this meaning of the condition (iii) by particular examples. We will see that this condition is not too restrictive: for a fixed method of constructing the fibre space V /S and a fixed "ambient" fibre space X/S the condition (iii) is satisfied by "almost all" families of fibre spaces V /S. In terms of numerical geometry of the varieties V and S the condition (iii) can be expressed in the following way. Let A * (V ) = dim V i=0 A i (V ) be the numerical Chow ring of the variety V , graded by codimension. Set A i (V ) = A dim V −i (V ) ⊗ R and denote by the symbol A mov i (V ) the closed cone in A i (V ), generated by the classes of mobile cycles, the families of which sweep out V , and by the symbol A + i (V ) the pseudoeffective cone in A i (V ), generated by the classes of effective cycles. Furthermore, by the symbol A i, j (V ) we denote the linear subspace in A i (V ), generated by the classes of subvarieties of dimension i, the image of which on S has dimension at most j. In the real space A i, j (V ) consider the closed cones A mov i, j (V ) of mobile and A + i, j (V ) of pseudoeffective classes. In a similar way we define the real vector space A i (S) and the closed cones A mov i (S) A + i (S). If δ = dim F is the dimension of the fibre of the projection π, then the operation of taking the preimage generates a linear map π * A i (S) → A δ+i, i (V ), whereas π * (A + i (S)) ⊂ A + δ+i, i (V ) and π * (A mov i (S)) ⊂ A mov δ+i, i (V ). Now let us consider the linear map γ: A 1 (S) → A δ, 1 (V ), defined by the formula z → −(K V · π * z). The condition (iii) means that the image of the cone γ(A mov 1 (S)) is contained in the boundary of the pseudoeffective cone A + δ, 1 (V ), that is, γ(A mov 1 (S)) ∩ Int A + δ, 1 (V ) = ∅. More precisely, for any class z ∈ A mov 1 (S) the intersection of the closed ray {γ(z) − t[F ] \| t ∈ R + } (where [F ] ∈ Int A + δ, 1 (V ) is the class of the fibre of the projection π) with the cone A + δ, 1 (V ) either is empty or consists of just one point γ(z). One may suggest that the condition (iii) is close to a precise one ("if and only if"), that is, its violation (or an essential deviation from this condition) implies the existence of another structure of a Fano-Mori fibre space on the variety V . The following remark gives an obvious way to check the condition (iii). Remark 0.1. Assume that on the variety V there is a numerically effective divisorial class L such that (L δ · F ) > 0 and the linear function (·L δ ) is non-positive on the cone γ(A mov 1 (S)), that is to say, for any mobile curve C on S the inequality L δ · K V · π −1 (C) 0(1) holds. Then the condition (iii) is obviously satisfied. The conditions (i) and (ii), however, are much more restrictive. They mean that all fibres of the projection π are varieties of sufficiently general position in their family. This implies that the dimension of the base for a fixed family of fibres is bounded from above (by a constant depending on the particular family, to which the fibres belong). In the examples considered in the present paper, for a sufficiently high dimension of the fibre δ = dim F the dimension of the base is bounded from above by a number of order 1 2 δ 2 . Recall that up to now not a single example was known of a fibration into higher dimensional Fano varieties over a base of dimension two and higher with just one structure of a rationally connected fibre space (for a brief historical survey, see subsection 0.5). 0.2. Fibrations into double spaces of index one. By the symbol P we denote the projective space P M , M 5. Let W = P(H 0 (P, O P (2M))) be the space of hypersurfaces of degree 2M in P. The following fact is true. Theorem 2. There exists a Zariski open subset W reg ⊂ W, such that for any hypersurface W ∈ W reg the double cover σ: F → P, branched over W , satisfies the conditions (i) and (ii) of Theorem 1, and moreover, the estimate codim((W \ W reg ) ⊂ W) (M − 4)(M − 1) 2 holds. An explicit description of the set W reg and a proof of Theorem 2 are given in §2. Fix a number M 5 and a non-singular rationally connected variety S of dimension dim S < 1 2 (M − 4)(M − 1). Let L be a locally free sheaf of rank M + 1 on S and X = P(L) = Proj ∞ ⊕ i=0 L ⊗i the corresponding P M -bundle. We may assume that L is generated by its sections, so that the sheaf O P(L) (1) is also generated by the sections. Let L ∈ Pic X be the class of that sheaf, so that Pic X = ZL ⊕ π * X Pic S, where π X : X → S is the natural projection. Take a general divisor U ∈ \|2(ML + π * X R)\|, where R ∈ Pic S is some class. If this system is sufficiently mobile, then by the assumption about the dimension of the base S and Theorem 2 we may assume that for every point s ∈ S the hypersurface U s = U ∩ π −1 X (s) ∈ W reg , and for that reason the double space branched over U s , satisfies the conditions (i) and (ii) of Theorem 1. Let σ: V → X the double cover branched over U. Set π = π X •σ: V → S, so that V is a fibration into Fano double spaces of index one over S. Recall that the divisor U ∈ \|2(ML + π * X R)\| is assumed to be sufficiently general. Theorem 3. Assume that the divisorial class (K S + R) is pseudoeffective. Then for the fibre space π: V → S the claims of Theorem 1 and Corollary 1 hold. In particular, Bir V = Aut V = Z/2Z is the cyclic group of order 2. Proof. Since the conditions (i) and (ii) of Theorem 1 are satisfied by construction of the variety V , it remains to check the condition (iii). Let us use Remark 0.1. Elementary computations show that the inequality (1) up to a positive factor is the inequality ((K S + R) · C) 0. Since the curve C belongs to a mobile family, sweeping out the base S, the last inequality holds if the class (K S + R) is pseudoeffective. Q.E.D. for the theorem. Example 0.2. Take S = P m , where m < 1 2 (M − 4)(M − 1), X = P M × P m and W X is a generic hypersurface of bidegree (2M, 2l), where l m + 1. Then for the double cover σ: V → X, branched over W X , the claims of Theorem 1 and Corollary 1 are true. Note that for l m on the double cover V there is another structure of a Fano fibre space: it is given by the projection π 1 : V → P M . Therefore, the condition (iii) of Theorem 1 and its realization in Theorem 3 turn out to be precise. Theorem 4. There is a Zariski open subset F reg ⊂ F , such that every hypersurface F ∈ F reg satisfies the conditions (i) and (ii) of Theorem 1, and the following estimate holds: codim((F \ F reg ) ⊂ F ) (M − 7)(M − 6) 2 − 5.(2) An explicit description of the subset F reg and a proof of Theorem 4 are given in §2-3. Fix a non-singular rationally connected variety S of dimension dim S < 1 2 (M − 7)(M − 6) − 5. As in subsection 0.2, let L be a locally free sheaf of rank M +1 on S and X = P(L) = Proj ∞ ⊕ i=0 L ⊗i the corresponding P M -bundle in the sense of Grothendieck; we assume that L is generated by global sections. Let π X : X → S be the projection, L ∈ Pic X the class of the sheaf O P(L) (1). Consider a general divisor V ∈ \|ML + π * X R\|, where R ∈ Pic S is some divisor on the base. By the assumption about the dimension of the base made above and Theorem 4 we may assume that the Fano fibre space π: V → S, where π = π X \| V , satisfies the conditions (i) and (ii) of Theorem 1. Theorem 5. Assume that the divisorial class (K S + 1 − 1 M R) is pseudoeffective. Then for the Fano fibre space π: V → S the claims of Theorem 1 and Corollary are true. In particular, the group Then the Fano fibre space V /P m satisfies all assumptions of Theorem 1 and therefore for this fibre space the claim of Theorem 1 and that of Corollary 1 are true. Note that for l m on the variety V there is another structure of a Fano fibre space, given by the projection V → P M . Note also that if we fix the dimension m of the base, then for M m the condition of Theorem 5 is close to the optimal one: it is satisfied for l m + 2, so that the only value of the integral parameter l, for which the problem of birational rigidity of the fibre space V /P m remains open, is l = m+1. In that case the projection V → P M is a K-trivial fibre space. 0.4. The structure of the paper. The present paper is organized in the following way. In §1 we prove Theorem 1. After that, in §2 we deal with the conditions of general position, which should be satisfied for every fibre of the fibre space V /S in order for the conditions (i) and (ii) of Theorem 1 to hold. The conditions of general position (regularity) are given for Fano double spaces of index one and Fano hypersurfaces of index one. This makes it possible to define the sets W reg and F reg and prove Theorem 2 and carry out the preparational work for the proof of Theorem 4, the main technical fact of the present paper, which implies Theorem 5, geometrically the most impressive result of this paper, in an obvious way. Bir V = Aut V is trivial. In §3 we complete the proof of Theorem 4, more precisely we show that the condition (ii) of Theorem 1 is satisfied for a regular Fano hypersurface F ∈ F reg . The proof makes use a combination of the technique of hypertangent divisors and the inversion of adjunction. Note that the approach of the present paper corresponds to the linear method of proving birational rigidity, see [31,Chapter 7]; the technique of the quadratic method (in the first place, the technique of counting multiplicities) is not used. The assumption in Theorem 1 that the base S of the fibre space V /S is nonsingular seems to be unnecessary and could be replaced by the condition that the singularities are at most terminal and (Q-)factorial. 0.5. Historical remarks and acknowledgements. The starting point of studying birational geometry of rationally connected fibre spaces seems to be the use of de Jonqiere transformations (see, for instance, [11]). In the modern algebraic geometry the objects of this type started to be systematically investigated in the works of V.A.Iskovskikh and M.Kh.Gizatullin about pencils of rational curves [12,13,6] over non-closed fields, which followed the investigation of the "absolute" case in the papers of Yu.I.Manin [16,17,18]. We also point out the paper of I.V.Dolgachev [4], which started (in the modern period) the study of K-trivial fibrations. After the breakthrough in three-dimensional birational geometry that was made in the classical paper of V.A.Iskovskikh and Yu.I.Manin on the three-dimensional quartic [14] the problems of the "relative" three-dimensional birational geometry were the next to be investigated, that is, the task was to describe birational maps of three-dimensional algebraic varieties, fibred into conics over a rational surface or into del Pezzo surfaces over P 1 . The famous Sarkisov theorem gave an almost complete solution of the question of birational rigidity for conic bundles [32,33]. A similar question for the pencils of del Pezzo surfaces remained absolutely open until 1996 [19]; see the introduction to the last paper about the reasons of those difficulties (the test class construction turned out to be unsuitable for studying the varieties of that type). The method of proving birational rigidity, realized in [19], generalized well into the arbitrary dimension, for varieties fibred into Fano varieties over P 1 . In a long series of papers [21,34,35,23,24,26,27,29] birational rigidity was shown for many classes of Fano fibre space over P 1 . At the same time, the birational geometry of the remaining families of three-dimensional varieties with a pencil of del Pezzo surfaces of degree 1 and 2 was investigated [7,8,9,10]; in that direction the results that were obtained were nearly exhaustive. However, the base of the fibre spaces under investigation remained one-dimensional and even Fano fibrations over surfaces seemed to be out of reach. The only exception in that series of results was the theorem about Fano direct products [25] and the papers about direct products that followed [28,2]. In those papers the Fano fibre spaces under consideration had the both the base and the fibre of arbitrary dimension. However, the fibre spaces themselves were very special (direct products) and could not pretend to be typical Fano fibre spaces. The present paper gives, at long last, numerous examples of typical birationally rigid Fano fibre spaces with the base and fibre of high dimension (for a fixed dimension of the fibre δ the dimension of the base is bounded by a constant of order ∼ 1 2 δ 2 ). Theorem 1 can be viewed as a realization of the well known principle: the "sufficient twistedness" of a fibre space over the base implies birational rigidity. This principle was many times confirmed in the class of fibrations over P 1 ; now it is extended to the the class of fibre spaces over a base of arbitrary dimension. The main object of study in this paper is a fibre space into Fano hypersurfaces of index one, so that it is a follow up of the paper [21]. From the technical viewpoint, the predecessors of this paper are [28,30], where the linear method of proving birational rigidity was developed. It is possible, however, that the quadratic techniques could be applied to the class of Fano fibre spaces over a base of arbitrary dimension as well. Various technical moments related to the arguments of the present paper were discussed by the author in his talks given in 2009-2014 at Steklov Institute of Mathematics. The author is grateful to the members of the divisions of Algebraic Geometry and Algebra and Number Theory for the interest in his work. The author also thanks his colleagues in the Algebraic Geometry research group at the University of Liverpool for the creative atmosphere and general support. Birationally rigid fibre spaces In this section we prove Theorem 1. We do it in three steps: first, assuming that the birational map χ: V V ′ is not fibre-wise, we prove the existence of a maximal singularity of the map χ, covering the base S ′ (subsection 1.1). After that, we construct such a sequence of blow ups of the base S + → S, that the image of every maximal singularity on S is a prime divisor (subsection 1.2). Finally, using a very mobile family of curves contracted by the projection π ′ , we obtain a contradiction with the condition (iii) of Theorem 1 (subsection 1.3). This implies that the map χ: V V ′ is fibre-wise, which completes the proof of Theorem 1. Maximal singularities of birational maps. In the notations of Theorem 1 fix a birational map χ: V V ′ onto the total space V ′ of a rationally connected fibre space π ′ : V ′ → S ′ . Consider any very ample linear system Σ ′ on S ′ . Let Σ ′ = (π ′ ) * Σ ′ be its pull back on V ′ , so that the divisors D ′ ∈ Σ ′ are composed from the fibres of the projection π ′ , and for that reason for any curve C ⊂ V ′ , contracted by the projection π ′ we have (D ′ · C) = 0. The linear system Σ ′ is obviously mobile. Let Σ = (χ −1 ) * Σ ′ ⊂ \| − nK V + π * Y \| be its strict transform on V , where n ∈ Z + . Lemma 1.1. For any mobile family of curves C ∈ C on S, sweeping out S, the inequality (C · Y ) 0 holds, that is to say, the numerical class of the divisor Y is non-negative on the cone A mov 1 (S). Proof. This is almost obvious. For a general divisor D ∈ Σ the cycle (D • π −1 (C)) is effective. Its class is −n(K V •π −1 (C))+(Y ·C)F , so that by the condition (iii) the claim of the lemma follows. Q.E.D. Obviously, the map χ is fibre-wise if and only if n = 0. Therefore, if n = 0, then the claim of Theorem 1 holds. So let us assume that n ≥ 1 and show that this assumption leads to a contradiction. The linear system Σ is mobile. Let us resolve the singularities of the map χ: let ϕ: V → V be a birational morphism (a composition of blow ups with non-singular centres), where V is non-singular and the composition χ•ϕ: V V ′ is regular. Furthermore, consider the set E of prime divisors on V , satisfying the following conditions: • every divisor E ∈ E is ϕ-exceptional, • for every E ∈ E the closed set χ • ϕ(E) ⊂ V ′ is a prime divisor on V ′ , • the set χ • ϕ(E) for every E ∈ E covers the base: π ′ [χ • ϕ(E)] = S ′ . Setting K = K V , write down Σ ⊂ \| − n K + (π * Y − E∈E ε(E)E) + Ξ\|, where Σ, as usual, is the strict transform of the mobile linear system Σ on V , ε(E) ∈ Z is some coefficient and Ξ stands for a linear combination of ϕ-exceptional divisors which do not belong to the set E. Definition 1.1. An exceptional divisor E ∈ E is called a maximal singularity of the map χ, if ε(E) > 0. Obviously, a maximal singularity satisfies the Noether-Fano inequality ord E ϕ * Σ > na(E), where a(E) = a(E, V ) is the discrepancy of the divisor E with respect to V . In this paper we somewhat modify the standard concept of a maximal singularity, requiring in addition that it is realized by a divisor on V ′ , covering the base. Let M ⊂ E be the set of all maximal singularities. Proposition 1.1. Maximal singularities do exist: M = ∅. Proof. Assume the converse, that is, for any E ∈ E the inequality ε(E) ≤ 0 holds. Let C ′ be a family of rational curves on V ′ , satisfying the following conditions: • the curves C ′ ∈ C ′ are contracted by the projection π ′ , • the curves C ′ ∈ C ′ sweep out a dense open subset in V ′ , • the curves C ′ ∈ C ′ do not intersect the set of points where the rational map (χ • ϕ) −1 : V ′ V is not well defined. Apart from that, we assume that a general curve C ′ ∈ C ′ intersects every divisor χ • ϕ(E), E ∈ E, transversally at points of general position. Such a family of curves we will call very mobile. Obviously, very mobile families of rational curves do exist. Let C ∼ = C ′ be the inverse image of the curve C ′ ∈ C ′ on V . Since the linear system Σ ′ is pulled back from the base, for a divisor D ∈ Σ we have the equality ( C · D) = 0. On the other hand, ( C · K) = (C ′ · K V ′ ) < 0 and ( C · (π * Y − E∈E ε(E)E)) 0, since by the condition (iii) of our theorem ( C ·π * Y ) 0 and by assumption −ε(E) ∈ Z + for all E ∈ E. Finally, the divisor Ξ (which is not necessarily effective) is a linear combination of such ϕ-exceptional divisors R ⊂ V , that π ′ [χ • ϕ(R)] is a proper closed subset of the base S ′ . So we have the equality ( C · Ξ) = 0. This implies that ( C · D) n > 0, which is a contradiction. Therefore, M = ∅. Q.E.D. for the proposition. Proposition 1.2. For any maximal singularity E ⊂ M its center centre(E, V ) = ϕ(E) on V does not cover the base: π(centre(E, V )) ⊂ S is a proper closed subset of the variety S. Proof. Assume the converse: the centre of some maximal singularity E ∈ M covers the base: π(centre(E, V )) = S. Let F = π −1 (s), s ∈ S be a fibre of general position. By assumption the strict transform F of the fibre F on V has a nonempty intersection with E, and for that reason every irreducible component of the intersection F ∩ E is a maximal singularity of the mobile linear system Σ F = Σ\| F ⊂ \| − nK F \|. However, by the condition (ii) of Theorem 1 on the variety F there are no mobile linear systems with a maximal singularity. This contradiction proves the proposition. 1.2. The birational modification of the base of the fibre space V /S. Now let us construct a sequence of blow ups of the base, the composition of which is a birational morphism σ S : S + → S, and the corresponding sequence of blow ups of the variety V , the composition of which is a birational morphism σ: V + → V , where V + = V × S S + , so that the following diagram commutes V + σ → V π + ↓ ↓ π S + σ S → S. The birational morphism σ S is constructed inductively as a composition of elementary blow ups σ i : S i → S i−1 , i = 1, . . ., where S 0 = S. Assume that σ i are already constructed for i k (if k = 0, then we start with the base S). Set V k = V × S S k and let π k : V k → S k be the projection. Consider the irreducible closed subsets π k (centre(E, V k )) ⊂ S k ,(3) where E runs through the set M. By Proposition 1.2, all these subsets are proper subsets of the base S k . If all of them are prime divisors on S k , we stop the procedure: set S + = S k and V + = V k . Otherwise, for σ k+1 we take the blow up of any inclusionminimal set (3) for all E ∈ M. It is easy to check that the sequence of blow ups σ terminates. Indeed, set α k = E∈M a(E, V k ). Since the birational morphism σ k : V k → V k−1 is the blow up of a closed irreducible subset, containing the centre of one of the divisors E ∈ M on V k−1 , we get the inequality α k+1 < α k . The numbers α i are by construction non-negative, which implies that the sequence of blow ups σ i is finite. Therefore, for any maximal singularity E ∈ M the closed subset π + (centre(E, V + )) ⊂ S + is a prime divisor. 1.3. The mobile family of curves. Again let us consider a very mobile family of curves C ′ on V ′ and its strict transform C + on V + . Let C + ∈ C + be a general curve and C + = π + (C + ) the corresponding curve of the family C + on S + . Furthermore, let Σ + be the strict transform of the linear system Σ on V + . For some class of divisors Y + on S + we have: Σ + ⊂ \| − nK + + π * + Y + \|, where for simplicity of notation K + = K V + . Note that even if Y is an effective or mobile class on S, in this case Y + is not its strict transform on S + , that is to say, we violate the principle of notations. The following observation is crucial. Proposition 1.3. The inequality (C + · Y + ) < 0 holds. In particular, the class Y + is not pseudoeffective. Proof. Assume the converse: (C + · π * Y + ) = (C + · Y + ) 0. We may assume that the resolution of singularities ϕ of the map χ filters through the sequence of blow ups σ: V + → V , so that for the strict transform Σ of the linear system Σ on V we have Σ ⊂ \| − n K + (π * + Y + − E∈E ε(E)E) + Ξ\|, where K = K V , ε(E) ∈ Z and Ξ is a linear combination of exceptional divisors of the birational morphism V → V + , which are not in the set E. For the strict transform C ∈ C of the curve C + ∈ C + and the divisor D ∈ Σ we have, as in the proof of Proposition 1.1, the equality ( C · D) = 0. By the construction of the divisor Ξ we have ( C · Ξ) = 0. Finally, ( C · K) < 0, whence we conclude that ( C · (π * + Y + − E∈E ε(E)E) < 0. By our assumption for at least one divisor E ∈ E we have the inequality ε(E) > 0. This divisor is automatically a maximal singularity, E ∈ M. By our construction, however, we can say more: E is a maximal singularity for the mobile linear system Σ + as well, that is, the pair V + , 1 n Σ + is not canonical and E realizes a noncanonical singularity of that pair. However, π + (centre(E, V + )) = E ⊂ S + is a prime divisor, so that π −1 + (E) ⊂ V + is also a prime divisor. The linear system Σ + has no fixed components, therefore for a general point s ∈ E and the corresponding fibre F = π −1 + (s) ⊂ V + we have: the linear system Σ F = Σ + \| F ⊂ \| − nK F \| is non-empty and for D F ∈ Σ F the pair F, 1 n D F is non log canonical by the inversion of adjunction (see [15]). This contradicts the condition (ii) of our theorem. Proposition 1.3 is shown. Q.E.D. Finally, let us complete the proof of Theorem 1. Let us write down explicitly the divisor π * + Y + in terms of the partial resolution σ. Let E + be the set of all exceptional divisors of the morphism σ, the image of which on V ′ is a divisor and covers the base S ′ . Therefore, E + can be identified with a subset of the set E. In the course of the proof of Proposition 1.3 we established that M + = M ∩ E + = ∅. Now we write π * + Y + = π * Y − E∈E + ε + (E)E + Ξ + . Besides, we have K + = σ * K V + E∈E + a + (E)E + Ξ K , where all coefficients a + (E) are positive and the divisor Ξ K is effective, pulled back from the base S + and the image of each of its irreducible component on V ′ has codimension at least 2, so that the general curve C + ∈ C + does not intersect the support of the divisor Ξ K . Let C ∈ C be its image on the original variety V and C = π(C) ∈ C the projection of the curve C on the base S. For a general divisor D ∈ Σ and its strict transform D + ∈ Σ + on V + the scheme-theoretic intersection (D + • π −1 + (C + )) is well defined, it is an effective cycle of dimension δ = dim F on V + . For its numerical class we have the presentation (D + • π −1 + (C + )) ∼ −n(σ * K V • π −1 + (C + ))+ + E∈E + (−na + (E) − ε + (E))E · C + F.(4) Since (C · Y ) 0 and (C + · π * + Y + ) < 0, we have − E∈E + ε + (E)E · C + < 0, so that in the formula (4) the intersection of the divisor in square brackets with C + is negative. Therefore, σ * (D + • π −1 + (C + )) ∼ −n(K V • π −1 (C)) + bF, where b < 0. Since on the left we have an effective cycle of dimension δ on V , we obtain a contradiction with the condition (iii) of our theorem. Proof of Theorem 1 is complete. Q.E.D. Varieties of general position In this section we state the explicit local conditions of general position for the double spaces (subsection 2.1) and hypersurfaces (subsection 2.2), defining the sets W reg ⊂ W and F reg ⊂ F . In subsection 2.1 we prove Theorem 2. In subsection 2.3-2.5 we prove a part of the claim of Theorem 4: the estimate for the codimension of the complement F \ F reg ; in subsection 2.5 we also consider some immediate geometric implications of the conditions of general position. Proof is obvious Q.E.D. Now we define the subset W reg ⊂ W, requiring that W ∈ W reg satisfies the condition (W1) at every non-singular and the condition (W2) at every singular point. Obviously, W reg ⊂ W is a Zariski open subset (possibly, empty). Proposition 2.3. The following estimate holds: codim((W \ W reg ) ⊂ W) (M − 4)(M − 1) 2 . Proof is obtained by the standard arguments, see [31,Chapter 3]: one considers the incidence subvariety I = {(o, W ) \| o ∈ W } ⊂ P × W; for a fixed point o ∈ P the codimension of the set of hypersurfaces W non−reg (o), containing that point and non-regular in it, is given by Propositions 2.1 and 2.2 (in the singular case M more independent conditions are added as q 1 ≡ 0). After that one computes the dimension of the set I non−reg = o∈P {o} × W non−reg (o) and considers the projection onto W. This completes the proof. Q.E.D. Obviously, for any hypersurface W ∈ W reg the double cover F → P, branched over W , is an irreducible algebraic variety. Moreover, by the condition (W2) the variety F belongs to the class of varieties with quadratic singularities of rank at least 5 [5]. Recall that a variety X is a variety with quadratic singularities of rank at least r, if in a neighborhood of every point o ∈ X the variety X can be realized as a hypersurface in a non-singular variety Y, and the local equation X at the point o is of the form β 1 (u * ) + β 2 (u * ) + . . . = 0, where (u * ) is a system of local parameters at the point o ∈ Y, and either β 1 ≡ 0, or β 1 ≡ 0 and rk β 2 r. It is clear that codim(Sing X ⊂ X ) r − 1, so that the variety F is factorial [1]. Furthermore, it is easy to show (see [5]), that the class of quadratic singularities of rank at least r is stable with respect to blow ups in the following sense. Let B ⊂ X be an irreducible subvariety. Then there exists an open set U ⊂ Y, such that U ∩ B = ∅, U ∩ B is a non-singular algebraic variety and for its blow up σ B : U + → U we have that (X ∩ U) + ⊂ U + is a variety of quadratic singularities of rank at least r. In order to see this, note the following simple fact: if Z ∋ o is a non-singular divisor on Y, where Z = X and the scheme-theoretic restriction X \| Z has at the point o a quadratic singularity of rank l, then X has at the point o a quadratic singularity of rank at least l. Now if B ⊂ Sing X , then the claim about stability is obvious. Therefore, we may assume that B ⊂ Sing X . The open set U ⊂ Y can be chosen in such a way that B ∩ U is a non-singular subvariety and the rank of quadratic points o ∈ B ∩ U is constant and equal to l r. But then in the exceptional divisor E = σ −1 B (B ∩ U) the divisor (X ∩ U) + ∩ E is a fibration into quadrics of rank l, so that (X ∩ U) + ∩ E has at most quadratic singularities of rank at least l. Therefore, (X ∩ U) + ⊂ U + has quadratic singularities of rank at least r as well, according to the remark above. For an explicit analytic proof, see [5]. The stability with respect to blow ups implies that the singularities of the variety F are terminal (for the particular case of one blow up it is obvious: the discrepancy of an irreducible exceptional divisor (X ∩ U) + ∩ E with respect to X is positive; every exceptional divisor over X can be realized by a sequence of blow ups of the centres). Finally, F satisfies the condition (ii) of Theorem 1, that is, the condition of divisorial canonicity, see the proof of part (ii) of Theorem 2 in [25] and Theorem 4 in [30]. This completes the proof of Theorem 2. Q.E.D. on the quadric {q 2 = 0} is not a sum of three (not necessarily distinct) hyperplane sections of this quadric, taken from the same linear pencil. Now arguing in the word for word the same way as in subsection . 2.1, we conclude that any hypersurface F ∈ F reg is an irreducible projective variety with factorial terminal singularities. Obviously, K F = −H F and Pic F = ZH F , where H F is the class of a hyperplane section F ⊂ P, that is, F is a Fano variety of index one. In order to prove Theorem 4, we have to show the following two facts: -the inequality (2), -the divisorial log-canonicity of the hypersurface F ∈ F reg , that is, the condition (ii) of Theorem 1 for the variety F . These two tasks are dealt with in the remaining part of this section and §3, respectively. The conditions of general position at a non-singular point. Let o ∈ F be a non-singular point. Fix an arbitrary non-zero linear form q 1 and consider the affine space of polynomials q 1 + P sing = {q 1 + q 2 + . . . + q M }, where P sing is the space of polynomials of the form f = q 2 + q 3 + . . . + q M . Let P i ⊂ {q 1 + P sing }, i = 1, 2, 3, be the closures of the subsets, consisting of such polynomials f , which do not satisfy the condition (R1.i), respectively. Set c i = codim(P i ⊂ {q 1 + P sing }). Proposition 2.4. For M 8 the following equality holds: min{c 1 , c 2 , c 3 } = c 2 = (M − 6)(M − 5) 2 . Proof is easy to obtain by elementary methods. First of all, by Lemma 2.1, shown below (where one must replace M by (M − 1)), we obtain c 1 = (M − 1)(M − 2) 2 + 2. Furthermore, a violation of the condition rkq 2 6 imposes on the coefficients of the quadratic form q 2 (M − 6)(M − 5) 2 < c 1 independent conditions. Assuming the condition rkq 2 6 to be satisfied, we obtain that the quadric {q 2 = 0} is factorial. It is easy to check that reducibility or nonreducedness of the divisor q 3 \| {q 2 =0} on this quadric gives M 3 − 6M 2 − 7M + 54 6 > (M − 6)(M − 5) 2 independent conditions on the coefficients of the cubic form q 3 . Finally, let us consider a hyperplane P = T o F and the quadratic hypersurface q 2 \| P ∩{q 1 =0} = 0. Its rank is at least 5, so it is still factorial. Let us estimate from below the number of independent conditions, which are imposed on the coefficients of the polynomials f (j, µ) = j + M − 1 M − 1 − j + M − 3 M − 1 − v(µ) + v(max(0, µ − 2)). Now, using the factoriality of the quadric, we obtain the estimate , 0)) . c 3 f (M, 3) − (M − 2)− − max max M −1 j 2 (f (j, 2) + f (M − j, 1)), max M −1 j 3 (f (j, 3) + f (M − j An elementary check shows that the minimum of the right hand side is strictly higher than c 2 (and for M → ∞ grows exponentially). Q.E.D. for the proposition. Proof is obtained by the standard methods [31,Chapter 3]. We just remind the scheme of arguments. Fix the first moment when the sequence of polynomials q 2 , . . . , q M becomes non-regular: assume that the regularity is first violated for q k , that is, the closed set {q 2 = . . . = q k−1 = 0} has the "correct" codimension k − 2 and q k vanishes on one of the components of that set. For k M − 1 we apply the method of [20] and obtain that violation of the regularity condition imposes on the coefficients of the polynomial f at least M + 1 k (M + 1)M 2 independent conditions; the right hand side of the last inequality is strictly higher than (6), which is what we need. Let us consider the last option: {q 2 = . . . = q M −1 = 0} ⊂ P M −1 is a one-dimensional closed set and q M vanishes on one of its irreducible components, say B. The case when B ⊂ P M −1 is a line is a special one: it is easy to check that vanishing on a line in P M −1 imposes on the polynomials q 2 , . . . , q M in total precisely (6) independent conditions. Therefore, we may assume that B is not a line, that is, dim B < B = k 2. Now we apply the method suggested in [22], Proposition 2.6. The following estimate holds: codim((F \ F reg ) ⊂ F ) (M − 7)(M − 6) 2 − 5. Proof is completely similar to the proof of Proposition 2.3 and follows from Propositions 2.4 and 2.5. Now let us consider some geometric facts which follow immediately from the conditions of general position. These facts will be needed in §3 to exclude log maximal singularities. In [25] it was shown that for any effective divisor D ∼ nH on F (where we write H in stead of H F to simplify the notations) the pair (F, 1 n D) is canonical at non-singular points o ∈ F . This fact will be used without special references. Now let D 2 = {q 2 \| F = 0} be the first hypertangent divisor, so that we have D + 2 ∈ \|2H − 3E\|. Recall that E ⊂ P M −1 is an irreducible quadric of rank at least 8. Obviously, the divisor D 2 ∈ \|2H\| satisfies the equality mult o deg D 2 = 3 M . Here and below the symbol mult o / deg means the ratio of multiplicity at the point o to the degree. Lemma 2.2. Let P ⊂ F be the section of the hypersurface F by an arbitrary linear subspace in P of codimension two, containing the point o. Then the restriction D 2 \| P is an irreducible reduced divisor on the hypersurface P ⊂ P M −2 . Proof. The variety P has at most quadratic singularities of rank at least 6 and for that reason it is factorial. Therefore, reducibility or non-reducedness of the divisor D 2 \| P means, that the equality D 2 \| P = H 1 + H 2 holds, where H i are possibly coinciding hyperplane sections of P . By the condition (R2.2) the equalities mult o H i = 2 hold. However, mult o D 2 \| P = 6. Therefore, D 2 \| P can not break into two hyperplane sections. Q.E.D. for the lemma. Proposition 2.7. The pair (F, 1 2 D 2 ) has no non log canonical singularities, the centre of which on F contains the point o: LCS(F, 1 2 D 2 ) ∋ o. Proof. Assume the converse. In any case codim(LCS F, 1 2 D 2 ⊂ F ) 6, so that consider the section P ⊂ F of the hypersurface F by a generic linear subspace of dimension 5, containing the point o. Then the pair (P, 1 2 D 2 \| P ) has the point o as an isolated centre of a non log canonical singularity. Let σ P : P + → P be the blow up of the non-degenerate quadratic singularity o ∈ P so that E P = E ∩ P + is a non-singular exceptional quadric in P 4 . Since 1 2 (D 2 \| P ) + ∼ H P − 3 2 E P and a(E P , P ) = 2 > 3 2 (where H P is the class of a hyperplane section of P ⊂ P 5 ), the pair (P + , 1 2 (D 2 \| P ) + ) is not log canonical. The union LCS(P + , 1 2 (D 2 \| P ) + ) of all centres of non log canonical singularities of that pair, intersecting E P , is a connected closed subset of the exceptional quadric E P , every irreducible component S P of which satisfies the inequality mult S P (D 2 \| P ) + 3. Coming back to the original pair (F, 1 2 D 2 ), we see that for some irreducible subvariety S ⊂ E the inequality mult S D + 2 3 holds, where S ∩ P + = S P , so that codim(S ⊂ E) ∈ {1, 2, 3}. However, the case codim(S ⊂ E) = 3 is impossible: by the connectedness principle this equality means that S P is a point, and then S ⊂ E is a linear subspace of codimension 3, which is impossible if rk q 2 8 (a 7-dimensional non-singular quadric does not contain linear subspaces of codimension 3). Consider the case codim(S ⊂ E) = 2. Let Π ⊂ E be a general linear subspace of maximal dimension. Then D + 2 \| Π is a cubic hypersurface that has multiplicity 3 along an irreducible subvariety S Π = S ∩ Π of codimension 2. Therefore, D + 2 \| Π is a sum of three (not necessarily distinct) hyperplanes in Π, containing the linear subspace S Π ⊂ Π of codimension 2, and for that reason D + 2 \| E is a sum of three (not necessarily distinct) hyperplane sections from the same linear pencil as well, and S is the intersection of the quadric E and a linear subspace of codimension 2. However, this is impossible by the condition (R2.3). Finally, if codim(S ⊂ E) = 1, then D + 2 \| E = 3S is a triple hyperplane section of the quadric E, which is impossible by the condition (R2.3). This completes the proof of Proposition 2.7. Q.E.D. Here is one more fact that will be useful later. Proposition 2.8. For any hyperplane section ∆ ∋ o of the hypersurface F the pair (F, ∆) is log canonical. Proof. This follows from a well known fact (see, for instance, [3,30]): if (p ∈ X) is a germ of a non-degenerate quadratic three-dimensional singularity, σ: X → X its resolution with the exceptional quadric E X ∼ = P 1 × P 1 and D X a germ of an effective divisor such that o ∈ D X and D X ∼ −βE X , then the pair (X, 1 β D X ) is log canonical at the point o. Q.E.D. Exclusion of maximal singularities In this section we complete the proof of Theorem 4. The symbol F stands for a fixed hypersurface of degree M in P, satisfying the regularity conditions: F ∈ F reg . As we mentioned in §2, in [25] it was shown that the pair (F, 1 n D) has no maximal singularities, the centre of which is not contained in the closed set Sing F , for every effective divisor D ∼ nH. In [5] it was shown that for any mobile linear system Σ ⊂ \|nH\| the pair (F, 1 n D) is canonical for a general divisor D ∈ Σ, that is, Σ has no maximal singularities. Therefore, in order to complete the proof of Theorem 4 it is sufficient to show that for any effective divisor D ∼ nH the pair (F, 1 n D) is log canonical, and we may assume only those log maximal singularities, the centre of which is contained in Sing F . In subsection 3.1 we carry out preparatory work: by means of the technique of hypertangent divisors we obtain estimates for the ratio mult o / deg for certain classes of irreducible subvarieties of the hypersurface F . After that we fix a pair (F, 1 n D) and assume that it is not log canonical. The aim is to bring this assumption to a contradiction. Let B * ⊂ Sing F be the centre of the log maximal singularity of the divisor D, o ∈ B * a point of general position, F + → F its blow up, D + the strict transform of the divisor D. In subsection 3.2 we study the properties of the pair (F + , 1 n D + ): we show that this pair has a non log canonical singularity, the centre of which is a subvariety of the exceptional divisor of the blow up of the point o. After that in subsections 3.2 and 3.3 we show that this is impossible, which completes the proof of Theorem 4. (i) For every irreducible subvariety of codimension 2 Y ⊂ F the following inequality holds: mult o deg Y 4 M . (ii) Let ∆ ∋ o be an arbitrary hyperplane section of the hypersurface F . For every prime divisor Y ⊂ ∆ the following inequality holds: mult o deg Y 3 M . (iii) Let P ∋ o be the section of the hypersurface F by an arbitrary linear subspace of codimension two. For every prime divisor Y ⊂ P the following inequality holds: mult o deg Y 4 M . Proof (7) we have Y i ⊂ D i+2 for a general hypertangent divisor D i+2 . By the construction of hypertangent linear system, at every step of our procedure the inequality mult o deg Y i+1 i + 3 i + 2 · mult o deg Y i holds, so that for the curve Y M −2 we have the estimates 1 mult o deg Y M −2 mult o deg Y · 5 4 · 6 5 · . . . · M M − 1 , which implies the claim (i). Let us prove the claim (ii). By Lemma 2.2, the divisor D 2 \| ∆ is irreducible and reduced, and by the condition (R2.1) it satisfies the equality mult o deg D 2 \| ∆ = 3 M . Therefore, we may assume that Y = D 2 \| ∆ , so that Y ⊂ D 2 and the effective cycle of codimension two (Y • D 2 ) on ∆ is well defined and satisfies the inequality mult o deg (Y • D 2 ) 3 2 · mult o deg Y. Let Y 2 be an irreducible component of that cycle with the maximal value of the ratio mult o / deg. Applying to Y 2 the technique of hypertangent divisors in precisely the same way as in the part (i) above, we see that by the condition (R2.1) the intersection Y 2 ∩ D 4 \| ∆ ∩ D 5 \| ∆ ∩ . . . ∩ D M −2 \| ∆ in a neighborhood of the point o is a one-dimensional closed set, where D 4 ∈ Λ 4 , . . ., D M −2 ∈ Λ M −2 are general hypertangent divisors (note that the last hypertangent divisor is D M −2 , and not D M −1 , as in the part (i), because the dimension of ∆ is one less than the dimension of F and the condition (R2.1) provides the regularity of the truncated sequence q 2 \| Π , . . . , q M −1 \| Π , where in this case Π is a hyperplane, cutting out ∆ on F ). Now, arguing in the word for word same way as in the proof of the claim (i), we obtain the estimate 1 mult o deg Y · 3 2 · 5 4 · 6 5 · . . . · M − 1 M − 2 , which implies that mult o deg Y 8 3(M − 1) . For M 9 the right hand side of the inequality does not exceed 3/M, which proves the claim (ii). Let us show the claim (iii). We argue in the word for word same way as in the proof of the part (ii), with the only difference: in order to estimate the multiplicity of the cycle (Y • D 2 \| P ) at the point o we use the hypertangent divisors D 4 \| P , D 5 \| P , . . . , D M −3 \| P (one less than above), so that we get the estimate 1 mult o deg Y · 3 2 · 5 4 · 6 5 · . . . · M − 2 M − 3 , which implies that mult o deg Y 8 3(M − 2) . For M 6 the right hand side of the inequality does not exceed 4/M, which proves the claim (iii). Proof of Proposition 3.1 is complete. Q.E.D. Let us resume the proof of Theorem 4. 3.2. The blow up of a singular point. Assume that the pair (F, 1 n D) is not log canonical for some divisor D ∈ \|nH\|, that is, for some prime divisor E * over F , that is, a prime divisor E * ⊂ F , where ψ: F → F is some birational morphism, F is non-singular and projective, the log Noether-Fano inequality holds: ord E * ψ * D > n(a(E * ) + 1). By linearity of the inequality in D and n we may assume the divisor D to be prime. Let B * = ψ(E * ) ⊂ F be the centre of the log maximal singularity E * . We known that B * ⊂ Sing F ; in particular, codim(B * ⊂ F ) 7. Let o ∈ B * be a point of general position, ϕ: F + → F its blow up, E ⊂ F + the exceptional quadric. Consider the first hypertangent divisor D 2 ∈ \|2H\| at the point o. By Lemma 2.2, the divisor D 2 is irreducible and reduced, and by Proposition 2.7, the pair (F, 1 2 D 2 ) is log canonical at the point o. Therefore, D = D 2 . Proposition 3.2. The following inequality holds mult o D 8 3 n. Proof. Consider the effective cycle (D • D 2 ) of codimension two. Obviously, Write down D + ∼ nH − νE, where ν 4 3 n. Let us consider the section P of the hypersurface F by a general 5-dimensional linear subspace, containing the point o. Let P + be the strict transform of P on F + and E P = P + ∩ E a non-singular three-dimensional quadric. Set also D P = D\| P . Obviously, the pair (P, 1 n D P ) has the point o as an isolated centre of a non log canonical singularity. Since a(E P ) = 2 and D + P ∼ nH P − νE P (where H P is the class of a hyperplane section of the variety P ), where ν 4 3 n < 2n, the pair (P + , 1 n D + P ) is not log canonical and the union LCS E (P + , 1 n D + P ) of centres of all non log canonical singularities of that pair, intersecting E P , is a connected closed subset of the exceptional quadric E P . Let S P be an irreducible component of that set. Obviously, the inequality mult S P D + P > n holds. Furthermore, codim(S P ⊂ E P ) ∈ {1, 2, 3}. Returning to the original pair (F, 1 n D), we see that there is a non log canonical singularity of the pair (F + , 1 n D + ), the centre of which is a subvariety S ⊂ E, such that S ∩ E P = S P and, in particular, codim(S ⊂ E) ∈ {1, 2, 3}. mult o deg (D • D 2 ) 3 2 · mult o deg D, Note at once that the case codim(S ⊂ E) = 3 is impossible: by the connectedness principle in that case S P is a point and for that reason S is a linear subspace of codimension 3 on the quadric E of rank at least 8, which is impossible. It is not hard to exclude the case codim(S ⊂ E) = 1, either. Assume that it does take place. Then the divisor S is cut out on E by a hypersurface of degree d S 1. Let H E be the class of a hyperplane section of the quadric E. The divisor 3.3. The case of codimension two. Starting from this moment, assume that codim(S ⊂ E) = 2. D + \| E ∼ νH E , Lemma 3.1. The subvariety S is contained in some hyperplane section of the quadric E. Proof. Since mult S D + > n and D + \| E ∼ νH E with ν 1) the secant lines of the set S Π sweep out a hyperplane in Π, 2) S Π ⊂ Π is a linear subspace of codimension two. In the first case the secant lines L ⊂ E of the set S sweep out a divisor on E, which ca only be a hyerplane section of the quadric E. In the second case S contains all its secant lines and is a section of E by a linear subspace of codimension two. Q.E.D. for the lemma. As we have just shown, one of the two options takes place: either there is a unique hyperplane section Λ of the quadric E, containing S (Case 1), or S = E ∩ Θ, where Θ is a linear subspace of codimension two (Case 2). Let us study them separately. Assume that Case 1 takes place. Then S is cut out on Λ by a hypersurface of degree d S 2. Set µ = mult S D + and γ = mult Λ D + , where µ > n and µ 2ν 8 3 n. Lemma 3.2. The following inequality holds: γ 2µ − ν 3 . Proof is easy to obtain in the same way as the short proof of Lemma 3.5 in [21, subsection 3.7]. Let L ⊂ Λ be a general secant line of the set S. Consider the section P of the hypersurface F by a general 4-plane in P, such that P ∋ o and P + ∩ E contains the line L. Obviously, o ∈ P is a non-degenerate quadratic point and E P = P + ∩ E ∼ = P 1 × P 1 is a non-singular quadric in P 3 . Set D P = D\| P . Obviously, γ = mult L D + P . Let σ L : P L → P + be the blow up of the line L, E L = σ −1 L (L) the exceptional divisor; since N L/P + ∼ = O ⊕ O L (−1), the exceptional surface E L is a ruled surface of the type F 1 , so that Pic E L = Zs ⊕ Zf, where s and f are the classes of the exceptional section and the fibre, respectively. Furthermore, E L \| E L = −s − f . Let D L be the strict transform of D + P on P L . Obviously, D L ∼ nH P − νE P − γE L (where H P is the class of a hyperplane section of P ), so that D L \| E L ∼ γs + (γ + ν)f. On the other hand, L is a general secant line of the set S and for that reason L contains at least two distinct points p, q ∈ S. Therefore, the divisor D + P has at the points p, q ∈ L the multiplicity µ and for that reason the effective 1-cycle D L \| E L contains the corresponding fibres σ −1 L (p) and σ −1 L (q) over those points with multiplicity (µ − γ). Therefore, γ + ν 2(µ − γ), whence follows the claim of the lemma. Q.E.D. Now let us consider the uniquely determined hyperplane section ∆ of the hypersurface F ⊂ P, such that ∆ ∋ o and ∆ + ∩ E = Λ. Set D ∆ = D\| ∆ . Write down D + \| ∆ + = D + ∆ + aΛ. Obviously, mult o D ∆ = 2(ν + a) 2ν + 2 2µ − ν 3 = 4 3 (µ + ν) > 8 3 n. Since as we noted above, the subvariety S is cut out on the quadric Λ by a hypersurface of degree d S 2, the divisor D + ∆ ∼ nH ∆ − (ν + a)Λ can not contain S with multiplicity higher than 1 d S (ν + a) ν + a 2 . Since the pair (F + , 1 n D + ) has a non log canonical singularity with the centre at S, the inversion of adjunction implies that the pair = (∆ + , 1 n (D + ∆ +aΛ)) has a non log canonical singularity with the centre at S as well. Recall that codim(S ⊂ ∆ + ) = 2. Consider the blow up σ S : ∆ → ∆ + of the subvariety S and denote by the symbol E S the exceptional divisor σ −1 S (S). The following fact is well known. Proposition 3.3. For some irreducible divisor S 1 ⊂ E S , such that the projection σ S \| S 1 is birational, the inequality mult S (D + ∆ + aΛ) + mult S 1 ( D ∆ + a Λ) > 2n (8) holds, where D ∆ and Λ are the strict transforms of D + ∆ and Λ on ∆, respectively. Proof: see Proposition 9 in [25]. Set µ S = mult S D + ∆ and β = mult S 1 D ∆ . Consider first the case of general position: S 1 = E S ∩ Λ. In that case S 1 ⊂ Λ and the inequality (8) takes the following form: µ S + β + a > 2n. Since µ S β, the more so 2µ S + a > 2n. On the other hand, we noted above that 2µ S ν + a. As a result, we obtain the estimate ν + 2a > 2n. Therefore, mult o D ∆ > ν + 2n > 3n. However, D ∆ ∼ nH ∆ is an effective divisor on the hyperplane section ∆ and by Proposition 3.1, (ii), it satisfies the inequality mult o deg D ∆ 3 M . This contradiction excludes the case of general position. Therefore, we are left with the only option: S 1 = E S ∩ Λ. In that case the inequality (8) takes the following form: µ S + β + 2a > 2n. This inequality is weaker than the corresponding estimate in the case of general position, but as a compensation we obtain the additional inequality 2µ S + 2β ν + a (the restriction D + ∆ \| Λ is cut out by a hypersurface of degree (ν + a) and contains the divisor S ∼ d S H Λ with multiplicity at least µ S + β). Combining the last two estimates, we obtain the inequality ν + 5a > 4n, which implies that 5(ν + a) > 8n and so mult o D Λ > 16 5 n; as we mentioned above, this contradicts Proposition 3.1, (ii). This completes the exclusion of Case 1. Therefore, Case 2 takes place: S = E ∩ Θ, where Θ ⊂ P M −1 is a linear subspace of codimension two. Let P ⊂ F be the section of the hypersurface F by the linear subspace of codimension two in P, which is uniquely determined by the conditions P ∋ o and P + ∩ E = S. Furthermore, let \|H − P \| be the pencil of hyperplane sections of F , containing P . For a general hyperplane section ∆ ∈ \|H − P \| we have: • the divisor D does not contain ∆ as a component, so that the effective cycle (D •∆) = D ∆ of codimension two on F is well defined; this cycle can be looked at as an effective divisor D ∆ ∈ \|nH ∆ \| on the hypersurface ∆ ⊂ P M −1 , • for the strict transform D + ∆ on F + the equality mult S D + ∆ = mult S D + holds. Of course, the divisor D ∆ may contain P as a component. Write down D ∆ = G + aP , where a ∈ Z + and G is an effective divisor that does not contain P as a component, G ∈ \|(n − a)H ∆ \|. Obviously, G + ∼ (n − a)H ∆ − (ν − a)E ∆ , where E ∆ = ∆ + ∩ E, and, besides, mult S G + = mult S D + − a > n − a. Set m = n − a. The effective cycle of codimension two G P = (F • P ) on ∆ is well defined and can be considered as an effective divisor G P ∈ \|mH P \| on the hypersurface P ⊂ P M −2 . The divisor G P satisfies the inequality mult o G P 2(ν − a) + 2 mult S G + > 4m. This is impossible by Proposition 3.1, (iii). Therefore, the assumption that the pair (F, 1 n D) is not log canonical for some divisor D ∼ nH, leads to a contradiction. Proof of Theorem 4 is complete. 0. 3 . 3Fibrations into Fano hypersurfaces of index one. The symbol P still stands for the projective space P M , M 10. Fix M. Let F = P(H 0 (P, O P (M))) be the space of hypersurfaces of degree M in P. The following fact is true. Proof. The conditions (i) and (ii) of Theorem 1 are satisfied by the generality of the divisor V . The inequality (1) up to a positive factor is the same as the inequality ((MK S + (M − 1)R) · C) 0. Therefore, by Remark 0.1, the condition (iii) of Theorem 1 also holds. Q.E.D. for the theorem. Example 0.2. Take S = P m , where m 1 2 (M − 7)(M − 6) − 6, X = P M × P m and V ⊂ X is a sufficiently general hypersurface of bidegree (M, l), where l satisfies 2. 1 . 1The double spaces of general position. The open subset W reg ⊂ W of hypersurfaces of degree 2M in P = P M is defined by local conditions, which a hypersurface W ∈ W reg must satisfy at every point o ∈ W . These conditions depend on whether the point o ∈ W is non-singular or singular. First, let us consider the condition of general position for a non-singular point o ∈ W . Let (z 1 , . . . , z M ) be a system of affine coordinates with the origin at the point o and w = q 1 + q 2 + . . . + q 2M the affine equation of the branch hypersurface W , where the polynomials q i (z * ) are homogeneous of degree i = 1, . . . , 2M. At a non-singular point o ∈ W (that is, q 1 ≡ 0) the hypersurface W must satisfy the condition (W1) the rank of the quadratic form q 2 \| {q 1 =0} is at least 2. Proposition 2.1. Violation of the condition (W1) imposes on the coefficients of the quadratic form q 2 (with the linear form q 1 fixed) (M − 2)(M − 1) 2 independent conditions. Proof is obvious. Q.E.D. Now let us consider the condition of general position for a singular point o ∈ W . Let w = q 2 + q 3 + . . . + q 2M be the affine equation of the branch hypersurface W with respect to a system of affine coordinates (z 1 , . . . , z M ) with the origin at the point o. At a singular point o the hypersurface W must satisfy the condition (W2) the rank of the quadratic form q 2 is at least 4. Proposition 2.2. Violation of the condition (W2) imposes on the coefficients of the quadratic form q 2 (M − 2)(M − 1) 2 independent conditions. 2. 2 . 2Fano hypersurfaces of general position. As in the case of double space, the open subset F reg ⊂ F of hypersurfaces of degree M in P = P M is defined by the local conditions, which a hypersurface F ∈ F reg should satisfy at every point o ∈ F . Again these conditions are different for non-singular and singular points o ∈ F . Consider first the conditions of general position for a non-singular point o ∈ F . Let (z 1 , . . . , z M ) be a system of affine coordinates with the origin at the point o and w = q 1 + q 2 + q 3 + . . . + q M the affine equation of the hypersurface F , where the polynomials q i (z * ) are homogeneous of degree i = 1, . . . , M. Here is the list of conditions of general position, which a hypersurface F should satisfy at a non-singular point o. (R1.1) The sequence q 1 , q 2 , . . . , q M −1 is regular in the local ring O o,P , that is, the system of equations q 1 = q 2 = . . . = q M −1 = 0 defines a one-dimensional subset, a finite set of lines in P, passing through the point o. In particular, q 1 ≡ 0. The equation q 1 = 0 defines the tangent space T o F (which we, depending on what we need, will consider either a linear subspace in C M , or as its closure, a hyperplane in P). Now setq i = q i \| {q 1 =0} for i = 2, . . . , M: these are polynomials on the linear space T o F ∼ = C M −1 . The condition (R1.1) means the regularity of the sequenceq in C M is the hyperplane {q 1 = 0}, that is, the tangent hyperplane T o F . An equivalent wording of this condition: every irreducible component of the closed set {q 2 =q 3 = 0} in P M −2 = P({q 1 = 0}) is non-degenerate. (R1.3) For any hyperplane P ⊂ P, P ∋ o, different from the tangent hyperplane T o F ⊂ P, the algebraic cycle of scheme-theoretic intersection of hyperplanes P , T o F , the projective quadric {q 2 = 0} ⊂ P and F , that is, the cycle, (P • {q 1 = 0} • {q 2 = 0} • F ), is irreducible and reduced. (The line above means the closure in P and the operation • of taking the cycle of scheme-theoretic intersection is considered here on the space P, too.) Now let us consider the conditions of general position for a singular point o ∈ F . Let (z 1 , . . . , z M ) be a system of affine coordinates with the origin at the point o and f = q 2 + q 3 + . . . + q M the affine equation of the hypersurface F , where the polynomials q i (z * ) are homogeneous of degree i = 2, . . . , M. Let us list the conditions of general position which must be satisfied for the hypersurface F at a singular point o. (R2.1) For any linear subspace Π ⊂ C M of codimension c ∈ {0, 1, 2} the sequence q 2 \| Π , . . . , q M −c \| Π (5) is regular in the ring O o,Π , that is, the system of equations q 2 \| P(Π) = . . . = q M −c \| P(Π) = 0 defines in the space P(Π) ∼ = P M −c−1 a finite set of points. (R2.2) The quadratic form q 2 (z * ) is of rank at least 8. (R2.3) Now let us consider (z 1 , . . . , z M ) as homogeneous coordinates (z 1 : . . . : z M ) on P M −1 . The divisor {q 3 \| {q 2 =0} = 0} q 3 , . . . , q M if the condition (R1.3) is violated. Define the values v(µ), µ = 0, 1, 2, 3, by the table µ 0 1 2 3 v(µ) 0 1 M 1 2 M(M + 1) − 1 and set 2. 4 . 4The conditions of general position at a singular point. Recall that P sing is the space of polynomials of the formf = q 2 + q 3 + . . . + q M in the variables z * = (z 1 , . . . , z M ), where q i (z * ) are homogeneous of degree i. LetP sing reg ⊂ P sing be the subset of polynomials satisfying the conditions (R2.1-R2.3). Proposition 2.5. The following estimate holds: codim((P sing \P sing reg ) ⊂ P sing ) = (M − 7)(M − 6) 2 . Proof. It is sufficient to show that violation of each of the conditions (R2.1-R2.3) at the point o = (0, . . . , 0) separately imposes on the polynomial f at least (M − 7)(M − 6)/2 independent conditions. It is easy to check that violation of the condition (R2.2) imposes on the coefficients of the quadratic form q 2 (z * ) precisely (M − 7)(M − 6)/2 independent conditions. Therefore, considering the condition (R2.3), we may assume that the condition (R2.2) is satisfied; in particular, the quadric {q 2 = 0}) is factorial and violation of the condition (R2.3) imposes on the coefficients of the cubic form q 3 (z * ) (with the polynomial q 2 fixed) for M 4. It remains to consider the case when the condition (R2.1) is violated. Lemma 2.1. Violation of the condition (R2.1) for one value of the parameter c = 0 imposes on the coefficients of the polynomial f M(M − 1 fixing k and the linear subspace B . To begin with, consider the case k M − 2. In that case there are indices i 1 , . . . , i k−1 ∈ {2, . . . , M − 1},such that the restrictions q i 1 \| B , . . . , q i k−1 \| B form a good sequence and B is one of its associated subvarieties (see[22, Sec.3, Proposition 4], the details of this procedure are described in the proof of the cited proposition). Taking into account that B ⊂ B is by construction a non-degenerate curve, we see that decomposable polynomials of the form l 1 . . . l a , where l i are linear forms on B ∼ = P k , can not vanish on B. This gives jk + 1 independent conditions for each of the polynomials q j for j ∈ {i 1 , . . . , i k−1 }, so that in total we get at leastk(M − k)(M − k + 1) 2 + M − 2k − 1 independentconditions for these polynomials (the minimum is attained for i 1 = M − k + 1, . . ., i k−1 = M − 1). Taking into account the condition q M \| B ≡ 0 and the dimension of the Grassmanian of k-dimensional subspaces in P M −1 , we obtain at least M 2 − kM + k 2 − M + k + 1 independent conditions for f . It is easy to check that the last number is not smaller than (6). Finally, if k = M − 1, that is, B is a non-degenerate curve in P M −1 , then the condition q M \| B ≡ 0 gives at least M(M − 1) + 1 independent conditions for q M . Proof of Lemma 2.1 is complete. Q.E.D. Now let us complete the proof of Proposition 2.5. For a fixed linear subspace Π ⊂ C M of codimension c ∈ {0, 1, 2} violation of regularity of the sequence (5) imposes on the polynomial f at least (M − c)(M − c − 1)/2 + 2 independent conditions. Subtracting the dimension of the Grassmanian of subspaces of codimension c in C M , we get the least value (M − 3)(M − 6)/2 for c = 2. This completes the proof of Proposition 2.5. Q.E.D. 2.5. Estimating the codimension of the complement to the set F reg . Recall that F ∈ F reg if and only if at every non-singular point o ∈ F the conditions (R1.1-3) are satisfied, and at every singular point o ∈ F the conditions (R2.1-3) are satisfied. Propositions 2.4 and 2.5 imply the following fact. 3. 1 . 1The method of hypertangent divisors. Fix a singular point o ∈ F , a system of coordinates (z 1 , . . . , z M ) on P with the origin at that point and the equation f = q 2 + . . . q M of the hypersurface F . Proposition 3.1. Assume that the variety F satisfies the conditions (R2.1, R2.2) at the singular point o. Then the following claims hold. is obtained by means of the method of hypertangent divisors[31, Chapter 3]. For k = 2, . . . , M − 1 letΛ k = k i=2 s k−i (q 2 + . . . + q i )\| F = 0be the k-th hypertangent linear system, where s j (z * ) are all possible homogeneous polynomials of degree j. For the blow up σ: F + → F of the point o with the exceptional divisor E = σ −1 (o), naturally realized as a quadric in P M −1 , we haveΛ + k ⊂ \|kH − (k + 1)E\| (where Λ + k is the strict transform of the system Λ k on F + ). Let D k ∈ Λ k , k = 2, . . . , M − 1 be general hypertangent divisors.Let us show the claim (i). By the condition (R2.1) the equalitycodim o (Bs Λ k ⊂ F ) = k − 1(7)holds, where the symbol codim o means the codimension in a neighborhood of the point o; therefore, Y ∩ D 4 ∩ D 5 ∩ . . . ∩ D M −1 in a neighborhood of the point o is a closed one-dimensional set. We construct a sequence of irreducible subvarieties Y i ⊂ F of codimension i: Y 2 = Y and Y i+1 is an irreducible component of the effective cycle (Y i • D i+2 ) with the maximal value of the ratio mult o / deg. The cycle (Y i • D i+2 ) every time is well defined, because by the inequality however, by Proposition 3.1, (i), the left hand part of this inequality does not exceed (4/M). Since deg D = nM, Proposition 3.2 is shown. Q.E.D. for every secant line L ⊂ E of the subvariety S we have L ⊂ D + . Let Π ⊂ E be a linear space of maximal dimension and of general position and S Π = S ∩ Π. The secant lines of the closed set S Π ⊂ Π of codimension two can not sweep out Π, since E ⊂ D + . Therefore, there are two options (see [30, Lemma 2.3]): and for that reason S is a hyperplane section of the quadric E. Let ∆ ∈ \|H\| be the uniquely determined hyperplane section of the hypersurface F , such that ∆ ∋ o and ∆ + ∩ E = S. The pair (F + , ∆ + ) is log canonical and for that reason D = ∆. For the effective cycle (D • ∆) of codimension two on F we have mult o (D • ∆) 2ν + 2 mult S D + > 4n, which contradicts Proposition 3.1. This excludes the case of a divisorial centre.so that 4 3 n ν > nd S so that mult o deg (D • ∆) > 4 M , ,q 3 , . . . ,q M −1 . 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[ "LAT Flux (photons/s/cm 2", "LAT Flux (photons/s/cm 2" ]	[ "P H T Tam ", "P S Pal ", "Y D Cui ", "N Jiang ", "Y Sotnikova ", "C W Yang ", "L Z Wang ", "B T Tang ", "Y B Li ", "J Mao ", "A K H Kong ", "Z H Zhong ", "J Ding ", "T Mufakharov ", "J F Fan ", "L M Dou ", "R F Shen ", "Y L Ai ", "\nSchool of Physics and Astronomy\nSun Yat-sen University\n510275GuangzhouPeople's Republic of China\n", "\nCAS Key Laboratory for Researches in Galaxies and Cosmology\nUniversity of Sciences and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n", "\nSpecial Astrophysical Observatory\nRussian Academy of Sciences\nRussian Federation. d Polar Research Institute of China\n451 Jinqiao Road369167, 200136ShanghaiNizhnij ArkhyzPeople's Republic of China\n", "\nJoint Center for Astronomy\nChinese Academy of Sciences South America Center for Astronomy\nCamino El Observatorio #1515China, Chile\n", "\nLas Condes\nSantiagoRepublic of Chile\n", "\nYunnan Observatories\nChinese Academy of Sciences\n650011KunmingPeople's Republic of China\n", "\nCenter for Astronomical Mega-Science\nChinese Academy of Sciences\n20A Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n", "\nUniversity of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China\n", "\nKey Laboratory for the Structure and Evolution of Celestial Objects\nChinese Academy of Sciences\n650011KunmingPeople's Republic of China\n", "\nInstitute of Astronomy\nNational Tsing Hua University\nSection 2. Kuang-Fu Road101, 30013HsinchuTaiwan, R.O.C\n", "\nDepartment of Astronomy & Astrophysics\nUniversity of California Santa Cruz\n1156 High Street95060Santa CruzCAUSA\n", "\nShanghai Astronomical Observatory\nChinese Academy of Sciences\n200030ShanghaiPeople's Republic of China\n", "\nRussian Federation. n Center for Astrophysics\nKazan Federal University\n18 Kremlyovskaya St420008Kazan\n", "\nGuangzhou University\n\n" ]	[ "School of Physics and Astronomy\nSun Yat-sen University\n510275GuangzhouPeople's Republic of China", "CAS Key Laboratory for Researches in Galaxies and Cosmology\nUniversity of Sciences and Technology of China\n230026HefeiAnhuiPeople's Republic of China", "Special Astrophysical Observatory\nRussian Academy of Sciences\nRussian Federation. d Polar Research Institute of China\n451 Jinqiao Road369167, 200136ShanghaiNizhnij ArkhyzPeople's Republic of China", "Joint Center for Astronomy\nChinese Academy of Sciences South America Center for Astronomy\nCamino El Observatorio #1515China, Chile", "Las Condes\nSantiagoRepublic of Chile", "Yunnan Observatories\nChinese Academy of Sciences\n650011KunmingPeople's Republic of China", "Center for Astronomical Mega-Science\nChinese Academy of Sciences\n20A Datun Road100012Chaoyang District, BeijingPeople's Republic of China", "University of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China", "Key Laboratory for the Structure and Evolution of Celestial Objects\nChinese Academy of Sciences\n650011KunmingPeople's Republic of China", "Institute of Astronomy\nNational Tsing Hua University\nSection 2. Kuang-Fu Road101, 30013HsinchuTaiwan, R.O.C", "Department of Astronomy & Astrophysics\nUniversity of California Santa Cruz\n1156 High Street95060Santa CruzCAUSA", "Shanghai Astronomical Observatory\nChinese Academy of Sciences\n200030ShanghaiPeople's Republic of China", "Russian Federation. n Center for Astrophysics\nKazan Federal University\n18 Kremlyovskaya St420008Kazan", "Guangzhou University\n" ]	[]	We report observations of a transient source Fermi J1544-0649 from radio to γrays. Fermi J1544-0649 was discovered by the Fermi-LAT in May 2017. Followup Swift-XRT observations revealed three flaring episodes through March 2018, and the peak X-ray flux is about 10 3 higher than the ROSAT all-sky survey (RASS) flux upper limit. Optical spectral measurements taken by the Magellan 6.5-m telescope and the Lick-Shane telescope both show a largely featureless spectrum, strengthening the BL Lac interpretation first proposed byBruni et al. (2018). The optical and mid-infrared (MIR) emission goes to a higher state in 2018, when the flux in high energies goes down to a lower level. Our RATAN-600m measurements at 4.8 GHz and 8.2 GHz do not indicate any significant radio flux variation over the monitoring seasons in 2017 and 2018, nor deviate from the archival NVSS flux level. During GeV flaring times, the spectrum is very hard (Γ γ ∼1.7) in the GeV band and at times also very hard ((Γ X ∼ < 2) in the X-rays, similar to a highsynchrotron-peak (or even an extreme) BL Lac object, making Fermi J1544-0649 a good target for ground-based Cherenkov telescopes.AbstractWe report observations of a transient source Fermi J1544-0649 from radio to γrays. Fermi J1544-0649 was discovered by the Fermi-LAT in May 2017. Followup Swift-XRT observations revealed three flaring episodes through March 2018, and the peak X-ray flux is about 10 3 higher than the ROSAT all-sky survey (RASS) flux upper limit. Optical spectral measurements taken by the Magellan 6.5-m telescope and the Lick-Shane telescope both show a largely featureless spectrum, strengthening the BL Lac interpretation first proposed byBruni et al. (2018). The optical and mid-infrared (MIR) emission goes to a higher state in 2018, when the flux in high energies goes down to a lower level. Our RATAN-600m measurements at 4.8 GHz and 8.2 GHz do not indicate any significant radio flux variation over the monitoring seasons in 2017 and 2018, nor deviate from the archival NVSS flux level. During GeV flaring times, the spectrum is very hard (Γ γ ∼1.7) in the GeV band and at times also very hard ((Γ X ∼ < 2) in the X-rays, similar to a highsynchrotron-peak (or even an extreme) BL Lac object, making Fermi J1544-0649 a good target for ground-based Cherenkov telescopes.	10.1016/j.jheap.2020.02.004	[ "https://arxiv.org/pdf/2001.11772v1.pdf" ]	211,003,845	2001.11772	dff3dc2c28b5f8517361bda6ef817c384bf80038	LAT Flux (photons/s/cm 2 2018 Mar 11 2018 Jun 19 P H T Tam P S Pal Y D Cui N Jiang Y Sotnikova C W Yang L Z Wang B T Tang Y B Li J Mao A K H Kong Z H Zhong J Ding T Mufakharov J F Fan L M Dou R F Shen Y L Ai School of Physics and Astronomy Sun Yat-sen University 510275GuangzhouPeople's Republic of China CAS Key Laboratory for Researches in Galaxies and Cosmology University of Sciences and Technology of China 230026HefeiAnhuiPeople's Republic of China Special Astrophysical Observatory Russian Academy of Sciences Russian Federation. d Polar Research Institute of China 451 Jinqiao Road369167, 200136ShanghaiNizhnij ArkhyzPeople's Republic of China Joint Center for Astronomy Chinese Academy of Sciences South America Center for Astronomy Camino El Observatorio #1515China, Chile Las Condes SantiagoRepublic of Chile Yunnan Observatories Chinese Academy of Sciences 650011KunmingPeople's Republic of China Center for Astronomical Mega-Science Chinese Academy of Sciences 20A Datun Road100012Chaoyang District, BeijingPeople's Republic of China University of Chinese Academy of Sciences 100049BeijingPeople's Republic of China Key Laboratory for the Structure and Evolution of Celestial Objects Chinese Academy of Sciences 650011KunmingPeople's Republic of China Institute of Astronomy National Tsing Hua University Section 2. Kuang-Fu Road101, 30013HsinchuTaiwan, R.O.C Department of Astronomy & Astrophysics University of California Santa Cruz 1156 High Street95060Santa CruzCAUSA Shanghai Astronomical Observatory Chinese Academy of Sciences 200030ShanghaiPeople's Republic of China Russian Federation. n Center for Astrophysics Kazan Federal University 18 Kremlyovskaya St420008Kazan Guangzhou University LAT Flux (photons/s/cm 2 2018 Mar 11 2018 Jun 19. Graphical Abstract Multi-wavelength observations of the BL Lac object Fermi J1544-0649: one year after its awakening We report observations of a transient source Fermi J1544-0649 from radio to γrays. Fermi J1544-0649 was discovered by the Fermi-LAT in May 2017. Followup Swift-XRT observations revealed three flaring episodes through March 2018, and the peak X-ray flux is about 10 3 higher than the ROSAT all-sky survey (RASS) flux upper limit. Optical spectral measurements taken by the Magellan 6.5-m telescope and the Lick-Shane telescope both show a largely featureless spectrum, strengthening the BL Lac interpretation first proposed byBruni et al. (2018). The optical and mid-infrared (MIR) emission goes to a higher state in 2018, when the flux in high energies goes down to a lower level. Our RATAN-600m measurements at 4.8 GHz and 8.2 GHz do not indicate any significant radio flux variation over the monitoring seasons in 2017 and 2018, nor deviate from the archival NVSS flux level. During GeV flaring times, the spectrum is very hard (Γ γ ∼1.7) in the GeV band and at times also very hard ((Γ X ∼ < 2) in the X-rays, similar to a highsynchrotron-peak (or even an extreme) BL Lac object, making Fermi J1544-0649 a good target for ground-based Cherenkov telescopes.AbstractWe report observations of a transient source Fermi J1544-0649 from radio to γrays. Fermi J1544-0649 was discovered by the Fermi-LAT in May 2017. Followup Swift-XRT observations revealed three flaring episodes through March 2018, and the peak X-ray flux is about 10 3 higher than the ROSAT all-sky survey (RASS) flux upper limit. Optical spectral measurements taken by the Magellan 6.5-m telescope and the Lick-Shane telescope both show a largely featureless spectrum, strengthening the BL Lac interpretation first proposed byBruni et al. (2018). The optical and mid-infrared (MIR) emission goes to a higher state in 2018, when the flux in high energies goes down to a lower level. Our RATAN-600m measurements at 4.8 GHz and 8.2 GHz do not indicate any significant radio flux variation over the monitoring seasons in 2017 and 2018, nor deviate from the archival NVSS flux level. During GeV flaring times, the spectrum is very hard (Γ γ ∼1.7) in the GeV band and at times also very hard ((Γ X ∼ < 2) in the X-rays, similar to a highsynchrotron-peak (or even an extreme) BL Lac object, making Fermi J1544-0649 a good target for ground-based Cherenkov telescopes. Multi-wavelength observations of the BL Lac object Fermi J1544-0649: one year after its awakening Introduction Super-massive black holes (SMBHs; with mass ∼ >10 6 M ) locate at the centres of galaxies. By accreting a large enough amount of mainly gaseous materials, a SMBH can generate luminous emission over the whole electromagnetic spectrum, referred as an Active Galactic Nuclei (AGN; Lynden-Bell, 1969). Sometimes, an AGN could generate powerful jets, when the jet direction nearly coaligns with the line of sight, this AGN is seen as a blazar from the Earth -characterized by large variability at all wavelengths and usually accompanied by γray emission (e.g., Urry and Padovani, 1995). Variability at all wavelengths is a defining feature of blazars, so some blazars may be seen as a transient. They may remain quiet and only become bright in a relatively short time scale (e.g., months), and all-sky high-energy monitors like Fermi-LAT and MAXI may catch such rare events. An optical transient, ASASSN-17gs, or AT2017egv, was detected at V=17.3 mag on 2017-05-25 09:36 UT (i.e., contemporaneous to the LAT transient detections). The host galaxy, 2MASX J15441967-0649156, was found to be at a spectroscopic Redshift of z=0.171, using the MDM 2.4m Hiltner telescope on the night of 2017 June 14 UT (Chornock and Margutti, 2017). The persistent radio source at the same position, NVSS J154419-064913, has a flux density of 46.6 mJy at 1.4 GHz in 1996/1997 (Condon et al., 1998). This flux density, at a Redshift of 0.171, corresponds to 4×10 31 erg s −1 Hz −1 , a radio luminosity which is above most of known radio-loud AGN (see, e.g., Fig. 11 of Heckman and Best, 2014). GMRT observed 66.6±8.4 mJy at 150 MHz between April 2010 and March 2012 (Intema et al., 2017). In this work, we present detailed data analysis in γ-rays and X-rays, optical photometry and spectroscopy, radio flux monitoring in following sections. We further discuss our main findings in Section. 5, including the characteristic blazar SED peaked at X/γ-ray, the fast Xray variation with a time scale down to 1 hour, the mysterious continuum optical component with week-scale variations. Observed Evolution of the High-Energy Emission γ-ray Emission The LAT detector is an all-sky monitor at energies from several tens of MeV to more than 300 GeV (Atwood et al., 2009). The γ-ray data 1 used in this work were obtained using the Fermi-LAT between 2008 August 4 and 2018 August 15. We used the Fermi Science Tools v10r0p5 package to reduce and analyze the data. Pass 8 data classified as "source" events were used. To reduce the contamination from Earth albedo γ-rays, events with zenith angles greater than 100 • were excluded. The instrument response functions "P8R2 SOURCE V6" were used. To constrain the normalization of diffuse background and the spectral parameters of nearby sources for latter shorter-duration analysis, we first carried out a binned maximum-likelihood analysis (gtlike) of a rectangular region of 21 • ×21 • centered on the position of Fermi J1544-0649, using 9-years of data. To this end, we subtracted the background contribution by including the Galactic diffuse model (gll iem v06.fits) and the isotropic background (iso P8R2 SOURCE V6 v06.txt), as well as the third Fermi-LAT catalog (3FGL; Acero et al., 2015) sources within 25 • away from Fermi J1544-0649. The recommended spectral model for each source as in the 3FGL catalog was used, while we modeled a putative source at the position of Fermi J1544-0649 with a power-law (PL): dN dE = N 0 E E 0 −Γ ,(1) where the normalization N 0 and spectral index Γ were allowed to vary. The normalization parameter values for the Galactic and isotropic diffuse components, and sources within 6 • from Fermi J1544-0649 were allowed to vary as well. Other parameters were held fixed. Using the whole data set from the first 8.6 years, we did not detect any source at the Fermi J1544-0649 position. γ-ray flux over monthly time bins were also deduced by letting the normalization and photon index to vary in the iteration. No significant detection (i.e., above TS=12) was found until May 2017. The same was done for year time scale, and only during the last two years was the source detected significantly (see Fig. 1). We thus confirm this transient nature as a recent event. With the 8.6-year background model at hand, we carried out maximum likelihood analysis on 3-day/6-day bins from April 2017 to August 2018, and the results are plotted in Fig. 2. The average γ-ray photon index during the Fermi flares is about 1.7. Flux upper limits were deduced and plotted whenever TS<9. It can be seen that the flaring period lasts for 180 days since MJD 57888, and is composed of two major flares at May to June and August to September 2017. There is a third major flare, though smaller in magnitude, in March 2018. All three major γ-ray flares are accompanied by X-ray flare seen by Swift-XRT. The γ-ray flux goes to a lower level of activity in 2018, as compared to May through October in 2017. 2.2. X-ray Emission 2.2.1. XMM-Newton observation XMM-Newton DDT observation of Fermi J1544-0649 (obs-id: 0811213301) was performed on 21st February, 2018 (MJD 58170) for about 58 ksec. We used EPIC-PN data for the X-ray analysis, as they have higher sensitivity than EPIC-MOS data. We verify that the MOS data return consistent results as the pn data. The data reduction was performed with the software SAS (version 16.1), using the most updated calibration files (updated on May 2018). The event files were processed using 'epproc' with 'bad' (e.g., 'hot', 'dead', 'flickering') pixels removed. The periods with high background events were examined and excluded by inspecting the light curves in the energy band 10-12 keV. As the X-rays of Figure 1: The γ-ray, optical and MIR light curves of Fermi J1544-0649. The Fermi-LAT γ-ray photon flux (orange) show an increase in 2017 and 2018 relative to previous years. The V -band data are collected from public releases of CRTS (grey) and ASAS-SN (cyan); the MIR data are drawn from WISE database in W1 (blue) and W2 (red). The magenta line indicates MJD 57888 (i.e., 2017 May 15), when Fermi J1544-0649 was discovered and so the Fermi flux seen in the second last data is mostly from thereafter. Fermi J1544-0649 is bright and the pile-up effect is apparent in the source center, we extracted the source events from an annular region with inner radius of 7.5 and outer radius of 40 , using single and double events (PATTERN≤4, FLAG=0). The background events were collected from a source-free circular region of radius 40 within net exposure time of 29.93 ks, and are composed of a total of 165 thousand net source counts in 0.3-10 keV band. We grouped the pn spectra to have at least 25 counts in each bin, and we adopt the χ 2 statistic for the spectral fits. The fitted pn spectra are shown in Fig. 3. The spectral analysis were performed using XSPEC(v12.9.1m). The uncertainties are given at 90% confidence levels for one parameter. At first, a simple neutral-hydrogen absorbed power-law (PL) model (tbabs × zpo) was used, and we obtained χ 2 /dof = 946/900 with n H = (14.7 ± 0.27) × 10 20 cm −2 . To understand the absorption and the spectrum of the object, we compared different models. A simple neutral-hydrogen absorbed PL model (tbabs × zpo), in which the neutral hydrogen column density is fixed at the Galactic value of 8.98 × 10 20 cm −2 (Kalberla et al., 2005), is used. It results in χ 2 /dof = 2397/901, showing that the model does not work for the data. We then added an intrinsic absorber (ztbabs) into the model, where the Redshift is fixed at 0.17 (Chornock and Margutti, 2017). The fit is then much improved (χ 2 /dof =1015/900), and results in an intrinsic absorber with neutral hydrogen column density of (7.0 ± 2.0) × 10 20 cm −2 . However, there is some systematics in the lowest (0.3-0.8 keV) and highest energy (>7 keV). When trying an ionized absorber model (Zdziarski et al., 1995), the ionisation parameter is essentially zero, indicating that the absorber is not heavily ionized. We have also used a log-parabolic (LP) model (eplogpar, in which N (E) = 10 −b(log(E/Ep)) 2 /E 2 often used for blazars; Tramacere et al., 2007). Here we fix the column density at the Galactic value. We obtained (χ 2 /dof = 1121/901) for this model with peak energy E p = (0.85 ± 0.03) keV and a curvature b of 0.40 ± 0.02. These results are shown in Table. 1. We also tried to allow the column density to vary in the LP model, but the fit parameters are not stable. Based on the above analysis, the PL model (with intrinsic absorption) and the LP model (without intrinsic absorption) can both describe the whole data set well. We then looked into the timing analysis. The EPIC-PN cleaned light curve in 0.3-10.0 keV band in 100 s bin is shown in Fig. 4. The background was subtracted and the instrumental effect was corrected using the task 'epicclorr'. The X-ray light curve shows that Fermi J1544-0649 varies on timescales of a few ks. This prompted us to perform time-resolved spectral analysis. We divided the whole observation into 40 Bayesian blocks (Scargle et al., 2013), calculated with 95% statistical significance using Python module Astropy (Astropy Collaboration et al., 2013Collaboration et al., , 2018. In Fig. 5 we show the spectra for the two different block intervals. In Table. 2) as follows: black -736-3328 sec, red -42016-56064 sec. resolved spectral analysis we ignored the blocks with lesser number of photons. We used 13 block intervals for time dependent spectral analysis. We employed the PL model (tbabs × zpo) first. We also used the LP model in the time-resolved spectra. The time-resolved spectral analysis results are shown in Table. 2. It can be seen that the spectrum becomes harder when brighter, i.e., when the count rate decreases, the softness ratio increases and the peak energy (in the LP model) decreases (see, third panel Fig. 4). This shows that the goodness-of-fit (i.e., reduced χ 2 ) in the time-integrated fits is affected by the changing spectrum. To conclude, the PL model is as good as the LP model based on the goodness-of-fit (especially for the short time interval spectral fits). Based on the goodness-of-fit for timeresolved XMM-Newton spectra, we found that the LP model and the PL model can both describe the data well. However, from the broad-band SED, the X-rays represent the synchrotron bump, and it is anticipated that the X-ray spectrum is curved. Swift-XRT observations Since 2017 May 26, Swift monitoring observations have been performed. Here we present all XRT results obtained until 2018 July 25. For Swift-XRT (114) data reduction, the level 2 cleaned event files of SWIFT-XRT were obtained from the events of photon counting (PC) mode data with xrtpipeline. The spectra were extracted from a circular region in the best source position with 20 radius. The background was estimated from an annular region in the same position with radii from 30 to 60 . The ancillary response files (arfs) were extracted with xrtmkarf. The PC redistribution matrix file (rmf) version (v.12) was used in the spectral fits. XRT spectra are grouped with 5 counts per bin. XRT spectrum is then analysed with XSPEC(v12.9.1m) in the similar process as the XMM-Newton EPIC-PN spectra. We here fix the absorption column density to be n H = (14.7±0.27)×10 20 cm −2 , the value found from the XMM-Newton analysis. From the fitted spectra, unabsorbed flux values were calculated from 0.3-10 keV in cgs units for all observations. The Swift-XRT light curve and evolution of the power-law index are plotted in Fig. 2. Some but not all Swift spectra can be fitted with the LP model (tbabszashifteplogpar). After the discovery of the first major flare in 2017 May 26, the Swift X-ray light curve shows a second major flaring episode in August and September 2017. After that, the high energy flux has decreased to a lower level, besides a third flaring episode in February to March 2018 (see Fig. 2). X-ray correlation properties We perform correlation studies among the flux, PL and LP model parameters to gain insights on the radiation process (see Fig. 6). For all cases linear correlation gives better fit statistics than constant correlation. We calculate Pearson's Correlation Coefficient (Pearson, 1896) along with standard deviation (Bowley, 1928) for these model parameters using Python module Scipy (Virtanen et al., 2019). In Fig. 6(a), we compare the X-ray flux with the power-law index and find the following relation: f or XMM − Newton : Γ = (−0.11 ± 0.02) × (flux) + (3.05 ± 0.11)[χ 2 ν = 3.09(11)], (2) f or Swift − XRT : Γ = (−0.08 ± 0.01) × (flux) + (2.55 ± 0.09)[χ 2 ν = 1.9(39)].(3) The correlation coefficient of the X-ray flux and power-law index are r = −0.86 ± 0.08 (for XMM-Newton data) and r = −0.55 ± 0.11 (Swift data) respectively. These results indicate a harder-when-brighter behavior. In Fig. 6(b), we compare the X-ray flux with the peak energy. The corresponding correlation coefficient is r = 0.81 ± 0.13 (for XMM-Newton data) and r = 0.83 ± 0.08 (Swift data), confirming the harder-when-brighter behavior. Indeed, we find for XMM-Newton and Swift the following relation: f or XMM − Newton : E p = (0.36 ± 0.05) × (flux) − (1.30 ± 0.28)[χ 2 ν = 1.2(11)],(4)f or Swift − XRT : E p = (0.14 ± 0.04) × (flux) + (0.95 ± 0.17)[χ 2 ν = 1.0(16)].(5) Next, we study the relations between the peak energy (E p ), SED peak value (S p ) and spectral curvature b, in a similar manner as in Tramacere et al. (2007). For XMM-Newton and Swift data, we obtain f or XMM − Newton : ln S p = (0.46 ± 0.06) * ln E p + (0.77 ± 0.02)[χ 2 ν = 0.002 (7)], The unit of S p is 10 −11 erg cm −2 s −1 and E p is in keV. The Pearson's correlation coefficient between ln S p and ln E p is r = 0.86 ± 0.10 (for XMM-Newton data) and r = 0.71 ± 0.12 (Swift data), showing strong positive correlation. Within the context of the synchrotron emission from one dominant component, S p depends on E p as: S p ∝ E α p . The value of α ∼0.5-0.9, we find here is smaller than unity, indicating that the spectral change should be caused by variation of the electron average energy (α = 1.5), or to the magnetic field change (α = 2), but not due to change in the beaming factor (α = 4; Tramacere et al., 2007). The result is shown in Fig. 6(c). The correlation between ln b and ln E p is significant (r = −0.50 ± 0.29) for XMM-Newton data but not for Swift data (r = −0.66 ± 0.14) (with b ∝ E −0.45 p , and such a negative correlation is expected in statistical or stochastic acceleration; Tramacere et al., 2007, one should note that Swift data were taken over a long time span (i.e., more than a year) while the XMM-Newton observation was taken within a day). Therefore, it is plausible that different physics is driving the spectral shape (referring here to the peak energy and curvature) at different time scales. The result is shown in Fig. 6(d). In summary, owing to the high sensitivity of XMM-Newton, we have found the rapid X-ray variation from Fermi J1544-0649 with timescale down to ∼1 hour, and a hardening X-ray spectrum following the rise of the X-ray flux. Both of these findings support a blazar scenario, in which the X-ray emission is dominated by synchrotron emission of a relativistic jet component. Other X-ray observations MAXI-GSC 2-20 keV light curve with 1 day time bin was obtained from a circular region at the best source position with 1.6 • radius from MAXI online data reduction system 2 . Some excess can be seen in the light curve during and shortly after the two major flares in May and August 2017, respectively. ROSAT-PSPC observed this position of sky for a total exposure ∼480 s during the all-sky survey (RASS). Since there is no detection in this position, and taking 5 counts as a minimum for a detection, the upper limit of the count rate is approximately 0.01. The upper limit of energy flux is taken to be about 10 −13 erg cm −2 s −1 (2RXS; Boller et al., 2016). Swift-XRT observations thus revealed a peak X-ray flux more than 10 3 higher than this upper limit. These values are indicated by dashed lines in the XRT count rate and light curve panels in Fig. 2. 3. UV, Optical, and IR properties UV and optical photometric measurements For Swift-UVOT data reduction, all extensions of sky images were stacked with uvotimsum. The source magnitudes were derived with 3-σ significance level from the circular region of 5 radius in the best source position of the stacked sky images from all the filters with uvotsource. The background was estimated from an annular region in the same position with radii from 10 to 20 . The Swift-UVOT light curves (extinction not corrected) of different filters are plotted in Fig. 2. To check any color change, we also applied interstellar de-reddening on the U, B, and V-magnitudes. The value of extinction was estimated using the web-based calculator maintained by the NASA/IPAC Infrared Science Archive 3 (Schlafly and Finkbeiner, 2011), and the corrected values are shown in Fig. 7. It can be seen that there is no color change against different U-band flux. In particular, no bluer-when-brighter behavior is seen. On 2018 February 21, XMM-OM observed Fermi J1544-0649 in FAST mode for 12 exposures with different filters. For XMM-OM data reduction, all exposures of sky images are extracted with omfchain. The source magnitudes are derived with 3-σ significance level from the circular region of 5 radius in the best source position of the stacked sky images from all the filters with omdetect. The background is estimated from an annular region in the same position with radii from 10 to 20 . The absolute magnitudes obtained from the analysis are shown in Table. 3. In summary, as seen in Fig. 2, the evolution of the optical emission from Fermi J1544-0649 is independent of that of the high-energy (i.e., X/γ-ray) emission. We did several tests (including a zDCF code Alexander, 1997) and did not find a correlation between γ-ray or X-ray flux with optical/NIR flux. In particular, the average optical flux in 2018 is higher than that in 2017, but the average X/γray flux is higher in 2017 than in 2018. This may be due to two emission regions not directly related to each other. Long-term Optical and Mid-infrared light curves A comprehensive examination of the long-term variability of Fermi J1544-0649 in every other available band is helpful for us to understand its nature. We checked its optical and mid-infrared (MIR) light curves as shown in Fig. 1. The Sesar et al., 2007). The ASAS-SN data possess even larger errors due to its shallow survey depth. Despite that, we can still see a significant (i.e., at the 4-σ level) brightening in the latest two epochs (∼ 0.7 mag). In addition to the ground-based optical time-domain surveys, the Wide-field Infrared Survey Explorer (WISE Wright et al., 2010;Mainzer et al., 2014) has scanned a specific sky area every half year at 3.4 and 4.6 µm (labeled W1 and W2) since 2010 Feb (except for a gap between 2011 Feb and 2013 Dec) and thus yielded 12-13 times of observations for each object up to now. We downloaded all of the public WISE data of Fermi J1544-0649 up to the end of 2018 July, distribut-ing over 12 epochs at intervals of half year. For each epoch, there are typically 12 individual exposures within one day. Hence the WISE database allows us to study both its long-term and intra-day MIR variability. First, we binned the data every half year (as we have done in Jiang et al., 2016;Dou et al., 2016), which displays an obvious and continuous trend of brightening since 2017 February. The latest exposures taken in 2018 July has brightened by ∼1.3 and ∼1.5 magnitudes in W1 and W2, respectively, in comparison with two years earlier; such an increase is even larger than in the optical band. We have also tried to explore the possibility of intra-day variability in each epoch following Jiang et al. (2012);Jiang (2018), which may provide a direct evidence for the jet toward us. Nevertheless, the short-timescale variability is insignificant. In summary, there are long-term variations in both optical and MIR bands, that are indicators of past AGN activity. Moreover, both bands show a trend of recent brightening, especially in 2018 when the high-energy emission goes down to a lower state. Optical spectroscopy To look for any spectral feature in optical, we obtained three spectra in 2017 and 2018. We carried out an observation using the IMACS (f/2) spectrograph on the 6.5-m Magellan telescope on 2017 September 7, with a total exposure time of 800s. Two standard stars and He-Ne-Ar lamp spectra were taken before and after the exposure for flux and wavelength calibration. The raw two-dimensional data reduction and spectral extraction were accomplished using standard routines in IRAF. To extract the nuclear spectra, we used the APALL task and chose an aperture of 2 . We also performed a spectroscopic observation of Fermi J1544-0649 by the Yunnan Faint Object Spectrograph and Camera (YFOSC) on the 2.4m telescope, located at the Lijiang Station of Yunnan Observatories (longitude = 100 • 01 51 , latitude = 26 • 42 32 N, altitude = 3193 m) of the Chinese Academy of Sciences on 2018 February 28. Grism #14 of YFOSC, which has a resolution of 1.67Å pixel −1 and wavelength coverage of 3200-7500Å, was used. Given the seeing conditions, we employed a slit with a width of 2. 5. The total exposure time is 3300 s to achieve a high signal-to-noise (S/N) ratio. The spectroscopic data were reduced following the standard procedures using IRAF, including bias and flat correction, cosmic ray rejection, spectrum extraction, wavelength calibration, and flux calibration. When extracting the spectrum, the aperture was selected to reach 2% of the peak value to include most of the light from the source; a good S/N ratio can therefore be obtained. The emission line of He-Ne lamp was used for wavelength calibration. The aperture of the lamp spectrum was identical to the aperture of the source, ensuring that function between wavelength and the position corresponds to the aperture of the source. BD+33d2642 is used as the spectroscopic standard star to calibrate the flux of the object. Considering the airmass of the standard star and the extinction coefficient at Lijiang Station, the sensitivity function can be determined by using the counts and the flux at each wavelength for the standard star. Then the sensitivity function is applied to Fermi J1544-0649 to convert the counts back to flux for Fermi J1544-0649. The airmass of the object, which is different from the standard star, is also considered. On 2018 May 11, we obtained a medium resolution (R∼2000) spectrum using the Kast double spectrograph (consisting of red and blue channels) on the 3-m Shane telescope at the Lick Observatory. We used the 600/4310 grism on the blue size and 600/5000 grating on the red side with a wavelength coverage approximately 3300-5500Å and 5500-8000Å. We apply a 1. 5 slit aligned at parallactic angle for the observation with a 30-minute exposure for both channels. The flux calibration is based on the spectrophotometric standard star Feige 67. The Magellan and Lick spectra obtained are largely featureless with only weak absorption lines (Fig. 8). These spectra are consistent with Fermi J1544-0649 being a BL Lac object. Along with Lijiang observation, these three optical spectra (taken at times shown by the dashed lines of Fig. 2) has shown strong variation which is not correlated with the X&γ-ray variation. We also checked that the stellar absorption lines from host galaxy (e.g. CaII, K, MgI and NaI Paiano et al., 2017) in the Lijiang spectrum have a redshift consistent with 0.17 (Chornock and Margutti, 2017;Bruni et al., 2018). The Lijiang spectrum was taken on 2018 February 28. Surprisingly, in the blue end, there are two unidentified BAL-like features at around 4000Å and 4200Å. If it is true, such BAL feature would indicate fast gas outflows blocking the line of sight towards the unknown optical source. This phenomenon is often observed in quasars (Weymann et al., 1991), and has been seen in at least one BL Lac object, PKS B0138-097 (Zhang et al., 2011). Such BALlike feature does appear just a week after the XMM-OM observation (21 February 2018, MJD 58170) when the optical spectrum (as seen by the magnitudes in different filters) is very blue, as compared to those taken in other epochs (Fig. 2 fifth panel); thus the BAL-like feature is seen during a unique optical color-changing state. The Lijiang observation is performed at the second part of the night before the early morning, sometime, the observation condition changes quickly due to the frog or wet air. However, according to the note by the Lijiang observer, there is no evidence of either quick air change, or instrumental malfunction during the Figure 8: The Lijiang 2.4m spectrum taken on 2018 February 28 (where two possible broad absorption lines (BALs) at around 4000Å and 4200Å are evident), is compared with those taken by the Magellan 6.5m telescope on 2017 September 7 and the Lick-Shane telescope on 2018 May 12. The two crosses mark the position of telluric absorption. Indicated by vertical dashed lines are those of CaII and K, MgI and NaI at z=0.171 (from left to right), but the NaI position is very close to a telluric feature and thus the identification is only tentative. The Magellan spectrum is smoothed by 3 pixels and the Lijiang and Lick-Shane spectrum by 5 pixels for display purpose. observation time. Fermi J1544-0649 was observed at 2018-02-28 20:41:55.601 for 3300s, and the airmass is 1.359291. The standard star bd332642 was observed at 2018-02-28 20:33:05.119 for 200s, and the AIRMASS is 1.12949. Of course, we could not exclude any undiscovered problems related to the instrument. RATAN 600-meter radio observations The measurements of the fluxes were obtained with the RATAN-600m radio telescope in transit mode by observing simultaneously at 1. 2, 2.3, 4.8, 8.2, 11.2, and 21.7 GHz. The observations were carried out during October and December 2017, and January, February-April and July 2018. The parameters of the antenna and receivers are listed in Table. 4, where f is the central frequency, ∆f is the bandwidth, ∆F is the flux density detection limit per beam, and BW -beam width (full width at half-maximum in RA). The detection limit for the RATAN single sector is approximately 5 mJy (the time of integration is 3 s) under good conditions at the frequency of 4.8 GHz and at an average antenna elevation. We averaged the data of observations for 2-25 days in order to get a reliable values of the flux density. Data were reduced using the RATAN standard software FADPS (Flexible Astronomical Data Processing System) reduction package (Verkhodanov, 1997). The flux density measurement procedure is described by Mingaliev et al. (2012Mingaliev et al. ( , 2014; Udovitskiy et al. (2016);Mingaliev et al. (2017). The following flux density calibrators were applied to obtain the calibration coefficients in the scale by Baars et al. (1977): 3C48, 3C147, 3C161, 3C286 and NGC7027. We also used the traditional RATAN flux density calibrators: J0237−23, 3C138, J1154−35, and J1347+12. Measurements of some calibrators were corrected for linear polarization and angular size, following the data from Ott et al. (1994) and Tabara and Inoue (1980). The systematic uncertainty of the absolute flux scale (3-10% at different RATAN frequencies) is not included in the flux error. The total error in the flux density includes the uncertainty of RATAN calibration curve and the error in the antenna temperature measurement. The radio emission was detected at 4.8 GHz and 8.2 GHz only. We measured the flux density at 4.8 GHz for each single scan. At the frequency of 8.2 GHz we used all scans in each observation epoch (Fig. 9) to get average flux density. We did not find any significant variation of the flux density in the 2017-2018 measurements. The average flux densities at 4.8 and 8.2 GHz and number of observations in each month are presented in Table. 5. Figure 9: The light curves for Fermi J1544-0649 at 4.8 and 8.2 GHz, obtained by RATAN-600m observations. The most enigmatic behavior of Fermi J1544-0649 is its recent 'turn-on' highenergy emission (X-rays and γ-rays) for about a year, while it remained quiescent for the past decade or so (see Fig. 1). During this 'turn-on' state, the γ-ray flux shows variabilities with a minimum time scale down to weeks. Bruni et al. (2018) suggest that Fermi J1544-0649 is a BL Lac object. Blazars are well known sources that show variability at all time scales from decades down to intra-day (i.e., IDV). And indeed, with XMM-Newton, who has a much better sensitivity than Fermi, we have discovered < 1 hour variation during a GeV low state. If Fermi J1544-0649 is a previously unknown blazar, its high-energy flux is constrained to be very low by all-sky monitors like Fermi-LAT and MAXI for around a decade before 2017, as well as ROSAT in the 1990s. The high-energy flares that began in May 2017 thus indicates a high-state never seen before for Fermi J1544-0649. Particularly in γ-rays, Fermi J1544-0649 remains in the quiescent state for nearly 9 years, which is a rather long period for a Fermi blazar. When comparing the γ-ray spectrum of Fermi J1544-0649 with sources in the second LAT AGN catalog (2LAC; Ackermann et al., 2011), the mean photon index of the three major categories of γ-ray BL Lac objects, high-synchrotron-peak (HSP), intermediate-synchrotron-peak, and low-synchrotron-peak, is 1.84, 2.08, and 2.32, respectively. With an average photon index of about 1.7, Fermi J1544-0649 (during its flares) has a spectral index at γ-rays well within that of HSP BL Lac objects. In both the low and flaring state, the SED of Fermi J1544-0649 remains a typical blazar SED, which consists mainly of a synchrotron peak in X-rays and an inverse-Compton peak in γ-rays. During the high state in May 2017, the γ-ray peak indicates Fermi J1544-0649 to be a high-frequency-peaked BL Lac object. The changing X-ray photon index of Fermi J1544-0649 is around Γ ∼ 2.0 during the major flaring period. i.e., 1.63 ± 0.13 on MJD 57936, 1.67 ± 0.0.07 on MJD 57991, and 1.63 ± 0.09 on MJD 58009, making Fermi J1544-0649 to be an extreme blazar candidate (at times) based on the synchrotron peak frequency (Abdo et al., 2010;Fan et al., 2016). Extreme high-frequency-peaked BL Lac objects (EHBLs), like 1ES 0229+200 and 1ES 1101-232, are blazars with a very high synchrotron peak (c.f., >1 keV; Costamante et al., 2001) and usually exhibit exceptionally hard TeV spectra, and they are good probes of the Extra-galactic Background Light (EBL) and Extra-galactic Magnetic Field (EGMF). Yet the sample of extreme blazars remains small (Costamante et al., 2018). Mkn 501 behaved like an EHBL throughout the 2012 observing season, with low and high-energy components peaked above 5 keV and 0.5 TeV, respectively. This suggests that being an EHBL may not be a permanent characteristic of a blazar, but rather a state which may change over time (Pian et al., 1998;Ahnen et al., 2018). Future Xray/TeV measurements may help us, not just to further probe the synchrotron/IC peak of Fermi J1544-0649, but also to unveil its rapid variations, even during a GeV low state. A simple SSC model for this blazar candidate -Fermi J1544-0649 Following the hypothesis that Fermi J1544-0649 is a blazar, we employ a simple Synchrotron Self-Compton (SSC) model to estimate the physics in Fermi J1544-0649. This SSC zone could be either from the main jet in a typical blazar scenario, or from a mini-jet in a misaligned blazar scenario. Since the source is varying by a large factor, spectral energy distribution (SED) are shown for two representative dates: 2017 May 26 (during the first flare) and 2017 July 1 (between the first and second flare), as shown in Fig. 10. It has a broad-band SED consistent with a BL Lac object. During the SSC fitting of the SED of Fermi J1544-0649, the first constraint comes from the relatively low UV emission when compared to the X-ray emission, clearly the UV emission is supposed to be mainly from the host galaxy and other sources rather than this synchrotron emission, in case of Fermi J1544-0649 it is particularly dominated by an unknown continuum component that varies independent from the X&γ-ray emission; therefore, the observed UV emission can Table 6: Parameters used in the SED fits. Here E e only represents the total electron energy currently in the GeV emitting region, it is a free parameter, meanwhile the total energy output per second for a jet is around 10 45 erg/s. only serve as an upper limit in our spectrum fitting, and a very hard electron spectral index of −1.4 is introduced in both the high state and the low state fittings. Noticeably, the intrinsic UV flux accompanied by the γ−ray flare could be much higher than this limit, due to the unknown extinction. Epoch B R b δ p E cut E e (G)( With such a hard electron spectrum, the inverse Compton (IC) fitting of the low state requires an electron cutoff energy of a quite low energy (8 GeV in Fig 10), in order to constrain the IC peak at <10 25 Hz; As a consequence of choosing such a low cutoff energy, a very compact gamma-ray emitting region is needed to boost up the IC flux. In Fig. 10, we adopt a R b size of merely 2.3×10 14 cm in the comoving frame, which corresponds to a minimum GeV variability of 10 minutes when using a Doppler factor of 25 (which is in the range of Doppler factors for blazars, e.g., Savolainen et al., 2010;Fan et al., 2014;Liodakis et al., 2017). As a comparison, the radius of the Innermost Stable Circular Orbit of a 3 × 10 8 M black hole (Bruni et al., 2018) is roughly 2.4×10 14 cm. The observed GeV spectrum of the high state is very hard and it allows the electron cutoff energy to move freely above 10 GeV during the fitting. This large parameter space of fitting the high state is also due to the lack of direct observational constraints on the magnetic field B, Doppler factor δ, and the size of the gamma-ray emitting region R b . In Fig. 10, we have shown one of the many fitting results with an electron cutoff energy of 25 GeV. More parameter details about our fitting can be found in Table. 6. The mysterious optical variation and MIR flare The strong optical variation of Fermi J1544-0649, as seen in Fig. 2, does not show one-to-one consistency with the X-/γ-rayvariation. Thus, besides the X/γray flare component and the host galaxy, alternative sources are likely to dominate the optical band, e.g., other blobs in the jet or even from the core region. γ-ray sources in BL Lac objects are normally considered as the relativistic shocks inside the jet. It is well accepted that the jet flow is very likely to be intermittent (Wang and Zhou, 2009). When an injected flow catches up with a slower flow or hits some over-dense medium, a shock is formed. The observed blobs in the jet, as seen in those best observed relativistic jets, e.g., M87 (Owen et al., 1989) and 3C 120 (Casadio et al., 2015), are commonly interpreted as shock waves moving along the jet. In case of an intermittent power injection, the shock could slow down and the opening angle of the beaming emission will be widening, the rise and the decay of an internal shock could naturally cause a variety of light curves based on different viewing angles. Clearly, each shock (knot) of the jet could result different non-thermal emissions, see e.g. the well observed jet of 3C 273 and M87 by HST (Biretta et al., 1999;Bahcall et al., 1995) and Chandra (Wilson and Yang, 2002;Sambruna et al., 2001), which has shown clearly resolved knots along the jet, and their peak energies gradually move from X-ray to optical with increasing distances to the nucleus. To explain the violent variation of this unknown optical source (with timescale down to 1 week) of Fermi J1544-0649, an intrinsic rise & decay of the accelerator close to the BH could cause the optical flare. Additionally, many magnetic launching models suggests a jet field of helix structure, which is also been supported by the polarization observations (e.g., Gabuzda et al., 2004). In the helical model, strongest emission is obtained when the shock wave reaches the bent regions towards the observer (Gomez et al., 1994). In the case where an alternative blob dominating the optical flare through its synchrotron emission, correlated infrared flares are likely to be observed (which is indeed observed in 2018, Fig. 1). In our simple SSC model above, the optical observation only functions as an upper-limit. Clearly, one-zone jet models, (see, e.g., Bruni et al., 2018), which includes a relativistic jet with a single emitting zone, an accretion disk, and the host galaxy, will face difficulties to explain the observed UV excess, the strong optical variation and MIR flare seen in 2018. Conclusion and Outlook The high-energy transient Fermi J1544-0649 is likely due to a sudden energy release (geometrical beaming may play a role as well) of a previously unknown BL Lac object, as first suggested by Bruni et al. (2018) -no high-energy emission was ever seen over the ∼9 years' lifetime of the Fermi satellite (nor by MAXI in X-rays) before April 2017. It is important to understand the mechanism that causes the sudden increase of radiation recently. We argue that a shock-in-jet model combined with viewing angle effects may explain the high-energy flares. Fermi J1544-0649 displays the typical blazar characteristics including strong X/γ-ray flares after a decade-long quiescent period, a typical blazar SED, and rapid variations with timescale down to <1 hour, The optical flux of Fermi J1544-0649 does not vary at the same time with the X/γ-ray flux. Thus, besides the X/γray flare component and the host galaxy, alternative sources are likely to dominate the optical band. Being a HSP BL Lac object, Fermi J1544-0649 is likely a TeV-emitting blazar, and it shows huge variation in X-rays and γ-rays. Observing Fermi J1544-0649 by current and/or upcoming Cherenkov Telescope Array, e.g., CTA will tell us about the position of the Compton peak, that in turn will constrain the jet physics. At times being an extreme blazar, it would also be another blazar to probe extreme particle acceleration, EBL, and/or EGMF. Acknowledgments We thank Nidia Morrell for helping during the Magellan observation, and Da-Hai Yan for useful discussion. PHT is supported by the National Science Foundation of China (NSFC) grants 11633007, 11661161010, and U1731136. JM is supported by the NSFC grants 11673062, the Hundred Talent Program of Chinese Academy of Sciences, the Major Program of Chinese Academy of Sciences (KJZD-EW-M06), and the Oversea Talent Program of Yunnan Province. JHF is supported by NSFC grants 11733001 and U1531245. This work is sponsored (in part) by the Chinese Academy of Sciences (CAS), through a grant to the CAS South America Center for Astronomy (CASSACA) in Santiago, Chile. This research made use of data supplied by the High Energy Astrophysics Science Archive Research Center (HEASARC) at NASA's Goddard Space Flight Center, and the UK Swift Science Data Centre at the University of Leicester. This publication makes use of data products from the Wide-field Infrared Survey Explorer (WISE), which is a joint project of the UCLA, and JPL/California Institute of Technology, funded by NASA. This publication also makes use of data products from NEOWISE, which is a project of the JPL/California Institute of Technology, funded by the Planetary Science Division of NASA. This work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. This paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile. Figure 2 : 2From top to bottom: The Fermi-LAT 0.1-300 GeV photon flux, Swift-XRT 0.3-10 keV count rate, energy flux, photon index, Swift-UVOT magnitudes of various filters, and WISE magnitudes of Fermi J1544-0649 as seen between February 2017 and August 2018. The three arrows in the top panel indicates the dates of spectroscopic observations. When the source is not detected by Fermi-LAT, 90% confidence level upper limits were derived and are plotted in grey. The error bars in the second, fifth and sixth panels are smaller than the symbols. For comparison, the dashed, horizontal lines show the upper limits (a count rate of 0.01 cts/s and an energy flux of 10 −13 erg cm −2 s −1 ) estimated from the RASS observations. A large increase in X-ray flux (up to three orders-of-magnitude) is clearly seen. In the second, third, and fourth panel the black diamonds represent the XMM-EPIC PN result from the 2018 February 21 observation. In the second panel the shown (SWIFT-XRT equivalent) count rate is converted from XMM-EPIC PN count rate with the help of WebPIMMS. Figure 3 : 3EPIC-pn spectrum fitted with different models for the whole observation Figure 4 :Figure 5 : 45Fig. 4top panel blocks are shown with red color along with the cleaned light curve. In the second panel we have calculated the softness ratio between (0.3-2.0) keV and (2.0-10.0) keV energy bands for the Bayesian block intervals. For time XMM-Newton EPIC-PN cleaned light curve for 0.3-10.0 keV (first panel) energy bands with 100 s time bins and the Bayesian block intervals with 95% statistical significance in red color, along with the softness ratio (second panel) and evolution of the peak energy (E p ) in the log-parabola model (third panel). XMM-Newton EPIC-PN cleaned time-resolved spectra for comparison. The color codes represent the time intervals (as defined in Figure 6 : 6Correlation between model parameters. XMM-Newton time-resolved data are indicated by red diamonds, and Swift data by black circles. (a) Power-law index versus count rate. (b) Peak energy (E p ) versus count rate. (c) SED peak value (S p ) versus SED peak energy (E p ). (d) Spectral curvature b versus peak energy (E p ). ln b = (−0.73 ± 0.42) * ln E p − (0.95 ± 0.16)[χ 2 ν = 0.01(7)], (7) f or Swift − XRT : ln S p = (0.92 ± 0.19) * ln E p − (0.07 ± 0.13)[χ 2 ν = 0.003(16)], (8) ln b = (−0.41 ± 0.14) * ln E p − (0.16 ± 0.07)[χ 2 ν = 0.03(16)]. Figure 7 : 7Left: the flux-color plot for optical observations. The x-and y-axis is the U-band magnitude and B −V color index, respectively. Right: B-or V-band magnitude versus U-band magnitude for various optical flux. The black straight line has a slope of 0.83 ± 0.03, and the red line a slope of 0.81 ± 0.05. Galactic extinction is corrected for in plotting these figures (see Section. 3.1). Figure 10 : 10The SED for two representative dates: 2017 May 26 (left panel for the first flare) and 2017 July 1 (right panel, between first and second flare). The model lines are from a one-zone SSC model described in the text. The Fermi spectra were derived from 2017 May 25-27 and 2017 June 30-July 2, respectively. The 4.8 and 8.2 GHz data are from RATAN-600m observations taken in September 2017. Galactic extinction is not corrected for the UV/optical/NIR data shown. Table 1 : 1XMM-Newton EPIC-pn spectral analysis result of Fermi J1544-0649 on 2018 February 21. (I) PL represents tbabs×zpo, (II) PL represents tbabs×ztbabs×zpo and (III) LP represents tbabs×zashift×eplogpar model components.Models n H (Galactic) n H Γ Ep b 10 11 ×Flux χ 2 ν (10 22 cm −2 ) (10 22 cm −2 ) (keV) (erg s −1 cm −2 ) (dof) (I) PL . . . 0.14 ± 0.003 2.47 ± 0.01 . . . . . . 4.68 ± 0.02 1.05(900) (II) PL 0.0898(fixed) 0.07 ± 0.01 2.45 ± 0.01 . . . . . . 4.57 ± 0.02 1.128(900) (III) LP 0.0898(fixed) . . . . . . 0.85 ± 0.03 0.40 ± 0.02 5.13 ± 0.02 1.246(901) Table 2 : 2XMM-Newton EPIC-PN time-resolved spectral analysis results of Fermi J1544-0649 on 2018 February 21Model (I): Power law Interval n H Γ 10 11 × F lux χ 2 ν (s) (10 22 cm −2 ) (erg s −1 cm −2 ) (dof) 0-736 0.14 ± 0.02 2.15 ± 0.07 8.12 ± 0.18 0.85(179) 736-3328 0.13 ± 0.01 2.16 ± 0.03 8.55 ± 0.10 1.17(458) 3424-4320 0.11 ± 0.02 2.12 ± 0.07 6.86 ± 0.14 1.05(196) 4640-6240 0.13 ± 0.01 2.17 ± 0.05 6.40 ± 0.11 0.98(302) 6240-6816 0.17 ± 0.03 2.40 ± 0.11 6.06 ± 0.18 0.96(108) 6816-7936 0.14 ± 0.02 2.27 ± 0.07 6.30 ± 0.13 0.94(208) 8128-8640 0.14 ± 0.03 2.24 ± 0.11 6.61 ± 0.19 1.23(111) 12640-13120 0.16 ± 0.04 2.51 ± 0.15 5.10 ± 0.18 0.94(79) 30624-32512 0.15 ± 0.04 2.66 ± 0.10 4.36 ± 0.10 1.06(166) 33120-35136 0.13 ± 0.09 2.66 ± 0.11 3.78 ± 0.09 0.98(155) 35232-36640 0.21 ± 0.05 2.90 ± 0.15 4.91 ± 0.14 1.34(114) 37632-40832 0.17 ± 0.04 2.83 ± 0.09 3.93 ± 0.08 0.87(210) 42016-56064 0.14 ± 0.02 2.65 ± 0.04 3.47 ± 0.03 1.01(451) Model (III): Log-parabolic (absorption fixed at the Galactic value) Interval E p b 10 11 × S p χ 2 ν (s) (keV) (erg s −1 cm −2 ) (dof) 0-736 2.10 ± 0.31 0.32 ± 0.12 3.04 ± 0.50 0.86(179) 736-3328 1.64 ± 0.17 0.24 ± 0.06 3.26 ± 0.50 1.15(458) 3424-4320 1.26 ± 0.74 0.15 ± 0.12 2.67 ± 0.50 1.05(196) 4640-6240 1.47 ± 0.30 0.21 ± 0.09 2.44 ± 0.50 0.98(302) 6240-6816 1.40 ± 0.22 0.53 ± 0.19 2.16 ± 0.50 0.97(108) 6816-7936 1.28 ± 0.24 0.33 ± 0.12 2.40 ± 0.50 0.93(208) 8128-8640 1.19 ± 0.63 0.23 ± 0.18 2.49 ± 0.50 1.26(111) 12640-13120 0.93 ± 0.34 0.49 ± 0.25 1.93 ± 0.50 0.93(79) 35232-36640 0.62 ± 0.47 0.60 ± 0.30 1.76 ± 0.51 1.35 Table 3 : 3XMM-Newton OM analysis results of Fermi J1544-0649 on 2018 February 21Exposure Exposure OM Absolute identifier (s) Filter magnitude S014 4400 V 16.96±0.02 S015 4400 U 16.86±0.01 S016 4400 B 17.69±0.01 S017 4400 B 17.63±0.01 S018 4400 UVW1 16.57±0.02 S019 4400 UVW1 16.48±0.01 S020 4400 UVM2 16.71±0.04 S021 4400 UVM2 16.80±0.04 S022 4400 UVM2 16.71±0.04 S023 4400 UVW2 16.87±0.07 S024 4400 UVW2 16.75±0.06 S025 3340 UVW2 16.89±0.08 optical (V -band) data are retrieved from public searching server of Cataline Real- Time Transient Survey 4 (CRTS; Drake et al., 2009) and All-Sky Automated Survey for Supernova (ASAS-SN Shappee et al., 2014; Kochanek et al., 2017) 5 . Although with large photometric errors, a long-term variation is clearly visible, indicative of an AGN. After measurement errors are taken into account, the CRTS variability amplitude (∆V ) is ∼ 0.07 mag (e.g., Equation 6 in Table 4 : 4Parameters of the RATAN-600m antenna and radiometersf ∆f ∆F FWHM x GHz GHz mJy beam −1 arcsec 21.7 2.5 50 11 11.2 1.4 15 16 8.2 1.0 10 22 4.8 0.6 5 35 2.25 0.08 40 80 1.28 0.08 200 110 Table 5 : 5The monthly-average flux densities of Fermi J1544-0649 obtained with the RATAN-600m.Epoch N obs S 8.2GHz S 4.8GHz (Jy) (Jy) September 2017 2457998-2458000 3 0.044±0.005 0.039±0.002 December 2017 2458097-2458114 11 0.033±0.005 0.038±0.003 January 2018 2458138-2458144 6 0.037±0.005 0.040±0.002 February 2018 2458155-2458178 21 0.028±0.005 0.039±0.002 March 2018 2458189-2458211 16 0.042±0.006 0.032±0.002 April 2018 2458222-2458239 11 0.035±0.005 0.030±0.002 May 2018 2458240-2458252 10 0.021±0.005 0.031±0.002 July 2018 2458302-2458318 5 0.030±0.004 0.040±0.003 provided by the FSSC at http://fermi.gsfc.nasa.gov/ssc/ MAXI on-demand queries, http://134.160.243.88/mxondem/ 3 http://irsa.ipac.caltech.edu/applications/DUST/ http://nunuku.caltech.edu/cgi-bin/getcssconedb_release_img. cgi 5 https://asas-sn.osu.edu/ The Spectral Energy Distribution of Fermi Bright Blazars. 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[ "A Value Distribution Result and Some Normality Criteria using Partial Sharing of Small Functions", "A Value Distribution Result and Some Normality Criteria using Partial Sharing of Small Functions" ]	[ "K S Charak \nDepartment of Mathematics\nUniversity of Jammu\nJammu-180 006INDIA\n", "Shittal Sharma \nDepartment of Mathematics\nUniversity of Jammu\nJammu-180 006INDIA\n" ]	[ "Department of Mathematics\nUniversity of Jammu\nJammu-180 006INDIA", "Department of Mathematics\nUniversity of Jammu\nJammu-180 006INDIA" ]	[]	In this paper, we first generalize a value distribution result of Lahiri and Dewan [4] and as an application of this result we prove a normality criterion using partial sharing of small functions. Further, in sequel normality criteria of Hu and Meng [3] and Ding, Ding and Yuan [1] are improved and generalized when the domain D := {z : \|z\| < R, 0 < R ≤ ∞}.	null	[ "https://arxiv.org/pdf/1412.8274v1.pdf" ]	118,776,318	1412.8274	5c43eed16278dd9c20b09dd01b9668890d53efd9	A Value Distribution Result and Some Normality Criteria using Partial Sharing of Small Functions 29 Dec 2014 K S Charak Department of Mathematics University of Jammu Jammu-180 006INDIA Shittal Sharma Department of Mathematics University of Jammu Jammu-180 006INDIA A Value Distribution Result and Some Normality Criteria using Partial Sharing of Small Functions 29 Dec 2014Normal FamiliesMeromorphic FunctionsDifferential Monomi- alsSharing of values AMS subject classification: 30D3530D45 In this paper, we first generalize a value distribution result of Lahiri and Dewan [4] and as an application of this result we prove a normality criterion using partial sharing of small functions. Further, in sequel normality criteria of Hu and Meng [3] and Ding, Ding and Yuan [1] are improved and generalized when the domain D := {z : \|z\| < R, 0 < R ≤ ∞}. Introduction and Main Results We assume that the reader is familiar with the theory of normal families of meromorphic functions on a domain D ⊆ C, one may refer to [6]. The idea of sharing of values was introduced in the study of normality of families of meromorphic functions, for the first time, by W. Schwick [7] in 1989. Two non-constant meromorphic functions f and g are said to share a value ω ∈ C IM(Ignoring multiplicities) if f and g have the same ω−points counted with ignoring multiplicities. If multiplicities of ω−points of f and g are counted, then f and g are said to share the value ω CM. For deeper insight into the sharing of values by meromorphic functions, one may refer to [10]. In this paper all meromorphic functions are considered on D := {z : \|z\| < R, 0 < R ≤ ∞} excepting Theorem A and Theorem 1.1, where the domain is the whole complex plane. A meromorphic function ω(z) is said to be a small function of a meromorphic function f (z) if T (r, ω) = o (T (r, f )) as r −→ R. Further, we say that a meromorphic function f share a small function ω partially with a meromorphic function g if E(ω, f ) = {z ∈ C : f (z) − ω(z) = 0} ⊆ E(ω, g) = {z ∈ C : g(z) − ω(z) = 0}, where E(ω, φ) denotes the set of zeros of φ − ω counted with ignoring multiplicities. The function of the form M [f ] = f n0 (f ′ ) n1 · · · (f (k) ) n k is called a differential monomial of f of degree d = n 0 + n 1 + · · · + n k , where n 0 , n 1 , · · · , n k are non-negative integers. In the present discussion, we have used the idea of partial sharing of small functions in the study of normality of families of meromorphic functions. One can verify that a good amount of results on normal families proved by using the sharing of values can be proved under weaker hypothesis of partial sharing of values or small functions. Lahiri and Dewan [4] proved the following result: Theorem A Let f be a transcendental meromorphic function and F = (f ) n0 (f (k) ) n1 , where n 0 (≥ 2), n 1 and k are positive integers such that n 0 (n 0 −1)+(1+k)(n 0 n 1 − n 0 − n 1 ) > 0. Then 1 − 1 + k n 0 + k − n 0 (1 + k) (n 0 + k){n 0 + (1 + k)n 1 } T (r, F ) ≤ N r, 1 F − ω + S(r, F ) for any small function ω( ≡ 0, ∞) of f . This is natural to ask whether Theorem A remains valid for a general class of monomials. In this direction, we have proved that it does hold for a larger class of monomials. Precisely, we have Theorem 1.1. Let f be a transcendental meromorphic function. Let F = f n0 (f ′ ) n1 · · · (f (k) ) n k , (1.1) where k, n 0 , n 1 , · · · , n k are non-negative integers with k ≥ 1, n 0 ≥ 2 and n k ≥ 1 such that n 0 (n 0 − 1) + k j=1 (j + 1)(n 0 n j − n j − n 0 ) + (k − 1)n 0 > 0. (1.2) Then 1 − 1 + k(k+1) 2 n 0 + k(k+1) 2 − n 0 (1 + k(k+1) 2 ) {n 0 + k(k+1) 2 }{n 0 + k j=1 (j + 1)n j } + o(1) T (r, F ) ≤ N r, 1 F − ω + S(r, F ) (1.3) for any small function ω( ≡ 0, ∞) of f. Note: When f has no poles then Theorem 1.1 holds without the condition (1.2). As an application of Theorem 1.1, we prove a normality criterion using the idea of partial sharing of small functions. Theorem 1.2. Let F be a family of meromorphic functions such that each f ∈ F has only zeros of multiplicity at least k ≥ 2. Let n 0 , n 1 , · · · , n k be nonnegative integers with n 0 ≥ 2, n k ≥ 1 such that n 0 (n 0 − 1) + k j=1 (j + 1)(n 0 n j − n 0 − n j ) + (k − 1)n 0 > 0 Let ω(z) be a small function of each f ∈ F having no zeros and poles at the origin. If there exists f ∈ F such that M [f ] share ω partially with M [ f ], for every f ∈ F , then F is a normal family. Further, one can see that Theorem 4.1 of Hu and Meng [3] may be generalized to a class of monomials as Theorem 1.3. Let k ∈ N and F be a family of non-constant meromorphic functions such that each f ∈ F has only zeros of multiplicity at least k. Let n 0 , n 1 , · · · , n k be non-negative integers with n 0 ≥ 2, n k ≥ 1 such that n 0 (n 0 − 1) + k j=1 (j + 1)(n 0 n j − n 0 − n j ) + (k − 1)n 0 > 0. Let ω(z) be a small function of each f ∈ F having no zeros and poles at the origin. If, for each f ∈ F , (M [f ] − ω) (z) = 0 implies \|f (k) (z)\| ≤ A, for some A > 0, then F is a normal family. Proof of Main Results Proof of Theorem 1.1: Since (see [8]) T (r, f ) + S(r, f ) ≤ CT (r, F ) + S(r, F ) and T (r, F ) ≤   n 0 + k j=1 (j + 1)n j   T (r, f ) + S(r, f ), where C is a constant, it follows that T (r, ω) = S(r, F ) as r −→ ∞. Precisely, ω is a small function of f iff ω is a small function of F . Now, by Second Fundamental Theorem of Nevanlinna for three small functions(see [2] pp. 47), we have [1 + o(1)]T (r, F ) ≤ N (r, F ) + N (r, 1 F ) + N r, 1 F − ω + S(r, F ). (2.1) Next, we have N (r, 1 F ) ≤ N (r, 1 f ) + k j=1 N 0 (r, 1 f (j) ) ≤ N (r, 1 f ) + k j=1 j N (r, 1 f ) + N (r, f ) + S(r, f ) = N (r, 1 f ) + k(k + 1) 2 N (r, 1 f ) + N (r, f ) + S(r, f ), where N 0 (r, 1 f (j) ) is the number of those zeros of f (j) in \|z\| ≤ r which are not the zeros of f . That is, N (r, 1 F ) ≤ 1 + k(k + 1) 2 N (r, 1 f ) + k(k + 1) 2 N (r, f ) + S(r, f ). (2.2) Also, we can see that N (r, 1 F )−N (r, 1 F ) ≥   (k + 1)n 0 + k j=1 n j − 1   N (k+1 (r, 1 f )+(n 0 −1)N k) (r, 1 f ), (2.3) where N (k+1 (r, 1 f ) and N k) (r, 1 f ) are the counting functions ignoring multiplicities of those zeros of f whose multiplicity is ≥ k + 1 and ≤ k respectively. Now from (2.2) and (2.3), we get N (r, 1 F ) ≤ 1 + k(k + 1) 2 N (k+1 (r, 1 f ) + 1 + k(k+1) 2 n 0 − 1   N (r, 1 F ) − N (r, 1 F ) −   (k + 1)n 0 + k j=1 n j − 1   N (k+1 (r, 1 f )   + k(k + 1) 2 N (r, f ) + S(r, f ). That is,   1 + 1 + k(k+1) 2 n 0 − 1   N (r, 1 F ) ≤ 1 + k(k + 1) 2 1 − (k + 1)n 0 + k j=1 n j − 1 n 0 − 1 N (k+1 (r, 1 f ) + 1 + k(k+1) 2 n 0 − 1 N (r, 1 F ) + k(k + 1) 2 N (r, f ) + S(r, f ). Since N (r, f ) = N (r, F ) and S(r, f ) = S(r, F ), we have N (r, 1 F ) ≤ 1 + k(k+1) 2 n 0 + k(k+1) 2 N (r, 1 F ) + ( k(k+1) 2 )(n 0 − 1) n 0 + k(k+1) 2 N (r, f ) + S(r, f ) = 1 + k(k+1) 2 n 0 + k(k+1) 2 N (r, 1 F ) + ( k(k+1) 2 )(n 0 − 1) n 0 + k(k+1) 2 N (r, F ) + S(r, F ). Therefore, (2.1) yields [1+o(1)]T (r, F ) ≤ N r, 1 F − ω + 1 + k(k+1) 2 n 0 + k(k+1) 2 N (r, 1 F )+ n 0 (1 + k(k+1) 2 ) n 0 + k(k+1) 2 N (r, F )+S(r, F ). (2.4) Also, if f has a pole of multiplicity p, then F has a pole of multiplicity n 0 p + n 1 (p + 1) + · · · + n k (p + k) ≥ n 0 + 2n 1 + · · · + (k + 1)n k = n 0 + k j=1 (j + 1)n j and therefore, N (r, F ) ≥   n 0 + k j=1 (j + 1)n j   N (r, F ). (2.5) Finally, from (2.4) and (2.5), we find that [1 + o(1)]T (r, F ) ≤ N r, 1 F − ω + 1 + k(k+1) 2 n 0 + k(k+1) 2 N r, 1 F + n 0 (1 + k(k+1) 2 ) (n 0 + k(k+1) 2 )(n 0 + k j=1 (j + 1)n j ) N (r, F ) + S(r, F ). That is, 1 − 1 + k(k+1) 2 n 0 + k(k+1) 2 − n 0 (1 + k(k+1) 2 ) (n 0 + k(k+1) 2 )(n 0 + k j=1 (j + 1)n j ) + o(1) T (r, F ) ≤ N r, 1 F − ω + S(r, F ). For the proof of Theorem 1.2, besides Theorem 1.1, we also need the following lemma which is a straight forward generalization of Lemma 3 in [1]. Lemma 2.1. Let f be a non-constant rational function with only zeros of multiplicity at least k, where k ≥ 2. Let n 0 , n 1 , n 2 , · · · , n k be non-negative integers with n 0 ≥ 2 and n k ≥ 1. Let ω = 0 be a finite complex number. Then M [f ] − ω has at least two distinct zeros. Proof of Theorem 1.2: Since normality is a local property, we may assume that D = D. Suppose F is not normal in D. In particular, suppose that F is not normal at z = 0. Then, by Zalcman's lemma (see [11]), there exist a sequence {f n } of functions in F , a sequence {z n } of complex numbers in D with z n −→ 0 as n −→ ∞, and a sequence {ρ n } of positive real numbers with ρ n −→ 0 as n −→ ∞ such that the sequence {g n } defined by g n (z) = ρ −α f n (z n + ρ n z); 0 ≤ α < k, converges locally uniformly to a non-constant meromorphic function g(z) in C with respect to the spherical metric. Moreover, g(z) is of order at most 2. By Hurwitz's theorem, the zeros of g(z) have multiplicity at least k. Let α = k j=1 jn j k j=0 n j < k. Then M [g n ](z) = (g n (z)) no (g ′ n (z)) n1 · · · g (k) n (z) n k = ρ −αn0 n (f n (z n + ρ n z)) n0 ρ −αn1+n1 n (f ′ n (z n + ρ n z)) n1 · · · ρ −αn k +kn k n f (k) n (z n + ρ n z) n k = ρ −α k j=0 nj + k j=1 jnj (f n (z n + ρ n z)) n0 (f ′ n (z n + ρ n z)) n1 · · · f (k) n (z n + ρ n z) n k = M [f n ](z n + ρ n z). On every compact subset of C that contains no poles of g, we have − ω 0 different from w 0 and v 0 respectively. Then, by Hurwitz's theorem, we see that for sufficiently large n, there exist points w n ∈ D(w 0 , r) and v n ∈ D(v 0 , r) such that M [f n ](z n + ρ n z) − ω(z n + ρ n z) = M [g n ](z) − ω(z n + ρ n z) −→ M [g](z) − ω 0 spherically uniformly, where ω 0 = ω(0). Since g is(M [f n ] − ω) (z n + ρ n w n ) = 0, and (M [f n ] − ω) (z n + ρ n v n ) = 0. Since by hypothesis, M [f n ] share ω partially with M [ f ], for every n, it follows that M [ f ] − ω (z n + ρ n w n ) = 0, and M [ f ] − ω (z n + ρ n v n ) = 0. By letting n −→ ∞, and noting that z n + ρ n w n −→ 0, z n + ρ n v n −→ 0, we find that M [ f ] − ω (0) = 0. Since the zeros of M [ f ] − ω have no accumulation point, z n + ρ n w n = 0 and z n + ρ n v n = 0 for sufficiently large n. That is, D(w 0 , r) ∩ D(v 0 , r) = φ, a contradiction. Proof of Theorem 1.3: As established in the proof of Theorem 1.2, we similarly find that M [g] ≡ ω 0 . By Theorem 1.1 and Lemma 2.6 in [12], M [g] − ω 0 has at least one zero w 0 , say. By Hurwitz's Theorem, there is a sequence of complex numbers {w n } such that w n −→ w 0 as n −→ ∞, and (M [f n ] − ω) (z n + ρ n w n ) = 0 Again, since k > α, \|g (k) n (w n )\| = ρ k−α n \|f (k) n (z n + ρ n w n )\| ≤ ρ (k−α) n A = Aρ k− k j=1 jn j k j−0 n j n −→ 0 as n −→ ∞. Therefore, g (k) (w 0 ) = lim n−→∞ g (k) n (w n ) = 0 ⇒ M [g](w 0 ) = 0 = ω 0 , which is a contradiction. Conclusions Though our results do generalize and improve the results of Hu and Meng [3] and Ding, Ding and Yuan [1] when the domain D is {z : \|z\| < R, 0, R ≤ ∞}, there seems no way of proving our results on arbitrary domain since the idea of small function on arbitrary domain is not available, as for as we know. However, by making certain modifications in the proofs of results of Hu and Meng [3] and Ding, Ding and Yuan [1], one can easily extend and improve these results on arbitrary domain with shared value being a non-zero complex value. Precisely, one obtains, Theorem 3.1. Let F be a family of non-constant meromorphic functions on a domain D with all zeros of each f ∈ F having multiplicity at least k, where k ≥ 2. Let ω = 0 be a finite complex number and n 0 , n 1 , · · · , n k be non-negative integers with n 0 ≥ 2 and n 1 + n 2 · · · + n k ≥ 1. If there exists f ∈ F such that M [f ] share ω partially with M [ f ] for every f ∈ F , then F is normal on D. The condition that f has only zeros of multiplicity atleast k in Theorem 3.1 is sharp. For example, consider the open unit disk D, an integer k ≥ 2, a non-zero complex number ω and the family F = {f m (z) = mz k−1 ; m = 1, 2, 3, · · · } Obviously, each f m ∈ F has only a zero of multiplicity k − 1, and for distinct positive integers m, and l; we find that share ω IM and F is not normal at z = 0. Also, ω = 0 in Theorem 3.1 is essential. For example, let F = {fm}, where f m (z) = 1 e mz +1 ; m = 1, 2, · · · and z ∈ D. Choose k = 2, n = 2, n 1 = 1, and n 2 = 0, we have M [f m ] = f 2 m f ′ m = − me mz (e mz + 1) 4 = 0. Thus, for any f, g ∈ F , M [f ] and M [g] share 0 IM. But we see that F is not normal in D. Theorem 3.2. Let F be a family of non-constant holomorphic functions on a domain D with all zeros of each f ∈ F having multiplicity at least k, where k ≥ 2. Let ω = 0 be a finite complex number and n 0 , n 1 , · · · , n k be non-negative integers with n 0 ≥ 1 and n 1 + n 2 · · · + n k ≥ 1. If there exists f ∈ F such that M [f ] share ω partially with M [ f ] for every f ∈ F , then F is normal on D. As an illustration of Theorem 3.2, we have the following example: Abstract In this article, we prove a distribution result for a certain class of differential polynomials and as a consequence prove a normality criterion concerning partially shared functions: Let F be a family of meromorphic functions in a domain D. Let m, k, n ≥ k + 1 be positive integers and h ≡ 0, ∞ be a meromorphic function having no zeros and poles at the origin. If, there exists f ∈ F such that f m (f n ) (k) share h partially with f m ( f n ) (k) , ∀f ∈ F, then F is normal in D, provided h ≡ f m ( f n ) (k) . Introduction and Main Results For normal families of meromorphic functions, one may refer to [4]. Further, we define a small function of a meromorphic function f in D R := {z : \|z\| ≤ R} to be a meromorphic function ω satisfying T (r, ω) = o (T (r, f )) as r −→ R. We say that f and g share a value a ∈ C IM if f and g have the same a−points counted with ignoring multiplicities. If multiplicities are counted, then they are said to share a CM (one may refer to [8]). In this paper, we use the idea of partial sharing of functions. A meromorphic function f is said to share a function ω partially with a meromorphic function g if Theorem A: Let k, n ≥ k + 1 be positive integers and f be a transcendental meromorphic function. Then (f n ) (k) assumes every finite non-zero value infinitely often. E(ω, f ) ⊆ E(ω, g), where E(ω, φ) = {z ∈ C : φ(z) − ω(z) = 0}, In 2009, Yuntong Li and Yongxing Gu [3] gave the corresponding distribution result for rational functions: Theorem B: Let k, n ≥ k + 2 be positive integers, a = 0 be a finite complex number and f be a non-constant rational function. Then (f n ) (k) − a has at least two distinct zeros. Corresponding to Theorem A and Theorem B, the normality criterion given by Yuntong Li and Yongxing Gu [3] is: Theorem C: Let F be a family of meromorphic functions in an arbitrary domain D. Let k, n ≥ k + 2 be positive integers and a = 0 be a finite complex number. If (f n ) (k) and (g n ) (k) share a in D for every pair of functions f, g ∈ F , then F is normal in D. It is natural to ask whether Theorem A, Theorem B and Theorem C can be generalized for functions instead of constants and the sharing can be replaced by partial sharing. Yes, we have been able to answer these questions as an application of the following value distribution result for differential polynomials. Then k 2(2k + 2) + o(1) T (r, F ) ≤ N r, 1 F − ω + S(r, F ) for any small function ω( ≡ 0, ∞) of f . Theorem 1.2. Let m, k, n ≥ k + 1 be positive integers and ω = 0 be a finite complex number, and f be a non-constant rational function, then f m (f n ) (k) − ω has at least two distinct zeros. As an application of Theorem 1.1 and Theorem 1.2, we prove the following two normality criteria: f ∈ F such that f m (f n ) (k) share h partially with f m ( f n ) (k) , ∀f ∈ F , then F is normal in D, provided h ≡ f m ( f n ) (k) . Remark: The condition h ≡ f m ( f n ) (k) can be omitted in Theorem 1.3 in case h is a small function of f on D R . Theorem 1.4. Let F be a family of meromorphic functions in D. Let m, k, n ≥ k + 1 be positive integers and h ≡ 0, ∞ be a meromorphic function having no zeros and poles at the origin. If, for each f ∈ F , f m (f n ) (k) − h (z) = 0 implies \| (f n ) (k) (z)\| ≤ A, for some A > 0, then F is normal in D. Proof of Main Results Proof of Theorem 1.1: Since F is a homogeneous differential polynomial in f of degree n + m, where exponents of f are positive integers, from [6],we have T (r, f ) + S(r, f ) ≤ CT (r, F ) + S(r, F ) and T (r, F ) ≤ BT (r, f ) + S(r, f ), where B and C are constants, hence T (r, ω) = S(r, F ) as r −→ ∞. Therefore, ω is a small function of f iff ω is a small function of F . Now, by Second Fundamental Theorem of Nevanlinna for three small functions (see [1] pp.47), we have [1 + o(1)] T (r, F ) ≤ N (r, F ) + N (r, 1 F ) + N r, 1 F − ω + S(r, F ) (2.1) Next,by using a result of Lahiri and Dewan( see [2], Lemma), we have N r, 1 F = N r, 1 f m (f n ) (k) ≤ N r, 1 f + N 0 r, 1 (f n ) (k) ≤ N r, 1 f + k N r, 1 f + N (r, f ) + S(r, f ) = (1 + k)N r, 1 f + kN (r, f ) + S(r, f ),(2.2) where N 0 r, 1 (f n ) (k) is the counting function ignoring multiplicity of those zeros of (f n ) (k) in \|z\| ≤ r which are not the zeros of f n and hence f . Also, if z 0 is a zero of f of order p ≤ k, then z 0 is a zero of F of order pn − k + mp ≥ 2(n + m) − k ≥ k + 2 + 2m > k + 3 and if z 0 is a zero of f of order p ≥ k + 1, then z 0 is a zero of F of order np − k + mp ≥ (k + 1)(m + n) − k ≥ nk + (k + 1)m + 1 > k(k + 1) + 2. Thus, it follows that N r, 1 F −N r, 1 F ≥ (k+2)N k) r, 1 f +[k(k+1)+1]N (k+1 r, 1 f , (2.3) where N (k+1 r, 1 f and N k) r, 1 f are the counting functions ignoring multiplicities of those zeros of f whose multiplicity is at least k + 1 and at most k respectively. From (2.2) and (2.3), we obtain N r, 1 F ≤ (k + 1)N (k+1 r, 1 f + k + 1 k + 2 N r, 1 F − N r, 1 F − k + 1 k + 2 kk + 1 + 1 N (k+1 r, 1 f + kN (r, f ) + S(r, f ) ≤ N r, 1 F − N r, 1 F + kN (r, f ) + S(r, f ). That is, N r, 1 F ≤ 1 2 N r, 1 F + k 2 N (r, f ) + S(r, f ). Since N (r, F ) = N (r, f ) and S(r, F ) = S(r, f ), we have N r, 1 F ≤ 1 2 N r, 1 F + k 2 N (r, F ) + S(r, F ). Therefore (2.1) yields [1+o(1)]T (r, F ) ≤ 1 2 N r, 1 F + k + 2 2 N (r, F )+N r, 1 F − ω +S(r, F ). (2.4) Also, if z 0 is a pole of f of multiplicity p, then z 0 is a pole of F of multiplicity np + k + mp ≥ 2k + 2 and therefore, N (r, F ) ≥ (2k + 2)N (r, F ). (2.5) Finally, from (2.4) and (2.5), we find that k 2(2k + 2) + o(1) T (r, F ) ≤ N r, 1 F − ω + S(r, F ). Proof of Theorem 1.2: If f is a polynomial, then f m (f n ) (k) has at least one multiple zero, since n ≥ k + 1. By Fundamental Theorem of Algebra, f m (f n ) (k) − ω has at least one zero. Supposef m (f n ) (k) − ω has exactly one zero, say z 0 . Then f m (f n ) (k) (z) = ω + A(z − z 0 ) l , where 0 = A is constant and l > 0 Since ω = 0, f m (f n ) (k) − ω has simple zeros only, which is not the case. Hence f m (f n ) (k) − ω has at least two distinct zeros. Now, consider the case when f is rational but not polynomial. Suppose on the contrary that f m (f n ) (k) − ω has no distinct zeros. Then f m (f n ) (k) − ω has either exactly one zero or no zero. First, we consider the case when f m (f n ) (k) − ω has exactly one zero. Let f (z) = A s i=1 (z − α i ) mi t h=1 (z − β h ) l h , (2.6) where A is a non-zero constant, m i ≥ 1(i = 1, 2, · · · , s) and l h ≥ 1(h = 1, 2, · · · , t). Put M = s i=1 m i ≥ s and N = t h=1 l h ≥ t. (2.7) Then (f n ) (k) = A n s i=1 (z − α i ) nmi−k t h=1 (z − β h ) nl h +k g k (z), (2.8) where g k (z) = n(M −N ) [n(M − N ) − 1] [n(M − N ) − 2] · · · [n(M − N ) − k + 1] z k(s+t−1) +· · · is a polynomial of degree at most k(s + t − 1). Thus, f m (f n ) (k) = A m+n s i=1 (z − α i ) (m+n)mi−k t h=1 (z − β h ) (m+n)l h +k g k (z) = P (z) Q(z) , say. (2.9) Since f m (f n ) (k) − ω has exactly one zero, z 0 say, from (2.9), we obtain f m (f n ) (k) = ω + B(z − z 0 ) l t h=1 (z − β h ) (m+n)l h +k ,(2.10) where l is a positive integer and B = 0 is a constant. Again, from (2.9), we have f m (f n ) (k) ′ = A m+n s i=1 (z − α i ) (m+n)mi−(k+1) t h=1 (z − β h ) (m+n)l h +(k+1)g (z),(2.11) whereg is a polynomial with degg ≤ (k + 1)(s + t − 1). Consequently (2.10), yields f m (f n ) (k) ′ = A m+n (z − z 0 ) l−1 t h=1 (z − β h ) (m+n)l h +(k+1)ĝ (z),(2.12) whereĝ(z) = l − [(m + n)N + kt] z t + · · · is a polynomial. Case-I: Suppose l = (m + n)N + kt. Then from (2.10) and using (2.9), we have deg P ≥ deg Q Case-II: Suppose l = (m + n)N + kt. It is sufficient to discuss the case M ≤ N here. By comparing (2.11) and (2.12), we get l − 1 ≤ degg ≤ (k + 1)(s + t − 1) and hence (m+n)N = l−kt ≤ degg+1−kt ≤ (k+1)(s+t−1)+1−kt ≤ (k+2)N ≤ (m+n)N i.e. N < N, which is again absurd. Finally, suppose f m (f n ) (k) − ω has no zero at all. Then l = 0 in (2.10), yields f m (f n ) (k) = ω + B t h=1 (z − β h ) (m+n)l h +k (2.13) and so f m (f n ) (k) ′ = BH(z) t h=1 (z − β h ) (m+n)l h +(k+1) ,(2.14) where H(z) is a polynomial of degree t − 1 < t. Proceeding as in the proof for Case-I, we again get a contradiction. This completes the proof. Proof of the Theorem 1.3: Since normality is a local property, we may assume that D = D. Suppose F is not normal in D. Then there exists at least one z 0 ∈ D a non-constant meromorphic function of order at most 2 and ω 0 = 0, ∞, it immediately follows that M [g] ≡ ω 0 . Using Theorem 1.1 and Lemma 2.1, M [g]−ω 0 has at least two distinct zeros, say, w 0 and v 0 . Choose r > 0 such that the open disks D(w 0 , r) = {z : \|z − w 0 \| < r} and D(v 0 , r) = {z : \|z − v 0 \| < r} are disjoint and their union contains no zeros of M [g] Example 3 . 3 . 33Consider F = {f m (z) = me z m : m ∈ N}, defined on C. Take k = 2, n = 1, n 1 = 0, and n 2 = 1.ThenM [f m ] = f m f ′′ m each m ≥ 2, M [f m ] share 1 partially with M [f 1 ].Next, we have ∀z, \|z\| ≤ r, r > 0; \|f m (z)M, say, where M > 0 depends on r and this is true for each m ∈ N. That is, F is locally bounded on C and hence by Montel Theorem F is normal. the set of zeros of φ − ω counted with ignoring multiplicities. In 1998, Y.Wang and M.Fang [7] proved: 2010 Mathematics Subject Classification: 30D30, 30D35, 30D45. Keywords: Distribution of Values, Normal Families, Meromorphic Functions, Differential Polynomials, Sharing of values. The work of second author is supported by University Grants Commission(UGC), INDIA (No.F.17-77/08(SA-1)) . Theorem 1 . 1 . 11Let f be a transcendental meromorphic function and m, k, n ≥ k + 1 be positive integers. Let F = f m (f n ) (k) . Theorem 1 . 3 . 13Let F be a family of meromorphic functions in an arbitrary domain D. Let m, k, n ≥ k + 1 be positive integers and h ≡ 0, ∞ be a meromorphic function having no zeros and poles at the origin. If, there exists ⇒ (m + n)M − ks + deg g k ≥ (m + n)N + kt ⇒ (m + n)M − ks + k(s + t − 1) ≥ (m + n)N + kt ⇒ (m + n)N ≤ (m + n)M − k < (m + n)M i.e.M > N.Noting that z 0 = α i ; ∀i, from (2.7),( 2.11) and (2.12), we obtains i=1 [(m + n)m i − (k + 1)] ≤ degĝ = t ⇒ (m + n)M − (k + 1)s ≤ t ⇒ (m + n)M ≤ (k + 1)s + t ≤ (k + 1)M + N < (k + 2)M ≤ (m + n)Mi.e.M < M, which is absurd. By Zalcman's Lemma, there exists a sequence {f j } of functions in F ; a sequence {z j } of complex numbers in D with z j −→ 0 as j −→ ∞; and a sequence {ρ j } of positive real numbers with ρ j −→ 0 as j −→ ∞ such that the sequence {g j } of scaled functionswhere 0 ≤ α < k; converges locally uniformly to a non-constant meromorphic function g(z) in C with respect to the spherical metric. Moreover, g(z) is of order at most 2.Put α = k m+n (< 1). ThenOn every compact subset of C that contains no poles of g, we getNow, g is a non-constant meromorphic function and g m (g n ) (k) is a homogeneous differential polynomial with exponents of g positive in each monomial. It follows that g and g m (g n ) (k) have the same order and hence g m (g n ) (k) ≡ h 0 . Thus, by Theorem 1.1 and Theorem 1.2, we find that g m (g n ) (k) − h 0 has at least two distinct zeros, say u 0 and v 0 . Since zeros are isolated, we can find two nonintersecting open disks D(u 0 , r) and D(v 0 , r) such that D(u 0 , r) ∪ D(v 0 , r) does not contain any zero of g m (g n ) (k) − h 0 different from u 0 and v 0 . Thus, by Hurwitz's theorem, we see that, for sufficiently large values of j, there exist points u j ∈ D(u 0 , r) and v j ∈ D(v 0 , r) such thatSince by hypothesis f m, for some f ∈ F , for every j, it follows thatSince z j + ρ j u j −→ 0 and z j + ρ j v j −→ 0 as j −→ ∞, we find thatSince the zeros of f m f n(k)− h have no accumulation point, it follows that z j +ρ j u j = 0 and z j +ρ j v j = 0, for sufficiently large j, which is a contradiction to the fact that D(u 0 , r) and D(v 0 , r) are non-intersecting.Proof of the Theorem 1.4: Proceeding as in the proof of the Theorem 1.3, we similarly find that g m (g n ) (k) ≡ h 0 . By Theorem 1.1 and Theorem 1.2,Thus (g n ) (k) (w 0 ) = lim j−→∞ g n j (k) (w j ) = 0 ⇒ g m (g n ) (k) (w 0 ) = 0 = h 0 , which is not possible. Normal families of meromorphic functions concerning shared values. J J Ding, L W Ding, W J Yuan, Complex Variables and Elliptic Equations. 58J.J. Ding, L.W. Ding and W.J. Yuan, Normal families of meromorphic functions concerning shared values, Complex Variables and Elliptic Equa- tions 58(2013), 113-121. Meromorphic fucntion. W K Hayman, Clarendon PressOxfordW.K. Hayman, Meromorphic fucntion, Oxford: Clarendon Press 1964. Normality criteria of meromorphic functions with multiple zeros. P C Hu, D W Meng, J. Math. Anal. Appl. 357P.C. Hu and D.W. Meng, Normality criteria of meromorphic functions with multiple zeros, J. Math. Anal. Appl. 357(2009), 323-329. Value distribution of the product of a meromorphic function and its derivative. I Lahiri, S Dewan, Kodai Math. J. 26I. Lahiri and S. Dewan, Value distribution of the product of a meromorphic function and its derivative, Kodai Math. J. 26(2003), 95-100. Normality criteria of meromorphic functions sharing one value. D W Meng, P C Hu, J. Math. Anal. Appl. 381D.W. Meng and P.C. Hu, Normality criteria of meromorphic functions sharing one value, J. Math. Anal. Appl. 381(2011), 724-731. J Schiff, Normal Families. BerlinSpringer-VerlagJ. Schiff, Normal Families, Springer-Verlag, Berlin,1993. Normal criteria for family of meromorphic fucntions. W Schwick, J. Anal. Math. 52W. Schwick, Normal criteria for family of meromorphic fucntions, J. Anal. Math. 52(1989), 241-289. On order of homogeneous differential polynomials. A P Singh, Indian J. Pure Appl. Math. 16A.P. Singh, On order of homogeneous differential polynomials, Indian J. Pure Appl. Math. 16(1985), 791-795. On the value distribution of f f (k) , Kodai Math. C C Yang, P C Hu, J. 192C.C. Yang and P.C. Hu, On the value distribution of f f (k) , Kodai Math.J. 19(2)(1996), 157-167. C C Yang, H X Yi, Uniqueness Theory of Meromorphic Functions. Beijing, New YorkKluwer AcademicC.C. Yang and H.X. Yi, Uniqueness Theory of Meromorphic Functions, Science Press, Kluwer Academic, Beijing, New York, 2003. Normal Families: New Perspectives. L Zalcman, Amer. Math. Soc. 35L. Zalcman, Normal Families: New Perspectives, Amer. Math. Soc. 35(1998), 215-230. Normality Criteria of Lahiri's Type and Their Applications. Xiao-Bin Zhang, Jun-Feng Xu, Hong-Xun Yi, ID 873184Journal of Inequalities and Applications. 16Xiao-Bin Zhang, Jun-Feng Xu and Hong-Xun Yi, Normality Criteria of Lahiri's Type and Their Applications, Journal of Inequalities and Applica- tions Volume 2011, Article ID 873184, 16 pages. Meromorphic fucntion. W K Hayman, Clarendon PressOxfordHayman, W.K., Meromorphic fucntion, Oxford: Clarendon Press 1964. Value distribution of the product of a meromorphic function and its derivative. I Lahiri, S Dewan, Kodai Math. J. 26Lahiri, I. and Dewan, S., Value distribution of the product of a meromorphic function and its derivative, Kodai Math. J. 26(2003), 95-100. On normal families of meromorphic functions. Y Li, Y Gu, J.Math. Anal. Appl. 354Li, Y. and Gu, Y., On normal families of meromorphic functions, J.Math. Anal. Appl. 354(2009),421-425. Normal Families. J Schiff, Springer-VerlagBerlinSchiff, J., Normal Families, Springer-Verlag, Berlin,1993. Normal criteria for family of meromorphic fucntions. W Schwick, J. Anal. Math. 52Schwick, W., Normal criteria for family of meromorphic fucntions, J. Anal. Math. 52(1989), 241-289. On order of homogeneous differential polynomials. A P Singh, Indian J. Pure Appl. Math. 16Singh, A.P., On order of homogeneous differential polynomials, Indian J. Pure Appl. Math. 16(1985), 791-795. Picard values and normal families of meromorphic functions with multiple zeros. Y F Wang, M L Fang, Acta Math. Sinica(Chin. Ser.). 14Wang, Y.F. and Fang, M.L., Picard values and normal families of meromor- phic functions with multiple zeros, Acta Math. Sinica(Chin. Ser.)14(1998), 743-748. Uniqueness Theory of Meromorphic Functions. C C Yang, H X Yi, Jammu-180 006Science PressKluwer Academic, Beijing, New York; INDIADepartment of Mathematics, University of JammuYang, C.C. and Yi, H.X., Uniqueness Theory of Meromorphic Functions, Science Press, Kluwer Academic, Beijing, New York, 2003. Department of Mathematics, University of Jammu, Jammu-180 006, INDIA. E-mail: [email protected] 2 E-mail: shittalsharma mat07@rediffmail. E-mail: [email protected] 2 E-mail: shittalsharma [email protected]	[]
[ "AN ENDLINE BILINEAR RESTRICTION ESTIMATE FOR PARABOLOIDS", "AN ENDLINE BILINEAR RESTRICTION ESTIMATE FOR PARABOLOIDS" ]	[ "Urbain Jianwei ", "Yang " ]	[]	[]	We prove an L 2 × L 2 → L q t L r x bilinear adjoint Fourier restriction estimate for n-dimensional elliptic paraboloids, with n ≥ 2 and 1 ≤ q ≤ ∞, 1 ≤ r ≤ 2 being on the endline 1 q = n+1 2 1 − 1 r except for the critical index. This includes the endpoint case when q = r = n+3 n+1 , a question left unsettled in Tao [33]. Apart from the critical index, it improves the sharp non-endline result of Lee-Vargas [25] to the full range, confirming a conjecture in the spirit of Foschi and Klainerman [13] on the elliptic paraboloid. Our proof is accomplished by uniting the profound induction-on-scale tactics based on the wave-table theory and the method of descent both stemming from [32]. 2010 Mathematics Subject Classification. 42B15, 42B20, 42B37.	null	[ "https://arxiv.org/pdf/2202.13905v2.pdf" ]	247,158,330	2202.13905	6a4faf4719e9668ea7796476c054931f03ba7e1c	AN ENDLINE BILINEAR RESTRICTION ESTIMATE FOR PARABOLOIDS 21 May 2022 Urbain Jianwei Yang AN ENDLINE BILINEAR RESTRICTION ESTIMATE FOR PARABOLOIDS 21 May 2022 We prove an L 2 × L 2 → L q t L r x bilinear adjoint Fourier restriction estimate for n-dimensional elliptic paraboloids, with n ≥ 2 and 1 ≤ q ≤ ∞, 1 ≤ r ≤ 2 being on the endline 1 q = n+1 2 1 − 1 r except for the critical index. This includes the endpoint case when q = r = n+3 n+1 , a question left unsettled in Tao [33]. Apart from the critical index, it improves the sharp non-endline result of Lee-Vargas [25] to the full range, confirming a conjecture in the spirit of Foschi and Klainerman [13] on the elliptic paraboloid. Our proof is accomplished by uniting the profound induction-on-scale tactics based on the wave-table theory and the method of descent both stemming from [32]. 2010 Mathematics Subject Classification. 42B15, 42B20, 42B37. Introduction Let n ≥ 2 be an integer and Σ be the elliptic paraboloid Σ = (ξ, τ ) ∈ R n+1 ; τ = − 1 2 \|ξ\| 2 with the surface measure dσ on Σ. For any test function f on Σ, define the adjoint Fourier restriction operator f dσ(x, t) as f dσ(x, t) = Σ e 2πi(x·ξ+tτ ) f (ξ, τ ) dσ(ξ, τ ). For any two smooth compact hypersurfaces Σ 1 and Σ 2 being transverse subsets of Σ, with induced surface measures dσ 1 and dσ 2 respectively, it is proved in [25] that for 1 < q, r ≤ ∞ such that 1 q < min n+1 4 , n+1 2 1 − 1 r , there exists a finite constant C = C q,r,Σ1,Σ2 > 0 such that j=1,2 f j dσ j L q (Rt;L r (R n x )) ≤ C j=1,2 f j L 2 (Σj dσj ) (1.1) holds for all test functions f 1 and f 2 supported on Σ 1 and Σ 2 respectively. This is an extension of the previous result of Tao [33] in the case q = r > n+3 n+1 to the mixed-norms. Moreover, it is pointed out in [25, Section 2.1] that 1 q ≤ n+1 2 1 − 1 r is necessary and a natural conjecture is that if we let Γ = (q, r); 1 q = n + 1 2 1 − 1 r , 1 ≤ q ≤ ∞, 1 ≤ r ≤ 2 ,(1.2) then for any (q, r) ∈ Γ, there is a constant C depending on q, r, Σ 1 and Σ 2 such that (1.1) holds. Notice that Γ is the borderline of the range for (q, r) such that (1.1) could be valid when 1 ≤ r ≤ 2. This conjecture, if true, is an endline version of [25] and it includes the endpoint estimate of [33] as a special case. By using Bernstein's inequality, the endline result would imply all the other cases. We call the left endpoint of the endline Γ in (1.2) the critical index for (1.1) (q c , r c ) := ( 4 3 , 2) , n = 2 , 1, n+1 n−1 , n ≥ 3 . The endline estimates can be reduced to the strongest estimate corresponding to the critical index. Indeed, if (1.1) were true for the critical index (q, r) = (q c , r c ), then we would be able to obtain the bilinear estimate with (q, r) in the full range (1.2) by interpolation with the energy estimates. This conjecture on the bilinear restriction estimates is connected in a deep way to the null form estimates of wave equations, an important device in the study of nonlinear wave equations (c.f. [13,18,19,32,38,24,25] ). It interacts dynamically with the linear restriction problems posed by Stein [30]. The bilinear approach was initiated from Bourgain [4] improving the boundedness of Mockenhaupt's cone multiplier [27] and developed further in [35,36,37]. Nowadays, it has become such a highly active research area that it is almost impossible to give a comprehensive summary for all the up-to-date works in a limited space and time. On the other hand, there already exist so many excellent survey articles and monographs, we refer to [5,42,34,15,10] and references therein for a panorama of this domain. Moreover, it turns out that the endpoint bilinear restriction estimates become more and more important in applications to PDEs. We only mention here, among other things, its connexion to the uniqueness in Calderón's inverse conductivity problem [16,17,12], where the bilinear restriction estimate along with its extensions played an essential role and it is pointed out in [17] that further improvements would be available provided one had the endpoint results. For more applications of these bilinear estimates to nonlinear dispersive equations, we refer to [11]. In the case when q = r > n+3 n+1 , the sharp (non-endpoint) bilinear estimate (1.1) was established by Tao [33] by adapting the mild induction-on-scale argument due to Wolff [41] for the sharp L 2 −bilinear estimate on the cone. The endpoint case q = r = n+3 n+1 was left open in [33] and is recently investigated by J. Lee [20]. We note that a new approach towards the Fourier restriction problems is proposed by Muscalu and Oliveira [28] relating the restriction theory with the multilinear harmonic analysis, where the authors obtained sharp linear and multilinear restriction theorems provided certain tensor product conditions on the input functions are satisfied. Although it was observed in [28,Section 9] that one may extend the admissible range of the exponents for (1.1) under extra hypothesis on the amount of transversality, the endpoint case remains unsettled even if the tensor-product condition is fulfilled. The purpose of this paper is to prove the bilinear estimate (1.1) for all (q, r) on the endline Γ except for the critical index (q c , r c ) for all n ≥ 2 and our main result reads Theorem 1.1. Let n ≥ 2 and Σ 1 , Σ 2 be two disjoint compact subsets of Σ. Then, for any (q, r) ∈ Γ\{(q c , r c )}, there is a finite constant C = C q,r,Σ1,Σ2 > 0 depending only on q, r, Σ 1 , Σ 2 such that (1.1) holds for all test functions f 1 and f 2 defined on Σ 1 and Σ 2 respectively. Remark 1.2. In case of q = r, the conjecture is usually refered as the Machedon-Klainerman conjecture [13,33,32,41], especially in three dimensions. In general dimensions, Foschi and Klainerman provided a tentative description on possible bilinear estimates of this form [13]. The mixed-norm extension is due to Lee and Vargas [25]. To highlight the origin of the question, we cautiously refer to it as the Foschi-Klainerman conjecture. We briefly describe our proof for Theorem 1.1. In [41], Wolff proved the sharp L 2 × L 2 → L q t,x bilinear estimate on the cone for all q > n+3 n+1 , by means of his celebrated induction on scale argument, to which we would refer as a mild version. To resolve the endpoint case q = n+3 n+1 , Tao introduced in [32] a profound version of the induction argument which enhanced the method in [41]. By building up an effective wave-table theory and exploring the possible ways that local energy could concentrate, the endpoint bilinear estimate on the cone was proved in [32] in the symmetric norms and then extended to the mixed-norms by Temur [39] ( see also [25,Section 2.1] for the non-endline case), as well as to the variable coefficient setting by J. Lee [21] (in the symmetric norms). Moreover, the endpoint result in [32] is also generalized to the case when one of the waves has large frequency in order to develop sharp null form estimates, which is nearly optimal due to its connexion with the (back then) unsettled endpoint bilinear restriction estimates on the paraboloid [32,Section 17]. See also [25,Section 2.1] for the version of mixednorms. For extensions of Tao's results to some second order hyperbolic equations with rough coefficients, we refer to Tataru [38]. A crucial geometric fact utilized in the proof of the endpoint bilinear estimates is that at any point of the cone, the normal vector is always in the lightray directions, due to the single vanishing principle curvature on the cone along the radiative null direction. This may be regarded as a lightcone version of the Kakeya compression phenomenon, compared to those observed by Bourgain [3] and Bourgain-Guth [6], where distorted tubes contained in a neighborhood of subvarieties are considered. Combining this fact with the energy estimates on lightcones of opposite colour [32,Section 13], one is able to dispose of the energy-concentrated case in order to close the enhanced induction for the endpoint problem. This geometric property fails on the paraboloid Σ since the Gaussian curvature is nowhere vanishing. See [20] for a study on the endpoint case, by using a new energy concentration argument, where (1.1) is still considered in the symmetric case, i.e. q = r = n+3 n+1 . The idea of this paper is different from [20]. We retain the original induction scheme of Tao [32] using the same energy concentration, and prove the bilinear estimate in the mixed-norms on the whole endline apart from the critical index. A novel ingredient that we take in is the use of the method of descent proposed in the same paper [32]. To illustrate the idea of this method, let us start with the three dimensional spacetime. As a well-known fact, a 2-plane parallel to a generatrix of a (2-dim) cone in R 3 , and not passing through its vertex, intersects the cone in a (1dim) parabola [1]. This elementary fact is readily generalized to higher dimensions. Indeed, for n ≥ 2, consider the (n + 1)-dimensional backward cone in R n+2 V := (x, t) ∈ R n+1 × R; t = −\|(x, x n+1 )\| , with x = (x, x n+1 ), x n+1 being the auxiliary variable, \|(x, x n+1 )\| = \|x\| 2 + x 2 n+1 . Denote e ± = en+1±en+2 √ 2 , where e j = (0, . . . , 0, 1 j−th , 0, . . . , 0), ∀ j ∈ {1, . . . , n + 2}. Let λ > 1 and Π λ be the (n+1)−dimensional hyperplane passing through the point −λe + and being normal to e + . Then P λ := Π λ ∩ V is an n-dimensional elliptic paraboloid in Π λ , symmetric around the axis passing through −λe + in direction of e − , and parametrized by the circular variables x ∈ R n x = span(e 1 , e 2 , . . . , e n ). The drawback of this fact is that the focal point of P λ depends on the varying parameter λ. To overcome this obstacle, one may stretch the integration along the e − direction by λ so that Σ can be treated as a limiting surface after scaling back along e − direction and letting λ → +∞. To match this change of variable, a negative power of λ will be involved, which is related to the null form estimates from dimensional analysis. This method was introduced as an intermediate step to demonstrate that the conjectured null form estimate (see (85) of Problem 17.1 [32]) implies the Machedon-Klainerman conjecture for paraboloids in the symmetric norm, i.e. q = r in (1.1), integrating out the one dimensional auxiliary variable x n+1 after taking limit. Our strategy towards Theorem 1.1 is to show that the (unlabelled) bilinear estimate on P. 260 of [32] φψ p R 1/p f 2 g 2 , as a consequence of the unsettled stronger estimate (85) in [32,Section 17], and employed in the intermediate step to get the endpoint bilinear estimate on paraboloids, can be indeed proved directly by suitably modifying the profound induction on scale argument of [32], not only in the symmetric norms, but also in the mixed L q t L r x norms for all q, r on the endline Γ \ {(q c , r c )}, without first resolving the much more difficult question on the null form conjecture. Since we will work essentially with the O(1)−neighbourhood of the cross section P λ on the cone, the above mentioned Kakeya compression property remains valid, replacing the the spatial unit sphere in which circular components used to be resident [32], with a subset of the paraboloid Σ. The distribution of the directions of the tubes associated to the careful wave-packet decomposition become congregated by a ratio O(λ −1 ), which will be compensated by the λ −1/q −factor arising from an average along the λ−stretched direction e − . The only missing answer for (1.1) in Theorem 1.1 is the critical index (q c , r c ), which is out of reach by the current method. In fact, this problem shares a level of the same difficulty concerning the endpoint multilinear restriction theorem of Bonnett-Carbery-Tao [2], a very difficult open question. Even as a weaker result, the endpoint multilinear Kakeya inequality can only be established through the intricate algebraic topological method by Guth [14]. The paper is organized as follows. In Section 2, we first introduce a λ−dependent operator S λ in a similar fashion to that of [32] and study the basic properties associated to the corresponding dispersive equation, emphasizing the energy estimates for the waves on conic sets of opposite colour. We then prove the careful wave packet decomposition and construct the wave tables on stretched spacetime cubes for the red and blue waves, to be specified in the context below. The crucial property that the red and blue waves on a stretched cube can be effectively approximated via C 0 −quilts of the wave tables on a quantitative interior of a proper enlargement of the cube in spirit of [32] will be proved. The proof is reduced to a tamed bilinear L 2 −Kakeya type estimate, which eradicates the logarithmic loss as reminded in the last section of [33]. In Section 3, we introduce the spatial localization operator P D in the S λ -operator version and use these operators to capture the energy concentration of waves. In Section 4, we introduce the core quantity A λ (R) to bootstrap with respect to the scales R ≤ λ. To this end, an auxiliary quantity A λ (R, r, r ′ ) will get involved, which is defined based on the notion of energy concentration. Here, the two parameters r, r ′ are scales for measuring the level of the concentration of energy. This is crucial for the endpoint estimate as in [32] in order to wrest in a universal constant strictly less than one. To close the induction, A λ needs to be controlled by A λ up to a constant very close to one, which is easy in the non-concentrated case. The difficult part is the case when the energy is highly concentrated, for which we make use of the Kakeya compression and a non-optimal control on the exterior energy in terms of A λ by inductive hypothesis. Finally, in Section 5, we close the induction on A λ (R) and complete the proof. To end up this section, we remark that it seems that the same argument should work also for general hyperbolic paraboloids for the endpoint problems of bilinear restriction estimate left open in [23,40,25]. One might also be able to get the endline result of the Wave-Schrödinger bilinear restriction estimates, where the off-endline case is established by Candy [8] and applied to the wave-Schrödinger interactions in the Zakharov system [9]. Moreover, it is probably more interesting to tackle the endpoint case of the bilinear estimates related to Klein-Gordon equations, which have been investigated by Bruce et al [7] and Candy [8] in the non-endpoint case. Finally, it is plausible that the ε−loss of the bilinear oscillatory integral estimates in [22] can also be removed. Notations. For any fixed (q, r) ∈ Γ \ {(q c , r c )}, let N ≥ 1 be a sufficiently large integer depending only on n, q, r and let C 0 = C 0 (ε, N ) = 2 ⌊ N ε ⌋ 10 , where ε > 0 will be taken small when necessary in the process of the proof, but it will never tend to zero. We use A B, A = O(B) or A = O(B) to denote A ≤ CB for some C > 0, which may change from line to line and depends only on n, ε, q, r, but not explicitly on C 0 . We use A ≪ B to denote A ≤ C −1 B for some sufficiently large constant C. Acknowledgements. The author is supported by NSFC grant No. 11901032 and Research fund program for young scholars of Beijing Institute of Technology. The author is also grateful to LAGA in Université de Sorbone Paris Nord, where part of this work was done. Preliminaries 2.1. The S λ −propagator and its basic properties. For each j = 1, 2, let V j be the projection from Σ j to the ξ−variables. Let e 1 = (1, 0, . . . , 0) ∈ R n . By compactness and a finite partition of Σ 1 , Σ 2 , we may assume V 1 = ξ ∈ R n : ξ − e 1 ≤ 1 200n , V 2 = ξ ∈ R n : ξ ≤ 1 200n , after using a suitable rotation, scaling and the Galilean transformation. Due to technical reasons, we also need the slightly enlarged version of V j , namely V j = ξ ∈ R n : dist(ξ, V j ) ≤ 1 100n , j = 1, 2. We denote B := {(ξ, s) : \|ξ\| ≤ 2, \|s\| ≤ 2} for short. By Plancherel's theorem, it is more convenient to work in the language of dispersive equations. For any λ ≥ 2 C0 , we introduce the S λ (t) operator : Definition 2.1. For any f j ∈ S(R n+1 ) with j ∈ {1, 2} such that f j ∈ C ∞ 0 ( V j × I) where I = [−2, 2], let S λ j (t)f j (x) = e 2πi x·ξ+xn+1s− t 2 \|ξ\| 2 λ+s a j (ξ, s) f j (ξ, s) dξds, where we denote x = (x, x n+1 ) for brevity and f j → f j is the Fourier transform on R n+1 and a j ∈ C ∞ c (R n+1 ) such that a j equals to one on V j × I and that a j vanishes outside {(ξ, s); dist((ξ, s), V j × I) ≤ (100n) −1 }. Proposition 2.2. For each j = 1, 2, let Ξ λ j = ξ s+λ ; (ξ, s) ∈ supp a j . Define K λ j (x, t) = e 2πi x·ξ+xn+1s− t 2 \|ξ\| 2 λ+s a j (ξ, s) dξds. Then, S λ j (t)f (x) = K λ j (·, t) * f (x), (2.1) with K λ j (x, t) M 1 + dist (x, t), Λ λ j −M (2.2) for all (x, t) ∈ R n+2 and all integers M ≥ 1, where Λ λ j := ℓ∈R ℓ v, − \|v\| 2 2 , 1 ; v ∈ 2 Ξ λ j , with 2 Ξ λ j being the set with the same center of Ξ λ j but with the double diameter. Here, Λ λ j is a conic hypersurface in R n+2 with dim(Λ λ j ) = n + 1. Proof. By definition, (2.1) is clear. Moreover, we have (2.2) by using the trivial estimate if (x, t) is in a C−neighbourhood of Λ λ j . Next, assume that (x, t) is C away from Λ λ j for C ≫ 1. Letting L = 1 + (2πi) −1 (x − t(s + λ) −1 ξ) · ∂ ξ + (2πi) −1 (x n+1 + t 2 (s + λ) −2 \|ξ\| 2 )∂ s 1 + \|x − t(s + λ) −1 ξ\| 2 + x n+1 + t 2 (s + λ) −2 \|ξ\| 2 2 , and integrating by parts using L M e 2πi x·ξ+xn+1s− t\|c γ,M (x, t; ξ, s+λ)\| M 1+ t ξ λ M−m 1+ x− t ξ s + λ + x n+1 + t \|ξ\| 2 2(s + λ) 2 −2M+m . To see this, denote s = (1 + \|s\| 2 ) 1 2 and let Z = x − tv with v = ξ s+λ , − \|ξ\| 2 2(s+λ) 2 . The adjoint operator L * of L can be written into the form L * = α · ∂ ξ,s + β where α = α(x, t; ξ, s + λ) ∈ C n+1 and β = β(x, t; ξ, s + λ) ∈ C are smooth functions such that on supp a j , we have \|α\| Z −1 and \|β\| tξ/λ Z −2 and for all γ with \|γ\| ≥ 1, we have \|∂ γ α\| ∼ \|∂γβ\| for someγ with \|γ\| + 1 = \|γ\|. Moreover, we have Since \|x − t λ v λ \| t/λ with v λ = λξ s+λ , − λ\|ξ\| 2 2(s+λ) 2 for all (ξ, s) ∈ supp a j , by combining this with the bound on c γ,M , we find that K λ j (x, t) can be bounded with sup (ξ,s)∈supp aj 1 + x − t(s + λ) −1 ξ + x n+1 + t 2 (s + λ) −2 \|ξ\| 2 −M 1 + dist (x, t), Λ λ j −M . The proof is complete. It is convenient to call F λ j (x, t) := S λ j (t)f j (x) the red and blue waves respectively for j = 1, 2. For each j, the energy of F λ j (x, t) is defined as E(F λ j ) := F λ j (·, 0) 2 L 2 (R n+1 x ) . When there is no need to distinguish the color, we shall simply call F λ a wave. The following energy estimates on conic sets of opposite colour is crucial. Lemma 2.3. For j = 1, 2, let Λ λ j (z 0 , r) be an O(r)−neighbourhood of Λ λ j + z 0 with z 0 = (x 0 , t 0 ) ∈ R n+2 and r ≥ 1. Then, we have F λ j L 2 (Λ λ k (z0,r)) (λr) 1/2 E(F λ j ) 1/2 , ∀ j, k ∈ {1, 2}, j = k ,(2.3) for all z 0 and r ≥ 1. Proof. The argument is similar to [32]. By translation invariance which is clear from modulation of the input function depending on z 0 in the frequency space, we may take z 0 = (0, 0) ∈ R n+1 × R. By symmetry, we only consider (j, k) = (1, 2). Let D λ,t 2 = x ∈ R n+1 ; dist((x, t), Λ λ 2 ) r . Let S λ, * 1 be the adjoint of S λ 1 . By the T T * principle, it suffices to show S λ, * 1 (t) 1 1 D λ,t 2 H(·, t) dt L 2 (R n+1 ) (λr) 1/2 H L 2 (R n+2 ) for all H ∈ L 2 (R n+2 ). Taking squares and multiplying out, we have S λ, * 1 (t) 1 1 D λ,t 2 H(·, t) dt 2 L 2 (R n+1 ) 1 1 D λ,t 2 (x) K λ 1 (x − x,t − t) 1 1 D λ,t 2 (x) H(x, t) H(x,t) dxdxdtdt, (2.4) where K λ 1 is given by Proposition 2.2 with a 1 replaced by a 2 1 . By Cauchy-Schwarz and the L 2 −boundedness sup t,t 1 1 D λ,t 2 S λ 1 (t) • S λ, * 1 (t)1 1 D λ,t 2 L 2 (R n+1 )→L 2 (R n+1 ) = O(1), the \|t −t\| λr part of the integral (2.4) is bounded by λr H 2 2 . Next, we estimate the \|t −t\| ≫ λr part of the integral. For any u ∈ Ξ λ 1 and v ∈ Ξ λ 2 , let L(ṽ, u) ⊂ R n+1 be the straight line passing through the origin along the direction ṽ −u, − 1 2 \|ṽ\| 2 + 1 2 \|u\| 2 . Define L(ṽ, v) for v ∈ Ξ λ 2 in the same way. If we let L ∆ (ṽ, u) be the two ends on L(ṽ, u) outside the ball in the spacetime R n+2 of radius ∆ with ∆ ≫ r and centered at the origin, then we have dist L ∆ (ṽ, u), L(ṽ, v) ∆ for all u ∈ Ξ λ 1 and v,ṽ ∈ Ξ λ 2 , thanks to the non-vanishing Gaussian curvature of Σ. Indeed, this is clear if the directions of (u −ṽ) and (v −ṽ) are separated by a fix small constant 0 < θ ≪ 1. Otherwise, there isũ having the property that \|ũ − u\| λ −1 and (ũ, − 1 2 \|ũ\| 2 ) belongs to the two-plane Π = Π(v,ṽ) passing through the origin such that L(ṽ, v) ⊂ Π and (0, . . . , 0 n times , 1) ∈ Π (the co-planar case). Using the condition that diam( V 1 ), diam( V 2 ) ≪ dist( V 1 , V 2 ) and the strict convexity of the parabola (curvature property), it is easy to deduce that ( by using Taylor's expansion say) the two vectors (ũ −ṽ, − 1 2 \|ũ\| 2 + 1 2 \|ṽ\| 2 ) and (v −ṽ, − 1 2 \|v\| 2 + 1 2 \|ṽ\| 2 ) are separated by an angle ϕ 1 which depends only on V 1 , V 2 . Simple solid geometric comparison inequalities yield the result. Using this fact, dist(Ξ λ 1 , Ξ λ 2 ) λ −1 and the D λ,t 2 , D λ,t 2 constraints for (x, t), (x,t): x = tv, − t 2 \|v\| 2 + O(r),x = tṽ , −t 2 \|ṽ\| 2 + O(r) for some v,ṽ ∈ Ξ λ 2 , one easily deduces that by using triangle inequality ∇ ξ,s x − x, (ξ, s) − (t − t)(2(λ + s)) −1 \|ξ\| 2 λ −1 \|t −t\|,(2.5) for all (ξ, s) ∈ supp a 1 . Using (2.2) and a non-stationary phase (integration by parts) argument yield (2.4) N (1 + \|t −t\|/λr) −N H(·, t) 2 \|H(·,t) 2 dtdt, concluding the proof by using Schur's test. We remark that one may need normalize (2.5) by dividing r on its both sides when defining the invariant differential operator akin to L as in the proof of Proposition 2.2. 2.2. The (λ, ̟, ̺)−wavepacket decomposition. Lemma 2.4. Let λ ≥ 2 10C0 , 0 < ̟ ≤ 2 −C0 and ̺ ∈ [2 C0/2 , λ] with C 0 large. Define L = ̟ −2 ̺ Z n+1 and Γ = ̺ −1 Z n . For any function f ∈ S(R n+1 ) such that f is supported in B, the following statement holds: For each (v, µ) ∈ L × Γ , there is a wave F λ v,µ such that we have S λ (t)f (x) = (v,µ)∈L×Γ F λ v,µ (x, t), ∀ (x, t) ∈ R n+1 × R. (2.6) Moreover, for any B ≫ 1, there are c v,µ,k > 0 and φ v,µ,k ∈ C ∞ (R n+2 ) with k ∈ Z such that we may decompose further F λ v,µ (x, t) = k∈Z c v,µ,k φ v,µ,k (x, t), for all x ∈ R n+1 and t ∈ R, and that there is a constant C n > 0, only depending on n, for which we have ̺ n (v,µ)∈L×Γ k∈Z c 2 v,µ,k B ̟ −Cn E(S λ f ). (2.7) For any t 0 ∈ R and v = (ν, ν n+1 ) ∈ (R n × R) ∩ L, we have for any integer M ≥ 1 \|φ v,µ,k (x, t)\| B,M ̟ −O(M) 1 + ̟ 2 ̺ −1 k − ν n+1 −B × 1 + ̺ −1 x − ν − t µ λ + x n+1 − k + t \|µ\| 2 2λ 2 −M , (2.8) for all x = (x, x n+1 ) ∈ R n+1 and \|t − t 0 \| λ̺ 2 . Finally, the Bessel type inequality holds ∆ sup t (v,µ)∈L×Γ m ∆ v,µ F λ v,µ (·, t) 2 L 2 (R n+1 ) 1 2 ≤ (1 + C n ̟) f L 2 , (2.9) for all m ∆ v,µ ≥ 0 such that sup (v,µ)∈L×Γ ∆ m ∆ v,µ ≤ 1 where ∆ is summing over a finite number of ∆'s. Proof. By translation in the physical spacetime and the modulation in the frequency space, we may take t 0 = 0 without loss of generality. Let Υ 0 ∈ S(R n+1 ) be a non-negative Schwartz function such that Υ 0 is supported in U := {(ξ, s) ∈ R n+1 ; \|(ξ, s)\| ≤ 1/10} and that Υ 0 equals to one on 1 2 U . Put Υ v (x) = Υ 0 (̟ 2 ̺ −1 (x − v)), v ∈ L. By the Poisson summation, we have v∈L Υ v (x) = 1 for all x. Let ✷ = [−1/2, 1/2) × · · · × [−1/2, 1/2) n times and 1 1 ✷ be the characteristic function of the unit box ✷. For each (v, µ) ∈ L × Γ , let a v,µ (x, ξ) = Υ v (x) 1 1 ✷ * 1 1 ✷ (̺(ξ − µ)) . For any f ∈ S(R n+1 ) such that supp f (ξ, s) ⊂ B, define f v,µ (x) = B e 2πi x,(ξ,s) a v,µ (x, ξ) f (ξ, s) dξds. Then, by Fubini's theorem, we have f (x) = (v,µ)∈L×Γ f v,µ (x) for all x ∈ R n+1 . By linearity of S λ (t), we have S λ (t)f (x) = (v,µ)∈L×Γ S λ (t)f v,µ (x) . (2.10) Denoting F λ (t) = S λ (t)f and F λ v,µ (t) = S λ (t)f v,µ , we get (2.6) . To obtain the further decomposition, let α ∈ C ∞ c (R n ) be such that α equals to one on {ξ ∈ R n ; \|ξ\| ≤ 50n} and vanishes outside an O(1)−neighborhood of this set. Let β ∈ C ∞ c (R) be a similar function such that β equals to one on [−50, 50]. Put p(ξ, s) = α(ξ)β(s) and define K λ,̺ t,µ (x) = R n+1 e 2πi x·ξ+xn+1s− t 2 \|ξ\| 2 λ+s p ̺(ξ − µ), s dξds. We have F λ v,µ (t) = K λ,̺ t,µ * f v,µ . Changing variables, we have K λ,̺ t,µ (x) = e 2πix·µ ̺ n R n+1 e 2πi ̺ −1 x·ξ+xn+1s− t 2 \|µ+̺ −1 ξ\| 2 λ+s p(ξ, s) dξds. Expanding \|µ+̺ −1 ξ\| 2 2(λ+s) with respect to (ξ, s) using (1 + θ) −1 = 1 − θ + θ 2 E(θ) with E(θ) = (1 + θ) −1 , we have K λ,̺ t,µ (x) = ̺ −n e 2πix·µ e −πitλ −1 \|µ\| 2 × R n+1 e 2πi ̺ −1 (x−t µ λ )·ξ+(xn+1+t \|µ\| 2 2λ 2 )s+t E λ,̺ µ (ξ,s) p(ξ, s) dξds, (2.11) where E λ,̺ µ is a smooth function on the support of p and bounded along with all its derivatives by O(λ −1 ̺ −2 ) for all µ, thanks to ̺ ≤ λ. In fact, write \|µ + ̺ −1 ξ\| 2 2(λ + s) = ̺ −2 \|ξ\| 2 + 2̺ −1 ξ, µ + \|µ\| 2 2λ 1 + s λ −1 , with \|s\|/λ 2 −10C0 ≪ C −1000 0 for large C 0 . Elementary algebraic manipulations lead to (2.11) with E λ,̺ µ (ξ, s) = − \|ξ\| 2 2λ̺ 2 + \|ξ\| 2 s 2λ 2 ̺ 2 + ξ, µ s λ 2 ̺ − \|ξ\| 2 s 2 2λ 3 ̺ 2 E s λ − ξ, µ s 2 ̺λ 3 E s λ − \|µ\| 2 s 2 2λ 3 E s λ . To treat E λ,̺ µ as an error term, we need the stability condition ̺ ≤ λ. Letting L = 1 + (2πi) −1 ̺ −1 (x − t µ λ ) + t∂ ξ E λ,̺ µ · ∂ ξ + (2πi) −1 (x n+1 + t \|µ\| 2 2λ 2 + t∂ s E λ,̺ µ )∂ s 1 + ̺ −1 (x − t µ λ ) + t ∂ ξ E λ,̺ µ 2 + x n+1 + t \|µ\| 2 2λ 2 + t ∂ s E λ,̺ µ 2 , such that for any integer M ≥ 1, we have L M e 2πi ̺ −1 (x−t µ λ )·ξ+(xn+1+t \|µ\| 2 2λ 2 )s+t E λ,̺ µ = e 2πi ̺ −1 (x−t µ λ )·ξ+(xn+1+t \|µ\| 2 2λ 2 )s+t E λ,̺ µ . Noting that 1 + ̺ −1 (x − t µ λ ) + t ∂ ξ E λ,̺ µ (ξ, s) + x n+1 + t \|µ\| 2 2λ 2 + t ∂ s E λ,̺ µ (ξ, s) ̺ −1 (x − t µ λ ) + x n+1 + t \|µ\| 2 2λ 2 holds for all (ξ, s) ∈ supp p and all \|t\| λ̺ 2 , we have by the non-stationary phase argument (M -fold integration by parts, see also the formula for K λ j in the proof of Proposition 2.2 ) K λ,̺ t,µ (x) M ̺ −n 1 + ̺ −1 x − t µ λ + x n+1 + t \|µ\| 2 2λ 2 −M . (2.12) In other words, for t being contained in an interval of length O(λ̺ 2 ), the kernel function K λ,̺ t,µ (x) is concentrated on a ̺ × · · · × ̺ n times ×1 × ̺ 2 λ plate, denoted as V λ,̺ µ , which is oriented along the direction ( µ λ , − \|µ\| 2 2λ 2 , 1) with thickness being approximately one in the x n+1 direction and of width ̺ in the circular directions. We call V λ,̺ µ the concentration plate for K λ,̺ t,µ and it is clear that V λ,̺ µ is contained in an O(̺)−neighbourhood of Λ λ j if we have the condition that µ ∈ Γ ∩ V j . Let η 0 ∈ S(R) have the same property as Υ 0 such that if we put η k (s) = η 0 (s−k), we have the partition of unity k∈Z η k (s) = 1 for all s ∈ R. Writing f v,µ (x) = k∈Z f v,µ,k with f v,µ,k (x) := η k (x n+1 )f v,µ (x), we have F λ v,µ (t) = k∈Z F λ v,µ,k (t) with F λ v,µ,k (t) := K λ,̺ t,µ * f v,µ,k . Now, we need the plate maximal function f → M ̟,̺ f (x) = sup r>0 1 \|R ̟,̺ r \| R ̟,̺ r \|f (x − x ′ )\| dx ′ where R ̟,̺ r = (x 1 , . . . , x n+1 ) ∈ R n+1 ; \|x 1 \|, . . . , \|x n \| ≤ r̟ −2 ̺, \|x n+1 \| ≤ r . Let f µ = v f v,µ . For each v = (ν, ν n+1 ) and k ∈ Z, we define c v,µ,k := 1 + ̟ 2 ̺ −1 \|k − ν n+1 \| −B M ̟,̺ f µ (ν, k), φ v,µ,k (t) := 1 c v,µ,k F λ v,µ,k (t) . Then, we have for any M ≥ 1 \|φ v,µ,k (x, t)\| B,M ̟ −O(M) 1 + ̟ 2 ̺ −1 \|k − ν n+1 \| −B × 1 + ̺ −1 x − ν − t µ λ + x n+1 − k + t \|µ\| 2 2λ 2 −M (2.13) for all v, µ, k and x and all t which is contained in an interval of length λ̺ 2 . The argument for (2.13) is standard by using dyadic decomposition. We sketch it briefly. Let (a λ µ , b λ µ ) = x − ν − tλ −1 µ, x n+1 − k + t 2 λ −2 \|µ\| 2 . Consider first \|k − ν n+1 \| ̟ −2 ̺. If \|a λ µ \| ̟ −2 ̺ and \|b λ µ \| ̟ −2 , we use (2.12) with a (different) sufficiently large M , and F λ v,µ,k (t) = K λ,̺ t,µ (x − x ′ )f v,µ,k (x ′ )dx ′ incorporated with the concentration property of Υ v that it is concentrated on a ball of radius ̟ −2 ̺ centered at v, and with η k on an interval of length being roughly one centered at k. Here, these variables are referred to be the x ′ in the convolution F λ v,µ,k and should not be confused with the fixed x = (x, x n+1 ) in (a λ µ , b λ µ ). Thus, in this case, (2.13) follows from the trivial averaging argument by using the telescoping decomposition R n+1 = R ̟,̺ 1 ∪ ∞ h=1 R ̟,̺ 2 h \ R ̟,̺ 2 h−1 . Next, consider \|a λ µ \| ≫ ̟ −2 ̺ or \|b λ µ \| ≫ ̟ −2 . We only take the case \|a λ µ \| ≫ ̟ −2 ̺ and \|b λ µ \| ̟ −2 to illustrate the idea and the other cases are tackled in the same way. By Fubini theorem, we integrate first w.r.t. the x−component. Split the integration over R n x into the union of the ball {\|x\| ≤ ̟ −2 ̺} and dyadic annuli {x : 2 k ̟ −2 ̺ ≤ \|x\| ≤ 2 k+1 ̟ −2 ̺} for k ≥ 1. Let C ≫ 1 be a universal constant and consider K a λ µ := {k ≥ 5C; 2 k−C ≤ ̟ 2 ̺ −1 \|a λ µ \| ≤ 2 k+C } where clearly we have card K a λ µ C. For all k ∈ K a λ µ , we use the fast decay of Υ 0 fixed at the beginning of the proof and 2 k ≈ C \|a λ µ \|̟ 2 ̺ −1 to conclude the proof. For k ∈ K a λ µ , consider if 2 k ≤ 2 −C \|a λ µ \|̺ −1 ̟ 2 , we use \|a λ µ − x\| \|a λ µ \| and the rapid decay of Υ 0 to conclude the proof; if 2 k ≥ 2 C \|a λ µ \|̺ −1 ̟ 2 , then we use \|a λ µ − x\| \|x\| \|a λ µ \| and the same argument as above to conclude the result. The same dyadic decomposition argument implies the desired result for the other two cases. For more details, one may consult [33,23,26]. Consider next when \|k − ν n+1 \| ≫ ̟ −2 ̺. Using the rapid decay of η 0 and Υ 0 so that for any B > 0, one can bring in a factor B 1 + ̟ 2 ̺ −1 \|k − ν n+1 \| −10B and the rest part of the proof is the same. To show (2.7), using supp f µ (ξ, s) ⊂ {(ξ, s); \|ξ − µ\| ≪ ̺ −1 , s ∈ [−10, 10]} for all µ, we claim that for any fixed C ≥ 1, one has M ̟,̺ f µ (ν, k) ≤ ̟ −O(1) M ̟,̺ f µ (x), (2.14) for all x ∈ (ν, k) + CR ̟,̺ 1 and all k ∈ Z, (v, µ) ∈ L × Γ , where the implicit constant in O(1) depends only on C and n. Squaring both sides of (2.14) and integrating on (ν, k) + CR ̟,̺ 1 then summing over v, k, we obtain (2.7) by the L 2 −boundedness of mutli-parameter maximal functions over all rectangles with sides parallel to axes (c.f. Chapter 2 of Stein [31]) and then summing over µ, by Plancherel and almost orthogonality in the frequency space. When changing orders in summing over v, µ, k, one needs to take advantagne of the fact that when k is at a distance ≈ 2 γ ̟ −2 ̺ away from ν n+1 , for some γ ≥ 1, there is a factor 2 −Bγ with B ≫ 1 that ensures the convergence of the geometric series. Thus, on each dyadic level 2 γ , one may classify ν n+1 into arithmetic progressions of length ≈ 2 γ so that the essential finite overlappedness occurs on each class. The O(2 γ )−loss is eaten by 2 −Bγ with B ≫ 1. It remains to show (2.14), which is deduced by the same argument of [33] based on the uncertainty principle. We leave the proof to Appendix A. We next prove the Bessel type inequality (2.9). For any ξ ′ ∈ R n , define P ξ ′ µ,̺ f (x) = e 2πi(x·ξ+xn+1s) 1 1 ✷ (̺(ξ − µ − ξ ′ )) f (ξ, s) dξds. Then, f µ (x) is the average of P ξ ′ µ,̺ f (x) over ̺ −1 ✷ with respect to ξ ′ . By Plancherel's theorem and Minkowski's inequality, we have the left side of (2.9) where ≤ ̺ n ̺ −1 ✷ ∆ (v,µ)∈L×Γ m ∆ v,µ Υ v (·) P ξ ′ µ,̺ f (·) 2 2 1 2 dξ ′ . (2.15) For each µ ∈ Γ , define B ̺,µ = µ + 1 ̺ ✷ and let O = µ∈Γ ξ ∈ B ̺,µ ; dist(ξ, R n \ B ̺,µ ) ≥ ̟ 2 ̺ −1 . For any ξ ′ ∈ ̺ −1 ✷, define Π O+ξ ′ : f (x) → e 2πi(x·ξ+xn+1s) 1 1 {(ξ,s) ; ξ∈O+ξ ′ } (ξ, s) f (ξ, s) dξds. Splitting f = Π O+ξ ′ f + id − Π O+ξ ′ fI = ̺ n ̺ −1 ✷ ∆ (v,µ)∈L×Γ m ∆ v,µ Υ v (·) P ξ ′ µ,̺ • Π O+ξ ′ f (·) 2 2 1 2 dξ ′ , (2.16) II = ̺ n ̺ −1 ✷ ∆ (v,µ)∈L×Γ m ∆ v,µ Υ v (·) P ξ ′ µ,̺ • id − Π O+ξ ′ f (·) 2 2 1 2 dξ ′ . (2.17) To deal with I, we use the Plancherel theorem and the strict orthogonality from the pairwise ̟ 2 ̺ −1 -separateness between the simply connected components of O, which allows a petite amplification in the frequency space caused by convolution with Υ v . Note that the enlargement on the support of the s−variable in the frequency space does not affect the disjointness of the cylindrical sets {(ξ, s); ξ ∈ B ̺,µ , dist(ξ, R n \ B ̺,µ ) ≥ ̟ 2 ̺ −1 /100} as µ ranges in Γ . We have I ≤ ̺ n ̺ −1 ✷ µ ∆,v m ∆ v,µ Υ v (·) P ξ ′ µ,̺ • Π O+ξ ′ f (·) 2 2 1 2 dξ ′ ≤ ̺ n ̺ −1 ✷ µ P ξ ′ µ,̺ • Π O+ξ ′ f (·) 2 2 1 2 dξ ′ ≤ ̺ n ̺ −1 ✷ µ 1 1 ✷ ̺(· − µ − ξ ′ ) Π O+ξ ′ f (·, s) 2 L 2 (R n ξ ) ds 1 2 dξ ′ ≤ f L 2 , where we have used ℓ 1 ⊂ ℓ 2 to get the first inequality and sup µ ∆,v m ∆ v,µ Υ v ≤ v Υ v ≤ 1, for the second estimate. We used the strict orthogonality again in the last step. For II, we have no strict orthogonality in the Fourier side anymore since the frequencies are located at the ̟ 2 ̺ −1 −neighbourhood of the boundary inside B ̺,µ . In particular, no disjointness of the frequency variables can be used. Instead, by using the Plancherel theorem and the almost orthogonality followed with Cauchy-Schwarz as in dealing with I above, we have II n ̺ n ̺ −1 ✷ µ ∆,v m ∆ v,µ Υ v (·) P ξ ′ µ,̺ • id − Π O+ξ ′ f (·) 2 2 1 2 dξ ′ n ̺ n ̺ −1 ✷ id − Π O+ξ ′ f 2 L 2 dξ ′ 1 2 n ̟ f L 2 , where in the last estimate we have used the Fubini theorem and that sup ξ∈R n ̺ n ̺ −1 ✷ 1 − 1 1 O+ξ ′ (ξ) dξ ′ ≤ C n ̟ 2 , which is an obvious fact by noting that ξ ′ in the above integrand is restricted inside the intersection of a cube of size ̺ −1 and an O( ̟ 2 ̺ −1 )−neighbordhood of ∪ µ ∂ B ̺,µ , union of the boundaries of the B ̺,µ 's. The proof is complete. 2.3. Construction of the wave tables. We set off the construction for the S λ −version of the wave table theory akin to [32]. The advantage of using wave tables is to eradicate the logarithmic loss arising from the repeatedly used dyadic pigeonhole principle [33,41], which blocked the approach to the endpoint results. From this section on, we start adopting new notations F λ and G λ to denote respectively the red and blue waves. We apply Lemma 2.4 with ̺ = R 1/2 F λ (x, t) = (v,µ)∈L×Γ1 F λ v,µ (x, t), G λ (x, t) = (v,µ)∈L×Γ2 G λ v,µ (x, t), where for each j ∈ {1, 2}, Γ j := Γ ∩ V j and t is always assumed to be contained in an interval of length O(λR). Moreover, for each (v, µ) ∈ L × Γ 1 , the wave packet F λ v,µ = k F λ v,µ,k (and similarly for G λ v,µ ) is tightly concentrated on a tube T λ v,µ , which is the union of the plates V λ ν,µ,k , on which F λ v,µ,k is concentrated in the sense of (2.8), for k satisfying \|k − ν n+1 \| ̟ −2 ̺. For each of those k's with the property that \|k − ν n+1 \| ≫ ̟ −2 ̺, F λ v,µ,k decreases very fast in the wave envelope F λ v,µ = k F λ v,µ,k in view of the ℓ 2 −summability (2.7) and (2.8). Note that each V λ v,µ,k is oriented in the direction of ( µ λ , − \|µ\| 2 2λ 2 , 1), parallel to T λ v,µ for the fixed µ. We shall say T λ v,µ is parametrized by (v, µ). Moreover, each tube T λ v,µ is of dimensions √ R × · · · × √ R (n+1) times ×λR. Denoting F λ T = F λ v,µ with T = T λ v,µ , we rewrite the decomposition for F λ into the form F λ = T1∈T1 F λ T1 , where T 1 is the the collection of T 1 tubes associated to the red waves in the above sense. For each T 1 , we write F λ T1 = V λ v,µ,k ⊂T1 F λ v,µ,k + V λ v,µ,k ⊂T1 F λ v,µ,k := F λ,g T1 + F λ,b T1 , where F λ,b T1 is the (global) part corresponding to the Schwartz tails. Similarly, we have the decomposition for the blue wave G λ = T2∈T2 G λ T2 with the local/global decomposition G λ T2 = G λ,g T2 + G λ,b T2 for each T 2 ∈ T 2 . See also Section 3 of [41] for the same decomposition. For each T j ∈ T j with j ∈ {1, 2}, we use ψ Tj to denote the bump function ψ Tj (x, t) = min 1, dist((x, t), T j ) −N adapted to T j . When we say a spacetime cube, we mean a cube with sides parallel to the axes. A λ−stretched cube of size R is a cube Q λ R ⊂ R n+2 x,t such that the length in the vertical direction equals to λR, i.e. along the t−axis, and having sides R in the horizontal components R n+1 x . We denote Q C0 (Q λ R ) to be the cubes obtained by bisecting each side of Q λ R consecutively such that every ∆ ∈ Q C0 (Q λ R ) is a λ−stretched cube of size 2 −C0 R. Fix ∆ ∈ Q C0 (Q λ R ), let K Q λ R (∆) = q ⊂ ∆; q ∈ Q J (Q λ R ) with J ≈ log R. Each q is a λ−stretched cube of size ≈ √ R. Let χ ∈ S(R n+2 ) be such that χ is compactly supported in a small neighbourhood of the origin and χ ≥ 1 on double of the unit ball. Let A q be the affine transform sending the John ellipsoid inside q to the unit ball such that if we let χ q = χ • A q , then we have χ q ≥ 1 1 q , where 1 1 q is the characteristic function of q . Definition 2.5. Let Q = Q λ R . For each ∆ ∈ Q C0 (Q) and T 1 ∈ T 1 , define m G λ , ∆ T1 = q∈KQ(∆) T2∈T2 χ q ψ T1 ψ −50 T2 G λ T2 2 L 2 (R n+2 ) , and set m G λ T1 = ∆∈QC 0 (Q) m G λ , ∆ T1 . The (λ, ̟, R 1/2 )−wave table F λ for F λ with respect to G λ over Q is defined as the vector-valued function F λ = F λ ̟,R 1/2 = F λ, ∆ ̟ ∆∈QC 0 (Q) , with F λ,∆ ̟ (x, t) := T1∈T1 m G λ , ∆ T1 m G λ T1 F λ T1 (x, t). Similarly, we define the (λ, ̟, R 1/2 )−wave table G λ for G λ with respect to F λ over Q in the symmetric way: G λ = G λ ̟,R 1/2 = G λ, ∆ ̟ ∆∈QC 0 (Q) , with G λ,∆ ̟ (x, t) := T2∈T2 m F λ , ∆ T2 m F λ T2 G λ T2 (x, t), where m F λ , ∆ T2 = q∈KQ(∆) T1∈T1 χ q ψ T2 ψ −50 T1 F λ T1 2 L 2 (R n+2 ) , and m F λ T2 = ∆∈QC 0 (Q) m F λ , ∆ T2 . Clearly, we have F λ = ∆∈QC 0 (Q) F λ, ∆ ̟ , G λ = ∆∈QC 0 (Q) G λ, ∆ ̟ . Remark 2.6. Note that the weights m G λ ,∆ T1 and m F λ ,∆ T2 appear different from that of [32] but closer to the form of those in [20]. However, these two forms are essentially equivalent and the definition we adopt here is more convenient when dealing with the paraboloid. We have also refined this definition by inserting ψ −50 Tj to the L 2 −integral for a technical reason. Remark 2.7. To lighten notations, we will suppress the subscript ̟ and omit the G λ , F λ on the shoulders of m G λ ,∆ T1 and m F λ ,∆ T2 respectively. The dependence on various of these parameters will be clear from the context. Note that by the linearity of the operator S λ (t), for any ∆, we find that F λ,∆ and G λ,∆ are red and blue waves respectively, and one may define the energy E(F λ,∆ ) and E(G λ,∆ ) as in the beginning of Section 2. By using the Bessel type inequality (2.9), we have Lemma 2.8. There is a constant C n depending only on n such that we have E(F λ ) 1/2 := ∆∈QC 0 (Q) E F λ,∆ 1 2 ≤ (1 + C n ̟)E(F λ ) 1/2 , (2.18) E(G λ ) 1/2 := ∆∈QC 0 (Q) E G λ,∆ 1 2 ≤ (1 + C n ̟)E(G λ ) 1/2 ,(2. 19) for any (λ, ̟, R 1/2 ) wave tables F λ , G λ over a spacetime cube Q. Next, we define the C 0 −quilts of F λ and G λ on Q = Q λ R as F λ C0 = ∆∈QC 0 (Q) 1 1 ∆ F λ, ∆ , G λ C0 = ∆∈QC 0 (Q) 1 1 ∆ G λ, ∆ . The (̟, C 0 )−interior of Q is defined as I ̟, C0 (Q) = ∆∈QC 0 (Q) (1 − ̟)∆. Here (1 − ̟)∆ is the stretched cube of the same center with ∆, but with side length multiplied by the constant (1 − ̟) with 0 < ̟ ≪ 1. These cubes play a crucial role to obtain the effective approximation to the product of red and blue waves via the C 0 −quilts. Let z 0 = (x 0 , t 0 ) ∈ R n+2 and define the conic set C λ (z 0 , r) = Λ λ 1 (z 0 , r) ∪ Λ λ 2 (z 0 , r), with Λ λ j (z 0 , r) given in Lemma 2.3 for j = 1, 2. Let X ̟,r z0 (Q) = I ̟,C0 (Q)∩ C λ (z 0 , r). For any u ∈ L ∞ loc (R n+1 x × R t ) and any measurable subset Ω ⊂ R n+2 , such that Ω = t∈I Π t × {t} for some I ⊂ R, we denote u Z(Ω) := I Πt \|u(x, t)\| s dx q s dt 1 q , with (q, s) = q + c , r − c ∈ Γ, where for any γ ∈ R, we denote γ + ( resp. γ − ) as a real number greater (resp. smaller) than but sufficiently close to γ. Thus, the sense of Z I ̟,C0 (Q) and Z X ̟,r z0 (Q) is clearly understood. The effective approximation of F λ G λ Z(Q) via C 0 −quilts below plays a fundamental role in the endpoint theory of bilinear estimates [32]. Proposition 2.9. For any R ∈ [2 10C0 , λ] , ̟ ∈ (0, 2 −C0 ] and Q = Q λ R , there exists a constant C, depending only on n and independent of C 0 , such that if F λ and G λ are red and blue waves with E(F λ ) = E(G λ ) = 1, and F λ , G λ are the (λ, ̟, R 1/2 )−wave tables for F λ and G λ over Q * := CQ respectively, we have F λ G λ Z(Q) ≤ (1 + C̟) F λ C0 G λ C0 Z(I ̟,C 0 (Q * )) + λ 1 q ̟ −O(1) (2.20) and there exists κ = κ Z > 0 such that F λ G λ Z(Q∩ C λ (z0,r)) ≤ (1 + C̟) F λ C0 G λ C0 Z(X ̟,r z 0 (Q * )) + λ 1 q ̟ −O(1) 1 + R r −κ , (2.21) holds for all z 0 ∈ R n+2 . Remark 2.10. As pointed out in [32], this result is a pigeonhole-free version of the arguments in Wolff [41]. For this reason, we refer to the method of [32] as a profound version of the induction on scale argument. Proof. Step 1. Reduction to a tamed bilinear L 2 −Kakeya type estimate. We omit z 0 in X ̟,r z0 for brevity. By the averaging argument in [32,25,39], there is a universal constant C such that we have F λ G λ Z(Q) ≤ (1 + C̟) F λ G λ Z(I ̟,C 0 (Q * )) F λ G λ Z(Q∩C λ (z0,r)) ≤ (1 + C̟) F λ G λ Z(X ̟,r (Q * )) . To get these two estimates, we note that the same method in [32] is directly applied to the stretched cubes in symmetric norms, and the mixed norms follows from the duality argument [25]. Writing F λ G λ = F λ C0 G λ C0 + F λ C0 G λ − G λ C0 + F λ − F λ C0 G λ , we are reduced to F λ C0 G λ − G λ C0 Z(I ̟,C 0 (Q * )) ≤ ̟ −O(1) λ 1 q , (2.22) F λ − F λ C0 G λ Z(I ̟,C 0 (Q * )) ≤ ̟ −O(1) λ 1 q , (2.23) F λ C0 G λ − G λ C0 Z(X ̟,r (Q * )) ≤ ̟ −O(1) 1 + R r −κ λ 1 q , (2.24) F λ − F λ C0 G λ Z(X ̟,r (Q * )) ≤ ̟ −O(1) 1 + R r −κ λ 1 q .(2.F λ,∆ G λ,∆ ′ Z(I ̟,C 0 (Q * )\{∆ ′ }) ≤ ̟ −O(1) λ 1 q ,(2.26) and max ∆,∆ ′ ∈QC 0 (Q * ) To show this claim, using Cauchy-Schwarz inequalities and Lemma 2.8, we get F λ,∆ G λ,∆ ′ Z(X ̟,r (Q * )\{∆ ′ }) ≤ ̟ −O(1) 1 + R r −κ λ 1 q . (2.27) Claim. If max ∆,∆ ′ ∈QC 0 (Q * ) F λ,∆ G λ,∆ ′ L 2 t,x (I ̟,C 0 (Q * )\{∆ ′ }) ≤ ̟ −O(1) λ 1/2 R − n−1 4 ,(2.max ∆,∆ ′ ∈QC 0 (Q * ) F λ,∆ G λ,∆ ′ L q t L 1 x I ̟,C 0 (Q * ) (λR) 1/q . (2.29) Similarly, applying Lemma 2.3 to F λ,∆ , G λ,∆ ′ in place of F λ 1 , F λ 2 there and Minkowski inequality, we have by interpolation with the energy estimates in Lemma 2.8 max ∆,∆ ′ ∈QC 0 (Q * ) F λ,∆ G λ,∆ ′ L q t L 1 x X ̟,r (Q * ) (λR) 1/q 1 + R r − 1 2q . (2.30) From (2.28) and Hölder inequality, we have (1) , no log R -loss involved compared to [33]. See [32] for the cone case. max ∆,∆ ′ ∈QC 0 (Q * ) F λ,∆ G λ,∆ ′ L q t L 2 x I ̟,C 0 (Q * )\{∆ ′ } ≤ ̟ −O(1) λ 1/q R 1 q − n+1 4 .( Step 2. Bilinear L 2 −reduction for (2.28). For any ∆, ∆ ′ , by definition of I ̟,C0 (Q * ) F λ,∆ G λ,∆ ′ 2 L 2 t,x (I ̟,C 0 (Q * )\{∆ ′ }) ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } q∈K Q * (∆ ′′ ) F λ,∆ G λ,∆ ′ 2 L 2 (q) 2 O(C0) max ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } q∈K Q * (∆ ′′ ) F λ,∆ G λ,∆ ′ 2 L 2 (q) . Recall F λ,∆ (x, t) = T1∈T1 m ∆ T1 m T1 F λ T1 (x, t), G λ,∆ ′ (x, t) = T2∈T2 m ∆ ′ T2 m T2 G λ T2 (x, t). For any q ∈ K Q * (∆ ′′ ) with ∆ ′′ ∈ Q C0 (Q * ) \ {∆ ′ }, we have F λ,∆ G λ,∆ ′ 2 L 2 (q) T1∈T1,T2∈T2 T1∩100CQ λ R =∅,T2∩100CQ λ R =∅ m ∆ T1 m T1 F λ T1 m ∆ ′ T2 m T2 G λ T2 2 L 2 (q) + T1∈T1,T2∈T2 T1∩100CQ λ R =∅, or T2∩100CQ λ R =∅ m ∆ T1 m T1 F λ T1 m ∆ ′ T2 m T2 G λ T2 2 L 2 (q) , where the second term is bounded by O(R −N ) using the rapid decay of the wave packets F λ T1 = F λ,g T1 + F λ,b T1 , G λ T2 = G λ,g T2 + G λ,b T2 away from T 1 and T 2 along with the ℓ 2 −summation (2.7). Indeed, if T 1 ∩ 100CQ λ R = ∅, then on any q ⊂ CQ λ R , we have \|F λ,g T1 \| N R −N since each V λ ⊂ T 1 is at a distance R away from q. The F λ,b T1 term is small as well by using the additional decaying factor in (2.8) and the above distance condition. The same argument applies to G λ T2 . For the first term, squaring it out, we are led to estimating q∈K Q * (∆ ′′ ) ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } (T1,T2),(T1,T2)∈T1×T2 Tj ∩100CQ λ R =∅,Tj ∩100CQ λ R =∅, j=1,2 m ∆ T1 m ∆ ′ T2 m ∆ T1 m ∆ ′ T2 m T1 m T2 mT 1 mT 2 χ 8 q (x, t) F λ T1 (x, t) G λ T2 (x, t) F λ T1 (x, t) G λ T2 (x, t) dxdt :=I λ (q,T1,T2,T1,T2) . For each q and T 1 , T 2 ,T 1 ,T 2 in the above summand, I λ (q, T 1 , T 2 ,T 1 ,T 2 ) equals to R n+2 χ 2 q F λ T1 * χ 2 q G λ T2 (ξ, s, τ ) χ 2 q F λ T1 * χ 2 q G λ T2 (ξ, s, τ ) dξdsdτ (2.32) by using Parseval's identity. For any T , belonging either to T 1 or T 2 , we let µ T be such that the direction of T is given by ( µT λ , − \|µT \| 2 2λ 2 , 1) and define J µT ,λ = (ξ, τ ) ; ξ = µ T + O(R −1/2 ), τ = − \|µ T \| 2 2λ + O λ −1 R −1/2 . Then, using supp(u * v) ⊂ supp(u) + supp(v) for any distributions u and v, it is easy to see that by adjusting the implicit constant in the perturbation terms O(· · ·) if necessary, χ 2 q F λ T (ξ, s, τ ) vanishes if (ξ, τ ) ∈ J µT ,λ for T being T 1 ,T 1 . Likewise for χ 2 q G λ T (ξ, s, τ ) with T being T 2 ,T 2 . In order to see this, we note that the support of χ q is contained in a plate of dimensions R −1/2 × · · · × R −1/2 × λ −1 R −1/2 and use a similar expansion of the phase function as in the proof of the wavepacket deomposition once more with R ≤ λ 2 . The condition I λ (q, T 1 , T 2 ,T 1 ,T 2 ) = 0 entails J µT 1 ,λ + J µT 2 ,λ ∩ J µT 1 ,λ + J µT 2 ,λ = ∅ . Hence µ T1 , µ T2 , µT 1 , µT 2 must belong to S R −1/2 := (µ 1 , µ 2 ,μ 1 ,μ 2 ) ∈ Γ 1 × Γ 2 × Γ 1 × Γ 2 ; µ 1 + µ 2 =μ 1 +μ 2 + O(R −1/2 ), \|µ 1 \| 2 + \|µ 2 \| 2 = \|μ 1 \| 2 + \|μ 2 \| 2 + O(R −1/2 ) Given µ 1 andμ 2 , if we let Π µ1,μ2 = ξ ∈ R n ; ξ −μ 2 , µ 1 −μ 2 = 0 and let Π R −1/2 µ1,μ2 be the O(R −1/2 )−neighborhood of Π µ1,μ2 , then µ 2 ∈ Π R −1/2 µ1,μ2 in order that (µ 1 , µ 2 ,μ 1 ,μ 2 ) ∈ S R −1/2 for someμ 1 ∈ Γ 1 , and this is the observation made in [33]. Consequently, for fixed µ 1 , µ 2 ,μ 2 , if we denoteμ 1 ≺ (µ 1 , µ 2 ,μ 2 ) as theμ 1 's determined by the S R −1/2 relation, then card{μ 1 ∈ Γ 1 :μ 1 ≺ (µ 1 , µ 2 ,μ 2 )} = O(1) for all (µ 1 , µ 2 ,μ 2 ). Using m G λ , ∆ T1 m F λ , ∆ ′ T2 m G λ , ∆ T1 m F λ , ∆ ′ T2 m G λ T1 m F λ T2 m G λ T1 m F λ T2 ≤ m F λ , ∆ ′ T2 m F λ T2 1/2 m F λ , ∆ ′ T2 m F λ T2 1/2 and Cauchy-Schwarz, we have as in [20] that max ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } q∈K Q * (∆ ′′ ) T1∈T1,T2∈T2 T1∩CQ λ R =∅,T2∩CQ λ R =∅ m G λ , ∆ T1 m G λ T1 F λ T1 m F λ , ∆ ′ T2 m F λ T2 G λ T2 2 L 2 (q) is bounded by the product of max ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } q∈K Q * (∆ ′′ ) T1,T1∈T1,T2,T2∈T2, (µT 1 , µT 2 , µT 1 , µT 2 )∈S R −1/2 T1,T1∩CQ λ R =∅, T2,T2∩CQ λ R =∅ χ 4 q (x, t)ψ T2 (x q , t q )ψT 1 (x, t) (ψ −1 T1 F λ T1 )(x, t)G λ T2 (x, t) ψT 2 (x q , t q ) 2 m F λ , ∆ ′ T2 m F λ T2 dxdt 1/2 and max ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } q∈K Q * (∆ ′′ ) T1,T1∈T1,T2,T2∈T2, (µT 1 , µT 2 , µT 1 , µT 2 )∈S R −1/2 T1,T1∩CQ λ R =∅, T2,T2∩CQ λ R =∅ χ 4 q (x, t)ψT 2 (x q , t q )ψ T1 (x, t) (ψ −1 T1 F λ T1 )(x, t)G λ T2 (x, t) ψ T2 (x q , t q ) 2 m F λ , ∆ ′ T2 m F λ T2 dxdt 1/2 where (x q , t q ) is the center of q and we have abused notation replacing 100C with C which does not affect the proof. By symmetry, we only estimate the first one, and the second term is handled by using the same method. Step 3. End of the proof. Squaring the first term at the end of Step 2 and using ℓ 1 ⊂ ℓ 2 , we need to estimate max ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } q∈K Q * (∆ ′′ ) T1∈T1,T2∈T2, T1∩CQ λ R =∅,T2∩CQ λ R =∅ 1 m F λ T2 R n+2 χ 8 q (x, t) (ψ −1 T1 F λ T1 )(x, t) G λ T2 (x, t) ψT 2 (x q , t q ) 2 T2∈T2, µT 2 ∈Π R −1/2 µ T 1 ,µT 2 T2∩CQ λ R =∅ m F λ ,∆ ′ T2 ψ 2 T2 (x q , t q ) T 1 ∈T1, µT 1 ≺(µT 1 ,µT 2 ,µT 2 ) T1∩CQ λ R =∅ ψT 1 (x, t) 2 dxdt. By using the uniform estimate card{μ 1 :μ 1 ≺ (µ 1 , µ 2 ,μ 2 )} = O(1) for all (µ 1 , µ 2 ,μ 2 ) and the concentration property of bump function ψT 1 , we have the uniform multiplicity estimate max T1,T2,T2 T 1∈T1 , µT 1 ≺(µT 1 ,µT 2 ,µT 2 ) T1∩CQ λ R =∅ ψT 1 L ∞ ̟ −O(1) . Dropping this term in the integral, we are reduced to estimating the product of max ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } q∈K Q * (∆ ′′ ) max T1∈T1,T2∈T2, T2∈T2, µT 2 ∈Π R −1/2 µ T 1 ,µT 2 T2∩CQ λ R =∅ m F λ ,∆ ′ T2 ψ 2 T2 (x q , t q ) (2.33) and ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } q∈K Q * (∆ ′′ ) T1∈T1,T2∈T2 T1∩CQ λ R =∅,T2∩CQ λ R =∅ 1 m F λ T2 χ 8 q (ψ −1 T1 F λ T1 )G λ T2 2 ψT 2 (x q , t q ) 2 dxdt. (2.34) To estimate (2.33), plug m F λ , ∆ ′ T2 = q ′ ∈K Q * (∆ ′ ) T ′ 1 ∈T1 χ q ′ ψ T2 ψ −50 T ′ 1 F λ T ′ 1 2 L 2 (R n+2 ) into (2.33). Let χ ∆ ′ := q ′ ∈K Q * (∆ ′ ) χ q ′ and W λ,R 1/2 q,∆ ′ ,T1,T2 (x, t) := χ ∆ ′ (x, t) T2∈T2, µT 2 ∈Π R −1/2 µ T 1 ,µT 2 T2∩CQ λ R =∅ ψ T2 (x q , t q )ψ T2 (x, t) 2 . Rearranging the order of summation, we find that (2.33) can be bounded with max ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } q∈K Q * (∆ ′′ ) max T1∈T1,T2∈T2, T ′ 1 ∈T1 χ ∆ ′ (x, t) ψ T ′ 1 (x, t) −50 F λ T ′ 1 (x, t) 2 W λ,R 1/2 q,∆ ′ ,T1,T2 (x, t) dxdt . (2.35) To estimate (2.35), we note that for any ∆ ′′ ∈ Q C0 (Q * ) \ {∆ ′ } , ∆ ′′ is either separated from ∆ ′ along the vertical time-direction by O(̟2 −C0 λR) or horizontally in the x−direction separated away from ∆ ′ by a distance ̟2 −C0 R, where in the second case, the projection of ∆ ′′ to the temporal component coincides with that of ∆ ′ . If we think of ψ T2 as the indicator function of T 2 and likewise for χ ∆ ′ , neglecting the Schwartz tails for the moment, then the above separation property implies that W λ,R 1/2 q,∆ ′ ,T1,T2 is bounded by the characteristic function of the intersection of ∆ ′ and T2∈T2, µT 2 ∈Π R −1/2 µ T 1 ,µT 2 zq∈T2∩CQ λ R =∅ T 2 (2.36) which is contained in an O(R 1/2 )−neighbourhood of Λ λ 2 + z q where z q = (x q , t q ) . This is ensured by the above separateness condition between ∆ ′′ and ∆ ′ . In fact, in the first case, since the directions of the T 2 -tubes are O(R −1/2 λ −1 ) separated, any point in ∆ ′ belongs to at most O(̟ −O(1) ) many T 2 tubes passing through z q (a bush at z q ). In the second case, since q ⊂ ∆ ′′ , if one starts from x q and travels along the direction ( Rλ to arrive at the image of the projection from ∆ ′ to R n+1 x . Consequently, these tubes barely meets ∆ ′ . By the R −1/2 −separateness of the µ T2 's, we obtain a uniform O(̟ −O(1) ) bound on the multiplicity of overlappings. Thus, by affording a constant ̟ −O(1) , the characteristic function of the set (2.36) is bounded by that of Λ λ 2 (z q , R 1/2 ). To clarify the idea of the proof based on this observation, let us think of ψ −50 T ′ 1 F λ T ′ 1 as the characteristic function of the tube T ′ 1 up to some constants, which is reasonable since F λ T ′ 1 has sufficent decay to eat the growth of ψ −50 T ′ 1 in view of (2.8). Using the decomposition F λ T ′ 1 = k F λ T ′ 1 ,k , where we neglect the F λ,b T ′ 1 part by confining k in the summation such that \|k − ν n+1 \| ̟ −2 ̺, we may partition T ′ 1 = ∪ V⊂T ′ 1 V with V being parallel plates and of dimensions 1 × R 1/2 × · · · × R 1/2 n times ×Rλ such that they are in direction of T ′ 1 . Then the transversality condition implies V ∩ T 2 is contained in a rectangle of dimension 1 × R 1/2 × · · · × R 1/2 n times ×λR 1/2 for any T 2 in (2.36) and any V ⊂ T ′ 1 in the above sense. Next, noting that by the more demanding transversality condition that ∀ T 2 in (2.36), µ T2 must satisfy µ T2 ∈ Π R −1/2 µT 1 ,µT 2 for the given T 1 ,T 2 , we claim that there are at most O(1) many T 2 's in (2.36), which intersect in a common sector of length λR 1/2 with any fixed V, taken from the partition of any tube T ′ 1 as above. This is because, Π µT 1 ,µT 2 ⊂ R n ξ is an (n − 1) dimensinal hyperplane passing through µT 2 and orthogonal to the vector (µ T1 − µT 2 ), from which one is convinced with the claim by noting the nowhere vanishing curvature of paraboloids. Indeed, if we let ℓ T be the axis of a given tube T , then in view of the O(R 1/2 ) perturbation, namely T ⊂ ℓ R 1/2 T where ℓ R 1/2 is the O(R 1/2 ) neighborhood of ℓ. For any T ′ 1 and V ⊂ T ′ 1 , partition V into the union of sectors {S j } j=1,...,O( √ R) of length λR 1/2 , i.e. V = ∪ j S j . For any T 2 , T ′ 2 taken from the bush (2.36), such that for some fixed j, we have T 2 ∩ S j = ∅ and T ′ 2 ∩ S j = ∅, then we have ℓ CR 1/2 T ′ 1 ∩ ℓ CR 1/2 T2 ∩ ℓ CR 1/2 T ′ 2 = ∅, for some fixed large C, which entails that µ T2 , µ T ′ 2 and µ T ′ 1 must be almost colinear. Let ℓ * = ℓ(µ T2 , µ T ′ 2 , µ T ′ 1 ) ⊂ R n be the line such that µ T2 , µ T ′ 2 , µ T ′ 1 ∈ ℓ CR −1/2 * . Then ℓ * intersects with Π µT 1 ,µT 2 transversely and the claim follows immediately. We have thus concluded the result, provided that it is legitimate to neglect the effects from Schwartz tails. For the general case, one may incorporate the Schwartz tails by the standard dyadic decomposition exploring the rapid decay of F λ,b T ′ 1 , ψ T2 and χ ∆ ′ etc. away from the geometric objects where concentration occurs. We refer to [41] T ′ 1 k: V λ ν,µ T ′ 1 ,k ⊂T ′ 1 c 2 v,µ T ′ 1 ,k χ ∆ ′ ψ T ′ 1 (x, t) −100 φ v,µ T ′ 1 ,k (x, t)W λ,R 1/2 q,∆ ′ ,T1,T2 (x, t)dxdt bounded by O(λR n+1 2 ) λR 1/2 R n 2 v,µ,k c 2 v,µ,k λR 1/2 ̟ −O(1) (2.37) where we have used (2.7) in the last estimate. For the F λ,b T ′ 1 , one use the decay in (2.8) and dyadically decompose for each T ′ 1 , the summation over k into segments \|k − ν n+1 \| ∼ 2 γ ̟ −2 ̺ for γ ≥ 1 and apply the same argument as for F λ,g T ′ 1 and summing over the dyadic pieces to complete the proof. χ 3 q ψ −1 T1 F λ T1 L ∞ R − n+1 4 λ −1/2 ψ −50 T1 F λ T1 χ q L 2 . In fact, the support F λ T1 * χ q is contained in a 1 × R −1/2 × · · · × R −1/2 n times ×λ −1 R −1/2 rectangle. Reproducing χ q F λ T1 = χ q F λ T1 * ρ λ R 1/2 for some L 1 −normalized Schwartz function ρ λ R 1/2 essentially supported in a box of volume λR n+1 2 , it is easy to see the L 2 → L ∞ norm of the operator associated to the kernel H (x, t; x ′ , t ′ ) := χ 2 q (x, t)ψ −1 T1 (x, t)ρ λ R 1/2 (x − x ′ , t − t ′ )ψ 50 T1 (x ′ , t ′ ) is bounded by O(λ − 1 2 R − n+1 4 ). Thus (2.34) is bounded by R − n+1 2 λ −1 ∆ ′′ ∈QC 0 (Q * )\{∆ ′ } q∈K Q * (∆ ′′ ) T1∈T1,T2∈T2 T1∩CQ λ R =∅,T2∩CQ λ R =∅ ψT 2 (x q , t q ) 2 ψ −50 T1 F λ T1 χ q 2 L 2 /m F λ T2 χ 2 q G λ T2 2 ψT 2 (x q , t q ) −10 dxdt, which is bounded by O(R − n 2 ), once we evoke the definition of m F λ T2 along with the concentration property of G λ T2 , leading to the estimate T 2 sup q χ q G λ T2 2 2 ψT 2 (x q , t q ) −10 ≤ ̟ −O(1) λR 1/2 . This estimate is readily deduced by using the wave packet decomposition in Lemma 2.4 and the same argument in Appendix I of [32]. We only remind the reader that the lifespan of q here is multiplied by λ which is the sole difference between the proof of the above estimate and (62) of [32]. Collecting the estimates on (2.33) and (2.34), we obtain (2.28) and the proof is complete. The Huygens principle: spatial localization In this section, we summarize some properties of the spatial localization operators introduced in [32] to capture the energy concentration of red and blue waves. We start with some notations. A disk is a subset D resident in R n+2 x,t of the form D = D(x D , t D ; r D ) = {(x, t D ) : \|x − x D \| ≤ r D }, for some (x D , t D ) ∈ R n+2 and r D > 0. We call t D the time coordinate of D and r D the radius of D. The indicator function of D is defined as 1 1 D (x) = 1 , (x, t D ) ∈ D , 0 , (x, t D ) ∈ D . Let D ± := D x D , t D ; r D (1 ± r − 1 2N D ) . For any c > 0, let cD := D(x D , t D ; c r D ). Define the disk exterior of D as D ext = D ext (x D , t D ; r D ) = {(x, t D ) : \|x − x D \| > r D }. (3.1) For any u ∈ L ∞ loc (R n+2 ) and disk D, we write u L 2 (D) := \|x−xD\|≤rD u(x, t D ) 2 dx 1 2 , u L 2 (D ext ) := \|x−xD\|>rD u(x, t D ) 2 dx 1 2 . 3.1. The localization operator P D . In order to localize the red and blue waves in the physical space, we introduce the localization operator P D as in [32]. Let Υ 0 (x) ≥ 0 be the Schwartz function in the proof of Lemma 2.4. For every r > 0, we set Υ r (x) = r −(n+1) Υ 0 (r −1 x). Definition 3.1. Let F λ (t) = S λ (t)f with f ∈ S(R n+1 ) such that supp f ⊂ B. For any disk D = D(x D , t D ; r D ), we define P D F λ at time t D as P D F λ (t D ) = 1 1 D * Υ r 1− 1 N D F λ (t D ), and ∀ t ∈ R P D F λ (t) = S λ (t − t D ) P D F λ (t D ) . The localization operator P D behaves almost like a sharp cut-off function, by admitting small mismatching between the margins of D and D ext . In particular, P D localizes a wave to D + and 1 − P D localizes to the exterior of D − . Lemma 3.2. Let D be a disk with radius r D = r ≥ C 0 . Then, P D F λ satisfies the following local energy estimates P D F λ L 2 (D ext + ) r −N E(F λ ) 1/2 (3.2) (1 − P D )F λ L 2 (D−) r −N E(F λ ) 1/2 (3.3) sup t P D F λ (t) 2 L 2 (R n+1 ) ≤ F λ 2 L 2 (D+) + O(r −N E(F λ )) (3.4) sup t (1 − P D )F λ (t) 2 L 2 (R n+1 ) ≤ F λ 2 L 2 (D ext − ) + O(r −N E(F λ )) (3.5) sup t (1 − P D )F λ (t) L 2 (R d ) ≤ E(F λ ) 1 2 , sup t P D F λ (t) L 2 (R 2 ) ≤ E(F λ ) 1 2 , (3.6) where D ext ± is the exterior of D ± in the sense of (3.1). Proof. The argument is exactly same to [32] and we sketch it below. To see (3.2), for any x ∈ D ext + and any x ′ such that (x ′ , t D ) ∈ D, one has \|x − x ′ \| ≥ \|x − x D \| − \|x ′ − x D \| ≥ r 1− 1 2N . The rapid decay of Υ 0 ∈ S(R n+1 ) entails sup x: (x,tD )∈D ext + 1 1 D * Υ r 1− 1 N (x) r −N . By using the Plancherel theorem, we get (3.2). To show (3.3), we use Υ 0 = 1 and the rapid decay of Υ 0 to get 1 − 1 1 D * Υ r 1− 1 N (x) = 1 − 1 1 D (x − x ′ ) Υ r 1− 1 N (x ′ )dx ′ M 1 + r −1+ 1 N dist x, D ext −M , (3.7) where for any x ∈ D − , we have that 1 1 D (x − x ′ ) = 0 implies \|x ′ \| ≥ \|x − x ′ − x D \| − \|x D − x\| ≥ r D − \|x − x D \| ≥ dist(x, D ext ). Using (3.7), we obtain (3.3) by taking M sufficiently large. Next, splitting P D F λ (t D ) 2 L 2 = P D F λ 2 L 2 (D+) + P D F λ 2 L 2 (D ext + ) , where the second term is bounded by O(r −N E(F λ )) due to (3.2), we get (3.4) by using 0 ≤ 1 1 D * Υ r 1− 1 N ≤ 1. Similarly, we get (3.5). Finally, (3.6) is obvious. The proof is complete. Concentration of red and blue waves. In this subsection, we use the localization property of P D to characterize the energy concentration of red and blue waves as in [32]. . For any red and blue waves F λ (t) = S λ (t)f and G λ (t) = S λ (t)g where f and g are supported in V 1 × I and V 2 × I respectively, we have P D F λ L ∞ R n+2 x,t \Λ λ 1 (zD, r(1+r − 1 2N )) M r −M E(F λ ) 1/2 , (3.8) P D G λ L ∞ R n+2 x,t \Λ λ 2 (zD, r(1+r − 1 2N )) M r −M E(G λ ) 1/2 , (3.9) for all M ≥ 1 where for j = 1, 2, Λ λ j (z D , r(1 + r − 1 2N )) is defined to be the conic O(r(1 + r − 1 2N ))−neighbourhood of Λ λ j + z D as in Lemma 2.3. Proof. By symmetry, we only show the first estimate (3.8). The argument is the same with (3.2) in Lemma 3.2. Let K λ j (x, t) be as in Proposition 2.2 and put A D 1,t (x, x ′ ) = K λ 1 (x − x ′ , t − t D ) 1 1 D * Υ r 1− 1 N (x ′ ) . Then P D F λ (t, x) = A D 1,t (x, x ′ ) F λ (x ′ , t D ) dx ′ . (3.10) By (2.2), we have \|K λ 1 (x − x ′ , t − t D )\| M 1 + dist (x − x ′ , t − t D ) , Λ λ 1 −100MN . (3.11) Assume that (x, t) / ∈ Λ λ 1 (z D , r(1 + r − 1 2N )). Then, we have x − x D − (t − t D ) v, − \|v\| 2 2 r(1 + r − 1 2N ), ∀ v ∈ 2 Ξ λ 1 . (3.12) Decompose the domain of integration for x ′ in (3.10) into the local part where for some large constant C, we have dist (x − x ′ , t − t D ), Λ λ 1 ≤ C −1 r(1 + r − 1 2N ) and the global part dist (x − x ′ , t − t D ), Λ λ 1 ≥ C −1 r(1 + r − 1 2N ). For the global part, (3.8) follows from using (3.11) and the Cauchy-Schwarz inequality. For the local part, we have for some v 1 ∈ 2 Ξ λ 1 x ′ − x + (t − t D ) v 1 , − \|v 1 \| 2 2 C −1 r(1 + r − 1 2N ) for C sufficiently large. Writing 1 1 D * Υ r 1− 1 N (x ′ ) = 1 1 D (x ′′ )Υ r 1− 1 N (x ′ − x ′′ )dx ′′ , and noting that by (3.12) \|x ′ − x ′′ \| ≥ \|x ′ − x D \| − \|x ′′ − x D \| x − x D − (t − t D ) v 1 , − \|v 1 \| 2 2 − C −1 r(1 + r − 1 2N ) − r r 1− 1 2N , we obtain (3.8) by using the rapid decay of Υ 0 . The proof is complete. Remark 3.4. The above lemma is analogous to Lemma 10.3 of [32]. We write a λ−stretched cube Q λ = Q λ (x Q , t Q ; r Q ) of side-length r Q centered at (x Q , t Q ) ∈ R n+2 . For any C > 0, we write CQ λ = Q λ (x Q , t Q ; Cr Q ). Lemma 3.5. Let r ≥ C 0 and D = D(z D ; r) with z D = (x D , t D ). For any red and blue waves F λ and G λ , we have for large C (1 − P D )F λ L ∞ (Q(zD, r/C)) r −N E(F λ ) 1/2 , (3.13) (1 − P D )G λ L ∞ (Q(zD, r/C)) r −N E(G λ ) 1/2 . (3.14) Proof. We only show the first estimate (3.13) by symmetry. Let N λ j,t (x) be given by Proposition 2.2. Write (1 − P D )F λ (t, x) = N λ 1,t−tD (x − x ′ ) 1 − (1 1 D * Υ r 1− 1 N )(x ′ ) F λ (t D , x ′ ) dx ′ . For any (x, t) ∈ Q(z D , r/C), we may write x ′ − x + (t − t D )(v, −\|v\| 2 /2) = (x ′ − x D ) − (x − x D ) + (t − t D )(v, −\|v\| 2 /2) , with v ∈ 2 Ξ λ 1 . In view of (3.11), we may assume that x ′ ∈ 1 2 D, otherwise, by using \|x − x D \| ≤ C −1 r, \|t − t D \| ≤ C −1 λr and the rapid decay of the kernel N λ 1,t−tD away from Λ λ 1 , we obtain an upper bound O(r −N E(F λ )). Next, by (3.7) 1 − 1 1 D * Υ r 1− 1 N (x ′ ) M 1 + r −1+ 1 N dist x ′ , D ext −M M r −M/N , for all x ′ ∈ 1 2 D. The proof is complete by taking M large. 3.3. A non-endpoint bilinear estimate. For 0 < r 1 < r 2 < +∞, we define the cubical annulus as Q ann (x Q , t Q ; r 1 , r 2 ) = Q λ (x Q , t Q ; r 2 ) \ Q λ (x Q , t Q ; r 1 ). We show a non-endpoint bilinear estimate for localized blue or red waves, on a dyadic annulus. This corresponds to Lemma 11.1 of Tao [32], which will be used to handle the case when the energy is highly concentrated. Lemma 3.6. Let 2 10C0 ≤ R ≤ λ, C 0 ≤ r ≤ R 1 2 + 4 N , and D = D(z D , r) with z D = (x D , t D ) . Then, there exists b > 0, depending only on Z and n, such that for any red and blue waves F λ , G λ with E(F λ ) = E(G λ ) = 1, we have (P D F λ )G λ Z(Q ann (zD;R,2R)) , F λ (P D G λ ) Z(Q ann (zD;R,2R)) 2 O(C0) λ 1 q R −b . (3.15) Proof. We only show the estimate for (P D F λ )G λ . By translation and modulation, we may take (x D , t D ) = (0, 0) without loss of generality. By interpolation and taking N large enough, it suffices to show (see Section 2) (P D F λ )G λ L 1 (Q ann (0,0;R,2R)) λR 3 4 + 2 N , (3.16) (P D F λ )G λ L 2 (Q ann (0,0;R,2R)) 2 O(C0) λ 1/2 R − n−1 4 R C/N . (3.17) We may focus on Ω λ r,R := Q ann (0, 0; R, 2R) ∩ Λ λ 1 (0, 0; Cr + R 1 N ) with C ≫ 1 fixed, since by Lemma 3.3, (P D F λ ) is bounded by O(R −N ) outside Λ λ 1 (0, 0; Cr + R 1 N ) . To get the L 1 (Ω λ r,R ) estimate, we have by using Lemma 2.3 and energy estimate ( P D F λ )G λ L 1 (Ω λ r,R ) λ((r + R 1 N )R) 1 2 λR 3 4 + 2 N . To get the L 2 −estimate (3.17) on Ω λ r,R , we use the wave packet decomposition for P D F λ Lemma 2.4 in Section 2 with ̺ = R 1/2 and ̟ = 2 −O(C0) P D F λ = T1∈T1 P D F λ T1 . Using the rapid decay property of the wave packets away from plates V λ T1 ⊂ T 1 (see (2.8)), we have (P D F λ )G λ L 2 (Ω λ r,R ) T1∈T1 dist(T1,(0,0)) R 1 2 + 50 N (P D F λ ) T1 G λ L 2 (Q ann (0,0;R,2R)) + R −N . Note that the directions of the tubes T 1 are O(λ −1 R −1/2 )-separated. By crude estimates, the multiplicities of the tubes, that are O(R 1/2+50/N )−close to the origin, over Q ann (0, 0; R, 2R) is bounded by O(R C/N ) with C depending only on n. By Cauchy-Schwarz, we have T1∈T1 dist(T1,(0,0)) R 1 2 + 1 N (P D F λ ) T1 G λ L 2 (Q ann (0,0;R,2R)) R C N T1∈T1 (P D F λ ) T1 G λ 2 L 2 1 2 . By the same partition of tubes into plates using (2.6) as we did in Section 2, where S λ (t)f is replaced by P D F λ , we have (P D F λ ) T1 = k: V λ v,µ T 1 ,k ⊂T1c v,µT 1 ,kφv, µT 1 , k + k: V λ v,µ T 1 ,k ⊂T1c v,µT 1 ,kφv, µT 1 , k and by Cauchy-Schwarz (P D F λ ) T1 G λ 2 L 2 kc 2 v,µT 1 ,k φ v,µT 1 ,k G λ 2 L 2 . Here, we usec andφ to stress that they are the coefficients and wave packets for P D F λ . Noting thatφ v,µT 1 is essentially concentrated in an O(R 1/2 )−neighbourhood of Λ λ 1 satisfying the same formula (2.8), we have φ v,µT 1 ,k G λ 2 L 2 ̟ −O(1) λR 1/2 by using Lemma 2.3 and standard dyadic decomposition to incorporate Schwartz tails. The coefficients (c v,µ,k ) v,µ,k also obeying the ℓ 2 −summation formula (2.7), summing over v, µ, k, we obtain (3.17). The proof is complete. Remark 3.7. It is in this lemma that we need take N depending on Z, whereas in [32], there is no need to do so. Explore the energy concentration We shall use the method of induction on scales in [32]. To this end, we consider a subclass of the red and blue waves by imposing a margin condition. Let Σ λ = (ξ, s, τ ) : τ = − \|ξ\| 2 2(λ + s) and for j = 1, 2 Σ λ j = (ξ, s, τ ) : τ = − \|ξ\| 2 2(λ + s) , (ξ, s) ∈ V j × I . We shall say Σ λ j is the lift of V j × I to Σ λ . For any R ≥ 2 C0 , we say F λ (t, x) is a R λ R -wave if it is a red wave with the spacetime Fourier transform F (ξ, s, τ ) being an L 2 measure on Σ λ 1 and satisfying the margin condition marg(F λ ) := dist supp( F λ ), Σ λ \ Σ λ 1 ≥ (100n) −1 − R − 1 N . Similarly, we define B λ R to be the subset of blue waves of functions G λ such that supp G λ ⊂ Σ λ 2 and satisfying the margin condition marg(G λ ) := dist supp( G λ ), Σ λ \ Σ λ 2 ≥ (100n) −1 − R − 1 N . Definition 4.1. Fix λ ≥ 2 10C0 . For any R ∈ [2 10C0 , λ], fix Q λ R ⊂ R n+2 a λ−stretched spacetime cube of size R. Let A λ (R) be the optimal constant C such that F λ G λ Z(Q λ R ) ≤ Cλ 1/q E(F λ ) 1/2 E(G λ ) 1/2 ,(4.1) holds for all F λ ∈ R λ R , G λ ∈ B λ R and all Q λ R . By translation in the physical spacetime and modulating the frequency variables resp., A λ (R) is independent of the particular choice of Q λ R . To show Theorem 1.1, we shall show that there is a fixed constant C * depending only on n, ε, V 1 , V 2 and the (q, s) exponent in Z−norm taken sufficiently close to the critical index (q c , r c ), such that A λ (R) ≤ C * holds for all R ≤ λ and all λ ≥ 2 10C0 . We may assume A λ (R) ≥ 1. We set for any 2 C0 ≤ R ≤ λ A λ (R) = sup 2 C 0 ≤λ ′ ≤λ sup 2 C 0 ≤R ′ ≤min(R,λ ′ ) A λ ′ (R ′ ) for a technical issue. We need to introduce an auxiliary quantity, a crucial innovation made in [32]. We call LS(Q λ ) : = [t Q λ −λ r Q λ 2 , t Q λ +λ r Q λ 2 ] the lifespan of Q λ = Q λ (x Q λ , t Q λ ; r Q λ ). In particular, the length of LS(Q λ R ) equals to λR. Definition 4.2. For any R ≥ 2 10C0 and r, r ′ > 0, we define A λ (R, r, r ′ ) to be the optimal constant C such that F λ G λ Z(Q λ R ∩ C λ (z0,r ′ )) ≤ Cλ 1/q E(F λ )E(G λ ) 1 2q E r,C0Q λ R (F λ , G λ ) 1 q ′ , holds for all F λ ∈ R λ R , G λ ∈ B λ R and all cubes Q λ R being as in Definition 4.1 and all z 0 = (x 0 , t 0 ) ∈ R n+2 . Recall that C λ (z 0 , r ′ ) = Λ λ 1 (z 0 , r ′ ) ∪ Λ λ 2 (z 0 , r ′ ) , and q is given by the Z−norm, q ′ = q q−1 . Here, E r,Q λ R is the energy concentration defined in the same way as [32] by letting E r,Q λ R (F λ , G λ ) = max 1 2 E(F λ ) 1/2 E(G λ ) 1/2 , sup D F λ L 2 (D) G λ L 2 (D) , where D ranges over all disks of radius r with the time coordinate t D ∈ LS(Q λ R ). We remark here that it is C 0 Q λ R rather than Q λ R in the definition of A λ in order to cover the structural constants by taking C 0 large. 4.1. Persistence of the non-concentration of energy. The following result will be used to handle the energy-concentrated case. = r(1 − C 0 r − 1 3N ). There exists a constant C > 0, such that if F λ ∈ R λ R , G λ ∈ B λ R with E(F λ ) = E(G λ ) = 1 and F λ , G λ are (λ, ̟, R 1/2 )-wave tables for F λ , G λ over Q, then sup ∆∈QC 0 (Q) E r # ,5Q F λ,∆ , G λ,∆ ≤ (1 + C̟) E r,5Q (F λ , G λ ) + O ̟ −O(1) R −N/2 holds for all ̟ ∈ (0, 2 −C0 ) and all Q. Proof. The argument is the same to [32] and we only sketch it. By (2.9), it suffices to show there is a universal constant C such that (D(z,r)) G λ L 2 (D(z,r)) + O(r −100N ) holds for all ̟ ∈ (0, 2 −C0 ) and all z, Q. sup △∈QC 0 (Q) F λ,∆ L 2 (D(z,r # )) G λ,∆ L 2 (D(z,r # )) ≤ (1 + C̟) F λ L 2 Following the proof of (56) in [32], we let D = D(z, r), D # =D(z, r # ), D ♭ = D(z, r ♭ ), r ♭ = r 1 − C 0 2 r − 1 3N . Then, we have D # D ♭ − D ♭ D ♭ + D by taking C 0 sufficiently large. We only deal with F λ and the same arguments works for G λ . Write F λ, ∆ int (t) = S λ (t − t D # ) T1∈T1 m ∆ T1 m T1 P D ♭ F λ T1 (t D # ), F λ, ∆ ext (t) = S λ (t − t D # ) T1∈T1 m ∆ T1 m T1 (1 − P D ♭ ) F λ T1 (t D # ). Then F λ,∆ (t) = F λ, ∆ int (t) + F λ, ∆ ext (t) . By linearity of P D ♭ , applying Lemma 2.8 and then using (3.4), we get F λ,∆ int (t D # ) 2 L 2 (R n+1 ) ≤ (1 + C̟) P D ♭ F λ (t D # ) 2 L 2 (R n+1 ) ≤ (1 + C̟) F λ 2 L 2 (D ♭ + ) + O(r −N ) ≤ (1 + C̟) F λ 2 L 2 (D) + O(r −N ) where we used D ♭ + ⊂ D in the last estimate. On the other hand, using the fast decay of wavepackets away from CQ, we have T1 m ∆ T1 m T1 (1 − P D ♭ )F λ T1 (t D # ) 2 L 2 (D # ) ≤ T1: T1∩CQ =∅ m ∆ T1 m T1 (1 − P D ♭ )F λ T1 (t D # ) 2 L 2 (D # ) + O( r −N ). for some large fixed constant C. By Minkowski's inequality, we are reduced to max T1∈T1 (1 − P D ♭ )F λ T1 (t D # ) L 2 (D # ) r −100N . (4.2) To this end, we condider the two cases • Case A. dist(T 1 , D # ) ≥ R 1 2 + 1 100N , • Case B. dist(T 1 , D # ) ≤ R 1 2 + 1 100N , For Case A, (4.2) follows from the concentration property (2.8) and the summability estimate (2.7). To handle Case B, we recall from the wave-packet decomposition that if we let F λ ext = (1 − P D ♭ )F λ , then E F λ ext T1 ̟ −O(1) ψ T1 (t D # )F λ ext (t D # ) 2 L 2 + r −100N . Using the assumption r ≥ R dist x T1 + t D # µ T1 λ , − \|µ T1 \| 2 λ 2 , 1 := X , x D # r # + R 1 2 + 1 100N , where z = (x D # , t D # ) and T 1 is parametrized by (x T1 , µ T1 ). Simple calculation yields that the disk centered at X of radius R 1 2 + 1 100N is contained in D ♭ − . Using the rapid decay of ψ T1 , (3.3) and D # ⊂ D ♭ − , we obtain (4.2). By the same argument, we have similar estimates for G λ,∆ . Collecting all of these estimates, we obtain the desired result by suitably adjusting the constant C. Control of A λ by A λ . Proposition 4.4. There is a constant C > 0 depending only on n and Z, but not explicitly on C 0 , such that we have for all R ∈ [2 10C0 , λ] and all r ≥ R 1 2 + 4 N A λ (R, r, C 0 (r + 1)) ≤ (1 + C2 −C0 )A λ (R) + 2 CC0 . (4.3) We divide the proof into three steps. F j q Z ≤ k j=1 F j q Z , for all (q, s) ∈ Γ close to the critical index (q c , r c ) . Please see Lemma 5.3 of [39] for the proof. Proposition 4.6. There is a constant C > 0 such that for any R ∈ [2 10C0 , λ], we have for all r ≥ C 0 R and r ′ > 0 A λ (R, r, r ′ ) ≤ (1 + C̟)A λ (R) + ̟ −C , for all 0 < ̟ ≤ 2 −C0 . Proof. Let F λ ∈ R λ R , G λ ∈ B λ R be red and blue waves with normalized energy. For any Q = Q λ R , let (x Q , t Q ) be the center of Q. Let D = D(z D , r/2) with z D = (x Q , t Q ) and write F λ = P D F λ + (1 − P D )F λ , G λ = P D G λ + (1 − P D )G λ . Using Lemma 3.5, we have (1 − P D )F λ G λ Z(Q λ R ) , (P D F λ )(1 − P D )G λ Z(Q λ R ) ≤ λ 1/q ̟ −O(1) . We are reduced to λ −1/q P D F λ P D G λ Z(Q) ≤ (1 + C̟)A λ (R) E r,C0Q (F λ , G λ ) 1/q ′ + ̟ −O(1) . (4.4) To see this is the case, let F λ D and G λ D be the wave tables for the red and blue waves P D F λ and P D G λ on an enlarged cube Q * containing Q and apply Proposition 2.9 so that we have P D F λ P D G λ Z(Q λ R ) ≤ (1 + C̟) [F λ D ] C0 [G λ D ] C0 Z(I ̟,C 0 (Q * )) + λ 1/q ̟ −O(1) . Applying Lemma 4.5 and the definition of A λ (R), we get λ −1/q [F λ D ] C0 [G λ D ] C0 Z(I ̟,C 0 (Q * )) ≤ ∆∈QC 0 (Q * ) λ −1 F λ,∆ D G λ,∆ D q Z(∆) 1/q ≤ A λ (2 −C0 R) △∈QC 0 (Q * ) E(F λ,∆ D ) q/2 E(G λ,∆ D ) q/2 1/q , (4.5) where we have used F λ,∆ D ∈ R λ 2 −C 0 R , G λ,∆ D ∈ B λ 2 −C 0 R . Using Cauchy-Schwarz, E(F λ D ), E(G λ D ) ≤ 1 + C̟> 0 with R 1/2+3/N ≤ r ≤ C 0 R. A λ (R, r, r ′ ) ≤ (1 + C̟)A λ (R/C 0 , r # , r ′ ) + ̟ −C 1 + R r ′ −θ with r # = r(1 − C 0 r − 1 3N ) holds for all 0 < ̟ ≤ 2 −C0 . Proof. Let F λ ∈ R λ R , G λ ∈ B λ R be red and blue waves with E(F λ ) = E(G λ ) = 1. For any Q = Q λ R , by using Proposition 2.9, we have for all z F λ G λ Z(Q∩C λ (z,r ′ )) ≤ (1 + C̟) [F λ ] C0 [G λ ] C0 Z(I ̟,C 0 (Q * )∩C λ (z,r ′ )) + λ 1/q ̟ −C 1 + R r ′ −κ . We are reduced to showing λ −1/q [F λ ] C0 [G λ ] C0 Z(I ̟,C 0 (Q * )∩C(z,r ′ )) ≤ (1 + C̟)A λ (R/C 0 , r # , r ′ ) E r,C0Q (F λ , G λ ) 1/q ′ + ̟ −C R −N/2 . (4.6) Using the definition of A λ (R, r, r ′ ), we have for all ∆ λ −1/q F λ,∆ G λ,∆ Z(∆∩C(z,r ′ )) ≤ A λ (R/C 0 , r # , r ′ ) E r # ,C0∆ (F λ,∆ , G λ,∆ ) 1/q ′ (E(F λ,∆ )E(G λ,∆ )) 1/(2q) By using Lemma 4.5, Proposition 4.3 with 2 −C0 CR ≪ C −1 0 R so that C 0 ∆ ⊂ 2Q, we obtain by Cauchy-Schwarz λ −1 [F λ ] C0 [G λ ] C0 q Z(I ̟,C 0 (Q * )∩C(z,r ′ )) ≤ λ −1 ∆∈QC 0 (Q * ) F λ,∆ G λ,∆ q Z(I ̟,C 0 (Q * )∩C λ (z,r ′ )) ≤ (1 + C̟)A λ (R/C 0 , r # , r ′ ) q E r,C0Q (F λ , G λ ) q/q ′ + ̟ −C R −qN/2 , (4.7) and (4.6) follows by adjusting the constant C. 4.2.3. Step 3. Proof of Proposition 4.4. With Proposition 4.6 and 4.7, we may complete the proof of Proposition 4.4. For the nonconcentrated case r ≥ C 0 R, letting r ′ = C 0 (r + 1) and ̟ = 2 −C0 in Proposition 4.6, we are done. In the high concentrated case, we follow [32] by letting J be the smallest integer such that r ≥ 2 −J C 0 R and define r := r 0 > r 1 > · · · > r J inductively by letting r j+1 = r # j which leads to r J = r + O(Jr − 1 4N ). Iterating Proposition 4.7 yields A λ 2 −j R, r j , C 0 (r + 1) ≤ (1 + C̟ j )A λ 2 −(j+1) R, r j+1 , C 0 (r + 1) + ̟ −C j 1 + R 2 j r −θ with ̟ j = ̟2 −(J−j)θ/C• for some fixed large C • ≫ C so that (see [21, Section 9]) J j=0 (1 + C̟ j ) ≤ e C J j=0 ̟j ≤ 1 + C̟, J j=0 ̟ −C j 2 −(J−j)θ ≤ ̟ −O(1) , where C is a universal constant. Therefore, we arrive at A λ (R, r, C 0 (r + 1)) ≤ (1 + C̟)A λ (2 −J R, r J , C 0 (r + 1)) + ̟ −O(1) . Using Proposition 4.6, we are done by suitably adjusting the constant C. End of the proof We start the induction by fixing a pair of red and blue waves F λ ∈ R λ R , G λ ∈ B λ R with E(F λ ) = E(G λ ) = 1. We are to show there is a universal constant C * independent of F λ and G λ such that F λ G λ Z(Q λ R ) ≤ C * λ 1/q holds for all Q λ R . To this end, we will show there is a universal constant δ > 0 small depending only on C 0 and possibly some other structural constants, such that λ −1/q F λ G λ Z(Q λ R ) ≤ (1 − δ)A λ (R) + 2 O(C0) holds for all Q λ R . Here O(C 0 ) is a universal constant as well. Taking suprema with respect to F λ , G λ satisfying the above conditions, we close the induction by definition of A λ (R) and the monotonicity of A λ (R) with respect to R and λ. Let A λ be given by Definition 4.2. We first prove the essential concentration of waves on the conic sets and then finish the proof of Theorem 1.1. Essential concentration along conic regions. The following property is for the use of the Kakeya compression property. Proposition 5.1. Let F λ ∈ R λ R and G λ ∈ B λ R be the red and blue waves fixed at the beginning of this section with E(F λ ) = E(G λ ) = 1. There exists a constant C > 0 depending only on n, ε, q, s such that for any R ∈ [2 10C0 , λ] and δ ∈ (0, 1/2), if Q λ R satisfies λ −1/q F λ G λ Z(Q λ R ) ≥ A λ (R)/2,(5.1) and we let r δ be the supremum of all radii r ≥ C 0 such that E r,C0Q λ R (F λ , G λ ) ≤ 1 − δ (5.2) holds and let r δ = C 0 if no such radius exists, then there exists a cube Q λ R δ of size R δ ∈ [2 C0 , R] and z δ ∈ R n+2 such that R 1/2+4/N δ ≤ r δ when r δ ≥ 2 2C0 , and we have F λ G λ Z(Q λ R ) ≤ (1 − C(δ + C −C 0 ) q ) −2/q F λ G λ Z(Ω λ δ ) + λ 1/q 2 CC0 ,(5.3) where Ω λ δ := Q λ R δ ∩ Λ λ δ with Λ λ δ = C λ (z δ , C 0 (r δ + 1)). The proof is divided into two cases: r δ ≥ R 1/2+4/N and r δ ≤ R 1/2+4/N which are treated in the following two subsubsections. 5.1.1. The medium or low concentration case: r δ ≥ R 1/2+4/N . In this case, we show there is a constant C such that for some z δ ∈ R n+2 , we have F λ G λ Z(Q λ R ) ≤ 1 − C(δ + C −C 0 ) q −1/q F λ G λ Z(Q λ R ∩Λ λ δ ) (5.4) holds with C independent of Q λ R and z δ . In particular, we have in this case R δ = R and Q λ R δ = Q λ R . By definition, there is D δ = D(z δ , r δ ) with z δ = (x 0 , t 0 ) and t 0 ∈ LS(C 0 Q λ R ), such that we have min F λ 2 L 2 (D δ ) , G λ 2 L 2 (D δ ) ≥ 1 − 2δ . (5.5) Let D ♮ = C 1/2 0 D δ = D(z δ , C 1/2 0 r δ ) and write F λ = P D ♮ F λ + (1 − P D ♮ )F λ , G λ = P D ♮ G λ + (1 − P D ♮ )G λ . By using Lemma 4.5 and the condition 1 ≤ A λ (R) ≤ 2 λ − 1 q F λ G λ Z(Q λ R ) , it suffices to show for some universal constant C > 0, we have (P D ♮ F λ ) G λ Z(Q λ R \Λ λ δ ) C −C 0 λ 1/q , (5.6) (1 − P D ♮ )F λ P D ♮ G λ Z(Q λ R \Λ λ δ ) C −C 0 λ 1/q ,(5.7) and (1 − P D ♮ )F λ (1 − P D ♮ )G λ Z(Q λ R ) (δ + C −C 0 ) λ 1 q A λ (R). (5.8) The proofs of (5.6) and (5.7) are the same. To see that these estimates are true, by using the energy estimate, we are reduced to P D ♮ F λ L ∞ (Q λ R \Λ λ δ ) N R −N/2 , P D ♮ G λ L ∞ (Q λ R \Λ λ δ ) N R −N/2 ,(5.9) which is obvious in view of Lemma 3.3 and r δ ≥ R 1/2+4/N . To show (5.8), we use the induction argument. Note that by using (5.2), (3.5), (3.6) and the assumption on r δ , we have E (1 − P D ♮ )F λ δ + R −N/2 , E (1 − P D ♮ )G λ δ + R −N/2 . It is easy to verify that we have (1)) . In fact, R ′ = R(1 + 50 2 5C 0 ) −N will do the job. This yields (5.8) by finitely partitioning Q λ R and using the definition of A λ (R ′ ) and the monotonicity of A λ (R). The proof is complete for this case. (1 − P D ♮ )F λ ∈ R λ R ′ and (1 − P D ♮ )G λ ∈ B λ R ′ with R ′ = R (1+o 5.1.2. The high concentration case: r δ ≤ R 1/2+4/N . We turn to the case where the blue and red waves are highly concentrated. Define R δ = max 2 2C0 , r 1/(1/2+4/N ) δ . Consider the case R δ > 2 2C0 . In this case, we necessarily have r δ > 2 C0 and there is z δ such that we have (5.5). Let Q = Q z δ ,λ R δ be the λ−stretched cube of size R δ centered at z δ . By splitting Q λ R = Q λ R ∩ Q ∪ Q λ R \ Q and using Lemma 4.5, we have F λ G λ q Z(Q λ R ) ≤ F λ G λ q Z( Q) + F λ G λ q Z(Q λ R \ Q) . For the first term on Q, the argument as in the medium or low concentration case leads to an estimate of the form (5.4) with Q λ R there replaced by Q. For the second term, write as before F λ G λ = (P D ♮ F λ )G λ :=I + (1 − P D ♮ )F λ P D ♮ G λ :=II + (1 − P D ♮ )F λ (1 − P D ♮ )G λ :=III For I and II, dyadic decomposing Q λ R \ Q into annuli around z δ of the form Q ann (z δ ; 2 j , 2 j+1 ) with 2 j R δ . Taking C 0 large and applying Lemma 3.6 then summing over dyadic 2 −jb , we are done. It remains to handle the III−term. Denote F λ = (1 − P D ♮ )F λ ,G λ = (1 − P D ♮ )G λ . Note thatF λ ,G λ are red and blue waves without the relaxed margin conditions required in R λ R and B λ R . Thus, we can not apply the inductive argument as in the case when r δ ≥ R 1/2+4/N . However, we may use the smallness of the exterior energy ofF λ ,G λ from the definition of r δ and a non-optimal estimate in terms of A λ (R). To this end, one needs to apply Galilean transforms sending the ξ−variables to a neighborhood of the orgin and a mild scaling so that by modifying the input functions, they meets the required margin conditions. To deal with the mixed-norm where Galilean transform does not directly apply, one needs to apply the duality argument and also covering C 0 Q λ R by a larger cube of the same shape CC 0 Q λ R for some fixed large C, then partition it into O(1) many cubes of size R, so that we can apply the definiton of A λ (R). Taking the inverse transform and affording a fixed universal constant, we have III Z(Q λ R ) δλ 1/q A λ (R) + λ 1/q 2 O(C0) (5.10) Thus, by using the (5.1) condition III Z(Q λ R ) δ F λ G λ Z(Q λ R ) + λ 1/q 2 O(C0) , Plugging this back we are done. During this process when using Galilean transform, the phase function is somehow distorted, however, this obstacle can be overcome by using the same argument as done for the proof of Lemma 2.4, so that the error term of the phase function can be handled using the standard trick in [36,37], by using Taylor expansions switching to the discretized version, applying the inductive argument based on A λ (R) and then summing over the absolutely convergent series. This is a tedious but very standard procedure, we refer to [37,Section 5], or Appendex B for an outline of the argument. It is because of this term III Z(Q λ R ) that we need the condition R ≤ λ. It remains to consider the case R δ = 2 2C0 . In this case, the energy is concentrated in a scale ≤ 2 C0 , we use the same argument as above using the trivial energy estimate for F λ G λ Z( Q) λ 1/q 2 O(C0) . Collecting all these estimates, we obtain (5.3) and the proof of Proposition 5.1 is complete. 5.2. Proof of Theorem 1.1. We are ready to show Theorem 1.1. Let C 1 and C 2 be the structural constants given by Proposition 5.1 and Proposition 4.4 respectively. We may take C 1 large so that C 1 ≥ 10000n. Next, we take δ = C −C1/100 0 . Let F λ ∈ R λ R , G λ ∈ B λ R . For any λ−stretched spacetime cube Q λ R , if it satisfies the condition (5.1), we let z δ , r δ , D δ be given by Proposition 5.1. If R δ > 2 C0 , then by using the definition of A λ (R δ , r δ , C 0 (r δ + 1)) and (5.3), we get λ −1/q F λ G λ Z(Q λ R ) ≤ 1 − C 1 (δ + C −C1 0 ) q −2/q A λ (R δ , r δ , C 0 (r δ + 1)) E r δ ,C0Q λ R (F λ , G λ ) 1/q ′ + 2 O(C0) , which entails by Proposition 4.4 and (5.2) λ −1/q F λ G λ Z(Q λ R ) ≤ 1 − C 1 (δ + C −C1 0 ) q −2/q (1 − δ) 1/q ′ (1 + C 2 2 −C0 )A λ (R) + 2 C2C0 + 2 O(C0) . Using q > 1, and taking C 0 large if necessary (depending only on q, C 1 ), one has ∃ δ ∈ (0, 1/10) and 0 < C < ∞, depending only on C 0 , C 1 , C 2 and q, such that λ −1/q F λ G λ Z(Q λ R ) ≤ (1 − δ ) A λ (R) + C . If R δ = 2 C0 , then we have the trivial estimate F λ G λ Z(Q λ R ) ≤ λ 1/q C by recalling the proof in the last subsection. Thus, we have max Q λ R λ −1/q F λ G λ Z(Q λ R ) ≤ max max Q λ R :(5.1)holds λ −1/q F λ G λ Z(Q λ R ) , max Q λ R :(5.1) fails λ −1/q F λ G λ Z(Q λ R ) + C ≤ max (1 − δ )A λ (R) + C , 1 2 A λ (R) + C ≤ (1 − δ )A λ (R) + 2C . Since the right side is independent of F λ ∈ R λ R and G λ ∈ B λ R , we get A λ (R) ≤ (1 − δ )A λ (R) + 2C . Taking suprema, we obtain A λ (R) ≤ 2δ −1 C . Finally, fix η ∈ S(R) with supp η ⊂ [−1, 1] so that for any f 1 and f 2 being test functions supported in V 1 and V 2 , if we letf j (ξ, s) = f j (ξ)η(s), then for R ≥ 2 100C0 satisfies the conditions in R R R and B R R for j = 1, 2 respectively. Applying the uniform estimate on A R (R) to F 1 and F 2 , we get F 1 F 2 Z(Q R R ) η R 1/q f 1 2 f 2 2 . Changing variables t → R t and letting R → +∞, we get (1.1) by using Lebesgue's dominated convergence and then Fatou's theorem followed with integrating x n+1 out, c.f. [32]. The proof is complete. Lemma 2.4. Then, using the reproducing formula f µ (ξ, s) = p(̺(ξ − µ), s) f µ (ξ, s) and taking inverse Fourier transform, we have f µ = Ψ µ,̺ * f µ where Ψ µ,̺ (x) = ̺ −n p ∨ (̺ −1 x, x n+1 )e 2πix·µ , x = (x, x n+1 ). By Minkowski inequality, the average of f µ over (ν, k) + R ̟,̺ r is bounded by (̟ −2 ̺) −n r −(n+1) \|f µ (x ′ )\| (ν,k)+R ̟,̺ r Ψ µ,̺ (x ′′ − x ′ ) dx ′′ dx ′ . (A.1) Splitting R n+1 x ′ = x + R ̟,̺ 1 ∪ k≥1 x + R ̟,̺ 2 k \ R ̟,̺ 2 k−1 , we have (A.1) ≤ k≥0 I k , where I 0 = (̟ −2 ̺) −n r −(n+1) x+R ̟,̺ 1 \|f µ (x ′ )\| (ν,k)+R ̟,̺ r Ψ µ,̺ (x ′′ − x ′ ) dx ′′ dx ′ , and for k ≥ 1 I k = (̟ −2 ̺) −n r −(n+1) x+R ̟,̺ 2 k \R ̟,̺ 2 k−1 \|f µ (x ′ )\| (ν,k)+R ̟,̺ r Ψ µ,̺ (x ′′ − x ′ ) dx ′′ dx ′ . For 0≤k≤̟ −O(1) I k , we use the boundedness of p ∨ to get 0≤k≤̟ −O(1) I k ̟ −O(1) M ̟,̺ f µ (x), For k≥̟ −O(1) I k . By using the elementary identity of sets (A 1 × B 1 ) \ (A 2 × B 2 ) = (A 1 \ A 2 ) × B 1 ∪ (A 1 ∩ A 2 ) × (B 1 \ B 2 ) , and r ≤ C, we have that for any k ≥ ̟ −O (1) , consider x ′′ ∈ (ν, k) + R ̟,̺ r and x ′ ∈ x + R ̟,̺ 2 k \ R ̟,̺ 2 k−1 , then either ̺ −1 \|x ′′ − x ′ \| ≥ ̺ −1 \|x − x ′ \| − ̺ −1 \|x − ν\| − ̺ −1 \|x ′′ − ν\| ≥ 2 k−1 ̟ −2 − C̟ −2 − Cr̟ −2 ≥ ̟ −2 (2 ̟ −O(1) −1 − 2(C + 1) 2 ) ≥ C̟ −O(1) or \|x ′′ n+1 − x ′ n+1 \| ≥ \|x n+1 − x ′ n+1 \| − \|x n+1 − k\| − \|x ′′ n+1 − k\| ≥ 2 k−1 − C − C 2 ≥ ̟ −O(1) − 2(C + 1) 2 ≥ C by taking C 0 large if necessary. Thus, by refining the above two estimates replacing the lower bound with 2 k , we have ̺ −1 \|x ′′ − x ′ \| + \|x ′′ n+1 − x ′ n+1 \| 2 k , ∀ k ≥ ̟ −O(1) . Hence I k M 2 −kM ̺ −n x+R ̟,̺ 2 k \R ̟,̺ 2 k−1 \|f µ (x ′ )\| dx ′ ̟ −O(1) 2 −k M ̟,̺ f µ (x) . Summing up k ≥ ̟ −O(1) , we are done. Combining the above two cases and taking suprema, we complete the proof of (2.14). Appendix B. The exterior energy estimate We outline the proof for the exterior energy induction (5.10), which one may compare with that for the formula (48) P. 239 of [32]. By translation invariance, we take t Q λ R = 0. Let r := (C 1/2 0 r δ ) −(1− 1 N ) + 20R −1 . By direct computation and λ ≥ R, we have marg(F λ ) ≥ marg(F λ ) − r, marg(G λ ) ≥ marg(G λ ) − r. Let r ′ = (200n) −1 + 2R − 1 N + r. and define O 1 = ξ ∈ R n : ξ − e 1 ≤ r ′ , O 2 = ξ ∈ R n : \|ξ\| ≤ r ′ . Let O ♭ 1 = ξ ∈ R n : \|ξ − e 1 \| ≤ r ′ − 2r and Γ be the conic set such that O ♭ 1 ⊂ Γ with the boundary of O ♭ 1 being tangent to that of Γ. Decomposing O 1 = O ′ 1 ∪ O ′′ 1 with O ′ 1 = O 1 ∩ Γ ∩ {ξ : \|ξ\| ≥ 1} and O ′′ 1 = O 1 \ O ′ 1 , we may write correspondinglẙ F λG λ =F λ, ′G λ +F λ, ′′G λ , where F λ, ′ is supported in the lift of O ′ 1 × [−d ′ , d ′ ] to Σ λ and likewise forF λ, ′′ . To handle the first term, we let κ 1 = 1 + r ′ − r 1 + r ′ , κ 2 = d d ′ , and make change of variables (ξ, s) → κ −1 1 ξ, κ −1 2 s forF λ, ′G λ to meet the margin condition in the new variables after modifying the initial data. The result follows from the inductive hypothesis. To handle the second term, we decompose O ′′ 1 = O ′′ 1 ∪ O ′′ 1 where O ′′ 1 = O ′′ 1 \ Γ and O ′′ 1 = O ′′ 1 \ O ′′ 1 . Write correspondinglẙ F λ, ′′G λ =F λ, ′′G λ +F λ, ′′G λ . For the first term, we partition O ′′ 1 into the union of O(n) many sectors ∆, i.e. O ′′ 1 = ∪ ∆ ∆ and writeF λ, ′′G λ = ∆F λ, ′′ ∆G λ . For each ∆, we rotate ∆ to ∆ ′ such that ∆ ′ is centered at e 1 so that after doing this rotation and changing variable s → κ −1 2 s, we haveF λ, ′′ ∆ ′G λ fulfills the margin condition and obtain the result by using the inductive hypothesis. To deal with theF λ, ′′G λ term, we decompose futher O 2 = O ′ 2 ∪ O ′′ 2 where O ′ 2 = {ξ : \|ξ\| ≤ r ′ − 2r}, O ′′ 2 = O 2 \ O ′ 2 . Write correspondingly, F λ, ′′G λ =F λ, ′′G λ, ′ +F λ, ′′G λ, ′′ . ForF λ, ′′G λ, ′ , we make angular partition as above for O ′′ 1 = ∪ S ′′ S ′′ , and write F λ, ′′G λ, ′ = S ′′F λ,S ′′G λ, ′ . Note that for each S ′′ , we first rotate S ′′ to be centered in the e 1 direction and then changing variables (ξ, s) → (κ ′ 1 ξ, κ −1 2 s) with 0 < κ ′ 1 < 1 depending only on κ 1 and r ′ so that we recover the margin condition in order to use the inductive hypothesis. In remains to handleF λ, ′′G λ, ′′ . We make angular partition for O ′′ 1 = ∪ S ′′ S ′′ as above and decompose O ′′ 2 = ∪ D ′′ D ′′ into O(n) many pieces D ′′ . We write correspondinglyF λ, ′′G λ, ′′ = S ′′ D ′′F λ,S ′′G λ,D ′′ . For each D ′′ and S ′′ , we first translate D ′′ to be centered at the origin, and S ′′ is accordingly translated toS ′′ . We then rotateS ′′ to the e 1 direction and apply the mild scaling as above to recover the margin condition so that the result follows from the induction. Unlike all the other above cases, we need to translate the center of D ′′ to the origin which makes the argument technically involved. To see this, we let ξ D ′′ be the center of D ′′ . After changing variables ξ → ξ D ′′ + ξ, we may write F λ,S ′′ (x, t)G λ,D ′′ (x, t) = e 4πi x·ξ D ′′ − t 2λ \|ξ D ′′ \| 2 k1,k2≥0 (t/λ 2 ) k1 k 1 ! (t/λ 2 ) k2 k 2 ! × S λ (t) f λ,S ′′ D ′′ ,k1 x − tλ −1 ξ D ′′ , x n+1 S λ (t) g λ D ′′ ,k2 x − tλ −1 ξ D ′′ , x n+1 , where f λ,S ′′ D ′′ ,k1 (ξ, s) = E λ (ξ, s; ξ D ′′ ) k1 F λ,S ′′ (ξ D ′′ + ξ, s, 0), g λ D ′′ ,k2 (ξ, s) = E λ (ξ, s; ξ D ′′ ) k2 G λ,D ′′ (ξ D ′′ + ξ, s, 0), with E λ ξ, s ; ξ D ′′ = λs λ+s ξ, ξ D ′′ + \|ξ D ′′ \| 2 2 . Noting that \|t\| λR, we may use Minkowski's inequality and then cover Q λ R with an enlarged CQ λ R , changing variables x → x + λ −1 tξ D ′′ to get F λ,S ′′G λ,D ′′ Z(Q λ R ) k1,k2≥0 1 k 1 !k 2 ! S λ f λ,S ′′ D ′′ ,k1 · S λ g λ D ′′ ,k2 Z(CQ λ R ) where we have used R ≤ λ. For each k 1 , k 2 , we have E S λ f λ,S ′′ D ′′ ,k1 δ, E S λ g λ D ′′ ,k2 δ . We first rotate the support of f λ,S ′′ D ′′ ,k1 (·, s) to the e 1 direction and then changing variables (ξ, s) → (κ ′′ 1 ξ, κ 2 s) for some appropriate κ ′′ 1 to recover the margin condition. The proof is complete by summing up k 1 , k 2 . c γ,M (x, t; ξ, s+λ) ∂ γ a j (ξ, s)dξds, where {c γ,M } γ are smooth functions, satisfying that for all 0 ≤ m ≤ M and all multi-indices γ with \|γ\| = m \|∂ γ β\| tξ/λ Z −2−\|γ\| . Here ∂ = ∂ ξ,s refers to taking derivatives only in the frequency variables (ξ, s). For any 0 ≤ m ≤ M and γ with \|γ\| = m, one easily finds that c γ,M is homogeneous of order M , where the exponent of the β-factor is at most (M − m). and using the triangle inequality, we have the right side of (2.15) ≤ I + II, ), then one needs spend for a period of time at least of length Lemma 3 . 3 . 33Let r ≥ C 0 and D = D(z D ; r) with z D = (x D , t D ) Proposition 4. 3 . 3Let 2 10C0 ≤ R ≤ λ and Q = Q λ R be a λ−stretched spacetime cube of size R. For each r ≥ R 1/2+1/N , we define r # : Denote d = 2 ( 1 + 21R − 1 N ) − (100n) −1 and d ′ = d + r. Then, we have supp( F λ ) ⊂ (ξ, s, τ ) : τ = − \|ξ\| 2 2(λ + s) , ξ ∈ O 1 , \|s\| ≤ d ′ , supp( G λ ) ⊂ (ξ, s, τ ) : τ = − \|ξ\| 2 2(λ + s) , ξ ∈ O 2 , \|s\| ≤ d ′ . < s < 2. Thus the claim is verified. It remains to prove (2.28), to which we refer as a tamed bilinear L 2 -Kakeya type estimate for except ̟ −O2.31) Interpolating (2.31) with (2.29) and (2.30) respectively, we obtain (2.26) and (2.27), where κ = 1 q 1 s − 1 2 > 0 with 1 To estimate (2.34), we use Wolff's Bernstein type inequality[41, Lemma 3.2] (x, t) = k:V λ ν,µ T ′ 1 ,k ⊂T ′ 1 c v,µ T ′ 1 ,k φ v,µ T ′ 1 ,k (x, t) F λ,g T ′ 1 +F λ,b T ′ 1 . Appendix A. On the locally constant property for the plate maximal function: proof of (2.14)We follow the standard argument in[33,26].Cr whenever x ∈ (ν, k) + CR ̟,̺ 1 , the average of f µ on (ν, k) + R ̟,̺ r is bounded by C M ̟,̺ f µ (x). Next, we assume 0 < r ≤ C. Note that the Fourier transform of f µ is supported in the set {(ξ, s); \|ξ − µ\| ≤ 10n̺ −1 , \|s\| ≤ 10}. Let p(ξ, s) be as in the proof of Edited and with a foreword by Ilia Itenberg, Viatcheslav Kharlamov and Eugenii I. Shustin. Unitext, 66. La Matematica per il 3+2. 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[ "Combination of Run-1 Exotic Searches in Diboson Final States at the LHC", "Combination of Run-1 Exotic Searches in Diboson Final States at the LHC" ]	[ "F Dias \nUniversity of Edinburgh\nEdinburghUK\n", "S Gadatsch [email protected] \nCERN\nGenevaSwitzerland\n", "M Gouzevich [email protected] \nInstitut de Physique Nucleaire de Lyon\nUniversite de Lyon\nUniversite Claude Bernard Lyon 1\n\nCNRS-IN2P3\nVilleurbanneFrance\n", "C Leonidopoulos [email protected] \nUniversity of Edinburgh\nEdinburghUK\n", "S F Novaes \nUniversidade Estadual Paulista\nSao PauloBrazil\n", "A Oliveira \nUniversita e INFN\nPadovaItaly\n", "M Pierini \nCERN\nGenevaSwitzerland\n", "T Tomei \nUniversidade Estadual Paulista\nSao PauloBrazil\n", "Sergio Novaes@cern Ch ", "Alexandra Oliveira@cern ", "Maurizio Ch ", "Pierini@cern ", "Thiago Ch ", "Tomei@cern ", "Ch " ]	[ "University of Edinburgh\nEdinburghUK", "CERN\nGenevaSwitzerland", "Institut de Physique Nucleaire de Lyon\nUniversite de Lyon\nUniversite Claude Bernard Lyon 1", "CNRS-IN2P3\nVilleurbanneFrance", "University of Edinburgh\nEdinburghUK", "Universidade Estadual Paulista\nSao PauloBrazil", "Universita e INFN\nPadovaItaly", "CERN\nGenevaSwitzerland", "Universidade Estadual Paulista\nSao PauloBrazil" ]	[ "JHEP04" ]	We perform a statistical combination of the ATLAS and CMS results for the search of a heavy resonance decaying to a pair of vector bosons with the √ s = 8 TeV datasets collected at the LHC. We take into account six searches in hadronic and semileptonic final states carried out by the two collaborations. We consider only public information provided by ATLAS and CMS in the HEPDATA database and in papers published in refereed journals. We interpret the combined results within the context of a few benchmark new physics models, such as models predicting the existence of a W or a bulk Randall-Sundrum spin-2 resonance, for which we present exclusion limits, significances, p-values and best-fit cross sections. A heavy diboson resonance with a production cross section of ∼4-5 fb and mass between 1.9 and 2.0 TeV is the exotic scenario most consistent with the experimental results. Models in which a heavy resonance decays preferentially to a WW final state are disfavoured.	10.1007/jhep04(2016)155	[ "https://arxiv.org/pdf/1512.03371v3.pdf" ]	53,007,406	1512.03371	5a9d904cf5e561b67dcb26acd09991eb9c855e8a	Combination of Run-1 Exotic Searches in Diboson Final States at the LHC 2016 F Dias University of Edinburgh EdinburghUK S Gadatsch [email protected] CERN GenevaSwitzerland M Gouzevich [email protected] Institut de Physique Nucleaire de Lyon Universite de Lyon Universite Claude Bernard Lyon 1 CNRS-IN2P3 VilleurbanneFrance C Leonidopoulos [email protected] University of Edinburgh EdinburghUK S F Novaes Universidade Estadual Paulista Sao PauloBrazil A Oliveira Universita e INFN PadovaItaly M Pierini CERN GenevaSwitzerland T Tomei Universidade Estadual Paulista Sao PauloBrazil Sergio Novaes@cern Ch Alexandra Oliveira@cern Maurizio Ch Pierini@cern Thiago Ch Tomei@cern Ch Combination of Run-1 Exotic Searches in Diboson Final States at the LHC JHEP04 1552016 We perform a statistical combination of the ATLAS and CMS results for the search of a heavy resonance decaying to a pair of vector bosons with the √ s = 8 TeV datasets collected at the LHC. We take into account six searches in hadronic and semileptonic final states carried out by the two collaborations. We consider only public information provided by ATLAS and CMS in the HEPDATA database and in papers published in refereed journals. We interpret the combined results within the context of a few benchmark new physics models, such as models predicting the existence of a W or a bulk Randall-Sundrum spin-2 resonance, for which we present exclusion limits, significances, p-values and best-fit cross sections. A heavy diboson resonance with a production cross section of ∼4-5 fb and mass between 1.9 and 2.0 TeV is the exotic scenario most consistent with the experimental results. Models in which a heavy resonance decays preferentially to a WW final state are disfavoured. Introduction Searches for new heavy resonances are one of the major components of the ATLAS and CMS physics programmes at the Large Hadron Collider (LHC) at CERN. Of particular interest is the coupling of new resonances to pairs of vector bosons. Models with Vectorial heavy resonances (i.e. W -like and Z -like bosons) are commonly considered as possible extensions of the SM, either in weakly coupled (see [1][2][3]) or strongly coupled versions, the so-called composite Higgs scenarios [4,5]. In these scenarios, the existence of new resonances is introduced to alleviate the hierarchy problem in the SM. Another common SM extension is the Warped Extra Dimensions or Randall-Sundrum (RS) model [6], which is an example of a class of models predicting neutral spin-2 resonances as Kaluza-Klein (KK) excitations of the graviton field (G * ). Two types of models are usually considered: the original version, in which only gravity is allowed to propagate into the extra-dimensional bulk ("RS1" models, see Ref. [7]) and variants of the original model, in which the SM fields are also allowed to propagate into the extra dimensional bulk ("bulk RS" models, see for example Ref. [8]). RS1 models favour the decay of G * to qq, + − and γγ final states, whereas in bulk RS models its decay to vector bosons. After a number of direct and indirect bounds from previous experiments, and in particular, the stringent constraints from the electro-weak precision measurements carried out at LEP [9] 1 , nowadays searches for heavy exotic resonances decaying to pairs of vector bosons typically focus on resonance masses above 1 TeV. When produced and decayed at the LHC, these particles would generate vector bosons with O(1 TeV) transverse momenta, requiring special reconstruction strategies. In particular, the quarks from a hadronically-decaying vector boson are very close to each other in the η −φ space. In their showering and hadronisation process they produce highly overlapping jets, in a so-called boosted topology. ATLAS and CMS handle this experimental signature by reconstructing the two partially overlapping jets as a single massive (or "fat") jet, noted in this paper as "J". One then exploits the jet mass m J and the momentum flow around the jet axis to distinguish these special jets from those originating from quark or gluon production [12][13][14][15][16][17]. A typical boosted longitudinally polarised and hadronically-decaying V boson 2 can be identified by a tagger with an efficiency of ∼ 50% and with a false-positive rate for light quarks or gluons of 2% [18,19]. The ATLAS and CMS collaborations have employed hadronic boson taggers in searches for heavy resonances in diboson final states with the proton-proton collision data collected in 2012 at a centre-of-mass energy of 8 TeV. In particular, the ATLAS search in the fully hadronic final state [20] has generated significant interest due to an excess of diboson events with invariant mass mass around 1.9 TeV. Small deviations in the same mass region are observed in other channels as well, e.g. the CMS search in the Z( + − ) V (qq) channel with = e, µ [21], and the CMS search in the fully hadronic V (qq) V (qq) final state [22]. Other analyses, e.g. the ATLAS and CMS searches in the W ( ν) V (qq) channel see no evidence of a deviation, indicating a possible tension between these experimental results in the scenario of a heavy exotic resonance. Additional results with potentially interesting deviations in the same mass region include a moderate excess (≈ 1 − 2σ of local significance) reported in the ATLAS [23,24] and CMS [25,26] searches in the dijet channel, as well as in the CMS search in the dilepton channel [27]. In addition, a search for right-handed W (and heavy neutrinos) [28] by CMS has reported a small excess in the electron channel [29] (however, this excess is not confirmed by a similar ATLAS analysis [30]). Finally, a CMS search for W( ν) H(bb) resonances reported an excess of ≈ 2σ, originating from a stronger excess in the electron channel and no evidence of a deviation in the muon channel [31]. At the same time, the CMS searches for WH or ZH resonances in the fully hadronic channel were inconclusive, with a mild upward fluctuation around 1.8 TeV and a lack of events around 2 TeV [32]. The dedicated searches for Z(qq) H(τ + τ − ) and H(bb) H(τ + τ − ), H(bb) H(bb) final states showed no excess [33,34]. Several attempts to provide a possible interpretation for this excess have been made during the last months. The deviation has been associated to possible signatures of various beyond-the-SM models, e.g. models with new W and Z vector bosons (see for example [35][36][37][38][39][40][41][42][43][44]), models involving new resonances with different spins (see for example [45][46][47][48][49][50][51][52]), composite and technicolor models (see for example [53][54][55][56][57][58]) and new and composite Higgs states (see for example [59][60][61][62][63][64][65][66][67]). A review of the different models offering an interpretation of the deviations reported in the ATLAS and CMS searches has been made in Ref. [68]. A natural next step would be to carry out a systematic comparison of the results reported by ATLAS and CMS in various channels, and examine if the apparent deviations work in a synergistic way towards a coherent picture. In particular, the goal is to quantify the level of agreement among the different results, and by using an exotic signal hypothesis for the interpretation of these deviations, to calculate the corresponding production cross section. We hereby present the first step in addressing this question, starting with the statistical combination of the results of the ATLAS and CMS Run-1 searches for vector boson pair resonances. The exotic models considered by the experiments are usually connected with the electroweak sector, with the predicted resonances mainly coupling to longitudinally polarised vector bosons V L . We consider the experimental results of the searches for heavy resonances decaying to three final states: Z L Z L , W L W L and W L Z L . We combine the results and interpret the derived exclusion limits in the context of a (W -like) spin-1 charged particle decaying to a W L Z L boson pair, and a neutral spin-2 particle (G bulk ). For the latter case, we only consider bulk RS scenarios, namely particles decaying to the Z L Z L , W L W L final states 3 . The paper is organised as follows: in Section 2 we present a general overview of the methodology used to emulate the ATLAS and CMS analyses; Sections 3 and 4 discuss the emulation of the hadronic and semileptonic analyses, respectively. Each section covers the individual searches by ATLAS and CMS, and their combination; in Section 5 we combine the Run-1 results provided by the two collaborations and discuss their interpretation in a 3 Models in which the exotic resonances have stronger couplings to transverse vector bosons (VT) than longitudinal ones (VL) typically have larger branching fractions to dilepton and dijet final states. It should be noted that boosted boson taggers are more efficient with VL than VT bosons [18]. This topic will be addressed in a future publication. few benchmark models considered in this study; we present the summary of the findings, along with the conclusions in Section 6. A brief note on the compatibility of the findings of this study with the preliminary Run-2 search results reported by ATLAS and CMS in December 2015 has been added in v2 of this paper and is presented after the conclusions. Additional information on the determination of the background and signal modelling for the ATLAS search in the fully hadronic channel is given in Appendices A, B. General methodology All exotic searches considered in this paper are looking for a diboson mass peak emerging on top of a falling background spectrum. In order to evaluate the significance of a deviation observed in the data, we need as input the shapes of the signal and background distributions, the total number of expected background events, the signal efficiency, and the experimentally measured distribution (data). This study is based exclusively on the public information provided by the two experimental collaborations in the HEPDATA database [69] and the cited papers (published in refereed journals). In particular, we employ the expected backgrounds with their corresponding uncertainties, as they have been estimated directly by ATLAS and CMS, wherever possible. The modelled signal distributions (namely, shapes and signal efficiencies for a few benchmark models and mass values) are also taken from the information publicly provided by the experiments, when available 4 . In order to emulate signal distributions for additional mass values, we carry out linear interpolations of the available models within the benchmark mass points. We derive exclusion limits on hypothetical signals by performing binned templated fits of the data distributions with linear combinations of the signal and background distributions. These calculations are carried out with the open-source statistical framework THETA [70] which uses the asymptotic approximation [71] of the CLs method [72,73]. In a few cases, the information published by ATLAS and CMS is not sufficient for this simple approach to produce satisfactory results. For example, uncertainty correlations that affect the background determination, or the mass-dependence of an important systematic uncertainty are not always properly documented. In these cases, we fit the data distributions to the functional form documented in the published analysis, e.g. the function used in the hadronic searches or an exponential function for the leptonic channels. Details about these fits are given in the corresponding sections of the paper, where we also discuss the agreement achieved in the background modelling. When it is necessary to model a signal distribution ourselves, we either use a Gaussian approximation with a resolution inferred from the relevant experimental paper, or we generate Monte Carlo (MC) samples using the Madgraph5 matrix-element event generator [74], matched to Pythia8 [75] for the hadronisation process. For the G bulk signal we use the Madgraph5 model files as presented in Ref. [76], while for the spin-1 signal W the ones described in Ref. [77]. These approximations are mainly motivated by our familiarity with the diboson and similar searches by ATLAS and CMS. The described procedure is validated using the nominal published results as benchmarks, as well as the comparison of our own calculations of the per-experiment combinations against the official combination of diboson searches [21,78]. We are able to reproduce the exclusion limits of each analysis individually and their combinations with an agreement of better than 20% in the region of interest for all channels, with the exception of the fully hadronic search in ATLAS (see Appendix A). Our methodology can be used as a set of guidelines for model builders in the absence of official combined results published by the two experiments. All diboson final states considered in this study contain at least one vector boson (W or Z) decaying hadronically. Because of the limited hadronic detector resolution, it is not possible to distinguish between hadronic W and hadronic Z jets. When interpreting an experimental result, special care is needed to account for possible cross-channel contamination of the final state under consideration. For example, a neutral heavy resonance decaying to a pair of vector bosons is expected to decay to both WW and ZZ final states. We consider models in which the relative branching fractions of neutral particle decays to WW and ZZ can vary, in order to study the relative importance of the different bosonic sub-channels to the combined result. We quantify this dependence by introducing as a free parameter the ratio r of the corresponding branching fractions: r ≡ B(X → WW) B(X → ZZ) (2.1) with r = 2 being the default ratio in the baseline bulk RS scenario. The full list of channels that we consider in this study is as follows: the fully hadronic searches X → V (qq) V (qq) (labelled "JJ"), searches including a W decaying leptonically X → W( ν) V (qq) (labelled " νJ"), and searches including a Z decaying leptonically X → Z( ) V (qq) (labelled " J"). Table 1 summarises the methods that have been used to emulate each of the analyses considered. Details of the individual analyses are given in the sections that follow. In this Section we discuss the analysis of the ATLAS and CMS searches in the VV → JJ channel. We first present the results of our analysis for the two searches separately, followed by their combination and a summary of the findings. Emulation of ATLAS search Description of the ATLAS analysis The ATLAS fully hadronic search analyses calorimetric dijet events. The main irreducible background is dijet production in QCD, which is dominated by 2 → 2 t-channel processes involving quarks and gluons. The contribution of these processes is minimised by restricting the jet acceptance to \|η\| < 2.0 and the rapidity difference between those two jets to \|∆η\| < 1.2. The events are required to have low missing transverse momentum and a rather symmetric dijet topology (similar p T for the two leading jets) to reduce the detector noise. After this selection, the efficiency is approximately 70-80% for a heavy vector boson signal, and above 80% for a G bulk signal. To further reduce the multijet background, two fat jets are reconstructed using the Cambridge-Aachen algorithm [81,82] with radius parameter R = 1.2. The mass-drop filtering algorithm [12] is applied to each of these jets for the identification of the subjets and grooming. Events are kept if each of the two leading jets satisfies the following conditions: have two sub-jets with similar transverse momentum, have less than 30 tracks matched to it, and have a pruned mass within a ±13 GeV window either around 82.4 GeV (for W tagging) or around 92.8 GeV (for Z tagging). The selection efficiency of the grooming algorithm for fat jets from a W resonance is between 30% and 40%. The events are subsequently classified into three non-mutually-exclusive categories, based on the jet-mass values: WW, WZ and ZZ. The overall product of the geometric acceptance with the signal efficiency for this analysis is typically 10-20%. Statistical analysis The analysis uses the smoothness test ("bump search") approach: the background is approximated by a steeply falling function, while the signal template is taken from simulation. The sum of the two components is then fitted to the data. The background function used by the ATLAS collaboration is: f (m JJ ) = p 0 (1 − m VV ) p 1 −ξp 2 m p 2 VV (3.1) where p 0 , p 1 and p 2 are free parameters and m JJ is the dijet invariant mass; ATLAS has also made the signal templates used in the analysis public. We employ the same function for the background description, but recalculate the background uncertainties in order to better account for the large scale correlations in m JJ . To this end, we refit the data in each of the three categories above using the aforementioned background parametrisation. We diagonalise the uncertainty matrix and obtain three uncertainty eigenvectors (σ λ i , with i = 0, 1, 2). Our fit result produces a background estimate which agrees with the nominal background within 10%, which is well within the uncertainties (see Appendix A). This background is subsequently used together with the associated uncertainties in our statistical analysis (see Fig. 1). We consider the following systematic uncertainties, treated as fully correlated across m JJ histogram bins: • Background uncertainty, obtained as described above. • Signal normalisation uncertainty, which is separated into two further sub-categories: a common-across-channels systematic uncertainty corresponding to the luminosity measurement (2.8%), and an additional term applicable to the JJ channel that covers V-tagging uncertainties as well as jet systematics. • Signal jet energy scale uncertainty, which includes jet transverse momentum and mass uncertainties (with a ±2% and ±5% impact on m JJ , respectively). An additional jet energy resolution uncertainty is known to have a negligible effect on the signal shape and is ignored in this study. Our statistical analysis produces expected exclusion limits that are typically 50% more stringent than the ones publicly provided by ATLAS. This discrepancy, discussed in detail in Appendix A, is corrected for with the introduction of a fudge factor, defined as the ratio of the ATLAS expected exclusion limits and the ones from this study obtained with the THETA statistical framework (see Fig. 2). With this correction, our calculated exclusion limits are in good agreement with the public ATLAS results (see Fig. 3). Results with WW, WZ and ZZ signal hypotheses As discussed above, due to the finite detector resolution, the V-tagging tool is not capable to differentiate between fat jets originating from W or Z bosons. However, there is a significant performance difference between W and Z tagging efficiencies of up to ≈ 30%, mainly as • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • WW ATLAS JJ W L W L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • ATLAS JJ W L Z L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • ATLAS JJ Z L Z L Figure 3. ATLAS hadronic search: Observed exclusion limits on exotic production cross section as a function of the resonance mass m X obtained with this study, with (black) and without (red) the correction discussed in the text ("fudge"), and comparison with the official ATLAS results (grey) for G bulk → W L W L (left), W → W L Z L (middle) and G bulk → Z L Z L (right) signal hypotheses and tagging selections. The green and yellow bands represent the one and two sigma variations around the median expected limits (dashed lines) calculated with the same fudge factor. a result of the different boson masses. By using the mass distribution of longitudinal Vjets, as documented in Fig. 1 of Ref. [20], and by taking into account the different W and Z efficiencies, we can calculate the efficiency of tagging selections for different signal hypotheses (WW, WZ, ZZ). The comparison of the tagging selection efficiencies can be found in Table 2. The effect of applying the different tagging selections to the WW, WZ and ZZ signal hypotheses as a function of the resonance mass is shown in Fig. 4. We assume that the m JJ spectrum is not affected by the mass window difference in the tagging selections, i.e. that the same distribution describes the three tagging categories WW, WZ and ZZ. Since the three categories have common events, they cannot be combined as if they were statistically independent. Instead, for each theoretical model under consideration we choose the tagging category that gives the best expected exclusion limits. For the W model the WZ tagging selection gives the best result, whereas for the G bulk graviton model in the W L W L and Z L Z L final states the ZZ tagging selection has the best performance. • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • WW window ■ ZZ window ◆ WZ ATLAS JJ W L W L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ • WZ window ■ WWσ(pp>X>VV) (fb) ATLAS JJ W L Z L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ • ZZ window ■ WW Emulation of CMS search Description of the CMS analysis The jet acceptance is restricted to \|η\| < 2.5 and \|∆η\| < 1.3 in order to reduce the contamination from multijet events. The detector noise is removed by requiring tight quality criteria on the jets. The pruning algorithm [13] is used to clean up the jet from soft and large-angle radiation. The mass of the resulting fat jet is constrained in the70 < m J < 100 GeV range. Finally, the signal-to-background ratio is enhanced by exploiting the jet N-subjettiness [14][15][16] variable τ N . This variable is used to quantify how well the jet constituents can be arranged into N sub-jets, i.e. in a consistency check with the hadronic V boson hypothesis. The ratio τ 12 = τ 2 /τ 1 is built with the two leading jets: the smaller the ratio, the larger the probability that the jet consists of two sub-jets. The analysis considers two categories: the high purity (HP) one, defined by requiring τ 12 < 0.5 for both jets, and the low purity (LP) one, defined by requiring one jet with τ 12 < 0.5 and the other one with 0.5 < τ 12 < 0.75. The HP category is characterised by a smaller background contamination. The LP category captures signal events with asymmetric decays of the vector-boson candidates in the laboratory frame. Dividing the event sample into the LP and HP categories improves the sensitivity of the analysis in the mass range between 1 TeV and 2 TeV, while avoiding the inefficiency of a tight τ 12 selection at large jet momenta. The product of the geometrical acceptance with the signal efficiency is similar to the one in the ATLAS search, ranging between 10% and 20%. Statistical analysis The CMS collaboration provides the binned data and background distributions with the associated uncertainties in the HEPDATA database (see Fig. 5), as well as the signal distributions for three different models along with their efficiencies [22]: W → W L Z L and G bulk decaying exclusively to Z L Z L or W L W L . We consider the following systematic uncertainties: • Background uncertainty, provided by CMS (in HEPDATA) and considered as fully correlated across the bins of the m JJ distribution. • Signal normalisation uncertainty, which is separated further into two sub-categories: a common-across-channels systematic uncertainty corresponding to the luminosity measurement (2.2%), and an additional term applicable to the JJ channel that covers V-tagging uncertainties, such as p T , pile-up and PDF dependencies (13%). The τ 12 uncertainties are treated separately in the category below. • Signal purity category migration uncertainty, which covers the effects of events "migrating" from the HP to the LP category, or vice-versa. This uncertainty amounts to 7.5% and 54 %, respectively. • Signal jet energy scale uncertainty, propagates to ±1% of uncertainty on m JJ ; It is treated in the same way as in the ATLAS case. All systematic uncertainties are treated as fully correlated across different m JJ bins. They are also considered as fully correlated between the LP and the HP categories, with the exception of the "purity category migration" uncertainty, which is treated as fully anticorrelated. Our statistical analysis for W → W L Z L , G bulk → W L W L and G bulk → Z L Z L models produces exclusion limits that are in very good agreement with the ones publicly provided by CMS. An example of this agreement can been seen in the left plot of Fig. 6. The exclusion limits calculated in a few benchmark models can be seen in the right plot of Fig. 6. The most stringent limits are obtained for the G bulk → Z L Z L hypothesis, thanks to the higher V-tagging efficiency for Z bosons. W → W L Z L (brown), G bulk → W L W L (red) and G bulk → Z L Z L (black) signal hypotheses. Combined LHC results of hadronic searches This section describes the combination of the ATLAS and CMS searches in the fully hadronic channel JJ and the interpretation of the results under different signal hypotheses. As a first step we note that ATLAS assumes a wide resonance in its JJ searches, whereas CMS assumes a narrow one. To ensure a consistent treatment of the search in the hadronic channel between the two experiments we introduce a +10% scale factor in the ATLAS selection efficiency. A description of the derivation of the scale factor and its impact on the search sensitivity is discussed in Appendix B. For every signal hypothesis under consideration we use the optimal mass selection windows as defined by ATLAS. We proceed by combining the THETA data cards of the individual ATLAS and CMS searches. The results of the statistical combination for the W L Z L , W L W L , and Z L Z L signal hypotheses can be seen in Fig. 7. In the 1.7 < m X < 2.2 TeV region we observe the largest discrepancy between expected and observed exclusion limits due to the presence of the excess in the m JJ spectrum. The excess is much smaller in the CMS analysis, which forces the combined results to lie between the ATLAS and the CMS curves. The sensitivity of the combined search as we move away from the deviation region is driven by the CMS analysis. The impact of the individual experimental results on the combination can be seen in the distribution of p-values (obtained using Wilks' theorem) depicted in Fig. 8. The CMS z-value or significance 5 in the excess region is of the order of 1σ, independently of the considered model and corresponding selections. The ATLAS significance ranges from less than 3σ for the W L W L selection to nearly 4σ for the Z L Z L selection, as a result of the different W and Z mass selection windows. The statistical significance of the combined result is very close to the one obtained with the ATLAS result alone, although slightly reduced. In fact, the ATLAS and CMS results are not contradictory: due to the small CMS excess observed in the same mass region, the CMS result cannot exclude the larger ATLAS excess. In order to further characterise the interplay between the ATLAS and the CMS results in the combination, we show in Fig. 9 the best-fit exotic signal cross section as a function of the resonance mass m X value for a few benchmark models and corresponding selections: W L Z L , W L W L and Z L Z L . The best-fitted cross section values are shown separately for the emulation of ATLAS and CMS searches, and their combination. The largest excess for the W L Z L and W L W L signal hypotheses is observed in the 1.9 < m X < 2.1 TeV mass range, while the excess extends down to m X = 1.8 TeV for the Z L Z L signal hypothesis. In these mass ranges, the ATLAS data suggests a production cross section of ≈ 10 fb, whereas the CMS data favours smaller values (≈ 3 fb) and is more consistent with the no-signal hypothesis. The m X profile of the fitted exotic signal cross section is essential identical to the one obtained from the ATLAS search emulation. Further tests of the compatibility between the ATLAS and CMS results can be seen in Fig. 10, showing scans of the profiled likelihood as a function of the exotic production cross section for m X = 2 TeV (mass value of largest excess). Due to the large uncertainties of the fit, the best-fit cross-section values by ATLAS and CMS are compatible within ±1σ for the W L Z L and W L W L hypotheses. The compatibility of the results from the two experiments is slightly reduced in the Z L Z L scenario. The dependence of these results on r ≡ B(X → W L W L )/B(X → Z L Z L ) can be seen in Fig. 11. The conclusions discussed above remain mostly unchanged. In summary, in the combination of fully hadronic results the small CMS excess results in a slight reduction of the larger ATLAS excess. However, the combined-search statistical significance stays well above 3σ for the W L Z L and Z L Z L hypotheses and close to 3σ for the W L W L hypotheses. The preferred mass range for a hypothetical exotic signal is 1.9 < m X < 2.0 TeV, with the corresponding production cross section in the 8-12 fb region. • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • ATLAS+CMS ■ ATLAS ◆ CMS 1σ (ATLAS + CMS) 2σ (ATLAS + CMS) JJ W L W L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • ATLAS+CMS ■ ATLAS ◆ CMS 1σ (ATLAS + CMS) 2σ (ATLAS + CMS) The results include the 10% scale factor discussed in the text. The dashed black curve corresponds to the combined search without the 10% scale factor discussed in the text. In this Section we discuss the analysis of the ATLAS and CMS searches in the WV → νJ and ZV → J channels. We follow the discussion pattern of the fully hadronic section: we first present the results of our analysis for the two searches separately, followed by their combination and a summary of our findings. JJ W L Z L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • ATLAS+CMS ■ ATLAS ◆ CMS 1σ (ATLAS +• • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ • ATLAS+CMS ■ ATLAS ◆ CMS m X (TeV) p-value JJ W L W L 1σ 2σ 3σ 4σ • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ • ATLAS+CMS ■ ATLAS ◆ CMS m X (TeV) p-value JJ W L Z L 1σ 2σ 3σ 4σ • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ • ATLAS+CMS ■ ATLAS ◆ CMS • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆σ(pp>X>WW) (fb) JJ W L W L • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆σ(pp>X>WZ) (fb) JJ W L Z L • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ATLAS+CMS JJ W L W L + Z L Z L ATLAS hadronic ZZ selection • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ • ZZ ■ r=0.5 ◆ r=1 ▲ r=2 ▼ WW ATLAS+CMS JJ W L W L + Z L Z L 1σ 2σ 3σ 4σ ATLAS hadronic ZZ selection • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ • ZZ ■ r=0.5 ◆ r=1 ▲ r=2 ▼WW Emulation of ATLAS search Description of the ATLAS analysis The ATLAS semileptonic search considers both the case in which the two quarks from the vector boson decay are reconstructed as a single merged jet (boosted regime), and the case in which they are reconstructed as two distinct jets (resolved regime). In this study, we focus on resonances heavier than 1.5 TeV, for which the merged regime largely drives the sensitivity. Thus we consider only the Merged Region (MR) categories of Refs. [79,80]. In both ZV → J and WV → νJ searches, the boosted jet is identified using the mass-drop filtering algorithm (as in the VV → JJ search). In addition, two same-flavour opposite-sign leptons, or one charged lepton and missing transverse energy (MET) are required. The events are selected online by single-or double-lepton based triggers. The detector coverage includes the tracker volume (\|η\| < 2.5) and the fiducial region of the electromagnetic calorimeter (for electrons) or the muon detector. The typical p T threshold for the charged leptons and for MET is 25 GeV. The main backgrounds are inclusive V production (i.e. Z +jets for the J channel and W +jets for the νJ channel), as well as tt production. Statistical analysis We build the likelihood for the ATLAS semileptonic searches using the information documented in the HEPDATA database. The ATLAS collaboration estimates the background uncertainties separately for each lepton category. The electron p T resolution is better than that of the muon in the high-p T region. The systematic uncertainties associated with different background sources (tt and electroweak components) are also treated separately. Nevertheless, the background distributions documented in the HEPDATA database (see Fig. 12) are presented jointly for electrons and muons. We model the signal distributions in the diboson mass spectrum with a Gaussian function, centred at the assumed resonance mass and with a width reflecting the experimental resolution. We assume a fixed value of 4% resolution in the J channel for all mass values 6 . Similarly, we assume a fixed value of 10% resolution in the νJ channel for all mass values 7 (see Fig. 1 in Ref. [79]). The signal distributions are normalised to the expected yield, as calculated from the theoretical cross section and the selection efficiency provided by the ATLAS collaboration. We consider the following systematic uncertainties, treated as fully correlated across m JJ histogram bins: • Background uncertainty, provided by the ATLAS experiment (in HEPDATA). • Signal normalisation uncertainty, which is separated into two further sub-categories: a common-across-channels systematic uncertainty corresponding to the luminosity measurement (2.8%), and an additional term accounting for all types of scale and efficiency systematic effects (10%). The latter is treated as uncorrelated between the J and νJ channels. Given the approximations that we have introduced to model the signal, we do not expect our statistical analysis to produce results matching with high accuracy the public ATLAS results. Similarly to the procedure followed for the emulation of the fully hadronic ATLAS search, we introduce a fudge factor to reduce this discrepancy. The value of the 6 The signal resolution for a mX = 2 TeV resonance in the J channel is 4%, decreasing to 3% for lower masses [80]. We assume a fixed resolution to simplify the analysis. 7 In the case of the νJ channel, the reconstruction of the resonance mass requires an assumption on the longitudinal momentum of the outgoing neutrino that is not detected. In practice, this is estimated from the MET measurement combined with a W mass constraint. The diboson resonance mass is subsequently computed using the jet, lepton and calculated neutrino momenta. The mass resolution in this channel is degraded compared to the J channel. fudge factor is chosen such that the expected exclusion limits produced by this study agree with the official limits by ATLAS. It is found to be between 0.8 and 1.2 in the resonance mass range of interest, slowly decreasing for larger mass values (Fig. 13). With this correction, our calculated exclusion limits are in good agreement with the public ATLAS results (Fig. 14). • • • • • • ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ • WZ Emulation of CMS search Description of the CMS analysis The CMS semileptonic analyses [21] are performed with data collected by single-lepton triggers for the νJ channel and double-lepton triggers for the J channel. Jets are identified as boosted vector bosons using the same algorithm employed for the fully hadronic search (see Section 3). Similarly to the strategy developed in the fully hadronic search, LP and HP categories are introduced, based on the value of τ 21 , to increase the analysis sensitivity. The analysis is performed by using a G bulk graviton as the benchmark signal model. In order to facilitate the interpretation of the search results in other theoretical models, the CMS collaboration provides the reconstruction efficiencies of leptonic and hadronic W L and Z L in the HP category, as function of the boson's p T and η. Those 2D efficiency maps include the effects of the pruned jet mass and τ 21 selections, as well as the resonance mass reconstruction. ATLAS llJ W L Z L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ • this study ■ official result ATLAS lνJ W L W L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ • Statistical analysis The background model is extracted by fitting the m VV data distributions for each lepton flavour with a levelled exponential f (m VV ) = N exp − m VV σ + k · m VV (4.1) where N , k and σ are free parameters. This function saturates in the high m VV region, and is meant to describe events where m VV was significantly mismeasured. For example, this may happen if a high p T muon leaves a nearly straight track barely bent by the magnetic field, or if the calculation of the neutrino momentum fails. In practice, this function is used in Ref. [21] to model the HP category with k as a free parameter, whereas for the LP category k can be set to 0. In the J channel we focus on the m VV > 700 GeV region, and we merge the contents of the (publicly available) 50 GeV wide bins to obtain a uniform, 100-GeV-wide binning for the m VV distribution. We use the diagonalised uncertainties from the fit (σ λ i , with i = 0, 1, 2) as background uncertainties. Figs. 15 and 16 show the comparison between the fits produced in this study and the official CMS fits on the data distributions. We model the signal distributions in the diboson mass spectrum with a Gaussian function. The HP signal yield is calculated from the theoretical cross section and the selection efficiency obtained from the algorithm described in Ref. [21]. The first step in this process is the generation of signal samples with the Madgraph5 generator as described in Sec. 2. We then apply acceptance selections on the leptons and generator-level jets, and use the 2D efficiency maps to emulate the V-boson reconstruction and tagging processes. Finally, we apply a 90% correction to account for b-jet veto inefficiencies. Considering the approximations made, this procedure is expected to reproduce the official CMS results within a 10% accuracy. The HP-category efficiencies that we obtain are consistent with the nominal G bulk → W L W L efficiencies for m X = 1.2 TeV within 6%. The LP category signal efficiencies are generally not provided, but examples of the LP/HP efficiency ratios are given for a G bulk signal with m X = 1.2 TeV. The ratio is 0.47 (0.25) for the J ( νJ) channel. The reason for the efficiency difference between the two cases lies in the different boosted jet selection applied in the two channels. We make the assumption that we can use the same LP/HP ratio for all mass points under consideration in this study, and use the values above to estimate the expected signal yields in the LP category. Finally, the τ 21 categorisation is not sensitive to the nature of the resonance 8 , therefore we use the same LP/HP ratio also for the W signal hypothesis. We consider the following systematic uncertainties, treated as fully correlated across m JJ histogram bins: • Background uncertainty, extracted from our fit to the data distributions. • Signal normalisation uncertainty, which is separated into two further sub-categories: a common-across-channels systematic uncertainty corresponding to the luminosity measurement (2.2%), and an additional uncertainty covering all lepton-related uncertainties (3.7% for electrons, 3% for muons), applied separately for the J and νJ channels. • Signal purity category migration uncertainty, which covers the effects of events "migrating" from the HP to the LP category, or vice-versa. This uncertainty amounts to 9% and 24%, respectively. As already discussed in previous sections, we apply a fudge factor to account for differences between our background description and the one from the public CMS result, as well as for the approximations introduced in the signal modelling (Fig. 17). With this correction, our calculated exclusion limits are in good agreement with the public CMS results (Fig. 18). The statistical uncertainties (one-and two-sigma coverage bands) are ≈ 50% smaller than expected, as they have been calculated with the asymptotic CLs method, which is known to underestimate uncertainties in tests with small statistics. We use the same procedure to recast the results in the context of a W → W L Z L signal search, with the results presented in Fig. 19. The jet mass selection for the J channel is 70 < m J < 110 GeV, to be compared with 65 < m J < 105 GeV for the νJ analysis. This choice was made in order to optimise the search for a neutral resonance (at the expense of the search for a charged one). Since the νJ channel mass window is shifted to a region with more background, the signal sensitivity for the νJ channel is reduced. Combined LHC results of semi-leptonic searches Here we discuss the combination of the ATLAS and CMS searches in the semileptonic channels ( νJ and J) and the interpretation of the results under different signal hypothe- ses, with final states including a leptonic W (→ ν) or Z (→ ) decay. The results are summarised in Fig. 20. Under the hypothesis of a Z L Z L benchmark model, only the J searches are relevant. In this channel, CMS observes a small excess (≈ 1σ) between 1.7 and 1.9 TeV, while ATLAS a < 1σ excess between 1.9 and 2.0 TeV, driven by the presence of one event in the highest bin of the merged analysis distribution. The combination of the two channels results in a more stringent limit and a moderate excess of the order of 1σ around 1.9 TeV. Above 2 TeV, ATLAS has not published their search results and the limit considered here is the one provided by CMS. While the significance of the observed deviation is too small to cause any excitement, the sensitivity of this analysis is strongly reduced. This has implications for the combination result discussed in Sec. 5. On the contrary, under the hypothesis of a W L W L benchmark model, only the νJ searches are relevant. An observed upward fluctuation around m VV = 1.8 TeV in the CMS data spectrum is compensated by a downward fluctuation in the same region for the ATLAS data. The two deviations effectively cancel each other, resulting into observed exclusion limits which are consistent with the experimental sensitivity and the backgroundonly hypothesis expectations. For the W L Z L benchmark model, we are able to combine the experimental results in the J and νJ channels. The sensitivity and the relative weight of the J channel is larger than those of the νJ channel in the combination. Similar to the interpretation of the search results in the Z L Z L signal hypothesis, we observe here that the combined results give a small excess (≈ 1σ) around m VV = 1.9 TeV. Combination of hadronic and semi-leptonic channels This section is dedicated to the combination of both hadronic and semileptonic channels by ATLAS and CMS under different signal hypotheses. The searches in the JJ and νJ channels contribute to constrain a hypothetical G bulk → W L W L production; searches in the JJ and J channels enter the combination for the interpretation of the results in a G bulk → Z L Z L signal scenario. Finally, all six searches (i.e. results in three channels by two experiments) enter the combination in the W → W L Z L signal hypothesis. The exclusion limits on production cross section, likelihood-ratio p-values, and best-fit cross sections as a function of a hypothetical resonance mass are summarised in Fig. 21. Scans of the profile likelihood as a function of the exotic production cross section for m X = 1.9 and 2.0 TeV (mass values of largest excesses for the benchmark models considered) are given in Fig. 22. The sensitivity of the search in the G bulk → Z L Z L signal hypothesis is dominated by the semileptonic analyses below 1.9 -2.0 TeV and the fully hadronic searches at higher mass ranges. The largest deviation is observed at m X = 1.9 TeV, driven by the ATLAS excess in the VV → JJ channel. The overall significance remains above 3σ. The preferred cross section for a hypothetical G bulk → Z L Z L signal as calculated in the J channel is ≈ 2 fb and increases to ≈ 9 fb for the JJ channel. When combined, the estimated cross section is 5 fb. The combination of the two channels reduces the exotic cross section favoured by the JJ results, and alleviates the potential disagreement between different channels, without reducing the overall significance of the excess. In other words, the combination of the two channels leads to a more coherent picture of the results by the two experiments. This is also evident from the profile likelihood scans shown in Fig. 21: given the uncertainty on the best-fit exotic production cross section, and contrary to what one might expect by considering the individual exclusion limits, the results obtained in different final states are not in tension with each other. In addition, the combination pushes the excess to mass values below 2 TeV. The picture is quite different in the G bulk → W L W L signal interpretation. The lack of a significant excess in the νJ channels is strong enough to reduce the significance of the JJ excess below the 1σ threshold. The combination of the ATLAS and CMS results disfavours the hypothesis of a resonance decaying exclusively to WW (an interpretation which in any case would be difficult to justify phenomenologically). Finally, the interpretation of the results in the context of a W signal hypothesis lies between the G bulk → Z L Z L and G bulk → W L W L scenarios: the νJ analyses are more sensitive than the fully hadronic ones, but their contribution is not as dominant as in the G bulk → W L W L case. Nevertheless, the excess survives above the 3σ threshold, thanks to the presence of a moderate excess in the J search around the same mass region. Overall, the estimated cross section of a hypothetical exotic signal is strongly reduced: the best-fit value changes from ≈ 10 fb (when using the JJ channel results only) to ≈ 5 fb (when combining the JJ, νJ and J channels). At this smaller cross section value, the outcome of the searches in the different channels is quite coherent, as shown in the profile likelihood scans depicted in Fig. 21. The mitigating effect of the J result is evident if one compares the νJ-and-J combined likelihood scan for the W combination to the likelihood scans in the semileptonic searches. The W L Z L curve is much more similar to the Z L Z L curve in the J channel than to the W L W L curve in the νJ channel. In conclusion, a resonance with a production cross section of ∼5 fb and mass between 1.9 and 2.0 TeV is the scenario most consistent with the experimental results out of all benchmark models considered in this study, as long as it does not decay exclusively to a W L W L final state. An example of the model independent combination of the Z L Z L and W L W L channels is shown in Fig. 23. In this case, one considers a resonance that can decay to both W L W L and Z L Z L , with the relative branching fraction determined by the r parameter introduced in Eq.(2.1). For r → 0 one recovers the G bulk → Z L Z L case, while for r → ∞ one recovers the G bulk → W L W L limit. It should be noted that for this combination we use a common mass window for the ATLAS analyses, namely the one that corresponds to the ZZ search, giving the best overall sensitivity (see Sec. 3). Therefore, the results obtained here on the W L W L exclusion limits and p-values are somewhat different than the ones presented in Fig. 21. The results obtained for generic values of r are similar to the Z L Z L case, i.e. they point to an overall excess. The size of the excess is reduced to 2σ, with a best-fit exotic production cross section around 4 fb. Particularly interesting is the r = 2 case, corresponding to a resonance with universal couplings to the pseudo-Goldstone bosons. In this case, despite the fact that B(X → W L W L ) = 2B(X → Z L Z L ), the combined deviation is found to have a ≈ 2.4σ significance for a cross section of ≈ 4 fb. It may be interesting to comment here on how the statistical methods that we have employed in this study compare with the simplified practices used by the theoretical community. A standard technique employed in many theoretical papers is to assume Gaussian likelihoods for the cross section of hypothetical signals, taking as central value the difference between the observed and expected limits, and as standard deviation the expected (95% C.L.) limit divided by 1.96. Then, one can use the cross sections and uncertainties as estimated in the various channels, and calculate a weighted average. This method should, in principle, work well for cases in which the fitted cross section comes with a relatively small uncertainty (which is, typically, not the case in most searches) and the systematic uncertainties can be considered as uncorrelated among channels and experiments (which may, or may not be the case). As an example, we note that the simplified combination of the search results in the G bulk → W L W L interpretation yields a best-fit cross section of 2.5 ± 1.6 fb (2.5 ± 1.4 fb) at m X = 1.9 TeV (2.0 TeV), to be compared with our result of 0.75 +1. 67 −0.75 fb (1.1 +1.4 −1.1 fb). Similarly, the simplified combination in the G bulk → Z L Z L interpretation yields a best-fit cross section of 4.7 ± 1.9 fb (4.4 ± 1.8 fb) at m X = 1.9 TeV (2.0 TeV), to be compared with our result of 5.2 +2.1 −1.6 fb (4.2 +1.9 −1.2 fb). While more data is needed to clarify the situation, the results from the analysis of the diboson searches is unquestionably one of the most interesting outcomes of the ATLAS and CMS exotic programmes during the first LHC run. The situation is even more intriguing if one adds to the picture the ≈ 2σ excess at 1.8-1.9 TeV observed by CMS in a WH resonance search. The W results shown in Fig. 21 emerge as the most promising hint in the quest for a new heavy resonance in the ATLAS and CMS data, as already pointed out in Ref. [39]. Conclusions We have performed a combination of the ATLAS and CMS searches for a heavy resonance decaying to a diboson final state, derived from the public information available for the six relevant analyses [20-22, 79, 80]. We have developed a methodology for the combination procedure, which begins with the work to emulate the public results by ATLAS and CMS for each individual analysis. This process is adjusted when necessary with correction factors to account for unknown uncertainties, and has been validated by reproducing the official results by the two experiments. We have presented combinations of the ATLAS and CMS searches for individual decay modes in various simplified models. At each step, the 95% CL limits, the likelihood ratio p-values, the profile likelihood scans, and the maximum likelihood fits of the production cross section as function of the resonance mass m X are provided. The combination is obtained in three scenarios: W → W L Z L , G bulk → W L W L , and G bulk → Z L Z L . We also obtain the full combination results for a G bulk resonance with generic W L W L and Z L Z L branching fractions. Out of all benchmark models considered, the combination favours the hypothesis of a resonance with mass 1.9-2.0 TeV and a production cross section ≈ 5 fb, as long as the resonance does not decay exclusively to W L W L final states. Depending on the details of the resonance model, a signal significance between 2.4 and 3.4σ is obtained for notable benchmark scenarios (see Table 3). In particular, the possibility of a W resonance, suggested by other searches in different final states, is corroborated by the diboson searches, with a significance of ≈ 3σ for a resonance mass of 1.9 TeV. Note added in v2 of the paper While preparing this manuscript for submission, ATLAS and CMS presented preliminary results in searches for diboson resonances with the first √ s = 13 TeV pp collision data. They include results in the W ( ν) V (qq) [83], Z( + − ) V (qq) [84], V (qq) V (qq) [85] and Z(νν) V (qq) [86] channels by ATLAS, and the W ( ν) V (qq) and V (qq) V (qq) channels by CMS [87]. No significant excess above the SM expectations is observed, however the experimental sensitivity is, in most cases, not comparable with the one from Run-1 yet. The notable exception is the newly added Z(νν) V (qq) channel. The most stringent exclusion limits in the preliminary analysis of Run-2 data are obtained in the following channels: • (HVT) W → W L Z L : 25 fb (20 fb) for m X = 1.9 TeV (2.0 TeV) in the W (qq) Z(νν) channel by ATLAS, and the combination of the two channels considered by CMS. • G bulk → W L W L : 15 fb (12 fb) for m X = 1.9 TeV (2.0 TeV) in the W ( ν) W (qq) channel by ATLAS. • G bulk → Z L Z L : 21 fb (15 fb) for m X = 1.9 TeV (2.0 TeV) in the Z(νν) Z(qq) channel by ATLAS. In assessing the compatibility of the Run-2 exclusion limits with the results obtained in this study (summarised in Table 3) we use parton luminosity ratio values of 13 (15) for m X = 1.9 TeV (2.0 TeV) for gg production (G bulk → W L W L and G bulk → Z L Z L channels) and 8 (8.5) for m X = 1.9 TeV (2.0 TeV) for qq production (W → W L Z L channels) [88] to calculate the increase in the exotic signal production cross section from 8 to 13 TeV. We observe that the absence of a significant deviation in the Run-2 data • creates a ∼ 2 − 3σ tension with the best-fit cross section derived in this paper in the G bulk → Z L Z L channel, • is consistent (within 1σ) with the (consistent-with-zero) result we obtain in the G bulk → W L W L channel, and • is also consistent (within 1σ) with the best-fit cross section that we have derived in the W → W L Z L channel. We, therefore, conclude that the preliminary analysis of the Run-2 data by ATLAS and CMS does not rule out the small deviation reported in the W → W L Z L channel of the Run-1 diboson searches. It is widely expected that a clear picture will emerge with the analysis of the larger 13 TeV datasets. Acknowledgments We would like to thank our colleagues at the ATLAS and CMS collaborations for their exemplary work and publication of a large number of papers on exotic searches. We thank Andreas Hinzmann for his precious help in the implementation of the CMS search in the X → VV → JJ channel. We also thank Goran Senjanović A Comparison of different approaches to emulate ATLAS VV → JJ analysis σ(pp>X>ZZ) (fb) llJ Z L Z L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ • CMS+ATLAS p-value llJ Z L Z L 1σ 2σ 3σ 4σ • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • ATLAS ■ CMS ◆ ATLAS+CMSσ(pp>X>WW) (fb) lvJ W L W L • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ • ATLAS ■ CMS ATLAS+CMS Z L Z L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ • lνJ ■ JJ ◆ JJ +lνJATLAS+CMS W L W L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ • JJ +lνJ ■ lνJ ◆ JJ ▲ noATLAS+CMS W L W L 1σ 2σ 3σ 4σ • • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ • JJ + llJATLAS+CMS W L Z L • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ The expected limits obtained in the emulation of the ATLAS VV → JJ channel show a 40% discrepancy with respect to the official results (see Sec. 3). This is the largest discrepancy observed among all the channels considered in this study. We have considered alternative approaches in our strategy and carried out several cross-checks, which are summarised here: • Nominal background: ATLAS publishes a background description with a total background uncertainty. This information can be used directly as an input to our analysis. The disadvantage of this approach is that it combines all systematic uncertainties into a single contribution, implying a correlation model that may not reflect the accuracy of the fit performed by the ATLAS collaboration. • Pure fitting: We have repeated the fit on the data distribution provided by the ATLAS collaboration. The fitting procedure naturally yields a covariance matrix for the shape parameters, which allows to adopt a more realistic correlation model. • Rescaling: This is a mixed approach in which the fit is performed over the data distribution to obtain the covariance matrix of the fitting function parameters, but the resulting background prediction and the corresponding uncertainties are then rescaled to match those provided by ATLAS. In this approach, the official ATLAS background prediction is used and our fit is only used to model the uncertainties and their correlations. • Sidebands: In this case we repeat the fit procedure described above, after excluding the region of the largest deviation (1700-2300 GeV) from the fit range, in order to exclude the possibility that it could bias the fit. Fig. 24 shows the ratio of the observed exclusion limits to the ones from the official ATLAS results for the different approaches summarised above. In all cases the differences are very small, which suggests that the explanation for the observed discrepancy should be attributed to a factor other than the background determination procedure. The discrepancy is absorbed in the fudge factor which, when tuned to deliver the official expected exclusion limits, remarkably removes (to a large extent) the differences in the observed limits. One should note that the decision to employ these correction factors in our analysis (for this and other channels) does not change qualitatively the conclusions of this study. This can be seen, for example, in the middle plot of Fig. 21, where it is shown that the two different approaches yield significances that differer typically by 0.5σ. B Narrow width approximation The CMS collaboration assumes a signal with negligible width, whereas the ATLAS collaboration simulates signal distributions with a model-dependent width of ≈ 7% of the resonance mass (see Table 1 of Ref. [20]). In this appendix we estimate the effect of this difference in the final exclusion limits and provide a recipe for obtaining the ATLAS results in the narrow-width approximation. The large width hypothesis used by the ATLAS collaboration impacts the limits through the modification of the signal shapes. In the JJ channel it widens the core for the signal distribution and creates a large left tail due to the interplay between proton PDFs [89] and the natural width of the resonance, as one can see in the left plot of Fig. 25. In practice, for a given total cross section we have events leaking outside the ±10% window around m X . This value corresponds typically to the experimental resolution of this channel. The amount of this leakage, f l is provided in Ref. [20] and corresponds typically to 15% in the region under study in this paper. We expect the events in the left tail to have no significant impact on the exclusion limits. A test was performed by truncating the signal to m X ± 200 GeV and repeating the JJ limit-setting procedure for the W hypothesis. As one can see in the right plot of Fig. 25, the difference in the expected exclusion limits does not exceed 2%. To map the ATLAS limits into a narrow width hypothesis we make the following approximation: The main difference between the wide and narrow resonances is the presence of leaking events in the right tail or under the peak. Consequently, by multiplying the signal efficiency of ATLAS by 1/f l we recover most of the properties of the narrow signal. In conclusion, we approximate the narrow signal hypothesis for ATLAS analyses by scaling the fully hadronic and semi-leptonic signals by a factor of 1.1 (i.e. by increasing the signal yield by 10%). Figure 1 . 1ATLAS hadronic search: Comparison between the official ATLAS fit (blue line) and the fit of this study with uncertainties as described in the text (coloured bands), with the overlaid data of the m JJ spectrum for the WW (left), WZ (middle) and ZZ (right) tagging selections. Figure 2 . 2ATLAS hadronic search: Ratio of observed exclusion limits obtained with this study to the ones of the official ATLAS result, as a function of the mass m X of the exotic resonance for the WW (black), ZZ (red) and WZ (magenta) tagging selections. mFigure 4 . 4X (TeV) σ(pp>X>VV) (fb) ATLAS JJ Z L Z L ATLAS hadronic search: Expected exclusion limits for different tagging and masswindow selections, as a function of the mass m X of the exotic resonance for G bulk → W L W L (left), W → W L Z L (middle) and G bulk → Z L Z L (right) signal hypotheses. The results have been obtained with the correction discussed in the text. Figure 5 .mFigure 6 . 56CMS hadronic search: m JJ data distribution overlaid with the background fit employed in this study with uncertainties for High (left) and Low (right) Purity samples. See text for details. X (TeV) σ(pp>X>VV) (fb)CMS JJV L V L CMS hadronic search. Left: Expected (dashed lines) and observed (continuous lines) exclusion limits on W → W L Z L production cross sections as a function of the resonance mass m X obtained with this study (black), and comparison with the official CMS results (red). The green and yellow bands (dashed lines) represent the one and two sigma variations around the median expected limits calculated in this study (by CMS). Right: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section as a function of the resonance mass m X obtained with this study for Figure 7 . 7Combination of hadronic searches: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section as a function of the resonance mass m X obtained with the emulation of the ATLAS (red) and CMS (blue) searches and their combination (black) for W L W L (left), W L Z L (middle) and Z L Z L (right) selections and signal hypotheses. The green and yellow bands represent the one and two sigma variations around the median expected limits. Figure 8 . 8Combination of hadronic searches: likelihood ratio p-values as a function of the exotic resonance mass m X obtained with the emulation of the ATLAS (red) and CMS (blue) searches and their combination (continuous black) for W L W L (left), W L Z L (middle) and Z L Z L (right) selections. Figure 9 . 1σ m X = 2 1σ Figure 10 . 91σ21σ10Combination of hadronic searches: Best fitted exotic production cross section as a function of the resonance mass m X obtained with the emulation of the ATLAS (red) and CMS (blue) searches and their combination (black) for W L W L (left), W L Z L (middle) and Z L Z L (right) selections and signal hypotheses. The green and yellow bands represent the one and two sigma variations around the median values. The results include the 10% scale factor discussed in the text. pp>X>WZ) (fb) log(L Best fit )-log(L) JJ W L Z L pp>X>ZZ) (fb) log(L Best fit )-log(L) JJ Z L Z L Combination of hadronic searches: Scans of the profile likelihood as a function of the exotic production cross section for a m X = 2 TeV signal (mass value of largest excess) for the emulation of the ATLAS (red) and CMS (blue) searches and their combination (black) for W L W L (left), W L Z L (middle) and Z L Z L (right) selections and signal hypotheses. 4 Semi-leptonic searches: WV → νJ and ZV → J Figure 11 . 11W L W L + Z L Z L ATLAS hadronic ZZ selection Combination of hadronic searches, and dependence of results obtained in this study on the r ≡ B(X → WW)/B(X → ZZ) parameter for a neutral bulk RS-like spin-2 particle hypothesis, and as a function of the resonance mass m X . Left: expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section. Middle: likelihood-ratio p-values. Right: best fitted exotic production cross section. Figure 12 . 12ATLAS ZV → J (left) and WV → νJ (right) searches: Comparison between the official ATLAS background (blue line) and its uncertainties (purple band) with the overlaid data of the m JJ spectrum for the Merged Region (of the vector boson hadronic reconstruction) category. Figure 13 . 13ATLAS semileptonic searches: Fudge factor as a function of the mass m X of the exotic resonance, calculated via the ratio of observed exclusion limits obtained with this study to the ones of the official ATLAS result, for the W → W L Z L (red) and G bulk → Z L Z L (black) signal hypotheses in the J channel, and for the W → W L Z L (magenta) and G bulk → W L W L (orange) signal hypotheses in the νJ channel. m X (TeV) σ(pp>X>WW) (fb) mFigure 14 . 14X (TeV) σ(pp>X>WZ) (fb) ATLAS lνJ W L Z L ATLAS semileptonic searches: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross sections as a function of the resonance mass m X obtained with this study (black), and comparison with the official CMS results (red) for G bulk → Z L Z L (top left), W → W L Z L (top right), G bulk → W L W L (bottom left) and W → W L Z L (bottom right) signal hypotheses in the J (top) and νJ (bottom) channels. The green and yellow bands represent the one and two sigma variations around the median expected limits calculated in this study, with all the corrections described in the text included. Figure 15 . 15CMS WV → νJ search: Comparison between the official CMS background (blue line) and the background modelling with uncertainties employed by this study (coloured bands), with the overlaid data of the m JJ spectrum for the HP (left-hand side) and LP (right-hand side) categories, plotted separately for the electron (top) and the muon (bottom) channels. Figure 16 . 16CMS ZV → J search: Comparison between the official CMS background (blue line) and the background modelling with uncertainties employed by this study (coloured bands), with the overlaid data of the m JJ spectrum for the HP (left-hand side) and LP (right-hand side) categories, plotted separately for the electron (top) and the muon (bottom) channels. Figure 17 .mFigure 18 .Figure 19 . 171819CMS semileptonic searches: Fudge factor as a function of the mass m X of the exotic resonance, calculated via the ratio of observed exclusion limits obtained with this study to the ones of the official CMS result for the G bulk → W L W L (red) and G bulk → Z L Z L (black) semileptonic analyses. X (TeV) σ(pp>X>ZZ) (fb)CMS llJZ L Z L CMS semileptonic searches: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross sections as a function of the resonance mass m X obtained with this study (black), and comparison with the official CMS results (red) for the G bulk → W L W L search in the νJ channel (left) and the G bulk → Z L Z L search in the J channel (right). The green and yellow bands represent the one and two sigma variations around the median expected limits calculated in this study, with all the corrections described in the text included. CMS semileptonic searches: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section as a function of the resonance mass m X obtained with this study for the G bulk → W L W L (red) and W (black) signal hypotheses in the νJ channel (left) and for the G bulk → Z L Z L (red) and W (black) signal hypotheses in the J channel (right). and Andrea Wulzer for fruitful discussions and valuable suggestions. A.O. thanks the CERN theory group for their hospitality. This material is based upon work partially supported by the Cooperation Agreement (SPRINT Program) between the São Paulo Research Foundation (FAPESP) and the University of Edinburgh, under Grant No. 2014/50208-0. A.O. is supported by the MIURFIRB RBFR12H1MW grant. The work of F. D. and C.L. is supported by the Science and Technology Facilities Council (STFC) in the UK. Figure 20 .m 20Combination of semileptonic searches for G bulk → Z L Z L (top), G bulk → W L W L (middle) and W → W L Z L (bottom) selections and signal hypotheses, and as a function of the resonance mass m X obtained with the emulation of the ATLAS (red) and CMS (blue) searches and their combination (black). Left: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section. The green and yellow bands represent the one and two sigma variations around the median expected limits. The results include the correction factors discussed in the text. Right: Likelihood ratio p-values. The dashed black curve corresponds to the combined search without the corrections discussed in the text. X (TeV) σ(pp>X>ZZ) (fb) mFigure 21 . 1σ Figure 22 .mmFigure 23 . 211σ2223X (TeV) σ(pp>X>WW) (fb) ATLAS+CMS W L Z L Combination of all ATLAS and CMS resonance searches for G bulk → Z L Z L (top), G bulk → W L W L (middle) and W → W L Z L (bottom) selections and signal hypotheses, and as a function of the resonance mass m X carried out in the hadronic (red) and semileptonic (blue) channels and their combination (black). The results include all correction factors discussed in the text. Left: Expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section. The green and yellow bands represent the one and two sigma variations around the median expected limits. Middle: Likelihood ratio p-values. The dashed black curve corresponds to the combined search without the corrections discussed in the text. Right: Best fitted exotic production cross section. The green and yellow bands represent the one and two sigma variations around the median values. Best fit )-log(L) ATLAS+CMS Z L Z L Combination of all ATLAS and CMS resonance searches: Scans of the profile likelihood as a function of the production cross section for a m X = 2.0 (1.9) TeV signal shown with continuous (dashed) lines in the hadronic (red) and semileptonic (blue) channels and their combination (black) for W L W L (left), W L Z L (middle) and Z L Z L (right) selections and signal hypotheses. ATLAS+CMS W L W L + Z L Z L X (TeV) p-value ATLAS+CMS W L W L + Z L Z L X (TeV) σ(pp>X>VV) (fb) ATLAS+CMS W L W L + Z L Z LATLAS hadronic ZZ selection Combination of all ATLAS and CMS resonance searches, and dependence of results obtained in this study on the r ≡ B(X → WW)/B(X → ZZ) parameter for a neutral bulk RS-like spin-2 particle hypothesis, and as a function of the resonance mass m X . Left: expected (dashed lines) and observed (continuous lines) exclusion limits on exotic production cross section. Middle: likelihood-ratio p-values. Right: best fitted exotic production cross section. Figure 24 . 24Emulation of ATLAS VV → JJ search and comparison of the alternative approaches for the background prediction considered: Fudge factors as a function of the resonance mass m X , determined via the ratio of the expected limits obtained with different background estimation techniques (black: "pure fitting", red: "nominal background", blue: "rescaling", magenta: "sidebands") over those in the official ATLAS result for the W L W L (left), W L Z L (middle) and Z L Z L channels (right). See text for details. Figure 25 . 25W L Z L m VV ⊂ m W ' ± 200GeV Narrow-width approximation. Left: Signal distribution in the diboson invariant mass for a 2 TeV W signal. The hatched ±200 GeV region around the signal represents the narrowwidth approximation. 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Public plots Yes JJ [22] HEPDATA HEPDATA HEPDATA HEPDATA No CMS νJ [21] Fit Fit MC Public plots & MC Yes J [21] Fit Fit MC Public plots & MC Yes Table 2 . 2Relative efficiencies for WW, WZ, ZZ signal hypotheses for tagging selection using different mass windows.Signal hypothesis Tagging selection WW WZ ZZ WW window 1.00 0.65 0.42 WZ window 0.84 1.00 0.65 ZZ window 0.70 0.84 1.00 Table 3 . 3Summary of results obtained in this study: significance, p-values and best-fit cross sections for different model interpretations at m X = 1.9 and m X = 2.0 TeV, i.e. the mass values where the largest excesses have been observed for different models. Our main results contain corrections that have been introduced to account for unknown uncertainties in the official results. (Additional results calculated without these correction factors are given inside the parentheses.)Signal hypothesis mX (TeV) Significance p-value Best-fit cross section (fb) 1.9 2.5 (3.1) 6.5 (1.0) ×10 −3 5.3 +2.3 −2.0 (5.5 +2.0 −1.6 ) W → WL ZL 2.0 2.5 (3.2) 7.0 (0.8) ×10 −3 4.3 +2.1 −1.5 (4.7 +1.8 −1.3 ) 1.9 0.49 (0.83) 0.30 (0.20) 0.75 +1.67 −0.75 (1.4 +1.7 −1.4 ) G bulk → WLWL 2.0 0.88 (1.33) 0.20 (0.092) 1.1 +1.4 −1.1 (1.8 +1.8 −1.4 ) 1.9 3.4 (3.8) 3.2 (0.65) ×10 −4 5.2 +2.1 −1.6 (4.7 +1.8 −1.2 ) G bulk → ZLZL 2.0 3.0 (3.5) 1.2 (0.24) ×10 −3 4.2 +1.9 −1.2 (3.9 +1.6 −1.0 ) 1.9 2.6 (3.4) 5.2 (0.40) ×10 −3 3.9 +2.4 −1.5 (4.9 +2.0 −1.7 ) G bulk (r=2) 2.0 2.4 (3.1) 8.8 (0.89) ×10 −3 3.1 +1.8 −1.3 (3.9 +1.6 −1.4 ) For recent analyses, including the LHC discovery of the Higgs boson, see for instance[10,11].2 In this paper we refer to a vector boson (W or Z) decaying hadronically by the generic label V. 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