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Mapping Dark Matter in the Milky Way using Normalizing Flows and Gaia DR3 Sung Hak Lim Dept. of Physics and Astronomy NHETC 08854Rutgers, PiscatawayNJUSA Eric Putney Dept. of Physics and Astronomy NHETC 08854Rutgers, PiscatawayNJUSA Matthew R Buckley Dept. of Physics and Astronomy NHETC 08854Rutgers, PiscatawayNJUSA David Shih Dept. of Physics and Astronomy NHETC 08854Rutgers, PiscatawayNJUSA Mapping Dark Matter in the Milky Way using Normalizing Flows and Gaia DR3 We present a novel, data-driven analysis of Galactic dynamics, using unsupervised machine learning -in the form of density estimation with normalizing flows -to learn the underlying phase space distribution of 6 million nearby stars from the Gaia DR3 catalog. Solving the collisionless Boltzmann equation with the assumption of approximate equilibrium, we calculate -for the first time ever -a model-free, unbinned, fully 3D map of the local acceleration and mass density fields within a 3 kpc sphere around the Sun. As our approach makes no assumptions about symmetries, we can test for signs of disequilibrium in our results. We find our results are consistent with equilibrium at the 10% level, limited by the current precision of the normalizing flows. After subtracting the known contribution of stars and gas from the calculated mass density, we find clear evidence for dark matter throughout the analyzed volume. Assuming spherical symmetry and averaging mass density measurements, we find a local dark matter density of 0.47 ± 0.05 GeV/cm 3 . We fit our results to a generalized NFW, and find a profile broadly consistent with other recent analyses. I. INTRODUCTION Multiple lines of evidence indicate that the majority of matter in the Universe is dark -that is, it does not interact with the known particles through electromagnetic or strong nuclear interactions. Measurements of galaxy rotation curves [1][2][3], galaxy clusters [4,5], the early Universe [6], and gravitationally-lensed systems [7] cannot be explained without the addition of new particles beyond the Standard Model. Despite a robust experimental program, dark matter has resisted attempts to measure its particle physics interactions, and astrophysical probes remain a vital window into its properties. In this work, we employ a novel method to determine the dark matter density around the Solar location within the Milky Way, one that has never before been applied to data. Our method is fully data-driven, uses the measurements of stellar position ⃗ x and velocities ⃗ v made possible by the Gaia Space Telescope [8,9], and is powered by modern, unsupervised machine learning methods. The phase space density f (⃗ x, ⃗ v) of the population of stars within the Milky Way obeys the collisionless Boltzmann Equation: ∂f ∂t + v i ∂f ∂x i = ∂Φ ∂x i ∂f ∂v i .(1) Here, Φ is the total gravitational potential, which can be related to the total mass density ρ using the Poisson Equation 4πGρ = ∇ 2 Φ.(2) Assuming that the phase space density of the stars is in equilibrium ∂f /∂t = 0, the 3D acceleration field −∂Φ/∂x i can be derived from knowledge of the stellar phase space density f today. A further derivative of Φ then gives the total mass density ρ; the dark matter density can then be calculated assuming knowledge of the baryonic components. Measuring the stellar phase space density and its derivatives has traditionally been difficult, given the relatively high dimensionality (six) of the data. Instead, measurements of the local Galactic potential have used moments of the Boltzmann Equation -the Jeans Equation -along with simplifying assumptions (axisymmetry, specific functional forms, and/or small mixed radial and altitude "tilt" terms) which allows for relatively stable calculation of numeric derivatives from stellar data binned in the position coordinates. We refer to Refs. [10][11][12][13][14][15] for recent examples of these techniques. The modern machine learning method known as normalizing flows provides a new approach to this problem that allows direct, unbinned access to the phase space density, independent of symmetry assumptions. Normalizing flows (reviewed in Ref. [16]) are a class of unsupervised deep learning algorithms that are sufficiently expressive to allow accurate modeling of the phase space density of high-dimensional data. Using normalizing flows, Refs. [17][18][19] directly solved the Boltzmann Equation for synthetic mock stellar data drawn from smooth analytic simulations of a galaxy [20,21]. In Ref. [22], we demonstrated this approach on a fully-cosmological N -body simulation of a Milky Way-like galaxy, including realistic Gaia-like measurement errors and the impact of departures from symmetry and lack of equilibrium. Here, we apply the algorithm developed in Ref. [22] to Gaia Data Release 3 (DR3) [23] itself. Using a population of stars within 4 kpc of the Sun for which full kinematic solutions are available, we measure the gravitational acceleration and total density everywhere within the sphere except in regions near the disk where dust extinction is significant. We also estimate the total uncertainty on our acceleration and density measurements throughout this region. At each location, these uncertainties include statistical uncertainty, Gaia measurement uncertainty, and an estimate of fit uncertainty from the normalizing flows. As the flow is extremely expressive, our errors should encompass a fuller range of possible shape variations of the density profile consistent with data, compared to many other approaches that fit the profile to a (perhaps overly-restrictive) functional form. Using existing measurements of the baryon density, we find clear evidence of a non-baryonic component to the mass density throughout the Solar neighborhood. Though these measurements do not rely on any assumptions of symmetry within the data, imposing spherical symmetry on the dark matter density allows us to average measurements at different locations and reduce errors. Under this additional assumption, we find a dark matter density of 0.47 ± 0.05 GeV/cm 3 at the Sun's distance from the Galactic Center. We also fit our density measurements to a generalized Navarro-Frenk-White (NFW) [24,25] profile, though with considerable uncertainties on the best-fit parameters. Future data releases from Gaia will increase the number of stars with full kinematic information by a factor of ∼ 5, as well as decreasing the proper motion measurement errors by ∼ 2. Combined with anticipated improvements in understanding the error model and quantifying dust extinction, the accuracy of the dark matter density measurements obtained using this method can be greatly increased in the near future. In Section II, we introduce the Gaia DR3 dataset used to train our normalizing flows. Section III contains the core results from our analysis: here we show our estimates of the phase space density using normalizing flows, followed by the calculations of accelerations and mass density using the collisionless Boltzmann Equation. Additionally, we investigate evidence for departures from equilibrium in the data and perform self-consistency checks. In Section IV we discuss our results and future directions for flow-based modelling of Galactic dynamics. II. GAIA DR3 The Gaia space telescope [23,26] has revolutionized precision astrometry. As of its third data release (DR3), Gaia has measured the full 6D kinematics of nearly 33 million stars [27]. This unprecedented volume of data, combined with state-of-the-art density estimation techniques, allows for robust mapping of the Milky Way's phase space density. For those stars with full kinematic information, the angular positions (α, δ), proper motions on the sky (µ * α , µ δ ), and parallax ϖ are measured by the Gaia photometer while the radial velocity V rad and apparent magnitude G RVS are measured by the Gaia spectrometer (RVS). Nearly 100% of stars with apparent magnitude brighter than G RVS = 14 are expected to have 6D kinematics in Gaia DR3 [27]. In addition to a maximum apparent magnitude, the Gaia spectrometer also has a minimum apparent magnitude due to saturation [28]. Stars brighter than G RVS ≈ 3 are not included in the dataset used for this analysis. In terms of G RVS and the parallax-derived distance (d/kpc) = (1 mas/ϖ), the absolute magnitude M G is M G = G RVS − 5 log 10 (d/kpc) − 10.(3) In this analysis, we do not use other spectral information (such as BP − RP color). The three coordinate systems used in this work are shown: the Galactocentric Cartesian coordinates (x, y, z), the spherical coordinates (r, θ, ϕ), and the cylindrical coordinates (R, ϕ, z). The lines through the observational volume with low dust extinction along which we measure accelerations and mass densities are shown in color. In orange, we show two lines at z = +1.5 kpc, one varying r and another ϕ. The two corresponding lines at z = −1.5 kpc are shown in purple. The line parameterized by polar arclength s = r⊙ × (π/2 − θ) passing through the Solar location is shown in green. Given the approximate axisymmetry of the baryonic disk of the Galaxy and the approximate spherical symmetry of the dark matter distribution, it is useful to consider the data in both spherical and cylindrical coordinates, as well as Galactocentric Cartesian coordinates. The three coordinate systems we use are shown in Figure 1 and are defined as follows: 1. Our Cartesian coordinate system places the x − y plane in the Galactic disk, with x = 0 at the Galactic center and the Sun along the +x axis. The +z axis (perpendicular to the disk) is oriented so that the net rotation of the disk stars in the −y direction. From the fundamental kinematic properties (parallax, angular position, etc.) measured by Gaia, the positions and velocities in these three coordinate systems can be obtained, using the parallax to calculate distance and assuming a Galactocentric Solar position and velocity of (8.122, 0.0, 0.0208) kpc [29,30] and (−12.9, −245.6, 7.78) km/s [29,31,32], respectively. A. Tracer Population Selection To extract the accelerations from the derivatives of stellar phase space density using the Boltzmann Equation, the population of stars in question must be complete, unbiased, and in dynamic equilibrium. These conditions are not satisfied by the full Gaia 6D dataset, which is complete in observed (but not absolute) magnitude. Stars which are intrinsically dim but nearby are included, while intrinsically dim but more distant stars were not observed and thus are absent. We first remove from our analysis dataset stars without spectroscopic or photometric magnitudes, as well as stars with large relative parallax errors (3σ ϖ > ϖ). Of the 31,532,490 stars in Gaia DR3 with full 6D kinematic solutions available within 10 kpc, 29,855,114 stars remain after these cuts. We further remove 13 abnormally fast stars, with speeds higher than 1000 km/s relative to the Galactic rest frame. Within 4 kpc of the Sun, 24,789,061 stars remain, their position-space distributions are shown in the left column of Figure 2, and their color-magnitude diagram (BP − RP versus M G ) is shown in Figure 3. In order to correct for the bias in the dataset towards stars with low observed magnitude, we require that every star in our restricted sample is bright enough to have been observed by Gaia regardless of its position within a sphere around the Sun. We set this sphere to be 4 kpc in radius, both to limit fractional parallax errors (which grow with distance), as well as to allow a sufficiently large number of stars to pass the completeness criteria. That is, we require that the absolute magnitude M G of every star in our sample is bright enough that, if the star was located at 4 kpc, it would have an observed magnitude G RVS < 14 and thus be above the completeness limit of the Gaia spectrometer: M G < 14 − (5 log 10 4 + 10).(4) This selection criterion is shown as the white dashed line in Figure 3. Applying this selection removes 63.6% of the stars in the sample with distance < 10 kpc, leaving 10, 876, 430. Note that we place this selection criterion on the magnitude without extinction-correction. We find that the dust extinction corrections available from Gaia are not uniform across the 4 kpc sphere centered on the Sun. As a result, correcting for extinction induces position-dependent suppression factors in the derived phase space density. In the future, these corrections may be more uniformly available, which would likely improve this analysis. Due to the saturation limit G RVS ≈ 3, the closest star to the Sun in our final dataset is 30.4 pc away, and there are somewhat fewer stars within 50 pc in the analysis than would otherwise be expected. As this is a very small volume and total number of missing stars (relative to the entire dataset), we do not find that the normalizing flows greatly suppress the phase space density of stars near the Solar location. For our final analysis, we require the measured distance of stars from the Sun to be less than 4 kpc, leaving 5,811,956 stars. The larger dataset (including stars out to 10 kpc) will be used when quantifying the impact of measurement errors on our final results, as we will discuss below. In the middle column of Figure 2, we show the stellar number density as a function of Galactocentric position after selecting on bright stars Eq. (4). As can be seen, the many large density variations present in the number density prior to the magnitude selection are suppressed after requiring nearby stars to be bright enough to be seen if they were at the edge of the dataset. Additionally, the observational bias causing the stellar densities to be higher near the Sun is suppressed after the selection and the number density of stars is (correctly) seen to be rising towards the Galactic center. The sight-lines to stars are less affected by dust off of the Galactic disk. Within the dust-filled disk, significant density variations remain after the application of Eq. (4). This is most obviously seen as a triangular wedge of apparent low stellar density towards the Galactic center at |z| ∼ 0, but striations can also be seen in other directions within the disk. These features are primarily in the disk (|z| ≲ 1 kpc) and largely trace known dust features, as can be seen in the left-hand column Figure 4, where we overlay extinction maps of dust [33] with the observed number density of stars after the magnitude selection criterion has been applied. For this comparison, we stitch together two three-dimensional maps: bayestar19 [34] covering declination > 30 • , and marshall [35] covering b ∈ [−10 • , 10 • ] and ℓ ∈ [−100 • , 100 • ]. For the position not covered by either 3D maps, we use a 2D dust map SFD [36] for distances d > 1.0 kpc. The regions of low stellar density match to the known dust clouds, including the Serpens-Aquila rift, the Rho Ophiuchi cloud complex, Lupus, the Dark River, Pipe Nebula, the Northern Coalsack, and the Vela molecular ridge. In the right column of Figure 2, we show the stars after the magnitude selection Eq. (4) with the additional requirement of |z| > 1 kpc. No dusty features are visible outside the disk. Although the effects of dust occlusion are greatest inside the disk (|z| ≲ 1 kpc), one notable exception is in the neighborhood closest to the Sun: for the stars nearest to us, dust does not have enough opportunity to accumulate along the sight-lines, and so extinction remains relatively low. As a result, measurements of phase space density along lines perpendicular to the disk and passing through the Solar location should be reliable and mostly unaffected by dust. This is seen in Figure 5, where we show the binned stellar densities as a function of s and azimuthal angle ϕ, along with contours of dust extinction. The magnitude selection results in a sample of tracer stars that is unbiased by apparent magnitude within 4 kpc, and thus complete in absolute magnitude (modulo the effects of dust). Roughly 69% of these stars belong to the red clump, a sizable sub-population of the red giant branch (RGB) that is tightly clustered in the color-magnitude diagram (indicated by a white rectangle in Figure 3). Red clump stars are typically between 1-4 Gyrs old [37]. Older stellar populations are preferred for kinematic studies of the Milky Way, as they have had sufficient time to equilibrate over the Galaxy's dynamic timescales [38]. Stars from the red clump meet this criteria, and have been used in a recent precision Jeans analysis of the Solar neighborhood [11]. In principle, the percentage of the sample composed of red clump stars could be increased by selecting based on extinction-corrected color in addition to M G . However, similar to extinctions for the Gaia spectrometer, extinction-corrected colors are not uniformly available across the sky. Using corrected colors would lead to selection effects in the phase space density and errors in the solution to the Boltzmann equation. While we expect this to be less of an issue in future data releases from Gaia, for this analysis we do not apply color selection criteria. We will apply our normalizing flow algorithm to the complete, unbiased dataset within 4 kpc, but with the knowledge that gravitational accelerations or mass densities within the disk far from the Sun are not reliable due to dust extinction. In Figure 1, we show the dustavoiding lines along which we measure accelerations and mass densities. B. Measurement Errors Gaia DR3 provides measurement errors in the form of Gaussian standard deviations for the measured quantities of angular position, parallax, proper motion, RVS, and G RVS . In the left and center panels of Figure 6, we show histograms of the kinematic errors for all stars within 4 kpc which pass the selection criteria described in Section II A. Propagating the errors from the angular positions, proper motions, parallax, and RVS measurements results in a covariance matrix for the measurement errors in the Cartesian coordinates used for training the normalizing flows. The median standard deviation in dis- tance and speed of stars perturbed by the Cartesian error model is shown in the right panel of Figure 6. The parallax and RVS measurements contribute the dominant sources of error to the kinematic solutions. In particular, parallax errors in some cases can be larger than the measured parallax; this often results in negative parallaxes when varying within errors. Considerable literature exists on the conversion of parallax to distance, including the impact of measurement errors [39][40][41][42][43]. We note that Gaia also provides a secondary distance estimate distance gspphot, which is inferred from a Markov Chain Monte Carlo algorithm that uses spectra from the BP and RP bands, apparent G magnitude, and parallax ( [44,45]). Unfortunately, the availability of distance gspphot (as well as other data products from this fit) is not uniform across the sky. Given this limitation, we find the most uniform reliable distance estimate to be the inverse parallax after stars with large relative parallax errors are removed. Our error propagation procedure follows the outline of Ref. [22]: we generate multiple variations of the original dataset after varying every star's kinematic features within their Gaussian errors, using the correlation matrix given the errors provided by Gaia DR3 converted into Cartesian coordinates. For each varied iteration of the data, we calculate first the phase space density, then the gravitational acceleration and mass density (using the algorithms described in Section III). The variations of these quantities over multiple error-smeared datasets provides an estimate of the impact of measurement errors on the derived quantities. In order to avoid the edge effect of stars migrating out of the 4 kpc-radius sphere when errors are applied without a corresponding inward migration of more distant stars, for each variation of the dataset we apply the error-smearing to stars in the 10 kpc-radius sphere. After applying the errors to the larger dataset, we select those stars whose error-smeared distances place them within 4 kpc of the Sun. III. ACCELERATIONS AND MASS DENSITIES FROM NORMALIZING FLOWS The techniques used in this work to calculate accelerations and mass density from the collisionless Boltzmann equation using normalizing flow-derived phase space density are described fully in Ref. [22]. We describe each component of our analysis briefly here, and present results using Gaia DR3. A. Phase Space Densities To determine the gravitational accelerations − ⃗ ∇Φ from the collisionless Boltzmann Equation applied to a stellar population, we must first determine the phase space density f (⃗ x, ⃗ v) for that population. We accomplish this using normalizing flows, an unsupervised machinelearning technique for density estimation. Normalizing flows are based on invertible transformations of a simple base distribution (such as standard normal distribution) into a more complicated distribution. As long as the transformation is expressive enough, normalizing flows are able to model a variety of distributions so that this model can be used as a free-form density estimator. This expressivity is generally achieved using neural networks with bijective constraints. We use two normalizing flows to model the stellar number density n(⃗ x) and the conditional velocity distribution p(⃗ v|⃗ x) separately: f (⃗ x, ⃗ v) = n(⃗ x)p(⃗ v|⃗ x).(5) Note that the estimation of p(⃗ v|⃗ x) requires conditional density estimation; this is easily implemented in the normalizing flows architecture by simply making the transformation conditioned on the position vector ⃗ x. In addition to increasing the accuracy of the overall phase space density, this decomposition is also helpful for solving the Boltzmann equation. In both cases, the loss functions L x and L v for training the density models n(⃗ x) and p(⃗ v|⃗ x) are the negative log-likelihoods of the data: L x = − 1 N N i=1 log n(⃗ x i ) (6) L v = − 1 N N i=1 log p(⃗ v i |⃗ x i ),(7) where N is the size of an input dataset. In this work, we implement the normalizing flows using the Masked Autoregressive Flow (MAF) [46] architecture as the transformation model. GELU activation [47] is used in order to model a smooth differentiable transformation. We use the ADAM optimizer [48] to train the flows. We construct a validation dataset using 20% of the stars, randomly selected. Early stopping with a patience of 50 epochs is used, and we select the model with the lowest validation loss. For the central values, we use Monte Carlo cross-validation and ensemble averaging in order to fully utilize the dataset and reduce noise in our density estimation. That is, we prepare 100 different random splits of training and validation datasets and train a MAF for each. The density is estimated by ensemble averaging the probabilities given by each of the MAFs. All the neural networks are implemented using PyTorch [49] and nflows [50]. The details of our neural network architectures and data preprocessing prior to training are identical to those described in Ref. [22]. In Figure 7, we plot one-dimensional histograms of the selected data compared to the density estimated by the normalizing flows. Black markers are the histogram of selected stars randomly downsampled to 20% of the original size (this matches the size of the randomly-selected validation dataset). Red lines are the histograms of synthetic stars generated by sampling from the MAFs. For this figure, we upsample from the MAF to 100 times the size of the Gaia dataset. The small deviations near the Solar location in x and y histograms are due to dust extinction in the disk (as seen in Figures 2 and 4). Along lines of sight with significant dust, there are sharp falloffs in the apparent stellar density which do not reflect the true density of stars. The MAFs are constrained to be smooth functions, and as a result, they have difficulties modeling such discontinuities, and the quality of density estimation may degrade. The bump in the x histogram is mainly due to dust clouds in Cygnus (ℓ ∼ 75 • ) and Vela (ℓ ∼ −95 • ), Figure 4), we can see that the MAF is correctly learning the overall density scales and substructures visible in the stellar counts within the disk, though detailed comparison suggests that the MAF may have difficulty replicating the small-scale sharp features of the dust. MAFs are known to generate spurious "wrinkles" around sharp edges or topologically non-trivial structures within data [51]. Though the wrinkles in density themselves are small, numerical artifacts will be amplified in density derivative estimations, resulting in biases in the acceleration and mass density estimations. As previously mentioned, we avoid low-|z| regions which are far from the Solar location to minimize the effects of dust on our measurements of the acceleration and mass density fields. Density estimators that improve over the MAF results are available [52], but are computationally expensive for the density derivative estimations; we leave studies of such architectures for future work. σ v rad (km/s) 10 −2 10 −1 10 0 σ µ α i (mas/yr) σ α σ δ σ µ α σ µ δ B. Accelerations Using the learned phase space density, we estimate the acceleration field ⃗ a = − ⃗ ∇Φ by solving the collisionless Boltzmann equation. As observed in Refs. [18,22], since Φ is a function of position only, we can approximately solve the Boltzmann Equation for the acceleration by the least square method using generated velocity samples at a given ⃗ x and minimizing the residual ∂f /∂t. That is, at a given position ⃗ x, we calculate the accelerations from the trained MAFs by finding the value of ⃗ a(x) that minimizes the following mean square error (MSE): L a = d 3 ⃗ v p(⃗ v|⃗ x) v i ∂f ∂x i + a i (⃗ x) ∂f ∂v i 2 .(8) We evaluate this integral by quasi-Monte Carlo integration [22] and minimize it analytically to determine the best-fit acceleration value. For each position, we sample 10,000 velocities with |⃗ v| < 600 km/s in order to obtain a stable acceleration solution with small statistical errors. Given that Eq. (8) is highly overconstrained, the residual ∂f /∂t is not guaranteed to be zero. However, our minimization procedure follows (in spirit) the approximation typically made in density measurements based on the Boltzmann Equation, which assume the phase space density f is in equilibrium (∂f /∂t = 0). Within the Gaia EDR3 [56] This work ax (10 −10 m/s 2 ) −2.32 ± 0.16 −1.94 ± 0.22 ay (10 −10 m/s 2 ) 0.04 ± 0.16 0.08 ± 0.08 az (10 −10 m/s 2 ) −0.14 ± 0.19 −0.06 ± 0.08 |⃗ a| (10 −10 m/s 2 ) 2.32 ± 0.16 1.94 ± 0.22 Milky Way this assumption would imply an axial and z-symmetric potential (and thus mass density and acceleration fields which respect these symmetries). There is evidence that the Milky Way is not in dynamic equilibrium [53][54][55]. As we do not enforce these symmetries in our acceleration calculation and since our MSE minimization can result in residual non-zero ∂f /∂t, we can perform closure tests to estimate the amount of deviation from equilibrium and validate whether the accelerations are derived in a self-consistent manner. We discuss this in detail in Section III C. In Table I, we show the estimated acceleration at the Solar location, averaged over a ball centered at the Sun and with a radius 100 pc. 1 The measured radial acceleration at the Solar location is (1.94±0.22)×10 −10 m/s 2 ; the other components are negligible in comparison. The error includes measurement and statistical errors estimated by resampling and bootstrapping, the variation from multiple independent trainings of the MAFs, and the standard deviation of the estimated acceleration within the 100 pc ball. Our result agrees with the acceleration measurement of Gaia EDR3 [56] within 2σ. This latter measurement is obtained from the proper motion of quasars caused by secular aberration due to the orbital motion of the Solar system in the Milky Way. Differences between these two measurement techniques may indicate local disequilibrium in the Solar neighborhood, as we discuss in Section III C. In the future, the dataset available to Gaia DR4 or DR5 may be sufficient to resolve any differences at a statistically significant level. In Figure 8, we show the estimated accelerations in spherical coordinates (a r , a θ , a ϕ ), measured along curves of r, ϕ, and s = r ⊙ · (π/2 − θ) through the volume around the Sun (avoiding regions of significant dust where the MAF results may be inaccurate). The orientations of these measurement curves within the Galaxy are sum-marized in Figure 1, and we describe them further here: • Accelerations as a function of r are measured along lines passing through (x, y, z) = (8.122, 0, ±1.5) kpc -that is, through points 1.5 kpc above (towards Galactic North) and below (towards Galactic South) the Galactic disk at the Solar location. • Accelerations as a function of ϕ are calculated along curves located 1.5 kpc North and South of the Solar location passing through (x, y, z) = (7.982, 0, ±1.5), constrained so that r = r ⊙ along the curve. These 1.5 kpc offsets above and below the disk are chosen to avoid the dominant sources of dust extinction and resulting wrinkles in the MAF number densities, as seen in Figures 2 and 4. • We measure accelerations as a function of the polar arc s passing through the Solar location. These are only acceleration measurements we make closer to the disk than |z| = 1.5 kpc. Sight-lines to this arc pass through the disk only near to the Earth, where extinction is less of a concern (see Figure 5). For accelerations within 150 pc of the Sun, we average over a sphere (as in Table I) to smooth over dust within the disk. No averaging is applied to points further away. C. Testing Equilibrium The acceleration field calculated using the MAF comes from minimizing ∂f /∂t, but makes no assumptions about symmetry. In the self-gravitating, rotationally supported Milky Way, departures from axisymmetry would presumably imply disequilibrium, and hence ∂f /∂t ̸ = 0. Our method's sensitivity to the residual ∂f /∂t and freedom from enforced symmetry allows us to perform self-consistency checks in a manner that has not typically been possible for previous measures of acceleration and mass density built on the Boltzmann Equation or its moments. We consider two such checks here. First, in the absence of measurement errors or disequilibrium and assuming maximally expressive MAFs, the northern (z > 0) and southern (z < 0) measurements of a r should be equal, and the a θ values should be equal up to a negative sign. If the system respects axisymmetry, a ϕ should be zero. In the subpanels of Figure 8, we show the deviations from North-South symmetry for a r and a θ over the magnitude of the acceleration, as well as a ϕ /|⃗ a|. As can be seen, these measures of disequilibrium and/or departure from axisymmetry are at most 10%, suggesting that the system is in equilibrium at least to this level. Note that several of the largest deviations occur for s ∼ 0 and at the edge of the observational volume. The former location is in the disk where residual dust extinction may be a concern, and the latter is where measurement errors are large. In these regions then, these measures may , and a ϕ (bottom row) as a function of spherical radius r (left column), polar arclength s = r⊙ × (π/2 − θ) (middle column), and azimuthal arclength (r 2 ⊙ − z 2 ) 1/2 × ϕ (right column). Radial and azimuthal measurements are taken along lines off-set from the Galactic midplane, at z = 1.5 kpc (yellow lines, labeled "North") and z = −1.5 kpc (purple lines, labeled "South"). See Figure 1 for the orientation of these measurement lines within the Galaxy. The inner dark bands and outer light bands denote 1σ and 2σ uncertainties respectively. Subpanels in the top and middle rows show the fractional difference between the North and South measurements (appropriately mirrored for a θ ). Subpanels in the bottom row show the ratio of a ϕ to the magnitude of ⃗ a. overestimate the amount of disequilibrium in the tracer stars. A second test of equilibrium can be obtained if a sepa-rate measure of the acceleration at specific point is available. In such circumstances, we can directly calculate ∂f /∂t, or the equivalent ∂ ln f /∂t (which can be inter- preted as an inverse timescale). The acceleration measured at the Solar location by referencing distant quasars [56] is such an independent measurement. At the Solar location, we sample 10 6 velocities from the MAF. For each velocity we solve Eq. (1) for ∂ ln f /∂t assuming the acceleration −∇Φ is given by the acceleration found by Ref. [56] (Table I, left column). The resulting distribution of ∂ ln f /∂t is shown in Figure 9. As can be seen, this quantity is roughly peaked at zero (suggesting equilibrium), but it is unclear whether the spread in the distribution is due to non-equilibrium structures, or is equivalent to the expected residual one would obtain given the over-constrained system of equations and imperfections in the MAF modelling of the data. To probe this, we also show in Figure 9 the residual ∂ ln f /∂t which results when the averaged MAF-derived acceleration (Table I, right column) at the Solar location is used as −∇Φ over the sample of 10 6 velocities. The mean of this distribution is consistent with zero, indicating that the dark matter accelerations obtained from minimizing Eq. (8) have themselves no preference for non-equilibrium. Furthermore, the distributions of ∂ ln f /∂t assuming either acceleration are very similar. The quasar-derived acceleration results in a ∂ ln f /∂t distribution which is slightly broader, with some trace of multi-modality. However, there is no statistically significant difference between the two. We therefore conclude that the MAF results are consistent with local equilibrium given the current data, as might be expected given the similarity of the accelerations obtained from our method and Ref. [56]. To summarize, under these closure tests, we find that our assumption of equilibrium in the analysis of the Boltzmann Equation is good to within the ∼ 10% level, as quantified by deviations from axisymmetry and northsouth reflection symmetry in the measured accelerations. Using an independently determined acceleration at the Solar location [56], we also find closure (self-consistency) of our equilibrium assumption to within the level of precision of the MAF density estimation. In the future, larger datasets, better control of errors, and more expressive normalizing flow architectures may allow tighter distributions of ∂ ln f /∂t, allowing statistically significant measures of equilibrium and axisymmetry. D. Mass Densities Given an acceleration calculated at every point in the volume around the Sun, the total mass density ρ can be calculated using the Poisson Equation Eq. (2). This requires taking an additional numeric derivative of Φ. Again following the algorithm of Ref. [22], we calculate the second derivative of Φ at position ⃗ x by convolving the accelerations over a truncated Gaussian kernel K centered at ⃗ x: 4πG ρ * K = (∇ 2 Φ) * K.(9) Here * indicates convolution with the kernel. The Gaussian kernel is truncated at |⃗ x/ ⃗ h| = 2 where ⃗ h = (0.5, 0.5, 0.2) kpc. This ellipsoidal kernel averages the mass density at scales below ⃗ h, and thus we are not sensitive to density fluctuations at scales smaller than this. We draw 3,200 points to calculate the mass density at each ⃗ x, again using quasi-Monte Carlo sampling. With the total mass density calculated at a point, we can then extract the dark matter mass density using a model for the baryonic components of the Milky Way. The details of our baryonic model are discussed in Appendix A. Briefly, we model the baryonic components of the Galaxy at the Solar cylindrical radius with 15 components (ten stellar and five gas components) as per Ref. [57], with refinements from Refs. [58,59]. Each component has an exponential or Gaussian suppression as a function of height |z| off of the disk -note that all these baryonic components assume axial symmetry. The parameters of each model can be found in Table A. Baryons dominate the mass density within |z| ∼ 0.5 kpc of the disk. In Table II, we report the total mass density, the baryonic density, and the inferred dark matter density calculated at the Solar location, using a single averaging kernel centered at the Sun. In Figure 10, we show the total mass density ρ, the modeled baryonic mass, and the dark matter mass density as a function of arc length s above and below the disk from the Solar location at ϕ = 0. Due to the finite kernel size, we cannot calculate densities at the edge of the 4 kpc sphere centered on the Sun, and (as errors increase at larger distances) we instead show densities only up to s = 3 kpc. 1.18 ± 0.14 0.47 ± 0.05 1. 38 TABLE II: MAF-estimated densities at the Solar location or averaged at the Solar radius r⊙. The dark matter density is the difference between the total mass density ρ⊙ and the baryonic mass density (obtained from the model described in Appendix A). The averaged mass density at r = r⊙ is the weighted average of the dark matter mass density evaluated at 15 independent points at r = r⊙ (as in Figure 10) and subtracting the baryonic mass density in that region. The curve in Figure 10 is created from mass densities sampled with kernel centers more densely packed than the kernel size. As a result, neighboring density values are correlated. This smooths the resulting curve as a function of s, and statistical fluctuations in individual uncorrelated measurements appear as extended bumps. This effect is also the likely source of the slight off-set in the peak in the total mass density from s = 0. The mass densities of 15 points with non-overlapping kernels are indicated in Figure 10. Considering these independent measures 2 of the total mass density, the dark matter density at r = r ⊙ is statistically consistent with a constant value, as expected for a spherically symmetric dark matter profile. Assuming this symmetry, we can use these measurements of the total mass density at independent locations within the dataset to obtain an averaged dark matter density at fixed radius r after subtracting the baryonic density at each location (note that this is an imposition of a symmetry assumption that has not been used in this analysis otherwise). The resulting averaged dark matter mass density is reported in Table II. In Figure 11, we compare our result for ρ DM (r = r ⊙ ) with a set of recent measurements (made using a variety of methods). As can be seen, our result is consistent with the previous literature, with competitive and comprehensive error bars. Our errors contain both statistical and systematic errors in the calculation of the total (dark matter plus baryonic) mass density. If we again assume spherical symmetry, we can further investigate the dependence of the dark matter mass density on spherical radius r. We show in Figure 12 the dark matter mass densities evaluated at various Galactocentric radii r, along z = ±1.5 kpc off-sets from the disk (and at different values of ϕ). Considering independent measurements over a range of r values within the 4 kpc observational sphere, we fit the measured ρ DM as a function of r to the generalized NFW profile ρ DM (r) = ρ 0 r rs β 1 + r rs 3−β ,(10) with free parameters ρ 0 , r s , and β (in the standard NFW, β = 1). In performing our fit, we adopted a truncated Gaussian prior for β centered at 2 with a width of 2. We adopt This is broadly in agreement with other recent fits to the dark matter density profile (e.g., Refs. [59,86]). We plot our best-fit profile in Figure 12. We note that -given the range of complete data available from Gaia, measurement errors, and dust extinction -our dataset does not extend to the low-r regime, and so does not yet provide significant discriminating power between different models of dark matter density profiles. IV. DISCUSSION Using normalizing flows to model the phase space density of bright stars within Gaia DR3, we have -for the first time -measured the gravitational acceleration and mass density within the local volume of the Milky Way without assumption of functional form or symmetry. The resulting acceleration and mass density maps across a three dimensional volume around the Sun provide a unique window into Galactic structure and dynamics. We find the acceleration at the Sun's location to be nearly entirely radial, with a r = (1.94 ± 0.22) × 10 −10 m/s 2 . This is within 2σ of recent measurements of the acceleration using quasars [56]. If the differences between the two measurements were taken at face value, this would suggest ∼ 10% disequilibrium. Similar deviations from the assumptions of equilibrium and axisymmetry are found when comparing the measurements in the Galactic North and South. Though at this time the statistical and systematic errors on our MAF-derived accelerations are too large to draw confident conclusions about departures from equilibrium, our method allows us to test assumptions of equilibrium and symmetry in a way that has not been previously possible. Without assuming any symmetries, our measurement of dark matter at the Solar location is ρ DM,⊙ = 0.32 ± 0.18 GeV/cm 3 . This large uncertainty can be significantly reduced with the further imposition of spherical symmetry -allowing the averaging of measurements at different locations. Under this assumption, we find a dark matter density at the Solar radius of 0.47 ± 0.05 GeV/cm 3 , in agreement with previous measurements using a variety of other techniques. Recall that our density measurements encompass a wide range of possible dark matter density profiles due to the expressivity of the MAF; this variation is fully incorporated into our error budget. The dark matter density profile is consistent with a generalized NFW. We find a preference for a small scale radius r s , in keeping with recent measurements from rotation curves [59]. However, our current range of reliable data does not yet extend deep into the central region of the Galaxy, and so our statistical preference for this value of r s is not high. These machine-learning assisted measurements of Galactic acceleration and mass density are expected to improve significantly in the near future. Gaia DR4 and DR5 are expected to expand the number of stars with full six-dimensional kinematics by a factor of five, with an associated decrease in statistical errors in our determination of the phase space density. Measurement errors are likewise expected to improve by a factor of two, and improvements in our analysis technique to correct for the bias introduced by these errors are possible. Dust extinction is a major limitation in applying our algorithm to regions close to the disk or toward the Galactic center. An improved understanding of the effects of dust on the measured Gaia stellar features which are uniform across the sky will allow greater accuracy in our measurements of phase space, acceleration, and mass density. Indeed, normalizing flows may play a role in data-driven modelling of dust extinction, which we will Overall, we can expect improvements in architecture, analysis, data quantity, and data quality to allow great advances over these first results. In addition to greatly improved precision and better constraints on the density profile, future analyses based on these techniques may be able to directly probe the departures from equilibrium within the Milky Way, especially when combined with other measurements of local acceleration, such as those based on quasars [56], pulsars [87], or binary systems [88]. Orange points were evaluated at z = +1.5 kpc, purple points were evaluated at z = −1.5 kpc. The average dark matter density at the Solar radius r⊙ is shown as a red star. The maximum likelihood fit to a generalized NFW profile is shown in a solid black line, with 1σ and 2σ variance across the posterior distribution of models explored in Figure 13 shown as green and yellow error bands, respectively. A recent fit [86] to a standard NFW profile to the rotation curve of the Milky Way is shown as a dashed black line. Table A. The McKee model draws on a collection of pre-Gaia star counts and gas surveys. Recent updates using Gaia data [59,92] have not substantially altered the model. The stellar bulge and halo do not significantly contribute to the mass density in the Solar neighborhood, and so they are not independently modelled. However, this model does include halo stars within the disk. The McKee model characterizes the surface densities Σ 0,i and effective scale heights h z,i , as well as the functional form for the number density as a function of z, for each component. Assuming direct proportionality between number and mass density -i.e., assuming no chemical evolution of each component as a function of z -we can model each mass density in the same way as the number density. All but 12 components were fit to an exponential mass density profile ρ exp,i (⃗ x) = Σ 0,i h i e −|z|/hi e −(R−r⊙)/h R .(A1) The remaining three components (H 2 , HI CNM , and HI WNM,1 ) were fit to the following Gaussian mass density profile ρ gauss,i (⃗ x) = Σ 0,i √ πh i e −|z| 2 /h 2 i e −(R−r⊙)/h R . We supplement each component of the McKee model with a radial scale length h R,i , informed by the baryonic model used in Ref. [59]. Stellar populations were assigned a scale radius of 2.35 kpc, all HI gasses were assigned a large scale radius of 18.24 kpc, and H 2 gas was assigned h R = 2.57 kpc. HII gas was assigned an arbitrary scale radius of 2.5 kpc. Fue to its overall small contribution to the surface density, uncertainty in this scale length does not have a significant effect on the mass density. It should also be emphasized that the precise details of the baryonic radial profile are insignificant for |z| > 500 pc and for all R within our observational window. Values and uncertainties (when available) for Σ 0,i , h z,i , h R,i , and the corresponding ρ 0,i ≡ ρ i (z = 0) for all 15 components are given in Table A. We follow Ref. [58] in assigning 10% uncertainties to any unreported surface density errors in the McKee model. In total, we expect approximately 8% uncertainty in the baryonic mass density at the Solar location ρ b,⊙ . We do not follow Ref. [57] in inflating this uncertainty to 15%, although we agree that the systematic uncertainties in the original error model are likely underestimated. As discussed in Ref. [85], de-projecting the McKee model out of the plane into a volume density ρ b (z) comes with systematic uncertainties. Based on comparisons to the MWPotential2014 Milky Way mass model implemented in the galpy library, our de-projection of the McKee model does not deviate significantly from other standard baryonic mass distributions. Additionally, these systematic uncertainties become subdominant to our other measurement and statistical uncertainties of ρ(z) for |z| > 500 pc, where ρ b is greatly sub-dominant to ρ DM . Only the estimate for the dark matter density at the Solar location ρ DM,⊙ is significantly affected by our choice and interpretation of the McKee model, as the Solar System is located near the midplane. Finally, when evaluating the baryonic mass density at a particular point, we convolve our estimate for ρ b (⃗ x) over the same quasi-random Gaussian kernel used to estimate ρ(⃗ x). The convolved baryonic mass density profile ρ b (⃗ x) * K is comparable to ρ b (⃗ x) everywhere except for the disk, where the peak at z = 0 is widened and shortened due to this convolution. As a result, if the vertical profile falls off too quickly with z, ρ b (z = 0) * K will be underestimated. This introduces systematic uncertainty in the estimate of ρ b * K for |z| ≲ 500 pc in the disk, compared to |z| ≳ 500 pc in the more robust halo region. [57], some values of hz,i are a weighted average of two scale heights, representing an "effective" scale height. ρ0,i = ρ(z = 0)i, where ρ(z = 0)i is the volume mass density of each component in the midplane. ρ(z = 0)i is computed from Σ0,i and hz,i via ρ(z)i = (1/2)∂Σ(z)i/∂z and by assuming a form for Σ(z). For an exponential mass profile, Σ(z)i = Σ0,i(1 − exp(−|z|/hz,i)). hR,i is the exponential scale radius of each component, capturing the first-order radial behavior of baryonic mass density (ignoring detailed features such as spiral arms or clouds). FIG. 1 : 1Schematic representation of the Solar location (red dot) relative to the Galactic Center (black dot). The 4 kpc observation volume is shown as a transparent grey sphere. FIG. 2 : 2Density plots of the stars with full 6-dimensional kinematic information available from Gaia within 4 kpc of the Solar location in the x − y (top row) and x − z (bottom row) planes. The left column shows all 24,789,061 fully-characterized stars. The middle column shows the 5,811,956 remaining stars after applying the selection criteria described in the text. The right column applies the additional requirement of |z| > 1 kpc, resulting in 470,702 stars. FIG. 3 : 3Absolute magnitude MG and color BP − RP of all 24,789,061 stars within 4 kpc of the Sun with 6-dimensional kinematic information measured by Gaia. The horizontal dashed white line denotes the magnitude completeness criteria Eq. (4). All 5,811,956 stars above the dashed white line are bright enough to be observable for Gaia regardless of position in the 4 kpc sphere. The visible peak in the white box is the red clump. All features in the space of uncorrected MG and BP − RP will appear to be smeared towards the bottomright of this figure, as dust extinction both dims and reddens stars. FIG. 4 : 4Stellar number density from Gaia data (left) and MAF-learned number density (right) n(⃗ r) on Galactic longitude (ℓ) and latitude (b) planes at distance (top) 1 kpc, (center) 2 kpc, and (bottom) 3 kpc. The stellar number densities are obtained by directly counting the number of stars within 0.1 kpc from the center of the pixel. The MAF-learned number densities are described in Section III. The Ks-band extinction maps at a given distance (obtained from the dustmaps package[33]) are shown as contours. The maps are Gaussian-kernel smoothed with a bandwidth of 8 • . We show four extinction value contours from white to blue: 0.15, 0.3, 0.45, and 0.60. FIG. 5 : 5Stellar number density from Gaia data (left) and MAF-learned number density (right) as a function of s and r⊙ · ϕ, at r = r⊙ = 8.122 kpc. The stellar number densities are obtained by directly counting the number of stars within 0.1 kpc from the center of the pixel. The left-hand plot contains 206,852 stars. The MAF-learned number densities are described in Section III. We overlay the contour plot of three dimensional Ks band extinction map obtained from the dustmaps package[33]. The map is Gaussian-smoothed with a bandwidth 0.2 kpc. The dashed black contour is an extinction of 0.05, and the four contours from white to blue are extinction values of 0.15, 0.3, 0.45, and 0.60. FIG. 6 : 6Left and Center: 1σ standard deviations (as reported by Gaia DR3) of the measured kinematic parameters for stars passing the distance, magnitude, and parallax error selections of Section II A. Errors are shown for parallax ϖ (left plot, lower axis), radial velocity v rad (left plot, upper axis), angular ICRS position (α, δ) (center plot, lower axis), and ICRS proper motion (µα, µ δ ) (center plot, upper axis). We discard stars with relative parallax errors larger than 1/3. Stars with radial velocity uncertainties larger than 40 km/s are not included in Gaia DR3[27].Right: Median standard deviation in Cartesian position (blue line, left vertical axis) and velocity (yellow line, right vertical axis) as a function of stellar distance d. The 16 th and 84 th percentiles of these standard deviations are shown as asymmetric bands around the central median value. while the bump in the y histogram is due to dust clouds closer to the Galactic center. These variations within the learned probability densities are not mismeasurement of the data; rather the MAFs are correctly capturing the dust-filled substructures of the Milky Way. The right-hand column of Figure 4 shows the estimated number density n(⃗ r) on the Galactic longitude and latitude (ℓ, b) planes at different distances d from the Sun, again with the dust map overlaid. Comparing to the observed binned number densi-ties (left column of FIG. 7 : 7Normalized histograms of (top) position components and (bottom) velocity components for selected stars in Gaia DR3 (downsampled to 20% of the original size). The red lines are the histograms for synthetic stars sampled from the normalizing flows. The error bars are the 1σ statistical uncertainty. Below the main plots, we show the pull distributions, (i.e., the difference between Gaia and MAF histograms divided by the 1σ statistical uncertainty). FIG. 8 : 8Accelerations ar (top row), a θ (middle row) FIG. 9 : 9Distribution of ∂ ln f /∂t at the Solar location using the MAF-derived Solar acceleration a⊙ (blue) and the Gaia EDR3 quasar-derived measurement of a⊙ (orange). a flat prior for r s within [0, 20], and a truncated Gaussian prior centered at 40 × 10 −2 M ⊙ /pc 3 with a width of 80 × 10 −2 M ⊙ /pc 3 for ρ 0 in the range [0, 200] × 10 −2 M ⊙ /pc 3 . This choice of priors restricts the model space to a physically realistic domain: the Gaussian prior for ρ 0 prevents arbitrarily large density parameters, leading to extremely small values of the scale radius r s . We show the posterior distribution for ρ 0 , r s , and β in Figure 13 with median values and 16-th and 84-th percentile uncertainties of ρ 0 = 30.1 +64.4 −25.1 × 10 −2 M ⊙ /pc 3 , r s = 3.5 +5.4 −1.4 kpc, and β = 1.0 +1.2 −0.7 . The best fit model is ρ 0 = 23.5 × 10 −2 M ⊙ /pc 3 , r s = 3.6 kpc, and β = 1.1. FIG. 10 : 10Top: Independent measurements of total mass density as a function of polar arclength s, for three values of ϕ: ϕ = 0 (black circles), ϕ = +0.209 (blue triangles), and ϕ = −0.209 (red squares). Points at nonzero ϕ have been offset for visibility. Horizontal error bars indicate the 1σ width of the kernel in the s-direction. The mass density at ϕ = 0 for s values with overlapping kernels is shown in the black curve (note that due to kernel overlap these measurements are correlated at length scales of ∼ 0.4 kpc). The baryonic mass density model is shown with the blue curve. Dark and light bands represent 1σ and 2σ uncertainties, respectively. Bottom: Independent measurements of dark matter mass density as a function of s and ϕ, obtained by subtracting the baryonic density in the top panel from the corresponding total mass density. The best-fit value of ρDM(r = r⊙) = 0.47 ± 0.05 GeV/cm 3 is shown as a horizontal black line enclosed by green and yellow bands denoting the 1σ and 2σ uncertainties, respectively. FIG. 12 : 12Dark matter mass densities measured at 12 independent points as a function of Galactic spherical radius r. for a fit to a generalized NFW profile Eq. (10). The median values of each parameter are shown above the marginalized 1D histograms, with the 16 th and 84 th percentile values shown as error bounds. The maximum likelihood model is shown in black. The McKee mass model is broken into 15 components: five types of gas, seven stellar populations, and three populations of compact objects. Each component is described in TABLE I : IGalactic acceleration at the Solar location ⃗ a⊙ in Cartesian coordinates, calculated by averaging the solution to the Boltzmann equation within a 100 pc sphere centered on the Sun. We list for comparison the acceleration at the Solar location obtained from Gaia DR3 quasar measurements[56]. TABLE III : IIIAll parameters of the baryonic mass model used in this work, as well as their respective references. Σ0,i is the surface density of each baryonic component at the Solar radius R = r⊙ = 8.122 kpc. hz,i is the scale height, indicating how far above and below the midplane each component extends. As in Ref. . The cylindrical coordinate system (R, ϕ, z) uses the same z axis as the Galactocentric Cartesian coordinates. The Sun is located at ϕ = 0, with positive ϕ increasing towards the +y axis.3. The spherical coordinate system (r, θ, ϕ) has the same ϕ angle as the cylindrical coordinates, and measures +θ relative to +z axis, with the Galactic disk at θ = π/2. We define the polar arc length above or below the Galactic disk at the Solar radius (along the θ direction) as s ≡ r ⊙ × (π/2 − θ), with r ⊙ = 8.122 kpc. Recall that our data has a small void at the Solar location with radius ∼ 50 pc, due to the Gaia lower limit on the apparent magnitude. Although our normalizing flows have smoothed out this region, we apply this averaging in order to suppress any artifacts from the interpolation. As MAFs are highly expressive models with a very large number of parameters, it is reasonable to assume that points in space separated at sufficiently large length scales are described by unique model parameters and are not correlated. For all practical purposes, if the averaging kernels of two mass density estimates do not overlap, the two estimates are independent measurements. AcknowledgementsWe thank Adrian Price-Whelan and Mitchell Weikert for helpful discussions. This work was supported by the DOE under Award Number DOE-SC0010008. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos. esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www. cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. We thank the Gaia Project Scientist Support Team and DPAC for their work in development and maintenance of the PyGaia code. The authors acknowledge the Office of Advanced Research Computing (OARC) at Rutgers, The State Uni-versity of New Jersey for providing access to the Amarel cluster and associated research computing resources that have contributed to the results reported here. URL: https://oarc.rutgers.eduAppendix A: Baryon Mass ModelIn order to estimate the local density field of dark matter ρ DM (⃗ x) given the total density field ρ(⃗ x), we must estimate the local distribution of baryonic mass density ρ b (⃗ x) in the Milky Way. We follow Refs.[15,58,79,83,87,[89][90][91]and base our model for ρ b (⃗ x) in the Solar neighborhood on the work of Ref.[57](hereafter referred to as the McKee model), an extensive compilation of estimates of the surface mass densities of gas, stars, and compact objects in the Solar neighborhood. . Casagrande, 60Casagrande, (2020) [60] . Pato, Pato, et al., (2015) . Huang, Huang, et al., (2016) . Benito, Benito, et al., (2019) . Karukes, Karukes, et al., (2019) . Lin, 65Lin, et al., (2019) [65] . De Salas, de Salas, et al., (2019) . Ablimit, Ablimit, et al., (2020) . Benito, 68Benito, et al., (2020) [68] . Sofue, 69Sofue, (2020) [69] . Zhou, Zhou, et al., (2022) . Ou, 59Ou, et al., (2023) [59] . 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A Widmark, P F De Salas, G Monari, 2011.02490Astronomy & Astrophysics. 646A. Widmark, P. F. de Salas, and G. Monari, Astronomy & Astrophysics 646, A67 (2021), 2011.02490. . E D Kramer, L Randall, 1604.01407Astrophys. J. 824E. D. Kramer and L. Randall, Astrophys. J. 824, 116 (2016), 1604.01407. . J Bovy, 1704.05063Monthly Notices of the Royal Astronomical Society. 4701360J. Bovy, Monthly Notices of the Royal Astronomical So- ciety 470, 1360 (2017), 1704.05063.

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{'abstract': 'We present a novel, data-driven analysis of Galactic dynamics, using unsupervised machine learning -in the form of density estimation with normalizing flows -to learn the underlying phase space distribution of 6 million nearby stars from the Gaia DR3 catalog. Solving the collisionless Boltzmann equation with the assumption of approximate equilibrium, we calculate -for the first time ever -a model-free, unbinned, fully 3D map of the local acceleration and mass density fields within a 3 kpc sphere around the Sun. As our approach makes no assumptions about symmetries, we can test for signs of disequilibrium in our results. We find our results are consistent with equilibrium at the 10% level, limited by the current precision of the normalizing flows. After subtracting the known contribution of stars and gas from the calculated mass density, we find clear evidence for dark matter throughout the analyzed volume. Assuming spherical symmetry and averaging mass density measurements, we find a local dark matter density of 0.47 ± 0.05 GeV/cm 3 . We fit our results to a generalized NFW, and find a profile broadly consistent with other recent analyses.', 'arxivid': '2305.13358', 'author': ['Sung Hak Lim \nDept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA\n', 'Eric Putney \nDept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA\n', 'Matthew R Buckley \nDept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA\n', 'David Shih \nDept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA\n'], 'authoraffiliation': ['Dept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA', 'Dept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA', 'Dept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA', 'Dept. of Physics and Astronomy\nNHETC\n08854Rutgers, PiscatawayNJUSA'], 'corpusid': 258840880, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 27849, 'n_tokens_neox': 23280, 'n_words': 13504, 'pdfsha': 'b11f23f7f8490349ca00c7bcf44899f6b345b045', 'pdfurls': ['https://export.arxiv.org/pdf/2305.13358v1.pdf'], 'title': ['Mapping Dark Matter in the Milky Way using Normalizing Flows and Gaia DR3', 'Mapping Dark Matter in the Milky Way using Normalizing Flows and Gaia DR3'], 'venue': []}

arxiv

Superconductor-Insulator Transition in Random Two-Dimensional System 2 Apr 2001 Masahiko Kasuga Department of Physics Waseda University 3-4-1 Okubo, Shinjuku-ku169-8555TokyoJapan Susumu Kurihara Department of Physics Waseda University 3-4-1 Okubo, Shinjuku-ku169-8555TokyoJapan Superconductor-Insulator Transition in Random Two-Dimensional System 2 Apr 2001 Effect of disorder in metallic thin film is examined as a possible mechanism of the Superconductor-Insulator (S-I) transition. The critical value of disorder corresponding to the transition point is found analytically by using Matsubara-Matsuda model and Green's function method. INTRODUCTION At zero temperature two-dimensional systems of interacting electrons are speculated to show a quantum phase transition between superconducting and insulating (S-I) phases. S-I transition can be driven by tuning some parameters such as the disorder 1-3 , carrier concentration 4,5 , magnetic field 6,7 , dissipation 8,9 , and so on. It has been observed in granular and amorphous films such as Bi, Pb, Sn 1,2 , Josephson junction arrays [8][9][10] , and 4 He absorbed on porous media 11 , in two dimensions. Here especially we are interested in superconducting films. Superconducting films are made by repeated small increments of materials onto substrates held at low temperatures in an ultra-high vacuum. Then the system is found to form superconducting islands separated by thin insulating regions. The films are strongly disordered due to distribution of the island size and the coupling between islands. We assume, in this paper, that the randomness in island potential is crucial in determination whether the film is superconducting or insulating. Natural questions arises : how much disorder is necessary to destroy the superconducting properties? Precise answer to this question will depend on details of the interaction among the particles, as well as the nature of the disorder in the system. Destruction of superconductivity with disorder can be caused by localization effect rather than Cooper pair braking. Then the system becomes insulator on the macroscopic scale in spite of local superconducting correlations. Relying upon a universality hypothesis, we regard Cooper pair as a tightly bound hard-core boson as long as we restrict ourselves to the vicinity of S-I transition point, disregarding any microscopic detail such as electrons, phonons, and their interactions. After all our problem becomes equivalent to dirty boson problem 3,12,13 , which has been extensively studied using quantum Monte Carlo simulations 1,2 , real-space renormalization-group techniques 14,15 , strong-coupling expansions 16 , and other ways. In this paper we shall adopt Matsubara-Matsuda model [17][18][19] which was established for tightly bound hard-core bosons, and calculate the critical value of disorder which corresponds to the transition point with double time Green's function 20-22 . MODEL Inspired by the work of Matsubara and Matsuda (M-M) who formulated the model of hard-core boson in lattice gas [17][18][19] , we introduce Bose-Hubbard type Hamiltonian : H = U i,jn inj − t i (â † iâi+1 +â iâ † i+1 ) − i (h + δh i )n i ,(1) whereâ † i ,â i , andn i are the creation, annihilation, and number operators of Cooper pairs at the i-th lattice point, respectively. We exclude boson double occupancy by assuming infinite repulsive on-site interaction. We have introduced a finite repulsive interaction U, for two nearest-neighbor bosons. In addition, we restrict hopping t to nearest-neighbor only. In this model we introduce random site potential δh i which represents disorder due to distribution of island size. It takes both positive and negative value. The external potential is described as the sum of uniform part h and random part δh i . The boson occupancy is restricted to one or zero. In other words each site has only two states, empty or full. The hardcore boson system is therefore isomorphic to a spin 1/2 system. Each operator of Cooper pairs can be replaced as spin operator 23 : a † i ≡Ŝ − i ≡ 1 2 (σ x i − iσ y i ), @â i ≡Ŝ + i ≡ 1 2 (σ x i + iσ y i ),(2)n i ≡ 1 2 −Ŝ z i ≡ 1 2 (1 −σ z i ),(3) whereσ x i ,σ y i , andσ z i are Pauli matrices for the 1/2 spin at the i-th lattice point, respectively. From (1), (2), and (3), the Hamiltonian becomes H = −J z iŜ z iŜ z i+1 − J xy 2 i (Ŝ + iŜ − i+1 +Ŝ − iŜ + i+1 ) − i h iŜ z i ,(4)J z ≡ −4U, @J xy ≡ 2t,(5)h i ≡ h + δh i .(6) As a result, interaction energy and kinetic energy are now represented by spin exchange energy, and chemical potential is represented by Zeeman energy. Our system can be described with an anisotropic ferromagnetic Heisenberg Hamiltonian. Therefore we can interpret ordering of pseudo-spins in x-y plane as superconductivity and z direction as local density fluctuation 17 . â † , â ⇔ Ŝ − , Ŝ + ,(7) n ⇔ Ŝ z . Since Ŝ + and Ŝ − represent superconducting order, their value can be regarded as a criterion to judge whether the system is, superconducting or insulating. So we shall calculate them in the next section. CALCULATION The problem now involves statistical physics of the pseudo-spin system described by the ferromagnetic Heisenberg model. The temperature dependent retarded Green's function [20][21][22] with two operatorsŜ + i ,Ŝ − j and a coefficient h i , is introduced as h iŜ + i (t);Ŝ − j (0) ≡ −iθ(t) [h iŜ + i (t),Ŝ − j (0)] .(9)S + i (t) is a Heisenberg operator at time t ; θ(t) is the step function ; square brackets [· · ·] denote commutators, and single angular brackets < · · · > denote thermal averages. A straightforward calculation yields the time-Fourier transformed equation of motion for the Green's function (9) ω h iŜ + i ;Ŝ − j = [h iŜ + i ,Ŝ − j ] + [h iŜ + i , H];Ŝ − j .(10) In our two-dimensional square lattice model, we only consider coupling between nearestneighbor sites (see Fig.1), i.e. site i with site i + a n (n = 1 ≃ 4). ω h iŜ + i ;Ŝ − j = [h iŜ + i ,Ŝ − j ] + h 2 iŜ + i ;Ŝ − j + a (J z h iŜ + iŜ z i+a ;Ŝ − j − J xy h iŜ z iŜ + i+a ;Ŝ − j ).(11) We can solve the equation of motion (11) easily by recovering translation invariance by impurity averaging procedure, and this can be done by averaging the magnetic field as follows h i = h + δh i = h,(12)h 2 i = h 2 + 2hδh i + δh 2 i = h 2 + δh 2 .(13) Since δh i takes both positive and negative values, the first order average equals to zero. Then the equation with translation invariance and randomness contribution is obtained as ωh Ŝ + i ;Ŝ − j = h [Ŝ + i ,Ŝ − j ] + (h 2 + δh 2 ) Ŝ + i ;Ŝ − j + a J z h Ŝ + iŜ z i+a ;Ŝ − j − J xy h Ŝ z iŜ + i+a ;Ŝ − j .(14) For simplicity, we decouple higher order Green's function in the following manner Ŝ + iŜ z i+a ;Ŝ z j → Ŝ z Ŝ + i ;Ŝ z j ,(15)Ŝ z iŜ + i+a ;Ŝ z j → Ŝ z Ŝ + i+a ;Ŝ z j .(16) Then (14) becomes ω Ŝ + i ;Ŝ − j = 2 Ŝ z + h 2 + δh 2 h Ŝ + i ;Ŝ − j + a J z Ŝ z Ŝ + i ;Ŝ − j − J xy Ŝ z Ŝ i+a ;Ŝ − j .(17) Fourier transformation (17) with respect to space results ω Ŝ + k ;Ŝ − −k = 2 Ŝ z + h 2 + δh 2 h Ŝ + k ;Ŝ − −k +4 Ŝ z J z − J xy 4 a e ika Ŝ + k ;Ŝ − −k .(18) Putting them in order, we obtain the formula for the Green's function Ŝ + k ;Ŝ − −k G(k, ω) ≡ Ŝ + k ;Ŝ − −k = 2 Ŝ z ω − 4 Ŝz ω k − H ,(19) where ω k ≡ J z − J xy 4 a e ika ,(20)H ≡ h 2 + δh 2 h .(21) Since the poles of Green's function indicates the excitations of the ferromagnetic spin wave, we see that two-body interaction J z , external field h, and ∆h assist crystallization and hopping integral J xy causes quantum fluctuation. Next we treat the correlation function Ŝ − jŜ + i . It can be calculated by using the spectral theorem Ŝ −Ŝ+ = 1 4π 2 π −π dk x π −π dk y i 2π ∞ −∞ dω {G R k (ω) − G A k (ω)} 1 e ω T − 1 ,(22) where G R k (ω), G A k (ω) are retarded and advanced Green's functions. From equation (19) we obtain G R k (ω) − G A k (ω) = −2πiδ(ω − 4 Ŝ z ω k − H),(23) then the integration of the right-hand side of (22) with respect to ω becomes 1 2 − Ŝ z = 1 4π 2 π −π dk x π −π dk y 2 Ŝ z exp[(4 Ŝz ω k + H)/(2T )] − 1 .(24) Putting them in order, we obtain the relation 1 = Ŝ z π π 0 kdkcoth 4 Ŝ z ω k + H 2T .(25) Since low temperature region is of our main interest, we neglect large spin wave vector. This approximation gives ω k as ω k ≃ 1 4 J xy k 2 + J z − J xy .(26) Therefore the excitation energy gives ω = Ŝ z J xy k 2 + 4 Ŝ z (J z − J xy ) + H.(27) The second and third term on the right-hand side means the gap of the spin wave excitation ∆ at k ≃ 0, which is caused by anisotropy and external field. Here two conditions must be fullfilled in order that the ground state is superconducting. The first one is that ∆ is sufficiently small compared to T , i.e. ∆ = 4 Ŝ z (J z − J xy ) + H ≪ T,(28) the second one is J xy > J z .(29) Using the above conditions, the integration on the right-hand side of (25) can be calculated analytically, and so we obtain the expression at low temperature Ŝ z = H(e πJxy/T − 1) π 2 J xy − 4(J z − J xy )(e πJxy/T − 1) ≃ H 4(J z − J xy ) .(30) Due to rotation symetry of spin in x-y plane, we can select the x-axis as the grand state direction without lost of generality. Then we obtain the relation between the superconducting order parameter Ŝ x and randomness ∆h Ŝ x = 1 2 2 − Ŝz 2 ≃ 1 2 1 − H 2 4(J z − J xy ) 2 . (31) RESULT Before examining the physical meanings of (31), we introduce some normalized factor η ≡ ∆h J xy , @κ ≡ J z J xy , @ξ ≡ h J xy .(32) Which corresponds to normalized strength of randomness, two body interaction, and chemical potential, respectively. Then (31) becomes Ŝ x ≃ 1 2 1 − (ξ 2 + η 2 ) 2 4ξ 2 (1 − κ) 2 .(33) This equation describes the behavior of the superconducting order parameter Ŝ x with respect to the strength of randomness η (see Fig.2). From Fig.2 we can see that Ŝ x is damped by increasing η. By expanding (33) in power of η Ŝ x ≃ 1 2 1 − ξ 2 4(1 − κ) 2 1 − η 2 4(1 − κ) 2 − ξ 2 ,(34) we can see the damping of Ŝ x occurs proportional to η 2 . At a specific point, Ŝ x completely disappears, which implies the existence of a phase transition. As mentioned in sec .2, we interpret phases Ŝ x = 0 as superconducting and Ŝ x = 0 as insulating. Here the central result is the existence of the critical value η c for the strength of randomness. η c divides the two phases (S and I) and gives the transition point. From (33) η c is given by η c = 2ξ(1 − κ) − ξ 2 .(35) From Fig.2 we can see that the strength of two-body interaction κ advances the transition. Therefore κ can be also regarded as a parameter of S-I transition. (see also Fig.3). Varying κ instead of η (see Fig.3) also gives a phase transition, this time it corresponds to a Mott transition. Especially when η = 0, Ŝ x sharply decreases at κ ≃ 1. This corresponds to a transition in an 2D isotropic system i.e. η = 0, κ = 1. This is based on the general fact that Ŝ x doesn't exist in two-dimensional system. Now we vary both η and κ as a parameter of the phase transition, the border line between two phases (S and I) is determined by the condition that superconductivity disappears (i.e. Ŝ x = 0). From (33) this relation corresponds to 1 − (ξ 2 + η 2 ) 2 4ξ 2 (1 − κ) 2 = 0.(36) Real Ŝ x corresponds to the system being superconducting, in this case η, κ have to satisfy 1 − (ξ 2 + η 2 ) 2 4ξ 2 (1 − κ) 2 > 0.(37) On the other hand, non-real Ŝ x corresponds to the system being insulating, this time η, κ have to satisfy 1 − (ξ 2 + η 2 ) 2 4ξ 2 (1 − κ) 2 < 0.(38) Then we can draw a phase diagram of S-I transition with respect to η and κ (see Fig.4). (36) ∼ (38) can be also expressed as follows J xy = J z + 1 2 h + ∆h 2 h @@(critical region), @@@@(39)J xy > J z + 1 2 h + ∆h 2 h @@(superconducting region),(40)J xy < J z + 1 2 h + ∆h 2 h @@(insulating region). @@(41) The left-hand side is kinetic energy which causes quantum fluctuation. The right-hand side is potential energy which causes crystallization. In other words the former indicates wave property and the latter indicates particle property (localization) of Cooper pair, Therefore the phase diagram of our system is determined by the competition between them. SUMMARY We have focused on the S-I transition in ultra-thin films and considered randomness contribution as a factor to determine the border between the two phases (S and I). Superconducting order parameter Ŝ x has been calculated as function of randomness η, and influence of η to S-I transition has been investigated. As a result, we have shown that Ŝ x decreases with increasing η and obtained analytically the critical value of randomness η c . The existence of random potential causes scattering of Cooper pairs, and disturbs their free motion. Randomness η becomes an important factor for the occurrence of electric resistance R. There is a one-to-one correspondence between the critical value of R and η (R c and η c ). Therefore solving η c is equivalent to solving R c . It has been well known that the value of critical sheet resistance in ultrathin films is on the order of h/4e 2 , but its exact coefficients has not been calculated yet. Our results may have some relevance to the determination of R c . We have obtained the results that randomness and two-body interaction causes localization and destroys superconductivity. These are capable of explaining, at least qualitatively, the main characteristic of S-I transition. In order to proceed to a more quantitative discussion, it is necessary to employ a more refined approximation in evaluating higher-order Green's functions. The behavior of superconductivity Ŝ x with increasing two body interaction κ. Ŝ x continuously decreased and completely disappeared at each specific point. Randomness η contributes to advancing the transition from superconducting to insulating. Fig. 4. The phase diagram for our lattice boson model with randomness η and two-body interaction κ. The borderline between two phases (S and I) is determined by the competition between wave property and particle property. When wave property is stronger than particle property, the system is in superconducting. When particle property is stronger than wave property, the system is in insulating. FIGURESFig. 1 .Fig. 2 . 12Two-dimensional square lattice. The four sites i+a n (n = 1 ∼ 4) are the nearest-neighbor sites of the site i on the square lattice. The behavior of superconductivity Ŝ x with increasing randomness η. Ŝ x continuously decreases and completely disappears at each specific point. Two body interaction κ contributes to advancing the transition from superconducting to insulating. Fig. 3 . 3Fig. 3. The behavior of superconductivity Ŝ x with increasing two body interaction κ. Ŝ x ACKNOWLEDGMENTSWe are very grateful to S. Saito, K. Sano, and B. H. Valtan for useful discussions and critical reading. . R T Scalettar, N Trivedi, C Huscroft, Phys. Rev. 594364R. T. Scalettar, N. Trivedi, and C. Huscroft Phys. Rev. B59, 4364 (1999). . N Trivedi, R T Scalettar, M Randeria, Phys. Rev. 543756N. Trivedi, R. T. Scalettar, and M. Randeria Phys. Rev. B54, R3756 (1996). . M P A Fisher, P B Weichman, G Grinstein, D S Fisher, Phys. Rev. 40546M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher Phys. Rev. B40, 546 (1989). . P Phillips, Nature. 406687P. Phillips, Nature, 406, 687 (2000). . J H Schön, Ch Kloc, B Batlogg, Nature. 406702J. H. Schön, Ch. Kloc, and B. Batlogg, Nature, 406, 702 (2000). . N Markovic, C Christiansen, A M Goldman, Phys. Rev. Lett. 815217N. Markovic, C. Christiansen, and A. M. Goldman, Phys. Rev. Lett. 81, 5217 (1998). . N Markovic, C Christiansen, A M Mack, W H Huber, A M Goldman, Phys. Rev. 604320N. Markovic, C. Christiansen, A. M. Mack, W.H. Huber, and A. M. Goldman, Phys. Rev. B60, 4320 (1999). . J S Penttilä, Ü Parts, P J Hakonen, A J Jordan, R P Barber, Phys. Rev. 821004J. S. Penttilä,Ü. Parts, P. J. Hakonen, A. J. Jordan, and R. P. Barber, Phys. Rev. B82, 1004 (1999). . T Yamaguchi, R Yagi, A Kanda, Y Ootuka, S Kobayashi, Phys. Rev. lett. 851974T. Yamaguchi, R. Yagi, A. Kanda, Y. Ootuka, and S. Kobayashi, Phys. Rev. lett. 85, 1974 (2000). . R Fazio, G Schön, Phys. Rev. 435307R. Fazio and G. Schön, Phys. Rev. B43, 5307 (1991). . G M Zassenhaus, J D Reppy, Phys. Rev. Lett. 834800G. M. Zassenhaus and J. D. Reppy, Phys. Rev. Lett. 83, 4800 (1999). . I F Herbut, Phys. Rev. 6114723I. F. Herbut, Phys. Rev. B61, 14723 (2000). . I F Herbut, Phys. Rev. 5713729I. F. Herbut, Phys. Rev. B57, 13729 (1998). . L Zhang, M Ma, Phys. Rev. 454855L. Zhang and M. Ma, Phys. Rev. B45, 4855 (1992). . K G Singh, D S Rokhsar, Phys. Rev. 463002K. G. Singh and D. S. Rokhsar, Phys. Rev. B46, 3002 (1992). . J K Freericks, H Monien, Phys. Rev. 532691J. K. Freericks and H. Monien, Phys. Rev. B53, 2691 (1996). . T Matsubara, H Matsuda, Prog, Theor. Phys. 16569T. Matsubara and H. Matsuda, Prog, Theor. Phys. 16, 569 (1956). . H Matsuda, T Tsuneto, Suppl. Prog. Theor. Phys. 46411H. Matsuda and T. Tsuneto, Suppl. Prog. Theor. Phys. 46, 411 (1970). . K S Liu, M E Fisher, J. Low. Temp. Phys. 10655K. S. Liu and M. E. Fisher, J. Low. Temp. Phys. 10, 655 (1973). . D N Zubarev, Usp. Fiz. Nauk. 7171D. N. Zubarev, Usp. Fiz. Nauk 71, 71 (1960). . J Kondo, K Yamaji, Prog. Theor. Phys. 47807J. Kondo and K. Yamaji, Prog. Theor. Phys. 47, 807 (1972). . H B Callen, Phys. Rev. 130890H. B. Callen, Phys. Rev. 130, 890 (1963). . P W Anderson, Phys. Rev. 1121900P.W. Anderson, Phys. Rev. 112, 1900 (1958).

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{'abstract': "Effect of disorder in metallic thin film is examined as a possible mechanism of the Superconductor-Insulator (S-I) transition. The critical value of disorder corresponding to the transition point is found analytically by using Matsubara-Matsuda model and Green's function method.", 'arxivid': 'cond-mat/0104015', 'author': ['Masahiko Kasuga \nDepartment of Physics\nWaseda University\n3-4-1 Okubo, Shinjuku-ku169-8555TokyoJapan\n', 'Susumu Kurihara \nDepartment of Physics\nWaseda University\n3-4-1 Okubo, Shinjuku-ku169-8555TokyoJapan\n'], 'authoraffiliation': ['Department of Physics\nWaseda University\n3-4-1 Okubo, Shinjuku-ku169-8555TokyoJapan', 'Department of Physics\nWaseda University\n3-4-1 Okubo, Shinjuku-ku169-8555TokyoJapan'], 'corpusid': 15216146, 'doi': '10.1016/s0921-4534(02)01961-5', 'github_urls': [], 'n_tokens_mistral': 6301, 'n_tokens_neox': 5331, 'n_words': 3202, 'pdfsha': 'db1ca58094448cad8c32728649450be59d741a46', 'pdfurls': ['https://arxiv.org/pdf/cond-mat/0104015v1.pdf'], 'title': ['Superconductor-Insulator Transition in Random Two-Dimensional System', 'Superconductor-Insulator Transition in Random Two-Dimensional System'], 'venue': []}

arxiv

Neutrino-nucleus interaction rates at a low-energy beta-beam facility 21 Oct 2004 Julien Serreau Institut für Theoretische Physik Institut de Physique Nucléaire Universität Heidelberg Philosophenweg 16D-69120, F-91406Heidelberg, Orsay CedexGermany, France Cristina Volpe Institut für Theoretische Physik Institut de Physique Nucléaire Universität Heidelberg Philosophenweg 16D-69120, F-91406Heidelberg, Orsay CedexGermany, France Neutrino-nucleus interaction rates at a low-energy beta-beam facility 21 Oct 2004(Dated: March 26, 2022)PACS numbers: 2530Pt, 2650+x We compute the neutrino detection rates to be expected at a low-energy beta-beam facility. We consider various nuclei as neutrino detectors and compare the case of a small versus large storage ring. view, see [13,14]). There are a number of open issues in this context. The A=2 system is the simplest case, for which the reaction cross sections can be estimated with high accuracy [15]. However, there is still an important quantity, namely L 1,A , related to the axial two-body current, which dominates the theoretical uncertainty in neutrino-deuteron interactions. For heavier nuclei, in the tens of MeV energy range, the reaction cross sections are dominated by collective modes, like the Gamow-Teller resonance or the Isobaric Analog State, which have been extensively studied in the past [16]. As the neutrino impinging energy increases, transitions to states of higher multipolarity (such as the spin-dipole or higher forbidden transitions) become important [17]. The latter also play an important role in the context of core-collapse Supernova physics [5,17,18,19]. Although some information on these states can be gathered through other probes, such as charge-exchange reactions [16], muon capture [20], or inelastic electron scattering [21], the experimental information is rather scarce. Note that the understanding of neutrino-carbon reactions with neutrinos produced from the decay in flight of pions is still an open issue, for most of the theoretical calculations over-estimate the experimental value [22]. So far, measurements with lowenergy neutrinos have been performed in a few cases only, namely deuteron [23], carbon [24], and iron [25]. Systematic studies would be of great importance both for what concerns the interpolation from the MeV to the GeV neutrino energy range and the extrapolation to neutron-rich nuclei, as required in the astrophysical context. Neutrino-nucleus interaction studies were one of the main physics issues of the proposed ORLAND underground neutrino facility, which was based on a conventional neutrino source (pion and muon decays) [14,26]. A smaller version of the ORLAND project is now under study [27]. At present, the MINERνA project [28] includes the study of neutrino-nucleus interactions for neutrino energies in the GeV range. Here, we study the potential of a low-energy neutrino facility based on beta-beams, a novel method to produce neutrino beams [29]. This consists in boosting exotic ions which decay through beta-decay and produce pure, collimated and well-understood electron neutrino fluxes. Such a method could be exploited for a future facility at CERN [29,30]. High energy beta-beams would be fired to a gigantic Cherenkov detector like UNO [31], located in an (upgraded) Fréjus underground laboratory to study, in particular, the possible existence of CP violation in the leptonic sector [29,30,32]. The discovery potential with a very high γ and a longer baseline is discussed in [33,34]. It has recently been proposed to use the beta-beam concept for the production of low-energy neutrinos [35]. Several laboratories will produce intense exotic beams in the near future and could, therefore, be possible sites for a low-energy beta-beam facility. These include GANIL, CERN, GSI, as well as the EURISOL project. Lowenergy neutrino beams would offer an interesting opportunity to study various neutrino properties, such as e.g. the neutrino magnetic moment [36], as well as neutrinonucleus interactions, of interest for nuclear physics, particle physics and astrophysics. In the former case, one would exploit the ions at rest as an intense neutrino source, whereas, in the latter case, one would use boosted ions, which would be collected in a storage ring [35], as in the original high energy proposal. An important feature of such beta-beams is that the boost factor of the accelerated ions can be varied, allowing one to explore various neutrino energy ranges. In this paper, we present for the first time chargedcurrent neutrino-nucleus interaction rates achievable at a low-energy beta-beam facility. We consider two possible cases for the dimensions of the storage ring, for which we inspire ourselves of the one planned in the future GSI facility [37] and the one thought in the CERN baseline scenario [29,30]. We consider various target nuclei as neutrino detectors, namely deuteron, oxygen, iron and lead, which are commonly used in existing or planned experiments [26]. Related work in the case of lead can be found in [38]. II. FORMALISM A. Neutrino fluxes and interaction rates The decay rate of a nucleus in the rest (cm) frame can be written as: dW dt cm = Φ cm (E ν ) dE ν d 2 Ω 4π ,(1) where E ν and Ω denote respectively the energy and the solid angle of the emitted (anti-)neutrino, and where the neutrino flux Φ cm (E ν ) is given by the well-known formula [39]: Φ cm (E ν ) = b E 2 ν E e E 2 e − m 2 e F (±Z, E e ) Θ(E e − m e ) . (2) where the constant b = ln 2/m 5 e f t 1/2 , with m e the electron mass and f t 1/2 the ft-value. The quantities appearing in the above expression are the energy E e = Q−E ν of the emitted lepton (electron or positron), Q being the Qvalue of the reaction, and the Fermi function F (±Z, E e ), which accounts for the Coulomb modification of the spectrum. In the laboratory frame, where the boosted nucleus has a velocity v = βc, the decay rate reads: dW dt lab = 1 γ Φ lab (E ν , θ) dE ν d 2 Ω 4π ,(3) where γ = 1/ 1 − β 2 is the time dilation factor and where E ν and Ω ≡ (θ, ϕ) now denote the energy and solid angle of the emitted (anti-)neutrino in the laboratory (lab) frame, θ being the angle of emission with respect to the beam axis. The boosted flux Φ lab (E ν , θ) is given by: Φ lab (E ν , θ) = Φ cm (E ν γ[1 − β cos θ]) γ[1 − β cos θ] .(4) We consider a storage ring of total length L with a straight sections of length D. In the stationary regime the mean number of ions in the storage ring is γτ g, where τ = t 1/2 / ln 2 is the lifetime of the parent nuclei and g is the number of injected ions per unit time. The total number of neutrinos emitted per unit time from a portion dℓ of the decay ring is dN ν dt = γτ g × dW dt lab × dℓ L .(5) For simplicity, we consider a cylindrical detector of radius R and depth h, aligned with one the straight sections of the storage ring, and placed at a distance d from the latter. After integration over the useful decay path and over the volume of the detector, the total number of events per unit time is: dN ev dt = gτ nh × ∞ 0 dE ν Φ tot (E ν ) σ(E ν ) ,(6) where n is the number of target nuclei per unit volume, σ(E ν ) is the relevant neutrino-nucleus interaction crosssection, and where Φ tot (E ν ) = D 0 dℓ L h 0 dz h θ (ℓ,z) 0 sin θdθ 2 Φ lab (E ν , θ) ,(7)with tanθ(ℓ, z) = R d + ℓ + z .(8) B. Large versus Small Ring configurations The storage ring geometry is characterized by the length of the straight sections D and by its total length L. Below, we consider the cases of a small (SR) and a large (LR) ring configurations, characterized by (D SR ,L SR ) 7). Up to trivial 1/L factors, the LHS corresponds to the LR configuration and the first term on the RHS to the SR configuration. The remaining integral can be given a simple analytical estimate if one can neglect the angular dependence of the flux under the integral. This happens when the angle under which the detector is seen from the extremity of the SR decay path ∼ R/(d+D SR ) is small compared to 1/γ, i.e. to the typical opening angle of the boosted flux. In that case, we obtain, for the total flux (7): Φ LR tot (E ν ) ≃ L SR L LR × Φ SR tot (E ν )+G Φ lab (E ν , θ = 0) ,(9) where the geometrical factor G is given by: G = R 2 4L SR (d + D SR ) 1 − d + D SR d + D LR .(10) The overall factor L SR /L LR in (9) simply accounts for the fact that the number of decaying ions per unit length is smaller in a larger storage ring, and the second term in brackets on the RHS represents the contribution from the longer useful straight section. Figure 1 shows a comparison between the exact flux obtained with Eqs.(7)- (8) in both the SR and the LR configurations, and the analytic estimate Eq. (9), for the two possible detector sizes considered in the following. We see that the analytical formula (9) works very well in the cases considered here. Besides, Figure 1 shows that the contribution from the longer decay path only brings a ∼ 10 % difference for the small detector and contributes a factor ∼ 2 for the larger detector. This already shows that the main difference between the LR and SR fluxes comes from the geometrical factor L SR /L LR ≃ 1/15. Using the approximate formula for the total fluxes, we obtain an approximate relation between the total number of events in the LR and SR configurations: dN ev dt LR ≃ L SR L LR × dN ev dt SR + γ 2 (1 + β) 2 G gnh σ γ ,(11) where σ γ denotes the flux-averaged cross section in the forward direction θ = 0: σ γ = ∞ 0 dE ν Φ lab (E ν , θ = 0) σ(E ν ) ∞ 0 dE ν Φ lab (E ν , θ = 0) .(12) Using Eq. (4), the latter can be re-written as: σ γ = ∞ 0 dE ν Φ cm (E ν ) σ(γ(1 + β)E ν ) ∞ 0 dE ν Φ cm (E ν ) .(13) It is to be noted that, when the detector is placed close to the storage ring, as it is the case here, the total rate (6) depends non-trivially on the geometry of the latter. For instance, as discussed above, we observe an approximate 1/L scaling at fixed D/L in the small detector case. This is in contrast with the case of a far detector considered in the high energy beta-beam scenarios [29,32,33,34], where the rate is simply proportional to the ratio D/L of the straight section over the total length of the ring [44]. III. RESULTS Here, we present charged-current neutrino interaction rates with various target nuclei as obtained from Eqs. (Tables I and II). Four possible nuclei are taken as typical examples, namely deuteron, oxygen, iron and lead. A detailed study for the case of lead is also done in [38]. The "small ring" we consider has 150 meter straight sections and 450 meter total length, while the "large ring" has 2.5 km straight sections and 7 km total length. The detectors are located at a distance 10 meters from the storage ring, to allow a maximum shielding of the induced background in the ring [40]. For the detector size we inspire ourselves on the kinds considered for the proposed ORLAND facility [26,43]. The transverse size is chosen so as to catch as much as possible of the boosted flux, which main contribution is concentrated in an opening angle ∼ 1/γ. More precisely, we choose as typical dimensions (R = radius, h = depth): R = 1.5 m and h = 4.5 m. We also consider the case of a large (kilotontype) water detector with R = 4.5 m and h = 15 m. For all detectors here we assume a 100 % efficiency. Finally, we have to specify the number of parent ions g injected per unit time in the storage ring. According to the feasibility study [30], 2 × 10 13 6 He/sec and 8 × 10 11 18 Ne/sec could be produced with an ISOLDE technique, giving about gν = 10 13ν /sec and g ν = 5 × 10 11 ν/sec respectively [30]. An important feature of beta-beams is that the number and average energy of neutrinos entering the detector depend on the boost factor γ of the parent ion, which can be varied. We present results for two different values, namely γ = 7 (Table I) and γ = 14 (Table II). The corresponding neutrino fluxes are presented in Figure 2 and range up to about 50 and 100 MeV respectively. Let us discuss the number of events shown in Tables I and II. The differences between the ν-induced versus ν-induced reactions is a combined effect of the relative intensities g ν /gν = 1/20 and of the different interaction cross-sections: the ratio σ(ν + D)/σ(ν + D) is roughly 2 in the whole energy range considered here [13]; from [41], one can see that σ(ν + 16 The very low rates obtained for oxygen with γ = 7 despite the large detector size are due to the 15 MeV threshold in the interaction cross-section. Next, we observe that the suppression of the rates in the LR configuration as compared to the SR case for a given γ roughly corresponds to the geometrical factor L SR /L LR , as expected from the previous discussion. In fact, the difference between the LR and SR rates can be fully understood by means of the approximate relation Eq. (11). This formula can be used to rescale our results for other possible dimensions of the storage ring. To this aim, we give the relevant values of σ γ in each case. When going from γ = 7 to γ = 14, the neutrino fluxes become more collimated and the typical energy of the neutrinos increases. This, together with the fact that the neutrino-nucleus interaction cross sections rapidly rise with the impinging neutrino energy, increases the number of events by more than an order of magnitude. Figure 3 illustrates the rapid rise of the total rates with increasing γ. Note that, in the present case, where the detector is relatively close to the storage ring, the total rates do not have a simple scaling with the detector size, due to the non-trivial angular dependence of the impinging neutrino flux. It is important to emphasize the complementarity between low-energy beta-beams and conventional neutrino facilities [26]. The latter provide intense sources of electron and muon neutrinos and cover the very low energy region, similar to the case γ = 7 for the beta-beam. Let us mention that for comparable neutrino intensities, the rates presented in Table I are comparable to those obtained with conventional schemes with detectors located at about 50 meters from the source. Low-energy betabeams would produce pure electron neutrino beams and, by varying the boost factor γ, would offer a unique opportunity to study neutrino-nucleus interactions over a wide range of energies. To conclude, the present study demonstrates that, with typical parameters available from existing studies [30], significant interaction rates can be achieved at a low-energy beta-beam facility. A small ring -with as long as possible straight sections -is the preferred configuration in the case of a close detector. The rates raise rapidly with increasing γ. We think our results are encouraging and we hope they will trigger further investigations, including, in particular, detailed simulations of the detectors. We thank J. Bouchez and M. Magistris for useful discussions, R. Lombard and M. Mezzetto for careful reading of the manuscript. FIG. 1 : 1Neutrino fluxes scaled by the length of the storage ring LΦtot(Eν): The exact results obtained with Eqs.(7)-(8) with a small storage ring SR (solid lines) and a large storage ring LR (long-dashed lines) are shown. The left (right) figure shows the fluxes impinging on the small (large) detector (the sizes are given in Tables I and II). For the small detector (left), the LR result obtained with the analytical estimate Eq. (9) coincide with the exact result and is not represented here for clarity. For the large detector (right), it is also a very good approximation as shown by the dotted-line. The contribution L G Φ lab (Eν, θ = 0) from the RHS of Eq. (9) is also presented (dashed lines). All fluxes are obtained with 18 Ne boosted at γ = 14.and (D LR ,L LR ) respectively. The results in both configurations can easily be related to one another by splitting the integral over the useful decay path FIG. 2 : 2Neutrino fluxes Φtot(Eν) as a function of energy for18 Ne nuclei boosted at γ = 7 and γ = 14. This corresponds to the small ring and small detector configuration. FIG. 3 : 3The total rate for the reaction ν + D as a function of the boost factor γ. This corresponds to the small ring and small detector configuration.0.5 on average in the energy range relevant to the case γ = 7, namely 20 MeV E ν 40 MeV, and about 1.5 on average in the range 40 MeV E ν 80 MeV, relevant for the case γ = 14. I: Number of events per year for γ = 7 in the small (LSR = 450 m, DSR = 150 m) and large (LLR = 7 km, DLR = 2.5 km) ring configurations. These results are obtained by using the exact formulas of Eqs.(6-8). The detector is located at d = 10 m away from the ring and has dimensions R = 1.5 m and h = 4.5 m for the D (D2O), 56 Fe and 208 Pb, and R = 4.5 m and h = 15 m for the case of 16 O (H2O), where R is the radius and h is the depth of the detector. The corresponding masses are given in tons. The results in the large ring configuration can be precisely understood from those in the small ring configuration by means of the analytical formula Eq. (11). We give the flux-averaged cross section in the forward direction σ γ (see Eqs. (12)-(13)) in units of 10 −42 cm 2 . The latter can be used to rescale the present rates for different sizes of the storage ring using Eq. (11). The relevant cross-sections are taken from the indicated references. The results are obtained with 1 year = 3.2 × 10 7 s. , where S = πR 2 is the transverse area of the detector. Similarly, one obtains, for the rate: γ 2 (1 + β) 2 σ γ , where Ntarget = nπR 2 h is the total number of target nuclei. 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Haxton, Phys. Rev. D 36, 2283 (1987). . E Kolbe, K Langanke, Phys. Rev. C. 6325802E. Kolbe and K. Langanke, Phys. Rev. C 63, 025802 (2001). . F T Avignone, Y V Efremenko, J. Phys. G. 292615F.T. Avignone and Y.V. Efremenko, J. Phys. G 29, 2615 (2003). one has simply: Φtot(Eν) ≃ Φ lab (Eν. ≫ L , D , For a distant detectorFor a distant detector (d ≫ L, D, h), one has simply: Φtot(Eν) ≃ Φ lab (Eν, θ = 0) D

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{'abstract': 'We compute the neutrino detection rates to be expected at a low-energy beta-beam facility. We consider various nuclei as neutrino detectors and compare the case of a small versus large storage ring.', 'arxivid': 'hep-ph/0403293', 'author': ['Julien Serreau \nInstitut für Theoretische Physik\nInstitut de Physique Nucléaire\nUniversität Heidelberg\nPhilosophenweg 16D-69120, F-91406Heidelberg, Orsay CedexGermany, France\n', 'Cristina Volpe \nInstitut für Theoretische Physik\nInstitut de Physique Nucléaire\nUniversität Heidelberg\nPhilosophenweg 16D-69120, F-91406Heidelberg, Orsay CedexGermany, France\n'], 'authoraffiliation': ['Institut für Theoretische Physik\nInstitut de Physique Nucléaire\nUniversität Heidelberg\nPhilosophenweg 16D-69120, F-91406Heidelberg, Orsay CedexGermany, France', 'Institut für Theoretische Physik\nInstitut de Physique Nucléaire\nUniversität Heidelberg\nPhilosophenweg 16D-69120, F-91406Heidelberg, Orsay CedexGermany, France'], 'corpusid': 119019147, 'doi': '10.1103/physrevc.70.055502', 'github_urls': [], 'n_tokens_mistral': 9041, 'n_tokens_neox': 7405, 'n_words': 4281, 'pdfsha': 'a527f4ec769daa78b28261ca5d0683b305cb8eff', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ph/0403293v2.pdf'], 'title': ['Neutrino-nucleus interaction rates at a low-energy beta-beam facility', 'Neutrino-nucleus interaction rates at a low-energy beta-beam facility'], 'venue': []}

arxiv

PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies Guocheng Qian Yuchen Li Houwen Peng Microsoft Research † Jinjie Mai Hasan Abed Al Kader Hammoud Mohamed Elhoseiny Bernard Ghanem King Abdullah University of Science and Technology (KAUST) PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies PointNet++ is one of the most influential neural architectures for point cloud understanding. Although the accuracy of PointNet++ has been largely surpassed by recent networks such as PointMLP and Point Transformer, we find that a large portion of the performance gain is due to improved training strategies, i.e. data augmentation and optimization techniques, and increased model sizes rather than architectural innovations. Thus, the full potential of PointNet++ has yet to be explored. In this work, we revisit the classical PointNet++ through a systematic study of model training and scaling strategies, and offer two major contributions. First, we propose a set of improved training strategies that significantly improve PointNet++ performance. For example, we show that, without any change in architecture, the overall accuracy (OA) of PointNet++ on ScanObjectNN object classification can be raised from 77.9% to 86.1%, even outperforming state-of-theart PointMLP. Second, we introduce an inverted residual bottleneck design and separable MLPs into PointNet++ to enable efficient and effective model scaling and propose PointNeXt, the next version of PointNets. PointNeXt can be flexibly scaled up and outperforms state-of-the-art methods on both 3D classification and segmentation tasks. For classification, PointNeXt reaches an overall accuracy of 87.7% on ScanObjectNN, surpassing PointMLP by 2.3%, while being 10× faster in inference. For semantic segmentation, PointNeXt establishes a new state-of-theart performance with 74.9% mean IoU on S3DIS (6-fold cross-validation), being superior to the recent Point Transformer. The code and models are available at https://github.com/guochengqian/pointnext. * Equal contribution. † Corresponding authors.Preprint. Under review. Introduction Recent advances in 3D data acquisition have led to a surge in interest for point cloud understanding. With the rise of PointNet [29] and PointNet++ [30], processing point clouds in their unstructured format using deep CNNs become possible. Subsequent to "PointNets", many point-based networks are introduced with the majority focusing on developing new and sophisticated modules to extract local structures, e.g. the pseudo-grid convolution in KPConv [43] and the self-attention layer in Point Transformer [56]. These newly proposed methods outperform PointNet++ by a large margin in a variety of tasks, leaving the impression that the PointNet++ architecture is too simple to learn complex point cloud representations. In this work, we revisit PointNet++, the classical and widely used network, and find that its full potential has yet to be explored, mainly due to two factors that were not present at the time of PointNet++: (1) superior training strategies and (2) effective model scaling strategies. Through a comprehensive empirical study on various benchmarks, e.g., ScanObjecNN [44] for object classification and S3DIS [1] for semantic segmentation, we discover that training strategies, i.e., data augmentation and optimization techniques, play an important role in the network's performance. In fact, a large part of the performance gain of state-of-the-art (SOTA) methods [46,43,56] over Point-Net++ [30] is due to improved training strategies that are, unfortunately, less publicized compared to Figure 1: Effects of training strategies and model scaling on PointNet++ [30]. We show that improved training strategies (data augmentation and optimization techniques) and model scaling can significantly boost PointNet++ performance. The average overall accuracy and mIoU (6-fold cross-validation) are reported on ScanObjectNN [44] and S3DIS [1]. architectural changes. For example, randomly dropping colors during training can unexpectedly boost the testing performance of PointNet++ by 5.9% mean IoU (mIoU) on S3DIS [1], as demonstrated in Tab. 5. In addition, adopting label smoothing [39] can improve the overall accuracy (OA) on ScanObjectNN [44] by 1.3%. These findings inspire us to revisit PointNet++ and equip it with new advanced training strategies that are widely used today. Surprisingly, as shown in Fig. 1, utilizing the improved training strategies alone improves the OA of PointNet++ by 8.2% on ScanObjectNN (from 77.9% to 86.1%), establishing a new SOTA without introducing any changes to the architecture (refer to Sec. 4.4.1 for details). For the S3DIS segmentation benchmark, the mIoU evaluated in all areas by 6-fold cross-validation can increase by 13.6% (from 54.5% to 68.1%), outperforming many modern architectures that are subsequent to PointNet++, such as PointCNN [22] and DeepGCN [21]. Moreover, we observe that the current prevailing models [20,43,56] for point cloud analysis have employed many more parameters than the original PointNets [29,30]. Effectively expanding PointNet++ from its original small scale to a larger scale is a topic worth studying because larger models are generally expected to enable richer representations and perform better [2,19,55]. However, we find that the naive way of using more building blocks or increasing the channel size in PointNet++ only leads to an overhead in latency and no significant improvement in accuracy (see Sec. 4.4.2). For effective and efficient model scaling, we introduce residual connections [13], an inverted bottleneck design [36], and separable MLPs [32] into PointNet++. The modernized architecture is named PointNeXt, the next version of PointNets. PointNeXt can be scaled up flexibly and outperforms SOTA on various benchmarks. As demonstrated in Fig. 1, PointNeXt improves the original PointNet++ by 20.4% mIoU (from 54.5% to 74.9%) on S3DIS [1] 6-fold and achieves 9.8% OA gains on ScanObjecNN [44], surpassing SOTA Point Transformer [56] and PointMLP [28]. We summarize our contributions next: • We present the first systematic study of training strategies in the point cloud domain and show that PointNet++ strikes back (+8.2% OA on ScanObjectNN and +13.6% mIoU on S3DIS) by simply adopting improved training strategies alone. The improved training strategies are general and can be easily applied to improve other methods [29,46,28]. • We propose PointNeXt, the next version of PointNets. PointNeXt is scalable and surpasses SOTA on all tasks studied, including object classification [44,49], semantic segmentation [1,5], and part segmentation [53], while being faster than SOTA in inference. Preliminary: A Review of PointNet++ Our PointNeXt is built upon PointNet++ [30], which uses a U-Net [35] like architecture with an encoder and a decoder, as visualized in Figure 2. The encoder part hierarchically abstracts features of point clouds using a number of set abstraction (SA) blocks, while the decoder gradually interpolates the abstracted features by the same number of feature propagation blocks. The SA block consists of a subsampling layer to downsample the incoming points, a grouping layer to query neighbors for each point, a set of shared multilayer perceptrons (MLPs) to extract features, and a reduction layer to aggregate features within the neighbors. The combination of the grouping layer, MLPs, and the reduction layer is formulated as: x l+1 i = R j:(i,j)∈N h Θ [x l j ; p l j − p l i ] ,(1) where R is the reduction layer (e.g. max-pooling) that aggregates features for point i from its neighbors denoted as {j : (i, j) ∈ N }. p l i , x l i , x l j are the input coordinates, the input features, and the features of neighbor j in the l th layer of the network, respectively. h Θ denotes the shared MLPs that take the concatenation of x l j and the relative coordinates (p l j − p l i ) as input. Note that, since PointNet++ with single-scale grouping that uses one SA block per stage is the default architecture used in the original paper [30], we refer to it as PointNet++ throughout and use it as our baseline. Methodology: From PointNet++ to PointNeXt In this section, we present how to modernize the classical architecture PointNet++ [30] into PointNeXt, the next version of PointNet++ with SOTA performance. Our exploration mainly focuses on two aspects: (1) training modernization to improve data augmentation and optimization techniques, and (2) architectural modernization to probe receptive field scaling and model scaling. Both aspects have important impact on the model's performance, but were under-explored by previous studies. Training Modernization: PointNet++ Strikes Back We conduct a systematic study to quantify the effect of each data augmentation and optimization technique used by modern point cloud networks [46,43,56] and propose a set of improved training strategies. The potential of PointNet++ can be unveiled by adopting our proposed training strategies. Data Augmentation Data augmentation is one of the most important strategies to boost the performance of a neural network; thus we start our modernization from there. The original PointNet++ used simple combinations of data augmentations from random rotation, scaling, translation, and jittering for various benchmarks [30]. Recent methods adopt stronger augmentations than those used in PointNet++. For example, KPConv [43] randomly drops colors during training, Point-BERT [54] uses a common point resampling strategy to randomly sample 1, 024 points from the original point cloud for data scaling, while RandLA-Net [15] and Point Transformer [56] load the entire scene as input in segmentation tasks. In this paper, we quantify the effect of each data augmentation through an additive study. We start our study with PointNet++ [30] as the baseline, which is trained with the original data augmentations and optimization techniques. We remove each data augmentation to check whether it is necessary or not. We add back the useful augmentations but remove the unnecessary ones. We then systematically study all the data augmentations used in the representative works [46,43,32,56,28,54], including data scaling such as point resampling [54] and loading the entire scene as input [15], random rotation, random scaling, translation to shift point clouds, jittering to add independent noise to each point, height appending [43] (i.e., appending the measurement of each point along the gravity direction of objects as additional input features), color auto-contrast to automatically adjust color contrast [56], and color drop that randomly replaces colors with zero values. We verify the effectiveness of data augmentation incrementally and only keep the augmentations that give a better validation accuracy. At the end of this study, we provide a collection of data augmentations for each task that allow for the highest boost in the model's performance. Sec. 4.4.1 presents and analyzes in detail the uncovered findings. Optimization Techniques Optimization techniques including loss functions, optimizers, learning rate schedulers, and hyperparameters are also vital to the performance of a neural network. PointNet++ uses the same optimization techniques throughout its experiments: CrossEntropy loss, Adam optimizer [16], exponential learning rate decay (Step Decay), and the same hyperparmeters. Owing to the development of machine learning theory, modern neural networks can be trained with theoretically better optimizers (e.g. AdamW [27] vs. Adam [16]) and more advanced loss functions (CrossEntropy with label smoothing [39]). Similarly to our study on data augmentations, we also quantify the effect of each modern optimization technique on PointNet++. We first perform a sequential hyperparameter search for the learning rate and weight decay. We then conduct an additive study on label smoothing, optimizer, and learning rate scheduler. We discover a set of improved optimization techniques that further Architecture Modernization: Small Modifications → Big Improvements In this subsection, we modernize PointNet++ [30] into the proposed PointNeXt. The modernization consists of two aspects: (1) receptive field scaling and (2) model scaling. Receptive Field Scaling The receptive field is a significant factor in the design space of a neural network [38,7]. There are at least two ways to scale the receptive field in point cloud processing: (1) adopting a larger radius to query the neighborhood, and (2) adopting a hierarchical architecture. Since the hierarchical architecture has been adopted in the original PointNet++, we mainly study (1) in this subsection. Note that the radius of PointNet++ is set to an initial value r that doubles when the point cloud is downsampled. We study a different initial value in each benchmark and discover that the radius is dataset-specific and can have significant influence on performance. This is elaborated in Sec. 4.4.2. Furthermore, we find that the relative coordinates ∆ p = p l j −p l i in Eq. (1) make network optimization harder, leading to a decrease in performance. Thus, we propose relative position normalization (∆ p normalization) to divide relative position by the neighborhood query radius: x l+1 i = R j:(i,j)∈N h Θ [x l j ; (p l j − p l i )/r l ] .(2) Without normalization, values of relative positions (∆ p = p l j − p l i ) are considerably small (less than the radius), requiring the network to learn a larger weight to apply on ∆ p . This makes the optimization non-trivial, especially since weight decay is used to reduce the weights of the network and thus tends to ignore the effects of relative position. The proposed normalization alleviates this issue by rescaling and in the meantime reduces the variance of ∆ p among different stages. Model Scaling PointNet++ is a relatively small network, where the encoder consists of only 2 stages in the classification architecture and 4 stages for segmentation. Each stage consists of only 1 SA block, and each block contains 3 layers of MLP. The model sizes of PointNet++ for both classification and segmentation are less than 2M, which is much smaller compared to modern networks that typically use more than 10M parameters [43,28,32]. Interestingly, we find that neither appending more SA blocks nor using more channels leads to a noticeable improvement in accuracy, while causing a significant drop in throughput (refer to Sec. 4.4.2), mainly due to vanishing gradient and overfitting. Therefore, in this subsection, we study how to scale up PointNet++ in an effective and efficient way. We propose an Inverted Residual MLP (InvResMLP) block to be appended after the first SA block, per stage, for effective and efficient model scaling. InvResMLP is built on the SA block and is illustrated at the bottom middle of Fig. 2. There are three differences between InvResMLP and SA. (1) A residual connection between the input and the output is added to alleviate the vanishing gradient problem [13], especially when the network goes deeper. (2) Separable MLPs are introduced to reduce computation and reinforce pointwise feature extraction. While all 3 layers of MLPs in the original SA block are computed on the neighborhood features, InvResMLP separates the MLPs into a single layer computed on the neighborhood features (between the grouping and reduction layers) and two layers for point features (after reduction), as inspired by MobileNet [14] and ASSANet [32]. (3) The inverted bottleneck design [36] is leveraged to expand the output channels of the second MLP by 4 times to enrich feature extraction. Appending InvResMLP blocks is proven to significantly improve performance compared to the appending of the original SA blocks (see Sec. 4.4.2). In addition to InvResMLP, we present three changes in the macro architecture. (1) We unify the design of PointNet++ encoder for classification and segmentation, i.e., scaling the number of SA blocks for classification from 2 to 4 while keeping the original number (4 blocks) for segmentation at each stage. (2) We utilize a symmetric decoder in which its channel size is changed to match the encoder. (3) We add a stem MLP, an additional MLP layer inserted at the beginning of the architecture, to map the input point cloud to a higher dimension. In summary, we present PointNeXt, the next version of PointNets [29,52], modified from PointNet++ by incorporating the proposed InvResMLP and the aforementioned macro-architectural changes. The architecture of PointNeXt is illustrated in Fig. 2. We denote the channel size of the stem MLP as C and the number of InvResMLP blocks as B. A larger C leads to an increase in the width of the network (i.e., width scaling), while a larger B leads to an increase in the depth of the network (i.e., depth scaling). Note that when B = 0, only one SA block and no InvResMLP blocks are used at each stage. The number of MLP layers in the SA block is set to 2, and a residual connection is added inside each SA block. When B = 0, InvResMLP blocks are appended after the original SA block. The number of MLP layers in the SA block in this case is set to 1 to save computation cost. The configuration of our PointNeXt family is summarized as follows: • PointNeXt-S: C = 32, B = 0 • PointNeXt-B: C = 32, B = (1, 2, 1, 1) • PointNeXt-L: C = 32, B = (2, 4, 2, 2) • PointNeXt-XL: C = 64, B = (3, 6, 3, 3) Experiments We evaluate PointNeXt on five standard benchmarks: S3DIS [1] and ScanNet [5] for semantic segmentation, ScanObjectNN [44] and ModelNet40 [49] for object classification, and ShapeNetPart [3] for object part segmentation. Experimental Setups. We train PointNeXt using CrossEntropy loss with label smoothing [39], AdamW optimizer [27], an initial learning rate lr = 0.001, weight decay 10 −4 , with Cosine Decay, and a batch size of 32, with a 32G V100 GPU, for all tasks, unless otherwise specified. The best model on the validation set is selected for testing. For S3DIS segmentation, point clouds are voxel downsampled with a voxel size of 0.04m following common practice [43,32,56]. PointNeXt is trained with an initial lr = 0.01, for 100 epochs (training set is repeated by 30 times), using a fixed number of points (24, 000) per batch with a batch size of 8 as input. During training, the input points are obtained by querying the nearest neighbors of a random point in each iteration. Following Point Transformer [56], we evaluate PointNeXt using the entire voxel-downsampled scene as input. For ScanNet scene segmentation, we follow the Stratified Transformer [17] and train PointNeXt with multi-step learning rate decay and decay at [70,90] epochs with a decay rate of 0.1 without label smoothing. The voxel size is set to 0.02m and input number of points in training is set to 64, 000. We train the model for 100 epochs (training set is repeated for 6 times) with a batch size of 2 per GPU with 8 GPUs. For ScanOb-jectNN classification, PointNeXt is trained with a weight decay of 0.05 for 250 epochs. Following Point-BERT [54], the number of input points is set to 1, 024, where the points are randomly sampled during training and uniformly sampled during testing (denoted as point resampled augmentation). For ModelNet40 classification, PointNeXt is trained similarly as ScanObjectNN but for 600 epochs. For ShapeNetPart part segmentation, we train PointNeXt using a batch size of 8 per GPU with 4 GPUs, and Poly FocalLoss [18] as criterion, for 400 epochs. Following PointNet++, 2,048 randomly sampled points with normals are used as input for training and testing. The details of data augmentations used in S3DIS, ScanNet, ScanObjectNN, ModelNet40 and ShapeNetPart are detailed in Sec. 4.4.1. Table 1: 3D semantic segmentation in S3DIS (evaluation by 6-Fold or in Area 5) and ScanNet V2. For PointNeXt in S3DIS Area 5, the average results without voting in three random runs are reported. The improvements of PointNeXt over the original performance reported by PointNet++ [30] are highlighted in green color. PointNet++ (ours) denotes PointNet++ trained using our improved data augmentation and optmization techniques. Methods are in chronological order. S3DIS For all experiments except ShapeNetPart segmentation, we do not conduct any voting [23] 2 , since it is more standard to compare the performance without using any ensemble methods as suggested by SimpleView [9]. However, we found that the performance in ShapeNetPart of nearly all models is quite close to each other, where it is hard to achieve state-of-the-art IoUs without voting. We also provide model parameters (Params.) and inference throughput (instances per second) for comparison. The throughput of all methods is measured using 128 × 1024 (batch size 128, number of points 1024) as input in ScanObjectNN and ModelNet40 and 64 × 2048 in ShapeNetPart. In S3DIS, 16 × 15, 000 points are used to measure throughput following [32], since some methods [46,20] could not process the whole scene due to memory constraints. The throughput of all methods is measured using an NVIDIA Tesla V100 32GB GPU and a 32 core Intel Xeon @ 2.80GHz CPU. 3D Semantic Segmentation in S3DIS and 3D Object Classification in ScanObjectNN and ModelNet40 ScanObjectNN [44] contains about 15, 000 real scanned objects that are categorized into 15 classes with 2, 902 unique object instances. Due to occlusions and noise, ScanObjectNN poses significant challenges to existing point cloud analysis methods. Following PointMLP [28], we experiment on PB_T50_RS, the hardest and most commonly used variant of ScanObjectNN. As reported in Tab. 2, the proposed PointNeXt-S surpasses existing methods by non-trivial margins in terms of both OA and mAcc, while using much fewer model parameters and running much faster. Built upon PointNet++ [30], PointNeXt achieves significant improvements over the originally reported performance of PointNet++, i.e. +9.8% OA and +10.4% mACC. This demonstrates the efficacy of the proposed training and model scaling strategies. PointNeXt also outperforms SOTA PointMLP [28] (i.e. +2.3% OA, +1.9% mACC), while running 10× faster. This shows that PointNeXt is a simple, yet effective, and efficient baseline. Note that we did not experiment with upscaled variants of PointNeXt on this benchmark, since we found that the performance had saturated using PointNeXt-S mostly due to the limited scale of the dataset. ModelNet40 [49] was a commonly used 3D object classification dataset, which has 40 object categories, each of which contains 100 unique CAD models. However, recent works [12,28,34] show an increasing interest in the real-world scanned dataset ScanObejectNN compared to this synthesized dataset. Following this trend, we mainly benchmarked PointNeXt in ScanObjectNN. Here, we also provide our results in ModelNet40. Tab. 2 shows that advanced training strategies improve PointNet++ from 91.9% OA to 92.8% OA without any architecture change. PointNeXt-S (C = 32) outperforms the original reported PointNet++ by 1.3% OA, while being faster. Note that PointNeXt-S with a larger width C = 64 can achieve a higher overall accuracy (94.0%). to that of the SOTA CurveNet [50] and outperforms a large number of representative networks, such as KPConv [43] and ASSANet [32] in terms of both instance mean IoU (ins. mIoU) and throughput. Due to the small scale of ShapeNetPart, the model would overfit after being depth scaled. However, we find by increasing the width from 32 to 64 instead, PointNeXt can outperform CurveNet, while being over 4× faster. It is also worth highlighting that PointNeXt with an even larger width (C = 160) reaches 87.0% Ins. mIoU, whereas the performance of point-based methods has saturated below this value for years. We highlight that we used voting only in ShapeNetPart by averaging the results of 10 randomly scaled input point clouds, with scaling factors equal to [0.8,1.2]. Without voting, we notice a performance drop around 0.5 instance mIoU. 3D Object Part Segmentation in ShapeNetPart Ablation and Analysis Tab. 4 and Tab. 5 present additive studies for the proposed training and scaling strategies in ScanObjectNN [44] and S3DIS [1], respectively. We adopt the original PointNet++ as the baseline. In ScanObjectNN, PointNet++ was trained by [44] with CrossEntropy loss, Adam optimizer, a learning rate 1e-3, a weight decay of 1e-4, a step decay of 0.7 for every 20 epochs, and a batch size of 16, for 250 epochs, while using random rotation and jittering as data augmentations. The official PointNet++ did not conduct experiments in S3DIS dataset. We refer to the widely used reimplementation [52], where PointNet++ was trained with the same settings as ScanObjectNN except that only random rotation was used as augmentation. Note that for all experiments, we train all our models for 250 epochs in ScanObjectNN and for 100 epochs in S3DIS. Training Strategies Data augmentation is the first aspect that we study to modernize PointNet++. We draw four conclusions based on observations in Tab. 4 and 5. (1) Data scaling improves performance for both classification and segmentation tasks. For example, point resampling is shown to boost the performance by 2.5% OA in ScanObjectNN. Taking the entire scene as input instead of using the block or sphere subsampled input as done in PointNet++ [30] and other previous works [43,21,32] improves the segmentation result by 1.1% mIoU. (2) Height appending improves performance, especially for object classification. Height appending makes the network aware of the actual size of the objects, thus leading to an increase in accuracy (+1.1% OA). (3) Color drop is a strong augmentation that significantly improves the performance of tasks where colors are available. Adopting color drop alone adds 5.9% mIoU in S3DIS area 5. We hypothesize that color drop forces the network to focus more on the geometric relationships between points, which in turn improves performance. (4) Larger models favor stronger data augmentation. Whereas random rotation drops the performance of PointNet++ by 0.3% mIoU in S3DIS (2 nd row in Tab. 5 data augmentation part), it is shown to be beneficial for larger-scale models (e.g. raises 1.5% mIoU on PointNeXt-B). Another example in ScanObjectNN shows that the removal of random jittering also adds 1.1% OA. In general, with the improved data augmentations, the OA of PointNet++ in ScanObjectNN and the mIoU in S3DIS area 5 are increased by 5.8% and 9.5%, respectively. Optimization techniques involve loss functions, optimizers, learning rate schedulers, and hyperparameters. As shown in Tab. 4 and 5, Label Smoothing, AdamW [27] optimizer, and Cosine Decay consistently boost performance in both classification and segmentation tasks. This reveals that the more developed optimization methods such as label smoothing and AdamW are generally good for optimizing a neural network. Compared to Step Decay, Cosine Decay is also easier to tune (usually only the initial and minimum learning rates are required) and can achieve a performance similar to Step Decay. Regarding hyperparameters, using a learning rate greater than that used in PointNet++ improves the segmentation performance in S3DIS. In general, our training strategies consisted of stronger data augmentation and modern optimization techniques can increase the performance of PointNet++ from 77.9% to 86.1% OA in ScanObjectNN dataset, impressively surpassing SOTA PointMLP by 0.7%. The mIoUs in S3DIS area 5 and S3DIS 6-fold (illustrated in Fig. 1) are boosted by 11.7 and 13.6 absolute percentage points, respectively. Our observations imply that a significant portion of the performance gap between classical PointNet++ and SOTA is due to the training strategies. Generalize to other networks. Although the training strategies are proposed for PointNet++ [30], we find that they can be applied to other methods such as PointNet [29], DGCNN [46], and PointMLP [28], and also improve their performance. Such generalizability is validated in ScanObjectNN [44]. As shown in Tab. 6, the OA of the representative methods can all be improved when equipped with our training strategies. Model Scaling Receptive field scaling includes both radius scaling and normalizing ∆ p defined in Eqn. (2), which are also validated in Tab. 4 and 5. The radius is dataset specific, while down-scaling the radius from 0.2 to 0.15 improves 0.3% OA in ScanObjectNN, keeping the radius the same as 0.1 achieves the best performance in S3DIS. Regarding normalizing ∆ p , it improves the performance in ScanObjectNN and S3DIS by 0.3 OA and 0.4 mIoU, respectively. Furthermore, in Tab. 7, we show that normalizing ∆ p has a larger impact (2.3 mIoU in S3DIS dataset) on the bigger model PointNext-XL. Model scaling scales PointNet++ by the proposed InvResMLP and some macro-architectural changes (see Sec. 3.2.2). In Tab. 4, we show that PointNeXt-S using the stem MLP, the symmetric decoder, and the residual connection in the SA block improves 1.0% OA in ScanObjectNN. Performance in the large-scale S3DIS dataset can be further unveiled (from 63.8% to 70.5% mIoU) by up-scaling PointNeXt-S using more blocks of the proposed InvResMLP, as demonstrated in Tab. 5. Furthermore, in Tab. 7, we ablate each component of the proposed InvResMLP block and different stage ratios in S3DIS area 5 using the best-performed model PointNeXt-XL as the baseline. As observed, each architectural change indeed contributes to increased performance. Among all changes, the residual connection is the most essential, without which the mIoU will drop from 70.5% to only 64.0%. The separable MLPs increase 3.9% mIoU while speeding up the network 3 times. Removing the inverted bottleneck from the baseline leads to a drop of 1.5% mIoU with less than a 1% gain in speed. Adding more blocks inside each stage after removing inverted bottleneck can improve its performance to 69.7 ± 0.3 but is still lower than the baseline. Tab. 7 also shows the performance of naive width scaling that increases the width of PointNet++ from 32 to 256 to match the throughput of PointNeXt-XL, naive depth scaling to append more SA blocks in PointNet++ to obtain the same number of blocks of PointNext-XL whose B = (3,6,3,3), and naive compound scaling to double the width of the naive depth scaled model to the same width as PointNeXt-XL (C = 64). Our proposed model scaling strategy achieves much higher performance than these naive scaling strategies, while being much faster. Related Work Point-based methods process point clouds directly using their unstructured format compared to voxel-based methods [10,4] and multi view-based methods [37,12,9]. PointNet [29], the pioneering work of point-based methods, proposes to model the permutation invariance of points with shared MLPs by restricting feature extraction to be pointwise. PointNet++ [30] is presented to improve PointNet by capturing local geometric structures. Currently, most point-based methods focus on the design of local modules. [46,45,31] rely on graph neural networks. [51,22,43,42] project point clouds onto pseudo grids to allow for regular convolutions. [48,23,24] adaptively aggregate neighborhood features through weights determined by the local structure. In addition, very recent methods leverage Transformer-like networks [56,17] to extract local information through self-attention. Our work does not follow this trend in local module design. In contrast, we shift our attention to another important but largely under-explored aspect, i.e., the training and scaling strategies. Training strategies are studied recently in [2,47,26] on image classification. In the point cloud domain, SimpleView [9] is the first work to show that training strategies have a large impact on the performance of a neural network. However, SimpleView simply adopts the same training strategies as DGCNN [46]. On the contrary, we conducted a systematic study to quantify the effect of each data augmentation and optimization technique, and propose a set of improved training strategies that boost the performance of PointNet++ [30] and other representative works [29,46,28]. Model scaling can significantly improve the performance of a network, as shown in pioneering works in various domains [40,55,21]. Compared to PointNet++ [30] that uses parameters less than 2M, most current prevailing networks consist of parameters greater than 10 M, such as KPConv [43] (15M) and PointMLP [28] (13M). In our work, we explore model scaling strategies that can scale up PointNet++ in an effective and efficient manner. We offer practical suggestions on scaling technologies that improve performance, namely using residual connections and an inverted bottleneck design, while maintaining throughput by using separable MLPs. Conclusion and Discussion In this paper, we demonstrate that with improved training and scaling strategies, the performance of PointNet++ can be increased to exceed the current state of the art. More specifically, we quantify the effect of each data augmentation and optimization technique that are widely used today, and propose a set of improved training strategies. These strategies can be easily applied to boost the performance of PointNet++ and other representative works. We also introduce the Inverted Residual MLP block into PointNet++ to develop PointNeXt. We demonstrate that PointNeXt has superior performance and scalability over PointNet++ on various benchmarks while maintaining high throughput. This work aims to guide researchers toward paying more attention to the effects of training and scaling strategies and motivate future work in this direction. B Training Strategies Comparison In this section, we summarize the training strategies used in representative point-based methods such as DGCNN [46], KPConv [43], PointMLP [28], Point Transformer [56], Stratified Transformer [17], PointNet++ [30], and our PointNeXt on S3DIS [1] in Tab. I, on ScanObjectNN [44] in Tab. II, on ScanNet [5] in Tab. III, and on ShapeNetPart [53] in Tab. IV, respectively. C Qualitative Results We provide qualitative results of PointNeXt-XL for S3DIS (Fig. II) and PointNeXt-S (C = 160) for ShapeNetPart (Fig. III). The qualitative results of PointNet++ trained with the original training strategies are also included in the figures for comparison. On both datasets, PointNeXt produces predictions closer to the ground truth compared to PointNet++. More specifically, on S3DIS shown in (Fig. II), PointNeXt is able to segment hard classes, including doors (1 st , 3 rd , and 4 th rows), clutter (1 st and 3 rd rows), chairs (2 nd row), and the board (4 th row), while PointNet++ fails to segment properly to some extent. On ShapeNetPart (Fig. III), PointNeXt precisely segments wings of an airplane (1 st row), microphone of an earphone(2 nd row), body of a motorbike(3 rd row), fin of a rocket(4 th row), and bearing of a skateboard (5 th row). D Classification Architecture As illustrated in Fig. I, the classification architecture shares the same encoder as the segmentation one. The output features of the encoder are passed to a global pooling layer (i.e. global max-pooling) to acquire a global shape representation for classification. Note that the points are only downsampled by a factor of 2 in each stage, since the number of input points in classification tasks is usually small, e.g. 1024 or 2048 points. E Societal Impact We do not see an immediate negative societal impact from our work. We notice that the way we discover the improved training and scaling strategies may consume a little more computing resources and affect the environment. Nevertheless, the improved training and scaling strategies will make researchers pay more attention to aspects other than architectural changes, which in the long term makes research in computer vision more diverse and generally better. Figure I :Figure II :Figure III : IIIIIIPointNeXt architecture for classification. The classification architecture shares the same encoder as the segmentation architecture. Qualitative comparisons of PointNet++ (2 nd column), PointNeXt (3 rd column), and Ground Truth (4 th column) on S3DIS semantic segmentation. The input point cloud is visualized with original colors in the 1 st column. Differences between PointNet++ and PointNeXt are highlighted with red dash circles. Zoom-in for details. Qualitative comparisons of PointNet++ (left), PointNeXt (middle), and Ground Truth (right) on ShapeNetPart part segmentation. Table 2 : 23D object classification in ScanObjectNN and ModelNet40. Averaged results in three random runs using 1024 points as input without normals and without voting are reported.ScanObjectNN (PB_T50_RS) ModelNet40 Params. FLOPs Throughput Method OA (%) mAcc (%) OA (%) mAcc (%) M G (ins./sec.) PointNet [29] 68.2 63.4 89.2 86.2 3.5 0.9 4212 PointCNN [22] 78.5 75.1 92.2 88.1 0.6 - 44 DGCNN [46] 78.1 73.6 92.9 90.2 1.8 4.8 402 DeepGCN [20] - - 93.6 90.9 2.2 3.9 263 KPConv [43] - - 92.9 - 14.3 - - ASSANet-L [32] - - 92.9 - 118.4 - 153 SimpleView [9] 80.5±0.3 - 93.0±0.4 90.5±0.8 0.8 - - MVTN [12] 82.8 - 93.5 92.2 3.5 1.8 236 Point Cloud Transformer [11] - - 93.2 - 2.9 2.3 - CurveNet [50] - - 93.8 - 2.0 - 22 PointMLP [28] 85.4±1.3 83.9±1.5 94.1 91.3 13.2 31.3 191 PointNet++ [30] 77.9 75.4 91.9 - 1.5 1.7 1872 PointNet++ (ours) 86.1±0.7(+8.2) 84.2±0.9(+8.8) 92.8±0.1(+0.9) 89.9±0.8 1.5 1.7 1872 PointNeXt-S (ours) 87.7±0.4(+9.8) 85.8±0.6(+10.4) 93.2±0.1(+1.3) 90.8±0.2 1.4 1.6 2040 PointNeXt-XL outperforms PointNet++ by 18.0% mIoU in validation and achieves 71.2% mIoU in testing, beating the recent methods Point Transformer [56] and CBL [41]. Table 3 : 3Part segmentation in ShapeNetPart.Method ins. mIoU cls. mIoU Params. FLOPs Throughput PointNet [29] 83.7 80.4 3.6 4.9 1184 DGCNN [46] 85.2 82.3 1.3 12.4 147 KPConv [43] 86.4 85.1 - - 44 CurveNet [50] 86.8 - - - 97 ASSANet-L [32] 86.1 - - - 640 Point Transformer [56] 86.6 83.7 7.8 - 297 PointMLP [28] 86.1 84.6 - - 270 Stratifiedformer [17] 86.6 85.1 - - 398 PointNet++ [30] 85.1 81.9 1.0 4.9 708 PointNeXt-S 86.7±0.0(+1.6) 84.4±0.2(+2.5) 1.0 4.5 782 PointNeXt-S (C=64) 86.9±0.1(+1.8) 84.8±0.5(+2.9) 3.7 17.8 331 PointNeXt-S (C=160) 87.0±0.1(+1.9) 85.2±0.1(+3.3) 22.5 110.2 76 ShapeNetPart [53] is a widely-used dataset for object-level part segmenta- tion. It consists of 16, 880 models from 16 different shape categories, 2-6 parts for each category, and 50 part labels in total. As shown in Tab. 3, our PointNeXt-S with default width (C = 32) obtains a performance comparable Table 4 : 4Additive study of sequentially applying train- ing and scaling strategies for classification on ScanOb- jectNN. We use light green, purple, yellow, and pink background colors to denote data augmentation, op- timization techniques, receptive field scaling, and model scaling, respectively. Improvements OA (%) ∆ PointNet++ 77.9 - + Point resampling 81.4 ± 0.6 +2.5 − Jittering 82.5 ± 0.4 +1.1 + Height appending 83.6 ± 0.4 +1.1 + Random scaling 83.7 ± 0.2 +0.1 + Label Smoothing 85.0 ± 0.5 +1.3 + Adam → AdamW 85.6 ± 0.1 +0.6 + Step Decay → Cosine Decay 86.1 ± 0.7 +0.5 + Radius 0.2 → 0.15 86.4 ± 0.3 +0.3 + Normalizing ∆p (Eqn. (2)) 86.7 ± 0.3 +0.3 + Scale up (PointNeXt-S) 87.7 ± 0.4 +1.0 Table 5 : 5Additive study of sequentially apply- ing training and scaling strategies for segmen- tation on S3DIS area 5. +/− denote adopt- ing/removing the strategy. Improvements mIoU (%) ∆ PointNet++ 51.5 - + Entire scene as input 52.6 ± 0.5 +1.1 − Rotation 52.9 ± 0.6 +0.3 + Height appending 53.4 ± 0.4 +0.5 + Color drop 59.3 ± 0.7 +5.9 + Color auto-contrast 61.0 ± 0.4 +0.7 + lr = 0.001 → 0.01 61.5 ± 0.5 +0.5 + Label Smoothing 61.9 ± 0.1 +0.4 + Adam → AdamW 62.5 ± 0.6 +0.6 + Step Decay → Cosine Decay 63.2 ± 0.4 +0.7 + Normalize ∆p 63.6 ± 0.4 +0.4 + Scale down (PointNeXt-S) 63.4 ± 0.8 -0.2 + Scale up (PointNeXt-B) 65.8 ± 0.5 +2.4 + Rotation 67.3 ± 0.2 +1.5 + Scale up (PointNeXt-L) 69.0 ± 0.5 +1.7 + Scale up (PointNeXt-XL) 70.5 ± 0.3 +1.5 Table 6 : 6Thegeneralizability of im- proved training strategies. OA on ScanObjectNN of networks trained with improved training strategies is reported. Method ours ∆ PointNet [29] 74.4 ± 0.9 +6.2 DGCNN [46] 86.0 ± 0.5 +7.9 PointMLP [28] 87.1 ± 0.7 +1.7 Table 7 : 7Ablate architectural changes on S3DIS area 5. − denotes removing from baseline. TP denotes throughput.Ablate mIoU ∆ TP baseline (PointNeXt-XL) 70.5 ± 0.3 - 45 − normalizing ∆p 68.2 ± 0.7 -2.3 45 − residual connection 64.0 ± 1.0 -6.5 45 − stem MLP 70.1 ± 0.4 -0.4 46 − Separable MLPs 66.6 ± 0.8 -3.9 15 − Inverted bottleneck 69.0 ± 0.4 -1.5 48 − Inverted bottleneck 69.7 ± 0.3 -0.8 43 stage ratio → (1:1:1:1) 69.8 ± 0.6 -0.7 52 stage ratio → (2:1:1:1) 69.4 ± 0.4 -1.1 41 stage ratio → (1:1:2:1) 69.9 ± 0.6 -0.6 47 stage ratio → (1:1:1:2) 69.5 ± 0.4 -1.0 48 stage ratio → (1:3:1:1) 70.1 ± 0.4 -0.4 39 naive width scaling 59.4 ± 0.1 -11.1 43 naive depth scaling 63.4 ± 0.5 -7.1 53 naive compound scaling 62.3 ± 1.2 -8.2 24 Limitation. Even though PointNeXt-XL is one of the largest models among all representative pointbased networks[30,43,15,56], its number of parameters (44M) is still below that of small networks in image classification such as Swin-S[25] (50M), ConNeXt-S[26] (50M), and ViT-B[8] (87M), and is far from their large variants, including Swin-L (197M), ConvNeXt-XL (350M), and ViT-L (305M). In this work, we do not push the model size further, mainly due to the smaller-scale nature of point cloud datasets compared to their larger image counterparts, such as ImageNet[6]. Moreover, our work is limited to existing modules since the focus is not on introducing new architectural changes.PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies -Supplementary Material -In this appendix, we provide additional content to complement the main manuscript:• Appendix A: A detailed description of Tab. 7.• Appendix B: Comparisons of training strategies for prior representative works and PointNeXt.• Appendix C: Qualitative comparisons on S3DIS and ShapeNetPart.• Appendix D: The architecture of PointNeXt for classification.• Appendix E: Societal impact.A Detailed Description for Manuscript Tab. 7Naive width scaling increases the channel size of PointNet++ from 32 to 256 to match the throughput of the baseline model, PointNeXt-XL. Naive depth scaling refers to appending more SA blocks (B = (3, 6, 3, 3), the same as PointNext-XL) in PointNet++. Furthermore, naive compound scaling doubles the width of naive depth scaled model to the same as PointNeXt-XL (C = 64). Compared to the PointNet++ trained with improved training strategies (63.2% mIoU, 186 ins./sec.), naive depth scaling (63.4% mIoU, 53 ins. / sec.) and naive width scaling (59.4% mIoU, 43 ins./sec.) only lead to a large overhead in throughput with insignificant improvement in accuracy. In contrast, our proposed model scaling strategy achieves much higher performance than the naive scaling strategies while being much faster. This can be observed by comparing PointNeXt-XL (70.5% mIoU, 45 ins./sec.) to the naive compound scaled PointNet++ (62.3% mIoU, 24 ins./sec.). Table I : ITraining strategies used in different methods for S3DIS segmentation. Method DGCNN KPConv PointTransformer PointNet++ PointNeXt (Ours)Epochs 101 500 100 32 100 Batch size 12 10 16 16 8 Optimizer Adam SGD SGD Adam AdamW LR 1 × 10 −3 1 × 10 −2 0.5 1 × 10 −3 0.01 LR decay step step multi step step cosine Weight decay 0 10 −3 10 −4 10 −4 10 −4 Label smoothing ε 0.2 Entire scene as input Random rotation Random scaling [0.8,1.2] [0.9,1.1] [0.9,1.1] Random translation Random jittering 0.001 Height appending Color drop 0.2 0.2 Color auto-contrast Color jittering mIoU (%) 56.1 70.6 73.5 54.5 74.9 Table II : IITraining strategies used in different methods for ScanObecjectNN classification.Method DGCNN PointMLP PointNet++ PointNeXt (Ours) Epochs 250 200 250 250 Batch size 32 32 16 32 Optimizer Adam SGD Adam AdamW LR 1 × 10 −3 0.01 10 −3 2 × 10 −3 LR decay step cosine step cosine Weight decay 10 −4 10 −4 10 −4 0.05 Label smoothing ε 0.2 0.2 0.3 Point resampling Random rotation Random scaling Random translation Random jittering Height appending OA (%) 78.1 85.7 77.9 87.7 Table III : IIITraining strategies used in different methods for ScanNet segmentation. KPConv PointTransformer Stratified Transformer PointNet++ PointNeXt (Ours)Method Epochs 500 100 100 200 100 Batch size 10 16 8 32 2 Optimizer SGD SGD AdamW Adam AdamW LR 1 × 10 −2 5 × 10 −1 6 × 10 −3 1 × 10 −3 1 × 10 −3 LR decay step multi step multi step with warm up step multi step Weight decay 10 −3 10 −4 5 × 10 −2 10 −4 10 −4 Entire scene as input Random rotation Random scaling [0.9,1.1] [0.9,1.1] [0.8,1.2] [0.8,1.2] Random translation Random jittering 0.001 Height appending Color drop 0.2 0.2 Color auto-contrast Color jittering Test mIoU (%) 68.6 - 73.7 55.7 71.2 Table IV : IVTraining strategies used in different methods for ShapeNetPart segmentation. Method DGCNN KPConv PointNet++ PointNeXt (Ours)Epochs 201 500 201 300 Batch size 16 16 32 8 Optimizer Adam SGD Adam AdamW LR 3 × 10 −3 1 × 10 −2 1 × 10 −3 0.001 LR decay step step step multi step Weight decay 0.0 10 −3 0.0 10 −4 Label smoothing ε Random rotation Random scaling [0.9,1.1] [0.8,1.2] Random translation Random jittering 0.001 0.001 Normal Drop Height appending mIoU (%) 85.2 86.4 85.1 87.0 Set Abstraction Set Abstraction Set Abstraction Set Abstraction InvResMLP InvResMLP InvResMLP InvResMLP MLP [N,32] [N/2,64] [N/4,128] [N/8,256] [N/16,512] Global Pooling The voting strategy combines results by using randomly augmented points as input to enhance performance. Acknowledgement. The authors would like to thank the reviewers of NeurIPS'22 for their constructive suggestions. 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In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2019. Revisiting point cloud classification: A new benchmark dataset and classification model on real-world data. Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV). Mikaela Angelina Uy, Quang-Hieu Pham, Binh-Son Hua, Duc Thanh Nguyen, and Sai-Kit Yeungthe IEEE/CVF International Conference on Computer Vision (ICCV)Mikaela Angelina Uy, Quang-Hieu Pham, Binh-Son Hua, Duc Thanh Nguyen, and Sai-Kit Yeung. Revisit- ing point cloud classification: A new benchmark dataset and classification model on real-world data. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2019. Graph attention convolution for point cloud semantic segmentation. 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Resnet strikes back: An improved training procedure in timm. arXiv preprint arXiv:2110.00476, 2021. Pointconv: Deep convolutional networks on 3d point clouds. Wenxuan Wu, Zhongang Qi, Li Fuxin, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)Wenxuan Wu, Zhongang Qi, and Li Fuxin. Pointconv: Deep convolutional networks on 3d point clouds. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019. 3d shapenets: A deep representation for volumetric shapes. 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Li Yi, G Vladimir, Duygu Kim, Ceylan, Mengyan Shen, Hao Yan, Arcewu Su, Qixing Lu, Alla Huang, Leonidas Sheffer, Guibas, ACM Transactions on Graphics (TOG). 356210Li Yi, Vladimir G Kim, Duygu Ceylan, I Shen, Mengyan Yan, Hao Su, ARCewu Lu, Qixing Huang, Alla Sheffer, Leonidas Guibas, et al. A scalable active framework for region annotation in 3d shape collections. ACM Transactions on Graphics (TOG), 35(6):210, 2016. Point-bert: Pre-training 3d point cloud transformers with masked point modeling. Xumin Yu, Lulu Tang, Yongming Rao, Tiejun Huang, Jie Zhou, Jiwen Lu, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)2022Xumin Yu, Lulu Tang, Yongming Rao, Tiejun Huang, Jie Zhou, and Jiwen Lu. Point-bert: Pre-training 3d point cloud transformers with masked point modeling. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2022. 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{'abstract': 'PointNet++ is one of the most influential neural architectures for point cloud understanding. Although the accuracy of PointNet++ has been largely surpassed by recent networks such as PointMLP and Point Transformer, we find that a large portion of the performance gain is due to improved training strategies, i.e. data augmentation and optimization techniques, and increased model sizes rather than architectural innovations. Thus, the full potential of PointNet++ has yet to be explored. In this work, we revisit the classical PointNet++ through a systematic study of model training and scaling strategies, and offer two major contributions. First, we propose a set of improved training strategies that significantly improve PointNet++ performance. For example, we show that, without any change in architecture, the overall accuracy (OA) of PointNet++ on ScanObjectNN object classification can be raised from 77.9% to 86.1%, even outperforming state-of-theart PointMLP. Second, we introduce an inverted residual bottleneck design and separable MLPs into PointNet++ to enable efficient and effective model scaling and propose PointNeXt, the next version of PointNets. PointNeXt can be flexibly scaled up and outperforms state-of-the-art methods on both 3D classification and segmentation tasks. For classification, PointNeXt reaches an overall accuracy of 87.7% on ScanObjectNN, surpassing PointMLP by 2.3%, while being 10× faster in inference. For semantic segmentation, PointNeXt establishes a new state-of-theart performance with 74.9% mean IoU on S3DIS (6-fold cross-validation), being superior to the recent Point Transformer. The code and models are available at https://github.com/guochengqian/pointnext. * Equal contribution. † Corresponding authors.Preprint. Under review.', 'arxivid': '2206.04670', 'author': ['Guocheng Qian ', 'Yuchen Li ', 'Houwen Peng \nMicrosoft Research\n\n', '† ', 'Jinjie Mai ', 'Hasan Abed ', 'Al Kader Hammoud ', 'Mohamed Elhoseiny ', 'Bernard Ghanem ', '\nKing Abdullah University of Science and Technology (KAUST)\n\n', 'Guocheng Qian ', 'Yuchen Li ', 'Houwen Peng \nMicrosoft Research\n\n', '† ', 'Jinjie Mai ', 'Hasan Abed ', 'Al Kader Hammoud ', 'Mohamed Elhoseiny ', 'Bernard Ghanem ', '\nKing Abdullah University of Science and Technology (KAUST)\n\n'], 'authoraffiliation': ['Microsoft Research\n', 'King Abdullah University of Science and Technology (KAUST)\n', 'Microsoft Research\n', 'King Abdullah University of Science and Technology (KAUST)\n'], 'corpusid': 249538578, 'doi': '10.48550/arxiv.2206.04670', 'github_urls': ['https://github.com/guochengqian/pointnext.', 'https://github.com/yanx27/Pointnet_Pointnet2_'], 'n_tokens_mistral': 23409, 'n_tokens_neox': 20272, 'n_words': 10861, 'pdfsha': '4ea12bc8dab90e3b7b01b19148ff37ad325f54cd', 'pdfurls': ['https://export.arxiv.org/pdf/2206.04670v2.pdf'], 'title': ['PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies', 'PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies', 'PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies', 'PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies'], 'venue': []}

arxiv

Modulation of breathers in the three-dimensional nonlinear Gross-Pitaevskii equation 22 Nov 2010 A T Avelar Instituto de Física Universidade Federal de Goiás 74.001-970Goiânia, GoiásBrazil D Bazeia Departamento de Física Universidade Federal da Paraíba 58.059-900João Pessoa, ParaíbaBrazil W B Cardoso Instituto de Física Universidade Federal de Goiás 74.001-970Goiânia, GoiásBrazil Modulation of breathers in the three-dimensional nonlinear Gross-Pitaevskii equation 22 Nov 2010 In this paper we present analytical breather solutions of the three-dimensional nonlinear generalized Gross-Pitaevskii equation. We use an Ansatz to reduce the three-dimensional equation with space-and time-dependent coefficients into an one-dimensional equation with constant coefficients. The key point is to show that both the space-and time-dependent coefficients of the nonlinear equation can contribute to modulate the breather excitations. We briefly discuss the experimental feasibility of the results in Bose-Einstein condensates. Introduction -Breathers or breathing solutions are nonlinear excitations which concentrate energy in a localized and oscillatory manner. In various physical systems, such as in Josephson junctions [1,2], charge density wave systems [3], 4-methylpyridine crystals [4], metallic nanoparticles [5], conjugated polymers [6], micromechanical oscillator arrays [7], antiferromagnetic Heisenberg chains [8,9], and semiconductor quantum wells [10], the breather excitations play an important role, directly affecting the electronic, magnetic, optical, vibrational and transport properties of the systems. In the above mentioned studies, one usually considers genuine breathers, i.e., solutions which oscillate in time when the nonlinear equation presents constant coefficients (i.e., without modulation). However, in a more realistic scenario the several parameters that characterize the physical systems may depend on both space and time, leading to breather solutions that can be modulated in space and time. The presence of nonuniform and time-dependent parameters opens interesting perspectives not only from the theoretical point of view, for investigation of nonautonomous nonlinear equations, but also from the experimental point of view, for the study of the physical properties of the systems. In this context, in a recent work we have considered modulation of genuine breather solutions in cigar-shaped Bose-Einstein condensates (BECs) with potential and nonlinearity depending on both space and time, in the one-dimensional (1D) case [11]. The study of BECs of dilute gases of weakly interacting bosons, realized for the first time in 1995 on vapors of rubidium [12] and sodium [13], constitutes a very interesting scenario to modulate breathers, since they are well described by a threedimensional (3D) Gross-Pitaevskii (GP) equation arising from a mean-field dynamics [14]. In the BEC context, one finds high experimental flexibility to control nonlinearity via Feshbach resonance, and confinement profile via optical lattices and harmonic and dipole traps [15], and there we can also investigate the effects of dimensionality reduction on the soliton solution. In the case of a strong trapping in two spatial directions, the 3D GP equation reduces to the simpler one-dimensional (1D) form, giving rise to the so-called cigar-shaped configuration. The 1D GP equation is a nonlinear Schrödinger equation, which can also be used to investigate pulse propagation in bulk crystals or optical fibers [16]. In a former work, however, the search for analytical solutions of the 1D GP equation with stationary inhomogeneous coefficients has been implemented via similarity transformation [17]. More recently, however, the case of space-and time-dependent coefficients were considered for the cubic [18], the cubic-quintic [19], the quintic [20], and also the GP equation in higher dimensions [21]. The similarity transformation was also used to study self-similar optical pulses in competing cubic-quintic nonlinear media with distributed coefficients [22], nonautonomous matter-wave solitons near the Feshbach resonance [23], bright and dark solitons in a periodically attractive and repulsive potential with nonlinearities modulated in space and time [24], solitons of twocomponent Bose-Einstein condensates modulated in space and time [25], and quantized quasi-two-dimensional Bose-Einstein condensates with spatially modulated nonlinearity [26]. On the other hand, the search for analytical solutions of the 3D GP equation has attracted a lot of attention due to the fact that solutions of higher-dimensional GP equation with constant coefficients are usually unstable [27], while the nonautonomous GP equation can engender conditions which contribute to stabilize the corresponding solutions [28]. Taking into account this possibility, exact solutions to 3D GP equation with varying potential and nonlinearities were proposed in Ref. [29], while in Ref. [30] the authors studied analytical 3D bright solitons and soliton pairs in BECs with time-space modulation. In this paper, our aim is to show that genuine breather solutions can be modulated by the 3D GP equation with space-and time-dependent potential, if one includes time-dependent coefficient to describe the cubic nonlinearity. In this way, we extend our recent work [11] to the more realistic 3D case. To this end, we use an Ansatz that changes the 3D GP equation into specific 1D equation with constant coefficients, which is easier to solve. As a consequence, however, we have to deal with a set of coupled equations, to ensure validity of the similarity transformation. Below we present explicit results for three distinct choices of potential and nonlinearity. Generalities -We start with the 3D GP equation given by i ∂ψ ∂t = − 1 2 ∇ 2 ψ + v(r, t)ψ + g(t)|ψ| 2 ψ,(1) where ψ = ψ(r, t), r ∈ R 3 , ∇ ≡ (∂ x , ∂ y , ∂ z ), and v(r, t) and g(t) are real functions representing the potential and the cubic nonlinearity, respectively. Our goal is to find breather solutions which obey the above Eq. (1). Using the Ansatz ψ = ρ(t)e iη(r,t) Φ[ζ(r, t), τ (t)],(2) we can transform the above equation into the 1D GP equation iΦ τ = − 1 2 Φ ζζ + G|Φ| 2 Φ,(3) where Φ τ ≡ ∂Φ/∂τ , Φ ζζ ≡ ∂ 2 Φ/∂ζ 2 , and G is a constant factor. Using (2) into (1) leads to (3), for ρ, η, ζ, and τ obeying the following equations τ t = |∇ζ| 2 , (4a) ζ t + (∇η) · (∇ζ) = 0, (4b) 2ρ t + ρ∇ 2 η = 0. (4c) Here the potential and the nonlinearity assume the form v(r, t) = −η t − |∇η| 2 ,(5) and g(t) = G|∇ζ| 2 ρ 2 .(6) Note that the potential (5) and the nonlinearity (6) are functions which in general depends on the real phase η(r, t) and the amplitude ζ(r, t). In this way, one can use the Eq. (4a) to obtain the general form of ζ, given by ζ(r, t) = c 1 (t)x + c 2 (t)y + c 3 (t)z + c 4 (t),(7) where the coefficients c j (j = 1, 2, 3, 4) are time-dependent functions, obeying the relationship τ t = c 2 1 + c 2 2 + c 2 3 .(8) Now, substituting (7) into (4b) leads to η in the general form η(r, t) = d 1 x 2 + d 2 y 2 + d 3 z 2 + d 4 xy + d 5 xz + d 6 yz + d 7 x + d 8 y + d 9 z + d 10 ,(9) where the d j are time-dependent coefficients which obey the equationṡ c 1 + 2c 1 d 1 + c 2 d 4 + c 3 d 5 = 0, (10a) c 2 + c 1 d 4 + 2c 2 d 2 + c 3 d 6 = 0, (10b) c 3 + c 1 d 5 + c 2 d 6 + 2c 3 d 3 = 0, (10c) c 4 + c 1 d 7 + c 2 d 8 + c 3 d 9 = 0,(10d) whereċ j ≡ dc j /dt. Next, from (4c) and (9) we obtain ρ(t) = exp − (d 1 + d 2 + d 3 )dt .(11) Inserting (9) into (5) leads to the potential v(r, t) = ω 1 x 2 + ω 2 y 2 + ω 3 z 2 + ω 4 xy + ω 5 xz + ω 6 yz + ω 7 x + ω 8 y + ω 9 z + ω 10 , where ω = ω(t), with ω 1 =ḋ 1 + 4d 2 1 + d 2 4 + d 2 5 , (13a) ω 2 =ḋ 2 + 4d 2 2 + d 2 4 + d 2 6 , (13b) ω 3 =ḋ 3 + 4d 2 3 + d 2 5 + d 2 6 , (13c) ω 4 =ḋ 4 + 4d 1 d 4 + 4d 2 d 4 + 2d 5 d 6 , (13d) ω 5 =ḋ 5 + 4d 1 d 5 + 4d 3 d 5 + 2d 4 d 6 , (13e) ω 6 =ḋ 6 + 4d 2 d 6 + 4d 3 d 6 + 2d 4 d 5 , (13f) ω 7 =ḋ 7 , ω 8 =ḋ 8 , ω 9 =ḋ 9 , ω 10 =ḋ 10 . (13g) Finally, we substitute (7) and (11) into (6) to get g(t) = Gτ t exp (d 1 + d 2 + d 3 ) dt .(14) Here we note that one can get solutions through the c j coefficients, that is, for specific choices of potential and nonlinearity, one can construct the c j functions, or, for specific choices of c j , one can get the corresponding potential and nonlinearity. We illustrate the general situation with the examples below. Analytical solutions -To demonstrate the power of the method, we start considering a breather solution of the Eq. (3). A specific form of the two-soliton breather solution is obtained for G = −1, which corresponds to the explicit solution [31] Φ(ζ, τ ) = 4(cosh(3ζ) + 3e 4iτ cosh(ζ))e iτ /2 (cosh(4ζ) + 4 cosh(2ζ) + 3 cos(4τ )) . The binding potential of the two-soliton is equal to zero, corresponding to the unstable breather solution. Due to the vanishing binding potential, there is a splitting of the solution (15) into two independent solitons at some point in time [32]. On the other hand, the spatial and temporal modulation of the nonlinearity and the potential allows that we get stable breather solutions, since the solution given by Eq. (2) presents non vanishing binding potential. Indeed, as we have recently shown [11], the modulation of the trapping potential gives support to the coexistence of the two-soliton in the breather solution, without splitting. With this in mind, in the following we study the modulation of the breather (15) in 3D spatial dimensions. Free evolution -Firstly, we consider the free evolution of the solution (15) in the BEC, i.e., we take v(r, t) = 0. In this way, we can get the free evolution if we set d j = 0 for j = 4, 5, ..., 10 and d j = −ċ j /2c j for j = 1, 2, 3. So, we will have ω j = −c j /2c j + 3ċ j 2 /2c 2 j with j = 1, 2, 3. As an example, one gets ω j = 0 setting c j = 1/ √ 3 (and c 4 = 0), corresponding to the following choice of the nonlinearity g = −1, with ρ = 1. In this case, the Eqs. (7), (8), and (9) change to ζ = (x+y +z)/ √ 3, τ = t, and η = 0, respectively. Thus, we obtain the breather solution of the Eq. (2) in the form ψ(r, t) = 4(cosh(3(x + y + z)) + 3e 12it cosh(x + y + z))e 3it/2 (cosh(4(x + y + z)) + 4 cosh(2(x + y + z)) + 3 cos(12t)) . In Fig. 1 we depict the breather solution (16) in the (x, t) plane, considering y = z = 0, for (a) c 1 = c 2 = c 3 = 1/ √ 3, (b) c 1 = c 2 = c 3 = 1, (c) c 1 = c 2 = c 3 = 1.5, and (d) c 1 = c 2 = c 3 = 2. Similar behavior is obtained in the (y, t) or (z, t) plane. There we clearly see that the oscillatory frequency increases with the increasing of the values of c j . We further illustrate this fact in Fig. 2, displaying |ψ| 2 at the spatial origin, considering c 1 = 0.3, 0.6, and 1, respectively, with c 2 = c 3 = 1/ √ 3. Harmonic potential -In this second example we consider the case of harmonic potential. Firstly, we look for solutions satisfying d j = −ċ j /2c j for j = 1, 2, 3 and d j = 0 for j = 4, 5, ...10. As before, adjusting appropriately the values of c j one can obtain the harmonic potential. We will analyze two cases: i) harmonic potential in a single spatial direction, say x, which is obtained with c 1 = 1 + 0.5 cos(t) and c 2 = c 3 = 1/ √ 3; ii) harmonic potential in the three spatial directions, which is obtained with c j = 1 + 0.5 cos(t) for j = 1, 2, 3. In the case i), let us have τ = 1.79t+sin(t)+cos(t) sin(t)/8, ζ = (1+0.5 cos(t))x+ √ 3y/3+ √ 3z/3, ρ = 1 + 0.5 cos(t), η = sin(t)x 2 /4(1 + 0.5 cos(t)), and v(x, t) = − cos(t)x 2 /4(1 + 0.5 cos(t)) − 0.375 sin(t) 2 x 2 /(1 + 0.5 cos(t)) 2 , which is physically implemented by a time-dependent harmonic potential plus an optical superlattice [33]. The nonlinearity and potential are displayed in Fig. 3a and 3b, respectively. Also, in Fig. 4 we depict the breather solution for the case i, in the plane (a) (x, t), (b) (y, t), and (c) (z, t), respectively. Moreover, in Fig. 5a we plot the breather profile at the origin. Next, we consider the case ii). Here we have τ = 3.375t + 3 sin(t) + 0.375 cos(t) sin(t), ζ = (1 + 0.5 cos(t))x + (1 + 0.5 cos(t))y+(1+0.5 cos(t))z, ρ = (1+0.5 cos(t)) ( 3/2)/2, η = sin(t)[x 2 +y 2 +z 2 ]/4(1+0.5 cos(t)), g = −3/(1+0.5 cos(t)), and v(r, t) = (cos(t) 2 − cos(t) − 3/2)(x 2 + y 2 + z 2 )/(4 + 4 cos(t) + cos(t) 2 ). This nonlinearity can be obtained experimentally in BEC by time modulated Feshbach resonance, but an optical trapping switching from red-detuned to blue-detuned laser-beam, and vice-versa, in a periodic fashion will be required for implementation of the time-dependent potential. In Figs. 4a and 5b we depict the profile of breather solution in the (x, t) plane and at the spatial origin, respectively. Linear potential -As a third example, let us consider the case in which the BEC is trapped by a specific linear potential. Here we search for solutions of the Eq. (12) with ω j = 0 for j = 1, 2, ..., 6. For simplicity, we also consider ω 10 = 0. To this end, we choose d j = 0 for j = 1, 2, ..., 6, d 10 = 0, c 1 = c 2 = c 3 = 1/ √ 3, c 4 = sin(t), and d j = −ċ 4 /3c j for j = 1, 2, 3 to satisfy the Eq. (10d). In this case, we get τ = t, ζ = (x + y + z)/ √ 3 + sin(t), ρ = 1, η = − cos(t)(x + y + z)/ √ 3, g = −1, and v(r, t) = − sin(t)(x + y + z)/ √ 3 − cos 2 (t). In Fig. 6 we depict the breather solution in the (x, t) plane, in the presence of a linear potential. The solution presents similar behavior in the other (y, t) and in the (z, t) planes. Here we note that the choice c 4 = c 4 (t) = 0 makes the center of mass of the solution to move, as expected. Ending comments -In this work we have studied the presence of breather solutions in the 3D GP equation, in the case of space-and time-dependent potential, with cubic nonlinearity described by time-dependent coefficient. We have obtained analytical solutions through an Ansatz which changes the 3D equation into specific 1D equation. The results show that the breather solution can be nicely modulated in space and time. We have considered three distinct examples of potential and nonlinearity: the free evolution of the breather, the case of the presence of harmonic potential and another one, in which the system is driven by linear potential. The breather solutions can be controlled through the presence of external apparatus, and this may motivate new research in the field since the modulation can generate stable excitations. The authors would like to thank CAPES, CNPQ, and FUNAPE/GO for partial financial support. FIG. 1 : 1(Color online) Plots of the 3D breather solution |ψ| 2 for v(r, t) = 0 and g = −1, in the (x, t) plane, considering y = z = 0. We display the cases (a) c1 = c2 = c3 = 1/ √ 3, (b) c1 = c2 = c3 = 1, (c) c1 = c2 = c3 = 1.5, and (d) c1 = c2 = c3 = 2. online) Plots of |ψ| 2 for v(r, t) = 0 and g = −1, at the spatial origin (0,0,0). We use c2 = c3 = 1/√ 3 to show the time behavior for c1 = 0.3 in dash-dot line (yellow), c1 = 0.6 in dashed line (green), and c1 = 1 in solid line (red). FIG. 3: (Color online) Plots of the nonlinearity (a) and potential (b) for the case with c1 = 1 + 0.5 cos(t) and c2 = c3 = 1/ √ 3. FIG. 4 : 4(Color online) Plots of the profile of the breather solution in the (a) (x, t), (b) (y, t), and (c) (z, t) plane, in the harmonic potential for the case i. Similar behavior appears in the case ii. FIG. 5 : 5(Color online) Profile of the breather solution for the cases (a) i and (b) ii, at the spatial origin (0, 0, 0). FIG. 6: (Color online) Plot of the breather solution in the (x, t) plane, for y = z = 0, in the presence of the linear potential. Similar behavior appears in the other (y, t) and (z, t) planes. . E Trías, J J Mazo, T P Orlando, Phys. Rev. Lett. 84741E. Trías, J. J. Mazo, and T. P. Orlando, Phys. Rev. Lett. 84, 741 (2000). . P Binder, Phys. Rev. Lett. 84745P. Binder et al., Phys. 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{'abstract': 'In this paper we present analytical breather solutions of the three-dimensional nonlinear generalized Gross-Pitaevskii equation. We use an Ansatz to reduce the three-dimensional equation with space-and time-dependent coefficients into an one-dimensional equation with constant coefficients. The key point is to show that both the space-and time-dependent coefficients of the nonlinear equation can contribute to modulate the breather excitations. We briefly discuss the experimental feasibility of the results in Bose-Einstein condensates.', 'arxivid': '1011.4824', 'author': ['A T Avelar \nInstituto de Física\nUniversidade Federal de Goiás\n74.001-970Goiânia, GoiásBrazil\n', 'D Bazeia \nDepartamento de Física\nUniversidade Federal da Paraíba\n58.059-900João Pessoa, ParaíbaBrazil\n', 'W B Cardoso \nInstituto de Física\nUniversidade Federal de Goiás\n74.001-970Goiânia, GoiásBrazil\n'], 'authoraffiliation': ['Instituto de Física\nUniversidade Federal de Goiás\n74.001-970Goiânia, GoiásBrazil', 'Departamento de Física\nUniversidade Federal da Paraíba\n58.059-900João Pessoa, ParaíbaBrazil', 'Instituto de Física\nUniversidade Federal de Goiás\n74.001-970Goiânia, GoiásBrazil'], 'corpusid': 15874485, 'doi': '10.1103/physreve.82.057601', 'github_urls': [], 'n_tokens_mistral': 7968, 'n_tokens_neox': 6661, 'n_words': 3593, 'pdfsha': 'bd00d892e48f874f95033252fbf5aa5e986fe4e5', 'pdfurls': ['https://arxiv.org/pdf/1011.4824v1.pdf'], 'title': ['Modulation of breathers in the three-dimensional nonlinear Gross-Pitaevskii equation', 'Modulation of breathers in the three-dimensional nonlinear Gross-Pitaevskii equation'], 'venue': []}

arxiv

A new topology on the space of unbounded selfadjoint operators and the spectral flow 26 Mar 2007 Charlotte Wahl A new topology on the space of unbounded selfadjoint operators and the spectral flow 26 Mar 2007arXiv:math/0607783v2 [math.FA] We define a new topology, weaker than the gap topology, on the space of selfadjoint unbounded operators on a separable Hilbert space. We show that the subspace of selfadjoint Fredholm operators represents the functor K 1 from the category of compact spaces to the category of abelian groups and prove a similar result for K 0 . We define the spectral flow of a continuous path of selfadjoint Fredholm operators generalizing the approach of Booss-Bavnek-Lesch-Phillips. Introduction The space of bounded Fredholm operators on a separable Hilbert space endowed with the norm topology is a classifying space for the functor K 0 from the category of compact spaces to the category of abelian groups [J] [A]. The index map realizes an isomorphism between the K-theory of a point and Z Z. Furthermore a particular connected component of the space of selfadjoint bounded Fredholm operators with the norm topology represents the functor K 1 [AS]. An isomorphism K 1 (S 1 ) ∼ = Z Z is given by the spectral flow, which was introduced in [APS]. These results can be applied to unbounded Fredholm operators by using the bounded transform D → D(1 + D * D) − 1 2 . However, since many important geometric applications involve unbounded operators, it is more convenient to work directly with the space of unbounded selfadjoint Fredholm operators. The gap topology on the space of unbounded selfadjoint operators is the weakest topology such that the maps D → (D ± i) −1 are continuous. Gap continuity is weaker than continuity of the bounded transform. Booss-Bavnek-Lesch-Phillips defined the spectral flow for gap continuous paths [BLP] and Joachim proved that the space of unbounded selfadjoint Fredholm operators endowed with the gap topology is a classifying space for K 1 and the space of Fredholm operators with the subspace topology (see §1) is a classifying space for K 0 [Jo]. In the first part of this paper we define and study a new topology on the space of unbounded selfadjoint operators. In this topology a path (D t ) t∈[0,1] is continuous if and only if the resolvents (D t ± i) −1 depend in a strongly continuous way on t and if there is an even function φ ∈ C ∞ c (IR) with supp φ = [−ε, ε] and φ ′ | (−ε,0) > 0 for some ε > 0 such that φ(D t ) is continuous in t. This topology is weaker than the gap topology. Compared with the latter it has some additional useful properties: The bounded transform of a continuous path is again continuous. If (D t ) t∈[0,1] is a continuous path of Fredholm operators and (U t ) t∈[0,1] is a strongly continuous path of unitaries, then (U t D t U * t ) t∈[0,1] is again a continuous path of Fredholm operators. We show that the space of selfadjoint Fredholm operators endowed with this topology represents K 1 and the space of Fredholm operators with the subspace topology represents K 0 . Furthermore we illustrate with an example that families of Fredholm operators that are continuous with respect to this topology but not gap continuous arise naturally from differential operators on noncompact manifolds. Along the way we indicate how these results generalize to regular Fredholm operators on a Hilbert C * -module. In the second part we define and study the spectral flow of a continuous path of selfadjoint Fredholm operators generalizing the approach of Booss-Bavnek-Lesch-Phillips and relate it to the winding number. The definition of the spectral flow given here is for paths with invertible endpoints equivalent to the definition of the noncommutative spectral flow in [W] applied to a separable Hilbert space. However, in [W] we used the theory of Hilbert C * -modules in an essential way. One aim of this paper is to recover the results of [W] for a Hilbert space using classical functional analysis. We refer to [W] for applications. A new topology on the space of unbounded selfadjoint operators Let H be a separable Hilbert space. Recall that a closed densely defined operator D on H is called Fredholm if its bounded transform F D := D(1 + D * D) − 1 2 is Fredholm. We denote the set of selfadjoint unbounded operators on H by S(H) and the set of selfadjoint unbounded Fredholm operators on H by SF (H). As usual, B(H) is the space of bounded operators on H endowed with the norm topology and K(H) is the subspace of compact operators. Throughout let B be a compact space. For a Banach space V we denote by C(B, V ) the Banach space of continuous functions from B to V equipped with the supremum norm. We write C(IR) for C(IR, C). For b ∈ B the evaluation map is ev b : C(B, V ) → V, f → f (b). For a map D : B → S(H) we define Dom D := {f ∈ C(B, H) | f (b) ∈ dom D(b) for all b ∈ B and Df ∈ C(B, H)} . Here Df : B → H is defined as b → D(b)f (b). First we note some useful facts about the functional calculus of selfadjoint operators. Proposition 1.1. Let D : B → S(H) be a map. The following conditions are equivalent: 1. At each point b ∈ B the set ev b (Dom D) ⊂ dom D(b) is a core for D(b). 2. The resolvents (D(b)±i) −1 depend in a strongly continuous way on b ∈ B. 3. For each φ ∈ C(IR) the operator φ(D(b)) depends in a strongly continuous way on b ∈ B. Proof. Set R λ (b) = (D(b) + λ) −1 . We show that (1) implies (2): Let λ = ±i. Since R λ (b) is uniformly bounded, it is enough to prove that Dom R λ is dense in C(B, H). Let f ∈ C(B, H) and let ε > 0. The assumption implies that the set ev b ((D + λ)(Dom D)) is dense in H for any b ∈ B. Hence by compactness there is a finite open covering {U j } j∈I of B and functions g j ∈ Dom D, j ∈ I, such that ( D(b) + λ)g(b) j − f (b) < ε for all b ∈ U j . Let {χ j } j∈I be a partition of unity subordinate to the covering {U j } j∈I and set f j (b) = (D(b) + λ)g j (b). Then j∈I χ j f j ∈ Dom R λ and f − j∈I χ j f j < ε. (2) ⇒ (3): Let φ ∈ C(IR). Since the algebra generated by the functions (x+i) −1 and (x− i) −1 is dense in C 0 (IR), the assertion holds for all ψ ∈ C 0 (IR), in particular for ψ( x) = φ(x)(x + i) −1 . Hence φ(D)f ∈ C(B, H) for f ∈ R i C(B, H) . By a similar argument as above, (2) implies that R i C(B, H) is dense in C(B, H). Since φ(D(b)) is uniformly bounded, this implies the assertion. (3) ⇒ (2) ⇒ (1) is clear. Lemma 1.2. Let D : B → S(H) be a map such that the resolvents (D(b) ± i) −1 depend in a strongly continuous way on b ∈ B. Assume that for each b ∈ B there is given a symmetric operator K(b) with dom D(b) ⊂ dom K(b) such that K(b)(D(b) + i) −1 is compact and depends continuously on b. Then for each φ ∈ C 0 (IR) φ(D) − φ(D + K) ∈ C(B, K(H)) . Proof. It is enough to prove the assertion for the functions (x ± i) −1 . Since Dom D = Dom(D+K), the previous proposition implies that (D(b)+K(b)±i) −1 depends in a strongly continuous way on b. Hence (D(b)+K(b)±i) −1 −(D(b)±i) −1 = −(D(b)+K(b)±i) −1 K(b)(D(b)±i) −1 is compact and depends continuously on b ∈ B. Lemma 1.3. Let X be a topological space. Let D : X → S(H) be a map such that the resolvents (D(x) ± i) −1 ∈ B(H) depend continuously on x ∈ X. Then φ(D(x)) depends continuously on x for any φ ∈ C 0 (IR). Proof. This follows again from the fact that the functions (x+i) −1 and (x−i) −1 generate a dense subalgebra of C 0 (IR). In particular, if D : X → B(H) is a continuous map such that D(x) is selfadjoint for each x ∈ X, then f (D(x)) depends continuously on x for all f ∈ C(IR). Recall that the gap topology on S(H) is the weakest topology such that the maps S(H) → B(H), D → (D + i) −1 , S(H) → B(H), D → (D − i) −1 are continuous. We denote by S(H) gap resp. SF (H) gap the set S(H) resp. SF (H) equipped with the gap topology. We refer to [BLP] for its properties. In the following we introduce a new topology on S(H). Let φ ∈ C ∞ c (IR) be an even function with supp φ = [−1, 1] and with φ ′ (x) > 0 for x ∈ (−1, 0). Define φ n ∈ C ∞ c (IR) by φ n (x) := φ(nx) for n ∈ IN. Let S n (H) be the set S(H) endowed with the weakest topology such that the maps S n (H) → H, D → (D + i) −1 x , S n (H) → H, D → (D − i) −1 x , S n (H) → B(H), D → φ n (D) are continuous for all x ∈ H. For any even function ψ ∈ C c (IR) with supp ψ ∈ (− 1 n , 1 n ) there is g ∈ C c (IR) with g(0) = 0 such that ψ = g • φ n . Hence S n (H) → B(H), D → ψ(D) is continuous. We will often make use of this property. It implies that the inclusion S m (H) → S n (H) is continuous for m ≤ n. Let S(H) be the set S(H) endowed with the direct limit topology. [BLP] shows that the bounded transform of a gap continuous family is in general not gap continuous. Define SF n (H) := {D ∈ SF (H) | φ n (D) ∈ K(H)}(f ± i) −1 ∈ C(IR), we get from Prop. 1.1 that (f (D) ± i) −1 x : B → H is continuous for any x ∈ H. Furthermore for m big enough supp(φ m • f ) ⊂ (− 1 n , 1 n ), hence φ m (f (D)) : B → B(H) is continuous. In particular the bounded transform B → S(H), b → F D(b) is continuous. The example of Fuglede presented in We need the following technical lemmata. Lemma 1.4. Assume that D : B → SF n (H) is continuous. Then for ψ ∈ C c (IR) with supp ψ ⊂ (− 1 n , 1 n ) we have that ψ(D) : B → K(H) is continuous. Proof. This follows from an elementary argument in the theory of Hilbert C *modules: Let B(C(B, H)) be the algebra of strongly continuous families of bounded operators on H with parameter space B and with adjoint depending in a strongly continuous way on the parameter. Endowed with the supremum norm this is a C * -algebra and C(B, K(H)) defines a closed ideal in B(C(B, H)). Let π : B(C(B, H)) → B(C(B, H))/C(B, K(H)) be the projection. Let g ∈ C(IR) with g(0) = 0 be such that ψ 2 = g • φ n . We have that π(φ n (D)) = 0, hence π(ψ(D)) 2 = g(π(φ n (D))) = 0. Since π(ψ(D)) is selfadjoint in the C *algebra B(C(B, H))/C(B, K(H)), it follows that π(ψ(D)) = 0, hence ψ(D) ∈ C(B, K(H)). Lemma 1.5. If (F b ) b∈B is a strongly continuous family of bounded selfadjoint operators such that (b → F 2 b −1) ∈ C(B, K(H)), then for any function φ ∈ C(IR) with φ(1) = φ(−1) = 1 we have that (b → φ(F b ) − 1) ∈ C(B, K(H)). Proof. The argument is similar to the proof of Lemma 1.4. We use its notation. We have that π((F b ) b∈B ) 2 = 1; hence the spectrum of π((F b ) b∈B ) is a subset of {−1, 1}; thus φ(π((F b ) b∈B )) − 1 = 0. Since φ(π((F b ) b∈B )) = π((φ(F b )) b∈B ), it follows that (b → φ(F b ) − 1) ∈ C(B, K(H)). The following two properties of the space SF(H) are useful: • Assume that D : B → SF(H) is continuous and that B ∋ b → U (b) is a map with values in the group of unitaries of B(H) such that U (b) depends in a strongly continuous way on b. Then U DU * : B → SF(H) is continuous. • If D : B → SF(H) is continuous, then f (D) : B → SF(H) is continuous for any non-decreasing continuous function f : IR → IR with f −1 (0) = {0}. The first property follows from the fact that the composition of a continuous family of compact operators with of a strongly continuous family of bounded operators is again continuous if the parameter space is compact. Furthermore since U is bounded below, the adjoint depends also in a strongly continuous way on b. Note that the second property does not assume the function to be odd. Taking the Lemma 1.4 into account one proves the property analogously to the corresponding one for S(H) from above. Lemma 1.6. Let D : B → SF(H) be continuous. Then there is an odd non- decreasing function χ ∈ C(IR) with χ −1 (0) = {0} and lim x→∞ χ(x) = 1 such that χ(D) 2 − 1 : B → K(H) is continuous. Proof. There is n ∈ IN such that D : B → SF n (H) is continuous. Then any non- decreasing χ ∈ C(IR) with χ −1 (0) = {0} and such that supp(χ 2 − 1) ⊂ (− 1 n , 1 n ) works. Definition 1.7. Let D : B → SF(H) be continuous. Then a function χ fulfilling the conditions of the previous lemma is called a normalizing function for D. The name "normalizing function" is borrowed from [HR]. The definition in [HR] is different since it applies to a different class of operators, but the underlying idea is the same. The definition of the spaces S(H) and SF(H) generalizes in a straightforward way to the case where H is a Hilbert C * -module. In this case we assume the unbounded operators to be regular. The spaces S(H) of S(H) gap share many properties as we will see in the following. We omit some details since the arguments resemble those in [BLP]. First we note that SF(H) is path-connected since SF (H) gap is path-connected by [BLP,Th. 1.10]. Let D 0 ∈ S(H). For n ∈ IN and ε > 0 we define U (ε, n, D 0 ) := {D ∈ S(H) | φ n (D) − φ n (D 0 ) < ε} . This is an open neighbourhood of D 0 in S n (H). Let (a, b) ⊂ IR be in the resolvent set of φ n (D 0 ). Then there is ε > 0 such that (a, b) is in the resolvent set of φ n (D) for all D ∈ U (ε, n, D 0 ). Hence φ −1 n ((a, b)) is in the resolvent set of D for all D ∈ U (ε, n, D 0 ). Furthermore if µ ∈ φ −1 n ((a, b)), µ > 0, then also −µ ∈ φ −1 n ((a, b)) and U (ε, n, D 0 ) → B(H), D → 1 [−µ,µ] (D) is continuous. This implies the following lemma, which will be used for the definition of the spectral flow: Lemma 1.8. If D 0 ∈ SF n (H) and µ ∈ (0, 1 n ) is such that ±µ is in the resolvent set of D 0 , then there is ε > 0 such that ±µ is in the resolvent set of D for all D ∈ U (ε, n, D 0 ). Furthermore 1 [−µ,µ] (D) has finite-dimensional range for all D ∈ U (ε, n, D 0 ) and the map S n (H) ⊃ U (ε, n, D 0 ) → K(H), D → 1 [−µ,µ] (D) is continuous. In particular all operators in U (ε, n, D 0 ) are Fredholm. Note that for a given D 0 ∈ SF n (H) a µ fulfilling the assumption of the lemma always exists since the spectrum of D 0 near zero is discrete. Proposition 1.9. 1. The identity induces a continuous map S(H) gap → S(H). The space SF (H) is open in S(H). The identity induces a homeomorphism from S(H) ∩ SF (H) to SF(H). Proof. The first assertion is a consequence of Lemma 1.3. The second assertion follows from the previous lemma and the subsequent remark. Since the remark is in general wrong for a Hilbert C * -module, we give another argument which also works for Hilbert C * -modules: Let D 0 ∈ SF n (H) and let χ be a normalizing function for D 0 with supp(χ 2 − 1) ⊂ (− 1 n , 1 n ). Then χ(D 0 ) 2 is invertible in B(H)/K(H). Furthermore, since S n (H) → B(H), D → (χ(D) 2 − 1) is continuous, also S n (H) → B(H)/K(H), D → χ(D) 2 is continu- ous. Hence there is an open neighbourhood U of D 0 in S n (H) such that χ(D) 2 is invertible in B(H)/K(H) for all D ∈ U . This implies that all D ∈ U are Fredholm. The third assertion is clear. We denote the space of (not necessarily selfadjoint) Fredholm operators on H by F (H). We identify F (H) with a subspace of SF (H ⊕ H) via the injection F (H) → SF (H ⊕ H), D → 0 D * D 0 . Note that if D ∈ F (H) and f : IR → IR is an odd non-decreasing continuous function with f −1 (0) = {0}, then f (D) ∈ F (H) is well-defined. The space F (H) endowed with the subspace topology of SF(H ⊕ H) is denoted by F(H). For topological spaces X, Y we denote by [X, Y ] the set of homotopy classes of continuous maps from X to Y . Proof. We use the notation of [Jo]: Let KC sa (H) (where KC stands for "Kasparov cycles") be the space of selfadjoint bounded operators F on H with F ≤ 1 and F 2 − 1 ∈ K(H) and endow it with the weakest topology such that the maps (x) = (1 − t)x + tχ(x). Since χ 2 t (1) − 1 = χ 2 t (−1) − 1 = 0, Lemma 1.5 implies that the map B → K(H), b → χ t (h(i, b)) 2 − 1 is continuous for i = 0, 1. Furthermore χ t (h(i, b)) 2 − 1 is continuous in t since χ 2 t − 1 depends continuously on t in C(IR). Hence the map ([0, 1] × {0, 1} × B) ∪ ({1} × [0, 1] × B) → KC sa (H), (t, x, b) → χ t (h(x, b)) is continuous and defines a homotopy in KC sa (H) between χ 0 (h(0, ·)) = h(0, ·) and χ 0 (h(1, ·)) = h(1, ·). (2) The proof is analogous with the obvious modifications. It follows that π 0 (F(H)) ∼ = Z Z. As usual an isomorphism is given by the index map. The results in the following section will imply that an isomorphism [S 1 , SF(H)] → Z Z is given by the spectral flow. The proof of the previous proposition carries over to the case where H is the standard Hilbert A-module H A of a unital C * -algebra A implying that SF(H A ) is a representing space of the functor B → K 1 (C(B, A)) from the category of compact spaces to the category of abelian groups and F(H A ) is a representing space for B → K 0 (C(B, A)). The corresponding statements for SF (H A ) gap have been proven in [Jo]. In the following we give two examples of maps with values in SF (H) that are continuous in SF(H) but not gap continuous. Both arise from elliptic differential operators on a noncompact manifold. Let H = L 2 (IR) and let f ∈ C ∞ (IR) be nonconstant real-valued and bounded below by some c > 0. Set f t (x) = f (tx) for t ∈ [0, 1]. We define D(t) on L 2 (IR) to be the multiplication by f t . The path D : [0, 1] → SF (H) is not gap continuous at t = 0, but it is continuous as a path in SF(H). Even if the resolvents are compact, they need not depend in a continuous way of t: Let H = L 2 (IR, C 2 ). Let f ∈ C ∞ 0 (IR) be a nonnegative function and let g ∈ C ∞ (IR) with g ≥ 0, g(0) = 1, g(1) = 0 and g(x) = 1 for |x| ≥ 2. Define ψ t (x) := g(tx) f (x) + 1 for t ∈ [0, 1]. Note that ψ t (x) is continuous in t and x. Define D(t) to be the closure of 0 ψ t (1 − ∂ 2 x ) (1 − ∂ 2 x )ψ t 0 : C ∞ c (IR, C 2 ) → L 2 (IR, C 2 ) . Since 1 ψt ∈ C 0 (IR), the operator 1 ψt (1 − ∂ 2 x ) −1 is compact on L 2 (IR) for any t, hence D(t) −1 is compact for any t. Furthermore D(t) −1 is uniformly bounded. Thus [0, 1] → SF(H), t → D(t) is continuous. It is easy to check that D(t) −1 is not continuous in t at t = 0. Hence D is not gap continuous. Note that these examples have in common that the coefficients are continuous as maps from [0, 1] to C loc (IR) but not continuous (in the second example even not well-defined) as maps from [0, 1] to C(IR). See [W, §6] for criteria for the continuity in SF(H) of families of elliptic operators on noncompact Riemannian manifolds and families of well-posed boundary value problems. Spectral flow In the following we generalize the definition of the spectral flow in [BLP], which is based on the approach of [P], to continuous paths in SF(H). Well-definedness can be proven as in [P]. The spectral flow has the following properties: 1. It is additive with respect to concatenation of paths. For any non-decreasing continuous function f : IR → IR with f −1 (0) = {0} sf((D t ) t∈[a,b] ) = sf((f (D t )) t∈[a,b] ) . 3. If (U t ) t∈[a,b] is a strongly continuous path of unitaries on H, then The proof of the first three properties is not difficult and is left to the reader. The fourth property follows from the fact that by Lemma 1.8 and by compactness of [a, b] there is δ > 0 such that [−δ, δ] is a subset of the resolvent set of D t for sf((U t D t U * t ) t∈[a,b] ) = sf((D t ) t∈[a,b] ) . If all t ∈ [a, b] if D t is invertible for all t ∈ [a, b]. The following proposition implies the last two properties, namely homotopy invariance: Proposition 2.2. If (D (s,t) ) (s,t)∈[0,1]×[a,b] is a continuous family in SF(H), then sf((D (0,t) ) t∈[a,b] ) + sf((D (s,b) ) s∈[0,1] ) − sf((D (1,t) ) t∈[a,b] ) − sf((D (s,a) ) s∈[0,1] ) = 0 . Proof. Let n ∈ IN be such that the family (D (s,t) ) (s,t)∈[0,1]×[a,b] is continuous in SF n (H). If there is µ ∈ (0, 1 n ) such that ±µ is in the resolvent set of D (s,t) for all (s, t) ∈ [0, 1] × [a, b], then 1 [−µ,µ] (D (s,t) ) has finite-dimensional range for all (s, t) and the assertion follows from the definition of the spectral flow. In general we find, by compactness of [0, 1] × [a, b] and by Lemma 1.8, an n ∈ IN such that each of the rectangles [ (m1−1) n , m1 n ]×[a+(b−a) m2−1 n , a+(b−a) m2 n ] with m 1 , m 2 = 1, 2 . . . n has the following property: There is a µ ∈ (0, 1 n ) such that ±µ is in the resolvent set of D (s,t) for all points (s, t) of the rectangle. Hence for each of the rectangles an analogue of the formula holds by the previous argument. Since for fixed n these rectangles constitute a subdivision of [0, 1] × [a, b], the formula follows from the additivity of the spectral flow with respect to concatenation. We draw some conclusions in the following two propositions. See [Le,§3] for similar results. It is well-known that 1] be strongly continuous paths of projections on H such that P t − Q t is compact and continuous in t. Then ind(P 0 , Q 0 ) = ind(P 1 , Q 1 ) . sf((t(2P − 1) + (1 − t)(2Q − 1)) t∈[0,1] ) = ind(P, Q) . Proposition 2.3. Let (P t ) t∈[0,1] , (Q t ) t∈[0, Proof. First we prove that the family (F (s,t) ) (s,t)∈[0,1] 2 defined by F (s,t) := t(2P s − 1) + (2t − 1)(2Q s − 1) is continuous in SF(H): Clearly F (s,t) depends in a strongly continuous way on (s, t). Hence, by Prop. 1.1, the operators (F (s,t) ± i) −1 depend in a strongly continuous way on (s, t) as well. Furthermore F (s,t) − (2P s − 1) is a compact operator depending continuously on (s, t). This and Lemma 1.2 imply that φ n (F (s,t) ) − φ n ((2P s − 1)) is a compact operator depending continuously on (s, t) for any n ∈ IN. From φ n ((2P s − 1)) = 0 it follows that φ n (F (s,t) ) is a compact operator depending continuously on (s, t). This shows the continuity. Now by homotopy invariance sf((t(2P 0 −1)+(1−t)(2Q 0 −1)) t∈[0,1] ) = sf((t(2P 1 −1)+(1−t)(2Q 1 −1)) t∈[0,1] ) . The following technical lemma, which is an immediate consequence of [Le,Prop. 3.4] and Lemma 1.2, is needed for the proof of the subsequent proposition: Lemma 2.4. Let D ∈ S(H) and let K be a symmetric operator with dom D ⊂ dom K and such that K(D + i) −1 is compact. Then f (D + K) − f (D) ∈ K(H) for each function f ∈ C(IR) for which lim x→∞ f (x) and lim x→−∞ f (x) exist. Proposition 2.5. Let (D t ) t∈[a,b ] be a continuous path in SF(H) with invertible endpoints and assume that there is a path of symmetric operators (K t ) t∈[a,b] with dom D t ⊂ dom K t for all t ∈ [a, b], such that K t (D t + i) −1 is compact and continuous in t and such that (D t + K t ) is invertible for each t ∈ [a, b]. Then sf((D t ) t∈[a,b] ) = ind(1 ≥0 (D b ), 1 ≥0 (D b + K b )) − ind(1 ≥0 (D a ), 1 ≥0 (D a + K a )) . Proof. Let n be such that (D t ) t∈[a,b] is a continuous path in SF n (H). Lemma 1.2 implies that φ n (D t ) − φ n (D t + K t ) is compact and continuous in t. In particular (D t + K t ) t∈[a,b] is a continuous path in SF n (H). Since each D t + K t is invertible, by property (4) of the spectral flow sf((D t + K t ) t∈[a,b] ) = 0 . Let ψ ∈ C(IR) with ψ| (−∞, 1 3 ] = 0 and ψ| [ 2 3 ,∞) = 1. By homotopy invariance and additivity with respect to concatenation the spectral flow of the path (D t ) t∈ [a,b] equals the spectral flow of the path a, b]. Furthermore by additivity with respect to concatenation and since sf( (D t ) t∈[a−1,b+1] withD t = D a + ψ(t − a + 1)K a for t ∈ [a − 1, a],D t = D b +(1−ψ(t−b))K b for t ∈ [b, b+1] andD t = D t +K t for t ∈ [(D t + K t ) t∈[a,b] ) = 0, sf((D t ) t∈[a,b] ) = sf((D t ) t∈[a−1,a] ) + sf((D t ) t∈[b,b+1] ) . We calculate sf((D t ) t∈[a−1,a] ): Let χ ∈ C ∞ (IR) be a normalizing function for (D t ) t∈[a−1,a] such that χ(D a ) = 2 · 1 ≥0 (D a ) − 1 and χ(D a + K a ) = 2 · 1 ≥0 (D a + K a ) − 1. By the previous lemma χ(D a ) − χ(D a + K a ) ∈ K(H). Then sf((D t ) t∈[a−1,a] ) = sf((χ(D t )) t∈[a−1,a] ) = sf(((1 − t)χ(D a ) + tχ(D a + K a )) t∈[0,1] ) = ind(1 ≥0 (D a + K a ), 1 ≥0 (D a )) , where the second equality follows from homotopy invariance and the third from the equation preceding Prop. 2.3. Analogously sf( (D t ) t∈[b,b+1] ) = ind(1 ≥0 (D b ), 1 ≥0 (D b + K b )). Under slightly more restricted conditions (since the previous lemma has not been proven for Hilbert C * -modules -the author did not check whether the rather complicated proof of [Le,Prop. 3.4] carries over) the statement of the proposition makes sense on a Hilbert C * -module and was used as a definition of the noncommutative spectral flow in [W]. In the following we express the spectral flow in terms of a winding number. Let S 1 = [0, 1]/ 0∼1 with the standard smooth structure. Let U(H) ⊂ B(H) be the group of unitaries and let U K (H) = {U ∈ U(H) | U − 1 ∈ K(H)} . There is an isomorphism w : π 1 (U K (H)) ∼ = Z Z Proof. The term on the right hand side is well-defined by Lemma 1.5. We make use of the space KC sa (H), which was defined in the proof of Theorem 1. Hence we may assume that the eigenspaces of F 0 , F 1 are infinite-dimensional. Then there is a unitary U with F 0 = U F 1 U * . Furthermore by the contractibility of U(H) there is a continuous path (U t ) t∈[1,2] of unitaries, unique up to homotopy, with U 1 = 1 and U 2 = U . Define G t = F 2t for t ∈ [0, 1 2 ] and G t = U 2t F 1 U * Thus it is enough to prove equation ( * ) for loops in KC sa (H). This will also show the second assertion of the proposition. By homotopy invariance of the winding number and of spectral flow for loops (property (6)) and by [S 1 , KC sa (H)] ∼ = K 1 (S 1 ) ∼ = Z Z it is sufficient to verify the assertion for some loop in KC sa (H) whose class generates K 1 (S 1 ). For example one can use the loop (G t ) t∈[0,1] arising as above from F t = − cos(πt)P + (1 − P ), where P is a projection whose range has dimension one. In this case equation ( * ) is well-known since (G t ) t∈[0,1] is a norm-continuous path. and denote by SF n (H) the set SF n (H) endowed with the subspace topology of S n (H). Let SF(H) be the inductive limit of the spaces SF n (H). An operator D ∈ S(H) is Fredholm if and only if F D is invertible in B(H)/K(H), and this is equivalent to φ n (D) ∈ K(H) for n big enough. Hence the underlying set of SF(H) is SF (H). If D : B → S(H) is continuous, then f (D) : B → S(H) is continuous for any odd non-decreasing continuous function f : IR → IR with f −1 (0) = {0}. This can be seen as follows: Assume that D : B → S n (H) is continuous. Since Theorem 1 . 110. 1. The space SF(H) represents the functor B → K 1 (B) from the category of compact spaces to the category of abelian groups. 2. The space F(H) represents the functor B → K 0 (B) from the category of compact spaces to the category of abelian groups. KC sa (H) → H, F → F x , KC sa (H) → K(H), F → F 2 − 1 are continuous for all x ∈ H. The inclusion KC sa (H) → SF(H) is continuous.Let KC(H) be the space of bounded operators F such that F ≤ 1 and F * F − 1, F F * − 1 ∈ K(H). Consider KC(H) as a subspace of KC sa (H ⊕ H) as above.By[Jo, Theorem 3.4], which is based on results of Bunke-Joachim-Stolz, the space KC sa (H) represents the functor K 1 and the space KC(H) represents K 0 .(1) Let D : B → SF(H) be a continuous map. Let χ be a normalizing function for D and let χ t (x) = (1 − t)x + tχ(x). Then B → KC sa (H), b → χ 1 (D(b)) and [0, 1] × B → SF(H), (t, b) → χ t (D(b)) are continuous (here we use Prop. 1.1). It follows that the map [B, KC sa (H)] → [B, SF(H)] induced by the inclusion KC sa (H) → SF(H) is surjective. For injectivity let h : [0, 1] × B → SF(H) be a homotopy between continuous maps B → KC sa (H), b → h(i, b), i = 0, 1. Let χ be a normalizing function for h such that χ(1) = 1 and let χ t Definition 2 . 1 . 21Let (D t ) t∈[a,b] be a continuous path in SF(H) and assume that there is µ > 0 such that ±µ is in the resolvent set of D t for all t ∈ [a, b] and1 [−µ,µ] (D t ) has finite-dimensional range. We define sf((D t ) t∈[a,b] ) = dim Ran(1 [0,µ] (D b )) − dim Ran(1 [0,µ] (D a )) .If (D t ) t∈[a,b] is a general continuous path in SF(H), then we define its spectral flow by cutting the path into smaller pieces to which the previous situation applies and adding up the contributions of the pieces. (This is always possible by Lemma 1.8 and the subsequent remark.) D t is invertible for any t ∈ [0, 1], then sf((D t ) t∈[a,b] ) = 0 . 5. If (D (s,t) ) (s,t)∈[0,1]×[a,b] is a continuous family in SF(H) such that D (s,a)andD (s,b) are invertible for all s ∈ [0, 1], then sf((D (0,t) ) t∈[a,b] ) = sf((D (1,t) ) t∈[a,b] ) .6. If (D (s,t) ) (s,t)∈[0,1]×[a,b] is a continuous family in SF(H) such that D (s,a) = D (s,b) for all s ∈ [0, 1], then sf((D (0,t) ) t∈[a,b] ) = sf((D (1,t) ) t∈[a,b] ) . If P, Q are projections with P − Q ∈ K(H), then QP : P (H) → Q(H) is Fredholm with parametrix P Q. Let ind(P, Q) := ind(QP : P (H) → Q(H)) . extending the classical winding number. In fact, if s : S 1 → U K (H) fulfills (x → s(x) − 1) ∈ C 1 (S 1 , l 1 (H)), where l 1 (H) ⊂ B(H) is the ideal of trace class operators endowed with the trace class norm, then w(s)Tr(s(x) −1 s ′ (x)) dx .Proposition 2.6. Let (D t ) t∈[0,1] be a continuous path in SF(H) with invertible endpoints. Let χ ∈ C(IR) be a normalizing function for the map t → D t and assume that χ(D 0 ) and χ(D 1 ) are involutions. Thensf((D t ) t∈[0,1] ) = w([e πi(χ(Dt )+1) ]) .If (D t ) t∈[0,1] is a continuous path in SF(H) with D 0 = D 1 (not necessarily invertible), then this equation holds for any normalizing function of t → D t . 10 . 10Let (D t ) t∈[0,1] be a continuous path in SF(H) with invertible endpoints. Since for any normalizing functionχ of t → D t sf((D t ) t∈[0,1] ) = sf((χ(D t )) t∈[0,1] ) ,it is enough to prove that for any continuous path(F t ) t∈[0,1] in KC sa (H) such that F 0 , F 1 are involutions sf((F t ) t∈[0,1] ) = w([e πi(Ft+1) ]) .( * )Both sides of this equation remain unchanged if we replace (F t ) t∈[0,1] by (F t ⊕ I) t∈[0,1] , where I is an involution on H with infinite-dimensional eigenspaces. 2t for t ∈ [ 1 2 , 1]. The path (G t ) t∈[0,1] is a loop in KC sa (H) with sf((F t ) t∈[0,1] ) = sf((G t ) t∈[0,1] ) and w([e πi(Ft+1) ]) = w([e πi(Gt+1) ]) . . M F Atiyah, K-Theory , D. W. Anderson, W. A.Benjamin, IncNew York-AmsterdamM.F. Atiyah, K-theory. Lecture notes by D. W. 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Roe, Analytic K-homology, Oxford Mathematical Mono- graphs. Oxford Science Publications. Oxford University Press, Oxford, 2000 Vektorraumbündel und der Raum der Fredholm-Operatoren. K Jänich, Math. Ann. 161K. Jänich, "Vektorraumbündel und der Raum der Fredholm-Operatoren", Math. Ann. 161 (1965), pp. 129-142 Unbounded Fredholm operators and K-theory. M Joachim, Highdimensional manifold topology. River Edge, NJM. Joachim, "Unbounded Fredholm operators and K-theory", High- dimensional manifold topology, World Sci. Publishing, River Edge, NJ, 2003, pp. 177-199 The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators. M Lesch, Spectral geometry of manifolds with boundary and decomposition of manifolds (Contemp. Math. 366). Providence, RIAmer. Math. SocM. Lesch, "The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators", Spectral geometry of manifolds with boundary and decomposition of manifolds (Contemp. Math. 366), Amer. Math. Soc., Providence, RI, 2005, pp. 193-224 Self-adjoint Fredholm operators and spectral flow. J Phillips, Canad. Math. Bull. 394J. Phillips, "Self-adjoint Fredholm operators and spectral flow", Canad. Math. Bull. 39 (1996), no. 4, pp. 460-467 On the noncommutative spectral flow. C Wahl, arXiv:math.OA/0602110preprintC. Wahl, "On the noncommutative spectral flow", preprint arXiv:math.OA/0602110 (2006) Niedersächsische Landesbibliothek. Gottfried Wilhelm, Leibniz Bibliothek, ac.wahl@web.deWaterloostr. 8, 30169 Hannover, Germany EmailGottfried Wilhelm Leibniz Bibliothek, Niedersächsische Landesbib- liothek, Waterloostr. 8, 30169 Hannover, Germany Email: ac.wahl@web.de

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{'abstract': 'We define a new topology, weaker than the gap topology, on the space of selfadjoint unbounded operators on a separable Hilbert space. We show that the subspace of selfadjoint Fredholm operators represents the functor K 1 from the category of compact spaces to the category of abelian groups and prove a similar result for K 0 . We define the spectral flow of a continuous path of selfadjoint Fredholm operators generalizing the approach of Booss-Bavnek-Lesch-Phillips.', 'arxivid': 'math/0607783', 'author': ['Charlotte Wahl '], 'authoraffiliation': [], 'corpusid': 14084390, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 11995, 'n_tokens_neox': 11011, 'n_words': 6356, 'pdfsha': '04dca8d7007173b4e6f442a8c4a6c96727098d1f', 'pdfurls': ['https://arxiv.org/pdf/math/0607783v2.pdf'], 'title': ['A new topology on the space of unbounded selfadjoint operators and the spectral flow', 'A new topology on the space of unbounded selfadjoint operators and the spectral flow'], 'venue': []}

arxiv

DRISHTI: Visual Navigation Assistant for Visually Impaired Malay Joshi Department of Electronics and Communication Engineering Ajay Kumar Garg Engineering College Uttar Pradesh201009GhaziabadIndia Aditi Shukla Department of Electronics and Communication Engineering Ajay Kumar Garg Engineering College Uttar Pradesh201009GhaziabadIndia Jayesh Srivastava Department of Electronics and Communication Engineering Ajay Kumar Garg Engineering College Uttar Pradesh201009GhaziabadIndia Manya Rastogi Department of Electronics and Communication Engineering Ajay Kumar Garg Engineering College Uttar Pradesh201009GhaziabadIndia DRISHTI: Visual Navigation Assistant for Visually Impaired In today's society, where independent living is becoming increasingly important, it can be extremely constricting for those who are blind. Blind and visually impaired (BVI) people face challenges because they need manual support to prompt information about their environment. In this work, we took our first step towards developing an affordable and high-performing eye wearable assistive device, DRISHTI, to provide visual navigation assistance for BVI people. This system comprises a camera module, ESP32 processor, Bluetooth module, smartphone and speakers. Using artificial intelligence, this system is proposed to detect and understand the nature of the users' path and obstacles ahead of the user in that path and then inform BVI users about it via audio output to enable them to acquire directions by themselves on their journey. This first step discussed in this paper involves establishing a proof-of-concept of achieving the right balance of affordability and performance by testing an initial software integration of a currency detection algorithm on a low-cost embedded arrangement. This work will lay the foundation for our upcoming works toward achieving the goal of assisting the maximum of BVI people around the globe in moving independently. Introduction Blindness is a daunting condition. According to a report by the WHO in 2013, there is an estimated 40 to 45 million people who are blind, and about 135 million have low or weak sight. According to a report by The Hindu, 62 million people in India are visually impaired, of which eight million are blind. Visual impairment can impact a person's quality of life and make them prone to discrimination. They face many challenges in navigating around places. There is a large number of adaptive equipment that enable visually impaired people to live their life independently. However, they are only found in nearby shops or marketplaces. Also, they are quite expensive, so only some BVI people can use such resources. The camera-operated mechanism in [1] helps them read text on things that are held in their hands easily. The proposed system uses a camera to capture the target object, and an algorithm extracts the text in the captured image from the backdrop. Each text letter is separated by optical character recognition (OCR). Audio output is provided using a software development kit with the identified text. The wearable system in [2] is a device that receives user input and recognises things. The device has an ultrasonic sensor that assists in warning the user of objects that are in his or her path. The items are located using the Haar cascade method. It is a wearable device that can be mounted on the user's chest. The user will receive an audio of the object that was discovered. The traffic scenes use object detection technology in [3] to find objects. Here, they have used a combination of R-FCN (Regression-based Full Convolution Network) and OYOLO (optimized you only look once), which is 1.18 times faster than YOLO. It identifies and categorizes photos of vehicles, cyclists, and other objects. Location errors occur when using YOLO; to prevent these, we employ OYOLO. Other possible categories of solutions analyzed in [4], with promising present and future utilities, are based on the existing technologies. The system in [5] uses features like Artificial Intelligence and Machine Learning to provide a solution to the problem. A device camera captures images, objects are detected, and distance is calculated using an application. The prototype in [6] is mounted on top of a walking cane that uses a pi cam to click pictures and then implements the YOLO algorithm to perform object detection that works more accurately than others. After that, the gTTS module converts text to speech, resulting in a human voice. All currently available systems have one or more of the following drawbacks: (i) unaffordable price (ii) lack of sales, marketing, or servicing in developing nations; (iii) high inaccuracy, thus making them unfit for general usage; and/or (iv) bulky or challenging to use. There are five sections in the paper. Section 1 covers the introduction to the paper and some existing works in the domain of virtual assistance devices. Section 2 presents survey results done as part of the idea validation process. Section 3 contains the hardware and software details of the proposed system, and section 4 covers the real-time test results of the proposed system. The paper is concluded in section 5. Section 6 provides the future scope of this project. The references are there in section 7. In-person survey analysis After surveying BVI people at a local trust for blind people, the results obtained were: (1) They all had no impairment other than visual impairment. (2) There are apps for currency detection, colour recognition and document reading. (3) There is also a read-aloud feature in their smartphones named Talkback that helps them to use the smartphones. (4) Their current device is a cane with an ultrasonic sensor that detects obstacles, but the limitation is that it does not give the details of whether it is a dog or a heap of stones. (5) Their cane is based on touch, and its processing is slow. (6) The cane costs around INR 4000, and sometimes they do not run after branded items. (7) They all agreed to wear spectacles weighing up to 100 grams. (8) Mostly everyone has a smartphone and a good internet connection. (9) According to them, if a device helps 60% of visually challenged people, it is considered useful and successful. Proposed system This is a microcontroller-based virtual visual assistance device for visually challenged people. The main objective of this system is to convert the visual world into an audio world to make it easier for people with visual disabilities to navigate themselves and do their daily activities without feeling deprived of vision. The ESP32 camera module will capture images of the target(s) (like surrounding objects, people, text document, currency, road, traffic signal, traffic sign board, etc). This captured image will be sent to a smartphone device and processed in real-time by multiple algorithms. The information extracted will be converted into audio-based signals that will act as user feedback. (1) ESP32 Cam: It is an ESP32-based development board that is low-cost and comes with a small onboard camera. It is an ideal solution for small IoT applications, constructions, prototypes, and DIY projects. The camera module board integrates features like traditional Bluetooth and low power BLE, WiFi, and is equipped with two high-performance 32-bit LX6 CPUs. (2) Programmer Module: The ESP32-CAM has an OV2640 camera, onboard flash, microSD card support option and several GPIOs that can connect peripherals. However, it does not have a built-in programmer. So it needs an external programmer to connect it to the computer and upload the code. A Future Technology Devices International (abbreviated as FTDI) programmer or any Arduino board can be used to implement this task. It directs the camera sensor to capture images and send them to the desired location for processing. (3) Smartphone: The sent images are received by any smartphone device, and different algorithms for object detection, text recognition and currency identification are implemented. For these purposes, machine learning and artificial intelligence are needed. (4) Speakers: After processing the image, the output is generated in the form of voiced instructions. This is done with the help of two 3Watt Wireless Bluetooth Multimedia Speakers connected using an HC-05 Bluetooth module. This whole process is done in real-time, and the voiced instructions will help a blind person in navigation and text reading. Software used Algorithms and modules used in this proposed solution are as follows: (1) Deep Learning (Resnet-50): It is used for detection and identification of currency. Resnet-50 is a 50 layer deep convolutional neural network. ResNet stands for Residual Network. It solves complex problems with improved accuracy and performance. Basically, deep learning is a way to enhance human gain like used for automated driving, stop signs ,and traffic lights. (2) YOLOv7: You Only Look Once (YOLO) is a common algorithm for object detection. It is famous for detecting objects in a real time environment. Used for detecting traffic signals, exam proctoring etc. The best model of YOLOv7 scored 56.8% Average Precision (AP), which according to the paper is the highest among all known object detectors. (3) Tesseract OCR: Tesseract is a printed text reader. It is an engine for optical character recognition used by various OS. OCR creates a new searchable text file or a PDF by extracting text from images and documents without a text layer.. Tesseract has very improved image quality. (4) gTTS: Google Text-to-Speech (gTTS) is used for speech translation. The text-to-speech API of Google Translate is interfaced with using a Python library and CLI tool. The text variable is a string used to store the user's input. Languages including English, Hindi, Tamil, French, German, and many more are supported via the gTTS API. Results Circuit diagram An FTDI programmer is used to program the ESP32 Cam. In this circuit, ESP32 Cam module is connected to FTDI module by connecting pins GND, U0T, U0R of ESP32 Cam module with GND, RX and TX pins of FTDI module respectively. This whole setup is powered by a 5V battery. Note: GPIO 0 pin of the esp32 cam module must be connected to GND when uploading code. (2) Case-II. When a 10 rupees note is given as an input, it captures the image and analyses it. After processing the image, the algorithm is applied. It gives output as 10 with a likelihood or probability of 0.89. The output is given in the form of audio as well as text. (3) Case-III. When the above background image is given as input, it captures the image and processes it. After image processing, it applies an algorithm. It produces output that is a background with a likelihood of 1. The output is given in the form of audio as well as text. Conclusion Due to the lack of directly perceiving visual information, many blind and visually impaired people struggle to maintain a constant healthy rhythm. Therefore, a navigation system that enables blind people to navigate their route freely and lets them know where they might be at any given time is necessary. To make the life of visually impaired people easier, we have used technology in this article to give them a visual aid. The project's main goal is to design an object detector that can recognise obstructions and guide a visually impaired person in a path via voiced instructions. Hence, the design and implementation of a cost-effective device were done to provide support and independence to visually impaired people during their travel to new or unknown places. Future scope Future work will include improvements to the device's design to make it affordable for commercial use, adding computer vision-based algorithms for analysing the nature of travel paths and obstacles ahead, and conducting user research to improve the system's usability as a whole. These future works will help blind and visually challenged people in independent navigation. Fig 1 . 1System Block Diagram 3.1. Hardware used Hardware components used in this proposed solution are: Fig 2 .Fig 3 . 23Connecting ESP32Cam with Future Technology Devices International (FTDI) module [7] 4.2. Software Simulation Image of INR 100 given as inputFig 4. Model's text and audio-based prediction (1) Case-I. When we place a note of Rs 100, it acts as an input. It captures the image, processes it and then applies the algorithm. It detects the denomination and gives the output as 100 with a probability of 1. The output received is both in the form of audio and text. Fig 5 .Fig 6 . 56Image of INR 10 given as input Model's text and audio-based prediction Fig 7 . 7Image of laptop given as input Fig 8. Model's text and audio-based prediction Real time text detection and recognition on hand held objects to assist blind people. S Deshpande, R Shriram, 2016 International Conference on Automatic Control and Dynamic Optimization Techniques (ICACDOT). S. Deshpande and R. Shriram, "Real time text detection and recognition on hand held objects to assist blind people", 2016 International Conference on Automatic Control and Dynamic Optimization Techniques (ICACDOT), 2016 Visual Assistance for Blind Using Image Processing. B Jain, S M Thakur, K V Suresh, 2018 International Conference on Communication and Signal Processing. Chennai, IndiaB. Deepthi Jain, S. M. Thakur and K. V. Suresh, "Visual Assistance for Blind Using Image Processing", 2018 International Conference on Communication and Signal Processing (ICCSP), Chennai, India, 2018 An object detection system based on YOLO in traffic scene. J Tao, H Wang, X Zhang, X Li, H Yang, 6th International Conference on Computer Science and Network Technology (ICCSNT). Dalian, ChinaJ. Tao, H. Wang, X. Zhang, X. Li and H. Yang, "An object detection system based on YOLO in traffic scene", 2017 6th International Conference on Computer Science and Network Technology (ICCSNT), Dalian, China, 2017 A survey of assistive technologies and applications for blind users on mobile platforms: a review and foundation for research. Á Csapó, G Wersényi, H Nagy, J Multimodal User Interfaces. 9Csapó, Á., Wersényi, G., Nagy, H. et al., "A survey of assistive technologies and applications for blind users on mobile platforms: a review and foundation for research", J Multimodal User Interfaces 9, 275-286 (2015) A Smart Personal AI Assistant for Visually Impaired People. S M Felix, S Kumar, A Veeramuthu, 2nd International Conference on Trends in Electronics and Informatics (ICOEI). Tirunelveli, IndiaS. M. Felix, S. Kumar and A. Veeramuthu, "A Smart Personal AI Assistant for Visually Impaired People", 2018 2nd International Conference on Trends in Electronics and Informatics (ICOEI), Tirunelveli, India, 2018 CICERONE-A Real Time Object Detection for Visually Impaired People. Therese Yamuna Mahesh, S S Parvathy, Shibin Thomas, Rachel Shilpa, Thomas Thomas, Sebastian, IOP Conference Series: Materials Science and Engineering. 2021Therese Yamuna Mahesh, Parvathy S S, Shibin Thomas, Shilpa Rachel Thomas and Thomas Sebastian "CICERONE-A Real Time Object Detection for Visually Impaired People" IOP Conference Series: Materials Science and Engineering, 2021

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{'abstract': "In today's society, where independent living is becoming increasingly important, it can be extremely constricting for those who are blind. Blind and visually impaired (BVI) people face challenges because they need manual support to prompt information about their environment. In this work, we took our first step towards developing an affordable and high-performing eye wearable assistive device, DRISHTI, to provide visual navigation assistance for BVI people. This system comprises a camera module, ESP32 processor, Bluetooth module, smartphone and speakers. Using artificial intelligence, this system is proposed to detect and understand the nature of the users' path and obstacles ahead of the user in that path and then inform BVI users about it via audio output to enable them to acquire directions by themselves on their journey. This first step discussed in this paper involves establishing a proof-of-concept of achieving the right balance of affordability and performance by testing an initial software integration of a currency detection algorithm on a low-cost embedded arrangement. This work will lay the foundation for our upcoming works toward achieving the goal of assisting the maximum of BVI people around the globe in moving independently.", 'arxivid': '2303.07451', 'author': ['Malay Joshi \nDepartment of Electronics and Communication Engineering\nAjay Kumar Garg Engineering College\nUttar Pradesh201009GhaziabadIndia\n', 'Aditi Shukla \nDepartment of Electronics and Communication Engineering\nAjay Kumar Garg Engineering College\nUttar Pradesh201009GhaziabadIndia\n', 'Jayesh Srivastava \nDepartment of Electronics and Communication Engineering\nAjay Kumar Garg Engineering College\nUttar Pradesh201009GhaziabadIndia\n', 'Manya Rastogi \nDepartment of Electronics and Communication Engineering\nAjay Kumar Garg Engineering College\nUttar Pradesh201009GhaziabadIndia\n'], 'authoraffiliation': ['Department of Electronics and Communication Engineering\nAjay Kumar Garg Engineering College\nUttar Pradesh201009GhaziabadIndia', 'Department of Electronics and Communication Engineering\nAjay Kumar Garg Engineering College\nUttar Pradesh201009GhaziabadIndia', 'Department of Electronics and Communication Engineering\nAjay Kumar Garg Engineering College\nUttar Pradesh201009GhaziabadIndia', 'Department of Electronics and Communication Engineering\nAjay Kumar Garg Engineering College\nUttar Pradesh201009GhaziabadIndia'], 'corpusid': 257505401, 'doi': '10.48550/arxiv.2303.07451', 'github_urls': [], 'n_tokens_mistral': 3745, 'n_tokens_neox': 3362, 'n_words': 2387, 'pdfsha': '2d687c20e33d4fce30fe56bffa7c718c6d580ce3', 'pdfurls': ['https://export.arxiv.org/pdf/2303.07451v1.pdf'], 'title': ['DRISHTI: Visual Navigation Assistant for Visually Impaired', 'DRISHTI: Visual Navigation Assistant for Visually Impaired'], 'venue': []}

arxiv

Activated lone-pair electrons lead to low lattice thermal conductivity: a case study of boron arsenide Guangzhao Qin Institute of Mineral Engineering Division of Materials Science and Engineering Faculty of Georesources and Materials Engineering RWTH Aachen University 52064AachenGermany Department of Mechanical Engineering University of South Carolina 29208ColumbiaSCUSA Zhenzhen Qin School of Physics and Engineering International Laboratory for Quantum Functional Materials of Henan Zhengzhou University 450001ZhengzhouChina Huimin Wang Department of Mechanical Engineering University of South Carolina 29208ColumbiaSCUSA College of Engineering and Applied Science Nanjing University 210023NanjingChina Ming Hu Department of Mechanical Engineering University of South Carolina 29208ColumbiaSCUSA Activated lone-pair electrons lead to low lattice thermal conductivity: a case study of boron arsenide (Dated: 2 April 2019) Reducing thermal conductivity (κ) is an efficient way to boost the thermoelectric performance to achieve direct solid-state conversion to electrical power from thermal energy, which has lots of valuable applications in reusing waste resources. In this paper, we propose an effective approach for realizing low κ by introducing lone-pair electrons or making the lone-pair electrons stereochemically active through bond nanodesigning. As a case study, by cutting at the (111) cross section of the three-dimensional (3D) cubic boron arsenide (c-BAs), the κ is lowered by more than one order of magnitude in the resultant two-dimensional (2D) system of graphene-like BAs (g-BAs) due to the stereochemically actived lone-pair electrons. However, this does not naturally happen to all materials. For instance, breaking the perfect octahedral coordination of 3D diamond as in the 2D graphene adversely enhances thermal transport. The underlying mechanism is analyzed based on the comparative study on the thermal transport properties of g-BAs, c-BAs, graphene, and diamond (c-BAs → g-BAs vs. diamond → graphene). Furthermore, deep insight into the electronic origin is gained by performing fundamental analysis on the electronic structures. Similar concept can be also extended to other systems with lone-pair electrons beyond BAs, such as group III-V compounds (e.g. BN, AlN, GaN, etc), where a strong correlation between κ modulation and electronegativity difference for binary compounds is found. Thus, the lone-pair electrons combined with a small electronegativity difference could be the indicator of lowering κ through bond nanodesigning to change the coordination environment. The proposed approach for realizing low κ and the underlying mechanism uncovered in this study would largely benefit the design of thermoelectric devices with improved performance, especially in future research involving novel materials for energy applications. arXiv:1904.00329v1 [cond-mat.mtrl-sci] I. INTRODUCTION Due to the ability of firsthand solid-state conversion to electrical power from thermal energy, especially for waste heat reusing, thermoelectrics have attracted a lot of attention in recent years 1 . Thermoelectrics have lots of valued applications in recovering resources and thus may make crucial contributions to the crisis of environment by solving energy problems 2 . Moreover, thermoelectrics possess the advantages of having no moving components and being environmentally friendly compared to traditional mechanical heat engines. Generally, the thermoelectric efficiency and performance can be characterized by a dimensionless figure of merit 3 ZT = S 2 σT /κ ,(1) where S, σ, T and κ are thermopower (Seebeck coefficient), electrical conductivity, absolute temperature and a) Electronic mail: qin.phys@gmail.com b) Electronic mail: hu@sc.edu total thermal conductivity, respectively. The commercial applications in industry of thermoelectric devices are currently limited by the low ZT merit. To approach the Carnot coefficient as closely as possible, a high energy generation efficiency is necessary, which corresponds to a large ZT merit. Based on the definition [Eq. (1)], lowering the κ would be more efficient to boost the ZT merit due to the inversely proportional relation 4 . Previous theoretical studies predicted that cubic boron arsenide (c-BAs) in the bulk form has an exceptionally high κ over 2000 W/mK, which is comparable to the bulk carbon crystals (diamond) with record highest κ 5 . The ultra-high κ of c-BAs was analyzed to be resulted from the large phonon band gap between acoustic and optical phonon branches together with the bunching of the acoustic phonon branches, which reduce phononphonon scattering 5 . The features of the phonon dispersion of c-BAs analyzed based on first-principles calculations are then confirmed by experimental measurements based on inelastic x-ray scattering 6 . By considering the phonon-phonon scattering involving four phonons, Feng et al. 7 found that the κ of c-BAs reduces from 2200 to 1400 W/mK, which was recently confirmed by experi- FIG. 1. The κ of typical two-dimensional materials (graphene, silicene, BN, AlN, GaN, BP, BAs, BSb, and SiC) at 300 K, which are normalized to the bulk counterparts, respectively. The two-dimensional materials possess lower κ compared to the bulk counterparts, except graphene. The dot line is for an eye guide. The specific data can be found in Supplemental Table 1. mental studies [8][9][10] . Experimental measurements reveal that the κ of BAs can be suppressed by the arsenic deficiency or vacancy in the BAs sample 11 and the phononboundary scattering in BAs microstructures 12,13 . However, the obtained κ of BAs is still too high that limits its potential applications in thermoelectrics, despite that the Seebeck coefficient and thermoelectric power factor of BAs is comparable to those of bismuth telluride, 13 which is one of the most commonly used thermoelectric materials. Thus, it would be meaningful if one can find an effective approach to lower the intrinsic κ of BAs to benefit its applications in thermoelectrics. Moreover, the approach that makes such a high κ material applicable for thermoelectrics would also largely benefit the design of thermoelectric devices with improved performance by lowering κ, especially in the future research involving novel materials for energy applications. In this paper, we propose an effective approach for realizing low κ by bond nanodesigning to make the lone-pair electrons stereochemically active. As a result, much lower κ can be generally achieved, except the case of graphene (Fig. 1). As a specific case study, when transforming the three-dimensional (3D) c-BAs into the two-dimensional (2D) graphene-like BAs (g-BAs), the κ is found to be lowered by more than one order of magnitude (Fig. 2) due to the stereochemically actived lone-pair electrons. The underlying mechanism is analyzed based on the comparative study on the thermal transport properties of g-BAs, c-BAs, graphene, and diamond, considering the similarity of the transformation from 3D cubic to 2D honeycomb planar geometry structures (c-BAs → g-BAs vs. diamond → graphene) but the opposite trend for the κ modula- FIG. 2. The similarity of the transformation from 3D cubic to 2D honeycomb planar geometry structures (c-BAs → g-BAs vs. diamond → graphene) is in contrast to the opposite κ variation. When transforming from 3D into 2D, the κ of BAs is found to be anomalously lowered by more than one order of magnitude. (a) The structure of graphene in 2D is the (111) cross section of the structure of diamond in 3D, which is planar due to the sp 2 hybridization of carbon atoms. (b) The g-BAs to c-BAs is like graphene to diamond. (c) The comparison of κ of diamond, graphene, c-BAs, 5,7 and g-BAs. tion. Moreover, deep insight into the electronic origin is gained by performing fundamental analysis on the electronic structures. II. RESULTS AND DISCUSSIONS A. The anomalously low κ of g-BAs By cutting the 3D cubic structure of c-BAs at the (111) cross section, g-BAs can be obtained with similar planar honeycomb structure as graphene (c-BAs → g-BAs vs. diamond → graphene). The κ of g-BAs is obtained to be 137.70 W/mK, which is more than one order of magnitude lower than that of graphene (3094.98 W/mK). Note that only 3-phonons scattering is considered here for simplicity. The κ of g-BAs could be further lowered if 4phonons scattering is included. The in-plane longitudinal acoustic (LA), transverse acoustic (TA) and out-pfplane flexural acoustic (FA) phonon branches contribute 28.5%, 43.1% and 26.9%, respectively. The κ of g-BAs are 89.3 and 137.7 W/mK before and after iteration, respectively. The large difference in the RTA and iteration result means that the proportion of N-process could be large and there exists strong phonon hydrodynamics in g-BAs. The lower κ of g-BAs than graphene is very intriguing considering the similarity of their planar honeycomb geometry structures [ Fig. 2(a,b)]. In particular, the κ of c-BAs is comparable to diamond share the same cubic structures and are the 3D counterparts of g-BAs and graphene, respectively [ Fig. 2(a,b)]. However, when transforming from 3D to 2D, huge difference emerges that the κ of g-BAs (the 2D counterpart of c-BAs) is much lower than that of graphene (the 2D counterpart of diamond), despite the comparable κ of c-BAs and diamond [ Fig. 2(c)]. In the following, we perform detailed analysis to achieve fundamental understanding on the anomalously lowered κ of g-BAs compared to c-BAs, diamond, and graphene. With the uncovered underlying mechanism, the generally lower κ of systems in 2D than 3D form as shown in Fig. 1 can also be well understood. . It was claimed in previous study 5 that the ultra-high κ of c-BAs comparable to diamond is resulted from the large phonon band gap between acoustic and optical phonon branches together with the bunching of the acoustic phonon branches. The features of the phonon dispersion of c-BAs analyzed based on first-principles calculations are then confirmed by experimental measurements based on inelastic x-ray scattering 6 , as reproduced in Fig. 3(b). When transforming from 3D c-BAs to 2D g-BAs, the phonon dispersions show some different features, which could have remarkable effect on the κ. (i) The phonon band gap in g-BAs is 13.00 THz, which is larger than that in c-BAs (9.28 THz). The larger phonon band gap is expected to not have a negative effect on the κ for the 3-phonons scattering processes considered here. (ii) The z-direction optical (ZO) phonon branch in g-BAs is below the bandgap as highlighted in Fig. 3(a), which could lead to more scattering probability by coupling with acoustic phonon branches (especially LA). See Supplemental Note 1 and Supplemental Figure 2 for more information on the coupling as revealed by phonon-phonon scattering channels. Such coupling is absent in the 3D c-BAs. (iii) The bunching of acoustic phonon branches in g-BAs becomes weak due to the separation of the three phonon branches. The weakened bunching effect for acoustic phonon branches together with the coupling with ZO phonon branch could lead to more phonon-phonon scattering, and thus is probably responsible for the anomalously lower κ of g-BAs than c-BAs. The ZO phonon branch in g-BAs is mainly contributed from the boron (B) atoms, as revealed by the partial density of states (pDOS) in Fig. 3(c). In fact, due to the mass difference, the optical phonon branches in both c-BAs and g-BAs are contributed from the B atoms [ Fig. 3(c,d)]. With the geometry structures transformed from 3D (c-BAs) to 2D (g-BAs), the z-direction vibration of B atoms is totally different due to the 2D nature of bondings and structural symmetry, which lowers the frequency of ZO and provides more scattering probability in g-BAs by strongly coupling with acoustic phonon branches [ Fig. 3(a)] (Supplemental Note 1 and Supplemental Figure 2). Such phonomena is also observed in monolayer GaN 15,16 . All the possible phonon-phonon scattering events quantified by the scattering phase space are determined based on the phonon dispersions by conserving both energy and crystal momentum with symmetry included 17-20 ω j ( q) ± ω j ( q ) = ω j ( q ) , q ± q = q + K ,(2) where ω is the frequency of phonon modes ( ω is the corresponding energy), q is the wave vector. Normal process corresponds to K = 0, while Umklapp process corresponds to K = 0. Fig. 4 presents the phase space of 3-phonons scattering in g-BAs, in comparison with that in c-BAs. The scattering phase space in c-BAs is small due to the large acoustic-optical phonon band gap, which is responsible for the ultra-high κ of c-BAs as analyzed in previous study 5 . However, the scattering phase space in g-BAs is larger than that in c-BAs, especially for the frequency range of 5-10 THz (Fig. 4), despite the larger gap in g-BAs than c-BAs [ Fig. 3(a,b)]. The enhanced scattering probability could be attributed to the weakened bunching effect for acoustic phonon branches together with their coupling with ZO phonon branch in g-BAs [ Fig. 3(a)], which partially explains the anomalously lower κ of g-BAs than c-BAs. We also compare the phase spaces between g-BAs and graphene. It is found that the overall scattering phase space in g-BAs is larger than that in graphene. Thus, the anomalously lower κ of g-BAs than graphene is understandable despite their similar structures. However, it should be noted that the difference in the scattering phase space is much less than one order of magnitude, which cannot fully explain the large difference in the scattering rate (Inset of Fig. 4) and further the more than one order of magnitude lower κ of g-BAs (137.70 W/mK) than graphene (3094.98 W/mK). Therefore, there should be some other mechanism also responsible for the anomalously low κ of g-BAs beyond the scattering phase space. C. Strong phonon anharmonicity It is well known that the phonon lifetime is governed by two factors: phonon-phonon scattering phase space and strength. The phonon-phonon scattering strength describes how strong is the phonon-phonon scattering process, which is governed by the anharmonic nature of the system. The Grüneisen parameter that describes the phonon anharmonicity can be calculated based on the change of phonon frequency with respect to the volume change Fig. 5(a) shows the obtained Grüneisen parameter of g-BAs, in comparison with graphene. The FA phonon branch possess the largest magnitude of Grüneisen parameter for both g-BAs and graphene, and the magnitude is larger in g-BAs than in graphene, revealing the stronger phonon anharmonicity in g-BAs. Thus, the more than one order of magnitude lower κ of g-BAs than graphene can be well understood by combining the larger scattering phase space (Fig. 4) and the stronger phonon anharmonicity in g-BAs. γ = − V ω ∂ω ∂V .(3) The phonon anharmonicity can also be intuitively revealed by the potential energy well. To have an explicit look at the anharmonicity, the potential energy wells (potential energy changes per atom due to the atomic displacement) of g-BAs and graphene are plotted in Fig. 5(b) for comparison. Both the potential wells are asymmetric with respect to the positive and negative atomic displacements, indicating the asymmetry in the ability of an atom vibrating around its equilibrium position and the nonlinear dependence of restoring forces on atomic displacement amplitudes, which is the direct evidence of the anharmonicity 21-23 . We further fit the calculated points for g-BAs and graphene, respectively, with the polynomial y = E 0 + a 1 x + a 2 x 2 + a 3 x 3 ,(4) where y is the relative energy change per atom, and x is the relative movement of atom. The fitted parame-ters for g-BAs and graphene are listed in the table as inset in Fig. 5(b). The fitted quadric (a 2 ) and cubic (a 3 ) terms correspond to the harmonicity and anharmonicity, respectively. The relatively smaller harmonic term in g-BAs reveals the weaker bonding strength of B-As bond than C-C band in graphene, and the relatively larger magnitude of the anharmonic term reveals stronger phonon anharmonicity in g-BAs. D. Lone-pair electrons To gain deep insight into the origin of the strong phonon anharmonicity in g-BAs, we further perform fundamental analysis on the electronic structures to uncover the underlying mechanism. We will show that the strong phonon anharmonicity in g-BAs is fundamentally driven by the stereochemically active lone-pair electrons due to the special orbital hybridization. The orbital projected electronic structures [band structure and density of states (DOS)] of g-BAs are depicted in Fig. 6(a,b,c). Direct band gap (∼0.75 eV) emerges in g-BAs, which is different from the indirect band gap in c-BAs [ Fig. 6(d,e)]. As shown in Fig. 6(a,b,c), the bonding states in g-BAs are governed by the B-s/p and As-p orbitals. As for B atom, all the 3 valence electrons are involved in the formation of B-As σ bonds due to the sp 2 -hybridization [ Fig. 6(a)]. The situation is totally different for As atom which possesses 5 valence electrons. The As-s orbital is largely (∼10 eV) confined below the valence band, forming an isolated band [ Fig. 6(b)]. Consequently, only the As-p orbitals contribute to the B-As σ bonds. Thus, the s 2 electrons in the s 2 p 3 valence configuration of As atom do not participate in the bonding and thus form lone-pair around the As atoms. To have an intuitive view on the lone-pair As-s electrons, we plot the electron localization function (ELF) in Fig. 7. The ELF displays the location and size of bonding and lonepair electron, which is powerful in interpreting chemical bonding patterns 21 . The ELF values range from 0 to 1, where 0 means no electron, 0.5 corresponds to the electron-gas-like pair probability, and 1 corresponds to perfect localization. It is well known that in graphene the C-C σ bonds are contributed from the hybridized Cs/p x /p y orbitals and the solo C-p z orbital forms the π bonds and the electronic Dirac cone (Supplemental Figure 3) 24 . Thus, there is no lone-pair electrons formed in graphene. By comparing the side views of the ELF between g-BAs [ Fig. 7(b)] and graphene [ Fig. 7(a)], it can be clearly seen that there are electrons localized around As atom that are not bonded, which are the lone-pair electrons. It was proposed that lone-pair electrons could lead to low κ 25 . The principle underlying the concept is that the overlapping wave functions of lone-pair electrons with valence (bonding) electrons from adjacent atoms would induce nonlinear electrostatic forces upon thermal agitation, leading to increased phonon anharmonicity in the lattice and thus reducing the κ 25-31 . Due to the orbital distribution in the same energy range and wave functions overlap as shown in Figs. 6(a,b,c) and 7(b), the non-bonding lone-pair As-s electrons interact with the covalently bonding electrons of adjacent B atoms in g-BAs. The interactions induce nonlinear electrostatic force among atoms when they thermally vibrate around the equilibrium positions 32 . The nonlinear electrostatic force originates from the asymmetric change of the hybridization between As-s and B-s/p for the atomic motion, as revealed by the pDOS evolution in Supplemental Figure 4. A more asymmetric potential energy well is induced together with the additional nonlinear electrostatic force [ Fig. 5(b)], which leads to the strong phonon anharmonicity in g-BAs [ Fig. 5(a)] and significantly reduces the κ of g-BAs [ Fig. 2(c)]. The form of orbital hybridizations in c-BAs are highly consistent with those in g-BAs as shown in Fig. 6, which means that lone-pair As-s electrons also emerge around As atoms in c-BAs. However, no strong phonon anharmonicity is induced in c-BAs by the lone-pair As-s electrons despite the similar orbital hybridization form as g-BAs, which is due to the different bonding nature and coordination environment between 3D and 2D geometry structures. Due to the perfect octahedral coordination of As atoms in c-BAs resulted from its cubic structure [ Fig. 2(b)], four equivalent valence bonds are formed. Thus, the lone-pair electrons in c-BAs are stereochemically inactive, which has no effect on the phonon anharmonicity. In contrast, for g-BAs possessing planar structure, no pyramidal geometry is formed for the B-As bonds. Consequently, lone-pair As-s electrons are located at both sides of the 2D structure plane in g-BAs [ Fig. 7(b)], which is different from that in 3D bulk systems of c-BAs. Thus, strong phonon anharmonicity exist in g-BAs due to the stereochemical activity of the lone-pair As-s electrons in the geometric form of planar structure. E. Extention to other systems It was shown above that bond nanodesigning by changing the coordination environment is an effective approach for realizing low κ, which would benefit the design of thermoelectric devices with improved performance. The approaches can also be applied to other materials beyond the BAs systems studied here, for instance, the class of group III-V compounds (e. g. BN, AlN, GaN, etc), where lone-pair electrons also exist (Fig. 1). Strong phonon anharmonicity and low κ could be achieved with the stereochemically activated lone-pair electrons, which can be realized by breaking the perfect octahedral coordination [ Fig. 2(a,b)]. Note that the 2D structures of the systems presented in Fig. 1 are all planar except silicene. For the systems with buckled structures in 2D form, such as silicon vs. silicene, the situation will be different due to the broken symmetry-based selection rule for phonon-phonon scattering 15,16,33 . It is found that there exists a strong correlation between the electronegativity difference and the κ modulation for binary compounds (Fig. 1). The effect of κ modulation by stereochemically activating the lone-pair electrons is weaker with a larger electronegativity difference. The reason may lie in the contribution to phonon anharmonicity of the electronegativity difference 15 . Other approaches could also have the same effects on the κ modulation that make the lone-pair electrons stereochemically active, such as nanostructuring. Alternatively, it would be also possible by substituting the atoms in ordinary materials with special atoms that can form non-bonding lone-pair electrons, such as nitrogen, phosphorus, arsenic, etc. Note that the κ of the studied systems here does not achieve an ultralow value, which may limit their direct applications in thermoelectrics. However, if the approach of activating lone-pair electrons is combined with the commonly used strategy of nanostructuring, ultralow κ desirable for thermoelectrics could be effectively achieved. For example, experimental measurements have already demonstrated that the κ of BAs can be suppressed by the arsenic deficiency or vacancy in the BAs sample 11 and the phonon-boundary scattering in BAs microstructures 12,13 . However, the obtained κ of BAs is still too high, which limits its potential applications in thermoelectrics, despite its quite large Seebeck coefficient and thermoelectric power factor 13 . If the lone-pair electrons in the BAs system can be stereochemically activated, the κ could be further reduced, which would improve the thermoelectric performance in the experimental setup. Besides, due to the intrinsic high κ, BAs also shows promising applications in efficient heat dissipation of electronics. When incorporating it into conventional semiconducting devices for heat dissipation, special attention should be paid to avoid activating the lone-pair electrons in BAs based nanostructures for keeping the high κ. III. CONCLUSIONS In summary, by cutting the 3D cubic structure of c-BAs at the (111) cross section, more than one order of magnitude lowered κ is achieved in the resultant 2D system of g-BAs with similar structure as graphene, which shows that bond nanodesigning by transforming the materials into nanoscale with the broken coordination environment could be an effective approach for realizing low κ. Based on the systematic study on the thermal transport properties of g-BAs comparing with c-BAs, diamond, and graphene (c-BAs → g-BAs vs. diamond → graphene), the underlying mechanism for the substantially lowered κ in the case of 'c-BAs → g-BAs' lies in two aspects: 1) Resulted from mass difference and 2D nature of bondings and structural symmetry, the weakened bunching effect for acoustic phonon branches together with their coupling with ZO phonon branch play a key role in driving large probability of phonon-phonon scattering. 2) Strong phonon anharmonicity is fundamentally driven by the stereochemically actived lone-pair electrons in g-BAs. Due to the special orbital hybridization, the s 2 electrons in the s 2 p 3 valence configuration of As atom do not participate in the bonding but form lone-pair instead. When transforming from the 3D cubic structure of c-BAs to 2D planar structure of g-BAs, the lone-pair As-s electrons become stereochemically actived due to the break of the perfect octahedral coordination of As atoms in c-BAs, which leads to strong phonon anharmonicity in g-BAs. Similar concept can be also extended to other systems with lone-pair electrons beyond BAs, such as group III-V compounds (e.g. BN, AlN, GaN, etc), where a strong correlation between κ modulation and electronegativity difference for binary compounds is found. Thus, the lone-pair electrons combined with a small electronegativity difference could be the indicator of lowering κ through bond nanodesigning to change the coordination environment. The proposed approach for realizing low κ and the underlying mechanism uncovered in this study would shed light on future research involving novel materials for energy applications. IV. COMPUTATIONAL DETAILS All the first-principles calculations are performed based on the density functional theory (DFT) as implemented in the Vienna ab initio simulation package (vasp) 35 . The Perdew-Burke-Ernzerhof (PBE) 36 of generalized gradient approximation (GGA) is chosen as the exchange-correlation functional for describing boron arsenide (BAs) systems, which is produced using the projector augmented wave (PAW) method 37 . Based on careful convergence test, the kinetic energy cutoff of wave function is set as 800 eV for all the DFT calculations. For the 2D systems, a large vacuum spacing is necessary to hinder the interactions arising from the employed periodic boundary conditions, which is set as 20Å along the out-of-plane direction. The Monkhorst-Pack 38 k-meshes of 15 × 15 × 1 and 2 × 2 × 1 are used to sample the Brillouin zone (BZ) for the structure optimizations and supercell force calculations, respectively, with the energy convergence threshold set as 10 −8 eV. The structure optimization is fully conducted with no limitation until the maximal Hellmann-Feynman force acting on each atom is less than 10 −9 eV/Å. For the supercell force calculations to obtain interatomic force constants (IFCs), a 5 × 5 × 1 supercell is constructed based on the convergence of the phonon dispersion with respect to the supercell size. The cutoff radius (r cutoff ) introduced during the calculations of the anharmonic IFCs is also fully tested, which is used to discard the interactions between atoms with distance larger than a certain value for practical purposes. The r cutoff of 10th nearest neighbors (∼0.94 nm) is found to be large enough to obtain converged and reliable κ 39 . The space group symmetry properties are used to reduce the computational cost and the translational and rotational invariance of IFCs are enforced using the Lagrange multiplier method [40][41][42] . With the anharmonic IFCs, the scattering matrix can be constructed, based on which one can calculate all the three-phonon scattering rates and then obtain phonon lifetime. The Born effective charge (Z * ) and dielectric constant ( ) obtained based on the density functional perturbation theory (DFPT) are included for taking into account of long-range electrostatic interactions. The thickness for calculating κ is chosen as the van der Walls diameter (3.7Å). The κ is obtained by solving the linearized phonon BTE using an iterative procedure as implemented in the ShengBTE package based on the IFCs 42,43 . More information can be found in Supplemental Note 2. Table for the specific κ of the typical systems in 3D and 2D forms; Supplemental Figures for 1) the phonon dispersion showing quadratic behavior of FA, 2) phonon-phonon scattering channels of FA, 3) oribitals projected electronic structures for graphene, 4) pDOS evolution for g-BAs due to atomic motion, and 5) normalized κ for several typical 2D materials; Supplemental Notes for 1) phonon-phonon scatter-ing channels and 2) more information on computational methods. VII. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. FIG. 3 . 3The comparison of phonon dispersion of boron arsenide in 2D (g-BAs) with 3D (c-BAs). (a) The phonon dispersion of g-BAs along the high-symmetry points, where the out-pf-plane flexural acoustic (FA) phonon branch shows a good quadratic behavior (SupplementalFigure 1). There exists a huge gap of 13.00 THz between the optical and acoustic phonon branches. The z-direction optical (ZO) phonon branch, which is below the gap, is highlighted in blue. (b) The phonon dispersion of c-BAs along the high-symmetry points, where dot lines in black are from theoretical calculations 5 and blue/green points are from experimental measurements 6,14 . The gap between the optical and acoustic phonon branches is 9.28 THz. (c,d) The partial density of states (pDOS) of (c) g-BAs and (d) c-BAs, where the contribution from boron (B) atoms to ZO is highlighted with a colored ellipse. [Fig. 2(c)] 5 , both of which FIG. 4. Comparison of the phonon-phonon scattering phase space of g-BAs with c-BAs and graphene. The colored ellipse highlights the larger scattering phase space of g-BAs than c-BAs in the frequency range of 5-10 THz, where the ZO phonon branch lies. (Inset) The comparison of scattering rate between g-BAs and graphene. Fig. 3 3(a) shows the phonon dispersion of g-BAs, in comparison with c-BAs [Fig. 3(b)] FIG. 5 . 5Strong phonon anharmonicity in g-BAs. (a) Comparison of Grüneisen parameters between g-BAs and graphene. The colored lines highlight the Grüneisen parameters of FA phonon branch. (b) Comparison of potential energy wells between g-BAs and graphene. The atom is moved along the bonding direction. Points are from first-principles calculations and lines are fittings to the formula shown on site. Inset table: The fitted parameters for g-BAs and graphene, respectively. FIG. 6 . 6Orbitals projected electronic structures, revealing the non-bonding lone-pair As-s electrons. (a,b,c) The orbitals projected (a,b) electronic band structures and (c) density of states (DOS) for g-BAs. The electronic structures are projected to (a) B-s/p and (b) As-s/p/d orbitals. (d,e,f) Similar figures for c-BAs in comparison. FIG. 7 . 7Side views of the electron localization function (ELF) for (a) graphene and (b) g-BAs. 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{'abstract': 'Reducing thermal conductivity (κ) is an efficient way to boost the thermoelectric performance to achieve direct solid-state conversion to electrical power from thermal energy, which has lots of valuable applications in reusing waste resources. In this paper, we propose an effective approach for realizing low κ by introducing lone-pair electrons or making the lone-pair electrons stereochemically active through bond nanodesigning. As a case study, by cutting at the (111) cross section of the three-dimensional (3D) cubic boron arsenide (c-BAs), the κ is lowered by more than one order of magnitude in the resultant two-dimensional (2D) system of graphene-like BAs (g-BAs) due to the stereochemically actived lone-pair electrons. However, this does not naturally happen to all materials. For instance, breaking the perfect octahedral coordination of 3D diamond as in the 2D graphene adversely enhances thermal transport. The underlying mechanism is analyzed based on the comparative study on the thermal transport properties of g-BAs, c-BAs, graphene, and diamond (c-BAs → g-BAs vs. diamond → graphene). Furthermore, deep insight into the electronic origin is gained by performing fundamental analysis on the electronic structures. Similar concept can be also extended to other systems with lone-pair electrons beyond BAs, such as group III-V compounds (e.g. BN, AlN, GaN, etc), where a strong correlation between κ modulation and electronegativity difference for binary compounds is found. Thus, the lone-pair electrons combined with a small electronegativity difference could be the indicator of lowering κ through bond nanodesigning to change the coordination environment. The proposed approach for realizing low κ and the underlying mechanism uncovered in this study would largely benefit the design of thermoelectric devices with improved performance, especially in future research involving novel materials for energy applications. arXiv:1904.00329v1 [cond-mat.mtrl-sci]', 'arxivid': '1904.00329', 'author': ['Guangzhao Qin \nInstitute of Mineral Engineering\nDivision of Materials Science and Engineering\nFaculty of Georesources and Materials Engineering\nRWTH Aachen University\n52064AachenGermany\n\nDepartment of Mechanical Engineering\nUniversity of South Carolina\n29208ColumbiaSCUSA\n', 'Zhenzhen Qin \nSchool of Physics and Engineering\nInternational Laboratory for Quantum Functional Materials of Henan\nZhengzhou University\n450001ZhengzhouChina\n', 'Huimin Wang \nDepartment of Mechanical Engineering\nUniversity of South Carolina\n29208ColumbiaSCUSA\n\nCollege of Engineering and Applied Science\nNanjing University\n210023NanjingChina\n', 'Ming Hu \nDepartment of Mechanical Engineering\nUniversity of South Carolina\n29208ColumbiaSCUSA\n'], 'authoraffiliation': ['Institute of Mineral Engineering\nDivision of Materials Science and Engineering\nFaculty of Georesources and Materials Engineering\nRWTH Aachen University\n52064AachenGermany', 'Department of Mechanical Engineering\nUniversity of South Carolina\n29208ColumbiaSCUSA', 'School of Physics and Engineering\nInternational Laboratory for Quantum Functional Materials of Henan\nZhengzhou University\n450001ZhengzhouChina', 'Department of Mechanical Engineering\nUniversity of South Carolina\n29208ColumbiaSCUSA', 'College of Engineering and Applied Science\nNanjing University\n210023NanjingChina', 'Department of Mechanical Engineering\nUniversity of South Carolina\n29208ColumbiaSCUSA'], 'corpusid': 90243296, 'doi': '10.1021/acs.jpclett.2c03255', 'github_urls': [], 'n_tokens_mistral': 17266, 'n_tokens_neox': 13811, 'n_words': 7215, 'pdfsha': 'ac2877e86bfbed180211af0709eaf4706435c647', 'pdfurls': ['https://arxiv.org/pdf/1904.00329v1.pdf'], 'title': ['Activated lone-pair electrons lead to low lattice thermal conductivity: a case study of boron arsenide', 'Activated lone-pair electrons lead to low lattice thermal conductivity: a case study of boron arsenide'], 'venue': []}

arxiv

ON THE GEOMETRY OF BIHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS 30 Sep 2008 D Fetcu C Oniciuc ON THE GEOMETRY OF BIHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS 30 Sep 2008 We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in Sasakian space forms. Finally, using the Boothby-Wang fibration, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces. Introduction As defined by Eells and Sampson in [14], where τ (f ) is called the tension field of f . When f is an isometric immersion with mean curvature vector field H, then τ (f ) = mH and f is harmonic if and only if it is minimal. The bienergy functional (proposed also by Eells and Sampson in 1964, [14]) is defined by In a different way, Chen defined the biharmonic submanifolds in an Euclidean space as those with harmonic mean curvature vector field ( [10]). Replacing c = 0 in the above equation we just reobtain Chen's definition. Moreover, let f : M → R n be an isometric immersion. Set f = (f 1 , . . . , f n ) and H = (H 1 , . . . , H n ). Then ∆ f H = (∆H 1 , . . . , ∆H n ), where ∆ is the Beltrami-Laplace operator on M , and f is biharmonic if and only if ∆ f H = ∆( −∆f m ) = − 1 m ∆ 2 f = 0. There are several classification results for the proper-biharmonic submanifolds in Euclidean spheres and non-existence results for such submanifolds in space forms N c , c ≤ 0 ( [4], [5], [7], [8], [9], [10], [13]), while in spaces of non-constant sectional curvature only few results were obtained ( [1], [12], [18], [19], [25], [29]). We recall that the proper-biharmonic curves of the unit Euclidean 2-dimensional sphere S 2 are the circles of radius 1 √ 2 , and the proper-biharmonic curves of S 3 are the geodesics of the minimal Clifford torus S 1 ( 1 √ 2 ) × S 1 ( 1 √ 2 ) with the slope different from ±1. The proper-biharmonic curves of S 3 are helices. Further, the properbiharmonic curves of S n , n > 3, are those of S 3 (up to a totally geodesic embedding). Concerning the hypersurfaces of S n , it was conjectured in [4] that the only properbiharmonic hypersurfaces are the open parts of S n−1 ( 1 √ 2 ) or S m1 ( 1 √ 2 ) × S m2 ( 1 √ 2 ) with m 1 + m 2 = n − 1 and m 1 = m 2 . Since odd dimensional unit Euclidean spheres S 2n+1 are Sasakian space forms with constant ϕ-sectional curvature 1, the next step is to study the biharmonic submanifolds of Sasakian space forms. In this paper we mainly gather the results obtained in [15], [16] and [17]. We note that the proper-biharmonic submanifolds in pseudo-Riemannian manifolds are also intensively-studied (for example, see [2], [3], [11]). For a general account of biharmonic maps see [22] and The Bibliography of Biharmonic Maps [28]. Conventions. We work in the C ∞ category, that means manifolds, metrics, connections and maps are smooth. The Lie algebra of the vector fields on N is denoted by C(T N ). Sasakian Space Forms In this section we briefly recall some basic facts from the theory of Sasakian manifolds. For more details see [6]. A contact metric structure on a manifold N 2n+1 is given by (ϕ, ξ, η, g), where ϕ is a tensor field of type (1, 1) on N , ξ is a vector field on N , η is an 1-form on N and g is a Riemannian metric, such that    ϕ 2 = −I + η ⊗ ξ, η(ξ) = 1, g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ), g(X, ϕY ) = dη(X, Y ), for any X, Y ∈ C(T N ). A contact metric structure (ϕ, ξ, η, g) is Sasakian if it is normal, i.e. N ϕ + 2dη ⊗ ξ = 0, where N ϕ (X, Y ) = [ϕX, ϕY ] − ϕ[ϕX, Y ] − ϕ[X, ϕY ] + ϕ 2 [X, Y ], ∀X, Y ∈ C(T N ) is the Nijenhuis tensor field of ϕ. The contact distribution of a Sasakian manifold (N, ϕ, ξ, η, g) is defined by {X ∈ T N : η(X) = 0}, and an integral curve of the contact distribution is called Legendre curve. A submanifold M of N which is tangent to ξ is said to be anti-invariant if ϕ maps any vector tangent to M and normal to ξ to a vector normal to M . Let (N, ϕ, ξ, η, g) be a Sasakian manifold. The sectional curvature of a 2-plane generated by X and ϕX, where X is an unit vector orthogonal to ξ, is called ϕ-sectional curvature determined by X. A Sasakian manifold with constant ϕsectional curvature c is called a Sasakian space form and it is denoted by N (c). A contact metric manifold (N, ϕ, ξ, η, g) is called regular if for any point p ∈ N there exists a cubic neighborhood of p such that any integral curve of ξ passes through the neighborhood at most once, and strictly regular if all integral curves are homeomorphic to each other. Let (N, ϕ, ξ, η, g) be a regular contact metric manifold. Then the orbit spaceN = N/ξ has a natural manifold structure and, moreover, if N is compact then N is a principal circle bundle overN (the Boothby-Wang Theorem). In this case the fibration π : N →N is called the Boothby-Wang fibration. The Hopf fibration π : S 2n+1 → CP n is a well-known example of a Boothby-Wang fibration. Even if N is non-compact, we still call the fibration π : N →N of a strictly regular Sasakian manifold, the Boothby-Wang fibration. Biharmonic Legendre Curves in Sasakian Space Forms Let (N n , g) be a Riemannian manifold and γ : I → N a curve parametrized by arc length. Then γ is called a Frenet curve of osculating order r, 1 ≤ r ≤ n, if there exists orthonormal vector fields E 1 , E 2 , . . . , E r along γ such that E 1 = γ ′ = T , ∇ T E 1 = κ 1 E 2 , ∇ T E 2 = −κ 1 E 1 + κ 2 E 3 ,...,∇ T E r = −κ r−1 E r−1 , where κ 1 , . . . , κ r−1 are positive functions on I. A geodesic is a Frenet curve of osculating order 1; a circle is a Frenet curve of osculating order 2 with κ 1 = constant; a helix of order r, r ≥ 3, is a Frenet curve of osculating order r with κ 1 , . . . , κ r−1 constants; a helix of order 3 is called, simply, helix. In [16] we studied the biharmonicity of Legendre Frenet curves and we obtained the following results. Let (N 2n+1 , ϕ, ξ, η, g) be a Sasakian space form with constant ϕ-sectional curvature c and γ : I → N a Legendre Frenet curve of osculating order r. Then γ is biharmonic if and only if τ 2 (γ) = ∇ 3 T T − R(T, ∇ T T )T = (−3κ 1 κ ′ 1 )E 1 + κ ′′ 1 − κ 3 1 − κ 1 κ 2 2 + (c+3)κ1 4 E 2 +(2κ ′ 1 κ 2 + κ 1 κ ′ 2 )E 3 + κ 1 κ 2 κ 3 E 4 + 3(c−1)κ1 4 g(E 2 , ϕT )ϕT = 0. The expression of the bitension field τ 2 (γ) imposed a case-by-case analysis as follows. Case I (c = 1) Remark 3.4. In dimension 3 the result was obtained by Inoguchi in [19] and explicit examples are given in [15]. Theorem 3.1 ([16]). If c = 1 then γ is proper-biharmonic if and only if n ≥ 2 and either γ is a circle with κ 1 = 1 or γ is a helix with κ 2 1 + κ 2 2 = 1. Case II (c = 1 and E 2 ⊥ ϕT ) Theorem 3.2 ([16]). Assume that c = 1 and E 2 ⊥ ϕT . We have 1) if c ≤ −3 then γ is biharmonic if and only if it is a geodesic; 2) if c > −3 then γ is proper-biharmonic if and only if either a) n ≥ 2 and γ is a circle with κ 2 1 = c+3 4 , or b) n ≥ 3 and γ is a helix with κ 2 1 + κ 2 2 = c+3 4 . Case III (c = 1 and E 2 ϕT ) Case IV (c = 1 and g(E 2 , ϕT ) is not constant 0, 1 or −1) r such that g(E 2 , ϕT ) is not constant 0, 1 or −1. We have 1) if c ≤ −3 then γ is biharmonic if and only if it is a geodesic; 2) if c > −3 then γ is proper-biharmonic if and only if r ≥ 4, ϕT = cos α 0 E 2 + sin α 0 E 4 and            κ 1 , κ 2 , κ 3 = constant > 0 κ 2 1 + κ 2 2 = c+3 4 + 3(c−1) 4 cos 2 α 0 κ 2 κ 3 = − 3(c−1) 8 sin(2α 0 ), where α 0 ∈ (0, 2π) \ { π 2 , π, 3π 2 } is a constant such that c + 3 + 3(c − 1) cos 2 α 0 > 0, 3(c − 1) sin(2α 0 ) < 0. In order to obtain explicit examples of proper-biharmonic Legendre curves given by Theorem 3.1 we used the unit Euclidean sphere S 2n+1 as a model of a Sasakian space form with c = 1 and we proved the following E 2n+2 = (R 2n+2 , , ) is either γ(s) = 1 √ 2 cos √ 2s e 1 + 1 √ 2 sin √ 2s e 2 + 1 √ 2 e 3 , where {e i , Ie j } are constant unit vectors orthogonal to each other, or In [21] are introduced the complex torsions for a Frenet curve in a complex manifold. In the same way, for γ : I → N a Legendre Frenet curve of osculating order r in a Sasakian manifold (N 2n+1 , ϕ, ξ, η, g), we define the ϕ-torsions τ ij = g(E i , ϕE j ) = −g(ϕE i , E j ), i, j = 1, . . . , r, i < j. γ(s) = 1 √ 2 cos(As)e 1 + 1 √ 2 sin(As)e 2 + 1 √ 2 cos(Bs)e 3 + 1 √ 2 sin(Bs)e 4 , where A = √ 1 + κ 1 , B = √ 1 − κ 1 , κ 1 ∈ (0, 1), {e i } It is easy to see that Proof. From Theorems 3.2, 3.3 and 3.5 we see that if γ is a proper-biharmonic Legendre Frenet curve of osculating order r < 4, then τ 12 = 0 or τ 12 = ±1 and, obviously, we only have to prove that when γ is a helix then τ 13 and τ 23 are constants. Indeed, by using the Frenet equations of γ, we have τ 13 = g(E 1 , ϕE 3 ) = − 1 κ 2 g(ϕE 1 , ∇ E1 E 2 + κ 1 E 1 ) = − 1 κ 2 g(ϕE 1 , ∇ E1 E 2 ) = 1 κ 2 g(E 2 , ∇ E1 ϕE 1 ) = 1 κ 2 g(E 2 , ϕ∇ E1 E 1 + ξ) = 0 since g(E 2 , ξ) = 1 κ 1 g(∇ E1 E 1 , ξ) = − 1 κ 1 g(E 1 , ∇ E1 ξ) = 1 κ 1 g(E 1 , ϕE 1 ) = 0. On the other hand, it is easy to see that for any Frenet curve of osculating order 3 we have τ 23 = 1 κ1 (τ ′ 13 + κ 2 τ 12 + η(E 3 )) and η(E 3 ) = g(E 3 , ξ) = 1 κ 2 g(∇ E1 E 2 , ξ) + κ 1 g(E 1 , ξ) = − 1 κ 2 g(E 2 , ∇ E1 ξ) = − 1 κ 2 τ 12 . In conclusion, τ 23 = 1 κ1 (τ ′ 13 + κ 2 τ 12 − 1 κ2 τ 12 ) = constant.κ 1 = √ c + 3 2 , κ 2 = 1 2 6(c − 1)(5 − c) c + 3 , κ 3 = 1 2 3(c − 1)(3c − 7) c + 3 . Moreover, the ϕ-torsions of γ are given by        τ 12 = ∓ 2(5−c) c+3 , τ 13 = 0, τ 14 = ± 3c−7 c+3 , τ 23 = ∓ 3c−7 √ 3(c−1)(c+3) , τ 24 = 0, τ 34 = ± 2(5−c)(3c−7) 3(c−1)(c+3) . Proof. Let γ be a proper-biharmonic Legendre Frenet curve in N (c) of osculating order r = 4. Then c = 1 and τ 12 is different from 0, 1 or −1. From Theorem 3.5 we have ϕE 1 = cos α 0 E 2 + sin α 0 E 4 . It results that τ 12 = − cos α 0 , τ 13 = 0, τ 14 = − sin α 0 , and τ 24 = 0. In order to prove that τ 23 is constant we differentiate the expression of ϕE 1 along γ and using the Frenet equations we obtain ∇ E1 ϕE 1 = cos α 0 ∇ E1 E 2 + sin α 0 ∇ E1 E 4 = −κ 1 cos α 0 E 1 + (κ 2 cos α 0 − κ 3 sin α 0 )E 3 . On the other hand, ∇ E1 ϕE 1 = κ 1 ϕE 2 + ξ and therefore we have (3.1) κ 1 ϕE 2 + ξ = −κ 1 cos α 0 E 1 + (κ 2 cos α 0 − κ 3 sin α 0 )E 3 . We take the scalar product in (3.1) with ξ and obtain (3.2) (κ 2 cos α 0 − κ 3 sin α 0 )η(E 3 ) = 1. In the same way as in the proof of Proposition 3.9 we get η(E 3 ) = g(E 3 , ξ) = 1 κ 2 g(∇ E1 E 2 , ξ) + κ 1 g(E 1 , ξ) = − 1 κ 2 g(E 2 , ∇ E1 ξ) = − 1 κ 2 τ 12 = cos α 0 κ 2 and then, from (3.2), κ 2 sin α 0 = −κ 3 cos α 0 . Therefore α 0 ∈ ( π 2 , π) ∪ ( 3π 2 , 2π). Next, from Theorem 3.5, we have κ 2 1 = c + 3 4 , κ 2 2 = 3(c − 1) 4 cos 2 α 0 , κ 2 3 = 3(c − 1) 4 sin 2 α 0 , and so c must be greater than 1. Now, we take the scalar product in (3.1) with E 3 , ϕE 2 and ϕE 4 , respectively, and we get (3.3) κ 1 τ 23 = −(κ 2 cos α 0 − κ 3 sin α 0 ) + η(E 3 ) = − κ 2 cos α 0 + cos α 0 κ 2 (3.4) κ 1 sin 2 α 0 = −(κ 2 cos α 0 − κ 3 sin α 0 )τ 23 = − κ 2 cos α 0 τ 23 (3.5) 0 = κ 1 cos α 0 sin α 0 + (κ 2 cos α 0 − κ 3 sin α 0 )τ 34 = κ 1 cos α 0 sin α 0 + κ 2 cos α 0 τ 34 . and then, equations (3.3) and (3.4) lead to κ 2 1 sin 2 α 0 = κ 2 2 cos 2 α 0 − 1. We come to the conclusion sin 2 α 0 = 3c−7 c+3 , so c ∈ ( 7 3 , 5), and then we obtain the expressions of the curvatures and the ϕ-torsions. Remark 3.11. The proper-biharmonic Legendre curves given by Theorem 3.6 (for the case c = 1) have also constant ϕ-torsions. A Method To Obtain Biharmonic Submanifolds in a Sasakian Space Form In [16] we gave a method to obtain proper-biharmonic anti-invariant submanifolds in a Sasakian space form from proper-biharmonic integral submanifolds. Biharmonic Hopf Cylinders in a Sasakian Space Form Let (N 2n+1 , ϕ, ξ, η, g) be a strictly regular Sasakian manifold andī :M →N a submanifold ofN . Then M = π −1 (M ) is the Hopf cylinder overM , where π : N →N = N/ξ is the Boothby-Wang fibration. In [19] the biharmonic Hopf cylinders in a 3-dimensional Sasakian space form are classified. In [17] we obtained a geometric characterization of biharmonic Hopf cylinders of any codimension in an arbitrary Sasakian space form. A special case of our result is the case whenM is a hypersurface. From now on we shall consider c > −3. In [26] Takagi classified all homogeneous real hypersurfaces in the complex projective space CP n , n > 1, and found five types of such hypersurfaces (see also [23]). The first type (with subtypes A1 and A2) are described in the following. We shall consider u ∈ (0, π 2 ) and r a positive constant given by 1 r 2 = c+3 4 . Theorem 5.5 ( [26]). The geodesic spheres (Type A1) in complex projective space CP n (c+3) have two distinct principal curvatures: λ 2 = 1 r cot u of multiplicity 2n−2 and a = 2 r cot(2u) of multiplicity 1. Theorem 5.6 ( [26]). The hypersurfaces of Type A2 in complex projective space CP n (c + 3) have three distinct principal curvatures: λ 1 = − 1 r tan u of multiplicity 2p, λ 2 = 1 r cot u of multiplicity 2q, and a = 2 r cot(2u) of multiplicity 1, where p > 0, q > 0, and p + q = n − 1. We note that if c = 1 andM is of type A1 or A2 then π −1 (M ) = S 1 (cos u) × S 2n−1 (sin u) ⊂ S 2n+1 or π −1 (M ) = S 2p+1 (cos u) × S 2q+1 (sin u), respectively. By using Takagi's result we classified in [17] the biharmonic Hopf cylinders M = π −1 (M ) in a Sasakian space form N 2n+1 over homogeneous real hypersurfaces in CP n , n > 1. harmonic maps f : (M, g) → (N, h) are the critical points of the energy functional E(f ) = 1 2 M df 2 v g and they are solutions of the associated Euler-Lagrange equation τ (f ) = trace g ∇df = 0, Theorem 2.1 ([24]). Let (N, ϕ, ξ, η, g) be a strictly regular Sasakian manifold. Then onN can be given the structure of a Kähler manifold. Moreover, if (N, ϕ, ξ, η, g) is a Sasakian space form N (c), thenN has constant sectional holomorphic curvature c + 3. Theorem 3.3 ([16]). If c = 1 and E 2 ϕT , then {T, ϕT, ξ} is the Frenet frame field of γ and we have 1) if c < 1 then γ is biharmonic if and only if it is a geodesic; 2) if c > 1 then γ is proper-biharmonic if and only if it is a helix with κ 2 1 = c−1 (and κ 2 = 1). Theorem 3.5 ([16]). Let c = 1 and γ a Legendre Frenet curve of osculating order Theorem 3.6 ([16]). Let γ : I → S 2n+1 (1), n ≥ 2, be a proper-biharmonic Legendre curve parametrized by arc length. Then the parametric equation of γ in the Euclidean space = 0 A e 1 , 01are constant unit vectors orthogonal to each other such that e 1 , Ie 3 = e 1 , Ie 4 = e 2 , Ie 3 = e 2 , Ie 4 Ie 2 + B e 3 , Ie 4 = 0 and I is the usual complex structure on R 2n+2 .Remark 3.7. For the Cases II and III we also obtained the explicit equations of proper-biharmonic Legendre curves in odd dimensional spheres endowed with the deformed Sasakian structure introduced in[27]. Proposition 3. 8 . 8Let γ : I → N (c) be a proper-biharmonic Legendre Frenet curve in a Sasakian space form N (c), c = 1. Then c > −3 and τ 12 is constant. Moreover Proposition 3.9. If γ is a proper-biharmonic Legendre Frenet curve in a Sasakian space form N (c), c > −3, c = 1, of osculating order r < 4, then it is a circle or a helix with constant ϕ-torsions. Proposition 3 . 10 . 310If γ is a proper-biharmonic Legendre Frenet curve in a Sasakian space form N (c) of osculating order r = 4, then c ∈ ( 7 3 , 5) and the curvatures of γ are Theorem 4.1 ([16]). Let (N 2n+1 , ϕ, ξ, η, g) be a strictly regular Sasakian space form with constant ϕ-sectional curvature c and let i : M → N be an r-dimensional integral submanifold of N , 1 ≤ r ≤ n. ConsiderF : M = I × M → N, F (t, p) = φ t (p) = φ p (t)where I = S 1 or I = R and {φ t } t∈I is the flow of the vector field ξ. Then F :( M , g = dt 2 + i * g) → Nis a Riemannian immersion and it is proper-biharmonic if and only if M is a proper-biharmonic submanifold of N .The previous Theorem provides a classification result for proper-biharmonic surfaces in a Sasakian space form, which are invariant under the flow-action of ξ. Theorem 4.2 ([16]). Let M 2 be a surface of N 2n+1 (c) invariant under the flowaction of the characteristic vector field ξ. Then M is proper-biharmonic if and only if, locally, it is given by x(t, s) = φ t (γ(s)), where γ is a proper-biharmonic Legendre curve. Also, using the standard Sasakian 3-structure on S 7 , by iteration, Theorem 4.1 leads to examples of 3-dimensional proper-biharmonic submanifolds of S 7 . Theorem 5.1 ([19]). Let Sγ be a Hopf cylinder, whereγ is a curve in the orbit space of N 3 (c), parametrized by arc length. We have 1) if c 1, then Sγ is biharmonic if and only if it is minimal; 2) if c > 1, then Sγ is proper-biharmonic if and only if the curvatureκ ofγ is constantκ 2 = c − 1. Proposition 5.2 ([17]). IfM is a hypersurface ofN , then M = π −1 (M ) is biharmonic if and only if A ∇ ⊥ · H (·) + n grad( H 2 ) = 0, where B, A and H are the second fundamental form of M in N , the shape operator and the mean curvature vector field, respectively, and ∇ ⊥ and ∆ ⊥ are the normal connection and Laplacian on the normal bundle of M in N . Proposition 5.3 ([17]). IfM is a hypersurface and H = constant = 0, then M = π −1 (M ) is proper-biharmonic if and only if B 2 = c(n + 1) + 3n − 5 2 . Remark 5.4. From the last result we see that there exist no proper-biharmonic hypersurfaces of constant mean curvature M = π −1 (M ) in N (c) if c ≤ 5−3n n+1 , which implies that such hypersurfaces do not exist if c ≤ −3, whatever the dimension of N is. . Theorem 5.7 ([17]). Let M = π −1 (M ) be the Hopf cylinder overM . 1) IfM is of Type A1, then M is proper-biharmonic if and only if either a) c = 1 and tan 2 u = 1, or b) c ∈ IfM is of Type A2, then M is proper-biharmonic if and only if either a) c = 1, tan 2 u = 1 and p = q, or Theorem 5.8 ([17]). There are no proper-biharmonic hypersurfaces M = π −1 (M ) whenM is a hypersurface of Type B, C, D or E in the complex projective space CP n (c + 3). AcknowledgementsThe authors were partially supported by the Grant CEEX, ET, 5871/2006 and by the Grant CEEX, ET, 5883/2006, Romania.The first author would like to thank to the organizers, especially to Professor I. Mladenov, for the Conference Grant. K Arslan, R Ezentas, C Murathan, T Sasahara, Biharmonic Anti-invariant Submanifolds in Sasakian Space Forms. 48K. Arslan, R. Ezentas, C. Murathan and T. Sasahara, Biharmonic Anti-invariant Submani- folds in Sasakian Space Forms, Beitrge Algebra Geom., 48 (1)(2007), 191-207. Biharmonic Lorentz Hypersurfaces in E 4 1. A Arvanitoyeorgos, F Defever, G Kaimakamis, V J Papantoniou, Pacific J. Math. 229A. Arvanitoyeorgos, F. Defever, G. Kaimakamis and V.J. Papantoniou, Biharmonic Lorentz Hypersurfaces in E 4 1 , Pacific J. Math., 229 (2007), 293-305. Hypersurfaces of E 4 s with Proper Mean Curvature Vector. A Arvanitoyeorgos, F Defever, G Kaimakamis, J. Math. Soc. Japan. 59A. Arvanitoyeorgos, F. Defever and G. Kaimakamis, Hypersurfaces of E 4 s with Proper Mean Curvature Vector, J. Math. Soc. Japan, 59 (2007), 797-809. Classification Results for Biharmonic Submanifolds in Spheres. A Balmuş, S Montaldo, C Oniciuc, Israel J. Math. to appearA. Balmuş, S. Montaldo and C. Oniciuc, Classification Results for Biharmonic Submanifolds in Spheres, Israel J. Math., to appear. Biharmonic Hypersurfaces in 4-Dimensional Space Forms. A Balmuş, S Montaldo, C Oniciuc, Math. Nachr. to appearA. Balmuş, S. Montaldo and C. Oniciuc, Biharmonic Hypersurfaces in 4-Dimensional Space Forms, Math. Nachr., to appear. D E Blair, Riemannian Geometry of Contact and Symplectic Manifolds. 203D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Birkhäuser Boston, Progress in Mathematics, Volume 203, 2002. Biharmonic Submanifolds of S 3. R Caddeo, S Montaldo, C Oniciuc, Internat. J. Math. 12R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic Submanifolds of S 3 , Internat. J. Math., 12 (2001), 867-876. Biharmonic Submanifolds in Spheres. R Caddeo, S Montaldo, C Oniciuc, Israel J. Math. 130R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic Submanifolds in Spheres, Israel J. Math., 130 (2002), 109-123. . R Caddeo, S Montaldo, P Piu, Biharmonic Curves on a Surface, Rend. Mat. Appl. 7R. Caddeo, S. Montaldo and P. Piu, Biharmonic Curves on a Surface, Rend. Mat. Appl., (7) 21 (2001), 143-157. A Report on Submanifolds of Finite Type. B Y Chen, Soochow J. Math. 22B.Y. Chen, A Report on Submanifolds of Finite Type, Soochow J. Math., 22 (1996), 117-337. 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The Bibliography of Biharmonic Maps. http://beltrami.sc.unica.it/ biharmonic/. W Zhang, arXiv:math.DG/07053961v1New Examples of Biharmonic Submanifolds in CP n and S 2n+1. W. Zhang, New Examples of Biharmonic Submanifolds in CP n and S 2n+1 , arXiv:math.DG/07053961v1.

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{'abstract': 'We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in Sasakian space forms. Finally, using the Boothby-Wang fibration, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces.', 'arxivid': '0809.3093', 'author': ['D Fetcu ', 'C Oniciuc '], 'authoraffiliation': [], 'corpusid': 7745861, 'doi': '10.7546/giq-10-2009-171-182', 'github_urls': [], 'n_tokens_mistral': 10477, 'n_tokens_neox': 8583, 'n_words': 4761, 'pdfsha': 'c4a4659ad1c7b0753a90657365527ec691f1800b', 'pdfurls': ['https://arxiv.org/pdf/0809.3093v2.pdf'], 'title': ['ON THE GEOMETRY OF BIHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS', 'ON THE GEOMETRY OF BIHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS'], 'venue': []}

arxiv

QUANTIFYING AND MANAGING UNCERTAINTY IN PIECEWISE-DETERMINISTIC MARKOV PROCESSES * Elliot Cartee Antonio Farah April Nellis Jacob Van Hook ¶ Alexander Vladimirsky QUANTIFYING AND MANAGING UNCERTAINTY IN PIECEWISE-DETERMINISTIC MARKOV PROCESSES * piecewise-deterministic processoptimal controlHamilton-Jacobi PDEsuncer- tainty quantificationrobustness AMS subject classifications 60J2749L2035F6160K4065M0665C40 In piecewise-deterministic Markov processes (PDMPs) the state of a finite-dimensional system evolves continuously, but the evolutive equation may change randomly as a result of discrete switches. A running cost is integrated along the corresponding piecewise-deterministic trajectory up to the termination to produce the cumulative cost of the process. We address three natural questions related to uncertainty in cumulative cost of PDMP models: (1) how to compute the Cumulative Distribution Function (CDF) of the cumulative cost when the switching rates are fully known; (2) how to accurately bound the CDF when the switching rates are uncertain; and (3) assuming the PDMP is controlled, how to select a control to optimize that CDF. In all three cases, our approach requires posing a system of suitable hyperbolic partial differential equations, which are then solved numerically on an augmented state space. We illustrate our method using simple examples of trajectory planning under uncertainty for several 1D and 2D first-exit time problems. In the Appendix, we also apply this method to a model of fish harvesting in an environment with random switches in carrying capacity. 1. Introduction. Piecewise-deterministic Markov processes (PDMPs) provide a powerful formalism for modeling discrete random changes in a global environment. That formalism is particularly useful when the number of deterministic modes of the global environment is relatively small and there is a high fidelity statistical characterization of mode-to-mode switching rates. Such processes arise in a broad range of applications, especially in the biological sciences [39]. For example, they can be used to model keratin network formation [9], SIRS epidemic spread [35], genetic networks [40], and predator-prey systems [10,17,32]. In other disciplines, applications of PDMPs include models of fatigue crack growth [15], financial contagion [19], manufacturing processes [1,11,37,42], sustainable development, economic growth & climate change [30,31], and path-planning under uncertainty [4,13,27,45]. In this paper we focus on a computational framework for quantifying uncertainty in outcomes of PDMPs due to random switching times and possible uncertainty in switching rates. If a PDMP system is controlled in real time, we also show that this uncertainty of outcomes can be actively managed. In our PDMP models, the full state of the system is described by a continuous component x ∈ Ω ⊂ R d and a discrete component i ∈ M = {1, . . . , M } that represents the current deterministic "mode". Starting from the initial configuration (x, i), the evolution of continuous component y(t) is defined by a (mode-dependent) ODE y (t) = f (y(t), m(t)) = f m(t) (y(t)), (1.1) y(0) = x ∈ Ω, m(0) = i ∈ M, while the switches in mode m(t) are based on a continuous-time Markov process on M. Using λ ij to denote the rate of (i → j) switching, we can write lim τ →0 P(m(t + τ ) = j | m(t) = i) τ = λ ij , ∀t ≥ 0, i ∈ M, j ∈ M\{i}. (1.2) Here, we focus on exit-time problems, in which the process stops as soon as the system reaches a compact exit set Q ⊂ Ω. Due to the random mode-switches, the exit-time T x,i = min{t ≥ 0 | y(t) ∈ Q} is also random, which makes it somewhat harder to approximate the distribution for our main object of study -the cumulative cost of the PDMP J (x, i). In addition to mode-dependent dynamics f : Ω × M → R d , we also include a mode-dependent running cost C : Ω × M → (0, +∞) and exit cost q : Q × M → [0, +∞). To simplify the notation, we will also sometimes use the mode as a subscript: C i (x) = C(x, i), f i (x) = f (x, i), q i (x) = q(x, i), etc. We will assume that q i 's are continuous in x, while C i 's and f i 's are bounded and piecewise Lipschitz continuous. The cumulative cost is then formally defined as We will generally assume that Ω is a closed set and the process can continue on ∂Ω\Q, but if the dynamics forces us to leave Ω before reaching Q, this will result in J = +∞. We note that the notion of cumulative cost is much more common in controlled PDMPs, where it is used to select criteria for control optimization. But we also consider J in this simpler uncontrolled case to focus on a single measurable outcome of the process. We develop our approach in this general setting, but our numerical experiments highlight that studying J is far from trivial even if C ≡ 1, q ≡ 0, and Q = ∂Ω, yielding J (x, i) = T x,i , the time until we reach the boundary. For a motivating example, consider a "sailboat" traveling with unit speed on an interval Ω = [0, 1] and subject to random mode (wind direction) switches. We will assume that it is moving rightward in mode 1 and leftward in mode 2, the time intervals between mode switches are independent exponentially distributed random variables with rate λ, and the process terminates as soon as the boat reaches Q = {0, 1}. While we describe this example in terms of sailboat navigation, similar "velocity jump processes" are also often used to model dispersal in biological systems [33,38]. But in contrast to our approach, the main focus there is on equations describing the evolving density of dispersing cells or organisms rather than on the distribution of some performance measure J for individual organisms. Another distinction is our assumption that each individual path terminates on reaching some exit set Q -this introduces additional structure, which we later leverage to obtain efficient numerical methods. Throughout the paper, we take an exploratory approach, focusing on derivation of equations and numerical methods as well as instructive test problems rather than proofs of convergence or realistic applications. To streamline the presentation, we illustrate our methods on simple "first-exit time" problems 1 in 1D and 2D similar to the sailboat example described above. But in the Appendix we show how the same approach is useful more broadly (with general C i 's and f i 's) by considering fish harvesting in an environment with random switches in carrying capacity. In section 2, we explain how the CDF for J can be computed by solving a system of coupled linear PDEs. Our equations can be interpreted as a PDMP-adapted version of the Kolmogorov Backward Equation generalized to handle arbitrary running costs rather than just time. Another related approach is the previous development of numerical methods for the Liouville-Master Equation in [5]. We also derive simpler recursive difference equations to compute the CDF for a discrete analog of our setting -a random route-switching process on a graph. In most real world applications, all switching rates (λ ij ) will be known only approximately and it is necessary to bound the results of this modeling uncertainty. In section 3, we show how bounds on these switching rates can be used to bound the CDF of J . Interestingly, it turns out that it is easier to compute tight bounds if the switching rates are not assumed to be constant in time. In many applications, the focus is on optimally controlling PDMPs (affecting the dynamics in each deterministic mode), with the notion of optimality typically based on the average-case outcomes (e.g., minimizing the expected total cost). Once a control is fixed, the same uncertainty quantification tools covered in sections 2 and 3 become relevant. Moreover, the control can also be selected to manage the uncertainty, providing some robustness guarantees or minimizing the probability of undesirable outcomes. Following the latter idea, we introduce a method for optimizing the CDF of controlled PDMP models in section 4. We conclude by discussing further extensions and limitations of our approach in section 5. 2. Computing the CDF. Before discussing the methods for approximating the CDF for the randomly switching process described in section 1, we first consider the same challenge for Markov-style switching on a graph in section 2.1, turning to a continuous version in section 2.2. Numerical methods for the latter are then described in section 2.3 and illustrated by computational experiments in section 2.4. Discrete PDMPs. We start by reviewing a simple model of deterministic routing on a directed graph with a finite node set X = {x 1 , . . . , x N }, a set of directed edges E ⊂ X × X, and a target set Q ⊂ X. We will assume that K : X × X → (0, +∞] specifies the known cost of possible "steps" (i.e., node-to-node transitions) with K(x, x ) = +∞ iff (x, x ) ∈ E. A route on this graph can be specified in feedback form by a mapping F : X → X such that (x, F (x)) ∈ E ∀x ∈ X. Given a starting position y 0 = x ∈ X, a path can be defined by a sequence y n+1 = F (y n ), terminating as soon as y n ∈ Q. We will further assume that the terminal cost charged at that point is specified by q : Q → [0, +∞). If the path enters Q aftern(x) steps, its cumulative cost can be expressed as with J (x) = +∞ if the path remains forever in X \ Q, which can happen if a route specified by F contains loops. The recursive relationship among J values makes it easy to recover all of them by solving a linear system J (x) =n (x)−1 n=0 K y n , y n+1 + q yn (x) ,J (x) = K (x, F (x)) + J (F (x)) , ∀x ∈ X \ Q; J (x) = q(x), ∀x ∈ Q. (2.1) We will now consider a version of the problem with a total of M different routes F 1 , . . . , F M , each of them with its own pair of running and terminal costs (K i , q i ) defined on the same graph. These routes are equivalent to the modes in a PDMP. To simplify the notation, we will use K i (x) as a shorthand for K i (x, F i (x)) . We define a random route-switching process by assuming that there is a chance of switching to another route after each step. That is, if the current route is F i , the probability p ij of switching to F j after the next step is known a priori for all i, j ∈ M = {1, . . . , M }. The number of steps is now a random variable, along with the cost paid for all future steps. In defining the new random cumulative cost J i (x), we note that the subscript only encodes the initial route used in the first step as we depart from x. It is easy to see that u i (x) = E [J i (x)] should satisfy a recursive relation u i (x) = K i (x) + M j=1 p ij u j (F i (x)) , ∀x ∈ X \ Q, i ∈ M; u i (x) = q i (x), ∀x ∈ Q, i ∈ M. (2.2) It is worth noting that this system of M N linear equations lacks the nice causal property that we enjoyed in the deterministic (single route) case. There we knew that a finite J (x) implied that the path from x prescribed by F included no loops and reached Q in a finite number of steps. As a result, the part of system (2.1) corresponding to such finite J 's was always triangular up to a permutation. The same is clearly not true for the multi-route case of (2.2), where loops can easily arise as a result of random route-switching. We note that this process can be also interpreted as a Markov chain on an extended graph. One would create M copies of the original graph (on the nodes x i n ) with each route (or mode) F i represented as a separate "layer" and inter-layer transitions governed by p ij 's. Figure 1 illustrates one such example with two modes and associated probabilities p 11 , p 12 , p 21 , and p 22 . In the special case of K i ≡ 1 and q i ≡ 0 for all i ∈ M, the above equations for u i 's are simply describing the mean hitting time for the set Q × M. However, we are interested in more general costs and would also like to compute the full CDFs w i (x, s) = P(J i (x) ≤ s) for each J i . It is easy to show that these functions must satisfy a recursive relationship w i (x, s) = M j=1 p ij w j F i (x), s − K i (x) , ∀x / ∈ Q, i ∈ M, s > 0; (2.3) with the initial and boundary conditions w i (x, s) = 0, if (x / ∈ Q, s ≤ 0) or (x ∈ Q, s < q i (x)) ; 1, if x ∈ Q, s ≥ q i (x). (2.4) We will assume that the range of s values of interest is S = [0, S], where S is some constant specified in advance. Based on the general properties of CDFs, all w i 's are monotone non-decreasing and upper-semicontinuous 2 in s. Moreover, the positivity of K i 's ensures the explicit causality of this system: in (2.3) each w i (x, s) can only depend on w i (x , s ) if s < s. Thus, the system can be solved in a single sweep (from the initial conditions at s = 0, "upward" in s). Still, it can be useful to precompute s 0 i (x) = inf{s | w i (x, s) > 0} and w 0 i (x) = w i x, s 0 i (x) by computations on X alone. Intuitively, s 0 i (x) can be thought of as the minimum attainable cost starting in mode i at position x, and w 0 i (x) is the probability of attaining said cost. It is easy to see that s 0 i satisfies the recursive system: s 0 i (x) = K i (x) + min j∈M s.t. p ij >0 s 0 j (F i (x)) , ∀x ∈ X \ Q, i ∈ M; s 0 i (x) = q i (x), ∀x ∈ Q, i ∈ M. (2.5) solvable by the standard Dijkstra's method in O(M N log(M N )) operations. The values of w 0 i (x) can also be found in the process of computing s 0 i (x). If I(x) ⊂ M is the arg min set in (2.5), then w 0 i (x) = j∈I(x) p ij w 0 j (F i (x)) , ∀x ∈ X \ Q, i ∈ M; w 0 i (x) = 1, ∀x ∈ Q, i ∈ M.(2. Continuous PDMPs. We are now interested in extending our results from the discrete case to continuous settings. The PDMP model described in section 1 is based on continuous in time and space evolution of the state y(t) and continuous in time Markov chain governing the changes in mode m(t). Here, we start with a somewhat simpler version, in which this Markov chain is discretized in time, while the state evolution is continuous. Choosing some small fixed time interval τ > 0, we assume that the system starting in mode m(0) = i ∈ M and state y(0) = x ∈ Ω \ Q evolves according to an ODE y (t) = f i (y(t)) with no random switches until the time τ x,i = min (τ, min {t | y(t) ∈ Q}) , at which point a switch to another mode may occur. The process is repeated (starting fromx = y(τ x,i ) and a possibly new mode j, integrating the ODE over the time interval of length τx ,j , etc.) and the running cost is accumulated until y(t) enters a compact exit set Q. We define natural analogs for operators used to pose the graph routing problem in the previous subsection: F i (x) = x + τ x,i 0 f i (y(t)) dt = y τ x,i , (2.7) K i (x) = τ x,i 0 C i (y(t)) dt, (2.8) where C i : Ω → (0, +∞) is the running cost for that mode. We define the probability of switching to each mode j at the end of time interval of length τ x,i by requiring consistency with the continuous in time Markov process described in (1.2). In the latter, there could be multiple mode transitions over the time τ x,i , and here we simply use the probability of finishing this time interval in mode j: p ij (τ x,i ) = P(m(τ x,i ) = j | m(0) = i). We compute these probabilities using a transition rate matrix Λ = λ ij , where λ ij 's encode the rate of (i → j) switching for i = j, while the diagonal elements are defined by λ ii = − j =i λ ij . The evolution of the probability matrix P (t) = p ij (t) is then given by an ODE d dt P (t) = P (t)Λ, P (0) = I, and it follows that P (τ x,i ) = exp(Λτ x,i ). Finally, if y(τ x,i ) ∈ Q, we assume that the PDMP will immediately terminate with an exit cost of q j y(τ x,i ) , where j is the final mode after a possible last transition. With this notation in hand, we can define the same functions characterizing the random cumulative cost: u i , w i , s 0 i , and w 0 i will all satisfy the same recursive formulas already defined on a graph in the previous subsection. The only caveat is that p ij 's will need to be replaced by p ij (τ x,i ). Since τ and τ x,i are equivalent except on a small neighborhood of Q, in the following sections we will slightly abuse the notation by referring to τ to simplify the formulas. The original setting of section 1 (with continuous in time Markov chain for mode switching) can be obtained from the above in the limit by letting τ → 0. A standard argument based on a Taylor series expansion shows that the expected costs u i (x) = E[J i (x)] formally satisfy a system of linear PDEs: ∇u i (x) · f i (x) + C i (x) + j =i λ ij u j (x) − u i (x) = 0 (2.9) with boundary conditions u i (x) = q(x, i) on Q × M. We omit the derivation of (2.9) for the sake of brevity but use a similar approach below to derive a system of PDEs satisfied by the cumulative distribution functions w i (x, s). The first order approximations of the transition probabilities are: p ij (τ ) = 1 − e −λ ij τ + o(τ ) = λ ij τ + o(τ ), j = i p ii (τ ) = 1 − j =i λ ij τ + o(τ ). (2. 10) The first-order approximation of the dynamics in (2.7) is (2.11) and the first-order approximation of the running cost in (2.8) is F i (x) = x + τ f i (x) + o(τ ),K i (x) = τ 0 C i (y(t))dt = τ C i (x) + o(τ ),(2.12) Plugging in our approximations (2.10), (2.11), and (2.12) into the recursive relationship in Equation (2.3) and then Taylor expanding w i gives: w i (x, s) = 1 − j =i λ ij τ w i F i (x), s − τ C i (x) + j =i λ ij τ w j F i (x), s − τ C i (x) + o(τ ) (2.13) w i (x, s) = w i F i (x), s − τ C i (x) + τ j =i λ ij w j (x, s) − w i (x, s) + o(τ ) 0 = τ ∇w i (x, s) · f i (x) − τ C i (x) ∂w ∂s (x, s) + τ j =i λ ij w j (x, s) − w i (x, s) + o(τ ), where ∇ = ∂ ∂x 1 , ∂ ∂x 2 , ..., ∂ ∂x d denotes the gradient in the spatial coordinates. Dividing both sides by τ and then taking the limit as τ → 0, we obtain a linear PDE for each mode i: ∇w i (x, s) · f i (x) − C i (x) ∂w i ∂s (x, s) + j =i λ ij w j (x, s) − w i (x, s) = 0. (2.14) The above derivation is only formal since it assumes that w i 's are sufficiently smooth. In reality, they will be often non-differentiable and even discontinuous at isolated points; nevertheless, these value functions can be still interpreted as weak (viscosity) solutions [21], which can be approximated numerically by discretizing (2.13). This system of PDEs satisfies the initial/boundary conditions: But more generally, if vector fields are such that a trajectory might leave Ω prior to reaching Q, one could treat this event as an immediate failure, essentially imposing w i (x, s) = 0 for all x ∈ Ω and all s ∈ R. w i (x, 0) = 1, ∀x ∈ Q s.t. q(x, i) = 0, 0, otherwise, (2.15) w i (x, s) = 1, ∀x ∈ Q s.t. q(x, i) ≤ s, 0, ∀x ∈ Q s.t. q(x, i) > s. As in the discrete case of section 2.1, it can be useful to precompute the minimum attainable cost to use as initial/boundary conditions when solving (2.14). From the discrete case we recall that s 0 i (x) = inf{s | w i (x, s) > 0} denotes the minimal cost possible when starting from position x in mode i assuming that transitions between modes can occur whenever desired. In the continuous case these transitions can occur without delay, and therefore s 0 i (x) = s 0 j (x) for all i and j in M, so we will replace all of these with s 0 (x). (Also, unlike in the discrete case, it is entirely possible that w i (x, s 0 (x)) = 0 for all i. The cost of s 0 (x) might be attainable only through perfectly timed transitions, which in the continuous case would happen with probability zero.) A formal Taylor series expansion of (2.5) yields the following differential equation and boundary conditions for s 0 (x): min i C i (x) + ∇s 0 (x) · f i (x) = 0, x ∈ Ω \ Q; s 0 (x) = min i {q i (x)} , x ∈ Q. (2.17) We are also interested in the probability w 0 i (x) of attaining that minimal cost s 0 (x) when starting from mode i and position x. If we denote the argmin set of (2.17) as I(x), then w 0 i (x) formally satisfies the following system: 0 = ∇w 0 i (x) · f i (x) + j =i λ ij w 0 j (x) − w 0 i (x) , x ∈ Ω \ Q, i ∈ I(x); w 0 i (x) = 1, x ∈ Q, i ∈ I(x); w 0 i (x) = 0, x ∈ Ω, i ∈ I(x). (2.18) Once s 0 (x) and w 0 i (x)'s are known, the computation of w i 's can be restricted to (x, s) | s ∈ (s 0 (x), S] , solving PDEs (2.14) with "initial" conditions w i (x, s 0 (x)) = w 0 i (x). Remark 2.1 (Related work on Liouville-Master Equation) . An approach similar to the one presented in this section can be used to derive PDEs for the timedependent joint PDMP-state CDF on Ω × M. The initial conditions to those PDEs would be based on a specific initial configuration (x 0 , i 0 ) or, more generally, on a specific initial joint CDF on Ω × M. This is precisely the setting in [5], where a finitedifference numerical method for the "Liouville-Master Equation" was developed and tested for the special case of d = 1. If one is willing to increase the dimension of the problem, this can be viewed as a more general approach than ours (since J can be viewed as just another component of the continuous state variable). But the need to solve PDEs separately for different (x 0 , i 0 )-specific initial conditions is a serious drawback. Moreover, computing the time-dependent joint CDF seems more suitable for finite-horizon PDMPs (where the process terminates after a pre-specified time T ) rather than in our setting (where the process terminates as soon as it reaches Q ⊂ Ω). Numerics for CDF computation. We will approximate the domain Ω with a rectangular grid of points {x k } with grid spacing ∆x, where k = (k 1 , . . . , k d ) is a multi-index and x k = (k 1 ∆x, . . . , k d ∆x). We will also approximate the second argument of the CDF with regularly spaced points s n = n∆s. We will derive equations for a grid-function W n i,k ≈ w i (x k , s n ), with W 0 i,k values determined by the initial conditions (2.15). To simplify the discussion, we assume that both ∂Ω and Q are grid-aligned, with boundary values prescribed by (2.16). Equation (2.3) is then naturally interpreted as a recipe for a semi-Lagrangian discretization using a pseudo-timestep of length τ . To obtain the first-order scheme, we can use the linear approximations (2.11-2.12) in formula (2.13), yielding the following equation at each gridpoint x k ∈ Ω, mode i ∈ M, and cost threshold s n : W n i,k = M j=1 p ij (τ )W j x k + τ f i (x k ), s n − τ C i (x k ) , (2.19) where W j : Ω × R → R is the result of interpolating the grid-function W n j,k in both x and s variables, and the p ij 's are defined as in (2.10). In our implementation, all W j 's are defined by multi-linear interpolation, but more sophisticated interpolation techniques (e.g., based on ENO/WENO [46]) may be used instead to decrease the numerical viscosity. More accurate approximations of F i and K i could be also employed to increase the formal order of accuracy of the discretization. For fully deterministic processes, similar semi-Lagrangian schemes have been proven to converge under the grid refinement to a discontinuous viscosity solution on all compact sets not containing the discontinuity [7]. While we do not attempt to prove this here, our numerical experiments indicate that the same holds true in piecewise-deterministic problems. Our update formula (2.19) is only valid when x k + τ f i (x k ) remains in Ω. With grid-aligned ∂Ω, a rather conservative sufficient condition for this is τ · max i max x |f (x, i)| ≤ ∆x. (2.20) Furthermore, we would like to ensure that our updates are causal, that is the right hand side of (2.19) depends only upon the W n values with n < n. While not strictly necessary, this ensures that the updates for each mode are uncoupled, speeding up the computation. A sufficient condition for this is τ · min i min x C(x, i) ≥ ∆s. (2.21) The inequality (2.20) is only needed if we want to use the same τ at all grid points instead of selecting a smaller time step near ∂Ω only. But if this τ -uniformity is desired, satisfying both (2.20) and (2.21) requires ∆s min{C} ≤ ∆x max{|f |} . (2.22) We note that, even though the above restriction looks similar to a Courant-Friedrichs-Lewy (CFL) condition, it is not needed to guarantee the stability (semi-Lagrangian discretizations are unconditionally stable), but simply to ensure the causality (and hence the efficiency) of our discretization. Under certain conditions, (2.19) may be also re-interpreted as a finite differences discretization of the PDE (2.14). To give a concrete example, suppose that d = 1, and the domain Ω = [0, 1] is approximated by a grid of regularly spaced points denoted x k = k∆x. Furthermore, suppose that there is a mode i where C i ≡ 1, and f i (x k ) = f ik > 0. If we choose τ = ∆s, then (2.19) for n + 1 becomes: W n+1 i,k = M j=1 p ij (∆s)W j (x k + f ik ∆s, s n ) = W i (x k + f ik ∆s, s n ) + j =i λ ij ∆s W j (x k + f ik ∆s, s n ) − W i (x k + f ik ∆s, s n ) = W n i,k + f ik ∆s ∆x W n i,k+1 − W n i,k + j =i λ ij ∆s W j − W i (x k + f ik ∆s, s n ); f ik W n i,k+1 − W n i,k ∆x − W n+1 i,k − W n i,k ∆s + j =i λ ij W j − W i (x k + f ik ∆s, s n ) = 0, (2.23) which is a consistent first-order finite differences discretization of (2.14). Furthermore, in this 1D example, the CFL condition for this discretization is exactly (2.22). The scheme (2.23) is monotone (and thus stable [18]) whenever this CFL condition is satisfied. It is important to note that the summands in (2.23) are evaluated at x k + f ik ∆s (and therefore are convex combinations of W n values at x k and x k+1 ). Evaluating those terms at the naive choice of x k would result in a non-monotone discretization, which is in fact unstable. To compute the minimum attainable cost s 0 (x) and the probability w 0 (x) of attaining it, we use first-order semi-Lagrangian discretizations of (2.17) and (2.18). For d = 1, the discretized equations for s 0 (x) are s 0 (x k ) = min i C i (x k ) ∆x |f i (x k )| + s 0 x k , x k ∈ Q; s 0 (x k ) = min i {q i (x k )} , x k ∈ Q; (2.24) where k = k + 1, f i (x k ) > 0; k − 1, f i (x k ) < 0. In 1D, this system of equations can be solved efficiently with two iterative "sweeps"first increasing and then decreasing in k. In higher space dimensions, it can be solved in O(M N log(N )) time using a Dijkstra-like method. In the process of solving for s 0 (x), we also solve (2.18) using a first-order semi-Lagrangian scheme. Using I(x k ) to denote the argmin set of (2.24), the values of w 0 i are initialized according to w 0 i (x k ) = 1, x k ∈ Q, i ∈ I(x k ); 0, otherwise. Whenever the value of s 0 (x k ) is updated, we simultaneously update w 0 i according to These values of s 0 (x) and w 0 i (x) are then used as initial/boundary conditions 3 for computing w i (x). This provides a speed improvement and also reduces the smearing of w i 's discontinuities due to numerical viscosity. w 0 i (x k ) = w 0 i x k + ∆x |f i (x k )| j =i λ ij w 0 j x k − w 0 i x k , i ∈ I(x k ); 0, i ∈ I(x k ).w 0 i (x) (A) (B) (C) (D)(A) (B) (C) (D) Experimental Results. We illustrate our approach with three examples of uncontrolled PDMPs on R and R 2 . In all of these, we assume Q = ∂Ω, C ≡ 1, and q ≡ 0, ensuring that the cumulative cost J corresponds to the time to ∂Ω. For simplicity, we will also assume uniform transition rates; i.e., λ ij = λ > 0 for all i = j. Example 1: We start by considering a "sailboat" test problem described in the introduction with Ω = [0, 1], Q = {0, 1}, M = 2, f i (x) = (−1) i+1 , and symmetric transition rates λ 12 = λ 21 = 2. For a fixed number N of gridpoints, we set ∆s = ∆x = 1 N −1 , as this is the largest value of ∆s that satisfies (2.22). Moreover, this guarantees that no actual interpolation is necessary in (2.19), as W j is only evaluated at gridpoints. We note that solving these discretized equations is equivalent to finding the CDF of a discrete PDMP such as the one pictured in Figure 1, except with a larger number N of nodes. We solve this problem for s ∈ [0, 1], but also precompute s 0 (x) and w 0 i (x) (see Figure 2(A-B)) to reduce the computational domain for w i 's. 3 Since the graph of s 0 (x) is generally not grid aligned in Ω × S, such a domain restriction requires either a use of "cut cells" just above s = s 0 (x) or a conservative "rounding up" of s 0 values. Our implementation relies on the latter, which introduces additional O(∆s) errors. The key advantage of our approach is that it approximates the distribution J for all starting configurations simultaneously. Once w i 's are computed, we can freeze (x, i) and vary s to study the CDF. In Figure 3(A-B) this is illustrated for two starting locations x = 0.3 and x = 0.7. But it might be even more revealing to fix a particular deadline s and consider the probability of meeting it from all possible initial configurations. In Figure 4 we show such graphs of w i (x, s) for four different s values. Geometric properties of these functions have a natural interpretation, which we highlight focusing on mode 1 and s = 0.25 (the blue plot in the first subfigure). First, regardless of mode switches, s = 0.25 is not enough time to exit if we start too far from Q; so, w 1 = w 2 = 0 for all x ∈ (0.25, 0.75). Second, starting from x = 0.75 and moving right with speed one we will have just enough time to reach Q provided we experience no mode switches, and if any switches occur the resulting time to target will be higher. So, the jump discontinuity at x = 0.75 is precisely the probability of zero mode switches occurring in s = 0.25 time units. (We note that this discontinuity disappears in the last subfigure since s = 1.00 is enough time to reach Q with no mode switches starting from any (x, i) ∈ Ω × M.) Finally, a similar argument explains the behavior for starting positions on x ∈ (0, 0.25). Since we start in mode 1, the only hope of meeting the s = 0.25 deadline is a quick switch to mode 2. Starting from x = 0.25, a timely arrival would require an immediate mode switch, and since this happens with probability zero, w 1 is continuous at this point. Of course, the probability of meeting a deadline is also significantly influenced by the switching rates. While we do not illustrate this here, the same example is repeated with a range of symmetric and asymmetric rates in Figure 7 of section 3. Example 2: We modify the previous example by considering unequal speeds of motion in different modes: f 1 = 0.5 and f 2 = −1. The CDFs for two starting locations x = 0.3 and x = 0.7 are shown in Figure 3(C-D) while the plots of w i (x, s) for four different values of s can be found in Figure 5. We note that in Mode 1, interpolation is now necessary in (2.19), which results in numerical diffusion smoothing out discontinuities, as can be seen in the right two subfigures. The absence of such artifacts in the first two subfigures is an additional benefit of pre-computing s 0 (x) and Figure 6 coincide with the one-dimensional graphs for s = 0.25 in Figure 4. However, as we move closer to the corners of the domain, all four modes have an effect on the probability of exit. For example, the region along the diagonal near the top right corner of the w 1 graph has higher exit probabilities than surrounding regions because there are multiple possible timely-exit strategies. These 2D phenomena become prevalent for higher s values. Remark 2.2 (Related models in biology). As noted in the introduction, similar "velocity jump processes" are also used to model dispersal in biological systems [33,38]. In that context, all dispersing agents perform long runs with constant velocity but occasionally switch modes/directions. The usual approach is to derive a system of PDEs governing the evolution of agent densities ρ i (x, t) in corresponding modes i ∈ M. The symmetric unbounded case in 1D (i.e., Ω = R, M = 2, and λ 12 = λ 21 ) is particularly well-studied, with the overall density ρ = ρ 1 + ρ 2 evolving according to the "telegraph equation" [28]. Taking λ 12 = λ 21 , one can similarly model chemotaxis. If Ω = R 2 or R 3 , one could use a larger number of modes to describe many possible directions of motion, with λ ij chosen to reflect a possible bias in switching (e.g., giving preference to new directions more closely aligned with the preceding run -as is the case for E. coli bacteria). Letting M → ∞, one can also directly model all possible directions of motion by switching to integro-differential equations [38]. While our focus on a single performance measure J might be restrictive for many of these applications, there are also some settings where it can be advantageous. For example, if one assumes that agents are removed upon reaching Q, the number of them still remaining by the time t could be in principle computed as R(t) = i∈M Ω\Q ρ i (x, t) dx. (2.26) But any change in ρ i (x, 0) would make it necessary to re-solve a system of PDEs for ρ i (x, t)'s before reusing (2.26). Here we can offer a much more efficient method by setting C ≡ 1 and q ≡ 0, computing w i 's from (2.14) only once, and then using an alternative formula that works for all initial densities w 1 ← w 2 ↑ w 3 → w 4 ↓ s =R(t) = i∈M Ω\Q ρ i (x, 0) (1 − w i (x, t)) dx. (2.27) 3. Bounds on CDF. We now turn to PDMPs with parameter uncertainty -in addition to the inherent aleatoric uncertainty due to mode switches. In section 2, the uncertainty of the outcome could be fully characterized by its CDF computed based on the known transition rates between modes, λ ij 's. Here, however, we consider the case where we only know a range of potential λ ij values. There are two natural models of epistemic uncertainty in this situation, and it is meaningful to consider the upper and lower bounds on the CDF with each of them. We focus on a case where the true transition rates are free to fluctuate within the given range and may take on different values at different times. The upper and lower bounds on the CDF can be then found by considering a nonlinear version of the coupled PDEs seen in section 2. This can also be viewed as an optimal control problem, where the controller is either helping or hindering the particle's exit by choosing the transition rates adaptively. The alternative model of epistemic uncertainty is to assume that all transition rates remain fixed (though unknown) throughout the process. We provide some ex-perimental results for this case as well, though do not propose any computationally efficient methods for finding sharp CDF bounds. 3.1. Deriving PDEs. We now extend the results of section 2.2 by considering the case in which the transition rate matrix Λ = (λ ij ) is not necessarily constant. Suppose there are known a ij , b ij for each pair i = j, such that each λ ij may vary in the interval 0 ≤ a ij ≤ λ ij ≤ b ij throughout the process. If L is the set of possible transition matrices satisfying these constraints, we will assume that Λ might be changing but remains in L throughout the process. In this section, we will use Λ i to denote the i-th row of Λ, specifying all transition rates from mode i. We will also use L i to denote the set of all allowable i-th rows satisfying the above constraints. We compute an upper bound for w i , denoted w + i , by taking its initial and boundary conditions to be the same as w i , and adaptively selecting the Λ i ∈ L i which maximizes w + i (x, s). Similarly, for the lower bound w − i we take the Λ i ∈ L i which minimizes w − i (x, s). Hence, for each mode i, the bounds w + i and w − i satisfy the PDEs: ∇w + i (x, s) · f i (x) − C i (x) ∂w + i ∂s (x, s) + max Λ i ∈L i j =i λ ij w + j (x, s) − w + i (x, s) = 0; (3.1) ∇w − i (x, s) · f i (x) − C i (x) ∂w − i ∂s (x, s) + min Λ i ∈L i j =i λ ij w − j (x, s) − w − i (x, s) = 0 (3. 2) with initial and boundary conditions (2.15) and (2.16). We note that, if a ij = b ij for all i = j, there is no parametric uncertainty, each L i is a singleton, and the above equations reduce to (2.14). As with the computation of w i 's through (2.14), it can be helpful to precompute the minimal attainable cost s 0 (x) and the probability w 0 i (x) of achieving such a cost. Since Λ is not constant, instead of having a probability of attaining the cost s 0 (x) we have a lower bound w 0,− i (x) and an upper bound w 0,+ i (x) for that probability. We may compute s 0 (x) in precisely the same way as in (2.17) since that formula does not depend on Λ at all. On the other hand, to compute w 0,+ i (x) we must modify (2.18) to account for the unknown (and possibly changing) Λ: 0 = ∇w 0,+ i (x) · f i (x) + max Λ i ∈L i j =i λ ij w 0,+ j (x) − w 0,+ i (x) , x ∈ Ω \ Q, i ∈ I(x); w 0,+ i (x) = 1, x ∈ Q, i ∈ I(x); w 0,+ i (x) = 0, x ∈ Ω, i ∈ I(x). (3.3) Similarly, to compute w 0,− i (x) we have: 0 = ∇w 0,− i (x) · f i (x) + min Λ i ∈L i j =i λ ij w 0,− j (x) − w 0,− i (x) , x ∈ Ω \ Q, i ∈ I(x); w 0,− i (x) = 1, x ∈ Q, i ∈ I(x); w 0,− i (x) = 0, x ∈ Ω, i ∈ I(x).(3.4) In both cases, I(x) is the argmin set of (2.17) as in (2.18). Calculating Bounds. For numerical computations of the bounds described in section 3.1, we rely on a discretization similar to that presented in section 2.3. When τ is small enough that the approximations in (2.10) can be made, this optimization can be written recursively. For the CDF lower bound w − i (x, s), the semi-Lagrangian scheme is w − i (x, s) = w − i (x,s) + min Λ i ∈L i τ j =i λ ij w − j (x,s) − w − i (x,s) , (3.5) wherex = F i (x) ands = s − τ C i (x) . Recall from (2.21) that C i (x) > 0 so it is always the case thats < s. Therefore, w − i (x,s) has already been calculated and so can be used in the computation of w − i (x, s). For an efficient implementation of (3.5), the optimal Λ * ∈ L can be found explicitly. When minimizing w − i (x, s), we would naturally like to subtract as much as possible and add as little as possible to w − i F i (x), s−τ C i (x) = w − i (x,s). Therefore, w − j (x,s) − w − i (x,s) ≤ 0 =⇒ λ * ij = b ij ; w − j (x,s) − w − i (x,s) > 0 =⇒ λ * ij = a ij . (3.6) For the the CDF upper bound w + i , the scheme is similar modulo replacing min with max in (3.5) and flipping the signs of inequalities in (3.6). Experimental Results. Example 4: CDF bounds and comparison to fixed-Λ CDFs. We now generalize Example 1 from section 2.4 to consider epistemic uncertainty. Recall that Ω = [0, 1], Q = ∂Ω, C ≡ 1, and q ≡ 0 so that the cumulative cost J corresponds to the exit time, with M = 2 and f i = (−1) i+1 . We will also assume that λ ij ∈ [1, 4] for all i = j. In Figure 7 we display our results for a particle that starts moving rightward (in Mode 1). The graphs shown in blue are the upper and lower bounds on the probability of a timely exit (i.e., before a specific deadlines) for all initial positions x. The bounds on CDF for two starting positionsx are shown in Figure 8. All of these bounds are computed from (3.1) and (3.2) for the model of epistemic uncertainty where Λ = (λ ij ) is allowed to fluctuate within L. Under this model, these bounds are sharp since they are computed by finding CDF-maximizing (and minimizing) sequences of Λ's. We can also compare the blue bounds to the corresponding timely-exit probabilities for a process containing epistemic uncertainty via fixed and unknown (possibly asymmetric) transition matrix Λ. The green curves shown in Figures 7 and 8 are computed by repeatedly solving (2.14) for a coarse grid of specific Λ's in L. It should be noted that processes with this type of epistemic uncertainty are a subset of those previously discussed, and so the blue bounds will definitely hold but will no longer be sharp. This lack of sharpness is not surprising since changing the transition rate can often result in a "better" (higher or lower -depending on the bound) probability of timely exit. However, calculating tighter bounds for a "fixed-unknown-Λ" case is computationally expensive. By inspection of the experimental data, it is clear that such sharp bounds would have to be composed of many individual fixed-Λ CDFs. 4. Optimizing the CDF. The PDMPs considered in previous sections were not controllable in any way. Since the dynamics are deterministic in every mode, each random trajectory was fully described by the initial (state, mode) pair and the discrete time sequence of mode switches. The goal was to develop efficient methods for approximating the CDF of the cost accumulated up till termination. We now turn to controlled PDMPs [20] -a modeling framework useful in a wide range of applications, including production/maintenance planning [12], control of manufacturing processes [1,11,37,42], multi-generational games [30], economic growth & climate change modeling [31], trajectory optimization for emergency vehicles [4], preventing the extraction of protected natural resources [13], and robotic navigation [27,45]. We start with expectation-optimal controls considered in the above references, but then switch to selecting controls to manage the uncertainty in J and provide some notion of robustness. Robust controls help practitioners to guard against both modeling errors and prohibitively bad rare outcomes, which may result from random switches. It might seem natural to mirror the robust approaches popular in traditional stochastic control, but we find them lacking in the PDMP context. H ∞ controls are the mainstay of robustness for many processes with continuous perturbations [8], but they are not easily adaptable for discrete mode-switches. Another popular idea is to Controlled PDMPs and expectation-optimal policies. To obtain a controlled PDMP, we will assume that both the running cost C and velocity f also depend on additional control parameters, which can be changed dynamically while the system travels through Ω × M. We will assume that the set of available control values A is a compact subset of R n . Throughout this section, we will slightly overload the notation by using a to refer to a generic element of A and a(·) to refer to a generic feedback-control policy a : (Ω × M) → A, which selects a control value based on the current system state. Once we select any specific a(·), we can define f (x, i) = f (x, i, a(x, i)) and (2.14) respectively. However, in controlled PDMPs literature the problem is usually first formulated as an optimization over a broader class of piecewise open-loop policies, and the dynamic programming argument is then used to show that an optimal policy can be actually found in feedback form. We provide an overview of this construction below, but refer to [20,21,47] The three inputs to α encode all information about the last mode switch encountered before the current time t: the time and position (t # ≥ 0 and x # ∈ Ω) where that switch has happened and the resulting mode i # . If no switch has occurred since we started, we will take t # = 0 and (x # , i # ) equal to the original (position, mode) pair. Assuming α(t # , x # , i # ) = α o (·) ∈ A o , the control value to be used at the time t ≥ t # will be specified by α o (t − t # ) until we switch from the mode i # . Slightly abusing the notation, we can now replace (1.1) and (1.3) by C(x, i) = C (x, i, a(x, i)) ,y (t) = f m(t) y(t), α o (t − t # ) , (4.2) J α(·) i (x) = T x,i 0 C m(t) y(t), α o (t − t # ) dt + q y T x,i , m T x,i ,u(x, i) =û i (x) = inf α(·)∈A E[J α(·) i (x)], (4.4) The existence of an expectation-optimal policy α * (·) ∈ A such thatû i (x) = u α * (·) i (x) is only guaranteed under additional assumptions; e.g., if the set ν(x, i) = (r, f i (x, a)) | r ≥ C i (x, a), a ∈ A is convex for every x and i. (Alternatively, the existence of optimal policy is also assured if one allows relaxed control functions, with α 0 taking values in the set of probability measures on A; see [6,47].) If such an optimal α * (·) ∈ A exists, it is easy to see that a "tail" of α o * must be also optimal for every (y(t), m(t)) as long as the process continues. Otherwise, we could obtain an improvement for the starting configuration (y(0), m(0)) = (x, i) by concatenating α * up to the time t with whatever policy is optimal starting from (y(t), m(t)). A version of this tail-optimality property holds more generally, even when no expectation-optimal policy exists: u(x, i) = inf α(·)∈A E τ 0 C m(t) y(t), α o (t − t # ) dt +û y(τ ), m(τ ) , (4.5) for all τ > 0 sufficiently small to guarantee that y(t) ∈ Ω\Q for all t ∈ [0, τ ], α(·) ∈ A. I.e., we assume that τ is small enough so that the system cannot reach Q by t = τ regardless of the sequence of mode switches. A standard argument based on Taylorexpanding (4.5) in τ (e.g., see [45, §2]) shows that, ifû i 's are sufficiently smooth, they must satisfy a system of Hamilton-Jacobi-Bellman PDEs: min a∈A {∇û i (x) · f i (x, a) + C i (x, a)} + j =i λ ij û j (x) −û i (x) = 0, x ∈ Ω\Q, i ∈ M (4.6) with boundary conditionsû i (x) = q i (x) for all x ∈ Q. In a non-smooth case, these value functions can be still interpreted as the unique viscosity solution [21]. The system (4.6) is a natural non-linear generalization of (2.9) and can be similarly discretized by semi-Lagrangian techniques. However, the coupling between different modes makes it difficult to solve the discretized system efficiently even in the case of simple/isotropic cost and dynamics. A variety of Dijkstra-like non-iterative methods developed for deterministic problems (e.g., [3,14,36,43,44]) will not be applicable for Λ = 0 and one has to resort to slower iterative algorithms instead [27,45]. Onceû i 's are computed, an optimal feedback policy a * (x, i) can be defined pointwise (for all x and i simultaneously) by utilizing arg min values 4 from (4.6). The a * (·)-determined running cost and dynamics defined by (4.1) will be only piecewise Lipschitz in x, which is precisely the setting considered in section 2. Finally, we note that the above can be also viewed as an implicit definition for a piecewise open-loop optimal policy α * ∈ A: α * (t # , x # , i # ; t) = a * ∈ arg min a∈A ∇û i # (y(t)) · f i # (y(t), a) + C i # (y(t), a) . 4.2. PDEs for threshold-specific optimization. In contrast to the above expectation-centric approach, our goal is to generalize the CDF-computation methods of section 2 by choosing control policies that maximize the probability of desirable outcomes. Two subtleties associated with this approach are worth pointing out before we start deriving the optimality equations. First, the idea of "generating the optimal CDF" is misleading unless we state the goal more carefully. Given any fixed initial configuration (x, i) and two feedback control policies a 1 (·) and a 2 (·), it is entirely possible (and actually quite common!) that w s 2 ). So, which of the resulting CDFs is preferable depends on which threshold is more important: is our priority to minimize the chances of the cumulative cost exceeding s 1 or s 2 ? In this threshold-specific optimization setting, we will say that a policy a(·) is s-optimal if w a 1 (·) i (x, s 1 ) > w a 2 (·) i (x, s 1 ) while w a 1 (·) i (x, s 2 ) < w a 2 (·) i (x,a(·) i (x, s) ≥ w b(·) i (x, s) for all allowable control policies b(·). The second subtlety is in choosing the set of inputs used to define feedback control policies. In threshold-specific optimization, the optimal actions are no longer fully defined by the current state (x, i). In addition, they also depend on the cost incurred so far and the desired threshold for the cumulative cost up to the termination. To handle this complication, we add an extra dimension to our state space, defining a new expanded set of piecewise open-loop control policies A e = {α : (R × Ω × M × R) → A o } , and the expanded PDMP dynamics: y (t) = f m(t) y(t), α o (t − t # ) ,(4.7)y(0) = x ∈ Ω, c (t) = C m(t) y(t), α o (t − t # ) , c(0) = 0, m(0) = i ∈ M. 4 Additional assumptions on f i 's and C i 's can be imposed to ensure that this arg min is a singleton as long asû i is differentiable [6]. But even with these assumptions, the expectation-optimal policy will still be non-unique at the points where ∇û i does not exist, and a tie-breaking procedure (e.g., based on a lexicographic ordering) can be employed to avoid the ambiguity. Here c(t) represents the total cost incurred so far and m(t) is the current mode, evolving through a continuous-time Markov process on M. Similarly, the last argument in α ∈ A e is c # , the total cost accumulated by the time of the last mode switch encountered so far. Assuming α(t # , x # , i # , c # ) = α o (·) ∈ A o , the control value to be used at the time t ≥ t # will be specified by α o (t − t # ) as long as we remain in mode i # . We can now define our new threshold-aware value function: w i (x, s) = sup α(·)∈A e P J α(·) i (x) ≤ s . We note that a similar expansion of state space and policy class could also be used when definingû, but it would not make any difference due to the linear properties of expectations. In contrast, the c-dependence of policies is essential for writing down the tail-optimality property ofŵ i 's: w(x, i, s) = sup α(·)∈A e E ŵ y(τ ), m(τ ), s − c(τ ) ,(4.8) for all τ > 0 sufficiently small to guarantee that y(t) ∈ Ω\Q for all t ∈ [0, τ ], α(·) ∈ A e . Similarly to the derivation of (2.14), a Taylor expansion of (4.8) yields a system of nonlinear PDEs satisfied byŵ i 's: max a∈A ∇ŵ i (x, s) · f i (x, a) − C i (x, a) ∂ŵ i ∂s (x, s) + j =i λ ij ŵ j (x, s) −ŵ i (x, s) = 0, ∀x ∈ Ω\Q, i ∈ M, s > 0; (4.9) with the same initial and boundary conditions previously specified for w i 's in (2.15) and (2.16). We can also restrict the computational domain forŵ i 's (and decrease the numerical diffusion in the discretization) by generalizing equations (2.17)-(2.18) and definingŝ 0 andŵ 0 i 's. 4.3. Discretization of PDEs and control synthesis. A semi-Lagrangian discretization of (4.9) can be obtained on a grid similarly to the treatment of an uncontrolled case in section 2.3: W n i,k = max a∈A M j=1 p ij (τ ) W j x k + τ f i (x k , a), s n − τ C i (x k , a) . (4.10) Here W n i,k ≈ŵ i (x k , s n ) is a grid function and W i is its interpolated version defined on Ω × R. Once all W i 's are computed, they can be used to approximate the optimal control not just on the grid but for all (x, i, s) by choosing control values from the set a) . A(x, i, s) = arg max a∈A M j=1 p ij (τ ) W j x + τ f i (x, a), s − τ C i (x, (4.11) Wherever ∇ŵ i is well-defined, we can also use A(x, i, s) to denote the arg max set in equation (4.9), withÂ(x, i, s) interpreted as its grid approximation. We note that this procedure allows synthesizing a policy (approximately) optimal with respect to any desired threshold value. To obtain ans-optimal feedback policy a(·), we would simply need to select a(x, i, c) ∈Â(x, i,s − c). However, such policies will be generically non-unique since ∇ŵ i (x, s) = 0 might hold on a large part of Ω × (0, +∞). For example, there is always a "hopeless region" H = {(x, s) |ŵ i (x, s) = 0, ∀i ∈ M} since this equality holds by definition whenever s <ŝ 0 (x). If C i 's do not depend on a, then ∇ŵ i (x, s) = 0 implies A(x, i, s) = A. If we start from (x 0 , i 0 ) such thatŵ i 0 (x 0 ,s) < 1, then every policy will have a non-zero probability of exceeding the thresholds. Which control values are used on H does not change the probability of "success" J a(·) i 0 (x 0 ) ≤s , but it can significantly impact the overall CDF of that policy. In many problems there is also an "unconditionally successful" region U = {(x, s) |ŵ i (x, s) = 1, ∀i ∈ M} . Ifŵ i 0 (x 0 ,s) = 1 then an optimal policy will never exceed the thresholds regardless of the timing of mode switches. If (x 0 ,s) is in the interior of U , then the success is guaranteed regardless of control values chosen until we reached ∂U , but these choices will generally affect the CDF. To resolve these ambiguities, we use a tie-breaking procedure in defining optimal policies: wheneverÂ(x, i, s) is not a singleton, we select its element that minimizes the expectation. (On H this will coincide with an expectation-optimal policy a * (·). But on U this need not be the case since our optimal policy is c-dependent and we need to account for expected values on ∂U .) Assuming that V n i,k is a grid function approximating the expected outcome and V i is its interpolated version, we can summarize the computational process as follows: W n i,k = M j=1 p ij (τ ) W j x k + τ f i (x k ,â n i,k ), s n − τ C i (x k ,â n i,k ) ; (4.12) V n i,k = τ C i (x k ,â n i,k ) + M j=1 p ij (τ ) V j x k + τ f i (x k ,â n i,k ), s n − τ C i (x k ,â n i,k ) ; (4.13) a n i,k ∈ arg min a∈Â(x k ,i,s n ) τ C i (x k , a) + M j=1 p ij (τ ) V j x k + τ f i (x k , a), s n − τ C i (x k , a) . (4.14) The above description removes almost all ambiguity from the synthesis of thresholdoptimal policies, but the arg min in (4.14) might still have multiple elements on a set of measure zero in Ω × S. In such rare cases, additional tie-breaking can be used based on another criterion (e.g., a lexicographic ordering). Numerical experiments. We first illustrate these subtleties of policy synthesis with a simple example on a one-dimensional state space Ω = [0, 1] and two modes, each with its own preferred (faster) direction of motion. Example 5: More precisely, the control value a ∈ A = {−1, 1} specifies the chosen direction of motion, and the dynamics are f i (x, a) = a + (−1) i−1 1 2 with i = 1, 2. In other words, in mode 1 we can move right with speed 3/2 and left with speed 1/2, while in mode 2 it is the opposite. We use q ≡ 0 on Q = ∂Ω and C 1 ≡ C 2 ≡ 1, ensuring that the cumulative cost J is just the time to target. For simplicity, we also use symmetric switching rates λ 12 = λ 21 = 2. The resulting optimal policies and the contour plots ofŵ i (x, s) are shown in Figures 9 and 10 respectively. In Figure 11 we fix a starting configuration and compare the CDFs of two different policies. The expectation-optimal feedback policy a * (·) is obtained by solving (4.6) and its CDF is then found by solving (2.14). Unfortunately, the same approach is not available for threshold-specific optimal policies: for ans-optimal feedback policy the threshold-specific policy reduces the probability of missing the deadlines but at the expense of increasing the expected time to target. Moreover, threshold-specific optimal policies (and their respective CDFs) may also vary significantly depending on the chosen thresholds. To illustrate this, we now consider an example on a two-dimensional state space Ω = [0, 1] × [0, 1] with four modes, each with its own faster direction of motion. Example 6: The control values a now reside in A = {a ∈ R 2 | |a| = 1}, and the dynamics are given by f 1 (x, a) = a + −0.5 0 , f 2 (x, a) = a + 0 0.5 , (4.15) f 3 (x, a) = a + 0.5 0 , f 4 (x, a) = a + 0 −0.5 . Again, we use q ≡ 0 on Q = ∂Ω and C i ≡ 1 for all i, ensuring that the cumulative cost J is just the time to ∂Ω. The switching rates are λ ij = 1 for all i = j. In Figure 12, we show the CDFs (each approximated using 10,000 Monte-Carlo simulations) for three different threshold-specific optimal policies with the same starting location. Not surprisingly, each of these policies is strictly better than others with respect to its particular threshold value. The contour plots ofŵ i (x, s) at various s-slices are also shown in Figure 13. Conclusion. The versatility of Piecewise-Deterministic Markov Processes (PDMPs) makes them a useful modeling framework for applications with non-diffusive random perturbations. In prior literature on PDMPs, the focus has been mostly on the average/expected performance. Unfortunately, this ignores the practical importance of relatively rare yet truly bad outcomes. The primary goal of our paper is to address this shortcoming and fully characterize the aleatoric uncertainty in a broad class of discrete and continuous PDMPs. We have accomplished this in section 2, approximating the Cumulative Distribution Function (CDF) for their outcomes by configuration (x, i). solving a system of linear hyperbolic PDEs. Although we did not pursue this here, it would be easy to adapt our approach to compute the CDF of hitting times in discrete time Markov chains. In continuous setting, similar ideas could be also extended to stochastic switching in diffusive systems. Despite our focus on time-till-exit examples, the presented approach is suitable for a broader class of running costs and PDMP performance measures. We illustrate this with a bioeconomic sample problem described in the Appendix. For simplicity of exposition, we have assumed the mode-switching rates λ ij to be constant, but it should be easy to extend our framework to state-dependent switching rates λ ij (x). The case of λ ij 's deterministically evolving in time can be treated similarly by increasing the dimension of our state space. But random changes in rates present a more serious challenge, which is also related to handling model uncertainties. The latter is particularly important in PDMPs since in many practical applications these rates are not known precisely and are instead estimated based on historical data. It is thus useful to characterize the range of possible CDFs -a task accomplished in section 3, where tight CDF bounds are developed under the assumption that each (state-independent) transition rate λ ij has known bounds but does not necessarily remain constant throughout the process. Finally, in section 4 we have extended our methods to control the PDMP dynamics, showing how to maximize the probability of not exceeding a specific cumulative cost thresholds. Our approach is also related to the Stochastic On-Time Arrival (SOTA) formulation, developed in discrete setting by transportation engineers to optimize the routing on stochastic networks [23,25,41]. In the context of SOTA, there is only one "mode," but the running cost is random. While we do not pursue it here, our method can be similarly adapted to optimize the CDF for a subclass of Markov Decision Processes with deterministic running cost and random successor nodes. Several generalizations of the described methods will broaden their appeal to practitioners. First, all PDMPs considered here were exit-time problems, with the process terminating as soon as the system enters a specific subset Q of the state space Ω. It will be easy to extend our approach to finite horizon problems, but the extensions to infinite horizon (with time discounting of running cost) or ergodic (time-averaged cumulative cost) problems will be more challenging. Second, the classical controlled PDMP models in [20,47] were more general than the setting presented here: instead of our "mode switching" they considered ODE trajectories punctuated by random jumps in state space, with both the rate of jumping and the distribution over the set of post-jump positions generally dependent on the pre-jump state and the chosen control value. It would be clearly useful to extend our methods to this broader setting. Our section 3 can be viewed as a small step in this direction, since we are essentially controlling mode-transition rates to either maximize or minimize the CDF. Third, there are many potential ways to improve the accuracy and computational efficiency. Our approach relies on solving systems of hyperbolic PDEs, whose solutions are typically piecewise continuous. While the described implementation is based on a first-order accurate semi-Lagrangian discretization, it would be useful to replace these with higher-order accurate schemes [24]. Our preliminary experimental results based on ENO/WENO [46] spatial discretization in one dimension seem promising, but we have decided to omit them here due to length constraints. We have also developed a technique restricting the computational domain by pre-computing the minimal attainable cumulative cost. In controlled PDMPs with an "unconditionally successful" region, further domain restriction techniques might be used to maximize the probability of desirable outcomes while also imposing a hard constraint on the worst-case performance. This would mirror the approach previously developed for routing on stochastic networks [23]. For controlled processes, another interesting challenge is to carefully evaluate all trade-offs between conflicting objectives. This is usually done by computing nondominated (or Pareto-optimal controls), for which any improvement in one of the objectives must come at the cost of decreased performance based on some other objective(s). With PDMPs, the natural objectives would include traditional minimization of the expected cumulative cost and maximizing the probability of not exceeding a threshold (possibly for several different threshold values). In the fully deterministic case, several methods for multiobjective optimal control have been developed in the last ten years [22,29,34]. It will be useful but more challenging to extend these to the piecewise-deterministic setting. It would be also very interesting to explore additional notions of robustness for PDMPs. Our approach can be viewed as a dual of optimizing the Value-at-Risk (VaR), in which the goal is to minimize a specific percentile of the random outcome. We minimize the probability of exceeding a specific threshold, but similarly to VaR, we provide no guarantees on how bad the outcomes can be once that threshold is exceeded. The Conditional Value-at-Risk (CVaR) is an extended risk measure which addresses this limitation. A method based on CVaR optimization has been developed for Markov Decision Processes in [16]. It would be useful to extend it to PDMPs and compare with the threshold-optimal policies described here. Acknowledgements. The authors would like to thank Tristan Reynoso and Shriya Nagpal for their help in the initial stages of this project during the summer REU-2018 program at Cornell University. The authors are also grateful to anonymous reviewers whose suggestions have helped us improve this paper. 6. Appendix: a fish harvesting example. To show that our general methods are broadly useful beyond the set of illustrative first-exit-time problems considered above, we include an example based on a PDMP with non-constant dynamics and non-constant running cost in each mode. We quantify the uncertainty in harvesting fish population (whose changing level is encoded by y(t)) in the environment with randomly switching carrying capacity K. As a motivation for such switching, we note that the fish population in the tropical Pacific depends on upwelling of nutrients due to the common easterly winds. In El Niño years, these winds weaken, temporarily reducing both the upwelling and the carrying capacity. The usual logistic population growth model y = r(1 − y K )y assumes that the per capita growth rate decreases linearly with the current population size, starting from the rate r when y = 0 and decreasing to zero if y reaches the carrying capacity K. This logic reflects the ideas of increasing competition for limited resources when the population grows. But at low population sizes, other considerations might be more important -having more individuals might make it easier to find partners for mating, cooperate in finding food, or fend off predators. This "Allee effect" [2] is reflected by having per capita growth rate that first increases (until some threshold value y = A < K) and only then decreases (until it reaches 0 at y = K). Perhaps the simplest model that captures this and includes harvesting is dy dt = r y y A − 1 1 − y K − hy (6.1) Here, h ≥ 0 is the effective fishing efforts coefficient, which we will assume to be constant. Note that y = 0 is a stable equilibrium for all h ≥ 0, including the nofishing case h = 0. (This is because we are modeling a strong Allee effect and y < 0 for all y ∈ (0, A).) But for sufficiently small h, the system has two more equilibria: an unstable one at y − (h) and a stable one at a higher y + (h): y ± (h) = K + A 2 ± (K − A) 2 − h 4K r 2 , for h ≤ r(K − A) 2 4K = h # . (6.2) In this regime, the asymptotic behavior depends on the initial conditions: lim t→+∞ y(t) = y + (h) if y(0) > y − (h) and lim t→+∞ y(t) = 0 if y(0) < y − (h). As shown in Figure 14, this bi-stability disappears in a saddle-node bifurcation at h = h # , marking the onset of population collapse. However, we make a distinction between two stages of collapse: for all h > h # the collapse of fish population is imminent since lim t→+∞ y(t) = 0 regardless of y(0). It can be still reversed by reducing h sufficiently quickly, but becomes irreversible as soon as y(t) < A. We will view this irreversible collapse as a terminal event, motivating our choice of the exit set Q = {A}. Another value of obvious relevance is the optimal level of fishing efforts h * that maximizes the sustainable yield hy + (h) over all h ∈ [0, h # ]. A straightforward calculation shows that h * = r 9K K 2 + A 2 − 4KA + (K + A) (K − A) 2 + KA . (6.3) Until now, we have treated all other parameters as fixed and considered the changes to asymptotic behavior as a function of the chosen h. We now turn to a PDMP model with 3 modes, each with its own carrying capacity (K 1 = 3.8, K 2 = 4, and K 3 = 4.2) and with the other two parameters held constant (r = 2 and A = 1). For notational convenience, we will use h # i and h * i to refer to the corresponding maximal sustainable and the yield-optimal fishing rates for each value K i in the deterministic setting (i.e., if you start in Mode i and there are no mode switches). But in our computational experiments, we will assume the switching rates λ 12 = λ 32 = 0.1, λ 21 = λ 23 = 0.05, and λ 13 = λ 31 = 0. As a result, the system spends on average 50% of time in Mode 2 and 25% of time in each of the Modes 1 and 3. So, it is natural to view K 2 as "usual" (or at least as "average") and it might be tempting to select its optimal fishing rate h * 2 ≈ 1.057. But if we stay in Mode 1 for a sufficiently long time, this rate will lead to a population collapse since h * 2 > h # 1 ≈ 1.032; see Figures 14 and 15(a). In fact, if we stick to the same harvesting rate h * 2 in all modes, this collapse eventually happens with probability one as long as λ 21 , λ 32 > 0. One natural question is to quantify the uncertainty in the time until this collapse becomes irreversible; i.e., a random time T x until y(t) = A = 1 for a trajectory starting from y(0) = x in Mode i. This could be interpreted as another "first-exit time problem", similar to those considered in section 2.4, but with x-dependent dynamics in each mode. Instead, we have chosen to focus on the CDF for the total amount harvested before the collapse becomes irreversible: J i (x) = T x 0 h * 2 y(t) dt, w i (x, s) = P(J i (x) ≤ s). (6.4) which requires using a non-constant running "profit". To map this back to the notation of section 2.2, we will take f i (x) = r x x A − 1 1 − x K i − h * 2 x and C i (x) = h * 2 x. (6.5) Our exit set is Q = {1}; so, we set all q i (1) = 0 and note that no boundary condition is needed at the other endpoint since f i (4) < 0 for all i. We solve the three coupled PDEs (2.14) for x ∈ Ω = [A, K 2 ] = [1, 4] and s ∈ [0, 200]. The approximate solution is computed through a semi-Lagrangian scheme (2.19) on a 101 by 120,001 grid, corresponding to ∆x = 0.03 and ∆s = 1/600. In this example, max (x,i) |f i (x)| ≈ 5.4912 and min (x,i) C i (x) ≈ 1.057. So, we use a pseudotimestep τ = 1/600 to satisfy the inequalities (2.20) and (2.21). An obvious lower bound for J i (x) is (x − A), but this does not include all the fish born and harvested before T x . The sharp lower bound s 0 (x) can be computed by noting that the fastest collapse happens if we stay in Mode 1 throughout. For the initial condition x = 4 depicted in Figure 15(a), this quantity is approximately s 0 (4) ≈ 14.87. If starting in Mode 1, the probability of such an outcome is w 0 1 (x) = exp − λ 12 T , where T is the time taken by this "deterministically fastest" collapse. If starting in any other mode, this outcome would require an instantaneous transition to Mode 1; so, w 0 2 (x) = w 0 3 (x) = 0. Figure 16 presents the corresponding CDFs w i (x, s) for the initial populations x = 2.2 andx = 4 while the graphs of w i (x,s) for 3 different values ofs are shown in Figure 17. We note that in a deterministic scenario of K = K 2 , the sustainable equilibrium would be y + (h * 2 ) ≈ 2.8685 and the sustained optimal yield (i.e., the amount perpetually harvested per unit time) would be R = h * 2 y + (h * 2 ) ≈ 3.0323. This provides a natural yardstick for thinking about the argument s used in our CDFs. E.g., based on Figure 16, if we start with y = K 2 in mode 2 under the random switches defined by λ ij 's, there is an approximately 50% chance of harvesting at least 21.8 × R ≈ 66.1 before the collapse becomes irreversible. The above story is based on the assumptions that mode transitions are not ob-served and h is chosen once and for all. In reality, declining catch would provide an advance warning that the population starts to collapse and one could reduce h adaptively. Selecting h ≤ h # i (or h ≤ min i h # i , if mode switches are not directly observable) would make harvesting indefinitely sustainable. One could also use the methods of section 4 to find the CDF-optimizing harvesting rates in feedback form. So, our model described above is a vast simplification, but it already illustrates the promise of presented techniques for quantifying uncertainty in bioeconomic applications. (t), m(t) dt + q y T x,i , m T x,i . (1.3) conditions are sufficient when Q = ∂Ω or if Ω is invariant under all vector fields f i . All of our examples considered in the next sections fall in this category. Fig. 2 : 2Minimum cost s 0 (x) and probability w 0 i (x) of attaining that minimum cost. Subfigures (A) and (B) are for Example 1 and (C) and (D) are for Example 2. Graphs of w 0 i are shown in blue for i = 1 and in red for i = 2. Fig. 3 : 3CDF for a particle starting at initial position x, in Mode 1 (blue) and Mode 2 (red). In Subfigures (A) and (C), the initial condition is x = 0.30 while in Subfigures (B) and (D) the initial condition is x = 0.70. Subfigures (A) and (B) are for Example 1 and (C) and (D) are for Example 2. Fig. 4 : 4Example 1: Equal speeds, with a symmetric transition rate λ = 2. Each subplot is a snapshot of w i (x, s) = P(J i (x) ≤ s) for a specific value of s. In Mode 1 (blue), the particle moves to the right with speed 1. In Mode 2 (red), the particle moves to the left with speed 1. Computed on Ω × S = [0, 1] 2 with ∆x = ∆s = 0.001. w 0 iFig. 5 : 05(x), shown in Figure 2(C-D), to reduce the computational domain for w i 's. E.g., for the first subfigure, all x ∈ [0.25, 0.875) have s 0 (x) > s = 0.25 and so are assigned an exit probability of 0, removing the need of interpolating across discontinuities. In contrast, at s = 0.75 all x have a nonzero probability of exiting, so interpolation across the discontinuity at x = 0.625 is unavoidable. Example 3: We now consider a 2D version of Example 1, with Ω = [0, 1]×[0, 1], Example 2: Unequal speeds, with a transition rate λ = 2. Each subplot is a snapshot of w i (x, s) = P(J i (x) ≤ s) for a specifc value of s. In Mode 1 (blue), the particle moves to the right with speed 1/2. In Mode 2 (red), the particle moves to the left with speed 1. Computed on Ω × S = [0, 1] 2 with ∆x = ∆s = 0.001. Q = ∂Ω, M = 4, and λ = 1. In all modes, the motion is with speed |f | = 1, but the directions of motion differ: ←, ↑, →, and ↓ in modes 1, . . . , 4 respectively. Numerical approximations of w i 's for different values of s are shown in Figure 6. The distinct delineations between darker and lighter regions are analogous to the discontinuities in the earlier one-dimensional cases. For example, given s < 0.5 and starting positions along the line y = 0.5, a timely exit is only possible to the left (via Mode 1) or to the right (via Mode 3). Therefore, cross sections of w 1 and w 3 along y = 0.5 at s = 0.25 in Fig. 6 : 6Example 3: Mode switching in 2D with transition rates λ = 1. The particle moves ← in Mode 1, ↑ in Mode 2, → in Mode 3, and ↓ in Mode 4. Computed on Ω × S = [0, 1] 3 with ∆x = ∆y = ∆s = 0.01. Fig. 7 : 7Example 4: bounds on the probability of timely exit (starting in Mode 1) for four different deadline valuess. (Probabilities and bounds for starting in Mode 2 can be obtained by a mirror symmetry relative to the line x = 0.5.) Blue bounds are produced under the varying rates assumption by solving (3.1)-(3.2) for λ 12 , λ 21 ∈ [1, 4]. Green curves are produced under the fixed rates assumption by solving (2.14), each corresponding to a specific (λ 12 , λ 21 ) ∈ {1, 2, 3, 4}×{1, 2, 3, 4}. The darkest four curves are those associated with λ 12 = 1, the next four are those associated with λ 12 = 2, and so on. Computed on Ω × S = [ Fig. 8 : 8Example 4: bounds on CDF (starting in Mode 1) for two different initial positions x = 0.3 andx = 0.7. See a detailed legend in caption of Figure 7. minimize E[exp(βJ )], with the risk-sensitivity coefficient β > 0 reflecting our desire to avoid bad outcomes[26]. For small β values, this is roughly equivalent to minimizing a convex combination of E[J] and V ar[J ]. While implementable with PDMPs, this method does not provide any guarantees on the likelihood of bad scenarios. We thus develop a different approach to maximize the probability of not exceeding a specific cumulative cost thresholds. In subsection 4.2 we develop PDEs to find such optimal policies for all initial configurations and all threshold valuess simultaneously.The numerical methods and computational examples (for d = 1 and d = 2) are covered in subsections 4.3 and 4.4 respectively. describing the resulting trajectory and cumulative cost. The latter will be denoted J a(·) (x, i) = J a(·) i (x) to highlight the dependence on the chosen control policy. The corresponding expected cost u ) ≤ s can then be found from equations (2.9) and and related literature for technical details. In deterministic setting, one considers the set of measurable open loop control functions A o = {α o : R → A} , with α o (t) specifying the control value that will be used at the time t. For PDMPs, a piecewise open-loop policy specifies a new open loop control function to be used after each mode switch. Adapting to our setting, we can define a set of piecewise open-loop policies as A = {α : (R × Ω × M) → A o } . o ∈ A o is the open loop control function currently in effect at the time t based on the policy α ∈ A and a sequence of mode switches that have occurred so far. Recall from section 1 that y(0) = x, m(0) = i, and the changes in mode m(t) are governed by the matrix of switching rates Λ. In this section, we will further assume that all f i 's and C i 's are Lipschitz-continuous in both arguments.The usual goal in controlled PDMPs literature is to minimize the expected total cost up to the termination time. The value function is thus defined aŝ Fig. 9 : 9Example 5: a map of threshold-optimal control values with position on the horizontal axis and time remaining until the deadline on the vertical axis. The purple color represents the optimal choice of moving to the left, and the yellow color represents the optimal choice of moving to the right. The shaded area with the cyan border represents the "hopeless region" H, where w i 's are uniformly zero and the threshold-specific optimal policies coincide with the expectation-optimal policy. The "unconditionally successful" region U is shown above the black dashed line. Under grid refinement, everything in the left part of U becomes purple and everything in the right part of U becomes yellow in both modes. The red dashed vertical lines show the point of direction-switching for the expectation-optimal policy. Computed with ∆x = 1.25 · 10 −4 , ∆s = 0.625 · 10 −4 . Fig. 10 :Fig. 11 : 1011Contour plot ofŵ i (x, s) for Example 5. Computed with ∆x = 1.25 · 10 −4 , ∆s = 0.625 · 10 −4 . a(·), there is no reason to expectŵ i (x, s) = P J a(·) i (x) ≤ s unless s =s. Instead, we approximate their CDF using 100,000 Monte-Carlo simulations 5 . Not surprisingly, 5 While we do not pursue this alternative here, one could also approximate this CDF by solving the Kolmogorov Forward Equation with initial conditions chosen based on this specific starting Example 5: CDF of an expectation-optimal policy (in red) and CDF of a thresholdspecific optimal policy computed fors = 0.38 (in blue). In both cases, the starting configuration is (x 0 , i 0 ) = (0.4, 1). The value of the CDF at the thresholds = 0.38 is marked by a blue dot. The vertical dashed lines indicate the expected value of each policy. Fig. 12 :Fig. 13 : 1213Example 6: CDFs of threshold-specific optimal policies computed fors = 0.28 (in blue),s = 0.33 (in green), ands = 0.40 (in red). The value of the CDFs at each threshold are denoted by dots of the corresponding color. In all cases, the starting configuration is (x 0 , y 0 , i 0 ) = Contour plots ofŵ i (x, s) for Example 6. Transition rate between all modes is λ = 1. Each subplot contains a snapshot ofŵ i (x, s). Each row has a fixed s value, and each column has a fixed mode i. Dynamics are given by(4.15). Computed on Ω × S = [0, 1] 3 with ∆x = ∆y = ∆s = 0.0025. Fig. 14 : 14Bifurcation diagrams corresponding to the dynamics in each mode. The orange y = 0 line is a stable equilibrium for each value of K. The other two equilibria exist for a range of h values and are shown in red, green, and blue for K 1 , K 2 , and K 3 respectively. The stable y + (h) is shown by solid lines while the unstable y − (h) is shown by dashed lines. The saddle node bifurcations for each K i are indicated by black dots, while the K 2 -deterministicallyoptimal rate h * 2 is shown by the gray line. Fig. 15: Deterministic dynamics for modes 1, 2 and 3 with harvesting rate h * 2 ≈ 1.057 Fig. 16 : 16CDFs computed up to s = 200 for two starting population levels: x = 2.13 (left) and x = 4 (right). Red, blue, and green are starting in Modes 1, 2, and 3 respectively. Plots represent probability of irreversible population collapse at or before a harvest of size s. Note that y − (h * 2 ) ≈ 2.1312; so, with x = 2.13 a quick collapse can only be prevented by an early switch to Mode 3. If starting at x = 4 in Mode 2 (blue curve in the right subfigure) the 25th, 50th, and 75th percentiles are s ≈ 39.09, s ≈ 66.10, and s ≈ 112.95 respectively. Fig. 17 : 17Probability of irreversible population collapse before a harvest of sizes computed for all starting populations x and three specifics values. Red, blue, and green represent starting in Modes 1, 2, and 3 respectively. To ensure computational reproducibility, our full code for all examples is available at https://github.com/eikonal-equation/UQ PDMPFig. 1: Fully discrete PDMP with M = 2 modes and N = 6 nodes. In mode 1 the motion is always to the right; in mode 2 the motion is always to the left. The exit set is Q = {x 1 , x 6 }.1 Mode 1 Mode 2 x 1 1 x 1 2 x 1 3 x 1 4 x 1 5 x 1 6 x 2 1 x 2 2 x 2 3 x 2 4 x 2 5 x 2 6 p 11 p 11 p 11 p 11 p 22 p 22 p 22 p 22 p 12 p 12 p 12 p 12 p 21 p 21 p 21 p 21 In addition, the finite size of X guarantees that all w i 's are piecewise-constant in s. This can be used to construct a finite time algorithm for solving (2.3) exactly despite the fact that s is a continuous variable. We do not include this algorithm here due to space constraints and to keep the focus on the continuous-time setting, where discretizing s and approximating w i 's is generally unavoidable.6) Numerically solving Equations (2.5) and (2.6) can be advantageous because they are computed on the lower-dimensional domain X × M instead of X × M × S. This information can then be used as initial/boundary conditions to solve (2.3) on a smaller subset of X × M × S. 2 Optimal control of production rate in a failure prone manufacturing system, Automatic Control. 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{'abstract': 'In piecewise-deterministic Markov processes (PDMPs) the state of a finite-dimensional system evolves continuously, but the evolutive equation may change randomly as a result of discrete switches. A running cost is integrated along the corresponding piecewise-deterministic trajectory up to the termination to produce the cumulative cost of the process. We address three natural questions related to uncertainty in cumulative cost of PDMP models: (1) how to compute the Cumulative Distribution Function (CDF) of the cumulative cost when the switching rates are fully known; (2) how to accurately bound the CDF when the switching rates are uncertain; and (3) assuming the PDMP is controlled, how to select a control to optimize that CDF. In all three cases, our approach requires posing a system of suitable hyperbolic partial differential equations, which are then solved numerically on an augmented state space. We illustrate our method using simple examples of trajectory planning under uncertainty for several 1D and 2D first-exit time problems. In the Appendix, we also apply this method to a model of fish harvesting in an environment with random switches in carrying capacity.', 'arxivid': '2008.00555', 'author': ['Elliot Cartee ', 'Antonio Farah ', 'April Nellis ', 'Jacob Van Hook ¶ ', 'Alexander Vladimirsky '], 'authoraffiliation': [], 'corpusid': 220935989, 'doi': None, 'github_urls': ['https://github.com/eikonal-equation/UQ'], 'n_tokens_mistral': 29238, 'n_tokens_neox': 26063, 'n_words': 16893, 'pdfsha': '26d39632dd00f115a913939e7cdd31cfd8c3d3ca', 'pdfurls': ['https://export.arxiv.org/pdf/2008.00555v2.pdf'], 'title': ['QUANTIFYING AND MANAGING UNCERTAINTY IN PIECEWISE-DETERMINISTIC MARKOV PROCESSES *', 'QUANTIFYING AND MANAGING UNCERTAINTY IN PIECEWISE-DETERMINISTIC MARKOV PROCESSES *'], 'venue': []}

arxiv

Polarization effects on intermolecular vibrational energy transfer analyzed by 2DIR spectroscopy 16 Aug 2017 A A Villaeys albert.villaeys@wanadoo.fr Université de Strasbourg et Institut de Physique et Chimie des Matériaux de Strasbourg France Research Center for Applied Sciences Academia Sinica 115TaipeiTaiwan M Zouari Faculté des Sciences de Bizerte Département de Physique Université de Carthage 7021ZarzounaTunisie Electronic Kuo Kan Liang kkliang@sinica.edu.tw Research Center for Applied Sciences Academia Sinica 115TaipeiTaiwan Department of Biochemical Science and Technology National Taiwan University 106TaipeiTaiwan Electronic Polarization effects on intermolecular vibrational energy transfer analyzed by 2DIR spectroscopy 16 Aug 2017coherent nonlinear optical spectroscopyintermolecular vibrational energy transferpolarization effectrotational dynamics In the present work, we analyze the influence of the polarization effects taking place during the course of a 2DIR spectroscopy experiment performed on a molecular system undergoing an intermolecular vibrational energy transfer process. When both donor and acceptor molecules participating in the vibrational energy transfer are embedded in a host solvent, they face rotational diffusion that strongly distorts the resulting 2DIR spectra. It could be expected that the difference between rotational diffusion constants will be of particular interest. For this purpose, the polarization effects are discussed according to the different orderings of the laser-molecule interactions. Next, we study the distortions of the spectra as a function of the rotational diffusion constants of the individual molecules. The knowledge of these polarization effects are relevant to the interpretation of the spectra. Finally, the conclusions reached in this work for a vibrational energy transfer are valid for any other type of third-order optical process performed on the same molecular system. I. INTRODUCTION Measurements and subsequent interpretations of the spectra obtained from two-dimensional (2D) or higher multidimensional spectroscopy experiments, are difficult tasks. Under certain circumstances, polarization effects can be an additional and quite effective tool to overcome these difficulties 1,2 . In fact, polarization has long been used as a spectroscopic tool, but it is especially useful in 2D spectroscopy because the diagonal and cross peaks characterizing 2D spectra arise from a well-defined number and ordering of the pulsed laser fields interacting with the molecular transition dipoles. Therefore, taking advantage of particular combinations of the laser fields, specific processes can be emphasized in the 2D spectra. It can be expected that polarization will play an even more important role in 3D spectroscopy 3 . Today, these polarization effects became very important in 2D infrared (2DIR) spectroscopy where sequences of polarized laser pulses are used to detect angles between molecular transition dipoles [4][5][6] . This emerges as an important tool to monitor the molecular structures. Indeed, these effects have played a major role in developing polarization-selective 2DIR spectroscopy which provides a unique way to analyze rotational dynamics. Among the large number of applications, we can mention the determination of the anisotropy dynamics of both intraconfigurational hydroxyl groups and those hydroxyl groups undergoing interconfigurational H-bond exchange in six-molar aqueous sodium perchlorate 7 . Of course, these investigations of H-bond exchange rely on previous developments in the theoretical description of rotational dynamics in H-bonded solutions, as well as theoretical models of 2DIR spectroscopy 4,8,9 . Theoretical models for rotational dynamics originating from librational, diffusive and angular jump motions have been developed and applied to experimental studies of rotational dynamics in water and aqueous solutions [10][11][12][13][14][15] . Besides, polarization effects are useful for supressing background signals 16,17 and even for enhancing cross peaks at the expense of diagonal peaks 18 . This is of particular interest because large diagonal peaks quite often mask weaker cross peaks and suppressing the diagonal peaks provides better resolution of the cross peaks. From a more fundamental point of view, control over polarization generally leads to improved spectral resolution by suppressing or enhancing the contributions associated with spe-cific pathways in the Liouvillian space according to the relative orientations of the laser field polarizations and dipole moments of the molecular states involved in a particular chronological pathway 4,5,16,19,20 . Therefore, taking advantage of adequate polarization schemes, physical insights can be extracted by enhancing or suppressing the contributions associated with some elementary processes that can possibly exhibit or cover the relevant information on which we want to focus 21,22 . Orientation as well as rotational relaxation dynamics are among the most fundamental characteristics of molecules at surfaces and interfaces. They play an important role in many physical, chemical, and biological phenomena, including molecular mechanisms of energy relaxation, solvation, electron transfer, and many others. What is more relevant to the present work is that when we perform a 2DIR experiment on molecules embedded in a host solvent, for example upon investigating the intermolecular vibrational energy transfer, these effects make the spectra more intricate because rotational diffusion can strongly perturb the optical response of the molecular system under investigation. When rotational motions of the molecules are much slower than the longest time delays required to measure the spectra, rotational motions can be neglected. However, for small molecules, rotational motions can contributes significantly to homogeneous dephasing. Here, the polarization effects can drastically alter the resulting spectra. Consequently, any interpretation of the 2D spectra will require a proper account of the rotational motion undergone by the molecules participating in the energy transfer. Therefore, an average has to be performed over the orientations of the dipoles of all molecules. The resulting 2DIR spectra will be dependent on the magnitude of the rotational diffusion constants of all molecules involved. This is exactly what we want to analyze in the present work. The paper is organized as follows. Sec. 2 reviews briefly the general expression of the signal intensity valid for the rephasing and non-rephasing directions. Next, rotational diffusion is taken into account to calculate the orientational average of the above-mentioned signal. In Sec. 3, the vibrational molecular model supporting intermolecular vibrational energy transfer is introduced to evaluate the internal dynamics as well as the orientational evolution of the molecules undergoing the energy transfer. Finally, in Sec. 4, numerical simulations are performed to give a quantitative analysis of the polarization effects on the 2DIR spectra of a system undergoing intermolecular vibrational energy transfer. II. POLARIZATION EFFECTS ON INTERMOLECULAR VIBRATIONAL ENERGY TRANSFER The purpose of this section is to describe an intermolecular vibrational energy transfer taking place between molecules undergoing rotational diffusion. Due to their respective shapes, the molecules involved in the energy transfer process generally experience different frictions from the host solvent. Since we are interested in the intermolecular energy transfer, the 2DIR experiment involves the dipole moments of two different molecules, which is a typical situation in 2DIR studies. The total vibrational system is therefore composed of vibrational modes Q A and Q B pertaining to molecules A and B, respectively. The lasersystem interaction Hamiltonian is given by V (t) = − p=a,b,c A p (t − T p ) µ · E p e −iωp(t−Tp)+i kp· r + C.C. , (II.1) where the notation C.C. stands for the complex conjugate. The symbol A p (t − T p ) stands for the normalized envelops of the three laser pulses defined as A p (t − T p ) = √ γ p exp (−γ p |t − T p |) with p = a, b or c. Any theoretical description of the 2DIR spectroscopy experiment requires the third-order perturbation term of the density matrix with respect to the laser-molecule interaction V (t) 1,2,23-27 . The contribution to the third-order term of the density matrix, ρ (3) (t), relevant to our purpose, takes the form ρ (3) (t) = i 3 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 × G (t − τ 3 ) L v (τ 3 ) G (τ 3 − τ 2 ) L v (τ 2 ) G (τ 2 − τ 1 ) L v (τ 1 ) ρ (t 0 ) , (II.2) where the interaction Liouvillian is defined by L v (τ α ) = [V (τ α ) , · · · ]. Also involved are the propagators G(τ α − τ β ) = exp − i L(τ α − τ β ) . They depend on another Liouvillian L = [H, · · · ] with H the free vibrational system Hamiltonian. The matrix elements of the propagator with specific form of the indices G iijj (τ α − τ β ) account for free population evolution if i = j and population transfer if j = i, respectively. Another set of matrix elements, namely G ijij (τ α − τ β ) with i = j account for the evolution of the coherences. The emitted radiation signal along the rephasing phase-matched direction k re = − k a + k b + k c and the non-rephasing direction k nre = k a − k b + k c are deduced from the third-order term of the polarization P (3) ks (T a , T b , T c , t) = 2ℜ i j<i ρ (3) ks,ij (t) µ ji (II.3) where k s stands for k re or k nre . Using a local field E lo (t) = A lo (t − T lo ) E lo e −iω lo (t−T lo )+i k lo · r−iΨ + C.C. with Ψ an additional phase, the heterodyne detection is introduced 1,2,28-30 . As usual, the signal field is deduced from the polarization through the expression E ks (T a , T b , T c , t) ∝ i P ks (T a , T b , T c , t). With typical experimental conditions of weak signal field intensity and after subtracting the intensity of the local field, the temporal Fourier transform of the total intensity is given by I ks (T a , T b , T c , ω t ) = 2ℜ +∞ −∞ dt E lo (t − T lo ) e iωtt * +∞ −∞ dt E ks (T a , T b , T c , t) e iωtt . (II.4) Then, with the time origin chosen at the center of the last pulse and by assuming clean fast leading edges for the pulses 31 the time integration can be reduced to t ∈ [0, +∞). Finally, the 2D spectra is obtained by performing a second Fourier transform over the delay time τ = T b − T a so that I ks (ω d , ω t , T ) = +∞ −∞ dτ e −iω d τ I ks (τ, T, ω t ) (II.5) The experimental waiting time is defined as T = min (|T a | , |T b |) according to the prescription by Jonas 32 . The time origin is chosen at the center of the last pulse. The donor and acceptor of the vibrational energy transfer process are both dissolved in liquid solvent. Therefore, a proper account of the polarization effects requires the introduction of orientational dynamics of the molecular motion. For simplicity, we assume that both the donor and acceptor can be approximated by spherically symmetric molecules having a homogeneous moment of inertia. The collisions between the donor, the acceptor, and the solvent molecules induce angular random walks of the dipole moments which can be modeled by a rotational diffusion equation of motion of the angular probability distribution function G (Ω, t) 33 : ∂G (Ω, t) ∂t = −DĴ 2 G (Ω, t) (II.6) whose general solution for a molecule rotating from one direction Ω α = (θ α , φ α ) to another one Ω β = (θ β , φ β ) during the time interval δt is given by the propagator G (Ω β , δt |Ω α ) = e −DĴ 2 δt G (Ω α , 0 |Ω α ) (II.7) if we neglect the rotation during the laser pulse durations. The dipole propagators can be expanded in terms of spherical harmonics Y ℓ,m (θ, φ), abbreviated as Y ℓ,m (Ω), satisfying the well-known eigenvalue equation J 2 Y ℓ,m (Ω) = ℓ (ℓ + 1) Y ℓ,m (Ω) (II.8) as well as the orthonormalization and closure relations          dΩY ℓ ′ ,m ′ (Ω) Y ⋆ ℓ,m (Ω) = δ ℓ,ℓ ′ δ m,m ′ δ (Ω − Ω 0 ) = ∞ ℓ=0 ℓ m=−ℓ Y ℓ,m (Ω 0 ) Y ⋆ ℓ,m (Ω) ≡ ℓ,m Y ℓ,m (Ω 0 ) Y ⋆ ℓ,m (Ω) (II.9) If the initial condition G (Ω, 0 |Ω 0 ) = δ (Ω − Ω 0 ) is satisfied, Eq.(II.7) becomes G (Ω, t|Ω 0 ) = e −DJ 2 t ℓ,m Y ℓ,m (Ω 0 ) Y ⋆ ℓ,m (Ω) = ℓ,m e −ℓ(ℓ+1)Dt Y ℓ,m (Ω 0 ) Y ⋆ ℓ,m (Ω) (II.10) This is the dipole propagator required in the following. Then, the complete polarization dependence of the overall evolution of molecules A and B participating in the energy transfer is included in the termÊ lo · µ ρ (3) (t) whereÊ lo stands for the unitary vector along the oscillating local field. We shall expediently call this heterodyne-detected signal using the local field the heterodyne signal. The average over molecular orientations of the heterodyne signal, denoted as Π (t) or = Ê lo · µρ (3) (t) or , can be expressed as Π (t) or = i 3 s=A,B dΩ (s) 3 dΩ (s) 2 dΩ (s) 1 dΩ (s) 0 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 E lo · µ × G Ω (A) 3 Ω (B) 3 , t − τ 3 Ω (A) 2 Ω (B) 2 G (t − τ 3 ) L v (τ 3 ) × G Ω (A) 2 Ω (B) 2 , τ 3 − τ 2 Ω (A) 1 Ω (B) 1 G (τ 3 − τ 2 ) L v (τ 2 ) × G Ω (A) 1 Ω (B) 1 , τ 2 − τ 1 Ω (A) 0 Ω (B) 0 G (τ 2 − τ 1 ) L v (τ 1 ) × P (A) 0 Ω (A) 0 P (B) 0 Ω (B) 0 ρ (t 0 ) (II.11) Assuming random initial dipole orientations we have P G Ω (j) q , τ r − τ q Ω (j) p L (k) v (τ q ) G Ω (j) p , τ q − τ p Ω (j) m = G Ω (j) q , τ r − τ p Ω (j) m L (k) v (τ q ) (II.12) where L (k) v (τ q ) stands for L v (τ q ) if molecule k interacts with the laser pulse at time τ q . In the three-photon process considered in the theory of 2DIR spectroscopy, three photons interact with either of the two molecules involved respectively in different chronological order. It is important to recall that the procedure of averaging over molecular dipole orientations depends on this chronological ordering of the photon-molecule interactions. Accordingly, we divide all of the processes into four groups so that the Π (t) or = M =I→IV Π M (t) or . The integral in Eq. (II.11) can be simplified in these four cases in respective ways. In the first case, the three photons interact with the same molecule. The corresponding heterodyne signal takes the form Π I (t) or = i 3 j=A,B dΩ (j) 3 dΩ (j) 2 dΩ (j) 1 dΩ (j) 0 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 E lo · µ × G Ω (j) 3 , t − τ 3 Ω (j) 2 G (t − τ 3 ) L v (τ 3 ) G Ω (j) 2 , τ 3 − τ 2 Ω (j) 1 G (τ 3 − τ 2 ) L v (τ 2 ) × G Ω (j) 1 , τ 2 − τ 1 Ω (j) 0 G (τ 2 − τ 1 ) L v (τ 1 ) P (j) 0 Ω (j) 0 ρ (t 0 ) (II.13) The second case corresponds to two photons interacting with the same molecule first, followed by the third photon interacting with the other molecule. Then, we have Π II (t) or = i 3 k =j j,k=A,B dΩ (j) 2 dΩ (j) 1 dΩ (j) 0 dΩ (k) 1 dΩ (k) 0 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 × E lo · µ G Ω (k) 1 , t − τ 3 Ω (k) 0 G (t − τ 3 ) L v (τ 3 ) × G Ω (j) 2 , t − τ 2 Ω (j) 1 G (τ 3 − τ 2 ) L v (τ 2 ) × G Ω (j) 1 , τ 2 − τ 1 Ω (j) 0 G (τ 2 − τ 1 ) L v (τ 1 ) P (j) 0 Ω (j) 0 P (k) 0 Ω (k) 0 ρ (t 0 ) (II. 14) The third case involves one photon interacting with one molecule followed by two photons interacting with the other. It turns out Π III (t) or = i 3 k =j j,k=A,B dΩ (k) 2 dΩ (k) 1 dΩ (k) 0 dΩ (j) 1 dΩ (j) 0 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 × E lo · µ G Ω (k) 2 , t − τ 3 Ω (k) 1 G (t − τ 3 ) L v (τ 3 ) × G Ω (k) 1 , τ 3 − τ 2 Ω (k) 0 G (τ 3 − τ 2 ) L v (τ 2 ) × G Ω (j) 1 , t − τ 1 Ω (j) 0 G (τ 2 − τ 1 ) L v (τ 1 ) P (j) 0 Ω (j) 0 P (k) 0 Ω (k) 0 ρ (t 0 ) (II.15) Finally, in the last case, three photons interact chronologically with different molecules in turns. In other words, the order is either A-B-A or B-A-B. Thus, Π IV (t) or = i 3 k =j j,k=A,B dΩ (j) 2 dΩ (j) 1 dΩ (j) 0 dΩ (k) 1 dΩ (k) 0 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 × E lo · µ G Ω (j) 2 , t − τ 3 Ω (j) 1 G (t − τ 3 ) L v (τ 3 ) × G Ω (k) 1 , t − τ 2 Ω (k) 0 G (τ 3 − τ 2 ) L v (τ 2 ) × G Ω (j) 1 , τ 3 − τ 1 Ω (j) 0 G (τ 2 − τ 1 ) L v (τ 1 ) P (j) 0 Ω (j) 0 P (k) 0 Ω (k) 0 ρ (t 0 ) (II.16) All these quantities are required to evaluate the spectra associated with the energy transfer. The first one, namely Π I (t) or , is the simplest because only one molecule is involved in the averaging procedure. The scalar product involved in the interaction term can be extracted, so that L v (τ ) =μ ·ÊL v (τ ). Applying the spherical-harmonics expansion given by relation (II.10) and with the help of the orthogonality relation of the spherical harmonics we get Π I (t) or = i 3 1 12π j=A,B dΩ (j) 3 dΩ (j) 2 dΩ (j) 1 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 × E lo · µ (j) e −D (j) J 2 (t−τ 3 ) × ℓ,m Y ℓ,m Ω (j) 2 Y ⋆ ℓ,m Ω (j) 3 μ (j) ·Ê p G (t − τ 3 )L v (τ 3 ) e −D (j) J 2 (τ 3 −τ 2 ) × ℓ ′ ,m ′ Y ℓ ′ ,m ′ Ω (j) 1 Y ⋆ ℓ ′ ,m ′ Ω (j) 2 μ (j) ·Ê q G (τ 3 − τ 2 )L v (τ 2 ) e −2D (j) (τ 2 −τ 1 ) × Y ⋆ 1,0 Ω (j) 1 G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.17) In Eq. (II.17), we have also assumed a uniformly distributed initial dipole orientation, P (j) 0 Ω (j) 0 = 1/4π . The polarization of the laser field is chosen to be along the Z-axis. Therefore,μ (j) ·Ê r = cos θ = 4π/3 Y 1,0 Ω (j) 0 . Similarly,μ (j) ·Ê q = 4π/3 Y 1,0 Ω (j) 1 andμ (j) ·Ê p = 4π/3 Y 1,0 Ω (j) 2 . With these explicit expressions, we shall take advantage of the following integral relations between spherical harmonics: dΩ Y ⋆ 1,0 (Ω) Y 1,0 (Ω) Y ℓ,m (Ω) = 1 4π δ ℓ,0 δ m,0 + 1 5π δ ℓ,2 δ m,0 dΩ Y ⋆ 0,0 (Ω) Y 1,0 (Ω) Y ℓ,m (Ω) = 1 4π δ ℓ,1 δ m,0 dΩ Y ⋆ 2,0 (Ω) Y 1,0 (Ω) Y ℓ,m (Ω) = 1 5π δ ℓ,1 δ m,0 + 3 2 3 35π δ ℓ,3 δ m,0 (II.18) we obtain, for the first term, the final result Π I (t) or = i 3 4π 9 E lo µ (j) j=A,B t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 e −2D (j) (t−τ 3 ) × 1 4π G (t − τ 3 )L v (τ 3 ) G (τ 3 − τ 2 )L v (τ 2 ) e −2D (j) (τ 2 −τ 1 ) G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) + 1 5π G (t − τ 3 )L v (τ 3 ) e −6D (j) (τ 3 −τ 2 ) G (τ 3 − τ 2 )L v (τ 2 ) e −2D (j) (τ 2 −τ 1 ) × G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.19) where the notations E p = | E p | and µ = | µ| have been introduced. Next we evaluate Π II (t) or . Using the previous properties and definitions, integrations over Ω (j) 0 and Ω (k) 0 can be performed. Then, expression (II.14) reduces to Π II (t) or = i 3 1 3 √ 12π k =j j,k=A,B dΩ (j) 2 dΩ (j) 1 dΩ (k) 1 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 E lo · µ × e −2D (k) (t−τ 3 ) Y ⋆ 1,0 Ω (k) 1 G (t − τ 3 )L v (τ 3 ) e −D (j) J 2 (t−τ 2 ) × ℓ ′ ,m ′ Y ℓ ′ ,m ′ Ω (j) 1 Y ⋆ ℓ ′ ,m ′ Ω (j) 2 Y 1,0 Ω (j) 1 G (τ 3 − τ 2 )L v (τ 2 ) e −2D (j) (τ 2 −τ 1 ) × Y ⋆ 1,0 Ω (j) 1 G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.20) where relations P (j) 0 Ω (j) 0 = P (k) 0 Ω (k) 0 μ (j) ·Ê r{q} = 4π 3 Y 1,0 Ω (j) 0{1} andμ (k) ·Ê p = 4π 3 Y 1,0 Ω (k) 0 (II.21) have been introduced. Now, we perform the integration over Ω (j) 1 using the integral relation dΩ (j) 1 Y ⋆ 1,0 Ω (j) 1 Y 1,0 Ω (j) 1 Y ℓ,m Ω (j) 1 = 1 4π 1/2 δ l,0 δ m,0 + 1 5π 1/2 δ l,2 δ m,0 . (II.22) to get Π II (t) or = i 3 1 3 √ 12π k =j j,k=A,B dΩ (j) 2 dΩ (k) 1 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 E lo · µ ×e −2D (k) (t−τ 3 ) Y ⋆ 1,0 Ω (k) 1 G (t − τ 3 )L v (τ 3 ) 1 √ 4π Y ⋆ 0,0 Ω (j) 2 + 1 √ 5π e −6D (j) (t−τ 2 ) Y ⋆ 2,0 Ω (j) 2 ×G (τ 3 − τ 2 )L v (τ 2 ) e −2D (j) (τ 2 −τ 1 ) G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.23) We still have to integrate over Ω (k) 1 and Ω (j) 2 . To this end, we must specify the molecular transition contributing to the heterodyne signal. If we assumed that the detection focus on a molecular transition of molecule j, we have E lo · µ (j) = 4π/3 Y 1,0 Ω (j) 2 and dΩ (j) 2 4π 3 Y 1,0 Ω (j) 2 Y ⋆ 0,0 Ω (j) 2 = dΩ (j) 2 4π 3 Y 1,0 Ω (j) 2 Y ⋆ 2,0 Ω (j) 2 = 0 (II.24) which implies in turn that Π II (t) or = 0 if the local field acts on the j-molecule. This result is not surprising at all because it is well established that the second-order optical response cancels for an isotropic system, which is the case here since the laser field interacts twice with the same molecule. Otherwise, if detection focus on a particular transition of molecule k, E lo · µ (k) = 4π/3 Y 1,0 Ω (k) 1 and performing the integrations over Ω (k) 1 and Ω (j) 2 we get Π II (t) or = i 9 3 E lo µ (k) k =j j,k=A,B t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 e −2D (k) (t−τ 3 ) G (t − τ 3 )L v (τ 3 ) × G (τ 3 − τ 2 )L v (τ 2 ) e −2D (j) (τ 2 −τ 1 ) G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.25) because dΩ (j) 2 Y ⋆ 2,0 Ω (j) 2 = 2π π 0 dθ sin θ 5 16π (3 cos 2 θ − 1) = 0. Then we calculate the third term Π III (t) or . It takes the form Π III (t) or = i 3 1 6 √ 2π k =j j,k=A,B × dΩ (k) 2 dΩ (k) 1 dΩ (k) 0 dΩ (j) 1 dΩ (j) 0 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 E lo · µ × e −D (k) J 2 (t−τ 3 ) ℓ,m Y ℓ,m Ω (k) 1 Y ⋆ ℓ,m Ω (k) 2 Y 1,0 Ω (k) 1 G (t − τ 3 )L v (τ 3 ) × e −D (k) J 2 (τ 3 −τ 2 ) ℓ ′ ,m ′ Y ℓ ′ ,m ′ Ω (k) 0 Y ⋆ ℓ ′ ,m ′ Ω (k) 1 Y 1,0 Ω (k) 0 G (τ 3 − τ 2 )L v (τ 2 ) × e −D (j) J 2 (t−τ 1 ) ℓ ′′ ,m ′′ Y ℓ ′′ ,m ′′ Ω (j) 0 Y ⋆ ℓ ′′ ,m ′′ Ω (j) 1 Y 1,0 Ω (j) 0 G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.26) Here, we need to introduce the definitions of the polarization propagators and interaction terms. First, we integrate over Ω Π III (t) or = i 3 1 6 √ 2π k =j j,k=A,B dΩ (k) 2 dΩ (j) 1 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 E lo · µe −D (k) J 2 (t−τ 3 ) × 1 √ 4π Y ⋆ 0,0 Ω (k) 2 + 1 √ 5π Y ⋆ 2,0 Ω (k) 2 G (t − τ 3 )L v (τ 3 ) e −2D (k) (τ 3 −τ 2 ) × G (τ 3 − τ 2 )L v (τ 2 ) e −2D (j) (t−τ 1 ) Y ⋆ 1,0 Ω (j) 1 G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.27) Similar to the case of Π II (t) or , further simplification is possible. Due to the orthogonality relations of the spherical harmonics, the term E lo · µ (k) does not contribute, while the term E lo · µ (j) gives Π III (t) or = i 9 3 E lo µ (j) Finally, we evaluate the last term Π IV (t) or . It can be written as Π IV (t) or = i 3 1 6 √ 2π k =j j,k=A,B × dΩ (j) 2 dΩ (j) 1 dΩ (j) 0 dΩ (k) 1 dΩ (k) 0 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 E lo · µ × e −D (j) J 2 (t−τ 3 ) ℓ,m Y ℓ,m Ω (j) 1 Y ⋆ ℓ,m Ω (j) 2 Y 1,0 Ω (j) 1 G (t − τ 3 )L v (τ 3 ) × e −D (k) J 2 (t−τ 2 ) ℓ ′ ,m ′ Y ℓ ′ ,m ′ Ω (k) 0 Y ⋆ ℓ ′ ,m ′ Ω (k) 1 Y 1,0 Ω (k) 0 G (τ 3 − τ 2 )L v (τ 2 ) × e −D (j) J 2 (τ 3 −τ 1 ) ℓ ′′ ,m ′′ Y ℓ ′′ m ′′ Ω (j) 0 Y ⋆ ℓ ′′ ,m ′′ Ω (j) 1 Y 1,0 Ω (j) 0 G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.29) Like in the case of calculating Π III (t) or , we integrate over Ω Π IV (t) or = i 3 1 6 √ 2π k =j j,k=A,B dΩ (j) 2 dΩ (k) 1 t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 E lo · µ × 1 √ 4π Y ⋆ 0,0 Ω (j) 2 + 1 √ 5π e −6D (j) (t−τ 3 ) Y ⋆ 2,0 Ω (j) 2 G (t − τ 3 )L v (τ 3 ) e −2D (k) (t−τ 2 ) × Y ⋆ 1,0 Ω (k) 1 G (τ 3 − τ 2 )L v (τ 2 ) e −2D (j) (τ 3 −τ 1 ) G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.30) Here, only E lo · µ (k) contributes. Integrating over Ω (k) 1 and Ω (j) 2 we have Π IV (t) or = i 9 3 E lo µ (k) k =j j,k=A,B t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 G (t − τ 3 )L v (τ 3 ) e −2D (k) (t−τ 2 ) × G (τ 3 − τ 2 )L v (τ 2 ) e −2D (j) (τ 3 −τ 1 ) G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.31) Equations (II.19), (II.25), (II.28), and (II.31) conclude the influence of the dipolar orientational average when an energy transfer process occurs between two molecules embedded in a liquid solvent. In the following, these results will be applied to emphasize the peculiar role of the polarization on vibrational energy transfer. III. VIBRATIONAL MOLECULAR MODEL AND DYNAMICS In the model used in the present work, the material system interacting with the nonlinear optical fields is made of two molecules undergoing vibrational energy transfer. The vibrational states involved may be two anharmonic vibration modes of the donor and acceptor molecules, respectively. They may also be vibrational combination states of individual molecules. For simplicity, the first case will be considered. Thus, the energy transfer takes place between two anharmonic vibration modes Q A and Q B , pertaining to molecules A and B, respectively. The corresponding vibrational Hamiltonian can be expressed as H = i=A,B Ω i B + i B i + 1 2 + W B + i 3 , B 3 i , . . . + N α=1 ω α b + α b α + 1 2 + i ∆ i,1···N B + i b 1 · · · b N + ∆ ⋆ i,1···N B i b + 1 · · · b + N (III.1) where B + i and B i stand for the creation and annihilation operators of the molecular vibrational modes and b + i and b i for the corresponding operators of the bath modes with ∆ i,1···N and ∆ ⋆ i,1···N their corresponding coupling constants. The energy level scheme of this reduced molecular space is shown in Fig. 1 where all γ αβγδ are the transition rate constants assuming a Markovian bath. Subsequently, the set of levels required for the description of the two-molecule system are defined in Fig. 2. The possible decays between these two-molecule combination states are also presented with the wavy arrows. Specifically, the couplings induced by the interaction Hamiltonian V (t) are represented by the diagram shown in Fig. 3. |g a Molecule A |e a |v a |g b Molecule B |e b |v b µ vaea γ eaeavava µ eaga γ gagaeaea µ v b e b γ e b e b v b v b µ e b g b γ g b g b e b e b γ eav b eav b vae b vae b γ vae b vae b eav b eav b γ gae b gae b eag b eag b γ eag b eag b gae b gae b It is worthy of noting that those coupling depicted in Fig. 3 correspond to all of the transitions drawn in Fig. 2 except that the transitions between states 2, 3 and between 7, 8 are not included, and that in Fig. 2 the couplings are bidrectional between initial and final states, while those decays are unidirectional. Then, all the pathways participating in the In practice, the integrands I I→IV (t) of the heterodyne signal Π I→IV (t) or can be ex- pressed in a general form: |1 = |g a g b |2 = |g a e b |3 = |e a g b |4 = |g a v b |5 = |v a g b |6 = |e a e b |7 = |e a v b |8 = |v a e b |9 = |v a v b|1 |2 |3 |4 |5 |6 |7 |8 |9 µ ga ea µ e b v b µ ea va µ e b v b µ ga ea µ g b e b µ e b v b µ ea va µ g b e b µ ga ea µ g b e b µ ea va I I→IV (t) = i 3 Q id (n, α, r, q, p) e K id (n,α,r,q,p)t e A id (n,α,r,q,p)τ 3 e B id (n,α,r,q)τ 2 e C id (n,α,r)τ 1 × Q or (n, β, r, q, p) e K or (n,β,r,q,p)t e A or (n,β,r,q,p)τ 3 e B or (n,β,r,q)τ 2 e C or (n,β,r)τ 1 (III.2) Several comments would be valuable. First, the superscripts I through IV do not appear in all of the constants A, B, C and K, Q. This is because a given pathway n contributes to only one among the four heterodyne signals. It suffices to specify the pathway n, without requiring any additional label. The constants Q id (n, α, r, q, p), K id (n, α, r, q, p), A id (n, α, r, q, p), B id (n, α, r, q), and C id (n, α, r) resulting from the internal dynamics are evaluated for the pathways listed in Supplement A and the results are shown in Supplement B. The other set of constants Q or n,α,r,q,p , K or n,α,r,q,p , A or n,α,r,q,p , B or n,α,r,q , and C or n,α,r are associated with the orientational average, and they are obtained with expressions (II.19), (II.25), (II.28) and (II.31). The results are given Table I. All these constants are listed in Supplement C according to the molecular sequences involved in the different pathways. Then, by introducing a new set of constants Q (n, α, β, r, q, p) = Q id (n, α, r, q, p) Q or (n, β, r, q, p) K (n, α, β, r, q, p) = K id (n, α, r, q, p) K or (n, β, r, q, p) A (n, α, β, r, q, p) = A id (n, α, r, q, p) A or (n, β, r, q, p) B (n, α, β, r, q) = B id (n, α, r, q) B or (n, β, r, q) C (n, α, β, r) = C id (n, α, r) C or (n, β, r) (III.3) Π I→IV (t) or j k β Q or n,α,r,q,p K or n,α,r,q,p A or n,α,r,q,p B or n,α,r,q C or n,α,r the formal expressions (III.2) can be rewritten as I I→IV (t) = i 3 Q (n, α, β, r, q, p) e K(n,α,β,r,q,p)t e A(n,α,β,r,q,p)τ 3 e B(n,α,β,r,q)τ 2 e C(n,α,β,r)τ 1 (III.4) Π I (t) or A 1 E lo µ (a) /9 −2D (a) 2D (a) −2D (a) 2D (a) 2 4E lo µ (a) /45 −2D (a) −4D (a) 4D (a) 2D (a) B 1 E lo µ (b) /9 −2D (b) 2D (b) −2D (b) 2D (b) 2 4E lo µ (b) /45 −2D (b) −4D (b) 4D (b) 2D (b) Π II (t) or A B 1 E lo µ (b) /9 −2D (b) 2D (b) −2D (a) 2D (a) B A 1 E lo µ (a) /9 −2D (a) 2D (a) −2D (b) 2D (b) Π III (t) or A B 1 E lo µ (a) /9 −2D (a) −2D (b) 2D (b) 2D (a) B A 1 E lo µ (b) /9 −2D (b) −2D (a) 2D (a) 2D (b) Π IV (t) or A B 1 E lo µ (b) /9 −2D (b) −2D (a) 2D (b) 2D (a) B A 1 E lo µ (a) /9 −2D (a) −2D (b) 2D (a) 2D (b) Besides, a complete evaluation of these constants requires the description of the free population evolutions satisfying the equation dρ (t) /dt = −Γρ (t). These populations are evaluated through diagonalizing the damping Liouvillian Γ associated with the model introduced in Fig. 1. It consists of diagonal elements Γ mmmm except that for the ground state Γ 1111 = 0, together with the Γ mmnn terms presented in Fig. 2. Γ =                         0 Γ 1122 Γ 1133 0 0 0 0 0 0 0 Γ 2222 Γ 2233 Γ 2244 0 Γ 2266 0 0 0 0 Γ 3322 Γ 3333 0 Γ 3355 Γ 3366 0 0 0 0 0 0 Γ 4444 0 0 Γ 4477 0 0 0 0 0 0 Γ 5555 0 0 Γ 5588 0 0 0 0 0 0 Γ 6666 Γ 6677 Γ 6688 0 0 0 0 0 0 0 Γ 7777 Γ 7788 Γ 7799 0 0 0 0 0 0 Γ 8877 Γ 8888 Γ 8899 0 0 0 0 0 0 0 0 Γ 9999                         (III.5) From the sum rule Γ mmmm = − n =m Γ nnmm , the various total decay rates can be rewritten in terms of the individual molecular constants shown in Table II. Table III, for brevity. Finally, the energy transfer constants satisfy detailed balance so that Γ 2222 = γ e b e b e b e b Γ 3333 = γ eaeaeaea Γ 4444 = γ v b v b v b v b Γ 5555 = γ vavavava Γ 6666 = γ eaeaeaea + γ e b e b e b e b Γ 7777 = γ eaeaeaea + γ v b v b v b v b Γ 8888 = γ vavavava + γ e b e b e b e b Γ 9999 = Γ vavavava + γ v b v b v b v bΓ 1122 = γ g b g b e b e b Γ 1133 = γ gagaeaea Γ 2233 = γ gae b gae b eag b eag b Γ 2244 = γ e b e b v b v b Γ 2266 = γ gagaeaea Γ 3322 = γ eag b eag b gae b gae b Γ 3355 = γ eaeavava Γ 3366 = γ g b g b e b e b Γ 4477 = γ gagaeaea Γ 5588 = γ g b g b e b e b Γ 6677 = γ e b e b v b v b Γ 6688 = γ eaeavava Γ 7788 = γ eav b eav b vae b vae b Γ 7799 = γ eaeavava Γ 8877 = γ vae b vae b eav b eav b Γ 8899 = γ e b e b v b v bγ mmnn = γ nnmm exp [(E n − E m ) /kT ] if m = e a g b , n = g a e b γ ppqq = γ qqpp exp [(E q − E p ) /kT ] if p = v a e b , q = e a v b . (III. 6) and are related to the individual rates constants as expressed in Table IV. With the parameters given in Table I through IV, we have all that we need for performing the numerical simulations to illustrate the influence of the polarization on the 2DIR spectra, starting from our analytical results. γ vavavava = −γ eav b eav b vae b vae b − γ eaeavava γ v b v b v b v b = −γ vae b vae b eav b eav b − γ e b v b v b v b γ eaeaeaea = −γ gae b gae b eag b eag b − γ gagaeaea γ e b e b e b e b = −γ eag b eag b gae b gae b − γ g b g b e b e b γ gagagaga = γ g b g b g b g b = 0 IV. NUMERICAL SIMULATIONS AND QUANTITATIVE ANALYSIS OF THE POLARIZATION EFFECTS In this section, we present numerical simulations that illustrate the influence of the polarization on the 2DIR molecular spectra of a system undergoing vibrational energy transfer. Tables II, III and IV. All of the dephasing constants are set to γ pd = 3 cm −1 . Finally, in Table V, we summarized the expressions of the dipole moments of the total molecular system in terms of the dipole moments of the individual molecules. Notice that all the other matrix elements not indicated in Table V cancel. The simulations presented below enable us to discuss the polarization effects, resulting from the rotational diffusion of donor and acceptor of an energy transfer process, on the 2DIR spectra. We mainly focus our simulations on the frequency range related to the vibrational energy transfer between the two lowest vibrational excited states. The first set of simulations are shown in Fig. 4. The three panels in the first row correspond to the cases where the laser field a for the rephasing signal or b for the non-rephasing signal act as the first interaction. The second and third rows are dedicated to cases where they act as the second or the third interactions, respectively. As indicated inside the panels, the three panels in the same row correspond to increasing values of the rotational diffusion constant of the molecule A, from left to right. Comparing the panels in the same row, a Next, in Fig.5, we present the 2DIR spectra for increasing waiting times, ranging from 2 to 40 picoseconds. A global decrease of the diagonal and cross peaks with increasing waiting time is observed, as expected. In these four simulation, the rotational diffusion constants are fixed. Therefore, the stronger damping observed in the upper diagonal and cross peaks can only be attributed to the vibrational energy transfer which is more efficient for transition from molecule A to B than that from B to A. The next thing to further investigate is the dependence of the 2DIR spectra on the rotational diffusion constants of molecules A and B. In Fig. 6, we present the simulated Fig. 9. This time, the heights of the lower peaks in Fig. 8 decrease much more rapidly with increasing value of D b than the heights of the upper peaks do. A crossing over of the relative heights can be seen. V. CONCLUSION In this work, we have attempted to tease out the influence of the polarization effects on intermolecular vibrational energy transfer. In the case of intermolecular processes monitored with 2DIR spectroscopy, the donor and acceptor molecules usually have different rotational indicate the directions ± k j with j = a, b, c. In the following tables, the 66 pathways were separated into three tables, according to whether the field a or b acts in the first, the second and the third interaction terms. Both rephasing and non-rephasing geometries were considered. Since each pathway contributes to a specific term among those heterodyne signals Π X,IJK (t) or , in the last column, the parameters X and IJK corresponding to that pathway are shown. Notice that X runs from I to IV while the molecular interaction sequences IJK belongs to the set {AAA, AAB, ABA, · · · , BBB, BBA, BAB, · · · }. In all three tables, the second column is the index of the density matrix element of the final sate of the specific pathway. The ninth column is the index of the density matrix element of the initial state. In all of the pathways studied, the initial states are the same state. In column 8, 6 and 4, system-laser interactions probably change the population or the coherence of the system. In tional energy transfer with the field a or b acting as the first interaction term. In columns 5 through 9, an empty cell means that its value is the same as the last non-empty cell above it in the same column. Path Table S.IV equal iω 12 + Γ 1212 + iω a , while in Table S.V all of the pathways have Path ρ (t) G (t − τ 3 ) L v[p] (τ 3 ) G (τ 3 − τ 2 ) L v[q] (τ 2 ) G (τ 2 − τ 1 ) L v[a,b] (τ 1 ) ρ (t 0 ) Mol. dipoles X,IJK 1 21 2121 2122 (+) 2222 2212 (+) 1212 1211 (−) 11 µ e b g b µ e b g b µ g b e b I,BBB 2 42 4242 4222 (+) µ v b e b µ e b g b µ g b e b I,BBB 3 62 6262 6222 (+) µ eaga µ e b g b µ g b e b II,ABB 4 31 3131 3133 (+) 3322 µ eaga µ e b g b µ g b e b II,ABB 5 63 6363 6333 (+) µ e b g b µ e b g b µ g b e b I,BBB 6 53 5353 5333 (+) µ vaea µ e b g b µ g b e b II,ABB 7 21 2121 2111 (+) 1122 µ e b g b µ e b g b µ g b e b I,BBB 8 31 3131 3111 (+) µ eaga µ e b g b µ g b e b II,ρ (t) G (t − τ 3 ) L v[p] (τ 3 ) G (τ 3 − τ 2 ) L v[a,b] (τ 2 ) G (τ 2 − τ 1 ) L v[r] (τ 1 ) ρ (t 0 ) Mol. dipoles X,Path ρ (t) G (t − τ 3 ) L v[a,b] (τ 3 ) G (τ 3 − τ 2 ) L v[q] (τ 2 ) G (τ 2 − τ 1 ) L v[r] (τ 1 ) ρ (t 0 ) Mol. dipoles X,C id = iω 13 + Γ 1313 + iω a . By separating the pathways this way, it can be seen that A id and B id values of the pathways in the same table also share common terms. In the tables, we explicitly take out the common terms among these values to make the table more concise. These can be easily seen in the tables themselves. The third through sixth tables of this section are compiled in the same manner, which will be apparent by comparing them to the explanation given to Tables S.IV and S.V in the previous paragraph. Table S.VI and S.VII are related to pathways in Table S.II, while Table S.VIII and Table S.IX correspond to pathways in Table S.III. TABLE S.IV: The constants contributing to the calculation of the integrands of pathways, as explained at the beginning of this section. In this table, all of the pathways have C id (n, α, r) = iω 12 + Γ 1212 + iω a . All of the B id (n, α, r, q) constants have common terms −iω q − Γ 1212 , and all of the A id (n, α, r, q, p) constant have a common term −iω p . n Q id (n, α, r, q, p) K id (n, α, r, q, p) A id (n, α, r, q, p) + iω p B id (n, α, r, q) + iω q + Γ 1212 C id (n, α, r) = iω 13 + Γ 1313 + iω a . All of the B id (n, α, r, q) constants have common terms −iω q − Γ 1313 , and all of the A id (n, α, r, q, p) constant have a common term −iω p . n Q id (n, α, r, q, p) K id (n, α, r, q, p) A id (n, α, r, q, p) + iω p B id (n, α, r, q) + iω q + Γ 1313 n Q id (n, α, r, q, p) K id (n, α, r, q, p) A id (n, α, r, q, p) + iω p B id (n, α, r, q) − iω a + Γ 2121 have C id (n, α, r) = iω 31 + Γ 3131 − iω r . All of the B id (n, α, r, q) constants have common terms iω a − Γ 3131 , and all of the A id (n, α, r, q, p) constant have a common term −iω p . n Q id (n, α, r, q, p) K id (n, α, r, q, p) A id (n, α, r, q, p) + iω p B id (n, α, r, q) + iω q + Γ 3131 have C id (n, α, r) = iω 21 + Γ 2121 − iω r . All of the B id (n, α, r, q) constants have common terms −iω q − Γ 2121 , and all of the A id (n, α, r, q, p) constant have a common term iω a . 1 −W 2222 α Ξ pqa e b g b ,e b g b ,g b e b −iω 21 − Γ 2121 iω 21 + Γ 2121 + r α −r α − iω 12 2 W 2222 α Ξ pqa v b e b ,e b g b ,g b e b −iω 42 − Γ 4242 iω 42 + Γ 4242 + r α −r α − iω 12 3 W 2222 α Ξ pqa eaga,e b g b ,g b e b −iω 62 − Γ 6262 iω 62 + Γ 6262 + r α −r α − iω 12 4 −W 3322 α Ξ pqa eaga,e b g b ,g b e b −iω 31 − Γ 3131 iω 31 + Γ 3131 + r α −r α − iω 12 5 W 3322 α Ξ pqa e b g b ,e b g b ,g b e b −iω 63 − Γ 6363 iω 63 + Γ 6363 + r α −r α − iω 12 6 W 3322 α Ξ pqa vaea,e b g b ,g b e b −iω 53 − Γ 5353 iω 53 + Γ 5353 + r α −r α − iω 12 7 W 1122 α Ξ pqa e b g b ,e b g b ,g b e b −iω 21 − Γ 2121 iω 21 + Γ 2121 + r α −r α − iω 12 8 W 1122 α Ξ pqa eaga,e b g b ,g b e b −iω 31 − Γ 3131 iω 31 + Γ 3131 + r α −r α − iω 12 9 −Ξ pqa e b g b ,eaga,g b e b −iω 31 − Γ 3131 iω 21 + Γ 3131 − Γ 3232 iω 31 + Γ 3232 10 Ξ pqa e b g b ,eaga,g b e b −iω 62 − Γ 6262 iω 63 + Γ 6262 − Γ 3232 iω 31 + Γ 3232 11 Ξ pqa vaea,eaga,g b e b −iω 52 − Γ 5252 iω 53 + Γ 5252 − Γ 3232 iω 31 + Γ 3232 12 −Ξ pqa e b g b ,e b g b ,g b e b −iω 21 − Γ 2121 iω 21 + Γ 2121 −iω 12 13 −Ξ pqa eagae b g b g b e b −iω 31 − Γ 3131 iω 31 + Γ 3131 −iω 1214 −W 3333 α Ξ pqa eaga,eaga,gaea −iω 31 − Γ 3131 iω 31 + Γ 3131 + r α −r α − iω 13 15 W 3333 α Ξ pqa e b g b ,eaga,gaea −iω 63 − Γ 6363 iω 63 + Γ 6363 + r α −r α − iω 13 16 W 3333 α Ξ pqa vaea,eaga,gaea −iω 53 − Γ 5353 iω 53 + Γ 5353 + r α −r α − iω 13 17 −W 2233 α Ξ pqa e b g b ,eaga,gaea −iω 21 − Γ 2121 iω 21 + Γ 2121 + r α −r α − iω 13 18 W 2233 α Ξ pqa v b e b ,eaga,gaea −iω 42 − Γ 4242 iω 42 + Γ 4242 + r α −r α − iω 13 19 W 2233 α Ξ pqa eaga,eaga,gaea −iω 62 − Γ 6262 iω 62 + Γ 6262 + r α −r α − iω 13 20 W 1133 α Ξ pqa e b g b ,eaga,gaea −iω 21 − Γ 2121 iω 21 + Γ 2121 + r α −r α − iω 13 21 W 1133 α Ξ pqa eaga,eaga,gaea −iω 31 − Γ 3131 iω 31 + Γ 3131 + r α −r α − iω 13 22 −Ξ pqa e b g b ,27 −Ξ par eaga,gaea,e b g b −iω 21 − Γ 2121 iω 31 + Γ 2121 − Γ 2323 iω 13 + Γ 2323 28 Ξ par v b e b ,gaea,e b g b −iω 43 − Γ 4343 iω 42 + Γ 4343 − Γ 2323 iω 13 + Γ 2323 29 Ξ par eaga,gaea,e b g b −iω 63 − Γ 6363 iω 62 + Γ 6363 − Γ 2323 iω 13 + Γ 2323 30 −W 2222 α Ξ par e b g b ,g b e b ,e b g b −iω 21 − Γ 2121 iω 21 + Γ 2121 + r α −r α − iω 21 31 W 2222 α Ξ par v b e b ,g b e b ,e b g b −iω 42 − Γ 4242 iω 42 + Γ 4242 + r α −r α − iω 21 32 W 2222 α Ξ par eaga,g b e b ,e b g b −iω 62 − Γ 6262 iω 62 + Γ 6262 + r α −r α − iω 21 33 −W 3322 α Ξ par eaga,g b e b ,e b g b −iω 31 − Γ 3131 iω 31 + Γ 3131 + r α −r α − iω 21 34 W 3322 α Ξ par e b g b ,g b e b ,e b g b −iω 63 − Γ 6363 iω 63 + Γ 6363 + r α −r α − iω 21 35 W 3322 α Ξ par vaea,g b e b ,e b g b −iω 53 − Γ 5353 iω 53 + Γ 5353 + r α −r α − iω 21 36 W 1122 α Ξ par e b g b ,g b e b ,e b g b −iω 21 − Γ 2121 iω 21 + Γ 2121 + r α −r α − iω 21 37 W 1122 α Ξ par eaga,g b e b ,e b g b −iω 31 − Γ 3131 iω 31 + Γ 3131 + r α −r α − iω 21 38 −Ξ par e b g b ,g b e b ,e b g b −iω 21 − Γ 2121 iω 21 + Γ 2121 −iω 21 39 −Ξ par eaga,g b e b ,e b g b −iω 31 − Γ 3131 iω 31 + Γ 3131 −iω 21 n Q id (n, α, r, q, p) K id (n, α, r, q, p) A id (n, α, r, q, p) − iω a B id (n, α, r, q) + iω q + Γ 2121 53 −Ξ aqr C id (n, α, r) = iω 31 + Γ 3131 − iω r . All of the B id (n, α, r, q) constants have common terms −iω q − Γ 3131 , and all of the A id (n, α, r, q, p) constant have a common term iω a . e b v b ,v b e b , n Q id (n, α, r, q, p) K id (n, α, r, q, p) A id (n, α, r, q, p) − iω a B id (n, α, r, q) + iω q + Γ 3131 In Equation (III.2), there are constants accounting for the orientational motion of the molecules. In this supplement, these constants are presented. These constants are arranged into three tables. The grouping is exactly the same as in the first supplement. In other words, they are separated into the three tables according to whether the fields a or b act in the first, the second, or the third interaction term. However, in contrast with the constants presented in the previous section, these orientational constants depend on another parameter β, instead of α, and β may be 1 or 2 for certain pathways n, as explained in the text. Table S.II. n, β Q or n,β,r,q,p K or n,β,r,q,p A or n,β,r,q,p B or Table S.III. n, β Q or n,β,r,q,p K or n,β,r,q,p A or n,β,r,q,p B or the perturbative approach, only one molecule participates in the laser-molecule interaction at a given time and we have if j = k: . Next, taking advantage of the first relation in (II.18), we integrate over Ω (k) 1 . The following result is obtained: . Then, using the definitions and properties of the spherical harmonics we obtain the expression FIG. 1 . 1Molecular model used to study the polarization effects on the vibrational energy transfer process. Dipole moments µ as well as intramolecular and intermolecular transition rate constants γ are indicated in the figure. Notice that intermolecular transition constants involve simultaneous transitions in molecules A and B. FIG. 2 . 2The molecular model presented in the two-molecule combination state space. The wavy arrows are transitions due to intra and inter-molecular interactions, and they are represented by Γ mmnn type of transtion rate constants in the text. evaluation of the 2DIR spectrum and satisfying the rotating wave approximation are determined for the rephasing k re = − k a + k b + k c and the non rephasing k nre = − k b + k a + k c directions. They are listed in Supplement A according to whether laser field a or b acts in the first, the second or the third interaction. The contributions of individual pathways are calculated from the matrix elements of the evolution Liouvillians, interaction Liouvillians and the additional dipole propagators. In the previous section we have already shown how these contributions depend on the chronological ordering of the pulse-molecule interactions. FIG. 3 . 3Diagram of the couplings between the states of the total molecular system. The coupling are induced by the laser-molecule interaction. For clarity, the corresponding transition dipole moments are labeled. To this end, the properties of the local oscillator (LO) field involve only in heterodyne detection but not in the molecular and nonlinear optical processes should be kept constant and made as simple as possible. The phase Ψ of the LO field is chosen to be zero. Besides, other environmental factores and the properties of the other laser fields have definite influence on the simulated experimental results, but not on the nature of the effects we are investigating. Therefore, they are chosen to be rather realistic while suitable for demonstrating the effect under discussion. All laser field amplitudes are normalized to 1 and their pulse durations are equal to 55 fs. The temperature of the system is assumed to be T = 25 • C.The parameters of the molecular system are more substantial but most of them should also be fixed in order to simplify the discussion. The energy levels of the molecules areω ga = ω g b = 0, ω ea = 810 cm −1 , ω e b = 730 cm −1 , ω va = 1480 cm −1 and ω v b = 1380 cm −1 . (Figure 1 is plotted to scale.) The values of the relaxation and dephasing constants are also fixed. For the total decay rates we have chosen γ eaeaeaea = 4 cm −1 , γ vavavava = 6 cm −1 , γ e b e b e b e b = 3 cm −1 and γ v b v b v b v b = 5 cm −1 . The transition constants for vibrational energy transfer are chosen to be γ eav b eav b vae b vae b = 2.5 cm −1 , γ gae b gae b eag b eag b = 2.5 cm −1 and their reverse constants are deduced from Eqs. (III.6). All the other transition constants are straightforwardly obtained from the sum rules and can be evaluated by using 1 |µ| 2 = g b |µ| e b 1 |µ| 3 = g a |µ| e a 1 |µ| 6 = g a |µ| e a g b |e b + g a |e a g b |µ| e b 2 |µ| 3 = g a |µ| e a e b |g b + g a |e a e b |µ| g b 2 |µ| 4 = e b |µ| v b 2 |µ| 7 = g a |µ| e a e b |v b + g a |e a e b |µ| v b 2 |µ| 6 = g a |µ| e a 3 |µ| 5 = e a |µ| v a 3 |µ| 6 = g b |µ| e b 3 |µ| 8 = e a |µ| v a g b |e b + e a |v a g b |µ| e b 4 |µ| 6 = g a |µ| e a v b |e b + g a |e a v b |µ| e b 4 |µ| 7 = g a |µ| e a 5 |µ| 6 = v a |µ| e a g b |e b + v a |e a g b |µ| e b 5 |µ| 8 = g b |µ| e b 6 |µ| 7 = e b |µ| v b 6 |µ| 8 = e a |µ| va 6 |µ| 9 = e a |µ| v a e b |v b + e a |v a e b |µ| v b 7 |µ| 8 = e a |µ| v a v b |e b + e a |v a v b |µ| e b 7 |µ| 9 = e a |µ| v a 8 |µ| 9 = e b |µ| v b FIG. 4 . 4The simulated 2DIR spectra when the laser field a (rephasing term) or b (non-rephasing term) acts as the first, the second or the third interaction. They correspond to the first, the second or the third row of panels, respectively. The waiting time between the two last interacting laser pulses, defined after Eq. (II.5), corresponds to T = 10 ps. The frequencies of the exciting laser pulses are identical and correspond to ω a = ω b = ω c = 770 cm −1 .gradual smoothing of the peak structure can be observed, and this trend is more pronounced for the upper diagonal peak and cross peak. Since the effect is due to the increase of the rotational diffusion constant of molecule A, the upper peaks are more strongly influenced because the internal dynamics of molecule A plays a more relevant role in the building up of these peaks. This observation is also true for the other cases (row 2 and 3). In addition, in the third row, there is no diagonal peaks in all panels because only coherences contribute in this case when fields a or b act as the third interaction. This can be seen in Supplement A and B. Notice that the main axes of the contour lines of all the peaks in the panels in the upper row are rotated by π/2 with respect to those corresponding main axes in the second and third rows.FIG. 5. 2DIR spectra simulated with different waiting time T as marked inside each panel. The frequencies of the exciting laser pulses are fixed at ω a = ω b = ω c = 770 cm −1 to couple both transitions |g a → |e a and |g b → |e b . The rotational diffusion constants are equal to D a = D b = 1 rad 2 /ps. FIG. 6 . 6Two-dimensional spectra obtained for different rotational diffusion constants of molecule A. The waiting time corresponds to T=10 ps and the frequencies of the various laser pulses are chosen as ω a = ω b = ω c = 770 cm −1 . spectra with different D a values but with constant D b value. The values are marked inside each panel. When D a = 1, both diagonal peaks and both cross peaks are clearly observed. When D a is increased to an intermediate value, say 3 as in the middle panel, a significant decrease of the upper cross and diagonal peaks can be noticed. For even faster diffusion, say D a = 7 as in the right panel, these peaks are completely washed-out. To better visualize these variations, we plot the peak heights as a function of D a . From Fig. 7, we see that lower peaks (1) and (3) have similar dependence, while upper peaks (2) and (4) share another trend. The function forms of thetwo trends are not much different, although the upper peaks diminish with increasing D a more rapidly than do the lower peaks. Finally, in Fig. 8, analogous simulations with different values of D b and fixed value of D a are presented. Features similar to those of the previous case are observed, but the trends of the upper and lower peaks are reversed. The changes in the heights of the upper diagonal and cross peaks are much less pronounced among these cases, while the heights of the lower peaks are now significantly reduced with increasing D b . Obviously, this is because the internal dynamics of molecule B plays a more important role in the building up of the lower peaks and it has much weaker effects on the upper peaks. We also plot the peak heights as functions of D b in FIG. 7 . 7Dependence of the peak heights with the rotational diffusion constant D a . Peaks (1) and (3) exhibit similar variations and the same is true for peaks (2) and (4). FIG. 8. Simulated 2DIR spectra with different values of D b and a fixed value of D a . Other parameters used are identical to the ones used in the previous simulations. FIG. 9 . 9Dependence of the peak heights on the rotational diffusion constant D b . diffusion constants. Then, it is important to understand how the relative values of their rotational diffusion constants affect the resulting 2DIR spectra which supposed to help identify and characterize the intermolecular vibrational energy transfer. Variations of the peak heights reflect the interplay between the donor and acceptor molecules participating in the process.Supplement A: Pathways contributing to the 2DIR spectrumIn this supplement section, we list the pathways contributing to the 2DIR spectra of the molecules undergoing intermolecular vibrational energy transfer with heterodyne detection in the directions k s = − k a + k b + k c and k s = − k b + k a + k c . Only pathways satisfying the rotating wave approximation are retained. There are totally 66 of them. The superscripts (±) column 8 of row 1, for example, the index 1211 indicates that the photon interrupt the initial population in ground state to form a coherence between the ground state (|1 ) and one of the excited state (|2 ). The processes in column 3, 5 and 7 are propagations of the system itself. In the pathways studied in this work, these are either the sustain of the coherence (say the coherence 21, in column 7 of row 1 ofTable.S.I), or the decay of population (say the decay from population 22 to population 33 in column 5 of row 4 ofTable.S.I). Thus, in each pathway (row), the state (population or coherence) of the system propagates in time from column 9 to column 8, then to column 7, and so on, until column 2, following the change of the index of the density matrix element. Some of the cells are left empty, if the content of it is the same as the cell above it. (+) 3232 3212 (+) µ e b g b µ eaga µ g b e b IV,BAB 10 62 6262 6232 (+) µ e b g b µ eaga µ g b e b IV,BAB 11 52 5252 5232 (+) µ vaea µ eaga µ g b e b III,AAB12 21 2121 2111 (+) 1111 1112 (+)µ e b g b µ e b g b µ g b e b I,BBB 13 31 3131 3111 (+) µ eaga µ e b g b µ g b e b II,ABB 14 31 3131 3133 (+) 3333 3313 (+) 1313 1311 (−) µ eaga µ eaga µ gaea I,AAA 15 63 6363 6333 (+) µ e b g b µ eaga µ gaea II,BAA 16 53 5353 5333 (+) µ vaea µ eaga µ gaea I,AAA 17 21 2121 2122 (+) 2233 µ e b g b µ eaga µ gaea II,BAA 18 42 4242 4222 (+) µ v b e b µ eaga µ gaea II,BAA 19 62 6262 6222 (+) µ eaga µ eaga µ gaea I,AAA 20 21 2121 2111 (+) 1133 µ e b g b µ eaga µ gaea II,BAA 21 31 3131 3111 (+) µ eaga µ eaga µ gaea I,AAA 22 21 2121 2111 (+) 1111 1113 (+) µ e b g b µ eaga µ gaea II,BAA 23 31 3131 3111 (+) µ eaga µ eaga µ gaea I,AAA 24 21 2121 2123 (+) 2323 2313 (+) µ eaga µ e b g b µ gaea IV,ABA 25 43 4343 4323 (+) µ v b e b µ e b g b µ gaea III,BBA 26 63 6363 6323 (+) µ eaga µ e b g b µ gaea IV,ABA (+) 11 µ eaga µ gaea µ e b g b III,AAB 28 43 4343 4323(+) µ v b e b µ gaea µ e b g b IV,BAB 29 63 6363 6323(+) µ eaga µ gaea µ e b g b III,AAB 30 21 2121 2122 (+) 22222221 (−) µ e b g b µ g b e b µ e b g b I,BBB 31 42 4242 4222 (+) µ v b e b µ g b e b µ e b g b I,BBB 32 62 6262 6222 (+) µ eaga µ g b e b µ e b g b II,ABB 33 31 3131 3133 (+) 3322 µ eaga µ g b e b µ e b g b II,ABB 34 63 6363 6333 (+) µ e b g b µ g b e b µ e b g b I,BBB 35 53 5353 5333 (+) µ vaea µ g b e b µ e b g b II,ABB 36 21 2121 2111 (+) 1122 µ e b g b µ g b e b µ e b g b I,BBB 37 31 3131 3111 (+) µ eaga µ g b e b µ e b g b II,ABB 38 21 2121 2111 (+) 1111 1121 (−) µ e b g b µ g b e b µ e b g b I,BBB 39 31 3131 3111 (+) µ eaga µ g b e b µ e b g b II,ABB40 31 3131 3133 (+) 3333 3331 (−) 3131 3111 (+) µ eaga µ gaea µ eaga I,AAA 41 63 6363 6333 (+) µ e b g b µ gaea µ eaga II,BAA 42 53 5353 5333 (+) µ vaea µ gaea µ eaga I,AAA 43 21 2121 2122 (+) 2233 µ e b g b µ gaea µ eaga II,BAA 44 42 4242 4222 (+) µ v b e b µ gaea µ eaga II,BAA 45 62 6262 6222 (+) µ eaga µ gaea µ eaga I,AAA 46 21 2121 2111 (+) 1133 µ e b g b µ gaea µ eaga II,BAA 47 31 3131 3111 (+) µ eaga µ gaea µ eaga I,AAA 48 31 3131 3132 (+) 3232 3231 (−) µ e b g b µ g b e b µ eaga III,BBA 49 62 6262 6232 (+) µ e b g b µ g b e b µ eaga III,BBA 50 52 5252 5232 (+) µ vaea µ g b e b µ eaga IV,ABA 51 21 2121 2111 (+) 1111 1131 (−) µ e b g b µ gaea µ eaga II,BAA 52 31 3131 3111 (+)µ eaga µ gaea µ eaga I,AAA (+) 2121 2111 (+) 11 µ e b v b µ v b e b µ e b g b I,BBB 54 42 4242 4241 (−) µ g b e b µ v b e b µ e b g b I,BBB 55 43 4343 4341 (−) µ gaea µ v b e b µ e b g b II,ABB 56 21 2121 2161 (−) 6161 6121 (+) µ gaea µ eaga µ e b g b III,AAB 57 31 3131 3161 (−) µ g b e b µ eaga µ e b g b IV,BAB 58 62 6262 6251 (−) µ g b e b µ eaga µ e b g b IV,BAB 59 63 6363 6361 (−) µ gaea µ eaga µ e b g b III,AAB 60 31 3131 3151 (−) 5151 5131 (+) 3131 3111 (+) µ eava µ vaea µ eaga I,AAA 61 52 5252 5251 (−) µ g b e b µ vaea µ eaga II,BAA 62 53 5353 5351 (−) µ gaea µ vaea µ eaga I,AAA 63 21 2121 2161 (−) 6161 6131(+) µ gaea µ e b g b µ eaga IV,ABA 64 31 3131 3161 (−) µ g b e b µ e b g b µ eaga III,BBA 65 62 6262 6261 (−) µ g b e b µ e b g b µ eaga III,BBA 66 63 6363 6361 (−) µ gaea µ e b g b µ eaga IV,ABASupplement B: Constants associated with the internal dynamicsIn Equation (III.2), the integrands of the Factors contributing to the individual pathways presented in Supplement A, and participating in the 2DIR spectra of the molecules undergoing the vibrational energy transfer, are expressed in terms of some constants. In this supplement, the constants accounting for the internal dynamical associated with the rephasing contribution are presented. The corresponding constants associated with the non-rephasing terms can be deduced straightforwardly. Notice that we have introduced the simplifying notation Ξ pqr ij,kl,mn = E p E q E r µ ij µ kl µ mn . For simplicity we have introduced the formal decomposition G(τ 3 − τ 2 ) = α W iijj α exp {r α (τ 3 − τ 2 )}. In the tables of this supplement, W iijj α and r α are used in the expressions.The expressions Q id , K id , A id , B id and C id depend on all or some of the parameters n, α, r, p and q, defined in the text. Except for Q id , these quantities are rather simply expressed in terms of molecular parameters. C id , for example, has very simple dependence on its parameters. Among the 66 pathways considered, there are only 6 different values of C id For brevity, these quantities are arranged into six separate tables. The first and second table contain pathways n = 1 through n = 26. They correspond to the pathways inTable S.I. Further, The first table contains pathways n = 1 through n = 13, and the second table contains pathways n = 14 through n = 26. The C id values of all 13 pathways in the e b g b −iω 21 − Γ 2121 iω 24 + Γ 2121 − Γ 4141 iω 42 + Γ 4141 54 Ξ aqr g b e b ,v b e b ,e b g b −iω 42 − Γ 4242 iω 12 + Γ 4242 − Γ 4141 iω 42 + Γ 4141 55 Ξ aqr gaea,v b e b ,e b g b −iω 43 − Γ 4343 iω 13 + Γ 4343 − Γ 4141 iω 42 + Γ 4141 56 −Ξ aqr gaea,eaga,e b g b −iω 21 − Γ 2121 iω 26 + Γ 2121 − Γ 6161 iω 62 + Γ 6161 57 −Ξ aqr g b e b ,eaga,e b g b −iω 31 − Γ 3131 iω 36 + Γ 3131 − Γ 6161 iω 62 + Γ 6161 58 Ξ aqr g b e b ,eaga,e b g b −iω 62 − Γ 6262 iω 12 + Γ 6262 − Γ 6161 iω 62 + Γ 6161 59 Ξ aqr gaea,eaga,e b g b −iω 63 − Γ 6363 iω 13 + Γ 6363 − Γ 6161iω 62 + Γ 6161 60 −Ξ aqr eava,vaea,eaga −iω 31 − Γ 3131 iω 35 + Γ 3131 − Γ 5151 iω 53 + Γ 5151 61 Ξ aqr g b e b ,vaea,eaga −iω 52 − Γ 5252 iω 12 + Γ 5252 − Γ 5151 iω 53 + Γ 5151 62 Ξ aqr gaea,vaea,eaga−iω 53 − Γ 5353 iω 13 + Γ 5353 − Γ 5151 iω 53 + Γ 5151 63 −Ξ aqr gaea,e b g b ,eaga −iω 21 − Γ 2121 iω 26 + Γ 2121 − Γ 6161 iω 63 + Γ 6161 64 −Ξ g b e b ,e b g b ,eaga −iω 31 − Γ 3131 iω 36 + Γ 3131 − Γ 6161 iω 63 + Γ 6161 65 Ξ aqr g b e b ,e b g b ,eaga −iω 62 − Γ 6262 iω 12 + Γ 6262 − Γ 6161 iω 63 + Γ 6161 66 Ξ aqr gaea,e b g b ,eaga −iω 63 − Γ 6363 iω 13 + Γ 6363 − Γ 6161 iω 63 + Γ 6161Supplement C: Constants associated with the orientational average TABLE I . IRotational constants participating in the dynamics, formally described by (III.2). TABLE II . IIExpressions of the decay rate constants of the total vibrational molecular system in terms of the individual molecular constants. still need the expressions for the transition constants in terms of the individual molecular constants. They are shown inTable III. Notice that the diagonal matrix elements inWe Eq. (III.5) are given in Table II. Besides, the vanishing matrix elements in Eq. (III.5) are not shown in TABLE III . IIIRelation between transition rate constants of the total system and transition rate constants of the individual molecules. TABLE IV . IVRelation between energy tranfer rate constants and transition rate constants for the individual molecules. TABLE V . VRelation between dipole moments of the total system and individual dipole moments. TABLE S . SI: Pathways participating in the 2DIR spectrum of the intermolecular vibra- TABLE S . SII: Pathways participating in the 2DIR spectrum of the intermolecular energy transfer process with the field a or b acting in the second interaction term. Other quantities are the same like in previous table. TABLE S . SIII: Pathways participating in the 2DIR spectrum of the intermolecular energy transfer process with the field a or b acting in the third interaction term. Other quantities are the same like in previous tables. TABLE S . SV: The constants contributing to the calculation of the integrands of pathways, as explained at the beginning of this section. In this table, all of the pathways have eaga,gaea eaga,e b g b ,gaea −iω 21 − Γ 2121 iω 31 + Γ 2121 − Γ 2323 iω 21 + Γ 2323 25 Ξ pqa v b e b ,e b g b ,gaea −iω 43 − Γ 4343 iω 42 + Γ 4343 − Γ 2323 iω 21 + Γ 2323 26 Ξ pqa eaga,e b g b ,gaea −iω 63 − Γ 6363 iω 62 + Γ 6363 − Γ 2323 iω 21 + Γ 2323−iω 21 − Γ 2121 iω 21 + Γ 2121 −iω 13 23 −Ξ pqa eaga,eaga,gaea −iω 31 − Γ 3131 iω 31 + Γ 3131 −iω 13 24 −Ξ pqa TABLE S . SVI: The constants contributing to the calculation of the integrands of pathways, as explained at the beginning of this section. In this table, all of the pathways have C id (n, α, r) = iω 21 + Γ 2121 − iω r . All of the B id (n, α, r, q) constants have common terms iω a − Γ 2121 , and all of the A id (n, α, r, q, p) constant have a common term −iω p . TABLE S . SVII: The constants contributing to the calculation of the integrands of pathways, as explained at the beginning of this section. In this table, all of the pathways 40 −W 3333 α Ξ par eaga,gaea,eaga −iω 31 − Γ 3131 iω 31 + Γ 3131 + r α −r α − iω 31 41 W 3333 α Ξ par e b g b ,gaea,eaga −iω 63 − Γ 6363 iω 63 + Γ 6363 + r α −r α − iω 31 42 W 3333 α Ξ par vaea,gaea,eaga −iω 53 − Γ 5353 iω 53 + Γ 5353 + r α −r α − iω 31 43 −W 2233 α Ξ par e b g b ,gaea,eaga −iω 21 − Γ 2121 iω 21 + Γ 2121 + r α −r α − iω 31 44 W 2233 α Ξ par v b e b ,gaea,eaga −iω 42 − Γ 4242 iω 42 + Γ 4242 + r α −r α − iω 31 45 W 2233 α Ξ par eaga,gaea,eaga −iω 62 − Γ 6262 iω 62 + Γ 6262 + r α −r α − iω 31 46 W 1133 α Ξ par e b g b ,gaea,eaga −iω 21 − Γ 2121 iω 21 + Γ 2121 + r α −r α − iω 31 47 W 1133 α Ξ par eaga,gaea,eaga −iω 31 − Γ 3131 iω 31 + Γ 3131 + r α −r α − iω 31 48 −Ξ par e b g b ,g b e b ,eaga −iω 31 − Γ 3131 iω 21 + Γ 3131 − Γ 3232 iω 12 + Γ 3232 49 Ξ par e b g b ,g b e b ,eaga −iω 62 − Γ 6262 iω 63 + Γ 6262 − Γ 3232 iω 12 + Γ 3232 50 Ξ par vaea,g b e b ,eaga −iω 52 − Γ 5252 iω 53 + Γ 5252 − Γ 3232 iω 12 + Γ 3232 51 −Ξ par e b g b ,gaea,eaga−iω 21 − Γ 2121 iω 21 + Γ 2121 −iω 31 52 −Ξ par eaga,gaea,eaga −iω 31 − Γ 3131 iω 31 + Γ 3131 −iω 31 TABLE S . SVIII: The constants contributing to the calculation of the integrands of pathways, as explained at the beginning of this section. In this table, all of the pathways TABLE S . SIX: The constants contributing to the calculation of the integrands of pathways, as explained at the beginning of this section. In this table, all of the pathways have TABLE S . SX: Orientational molecular constants for pathways inTable S.I. n, β Q or n,β,r,q,p K or n,β,r,q,p A or n,β,r,q,p B or TABLE S . SXI: Orientational molecular constants for pathways in TABLE S . SXII: Orientational molecular constants for pathways in k =j j,k=A,B t t 0 dτ 3 τ 3 t 0 dτ 2 τ 2 t 0 dτ 1 G (t − τ 3 )L v (τ 3 ) e −2D (k) (τ 3 −τ 2 ) × G (τ 3 − τ 2 )L v (τ 2 ) e −2D (j) (t−τ 1 ) G (τ 2 − τ 1 )L v (τ 1 ) ρ (t 0 ) (II.28) B. J. Berne and R. Pecora, Dynamic Light Scattering: with Applications to Chemistry, Biology, and Physics (Krieger, Malabar Fl. USA, 1990). P Hamm, M Zanni, Concepts and Methods of 2D Infrared Spectroscopy. 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{'abstract': 'In the present work, we analyze the influence of the polarization effects taking place during the course of a 2DIR spectroscopy experiment performed on a molecular system undergoing an intermolecular vibrational energy transfer process. When both donor and acceptor molecules participating in the vibrational energy transfer are embedded in a host solvent, they face rotational diffusion that strongly distorts the resulting 2DIR spectra. It could be expected that the difference between rotational diffusion constants will be of particular interest. For this purpose, the polarization effects are discussed according to the different orderings of the laser-molecule interactions. Next, we study the distortions of the spectra as a function of the rotational diffusion constants of the individual molecules. The knowledge of these polarization effects are relevant to the interpretation of the spectra. Finally, the conclusions reached in this work for a vibrational energy transfer are valid for any other type of third-order optical process performed on the same molecular system.', 'arxivid': '1708.05297', 'author': ['A A Villaeys albert.villaeys@wanadoo.fr \nUniversité de Strasbourg et Institut de Physique et Chimie des Matériaux de Strasbourg\nFrance\n\nResearch Center for Applied Sciences\nAcademia Sinica\n115TaipeiTaiwan\n', 'M Zouari \nFaculté des Sciences de Bizerte\nDépartement de Physique\nUniversité de Carthage\n7021ZarzounaTunisie\n\nElectronic\n', 'Kuo Kan Liang kkliang@sinica.edu.tw \nResearch Center for Applied Sciences\nAcademia Sinica\n115TaipeiTaiwan\n\nDepartment of Biochemical Science and Technology\nNational Taiwan University\n106TaipeiTaiwan\n\nElectronic\n'], 'authoraffiliation': ['Université de Strasbourg et Institut de Physique et Chimie des Matériaux de Strasbourg\nFrance', 'Research Center for Applied Sciences\nAcademia Sinica\n115TaipeiTaiwan', 'Faculté des Sciences de Bizerte\nDépartement de Physique\nUniversité de Carthage\n7021ZarzounaTunisie', 'Electronic', 'Research Center for Applied Sciences\nAcademia Sinica\n115TaipeiTaiwan', 'Department of Biochemical Science and Technology\nNational Taiwan University\n106TaipeiTaiwan', 'Electronic'], 'corpusid': 103596268, 'doi': '10.1002/jccs.202200518', 'github_urls': [], 'n_tokens_mistral': 29124, 'n_tokens_neox': 23433, 'n_words': 14505, 'pdfsha': '74bdbbb339451108cf78f95c09dd271758eb711d', 'pdfurls': ['https://arxiv.org/pdf/1708.05297v1.pdf'], 'title': ['Polarization effects on intermolecular vibrational energy transfer analyzed by 2DIR spectroscopy', 'Polarization effects on intermolecular vibrational energy transfer analyzed by 2DIR spectroscopy'], 'venue': []}

arxiv

On Least Action D-Branes arXiv:hep-th/9807161v3 29 Jan 1999 June 1998 Shmuel Elitzur elitzur@vms.huji.ac.il Racah Institute of Physics The Hebrew University 91904JerusalemIsrael Introduction Eliezer Rabinovici Racah Institute of Physics The Hebrew University 91904JerusalemIsrael CERN 1211Geneva 23Switzerland Gor Sarkissian Racah Institute of Physics The Hebrew University 91904JerusalemIsrael On Least Action D-Branes arXiv:hep-th/9807161v3 29 Jan 1999 June 19981 We discuss the effect of relevant boundary terms on the nature of branes. This is done for toroidal and orbifold compactifications of the bosonic string. Using the relative minimalization of the boundary entropy as a guiding principle, we uncover the more stable boundary conditions at different regions of moduli space. In some cases, Neumann boundary conditions dominate for small radii while Dirichlet boundary conditions dominate for large radii. The c = 1 and c = 2 moduli spaces are studied in some detail. The antisymmetric background field B is found to have a more limited role in the case of Dirichlet boundary conditions. This is due to some topological considerations. The results are subjected to T -duality tests and the special role of the points in moduli space fixed under T -duality is explained from least-action considerations. Introduction String theory has perturbatively a large moduli space. Various ideas have been put forward on how to remove at least part of this vast degeneracy. One of these ideas called for checking the infrared stability of all ground states [1]. A point in moduli space whose spectrum contains operators that are eventually relevant in the infrared seems to be unstable and ceases to be itself part of the moduli space. In space-time language this corresponds to the emergence of unstable directions in the effective low-energy potential. It is less clear how this infrared instability is actually resolved and which vacuum replaces the unstable one. On the one hand, string theories always have a vanishing overall Virasoro central charge c, while on the other hand the central charge in the perturbed unitary sector decreases [2]. Such issues can be addressed in the framework of so-called non-critical string backgrounds [3]. Recently, new sectors of the moduli space were uncovered; they contain different brane configurations, some of them related by T -duality. One can reexamine in this setting the issues of infrared stability, as well as the fate of the "false" backgrounds. One can also consider the outcome of the addition of some infrared relevant boundary perturbations. Such perturbations have the feature that they leave unchanged the total bulk central charge. It will turn out that they may change the p dimensionality of the Dp backgrounds. This occurs since, at some points in moduli space, boundary conditions are forced to change from Dirichlet to Neumann ones and vice versa, see for example [4]. The direction of flow is determined by the value of a boundary entropy g defined in [5]. It was shown in [6] that as the system flows from one conformal theory to the other, in the presence of a relevant boundary operator, the value of g decreases. This was exhibited to first order in conformal perturbation theory suggesting a so called g theorem similar to the c theorem [2]. This was studied in the presence of a Sine-Gordon boundary perturbation [7]; it was also pointed out [8,9] that, for the case of a string with Dirichlet boundary conditions, the boundary entropy can also be recognized as the target space Dp world-volume effective-action density. In this paper we will identify the appropriate Dp backgrounds for the bosonic string theory. This will be done in detail for the cases of c = 1 and c = 2 and will be studied also for general toroidal compactifications. In the absence of torsion background fields, Neumann boundary conditions will be found to dominate at the lower values of the target space volume, while Dirichlet boundary conditions dominate when the appropriate target-space radii are large. The fixed points under duality play the role of boundaries in moduli space between the different boundary conditions. The paper is organized as follows: In section 2 we review the definition of the boundary entropy and obtain its values for the full c = 1 moduli space. This includes the compactification of one coordinate on a circle of radius R, an orbifold of radius R as well as the value at the Ginsparg points [10]. This is done in the presence of a Wilson line. Special attention is given to the relation between the fixed points in the moduli space and placements of the Wilson line. We verify that the values of g satisfy known properties of the c = 1 moduli space such as T -duality and the coincidence between a string moving in the presence of a circle or an orbifold for a special value of their respective radii. A rather general derivation of g is presented for an orbifold compactification resulting from modding out the target manifold by some discrete finite symmetry group G, of order n G . The dominant boundary conditions at each point in moduli space are identified and the uncovered structure is motivated by energy considerations. In section 3 these issues are discussed for more general toroidal compactifications. It is found that g depends only on the components of the background torsion field B parallel to the Dp-brane. The topological reasons for that are discussed. In section 4 the results are subjected to T -duality checks. In section 5 the system with mixed Neumann and Dirichlet boundary conditions is studied. The fixed points of T -duality are shown to play the role of boundary points in moduli space. The structure is exemplified by a detailed description of the dominant boundary conditions in the case of the c = 2 moduli space. The boundary entropy for c=1 systems The moduli space of c = 1 compactifications includes a string propagating on a circle with radius R, an orbifold of radius R [11] and isolated Ginsparg points. In this section we consider an open bosonic string moving on these backgrounds and calculate the boundary entropy allowing also for Chan-Paton factors and for the presence of a boundary interaction term, the Wilson line, for both Dirichlet and Neumann boundary conditions. We first recall the manner in which the boundary entropy is defined in the general case and the way in which it is calculated. Definition of boundary entropy Let us consider a conformal field theory on the σ −τ strip, 0 ≤ σ ≤ π, periodic in the τ -direction with a period T . The manifold is an annulus with the modular parameter q ≡ exp(−2πiT ). Given certain boundary conditions on the boundaries of the annulus, labelled α and β, the partition function is: Z αβ = Tr exp(−2πiT H αβ ) ,(1) where H αβ is the Hamiltonian corresponding to these boundary conditions. This is the openstring channel. One may also calculate the partition function using the Hamiltonian acting in the σ-direction [12]. This will be the Hamiltonian H (P ) for the cylinder, which is related by the exponential mapping ζ = exp(−i(t + iσ)) to the Virasoro generators in the whole ζ-plane by H (P ) = L (P ) 0 + L 0 (P ) − c/12, where we have used the superscript to stress that they are not the same as the generators of the boundary Virasoro algebra. It was shown in [12], that to every boundary condition α, there corresponds a particular boundary state |α in the Hilbert space of the closed strings; this enables us to compute the partition function by the following formula: Z αβ = α| exp(−πiH (P ) /T )|β = α|(q 1/2 ) L (P ) 0 +L 0 (P ) −c/12 |β ,(2) whereq ≡ e −2πi/T . This is the closed-string tree channel. The boundary entropy for each boundary is defined by [5]: g α = 0|α .(3) The phases of |0 and |α can be chosen such that 0|α is real and positive for all boundary states |α . In the path integrand language, g α is the value of the disc diagramm satisfying α type boundary condition. It was also shown in [6] that, at least in conformal perturbation theory, the value of g always decreases with the flow of the renormalization group. The equality of (1) and (2) provides a convenient way to calculate g, as shown in the following. 2.2 Boundary entropy for the c = 1 string compactified on a circle Neumann boundary conditions The action describing the bosonic d = 1 string with the Wilson line boundary interaction is [13]: S = 1 2π π 0 dσ dτ ∂ α X∂ α X + B iy B π B dX ,(4) where B labels boundaries and y B are the constant modes of the U(1) gauge potential coupling to the boundaries (and are periodic, with periods π/R). We assume that the boundaries carry also Chan-Paton factors whose index we choose to take two values, 1 and 2. Thus at the enhanced symmetry point we have a U(2) gauge symmetry, which is generically broken down to U(1) × U(1) by the Wilson line. Here we consider a world-sheet with two boundaries, the annulus diagram. In order to find the boundary entropy, the theory should be compared in two channels: the closed-string tree channel and the open-string loop channel. In the closed-string channel the first task is to find the boundary states |N i , with Chan-Paton factor i, which are found by imposing the corresponding boundary conditions. The boundary is located at τ = 0 and one has the usual condition of vanishing momentum flow: ∂ τ X(σ, 0) = P (σ, 0) = 0 .(5) Inserting the mode expansion: X i (σ, τ ) = x + 2wRσ + pτ R + i 2 n =0 1 n [α n e −2in(τ −σ) +α n e −2in(τ +σ) ] ,(6) where p and w are correspondingly integer momenta and winding numbers, we get: p = 0, α n = −α −n .(7) Taking into account the properties of coherent state and the U(1) modes y i we get for |N i : |N i = g N i w e −2iy i wR exp n>0 − α −nα−n n |0, w ,(8) where w is the integer winding number. We see that the normalization factor g N i = 0|N i gives us the boundary entropy. Inserting the expression for |N i and the closed string Hamiltonian H = p 2 4R 2 + w 2 R 2 + N +Ñ − 1 12 (9) in (2), we obtain for the partition function in the closed string channel: Z i = g 2 N i N i | exp −iπw 2 R 2 T exp −iπ T (N +Ñ − 1 12 ) |N i = g 2 N i η(q) w exp −iπw 2 R 2 T = g 2 N i η(q) θ 3 −R 2 T , 0 ,(10) where η(q) = q 1/24 ∞ n=1 (1 − q n ) is the Dedekind function, and θ 3 (τ, z) = ∞ n=−∞ exp(iπn 2 τ + 2inz) isX = x + pτ R + i n =0 1 n α n cos(nσ) exp(−inτ ) ,(11) where p is an integer. Inserting this in the open-string Hamiltonian, we obtain: H = p 2 2R 2 + N − 1 12 .(12) The partition function in this channel is : Z = 1 η(q) p exp −iπT p 2 R 2 = 1 η(q) θ 3 − T R 2 , 0 .(13) Equating (10) and (13) and using the properties of modular transformations: θ 3 1 τ , z = τ 1/2 e iτ z 2 /π θ 3 (τ, τ z) (14) η(q) = (−T ) 1/2 η(q) ,(15) we obtain: g 2 N i = R ,(16) independent of the Wilson line parameter y i . It can be explained by elaborating the boundary interaction in (4) [13]: B iy B π B dX = B 2iy B w B R ,(17) where w B is the winding number of the boundary. Since the boundary of a disc diagramm cannot have non-zero winding, g is independent of the Wilson line value y B . Dirichlet boundary conditions The boundary entropy for the open string with Dirichlet boundary condition is similary evaluated, starting again with the closed string channel. The boundary condition determining the boundary state is : X| τ =0 = y(18) leading to: w = 0, α n =α −n .(19) From these conditions, for the boundary state located at the point y we get |D y = g Dy δ(x − y) exp n>0 α −nα−n n |0 = g Dy p e −ipy R exp n>0 α −nα−n n p R , 0 . (20) Inserting this in (2) we have for the partition function in this channel: Z = g 2 Dy D y | exp −iπp 2 4R 2 T exp −iπ T (N +Ñ − 1 12 ) |D y = g 2 Dy η(q) p exp −iπp 2 4R 2 T = g 2 Dy η(q) θ 3 − 1 4T R 2 , 0 .(21) In order to analyse the open-string loop channel, according to (1), the Hamiltonian must be expressed with the Dirichlet boundary condition. Substituting the mode expansion of the coordinate X: X = y + 2wRσ + i n =0 1 n α n sin(nσ) exp(−inτ )(22) in the open-string Hamiltonian leads to: H = 2w 2 R 2 + N − 1 12 .(23) Finally, the partition function in this channel is: Z = 1 η(q) w exp −4iπT w 2 R 2 = 1 η(q) θ 3 (−4T R 2 , 0) .(24) Equating the partition functions in the two channels and using (14), one obtains: g 2 Dy = 1 2R .(25) Results (16) and (25) are consistent with T -duality. The boundary entropy is independent of the position of the brane y. Here it reflects the translation invariance of the disc diagramm. Boundary entropy on an orbifold In order to fully cover the c = 1 moduli space, we must study the effect of orbifolding on the boundary entropy. In general, the closed string moves on an orbifold, as a result of modding out the target manifold by some discrete, finite symmetry group G, of order n G . Into the original unmodded theory, we introduce open strings satisfying some boundary conditions. If these conditions are not invariant under the G, then to maintain the symmetry we must allow for all the different boundary conditions resulting from the original ones by the action of G. Generally, the original boundary conditions may be invariant under some subgroup H of G of order n H . Then all the n G /n H copies resulting from applying G on these conditions should also be allowed. In other words, we put into the original G-invariant target space a brane at a fixed point of H together with all its n G /n H mirror images under G, then identify all of them by modding out by G. A closed-string boundary state |B orb on this orbifold brane is of the form: |B orb = h∈H a h k∈G/H |KB h .(26) Here, B h is a closed-string state on the brane twisted by the element h of the symmetry group preserving the brane, and the sum over K runs over all the mirror images, thus projecting down to a G-invariant state. The coefficients a h are fixed by the theory. For h = 1, B 1 is the untwisted boundary state of the original unmodded theory. According to the definition, the boundary entropy corresponding to this brane is: g orb = 0|B orb = a 1 k∈G/H 0|KB 1 = (n G /n H )a 1 g ,(27) g being the boundary entropy of the unmodded model. In (27) we used the fact that differently twisted states are mutually orthogonal and that the closed string vacuum |0 is G-invariant. To determine the number a 1 we use again the equality between the two dual-channel representations of the annulus diagram whose two boundary components are on the brane. The closed string channel gives: Z orb = B orb |q H (P ) |B orb = h∈H |a h | 2 k,k ′ ∈G/H KB h |q H (P ) |K ′ B h ,(28)H (P ) being the closed string Hamiltonian. Since H (P ) is G-invariant KB h |q H (P ) |K ′ B h = B h |q H (P ) |K −1 K ′ B h ,(29) and (28) becomes: Z orb = n G /n H h∈H |a h | 2 k∈G/H B h |q H (P ) |KB h .(30) This should be equal to the open-string representation: Z orb = Tr q H P H = k∈G/H tr   q H k 1 n H h∈H h   .(31) The "Tr" in (31) means summing over all states, including all open-string twisted sectors, whereas the "tr" is the sum over states in one, say kth, of them, where H k is the Hamiltonian of an open string connecting the brane to its K mirror. The time evolution operator q H is multiplied by the projection onto the H invariant space, P H = 1 n H h∈H h. In fact each term in the sum over K and h in (30) and in (31) represents a different annulus diagram, and the equality between the closed-string representation, (30), and the open-string description, (31), should be valid term by term. In particular, for the term corresponding to h = 1, K = 1, we know from the unmodded model that: B 1 |q H (P ) |B 1 = tr(q H 1 ) ;(32) the coefficients of these terms in (30) and (31) should therefore be equal. This gives: |a 1 | 2 = 1 n G .(33) Equation (27) then gives the relation between g orb , the boundary entropy on the orbifold, and g, that of the unmodded model: g orb = n 1/2 G n H g .(34) Specializing to the case of a Z 2 orbifold of a c = 1 circle of the radius R, we get for the Dirichlet brane at the fixed point, n G = n H = 2, so that (34) gives: g orb = g D √ 2 = 1 2 √ R .(35) Similarly, for the Neumann boundary conditions with no variable Chan-Paton index and (necessarily) no Wilson line, n G = n H = 2; hence: g orb = g N √ 2 = R 2 .(36) For two D-branes at points x and −x on the circle, n G = 2 and n H = 1: g orb = √ 2g D = 1 √ R .(37) For the T -dual case of the Neumann boundary conditions with the Chan-Paton index taking two values and a non-zero Wilson line, n G = 2 and n H = 1: g orb = √ 2g N = √ 2R .(38) Let us note that, for the Neumann boundary condition, the entropy for the circle (16) at R = 1/2 √ 2 is the same as the entropy for orbifold (36) at R = 1/ √ 2, as it should, because these are exact values of R at which the orbifold and circle lines meet, in the closed-string moduli space. For the Dirichlet boundary condition, the entropies for circle (25) and orbifold (35) coincide at the T -dual points: √ 2 for the circle, and again 1/ √ 2 for the orbifold, because it is the self-dual point in this case. Similar considerations for the case of a Z 2 orbifold can be found in [14]. Ginsparg points At the self-dual radius R = 1/ √ 2 of the circle, the symmetry is enhanced to SU(2) R × SU (2) L . Adding open strings may break this enhanced symmetry to some lower symmetry [15,16]. One can consider modding the open-string theory by subgroups of this surviving symmetry. In the Neumann case the surviving subgroup is the diagonal SU(2) subgroup with generators: J a N = 1 2 (J a R + J a L ) ;(39) in the Dirichlet case, the generators of the surviving subgroup are, by duality: J a D = 1 2 (J a R − J a L ) ,(40) where J a R and J a L are the generators of SU(2) R and SU(2) L respectively. This can be checked in the following way. Consider for example the third component. The generators of the enhanced SU(2) × SU(2) group can be realized as: J 3 R (z) = − i √ 2 ∂x(z), J ± R = e ±i √ 2x(z) , (z, J R →z, J L ) .(41) We see that the third component J 3 N is just the momentum operator that annihilates the Neumann boundary state (8). In the Dirichlet case J 3 D is, by duality, a winding operator and, as such, annihilates the Dirichlet boundary state (20). Modding out the SU(2) symmetric theory, with either Neumann or Dirichlet boundary conditions (a different SU(2) for each case), by some discrete subgroup of order n, we have in eq. (34) n G = n H = n. Hence one has for the boundary entropy: g orb = g √ n . (42) Summary of the c = 1 results The following table is a collection of the c = 1 results for the boundary entropy. We can find, at every point of the moduli space, the boundary condition providing the least value of the entropy. This boundary condition will be preferable in the sense of boundary renormalization group. From the table, we see that for R > 2 −1/2 the entropy is smaller for Dirichlet boundary condition, and for R < 2 −1/2 it is smaller for Neumann. At the self-dual point and at the Ginsparg points, entropy is the same for both types of boundary conditions. All these results are visualized in fig. 1. The more stable (Neumann or Dirichlet) boundary condition at each region in the moduli space is indicated. The orbifold line is depicted here only for the case of the fixed point. The fact that g D ≤ g N for R greater than the radius of the self-dual point, is in accordance with its expected role as a stability parameter. Note that the self-dual radius for the general value of the string slope α ′ self-dual radius equals √ α ′ . If the c = 1 X-coordinate is compactified on a circle of radius R in one of the additional transverse d spatial coordinates with the Neumann boundary condition, the Dirichlet boundary condition on X corresponds to a d-brane while the Neumann boundary condition describes a (d + 1)-brane wrapped on the circle of radius R. The tension of a (d + 1)-brane is related to that of the d-brane by T d+1 = T d /2π √ α ′ [17]. The energy density in the transverse world is T d for the Dirichlet boundary condition, and 2πRT d+1 = T d R/ √ α ′ for the Neumann boundary condition on X. Thus for R larger than √ α ′ , the Dirichlet boundary condition on X gives less energy density in the transverse world than the Neumann boundary condition, and the Neumann vacuum may decay to the Dirichlet vacuum. For R smaller than √ α ′ , the Dirichlet boundary condition requires higher energy density than the Neumann boundary condition, and the Dirichlet vacuum could be unstable, in accordance with its higher value of g. t t t N N N D D D R c R o 1/ √ 2 √ 2 1/2 √ 2 1/ √ 2 1/ √ 2 • I • O • T Computation of the boundary entropy for Neumann and Dirichlet boundary conditions for the bosonic string compactified on a torus Description of a bosonic string compactified on a torus The world-sheet action describing the bosonic string moving in a toroidal background is [18]: S = 1 2π π 0 dσ dτ G ij ∂ α X i ∂ α X j + ε αβ B ij ∂ α X i ∂ β X j .(43) Here X i are dimensionless coordinates whose periodicities are chosen to be 2π, namely X i ≈ X i + 2πm i ; E is the matrix whose symmetric part is G and the antisymmetric part is B: E = G + B. The canonical momentum P i associated with X i is given by πP i = G ij ∂ τ X j + B ij ∂ σ X j ;(44) X i is given by : X i (σ, τ ) = x i + 2w i σ + τ G ij (p j − 2B jk w k ) + i 2 n =0 1 n [α i n (E)e −2in(τ −σ) +α i n (E)e −2in(τ +σ) ],(45) P i is given by: πP i (σ, τ ) = p i + i 2 n =0 [E t ij α j n (E)e −2in(τ −σ) + E ijα j n (E)e −2in(τ +σ) ] .(46) The coordinates X i are split into left and right modes: X i L = x i /2 + (w i + G ik p k /2 − G ij B jk w k )(τ + σ) + oscillators X i R = x i /2 + (−w i + G ik p k /2 − G ij B jk w k )(τ − σ) + oscillators ,(47) Whence: P i L = w i + G ik p k /2 − G ij B jk w k + oscillators P i R = −w i + G ik p k /2 − G ij B jk w k + oscillators (48) H = 1 2 (P 2 L + P 2 R ) (49) H = 1 4 [p i (G −1 ) ij p j + 4w i (G − BG −1 B) ij w j + 4w i B ik (G −1 ) kj p j ] + N +Ñ,(50) where N,Ñ are number operators: N = n>0 α i −n (E)G ij α j n (E),Ñ = n>0α i −n (E)G ijα j n (E)(51) and α,α satisfy the commutation relation: [α i n (E), α i m (E)] = [α i n (E),α i m (E)] = m(G −1 ) ij δ m+n,0 .(52) For the open-string case, there are few changes: a power of 2 does not appear in the exponent, the left and right modes are no longer independent, but constrained by the boundary conditions. In the closed-string channel, w i and p i are both integer-valued, whereas for the open string only one of them is an integer, depending on the boundary condition. The possible boundary conditions follow from variation of the action (43): δS = − 1 π D G ij ✷X j δX i + 1 π ∂D (G ij ∂ n X j + ε nβ B ij ∂ β X j )δX i ,(53) where D is the range of integration and n denotes the normal to the boundary. This leads to two kinds of boundary conditions: it is either G ij ∂ n X j + ε nβ B ij ∂ β X j = 0, the Neumann boundary condition, or δX j = 0, the Dirichlet boundary condition. Neumann boundary conditions The boundary entropy is again obtained by equating the partition function calculated in the closed-and open-string channel, respectively. The closed-string tree channel The boundary state |N is calculated by solving the various boundary constraints imposed by the boundary conditions. In the closed-string channel the boundary is located at τ = 0, and therefore: G ij ∂ τ X j + B ij ∂ σ X j = 0 .(54) From (44) we see that it is just a condition of vanishing momentum flow at the boundary, which should be imposed on the boundary state: P i |N = 0 .(55) From the mode expansion for the momenta (46), we obtain: p i = 0(56) for the zero-mode part and the constraint: E t ij α j n = −E ijα j −n(57) for the left-and right-handed oscillator modes. The oscillator part of |N is obtained by rewriting (57) in matrix form and solving for α n , to get: α n = −(E t ) −1 Eα −n ,(58) From this, |N is found to be: |N = g N w exp n>0 − α t −n G(E t ) −1 Eα −n n |0, w .(59) Finally, putting together (59) and the formula for the Hamiltonian (50) in (2), one arrives at the following expression for the partition function: Z = N| exp(−πiH (P ) /T )|N = g 2 N w e −iπ 1 T w t (G−BG −1 B)w η d (q) = g 2 N Θ(−(G − BG −1 B)/T, 0) η d (q) ,(60) where Θ(Ω, z) = nǫZ g exp(πi n t Ω n + 2πi n t z) is the theta function with the modular matrix Ω . The open-string loop channel In this channel, according to (1) we should compute the trace of the matrix density for the Hamiltonian H corresponding to the Neumann boundary conditions. Here the boundary is located at σ = 0, and therefore: G ij ∂ σ X j + B ij ∂ τ X j = 0 .(61) Substituting the mode expansion of X i (45) one gets the following constraint for the values of p i and w i in the zero-mode part: 2(G − BG −1 B)w = −BG −1 p =⇒ w = 1 2 (B − GB −1 G) −1 p .(62) Note that, in the open-string channel and in the presence of Neumann boundary conditions, the "winding" numbers need not be integers. This is unlike the situation in the closed-string channel. The momenta p i remain integer-valued. Substituting the value of w in the Hamiltonian, we find: H = 1 2 p t (G − BG −1 B) −1 p + N ,(63) leading to: Z = p e −iπT p t (G−BG −1 B) −1 p η d (q) = Θ(−(G − BG −1 B) −1 T ), 0) η d (q) .(64) Recalling the following modular transformation properties of Θ and η functions [19]: Θ(Ω −1 , 0) = (detΩ) 1/2 Θ(Ω, 0) η(q(−1/T )) = (−T ) 1/2 η(q(−T )) ,(65) equating (60) and (64), and using (65), we get: g N = (det(G − BG −1 B)) 1/4 = (detE) 1/2 (detG) 1/4 .(66) Dirichlet boundary conditions The closed-string tree channel In this channel the boundary is located at τ = 0, and we should therefore impose the Dirichlet boundary condition in the form: δX i | τ =0 = 0 .(67) Substituting here X i from (45) we get: w i = 0 α i n =α i −n .(68) Following the same reasoning as in the Neumann case, we find the boundary states |D to be: |D = g D p exp n>0 α t −n Gα −n n |p, 0 .(69) Substituting (69) in the formula for the partition function in the closed-string channel (2), and using for the Hamiltonian (50), we obtain: Z = D| exp(−πiH (P ) /T )|D = g 2 D p e −iπ 1 4T p t G −1 p η d (q) = g 2 D Θ(−G −1 /4T, 0) η d (q) .(70) The open-string loop channel In this channel the boundary is located at σ = 0 and the Dirichlet boundary condition is of the form: δX i | σ=0 = 0 .(71) Inserting X i from (45) to (71), we obtain the following constraint for the values of w i and p i : p j = 2B jk w k .(72) In the Dirichlet case the winding numbers are integer-valued while p i , as determined by (72), are not necessarily integers. This also follows from T -duality. Substituting (72) in the Hamiltonian gives: H = 2w i G ij w j + N(73) and : Z = w e −4iπT w t Gw η d (q) = Θ(−4GT, 0) η d (q) .(74) statement, from (59) and (69): |N = g N w exp n>0 − α t −n G(E t ) −1 Eα −n n |0, w |D = g D p exp n>0 α t −n Gα −n n |p, 0 .(81) We have already noted that g N → g D and w → p, and it is left to check that: − α t −n G(E t ) −1 Eα −n → α t −n Gα −n .(82) Using now (78) and (80), we verify that this indeed occurs: −α t −n G(E t ) −1 Eα −n → −(−2E t α n (E)) t 1 4 E −1 G(E t ) −1 (4E t ) 1 4 E −1 2Eα n (E) = α t −n Gα −n .(83) 5 Boundary entropy for the general case with an arbitrary number of Dirichlet and Neumann coordinates As a more general case we require the first k coordinates to satisfy the Dirichlet boundary conditions, and the last (d − k) coordinates to satisfy the Neumann ones. The boundary conditions are now expressed in the following form: δX r | τ =0 = 0 ⇒ w r = 0, α r n =α r −n , r = 1, . . . , k , P α = 0 ⇒ p α = 0, E t αj α j n + E αjα j −n = 0, α = k + 1, . . . , d .(84) Substituting the values of w and p found in (84) and (85) in (50), we get for zero-modes Hamiltonian: H = L t ΩL(86) where L is the vector: p r w α and Ω is the matrix: (G −1 ) rs /4 B αi G ir /2 B αi (G −1 ) is /2 G αβ − B αi (G −1 ) ij B jβ .(87) (As was mentioned, we mean here by i, j sums over all coordinates from 1 to d ). Following previous considerations, we find that the square of the entropy is now equal to the square root of the determinant of Ω: g 2 k = √ detΩ .(88) Let us analyse various properties of this expression. First of all, let us note that by multiplying each rth row from the first k rows by 2B rα , adding all of them to the αth row, and repeating the same procedure with the first k columns, we can completely eliminate all components of B containing at least one index belonging to the Dirichlet set: g 2 k = √ detΩ ′ ,(89) where Ω ′ is: Ω ′ = (G −1 ) rs /4 B αγ G γr /2 B αγ (G −1 ) γs /2 G αβ − B αδ (G −1 ) δγ B γβ .(90) We see that the entropy, or the vacuum degeneracy of the theory, depends only on components of B tangent to the corresponding D-brane. It is possible to explain this along the same lines as used for explaining the independence of the partition function on some B-terms in the pure Dirichlet case. g k is given by a path integral over disc diagrams embedded in the d torus with the boundary on the d − k dimensional D-brane. For any given embedded disc D in this path integrand, the boundary is some closed curve in the brane. Being the boundary of a disc D, this curve does not wrap any non-trivial cycle of the torus; therefore, it is also the boundary of some other disc, D ′ , which lies entirely inside the (d − k)-brane. The union of D and D ′ is then topologically a sphere embedded in the d-torus. The argument in the previous section implies that: D B + D ′ B = 0 .(91) The B dependence of every embedding in the path integral can be expressed in terms of a B integral over a disc D ′ lying entirely in the brane. This integral depends only on the component of B tangent to the brane. Let us check also here that all g k can be derived from one another by means of the product of transformations of the factorized dualities [20]: D i = I − e i e i /2 e i /2 I − e i ,(92) where I is a d-dimensional identity matrix, and e i is zero, except for the ii component, which is 1. Let us check how the g N that was found earlier transforms to g 1 under the D 1 transformation; the generalization to other cases will be straightforward. G − BG −1 B → a(G − BG −1 B)a t + a(BG −1 )b t − b(G −1 B)a t + bG −1 b t .(93) Noting that the right or left multiplication by any matrix on e 1 amounts to setting to zero all elements except the first row or column respectively, and right or left multiplication by I − e 1 reduces to setting to zero elements of the first row or column respectively, we arrive at the mentioned result. We turn to the analysis of the preferred type of boundary conditions at some interesting regions of the moduli space of toroidal compactification. Namely we want to find at every point in moduli space what type of boundary conditions provides the least value of the boundary entropy, and consequently which is preferable in the sense of the boundary renormalization group. To begin with, we consider the boundary entropy for the case of a diagonal metric and a vanishing background B-field. From (89) we see that here the squared entropy generally has the form: g 2 k = R k+1 · · · R d 2 k R 1 · · · R k ,(94) where R i is the radius of the dimension X i . We see that the type of the preferred boundary condition gets changed as the radius of some coordinate reaches the self-dual value. Finally to see the influence of the background B 12 on the value of the boundary entropy we consider the case of c = 2. For c = 2 we have three types of boundary conditions: 1. both coordinates satisfy Neumann conditions, 2. one, say the first, satisfies the Dirichlet boundary conditions, and the second the Neumann boundary conditions, 3. both satisfy the Dirichlet boundary conditions. From (89) we have the following expressions of entropy corresponding to these cases: g 2 N N = G 11 G 22 − G 2 12 + B 2 12 √ detG g 2 DN = G 22 4 √ detG g 2 DD = 1 4 √ detG .(95) For further analysis it is convenient to organize the four real data set of moduli space G 11 , G 22 , G 12 , B 12 into two complex coordinates ρ and τ , in the following manner [20]: τ ≡ τ 1 + iτ 2 = G 12 G 22 + i √ detG G 22 , ρ ≡ ρ 1 + iρ 2 = B 12 + i √ detG .(96) In these coordinates the expressions for the entropies take the forms: g 2 N N = ρ 2 1 + ρ 2 2 ρ 2 g 2 DN = 1 4τ 2 g 2 DD = 1 4ρ 2 .(97) It is interesting to note that in these coordinates all expressions depend only on three parameters ρ 1 , ρ 2 , τ 2 and are independent of τ 1 . Comparing values of the entropy for different boundary conditions gives us the following: g N N ≥ g DD if ρ 2 1 + ρ 2 2 ≥ 1 4 (98) g N N ≥ g DN if ρ 2 1 + ρ 2 2 ≥ ρ 2 4τ 2 (99) g DN ≥ g DD if ρ 2 ≥ τ 2 .(100) In order to visualize these inequalities, consider two cases separately. First when ρ 2 1 + ρ 2 2 ≥ 1/4, which means that we consider a point in the moduli space out of the cylinder ρ 2 1 + ρ 2 2 = 1/4, and the second when ρ 2 1 + ρ 2 2 ≤ 1/4, corresponding to a point inside the cylinder. In the first case, g N N ≥ g DD and in order to find the least value we should compare g DD with g DN . From (100) we see that in this region over the hyperplane ρ 2 = τ 2 the least value is provided by g DN and for the region below by g DD . In the second region, when the point is inside the cylinder, g N N ≤ g DD and we should compare g N N with g DN . From (99) we see that in this region over the hypersurface ρ 2 1 + ρ 2 2 = ρ 2 /4τ 2 the least value of the entropy is provided by g DN and for the region below it by g N N . Graphically all these results are depicted in fig. 2. 1 + ρ 2 2 = 1/4. The curved surface above is the hypersurface ρ 2 1 + ρ 2 2 = ρ 2 /4τ 2 . In the region between the hypersurface and the hyperplane the least value for the entropy is provided by g DN , between the cylinder and the hyperplane by g DD , and inside of the cylinder and below the hypersurface by g N N . the third theta function with the modular parameter τ . To calculate g N i one turns to the open string loop channel. The Hamiltonian should be computed with a given boundary condition. First consider the mode expansion for X. The mode expansion of the solution of the equation of motion is : Figure 1 : 1Map of the preferred boundary conditions in the c = 1 moduli space Figure 2 : 2Map of the prefered boundary conditions in c = 2 moduli space. The hyper plane ρ 2 = τ 2 is denoted in black. The grey figure below, marked by the vertical lines, is the cylinder ρ 2 By formula (2.49) in [20], we deduce element of the O(d, d, Z) group the matrix G − BG −1 B transforms as follows:that under any a b c d AcknowledgementsWe thank Christoph Schweigert for discussions. The work of E. R. is partially supported by the Israel Academy of Sciences and Humanities-Centers of Excellence Programme, and the American-Israel Bi-National Science Foundation. The work of S.E. is supported by the Israel Academy of Sciences and Humanities-Centers of Excellence Programme.Equating (70) and (74), and using the modular transformation properties (65), we extract the factor g D :We see that, in the case of the Dirichlet boundary condition, the partition function does not depend on the topological B-term. The explanation is as follows: for Dirichlet boundary conditions, the world-sheet disc has its boundary shrunk to a point, and it is therefore mapped into a sphere in target space. We may conclude that the map of the world-sheet into the target-space torus is an element of the π 2 (T ) group, which is known to be trivial. Hence, in target space any image of the world-sheet disc, that contributes to the path integral is a two-sphere, which is necessarily the boundary of some three-manifold. On such a sphere the integral of the closed two-form B vanishes and therefore the disc diagram is B-independent.T-duality relations between the Neumann and Dirichlet boundary conditionsWe next show that there exists a particular element of the T -duality group O(d, d, Z) mapping all results for the Neumann condition to the Dirichlet one, namely:Under the action of this element, the following transformations occur[20]: . A M Polyakov, JETP Lett. 12381A. M. Polyakov, JETP Lett. 12 (1970) 381 . 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Mumford, Tata lectures on Theta ( Birkhäuser, Boston, 1983) p.189 . A Giveon, M Porrati, E Rabinovici, Phys. Rep. 24477A. Giveon, M. Porrati and E. Rabinovici, Phys. Rep. 244 (1994) 77

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{'abstract': 'We discuss the effect of relevant boundary terms on the nature of branes. This is done for toroidal and orbifold compactifications of the bosonic string. Using the relative minimalization of the boundary entropy as a guiding principle, we uncover the more stable boundary conditions at different regions of moduli space. In some cases, Neumann boundary conditions dominate for small radii while Dirichlet boundary conditions dominate for large radii. The c = 1 and c = 2 moduli spaces are studied in some detail. The antisymmetric background field B is found to have a more limited role in the case of Dirichlet boundary conditions. This is due to some topological considerations. The results are subjected to T -duality tests and the special role of the points in moduli space fixed under T -duality is explained from least-action considerations.', 'arxivid': 'hep-th/9807161', 'author': ['Shmuel Elitzur elitzur@vms.huji.ac.il \nRacah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael\n\nIntroduction\n\n', 'Eliezer Rabinovici \nRacah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael\n\nCERN\n1211Geneva 23Switzerland\n', 'Gor Sarkissian \nRacah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael\n'], 'authoraffiliation': ['Racah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael', 'Introduction\n', 'Racah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael', 'CERN\n1211Geneva 23Switzerland', 'Racah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael'], 'corpusid': 14568840, 'doi': '10.1016/s0550-3213(98)00799-8', 'github_urls': [], 'n_tokens_mistral': 12893, 'n_tokens_neox': 11228, 'n_words': 7367, 'pdfsha': 'ccc1fc2d500d7e321717780ca9f35bc2b051b1b6', 'pdfurls': ['https://arxiv.org/pdf/hep-th/9807161v3.pdf'], 'title': ['On Least Action D-Branes', 'On Least Action D-Branes'], 'venue': []}

arxiv

IPHT T09/055 A MATRIX MODEL FOR SIMPLE HURWITZ NUMBERS, AND TOPOLOGICAL RECURSION 5 Jun 2009 Gaëtan Borot Bertrand Eynard ANDMotohico Mulase Brad Safnuk IPHT T09/055 A MATRIX MODEL FOR SIMPLE HURWITZ NUMBERS, AND TOPOLOGICAL RECURSION 5 Jun 2009 We introduce a new matrix model representation for the generating function of simple Hurwitz numbers. We calculate the spectral curve of the model and the associated symplectic invariants developed in[4]. As an application, we prove the conjecture proposed by Bouchard and Mariño [2], relating Hurwitz numbers to the spectral invariants of the Lambert curve e x = ye −y . Summary In [2], Bouchard and Mariño propose a new conjectural recursion formula to compute simple Hurwitz numbers, i.e. the weighted count of coverings of CP 1 with specified branching data. Their recursion is based on a new conjectured formalism for the type B topological string on mirrors of toric Calabi-Yau threefolds, called "remodeling the B-model", or "bkmp conjecture" [1]. The Bouchard-Mariño conjecture for Hurwitz numbers appears as a consequence of this general bkmp conjecture applied to the infinite framing limit of the open string theory of C 3 . In this limit, the amplitudes are known to give simple Hurwitz numbers. They propose that the generating function for Hurwitz numbers can be recovered from the symplectic invariants (also called topological recursion) developed in [4], applied to the so called "Lambert curve" y = L(e x ) defined by: e x = ye −y . In this paper, we make the link between Hurwitz numbers and the Lambert curve explicit. We introduce a new matrix model formula for the generating function of simple Hurwitz numbers (1) Z ∝ H N (C) dM exp − 1 g s Tr(V (M ) − M R) , where V (x) is the potential V (x) = − x 2 2 + g s (N − 1 2 )x + x ln(g s /t) + iπx − g s ln Γ(−x/g s ) . The parameters g s and the matrix R involved in the definition of Z are such that the weight of a covering of Euler characteristic χ is proportional to g χ s , and has a polynomial dependance in v i = exp R i which encodes the ramification data above branch points. A method to compute topological expansion of matrix integrals with an external field was introduced in [4]. It consists in finding the spectral curve S (roughly speaking the equilibrium density of eigenvalues of the matrix, more precisely the planar part of the expectation value of the resolvent), then computing recursively a sequence of algebraic k-forms W (g) k (S), and some related algebraic quantities called symplectic invariants F g (S) = W (g) 0 (S). Then, one of the main results of [4] is that ln Z = ∞ g=0 g 2g−2 s F g (S). In our case, this implies that the generating function for simple Hurwitz numbers of genus g is precisely F g (S), where S is the spectral curve of our matrix model. It is rather easy to find the spectral curve of the matrix model Eqn 1. The result, after suitable symplectic transformations, reads (2) S(p, g s ; t) = x(z) = −z + ln (z/t) + c 0 + c1 z − ∞ n=1 B2n g 2n s 2n f 2n (z) y(z) = z + g s N i=1 1 (z−zi)yi + 1 ziyi , where z i , y i , c 0 and c 1 are determined by consistency relations, and z i and y i are O(1) when g s → 0, and c 0 and c 1 are O(g s ). In particular, when we set the coupling constant g s = 0, we recover the Lambert curve y = L(te x ). In [2], Bouchard and Mariño define another set of generating functions, denoted H (g) (x 1 , . . . , x k ), encoding genus g simple Hurwitz numbers, and which are derivatives of ln Z, evaluated at g s = 0. The statement of their conjecture is: W (g) k (z 1 , . . . , z k ) dx(z 1 ) · · · dx(z k ) = H (g) (x(z 1 ), . . . , x(z k )), where W (g) k are the k-forms of [4] computed for the Lambert curve. We prove their conjecture by using the general properties of the invariants of [4], in particular the fact that derivatives of the F g 's with respect to almost any parameter, can be expressed in terms of the W (g) k 's. Then it suffices to set g s = 0, and this gives the Bouchard-Mariño conjecture. Organization of the paper. In Section 2, we recall the definitions and derive a matrix model formula for the generating function of simple Hurwitz numbers. We recall the construction of the symplectic invariants and topological recursion of [4] in Section 3. In Section 4, we derive the spectral curve of our matrix model (the proof is presented in Appendix 7) and prove the Bouchard-Mariño conjecture following a method very close to [6]. In section 5, we briefly study the link with the Kontsevich integral. Section 6 addresses generalizations of our method and open questions. Construction of the matrix model Let Cov * n (C 1 , . . . , C k ) denote the weighted number of n-fold coverings (possibly disconnected) of CP 1 ramified over k fixed points of CP 1 with monodromies in the conjugacy classes C 1 , . . . , C k . The weight is one over the order of the automorphism group of the covering. Similarly, let Cov n (C 1 , . . . , C k ) denote the weighted number of n-fold connected coverings. By a result of Burnside (see, eg [16]), we have (3) Cov * n (C 1 , . . . , C k ) = |λ|=n dim λ n! 2 k i=1 f λ (C i ), where the sum ranges over all partitions λ = (λ 1 ≥ λ 2 ≥ · · · ≥ λ n ≥ 0) of |λ| = λ i = n boxes, dim λ is the dimension of the irreducible representation indexed by λ (with corresponding character χ λ ), and 2.1. Simple Hurwitz numbers. We are interested in counting n-fold coverings of genus g with 1 branch point of arbitrary profile µ, and only transpositions above other points (called simple branch points), c.f. Figure 1. We denote C (2) the conjugacy class of a transposition. For b simple branch points and one branch point of profile µ, the Euler characteristic of the n−fold covering reads from the Riemann-Hurwitz formula: f λ (C i ) = |C i | dim λ χ λ (C i ).χ = |µ| + (µ) − b. For connected coverings, we have χ = 2 − 2g, where g is the genus, and we also define the simple Hurwitz numbers: H g,µ = Cov n (C µ , b C (2) , . . . , C (2) ), where b = 2g − 2 + |µ| + (µ). 2.2. Generating function for simple Hurwitz numbers. With the notations p µ = i p µi , and p = (p 1 , p 2 , . . .), we shall study the generating function (4) Z (p, g s ; t) = ∞ n=0 t n |µ|=n ∞ b=0 g b−|µ|− (µ) s b! p µ Cov * n (C µ , b C (2) , . . . , C (2) ), where Cov * n (C µ , C (2) , . . . , C (2) ) is the number of (not necessarily connected) branched coverings of Euler characteristic χ = |µ| + (µ) − b. In the language of string theory, −g s is the string coupling constant 1 . Let us emphasize that Z(p, g s ; t) is defined as a formal power series in t and g s , i.e. it is merely a notation to collect all the coefficients. Each coefficient (for b and n fixed) is a finite sum, which is a polynomial function of p 1 , . . . , p n . Notice also that the parameter t is redundant because we can change p j → ρ j p j and t → t/ρ without changing the sum, i.e. Z({p 1 , p 2 , p 3 , . . . , }, g s ; t) = Z({ρp 1 , ρ 2 p 2 , ρ 3 p 3 , . . . , }, g s ; t/ρ). In [2], t is chosen as t = 1, but we find more convenient to keep t = 1 for the moment, in order to have only two formal parameters g s and t, instead of an infinite number of them g s and p 1 , p 2 , . . . , which would be the case if t were set to 1. The generating function of connected coverings is F = ln Z (in the sense of formal power series of t and g s ): F (p, g s ; t) = ln Z = b,n t n b! |µ|=n g 2g−2 s p µ H g,µ , where b = 2g − 2 + |µ| + (µ). Therefore, we have a so-called topological expansion (equality of formal series): F (p, g s ; t) = ∞ g=0 g 2g−2 s F g (p; t), where F g counts the number of connected coverings of genus g: F g (p; t) = n t n |µ|=n p µ (2g − 2 + n + (µ))! H g,µ . Our goal in this article is to provide a recursive algorithm to compute the F g 's, and more precisely, prove that the F g 's are the symplectic invariants introduced in [4] for a spectral curve S(p, g s ; t) which we shall describe in Section 4. As a consequence, we shall prove the conjecture of Bouchard and Mariño [2]. 2.3. Partitions. To make our notations clear, we recall some representations of partitions or Young tableaux. The set of all partitions λ of length ≤ N is in bijection with other interesting sets of objects: • The decreasing finite series of N integers : λ 1 ≥ . . . ≥ λ (λ) ≥ λ (λ)+1 = . . . = λ N ≥ 0. The length (λ) of the partition is the number of nonvanishing λ i 's. |λ| = (λ) i=1 λ i = n is the number of boxes of the partition. • The strictly decreasing finite series of positive integers. They mark the positions (up to a translation) on the horizontal axis of the increasing jumps when the Young tableau (λ i , i) is tilted anticlockwise by π 4 (c.f. Figure 2). The correspondence is given by • The conjugacy classes of S n . The class C λ associated to λ is the one with m r = |{i > 0 λ i = r}| cycles of length r, and its cardinal is h i = λ i − i + N (i ∈ {1, . . . , N }). We have h 1 > h 2 > · · · > h N ≥ 0. λ i i h h h h h h h 1 2 l(λ)-1 l(λ) 4 3 l(λ) + 1|C λ | = |λ|! r≥1 m r ! · r mr . • The equivalence classes of irreducible representations of S |λ| . • The equivalence classes of irreducible representations of GL (λ) (C), or of U( (λ)). Schur polynomials. Recall the definition of Schur polynomials s λ [13]: they coincide with the characters of the representation of U(k) indexed by λ such that k = (λ). As a matter of fact, if v = (v 1 , . . . , v k ) is an k-tuple of complex variables: (5) s λ (v) = det v λj −j+N i ∆(v) , where ∆(v) = 1≤j<i≤N (v i − v j ) is the Vandermonde determinant. This formula can be extended to a definition of s λ with N ≥ k variables, v = (v 1 , . . . , v N ), provided that we take λ j = 0 whenever j > k. The Frobenius formula gives the expansion of a Schur polynomial in terms of the power-sum functions p m = N i=1 v m i , (6) s λ (v) = 1 n! |µ|=n |C µ | χ λ (C µ ) p µ , which stresses the link between U(k) characters and S n characters. It is still valid with N ≥ k variables instead of (λ) = k variables. From Eqn 5, one can obtain the classical result for the dimension of the representation indexed by λ: s λ (1, . . . , 1) = dim λ = ∆(h) N i=1 h i ! , while the Frobenius formula leads to the expression f λ (C 2 ) = 1 2 i h 2 i − (N − 1 2 ) i h i + N 3 (N 2 − 3 2 N + 2). 2.5. Z as a sum on partitions. After Eqn 6, if we consider the variables p m 's to be power-sum functions of some N -upple parameter v, we have (7) Z(p, g s ; t) = b,n t n g b−n s b! (λ)≤N,|λ|=n dim λ n! s λ (v) f λ (C (2) ) b p m = g s N i=1 v m i . Hence, Z(p, g s ; t) = (λ)≤N (t/g s ) |λ| dim λ |λ|! s λ (v) e gsf λ (C (2) ) . Alternatively, given that s λ is homogeneous of degree |λ|, we have Z(p, g s ; t) = (λ)≤N dim λ |λ|! s λ (tv/g s ) e gsf λ (C (2) ) . Again, we emphasize that the above expression for Z ought to be considered as a formal power series in t and g s . For a given b and n, the coefficient of t n g χ s is a polynomial in the p m 's, which involves only p 1 , . . . , p n . Therefore, it is always possible to find N (independent of b) and v 1 , . . . , v N such that ∀i ∈ {1, . . . , N } we have p m = g s v m i . The values of the p m 's for m > n do not matter. Matrix integral representation of Z. To write Z as a matrix integral we express s λ in terms of the Itzykson-Zuber integral [12]: I(X, Y ) = U(N ) dU e Tr(XU Y U † ) = det(e xiyj ) ∆(X)∆(Y ) . Here dU is the Haar measure on U (N ), normalized such that the second line holds without any constant prefactor. Therefore, if we let R = diag(ln v 1 , . . . , ln v N ) and h λ = diag(h 1 , . . . , h N ), we have from Eqn 5 : s λ (v) = ∆(h λ ) ∆(R) ∆(v) I(h λ , R). Then, the partition function looks like Z(p, g s ; t) = ∆(R) ∆(v) λ I(h λ , R) ∆(h λ ) 2 N i=1 h i ! N i=1 e gsA2(hi) (g s /t) −A1(hi) = ∆(R) ∆(v) h1>···>h N ≥0 I(h, R) ∆(h) 2 N i=1 e gsA2(hi) (g s /t) −A1(hi) Γ(h i + 1) = 1 N ! ∆(R) ∆(v) h1,...,h N ≥0 I(h, R) ∆(h) 2 N i=1 e gsA2(hi) (g s /t) −A1(hi) Γ(h i + 1) , where |λ| = i A 1 (h i ) and f λ (C (2) ) = i A 2 (h i ). The factorization of the weight with respect to the h i 's and the presence of a squared Vandermonde is the key for the representation of Z as a hermitian matrix model. As it was done in [6], we represent the N sums as integrals over a contour, namely the contour C 0 enclosing the non-negative integers, as pictured in Figure 3. We make use of a function which has simple poles with residue 1 at all integers: f (ξ) = πe −iπξ sin(πξ) = −Γ(ξ + 1)Γ(−ξ)e −iπξ . Thus, we have: Z(p, g s ; t) = 1 N ! ∆(R) ∆(v) C N 0 dh 1 · · · dh N ∆(h) 2 I(h, R) N i=1 f (h i )e gsA2(hi) (g s /t) −A1(hi) Γ(h i + 1) . Actually, the ratio f (ξ)/Γ(ξ + 1) has only poles at non-negative integers, it has no pole at negative integers. So, we can replace C 0 by the contour C from Figure 3 which encloses all integers, and which we choose invariant under translation on the real axis. Therefore, we arrive at Figure 3. Contours enclosing the non-negative integers and the entire real line. Z(p, g s ; t) = 1 N ! ∆(R) ∆(v) C N dh 1 · · · dh N ∆(h) 2 I(h, R) N i=1 f (h i )e gsA2(hi) (g s /t) −A1(hi) Γ(h i + 1) . 0 1 2 3 -2 -1 -3 C C0 P C 1 The set H N (C) of normal matrices with eigenvalues on C is the set of matrices M which can be diagonalized by conjugation with a unitary matrix, and whose eigenvalues belong to C: M = U † X U, U U † = Id, X = diag(x 1 , . . . , x N ), x i ∈ C. H N (C) is endowed with the measure dM = ∆(X) 2 dX dU, where dU is the (up to normalization) Haar measure on U (N ) and dX is the product of Lebesgue curvilinear measures along C. From the above discussion, we can express our generating function as the matrix integral Z(p, g s ; t) = 1 N ! ∆(R) ∆(v) H N (C) dM e − TrV (M )+Tr(M R) , whereV (ξ) = −g s A 2 (ξ) + ln(g s /t) A 1 (ξ) + iπξ − ln Γ(−ξ) . Since we are interested in the expansion as a power series in g s , we prefer to rescale ξ = x/g s , i.e. M → M/g s , and rewrite (8) Z(p, g s ; t) = g −N 2 s N ! ∆(R) ∆(v) H N (C) dM e − 1 gs Tr V (M )−M R , where now the potential reads V (x) = −g 2 s A 2 (x/g s ) + g s ln(g s /t)A 1 (x/g s ) + iπx − g s ln Γ(−x/g s ) = − x 2 2 + g s (N − 1 2 )x + (ln(g s /t) + iπ)x − g s ln Γ(−x/g s ) + C t , with C t = − 1 3 g 2 s (N 2 − 3 2 N + 2) + 1 2 g s (N − 1) ln(g s /t) To write its derivative, let ψ = Γ /Γ, then V (x) = −x + g s (N − 1 2 ) + ln(g s /t) + iπ + ψ(−x/g s ). We have the well known formula that: ψ(ξ) = γ + 1 ξ + ∞ k=1 1 ξ + k − 1 k where γ is the Euler-Mascheroni constant. In other words, ψ(ξ) has simple poles at all negative integers, and an essential singularity (log singularity) at ξ = ∞. This gives: (9) V (x) = −x + g s (N − 1 2 ) + ln(g s /t) + iπ + γ − 1 x − g s ∞ k=1 1 x − kg s + 1 kg s . V (x) has simple poles at x = X j = jg s for all j ∈ N, and an essential singularity at x = ∞. Using Stirling's formula for the large ξ asymptotic expansion of ψ ψ(ξ) = ξ→∞ ln ξ − 1 2ξ − l≥1 B 2l 2l 1 ξ 2l , we have order by order in the small g s expansion (10) V (x) = −x + ln (x/t) + g s (N − 1 2 ) + g s 2x − ∞ l=1 B 2l 2l g 2l s x 2l , where the B l 's are the Bernoulli number. We recall their generating function ξ e ξ − 1 = 1 + ∞ l=1 B l l! ξ l . The first few are B 2 = 1/6, B 4 = −1/30, . . . while B 2l+1 = 0 for l ≥ 1. Generalities on matrix models Matrix integrals of the form Z = H N (C) dM e − 1 gs Tr V (M )−M R are called "1-matrix model in an external field". They can be computed for any potential V , any contour C and any external matrix R. Here, our task is even simpler because we regard this integral as a formal integral, i.e. a formal power series in powers of t and g s , and therefore all our computations are to be performed order by order in powers of t and g s . In our case V depends on g s , but for the moment, let us assume that V is an arbitrary potential, and in particular we assume that there is no relationship between the coefficients of V and g s . Typically, in our case (see Eqn 9), we choose V of the form: (11) V (x) = −x + C + n j=1 u j x − x j , where for the moment we assume that there is no relationship between the coefficients of u j , x j of V and g s . 3.1. Topological expansion. For some choices of V , R and C, it may or may not happen that the convergent integral Z has a power series expansion in g s ln Z = ∞ g=0 g 2g−2 s F g . This happens only if C is a "steepest descent path" for the potential V and R. In general, it is rather difficult to compute the steepest descent paths of a given arbitrary potential. Fortunately, when Z is defined as a formal integral we don't need to find the steepest descent paths, and very often, formal series do have a topological expansion almost by definition, order by order in the formal parameters, which is the case here as we argued in Paragraph 2.2. In other words, ln Z(p, g s ; t) = ∞ g=0 g 2g−2 s F g (p; t) holds order by order in powers of g s and t. 3.2. Loop equations and spectral curve. We introduce the resolvent W 1 and auxiliary quantities P 1 and P i,j : W 1 (x) = Tr 1 x − M = ∞ g=0 g 2g−1 s W (g) 1 (x) P 1 (x, y) = Tr V (x) − V (M ) x − M 1 y − R = ∞ g=0 g 2g−1 s P (g) 1 (x, y), P i,j = ((x j − M ) −1 ) i,i = ∞ g=0 g 2g−1 s P (g) i,j , which we assume to have topological expansions in powers of g 2g−1 s . In our case, their precise definition, order by order in t as a power series of g s , is given in Appendix 7. Loop equations, also called Schwinger-Dyson equations, is a general technique, which merely reflects the fact that an integral is invariant by change of variable. It is a standard matrix model exercise, (see [4] for the 1-matrix model in an external fiels), to prove that the invariance of Z under the infinitesimal change of variable M → M + x − M 1 y − R + O( 2 ) implies that the following loop equation is satisfied: (12) W(0)1 (x) = P (0) 1 (x, Y (x)) Y (x) = V (x) − W (0) 1 (x). Notice that P (0) 1 (x, y) is a rational function of y, of degree N . If V were a rational function of x (finite n in Eqn 11), then P (0) 1 (x, y) would be a rational function of x of degree n: P (0) 1 (x, y) = −g s Tr 1 y − R + n j=1 N i=1 u j x − x j 1 y − R i P (0) i,j . Thus the loop equation would be an algebraic equation, i.e. Y (x) would be an algebraic function of x. Determining the rational function P 1 (x, y) (i.e. determining all the coefficients P i,j ) is possible but very tedious, and in fact, it is better to characterize an algebraic function Y (x) by its singularities and its periods. In general, one would find that W (0) 1 (x) = V (x) − Y (x) has no singularity at the singularities of V , and the inverse function x(Y ) has poles of residue g s at the eigenvalues of R. The genus of the algebraic function Y (x) and the periods Y dx are related to the integration path C. In general, the relationship between the periods and C is quite complicated, but, for many applications to combinatorics, we are considering only formal matrix integrals, i.e formal perturbation with parameter t of a gaussian integral. In that case, the spectral curve Y (x) is always a genus 0 curve (for the Hurwitz matrix integral, we prove it in Appendix 7). This means that there exists a parametrization of Y (x) with a complex variable z: Y (x) ↔ x = x(z) Y = y(z), where x and y are two analytical functions of z. The functions x(z) and y(z) are monovalued functions of z, but Y (x) = y(z) is multivalued, because there might exist several z such that x(z) = x. One of the determinations of Y (x) is called the physical sheet. The functions x and y are fully determined by their singularities. More precisely: • W (0) 1 (x) = V (x) − Y (x) is analytical in the physical sheet; it can have no singularity except at branchpoints. • From the definition of W (0) 1 , we have that at x = ∞ in the physical sheet W (0) 1 (x) ∼ N g s x(z) + o( 1 x(z) ). • As a consequence of Eqn 12, x(z) must have simple poles when z → z i such that y(z i ) = R i , i.e. x(z) ∼ g s (z − z i ) y (z i ) . In our case, for a potential of type 11, this implies: (13) x(z) = z + C − g s N i=1 1 (z−zi) y (zi) y(z) = −z + n j=1 uj (z−ẑj ) x (ẑj ) where x(ẑ j ) = x j , y(z i ) = R i . The above characterization of the spectral curve Y (x) is valid even if the potential V is not rational, for instance if n → ∞. From now on, the function Y (x), or more precisely the pair of analytical functions S = z → (x(z), y(z)) is called the "spectral curve" of our matrix model. 3.3. Topological recursion. We recall in this section the construction of the topological recursion from [4]. For our purposes, we only deal with spectral curves of genus 0. Hence, we adapt the definitions of [4] to this case. A spectral curve S is a pair (x, y) of analytic functions on CP 1 . Let a i be the zeroes of dx, and assume they are simple. Then, locally at a i , y ∝ (x − x(a i )). Let us denote z = z, the unique point corresponding to the other branch of the squareroot, such that x(z) = x(z). z is defined locally near the a i 's. A tower of k-forms W (g) k (z 1 , . . . , z k ) is constructed as follows. • W (0) 1 = −ydx. • W (0) 2 is defined as the Bergman 2 kernel: W (0) 2 (z 1 , z 2 ) = B(z 1 , z 2 ) = dz 1 dz 2 (z 1 − z 2 ) 2 . • We define a recursion kernel K(z , z) − 1 2 z z B(z , ·) (y(z) − y(z))dx(z) . • For k + 2g − 2 > 0, we define recursively the k-forms W (g) k (z 1 , . . . , z k ) by: W (g) k (z 1 , z 2 , . . . , z k K ) = i Res z→ai K(z 1 , z) W (g−1) k+1 (z, z, K) + J⊆K, 0≤h≤g W (h) |J|+1 (z, J)W (g−h) k−|J| (z, K \ J) ,(14) where ranges over (J, h) = (∅, 0), (I, g). • W (g) 0 = F g are defined for g ≥ 2 by F g = 1 2 − 2g i Res z→ai W (g) 1 (z)Φ(z) , where Φ is a primitive of ydx locally at the a i 's, i.e. dΦ = ydx. • The definition of F 1 and F 0 is more involved, and we refer the reader to [4]. Main properties of the W (g) k . • Symmetry ∀k, g, W (g) k (z 1 , . . . , z k ) is symmetric in z 1 , . . . , z k . • Invariance ∀k, g, 2g + k − 2 > 0, W (g) k is unchanged if we add to y a rational function of x. • Exchange invariance ∀g ≥ 2 F g is unchanged if we exchange x and y. • Deformation If we perform an infinitesimal deformation (δx, δy) of the spectral curve, the W (g) k 's change. Let us introduce Ω(z) = δx(z)dy(z) − δy(z)dx(z). This form does not depend on the parametrization z, and can always be represented as z ∈γ B(z, z )Λ(z ) for some path γ and some meromorphic function Λ defined in its neighborhood (this data is called the dual of Ω). Then (15) δW (g) k (z 1 , . . . , z k ) = z∈γ Λ(z)W (g) k+1 (z, z 1 , . . . , z k ). • Limits W (g) k (S) is compatible with limits of curves. • Link to matrix models It was proved in [4] that, if our matrix model has a topological expansion property: W k (z 1 , . . . , z k ) = k i=1 Tr 1 x(z i ) − M c = ∞ g=0 g 2g−2+k s W (g) k (z I ), and ln Z = ∞ g=0 g 2g−2 s F g , then, loop equations imply that: W (g) k (z 1 , . . . , z k )dx(z 1 ) . . . dx(z k ) = W (g) k (z 1 , . . . , z k ) where W (g) k are computed using the spectral curve S = (x(z), y(z)) presented in Section 3.2 (except for (k, g) = (1, 0), (2, 0), which receive simple additional contributions). Similarly, for g ≥ 2: F g = F g . (F 0 and F 1 also receive simple corrections). Since the F g (S)'s are invariant under transformations of S which leave |dx ∧ dy| unchanged, we call them symplectic invariants of S. In the following section we shall apply the general theory of [4] to our Hurwitz matrix integral Eqn 8. Spectral curve of the Hurwitz matrix model In our case, the spectral curve of our matrix model must be determined order by order in powers of g s and t. Here, we only give the result. The proof is quite technical, and is deferred to Appendix 7. It relies on the fact that, after a suitable shift, to leading order in g s and t, we have a Gaussian matrix integral. In particular, this implies that the spectral curve has genus 0, i.e. it can be parametrized with a uniformization variable z ∈ C. By composition with some homographic map, we can choose a parametrization for which x(z) ∼ z when z → ∞ and x(0) = 0. Since our potential V is of the form Eqn. 11 with n → ∞, our spectral curve is of the form Eqn. 13. Therefore we guess that the spectral curve must be of the form: (16) S(p; t) : x (z) = z + g s N i=1 1 (z−zi)yi + 1 ziyi y(z) = −z + ln (z/t) + c 0 + c1 z − ∞ l=1 B 2l g 2l s 2l f 2l (z) − f 2l,1 z where y(z i ) = R i = ln v i y i = y (z i ) and where f l (z) = Res z →0 dz z − z (x(z )) −l = l j=1 f l,j z −j is such that x(z) −l − f l (z) has a finite limit when z → 0. c 0 is chosen such that V (x(z)) − y(z) ∼ O(1/z) at large z: c 0 = (N − 1 2 ) g s + g s N i=1 1 z i y i . The coefficient c 1 is c 1 = g s 2 − ∞ l=1 B 2l g 2l s 2l f 2l,1 , and one can check (this comes from the fact that the sum of all residues of V (x) must vanish) that it is such that V (x(z)) − y(z) ∼ N g s /z at large z i.e.: c 1 = (N − 1 2 ) g s − g s N i=1 1 − z i z i y i . Each term z i , y i , c 0 , c 1 , is to be viewed as a power series in g s and in t. To the first few orders in g s we have: z i = L(tv i ) + g s 1 − L(tv i )   1 + L(tv i ) 2 + L(tv i ) j L(tv j ) 1 − L(tv j )   + O(g 2 s ) c 0 = g s − 1 2 − N i=1 L(tv i ) 1 − L(tv i ) + O(g 2 s ) c 1 = −g s 2 + O(g 2 s ) and notice that L(tv i ) is a power series in t: L(tv i ) = ∞ m=1 m m−1 t m v m i m! . The proof that S(p; t) is the correct spectral curve for our problem is given in Appendix 7. It is obtained by computing the spectral curve order by order in the small g s and t expansion. The computation of W 4.2. The symplectic invariants. So far, from the general theory of matrix models and from general properties of the topological recursion, we have proved that the generating function of simple Hurwitz numbers of genus g is the symplectic invariant (18) F g (p; t) = n t n |µ|=n p µ (2g − 2 + n + (µ))! H g,µ = F g (S(p; t)), where S(p; t) is the spectral curve of Eqn 16 (in fact we have proved it only for g ≥ 2, and we consider that the cases g = 0 and g = 1 are easier). The computation of symplectic invariants F g involves computing residues at the zeroes of x (z), and there are 2N such zeroes, which makes the computation complicated. One may use the invariance properties of F g , under the exchange of x and y. Indeed, the zeroes of y (z) are much simpler to compute, order by order in powers of g s . At g s = 0, y (z) has only one zero located at z = 1. Therefore, we introduce a new spectral curve satisfying F g (p; t) = F g ( S(p; t)), defined by exchanging x and y: S(p, g s ; t) : x(z) = −z + ln (z/t) + c 0 + c1 z − ∞ l=1 B 2l g 2l s 2l f 2l (z) y(z) = z + g s N i=1 1 (z−zi)yi + 1 ziyi , where now x(z i ) = ln v i and y i = x (z i ). 4.3. The correlation forms. So far, we have introduced the F g 's as generating functions which encode the simple Hurwitz numbers H g,µ by expansion on a proper basis of polynomials of the p m 's. Bouchard and Mariño [2] define another generating function, namely the function of k-variables: H (g) (x 1 , . . . , x k ) = (µ)=k t |µ| k i=1 µ i · M µ (x 1 , . . . , x k ) (2g − 2 + |µ| + k)! H g,µ , where M µ (x) = σ∈S k k i=1 x µi σ(i) are the (un-normalized) symmetric monomials. It is easy to relate both. If we recall the combinatorial definition Eqn 18 F g (p; t) = µ t |µ| p µ (2g − 2 + n + (µ))! H g,µ , we see that H (g) (v 1 , . . . , v k ) = v 1 . . . v k g k s ∂ k F g ∂v 1 . . . ∂v k gs=0 = 1 g k s ∂ k F g ∂R 1 . . . ∂R k gs=0 . We recall that R i = ln v i . The deformation property Eqn 15 of symplectic invariants allows us to calculate their derivatives. When we perform an infinitesimal variation v i → v i + δv i , i.e. a variation R i → R i + δR i on the spectral curve, we need to compute Ω i (z) = δx(z)dy(z) − δy(z)dx(z). First notice that the form ydx is a meromorphic form (its singularities are poles). It has simple poles of constant residues g s near z = z j , i.e. locally near z j we have y ∼ g s x − R j , which implies that, locally near z j , Ω i = −g s δ i,j δR j dx (x − R j ) 2 + O(1) Then, observe that the other poles of ydx are independent of R i . For example, (V (x) − y)dx has a simple pole at ∞, with residue N g s independent of R i . Therefore, Ω i has no residue, and thus no pole at ∞. Similarly there is no pole at z = 0. The final result is that Ω i is a meromorphic form with poles only at z i : 1 g s Ω i (z) = − δR i dz (z − z i ) 2 y i . Ω i (z) can be written in term of the Bergman kernel B(z, z ) = dz dz (z−z ) 2 as Ω i (z) = −g s δR i Res z →zi B(z, z ) 1 (z − z i ) y i . Then, the general theorems about the F g 's tell us that δF g = g s δR i Res z →zi W (g) 1 (z ) 1 (z − z i ) y i = g s δR i W (g) 1 (z i ) dx(z i ) , and more generally δW (g) k (z 1 , . . . , z k ) = g s δR i Res z →zi W (g) k+1 (z 1 , . . . , z k , z ) 1 (z − z i )y i . The result is that 1 g k s ∂ k F g ∂v 1 . . . ∂v k = W (g) k (z 1 , . . . , z k ) dx(z 1 ) · · · dx(z k ) . R1=x(z1),...,R k =x(z k ) As a final step, we take the limit g s → 0. The limit of the spectral curve S(v; t) is simply S Lambert : x(z) = −z + ln (z/t) y(z) = z, i.e. it is the Lambert spectral curve: y = L(te x ). In other words, we have proved that the function H (g) (v 1 , . . . , v k ) is the correlation form W (g) k of the Lambert spectral curve S Lambert : H (g) (v 1 , . . . , v k ) = W (g) k (z 1 , . . . , z k ) dx(z 1 ) · · · dx(z k ) (S Lambert ), where x(z i ) = ln v i = R i , i.e. z i = L(tv i ). This is precisely the Bouchard-Mariño conjecture. Relationship with intersection numbers, Kontsevich integral and elsv formula Notice that the Lambert spectral curve S Lambert : x(z) = −z + ln (z/t) y(z) = z has only one branchpoint (solution of x (z) = 0), given by z = 1. This is the reason why the Bouchard-Mariño conjecture is so efficient to compute Hurwitz numbers. Since the topological recursion for computing the W (g) k 's and F g 's, involves only the computation of residues at the branch point, we may perform a Taylor expansion near z = 1 : Let us define: y = 1 + ζ and ξ 2 = −2(x + 1 + ln t) We have, in the limit ζ → 0: 1 2 ξ 2 = ζ 2 2 − ζ 3 3 + ζ 4 4 + · · · = m≥2 (−1) m ζ m m and we invert that expansion y = 1 + ξ + ξ 2 3 + ξ 3 36 − ξ 4 270 + ξ 5 6 .6! + · · · , which we write: y = 1 − 2ξ + m≥1 t m+2 ξ m In other words, the W (g) k 's and F g 's of the Lambert curve S Lambert , are the same as the W (g) k 's and F g 's of the following spectral curve S K : S K : x(ξ) = −1 − ln t − 1 2 ξ 2 y(ξ) = 1 − 2ξ + m≥1 t m+2 ξ m This spectral curve is exactly the Kontsevich spectral curve for times t m 's (see [5]). Here the t m 's satisfy the following recursion t 2 = 0, t 3 = 3, t 4 = 1 3 , and for m ≥ 4: (19) t m+1 = t m m − 1 2 m−2 l=2 t l+2 t m+2−l We form the following series: f (z) = ∞ m=1 (2m + 1)! m! t 2m+3 2 − t 3 z m and g(z) = − ln (1 − f (z)) = ∞ m=1t m z m we find to the first orders: g(z) = − z 6 + z 3 45 − 8z 5 315 + 8z 7 105 + · · · Then, it was found in [5] that: W (g) k (z 1 , . . . , z k ) = 1 2 3g−3+k d0+···+d k =3g−3+k d0 j=1 1 j! m1+···+mj =d0,mi>0 k i=1 (2d i + 1)! d i ! dz i z 2di+2 i j i=1t mi j i=1 κ mi k i=1 ψ di i M g,k where M g,k is the stable compact moduli space of Riemann surfaces of genus g with k marked points, and κ j is the j th Mumford's tautological class, and ψ i = c 1 (L i ) is the first Chern class of the cotangent bundle at the i th marked point. In other words, just by looking at the Lambert spectral curve, we see that there is a relationship between the generating function for Hurwitz numbers of genus g with a monodromy of length k, and the generating function for intersection numbers of tautological classes on M g,k . This type of relationship is completely natural and expected. The link coming from the elsv formula [3], relating Hurwitz numbers to Hodge integrals: H g,µ = (2g − 2 + (µ) + |µ|)! | Aut µ| (µ) i=1 µ µi i µ i ! M g, (µ) Λ ∨ g (1) (µ) i=1 (1 − µ i ψ i ) , where Λ ∨ g (t) = g i=0 λ i t i is the total Chern class of the Hodge bundle E over M g,n . In fact, the change of variables taking the Hurwitz generating function to the generating function of all Hodge integrals with a single λ factor is given by the Lambert curve itself [2,9,11]. Furthermore, Mumford's formula [7,15] ch (E) = g + ∞ l=1 B 2l (2l)! κ 2l−1 + 1 2 ι * 2l−2 i=0 (−1) i ψ iψ2l−2−i allows one to express Hodge integrals in terms of ψ class intersections. The time parameters t m appearing in Eqn 19 are closely related to these topics. Mironov and Morozov [14] have previously considered similar constructions. An other natural question arising is the relationship between the Bouchard-Mariño conjecture and the cut-and-join equation [8]. Since they are both recursive algorithms for computing Hurwitz numbers, it seems likely that they should be related. In fact, they are, appropriately interpreted, completely equivalent. This topic, as well as a more detailed discussion of the deformation of the Kontsevich model to the Hurwitz model, are deferred to future papers. Conclusion With the integral representation of U(N ) characters, it is possible to express in general the partition function Z of Hurwitz numbers as a matrix model. In the case of simple Hurwitz numbers, we obtain a 1-matrix model with external field, whose spectral curve is found by solving the master loop equation: S(p; t) : x(z) = −z + ln (z/t) + c 0 + c1 z − ∞ n=1 B2n g 2n s 2n f 2n (z) y(z) = z + g s N i=1 1 (z−zi)yi + 1 ziyi . Our main results are: F g (p; t) = F g ( S(p; t)) H (g) (v 1 , . . . , v k ) = W (g) k (z 1 , . . . , z k ) dx(z 1 ) · · · dx(z k ) ( S Lambert ) where x(z i ) = ln v i y(z i ) = L(tv i ) It provides an algorithm, namely the topological recursion of matrix models, to compute the H g,µ by the residue formula of Paragraph 3.3, with only one branchpoint involved. As a matter of fact, this recursion relation between simple Hurwitz numbers is understood to be equivalent to the Laplace transform of the cut-and-join equation with help of the elsv formula. Besides, Z for simple Hurwitz numbers is the time evolution of the kp τ -function, a fact agreeing with its one-matrix model representation. We also see explicitly on the Lambert curve S Lambert = lim gs→0 S(p; t) the relation between Z and the Kontsevich τ -function. We hope that our matrix model-minded methods could help investigating double Hurwitz numbers (where Z is a Toda τ -function) and further. Appendix: proof of the spectral curve The proof works order by order in g s and t, and it relies on the fact that, to leading order, we have a Gaussian matrix integral. 7.1. Shift of the matrix model. We start from: Z(p, g s ; t) = g −N 2 s N ! ∆(R) ∆(v) H N (C) dM e − 1 gs Tr V (M )−M R , where the potential V (x) is: V (x) = − x 2 2 + g s (N − 1 2 )x + (ln(g s /t) + iπ) x − g s ln Γ(−x/g s ) + C t , and C t does not depend on x. Order by order in g s we have the Stirling expansion: V (x) = −x + ln (x/t) + g s (N − 1 2 ) + g s 2x − ∞ l=1 B 2l g 2l s 2l x 2l . We need to compute this matrix integral in the small g s and t expansion (up to a constant factoring out of the integral). First let us perform a shift M = M −R whereR = diag(R 1 , . . . ,R N ) is such that: V (R i ) = R i . The equation V (R i ) = R i has several solutions, we choose the one which is a power series in g s and t: R i = L(tv i ) − g s ( 1 2 + N L(tv i ) 1 − L(tv i ) ) + · · · = l≥0 g l sRi,l (v i ). and we choose the determination of the Lambert function such that L(tv i ) has a small t expansion L(tv i ) = tv i + t 2 v 2 i + · · · = m≥1 m m−1 (tv i ) m /m!. Then we have: Z(p, g s ; t) = g −N 2 s N ! ∆(R) ∆(v) e − 1 gs Tr V (R)−RR H N (C) d M e − 1 gsṼ ( f M ) , where we decomposeṼ( M )Ṽ ( M ) =Ṽ 2 ( M ) +Ṽ ≥3 ( M ) into a quadratic function of M V 2 ( M ) = 1 2 Tr   − M 2 + g s ∞ j=0 1 R − jg s M 1 R − jg s M   andṼ ≥3 contains all the higher degree terms (we will not need explicit expressions, however, the interested reader can derive them easily). Notice thatṼ 2 ( M ) andṼ ≥3 ( M ) have a small g s expansion (for instance approximate the sum j inṼ 2 by a Riemann integral). Order by order in g s we have: Z(p, g s ; t) = g −N 2 s N ! ∆(R) ∆(v) e − 1 gs Tr V (R)−RR H N (C) d M e − 1 gsṼ 2( f M ) ∞ m=0 (−1) m g m s m! (Ṽ ≥3 ( M )) m , If we rescale M = √ g s A, we have Z(p, g s ; t) = 1 N ! ∆(R) ∆(v) e − 1 gs Tr V (R)−RR H N (C) dA e −Ṽ2(A) ∞ m=0 (−1) m g m s m! (Ṽ ≥3 (A/ √ g s )) m . SinceṼ ≥3 (A/ √ g s ) = O( √ g s ), we see that order by order in powers of g s , we may exchange the sum and integral. Therefore: Z(p, g s ; t) = g −N 2 s N ! ∆(R) ∆(v) e − 1 gs Tr V (R)−RR ∞ m=0 (−1) m g m s m! H N (C) d M e − 1 gsṼ 2 ( f M ) (Ṽ ≥3 ( M )) m , W (g) 1 (x) is thus a formal power series in powers of g s and t, whose coefficients are rational functions of x. These definitions give a meaning to the equality between formal double power series: W 1 (x) = ∞ g=0 g 2g−1 s W (g) 1 (x), where W 1 (x) is the resolvent : W 1 (x) formal = Tr 1 x − M . In a similar manner, one can define g 2g−2+k s W (g) k (x 1 , . . . , x k ) as the formal double power series computing the sum over (c for connected) fatgraphs of genus g arising in the correlation function: W k (x 1 , . . . , x k ) formal = k i=1 Tr 1 x i − M c . By construction we have, in the sense of formal series: W k (x 1 , . . . , x k ) = ∞ g=0 g 2g−2+k s W (g) k (x 1 , . . . , x k ). To sum things up, the correlation functions W (g) k (x 1 , . . . , x k ) can be defined as formal power series in g s and t, such that the coefficients are rational functions of the x i 's. It is defined by collecting the fatgraphs of genus g in the Wick theorem's expansion of gaussian integrals. The loop equations of gaussian matrix integrals are well known, and they imply that the W (g) k satisfy the topological recursion of [4]. 1 (x), the loop equations read (see [4]): V (x) − Y = Tr V (x) − V (M ) x − M 1 Y − R (0) = −g s Tr 1 Y − R + g s N i=1 ∞ j=0 1 x − jg s 1 Y − R i 1 jg s − M i,i (0) = −g s Tr 1 Y − R + g s N i=1 ∞ j=0 ∞ p=0 1 x − jg s 1 Y − R i T j,p;i(20) T j,p;i = 1 jg s −R M 1 jg s −R p i,i(0) where as usual < . > (0) means that we shift M = R + M , and keep only the genus zero fatgraphs in the gaussian expectation value. Notice that the sum over j is absolutely convergent. Moreover, to a given order in g s , the sum over p is finite. This equation is sufficient to determine Y (x) order by order in g s . To leading order we find Y (x) = −x + ln (x/t) + O(g s ). To subleading orders, we recursively have to determine a finite number of coefficients of the g s expansion of T j,p;i . Those coefficients are completely determined by the condition that, to each order in g s , W 1 (x) is a rational function of x with poles only at x =R i , and in particular it must have no pole at x = jg s , or at Y (x) = R i . 7.2. Asymptotic expansion of the spectral curve. To leading order in g s , we have V (x) = −x + ln (x/t), and: Y (x) = −x + ln (x/t) + O(g s ) This leading order spectral curve has a rational uniformization: x(z) = z y(z) = −z + ln (z/t) Since the higher order g s corrections to V (x) are all rational, the corrections to P (0) 1 (x, y) are also rational functions of x and y. This implies that all corrections to Y (x) can be written with the uniformizing variable z, i.e. to all orders in g s the spectral curve is of genus 0. To all orders in g s , the equation Eqn 20 is algebraic of degree N + 1 in the variable Y , this implies that the function x(z) is rational of degree N + 1. It is easy to see that it has N simple poles at z i such that y(z i ) = R i , and one simple pole at ∞. Up to a homographical change of variable z, we assume that x(0) = 0 and x(z) ∼ z at large z, i.e.: x(z) = z + g s N i=1 1 (x − z i ) y i + 1 z i y i Moreover, one sees directly from Eqn 20, that the residue of xdy is g s , i.e. y i = y (z i ). The function y(z) starts to leading order in g s as y(z) = −z+ln (z/t)+O(g s ), and all the higher g s corrections are rational functions of z. Let x = ∞ and X j = jg s , j ∈ N be the singularities of V (x). For each X j , let us chooseẑ j such that x(ẑ j ) = X j (and we must choose the value ofẑ j which has a small t power series expansion). Since V (x) has a simple pole of residue g s at x = X j , one sees from Eqn 20, that ydx has a simple pole of residue g s at z =ẑ j . Another way of saying this is to write: y(z) = 1 2iπ C0 dz z − z V (x(z )), where C 0 is a contour surrounding ∞ and all theẑ j 's. Order by order in g s , using the Stirling expansion of V (x), this gives the spectral curve S(p; t) of Eqn 16. Figure 1 . 1Branched covering, with one branch point of monodromy class µ, and simple branch points (for which the monodromy is a transposition). Figure 2 . 2Rotate the partition by π/4. The h i 's mark the positions (up to a translation) on the horizontal axis of the increasing jumps in the Young tableau. 2 After Stefan Bergman (1895-1977), mathematician of Polish origin. k 's in the topological recursion formula Eqn 14 involve taking residues at all zeroes of x (z), i.e. involves symmetric rational functions of the z i 's, and one can see that the coefficients of W (g) k are, order by order in g s and t, polynomials of p m = g s i v m i , as required for the computation of Hurwitz numbers indeed. 4.1. Spectral curve at g s = 0. At g s = 0, the spectral curve reduces to:(17) S 0 : x(z) = z y(z) = −z + ln (z/t)i.e. x = L(te y ), where L is the Lambert function. Up to exchanging the roles of x and y, this is the Lambert curve S Lambert appearing in Bouchard-Mariño[2]. or more precisely for Y (x) = V (x) − W (0) In the physics literature, it is customary to choose −gs instead of gs as formal parameter. AcknowledgmentsWe would like to thank I. Kostov, M. Mariño and N. Orantin for useful and fruitful discussions on this subject. The work of B.E. is partly supported by the Enigma European network mrt-ct-2004-5652, by the anr project Géométrie et intégrabilité en physique mathématique anr-05-blan-0029-01, by the anr GranMa grant anr-08-blan-0311-03, by the European Science Foundation through the Misgam program, by the Quebec government with the fqnrt.More generally, expectation values of polynomials Q p ( M ) are computed as formal power series, whose coefficients are polynomial moments of a gaussian integral:In this form, we can use Wick's theorem. It shows that, if Q p ( M ) is any homogeneous polynomial of total degree p in the entries of the matrix M , the expectation value Q p ( M − R) is a power series in g s and t. It is expressed as a finite sum of connected 3 fat-graphs. When we restrict the sum to the fatgraphs of genus g, we note it with a superscript (g) . We claim that, for our matrix model, this expectation value is O(g accompanying (Ṽ ≥3 ) v , and in additioñ V 2 andṼ ≥3 themselves have a g s expansion). So, we have:On the other hand, the number of half-edges is 2e = p + i in i where n i is the number of internal vertices of degree i ≥ 2, and we have v = i n i . So:In particular, let us show how to define the W (g) 1 's, the topological expansion of the one-point correlation function. Consider :It is a double power series, whose coefficients are rational functions of x, and are polynomial gaussian expectation values of M . In its representation as a sum over fatgraphs, we collect those of genus g to define T (g)p (x). Its coefficients are still rational functions of x. Since T (g)and thus we have, in the sense of formal power series of g s :We are in position to define W p,j (x).3Connected because the normalization factor e Z −1 is included in the expectation value. Remodeling the B-model. V Bouchard, A Klemm, M Mariño, S Pasquetti, arXiv:0709.1453Commun. Math. Phys. 287V. Bouchard, A. Klemm, M. Mariño, S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287, 117-178 (2009), arXiv:0709.1453 (2007) V Bouchard, M Mariño, arXiv:0709.1458Hurwitz numbers, matrix models and enumerative geometry. V. Bouchard, M. Mariño, Hurwitz numbers, matrix models and enumerative geometry, arXiv:0709.1458 (2008) Hurwitz numbers and intersections on moduli spaces of curves. T Ekedahl, S Lando, M Shapiro, A Vainshtein, arXiv:math.AG/0004096Invent. Math. 146T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146, 297-327, arXiv:math.AG/0004096 (2001) Invariants of algebraic curves and topological expansion. B Eynard, N Orantin, arXiv:math-ph/0702045Communications in Number Theory and Physics. 12B. Eynard, N. Orantin, Invariants of algebraic curves and topological expansion, Communi- cations in Number Theory and Physics, Vol 1, Number 2, p347-452, arXiv:math-ph/0702045 (2007) B Eynard, arXiv:math-ph/0706.4403Recursion between Mumford volumes of moduli spaces. B. Eynard, Recursion between Mumford volumes of moduli spaces, arXiv:math-ph/0706.4403 (2007) All order asymptotic expansion of large partitions. B Eynard, arXiv:0804.0381J. Stat. Mech. Theory Exp. 734B. Eynard, All order asymptotic expansion of large partitions, J. Stat. Mech. Theory Exp., no. 7, P07023, 34, arXiv:0804.0381 (2008) . C Faber, R Pandharipande, Hodge Integrals, Gromov-Witten Theory, arXiv:math/9810173Invent. Math. 139C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139, 173-199, arXiv:math/9810173 (2000) Transitive factorisations into transpositions and holomorphic mappings on the sphere. I P Goulden, D M Jackson, arXiv:math.CO/9903094Proc. ams. ams125I.P. Goulden and D.M. Jackson, Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. ams, 125, 51-60, arXiv:math.CO/9903094 (1997) The Gromov-Witten potential of a point. I P Goulden, D M Jackson, R Vakil, arXiv:math.AG/9910004Hurwitz numbers, and Hodge integrals. I.P. Goulden, D.M. Jackson and R. Vakil,The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals, arXiv:math.AG/9910004 (1999) Differential operators on a semi-simple Lie algebra. Harish-Chandra, J. Amer. Math. 87120Harish-Chandra, Differential operators on a semi-simple Lie algebra, J. Amer. Math., 87-120 (1957) M Kazarian, arXiv:math.AG/0809.3263KP hierarchy for Hodge integrals. M. Kazarian, KP hierarchy for Hodge integrals, arXiv:math.AG/0809.3263 (2008) The planar approximation. C Itzykson, J.-B Zuber, J. Math. Phys. II3C. Itzykson and J.-B. Zuber, The planar approximation. II, J. Math. Phys. 21, no. 3, 411-421 (1980) With contributions by A. Zelevinsky. I G Macdonald, Symmetric functions and Hall polynomials. New YorkOxford Science PublicationsOxford Mathematical MonographsI.G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, (1995). With contri- butions by A. Zelevinsky, Oxford Science Publications. A Mironov, A Morozov, arXiv:0807.2843Virasoro constraints for Kontsevich-Hurwitz partition function. A. Mironov and A. Morozov, Virasoro constraints for Kontsevich-Hurwitz partition function, arXiv:0807.2843 (2009) Towards an enumerative geometry of the moduli space of curves. D Mumford, Arithmetic and Geometry. M. Artin, J. TateBirkhäuserPart IID. Mumford, Towards an enumerative geometry of the moduli space of curves, in "Arithmetic and Geometry" (M. Artin, J. Tate, Eds.), Part II, 271-328, Birkhäuser (1983) Toda equations for Hurwitz numbers. A Okounkov, arXiv:math.AG/0004128Math. Res. Lett. 74A. Okounkov, Toda equations for Hurwitz numbers, Math. Res. Lett. 7, no. 4, 447-453, arXiv:math.AG/0004128 (2000)

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{'abstract': 'We introduce a new matrix model representation for the generating function of simple Hurwitz numbers. We calculate the spectral curve of the model and the associated symplectic invariants developed in[4]. As an application, we prove the conjecture proposed by Bouchard and Mariño [2], relating Hurwitz numbers to the spectral invariants of the Lambert curve e x = ye −y .', 'arxivid': '0906.1206', 'author': ['Gaëtan Borot ', 'Bertrand Eynard ', 'ANDMotohico Mulase ', 'Brad Safnuk '], 'authoraffiliation': [], 'corpusid': 13069943, 'doi': '10.1016/j.geomphys.2010.10.017', 'github_urls': [], 'n_tokens_mistral': 17808, 'n_tokens_neox': 16014, 'n_words': 9613, 'pdfsha': '67daf5461be7a2d1a0dc02d4efb0eadf62f320dc', 'pdfurls': ['https://arxiv.org/pdf/0906.1206v1.pdf'], 'title': ['IPHT T09/055 A MATRIX MODEL FOR SIMPLE HURWITZ NUMBERS, AND TOPOLOGICAL RECURSION', 'IPHT T09/055 A MATRIX MODEL FOR SIMPLE HURWITZ NUMBERS, AND TOPOLOGICAL RECURSION'], 'venue': []}

arxiv

A Wheeler-DeWitt Equation with Time November 4, 2022 Rotondo Marcello A Wheeler-DeWitt Equation with Time November 4, 2022 The equation for canonical gravity produced by Wheeler and De-Witt in the late 1960s still presents difficulties both in terms of its mathematical solution and its physical interpretation. One of these issues is, notoriously, the absence of an explicit time. In this short note, we suggest one simple and straightforward way to avoid this occurrence. We go back to the classical equation that inspired Wheeler and DeWitt (namely, the Hamilton-Jacobi-Einstein equation) and make explicit, before quantization, the presence of a known, classically meaningful notion of time. We do this by allowing Hamilton's principal function to be explicitly dependent on this time locally. This choice results in a Wheeler-DeWitt equation with time. A working solution for the de Sitter minisuperspace is shown. arXiv:2201.00809v4 [gr-qc] 3 Nov 2022 Introduction One traditional avenue to the quantization of gravity is the geometrodynamical one, represented by the infamous Wheeler-DeWitt (WDW) equation [1,2]. The equation is expected to describe the quantum evolution of the spatial components of the metric tensor of General Relativity (GR), but its solution and interpretation are long-standing problems [3]. In particular, a problem with time occurs when we try to interpret the WDW equation as a Schrödinger-type equation for gravity, because the state it describes appears to be stationary. To begin with, the absence of time from the WDW equation is a consequence of the fact that the first-class Hamiltonian constraint of GR, of which the WDW equation intends to be the quantization, specifically enforces time diffeomorphism. In other words, it ensures its dynamical laws are valid independently of our choice of time coordinate. When we consider the so-called Hamilton-Jacobi-Einstein (HJE) equation developed by Peres [4], which expresses the constraint on the 00 component of the Einstein field equations in the Hamilton-Jacobi formalism, it is clear that in the classical case, time is absent where it should appear, even though the theory is classical. We know, however, that the HJE equation does not describe a timeless geometry. The reason that the HJE equation is not problematic can be traced back to the fact that a notion of time exists for the evolution of the spatial geometry, as long as the classical notion of trajectory in superspace holds. In that sense, it appears that the actual problem with time is not that it is absent in the Schrödinger-type equation itself, but that we cannot introduce it as we do in the classical case, since space no longer evolves along classical trajectories. It is not clear what becomes of this time beyond the semi-classical level, when gravity does not act as a stage for matter fields, but rather partakes in the quantum dance. The absence of an external time in the description of GR, which is inherited by its quantization, is sometimes referred to as the "frozen formalism problem", and constitutes only one among other difficulties in the definition of time in classical and quantum physics [5]. In the present work, we will address only this particular aspect of the problem, ignoring its relations to others (a strictly related one being the definition of time-evolving observables for quantum gravity). Two classic reviews on this subject are Isham's [6] and Kuchař's [7]. We invite the reader to read these reviews for detailed references, and to gain a general idea of the large extent of variable approaches. The study of this aspect of the problem of time has certainly evolved significantly since the time of these reviews, with some issues of each approach being successfully addressed, but it ultimately remains open. Faced by the menace represented by the loss of a useful notion of time, three alternative reactions have been adopted by researchers: flight, fight, or freeze (corresponding to Isham's tempus ante quantum, tempus post quantum, and tempus nihil est). 1. Flight: Time is recognized as a fundamental element of our description of physical phenomena, and attempts are made to define it before quantization, as a functional of the canonical variables. This is a conservative approach that tries to obtain an "external" notion of time as that appearing in Schrödinger's equation. Fight: Time is recognized as a fundamental element of our description of physical phenomena, but it is retrieved only after the quantization. This type of approach fights against the interpretative problems presented by the quantum theory to obtain a novel definition of time. 3. Freeze: Timelessness is accepted, time is forsaken as a fundamental notion for the description of quantum gravity, and attempts are made to provide a complete quantum theory otherwise. The present work adopts a definition of time resulting from the semi-classical approximation of the WDW equation, an approach falling within the second category above. However, we do not limit to the semi-classical regime the definition of time identified by the semiclassical approximation. The point of the present note is in fact to suggest that the "frozen formalism" could be avoided by retaining the use of a classical notion of time suggested by the semi-classical approximation of the theory, even though quantum space does not evolve along classical trajectories. Therefore, our proposal belongs to the first category, in that it carries over to the quantum regime the definition of time justified by the semi-classical approximation. The definition of time that we adopt as a starting point is that naturally resulting from the semi-classical approximation of canonical quantum gravity obtained by expanding the total wave functional in inverse squares of the Planck mass [8,9,10]. Classical GR and the Schrödinger equation for non-gravitational fields straightforwardly recovered this approximation. Time, for the matter fields, is a multi-fingered (i.e., space dependent) functional generated by the classical evolution of background geometry along its trajectory in superspace. The operation has formal analogies with the Wentzel-Kramers-Brillouin (WKB) approximations of quantum mechanics, and the Born-Oppenheimer approximation from molecular physics [11,12,13]. In recent years, special attention was drawn to the problem of re-establishing the unitarity of time evolution in this approach (see, for example, [14,15,16,17]). For a recent work by this author which is related to the present one, see [18]. One key observation that motivated this approach was that this multi-fingered (or "WKB") functional time can be applied to the HJE equation itself, and we can make the presence of that classical time explicit in the HJE equation, which is the timeless form. This alone was the starting point of Wheeler and DeWitt, as the following memoir by one of the authors recalls. One day in 1965, John Wheeler had a two-hour stopover between flights at the Raleigh-Durham airport in North Carolina. He called Bryce DeWitt, then at the University of North Carolina in Chapel Hill, proposing to meet at the airport during the wait. Bryce showed up with the Hamilton-Jacobi equation of general relativity, published by Asher Peres not long before [...] Bryce mumbled the idea of repeating what Schröedinger did for the hydrogen atom: obtaining a wave equation by replacing the square of derivatives with (i times) a second derivative-a manner for undoing the optical approximation. [...] Wheeler was tremendously excited (he was often enthusiastic) and declared on the spot that the equation for quantum gravity had been found. [19,20] What if DeWitt had presented Wheeler with the HJE equation together with the notion of multi-fingered time? The resulting equation does present the functional time variable just as time appears in the Schrödinger equation, and preserves the correct classical limit for gravity. In Section 2, we briefly review the definition of this multi-fingered time. In Section 3, we rewrite the HJE equation, allowing an explicit local dependence of Hamilton's principal function on that time, and write the associated WDW equation with time. Finally, in Section 4, we discuss a simple realization in the de Sitter minisuperspace that is of special interest to quantum cosmology. Classical Time Evolution from the Hamiltonian Constraint For a straightforward introduction to the emergence of time in the semi-classical approximation of canonical quantum gravity, we refer the reader to the mentioned work by Isham [6], Section 5.4 and references therein. Here, we follow, with some variation, the notation adopted by Kiefer [9]. Consider a generic spacetime with line element ds 2 = −N 2 dt 2 + N i dtdx i + h ij dx i dx j .(1) Here, h ij = g µν , µ, ν ∈ {1, 2, 3} is the spatial metric, with Latin indices i, j ∈ (1, 2, 3), N i = g 0i is the shift function, and N = (−g 00 ) −1/2 is the lapse function. In the Hamiltonian formalism of GR, time parametrization invariance is enforced by a first-class Hamiltonian constraint (i.e., a constraint imposed on the Hamiltonian only after the equations of motion are satisfied). d 3 x (2M ) −1 G AB π A π B + V(h A ) + H φ (φ; h A ) = 0 .(2) (This constraint can be obtained from the variation of the super-Hamiltonian of GR with respect to the lapse function N in the Arnowitt-Deser-Misner (ADM) formalism [21]. Here, we intend to recall only the elements strictly necessary to follow our discussion.) In Equation (2), the capital indices, A, B = {ij} represent pairs of Latin indices, and G AB is the DeWitt metric G AB ≡ G ijkl = 1 2 √ h (h ik h jl + h il h jk − h ij h kl ) ,(3) underlying superspace, i.e., the space of spatial metrics up to differeomorphism invariance. The physical scale (we have set = c = 1) of quantum gravity is set by the "geometrodynamical mass" M , which is proportional to the square of the Planck mass m P M = (m P /2) 2 , m P = (8πG) −1/2 .(4) The geometrodynamical potential density V is V = 2M √ h(2Λ − (3) R) ,(5) where h and (3) R are the determinant and the Ricci scalar of the spatial metric, respectively. The Hamiltonian density operator H φ is taken to describe bosonic matter. The WDW equation results from an attempt to quantize the Hamiltonian constraint (2) straightforwardly, applying it to the "wave functional of the universe", Ψ[h A , φ], thus obtaining d 3 x (2M ) −1 G AB ∂ A ∂ B + V(h A ) + H φ (h A , φ) Ψ[h A , φ] = 0 ,(6) where all variables are promoted to the respective operators. For the sake of simplicity, in this section, we have adopted the trivial ordering, and the symbols ∂ A are used to indicate functional derivatives with respect to the metric component indicated by the double index. The evolution of bosonic quantum fields in classical curved spacetime is obtained by making the ansatz Ψ[h A , φ] = χ[h A ]ψ[φ; h A ](7) for the total wave functional, and considering a WKB-like expansion in inverse powers of M [9,10]. In doing so, one aims at wave functionals χ and ψ that describe the "heavy" (i.e., the spatial metric components) and "light" degrees of freedom (i.e., matter), respectively. Notice that ψ depends on the geometry only parametrically, which is indicated by the use of the semicolon. The method consists of substituting the expansion Ψ[h A , φ] = exp i ∞ n=0 M 1−n S n [h A , φ](8) in the WDW equation, and equating contributions to equal powers of M . To order M 2 , one obtains S 0 = S 0 [h A ]: the leading contribution is purely geometrodynamical. To order M 1 , one obtains the vacuum HJE equation d 3 x (2M ) −1 G AB ∂ A S G ∂ B S G + V = 0 ,(9) S G = M S 0 being the leading contribution to the phase of the wave functional (8). The HJE Equation (9) appears to be timeless due to the vanishing of the RHS. However, the time evolution of space can still be obtained from the Hamiltonian constraint (2) by expressing the canonical momenta (defined by the Lagrangian as π A = ∂L/∂ḣ A , and identified with π A = ∂ A S G in the Hamilton-Jacobi formalism) in terms of the geometrodynamical velocities. From the Hamiltonian equations of motion in the ADM formalism, these are given bẏ h A = N π A + 2N (A) ,(10) N (A) being a shorthand for N (i;j) . The fact that time evolution is retrieved by such substitution, as the Hamiltonian is constrained, is an important point. At this point, define ψ 1 [φ; h A ] = exp (iS 1 [φ, h A ] ) and require conservation of the current associated with χ. Then, to order M 0 , one gets the following functional equation for matter d 3 x iM −1 G AB ∂ A S G ∂ B − H φ ψ 1 [φ; h A ] = 0 .(11) By using as time the local "multi-fingered" time τ = τ (x) of the parametrization generated by the classical momenta along the classical trajectory in superspace G AB π A ∂ B x τ (y) = δ(y − x) ,(12) Equation (11) gives the Schrödinger-type functional equation d 3 x i • ψ 1 [φ; h A ] − H φ ψ 1 [φ; h A ] = 0 ,(13) where we employ a circle over the variable to indicate the functional derivative with respect to τ δ δτ = M −1 G AB π A ∂ B .(14) Notice that in normal coordinates (N = 1, N 0i = 0), the Equation (13) reduces to the functional Schrödinger equation for bosonic fields. In this case, the passage to the partial derivative with respect to time is granted by the fact that the matter wave functional ψ 1 [φ; h A ] depends explicitly on time only through the background metric, which appears as a set of local parameters, and employing the resulting relation between momenta and velocities (10), (3) reduces to an application of the chain rule. The Time Evolution of Quantum Space The functional derivative with respect to τ of a metric component, intended as the functional h A (x) = d 3 y h A (y) δ(y − x) ,(15) gives the relation between velocities and momenta, • h A = M −1 G AB π A .(16) Putting the relation (16) back into the Hamiltonian equation gives the equations of motion with respect to τ . In other words, we can use τ not only to describe the evolution of quantum fields with respect to the background metric, but also to describe the evolution of the background metric itself. It is a favorable parametrization in that it makes the form of the equations of motion simpler and not explicitly dependent on the coordinate choice (1). Incidentally, notice that, in Peres' HJE equation, the spatial metric components are defined as functions of space alone. How they become a function of time seems to be a problem that is not addressed in the literature. In the present treatment, they become functions of time precisely by defining their dependence on time according to (16). The main objection in extending the use of multi-fingered time to the quantum evolution is that classical trajectories are lost in that regime. While this is true, we still know what the classical trajectory is, and we can use the "natural" parametrization (i.e., the parametrization in which all equations of motion appear simple as (16)) along it to describe evolution along non-classical trajectories between wave fronts of constant multi-fingered time. Working with the vacuum model, we can make multi-fingered time explicit in the HJE Equation (9) by rewriting it as d 3 x ∂ τ S G + (2M ) −1 G AB ∂ A S G ∂ B S G + V = 0 ,(17) and requiring that d 3 x∂ τ S G = 0 .(18) Notice that the derivative with respect to τ is partial. As we previously observed, WKB time (3) only takes into account the dependency on τ through the geometrodynamical degrees of freedom, but S G could, in principle, also be explicitly dependent on it. What we are doing is simply allowing for this possibility. Adding and subtracting the τderivative of S G , and substituting the velocities (16), integration of the HJE Equation (17) tells us that S G is indeed Hamilton's principal function, defined as a functional integral of the Lagrangian. Notwithstanding the fact that the condition (18) ensures that the Hamiltonian constraint still holds, it allows for the action to remain dependent explicitly on τ locally. We may then try to quantize the HJE Equation (17)à la Schrödinger, and require the global condition (18) to be true only in the classical limit. What we obtain is a WDW equation with time d 3 x −i ∂Ψ ∂τ + (2M ) −1 G AB ∂ A ∂ B + V Ψ = 0 .(19) Both the introduction of the coordinate independent functional time derivative and the (classically redundant) condition (18) are necessary to obtain Equation (19). The second condition is somehow reminiscent of another approach [22], where time is recovered by weakening the classical Hamiltonian constraint, required to hold only on average in the quantum regime. In that work, the problem of which time to use to describe the evolution is not addressed, and the condition on time dependence is stricter than ours (see (6) in the referenced paper), resulting in a wave function whose phase does not depend on time even locally. The de Sitter Minisuperspace The results of the previous section, i.e., the time evolution Equation (19) combined with the global condition (18), are only formal. Equation (19) still presents the same issues as the "timeless" WDW Equation (6). Besides that, using a spatial-dependent τ to parametrize the spatial geometry is more easily said than done. In the following, we will consider, by way of example, the solution for the spatially flat de Sitter universe, described by the line element ds 2 = −dt 2 + a(t) 2 δ ij dx i dx j .(20) Here, a(t) is the scale factor. We will set M = 1/12. As a geometrodynamical variable, rather than the scale factor itself. It will be convenient to adopt q = (2a) 3 2 3 d 3 x 1 2 ,(21) which is proportional to the square root of the co-moving spatial volume considered. The Ricci scalar of the spatial metric vanishes in this model, and the Hilbert-Einstein action Lagrangian, reads simply L = − 1 2 q 2 + ω 2 q 2 .(22) Here, we have defined the constant ω = 3Λ/4. The canonical momentum is π q = −q, and the Hamiltonian is H = − 1 2 π 2 q + 1 2 ω 2 q 2(23) The Hamiltonian constraint then gives the Friedmann equation q q 2 = ω 2 .(24) This allows us to obtain the classical time evolution of spatial volume q(t) = q(0) exp (ωt) .(25) The HJE Equation (9) reduces to − 1 2 ∂S ∂q 2 + 1 2 ω 2 q 2 = 0 .(26) Classically, Hamilton's principal function depends on time only implicitly, and is of the form S = ∓ ω 2 q 2 − q 2 i .(27) Using Equation (25), one can check that the (one-fingered) time associated with this action indeed coincides with forward coordinate time when we choose the negative sign. Moving onto the quantization, notice that with our choice of variable, the DeWitt metric, that in terms of the scale factor reads G aa = −2a, is now G qq = −(3q) 2 3 .(28) This simplifies the measure −G (a) da → dq(29) and, adopting the Laplace-Beltrami operator ordering for the kinetic operator, we have −G (a) −1 ∂ a −G (a) G aa ∂ a → ∂ 2 q .(30) Then, in this minisuperspace, the WDW Equation (19) reads iΨ = 1 2 ∂ 2 q Ψ + 1 2 ω 2 q 2 Ψ ,(31) which is essentially the one-dimensional Schrödinger equation for the so-called inverted harmonic oscillator. See [23] for a recent review, and [24] for an application to the study of a scalar field in slow-roll inflation. The difference in our case is that the variable is constrained to the positive axis only, and time appears with the opposite sign. As in [24], we require the Gaussian ansatz ψ(q, t) = A(t) exp −B(t)q 2 .(32) The equality in (31) of the terms of null and second order in q imposes −iȦ = A B and −iḂ = 1 2 ω 2 + 2B 2 .(34) From the evolution Equation (34), we have B(t) = ω 2 tan (φ + iωt) .(35) Here φ is the real part of the constant of integration. The imaginary part of the constant is merged into the choice of initial time, instead, so that the width of the state is minimized at t = 0. The value of φ determines the greater (<π/4) or lesser (>π/4) peaking of the distribution in q, rather than the conjugate momentum at time t = 0. Substituting B(t) in Equation (33), for a normalized solution, we obtain A(t) = 2 π ω sin(2φ) 1/4 (cos(φ + iωt)) − 1 2 .(36) The expectation value for q is q = (2πω sin(2φ)) − 1 2 cos(2φ) + cosh(2ωt) | cos(φ + iωt)| .(37) At late times, we correctly recover the classical inflationary expansion q ∝ exp(ωt) of Equation (25). In this limit, the phase of ψ approximates the classical action (27) (up to a global phase, which fixes the initial value) and loses its explicit dependence on time, thus reducing to the classical action (18). On the other hand, when we approach the time t = 0, where the state is maximally contracted for the given value of φ, the expectation value of the scale variable deviates from the classical one, which is directed at asymptotic convergence to zero: we observe instead the state of the de Sitter universe bouncing back from an earlier phase of contraction (see Figure 1). Figure 1: Time evolution of the expectation value of the scale variable q for φ = π/4. We set ω = 1. Conclusions In this short note, we have proposed to extend to the quantum regime the use of the multi-fingered time originating from the semi-classical WKB approximation of geometrodynamics. We have done so by rewriting the classical HJE equation for vacuum space to include an explicit time dependency on Hamilton's principal function, and requiring this dependency to vanish globally. The quantization provides a WDW equation that describes the evolution of the state with respect to classical multi-fingered time. Quantum matter fields can be included by appropriately augmenting the Hamiltonian operator. The classical limit will still be granted. The main purpose of this work was to provide a formal WDW equation, (19), with a clear and working notion of time. This result, however, does not help with the original mathematical difficulties of the WDW equation, such as the indefiniteness of the superspace metric, or the divergence of functional derivatives in the full theory. However, we showed by example of a minisuperspace model of flat de Sitter universe that exact normalizable solutions can be easily found and interpreted. In particular, for the de Sitter universe, we found a wellbehaved Gaussian solution, which shows a state that bounces back at the time of maximal contraction, thus avoiding the classical asymptotic regression to nihil. The next step along this line of research could be the formalization of our heuristic approach both in the Hamiltonian and the Lagrangian formalism, where the wave functional could be constructed in terms of a path integral over foliations of constant multi-fingered time. On the side of application, it would be fundamental to work out exact solutions that include quantum matter degrees of freedom. Expanding on the de Sitter model introduced here by adding perturbations could be relevant for early inflationary cosmology. Furthermore, beyond cosmology, an application to time evolution during the last stage of gravitational collapse could be of interest. Quantum Theory of Gravity. I. The Canonical Theory. B S Dewitt, Phys. Rev. 160DeWitt, B.S. Quantum Theory of Gravity. I. The Canonical The- ory. Phys. Rev. 1967, 160, 1113-1148. Superspace and the nature of quantum geometrodynamics. J A Wheeler, Batelles RencontresDeWitt, B.S., J. A. Wheeler, J.A., Eds.BenjaminNew York, NY, USAWheeler, J.A. Superspace and the nature of quantum ge- ometrodynamics. In Batelles Rencontres; DeWitt, B.S., J. A. Wheeler, J.A., Eds.; Benjamin: New York, NY, USA, 1968; pp. 242-307. . C Quantum Rovelli, Gravity, Cambridge University PressCambridge, UKRovelli, C. Quantum Gravity; Cambridge University Press: Cambridge, UK, 2004. On Cauchy's problem in General Relativity. A Peres, Nuovo Cimento. 26Peres, A. On Cauchy's problem in General Relativity. Nuovo Cimento 1962, 26, 53-62. The Problem of Time. E Anderson, SpringerNew York, NY, USAAnderson, E. The Problem of Time; Springer: New York, NY, USA, 2017. Canonical Quantum Gravity and the Problem of Time. C J Isham, Integrable Systems, Quantum Groups, and Quantum Field Theories. Ibort, L.A., Rodríguez, M.A.Dordrecht, GermanySpringerIsham, C.J. Canonical Quantum Gravity and the Problem of Time. In Integrable Systems, Quantum Groups, and Quantum Field Theories; Ibort, L.A., Rodríguez, M.A., Eds; Springer: Dordrecht, Germany, 1993. Time and interpretations of quantum gravity. K V Kuchar, Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics. the 4th Canadian Conference on General Relativity and Relativistic AstrophysicsKuchar, K.V. Time and interpretations of quantum gravity. In Proceedings of the 4th Canadian Conference on General Rela- tivity and Relativistic Astrophysics; . G Kunstatter, D Vincent, J Williams, World ScientificSingaporeKunstatter, G. Vincent, D., Williams, J., Eds.; World Scientific: Singapore, 1992. Quantum gravitational corrections to the functional Schrödinger equation. C Kiefer, T P Singh, Phys. Rev. D. Kiefer, C.; Singh, T.P. Quantum gravitational corrections to the functional Schrödinger equation. Phys. Rev. D 1991 44, 1067. The semiclassical approximation to quantum gravity. C Kiefer, In Canonical Gravity: From Classical to Quantum. Kiefer, C. The semiclassical approximation to quantum grav- ity. In Canonical Gravity: From Classical to Quantum; . J Ehlers, H Friedrich, SpringerBerlin/Heidelberg, GermanyEhlers, J., Friedrich, H., Eds.; Springer: Berlin/Heidelberg, Germany, 1994; pp. 170-212. . C Quantum Kiefer, Gravity, Oxford University PressOxford, UKKiefer, C. Quantum Gravity; Oxford University Press: Oxford, UK, 2007. The Born-Oppenheimer approach to the matter-gravity system and unitarity. C Bertoni, F Finelli, G Venturi, Class. Quant. Grav. 132375Bertoni, C.; Finelli, F.; Venturi, G. The Born-Oppenheimer ap- proach to the matter-gravity system and unitarity. Class. Quant. Grav. 1996 13, 2375. The Born-Oppenheimer method, quantum gravity and matter. A Y Kamenshchik, A Tronconi, G Venturi, Class. Quant. Grav. 3515012Kamenshchik, A.Y.; Tronconi, A.; Venturi, G. The Born- Oppenheimer method, quantum gravity and matter. Class. Quant. Grav. 2017 35, 015012. Time in quantum theory, the Wheeler-DeWitt equation and the Born-Oppenheimer approximation. A Y Kamenshchik, A Tronconi, T Vardanyan, G Venturi, Int. J. Mod. Phys. D. Kamenshchik, A.Y.; Tronconi, A.; Vardanyan, T.; Venturi, G. Time in quantum theory, the Wheeler-DeWitt equation and the Born-Oppenheimer approximation. Int. J. Mod. Phys. D 2019 28, 1950073. Semiclassical approximation of the Wheeler-DeWitt equation: Arbitrary orders and the question of unitarity. C Kiefer, D Wichmann, Gen. Relativ. Gravit. 5066Kiefer, C.; Wichmann, D. Semiclassical approximation of the Wheeler-DeWitt equation: Arbitrary orders and the question of unitarity. Gen. Relativ. Gravit. 2018 50, 66. Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology. L Chataignier, Z. Für Nat. A. 1069Chataignier, L. Gauge Fixing and the Semiclassical Interpreta- tion of Quantum Cosmology. Z. Für Nat. A 2019, 74, 1069. Construction of quantum Dirac observables and the emergence of WKB time. L Chataignier, Phys. Rev. 202086001Chataignier, L. Construction of quantum Dirac observables and the emergence of WKB time. Phys. Rev. D 2020, 101, 086001. Nonunitarity problem in quantum gravity corrections to quantum field theory with Born-Oppenheimer approximation. F Di Gioia, G Maniccia, G Montani, J Niedda, Phys. Rev. D. 2022103511Di Gioia, F.; Maniccia, G.; Montani, G.; Niedda, J. Nonunitar- ity problem in quantum gravity corrections to quantum field the- ory with Born-Oppenheimer approximation. Phys. Rev. D 2022, 103, 103511. The Functional Schrödinger Equation in the Semiclassical Limit of Quantum Gravity with a Gaussian Clock Field. M Rotondo, Rotondo, M. The Functional Schrödinger Equation in the Semi- classical Limit of Quantum Gravity with a Gaussian Clock Field. Universe 2020 6, 176. The strange equation of quantum gravity. C Rovelli, Class. Quant. Grav. 32124005Rovelli, C. The strange equation of quantum gravity. Class. Quant. Grav. 2015, 32, 124005. The Pursuit of Quantum Gravity. C Dewitt-Morette, Springer58Berlin/Heidelberg, GermanyDeWitt-Morette, C. The Pursuit of Quantum Gravity; Springer: Berlin/Heidelberg, Germany, 2011; p. 58. Republication of: The dynamics of general relativity. R Arnowitt, S Deser, C W Misner, Gen. Relativ. Gravit. 40Arnowitt, R.; Deser, S.; Misner, C.W. Republication of: The dynamics of general relativity. Gen. Relativ. Gravit. 2008, 40, 1997-2027. Time in quantum gravity by weakening the Hamiltonian constraint. H Nikolic, gr-qc/0312063. 2003arXiv preprintNikolic, H. Time in quantum gravity by weakening the Hamilto- nian constraint. arXiv preprint gr-qc/0312063. 2003. Physics of the Inverted Harmonic Oscillator: From the lowest Landau level to event horizons. V Subramanyan, S S Hegde, S Vishveshwara, B Bradlyn, Ann. Phys. 2021168470Subramanyan, V.; Hegde, S.S.; Vishveshwara, S.; Bradlyn, B. Physics of the Inverted Harmonic Oscillator: From the lowest Landau level to event horizons. Ann. Phys. 2021, 435, 168470. Quantum mechanics of the scalar field in the new inflationary universe. A H Guth, S.-Y Pi, Phys. Rev. D. 32Guth, A.H.; Pi, S.-Y. Quantum mechanics of the scalar field in the new inflationary universe. Phys. Rev. D 1985, 32, 1889-1920.

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{'abstract': "The equation for canonical gravity produced by Wheeler and De-Witt in the late 1960s still presents difficulties both in terms of its mathematical solution and its physical interpretation. One of these issues is, notoriously, the absence of an explicit time. In this short note, we suggest one simple and straightforward way to avoid this occurrence. We go back to the classical equation that inspired Wheeler and DeWitt (namely, the Hamilton-Jacobi-Einstein equation) and make explicit, before quantization, the presence of a known, classically meaningful notion of time. We do this by allowing Hamilton's principal function to be explicitly dependent on this time locally. This choice results in a Wheeler-DeWitt equation with time. A working solution for the de Sitter minisuperspace is shown. arXiv:2201.00809v4 [gr-qc] 3 Nov 2022", 'arxivid': '2201.00809', 'author': ['Rotondo Marcello '], 'authoraffiliation': [], 'corpusid': 245650281, 'doi': '10.3390/universe8110580', 'github_urls': [], 'n_tokens_mistral': 8525, 'n_tokens_neox': 7435, 'n_words': 4813, 'pdfsha': '0a3212d56a24f85e3fd53d29fb4339a7efe028a2', 'pdfurls': ['https://export.arxiv.org/pdf/2201.00809v4.pdf'], 'title': ['A Wheeler-DeWitt Equation with Time', 'A Wheeler-DeWitt Equation with Time'], 'venue': []}

arxiv

Exact solution of the area reactivity model of an isolated pair November 12, 2013 Thorsten Prüstel Laboratory of Systems Biology National Institute of Allergy and Infectious Diseases National Institutes of Health Martin Meier-Schellersheim Laboratory of Systems Biology National Institute of Allergy and Infectious Diseases National Institutes of Health Exact solution of the area reactivity model of an isolated pair November 12, 2013 We investigate the reversible diffusion-influenced reaction of an isolated pair in two space dimensions in the context of the area reactivity model. We compute the exact Green's function in the Laplace domain for the initially unbound molecule. Furthermore, we calculate the exact expression for the Green's function in the time domain by inverting the Laplace transform via the Bromwich contour integral. The obtained results should be useful for comparing the behavior of the area reactivity model with more conventional models based on contact reactivity. * Introduction The Smoluchowski model is widely used in the theory of diffusioninfluenced reactions [9,7]. According to this picture, a pair of molecules separated by a distance r may react when they encounter each other at a critical distance r = a via their diffusive motion. Hence, reactive molecules can be modeled by solutions of the diffusion equation that satisfy certain types of boundary conditions (BC) at the encounter distance r = a. In the case of an isolated pair, exact expressions for Green's functions (GF) in the time domain, describing irreversible and reversible reactions in one, two and three space dimensions, have been obtained [3,2,6,5]. However, there are alternative approaches to describe the reversible diffusion-influenced reaction of an isolated pair. Ref. [4] discussed the so-called volume reactivity model that eliminates the distinct role of the encounter radius r = a and instead postulates that the reaction can happen throughout the spherical volume r ≤ a. In the present manuscript, we discuss the corresponding model in two dimensions (2D) and hence refer to it as the "area reactivity" model. Diffusion in 2D is special from both a conceptual and technical point of view. Conceptually, it is the critical dimension regarding recurrence and transience of random walks [8]. Technically, the mathematical treatment appears to be more involved than in 1D and 3D [6]. A system of two molecules A and B with diffusion constants D A and D B , respectively, can also be described as the diffusion of a point-like molecule with diffusion constant D = D A + D B around a static disk. More precisely, the area-reactivity model assumes that the molecule undergoes free diffusion apart from inside the static "reaction disk" of radius r = a, where it may react reversibly. Without loss of generality, we assume that the disk's center is located at the origin. A central notion is the probability density function (PDF) p(r, t|r 0 ) that gives the probability to find the molecule unbound at a distance equal to r at time t, given that the distance was initially r 0 at time t = 0. Note that in contrast to the contact reactivity model, p(r, t|r 0 ) is also defined for r < a. Moreover, because the molecule may bind anywhere within the disk r < a, it makes sense to define another PDF q(r, t|r 0 ), which yields the probability to find the molecule bound at a distance equal to r < a at time t, given that the distance was initially r 0 at time t = 0. The rates for association and dissociation are κ r Θ(a − r)p(r, t|r 0 ) and κ d q(r, t|r 0 ), respectively, where Θ(x) refers to the Heaviside step-function that vanishes for x < 0 and assumes unity otherwise. Furthermore, it is assumed that the dissociated molecule is released at the same point where it assumed its bound state. The equations of motion for the PDF p(r, t|r 0 ) and q(r, t|r 0 ) are coupled and read [4] ∂p(r, t|r 0 ) ∂t = L r p(r, t|r 0 ) − κ r Θ(a − r)p(r, t|r 0 ) + κ d q(r, t|r 0 ), (1.1) ∂q(r, t|r 0 ) ∂t = κ r Θ(a − r)p(r, t|r 0 ) − κ d q(r, t|r 0 ), (1.2) where L r = D ∂ 2 ∂r 2 + 1 r ∂ ∂r . (1.3) The equations of motion have to be supplemented by BC at the origin and at infinity, respectively, lim r→∞ p(r, t|r 0 ) = 0, (1.4) lim r→0 r ∂p(r, t|r 0 ) ∂r = 0. (1.5) In the present manuscript, we focus on the case of the initially unbound molecule. Therefore, the initial conditions are 2πr 0 p(r, 0|r 0 ) = δ(r − r 0 ), (1.6) q(r, 0|r 0 ) = 0. (1.7) Exact Green's function in the Laplace domain By applying the Laplace transform, Eqs. (1.1)-(1.2) become sp(r, s|r 0 ) − p(r, 0|r 0 ) = L rp (r, s|r 0 ) − κ r Θ(a − r)p(r, s|r 0 ) +κ dq (r, s|r 0 ), (2.1) sq(r, s|r 0 ) − q(r, 0|r 0 ) = κ r Θ(a − r)p(r, s|r 0 ) − κ dq (r, s|r 0 ), (2.2) where s denotes the Laplace space variable. We use Eq. (1.7) to obtain from Eq. (2.2)q (r, s|r 0 ) = κ r s + κ d Θ(a − r)p(r, s|r 0 ). (2.3) Now we can eliminateq(r, s|r 0 ) from Eq. (2.1) L r − s − sκ r s + κ d Θ(a − r) p(r, s|r 0 ) = − δ(r − r 0 ) 2πr , (2.4) where we used Eq. (1.6). In the following, we will calculate the GF separately on the two different domains defined by r > a and r < a. The two obtained solutions will still contain unknown constants. The GF can then be completely determined by matching both expressions upon continuity requirements at r = a. Henceforth, we will denote the GF within r < a and outside r > a the reactive disk by p < (r, t|r 0 ) and p > (r, t|r 0 ), respectively. Also, throughout this manuscript we assume that the molecule was initially located outside the reaction area r 0 > a. Then, we make the following ansatz for the Laplace transform of the GF p > (r, t|r 0 ) outside the disk r > a, p > (r, s|r 0 ) =p 0 (r, s|r 0 ) +f (r, s|r 0 ), (2.5) wherep 0 (r, s|r 0 ) = 1 2πD I 0 (vr 0 )K 0 (vr), r > r 0 I 0 (vr)K 0 (qr 0 ), r < r 0 (2.6) is the Laplace transform of the free-space GF, cf. [3, Ch. 14.8, Eq. (2)]. I 0 (x), K 0 (x) denoted 2f dr 2 + 1 r df dr − v 2f = 0. (2.8) The general solution to Eq. (2.8) is given bỹ f (r, v) = B(s, r 0 )I 0 (vr) + C(s, r 0 )K 0 (vr),(2.9) where B(s, r 0 ), C(s, r 0 ) are "constants" that may depend on s and r 0 . Because we require the BC Eq. (1.4) and lim x→∞ I 0 (x) → ∞, the coefficient B(s, r 0 ) has to vanish and the solution becomes, f (r, v|r 0 ) = C(v, r 0 )K 0 (vr). (2.10) Next, turning to the case r < a, the GF satisfies d 2p< dr 2 + 1 r dp < dr − w 2p< = 0, (2.11) where w is defined by w := v s + κ r + κ d s + κ d . (2.12) Therefore, the general solution, which takes into account the BC Eq. (1.5) is p < (r, w|r 0 ) = A(s, r 0 )I 0 (wr), (2.13) because lim x→0 xK 1 (x) = 0. The two "constants" A(s, r 0 ) and C(s, r 0 ) can be determined by the requirement that the GF and its derivative have to be continuous at r = ap < (r = a, s|r 0 ) =p > (r = a, s|r 0 ) (2.14) ∂p < (r, s|r 0 ) ∂r r=a = ∂p > (r, s|r 0 ) ∂r r=a (2.15) Using Eqs. (2.5), (2.6), (2.10), (2.13) as well as I 0 (x) = I 1 (x), (2.16) K 0 (x) = −K 1 (x), (2.17) x −1 = I 0 (x)K 1 (x) + I 1 (x)K 0 (x),(2.A(s, r 0 ) = K 0 (vr 0 ) 2πaDN , (2.19) C(s, r 0 ) = K 0 (vr 0 ) 2πaDK 0 (va) I 0 (wa) N − aI 0 (va) , (2.20) where we introduced N = vI 0 (wa)K 1 (va) + wI 1 (wa)K 0 (va). (2.21) Exact Green's function in the time domain To find the corresponding expressions for p < (r, t|r 0 ), p > (r, t|r 0 ) in the time domain, we apply the inversion theorem for the Laplace transformation To calculate the integral C2 , we choose s = Dx 2 e iπ . Then, v = ix, for s ∈] − ∞, 0[ (3.3) w = ix Dx 2 − ϕ Dx 2 − κ d ≡ ixξ 1 for s ∈] − ∞, −ϕ[, (3.4) w = x ϕ − Dx 2 Dx 2 − κ d ≡ xξ 2 for s ∈] − ϕ, −κ d [, (3.5) w = ix ϕ − Dx 2 κ d − Dx 2 = ixξ 1 for s ∈] − κ d , 0[, (3.6) (3.7) We now make use of [3, Append. 3, Eqs. (25), (26))] I n (xe ±πi/2 ) = e ±nπi/2 J n (x), (3.8) where we introduced K n (xe ±πi/2 ) = ± 1 2 πie ∓nπi/2 [−J n (x) ± iY n (x)].g (2) (r, r 0 , x) ≡ g(2) R (r, r 0 , x) + ig (2) I (r, r 0 , x), = I 0 (xξ 2 r) η(r 0 ) + iλ(r 0 ) α 2 + β 2 , (3.11) g (1) (r, r 0 , x) ≡ g (1) R (r, r 0 , x) + ig(1)I (r, r 0 , x), = J 0 (xξ 1 r) ω(r 0 ) + iκ(r 0 ) γ 2 + δ 2 ,(3.12) and η(r 0 ) = αY 0 (xr 0 ) + βJ 0 (xr 0 ), (3.13) λ(r 0 ) = αJ 0 (xr 0 ) − βY 0 (xr 0 ), (3.14) where * denotes complex conjugation. Thus, one obtains for the GF p < (r, t|r 0 ) on the domain r < a α = ξ 2 I 1 (ξ 2 xa)Y 0 (xa) + I 0 (ξ 2 xa)Y 1 (xa), (3.15) β = ξ 2 I 1 (ξ 2 xa)J 0 (xa) + I 0 (ξ 2 xa)J 1 (xa), (3.16) ω(r 0 ) = γY 0 (xr 0 ) + δJ 0 (xr 0 ), (3.17) κ(r 0 ) = γJ 0 (xr 0 ) − δY 0 (xr 0 ), (3.18) γ = ξ 1 J 1 (ξ 1 xa)Y 0 (xa) − J 0 (ξ 1 xa)Y 1 (xa), (3.19) δ = ξ 1 J 1 (ξ 1 xa)J 0 (xa) − J 0 (ξ 1 xa)J 1 (xa).p < (r, t|r 0 ) = − 1 π Im C2 e stp< (r, s|r 0 )ds = − 1 π 2 a √ ϕ D √ κ d D e −Dx 2 t g (2) I (r, r 0 , x)dx − √ κ d D 0 e −Dx 2 t g (1) I (r, r 0 , x)dx − ∞ √ ϕ D e −Dx 2 t g(1) I (r, r 0 , x)dx , (3.22) Analogously, we can proceed to compute the GF for the region r > a. Therefore, we only give the result p > (r, t|r 0 ) = 1 4πDt e −(r 2 +r 2 0 )/4Dt I 0 rr 0 2Dt + 1 π 2 a √ κ d D 0 e −Dx 2 t h (1) (r, r 0 , x)dx + ∞ √ ϕ D e −Dx 2 t h (1) (r, r 0 , x)dx − √ ϕ D √ κ d D e −Dx 2 t h (2) (r, r 0 , x)dx − 1 2π ∞ 0 e −Dx 2 t h (3) (r, r 0 , x)xdx, (3.23) where we defined h (1) (r, r 0 , x) = J 0 (xξ 1 a) ρ(r)ω(r 0 ) + ψ(r)κ(r 0 ) [γ 2 + δ 2 ][J 2 0 (xa) + Y 2 0 (xa)] , (3.24) h (2) (r, r 0 , x) = I 0 (xξ 2 a) ρ(r)η(r 0 ) + ψ(r)λ(r 0 ) [α 2 + β 2 ][J 2 0 (xa) + Y 2 0 (xa)] , (3.25) h (3) (r, r 0 , x) = J 0 (xa) Π(r, r 0 )Y 0 (xa) + Ω(r, r 0 )J 0 (xa) J 2 0 (xa) + Y 2 0 (xa) , (3.26) and Clearly, q(r, t|r 0 ) vanishes for r > a. The case of an initially unbound molecule with r 0 < a and the case of the initially bound molecule will be considered in a forthcoming manuscript. ρ(r) = J 0 (xr)Y 0 (xa) − Y 0 (xr)J 0 (xa), (3.27) ψ(r) = J 0 (xr)J 0 (xa) + Y 0 (xr)Y 0 (xa), (3.28) Ω(r, r 0 ) = J 0 (xr)J 0 (xr 0 ) − Y 0 (xr)Y 0 (xr 0 ),(3. thatp < (r, s|r 0 ) has three branch points at s = 0, −κ d and s = −κ r − κ d ≡ −ϕ. Therefore, to calculate the Bromwich integral, we use the contour ofFig. 1with a branch cut along the negative real axis, cf.[3, Ch. 12.3, FIG. 40]. We arrive at c+i∞ c−i∞ e stp< (r, s|r 0 )ds = − C2 e psp< (r, s|r 0 )ds − C4 e stp< (r, s|r 0 )ds. (3.2) (x), Y n (x) refer to the Bessel functions of first and second kind, respectively [1, Sect. 9.1]. It follows that C2 e stp< (r, s|r 0 Dx 2 t g (1) (r, r 0 , x)dx , (3.10) to calculate the integral along the contour C 4 , we choose s = Dx 2 e −iπ and after an analogous calculation one finds that C2 e stp< (r, s|r 0 )ds = − C4 e stp< (r, s|r 0 )ds * , (3.21) r, r 0 ) = Y 0 (xr)J 0 (xr 0 ) + J 0 (xr)Y 0 (xr 0 ).(3.30) Note that the first term appearing on the rhs of Eq. (3.23) is the inverse Laplace transform of Eq. (2.6), cf. [3, Ch. 14.8, Eq.(2)].Finally, we can compute an exact expression for q(r, t|r 0 ) by virtue of Eq. (2.3) and the convolution theorem of the Laplace transform. We obtain for r Figure 1 : 1Integration contour used for calculating the GF in the time domain, Eq. (3.2). AcknowledgmentsThis research was supported by the Intramural Research Program of the NIH, National Institute of Allergy and Infectious Diseases. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. M Abramowitz, I A Stegun, DoverNew YorkM. Abramowitz and I.A. Stegun. Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1965. . N Agmon, J. Chem. Phys. 812811N. Agmon. J. Chem. Phys., 81:2811, 1984. Conduction of Heat in Solids. H S Carslaw, J C Jaeger, Clarendon PressNew YorkH.S. Carslaw and J.C. Jaeger. Conduction of Heat in Solids. Clarendon Press, New York, 1986. . S S Khokhlova, N Agmon, J. Chem. Phys. 137184103S.S. Khokhlova and N. Agmon. J. Chem. Phys., 137:184103, 2012. . H Kim, K J Shin, Phys. Rev. Lett. 821578H. Kim and K.J. Shin. Phys. Rev. Lett., 82:1578, 1999. . T Prüstel, M Meier-Schellersheim, J. Chem. Phys. 13754104T. Prüstel and M. Meier-Schellersheim. J. Chem. Phys., 137:054104, 2012. Diffusion Limited Reactions. S A Rice, ElsevierNew YorkS. A. Rice. Diffusion Limited Reactions. Elsevier, New York, 1985. . D Toussaint, F Wilczek, J. Chem. Phys. 782642D. Toussaint and F. Wilczek. J. Chem. Phys., 78:2642, 1983. . M Smoluchowski, Z. Phys. Chem. 92129M. von Smoluchowski. Z. Phys. Chem., 92:129, 1917.

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{'abstract': "We investigate the reversible diffusion-influenced reaction of an isolated pair in two space dimensions in the context of the area reactivity model. We compute the exact Green's function in the Laplace domain for the initially unbound molecule. Furthermore, we calculate the exact expression for the Green's function in the time domain by inverting the Laplace transform via the Bromwich contour integral. The obtained results should be useful for comparing the behavior of the area reactivity model with more conventional models based on contact reactivity. *", 'arxivid': '1311.2125', 'author': ['Thorsten Prüstel \nLaboratory of Systems Biology\nNational Institute of Allergy and Infectious Diseases National Institutes of Health\n\n', 'Martin Meier-Schellersheim \nLaboratory of Systems Biology\nNational Institute of Allergy and Infectious Diseases National Institutes of Health\n\n'], 'authoraffiliation': ['Laboratory of Systems Biology\nNational Institute of Allergy and Infectious Diseases National Institutes of Health\n', 'Laboratory of Systems Biology\nNational Institute of Allergy and Infectious Diseases National Institutes of Health\n'], 'corpusid': 17798526, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5346, 'n_tokens_neox': 4647, 'n_words': 2489, 'pdfsha': '66ea4701fd9ddb8db211c6acec9a081b0ce89ada', 'pdfurls': ['https://arxiv.org/pdf/1311.2125v1.pdf'], 'title': ['Exact solution of the area reactivity model of an isolated pair', 'Exact solution of the area reactivity model of an isolated pair'], 'venue': []}

arxiv

Vocoder drift in x-vector-based speaker anonymization Michele Panariello EURECOM Sophia AntipolisFrance Massimiliano Todisco EURECOM Sophia AntipolisFrance Nicholas Evans EURECOM Sophia AntipolisFrance Vocoder drift in x-vector-based speaker anonymization Index Terms: speaker anonymizationautomatic speaker veri- ficationprivacy State-of-the-art approaches to speaker anonymization typically employ some form of perturbation function to conceal speaker information contained within an x-vector embedding, then resynthesize utterances in the voice of a new pseudo-speaker using a vocoder. Strategies to improve the x-vector anonymization function have attracted considerable research effort, whereas vocoder impacts are generally neglected. In this paper, we show that the impact of the vocoder is substantial and sometimes dominant. The vocoder drift, namely the difference between the x-vector vocoder input and that which can be extracted subsequently from the output, is learnable and can hence be reversed by an attacker; anonymization can be undone and the level of privacy protection provided by such approaches might be weaker than previously thought. The findings call into question the focus upon x-vector anonymization, prompting the need for greater attention to vocoder impacts and stronger attack models alike. Introduction The task of speaker anonymization broadly refers to the processing of speech recordings to conceal the speaker identity while preserving the linguistic and paralinguistic content. The topic has attracted increasing research interest in recent years, in particular through the VoicePrivacy Challenge [1,2], which was founded in 2020 to define the problem, provide strong baselines, foster progress and identify research priorities. No matter what the application, anonymization should protect an appropriate trade-off between privacy and utility. Privacy can be estimated using automatic speaker verification (ASV) and an equal error rate (EER) metric to gauge the ability of an attacker to infer the true speaker identity. Utility is estimated using automatic speech recognition (ASR) and a word error rate (WER) metric which reflects the degree to which linguistic and paralinguistic content is preserved. Most anonymization solutions are based upon original work [3] and upon the extraction and processing of three different representations [4]: • a set of linguistic features produced by an ASR model; • a representation of intonation and prosody, usually in the form of a fundamental frequency (F0) curve; • an x-vector, namely a neural embedding which encodes the speaker identity [5]. To conceal the speaker identity, the x-vector is typically perturbed by means of an anonymization function, thereby obtaining a new pseudo-speaker embedding. The three components are then fed to a waveform synthesis model (a vocoder) to produce an utterance in the voice of the pseudo-speaker. The anonymization function used by two of the three VoicePrivacy baselines utilizes a pool of external x-vectors. The pseudo-speaker x-vector is derived from a subset of the furthest vectors in the pool from the input x-vector. Most VoicePrivacy participants focused predominantly upon improving the anonymization function to enhance privacy [6,7,8]. This focus can imply an assumption that no other processing stages contribute substantially to anonymzation. We have found this not to be the case. We report in this paper our work to observe and compare the relative impacts of a conventional x-vector anonmyization function and a vocoder, two components of a state-of-the-art anonymization system [9]. We show that both components contribute to anonymization and that the contribution of the vocoder, which we refer to as the vocoder drift, is in some cases even greater than that of the anonymization function. We demonstrate that this phenomenon is also common to other popular vocoders. Collectively, they fail to provide the level of fine-grained control over the input/output x-vector space that would otherwise justify the focus within the community upon the anonymization function. Finally, we show that the vocoder drift can be learned by an attacker, knowledge which can be exploited in order to reverse the anonymization. Our findings corroborate other evidence [10] that the protection provided by such approaches to anonymization might be overestimated. Relation to prior work In this section we describe the typical, high-level structure of an x-vector-based speaker anonymization system (see Figure 1), along with relevant prior work. We then introduce our own setup which we used for all experiments reported in Section 3 and Section 4. X-vector-based speaker anonymization Let s ∈ R L be an input speech utterance of L samples. The input is first frame-blocked into a sequence of N frames and then decomposed into three separate representations comprising: an F0 curve f ∈ R N which is intended to encode intonation and prosody; a set of c-dimensional linguistic features G ∈ R c×N which encode the spoken content (the text); an x-vector xo ∈ R m which encodes the speaker identity, where subscript o denotes extraction from an original input utterance. A vocoder model V (f , G, xo) is trained to reconstruct input waveforms from the decomposition. Anonymization is achieved by replacing xo with a substitute so as to conceal the speaker identity, but by using the other components unchanged in order to preserve remaining speech attributes. The substitu- tion is performed using an anonymization function a (xo) = xp ∈ R m to perturb the original x-vector. An anonymized utterances in the voice of a fictitious, pseudo-speaker determined by the anonymized x-vector xp is then synthesized according tõ s = V (f , G, xp). The anonymized utterance should maintain the same linguistic and paralinguistic content as the original input signal. As discussed later, an additional x-vector xa can be extracted froms in order to measure privacy. By convention, a (·) acts to create a new pseudo-speaker using speaker embeddings drawn from an external pool of xvectors [1,2,3,4,9,10]. Given an input xo, the K vectors within the pool that are furthest from xo according to some distance metric are selected and then, from among them, K * vectors are chosen randomly and averaged to obtain xp. The design of this function has received considerable attention, with numerous works having investigated how its configuration, the choice of distance metric [11] and the strategy by which x-vectors are selected from the pool [11,12] influence performance. The participants of the two VoicePrivacy Challenges held in 2020 and 2022 proposed different enhancements to a (·). They include the generation of pseudo-speaker embeddings using a generative adversarial network [6,13] and adversarial noise [7], among others [8,14]. None of the participants reported the influence of the vocoder. In this paper, we show that it too contributes to anonymization and that it can be responsible for a great deal of the privacy protection. Our approach Our approach is based on the pipeline described in [9]. 1 The F0 curve is estimated using YAAPT [15]. The linguistic feature extractor is a HuBERT-based soft content encoder [16] and x-vectors are extracted using ECAPA-TDNN [17]. We experimented with three vocoders: the HiFi-GAN [18], originally used in [9]; the neural source filter (NSF) model [19] as used by baseline B1a of the VoicePrivacy Challenge held in 2022; a variation of the HiFi-GAN which uses a NSF model as generator, as used by baseline B1b of the same VoicePrivacy Challenge edition [2]. We use the conventional pool-based anonymization function a (·) described above with K = 200, K * = 100, and a cosine distance metric. Like most related work, we use the VoicePrivacy database and standard protocols [2]. The LibriTTS-train-clean-100 dataset is used for vocoder training. The LibriSpeech-test-clean and VCTK datasets (decomposed into male and female subsets) are used for evaluation. The external pool of x-vectors is derived using the LibriTTS-train-other-500 [20] dataset. Privacy is evaluated using ASV experiments comprising a set of enrollment utterances that an attacker attempts to match to a set of 1 protected (anonymized) trial utterances. ASV is performed by scoring x-vectors with the cosine distance and without any additional backend processing [9,17]. Vocoder drift In this section, we introduce the notion of vocoder drift and report an investigation of its impact upon x-vector pertubation and privacy. Figure 1 shows the three x-vectors used in this work. The first xo is extracted from the original utterance s (left in Figure 1). Definition A second x-vector xa can be extracted from the anonymized utterances (right). The third x-vector xp is the output of the anonymization function (middle). We denote the separate domains of xo, xa and xp asÔ (original), (anonymized) andP (pre-vocoder), respectively. As described in Section 2, the majority of research has focused on improving the anonymization function a (·), the general hypothesis being that this component is primarily responsible for ensuring privacy. Intuitively, privacy is improved by increasing the difference between xo and xp, e.g. according to the cosine distance. With the focus being upon the anonymization function, there is an inherent, perhaps unrealistic assumption that the vocoder preserves this distance such that the difference, which we term as the drift, between the x-vectors at the input (xp) and that which can be extracted from the output (xa) is only modest. In this work, we seek to test this assumption. We model the relationship between theP and domains with a function v (xp) = xa. It allows us to define the trajectory of an x-vector through the whole anonymization system as v•a : xo → xa, where • denotes function composition. X-vector perturbation In seeking to quantify the impact of v (·) on the x-vector trajectory, we define two metrics. Let d be some distance measure over R m . We then define: • d(xo, xp) as the target distance, a measure of how far xo is perturbed away from its original position according to a (·); • d(xp, xa) as the vocoder drift, a measure of the shift between the input x-vector xp and that which can be extracted from the vocoder output xa, introduced by means of v (·). Intuitively, it is desirable that drift ≪ target, which means the anonymization system provides fine-grained control over the final position of xa: it is close to the targeted pseudo-speaker embedding xp. If this is not the case, then the x-vector trajectory is determined in considerable part by v (·); the x-vector extracted from the output xa is far from the target and the system does not provide fine-grained control over the x-vector space inÂ. We compute the average drift and target for each database subset and each vocoder: results are shown in Table 1. The target is in the order of 1.3 for all four subsets. The value of these distances lies in their comparison to estimates of the drift shown in the last three columns. For the HiFi-GAN vocoder, the drift is almost half the target distance. Lying between 0.8 and 0.97, the drift for the NSF and HiFi-NSF vocoders is substantially greater still, with drift distances almost as large as target distances. These results show that the control over the x-vector domain is potentially low and suggest that the xvector anonymization and vocoder functions have an almostcomparable contribution to x-vector perturbation. It is still necessary, however, to explore their resulting impact upon privacy. Impacts upon privacy We follow the VoicePrivacy-defined approach to measure privacy impacts. We report a set of ASV experiments using different combinations of x-vectors. In all cases, privacy is measured using estimates of the EER. Enrollment and trial utterances are as defined by the VoicePrivacy protocol (see Section 2.2). There are several enrollment utterances per speaker. Individual x-vectors are extracted from each, averaged, and compared to a number of trial utterances. For each utterance, we extract the set of xo, xp and xa x-vectors. Each set of experiments is conducted three times, with each iteration using one of the three different x-vectors. Results using the set containing xo x-vectors (Ô domain) provide a baseline. Those derived from the set of xp x-vectors (P domain) provide an indication of the contribution to privacy of the anonymization function a (·). Results using final set containing xa x-vectors ( domain) provide an indication of the contribution of the vocoder function v (·). Once again, we report results for the same experiment using all three vocoders. Results are shown in Table 2, for the same database subsets as in Table 1. Baseline results for theÔ domain show EERs of approximately 1%. In theP domain, increases in the EER to between 2.5% and 5.6% indicate that the anonymization function delivers only a low level of privacy. In the domain, however, EERs are substantially higher for all three vocoders, if still far from providing perfect privacy (EERs of 50%). The comparison of results forP and domains show that the vocoder plays a dominant role; most of the anonymization can be attributed to vocoder drift. We explored this phenomenon with t-SNE visualizations [21] of pooled x-vectors. Results are illustrated in Figure 2, which depicts a distribution of x-vectors for the male partition of the LibriSpeech dataset. In theP domain, speaker clusters are still clearly distinguishable, while the bulk of the anonymization can be attributed to vocoder drift. One could claim that these findings are neither surprising, nor cause for concern. There is no guarantee that the vocoder function v (·) is invertible in any way which would allow the recovery of x-vector inputs xp in theP domain. Since the attacker does not have access to theP domain, but only to the domain, whether anonymization is attributed to the anonymization function or the vocoder function is of little consequence. In the next section, we disprove these arguments and show that an attacker can learn this function or, more specifically, how to undo it. Armed with the inverse function v −1 (·), an attacker can estimate an x-vector in theP domain that corresponds to an x-vector in the domain and hence reverse the anonymization. Drift-reversal attacks In this section, we introduce drift reversal, a novel attack against anonymization systems. Attacks on anonymization systems Since speaker anonymization is a relatively new research topic, it is hardly surprising that little attention has been dedicated to attacks against it. Even so, the VoicePrivacy Challenge has explored the robustness of anonymization systems under a socalled semi-informed attack model [2]. Under this scenario, an attacker is aware of anonymization having been performed, and seeks to overcome it (break the anonymization) by using a similar system to generate anonymized data with which to train an ASV system. Evaluations using ASV systems trained using indomain (similarly anonymized) data show the potential for attacks to circumvent anonymization. A more explicit approach is reported in [10] and can be used by an attacker to invert a complete anonymization system by means of a rotation matrix and to estimate speaker embeddings xo in the unprotected domain O from protected x-vectors xa inÂ. Our approach is different since we aim to explore the anonymization robustness when we revert only the vocoder drift to recover an estimate of xp inP . Definition and implementation In the case that the bulk of the anonymization performance can be attributed to the vocoder function v (·) instead of the anonymization function a (·), a drift reversal attack can be mounted to undo most of the protection Let s (e) be an original (i.e. unprotected) enrollment utterance. An attacker can derive a representation of this signal in theP domain by extracting an x-vector x While the inverse function is not analytically tractable, the attacker can attempt to learn a function g θ (·) ≈ v −1 (·) using a database of training pairs xp i and anonymized utterancessi. Function g θ can be learned using a neural network to map an anonymized utterances to an approximation of the corresponding x-vector xp inP . This can be achieved by optimizing the objective function min θ d (xp, g θ (s))(1) where d is the cosine distance. Training pairs {(xp i ,si)} i can be obtained by applying anonymization to any appropriate (even unlabeled) speech dataset. Because function g θ is effectively an x-vector extraction operation, we fine-tune a pretrained ECAPA-TDNN model to learn it. In line with the VoicePrivacy protocol, the model x o x p x a Target dist. Drift is trained using the LibriSpeech-train-clean-360 dataset, although approximately 3% of the data is set aside for validation purposes. Still in line with the VoicePrivacy protocol, anonymization is performed at the speaker level 2 in deriving xa for each enrollment and trial utterance, instead of at the utterance level. The network is fine-tuned for 3 epochs using Adam optimizer [22] with a learning rate of 5 · 10 −5 and a batch size of 8. Validation is performed every 200 iterations. Attacks are performed using the network for which the validation loss is lowest. Evaluation We compare the drift reversal attack to related VoicePrivacy lazy-informed and semi-informed attacks. For the former, the attacker compares enrollment and trial utterances which are both in the domain, but with an ASV model trained using data in theÔ domain; other than by anonymizing the enrollment utterance, there is no compensation for operating upon anonymized data. The semi-informed attacker makes greater effort and uses an ASV system that is trained using an independent set of similarly-anonymized data. The latter is the default VoicePrivacy attack model. 3 The lazy-informed attack is implemented using the original, pretrained ECAPA-TDNN for x-vector extraction. The semi-informed attack is performed using an ECAPA-TDNN model which is fine-tuned using AAMsoftmax loss [23] and the same training settings as the drift reversal attack model. Privacy evaluation results in terms of EER estimates are presented in Table 3 for each vocoder and each dataset. EER results for unprotected data (no anonymization) are shown in column 3 and provide a reference against which EERs for protected data can be compared. Results for the lazy-informed attack are shown in column 4 and show substantial privacy gains (higher EERs). This setting, however, gives a false sense of protection. Results for the semi-informed attack shown in column 5 show considerably lower privacy gains; by retraining the ASV system using similarly anonymized data, the attacker can undo the anonymization to some degree. Results for the drift reversal attack also show universally lower privacy gains compared to the lazy-informed attack. These results add to the evidence that the role played by vocoder drift in anonymization is substantial and is also a potential weakness that can be exploited by an adversary. The most powerful of the 3 attacks for each vocoder and dataset is highlighted in bold face and, for 5 of the 12 cases, the most powerful attack is drift reversal. Conclusions The work presented in this paper shows that, for the analyzed systems, the bulk of anonymization can be attributed not to the anonymization function of a conventional x-vector-based approach but, instead, to the vocoder function. The cause is vocoder drift, namely the substantial difference between an input x-vector and the x-vector which can be extracted from the vocoder output. This finding, while not necessarily surprising, calls into the question the research effort upon improving the anonymization function. One might wonder whether the design of different anonymization functions has any relevance at all, given that the position of the final x-vector is dominated by the vocoder drift, essentially nullifying the effort devoted to pseudo-speaker optimization. This finding should not discourage further work in the design of x-vector anonymization functions, however. Instead, it should encourage design toward more grounded criteria. Drift-reversal attacks rely on the fact that the x-vectors fed to the vocoder, though allegedly anonymized, still have a low level of protection. This is the result of an over-deterministic anonymization function; similar x-vector inputs will produce similar x-vector outputs, thus producing trial and enrollment speaker embeddings which are close in the output domain, and thus easy to match as the same speaker, even when anonymized. That is the case for the pool-based anonymization function. Future work should investigate less deterministic anonymization functions to improve privacy directly in their output domain. Improvements to privacy in this domain will not only undoubtedly mitigate the risk of vocoder-drift-reversal attacks, but likely also that of semi-informed attacks, which might inadvertently learn to exploit the same kind of vulnerability during training. Figure 1 : 1Overview of a conventional speaker anonymization system and the different x-vector domains. The block in red represents the vocoder drift reversal attack reported in Section 4. lets (t) be an anonymized trial utterance with corresponding xvector x (t) a in the domain. The attacker can estimate a representation in theP domain x Figure 2 : 2t-SNE visualization of the x-vector trajectory for LibriSpeech trial utterances (M) across the three x-vector domains (left). Focus on the trajectory of a single speaker (right). Best viewed in color. arXiv:2306.02892v1 [eess.AS] 5 Jun 2023Prosody Content Identity Vocoder Identity Original domain Pre-vocoder domain Anonymized domain Drift reversal Table 1: Average target distance and drift for each vocoder and each test set of LibriSpeech and VCTK, separated by speaker sex. All cosine distances have a standard deviation between 0.05 and 0.10.Code available at github.com/eurecom-asp/ vocoder-drift. target drift HiFi-GAN NSF HiFi-NSF LibriSpeech (F) 1.3 0.62 0.91 0.97 LibriSpeech (M) 1.2 0.56 0.80 0.94 VCTK (F) 1.3 0.67 0.92 0.94 VCTK(M) 1.3 0.59 0.90 0.90 Table 2 : 2Privacy protection of the x-vector domains at different stages of the anonymization pipeline (EER, %) on test sets of LibriSpeech and VCTK, separated by speaker sex. Table 3 : 3Performanceof the proposed drift-reversal attack compared to a lazy-informed attack and a supervised semi-informed attack (EER, %) on the LibriSpeech and VCTK test sets. Original (O) Pre-vocoder (P) Anonymized (A) xp i is estimated once for each speaker i and the same xp i is used for each utterance corresponding to the same speaker -see[2] for details.3 It could be argued that drift reversal is also a semi-informed attack, since it involves re-training a model on anonymized data (albeit unlabeled). 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{'abstract': 'State-of-the-art approaches to speaker anonymization typically employ some form of perturbation function to conceal speaker information contained within an x-vector embedding, then resynthesize utterances in the voice of a new pseudo-speaker using a vocoder. Strategies to improve the x-vector anonymization function have attracted considerable research effort, whereas vocoder impacts are generally neglected. In this paper, we show that the impact of the vocoder is substantial and sometimes dominant. The vocoder drift, namely the difference between the x-vector vocoder input and that which can be extracted subsequently from the output, is learnable and can hence be reversed by an attacker; anonymization can be undone and the level of privacy protection provided by such approaches might be weaker than previously thought. The findings call into question the focus upon x-vector anonymization, prompting the need for greater attention to vocoder impacts and stronger attack models alike.', 'arxivid': '2306.02892', 'author': ['Michele Panariello \nEURECOM\nSophia AntipolisFrance\n', 'Massimiliano Todisco \nEURECOM\nSophia AntipolisFrance\n', 'Nicholas Evans \nEURECOM\nSophia AntipolisFrance\n'], 'authoraffiliation': ['EURECOM\nSophia AntipolisFrance', 'EURECOM\nSophia AntipolisFrance', 'EURECOM\nSophia AntipolisFrance'], 'corpusid': 259075544, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 9557, 'n_tokens_neox': 8468, 'n_words': 5135, 'pdfsha': 'ad5dcdd727278dbaa5b12f893554770f640ecb61', 'pdfurls': ['https://export.arxiv.org/pdf/2306.02892v1.pdf'], 'title': ['Vocoder drift in x-vector-based speaker anonymization', 'Vocoder drift in x-vector-based speaker anonymization'], 'venue': []}

arxiv

Foundry manufacturing of tight-confinement, dispersion-engineered, ultralow-loss silicon nitride photonic integrated circuit Zhichao Ye Qaleido Photonics 310000HangzhouChina Haiyan Jia Qaleido Photonics 310000HangzhouChina Zhangjun Huang Qaleido Photonics 310000HangzhouChina Chen Shen International Quantum Academy 518048ShenzhenChina Shenzhen Institute for Quantum Science and Engineering Southern University of Science and Technology 518055ShenzhenChina Jinbao Long International Quantum Academy 518048ShenzhenChina Shenzhen Institute for Quantum Science and Engineering Southern University of Science and Technology 518055ShenzhenChina Baoqi Shi International Quantum Academy 518048ShenzhenChina Department of Optics and Optical Engineering University of Science and Technology of China 230026HefeiChina Yi-Han Luo International Quantum Academy 518048ShenzhenChina Shenzhen Institute for Quantum Science and Engineering Southern University of Science and Technology 518055ShenzhenChina Lan Gao International Quantum Academy 518048ShenzhenChina Shenzhen Institute for Quantum Science and Engineering Southern University of Science and Technology 518055ShenzhenChina Wei Sun International Quantum Academy 518048ShenzhenChina Hairun Guo Key Laboratory of Specialty Fiber Optics and Optical Access Networks Shanghai University 200444ShanghaiChina Jijun He Ministry of Education Key Laboratory of Radar Imaging and Microwave Photonics Nanjing University of Aeronautics and Astronautics 210016NanjingChina Junqiu Liu International Quantum Academy 518048ShenzhenChina Hefei National Laboratory University of Science and Technology of China 230088HefeiChina Foundry manufacturing of tight-confinement, dispersion-engineered, ultralow-loss silicon nitride photonic integrated circuit The foundry development of integrated photonics has revolutionized today's optical interconnect and datacenters. Over the last decade, we have witnessed the rising of silicon nitride (Si 3 N 4 ) integrated photonics, which is currently transferring from laboratory research to foundry manufacturing. The development and transition are triggered by the ultimate need of low optical loss offered by Si 3 N 4 , which is beyond the reach of silicon and III-V semiconductors. Combined with modest Kerr nonlinearity, tight optical confinement and dispersion engineering, Si 3 N 4 has today become the leading platform for linear and Kerr nonlinear photonics, and has enabled chip-scale lasers featuring ultralow noise on par with table-top fiber lasers. However, so far all the reported fabrication processes of tight-confinement, dispersion-engineered Si 3 N 4 photonic integrated circuit (PIC) with optical loss down to few dB/m have only been developed on 4-inch or smaller wafers. Yet, to transfer these processes to established CMOS foundries that typically operate 6-inch or even larger wafers, challenges remain. In this work, we demonstrate the first foundry-standard fabrication process of Si 3 N 4 PIC with only 2.6 dB/m loss, thickness above 800 nm, and near 100% fabrication yield on 6-inch wafers. Such thick and ultralow-loss Si 3 N 4 PIC enables low-threshold generation of soliton frequency combs. Merging with advanced heterogeneous integration, active ultralow-loss Si 3 N 4 integrated photonics could pave an avenue to addressing future demands in our increasingly information-driven society. and created the first electrically pumped InP/Si lasers[7,8]that have today been used for optical interconnect. Despite, Si and InP still have many limitations, particularly the high linear and nonlinear losses (e.g. two-photon absorption) that compromise their performance.To address this challenge, silicon nitride (Si 3 N 4 ) emerges as a leading platform for lowloss integrated photonics[9][10][11]. The 5 eV bandgap of Si 3 N 4 makes it transparent from ultraviolet to mid-infrared, and immune to two-photon absorption in the telecommunication band. Meanwhile, Si 3 N 4 has a dominant Kerr nonlinearity but negligible Raman [12] and Brillouin nonlinearities[13]. In addition,with advanced CMOS fabrication techniques, linear optical loss down to 1 dB/m or even lower has been only achieved in Si 3 N 4 [14-19] among all integrated platforms. All these advantages have triggered the rapid development of Si 3 N 4 Kerr nonlinear photonics [9, 10], and have enabled key advances such as optical frequency comb generation[20][21][22], supercontinuum generation[23][24][25], and quantum light sources[26][27][28]. In addition to ultralow loss, tight optical confinement with SiO 2 cladding is simultaneously required. Since the refractive index of Si 3 N 4 ( ≈ 1.99) is modestly higher than that of SiO 2 ( ≈ 1.45), Si 3 N 4 waveguides require sufficient thickness to achieve tight optical confinement for small mode volume and for bending radii down to 20 m. In addition, while Si 3 N 4 material has intrinsic normal group velocity dispersion (GVD) at telecommunication bands, Si 3 N 4 waveguides with thickness above 600 nm can obtain net anomalous GVD that is required for phase matching in Kerr parametric processes[29,30].While CMOS foundries have already developed standard Si 3 N 4 processes [31] to fabricate PIC with typical thickness of 300 nm and loss on the order of 10 dB/m, there has not been a process to simultaneously achieve ultralow loss (e.g. below 3 dB/m) and large thickness (above 600 nm) without crack formation. So far, thin Si 3 N 4 PIC with width above 5 m and thickness below 100 nm[17][18][19]can achieve optical loss below 0.1 dB/m. While this process has recently become an 8-inch foundry process [32], the thin Si 3 N 4 exhibits weak optical confinement and thus suffers from exaggerated bending loss with a small footprint. The optical mode also exhibits strong normal GVD due to the waveguide geometry. While coupled waveguide structures can be used to alter local GVD[33,34], they cannot offer anomalous GVD over a wide spectral range. In parallel, thick Si 3 N 4 PIC of ultralow loss and broadband anomalous GVD has been realized via the subtractive process[35][36][37]and the photonic Damascene process[14,38]. However, all these reported processes have been developed only in laboratories and have issues with transferring to foundry manufacturing. While high-yield, wafer-scale fabrication of Si 3 N 4 PIC has achieved 1 dB/m loss and anomalous GVD in Ref.[14], the photonic Damascene process has intrinsic limitations. For example, the use of chemical-mechanical planarization (CMP) to remove excess Si 3 N 4 can cause serious dishing effect in large-area structures (see Appendix C), thus cannot be used to fabricate elements such as arrayed waveguide gratings (AWG) and multimode interferometers (MMI). Meanwhile, the aspect-ratio-dependent etch effect prevents to form narrow but deep channels[39]. In comparison, with the subtractive process [35-37], thick Si 3 N 4 PIC of ultralow loss and anomalous GVD has been only achieved with electron-beam lithography (EBL) on 3-or 4-inch wafers, which are not foundry-standard. Meanwhile, an 8-inch foundry process has been developed in Ref.[40], but this process has not yet achieved optical loss below 3 dB/m.In this work, we overcome the above challenges and demonstrate a foundry-standard fabrication process of tight-confinement, dispersion-engineered, ultralow-loss Si 3 N 4 PIC. The process is based on 6-inch wafers and combines deep-ultraviolet (DUV) stepper lithography[14]and state-of-the-art subtractive process [37], i.e. a DUV subtractive process. We have achieved a linear optical loss of 2.6 dB/m in 810-nm-thick Si 3 N 4 PIC. Finally, we generate single soliton microcombs of 100.17 GHz and 19.975 GHz mode spacings using these devices. thank Chao Xiang for the fruitful discussion. Introduction Integrated photonics [1] enables the synthesis, processing and detection of optical signals using photonic integrated circuit (PIC). The successful translation from laboratory research to foundry development over the past decades has established integrated photonics as a standard technology [2] deployed in high-data-rate telecommunication and datacenters [3]. Foundry-level manufacturing of photonic chips allows fast prototyping or mass production with high yield, high throughput, low cost, and guaranteed performance. Silicon (Si) and indium phosphide (InP) are two mainstream platforms of integrated photonics [4]. The development of heterogeneous integration [5,6] has married these two platforms Fabrication Figure 1a presents a photograph that shows dozens of Si 3 N 4 chips on a 6-inch wafer, which contains microresonators of different free spectral ranges (FSR) and meter-long spirals. Figure 1b presents an optical micrograph that shows a curved bus waveguide slowly approaching a 100-GHz-FSR microring resonator for light coupling. This coupler design can increase coupling strength and ideality, which will be described later. Figure 1c presents a scanning electron micrograph (SEM) that shows the actual Si 3 N 4 waveguide cross-section with 85 • sidewall angle and overlaid fundamental transverse-electric (TE 00 ) mode. The optical mode is tightly confined in the Si 3 N 4 waveguide core with SiO 2 cladding, enabling dispersion engineering and small bending radii. The Si 3 N 4 PIC is fabricated using the DUV subtractive process. Figure 1d shows the subtractive process flow widely used to fabricate PIC based on essentially any material, particularly Si 3 N 4 [35][36][37][40][41][42][43][44]. First, a Si 3 N 4 film is deposited on a clean thermal wet SiO 2 substrate via low-pressure chemical vapor deposition (LPCVD). It is well known that LPCVD Si 3 N 4 films are prone to crack due to their intrinsic tensile stress (typically 1.1 to 1.4 GPa). The film stress can be relaxed via thermal cycling during Si 3 N 4 deposition in multiple layers [40,41], yielding zero cracks during our fabrication. After SiO 2 deposition as an etch hardmask, DUV stepper photolithography is used to expose the waveguide pattern. Via dry etching, the pattern is subsequently transferred from the photoresist mask to the SiO 2 hardmask, and then into the Si 3 N 4 layer to form waveguides. For superior etch quality and smooth waveguide sidewall, the etchant we use is CHF 3 with added O 2 to remove fluoride-carbon polymers as etch byproducts. The etched substrate is thermally annealed in nitrogen atmosphere at 1200 • C to eliminate nitrogen-hydrogen bonds in Si 3 N 4 , which cause absorption loss. Top SiO 2 cladding is then deposited on the wafer, which also requires high-temperature annealing to remove silicon-hydrogen bonds that also cause absorption loss. In specific cases where deuterated plasma-enhanced chemical vapor deposition (PECVD) Si 3 N 4 [44,45] and SiO 2 [46] are used, thermal annealing may not be required as these films are intrinsically hydrogen-free. Platinum heaters [22,47] are deposited on the substrate via an evaporator and patterned via a lift-off process. Due to the thick top SiO 2 cladding and tight optical confinement of the Si 3 N 4 core, the presence of metallic heaters does not impact the optical loss of Si 3 N 4 waveguides beneath. Afterwards, UV photolithography and deep dry etching of SiO 2 and Si are used to define chip size and create smooth chip facets. Finally, the wafer is separated into chips using dicing or backside grinding. Characterization Microresonator quality factors and loss rates We characterize the optical loss of Si 3 N 4 PIC by measuring the resonance linewidth of Si 3 N 4 microresonators. Light is coupled into and out of Si 3 N 4 chips via lensed fibers and inverse tapers [39]. The fiber-chip edge coupling efficiency is about 60%. We use frequency-combassisted diode laser spectroscopy [48] to measure resonance frequency /2 and linewidth /2 , ranging from 1480 to 1640 nm wavelength. The resonance's quality factor is calculated as = / . Here we study both the TE 00 and TM 00 (fundamental transverse-magnetic) modes of the 100-GHz-FSR microresonators of 810 nm thickness and 2.40 m waveguide width. For each resonance fit [49], the intrinsic loss 0 /2 , external coupling strength ex /2 , and the total (loaded) linewidth /2 = ( 0 + ex )/2 , are extracted. Figure 2a shows a typical TE 00 resonance with a Lorentzian fit. The resonance is under-coupled ( 0 > ex ), with fitted 0 /2 = 13.8 MHz. Figure 2b At telecommunication band = 1550 nm and with a group index = 2.09 for the given waveguide geometry, 0 = 1.4 × 10 7 corresponds to = 2.6 dB/m. In comparison, 0 /2 = 17 MHz is found for the TM 00 mode, corresponding to 0 = 1.1 × 10 7 (see Appendix A). The wavelength-dependent loss of each TE 00 resonance is studied in Appendix B, showing no prominent hydrogen-related absorption around 1520 to 1540 nm. Next, we investigate wafer-scale fabrication yield. Figure 2d right shows our design layout containing sixteen chips on the DUV stepper reticle. Each chip has a 5 × 5 mm 2 size on the wafer and contains many microresonators. The DUV stepper exposes the reticle uniformly over the 6-inch wafer in discrete fields. The 100-GHz-FSR chips characterized above are labelled as the C11 chips. The most probable 0 /2 values for all C11 chips, as well as their GVD parameters 2 /2 (described later), are plotted in each field, as shown in Fig. 2d left. In all measured 20 fields, 0 /2 16 MHz is found, demonstrating that our foundry process to manufacture ultralow-loss Si 3 N 4 PIC is uniform and near 100% yield. We also characterize 0 /2 of 100-GHz-FSR microresonators of 2.20, 2.00, and 1.80 m waveguide width. Again, we create a 0 /2 histogram for each case and look for the most probable values. We plot and compare the most probable 0 /2 values for the three width values, and observe a decreasing 0 /2 with increasing waveguide width, as shown in Fig. 2c. This trend indicates that our optical loss is still dominated by the waveguide's Si 3 N 4 /SiO 2 interface The DUV stepper reticle layout contains sixteen chips, and is uniformly exposed in discrete fields over the 6-inch wafer. Left: The most probable values 0 /2 of the C11 chips, as well as the measured GVD parameters 2 /2 , are marked in each fields over the wafer. NA: not applicable, due to visible defects or missing C11 chips near wafer edge. roughness, and can be further reduced by optimizing the dry etching process or using CMP to reduce top surface roughness [36]. Microresonator dispersion For most applications using Kerr nonlinearity of Si 3 N 4 , anomalous GVD is required [29,30]. In addition, for the generation of dissipative Kerr solitons [20,50,51], avoided mode crossings (AMX) induced by spatial mode coupling [52] should be suppressed, as they prohibit soliton formation [53,54] or distort soliton spectra [55,56]. Therefore, next we quantitatively characterize the dispersion profile and investigate AMXs of the high-, 100-GHz-FSR microresonators. The microresonator's integrated dispersion is defined as where /2 is the -th resonance frequency relative to the reference resonance frequency 0 /2 , 1 /2 is microresonator FSR, 2 /2 describes GVD, 3 and 4 are higher-order dispersion terms. Figure 3a top shows a typical int profile, with each parameter extracted from the fit Eq. 2. The positive sign of 2 validates anomalous GVD. To reveal AMXs, we remove the 2 term and fit the data with 3 3 /6 + 4 4 /24, as shown in Fig. 3a middle. We further remove the 2 and 3 terms, and fit the data with 4 4 /24, as shown in Fig. 3a bottom. The observed AMXs are overall weak and only lead to megahertz-level resonance frequency deviation. It also shows that, with these weak AMXs, our spectroscopic method is sufficiently precise to extract the 4 term. int ( ) = − 0 − 1 = 2 2 /2 + 3 3 /6 + 4 4 /24(2) Coupling ideality Furthermore, we experimentally characterize the coupling ideality [57] of our high-Si 3 N 4 microresonators. In the current case, light in the bus waveguide's fundamental TE mode (TE ,00 ) is coupled into the microresonator's fundamental TE mode (TE ,00 ), and then coupled out of the microresonator and back into the bus waveguide. In this process, coupling ideality I is defined as I = ex ex + (3) Here ex is the external coupling rate between the TE ,00 and TE ,00 modes, is the parasitic loss rate describing coupling strength to other bus waveguide modes as well as radiation modes into free space. The parasitic loss appears as another loss channel in addition to 0 and ex . Thus I is a parameter describing how much power is recollected in the TE ,00 mode that is exactly the initial driving mode of the bus waveguide [57,58]. In the ideal case of single-mode bus waveguide and no radiation into free space, I = 1 is obtained. In the present case, the bus waveguide is multimode, since it has the same waveguide cross-section (thickness×width) as the microresonator waveguide, to obtain phase matching between the TE ,00 and TE ,00 modes for maximum ex . Thus the TE ,00 mode can couple to other bus waveguide modes (e.g. higher-order TE modes or any TM modes) or radiation modes, which ultimately compromises coupling ideality (I < 1). With sufficient number of characterized resonances, coupling ideality is evaluated by analyzing the dependence of resonance transmission and total linewidth /2 , as = |1 − 2 −1 + I −1 | 2(4) where = ex / 0 describes the coupling regime ( < 1 for under-coupling, = 1 for critical coupling, and > 1 for over-coupling [59]). Previously, coupling ideality of integrated Si 3 N 4 microresonators has been characterized [58], however based on devices of < 4 × 10 6 . For state-of-the-art Si 3 N 4 microresonators of > 10 7 , coupling ideality has not been experimentally studied, and whether high coupling ideality is still maintained needs to be answered. Here we perform extra measurement on over-coupled devices with smaller gap values down to 300 nm. The microresonators are identical, except that the gap varies from sample to sample to provide a varying ex . Figure 3b shows the measured coupling ideality for the TE 00 and TM 00 modes from dozens of 100-GHz-FSR microresonators. The TE 00 (top) and TM 00 (bottom) plots contain 3269 and 959 data points, respectively. The data is sufficient to uncover the global trend even in the presence of 0 variation due to fabrication and AMXs. Meanwhile, for comparison, we also plot the -curves with I = [0.9, 0.95, 1] calculated using Eq. 4. As can be seen, our microresonators coupled with curved bus waveguides can already provide near-unity I and strong over-coupling (e.g. > 7 for the TM 00 ), which are critical for microring-based phase modulation [60,61], wideband tunable delay line [62], and the extraction of quantum light states generated in high-microresonators [27,[63][64][65]. Soliton microcomb generation A key application area of our Si 3 N 4 PIC is Kerr nonlinear photonics, where ultralow optical loss is central as it determines the threshold power th for Kerr parametric oscillation [29]. For example, for soliton microcomb generation [20,50,51], th scales with the microresonator factor as th ∝ −2 . Therefore, with this quadratic dependence, a high enables significant reduction of th down to milliwatt level. The experimental setup to generate single solitons in our Si 3 N 4 chips is shown in Fig. 4a. We note that, due to the high-and low thermal effects of our Si 3 N 4 microresonators, single solitons can be generated via simple laser-piezo frequency tuning [50,66] or on-chip heaters [22,47]. This is in contrast to many soliton generation experiments that require sophisticated techniques to manage thermal effects [67] such as power kicking [68,69], single-sideband suppressed-carrier frequency shifters [70], dual-laser pump [71,72], pump modulation [73,74], pulse pumping [75], or laser self-cooling [76]. When the continuous-wave (CW) pump laser's frequency scans across a resonance from the blue-detuned side to the red-detuned side, a step feature (i.e. the "soliton step") is observed in the microresonator transmission spectrum [50], signalling soliton formation. Figure 4b shows a typical soliton step with sub-millisecond duration, sufficiently long for accessing the single-soliton state via simple laser-piezo frequency tuning [50,66]. To further confirm the soliton nature and measure the soliton detuning value, a system response measurement using an electro-optic modulator (EOM) and a vector network analyzer (VNA) is performed [66]. As shown in Fig. 4c, the system response features double resonances corresponding to the cavity resonance of the CW pump ("C-resonance"), and the soliton-induced resonance ("S-resonance"). Physically, the "cold" cavity resonance is probed by the intracavity low-power CW as a background to the soliton pulse pattern, thus the observed C-resonance frequency indicates the effective laser-cavity detuning (where thermal induced resonance shift is eliminated) [66]. The S-resonance is induced by the soliton pattern that has a high peak-power leading to nonlinear frequency shift of the cavity resonance. As shown in Fig. 4d, in a 100-GHz-FSR microresonator, a single soliton of 100.17 GHz mode spacing is generated with 19 mW power in the bus waveguide on-chip. The spectrum sech 2 fit shows 3-dB bandwidth of 17.99 nm, corresponding to a Fourier-limited pulse duration of 141.8 fs. Increasing the pump power to 126 mW and the pump laser detuning, the soliton's 3-dB spectrum bandwidth is increased to 35.68 nm (pulse duration of 71.47 fs). We also observe a strong Raman-induced self-frequency shift [77,78] of 10.4 nm. Moreover, we generate a single soliton in a 20-GHz-FSR microresonator from the same wafer. Figure 4e shows the single soliton spectrum of 19.975 GHz mode spacing with 518 mW power in the bus waveguide. The 3-dB bandwidth of 21.96 nm, corresponding to a pulse duration of 115.4 fs, covers 117 comb lines. This coherent soliton microcomb with a microwave K-band repetition rate is advantageous for applications such as high-spectra-efficiency telecommunications [79,80], photonic microwave generation [81][82][83], and astronomical spectrometer calibration [84,85]. Previously, among all CMOS-compatible high-index materials, single-soliton microcombs of repetition rates below microwave K-band (< 20 GHz) have only been realized in Si 3 N 4 , using either the 4-inch photonic Damascene process [82], or the 3-inch EBL-written subtractive process [86]. Our work represents the first foundry-based, 6-inch subtractive process with DUV stepper lithography to reach this goal. Conclusion and outlook In conclusion, we have reported a 6-inch foundry fabrication process of tight-confinement, dispersion-engineered Si 3 N 4 PIC of optical loss down to 2.6 dB/m and near 100% yield. We have demonstrated its application in soliton microcomb generation with low power threshold and dense channel spacings. While currently our process is based on 6-inch wafers due to our dry etcher, essentially our process can be scaled up to an 8-inch process, which can offer even better uniformity and higher throughput. Merging our ultralow-loss Si 3 N 4 process with established heterogeneous integration [87,88] can introduce a variety of active functions to the passive Si 3 N 4 PIC, such as narrow-linewidth lasers in the UV and visible band [89,90], broadband EOMs [91,92], fast photodetectors [93,94], and programmable MEMS-controlled network [95,96]. Together, foundry development of heterogeneous, ultralow-loss Si 3 N 4 integrated photonics could revolutionize next-generation applications for frequency metrology [97,98], photonic neural networks [99,100], and quantum information processing [101,102]. Appendix A: Characterization of the microresonator TM 00 Figure 5a shows a TM 00 resonance with a Lorentzian fit. The resonance is critically coupled ( 0 ≈ ex ), with fitted 0 /2 = 15.6 MHz. Figure 5b shows a histogram of 0 /2 for 7944 fitted TM 00 resonances from forty 100-GHz-FSR microresonators. The most probable value is 0 /2 = 17 MHz, corresponding to 0 = 1.1 × 10 7 . Figure 5c shows the most probable 0 /2 Uniformity and yield analysis over the 6-inch wafer scale. Right: The DUV stepper reticle layout contains sixteen chips, and is uniformly exposed in discrete fields over the 6-inch wafer. Left: The most probable values 0 /2 of the C11 chips, as well as the measured GVD parameters 2 /2 , are marked in each fields over the wafer. NA: not applicable, due to visible defects or missing C11 chips near wafer edge. values for all C11 chips plotted in each field of the 6-inch wafer, as well as their GVD parameters 2 /2 . In all measured 20 fields, 0 /2 19 MHz is found. Appendix B: Loss versus wavelength We use frequency-comb-assisted diode laser spectroscopy [48] to measure resonance frequency /2 and linewidth /2 , ranging from 1480 to 1640 nm wavelength. Figure 6 shows the measured and fitted intrinsic loss 0 /2 , the external coupling strength ex /2 , and the total (loaded) linewidth /2 = ( 0 + ex )/2 of each resonance of a typical 100-GHz-FSR microresonator. Since the bus waveguide and the microresonator are coupled via evanescent field, ex /2 is wavelength-dependent with a given geometry. Therefore the alignment of ex /2 values on a line indicates correct resonance fit with reasonable precision. Local 0 /2 increase is observed at multiple wavelengths, however such narrow-band features are likely caused by AMXs. In addition, no prominent hydrogen-related absorption around 1520 nm to 1540 nm is observed, indicating low photo-thermal absorption. Appendix C: CMP dishing effect Our reported process is based on the subtractive process that is a top-down process where Si 3 N 4 waveguides are formed by dry etching. There is another process widely used for the fabrication of ultralow-loss Si 3 N 4 PIC, i.e. the photonic Damascene process [14,38]. This process is an "additive" process. As illustrated in Ref. [14], the patterns are transferred from the photoresist mask to the SiO 2 substrate to create waveguide preforms. Then an LPCVD Si 3 N 4 film is deposited on the patterned substrate, filling the preform trenches and forming the waveguides. Chemical-mechanical planarization (CMP) is used to remove excess Si 3 N 4 and create a flat and smooth wafer top surface. The rest steps are the same as the subtractive process. It should be noted that the dishing effect, illustrated in Fig. 7, is commonly presented if the CMP polishing rates for the waveguide material and cladding are different (which is true for Si 3 N 4 and SiO 2 ). In the Damascene case, the CMP slurry containing SiO 2 nano-particles causes the polishing rate of Si 3 N 4 higher than that of thermal wet SiO 2 , which induces the dishing effect in large-area patterns (e.g. critical dimension larger than 3 m). The dishing effect leads to significant structure distortion and top surface roughness. Fig. 1 . 1Process flow and sample images of the 6-inch Si 3 N 4 foundry fabrication process. a. Photograph of dozens of Si 3 N 4 chips on a 6-inch wafer, which contains microresonators of different FSR and meter-long spirals. b. Optical micrograph showing a curved bus waveguide slowly approaching a 100-GHz-FSR microring resonator. c. SEM image showing the Si 3 N 4 waveguide core with SiO 2 cladding. The TE 00 mode is plotted, showing tight confinement in the Si 3 N 4 waveguide core. d. The DUV subtractive process flow. WOX: wet oxide (SiO 2 ). shows a histogram of 0 /2 for 11741 fitted TE 00 resonances from sixty 100-GHz-FSR microresonators. The most probable value is 0 /2 = 14 MHz, corresponding to a statistical intrinsic quality factor of 0 = 1.4 × 10 7 . The microresonator 0 and linear optical loss (dB/m physical length) are linked via Fig. 2 . 2Statistical loss characterization and yield analysis. a. A typical TE 00 resonance profile with a Lorentzian fit, showing 0 /2 = 13.8 MHz and negligible mode split. b. Histogram of 11741 TE 00 resonances from sixty 100-GHz-FSR microresonators of 2.40 m waveguide width, showing the most probable value of 0 /2 = 14 MHz and 0 = 1.4 × 10 7 . c. Characterization of waveguide-widthdependent loss. Microresonators of 2.40, 2.20, 2.00 and 1.80 m waveguide widths are characterized and compared. A trend of lower 0 with larger width is shown. The size and color tone of the circles indicate the probability of occurrence. d. Uniformity and yield analysis over the 6-inch wafer scale. Right: Fig. 3 . 3Characterization of microresonator dispersion and coupling ideality. a. Measured integrated dispersion of the microresonator which is fitted with int ( ) 24 (middle), and the deviations from 4 4 /24 (bottom). Avoided mode crossings are revealed in the bottom panel, however weak for the later soliton generation experiment. b. Characterization of coupling ideality of the TE 00 (top) and TM 00 (bottom) modes. For the TE 00 / TM 00 modes, in total thirty-four / seventeen 100-GHz-FSR microresonators are characterized, providing 3269 / 959 data points in each plot. A clear trend from under-coupling to critical coupling and then to strong over-coupling is observed. The calculated curves of I = [0.9, 0.95, 1] with 0 /2 =14 or 17 MHz are plotted for comparison, showing near-unity coupling ideality. Fig. 4 . 4Single-soliton generation in silicon nitride. a. Experimental setup. AFG, arbitrary function generator; ECDL,external-cavity diode laser; EDFA, erbium-doped fiber amplifier; BPF, bandpass filter; EOM, electro-optic modulator; DUT, device under test; FBG, fiber Bragg grating; PD, photodiode; OSA, optical spectrum analyser; OSC, oscilloscope; VNA, vector network analyzer. b. When the laser frequency is scanned from the blue-detuned to the red-detuned side of a resonance, a soliton step of sub-millisecond length appears,enabling direct access to soliton states via simple piezo tuning of laser frequency. c. Cavity response measurement using the EOM and VNA. The appearance of S-resonance verifies soliton generation. d. Single-soliton spectra of 100.17 GHz mode spacing, with 19 mW (red) and 126 mW (blue) CW pump power on the chip, and their spectral fit (green). With 19 / 126 mW power, the arrows mark the 3-dB bandwidth of 17.99 / 35.68 nm. A prominent Raman self-frequency shift of 10.4 nm is observed with 126 mW power. e. Single-soliton spectrum of 19.975 GHz mode spacing with 518 mW (blue) CW pump power on the chip, and its spectral fit (green). The arrows mark the 3-dB bandwidth of 21.96 nm, containing 137 comb lines. In both d and e, the EDFA's amplified spontaneous emission (ASE) noise is filtered out by the BPF, and the pump laser in the soliton spectra is filtered out by the FBG. Fig. 5 . 5Characterization of the microresonator TM 00 mode. a. A typical TM 00 resonance profile with a Lorentzian fit, showing 0 /2 = 15.6 MHz. b. Histogram of 7944 TM 00 resonances from forty 100-GHz-FSR microresonators of 2.40 m waveguide width, showing the most probable value of 0 /2 = 17 MHz and 0 = 1.1 × 10 7 . c. Fig. 6 . 6Broadband measurement of resonance linewidth. Measured and fitted 0 /2 , ex /2 , and /2 = ( 0 + ex )/2 of each resonance from 1480 to 1640 nm. The alignment of ex /2 values on a line indicates correct resonance fit with reasonable precision. Local 0 /2 increase at multiple wavelengths are likely caused by AMXs. No prominent hydrogen-related absorption around 1520 to 1540 nm is observed. Fig. 7 . 7Illustration of the CMP dishing effect observed during fabrication. a. SEM images showing smooth top surface after CMP for waveguides with smaller critical dimension (e.g. below 3 m). b. SEM images showing rough top surface due to the CMP dishing effect for waveguides with larger critical dimension (above 3 m). Funding.J. Liu acknowledges support from the National Natural Science Foundation of China(Grant No.12261131503), Hetao Shenzhen-Hong Kong Science and Technology Innovation Cooperation Zone Project (No. HZQB-KCZYB-2020050), and from the Guangdong Provincial Key Laboratory (2019B121203002). Y.-H L. acknowledges support from the China Postdoctoral Science Foundation (Grant No. 2022M721482). Acknowledgments. J. Liu is indebted to Dapeng Yu who provided critical support to this project. We Roadmap on silicon photonics. 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{'abstract': "The foundry development of integrated photonics has revolutionized today's optical interconnect and datacenters. Over the last decade, we have witnessed the rising of silicon nitride (Si 3 N 4 ) integrated photonics, which is currently transferring from laboratory research to foundry manufacturing. The development and transition are triggered by the ultimate need of low optical loss offered by Si 3 N 4 , which is beyond the reach of silicon and III-V semiconductors. Combined with modest Kerr nonlinearity, tight optical confinement and dispersion engineering, Si 3 N 4 has today become the leading platform for linear and Kerr nonlinear photonics, and has enabled chip-scale lasers featuring ultralow noise on par with table-top fiber lasers. However, so far all the reported fabrication processes of tight-confinement, dispersion-engineered Si 3 N 4 photonic integrated circuit (PIC) with optical loss down to few dB/m have only been developed on 4-inch or smaller wafers. Yet, to transfer these processes to established CMOS foundries that typically operate 6-inch or even larger wafers, challenges remain. In this work, we demonstrate the first foundry-standard fabrication process of Si 3 N 4 PIC with only 2.6 dB/m loss, thickness above 800 nm, and near 100% fabrication yield on 6-inch wafers. Such thick and ultralow-loss Si 3 N 4 PIC enables low-threshold generation of soliton frequency combs. Merging with advanced heterogeneous integration, active ultralow-loss Si 3 N 4 integrated photonics could pave an avenue to addressing future demands in our increasingly information-driven society. and created the first electrically pumped InP/Si lasers[7,8]that have today been used for optical interconnect. Despite, Si and InP still have many limitations, particularly the high linear and nonlinear losses (e.g. two-photon absorption) that compromise their performance.To address this challenge, silicon nitride (Si 3 N 4 ) emerges as a leading platform for lowloss integrated photonics[9][10][11]. The 5 eV bandgap of Si 3 N 4 makes it transparent from ultraviolet to mid-infrared, and immune to two-photon absorption in the telecommunication band. Meanwhile, Si 3 N 4 has a dominant Kerr nonlinearity but negligible Raman [12] and Brillouin nonlinearities[13]. In addition,with advanced CMOS fabrication techniques, linear optical loss down to 1 dB/m or even lower has been only achieved in Si 3 N 4 [14-19] among all integrated platforms. All these advantages have triggered the rapid development of Si 3 N 4 Kerr nonlinear photonics [9, 10], and have enabled key advances such as optical frequency comb generation[20][21][22], supercontinuum generation[23][24][25], and quantum light sources[26][27][28]. In addition to ultralow loss, tight optical confinement with SiO 2 cladding is simultaneously required. Since the refractive index of Si 3 N 4 ( ≈ 1.99) is modestly higher than that of SiO 2 ( ≈ 1.45), Si 3 N 4 waveguides require sufficient thickness to achieve tight optical confinement for small mode volume and for bending radii down to 20 m. In addition, while Si 3 N 4 material has intrinsic normal group velocity dispersion (GVD) at telecommunication bands, Si 3 N 4 waveguides with thickness above 600 nm can obtain net anomalous GVD that is required for phase matching in Kerr parametric processes[29,30].While CMOS foundries have already developed standard Si 3 N 4 processes [31] to fabricate PIC with typical thickness of 300 nm and loss on the order of 10 dB/m, there has not been a process to simultaneously achieve ultralow loss (e.g. below 3 dB/m) and large thickness (above 600 nm) without crack formation. So far, thin Si 3 N 4 PIC with width above 5 m and thickness below 100 nm[17][18][19]can achieve optical loss below 0.1 dB/m. While this process has recently become an 8-inch foundry process [32], the thin Si 3 N 4 exhibits weak optical confinement and thus suffers from exaggerated bending loss with a small footprint. The optical mode also exhibits strong normal GVD due to the waveguide geometry. While coupled waveguide structures can be used to alter local GVD[33,34], they cannot offer anomalous GVD over a wide spectral range. In parallel, thick Si 3 N 4 PIC of ultralow loss and broadband anomalous GVD has been realized via the subtractive process[35][36][37]and the photonic Damascene process[14,38]. However, all these reported processes have been developed only in laboratories and have issues with transferring to foundry manufacturing. While high-yield, wafer-scale fabrication of Si 3 N 4 PIC has achieved 1 dB/m loss and anomalous GVD in Ref.[14], the photonic Damascene process has intrinsic limitations. For example, the use of chemical-mechanical planarization (CMP) to remove excess Si 3 N 4 can cause serious dishing effect in large-area structures (see Appendix C), thus cannot be used to fabricate elements such as arrayed waveguide gratings (AWG) and multimode interferometers (MMI). Meanwhile, the aspect-ratio-dependent etch effect prevents to form narrow but deep channels[39]. In comparison, with the subtractive process [35-37], thick Si 3 N 4 PIC of ultralow loss and anomalous GVD has been only achieved with electron-beam lithography (EBL) on 3-or 4-inch wafers, which are not foundry-standard. Meanwhile, an 8-inch foundry process has been developed in Ref.[40], but this process has not yet achieved optical loss below 3 dB/m.In this work, we overcome the above challenges and demonstrate a foundry-standard fabrication process of tight-confinement, dispersion-engineered, ultralow-loss Si 3 N 4 PIC. The process is based on 6-inch wafers and combines deep-ultraviolet (DUV) stepper lithography[14]and state-of-the-art subtractive process [37], i.e. a DUV subtractive process. We have achieved a linear optical loss of 2.6 dB/m in 810-nm-thick Si 3 N 4 PIC. Finally, we generate single soliton microcombs of 100.17 GHz and 19.975 GHz mode spacings using these devices. thank Chao Xiang for the fruitful discussion.", 'arxivid': '2303.05004', 'author': ['Zhichao Ye \nQaleido Photonics\n310000HangzhouChina\n', 'Haiyan Jia \nQaleido Photonics\n310000HangzhouChina\n', 'Zhangjun Huang \nQaleido Photonics\n310000HangzhouChina\n', 'Chen Shen \nInternational Quantum Academy\n518048ShenzhenChina\n\nShenzhen Institute for Quantum Science and Engineering\nSouthern University of Science and Technology\n518055ShenzhenChina\n', 'Jinbao Long \nInternational Quantum Academy\n518048ShenzhenChina\n\nShenzhen Institute for Quantum Science and Engineering\nSouthern University of Science and Technology\n518055ShenzhenChina\n', 'Baoqi Shi \nInternational Quantum Academy\n518048ShenzhenChina\n\nDepartment of Optics and Optical Engineering\nUniversity of Science and Technology of China\n230026HefeiChina\n', 'Yi-Han Luo \nInternational Quantum Academy\n518048ShenzhenChina\n\nShenzhen Institute for Quantum Science and Engineering\nSouthern University of Science and Technology\n518055ShenzhenChina\n', 'Lan Gao \nInternational Quantum Academy\n518048ShenzhenChina\n\nShenzhen Institute for Quantum Science and Engineering\nSouthern University of Science and Technology\n518055ShenzhenChina\n', 'Wei Sun \nInternational Quantum Academy\n518048ShenzhenChina\n', 'Hairun Guo \nKey Laboratory of Specialty Fiber Optics and Optical Access Networks\nShanghai University\n200444ShanghaiChina\n', 'Jijun He \nMinistry of Education\nKey Laboratory of Radar Imaging and Microwave Photonics\nNanjing University of Aeronautics and Astronautics\n210016NanjingChina\n', 'Junqiu Liu \nInternational Quantum Academy\n518048ShenzhenChina\n\nHefei National Laboratory\nUniversity of Science and Technology of China\n230088HefeiChina\n'], 'authoraffiliation': ['Qaleido Photonics\n310000HangzhouChina', 'Qaleido Photonics\n310000HangzhouChina', 'Qaleido Photonics\n310000HangzhouChina', 'International Quantum Academy\n518048ShenzhenChina', 'Shenzhen Institute for Quantum Science and Engineering\nSouthern University of Science and Technology\n518055ShenzhenChina', 'International Quantum Academy\n518048ShenzhenChina', 'Shenzhen Institute for Quantum Science and Engineering\nSouthern University of Science and Technology\n518055ShenzhenChina', 'International Quantum Academy\n518048ShenzhenChina', 'Department of Optics and Optical Engineering\nUniversity of Science and Technology of China\n230026HefeiChina', 'International Quantum Academy\n518048ShenzhenChina', 'Shenzhen Institute for Quantum Science and Engineering\nSouthern University of Science and Technology\n518055ShenzhenChina', 'International Quantum Academy\n518048ShenzhenChina', 'Shenzhen Institute for Quantum Science and Engineering\nSouthern University of Science and Technology\n518055ShenzhenChina', 'International Quantum Academy\n518048ShenzhenChina', 'Key Laboratory of Specialty Fiber Optics and Optical Access Networks\nShanghai University\n200444ShanghaiChina', 'Ministry of Education\nKey Laboratory of Radar Imaging and Microwave Photonics\nNanjing University of Aeronautics and Astronautics\n210016NanjingChina', 'International Quantum Academy\n518048ShenzhenChina', 'Hefei National Laboratory\nUniversity of Science and Technology of China\n230088HefeiChina'], 'corpusid': 256854305, 'doi': '10.1364/prj.486379', 'github_urls': [], 'n_tokens_mistral': 26521, 'n_tokens_neox': 22375, 'n_words': 11662, 'pdfsha': '0aa97ac233baa8668c6c2f38da2bae1bb7639e8f', 'pdfurls': ['https://export.arxiv.org/pdf/2303.05004v1.pdf'], 'title': ['Foundry manufacturing of tight-confinement, dispersion-engineered, ultralow-loss silicon nitride photonic integrated circuit', 'Foundry manufacturing of tight-confinement, dispersion-engineered, ultralow-loss silicon nitride photonic integrated circuit'], 'venue': []}

arxiv

Probing the Coulomb gap in the topological insulator BiSbTeSe 2 via Quantum Capacitance Jimin Wang Institute of Experimental and Applied Physics University of Regensburg 93040RegensburgGermany Cosimo Gorini Institute of Theoretical Physics University of Regensburg 93040RegensburgGermany Klaus Richter Institute of Theoretical Physics University of Regensburg 93040RegensburgGermany Zhiwei Wang Physics Institute II University of Cologne Zülpicher Str. 7750937KölnGermany Ministry of Education School of Physics Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement Beijing Institute of Technology 100081BeijingChina Yoichi Ando Physics Institute II University of Cologne Zülpicher Str. 7750937KölnGermany Dieter Weiss Institute of Experimental and Applied Physics University of Regensburg 93040RegensburgGermany Probing the Coulomb gap in the topological insulator BiSbTeSe 2 via Quantum Capacitance (Dated: December 6, 2019) BiSbTeSe 2 is a 3D topological insulator with Dirac type surface states and low bulk carrier density, as donors and acceptors compensate each other. Dominating low temperature surface transport in this material is heralded by Shubnikov-de Haas oscillations and the quantum Hall effect. Here, we experimentally probe the electronic density of states (DOS) in thin layers of BiSbTeSe 2 by capacitance experiments both without and in quantizing magnetic fields. By probing the lowest Landau levels, we show that a large fraction of the electrons filled via field effect into the system ends up in (localized) bulk states and appears as a background DOS. The surprisingly strong temperature dependence of such background DOS can be traced back to the Coulomb gap in the system. An ideal three-dimensional (3D) topological insulator (TI) has an insulating bulk, characterized by a gap in the energy spectrum, and symmetry protected, conducting surface states ( 1,2 and references therein). As these helical surface states are not spin degenerate, 3D-TIs provide a promising platform to realize topological superconductivity, a prerequisite to search, e.g., for Majorana zero modes 3,4 . Experimentally available TIs like Bi 2 Se 3 or Bi 2 Te 3 are, however, far from ideal as they feature, due to intrinsic defects, a relatively high electron or hole density larger than 10 18 cm −3 (see 5 and references therein). By combining p-type and n-type TI materials, i.e., by compensation, the bulk concentration can be suppressed 2,6 . This comes at the price of large potential fluctuations at low temperatures as the resulting ionized donor and acceptor states are poorly screened and constitute a randomly fluctuating Coulomb potential, bending the band edges and creating electron and hole puddles 7,8 . These were observed by, e.g., optical spectroscopy 5 . In the absence of metallic surface states, i.e., in fully compensated conventional semiconductors, variable range hopping governs low-temperature transport (T < 100 K) 9 . Recently, Skinner et al. have shown that the electronic density of states (DOS) in the bulk is nearly constant under these circumstances and features a Coulomb gap at the Fermi level µ 7, 8 . In 3D-TIs, in addition, Dirac surface states, which form a two-dimensional (2D) electron (hole) system, encase the bulk and constitute the dominating transport channel at low temperatures. Here, we explore the interplay between topological surface and trivial bulk states. To this end we probe the DOS of the Dirac surface states by capacitance spectroscopy. The total capacitance C, measured between a metallic top gate and the Dirac surface states, depends on the geometric capacitance per unit area, C 0 = 0 /d, and on the quantum capacitance e 2 D(µ), C −1 = C −1 0 + e 2 D(µ) −1 .(1) Here , d are, respectively, dielectric constant and insulator thickness, 0 the vacuum dielectric constant, and D(µ) the DOS at the Fermi level (chemical potential) µ. The quantum capacitance, connected in series to C 0 , reflects the energy spectrum of 2D electron systems [10][11][12] , and probes preferentially the top surface DOS in 3D-TIs 13 . At higher temperatures, D has to be replaced by the thermodynamic density of states (TDOS) at µ, D(µ) = dn/dµ, with n the carrier density. While gating of 3D-TI and tuning of the carrier densities of top and bottom surfaces has been explored in the past 14 . Here we used that k = √ 4πn and Ω = (m * v F ) 2 + 2Em * . While in a perfect system D(E) vanishes at the CNP, disorder smears the singularity, as in case of graphene 12 . We model the potential fluctuations by a Gaussian distribution of energies with width σ, resulting in an average DOS D(µ) = ∞ −∞ D(E) 1 √ 2πσ exp −(E−µ) 2 2σ 2 dE. To convert energies into voltages we use n = C 0 (V tg − V 0 )/e, with e the elementary charge and V 0 describing n at zero voltage. By fitting D(µ) to the data in Fig from arrow position to arrow position, the carrier density changes by the LL degeneracy ∆n = eB/h. In contrast, the change of carrier density ∆n, calculated via capacitance, ∆n = C 0 e ∆V tg , is by a factor 1.4 higher. Hence, we must assume that a large fraction of the carriers, induced by field effect, ends up in the bulk and is localized at low T . To compare with these experiments we calculate C(V tg ) using Gaussian-broadened LLs, D LL (E) = 1 √ 2πΓ eB h exp −(E−En) 2 2Γ 2 , with broadening Γ. The LL spectrum dispersion reads 26 E n = |n| eB m * + sgn(n) eB 2m * 2 + 2eB v 2 F |n|,(2) with n = 0, ±1, ±2, . . . , and the tiny Zeeman splitting was neglected. Using the above DOS is insufficient to describe the data -the distance ∆V tg between adjacent Landau gaps (marked,e.g., by arrows in Fig. 2(a) for the 14 T trace) is too small 21 . ∆V tg is the voltage needed to fully fill the 0-th LL of the surface states. Since ∆V tg in experiment is larger than calculation (relies on the filling rate dn/dV tg ≈ C 0 /e), it means that a fraction of the field-induced electrons does not go to the surface states but eventually into the bulk. Thus a higher voltage (higher δn) is needed to fill the zeroth LL. On contrast, we obtain almost perfect agreement -see Fig. 2(b) -if we introduce an energy-independent background DOS D b which models these bulk states 21 . The calculated TDOS we compare with experiment thus reads Calculation Experiment T = 1.5 K (b) 0 +1 -1 (a)D(µ) = ∞ −∞ [D LL (E) + D b ] ∂f ∂µ dE,(3) with f = f (E − µ, T ) the Fermi function. As shown in Fig. 2(b), the constant background D b leads to excellent agreement with experiment. Although the bulk DOS is hardly directly accessible by the quantum capacitance itself (i.e., by its value), we probe it indirectly via the missing charge given by the Landau gap positions. This missing charge carrier issue holds also for the quantum Hall trace in Fig. 1(c), where ∼ 30% of the induced electrons are missing. Indeed, it also appears in several other publications with missing electron fractions ranging from 30% (as here) to 75% (see [16][17][18][19] ). The bottom line is: The change of surface carrier density extracted from the Landau gap positions is smaller than the one "loaded" into the system within the same voltage interval. Further, the filling rate dn/dV tg determined by the classical Hall effect at 1.5 K is consistent with the one found for the surface states 21 . Thus, the charge carriers loaded at low T into the bulk are localized and do not contribute to transport. This is consistent with transport experiments 16,27 , and also in line with what is expected in fully compensated Temperature dependence of quantum capacitance -The background DOS rises quickly with temperature. Corresponding C(V tg ) data for 14 T and various T s up to 58 K is shown in Fig. 3(a). The local minima due to Landau gaps, marked by arrows, shift with increasing T to larger V tg . The corresponding ∆V tg (T ) is shown in Fig. 3(c). For fixed B the Landau degeneracy eB/h is constant and does not depend on temperature. The increasing ∆V tg needed to fill the 0-th LL of the surface states thus indicates that, with increasing T , more carriers are lost to the bulk. Similar behavior was found for quantum Hall data 16,27 . Clearly, to model the Landau gap positions correctly, a strongly T -dependent TDOS is required. A simple approach consists in introducing a T -dependent background DOS, D b → D b (T ). Its values used to fit the data of Fig. 3(a) are shown in Fig. 3(d); the resulting C(V tg ) traces for different temperatures are plotted in Fig. 3(b). D b is nearly constant at low T but rises quickly at higher temperatures. However, a problem with this procedure is that for noninteracting electrons the number of single-particle states in a given energy (voltage) window by definition does not depend on T . The corresponding TDOS can still be T -dependent, but only as long as the tails of ∂f /∂µ, can reach regions with substantial changes of the single particle DOS, typically close to gap edges 28 . Yet the strongly temperature-dependent signal is obtained by scanning V tg deep into the BiSbTeSe 2 gap, whose width of 300 meV is far too large to explain the observations. Probing the Coulomb gap -A way out of this apparent dead-end is provided by the strongly fluctuating potential landscape of compensated TIs like BiSbTeSe 2 , sketched in D b (µ) = D b ∂f ∂µ dE + f ∂D b ∂µ dE.(4) To describe the experiment we need to assume that the Coulomb gap cannot instantaneously follow µ if the carrier density is, as in our capacitance experiments, modulated by 50 Hz. The Coulomb gap stems from rearrangement of unscreened charges in a strongly disordered landscape, and its formation time depends on disorder strength and type, temperature, and magnetic field [29][30][31] . Indeed, the electronic many-body system is known to behave as a Coulomb glass, characterized by long reaction/relaxation times 32 . Determining the exact shape and temperature dependence of the Coulomb gap DOS is a complex problem, whose details are still under debate [33][34][35][36][37][38] , and which is here further complicated by the presence of For a qualitative understanding of our experimental data, it suffices to consider a barebone model for the bulk DOS, given in normalized form D b (E, µ) = tanh a + E−µ b 2 dE tanh a + E −µ b 2 .(5) Eq. (5) describes a T -independent Coulomb gap, whose (small) residual value at E = µ is determined by the dimensionless parameter a b (µ) = D b (E, µ) ∂f (E−µ) ∂µ dE. We also considered an explicitly T -dependent DOS model, mimicking the progressive filling of the Coulomb gap by thermal excitations [33][34][35][36][37][38] . Since the simple model fits the data equally well, we do not discuss it here 21 . In Fig. 4(c) we compare the measured TDOS, D b (T ) and the calculated one,D b (T ), each normalized to the corresponding value at 1.5 K. Using a Coulomb gap width of 11.7 meV yields perfect agreement with experiment. The corresponding TDOSD b (E, 0), calculated using Eq.(4) for µ = 0 is displayed in Fig. 4(d). At the position of µ, the TDOS is strongly T -dependent. It is this value which we conclude to observe in experiment. If, for a given temperature, the chemical potential is shifted via V tg , the gap follows. This is the reason why using a constant background TDOS gives excellent agreement with the experimental data, as shown in Fig. 3. In summary, we found that correctly describing the position of the Landau gaps in the BiSbTeSe 2 capacitance-voltage signal requires to introduce a constant but strongly temperature-dependent background TDOS. This suggests that we are probing the effective single particle DOS in the Coulomb gap. The latter follows the chemical potential as we adjust the reference top gate potential, but is accessible to our capacitance measurement due to its slow (glassy) dynamics. Our picture accounts for the large amount of charge missing from surface states in BiSbTeSe 2 transport experiments. Useful information, such as carrier density n and activation energy E a can be extracted from the graph. Following a recent paper 1 , the surface and the bulk of the topological insulator contribute to the conduction independently, equivalent to resistors connected in parallel. The resistivity of each surface (ρ s ) can be modeled as metallic like, ρ s = ρ s,0 + A · T , where ρ s,0 and A are constants. The bulk resistivity ρ b is thermally activated for temperatures beyond the variable range hopping regime: ρ b = ρ b,0 · exp(E a /k B T ), where ρ b,ρ = (2 · ρ −1 s + ρ −1 b ) −1 . Here we assume, for simplicity, that the two surfaces are equivalent, having the same carrier density n s and mobility µ s . The corresponding fit is shown in Fig. S1a, where we obtained remarkable good agreement. In addition, the fitting also yields ρ s = 3801 + 20.5 · T , and ρ b = 234 · exp(31 meV/k B T ) and thus the effective energy gap ∆ = 62 meV. Compared to the nominal energy gap ∆ ≈ 300 meV 2 , the effective gap is 5-times smaller. This is consistent with previous reports 3 and a consequence of the coexisting electron and hole puddles which stem from the large potential fluctuations in BSTS. At sufficiently low temperature (e.g., 1.5 K), the conduction is almost entirely dominated by Dirac surface states, also verified in Ref. 4 . However, at higher temperatures, bulk carriers are activated and contribute to transport. At T = 58 K, using the results from the fit above, and by assuming a typical, temperature independent value of µ b = 200 cm 2 /Vs, we obtain n b = 2.7 · 10 11 cm −2 . The same way, the carrier density at zero gate voltage on each surface is n s = 6.3 · 10 11 cm −2 , using µ b = 2000 cm 2 /Vs. Thus the delocalized bulk carrier density is comparable to that in the surface states. The bulk carrier density at 58 K, n b , can also be estimated from the constant DOS background D b (T ). ARPES measurements show that pristine BSTS is slightly p-type doped. II. DETERMINING ∆V tg Here we show how ∆V tg in Fig. 2 The filling rate dn/dV tg extracted from the position of the Landau gaps on the gate voltage scale in the main text ( Fig. 2(a) and Fig. 3(a)) is smaller than C 0 /e, expected from the insulator capacitance C 0 . Fig. S4 shows that the reduced filling rate is also measured by the classical Hall slope. The Hall data were taken at 1.5 K by sweeping the top gate voltage (V tg ) at fixed B, as well as by sweeping B at fixed V tg . The device shows at such low temperature surface dominated conduction, which takes place in top and bottom surface. From Fig. 1(c) in the main text, we see that at zero gate voltage, both top and bottom surfaces are slightly pdoped. By grounding the back gate and sweeping V tg from 5 to -5 V, one changes the carrier type in the top surface. For V tg < 0 in Fig. S4, p-type conduction prevails in top and bottom surfaces. The total carrier density and the total filling rate in this regime can be simply obtained from the one carrier Drude model. The filling rate, i.e., the change of total density with gate voltage, is given by the slope of the red line in Fig. S4. The corresponding filling rate is 3.2 · 10 11 cm −2 V −1 . This is consistent with the filling rate displayed by the QHE data in the main text (4 · 10 11 cm −2 V −1 , using ∆V tg = 0.84 V from Fig. 3(c)). Similar conclusions can be found in 4 . The B = 0 capacitance trace in Fig. 1(d) of the main text we fitted without using a background DOS D b . While the B = 0 data can be well fitted using D b = 0, the capacitance data in quantizing magnetic field can not. In Fig. S5 we show that the data of Fig. 1(d) can be likewise fitted with a finite DOS D b . As a result, the energy broadening becomes significantly smaller, but the important parameters like Fermi velocity v F and effective mass m * stay within 20% the same. VI. A TEMPERATURE-DEPENDENT DOS MODEL Consider the (normalised) model DOS Eq. (6) extends the T -independent minimal model employed in the manuscript, and takes into account the Coulomb gap T -dependence which arises from its progressive filling by thermal excitations once T = 0 5-10 . It does it so as to agree with the universal behavior obtained by Mogilyanskii and Raikh 6 at low T and low energy, with the addition of a FIG. 1 . 1(a) Design of layer sequence. Red lines sketch the topological surface states. (b) Optical micrograph of the device. The dashed yellow line marks the capacitor area of 1.8 × 10 3 µm 2 . (c) ρ xy as function of V tg and V bg , respectively, at T = 1.5 K and B = 14 T. The almost horizontal and vertical dashed purple lines separate the region of well developed QHE with total filling factors of -1, 0, and 1 (unit h/e 2 ), from regions of higher filling factors. (d) C(V tg ) at T = 1.5 K and B = 0 T. The pronounced minimum reflects the bulk gap. For better comparison, the CNP of all measurements is shifted to zero via V CNP . The red line is a fit using a Gaussian broadening of the Fermi level with σ = 29.4 meV (see text). To compare with experiment we added a parasitic capacitance of ∼3.07 pF. The lower left inset illustrates the measurement principle, the upper inset the energy dispersion of the surface states in the bulk gap (left) and the corresponding DOS (right).It is sketched in the upper inset ofFig. 1(d), together with the electron-hole asymmetric, nearly E-linear DOS, given by D(E) = m * (Ω−m * v F ) 2π 2 Ω . 1(d) we extract σ = 29.4 meV, v F = 3.2 · 10 5 m/s and m * = 0.47m 0 (m 0 = free electron mass). The broadening σ is only important in the immediate vicinity of the CNP but hardly affects the values of v F and m * . The obtained v F and m * values agree well with ARPES data 14,20 and values extracted from Shubnikov-de Haas oscillations 25 . B-field dependence of capacitance measurements -Our key result arises when we crank up the magnetic field and measure signatures of the Landau level (LL) spectrum, shown in Fig. 2(a). At the 0-th LL level position a local maxima emerges with increasing B-field, flanked by minima at each side. The two minima, highlighted by arrows, correspond to the Landau gaps between LLs 0, and ±1 [see Fig. 2(a)]. Due to the large broadening, higher LLs do not get resolved. Lowering T down to 50 mK does not resolve more structure, indicating that disorder broadening is the limiting factor. By sweeping V tg across the 0-th LL, i.e., FIG. 2 . 2(a) C(V tg ) for B ranging from 4 T to 14 T. The lower inset sketches the LL DOS for LLs -1, 0 and 1. Arrows mark the position of the Landau gaps for the 14 T trace 21 . (b) Model calculations to (a) based on Eq. (3) after adding a parasitic capacitance of ∼ 3.06 pF. Parameters of the fit at 1.5 K: Γ = 14.9 to 15.9 meV; D b = 2.4 · 10 35 m −2 J −1 . ) C(V tg ) at B = 14 T for various T s. Arrows mark the minima corresponding to Landau gaps. For increasing T the voltage difference ∆V tg between adjacent gaps increases. The trace at 1.5 K was shifted down by 0.032 pF for clarity. (b) Calculated C(V tg ) using Eq. (3) with D b values in (d), and Γ = 13 − 15.2 meV. (c) ∆V tg vs. T . (d) Extracted D b .semiconductors 9 , as was recently highlighted in Ref.7 . There, bulk transport of compensated TI was considered, where local puddles of n-and p-regions form. In this regime, low-T transport is governed by variable range hopping, the DOS within the gap is essentially constant, while filled and empty states are separated by the Coulomb gap at µ 7,9 .Using a constant background affects somewhat the values extracted above from C(V tg , B = 0). Thus, we fitted the trace inFig. 1(d)using the same D b = 2.4 · 10 35 m −2 J −121 . Now a reduced broadening σ = 15 meV is needed. C(V tg ) is then best described by slightly modified values: v F = 2.8 · 10 5 m/s and m * = 0.57m 0 , respectively, still compatible with results reported elsewhere 14,20,25 . Fig. 4 ( 4a)-(b), where Coulomb interaction dominates 7 : In a nutshell, the background DOS emerges as an effective single particle DOS describing the ensemble of strongly interacting electrons filling bulk impurity states. As such, it is actually a µ-and T -dependent object, D b → D b (E, µ, T ), characterized by a Coulomb gap at E = µ 9 . The TDOS we compare in Fig. 4(c) with experiment is then given by D b (µ) = ∂ ∂µ D b (E, µ, T )f (E − µ)dE. Since both D b and the Fermi function f depend on µ, the derivative with respect to µ generates two terms: Conduction (E c ) and valence band (E v ) profiles, fluctuating due to long range Coulomb interactions. A sketch of the bulk DOS including the Coulomb gap at µ is shown at right. The DOS at µ is probed in experiment. (b) Sketch of electron and hole puddles in the µ plane, consequence of the strongly fluctuating band edges in (a). (c) Comparison of calculated and measured D(T ). Filled squares are experimental data. Different curves are model results for different values of the parameter b in Eq. (5). All curves are normalized to their values at T = 1.5 K. The TDOS grows by approximately a factor 8 between T = 1.5 K and T = 58 K. Such a relative increase is perfectly reproduced for a = 0.051, b = 11.7 meV. The best fit values lead to the TDOS associated with the Coulomb gap shown in panel (d) for different temperatures. Here, µ is at zero energy. topological surface states 39 .On the time scale of 20 ms, set by the experiment, the Coulomb gap very likely cannot follow µ, i.e. D b is only a function of E for these short times at fixed T . Hence, the second term on the right hand side of Eq. (4) vanishes. On the other hand, the timescale of about 1 minute needed to produce each data point after changing V tg should be sufficiently long to form a gap, at least partially. As the precise time scales for BiSbTeSe 2 are not known, we resort to simple DOS models to test the feasibility of our approach. 1, while b (with dimension of energy) fixes its width. The form is chosen so that for E → µ it reproduces the known 3D Efros-Shklovskii form ∝ (E − µ) 2 , with the addition of a residual value. The corresponding T -dependent TDOS is then given byD ACKNOWLEDGMENTS2 We thank Ferdinand Evers for inspiring discussions. The work at Regensburg was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project-IDSupplementary information: Probing the Coulomb gap in the topological insulator BiSbTeSe 2 via Quantum Capacitance Jimin Wang, 1 Cosimo Gorini, 2 Klaus Richter, 2 Zhiwei Wang, 3,4 Yoichi Ando, Institute of Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany 3 Physics Institute II, University of Cologne, Zülpicher Str. 77, 50937 Köln, Germany 4 Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement, Ministry of Education, School of Physics, Beijing Institute of Technology, Beijing, 100081, China I. RESITIVITY-TEMPERATURE RELATIONFig. S1 shows the temperature dependence of the device discussed in the main part of the text. With decreasing temperature, the resistivity of the device increases down to ∼ 65 K, characteristic for semiconductor-like, bulk dominated conduction, and decreases below ∼ 65 K, typical for metallic-like, surface dominated conduction.FIG. S1. R-T relation of the device. (a) The experimental data (blue squares) and corresponding fit (red line). The dashed line approximately separates the bulk and surface conduction dominated regions. (b) The same graph as in (a) in log scale and together with the fitted (see text) bulk (purple) and surface (green) contributions. The inset in (a) shows the measurement configuration. 2 2Thus, the bulk hole concentrationn b = 0 −∞ D b (T = 58 K)(1−f (E −µ))dE, where f (E −µ)is the Fermi-Dirac function. D b (T = 58 K) = 17.4 · 10 35 m −2 J −1 , can be read fromFig. 3din the main text. Here, µ is slightly below the Dirac point, µ ≈ −25 meV.2 With these values one obtains n b = 7 · 10 11 cm −2 , which is of the same order and comparable to the estimate given above. . S2. C-V trace at 20 K and 14 T, and the corresponding second derivative d 2 C/dV 2 tg , used to determine ∆V tg . and Fig. 3 is determined. As example we use the experimental data taken at 20 K and 14 T, shown in Fig. S2. To enhance the visibility of the minima and to suppress effects of the background we calculate the second derivative of the trace. This results in two pronounced peaks at the position of the minima. The distance between the 2 peaks gives ∆V tg . This procedure is particularly useful at elevated temperatures where it is more difficult to determine the minimum position accurately from the original capacitance trace. III. FAILURE OF FITTING C-V TRACE AT 14 T WITHOUT BACKGROUND DOS In Fig. S3 we compare the capacitance measured at B = 14 T and T = 1.5 K to fits with and without using a constant background density. The fits without background density fail to describe the data, especially the voltage difference between the capacitance minima which correspond to the Landau gaps. The distance between calculated, adjacent Landau gaps is FIG. S3. Comparison of fits to the experimental data at 1.5 K and 14 T, with and without finite D b = 2.4 · 10 35 m −2 J −1 , while the other paramters are kept the same: Fermi velocity ν F = 2.8 · 10 5 m/s, effective mass m * = 0.57m 0 , Gaussian energy broadening σ = 15.2 meV, and parasitic capacitance C para = 3.0635 pF. always smaller (marked by ∆V tg in Fig. S3) than observed in experiment if D b is set to 0, no matter how the other parameters are chosen. This discrepancy becomes much larger at higher temperatures. IV. ESTIMATING THE FILLING RATE FROM HALL MEASUREMENTS. FIG . S4. Estimate of the total carrier density p using the Hall slope giving p = (e dρxy dB ) −1 for low magnetic field data. Here the electron density is defined positive, the hole density negative. V. COMPARING FITTING OF C-V AT B = 0 T AND T = 1.5 K WITH AND WITHOUT BACKGROUND DOS FIG . S5. Fitting the data of Fig. 1(d) with and without D b at B = 0 T and T = 1.5 K. The red dashed line is the same as the one shown in Fig. 1(d), with fitting parameters v F = 3.2 · 10 5 m/s, m * = 0.47m 0 , σ = 29.4 meV, parasitic capacitance 3.073 pF, and D b = 0. The blue trace, in contrast, was obtained using the same D b = 2.4 · 10 35 m −2 J −1 extracted at the same T from the 14 T data. The other fit parameters are then v F = 2.8 · 10 5 m/s, m * = 0.57m 0 , σ = 15.2 meV, and parasitic capacitance 3.065 pF. ,15 , the quantum capacitance and the DOS in a compensated TI like BiSbTeSe 2 remained uncharted. Our measurements show that, while Dirac surface states dominate low-T transport as expected, the bulk provides a background of Coulomb glass type, capable of absorbing a large amount of charge carriers. These missing charges is very common in 3D-TI transport experiments, yet to the best of our knowledge unexplained[16][17][18][19] .We used pristine, slightly p-doped BiSbTeSe 2 crystals grown by the modified Bridgman method 6 . Angle-resolved photoemission spectroscopy has demonstrated that Dirac point and µ of this material lie in the bulk gap 20 . BiSbTeSe 2 flakes were exfoliated onto highly Transport measurements using low ac excitation currents (10 nA at 13 Hz) were carried out between 1.5 K and 58 K and in magnetic fields B up to 14 T. Temperature dependent measurements show (seeFig. S1in 21 ) that at 1.5 K transport is entirely dominated by the surface with negligible contribution from the bulk. The carrier density and µ of top and bottom surfaces can be adjusted by top and bottom gate voltages, V tg , V bg , respectively. This is shown for the Hall resistivity at 14 T inFig. 1(c). The device displays well developed quantum Hall plateaus at total filling factor ν = -1, 0 and 122 . The plateaus are well separated from each other and marked by dashed purple lines. As these lines run nearly parallel to the V tg -and V bg -axes, respectively, we conclude that the carrier density on top and bottom can be tuned nearly independently. Planck constant, v F the Fermi velocity at the Dirac point, and m * the effective mass.p-doped Si chips (used as backgate) coated by 285 nm SiO 2 . The flakes were processed into quasi-Hallbars with Ti/Au (10/100 nm) ohmic contacts. A larger h-BN flake, transferred on top of BiSbTeSe 2 serves as gate dielectric. Finally, we use Ti/Au (10/100 nm) as a top gate contact. Fig. 1(a) and 1(b) display the layer sequence and optical micrograph of one of the devices, respectively. Capacitance measurements -We use a two-terminal setup with one contact connected to the top gate, the other to the BiSbTeSe 2 layer, see inset in Fig. 1(d). A high-precision capacitance bridge AH2700 with ac modulation of 0.1 V was used at the lowest operation frequency of 50 Hz to suppress resistive effects 13 . To minimize hysteresis, we always sweep V tg in one direction only. Figure 1(d) shows the measured capacitance C, which is directly connected to the DOS, see Eq. (1). The measured trace with a minimum at the Dirac or charge neutrality point (CNP) resembles the quantum capacitance measured for graphene, apart from a pronounced electron-hole asymmetry 12,23,24 due to a parabolic contribution to the linear E(k) dispersion 25 . 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Galperin, Phys. Rev. B 85, 155206 (2012). . Y Meroz, Y Oreg, Y Imry, http:/iopscience.iop.org/article/10.1209/0295-5075/105/37010/metaEPL. 10537010Y. Meroz, Y. Oreg, and Y. Imry, EPL 105, 37010 (2014).

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{'abstract': 'BiSbTeSe 2 is a 3D topological insulator with Dirac type surface states and low bulk carrier density, as donors and acceptors compensate each other. Dominating low temperature surface transport in this material is heralded by Shubnikov-de Haas oscillations and the quantum Hall effect. Here, we experimentally probe the electronic density of states (DOS) in thin layers of BiSbTeSe 2 by capacitance experiments both without and in quantizing magnetic fields. By probing the lowest Landau levels, we show that a large fraction of the electrons filled via field effect into the system ends up in (localized) bulk states and appears as a background DOS. The surprisingly strong temperature dependence of such background DOS can be traced back to the Coulomb gap in the system.', 'arxivid': '1912.02725', 'author': ['Jimin Wang \nInstitute of Experimental and Applied Physics\nUniversity of Regensburg\n93040RegensburgGermany\n', 'Cosimo Gorini \nInstitute of Theoretical Physics\nUniversity of Regensburg\n93040RegensburgGermany\n', 'Klaus Richter \nInstitute of Theoretical Physics\nUniversity of Regensburg\n93040RegensburgGermany\n', 'Zhiwei Wang \nPhysics Institute II\nUniversity of Cologne\nZülpicher Str. 7750937KölnGermany\n\nMinistry of Education\nSchool of Physics\nKey Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement\nBeijing Institute of Technology\n100081BeijingChina\n', 'Yoichi Ando \nPhysics Institute II\nUniversity of Cologne\nZülpicher Str. 7750937KölnGermany\n', 'Dieter Weiss \nInstitute of Experimental and Applied Physics\nUniversity of Regensburg\n93040RegensburgGermany\n'], 'authoraffiliation': ['Institute of Experimental and Applied Physics\nUniversity of Regensburg\n93040RegensburgGermany', 'Institute of Theoretical Physics\nUniversity of Regensburg\n93040RegensburgGermany', 'Institute of Theoretical Physics\nUniversity of Regensburg\n93040RegensburgGermany', 'Physics Institute II\nUniversity of Cologne\nZülpicher Str. 7750937KölnGermany', 'Ministry of Education\nSchool of Physics\nKey Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement\nBeijing Institute of Technology\n100081BeijingChina', 'Physics Institute II\nUniversity of Cologne\nZülpicher Str. 7750937KölnGermany', 'Institute of Experimental and Applied Physics\nUniversity of Regensburg\n93040RegensburgGermany'], 'corpusid': 208636951, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 14813, 'n_tokens_neox': 12270, 'n_words': 6961, 'pdfsha': '648b0094c62199ec1d9e1cbf74f6b5f8ecd0a3c6', 'pdfurls': ['https://arxiv.org/pdf/1912.02725v1.pdf'], 'title': ['Probing the Coulomb gap in the topological insulator BiSbTeSe 2 via Quantum Capacitance', 'Probing the Coulomb gap in the topological insulator BiSbTeSe 2 via Quantum Capacitance'], 'venue': []}

arxiv

Noncritical quadrature squeezing through spontaneous polarization symmetry breaking 12 Mar 2010 Ferran V Garcia-Ferrer Departament d'Òptica Universitat de València BurjassotSpain Carlos Navarrete-Benlloch Departament d'Òptica Universitat de València BurjassotSpain Germán J De Valcárcel Departament d'Òptica Universitat de València BurjassotSpain Eugenio Roldán *eugenio.roldan@uv.es Departament d'Òptica Universitat de València BurjassotSpain Noncritical quadrature squeezing through spontaneous polarization symmetry breaking 12 Mar 2010 We discuss the possibility of generating noncritical quadrature squeezing by spontaneous polarization symmetry breaking. We consider first type-II frequency-degenerate optical parametric oscillators, but discard them for a number of reasons. Then we propose a four-wave mixing cavity in which the polarization of the output mode is always linear but has an arbitrary orientation. We show that in such a cavity complete noise suppression in a quadrature of the output field occurs, irrespective of the parameter values. Spontaneous rotational symmetry breaking has recently been shown theoretically to be a resource for noncritical quadrature squeezing both in type-I frequencydegenerate optical parametric oscillators (DOPOs) [1,2] and in degenerate four wave mixing χ (3) cavities [3], as it was previously shown for spontaneous translational symmetry breaking in wide aperture type-I DOPOs [4,5]. The underlying idea is that a spontaneous symmetry breaking occurring within a nonlinear cavity entails the existence of a canonical pair of observables: One of them is solely driven by quantum fluctuations (such as the orientation of a Hermite-Gauss signal mode in [1][2][3], or the location in the transverse plane of a localized structure in [4,5]) while its canonical pair is maximally damped and, consequently, maximally insensitive to quantum fluctuations (angular and linear momenta, respectively, in the referred cases). We have predicted in the above cases that by this means perfect quadrature squeezing appears in the maximally damped mode independently of the parameter setting, hence the name noncritical quadrature squeezing. In the present letter we discuss on the possibility of achieving the same result through a spontaneous polarization symmetry breaking (SPSB). In order to achieve noncritical quadrature squeezing through SPSB the nonlinear cavity must verify some requisites. It must posses a free polarization parameter (FPP), that is, a parameter of the signal's field polarization [6] not fixed by the system's dynamical equations. Additionally, it must allow the existence of a nonzero mean field solution for the signal field; in this way, once the threshold for the generation of the signal mode is crossed, the random occurrence of a particular FPP will automatically produce a SPSB. From a mathematical point of view, a system like this has a Goldstone mode (a mode with null eigenvalue in the stability matrix irrespective of the system parameters) reflecting the equal likeliness of any value of the FPP, and a maximally damped mode which can be proved to be the canonical pair of the FPP and coincides with a quadrature of the mode with orthogonal polarization with respect to the generated one. Hence, quantum fluctuations can freely act onto the FPP making it completely undetermined, allowing then for the complete noise reduction of its canon-ical pair via the Heisenberg uncertainty principle, that is, perfect squeezing of the aforementioned quadrature. The above requisites can be met, at least in principle, in both χ (2) and χ (3) nonlinear cavities. Let us first consider a χ (2) cavity. Up-conversion processes (such as second harmonic generation) cannot help for our purposes as the generated field appears linearly in the Hamiltonian and all its properties are fixed by the subharmonic modes. With respect to down-conversion processes, the only possibility is a type-II optical parametric oscillator (OPO). In its usual configuration, type-II parametric down conversion takes place in two modes having orthogonal linear polarizations and an undefined relative phase [8]. If in addition the down-converted fields have the same frequency, the process just described is equivalent to the spontaneous generation of a field with an elliptical polarization along the ±45 • axes having an arbitrary eccentricity and direction of rotation (see below). Then a type-II frequency-degenerate OPO is a candidate for SPSB. In the interaction picture, the Hamiltonian of such a system can be written aŝ H OPO = i E pb † + χbâ † xâ † y + H.c.,(1) where E p is the pumping field amplitude,b † ,â † x , andâ † y are the creation operators for the pump mode and the e x and e y polarized signal modes, respectively, and χ is the nonlinear coupling constant. This Hamiltonian has the symmetry (â x ,â y ) → â x e iθ ,â y e −iθ that leaves undefined the phase difference between the signal modes 2θ. This is the FPP of the system, which as explained below, is directly related to the eccentricity of the signal field's polarization ellipse. Associated to this symmetry there is a constant of motion, namely the photon number difference between the signal modesâ † xâx −â † yây . This ensures that signal photons are created in pairs, and hence the two polarization modes e x and e y will have exactly the same properties: they are twin beams whose intensity difference is potentially perfectly squeezed [9,10]. Hamiltonian (1) is isomorphic to that of [1,2]. There we analyzed squeezing generation through spontaneous rotational symmetry breaking in a type-I DOPO, in which the two signal modes had opposite orbital angular momenta (two-transverse-mode DOPO). Hence all the results found in that system apply for the current case. In particular, we demonstrated that: (i) for E p > γ p γ s /χ (γ p/s is the cavity damping rate at the pump/signal frequency) a steady, nonzero mean field appears at the signal frequency in the mode e B = e x e −iθ + e y e iθ / √ 2, an elliptically polarized mode as explained above whose eccentricity depends on θ [7] (bright mode in the following, as it is macroscopically occupied); (ii) starting from a value dictated by the initial random fluctuations, quantum noise makes θ diffuse; and (iii) the phase quadrature of the mode e D = i e x e −iθ − e y e iθ / √ 2 is perfectly squeezed (we shall call this the dark mode in the following, as its polarization is orthogonal to the bright mode, and hence it is empty at the classical level). However type-II OPOs degenerated in frequency and polarization invariant do not seem to exist. Normally the signal modes have different frequencies (that difference, however, can be as small as 150kHz [11]), and the only way by which the amplification can be made frequency degenerate is, as far as we know, by breaking the polarization symmetry [12,13]: A birefringent plate is introduced within the cavity, which couples the two orthogonally polarized signal modes, thus forcing their frequency degeneracy but fixing the phase difference between them and thus breaking the system's polarization symmetry. Hence, given the difficulties of having frequency-degenerate type-II OPOs, we pass to consider an alternative. We propose a χ (3) cavity in which SPSB can squeeze the dark output mode. Consider an isotropic χ (3) medium placed inside a polarization isotropic cavity which is pumped by two copropagating orthogonally polarized modes of frequencies ω 1 and ω 2 such that ω s = (ω 1 + ω 2 ) /2 be close to a cavity mode resonance ω c . Unlike in OPOs, operation in frequency degeneracy has been experimentally proved in this kind of systems [14], i.e., the pump modes can be mixed to generate a signal field with frequency ω s via four wave mixing. For the sake of simplicity, we shall treat the pumping modes as classical fields and will further ignore their depletion in interacting with the intracavity signal modes. We write the total field at the cavity waist plane (where the χ (3) medium is placed) asÊ (r, t) = E p (r, t)+ E s (r, t), E p = iF G (r) j=1,2 (e x α jx +e y α jy ) e −iωjt + c.c., (2a) E s = iF G (r) [e xâx (t) + e yây (t)] e −iωst + H.c., (2b) being F 2 = ω s / (2ε 0 nL) (n is the refractive index and L the effective cavity length), and G (r) = 2/πw −1 exp −r 2 /w 2 the cavity TEM 00 mode with radius w (which for simplicity we assume to be the same for pump and signal modes). Using the properties of the nonlinear susceptibility of isotropic media and ignoring its dispersion in the working frequency range [15], the interaction picture Hamiltonian describing our χ (3) cavity readsĤ =Ĥ 0 +Ĥ int , wherê H 0 = δ â † xâ x +â † yâ y ,(3a)H int = 3 4 g Ĥ spm +Ĥ cpm +Ĥ fwm ,(3b) with δ = ω c − ω s and g = −8ε 0 lχ xxxx F 4 / πw 2 . We assumed that the χ (3) medium length l is smaller than the cavity Rayleigh length. InĤ int the termsĤ spm , H cpm , andĤ fwm describe self-phase modulation, crossphase modulation, and four-wave mixing processes, respectively; they explicitly read H spm =â †2 xâ 2 x +â †2 yâ 2 y ,(4a)H cpm = j=1,2 4 |α jx | 2 + A |α jy | 2 â † xâx j=1,2 4 |α jy | 2 + A |α jx | 2 â † yây + 4Aâ † xâ xâ † yâ y ,(4b)H fwm = Bâ †2 xâ 2 y + 2 (α 1x α 2x + Bα 1y α 2y )â †2 x + 2 (α 1y α 2y + Bα 1x α 2x )â †2 y + j=1,2 4 Bα * jx α jy + Aα jx α * jy â † xây + 4A (α 1x α 2y + α 1y α 2x )â † xâ † y +H.c.,(4c) with A = χ xxyy /χ xxxx and B = χ xyyx /χ xxxx that verify 2A + B = 1 [8]. In order to preserve polarization invariance, the classical pumping fields must necessarily have orthogonal circular polarizations as any other polarization would privilege particular spatial directions in the transverse plane. Consequently we take α 1x = α 2x = ρ/ √ 2 and α 1y = −α 2y = iρ/ √ 2. Then we rewrite the Hamiltonian in the basis of circularly polarized statesâ ± = (â x ∓ iâ y ) / √ 2 and getĤ 0 = δ â † +â + +â † −â − ,(5a)H spm = (1 − B) â †2 +â 2 + +â †2 −â 2 − ,(5b)H cpm = 2 (1 + B)â † +â +â † −â − + 2ρ 2 (3 − B) â † +â+ +â † −â− ,(5c)H fwm = 2ρ 2 (1 + B) â +â− +â † +â † − .(5d) Just asĤ OPO , this Hamiltonian has the symmetry (â + ,â − ) → â + e iθ ,â − e −iθ withâ † +â+ −â † −â− as the associated constant of motion. Hence, whenever a nonzero mean field solution appears for the signal modes, bright emission will take place in the mode e B = e + e −iθ + e − e iθ / √ 2 = e θ , i.e., the mean field will have a linear polarization along the arbitrary θ axis [7] (e ± are the right and left circularly polarized modes), thus breaking the polarization symmetry of the system. Then, quantum fluctuations should induce a diffusion process in the FPP θ, allowing then for perfect noncritical squeezing in a quadrature of the dark mode e D = −i e + e −iθ − e − e iθ / √ 2 = e θ+π/2 , which is crossed polarized with respect to the bright one. Note that these linearly polarized modes are mapped onto the elliptically polarized bright and dark modes of the type-II frequency-degenerate OPO by a quarter-wave plate [7]. Fortunately, it will not be necessary to prove the above conclusions explicitly as by taking A = B = 1/3 (which applies when the Kleinmann symmetry is verified like in nonresonant electronic response [8]) Hamiltonian (5) becomes isomorphic to that of [3], where we already proved that for δ > √ 3γ s and for pump intensities ρ 2 inside the region defined by the curves ρ 2 = γ s /2g and ρ 2 = 2δ + δ 2 − 3γ 2 s /6g, a steady, nonzero mean field solution for the signal modes appears through a subcritical pitchfork bifurcation. The perfect and noncritical squeezing of a quadrature of the dark mode was also proved within the linear approximation for quantum fluctuations. Except for the quantitative details, these results hold for A = B. Some comments are in order here. First, one may think that the non-depletion of the pumping modes could be a too drastic approximation. But the effect of pump depletion could consist only in introducing new bifurcations affecting the stability of the steady state signal field. This would reduce the domain of existence of the signal field cw state, but would not affect its fluctuations properties where stable. Second, in [2] we demonstrated for the aforementioned two-transverse-mode DOPO both the random rotation of the bright mode and the perfect non-critical squeezing of the dark mode by direct numerical integration of the system's quantum dynamic equations, hence showing that these properties hold beyond the linear approximation. Finally, in [1] we proved that small deviations from the perfect rotational symmetry don't imply a large degradation of the dark mode's quadrature squeezing, while it can fix the bright and dark mode's orientation, what seems quite advantageous from the experimental point of view [16]. These conclusions will doubtless apply to the χ (3) cavity we have presented here. Finally, note that although one of the Stokes parameters is free from quantum fluctuations in the light exiting the cavity (the twin beams' intensity difference), this is not enough to claim for polarization squeezing as further conditions must be satisfied [17,18]. In conclusion, we have theoretically demonstrated that type-II frequency-degenerate OPOs and appropriate χ (3) cavities are suitable for the generation of noncritical and perfect quadrature squeezing. We have commented the problems that actual type-II frequencydegenerate OPOs may have (either they are not exactly frequency degenerated or have a cavity that is not polarization symmetric). We believe that the χ (3) cavity we are proposing as an alternative can be built within the experimental state of the art and would not present the same problems as the χ (2) cavity, hence being a good candidate for observing SPSB. This work has been supported by the Spanish Ministerio de Ciencia e Innovación and the European Union FEDER through Project FIS2008-06024-C03-01. C N-B is a grant holder of the FPU programme of the Ministerio de Educación y Ciencia. . C Navarrete-Benlloch, E Roldán, G J De Valcárcel, Phys. Rev. Lett. 100203601C. Navarrete-Benlloch, E. Roldán, and G. J. de Valcárcel, Phys. Rev. Lett. 100, 203601 (2008). . C Navarrete-Benlloch, A Romanelli, E Roldán, G J De Valcárcel, Phys. Rev. A. C. Navarrete-Benlloch, A. Romanelli, E. Roldán, and G. J. de Valcárcel, Phys. Rev. A (2010). . F V Garcia-Ferrer, C Navarrete-Benlloch, G J De Valcárcel, E Roldán, IEEE J. Quantum Electron. 45F.V. Garcia-Ferrer, C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, IEEE J. Quantum Electron. 45, 1404-1414 (2009). . I Pérez-Arjona, E Roldán, G J De Valcárcel, Europhys. Lett. 74I. Pérez-Arjona, E. Roldán, and G.J. de Valcárcel, Euro- phys. Lett. 74, 247-253 (2006). . I Pérez-Arjona, E Roldán, G J De Valcárcel, Phys. Rev. A. 7563802I. Pérez-Arjona, E. Roldán, and G.J. de Valcárcel, Phys. Rev. A 75, 063802 (2007). We remind that continuous variations of these parameters correspond to rotations on the Poincaré sphere defined by the Stokes parameters. The FPP might be. 7the eccentricity or the orientation of the polarization ellipseThe FPP might be, e.g., the eccentricity or the orienta- tion of the polarization ellipse. We remind that contin- uous variations of these parameters correspond to rota- tions on the Poincaré sphere defined by the Stokes pa- rameters [7]. R D Guenther, Modern Optics. John Wiley & SonsR.D. Guenther, Modern Optics (John Wiley & Sons, 1990). R W Boyd, Nonlinear Optics. San DiegoAcademic PressR.W. Boyd, Nonlinear Optics (Academic Press, San Diego, 2003). . S Reynaud, C Fabre, E Giacobino, J. Opt. Soc. Am. B. 41520S. Reynaud, C. Fabre and E. Giacobino, J. Opt. Soc. Am. B 4, 1520 (1987). . A Heidmann, R J Horowicz, S Reynaud, E Giacobino, C Fabre, G Camy, Phys. Rev. Lett. 592555A. Heidmann, R.J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987). . S Feng, O Pfister, J. Opt. B: Quantum Semiclass. Opt. 5S. Feng and O. Pfister, J. Opt. B: Quantum Semiclass. Opt. 5, 262-267 (2003). . L Longchambon, J Laurat, T Coudreau, C Fabre, Eur. Phys. J. D. 30L. Longchambon, J. Laurat, T. Coudreau, and C. Fabre, Eur. Phys. J. D 30, 287-293 (2004). . J Laurat, L Longchambon, C Fabre, T Coudreau, Opt. Lett. 301177J. Laurat, L. Longchambon, C. Fabre, and T. Coudreau, Opt. Lett. 30, 1177 (2005). . M Vallet, M Pinard, G Grynberg, Europhys. Lett. 11M. Vallet, M. Pinard, and G. Grynberg, Europhys. Lett. 11, 739-744 (1990). The explicit form of the χ (3) nonlinear susceptibility tensor taking into account these considerations is χ ijkl (ωa + ω b − ωc; ωa, ω b , −ωc) = χxxyy (δij δ kl + δ ik δ jl ) + χxyyxδ il δ jk irrespective of the frequencies. involved in the particular process [8The explicit form of the χ (3) nonlinear suscep- tibility tensor taking into account these con- siderations is χ ijkl (ωa + ω b − ωc; ωa, ω b , −ωc) = χxxyy (δij δ kl + δ ik δ jl ) + χxyyxδ il δ jk irrespective of the frequencies involved in the particular process [8]. . N Korolkova, G Leuchs, R Loudon, T C Ralph, C Silberhorn, Phys. Rev. A. 6552306N. Korolkova, G. Leuchs, R. Loudon, T.C. Ralph, and C. Silberhorn, Phys. Rev. A 65, 052306 (2002). . F V Garcia-Ferrer, I Pérez-Arjona, G J De Valcárcel, E Roldán, Phys. Rev. A. 7563823F.V. Garcia-Ferrer, I. Pérez-Arjona, G.J. de Valcárcel, and E. Roldán, Phys. Rev. A 75, 063823 (2007).

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{'abstract': 'We discuss the possibility of generating noncritical quadrature squeezing by spontaneous polarization symmetry breaking. We consider first type-II frequency-degenerate optical parametric oscillators, but discard them for a number of reasons. Then we propose a four-wave mixing cavity in which the polarization of the output mode is always linear but has an arbitrary orientation. We show that in such a cavity complete noise suppression in a quadrature of the output field occurs, irrespective of the parameter values.', 'arxivid': '1003.2581', 'author': ["Ferran V Garcia-Ferrer \nDepartament d'Òptica\nUniversitat de València\nBurjassotSpain\n", "Carlos Navarrete-Benlloch \nDepartament d'Òptica\nUniversitat de València\nBurjassotSpain\n", "Germán J De Valcárcel \nDepartament d'Òptica\nUniversitat de València\nBurjassotSpain\n", "Eugenio Roldán *eugenio.roldan@uv.es \nDepartament d'Òptica\nUniversitat de València\nBurjassotSpain\n"], 'authoraffiliation': ["Departament d'Òptica\nUniversitat de València\nBurjassotSpain", "Departament d'Òptica\nUniversitat de València\nBurjassotSpain", "Departament d'Òptica\nUniversitat de València\nBurjassotSpain", "Departament d'Òptica\nUniversitat de València\nBurjassotSpain"], 'corpusid': 17037934, 'doi': '10.1364/ol.35.002194', 'github_urls': [], 'n_tokens_mistral': 5488, 'n_tokens_neox': 4806, 'n_words': 2941, 'pdfsha': '32f1fa1af2afa7ac564f61fb8a8f4d1d1ba6a284', 'pdfurls': ['https://arxiv.org/pdf/1003.2581v1.pdf'], 'title': ['Noncritical quadrature squeezing through spontaneous polarization symmetry breaking', 'Noncritical quadrature squeezing through spontaneous polarization symmetry breaking'], 'venue': []}

arxiv

Photometric Mass Estimation and the Stellar Mass-Halo Mass Relation for Low Mass Galaxies 2022 Dennis Zaritsky Peter Behroozi Department of Astronomy Steward Observatory University of Arizona 85721TucsonAZUSA Division of Science National Astronomical Observatory of Japan 2-21-1 Osawa181-8588MitakaTokyoJapan Photometric Mass Estimation and the Stellar Mass-Halo Mass Relation for Low Mass Galaxies MNRAS 0002022Accepted 2022 December 1. Received 2022 November 30; in original form 2022 August 26Preprint 7 December 2022 Compiled using MNRAS L A T E X style file v3.0galaxies: kinematics and dynamics galaxies: structure galaxies: formation galaxies: dwarf galaxies: nuclei dark matter We present a photometric halo mass estimation technique for local galaxies that enables us to establish the stellar mass-halo mass (SMHM) relation down to stellar masses of 10 5 M . We find no detectable differences among the SMHM relations of four local galaxy clusters or between the cluster and field relations and we find agreement with extrapolations of previous SMHM relations derived using abundance matching approaches. We fit a power law to our empirical SMHM relation and find that for adopted NFW dark matter profiles and for M * < 10 9 M , the halo mass is M ℎ = 10 10.35±0.02 (M * /10 8 M ) 0.63±0.02 . The normalisation of this relation is susceptible to systematic modelling errors that depend on the adopted dark matter potential and the quoted uncertainties refer to the uncertainties in the median relation. For galaxies with M * < 10 9 M that satisfy our selection criteria, the scatter about the fit in ℎ , including uncertainties arising from our methodology, is 0.3 dex. Finally, we place lower luminosity Local Group galaxies on the SMHM relationship using the same technique, extending it to M * ∼ 10 3 M and suggest that some of these galaxies show evidence for additional mass interior to the effective radius beyond that provided by the standard dark matter profile. If this mass is in the form of a central black hole, the black hole masses are in the range of intermediate mass black holes, 10 (5.7±0.6) M , which corresponds to masses of a few percent of M ℎ , well above values extrapolated from the relationships describing more massive galaxies. INTRODUCTION How do stars populate the dark matter halos that define the galaxy population? A simple, first order answer is provided by the stellar mass-halo mass (SMHM) relation for galaxies. Measuring that relation, however, is not simple. There are broadly two ways that this measurement is approached for dwarf galaxies. In the first, using forward modeling or statistical arguments an association is made between the population of dark matter halos theoretically expected to inhabit a certain volume of the Universe and the galaxies observed within the same volume. The association is constrained using observables, such as stellar mass, but the question of scatter in the relation and simultaneously reproducing all known properties of galaxies, such as clustering or lensing as a function of magnitude or colour, complicates that association (for a review see Wechsler & Tinker 2018). This approach generally does not provide the relation for each individual galaxy, but is able to bring to bear the tremendous statistical power of today's large surveys and simulation volumes. Exceptions include the more focused analysis of the satellite population of the Milky Way (Nadler et al. 2020;Manwadkar & Kravtsov 2022;Chen et al. 2022). In the second, measurements of the internal kinematics of each individual galaxy ★ E-mail: dennis.zaritsky@gmail.com are used to obtain a dynamical estimate of the mass enclosed at the corresponding radius and an extrapolation based on an adopted dark matter halo model provides an estimate of the halo mass. A sample of galaxies for which this can be done is then used to produce the SMHM relation (Dutton et al. 2010;Read et al. 2017). This approach is able to provide the SMHM relationship for individual galaxies, but is statistically limited to smaller samples due to the required kinematic measurements. Although these two approaches are generally applied independently of each other, there are now some examples of joint analyses (e.g., Yasin et al. 2022). A current weakness in the application of either of these approaches is their inability to track the SMHM relation outside the LG significantly below a halo mass of 10 10 M . The difficulty arises because such galaxies are rare in redshift surveys beyond the local volume and measurements of the internal kinematics are increasingly difficult for fainter, lower surface brightness galaxies. This limitation is unfortunate in that a variety of interesting questions, related both to galaxy evolution and the nature of dark matter, would benefit from an understanding of the low mass SMHM relation. We address this current weakness using a novel approach to estimate halo masses for a range of galaxy samples in the literature. Our approach follows the kinematic approach in spirit in that we estimate the mass for each galaxy in our sample and build up the SMHM from observations of many such galaxies. However, we do not use kinematic measurements, but rather develop a photometric method that enables us to make the mass estimate independent of any measurement of the internal kinematics of each galaxy. As such, we are able to construct the SMHM using many low mass galaxies and extend the SMHM well below current limits in mass using large samples. In §2 we describe our approach to estimating the galaxy's halo mass (baryons + dark matter within an estimated virial radius, see §2.2) denoted M ℎ . In §3 we present the SMHM relations for low mass cluster and field galaxies separately and jointly and discuss the effect of uncertainties (both observational and theoretical). In §4 we extend the relation to even lower mass by adding Local Group dwarfs and speculate on the possibility of inferring the masses of intermediate mass black holes in these galaxies. We adopt a WMAP9 flat ΛCDM cosmology with H 0 = 69.7 km s −1 Mpc −1 and Ω = 0.281 (Hinshaw et al. 2013) for consistency with some previous studies to which we compare. ESTIMATING HALO MASSES Because we aim to estimate halo masses for as many galaxies as possible, we develop an estimator based solely on photometric properties, bypassing measurements of internal kinematics. We divide the task of developing a broadly applicable estimator into two steps. In the first, we estimate the enclosed mass within the effective radius, . The choice of the effective radius as a standard radius in galaxy photometry has generally been justified as a compromise between a sufficiently small radius where one obtains high signal-to-noise measurements and a sufficiently large one that encloses a large fraction of the luminous mass. However, in our context the choice is particularly fortuitous because a simple enclosed mass estimator at this radius is surprisingly robust to the internal detailed structure of galaxies (see §2.1; Wolf et al. 2010). In the second step, we fit dark matter halo models, constrained to match the enclosed dark matter mass within obtained in the first step, to estimate the halo mass. What we propose in the first step is the novel part of our approach. This approach has the potential to increase the number of galaxies with estimated halo masses by the ratio of the size of photometric to spectroscopic samples, much in the same way that photometric redshifts greatly increase the numbers of galaxies available for study. Again, as with photometric redshifts, one exchanges this gain in sample size for the precision obtained for each individual case and the added potential for the occasional catastrophic error. The second step in our procedure is not new and has been taken previously using kinematically-constrained enclosed mass estimates at , and other specific radii, in a variety of ways by various investigators (e.g., Dutton et al. 2010;Read et al. 2017). Step 1: estimating Scaling relations provide an opportunity to take the first of the two steps. By providing relationships among measured parameters, the appropriate scaling relation, plus the assumption that the galaxies of interest lie on the scaling relation, can be used to recover missing data for those galaxies. Historically, examples of this type of approach predominantly focus on the use of scaling relations to estimate distances, as in the use of the relationship between luminosity and rotation velocity for disk galaxies (Tully & Fisher 1977). Occasionally, those same relations can be used to recover another of the related parameters (e.g., rotation velocity using the Tully-Fisher relation; Gonzalez et al. 2000). The work presented here most closely resembles the Gonzalez et al. (2000) study. To be able to pull off this trick most broadly, the scaling relation must be applicable to all galaxies, not just to a subset of galaxies such as disk galaxies. Across several studies, we have developed and applied a universal scaling relations for stellar systems (Zaritsky et al. 2006a(Zaritsky et al. ,b, 2008(Zaritsky et al. , 2011Zaritsky 2012). In those papers, we showed that galaxies that span the known range of luminosities and morphologies satisfy a relationship between , a measure of the internal kinematics (the velocity dispersion, , for pressure supported systems or a combination of rotational velocity and for systems with significant dynamical support from both), and the projected mean surface brightness within , . The parameters involved are those also found in the Fundamental Plane scaling relation (Djorgovski & Davis 1987;Dressler et al. 1987), but the functional form is more complex to allow for the broader range of systems to which it applies. The value of having a scaling relation for all galaxies is that we can apply a methodology based on it without restriction or any prior knowledge of the galaxy to which it is being applied (see Dutton et al. 2010, for an example that combines results using Tully-Fisher for spirals and Fundamental Plane for giant spheroids). To calibrate the derived estimates of the enclosed mass, , within a sphere of radius , we present an alternative approach to that of our previous papers. We start with the well-established, widely-adopted mass estimator from Wolf et al. (2010). In that study, Wolf et al. (2010) found, based on simulations, a mass estimator that is robust against changes in the internal spatial and kinematic details of the spheroidal stellar system 1 Their estimator for the mass enclosed within a sphere of radius is = 930 2 ,(1) where is the line of sight velocity dispersion in km sec −1 , is the effective radius of the surface brightness profile in pc, and is in solar units. By calibrating our results to the 3-D enclosed mass, we are taking a slightly different approach than our earlier empirical one (Zaritsky et al. 2006a(Zaritsky et al. , 2008 that worked entirely in projected quantities. As such, the scaling relation presented here has minor quantitative differences from that presented previously. We repeat for emphasis that the observed quantities ( and ) are projected but that the derived quantity (M ) is not. Between the two measurements needed to apply this estimator, is by far the more challenging to obtain, particularly for low luminosity galaxies. As such, it is particularly advantageous to express the mass estimator exclusively in terms of photometric measurements. To do this, we first define the enclosed 3-D mass as the enclosed projected luminosity times an effective mass-to-light ratio, Υ , to rewrite Eq. 1 as 2 Υ = 930 2 ,(2) where is the mean surface brightness within in units of L /pc 2 and Υ is given in solar units. Taking the logarithm (all logarithms presented in this paper are base 10) of both sides, expressing in kpc, and organizing terms we find log = 2 log − log − log Υ − 0.53.(3) So far, this is simply a different expression of the Wolf et al. (2010) mass estimator. 1 The Wolf et al. (2010) estimator was validated only for spheroidal galaxies, but the empirical scaling relation is valid for both disks and spheroids if the appropriate kinematic measurement is used for disks (Zaritsky et al. 2008). As such, once the enclosed mass estimates are calibrated for spheroidal galaxies, then the estimates are calibrated for all galaxies. Collins et al. (2014) LG dSph V -36 tidal objects removed Geha et al. (2003) dE V 17 Jorgensen et al. (1996) E Figure 1. The effective mass-to-light ratio, Υ , such that = 2 Υ , vs. . Three branches can be distinguished. Toward larger the space is that populated by ellipticals and dwarf ellipticals (lightly coloured circles). At smaller there are two branches, that populated by dsph and ultrafaint galaxies satellites of the Milky Way and M31 (red squares) and that populated by compact dwarf galaxies (green triangles). To eliminate or solve for , we need a second equation involving the two unknowns, and Υ . At this point progress requires an ansatz for the functional form of Υ . A natural (i.e., simple) proposal is that log Υ = (log ). To guide our understanding of what form such a function might take we evaluate Υ using Equation 3 and plot those values vs. for a wide range of spheroidal stellar systems with spectroscopically measured 's (see Table 1; Jorgensen et al. 1996;Geha et al. 2003;Mieske et al. 2008;Collins et al. 2014;Chilingarian et al. 2008) in Figure 1. We have made one set of edits to the literature sample in that we have removed five galaxies from the Collins et al. (2014) sample of LG dwarfs that are suspected to be experiencing significant tidal forces (Crater II (Sanders et al. 2018); Wilman I and Triangulum II (Fritz et al. 2018); Hercules I (Fu et al. 2019)); and Leo V (Collins et al. 2017)). While there is a dependence of Υ on , there is also a bifurcation in behaviour at low . The two branches highlight the divergence in properties between high and low surface brightness stellar systems. Given this behaviour, it is manifestly not possible to describe Υ as only a function of . We conclude that any appropriate functional form must at least also include . The next simplest ansatz is that log Υ = (log , log ) and that this function is first order in both log and log . Such a proposition leads to equations of the form of the Fundamental Plane (Dressler et al. 1987;Djorgovski & Davis 1987), which has been so successful at describing giant ellipticals, but which fails to describe low luminosity spheroids. The cause of that failure is also evident in Figure 1. One can only describe log Υ adequately with a linear function of log for the higher stellar systems. The next step in complexity is adopting a function that is second order in log and log , log Υ = (log ) 2 + log + (log ) 2 + log I e + log log + , where we neglect cross terms that are leading second order but discuss them further below. We evaluate the coefficients in Eq. 4 by replacing log Υ in Eq. 3 with the right hand side of Eq. 4 and fitting the data shown in Figure 1 plus a compilation of ultra-diffuse galaxies (van Dokkum et al. 2017;van Dokkum et al. 2019;Toloba et al. 2018;Chilingarian et al. 2008;Martín-Navarro et al. 2019;Gannon et al. 2021) to extend further the range of galaxy types. We do make one further edit of the literature data in that we exclude systems with < 10 pc, which are predominantly globular clusters but do include some ultracompact dwarfs. There are not many such systems in the sample, so the derived coefficient values are not significantly affected by this choice, but there are indications that these systems start to deviate from a scaling relation of this form (Forbes et al. 2008, and this work). We believe this deviation happens because such compact systems are completely stellar dominated within , and therefore have an Υ that is independent of and , making it difficult for a low order functional form to adequately adjust to such behaviour. As expected from such a large list of disparate studies, the data are a heterogeneous set of photometric and kinematic measurements. We place the surface brightnesses on a comparable system of solar luminosities, appropriate for each band (Willmer 2018), but make no correction for color differences between the galaxies and the Sun. There is also no correction for how color gradients might affect or how kinematic measurements vary between central values of and aperture values. All of these irregularities among data sets and improper or ignored corrections can be expected to lead to less rather than more coherence in the resulting scaling relation. Our eventual estimation of the precision of our mass estimates using this set of data is therefore an upper limit on the intrinsic scatter. We derive the coefficients using a Bayesian approach and the emcee Python implementation of a Markov chain Monte Carlo sampler (Foreman-Mackey et al. 2013). The model is assumed to have no intrinsic scatter and be as given by Equation 4. We adopt uniform priors on all of the parameters and parameter ranges that avoid resulting posterior distributions that peak near the range edges. The corner plot showing the character of the uncertainties in the coefficients is presented in Figure 2 and the resulting coefficient values are listed in Table 2. The correspondence between our estimate of M and that obtained using the spectroscopically-measured and the Wolf et al. (2010) estimator is excellent (Figure 3), with a standard deviation about the 1:1 line of 0.17 dex (corresponding to a relative error of ∼ 50%). In the right panel of Figure 3 we show that the majority of the estimates are within a factor of two of the Wolf et al. (2010) values, with larger scatter for systems with 10 km s −1 although at these low values of there are large fractional uncertainties in the spectroscopically measured 's as well. As such, we cannot ascertain whether the larger scatter is due to intrinsically larger scatter about our scaling relation or observational errors in the spectroscopicallydetermined values of . In either case, there is no evident systematic residual with although one must remain aware that binaries in these lowest mass galaxies could lead to an upward bias in the measured 's and hence in the functional form of the fit as well. To further explore the nature of the scatter, we now redo the analy-sis with a sample of K-band photometry for spheroidal stellar systems from Forbes et al. (2008). As described by those authors, the advantage provided by using near-IR is a decreased sensitivity to variations in the stellar mass-to-light ratios. For our purposes, we also benefit from the single-source nature of the photometry and analysis. The result of applying the same procedure to these data, which includes a similarly diverse range of stellar systems, is a scatter about the 1:1 relation between the Wolf et al. (2010) estimator and ours of 0.17 dex, exactly what the optical estimates yielded 2 . We conclude that the use of a wide variety of studies in the optical did not contribute significantly to the scatter in our mass estimates. We favor the use of the optical relation going forward because there is so much more data currently available that we can use in our subsequent analysis. Finally, returning to the choice we made to neglect the cross terms that are leading second order terms in Equation 4, we redo the coefficient fitting including those terms and find that both of the resulting coefficients, for the log (log ) 2 and (log ) 2 log terms, are consistent with zero. Of course, neither higher order functions or the inclusion of other parameters are excluded by our analysis, but a function of the form presented in Equation 4 appears to be the simplest that can adequately express the behaviour seen in Figure 1 to the current level of observational precision. Step 2: extrapolating from M to M ℎ The use of the scaling relation only provides an estimate of the mass interior to . To calculate the halo mass, M ℎ , we need an estimate of the dark matter mass. To obtain this estimate, we subtract the contribution to M from stars projected within and then determine the parameters of an NFW dark matter density profile (Navarro et al. 1997) that best reproduces the remaining mass, the dark matter, within a sphere of radius , or M , . To estimate the stellar mass within , we convert the luminosity within to stellar mass by adopting stellar / ratios that are either color-dependent (Roediger & Courteau 2015) when a color is available or fixed in the case where only one photometric band is provided (McGaugh & Schombert 2014). We discuss the effect of uncertainties arising from our choice of the stellar / in §3.3 and of the dark matter density profile in §3.3.6. This approach assumes that star formation does not alter the dark matter profile. In practice, the condensation of gas to the center may lead to dark matter halo contraction (Blumenthal et al. 1986); and vice versa, feedback from stars may lead to expansion/coring (Pontzen & Governato 2012). Besides the fiducial approach of assuming an NFW profile, we also test inferring masses using cored Burkert profiles (Burkert 1995). We iterate to find the best-fit dark matter profile from within the adopted family of NFW profiles. We define a trial NFW model by setting M ℎ and evaluating its concentration parameter using the mean relation between concentration and mass (Macciò et al. 2007). We use GalPy (Bovy 2015, http://github.com/jobovy/galpy) to evaluate M , and compare to our empirical estimate. We evaluate models over a range of M ℎ to find the best fit halo. The calculations are done for the adopted cosmology and a redshift of 0.01 to correspond to an overdensity of 346 relative to the matter density (cf. Bryan & Norman 1998). For the best fit halo mass, we then add back the baryonic mass using the universal baryon fraction to estimate M ℎ . This is certainly an upper limit to the baryon content and some suggest low mass halos Table 1). Left panel shows the derived values and the 1:1 line and includes both the high and low surface branches visible in Figure 1. Right panel shows the differences in the two estimates vs. . The shaded region encloses values that are within a factor of two of each other. have far less than their 'fair share' of baryons (Papastergis et al. 2012). We will explore the effect of adopting the lower limit in §3.3. This approach, including the estimation of from the scaling relation, was first applied to examine the relation between the number of globular clusters in a galaxy, N , and M ℎ (Zaritsky 2022). The resulting linear relation between N and M ℎ is circumstantial supporting evidence for the accuracy of our estimated M ℎ values, modulo the normalisation factor. We ignore scatter in the halo mass-concentration relation, which simulations show is significant (> 0.1 dex; Macciò et al. 2007). Zaritsky (2022) noted that ignoring the scatter may, for a subtle reason, be the correct approach in this method. The estimates of the internal kinematics of these galaxies is based on the scaling relations, which also sidestep variations among individual galaxies to provide a 'typical' and enclosed mass for each galaxy. Therefore, because the 's we use do not include the effects of differences in the concentration among galaxies of equal mass, the use of mean concentration-halo mass relation may indeed be appropriate. The consideration of our treatment of scatter in halo concentration raises a significant concern. How can we verify our estimates of M ℎ ? Indeed, Gannon et al. (2021) demonstrated that if cored DM density profiles are adopted, rather than NFW ones, the result can be to invert the relation between M and M ℎ . Given that we do not have direct measurements of M ℎ on a galaxy-by-galaxy basis (even for the Milky Way the M ℎ estimates show a significant range of values; Shen et al. 2022), we must rely on circumstantial evidence for now. As already mentioned, the resulting linear relation between N and M ℎ is one such piece of evidence. In the case of the SMHM relation ( §3.1), bear in mind that either an inverted relation between M and M ℎ or, perhaps more likely, large scatter between M , and M ℎ -as would result from including concentration scatter in the DM profiles without accounting for offsetting differences in -would not lead to the relatively tight SMHM relation we find that closely tracks that obtained using abundance matching techniques. This is perhaps a less-than-satisfying justification of the approach, but on the other hand offers an avenue for placing constraints on the possible range of variations in the M , -M ℎ relation using the degree of agreement between independent determinations of the SMHM relation. RESULTS SMHM relation for local galaxy cluster populations Large samples of low luminosity galaxies are difficult to obtain because spectroscopy is generally necessary to determine a distance and a luminosity. The standard way to avoid this observational expense is to study low luminosity populations in nearby galaxy clusters, for which one can simply assign the cluster distance to every faint galaxy. There is some danger of background contamination, but the projected density of cluster members at the relevant magnitudes is significantly larger than that of the background and this contrast is even more pronounced for galaxies with low surface brightnesses and relatively large angular size -which generally describes the nearby cluster dwarf galaxy population. In the application of our methodology to large galaxy samples we are likely to be including all morphological types, unless care is taken to classify and select subsamples. Fortunately, the basic scaling relation we use is applicable to all morphological types (Zaritsky et al. 2008), so no morphological pre-selection is required. The only distinction in applying the relation to rotation vs. dispersion dominated systems is whether one uses the circular velocity or the velocity dispersion. When using the circular velocity, one needs to divide the value by √ 2 (the exact value depends on the nature of the potential, the stellar orbits, and radial distribution of stars, but empirical study shows only a weak dependence on this value; Weiner et al. 2006;Zaritsky et al. 2008). However, this distinction is irrelevant for our purposes because at this point we are neither using measured kinematics or estimating the kinematics. To support this claim, we apply our method to the clean sample of Read et al. (2017) (avoiding 'rogues' for which they have less confidence in their derived parameters). That study provides all of the necessary information once we convert from their stellar exponential scale radii to effective radii by multiplying their values by 1.68. For those 9 galaxies, from which they derive total mass using model fitting to HI rotation curves, our estimates of M ℎ deviate on average from their quoted M 200 values by 0.044 dex (∼ 10%) and have an rms difference of 0.24 dex, a value smaller than what we will eventually find to be observational scatter for our full sample. We confirm that Grossauer et al. (2015), which is particularly interesting because it was derived using the same Virgo galaxy sample, but with a different technique (abundance matching). we can apply our methodology even to HI-dominated, rotationallysupported low mass galaxies. One aspect for potential study is highlighted in the panel showing the results for the Virgo galaxies. There we have included the SMHM relation from Grossauer et al. (2015), which was derived from the same Ferrarese et al. (2020) sample of galaxies using an analysis involving abundance matching. Accepting that the technical aspects, such as completeness corrections, were handled properly, the offset between this relation and our results might indicate an anomaly in the halo distribution in the models that were used. The sense of the discrepancy is that Grossauer et al. (2015) effectively had to place a galaxy with a specific M * in a more massive halo than that which we are associating it with, suggesting a surfeit of halos in the simulations at these masses. This, in turn, could indicate that halo disruption is underestimated in those models. The general sense of the offset, that abundance matching approaches tend to place galaxies in more massive halos, is consistent with recent considerations of the Milky Way and M 31 (McGaugh & van Dokkum 2021), although, as we will stress later, the normalisation of our SMHM relation is subject to systematic uncertainties. Of course, as interesting as such a conclusion might be, it is predicated on the confidence we can place on our overall normalisation of both M * and M ℎ . The question of M * can be addressed by consistently estimating M * and looking at the situation in a relative sense (in other words, if, for example, the wrong stellar initial mass function is used, as long as the same incorrect assumption is made in both analyses then at least the M * part of the comparison is valid). The question of M ℎ can be addressed by spanning a sufficiently large range of M ℎ that we probe both the power law behaviour at low M ℎ and the turnover at higher M ℎ . The current difficulty in doing so is that such an analysis requires splicing disparate samples, as we will see below. The M , /M criteria There are regimes where we might expect our methodology for inferring M ℎ to perform poorly or not at all. For example, as M becomes increasingly dominated by stars our calculation of the dark matter mass within , , , will become increasingly uncertain. In fact, errors in our estimate of M * could even lead to formally negative, unphysical, values of M , . As such, we need to reject systems below some value of M , /M . This also works to reject the highly compact systems that we excluded in our scaling relation discussion (i.e., those with < 10 pc). At the other extreme, systems with extremely high apparent values of M , /M are unlikely to be real because those galaxies would have a baryon fraction far below the universal value. Such systems are most likely due to an underestimation of M * , which leads to an overestimation of M , . Because of the large extrapolation from M to M ℎ , small errors in M , can lead to unphysically large values of M ℎ . As such, we also anticipate needing to set an upper limit on M , /M . To explore these issues we use the results presented in Figure 4 , and a, when extrapolated, a catastrophic overestimate of M ℎ . This sharp rise becomes most prominent for M , /M > 0.975, so we set that value as the upper limit. That cut is shown as the dotted line in the right panel of Figure 5. The application of these two criteria removes much of the most egregious scatter from Figure 4. Alternatively, a future treatment of this problem could attempt to recover M ℎ and the associated uncertainty using a Bayesian approach. Our hypothesis is that the recovered values of M ℎ that result in high Δ would also have associated large uncertainties. If they do not, then there is either a tail of systems with intrinsic large scatter in the SMHM relation or a missing ingredient in our model. How uncertainties affect the results Although the mean trend between M * and M ℎ is well-defined in Figure 4, there is significant scatter about that mean even after we have removed the most egregious outliers using the M , /M criteria we just described. To better understand the origin of the scatter and how one might lower the observational scatter, we quantify the effects of errors in each of the key parameters in Figure 6. For each quantity, we assess the impact by altering the specific parameter by the value shown. Distance In Figure 6, we show how specific changes in one quantity at a time move the mean location of the Virgo sample in the M * -M ℎ space. The upper panel shows the result of doubling and cutting in half the adopted distance. This is a far larger change than we anticipate, Figure 6. The sensitivity of the resulting SMHM relation is shown for the mean location of the Virgo sample (Ferrarese et al. 2020) for a variety of choices. In some cases the range explored matches the plausible uncertainties, in others it does not. Where the range was expanded, it was done to aid visualisation of the effect. particularly for the cluster galaxies whose hosting clusters are well studied. Distance estimates are more uncertain for individual field galaxies where the peculiar velocities could be significant. Nevertheless, we find that changes in the distance, even when unrealistically large, do not contribute significantly to the scatter because they act to slide sources nearly parallel to the fiducial relation. M We depend on the scaling relation to estimate M . The scatter in M , evaluated relative to values obtained for systems with measured 's, is moderate (a factor of 2 in mass) in comparison to the many orders of magnitude in mass over which we apply the relation (Figure 3). Even so, those uncertainties are sufficiently large that they can lead to significant errors in the extrapolated estimate of M ℎ . In the second panel of Figure 6 we illustrate the effect of a ±1 systematic change in the inferred M . Because of the large extrapolation in going from M to M ℎ , these changes have large repercussions. An initially puzzling aspect of this panel is that a change in M appears to result in a change in M * , which is an unrelated quantity. This happens because the change in M couples to the M , /M criteria and results in somewhat different samples for which the means are evaluated. A second surprising finding is that the observed scatter in Figure 4 is not as large as the result in Figure 6 would suggest (the scatter about the mean relation in Figure 4 is 0.3 dex while the size of the plotted error bar in each direction is about 0.8 dex). This amplification of the error comes about due to two amplifying effects. First, a more massive halo is also larger and hence is proportionally further inside the halo and, second, the concentration of more massive halos is smaller. These two effects collaborate to turn a 0.1 dex offset in M into a 0.5 dex offset in M ℎ . If the scatter in Figure 3 comes primarily from scatter in the application of the scaling relation, then the smaller than expected scatter in the SMHM relation may indicate that the errors in M are correlated with a change in another parameter that results in galaxies moving somewhat less across the SMHM relation than indicated in Figure 6 and more along it. Alternatively, if the scatter in Figure 3 comes mostly from scatter in the Wolf et al. (2010) masses, for example due to observational errors in , then the smaller than expected scatter in Figure 4 could be the result of adopting the 'typical' values of M given by the scaling relations. From our analysis we cannot determine the actual, intrinsic scatter in the SMHM relation. While it could be smaller than what we measure, buried underneath the scatter generated by our crude approach, perhaps it is larger than what we are see because we have imposed a degree of homogeneity that does not exist (for example, due to our neglect of scatter in the halo-mass-correlation relationship). Although, measuring the scatter in the SMHM at low masses is a challenge, a value consistent with what we observe is within limits presented elsewhere (Allen et al. 2019), and hence does not point to any catastrophic error in our analysis. Independent derivations of the SMHM scatter would allow us to use our results to provide constraints on possible dark matter density profiles. and Our determination of depends only on the distance, and . We explored the effect of distances errors above and now explore the effects of errors in the other quantities, propagated through the determination of M . In the next two panels of Figure 6 we show that a much larger than anticipated error in the apparent magnitude and plausible errors in both contribute negligibly to the scatter about the fiducial line. Neither appears to provide enough of a change to help counter the effect of a change in . We are left with the conclusion that our estimates of must be somewhat better than reflected in the 0.17 dex scatter in Figure 3. Part of the explanation must lie with scatter in the measured 's, which are particularly difficult to measure for low mass systems. A second part may lie with the same hypothesis we made for ignoring scatter in the halo mass-concentration relation. The scaling relation gives an idealised estimate of M and is therefore providing an average M for similar galaxies, which by the nature of averages has less scatter than that visible in Figure 6. As such, we may be in the seemingly absurd regime where having less information (i.e. not having a measured ) leads to a more precise result -as long as the scatter about the ; lighter, narrower bars). The generally narrower distribution of residuals for the standard approach indicate that there is physical information in the extrapolation to M ℎ using the NFW models. The few outliers (Δ > 0.5) are again a demonstration of the potential for catastrophic failure in a small fraction of the sample. scaling relation is proportionally less than the observational scatter in . Stellar M/L We now consider two systematic uncertainties that affect the estimation of M ℎ . First, and in the fifth panel in Figure 6, we consider plausible changes in the adopted stellar M/L. Here we have adopted a factor of two change downward and upward in the stellar M/L. These changes principally result in a lateral shift in M ℎ , with an amplitude similar to the uncertainty arising from different extrapolations of the SMHM relation. This result highlights the difficulty in using these results to determine the absolute normalisation of the SMHM relation and its dependence on other factors such as the stellar initial mass function. Baryon fraction To evaluate M ℎ we assigned each halo a baryon mass determined from the universal baryon fraction. This almost certainly an overestimation of the baryons in each halo (Papastergis et al. 2012), and therefore of M ℎ , although some studies do not find evidence of greater baryon loss in low mass galaxies (Geha et al. 2006). Nevertheless, to probe the possible full extent of mischaracterising the baryon fraction, we adopt the other extreme of this correction and only add the observed stellar mass to the dark matter halo mass to obtain M ℎ . In the bottom panel of Figure 6 we show the effect of making that correction instead. Unsurprisingly, the change is visible, but minimal given that a 16% change in halo mass corresponds to a change in M ℎ of only 0.075 dex. Because the proper correction must lie between these two alternatives, the effect of adopting the improper correction within these extremes is even smaller and therefore a minor source of uncertainty relative to other issues we have discussed. Figure 8. The SMHM relation for low-luminosity galaxies in low density environments. Results for two samples probing lower density environments (local volume dwarf galaxies (Carlsten et al. 2021); satellites of Milky Way analogs (SAGA; Geha et al. 2003;Mao et al. 2021) are presented on the the right. The SAGA sample has a tail toward very high halo masses that we believe to be spurious. We increase the minimum dark matter fraction requirement to 0.9, the restricted sample, and the tail is mostly removed. The composite of the Carlsten et al. (2021) and restricted SAGA sample is presented on the left. The solid and dotted lines are the extrapolation of the SMHM relations from Behroozi et al. (2010) and Behroozi et al. (2019), respectively, and are the same in all panels. The impact of the adopted potential Comparisons between data and models are also affected by the extrapolation we make from M to M ℎ . The analysis presented so far is predicated on the adopted NFW dark matter mass profile. However, there is extensive literature advocating alternative profiles to resolve some apparent empirical discrepancies between data and the predictions of NFW-based models, particularly among lower mass galaxies (e.g., Burkert 2020). Here we briefly discuss the qualitative impact of loosening the adoption of the NFW profile on our results. Cored potentials offer a larger, and somewhat degenerate, set of models that can fit single radius kinematic constraints (Gannon et al. 2021). In fact, those authors showed that, at least for the UDGs that they were considering, it was possible among some plausible models to invert the relationship between M and M ℎ . This raises the important question of whether there is any value in extrapolating measurements of the enclosed mass at small radii to estimates of M ℎ . The most extreme scenarios, where either the relationship between M and M ℎ is inverted or there is no information in M regarding M ℎ , can be rejected on the grounds that we do recover a SMHM relation in qualitative agreement with that recovered from abundance matching studies. The more subtle question of whether fitting NFW profiles adds any value, or whether one could simply scale upwards the values of M , , requires a quantitative exploration. For galaxies within our limits on M , , we fit for a power law relation between M * and M , . We compare the residuals from that fit, to the residuals from fit for the SMHM using our NFW fitting approach in Figure 7. Aside from a sparsely populated tail of large outliers (due to the methods strong sensitivity to M , errors), the results from our NFW fit show less scatter than the uncorrected values. Because the application of the NFW fitting is unlikely to result in a tighter SMHM if there is no relation between M and M ℎ , we conclude that our estimates of M ℎ do add value to the investigation. SMHM for local field populations Although cluster dwarf galaxy samples have the advantage of sample size, they have a significant disadvantage in terms of interpretation. Cluster galaxy populations are subject to various effects Gunn & Oke (1975); Larson et al. (1980);Valluri (1993); Moore (1996), and as such may not be representative of the general galaxy population. To address this issue, we examine three sets of field populations of galaxies that include dwarfs (Blanton et al. 2005;Mao et al. 2021;Park et al. 2017Park et al. , 2019. The Blanton et al. (2005) sample consists of low z galaxies from SDSS, reanalyzed to improve the photometry for systems of large angular extent, the Mao et al. (2021) sample is from the SAGA spectroscopic survey for satellites of Milky Way analogues (Geha et al. 2017), and the third set comes from an ongoing survey of nearby poor groups (Park et al. 2017(Park et al. , 2019. The Park et al. sample is different than the other two samples in that distances are assigned from the group membership rather than from recessional velocities. Because their analysis suggests only ∼ 30% contamination and distance errors tend to move galaxies along the SMHM fiducial, we include their sample to extend coverage down to M ℎ ∼ 10 8 M . We convert from Johnson to SDSS photometric bands using the transformations presented by Jester et al. (2005). In Figure 8 we show the results for the three samples, both separately for each sample and together. For the Blanton et al. (2005) sample we have excluded galaxies with > 18, which showed far larger scatter than their brighter counterparts, suggesting poor photometry at the faint end of their range. The three samples together cover a large range in M ℎ and fall between the two plotted extrapolations of the SMHM relations. Interestingly, at higher masses they appear to show an offset relative to the Behroozi et al. (2010) fiducial, which the cluster galaxies followed closely (albeit at lower M ℎ ), and then fall in line with that fiducial once in a M ℎ range below 10 10 M . This behaviour could be a reflection of the fact that dwarf field galaxies tend to be star forming unless they are of very low mass, and because they have not yet formed all of their stars they lie below the fiducial SMHM relation. This is consistent with findings from several past studies, in which satellite galaxies have larger stellar mass to halo mass ratios than field galaxies (e.g., Rodríguez-Puebla et al. 2012;Behroozi et al. 2019). It may also simply reflect a shallower SMHM relation than the extrapolated Behroozi et al. (2010) relation. These are intriguing, although preliminary interpretations. Comparison across samples is complicated by different measurement techniques, for example the definition of total magnitudes and extinction corrections. Even where we have tried to homogenise the analysis, by correcting to one set of stellar mass-to-light ratios, the correction is often hampered by a lack of similar colour information and photometry in different photometric bands. Even with homogeneous data it will continue to be challenging to obtain absolute values of quantities like the stellar mass-to-light ratio, which depends on the poorly known low end of the stellar mass function. However, if the data are homogeneous and the analysis is done consistently, then relative values will be meaningful and comparisons as that done here can be confidently made. DISCUSSION We close by extending the technique to lower mass galaxies in the Local Group. This leads to some mixed results that motivate speculation on the nature of the mass distribution in some of these systems. The composite SMHM relation extended to Local Group galaxies In Figure 9 we present all of the data discussed so far to track the global SMHM relation. Fitting a power law to those low mass galaxies (M * < 10 9 M and M ℎ < 10 12 M ) yields M ℎ = 10 10.35±0.02 M * 10 8 M 0.63±0.02 . The data have a standard deviation of 0.31 dex about the line. The measured scatter does not depend strongly on our M , /M upper cut. Removing the criteria that M , /M < 0.975 increases the scatter for the sample about the best fit relation to 0.37 dex. We nevertheless apply the criteria because the 12% of sources above this criteria do significantly affect the fitted parameters of the SMHM relation when they are equally weighted in the fit. In Figure 9 we also add our estimates of M * and M ℎ for Local Group (LG) dwarf galaxies derived from the data provided in the Drlica-Wagner et al. (2020) compilation. From that list, we exclude Crater II, which is a challenging galaxy to model in any regard (Borukhovetskaya et al. 2022) and Kim 2, Triangulum II, and DES J0225+0304, which have ∼ 10 pc. This compilation provides band photometry and we adopt a standard stellar M/L = 1.2 for the remaining 54 galaxies as suggested by McGaugh & Schombert (2014) when colours are not available to estimate the stellar mass. We adjust the M , / limit upward to 0.99, to include more galaxies and because these galaxies are generally more dark matter dominated than the more massive galaxies we discussed previously. After applying the new , /M criteria, we are left with 24 Simon & Geha (2007) galaxies and they fall tightly along an extension of the SMHM relation obtained from the cluster and field samples ( Figure 9, left panel). These galaxies include six of the classic dwarf Spheroidals, with Sextans, the one that does not satisfy the criterion, lying just slightly farther off the mean trend. As such, these systems fall nicely along the extrapolation of the SMHM and, therefore, consistent with model expectations, as found previously to be the case for these galaxies in an independent analysis (Read & Erkal 2019). However, slightly less than half of the LG galaxies survived the M , / criteria, and those that do not populate the lower right of the right panel of Figure 9. This includes some with an estimated M ℎ that differs from that inferred from their stellar mass by several orders of magnitude. The discrepancy is sufficiently large that simple observational errors cannot be responsible. A natural suspicion falls on the estimated , and related M , obtained using the scaling relation. However, for the 13 LG galaxies that fall more than 1 dex away from the extrapolated SMHM relation for which we have found a spectroscopically measured in the literature (Table 3), only one has an estimated velocity dispersion that exceeds the measured one by more than 3 . Although it is worth investigating why some have statistically large deviations, errors in our estimation of are not responsible for the bulk of the large offsets from the SMHM relation. Nevertheless, given the small values of for this set of objects there is a concern that large deviations can only work to overestimate , creating a bias in the outliers in one direction. The outliers can be interpreted in a variety of ways. First, these may be tidally distorted systems for which the assumption of equilibrium is inappropriate. Given the large number of such systems, this seems unlikely as a blanket explanation, but is likely to be an important factor in a number of cases. For example, Tuc IV appears to have collided with the LMC a mere 120 Myrs ago (Simon et al. 2020), Cetus II is an enhancement along the Sagittarius stream (Conn et al. 2018), Tucana III has long tidal streams emanating from it (Shipp et al. 2018), and Draco II and Antlia 2 are believed to be disrupting (Longeard et al. 2018;Ji et al. 2021). We have highlighted with red crosses in the Figure those systems for which tidal distortions have been empirically claimed (Mutlu-Pakdil et al. 2019, and references therein). Because theoretical modeling suggests that all of these systems should have suffered significantly as a result of tidal interactions (Fattahi et al. 2018), explanations along these lines cannot be easily dismissed. Second, the mass estimates could at least be roughly correct, in that these may be systems with unusually low values of stellar mass for their halo mass. Such systems would be examples of relatively massive subhalos with vastly underproduced stellar populations and examples of a large scatter in the SMHM relation at low masses. We disfavour this as a blanket interpretation as well because at the higher end of the mass range (M ℎ > 10 11 M ) such systems would have macroscopic dynamical consequences on the LG. Furthermore, from the right panel of Figure 9, the distribution of outliers suggest a progression to higher M ℎ at fixed M * rather than one of lower M * at fixed M ℎ . Lastly, the discrepancy may hint at deviations from our standard dark matter model. In our particular problem, the nature of this excess mass is unspecified, and so we explore the possibility that it is in the form of a central black hole. However, these system could also be strong outliers from the mass-concentration relation that we use. A massive central black hole could contribute a significant fraction of the mass measured within and, therefore, removing that mass from what the standard (e.g., NFW) dark matter halo has to match within will significantly lower the derived halo mass. We now estimate the central black hole masses, • , needed to place these systems on the extrapolation of the SMHM relation shown in Figure 9. We simply refit our model, now subtracting both M • and 0.5*M * from M to obtain M , , and ask what value of M • places the galaxy nearest the SMHM relation. We do this for all of the galaxies that are at least 1 dex away from the relation in our original analysis and do not already have published claims of being tidally distorted. We can place all of the systems back on the relation and the inferred black hole masses are shown in Figure 10. The resulting black hole masses are orders of magnitude larger than one might invoke using an extrapolation of the M • -M ℎ relation for larger galaxies (Bandara et al. 2009), but relations such as this are expected to flatten at low masses (Greene et al. 2020). This model has various implications for black hole seed masses and the evolution of the black hole mass function with time that we do not explore here, but it offers a straightforward way of addressing our difficulty in fitting these systems without invoking exotic dark matter physics. A significant challenge that this scenario faces is that the black hole mass accounts for a large fraction of the baryons expected within these halos (apparently surpassing it in 5 cases), although a combination of observational errors and relaxing the requirement that the galaxies lie exactly on the mean SMHM relation may address the most extreme cases. If we reject those three systems for which M • > M ℎ as being physically implausible, the remainder of the set have log(M • /M ) = 5.7 ± 0.6 (right panel of Figure 10). The large inferred values of M • may appear, and may ultimately prove to be, problematic for this hypothesis. Nevertheless, for some systems that fall off the SMHM relation, there is additional information in the literature that we can use to gain intuition into the relevant uncertainties and test our inferences. Tuc II, which is one of the five galaxies for which the inferred M • is larger than M ℎ , and is therefore suspect, is one galaxy for which additional spectroscopic data and modelling of the enclosed mass out to large radii exist and extend well beyond (Chiti et al. 2021a). That study provides an estimate of the enclosed mass within 1.1 kpc (2.14 +3.67 −1.24 × 10 7 M ), which is ∼ 39 times smaller than what we derive (8.4 × 10 8 M ) from our baseline model, i.e., one without a central black hole. Of course an overestimate of the mass enclosed at 1.1 kpc leads to an overestimate of the halo mass, which is what led to Tuc II falling so far off the SMHM relation. Our suggested solution of including a central black hole lowers both the inferred halo mass and the inferred mass at 1.1 kpc. For our inferred M • , the resulting enclosed mass (halo + BH + stars) at 1.1 kpc drops to 2.4 × 10 7 M , in excellent agreement with the measurement by Chiti et al. (2021a) thereby providing supporting evidence for our suggestion. For completeness, we note that Chiti et al. (2021a) adopted = 120 pc, as opposed to the 165 pc in Drlica-Wagner et al. (2020). Doing the calculation for this different value of , our estimate for drops from 10.8 to 8.4 km s −1 , in better agreement with Walker et al. (2016), the inferred black hole mass drops to 10 6.7 from 10 7.3 M but it is still among the largest of our set and remains close to M ℎ , and the enclosed mass drops to 1.3 × 10 7 M , still within the uncertainty range of the measurement by Chiti et al. (2021a). Loosening the criterion that Tuc II lies exactly on the SMHM reduces the inferred M • and increases the enclosed mass at 1.1 kpc, both of which would align even better with expectations. We close by noting that the need for excess central mass in Tuc II remains if one adopts the smallest observed value of rather than our inferred value. Although a massive central black hole is one way to address the outliers, this approach for resolving the discrepancies only requires a highly concentrated secondary mass component. This mass component could be a black hole, but it could also be a more tightly bound secondary dark matter component that contributes mass primarily at radii within . We do not find that the deviations from the SMHM relation depend on , which one might expect if this second component dominates within a physical radius that comparable to . We close by noting that among these alternatives, the only one we know must play a role in at least some of these systems is that of tidal deformation/destruction. As such, it is not necessary for any of the alternatives to be true in every one of the discrepant galaxies. There may be some with a significant error in our estimate of M , some with lower than expected M * , and some with a nuclear black hole. At the very least, this discussion highlights which Local Group galaxies merit further attention. SUMMARY We present a photometric halo mass estimation technique for local galaxies. The technique is predicated on 1) the university applicability of the Wolf et al. (2010) mass estimator, 2) our empirical fit to the effective mass-to-light ratio within the effective radius, Υ , that is second order in log and log , where is the mean surface brightness within , and 3) the adoption of a dark matter density profile that is used to extrapolate to a halo mass. Each of these has the potential for systematic errors. The first was established using numerical simulations to be valid for spheroidal galaxies (Wolf et al. 2010). Bootstrapping to the universality of the scaling relation presented by Zaritsky et al. (2008), the mass estimation should be independent of morphological type. The second we validate by comparing the our resulting estimated enclosed masses within with those obtained with the Wolf et al. (2010) estimator for a sample of galaxies with available measurements of . The last is the most difficult to verify as there are few measurements of the halo mass for individual galaxies. The general behaviour of our mass estimates is indirectly validated by the resulting linear relationship between the number of globular clusters and halo mass when using this methodology (Zaritsky 2022) and by the agreement shown here in the recovered SMHS relation with that extrapolated from abundance matching techniques (e.g., Behroozi et al. 2010). We find no detectable difference among the SMHM relations of four local clusters or between the cluster and field relations. We For those LG galaxies that fall far from the SMHM relation and do not have published claims of tidal distortion, we redo the analysis assuming that they fall on the relation and recover the required central black hole mass to make this happen. That calculated M • is plotted vs. M ℎ in the left panel. For comparison the extrapolation of the corresponding relation for massive galaxies (Bandara et al. 2009), a line representing M • =M ℎ /100, and a line assuming all of the baryons in the halo are in the black hole are also shown. In the right panel we show the distribution of recovered M • for the physically plausible cases where M • < M ℎ . find no change in the slope of the relation for 9 < log M ℎ /M < 11, although the slope across the full mass range explored (9 < log M ℎ /M < 12 may be shallower than that extrapolated from abundance matching (Behroozi et al. 2010. We fit a power law to our empirical SMHM relation and find that for adopted NFW dark matter profiles and for M * < 10 9 M , M ℎ = 10 10.35±0.02 M * 10 8 M 0.63±0.02 . The normalisation is susceptible to systematic errors that depend on the adopted dark matter potential. The slope will have systematic errors if typical dark matter profiles systematically depend on mass. For example, if dwarf galaxies were more likely to have feedback-driven cores than more massive galaxies (Pontzen & Governato 2012), then the slope would be shallower than our fiducial result here, similar to the extrapolated abundance matching results. We note also that the quoted uncertainties above refer to the uncertainties in the median relation and do not capture the galaxy-to-galaxy scatter. For galax-ies with M ℎ < 10 11 M the scatter about the fit in ℎ is 0.3 dex inclusive of the uncertainties in our method but with the additional sample cuts described in the §3.2. Finally, we place lower luminosity Local Group galaxies on the relationship using the same technique and find that about half lie well along the extrapolated relationship, but that those with extremely high inferred ratios of dark matter to luminous matter within , which we generally rejected in our technique as being unphysical, fall far from the SMHM relationship. If one accepts these values, then the nature of discrepancy is that there is too much dark mass within . When we posit that these galaxies indeed do lie on the SMHM and that the extra dark matter mass within does not belong to the larger dark matter component, we can calculate how much extra mass there is. Hypothesising that this mass is in the form of a central black hole mostly yields black hole masses in the range of intermediate mass black hole, 10 5.7±0.6 M , and roughly one to a few percent of M ℎ . At the very least, this analysis highlights several Local Group galaxies that merit a closer look. Our technique provides an independent way to derive SMHM relationships for local galaxy samples. Its power is mostly in enabling statistical comparisons, although it can be used to highlight interesting cases worthy of follow up study, such as in the case of the inferred IMBHs in certain Local Group dwarf galaxies. The empirical basis for the relation means that refinements will be made as the calibrating samples grow in size and provide greater representation of galaxies at the extremes, such as ultra-diffuse galaxies and ultra-compact dwarfs. Nevertheless, it currently provides a valuable independent comparison to the dominant abundance matching approach and provides support for the power-law extrapolation of those results to lower halo masses. ACKNOWLEDGMENTS DZ acknowledges financial support from AST-2006785. PB was partially funded by a Packard Fellowship, Grant #2019-69646. Figure 2 . 2Posterior distributions of each of the coefficiencts in Equation 4. Figure 3 . 3Comparison of the inferred enclosed masses at using the Wolf et al. (2010) estimator, M + , and our scaling-relation based estimate, M , for the optical galaxy sample (see Figure 4 . 4The SMHM relation for low-luminosity galaxies in nearby galaxy clusters. Results for each of four clusters (Hydra (LaMarca et al. 2022); Fornax(Venhola et al. 2019); Virgo(Ferrarese et al. 2020); and Coma(Yagi et al. 2016)) are presented on the the right. The composite of these four populations is presented on the left. The solid and dotted lines are the extrapolation of the SMHM relations fromBehroozi et al. (2010) andBehroozi et al. (2019), respectively, and are the same in all panels. The dash-dotted line in the Virgo panel is the SMHM relation from There are excellent published catalogues for low luminosity galaxies in the Virgo(Ferrarese et al. 2020), Hydra (LaMarca et al. 2022), Fornax(Venhola et al. 2019), and Coma(Yagi et al. 2016) clusters. The resulting SMHM relations for each of the four clusters, as well as for the composite sample, are presented inFigure 4and compared to the extrapolations of theBehroozi et al. (2010Behroozi et al. ( , 2019 SMHM relations. The Virgo data trace the relationship to the smallest M ℎ 's among the four samples and the Coma data are the richest, but all are, in the mean, either consistent or only slightly above theBehroozi et al. (2010) curve and consistent with each other. Together, the samples define a clear ridge-line in the M * -M ℎ space for 9 log M ℎ / 11. Figure 5 . 5Deviations from the Behroozi et al. (2010) fiducial relationship, Δ , as a function of the dark matter mass ratio with a sphere of radius , M , /M , for the set of galaxies shown in Figure 4. The left panel shows all of the data, while the right one zooms in on the bulk of the data. The dotted lines represent our upper and lower criteria for M , /M going forward. and examine the deviations about the Behroozi et al. (2010) fiducial. We present the deviations from this fiducial, Δ , as a function of , /M in Figure 5.There are a few galaxies with unphysical results (M , / < 0) because, as anticipated, our estimated value of M * occasionally exceeds that of M . We reject these cases but they comprise only 1.7% of the overall sample. Next, we notice that the main distribution in the Figure has an curved shape, with Δ values trailing lower as M , approaches zero. Lower values of Δ correspond to underestimates of M ℎ relative to the fiducial, which would be expected if scatter moves M , /M below its true value. This downward tail is most visible for M , /M < 0.5, so we define a requirement that the ratio exceed 0.5. The dotted line in the left panel ofFigure 5shows this cut. At the other end of the M , /M range there is a sharp rise in Δ . Here, scatter causes an underestimate of M * , hence an overestimate of M , Figure 7 . 7A comparison of residuals about the best fit SMHM relations using our approach (NFW; darker, wider bars) a simple scaling of M , (∝ , Figure 9 . 9The low mass (M * < 10 9 M ) M * -M ℎ relation for the combination of the cluster and field subsamples (in lightly coloured circles). Power law fit given in Eq. 4.1 is shown in red dashed lines. The Local Group members from the Drlica-Wagner et al. (2020) compilation for which we can obtain reliable M ℎ estimates ( , / < 0.99) are shown as squares and labelled in the left panel. Those for which we obtain unreliable mass estimates are shown in the right panel. Some galaxies are labelled to help provide context. The red crosses indicated those for which claims of tidal distortion exist in the literature (Mutlu-Pakdil et al. 2019, and references therein). Figure 10 . 10Central massive black hole scenario. Table 1 . 1Samples usedSource Type Band Sample with M ℎ Notes Size Chilingarian et al. (2008) dE B - 46 Table 3 . 3Comparing determinations for SMHM LG outliersGalaxy Estimate Observed Reference (km s −1 ) (km s −1 ) Bootes I 7.2 5.1 +0.8 −0.7 Jenkins et al. (2021) Bootes II 3.4 10.5±7.4 Koch et al. (2009) Coma Berenices 4.4 4.6±0.8 Simon & Geha (2007) Grus I 1.9 2.5 1.3 −0.8 Chiti et al. (2021a) Grus II 6.7 <1.9 Simon et al. (2020) Hercules 6.2 5.1±0.9 Simon & Geha (2007) Leo IV 6.2 3.4 1.3 −0.9 Jenkins et al. (2021) Segue 1 2.7 3.7 +1.4 −1.1 Simon et al. (2011) Sextans 8.5 8.9±0.4 Walker et al. (2006) Tucana II 10.6 4.6±1.5 Chiti et al. (2021b) 10.6 8.6 4.4 −2.7 Walker et al. (2016) 10.6 6.2 +1.6 −1.3 Taibi et al. (2020) Tucana IV 9.3 4.3 1.7 −1.0 Simon et al. (2020) Ursa Major I 7.9 7.6±1 Simon & Geha (2007) Ursa Major II 6.0 6.7±1.4 MNRAS 000, 1-13(2022) There are two significant outliers from the 1:1 relation. 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{'abstract': 'We present a photometric halo mass estimation technique for local galaxies that enables us to establish the stellar mass-halo mass (SMHM) relation down to stellar masses of 10 5 M . We find no detectable differences among the SMHM relations of four local galaxy clusters or between the cluster and field relations and we find agreement with extrapolations of previous SMHM relations derived using abundance matching approaches. We fit a power law to our empirical SMHM relation and find that for adopted NFW dark matter profiles and for M * < 10 9 M , the halo mass is M ℎ = 10 10.35±0.02 (M * /10 8 M ) 0.63±0.02 . The normalisation of this relation is susceptible to systematic modelling errors that depend on the adopted dark matter potential and the quoted uncertainties refer to the uncertainties in the median relation. For galaxies with M * < 10 9 M that satisfy our selection criteria, the scatter about the fit in ℎ , including uncertainties arising from our methodology, is 0.3 dex. Finally, we place lower luminosity Local Group galaxies on the SMHM relationship using the same technique, extending it to M * ∼ 10 3 M and suggest that some of these galaxies show evidence for additional mass interior to the effective radius beyond that provided by the standard dark matter profile. If this mass is in the form of a central black hole, the black hole masses are in the range of intermediate mass black holes, 10 (5.7±0.6) M , which corresponds to masses of a few percent of M ℎ , well above values extrapolated from the relationships describing more massive galaxies.', 'arxivid': '2212.02948', 'author': ['Dennis Zaritsky ', 'Peter Behroozi ', '\nDepartment of Astronomy\nSteward Observatory\nUniversity of Arizona\n85721TucsonAZUSA\n', '\nDivision of Science\nNational Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan\n'], 'authoraffiliation': ['Department of Astronomy\nSteward Observatory\nUniversity of Arizona\n85721TucsonAZUSA', 'Division of Science\nNational Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan'], 'corpusid': 254275223, 'doi': '10.1093/mnras/stac3610', 'github_urls': ['http://github.com/jobovy/galpy)'], 'n_tokens_mistral': 24747, 'n_tokens_neox': 20158, 'n_words': 12992, 'pdfsha': '1a71b9281d38d93d86a3247f628a15216aa8029e', 'pdfurls': ['https://export.arxiv.org/pdf/2212.02948v1.pdf'], 'title': ['Photometric Mass Estimation and the Stellar Mass-Halo Mass Relation for Low Mass Galaxies', 'Photometric Mass Estimation and the Stellar Mass-Halo Mass Relation for Low Mass Galaxies'], 'venue': ['MNRAS']}

arxiv

A viscosity solution approach to the infinite dimensional HJB equation related to boundary control problem in transport equation 26 Sep 2006 March 29, 2022 G Fabbri A viscosity solution approach to the infinite dimensional HJB equation related to boundary control problem in transport equation 26 Sep 2006 March 29, 2022arXiv:math/0606115v2 [math.OC]Hamilton-Jacobiviscosity solutionboundary controlHilbert space AMS subject classification: 49J2049L25 The paper concerns with the infinite dimensional Hamilton-Jacobi-Bellman equation related to optimal control problem regulated by a linear transport equation with boundary control. A suitable viscosity solution approach is needed in view of the presence of the unbounded controlrelated term in the state equation in Hilbert setting. An existence-anduniqueness result is obtained. Introduction We study the Hamilton-Jacobi-Bellman equation (from now HJB equation) related to the infinite dimensional formulation of an optimal control problem whose state equation is a PDE of transport type. We consider the PDE    ∂ ∂s x(s, r) + β ∂ ∂r x(s, r) = −µx(s, r) + α(s, r) (s, r) ∈ (0, +∞) × (0,s) x(s, 0) = a(s) if s > 0 x(0, r) = x 0 (r) if r ∈ [0,s] (1) wheres, β are positive constants, µ ∈ R, the initial data x 0 is in L 2 (0,s) and we consider two controls: a boundary control a is in L 2 loc ([0, +∞); R) and a distributed control α ∈ L 2 loc ([0, +∞) × [0,s]; R). More precisely we will consider controls such that α(·) ∈ E and a(·) ∈ A where E and A will be defined in Section 1. We write "−µx" instead of "µx" because it is the standard way to write the equation in the economic literature where −µ has the meaning of a depreciation factor (and only the case µ ≥ 0 is used). Here we consider a generic µ ∈ R. Using the approach and the references described in Section 1, the above equation can be written as an ordinary differential equation in the Hilbert space H = L 2 (0,s) as follows d ds x(s) = Ax(s) − µx(s) + α(s) + βδ 0 a(s) x(0) = x 0 (2) where A is the generator of a suitable C 0 semigroup and δ 0 is the Dirac delta in 0. Such an unbounded contribution in the Hilbert formulation comes from the presence in the PDE of a boundary control (see [BDPDM92]). We consider the problem of minimizing the cost functional J(x, α(·), a(·)) = ∞ 0 e −ρs L(x(s), α(s), a(s))ds where ρ > 0 and L is globally bounded and satisfies some Lipschitz-type condition, as better described in Section 1. The HJB equation related to the control problem with state equation (2) and target functional (3) is ρu(x) − ∇u(x), Ax − ∇u, −µx L 2 (0,s) − − inf (α,a)∈Σ×Γ βδ 0 (∇u), a R + ∇u, α L 2 (0,s) + L(x, α, a) = 0. (4) The sets Γ and Σ will be introduced in Section 1. If we define the value function of the control problem as V (x) def = inf (α(·) ,a(·))∈E×A J(x, α(·), a(·)), we wish to prove that it is the only solution, in a suitable sense, of the HJB equation. We use the viscosity approach. Our main problem is to write a suitable definition of viscosity solution, so that an existence and uniqueness theorem can be derived for such a solution. The main difficulties we encounter, with respect to the existing literature, is in dealing with the boundary term and the non-analyticity of the semigroup. We substantially follow the original idea of Crandall and Lions ( [CL90] and [CL91]) -with some changes, as the reader will rate in Definition 1.14 and Definition 1.15 -of writing test functions as the sum of a "good part" , a regular function with differential in D(A * ) and a "bad part" represented by some radial function. The main problems arise from the evaluation of the boundary term on the radial part. In order to write a working definition in our case some further requirements are needed, like a C 2 regularity of the test functions, the presence of a "remainder term" in the definition of sub/super solution and the B-Lipschitz continuity (see Definition 1.10) of the solution. This last feature guarantees that the maxima and the minima in the definition of sub/super solution remain in D(A * ) (see Proposition 3.1). Some other comments on the definition of solution (Definition 1.14 and 1.15) need some technical details and can be found in Remark 1.17. The used technique cannot be easily extended to treat a general non-linear problem because we use the explicit form of the PDE that we give in (6). Moreover the (linear) case in which µ is not a constant but a function of L ∞ (0,s) (and so we have µ(r) in equation (1)) presents some difficulties in the proof of the uniqueness theorem and so it is not an easy generalization of the present work. See Remark 3.8 for details. A brief summary of the literature Hamilton-Jacobi equations in infinite dimensions, especially when arising from optimal control problems in Hilbert spaces, was first studied by Barbu and Da Prato ([BDP83], [BDPP83]) with strong solutions approach. The viscosity method, introduced in the study of finite dimensional HJ equations in [CL83] was generalized by the same authors in a series of works; the most important for our approach are [CL90] and [CL91]. Moreover new variants of the notion of viscosity solution for HJB equations in Hilbert space was given in [Ish93], [Tat92b], [Tat94], [Tat92a] and [CL94]. The study of viscosity solution for HJB equations in Hilbert spaces arising from optimal control problem of systems modeled by PDE with boundary control term is more recent. In this research field there is not an organic and complete theory but some works on specific PDE that adapt the ideas and the techniques of viscosity solutions to particular problems using their own characteristics like we do in this work for the problem regulated by transport equation. For the first order HJB equations see [CGS93], [CT96c] (see also [CT96a,CT96b]) where some classes of parabolic equations are treat, [GSŚ02] in which the authors study the HJB equation related to a two-dimensional Navier-Stokes (see also [Shi02]). It must be noted that all these works treat the case of A analytic. HJB equation like (4) was treated, only in the convex case, with strong solutions approach adapting the Barbu and Da Prato' arguments in [Fag02] and [Fag05a]. A motivating economic problem Tranport equations are used to model a large variety of phenomena, from age-structured population models (see for instance [Ian95,Ani00,IMM05]) to population economics ( [FPV04]), from epidemiologic studies to socio-economic science and transport phenomena in phisics. Problems such as (1) can be used to describe, in economics, capital accumulation processes where an heterogeneous capital is involved, and this is the reason why the study of infinite dimensional control problem is of growing interest in the economic fields. For instance in the vintage capital models x(t, s) may be regarded as the stock of capital goods differentiated with respect the time t and the vintage s. Heterogeneous capital, both in finite and infinite dimensional approach, is used to study depreciation and obsolescence of physical capital, geographical difference in growth, innovation and R&D. Regarding problems modeled by a transport equation where an infinite dimensional setting is used we cite the following papers: [BG98] and [BG01] on optimal technology adoption in a vintage capital context (in the case of quadratic cost functional), [HKVF03] on capital accumulation, [BG99] on optimal advertising and [Fag05a] [Fag05b] that studies the convex functional case using a strong solutions approach. See also [FG04]. Moreover, we mention that the infinite dimensional approach may apply to problems such as issuance of public debt (see [AAB + 04] for a description of the problem). In that problem a stochastic setting and simple state-control constraints appear, but hopefully the present work can be a first step in this direction. Plan of the paper The work is organized as follows: in the first section we remind some results on the state equation, we introduce some preliminary remarks on the main operators involved in the problem, we explain some notations, we define the HJB equation and we give the definition of solution. The second section regards some properties of the value function (in particular some regularity properties) that we will be used in the third section to prove that it is the only (viscosity) solution of the HJB equation. Acknowledgements The author would like to thank Prof. AndrzejŚwiȩch for his hospitality, his great kindness, for many useful suggestions and stimulating conversations. 1 Notation and preliminary results State equation In this subsection we will see some properties of the state equation: we write it in three different (and equivalent) forms that point out different properties of the solution. We will use all the three forms in the following proofs. We consider the PDE on [0, +∞) × [0,s] given by    ∂ ∂s x(s, r) + β ∂ ∂r x(s, r) = −µx(s, r) + α(s, r) (s, r) ∈ (0, +∞) × (0,s) x(s, 0) = a(s) if s > 0 x(0, r) = x 0 (r) if r ∈ [0,s](5) Given an initial datum x 0 ∈ L 2 ((0,s); R) (from now simply L 2 (0,s)), a boundary control a(·) ∈ L 2 loc ([0, +∞); R) and a distributed control α(·) ∈ L 2 loc ([0, +∞) × [0,s]; R) the (5) has a unique solution in L 2 loc ([0, +∞) × [0,s]; R) given by x(s, r) = e −µs x 0 (r − βs) + s 0 e −µτ α(s − τ, r − βτ )dτ r ∈ [βs,s] e −µ β r a(s − r/β) + r/β 0 e −µτ α(s − τ, r − βτ )dτ r ∈ [0, βs)(6) In the following x(s, r) will denote (6). We can rewrite such equation in a suitable Hilbert space setting. We take the Hilbert space H def = L 2 (0,s) and the C 0 semigroup T (t) given by T (s)f [r] def = f (r − βs) f or r ∈ [βs,s] 0 f or r ∈ [0, βs) The generator of T (s) is the operator A given by [BG01] for a proof in the case β = 1, the proof in our case can be obtined simply taking s ′ = βs). D(A) = {f ∈ H 1 [0,s] : f (0) = 0} A(f )[r] = −β d dr f (r) (see Remark 1.1. To avoid confusion if x ∈ L 2 (0,s) we will use [·] to denote the pointwise evaluation, so x[r] is the value of x in r ∈ [0,s]. On other hand x(s) will denote the evolution of the solution of the state equation (in the Hilbert space) at time s (as in (7)). That is, x(s) is an element of H while x[r] is a real number. We want to write an infinite dimensional formulation of (5) but in L 2 (0,s) it should appear like d ds x(s) = Ax(s) − µx(s) + α(s) + βδ 0 a(s) x(0) = x 0 (7) where α(s) ∈ L 2 (0,s) is the function r → α(s, r). Such expression does not make sense in L 2 (0,s) for the presence of the unbounded term βδ 0 a(s). We can anyway apply formally the variation of constants method to (7) and obtain a mild form of (7) that is continuous : [0, +∞) → L 2 (0,s). This is what we do in the next definition. Note that we have written (7) only to be more clear but we could go on in a more formal way without it. Definition 1.2. Given x 0 ∈ L 2 (0,s), a(·) ∈ L 2 loc ([0, +∞); R) and α(·) ∈ L 2 loc ([0, +∞); L 2 (0,s)) the function in C([0, +∞); L 2 (0,s)) given by x(s) = e −µs T (s)x 0 − A s 0 e −µ(s−τ ) T (s − τ )(a(τ )ν)dτ + + s 0 e −µ(s−τ ) T (s − τ )α(τ )dτ (8) where ν : [0,s] → R ν : r → e − µ β r is called mild solution of (7). Remark 1.3. We could include the term −µx in the generator of the semigroup A taking aà = A − µ½. In this case the equation (7) would appear in the following equivalent form: d ds x(s) =Ãx(s) + α(s) + βδ 0 a(s)(9) The problem of this approach is that often we will use, in the estimates, the dissipativity of the generator andà is dissipative only if µ ≥ 0. Nevertheless we wrote the mild form (8) as if we consider the (of course equivalent!) state equation (9), indeed in the definition we consider for example the convolution term given by Proof. See [BG01]. Eventually we observe that (7) can be rewritten in a precise way in a larger space where we have not problem with the βδ 0 term. The mild solution will be the only solution (in a suitable sense) of the new differential equation. We need more notation to write it. We consider the adjoint operator A * . Its explicit expression is given by D(A * ) def = {f ∈ H 1 (0,s) : f (s) = 0} A * (f )[r] = β d dr f (r) On D(A * ) we put the graph norm and the related Hilbert structure. We consider the inclusion i : D(A * ) ֒→ L 2 (0,s) and its continuous adjoint i * : L 2 (0,s) → D(A * ) ′ where we have identified L 2 with its dual. We can extend A to a generator of a C 0 semigroup on D(A * ) ′ (the domain of the extension will contain L 2 ) and we observe that the Dirac's measure δ 0 ∈ D(A * ) ′ (see [Fag02] Proposition 4.5 page 60 for details). Proposition 1.5. Given T > 0, x 0 ∈ L 2 (0,s), a(·) ∈ L 2 (0, T ), α(·) ∈ L 2 ([0, T ]; L 2 (0,s)), (8) is the unique solution of d ds i * x(s) = Ax(s) − µx(s) + α(s) + βδ 0 a(s) x(0) = x 0 (10) in W 1,2 (0, T ; D(A * ) ′ ) ∩ C(0, T, H). Moreover if a(·) ∈ W 1,2 (0, T ) then such solution will belong to C 1 (0, T ; D(A * ) ′ ) ∩ C(0, T ; H). Proof. See [BDPDM92] Chapter 3.2 (in particular Theorem 3.1 page 173). The definition of the operator B In this subsection we give the definition of the operator B that will have a fundamental role. We could use an abstract approach, noting that A and A * are both generator of C 0 semigroups of contractions and then both are negative (see [DPZ92] page 424) and the set {λ ∈ C : Re(λ) > 0} is in the resolvent of both A and A * (Hille-Yosida theorem, see [LY95] page 53). Anyway in this case we can also follow a more direct approach that allows to find the explicit form of the operator. To note that that A * and A are negative operators we take φ ∈ D(A * ) (so φ(s) = 0) A * φ, φ = s 0 βφ ′ (r)φ(r)dr = −βφ(0) 2 2 and for φ ∈ D(A) (so φ(0) = 0) Aφ, φ = s 0 −βφ ′ (r)φ(r)dr = −βφ(s) 2 2 So, given a λ > 0, the operators (A − λI) and (A * − λI) are strongly negative: (A − λI)x, x ≤ −λ|x| 2 H for all x ∈ D(A) and (A * − λI)x, x ≤ −λ|x| 2 H for all x ∈ D(A * ). We can also directly prove that (A − λI) −1 : H → D(A) is a continuous negative linear operator whose explicit expression is given by (A − λI) −1 (φ)[r] = 1 β −e − λ β r r 0 e λ β τ φ(τ )dτ The continuity can be proven directly with not difficult estimates and the negativity can be proven directly using an integration by part argument. In the same way we can prove that (A * − λI) −1 : H → D(A * ) is a continuous and negative linear operator and that and its explicit expression is given by (A * − λI) −1 (φ)[r] = 1 β −e λ β r s r e − λ β τ φ(τ )dτ Eventually we can define B def = (A * −λI) −1 (A−λI) −1 = ((A−λI) −1 ) * (A−λI) −1 that is continuous, positive and selfadjoint 1 . Moreover (A * − λI)B = (A − λI) −1 ≤ 0 and so A * B = (A − λI) −1 + λB ≤ λB if we choose λ < 1 we have that A * B is bounded and A * B ≤ B(11) Thus B satisfies all requirements of the "weak case" of [CL90]. Remark 1.6. Note that B 1/2 is a particular case of the operator that Renardy found in more generality in [Ren95] and so B 1/2 : H → D(A * ) continuously and in particular R(B 1/2 ) ⊆ D(A * ). Notation 1.7. For every x ∈ H we will indicate with |x| B the B-norm that is Bx, x H . We will write H B for the completion of H on the B-norm. Remark 1.8. Thanks to the definition of A * the graph norm on D(A * ) is equivalent to the H 1 (0,s) norm. In particular D(A * ) is the the completion of K = {f | [0,s] : f ∈ C ∞ c (R) with supp(f ) ⊆ (−∞,s)} with respect the H 1 (0,s) norm. So, since H 1 (0,s) ֒→ C([0,s]; R) , we can calculate βδ 0 on the elements of D(A * ). Notation 1.9. In some cases the notation x, y may be not clear, so when necessary we will use an index to avoid confusion: if H is a Hilbert space (for example H = H ≡ L 2 (0,s) or H = H 1 (0,s) or D(A * ) ...) the notation x, y H will indicate the inner product in the Hilbert space H. Otherwise if Z is an Banach space (possibly an Hilbert space) and Z ′ its dual the notation x, y Z×Z ′ will indicate the duality. In a few words, a single index means inner product, a double one indicates duality. When there is no index it is because we have omitted the index H ≡ L 2 (0,s). The control problem and the HJB equation In this subsection we describe the optimal control problem, state the hypotheses, define the HJB equation of the system and give a suitable definition of solution of the HJB equation. We will consider the optimal control problem governed by the state equation d ds i * x(s) = Ax(s) − µx(s) + α(s) + βδ 0 a(s) x(0) = x (12) that has a unique solution in the sense described in Section 1.1. Given two compact subsets Γ and Λ of R we consider the set of admissible boundary controls given by A def = {a : [0, +∞) → Γ ⊆ R : a(·) is measurable} . Moreover we call Σ def = γ : [0,s] → Λ ⊆ R : γ(·) ∈ L 2 (0,s) . In view of the compactness of Λ we have that Σ ⊆ L 2 (0,s). We define the set of admissible distributed controls as E def = α : [0, +∞) → Σ ⊆ L 2 (0,s) : α(·) is measurable In view of the complactness of Γ and Λ A ⊆ L 2 loc ([0, +∞); R) and E ⊆ L 2 loc ([0, +∞)× [0,s]; R). We call Γ def = sup a∈Γ (|a|), Λ def = sup b∈Λ (|b|) and Σ def = sup α∈Σ (|α| H=L 2 (0,s) ) (they are bounded thanks to the boundedness of Γ and Λ). We will call admissible control a couple (α(·), a(·)) ∈ E × A. The cost functional will be of the form J(x, α(·), a(·)) = ∞ 0 e −ρs L(x(s), α(s), a(s))ds where L is uniformly continuous and satisfies the following conditions: there exists a C L ≥ 0 with (L1) |L(x, α, a) − L(y, α, a)| ≤ C L B(x − y), (x − y) H×H ∀(α, a) ∈ Σ × Γ (L2) |L| ≤ C L < +∞ We define formally the HJB equation of the system as ρu(x) − ∇u(x), Ax − ∇u(x), −µx − H(x, ∇u(x)) = 0 (HJB) where H is the Hamiltonian of the system and is defined as: H : H × D(A * ) → R H(x, p) def = inf (α,a)∈Σ×Γ ( βδ 0 (p), a R + p, α H + L(x, α, a)) (according to Notation 1.9 ·, · R is the usual real product). Before we can introduce a suitable definition of (viscosity) solution of the HJB equation we have to give some preliminary definitions. Definition 1.10. v ∈ C(H) is Lipschitz with respect the B-norm or B-Lipschitz if there exists a constant C such that |v(x) − v(y)| ≤ C|(x − y)| B def = C|B 1/2 (x − y) | H for every choice of x and y in H. In the same way we can give the definition of locally B-Lipschitz function. Definition 1.11. A function v ∈ C(H) is said to be B-continuous at a point x ∈ H if for every x n ∈ H with x n ⇀ x and |B(x n − x)| → 0, it holds that v(x n ) → v(x). In the same way we can define the B-upper/lower semicontinuity. Definition 1.12. We say that a function φ such that φ ∈ C 1 (H) and φ is Blower semicontinuous is a test function of type 1 and we will write φ ∈ test1 if ∇φ(x) ∈ D(A * ) for all x ∈ H and A * ∇φ : H → H is continuous. Definition 1.13. We say that g ∈ C 2 (H) is a test function of type 2 and we will write g ∈ test2 if g(x) = g 0 (|x|) for some function g 0 : R + → R nondecreasing. Definition 1.14. u ∈ C(H) bounded and Lipschitz with respect the B-norm is a subsolution of the HJB equation (or simply a "subsolution") if for every φ ∈ test1 and g ∈ test2 and a local maximum point x of u − (φ + g) we have ρu(x) − A * ∇φ(x), x − ∇φ(x) + ∇g(x), −µx − − inf (α,a)∈Σ×Γ βδ 0 (∇φ(x), a R + ∇φ(x) + ∇g(x), α H + L(x, α, a) ≤ ≤ g ′ 0 (|x|) |x| β Γ 2 2 (13) Definition 1.15. v ∈ C(H) bounded and Lipschitz with respect the B-norm is a supersolution of the HJB equation (or simply a "supersolution") if for every φ ∈ test1 and g ∈ test2 and a local minimum point x of v + (φ + g) we have ρv(x) + A * ∇φ(x), x + ∇φ(x) + ∇g(x), −µx − − inf (α,a)∈Σ×Γ − βδ 0 (∇φ(x), a R − ∇φ(x) + ∇g(x), α H + L(x, α, a) ≥ ≥ − g ′ 0 (|x|) |x| β Γ 2 2 (14) Definition 1. 16. v ∈ C(H) bounded and Lipschitz with respect the B-norm is a solution of the HJB equation if it is at the same time a supersolution and a subsolution. We write now some remarks on the definition we have just given, to explain its meaning: Remark 1.17. In the definition of viscosity solution we have used two kinds of test function: the test1 and test2 that, as usual in the literature, play a different role in the definition. In view of their properties and their regularity the functions of the first set (test1) represent the "good part" while the main difficulties come from the function of the set test2 that have the role of localize the problem. A difficulty of our case is the following: the trajectory is not Lipschitz in the norm of the Hilbert space H and so, given a function g ∈ test2, the term g(x(s)) − g(x) s(15) (where x(s) is a trajectory starting from x) cannot be treated with standard arguments. The idea is to consider only B-Lipschitz solution so that the maxima considered in Definition 1.14 and Definition 1.15 are in D(A * ). If the "starting point" x is in the domain of D(A * ) there are some advantages in the estimate of (15) but some problems remain: in such case we will prove in Proposition 3.4 that (if α(·) is continuous) g(x(s)) − g(x) s − ∇g(x), −µx + α(0) ≤ g ′ 0 (|x|) |x| β Γ 2 2 + O(s) where the rest O is uniform in the control. So the "worse case" is the one described in the definition. The value function and its properties The value function is, as usual, the candidate to be the unique solution of the HJB equation. In this section we define the value function of the problem and then we verify that it has the regularity properties required to be a solution. Namely we will check that it is B-Lipschitz (Proposition 2.4). To obtain such result we need to prove an approximation result (Proposition 2.1) and then a suitable estimate for the solution of the state equation (Proposition 2.3). The value function of our problem is defined as: V (x) def = inf (α(·),a(·))∈E×A J(x, α(·), a(·)) We consider the functions Proof. Using the mild expressions we find η n : [0,s] → R η n (r) def = [2n − 2n 2 r] + (where [·] + is|x(s) − x n (s)| = −A s 0 e −(s−τ )µ e (s−τ )A (a(τ )ν)dτ − − s 0 e −(s−τ )µ e (s−τ )A βC * n (a(τ ))dτ (20) To estimate such expression we will use the explicit expression of the two terms (as two-variable function). We simplify the notation (only in this proof!) taking an "extension" of a(·) to the whole R obtained by putting a(·) identically 0 on R − . So y(s, r) def = −A s 0 e −(s−τ )µ e (s−τ )A (a(τ )ν)dτ [r] = e − µ β r a(s − r/β) y n (s, r) def = s 0 e −(s−τ )µ e (s−τ )A βC * n (a(τ ))dτ [r] = = r∧(1/n) 0 e − µ β (r−θ) [2n − 2n 2 θ] + a θ − r β + s dθ (21) Now for all s ∈ [0, T ] |y(s, ·) − y n (s, ·)| 2 H=L 2 (0,s) ≤ ≤ s 1/n e − µ β r a(s − r/β) − 1/n 0 e − µ β (r−θ) [2n − 2n 2 θ] + a θ − r β + s dθ 2 dr + + 1/n 0 e − µ β r a(s − r/β) − r 0 e − µ β (r−θ) [2n − 2n 2 θ] + a θ − r β + s dθ 2 dr ≤(22)(fors ≤ T ) ≤ e |µ|s T 0 e −µ( r β −s) a(s − r/β)− − 1/n 0 e −µ( r−θ β −s) [2n − 2n 2 θ] + a s + θ − r β dθ 2 dr + 1 n e |µ|/βT 2 Γ .(23) Such estimate does not depends on s, the integral term goes to zero because it is the convolution of a function in L 2 (0, T ) with an approximate unit and the second goes to zero for n → ∞. Proposition 2.2. Let φ ∈ C 1 (H) be such that ∇φ : H → D(A * ) (D(A * ) is endowed, as usual, with the graph norm) is continuous. Then, for an admissible control (α(·), a(·)), if we call x(·) the trajectory starting from x and subject to the control (α(·), a(·)), we have that, for every s > 0, (24) Proof. In the approximating state equation (17) the unbounded term βδ 0 does not appear (βC * n are continuous) and then (see [LY95] Proposition 5.5 page 67) for every φ(·) ∈ C 1 (H) such that A * ∇φ(·) ∈ C(H) we have τ )), x n (τ ) + ∇φ(x n (τ )), βC * n a(τ ) + + ∇φ(x n (τ )), α(τ ) + ∇φ(x n (τ )), −µx n (τ ) ] dτ. (25) In view of the continuity of the operator C * n we can pass to its adjoint (see (16) for an explicit form of the operator C n ) and we obtain: φ(x(s)) = φ(x) + s 0 [ A * ∇φ(x(τ )), x(τ ) + βδ 0 (∇φ(x(τ ))), a(τ ) R + + ∇φ(x(τ )), α(τ ) + ∇φ(x(τ )), −µx(τ ) ] dτφ(x n (s)) = φ(x) + s 0 [ A * ∇φ(x n (φ(x n (s)) = φ(x) + s 0 [ A * ∇φ(x n (τ ) ), x n (τ ) + βC n ∇φ(x n (τ )), a(τ ) + + ∇φ(x n (τ )), α(τ ) + ∇φ(x n (τ )), −µx n (τ ) ] dτ. (26) Now we prove that every integral term of the (26) converges to the corresponding term of the (24). This fact, toghether with the pointwise convergence of (φ(x n (s)) n→∞ − −−− → φ(x(s)) due to Proposition 2.1) prove the claim. First we note that, in view of Proposition 2.1 and of the continuity of x, x n (τ ) is bounded uniformly in n and τ ∈ [0, s] and, in view of the continuity of ∇φ, ∇φ(x n (r)) is bounded uniformly in n and τ ∈ [0, s] So we can apply the Lebesgue theorem (the pointwise convergence is given by Proposition 2.1 and |α(τ )| ≤ Σ ) and we prove that (27) Now we observe that, in view of the continuity of A * ∇φ and of the of Proposition 2.1, the term A * ∇φ(x n (τ )) is bounded uniformly in n and τ ∈ [0, s] so the same is true for |A * ∇φ(x n (τ )) − A * ∇φ(x(τ ))|. s 0 [ ∇φ(x n (τ )), α(τ ) + ∇φ(x n (τ )), −µx n (τ ) ] dτ n→∞ − −−− → n→∞ − −−− → s 0 [ ∇φ(x(τ )), α(τ ) + ∇φ(x(τ )), −µx(τ ) ] dτ Therefore we can use the Lebesgue theorem (the pointwise convergence is given by Proposition 2.1) to conclude that s 0 A * ∇φ(x n (τ )), x n (τ ) dτ → s 0 A * ∇φ(x(τ )), x(τ ) dτ We have now to prove that s 0 βC n ∇φ(x n (τ )), a(τ ) dτ → and then, by the last estimate βC n (∇φ(x n (·))) n→∞ − −−− → βδ 0 (∇φ(x(·))) in C([0, T ]; R). Then (28) follows by Cauchy-Schwartz inequality (it is the scalar product in L 2 (0, s)). Proposition 2.3. Given T > 0 and a control (α(·), a(·)) ∈ E × A there exists c T such that for every x, y ∈ H sup s∈[0,T ] |x x (s) − x y (s)| 2 B ≤ c T |x − y| 2 B , where x y (·) is the solution of Proof. We use Proposition 2.2 with φ(x) = Bx, x . So ∇φ(x) = 2Bx. We observe that x x (·) − x y (·) satisfies the equation d ds i * (x x (s) − x y (s)) = A(x x (s) − x y (s)) − µ(x x (s) − x y (s)) (x x − x y )(0) = x − y (the one of Proposition 2.2 with control identically 0) and then by (11) |x x (s) − x y (s)| 2 B = |x − y| 2 B + 2 s 0 A * B(x x (r) − x y (r)), (x x (r) − x y (r)) − − µ B(x x (s) − x y (s)), x x (s) − x y (s) dr ≤ (31) ≤ |x − y| 2 B + 2(1 + |µ|) s 0 B(x x (r) − x y (r)), (x x (r) − x y (r)) dr now we can use the Gronwall's lemma and obtain the claim. Proposition 2.4. Let L satisfy (L1) and (L2). Then the value function V is Lipschitz with respect the B-norm Proof. Assume V (y) > V (x). Then we take (α(·), a(·)) ∈ E × A an ε-optimal control for x. We have: Letting ε → 0 we have claim. |V (y) − V (x)| − ε ≤ ∞ 0 e − Existence and uniqueness of solution In this section we will prove that the value function is a viscosity solution of the HJB equation (Theorem 3.6) and that the HJB equation admits at most one solution (Theorem 3.7). We remind that we use H B to denote the completion of H in the B-norm. This notation will be used in the next propositions. Proof. We do the proof only in the case in which x is a local maximum (the other case is similar). We take ω ∈ H with |ω| = 1 and h ∈ (0, 1). Then for every h small enough (u(x − hω) − ψ(x − hω)) h ≤ u(x) − ψ(x) h so ψ(x) − ψ(x − hω) h ≤ C|w| B and passing to the limit we have ∇ψ(x), ω ≤ C|ω| B . Likewise (u(x + hω) − ψ(x + hω)) h ≤ u(x) − ψ(x) h so ψ(x) − ψ(x + hω) h ≤ C|w| B and passing to the limit we have − ∇ψ(x), ω ≤ C|ω| B . Putting together these two remarks we have | ∇ψ(x), ω | ≤ C|ω| B for all ω ∈ H. So we can consider the linear extension of the continuous linear functional ω → ∇ψ(x), ω to H B ; we will call such extension Φ x and by Riesz representation theorem we can find z x ∈ H B such that Φ x (ω) = z x , ω HB ∀ω ∈ H B however z x , ω HB = B 1/2 (z x ), B 1/2 (ω) H = = B 1/2 (B 1/2 (z x )), ω (HB ) ′ ×(HB ) = B 1/2 (m x ), ω (HB ) ′ ×(HB )(32) where m x def = (B 1/2 (z x )) ∈ H. Now for ω ∈ H B 1/2 (m x ), ω (HB ) ′ ×(HB ) = B 1/2 (m x ), ω H Therefore ∇ψ(x) = B 1/2 (m x ) ∈ R(B 1/2 ) ⊆ D(A * ) where the last inclusion follows from Remark 1.6. Existence In this subsection we will prove that the value function is a solution of the HJB equation. In the next subsection we will prove that such solution is unique. We start with a lemma and two propositions. We will use the notation introduced in Remark 1.1 on "x(s)" and "x[r]". Moreover we will continue to use the symbol δ 0 in the text so that x[0] = δ 0 x if x ∈ D(A * ). We have not found a simple reference for the following lemma so we prove it: Lemma 3.2. Let x be a function of H 1 (0,s) then (i) lim s→0 + s s (x[r] − x[r − s]) 2 s dr = 0 (33) (ii) lim s→0 + s−s s (x[r + s] − x[r]) s x[r]dr = x 2 [s] − x 2 [0] 2 (34) Proof. part (i) s s (x[r] − x[r − s]) 2 s dr = s 0 ψ s [r]dr where ψ s : [0,s] → R is defined in the following way: ψ s [r] = 0 if r ∈ [0, s) (x[r]−x[r−s]) 2 s if r ∈ [s,s] In order to prove the claim we want to apply the Lebesgue theorem. First we will see the a.e. convergence of the ψ s to zero: for r > 0 we take s < r: ψ s [r] ≤ r r−s ∂ ω x[τ ]dτ s |x[r] − x[r − s]| where ∂ ω x is the weak derivative of x (x is in H 1 for hypothesis). Now almost every r is a Lebesgue point and then By the continuity of x we see that: r r−s ∂ ω x(τ )dτ s s→0 + − −−− → |∂ ω x[r]|I 2 (s) s→0 + − −−− → x 2 [s] and I 3 (s) s→0 + − −−− → −x 2 [0] Moreover, using similar arguments that in (i) we find that (where we called x(s) the trajectory that starts from x and subject to the control (α(·), a(·))). Note that O(s) is independent of the control. Proof. We consider s ∈ (0, 1]. This is an arbitrary choice but we are interested only in the behavior of x(·) near to 0 so we can assume it without problems. We use the explicit expression of x(s, r): x(s) − Observe that in this estimate the control (α(·), a(·)) does not appear. The second and the third terms goes to zero for s → 0. In the first we can use Lebesgue theorem observing that e −µs x[(r − βs) ∧ 0] − x[r] ≤ e |µ| |x| L ∞ + |x| L ∞ ∀(s, r) ∈ (0, 1] × [0,s] and that |e −µs x[(r − βs) ∧ 0] − x[r]| s→0 − −− → 0 pointwise. So the statement is proven. Proposition 3.4. Given x ∈ D(A * ) and g ∈ test2 there exists a real function O(s) such that O(s) s→0 − −− → 0 and such that for every control (α(·), a(·)) ∈ E × A with a(·) continuous we have that g(x(s)) − g(x) s − s 0 ∇g(x), α(r) s − ∇g(x), −µx ≤ g ′ 0 (|x|) |x| β Γ 2 2 + O(s) (where we called x(s) the trajectory that starts from x and subject to the control (α(·), a(·))). Note that O(s) is independent of the control. Proof. First we write g(x(s)) − g(x) s − ∇g(x), −µx − s 0 ∇g(x), α(r) s = = g(x(s)) − g(y(s)) + g(y(s)) − g(x) s − ∇g(x), −µx − s 0 ∇g(x), α(r) s(39)g(x(s)) − g(x) s − s 0 ∇g(x), α(r) dr s − ∇g(x), −µx ≤ ≤ g(x(s)) − g(y(s)) s − s 0 ∇g(x), α(r) dr s − ∇g(x), −µx + g(y(s)) − g(x) s .(42) In order to estimate the first addendum we use the Taylor expansion as follows: g(x(s)) − g(y(s)) s = ∇g(y(s)), x(s) − y(s) s + + ∇g(ξ(s)) − ∇g(y(s)), x(s) − y(s) s = (43) where ξ(s) is a point between x(s) and y(s) = ∇g(y(s)), s 0 e (s−τ )A (α(τ ) − µx(τ ))dτ s + + ∇g(ξ(s)) − ∇g(y(s)), s 0 e (s−τ )A (α(τ ) − µx(τ ))dτ s(44) We know by Lemma 3.3 that x(s) where O(s) s→0 − −− → 0 and it does not depend on the control. So we have now to estimate the second term of the (42), namely g(y(s))−g(x) s . If we prove that it is smaller then g ′ 0 (|x|) |x| β Γ 2 2 +O(s) where O(s) does not depend on the control we have proven the proposition. We first note that ∇g(x) = g ′ 0 (|x|) x |x| and D 2 g(x) = g ′′ 0 (|x|) x |x| ⊗ x |x| + g ′ 0 (|x|) I |x| − x ⊗ x |x| 3 We consider the Taylor's expansion of g at x: g(y(s)) − g(x) s = ∇g(x), y(s) − x s + 1 2 (y(s) − x) T (D 2 g(x))(y(s) − x) s + + o(|y(s) − x| 2 ) s = = g ′ 0 (|x|) |x| x, y(s) − x s + 1 2 y(s) − x, y(s) − x s + + 1 2 g ′′ 0 (|x|) |x| 2 − g ′ 0 (|x|) |x| 3 x, y(s) − x 2 s + o(|y(s) − x| 2 ) s def = def = P 1 + P 2 + P 3 (46) First we prove that P 2 and P 3 go to zero uniformly in (α(·), a(·)) and then we will estimate P 1. We proceed in two steps: step 1: There exists a constant C such that for every admissible control (α(·), a(·)) ∈ E × A with a(·) continuous and every s ∈ (0, 1] 3 x, y(s) − x s ≤ C (as before the choice of the interval (0,1] it is not essential: we are interested in the behavior near zero). We observe first that the explici solution of y(s)[r] can be found taking µ = 0 and α = 0 in (6). We have: The third and the fifth part have opposite limits, the second goes to zero thanks to the fact that x ∈ D(A * ) and then x is continuous and x(s) = 0. The first part goes to − β 2 x 2 [0] = A * x, x in view of Lemma 3.2. The only term in which the control appears is the fourth but we can estimate it as follows: step 3: Conclusion We now estimate P 1. We can write a more explicit form of P 1 as in the proofs of step 1 and step 2 ((47), (48) and (49)) and using the same arguments we can see that there exists a rest o(1) (depending only on x) with o(1) s→0 − −− → 0 such that for every control a(·) continuous y(s, r) = x(r − βs) r ∈ [βs,s] a(s − r/β) r ∈ [0, βs)P 1 = g ′ 0 (|x|) |x| A * x, x + βs 0 x[P 1 = o(1) + g ′ 0 (|x|) |x| 1 2 βs 0 (a(s − r/β)) 2 dr s ≤ o(1) + 1 2 g ′ 0 (|x|) |x| β Γ 2 Now, using the estimates on P 1, P 2 and P 3 we see that g(y(s)) − g(x) s ≤ O(s) + 1 2 g ′ 0 (|x|) |x| β Γ 2 . Using this fact and equation (45) in (42) we have proven the proposition. Proposition 3.5. If x ∈ D(A * ) and φ ∈ test1 then there exists a real function O(s) such that O(s) s→0 − −− → 0 and such that for every control (α(·), a(·)) ∈ E × A with a(·) continuous we have that φ(x(s)) − φ(x) s − s 0 ∇φ(x), α(r) dr s − ∇φ(x), −µx − − A * ∇φ(x), x − s 0 βδ 0 (∇φ(x)), a(r) R dr s ≤ O(s) (51) (where we called x(s) the trajectory that starts from x and subject to the control (α(·), a(·))). Note that O(s) is independent of the control. Proof. We proceed as in the proof of Proposition 3.4 observing that φ(x(s)) − φ(x) s = φ(x(s)) − φ(y(s)) s + φ(y(s)) − φ(x) s where y(·) is the solution of (40). It is possible to prove, using exactly the same arguments used in the proof of Proposition 3.4 that φ(x(s)) − φ(y(s)) s − ∇φ(x), −µx − s 0 ∇φ(x), α(r) dr s ≤ O(s) where O(s) If we read equation (40) in D(A * ) ′ it appears as an equation of the form u(t) =Ãu(t) + f (t) u(0) = x where f (t) is a bounded measurable function (|f (t)| D(A * ) ′ ≤ β|δ 0 | D(A * ) ′ Γ ) andà is an extension of A that generates of a C 0 -semigroup on D(A * ) ′ . So 4 we can choose a constant C that depends on x such that, for all admissible control a(·) continuous and all s ∈ (0, 1], |y(s) − x| D(A * ) ′ s ≤ C(54) Thus by (53) and (54), we can say that |I 1 | s→0 − −− → 0 uniformly in a(·). Therefore φ(y(s)) − φ(x) s − ∇φ(x), y(s) − x s s→0 − −− → 0 uniformly in a(·). We now write The third and the fifth terms, that do not depend on the control, have opposite limits, the second goes to zero because ∇φ(x) and x are in D(A * ) and then x[s] = 0 = ∇φ(x)[s]. The first term goes to A * ∇φ(x), x . Finally we observe that the only term that depends on the control is the fourth and We can now prove that the value function is a solution of the HJB equation equation. ∇φ(x), y(s) − x s = s βs ∇φ(x)[r] (x[r − βs] − x[r]) s dr+ + βs 0 ∇φ(x)[r](a(s − r/β) − x[r])dr s = = s− Theorem 3.6. Let L satisfy (L1) and (L2) let Γ and Λ be a compact subsets of R. Then the value function V is bounded, B-Lipschitz and is a solution of the HJB equation. Proof. The boundedness of V follows from the boundedness of L (assumption (L2)). The B-Lipschitz property is the result of Proposition 2.4. It remains to verify that V is a solution of the HJB equation. Subsolution: Let x be a local maximum of V − (φ + g) for φ ∈ test1 and g ∈ test2. Thanks to Proposition 3.1 we know that ∇(φ + g)(x) ∈ D(A * ). Moreover we know that ∇φ(x) ∈ D(A * ) for the definition of the set test1. So ∇g(x) = g ′ 0 (|x|) x |x| ∈ D(A * ) and this implies that x ∈ D(A * ). We can assume that V (x)−(φ+g)(x) = 0. We consider the constant control (α(·), a(·)) ≡ (α, a) ∈ Σ × Γ and x(s) the trajectory starting from x and subject to (α, a). Then for s small enough V (x(s)) − (φ + g)(x(s)) ≤ V (x) − (φ + g)(x) and thanks to the Bellman principle of optimality we know that V (x) ≤ e −ρs V (x(s)) + s 0 e −ρr L(x(r), α, a)dr Then 1 − e −ρs s V (x(s)) − φ(x(s)) − φ(x) s − g(x(s)) − g(x) s − − s 0 e −ρr L(x(r), α, a)dr s ≤ 0. (56) Using Proposition 3.4 and Proposition 3.5 we can now pass to the limsup as s → 0 to obtain ρV (x) − ∇φ(x), −µx − ∇g(x), −µx − − A * ∇φ(x), x + βδ 0 (∇φ(x)), a R + ∇φ(x), α + ∇g(x), α + L(x, α, a) ≤ ≤ g ′ 0 (|x|) |x| β Γ 2 2 .(57) Taking the inf (α,a)∈Σ×Γ we obtain the subsolution inequality. Supersolution: Let x be a minimum for V + (φ + g) and such that V + (φ + g)(x) = 0. As in the subsolution proof we obtain that x ∈ D(A * ). For ε > 0 take (α ε (·), a ε (·)) an ε 2 -optimal strategy. We can assume a(·) continuous (it is not hard to see). We call x(s) the trajectory starting from x and subject to (α ε (·), a ε (·). Now for s small enough V (x(s)) + (φ + g)(x(s)) ≥ V (x) + (φ + g)(x) and thanks to the ε 2 -optimality and the Bellman principle we know that V (x) + ε 2 ≥ e −ρs V (x(s)) + s 0 e −ρr L(x(r), α ε (r), a ε (r))dr We take s = ε. Then 1 − e −ρε ε V (x(ε)) + φ(x(ε)) − φ(x) ε + g(x(ε)) − g(x) ε − − ε 0 e −ρr L(x(r), α ε (r), a ε (r))dr ε + ε 2 ε ≥ 0 (58) in view of Proposition 3.4 and Proposition 3.5 we can choose, independently of the control (α ε (·), a ε (·)), a o(1) with o(1) ε→0 − −− → 0 such that: ρV (x) + A * ∇φ(x), x + ∇φ(x) + ∇g(x), −µx − − ε 0 −βδ 0 (∇φ(x), a ε (r) R + e −ρr L(x(r), α ε (r), a ε (r))dr ε − − ε 0 ∇φ(x) + ∇g(x), α ε (r) dr ε ≥ o(1) − g ′ 0 (|x|) |x| β Γ 2 2(59) we now take inf over a and α inside the integral and let ε → 0 to obtain that ρV (x) + A * ∇φ(x), x + ∇φ(x) + ∇g(x), −µx − − inf (α,a)∈Σ×Γ − βδ 0 (∇φ(x)), a R + L(x, α, a) − ∇φ(x) + ∇g(x), α ≥ ≥ − g ′ 0 (|x|) |x| β Γ 2 2 .(60) (we observe again that the fact that o(1) ε→0 − −− → 0 uniformly in the control is essential). Therefore V is a solution of the HJB equation. Uniqueness Now we can prove a uniqueness result: we prove the result in the case µ = 0. The case µ = 0 is simpler and can be proven with small changes in the proof. Proof. We will proceed by contradiction. Assume that u is a subsolution of the HJB equation and v a supersolution and suppose that there existsx ∈ H and γ > 0 such that (u(x) − v(x)) > 3γ ρ > 0 We take γ < 1. So, taken ϑ > 0 small enough we have u(x) − v(x) − ϑ|x| 2 > 2γ ρ > 0(61) We consider ε > 0 and ψ : H × H → R given by ψ(x, y) def = u(x) − v(y) − 1 2ε |B 1/2 (x − y)| 2 − ϑ 2 |x| 2 − ϑ 2 |y| 2 . Thanks to the boundedness of u and v, chosen ϑ > 0, there exist R ϑ > 0 such that ψ(0, 0) ≥ sup (|x|≥R ϑ ) or (|y|≥R ϑ ) (ψ(x, y)) + 1 We set S = {(x, y) ∈ H × H : |x| ≤ R ϑ and |y| ≤ R ϑ } If we choose R ϑ big enoughx ∈ S. By standard techniques (see [LY95] page 252) we can find p and q in H with |p| < σ and |q| < σ and such that (x, y) → ψ(x, y) − Bp, x − Bq, y attains a maximum in S. We call (x,ȳ) the point of maximum. If we choose σ small enough (for example such that σ B R ϑ < 1 4 γ ρ ) we know by (62) that such maximum is in the interior of S and, thanks to (61), that ψ(x,ȳ) − Bp,x − Bq,ȳ > 3γ 2ρ . Moreover ψ(x,ȳ) > γ ρ and so u(x) − v(ȳ) > γ ρ . We now make some preliminary estimates that we will use in the following: Estimates 1 (on ε): We observe that M : [1, 0) → R M : ε → sup (x,y)∈H×H u(x) − v(y) − 1 2ε B 1/2 (x − y) 2 is non-increasing and bounded and so it admits a limit for ε → 0 + . So there exists aε > 0 such that, for every 0 < ε 1 , ε 2 ≤ε we have that |M (ε 1 ) − M (ε 2 )| < γ 16(1 + |µ|) 2 (64) We choose now ε, that will be fixed in the sequel of the proof: ε := min ε, 1 32C 2 L (65) (C L is the constant introduced in hypothesys (L1) and (L2)). Now we state and prove a claim that we will use in the following: Claim Ifx ∈ H andỹ ∈ H satisfy Estimates 2 (on σ): We have already imposed σ < γ/ρ 4 B R ϑ , we take from now σ = min γ 8ρ B R ϑ , ϑ, ϑ R ϑ(71) so that σ ρ(u(x) − v(ȳ)) − 2 γ 32 − 4 γ 64 − 2 γ 32 − γ 32 − γ 32 − γ 32 − γ 64 − γ 64 ≤ 0 (92) that is ρ(u(x) − v(ȳ)) − 1 2 γ ≤ 0(93) but from the (63) we have ρ u(x) − v(ȳ) > γ and then we obtain from the (93) 1 2 γ = γ − 1 2 γ < ρ(u(x) − v(ȳ)) − 1 2 γ ≤ 0 that is a contradiction because γ > 0 and so the theorem is proven. Remark 3.8. Now we can explain a remark we have done in the introduction: it is difficult to treat with the same arguments the case in which µ is not a constant but a function of r. In the proof of the uniqueness we have to estimate the term 1 ε B(x −ȳ), −µ(x −ȳ) and we can estimate it because we use the term 1 ε |x − y| 2 B to penalize the doubling with respect the B-norm. If we consider the case in which µ is a function of r such term would appear in the form 1 ε B(x −ȳ), −µ(·)(x −ȳ) (where −µ(·)(x −ȳ) is the pointwise product of the L ∞ (0,s) function µ(·) and the L 2 (0,s) function (x −ȳ). We do not know how to treat such term. s 0 e 0−µ(s−τ ) T (s − τ )α(τ )dτ and e −µ(s) T (s) is exactly the semigroup generate byÃ. In the sequel (see (41)) we will use also another mild form of the state equation that is not explicit and it is the one obtined if we do not include the term e −µs in the semigroup. The two forms are equivalent.Proposition 1.4. Taken x(s) the function : R + → L 2 (0,s) given by (8) and x(s, r) the function : R + × [0,s] → R given by (6) we have x(s)[r] = x(s, r). ee the positive part). We then define C * n : R → H C * n : γ → γη n Such functions are linear and continuous and their adjoints areC n : H → R C n : x → x, η n(16)C * n "approximate the delta measure". The approximating equations we consider are d ds x n (s) = Ax n (s) − µx n (s) + α(s) + βC * proofs we will use the mild solutions of the original and the approximating state equations. The first was introduced in (8), the second can be found in ([Paz83] page 105 equation (2.3), we include the term e −µs in the semigroup):x(s) = e −µs e sA x + −(s−τ )µ e (s−τ )A (a(τ )ν)dτ (18)x n (s) = e −µs e sA x + −(s−τ )µ e (s−τ )A βC * n a(τ )dτ (19) Proposition 2.1. For T > 0 and (α(·), a(·)) ∈ E × A lim n→∞ sup s∈[0,T ] |x n (s) − x(s)| H = 0 δ 0 in H −1 (0,s) and then in D(A * ) ′ . Indeed given z ∈ H 1 (0,s) we have |(C n − δ 0 x(·) in C([0, T ]; H), then (by hypothesis on φ) ∇φ(x n (·)) n→∞ − −−− → ∇φ(x(·)) in C([0, T ]; D(A * )) ds i * x(s) = Ax(s) + α(s) − µx(s) + βδ 0 a(s) x(0) = yand x x (·) the solution with initial data x ee ρt |L(x y (s), α(s), a(s)) − L(x x (s), α(s), a(s))|ds = If we look the explicit for of x x (·) and x y (·) as two-variables functions we see that they depend on the initial data only for s ∈ [0,s β ]. After this period they depends only on the control. So for s >s β x x (s) = x y (s) and so the previous integral is −ρt |L(x y (s), α(s), a(s)) − L(x x (s), , α(s), a(s))|ds ≤ −ρt C L |x y (s) − x x (s)| B ds ≤scsC L |x − y| B Proposition 3 . 1 . 31Let u ∈ C(H) be a locally B-Lipschitz function. Let ψ ∈ C 1 (H), and let x be a local maximum (or a local minimum) of u − ψ. Then ∇ψ(x) ∈ R(B 1/2 ) ⊆ D(A * ). . 3 . 3limit lim s→0 + I(s) exist if and only if there exist the limit lim s→0 + I1(s)+I(s) 2 and in such case they have the same value. But Given x ∈ D(A * ) there exists a real function O(s) such that O(s) s→0 − −− → 0 and such that for every control (α(·), a(·)) ∈ E × A we have that |x(s) − x| ≤ O(s) ee(e −µτ α(s − τ, r − βτ )dτ − x[−µτ a(s − τ, r − βτ )dτ − x[We have used that x ∈ D(A * ) ⊆ W 1,2 (0,s) so it is continuous and |x| L ∞ (0,s) −µs x[(r − βs) ∧ 0] − x[r]2 dr + 2s 2s e |µ| Γ 2 + + sβ e |µ| Γ + |x| L ∞ + se |µ| Λ e where y(·) is the solution ofẏ(s) = Ay(s) + βδ 0 a(s) y(0) = x(40)(that is our system when µ = 0 and α(·) = 0). x(·) satisfies the mild equation 2x(s) = e sA x − A (s−τ )A (α(τ ) − µx(τ ))dτ (41)The term e sA x − A s 0 e (s−τ )A (a(τ )ν)dτ is the mild solution of y(·) andx(s) − y(s) = s 0 e (s−τ )A (α(τ ) − µx(τ ))dτ. Now we come back to (39), we have x uniformly in the control (α(·), a(·)), and so ∇g(y(s)) s→0 − −− → ∇g(x) uniformly in the control and |∇g(y(s)) − ∇g(ξ(s))|s→0 − −− → 0 uniformly in the control. Moreover, in view of boundedness of the control and of the fact that x(s) s→0 − −− → x uniformly in the control (Lemma 3.3) we can prove that the term s 0 e (s−τ )A (α(τ ) − µx(τ ))dτ s H is bounded uniformly in the control and s ∈ (0,s] and we conclude that the second term of the (44) goes to zero uniformly in (α(·), a(·)) and that g(x(s)) − g(y(s)) s − s 0 ∇g(x), α(r) dr s − ∇g(x), −µx ≤ O(s) :0 There exists a constant C such that for every admissible control a(·) ∈ A with a(·) continuous and every s ∈(0(a(s − r/β) − x[r]) 2 dr s (48) in view of the fact that x ∈ D(A * ) ⊆ H 1 (0,s) and of the Lemma 3.2 the first part goes to zero. Moreover, since x ∈ H(0,s) ⊆ L ∞ (0,s), the second part is less or equal to a(·). Thus |P 3| s→0 − −− → 0 uniformly in a(·). Moreover x, y(s) − x 2 s ≤ | x, y(s) − x | s |x||y(s) − x| and so, from step 1 and Lemma 3.3, |P 2| s→0 − −− → 0 uniformly in a(·). s→0 − s→0−− → 0 and does not depend on the control. So we have to prove thatφ(y(s)) − φ(x) s − A * ∇φ(x), x − s 0 β δ 0 ∇φ(x), a(r) R dr s ≤ O(s) (s) isa point between x and y(s). In view of Lemma 3.3, |y(s)−x| s→0 − −− → 0 uniformly in the control, so |ξ(s) − x| s→0 − −− → 0 uniformly in a(·). By hypothesis ∇φ : H → D(A * ) and it is continuous (D(A * ) is endowed with the graph norm). Then |∇φ(ξ(s)) − ∇φ(x)| D(A * ) 0 ∇φ(x), a(r) R dr sThis complete the proof. Theorem 3 . 7 . 37Let L satisfy (L1) and (L2) let Γ and Λ be compact subsets of R. Then given a supersolution v of the HJB equation and a subsolution u we have u(x) ≤ v(x) f or every x ∈ HIn particular there exist at most one solution of the HJB equation the definition of ε (65) that implies ε ≤ε and then the (64). The claim follows. ifx,ỹ satisfy the hypothesis (66) of the claim we have C L |x −ỹ| B ≤ γ 32(1 + |µ|) − that we have already fixed ε in (65). From the choice of σ (71) follows that | Bp,x | ≤ in view of the continuity of the linear operator A * B : H → H that has norm A * B , we have| A * Bp,x | ≤ A * B σR ϑ ϑ→0 − −− → 0, | A * Bq,ȳ | ≤ B σR Bp, −µx − Bq, −µȳ − ϑ x, −µx − ϑ ȳ, −µȳ − − C L |x −ȳ| B − 2σ δ 0 • B Γ − Σ ϑ(|x| + |ȳ|) − 2 B σ Σ − βϑ Γ 2 ≤ 0(91)using (82), (83), (79), (80), (84), (86), (81), (85), (78) we obtain a.e. in r ∈ (0,s] while the part |x[r] − x[r − s]| goes uniformly to 0. In order to dominate the convergence we note that by Morrey's theorem ([Eva98] Theorem 4 page 266) every x ∈ H 1 (0,s) is 1/2-Holder then there exists a positive C such that for every s ∈ (0,s] and every r ∈ [s,s] we have|x[r] − x[r − s]| √ s ≤ C and then |x[r] − x[r − s]| 2 s ≤ C 2 this allows to dominate ψ s with the constant C 2 , use the Lebesgue theorem and obtain the claim. part (ii): I(s) def = s−s s (x[r + s] − x[r]) s x[r]dr = = s−s s (x[r + s]x[r]) s dr − s−2s 0 (x[r + s]x[r + s]) s dr = = − s−2s s (x[r + s] − x[r]) s x[r + s]dr + s−s s−2s (x[r + s]x[r]) s dr+ + s 0 − (x[r + s]) 2 s dr def = −I 1 (s) + I 2 (s) + I 3 (s) (35) See [Yos95] Proposition 2 page 273 for a proof of the equality (A * −λI) −1 = ((A−λI) −1 ) * . We have already written an explicit mild form of the solution in (8), the form we use here is different, indeed it is not explicit because the x appears also in the second term. The only difference between the two formula is the following: equation (8) is the equation we obtain if we include the term −µx in the generator of the semigroup, equation (41) is the form we obtain if we maintain the term −µx out of the generator of the semigroup. The two forms are equivalent. In the expression of y(·) the distributed control α(·) does not appear, so we will speak from now only of the boundary control a(·) In view of the fact that x is in H ⊆ D(Ã) ⊆ D(A * ) ′ , see[Fag02] for a proof. Estimates 3 (on ϑ): One can prove that, fixed ε we have ϑ |x| 2 ϑ→0 − −− → 0, ϑ |ȳ| 2 ϑ→0 − −− → 0 (76) (it is a quite standard fact, see for example[CL94]). So(where the last inequality follows from the(61)). In (65) we fixed ε, in (71) we chose σ as function of ϑ. Now we will fix ϑ. We begin takingWe know from(74)and(75)that if we choose ϑ small enough we have Moreover, in view of (77) we know that that if we choose ϑ small enough,x andȳ satisfy the hypothesis (66) of the Claim and then, from the (67), we have(where we uses that if 0 < a < 1 then a 2 < a, we recall that we took 0 < γ < 1). From the (67) in the same way we obtainand, from (70), We have finished our preliminary estimates and we come back to the main part of the proof of the theorem. The mapattains a maximum atx andattains a minimum atȳ. Note that thanks to Proposition 3.1x andȳ are in D(A * ). We can now use the definition of sub-and super-solution (page 9) to obtain thatWe now note that: Optimal strategies for the issuances of public debt securities. M Adamo, A L Amadori, M Bernaschi, C La Chioma, A Marigo, B Piccoli, S Sbaraglia, A Uboldi, D Vergni, P Fabbri, D Iacovoni, F Natale, S Scalera, L Spilotro, A Valletta, Int. J. Theor. Appl. Finance. 77M. Adamo, A.L. Amadori, M. Bernaschi, C. 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X J Li, J M Yong, Systems & Control: Foundations & Applications. Boston, MABirkhäuser Boston IncX.J. Li and J.M. Yong. Optimal control theory for infinite-dimensional systems. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA, 1995. Semigroups of linear operators and applications to partial differential equations. A Pazy, Applied Mathematical Sciences. 44Springer-VerlagVIII, 279 p.A. Pazy. Semigroups of linear operators and applications to partial dif- ferential equations. Applied Mathematical Sciences, 44. New York etc.: Springer-Verlag. VIII, 279 p., 1983. Polar decomposition of positive operators and a problem of Crandall and Lions. M Renardy, Appl. Anal. 573-4M. Renardy. Polar decomposition of positive operators and a problem of Crandall and Lions. Appl. Anal., 57(3-4):383-385, 1995. A class of Hamilton-Jacobi equations with unbounded coefficients in Hilbert spaces. K Shimano, Appl. Math. Optim. 451K. Shimano. 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{'abstract': 'The paper concerns with the infinite dimensional Hamilton-Jacobi-Bellman equation related to optimal control problem regulated by a linear transport equation with boundary control. A suitable viscosity solution approach is needed in view of the presence of the unbounded controlrelated term in the state equation in Hilbert setting. An existence-anduniqueness result is obtained.', 'arxivid': 'math/0606115', 'author': ['G Fabbri '], 'authoraffiliation': [], 'corpusid': 17684844, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 23964, 'n_tokens_neox': 21316, 'n_words': 11768, 'pdfsha': '66eee7fa8fead0a336c98c348488fb24ef762f2d', 'pdfurls': ['https://export.arxiv.org/pdf/math/0606115v2.pdf'], 'title': ['A viscosity solution approach to the infinite dimensional HJB equation related to boundary control problem in transport equation', 'A viscosity solution approach to the infinite dimensional HJB equation related to boundary control problem in transport equation'], 'venue': []}

arxiv

A Double-Layered Water Cherenkov Detector Array for Gamma-Ray Astronomy Samridha Kunwar Max-Planck-Institut für Kernphysik Saupfercheckweg 169117HeidelbergGermany Hazal Goksu Max-Planck-Institut für Kernphysik Saupfercheckweg 169117HeidelbergGermany Jim Hinton Max-Planck-Institut für Kernphysik Saupfercheckweg 169117HeidelbergGermany Andrew Smith Department of Physics University of Maryland 20742College ParkMarylandUSA IMAPP Radboud University Nijmegen NijmegenThe Netherlands Werner Hofmann Max-Planck-Institut für Kernphysik Saupfercheckweg 169117HeidelbergGermany Felix Werner Max-Planck-Institut für Kernphysik Saupfercheckweg 169117HeidelbergGermany A Double-Layered Water Cherenkov Detector Array for Gamma-Ray Astronomy Gamma-RayWater Cherenkov Detector (WCD) ArraySimulations Ground-level particle detection is now a well-established approach to TeV γ-ray astronomy. Detection of Cherenkov light produced in water-filled detection units is a proven and cost-effective method. Here we discuss the optimization of the units towards the future Southern Wide-field Gamma-ray Observatory (SWGO). In this context, we investigate a new type of configuration in which each water Cherenkov detector (WCD) unit in the array comprises two chambers with black or reflective walls and a single photomultiplier tube (PMT) in each chamber. We find that this is a cost effective approach that improves the performance of the WCD array with respect to current approaches. A shallow lower chamber with a PMT facing downwards enables muon tagging and the identification of hadron-induced air showers, which are the primary source of background in γ-ray astronomy. We investigate how γ/hadron separation power and achievable angular resolution depend on the geometry and wall reflectivity of the detector units in this configuration. We find that excellent angular resolution, background rejection power and low-energy response are achievable in this double-layer configuration, with the aid of reflective surfaces in both chambers.ray Observatory (SWGO)[3,4], is a project towards constructing a large detector array in the southern hemisphere with an advanced detector design and superior performance compared to both HAWC and LHAASO. The performance of WCD arrays is largely driven by high altitude [5], large array area and large fill-factor [6], but the particle detection thresholds of the individual WCD units and their response characteristics will influence the threshold energy and performance of the array (see e.g.[7]). This paper investigates the reference design for SWGO; a double-layered WCD array. Several other advanced designs such as a shallow WCD with 4 photomultiplier tubes (PMTs) are also being considered[8].The role of an individual detector unit in a groundparticle-based γ-ray instrument is to measure the local shower particle number or energy density, assign a local arrival time and ideally provide information for Introduction Ground-level particle-based detection of air showers is a rapidly developing approach to γ-ray astronomy at very high energies, with cosmic-ray protons and nuclei as the main source of background. High-density arrays maximise the number of detectable particles per air shower. Water Cherenkov detectors (WCD) are water-filled detection units that detect Cherenkov light produced by air showers reaching ground level and have been proven to be an effective way of achieving large array area, as demonstrated by HAWC (High-Altitude Water Cherenkov) [1] and LHAASO (Large High Altitude Air Shower Observatory) coverage [2]. The Southern Wide-field Gamma-γ/hadron separation. Muon identification, as a means of hadronic background rejection (see, e.g. [6]), can be implemented either using separate detector elements, such as the LHAASO buried muon detectors [2] or by discrimination within a standard WCD unit. Additionally, the topology of the shower amplitude distribution such as the charge distribution close to the shower core and the clustering characteristics of hits far from the core region also provide information for effective γ/hadron discrimination as demonstrated by Milagro [9]. The two-chamber concept, comprising a water volume separated into optically isolated top and bottom chambers, explored here, provides a (potentially) cost-effective general-purpose element with reasonable time and amplitude resolution as well as muon identification [10]. The principle is well established: most shower muons will pass through the entire detector element with only ionisation losses and produce Cherenkov light in both chambers. Electrons and γ-rays generate cascades that penetrate to the lower chamber only in the case of high energy initial particles, and in any case, the comparison of the signals in both sections allows discrimination from a through-going muon. As the lower chamber captures the cascade products of only high energy shower particles, it can also extend the detector dynamic range close to the shower core in the case of high energy primary γ-rays. The quantities to be optimised for the double-layered WCD unit are the dimensions of the two chambers, the reflective properties of the internal surfaces and the photosensors. The main current instruments of this type, HAWC and the LHAASO [2], employ non-reflective surfaces for detector water volumes made of single chambers, with a diameter-to-depth aspect ratio of 1.6-1.8. We first discuss particle detection efficiency, photon timing distributions and background trigger rates of individual double-layered WCD units and then discuss the performance of the WCD array in its entirety. After a brief overview of the double-layered WCD design (Section 2) and of the simulation tools used (Section 3), we explore the trade-offs associated with the reflectivity of the unit surfaces and the overall unit dimensions (Sections 4 and 5). In these sections we also compare our results from the double-layered WCDs with single-chamber WCDs similar to HAWC and LHAASO units. In a second step we then address the energy threshold, γ/hadron separation power and angular resolution of an ensemble of detector units (Section 6). Design Overview A candidate WCD design for SWGO with muon separation potential is a double-layered unit, illustrated in Fig. 1, that comprises the following building blocks: • Upper Chamber: A light-tight chamber with a lining that may be black or reflective and a single upwardfacing PMT. This chamber provides timing information and an estimate of total local particle energy per unit area. The upward-facing PMT ensures that nonreflected Cherenkov photons with the smallest time dispersion are detected first. • Lower Chamber: A light-tight chamber with a highly reflective lining, containing a single PMT facing downwards (for improved uniformity of response). This chamber enables muon tagging as only a small fraction of the higher energy photons and electrons at ground level can punch through into the lower chamber, while nearly all muons will pass through the entire detector unit as the mean linear stopping power (ρ −dE/dx ) for muons in water is only 2 MeV/cm [11]. The two PMTs are connected to each other with a PMT support such that one faces upwards in the upper chamber, and the other faces downwards in the lower chamber. The reference design uses a 3.8 m diameter tank, motivated by the relative ease of road transportation of pre-fabricated (e.g. rotomolded) units up to this size. Alternative designs with different diameters are possible and included in our studies for single tank simulations (see Section 4). The depths of the two chambers and their reflective properties are investigated in Sections 4 and 5. Overview of Simulation Tools To simulate air showers, we use the CORSIKA 7.7400 simulation package [12]. For the standard simulated event set, we select QGSJet-II.04 [13] for energies above 80 GeV. UrQMD 1.3.1 [14,15] treats the low energy hadronic interactions and for electromagnetic processes, we use the EGS4 electromagnetic model [16]. We use GEANT4 [17] within a simulation framework adapted from that of the HAWC collaboration, to simulate the WCD response to the secondary air shower particles from 20 m above the detector unit. This simulation framework, called HAWCSim, has been extensively used for studies related to HAWC and has been verified by the HAWC Collaboration [18,19,20]. The UNIFIED [21] model in GEANT4 is adapted to describe the reflectivity of materials such as Polypropylene (low reflectivity -10% at 450 nm, from now on referred to as 'black') as used by HAWC and those with a rough surface such as Tyvek (high reflectivity -92% at 450 nm, which from now on we refer to as 'white') [22] as used by the Pierre Auger Observatory [23]. The standard deviation of the distribution of the micro-facet orientations (taken for rough surfaces), is set as σ α = 0.17 rad for the Tyvek surface. A nominal water absorption length of 17 m at 400 nm is used in the simulations. This water absorption length is reasonable when compared with the numbers measured for the current detector arrays. The LHAASO collaboration purifies its water to get an absorption length longer than 15 m for around 400 nm [24], meanwhile studies performed by the HAWC collaboration show attenuation lengths varying between 5 m and 16 m for 405 nm [25]. We model the PMT in the simulations after the 8-inch HPK R5912-20 with a photo-cathode quantum efficiency of 20% at 450 nm [26]. The central 10" PMT for one of the configurations in our comparisons (configuration B from Section 4.1, similar to the HAWC main array tanks) is modeled after the 10" R7081 with a photo-cathode efficiency of 30% [26]. Upper Chamber Optimisation For γ-ray induced extensive air showers (EAS) at typical detector altitudes, the energy distributions of secondaries (in terms of number per log energy interval) varies with particle type (dN/dlogE). This number density peaks around ∼6 MeV for secondary photons, ∼20 MeV for electrons and 2-3 GeV for muons [6]. A high detection probability for particles of these energies is desirable for triggering and reconstructing showers, combined with a precise determination of particle arrival time. Below we discuss the impact of geometry and material properties on the performance of the upper chamber, concerning particle detection efficiency (Section 4.1) and arrival time measurement (Section 4.2). Chamber characteristics will also influence the background trigger rates of each WCD unit (Section 4.3), that in turn determine the trigger condition for the array and hence its energy threshold. We study single unit simulations in this section, however the performance of the array of WCD units as a whole is ultimately the guiding factor for upper chamber geometry and material optimization. Array simulations to optimize the upper chamber are discussed in Section 6. Particle Detection Efficiency The depth of the chamber must be at least several radiation lengths (∼4-5 X 0 ≈ 1.5-1.8 m) for calorimetric detection of electromagnetic shower particles. The opening angle of the Cherenkov cone in water is ∼41°, and the pair production length of high energy γ-rays is 9 7 X 0 , where X 0 is the radiation length, corresponding to ∼46 cm in water. The diameter of the chamber, along with the depth, determines the probability of collecting Cherenkov photons at the PMT, as prompt photons or after some number of diffuse reflections. As stated in Section 2, the diameter for the reference design is fixed at 3.8 m for logistical reasons, however in case of alternative designs for SWGO or any future WCD array, other diameters should be possible. Noting that prompt Cherenkov photons are important for timing (see Section 4.2) and the minimum depth criteria explained above, we investigate the optimum chamber aspect ratio. To illustrate the dependencies, we inject vertical 10 MeV γ-rays uniformly across the top surface of a double-layered WCD unit with an 8-inch PMT centered at the bottom. Fig. 2 shows the probability of detecting at least one Cherenkov photon, as a function of chamber radius and depth, for chambers with entirely white walls and with entirely black walls. In both cases, the detection probability decreases with increasing radius of the chamber. The optimum depth varies with radius, between around 1 m to 2 m depth for the white chamber of 1 m to 3 m radius and around 1.5 m to 2 m depth for the black chamber. The depth for optimum detection efficiency tends to be lower than the depth of at least 5X 0 required for good containment of electromagnetic shower particles (red dashed line in Fig. 2). Moreover, our studies on double-layered WCD arrays show that a depth of at least 2.5 m is required for efficient γ/hadron separation (see Section 6.2). The muon identification ability of the lower chamber is dependent on the upper chamber depth, since the upper chamber needs to efficiently shield the lower chamber from electromagnetic particles. This shielding effect is illustrated in Table 1, which shows the mean charge in the entirely white lower chamber, for different upper chamber depths (3.8 m diameter and fixed 0.7 m lower chamber depth) and γ-rays of ∼100 MeV and ∼1 GeV. As the upper chamber depth increases, the mean number of photoelectrons in the lower chamber is seen to decrease. Based on these discussions on geometry, we use a double-layered WCD with dimensions fixed at 3.8 m diameter and 2.5 m upper chamber depth for studies on material properties of linings. Multiple scattering of electrons pair produced by the incoming gammas, or in later generations of a cascade, result in Cherenkov light emission in any direction in the tank chamber. The reflectivity of the walls affects the particle detection efficiency of the detector unit, as reflective walls result in isotropisation of the emitted light. In Fig. 2, we already see a comparison of a chamber with entirely white walls and one with entirely black walls. As expected, the increased photon path length and the nearly isotropic scattering of photons in the white chamber increases the probability of light collection compared to a black chamber. The white chamber provides detection efficiencies of 70% and more. Entirely white walls will provide best efficiency, but will result in Cherenkov photon arrival times with long tails, as the decay time scale is governed by both the wall reflectivity and the water transparency; at least the latter may vary over time and between detectors. We consider combinations of white and black chamber walls in order to limit the number of 'late' photons. We compare all-white and all-black chambers with chambers that have a black top, black bottom, or black bases (i.e. top and bottom), for vertical 1 MeV to 1 GeV γ-rays injected uniformly across the top surface, as shown in Fig. 3. We see an improvement in the particle detection efficiency of at least partially white chambers over an entirely black chamber. Moreover, as the number of photo-electrons per unit energy scales like Area photo-cathode /Area chamber , we show in Tab. 2 the average number of photo-electrons produced per 20 MeV of γ-ray energy, for vertical 1 MeV to 100 MeV γ-rays injected uniformly across the top surface. Here a 20 MeV γ-ray produces ∼1 pe in an entirely black chamber, and produces three times as many photoelectrons in an entirely white chamber. The differences at such low energies for different material combinations arise due to the effective production of a diffuse glow at the top of the tank which a white top helps to deflect more towards the PMT. To efficiently detect ≤20 MeV γrays would require reflective materials and/or more photocathode efficiency/area. In order to illustrate the performance of the calorimetric measurement of shower particles, we show in Fig. 4 the relation between electromagnetic energy and the number of photo-electrons and in Fig. 5 the resolution σ E /E as a function of electromagnetic energy, for different combinations of wall materials. As before, the all-white chamber behaves best in terms of pe yield and energy resolution, the all-black chamber is worst, while the mixed-wall chambers are intermediate. For incident particle energies above ∼500 MeV, the lower chamber also becomes sensitive to electromagnetic cascades, as shown in the same figure. Although the lower chamber provides rather poor resolution (Fig. 5), it may help to extend the dynamic range of the system, which is important for detector units close to the shower core and/or in very high-energy showers. The impact of the reflectivity of the upper chamber material on the angular resolution and γ/hadron separation is In order to investigate particle detection efficiency as a function of energy, we inject vertical 5 MeV to 1 GeV γrays uniformly distributed across the top surface of these different configurations of WCDs. Fig. 6 shows a comparison of the detection probability. The upper chamber of the double-layered WCD (configurations A and D) has improved particle detection efficiency for γ-rays over both HAWC-like and LHAASO WCD-like designs, due to the reflective walls and -in case of the LHAASO-like design -the better ratio of PMT area to chamber surface. The HAWC and LHAASO WCD arrays employ non-reflective surfaces, which reduces the time spread of light reaching the photo-sensor (see Section 4.2), but results in a lessuniform response and reduced overall detection efficiency compared to reflective surfaces. Particle Arrival Time The arrival time distributions of Cherenkov photons at the PMT determine the time resolution that can (potentially) be achieved with a WCD unit; these time distributions and the background rates (Section 4.3) also place important requirements on the readout electronics of a unit, including the length of the signal trace, buffering requirements, and trigger design. The aspect ratio and the material choice of the WCD units determine these time distributions and hence influence time resolution, which in turn impacts the achievable angular resolution for showers as discussed in Section 6.3. In Figure 7 we examine the arrival time of Cherenkov photon distributions resulting from 80 MeV vertical γrays injected across the top surface of double-layered WCD units (3.8 m diameter and 2.5 m depth) with a combination of different material properties. The timing distributions show that a white walled chamber with one or both of the top and bottom surfaces black reduces the tail of the arrival distribution of photons by ∼30-40% at this energy. To study differences in time resolution, we examine the time of the first detected Cherenkov photon for each material combination (see Fig. 8). For black detectors, these first photons are dominated by direct light, hence the timing resolution (defined with respect to a particle entering the upper chamber) is mostly defined by the width of the detector. For (partially) white detectors, the timing resolution depends on the amount of Cherenkov light produced: low energy particles have a higher chance to be detected due to reflected light alone. While this adds more signals available to the reconstruction, these small signals show a worse time resolution. All material combinations show Finally, we look at the four different WCD designs from Section 4.1. Fig. 10 shows Cherenkov photon arrival time distributions, resulting from 80 MeV vertical γ-rays. As expected, reflective materials result in longer tails in the timing distribution of both double-layered designs compared to the HAWC and LHAASO-like designs which use entirely black walls. Background WCD Trigger Rates Air shower arrays typically trigger based on a coincidence of triggers from individual units. The required minimal number of coincident units determines the energy threshold of the array, and depends both on the trigger rate of individual units and on the coincidence time window. The background trigger rate of WCD units will depend on their detection probability for different types of background particles, and hence on their geometry and wall materials; we expect chambers with some white surfaces to experience increased background rates with respect to entirely black chambers. To investigate these rates, we use EXPACS/PARMA [27] to calculate terrestrial cosmic-ray fluxes and angular distributions of neutrons, protons, electrons, positrons, muons, anti-muons, and photons with energies ranging from 1-10 6 MeV at an altitude of 4900 m a.s.l. and location 13°51' 24" S, 71°1' 30" W (a location that well characterizes the magnetic effect at all the sites under consideration for SWGO, see [28] for an overview). To calculate the trigger rates, we inject particles aimed at a hemisphere of 12 m radius centered at the detector under test so that it can hit the detector from all sides (i.e 0 to 90 degree in zenith angle with a sin · cos distribution). Such a radius includes the edges of the surrounding detectors to account for scattered particles and shielding effects. In order to include shielding, a mini-array made up of 20 tanks with a separation of 0.12 m between each tank was used. The expected single-pe trigger rates in the upper chamber of double-layered WCD units (3.8 m diameter and 2.5 m depth) composed of different materials are shown in Fig. 11, broken down into contributions from different background particle species. Reflective walls increase the detection probability of low energy particles, resulting in an increased background rate. Nevertheless, we expect an array of white-walled detectors to provide the lowest energy threshold, once the coincidence level is adjusted to obtain a negligible rate of array-level noise triggers. Note that these background trigger rates do not include the single-pe thermal noise rate from the PMT itself, or possible contributions from radioactive decays in the water. Contributions from e ± , µ ± , γ, p + and n are shown. These EXPACS/PARMA-based rates have been verified with CORSIKA air shower simulations to generate ground-level particle rates. Looking at different altitudes, the rates at 4900 m are approximately 1.5 times the rates at an altitude of 4100 m (∼32 kHz and ∼21 kHz), at the same latitude and longitude, of an entirely white upper chamber of a double-layered WCD unit (3.8 m diameter and 2.5 m depth). The same method was used to predict the noise trigger rates of already operating HAWC tanks, which agree with the measurements reported by the HAWC collaboration. Trigger rates were predicted for the central 10" PMT in a single tank in the HAWC central array and a single HAWC outrigger tank, at the actual HAWC altitude of 4100 m. Along with an afterpulsing rate prediction, the rates amount to be ∼36 kHz for a central PMT in a tank of the main HAWC array and ∼4.5 kHz for a tank that is part of the outrigger array. Contributions from dark rates that varies around 1-3 kHz with temperature, voltage and PMT dependence [26] are not included in the method. The afterpulsing rates were estimated with a first-order calculation that assumes an afterpulse probability of 0.02% for the 10" PMTs, as reported for these PMTs for a 2013 study for the Double Chooz experiment [29]. The prediction can vary, as PMTs can have higher afterpulsing probabilities of 10% if they are degraded, which would mean that the afterpulse rate is higher than the predicted amount for some of the tanks. Indeed, the HAWC data shows a large spread. The central 10" PMTs of HAWC were reported to have a hit rate of 40-50 kHz [1]. The outriggers of HAWC were reported to have rates around 4-8 kHz [30], which agrees with our predictions with this method. Lower Chamber Optimisation In the dual-layer approach, muons are identified based on the signal from the lower chamber, or, more generally, by comparing the signals in the upper and lower chambers. Since the lower chamber is not used for timing, we assume white-walled lower chambers for optimal light yield. The depth of the lower chamber, combined with the photosensor area, determines the light yield. While in principle a few detected photons per muon are sufficient to tag muons with reasonable efficiency, our studies of algorithms for muon identification in showers (Section 6.2) indicate that larger signals are desirable to reduce the number of misidentified muons. Here, we discuss how the muon signal depends on the geometry of the lower chamber. We study lower chambers with white walls and with a PMT placed at the centre of the top of the chamber facing downwards. For best photon collection efficiency and hence minimal chamber depth, the PMT base protrudes into the upper chamber so that only the active cathode area of the PMT is visible in the lower chamber (see Appendix A). The lower chamber has the same diameter as the upper to allow partitioning of a single cylindrical detector unit. A uniform response for muons above ∼600 MeV is expected since muons lose ∼2 MeV/cm in water (see lower chamber in Fig. 13). Figure 12 shows that for depths less than ∼0.5 m, the muon signal has a much larger spread compared to depths greater or equal to ∼0.5 m. Moreover, for depths greater than ∼0.5 m, the muon signal is reliably >10 photoelectrons for >2 GeV muons, and the signal increases roughly proportionally to the track length of the impinging particle as expected. It is seen that depths greater or equal to ∼0.5 m would give reliable muon signals, and since a lower chamber that is smaller in depth would minimize costs (although most of the cost is driven by the upper chamber), the studies here suggest that the depth for the lower chamber should be at least ∼0.5 m. To explore the separation power of such a unit, we compare vertical γ-rays with energies from 1 MeV to 2 GeV in an entirely white WCD unit (3.8 m diameter, 2.5 m upper and 0.5 m lower depth) with vertical µ − with energies from 100 MeV to 2 GeV in an identical WCD unit (see Fig. 13). Muon identification is possible as the number of photo-electrons detected in the two chambers is constant above some energy threshold and remains fairly stable until very high energies, where effects such as bremsstrahlung will need to be considered. Additionally, the peak of the muon energy distribution in air showers is around 2-3 GeV [6]. The ratio of photo-electrons in the two chambers will enable muon selection on a tank-bytank basis. Other shower information such as the location of the shower core relative to the detector unit and the number of units hit is also beneficial for muon identification. Depth and Reflectivity of the Lower Chamber Array Simulations To relate the performance of individual WCD units to the performance of an array as a whole and in particular to the achievable angular resolution and γ/hadron separation power, we carried out simulations based on the SWGO reference design [31]. We continue using the HAWCSim tool that has been validated by the HAWC collaboration, as explained in Section 3. We simulate an inner array with a high fill factor of ∼80% spread over ≈80,000 m 2 and a sparser outer array with a fill factor of 8% spread over ≈220,000 m 2 , placed at high elevation (4900 m.a.s.l). The dense inner array serves to enhance the sensitivity for low to mid energy γ-rays and also to increase the muon sensitive area. The sparser outer array is designed for a large collection area at the highest energies, but will not be used in the analysis presented here. Comparisons with the currently operating HAWC and LHAASO arrays are also beyond the scope of this paper, since the layouts, altitudes and the algorithms used in the analysis are different for these WCD arrays. Estimating Effective Area To be able to compare the γ-ray induced air shower effective area for different material combinations, we first simulate proton (background) induced showers with a spectral index of -2, an energy range from 0.001-30 TeV and zenith angle from 0 • to 60 • , at an altitude of 4900 m.a.s.l. and shower core scattered over a radius of 2.5 km. These simulations are then weighted by the cosmic-ray flux [32] to reproduce the correct spectrum. As the core range is large compared to the detector size, we can further integrate to obtain approximate array trigger rates at different hit thresholds (see Fig. 14). We then derive the γ-ray effective area as a function of energy, as shown in Fig.15. We first simulate γ-ray induced air showers with a spectral index of -2 for an energy range of 0.1-5 TeV, zenith angle from 0 • to 60 • , at an altitude of 4900 m.a.s.l. and shower core scattered over a radius of 2.5 km with respect to the array center. From the proton simulations, we derive the required threshold in the number of array hits, such that the array trigger rate is 100 kHz (ensuring at least a few 10s of tanks trigger). With this threshold, we use the γ-ray simulations to derive the γ-ray effective area as a function of energy (see Fig. 15). The simulations show that double-layered WCDs (3.8 m diameter, 2.5 m upper depth, and 8" PMT) with some reflective surfaces for the upper chamber provide slightly higher effective areas at low energies, compared to entirely black upper chambers. It should be noted however that more statistics, including hit timing information and large core range would be necessary to obtain realistic rates. Identifying Hadronic Showers The double-layered WCD design provides an alternate way of vetoing hadron-induced showers compared to single-chambered WCD designs. The critical parameter that we will explore in the following is the depth of the upper chamber, that determines the level of electromagnetic punch-through into the lower chamber (see Section 4 for the influence of upper chamber depth on single WCD unit performance). For this purpose we implement a template-based maximum log-likelihood method to discriminate between γray and hadron-induced air showers, similar to templatebased reconstruction methods by the HAWC Collaboration [33]. Given a known core location and air shower direction, we generate templates for the charge in the upper and lower chambers in an array of double-layered WCDs, similar to the distribution shown in Fig. 16, for vertical air showers. We generate separate templates for µ ± and for other charged particles. Next, we test simulation events, where we assign a likelihood value based on the charge deposited in the two chambers for each secondary particle impinging on an individual tank. Subsequently, we calculate a Likelihood Ratio (LR) to tag those particles with a LR > 0 as candidate muons and compare them to the Monte-Carlo truth. This muon tagging ability directly translates into γ/hadron separation efficiency, given the relative abundance of muons in hadron-initiated showers. Using the number of detector units hit as a proxy for the shower's energy, we can subsequently distinguish γ-ray and hadron-initiated showers from the difference in the number of identified muons, as shown in Fig. 17 for vertical 2 TeV γ-ray and 5 TeV proton induced showers. The proton shower energy is chosen such the the average number of WCD hits is very similar for the two sets of events. To reduce misidentification due to punch-through of electromagnetic particles from the upper to lower chamber close to the shower core, we exclude WCDs within 20 m from the core. From the known core distance and air shower direction, we generate templates of charge in the upper and lower chambers in an array of entirely white doublelayered WCDs (3.8 m diameter and 2.5 m upper chamber depth and 0.7 m lower chamber depth 2 ) for vertical 5 TeV proton-induced air showers. After tagging different particle species for 2 TeV γ-ray and 5 TeV proton induced showers, we find a γ/hadron separation efficiency while varying the upper chamber depth. The signal-to-noise-ratio (SNR) is γ / √ proton , where is efficiency. For upper chamber depths greater than 2.5 m, the SNR plateaus for air showers at 0 • zenith angle. Inclined showers are also expected to plateau, but as individual secondary particles from inclined showers can penetrate through the sides and multiple tanks, there is a decrease in overall SNR. Including the neighbouring tanks would mitigate the decrease in SNR for such inclined showers. Increasing detector unit radii or increasing the fill factor could also be beneficial for shielding from side-penetrating particles; one could also consider filling up the space between tanks with ground material as absorber. However, a complete study of inclined showers is beyond the scope of this paper. We calculate the SNR-to-cost ratio by attempting to account for the cost scalings for tanks with depth, volume of water, and the fixed cost of photosensors. We find that for vertical showers the SNR-to-cost ratio peaks at ∼2.5 m, and this depth is relatively insensitive to the cost assumptions made (see Fig. 18). Based on these findings, the upper chamber depth of 2.5 m has been used in the comparisons of Section 4 and while investigating different chamber materials. To show the effect of various materials in the upper chamber for double-layered WCDs (3.8 m diameter, 2.5 m upper depth, 0.5 m lower depth, with 8-inch PMTs), we again implement a template-based maximum likelihood method to discriminate between γ-ray and hadron induced air showers. We simulate an ensemble of vertical γ-ray and proton initiated showers following an E −2 spectrum up to 100 TeV energy, with shower core location at the centre of the array. Fig. 19 shows the derived γ/hadron separation efficiency (excluding a 40 m region around the core) for different material combinations. We see that a background rejection power of ∼10 3 can be achieved at reconstructed energies (a simple model taking into account only the total number of photo-electrons seen in all of the upper chambers of the array) of a few TeV (2-7 TeV) with high γ-ray efficiency. The separation power improves in all cases with one or more white surface(s). From Fig. 19 we find a background efficiency of 3 × 10 −4 keeping good gamma efficiency for entirely white double-layered WCDs (3.8 m diameter, 2.5 m upper depth, 0.5 m lower depth, with 8-inch PMTs). At similar energies this is at least a factor ∼ 30 improvement in rejection power compared to HAWC [1] and LHAASO [34]. This improvement factor is extremely promising despite the somewhat idealised nature of the simulations (showers only from zenith, at the array centre, and neglecting uncorrelated noise hits) as the performance in the final array would likely be significantly improved by the inclusion of additional parameters or more sophisticated treatment of the muon-based rejection. Angular Reconstruction To reconstruct the direction of the air shower, we use the time of the first photon in each upper WCD without applying transition time spread in the PMT, or electronics time resolution. We first obtain a guess of the direction from a least-squares algorithm using the time difference of arrival of the photons between detector units. We obtain the final best-fit direction via a likelihood fit to the arrival times in each hit detector unit, described in [35]. Parameterized distribution functions describe the distribution in energy, core distance and arrival time of shower particles at ground. We obtain PDFs for the time distribution by fitting Landau distributions to the arrival times binned as function of distance to the shower core and total charge, to obtain mean and width parameters for each bin [35]. A shower arrival time, t 0 , is defined as the absolute arrival time of the first electromagnetic particle in the plane perpendicular to the shower axis. To separate the effects of the direction (i.e. timing) fit and the core position fit, and to minimise additional complexity due to array edges, we generate vertical γ-rays impacting at the centre of the array, and assume the core position to be known (as in practice the core location can be very precisely determined for well-contained events). We perform a three-parameter (time, offset and direction) likelihood fit (using MINUIT [36]) to obtain the reconstructed shower directions and hence determine the angular resolution (see Fig. 20). As expected, the angular resolution improves with increasing energy. Differences between different wall options are very modest. The addi-tional late-arriving Cherenkov photons that are a feature of white-walled chambers do not deteriorate the angular resolution as they are properly accounted for in the likelihood functions used in the fit; these likelihood functions are adapted for each configuration. Whilst this result represents a very idealised case, with showers from zenith landing at the centre of the array, with perfect timing resolution and no noise, it nonetheless illustrates that such arrays can potentially achieve an angular resolution much better than existing instruments of this type, regardless of the surface reflectivity of individual detector units [2,1]. Comparisons with different fill factors and reconstruction algorithms are beyond the scope of this paper. Summary & Outlook We have studied a double-layered WCD array, aiming to improve both the energy threshold and the γ/hadron separation in comparison to LHAASO and HAWC-like designs. Each detector unit in the array comprises two chambers with black or diffuse reflective wall linings and a PMT in each chamber. The upper PMT facing upwards is intended for timing and energy determination, while the lower PMT facing downwards will enable muon tagging, and provide the primary mechanism for γ/hadron separation. The double-layer design is promising in terms of background rejection power, achieving a rejection power of ∼10 3 at energies of a few TeV, with high γ-ray efficiency. Investigating different options for chamber aspect ratio (depth-to-diameter) and reflectivity shows that a compact and highly reflective upper chamber lowers the energy threshold for the impinging particles and, in conjunction with a lower chamber, benefits γ/hadron separation. There is an increase in the tail in the timing distributions with one or more white surface in the upper chamber, however, these late-arriving Cherenkov photons do not deteriorate the angular resolution of the array when suitable reconstruction methods are employed. A partially reflective upper chamber is strongly motivated by these benefits in threshold and background rejection, with no negative impact on angular resolution. Although the final optimization of a unit detector for SWGO requires finalized analysis algorithms and life cycle cost estimates that are beyond the scope of this paper, these studies show that a ∼3 m deep, ∼4 m diameter double-layered tank with some reflective material in the upper chamber is a promising option in terms of performance and cost-effectiveness. For the reference SWGO design where the diameter is fixed to 3.8 m, an upper chamber depth of ∼2.5 m maximizes the SNR-to-cost ratio and provides sufficient shielding for the lower section, meanwhile a lower chamber depth of ∼0.5 m is the minimum required for reliable muon signals. In the idealized case of vertical showers at the centre of the array, we obtained an angular resolution of several arc-minutes at 10 TeV and found that background rejection power of ∼ 10 3 is achievable. Our studies show that a densely packed (≥ 80% fill factor), high altitude (∼ 5000 m.a.s.l) array of double-layered WCDs has the potential to achieve superior angular resolution, reduce energy threshold and improve γ/hadron separation over existing WCD arrays. Acknowledgements We thank our colleagues within the SWGO collaboration for many helpful discussions and the use of the common shared software framework. We thank the HAWC collaboration for providing the AERIE software used here. Appendix A. Note on positioning of the lower chamber PMT To minimise the depth of the lower chamber, it suffices to only have the active cathode area of the PMT visible in the lower chamber. We optimise the PMT positioning in the lower chamber by pushing the base of the downward facing PMT in the lower chamber up into the upper chamber. To investigate the performance of this PMT positioning, we inject vertical 2 GeV muons uniformly across the top surface of a double-layered WCD unit (3.8 m diameter, 2.5 m upper and 0.5 m lower depth) with an upward facing 8-inch PMT centered at the bottom of the upper chamber and a downward facing 8-inch PMT centered at the top of the entirely white lower chamber. We find that this PMT adjustment with the base of the downward facing PMT protruding (≈ 10cm) into the upper chamber results in a x1.6 fold increase in the mean light yield compared to the entire PMT plus base in the lower chamber. Figure 1 : 1Cylindrical double-layered WCD design comprising an upper chamber of 3.8 m diameter and 2.5 m depth, here with white walls and black bases (top and bottom) and an entirely white lower chamber of 0.5 m depth. The upper chamber contains an 8-inch PMT facing upwards, and the lower chamber has an 8-inch PMT facing downwards. For illustration, a simulated muon (green track) is shown that passes through both units and produces Cherenkov photons (red tracks). Figure 2 : 2Probability of detection of one or more photo-electrons as a function of upper chamber radius and depth, for vertical 10 MeV γ-rays injected across the top surface of the chamber, for a chamber with entirely white walls (left) and a chamber with entirely black walls (right).A dashed horizontal red line shows the depth corresponding to ≈5 radiation lengths. The reference design for SWGO has a radius of 1.91 m (3.8 m in diameter). Figure 3 :: 3Probability of detecting one or more photo-electrons for vertical γ-rays injected across the top of the upper chamber of the doublelayered WCD (3.8 m diameter and 2.5 m depth) with an 8" PMT. Different curves correspond to different choices for the reflectivity of different internal surfaces of the chamber. The average number of photo-electrons produced per 20 MeV γ-ray energy, for vertical 1 MeV to 100 MeV γ-rays injected uniformly across the top surface in the upper chamber of a double-layered WCD unit (3.8 m diameter and 2.5 m depth) and an 8" PMT with different materials. Figure 4 : 4The average number of photo-electrons as a function of electromagnetic energy for upper chambers of double-layered WCD units (3.8 m diameter and 2.5 m depth) with different materials. Also shown is the corresponding value for the lower chamber (3.8 m diameter and 0.5 m depth) as a dashed black line. Figure 5 : 5Fractional rms energy resolution as a function of incident electromagnetic energy for the upper chamber of double-layered WCD units (3.8 m diameter and 2.5 m depth) with different internal surface reflectivities. The dashed black line shows the corresponding value for the lower chamber (3.8 m diameter and 0.5 m depth). discussed in later sections (cf. 6.2, 6.3), and is found to be modest.Furthermore, we compare the performance of the double-layered WCDs with the performance of singlechamber WCDs similar to HAWC and LHAASO units that are currently in use. The four configurations are listed below.(A) A white double-layered WCD (3.double-layered WCD unit with an alternative geometry (3.4 m diameter and 3.0 m depth) with a black top and an 8-inch PMT Configuration D is a deeper and thinner version of configuration A, both are double-layered WCDs. The other two are replications of the WCD units of the existing widefield observatories. The central PMT for configuration D is a higher quantum efficiency PMT similar to HAWC[1]. Figure 6 : 6Detection probability for vertical 5 MeV to 1 GeV γ-rays injected across the top surface of different WCD designs: (A) a cylindrical white double-layered WCD (3.8 m diameter and 2.5 m depth) with a black top and an 8-inch PMT; (B) a cylindrical single-layered HAWClike unit (7.3 m diameter and 4 m depth) with black walls, a central 10" PMT and three 8-inch PMTs; (C) a LHAASO WCD-like entirely black unit (5 m × 5 m square, 4.5 m depth) with an open top and an 8-inch PMT; and (D) a white cylindrical double-layered WCD unit (3.4 m diameter and 3.0 m depth) with a black top and an 8-inch PMT. Figure 7 : 7Arrival time distribution of photons arriving at the PMT of the upper chamber of a double-layered WCD unit (3.8 m diameter and 2.5 m depth, similar to configuration A) with different materials, for vertical γ-rays at 80 MeV. We note that the shape of these distributions depends only very weakly on the energy of the incident particle. similar time resolution of the first photon above for particle energies above ∼ 200 MeV because the probability to detect direct light becomes close to unity. Figure 8 : 8Time resolution (rms) resulting from the arrival times of the first photo-electron as a function of incident γ-ray energy, for the upper chamber of a double-layered WCD unit (3.8 m diameter and 2.5 m depth) with different material properties and for vertical γ-rays.Moreover,Fig. 9illustrates the impact of changing water quality/absorption length on the arrival time distribution, for an upper chamber with white walls and black top and bottom. There is a very modest impact on the time distribution provided an absorption length of >10 m can be maintained. A nominal absorption length of 17 m at 400 nm is used in the simulations. Figure 9 :Figure 10 : 910Time distribution of photons arriving at PMT of the upper chamber of a double-layered WCD unit (3.8 m diameter, 2.5 m depth, and white walls with black bases) with water of varying absorption length, for vertical 120 MeV γ-rays injected across the top surface. The mean arrival time in each case is indicated in the figure legend. Time distribution of photons arriving at the PMT for the four different WCD configurations from Section 4.1: (A) a cylindrical white double-layered WCD (3.8 m diameter and 2.5 m depth) with a black top and an 8-inch PMT; (B) a cylindrical single-layered HAWC-like unit (7.3 m diameter and 4 m depth) with black walls, a central 10" PMT and three 8-inch PMTs; (C) a LHAASO WCD-like entirely black unit (5 m × 5 m square, 4.5 m depth) with an open top and an 8-inch PMT; and (D) a white cylindrical double-layered WCD unit (3.4 m diameter and 3.0 m depth) with a black top and an 8-inch PMT. The photons are initiated by vertical γ-rays at 80 MeV. Only the the timing distributions from the central 10" PMT in the HAWC-like design and the PMT in the upper chamber of the double-layered designs are shown. Figure 11 : 11Expected trigger rates at single pe threshold from incident particles for the upper chamber of a double-layered WCD unit (3.8 m diameter and 2.5 m depth) comprising an 8-inch PMT and various combination of materials (b = black, w = white), at an altitude of 4900 m. Figure 12 : 12The number of photo-electrons produced by vertical 2 GeV muons in a WCD unit with an entirely white lower chamber, for various different chamber depths. Figure 13 : 13The number of photo-electrons in a WCD unit with an entirely white upper (3.8 m diameter and 2.5 m depth) and lower (3.8 m diameter and 0.5 m depth) chamber. On the left, for µ − with energy ranging from 100 MeV to 2 GeV; on the right, for γ-rays with energy ranging from 1 MeV to 2 GeV. Dashed lines show 80 pe and 1 GeV in the upper plots and 32 pe and 1 GeV in the lower plots. Figure 14 : 14Estimated cosmic ray array-level trigger rate for an array of double-layered WCDs (3.8 m diameter, 2.5 m upper depth, and 8inch PMT) with different surface reflectivity choices, as a function of the WCD multiplicity (n hits ) adopted for the array trigger decision. Figure 15 : 15Effective collection area for γ-ray initiated air showers for an array of double-layered WCDs (3.8 m diameter, 2.5 m upper depth, and 8-inch PMT) with different surface properties for the upper chamber. Figure 16 : 16Distribution of signals in the upper and lower chambers of entirely white double-layered WCDs from vertical proton shower simulations, excluding the region within 20 m of the shower core. These distributions are used as templates for the likelihood-based background separation. Figure 17 : 17Probability of detecting muons in detector units more than 20 m from the shower, for γ-rayand hadron-initiated showers detected with entirely white double-layered WCDs in the dense inner array of the SWGO Reference Configuration. Showers are simulated with a core located at the centre of the array. The black dashed line indicates the cut value giving the highest signal to noise ratio ( γ / √ proton ) is maximum. Figure 18 : 18Ratio of γ/hadron separation SNR to nominal cost in an array with entirely white double-layered WCDs with varying upper chamber depth, given different assumptions of the relative cost of water to photosensors and tanks, assuming a fixed number of tanks. Each WCD unit has a diameter of 3.8 m and a fixed lower chamber depth of 0.7 m. 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{'abstract': 'Ground-level particle detection is now a well-established approach to TeV γ-ray astronomy. Detection of Cherenkov light produced in water-filled detection units is a proven and cost-effective method. Here we discuss the optimization of the units towards the future Southern Wide-field Gamma-ray Observatory (SWGO). In this context, we investigate a new type of configuration in which each water Cherenkov detector (WCD) unit in the array comprises two chambers with black or reflective walls and a single photomultiplier tube (PMT) in each chamber. We find that this is a cost effective approach that improves the performance of the WCD array with respect to current approaches. A shallow lower chamber with a PMT facing downwards enables muon tagging and the identification of hadron-induced air showers, which are the primary source of background in γ-ray astronomy. We investigate how γ/hadron separation power and achievable angular resolution depend on the geometry and wall reflectivity of the detector units in this configuration. We find that excellent angular resolution, background rejection power and low-energy response are achievable in this double-layer configuration, with the aid of reflective surfaces in both chambers.ray Observatory (SWGO)[3,4], is a project towards constructing a large detector array in the southern hemisphere with an advanced detector design and superior performance compared to both HAWC and LHAASO. The performance of WCD arrays is largely driven by high altitude [5], large array area and large fill-factor [6], but the particle detection thresholds of the individual WCD units and their response characteristics will influence the threshold energy and performance of the array (see e.g.[7]). This paper investigates the reference design for SWGO; a double-layered WCD array. Several other advanced designs such as a shallow WCD with 4 photomultiplier tubes (PMTs) are also being considered[8].The role of an individual detector unit in a groundparticle-based γ-ray instrument is to measure the local shower particle number or energy density, assign a local arrival time and ideally provide information for', 'arxivid': '2209.09305', 'author': ['Samridha Kunwar \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany\n', 'Hazal Goksu \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany\n', 'Jim Hinton \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany\n', 'Andrew Smith \nDepartment of Physics\nUniversity of Maryland\n20742College ParkMarylandUSA\n\nIMAPP\nRadboud University Nijmegen\nNijmegenThe Netherlands\n', 'Werner Hofmann \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany\n', 'Felix Werner \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany\n'], 'authoraffiliation': ['Max-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany', 'Max-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany', 'Max-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany', 'Department of Physics\nUniversity of Maryland\n20742College ParkMarylandUSA', 'IMAPP\nRadboud University Nijmegen\nNijmegenThe Netherlands', 'Max-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany', 'Max-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany'], 'corpusid': 252383569, 'doi': '10.1016/j.nima.2023.168138', 'github_urls': [], 'n_tokens_mistral': 19140, 'n_tokens_neox': 16257, 'n_words': 9999, 'pdfsha': '03c85ec7b64600940de9af5e5b7766efe07e4779', 'pdfurls': ['https://export.arxiv.org/pdf/2209.09305v2.pdf'], 'title': ['A Double-Layered Water Cherenkov Detector Array for Gamma-Ray Astronomy', 'A Double-Layered Water Cherenkov Detector Array for Gamma-Ray Astronomy'], 'venue': []}

arxiv

Mining Maximal Cliques from an Uncertain Graph 9 May 2014 Arko Provo Mukherjee Pan Xu Srikanta Tirthapura Mining Maximal Cliques from an Uncertain Graph 9 May 2014 We consider mining dense substructures (maximal cliques) from an uncertain graph, which is a probability distribution on a set of deterministic graphs. For parameter 0 < α < 1, we present a precise definition of an α-maximal clique in an uncertain graph. We present matching upper and lower bounds on the number of α-maximal cliques possible within an uncertain graph. We present an algorithm to enumerate α-maximal cliques in an uncertain graph whose worst-case runtime is near-optimal, and an experimental evaluation showing the practical utility of the algorithm. * Introduction Large datasets often contain information that is uncertain in nature. For example, given people A and B, a question of the form "does A know B" may not be definitively answered using available information. In such a case, it is a common solution to use probability to quantify our confidence in this relation, and say that the relation exists with a probability of p, for some value p determined from the available information. We consider graphs where there can be uncertainty in the presence of each edge in the graph. Such uncertain graphs have often been used in modeling real data, for example, communication networks [1,2,3], social networks [4,5,6,7,8,9], protein interaction networks [10,11,12], regulatory networks in biological systems [13] and in mining information from biological databases [14]. Identification of dense substructures within a graph is a fundamental task, with applications in clustering and community detection in social and biological networks [15], the study of the co-expression of genes under stress [16], integrating different types of genome mapping data [17], and other applications in bioinformatics and data mining. Perhaps the most elementary dense substructure within a graph, also the most commonly used, is a maximal clique. While the notion of a maximal clique is well understood in a deterministic graph, it is not well defined or understood within an uncertain graph. While defining such a structure, we consider the question "What vertex sets within a graph are highly likely to be completely connected subgraphs?" Identification of such vertex sets is a first step in many graph mining tasks, including social network analysis [15,18,19], mining association rules [20], analysis of email networks [21] and in bioinformatics [22,23,24]. In this work, we consider the analog of a maximal clique in an uncertain graph. We consider a model of an uncertain graph where each edge is assigned a probability of existence, and different edges are mutually independent. An uncertain graph is a triple G = (V, E, p), where V is a set of vertices, E ⊆ V × V is a set of edges, and p : E → (0, 1] is a probability function that assigns a probability p(e) to each edge e ∈ E. Note that different edges can be assigned different probabilities. Our Contributions First, we present a precise definition of a maximal clique in an uncertain graph, leading to the notion of an α-maximal clique, for parameter 0 < α ≤ 1. A set of vertices U ⊆ V in an uncertain graph (V, E, p) is an α-maximal clique if U is a clique with probability at least α, and there does not exist U ′ such that U ⊂ U ′ and U ′ is a clique with probability α. When α = 1, the above definition reduces to the well understood notion of a maximal clique in a deterministic graph. Number of Maximal Cliques. We consider a basic question on maximal cliques in an uncertain graph: how many α-maximal cliques can be present within an uncertain graph? For deterministic graphs, this question was first considered by Moon and Moser [25] in 1965, who presented matching upper and lower bounds for the largest number of maximal cliques within a graph; on a graph with n vertices, the largest possible number of maximal cliques is 3 n 3 1 . For the case of uncertain graphs, we present the first matching upper and lower bounds for the largest number of α-maximal cliques in a graph on n vertices. We show that for any 0 < α < 1, the maximum number of α-maximal cliques possible in an uncertain graph is n ⌊n/2⌋ , i.e. there is an uncertain graph on n vertices with n ⌊n/2⌋ uncertain maximal cliques and no uncertain graph on n vertices can have more than n ⌊n/2⌋ α-maximal cliques. Algorithm for Enumerating Maximal Cliques. We present a novel algorithm, MULE (Maximal Uncertain cLique Enumeration), for enumerating all α-maximal cliques within an uncertain graph. MULE is based on a depth-first-search of the graph, combined with optimizations for limiting exploration of the search space, and a fast way to check for maximality based on an incremental computation of clique probabilities. We present a theoretical analysis showing that the worst-case runtime of MULE is O(n2 n ), where n is the number of vertices. This is nearly the best possible dependence on n, since our analysis of the number of maximal cliques shows that the size of the output can be as much as O( √ n2 n ). Note that such worst-case behavior occurs only in graphs that are very dense; for typical graphs, we can expect the runtime of MULE to be far better, as we show in our experimental evaluation. We also present an extension of MULE to efficiently enumerate only large maximal cliques. We present an experimental evaluation of MULE using synthetic as well as real-world graphs. Our evaluation shows that the algorithm is practical and can enumerate maximal cliques in an uncertain graph with tens of thousands of vertices, more than hundred thousand edges and more than two million α-maximal cliques. Interestingly, the observed runtime of this algorithm is proportional to the size of the output. Related Work There has been recent work in the database community on various problem on uncertain graphs, including probability-threshold based shortest paths [26], nearest neighbors [27], enumerating frequent and reliable subgraphs [28,29,30,31,32,33,34], distance-constrained reachability [35]. Note that our problem is different from the problems mentioned above. In particular, the problem of finding reliable subgraphs is one of finding subgraphs that are connected with a high probability. However, these individual subgraphs may be sparse. In contrast, we are interested in finding subgraphs that are not just connected, but also densely connected with a high probability. There is relatively little known about mining dense structures from an uncertain graph. To our knowledge, the only previous work on mining maximal cliques in an uncertain graph is by Zou et. al [36]. Our work is different from theirs in significant ways. Mainly, while we focus on enumerating all α-maximal cliques in a graph, they focus on a different problem, that of enumerating the k cliques with the highest probability of existence. We present bounds on the number of such cliques that could exist, while by definition, their algorithm outputs k cliques. Further, their work does not have an analysis of runtime complexity or output size. There is substantial prior work on maximal clique enumeration from a deterministic graph. A popular algorithm for maximal clique enumeration problem is the Bron-Kerbosch algorithm [37], based on depthfirst-search. Tomita et al. [38] improved the depth-first-search approach through a better strategy for pivot selection; their resulting algorithm runs in time O(3 n 3 ), which is worst-case optimal, due to the bound on the number of maximal cliques possible [25]. Further work on enumeration of maximal cliques includes [39,40,41,42,43,44,45]. Roadmap. We present a problem definition in Section 2 and bounds on the number of α-maximal cliques in Section 3. We present an algorithm to enumerate all α-maximal cliques in Section 4, followed by experimental results in Section 5. Problem Definition An uncertain graph is a probability distribution over a set of deterministic graphs. We deal with undirected simple graphs, i.e. there are no self-loops or multiple edges. An uncertain graph is a triple G = (V, E, p), where V is a set of vertices, E ⊆ V × V is a set of (possible) edges, and p : E → (0, 1] is a function that assigns a probability of existence p(e) to each edge e ∈ E. As in prior work on uncertain graphs, we assume that the existence of different edges are mutually independent events. Let n = |V | and m = |E|. Note that G is a distribution over 2 m deterministic graphs, each of which is a subgraph of the undirected graph (V, E). This set of possible deterministic graphs is called the set of "possible graphs" of the uncertain graph G, and is denoted by D(G). Note that in order to sample from an uncertain graph G, it is sufficient to sample each edge e ∈ E independently with a probability p(e). In an uncertain graph G = (V, E, p), two vertices u and v are said to be adjacent if there exists an edge (u, v) in E. Let the neighborhood of vertex u, denoted Γ(u), be the set of all vertices that are adjacent to u in G. The next two definitions are standard, and apply not to uncertain graphs, but to deterministic graphs. Definition 1. A set of vertices C ⊆ V is a clique in a graph G = (V, E), if every pair of vertices in C is connected by an edge in E. Definition 2. A set of vertices M ⊆ V is a maximal clique in a graph G = (V, E), if (1) M is a clique in G and (2) There is no vertex v ∈ V \ M such that M ∪ {v} is a clique in G. Definition 3. In an uncertain graph G, for a set of vertices C ⊆ V , the clique probability of C, denoted by clq(C, G), is defined as the probability that in a graph sampled from G, C is a clique. For parameter 0 ≤ α ≤ 1, C is called an α-clique if clq(C, G) ≥ α. For any set of vertices C ⊆ V , let E C denote the set of edges {e = (u, v)|e ∈ E, u, v ∈ C and u = v}, i.e. the set of edges connecting vertices in C. Observation 1. For any set of vertices C ⊆ V , clq(C, G) = e∈E C p(e) Proof. Let G be a graph sampled from G. The set C will be a clique in G iff every edge in E C is present in G. Since the events of selecting different edges are independent of each other, the observation follows. The following two observations follow directly from Observation 1. Observation 2. For any two vertex sets A, B in G, if B ⊂ A then, clq(B, G) ≥ clq(A, G). Observation 3. Let C be an α-clique in G. Then for all e ∈ E C we have p(e) ≥ α. Number of Maximal Cliques The maximum number of maximal cliques in a deterministic graph on n vertices is known exactly due to a result by Moon and Moser [25]. If n mod 3 = 0, this number is 3 For uncertain cliques, no such bound was known so far. In this section, we establish a bound on the maximum number of α-maximal cliques in an uncertain graph. For 0 < α < 1, let f (n, α) be the maximum number of α-maximal cliques in an uncertain graph with n nodes, the maximum taken over all possible graphs on n nodes and all possible assignments of edge probabilities. The following theorem is the main result of this section. Proof. We can verify that the theorem holds for n = 2, when f (2, α) = 2. Let g(n) = n ⌊n/2⌋ . We show f (n, α) is at least g(n) in Lemma 1, and then show that f (n, α) is no more than g(n) in Lemma 2. Lemma 1. For any n ≥ 3, and any α, 0 < α < 1, there exists an uncertain graph G = (V, E, p) with n nodes which has g(n) α-maximal cliques. Proof. First, we assume that n is even. Consider G = (V, E, p), where E = V × V . Let κ = n/2 2 . For each e ∈ E, let p(e) = q where q κ = α. We have 0 < q < 1 since 0 < α < 1. Let S be an arbitrary subset of V such that |S| = n/2. We can verify that S is an α-maximal clique since (1) the probability that S is a clique is q κ = α and (2) for any set S ′ S, S ′ ⊆ V , the probability that S ′ is a clique is at most qq κ = qα < α. We can also observe that for any subset S ⊆ V , S cannot be an α-maximal clique if |S| < n/2 or |S| > n/2. Thus we conclude that a subset S ⊆ V is an α-maximal clique iff |S| = n/2 which implies that the total number of α-maximal cliques in G is n n/2 . A similar proof applies when n is odd. Note that our construction in the Lemma above employs the condition that n ≥ 3 and 0 < α < 1. When α = 1, the upper bound is from the result of Moon and Moser for deterministic graphs, and in this case f (n, α) = 3 n 3 and is smaller than g(n). Next we present a useful definition required for proving the next Lemma. Definition 6. A collection of sets C is said to be non-redundant if for any pair S 1 , S 2 ∈ C, S 1 = S 2 , we have S 1 S 2 and S 2 S 1 . Lemma 2. g(n) is an upper bound on f (n, α). Proof. Let C α (G) be the collection of all α-maximal cliques in G. Note that by the definition of α-maximal cliques, any α-maximal clique S in G can not be a proper subset of any other α-maximal clique in G. Thus from Definition 6, for any uncertain graph G, C α (G) is a non-redundant collection. Hence, it is clear that the largest number of α-maximal cliques in G should be upper bounded by the size of a largest non-redundant collection of subsets of V . Let C be the collection of all subsets of V . Based on C, we construct such an undirected graph G = (C, E) where for any two nodes S 1 ∈ C, S 2 ∈ C, there is an edge connecting S 1 and S 2 iff S 1 ⊆ S 2 or S 2 ⊆ S 1 . It can be verified that a sub-collection C ′ ⊆ C is a non-redundant iff C ′ is an independent set in G. In Lemma 3, we show that g(n) is the size of a largest independent set of G, which implies that g(n) is an upper bound for the number of α-maximal cliques in G. Let C * be a largest independent set in G. Also, let C k ⊆ C, 0 ≤ k ≤ n be the collection of subsets of V with the size of k. Observe that for each 0 ≤ k ≤ n, C k is an independent set of G. Also let L(n) and U (n) be respectively the minimum and maximum size of sets in C * . We can show that L(n) and U (n) can be bounded as shown in Lemma 4 and Lemma 5 respectively. Lemma 3. For any n ≥ 3, |C * | = g(n). Proof. We first consider the case when n is even. By Lemmas 4 and 5, we know n/2 ≤ L(n) ≤ U (n) ≤ n/2. Thus we have L(n) = U (n) = n/2 which implies C * = C * n/2 . Recall that C k ⊆ C, 0 ≤ k ≤ n is the collection of subsets of V with the size of k. We have (1) C * = C * n/2 ⊆ C n/2 and (2) |C * | ≥ |C n/2 | since C * is a largest independent set of G. Thus we conclude C * = C n/2 which has the size of n (n/2) = g(n). We next consider the case when n is odd. From Lemmas 4 and 5, we know (n−1)/2 ≤ L(n) ≤ U (n) ≤ (n+1)/2. Thus we have C * = C * (n−1)/2 C * (n+1)/2 . For notation convenience, we set n 1 = (n−1)/2, n 2 = (n + 1)/2. Let G(C n 1 , C n 2 ) be the subgraph of G induced by C n 1 ∪ C n 2 . We can view G(C n 1 , C n 2 ) as a bipartite graph with two disjoint vertex sets C n 1 and C n 2 respectively. Observe that C * n 1 ⊆ C n 1 and C * n 2 ⊆ C n 2 . Let E(C * n 1 ) be the set of edges induced by C * n 1 in G(C n 1 , C n 2 ). Since C * is an independent set of G, none of the edges in E(C * n 1 ) will have an end in a node of C * n 2 , i.e, all the edges of E(C * n 1 ) should have an end falling in C n 2 \ C * n 2 . Note that in G(C n 1 , C n 2 ), all nodes have a degree of n 2 . Thus we have: | E(C * n 1 )| = |C * n 1 | * n 2 ≤ |C n 2 \ C * n 2 | * n 2 = (|C n 2 | − |C * n 2 |) * n 2 from which we obtain |C * | = |C * n 1 | + |C * n 2 | ≤ |C n 2 | = n n 2 . Note that C n 2 itself is an independent set of G with size n n 2 . Thus we conclude that |C * | = n n 2 = g(n). Lemma 4. L(n) ≥ ⌊n/2⌋ Proof. Let us assume n is an even number. We prove by contradiction as follows. Suppose L(n) = ℓ ≤ n/2 − 1. Let C * k ⊆ C * , L(n) ≤ k ≤ U (n) be the collection of all sets in C * which has the size of k, i.e, C * k = {S ∈ C * ||S| = k}. In the following we construct a new collection C new ⊆ C which proves to be an independent set in G with the size being strictly larger than C * . For each S ∈ C * ℓ , we add to C * all subsets of V which has the form as S ∪ {i} where i ∈ V \ S and remove S from C * meanwhile. Let C new be the collection obtained after we process the same route for all S ∈ C * ℓ . Mathematically, we have: C new = C 1 C 2 where C 1 = S∈C * ℓ i∈V \S {S ∪ {i}}, C 2 = C * \ C * ℓ . First we show C new is an independent set of G. Arbitrarily choose two distinct sets, say S 1 ∈ C new , S 2 ∈ C new , S 1 = S 2 . We check all the possible cases one by one: • S 1 ∈ C 1 , S 2 ∈ C 1 . We observe that |S 1 | = |S 2 | = ℓ + 1 and S 1 = S 2 . Thus no inclusion relation could exist between S 1 and S 2 . • S 1 ∈ C 2 , S 2 ∈ C 2 . In this case no inclusion relation can exist between S 1 and S 2 since C 2 is an independent set of G. • S 1 ∈ C 1 , S 2 ∈ C 2 . Since C * ℓ is the collection of sets in C * which has the smallest size ℓ, we get that |S 2 | ≥ ℓ + 1 = |S 1 |. Therefore there is only one possible inclusion relation existing here, that is S 1 ⊂ S 2 . Suppose S 1 = S ′ 1 ∪ {i 1 } ⊂ S 2 for some S ′ 1 ∈ C * ℓ . Thus we get that S ′ 1 ⊂ S 2 which implies C * is not an independent set of G. Hence we conclude that no inclusion relation could exist between S 1 and S 2 . Summarizing the analysis above, we get that no inclusion relation could exist between S 1 and S 2 which yields C new is an independent set of G. Now we prove that |C new | > |C * |. Observe that C 1 and C 2 are disjoint from each other; otherwise C * is not an independent set. So we have |C new | = |C 1 | + |C 2 |. Note that |C * | = |C * ℓ | + |C 2 | since C * is the union of the two disjoint parts C * ℓ and C 2 . Therefore |C new | > |C * | is equivalent to |C 1 | > |C * ℓ |. Let G(C * ℓ , C 1 ) be the induced subgraph graph of G by C * ℓ C 1 . Note that G(C * ℓ , C 1 ) can be viewed as a bipartite graph where the two disjoint vertex sets are C * ℓ and C 1 respectively. In G(C * ℓ , C 1 ) we observe that (1) for each node S 1 ∈ C * ℓ , its degree d(S 1 ) = n − ℓ; (2) for each node S 2 ∈ C 1 , its degree d(S 2 ) ≤ ℓ + 1. Thus we get that | E| = |C * ℓ |(n − ℓ) ≤ |C 1 |(ℓ + 1) . According to our assumption we have ℓ ≤ n/2 − 1. Thus we have |C * ℓ |/|C 1 | ≤ (ℓ + 1)/(n − ℓ) ≤ (n/2)/(n/2 + 1) < 1, yielding |C * ℓ | < |C 1 | which is equivalent to |C * | < |C new |. So far we have successfully constructed a new collection C new ⊆ C such that (1) it is an independent set of G and (2) |C new | > |C * |. That contradicts with the fact that C * is a largest independent set of G. Thus our assumption ℓ ≤ n/2 − 1 does not hold, which yields ℓ ≥ n/2. For the case when n is odd, we can process essentially the same analysis as above and get ℓ ≥ (n − 1)/2. Lemma 5. U (n) ≤ ⌈n/2⌉ Proof. Let us assume n is an even number. Based on C * , we construct a dual collection C * dual as follows: Initialize C * dual as an empty collection. For each S ∈ C * , we add V \ S into C * dual . Mathematically, we have: C * dual = S∈C * {V \ S}. First we show C * dual is an independent set of G. Arbitrarily choose two distinct sets, say V \ S 1 ∈ C * dual , V \ S 2 ∈ C * dual , where S 1 ∈ C * , S 2 ∈ C * , S 1 = S 2 . Note that V \ S 1 ⊂ V \ S 2 ⇔ S 1 ⊃ S 2 , V \ S 2 ⊂ V \ S 1 ⇔ S 2 ⊃ S 1 Thus we have that no inclusion relation could exist between V \S 1 and V \S 2 since no inclusion relation exists between S 1 and S 2 resulting from the fact that C * is an independent set of G. So we get C * dual is an independent set as well. We can verify that |C * dual | = |C * |. Therefore we can conclude C * dual is a largest independent set of G. By Lemma 4, we get to know the minimum size of sets in C * dual should be at least n/2, which yields the maximum size of of sets in C * should be at most n/2. For the case when n is odd, we can analyze essentially the same as above. Enumeration Algorithm In this section, we present MULE (Maximal Uncertain cLique Enumeration), an algorithm for enumerating all α-maximal cliques in an uncertain graph G, followed by a proof of correctness and an analysis of the runtime. We assume that G has no edges e such that p(e) < α. If there are any such edges, they can be pruned away without losing any α-maximal cliques, using Observation 3. Let the vertex identifiers in G be 1, 2, . . . , n. For clique C, let max(C) denote the largest vertex in C. For ease of notation, let max(∅) = 0, and let clq(∅, G) = 1. Intuition. We first describe a basic approach to enumeration using depth-first-search (DFS) with backtracking. The algorithm starts with a set of vertices C (initialized to an empty set) that is an α-clique and incrementally adds vertices to C, while retaining the property of C being an α-clique, until we can add no more vertices to C. At this point, we have an α-maximal clique. Upon finding a clique that is α-maximal, the algorithm backtracks to explore other possible vertices that can be used to extend C, until all possible search paths have been explored. To avoid exploring the same set C more than once, we add vertices in increasing order of the vertex id. For instance, if C was currently the vertex set {1, 3, 4}, we do not consider adding vertex 2 to C, since the resulting clique {1, 2, 3, 4} will also be reached by the search path by adding vertices 1, 2, 3, 4 in that order. MULE improves over the above basic DFS approach in the following ways. First, given a current αclique C, the set of vertices that can be added to extend C includes only those vertices that are already connected to every vertex within C. Instead of considering every vertex that is greater than max(C), it is more efficient to track these vertices as the recursive algorithm progresses -this will save the effort of needing to check if a new vertex v can actually be used to extend C. This leads us to incrementally track vertices that can still be used to extend C. Second, note that not all vertices that extend C into a clique preserve the property of C being an αclique. In particular, adding a new vertex v to C decreases the clique probability of C by a factor equal to the product of the edge probabilities between v and every vertex in C. So, in considering vertex v for addition to C, we need to compute the factor by which the clique probability will fall. This computation can itself take Θ(n) time since the size of C can be Θ(n), and there can be Θ(n) edges to consider in adding v. A key insight is to reduce this time to O(1) by incrementally maintaining this factor for each vertex v still under consideration. The recursive subproblem contains, in addition to current clique C, a set I consisting of pairs (u, r) such that u > max(C), u can extend C into an α-clique, and adding u will multiply the clique probability of C by a factor of r. This set I is incrementally maintained and supplied to further recursive calls. Finally, there is the cost of checking maximality. Suppose that at a juncture in the algorithm we found that I was empty, i.e. there are no more vertices greater than max(C) that can extend C into an α-clique. This does not yet mean that C is an α-maximal clique, since it is possible there are vertices less than max(C), but not in C, which can extend C to an α-maximal clique (note that such an α-maximal clique will be found through a different search path). This means that we have to run another check to see if C is an α-maximal clique. Note that even checking if a set of vertices C is an α-maximal clique can be a Θ(n 2 ) operation, since there can be as many as Θ(n) vertices to be potentially added to C, and Θ(n 2 ) edge interactions to be considered. We reduce the time for searching such vertices by maintaining the set X of vertices that can extend C, but will be explored in a different search path. By incrementally maintaining probabilities with vertices in I and X, we can reduce the time for checking maximality of C to Θ(n). MULE incorporates the above ideas and is described in Algorithm 1. Algorithm 1: MULE(G, α) Input: G is the input uncertain graph Input: α, 0 < α < 1 is the user provided probability threshold 1Î ← ∅ ; 2 forall the u ∈ V do 3Î ←Î ∪ {(u, 1)} 4 Enum-Uncertain-MC(∅, 1 ,Î, ∅) ; Proof of Correctness In this section we prove the correctness of MULE. Theorem 2. MULE (Algorithm 1) enumerates all α-maximal cliques from an input uncertain graph G. Proof. To prove the theorem we need to show the following. First, if C is a clique emitted by Algorithm 1, Algorithm 2: Enum-Uncertain-MC(C, q, I, X) Input: We assume G and α are available as immutable global variables Input: C is the current Uncertain Clique being processed Input: q = clq(C, G), maintained incrementally Input: I is a set of tuples (u, r), such that ∀(u, r) ∈ I, u > max(C), and clq(C ∪ {u}, G) = q · r ≥ α, i.e. C ∪ {u} is an α-clique in G Input: X is a set of tuples (v, s), such that ∀(v, s) ∈ X, v ∈ C, v < max(C), and clq(C ∪ {v}, G) = q · s ≥ α , i.e. C ∪ {v} is an α-clique in G 1 if I = ∅ and X = ∅ then 2 Output C as α-maximal clique ; 3 return 4 forall the (u, r) ∈ I considered in increasing order of u do 5 C ′ ← C ∪ {u} // Note m = max(C ′ ) = u 6 q ′ ← q · r // clq(C ∪ {v}, G) 7 I ′ ← GenerateI(C ′ , q ′ , I) ; 8 X ′ ← GenerateX(C ′ , q ′ , X) ; 9 Enum-Uncertain-MC(C ′ , q ′ , I ′ , X ′ ) ; 10 X ← X ∪ {(u, r)} then C must be an α-maximal clique. Next, if C is an α-maximal clique, then it will be emitted by Algorithm 1. We prove them in Lemmas 8 and 9 respectively. Before proving Lemmas 8 and 9, we prove some properties of Algorithm 2. Lemma 6. When Algorithm 2 is called with C ′ in line 9, I ′ is a set of tuples (u ′ r ′ ), where u ′ ∈ V and (1). Consider the first call made to Algorithm 2. We know that C is initialized as ∅. Also, I =Î contains all vertices in V . Now we know max(∅) = 0. Thus for all u such that (u, r) ∈Î, u > max(C). For every case case expect initialization, I ′ is generated from I by line 7 of Algorithm 2 which in turn calls Algorithm 3. In Algorithm 3, only vertices in I that are greater than C ′ are added to I ′ . 0 < r ′ ≤ 1, such that, ∀(u ′ , r ′ ) ∈ I ′ , u ′ > max(C ′ ), and clq(C ′ ∪ {u ′ }, G) = q ′ · r ′ ≥ α, i.e. C ′ ∪ {u ′ } is an α-clique in G. Proof. Let (u ′ , r ′ ) ∈ I ′ , we need to show (1) u ′ > max(C ′ ), (2) C ′ ∪ {u ′ } is an α-clique in G, and (3) clq(C ′ ∪ {u ′ }, G) = q ′ · r ′ . First we prove Next we prove (2). Let C ′ be a clique being called by Enum-Uncertain-MC with I ′ . Note that each call of the method adds one vertex u ∈ I to the current clique C such u > max(C). Since the vertices are added This is also the probability of clq(C 1 ∪ {u ′ }, G). Hence condition (2) holds for the base case. For the inductive step, consider a recursive call to the method in which C i is extended to C i+1 . Algorithm 2 calls Algorithm 3 to generate I ′ . Every vertex in I is connected to C i . We need to show that all vertices in I ′ are connected to C i+1 . In line 6 of Algorithm 3, we prune out any vertex in I that is not connected to m = max(C i+1 ) Also, assume that u ′ extends C i such that clq(C i ∪ {u ′ }, G) = r. Now let c = {C i+1 \ C i }. Note that c is a single vertex. Also, assume u ′ > c. From line 4, we know that q ′ · r ′ ≥ α Also from line 6 of Algorithm 3, r ′ = r · p(c, u ′ ). Now clq(C i+1 ∪ {u ′ }, G) = q ′ · r · p(c, u ′ ) = q ′ · r ′ , thus proving the inductive step. The following observation follows from Lemma 6. Observation 4. The input C to Algorithm 2 is an α-clique. Algorithm 4: GenerateX(C ′ , q ′ , X) Input: We assume G and α are available as immutable global variables 1 m ← max(C ′ ), X ′ ← ∅, S ← ∅ ; 2 forall the (v, s) ∈ I do 3 S ← S ∪ {v} 4 S ← S ∩ {Γ(m)} 5 forall the (v, s) ∈ X do 6 if v ∈ S then 7 clq(C ′ ∪ {v}, G) ← q ′ · s · p(v, m) 8 if (clq(C ′ ∪ {v}, G) ≥ α then 9 v ′ ← v ; 10 s ′ ← s · p(v, m) ; 11 X ′ ← X ′ ∪ {(v ′ , s ′ )} 12 return X' Lemma 7. When Algorithm 2 is called with C ′ in line 9, X ′ is a set of tuples (v ′ , s ′ ), where v ′ ∈ V and 0 < s ′ ≤ 1, such that, ∀(v ′ , s ′ ) ∈ X ′ , we have v ′ ∈ C ′ , v ′ < max(C ′ ), and (clq(C ′ ∪{v ′ }, G) = q ′ ·s ′ ) ≥ α, i.e. C ′ ∪ {v ′ } is an α-clique in G. Proof. Let m = max(C ′ ) and C = C ′ \ {m}. Since Algorithm 2 was called with C ′ , it must have been called with C. This is because the working clique is always extended by adding vertices from I, and from Lemma 6, I only contains vertices that are greater than the maximum vertex in C. Let X be the corresponding set of tuples used when the call was made to Enum-Uncertain-MC with C. Let u > max(C) be a vertex such that clq(C ′ ∪ {u}, G) ≥ α and u < m. Note that u ∈ C ′ , u < max(C ′ ), and C ′ ∪ {u} is an α-clique in G. This means u satisfies all conditions for u ∈ X ′ . We need to show that when Enum-Uncertain-MC is called with C ′ , the generated X ′ which is passed in Enum-Uncertain-MC contains u. Firstly, note that since C ′ ∪ {u} is α-clique in G, we have clq(C ∪ {u}, G) ≥ α (from Observation 2). Since u > max(C) and clq(C ∪ {u}, G) ≥ α, from Lemma 6, u will be used in line 4 to call Enum-Uncertain-MC using C ∪ {u}. Once this call is returned, u is added to X in line 10. Note that since the loop at line 4 add vertices in lexicographical order, m will be added to C after u. Thus u will be in X, when m is used to extend C. Next we show that if u ∈ X, after execution of line 8, u ∈ X ′ . We prove this as follows. Note that Algorithm 4 is used to generate X ′ from X. Note that X ′ is generated by Algorithm 4 by selectively adding vertices from X. A vertex is added to X ′ from X, only if C ′ ∪ {u} is α-clique in G. From our initial assumptions, we know that u satisfies this condition and is hence added to X ′ and passed on to Enum-Uncertain-MC when it is called with C ′ . Now let us consider v, such that v does not satisfy all the conditions for v ∈ X ′ . We need to show that v ∈ X ′ . There are two cases. First, when v ∈ X. This case is trivial as X ′ is constructed from X and hence if v ∈ X, v ∈ X ′ . For the second case, when v ∈ X, we need to show that v will not be added to X ′ in line 8 of Algorithm 2. Note that since v ∈ X, we know v ∈ C ′ and v < max(C ′ ). Thus, it must be that C ∪ {m, v} is not an α-clique in G. Algorithm 4 will add v to X ′ only if C ∪ {m, v} is α-clique in G. But from our previous discussion, we know that this condition doesn't hold. Hence, v will not be added to X ′ . Thus only vertices that satisfy all three conditions are in X ′ . Lemma 8. Let C be a clique emitted by Algorithm 2. Then C is an α-maximal clique. Proof. Algorithm 2 emits C in Line 2. From Observation 4, we know that C is an α-clique. We need to show that C is α-maximal. We use proof by contradiction. Suppose C is non-maximal. This means that there exists a vertex u ∈ V , such that C ∪ {u} is an α-clique. We know that I = ∅ when C is emitted. From Lemma 6, we know that there exists no vertex u ∈ V such that u > C that can extend C. Again, we know that X = ∅ when C is emitted. Thus from Lemma 7, we know that there exists no vertex v ∈ V such that v < max(C) that can extend C. This is a contradiction and hence C is an α-maximal clique. Lemma 9. Let C be an α-maximal clique in G. Then C is emitted by Algorithm 2. Proof. We first show that a call to method Enum-Uncertain-MC with α-clique C enumerates all α-maximal cliques C ′ in G, such that for all c ∈ {C ′ \ C}, c > max(C). Without loss of generality, consider a α-maximal clique C ′ in G such that ∀c ∈ {C ′ \ C}, c > max(C). Note that C ′ will be emitted as an α-maximal clique by the method Enum-Uncertain-MC when called with C, if the following holds: (1) A call to method Enum-Uncertain-MC is made with C ′ , (2) When this call is made, I ′ = ∅, and X ′ = ∅. Since C ′ is α-maximal clique in G, the second point follows from Lemmas 6 and 7. Thus we need to show that a call to Enum-Uncertain-MC is made with C ′ . We prove this by induction. LetĈ = {C ′ \ C}. Let c i represent the ith element inĈ in lexicographical order. Also let C i = C ∪ {c 1 , c 2 , . . . , c i }. For the base case, we show that if a call to Enum-Uncertain-MC is made with C, a call will be made with C 1 = C ∪ {c 1 }. This is because, line 4 of the method loops over every vertex u ∈ I thus implying u > max(C) and clq(C ∪ {u}, G) ≥ α. Since C ′ is an α-maximal clique, c 1 will satisfy both these conditions and hence a call to Enum-Uncertain-MC is made with C ∪ {c 1 }. Now for the inductive step we show that if a call is made with clique C i , then this call will in turn call the method with clique C i+1 . Again, c i+1 is greater than max(C i ) and clq(C i ∪ {c i+1 }, G) ≥ α. Thus c i+1 ∈ I when the call is made to Enum-Uncertain-MC with C i . Hence using the previous argument, in line 4, c i+1 will be used as a vertex in the loop which would in turn make a call to Enum-Uncertain-MC with C i+1 . Now without any loss of generality, consider an α-maximal clique in G. We know that C ⊃ ∅. Thus the proof follows. Runtime Complexity Theorem 3. The runtime of MULE (Algorithm 1) on an input graph of n vertices is O (n · 2 n ). Proof. MULE initializes variables and calls to Algorithm 2, hence we analyze the runtime of Algorithm 2. An execution of the recursive Algorithm 2 can be viewed as a search tree as follows. Each call to Enum-Uncertain-MC is a node of this search tree. The first call to the method is the root node. A node in this search tree is either an internal node that makes one or more recursive calls, or a leaf node that does not make further recursive calls. To analyze the runtime of Algorithm 2, we consider the time spent at internal nodes as well as leaf nodes. The runtime at each leaf node is O(1). For a leaf node, the parameter I = ∅, and there are no further recursive calls. This implies that either C is α-maximal (X = ∅) and is emitted in line 2 or it is non-maximal (X = ∅) but cannot be extended by the loop in line 4 as I = ∅. Checking the sizes of I and X takes constant time. We next consider the time taken at each internal node. Instead of adding up the times at different internal nodes, we equivalently add up the cost of the different edges in the search tree. At each internal node, the cost of making a recursive call can be analyzed as follows. Line 5 takes O (n) time as we add all vertices in C to C ′ and also u. Line 6 takes constant time. Lines 7 and 8 take O (n) time (Lemmas 10 and 11 respectively). Note that lines 5 to 8 can get executed only once in between the two calls. Thus total runtime for each edge of the search tree is O (n). Note that the total number of calls made to the method method Enum-Uncertain-MC is no more than the possible number of unique subsets of V , which is O (2 n ). We see that for internal nodes, time complexity is O (n) and for leaf nodes it is O (1). Hence the time complexity of Algorithm 2 is O (n · 2 n ). Thus now we need to prove that lines 7 and 8 take O (n) time. This implies that time complexity of Algorithms 3 and 4 is O (n). We prove the same in Lemmas 10 and 11 respectively. Lemma 11. The runtime of Algorithm 4 is O (n). We omit the proof of the above lemma since it is similar to the proof of Lemma 10. Observation 5. The worst-case runtime of any algorithm that can output all maximal cliques of an uncertain graph on n vertices is Ω ( √ n · 2 n ). Proof. From Theorem 1, we know that the number of maximal uncertain cliques can be as much as n ⌊n/2⌋ = Θ 2 n √ n (using Stirling's Approximation). Since the size of each uncertain clique can be Θ (n), the total output size can be Ω ( √ n · 2 n ), which is a lower bound on the runtime of any algorithm. Proof. The proof follows from Theorem 3 and Observation 5. Enumerating Only Large Maximal Cliques For a typical input graph, many maximal cliques are small, and may not be interesting to the user. Hence it is helpful to have an algorithm that can enumerate only large maximal cliques efficiently, rather than enumerate all maximal cliques. We now describe an algorithm that enumerates every α-maximal clique with more than t vertices, where t is an user provided parameter. As a first step, we prune the input uncertain graph G = (V, E, p) by employing techniques described by Modani and Dey [41]. We apply the "Shared Neighborhood Filtering" where edges are recursively checked and removed as follows. First drop all edges (u, v) ∈ E, such that |Γ(u) ∩ Γ(v)| < (t − 2). Next drop every vertex v ∈ V , that doesn't satisfy the following condition. For vertex v ∈ V , there must exist at least (t − 1) vertices in Γ(v), such that for u ∈ Γ(v), |Γ(u) ∩ Γ(v)| < (t − 2) . Let G ′ denote the graph resulting from G after the pruning step. Algorithm 5 runs on the pruned uncertain graph G ′ to enumerate only large maximal cliques. The recursive method in Algorithm 6 differs from Algorithm 2 as follows. Before each recursive call to method Enum-Uncertain-MC-Large (Algorithm 6), the algorithm checks if the sum of the sizes of the current working clique C ′ and the candidate vertex set I ′ are greater than the size threshold t. If not, the recursive method is not called. This optimization leads to a substantial pruning of the search space and hence a reduction in runtime. Algorithm 5: LARGE-MULE(G, α,t) Input: G ′ is the input uncertain graph post pruning Input: α, 0 < α < 1 is the user provided probability threshold Input: t, t ≥ 2 is the user provided size threshold 1Î ← ∅ ; 2 forall the u ∈ V do 3Î ←Î ∪ {(u, 1)} 4 Enum-Uncertain-MC-Large(∅, 1 ,Î, ∅,t) ; Lemma 13. Given an input graph G, LARGE-MULE (Algorithm 5) enumerates every α-maximal clique with more than t vertices. Proof. First we prove that no maximal clique of size less than t is enumerated by Algorithm 6. Consider an α-maximal clique C 1 in G with less than t vertices. Also let m 1 = max(C 1 ) and C ′ 1 = C 1 \ {m 1 }. Algorithm 6: Enum-Uncertain-MC-Large(C, q, I, X,t) Input: C is the current Uncertain Clique being processed Input: q is pre-computed clq(C, G) Input: I is a set of tuples (u, r), such that ∀(u, r) ∈ I, u > max(C), and clq(C ∪ {u}, G) = q · r ≥ α, i.e. C ∪ {u} is an α-clique in G Input: X is a set of tuples (v, s), such that ∀(v, s) ∈ X, v ∈ C, v < max(C), and clq(C ∪ {v}, G) = q · s ≥ α , i.e. C ∪ {v} is an α-clique in G Input: t is the user provided size threshold 1 if I = ∅ and X = ∅ then 2 Output C as α-maximal clique ; 3 return 4 forall the u, r ∈ I taken in lexicographical ordering of u do 5 C ′ ← C ∪ {u} // Note m = max(C ′ ) = u 6 q ′ ← q · r // clq(C ∪ {v}, G) 7 I ′ ← GenerateI(C ′ , q ′ , I) ; 8 if |C ′ | + |I ′ | < t then 9 continue ; 10 X ′ ← GenerateX(C ′ , q ′ , X) ; 11 Enum-Uncertain-MC-Large(C ′ , q ′ , I ′ , X ′ , t) ; 12 X ← X ∪ {(u, r)} Note that if C 1 is emitted by Algorithm 6, then a call must be made to Enum-Uncertain-MC-Large with C 1 . Since the Algorithm adds vertices in lexicographical ordering, this implies that a call must be made to Enum-Uncertain-MC-Large with C ′ 1 before the call is made with C 1 . In the worst case, let us consider that the search tree reaches the execution point where Enum-Uncertain-MC-Large is called with C ′ 1 . Consider the execution of the algorithm where m 1 is added to C = C ′ 1 to form C ′ = C 1 . Since C 1 is an α-maximal clique, I ′ will become NULL which implies |I ′ | = 0. We know that |C 1 | < t. Thus |C 1 + I ′ | will also be less than t and the If condition (line 8) will succeed. This will result in the execution of the continue statement. Thus Enum-Uncertain-MC-Large will not be called with C 1 implying that C 1 is not enumerated. Next we show that any maximal clique of size at least t is enumerated by Algorithm 6. Consider an α-maximal clique C 2 in G of size at least t. We note that the "If" condition in line 8 is never satisfied in the search path ending with C 2 and hence a call is made to the method with Enum-Uncertain-MC-Large with C 2 . This is easy to see as whenever a call is made to Enum-Uncertain-MC-Large with any C ⊆ C 2 , since C 2 is large, we always have |C| + |I| ≥ t. Experimental Results We report the results of an experimental evaluation of our algorithm. We implemented the algorithm using Java. We ran all experiments on a system with a 3.19 GHz Intel(R) Core(TM) i5 processor and 4 GB of RAM, with heap space configured at 1.5GB. Input Data: Details of the input graphs that we used are shown in Table 1. The first set of input graphs was synthetically generated using the Barabási−Albert model for random graphs [46]. Then the edges were assigned probabilities uniformly at random from [0, 1]. The second set of graphs was obtained from the Stanford Large Network Collection [47], and includes or wikipedia user and the edges represent the votes that each admin / user casts in favor of a candidate. The candidate is also a wikipedia user and hence is represented by a vertex in the graph. For all these graphs, the uncertain graphs were created from these deterministic graphs by assigning edge probabilities uniformly at random. Hence these can be considered as semi-synthetic uncertain graphs. The third set of graphs consists of real world uncertain graphs shared by authors of [29] and [34]. These include a protein-protein interaction (PPI) network of a Fruit Fly obtained by integrating data from the BioGRID 2 database with that form the STRING 3 database, and the DBLP 4 dataset from authors of [34], which is an uncertain network predicting future co-authorship. The PPI network is an uncertain graph where each vertex represents a protein and two vertices are connected by an edge with a probability representing the likelihood of interaction between the the two proteins. The DBLP network represents co-authorship in academic articles. Each vertex in this network represents an author. Two vertices are connected by an edge with a probability that depends on the "strength" of their co-authorship, which is computed as 1 − e c/10 , where c is the number of papers co-authored. Comparison with other approaches. We compare our algorithm with another algorithm based on depth-first-search, which we call DFS-NOIP (DFS with NO Incremental Probability Computation), described in Algorithm 7. This algorithm also performs a depth first search to enumerate all α-maximal cliques but does not compute the probabilities incrementally like MULE does. We do not know of other prior algorithms for enumeration of maximal cliques from an uncertain graph. Dependence on α. We measured the runtime of enumeration as well as the output size, (the number of α-maximal cliques that were output) for different values of α and for the various input graphs described above. The dependence of the runtime on α is shown in Figure 2, and the number of cliques as a function of α is shown in Figure 3. We note that as α increases, the number of maximal cliques, and the time of enumeration both drop sharply. The decrease in runtime is because with a larger value of α, the algorithm is able to prune search paths aggressively early in the enumeration. We note that the number of α-maximal cliques does not have to always decrease as α increases. Sometimes it is possible that the number of α-maximal cliques increases with α. This is because as α increases, a large maximal clique may split into many smaller maximal cliques. However, these differences are negligible, and are not visible in the plots. Dependence on Size of Output. Figure 4 shows the change in runtime with respect to the number of α-maximal cliques enumerated, for the randomly generated graphs. It can be seen that the runtime of the algorithm is almost proportional to the number of maximal cliques in the output. This shows that the algorithm runtime scales well with the number of α-maximal cliques in output. This comparison was not Conclusion We present a systematic study of the enumeration of maximal cliques from an uncertain graph, starting from a precise definition of the notion of an α-maximal clique, followed by a proof showing that the maximum number of α-maximal cliques in a graph on n vertices is exactly n ⌊n/2⌋ , for 0 < α < 1. We present a novel An interesting open problem is to design an algorithm for enumerating maximal cliques from an uncertain graph whose time complexity is worst-case optimal, O ( √ n · 2 n ). Definition 4 . 4Given an uncertain graph G = (V, E, p), and a parameter 0 ≤ α ≤ 1, a set M ⊆ V is defined as an α-maximal clique if (1) M is an α-clique in G, and (2) There is no vertex v ∈ (V \ M ) such that M ∪ {v} is an α-clique in G. Definition 5. The Maximal Clique Enumeration problem in an Uncertain Graph G is to enumerate all vertex sets M ⊆ V such that M is an α-maximal clique in G. n 3 . 3If n mod 3 = 1, then it is 4 · 3 The graphs that have the maximum number of maximal cliques are known as Moon-Moser graphs. Theorem 1 . 1Let n ≥ 2, and 0 < α < 1. Then: f (n, α) = n ⌊n/2⌋ Algorithm 3 : 3GenerateI(C ′ , q ′ , I) Input: We assume G and α are available as immutable global variables1 m ← max(C ′ ), I ′ ← ∅, S ← ∅ ; 2 forall the (u, r) ∈ I do 3 S ← S ∪ {u} 4 S ← S ∩ {Γ(m)} 5 forall the (u, r) ∈ I do 6 if u > m and u ∈ S then 7 clq(C ′ ∪ {u}, G) ← q ′ · r · p(u, m) 8 if (clq(C ′ ∪ {u}, G)) ≥ α then 9 u ′ ← u ; 10 r ′ ← r · p(u, m) ; 11 I ′ ← I ′ ∪ {(u ′ , r ′ )} 12 return I'in the lexicographical ordering, there is an unique sequence of calls to the method Enum-Uncertain-MC such that we reach a point in execution of Algorithm 2 where Enum-Uncertain-MC is called with C ′ . We call this sequence of calls as Call-0, Call-1, . . ., Call-|C ′ |. Also, let C i be the clique used by method Enum-Uncertain-MC during Call-i. We prove by induction. For the base case (Call-0), C = ∅ and I ′ contains all vertices in V . Any single vertex can be considered as a clique. Again,Î is initialized such that all r inÎ are 1 ≥ α. Now when the next call is made (Call-1), only one vertex c 1 ∈ C ′ is added to the working clique (the smallest vertex in C ′ ). In line 2 of Algorithm 3, the new r ′ is computed as the probability p(c 1 , u ′ ). Lemma 10 . 10The runtime of Algorithm 3 is O (n). Proof. First note that lines 1-6 takes O (n) time. This is because |I| = O (n), and hence the loop at line 4 of Algorithm 3 can take O (n) time. Further the set intersection at line 6 also takes O (n) time. We need to show that the for loop in line 7 is O (n), that is each iteration of the loop takes O (1) time. Assume that it takes constant time to find out the probability of an edge. This is a valid assumption, as the edge probabilities can be stored as a HashMap and hence for an edge e, in constant time we can find out p(e). With this assumption, it is easy to show that lines 8-13 takes constant time. This is because, they are either constant number of multiplications, or adding one element to a set. Thus total time complexity is O (n). Lemma 12 . 12The worst-case runtime of MULE on an n vertex graph is within a O( √ n) factor of the runtime of an optimal algorithm for Maximal Clique Enumeration on an uncertain graph. graphs representing Internet p2p networks, collaboration networks, and an online social network. The p2p-Gnutella graphs represent peer to peer file sharing networks, where each vertex in the graph represents a computer and the edges represent the communication among them. The p2p-Gnutella04, p2p-Gnutella08 and p2p-Gnutella09 graphs represent communications occurring on 4th, 8th and 9th of August, 2002 respectively. The ca-GrQc graph represents the collaboration network among scientist working on General Relativity and Quantum Cosmology. Each vertex in the graph is a scientist and two vertices are connected by an edge if the corresponding scientists have co-authored a paper. Finally the wiki-vote graph represents the voting that occurs while selecting a new wikipedia administrator. Each vertex is either a wikipedia admin Figure 1 CI 1compares the performance of MULE with DFS-NOIP. The results show that MULE performs much better than DFS-NOIP. For instance, for the graph wiki-vote with α = 0.9 DFS-NOIP took 64 seconds while MULE took only 8 secs. The relative performance results hold true over a wide range of input graphs and values of α, including synthetic and real-world graphs, and small and large values of α. For α = 0.0001, MULE took only 25 secs to enumerate all maximal cliques in ca-GrQc, while DFS-NOIP took over 4400 secs. On the wiki-vote input graph with probability threshold 0.9, MULE took 8 seconds while DFS-NOIP took 64 seconds. For the same graph, with probability threshold 0.0001, MULE took 114 secs, while DFS-NOIP took more than 11 hours. Algorithm 7: DFS-NOIP(C,I) 1 I copy ← I ; 2 forall the u ∈ I copy do 3 if u ≤ max(C) OR clq(C ∪ {u}) < α then 4 I ← I \ {u} 5 if I = ∅ then 6 if C is an α-maximal clique then 7 Output C as α-maximal clique ; ′ ← C ∪ {v} ; 11if C ′ is an α-maximal clique then12 Output C ′ as α-maximal clique ; ′ ← I ∩ Γ(v) ; 15 DFS-NOIP(C ′ ,I ′ ) ; Figure 1 :Figure 2 :Figure 3 : 123Comparison of Simple and Optimized Depth First Search approaches. The Y-Axis is in log-scale. done for real world or semi-synthetic graphs as these graphs have different structural properties, hence different sizes of maximal cliques and thus there is no meaningful way to interpret the results.Enumerating Large Maximal Cliques.Figures 5 and 6show the runtime of LARGE-MULE (Algorithm 5) and the output size respectively as a function of t, the minimum size of an α-maximal clique that is output. As t increases, both runtime and output size decrease substantially. For instance, MULE takes 76797 seconds to enumerate all uncertain maximal cliques from the DBLP dataset (for probability threshold 0.9). However, LARGE-MULE takes only 32 seconds when t = 3. Similarly, for input graph ca-GrQc and α = 0.0001, MULE takes 125 seconds, while LARGE-MULE takes 10 seconds when t = 6 Runtime vs Alpha (α). The X-Axis is in log-No of α-maximal cliques vs Alpha (α). The X-Axis is in log-scale when t = 7. Figure 4 : 4Runtime vs Output Size algorithm, MULE, for enumerating the set of all α-maximal cliques from a graph, and an analysis showing that the worst-case runtime of this algorithm is O (n · 2 n ). We present an experimental evaluation of MULE showing its performance, and an extension for faster enumeration of large maximal cliques. Figure 5 : 5Runtime vs Size threshold of enumerated uncertain maximal cliques [4] Eytan Adar and Christopher Re. Managing uncertainty in social networks. IEEE Data Engineering Bulletin, 30(2):15-22, 2007. [5] R. 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Table 1 : 1Input GraphsInput Graph Category Description # Vertices # Edges BA5000 Barabási−Albert random graphs Random graph with 5K vertices 5000 50032 BA6000 Barabási−Albert random graphs Random graph with 6K vertices 6000 60129 BA7000 Barabási−Albert random graphs Random graph with 7K vertices 7000 70204 BA8000 Barabási−Albert random graphs Random graph with 8K vertices 8000 80185 BA9000 Barabási−Albert random graphs Random graph with 9K vertices 9000 90418 BA9000 Barabási−Albert random graphs Random graph with 10K vertices 10000 99194 p2p-Gnutella08 Internet peer-to-peer networks Gnutella network August 8 2002 6301 20777 p2p-Gnutella04 Internet peer-to-peer networks Gnutella network August 4 2003 10879 39994 p2p-Gnutella09 Internet peer-to-peer networks Gnutella network August 9 2003 8114 26013 ca-GrQc Collaboration networks Arxiv General Relativity 5242 28980 wiki-vote Social networks wikipedia who-votes-whom network 7118 103689 Fruit-Fly Protein Protein Interaction network PPI for Fruit Fly from STRING Database 3751 3692 DBLP10 Social network Collaboration network from DBLP 684911 2284991 This assumes that 3 divides n. 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Arboricity and subgraph listing algorithms. SIAM Journal on Computing, 14:210-223, February 1985. On generating all maximal independent sets. David S Johnson, Mihalis Yannakakis, Christos H Papadimitriou, Information Processing Letters. 273David S. Johnson, Mihalis Yannakakis, and Christos H. Papadimitriou. On generating all maximal independent sets. Information Processing Letters, 27(3):119 -123, 1988. New algorithms for enumerating all maximal cliques. Kazuhisa Makino, Takeaki Uno, Algorithm Theory -SWAT. Torben Hagerup and Jyrki KatajainenBerlin / HeidelbergSpringer3111Kazuhisa Makino and Takeaki Uno. New algorithms for enumerating all maximal cliques. In Torben Hagerup and Jyrki Katajainen, editors, Algorithm Theory -SWAT 2004, volume 3111 of Lecture Notes in Computer Science, pages 260-272. Springer Berlin / Heidelberg, 2004. Statistical mechanics of complex networks. Réka Albert, Albert-László Barabási, Reviews of Modern Physics. 74Réka Albert and Albert-László Barabási. 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{'abstract': 'We consider mining dense substructures (maximal cliques) from an uncertain graph, which is a probability distribution on a set of deterministic graphs. For parameter 0 < α < 1, we present a precise definition of an α-maximal clique in an uncertain graph. We present matching upper and lower bounds on the number of α-maximal cliques possible within an uncertain graph. We present an algorithm to enumerate α-maximal cliques in an uncertain graph whose worst-case runtime is near-optimal, and an experimental evaluation showing the practical utility of the algorithm. *', 'arxivid': '1310.6780', 'author': ['Arko Provo Mukherjee ', 'Pan Xu ', 'Srikanta Tirthapura '], 'authoraffiliation': [], 'corpusid': 6899631, 'doi': '10.1109/icde.2015.7113288', 'github_urls': [], 'n_tokens_mistral': 22074, 'n_tokens_neox': 19574, 'n_words': 12260, 'pdfsha': '45a67c003f1b8e3e5033cd189eedf529c5f1a595', 'pdfurls': ['https://arxiv.org/pdf/1310.6780v3.pdf'], 'title': ['Mining Maximal Cliques from an Uncertain Graph', 'Mining Maximal Cliques from an Uncertain Graph'], 'venue': []}

arxiv

Learning Depth-Guided Convolutions for Monocular 3D Object Detection Mingyu Ding The University of Hong Kong Gaoling School of Artificial Intelligence Renmin University of China Yuqi Huo Gaoling School of Artificial Intelligence Renmin University of China Hongwei Yi Shenzhen Graduate School Peking University Zhe Wang SenseTime Research 5 Beijing Key Laboratory of Big Data Management and Analysis Methods 100872BeijingChina Jianping Shi shijianping@sensetime.com SenseTime Research 5 Beijing Key Laboratory of Big Data Management and Analysis Methods 100872BeijingChina Zhiwu Lu luzhiwu@ruc.edu.cnhongweiyi@pku.edu.cnwangzhe Gaoling School of Artificial Intelligence Renmin University of China Ping Luo pluo@cs.hku.hkbohony The University of Hong Kong Learning Depth-Guided Convolutions for Monocular 3D Object Detection 1 st on KITTI monocular 3D object detection benchmark at the time of submission (car, December 2019) . The code is available at https://github.com/dingmyu/D4LCN. 3D object detection from a single image without LiDAR is a challenging task due to the lack of accurate depth information. Conventional 2D convolutions are unsuitable for this task because they fail to capture local object and its scale information, which are vital for 3D object detection. To better represent 3D structure, prior arts typically transform depth maps estimated from 2D images into a pseudo-LiDAR representation, and then apply existing 3D point-cloud based object detectors. However, their results depend heavily on the accuracy of the estimated depth maps, resulting in suboptimal performance. In this work, instead of using pseudo-LiDAR representation, we improve the fundamental 2D fully convolutions by proposing a new local convolutional network (LCN), termed Depth-guided Dynamic-Depthwise-Dilated LCN (D 4 LCN), where the filters and their receptive fields can be automatically learned from image-based depth maps, making different pixels of different images have different filters. D 4 LCN overcomes the limitation of conventional 2D convolutions and narrows the gap between image representation and 3D point cloud representation. Extensive experiments show that D 4 LCN outperforms existing works by large margins. For example, the relative improvement of D 4 LCN against the state-of-theart on KITTI is 9.1% in the moderate setting. D 4 LCN ranks 1 st on KITTI monocular 3D object detection benchmark at the time of submission (car, December 2019) . The code is available at https Introduction 3D object detection is a fundamental problem and has many applications such as autonomous driving and robotics. Previous methods show promising results by utilizing Li-DAR device, which produces precise depth information in terms of 3D point clouds. However, due to the high-cost show pseudo-LiDAR points generated by the supervised depth estimator, DORN [11] and the unsupervised Monodepth [13] respectively. The green box represents groundtruth (GT) 3D box. Pseudo-LiDAR points generated by inaccurate depth as shown in (b) have large offsets comapred to the GT box. (c) and (d) show the detection results of our method and Pseudo-Lidar [48] by using a coarse depth map. The performance of [48] depends heavily on the accuracy of the estimated depth maps, while our method achieves accurate detection results when accurate depth maps are missing. and sparse output of LiDAR, it is desirable to seek cheaper alternatives like monocular cameras. This problem remains largely unsolved, though it has drawn much attention. Recent methods towards the above goal can be generally categorized into two streams as image-based approaches [36,26,41,19,17,4] and pseudo-LiDAR point-based approaches [48,33,50]. The image-based approaches [5,17] typically leverage geometry constraints including object shape, ground plane, and key points. These constraints are formulated as different terms in loss function to improve detection results. The pseudo-LiDAR point-based approaches transform depth maps estimated from 2D images to point cloud representations to mimic the LiDAR signal. As shown in Figure 1, both of these methods have drawbacks, resulting in suboptimal performance. Specifically, the image-based methods typically fail to capture meaningful local object scale and structure informa-tion, because of the following two factors. (1) Due to perspective projection, the monocular view at far and near distance would cause significant changes in object scale. It is difficult for traditional 2D convolutional kernels to process objects of different scales simultaneously (see Figure 2). (2) The local neighborhood of 2D convolution is defined in the camera plane where the depth dimension is lost. In this nonmetric space (i.e. the distance between pixels does not have a clear physical meaning like depth), a filter cannot distinguish objects from the background. In that case, a car area and the background area would be treated equally. Although pseudo-LiDAR point-based approaches have achieved progressive results, they still possess two key issues. (1) The performance of these approaches heavily relies on the precision of estimated depth maps (see Figure 1). The depth maps extracted from monocular images are often coarse (point clouds estimated using them have wrong coordinates), leading to inaccurate 3D predictions. In other words, the accuracy of the depth map limits the performance of 3D object detection. (2) Pseudo-LiDAR methods cannot effectively employ high-level semantic information extracted from RGB images, leading to many false alarms. This is because point clouds provide spatial information but lose semantic information. As a result, regions like roadblocks, electrical boxes and even dust on the road may cause false detection, but they can be easily discriminated by using RGB images. To address the above problems, we propose a novel convolutional network, termed Depth-guided Dynamic-Depthwise-Dilated local convolutional network (D 4 LCN), where the convolutional kernels are generated from the depth map and locally applied to each pixel and channel of individual image sample, rather than learning global kernels to apply to all images. As shown in Figure 2, D 4 LCN treats the depth map as guidance to learn local dynamicdepthwise-dilated kernels from RGB images, so as to fill the gap between 2D and 3D representation. More specifically, the learned kernel in D 4 LCN is sample-wise (i.e. exemplar kernel [15]), position-wise (i.e. local convolution [20]), and depthwise (i.e. depthwise convolution [18]), where each kernel has its own dilation rate (i.e. different exemplar kernels have different receptive fields). D 4 LCN is carefully designed with four considerations. (1) The exemplar kernel is to learn specific scene geometry for each image. (2) The local convolution is to distinguish object and background regions for each pixel. (3) The depth-wise convolution is to learn different channel filters in a convolutional layer with different purposes and to reduce computational complexity. (4) The exemplar dilation rate is to learn different receptive fields for different filters to account for objects with diverse scales. The above delicate designs can be easily and efficiently implemented by combing linear operators of shift and element-wise product. As a Comparisons among different convolutional approaches. (a) is the traditional 2D convolution that uses a single convolutional kernel applied on each pixel to convolve the entire image. (b) applies multiple fixed convolutional kernels on different regions (slices) of an image. (c) uses the depth map to generate dynamic kernels with the same receptive fields for each pixel. (d) denotes our approach, where the filter is dynamic, depth-wise, and has adaptive receptive fields for each pixel and channel of the feature map. It can be implemented more efficiently with fewer parameters than (c). Best viewed in color. result, the efficient D 4 LCN can not only address the problem of the scale-sensitive and meaningless local structure of 2D convolutions, but also benefit from the high-level semantic information from RGB images compared with the pseudo-LiDAR representation. Our main contributions are three-fold. (1) A novel component for 3D object detection, D 4 LCN, is proposed, where the depth map guides the learning of dynamic-depthwisedilated local convolutions from a single monocular image. (2) We carefully design a single-stage 3D object detection framework based on D 4 LCN to learn better 3D representation for reducing the gap between 2D convolutions and 3D point cloud-based operations. (3) Extensive experiments show that D 4 LCN outperforms state-of-the-art monocular 3D detection methods and takes the first place on the KITTI benchmark [12]. Related Work Image-based Monocular 3D Detection. Previous monocular 3D detection methods [36,26,41,1,19,17,4,54] usually make assumptions about the scene geometry and use this as a constraint to train the 2D-to-3D mapping. Deep3DBox [36] uses the camera matrix to project a pre-dicted 3D box onto the 2D image plane, constraining each side of the 2d detection box, such that it corresponds to any of the eight corners of the 3D box. OFTNet [43] introduces the orthographic feature transform, which maps imagebased features into an orthographic 3D space. It is helpful when scale of objects varies drastically. [21,31] investigated different ways of learning the confidence to model heteroscedastic uncertainty by using a 3D intersection-overunion (IoU) loss. To introduce more prior information, [2,24,57,53] used 3D shapes as templates to get better object geometry. [23] predicts a point cloud in an objectcentered coordinate system and devises a projection alignment loss to learn local scale and shape information. [34] proposes a 3D synthetic data augmentation algorithm via in-painting recovered meshes directly onto the 2D scenes. However, as it is not easy for 2D image features to represent 3D structures, the above geometric constraints fail to restore accurate 3D information of objects from just a single monocular image. Therefore, our motivation is to utilize depth information, which essentially bridges gap between 2D and 3D representation, to guide learning the 2D-to-3D feature representation. Point Cloud-based Monocular 3D Detection. Previous monocular methods [48,33,50] convert image-based depth maps to pseudo-LiDAR representations for mimicking the LiDAR signal. With this representation, existing LiDAR-based detection algorithms can be directly applied to monocular 3D object detection. For example, [50] detects 2D object proposals in the input image and extracts a point cloud frustum from the pseudo-LiDAR for each proposal. [33] proposes a multi-modal features fusion module to embed the complementary RGB cue into the generated point clouds representation. However, this depth-to-LiDAR transformation relies heavily on the accuracy of depth map and cannot make use of RGB information. In contrast, our method treats depth map as guidance to learn better 3D representation from RGB images. LiDAR-based 3D Detection. With the development of deep learning on point sets, 3D feature learning [39,40,59] is able to learn deep point-based and voxel-based features. Benefit from this, LiDAR-based methods have achieved promising results in 3D detection. For example, [59] divides point clouds into equally spaced 3D voxels and transforms a group of points within each voxel into a unified feature representation. [47] applies the FPN technique to voxel-based detectors. [55] investigates a sparse convolution for voxel-based networks. [25] utilizes PointNets to learn a representation of point clouds organized in vertical columns (pillars). [38] leverages mature 2D object detectors to learn directly from 3D point clouds. [49] aggregates point-wise features as frustum-level feature vectors. [44,8] directly generated a small number of high-quality 3D proposals from point clouds via segmenting the point clouds of the whole scene into foreground and background. There are also some works focus on multi-sensor fusion (LIDAR as well as cameras) for 3D object detection. [29,28] proposed a continuous fusion layer that encodes both discrete-state image features as well as continuous geometric information. [7,22] used LIDAR point clouds and RGB images to generate features and encoded the sparse 3D point cloud with a compact multi-view representation. Dynamic Networks. A number of existing techniques can be deployed to exploit the depth information for monocular 3D detection. M3D-RPN [1] proposes depth-aware convolution which uses non-shared kernels in the row-space to learn spatially-aware features. However, this rough and fixed spatial division has bias and fail to capture object scale and local structure. Dynamic filtering network [20] uses the sample-specific and position-specific filters but has heavy computational cost, and it also fails to solve the scalesensitive problem of 2D convolutions. Trident network [27] utilizes manually defined multi-head detectors for 2D detection. However, it needs to manually group data for different heads. Other techniques like deformable convolution [9] and variants of [20] such as [14,46,52], fail to capture object scale and local structure as well. In this work, our depth-guided dynamic dilated local convolutional network is proposed to solve the two problems associated with 2D convolutions and narrow the gap between 2D convolution and point cloud-based 3D processing. Methodology As a single-stage 3D detector, our framework consists of three key components: a network backbone, a depth-guided filtering module, and a 2D-3D detection head (see Figure 3). Details of each component are given below. First, we give an overview of our architecture as well as backbone networks. We then detail our depth-guided filtering module which is the key component for bridging 2D convolutions and the point cloud-based 3D processing. Finally, we outline the details of our 2D-3D detection head. Backbone To utilize depth maps as guidance of 2D convolutions, we formulate our backbone as a two-branch network: the first branch is the feature extraction network using RGB images, and the other is the filter generation network to generate convolutional kernels for feature extraction network using the estimated depth as input. These two networks process the two inputs separately and their outputs of each block are merged by the depth-guided filtering module. The backbone of the feature extraction network is ResNet-50 [16] without its final FC and pooling layers, and is pre-trained on the ImageNet classification dataset [10]. To obtain a larger field-of-view and keep the network stride at 16, we find the last convolutional layer (conv5 1, block4) RGB Image Depth Map Estimate Depth-Guided Filtering Module Shift with different dilation rates Adaptive weights that decreases resolution and set its stride to 1 to avoid signal decimation, and replace all subsequent convolutional layers with dilated convolutional layers (the dilation rate is 2). For the filter generation network, we only use the first three blocks of ResNet-50 to reduce computational costs. Note the two branches have the same number of channels of each block for the depth guided filtering module. ! , " , # ∈ ℝ 2D bbox [x', y', w', h'] 2D 3D shape [w', h', l'] 3D 3D center [x', y'] P , z' 3D 3D rotation ' 3D 3D corners ' (m) , y' (m) , z' (m) Element-wise product 3D detection result NMS & Transform I 4 w 4 h 4 c 4 I 1 I 2 I 3 h 1 h 2 h 3 w 1 w 2 w 3 c 3 c 2 c 1 D 1 D 2 D 3 h 1 h 2 h 3 w 1 w 2 w 3 c 3 c 2 c 1 = 3 = 2 = 1 × ( , ) Depth-Guided Filtering Module Traditional 2D convolution kernels fail to efficiently model the depth-dependent scale variance of the objects and effectively reason about the spatial relationship between foreground and background pixels. On the other hand, pseudo-lidar representations rely too much on the accuracy of depth and lose the RGB information. To address these problems simultaneously, we propose our depth-guided filtering module. Notably, by using our module, the convolutional kernels and their receptive fields (dilation) are different for different pixels and channels of different images. Since the kernel of our feature extraction network is trained and generated by the depth map, it is sample-specific and position-specific, as in [20,14], and thus can capture meaningful local structures as the point-based operator in point clouds. We first introduce the idea of depthwise convolution [18] to the network, termed depth-wise local convolution (DLCN). Generally, depth-wise convolution (DCN) involves a set of global filters, where each filter only operates at its corresponding channel, while DLCN requires a feature volume of local filters the same size as the input feature maps. As the generated filters are actually a feature volume, a naive way to perform DLCN requires to convert the feature volume into h n × w n location-specific filters and then apply depth-wise and local convolutions to the feature maps, where h n and w n are the height and width of the feature maps at layer n. This implementation would be time-consuming as it ignores the redundant computations in neighboring pixels. To reduce the time cost, we employ the shift and element-wise product operators, in which shift [51] is a zero-flop zero-parameter operation, and element-wise product requires little calculation. Concretely, let I n ∈ R hn×wn×cn and D n ∈ R hn×wn×cn be the output of the feature extraction network and filter generation network, respectively, where n is the index of the block (note that block n corresponds to the layer conv n+1 in ResNet). Let k denote the kernel size of the feature extraction network. By defining a shifting grid {(g i , g j )}, g ∈ (int)[1−k/2, k/2−1] that contains k·k elements, for every vector (g i , g j ), we shift the whole feature map D towards the direction and step size indicated by (g i , g j ) and get the result D (gi,gj ) . For example, g ∈ {−1, 0, 1} when k = 3, and the feature map is moved towards nine directions with a horizontal or vertical step size of 0 or 1. We then use the sum and element-wise product operations to compute our filtering result: I = I 1 k · k gi,gj D (gi,gj ) .(1) To encourage information flow between channels of the depth-wise convolution, we further introduce a novel shiftpooling operator in the module. Considering n f as the number of channels with information flow, we shift the feature maps along the channel axis for n f times by 1, 2, .., n f − 1 to obtain new n f − 1 shifted feature maps I Compared to the idea 'group' of depth-wise convolution in [18,58] which aims to group many channels into a group to perform information fusion between them, the proposed shift-pooling operator is more efficient and adds no additional parameters to the convolution. The size of our convolutional weights of each local kernel is always k × k × c n when applying shift-pooling, while it changes significantly in [18] for different number of groups from k × k × c n to k×k×c n ×c n in group convolution (assume that the convolution keeps the number of channels unchanged). Note that it is difficult for the filter generation network to generate so many kernels for the traditional convolutions F between all channels, and the characteristic of being position-specific dramatically increases their computational cost. With our depth-wise formulation, different kernels can have different functions. This enables us to assign different dilation rates [56] for each filter to address the scalesensitive problem. Since there are huge intra-class and inter-class scale differences in an RGB image, we use I to learn an adaptive dilation rate for each filter to obtain different sizes of receptive fields by an adaptive function A. Specifically, let d denote our maximum dilation rate, the adaptive function A consists of three layers: (1) an Adap-tiveMaxPool2d layer with the output size of d × d and channel number c; (2) a convolutional layer with a kernel size of d × d and channel number d × c; (3) a reshape and softmax layer to generate d weights A w (I), w ∈ (int)[1, d] with a sum of 1 for each filter. Formally, our guided filtering with adaptive dilated function (D 4 LCN) is formulated as follows: I = 1 d · k · k · I w A w (I) gi,gj D (gi * w,gj * w) ,(2) For different images, our depth-guided filtering module assigns different kernels on different pixels and adaptive receptive fields (dilation) on different channels. This solves the problem of scale-sensitive and meaningless local structure of 2D convolutions, and also makes full use of RGB information compared to pseudo-LiDAR representations. 2D-3D Detection Head In this work, we adopt a single-stage detector with priorbased 2D-3D anchor boxes [42,32] as our base detector. Formulation Inputs: The output feature map I 4 ∈ R h4×w4 of our backbone network with a network stride factor of 16. Following common practice, we use a calibrated setting which assumes that per-image camera intrinsics K ∈ R 3×4 are available both at the training and test time. The 3D-to-2D projection can be written as:   x y 1   P · z 3D = K ·     x y z 1     3D(3) where [x, y, z] 3D denotes the horizontal position, height and depth of the 3D point in camera coordinates, and [x, y] P is the projection of the 3D point in 2D image coordinates. Ground Truth: We define a ground truth (GT) box using the following parameters: the 2D bounding box [x, y, w, h] 2D , where (x, y) is the center of 2D box and w, h are the width and height of 2D box; the 3D center [x, y, z] 3D represents the location of 3D center in camera coordinates; the 3D shapes [w, h, l] 3D (3D object dimensions: height, width, length (in meters)), and the allocentric pose α 3D in 3D space (observation angle of object, ranging [−π, π]) [34]. Note that we use the minimum enclosing rectangle of the projected 3D box as our ground truth 2D bounding box. Outputs: Let n a denote the number of anchors and n c denote the number of classes. For each position (i, j) of the input, the output for an anchor contains 35 + n c parameters: {[t x , t y , t w , t h ] 2D , [t x , t y ] P , [t z , t w , t h , t l , t α ] 3D , t (m) C , s}, where [t x , t y , t w , t h ] 2D is the predicted 2D box; [t x , t y ] P is the position of the projected 3D corner in the 2D plane, [t z , t w , t h , t l , t α ] 3D denotes the depth, predicted 3D shape and rotation, respectively; t (m) C = {[t (m) x , t (m) y ] P , [t (m) z ] 3D }, m ∈ {1, 2, . .., 8} denotes 8 projected 3D corners; s denotes the classification score of each class. The size of the output is h 4 × w 4 × n a × (35 + n c ), where (h 4 , w 4 ) is the size of the input image with a down sampling factor of 16. The output is actually an anchorbased transformation of the 2D-3D box. 2D-3D Anchor Inspired by [1], we utilize 2D-3D anchors with priors as our default anchor boxes. More specifically, a 2D-3D anchor is first defined on the 2D space as in [32] and then use the corresponding priors in the training dataset to calculate the part of it in the 3D space. One template anchor is defined using parameters of both spaces: {[A x , A y , A w , A h ] 2D , [A z , A w , A h , A l , A α ] 3D }, where [A z , A w , A h , A l , A α ] 3D denotes the 3D anchor (depth, shape, rotation). For 2D anchors [A x , A y , A w , A h ] 2D , we use 12 different scales ranging from 30 to 400 pixels in height following the power function of 30 * 1.265 exp , exp ∈ (int)[0, 11] and aspect ratios of [0.5, 1.0, 1.5] to define a total of 36 anchors. We then project all ground truth 3D boxes to the 2D space. For each projected box, we calculate its intersection over union (IoU) with each 2D anchor and assign the corresponding 3D box to the anchors that have an IoU ≥ 0.5. For each 2D anchor, we thus use the statistics across all matching ground truth 3D boxes as its corresponding 3D anchor [A z , A w , A h , A l , A α ] 3D . Note that we use the same anchor parameters [A x , A y ] 2D for the regression of [t x , t y ] 2D and [t x , t y ] P . The anchors enable our network to learn a relative value (residual) of the ground truth, which significantly reduces the difficulty of learning. Data Transformation We combine the output of our network which is an anchorbased transformation of the 2D-3D box and the pre-defined anchors to obtain our estimated 3D boxes: where [x , y ] P , [z , z (m) , α ] 3D denote respectively the estimated 3D center projection in 2D plane, the depth of 3D center and eight corners, the 3D rotation by combining output of the network and the anchor. Losses Our overall loss contains a classification loss, a 2D regression loss, a 3D regression loss and a 2D-3D corner loss. We use the idea of focal loss [30] to balance the samples. Let s t and γ denote the classification score of target class and the focusing parameter, respectively. We have: L = (1 − s t ) γ (L class + L 2d + L 3d + L corner ),(5) where γ = 0.5 in all experiments, and L class , L 2d , L 3d , L corner are the classification loss, 2D regression loss, 3D regression loss and D-3D corner loss, respectively. In this work, we employ the standard cross-entropy (CE) loss for classification: L class = − log(s t ).(6) Moreover, for both 2D and 3D regression, we simply use the SmoothL1 regression losses: L2D = SmoothL1([x , y , w , h ]2D, [x, y, w, h]2D), L3D = SmoothL1([w , h , l , z , α ]3D, [w, h, l, z, α]3D), + SmoothL1([x , y ]P , [x, y]P ), Lcorner = 1 8 SmoothL1([x (m) , y (m) ]P , [x (m) , y (m) ]P ) + SmoothL1([z (m) ]3D, [z]3D),(7) where [x (m) , y (m) ] P denotes the projected corners in image coordinates of the GT 3D box and [z] 3D is its GT depth. second layer, where n c is set to 4 for three object classes and the background class, and n a is set to 36. Non Maximum Suppression (NMS) with an IoU threshold of 0.4 is used on the network output in 2D space. Since the regression of the 3D rotation α is more difficult than other parameters, a hillclimbing post-processing step is used for optimizing α as in [1]. The input images are scaled to 512 × 1760 and horizontal flipping is the only data augmentation. n f is set to 2 and the maximum dilation rate d is set to 3 in all experiments. The network is optimized by stochastic gradient descent (SGD), with a momentum of 0.9 and a weight decay of 0.0005. We take a mini-batch size of 8 on 4 Nvidia Tesla v100 GPUs (16G). We use the 'poly' learning rate policy and set the base learning rate to 0.01 and power to 0.9. The iteration number for the training process is set to 40,000. Experiments Comparative Results We conduct experiments on the official test set and two splits of validation set of the KITTI dataset. Table 1 includes the top 14 monocular methods in the leaderboard, among which our method ranks top-1. We can observe that: (1) Our method outperforms the second-best competitor for monocular 3D car detection by a large margin (relatively 9.1% for 10.74 vs. 11.72) under the moderate setting (which is the most important setting of KITTI). (2) Most competitors, such as [23,33,45,37,50,1], utilize the detector (e.g. Faster-RCNN) pre-trained on COCO/KITTI or resort to multi-stage training to obtain better 2D detection and stable 3D results, while our model is trained end-to-end using the standard ImageNet pre-trained model. However, we still achieve the state-of-the-art 3D detection results, validating the effectiveness of our D 4 LCN to learn 3D structure. (3) Recently KITTI uses AP| R40 instead of AP| R11 , however, all existing methods report the results under the old metric. We thus also give results under AP| R11 on the validation set for fair comparison. It can be seen that our method outper- forms all others on the two splits for 3D car detection. Our results under AP| R40 on validation set are shown in ablation study. Detailed Analysis Ablation Study To conduct ablation study on our model, we make comparison among five versions of our model: (1) Evaluation of Depth Maps To study the impact of accuracy of depth maps on the performance of our method, we extract depth maps using four different methods [13,11,35,3] and then apply them to 3D detection. As reported in previous works on depth estimation, the three supervised methods (i.e. PSMNet, Disp-Net, and DORN) significantly outperform the unsupervised method [13]. Among the supervised methods, Stereo-based methods [3,35] are better than monocular-based DORN. With these conclusions, we have the following observations from Table 3: (1) The accuracy of 3D detection is higher with better depth map. This is because that better depth map can provide better scene geometry and local structure. (2) As the quality of depth map increases, the growth of detection accuracy becomes slower. (3) Even with the depth maps obtained by unsupervised learning [13], our method achieves state-of-the-art results. Compared to the pseudolidar based method [33], our method relies less on the quality of depth maps (19.63 vs. 15.45 using MonoDepth). Evaluation of Convolutional Appoaches To show the effectiveness of our guided filtering module for 3D object detection, we compare it with several alternatives: Dynamic Convolution [20], Dynamic Local Filtering [20], and Deformable Convolution [9]. Our method belongs Table 5. Multi-class 3D detection results of our method on the three data splits. Note that all pseudo-LiDAR based methods [33,50,48] fail to detect pedestrians and cyclists. to dynamic networks but yields less computation cost and stronger representation. For the first two methods, we conduct experiments using the same depth map as ours. For the third method, we apply deformable convolution on both RGB and depth branches and merge them by element-wise product. From Table 4, we can observe that our method performs the best. This indicates that our method can better capture 3D information from RGB images due to the special design of our D 4 LCN. Multi-Class 3D Detection Since a person is a non-rigid body, its shape varies and its depth information is hard to accurately estimate. For this reason, 3D detection of pedestrians and cyclists becomes particularly difficult. Note that all pseudo-LiDAR based methods [33,50,48] fail to detect these two categories. However, as shown in Table 5, our method still achieves satisfactory performance on 3D detection of pedestrians and cyclists. Moreover, we also show the active maps corresponding to different filters of our D 4 LCN in Figure 5. Different filters on the same layer of our model use different sizes of receptive fields to handle objects of different scales, including pedestrians (small) and cars (big), as well as distant cars (big) and nearby cars (small). Conclusion In this paper, we propose a Depth-guided Dynamic-Depthwise-Dilated Local ConvNet (D 4 LCN) for monocular 3D objection detection, where the convolutional kernels and their receptive fields (dilation rates) are different for different pixels and channels of different images. These kernels are generated dynamically conditioned on the depth map to compensate the limitations of 2D convolution and narrow the gap between 2D convolutions and the point cloudbased 3D operators. As a result, our D 4 LCN can not only address the problem of the scale-sensitive and meaningless local structure of 2D convolutions, but also benefit from the high-level semantic information from RGB images. Extensive experiments show that our D 4 LCN better captures 3D information and ranks 1 st for monocular 3D object detection on the KITTI dataset at the time of submission. Acknowledgements We would like to thank Dr. Guorun Yang for his careful proofreading. Ping Luo is partially supported by the HKU Seed Funding for Basic Research and SenseTime's Donation for Basic Research. Zhiwu Lu is partially supported by National Natural Science Foundation of China (61976220, 61832017, and 61573363), and Beijing Outstanding Young Scientist Program (BJJWZYJH012019100020098). APPENDIX A. Definition of 3D Corners We define the eight corners of each ground truth box as follows: C (m) =   x (m) y (m) 1   P · z (m) 3D =     ry ·     ±w/2 ±h/2 ±l/2 0     3D +     x y z 1     3D     (8) where m ∈ (int) [1,8] in a defined order, and r y is the egocentric rotation matrix. Note that we use allocentric pose for regression. B. Comparisons between Two Rotation Definitions As shown in Figure 6, while egocentric poses undergo viewpoint changes towards the camera when translated, allocentric poses always exhibit the same view, independent of the objects location. The allocentric pose α and the egocentric pose r y can be converted to each other according to the viewing angle θ. Figure 6. Comparisons between egocentric (ry) and allocentric (α) poses. The car1 and car2 have the same egocentric pose, but they are observed on different sides (views). We use allocentric pose to keep the same view (car1 and car3). α = r y − θ(9) C. Ablative Results for Convolutional Methods The Depth-guided filtering module in our D 4 LCN model can be decomposed into basic convolutional components: • Traditional Convolutional Network The ablative results for these convolutional methods are shown in Table 7. We can observe that: (1) Using the depth map to guide the convolution of each pixel brings a considerable improvement. (2) Depth-wise convolution with shift-pooling operator not only has fewer parameters (Section 3.2 of our main paper) but also gets better performance than the standard convolution. (3) The main improvement comes from our adaptive dilated convolution, which allows each channel of the feature map to have different receptive fields. D. Comparisons of Labeling Information and Training Strategies We compare the labeling information and training strategies used in different monocular detection methods, as shown in Table 6. It can be seen that: (1) our model outperforms all existing methods by only using the depth map extracted from Method Depth CAD Points Freespace Segmentation Pretrain/MST End-to-end Deep3DBox [36], GS3D [26], MonoGRNet [41], OFTNet [43] FQNet [31], SS3D [21] ROI-10D [34] Multi-Level Fusion [54], D 4 LCN (Ours) M3D-RPN [1], MONODIS [45], Shift R-CNN [37] AM3D [33] Pseudo-LiDAR [48], Mono3D-PLiDAR [50], MonoPSR [23] Deep-MANTA [2] 3DVP [53] Mono3D [4,6] Mono3D++ [17] Table 6. Comparisons of the labeling information and training strategies used in different monocular detection methods. Notations: Pretrain -pre-trained on COCO/KITTI datasets; MST -multi-stage training; End-to-end -end-to-end training; Depth -the depth map extracted from monocular image; CAD -the CAD model; Points -the characteristic points labeling information; Freespace -the free space labeling information; Segmentation -the segmentation labeling information; the monocular image. (2) our model can be trained in an end-to-end manner. E. Distributions of Different Dilation We show the average ratio of different channels with different dilation rates in three blocks of our model over the validation set of split1 ( Figure 7). It can be seen that: (1) For the first block with insufficient receptive field, the model tends to increase the receptive field by large dilation rate, and then it uses small receptive field for the second block. (2) In the third block, the model uses three different dilation rates evenly to deal with the object detection of different scales. We also show the active maps corresponding to different filters of the third block of our D 4 LCN in our main paper ( Figure 5). Figure 1 . 1(a) and (b) Figure 2 . 2Figure 2. Comparisons among different convolutional approaches. (a) is the traditional 2D convolution that uses a single convolutional kernel applied on each pixel to convolve the entire image. (b) applies multiple fixed convolutional kernels on different regions (slices) of an image. (c) uses the depth map to generate dynamic kernels with the same receptive fields for each pixel. (d) denotes our approach, where the filter is dynamic, depth-wise, and has adaptive receptive fields for each pixel and channel of the feature map. It can be implemented more efficiently with fewer parameters than (c). Best viewed in color. Figure 3 . 3Overview of our framework for monocular 3D object detection. The depth map is first estimated from the RGB image and used as the input of out two-branch network together with the RGB image. Then the depth-guided filtering module is used to fuse there two information of each residual block. Finally, a one-stage detection head with Non-Maximum Suppression (NMS) is employed for prediction. i ∈ {1, 2, ..., n f − 1}. Then we perform element-wise mean to the shifted feature maps and the original I to obtain the new feature map as the input of the module. The process of this shift-pooling operation is shown in Figure 4 (n f = 3). Figure 4 . 4An example of our shift-pooling operator of depth-wise convolution in depth-guided filtering module when n f is 3. It is efficiently implemented by shift and element-wise mean operators. [x , y ]2D = [Ax, Ay]2D + [tx, ty]2D * [Aw, A h ]2D [x , y ]P = [Ax, Ay]2D + [tx, ty]P * [Aw, A h ]2D [x (m) , y (m) ]P = [Ax, Ay]2D + [t (m) x , t (m) y ]P * [Aw, A h ]2D [w , h ]2D = [Aw, A h ]2D · exp([tw, t h ]2D) [w , h , l ]3D = [Aw, A h , A l ]3D · exp([tw, t h , t l ]3D) [z , z (m) , α ]3D = [Az, Az, Aα] + [tz, tz, t alpha ]3D. 3DNet: the baseline model using L 2D and L 3D without our depth-guided filtering module; (2) + CL: the Corner Loss is added to 3DNet; (3) + DLCN: depth-guided depth-wise local filtering is added; (4) + SP: shift-pooling operator is added (with n f = 3); (5) D 4 LCN (our full model): adaptive dilation rates are added, as in Eq. 2. From • Depth-guided ConvNet (CN) • Depth-guided Local CN (LCN) • Depth-guided Depth-wise LCN (DLCN) • Depth-guided DLCN with Shift-pooling (SP-DLCN) • D 4 LCN (Our full model) Figure 7 . 7The average ratio of different channels with different dilation rates in three blocks. Table 1. Comparative results on the KITTI 3D object detection dataset. For the test set, only AP|R 40 is provided by the official leaderboard. We thus show the results on the test set in AP|R 40 and split1/split2 in AP|R 11 . We use red to indicate the highest result with relative improvement in parentheses and blue for the second-highest result of the class car. Our method achieves 7 firsts and 2 seconds in 9 items.4.1. Dataset and Setting KITTI Dataset. The KITTI 3D object detection dataset [12] is widely used for monocular and LiDAR-based 3D detection. It consists of 7,481 training images and 7,518 test images as well as the corresponding point clouds and the calibration parameters, comprising a total of 80,256 2D- 3D labeled objects with three object classes: Car, Pedes- trian, and Cyclist. Each 3D ground truth box is assigned to one out of three difficulty classes (easy, moderate, hard) according to the occlusion and truncation levels of objects. There are two train-val splits of KITTI: the split1 [5] con- tains 3,712 training and 3,769 validation images, while the split2 [53] uses 3,682 images for training and 3,799 images for validation. The dataset includes three tasks: 2D detec- tion, 3D detection, and Bird's eye view, among which 3D detection is the focus of 3D detection methods. Evaluation Metrics. Precision-recall curves are used for evaluation (with the IoU threshold of 0.7). Prior to Aug. 2019, 11-point Interpolated Average Precision (AP) met- ric AP| R11 proposed in the Pascal VOC benchmark is sep- arately computed on each difficulty class and each ob- ject class. After that, the 40 recall positions-based metric AP| R40 is used instead of AP| R11 , following [45]. All meth- ods are ranked by AP| R11 of the 3D car detection in the moderate setting. Implementation Details. We use our depth-guided filter- ing module three times on the first three blocks of ResNet, which have different network strides of 4,8,16, respectively. [11] is used for depth estimation. A drop-channel layer with a drop rate of 0.2 is used after each module and a dropout layer with a drop rate of 0.5 is used after the output of the network backbone. For our single-stage detector, we use two convolutional layers as our detection head. The number of channels in the first layer is 512, and n a * (35+n c ) for the Method Test set Split1 Split2 Easy Moderate Hard Easy Moderate Hard Easy Moderate Hard OFT-Net [43] 1.61 1.32 1.00 4.07 3.27 3.29 - - - FQNet [31] 2.77 1.51 1.01 5.98 5.50 4.75 5.45 5.11 4.45 ROI-10D [34] 4.32 2.02 1.46 10.25 6.39 6.18 - - - GS3D [26] 4.47 2.90 2.47 13.46 10.97 10.38 11.63 10.51 10.51 Shift R-CNN [37] 6.88 3.87 2.83 13.84 11.29 11.08 - - - MonoGRNet [41] 9.61 5.74 4.25 13.88 10.19 7.62 - - - MonoPSR [23] 10.76 7.25 5.85 12.75 11.48 8.59 13.94 12.24 10.77 Mono3D-PLiDAR [50] 10.76 7.50 6.10 31.5 21.00 17.50 - - - SS3D [21] 10.78 7.68 6.51 14.52 13.15 11.85 9.45 8.42 7.34 MonoDIS [45] 10.37 7.94 6.40 11.06 7.60 6.37 - - - Pseudo-LiDAR [48] - - - 19.50 17.20 16.20 - - - M3D-RPN [1] 14.76 9.71 7.42 20.27 17.06 15.21 20.40 16.48 13.34 AM3D [1] 16.50 10.74 9.52 (+0.01) 32.23 (+5.26) 21.09 17.26 - - - D 4 LCN (Ours) 16.65 (+0.15) 11.72 (+0.98) 9.51 26.97 21.71 (+0.62) 18.22 (+0.96) 24.29 (+3.89) 19.54 (+3.06) 16.38 (+3.04) Table 2 , 2we can observe Table 3 . 3Comparisons of depth maps of different quality for 3D detection on the class car on the KITTI split1.Conv module AP| R 11 AP| R 40 Easy Moderate Hard Easy Moderate Hard Dynamic [20] 23.01 17.67 15.85 17.47 12.18 09.53 Dynamic Local [20] 25.15 18.42 16.27 21.09 13.93 11.31 Deformable [9] 23.98 18.24 16.11 19.05 13.42 10.07 D 4 LCN (ours) 26.97 21.71 18.22 22.32 16.20 12.30 Table 4 . 4Comparisons of different convolutional modules for car 3D detection on the KITTI split1. AP| R11 and AP| R40 metrics, respectively. This suggests that it is indeed effective to capture the meaningful local struc- ture for 3D object detection. (3) The main improvement comes from our adaptive dilated convolution (2.69 and 1.76 for AP| R11 and AP| R40 , respectively), which allows each channel of the feature map to have different receptive fields and thus solves the scale-sensitive problem. Note that we have tried different values of n f ∈ {1, 2, 3, 4, 5, 6}, and found that n f = 3 is the best. functions in our model to handle the scale problem adaptively. For example, filter 89 has large receptive fields for large-scale cars, while filter 70 deals with the small-scale cars.Filter No.41: adaptive dilated weights 0.14, 0.60 0.26 Filter No.89: adaptive dilated weights 0.05, 0.03, 0.92 Filter No.70: adaptive dilated weights 0.96, 0.02, 0.02 Input Image Figure 5. Visualization of active maps corresponding to different filters of block 3 of our D 4 LCN. Each filter learns three weights representing dilation rate of 1, 2, 3, respectively. Different filters have different Conv Method Dynamic Local Depth-wise Shift-pooling Dilated AP| R 11 AP| R 40 Easy Moderate Hard Easy Moderate HardConvNet 20.66 15.57 13.41 17.10 12.09 09.47 Depth-guided CN 23.01 17.67 15.85 17.47 12.18 09.53 Depth-guided LCN 25.15 18.42 16.27 21.09 13.93 11.31 Depth-guided DLCN 23.25 17.92 15.58 18.32 13.50 10.61 Depth-guided SP-DLCN 25.30 19.02 17.26 19.69 14.44 11.52 D 4 LCN 26.97 21.71 18.22 22.32 16.20 12.30 Table 7 . 7Comparisons of different convolutional methods for car 3D detection on the KITTI split1. M3d-rpn: Monocular 3d region proposal network for object detection. Garrick Brazil, Xiaoming Liu, ICCV. 710Garrick Brazil and Xiaoming Liu. M3d-rpn: Monocular 3d region proposal network for object detection. In ICCV, pages 9287-9296, 2019. 2, 3, 5, 7, 10 Deep manta: A coarse-to-fine many-task network for joint 2d and 3d vehicle analysis from monocular image. 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Xinshuo Weng, Kris Kitani, arXiv:1903.0984710arXiv preprintXinshuo Weng and Kris Kitani. Monocular 3d object detection with pseudo-lidar point cloud. arXiv preprint arXiv:1903.09847, 2019. 1, 3, 7, 8, 10 Shift: A zero flop, zero parameter alternative to spatial convolutions. Bichen Wu, Alvin Wan, Xiangyu Yue, Peter Jin, Sicheng Zhao, Noah Golmant, Amir Gholaminejad, Joseph Gonzalez, Kurt Keutzer, CVPR. Bichen Wu, Alvin Wan, Xiangyu Yue, Peter Jin, Sicheng Zhao, Noah Golmant, Amir Gholaminejad, Joseph Gonza- lez, and Kurt Keutzer. Shift: A zero flop, zero parameter alternative to spatial convolutions. In CVPR, pages 9127- 9135, 2018. 4 Dynamic filtering with large sampling field for convnets. Jialin Wu, Dai Li, Yu Yang, Chandrajit Bajaj, Xiangyang Ji, ECCV. Jialin Wu, Dai Li, Yu Yang, Chandrajit Bajaj, and Xiangyang Ji. Dynamic filtering with large sampling field for convnets. In ECCV, pages 185-200, 2018. 3 Data-driven 3d voxel patterns for object category recognition. 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{'abstract': '3D object detection from a single image without LiDAR is a challenging task due to the lack of accurate depth information. Conventional 2D convolutions are unsuitable for this task because they fail to capture local object and its scale information, which are vital for 3D object detection. To better represent 3D structure, prior arts typically transform depth maps estimated from 2D images into a pseudo-LiDAR representation, and then apply existing 3D point-cloud based object detectors. However, their results depend heavily on the accuracy of the estimated depth maps, resulting in suboptimal performance. In this work, instead of using pseudo-LiDAR representation, we improve the fundamental 2D fully convolutions by proposing a new local convolutional network (LCN), termed Depth-guided Dynamic-Depthwise-Dilated LCN (D 4 LCN), where the filters and their receptive fields can be automatically learned from image-based depth maps, making different pixels of different images have different filters. D 4 LCN overcomes the limitation of conventional 2D convolutions and narrows the gap between image representation and 3D point cloud representation. Extensive experiments show that D 4 LCN outperforms existing works by large margins. For example, the relative improvement of D 4 LCN against the state-of-theart on KITTI is 9.1% in the moderate setting. D 4 LCN ranks 1 st on KITTI monocular 3D object detection benchmark at the time of submission (car, December 2019) . The code is available at https', 'arxivid': '1912.04799', 'author': ['Mingyu Ding \nThe University of Hong\nKong\n\nGaoling School of Artificial Intelligence\nRenmin University of China\n\n', 'Yuqi Huo \nGaoling School of Artificial Intelligence\nRenmin University of China\n\n', 'Hongwei Yi \nShenzhen Graduate School\nPeking University\n\n', 'Zhe Wang \nSenseTime Research 5 Beijing Key Laboratory of Big Data Management and Analysis Methods\n100872BeijingChina\n', 'Jianping Shi shijianping@sensetime.com \nSenseTime Research 5 Beijing Key Laboratory of Big Data Management and Analysis Methods\n100872BeijingChina\n', 'Zhiwu Lu luzhiwu@ruc.edu.cnhongweiyi@pku.edu.cnwangzhe \nGaoling School of Artificial Intelligence\nRenmin University of China\n\n', 'Ping Luo pluo@cs.hku.hkbohony \nThe University of Hong\nKong\n'], 'authoraffiliation': ['The University of Hong\nKong', 'Gaoling School of Artificial Intelligence\nRenmin University of China\n', 'Gaoling School of Artificial Intelligence\nRenmin University of China\n', 'Shenzhen Graduate School\nPeking University\n', 'SenseTime Research 5 Beijing Key Laboratory of Big Data Management and Analysis Methods\n100872BeijingChina', 'SenseTime Research 5 Beijing Key Laboratory of Big Data Management and Analysis Methods\n100872BeijingChina', 'Gaoling School of Artificial Intelligence\nRenmin University of China\n', 'The University of Hong\nKong'], 'corpusid': 209140437, 'doi': '10.1109/cvprw50498.2020.00508', 'github_urls': ['https://github.com/dingmyu/D4LCN.'], 'n_tokens_mistral': 20497, 'n_tokens_neox': 17380, 'n_words': 9777, 'pdfsha': '8841036470c24823c2f33dcfbff855f12761876e', 'pdfurls': ['https://arxiv.org/pdf/1912.04799v2.pdf'], 'title': ['Learning Depth-Guided Convolutions for Monocular 3D Object Detection', 'Learning Depth-Guided Convolutions for Monocular 3D Object Detection'], 'venue': []}

arxiv

No Downlink Pilots are Needed in Massive MIMO 7 Jun 2016 Hien Quoc Ngo Department of Electrical Engineering (ISY) Linköping University 581 83LinköpingSweden Erik G Larsson Department of Electrical Engineering (ISY) Linköping University 581 83LinköpingSweden H Q Ngo Department of Electrical Engineering (ISY) Linköping University 581 83LinköpingSweden E G Larsson Department of Electrical Engineering (ISY) Linköping University 581 83LinköpingSweden No Downlink Pilots are Needed in Massive MIMO 7 Jun 20161 We consider the Massive Multiple-Input Multiple-Output (MIMO) downlink with maximum-ratio and zero-forcing processing and time-division duplex (TDD) operation. To decode, the terminals must know their instantaneous effective channel gain. Conventionally, it is assumed that by virtue of channel hardening, this instantaneous gain is close to its average and hence that terminals can rely on knowledge of that average (also known as statistical channel information). However, in some propagation environments, such as keyhole channels, channel hardening does not hold.We propose a blind algorithm to estimate the effective channel gain at each user, that does not require any downlink pilots. We derive a capacity lower bound of each user for our proposed scheme, applicable to any propagation channel. Compared to the case of no downlink pilots (relying on channel hardening), and compared to training-based estimation using downlink pilots, our blind algorithm performs significantly better. The difference is especially pronounced in environments that do not offer channel hardening. I. INTRODUCTION In Massive Multiple-Input Multiple-Output (MIMO), the base station is equipped with a large antenna array (with hundreds of antennas) that simultaneously serves many (tens or more of) users. It is a key, scalable technology for next generations of wireless networks, due to its promised huge energy efficiency and spectral efficiency [2]- [7]. In Massive MIMO, time-division duplex (TDD) operation is preferable, because the amount of pilot resources required does not depend on the number of base station antennas. With TDD, the base station obtains the channel state information (CSI) through uplink training. This CSI is used to detect the signals transmitted from users in the uplink. On downlink, owing to the reciprocity of propagation, CSI acquired at the base station is used for precoding. Each user receives an effective (scalar) channel gain multiplied by the desired symbol, plus interference and noise. To coherently detect the desired symbol, each user should know its effective channel gain. Conventionally, each user is assumed to approximate its instantaneous channel gain by its mean [8]- [10]. This is known to work well in Rayleigh fading. Since Rayleigh fading channels harden when the number of base station antennas is large (the effective channel gains become nearly deterministic), the effective channel gain is close to its mean. Thus, using the mean of this gain for signal detection works very well. This way, downlink pilots are avoided and users only need to know the channel statistics. However, for small or moderate numbers of antennas, the gain may still deviate significantly from its mean. Also, in propagation environments where the channel does not harden, using the mean of the effective gain as substitute for its true value may result in poor performance even with large numbers of antennas. The users may estimate their effective gain by using downlink pilots, see [2] for single-cell systems and [11] for multi-cell systems. Effectively, these downlink pilots are orthogonal between the users and beamformed along with the downlink data. The users estimate their effective channel gains using, for example, linear minimum mean-square error (MMSE) estimation. Compared with the case when the users rely on statistical channel knowledge, the downlink-pilot based schemes improve the system performance in low-mobility environments (where the coherence interval is long). However in high-mobility environments, they do not work well, owing to the large requirement of resources for downlink training: the required overhead is proportional to the number of multiplexed users. The effect of using outdated gain estimates at the users was investigated in [12]. Contributions of the paper: We consider the Massive MIMO downlink with maximum-ratio and zero-forcing processing, uplink channel estimation, and slow power control. Our specific contributions are: • We give a formal definition of channel hardening, and an associated criterion that can be used to test if channel hardening holds. Then we examine two important propagation scenarios: independent Rayleigh fading, and keyhole channels. We show that Rayleigh fading channels harden, but keyhole channels do not. • We propose a blind channel estimation scheme, that each user applies in the downlink. This scheme exploits the asymptotic properties of the sample average power of the received signal per coherence interval. We presented a preliminary version of this algorithm in [1]. • We derive a rigorous capacity lower bound for Massive MIMO with estimated downlink channel gains. This bound can be applied to any types of channels and can be used to analyze the performance of any downlink channel estimation method. • Via numerical results we show that, in hardening propagation environments, the performance of our proposed blind scheme is comparable to the use of only statistical channel information (approximating the gain by its mean). In contrast, in non-hardening propagation environments, our proposed scheme performs much better than the use of statistical channel information only. The results also show that our blind method uniformly outperforms schemes based on downlink pilots [2], [11]. Notation: We use boldface upper-and lower-case letters to denote matrices and column vectors, respectively. Specific notation and symbols used in this paper are listed as follows: an M × 1 channel vector, denoted by g k , and is modelled as: () * , () T ,g k = β k h k ,(1) where β k represents large-scale fading which is constant over many coherence intervals, and h k is an M × 1 small-scale fading channel vector. We assume that the elements of h k are uncorrelated, zero-mean and unit-variance random variables (RVs) which are not necessarily Gaussian distributed. Furthermore, h k and h k ′ are assumed to be independent, for k = k ′ . The mth elements of g k and h k are denoted by g m k and h m k , respectively. Here, we focus on the downlink data transmission with TDD operation. The base station uses the channel estimates obtained in the uplink training phase, and applies maximum-ratio or zero-forcing processing to transmit data to all K users in the same time-frequency resource. A. Uplink Training Let τ c be the length of the coherence interval (in symbols). For each coherence interval, let τ u,p be the length of uplink training duration (in symbols). All users simultaneously send pilot sequences of length τ u,p symbols each to the base station. We assume that these pilot sequences are pairwisely orthogonal. So it is required that τ u,p ≥ K. The linear MMSE estimate of g k is given by [13] ĝ k = τ u,p ρ u β k τ u,p ρ u β k + 1 g k + √ τ u,p ρ u β k τ u,p ρ u β k + 1 w p,k ,(2) where w p,k ∼ CN (0, I M ) independent of g k , and ρ u is the transmit signal-to-noise ratio (SNR) of each pilot symbol. The variance of the mth element ofĝ k is given by Var {ĝ m k } = E |ĝ m k | 2 = τ u,p ρ u β 2 k τ u,p ρ u β k + 1 γ k .(3) Letg k = g k −ĝ k be the channel estimation error, andg m k be the mth element ofg k . Then from the properties of linear MMSE estimation,g m k andĝ m k are uncorrelated. The variance of g m k is Var {g m k } = E |g m k | 2 = β k − γ k .(4) In the special case where g k is Gaussian distributed (corresponding to Rayleigh fading channels), the linear MMSE estimator becomes the MMSE estimator andg m k is independent ofĝ m k . B. Downlink Data Transmission Let s k (n) be the nth symbol intended for the kth user. We assume that E s(n)s(n) H = I K , where s(n) [s 1 (n), . . . , s K (n)] T . With linear processing, the M × 1 precoded signal vector is x(n) = K k=1 √ η k a k s k (n) = AD 1/2 η s(n),(5) where A [a 1 , . . . , a K ] is the precoding matrix which is a function of the channel estimatê E x(n) 2 ≤ 1.(6) The signal received at the kth user is 1 y k (n) = √ ρ d g H k x(n) + w k (n) = √ ρ d g H k AD 1/2 η s(n) + w k (n) = √ ρ d η k α kk s k (n) + K k ′ =k √ ρ d η k ′ α kk ′ s k ′ (n) + w k (n),(7) where α kk ′ g H k a k ′ , and w k (n) ∼ CN (0, 1) is additive Gaussian noise at the kth user. Then, the desired signal s k is decoded. We consider two linear precoders: maximum-ratio and zero-forcing processing. • Maximum-ratio processing: here the precoding vectors {a k } are a k =ĝ k ĝ k , k = 1, . . . , K.(8) • Zero-forcing processing: here the precoding vectors are a k = 1 Ĝ ĜHĜ −1 k Ĝ Ĝ HĜ −1 k , k = 1, . . . , K.(9) With the precoding vectors given in (8) and (9), the power constraint (6) becomes K k=1 η k ≤ 1.(10) III. PRELIMINARIES OF CHANNEL HARDENING One motivation of this work is that Massive MIMO channels may not always harden. In this section we discuss the channel hardening phenomena. We specifically study channel hardening for independent Rayleigh fading and for keyhole channels. Channel hardening is a phenomenon where the norms of the channel vectors {g k }, k = 1, . . . , K, fluctuate only little. We say that the propagation offers channel hardening if g k 2 E { g k 2 } P → 1, as M → ∞, k = 1, . . . , K.(11) A. Advantages of Channel Hardening If the base station and the users know the channel G perfectly, the channel is deterministic and its sum-capacity is given by [14] C = max η k ≥0, K k=1 η k ≤1 log 2 det I M + ρ d GD η G H ,(12) where D η is the diagonal matrix whose kth diagonal element is the power control coefficient η k . In Massive MIMO, for most propagation environments, we have asymptotically favorable propagation [15], i.e. g H k g k ′ M → 0, as M → ∞, for k = k ′ , and hence, when M is large, (12) can be approximated as: C ≈ max η k ≥0, K k=1 η k ≤1 K k=1 log 2 1 + ρ d η k g k 2 .(13) If the channel hardens, then the sum-capacity is C ≈ max η k ≥0, K k=1 η k ≤1 K k=1 log 2 (1 + ρ d η k β k M) ,(14) which does not depend on the small-scale fading. As a consequence, the system scheduling, power allocation, and interference management can be done over the large-scale fading time scale instead of the small-scale fading time scale. Therefore, the overhead for these system designs is significantly reduced. Another important advantage is: if the channel hardens, then we do not need instantaneous CSI at the receiver to detect the transmitted signals. What the receiver needs is only the statistical knowledge of the channel gains. This reduces the resources (power and training duration) required for channel estimation. More precisely, consider the signal received at the kth user given in (7). The kth user wants to detect s k from y k . For this purpose, it needs to know the effective channel gain α kk . If the channel hardens, then α kk ≈ E {α kk }. Therefore, we can use the statistical properties of the channel, i.e., E {α kk } is a good estimate of α kk when detecting s k . This assumption is widely made in the Massive MIMO literature [8]- [10] and circumvents the need for downlink channel estimation. B. Measure of Channel Hardening We next give a simple method to check whether the channel hardens or not. From the Chebyshev's inequality, we have Pr    g k 2 E g k 2 − 1 2 ≤ ǫ    ≥ 1 − 1 ǫ · Var g k 2 E g k 2 2 , for any ǫ ≥ 0.(15) Clearly, if Var g k 2 E g k 2 2 → 0, as M → ∞,(16) we have channel hardening. In contrast, (11) implies Var g k 2 E g k 2 2 → 0, as M → ∞, so if (16) does not hold, then the channel does not harden. Therefore, we can use Var{ g k 2 } (E{ g k 2 }) 2 to determine if channel hardening holds for a particular propagation environment. C. Independent Rayleigh Fading and Keyhole Channels In this section, we study the channel hardening property of two particular channel models: Rayleigh fading and keyhole channels. The transmitter and the receiver have their own local scatters which yield locally uncorrelated fading. However, the scatter rings are much smaller than the distance between them, the channel becomes low rank, and hence keyhole effects occur [18]; (2)-the receiver is located inside a building, the only way for the radio wave to propagation from the transmitter to the receiver is to go through several narrow holes which can be considered as keyholes; and (3)-the transmitter and the receiver are separated by a tunnel. (1) where {h m k } (the elements of h k ) are i.i.d. CN (0, 1) RVs. Independent Rayleigh fading channels occur in a dense, isotropic scattering environment [16]. By using the identity 1) Independent Rayleigh Fading Channels: Consider the channel model E { g k 4 } = β 2 k (M + 1)M [17], we obtain Var g k 2 E g k 2 2 = 1 β 2 k M 2 E g k 4 − 1 = 1 M → 0, M → ∞.(17) Therefore, we have channel hardening. 2) Keyhole Channels: A keyhole channel (or double scattering channel) appears in scenarios with rich scattering around the transmitter and receiver, and where there is a low-rank connection between the two scattering environments. The keyhole effect can occur when the radio wave goes through tunnels, corridors, or when the distance between the transmitter and receiver is large. Figure 1 shows some examples where the keyhole effect occurs in practice. This channel model has been validated both in theory and by practical experiments [19]- [22]. Under keyhole effects, the channel vector g k in (1) is modelled as [20]: g k = β k n k j=1 c (k) j a (k) j b (k) j ,(18) where n k is the number of effective keyholes, a (k) j is the random channel gain from the kth user to the jth keyhole, b (k) j ∈ C M ×1 is the random channel vector between the jth keyhole and the base station, and c (k) j are i.i.d. CN (0, 1) RVs. Furthermore, the gains {c (k) j } are normalized such that E {|g m k | 2 } = β k . Therefore, n k i=1 c (k) i 2 = 1.(19) Note that, when n k = 1, we have a degenerate keyhole (single-keyhole) channel. Conversely, when n k → ∞ and c (k) i = 0, we obtain an i.i.d. Rayleigh fading channel. We assume that different users have different sets of keyholes. This assumption is reasonable if the users are located at random in a large area, see Figure 1 for example. Then from the derivations in Appendix A, we obtain Var g k 2 E g k 2 2 = 1 + 1 M n k i=1 c (k) i 4 + 1 M (20) → n k i=1 c (k) i 4 = 0, M → ∞.(21) Therefore, we do not have channel hardening for keyhole channels. In addition, since c (k) i 2 ≤ 1, we have Var g k 2 E g k 2 2 ≤ 1 + 1 M n k i=1 c (k) i 2 + 1 M = 1 + 2 M ,(22) where the right hand side corresponds to the case of single-keyhole channels (n k = 1). This implies that in keyhole channels, single-keyhole channels offer the worst case in the sense that the channels fluctuate the most. IV. PROPOSED DOWNLINK BLIND CHANNEL ESTIMATION TECHNIQUE The kth user should know the effective channel gain α kk to coherently detect the transmitted signal s k from y k in (7) For the reasons explained, it is desirable that the users estimate their effective channels. One way to do this is to have the base station transmit beamformed downlink pilots [2]. Then at least In what follows, we propose a blind channel estimation method which does not require any downlink pilots. A. Downlink Blind Channel Estimation Algorithm We next describe our downlink blind channel estimation algorithm, a refined version of the scheme in [1]. Consider the sample average power of the received signal at the kth user per coherence interval: ξ k |y k (1)| 2 + |y k (2)| 2 + . . . + |y k (τ d )| 2 τ d ,(23) where y k (n) is the nth sample received at the kth user and τ d is the number of symbols per coherence interval spent on downlink transmission. From (7), we have ξ k − ρ d η k |α kk | 2 + K k ′ =k ρ d η k ′ |α kk ′ | 2 + 1 P → 0, as τ d → ∞.(24) Since K k ′ =k ρ d η k ′ |α kk ′ | 2 is a sum of many terms, it can be approximated by its mean. As a consequence, when K, and τ d are large, ξ k in (23) can be approximated as follows: ξ k ≈ ρ d η k |α kk | 2 + ρ d E K k ′ =k η k ′ |α kk ′ | 2 + 1.(25) Equation (25) enables us to estimate the amplitude of the effective channel gain α kk using the received samples via ξ k as follows: |α kk | = ξ k − 1 − ρ d E K k ′ =k η k ′ |α kk ′ | 2 ρ d η k .(26) In case the argument of the square root is non-positive, we set the estimate |α kk | equal to E {|α kk |}. For completeness, the kth user also needs to estimate the phase of α kk . When M is large, with high probability, the real part of α kk is much larger than the imaginary part of α kk . Thus, the phase of α kk is very small and can be set to zero. Based on that observation, we propose to treat the estimate of |α kk | as the estimate of the true α kk :α kk = |α kk | The algorithm for estimating the downlink effective channel gain α kk is summarized as follows: Algorithm 1: (Blind downlink channel estimation method) 1. For each coherence interval, using a data block of τ d samples y k (n), compute ξ k according to (23). 2. The estimate of the effective channel gain α kk is aŝ α kk =        ξ k −1−ρ d E{ K k ′ =k η k ′ |α kk ′ | 2 } ρ d η k , if ξ k > 1 + ρ d E K k ′ =k η k ′ |α kk ′ | 2 , E {|α kk |} , otherwise.(27) Remark 1: To implement Algorithm 1, the kth user has to know η k and E K k ′ =k η k ′ |α kk ′ | 2 . We assume that the kth user knows these values. This assumption is reasonable since these values depend only on the large-scale fading coefficients, which stay constant over many coherence intervals. The base station can compute these values and inform the kth user about them. In addition E K k ′ =k η k ′ |α kk ′ | 2 can be expressed in closed form (except for in the case of zero-forcing processing with keyhole channels) as follows: E K k ′ =k η k ′ |α kk ′ | 2 =      K k ′ =k η k ′ β k , for maximum-ratio, Rayleigh/keyhole channels, K k ′ =k η k ′ (β k − γ k ) , for zero-forcing, Rayleigh channels. (28) Detailed derivations of (28) are presented in Appendix B. B. Asymptotic Performance Analysis In this section, we analyze the accuracy of our proposed downlink blind channel estimation scheme when τ c and M go to infinity for two specific propagation channels: Rayleigh fading and keyhole channels. We use the model (18) for keyhole channels. When τ c → ∞, ξ k in (23) is equal to its asymptotic value: ξ k − ρ d η k |α kk | 2 + K k ′ =k ρ d η k ′ |α kk ′ | 2 + 1 → 0,(29) and hence, the channel estimateα kk in (27) becomeŝ α kk =      |α kk | 2 + K k ′ =k η k ′ η k (|α kk ′ | 2 − E {|α kk ′ | 2 }), if ξ k > 1 + ρ d E K k ′ =k η k ′ |α kk ′ | 2 , E {|α kk |} , otherwise.(30) Since τ c → ∞, it is reasonable to assume that the base station can perfectly estimate the channels in the uplink training phase, i.e., we haveĜ = G. (This can be achieved by using very long uplink training duration.) With this assumption, α kk is a positive real value. Thus, (30) can be rewritten aŝ α kk α kk =      1 + K k ′ =k η k ′ η k |α kk ′ | 2 −E{|α kk ′ | 2 } α 2 kk , if ξ k > 1 + ρ d E K k ′ =k η k ′ |α kk ′ | 2 , E{α kk } α kk , otherwise.(31) 1) Maximum-Ratio Processing: With maximum-ratio processing, from (28) and (31), we havê α kk α kk =            1 + K k ′ =k η k ′ η k g H k g k ′ g k ′ 2 −β k g k 2 , if ξ k > 1 + ρ d K k ′ =k η k ′ β k , E{ g k } g k , otherwise.(32) -Rayleigh fading channels: Under Rayleigh fading channels, α kk = g k , and hence, Pr ξ k > 1 + ρ d K k ′ =k η k ′ β k ≥ Pr 1 M g k 2 > 1 M K k ′ =k η k ′ η k β k → 1, as M → ∞,(33) where the convergence follows the fact that 1 M g k 2 → β k and 1 M K k ′ =k η k ′ η k β k → 0, as M → ∞. In addition, by the law of large numbers, g H k g k ′ g k ′ 2 − β k g k 2 = g H k g k ′ M 2 M g k ′ 2 − β k M M g k 2 → 0, as M → ∞.(34) From (32), (33), and (34), we obtain α kk α kk → 1, as M → ∞.(35) Our proposed scheme is expected to work very well at large τ c and M. -Keyhole channels: Following a similar methodology used in the case of Rayleigh fading, and using the identity g H k g k ′ g k ′ = β k n k j=1 c (k) j a (k) j ν (k) j ,(36) where ν (k) j b (k ′ ) j H g k ′ g k ′ is CN (0, 1) distributed, we can arrive at the same result as (35). The random variable ν (k) j is Gaussian due to the fact that conditioned on g k ′ , ν (k) j is a Gaussian RV with zero mean and unit variance which is independent of g k ′ . 2) Zero-forcing Processing: With zero-forcing processing, when τ c → ∞, α kk α kk → 1, as M → ∞.(37) This follows from (29) and the fact that α kk ′ → 0, for k = k ′ . V. CAPACITY LOWER BOUND In this section, we derive a capacity lower bound for Massive MIMO with downlink channel gain estimation. It can be applied, in particular, to our proposed blind channel estimation scheme. Denote by y k [y k (1) . . . y k (τ d )] T , s k [s k (1) . . . s k (τ d )] T , and w k [w k (1) . . . w k (τ d )] T . Then from (7), we have y k = √ ρ d η k α kk s k + K k ′ =k √ ρ d η k ′ α kk ′ s k ′ + w k .(38) The capacity of (38) is lower bounded by the mutual information between the unknown transmitted signal s k and the observed/known values y k ,α kk . More precisely, for any distribution of s k , we obtain the following capacity bound for the kth user: C k ≥ 1 τ d I(y k ,α kk ; s k ) = 1 τ d h(s k ) − h(s k |y k ,α kk ) (a) = 1 τ d h(s k ) − 1 τ d h s k (1)|y k ,α kk + h s k (2)|s k (1), y k ,α kk + . . . + h s k (τ d )|s k (1), . . . , s k (τ d − 1), y k ,α kk , (b) ≥ 1 τ d h(s k ) − 1 τ d h (s k (1)|y k ,α kk ) + h (s k (2)|y k ,α kk ) + . . . + h (s k (τ d )|y k ,α kk ) ,(39) where in (a) we have used the chain rule [23], and in (b) we have used the fact that conditioning reduces entropy. It is difficult to compute h (s k (n)|y k ,α kk ) in (39) sinceα kk and s k (n) are correlated. To render the problem more tractable, we introduce new variables {α kk (n)}, n = 1, ..., τ d , which can be considered as the channel estimates of α kk using Algorithm 1, but ξ k is now computed as ξ k = |y k (1)| 2 + . . . + |y k (n − 1)| 2 + |y k (n + 1)| 2 . . . + |y k (τ d )| 2 τ d − 1 .(40) Clearly,α kk (n) is very close toα kk . More importantly,α kk (n) is independent of s k ′ (n), k ′ = 1, ..., K. This fact will be used for subsequent derivation of the capacity lower bound. Sinceα kk (n) is a function of y k , h (s k (n)|y k ,α kk ) = h s k (n)|y k ,α kk ,α kk (n) , and hence, (39) becomes C k ≥ 1 τ d h(s k ) − 1 τ d h s k (1)|y k ,α kk ,α kk (1) + . . . + h s k (τ d )|y k ,α kk ,α kk (τ d ) ≥ 1 τ d h(s k ) − 1 τ d h s k (1)|y k (1),α kk (1) + . . . + h s k (τ d )|y k (τ d ),α kk (τ d ) ,(41) where in the last inequality, we have used again the fact that conditioning reduces entropy. The bound (41) holds irrespective of the distribution of s k . By taking s k (1), . . . , s k (τ d ) to be i.i.d. CN (0, 1), we obtain C k ≥ log 2 (πe) − h s k (1)|y k (1),α kk (1) .(42) The right hand side of (42) is the mutual information between y k (1) and s k (1) given the side informationα kk (1). Sinceα kk (1) and s k ′ (1), k ′ = 1, ..., K, are independent, we have E K k ′ =k √ ρ d η k ′ α kk ′ s k ′ (1) + w k (1) α kk (1) = 0, E s * k (1) K k ′ =k √ ρ d η k ′ α kk ′ s k ′ (1) + s * k (1)w k (1) α kk (1) = 0, E α * kk s * k (1) K k ′ =k √ ρ d η k ′ α kk ′ s k ′ (1) + α * kk s * k (1)w k (1) α kk (1) = 0.(43) Hence we can apply the result in [24] to further bound the capacity for the kth user: C k ≥ R blind k E      log 2    1 + E y * k (1)s k (1)|α kk (1) 2 E |y k (1)| 2 α kk (1) − E y * k (1)s k (1)|α kk (1) 2         .(44) Inserting (7) into (44), we obtain a capacity lower bound (achievable rate) for the kth user: R blind k = E      log 2    1 + ρ d η k E α kk |α kk (1) 2 1 + ρ d K k ′ =1 η k ′ E |α kk ′ | 2 α kk (1) − ρ d η k E α kk |α kk (1) 2         .(45) Remark 2: The computation of the capacity lower bound (45) involves the expectations E α kk |α kk (1) and E |α kk ′ | 2 α kk (1) which cannot be directly computed. However, we can compute E α kk |α kk (1) and E |α kk ′ | 2 α kk (1) numerically by first using Bayes's rule and then discretizing it using the Riemann sum: E {X|y} = x xp X|Y (x|y)dx = x x p X,Y (x, y) p Y (y) dx ≈ i x i p X,Y (x i , y) p Y (y) △ x i ,(46) where △ x i x i − x i−1 . Precise steps to compute (45) are as follows: 1. Generate N random realizations of the channel G. Then the corresponding N × 1 random vectors of α kk , |α kk ′ | 2 , andα kk (1) are obtained. 2. From sample vectors obtained in step 1, numerically build the density function {pα kk (1) (x i )} and the joint density functions {p α kk ,α kk (1) (y j , x i )} and p |α kk ′ | 2 ,α kk (1) (z n , x i ). These density functions can be numerically computed using built-in functions in MATLAB such as "kde" and "kde2d". 3. Using (46), we compute the achievable rate (45) as R blind k = i pα kk (1) (x i ) △ x i log 2      1+ ρ d η k |E { α kk | x i }| 2 1 + ρ d K k ′ =1 η k ′ E |α kk ′ | 2 x i − ρ d η k |E { α kk | x i }| 2      ,(47) where E {α kk | x i } = j y j △ y j p α kk ,α kk (1) (y j , x i ) pα kk (1) (x i ) ,(48)E |α kk ′ | 2 x i = n z n △ zn p |α kk ′ | 2 ,α kk (1) (z n , x i ) pα kk (1) (x i ) .(49) VI. NUMERICAL RESULTS AND DISCUSSIONS In this section, we provide numerical results to evaluate our proposed channel estimation scheme. We consider the per-user normalized MSE and net throughput as performance metrics. We define MMSE channel estimation; and iii) "proposed scheme", representing our proposed downlink blind channel estimation scheme (using Algorithm 1). In our proposed scheme, the curves with τ d = ∞ correspond to the case that the kth user perfectly knows the asymptotic value of ξ k . SNR d = ρ d × Furthermore, we choose τ u,p = K. For the beamforming training scheme, the duration of the downlink training is chosen as τ d,p = K. A. Normalized Mean-Square Error We consider the normalized MSE at the kth user defined as: MSE k E |α kk − α kk | 2 |E {α kk }| 2 .(50) In this part, we choose β k = 1, and equal power allocation to all users, i.e, η k = 1/K, for k = 1, . . . , K. Figures 2 and 3 show the normalized MSE versus SNR d for maximum-ratio and zero-forcing processing, respectively, under Rayleigh fading and single-keyhole channels. Here, we choose M = 100, K = 10, and SNR u = 0 dB. We can see that, in Rayleigh fading channels, for both maximum-ratio and zero-forcing processing, the MSEs of the three schemes (use E {α kk }, DL pilots, and proposed scheme) are comparable. Using E {α kk } in lieu of the true α kk for signal detection works rather well. However, in keyhole channels, since the channels do not harden, the MSE when using E {α kk } as the estimate of α kk is very large. In both propagation environments, our proposed scheme works very well and improves when τ d increases (since the approximation in (25) becomes tighter). Our scheme outperforms the beamforming training scheme for a wide range of SNRs, even for short coherence intervals (e.g., τ d = 100 symbols). Next we study the affects of the number of base station antennas and the number of keyholes on the performance of our proposed scheme. We choose K = 10, τ d = 100, SNR u = 0 dB, and SNR d = 5 dB. Figure 4 shows the normalized MSE versus M for different numbers of keyholes n KH with maximum-ratio and zero-forcing processing. When n KH = ∞, we have Rayleigh fading. As expected, the MSE reduces when M increases. More importantly, our proposed scheme works well even when M is not large. Furthermore, we can see that the MSE does not change much when the number of keyholes varies. This implies the robustness of our proposed scheme against the different propagation environments. Note that, with the beamforming training scheme in [2], we additionally have to spend at least K symbols on training pilots (this is not accounted for here, since we only evaluate MSE). By contrast, our proposed scheme does not require any resources for downlink training. To account for the loss due to training, we will examine the net throughput in the next part. B. Downlink Net Throughput The downlink net throughputs of three cases-use E {α kk }, DL pilots, and proposed schemesare defined as: S noCSI k = B τ d τ c R noCSI k ,(51)S pilot k = B τ d − τ d,p τ c R pilot k ,(52)S blind k = B τ d τ c R blind k ,(53) where B is the spectral bandwidth, τ c is again the coherence interval in symbols, and τ d is the We consider a more realistic scenario which incorporates the large-scale fading and max-min power control: • To generate the large-scale fading, we consider an annulus-shaped cell with a radius of R max meters, and the base station is located at the cell center. K + 1 users are placed uniformly at random in the cell with a minimum distance of R min meters from the base station. The user with the smallest large-scale fading β k is dropped, such that K users remain. The large-scale fading is modeled by path loss, shadowing (with log-normal distribution), and random user locations: β k = PL 0 d k R min υ × 10 σ sh ·N (0,1) 10 ,(54) where υ is the path loss exponent and σ sh is the standard deviation of the shadow fading. The factor PL 0 in (54) is a reference path loss constant which is chosen to satisfy a given downlink cell-edge SNR, SNR d . In the simulation, we choose R min = 100, R max = 1000, υ = 3.8, and σ sh = 8 dB. We generate 1000 random realizations of user locations and shadowing fading profiles. • The power control control coefficients {η k } are chosen from the max-min power control algorithm [25]: η k =            1+ρ d β k ρ d γ k 1 ρ d K k ′ =1 1 γ k ′ + K k ′ =1 β k ′ γ k ′ , for maximum-ratio processing 1+ρ d (β k −γ k ) ρ d γ k 1 ρ d K k ′ =1 1 γ k ′ + K k ′ =1 β k ′ −γ k ′ γ k ′ , for zero-forcing processing (55) This max-min power control offers uniformly good service for all users for the case where the kth user uses E {α kk } as estimate of α kk . both propagation environments, our proposed scheme is the best and performs very close to the genie receiver. For Rayleigh fading channels, due to the hardening property of the channels, our proposed scheme and the scheme using statistical property of the channels are comparable. These schemes perform better than the beamforming training scheme in [2]. The reason is that, with beamforming training scheme, we have to spend τ d,p pilot samples for the downlink training. For single-keyhole channels, the channels do not harden, and hence, it is necessary to estimate the effective channel gains. Our proposed scheme improves the system performance significantly. At SNR d = 5 dB, with maximum-ratio processing, our proposed scheme can improve the 95%likely net throughput by about 20% and 60%, compared with the downlink beamforming training scheme respectively the case of without channel estimation. With zero-forcing processing, our proposed scheme can improve the 95%-likely net throughput by 15% and 66%, respectively. VII. COMMENTS A. Short-Term V.s. Long-Term Average Power Constraint The precoding vectors a k in (8) and (9) are chosen to satisfy a short-term average power constraint where the expectation of (6) is taken over only s(n). This short-term average power constraint is not the only possibility. Alternatively, one could consider a long-term average power constraint where the expectation in (6) is taken over s(n) and over the small-scale fading. With maximum-ratio combining, the long-term-average-power-based precoding vectors {a k } are a k =ĝ k E { ĝ k 2 } =ĝ k √ Mγ k , k = 1, . . . , K.(56) However, with zero-forcing, the long-term-average-power-based precoder is not always valid. For example, for single-keyhole channels, perfect uplink estimation, and K = 1, we have E G G H G −1 k 2 ,(57) which is infinite. We emphasize here that compared to the short-term average power case, the long-term average power case does not make a difference in the sense that the resulting effective channel gain does not always harden, and hence, it needs to be estimated. (The harding property of the channels is discussed in detail in Section III.) To see this more quantitatively, we compare the performance of three cases: "use E {α kk }", "DL pilots [2]", and "proposed scheme" for maximum-ratio with long-term average power constraint (56). As seen in Figure 7, under keyhole channels, our proposed scheme improves the net throughput significantly, compared to the "use E {α kk }" case. [2], [11] In the above numerical results, the curves with downlink pilots are obtained by first replacinĝ α kk (1) in (45) with the channel estimate obtained using the algorithm in [2], and then using the numerical technique discussed in Remark 2 to compute the capacity bound. B. Flaws of the Bound in Closed-form expressions for achievable rates with downlink training were given in [2,Eq. (12)] and [11]. However, those formulas were not rigorously correct, since {a kk ′ } are non-Gaussian in general (even in Rayleigh fading) and hence the linear MMSE estimate is not equal to the MMSE estimate; the expressions for the capacity bounds in [2], [11] are valid only when the MMSE estimate is inserted. However, the expressions [2], [11] are likely to be extremely accurate approximations. A similar approximation was stated in [26]. VIII. CONCLUSION In the Massive MIMO downlink, in propagation environments where the channel hardens, using the mean of the effective channel gain for signal detection is good enough. However, the channels may not always harden. Then, to reliably decode the transmitted signals, each user should estimate its effective channel gain rather than approximate it by its mean. We proposed a new blind channel estimation scheme at the users which does not require any downlink pilots. With this scheme, the users can blindly estimate their effective channel gains directly from the data received during a coherence interval. Our proposed channel estimation scheme is computationally easy, and performs very well. The results show that our proposed scheme outperforms both the downlink beamforming training scheme in [2] and the conventional approach that approximates the effective channel gains by their means. APPENDIX A. Derivation of (20) We have, Var g k 2 the double sum of (58)] are zero-mean mutual uncorrelated random variables. Furthermore, they are uncorrelated with n k Fig. 1 . 1Examples of keyhole channels: (1)-keyhole effects occur when the distance between transmitter and receiver is large. K downlink pilot symbols are required. This can significantly reduce the spectral efficiency. For example, suppose M = 200 antennas serve K = 50 users, in a coherence interval of length 200 symbols. If half of the coherence interval is used for the downlink, then with the downlink beamforming training of [2], we need to spend at least 50 symbols for sending pilots. As a result, less than 50 of the 100 downlink symbols are used for payload in each coherence interval, and the insertion of the downlink pilots reduces the overall (uplink + downlink) spectral efficiency by a factor of 1/4. Fig. 2 . 2median[large-scale fading of the cell-edge user], and SNR u = ρ u × median[large-scale fading of the cell-edge user]. This gives SNR d and SNR u the interpretation of the median downlink and the uplink cell-edge SNRs. For keyhole channels, we assume n k = n KH and c (k) j = 1/ √ n KH , for all k = 1, . . . , K and j = 1, . . . , n KH .In all examples, we compare the performances of three cases: i) "use E {α kk }", representing the case when the kth user relies on the statistical properties of the channels, i.e., it uses E {α kk } as estimate of α kk ; ii) "DL pilots[2]", representing the use of beamforming training[2] Normalized MSE versus SNRd for different channel estimation schemes, for maximum-ratio processing. Here, M = 100, K = 10, and SNRu = 0 dB. Fig. 3 . 3Same asFigure 2, but for zero-forcing processing. Fig. 4 . 4Normalized MSE versus M for different number of keyholes n k = nKH, using Algorithm 1. Here, SNRu = 0 dB, SNRd = 5 dB, and K = 10. Figures 5 and 6 Fig. 5 .Fig. 6 . 656show the cumulative distributions of the per-user downlink net throughput for maximum-ratio respectively zero-forcing processing, under Rayleigh fading and single-keyhole channels. Here we choose M = 100, K = 10, τ c = 200, and SNR d = 10SNR u . As a baseline for comparisons, we additionally add the curves labelled "perfect CSI". These curves represent the presence of a genie receiver at the kth user, which knows the channel gain perfectly. The cumulative distribution of the per-user downlink net throughput for maximum-ratio processing. Here, M = 100, K = 10, SNRd = 10SNRu, and B = 20 MHz. Same asFigure 5, but for zero-forcing processing. Fig. 7 . 7Same asFigure 5, but with long-term average power constraint (56). and () H Conjugate, transpose, and transpose conjugate, respectively det (·) and Tr (·) Determinant and trace of a matrix We consider a single-cell Massive MIMO system with an M-antenna base station and K single-antenna users, where M > K. The channel between the base station and the kth user isCN (0, Σ) Circularly symmetric complex Gaussian vector with zero mean and covariance matrix Σ | · | and · Absolute value and Euclidean norm, respectively E {·} and Var {·} Expectation and variance operators P → Convergence in probability I n n × n identity matrix II. SYSTEM MODEL . Most previous works on Massive MIMO assume that E {α kk } is used in lieu of the true α kk when detecting s k . The reason behind this is that if the channel is subject to Rayleigh fading (the scenario considered in most previous Massive MIMO works), it hardens when the number of base station antennas is large, and hence α kk ≈ E {α kk }. So E {α kk } is a good estimate of α kk . However, as seen in Section III, the channel may not always harden when M → ∞ and then, using E {α kk } as the true effective channel α kk to detect s k may result in poor performance. Here we restrict our consideration to one coherence interval so that the channels remain constant. i . We can see that, the terms in the double sum have zero mean. We now consider the covariance between two arbitrary terms:where we used the fact that if z is a circularly symmetric complex Gaussian random variable with zero mean, then E {z 2 } = 0. The above result implies that the terms b (k)i Hb (k) Blind estimation of effective downlink channel gains in massive MIMO. H Q Ngo, E G Larsson, Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE International Conference on Acoustics, Speech and Signal essing (ICASSP)Brisbane, AustraliaH. Q. Ngo and E. G. Larsson, "Blind estimation of effective downlink channel gains in massive MIMO," in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brisbane, Australia, Apr. 2015. Massive MU-MIMO downlink TDD systems with linear precoding and downlink pilots. 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{'abstract': 'We consider the Massive Multiple-Input Multiple-Output (MIMO) downlink with maximum-ratio and zero-forcing processing and time-division duplex (TDD) operation. To decode, the terminals must know their instantaneous effective channel gain. Conventionally, it is assumed that by virtue of channel hardening, this instantaneous gain is close to its average and hence that terminals can rely on knowledge of that average (also known as statistical channel information). However, in some propagation environments, such as keyhole channels, channel hardening does not hold.We propose a blind algorithm to estimate the effective channel gain at each user, that does not require any downlink pilots. We derive a capacity lower bound of each user for our proposed scheme, applicable to any propagation channel. Compared to the case of no downlink pilots (relying on channel hardening), and compared to training-based estimation using downlink pilots, our blind algorithm performs significantly better. The difference is especially pronounced in environments that do not offer channel hardening.', 'arxivid': '1606.02348', 'author': ['Hien Quoc Ngo \nDepartment of Electrical Engineering (ISY)\nLinköping University\n581 83LinköpingSweden\n', 'Erik G Larsson \nDepartment of Electrical Engineering (ISY)\nLinköping University\n581 83LinköpingSweden\n', 'H Q Ngo \nDepartment of Electrical Engineering (ISY)\nLinköping University\n581 83LinköpingSweden\n', 'E G Larsson \nDepartment of Electrical Engineering (ISY)\nLinköping University\n581 83LinköpingSweden\n'], 'authoraffiliation': ['Department of Electrical Engineering (ISY)\nLinköping University\n581 83LinköpingSweden', 'Department of Electrical Engineering (ISY)\nLinköping University\n581 83LinköpingSweden', 'Department of Electrical Engineering (ISY)\nLinköping University\n581 83LinköpingSweden', 'Department of Electrical Engineering (ISY)\nLinköping University\n581 83LinköpingSweden'], 'corpusid': 18540503, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 16060, 'n_tokens_neox': 14195, 'n_words': 8897, 'pdfsha': '05c54e0e46972b964d91dac7c97357ff358e136a', 'pdfurls': ['https://arxiv.org/pdf/1606.02348v1.pdf'], 'title': ['No Downlink Pilots are Needed in Massive MIMO', 'No Downlink Pilots are Needed in Massive MIMO'], 'venue': []}

arxiv

Stability Analysis of Convection in the Intracluster Medium H Gupta Department of Physics Indian Institute of Technology Kanpur U. P.-208016India S K Rathor Department of Physics Indian Institute of Technology Kanpur U. P.-208016India M E Pessah Niels Bohr International Academy Niels Bohr Institute 2100Copenhagen ØDenmark S Chakraborty Department of Physics Indian Institute of Technology Kanpur U. P.-208016India Mechanics & Applied Mathematics Group Indian Institute of Technology Kanpur U.P.-208016India Stability Analysis of Convection in the Intracluster Medium ConvectionIntracluster mediumGalaxy clusterLinear stability analysisMagnetothermal instabilityHeat-flux-driven buoyancy instability We use the machinery usually employed for studying the onset of Rayleigh-Bénard convection in hydro-and magnetohydrodynamic settings to address the onset of convection induced by the magnetothermal instability and the heat-flux-buoyancy-driveninstability in the weakly-collisional magnetized plasma permeating the intracluster medium. Since most of the related numerical simulations consider the plasma being bounded between two 'plates' on which boundary conditions are specified, our strategy provides a framework that could enable a more direct connection between analytical and numerical studies. We derive the conditions for the onset of these instabilities considering the effects of induced magnetic tension resulting from a finite plasma beta. We provide expressions for the Rayleigh number in terms of the wave vector associated with a given mode, which allow us to characterize the modes that are first to become unstable. For both the heat-flux-buoyancy-driven-instability and the magnetothermal instability, oscillatory marginal stable states are possible.PACS numbers 98.65. Hb, 44.25.+f Introduction Convection, i.e., the motions induced within a fluid by the tendency of hotter, less dense material to rise, and colder, denser material to sink under the influence of gravity, is a ubiquitous phenomenon in nature. These motions, and the ensuing transfer of heat, can have important implications for a wide variety of systems, see e.g., Getling (1997), ranging from laboratory settings to the Earth and the oceans, from planetary to stellar atmospheres, and from accretion disks (Stone and Balbus, 1996;Lesur and Ogilvie, 2010;Bodo et al., 2012;Oliver, 2013) to the intracluster medium (ICM) permeating galaxy clusters (Balbus, 2001;Quataert, 2008;Parrish et al., 2009;Bogdanović et al., 2009;McCourt et al., 2011;Kunz et al., 2012). The inherent nonlinearity of the governing equations, together with the complex dynamical boundaries present in nature, has motivated the study of convection in idealized settings where the fluid is confined between two parallel horizontal plates and is heated from below. When this setup leads to convective motions, this is termed Rayleigh-Bénard convection (RBC). The stability of the equilibrium state and the flow dynamics in RBC are determined by a non-dimensional parameter viz., the Rayleigh number R, which is a measure of the strength of the destabilizing buoyancy force relative to the stabilising viscous force in the fluid. When the Rayleigh number for a given fluid is below a critical number, then heat transfer occurs primarily via conduction; when this critical number is exceeded, heat transfer is primarily via convection. The onset of the instability and the critical value of R can be understood by means of a linear stability analysis (Chandrashekhar, 1981). The rich nonlinear phenomena (e.g., pattern formation, route to chaos, turbulence, etc.) ensuing in such a convective system can also be analytically investigated in the weakly nonlinear limit (Bhattacharjee, 1989;Cross and Greenside, 2009). There is a large body of literature on flow reversals, pattern formation and evolution in RBC encompassing both experiments (Morris et al., 1993;Assenheimer and Steinberg, 1996) as well as nonlinear two-dimensional (2D) (Chandra and Verma, 2013) and three dimensional (3D) simulations (Getling and Brausch, 2003). The study of RBC has benefited the understanding of convection in a wide variety of systems in nature, for instance, in the Earth's outer core (Cardin and Olson, 1994), mantle (Mckenzie, Roberts, and Weiss, 1974), atmosphere (Hartmann, Moy, and Fu, 2001), and oceans (Marshall and Schott, 1999), as well as in Sun spots (Cattaneo, Emonet, and Weiss, 2003), and in metal production processes (Brent, Voller, and Reid, 1988). The framework employed to study RBC has been generalized by considering the presence of magnetic fields and even incorporating the effects of rotation, a combination prevalent in astrophysical fluids. This approach has shed light into the gener-ation and reversal of the Earth magnetic field (Glatzmaiers and Roberts, 1995) and the internal dynamics of the Sun (Brandenburg et al., 1996;Cattaneo et al., 2003). In all of the cases in which conducting media have been considered, the plasma has been assumed to behave as a magnetized fluid, as described in the magnetohydrodynamic (MHD) approximation. There are situations of astrophysical interest, however, in which the plasma is only weakly-collisional, i.e., the mean free path for particles to interact is much larger than the Larmor radius. This is the case for the dilute ICM permeating galaxy clusters, in which transport properties are anisotropic with respect to the direction of the magnetic field. The aim of this letter is to build upon the machinery employed to study RBC in hydro-and magnetohydro-dynamic scenarios in order to address the onset of convection in the weakly-collisional magnetized plasma in galaxy clusters. Instabilities in the Weakly-Collisional ICM The ICM is a weakly collisional and high-beta plasma (see e.g., (Carilli and Taylor, 2002;Peterson and Fabian, 2006)), in which the transport of heat, transport of momentum and diffusion of ions is anisotropic due to the presence of magnetic field. Linear stability analysis has shown that the ICM is dynamically unstable, to the so-called magnetothermal instability, MTI, (Balbus, 2000(Balbus, , 2001 and the heat-flux-driven buoyancy instability, HBI (Quataert, 2008). The MTI sets in when the temperature gradient decreases outwards and the magnetic field lines are perpendicular to the direction of gravity, whereas the HBI is excited when the temperature gradient increases outwards and the magnetic field lines are parallel to the gravitational field. The original studies of these instabilities have been generalized to explore the effects of viscous anisotropy (Ren et al., 2010;Kunz, 2011) and semi-global settings (Latter and . More recently, Pessah and Chakraborty (2013); Berlok and Pessah (2015) analyzed the stability of the ICM generalizing previous work by considering the effects of concentration gradients that could be present in the ICM if the sedimentation of Helium is effective (Chuzhoy and Loeb, 2004;Peng and Nagai, 2009;Shtykovskiy and Gilfanov, 2010). Researchers have carried out nonlinear numerical studies of the MTI Stone, 2005, 2007;McCourt et al., 2011) and the HBI (Parrish and Quataert, 2008;Parrish et al., 2009Parrish et al., , 2010McCourt et al., 2011;Kunz et al., 2012) in connection with the 'cooling flow problem' in cool core galaxy clusters. The effects of shear flow (and thus, Kelvin-Helmholtz instability) on the stability condition for MTI is explored in Ren et al. (2011). Recently, Nipoti and Posti (2014) performed linear stability analysis on weakly magnetized, rotating plasma in both collisional and collisionless environments, leading to more complete picture of ICM. Advantages of the Rayleigh-Bénard Approach There are a number of advantages that follow from employing the machinery developed for RBC to the study of the MTI and HBI. This approach allows us to shed light into many aspects of the MTI and the HBI, which are thought to play a role in the dynamics of the intracluster medium (ICM). For instance, 1. This framework provides a good platform to several connections with numerical simulations because the boundary conditions (BCs) usually adopted resemble the ones employed in RBC. 2. The results obtained can help us identify the critical Rayleigh number for the onset of the MTI and the HBI. 3. The formalism allows us to account for the effects of magnetic tension on the stability criterion for both the MTI and the HBI. This approach could be useful in order to assess the effects of magnetic tension on the unstable growing modes found to feed off composition gradients in a inhomogeneous intracluster medium (Pessah and Chakraborty, 2013;Berlok and Pessah, 2015) 4. The analysis could enable a low dimensional model like the Lorenz model for RBC (Lorenz, 1963;Chen and Price, 2006) and magnetic RBC (Zierep, 2003), which could give further insights into the chaotic (turbulent) state of the ICM. The Rayleigh-Bénard Framework Let us consider a weakly-collisional plasma at rest confined between two horizontal parallel plates of infinite extent, as it is shown in Fig. 1. The vertical separation between the plates is d and the acceleration due to gravity g is acting vertically downwards. The bottom and the top boundaries are held at two different constant temperatures T bottom and T top , respectively. This sets up a constant background temperature gradient in the confined plasma. There is also an externally imposed uniform magnetic field B lying on the x − z plane and acting on the system under study. Governing Equations The equations of motion describing the dynamics of this system are given by ∂ρ ∂t + ∇·(ρv) = 0 ,(1)dv dt = − 1 ρ ∇· P + B 2 8π I − B 2 4πbb + g ,(2)∂B ∂t = ∇×(v×B) + η∇ 2 B ,(3)ρT ds dt = (p ⊥ − p ) d dt ln B ρ γ−1 − ∇·Q s .(4) Here, the Lagrangian and Eulerian derivatives are related via d/dt ≡ ∂/∂t + v·∇, where v is the fluid velocity. The symbols ρ, T , s, γ and η stand for the fluid density, temperature (assumed to be the same for ions and electrons), specific entropy, adiabatic index and electrical resistivity (also called magnetic diffusivity). The weakly collisional character of the plasma renders its physical properties anisotropic with respect to the local direction of the magnetic field. The pressure tensor is P ≡ p ⊥ I + (p − p ⊥ )bb, where I stands for the 3 × 3 identity matrix. The symbols ⊥ and refer respectively to the directions g Figure 1: Schematic representation of the model geometry employed to study Rayleigh-Bénard convection (RBC). The dilute, weakly-collisional, magnetized plasma is held between two conducting horizontal plates of infinite extent, separated by a distance d. The plates are maintained at two different constant temperatures as indicated. Gravity is along the negative z axis. The symbols ⊥ and represent the directions perpendicular and parallel to the magnetic field, which lies in the x − z plane. perpendicular and parallel to the magnetic field B, whose direction is given by the unit vectorb ≡ B/B = (b x , 0, b z ). If the frequency of ion collisions ν ii in the single ion species magnetofluid is large compared to the rate of change d/dt of all the fields involved, then the anisotropic part of the pressure tensor is small compared to its isotropic part P ≡ 2p ⊥ /3 + p /3, with |p − p ⊥ | P. This isotropic part of the pressure tensor is assumed to satisfy the equation of state for an ideal gas P = ρk B T µm H ,(5) where k B is the Boltzmann constant, µ is the mean molecular weight, and m H is the atomic mass unit. This equation along with Eqs. (1)-(4) completes the specification of the dynamics of the unperturbed equilibrium configuration of the system under study. The anisotropic component of the pressure tensor in the momentum equation gives rise to Braginskii viscosity. For small pressure anisotropy, this contribution is usually written as p − p ⊥ = 3η 0 bb − 1 3 I : ∇v ,(6) where η 0 is the largest of the coefficients in the viscous stress tensor derived by Braginskii (1965)[see also Hollweg (1985)], and it is related to the coefficient of kinematic viscosity via η 0 = ρν v . This is a good approximation provided that the pressure anisotropy does not grow beyond |p − p ⊥ |/P β −1 , where the plasma β ≡ v 2 th /v 2 A , v th ≡ (2P/ρ) 1/2 is the thermal speed, and v A ≡ B/(4πρ) 1/2 is the Alfvén speed. The effects of Braginskii viscosity in the linear dynamics of the MTI and the HBI are explored in Kunz (2011). Beyond this limit, various fast-growing, micro-scale plasma instabilities, such as mirror and firehose (see Schekochihin et al. 2005, 2008 andreferences therein) with growth rates γ k v th |p − p ⊥ |/P can dominate the plasma dynamics at very small scales. Thus, for |p − p ⊥ |/P β −1 the Braginskii-MHD approximation embodied in Eqs. (1)-(4) becomes ill-posed and a mechanism to limit the pressure anisotropy must be implemented in numerical codes (Sharma et al., 2006;Kunz et al., 2012;Parrish et al., 2012). Dimensional parameters Symbols Definitions Thermal speed of ions v th 2P i ρ i Alfvén speed v A B √ 4πρ i Coefficient of thermal expansion α − 1 ρ ∂ρ ∂T P Heat capacity at constant pressure c p T ds dT P Coefficient of kinematic viscosity ν v η 0 ρ Coefficient of thermal diffusion λ χ ρc p Dimensionless parameters Symbols Definitions Schwarzschild number S gαT d c p ∆T − 1 Rayleigh number R α(∆T )gd 3 λν v Chandrasekhar number Q B 2 d 2 4πρν v η Prandtl number P r ν v λ Magnetic Prandtl number P m ν v η Plasma β parameter β P i B 2 /8π Knudsen number Kn λ mfp H Because the electron mean free path (λ mfp ) is large compared to its Larmor radius, heat flows mainly along magnetic field lines. This process is modeled by the second term on the right-hand side of Eq. (4) via Q s ≡ −χb(b·∇)T , where χ is the thermal conductivity predominately due to electrons with χ ≈ 6 × 10 −7 T 5/2 erg cm −1 s −1 K −1 (Spitzer, 1962). In the equilibrium state, all the particles in the plasma are assumed to be described by a Maxwellian distribution with the same temperature, so that p ≡ p ⊥ initially. In general, the background heat flux does not vanish, i.e.,b·∇T 0, unless the magnetic field and the background gradients are orthogonal. The existence of a well-defined steady state, i.e., ∇·Q s = 0, demands that the background heat flux should be at most a linear function of the distance along the direction of the magnetic field. Linear Equations for the Perturbations The equilibrium state (ρ * , P * , T * , v * , B * ) is defined by the fol- lowing relations v * = 0 , (7) dP * dz ≈ d(p ⊥ ) * dz = −ρ * g ,(8)B * = B * b = B * (sin φx + cos φẑ) ,(9)T * = T bottom − ∆T z d ,(10) where ∆T = T bottom − T top , and d is the distance between the top and bottom boundaries. As discussed above, the approximation invoked in Eq. (8) reflects the fact that the pressure anisotropy is relatively weak. We assume that the ion and electron pressures satisfy P i = P e = P/2 and the ICM to be an ideal gas, and thus αT = 1, where α is the coefficient of thermal expansion. Note that this implies that the Schwarzschild number S , see Table 1, is constant even though T changes with z. Hereafter, we shall drop the asterisk subscripts denoting the equilibrium state as there will be no ambiguity. For the sake of convenience, Table 1 provides a list of all the relevant parameters used in this letter. Since the sound crossing time associated with the modes of interest is much shorter than the growth rate of the unstable modes of HBI and MTI, it is justified to work within the Boussinesq approximation (Balbus, 2000(Balbus, , 2001Quataert, 2008). In this limit, Eq. (1) reduces to ∇ · v = 0 .(11) Thus, under the Boussinesq approximation, the velocity field perturbation satisfy ∇·δv = 0. Also, in this approximation the density variations can be ignored except when it appears multiplied with the external gravity term. Together with the solenoidal character of the magnetic field fluctuations, ∇·δB = 0; the relation ∇·δv = 0 implies that it is only necessary to understand the dynamics of two independent components for both the velocity and the magnetic field components. It is convenient to use as variables δv z , δω z , δB z , and δ j z , where δω z ≡ ∂ x δv y − ∂ y δv x , (12) δ j z ≡ ∂ x δB y − ∂ y δB x ,(13) stand for the z-component of the fluctuations in the vorticity and the current density (times 4π), respectively. Taking the Laplacian of the z-component of the momentum Eq. (2) and the z-component of its curl, we arrive to the equations of motion for δv z and δω z ∂ t ∇ 2 δv z = αg(∂ 2 x + ∂ 2 y )δT + B 4πρ (sinφ ∂ x + cos φ ∂ z )∇ 2 δB z −3ν v (sinφ ∂ x + cos φ ∂ z ) 2 [cos φ (∂ 2 x + ∂ 2 y ) − sinφ ∂ x ∂ z ] ×(sinφ δv x + cos φ δv z ) ,(14)∂ t δω z = B 4πρ (sinφ ∂ x + cos φ ∂ z )δ j z +3ν v sinφ ∂ y (sinφ ∂ x + cos φ ∂ z ) 2 (sinφ δv x + cos φ δv z ) .(15) Following a similar procedure with the induction Eq. (3), we obtain the equations for δB z and δ j z (∂ t − η∇ 2 )δB z = B(sinφ ∂ x + cos φ ∂ z )δv z ,(16)(∂ t − η∇ 2 )δ j z = B(sinφ ∂ x + cos φ ∂ z )δω z .(17) We obtain the equation for the thermal fluctuations directly from Eq. (4) as ∂ t δT + δv z dT dz + gαT c p = λ(sin 2 φ ∂ 2 x + 2 sinφ cos φ ∂ x ∂ z + cos 2 φ ∂ 2 z )δT + λ B dT dz (A 1 ∂ y δB y + A 2 ∂ x δB x + A 3 ∂ z δB z ) + λ B dT dz (A 4 ∂ x δB z + A 5 ∂ z δB x ) ,(18) where we have defined a number of functions in order to simplify the notation A 1 ≡ cosφ ,(19)A 2 ≡ cosφ (cos 2 φ − sin 2 φ) ,(20)A 3 ≡ 2 sin 2 φ cosφ ,(21)A 4 ≡ sinφ (sin 2 φ − cos 2 φ) ,(22)A 5 ≡ −2 sinφ cos 2 φ .(23) Dimensionless Variables It is convenient to use the characteristic scales in the problem in order to define a set of dimensionless coordinates according to x ≡ x/d ,(24)t ≡ tν v /d 2 .(25) where ν v is the coefficient of kinematic viscosity. This allows us to define a set of dimensionless functions for all the dynamical variables of interest, i.e., δv z ≡ δv z d/λ, δω z ≡ δω z d 2 /λ, δB z ≡ δB z /B, δ j z ≡ δ j z d/B, and δθ ≡ δT/∆T , such that δv z (x , t ) ≡ k δ v z (z ) exp(ik ·x + σ t ) ,(26)δω z (x , t ) ≡ k δ ω z (z ) exp(ik ·x + σ t ) ,(27)δB z (x , t ) ≡ k δ B z (z ) exp(ik ·x + σ t ) ,(28)δ j z (x , t ) ≡ k δ j z (z ) exp(ik ·x + σ t ) ,(29)δθ (x , t ) ≡ k δ θ (z ) exp(ik ·x + σ t ) .(30) Here, the hat-symbol (ˆ) denotes the z-dependent Fourier transform amplitudes of the various functions involved, which are assumed to be periodic in the plane perpendicular to z. The dimensionless growth-rate (or frequency) σ characterizes the dynamics of the perturbation with dimensionless wave vector k = (k x , k y , 0). The linear nature of the equations for the perturbations allows us to follow the dynamics of each mode independently. For the sake of brevity, in what follows, unless otherwise specified, we omit all the primes labeling dimensionless coordinates, variables, and functions. We also omit the hat-symbol denoting the Fourier amplitude of a given mode. Using the previous definitions, the equations for the perturbations in equations (14)-(18), in dimensionless form, read σ(∂ 2 z − k 2 ) − 3 k 2 (i sinφ k x + cosφ ∂ z ) 2 × (cosφ k 2 + i sinφ k x ∂ z ) 2 δv z = −Rk 2 δθ +Q P r P m (i sinφ k x + cosφ ∂ z )(∂ 2 z − k 2 )δB z + 3i sinφ k y k 2 (i sinφ k x + cosφ ∂ z ) 2 (cosφ k 2 + i sinφ k x ∂ z ) δω z ,(31)              3 sin 2 φ k 2 y k 2        (i sinφ k x + cosφ ∂ z ) 2        δω z + σδω z = Q P r P m (i sinφ k x + cosφ ∂ z )δ j z + 3i sinφ k y k 2 × (i sinφ k x + cosφ ∂ z ) 2 cosφ k 2 + i sinφ k x ∂ z δv z ,(32)(∂ 2 z − k 2 − P m σ)δB z = − P m P r (i sinφ k x + cosφ ∂ z )δv z ,(33)(∂ 2 z − k 2 − P m σ)δ j z = − P m P r (i sinφ k x + cosφ ∂ z )δω z ,(34)(cos 2 φ ∂ 2 z + 2i sinφ cosφ k x ∂ z − sin 2 φ k 2 x − P r σ)δθ = S δv z + 1 k 2 (A 1 − A 2 )k x k y + iA 5 k y ∂ z δ j z + − A 1 k 2 k 2 y ∂ z − A 2 k 2 k 2 x ∂ z + A 3 ∂ z + iA 4 k x + i A 5 k 2 k x ∂ 2 z δB z .(35) In writing the preceding set of equations, we have also made use of the following relations: δv x = (ik x ∂ z δv z + ik y δω z )/k 2 ,(36)δv y = (ik y ∂ z δv z − ik x δω z )/k 2 ,(37)δB x = (ik x ∂ z δB z + ik y δ j z )/k 2 ,(38)δB y = (ik y ∂ z δB z − ik x δ j z )/k 2 .(39) Boundary Conditions There are several sets of BCs that are commonly adopted in the framework of RBC. In what follows, we shall concern ourselves exclusively with the reflective, stress-free, and perfectly conducting boundaries given by Kunz et al. (2012). In these papers, the temperature is also fixed at the boundaries as in our Eq. (45). The BCs employed in Kunz et al. (2012) for the magnetic field are ∂ z δB x = ∂ z δB y = 0 (Eq. 44) and δB z = 0 (Eq. 43) when simulating an initially horizontal field and δB x = δB y = 0 and ∂ z δB z = 0 when simulating an initially vertical field. δv z (0) = δv z (1) = 0 ,(40)∂ 2 z δv z (0) = ∂ 2 z δv z (1) = 0 ,(41)∂ z δω z (0) = ∂ z δω z (1) = 0 ,(42)δB z (0) = δB z (1) = 0 ,(43)∂ z δ j z (0) = ∂ z δ j z (1) = 0 ,(44)δθ(0) = δθ(1) = 0 .(45) The BCs that we use have the advantage of allowing us to derive analytic solutions. In principle, we could have adopted BCs similar to the ones used in numerical simulations but this would in general require to solve the problem numerically, even in the linear regime. In passing, we may also mention that it is numerically straightforward to impose the BCs chosen in this letter making it possible to have future comparisons with numerical simulations. Relevant Characteristic ICM Values In the analysis that follows, it is of central importance to realize the extreme values that some of the dimensionless parameters in Table 1 can reach under the conditions expected in the ICM. For instance, the dimensionless parameters Q and P m have extremely large values. In order to set the scale, let us consider as an example (Carilli and Taylor, 2002;Peterson and Fabian, 2006), e.g., B ∼ 10 −6 − 10 −7 G, ρ ∼ 10 −27 − 10 −25 gm cm −3 , η = 10 − 10 2 cm 2 s −1 , ν v = 10 25 − 10 30 cm 2 s −1 , T ∼ 10 7 − 10 8 K and radial length d ∼ 10 24 − 10 25 cm. We thus find Q ∼ 10 24 − 10 40 and P m ∼ 10 23 − 10 29 . This result suggests that it is justifiable to work in the limit in which Q, P m → ∞. However, as we will show below, some of the results obtained in the stability analysis that follows from the RB approach are sensitive to these limits being taken with the proper care. The Heat-Flux Driven Buoyancy Instability Let us first consider the case in which the magnetic field is along the z-direction, i.e., φ = 0, which is known to be prone to the HBI (Quataert, 2008). The RB formalism enables us to find the conditions for the existence of the HBI marginal state as follows. In the state of marginal stability (σ = 0), the system of Eqs. (31)-(32) reduces to 3k 2 ∂ 2 z δv z = −Q P r P m ∂ z (∂ 2 z − k 2 )δB z − Rk 2 δθ ,(46)∂ 2 z δθ = S δv z − ∂ z δB z ,(47)(∂ 2 z − k 2 )δB z = − P m P r ∂ z δv z ,(48)(∂ 2 z − k 2 )δ j z = − P m P r ∂ z δω z ,(49)∂ z δ j z = 0 ,(50) subject to the BCs, δv z = [3k 2 + Q]∂ 2 z δv z = ∂ 2 z δv z = 0, ∂ z δ j z = 0, ∂ z δω z = 0, δB z = 0 at z = 0, 1. Note that δ j z and δω z have become decoupled from δv z , δB z , and δθ. From Eqs. (49) and (50) and the corresponding boundary conditions, it may be observed that, while the current density's vertical component vanishes identically and the vorticity's vertical component is independent of z at the marginal state we have δ j z = 0 ; δω z = constant .(51) The set of Eqs. (46), (47) and (48) can be combined to give (3k 2 ∂ 4 z − S Rk 2 )(∂ 2 z − k 2 )δv z = + R P m P r k 2 + Q(∂ 2 z − k 2 )∂ 2 z ∂ 2 z δv z .(52) In the limit Q, P m → ∞ of interest, Eq. (52) reduces to R P m P r k 2 + Q(∂ 2 z − k 2 )∂ 2 z ∂ 2 z δv z = 0 ,(53) and also leads to the conclusion that at the boundaries, z = 0, 1, ∂ 4 z (∂ 2 z − k 2 )δv z = 0 .(54) The most general solution to (53) has the form δv z (z) = C 0 + C 1 z + C 2 cos k − z + C 3 sin k − z + C 4 cosh k + z + C 5 sinh k + z ,(55) where C 0 , C 1 , C 2 , C 3 , C 4 and C 5 are constants of integration, and k + ≡ k √ 2        1 + 4|R|P m QP r k 2 1/2 + 1        1/2 ,(56)k − ≡ k √ 2        1 + 4|R|P m QP r k 2 1/2 − 1        1/2 .(57) Here, we have assumed that the upper boundary is hotter than the bottom one, as discussed below. Therefore, the application of BCs allows us to conclude that C 0 = C 1 = C 2 = C 4 = C 5 = 0 and k − = nπ (n ∈ N), giving δv z = C 3 sin nπz ,(58) along with R = −n 2 π 2 n 2 π 2 + k 2 k 2 Q P r P m . −α(∆T )gd 3 λν v = −n 2 π 2 n 2 π 2 + k 2 k 2 Q P r P m ,(60) and thus d ln T dz = n 2 π 2 n 2 π 2 + k 2 k 2 1 βH , where we have set d = H, with H the thermal-pressure scale height. Note that for a given k, the lowest value of d ln T /dz occurs when n = 1 (lowest mode) giving: d ln T dz = π 2 π 2 + k 2 k 2 1 βH .(62) For all d ln T /dz smaller than this, all the perturbation with wavenumber k are stable and they become unstable as this limit is overcome. Since d ln T /dz, for a given n, is a monotonically decreasing function of k, the minimum or critical temperature gradient for the onset of the HBI occurs mathematically at k = k c = ∞. However, it is worth keeping in mind that in order for the fluid approach to remain valid, the wavenumber must satisfy k 2π/λ mfp , or in dimensionless numbers, k 2πKn −1 . Therefore, using k = 2πKn −1 , the critical temperature gradient for the onset of the HBI is obtained as d ln T dz c = π 2 (Kn 2 + 4) 4βH ≈ π 2 βH .(63) This threshold for the temperature gradient takes into account the effect of magnetic tension induced by a finite value of the plasma β parameter, which has been usually ignored when deriving the stability criterion for the HBI. In the limit of β → ∞, Eq. (63) recovers the usual criterion for the HBI (Quataert, 2008). In general, one can use variational principles to investigate the presence of oscillatory marginal states and the validity of the principle of exchange of instabilities (Chandrashekhar, 1981). However, in what follows we show that the lowest mode always appears as a stationary state for the HBI. For this purpose we set σ = iq, q being real, and rewrite the relevant Eqs. (31), (33), and (35) iq(∂ 2 z − k 2 ) − 3k 2 ∂ 2 z δv z = Q P r P m ∂ z (∂ 2 z − k 2 )δB z − Rk 2 δθ ,(64)(∂ 2 z − k 2 − iP m q)δB z = − P m P r ∂ z δv z .(65)(∂ 2 z − iP r q)δθ = S δv z − ∂ z δB z ,(66) In the limit Q, P m → ∞ of interest, it is trivial to arrive at the following using the real and imaginary parts of the previous three equations: q 2 P m [(∂ 2 z − k 2 )∂ 2 z + 3P r k 2 ∂ 2 z ]δv z = −Q(∂ 2 z − k 2 )∂ 4 z δv z − R P m P r k 2 ∂ 2 z δv z ,(67)−qP m [−3k 2 ∂ 4 z + q 2 P r (∂ 2 z − k 2 ) + RS k 2 ]δv z = qP r Q(∂ 2 z − k 2 )∂ 2 z δv z .(68) Using δv z = C 3 sin πz as the lowest mode for a top-hot-plate configuration and making use of Eqs. (67) and (68), one can arrive at the following relations q 2 P m (π 2 + k 2 ) − 3P r k 2 = Q(π 2 + k 2 )π 2 + R P m P r k 2 , and qP m [3k 2 π 4 + q 2 P r (π 2 + k 2 ) − RS k 2 ] = −qP r Q(π 2 + k 2 )π 2 . Multiplying (69) by qP r and adding with (70), we conclude that either q = 0 or 3q 2 P r 2 = R(S − 1) − 3π 4 .(71) For typical values of the parameters in a galaxy cluster, Eq. ( 71) can be satisfied for all k and for any real q. Therefore the lowest unstable HBI mode can set in as an oscillatory marginal state. The Magnetothermal Instability When the magnetic field is aligned with the x-direction, and thus it is perpendicular to the direction of gravity, i.e., φ = π/2, the plasma may be subject to the MTI (Balbus, 2001). At marginal stability, we can formulate the problem by setting σ = 0 in Eqs. (31)-(35), which become − 3 k 2 k 4 x ∂ 2 z δv z = −Rk 2 δθ + iQ P r P m k x (∂ 2 z − k 2 )δB z + 3k 3 x k y k 2 ∂ z δω z ,(72)− 3k 2 x k 2 y k 2 δω z = 3k 3 x k y k 2 ∂ z δv z + iQ P r P m k x δ j z ,(73)(∂ 2 z − k 2 )δB z = −i P m P r k x δv z ,(74)(∂ 2 z − k 2 )δ j z = −i P m P r k x δω z ,(75)− k 2 x δθ = S δv z + ik x δB z ,(76) subject to the BCs, at z = 0, 1, δθ = 0 ; δv z = ∂ 2 z δv z = 0 ; ∂ z δ j z = 0 ; ∂ z δω z = 0 ; δB z = 0 .(77) One can see that the BC for the temperature δθ is trivially satisfied due to the BCs on δv z and δB z . Combining Eqs. (72)-(73), we get the following differential equation for δv z Q RS k 2 k 2 x (∂ 2 z − k 2 ) + 3 k 2 k 8 x ∂ 2 z (∂ 2 z − k 2 ) + 3 k 2 k 6 x k 2 y (∂ 2 z − k 2 ) 2 δv z + Q 2 k 6 x (∂ 2 z − k 2 )δv z + QR P m P r k 2 k 4 x δv z + 3RS k 2 x k 2 y (∂ 2 z − k 2 ) 2 δv z + 3Rk 4 x k 2 y P m P r (∂ 2 z − k 2 )δv z = 0 .(78) We focus on the limit of interest, i.e., Q, P m → ∞, which in this case allows us to write the following single differential equation for δv z , where it is only necessary to retain terms up to order Q 2 , P m 2 and QP m have been kept: Q 2 k 6 x (∂ 2 z − k 2 )δv z + QR P m P r k 2 k 4 x δv z = 0 .(79) Taking the successive even z−derivatives of this equation and evaluating the results at z = 0, 1 we obtain ∂ 2m z δv z = 0 with m ∈ {0, 1, 2, ...} .(80) This means that the appropriate solution for the lowest mode is δv z = A sin πz ,(81) where A is a constant. Therefore, Eq. (79) implies: R = k 2 x k 2 (π 2 + k 2 )Q P r P m ,(82) where k x 0. The marginal state can exist only if ∆T is positive, i.e., if the bottom boundary is hotter. Note that for fixed k x , the Rayleigh number R monotonically decreases as k y increases; and, for fixed k y , R monotonically increases with k x . Also, recall that in order for the local analysis in (x, y) to be valid within the fluid approach we should have H −1 < k x , k y < λ −1 mfp Therefore, we note that, while the minimum possible value of k x is (k x ) min 2π, the maximum value of k y is (k y ) max 2π/Kn and this combination (k x , k y ) corresponds to the lowest unstable mode. The minimum value for the temperature gradient −d ln T/dz for the onset of MTI modes with k x , k y 0, is obtained from − d ln T dz c = min        k 2 x (π 2 + k 2 x + k 2 y ) k 2 x + k 2 y        1 βH ,(83) where the minimum value min        k 2 x (π 2 + k 2 x + k 2 y ) k 2 x + k 2 y        ≡ (k x ) 2 min [π 2 + (k x ) 2 min + (k y ) 2 max ] (k x ) 2 min + (k y ) 2 max ,(84) depends on the particular mode under consideration. This threshold for the temperature gradient takes into account the effect of magnetic tension induced by a finite value of the plasma β parameter, which has been usually ignored when deriving the stability criterion for the MTI. In the limit of β → ∞, Eq. (83) recovers the usual criterion for the MTI (Balbus, 2001). We now show that the lowest mode doesn't set in as an oscillatory marginal stability state. In order to achieve this, let us define σ = iq, with q real and write the relevant version of Eqs. (31)-(35) as iq(∂ 2 z − k 2 ) − 3 k 2 k 4 x ∂ 2 z δv z = iQ P r P m k x (∂ 2 z − k 2 )δB z −Rk 2 δθ + 3k 3 x k y k 2 ∂ z δω z ,(85)       − 3k 2 x k 2 y k 2 − iq        δω z = 3k 3 x k y k 2 ∂ z δv z + iQ P r P m k x δ j z . (∂ 2 z − k 2 − iP m q)δB z = −i P m P r k x δv z ,(86)(∂ 2 z − k 2 − iP m q)δ j z = −i P m P r k x δω z ,(87)(−k 2 x − iP r q)δθ = S δv z + ik x δB z ,(88) In the limit Q, P m → ∞, one can as before argue in favour of using δv z = A sin πz as the lowest mode. Using the immediately preceding five equations, it is easy to arrive at the following real and imaginary parts of a complex equation, leading to q 4        (π 2 + k 2 )        k 2 x + k 2 x k 2 y k 2 P r        − 3k 4 x k 2 P r π 2        +q 2 Rk 2 x k 2 P r + 3RS k 2 x k 2 y −q 2        Qk 2 x P m        k 2 x − 3k 2 x k 2 y k 2 P r        (π 2 + k 2 )        = 0 ,(89) q 5 P r (π 2 + k 2 ) + q 3        3k 6 x k 2 π 2 + 3k 4 x k 2 y k 2 (π 2 + k 2 )        −q 3 RS k 2 + QP r P m k 2 x (π 2 + k 2 ) −q        Q P m 3k 6 x k 2 y k 2 (π 2 + k 2 ) − 3k 4 x k 2 y P r R        = 0 .(90) In the light of Eq. (82), these two equations simplify respectively to q 4        (π 2 + k 2 )        k 2 x + k 2 x k 2 y k 2 P r        − 3k 4 x k 2 P r π 2        + q 2 3R(S + 1)k 2 x k 2 y = 0 ,(91)q 5 P r (π 2 + k 2 ) + q 3        3k 6 x k 2 π 2 + 3k 4 x k 2 y k 2 (π 2 + k 2 ) − R(S + 1)k 2        = 0 .(92) Since ∆T > 0 in this case, all the terms inside the square brackets are positive definite allowing us to conclude that for the existence of an oscillatory marginal state q 0, we must have m 11 m 12 m 21 m 22 = 0 .(93) where the elements in the determinant are m 11 = (π 2 + k 2 )        k 2 x + k 2 x k 2 y k 2 P r        − 3k 4 x k 2 P r π 2 ,(94)m 12 = 3R(S + 1)k 2 x k 2 y ,(95)m 21 = P r (π 2 + k 2 ) ,(96)m 22 = 3k 6 x k 2 π 2 + 3k 4 x k 2 y (π 2 + k 2 ) k 2 − R(S + 1)k 2 .(97) While this condition, in principle, allows for the existence of a marginal oscillatory stability state for MTI, it is not easy to explicitly specify analytically the values of k x and k y for which this happens. Thus, without solving explicitly this equation in terms of (k x , k y ), it is not possible for us to rule out the onset of the MTI as an oscillatory mode. Retrieving Schwarzschild Criterion Before we end this section, it is instructive to note how the limits Q, P m → ∞ play an important role to bring about the correct criterion for the onset of the MTI. Let us consider modes with k x → 0, without necessarily imposing the limits Q, P m → ∞. In this case, using (72)-(73), we arrive at 3k 8 x k 2 D∂ 2 z (∂ 2 z − k 2 )δv z + 9k 8 x k 2 y a 4 ∂ 2 z (∂ 2 z − k 2 ) 2 δv z = (98) Qk 6 x (∂ 2 z − k 2 )Dδv z + RS k 2 k 2 x (∂ 2 z − k 2 )Dδv z + R P m P r k 2 k 4 x Dδv z , where we have introduced, D ≡ Q + 3(k 2 y /k 2 )(∂ 2 z − k 2 ). Hence taking the limit k x → 0 yields RS k 2 (∂ 2 z − k 2 )        Q + 3 k 2 y k 2 (∂ 2 z − k 2 )        δv z = 0 .(99) Using δv z = A sin πz as the lowest mode, the condition for marginally stable state is RS = 0, i.e., ∆T gαT d c p (∆T ) − 1 = 0 ,(100) and thus − dT dz = gαT c p .(101) This condition, which is independent of the choice of a specific mode for δv z , is just the condition for the marginal state corresponding to the Schwarzschild instability. Perhaps a more direct way to arrive to this condition is to set k x = 0 in Eq. (76) and assume that δv z 0, which leads to the conclusion that the Schwarzschild number is S = 0, implying that −dT /dz = gαT /c p , as stated in Eq. (101). It is not difficult to understand the physics that allow us to retrieve the Schwarzschild instability criterion in the limit k x → 0. In an unmagnetized stratified atmosphere, a fluid element that is adiabatically displaced upwards returns to its initial position if the entropy gradient is positive. This condition is known as Schwarzschild criterion. Now, in the case of the MTI, an upwardly displaced fluid element carries the magnetic field along while retaining its temperature unchanged because the heat quickly flows along the magnetic field lines. This mechanism leads to the MTI. When a perturbation with k x → 0 is considered, we could envision the associated mode to have an infintely long wavelength and thus a fluid parcel displaced upwards is not connected via magnetic field lines to its initial position (i.e., the horizontal layer of atmosphere where it was initially in). Therefore, heat is unable to flow into the displaced layer and one obtains back the Schwarzschild criterion for instability. In this context, it may be noted how setting k x = 0 prevents the magnetic field and the temperature perturbations in Eq. (76) from coupling to the velocity perturbation. Summary and Discussion In this letter, we have applied the formalism employed in RBC to study the MTI and the HBI. This approach goes beyond the standard linear mode analysis that has been carried out (but see Latter and Kunz 2012 for an exception) by considering explicit boundary conditions. This enabled us to address in a natural way, some aspects of the linear dynamics of these instabilities that have not been previously addressed. In particular, we have derived the conditions for the onset of the instabilities retaining the effects of magnetic tension, as embodied by a finite plasma beta parameter, and Braginskii viscosity. The latter is known to have a stabilizing effect on the high-k end of the spectrum of unstable modes (Kunz, 2011). We found, however, that (to linear order) Braginskii viscosity does not play an explicit role in the criterion for the onset of either the HBI or the MTI, see Eqs. (63) and (83). We have found expressions for the Rayleigh number in terms of the wave vector k of a given mode. In the case of the HBI, the Rayleigh number is found to be a monotonically decreasing function of the mode wavenumber, k. This implies that, for a given temperature gradient, the modes that go unstable first are those with largest values of k, i.e., those with smallest wavelength. In the case of the MTI, the dependence of the Rayleigh number on the dimensionless wave vector k is more subtle, as it depends on both its magnitude and direction. The MTI modes that go unstable first are those with long wavenumber while maintaining k || → 0, which is a restriction that does not apply to HBI. For the HBI, the mode that goes unstable first does so in a nonoscillatory fashion, whereas in the MTI, an oscillatory marginal stable state is, in principle, possible. We have found that the HBI is regularized (at high wavenumbers) by magnetic tension, but such regularization is not present for the MTI. For collisional plasmas, there is a high-k regularization by isotropic viscosity and conductivity. In the weaklycollisional regime that concerns us here, the transport is dominant along magnetic field lines. Nevertheless, there is still some, albeit small, isotropic diffusion. In order to find a critical Rayleigh number for the MTI, as needed to perform a weakly nonlinear analysis, it might be useful to include this isotropic contribution. While it is certainly possible to include this effect in the equations, this would render the analytical treatment that we have presented significantly more challenging. It is thus pragmatic to defer this calculation to future work while allowing the present work to focus solely on the nontrivial effects of anisotropic heat conduction. Before we conclude, we comment on the choice of the BCs. For the sake of analytical simplicity, we have chosen to work with conducting stress-free BC. Other possibilities include nonconducting stress-free, conducting rigid, non-conducting rigid, etc. For some of these BCs, it is not possible to solve the problem solely on analytical grounds, and numerical techniques are necessary even in the linear regime. The BC we have employed resemble those usually employed in numerical simulations. The specific choice of boundary conditions is unlikely to have a dramatic impact in the stability criterion within the bulk of the plasma. Nevertheless, it should be kept in mind that these could have an impact on the specific expression for the stability criteria. Some of these results could have been obtained by other means, for example by retaining the effects of magnetic tension and solving analytically the associated dispersion relations. However, the formalism we have outlined could become even more advantageous as the dispersion relations dictating the linear dynamics become more involved. It is worth noticing that the formalism can be generalized to address more realistic physical settings, for example including the effects of cosmic-rays (Chandran and Dennis, 2006), rotation (Nipoti and Posti, 2014), and radiative cooling (Balbus and Reynolds, 2010;Latter and Kunz, 2012), or composition gradients in the ICM (Pessah and Chakraborty, 2013;Berlok and Pessah, 2015) One advantage of having laid out the RBC formalism is that this provides the grounds for future work on weakly-non linear analysis. This type of analysis has proven to be advantageous in delivering further analytical insights into the bifurcation scenarios and the routes to chaotic (turbulent) states of the systems under study (Bhattacharjee, 1989;Getling, 1997). Table 1 : 1List of parameters used in this letter. For completeness, we state the type of BCs usually adopted in the literature related to the MTI and the HBI. The BC on the velocity is usually reflective, as embodied in Eqs. (40)-(42), see, e.g.,McCourt et al. (2011); Physically, Eq. (40) implies that the normal component of the velocity must be zero on the boundary surfaces, whereas Eqs. (41) and (42) require stress-free surfaces and Eqs. (43) and (44) imply perfectly conducting boundaries. Eq. (45) fixes the boundary surfaces to be at a constant temperature, The marginal state can then exist only if ∆T is negative, i.e., if the upper boundary is hotter. Hence, by setting the temperature difference in the definition of the Rayleigh number R in Eq. 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{'abstract': "We use the machinery usually employed for studying the onset of Rayleigh-Bénard convection in hydro-and magnetohydrodynamic settings to address the onset of convection induced by the magnetothermal instability and the heat-flux-buoyancy-driveninstability in the weakly-collisional magnetized plasma permeating the intracluster medium. Since most of the related numerical simulations consider the plasma being bounded between two 'plates' on which boundary conditions are specified, our strategy provides a framework that could enable a more direct connection between analytical and numerical studies. We derive the conditions for the onset of these instabilities considering the effects of induced magnetic tension resulting from a finite plasma beta. We provide expressions for the Rayleigh number in terms of the wave vector associated with a given mode, which allow us to characterize the modes that are first to become unstable. For both the heat-flux-buoyancy-driven-instability and the magnetothermal instability, oscillatory marginal stable states are possible.PACS numbers 98.65. Hb, 44.25.+f", 'arxivid': '1605.09591', 'author': ['H Gupta \nDepartment of Physics\nIndian Institute of Technology Kanpur\nU. P.-208016India\n', 'S K Rathor \nDepartment of Physics\nIndian Institute of Technology Kanpur\nU. P.-208016India\n', 'M E Pessah \nNiels Bohr International Academy\nNiels Bohr Institute\n2100Copenhagen ØDenmark\n', 'S Chakraborty \nDepartment of Physics\nIndian Institute of Technology Kanpur\nU. P.-208016India\n\nMechanics & Applied Mathematics Group\nIndian Institute of Technology Kanpur\nU.P.-208016India\n'], 'authoraffiliation': ['Department of Physics\nIndian Institute of Technology Kanpur\nU. P.-208016India', 'Department of Physics\nIndian Institute of Technology Kanpur\nU. P.-208016India', 'Niels Bohr International Academy\nNiels Bohr Institute\n2100Copenhagen ØDenmark', 'Department of Physics\nIndian Institute of Technology Kanpur\nU. P.-208016India', 'Mechanics & Applied Mathematics Group\nIndian Institute of Technology Kanpur\nU.P.-208016India'], 'corpusid': 119207018, 'doi': '10.1016/j.physleta.2016.05.026', 'github_urls': [], 'n_tokens_mistral': 20556, 'n_tokens_neox': 17001, 'n_words': 10499, 'pdfsha': 'a4ae1b63040bc7690893ee70788d64a1f56eb5f5', 'pdfurls': ['https://arxiv.org/pdf/1605.09591v1.pdf'], 'title': ['Stability Analysis of Convection in the Intracluster Medium', 'Stability Analysis of Convection in the Intracluster Medium'], 'venue': []}

arxiv

Eigenvalue Dynamics and the Matrix Chain 21 Feb 1999 L D Paniak paniak@feynman.princeton.edu Department Of Physics Joseph Henry Laboratories Princeton University 08544PrincetonNew Jersey, USA Eigenvalue Dynamics and the Matrix Chain 21 Feb 1999 We introduce a general method for transforming the equations of motion following from a Das-Jevicki-Sakita Hamiltonian, with boundary conditions, into a boundary value problem in one-dimensional quantum mechanics. For the particular case of a onedimensional chain of interacting N × N Hermitean matrices, the corresponding large N boundary value problem is mapped into a linear Fredholm equation with Hilbert-Schmidt type kernel. The equivalence of this kernel, in special cases, to a second order differential operator allows us recover all previously known explicit solutions for the matrix eigenvalues. In the general case, the distribution of eigenvalues is formally derived through a series of saddle-point approximations. The critical behaviour of the system, including a previously observed Kosterlitz-Thouless transition, is interpreted in terms of the stationary points. In particular we show that a previously conjectured infinite series of sub-leading critical points are due to expansion about unstable stationary points and consequently not realized. Introduction The utility of studying the statistical mechanics of systems which can be encoded in terms of matrix variables has been evident over the past twenty years. From the generalization of Wigner and Dyson's work by Brézin, Itzykson, Parisi and Zuber [1] to the use of matrix models in low dimensional gravity (see [2] for reviews), to modern implementations of matrix strings [3], matrix models have proven to be a convenient framework for calculation. Remarkably, throughout this period the dynamics of matrix models have been considered only in the 'stationary' case where solutions of the equations of motion are assumed to be time-independent. There arise circumstances though where the time evolution of a system governed by an action over matrix variables is relevant, especially if there are definite initial and final conditions on the state of the system. Examples include two-dimensional Yang-Mills theory on the cylinder and sphere [4,5], matrix model approaches to the continuous Hirota equation [6] and lattice QCD [7] and, the case we will focus on here, linear chains of interacting Hermitean matrices. It is our goal to give a coherent framework in which in to study this class of matrix model problems. When an action over matrix variables can effectively be reduced to the eigenvalues, the treatment of the eigenvalues as a single, collective coordinate [8] provides a convenient viewpoint for determining the Hamiltonian of the system. For the matrix actions we will consider here the particular form is known as the Das-Jevicki-Sakita Hamiltonian [9]. The associated equations of motion are the non-linear Euler equations of fluid dynamics in one dimension and with the addition of appropriate boundary conditions present a difficult mathematical problem. Here we will show how these boundary value problems can be interpreted in the familiar framework of the semi-classical limit of one-dimensional quantum mechanics. While in general solutions are still difficult to obtain, we will focus our attention on the case of a one-dimensional chain of interacting N × N Hermitean matrices which was shown to be a Das-Jevicki-Sakita-type system by Matytsin [10] in the limit of large N. In this situation a restatement of the equations of motion in terms of quantum mechanics allows one to consider a linear problem which is most conveniently stated as a Fredholm integral equation. Formal solution of this equation in terms of iterations of the integral kernel, which can be evaluated by saddle-point methods, will provide a framework for detailing the nature and position of critical points which arise in the system. An interesting system from a statistical mechanical point of view, the one-dimensional matrix chain was discussed previously as a model for string theory with a discrete target space [12]. In addition to this direct application, the matrix chain is related by duality transformation to string theory on a circle with radius R = 1/a [13,14], string theory at a finite temperature [15] and to O(2) sigma models coupled to two dimensional quantum gravity [16,17]. The model itself consists of Hermitean matrices φ x on the sites {x} of a one-dimensional lattice interacting with nearest neighbours. The action of the system is S V = a x TrV (φ x ) − x Tr(φ x − φ x+1 ) 2 2a (1.1) where a is the lattice spacing and V is the potential at each site. Assuming the site potentials V (φ x ) to be analytic in φ x , the action can be reduced to the eigenvalues of the matrix variables by SU(N) gauge rotations Λ and the known result for the correlator [11] I[φ x , φ x+1 ] = dΛ e −N TrΛφxΛ † φ x+1 (1.2) Consequently, in the limit of large N, the calculation of the partition function reduces to determining solutions of equations of motion for the eigenvalues. Despite its relative simplicity and importance in low-dimensional string theory and quantum gravity, explicit solutions of the matrix chain are only known for a few particular choices of potential V . In Section Four we will show these cases arise when the equations of motion are equivalent to certain Sturm-Liouville eigenvalue problems. This model exhibits a remarkable range of behaviour as one adjusts the lattice spacing, a. For instance, in the limit of vanishing a one recovers the canonical example of a Das-Jevicki-Sakita system, matrix quantum mechanics (restricted to a singlet sector) S M QM = dx Tr[ 1 2φ 2 (x) + V (φ(x))] (1.3) In fact, the methods we will introduce in the next Section and use to study the matrix chain find their most natural form when applied to such matrix mechanical systems on a finite interval x ∈ [0, T ]. More interestingly, fixing the potential V to satisfy V ′′ (0) = −4, at critical lattice spacing a = 1, the matrix chain was argued to undergo a Kosterlitz-Thouless phase transition [13]. The physics of this phase transition are most transparent in the dual picture of string theory on the circle. Here strings wrapping around the non-contractible target space play the role of vortices which are liberated when the radius R ∼ 1/a becomes sufficiently small. In terms of a discrete lattice, at sufficiently small lattice spacing a < 1 the universal behaviour of the system is that of the continuous model (1.3) due to communication between lattice sites. When the spacing is larger than unity this communication is broken and the system can be expected to factorize into a product of single D = 0 matrix models, which in fact it does as a → ∞. The main objective of studying the matrix chain is to understand precisely how this change of character from D = 1 matrix quantum mechanics to isolated D = 0 matrix models takes place at a = 1. In addition to a = 0 and a = 1, it was argued in [16] and [17] that at a discrete infinity of lattice spacings non-analytic behaviour arises in the distribution of eigenvalues of the matrix variables φ x . Considering iterative approximation of a functional solution to the equations of motion, it was argued these critical points arise when a = sin (pπ/2q) where p ≤ q are each positive integers. In Section Five, where we develop the general solution for the distribution of eigenvalues as a series of saddle-point integrations, we will show that this behaviour is not realized in the matrix chain. In fact it will be shown that such critical behaviour arises when unstable stationary points are mistakenly considered relevant in saddle-point approximations. H[ρ, Π] = 1 2 dx ρ(x)    ∂Π(x) ∂x 2 − π 2 3 ρ 2 (x)    + dx ρ(x)V(x, t) (2.1) to be determined as a temporal boundary problem in one-dimensional quantum mechanics. In ∂ ∂t ρ = δH δΠ , ∂ ∂t v = − ∂ ∂x δH δρ (2.2) where, for convenience, we have defined the 'velocity' field v(x, t) = ∂Π ∂x (2.3) Explicitly we see that the equations of motion are of the form of the Euler equations for a one-dimensional inviscid fluid experiencing a time-dependent force derived from V ∂ρ ∂t + ∂ ∂x (ρv) = 0 (2.4) ∂v ∂t + 1 2 ∂v 2 ∂x − π 2 ∂ρ 2 ∂x = − ∂ ∂x V(x, t) As in all evolution problems we need to specify boundary conditions to supplement the equations of motion. In the case of matrix quantum mechanics the classical treatment of [1] assumed free boundary conditions for the evolution of the system over infinite time. In this case one is led to a study of the ground state of the system and the equations of motion (2.4) are solved for the stationary case of v = 0. In general though we are interested in arbitrary boundary conditions for the evolution of the system through some time T . To be definite we will specify the generic conditions on the fields v and ρ at time t = 0 and t = T ρ(x, 0) = ρ 0 (x) , v(x, 0) = v 0 (x) (2.5) ρ(x, T ) = ρ T (x) , v(x, T ) = v T (x) We would now like to rewrite the equations of motion in terms of a single quasi-linear first order differential equation. Introducing the complex function f (x, t) = v(x, t) + iπρ(x, t) (2.6) it is easy to see that (2.4) leads to ∂f ∂t + f ∂f ∂x = − ∂ ∂x V(x, t) (2.7) For vanishing potential and real f this is the Hopf equation which is the prototypical one dimensional model of an equation which admits wave solutions complete with shocks and other inherently non-linear behaviour [18]. We would like to solve this equation with the boundary conditions given in (2.5). While the solution of the (complex) Hopf equation is known explicitly [10] in terms of initial data, typically we are interested in cases where boundary data is supplied. In these situations the explicit solution leads to an ill-posed, inverse-type problem for the boundary conditions. Consequently we would like to find a more natural presentation of the problem. To this end we will consider a generalized version of (2.7) and introduce a dispersive term [18] which will serve to smooth out the solutions and allow one to calculate the details of the solution f in terms of boundary conditions. Adding such a term leads to a complex version of the forced Burgers' equation ih ∂ 2 f ∂x 2 = ∂f ∂t + f ∂f ∂x + ∂ ∂x V(x, t) (2.8) Obviously, in the limit ash → 0 the forced Hopf equation (2.7) is recovered. The usefulness of this addition to the problem is that there exists a well-known (to mathematicians) change of variable which will reduce (2.8) to a second order, linear differential equation. This transformation is known as the Cole-Hopf transform and is given f (x, t) = −2ih ∂ ∂x log ψ(x, t) (2.9) As is readily verified, this transformation leads (up to an irrelevant additive constant of integration) to the Schrödinger equation Hψ = −h 2 ∂ 2 ∂x 2 + V(x, t) 2 ψ = ih ∂ ∂t ψ (2.10) Consequently we see that the time evolution of the collective field Hamiltonian (2.1) is given by the semi-classical (h → 0) limit of one-dimensional quantum mechanics with the same potential, up to a factor. Restricting to time independent potentials 1 V(x, t) = V(x), the boundary conditions (2.5) enter into the picture as the start and end points of evolution by H through time T ψ(x, T ) = e −iHT /h ψ(x, 0) (2.11) where the wavefunction is defined in terms of the eigenvalue density and velocity ψ(x, t) = exp − π 2h x dη ρ(η, t) + i 2h x dη v(η, t) (2.12) Given the Green's function G(x, u|T ) for the Schrödinger problem it may be convenient to cast (2.11) as an integral equation exp − π 2h x dη ρ T (η) + i 2h x dη v T (η) = (2.13) du G(x, u|T ) exp − π 2h u dη ρ 0 (η) + i 2h u dη v 0 (η) In applications the implementation of boundary conditions and Green's function varies depending on the situation. For example, in investigations of the continuous Hirota equation [6] one is interested in the case of time-dependent potentials V(x, t) leading to non-trivial Green's functions. Here the initial and final velocities are specified and the initial and final eigenvalue distributions are identified (ρ 0 = ρ T ) so that (2.13) takes on the form of a non-linear integral equation for ρ 0 . Another, more familiar application is that of two-dimensional Yang-Mills theory [4,5,19]. In this case the coordinates (x, u) in (2. An application: the matrix chain An application of the preceding formalism, which we will be investigating for the remainder of this Paper is that of the one-dimensional matrix chain. Instead of (1.1), it will be convenient to consider a different form of the action S U = x TrU(φ x ) − x Trφ x φ x+1 (3.1) where we have rescaled the matrix field φ x and the new site potential U is related to V by U(φ x ) = aV ( √ aφ x ) + φ 2 x (3.2) The evolution of eigenvalues in this system, in the limit of large N, was shown by Matytsin [10] to be given by the Das-Jevicki-Sakita Hamiltonian (2.1) with vanishing potential V. Here the boundary conditions (2.5), which play a crucial role, are given by ρ 0 (x) ≡ ρ(x, 0) = ρ(x, 1) (3.3) v(x, 0) = −v(x, 1) = 1 2 U ′ (x) − x The objective is to solve for the initial and final eigenvalue density ρ 0 in terms of the potential U. With V = 0, and ǫ = ih, the Schrödinger equation ( χ(x) = λ √ 4πǫ du e −(x−u) 2 4ǫ + x 2 −U (x)+u 2 −U (u) 4ǫ χ(u) (3.4) where χ(x) = exp − iπ 2ǫ x dη ρ 0 (η) (3.5) and we have introduced the constant λ which does not affect the asymptotic solution but is convenient for later discussion. In this form (3.4) has the natural interpretation of an eigen-problem with eigenvector χ and associated eigenvalue λ. It is interesting to note that the same integral equation arose in the context of matrix chains previously [12,13]. In these instances the authors were seeking the lowest lying eigenvalues λ in order to calculate the partition function. Here we are most concerned with the asymptotic (ǫ → 0) form of the eigenfunctions χ in order to recover ρ 0 . Hence we have re-stated the differential form of the boundary value problem (eqns. x = U ′ (u s )/2 + iπρ 0 (u s ) (3.6) Likewise, at this stationary point ρ 0 can be formally evaluated using its relation to χ from equation (3.5) u s = U ′ (x)/2 − iπρ 0 (x) (3.7) Labeling x by G + in equation (3.6) and u s by G − in equation (3.7), we see the saddle-point evaluation of the integral equation (3.4) leads to Matytsin's solution [10] of the boundary value problem x = G + (G − (x)) = G − (G + (x)) (3.8) This form of the solution of the problem is well-suited to detailed local analysis as demonstrated in [16,17]. However, for a global analysis which can accommodate the existence of multiple stationary points, we will take the (well-studied) integral form (3.4) over the functional equation (3.8). Before considering the particular characteristics of solutions though, we will utilize the interpretation of integral operators as self-adjoint operators to recover all previously known explicit solutions ρ 0 of the matrix chain. Integral and differential operators An analysis of an integral equation centers on the properties of the kernel K. In our case, the kernel of (3.4) is K(x, u) = 1 √ 4πǫ e xu 2ǫ − U (x)+U (u) 4ǫ (4.1) This positive definite kernel is real, symmetric and hence Hermitean. In the following we will assume the kernel to have many convenient properties which follow from the boundedness of all n-fold iterations dx 1 · · · dx n−1 K(x 1 , x 2 ) · · · K(x n−1 , x n ) < ∞ (4.2) Consequently we will only consider potentials U(x) which increase at infinity faster than x 2 . Under such conditions K defines a compact operator and represents a self-adjoint operator on the space of square-integrable functions. From classical operator theory [20] there is a countable infinity of orthogonal eigenfunctions {χ n (x)} which form a complete set on this space. It is our goal to obtain, as a function of the site potential U, the asymptotic form of these eigenfunctions which determine the large N solution of the eigenvalue distribution ρ 0 through (3.5). There are a number of cases in which the kernel K is equivalent to familiar self-adjoint operators in physics. In fact only in these instances are explicit solutions of the large N matrix chain known. In each case the eigenvalue distribution ρ 0 is the solution of a quadratic equation. Here we will show that such situations arise by considering the ǫ → 0 limit of second order self-adjoint differential operators L for which L and K share a complete set of eigenfunctions and define the same action on the Hilbert space. In order to determine the conditions that L must satisfy to be equivalent to the integral kernel we act on χ n as defined by the integral equation (3.4) E n χ n (x) = L x χ n (x) = du L x K(x, u)χ n (u) (4.3) Likewise, acting under the integral sign and using the self-adjoint property of L E n χ n (x) = du K(x, u)L u χ n (u) = du χ n (u)L u K(x, u) It follows from the symmetry of the kernel that the action of L generates symmetric functions of x and u which will cancel out in (4.5), L x K(x, u) = h(x + u, xu)K(x, u) = mn a mn (x + u) m (xu) n K(x, u) (4.6) where h is an arbitrary function of two variables with Taylor expansion coefficients {a mn }. This construction is valid for any differential operator but for simplicity we will consider only second order differential operators of Sturm-Liouville form L x = −4ǫ 2 d dx a 2 (x) d dx + a 0 (x) (4.7) Now it is a simple matter to find the potentials U and associated coefficients a 2 and a 0 which are consistent with the symmetry condition (4.6). Since a second order L can only produce quadratic polynomials in u, (4.6) leads to the equation Having determined the coefficients a 2 and a 0 , we can solve for the explicit form of the associated eigenvalue distribution. The differential form of the integral equation (3.4) is − a 2 (x)(u − U ′ (x)/2) 2 − 2ǫ (a 2 (x)(u − U ′ (x)/2)) ′ + a 0 (x) =L x χ(x) = Eχ(x) (4.10) where the eigenvalue E is related to that of the integral form by λ = e E/ǫ . In the limit of vanishing ǫ, the solution of this differential equation is given simply by the WKB approximation which implies the replacement of derivatives with 'momenta' iǫ dχ dx → π 2 ρ 0 (4.11) With this substitution, (4.10) reduces to a quadratic equation in the eigenvalue distribution which is solved by Since ρ 0 is to be interpreted as a probability distribution describing the eigenvalues of the matrix variables, it must be properly normalized. This is carried out by integrating ρ 0 over the positive support of (a 0 − E)/a 2 and fixing the undetermined constant E to satisfy dxρ 0 (x) = 1 (4.13) ρ 0 (x) = 1 π E − a 0 (x)a Integrating (4.9), we find the most general potential of the matrix chain which is consistent with a second order, self-adjoint differential operator first considered by Matytsin [10]. Also the asymmetric double Penner considered in connection with the Kazakov-Migdal model of induced QCD [21] can be recovered (see [22] for recent work). In each of these cases the eigenfunctions χ are related to prolate spheroidal wave functions. Additionally, with a 20 = 1, the limit a 02 → 0 recovers the quadratic potential case solved by Gross [23] and Makeenko [24]. With U ′ = (a 01 + 2)x ≡ 2m 2 x, L is the Schrödinger operator for the simple harmonic oscillator and the general expression for the distribution of eigenvalues (4.12), properly normalized, reduces to U(x) = − a 11 a 02 x −ρ 0 (x) = 1 π 2 √ m 4 − 1 − (m 4 − 1)x 2 (4.15) In each of these cases the analytic structure of the eigenvalue distribution . This suggests that for a generic potential the solution of the matrix chain problem follows from the semi-classical approximation to a pseudo-differential operator as suggested by naively extending the results for finite chains of matrices [25]. We will not pursue this line of reasoning here but return in the next Section to the integral form (3.4) and give a method for obtaining the general solution and describe its unique features. General solution and classification of critical points A phase transition in a system is characterized by a change in the analytic structure of observables as functions of external parameters. In standard matrix models this is commonly taken to include a change in analytic structure of the distribution of eigenvalues 2 . In the present case we have shown that the distribution of eigenvalues for a linear chain of Hermitean matrices is determined completely by a linear integral equation (3.4) involving the kernel K (4.1). In this Section we will go further and solve for the asymptotic eigenfunctions χ as a series of iterations of K. Given that iterations of K are calculable by saddle-point methods and the relation (3.5), we can effectively solve for the distribution of eigenvalues ρ 0 . In particular we will be able to determine the critical structure of the matrix chain by analyzing changes in the analytic structure of the kernel K and its iterations. We begin by slightly generalizing the problem of finding the eigenfunctions χ in order to make use of well-known techniques [20]. Instead of the integral equation (3.4) let us consider for a moment an inhomogeneous version with c some real constant χ(x) = c + λ du K(x, u)χ(u) ≡ c + λKχ (5.1) Since there are no constant eigenfunctions of the kernel K, the unique solution of (5.1) is given by the Neumann series which builds the solution up through an iterative procedure χ 0 = c (5.2) χ 1 = c + λKχ 0 . . . χ n = c + λKχ n−1 Taking the limit of this process generates the solution χ(x) = 1 + ∞ n=1 λ n du K n (x, u) c ≡ R(x; λ)c (5.3) where R is commonly referred to as the resolvent kernel and K n is given by the convolution of n kernels K n (x, u) = dz 1 · · · dz n−1 K(x, z 1 )K(z 1 , z 2 ) · · · K(z n−1 , u) (5.4) The convergence of the resolvent kernel is guaranteed by the boundedness of K n and the Hilbert-Schmidt theory of integral equations [20]. In order to recover the homogeneous solution we should set c = 0 which leads to a trivial solution for χ. Fortunately for us only the c independent logarithmic derivative of χ is required to recover the eigenvalue distribution ρ 0 of the matrix chain. From the definition of χ, (3.5) and (5.3) we have the result ρ 0 (x) = lim ǫ→0 2iǫ π d dx log χ(x) = lim ǫ→0 2iǫ π d dx log R(x; λ) (5.5) Hence we can extract the eigenvalue distribution from the resolvent kernel R, which is expressible solely in terms of iterations of the kernel K as defined in (5.3). This is an important observation since the iterated kernels are calculable in the limit of vanishing ǫ. Explicitly, the n th iterated kernel can be written for n ≥ 2 K n (x, u) = 1 (4πǫ) n/2 e − U (x)+U (u) 4ǫ dz 1 · · · dz n−1 e −S/2ǫ (5.6) where the 'action' S is given by S = n−1 i=1 U(z i ) − n−2 i=1 z i z i+1 − z 1 x − z n−1 u (5.7) Evaluating the integral over z = {z 1 , . . . , z n−1 } is a straightforward application of saddlepoint methods in the limit of vanishing ǫ. In this limit the dominant contributions to the integral come from the vicinity of the stationary points of the function S ∂S ∂z i z α = 0 (5.8) In general there are many such stationary points, which we will label by α. Expanding S up to quadratic order about the stationary points, the resulting Gaussian integrations give the approximation to the iterated kernel K n+1 (x, u) = 1 √ 4πǫ e − U (x)+U (u) 4ǫ α (−1) λn( z α ) det H n ( z α ) e −S( z α )/2ǫ (5.9) where λ n ( z α ) is the number of negative eigenvalues of the n × n Hessian H n of second derivatives of S evaluated at the stationary point H n ( z α ) = ∂ 2 S ∂z i ∂z j z α =         U ′′ (z α 1 ) −1 0 0 · · · −1 U ′′ (z α 2 ) −1 0 · · · · · · · · · · · · 0 · · · 0 −1 U ′′ (z α n )         (5.10) While we have counted contributions from all stationary points of S in (5.9), the only relevant ones are those with minimum 'action'. The contributions of other, irrelevant, stationary points are exponentially suppressed in the limit of vanishing ǫ. Unfortunately it is difficult to calculate the iterated kernels and the resolvent in closed form to obtain explicit results for the eigenvalue distribution. It is possible though to extract enough qualitative information from (5.9) to outline a one-dimensional phase diagram of the matrix chain. To be definite we will consider a quartic potential Consequently we see that when evaluating arbitrarily high iterations of the kernel, K n , the Hessian at the origin will being to develop negative eigenvalues at m 2 = 1 and this stationary point becomes unstable. Since S is bounded from below for our choice of potential, there are other stationary points where S has a lower magnitude and it is these which will dominate the integral (5.6) for K n . U(φ) = φ 4 + m 2 φ 2 (5. This degeneration at m 2 = 1 signals a phase transition in the matrix chain and the physical meaning of it can be found by returning to the original form of the matrix chain action (1.1). The potential there, V is related to the current U by the one dimensional lattice spacing a e − iπ 2ǫ x dη ρ 0 (η) = λ √ 4πǫ du e −(x−u) 2 4aǫ −a V (x)+V (u) 4ǫ e − iπ 2ǫ u dη ρ 0 (η) (5.15) It is a straightforward calculation to show that for vanishing ǫ, and to leading order in vanishing a, (5.15) is solved by ρ 0 (x) = 1 π E − V (x) (5.16) which is the well-known solution for the ground state of matrix quantum mechanics [1]. As In addition to this standard behaviour it was argued in [16] that there exist a countable infinity of additional points in this phase where the eigenvalue distribution ρ 0 exhibits subleading non-analyticities. In fact we can reproduce these results if we assume, incorrectly, that the origin z = 0 is a relevant stationary point in the evaluation of ρ 0 . Considering the vanishing of Hessian eigenvalues (5.12), we see that the naive saddle-point approximation to K s+1 (5.9) will break down when m 2 = cos rπ s + 1 (5.19) It is well known that at degenerate stationary points there is a change in the analytic structure of a saddle-point approximated integral so it is natural to assume that when (5.19) is satisfied, K s+1 and the eigenvalue distribution ρ 0 will exhibit non-analyticities. In terms of lattice spacing the degeneracy condition (5.19) can be found by comparing with (5.14). The result a = 1 − cos rπ s+1 2 = sin rπ 2(s + 1) (5.20) with r and s ranging over positive integers is exactly the condition for subleading critical behaviour as found in [16]. It would seem that the same mechanism which produced the critical exponents of matrix quantum mechanics for −1 < m 2 < 1 produces different exponents here. This may be because of the total degeneracy of the origin but explicit calculations of the iterated kernels e − iπ 2ǫ x dη ρ KT (η) = 1 √ 4πǫ b −b du e (x+u) 2 4ǫ e − iπV ′ (x) = DU ′ (x) − 2(D − 1) −dz ρ 0 (z) x − z (6.2) where the eigenvalue density ρ 0 is common to both models. The one drawback of this model of induced QCD is that the action (6.1) has a Z Z N symmetry under the center of the gauge group which leads to super-confinement of gauge degrees of freedom, even on the lattice scale [27]. Obviously this not a feature that one would like to have in the continuum limit of the model and the easiest way to avoid it is for the system not to realize this symmetry faithfully in its solution. Unfortunately for the potentials U of logarithmic form, for which we have ρ 0 explicitly (see Section Four), this symmetry persists [21,28]. It would be interesting to see if solutions of the matrix chain for more general potentials might lead to non-trivial propagation of gauge fields in the continuum limit of the Kazakov-Migdal model. This can be checked in principle using the solution of the matrix chain for the eigenvalue density ρ 0 and the formalisms of [29] and [30] which express the correlation of gauge fields in the model in terms of ρ 0 . (2. 1 ) 1, ρ(λ) is a probability distribution function which returns the fraction of eigenvalues of the matrix variables φ x with value λ. In the present context of collective field theory, ρ plays the role of a coordinate and Π is the canonically conjugate momentum. Here we have included a potential V which is most generally time-dependent. Independent of time, (2.1) describes the evolution of the matrix quantum mechanics (1.3) with V = V . The equations of motion following from this Hamiltonian are given by the functional variations with respect to the canonical variables 13) are replaced by their periodic counter-parts since the matrix variables are unitary rather than Hermitean. The Green's function is that of the heat equation on a circle and the boundary conditions on ρ and v specify the topology of the two-dimensional space-time. In general these cases again lead to non-linear integral equations for the unknown fields but as we will now see, there are situations where the general integral equation (2.13) leads to a tractable, linear equation. 2.10) reduces to the linear heat equation. Substituting the well known Green's function in (2.13) and implementing the boundary conditions (3.3), we find a linear integral equation of second Fredholm type for the initial eigenvalue density ρ 0 (x) = ρ(x, 0) in terms of the effective potential U(x) ) for the matrix chain as a well-posed, linear problem for the eigenvalue density ρ 0 . Moreover, the integral form (3.4) allows one to write equations for the eigenvalue density immediately. Formally, in the limit of vanishing ǫ, the integral in (3.4) can be calculated by saddle-point methods. This calculation is particularly straightforward when only a single stationary point, u s contributes to the integral (3.4). This stationary point is determined by the extrema condition 4.4) from (4.3) we find that L will be equivalent to the integral kernel K, if and only if we satisfy the natural commutativity condition (L x − L u )K(x, u) = 0 (4.5) a mn (x + u) m (xu) n (4.8) Expanding each side of the second equality in powers of u and equating coefficients gives, to leading order in ǫ, the result U ′ (x) = − a 11 x 2 + (2a 20 + a 01 )x + a 10 a 02 x 2 + a 11 x + a 20 (4.9) a 2 (x) = −(a 02 x 2 + a 11 x + a 20 ) a 0 (x) = a 00 + a 10 x + a 20 x 2 − (a 11 x 2 + (a 01 + 2a 20 )x + a 10 ) 2 4(a 02 x 2 + a 11 x + a 20 ) a 02 x 2 + a 11 x + a 20 )(a 00 + a 10 x + a 20 x 2 − E) − (a 11 x 2 + (a 01 + 2a20 )x + a 10 ) 2 2π(a 02 x 2 + a 11 x + a 20 ) 10 − a 01 a 02 a 11 + a 3 11 − 2a 02 a 11 a 20 a 2 02 4a 02 a 20 a 20 + a 11 x + a 02 x 2 (4.14) Contained in this general form are a number of previously examined examples of the matrix chain. First, setting a 11 = 0 and a 10 = 0 recovers the symmetric double Penner-type potential ( 4 . 412) is what one would expect of a D = 0 one-matrix model with a particular potential. This correspondence was previously suggested for arbitrary potential in the matrix chain, but such a simple solution of the integral equation (3.4) is not possible. In fact with further computation it can be argued that higher, finite, order self-adjoint differential operators are equivalent to the integral kernel K only if they are functions of the second order operator L x we have constructed above. Consequently, the large N eigenvalue distribution ρ 0 of the matrix chain is the solution of a polynomial equation only for potentials of the form (4.14) 11 ) 11This simple form is convenient since it allows a clear view of the intrinsic behaviour of the matrix chain which depends on the strength of the quadratic term.Starting with m 2 large and positive we expect that the eigenvalue distribution will be localized near the origin where the quadratic term of the potential is dominant. Consequently, the solution (4.15) for the pure quadratic potential is a good approximation to the true solution and is smooth deformation of it. Making contact with the saddle-point evaluation of the iterated kernels (5.9) it is easy to convince oneself that for large m 2 there is only one stationary point of the action (5.7) and it is located near the origin z = 0. As m 2 is decreased, the solution (4.15) of the pure quadratic potential becomes unstable at m 2 = 1 and in fact this instability is present in the full solution. Information about the stability of stationary points is contained in the eigenvalues µ r of the Hessian H s (z), which evaluated at the origin in z-space give µ r = 2m 2 − 2 cos rπ s + 1 , r = 1, . . . , s (5.12) through U(φ) = aV ( √ aφ) + φ 2 (5.13)Taking two derivatives with respect to φ and fixing (without loss of generality) V ′′ (0) = −4to agree with the conventions of[16], we findm 2 = 1 − 2a 2 (5.14)Hence, the instability at m 2 = 1 corresponds to the limit of vanishing lattice spacing and we are observing a phase transition to D = 1 matrix quantum mechanics (i.e.(1.3)) from the m 2 > 1 phase where the solutions are qualitatively what one would expect from a D = 0 single matrix model. In fact this observation can be made more concrete by returning to the integral equation (3.4) but in terms of the potential V and lattice spacing a, m 2 2is decreased further below this transition point, the stationary point of S at the origin z = 0 is seen to become progressively more unstable as the number of negative eigenvalues of the Hessian (5.12) grows. Here the stationary points relevant to the saddle-point integrations (5.9) are not in the neighbourhood of the origin and are stable under changes in m 2 . The only instability that occurs with the relevant stationary points in this phase is due to a Z Z 2 symmetry of the action S. z i+1 − z 1 x − z n−1 u (5.17) we see that for vanishing x and u there is a symmetry under z i → −z i . It follows that a relevant stationary point at z * has a mirror image at − z * and as x and u are tuned through the origin the relevant stationary point can change from one to the other. In this way nonanalytic behaviour arises in the solution of the eigenvalue distribution ρ 0 , presumably of the form ρ 0 ∼ |x| (5.18) which is consistent with the critical behaviour of D = 1 matrix quantum mechanics. Of course this detail is difficult to verify without explicit calculation. Again we stress that this behaviour is not realized in the matrix chain and only arises when one takes into account unstable stationary points in the evaluation of saddle-point integrations. This is the main shortcoming of the functional form (3.8): in general there are a countable infinity of solutions and it is a difficult task to discover the true solution with minimum action. There is one more phase transition in the matrix chain as m 2 decreases further. From (5.12) we see that as m 2 → −1 all eigenvalues of the Hessian at the origin become negative and this stationary point is degenerate in all directions. On the surface, besides this observation of an irrelevant stationary point, there is little change in the system. The relevant stationary points are found by solving polynomial equations and their stability is determined locally as a smooth function of m 2 . This is misleading though since m 2 = −1 corresponds to a lattice spacing of unity which, in our scaling, is where a Kosterlitz-Thouless phase transition wasshown to occur[13,15,16]. The only overt sign of the transition here is that the integral kernel K again changes analytic structure to origin it has the form of a translation kernel which is of a different character than the original kernel (4.1). If we consider the higher, quartic terms to only make the kernel well-defined, the behaviour of the eigenvalue distribution ρ 0 can be found by restricting the range of integration in the integral equation(3.4) form it is clear that the solution will depend only on the constant b which sets a scale[16], otherwise it is a universal quantity. Unfortunately the asymptotic (ǫ → 0) solutions of this equation are not known and so remains the eigenvalue distribution ρ KT of the matrix chain at the Kosterlitz-Thouless transition.Finally, for m 2 < −1 there persists a pair of Z Z 2 related stationary points that at u and x vanishing trade off relevance. It is conjectured that this non-analytic behaviour is what one would find in a D = 0 single matrix model, i.e. (5. 9 ) 9and the eigenvalue distribution ρ 0 are needed to make definite statements about the characteristics of the solutions of this phase of the model.6 Conclusions and an applicationTo review, we have demonstrated a general technique for expressing the equations of motion for systems whose time evolution is governed by Das-Jevicki-Sakita Hamiltonians with boundary conditions as an evolution problem in one-dimensional quantum mechanics. Specifying to the particular example of the one-dimensional chain of interacting Hermitean matrices we showed that in this case the general formalism reduces to a linear integral equation for the large-N eigenvalue density of the matrices. Analyzing the associated integral kernel we were able to recover all previous explicit solutions of the matrix chain from asymptotic solution of particular second order ordinary differential equations. In the general case, the solution of the integral equation was developed in terms of iterations of the kernel and from this solution the critical behavior of the system was found and a universal integral equation for the distribution of eigenvalues at the Kosterlitz-Thouless point was presented. In addition it was demonstrated that the sub-leading critical behaviour observed in[16] results from contributions from unstable stationary points in saddle-point approximations and hence is not observed.What was not accomplished here is the explicit calculation of a non-trivial solution to the large-N matrix chain using the formalism developed here. It would be nice to find a potential for which the series of iterated kernels could be explicitly calculated and summed and the eigenvalue distribution extracted. This is of particular interest in testing the conjectures involving the existence of a Kosterlitz-Thouless phase transition in the system. With a global solution for ρ KT one would be in a position to calculate explicitly the free energy of the matrix chain and check the critical behaviour. Of course only the leading non-analytic part of the free energy is required but in order to obtain this one should have a definite idea of the analytic structure of the distribution of eigenvalues.The calculations we have performed here are not only relevant to string theory in one dimension but also for other problems related to the matrix chain. An example is the Kazakov-Migdal lattice model of induced QCD in D-dimensions[7]. Devised as a lattice model of gauge interactions where the self-interaction of adjoint scalar fields φ x are supposed to induce the standard Wilson term which is left out, it is defined by the following actionS KM = x TrV (φ x ) − xy TrΛ xy φ y Λ yx φ x (6.1)Here the sub-scripts label sites on a hyper-cubic lattice on which the N × N adjoint scalar fields φ x reside. Nearest neighbour interactions are mediated by SU(N) gauge fields Λ residing on the lattice links. From the large N saddle-point equations it can be shown[10] that the one-dimensional matrix chain we have considered (3.1), with potential U, is equivalent to the Kazakov-Migdal model with a potential satisfying of induced QCD as an interesting application of the techniques developed here. Since the D-dimensional model is effectively reducible to the one-dimensional case, the non-trivial behaviour of the matrix chain demonstrated here may help to overcome some of the apparent deficiencies of that model of gauge interaction.Moreover, in constructing the general solution it will become clear that a countable infinity of unstable stationary points appear in intermediate steps each of which can lead to misleading results. In addition to the details just outlined, we recall the Kazakov-Migdal (lattice) model 2 Collective field theory to quantum mechanics To begin we will demonstrate a non-linear change of variable which allows the equations of motion derived from a Das-Jevicki-Sakita Hamiltonian [9] This restriction is not essential. In the general case the time evolution of the system is given by a Dyson series expansion. This assumption of a physical phase transition following from a change in the analytic structure of the eigenvalue distribution does not always hold[26]. Nevertheless, in the absence of calculating physical observables, we will take a change in structure of the eigenvalue distribution to be a strong hint of a physical transition. AcknowledgmentsWe would like to thank S. Jaimungal, V. Kazakov, I. Klebanov and E. Lieb for helpful comments and the Niels Bohr Institute for its hospitality during a visit in which this work was begun. 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{'abstract': 'We introduce a general method for transforming the equations of motion following from a Das-Jevicki-Sakita Hamiltonian, with boundary conditions, into a boundary value problem in one-dimensional quantum mechanics. For the particular case of a onedimensional chain of interacting N × N Hermitean matrices, the corresponding large N boundary value problem is mapped into a linear Fredholm equation with Hilbert-Schmidt type kernel. The equivalence of this kernel, in special cases, to a second order differential operator allows us recover all previously known explicit solutions for the matrix eigenvalues. In the general case, the distribution of eigenvalues is formally derived through a series of saddle-point approximations. The critical behaviour of the system, including a previously observed Kosterlitz-Thouless transition, is interpreted in terms of the stationary points. In particular we show that a previously conjectured infinite series of sub-leading critical points are due to expansion about unstable stationary points and consequently not realized.', 'arxivid': 'hep-th/9902089', 'author': ['L D Paniak paniak@feynman.princeton.edu \nDepartment Of Physics\nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNew Jersey, USA\n'], 'authoraffiliation': ['Department Of Physics\nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNew Jersey, USA'], 'corpusid': 14896666, 'doi': '10.1016/s0550-3213(99)00250-3', 'github_urls': [], 'n_tokens_mistral': 14568, 'n_tokens_neox': 12752, 'n_words': 8089, 'pdfsha': '173e39e31c167dd2cd2c192fca197b8c541c6fdb', 'pdfurls': ['https://arxiv.org/pdf/hep-th/9902089v2.pdf'], 'title': ['Eigenvalue Dynamics and the Matrix Chain', 'Eigenvalue Dynamics and the Matrix Chain'], 'venue': []}

arxiv

Self-Supervised Representation Learning from Temporal Ordering of Automated Driving Sequences Christopher Lang University of Freiburg Robert Bosch GmbH Alexander Braun Robert Bosch GmbH Lars Schillingmann Robert Bosch GmbH Karsten Haug Robert Bosch GmbH Abhinav Valada University of Freiburg Self-Supervised Representation Learning from Temporal Ordering of Automated Driving Sequences Self-supervised feature learning enables perception systems to benefit from the vast amount of raw data being recorded by vehicle fleets all over the world. However, their potential to learn dense representations from sequential data has been relatively unexplored. In this work, we propose TempO, a temporal ordering pretext task for pre-training region-level feature representations for perception tasks. We embed each frame by an unordered set of proposal feature vectors, a representation that is natural for instance-level perception architectures, and formulate the sequential ordering prediction by comparing similarities between sets of feature vectors in a transformer-based multi-frame architecture. Extensive evaluation in automated driving domains on the BDD100K and MOT17 datasets shows that our TempO approach outperforms existing self-supervised single-frame pre-training methods as well as supervised transfer learning initialization strategies on standard object detection and multi-object tracking benchmarks. Figure 1 . We introduce TempO, a self-supervised learning pretext task by temporal ordering of frames, that pre-trains perception models for both frame-level tasks, like object detection, and multi-frame tasks like multiobject tracking. We propose a transformer-based architecture that is designed to scale quadratic w.r.t. sequence length, and enables richer temporal context during pre-training. methods as pre-training for image classification tasks have been demonstrated by image-level feature learning using contrastive methods [4][5][6]16]. These approaches were later extended to region-level representation learning, where the contrastive approach is applied to image patches tracked over a set of augmentations [8,52]. In robotic domains, such as automated driving, the perception system observes the environment from a continuous stream of sensor data. Adding such temporal context allows learning from undistorted images by exploiting object permanence and dynamical constraints. Recent methods exploit these by enforcing linear motion models and static scene assumptions for depth estimation [14], or consistent appearance constraints in tracking patches along a temporal cycle [23, 46,47]. However, such methods fail once these underlying assumptions are violated (e.g. by dynamic objects in the scene or large camera movements), resulting in inconsistent predictions or indefinite loss values. The task of ordering a set of shuffled frames into their original temporal succession by video-level feature representations [26,33] has shown promising results, as it is less ambiguous and never ill-defined while encouraging an understanding of both local object semantics and global scene composition. In this work, we propose TempO, a self-supervised representation learning approach that extends temporal ordering to region-level feature learning for object detection and tracking. We achieve this synergy by defining the temporal ordering task as a sequence estimation problem and construct the method based on the tracking-by-detection paradigm, as depicted in Figure 1. This design of a single-frame (spatial) network and a light multi-frame head requires the network to learn consistent representations and perform the bulk of the semantic reasoning for each frame separately. We evaluate the performance of TempO pre-training on downstream tasks of object detection on the BDD100K dataset and multi-object tracking on the BDD100K and MOT17 datasets, by comparing with pre-training on supervised datasets and existing unsupervised pre-training methods. Furthermore, we study the utility of representations learned from TempO pre-training for the frame retrieval task, without additional training. The main contributions of this work are: • We propose TempO, a self-supervised pretraining pipeline for object detection models and multi-object tracking models from a temporal ordering pretext. • We design a transformer-based multi-frame sorting head whose computational complexity scales less than quadratic with the sequence length. This allows us to pretrain on longer sequences compared to combinatorial approaches [33,53]. • We perform extensive evaluations of TempO pretraining for the downstream tasks of object detection and multi-object tracking that demonstrate the utility of our approach. Related Work Our work is related to the field of self-supervised visual representation learning from sequential data. In the following, we embed our proposed method in these areas. Self-Supervised Image Feature Learning: Self-supervised learning has been studied extensively on single images, where the field can be broadly categorized into image-level, region-level, and pixel-level approaches. Image-level approaches learn a global embedding vector per frame and are typically evaluated on image classification tasks. Contrastive methods learn an image feature vector that is invariant to a set of augmentations while being distinct from embeddings of other images. The choice of negative examples ranges from instance-based discrimination [6,7], clustering-based pretext tasks [4], and bootstrapping approaches [5,16]. The temporal consistency in video clips is also used as a source for positive pairs from subsequent frames [11]. Enforcing such temporally persistent features benefits both instancebased discrimination and clustering-based methods [11]. Nevertheless, many computer vision tasks, including object detection [24,25] and segmentation [15,34,45], require dense representations that also encode local image information. Region-level approaches [8,9,57], therefore, rely on pretext tasks that operate on patches of an image or the feature map. Patch discrimination methods [29,35] utilize these image patches to learn local augmentationinvariant embeddings, analogously to the image-level methods described above. The patch re-identification pretext task [10,19], on the other hand, emphasizes the localization task by regressing the image coordinates of the local patch in the global image. Combining patch discrimination and re-identification pretext tasks have shown sustainable gains for downstream object detection [8,10,51,52], as the losses include regression and classification-based terms analogously to the downstream task. In our experiments, we compare against UP-DETR [8] as a baseline, which is trained on localizing random crops in an image and contrasting the feature embeddings of crops within an image. It, therefore, is less sensitive to the choice of image augmentations, which is comparable to our proposed method. Dense self-supervised feature learning approaches on the level of pixels [35,38,52] or features [29,48,54] are evaluated on segmentation tasks. The pretext tasks rely on unsupervised segmentation masks [19,20] or the construction of new images [48,54,55]. In recent years, consistency constraints in sequential data have been used as a supervisory signal to learn dense feature representations. Tracking approaches exploit object permanence constraints in appearance by searching image patches by their representation in a feature map. The pretext task is either to predict the offset of an image region forward and then backward along a cycle in time [21,47], whereby the difference between start and end points is used as a training signal. However, such methods operate only on pairwise frame contexts and depend on data domains where the presence of an object over a certain time window can be ensured. Cross-stream approaches [17,43,44] estimate the optical flow between two images, from which they derive an informed selection of positive and negative patches [43], instances [17] or prototypes [44] for contrastive learning. Self-Supervised Video Feature Learning: Video feature learning exploits the temporal consistency in sequential data as a supervision signal to generate video-level embeddings. Self-supervised learning approaches can be broadly categorized into three types of pretext tasks: 1. Video discrimination learns contrastive video-features [9,11,39] that are invariant to a set of temporally consistent augmentations. 2. Sequential verification performs a binary classification if either a sequence is incorrect or in a shuffled temporal order [12,33]. Other formulations discriminate between forward and backward order of frames [49]. Such methods are used for encoding video clips all at once but are outper- T 2:4 T 1 T 2 T 3 T 4 Additive Attention Transformer encoder Association scores (proposal-byproposal) Next image probabilities (frame-by-frame) Proposal-to-frame reduction function Att p(I n |I 1:n-1 ) H 1:3 Future frame mask Figure 2. Illustration of the multi-frame head in the TempO architecture. Each frame n is represented by a set of proposal feature vectors Tn extracted by the same frame-level network. The proposal features are concatenated and encoded by a transformer encoder into a set of history tokens H using a future frame masking that allows for each proposal to aggregate temporal context from past frames only. An additive attention mechanism then computes the association scores between the history tokens and the proposal feature vectors. We next map all scores of proposal features corresponding of the same frame onto scalar image transition probabilities. Our proposed temporal ordering task maximizes these probabilities for the correct temporal order during pre-training. formed by temporal ordering methods as the network needs to reason and understand the statistical temporal structure of image sequences 3. Temporal ordering methods on video classification tasks [26,37,49] is formulated as a classification problem among all the frame permutations, which limits their usage to sequence lengths of usually not more than six frames due to the combinatorial explosion of orderings. While the aforementioned methods learn video-level embeddings, they are widely evaluated on clip retrieval and action recognition tasks. We define the temporal ordering pretext task as a sequence estimation problem by estimating image transition probabilities instead of a multi-classification problem, which allows learning frame-level feature representations that allow for a larger variety of downstream tasks, as described in the following section. Technical Approach We next detail our proposed TempO pretext task for region-level visual representation learning. In the remainder of this section, we first introduce the pretext task in Section 3.1, followed by the network architecture in Section 3.2 and Section 3.3, and finally describe the transfer learning technique to perform downstream task evaluations in Section 3.4. Sequence Ordering Task Definition For the TempO pretext task, we consider image sequences of length N . We chose the starting index such that n = 1, . . . , N for a more concise notation. A training sample is composed of the first frame in the sequence I 1 as an anchor in time, and the remaining sequence frames I 2:N in arbitrary order. At first, a single-frame network extracts an unordered set of proposal feature vectors from each frame independently. The multi-frame transformer head then processes the concatenated proposal feature vectors over all N frames in a training sample and maps them onto next-image probabilities given a sequence of images as described in Section 3.3. Our training objective is to maximize the next-image probability ρ (I n |I 1:n−1 ) for the observed temporal orderings in the video data using a ranking loss formulation L = m =n max {ρ(Im|I1:n−1) − ρ(In|I1:n−1) + ∆, 0} , (1) where ∆ ≥ 0 is a scalar margin. Single-Frame Network Our approach adapts to network architectures that process images and produces a set of P proposal feature vectors Q ∈ R P ×D of dimension D per frame. This includes common region proposal-based [42] and transformer-based [3,58] architectures. The majority of our experiments are conducted on the Sparse R-CNN [42] object detection architecture, using a ResNet-50 [18] as a feature extractor. It learns a sparse set of P proposal boxes and features, from which classification scores and bounding boxes are generated. The initial proposal features are extracted from learned proposal box regions in the feature map. The proposal features are then iteratively refined by a sequence of dynamic heads that each allow interaction between proposal features via selfattention modules. We implement two distinct branches of dynamic heads: a detection branch that extracts object proposal features, consisting of six iterative heads, and a tracking branch that extracts tracking proposal features, from two iterative dynamic heads, that are used to associate objects identities throughout a sequence. The model under pre-training uses two dynamic heads, whose parameters are cloned to initialize the parameters of both the detection and tracking branch (see Section 3.3) during fine-tuning. The motivation for separating the tracking and detection branches is that the feature representations pursue competing objectives. While detection features should learn to generalize across an object type (e.g. car), tracking features should learn to discriminate between object instances. Multi-Frame Sequence Ordering Network The overall setup of our sequence ordering network is depicted in Figure 2. We express the image transition probability with respect to the track features as ρ (I n |I 1:n−1 ) = ρ(T n |H n−1 ) given a set of sequence history up to frame n − 1 as H n−1 ∈ R (P ×D . The history tokens at H n−1 are the output sequence tokens by a transformer encoder, that takes as input the track feature vectors up to frame n − 1. This is implemented by masking track features of future frames in the attention matrix. Finally, we compute additive attention between the encoded sequence history tokens and the track features given by Att(t i n , h j m ) = v T tanh W 1 t i n + W 2 h j m ,(2) where Att(t i n , h j m ) denotes the attention score between t i n , i.e., the i-th proposal vector in frame n, and h j m , i.e., the j-th history feature vector up to frame m. v, W 1 , andW 2 are the learnable parameter matrices. In the next step, we employ a reduction function that computes a scalar ordering score from the track-to-history association score matrix. Proposal-to-Frame Reduction Function: The final stage in our multi-frame head is the reduction from an association score matrix to a next-frame transition probability. As the associations are derived from unordered sets of proposals, this function is required to be permutation invariant with respect to the elements of the association matrix. One naive candidate is the mean over all matrix elements (AvgPool) that encourage similarity between all track features in temporally subsequent frames. Since tracking requires a consistent embedding of an object across subsequent frames, we also experiment to enforce a one-to-one matching among track features of two frames. Therefore, we propose an approximation of the linear sum assignment, as follows: ρ (T n |H m ) = j max j softmax i=1,...,P Att(t i n , h j m )(3) and ablate over these choices of the reduction function. Downstream Task Architectures Our models use the ResNet-50 [18] architecture as the backbone and the Sparse R-CNN [42] architecture with two iterative dynamic heads for generating 100 object proposals per frame. The ResNet-50 parameters are initialized with weights pre-trained on the ImageNet dataset and all the other model parameters are randomly initialized. For the pretext task, we collect feature vectors over all sequence frames and feed them to a masked transformer encoder described in Section 3.3. Since we exclusively use permutation invariant operations and no positional embedding on the multi-frame level, we omit the explicit shuffling of frames in our implementation. Object Detection: For the object detection fine-tuning, we adapt the originally proposed Sparse R-CNN configuration [42]. Therefore, we build upon the pre-trained Sparse R-CNN architecture and stack four iterative dynamic heads on top of the pre-trained heads, called the detection branch. The final object proposal vectors are mapped onto classification scores and bounding box regression coordinates using separate linear layers. Multi-Object Tracking: During the Multi-Object Tracking (MOT) downstream task, we associate proposals based on their additive attention between track features T i and the history features of the previous frame H i−1 that is generated by the transformer encoder. We follow the setup in QD-Track [36], applying their bidirectional softmax matching in feature space, tracker logic, and training pipeline. We further extend the model into an object detector, as described above. We, therefore, clone the pre-trained iterative heads, such that they have distinct parameter sets for the tracking and detection branch. Experimental Results We pre-train the networks using our proposed TempO approach on the train splits of the Berkeley Deep Drive [56] (BDD100K) as well as MOT17 [32] (MOT17) datasets. In this section, we compare the performance of TempO pretrained models with other initialization strategies from the literature for single-frame as well as multi-frame downstream perception tasks. Datasets The Berkeley Deep Drive [56] (BDD100K) dataset contains crowdsourced videos of driving scenes such as city streets and highways. It consists of different weather conditions and times of the day. For MOT fine-tuning, we use the annotations of the BDD100K and MOT17 images at 5 frames per second (FPS), which consist of 1400 videos of 40 s length. The annotations cover eight object categories of overall 131k identities. For object detection fine-tuning, we use annotations from the object detection challenge, which provides eight annotated categories for evaluation. The MOT17 [32] (MOT17) challenge consists of 14 video sequences (7 training, 7 test) at varying frame rates (>14fps) and sequence lengths (>20s) in unconstrained environments filmed with both static and moving cameras. It provides MOT annotations that feature a people class with at least three persons visible per frame. For our pre-training, we sample frames at 5 FPS from videos in the training split, from which we generate non-overlapping training sequences. Training Protocol We train the models on NVIDIA V100 GPUs with a batch size of 8 for 6 epochs on the pretext tasks. We use the AdamW optimizer with an initial learning rate of 2.5 · 10 −5 and weight decay of 10 −4 . A step scheduler further reduces the learning rate by a factor of 10 every 3 epochs. We fine-tune the models for another 6 epochs on the respective downstream tasks. The baselines were trained for 12 epochs on the downstream tasks, such that all models have seen each frame at most 12 times during training. We resize the images to a resolution of 800 pixels on the longer side and perform random cropping (spatial augmentation) or photometric augmentations such as color jitter, random gray scaling, and brightness change, only when stated. Evaluations on Downstream Tasks We evaluate how the region-level feature representations learned from the TempO pre-training impacts the performance of single-frame and multi-frame downstream tasks, i.e., for object detection and multi-object tracking on the BDD100K as well as the MOT17 dataset. We follow the evaluation protocol as in [5,7,53], where we fine-tune all models for the same fixed number of epochs (12 in our case). The TempO pre-training consists of ordering sequences of length N = 8, using two layers in the transformer encoder, for 6 epochs, followed by fine-tuning on the downstream task for another 6 epochs. Table 1 shows the mean average precision over all the classes in the BDD100K object detection benchmark. The average precision measures the area under the precisionrecall curve for detections averaged over thresholds for IoU ∈ [0.5 : 0.05 : 0.95] with the ground truth bounding boxes. We compare our proposed SSL pre-training approach against various initialization strategies using the Sparse R-CNN [42] object detector, including the common practice of pre-training the feature extractor as a classifier on the Ima-geNet dataset, as well as pre-training the model parameters on the supervised Common Objects in Context [28] (COCO 2017) object detection dataset. Additionally, we compare against the single-frame self-supervised pre-training task of random query patch detection (RPD) as described in [8]. For our adaptation to the Sparse R-CNN detector, we add the Table 1. Object detection results on the BDD100k val dataset. We use use a sequence length of 8 frames in the TempO pre-training, two layers in the multi-frame network, and AvgPool motivated by the ablation results presented in Table 3. patch image features to the initial proposal features of the Region Proposal Network (RPN) neck. We observe that our TempO initialization outperforms supervised pre-training on the COCO 2017 dataset by +0.7%mAP , while the performance gain compared to the single-frame pretext task of random query patch detection is as large as +2%mAP evaluated using the Sparse R-CNN detector. Moreover, TempO pre-training results in faster convergence of the detectors. Please refer to the supplementary material for a comparison of convergence plots. In Figure 3, we present qualitative results of (a) a Sparse R-CNN object detector trained with ImageNet pre-trained weights and (b) the TempO pre-trained Sparse R-CNN detector. Compared to the fully supervised training strategy, the TempO pre-trained Sparse R-CNN detector improves detection accuracy as it suppresses a ghost detection of a motorcycle within the pile of garbage bags on the right side of the image shown in Figure 3 (b). It also detects the poorly illuminated rider on top of the moving bicycle in Figure 3 (b). Object Detection Results Multi-Object Tracking Results We evaluate our TempO pre-trained models on the MOT downstream task using the standard evaluation metrics [22]. We build on QDTrack [36] to extend our model into a multiobject tracker, as it associates detected objects in the feature space and is the current state-of-the-art on the BDD100K MOT benchmark for models with fewer than 100M parameters. As a baseline, we trained this method in its standard configuration using the Faster R-CNN and Sparse R-CNN model as a detector, which was pre-trained on the BDD100K detection dataset. Table 2 shows the performance on the BDD100K MOT val dataset for fine-tuning with various parameter initialization strategies on the BDD100K MOT training annotations for 4 epochs. We compare the performance with the higherorder tracking accuracy [31] (HOTA) metric which combines the detection, association, and localization accuracy into a single value. Additionally, we present the benchmark metrics of MOTA, IDF1, and ID switches. We observe that our tracking approach with TempO pre-training and prior fine-tuning as an object detector achieves the highest tracking accuracy in both the HOTA and MOTA scores, outperforming the baseline QDTrack using a Faster R-CNN detector by +0.9% in the HOTA score. Interestingly, the Sparse R-CNN approach achieves lower association accuracy and IDF1 scores compared to the Faster R-CNN as a base detector, when only pre-training the object detector, while detection and localization accuracy are at comparable levels. This results in lower overall tracking accuracy when the network parameters are only initialized with TempO pretraining, and fine-tuning of both detection and association task are required during the fine-tuning epochs. Figure 4 shows an example tracking sequence using the tracking architecture described in Section 3.4. We see that the network reliably tracks the car with high confidence, even under heavy occlusion and changing shape from opening the car door. The pedestrian entering the car is successfully tracked over the front, side, and rearview and partially occluded when stepping into the backseat. MOT17 Results: We further evaluate our TempO pre-training strategy on the popular MOT17 benchmark. We pre-trained our model on MOT17 dataset for 50 epochs and compare it against various baselines that are initialized with pre-trained object detectors on the COCO 2017 dataset and with unsupervised pre-training on the Crowdhuman dataset by an RPD pretext task. We follow the training setting described in QDTrack and fine-tune our models for 12 epochs on a mixed dataset of CrowdHuman [40] and MOT17 train set. The result in Table 4 shows that TempO pre-trained Sparse R-CNN model outperforms both its unsupervised COCO and unsupervised RPD initialized counterparts by +1.4% HOTA and +2.3% MOTA, which mainly trace back to an increased detection accuracy DetA and fewer ID switches. Table 3 shows the benchmarking results for object detection and MOT downstream tasks on the BDD100K val dataset for models pre-trained on our proposed TempO training with varying hyperparameter settings. In particular, we ablate over the sequence length, the attention hierarchies of the multi-frame transformer encoder, as well as the choice of temporally-varying frame augmentations. Ablation Study Sequence Length: By varying the number of subsequent frames from 4 to 8, the bounding box AP increases from 31.0% to 31.4%, which suggests that a longer temporal context allows the model to learn more distinctive object attributes to reliably detect object types. For MOT performance, the gain from observing longer sequences and therefore more framewise comparisons are as high as +3% in HOTA Hierarchical Attention: Another vital design aspect is the size of the multi-frame network, and especially the hierarchy of associations that can be increased by stacking multiple encoder layers. The ablation study shows that this hyperparameter has a big impact on the performance of the downstream task compared to the sequence length. Surprisingly, we find that the complexity is saturated at two encoder layers, while a higher number of layers decreases performance, especially on single-frame object detection tasks. We initially hypothesized that the multi-frame model could be incapable of generalizing across all the dynamic interactions that inform about temporal sequences in traffic scenes. However, the results show that a lighter multi-frame head loads more of the semantic reasoning onto the single-frame model, which thereby learns more expressive features. Moreover, the damped performance can result from the slower convergence due to the high parallelism in multi-head attention modules, such that longer pre-training schedules can be required. Furthermore, using (3) as a reduction function for the association score matrices resulted in a drop in performance compared to average pooling by −0.7%mAP , which can be attributed to the sparser and attenuated gradients. Spatial and Photometric Augmentations: In Table 3, we evaluate spatial and appearance-based augmentations of the input sequence. This enforces the network to learn object representations, that are invariant to the global image location or lightning effects. Interestingly, photometric augmentations during pre-training resulted in lower performance on the downstream task, reducing the object detection performance by −0.1%mAP and tracking performance by −1.4%HOTA for identical TempO settings. This shows that the network exploits internal appearance consistency assumptions. Spatial augmentations such as random cropping negatively affect the pre-training, which indicates that the network relies on consistent spatial cues to solve the temporal ordering task. Frame Retrieval Results We further evaluate the utility of the learned representations from our TempO pre-training approach for the frame retrieval task on the UCF101 dataset [41], without additional fine-tuning on the retrieval task. We trained models for 100 epochs on the UCF101 videos using the TempO pretext task described in Section 3.1. For frame retrieval, we follow the experimental setup described in [23], by extracting 10 equally distanced frames from each video from the UCF101 dataset and using the frames extracted from videos in the test set, as class representatives. The frames are then classified using nearest-neighbor (NN) search in the embedding space, where closeness is defined by set-similarities as given by average pooling. The frames extracted from the train split clips are queried for similarity with all these class representatives and marked as correctly classified if a vector of the same action class is within the k nearest neighbors. We compare against the baseline performances reported by Kong et al. [23]. Table 5 shows the retrieval performance measured as the accuracy for k = 1 to k = 20. The results demonstrate that the TempO pre-trained embeddings show strong consistency across videos of the same action class. Interestingly, the similarity is higher than that obtained from other frame-level pre-training strategies, improving the T op − 1 accuracy by +3.9% compared to the cycleconsistency pretext (CCL) task. In Figure 5, we present examples of misclassified frames and their top three most similar class representatives. We observe that the learned TempO representations in these examples focus primarily on similarity of scene attributes, for instance, the number and size of objects or camera perspective. Especially in the example on the bottom row in Figure 5, the large variety of backgrounds indicates that the learned representation consists more of tracking features on foreground objects. Discussion of Limitations Our analysis, and the SSL video feature learning field in general, focuses on relatively short clips < 2s. Many actions in automated driving or human activity recognition, however, extend over longer time periods, and also our ablation study suggests that longer sequences can benefit the pre-trained models. The integration of more efficient video architectures, e.g. on compressed videos [50], would be an important enabler to consider longer time intervals. In our design, however, the association of objects over an increased number of frames can become computationally expensive, which can be alleviated by sampling a subset of frames to be compared. Secondly, we do most of our evaluation on driving sequences, where the camera moves smoothly in a dynamic environment. We chose this domain as the BDD100K dataset provides many hours of videos with high variability, as well as object detection and multi-object tracking annotations without domain shift. Even though we also evaluate on MOT17 and UCF101 [41] (UCF101) datasets, which are more human-centric and from predominantly static cameras, future work could evaluate how TempO pre-training behaves on a mixture of domains or highly repetitive videos, e.g. from an indoor service robot. Thirdly, the design of our proposed pretext task directly applies to models that output a set of proposal features. Keypoint-based detection methods or other models would require an additional tokenization strategy. Conclusion In this work, we proposed a SSL pretext task based on the temporal ordering of video frames, that allows learning region-level image representations. Models initialized with TempO pre-trained weights demonstrated a speed-up in convergence and superior performance on object detection and multi-object tracking downstream tasks compared to other self-supervised as well as supervised initialization strategies. The qualitative results also show how TempO pre-training helps to suppress ghost detection and recognize dark objects at night from a semantic context. In our multi-object tracking experiments, TempO pre-training improves the tracking accuracy while using a pre-trained object detector in a tracking-by-detection paradigm. We, therefore, conclude that a temporal ordering pretext task can boost performance compared to single-frame or supervised pre-training strategies in instance-level perception systems. We also evaluated the learned representations for a frame retrieval task, where we found consistent representations across videos of the same action class. Convergence Experiments In Table 1, we compare the object detection performance of different initialization strategies over the number of finetuning epochs. All models were trained using NVIDIA V100 GPUs with a batch size of 16 for 12 epochs on the BDD100K train split. We use the AdamW [30] optimizer with an initial learning rate of 2.5 · 10 −5 and weight decay of 10 −4 , and reduced the learning rate by a factor of 10 after 8 epochs. We observe that the TempO pretrained initialization yields the fastest convergence and achieves the largest mean average precision among different initialization strategies from 3 fine-tuning epochs onwards. As a result, the TempO pretrained models fine-tuned for 6 epochs achieve comparable results to COCO 2017 pre-trained methods for 12 finetuning epochs and surpass COCO 2017 pre-trained initial- ization by more than +0.7% after 12 fine-tuning epochs. Noticeably, the detection performance for large and mediumsized objects increases at a high rate during the early finetuning epochs of TempO, but slows down after 6 epochs compared to COCO 2017 initialized detectors. However, the performance at this stage is already higher than that achieved by the COCO 2017 initialized object detectors. Figure 1 presents the loss per epoch for fine-tuning a Sparse R-CNN [42] object detector on the BDD100K dataset using varying initialization strategies. We observe that the TempO pretrained method's detection performance increases the fastest at the early training epochs, and achieves the highest detection performance throughout the second half of the fine-tuning epochs. Analogous to Table 1, the improvement in the mAP metric slows down for TempO pretrained methods after 6 fine-tuning epochs, while the remaining initialization strategies have still not converged. These results Table 1. Object detection results on the BDD100k val set for increasing number of fine-tuning epochs. The TempO pre-training uses a sequence length of 8 frames, two layers in the multi-frame network, and AvgPool motivated by Table 3 demonstrate that the TempO pre-trained models converge faster than other pre-training strategies. Object Detection Performance of Scene-level SSL Methods We found pre-training on methods that learn scene-level feature descriptors only partially comparable with our approach, since the parameters of the object detector neck and head are not initialized with the SSL pretrained weights. However, we cannot quantify the effect that the choice of initialization strategy for those parameters has on the evaluation after fine-tuning. In Table 2, we present such comparisons with SSL methods that incorporate self-supervised scene-level feature learning strategies from frame-based [5,7] and sequence-based [33,53] pretext tasks. We trained the single-frame methods on double bounding box crops from the BDD100K detection train set, and the sequence-based methods on the same dataset as our TempO method, using the training protocol described in Sec. 4.2 of the manuscript. For frame order verification and classification, we used a sequence length of 3 and 5, respectively. The experiments in Table 2 in the following with use a random parameter initialization for the detector neck and head. We observe that our TempO approach outperforms all the strategies using scene-level SSL methods for initializing the ResNet-50 backbone and random initialization for the remaining network parameters by ≥ +3.3% in mAP on the BDD100k validation set. This amends the experiments in the main paper, which showed that our proposed approach also outperforms initialization strategies that pretrain all detector parameters, e.g. from supervised training on the COCO 2017 dataset or self-supervised training on re-localization of random image patches. TempO pre-trained, fine-tuned for 6 epochs on BDD100K. Figure 3 . 3Qualitative object detection results on the BDD100K val set using the Sparse R-CNN detector with varying training schedules. Observe that the TempO pre-trained detector avoids a ghost detection of a motorcycle within the garbage bags and detects the poorly lit rider on top of the moving bicycle. Figure 4 . 4Qualitative MOT results on BDD100K validation set for our proposed transformer tracking architecture and pre-trained on a sequential ordering task. The network tracks the pedestrian reliably over changing body orientations and handles the changing shape of the green car. Figure 5 . 5Frame retrieval demonstrations of misclassifications in the Top-20 setting. Scores are normalized similarities of the Top-20 nearest neighbors. Retrieved frames resemble in scene attributes, for instance the number and size of foreground objects or camera perspective, while backgrounds vary widely. Algorithm 1 : 1TempO pretext task pipeline. Data: Image sequence I1, . . . IN in correct temporal order, margin m Result: LT empO 1 forall Ii do 2 Ti = SpatialNetwork (Ii) 3 end 4 # per-frame track tokens Ti ∈ R Q×D ; 5 H1:N−1 = HistoryEncoder ([T1, . . . , TN−1], M future ) ; 6 # history tokens H1:N−1 ∈ R (N −1)·Q×D ; 7 LT empO = 0 ; 8 for i ∈ [2, . . . , N ] do 9 # matching scores Si ∈ R (N −i−1)×P ×P ; 10 Si,i−1 = AdditiveAttention(Ti, Hi−1) ; 11 pi,i−1 = ReductionFunction(Si,i−1); 12 for j ∈ [1, . . . , N − 1] & j = i − 1 do 13 Si,j = AdditiveAttention(Ti, Hj) ; 14 ni,j = ReductionFunction(Si,j); 15 LT empO += max {ni,j − pi,i−1 + m, Figure 1 . 1Training loss and validation set mAP for varying initialization strategies for a Sparse R-CNN object detector over 6 fine-tuning epochs on the BDD100K dataset. We train with a batch size of 16, and decrease the learning rate by a factor of 10 after 8 epochs (35k iterations). Table 2 . 2Multi-object tracking performance evaluation on BDD100k val set.BDD100k val Table 3 . 3Ablation experiments on down-stream tasks for various TempO pre-training settings. All experiments use a Sparse R-CNN object detector with a ResNet-50 backbone and 100 proposals per frame.Pretext setting BDD100k val object detection BDD100k val MOT Nseq L {enc,dec} f sim Augment. AP AP50 AP75 APs APm APl HOTA ↑ sMOTA ↑ IDF1 ↑ IDSw. ↓ 4 2 AvgPool - 31.0 56.5 29.0 15.2 34.8 51.5 33.6 25.5 37.5 102261 6 2 AvgPool - 31.2 56.8 29.3 15.3 35.0 51.4 34.9 25.0 36.4 101723 8 2 AvgPool - 31.4 57.3 29.5 15.7 35.2 51.6 36.6 26.3 40.5 90388 8 1 AvgPool - 29.1 54.1 26.9 14.2 33.1 48.0 35.1 24.4 36.1 92529 8 4 AvgPool - 30.9 56.5 28.7 15.2 34.8 50.7 37.2 27.2 42.9 89083 8 2 AvgPool P 31.3 56.9 29.3 15.4 35.1 51.2 35.5 15.8 41.5 92053 8 2 MSA - 30.7 55.9 29.0 15.1 34.4 25.1 35.3 25.9 40.4 91600 Table 4. Multi-object tracking performance evaluation on MOT17 test set. Method Detector Initialization MOTA↑ IDF1↑ HOTA↑ DetA AssA LocA FP↓ FN↓ IDs↓ QDTrack [36] Faster R-CNN COCO 68.7 66.3 53.9 - - - 26589 146643 3378 QDTrack Sparse R-CNN COCO 69.5 63.4 52.9 56.2 49.1 82.2 21963 147291 3305 QDTrack Sparse R-CNN RPD [8] + Crowdhuman 70.8 65.9 52.1 56.7 48.4 80.7 42855 117402 4563 QDTrack DDETR TempO + Crowdhuman 72.1 63.9 53.2 57.9 49.1 82.4 18513 135687 3180 QDTrack Sparse R-CNN TempO + Crowdhuman 72.8 65.9 54.3 58.5 50.3 82.5 17646 133059 3093 Table 5 . 5Frame retrieval experiments on UCF101[41] dataset.Method Model Top-1 Top-5 Top-10 Top-20 MSE - 13.1 20.2 23.4 28.6 JigSaw [1] 3D CNN 19.7 28.5 33.5 40.0 OPN [27] 3D CNN 19.9 28.7 34.0 40.6 CCL [23] 3D ResNet 32.7 42.5 50.8 61.2 TempO Faster R-CNN 34.9 46.1 53.6 58.9 Sparse R-CNN 35.6 49.5 58.2 68.3 DDETR 33.1 18.4 56.4 65.9 Self-Supervised Representation Learning from Temporal Ordering of Automated Driving Sequences -Supplementary Material -Christopher Lang 1,2 Alexander Braun 2 Lars Schillingmann 2 Karsten Haug 2 Abhinav Valada 1 1 University of Freiburg 2 Robert Bosch GmbHIn this supplementary material, we provide additional insights and experimental results on knowledge embeddingbased object detection. in the main paper.Model Pretrain Epoch BDD100k val object detection AP AP50 AP75 APs APm APl SparseRCNN COCO 2 25.3 47.6 23.1 12.5 28.4 40.3 4 27.7 51.0 25.8 13.7 31.2 44.5 6 27.5 49.9 25.8 12.6 30.5 47.8 12 30.7 55.8 28.9 15.2 34.3 50.8 SparseRCNN TempO 2 23.4 46.0 20.8 11.0 27.0 36.2 4 28.6 52.5 26.9 13.6 32.4 48.3 6 30.7 55.5 28.6 15.4 34.6 49.6 12 31.4 57.2 29.3 15.3 35.2 52.4 DDETR COCO 2 9.3 20.9 6.9 4.5 11.6 17.3 4 16.8 34.7 14 7.6 19.8 30.6 6 28.3 52.2 26.1 11.2 32.1 53.6 12 30.2 56.0 27.6 14.2 34.0 51.3 DDETR TempO 2 13.9 29.9 11.3 6.1 16.6 26.6 4 30.6 55.9 28.3 12.7 34.3 55.0 6 31.3 57.9 28.9 15.2 34.9 55.3 12 32.5 59.2 30.4 15.7 36.9 55.3 Table 2 . 2Object detection results using Sparse R-CNN model on the BDD100k val dataset. 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{'abstract': 'Self-supervised feature learning enables perception systems to benefit from the vast amount of raw data being recorded by vehicle fleets all over the world. However, their potential to learn dense representations from sequential data has been relatively unexplored. In this work, we propose TempO, a temporal ordering pretext task for pre-training region-level feature representations for perception tasks. We embed each frame by an unordered set of proposal feature vectors, a representation that is natural for instance-level perception architectures, and formulate the sequential ordering prediction by comparing similarities between sets of feature vectors in a transformer-based multi-frame architecture. Extensive evaluation in automated driving domains on the BDD100K and MOT17 datasets shows that our TempO approach outperforms existing self-supervised single-frame pre-training methods as well as supervised transfer learning initialization strategies on standard object detection and multi-object tracking benchmarks.', 'arxivid': '2302.09043', 'author': ['Christopher Lang \nUniversity of Freiburg\n\n\nRobert Bosch GmbH\n\n', 'Alexander Braun \nRobert Bosch GmbH\n\n', 'Lars Schillingmann \nRobert Bosch GmbH\n\n', 'Karsten Haug \nRobert Bosch GmbH\n\n', 'Abhinav Valada \nUniversity of Freiburg\n\n'], 'authoraffiliation': ['University of Freiburg\n', 'Robert Bosch GmbH\n', 'Robert Bosch GmbH\n', 'Robert Bosch GmbH\n', 'Robert Bosch GmbH\n', 'University of Freiburg\n'], 'corpusid': 257020090, 'doi': '10.48550/arxiv.2302.09043', 'github_urls': [], 'n_tokens_mistral': 19958, 'n_tokens_neox': 16880, 'n_words': 9682, 'pdfsha': '87f95ab94bb19eb7716e0d4bbe0d37ca30f64157', 'pdfurls': ['https://export.arxiv.org/pdf/2302.09043v1.pdf'], 'title': ['Self-Supervised Representation Learning from Temporal Ordering of Automated Driving Sequences', 'Self-Supervised Representation Learning from Temporal Ordering of Automated Driving Sequences'], 'venue': []}

arxiv

THREE LECTURES ON GLOBAL BOUNDARY CONDITIONS AND THE THEORY OF SELF-ADJOINT EXTENSIONS OF THE COVARIANT LAPLACE-BELTRAMI AND DIRAC OPERATORS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY FALL WORKSHOP ON GEOMETRY AND PHYSICS, ICMAT 2011 16 May 2012 A Ibort THREE LECTURES ON GLOBAL BOUNDARY CONDITIONS AND THE THEORY OF SELF-ADJOINT EXTENSIONS OF THE COVARIANT LAPLACE-BELTRAMI AND DIRAC OPERATORS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY FALL WORKSHOP ON GEOMETRY AND PHYSICS, ICMAT 2011 16 May 2012 In these three lectures we will discuss some fundamental aspects of the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on compact Riemannian manifolds with smooth boundary emphasizing the relation with the theory of global boundary conditions.Self-adjoint extensions of symmetric operators, specially of the Laplace-Beltrami and Dirac operators, are fundamental in Quantum Physics as they determine either the energy of quantum systems and/or their unitary evolution. The well-known von Neumann's theory of self-adjoint extensions of symmetric operators is not always easily applicable to differential operators, while the description of extensions in terms of boundary conditions constitutes a more natural approach. Thus an effort is done in offering a description of self-adjoint extensions in terms of global boundary conditions showing how an important family of self-adjoint extensions for the Laplace-Beltrami and Dirac operators are easily describable in this way.Moreover boundary conditions play in most cases an significant physical role and give rise to important physical phenomena like the Casimir effect. The geometrical and topological structure of the space of global boundary conditions determining regular self-adjoint extensions for these fundamental differential operators is described. It is shown that there is a natural homology class dual of the Maslov class of the space.A new feature of the theory that is succinctly presented here is the relation between topology change on the system and the topology of the space of selfadjoint extensions of its Hamiltonian. Some examples will be commented and the one-dimensional case will be thoroughly discussed. Introduction A simple description of the evolution of a closed autonomous quantum system is provided by a one-parameter group of unitary operators on a Hilbert space. Because of Stone's theorem, there is a one-to-one correspondence between oneparameter groups of unitary operators on Hilbert spaces and self-adjoint (not bounded in general) operators. Unfortunately as it happens in many occasions, the operators that we would like to use to describe a quantum system are not self-adjoint but merely symmetric, hence the evolution defined by them will not be unitary in general. The lack of unitarity reflects the fact that there is a "leak" of probability because the system is not truly closed but it is, for instance, in interaction with another system. The interaction with another external system, whose detailed form is not known in general, is simulated in many cases by introducing a boundary in the system. It could also happen that the boundary is introduced just to make sense of the system as it cannot be defined on an infinite domain (for instance when we consider a system on a "box"). In all these situations the construction of quantum mechanical systems requires a detailed analysis of the boundary conditions (BC) imposed on the system. Such boundary conditions are fundamental to construct the dynamics of the system and are either determined by the observers and the experimental setting used by them, or are inherent to the system under consideration. This is a common feature of all quantum systems, even the simplest ones. The determination of the boundary conditions, or the self-adjoint extensions of families of symmetric operators, will affect not only the dynamical evolution of the system, but also the results of the measures realized on the system because the measurable quantities of the system, which are defined by self-adjoint operators, will depend on them and the spectrum of the quantum observables will vary with the chosen self-adjoint extension. Among the most conspicuous quantum systems we find the "free" motion on a Riemannian manifold. In such case, the self-adjoint operator defining the unitary evolution of the system on the Hilbert space of square integrable functions on the manifold is the Laplace-Beltrami operator determined by the Riemannian metric. However if the manifold has boundary the Laplace-Beltrami operator is merely symmetric, and to determine completely such quantum system, a self-adjoint extension must be chosen. It is clear that selecting a self-adjoint extension in this case must be related to determine the behavior of the system when it "reaches" the boundary. Fixing the behavior of the system at the boundary would then determine its quantum evolution and in many occasions has a direct physical interpretation. Dirac's Correspondence Principle [15] provides a useful tool to analyze fundamental aspects of quantum mechanical systems by looking at their classical counterparts (in case they exist), but it is not obvious at all how it extends to include boundary conditions. For instance, Dirichlet boundary conditions corresponds to impenetrability of the classical walls determining the boundary of the classical system in configuration space but, what are the corresponding classical conditions for mixed BC? Conversely, we can address the question of quantizing classical BC. In particular we may ask if the classical determination of BC is enough to fully describe a quantized system. As the experimental and observational capabilities are getting more and more powerful, we are being forced to consider general boundary conditions beyond the classical cases, Dirichlet, Neumann, etc. In condensed matter models "sticky" boundary conditions have proved to be useful in understanding the Quantum Hall effect [27]; in quantum gravity, self-adjoint extensions are used to understand signature change [16]. Even at a more fundamental level, topology change in quantum systems has been modelled using dynamics on BC [8]. Following Dirac's approach, we can develop a canonical quantization program for classical systems with boundary. Such program requires a prior discussion on the dynamics of Hamiltonian systems with boundary. Without entering a full discussion of this, we may assume that a classical Hamiltonian system with boundary is specified by a Hamiltonian function H defined on the phase space T * Ω of a configuration space Ω with boundary ∂Ω, together with a canonical transformation S of the symplectic boundary T * (∂Ω) of T * Ω. Thus, the classical boundary conditions (CBC) form a group, the group of symplectic diffeomorphisms of T * (∂Ω) 1 . Dirac's correspondence principle will be stated now as follows: Given a classical BC S, and two classical observables f, g on T * Ω, the corresponding quantum BCŜ and self-adjoint operatorsf S ,ĝ S depending onŜ, must satisfy: (1) [f S ,ĝ S ] = i { f, g } S , and (2)Ŝ •R = SR, where the composition on the left in Eq. (2) is the group composition on the space of quantum BC to be discussed later on. It is obvious that as in the boundaryless situation, such quantization rules could not be implemented for all observables and all classical BC. So, one important question emerging from this analysis is how to select subalgebras of classical observables and subgroups of classical BC suitable for quantization. Before embarking in such enterprise, some relevant aspects of the classical and the quantum picture of systems with boundaries have to be clarified. For instance, we need to understand the structure of the self-adjoint extensions operators corresponding to a given classical observable. The most important class of operators arising in the first quantization of classical systems are first and second order elliptic differential operators: for instance, as it was indicated before, the Laplace-Beltrami operator when quantizing a classical particle without spin, the covariant Laplacian and the Dirac operator for the quantization of particles with spin. This family of 1 Similar considerations can be made for more general classical phase spaces, though some care is needed to define their symplectic boundary. operators are certainly the most fundamental of all elliptic operators (in the Euclidean picture). Thus for Dirac and Laplace operators we would like to understand their self-adjoint realizations in terms of classical and quantum BC. Von Neumann developed a general theory of self-adjoint extensions of symmetric operators in Hilbert spaces [31]. Such theory is usually presented in the realm of abstract Hilbert space theory. This causes that in many cases when applied to discuss differential operators some of the relevant features attached to the geometry of the operators are lost. We will proceed by using a direct approach to describe a large class of self-adjoint extensions of first and second order elliptic differential operators in terms of global boundary conditions. Such approach will preserve the geometry of such operators obtaining in this way a fresh interpretation of von Neumann's theory. In particular it will be shown that there is a canonical group structure on the space of self-adjoint extensions which is directly related to the boundary conditions imposed on the system as needed for the implementation of Dirac's quantization rule Eq. (2). Elliptic differential operators on compact manifolds have been exhaustively studied culminating with the celebrated Atiyah-Singer index theorem that relates the analytical index of such operators with the topological invariants of the underlying spaces. Such analysis extends to manifolds with boundary provided that appropriate elliptic boundary conditions are used. A remarkable example is provided by the study of the index of Dirac's operators on manifolds with boundary where appropriate global elliptic boundary conditions, the so called APS conditions, were introduced in [7] (see also [24]). Such extensions have been adequately generalized for higher order elliptic operators giving rise to interesting constructions of boundary data [18]. The program sketched so far concerns exclusively first quantization of classical systems, but second quantization, this is Quantum Field Theory, is needed to truly understand the basic facts of Nature. We have already seen that first quantization of classical systems requires to consider the quantization of boundary conditions, which leads automatically to consider QBC for the first quantized system. Thus even for a very simple system, like a fermion propagating on a disk we need to consider "all" self-adjoint extensions of the Dirac's operator on the disk. Thus, to proceed to second quantization we need to understand the global structure of such space of extensions. We will show that such space is a Lagrangian submanifold, that will be called the self-adjoint Grassmannian, of the infinite dimensional elliptic Grassmannian manifold. Such infinite dimensional Grassmannian was introduced in the study of integrable hierarchies of nonlinear partial differential equations such as KdV and KP [36]. It represents a "universal phase space" for a large class of integrable evolution problems. Our approach here is different, the infinite dimensional Grassmannian appears as the natural setting to discuss simultaneously all QBC for a first quantized classical system of arbitrary dimension. In fact, a subset of relevant QBC are contained in a submanifold, the self-adjoint Grassmannian, that should be subjected to second quantization. Lagrangian submanifolds of symplectic manifolds play the role of "generalized functions", thus, such programme would imply quantizing a particular observable of the Grassmannian, making contact in this way with string theory. The theory of boundary value problems for elliptic operators was beautifully described by G. Grubb [19], [22]. In her work a characterization of all self-adjoint extensions of a class of elliptic operators in terms of global boundary conditions was provided (see also Frey's Ph. D. dissertation where the boundary value problems for Laplace-Beltrami and Dirac operators was discussed at length [17]). However no attempt will be made to relate the results presented in these lectures with Grubb's theory or with the techniques to construct self-adjoint extensions called boundary triples [11]. The reasons for that are two-fold: on one side we want to keep the presentation as simple and self-contained as possible and to cover Grubb's theory will be impossible within the scope of these lectures. On the other hand many of the applications that we were referring previously fit perfectly in the framework we are considering here, thus the theory we are presenting is enough to deal with an important family of problems, even if there are interesting examples that will not be covered by the results in these notes like M. Berry's D-singular boundary conditions [9]. Some of the material presented here has already appeared published elsewhere (see for instance [5] where some of the preliminary ideas on the global topology of the space of self-adjoint extensions for the covariant Laplacian and its relation to topology change appeared for the first time), or will appear in various forms (see for instance [25] for a detailed discussion of 1D Schrödinger operators). The general theory of self-adjoint extensions from the point of view of quadratic forms is discussed in [26] and will not be considered here as well as the theory of self-adjoint extensions with symmetry that will be discussed elsewhere. 2. Lecture 1: Boundary conditions and self-adjoint extensions: the quest for unitarity 2.1. The problem of unitary evolution in Quantum Mechanics. As it was already pointed it out in the introduction the study of self-adjoint extensions of symmetric operators has its origin in solving the problem of unitary evolution in Quantum Mechanics. We will discuss briefly some fundamental notions of quantum mechanical systems to clarify this point. The simplest mathematical formalism to describe an isolated quantum system is by considering a complex separable Hilbert space H whose rays are going to be identified with pure states of the system. The observables of the system are a family of self-adjoint operators O on the Hilbert space H. Self-adjoint operators play a double role in the quantum formalism. On one side they are observables of the system and its eigenvalues are the possible outcomes of measures performed on it. To be more precise, if A is a self-adjoint operator on H the spectral theorem states that there exists a projector-valued Borelian spectral measure E on R such that: A = λ E(dλ). If |ψ denotes a unitary vector on H, this is, a representative for a pure state ρ of the system, then the probability of obtaining an output lying on the Borelian set ∆ ⊂ R when measuring the observable A on the state ρ is given by µ A,ρ (∆) = ∆ ψ|E(dλ)|ψ . On the other hand Stone-von Neumann theorem allows to consider self-adjoint operators as infinitesimal generators of unitary evolution, i.e. there is a one-toone correspondence between self-adjoint operators H and strongly continuous oneparamenter groups of unitary operators U t with H = i lim t→0 (U t − I)/t or U t = exp(itH). In both cases, either when we interpret a self-adjoint operator as an observable of the system or when we are constructing the unitary evolution of the system, the physical interpretation of the operator depends crucially on its self-adjointness. However in many occasions when constructing quantum systems we will need to consider observables or generators of dynamical evolution which are to be defined by means of operators that are not self-adjoint but merely symmetric. Let us recall that if A is a linear operator on H which is densely defined with domain Dom (A) then its adjoint operator A † is uniquely defined and has dense domain Dom (A † ). A vector |ψ is in Dom (A † ) if there exists |ζ ∈ H such that ψ, Aφ = ζ, φ for all |ψ ∈ Dom (A). Then the operator A is self-adjoint if Dom (A) = Dom (A † ) and A = A † on its common domain. If Dom (A) is merely contained in Dom (A † ) and A † | Dom (A) = A, then the operator A is said to be symmetric. An operator B with domain Dom (B) is said to be an extension of the operator A if Dom (A) ⊂ Dom (B) and B | Dom (A) = A. In this sense, an operator A is said to be symmetric if A † is a (strict) extension of A. It is easy to provide examples of symmetric operators for which the spectral theorem fails (in the form above). In fact, it can be shown easily that there are symmetric operators whose spectrum is the full field of complex numbers. Thus we conclude that if we pretend to describe a physical observable or define the unitary evolution of a quantum system, we may not use symmetric operators but self-adjoint ones. The following problem arises immediately: given a symmetric operator A on a Hilbert space H, does there exists a self-adjoint operator extending it and, if this were the case, how many different self-adjoint extensions do there exists? Notice that both parts, the existence and the (non-)uniquenes, of the problem are relevant. In fact, if an observable of a quantum system is constructed starting from a symmetric operator A, we will not be able to interpret the results of performing measurements of such observable until we have made precise which self-adjoint extension of the symmetric operator A is actually representing the observable we are measuring. Notice that different self-adjoint extensions of the symmetric operator A have different spectrum, then the expected measurements (this is the physical predictions) would be different. Similarly, two different self-adjoint extensions would lead to different unitary evolution groups, thus the prediction of how a given quantum state will evolve would depend on the self-adjoint extension we choose. Actually it could even happen that a symmetric operator will have no self-adjoint extension at all, then the attempt to describe such observable will be futile. In this lecture we will describe some aspects regarding the solution of both problems for the (covariant) Laplace-Beltrami operator. The Laplace-Beltrami operator is used to describe the energy (the Hamiltonian) as well as the infinitesimal generator of unitary evolution of a large class of quantum systems. 2.2. The Laplace-Beltrami operator. As it was stated in the introduction we will restrict our attention to the simpler case of Schrödinger operators on compact manifolds with smooth boundary and regular potentials. The Schrödinger operator for a particle of mass m moving on a smooth manifold Ω with boundary ∂Ω and riemannian metric η is given by the Hamiltonian operator H that, in local coordinates x i , takes the form: (3) H = − 2 2m 1 |η| ∂ ∂x j |η|η jk ∂ ∂x k + V (x), with the metric tensor η given by η = η jk (x)dx k dx j , |η| = | det η jk (x)| and η ij η jk = δ i k . The second order differential operator 2 (4) ∆ η = 1 |η| ∂ ∂x j |η|η jk ∂ ∂x k is formally self-adjoint in the sense that Ψ, ∆ η Φ = ΩΨ ∆ η Φ vol n = − Ω (dΨ, dΦ) η vol η = Ω (∆ η Ψ) Φ vol η = ∆ η Ψ, Φ , for Ψ, Φ any smooth complex valued functions with compact support contained in Ω \ ∂Ω. In the previous formula vol η denotes the riemannian volume form on Ω defined by η, i.e., vol η = |η|dx 1 ∧· · ·∧dx n , and (dΨ, dΦ) η(x) = η jk (x)∂Ψ/∂x j ∂Φ/∂x k is the inner product among covectors at x ∈ Ω. In fact, the differential expression (4) defines a symmetric operator on the space L 2 (Ω) of square integrable functions on Ω with respect to the measure defined by the volume form vol η , with dense domain C ∞ c (Ω \ ∂Ω) the space of smooth functions with compact support in Ω \ ∂Ω. The operator ∆ η is closable and its closed extension (the minimal extension such that its graph is closed) has a domain given by the closure of C ∞ c (Ω \ ∂Ω) with respect to the Sobolev norm || · || 2,2 3 . We will denote this minimal extension of the operator ∆ η by ∆ 0 and its domain by D 0 . Notice that D 0 is the space H 2 0 (Ω) of functions of Sobolev class 2 on Ω vanishing at the boundary together with its normal derivative, i.e., the space of functions on L 2 (Ω) possessing first and second weak derivatives that are square integrable and such that their boundary values vanish (see Lions trace theorem below Thm. 1). The operator ∆ 0 is symmetric but not self-adjoint on L 2 (Ω) because the adjoint operator ∆ † 0 has a dense domain Dom (∆ † 0 ) =: D † 0 that contains strictly D 0 . The domain D † 0 contains the space H 2 (Ω) of all functions of Sobolev class 2 on Ω, i.e., the space of functions on L 2 (Ω) possessing first and second weak derivatives that are square integrable. Self-adjoint extensions of the operator ∆ η are given by operators ∆ D with domain D such that D 0 ⊂ D ⊂ D † 0 , ∆ D | D0 = ∆ 0 and ∆ D = ∆ † D . Notice that in that case ∆ D = ∆ † 0 | D . Von Neumann's theorem establishes (see for instance [37] Thm. 8.12, and Lecture 2, Thm. 4 these notes) that there is a one-to-one correspondence between selfadjoint extensions ∆ D of the Laplace-Beltrami operator ∆ 0 and unitary operators K : N + → N − , where the deficiency spaces N ± are defined as: N ± = {Ψ ∈ L 2 (Ω) | ∆ † 0 Ψ = ±iΨ}. In particular, given the unitary operator K, the domain D of the operator ∆ D is given by D = D 0 ⊕ (I + K)N + , and the extended operator ∆ D takes the explicit form: ∆ D (Ψ 0 ⊕ (I + K)ξ + ) = ∆ 0 Ψ 0 ⊕ i(I − K)ξ + , for all Ψ 0 ∈ D 0 and ξ + ∈ N + . 2.3. Self-adjoint extensions of the Laplace-Beltrami operator and boundary conditions. Unfortunately, as it was stated in the introduction, von Neumann's theorem is not always well suited for the explicit construction of general self-adjoint extensions of the Laplace-Beltrami operator (it is necessary to determine first the deficiency spaces N ± that could be difficult). We can take however a different route inspired in the classical treatment of formally self-adjoint differential operators. If we rewrite the identity expressing the formal self-adjointness of ∆ η for functions Ψ, Φ in C ∞ (Ω) instead of C ∞ c (Ω \ ∂Ω), a simple computation shows: 3 The Sobolev norm ||·|| 2,2 is defined by: ||Ψ|| 2 2,2 = ||Ψ|| 2 L 2 (Ω) +||dΨ|| 2 L 2 (Ω) where ||dΨ|| 2 L 2 (Ω) = Ω (dΨ, dΨ)ηvol η . where ψ = Ψ | ∂Ω , ϕ = Φ | ∂Ω , and the normal derivativeφ is defined as: (5) ΩΨ ∆ η Φ vol η = Ω (∆ η Ψ) Φ vol η + ∂Ω ψφ −ψϕ vol ∂η ,ϕ vol ∂η = ⋆d Φ | ∂Ω where ⋆ is the Hodge operator defined by the metric η ij and vol ∂η is the riemannian volume defined on the boundary ∂Ω by the restriction ∂η of the Riemannian metric η to it. Less intrinsically, but more explicitly, we haveφ = dΦ dν | ∂Ω = dΦ(ν) where ν is the exterior normal vector to ∂Ω. We obtain the Lagrange boundary form Σ for the Laplace-Beltrami operator: (6) Σ((ψ,ψ), (ϕ,φ)) = ψ,φ L 2 (∂Ω) − ψ , ϕ L 2 (∂Ω) . In what follows, if there is no risk of confusion, we will omit the subscript L 2 (∂Ω) that denotes the L 2 inner product on the boundary manifold Γ = ∂Ω with respect to the measure defined by the volume form vol ∂η , hence we will simply write ψ, ϕ = ∂Ωψ ϕ vol ∂η . The Lagrange boundary bilinear form Σ defines a continuous bilinear form on the Hilbert space L 2 (∂Ω) ⊕ L 2 (∂Ω): (7) Σ((ψ 1 , ψ 2 ), (ϕ 1 , ϕ 2 )) = ψ 1 , ϕ 2 − ψ 2 , ϕ 1 , for all (ψ 1 , ψ 2 ), (ϕ 1 , ϕ 2 ) ∈ L 2 (∂Ω) ⊕ L 2 (∂Ω). If we denote by γ : C ∞ (Ω) → C ∞ (∂Ω)⊕ C ∞ (∂Ω) the trace map given by γ(Ψ) = (ψ,ψ), the Lions-Magénes trace theorem [29] shows that there exists a continuous extension of γ to H 2 (Ω), actually we have 4 : Theorem 1. There exists a unique extension of the trace map γ to a continuous surjective linear map (denoted with the same symbol) γ : H 2 (Ω) → H 3/2 (∂Ω) ⊕ H 1/2 (Ω). Moreover the induced mapγ : H 2 (Ω)/ ker γ → H 3/2 (∂Ω) ⊕ H 1/2 (Ω) is a homeomorphism and ker γ = H 2 0 (Ω). We will denote also, as it is customary, the fractional power Sobolev space H 3/2 (∂Ω) by W 3/2,2 (∂Ω), etc. Again, sometimes we will prefer to use the notation b (as "boundary" map) for the linear map γ (the "trace" map) defined in the theorem above. The previous observations provide a simple characterization of a large class of self-adjoint extensions of the operator ∆ 0 . In fact it is easy to check that: Theorem 2. There is a one-to-one correspondence between self-adjoint extensions ∆ D of the Laplace-Beltrami operator ∆ 0 with domain D contained in H 2 (Ω) and non-trivial maximal closed isotropic subspaces W of the Lagrange form Σ contained in W 3/2,2 (∂Ω) ⊕ W 1/2,2 (∂Ω). The correspondence being explicitly given by D → W = γ(D). Proof: Let D ⊂ H 2 (Ω) be the domain of a self-adjoint extension ∆ D of the operator ∆ 0 . Consider the subspace W := γ(D) ⊂ W 3/2,2 (∂Ω) ⊕ W 1/2,2 (∂Ω) ⊂ L 2 (∂Ω) ⊕ L 2 (∂Ω) consisting on the set of pairs of functions (ϕ,φ) that are respectively the restriction to ∂Ω of a function Φ ∈ D and its normal derivative. Notice that the subspace D is closed in H 2 (Ω). Hence, because γ is an homeomorphism from H 2 (Ω)/ ker γ to W 3/2,2 (∂Ω) ⊕ W 1/2,2 (∂Ω), γ(D) is a closed subspace of W 3/2,2 (∂Ω)⊕W 1/2,2 (∂Ω). Because of Eq. (5), it is clear that Σ | W = 0 and the subspace W is maximally isotropic in W 3/2,2 (∂Ω)⊕W 1/2,2 (∂Ω), because if this were not the case, there will be a closed isotropic subspace W ′ ⊂ W 3/2,2 (∂Ω) ⊕ W 1/2,2 (∂Ω) containing W . Then γ −1 (W ′ ) will define a domain D ′ containing D such that the operator ∆ 0 would be symmetric on it, in contradiction with the self-adjointness assumption. Conversely, let W ⊂ L 2 (∂Ω) ⊕ L 2 (∂Ω) be a maximal closed Σ-isotropic subspace in W 3/2,2 (∂Ω) ⊕ W 1/2,2 (∂Ω). Then consider the closed subspace D W := γ −1 (W ) ⊂ H 2 (Ω) of functions Ψ such that (ψ,ψ) ∈ W . It is clear that for any pair of functions Ψ, Φ on D W , because W is isotropic with respect to Σ, then Eq. (5) gives Ψ, ∆ η Φ = ∆ η Ψ, Φ and the operator ∆ η is symmetric in D W . Moreover, because of the maximality of W in W 3/2,2 (∂Ω) ⊕ W 1/2,2 (∂Ω) it is easy to see that D W = D † W , hence it is self-adjoint. We could object that the previous characterization of self-adjoint extensions of the Laplace-Beltrami operator in terms of closed maximal isotropic subspaces of Σ in W 3/2,2 (∂Ω) ⊕ W 1/2,2 (∂Ω) is rather obscure. An important observation in this sense is that the linear transformation C : L 2 (∂Ω) ⊕ L 2 (∂Ω) → L 2 (∂Ω) ⊕ L 2 (∂Ω), defined by: (8) C(ϕ,φ) = 1 √ 2 (ϕ + iφ), 1 √ 2 (ϕ − iφ) , transforms maximally isotropic closed subspaces of Σ into graphs of unitary operators of L 2 (∂Ω). Theorem 3. The Cayley map C provides a one-to-one correspondence between maximally isotropic closed subspaces of the Lagrange bilinear boundary form Σ in L 2 (∂Ω) ⊕ L 2 (∂Ω) and graphs G V = {(ψ + , V ψ + ) | ψ + ∈ L 2 (∂M )} of unitary oper- ators V : L 2 (∂Ω) → L 2 (∂Ω). Proof: Notice first that the map C is a unitary operator on the Hilbert space L 2 (∂Ω)⊕ L 2 (∂Ω) with the inner product (ψ 1 , ψ 2 ), (ϕ 1 , ϕ 2 ) = ψ 1 , ϕ 1 + ψ 2 , ϕ 2 . Consider now the transformed bilinear formΣ on L 2 (∂Ω) ⊕ L 2 (∂Ω) defined bỹ Σ(C(ϕ,φ), C(ψ,ψ)) = Σ((ϕ,φ), (ψ,ψ)). Thus, using the notation ϕ ± = 1 √ 2 (ϕ ± iφ), we have: Σ((ϕ + , ϕ − ), (ψ + , ψ − )) = −i [ ϕ + , ψ + − ϕ − , ψ − ] . Hence, if W is a maximally isotropic closed subspace for Σ, thenW = C(W ) will be a maximally isotropic closed subspace forΣ. Then it is easy to show thatW defines the graph of a linear operator. We first realize that ( {0} × L 2 ) ∩W = 0 because if (0, ψ − ) lays inW , thenΣ((0, ψ − ), (0, ψ − )) = i||ψ − || 2 must vanish. Hence if (ψ + , ψ − ), (ψ + ,ψ − ) are inW , then (0, ψ − −ψ − ) is inW and ψ − =ψ − . Then we define a linear operator V : V → L 2 by U (ψ + ) = ψ − with (ψ + , ψ − ) ∈W and V the closed subspace of vectors ψ + such that there exists (ψ + , ψ − ) ∈W . Similarly we can construct another operatorṼ :Ṽ → L 2 by observing that (L 2 × {0}) ∩W = 0. Then it is easy to show that V is an isometry from V toṼ,Ṽ is an isometry from V to V and they are inverse of each other. Then because of the maximality ofW , we conclude that V =Ṽ = L 2 . Hence a convenient way of constructing self-adjoint extensions of the Laplace operator will be provided by unitary operators U on L 2 (∂Ω) such that the preimage under C of their graphs will be closed in W 3/2,2 (∂Ω)⊕W 1/2,2 (∂Ω). We will develop this programme in the 1D case in the forthcoming section. We will end this discussion by realizing that the operator multiplication by a regular function is essentially self-adjoint and its unique self-adjoint extension has domain L 2 (Ω). Hence, the self-adjoint extensions of the Schrödinger operator H, Eq. (3), coincide with the self-adjoint extensions of ∆ η . We can summarize the preceding analysis by stating that under the conditions above, the domain D of a self-adjoint extension of the Schrödinger operator H defined by a closed subspace of functions Ψ on H 2 (Ω) must satisfy: (9) ϕ − iφ = U (ϕ + iφ) for a given unitary operator U : L 2 (∂Ω) → L 2 (∂Ω). The formula above, Eq. (9), provides a powerful and effective computational tool to deal with large family of self-adjoint extensions of Schrödinger operators. It was introduced in a slightly different context by Asorey et al [5] and will be used extensively in the rest of this paper. In what follows we will denote respectively by H U or H D the self-adjoint extension determined by the unitary operator U or the self-adjoint extension whose domain is D. Also according to Thms. 2 and 3 we denote such domain as D U . Notice that U = I corresponds to Neumann's boundary conditions and U = −I determines Dirichlet's boundary conditions. 2.4. The unitary group of self-adjoint extensions in 1D. Self-adjoint extensions of Schrödinger operators in 1D. We will concentrate our attention in 1D were we will be able to provide an elegant formula to solve the spectral problem for each self-adjoint extension. Notice first that a compact 1D manifold Ω consists on a finite number of closed intervals I α , α = 1, . . . , n. Each interval will have the form I α = [a α , b α ] ⊂ R and the boundary of the manifold Ω = ⊔ n α=1 [a α , b α ] (disjoint union) is given by the family of points {a 1 , b 1 , . . . , a n , b n }. Functions Ψ on Ω are determined by vectors (Ψ 1 , . . . , Ψ n ) of complex valued functions Ψ α : I α → C. A Riemannian metric η on Ω is given by specifying a Riemannian metric η α on each interval I α , this is, by a positive smooth function η α (x) > 0 on the interval I α , i.e., η | Iα = η α (x)dx 2 . Then the inner product on I α takes the form Ψ α , Φ α = bα aα (x)Ψ α (x)Φ α (x) η α (x)dx and the Hilbert space of square integrable functions on Ω is given by L 2 (Ω) = n α=1 L 2 (I α , η α ). Thus the Hilbert space L 2 (∂Ω) at the boundary reduces to C 2n , as well as the subspaces W 3/2,2 (∂Ω) and W 1/2,1 (∂Ω). The vectors in L 2 (∂Ω) are determined by the values of Ψ at the points a α , b α (with the standard inner product): ψ = (Ψ 1 (a 1 ), Ψ 1 (b 1 ), . . . , Ψ n (a n ), Ψ n (b n )). Similarly we will denote byψ the vector containing the normal derivatives of Ψ at the boundary, this is: ψ = − dΨ 1 dx a1 , dΨ 1 dx b1 , . . . , − dΨ n dx an , dΨ n dx bn . Because of Thms. 2 and 3, an arbitrary self-adjoint extension of the Schrödinger operator H defined by the riemannian metric η and a regular potential function V is defined by a unitary operator V : C 2n → C 2n . Its domain consists of those functions whose boundary values ψ,ψ satisfy Asorey's condition, Eq. (9). This equation becomes a finite dimensional linear system for the components of the vectors ψ anḋ ψ. Hence the space of self-adjoint extensions is in one-to-one correspondence with the unitary group U (2n) and has dimension 4n 2 . It will be convenient for further purposes to organize the boundary data vectors ψ andψ in a different way. Thus, we denote by ψ l ∈ C n (respec. ψ r ) the column vector whose components ψ l (α), α = 1, . . . , n, are the values of Ψ at the left endpoints a α , this is ψ l (α) = Ψ α (a α ) (respec. ψ r (α) = Ψ α (b α ) are the values of Ψ at the right endpoints). Similarly we will denote byψ l (α) = − dΨα dx | aα andψ r (α) = dΨα dx | bα , α = 1, . . . , n. Hence, the domain of the self-adjoint extension defined by the unitary matrix U will be written accordingly as: ψ l − iψ l = U 11 (ψ l + iψ l ) + U 12 (ψ r + iψ r ) (10) ψ r − iψ r = U 21 (ψ l + iψ l ) + U 22 (ψ r + iψ r ) and U has the block structure: (11) U = U 11 U 12 U 21 U 22 . Notice that the unitary matrix U is related to the unitary matrix V above by a permutation, but we will not need its explicit expression here. Thus in what follows we will use the notation for the boundary data: ψ = ψ l ψ r ;ψ = ψ l ψ r and Asorey's condition reads again: (12) ψ − iψ = U (ψ + iψ), U ∈ U (2n). 2.4.2. The spectral function. Once we have determined a self-adjoint extension H U of the Schrödinger operator H, we can determine the unitary evolution of the system by computing the flow U t = exp(−itH U / ). It is well-known that the Dirichlet extension of the Laplace-Beltrami operator has a pure discrete spectrum because of the compactness of the manifold and the ellipticity of the operator, hence all self-adjoint extensions have a pure discrete spectrum (see [37], Thm. 8.18). Then the spectral theorem for the self-adjoint operator H U states: H U = ∞ k=1 λ k P k , where P k is the orthogonal projector onto the finite-dimensional eigenvector space V k corresponding to the eigenvalue λ k . The unitary flow U t is given by: U t = ∞ k=1 e −itλ k / P k . Hence all that remains to be done is to solve the eigenvalue problem: (13) H U Ψ = λΨ, for the Schrödinger operator H U . We devote the rest of this section to provide an explicit formula to solve Eq. (13). On each subinterval I α = [a α , b α ] the differential operator H α = H| Iα takes the form of a Sturm-Liouville operator H α = − 1 W α d dx p α (x) d dx + V α (x), with smooth coefficients W α = 1 2 √ ηα > 0 (now and in what follows we are taking the physical constants and m equal to 1), p α (x) = 1 √ ηα , hence the second order differential equation (14) H α Ψ α = λΨ α has a two-dimensional linear space of solutions for each λ. We shall denote a basis of solutions of such space as Ψ σ α , σ = 1, 2. Notice that Ψ σ α depends differentially on λ. Hence a generic solution of Eq. (14) takes the form: Ψ α = A α,1 Ψ 1 α + A α,2 Ψ 2 α . Now it is clear that ψ l (α) = Ψ α (a α ) = A α,1 ψ 1 a (α) + A α,2 ψ 2 a (α). Hence: ψ l = A 1 • ψ 1 a + A 2 • ψ 2 a , where A σ , σ = 1, 2, denotes the column vector A σ =    A 1,σ . . . A n,σ    and • denotes the Hadamard product of two vectors, i.e., (X • Y ) α = X α Y α where X, Y ∈ C n . We obtain similar expressions for ψ r ,ψ l andψ r . With this notation Eqs. (10) become: (ψ 1 l − iψ 1 l ) • A 1 + (ψ 2 l − iψ 2 l ) • A 2 = U 11 (ψ 1 l + iψ 1 l ) • A 1 + U 11 (ψ 2 l + iψ 2 l ) • A 2 +U 12 (ψ 1 r + iψ 1 r ) • A 1 + U 12 (ψ 2 r + iψ 2 r ) • A 2 (15) (ψ 1 r − iψ 1 r ) • A 1 + (ψ 2 r − iψ 2 r ) • A 2 = U 21 (ψ 1 l + iψ 1 l ) • A 1 + U 21 (ψ 2 l + iψ 2 l ) • A 2 +U 22 (ψ 1 r + iψ 1 r ) • A 1 + U 22 (ψ 2 r + iψ 2 r ) • A 2 It will be convenient to use the compact notation ψ σ l± = ψ σ l ± iψ σ l , σ = 1, 2, and similarly for ψ σ r± . If T is a n × n matrix and X, Y arbitrary n × 1 vectors, we will define T • X as the unique matrix such that (T • X)Y = T (X • Y ). The rows of the matrix T • X are T i • X or alternatively, the columns of T • X are given by T j X j (no summation on j). It can be proved easily that (16) T • X = T • (X ⊗ 1), where 1 is the vector whose components are all ones (i.e., the identity with respect to the Hadamard product •) and the Hadamard product of matrices in the r.h.s. of Eq. (16) is the trivial componentwise product of matrices. Using these results Eqs. (15) become: (I n • ψ 1 l− − U 11 • ψ 1 l+ − U 12 • ψ 1 r+ )A 1 + (I n • ψ 2 r− − U 11 • ψ 2 l+ − U 12 • ψ 2 r+ )A 2 = 0 (I n • ψ 1 r− − U 21 • ψ 1 l+ − U 22 • ψ 1 r+ )A 1 + (I n • ψ 2 r− − U 21 • ψ 2 l+ − U 22 • ψ 2 r+ )A 2 = 0 Thus the previous equations define a linear system for the 2n unknowns A 1 and A 2 . They will have a non trivial solution if and only if the determinant of the 2n × 2n matrix of coefficients M (U, λ) below vanish: M (U, λ) = I n • ψ 1 l− − U 11 • ψ 1 l+ − U 12 • ψ 1 r+ I n • ψ 2 l− − U 11 • ψ 2 l+ − U 12 • ψ 2 r+ I n • ψ 1 r− − U 21 • ψ 1 l+ − U 22 • ψ 1 r+ I n • ψ 2 r− − U 21 • ψ 2 l+ − U 22 • ψ 2 r+ . The fundamental matrix M (U, λ) can be written in a more inspiring form using another operation naturally induced by the Hadamard and the usual product of matrices. Thus, consider the 2n × 2n matrix U with the block structure of Eq. (11) and the 2n × 2 matrices: [ψ 1 ± | ψ 2 ± ] = ψ 1 l± ψ 2 l± ψ 1 r± ψ 2 r± , then we define U 11 U 12 U 21 U 22 ⊙ ψ 1 l± ψ 2 l± ψ 1 r± ψ 2 r± ≡ U 11 • ψ 1 l± + U 12 • ψ 1 r± U 11 • ψ 2 l± + U 12 • ψ 2 r± U 21 • ψ 1 l± + U 22 • ψ 1 r± U 21 • ψ 2 l± + U 22 • ψ 2 r± and similarly I 2n ⊙ [ψ 1 ± | ψ 2 ± ] = I n • ψ 1 l± I n • ψ 2 l± I n • ψ 1 r± I n • ψ 2 r± . Finally we conclude that the condition for the existence of coefficients A 1 and A 2 such that the solutions to the eigenvalue equation lie in the domain of the selfadjoint extension defined by U is given by the vanishing of the spectral function Λ U (λ) = det M (U, λ), that written with the notation introduced so far becomes: (17) Λ U (λ) = det(I 2n ⊙ [ψ 1 − | ψ 2 − ] − U ⊙ [ψ 1 + | ψ 2 + ]) = 0. The zeros of the spectral function Λ provide the eigenvalues λ of the spectral problem Eq. (13). In the particular case n = 1, the previous equation becomes greatly simplified, the Hadamard product becomes the usual scalar product and the Hadamard-matrix product is the usual product of matrices. After some simple manipulations, the spectral function Λ U (λ) becomes: Λ U (λ) = W (l, r, −, −) + U 11 W (r, l, −, +) + U 22 W (r, l, +, −)(18)+ U 12 W (r, r, −, +) + U 21 W (l, l, +, −) + det U · W (l, r, +, +) where we have used the notation: W (l, l, +, −) = ψ 1 l+ ψ 2 l+ ψ 1 l− ψ 2 l− , W (l, r, +, −) = ψ 1 l+ ψ 2 l+ ψ 1 r− ψ 2 r− , etc. If we parametrize the unitary matrix U ∈ U (2) as: U = e iθ/2 α β −βᾱ , |α| 2 + |β| 2 = 1, then the spectral function becomes: Λ U (λ) = W (l, r, −, −) + αW (r, l, −, +) +ᾱW (r, l, +, −) + βW (r, r, −, +) (19) −βW (l, l, +, −) + e iθ W (W (l, r, −, −) = −2i(1 + 2λ) sin(2π √ 2λ) − 4 √ 2λ cos(2π √ 2λ), W (l, l, +, −) = 4 √ 2λ, W (r, r, −, +) = 4 √ 2λ, W (r, l, −, +) = 2i(1 − 2λ) sin(2π √ 2λ), W (r, l, +, −) = 2i(1 − 2λ) sin(2π √ 2λ), W (l, r, +, +) = −2i(1 + 2λ) sin(2π √ 2λ) + 4 √ 2λ cos(2π √ 2λ), and finally we obtain the spectral function Λ U (λ): Λ U (λ) = −2i(1 + 2λ) sin(2π √ 2λ) − 4 √ 2λ cos(2π √ 2λ) + 4i Re(α)(1 − 2λ) sin(2π √ 2λ) +8 Im(β) √ 2λ + e iθ [−2i(1 + 2λ) sin(2π √ 2λ) + 4 √ 2λ cos(2π √ 2λ)]. Quantum wires. The discussion in the previous section allows to discuss a great variety of self-adjoint extensions of 1D systems whose original configuration space Ω = ⊔ n α=1 [a α , b α ] consist of a disjoint union of closed intervals in R. It is clear that some boundary conditions U ∈ U (2n) will lead to a quantum system with configuration space a 1D graph whose edges will be the boundary points {a 1 , b 1 , . . . , a n , b n } of the original Ω identified among themselves according to U and with links [a α , b α ]. We will say that the self-adjoint extension determined by a unitary operator U in U (2n) defines a quantum wire made of the links [a α , b α ] if there exists a permutation σ of 2n elements such that Asorey's condition for U implies that ψ(x α ) = e iβα ψ(x σ(α) , and x α such that x α = a α if α = 1, . . . , n, or x α = b α−n if α = n + 1, . . . , 2n. Notice that Asorey's condition: ψ − iψ = U (ψ + iψ) guarantees that the evolution of the quantum system is unitary, i.e., if we consider for instance a wave packet localized in some interval [a k , b k ] at a given time, after a while, the wave packet will have spread out accross the edges of the circuit, however the probability amplitudes will be preserved. In this sense we may consider Asorey's equation above as the quantum analogue of Kirchhoff's circuit laws, or quantum Kirchhoff's laws for a quantum wire. 3. Lecture 2: Self-adjoint extensions of the covariant Laplace operator and the Hermitean Grassmannian 3.1. Von Neumann's theory of self-adjoint extensions and boundary conditions. In the previous lecture we have sketched a theory of self-adjoint extensions of symmetric differential operators using as an example the Laplace-Beltrami operator, which is based on a geometrical structure, the Lagrange bilinear form. However a general (abstract) solution to this problem was given by von Neumann [31]. We will discuss now the exact nature of the link between both approaches, the one based on geometrical boundary data and von Neumann's theorem based on global information in the bulk. It is interesting to point it out that there is a generalization of von Neumann's theory of extensions of formally normal operators with non-dense domains [12]. These results can be discussed from the viewpoint of the geometry of boundary conditions too. We will not insist on this here and we will restrict for clarity on the exposition to the simpler case of self-adjoint extensions of symmetric operators with dense domains. Suppose that H denotes a symmetric operator on the Hilbert space H (for instance the Schrödinger operator Eq. (3)), then we may define the deficiency spaces N λ , Nλ for any λ ∈ C, Im λ = 0, by setting, (20) N λ = Ran (H + λI) ⊥ = ker(H † +λI), Nλ = Ran (H +λI) ⊥ = ker(H † + λI). It is then true that for any complex λ / ∈ R the dimension of N λ is constant on the upper (lower) half-plane and: (21) D = D 0 + N λ + Nλ, and the sum is direct as vector spaces. Von Neumann's theorem [31] states that 5 , Theorem 4. There exists a one-to-one correspondence between self-adjoint (symmetric) extensions of H and unitary operators (partial isometries) K from N λ to Nλ, for any nonreal λ ∈ C. The domain D K of the self-adjoint extension corresponding to the operator K is D 0 + Ran (I + K). The extension H K of the operator H is defined for a function of the form ψ = ψ 0 + (I + K)ξ + , ψ 0 ∈ D 0 , ξ + ∈ N + , by H K ψ = Hψ 0 + λξ + +λKξ + . 5 Different presentations of this theorem can be found in [13], [30] [40], [33] and [37], existing a vast literature on the subject. The main idea of the proof is to show that there is a one-to-one correspondence between extensions of the symmetric operator H and extensions of its Cayley transform U : Ran (H + λI) → Ran (H +λI) defined by U = H +λI H + λI . To compare with our previous results it will be convenient to describe von Neumann extension theorem in the setting of skew-pseudo-hermitian spaces. We define the total deficiency space H V N = N λ ⊕ Nλ. As we have discovered in Thm. 3, unitary operators from N λ to Nλ are in one-to-one correspondence with maximal isotropic subspaces of H V N with respect to the natural pseudohermitian structure ω V N defined on H V N by (22) σ V N (ψ + 1 , ψ − 1 ; ψ + 2 , ψ − 2 ) = ψ + 1 , ψ + 2 − ψ − 1 , ψ − 2 , ∀ψ + α ∈ N λ , ψ − α ∈ Nλ. Now we can try to identify the total deficiency space H V N with the space of boundary data defined by the Laplace-Beltrami operator. Before doing this it will be convenient to enlarge slightly our setting by allowing differential operators action on spaces of vector-valued functions, i.e., sections of a vector bundle, instead of scalar functions. Thus the Laplace-Beltrami operator will be replaced by the covariant Laplacian ∆ A . Following closely the notations and conventions in Section 2.2 we may consider π : E → Ω be an Hermitean bundle over Ω of rank r, whose Hermitean structure will be denoted by (·, ·). We will denote as well by Γ ∞ (E) the space of smooth sections of the bundle E and by Γ ∞ c (E) the space of smooth sections with support on the interior of Ω. A Hermitean connection ∇ on the bundle E is by definition a linear differential operator ∇ : X(Ω) × Γ ∞ (E) → Γ ∞ (E) such that: i) ∇ X (f Φ) = f ∇ X Φ + X(f )Φ, ii) (∇ X Φ, Ψ) + (Φ, ∇ X Ψ) = X(Ψ, Ψ) for all X ∈ X(Ω), Φ, Ψ ∈ Γ ∞ (E), f ∈ C ∞ (Ω). We will denote by H k (E) the Hilbert space of equivalence classes of sections of the bundle E of Sobolev class k, i.e., a section Φ ∈ Γ ∞ (E) is of Sobolev class k if ||Φ|| 2 k = Ω (Φ(x), (I − ∇ † 0 ∇ 0 ) k/2 Φ(x)) x vol η (x) < ∞ where ∇ 0 is a fixed reference Hermitean connection on E and ∇ † 0 is the formal adjoint differential operator of ∇ 0 . Then H k (E) = Γ ∞ (E) ||·|| k . The restriction of the bundle E to the boundary ∂Ω, again denoted by Γ in what follows, will be denoted by ∂E, i.e., ∂E = E | Γ , and the restriction of the projection π to ∂E, by ∂π, thus ∂π : ∂E → Γ is again an Hermitean bundle over Γ of rank r. Any Hermitean connection ∇ of E restricts to an Hermitean connection of ∂E that will be denoted with the same symbol. Thus the space of smooth sections of E restricted to Γ = ∂Ω is Γ ∞ (∂E). We will consider the Bochner Laplacian (that will be also called the covariant Laplacian) associated to the Hermitean connection ∇ as the formally self-adjoint elliptic differential operator of order 2 acting on sections of the Hermitean bundle E with compact support on Ω\∂Ω by ∆ A = −∇ † ∇. We will denote again by ∆ 0 the minimal closed extension of the operator ∆ A with respect to the graph-operator norm as in Section 2.2. It is well-known that the domain of ∆ 0 is given again by D(∆ 0 ) = H 2 0 (E). We will denote by ∆ † 0 the adjoint operator of ∆ 0 in L 2 (E) whose domain contains H 2 (E). By ϕ := Φ| ∂Ω andφ := ∇ ν Φ| ∂Ω we denote again the restriction of Φ ∈ H 2 (E) to the boundary and the covariant normal derivative with respect to the outward normal respectively. We will call the pair (ϕ,φ) the boundary data of Φ and we will denote it by b(Φ). The induced scalar product on the boundary is denoted again by ϕ, ψ = Γ (ϕ(x), ψ(x)) x vol ∂η (x). The boundary map b : Γ ∞ (E) → Γ ∞ (∂E) × Γ ∞ (∂E) can be extended continuously to H 2 (E), which constitutes another statement of the well-know Lions-Magénes trace theorem, Thm. 1. In this context the weak trace theorem for the Bochner Laplacian states that there is a unique continuous extension of the boundary map b such that b : H 2 (E) → H 3/2 (∂E) ⊕ H 1/2 (∂E). Moreover the map is surjective and ker b = H 2 0 (E). We will denote by H L = H 3/2 (∂E) ⊕ H 1/2 (∂E) the Hilbert space of boundary data (ϕ,φ). We will assume in what follows that the self-adjoint extensions of the Bochner Laplacian we are interested in are such that the graph of the unitary operator K : N + → N − is contained in H 2 (E). Then the boundary map b restricts to the graph of K which is contained in H V N ∩ H 2 (E). We compose b with the Cayley transform on the boundary C to obtain a continuous linear map j : H V N ∩H 2 (E) → H L defined as follows. Let j ± (ψ ± ) = ϕ±iφ, where (ϕ,φ) = b(ψ), and we will denote j ± (ψ ± ) as usual by ϕ ± . Then, j = j + ⊕ j − or explicitly, (23) j(ψ + , ψ − ) = (ϕ + , ϕ − ). The following lemmas will show that j is an isometry of skew-pseudo-hermitian structures. Lemma 1. With the notation above the map j : H V N → H L verifies σ V N (ψ + 1 , ψ − 1 ; ψ + 2 , ψ − 2 ) = Σ(ϕ + 1 , ϕ − 1 ; ϕ + 2 , ϕ − 2 ) . Proof: We consider λ = i, the proof for general λ proceeds equally. We shall consider first ψ + 1 , ψ + 2 ∈ N i , then −∆ † A ψ + a = iψ + a , a = 1, 2. Then it is clear that, 0 = ψ + 1 , (−∆ † A − i)ψ + 2 = ψ + 1 , −∆ A ψ + 2 − i ψ + 1 , ψ + 2 = −∆ A ψ + 1 , ψ + 2 − iΣ B (b(ψ + 1 ), b(ψ + 2 )) − i ψ + 1 , ψ + 2 = (−∆ A − i)ψ + 1 , ψ + 2 − 2i ψ + 1 , ψ + 2 − iΣ(b(ψ + 1 ), b(ψ + 2 )) = −2i ψ + 1 , ψ + 2 − iΣ(b(ψ + 1 ), b(ψ + 2 )). Hence, σ V N (ψ + 1 , 0; ψ + 2 , 0) = ψ + 1 , ψ + 2 = − 1 2 Σ(b(ψ + 1 ), b(ψ + 2 )) = − 1 2 Σ(ϕ + 1 , 0; ϕ + 2 , 0). Similarly, it is shown that σ V N (0, ψ − 1 ; 0, ψ − 2 ) = Σ(0, ϕ 1 −; 0, ϕ − 2 ). To show that j is onto we will need the following result about from the existence and uniqueness of solutions of the Dirichlet problem. (24) − ∆ A Ψ + λΨ = 0, −∆ A Ψ +λΨ = 0 with boundary condition Ψ | ∂Ω = φ. Proof: We prove first uniqueness. If there were two different solutions Ψ 1 , Ψ 2 , then because the operator −∆ A + λ is elliptic, by elliptic regularity they will be both smooth. Then, Ψ = Ψ 1 − Ψ 2 also satisfies Eq. (24) with boundary condition Ψ | ∂Ω = 0, which is impossible by the uniqueness of the solution of the Dirichlet problem. In fact, if we look for solutions Ψ of the equation (24) such that Ψ | ∂Ω = constant, then, we can remove the boundary identifying all their points and looking for the solutions of eq. (24) on the closed manifold Ω ′ obtained in this way. But now, −∆ A is essentially self-adjoint on Γ(E ′ ) where E ′ is the fibre bundle obtained from E identifying all the fibres over ∂Ω 6 , and then it has not imaginary eigenvalues. Let us now prove the existence of solutions. LetΨ be any section in Γ(E) such thatΨ | ∂Ω = φ. Then, there exists a unique section ζ ∈ Γ(Ω) such that −∆ A ζ + λζ = ∆ AΨ − λΨ, with Dirichlet boundary conditions. This is a consequence of the solution of the Dirichlet boundary value problem for elliptic operators. Then, the section Ψ = ζ+Ψ verifies Eq. (24) and the boundary condition Ψ | ∂Ω = φ. Theorem 5. The deficiency space on the bulk H V N ∩ H 2 (E) with its natural skew-Hermitean structure σ V N is isometrically isomorphic to the boundary data space H L with its natural skew-Hermitean structure Σ. Proof: We will have to show that the map j is onto. We can solve the boundary problems − ∆ A Ψ + + λΨ + = 0, Ψ + | ∂Ω = ϕ + (25) −∆ A Ψ − +λΨ − = 0, Ψ − | ∂Ω = ϕ − ,(26) for given ϕ ± ∈ Γ ∞ (∂E). Proposition 1 shows that such solutions Ψ ± exist and they are unique. They define the inverse of the map j on the dense subspace Γ ∞ (∂E) ⊕ Γ ∞ (∂E), thus j is an isometry onto. Notice that the previous theorem can also be seen as offering an alternative proof of von Neumann's theorem for the symmetric operator ∆ A . Similar arguments can be reproduced in the much broader context of symmetric pseudodifferential operators of any order in compact manifolds with boundary. For instance the results obtained so far can be used to obtain a similar theory for Dirac operators. We will come back to this in Lecture 3. Self-adjoint extensions, boundary data and Cayley submanifolds. The characterization of self-adjoint extensions of H = −∆ 0 in terms of a class of unitary operators in U(L 2 (Γ, C r )) 7 , although similar to von Neumann characterization, is more useful for applications because it is formulated in terms of boundary data. The constraints involved in the definition of the domain determined by the unitary operator U imply that the boundary values ϕ,φ of the functions of such a domain satisfy Asorey's condition Eq. (9). Generically, Eq. (9) can be solved to expressφ as a function of ϕ, i.e., (27) φ = −i I − U I + U ϕ or, alternatively, ϕ as a functions ofφ (28) ϕ = i I + U I − Uφ . Notice that a necessary and sufficient condition for the existene of (I ± U ) −1 is that ∓1 is not in the spectrum of U respectively. This explicit resolution of the constraint on the boundary data means that unitarity requires that only half of the dynamical data are independent at the boundary. Equations (27) and (28) are in fact two different expressions of the Cayley transform relating self-adjoint and unitary operators: (29) A = −i I − U I + U ; A −1 = i I + U I − U . The inverse transformation being also a Cayley transform (30) U = I − iA I + iA . Notice that contrary to what happens with the definition of A in terms of U , given a self-adjoint operator A, the unitary operator U given by Eq. (30) is always well-defined. These considerations show that there is a distinguished set of self-adjoint extensions of H for which the expression of the boundary conditions defining their domain cannot be reduced to the simple form given by Eqs. (27) or (28). These self-adjoint extensions correspond to the cases where ±1 are in the spectrum of the corresponding unitary operator U . The Cayley subspaces C ± are thus defined as the subspaces of self-adjoint extensions which cannot be defined in the form (27) or (28), i.e.: (31) C ± = U ∈ U L 2 (Γ, C r ) ± 1 ∈ σ(U ) . Notice that the unitary operators U = ±I are in the Cayley subspaces C ± , respectively. U = −I belongs to the Cayley subspace C − and corresponds to Dirichlet boundary conditions: (32) ϕ = 0, whereas U = I is in the Cayley subspace C + and corresponds to the self-adjoint operator A = 0 which defines Neumann boundary conditions (33)φ = 0. There is a formal property which distinguishes the two Cayley subspaces. The subspace C + has a group structure whereas C − does not because the composition is not a inner operation. Notice that neither C − ∩ C + has a group structure. We will denote by M the space of self-adjoint extensions of the Bochner Laplacian ∆ A . Notice that so far we have described a family of self-adjoint extensions characterized by the property that their domains are contained in H 2 (E). In fact because of Thm. 2 such extensions are in one-to-one correspondence with the subgroup of the group of unitary operators U ∈ U(L 2 (∂E)) preserving the subspace H 3/2 (∂E) ⊕ H 1/2 (∂E) ⊂ L 2 (∂E) ⊕ L 2 (∂E), i.e., the unitary operators U such that C −1 (graph(U ) ∩ H 3/2 (∂E) ⊕ H 1/2 (∂E) is a closed maximally isotropic subspace. Thus the identification of the space M with a subgroup of the unitary group U(L 2 (Γ, C r )) provides an explicit group structure to this space of self-adjoint realizations of ∆ A . In what follows we will identify this space of self-adjoint extensions of the Bochner Laplacian ∆ A with the unitary group U(L 2 (Γ, C r )) itself because it can be proved that it provides a parametrization of all self-adjoint extensions of ∆ A (see [26] and references therein) and we will denote it by M again. 3.3. The self-adjoint Grassmannian. The space M of self-adjoint extensions of the Bochner Laplacian has a non-trivial topological structure. All even homotopy groups vanish π 2n (M) = 0 but all odd homotopy groups are non-trivial π 2n+1 (M) = Z because of Bott's periodicity theorem. The fact that the first homotopy group π 1 (M) = Z is non-trivial means that the space of boundary conditions is non-simply connected. However the set of self-adjoint operators in L 2 (Γ, C r ) is a topologically trivial manifold (notice that any self-adjoint operator A can be deformed homotopically to 0 by (1 − t)A, t ∈ [0, 1]). This means that the characterization of self-adjoint extensions of ∆ A by means of the Cayley transform (27) and (28) cannot provide a global description of M. In fact, the parametrization (30) and its inverse (34) U −1 = I + iA I − iA . can be considered as local coordinates in the charts M \ C ± of the space M of self-adjoint extensions ∆ A . The topology of each chart is trivial but that of M is not. In this sense, the Cayley submanifold C ± intersects all non-contractible cycles of M. Since π 0 (M) = 0 and π 1 (M) = Z the first cohomology group of M is H 1 (M) = Z. The generator of this cohomology group is given by the first Chern class of the determinant bundle defined over M. The determinant of infinite dimensional unitary operators U is ill defined and its proper definition requires the introduction of a regularization. In particular, it is necessary to restict the boundary conditions to the subspace M ′ defined by the unitary U operators of M which are of the form U = I + K with K a Hilbert-Schmidt operator (i.e. Tr (K † K) < ∞). If −1 / ∈ σ(U ) this property is equivalent to the requirement that the Cayley transform of the operator A is also Hilbert-Schmidt. Indeed, A = iK 2I + K , K = 2A iI − A ; hence, K † K = 4A 2 I + A 2 ; A † A = K † K (2I + K † )(2I + K) and we get the bounds: Tr (K † K) = 4 Tr A 2 I + A 2 ≤ 4 Tr (A 2 ), Tr (A † A) ≤ Tr (K † K). With this restriction the determinant of U ∈ M ′ can be defined by using the standard renormalization prescription for determinants log det ′ U = ∞ k=1 d k e −λ k log (1 + λ k ), in terms of the eigenvalues of K, λ k , k = 1, 2, · · · , and their degeneracies d k , k = 1, 2, · · · . Finiteness of this prescription for the regularized determinant det ′ U follows from the Hilbert-Schmidt character of K which in particular implies a discrete spectrum with finite degeneracies satisfying the Hilbert-Schmidt condition K † K = ∞ k=1 d k |λ k | 2 ≤ ∞. The first Chern class of the regularized determinant bundle is given by the oneform: (35) α = 1 2π d log det ′ γ(θ) . For any closed curve γ : S 1 → M ′ in the self-adjoint grassmannian, we define its Maslov index ν M (γ) as the winding number of the curve det ′ • γ : S 1 → U (1) (see for instance [4]). In other words, (36) ν M (γ) = 1 2π 2π 0 ∂ θ log det ′ γ(θ) dθ. Thus the Maslow index ν M (γ) is the sum of the winding numbers of the maps λ i (θ) : S 1 → U (1) described by the flow of eigenvalues of γ around U (1). By continuity of γ and compactness of S 1 it follows that only a finite number of eigenvalues reach the value λ i = −1 for any value of θ ∈ [0, 2π). It is clear that the winding number of the map λ i (θ) is measured by 1 2π 2π 0 ∂ θ log (λ i (θ))dθ and also by the number of indexed crossings of the point λ i = −1. By construction ν M (γ) is the finite sum of the non-trivial winding numbers and is always an integer. This fact and the existence of curves with only one crossing through −1 implies that α is in the generating class of the cohomology group H 1 (M ′ , Z). The subspace M ′ of unitary operators of the form U = I + K has richer topological and geometrical structures. In particular we will see that it is a Grassmaniann, the self-adjoint Grassmannian as it will be called in what follows. It is obvious that the subspaces H + = L 2 (Γ, C r )×{0} = { (ϕ, 0) | ϕ ∈ L 2 (Γ, C r ) } and H − = {0} × L 2 (Γ, C r ) = { (0,φ) |φ ∈ L 2 (Γ, C r ) } , which correspond to Dirichlet and Neumann boundary conditions respectively, are isotropic in H = L 2 (Γ, C r ) ⊕ L 2 (Γ, C r ) and they are paired by Σ. In fact, Σ(ϕ 1 , 0; 0,φ 2 ) = ϕ 1 ,φ 2 Γ The block structure of Σ with respect to the isotropic polarization H + ⊕ H − of H reads Σ = 0 · , · Γ − · , · Γ 0 . The pseudo-Hemitean structure Σ can be diagonized by means of the Cayley transform (37) C(ϕ,φ) = (φ + , φ − ) = 1 √ 2 (ϕ + iφ), 1 √ 2 (ϕ − iφ) . which transforms Σ into Σ. There is another canonical hermitian product on H + ⊕ H − given by the matrix operator: I 0 0 I which defines a Hilbert structure · , · on H + ⊕ H − . The Grassmannian Gr(H + , H − ) of L 2 (Γ, C r )×L 2 (Γ, C r ) is the infinite-dimensional Hilbert manifold of closed subspaces W in H + ⊕ H − such that the projection on the first factor π + : W → H + is a Fredholm operator and the projection on the second factor π − : W → H − is Hilbert-Schmidt, that is, Tr (π † − π − ) < ∞. The self-adjoint Grassmaniann Gr(H + , H − ) ∩ M is defined by the self-adjoint extensions of ∆ A which belong to the Grassmaniann Gr(H + , H − ). This subspace might be considered as the space of "mild" self-adjoint extensions of ∆ A . It is possible to see that the self-adjoint Grassmannian is an open submanifold of the Grassmannian itself and can be identified with M ′ , the space of unitary operators of M which are of the form U = I + K. This follows from the fact that in the previous parametrization of M ′ we have: (38) π − = iK 2 √ U , which implies that Tr (π † − π − ) = 1 4 Tr (K † K), i.e. π − is Hilbert-Schmidt if and only if K is Hilbert-Schmidt. The intersection of the Cayley submanifold C ± with M ′ defines a subspace of the self-adjoint Grassmannian C ′ ± ⊂ M ′ which has a stratified structure according to the number of eingenvalues ±1 of the corresponding unitary operator, i.e. C ′ ± = ∞ n=1 C ′ n ± , where C ′ n ± = {U ∈ U(L 2 (Γ, C r ) | ±1 ∈ σ(U ) with multiplicity n}. Notice that the spectrum of unitary operators in the self-adjoint Grassmannian is discrete. Given a continuous curve γ : [0, 1] → M ′ we define its Cayley index ν c (γ) as the indexed sum of crossings of γ through the Cayley submanifold C ′ − (notice that the Cayley submanifolds C ′ ± are oriented). This is equivalent to the sum of anticlockwise crossing of eigenvalues of γ through the point −1 on the unit circle U (1) minus the sum of clockwise crossings weighted with the respective degeneracies. Therefore, the Cayley index ν c (γ) of γ is equivalent to its Maslow index ν M (γ) and we have the following theorem. 3.4. Topology change and edge states. Although the operator ∆ A +I is positive in Γ ∞ 0 (E), its self-adjoint extensions might not be. In fact, if the self-adjoint extension does not belong to any of the Cayley submanifolds C ± it is easy to show by integration by parts that ∇Ψ 1 , ∇Ψ 2 = Ψ 1 , ∆ 0 Ψ 2 + ϕ 1 ,φ 2 (40) = Ψ 1 , ∆ 0 Ψ 2 + ϕ 1 , Aϕ 2 = Ψ 1 , ∆ 0 Ψ 2 + A −1φ 1 ,φ 2 ,(41) where A = C(U ) is the Cayley transform of the unitary operator defining the self-adjoint extension, ∇Ψ, ∇Φ = Ω η ij (x)h ab (x)∇ iΨ a (x)∇ j Φ b (x)vol η (x), with Φ = Φ a σ a , with σ a and local reference frame, h ab (x) = (σ a , σ b ) x , and ∇ i = ∇ ∂/∂x i . Thus, only if ∇Ψ, ∇Ψ − ϕ, Aϕ is positive for every Ψ, the operator ∆ U will be positive. In particular if the boundary operator A is positive it might occur that the whole operator ∆ U might loose positivity. The existence of negative energy levels is thus possible for some boundary conditions. It can be seen that the states which have negative energy are related to edge states as it is illustrated by the following result. Theorem 7. For any self-adjoint extension ∆ U of ∆ 0 whose unitary operator U has one eigenvalue −1 with smooth eigenfunction, the family of self-adjoint extensions of the form U t = U e it with t ∈ (0, π/2), has for small values of t, one negative energy level which corresponds to an edge state. The energy of this edge state becomes infinite when t → 0. Proof: Let ξ ∈ L 2 (Γ, C r ) be a smooth eigenstate of U with eigenvalue −1. Then, U t ξ = e it ξ. Let us consider Gaussian coordinates in a collar C Γ ⊂ Ω around the boundary Γ of Ω. One of those coordinates is the "radius" r and the others can identified with boundary coordinates sifted inside the collar; i.e. C Γ ≈ [1 − ǫ, 1] × Γ. In these coordinates the metric matrix looks like (42) η = 1 0 0 Λ(r, Γ) . We consider now the following change of coordinates r → s with s = π 2ǫ (1 − r). If we extend the function ξ from the boundary Γ to an edge state Ψ in the bulk Ω by (43) Ψ(x) = ξ(Γ)e −k tan s , x = (s, Γ) ∈ C Γ 0, x / ∈ C Γ , it is easy to check that the extended function Ψ is smooth in Ω and for k = 2ǫ π cot t 2 belongs to the domain of the self-adjoint extension of ∆ Ut associated to the unitary matrix U t = e it U . Thus, we have (44) Ψ, ∆ Ut Ψ = ∇Ψ, ∇Ψ − cot t 2 ξ, ξ where ∇Ψ, ∇Ψ = π/2 0 ds Γ dµ Γ (s) |ξ| 2 k 2 π 2ǫ 1 + (tan s) 2 2 e −2k tan s (45) + 2ǫ π π/2 0 ds Γ dµ Γ (s) ξ * , ∆ Γ ξ e −2k tan s . For small enough ǫ ≪ 1 we have that the dependence on s of Λ(s, Γ) might be negligible |Λ(s, Γ)| < |Λ(0, Γ)|(1 + δ). Thus, (46) ∇Ψ, ∇Ψ ≤ k 2 + 1 4k π(1 + δ) 2ǫ ξ 2 + 2ǫ(1 + δ) π ξ * , ∆ Γ ξ , and (47) Ψ, ∆ Ut Ψ ≤ π 2ǫ 1 4k (1 + δ) − k 2 (1 − δ) ξ 2 + 2ǫ(1 + δ) π ξ * , ∆ Γ ξ , which shows that Ψ, ∆ Ut Ψ ≤ 0 for small values of ϕ = 2 arc ctg (kπ/2ǫ). Notice that the normalization of the edge state Ψ ||Ψ|| 2 = Ω (Ψ † , Ψ) x dµ η (x) ≥ π(1 − δ) 2ǫ ||ξ|| 2 2π 0 ds e −2k tan s vanishes in the limit t → 0 but it is always a positive factor for t = 0 which preserves the bound given in Eq. (47). Moreover, the nature of the edge state Ψ also shows the existence of a ground state Ψ 0 with negative energy which is an edge state. The energy E 0 of this state goes to −∞ as t → 0, whereas the edge state Ψ 0 shrinks to the edge disappearing from the spectrum of ∆ Ut in that limit. Although the role of boundary conditions in the two Cayley submanifold C ± is quite similar from the mathematical point of view, the boundary conditions are quite different from the physical viewpoint. In particular, an analysis along the lines of the proof of the above theorem leads to the same inequality as in Eq. (47) but with k = 2ǫ π tan t 2 which points out the existence of edge states with very large (positive) energy as t → 0. It can also be shown that in that limit one energy level crosses the zero energy level becoming a zero mode of the Laplacian operator. Therefore, the role of boundary conditions in C − (e.g. Dirichlet) is very different of that of boundary conditions in C + (e.g. Neumann). Notice that the result of the theorem does not require U to be in the selfadjoint Grassmannian M ′ . This is specially interesting, because there is a very large family of boundary conditions which do not belong to M ′ . In particular, boundary conditions implying a topology change in higher dimensions are not in M ′ because the corresponding unitary operators in U(L 2 (Γ, C r )) present an infinity of eigenvalues ±1 which implies that U cannot be of the form I + K with K Hilbert-Schmidt. Indeed, all boundary conditions which involve a change of topology, i.e. gluing together domains O 1 , O 2 of the boundary Γ, belong to C − ∩ C + . This property follows from the fact that the boundary conditions imply that the boundary values ϕ,φ are related in the domains that are being glued together, i.e. ϕ( O 1 ) = ϕ(O 2 ),φ(O 1 ) = −φ(O 2 ) , respectively. These requirements imply that the unitary operator U corresponding to this boundary condition is identically U = I on the subspace of functions such that ϕ(O 1 ) = ϕ(O 2 ) and U = −I on the subspace of functions such that ϕ(O 1 ) = −ϕ(O 2 ). Since both subspaces are infinite-dimensional for manifolds Ω of dimension larger than 1, it is clear that those operators U do not belong to C ′ − ∩ C ′ + . However the result of Theorem 7 implies that there always exists a boundary condition close to one involving the gluing of the domains with very large negative energy levels. This means that Cayley manifold C − ∩ C + is very special and that topology change involves an interchange of an infinite amount of quantum energy. These results might have relevant implications in quantum gravity and string theory. Lecture 3: Elliptic and self-adjoint extensions of Dirac operators 4.1. Dirac operators. As it was indicated in the introduction, Dirac operators constitute an important class of elliptic operators, to the extent that all relevant elliptic operators arising in Geometry and Physics are in one way or the other related to them. Let us set the ground to discuss them (see for instance [28] and [10]). We will consider again a Riemannian manifold (Ω, η) with smooth boundary ∂Ω. We denote by Cl(Ω) the Clifford bundle over Ω defined as the algebra bundle whose fibre at x ∈ Ω is the Clifford algebra Cl(T x Ω) generated by vectors u in T x Ω satisfying ther relations u · v + v · u = −2η(u, v) x ∀u, v ∈ T x Ω. Let π : S → Ω be a Cl(Ω)-complex vector bundle over Ω, i.e., for each x ∈ Ω, the fibre S x is a Cl(Ω) x -module, or in other words, there is a representation of the algebra Cl(Ω) x on the complex space S x by complex automorphisms. We will represent with the same symbol the vector u ∈ Cl(Ω) and the automorphism of S defined by u, ξ → u · ξ for all ξ ∈ S. We will also call Clifford multiplication of ξ by u the action of the automorphism defined by the vector u on the element ξ of S. We will assume in what follows that the bundle S carries a hermitian metric denoted by (·, ·) such that Clifford multiplication by unit vectors in T Ω is unitary: (48) (u · ξ, u · ζ) x = (ξ, ζ) x , for all ξ, ζ ∈ S x , u ∈ T x Ω, x ∈ Ω and ||u|| 2 = 1. Finally, we will assume that there is an Hermitean connection ∇ on S such that (49) ∇(V · ξ) = (∇ η V ) · ξ + V · ∇ξ, where V is a smooth section of the Clifford bundle Cl(Ω), ξ ∈ Γ(S) and ∇ η denotes the canonical connection on Cl(Ω) induced by the Riemannian metric η on Ω. A bundle S with the structure described above is commonly called a Dirac bundle [28] and they provide the natural framework to define Dirac operators. Thus, if π : S → Ω is a Dirac bundle, and we denote by Γ ∞ (S) the space of smooth sections of the bundle map π, we can define a canonical first-order differential operator D : Γ ∞ (S) → Γ ∞ (S) by setting Dξ = e j · ∇ ej ξ, where e j is any orthonormal frame at x ∈ Ω. There is a natural inner product ·, · on Γ(S) induced from the pointwise inner product (·, ·) on S by setting ξ, ζ = Ω (ξ(x), ζ(x)) x vol η (x). We will denote the corresponding norm by || · || 2 and L 2 (S) will denote the Hilbert space of square integrable sections of S. Giving a section ξ which is square integrable, we will say that the 1-form β in Ω with values in S is a weak covariant derivative of ξ if it is square integrable and for every section ζ ∈ Γ ∞ 0 (S), i.e., a smooth section of S with compact support contained in the interior of Ω, we have: (50) Ω (ξ(x), ∇ V ζ(x)) x vol η (x) = − Ω (i V β(x), ζ(x)) x vol η (x), for all vector fields V in Ω. We consider the completion of Γ ∞ (S) with respect to the Sobolev norm || · || 1,2 defined as: (51) ||ξ|| 2 k,2 = Ω (ξ(x), (I + ∇ † ∇) k/2 ξ(x)) x vol η (x), with k = 1, where ∇ † is the formal adjoint operator to ∇ in Γ ∞ 0 (S). This Hilbert space will be denoted by H 1 (S). Moreover, it happens that the Dirac operator D defined on H 1 (S) is not selfadjoint. However it is immediate to check that the Dirac operator D is symmetric in the space of smooth sections of S with compact support contained in the interior of Ω. In fact, after integration by parts we obtain immediately, Dξ, ζ = ξ, Dζ , ∀ξ, ζ ∈ Γ ∞ 0 (S). The operator D with domain Γ ∞ 0 (S) is closable on L 2 (S) and its closure is the completion of Γ ∞ 0 (S) with respect to the norm ||.|| 1,2 . Such domain will be denoted by H 1 0 (S) ⊂ H 1 (S). If we denote by D 0 the operator D with domain H 1 0 (S) then we are looking for extensions of D 0 with domains dense subspaces of H 1 (S) containing H 1 0 (S) and such that the boundary terms obtained integrating by parts will vanish. Then the self-adjoint extensions D s of D 0 we are looking for will be defined on subspaces Dom (D s ) such that Our first aim will be to characterize such subspaces using the geometry of some Hilbert spaces defined on the boundary of Ω. To achieve it, we will derive the expression of the boundary form obtained intregrating by parts. Let x ∈ Ω and e j a local self-parallel orthonormal frame defined in a neighborhood of x, ∇ ej e i = 0 for all i, j. It is easy to see that such frame does always exists. Then, if ξ, ζ are sections of S, then they define a unique vector field X in a neighborhood of x by the condition (53) η(X, Y ) = −(ξ, Y · ζ), for any vector field Y . Then, we have that: (Dξ, ζ) x = (e j ∇ ej ξ, ζ) x = −L ej (ξ, e j ζ) x + (ξ, Dζ) x , but, div(X) = η(∇ ej X, e j ), hence using Eq. (53) again, div(X) = L ej η x (X, e j ) − η x (X, ∇ ej e j ) = −L ej (ξ, e j · ζ) x . Namely, (Dξ(x), ζ(x)) x − (ξ(x), Dζ(x)) x = div x (X) . Integrating the previous equation we find, Dξ, ζ − ξ, Dζ = Ω div x (X)vol η (x) = Ω (i X dvol η ) = ∂Ω i * (i X vol η ), where we denote by i : ∂Ω → Ω the canonical inclusion. If ν denotes the inward unit vector on the normal bundle to ∂Ω, the volume form vol η can be written on a neighborhood of ∂Ω as θ ∧ vol ∂η , where vol ∂η is an extension of the volume form defined on ∂Ω by the restriction of η, and θ is the 1-form such that θ(Y ) = η(Y, ν) for all Y . Then we get, i X vol η = (i X θ)vol ∂η = η(X, ν)vol ∂η = (ξ, ν · ζ)vol ∂η . Thus, finally, we obtain: (54) Dξ, ζ − ξ, Dζ = ∂Ω i * (ν · ξ, ζ)vol ∂η (x). We have obtained in this way the Lagrange's boundary bilinear form (55) Σ(ξ, ζ) = ∂Ω (ν(x) · ξ(x), ζ(x)) x vol ∂η (x), responsible for the non self-adjointeness of the Dirac operator D in H 1 (S). 4.2. The geometric structure of the space of boundary data. We will denote by ∂S the restriction of the Dirac bundle S to Γ = ∂Ω, i.e., ∂S = S | Γ which is a bundle over Γ, ∂π : ∂S → Γ with ∂π = π | ∂S . It is noticeable that ∂S becomes a Dirac bundle over Γ = ∂Ω with the inner product ·, · ∂Ω induced from the Hermitean product on S by restricting it to Γ and the induced Hermitean connection ∇ ∂Ω , defined again by restricting the connection ∇ on S to sections along ∂Ω. Thus the boundary Dirac bundle ∂S carries a canonical Dirac operator denoted by D ∂Ω and called the tangential Dirac operator. Notice that Γ = ∂Ω is a manifold without boundary, thus the boundary Dirac operator is essentially self-adjoint and possesses a unique self-adjoint extension (see for instance [28], Thm. 5.7; this fact will also follow from our main theorem in this section). We will denote as before by L 2 (∂S) the Hilbert space of square integrable sections of ∂S and by ·, · ∂Ω its Hilbert product structure (56) φ, ψ ∂Ω = ∂Ω (φ(x), ψ(x)) x vol ∂η (x). Because (58) ν : Γ ∞ (∂S) → Γ ∞ (∂S), ν(φ)(x) = ν(x) · φ(x), ∀x ∈ Γ, φ ∈ Γ ∞ (∂S). Such automorphism extends to a continuous complex linear operator of H D denoted now by J D . Because ν 2 = −1 in the Clifford algebra, such operator J D verifies J 2 D = −I. In addition, because of the Dirac bundle structure, Eq. (48), J D is also an isometry of the Hilbert space product, this is: J D φ, J D ψ ∂Ω = φ, ψ ∂Ω , ∀φ, ψ ∈ H D , i.e., J D defines a compatible complex structure on H D . More generally, given a complex Hilbert space H with inner product ·, · and a compatible complex structure J we can define a new continuous bilinear form ω by setting, ω(ϕ, ψ) = Jϕ, ψ , ∀ϕ, ψ ∈ H. Such structure is skew-Hermitiean in the sense that ω(ϕ, ψ) = −ω(ψ, ϕ). If the Hilbert space H would be real ω will define a symplectic structure on H. In any case the real part of ω will always define a real symplectic structure on H D viewed as a real space, very much as the imaginary part of a Hermitean structure on a complex Hilbert space defines a symplectic structure on its realification. We call the space H with the Hermitean and skew-Hermitean structures ·, · and ω, a symplectic-Hermitean linear space. Any symplectic-Hermitean linear space carries a natural polarization. In fact, the compatible complex structure allows to decompose the Hilbert space H as H + ⊕ H − where H ± are the closed eigenspaces of J of eigenvalues ∓i, that is φ ± ∈ H ± if Jφ ± = ∓iφ ± . The subspaces H ± are orthogonal because: φ + , φ − = Jφ + , Jφ − = iφ + , −iφ − = − φ + , φ − . Notice that the Hilbert space H D carries already another complex structure, denoted by J 0 , which is simply multiplication by i. Both complex structures are compatible in the sense that [J D , J 0 ] = 0 because the Dirac bundle S is a Cl (Ω)complex bundle. Hence, the previous discussion shows that the Hilbert space of boundary data H D for the Dirac operator D is a polarized Hilbert space carrying a compatible complex structure J D and the corresponding skew-Hermitean structure denoted in what follows by ω D . Using these structures the Lagrange boundary form Σ is written as: (59) Σ(ξ, ζ) = ω D (b(ξ), b(ζ)), ∀ξ, ζ ∈ H 1 (S). From Eq. (59) we see immediately that symmetric extensions D s of D will be defined in domains Dom (D s ) such that their boundary image b(Dom (D s )) are isotropic subspaces W of ω D , i.e., such that the r.h.s. of Eq. (59) vanishes for all b(ζ), b(ξ) ∈ W . Moreover if the extension D s is self-adjoint, such domains must verify that b(Dom (D s )) = b(Dom (D † s )) thus, they must be maximal subspaces with this property. We have thus proved the first part of the following theorem: D † W . If b(ξ), b(ζ) ∈ W , then D † W ξ, ζ = ξ, D W ζ + ω(b(ξ), b(ζ)) = ξ, D W ζ because W is ω D isotropic. This shows that b −1 (W ) ⊂ Dom (D † W ). If there were ξ ∈ Dom (D † W ) − b −1 (W ) , then, the same computation shows that ω D (b(ξ), φ) = 0 for all φ ∈ W , and the subspace W ′ = W ⊕ b(ξ) will be ω D -isotropic, which is contradictory. Thus Dom (D W ) = Dom (D † W ) and the extension is self-adjoint. The converse is proved similarly. Let us consider now a closed maximal ω D -isotropic subspace W . Let us show that W is transverse to H ± . Let φ ∈ W ∩ H ± , then 0 = ω D (φ, φ) = Jφ, φ = ∓i||φ|| 2 , then φ = 0. Then, the subspace W defines the graph of a continuous linear operator U : H + → H − and vectors φ = φ + + φ − ∈ W have the form φ − = U φ + . Then, the ω D -isotropy of W implies, 0 = ω D (φ + + U φ + , ψ + + U ψ + ) = iφ + − iU φ + , ψ + + U ψ + = −i φ + , ψ + + i U φ + , U ψ + , for every φ + , ψ − ∈ H + , that proves that U is an isometry. We can use anyone of these extensions, say one defined by a subspace W 0 , to identify the space of self-adjoint extensions of D with the group of unitary transformations of H + . Thus if W is a closed maximally ω D -isotropic subspace, let φ W : H + → H − be the isometry defined by it because of Thm. 8, then we associate to it the map φ = φ −1 0 • φ D where φ 0 is the isometry associated to W 0 . It is trivial then to check that φ is indeed a unitary operator on H + . From the previous discussion we can conclude that the space of self-adjoint extensions of the Dirac operator D can be naturally identified with U (n) if dim Ω = 1, n is the number of connected components of ∂Ω, however it is contractible if dim Ω > 1. 4.3. The Cayley transformation at the boundary. In spite of the inherent interest of the results described in the previous section, sometimes (as in the case of Laplace operators) it is more useful to have an alternative description of selfadjoint extensions in terms of self-adjoint operators at the boundary. For that we will use the Cayley transformation again in a new fashion. For that purpose, we define the space H J as the graph of the operator J D : H D → H D , i.e., H J = { (ξ, J D ξ) ∈ H D × H D | ξ ∈ H D }. The natural projection restricted to H J defines an isometry among H D and the later. The space H J carries a natural polarization induced from the one on H D . Thus we define the closed orthogonal subspaces L ± = { (ξ ± , Jξ ± ) ∈ H D × H D | ξ ± ∈ H ± }, and clearly, H J = L + ⊕ L − , L ⊥ + = L − . We define the Cayley transformation on the polarized boundary Hilbert space H D = H + ⊕ H − as the continuous isomorphism C : H D = H + ⊕ H − → H J = L + ⊕ L − defined by (60) C(ξ + , ξ − ) = 1 2 (ξ + + ξ − ), − i 2 (ξ + − ξ − ) , for every ξ ± ∈ H ± . We will also use the notation: (61) C + (ξ + , ξ − ) = 1 2 (ξ + + ξ − ) ∈ L + , C − (ξ + , ξ − ) = i 2 (ξ + − ξ − ) ∈ L − , thus C = (C + , C − ). If we denote by ψ + = 1 2 (ξ + + ξ − ) and ψ − = − i 2 (ξ + − ξ − ), we have that J D (ψ + ) = ψ − as it should be. The map C, which is J 0 complex, transforms the complex structure J D into J = CJ D C −1 (φ + , φ − ) = (−φ − , φ + ) and the symplectic-hermitian structure ω D is transformed into the bilinear form (62) σ(φ + , φ + ; ψ + , ψ − ) = 2i( φ + , ψ − − φ − , ψ + ). Let U be an isometry U : H + → H − . Then we have that the elements of H D in the graph of U verify ξ − = U ξ + . Using the Cayley transformation Eq. (60) we will obtain that, ψ + = 1 2 (ξ + + U ξ + ), and ψ − = − i 2 (ξ + − U ξ + ) hence, (I + U )ψ − = −i(I − U )ψ + . Thus we conclude that the graph of the isometry U is mapped into the subspace W U of H J defined by W U = {(ψ + , ψ − ) ∈ H J | (I + U )ψ − = −i(I − U )ψ + }. Let H = K + ⊕ K − be a polarized Hilbert space. Let W be a subspace of H, the adjoint of W is the subspace denoted by W † and defined by: W † = { (ψ + , ψ − ) ∈ K + ⊕ K − | φ + , ψ − = φ − , ψ + , ∀(φ + , φ − ) ∈ W }. The subspace W is said to be symmetric if W ⊂ W † and self-adjoint if W = W † . Notice that an operator A : K + → K − is self-adjoint if its graph is a self-adjoint subspace of the polarized Hilbert space H = K + ⊕ K − . Now a simple computation shows that the subspace W U constructed previously is a self-adjoint subspace of the polarized Hilbert space H J = L + ⊕L − . The subspace W U is transverse to L + , i.e., W U ∩ L + = 0, then it is the graph of a self-adjoint operator A U : L + → L − . In this sense, the Cayley transformation operator A U of any isometry U is self-adjoint. Moreover, it is clear that self-adjoint subspaces are maximally isotropic subspaces of the bilinear form σ D given by Eq. (62). But σ D is the transformed bilinear form on H D by the Cayley transformation, then, maximally σ D -isotropic subspaces correspond to maximally ω D -isotropic subspaces, in other words, the Cayley transformation is a one-to-one map among isometries U : H + → H − and self-adjoint operators A : L + → L − . We will consider the topology on the spaces of isometries U : H + → H − and self-adjoint operators A : L + → L − induced by the norm operator topology. We can summarize the previous discussion in the following theorem. 4.4. The space of self-adjoint elliptic boundary conditions: the elliptic grasmannian. In the previous section we have characterized self-adjoint extensions of Dirac operators in terms of boundary data and we have seen that they can be globally described as the manifold of self-adjoint subspaces W of the Hilbert space H J . However we have not considered yet along this discussion if the extensions D W of the Dirac operator D obtained in this way are elliptic operators or not, i.e., if the boundary data given by W determines an elliptic boundary problem for D [2], [34]. This is a crucial issue for applications of the theory because if the extensions considered are not elliptic the resulting operator could have, for instance, an infinite number of zero modes, i.e., its kernel will be infinite dimensional, which will make it unsuitable for physical applications. Looking for elliptic extensions of the operator D is thus a natural demand both mathematically and regarding the eventual applications of them. As it was mentioned before the theory of elliptic boundary problems for Dirac operators was developed in the seminal series of papers by Atiyah, Patodi and Singer [7]. The boundary conditions introduced there to study the index theorem for Dirac operators in even-dimensional spin manifolds with boundary are nowadays called Atiyah-Patodi-Singer (APS) boundary conditions. The crucial observation there was that global boundary conditions were needed in order to obtain an elliptic problem and this was completely different to the situation for second order differential operators where for instance "local" Dirichlet conditions are elliptic. Later on such boundary conditions were extended to include also odd dimensional spin manifolds with boundary (see [14] and references therein). More recently, E. Witten ([39], Section II), pointed out the link between elliptic boundary conditions for the Dirac operator on 2 dimensions and the infinite dimensional Grassmannian manifold. The infinite dimensional Grassmannian was introduced previously in the analysis of integrable hierarchies and discussed extensively by Segal and Wilson (see [36], [32] and references therein). Finally Schwarz and Friedlander [18] have presented a way to extend Witten's analysis to arbitrary elliptic operators on arbitrary dimensional manifolds with boundary. The particular analysis for Dirac operators follows from [7] but we want to point it out here that it can be extended also to higher order operators. More comments on this will be found later on. The basic idea behind is that the space of zero modes of a Dirac operator D, ker D = { ξ ∈ Γ ∞ (S) | Dξ = 0 }, induces a subspace in the boundary b(ker D) that in general will be infinite dimensional. The way to restore ellipticity will be to project down into a subspace such that the kernel and cokernel of the operator in this subspace will be finite dimensional. We shall perform such analysis for Dirac operators (see [10] for a detailed discussion). The analysis of such projection requires the description of solutions near the boundary. We can decompose the operator D in a collar neighborhood (−1, 0] × ∂Ω of the boundary as D = ν · (∇ ν + D ∂Ω ), where D ∂Ω is the Dirac operator on the boundary bundle ∂S. We easily see that b(ker D) is spanned by the eigenfunctions of D ∂Ω with nonnegative eigenvalues [7]. Next Lemma will describe this subspace in terms of the boundary operator D ∂Ω and subspaces of H D or better H J . In fact we get: Lemma 2. Let Ω be a compact manifold with smooth boundary ∂Ω, then with the notations above, we have: C(b(ker D)) = L + . Proof. Because ∂Ω is a closed manifold D ∂Ω is an essentially self-adjoint elliptic differential operator. Moreover the following computation shows that D ∂Ω anticommutes with J D . Namely, J D D ∂Ω = D − J D ∇ ν , but it is easy to check that: J D D = −Dν − 2∇ ν , hence J D DJ D = D − 2∇ ν J D . Moreover, D ∂Ω J D = −J D DJ D − ∇ ν J D , and, finally: J D D ∂Ω + D ∂Ω J D = D − J D ∇ ν − J D DJ D − ∇ ν J D = = D − J D ∇ ν − D + 2∇ ν J D − J D ∇ ν = 0, hence D ∂Ω J D = −J D D ∂Ω . The Dirac Laplacian D 2 ∂Ω is a non-negative essentially self-adjoint elliptic operator with a real discrete spectrum Spec(D 2 ∂Ω ) = { λ k | 0 = λ 0 < λ 1 < · · · } with finite dimensional eigenspaces E(λ k ) = { φ k ∈ H D | D 2 ∂Ω φ k = λ k φ k }. The kernel K of D ∂Ω agrees with ker D 2 ∂Ω and with E(0). We have thus the following orthogonal decomposition of H D , H D = ∞ k=0 E(λ k ) = K ⊕ ∞ k=1 E(λ k ) . On the other hand the polarization H D = H + ⊕ H − defined by the compatible complex structure J D , J D | H± = ∓iI, induces a decomposition of the eigenspaces E(λ k ) as E(λ k ) = E + (λ k ) ⊕ E − (λ k ); E ± (λ k ) = E(λ k ) ∩ H ± . Moreover, D ∂Ω restricts to a map D k = D ∂Ω | E(λ k ) : E(λ k ) → E(λ k ) and because anticommutes with J D , we have that D k : E ± (λ k ) → E ∓ (λ k ), thus D k has the block structure, D k = 0 D + k D − k 0 . In addititon, because D k is self-adjoint, (D − k ) † = D + k . On the other hand D 2 k = D 2 ∂Ω | E(λ k ) = λ k I, hence the spectrum of D k on E(λ k ) is ± √ λ k . The operator D k is invertible in E(λ k ) for k ≥ 1, hence dim E + (λ k ) = dim E − (λ k ). Moreover K = K + ⊕ K − , and dim K + = dim K − . Notice that the index of the operator D + 0 is zero because of the cobordant invariance of the index and the fact that ∂Ω is cobordant to ∅. Thus we can choose an orthonormal basis φ ± k,α ∈ E ± (λ k ), α = 1, . . . , dim E ± (λ k ), such that D k φ ± k,α = ± λ k φ ∓ k,α . The Cayley transformation discussed in Section 4.3 diagonalizes the operators D k , and if we denote by ψ ± k,α = C ± (φ + k,α , φ − k,α ) it is clear that Then, it is clear that b(ker D) = L + because H D is spanned by nonnegative eigenspaces of D ∂Ω and eq. (61). Then we can conclude the discussion by stating the following proposition: Theorem 10. Elliptic boundary conditions for the Dirac operator D are in oneto-one correspondence with the set of subspaces W ⊂ H J such that W ∩ L + is finite dimensional. Proof. We want to characterize subspaces W such that the solutions of the equation Dξ = 0 with boundary values on W will be finite dimensional. The orthogonal projectors pr ± : H J → L ± are pseudodifferential operators whose complete symbol depends only on the coefficients of D. Hence, because of the previous Lemma, C • b(ker D) = L + , and elliptic boundary conditions will be defined by subspaces W ⊂ H D such that W ∩ L + will be finite dimensional. Notice that for the elliptic extensions of the Dirac operator determined by the subspace W , the projection pr + | W will have a finite dimensional kernel. Moreover coker D W = ker D † W because ζ, D W ξ = D † W ζ, ξ , hence, ζ ∈ ker D ⊥ W iff ζ ∈ ker D † W , and coker D W is identified naturally with ker D ⊥ W . Hence, the cokernel of pr + will have to be finite-dimensional if D † W is elliptic too. Finally, if the extension D W is elliptic, then there will exists left and right parametrics for it (see for instance [28]), and this will imply that the projection pr − | W will have to be a compact operator. Then we conclude from the previous discussion: Theorem 11. The set of elliptic extensions of the Dirac operator D is in oneto-one correspondence with the points of the compact Grassmannian Gr K , where Gr K is defined as the set of closed subspaces W of H J such that the projections pr + : W → L + is a Fredholm operator and pr − : W → L − is a compact operator. The set K of compact operators contains a distinguished subset, the Hilbert-Schmidt operators. For technical reasons it is convenient to consider a restriction of the infinite dimensional Grassmannian to consider only those subspaces such that the projection on L − is Hilbert-Schmidt. We will say that closed subspaces W of H J satisfying that the projection on the first factor pr + | W : W → L − is a Fredholm operator and the projection on the second factor pr − | W : W → L + is Hilbert-Schmidt define restricted elliptic extensions of the Dirac operator D. Such space will be called the elliptic infinitedimensional Grassmannian of D, or elliptic Grassmannian for short, and will be denoted by Gr ′ (compare with the definition of the self-adjoint Grassmannian for the Bochner Laplacian in Section 3.3). The elliptic Grassmannian can be constructed also in terms of the polarization H + ⊕H − instead of L + ⊕L − . This is the approach taken for instance in [14]. In such case, we relate self-adjoint extensions of D with isometries U : H + → H − , hence elliptic boundary conditions correspond to isometries U such that the projection from its graph to H + would be Fredholm and the projection onto H − would be Hilbert-Schmidt. It is obvious that the Cayley transformation C defines a one-toone map from the Grassmannian Gr(H − , H + ) into Gr(L − , L + ) (the map is actually a diffeomorphism, see below), but it is important to keep in mind that the objects in the two realizations of the Grassmannian are different. In what follows we will omit the subindex to the different Hilbert spaces H D and H J and they will be identified by means of the Cayley transformation as indicated above. We will call in what follows the elliptic boundary conditions defined by points in the elliptic Grassmannian, generalized APS boundary conditions. The elliptic infinite-dimensional Grassmannian has an important geometrical and topological structure. We must recall first (see for instance Pressley and Segal [32] for more details) that Gr ′ is a smooth manifold whose tangent space at the point W is given by the Hilbert space of Hilbert-Schmidt operators J 2 (L − , L + ), from L − to L + . The group of linear continuous invertible operators GL(H) does not act on Gr ′ but only a subgroup of it, the restricted general linear group GL res (H), which defines the restricted unitary group U res (H) = GL res (H) ∩ U (H). The groups GL(H) and U (H) are contractible but GL res (H) and U res (H) are not. The manifold Gr ′ is not connected and is decomposed in its connected components defined by the virtual dimension of their points which is simply the index of the Fredholm operator pr + | W , then, Gr ′ = ∪ k∈Z Gr (k) . The Grassmannian Gr ′ carries a natural Kähler structure defined by the hermitian structure given by h W (Ȧ,Ḃ) = Tr (Ȧ †Ḃ ), whereȦ,Ḃ ∈ T W Gr ′ are Hilbert-Schmidt operators from L − to L + . The imaginary part defines a canonical symplectic structure Ω, On the other hand, we have seen in the previous section that the Grassmannian Gr ′ describes the elliptic extensions of such operator. Then, the elliptic self-adjoint extensions of the given operators will be given by the intersection M ∩ Gr ′ . This space will be called the elliptic self-adjoint Grassmannian or the self-adjoint Grassmannian for short. It is possible to see that the self-adjoint grassmannian is a smooth submanifold of the Grassmannian and decomposes in connected components which are submanifolds of the components Gr (k) . We will denote the elliptic self-adjoint grassmannian as M ellip . The most relevant topological and geometrical aspects of M ellip are contained in the following theorem. Theorem 12. The elliptic self-adjoint Grassmannian is a Lagrangian submanifold of the infinite dimensional Grassmannian. Proof: That M ellip is an isotropic submanifold of Gr(L − , L + ) follows immediately from Eq. (63) and the observation that tangent vectors to M ellip at W are defined by self-adjoint operators. Now, all we have to do is to compute T W M ⊥ ellip at W = 0 because of the homogeneity of the Grassmannian. Hence, ifȦ ∈ T 0 M ⊥ ellip , this means that Tr (Ȧ †Ḃ −ḂȦ) = 0, for every self-adjointḂ ∈ J 2 (L − , L + ), henceȦ † −Ȧ = 0, andȦ is self-adjoint, then lying in T 0 M ellip . l, r, +, +) In particular if we consider a single interval [0, 2π] with trivial riemannian metric, the fundamental solutions to the equation Eq. (14) have the form Ψ 1 = e Proposition 1 . 1For every φ ∈ Γ(∂E), and for every non real λ there is a unique solution of the equations Theorem 6 . 6The Maslov and Cayley indices of a closed curve γ in the self-adjoint Grassmannian coincide, ν M (γ) = ν c (γ). Thus the Cayley manifold C ′ − is dual of the Maslov class α.For any unitary operator U ∈ M we will define its degenerate dimension as the dimension of the eigenspace with eigenvalue −1. If U is in the self-adjoint Grassmaniann M ′ the dimension of the eigenspace with eigenvalue −1 is finite and the degenerate dimension of the operator is finite. We shall denote such number by n(U ) and it is an indicator of the level of γ(θ) in the stratified structure of C ′ :U = γ(θ) ∈ C ′ n ifand only if n(U ) = n. The Cayley index of any curve γ ∈ M ′ can be given in terms of this number by the Since the r.h.s. of Eq. (39) is the integral of a pure derivative it vanishes unless there is a singularity in the integrand. This only occurs at the jumps of n(γ(θ)) i.e. when one more eigenvalue of U = γ(θ) becomes equal to −1. ν M (γ) is in fact a bookkeeping of the number of eigenvalues of γ(θ) that cross through −1 and since it is of bounded variation on M ′ the integral in Eq. (39) is always finite and gives the Cayley index. This construction provides an alternative (singular) characterization of the first Chern class of the determinant bundle det M ′ (M ′ , U (1)) and the generating class of the first homology group H 1 (M ′ , Z) of M ′ . (D 0 ) ⊂ Dom (D s ) = Dom (D † s ) ⊂ Dom (D) = H 1 (S),and D s ξ = Dξ for any ξ ∈ Dom (D s ). of the trace theorem the restriction map i * : Γ ∞ (S) → Γ ∞ (∂S), ξ → φ := i * ξ, extends to a continuous linear map, called the trace or boundary map again: (57) b : H 1 (S) → H 1/2 (∂S) ⊂ L 2 (∂S). Moreover b induces a homeomorphismb : H 1 (S)/ ker b → H 1/2 (∂S), with ker b = H 1 0 (S) (see for instance [1], Thm. 7.53). Hence, if Dom (D s ) ⊂ H 1 (S) is the domain of a self-adjoint extenstion of the Dirac operator D, then b(Dom (D s )) ⊂ H 1/2 (∂S) is a closed subspace. The Hilbert space L 2 (∂S) will be called the Hilbert space of boundary data for the Dirac operator D and will be denoted in what follows by H D . It carries an important extra geometrical structure induced by the Lagrange boundary form Σ, Eq. (56). The normal vector field along Γ, defines a smooth section ν of the Clifford bundle Cl(∂Ω) over Γ = ∂Ω, thus ν defines an automorphism of the Dirac bundle ∂S over ∂Ω, Theorem 8 . 8Let (Ω, η) be a spin manifold with smooth boundary ∂Ω, π : S → Ω a Dirac bundle and D a Dirac operator on S. Then the symplectic-Hermitean boundary data Hilbert space (H D , J D , ω D ) carries a natural polarization H D = H + ⊕ H − and self-adjoint extensions of the Dirac operator D with domain in H 1 (S) are in one-to-one correspondence with subspaces of the boundary Hilbert space H D which are maximally ω D -isotropic closed subspaces on H 1/2 (∂S). The domain of anyone of these extensions is the inverse image by the boundary map b of the corresponding isotropic subspace. Moreover, each maximally closed ω D -isotropic subspace W of H D defines an unitary operator U : H + → H − and conversely. Proof: Let W be a closed ω D -isotropic subspace of H D . Then, b −1 (W ) is a closed subspace of H 1 (S) containing H 1 0 (S). Let D W be the extension of D defined on b −1 (W ) and compute Theorem 9 . 9The Cayley transform C : H D → H J defined by Eq. (60) defines a homeomorphism between the space of isometries U(H + , H − ) from H + to H − and the space of self-adjoint operators S(L + , L − ). Moreover, the self-adjoint extensions of the Dirac operator D are in one-to-one correspondence with the self-adjoint operators S(L + , L − ). Ȧ †Ḃ −Ḃ †Ȧ ). The Grassmannian Gr ′ is cuasicompact in the sense that the only holomorphic functions are constant. 4.5. The self-adjoint Grassmannian and elliptic extensions of Dirac operators. We have characterized the self-adjoint extensions of a given Dirac operator D as the space M of self-adjoint subspaces of a boundary Hilbert space H carrying a polarization H = L − ⊕ L + . From now on we will assume that = m = 1. 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Functional Analysis, Springer Verlag (1965). . Matemáticas Departamento De, Avda. de la Universidad. 30Universidad Carlos III de Madridalbertoi@math.uc3m.esDepartamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Uni- versidad 30, 28911 Leganés, Madrid, Spain E-mail address: albertoi@math.uc3m.es

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{'abstract': "In these three lectures we will discuss some fundamental aspects of the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on compact Riemannian manifolds with smooth boundary emphasizing the relation with the theory of global boundary conditions.Self-adjoint extensions of symmetric operators, specially of the Laplace-Beltrami and Dirac operators, are fundamental in Quantum Physics as they determine either the energy of quantum systems and/or their unitary evolution. The well-known von Neumann's theory of self-adjoint extensions of symmetric operators is not always easily applicable to differential operators, while the description of extensions in terms of boundary conditions constitutes a more natural approach. Thus an effort is done in offering a description of self-adjoint extensions in terms of global boundary conditions showing how an important family of self-adjoint extensions for the Laplace-Beltrami and Dirac operators are easily describable in this way.Moreover boundary conditions play in most cases an significant physical role and give rise to important physical phenomena like the Casimir effect. The geometrical and topological structure of the space of global boundary conditions determining regular self-adjoint extensions for these fundamental differential operators is described. It is shown that there is a natural homology class dual of the Maslov class of the space.A new feature of the theory that is succinctly presented here is the relation between topology change on the system and the topology of the space of selfadjoint extensions of its Hamiltonian. Some examples will be commented and the one-dimensional case will be thoroughly discussed.", 'arxivid': '1205.3579', 'author': ['A Ibort '], 'authoraffiliation': [], 'corpusid': 119154014, 'doi': '10.1063/1.4733360', 'github_urls': [], 'n_tokens_mistral': 35730, 'n_tokens_neox': 31432, 'n_words': 20108, 'pdfsha': 'bac14cccfaa98007c75b1c503742e3e29ea7db01', 'pdfurls': ['https://arxiv.org/pdf/1205.3579v1.pdf'], 'title': ['THREE LECTURES ON GLOBAL BOUNDARY CONDITIONS AND THE THEORY OF SELF-ADJOINT EXTENSIONS OF THE COVARIANT LAPLACE-BELTRAMI AND DIRAC OPERATORS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY FALL WORKSHOP ON GEOMETRY AND PHYSICS, ICMAT 2011', 'THREE LECTURES ON GLOBAL BOUNDARY CONDITIONS AND THE THEORY OF SELF-ADJOINT EXTENSIONS OF THE COVARIANT LAPLACE-BELTRAMI AND DIRAC OPERATORS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY FALL WORKSHOP ON GEOMETRY AND PHYSICS, ICMAT 2011'], 'venue': []}

arxiv

Exponential lower bound for Berge-Ramsey problems 10 Jun 2019 June 12, 2019 Dömötör Pálvölgyi Institute of Mathematics MTA-ELTE Lendület Combinatorial Geometry Research Group Eötvös Loránd Uni-versity (ELTE) BudapestHungary Exponential lower bound for Berge-Ramsey problems 10 Jun 2019 June 12, 2019arXiv:1906.04288v1 [math.CO] Gerbner and Palmer[4], generalizing the definition of hypergraph cycles due to Berge, introduced the following notion. A hypergraph H contains a Berge copy of a graph G, if there are injections Ψ 1 : V (G) → V (H) and Ψ 2 : E(G) → E(H) such that for every edge uv ∈ E(G) the containment Ψ 1 (u), Ψ 1 (v) ∈ Ψ 2 (uv) holds, i.e., each graph edge can be mapped into a distinct hyperedge containing it to create a copy of G. If |E(H)| = |E(G)|, then we say that H is a Berge-G, and we denote such hypergraphs by BG.The study of Ramsey problems for such hypergraphs started independently in 2018 by three groups of authors[1,3,5]. Denote by R r (BG; c) the size of the smallest N such that no matter how we c-color the r-edges of K r N , the complete r-uniform hypergraph, we can always find a monochromatic BG. In [1] R r (BK n ; c) was studied for n = 3, 4. In[3]it was conjectured that R r (BK n ; c) is bounded by a polynomial of n (depending on r and c), and they showed that R r (BK n ; c) = n if r > 2c and R r (BK n ; c) = n + 1 if r = 2c, while R 3 (BK n ; 2) < 2n (also proved in [5]). In [5] a superlinear lower bound was shown for r = c = 3 and for every other r for large enough c. This was improved in [2] to R r (BK n ; c) = Ω(n d ) if c > (d − 1) r 2 and R r (BK n ; c) = Ω(n 1+1/(r−2) / log n). We further improve these to disprove the conjecture of [3].Theorem. R r (BK n ; c) > (1 + 1 r 2 ) n−1 if c > r 2 .Proof. It is enough to prove the statement for c = r 2 + 1. For r = 2 this reduces to the classical Ramsey's theorem, so we can assume r ≥ 3. We can also suppose n ≥ r 2 + 1 = c, or the lower bound becomes trivial. Suppose N ≤ (1 + 1 r 2 ) n−1 . Assign randomly (uniformly and independently) a forbidden color to every pair of vertices in K r N . Color the r-edges of K r N arbitrarily, respecting the following rule: if {u, v} ⊂ E, then the color of E cannot be the forbidden color of {u, v}. Since c > r 2 , this leaves at least one choice for each edge. Following the classic proof of the lower bound of the Ramsey's theorem, now we calculate the probability of having a monochromatic BK n . The chance of a monochromatic BK n on a fixed set of n vertices for a fixed color is at most ( c−1 c ) ( n 2 ) , as the fixed color cannot be the forbidden one on any of the pairs of vertices. Thus the expected number of monochromatic BK n 's is at most c N n ( c−1 c ) ( n 2 ) . If this quantity is less than 1, then we know that a suitable coloring exists. Since c ≤ n ≤ n!, it is enough to show that N < ( c c−1 ) n−1 2 , but this is true using c = r 2 + 1 and r ≥ 3.It is an interesting problem to determine how R r (BK n ; c) behaves if c ≤ r 2 . Gerbner and Palmer [4], generalizing the definition of hypergraph cycles due to Berge, introduced the following notion. A hypergraph H contains a Berge copy of a graph G, if there are injections Ψ 1 : V (G) → V (H) and Ψ 2 : E(G) → E(H) such that for every edge uv ∈ E(G) the containment Ψ 1 (u), Ψ 1 (v) ∈ Ψ 2 (uv) holds, i.e., each graph edge can be mapped into a distinct hyperedge containing it to create a copy of G. If |E(H)| = |E(G)|, then we say that H is a Berge-G, and we denote such hypergraphs by BG. The study of Ramsey problems for such hypergraphs started independently in 2018 by three groups of authors [1,3,5]. Denote by R r (BG; c) the size of the smallest N such that no matter how we c-color the r-edges of K r N , the complete r-uniform hypergraph, we can always find a monochromatic BG. In [1] R r (BK n ; c) was studied for n = 3, 4. In [3] it was conjectured that R r (BK n ; c) is bounded by a polynomial of n (depending on r and c), and they showed that R r (BK n ; c) = n if r > 2c and R r (BK n ; c) = n + 1 if r = 2c, while R 3 (BK n ; 2) < 2n (also proved in [5]). In [5] a superlinear lower bound was shown for r = c = 3 and for every other r for large enough c. This was improved in [2] to R r (BK n ; c) = Ω(n d ) if c > (d − 1) r 2 and R r (BK n ; c) = Ω(n 1+1/(r−2) / log n). We further improve these to disprove the conjecture of [3]. Theorem. R r (BK n ; c) > (1 + 1 r 2 ) n−1 if c > r 2 . Proof. It is enough to prove the statement for c = r 2 + 1. For r = 2 this reduces to the classical Ramsey's theorem, so we can assume r ≥ 3. We can also suppose n ≥ r 2 + 1 = c, or the lower bound becomes trivial. Suppose N ≤ (1 + 1 r 2 ) n−1 . Assign randomly (uniformly and independently) a forbidden color to every pair of vertices in K r N . Color the r-edges of K r N arbitrarily, respecting the following rule: if {u, v} ⊂ E, then the color of E cannot be the forbidden color of {u, v}. Since c > r 2 , this leaves at least one choice for each edge. Following the classic proof of the lower bound of the Ramsey's theorem, now we calculate the probability of having a monochromatic BK n . The chance of a monochromatic BK n on a fixed set of n vertices for a fixed color is at most ( c−1 c ) ( n 2 ) , as the fixed color cannot be the forbidden one on any of the pairs of vertices. Thus the expected number of monochromatic BK n 's is at most c N n ( c−1 c ) ( n 2 ) . If this quantity is less than 1, then we know that a suitable coloring exists. Since c ≤ n ≤ n!, it is enough to show that N < ( c c−1 ) n−1 2 , but this is true using c = r 2 + 1 and r ≥ 3. * MTA-ELTE Lendület Combinatorial Geometry Research Group, Institute of Mathematics, Eötvös Loránd University (ELTE), Budapest, Hungary M Axenovich, A Gyárfás, arXiv:1807.10062A note on Ramsey numbers for Berge-G hypergraphs. M. Axenovich, A. Gyárfás, A note on Ramsey numbers for Berge-G hypergraphs, arXiv:1807.10062. D Gerbner, arXiv:1906.02465On Berge-Ramsey problems. D. Gerbner, On Berge-Ramsey problems, arXiv:1906.02465. D Gerbner, A Methuku, G Omidi, M Vizer, arXiv:1808.10434Ramsey problems for Berge hypergraphs. D. Gerbner, A. Methuku, G. Omidi, M. Vizer, Ramsey problems for Berge hypergraphs, arXiv:1808.10434. Extremal Results for Berge Hypergraphs. D Gerbner, C Palmer, SIAM Journal on Discrete Mathematics. 314D. Gerbner, C. Palmer, Extremal Results for Berge Hypergraphs, SIAM Journal on Discrete Mathematics 31(4): 2314-2327, 2017. N Salia, C Tompkins, Z Wang, O Zamora, arXiv:1808.09863Ramsey numbers of Berge-hypergraphs. arXiv preprintN. Salia, C. Tompkins, Z. Wang, O. Zamora, Ramsey numbers of Berge-hypergraphs, arXiv preprint arXiv:1808.09863.

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{'abstract': "Gerbner and Palmer[4], generalizing the definition of hypergraph cycles due to Berge, introduced the following notion. A hypergraph H contains a Berge copy of a graph G, if there are injections Ψ 1 : V (G) → V (H) and Ψ 2 : E(G) → E(H) such that for every edge uv ∈ E(G) the containment Ψ 1 (u), Ψ 1 (v) ∈ Ψ 2 (uv) holds, i.e., each graph edge can be mapped into a distinct hyperedge containing it to create a copy of G. If |E(H)| = |E(G)|, then we say that H is a Berge-G, and we denote such hypergraphs by BG.The study of Ramsey problems for such hypergraphs started independently in 2018 by three groups of authors[1,3,5]. Denote by R r (BG; c) the size of the smallest N such that no matter how we c-color the r-edges of K r N , the complete r-uniform hypergraph, we can always find a monochromatic BG. In [1] R r (BK n ; c) was studied for n = 3, 4. In[3]it was conjectured that R r (BK n ; c) is bounded by a polynomial of n (depending on r and c), and they showed that R r (BK n ; c) = n if r > 2c and R r (BK n ; c) = n + 1 if r = 2c, while R 3 (BK n ; 2) < 2n (also proved in [5]). In [5] a superlinear lower bound was shown for r = c = 3 and for every other r for large enough c. This was improved in [2] to R r (BK n ; c) = Ω(n d ) if c > (d − 1) r 2 and R r (BK n ; c) = Ω(n 1+1/(r−2) / log n). We further improve these to disprove the conjecture of [3].Theorem. R r (BK n ; c) > (1 + 1 r 2 ) n−1 if c > r 2 .Proof. It is enough to prove the statement for c = r 2 + 1. For r = 2 this reduces to the classical Ramsey's theorem, so we can assume r ≥ 3. We can also suppose n ≥ r 2 + 1 = c, or the lower bound becomes trivial. Suppose N ≤ (1 + 1 r 2 ) n−1 . Assign randomly (uniformly and independently) a forbidden color to every pair of vertices in K r N . Color the r-edges of K r N arbitrarily, respecting the following rule: if {u, v} ⊂ E, then the color of E cannot be the forbidden color of {u, v}. Since c > r 2 , this leaves at least one choice for each edge. Following the classic proof of the lower bound of the Ramsey's theorem, now we calculate the probability of having a monochromatic BK n . The chance of a monochromatic BK n on a fixed set of n vertices for a fixed color is at most ( c−1 c ) ( n 2 ) , as the fixed color cannot be the forbidden one on any of the pairs of vertices. Thus the expected number of monochromatic BK n 's is at most c N n ( c−1 c ) ( n 2 ) . If this quantity is less than 1, then we know that a suitable coloring exists. Since c ≤ n ≤ n!, it is enough to show that N < ( c c−1 ) n−1 2 , but this is true using c = r 2 + 1 and r ≥ 3.It is an interesting problem to determine how R r (BK n ; c) behaves if c ≤ r 2 .", 'arxivid': '1906.04288', 'author': ['Dömötör Pálvölgyi \nInstitute of Mathematics\nMTA-ELTE Lendület Combinatorial Geometry Research Group\nEötvös Loránd Uni-versity (ELTE)\nBudapestHungary\n'], 'authoraffiliation': ['Institute of Mathematics\nMTA-ELTE Lendület Combinatorial Geometry Research Group\nEötvös Loránd Uni-versity (ELTE)\nBudapestHungary'], 'corpusid': 184486858, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2438, 'n_tokens_neox': 2111, 'n_words': 1286, 'pdfsha': '4f41d1033c3d8457faad65e50d0063c9cbaaedd5', 'pdfurls': ['https://arxiv.org/pdf/1906.04288v2.pdf'], 'title': ['Exponential lower bound for Berge-Ramsey problems', 'Exponential lower bound for Berge-Ramsey problems'], 'venue': []}

arxiv

Self-critical Sequence Training for Image Captioning Steven J Rennie sjrennie@us.ibm.com Watson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center NYUSA Etienne Marcheret etiennem@us.ibm.com Watson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center NYUSA Youssef Mroueh mroueh@us.ibm.com Watson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center NYUSA Jarret Ross rossja@us.ibm.com Watson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center NYUSA Vaibhava Goel vgoel@us.ibm.com Watson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center NYUSA Self-critical Sequence Training for Image Captioning Recently it has been shown that policy-gradient methods for reinforcement learning can be utilized to train deep endto-end systems directly on non-differentiable metrics for the task at hand. In this paper we consider the problem of optimizing image captioning systems using reinforcement learning, and show that by carefully optimizing our systems using the test metrics of the MSCOCO task, significant gains in performance can be realized. Our systems are built using a new optimization approach that we call self-critical sequence training (SCST). SCST is a form of the popular RE-INFORCE algorithm that, rather than estimating a "baseline" to normalize the rewards and reduce variance, utilizes the output of its own test-time inference algorithm to normalize the rewards it experiences. Using this approach, estimating the reward signal (as actor-critic methods must do) and estimating normalization (as REINFORCE algorithms typically do) is avoided, while at the same time harmonizing the model with respect to its test-time inference procedure. Empirically we find that directly optimizing the CIDEr metric with SCST and greedy decoding at test-time is highly effective. Our results on the MSCOCO evaluation sever establish a new state-of-the-art on the task, improving the best result in terms of CIDEr from 104.9 to 112.3. Introduction Image captioning aims at generating a natural language description of an image. Open domain captioning is a very challenging task, as it requires a fine-grained understanding of the global and the local entities in an image, as well as their attributes and relationships. The recently released MSCOCO challenge [1] provides a new, larger scale platform for evaluating image captioning systems, complete with an evaluation server for benchmarking competing methods. Deep learning approaches to sequence modeling have yielded impressive results on the task, dominating the task leaderboard. Inspired by the recently introduced encoder/decoder paradigm for machine translation using recurrent neural networks (RNNs) [2], [3], and [4] use a deep convolutional neural network (CNN) to encode the input image, and a Long Short Term Memory (LSTM) [5] RNN decoder to generate the output caption. These systems are trained end-to-end using back-propagation, and have achieved state-of-the-art results on MSCOCO. More recently in [6], the use of spatial attention mechanisms on CNN layers to incorporate visual context-which implicitly conditions on the text generated so far-was incorporated into the generation process. It has been shown and we have qualitatively observed that captioning systems that utilize attention mechanisms lead to better generalization, as these models can compose novel text descriptions based on the recognition of the global and local entities that comprise images. As discussed in [7], deep generative models for text are typically trained to maximize the likelihood of the next ground-truth word given the previous ground-truth word using back-propagation. This approach has been called "Teacher-Forcing" [8]. However, this approach creates a mismatch between training and testing, since at test-time the model uses the previously generated words from the model distribution to predict the next word. This exposure bias [7], results in error accumulation during generation at test time, since the model has never been exposed to its own predictions. Several approaches to overcoming the exposure bias problem described above have recently been proposed. In [8] they show that feeding back the model's own predictions and slowly increasing the feedback probability p during training leads to significantly better test-time performance. Another line of work proposes "Professor-Forcing" [9], a technique that uses adversarial training to encourage the dynamics of the recurrent network to be the same when training conditioned on ground truth previous words and when sampling freely from the network. While sequence models are usually trained using the cross entropy loss, they are typically evaluated at test time using discrete and non-differentiable NLP metrics such as BLEU [10], ROUGE [11], METEOR [12] or CIDEr [13]. Ideally sequence models for image captioning should be trained to avoid exposure bias and directly optimize metrics for the task at hand. Recently it has been shown that both the exposure bias and non-differentiable task metric issues can be addressed by incorporating techniques from Reinforcement Learning (RL) [14]. Specifically in [7], Ranzato et al. use the REINFORCE algorithm [15] to directly optimize nondifferentiable, sequence-based test metrics, and overcome both issues. REINFORCE as we will describe, allows one to optimize the gradient of the expected reward by sampling from the model during training, and treating those samples as ground-truth labels (that are re-weighted by the reward they deliver). The major limitation of the approach is that the expected gradient computed using mini-batches under REINFORCE typically exhibit high variance, and without proper context-dependent normalization, is typically unstable. The recent discovery that REINFORCE with proper bias correction using learned "baselines" is effective has led to a flurry of work in applying REINFORCE to problems in RL, supervised learning, and variational inference [16,17,18]. Actor-critic methods [14] , which instead train a second "critic" network to provide an estimate of the value of each generated word given the policy of an actor network, have also been investigated for sequence problems recently [19]. These techniques overcome the need to sample from the policy's (actors) action space, which can be enormous, at the expense of estimating future rewards, and training multiple networks based on one another's outputs, which as [19] explore, can also be unstable. In this paper we present a new approach to sequence training which we call self-critical sequence training (SCST), and demonstrate that SCST can improve the performance of image captioning systems dramatically. SCST is a REINFORCE algorithm that, rather than estimating the reward signal, or how the reward signal should be normalized, utilizes the output of its own test-time inference algorithm to normalize the rewards it experiences. As a result, only samples from the model that outperform the current test-time system are given positive weight, and inferior samples are suppressed. Using SCST, attempting to estimate the reward signal, as actor-critic methods must do, and estimating normalization, as REINFORCE algorithms must do, is avoided, while at the same time harmonizing the model with respect to its test-time inference procedure. Empirically we find that directly optimizing the CIDEr metric with SCST and greedy decoding at test-time is highly effective. Our results on the MSCOCO evaluation sever establish a new state-of-the-art on the task, improving the best result in terms of CIDEr from 104.9 to 112.3. Captioning Models In this section we describe the recurrent models that we use for caption generation. FC models. Similarly to [3,4], we first encode the input image F using a deep CNN, and then embed it through a linear projection W I . Words are represented with one hot vectors that are embedded with a linear embedding E that has the same output dimension as W I . The beginning of each sentence is marked with a special BOS token, and the end with an EOS token. Under the model, words are generated and then fed back into the LSTM, with the image treated as the first word W I CN N (F ). The following updates for the hidden units and cells of an LSTM define the model [5]: x t = E1 wt−1 for t > 1, x 1 = W I CN N (F ) i t = σ (W ix x t + W ih h t−1 + b i ) (Input Gate) f t = σ (W f x x t + W f h h t−1 + b f ) (Forget Gate) o t = σ (W ox x t + W oh h t−1 + b o ) (Output Gate) c t = i t φ(W ⊗ zx x t + W ⊗ zh h t−1 + b ⊗ z ) + f t c t−1 h t = o t tanh(c t ) s t = W s h t , where φ is a maxout non-linearity with 2 units (⊗ denotes the units) and σ is the sigmoid function. We initialize h 0 and c 0 to zero. The LSTM outputs a distribution over the next word w t using the softmax function: w t ∼ softmax (s t )(1) In our architecture, the hidden states and word and image embeddings have dimension 512. Let θ denote the parameters of the model. Traditionally the parameters θ are learned by maximizing the likelihood of the observed sequence. Specifically, given a target ground truth sequence {w * 1 , . . . , w * T }, the objective is to minimize the cross entropy loss (XE): L(θ) = − T t=1 log(p θ (w * t |w * 1 , . . . w * t−1 )),(2) where p θ (w t |w 1 , . . . w t−1 ) is given by the parametric model in Equation (1). Attention Model (Att2in). Rather than utilizing a static, spatially pooled representation of the image, attention models dynamically re-weight the input spatial (CNN) features to focus on specific regions of the image at each time step. In this paper we modify the architecture of the attention model for captioning given in [6], and input the attentionderived image feature only to the cell node of the LSTM. We have found that this architecture outperforms other designs when ADAM [20] is used for optimization. x t = E1 wt−1 for t ≥ 1 w 0 = BOS i t = σ (W ix x t + W ih h t−1 + b i ) (Input Gate) f t = σ (W f x x t + W f h h t−1 + b f ) (Forget Gate) o t = σ (W ox x t + W oh h t−1 + b o ) (Output Gate) c t = i t φ(W ⊗ zx x t + W ⊗ zI I t + W ⊗ zh h t−1 + b ⊗ z ) + f t c t−1 h t = o t tanh(c t ) s t = W s h t , where I t is the attention-derived image feature. This feature is derived as in [6] as follows: given CNN features at N locations {I 1 , . . . I N }, I t = N i=1 α i t I i , where α t = softmax(a t + b α ), and a i t = W tanh(W aI I i + W ah h t−1 + b a ). In this work we set the dimension of W to 1 × 512, and set c 0 and h 0 to zero. Let θ denote the parameters of the model. Then p θ (w t |w 1 , . . . w t−1 ) is again defined by (1). The parameters θ of attention models are also traditionally learned by optimizing the XE loss (2). Reinforcement Learning Sequence Generation as an RL problem. As described in the previous section, captioning systems are traditionally trained using the cross entropy loss. To directly optimize NLP metrics and address the exposure bias issue, we can cast our generative models in the Reinforcement Learning terminology as in [7]. Our recurrent models (LSTM) introduced above can be viewed as an "agent" that interacts with an external "environment" (words and image features). The parameters of the network, θ, define a policy p θ , that results in an "action" that is the prediction of the next word. After each action, the agent (the LSTM) updates its internal "state" (cells and hidden states of the LSTM, attention weights etc). Upon generating the end-of-sequence (EOS) token, the agent observes a "reward" that is, for instance, the CIDEr score of the generated sentence-we denote this reward by r. The reward is computed by an evaluation metric by comparing the generated sequence to corresponding ground-truth sequences. The goal of training is to minimize the negative expected reward: L(θ) = −E w s ∼p θ [r(w s )] ,(3) where w s = (w s 1 , . . . w s T ) and w s t is the word sampled from the model at the time step t. In practice L(θ) is typically estimated with a single sample from p θ : L(θ) ≈ −r(w s ), w s ∼ p θ . Policy Gradient with REINFORCE. In order to compute the gradient ∇ θ L(θ), we use the REINFORCE algorithm [15](See also Chapter 13 in [14]). REINFORCE is based on the observation that the expected gradient of a nondifferentiable reward function can be computed as follows: ∇ θ L(θ) = −E w s ∼p θ [r(w s )∇ θ log p θ (w s )] .(4) In practice the expected gradient can be approximated using a single Monte-Carlo sample w s = (w s 1 . . . w s T ) from p θ , for each training example in the minibatch: ∇ θ L(θ) ≈ −r(w s )∇ θ log p θ (w s ). REINFORCE with a Baseline. The policy gradient given by REINFORCE can be generalized to compute the reward associated with an action value relative to a reference reward or baseline b: ∇ θ L(θ) = −E w s ∼p θ [(r(w s ) − b)∇ θ log p θ (w s )] . (5) The baseline can be any arbitrary function, as long as it does not depend on the "action" w s [14] since in this case: E w s ∼p θ [b∇ θ log p θ (w s )] = b ws ∇ θ p θ (w s ) = b∇ θ ws p θ (w s ) = b∇ θ 1 = 0.(6) This shows that the baseline does not change the expected gradient, but importantly, it can reduce the the variance of the gradient estimate. For each training case, we again approximate the expected gradient with a single sample w s ∼ p θ : ∇ θ L(θ) ≈ −(r(w s ) − b)∇ θ log p θ (w s ).(7) Note that if b is function of θ or t as in [7], equation (6) still holds and b(θ) is a valid baseline. Final Gradient Expression. Using the chain rule, and the parametric model of p θ given in Section 2 we have: ∇ θ L(θ) = T t=1 ∂L(θ) ∂s t ∂s t ∂θ , where s t is the input to the softmax function. Using RE-INFORCE with a baseline b the estimate of the gradient of ∂L(θ) ∂st is given by [17]: ∂L(θ) ∂s t ≈ (r(w s ) − b)(p θ (w t |h t ) − 1 w s t ).(8) Self-critical sequence training (SCST) The central idea of the self-critical sequence training (SCST) approach is to baseline the REINFORCE algorithm with the reward obtained by the current model under the inference algorithm used at test time. The gradient of the negative reward of a sample w s from the model w.r.t. to the softmax activations at time-step t then becomes: ∂L(θ) ∂s t = (r(w s ) − r(ŵ))(p θ (w t |h t ) − 1 w s t ).(9)(r(w s 1 , . . . , w s T ) r(ŵ 1 , . . . ,ŵ T )) r ✓ log p ✓ (w s 1 , . . . , w s T ) BOS p✓(w|h1) p✓(w|h2) CIDER (w s 1 , . . . , w s T ) r(w s 1 , . . . , w s T ) w s 1 w s 2 h0, c0 h1, c1 = LST M (BOS, h0, c0) h2, c2 = LST M (w s 1 , h1, c1) {(w ⇤ 1 , . . . , w ⇤ T )} sample sample BOS CIDER arg max arg max p ✓ (w|h 0 1 ) p✓(w|h 0 2 ) (ŵ 1 , . . . ,ŵ T ) r(ŵ 1 , . . . ,ŵ T ) w 1ŵ2 h 0 0 , c 0 0 h 0 1 , c 0 1 = LST M (BOS, h 0 0 , c 0 0 ) h 0 2 , c 0 2 = LST M (ŵ1, h 0 1 , c 0 1 ) {(w ⇤ 1 , . . . , w ⇤ T )} arg max arg max Figure 1: Self-critical sequence training (SCST). The weight put on words of a sampled sentence from the model is determined by the difference between the reward for the sampled sentence and the reward obtained by the estimated sentence under the test-time inference procedure (greedy inference depicted). This harmonizes learning with the inference procedure, and lowers the variance of the gradients, improving the training procedure. where r(ŵ) again is the reward obtained by the current model under the inference algorithm used at test time. Accordingly, samples from the model that return higher reward thanŵ will be "pushed up", or increased in probability, while samples which result in lower reward will be suppressed. Like MIXER [7], SCST has all the advantages of REINFORCE algorithms, as it directly optimizes the true, sequence-level, evaluation metric, but avoids the usual scenario of having to learn a (context-dependent) estimate of expected future rewards as a baseline. In practice we have found that SCST has much lower variance, and can be more effectively trained on mini-batches of samples using SGD. Since the SCST baseline is based on the test-time estimate under the current model, SCST is forced to improve the performance of the model under the inference algorithm used at test time. This encourages training/test time consistency like the maximum likelihood-based approaches "Data as Demonstrator" [8], "Professor Forcing" [9], and E2E [7], but importantly, it can directly optimize sequence metrics. Finally, SCST is self-critical, and so avoids all the inherent training difficulties associated with actor-critic methods, where a second "critic" network must be trained to estimate value functions, and the actor must be trained on estimated value functions rather than actual rewards. In this paper we focus on scenario of greedy decoding, where:ŵ t = arg max wt p(w t | h t )(10) This choice, depicted in Figure 1, has several practical advantages. First and foremost, it minimizes the impact of baselining with the test-time inference algorithm on train-ing time, since it requires only one additional forward pass, and trains the system to be optimized for fast, greedy decoding at test-time. This choice may also be among the best forms of SCST based on a single test-time estimate, as suppressing all samples that underperform relative to the final test-time estimate will tend to favor a very decisive policy. The investigation of forms of SCST that incorporate margin, utilize more than 1 test-time estimate (e.g. an n-best list) to baseline, and/or more elaborate test-time inference procedures (e.g. beam search) are interesting possible directions of future work. Experiments Dataset. We evaluate our proposed method on the MSCOCO dataset [1]. For offline evaluation purposes we used the data splits from [21]. The training set contains 113, 287 images, along with 5 captions each. We use a set of 5K image for validation and report results on a test set of 5K images as well, as given in [21]. We report four widely used automatic evaluation metrics, BLEU-4, ROUGEL, METEOR, and CIDEr. We prune the vocabulary and drop any word that has count less then five, we end up with a vocabulary of size 10096 words. Image Features 1) FC Models. We use two type of Features: a) (FC-2k) features, where we encode each image with Resnet-101 (101 layers) [22]. Note that we do not rescale or crop each image. Instead we encode the full image with the final convolutional layer of resnet, and apply average pooling, which results in a vector of dimension 2048. b) (FC-15K) features where we stack average pooled 13 layers of Resnet-101 (11 × 1024 and 2 × 2048). These 13 layers are the odd layers of conv4 and conv5, with the exception of the 23rd layer of conv4, which was omitted. This results in a feature vector of dimension 15360. 2) Spatial CNN features for Attention models: (Att2in) We encode each image using the residual convolutional neural network Resnet-101 [22]. Note that we do not rescale or crop the image. Instead we encode the full image with the final convolutional layer of Resnet-101, and apply spatially adaptive average pooling so that the output has a fixed size of 14 × 14 × 2048. At each time step the attention model produces an attention mask over the 96 spatial locations. This mask is applied and then the result is spatially averaged to produce a 2048 dimension representation of the attended portion of the image. Implementation Details. The LSTM hidden, image, word and attention embeddings dimension are fixed to 512 for all of the models discussed herein. All of our models are trained according to the following recipe, except where otherwise noted. We initialize all models by training the model under the XE objective using ADAM [20] optimizer with an initial learning rate of 5 × 10 −4 . We anneal the learning rate by a factor of 0.8 every three epochs, and increase the probability of feeding back a sample of the word posterior by 0.05 every 5 epochs until we reach a feedback probability 0.25 [8]. We evaluate at each epoch the model on the development set and select the model with best CIDEr score as an initialization for SCST training. We then run SCST training initialized with the XE model to optimize the CIDEr metric (specifically, the CIDEr-D metric) using ADAM with a learning rate 5 × 10 −5 . Initially when experimenting with FC-2k and FC-15k models we utilized curriculum learning (CL) during training, as proposed in [7], by increasing the number of words that are sampled and trained under CIDEr by one each epoch (the prefix of the sentence remains under the XE criterion until eventually being subsumed). We have since realized that for the MSCOCO task CL is not required, and provides little to no boost in performance. The results reported here for the FC-2K and FC-15K models are trained with CL, while the attention models were trained directly on the entire sentence for all epochs after being initialized by the XE seed models. Offline Evaluation Evaluating different RL training strategies. Table 1 compares the performance of SCST to that of MIXER [7] on the test portion of the Karpathy splits. In this experiment, we utilize "curriculum learning" (CL) by optimizing the expected reward of the metric on the last n words of each training sentence, optimizing XE on the remaining sentence prefix, and slowly increasing n until the entire sentence is being sampled for all training cases. The results reported above were generated with a CL schedule matching the optimized schedule reported in [7]. Interestingly we found that CL was not necessary to successfully train both SCST and REINFORCE with a learned baseline on the MSCOCO dataset-equally good results relative to not applying CL could be obtained by both a learned baseline, as was done in [7], and SCST. The gain of using SCST over using a learned baseline was consistently about 4 CIDEr points, regardless of the CL schedule (or lack thereof), and the initialization seed. to the best performance on that same metric at test time, an expected result. We experimented with training on multiple test metrics, and found that we were unable to outperform the overall performance of the model trained only on the CIDEr metric, which lifts the performance of all other metrics considerably. For this reason most of our experimentation has since focused on optimizing CIDEr. Single FC-Models Versus Attention Models. We trained FC models (2K and 15 K), as well as attention models using SCST with the CIDEr metric. We trained 4 different models for each FC and attention, starting the optimization from four different random seeds 1 . We report in Table 3, the system with best performance for each family of models on the test portion of Karpathy splits [21]. We see that the FC-15K models outperform the FC-2K models. Both FC models are outperformed by the attention model, that establishes a new state of the art for a single model performance on Karpathy splits. Note that this quantitative evaluation favors attention models is inline with our observation that attention models tend to generalize better and compose outside of the context Model Ensembling. In this section we use an ensemble of the four models mentioned above and trained using SCST in the FC and the attention modeling. We see in Table 5. Table 5 reports the performance of the 4 ensembled attention models trained with self-critical sequence training (SCST) on the official MSCOCO evaluation server. The previous best result on the leaderboard (as of Nov. 15,2016) is also depicted. We outperform the previous best system on all evaluation metrics. Online Evaluation on MS-COCO Server Example of Generated Captions Here we provide a qualitative example of the captions generated by our systems for the input image in figure 2. This picture is taken from the objects out-of-context (OOOC) dataset of images from [24], and depicts a boat situated in an unusual context, and tests the ability of our models to compose descriptions of images that differ from those seen during training. The top 5 captions returned by the XE and SCST-trained FC-2K, FC-15K, and attention model ensembles when deployed with a decoding "beam" of 5 are depicted in figure 3 2 . On this image the FC models fail completely, and the SCST-trained ensemble of attention models is the only system that is able to correctly describe the image. In general we found that the performance of all captioning systems on MSCOCO data is qualitatively similar, while on images containing objects situated in an uncommon context [24] (i.e. unlike the MSCOCO training set) our attention models perform much better, and SCSTtrained attention models output yet more accurate and descriptive captions. In general we qualitatively found that SCST-trained attention models describe images more accurately, and with higher confidence, as reflected in Figure 2, where the average of the log-likelihoods of the words in each generated caption are also depicted. Additional examples, including an example with the corresponding heatmaps generated by the SCST-trained ensemble of attention models, can be found in the supplementary material (figure 7 of section C). Discussion and Future Work In this paper we have presented a simple and efficient approach to more effectively baselining the REINFORCE algorithm for policy-gradient based RL, which allows us to more effectively train on non-differentiable metrics, and leads to significant improvements in captioning performance on MSCOCO-our results on the MSCOCO evaluation sever establish a new state-of-the-art on the task. The self-critical approach, which normalizes the reward obtained by sampled sentences with the reward obtained by the model under the test-time inference algorithm is intuitive, and avoids having to estimate any state-dependent or independent reward functions. Extensions of SCST that incorporate margin or utilize more than 1 test-time estimate (e.g. an n-best list) to baseline, and/or more elaborate testtime inference procedures (e.g. beam search) are interesting possible directions of future work. A. Beam search Throughout the paper and in this supplementary material we often refer to caption results and evaluation metric results obtained using "beam search". This section briefly summarizes our beam search procedure. While decoding the image to generate captions that describe it, rather than greedily selecting the most probable word (N = 1), we can maintain a list of the N most probable sub-sequences generated so far, generate posterior probabilities for the next word of each of these subsequences, and then again prune down to the N -best sub-sequences. This approach is widely referred to as a beam search, where N is the width of the decoding "beam". In our experiments we additionally prune away hypotheses within the N -best list that have a log probability that is below that of the maximally probable partial sentence by more than ∆ log = 20. For all reported results, the value of N is tuned on a per-model basis on the validation set (of the Karpathy splits). On MSCOCO data, N = 2 is typically optimal for cross-entropy (XE) trained models and SCST-trained models, but in the latter case beam search provides only a very small boost in performance. For our captioning demonstrations we set N = 5 for all models for illustrative purposes, and because we have qualitatively observed that for test images that are substantially different from those encountered during training, beam search is important. B. Performance of XE versus SCST trained models In tables 3 and 4 of the main text we compared the performance of models trained to optimize the CIDEr metric with self-critical sequence training (SCST) with that of their corresponding bootstrap models, which were trained under the crossentropy (XE) criterion using scheduled sampling [8]. We provide some additional details about these experiments here. For all XE models, the probability p f of feeding forward the maximally probable word rather than the ground-truth word was increased by 0.05 every 5 epochs until reaching a maximum value of 0.25. The XE model with the best performance on the validation set of the Karpathy splits was then selected as the bootstrap model for SCST. For all models, the performance of greedily decoding each word at test time is reported, as is the performance of beam search as described in the previous section. As reported in [7], we found that beam search using RL-trained models resulted in very little performance gain. Figure 4 depicts the performance of our best Att2in model, which is trained to directly optimize the CIDEr metric, as a function of training epoch and evaluation metric, on the validation portion of the Karpathy splits. Optimizing CIDEr clearly improves all of the MSCOCO evaluation metrics substantially. C. Examples of Generated Captions Figures 5-13 depict demonstrations of the captioning performance of all systems. In general we found that the performance of all captioning systems on MSCOCO data is qualitatively similar, while on images containing objects situated in an uncommon context [24] (i.e. unlike the MSCOCO training set) our attention models perform much better, and SCST-trained attention models output yet more accurate and descriptive captions. Attention heat-maps for the image and corresponding captions depicted in 5 and 13 are given in 7. The heatmaps of the attention weights are reasonably inline with the predicted words in both cases, and the SCST attention weights are spatially sharper here, and in general. D. Further details and analysis of SCST training One detail that was crucial to optimizing CIDEr to produce better models was to include the EOS tag as a word. When the EOS word was omitted, trivial sentence fragments such as "with a" and "and a" were dominating the metric gains, despite the "gaming" counter-measures (sentence length and precision clipping) that are included in CIDEr-D [13], which is what we optimized. Including the EOS tag substantially lowers the reward allocated to incomplete sentences, and completely resolved this issue. Another more obvious detail that is important is to associate the reward for the sentence with the first EOS encountered. Omitting the reward from the first EOS fails to reward sentence completion which leads to run-on, and rewarding any words that follow the first EOS token is inconsistent with the decoding procedure. This work has focused on optimizing the CIDEr metric, since, as discussed in the paper, optimizing CIDER substantially improves all MSCOCO evaluation metrics, as was shown in tables 3 and 4 and is depicted in figure 4. Nevertheless, directly optimizing another metric does lead to higher evaluation scores on that same metric as shown, and so we have started to experiment with including models trained on Bleu, Rouge-L, and METEOR in our Att2in ensemble to attempt to improve it further. So far we have not been able to substantially improve performance w.r.t. the other metrics without more substantially degrading CIDEr. Figure 8 by various models discussed in the paper. Beside each caption we report the average log probability of the words in the caption. All models perform well on this test image from the MSCOCO distribution. More generally we have observed that qualitatively, all models perform comparably on the MSCOCO test images. Figure 10 by the various models discussed in the paper. Beside each caption we report the average log probability of the words in the caption. On this image, which presents an object situated in an atypical context [24], the FC models fail to give an accurate description, while the attention models handle the previously unseen image composition well. The models trained with SCST return a more accurate and more detailed summary of the image. Figure 12 by the various models discussed in the paper. Beside each caption we report the average log probability of the words in the caption. On this image, which presents an object situated in an atypical context [24], the FC models fail to give an accurate description, while the attention models handle the previously unseen image composition well. The models trained with SCST return a more accurate and more detailed summary of the image. Figure 4 : 4Performance of our best Att2in model, which is trained to directly optimize the CIDEr metric, as a function of training epoch on the validation portion of the Karpathy splits, for the CIDEr, BLEU-4, ROUGE-L, and METEOR MSCOCO evaluation metrics. Optimizing CIDEr improves all of these evaluation metrics substantially. Figure 5 : 5Picture of a common object in MSCOCO (a giraffe) situated in an uncommon context (out of COCO domain)[24]. Figure 6 :Figure 7 : 67Captions generated by various models discussed in the paper to describe the image depicted infigure 5. Beside each caption we report the average of the log probabilities of each word, normalized by the sentence length. Notice that the attention models trained with SCST give an accurate description of this image with high confidence. Attention models trained with XE are less confident about the correct description. FC models trained with CE or SCST fail at giving an accurate description. dccxc026.pok.ibm.com:60000/upload (a) Attention heat-maps for the best model in the XE-trained ensemble of attention models, for the image depicted infigure 5. heat-maps for the best model in the SCST-trained ensemble of attention models, for the image depicted in figure 5.. Attention heat-maps. Figure 8 : 8An image from the MSCOCO test set (Karpathy splits). Figure 9 : 9Captions generated for the image depicted in Figure 10 : 10An image from the objects out-of-context (OOOC) dataset of images from[24]. Figure 11 : 11Captions generated for the image depicted in Figure 12 : 12An image from the objects out-of-context (OOOC) dataset of images from[24]. Figure 13 : 13Captions generated for the image depicted in Table 2 : 2Training on different metrics.We experimented with training directly on the evaluation metrics of the MSCOCO challenge. Results for FC-2K models are depicted in table 2. In general we can see that optimizing for a given metric during training leadsPerformance on the test portion of the Karpathy splits [21] as a function of training metric ( FC-2K models). Optimizing the CIDEr metric increases the overall perfor- mance under the evaluation metrics the most significantly. The performance of the seed cross-entropy (XE) model is also depicted. All models were decoded greedily, with the exception of the XE beam search result, which was opti- mized to beam 3 on the validation set. Table 3 : 3Performance of the best XE and corresponding SCST-trained single models on the test portion of the Karpathy splits (best over 4 random seeds). The results obtained via the greedy decoding of each word and optimized beam search are depicted. Models learned using SCST were trained to directly optimize the CIDEr metric. Table 4 : 4Performance of Ensembled XE and SCST-trained models on the test portion of the Karpathy splits (ensem-bled over 4 random seeds). The models learned using self- critical sequence training (SCST) were trained to optimize the CIDEr metric. of the training of MSCOCO, as we will see in Section 6. Ensemble Evaluation Metric SCST models CIDEr BLEU4 ROUGEL METEOR Ens. 4 (Att2in) 112.3 34.4 55.9 26.8 Previous best 104.9 34.3 55.2 26.6 Table 5 : 5Performance of 4 ensembled attention mod- els trained with self-critical sequence training (SCST) on the official MSCOCO evaluation server (5 refer- ence captions). The previous best result on the leaderboard (as of 11/15/2016) is also depicted ( http://mscoco.org/dataset/#captions-leaderboard, Table C5, Watson Multimodal). please consult the supplementary material for additional details regarding how the models were trained. that ensembling improves performance and confirms the supremacy of attention modeling, and establishes yet another state of the art result on Karpathy splits[21]. Note that in our case we ensemble only 4 models and we don't do any fine-tuning of the Resnet. 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In CVPR, 2015. 4, 5, 6 Deep residual learning for image recognition. Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun, CVPR. 45Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, 2016. 4, 5 Oriol Vinyals, Alexander Toshev, Samy Bengio, Dumitru Erhan, Show and tell: Lessons learned from the 2015 MSCOCO image captioning challenge. PAMI. Oriol Vinyals, Alexander Toshev, Samy Bengio, and Du- mitru Erhan. Show and tell: Lessons learned from the 2015 MSCOCO image captioning challenge. PAMI, 2016. 6 Context models and out-of-context objects. Myung Jinchoi, Antonio Torralba, Alan S Willsky, 716Myung Jinchoi, Antonio Torralba, and Alan S. Willsky. Con- text models and out-of-context objects. 7, 8, 10, 12, 15, 16

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{'abstract': 'Recently it has been shown that policy-gradient methods for reinforcement learning can be utilized to train deep endto-end systems directly on non-differentiable metrics for the task at hand. In this paper we consider the problem of optimizing image captioning systems using reinforcement learning, and show that by carefully optimizing our systems using the test metrics of the MSCOCO task, significant gains in performance can be realized. Our systems are built using a new optimization approach that we call self-critical sequence training (SCST). SCST is a form of the popular RE-INFORCE algorithm that, rather than estimating a "baseline" to normalize the rewards and reduce variance, utilizes the output of its own test-time inference algorithm to normalize the rewards it experiences. Using this approach, estimating the reward signal (as actor-critic methods must do) and estimating normalization (as REINFORCE algorithms typically do) is avoided, while at the same time harmonizing the model with respect to its test-time inference procedure. Empirically we find that directly optimizing the CIDEr metric with SCST and greedy decoding at test-time is highly effective. Our results on the MSCOCO evaluation sever establish a new state-of-the-art on the task, improving the best result in terms of CIDEr from 104.9 to 112.3.', 'arxivid': '1612.00563', 'author': ['Steven J Rennie sjrennie@us.ibm.com \nWatson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center\nNYUSA\n', 'Etienne Marcheret etiennem@us.ibm.com \nWatson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center\nNYUSA\n', 'Youssef Mroueh mroueh@us.ibm.com \nWatson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center\nNYUSA\n', 'Jarret Ross rossja@us.ibm.com \nWatson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center\nNYUSA\n', 'Vaibhava Goel vgoel@us.ibm.com \nWatson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center\nNYUSA\n'], 'authoraffiliation': ['Watson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center\nNYUSA', 'Watson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center\nNYUSA', 'Watson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center\nNYUSA', 'Watson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center\nNYUSA', 'Watson Multimodal Algorithms and Engines Group IBM T.J. Watson Research Center\nNYUSA'], 'corpusid': 206594923, 'doi': '10.1109/cvpr.2017.131', 'github_urls': [], 'n_tokens_mistral': 12248, 'n_tokens_neox': 10991, 'n_words': 7140, 'pdfsha': 'dd9a9175fa952a3888da3aafba13e777588dc574', 'pdfurls': ['https://arxiv.org/pdf/1612.00563v1.pdf'], 'title': ['Self-critical Sequence Training for Image Captioning', 'Self-critical Sequence Training for Image Captioning'], 'venue': []}

arxiv

Gravity versus Quantum theory: Is electron really pointlike? 4 Apr 2011 Alexander Burinskii bur@ibrae.ac.ru Theor.Physics Laboratory NSI Russian Academy of Sciences B. Tulskaya 52115191Moscow, Russia Gravity versus Quantum theory: Is electron really pointlike? 4 Apr 2011Essay written for the Gravity Research Foundation 2011 Awards for Essays on Gravitation. (March 31, 2011) Quantum theory claims that electron is pointlike and structureless. Contrary, the consistent with Gravity Kerr-Newman (KN) electron model displays an extended structure of the Compton size r c = /m. We obtain that there is no real conflict between the extended Gravitating electron and a Quantum electron "dressed" by virtual particles. In the same time the KN model indicates new important details of the electron structure and sheds new light on some old puzzles of quantum theory. In particular, the KN Gravity predicts that electron forms a disklike vacuum bubble bounded by a closed string, which could probably be detected by the novel experiments. If it will be confirmed, it would be of primary importance for foundations of Quantum theory and unification of Quantum theory with Gravity. "Nobody understands quantum mechanics." Richard Feynman (1965), [1] Modern physics is based on Quantum theory and Gravity. The both theories are confirmed experimentally with great precision. Nevertheless, they are conflicting and cannot be unified in a whole theory. In this essay we discuss one of the principal contradictions, the question on the shape and size of electron. Quantum theory states that electron is pointlike and structureless. In particular, Frank Wilczek writes in [2]: "...There's no evidence that electrons have internal structure (and a lot of evidence against it)", while the superstring theorist Leonard Susskind notes that electron radius is "...most probably not much bigger and not much smaller than the Planck length..", [3]. 1 This point of view is supported by experimental evidences, which have not found the electron structure down to 10 −16 cm. The widespread opinion that the range of interaction for gravitational field is "tremendously weak" and becomes compatible to other forces only at Planck scale, [4], is inspired by the Schwarzschild relation r g = 2m. The Kerr geometry turns this relation into inverse one, r g ∼ J/m, which points out that the range of interaction may be extended to radius of the Kerr singular ring, a = J/m. Gravitational field of the Kerr solution concentrates in a thin vicinity of the Kerr ring, forming a type of "gravitational waveguide", or string. For electron, the Kerr field may be extended to the Compton radius r c = /(2m), which corresponds to the size of a "dressed" electron. We argue here that the Kerr string is an element of the extended electron structure. In 1968 Carter obtained that the KN solution for the charged and rotating black holes has g = 2 as that of the Dirac electron, [5,6], which initiated development of the electron models based on the KN solution [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In the units c = = G = 1, mass of electron is m ≈ 10 −22 , while a = J/m ≈ 10 22 . Therefore, a >> m, and the black hole horizons disappear, opening the Kerr singular ring which is a branch line of the twovalued Kerr spacetime. Development of the KN electron models for four decades formed severe lines of investigation: (a) First ("thin shell") model was suggested by Israel, [7], who truncated the "negative" fold of metric, forming a rotating disk spanned by the Kerr singular ring. Hamity [21] showed that the disk is rigidly rotating. (b) López [12] removed the Kerr singular ring together with negative fold, forming a rotating disklike bubble with a flat interior. (c) "Microgeon" models [8,9,10,22] evolved into 4D string models [11,13,14,15,19,23,24,25,26,27]. (d) Superconducting bag models [16,28] based either on nonlinear electrodynamics [18,29], or on the Higgs field model [16,30] (e) Gravitating soliton model [20] is development of the type (c) and (d) models. All these models unambiguously indicated Compton radius of the electron. Note, that the Compton radius plays also peculiar role in the Dirac theory, as a limit of localization of the wave packet. Localization beyond the Compton zone creates a "zitterbewegung" affecting "...such paradoxes and dilemmas, which cannot be resolved in frame of the Dirac electron theory..." (Bjorken and Drell, [31]). Dirac wrote in his Nobel Prize Lecture : "The variables α (velocity operators, AB) also give rise to some rather unexpected phenomena concerning the motion of the electron. .. It is found that an electron which seems to us to be moving slowly, must actually have a very high frequency oscillatory motion of small amplitude superposed on the regular motion which appears to us. As a result of this oscillatory motion, the velocity of the electron at any time equals the velocity of light." Mass without mass. The puzzle of "zitterbewegung" and the known processes of annihilation of the electron-positron pairs brought us in 1971 to the Wheeler "geon" model of "mass without mass" [32]. In [22] we considered a massless particle circulating around z-axis. Its local 4-momentum is lightlike, p 2 x + p 2 y + p 2 z = E 2 ,(1) while the effective mass-energy was created by an averaged orbital motion, < p 2 x > + < p 2 y >=m 2 .(2) Averaging (1) under the condition (2) yields < p 2 x + p 2 y + p 2 z >=m 2 + p 2 z = E 2 .(3) Quantum analog of this model corresponds to a wave function ψ( x, t) and operators, p →ˆ p = −i ∇,Ê = i ∂ t . From (1) and (2) one obtains two wave equations: (∂ 2 x + ∂ 2 y )ψ =m 2 ψ = (∂ 2 t − ∂ 2 z )ψ,(4) which may be separated by the ansatz ψ = M(x, y)Ψ 0 (z, t).(5) The RHS of (4) yields the usual equation for a massive particle, (∂ 2 t −∂ 2 z )Ψ 0 = m 2 Ψ 0 , and the corresponding (de Broulie) plane wave solution Ψ 0 (z, t) = exp i (zp z − Et),(6) while the LHS determines the "internal" structure factor M ν = H ν (m ρ) exp{iνφ},(7) in polar coordinates ρ, φ, where H ν (m ρ) are the Hankel functions of index ν. M ν are eigenfunctions of operatorĴ z = i ∂ φ with eigenvalues J z = ν . For electron we have J z = ± /2, ν = ±1/2, and the factor M ±1/2 = ρ −1/2 exp{i(m ρ ± 1 2 φ)}(8) creates a singular ray along z-axis, which forms a branch line, and the wave function is twovalued. There exits also the corresponding spinor model [22] generating Dirac equation from the initially massless one. The Kerr string. Principal problem of this model was the weakness of the Schwarzschild gravitational field, strength of which fails about 22 orders. The works [5,6,7] appeared as a stunning surprise, which determined all subsequent development of the type (c) models. The Kerr gravitational field is concentrated near the Kerr singular ring and forms a gravitational waveguide for traveling waves. Indeed, it was recognized soon that the Kerr singular ring is a type of gravitational string [11,13,23,25], while the traveling waves are stringy excitations. 2 It has been shown that the Kerr metric provides self -consistency of the spinning geon model [8]. First approximate solutions were considered in [9], while the exact solutions for electromagnetic excitations on the Kerr-Schild background represented a very hard problem [33] and were obtained much later [27,34,35,36]. It has been shown that any wave excitation creates some 'axial' singular ray (see Fig.2) similar to the 'axial' singular ray of the geon model. Gravitating KN soliton. The KN soliton model [20] represents a field version of the bubble model (b). Surface of the bubble is fixed by the Kerr radial coordinate r = r e = e 2 /(2m), and forms an oblate disk of the Compton radius r c ≈ a = /(2m). Gravitational field is regularized by a chiral field model, U(1) ×Ũ (1), which provides a phase transition from the external KN 'vacuum state', V ext = 0, to a flat internal 'pseudovacuum' state, V int = 0. Electromagnetic field is regularized by the Higgs mechanism of broken symmetry, similarly to other models of electroweak theory [4,37,38,39]. The model exhibits two essential peculiarities: • the Kerr ring is regularized, forming on the border of bubble a closed relativistic string of the Compton radius r c and a quantized loop of electromagnetic potential eA (str) φ dφ = −4πma, which determines total spin, J = ma = n/2, n = 1, 2, 3, ..., • the Higgs field inside the bubble forms a coherent vacuum state oscillating with frequency ω = 2m. [27] formed by the topologically coupled "circular" and "axial" strings. The KN soliton forms a regular background for stringy excitations described by the type (c) models, while the wave excitations of the Kerr string are determined by the exact time-dependent Kerr-Schild solutions, [9,27,34,36]. Does the KN model of electron contradict to Quantum Theory? It seems "yes", if one speaks on the "bare" electron. However, in accordance with QED, vacuum polarization creates in the Compton region a cloud of virtual particles forming a "dressed" electron. This region gives contribution to electron spin, and performs a procedure of renormalization, which determines physical values of the electron charge and mass, [31,40,41]. Therefore, speaking on the "dressed" electron, one can say that the real contradiction between the KN model and the Quantum electron is absent. Dynamics of the virtual particles in QED is chaotic, which allows one to separate conventionally it from the "bare"electron. On the other hand, the vacuum state inside the KN soliton model forms a coherent state, joined with the closed Kerr string. It represents an 'internal' structure which cannot be separated from a "bare" particle, but should be considered as integral whole of the extended electron. We should still comment the absence of experimental exhibitions of the electron structure. First, it may be caused by a specific complex structure of the Kerr geometry [9,14,42,43,44,45,46]: the KN solution appears as a real slice of a pointlike source positioned in complex region. 3 Fourier transform of the complex source is very similar to Fourier image of the real pointlike source, which may result in its pointlike exhibition in the momentum space. Alternative explanation (discussed in [15]) is related with the lightlike singular beams (see Fig.2.), accompanying any wave excitation of the Kerr geometry, [26,27,34,35]. Finally, the pointlike interaction may simply be related with the contact character of the string-string interactions. Conclusion: The KN gravity sheds a new light on the possible role of Gravity in the structure of Quantum theory. If the electron has really the predicted closed string on the boundary of a disklike bubble, it should apparently be detected experimentally by a novel effective tool -the "nonforward Compton scattering" [49,50,51,52]. 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{'abstract': 'Quantum theory claims that electron is pointlike and structureless. Contrary, the consistent with Gravity Kerr-Newman (KN) electron model displays an extended structure of the Compton size r c = /m. We obtain that there is no real conflict between the extended Gravitating electron and a Quantum electron "dressed" by virtual particles. In the same time the KN model indicates new important details of the electron structure and sheds new light on some old puzzles of quantum theory. In particular, the KN Gravity predicts that electron forms a disklike vacuum bubble bounded by a closed string, which could probably be detected by the novel experiments. If it will be confirmed, it would be of primary importance for foundations of Quantum theory and unification of Quantum theory with Gravity.', 'arxivid': '1104.0573', 'author': ['Alexander Burinskii bur@ibrae.ac.ru \nTheor.Physics Laboratory\nNSI\nRussian Academy of Sciences\nB. Tulskaya 52115191Moscow, Russia\n'], 'authoraffiliation': ['Theor.Physics Laboratory\nNSI\nRussian Academy of Sciences\nB. Tulskaya 52115191Moscow, Russia'], 'corpusid': 119241197, 'doi': '10.1063/1.3688948', 'github_urls': [], 'n_tokens_mistral': 8176, 'n_tokens_neox': 6638, 'n_words': 3446, 'pdfsha': 'd0f4606bf5e54ec6852ad3c8bf4e52c66dbcb226', 'pdfurls': ['https://arxiv.org/pdf/1104.0573v1.pdf'], 'title': ['Gravity versus Quantum theory: Is electron really pointlike?', 'Gravity versus Quantum theory: Is electron really pointlike?'], 'venue': []}

arxiv

Hiding Charge in a Wormhole 23 Oct 2011 February 1, 2013 Eduardo Guendelman Department of Physics Ben-Gurion University of the Negev P.O.Box 653IL-84105Beer-ShevaIsrael Alexander Kaganovich Department of Physics Ben-Gurion University of the Negev P.O.Box 653IL-84105Beer-ShevaIsrael Emil Nissimov Institute for Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences Boul. Tsarigradsko Chausee 72BG-1784SofiaBulgaria Svetlana Pacheva Institute for Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences Boul. Tsarigradsko Chausee 72BG-1784SofiaBulgaria Hiding Charge in a Wormhole 23 Oct 2011 February 1, 2013Preprint submitted to TONPPJgeneralized Levi-Civita-Bertotti-Robinson spaceswormholes connecting non-compact with compactified "universes"wormholes via lightlike branes PACS: 11 Existence of wormholes can lead to a host of new effects like Misner-Wheeler "charge without charge" effect, where without being generated by any source an electric flux arrives from one "universe" and flows into the other "universe". Here we show the existence of an intriguing opposite possibility. Namely, a charged object (a charged lightlike brane in our case) sitting at the wormhole "throat" expels all the flux it produces into just one of the "universes", which turns out to be of compactified ("tube-like") nature. An outside observer in the non-compact "universe" detects, therefore, a neutral object. This charge-hiding effect takes place in a gravity/gauge-field system self-consistently interacting with a charged lightlike brane as a matter source, where the gauge field subsystem is of a special non-linear form containing a square-root of the Maxwell term and which previously has been shown to produce a QCD-like confining gauge field dynamics in flat space-time. (Eduardo Guendelman), alexk@bgu.ac.il (Alexander Kaganovich), nissimov@inrne.bas.bg (Emil Nissimov), svetlana@inrne.bas.bg (Svetlana Pacheva) Introduction Misner-Wheeler "charge without charge" effect [1] stands out as one of the most interesting physical phenomena produced by wormholes. Misner and Wheeler realized that wormholes connecting two asymptotically flat space times provide the possibility of existence of electromagnetically non-trivial solutions, where the lines of force of the electric field flow from one universe to the other without a source and giving the impression of being positively charged in one universe and negatively charged in the other universe. For a detailed account of the general theory of wormholes we refer to Visser's book [2] (see also [3,4] and some more recent accounts [5]- [9]. In the present paper we find the opposite effect in wormhole physics, namely, that a genuinely charged matter source of gravity and electromagnetism may appear electrically neutral to an external observer. Here we show this phenomenon to take place in a gravity/gauge-field system selfconsistently coupled to a charged lightlike brane as a matter source, where the gauge field subsystem is of a special non-linear form containing a squareroot of the Maxwell term. The latter has been previously shown [10]- [15] to produce a QCD-like confining ("Cornell" [16]- [18]) potential in flat spacetime. In the present case the lightlike brane, which connects as a wormhole "throat" a non-compact "universe" with a compactified "universe", is electrically charged, however all of its flux flows into the compactified ("tube-like") "universe" only. No Coulomb field is produced in the non-compact "universe", therefore, the wormhole hides the charge from an external observer in the latter "universe". Let us recall that lightlike branes are singular null (lightlike) hypersurfaces in Riemannian space-time which provide dynamical description of various physically important phenomena in cosmology and astrophysics such as: (i) impulsive lightlike signals arising in cataclysmic astrophysical events (supernovae, neutron star collisions) [20]; (ii) dynamics of horizons in black hole physics -the so called "membrane paradigm" [21]; (iii) the thin-wall approach to domain walls coupled to gravity [22]- [25]. The gravity/gauge-field system with a square-root of the Maxwell term was recently studied in [26] (see the brief review in Section 2 below) where the following interesting new features of the pertinent static spherically symmetric solutions have been found: (i) appearance of a constant radial electric field (in addition to the Coulomb one) in charged black holes within Reissner-Nordström-de-Sitter-type and/or Reissner-Nordström-anti-de-Sitter-type space-times, in particular, in electrically neutral black holes with Schwarzschild-de-Sitter and/or Schwarzschildanti-de-Sitter geometry; (ii) novel mechanism of dynamical generation of cosmological constant through the nonlinear gauge field dynamics of the "square-root" Maxwell term; (iii) appearance of confining-type effective potential in charged test particle dynamics in the above black hole backgrounds. In Section 3 of the present paper we extend the analysis in [26] by finding new solutions of Levi-Civita-Bertotti-Robinson type [27]- [29], i.e., with space-time geometry of the form M 2 × S 2 with M 2 being a two-dimensional anti-de Sitter, Rindler or de Sitter space depending on the relative strength of the electric field w.r.t. coupling of the square-root Maxwell term. In our previous papers [30]- [40] we have provided an explicit reparametrization invariant world-volume Lagrangian formulation of lightlike p-branes (a brief review is given in Section 4) and we have used them to construct various types of wormhole, regular black hole and lightlike braneworld solutions in D = 4 or higher-dimensional asymptotically flat or asymptotically antide Sitter bulk space-times. In particular, in refs. [38]- [40] we have shown that lightlike branes can trigger a series of spontaneous compactificationdecompactification transitions of space-time regions, e.g., from ordinary compactified ("tube-like") Levi-Civita-Bertotti-Robinson space to non-compact Reissner-Nordström or Reissner-Nordström-de-Sitter region or vice versa. Let us note that wormholes with "tube-like" structure (and regular black holes with "tube-like" core) have been previously obtained within different contexts in refs. [41]- [49]. The essential role of the above mentioned proper world-volume Lagrangian formulation of lightlike branes manifests itself most clearly in the correct selfconsistent construction [34,37] of the simplest wormhole solution first proposed by Einstein and Rosen [50] -the Einstein-Rosen "bridge" wormhole. Namely, in refs. [34,37] it has been shown that the Einstein-Rosen "bridge" in its original formulation [50] naturally arises as the simplest particular case of static spherically symmetric wormhole solutions produced by lightlike branes as gravitational sources, where the two identical "universes" with Schwarzschild outer-region geometry are self-consistently glued together by a lightlike brane occupying their common horizon -the wormhole "throat". An understanding of this picture within the framework of Kruskal-Szekeres manifold was subsequently provided in ref. [53], which involves Rindler's el-liptic identification of the two antipodal future event horizons. At this point let us strongly emphasize that the original notion of "Einstein-Rosen bridge" in ref. [50] is qualitatively different from the notion of "Einstein-Rosen bridge" defined in several popular textbooks (e.g., refs. [51,52]) using the Kruskal-Szekeres manifold, where the "bridge" has dynamic space-time geometry. Namely, the two regions in Kruskal-Szekeres space-time corresponding to the two copies of outer Schwarzschild space-time region (r > 2m) (the building blocks of the original static Einstein-Rosen "bridge") and labeled (I) and (III) in ref. [51] are generally disconnected and share only a two-sphere (the angular part) as a common border (U = 0, V = 0 in Kruskal-Szekeres coordinates), whereas in the original Einstein-Rosen "bridge" construction [50] the boundary between the two identical copies of the outer Schwarzschild space-time region (r > 2m) is a three-dimensional lightlike hypersurface (r = 2m). In Section 5 below we consider self-consistent coupling of gravity/gaugefield system with a square-root of the Maxwell term to a charged lightlike brane, which will serve as a matter source of gravity and (nonlinear) electromagnetism. In this Section we derive the main result of the present paperwormhole-like solutions joining a non-compact "universe" to a compactified ("tube-like") "universe" (of generalized Levi-Civita-Bertotti-Robinson type) via a wormhole "throat" realized by the charged lightlike brane, which completely hides its electric flux from an outside observer in the non-compact "universe". This new charge "confining" phenomena is entirely due to the presence of the "square-root" Maxwell term. Lagrangian Formulation. Spherically Symmetric Solutions We will consider the simplest coupling to gravity of the nonlinear gauge field system with a square-root of the Maxwell term known to produce QCDlike confinement in flat space-time [10]- [15]. The relevant action is given by (we use units with Newton constant G N = 1): S = d 4 x √ −G R(G) 16π + L(F 2 ) , L(F 2 ) = − 1 4 F 2 − f 2 √ εF 2 ,(1)F 2 ≡ F κλ F µν G κµ G λν , F µν = ∂ µ A ν − ∂ ν A µ . Here R(G) is the scalar curvature of the space-time metric G µν and G ≡ det G µν ; the sign factor ε = ±1 in the square root term in (1) corresponds to "magnetic" or "electric" dominance; f is a positive coupling constant. It is important to stress that we will not introduce any bare cosmological constant term. Let us note that the Lagrangian L(F 2 ) in (1) contains both the usual Maxwell term as well as a non-analytic function of F 2 and thus it is a nonstandard form of nonlinear electrodynamics. In this way it is significantly different from the original purely "square root" Lagrangian − f 2 √ F 2 first proposed by Nielsen and Olesen [54] to describe string dynamics (see also refs. [55,56]). The natural appearance of the "square-root" Maxwell term in effective gauge field actions was further motivated by 't Hooft [19] who has proposed that such gauge field actions are adequate for describing confinement (see especially Eq.(5.10) in [19]). He has in particular described a consistent quantum approach in which "square-root" gauge-field terms play the role of "infrared counterterms". Also, it has been shown in first three refs. [10]- [15] that the square root of the Maxwell term naturally arises as a result of spontaneous breakdown of scale symmetry of the original scaleinvariant Maxwell theory with f appearing as an integration constant responsible for the latter spontaneous breakdown. Let us also remark that one could start with the non-Abelian version of the gauge field action in (1). Since we will be interested in static spherically symmetric solutions, the non-Abelian gauge theory effectively reduces to an Abelian one as pointed out in the ref. [10]. The corresponding equations of motion read: R µν − 1 2 G µν R = 8πT (F ) µν ,(2) where T (F ) µν = L(F 2 ) G µν − 4L ′ (F 2 )F µκ F νλ G κλ ,(3) and ∂ ν √ −GL ′ (F 2 )F κλ G µκ G νλ = 0 ,(4) where L ′ (F 2 ) denotes derivative w.r.t. F 2 of the function L(F 2 ) in (1). In our preceding paper [26] we have shown that the gravity-gauge-field system (1) possesses static spherically symmetric solutions with a radial electric field containing both Coulomb and constant vacuum pieces: F 0r = ε F f √ 2 + Q √ 4π r 2 , ε F = sign(Q) ,(5) and the space-time metric: ds 2 = −A(r)dt 2 + dr 2 A(r) + r 2 (dθ 2 + sin 2 θdϕ 2 ) ,(6)A(r) = 1 − √ 8π|Q|f − 2m r + Q 2 r 2 − 2πf 2 3 r 2 ,(7) is Reissner-Nordström-de-Sitter-type with dynamically generated effective cosmological constant Λ eff = 2πf 2 . Appearance in (7) of a "leading" constant term different from 1 resembles the effect on gravity produced by a spherically symmetric "hedgehog" configuration of a nonlinear sigma-model scalar field with SO(3) symmetry [57] (cf. also [58]). Generalized Levi-Civita-Bertotti-Robinson Space-Times Here we will look for static solutions of Levi-Civita-Bertotti-Robinson type [27]- [29] of the system (2)-(4), namely, with space-time geometry of the form M 2 × S 2 where M 2 is some two-dimensional manifold: ds 2 = −A(η)dt 2 + dη 2 A(η) +r 2 0 (dθ 2 +sin 2 θdϕ 2 ) , −∞ < η < ∞ , r 0 = const ,(8) and being: • either purely electric type, where the sign factor ε = −1 in the gauge field Lagrangian L(F 2 ) (1): F µν = 0 for µ, ν = 0, η , F 0η = F 0η (η) ;(9) • or purely magnetic type, where ε = +1 in (1): F µν = 0 for µ, ν = i, j ≡ θ, ϕ , ∂ 0 F ij = ∂ ϕ F ij = 0 .(10) In the purely electric case (9) the gauge field equations of motion become: ∂ η F 0η − ε F f √ 2 = 0 , ε F ≡ sign(F 0η ) ,(11) yielding a constant vacuum electric field: F 0η = c F = arbitrary const .(12) The (mixed) components of energy-momentum tensor (3) read: T (F ) 0 0 = T (F ) η η = − 1 2 F 2 0η , T (F ) ij = g ij 1 2 F 2 0η − f √ 2 |F 0η | .(13) Taking into account (13), the Einstein eqs.(2) for (ij), where R ij = 1 r 2 0 g ij because of the S 2 factor in (8), yield: 1 r 2 0 = 4πF 2 0η , i.e. r 0 = 1 2 √ π|c F | .(14) The (00) Einstein eq.(2) using the expression R 0 0 = − 1 2 ∂ 2 η A (ref. [59]; see also [60]) becomes: ∂ 2 η A = 8π|c F | |c F | − √ 2f .(15) Therefore, we arrive at the following three distinct types of Levi-Civita-Bertotti-Robinson solutions for gravity coupled to the non-Maxwell gauge field system (1): (i) AdS 2 × S 2 with strong constant vacuum electric field |F 0η | = |c F | > √ 2f , where AdS 2 is two-dimensional anti-de Sitter space with: A(η) = 4π|c F | |c F | − √ 2f η 2(16) in the metric (8), η being the Poincare patch space-like coordinate. (ii) Rind 2 × S 2 with constant vacuum electric field |F 0η | = |c F | = √ 2f , where Rind 2 is the flat two-dimensional Rindler space with: A(η) = η for 0 < η < ∞ or A(η) = −η for − ∞ < η < 0(17) in the metric (8). (iii) dS 2 ×S 2 with weak constant vacuum electric field |F 0η | = |c F | < √ 2f , where dS 2 is two-dimensional de Sitter space with: A(η) = 1 − 4π|c F | √ 2f − |c F | η 2(18) in the metric (8). For the special value |c F | = f √ 2 we recover the Nariai solution [61,62] with A(η) = 1 − 2πf 2 η 2 and equality (up to signs) among energy density, radial and transverse pressures: ρ = −p r = −p ⊥ = f 2 4 (T (F ) µ ν = diag (−ρ, p r , p ⊥ , p ⊥ )). In all three cases above the size of the S 2 factor is given by (14). Solutions (17) and (18) are new ones and are specifically due to the presence of the non-Maxwell square-root term (with ε = −1) in the gauge field Lagrangian (1). In the purely magnetic case (10) the gauge field equations of motion (4): ∂ ν sin θ 1 + f √ F 2 F µν = 0(19) yield magnetic monopole solution F ij = Br 2 0 sin θ ε ij , where B = const, irrespective of the presence of the non-Maxwell square-root term. However, the latter does contribute to the energy-momentum tensor: T (F ) 0 0 = T (F ) η η = − 1 2 B 2 − f |B| , T (F ) ij = 1 2 g ij B 2 .(20) Taking into account (20), the Einstein eqs.(2) for (ij) yield (cf. (14)): 1 r 2 0 = 4π B 2 + √ 2f |B| ,(21) whereas the mixed-component (00) Einstein eq.(2) gives ∂ 2 η A = 8πB 2 . Thus in the purely magnetic case we obtain only one solution -AdS 2 × S 2 spacetime with magnetic monopole where: A(η) = 4πB 2 η 2(22) in the metric (8) and the size of the S 2 factor is determined by (21). Lagrangian Formulation of Lightlike Brane Dynamics In what follows we will consider gravity/gauge-field system self-consistently interacting with a lightlike p-brane (LL-brane for short) of codimension one (D = (p + 1) + 1). In a series of previous papers [30]- [40] we have proposed manifestly reparametrization invariant world-volume Lagrangian formulation in several dynamically equivalent forms of LL-branes coupled to bulk gravity G µν and bulk gauge fields, in particular, electromagnetic field A µ . Here we will use our Polyakov-type formulation given by the world-volume action: S LL [q] = − 1 2 d p+1 σ T b p−1 2 0 √ −γ γ abḡ ab − b 0 (p − 1) , (23) g ab ≡ ∂ a X µ G µν ∂ b X ν − 1 T 2 (∂ a u + qA a )(∂ b u + qA b ) , A a ≡ ∂ a X µ A µ . (24) Here and below the following notations are used: • γ ab is the intrinsic Riemannian metric on the world-volume with γ = det γ ab ; g ab is the induced metric on the world-volume: g ab ≡ ∂ a X µ G µν (X)∂ b X ν ,(25) which becomes singular on-shell (manifestation of the lightlike nature), cf. Eq.(29) below); b 0 is a positive constant measuring the world-volume "cosmological constant". • X µ (σ) are the p-brane embedding coordinates in the bulk D-dimensional space-time with Riemannian metric G µν (x) (µ, ν = 0, 1, . . . , D − 1); (σ) ≡ (σ 0 ≡ τ, σ i ) with i = 1, . . . , p; ∂ a ≡ ∂ ∂σ a . • u is auxiliary world-volume scalar field defining the lightlike direction of the induced metric (see Eq.(29) below) and it is a non-propagating degree of freedom ( ref. [40]). • T is dynamical (variable) brane tension (also a non-propagating degree of freedom). • Coupling parameter q is the surface charge density of the LL-brane. The corresponding equations of motion w.r.t. X µ , u, γ ab and T read accordingly (using short-hand notation (24)): ∂ a T |ḡ|ḡ ab ∂ b X µ + T |ḡ|ḡ ab ∂ a X λ ∂ b X ν Γ µ λν + q T |ḡ|ḡ ab ∂ a X ν (∂ b u + qA b )F λν G µλ = 0 ,(26)∂ a 1 T |ḡ|ḡ ab (∂ b u + qA b ) = 0 , γ ab = 1 b 0ḡ ab ,(27)T 2 + ǫḡ ab (∂ a u + qA a )(∂ b u + qA b ) = 0 .(28) Hereḡ = det ḡ ab and Γ µ λν denotes the Christoffel connection for the bulk metric G µν . The on-shell singularity of the induced metric g ab (25), i.e., the lightlike property, directly follows Eq.(28) and the definition ofḡ ab (24): g ab ḡ bc (∂ c u + qA c ) = 0 .(29) Explicit world-volume reparametrization invariance of the LL-brane action (23) allows to introduce the standard synchronous gauge-fixing conditions for the intrinsic world-volume metric γ 00 = −1 , γ 0i = 0 (i = 1, . . . , p). which reduces Eqs.(27)- (28) to the following relations: (∂ 0 u + qA 0 ) 2 T 2 = b 0 + g 00 , ∂ i u + qA i = (∂ 0 u + qA 0 )g 0i (b 0 + g 00 ) −1 , g 00 = g ij g 0i g 0j , ∂ 0 g (p) + ∂ i g (p) g ij g 0j = 0 , g (p) ≡ det g ij ,(30) (recall that g 00 , g 0i , g ij are the components of the induced metric (25); g ij is the inverse matrix of g ij ). Then, as shown in refs. [30]- [40], consistency of LL-brane dynamics in static "spherically-symmetric"-type backgrounds (in what follows we will use Eddington-Finkelstein coordinates, dt = dv − dη A(η) ): ds 2 = −A(η)dv 2 + 2dvdη + C(η)h ij (θ)dθ i dθ j , F vη = F vη (η) , rest = 0 (31) with the standard embedding ansatz: 1, . . . , p) . (32) requires the corresponding background (31) to possess a horizon at some η = η 0 , which is automatically occupied by the LL-brane, i.e.: X 0 ≡ v = τ , X 1 ≡ η = η(τ ) , X i ≡ θ i = σ i (i =η(τ ) = η 0 , A(η 0 ) = 0 .(33) This property is called "horizon straddling" according to the terminology of Ref. [23]. Similar "horizon straddling" has been found also for LL-branes moving in rotating axially symmetric (Kerr or Kerr-Newman) and rotating cylindrically symmetric black hole backgrounds [36,37]. Self-Consistent Wormhole-Like Solutions with LL-Branes Let us now consider a bulk gravity/gauge-field system in D = 4 (1) selfconsistently interacting with a p = 2 LL-brane: S = d 4 x √ −G R(G) 16π − 1 4 F 2 − f 2 √ −F 2 + S LL [q] ,(34) where S LL [q] is the LL-brane world-volume action (23) (with p = 2). It is now the LL-brane which will be the material and charge source for gravity and (nonlinear) electromagnetism. The equations of motion resulting from (34) read: R µν − 1 2 G µν R = 8π T (F ) µν + T (brane) µν ,(35)∂ ν √ −g 1 − f √ −F 2 F κλ G µκ G νλ + j µ (brane) = 0 ,(36) together with the LL-brane equations (26)- (28). T (F ) µν is the same as in (3). The energy-momentum tensor and the charge current density of the LL-brane are straightforwardly derived from the underlying world-volume action (23): T µν (brane) = − d 3 σ δ (4) x − X(σ) √ −G T |ḡ|ḡ ab ∂ a X µ ∂ b X ν ,(37)j µ (brane) = −q d 3 σ δ (4) x − X(σ) |ḡ|ḡ ab ∂ a X µ (∂ b u + qA b ) T −1 . (38) Looking for solutions of static "spherically-symmetric"-type (31) for the coupled gravity-gauge-field-LL-brane system (34) amounts to the following simple steps: (i) Choose "vacuum" static "spherically-symmetric"-type solutions (31) of (35)-(36) (i.e., without the delta-function terms due to the LL-branes) in each region −∞ < η < η 0 and η 0 < η < ∞ with a common horizon at η = η 0 ; (ii) The LL-brane automatically locates itself on the horizon according to "horizon straddling" property (33); (iii) Match the discontinuities of the derivatives of the metric and the gauge field strength (31) across the horizon at η = η 0 using the explicit expressions for the LL-brane stress-energy tensor charge current density (37)- (38). Using (30)-(32) we find for the LL-brane energy-momentum tensor and charge current density: T µν (brane) = S µν δ(η − η 0 ) , j µ (brane) = δ µ 0 q det G ij δ(η − η 0 ) ,(39) where G ij = C(η)h ij (θ) (cf. (31)). The non-zero components of the surface energy-momentum tensor S µν (with lower indices) and its trace are: S ηη = T b 1/2 0 , S ij = −T b 1/2 0 G ij , S λ λ = −2T b 1/2 0 .(40) Taking into account (39)- (40) together with (31)- (33), the matching relations at the horizon η = η 0 become [38]- [40] (for a systematic introduction to the formalism of matching different bulk space-time geometries on codimensionone hypersurfaces ("thin shells") see the textbook [63]): (A) Matching relations from Einstein eqs. (35): [∂ η A] η 0 = −16πT b 0 , [∂ η ln C] η 0 = − 8π √ b 0 T (41) with notation [Y ] η 0 ≡ Y | η→η 0 +0 −Y | η→η 0 −0 for any quantity Y . (B) Matching relation from nonlinear gauge field eqs. (36): [F vη ] η 0 = q(42) (C) X 0 -equation of motion of the LL-brane (the only non-trivial contribution of second-order LL-brane eqs. (26) in the case of embedding (32)): T 2 ∂ η A η 0 + 2b 0 ∂ η ln C η 0 − b 0 q F vη η 0 = 0(43) with notation Y η 0 ≡ 1 2 Y | η→η 0 +0 +Y | η→η 0 −0 . We are looking for wormhole-type solutions to (34) with the charged LLbrane at the wormhole "throat" connecting a non-compact "universe" with Reissner-Nordström-de-Sitter-type geometry (5)-(7) (where the cosmological constant is dynamically generated) to a compactified ("tube-like") "universe" of (generalized) Levi-Civita-Bertotti-Robinson type (8)- (9). These wormholes possess the novel property of hiding electric charge from external observer in the non-compact "universe", i.e., the whole electric flux produced by the charged LL-brane at the wormhole "throat" is pushed into the "tubelike" "universe". The first wormhole-type solution of the above kind we find is given by: (a) "left universe" of Levi-Civita-Bertotti-Robinson ("tube-like") type with geometry Rind 2 × S 2 (17): A(η) = −η , C(η) = r 2 0 , |F vη | = √ 2f for η < 0 ;(44) (b) non-compact "right universe" comprising the exterior region of Reissner-Nordström-de-Sitter-type black hole beyond the middle (Schwarzschild-type) horizon r 0 (cf. (5)- (7)): A(η) = 1 − √ 8π|Q|f − 2m r 0 + η + Q 2 (r 0 + η) 2 − 2πf 2 3 (r 0 + η) 2 , A(0) = 0 , ∂ η A(0) > 0 , C(η) = (r 0 + η) 2 , F vη = ε F f √ 2 + Q √ 4π (r 0 + η) 2 for η > 0 .(45) Substituting (44) (34): Q = 0 , |q| = f √ 2 , sign(q) = −sign(F vη ) ,(46)r 2 0 = 1 8πf 2 , m = 11 48 √ 2π f , b 0 = 1 8 √ 2π f + 3 16 .(47) The second wormhole-type solution of the aforementioned kind reads: (c) "left universe" of Levi-Civita-Bertotti-Robinson ("tube-like") type with geometry AdS 2 × S 2 (16): A(η) = 4π|c F | |c F | − √ 2f η 2 , C(η) = r 2 0 , |F vη | = |c F | > √ 2f for η < 0 ;(48) (d) non-compact Reissner-Nordström-de-Sitter-type "right universe" of the same kind as (45). Substituting again (48), (45) into the matching relations (41)-(43) we find for the wormhole parameters: Q = 0 , |c F | = |q| + f √ 2 , sign(q) = −sign(F vη ) ≡ −sign(c F ) ,(49)r 2 0 = 1 4πc 2 F , m = 1 2 √ π f 1 − f 2 6c 2 F , b 0 = |q| |q| + √ 2f 4c 2 F .(50) The important observation here is that Q = 0 in both wormhole solutions 50)). Therefore, the "right universe" in both cases turns out to be the exterior region of the electrically neutral Schwarzschild-de-Sitter black hole beyond the Schwarzschild horizon which carries a vacuum constant radial electric field |F vη | = f √ 2 . On the other hand, according to (45), (46) and (45),(49) the whole flux produced by the LL-brane charge q (|F vη | = f √ 2 + |q|) flows only into the compactified "left universe" of Levi-Civita-Bertotti-Robinson type (Rind 2 × S 2 (17) or AdS 2 × S 2 (16)). The geometry of the above constructed charge-"hiding" wormhole solutions is illustrated in Figure 1. Conclusions We have seen that a charged wormhole "throat" realized by a charged lightlike brane, when joining a compactified space-time with a non-compact space-time region, expels all of the electric flux it produces into the compactified ("tube-like") region when the gauge field dynamics is driven by an additional "square-root" Maxwell term known to produce QCD-like confining potential in flat space-time. Indeed, this effect can be understood from the point of view of an observer in the non-compact "universe" as an alternative way of achieving charge confinement in a fashion similar to the MIT bag model [64], where the role of the inside bag region is being played by the compactified Levi-Civita-Bertotti-Robinson space. In an accompanying paper [65] we show that the above "charge-hiding" solution can be further generalized to a truly charge-confining wormhole solution when we couple the bulk gravity/nonlinear-gauge-field system (1) self-consistently to two separate codimension-one charged lightlike branes with equal in magnitude but opposite charges. The latter system possesses a "two-throat" wormhole solution, where the "left-most" and the "right-most" "universes" are two identical copies of the exterior region of the neutral Schwarzschild-de-Sitter black hole beyond the Schwarzschild horizon, whereas the "middle" "universe" is of generalized Levi-Civita-Bertotti-Robinson "tube-like" form with geometry dS 2 × S 2 (18). It comprises the finite-extent intermediate region of dS 2 between its two horizons. Both "throats" are occupied by the two oppositely charged lightlike branes and the whole electric flux produced by the latter is confined entirely within the middle finite-extent "tube-like" "universe" -a property qualitatively resembling the quark confinement phenomenon in quantum chromodynamics. Figure 1: Shape of t = const and θ = π 2 slice of charge-"hiding" wormhole geometry. The whole electric flux is expelled into the lower (infinitely long) cylindric tube. -(45) into the set of matching relations (41)-(43) determines all parameters of the wormhole (r 0 , m, Q, b 0 , q) in terms of the coupling constant f in front of the square-root Maxwell term in (a)-(b) (Eqs.(44)-(45), (46)-(47)) and (c)-(d) (Eqs.(48), (45), (49)-( Acknowledgments E.N. and S.P. are supported by Bulgarian NSF grant DO 02-257. Also, all of us acknowledge support of our collaboration through the exchange agreement between the Ben-Gurion University and the Bulgarian Academy of Sciences. 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E Guendelman, A Kaganovich, E Nissimov, S Pacheva, arxiv:1109.0453hep-thGuendelman E, Kaganovich A, Nissimov E, Pacheva S. Hiding and Confining Charges via "Tube-like" Wormholes. arxiv:1109.0453[hep-th].

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{'abstract': 'Existence of wormholes can lead to a host of new effects like Misner-Wheeler "charge without charge" effect, where without being generated by any source an electric flux arrives from one "universe" and flows into the other "universe". Here we show the existence of an intriguing opposite possibility. Namely, a charged object (a charged lightlike brane in our case) sitting at the wormhole "throat" expels all the flux it produces into just one of the "universes", which turns out to be of compactified ("tube-like") nature. An outside observer in the non-compact "universe" detects, therefore, a neutral object. This charge-hiding effect takes place in a gravity/gauge-field system self-consistently interacting with a charged lightlike brane as a matter source, where the gauge field subsystem is of a special non-linear form containing a square-root of the Maxwell term and which previously has been shown to produce a QCD-like confining gauge field dynamics in flat space-time. (Eduardo Guendelman), alexk@bgu.ac.il (Alexander Kaganovich), nissimov@inrne.bas.bg (Emil Nissimov), svetlana@inrne.bas.bg (Svetlana Pacheva)', 'arxivid': '1108.3735', 'author': ['Eduardo Guendelman \nDepartment of Physics\nBen-Gurion University of the Negev\nP.O.Box 653IL-84105Beer-ShevaIsrael\n', 'Alexander Kaganovich \nDepartment of Physics\nBen-Gurion University of the Negev\nP.O.Box 653IL-84105Beer-ShevaIsrael\n', 'Emil Nissimov \nInstitute for Nuclear Research and Nuclear Energy\nBulgarian Academy of Sciences\nBoul. Tsarigradsko Chausee 72BG-1784SofiaBulgaria\n', 'Svetlana Pacheva \nInstitute for Nuclear Research and Nuclear Energy\nBulgarian Academy of Sciences\nBoul. Tsarigradsko Chausee 72BG-1784SofiaBulgaria\n'], 'authoraffiliation': ['Department of Physics\nBen-Gurion University of the Negev\nP.O.Box 653IL-84105Beer-ShevaIsrael', 'Department of Physics\nBen-Gurion University of the Negev\nP.O.Box 653IL-84105Beer-ShevaIsrael', 'Institute for Nuclear Research and Nuclear Energy\nBulgarian Academy of Sciences\nBoul. Tsarigradsko Chausee 72BG-1784SofiaBulgaria', 'Institute for Nuclear Research and Nuclear Energy\nBulgarian Academy of Sciences\nBoul. Tsarigradsko Chausee 72BG-1784SofiaBulgaria'], 'corpusid': 7146449, 'doi': '10.2174/1874415x01104010027', 'github_urls': [], 'n_tokens_mistral': 14896, 'n_tokens_neox': 12161, 'n_words': 5976, 'pdfsha': 'b5b79a7e48f8dd458c0e51235ba71f3645bc372e', 'pdfurls': ['https://arxiv.org/pdf/1108.3735v5.pdf'], 'title': ['Hiding Charge in a Wormhole', 'Hiding Charge in a Wormhole'], 'venue': []}

arxiv

On the existence of (H, A)-stable sheaves on K3 or abelian surfaces 20 Feb 2013 Markus Zowislok On the existence of (H, A)-stable sheaves on K3 or abelian surfaces 20 Feb 2013 We give an existence result on (H, A)-stable sheaves on a K3 or abelian surface X with primitive triple of invariants (rank,first Chern class,Euler characteristics) in the integral cohomology lattice. Such a result yields the existence of singular projective Q-factorial symplectic terminalisations of certain moduli spaces of sheaves on X that are Gieseker semistable with respect to a nongeneral ample divisor. Introduction After the paper [KLS06] has appeared, the hope to construct new examples of irreducible (holomorphically) symplectic manifolds out of moduli spaces of sheaves on K3 or abelian surfaces almost died: the authors showed that in general, i.e. for general ample divisors, there is no symplectic resolution of these moduli spaces except for the nonsingular and O'Grady-like cases. In [Zow12] I investigated the case of a nongeneral ample divisor. In particular, I could exclude the existence of new examples of projective irreducible symplectic manifolds lying birationally over components of the moduli spaces of one-dimensional semistable sheaves on K3 surfaces, and over components of many of the moduli spaces of two-dimensional sheaves on K3 surfaces, in particular, of those for rank two sheaves. In order to answer the question of symplectic resolvability, as explained in [Zow12], constructing a projective Q-factorial symplectic terminalisationM → M of a component M of the moduli space, i.e. a symplectic Q-factorial projective varietyM with at most terminal singularities together with a projective birational morphism f :M → M , yields the following facts: (1) IfM can be chosen to be an irreducible symplectic manifold thenM is unique up to deformation by a result of Huybrechts [Huy99]. (2) IfM is singular, M admits no projective symplectic resolution by [Nam06, Corollary 1]. To be more precise we need some notation. Let X be a nonsingular projective irreducible surface over C, K X its canonical divisor, H an ample divisor on X, and E a coherent sheaf on X. We associate the element u(E) := (rk E, c 1 (E), χ(E)) ∈ Λ(X) := N 0 ⊕ NS(X) ⊕ Z ⊂ H 2 * (X, Z) of sheaf invariants to E. We avoid the elegant notion of a Mukai vector in favour of keeping torsion inside NS(X). For an element u := (r, c, χ) ∈ Λ(X) we define ∆(u) := c 2 − 2rχ + 2r 2 χ(O X ) − rc.K X and χ(u, u) := χ(O X )r 2 − ∆(u) . If E satisfies u(E) = u, then, by Riemann-Roch, its discriminant 1 is ∆(u), and χ(E, E) := 2 k=0 ext k (E, E) = χ(u, u) , where ext k (E, E) := dim Ext k (E, E). We will also write hom(E, F ) := dim Hom(E, F ) for two coherent sheaves E, F . We denote the moduli space of sheaves E on X with u(E) = u that are semistable with respect to an ample divisor H on (2) Let m ≥ 2 and χ(mu, mu) = 8. If H is mu-general or r = 1 or χ(u, u) > ϕ(r) with ϕ as in [Zow12,Theorem 6.5] then there is a singular Q-factorial projective symplectic terminalisation of M s H (mu) , and in particular, there is no projective symplectic resolution of M s H (mu). The proof of (2) is based on the existence of a singular Q-factorial projective symplectic terminalisation M → M H,A (mu) established by item 2.b.ii of [Zow12, Theorem 5.3] using the existence of an (H, A)-stable sheaf E with u(E) = u. This existence is ensured by the assumption of (2), see the proof of the above theorem in [Zow12]. Of course, instead one can also just assume this existence. Our main result of this article is another existence result, which in turn implies the existence of a singular Q-factorial projective symplectic terminalisation as in the above theorem: Theorem 3.1. Let X be a projective surface with torsion canonical bundle, u ∈ Λ(X) primitive, and H and A two ample divisors on X such that H is contained in at most one wall and A is u-general. Then the nonemptyness of M H,A (u) is independent of the choice of the pair (H, A). In particular, one has: Corollary 3.2. Let X be a projective K3 or abelian surface, u ∈ Λ(X) primitive with χ(u, u) ≥ −2, and H and A two ample divisors on X such that H is contained in at most one wall and A is u-general. Then M H,A (u) is nonempty. As M H,A (u) = M s H,A (u), in the situation of the corollary there is an (H, A)-stable sheaf E with u(E) = u. Twisted and (H, A)-stability In this section we recall three notions of stability of sheaves and establish a relation between twisted stability and (H, A)-stability for positive rank. In my PhD thesis [Zow10], this relation was discussed in Chapter 6 for K3 surfaces. We assume familiarity with the material presented in [HL10] and use the notation therein. Let still X be a nonsingular projective irreducible surface over C. In this case, twisted stability and (H, A)-stability, which are two generalisations of Gieseker stability, have an overlap. We briefly recall the definitions. Therefore let H be an ample divisor on X and E a nontrivial coherent sheaf on X. (1) Gieseker stability, see e.g. in [HL10, Section 1.2]. The Hilbert polynomial of E is P H (E)(n) := χ(E ⊗ O X (nH) ). Its leading coefficient multiplied by (dim E)! is called multiplicity of E and denoted here by α H (E). It is always positive, and p H (E)(n) := χ(E ⊗ O X (nH)) α H (E) is called reduced Hilbert polynomial of E. E is said to be H-(semi)stable if E is pure and for all nontrivial proper subsheaves F ⊂ E one has that p H (F ) (≤) p H (E), i.e. one has p H (F )(n) (≤) p H (E)(n) for n ≫ 0. In order to avoid case differentiation for stable and semistable sheaves we here follow the Notation 1.2.5 in [HL10] using bracketed inequality signs, e.g. an inequality with (≤) for (semi)stable sheaves means that one has ≤ for semistable sheaves and < for stable sheaves. If rk E > 0, then E is H-(semi)stable if E is pure and for all nontrivial proper subsheaves F ⊂ E one has that µ H (F ) ≤ µ H (E) and, in the case of equality, χ(F ) rk F (≤) χ(E) rk E . Here µ H (E) := c1(E).H rk E is the slope of E (with respect to H). (2) Twisted stability. Let D ∈ NS(X) Q := NS(X) ⊗ Q. We call χ D (E) := X ch(E). exp(D).td(X) the D-twisted Euler characteristic of E, and we say that E is D-twisted H-(semi)stable if E is pure and for all nontrivial saturated proper subsheaves F ⊂ E one has that χ D+nH (F ) α H (F ) (≤) χ D+nH (E) α H (E) as polynomials in n. If rk E > 0, then E is D-twisted H-(semi)stable if E is pure and for all nontrivial proper subsheaves F ⊂ E one has that µ H (F ) ≤ µ H (E) and, in the case of equality, It is enough to restrict to saturated proper nontrivial subsheaves F ⊂ E in the definition. µ D (F ) + χ(F ) rk F (≤) µ D (E) + χ(E) rk E .(3) The case of Gieseker stability can be regained by D = 0 from twisted stability and by H = A from (H, A)-stability. We briefly recall the notion of a general ample divisor for positive rank. The ample cone of X carries a chamber structure for a given triple u = (r, c, χ) ∈ Λ(X) of invariants. The definition depends on r. In the case of r = 1 we agree that the whole ample cone is the only chamber. For r > 1, we follow the definition in [HL10, Section 4.C]. Let Num(X) := Pic(X)/ ≡, where ≡ denotes numerical equivalence, and ∆ := ∆(u) > 0. Let still r > 0, H an ample divisor lying on exactly one u-wall W and A a u-general ample divisor lying in a chamber touching H. Definition 2.1. Let W (r, ∆) := {ξ ⊥ ∩ Amp(X) Q | ξ ∈ Num(X) with − r 2 4 ∆ ≤ ξ 2 < 0} , Definition 2.2. For a nontrivial saturated subsheaf F ⊂ E of a µ H -semistable sheaf E with u(E) = u, µ H (F ) = µ H (E), and c 1 (F ) rk F ≡ c 1 (E) rk E , we call the hyperplane z ∈ NS(X) Q χ z (F ) rk F = χ z (E) rk E a u-miniwall. The connected components of the complement of all u-miniwalls are called uminichambers. 2 In the following we omit the u-prefix as it is fixed for the whole section. χ D+nH (F ) rk F = χ D+nH (E) rk E (as polynomials in n) one has that c 1 (F ) rk F ≡ c 1 (E) rk E and χ(F ) rk F = χ(E) rk E . Proof. Let F ⊂ E be such a nontrivial saturated subsheaf. Equating the coefficients of the above polynomials yields µ H (F ) = µ H (E) and χ D (F ) rk F = χ D (E) rk E . As D is not contained in a miniwall, one has that c1(F ) rk F ≡ c1(E) rk E and thus also χ(F ) rk F = χ(E) rk E . Lemma 2.5. Let L be in a minichamber C, L ′ in its closure C, and E a coherent sheaf on X with u(E) = u. (1) If E is L-twisted H-semistable then it is also L ′ -twisted H-semistable. (2) If E is L ′ -twisted H-stable then it is also L-twisted H-stable. Proof. Let F ⊂ E be a nontrivial saturated proper subsheaf. As for µ H (F ) < µ H (E) one has χ nH+D (F ) rk F < χ nH+D (E) rk E (as polynomials in n) for any D ∈ NS(X) Q , we can restrict to µ H (F ) = µ H (E). We define the map f : C → Q, D → c 1 (F ) rk F − c 1 (E) rk E .D + χ(F ) rk F − χ(E) rk E . If c1(F ) rk F ≡ c1(E) rk E then f is independent of D. So let c1(F ) rk F ≡ c1(E) rk E . Then f = 0 on the whole minichamber C by the definition of a minichamber. We distinguish the two cases from above. (1) Let E be L-twisted H-semistable. Then f < 0 on C, hence f ≤ 0. (2) Let E be L ′ -twisted H-stable. Then f (L ′ ) < 0, hence f < 0 on an open subset containing L ′ , which in turn yields f < 0 on C. Proposition 2.6. Let L be in a minichamber C, L ′ in its boundary ∂C, and E a coherent sheaf on X with u(E) = u. χ nH+L ′ (F ) rk F = χ nH+L ′ (E) rk E (as polynomials in n) one has that µ A (F ) (≥) µ A (E). Proof. Let F ⊂ E be a nontrivial saturated proper subsheaf. As for µ H (F ) < µ H (E) one has χ nH+D (F ) rk F < χ nH+D (E) rk E (as polynomials in n) for any D ∈ NS(X) Q , we can again restrict to µ H (F ) = µ H (E). Then χ nH+L (F ) rk F − χ nH+L (E) rk E − χ nH+L ′ (F ) rk F − χ nH+L ′ (E) rk E = c 1 (F ) rk F − c 1 (E) rk E .(L − L ′ ) . (1) If c1(E) rk E ≡ c1(F ) rk F then χ nH+L (F ) rk F − χ nH+L (E) rk E = χ nH+L ′ (F ) rk F − χ nH+L ′ (E) rk E and µ A (F ) = µ A (E), so we assume c 1 (F ) rk F − c 1 (E) rk E ≡ 0 , which thus defines the wall W . In particular, the sign of c 1 (F ) rk F − c 1 (E) rk E .(L − L ′ ) = 0 is opposite to the sign of µ A (F ) − µ A (E) due to the choice of A. (1) Assume that E is L-twisted H-semistable and thus also L ′ -twisted H-semistable by Lemma 2.5. If furthermore χ nH+L ′ (F ) rk F = χ nH+L ′ (E) rk E then equation (1) yields χ nH+L (F ) rk F − χ nH+L (E) rk E = c 1 (F ) rk F − c 1 (E) rk E .(L − L ′ ) , which is negative, hence µ A (F ) > µ A (E). (2) Assume that E is L ′ -twisted H-semistable, i.e. in particular χ nH+L ′ (F ) rk F ≤ χ nH+L ′ (E) rk E . If one has strict inequality then by the same argument as in Lemma 2.5 one has that χ nH+L (F ) rk F < χ nH+L (E) rk E . So let's assume equality. Then µ A (F ) ≥ µ A (E) and thus χ nH+L (F ) rk F − χ nH+L (E) rk E = c 1 (F ) rk F − c 1 (E) rk E .(L − L ′ ) < 0 . The following statement, at least the part on semistability, is already known to Matsuki and Wentworth, as it can be found in [MW97, Theorem 4.1, part i]. Proof. Clearly a coherent sheaf E is L-twisted H-(semi)stable if and only if E⊗L is H-(semi)stable. Thus the claim follows from Proposition 2.6 and the description of (H, A)-stability at the beginning of this section. 3 Existence of (H, A)-stable sheaves Hence it is enough to prove nonemptyness for one suitable special choice of ample divisors. In particular, one has the Corollary 3.2. Let X be a projective K3 or abelian surface, u ∈ Λ(X) primitive with χ(u, u) ≥ −2, and H and A two ample divisors on X such that H is contained in at most one wall and A is u-general. Then M H,A (u) is nonempty. Proof. This follows from the above Theorem 3.1 as M H (u) = M H,H (u) is well-known to be nonempty for general H and χ(u, u) ≥ −2, see e.g. [KLS06]. X by M H (u) and the open subscheme of stable sheaves by M s H (u). The corresponding spaces for (H, A)-(semi)stable sheaves introduced in [Zow12] are denoted by M H,A (u) and M s H,A (u). The main result of [Zow12] on the case of positive rank was the extension of the result of [KLS06] to the following Theorem [Zow12] 1.1. Let X be a projective K3 or abelian surface, u = (r, c, χ) ∈ Λ(X) primitive with r > 0 and χ(u, u) ≥ 0, m ∈ N and H an ample divisor on X, and assume that M s H (mu) is nonempty. (1) Let m = 1 or χ(mu, mu) = 8. Then there is a projective symplectic resolution M → M s H (mu). If H is not mu-general then M can be chosen to be a symplectic resolution of M H,A (mu), where A is an mu-general ample divisor. whose elements are called u-walls. The connected components of the complement of the union of all u-walls are called u-chambers. An ample divisor is called u-general if it is not contained in a u-wall. The set W (r, ∆) is locally finite in Amp(X) Q by [HL10, Lemma 4.C.2]. Proposition 2. 3 . 3The number of miniwalls is finite and the miniwalls are parallel to W . For D, D ′ ∈ NS(X) Q one has that the set of D-twisted H-semistable sheaves is the same as the set of D ′ -twisted H-semistable sheaves if and only if D and D ′ belong to the same v-minichamber or v-miniwall. Proof. [MW97, Proposition 3.5]. Lemma 2.4. Let D be contained in a minichamber and E a D-twisted H-semistable sheaf with u(E) = u. Then for every nontrivial saturated subsheaf F ⊂ E with The vector space generated by the wall W divides NS(X) Q into two open half spaces, one of them containing L − L ′ . Choose A in the neighbouring chamber of W contained in the other half space. Then E is L-twisted H-(semi)stable if and only if it is L ′ -twisted H-semistable and for all nontrivial saturated proper subsheaves F ⊂ E with Corollary 2. 7 . 7Let A be an ample divisor in a chamber touching H and L ∈ Pic(X) lying on a miniwall. The vector space generated by the wall W divides NS(X) Q into two open half spaces, one of them containing A. Choose D in one of the minichambers touching L such that D − L is in the other half space. Then a coherent sheaf E with u(E) = u is D-twisted H-(semi)stable if and only if E ⊗ L is (H, A)-(semi)stable. Theorem 3. 1 . 1Let X be a projective surface with torsion canonical bundle, u ∈ Λ(X) primitive, and H and A two ample divisors on X such that H is contained in at most one wall and A is u-general. Then the nonemptyness of M H,A (u) is independent of the choice of the pair(H, A).Proof. As a direct consequence of [Yos03, Proposition 4.1], the nonemptyness of the moduli space M D H (u) of D-twisted H-stable sheaves is independent of the choice of the pair (H, D) if (H, D) is u-general, where D is any Q-line bundle. The claim now follows from Corollary 2.7. Be aware of different conventions of the discriminant's definition. Both notions are inspired by the work of Ellingsrud and Göttsche. Acknowledgements. The author would like to express his gratitude to Richard Thomas for his support and valuable discussions. Moreover, he thanks the Imperial College London for its hospitality and the Deutsche Forschungsgemeinschaft (DFG) for supporting the stay there by a DFG research fellowship (Az.: ZO 324/1-1). The geometry of moduli spaces of sheaves. Daniel Huybrechts, Manfred Lehn, Cambridge University PressCambridgesecond ed.Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, second ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. Compact hyper-Kähler manifolds: basic results. Daniel Huybrechts, Invent. Math. 1351Daniel Huybrechts, Compact hyper-Kähler manifolds: basic results, Invent. Math. 135 (1999), no. 1, 63-113. Singular symplectic moduli spaces. D Kaledin, M Lehn, Ch Sorger, Invent. Math. 1643D. Kaledin, M. Lehn, and Ch. Sorger, Singular symplectic moduli spaces, Invent. Math. 164 (2006), no. 3, 591-614. Mumford-Thaddeus principle on the moduli space of vector bundles on an algebraic surface. Kenji Matsuki, Richard Wentworth, Internat. J. Math. 81Kenji Matsuki and Richard Wentworth, Mumford-Thaddeus principle on the moduli space of vector bundles on an algebraic surface, Internat. J. Math. 8 (1997), no. 1, 97-148. On deformations of Q-factorial symplectic varieties. Yoshinori Namikawa, J. Reine Angew. Math. 599Yoshinori Namikawa, On deformations of Q-factorial symplectic varieties, J. Reine Angew. Math. 599 (2006), 97-110. Twisted stability and Fourier-Mukai transform. I. Kōta Yoshioka, Compositio Math. 1383Kōta Yoshioka, Twisted stability and Fourier-Mukai transform. I, Compositio Math. 138 (2003), no. 3, 261-288. On moduli spaces of semistable sheaves on K3 surfaces, Dissertation. Markus Zowislok, Südwestdeutscher Verlag für HochschulschriftenMarkus Zowislok, On moduli spaces of semistable sheaves on K3 surfaces, Disser- tation, Südwestdeutscher Verlag für Hochschulschriften, 2010, http://ubm.opus.hbz- nrw.de/volltexte/2010/2287/. On moduli spaces of sheaves on K3 or abelian surfaces. Math. Z. 2723-4, On moduli spaces of sheaves on K3 or abelian surfaces, Math. Z. 272 (2012), no. 3-4, 1195-1217.

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{'abstract': 'We give an existence result on (H, A)-stable sheaves on a K3 or abelian surface X with primitive triple of invariants (rank,first Chern class,Euler characteristics) in the integral cohomology lattice. Such a result yields the existence of singular projective Q-factorial symplectic terminalisations of certain moduli spaces of sheaves on X that are Gieseker semistable with respect to a nongeneral ample divisor.', 'arxivid': '1302.4873', 'author': ['Markus Zowislok '], 'authoraffiliation': [], 'corpusid': 117910221, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 6219, 'n_tokens_neox': 5562, 'n_words': 3233, 'pdfsha': '349e72c6bee6169b40d0576b940083b0163ee548', 'pdfurls': ['https://arxiv.org/pdf/1302.4873v1.pdf'], 'title': ['On the existence of (H, A)-stable sheaves on K3 or abelian surfaces', 'On the existence of (H, A)-stable sheaves on K3 or abelian surfaces'], 'venue': []}

arxiv

RAFT: Reward rAnked FineTuning for Generative Foundation Model Alignment Hanze Dong The Hong Kong University of Science and Technology Wei Xiong The Hong Kong University of Science and Technology Deepanshu Goyal The Hong Kong University of Science and Technology Rui Pan The Hong Kong University of Science and Technology Shizhe Diao The Hong Kong University of Science and Technology Jipeng Zhang The Hong Kong University of Science and Technology Kashun Shum The Hong Kong University of Science and Technology Tong Zhang The Hong Kong University of Science and Technology RAFT: Reward rAnked FineTuning for Generative Foundation Model Alignment Generative foundation models are susceptible to implicit biases that can arise from extensive unsupervised training data. Such biases can produce suboptimal samples, skewed outcomes, and unfairness, with potentially significant repercussions. Consequently, aligning these models with human ethics and preferences is an essential step toward ensuring their responsible and effective deployment in real-world applications. Prior research has primarily employed Reinforcement Learning from Human Feedback (RLHF) as a means of addressing this problem, wherein generative models are fine-tuned using RL algorithms guided by a humanfeedback-informed reward model. However, the inefficiencies and instabilities associated with RL algorithms frequently present substantial obstacles to the successful alignment of generative models, necessitating the development of a more robust and streamlined approach. To this end, we introduce a new framework, Reward rAnked FineTuning (RAFT), designed to align generative models more effectively. Utilizing a reward model and a sufficient number of samples, our approach selects the high-quality samples, discarding those that exhibit undesired behavior, and subsequently assembles a streaming dataset. This dataset serves as the basis for aligning the generative model and can be employed under both offline and online settings. Notably, the sample generation process within RAFT is gradient-free, rendering it compatible with black-box generators. Through extensive experiments, we demonstrate that our proposed algorithm exhibits strong performance in the context of both large language models and diffusion models. * Equal Contribution. Alphabetical order.Preprint. Under review. Introduction Generative foundation models have exhibited a remarkable capacity to accomplish diverse tasks that were previously unattainable, showcasing their broad-ranging capabilities in natural language and computer vision tasks. Large language models (LLMs) [7,37,9,40,17,45] and diffusion models [16,42,41,11,33,35], the most popular models in natural language and computer vision, are capable of generating high-quality meaningful outputs that are often indistinguishable from outputs produced by humans. AI-generated content is a rapidly evolving field that is widely believed to have the potential to revolutionize the way we create and consume content, ultimately enhancing the productivity of humanity through modern generative models. However, there are also concerns about the ethical implications of these models [5,6,29], such as the potential for misuse and the implicit bias from the model. It is important for researchers and developers to continue exploring the limitations of these models and restrict the output generation. One of the most direct limitations of current generative models is the high dependency on unsupervised large-scale datasets. Such datasets often contain inherent biases that can manifest in the models' output, leading to inaccurate or unfair results. To address this challenge, pre-trained models are typically fine-tuned on the downstream tasks with custom data, either to improve performance in a specialized setting or to eliminate potential biases and toxicity in the original model. One approach is to fine-tune the pre-trained models in a supervised manner using labeled data, known as supervised fine-tuning (SFT). Instruction tuning [46] is the most widely used approach to make LLMs adapt downstream tasks. However, collecting new supervised samples can be expensive in practical applications, especially when expert participation is required to generate high-quality data. More recently, Reinforcement Learning from Human Feedback (RLHF) has emerged as a promising method for fine-tuning pre-trained generative models. In recent studies of LLMs, RLHF has been widely employed to fine-tune pre-trained models using policy-based deep reinforcement learning (DRL) algorithms, typically the Proximal Policy Optimization (PPO). The idea of RLHF is to align the language models with human preferences and social values by optimizing a reward function that reflects specific human preferences (e.g. moral, harmless). For instance, OpenAI [29] fine-tuned a version of GPT-3 using RLHF with a reward function that emphasized certain human values. It is noteworthy to indicate that the alignment process often exerts a deleterious effect on the performance of generation, commonly referred to as the "alignment tax" in the literature [1]. Specifically, when the reward model assesses only certain specific aspects, it may neglect the quality of the generated output. There has also been another line of work attempting to execute RLHF on visual generative models [15,22,50]. This alignment process can be achieved through prompt learning or fine-tuning the diffusion model. Unlike the LLMs, the image generation process is typically not sequential: the pixels are generated simultaneously. Consequently, PPO is not well adapted to the vision task, and numerous adaptations are required in these works to align the visual generative models. Although PPO is a well-established DRL method with numerous studies showcasing its effectiveness [39,12], it learns in a trial-and-error fashion by interacting with the environment and is generally significantly less stable and less efficient as compared to supervised learning [8]. On the other hand, for fine-tuning generative models using a reward function rather than a pre-determined supervised dataset, collecting (high-quality) samples can be more feasible. In particular, the model can generate a large number of samples that can be used for training. Meanwhile, the reward function provides a useful criterion for selecting high-quality samples. Against this backdrop, we aim to pose a question: Can we enhance the alignment process by leveraging the generative model to generate more training samples and selectively filter them using a reward model? This forms the fundamental motivation behind our algorithm. We propose an alignment framework, RAFT (Reward rAnked FineTuning), which fine-tunes generative models using samples ranked by reward functions in an efficient manner. It is important to note that SFT-like training is more stable and robust compared to PPO, as it directly guides the training process with concrete samples. This type of training facilitates faster learning, with lower sample complexity compared to PPO, resulting in more efficient and expedient convergence. Additionally, PPO training typically requires the simultaneous involvement of three models: the model being trained, the reference model, and the reward model. This places significant pressure on memory resources and entails a substantial burden of gradient computations. In contrast, our model achieves a complete separation between reward-based sample filtering and model training, eliminating the need for gradient computations during the filtering process. As a result, we can better allocate memory and save computational resources. This even enables us to consider incorporating training data from multiple sources, expanding the alignment process beyond the confines of an RL setting. Moreover, our algorithm introduces a limited number of hyperparameters, making it more convenient to adjust. The flexibility empowers our algorithm to perform alignment on arbitrary generative models, including LLMs and diffusion models. Contributions. We propose a novel alignment scheme for generative models called RAFT, which utilizes a reward model to rank the output of the generative model, allowing us to continue training using SFT-like techniques with the selected samples. This approach encourages the generative model to prioritize samples with higher rewards. Compared to PPO, our method offers significant computational advantages, leading to substantial savings in memory and gradient computations. Furthermore, due to the stability of SFT-like training, our approach exhibits lower sample complexity and requires fewer adjustable parameters, which can be implemented to any generative model. We believe this represents a novel and competitive alignment algorithm that contributes to the wellbehaved behavior of generative models. Our experiments on both LLMs and diffusion models have justified the effectiveness of our algorithm, thereby validating its efficacy in the respective application domains. Additionally, these results support the notion that RAFT has the potential to be successful in other related generative models. Related Work Generative foundation model. Foundation models [6] are generally pre-trained on large data and adapted to a broad range of downstream tasks. The roadmap towards the foundation model reveals a transition pattern from discriminative models (e.g., BERT) to generative models (e.g., GPT-3) due to their great scalability. Generative foundation models have reshaped the landscape of natural language processing (NLP), some of which even demonstrate emergent capabilities [47] in complex reasoning tasks. Similar trends are observed in image generation, where diffusion models [5,6,29] have shown great text-to-image generation abilities with the increase of high-quality data and training compute. In particular, diffusion models captures the path from standard Gaussian distribution to the data distribution, which is proven to be successful in a variety of vision tasks, such as image inpainting, super-resolution, text-to-image generation, image denoising [16,11]. Although generative foundation models have pushed the state-of-the-art on various language and vision tasks, respectively, they are suffering from implicit biases, leading to inaccurate or unfair results. Alignment of generative models. Alignment [23] is first proposed to build agents that behave in accordance with the human's intention. By communicating with human, agents can get accurate supervised signals [52] by applying several scalable reward learning methods [23,10,21]. Alignment benefits many recent generative foundation models, like InstructGPT [29], Claude [3] and Sparrow [14], in achieving better performance. In language foundation model training [29,43,28,2,3,14,52,49,38], alignment is often achieved by Reinforcement Learning from Human Feedback (RLHF). The main idea is learning a reward function to reflect human preferences with human annotations and optimize language models by RL methods like proximal policy optimization (PPO) [39]. By incorporating supervised finetuning (SFT), InstructGPT [29] successfully achieved alignment for GPT-3 [7]. Besides, Claude [1,3] and Sparrow [14] stressed aligning language foundation models from helpful, honest, and harmless (HHH) human feedbacks. In visual generative models, several works [15,22,50] studied aligning them with human feedbacks. Models are expected to understand specific visual control signals like colors, counts, and backgrounds [22] more accurately after alignment. It is still challenging to achieve tradeoffs between aligning human preferences and generating high-fidelity images. RRHF [51] is an independent work that is contemporaneous with ours, which also filters samples generated by the generator to serve as training samples for the generative model. It is important to note that RRHF has a diverse range of sources for the generator, whereas our primary focus lies in the online generated samples of the trained model itself, consistent with the setup of RL. Moreover, we also validate the possibility of RAFT on diffusion models beyond the LLMs. Algorithm Problem Setup We adopt the standard RL setting for a clear presentation. We consider an initial generative model G 0 = g(w 0 , x) with model parameter w 0 , which can take input x and generate a random output y according to a distribution p α G0 (y|x), where α is a temperature parameter to control the diversity. We also assume that we have a reward function r(x, y), which returns a reward for any input-output pair (x, y). Due to common usage conventions, we refer to the input as the "prompt". We will use the reward function to guide the outputs of g(w, x). Specifically, if we denote p g (y|w, x) as the conditional distribution of g(w, x), and consider a distribution D of the training input x, the objective of reward optimization is max w E x∼D,y∼pg(·|w,x) r(x, y). (1) RAFT: Reward rAnked FineTuning In this subsection, we introduce our approach RAFT (Reward rAnked FineTuning), which is based on the combination of ranking samples by rewards and supervised fine-tuning. While SFT is widely used in the study of generative models, its effectiveness highly depends on the quality of the dataset utilized. However, achieving a high-quality dataset often necessitates the involvement of domain experts, leading to significant costs and resource requirements. In our problem setup, we have access to a task-specific reward function that can assign a reward signal to each input-response pair (x, y). This provides us with a selective approach to extract only high-reward samples for subsequent fine-tuning. The presence of such a reward function allows us to generate favorable samples at a lower cost and enables us to continuously improve the generative models via supervised fine-tuning on these chosen samples. In view of this, we present a generic alignment algorithm called RAFT (Reward rAnked Fine-Tuning), which can be applied to any tunable generative model. We now formally formulate the above idea by describing the procedure of RAFT. Learning process. Let X = {x 1 , . . . , x n } be a set of n training prompts. Given an initial model g(w 0 , ·), RAFT iteratively updates w 0 as in Algorithm 1. At each stage t, RAFT samples a batch of prompts and generates responses by g(w t−1 , ·). The associated reward of these samples is then computed using the reward function. RAFT subsequently ranks the collected samples and selects the 1/k percent of samples with the highest reward as the training samples B. The current generative model is then fine-tuned on this dataset, and the next stage begins. In this process, the sampling process of training data and the model training are completely decoupled. Moreover, the sampling process does not require any gradient computations, allowing for convenient handling of the sampling procedure and efficient management of computational resources and memory during training. Algorithm 1 RAFT: Reward rAnked FineTuning 1: Input: Prompt set X = {x 1 , . . . , x n }, reward function r(·), initial model G 0 = g(w 0 , ·), acceptance ratio 1/k, batch size b, temperature parameter α. Generate y ∼ p α Gt−1 and compute r(x, y). 3. Model fine-tuning. Fine-tune w t−1 on B to obtain G t = g(w t , ·). 9: end for It is worth mentioning that the three steps of RAFT, i.e., data collection, data ranking, and model fine-tuning, can be implemented and executed separately. Therefore, as long as the computation source and memory source permit SFT on some specific model, the alignment process can be done with RAFT. Moreover, in the practical implementation of RAFT, one can use batch inference and model parallelism to accelerate data collection. Experiments Language Models We consider three tasks for language models. For all the experiments, We use the AdamW optimizer [26] with fixed learning rates, where the details are described for each experiment and each model. We use the PPO algorithm as a baseline, which is implemented via the public package TRL (https: //github.com/lvwerra/trl). The reported metrics are evaluated on a hand-out test set, and we keep the test config for all methods for a fair comparison. The detailed config is also described for each experiment. Movie review completion. The first task is the text continuation on the IMDB dataset 2 , where we hope the generated texts are of positive sentiment. We use the 25K training samples as the training set and randomly sample 3.2K samples from validation samples as the test set. To generate the prompts, we truncate the first 64 tokens of the review as the input prompt. We then use a language model to generate 48 tokens, with the goal of conveying positive sentiment while ensuring maximum fluency. For evaluating the positivity of the generated text, we use a distilled version of the BERT base model [36], which can provide a sentiment score for our experiment. We conduct the experiment with language model LLaMA-7B [45]. To improve the model performance, we first fine-tune the LLaMA-7B model with the 25K training set for one epoch to obtain LLaMA-7B-IMDB as the starting point to be aligned. Meanwhile, we also use a model fine-tuned on the 12.5K training samples with positive labels as a baseline. The hyper-parameters used for the IMDB experiment are described in Table 10, where the parameters of PPO mainly follow from the setup in the TRL package, but we use a smaller KL coefficient because we find that it is hard for PPO to converge with the original choice possibly due to a longer output length we adopt. To facilitate a discussion of the computation complexity, we introduce the concept of cost as, cost = forward × 1 + backward × 2. This is because batch inference in PPO leads to a failure estimation of the KL penalty (described in Section 5). This temporary issue of PPO can be fixed in the future. For a fair comparison, we adopt cost as the metric for evaluation. For instance, if we query a batch of 64 prompts, and train on this set for 4 epochs, then the total cost is 64 × 1 + 64 × 4 × 2 = 576. The backward operation is considered to be more expansive because we typically can take a much larger batch size in the forward process. The relationship between reward and cost is reported in the left figure of Figure 1. As we expect, the SFT-like RAFT is more efficient than PPO in terms of reward learning. While we do not impose any explicit constraint on the policy update, we observe that after the reward exceeds a certain threshold, the fluency of the language model and the diversity of the output degrade significantly. Human investigation suggests that models with reward ∼ 0.8 achieve the best performance as a model with a larger reward (e.g., > 1.5) will generate positive words (e.g.:"I recommend it to all", "The acting is great") regardless of the prompt. Table 2 reports the result of the review completion task. Meanwhile, we report the relationship between perplexity and reward in the right figure of Figure 1, where we take the mean reward as the representative value for models with the same perplexity. Compared to the PPO-aligned model, we observe that RAFT achieves a better balance between reward and perplexity. It turns out that this may arise from the choice of a small learning rate for RAFT since when we use a learning rate identical to PPO (referred to as the RAFT with large lr in the figure), the perplexity is worse than that of PPO. Unfortunately, with more than 10 rounds of experiments, it proved challenging for PPO to converge using the same learning rate as RAFT, as depicted in the left figure of Figure 1. This is due to the noisy learning signals of the RL, which makes it necessary to select a relatively larger learning rate for PPO. Compared to the SFT-aligned model, RAFT achieves a higher reward even if being trained with fewer samples. This improvement can be attributed to the reward function's role in selecting the "core set" of generated reviews, which enables the model to adapt more rapidly to the reward function. Daily dialogue. The original LLaMA-7B model exhibits limitations in responding to daily dialogue, often resulting in inadequate or even nonexistent responses. To address this issue, we aim to align the model with the DailyDialog dataset 3 [24] for dialogue generation. The dataset consists of conversations from humans in daily life and each utterance contains labels of intent and emotional information. For our needs, we opt for a context window of 3 or 5 as the prompt and ask the model to respond given the context. This preprocessing leads to a training set of 35K samples and a test set of 2K samples. We also additionally add prefixes "###Human:" and "###Assistant:" to help to model understand when to respond given the context. Following [32], we use a linear combination of meteor score [4] and intent score as the reward function where the dataset responses are taken as the reference texts. The intent score measures whether the generated text's intent matches the reference's intent. To this end, we train an intent classifier (fine-tuned RoBERTa) that classifies the text into intent categories {inform, question, directive, commissive}. The intent score is then given by the score of the reference's intent from the classifier. A high intent score indicates that the model provides contextually appropriate responses. The hyper-parameters and more implementation details can be found in the Appendix D. We run the experiments with three random seeds and report the best models for each method in Table 3. We can see that the RAFT-aligned model achieves higher rewards than the PPO-aligned one, while maintaining similar levels of diversity metrics and lexical accuracy, and semantic quality. Moreover, the best model of RAFT is obtained at iteration 10 (trained on 2.56K samples in total), while PPO gives the best model when trained on the whole training set (35K samples) for one epoch. The SFT-aligned model by the training set achieves the best performance in this case because the reward function used for this experiment takes the responses in the dataset as the reference texts, so all the samples in the training set achieve the highest rewards even without ranking. Human preference data about helpfulness and harmlessness. The last task uses the HH-RLHF (Helpful and Harmless) dataset 4 [2], which is collected for model alignment according to human preferences. The dataset consists of 112K training samples and 12.5K test samples. Each sample of the HH-RLHF dataset consists of a prompt x and two responses y w and y l where y w is the preferred compared to y l . Due to the pairwise structure of the HH-RLHF dataset, we can follow [29] to improve the model by following the steps: SFT, reward modeling, and RLHF. Due to space constraints, we defer a thorough description of this experiment to Appendix A but first present the evaluation metrics of the aligned model in Table 4. As we can see, both RAFT and PPO significantly outperform the SFT in terms of reward learning. The diversity metrics of RAFT and PPO are worse than the original LLaMA-7B and LLaMA-7B-SFT as the alignment tax. RAFT achieves the best perplexity because after SFT, with appropriate generation config, the model LLaMA-7B-SFT can generate high-quality responses to further improve the model performance. Meanwhile, the case study shows that the RAFT-aligned model tends to respond with more details, thus leading to longer responses and high metrics of unique-1 and unique-2. The PPO-small-lr uses a learning rate identical to that of RAFT and is plotted to illustrate our choice of the learning rate for PPO; (2) The second figure reports the relationship between reward and model perplexity for one representative experiment but the idea remains the same for other random seeds. If one perplexity value corresponds to multiple models, we use the mean reward as the representative value. Diffusion Models We consider to use Stable-diffusion v1.5 (SD-1.5) as our visual generative model (https:// huggingface.co/runwayml/stable-diffusion-v1-5). For all experiments, we use AdamW optimizer with fixed learning rate. It should be noted that for image-related tasks, CLIP [31,20], as a text-image matching score function, can be effectively utilized as a reward function to evaluate the degree of a certain concept. For efficient fine-tuning, we use LoRA [19] in our experiments. Resolution adaptation. Although Stable diffusion was initially trained on a resolution of 256 × 256, due to catastrophic forgetting, SD-1.5 struggles to generate images at this resolution. However, we emphasize that by using a small number of generated samples and the RAFT algorithm, we can restore SD's ability to generate images at 256 × 256 resolution. The reward function is chosen as the CLIP-based aesthetic predictor (https://github.com/LAION-AI/aesthetic-predictor). We use the CIFAR-10 labels as our prompts (airplane, automobile, bird, cat, deer, dog, frog, horse, ship, truck). Figure 2 has clearly demonstrated that with proper reward function, RAFT algorithm can improve the 256 × 256 image quality significantly. We also show that the out-of-domain prompts (such as CIFAR-100 labels) can also be improved significantly (refer to Appendix). Text-Image alignment. For 512 × 512 resolution, SD-1.5 generally produces satisfactory outcomes. The main determinant affecting the generated outputs of SD-1.5 lies in the presentation method SD-1.5 SAMPLES RAFT-ALIGNED SD-1.5 SAMPLES "Edward Hopper style vase" "Edward Hopper style vase" "A golden compass" "A golden compass" "Monet style cat" "Monet style cat" "Da Vinci style apple" "Da Vinci style apple" "An astronaut holding a fish" "An astronaut holding a fish" of prompts, which is because of the inductive bias in training data. Thus, the observed bias in the generated samples is more directly associated with the prompt delivery process. For example, the generator usually puts too much importance on the "style" information and ignore the objects. In such cases, we employ CLIP to evaluate the generated results and utilize the RAFT algorithm to achieve better alignment between the image and text prompts. Specifically, we use the OpenCLIP score with prompt input as the reward function (https://github.com/mlfoundations/open_clip). Figure 3 provide an illustrative case to demonstrate the lack of proper alignment between SD- 1.5 and textual data. It is fortunate that our proposed RAFT algorithm can facilitate the attainment of well-aligned outputs through fine-tuning. Discussion and Extension Simple parameter configuration. Traditionally, the PPO algorithm [39] has been the prominent approach in the area of RLHF for maximizing rewards (1). In practice, to prevent the model from moving too far from the initial model w 0 , PPO optimizes the following modified reward function: r(x, a) = r(x, a) − β log p α Gt (y|x) p α G0 (y|x) ,(2) where β > 0 is a hyper-parameter to control the penalization of the KL divergence between the current policy and the initial policy. Despite that PPO is a well-established DRL method, it requires intricate parameter configurations and extensive code-level tricks to converge. Meanwhile, PPO learns in a trial-and-error manner by interacting with the environment, which is less stable and efficient than supervised learning [8]. On the contrary, RAFT is easy to implement and relies on much fewer hyper-parameters, as summarized in Table 1. Specifically, RAFT's hyper-parameters include (1) batch-size, (2) acceptance ratio, which determines the degree of reward preference, with a smaller ratio leading to the acceptance of only highly rewarding samples, (3) temperature, which regulates the level of randomness, with a larger temperature leading to more diverse samples. In general, the hyper-parameters of RAFT are explainable and clean. Computational consideration. Although RAFT is less sample-efficient in data collection (forward operation) because it discards most of the generated samples, this actually benefits backward propagation because we only train on a subset of the generated dataset. Since backward propagation is typically more computationally expensive than forward operation, this can provide some computational advantages for RAFT as a by-product. Another advantage of RAFT is that its three steps -data collection, data ranking, and model fine-tuning -can be performed separately. As a result, RAFT only needs to load one model at a time, which reduces the GPU memory requirement. In contrast, the PPO-based alignment approach used in LLM typically requires loading three models: the language model, the reference model (for KL estimation), and the reward model (when the reward model itself is an LLM). Therefore, PPO usually requires much more GPU memory to align the generative model. Advanced generation strategy. It turns out that the performance of RAFT heavily relies on the quality of generated samples. In Section 3, we mainly adjust the temperature parameter in our experimental setup. As an extension, we may consider more advanced search methods, including the beam search [34], top-k sampling [13], top-p sampling [18], contrastive search [44], best-of-n sampling [28]. We have adopted the top-k sampling as well as a lower temperature in the test settings for all methods to generate samples with higher quality (see Appendix D for details). Regularization. In practice, the reward function is usually imperfect, and overfitting the reward can be at the cost of the quality of response measured by other metrics such as fluency or diversity. We may leverage a KL penalty as in (2) to obtain a modified reward target. We defer a detailed discussion of these extensions in Appendix C and leave a comprehensive study of their impacts for future study. Conclusion In this paper, we introduced Reward rAnked FineTuning (RAFT), an alternative method for aligning generative models to human preference using a reward function. As compared to the popular PPO algorithm, RAFT is clean and easy to implement with a simple parameter configuration. Moreover, the SFT-like RAFT typically converges more robustly and faster than the DRL approach PPO in reward learning. Notably, since the data generation and fine-tuning of RAFT are decoupled, RAFT can be implemented with less GPU memory source and enjoys greater flexibility in terms of data sources. We hope that RAFT can enrich the toolbox of RLHF and inspire further research and development in the model alignment of building foundational generative models. One potential limitation is that the reward model is usually not robust enough to capture the ground truth reward. During the alignment process, it becomes essential to prevent the model from falling into adversarial examples that can deceive the reward model. This necessitates the incorporation of early stopping to a certain extent. We emphasize the need for further investigation into the implications arising from the imperfect nature of the reward model. A More Experimental Results with LLMs All the experiments are conducted using 8×A100 (40G) with 600G RAM, and half-precision training (bf16). The code will be publicly available on GitHub in the final version. In Section 5, we discuss the implementation of PPO, which necessitates the loading of multiple LLMs concurrently and thus requires a significant amount of memory. Even with half-precision training, the out-of-memory error happens when we compute intermediate values during the training (e.g. attention scores). Following TRL, we use Parameter-Efficient Fine-Tuning (PEFT) in our experiment with the peft library, and perform Low-Rank Adaptation (LoRA) [19] for PPO with all the experiments. We will also conduct a case study for RAFT with LoRA training, using the HH-RLHF dataset, to obtain a more comprehensive comparison. In the rest of this section, we present a thorough description of the experiments on the HH-RLHF dataset. A.1 SFT and Reward Modeling on HH-RLHF Dataset We follow the training procedure outlined by Ouyang et al. [29]. First, we perform SFT on the 112K positive training samples of the HH-RLHF dataset. Then, we use 100K pairwise samples as the training set and a handout evaluation set of 12.5K samples to train the reward model, by adopting the following loss: loss(θ) = −E x,yw,y l ∼Dtrain log(σ(r θ (x, y w ) − r θ (x, y l ))) , where r θ (x, y) is the predicted reward of the model for prompt x and response y, and D train is the empirical distribution of the training set. Here y w is the preferred response, and σ(·) is the sigmoid function. We report the hyper-parameters in Table 5 where we adopt the same parameters for two reward models and report evaluation loss and accuracy in Figure 4. A.2 Reward Maximization: Setup and Implementation Details Prompt dataset. We use a context window of 256 tokens and discard the prompts with more tokens to reduce the burden on the GPU memory resources. This results in a prompt set of 82,147 samples (originally 112K). Parameter tuning. For the RAFT algorithm, we start to search the learning rate with 2 × 10 −5 and ultimately adopt it since it already provides satisfactory alignment performance. We suspect that this is because LLaMA-7B-SFT can already generate high-quality samples and SFT on the generated samples does not hurt the fluency and diversity that much so a larger learning rate also works. For the PPO algorithm, we adopt most of the parameter settings in the IMDB and Daily dialogue experiments, which prove to converge nicely. We primarily tune the weight of the KL penalty due to the different output lengths. We also tune the learning rate. To this end, we start with an experiment without KL penalty, and find that the KL estimation can be of the order ∼ 400 at most when it achieves the highest training reward. Therefore, we search the KL coefficient β in {0.001, 0.01}, where we run each experiment three times independently for up to 80 iterations (i.e., one epoch of the prompt set, roughly 20 hours) and takes β = 0.001 eventually (the same as [2]). For the KL regularization, we follow [52] to set the KL coefficient to be dynamically adapted (the default setting of TRL package). For the learning rate, we experiment with {1 × 10 −5 , 5 × 10 −6 } but find that the smaller learning rate does not bring improvement on the training stability or model performance. Thus, we eventually use 1 × 10 −5 . For the generation configuration, we allow the model to generate up to 128 new tokens given the prompt. For RAFT algorithm, we use a lower temperature 0.85 for a more stable generation quality. For PPO algorithm, we follow the setting in TRL package and do not tune the generation configuration because it seems that the KL estimation can fail when we use a more complicated generation configuration. For instance, we attempt to force the model to generate at least 60 tokens to provide more details, but almost half of the KL estimation becomes negative. We also follow the setting in TRL to use a random generation length for PPO, which is randomly sampled from [64, 128] because we find that a fixed generation length can more easily employ the imperfections of the reward modeling as detailed below. Imperfect reward modeling. It proves challenging to optimize the reward signal while improving the model performance. This is because the reward model we use for the experiment is far from perfect (with 68.27% evaluation accuracy). While we observe that RAFT achieves good performance (in both reward learning and human investigation) in all the experiments with appropriate hyper-parameters and data cleaning process, the PPO-aligned model may overfit the reward function by exploiting these imperfections, which leads to undesired behaviors. Specifically, during the training process of RAFT, we observe that the training reward is positively correlated with response length, which may suggest that the used reward function prefers responses with more details. However, the PPO models can overfit this feature of the reward model by using repetitions of ":):):)" or ";););)" for a higher reward. To address this issue, we monitor the diversity metric for each iteration and such overfitting behavior can be detected as a sudden decrease in these metrics (due to the repetition of the same strings). Once such an undesired pattern happens, we consider the training fails and restart the training. Moreover, we impose the following postprocessing procedure for both RAFT and PPO. Postprocessing. To improve the model performance, we adopt a maximum new token parameter of 128. In some cases, the initial checkpoint LLaMA-7B-SFT replies to the prompt (chat history) with multiple rounds of conversation. We adopt the same postprocessing for both RAFT and PPO by splitting the responses according to ###Human and deleting all the special notations at the end of the responses. Meanwhile, we truncate the trajectory of MDP for the PPO accordingly. We further filter the collected dataset of RAFT to delete the repetition of #, which is applicable because of the decoupled nature between the data generation and fine-tuning in RAFT. RAFT variant. Algorithm 2 presents a variant of RAFT, where we generate k responses for each input prompt and select the response with the highest reward. This is the first version of RAFT, primarily motivated by the best-of-k sampling [28]. Since a global ranking process in Algorithm 1 tends to be more efficient than such a local ranking process within a prompt, we adopt the global ranking as our main algorithm. However, since we model the reward function for the HH-RLHF dataset by a comparison of responses within a prompt, such a local ranking is more appropriate, and we adopt it for this experiment. Since we adopt different temperature parameters for two algorithms, their training rewards are not comparable directly. Therefore, we normalize both reward sequences for the figure to illustrate the convergence behaviors; (2) The second figure reports the reward curves of RAFT, including the training reward, the reward of the best-of-k policy (i.e. the mean reward of the samples used for SFT), and the evaluation reward on the hand-out dataset. Note that we use a smaller temperature for evaluation thus leading to a higher reward. Algorithm 2 RAFT with local ranking 1: Input: Prompt set X = {x 1 , . . . , x n }, reward function r(·), initial model G 0 = g(w 0 , ·), acceptance ratio 1/k, batch size b, temperature parameter α. 2. Data ranking. Generate y 1 , . . . , y k ∼ p α Gt Let y be the output y j with the largest reward r(x, y j ) and insert (x, y) into B; 3. Model fine-tuning. Fine-tune w t−1 on B to obtain G t = g(w t , ·). 9: end for A.3 Result and Discussion Model summary. Table 6 lists the metrics of the representative models of each method, where the models are considered the best models across all independent runs of experiments. As we can see, both RAFT and PPO achieve high rewards and outperform the SFT-aligned model and also the original LLaMA model. In comparison, RAFT tends to reply with more details, as the response of RAFT is usually longer (further shown in the case study presented in Table 9). RAFT also achieves a better perplexity, even though with a learning rate comparable to PPO. We suspect that this is because the base model LLaMA-7B-SFT can already output samples of high quality and fine-tuning on these samples does not hurt the fluency. This can be partially proved by the model improvement from the original LLaMA to the SFT-aligned model. Finally, as the alignment tax, the diversity of the output usually decreases for both methods compared to the starting point. RAFT achieves a competitive trade-off between output diversity and reward learning, including both the model from full training and the model from LoRA training. Training curves. We report the reward curves of PPO and RAFT in Figure 5. The left figure plots the normalized training rewards of PPO and RAFT in terms of the cost (introduced in Section A), averaged across three random seeds. The SFT-like RAFT converges faster than PPO and this partially compensates for the cost of additional generated samples at each iteration, leading to a lower cost. It is worth noting that the practical gap between PPO and RAFT is much larger than their difference in cost. This is because we can set per-GPU batch size as 8 for RAFT, while we can only set it as 3 for PPO, leading to a much slower speed of PPO. We suspect this is because the implementation of PPO is far more complicated than RAFT and requires more intermediate variables. To better illustrate the convergence behavior of RAFT, we report both the reward of the trained model (training reward and eval reward) and the best-of-k model in the right figure. As we can see, the trained model improves steadily by SFT on the subset generated by the best-of-k model and filtered by the reward model. The best-of-k model also benefits from such an improvement of the trained model in turn. Eventually, both of trained model and the best-of-k model converge. Learning stability. RAFT converges nicely for all the experiments with the hyper-parameters specified in Table 12, including both the full training and LoRA training, demonstrating the stability of SFT-based learning. In contrast, the reported results of PPO in this section were averaged across three successful experiments. We observe that the PPO training can fail for one of the following reasons. First, the training reward of PPO sometimes collapses suddenly in the middle of training, where the responses become simple repetitions of certain strings, or the model refuses to reply to the prompt. Second, the KL estimation fails easily when we modify the generation configuration (e.g. an attempt to force the model to generate longer responses by specifying the "minimum length"), which may suggest that we should use the simple generation strategy for PPO training. Finally, as we mentioned in the last subsection, the PPO-aligned model sometimes will converge to the model that achieves a high reward by repeating ":)". We detect such a phenomenon by a sudden drop in the diversity metric and restart the training. We notice that these issues mainly arise due to the difference between the standard RL setting and RLHF for LLMs, where the former always pursues a higher reward, and the latter faced an imperfect RL formulation, so a higher reward might not necessarily lead to better performance. Meanwhile, since the generation and backward propagation of PPO are tightly coupled together, the additional data-cleaning processes (generation configuration tuning, special notations cleaning) are hard to apply. However, when they enter the camp, they all find themselves trapped in this cre LLaMA-PPO: was one of them. I'm not sure if the movie was supposed to be funny when the whole cast of zombies were strolling around in the street, but I was laughing all the way through it. I recommend LLaMA-RAFT: will make you believe that it is inevitable! The scenery, the acting, the story -perfect! I'm not a big fan of horror movies, but this one really got to me. I can't say too much Prompt 2: First of all let us discuss about the story. It is a copy of the movie "Hitch" with an added Indian Flavor to it. One guy, who is a Love guru, and another man who is seemingly a sucker when it comes to ladies, and how this seemingly Responses: Reference: sucker becomes a charmer with the help of the love guru forms the story. Salman Khan is the love guru, and Govinda is the lame guy. Now coming to artists' performance, Salman Khan overacts throughout the movie, he tries to be funny, but fails big time. You can see Salman shouting throughout the movie, no real acting is seen in his performance. Govinda pairs opposite Katrina Kaif(Oh, my god, she is one heck of a girl. LLaMA: sucker gains confidence and becomes a match made in heaven for the love guru. The story is very predictable and you can guess what is going to happen in the end, but the twists and turns in between make it LLaMA-SFT: sucker, in fact is the love guru in disguise. It is the typical Hindi movie story. And you know what, that is OK. Director, Aditya Chopra, has not tried to make an LLaMA-PPO: ordinary guy helps him with his love problems. It is a very funny movie with a superb performance by the character actor Nitin. He is a treat to watch in this movie. It is a must watch movie of the LLaMA-RAFT: sucker is transformed into a Love guru. I found the movie funny, and also very entertaining. The movie has a good story line, and the acting by the lead actors is also good Prompt 3: This low budget crocodile movie really delivers the goods. The fact that it was inspired by true events would mean little if you wound up with a fake looking crocodile, bad C.G.I., or an obvious studio setting. Fortunately none of the above are involved with this ter Responses: Reference: rific, very realistic film. The crocodile is real, there is no C.G.I., and the on location filming takes place in an actual Australian swamp. The actors were obviously inspired to create as much realism as possible in their performances, and they succeed. LLaMA: rific movie. The movie opens with a film crew arriving at the beach to film a documentary about crocodiles. The main character is a photographer who is doing research on the creatures. The crew are surprised when LLaMA-SFT: rific movie. It has a very dark atmosphere, which makes the movie very spooky and thrilling. It's a great story as well, with many twists and shocks. I loved this movie, it's LLaMA-PPO: rific film. It has authentic looking Africa settings and a very impressive crocodile. It has a great story with lots of suspense and tense moments. It has an excellent cast with seasoned veterans and up and com LLaMA-RAFT: rific movie. The crocodile is not only convincing, but looks real enough to be scary. The action is fast paced, and the script is full of great one liners. The cast is excellent, and Table 7: Representative examples of IMDB experiments with randomly sampled prompts. B.2 Diffusion Model Samples All the experiments of diffusion models are performed with Nvidia-3090 with 256G RAM. We will release the code and demos for our paper. Specifically, Figure 6 depicts the samples generated during resolution adaptation without any cherrypicking involved. It is evident that our approach has significantly improved the quality of generated samples. (a) SD- 1.5 (b) SD-1.5 + RAFT Figure 6: Random 256 × 256 generated results of SD-1. 5. Black samples indicate failure cases. It is worth noting that in our experiments conducted at a resolution of 256×256, significant improvements were observed not only for the prompts used during training but also for other prompts. For instance, when using CIFAR-10 labels as samples, notable improvements in the generated quality were observed when utilizing CIFAR-100 labels (Figure 7). This observation highlights the generalization capability of our RAFT algorithm in enhancing sample quality during the alignment process. RAFT-ALIGNED SD-1.5 SAMPLES "Rembrandt style car" "Rembrandt style car" "Submarine" "Submarine" "Paul Delaroche style fish" "Paul Delaroche style fish" SD-1.5 + RAFT "Jacques-Louis David style big ben" "Jacques-Louis David style big ben" "Van Gogh style astronaut" "Van Gogh style astronaut" D Parameter Settings E Usage of RAFT in LMFlow LMFlow (https://github.com/OptimalScale/LMFlow) is a public package, which aims to provide a general and easy-to-use framework for researchers and engineers to finetune/align models. To run example code of RAFT alignment in LMFlow, one may simply execute: ./scripts/run_raft_align.sh By default this aligns GPT-2 base model [30] with the proposed RAFT algorithm on IMDB dataset [27]. To specify LLaMA as the model, one can change the following option in the script: --model_name_or_path {path-to-downloaded-llama-model} with an extra option "--use_lora 1" if LoRA is applied during the alignment process. We also added the diffusion demos in the LMFlow package. 2: for Stage t = 1, . . . , T do 3: 1 . Data collection. Sample a batch D t from X of size b; Figure 1 : 1Training reward of movie review completion on IMDB dataset: (1) The first figure plots the reward with respect to the cost, where the results are averaged over 5 random seeds. Figure 2 : 2(a) SD-1.5 (256 × 256 resolution, aesthetic score = 5.23) (b) RAFT-aligned SD-1.5 (256 × 256 resolution, aesthetic score = 6.55) Resolution Adaptation. (RAFT-aligned models can generate proper 256 × 256 samples) Figure 3 : 3Text-Image Alignment with RAFT. (512×512 resolution) Figure 4 : 4Evaluation loss and evaluation accuracy of reward modeling. The best LLaMA-7B model achieves an accuracy of 79.52% on the 12.5K validation samples, while the best GPE-NEO-2.7B model achieves an accuracy of 68.27%. Figure 5 : 5Reward curves of RAFT and PPO for the HH-RLHF experiment where the results are averaged over 3 random seeds: (1) The first figure plots the training reward curve of PPO and RAFT with respect to the cost. 2: for Stage t = 1, . . . , T do 3: 1 . Data collection. Sample a batch D t from X of size b; Figure 7 : 7Resolution Adaptation (256 × 256 generated results of CIFAR-100 out-of-domain prompts) SD-1.5 SAMPLES Figure 8 : 8Text-Image Alignment with RAFT Furthermore, we have included additional examples of Text-Image Alignment inFigure 8, further demonstrating the crucial role of RAFT alignment in diffusion models. Table 1 : 1Hyper-parameters of RAFT.HYPER-PARAMETER DEFINITION COMMENTS b BATCH SIZE PARALLEL THE TRAINING PROCESS 1/k ACCEPTANCE RATIO LARGE k: HIGHER REWARD PREFERENCE 1/α TEMPERATURE SMALL α: DIVERSE GENERATION Table 2 : 2Result of review completion on IMDB dataset. The results are tested on the 3.2K test samples with 8 random seeds. The model LLaMA-7B-IMDB is pre-trained on the whole IMDB dataset, while the SFT approach only uses the positive samples.MODEL ALIGNMENT MEAN REWARD PPL MSTTR-100 DISTINCT 1 DISTINCT 2 UNIQUE1 UNIQUE 2 LLAMA-7B-IMDB - −0.027 ± 0.010 10.5625 0.653 ± 0.002 0.071 ± 0.003 0.351 ± 0.002 5367 ± 64.4 34414 ± 197 LLAMA-7B SFT 0.250 ± 0.017 10.75 0.653 ± 0.001 0.072 ± 0.001 0.351 ± 0.001 5398 ± 59.9 34501 ± 72.8 LLAMA-7B-IMDB PPO 0.783 ± 0.014 12 0.636 ± 0.001 0.066 ± 0.001 0.322 ± 0.002 4892 ± 38.9 31043 ± 158 LLAMA-7B-IMDB RAFT 0.781 ± 0.018 10.75 0.646 ± 0.001 0.064 ± 0.001 0.324 ± 0.001 4656 ± 71.5 30194 ± 91.2 Table 3 : 3Result of daily dialogue. The results are tested on the 2K test samples and are averaged on 8 random seeds. We omit the standard deviation except for the reward metrics due to space constraints. The SFT method achieves the best performance because the reward function takes the dataset as reference texts to compute the intent and meteor scores.BASE MODEL ALIGNMENT INTENT METEOR BEST SCORE ROUGE1 ROUGE2 BLEU MSTTR-100 DISTINCT 1 DISTINCT 2 LLAMA-7B - 0.437 ± 0.002 0.147 ± 0.002 0.859 0.150 0.030 0.005 0.607 0.102 0.457 LLAMA-7B SFT 0.481 ± 0.007 0.170 ± 0.003 0.866 0.177 0.044 0.012 0.606 0.098 0.452 LLAMA-7B PPO 0.471 ± 0.003 0.161 ± 0.002 0.861 0.158 0.037 0.009 0.615 0.093 0.447 LLAMA-7B RAFT 0.489 ± 0.001 0.161 ± 0.004 0.864 0.170 0.041 0.008 0.600 0.094 0.437 Table 4 : 4Result of experiments on HH-RLHF dataset. The results are tested on the 2K test samples and are averaged on 8 random seeds.BASE MODEL ALIGNMENT MEAN REWARD PPL MSTTR-100 DISTINCT 1 DISTINCT 2 UNIQUE 1 UNIQUE 2 PRED LENGTH LLAMA-7B - 1.724 ± 0.027 4.656 0.588 ± 0.003 0.092 ± 0.001 0.412 ± 0.004 3699 23484 39.7 LLAMA-7B SFT 2.781 ± 0.024 3.031 0.622 ± 0.002 0.081 ± 0.001 0.414 ± 0.002 4689 37303 62.3 LLAMA-7B-SFT PPO 3.448 ± 0.024 3.828 0.596 ± 0.004 0.075 ± 0.001 0.354 ± 0.003 3893 29486 55.5 LLAMA-7B-SFT RAFT 3.458 ± 0.023 3.125 0.599 ± 0.002 0.072 ± 0.001 0.390 ± 0.002 4358 38323 70.1 [ 50 ] 50Xiaoshi Wu, Keqiang Sun, Feng Zhu, Rui Zhao, and HongshengLi. Better aligning text-toimage models with human preference. arXiv preprint arXiv:2303.14420, 2023.[51] Zheng Yuan, Hongyi Yuan, Chuanqi Tan, Wei Wang, Songfang Huang, and Fei Huang. Rrhf:Rank responses to align language models with human feedback without tears.arXiv preprint arXiv:2304.05302, 2023. [52] Daniel M Ziegler, Nisan Stiennon, Jeffrey Wu, Tom B Brown, Alec Radford, Dario Amodei, Paul Christiano, and Geoffrey Irving. Fine-tuning language models from human preferences. arXiv preprint arXiv:1909.08593, 2019. Table 5 : 5Hyper-parameters reward modeling on HH-RLHF dataset.MODELS HYPER-PARAMETER VALUE LEARNING RATE 2 × 10 −5 SFT DECAY MODE LINEAR DECAY EPOCH 1 BATCH SIZE 32 LORAL RANK r 16 LORA ALPHA 32 REWARD MODELING LORA DROPOUT 0.1 LEARNING RATE 3 × 10 −5 DECAY MODE LINEAR DECAY EPOCH 1 GPT-Neo-2.7B achieves an evaluation accuracy of 68.27%, comparable to the GPT-J-6B model 5 . Moreover, LLaMA-7B achieves an evaluation accuracy of 79.52% that significantly outperforms GPT-Neo-2.7B. However, the PPO model requires loading the language model and reward model at the same time. During our current implementation with TRL, we encountered an out-of-memory error when attempting to train the model using 8×A100 (40G). It became evident that this issue arose during the computation of certain intermediate variables, rendering the training process infeasible. Therefore, we choose the GPT-Neo-2.7B as our reward model in this experiment. Notably, since the data generation, data ranking, and SFT in RAFT can be performed separately, as long as we can fine-tune the model, we can also align the model with RAFT. Table 6 : 6Complete table of results on HH-RLHF dataset.The results are tested on the 2K test samples and are averaged on 8 random seeds. The LLaMA-7B-SFT is the SFT-aligned model.BASE MODEL ALIGNMENT MEAN REWARD PPL MSTTR-100 DISTINCT 1 DISTINCT 2 UNIQUE 1 UNIQUE 2 PRED LENGTH LLAMA-7B - 1.724 ± 0.027 4.656 0.588 ± 0.003 0.092 ± 0.001 0.412 ± 0.004 3699 23484 39.7 LLAMA-7B SFT 2.781 ± 0.024 3.031 0.622 ± 0.002 0.081 ± 0.001 0.414 ± 0.002 4689 37303 62.3 LLAMA-7B-SFT PPO 3.448 ± 0.024 3.828 0.596 ± 0.004 0.075 ± 0.001 0.354 ± 0.003 3893 29486 55.5 LLAMA-7B-SFT RAFT-LORA 3.444 ± 0.028 3.156 0.601 ± 0.002 0.073 ± 0.001 0.389 ± 0.002 4401 37586 69.0 LLAMA-7B-SFT RAFT 3.451 ± 0.025 3.281 0.609 ± 0.002 0.074 ± 0.001 0.396 ± 0.002 4703 40920 72.6 LLAMA-7B-SFT RAFT 3.458 ± 0.023 3.125 0.599 ± 0.002 0.072 ± 0.001 0.390 ± 0.002 4358 38323 70.1 For LLM, we provide examples of experiments on IMDB, Daily Dialogue, and HH-RLHF. Just saw this movie yesterday night and I almost cried. No, it wasn't because it got me utterly petrified, no. It was absolutely HORRENDOUS! Sometimes, you see movies that make you wonder what will become of the human race in the near future -this movieResponses:Reference: is one of those. It's as though the writer, actors, director, et al, just came together and copied and pasted scenes of their favorite horror flicks, zipped it all together and said "hey, here's Satan's whip!!!" After seeing this movie, I could not help but be tormented by the sight of people whom call themselves "actors"; waltzing around like they're some kind of talented artistic interpreters.LLaMA: is the answer. It was a movie that made me laugh, cry, and terrified me to the core. I was literally screaming and crying throughout the movie. I have watched many horror movies in my life LLaMA-SFT: is one of those. It's a story about a group of students that go to the school camp and have asessionön surviving.B Examples B.1 LLM Samples IMDB Movie Review Examples Prompt 1: Table 10 : 10Hyper-parameters-IMDB for fine-tuning LLaMA-7B. MAX NEW WOKEN 48 LORA RANK, ALPHA, DROPOUT (16, 32, 0.05)MODELS HYPER-PARAMETER VALUE LEARNING RATE 2 × 10 −5 SFT DECAY MODE LINEAR DECAY EPOCH 1 BATCH SIZE 32 BATCH SIZE b 1280 UPDATE EPOCHS FOR EACH STAGE 4 RAFT LEARNING RATE 2 × 10 −6 ACCEPTANCE RATIO 1/k 0.2 TEMPERATURE 1/α 1 MAX NEW WOKEN 128 STEPS PER UPDATE 1024 UPDATE EPOCHS FOR EACH STAGE 4 LEARNING RATE 1 × 10 −5 KL COEFFICIENT 0.05 DISCOUNT FACTOR 1 PPO CLIP RATIO 0.2 GAE PARAMETER 0.95 TEMPERATURE 1/α 1 TOP K 0.0 TEST SETTINGS TEMPERATURE 1/α 0.7 MAX NEW TOKEN 48 DO SAMPLE TRUE Table 11 : 11Hyper-parameters-Daily-Dialogue for fine-tuning LLaMA-7B. We adopt the same strategy for all models to truncate the responses so that only take one round of conversation as the model output for both the training and testing.MODELS HYPER-PARAMETER VALUE BATCH SIZE b 1280 UPDATE EPOCHS FOR EACH STAGE 4 RAFT LEARNING RATE 2 × 10 −6 ACCEPTANCE RATIO 1/k 0.2 TEMPERATURE 1/α 1 MAX NEW WOKEN 48 STEPS PER UPDATE 1024 UPDATE EPOCHS FOR EACH STAGE 4 LEARNING RATE 2 × 10 −5 PPO KL COEFFICIENT 0.05 DISCOUNT FACTOR 1 CLIP RATIO 0.2 GAE PARAMETER 0.95 TEMPERATURE 1/α 1 MAX NEW WOKEN 48 LORA RANK, ALPHA, DROPOUT (16, 32, 0.05) TOP K 20 TEST SETTINGS TEMPERATURE 1/α 0.7 MAX NEW TOKEN 48 DO SAMPLE TRUE Table 12 : 12Hyper-parameters of HH-RLHF experiment for fine-tuning LLaMA-7B.MODELS HYPER-PARAMETER VALUE BATCH SIZE b 1024 × 8 UPDATE EPOCHS FOR EACH STAGE 3 RAFT LEARNING RATE 2 × 10 −5 ACCEPTANCE RATIO 1/k 1/8 TEMPERATURE 1/α 0.85 MAX NEW WOKEN 128 RAFT-LORA LORA RANK, ALPHA, DROPOUT (16, 32, 0.05) OTHER HYPER-PARAMETERS SAME AS RAFT STEPS PER UPDATE 1024 UPDATE EPOCHS FOR EACH STAGE 4 LEARNING RATE 2 × 10 −5 PPO KL COEFFICIENT 0.001 DISCOUNT FACTOR 1 CLIP RATIO 0.2 GAE PARAMETER 0.95 TEMPERATURE 1/α 1 MAX NEW WOKEN [64, 128] LORA RANK, ALPHA, DROPOUT (16, 32, 0.05) TOP K 40 TEST SETTINGS TEMPERATURE 1/α 0.7 MAX NEW TOKEN 128 DO SAMPLE TRUE Table 13 : 13Hyper-parameters for fine-tuning SD-1.5.BATCH SIZE b 10 RESOLUTION NO. OF ITERATIONS FOR EACH STAGE BATCH SIZE b 1 TEXT-IMAGE NO. OF ITERATIONS FOR EACH STAGETASK HYPER-PARAMETER VALUE 100 ADAPTATION LEARNING RATE 6 × 10 −6 ACCEPTANCE RATIO 1/k 0.05 800 ALIGNMENT LEARNING RATE 3 × 10 −6 ACCEPTANCE RATIO 1/k 0.05 https://huggingface.co/datasets/imdb 3 https://huggingface.co/datasets/daily_dialog https://huggingface.co/datasets/Dahoas/full-hh-rlhf https://huggingface.co/Dahoas/gptj-rm-static C.2 Potential ExtensionsIt turns out that the performance of the our SFT-like algorithm RAFT heavily relies on the quality of the generated data. In what follows, we discuss several potential approaches to improve the quality of the generated samples for future study.Expert Generator as Data Source. The discussion in this paper follows from the standard RL workflow for a better understanding. Thanks to the decoupled nature of data generation and finetuning in RAFT, we can also incorporate other data sources in addition to the trained model itself. For instance, we can leverage some expert generators (e.g. GPT4 or human) to generate (part of) the responses given the prompt. A more straightforward approach is to perform some prompt engineering in the data generation process, where there is rich literature showcasing that it can largely improve the generation quality[25]. It is known that in-context learning[7,48]improves LLM performance, especially for those challenging logical reasoning tasks. Given the input prompt is x, instead of using x directly, we may add some additional context and input the new promptx to the model and get the response y. In other words, we can obtain an "expert" generator through proper prompt engineering. For diffusion models, it is also applicable that powerful models (e.g. Midjourney) and proper prompts can provide better generation quality.Regularization. We note that we can also incorporate a KL penalty into our reward function. 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Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yogatama, Maarten Bosma, Denny Zhou, Donald Metzler, Ed H Chi, Tatsunori Hashimoto, Oriol Vinyals, Percy Liang, Jeff Dean, William Fedus, Transactions on Machine Learning Research. Survey CertificationJason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yogatama, Maarten Bosma, Denny Zhou, Donald Metzler, Ed H. Chi, Tatsunori Hashimoto, Oriol Vinyals, Percy Liang, Jeff Dean, and William Fedus. Emergent abilities of large language models. Transactions on Machine Learning Research, 2022. URL https://openreview. net/forum?id=yzkSU5zdwD. Survey Certification. Chain of thought prompting elicits reasoning in large language models. Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Ed Chi, Quoc Le, Denny Zhou, arXiv:2201.11903arXiv preprintJason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Ed Chi, Quoc Le, and Denny Zhou. Chain of thought prompting elicits reasoning in large language models. arXiv preprint arXiv:2201.11903, 2022. Jeff Wu, Long Ouyang, M Daniel, Nisan Ziegler, Ryan Stiennon, Jan Lowe, Paul Leike, Christiano, arXiv:2109.10862Recursively summarizing books with human feedback. arXiv preprintJeff Wu, Long Ouyang, Daniel M Ziegler, Nisan Stiennon, Ryan Lowe, Jan Leike, and Paul Christiano. Recursively summarizing books with human feedback. arXiv preprint arXiv:2109.10862, 2021. LLaMA-SFT: Yes, exactly. In general, it's important to be clear with your child about why they're doing something. It's also important to offer them incentives that are meaningful to them. For example, if you tell your child that they have to do their homework because you said so, that's not a very good incentive for them. However, if you tell them that they'll be able to use the LLaMA-PPO: Yes, that's one strategy you can use. It can help motivate them to do their homework. Good luck with your students! I hope this helps. Have a great day! LLaMA-RAFT: Yes, that's a good strategy! Kids often think about what they want right now, so offering them something in exchange for doing their homework can be a powerful motivator. You might also want to offer them some kind of reward if they do well on their homework, to encourage them to do their best. Offering them some kind of incentive can help them want to do well, and also it can help you both stay on track. Daily Dialogue Examples Prompt 1: Human: A reasonable number of sick days will be covered by the company. Any extended illness will be covered by insurance. Have you read the other terms of the contract? Assistant: Yes, I have. Human: Do you have any other questions? Assistant: Just one. I noticed an item about flex-time. Is that a possibility for me? Human: Yes, it is, but you'll have to discuss the details with your manager. Assistant: Responses: Reference: That's acceptable. U.S.; U.S3LLaMA-SFT: I could search for more information, but I'll just share what I've found. Here's a review of the conditions in 1935 and 1938. It looks like the storms had similar tracks, but 1938 was stronger in some ways. In 1938 there were more people living on the coast, and the hurricane was more intense. LLaMA-PPO: Yes, the hurricane in 1935 was stronger than Katrina. The 1938 storm was bigger. I hope this helps. Good luck with your project! LLaMA-RAFT: Yes, the 1938 hurricane was the most intense storm that year, and the second-most intense storm ever recorded. It caused over 6,000 deaths. I'm not sure about the Florida Keys storm you mentioned, but it sounds like it was a major storm as well. It's very interesting that there were two such intense storms in such a short time periodDaily Dialogue Examples Prompt 1: Human: A reasonable number of sick days will be covered by the company. Any extended illness will be covered by insurance. Have you read the other terms of the contract? Assistant: Yes, I have. Human: Do you have any other questions? Assistant: Just one. I noticed an item about flex-time. Is that a possibility for me? Human: Yes, it is, but you'll have to discuss the details with your manager. Assistant: Responses: Reference: That's acceptable. LLaMA: I understand. I'll see him tomorrow. LLaMA-SFT: That's fine. I'm very satisfied with the contract. Thanks for your help. LLaMA-PPO: I understand. I hope I will be able to take advantage of flex-time. LLaMA-RAFT: That's fine. I'm sure we can work something out. Prompt 2: Human: Hello, Mr. Black, how are you ? Assistant: Fine, thank you, and how are you ? Human: Very well, thank you. It's nice to meet you again. I am awfully sorry for not being able to meet you at the airport, as I was busy yesterday. I do hope you had a very enjoyable trip from New York. Assistant: Responses: Reference: Yes, a very pleasant journey indeed. Thank you. LLaMA: Thank you very much. I had a very good time in New York myself. LLaMA-SFT: Yes, I did. But I'm glad to meet you again. LLaMA-PPO: It was very pleasant. I had a good time . LLaMA-RAFT: Yes, it was very comfortable. The flight was very pleasant. Prompt 3: Human: Anything else ? Assistant: Yes, one last thing. How much holiday time do you give your employees every year ? Human: Our employees receive 15 days of paid vacation every year. If you don't use the full 15 days, they carry over to the following year . Assistant: Responses: Reference: How about sick days? LLaMA: Is there an option for unpaid sick days ? LLaMA-SFT: Do you get an additional 10 days of sick leave? LLaMA-PPO: How about your employees ? LLaMA-RAFT: That's pretty good. How about sick leave? Table 8: Representative examples of daily dialogue experiments with randomly sampled prompts. HH-RLHF Examples Prompt 1: Human: How can I get my girlfriend to cook more? Assistant: Have you tried reminding her of how nice the food tastes? Human: I could do it more. Assistant: Responses: LLaMA: When you cook do you use a lot of spices? LLaMA-SFT: That's an easy one, so it may not be the most effective strategy. Do you know of any other strategies? LLaMA-PPO: That might help. Also, make sure the kitchen is always clean and organized. This will help keep things fresh for cooking. Good luck! LLaMA-RAFT: That's a good idea! You can also try complimenting her on her cooking, and offering to help her in the kitchen. She might enjoy cooking more if she feels appreciated. You could also suggest recipes that she might like, or help her create a menu. Cooking can be a lot of fun, and it's a great way to spend time together. Prompt 2: Human: How do you get a child to do homework? Assistant: There are many strategies, but you can try asking them why they should do it. Ask them if there's anything they'd like to gain from doing their homework. Try asking if there's anything you can do for them, and use those as motivation for them to do their homework. You can also remind them that they'll be rewarded with extra privileges, like video games, if they do their homework. Alternatively, you can incentivize them by including them in your plans, and offering them rewards if they get good grades. Human: So make it so there is something in it for them? Assistant: There was another one, called the Great Hurricane of 1938, that was bigger and stronger. Human: Are you sure the date is correct? I know there was a strong storm in 1935 that hit the Florida Keys but that was before they started naming storms. Is that the one you are referring to? Assistant: Responses: LLaMA: Yes. You can remind them that they can do anything they want, but if they do their homework, you can do things with them. You can also try to make your time together enjoyable if they do their homework and get good grades. LLaMA-SFT: Yes, exactly. In general, it's important to be clear with your child about why they're doing something. It's also important to offer them incentives that are meaningful to them. For example, if you tell your child that they have to do their homework because you said so, that's not a very good incentive for them. However, if you tell them that they'll be able to use the LLaMA-PPO: Yes, that's one strategy you can use. It can help motivate them to do their homework. Good luck with your students! I hope this helps. Have a great day! LLaMA-RAFT: Yes, that's a good strategy! Kids often think about what they want right now, so offering them something in exchange for doing their homework can be a powerful motivator. You might also want to offer them some kind of reward if they do well on their homework, to encourage them to do their best. Offering them some kind of incentive can help them want to do well, and also it can help you both stay on track. Prompt 3: Human: I am doing a project on weather disasters in the United States. Can you tell me what the strongest hurricane was to ever hit the U.S. was? Assistant: I'm going to guess you're thinking of the Hurricane Katrina in 2005. Human: Well, I'm not really thinking of anything in particular. Was Katrina the strongest hurricane to ever hit the U.S.? Assistant: There was another one, called the Great Hurricane of 1938, that was bigger and stronger. Human: Are you sure the date is correct? I know there was a strong storm in 1935 that hit the Florida Keys but that was before they started naming storms. Is that the one you are referring to? Assistant: Responses: LLaMA: Yeah, the Great Hurricane of 1938. LLaMA-SFT: I could search for more information, but I'll just share what I've found. Here's a review of the conditions in 1935 and 1938. It looks like the storms had similar tracks, but 1938 was stronger in some ways. In 1938 there were more people living on the coast, and the hurricane was more intense. LLaMA-PPO: Yes, the hurricane in 1935 was stronger than Katrina. The 1938 storm was bigger. I hope this helps. Good luck with your project! LLaMA-RAFT: Yes, the 1938 hurricane was the most intense storm that year, and the second-most intense storm ever recorded. It caused over 6,000 deaths. I'm not sure about the Florida Keys storm you mentioned, but it sounds like it was a major storm as well. It's very interesting that there were two such intense storms in such a short time period. Representative examples of HH-RLHF experiments with randomly sampled prompts. 9Table 9: Representative examples of HH-RLHF experiments with randomly sampled prompts. (x n , y n )}, r(·), G 0 = g(w 0 , ·), k, b. 2: Randomly choose k samples from D as B, and save their rewards. 3: for Stage t = 1. . , Offline RAFT 1: Input: D = {(x 1 , y 1 ). T do 4: Sample a batch B b from D of size bOffline RAFT 1: Input: D = {(x 1 , y 1 ), . . . , (x n , y n )}, r(·), G 0 = g(w 0 , ·), k, b. 2: Randomly choose k samples from D as B, and save their rewards. 3: for Stage t = 1, . . . , T do 4: Sample a batch B b from D of size b; 7: if r > min (x ′ ,y ′ )∈B r(x ′ , y ′ ) then 8: Remove argmin (x ′ ,y ′ )∈B r(x ′ , y ′ ) from B; 9: Insert (x, y) and r(x, y) to B. Compute reward r(x, y). 6to obtain G t = g(w t , ·). 13: end for5: for x, y ∈ B b do 6: Compute reward r(x, y). 7: if r > min (x ′ ,y ′ )∈B r(x ′ , y ′ ) then 8: Remove argmin (x ′ ,y ′ )∈B r(x ′ , y ′ ) from B; 9: Insert (x, y) and r(x, y) to B. to obtain G t = g(w t , ·). 13: end for

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{'abstract': 'Generative foundation models are susceptible to implicit biases that can arise from extensive unsupervised training data. Such biases can produce suboptimal samples, skewed outcomes, and unfairness, with potentially significant repercussions. Consequently, aligning these models with human ethics and preferences is an essential step toward ensuring their responsible and effective deployment in real-world applications. Prior research has primarily employed Reinforcement Learning from Human Feedback (RLHF) as a means of addressing this problem, wherein generative models are fine-tuned using RL algorithms guided by a humanfeedback-informed reward model. However, the inefficiencies and instabilities associated with RL algorithms frequently present substantial obstacles to the successful alignment of generative models, necessitating the development of a more robust and streamlined approach. To this end, we introduce a new framework, Reward rAnked FineTuning (RAFT), designed to align generative models more effectively. Utilizing a reward model and a sufficient number of samples, our approach selects the high-quality samples, discarding those that exhibit undesired behavior, and subsequently assembles a streaming dataset. This dataset serves as the basis for aligning the generative model and can be employed under both offline and online settings. Notably, the sample generation process within RAFT is gradient-free, rendering it compatible with black-box generators. Through extensive experiments, we demonstrate that our proposed algorithm exhibits strong performance in the context of both large language models and diffusion models. * Equal Contribution. Alphabetical order.Preprint. Under review.', 'arxivid': '2304.06767', 'author': ['Hanze Dong \nThe Hong Kong University of Science and Technology\n\n', 'Wei Xiong \nThe Hong Kong University of Science and Technology\n\n', 'Deepanshu Goyal \nThe Hong Kong University of Science and Technology\n\n', 'Rui Pan \nThe Hong Kong University of Science and Technology\n\n', 'Shizhe Diao \nThe Hong Kong University of Science and Technology\n\n', 'Jipeng Zhang \nThe Hong Kong University of Science and Technology\n\n', 'Kashun Shum \nThe Hong Kong University of Science and Technology\n\n', 'Tong Zhang \nThe Hong Kong University of Science and Technology\n\n'], 'authoraffiliation': ['The Hong Kong University of Science and Technology\n', 'The Hong Kong University of Science and Technology\n', 'The Hong Kong University of Science and Technology\n', 'The Hong Kong University of Science and Technology\n', 'The Hong Kong University of Science and Technology\n', 'The Hong Kong University of Science and Technology\n', 'The Hong Kong University of Science and Technology\n', 'The Hong Kong University of Science and Technology\n'], 'corpusid': 258170300, 'doi': '10.48550/arxiv.2304.06767', 'github_urls': ['https://github.com/LAION-AI/aesthetic-predictor).', 'https://github.com/mlfoundations/open_clip).', 'https://github.com/OptimalScale/LMFlow)'], 'n_tokens_mistral': 27340, 'n_tokens_neox': 23743, 'n_words': 14089, 'pdfsha': '0a5cf81763d90c23cfb5180894c791debf8f807e', 'pdfurls': ['https://export.arxiv.org/pdf/2304.06767v2.pdf'], 'title': ['RAFT: Reward rAnked FineTuning for Generative Foundation Model Alignment', 'RAFT: Reward rAnked FineTuning for Generative Foundation Model Alignment'], 'venue': []}

arxiv

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH Hadron Transverse Momentum Distributions in Muon Deep Inelastic Scattering at 160 GeV/c The COMPASS Collaboration The COMPASS Collaboration 18000. 16636 C Adolph Physikalisches Institut Universität Erlangen-Nürnberg 91054ErlangenGermany M G Alekseev Trieste Section of INFN 34127TriesteItaly V Yu Alexakhin Joint Institute for Nuclear Research Moscow region141980DubnaRussia Yu Alexandrov Lebedev Physical Institute 119991MoscowRussia G D Alexeev Joint Institute for Nuclear Research Moscow region141980DubnaRussia A Amoroso Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly V Andrieux A Austregesilo CERN 1211Geneva 23Switzerland Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France B Badełek Faculty of Physics University of Warsaw 00-681WarsawPoland F Balestra Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly J Barth Physikalisches Institut Universität Bonn 53115BonnGermany G Baum Universität Bielefeld Fakultät für Physik 33501BielefeldGermany Y Bedfer A Berlin Institut für Experimentalphysik Universität Bochum 44780BochumGermany J Bernhard Institut für Kernphysik Universität Mainz 55099MainzGermany R Bertini Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly K Bicker CERN 1211Geneva 23Switzerland Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France J Bieling Physikalisches Institut Universität Bonn 53115BonnGermany R Birsa Trieste Section of INFN 34127TriesteItaly J Bisplinghoff Helmholtz-Institut für Strahlen-und Kernphysik Universität Bonn 53115BonnGermany M Boer P Bordalo LIP 1000-149LisbonPortugal F Bradamante Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly C Braun Physikalisches Institut Universität Erlangen-Nürnberg 91054ErlangenGermany A Bravar Trieste Section of INFN 34127TriesteItaly A Bressan Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly M Büchele Physikalisches Institut Universität Freiburg 79104FreiburgGermany E Burtin L Capozza M Chiosso Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly S U Chung Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France A Cicuttin Abdus Salam ICTP and Trieste Section of INFN 34127TriesteItaly M L Crespo Abdus Salam ICTP and Trieste Section of INFN 34127TriesteItaly S Dalla Torre Trieste Section of INFN 34127TriesteItaly S S Dasgupta Matrivani Institute of Experimental Research & Education Calcutta-700 030India S Dasgupta Trieste Section of INFN 34127TriesteItaly O Yu Denisov Torino Section of INFN 10125TurinItaly S V Donskov N Doshita Yamagata University 992-8510YamagataJapan Universidade Técnica de Lisboa LisbonPortugal V Duic Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly W Dünnweber Department für Physik Ludwig-Maximilians-Universität München 80799MunichGermany io M Dziewiecki Institute of Radioelectronics Warsaw University of Technology 00-665WarsawPoland A Efremov Joint Institute for Nuclear Research Moscow region141980DubnaRussia C Elia Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly P D Eversheim Helmholtz-Institut für Strahlen-und Kernphysik Universität Bonn 53115BonnGermany W Eyrich Physikalisches Institut Universität Erlangen-Nürnberg 91054ErlangenGermany M Faessler Department für Physik Ludwig-Maximilians-Universität München 80799MunichGermany io A Ferrero A Filin M Finger M. Finger jrH 19 Fischer Physikalisches Institut Universität Freiburg 79104FreiburgGermany C Franco LIP 1000-149LisbonPortugal N Du Fresne Von Hohenesche CERN 1211Geneva 23Switzerland Institut für Kernphysik Universität Mainz 55099MainzGermany J M Friedrich Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France V Frolov CERN 1211Geneva 23Switzerland R Garfagnini Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly F Gautheron Institut für Experimentalphysik Universität Bochum 44780BochumGermany O P Gavrichtchouk Joint Institute for Nuclear Research Moscow region141980DubnaRussia S Gerassimov Lebedev Physical Institute 119991MoscowRussia Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France R Geyer Department für Physik Ludwig-Maximilians-Universität München 80799MunichGermany io M Giorgi Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly I Gnesi Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly B Gobbo Trieste Section of INFN 34127TriesteItaly S Goertz Physikalisches Institut Universität Bonn 53115BonnGermany S Grabmüller Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France A Grasso Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly B Grube Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France R Gushterski Joint Institute for Nuclear Research Moscow region141980DubnaRussia A Guskov Joint Institute for Nuclear Research Moscow region141980DubnaRussia T Guthörl Physikalisches Institut Universität Freiburg 79104FreiburgGermany F Haas Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France D Von Harrach Institut für Kernphysik Universität Mainz 55099MainzGermany F H Heinsius Physikalisches Institut Universität Freiburg 79104FreiburgGermany F Herrmann Physikalisches Institut Universität Freiburg 79104FreiburgGermany C Heß Institut für Experimentalphysik Universität Bochum 44780BochumGermany F Hinterberger Helmholtz-Institut für Strahlen-und Kernphysik Universität Bonn 53115BonnGermany Ch Höppner Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France N Horikawa Also at Chubu University 487-8501KasugaiAichiJapan n N D&apos;hose Also at KEK 1-1 Oho, Tsukuba, Ibaraki305-0801Japan S Huber Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France S Ishimoto Yamagata University 992-8510YamagataJapan Universidade Técnica de Lisboa LisbonPortugal Also at KEK 1-1 Oho, Tsukuba, Ibaraki305-0801Japan Yu Ivanshin Joint Institute for Nuclear Research Moscow region141980DubnaRussia T Iwata Yamagata University 992-8510YamagataJapan Universidade Técnica de Lisboa LisbonPortugal R Jahn Helmholtz-Institut für Strahlen-und Kernphysik Universität Bonn 53115BonnGermany V Jary P Jasinski Institut für Kernphysik Universität Mainz 55099MainzGermany R Joosten Helmholtz-Institut für Strahlen-und Kernphysik Universität Bonn 53115BonnGermany E Kabuß Institut für Kernphysik Universität Mainz 55099MainzGermany D Kang Institut für Kernphysik Universität Mainz 55099MainzGermany B Ketzer Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France G V Khaustov Yu A Khokhlov Also at Moscow Institute of Physics and Technology, Moscow Region III. Physikalisches Institut RWTH Aachen University 141700, 52056AachenRussia, Germany Yu Kisselev Institut für Experimentalphysik Universität Bochum 44780BochumGermany F Klein Physikalisches Institut Universität Bonn 53115BonnGermany K Klimaszewski National Centre for Nuclear Research 00-681WarsawPoland J H Koivuniemi Institut für Experimentalphysik Universität Bochum 44780BochumGermany V N Kolosov K Kondo Yamagata University 992-8510YamagataJapan Universidade Técnica de Lisboa LisbonPortugal K Königsmann Physikalisches Institut Universität Freiburg 79104FreiburgGermany I Konorov Lebedev Physical Institute 119991MoscowRussia Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France V F Konstantinov A M Kotzinian Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly O Kouznetsov Joint Institute for Nuclear Research Moscow region141980DubnaRussia M Krämer Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France Z V Kroumchtein Joint Institute for Nuclear Research Moscow region141980DubnaRussia N Kuchinski Joint Institute for Nuclear Research Moscow region141980DubnaRussia F Kunne K Kurek National Centre for Nuclear Research 00-681WarsawPoland R P Kurjata Institute of Radioelectronics Warsaw University of Technology 00-665WarsawPoland A A Lednev A Lehmann Physikalisches Institut Universität Erlangen-Nürnberg 91054ErlangenGermany S Levorato Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly J Lichtenstadt School of Physics and Astronomy Tel Aviv University 69978 A Maggiora Torino Section of INFN 10125TurinItaly A Magnon N Makke Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly G K Mallot CERN 1211Geneva 23Switzerland A Mann Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France C Marchand A Martin Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly J Marzec Institute of Radioelectronics Warsaw University of Technology 00-665WarsawPoland H Matsuda Yamagata University 992-8510YamagataJapan Universidade Técnica de Lisboa LisbonPortugal T Matsuda University of Miyazaki 889-2192MiyazakiJapan G Meshcheryakov Joint Institute for Nuclear Research Moscow region141980DubnaRussia W Meyer Institut für Experimentalphysik Universität Bochum 44780BochumGermany T Michigami Yamagata University 992-8510YamagataJapan Universidade Técnica de Lisboa LisbonPortugal Yu V Mikhailov Y Miyachi Yamagata University 992-8510YamagataJapan Universidade Técnica de Lisboa LisbonPortugal A Morreale A Nagaytsev Joint Institute for Nuclear Research Moscow region141980DubnaRussia T Nagel Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France F Nerling Physikalisches Institut Universität Freiburg 79104FreiburgGermany S Neubert Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France D Neyret V I Nikolaenko J Novy W.-D Nowak Physikalisches Institut Universität Freiburg 79104FreiburgGermany A S Nunes LIP 1000-149LisbonPortugal A G Olshevsky Joint Institute for Nuclear Research Moscow region141980DubnaRussia M Ostrick Institut für Kernphysik Universität Mainz 55099MainzGermany R Panknin Physikalisches Institut Universität Bonn 53115BonnGermany D Panzieri Torino Section of INFN University of Eastern Piedmont 15100, 10125Alessandria, TurinItaly B Parsamyan Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly S Paul Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France G Piragino Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly S Platchkov J Pochodzalla Institut für Kernphysik Universität Mainz 55099MainzGermany J Polak Technical University in Liberec 46117LiberecCzech Republic Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly V A Polyakov J Pretz Physikalisches Institut Universität Bonn 53115BonnGermany M Quaresma LIP 1000-149LisbonPortugal C Quintans LIP 1000-149LisbonPortugal J.-F Rajotte Department für Physik Ludwig-Maximilians-Universität München 80799MunichGermany io S Ramos LIP 1000-149LisbonPortugal G Reicherz Institut für Experimentalphysik Universität Bochum 44780BochumGermany E Rocco CERN 1211Geneva 23Switzerland V Rodionov Joint Institute for Nuclear Research Moscow region141980DubnaRussia E Rondio National Centre for Nuclear Research 00-681WarsawPoland N S Rossiyskaya Joint Institute for Nuclear Research Moscow region141980DubnaRussia D I Ryabchikov V D Samoylenko A Sandacz National Centre for Nuclear Research 00-681WarsawPoland M G Sapozhnikov Joint Institute for Nuclear Research Moscow region141980DubnaRussia S Sarkar Matrivani Institute of Experimental Research & Education Calcutta-700 030India I A Savin Joint Institute for Nuclear Research Moscow region141980DubnaRussia G Sbrizzai Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly P Schiavon Department of Physics and Trieste Section of INFN University of Trieste 34127TriesteItaly C Schill Physikalisches Institut Universität Freiburg 79104FreiburgGermany T Schlüter Department für Physik Ludwig-Maximilians-Universität München 80799MunichGermany io A Schmidt Physikalisches Institut Universität Erlangen-Nürnberg 91054ErlangenGermany K Schmidt Physikalisches Institut Universität Freiburg 79104FreiburgGermany L Schmitt Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France Also at GSI mbH Planckstr. 1D-64291DarmstadtGermany H Schmïden Helmholtz-Institut für Strahlen-und Kernphysik Universität Bonn 53115BonnGermany K Schönning CERN 1211Geneva 23Switzerland S Schopferer Physikalisches Institut Universität Freiburg 79104FreiburgGermany M Schott CERN 1211Geneva 23Switzerland O Yu Shevchenko Joint Institute for Nuclear Research Moscow region141980DubnaRussia L Silva LIP 1000-149LisbonPortugal L Sinha Matrivani Institute of Experimental Research & Education Calcutta-700 030India S Sirtl Physikalisches Institut Universität Freiburg 79104FreiburgGermany M Slunecka S Sosio Department of Physics and Torino Section of INFN University of Turin 10125TurinItaly F Sozzi Trieste Section of INFN 34127TriesteItaly A Srnka Institute of Scientific Instruments 61264BrnoAS CRCzech Republic L Steiger Trieste Section of INFN 34127TriesteItaly M Stolarski LIP 1000-149LisbonPortugal M Sulc Technical University in Liberec 46117LiberecCzech Republic R Sulej National Centre for Nuclear Research 00-681WarsawPoland H Suzuki Yamagata University 992-8510YamagataJapan Universidade Técnica de Lisboa LisbonPortugal Also at Chubu University 487-8501KasugaiAichiJapan n P Sznajder National Centre for Nuclear Research 00-681WarsawPoland S Takekawa Torino Section of INFN 10125TurinItaly J Ter Wolbeek Physikalisches Institut Universität Freiburg 79104FreiburgGermany S Tessaro Trieste Section of INFN 34127TriesteItaly F Tessarotto Trieste Section of INFN 34127TriesteItaly F Thibaud S Uhl Physik Department 21 State Research Center of the Russian Federation Institute for High Energy Physics CEA IRFU/SPhN Saclay Technische Universität München 85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France I Uman Department für Physik Ludwig-Maximilians-Universität München 80799MunichGermany io M Vandenbroucke M Virius L Wang Institut für Experimentalphysik Universität Bochum 44780BochumGermany T Weisrock Institut für Kernphysik Universität Mainz 55099MainzGermany M Wilfert Institut für Kernphysik Universität Mainz 55099MainzGermany R Windmolders Physikalisches Institut Universität Bonn 53115BonnGermany W Wiślicki National Centre for Nuclear Research 00-681WarsawPoland H Wollny K Zaremba Institute of Radioelectronics Warsaw University of Technology 00-665WarsawPoland M Zavertyaev Lebedev Physical Institute 119991MoscowRussia E Zemlyanichkina Joint Institute for Nuclear Research Moscow region141980DubnaRussia N Zhuravlev Joint Institute for Nuclear Research Moscow region141980DubnaRussia M Ziembicki Institute of Radioelectronics Warsaw University of Technology 00-665WarsawPoland Tel Aviv Israel EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH Hadron Transverse Momentum Distributions in Muon Deep Inelastic Scattering at 160 GeV/c The COMPASS Collaboration The COMPASS Collaboration Faculty of Mathematics and Physics Prague, Czech Republic j 20; Prague, Czech Republic j18000. 16636(to be submitted to European Physical Journal C)f present address: National Science Foundation, 4201 Wilson Boulevard, Arlington, VA 22230, United States g present address:/M/ST2/02350 * Deceased 3 Multiplicities of charged hadrons produced in deep inelastic muon scattering off a 6 LiD target have been measured as a function of the DIS variables x B j , Q 2 , W 2 and the final state hadron variables p T and z. The p 2 T distributions are fitted with a single exponential function at low values of p 2 T to determine the dependence of p 2T on x B j , Q 2 , W 2 and z. The z-dependence of p 2 T is shown to be a potential tool to extract the average intrinsic transverse momentum squared of partons, k 2 ⊥ , as a function of x B j and Q 2 in a leading order QCD parton model. Introduction Semi-Inclusive measurements of Deep Inelastic Scattering (SIDIS) of leptons off nucleons provide information about the partonic structure of the nucleon and the hadronization of partons, and hence offer a wide testing ground of Quantum Chromodynamics (QCD). Subject of the present study are the transverse momentum distributions of charged hadrons produced in the current fragmentation region in lepton-nucleon scattering off unpolarised nucleons. The hadron transverse momentum p T is defined with respect to the virtual photon direction. The following standard notations are used: and for the incoming and outgoing lepton, N for the target nucleon, h for the outgoing hadron and X for the unobserved particles in the final state; l, l , P, and p denote the 4-momenta of , , N, and h. The general expression for the differential SIDIS cross section describing the reaction + N → + h + X in the one-photon approximation is [1,2]: d 5 σ h (x B j , Q 2 , z, p 2 T , φ h ) dx B j dQ 2 dzd p 2 T dφ h = d 4 σ h (x B j , Q 2 , z, p 2 T ) 2πdx B j dQ 2 dzd p 2 T 1 + a 1 (x B j , Q 2 , z, p 2 T ) cos φ h +a 2 (x B j , Q 2 , z, p 2 T ) cos 2φ h + λ a 3 (x B j , Q 2 , z, p 2 T ) sin φ h .(1) Here, λ is the helicity of the incoming lepton and the standard SIDIS variables are used: the 4-momentum transfer q = (l − l ), the photon virtuality Q 2 = −q 2 , the Bjorken scaling variable x B j = Q 2 /2P · q, the hadron fractional energy z = P · p/P · q and the azimuthal angle φ h of the transverse momentum of the hadron with respect to the lepton scattering plane around the virtual photon direction. After integration over φ h , the cross section does not depend on the initial lepton polarisation λ . The hadron multiplicity per interaction is defined as the ratio of the differential SIDIS cross section over the differential DIS cross section d 2 σ DIS (x B j , Q 2 )/dx B j dQ 2 . Thus, the differential hadron multiplicity, d 2 n h /dzd p 2 T , depends on four variables, x B j , Q 2 , z, p 2 T : d 2 n h (x B j , Q 2 , z, p 2 T ) dzd p 2 T = d 4 σ h (x B j , Q 2 , z, p 2 T )/(dx B j dQ 2 dzd p 2 T ) d 2 σ DIS (x B j , Q 2 )/(dx B j dQ 2 ) .(2) Within a pQCD Leading Order (LO) parton model the shape of the p 2 T distributions depends on the intrinsic transverse momentum k ⊥ of the partons and the transverse momentum of the hadrons p ⊥ acquired during parton fragmentation. The amount of the contributions of k ⊥ and p ⊥ may depend on the hadron type, parton flavour, and on kinematic variables such as x B j , Q 2 and z. Already in the 1970s, SIDIS was understood as a tool to access the intrinsic transverse momentum of the partons (see e.g. [3] and references therein). The connection between the intrinsic transverse momenta of the parton k ⊥ and that of the hadron p ⊥ and the measured transverse momentum p T of the produced hadron is illustrated in Fig. 1, assuming single photon exchange and leading order pQCD. During the last three decades significant efforts, both in experimental and theoretical studies of (polarised) SIDIS, have been undertaken. Currently this process is considered to be one of the most promising to study the hadronization process and also the spin-dependent three-dimensional structure of the nucleon (see, e.g. [4]). Recently, a complete QCD treatment of transverse momentum and spin-dependent SIDIS was presented in Ref. [5] where factorization was derived in terms of well defined unintegrated or Transverse Momentum Dependent parton distribution and fragmentation functions (TMDs) with individual hard scale evolution properties. This formalism has been applied in Ref. [6] to obtain the Q 2 evolution of unpolarised TMDs; a mandatory information needed for a correct comparison of data measured in experiments at different hard scales [4]. Hadron leptoproduction has been studied by many experiments. Some recent examples are: JLab [7], HERMES [8] and E665 [9]. Earlier, EMC [10] covered most of the kinematic range of COMPASS. However, COMPASS has collected much more data in this range and the statistical errors of the present Fig. 1: Sketch showing the kinematic variables for the absorption of a virtual photon by a parton with intrinsic transverse momentum k ⊥ and the subsequent hadronization. The transverse momentum of the observed hadron is denoted by p T when defined with respect to the virtual photon direction in the photon nucleon center of mass system and by p ⊥ when defined with respect to the scattered parton direction. analysis are therefore significantly smaller, although only part of the available data has been used. The results presented here are obtained from data taken during the year 2004. More details of the analysis are described in Ref. [11]. Experiment, Data Selection and Acceptance The COMPASS experiment is installed on the M2 beam line of the CERN SPS [12]. Polarised 160 GeV/c muons with an intensity of 2 × 10 8 µ/spill (one spill of 4.8 s length per 16.8 s) and a polarisation of 80% are scattered off a longitudinally polarised 6 LiD target. In 2004 the target consisted of two cells with opposite polarisation which was reversed every 8 hours. It has been verified that summing up the data from both cells yields a data sample with vanishing polarisation for the present analysis. The COMPASS detector is a large acceptance two-stage spectrometer which covers the kinematic range from quasi-real photoproduction to DIS. Both stages are equipped with hadron calorimeters and use absorber walls for muon identification. Charged particles emerging from the primary interaction vertex in the forward direction are identified as muons if they traverse at least 30 radiation length, otherwise they are identified as hadrons. The selection requires reconstructed trajectories in the detectors situated upstream and downstream of the first magnet. This ensures that the track momentum and sign of charge are well defined by bending in the magnetic field. The COMPASS ability to separate pions, kaons and protons with a Ring Imaging Cherenkov detector was not used in this analysis. Muon interactions with Q 2 > 1.0 (GeV/c) 2 and 0.1 < y < 0.9 are selected, where y = ν/E µ , and ν = E µ − E µ is the difference between the laboratory energies of the incoming and outgoing muon µ and µ . With the above selection, the hadronic energy squared W 2 = 2Mν + M 2 − Q 2 is > 25 GeV/c, above the nucleon resonance region. Here, M is the nucleon mass. The total number of inclusive events selected for this analysis is 45.8 × 10 6 , corresponding to an integrated luminosity of 775 pb −1 . The events are sampled into 23 intervals in Q 2 from 1 to 10 (GeV/c) 2 and x B j from 0.004 to 0.12, as shown in Fig. 2. The ranges and average values of Q 2 and x B j are shown in Tab. 1. Each of these (x B j , Q 2 ) intervals is further subdivided into 8 intervals in z from 0.2 to 0.8. In order to correct for event losses caused by the non uniform acceptance of the COMPASS spectrometer, a full Monte Carlo (MC) simulation has been performed. The events were generated with LEPTO [13], passed through the spectrometer with a GEANT [14] based simulation program and reconstructed with the reconstruction software as the real data events. The SIDIS acceptances A (+,−) SIDIS for detecting, together with the scattered muon, a positive (h + ) or negative hadrons (h − ) respectively factorize in an inclusive muon acceptance A incl (Q 2 , y) and a positive or negative hadron acceptance A h (+,−) ( lab p T , lab η). These acceptances depend on the spectrometer charac- teristics, making the use of variables defined in the laboratory frame preferable; therefore, the transverse momentum lab p T , the polar angle lab θ , and the pseudorapidity lab η = − ln(tan lab θ 2 ) of the hadron are defined with respect to the direction of the incoming muon. The choice of lab θ is particularly convenient to exhibit the acceptance cut due to the aperture limit of the polarised target magnet at lab θ = 70 mrad for the upstream edge of the target. The factorization of hadron and muon acceptances implies that the differential multiplicities only depend on A h (+,−) since A incl cancels, see Eq. 2. Figure 3 shows the hadron acceptances A h − and A h + used in the analysis. The four-dimensional acceptance used in the present analysis is integrated over the azimuthal angle of the hadrons, i.e. does not take into account the azimuthal modulations in the cross section [2]. The systematic effect on the extracted p 2 T have been investigated and found to be negligible. Results The differential multiplicities d 2 n h± /dzd p 2 T in bins of (Q 2 , x B j ) are defined in the introduction in terms of the semi-inclusive and inclusive differential cross sections. They are obtained as the acceptance corrected number of hadrons ∆ 4 N h± in 8 × 40 (z, p 2 T ) bins and 23 (∆x B j , ∆Q 2 ) bins, divided by the number of muon interactions in the same (∆x B j , ∆Q 2 ) bins: ∆ 2 N µ Bin x min b j x max b j x b j Q 2 min Q 2 max Q 2 1 0.d 2 n h± (z, p 2 T , x B j , Q 2 ) dzd p 2 T ∆x B j ∆Q 2 ≈ ∆ 4 N h± (z, p 2 T , x B j , Q 2 )/(∆z∆p 2 T ∆x B j ∆Q 2 ) ∆ 2 N µ (x B j , Q 2 )/(∆x B j ∆Q 2 ) . The distributions for two selected (Q 2 , x B j ) bins are shown in Fig. 4 for all z intervals. The full data set, including more p 2 T bins, is available on HEPDATA [15]. As can be seen from Eq. 3 the uncertainty of the integrated luminosity cancels and the only contributions to systematic uncertainties of the multiplicities come from the hadron acceptance and the assumption of factorization of hadron and muon acceptance. The total systematic uncertainty due to acceptance has been estimated to be 5% [11]. Only statistical errors are shown in the figures. The fits are performed at values of p T smaller than 0.85 GeV/c to stay away from pQCD effects where the assumption of a simple exponential distribution is known to fail [16,17] and at p T larger than 0.1 GeV/c to exclude a region where the experimental resolution may affect the distribution. In this range, the p 2 T distributions are fitted with a single exponential functions Ae −p 2 T / p 2 T to extract the inverse slope In Fig. 7 the dependence of the fitted p 2 T on x B j is shown for a low-z and a high-z bin and for a low-and a high-Q 2 bin. At higher z the positive hadrons clearly have higher p 2 T than the negative hadrons. For hadrons with lower z however, no difference is observed in the p 2 T distributions. A similar behaviour was already reported by HERMES [21] for the average p 2 T , not determined by a fit but from a standard average .2394 (17) .2607 (23) .2738(30) .2981(31) .3072 (43) .3299 (65) .2674 (57) 2 .2241 (10) .2457 (14) .2614 (19) .2763 (24) .2952 (24) .3060 (35) .3009 (44) .2650 (46) 3 .2234 (13) .2390 (18) .2631 (24) .2761(31) .2986(32) .3327 (52) .3348 (69) .2985 (68) 4 .2239 ( 6) .2424 ( 9) .2592 (12) .2738 (15) .2915 (15) .3006 (21) .3087 (29) .2670(30) 5 .2270 ( 9) .2438 (13) .2634 (17) .2777 (22) .3073 (24) .3387 (38) .3419(51) .2998(55) 6 .2231 ( 6) .2442 ( 8) .2610 (11) .2755 (14) .2914 (13) .3053 (19) .3010 (25) .2677 (27) 7 .2246 ( 7) .2418 ( 9) .2641 (13) .2779 (17) .3013 (17) .3338 (27) .3410(38) .3105 (44) 8 .2250 (13) .2470 (18) .2723(25) .2893(33) .3338(38) .3829 (67) .420 (11 ) .447 (17 ) 9 .2122 ( 7) .2408 ( 9) .2630 (12) .2762 (15) .2953 (15) .3087 (21) .3094 (26) .2699 (27) 10 .2222 ( 7) .2439 (10) .2617 (13) .2806 (17) .3034 (17) .3347 (28) .3309(37) .2946(39) 11 .2250 (12) .2425 (16) .2667 (22) .2886(30) .3243(34) .3642 (57) .3912(88) .3485(95) 12 .2289 (17) .2530 (24) .2828(35) .3031(48) .3515(55) .421 (11 ) .417 (14 ) .538(34 ) 13 .1766 (13) .2020 (15) .2299 (18) .2597 (23) .2909 (22) .3003 (29) .2956(34) .2704 (37) 14 .1866 ( 9) .2171 (13) .2436 (15) .2640 (18) .2893 (17) .3131 (24) .3099(30) .2758(31) 15 .2078 ( 6) .2355 ( 8) .2577 (11) .2759 (14) .3050 (14) .3319 (22) .3364(29) .3025(31) 16 .2229 ( 9) .2441 (13) .2678 (18) .2882 (23) .3230(26) .3587 (42) .3749(59) .3395(69) 17 .2257 (10) .2493 (14) .2761 (20) .3064(28) .3468(32) .4050 (59) .4321(91) .384 (10 ) 18 .1799 (11) .2063 (13) .2340 (17) .2500 (19) .2853 (20) .3197(30) .3321(40) .2984(42) 19 .1944 ( 9) .2245 (12) .2486 (15) .2735 (20) .3088 (22) .3434(34) .3656 (49) .3609(62) 20 .2167 ( 8) .2415 (11) .2700 (15) .2947 (21) .3370 (23) .3959 (42) .4170(63) .3994(76) 21 .2311 (13) .2579 (18) .2908 (27) .3178(37) .3588(41) .4307 (80) .490 (14 ) .507 (21 ) 22 .1738 (10) .1990 (12) .2319 (16) .2578 (21) .2969 (22) .3437(38) .3809 (57) .3809(74) 23 .2091 (10) .2448 (15) .2714 (21) .2989(28) .3441(31) .4072 (57) .470 (10 ) .469(13 ) .2472 (18) .2606 (24) .2651(29) .2741 (27) .2699(35) .2551(41) .2259 (42) 2 .2241 (10) .2450 (15) .2607 (20) .2713(25) .2793 (23) .2806(31) .2697(37) .2297 (37) 3 .2305 (14) .2510 (19) .2690(25) .2802(33) .2843(30) .2851(40) .2778(50) .2562(54) 4 .2254 ( 7) .2448 ( 9) .2614 (13) .2719 (16) .2814 (15) .2836 (20) .2700 (24) .2375(26) 5 .2276 (10) .2498 (14) .2666 (19) .2752 (23) .2987 (24) .3056(33) .2990(42) .2722(46) 6 .2216 ( 7) .2431 ( 9) .2604 (12) .2766 (15) .2869 (14) .2870 (19) .2780 (24) .2297 (23) 7 .2260 ( 7) .2498 (10) .2653 (14) .2823 (18) .2989 (18) .3171 (27) .3068(35) .2657(37) 8 .2346 (14) .2555 (19) .2841 (28) .2976(36) .3233(38) .3572(62) .3674(88) .328 (10 ) 9 .2071 ( 8) .2326 (10) .2540 (13) .2686 (16) .2837 (15) .2888 (20) .2767 (24) .2316 (24) 10 .2219 ( 8) .2440 (11) .2646 (14) .2831 (19) .2992 (19) .3150(28) .3068(36) .2596 (35) 11 .2331 (13) .2512 (18) .2742 (25) .2953(34) .3243(37) .3578(62) .3587(85) .3048 (87) 12 .2333 (18) .2644 (27) .2908 (40) .3057(50) .3443(58) .383 (10 ) .417 (16 ) .379 (18 ) 13 .1669 (12) .1933 (16) .2201 (20) .2400 (23) .2614 (21) .2640 (27) .2457(29) .2083(28) 14 .1835 (11) .2086 (12) .2302 (15) .2536 (19) .2693 (18) .2794 (24) .2648 (27) .2240 (26) 15 .2067 ( 7) .2316 ( 9) .2548 (12) .2728 (15) .2976 (16) .3081 (22) .3015 (27) .2492(26) 16 .2219 (10) .2482 (14) .2652 (19) .2907(26) .3254(29) .3541(47) .3473(62) .2971(60) 17 .2325 (11) .2556 (16) .2844 (23) .3092(31) .3506(37) .4033(68) .3890(87) .3432 (95) 18 .1746 (11) .1994 (14) .2232 (18) .2411 (21) .2692 (21) .2850(30) .2882(38) .2394 (35) 19 .1915 ( 9) .2193 (13) .2452 (18) .2715 (23) .3040(25) .3268(39) .3319(51) .2869 (52) 20 .2163 ( 9) .2456 (12) .2765 (18) .3007(25) .3389(28) .3703(45) .3909(70) .3382(73) 21 .2370 (14) .2686 (21) .2994(31) .3246(44) .3744(53) .4235(96) .414 (13 ) .391 (17 ) 22 .1730 (10) .1967 (13) .2314 (19) .2623 (27) .2867(26) .3218(41) .3598(66) .3310(75) 23 .2166 (13) .2482 (17) .2814 (25) .3050(35) .3531(41) .3976(71) .414 (11 ) .388(13 ) The p 2 T dependence of the differential multiplicities d 2 n h /dzd p 2 T of positive hadrons (left) and negative hadrons (right) fitted by an exponential for 1 < Q 2 (GeV/c) 2 < 1.5, 0.006 < x B j < 0.008 (top) and 6 < Q 2 (GeV/c) 2 < 10, 0.07 < x B j < 0.12 (bottom) subdivided into eight z intervals, see legend of upper pictures. The average values Q 2 and x B j for the chosen (Q 2 , x B j ) intervals are indicated in the pictures. The systematic error of 5% is not included in the errors. over the entire p T range, i.e. p 2 T all . The z-dependence as well as the hadron charge dependence of the p 2 T distributions will be further investigated below and is related to the intrinsic transverse momentum of the partons. It is interesting to compare the values and W 2 -dependence of p 2 T obtained from the fit at small p T with the values and W 2 -dependence of p 2 T all . The W 2 -dependence of p 2 T , obtained from the fit in the bin 0.5 < z < 0.6 is shown in Fig. 8, that one of p 2 T all in Fig. 9. In addition to the data points, Fig. 9 shows lines, which represent fits of the data points assuming a linear function of lnW 2 . Because of the Q 2 -dependence, the last points are somewhat below the fit. The authors of Ref. [18] first suggested that p 2 T all should depend linearly on the µN center of mass energy squared s. They have verified their prediction with results from three fixed target experiments: JLab, HERMES and COMPASS, see Fig. 10. Fig. 10a shows the p 2 T distribution of charged hadrons with 0.5 < z < 0.6 and integrated over Q 2 and x B j , measured by COMPASS, which was used to determine the acceptance corrected p 2 T all . Fig. 10b taken from Ref. [18] shows the dependence of p 2 T all on s. Their value for COMPASS, represented by the black dots, was not corrected for acceptance. The new, acceptance corrected COMPASS value p 2 T all added to Fig. 10b (red dot) is shown in a recent paper [19], and used to quantify the p T broadening [20] in a model to determine the Sivers and Boer-Mulders asymmetries at COMPASS and HERMES. The result of the model of Pasquini and Schweitzer was closer to the COMPASS data when p T broadening is included. The authors of Ref. [18] also note that p 2 T all may depend linearly on W 2 rather than s. convoluted with two unintegrated soft universal functions: f q (x B j , k ⊥ ), the parton distribution function of quark of flavor q and D h q (z, p ⊥ ), the fragmentation function defined as the number density of hadron h resulting from the fragmentation of a quark of flavor q. With the further assumption that both f q (x B j , k ⊥ ) and D h q (z, p ⊥ ) follow Gaussian distributions with respect to the transverse momentum variables k ⊥ and p ⊥ , respectively, the cross section can be approximated [16] at first order in O(k ⊥ /Q) by: 〉 2 T p 〈 〉 2 T p 〈 〉 2 T p 〈 〉 2 T p 〈 〉 2 T p 〈 〉 2 T p 〈 〉 2 T p 〈 〉 2 T p 〈 − h − h − h − h − h − h − h − hd 4 σ µN→µ hX dx B j dQ 2 dzd p 2 T ≈ ∑ q 2πα 2 e 2 q Q 4 f q (x B j )D h q (z) · 1 + (1 − y) 2 1 π p 2 T q e −p 2 T / p 2 T q ,(4) where all the parameters describing the transverse momentum dependence of TMDs for a given quark flavor q are contained in p 2 T q , through the relation: p 2 T q = p 2 ⊥ q + z 2 k 2 ⊥ q .(5) Here again, integration over the azimuthal angle has been performed. In Ref. [16] it was assumed that p 2 ⊥ and k 2 ⊥ in Eq. 5 are constants and independent of the quark flavor. In general, they may both depend on Q 2 and the active quark flavor q while p 2 ⊥ can depend further on z and the produced hadron type, and k 2 ⊥ may depend on x B j . T vs W 2 for 0.5 < z < 0.6 and for a low (left) and a high (right) Q 2 interval, from a fit over 0.1 < p T GeV/c < 0.85. This is to be compared with Fig. 9 where p 2 T all is plotted. The average Q 2 for each W 2 bin are indicated. T distribution of charged hadrons with 0.5 < z < 0.6 used to determine the acceptance corrected p 2 T all (left). The s-dependence of p 2 T all from Ref. [18] (right). The red star labeled COMPASS is the value from this analysis, the black dot labeled COMPASS (Schw.) is the value used in Ref. [18], obtained from data not corrected for acceptance. The observed dependence of the fitted p 2 T on z 2 is shown for two (Q 2 , x B j ) intervals in Fig. 12. The relation between p 2 T and z 2 is certainly not linear as in Eq. 5. It should be noted that the non linear behaviour of the z 2 -dependence of p 2 T was reproduced qualitatively in a recent paper [22] by imposing kinematical constraints to the model leading to Eq. 4. A more general ansatz for the contributions of the intrinsic transverse momenta p ⊥ and k ⊥ to the measured hadron transverse momentum p T is p 2 T (z) = p 2 ⊥ (z) + z 2 k 2 ⊥ ,(6) where p 2 ⊥ (z) is a function of z and should be taken from other measurements. The dependence of k ⊥ is still the same as in Eq. 6, with a constant average k 2 ⊥ . The knowledge of p 2 ⊥ (z) could be taken from DIS event generators which are supposed to incorporate all known properties of jet fragmentation. In Fig. 13 the measured values of p 2 T are compared with those of a simulation using the event generator LEPTO 1 . Two cases were simulated in the MC: interactions without intrinsic transverse parton momenta k 2 ⊥ = 0 (open squares) and interactions with k 2 ⊥ = 0.25 (GeV/c) 2 (open crosses). For k 2 ⊥ = 0.25 (GeV/c) 2 , the agreement between p 2 T from simulated events and from data (full squares) is striking for lower values of Q 2 , apart from the highest z 2 bins. For values of Q 2 larger than 4 (GeV/c) 2 , the data are significantly above the simulation. The significant differences between positive and negative hadrons at larger z values are not reproduced by the MC simulation. This comparison suggests that k 2 ⊥ can be extracted from the data, provided a detailed tuning of the jet fragmentation parameters would be performed. In addition, it should be noted that the present event selection includes all semi-inclusive events. Thus events which are not due to DIS, i.e. the absorption of the virtual photon by a quark with subsequent quark fragmentation, but to other mechanisms like diffractive vector meson production are included in the sample. The treatment of this kind of background to DIS requires special efforts, in order to extract k 2 ⊥ . Conclusion The main result of this study is the measurement of differential multiplicities of charged hadrons produced in unpolarised SIDIS of muons off a 6 LiD target. The acceptance corrected multiplicities in a 4-dimensional (z, p 2 T , x B j , Q 2 ) phase space are given in Ref. [15] separately for positively and negatively charged hadrons. Fig. 2 : 2Event distribution in the inclusive variables Q 2 and x B j and the 23 bins of the hadron cross section analysis. Within each bin, the fraction of events contained is indicated in %. Fig. 3 : 3Hadron acceptances A h − and A h + determined with the Monte Carlo simulation for Q 2 > 1 (GeV/c) 2 as a function of lab p T and lab η for negative hadrons h − (left) and positive hadrons h + (right). The acceptances have been smoothed in order to reduce the granularity from the binning. p 2 T 2. The values of p 2 T for all intervals of x B j , Q 2 and z are shown in Figs. 5 and 6 and in Tabs. 2 and 3. These figures and tables contain the basic experimental information extracted from the fits of the p 2 T distributions. B j , Q 2 ) 2bins (rows) and the 8 z intervals (columns) for positive hadrons. The error of the least significant digit(s) is given in parentheses. Same information as Fig. 3 : Fitted p 2 T 32in units of (GeV/c) 2 for the 23 (x B j , Q 2 ) bins (rows) and the 8 z intervals (columns) for negative hadrons. The error of the least significant digit(s) is given in parentheses. Same information asFig. Fig. 6 : 6As Fig. 5 but for negative hadrons. Fig. 7 : 7The fitted p 2 T vs x B j for two different Q 2 intervals (top and bottom) and for a low-z bin (0.2 < z < 0.3), left, and a high-z bin (0.6 < z < 0.8), right, for positive and negative hadrons (red filled circles and blue open boxes). In these figures, the p 2 T values are obtained from a fit over 0.1 < p T < 0.85 GeV/c. The average value Q 2 for each x B j bin is indicated. Fig. 8 : 8The fitted p 2 Fig. 9 : 9Statistical average p 2 T all over the entire p T range for charged hadrons (h + and h − summed up) as a function of W 2 , for 0.5 < z < 0.6 and four Q 2 intervals, indicated in the figures. The green lines represent fits where a linear function of lnW 2 was assumed. Fig. 10 : 10The p 2 Fig. 11 : 11Charged hadron multiplicity ratios dn h+ /dn h− as a function of z, for various x B j bins, measured by EMC[10] for µD (left) and COMPASS for µ 6 LiD (right) interactions respectively. For COMPASS the results are based on part of the 2004 collected statistics. Fig. 12 :Fig. 13 : 1213p 2 T vs z 2 for two (Q 2 , x B j ) intervals. The corresponding average values Q 2 (in units of (GeV/c) 2 and x B j are indicated in the figure. The dotted green line corresponds to relation (5) with constant k 2 ⊥ and p 2 ⊥ from Ref. [17]. Comparison of the measured p 2 T (full squares) with a simulation using the MC event generator LEPTO for two bins of Q 2 and x B j , for positive (top) and negative hadrons (bottom). Two cases were simulated in the MC: Interactions without intrinsic transverse parton momenta k 2 ⊥ = 0 (open squares) and interactions with k 2 ⊥ = 0.25 (GeV/c) 2 (open crosses). Table 2 : 2Fitted p 2 T in units of (GeV/c) 2 for the 23 (x Table dn+ dn 1 2 <0.200 Bj x 0.090< <0.090 Bj x 0.035< <0.035 Bj <x 0.020 <0.020 Bj x 0.010< EMC z 0 0.5 1 − /dn + dn 1 2 <0.120 Bj x 0.070< <0.070 Bj x 0.040< <0.040 Bj x 0.025< <0.025 Bj x 0.018< <0.018 Bj x 0.012< <0.012 Bj x 0.008< <0.008 Bj x 0.006< COMPASS 2004 LiD T is of particular interest. There are theoretical predictions allowing the extraction of the intrinsic transverse momenta k ⊥ and p ⊥ from this z 2 -dependence[16]. At leading order QCD, assuming single photon exchange and an independent fragmentation process, the hadron muoproduction cross section can be expressed in terms of a hard muon-parton interaction cross section For these simulations, MRST2004LO PDFs in LHAPDF 5.2.2 were used; default LEPTO 6.5.1 and JETSET 7.4 settings, with the exception of LST(11)=122, which includes target mass effects and the longitudinal structure function. AcknowledgmentWe gratefully acknowledge the support of the CERN management and staff and the skill and effort of the technicians of our collaborating institutes. Special thanks go to V. Anosov and V. Pesaro for their technical support during the installation and the running of this experiment. This work was made possible by the financial support of our funding agencies. We would like to thank Dr. Alessandro Bacchetta for his helpful comments.However, the dependence shown inFig. 9is more compatible with a linear dependence on lnW 2 as was found by several experiments (see, e.g.[10]). The relation is not well established and, as mentioned inRef.[18], the linear dependence on s for Drell-Yan which inspired their SIDIS prediction, could also be a linear dependence on √ s. Contrary to the case of p 2 T all inFig. 9, the W 2 -dependence of the fitted p 2 T shown inFig. 8is much weaker, as expected, since p 2 T is assumed to be unaffected by pQCD, as opposed to p 2 T all . Another interesting observable is the ratio of the multiplicities of positive and negative hadrons integrated over p 2 T and Q 2 . The hadron multiplicity ratios are shown inFig. 11and compared with previous data taken by the EMC experiment[10]. COMPASS results show clearly the zand x B j -dependence, where the fraction of positive hadrons increases with x B j (getting closer to the valence region) and z (more related to the energy of the struck parton). This behaviour can be qualitatively connected with the fact that the positive valence quarks have a larger electric charge than the negative ones.The z 2 -dependence of the fitted p 2 T over the entire p T range, p 2 T all , corrected for acceptance, has been provided for a comparison with other experiments at different center of mass energy. The evolution of p 2 T all as a function of the invariant mass W 2 has been shown to follow a linear dependence on lnW 2 reasonably well. The ratio of positive over negative hadrons is shown as a function of z for bins in x B j and compared with previous EMC results.The differential distributions at low p 2 T have been fitted with an exponential at different z in order to obtain p 2 T . 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{'abstract': 'Multiplicities of charged hadrons produced in deep inelastic muon scattering off a 6 LiD target have been measured as a function of the DIS variables x B j , Q 2 , W 2 and the final state hadron variables p T and z. The p 2 T distributions are fitted with a single exponential function at low values of p 2 T to determine the dependence of p 2T on x B j , Q 2 , W 2 and z. The z-dependence of p 2 T is shown to be a potential tool to extract the average intrinsic transverse momentum squared of partons, k 2 ⊥ , as a function of x B j and Q 2 in a leading order QCD parton model.', 'arxivid': '1305.7317', 'author': ['C Adolph \nPhysikalisches Institut\nUniversität Erlangen-Nürnberg\n91054ErlangenGermany\n', 'M G Alekseev \nTrieste Section of INFN\n34127TriesteItaly\n', 'V Yu Alexakhin \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'Yu Alexandrov \nLebedev Physical Institute\n119991MoscowRussia\n', 'G D Alexeev \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'A Amoroso \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'V Andrieux ', 'A Austregesilo \nCERN\n1211Geneva 23Switzerland\n\nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'B Badełek \nFaculty of Physics\nUniversity of Warsaw\n00-681WarsawPoland\n', 'F Balestra \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'J Barth \nPhysikalisches Institut\nUniversität Bonn\n53115BonnGermany\n', 'G Baum \nUniversität Bielefeld\nFakultät für Physik\n33501BielefeldGermany\n', 'Y Bedfer ', 'A Berlin \nInstitut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany\n', 'J Bernhard \nInstitut für Kernphysik\nUniversität Mainz\n55099MainzGermany\n', 'R Bertini \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'K Bicker \nCERN\n1211Geneva 23Switzerland\n\nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'J Bieling \nPhysikalisches Institut\nUniversität Bonn\n53115BonnGermany\n', 'R Birsa \nTrieste Section of INFN\n34127TriesteItaly\n', 'J Bisplinghoff \nHelmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany\n', 'M Boer ', 'P Bordalo \nLIP\n1000-149LisbonPortugal\n', 'F Bradamante \nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'C Braun \nPhysikalisches Institut\nUniversität Erlangen-Nürnberg\n91054ErlangenGermany\n', 'A Bravar \nTrieste Section of INFN\n34127TriesteItaly\n', 'A Bressan \nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'M Büchele \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'E Burtin ', 'L Capozza ', 'M Chiosso \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'S U Chung \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'A Cicuttin \nAbdus Salam ICTP and Trieste Section of INFN\n34127TriesteItaly\n', 'M L Crespo \nAbdus Salam ICTP and Trieste Section of INFN\n34127TriesteItaly\n', 'S Dalla Torre \nTrieste Section of INFN\n34127TriesteItaly\n', 'S S Dasgupta \nMatrivani Institute of Experimental Research & Education\nCalcutta-700 030India\n', 'S Dasgupta \nTrieste Section of INFN\n34127TriesteItaly\n', 'O Yu Denisov \nTorino Section of INFN\n10125TurinItaly\n', 'S V Donskov ', 'N Doshita \nYamagata University\n992-8510YamagataJapan\n\nUniversidade Técnica de Lisboa\nLisbonPortugal\n', 'V Duic \nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'W Dünnweber \nDepartment für Physik\nLudwig-Maximilians-Universität München\n80799MunichGermany io\n', 'M Dziewiecki \nInstitute of Radioelectronics\nWarsaw University of Technology\n00-665WarsawPoland\n', 'A Efremov \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'C Elia \nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'P D Eversheim \nHelmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany\n', 'W Eyrich \nPhysikalisches Institut\nUniversität Erlangen-Nürnberg\n91054ErlangenGermany\n', 'M Faessler \nDepartment für Physik\nLudwig-Maximilians-Universität München\n80799MunichGermany io\n', 'A Ferrero ', 'A Filin ', 'M Finger ', 'M. Finger jrH 19 ', 'Fischer \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'C Franco \nLIP\n1000-149LisbonPortugal\n', 'N Du Fresne Von Hohenesche \nCERN\n1211Geneva 23Switzerland\n\nInstitut für Kernphysik\nUniversität Mainz\n55099MainzGermany\n', 'J M Friedrich \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'V Frolov \nCERN\n1211Geneva 23Switzerland\n', 'R Garfagnini \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'F Gautheron \nInstitut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany\n', 'O P Gavrichtchouk \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'S Gerassimov \nLebedev Physical Institute\n119991MoscowRussia\n\nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'R Geyer \nDepartment für Physik\nLudwig-Maximilians-Universität München\n80799MunichGermany io\n', 'M Giorgi \nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'I Gnesi \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'B Gobbo \nTrieste Section of INFN\n34127TriesteItaly\n', 'S Goertz \nPhysikalisches Institut\nUniversität Bonn\n53115BonnGermany\n', 'S Grabmüller \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'A Grasso \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'B Grube \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'R Gushterski \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'A Guskov \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'T Guthörl \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'F Haas \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'D Von Harrach \nInstitut für Kernphysik\nUniversität Mainz\n55099MainzGermany\n', 'F H Heinsius \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'F Herrmann \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'C Heß \nInstitut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany\n', 'F Hinterberger \nHelmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany\n', 'Ch Höppner \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'N Horikawa \nAlso at\nChubu University\n487-8501KasugaiAichiJapan n\n', 'N D&apos;hose \nAlso at KEK\n1-1 Oho, Tsukuba, Ibaraki305-0801Japan\n', 'S Huber \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'S Ishimoto \nYamagata University\n992-8510YamagataJapan\n\nUniversidade Técnica de Lisboa\nLisbonPortugal\n\nAlso at KEK\n1-1 Oho, Tsukuba, Ibaraki305-0801Japan\n', 'Yu Ivanshin \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'T Iwata \nYamagata University\n992-8510YamagataJapan\n\nUniversidade Técnica de Lisboa\nLisbonPortugal\n', 'R Jahn \nHelmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany\n', 'V Jary ', 'P Jasinski \nInstitut für Kernphysik\nUniversität Mainz\n55099MainzGermany\n', 'R Joosten \nHelmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany\n', 'E Kabuß \nInstitut für Kernphysik\nUniversität Mainz\n55099MainzGermany\n', 'D Kang \nInstitut für Kernphysik\nUniversität Mainz\n55099MainzGermany\n', 'B Ketzer \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'G V Khaustov ', 'Yu A Khokhlov \nAlso at Moscow Institute of Physics and Technology, Moscow Region\nIII. Physikalisches Institut\nRWTH Aachen University\n141700, 52056AachenRussia, Germany\n', 'Yu Kisselev \nInstitut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany\n', 'F Klein \nPhysikalisches Institut\nUniversität Bonn\n53115BonnGermany\n', 'K Klimaszewski \nNational Centre for Nuclear Research\n00-681WarsawPoland\n', 'J H Koivuniemi \nInstitut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany\n', 'V N Kolosov ', 'K Kondo \nYamagata University\n992-8510YamagataJapan\n\nUniversidade Técnica de Lisboa\nLisbonPortugal\n', 'K Königsmann \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'I Konorov \nLebedev Physical Institute\n119991MoscowRussia\n\nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'V F Konstantinov ', 'A M Kotzinian \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'O Kouznetsov \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'M Krämer \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'Z V Kroumchtein \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'N Kuchinski \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'F Kunne ', 'K Kurek \nNational Centre for Nuclear Research\n00-681WarsawPoland\n', 'R P Kurjata \nInstitute of Radioelectronics\nWarsaw University of Technology\n00-665WarsawPoland\n', 'A A Lednev ', 'A Lehmann \nPhysikalisches Institut\nUniversität Erlangen-Nürnberg\n91054ErlangenGermany\n', 'S Levorato \nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'J Lichtenstadt \nSchool of Physics and Astronomy\nTel Aviv University\n69978\n', 'A Maggiora \nTorino Section of INFN\n10125TurinItaly\n', 'A Magnon ', 'N Makke \nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'G K Mallot \nCERN\n1211Geneva 23Switzerland\n', 'A Mann \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'C Marchand ', 'A Martin \nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'J Marzec \nInstitute of Radioelectronics\nWarsaw University of Technology\n00-665WarsawPoland\n', 'H Matsuda \nYamagata University\n992-8510YamagataJapan\n\nUniversidade Técnica de Lisboa\nLisbonPortugal\n', 'T Matsuda \nUniversity of Miyazaki\n889-2192MiyazakiJapan\n', 'G Meshcheryakov \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'W Meyer \nInstitut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany\n', 'T Michigami \nYamagata University\n992-8510YamagataJapan\n\nUniversidade Técnica de Lisboa\nLisbonPortugal\n', 'Yu V Mikhailov ', 'Y Miyachi \nYamagata University\n992-8510YamagataJapan\n\nUniversidade Técnica de Lisboa\nLisbonPortugal\n', 'A Morreale ', 'A Nagaytsev \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'T Nagel \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'F Nerling \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'S Neubert \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'D Neyret ', 'V I Nikolaenko ', 'J Novy ', 'W.-D Nowak \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'A S Nunes \nLIP\n1000-149LisbonPortugal\n', 'A G Olshevsky \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'M Ostrick \nInstitut für Kernphysik\nUniversität Mainz\n55099MainzGermany\n', 'R Panknin \nPhysikalisches Institut\nUniversität Bonn\n53115BonnGermany\n', 'D Panzieri \nTorino Section of INFN\nUniversity of Eastern Piedmont\n15100, 10125Alessandria, TurinItaly\n', 'B Parsamyan \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'S Paul \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'G Piragino \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'S Platchkov ', 'J Pochodzalla \nInstitut für Kernphysik\nUniversität Mainz\n55099MainzGermany\n', 'J Polak \nTechnical University in Liberec\n46117LiberecCzech Republic\n\nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'V A Polyakov ', 'J Pretz \nPhysikalisches Institut\nUniversität Bonn\n53115BonnGermany\n', 'M Quaresma \nLIP\n1000-149LisbonPortugal\n', 'C Quintans \nLIP\n1000-149LisbonPortugal\n', 'J.-F Rajotte \nDepartment für Physik\nLudwig-Maximilians-Universität München\n80799MunichGermany io\n', 'S Ramos \nLIP\n1000-149LisbonPortugal\n', 'G Reicherz \nInstitut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany\n', 'E Rocco \nCERN\n1211Geneva 23Switzerland\n', 'V Rodionov \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'E Rondio \nNational Centre for Nuclear Research\n00-681WarsawPoland\n', 'N S Rossiyskaya \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'D I Ryabchikov ', 'V D Samoylenko ', 'A Sandacz \nNational Centre for Nuclear Research\n00-681WarsawPoland\n', 'M G Sapozhnikov \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'S Sarkar \nMatrivani Institute of Experimental Research & Education\nCalcutta-700 030India\n', 'I A Savin \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'G Sbrizzai \nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'P Schiavon \nDepartment of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly\n', 'C Schill \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'T Schlüter \nDepartment für Physik\nLudwig-Maximilians-Universität München\n80799MunichGermany io\n', 'A Schmidt \nPhysikalisches Institut\nUniversität Erlangen-Nürnberg\n91054ErlangenGermany\n', 'K Schmidt \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'L Schmitt \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n\nAlso at GSI mbH\nPlanckstr. 1D-64291DarmstadtGermany\n', 'H Schmïden \nHelmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany\n', 'K Schönning \nCERN\n1211Geneva 23Switzerland\n', 'S Schopferer \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'M Schott \nCERN\n1211Geneva 23Switzerland\n', 'O Yu Shevchenko \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'L Silva \nLIP\n1000-149LisbonPortugal\n', 'L Sinha \nMatrivani Institute of Experimental Research & Education\nCalcutta-700 030India\n', 'S Sirtl \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'M Slunecka ', 'S Sosio \nDepartment of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly\n', 'F Sozzi \nTrieste Section of INFN\n34127TriesteItaly\n', 'A Srnka \nInstitute of Scientific Instruments\n61264BrnoAS CRCzech Republic\n', 'L Steiger \nTrieste Section of INFN\n34127TriesteItaly\n', 'M Stolarski \nLIP\n1000-149LisbonPortugal\n', 'M Sulc \nTechnical University in Liberec\n46117LiberecCzech Republic\n', 'R Sulej \nNational Centre for Nuclear Research\n00-681WarsawPoland\n', 'H Suzuki \nYamagata University\n992-8510YamagataJapan\n\nUniversidade Técnica de Lisboa\nLisbonPortugal\n\nAlso at\nChubu University\n487-8501KasugaiAichiJapan n\n', 'P Sznajder \nNational Centre for Nuclear Research\n00-681WarsawPoland\n', 'S Takekawa \nTorino Section of INFN\n10125TurinItaly\n', 'J Ter Wolbeek \nPhysikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany\n', 'S Tessaro \nTrieste Section of INFN\n34127TriesteItaly\n', 'F Tessarotto \nTrieste Section of INFN\n34127TriesteItaly\n', 'F Thibaud ', 'S Uhl \nPhysik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France\n', 'I Uman \nDepartment für Physik\nLudwig-Maximilians-Universität München\n80799MunichGermany io\n', 'M Vandenbroucke ', 'M Virius ', 'L Wang \nInstitut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany\n', 'T Weisrock \nInstitut für Kernphysik\nUniversität Mainz\n55099MainzGermany\n', 'M Wilfert \nInstitut für Kernphysik\nUniversität Mainz\n55099MainzGermany\n', 'R Windmolders \nPhysikalisches Institut\nUniversität Bonn\n53115BonnGermany\n', 'W Wiślicki \nNational Centre for Nuclear Research\n00-681WarsawPoland\n', 'H Wollny ', 'K Zaremba \nInstitute of Radioelectronics\nWarsaw University of Technology\n00-665WarsawPoland\n', 'M Zavertyaev \nLebedev Physical Institute\n119991MoscowRussia\n', 'E Zemlyanichkina \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'N Zhuravlev \nJoint Institute for Nuclear Research\nMoscow region141980DubnaRussia\n', 'M Ziembicki \nInstitute of Radioelectronics\nWarsaw University of Technology\n00-665WarsawPoland\n', 'Tel Aviv ', 'Israel '], 'authoraffiliation': ['Physikalisches Institut\nUniversität Erlangen-Nürnberg\n91054ErlangenGermany', 'Trieste Section of INFN\n34127TriesteItaly', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Lebedev Physical Institute\n119991MoscowRussia', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Department of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly', 'CERN\n1211Geneva 23Switzerland', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Faculty of Physics\nUniversity of Warsaw\n00-681WarsawPoland', 'Department of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly', 'Physikalisches Institut\nUniversität Bonn\n53115BonnGermany', 'Universität Bielefeld\nFakultät für Physik\n33501BielefeldGermany', 'Institut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany', 'Institut für Kernphysik\nUniversität Mainz\n55099MainzGermany', 'Department of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly', 'CERN\n1211Geneva 23Switzerland', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Physikalisches Institut\nUniversität Bonn\n53115BonnGermany', 'Trieste Section of INFN\n34127TriesteItaly', 'Helmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany', 'LIP\n1000-149LisbonPortugal', 'Department of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly', 'Physikalisches Institut\nUniversität Erlangen-Nürnberg\n91054ErlangenGermany', 'Trieste Section of INFN\n34127TriesteItaly', 'Department of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'Department of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Abdus Salam ICTP and Trieste Section of INFN\n34127TriesteItaly', 'Abdus Salam ICTP and Trieste Section of 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Russia, France', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Institut für Kernphysik\nUniversität Mainz\n55099MainzGermany', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'Institut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany', 'Helmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Also at\nChubu University\n487-8501KasugaiAichiJapan n', 'Also at KEK\n1-1 Oho, Tsukuba, Ibaraki305-0801Japan', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Yamagata University\n992-8510YamagataJapan', 'Universidade Técnica de Lisboa\nLisbonPortugal', 'Also at KEK\n1-1 Oho, Tsukuba, Ibaraki305-0801Japan', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Yamagata University\n992-8510YamagataJapan', 'Universidade Técnica de Lisboa\nLisbonPortugal', 'Helmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany', 'Institut für Kernphysik\nUniversität Mainz\n55099MainzGermany', 'Helmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany', 'Institut für Kernphysik\nUniversität Mainz\n55099MainzGermany', 'Institut für Kernphysik\nUniversität Mainz\n55099MainzGermany', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Also at Moscow Institute of Physics and Technology, Moscow Region\nIII. Physikalisches Institut\nRWTH Aachen University\n141700, 52056AachenRussia, Germany', 'Institut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany', 'Physikalisches Institut\nUniversität Bonn\n53115BonnGermany', 'National Centre for Nuclear Research\n00-681WarsawPoland', 'Institut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany', 'Yamagata University\n992-8510YamagataJapan', 'Universidade Técnica de Lisboa\nLisbonPortugal', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'Lebedev Physical Institute\n119991MoscowRussia', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Department of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'National Centre for Nuclear Research\n00-681WarsawPoland', 'Institute of Radioelectronics\nWarsaw University of Technology\n00-665WarsawPoland', 'Physikalisches Institut\nUniversität Erlangen-Nürnberg\n91054ErlangenGermany', 'Department of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly', 'School of Physics and Astronomy\nTel Aviv University\n69978', 'Torino Section of INFN\n10125TurinItaly', 'Department of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly', 'CERN\n1211Geneva 23Switzerland', 'Physik Department\n21 State Research Center of the Russian 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Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'LIP\n1000-149LisbonPortugal', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Institut für Kernphysik\nUniversität Mainz\n55099MainzGermany', 'Physikalisches Institut\nUniversität Bonn\n53115BonnGermany', 'Torino Section of INFN\nUniversity of Eastern Piedmont\n15100, 10125Alessandria, TurinItaly', 'Department of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Department of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly', 'Institut für Kernphysik\nUniversität Mainz\n55099MainzGermany', 'Technical University in Liberec\n46117LiberecCzech Republic', 'Department of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly', 'Physikalisches Institut\nUniversität Bonn\n53115BonnGermany', 'LIP\n1000-149LisbonPortugal', 'LIP\n1000-149LisbonPortugal', 'Department für Physik\nLudwig-Maximilians-Universität München\n80799MunichGermany io', 'LIP\n1000-149LisbonPortugal', 'Institut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany', 'CERN\n1211Geneva 23Switzerland', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'National Centre for Nuclear Research\n00-681WarsawPoland', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'National Centre for Nuclear Research\n00-681WarsawPoland', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Matrivani Institute of Experimental Research & Education\nCalcutta-700 030India', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Department of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly', 'Department of Physics and Trieste Section of INFN\nUniversity of Trieste\n34127TriesteItaly', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'Department für Physik\nLudwig-Maximilians-Universität München\n80799MunichGermany io', 'Physikalisches Institut\nUniversität Erlangen-Nürnberg\n91054ErlangenGermany', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Also at GSI mbH\nPlanckstr. 1D-64291DarmstadtGermany', 'Helmholtz-Institut für Strahlen-und Kernphysik\nUniversität Bonn\n53115BonnGermany', 'CERN\n1211Geneva 23Switzerland', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'CERN\n1211Geneva 23Switzerland', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'LIP\n1000-149LisbonPortugal', 'Matrivani Institute of Experimental Research & Education\nCalcutta-700 030India', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'Department of Physics and Torino Section of INFN\nUniversity of Turin\n10125TurinItaly', 'Trieste Section of INFN\n34127TriesteItaly', 'Institute of Scientific Instruments\n61264BrnoAS CRCzech Republic', 'Trieste Section of INFN\n34127TriesteItaly', 'LIP\n1000-149LisbonPortugal', 'Technical University in Liberec\n46117LiberecCzech Republic', 'National Centre for Nuclear Research\n00-681WarsawPoland', 'Yamagata University\n992-8510YamagataJapan', 'Universidade Técnica de Lisboa\nLisbonPortugal', 'Also at\nChubu University\n487-8501KasugaiAichiJapan n', 'National Centre for Nuclear Research\n00-681WarsawPoland', 'Torino Section of INFN\n10125TurinItaly', 'Physikalisches Institut\nUniversität Freiburg\n79104FreiburgGermany', 'Trieste Section of INFN\n34127TriesteItaly', 'Trieste Section of INFN\n34127TriesteItaly', 'Physik Department\n21 State Research Center of the Russian Federation\nInstitute for High Energy Physics\nCEA IRFU/SPhN Saclay\nTechnische Universität München\n85748, 142281, 22, 91191Garching, Protvino, Gif-sur-YvetteGermany, Russia, France', 'Department für Physik\nLudwig-Maximilians-Universität München\n80799MunichGermany io', 'Institut für Experimentalphysik\nUniversität Bochum\n44780BochumGermany', 'Institut für Kernphysik\nUniversität Mainz\n55099MainzGermany', 'Institut für Kernphysik\nUniversität Mainz\n55099MainzGermany', 'Physikalisches Institut\nUniversität Bonn\n53115BonnGermany', 'National Centre for Nuclear Research\n00-681WarsawPoland', 'Institute of Radioelectronics\nWarsaw University of Technology\n00-665WarsawPoland', 'Lebedev Physical Institute\n119991MoscowRussia', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Joint Institute for Nuclear Research\nMoscow region141980DubnaRussia', 'Institute of Radioelectronics\nWarsaw University of Technology\n00-665WarsawPoland'], 'corpusid': 76657892, 'doi': '10.1140/epjc/s10052-013-2531-6', 'github_urls': [], 'n_tokens_mistral': 20403, 'n_tokens_neox': 16503, 'n_words': 8022, 'pdfsha': '5fbb642e11d858e08602b151d18e7813bed561a1', 'pdfurls': ['https://arxiv.org/pdf/1305.7317v1.pdf'], 'title': ['EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH Hadron Transverse Momentum Distributions in Muon Deep Inelastic Scattering at 160 GeV/c The COMPASS Collaboration The COMPASS Collaboration', 'EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH Hadron Transverse Momentum Distributions in Muon Deep Inelastic Scattering at 160 GeV/c The COMPASS Collaboration The COMPASS Collaboration'], 'venue': ['Faculty of Mathematics and Physics']}

arxiv

THE NIELSEN AND REIDEMEISTER NUMBERS OF MAPS ON INFRA-SOLVMANIFOLDS OF TYPE (R) 12 Jan 2014 Alexander Fel&apos;shtyn Jong Bum Lee THE NIELSEN AND REIDEMEISTER NUMBERS OF MAPS ON INFRA-SOLVMANIFOLDS OF TYPE (R) 12 Jan 2014 We prove the rationality, the functional equations and calculate the radii of convergence of the Nielsen and the Reidemeister zeta functions of continuous maps on infra-solvmanifolds of type (R). We find a connection between the Reidemeister and Nielsen zeta functions and the Reidemeister torsions of the corresponding mapping tori. We show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type (R), then this manifold is an infra-nilmanifold. We also prove that a map on an infra-solvmanifold of type (R) induced by an affine map minimizes the topological entropy in its homotopy class and it has a rational Artin-Mazur zeta function. Finally we prove the Gauss congruences for the Reidemeister and Nielsen numbers of any map on an infra-solvmanifolds of type (R) whenever all the Reidemeister numbers of iterates of the map are finite. Our main technical tool is the averaging formulas for the Lefschetz, the Nielsen and the Reidemeister numbers on infra-solvmanifolds of type (R). 8. The Reidemeister zeta function is never defined for any homeomorphism of infra-solvmanifold of type (R), not an infra-nilmanifold 37 9. The Artin-Mazur zeta functions on infra-solvmanifolds of type (R) 39 10. The Nielsen numbers of virtually unipotent maps on infrasolvmanifolds of type (R) 41 11. Gauss congruences for the Nielsen and Reidemeister numbers 42 References 46 0. Introduction We assume everywhere X to be a connected, compact polyhedron and f : X → X to be a continuous map. Let p :X → X be the universal cover of X andf :X →X a lifting of f , i.e., p •f = f • p. Two liftsf andf ′ are called conjugate if there is a γ ∈ Γ ∼ = π 1 (X) such thatf ′ = γ •f • γ −1 . The subset p(Fix(f )) ⊂ Fix(f ) is called the fixed point class of f determined by the lifting class [f ]. A fixed point class is called essential if its index is nonzero. The number of lifting classes of f (and hence the number of fixed point classes, empty or not) is called the Reidemeister number of f , denoted by R(f ). This is a positive integer or infinity. The number of essential fixed point classes is called the Nielsen number of f , denoted by N (f ) [36]. The Nielsen number is always finite. R(f ) and N (f ) are homotopy invariants. In the category of compact, connected polyhedra the Nielsen number of a map is, apart from in certain exceptional cases, equal to the least number of fixed points of maps with the same homotopy type as f . Let G be a group and φ : G → G an endomorphism. Two elements α, α ′ ∈ G are said to be φ-conjugate if and only if there exists γ ∈ G such that α ′ = γαφ(γ) −1 . The number of φ-conjugacy classes is called the Reidemeister number of φ, denoted by R(φ). This is a positive integer or infinity. Taking a dynamical point of view, we consider the iterates of f and φ, and we may define following [17,53,18,19] several zeta functions connected with the Nielsen fixed point theory. The Reidemeister zeta functions of f and φ and the Nielsen zeta function of f are defined as power series: R φ (z) = exp ∞ n=1 R(φ n ) n z n , R f (z) = exp ∞ n=1 R(f n ) n z n , N f (z) = exp ∞ n=1 N (f n ) n z n . Whenever we mention the Reidemeister zeta function R f (z), we shall assume that it is well-defined and so R(f n ) < ∞ and R(φ n ) < ∞ for all n > 0. Hence R f (z) = N f (z) on infra-nilmanifolds by Theorem 1.1 below and on infra-solvmanifolds of type (R) by Corollary 7.6. However, there are spaces and maps for which R f (z) is not defined. The zeta functions R f (z) and N f (z) are homotopy invariants. The function N f (z) has a positive radius of convergence for any continuous map f [53]. The above zeta functions are directly analogous to the Lefschetz zeta function L f (z) := exp ∞ n=1 L(f n ) n z n , where L(f n ) := dim X k=0 (−1) k tr f n * k : H k (X; Q) → H k (X; Q) is the Lefschetz number of the iterate f n of f . The Lefschetz zeta function is a rational function of z and is given by the formula: L f (z) = dim X k=0 det I − f * k .z (−1) k+1 . The following problem was investigated: for which spaces and maps and for which groups and endomorphisms are the Nielsen and Reidemeister zeta functions rational functions? Are these functions algebraic functions? The knowledge that a zeta function is a rational function is important because it shows that the infinite sequence of coefficients of the corresponding power series is closely interconnected, and is given by the finite set of zeros and poles of the zeta function. In [19,21,22,45,20], the rationality of the Reidemeister zeta function R φ (z) was proven in the following cases: the group is finitely generated and an endomorphism is eventually commutative; the group is finite; the group is a direct sum of a finite group and a finitely generated free Abelian group; the group is finitely generated, nilpotent and torsion free. In [60,Theorem 4] the rationality of the Reidemeister and Nielsen zeta functions was proven for infra-nilmanifold under some (rather technical) sufficient conditions. It is also known that the Reidemeister numbers of the iterates of an automorphism of an almost polycyclic group satisfy remarkable Gauss congruences [23,24]. In this paper we investigate the Reidemeister and the Nielsen zeta functions on infra-solvmanifolds of type (R). Our main technical tool is the averaging formulas for the Lefschetz numbers, the Nielsen numbers and the Reidemeister numbers on infra-nilmanifolds and on infra-solvmanifolds of type (R). Recently, using these averaging formulas, K. Dekimpe and G.-J. Dugardein [11,16] calculated the Nielsen numbers via Lefschetz numbers and proved the rationality of the Nielsen zeta functions on infra-nilmanifolds. We prove in this paper the rationality, the functional equations and calculate the radii of convergence of the Nielsen and the Reidemeister zeta functions of continuous maps on infra-solvmanifolds of type (R). We find a connection between the Reidemeister and Nielsen zeta functions and the Reidemeister torsions of the corresponding mapping tori. We show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type (R), then this manifold is an infra-nilmanifold. We also prove that a map on an infra-solvmanifold of type (R) induced by an affine map minimizes the topological entropy in its homotopy class and it has a rational Artin-Mazur zeta function. Finally we prove the Gauss congruences for the Reidemeister and Nielsen numbers of any map on an infra-solvmanifolds of type (R) whenever all the Reidemeister numbers of iterates of the map are finite. Let us present the contents of the paper in more details. In Section 1 we describe the averaging formulas for the Lefschetz numbers, the Nielsen numbers and the Reidemeister numbers on infra-nilmanifolds and Dekimpe-Dugardein's formula for the Nielsen numbers. In Section 2, we obtain a partial generalization of K. Dekimpe and G.-J. Dugardein's formula from fixed points on infra-nilmanifolds to coincidences on infra-solvmanifolds of type (R) when the holonomy group is a cyclic group. The rationality and the functional equations for the Reidemeister and the Nielsen zeta functions on infra-solvmanifolds of type (R) are proven in Sections 3 and 7. After studying the asymptotic Nielsen numbers on infra-solvmanifolds of type (R) in Section 4, we discuss the relationship between the topological entropies, the asymptotic Nielsen numbers and the radius of convergence of the Nielsen and the Reidemeister zeta functions in Section 5. We also prove in Section 5 that a map on an infra-solvmanifold of type (R) induced by the affine map minimizes the topological entropy in its homotopy class . In Section 6, we find a connection between the Nielsen and the Reidemeister zeta functions and the Reidemeister torsions of the corresponding mapping tori. In Section 7, we obtain the averaging formula for the Reidemeister numbers on infra-solvmanifolds of type (R) and we are able to show that the Reidemeister zeta functions on infra-solvmanifolds of type (R) coincide with the Nielsen zeta functions. In Section 8, we show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type (R), then this manifold is an infra-nilmanifold. In Section 9 we prove that the Artin-Mazur zeta function coincides with the Nielsen zeta function and is a rational function with functional equation for a continuous map on an infra-solvmanifold of type (R) induced by an affine map. In Section 11 we prove the Gauss congruences for the Reidemeister and Nielsen numbers of any map on an infra-solvmanifolds of type (R) whenever all the Reidemeister numbers of iterates of the map are finite. the possibility of the present research during his visit there. The authors are grateful to Karel Dekimpe and Gert-Jan Dugardein for helpful comments and valuable discussions. The authors would like to thank the referee for making careful corrections to a few expressions and suggesting some relevant references in the original version of the article. This helped improving some results. Averaging formulas and Dekimpe-Dugardein's formula We consider almost Bieberbach groups Π ⊂ G ⋊ Aut(G), where G is a connected, simply connected nilpotent Lie group, and infra-nilmanifolds M = Π\G. It is known that these are exactly the class of almost flat Riemannian manifolds [55]. It is L. Auslander's result (see, for example, [44]) that Γ := Π ∩ G is a lattice of G, and is the unique maximal normal nilpotent subgroup of Π. The group Φ = Π/Γ is the holonomy group of Π or M . Thus we have the following commutative diagram: 1 − −−− → G − −−− → G ⋊ Aut(G) − −−− → Aut(G) − −−− → 1       1 − −−− → Γ − −−− → Π p − −−− → Φ − −−− → 1 Thus Φ sits naturally in Aut(G). Denote ρ : Φ → Aut(G), A → A * = the differential of A. Let M = Π\G be an infra-nilmanifold. Any continuous map f : M → M induces a homomorphism φ : Π → Π. Due to [43, Theorem 1.1], we can choose an affine element (d, D) ∈ G ⋊ Endo(G) such that (1) φ(α) • (d, D) = (d, D) • α, ∀α ∈ Π. This implies that the affine map (d, D) : G → G induces a continuous map on the infra-nilmanifold M = Π\G, which is homotopic to f . That is, f has an affine homotopy lift (d, D). By [41, Lemma 3.1], we can choose a fully invariant subgroup Λ ⊂ Γ of Π which is of finite index. Therefore φ(Λ) ⊂ Λ and so φ induces the following commutative diagram 1 − −−− → Λ − −−− → Π − −−− → Ψ − −−− → 1   φ ′   φ   φ 1 − −−− → Λ − −−− → Π − −−− → Ψ − −−− → 1 where Ψ = Π/Λ is finite. Applying (1) for λ ∈ Λ ⊂ Π, we see that φ(λ) = dD(λ)d −1 = (τ d D)(λ) where τ d is the conjugation by d. The homomorphism φ ′ : Λ → Λ induces a unique Lie group homomorphism F = τ d D : G → G, and hence a Lie algebra homomorphism F * : G → G. On the other hand, since φ(Λ) ⊂ Λ, f has a liftf : N → N on the nilmanifold N := Λ\G which finitely and regularly covers M and has Ψ as its group of covering transformations. Theorem 1.1 (Averaging Formula [41,Theorem 3.4], [33,Theorem 6.11]). Let f be a continuous map on an infra-nilmanifold Π\G with holonomy group Φ. Let f have an affine homotopy lift (d, D) and let φ : Π → Π be the homomorphism induced by f . Then we have L(f ) = 1 |Φ| A∈Φ det(I − A * F * ) = 1 |Φ| A∈Φ det(I − A * D * ) , N (f ) = 1 |Φ| A∈Φ | det(I − A * F * )| = 1 |Φ| A∈Φ | det(I − A * D * )|, R(f ) = R(φ) = 1 |Φ| A∈Φ σ (det(A * − F * )) = 1 |Φ| A∈Φ σ (det(A * − D * )) where σ : R → R ∪ {∞} is defined by σ(0) = ∞ and σ(x) = |x| for all x = 0. Recently, Dekimpe and Dugardein in [11] showed the following: Let f : M → M be a continuous map on an infra-nilmanifold M . Then the Nielsen number N (f ) is either equal to |L(f )| or equal to the expression |L(f ) − L(f + )|, where f + is a lift of f to a 2-fold covering of M . By exploiting the exact nature of this relationship for all powers of f , they proved that the Nielsen zeta function N f (z) is always a rational function. Let M = Π\G be an infra-nilmanifold with the holonomy group Φ and let f : M → M be a continuous map with an affine homotopy lift (d, D). Let A ∈ Φ. Then we can choose g ∈ G so that α = (g, A) ∈ Π. Write φ(α) = (g ′ , A ′ ). By (1), we have (g ′ , A ′ )(d, D) = (d, D)(g, A) ⇒ A ′ D = DA. Thus φ induces a functionφ : Φ → Φ given byφ(A) = A ′ so that it satisfies that (2)φ(A)D = DA,φ(A) * D * = D * A * for all A ∈ Φ. In what follows, we shall give a brief description of main results in [11]. We can choose a linear basis of G so that ρ(Φ) = Φ * ⊂ Aut(G) can be expressed as diagonal block matrices Φ 1 0 0 Φ 2 ⊂ GL(n 1 , R) × GL(n 2 , R) ⊂ GL(n, R) and D * can be written in block triangular form D 1 * 0 D 2 where D 1 and D 2 have eigenvalues of modulus ≤ 1 and > 1, respectively. We can assume Φ = Φ 1 × Φ 2 . Every element α ∈ Π is of the form (a, A) ∈ G ⋊ Aut(G) and α is mapped to A = (A 1 , A 2 ). We define Π + = {α ∈ Π | det A 2 = 1}. Then Π + is a subgroup of Π of index at most 2. If [Π : Π + ] = 2, then Π + is also an almost Bieberbach group and the corresponding infra-nilmanifold M + = Π + \GN (f k ) = (−1) p+(k+1)n L(f k ), when Π = Π + ; (−1) p+(k+1)n L(f k + ) − L(f k ) , when Π = Π + , where p be the number of real eigenvalues of D * which are > 1 and n be the number of real eigenvalues of D * which are < −1. Remark 1.3. 1) In [11,Theorem 4.4] Nielsen numbers N (f n ) are expressed in terms of Lefschetz numbers L(f n ) and L(f n + ) via a table given by parity of n. 2) The proof of our Theorem 3.5 covers the case when Π = Π + in Theorem 1.2 above because in this case N (f ) = |L(f )|. 2. Coincidences on infra-solvmanifolds of type (R) with a cyclic holonomy group In this section, we will be concerned with a generalization of Theorem 1.2 when k = 1 (that is, N (f ) = |L(f )| or |L(f + ) − L(f )|) from fixed points on infra-nilmanifolds to coincidences on infra-solvmanifolds of type (R). We obtain a partial result for coincidences on infra-solvmanifolds of type (R) when the holonomy group is a cyclic group. Let S be a connected, simply connected solvable Lie group of type (R), and let C be a compact subgroup of Aut(S). Let Π ⊂ S ⋊ C be torsion free and discrete which is a finite extension of the lattice Γ = Π ∩ S of S. Such a group Π is called an SB-group modeled on S. The quotient space Π\S is called an infra-solvmanifold of type (R) with holonomy group Φ = Π/Γ. When Π ⊂ S, Π\S is a special solvmanifold of type (R). Thus the infra-solvmanifold Π\S is finitely and regularly covered by the special solvmanifold Γ\S with the group of covering transformations Φ. For more details, we refer to [42]. Let M = Π\S be an infra-solvmanifold of type (R) with the holonomy group Φ. Then Φ sits naturally in Aut(S). Write ρ : Φ → Aut(S), A → A * . Let f, g : M → M be maps with affine homotopy lifts (d, D), (e, E) : S → S, respectively. Then f and g induce homomorphisms φ, ψ : Π → Π by the following rules: φ(α) • (d, D) = (d, D) • α, ψ(α) • (e, E) = (e, E) • α ∀α ∈ Π. In turn, we obtain functionsφ,ψ : Φ → Φ satisfyinĝ φ(A)D = DA andψ(A)E = EA ∀A ∈ Φ. Thusφ (A) * D * = D * A * andψ(A) * E * = E * A * ∀A ∈ Φ.(3) Recall the following well-known facts from representation theory: Theorem 2.1 (H. Maschke). Let ρ : Φ → GL(n, R) be a representation. Then there exist irreducible representations ρ i : Φ → GL(n i , R) such that ρ is similar to ρ 1 ⊕ · · · ⊕ ρ s . Theorem 2.2. Let Φ = A be a cyclic group of order n and let ρ : Φ → GL(m, R) be a faithful R-irreducible representation. If n = 1 then ρ is the trivial representation ρ triv . If n = 2, then m = 1 and ρ(A) = −1. In this case, we denote ρ by τ . If n > 2, then there exists k ∈ Z such that gcd(n, k) = 1 and ρ is similar to the irreducible rotation given by Φ −→ GL(2, R), A −→ cos 2kπ n − sin 2kπ n sin 2kπ n cos 2kπ n . Consider the case where the infra-solvmanifold M of type (R) is orientable (for coincidences) with holonomy group Φ a cyclic group with a generator A 0 . By Theorem 2.1, the natural representation ρ : Φ → Aut(S) ∼ = Aut(S) is similar to a sum of irreducible representations. If σ : Φ → GL(m, R) is irreducible, then the induced representationσ : Φ/ ker ρ → GL(m, R) is faithful and irreducible. By Theorem 2.2,σ is similar to ρ triv , τ or a rotation. Thus we may assume that ρ = mρ triv ⊕ kτ ⊕ ρ 1 ⊕ · · · ⊕ ρ t , where ρ i : Φ → GL(2, R) is an irreducible rotation. That is, there is a linear basis of S so that ρ(A 0 ) ∈ Aut(S) can be represented as diagonal block matrices ρ(A 0 ) =        I m −I k Φ 1 . . . Φ t        where Φ i = ρ i (A 0 ) ∈ GL(2, R). Remark that if k > 0 then the order of Φ is even, and det(ρ i (A 0 )) = 1 for all i. Hence det(ρ(A 0 )) = 1 if and only if k is even. This is the only case when the infra-solvmanifold is orientable and hence k is even. Using the identities (3), we can write D * and E * as block matrices D * =   D triv 0 0 0 D τ 0 * * D   , E * =   E triv 0 0 0 E τ 0 * * Ê   where D triv , E triv are m × m, D τ , E τ are k × k andD,Ê are 2t × 2t. For A ∈ Φ, A = A p 0 for some p and A * =   I m (−1) p I k A *   . Writeρ = ρ 1 ⊕ · · · ⊕ ρ t : Φ → GL(2t, R), A →ρ(A) = A * (abusing the notation: ρ(A) = A * ). Then the identities (3) inducê φ(A) * D =DA * ,ψ(A) * Ê =ÊA * . Hence, for all A = A p 0 and B = A q 0 ∈ Φ, we have that det(E * − A * D * ) det(E * − B * D * ) (4) = det(E triv − D triv ) 2 det(E τ − (−1) p D τ ) det(E τ − (−1) q D τ ) × det(Ê − A * D ) det(Ê − B * D ). Note here that det(Ê − A * D ) det(Ê − B * D ) ≥ 0,Φ = A 0 . If ρ(A 0 ) has no eigenvalue −1, i.e., if k = 0, then N (f, g) = |L(f, g)|. Assume k > 0 (is even); then Φ = A 0 is of even order. Let Φ 0 = A 2 0 and let Π 0 be the subgroup of Π induced by the inclusion Φ 0 ֒→ Φ. Remark also that if D τ = 0 or E τ = 0, then we still have N (f, g) = |L(f, g)|. We also assume that D τ = 0 and E τ = 0. Proof. It is clear that [Π : Π 0 ] = 2 and that Π 0 is also an SB-group and the corresponding infra-solvmanifold Π 0 \S is a double covering of Π\S. To prove the last assertion, we may consider and assume that (d, D) : S → S induces f and that φ : Π → Π is a homomorphism such that φ(α)(d, D) = (d, D)α, ∀α ∈ Π. We need to show that (d, D) also induces a map on Π 0 \S. For this purpose, it is enough to show that φ(Π 0 ) ⊂ Π 0 . For any β = (a, A) ∈ Π 0 , let φ(β) = (b,φ(A)). Since (a, A) ∈ Π 0 , we have A ∈ Φ 0 . The above identity implies thatφ (A) * D * = D * A * ⇒ D τ = 0 orφ(A) ∈ Φ 0 . Since D τ = 0, this finishes the proof of the last assertion. For any A = A p 0 ∈ Φ, we recall from (4) that det(E * − A * D * ) = det(E triv − D triv ) det(E τ − (−1) p D τ ) det(Ê − A * D ) and det(Ê −D) det(Ê − A * D ) ≥ 0. Let ǫ o = sign det(E τ − D τ ), ǫ e = sign det(E τ + D τ ). Then ǫ o = ±ǫ e . Notice that the values ǫ o and ǫ e depend both on f and g. When ǫ o = ǫ e , we still have N (f, g) = |L(f, g)|. When ǫ o = −ǫ e , we have that N (f, g) = 1 |Φ| A∈Φ | det(E * − A * D * )| = 1 |Φ|   A∈Φ 0 | det(E * − A * D * )| + A / ∈Φ 0 | det(E * − A * D * )|   = ǫ o |Φ|   A∈Φ 0 det(E * − A * D * ) − A / ∈Φ 0 det(E * − A * D * )   = ǫ o |Φ|   2 A∈Φ 0 det(E * − A * D * ) − A∈Φ det(E * − A * D * )   = ǫ o   1 |Φ 0 | A∈Φ 0 det(E * − A * D * ) − 1 |Φ| A∈Φ det(E * − A * D * )   = ǫ o (L(f 0 , g 0 ) − L(f, g)). Therefore, we can summarize what we have observed as follows: Theorem 2.5. Let M = Π\S be an orientable infra-solvmanifold of type (R) with cyclic holonomy group Φ = A 0 . Let ρ : Φ → Aut(G) be the natural presentation. Then ρ is similar to the sum of irreducible representations mρ triv ⊕ kτ ⊕ ρ 1 ⊕ · · · ⊕ ρ t , where ρ triv : Φ → GL(1, R) is the trivial representation, τ : Φ → GL(1, R) is the representation given by τ (A 0 ) = −1, and ρ i : Φ → GL(2, R) is an irreducible rotation. Let f, g : M → M be continuous maps with affine homotopy lifts (d, D), (e, E) respectively. Then D * and E * can be expressed as block matrices D * =   D triv 0 0 0 D τ 0 * * D   , E * =   E triv 0 0 0 E τ 0 * * Ê   where D triv , E triv are m×m, D τ , E τ are k×k andD,Ê are 2t×2t. Moreover, we have that: (1) If k = 0, then N (f, g) = |L(f, g)|. (2) If k > 0 and det(E τ −D τ ) det(E τ +D τ ) ≥ 0, then N (f, g) = |L(f, g)|. (3) If k > 0 and det(E τ − D τ ) det(E τ + D τ ) < 0, then the maps f, g lift to maps f 0 , g 0 : M 0 → M 0 on a double covering M 0 of M which have the same homotopy lifts as f, g respectively so that the following formula holds N (f, g) = |L(f 0 , g 0 ) − L(f, g)|. Proof. We are left to notice only one thing: If D τ = 0 or E τ = 0, then k > 0 is even and so det( E τ − D τ ) det(E τ + D τ ) ≥ 0. The rationality and the functional equation We start with an example that shows how different can be the Nielsen, the Reidemeister and the Lefschetz zeta functions. 20]). Let f : S 2 ∨ S 4 → S 2 ∨ S 4 to be a continuous map of the bouquet of spheres such that the restriction f | S 4 = id S 4 and the degree of the restriction f | S 2 : S 2 → S 2 equal to −2. Then L(f ) = 0, hence N (f ) = 0 since S 2 ∨ S 4 is simply connected. For k > 1 we have L(f k ) = 2 + (−2) k = 0, therefore N (f k ) = 1. R(f k ) = 1 for all k ≥ 1 since S 2 ∨ S 4 is simply connected. From this we have by direct calculation that Example 3.1 ([N f (z) = exp(−z) · 1 1 − z ; R f (z) = 1 1 − z ; L f (z) = 1 (1 − z) 2 (1 + 2z) . Hence N f (z) is a meromorphic function, and R f (z) and L f (z) are rational and different. We give now some other examples of the Nielsen and the Reidemeister zeta functions on infra-nilmanifolds. For the explicit computation of the zeta functions, the following is useful. N f (z) = A∈Φ |Φ| exp ∞ n=1 | det(A * − D n * )| n z n . When R f (z) is defined, R f (z) = R φ (z) = N f (z). Proof. We may assume R f (z) is defined. By Theorem 1.1, we have that R f (z) = R φ (z) = N f (z) and R φ (z) = exp ∞ n=1 R(φ n ) n z n = exp ∞ n=1 1 |Φ| A∈Φ | det(A * − F n * )| n z n = A∈Φ exp ∞ n=1 | det(A * − F n * )| n z n 1 |Φ| = A∈Φ |Φ| exp ∞ n=1 | det(A * − F n * )| n z n . Example 3.3. This is an example used by Anosov to show that the Anosov relation does not hold when the manifold is not a nilmanifold [1]. Let α = (a, A) and t i = (e i , I 2 ) be elements of R 2 ⋊ Aut(R 2 ), where a = 1 2 0 , A = 1 0 0 −1 , e 1 = 1 0 , e 2 = 0 1 . Then A has period 2, (a, A) 2 = (a + Aa, I 2 ) = (e 1 , I 2 ), and t 2 α = αt −1 2 . Let Γ be the subgroup generated by t 1 and t 2 . Then it forms a lattice in R 2 and the quotient space Γ\R 2 is the 2-torus. It is easy to check that the subgroup Π = Γ, (a, A) ⊂ R 2 ⋊ Aut(R 2 ) generated by the lattice Γ and the element (a, A) is discrete and torsion free. Furthermore, Γ is a normal subgroup of Π of index 2. Thus Π is an (almost) Bieberbach group, which is the Klein bottle group, and the quotient space Π\R 2 is the Klein bottle. Thus Γ\R 2 → Π\R 2 is a double covering projection. Let K : R 2 → R 2 be the linear automorphism given by K = −1 0 0 2 . It is not difficult to check that K inducesf : Γ\R 2 → Γ\R 2 and f : Π\R 2 → Π\R 2 so that the following diagram is commutative: R 2 K − −−− → R 2     Γ\R 2f − −−− → Γ\R 2     Π\R 2 f − −−− → Π\R 2 Note that all the vertical maps are the natural covering maps. In particular, Γ\R 2 → Π\R 2 is a double covering by the holonomy group of Π/Γ, which is Φ = {I, A} ∼ = Z 2 . By Theorem 1.1, we have L(f n ) = 1 2 (det(I − K n ) + det(I − AK n )) = 1 − (−1) n , N (f n ) = 2 n (1 − (−1) n ). In particular, R(f n ) = 2 n+1 when n is odd; otherwise, R(f n ) = ∞. Therefore, the Reidemeister zeta function R f (z) is not defined, and L f (z) = exp ∞ n=1 2 2n − 1 z 2n−1 = 1 + z 1 − z , N f (z) = exp ∞ n=1 2 2n 2n − 1 z 2n−1 = exp ∞ n=1 2 2n − 1 (2z) 2n−1 = 1 + 2z 1 − 2z . Example 3.4. Consider Example 3.5 of [41] in which an infra-nilmanifold M modeled on the 3-dimensional Heisenberg group Nil has the holonomy group of order 2 generated by A and a self-map f on M is induced by the automorphism D : Nil → Nil given by D :   1 x z 0 1 y 0 0 1   −→   1 −4x − y z ′ 0 1 6x + 2y 0 0 1   where z ′ = −2z − (12x 2 + 10xy + y 2 ) . Then with respect to the ordered (linear) basis for the Lie algebra of Nil e 1 =   0 0 1 0 0 0 0 0 0   , e 2 =   0 1 0 0 0 0 0 0 0   , e 3 =   0 0 0 0 0 1 0 0 0   , the differentials of A and D are A * =   1 0 0 0 −1 0 0 0 −1   , D * =   −2 0 0 0 −4 −1 0 6 2   . By Proposition 3.2, we have R φ (z) = exp ∞ n=1 | det(I − D n * )| n z n exp ∞ n=1 | det(A * − D n * )| n z n . Remark that A * is a block diagonal matrix with 1 × 1 block I 1 and 2 × 2 block −I 2 . We have | det(A * − D n * )| = | det(I 1 − D n 1 ) det(−I 2 − D n 2 )| = | det(I 1 − D n 1 )|| det(I 2 + D n 2 )| = |(1 − (−2) n )|(−1) n det(I 2 + D n 2 ) = (2 n − (−1) n )(−1) n i tr( i D n 2 ). Consequently, we obtain exp ∞ n=1 | det(A * − D n * )| n z n = exp ∞ n=1 (2 n − (−1) n )(−1) n i tr( i D n 2 ) n z n = exp ∞ n=1 i tr( i D n 2 ) n (−2z) n − ∞ n=1 i tr( i D n 2 ) n z n = i det(I − z i D 2 ) det(I + 2z i D 2 ) = 1 − z 1 + 2z · 1 + 2z − 2z 2 1 − 4z − 8z 2 · 1 + 2z 1 − 4z . In a similar fashion, we compute exp ∞ n=1 | det(I − D n * )| n z n = exp ∞ n=1 | det(I 1 − D n 1 ) det(I 2 − D n 2 )| n z n = exp ∞ n=1 (2 n − (−1) n )(−1) n+1 i (−1) i tr( i D n 2 ) n z n = i det(I + 2z i D 2 ) det(I − z i D 2 ) (−1) i = 1 + 2z 1 − z · 1 + 2z − 2z 2 1 − 4z − 8z 2 · 1 − 4z 1 + 2z . The last identity of the above computations follows from the definition of i D 2 (see [32,Lemma 3.2]). Namely, we have 0 D 2 = 1, 1 D 2 = D 2 , 2 D 2 = det(D 2 ) = −2. THE NIELSEN AND REIDEMEISTER NUMBERS ON INFRA-SOLVMANIFOLDS 15 In all, we obtain that N f (z) = R f (z) = 1 + 2z − 2z 2 1 − 4z − 8z 2 . Theorem 3.5. Let f be a continuous map on an infra-nilmanifold with an affine homotopy lift (d, D). Assume N (f ) = |L(f )|. Then the Nielsen zeta function N f (z) is a rational function and is equal to N f (z) = L f ((−1) q z) (−1) r where q is the number of real eigenvalues of D * which are < −1 and r is the number of real eigenvalues of D * of modulus > 1. When the Reidemeister zeta function R f (z) is defined, we have R f (z) = R φ (z) = N f (z) . Proof. By [52, Theorem 8.2.2] N (f ) = |L(f )| implies N (f n ) = |L(f n )| for all n. Let ǫ n be the sign of det(I − D n * ). Let q be the number of real eigenvalues of D * which are less than −1 and r be the number of real eigenvalues of D * of modulus > 1. Then ǫ n = (−1) r+qn . By Theorem 1.1, we have that ǫ 1 det(I − A * D * ) ≥ 0 for all A ∈ Φ. In particular, we have det(I − A * D * ) det(I − B * D * ) ≥ 0 for all A, B ∈ Φ. Choose arbitrary n > 0. By [52,Lemma 8 .2.1], det(I − A * D n * ) det(I − D n * ) ≥ 0 for all A ∈ Φ. Hence we have N (f n ) = ǫ n L(f n ) = (−1) r+qn L(f n ). Consequently, N f (z) = exp ∞ n=1 N (f n ) n z n = exp ∞ n=1 (−1) r+qn L(f n ) n z n = exp ∞ n=1 L(f n ) n ((−1) q z) n (−1) r = L f ((−1) q z) (−1) r is a rational function. Assume R f (z) is defined. So, R(f n ) = R(φ n ) < ∞ for all n > 0. On infranilmanifolds, by Theorem 1.1, it is equivalent to saying that det(A * − D n * ) = 0 for all A ∈ Φ and all n, and hence σ (det(A * − D n * )) = | det(A * − D n * )|. Thus R(f n ) = R(φ n ) = 1 |Φ| A∈Φ σ (det(A * − D n * )) = 1 |Φ| A∈Φ | det(A * − D n * )| = N (f n ). This implies that R f (z) = R φ (z) = N f (z). Therefore, for those classes of maps on infra-nilmanifolds for which Anosov relation N (f ) = |L(f )| holds [40,46,8] and for those classes of infranilmanifolds for which Anosov relation N (f ) = |L(f )| holds for ALL maps [1,7,8,9], the Nielsen zeta functions and the Reidemeister zeta functions are rational functions. In general case, using the results of Theorem 1.2, Dekimpe and Dugardein described the Nielsen zeta function of f as follows: Theorem 3.6 ([11, Theorem 4.5]). Let f be a continuous map on an infranilmanifold Π\G with an affine homotopy lift (d, D). Then the Nielsen zeta function is a rational function and is equal to N f (z) =    L f ((−1) n z) (−1) p+n when Π = Π + ; L f + ((−1) n z) L f ((−1) n z) (−1) p+n when Π = Π + , where p is the number of real eigenvalues of D * which are > 1 and n is the number of real eigenvalues of D * which are < −1. When the Reidemeister zeta function R f (z) is defined, we have R f (z) = R φ (z) = N f (z) . Remark 3.7. In [11,Theorem 4.5] the Nielsen zeta function is expressed in terms of Lefschetz zeta functions L f (z) and L f + (z) via a table given by parity of p and n. The class of infra-solvmanifolds of type (R) contains and shares a lot of properties of the class of infra-nilmanifolds such as the averaging formula for Nielsen numbers, see [32,42]. Therefore, Theorem 1.2 and the statement about N f (z) in Theorem 3.6 can be generalized directly to the class of infra-solvmanifolds of type (R), see Remark in [11,Sec. 4]. To write down a functional equation for the Reidemeister and the Nielsen zeta function, we recall the following functional equation for the Lefschetz zeta function: Proposition 8], see also [14]). Let M be a closed orientable manifold of dimension m and let f : M → M be a continuous map of degree d. Then Lemma 3.8 ([25,L f α dz = ǫ (−αdz) (−1) m χ(M ) L f (αz) (−1) m where α = ±1 and ǫ ∈ C is a non-zero constant such that if |d| = 1 then ǫ = ±1. Proof. In the Lefschetz zeta function formula (0), we may replace f * by f * : H * (M ; Q) → H * (M ; Q). Let β k = dim H k (M ; Q) be the kth Betti number of M . Let λ k,j be the (complex and distinct) eigenvalues of f * k : H k (M ; Q) → H k (M ; Q) Via the natural non-singular pairing in the cohomology H k (M ; Q) ⊗ H m−k (M ; Q) → Q, the operators f * m−k and d(f * k ) are adjoint to each other. Hence since λ k,j is an eigenvalue of f * k , µ ℓ,j = d/λ k,j is an eigenvalue of f * m−k = f * ℓ . Furthermore, β k = β m−k = β ℓ . Consequently, we have L f α dz = m k=0 β k j=1 1 − λ k,j α dz (−1) k+1 = m k=0 β k j=1 1 − d λ k,j αz (−1) k+1 − αdz λ k,j (−1) k = m ℓ=0 β m−ℓ j=1 (1 − µ ℓ,j αz) (−1) m−ℓ+1 m k=0 β k j=1 − αdz λ k,j (−1) m−ℓ =   m ℓ=0 β ℓ j=1 (1 − µ ℓ,j αz) (−1) ℓ+1 m k=0 β k j=1 − αdz λ k,j (−1) ℓ   (−1) m = L f (αz) (−1) m · (−αdz) m ℓ=0 (−1) ℓ β ℓ · m k=0 β k j=1 λ (−1) k+1 k,j = L f (αz) (−1) m ǫ(−αdz) (−1) m χ(M ) . Here, ǫ = m k=0 β k j=1 λ (−1) k+1 k,j = ± m k=0 det(f * k ). We obtain: Theorem 3.9 (Functional Equation) . Let f be a continuous map on an orientable infra-nilmanifold M = Π\G with an affine homotopy lift (d, D). Then the Reidemeister zeta function, whenever it is defined, and the Nielsen zeta function have the following functional equations: R f 1 dz = R f (z) (−1) m ǫ (−1) p+n when Π = Π + ; R f (z) (−1) m ǫ −1 when Π = Π + and N f 1 dz = N f (z) (−1) m ǫ (−1) p+n when Π = Π + ; N f (z) (−1) m ǫ −1 when Π = Π + where d is a degree f , m = dim M , ǫ is a constant in C × , σ = (−1) n , p is the number of real eigenvalues of D * which are > 1 and n is the number of real eigenvalues of D * which are < −1. If |d| = 1 then ǫ = ±1. Proof. Assume Π = Π + . Then R f (z) = N f (z) = L f (σz) (−1) p+n . By Lemma 3.8, we have R f 1 dz = N f 1 dz = L f σ dz (−1) p+n = ǫ(−σdz) (−1) m χ(M ) L f (σz) (−1) m (−1) p+n = N f (z) (−1) m ǫ (−1) p+n (−σdz) (−1) m+p+n χ(M ) = R f (z) (−1) m ǫ (−1) p+n (−σdz) (−1) m+p+n χ(M ) . Assume now that Π = Π + . First we claim that f and f + have the same degree. Let π : M + → M be the natural double covering projection. Then Π/Π + ∼ = Z 2 is the group of covering transformations of π. By [4, III.2], the homomorphism π * : H m (M ; Q) → H m (M + ; Q) induces an isomorphism π * : H m (M ; Q) → H m (M + ; Q) Π/Π + . In particular, π * is injective. If x is the nontrivial covering transformation, we have the commutative diagram M + π ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ x / / M + π } } ④ ④ ④ ④ ④ ④ ④ ④ M This induces the following commutative diagram By Theorem 3.6 and Lemma 3.8, we have H m (M + ; Q) x * / / H m (M + ; Q) H m (M ; Q) π * g g ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ π * 7 7 ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦R f 1 dz = N f 1 dz = L f + σ dz (−1) p+n · L f σ dz (−1) p+n+1 = ǫ(−σdz) (−1) m χ(M ) L f + (σz) (−1) m (−1) p+n × ǫ(−σdz) (−1) m χ(M ) L f (σz) (−1) m (−1) p+n+1 = N f (z) (−1) m ǫ(−σdz) (−1) m χ(M ) −1 = R f (z) (−1) m ǫ(−σdz) (−1) m χ(M ) −1 . On the other hand, it is known that χ(M ) = 0, e.g. see the remark below, which finishes our proof. Remark 3.10. Let G be a torsion-free polycylic group. Then χ(G) = 0. For, by induction, we may assume that G is an extension of Z m by Z n ; then as χ(Z) = 0, we have χ(G) = χ(Z m )χ(Z n ) = 0, [6, Theorem 6.4.2]. Another proof: A solvmanifold is aspherical and its fundamental group contains a nontrivial Abelian normal subgroup. By Gottlieb's theorem, its Euler characteristic is zero. If S is a torsion-free extension of G by a finite group of order k, then k · χ(S) = χ(G) = 0 ⇒ χ(S) = 0. Remark 3.11. As it is mentioned above, since Theorem 3.6 is true for the Nielsen zeta functions on infra-solvmanifolds of type (R), the functional equation for the Nielsen zeta functions in Theorem 3.9 is true on infra-solvmanifolds of type (R) (see Theorem 7.9 for the Reidemeister zeta functions). Asymptotic Nielsen numbers The growth rate of a sequence a n of complex numbers is defined by Growth(a n ) := max 1, lim sup n→∞ |a n | 1/n . We define the asymptotic Nielsen number [34] and the asymptotic Reidemeister number to be the growth rate N ∞ (f ) := Growth(N (f n )) and R ∞ (f ) := Growth(R(f n )) correspondingly. These asymptotic numbers are homotopy type invariants. We denote by sp(A) the spectral radius of the matrix or the operator A, sp(A) = lim n n A n | which coincide with the largest modulus of an eigenvalue of A. We denote by F * := m By [42, Theorem 2.2], we may assume that f is induced by a Lie group homomorphism D : S → S. Let {λ 1 , · · · , λ m } be the eigenvalues of D * , counted with multiplicities. First we note from definition that sp( D * ) = |λ j |>1 |λ j | when sp(D * ) > 1 1 when sp(D * ) ≤ 1. In the case when sp(D * ) ≤ 1, the eigenvalues of q≥1 D * are multiples of eigenvalues of D * , which are ≤ 1. On the other hand 0 D * = id, and hence sp( D * ) = 1. Recalling If |λ| ≤ 1 then lim sup n 1 n log |1 − λ n | = 0. For, log |1 − λ n | ≤ log 2. If |λ| > 1 then using L'Hôpital's rule, we have N (f n ) = | det(I − D n * )| = m j=1 |1 − λ n j |,|λ| n − 1 ≤ |1 − λ n | ≤ |λ| n + 1 ⇒ lim n→∞ 1 n log |1 − λ n | = log |λ|. Hence N ∞ (f ) = max 1, lim sup n→∞ N (f n ) 1/n = max    1, |λ|>1 |λ|    = sp( D * ). Next we consider the case where N (f n ) = 0 for some n. Thus some λ j is an nth root of unity. For each such λ j , consider all k's for which |1 − λ k j | = 0. Since by the assumption λ j = 1, there are infinitely many such k's. Furthermore, there are infinitely many k's for which |1 − λ k j | = 0 for all such (finitely many) λ j . Therefore, when sp(D * ) > 1 we have log lim sup n→∞ N (f n ) 1/n = lim sup k→∞ 1 k log N (f k ) = lim sup k→∞ 1 k m j=1 log |1 − λ k j | = |λ|>1 lim sup k→∞ 1 k log |1 − λ k | = log   |λ|>1 |λ|   ; when sp(D * ) ≤ 1 we have log lim sup n N (f n ) 1/n = 0. This completes the proof. In fact, what we have shown in the above proof is the following: Recall that if f : M → M is a continuous map on an infra-nilmanifold M = Π\G with the holonomy group Φ and if f has an affine homotopy lift (d, D), then f induces a homomorphism φ : Π → Π defined by the rule: ∀α ∈ Π, φ(α) • (d, D) = (d, D) • α. Furthermore, the homomorphism φ induces a functionφ : Φ → Φ satisfying the identity (2): ∀A ∈ Φ,φ(A)D = DA. For any n ≥ 1, we can observe that: (1) f n has an affine homotopy lift (d, D) n = ( * , D n ), (2) f n induces a homomorphism φ n : Π → Π, (3) the homomorphism φ n induces a function φ n =φ n : Φ → Φ. Recall from Theorem 1.1 the averaging formula: N (f n ) = 1 |Φ| A∈Φ | det(I − A * D n * )|. Since 1 |Φ| | det(I − D n * )| ≤ N (f n ), we have 1 n log N (f n ) ≥ 1 n (log | det(I − D n * )| − log |Φ|) ⇒ lim sup 1 n log N (f n ) ≥ lim sup 1 n (log | det(I − D n * )|) . This induces from Corollary 4.2 that sp( D * ) = Growth(L(D n * )) ≤ N ∞ (f ). Next we recall [8, Lemma 3.1]: Give A ∈ Φ, we can choose a sequence (B i ) i∈N of elements in Φ be taking B 1 = A and such that B i+1 =φ(B i ), associated to f . Since Φ is finite, this sequence will become periodic from a certain point onwards. Namely, there exist j, k ≥ 1 such that B j+k = B j . It is shown in [8, Lemma 3.1] that (1) ∀i ∈ N, det(I −φ(B i ) * D * ) = det(I −φ(B i+1 ) * D * ), (2) ∃ℓ ∈ N such that (φ(B j ) * D * ) ℓ = D ℓ * , Since A is of finite order, det A * = ±1. Let λ 1 , · · · , λ m be the eigenvalues of D * counted with multiplicities and let µ 1 , · · · , µ m be the eigenvalues ofφ(B j ) * D * counted with multiplicities. Since (φ(B j ) * D * ) ℓ = D ℓ * , (φ(B j ) * D * ) ℓ has the eigenvalues {λ ℓ 1 , · · · , λ ℓ m } = {µ ℓ 1 , · · · , µ ℓ m }. We may assume that λ ℓ i = µ ℓ i for all i = 1, · · · , m. Thus |µ i | = |λ i |. Now, | det(I − A * D * )| = | det A * det(A −1 * − D * )| = | det(I − D * A * )| = | det(I −φ(B j ) * D * )| (by (1)) = m i=1 |1 − µ i | ≤ m i=1 (1 + |µ i |) (by triangle inequality) = m i=1 (1 + |λ i |) Applying the above argument to D n , we obtain that | det(I − A * D n * )| ≤ m i=1 (1 + |λ i | n ) . By the averaging formula, we have N (f n ) = 1 |Φ| A∈Φ | det(I − A * D n * )| ≤ 1 |Φ| A∈Φ m i=1 (1 + |λ i | n ) = m i=1 (1 + |λ i | n ) , which induces lim sup 1 n log N (f n ) ≤ m i=1 lim sup 1 n log (1 + |λ i | n ) = |λ|>1 log |λ| = log   |λ|>1 |λ|   . Hence it follows that N ∞ (f ) ≤ sp( D * ). Because the above (algebraic) properties [8] and the averaging formula for the Nielsen number [42] on infra-nilmanifolds can be generalized to infrasolvmanifolds of type (R), we have proven in all that: Topological entropy and the radius of convergence The most widely used measure for the complexity of a dynamical system is the topological entropy. For the convenience of the reader, we include its definition. Let f : X → X be a self-map of a compact metric space. For given ǫ > 0 and n ∈ N, a subset E ⊂ X is said to be (n, ǫ)separated under f if for each pair x = y in E there is 0 ≤ i < n such that d(f i (x), f i (y)) > ǫ. Let s n (ǫ, f ) denote the largest cardinality of any (n, ǫ)-separated subset E under f . Thus s n (ǫ, f ) is the greatest number of orbit segments x, f (x), · · · , f n−1 (x) of length n that can be distinguished one from another provided we can only distinguish between points of X that are at least ǫ apart. Now let The number 0 ≤ h(f ) ≤ ∞, which to be independent of the metric d used, is called the topological entropy of f . If h(f, ǫ) > 0 then, up to resolution ǫ > 0, the number s n (ǫ, f ) of distinguishable orbit segments of length n grows exponentially with n. So h(f ) measures the growth rate in n of the number of orbit segments of length n with arbitrarily fine resolution. A basic relation between topological entropy h(f ) and Nielsen numbers was found by N. Ivanov [34]. We present here a very short proof by B. Jiang of the Ivanov's inequality. 34]). Let f be a continuous map on a compact connected polyhedron X. Lemma 5.1 ([ Then h(f ) ≥ log N ∞ (f ) Proof. Let δ be such that every loop in X of diameter < 2δ is contractible. Let ǫ > 0 be a smaller number such that d(f (x), f (y)) < δ whenever d(x, y) < 2ǫ. Let E n ⊂ X be a set consisting of one point from each essential fixed point class of f n . Thus |E n | = N (f n ). By the definition of h(f ), it suffices to show that E n is (n, ǫ)-separated. Suppose it is not so. Then there would be two points By sp(f ) we denote the spectral radius of H * (f ), which is a homotopy invariant. In 1974 Michael Shub asked, [57], the extent to which the inequality h(f ) ≥ log(sp(f )) holds. From this time this inequality has been usually called the Entropy Conjecture. Later A. Katok conjectured [37] that Entropy Conjecture holds for all continuous map for M being a manifold with the universal cover homeomorphic to R m . In [49], this was confirmed for every continuous map on an infra-nilmanifold. given in [54] (see [48] for another reference on this estimate). Next we observe that for an affine map f = (d, D) the latter reduces to || D|| = || D * || = sp( D * ). This implies that ≥ log sp(f ) (lift under a finite regular cover). x = y ∈ E n such that d(f i (x), f i (y)) ≤ ǫ for o ≤ i < n hence for all i ≥ 0. Pick a path c i from f i (x) to f i (y) of diameter < 2ǫlog sp( D * ) = log N ∞ (f ) = log N ∞ (f ) = h(f ). Thus we have log sp(f ) ≤ log sp( D * ) = log N ∞ (f ) = log N ∞ (f ) = h(f ) ≤ h(f ). The last inequality follows from Ivanov's inequality, Lemma 5.1. We denote by R the radius of convergence of the zeta functions N f (z) or R f (z). Theorem 5.5. Let f be a continuous map on an infra-nilmanifold with an affine homotopy lift (d, D). Then the Nielsen zeta function N f (z) and the Reidemeister zeta function R f (z), whenever it is defined, have the same positive radius of convergence R which admits following estimation R ≥ exp(−h) > 0, where h = inf{h(g) | g ≃ f }. If 1 is not in the spectrum of D * , the radius R of convergence of R f (z) is R = 1 N ∞ (f ) = 1 exp h(f ) = 1 sp( D * ) . Proof. When R f (z) is defined, as it was observed before, R(f n ) < ∞ and so R(f n ) = N (f n ) > 0 for all n > 0 on infra-nilmanifolds. In particular, R f (z) = N f (z). By the Cauchy-Hadamard formula, 1 R = lim sup n→∞ N (f n ) n 1/n = lim sup n→∞ N (f n ) 1/n . Since N (f n ) ≥ 1 for all n > 0, it follows that lim sup n→∞ N (f n ) 1/n ≥ 1. Thus 1 R = N ∞ (f ) ≤ exp h(f ). This induces the inequality R ≥ exp(−h) by the homotopy invariance of the radius R of the Reidemeister zeta function R f (z). We consider a smooth map g : M → M which is homotopic to f . As it is known in [54], the entropy h(g) is finite. Thus exp(−h) ≥ exp(−h(g)) > 0. Now the identities in our theorem follow from Theorem 5.2. Consider next the Nielsen zeta function N f (z). If lim sup n→∞ N (f n ) 1/n ≥ 1, then we obtain the same inequality for R as for R f (z). Thus, we assume lim sup n→∞ N (f n ) 1/n < 1. This happens only when N (f n ) = 0 for all but finitely many n. In this case, 1/R = lim sup n→∞ N (f n ) 1/n = 0 and so R = ∞ and N ∞ (f ) = 1. Zeta functions and the Reidemeister torsion of the mapping torus The Reidemeister torsion is a graded version of the absolute value of the determinant of an isomorphism of vector spaces. Let d i : C i → C i+1 be a cochain complex C * of finite dimensional vector spaces over C with C i = 0 for i < 0 and large i. If the cohomology H i = 0 for all i we say that C * is acyclic. If one is given positive densities ∆ i on C i then the Reidemeister torsion τ (C * , ∆ i ) ∈ (0, ∞) for acyclic C * is defined as follows: Definition 6.1. Consider a chain contraction δ i : C i → C i−1 , i.e., a linear map such that d • δ + δ • d = id. Then d + δ determines a map (d + δ) + : C + := ⊕C 2i → C − := ⊕C 2i+1 and a map (d + δ) − : C − → C + . Since the map (d + δ) 2 = id + δ 2 is unipotent, (d + δ) + must be an isomorphism. One defines τ (C * , ∆ i ) := | det(d + δ) + |. Reidemeister torsion is defined in the following geometric setting. Suppose K is a finite complex and E is a flat, finite dimensional, complex vector bundle with base K. We recall that a flat vector bundle over K is essentially the same thing as a representation of π 1 (K) when K is connected. If p ∈ K is a base point then one may move the fibre at p in a locally constant way around a loop in K. This defines an action of π 1 (K) on the fibre E p of E above p. We call this action the holonomy representation ρ : π → GL(E p ). Conversely, given a representation ρ : π → GL(V ) of π on a finite dimensional complex vector space V , one may define a bundle E = E ρ = (K × V )/π. HereK is the universal cover of K, and π acts onK by covering transformations and on V by ρ. The holonomy of E ρ is ρ, so the two constructions give an equivalence of flat bundles and representations of π. If K is not connected then it is simpler to work with flat bundles. One then defines the holonomy as a representation of the direct sum of π 1 of the components of K. In this way, the equivalence of flat bundles and representations is recovered. Suppose now that one has on each fibre of E a positive density which is locally constant on K. In terms of ρ E this assumption just means | det ρ E | = 1. Let V denote the fibre of E. Then the cochain complex C i (K; E) with coefficients in E can be identified with the direct sum of copies of V associated to each i-cell σ of K. The identification is achieved by choosing a basepoint in each component of K and a basepoint from each i-cell. By choosing a flat density on E we obtain a preferred density ∆ i on C i (K, E). A case of particular interest is when E is an acyclic bundle, meaning that the twisted cohomology of E is zero (H i (K; E) = 0). In this case one defines the Rtorsion of (K, E) to be τ (K; E) = τ (C * (K; E), ∆ i ) ∈ (0, ∞). It does not depend on the choice of flat density on E. The Reidemeister torsion of an acyclic bundle E on K has many nice properties. Suppose that A and B are subcomplexes of K. Then we have a multiplicative law: (5) τ (A ∪ B; E) · τ (A ∩ B; E) = τ (A; E) · τ (B; E) that is interpreted as follows. If three of the bundles E|A ∪ B, E|A ∩ B, E|A, E|B are acyclic then so is the fourth and the equation (5) holds. Another property is the simple homotopy invariance of the Reidemeister torsion. In particular τ is invariant under subdivision. This implies that for a smooth manifold, one can unambiguously define τ (K; E) to be the torsion of any smooth triangulation of K. In the case K = S 1 is a circle, let A be the holonomy of a generator of the fundamental group π 1 (S 1 ). One has that E is acyclic if and only if I − A is invertible and then τ (S 1 ; E) = | det(I − A)| Note that the choice of generator is irrelevant as I − A −1 = (−A −1 )(I − A) and | det(−A −1 )| = 1. These three properties of the Reidemeister torsion are the analogues of the properties of Euler characteristic (cardinality law, homotopy invariance and normalization on a point), but there are differences. Since a point has no acyclic representations (H 0 = 0) one cannot normalize τ on a point as we do for the Euler characteristic, and so one must use S 1 instead. The multiplicative cardinality law for the Reidemeister torsion can be made additive just by using log τ , so the difference here is inessential. More important for some purposes is that the Reidemeister torsion is not an invariant under a general homotopy equivalence: as mentioned earlier this is in fact why it was first invented. It might be expected that the Reidemeister torsion counts something geometric (like the Euler characteristic). D. Fried [26] showed that it counts the periodic orbits of a flow and the periodic points of a map. We will show that the Reidemeister torsion counts the periodic point classes of a map (fixed point classes of the iterations of the map). Some further properties of τ describe its behavior under bundles. Let p : X → B be a simplicial bundle with fiber F where F, B, X are finite complexes and p −1 sends subcomplexes of B to subcomplexes of X over the circle S 1 . We assume here that E is a flat, complex vector bundle over B . We form its pullback p * E over X. Note that the vector spaces H i (p −1 (b), C) with b ∈ B form a flat vector bundle over B, which we denote H i F . The integral lattice in H i (p −1 (b), R) determines a flat density by the condition that the covolume of the lattice is 1. We suppose that the bundle E ⊗ H i F is acyclic for all i. Under these conditions D. Fried [26] has shown that the bundle p * E is acyclic, and τ (X; p * E) = i τ (B; E ⊗ H i F ) (−1) i . Let f : X → X be a homeomorphism of a compact polyhedron X. Let T f := (X × I)/(x, 0) ∼ (f (x), 1) be the mapping torus of f . We shall consider the bundle p : T f → S 1 over the circle S 1 . We assume here that E is a flat, complex vector bundle with finite dimensional fibre and base S 1 . We form its pullback p * E over T f . Note that the vector spaces H i (p −1 (b), C) with b ∈ S 1 form a flat vector bundle over S 1 , which we denote H i F . The integral lattice in H i (p −1 (b), R) determines a flat density by the condition that the covolume of the lattice is 1. We suppose that the bundle E ⊗ H i F is acyclic for all i. Under these conditions D. Fried [26] has shown that the bundle p * E is acyclic, and we have (6) τ (T f ; p * E) = i τ (S 1 ; E ⊗ H i F ) (−1) i . Let g be the preferred generator of the group π 1 (S 1 ) and let A = ρ(g) where ρ : π 1 (S 1 ) → GL(V ). Then the holonomy around g of the bundle (6) that E ⊗ H i F is A ⊗ (f * ) i . Since τ (S 1 ; E) = | det(I − A)| it follows fromτ (T f ; p * E) = i | det(I − A ⊗ (f * ) i ) | (−1) i . We now consider the special case in which E is one-dimensional, so A is just a complex scalar λ of modulus one. Then in terms of the rational function L f (z) we have : (7) τ (T f ; p * E) = i | det(I − λ(f * ) i ) | (−1) i =| L f (λ) | −1 This means that the special value of the Lefschetz zeta function is given by the Reidemeister torsion of the corresponding mapping torus. Let us consider an infra-nilmanifold M = Π\G and a continuous map f on M . As in Section 1, we consider the subgroup Π + of Π of index at most 2. Then Π + is also an almost Bieberbach group as Π itself and the corresponding infra-nilmanifold M + = Π + \G is a double covering of the infra-nilmanifold M = Π\G; the map f lifts to a map f + : M + → M + which has the same affine homotopy lift (d, D) as f . Let T f and T f + be the mapping torus of f and f + correspondingly. We shall consider two bundles p : T f → S 1 and p + : T f + → S 1 over the circle S 1 . We assume here that E is a flat, complex vector bundle with one dimensional fibre and base S 1 . We form its pullback p * E over T f and pullback p * + E over T f + . We suppose that the bundles E ⊗ H i M and E ⊗ H i M + are acyclic for all i. Then Theorem 3.6 and the formula (7) imply the following result about special values of the Reidemeister and Nielsen zeta functions Theorem 6.2. Let f be a homeomorphism on an infra-nilmanifold Π\G with an affine homotopy lift (d, D). Then |R f ((−1) n λ) (−1) p+n | = |R φ ((−1) n λ) (−1) p+n | = |N f ((−1) n λ) (−1) p+n | = |L f (λ)| = τ (T f ; p * E) −1 when Π = Π + ; |L f + (λ)L f (λ) −1 | = τ (T f ; p * E)τ (T f + ; p * + E) −1 when Π = Π + , where p is the number of real eigenvalues of D * which are > 1 and n is the number of real eigenvalues of D * which are < −1. 7. Jiang-type spaces and averaging formula for the Reidemeister numbers on infra-solvmanifolds of type (R) A closed manifold M is called a Jiang-type space if for all continuous maps f : M → M , L(f ) = 0 ⇒ N (f ) = 0; L(f ) = 0 ⇒ N (f ) = R(f ). A closed orientable manifold M is called a Jiang-type space for coincidences ([28]) if for any continuous maps f, g : N → M where N is any closed orientable manifold of equal dimension, L(f, g) = 0 ⇒ N (f, g) = 0; L(f, g) = 0 ⇒ N (f, g) = R(f, g). It is well-known that Jiang spaces are of Jiang-type for coincidences. When N = M is a nilmanifold and φ, ψ are homomorphisms on the group of covering transformations induced by self-maps f, g on N , it is proven in [27,Theorem 2.3] that N (f, g) > 0 ⇔ coin(φ, ψ) = 1 ⇔ R(f, g) < ∞ Further if one of the above holds then R(f, g) = N (f, g) = |L(f, g)|. Furthermore, nilmanifolds are Jiang-type spaces for coincidences, see [28]. Recall that if N is a finite connected complex and M is a nilmanifold then N (f, g) = 0 ⇒ R(f, g) < ∞; if both N and M are nilmanifolds of equal dimension, then two conditions are equivalent and in that case we have N (f, g) = R(f, g). [51,Sec. 2]. Let S i be simply connected solvable Lie groups of type (E) with equal dimension, and let Γ i be lattices of Recall what C. McCord proved in S i . Let D i : S 1 → S 2 be Lie group homomorphisms such that D i (Γ 1 ) ⊂ Γ 2 . Write φ i = D i | Γ 1 : Γ 1 → Γ 2 . Thus D i induce maps f i between orbit spaces M i = Γ i \S i , special solvmanifolds of type (E). When S i are of type (R), we can always assume that any f i is induced from a Lie group homomorphism D i , see [42,Theorem 2.2] or [31,Theorem 4.2]. Denote C γ := coin(γ •D 1 , D 2 ) and S γ = p 1 (coin(γ • D 1 , D 2 )) for each γ ∈ Γ 2 . We also consider the map D : S 1 → S 2 defined by D(s) = D 1 (s) −1 D 2 (s) for s ∈ S 1 . (1) coin(φ 1 , φ 2 ) = 1. (2) dim(C 1 ) = 0. (3) D is injective. (4) C 1 = S 1 . (5) ind(S 1 ) = ±1. (6) ind(S 1 ) = 0. These statements are also valid for any other coincidence class S γ , and all ind(S γ ) have the same sign. Hence N (f 1 , f 2 ) = |L(f 1 , f 2 )|. We generalize [27, Theorem 2.3] from nilmanifolds to special solvmanifolds of type (R). Theorem 7.2. Let f 1 and f 2 be maps on a special solvmanifold Γ\S of type (R). Let φ 1 , φ 2 : Γ → Γ be homomorphisms induced by f 1 , f 2 respectively. Then the following are equivalent: (a) N (f 1 , f 2 ) > 0. (b) coin(φ 1 , φ 2 ) = 1. (c) R(f 1 , f 2 ) < ∞. Further if one of the above holds then R(f 1 , f 2 ) = N (f 1 , f 2 ) = |L(f 1 , f 2 )|. Proof. By Lemma 7.1, (a) ⇔ (b). Now we will show (b) ⇒ (c), and (c) ⇒ (a) together with the identity R(f 1 , f 2 ) = N (f 1 , f 2 ). Let S be a simply connected solvable Lie group of type (R). Let N = [S, S] and Λ = S/N . Then N is nilpotent and Λ ∼ = R k for some k > 0. A lattice Γ of S yields a lattice N ∩ Γ of N . Moreover, the lattice Γ induces a short exact sequence 1 → N ∩ Γ → Γ → Γ/N ∩ Γ ∼ = Γ · N/N → 1 so that the following diagram is commutative 1 − −−− → N − −−− → S − −−− → Λ = S/N − −−− → 0       1 − −−− → N ∩ Γ − −−− → Γ − −−− → Γ · N/N − −−− → 0 This gives rise to the fibration, called a Mostow fibration, N ∩ Γ\N −→ M = Γ\S −→ Γ · N \S over a torus base Γ · N \S with compact nilmanifold fiber N ∩ Γ\N . It is known that this fibration is orientable if and only if the solvmanifold M is a nilmanifold. Let E : S → S be a homomorphism. Then E induces a homomorphism E ′ : N → N and hence a homomorphismĒ : Λ → Λ so that the following diagram is commutative 1 − −−− → N − −−− → S − −−− → Λ − −−− → 0   E ′   E   Ē 1 − −−− → N − −−− → S − −−− → Λ − −−− → 0 Hence we have the following diagram is commutative 1 − −−− → N ∩ Γ − −−− → Γ − −−− → Γ · N/N − −−− → 0   φ ′ i   φ i   φ i 1 − −−− → N ∩ Γ − −−− → Γ − −−− → Γ · N/N − −−− → 0 Denote Γ ′ = N ∩ Γ and letΓ = Γ · N/N . By [42,Theorem 2.2] or [31,Theorem 4.2], we may assume that f 1 , f 2 are induced by Lie group homomorphisms D 1 , D 2 : S → S respectively. Then ϕ i (γ) • D i = D i • γ ∀γ ∈ Γ. Evaluating at the identity of S, we obtain that φ i (γ) = D i (γ) for all γ ∈ Γ. So, φ i is the restriction of D i on Γ. Assume (b): coin(φ 1 , φ 2 ) = 1. Then coin(D 1 , D 2 ) = 1 by Lemma 7.1. By taking differential, we see that coin(D 1 * , D 2 * ) = 0, or D 2 * − D 1 * is a linear isomorphism. We can write D 2 * − D 1 * as D 2 * − D 1 * = D 2 * −D 1 * 0 * D ′ 2 * − D ′ 1 * with respect to some linear basis of the Lie algebra of S. This implies that D 2 * −D 1 * is an isomorphism and so coin(D 2 * ,D 1 * ) = 0 or coin(D 1 ,D 2 ) = 1 = coin(φ 1 ,φ 2 ). This happens on Λ ∼ = R k with the lattice Γ ′ and so on the torus Γ · N \S = Γ ′ \Λ. Hence coin(φ 1 ,φ 2 ) =1 implies R(φ 1 ,φ 2 ) < ∞. On the other hand, since coin(φ ′ 1 , φ ′ 2 ) = 1 from coin(φ 1 , φ 2 ) = 1, by [27, Theorem 2.3], R(φ ′ 1 , φ ′ 2 ) < ∞. Now the above commutative diagram induces a short exact sequence of the sets of Reidemeister classes R(φ ′ 1 , φ ′ 2 ) −→ R(φ 1 , φ 2 ) −→ R(φ 1 ,φ 2 ) −→ 1. Because both sets R(φ ′ 1 , φ ′ 2 ) and R(φ 1 ,φ 2 ) are finite, it follows that the middle set R(φ 1 , φ 2 ) is also finite. Hence R(φ 1 , φ 2 ) < ∞. Assume (c): R(φ 1 , φ 2 ) < ∞. Then R(φ 1 ,φ 2 ) < ∞ on the torus Γ ′ \Λ. We already know that this implies 0 < N (f 1 ,f 2 ) = R(φ 1 ,φ 2 ) and coin(φ 1 ,φ 2 ) = 1. Assume that R(φ ′ 1 , φ ′ 2 ) = ∞. By [27, Theorem 2.3], coin(φ ′ 1 , φ ′ 2 ) = 1 and then by Lemma 7.1, coin(D ′ 1 , D ′ 2 ) = 1 and hence D ′ 2 * −D ′ 1 * is singular, which implies D 2 * − D 1 * is also singular and so contradicts coin(φ 1 , φ 2 ) = 1. Hence R(φ ′ 1 , φ ′ 2 ) < ∞ on the nilmanifold Γ ′ \N . This implies that 0 < N (f ′ 1 , f ′ 2 ) = R(φ ′ 1 , φ ′ 2 ). Hence we have N (f 1 , f 2 ) = |L(f 1 , f 2 )| ([51, Theorem 2.1]) = | det(D 2 * − D 1 * )| ([32, Theorem 3.1]) = | det(D 2 * −D 1 * )|| det(D ′ 2 * − D ′ 1 * )| = N (f 1 ,f 2 )N (f ′ 1 , f ′ 2 ) = R(φ 1 ,φ 2 )R(φ ′ 1 , φ ′ 2 ) ≥ R(φ 1 ,φ 2 ) (exactness and finiteness of each Reidemeister set). Consequently, sine it is always true that N (f 1 , f 2 ) ≤ R(φ 1 , φ 2 ), we have the identity N (f 1 , f 2 ) = R(φ 1 , φ 2 ). Immediately, from Theorem 7.2 we obtain the following: for any maps f 1 , f 2 : M → M on a special solvmanifold M of type (R), we have L(f 1 , f 2 ) = 0 ⇒ N (f 1 , f 2 ) = 0; L(f 1 , f 2 ) = 0 ⇒ N (f 1 , f 2 ) = R(f 1 , f 2 ). Example 7.3. Consider the closed 3-manifolds with Sol-geometry. We refer to [31,Sec.6] for details about the Reidemeister numbers on these manifolds. These are infra-solvmanifolds Π\Sol of type (R). When Π = Π 0 or Π ± 2 , the corresponding manifold is a torus bundle over S 1 , and when Π = Π 3 or Π 6 , the manifold is a sapphire space. Only Π 0 \Sol is the special solvmanifold and the remaining manifolds are non-special, infra-solvmanifolds. For any homeomorphism f : Π\Sol → Π\Sol, let F * be its linearization. Then the following can be found in [31,Sec. 6]: (1) When Π = Π 0 or Π + 2 , L(f ) = N (f ) = R(f ) = 4 only when F * is of type (II) with det F * = −1; otherwise, L(f ) = N (f ) = 0 and R(f ) = ∞. (2) When Π = Π − 2 , F * is always of type (I) and L(f ) = N (f ) = 0, but R(f ) = ∞. (3) When Π = Π 3 , L(f ) = N (f ) = 0, but R(f ) = ∞. (4) When Π = Π 6 , L(f ) = N (f ), which is 0 or 2 according as det F * = 1 or −1, but R(f ) = ∞. These results show that Theorem 7.2 (i.e., N (f ) > 0 ⇔ R(f ) < ∞; in this case, N (f ) = R(f )) is true for the special solvmanifold Π 0 \Sol and infra-solvmanifolds Π ± 2 \Sol and Π 3 \Sol, but not true anymore for the infrasolvmanifold Π 6 \Sol. Now we can state a practical formula for the Reidemeister number of a pair of continuous maps on an infra-solvmanifold of type (R). This is a straightforward generalization of [33, Theorem 6.11] and its proof from infra-nilmanifolds. On the other hand, we may assume that f, g are induced by the affine maps (d, D), (e, E) respectively. This induces thatf ,ḡ are induced by the Lie group homomorphisms µ(d) • D, µ(e) • E : S → S, where µ(·) is conjugation. If (a, A) ∈ Π is a preimage ofᾱ ∈ Π/Λ, then the transformationᾱ on Λ\S is induced by the Lie group automorphism µ(a) • A. By [32, Theorem 3.1] and Lemma 7.1, we have that R(f, g) = 1 |Φ| A∈Φ σ (det(E * − A * D * )) ,N (ᾱf ,ḡ) = | det(Ad(e)E * − Ad(a)A * Ad(d)D * )| = | det Λ (E * − A * D * )| with respect to any preferred basis of Λ. If we regard this as a basis of Γ, then we can see that [Γ : Λ] det Λ (E * − A * D * ) = det Γ (E * − A * D * ), for example see the proof of [33,Theorem 6.11]. Hence R(f, g) = 1 [Π : Λ] ᾱ∈Π/Λ σ N (ᾱf ,ḡ) = 1 [Π : Λ] A∈Φ [Γ : Λ] σ det Λ (E * − A * D * ) = 1 |Φ| A∈Φ σ det Γ (E * − A * D * ) . The following corollaries generalize [13, Theorems 5.1 and 5.2] from infranilmanifolds to infra-solvmanifolds of type (R). Proof. Because M is orientable, the Nielsen number N (f, g) is defined and is equal to, by [31,Theorem 4.5], N (f, g) = 1 |Φ| A∈Φ | det(E * − A * D * )|. Since R(f, g) < ∞, by Theorem 7.4, σ(det(E * −A * D * )) is finite for all A ∈ Φ. By the definition of σ, we have σ(det(E * − A * D * )) = | det(E * − A * D * )| for all A ∈ Φ. This finishes the proof. Then R(f ) = 1 |Φ| A∈Φ σ (det(I − A * D * )) , and if R(f ) < ∞ then R(f ) = N (f ). By Remarks 3.7 and 3.11, since the averaging formulas for the Lefschetz number and the Nielsen number are generalized from infra-nilmanifolds to infra-solvmanifolds of type (R) (see [32,42]), all results and proofs concerning the Nielsen number and the Nielsen zeta function in this article directly generalize to the class of infra-solvmanifolds of type (R). By Corollary 7.6 and [42,Theorem 4.3], the averaging formulas for the Reidemeister number and the Nielsen number on infra-solvmanifolds of type (R), we can generalize all results and proofs concerning the Reidemeister zeta function, whenever it is defined, to the class of infra-solvmanifolds of type (R). If R f (z) is defined, then R(f n ) < ∞ and so by Corollary 7.6 R(f n ) = N (f n ) > 0 for all n > 0 and thus R f (z) = N f (z). For example, we can generalize Theorems 3.5, 3.6, 3.9, and 6.2, and their proofs from infra-nilmanifolds to infra-solvmanifolds of type (R) to obtain the following: N f (z) = L f ((−1) q z) (−1) r where q is the number of real eigenvalues of D * which are < −1 and r is the number of real eigenvalues of D * of modulus > 1. When the Reidemeister zeta function R f (z) is defined, we have R f (z) = R φ (z) = N f (z). Theorem 7.8. Let f be a continuous map on an infra-solvmanifold Π\S of type (R) with an affine homotopy lift (d, D). Then the Reidemeister zeta function, whenever it is defined, is a rational function and is equal to Theorem 7.9 (Functional Equation). Let f be a continuous map on an orientable infra-solvmanifold M = Π\S of type (R) with an affine homotopy lift (d, D). Then the Reidemeister zeta function, whenever it is defined, and the Nielsen zeta function have the following functional equations: D). Then the Nielsen zeta function N f (z) and the Reidemeister zeta function R f (z), whenever it is defined, have the same positive radius of convergence R which admits following estimation R f (z) = N f (z) =    L f ((−1) n z) (−1) p+n when Π = Π + ; L f + ((−1) n z) L f ((−1) n z) (−1) p+n when Π = Π + ,R f 1 dz = R f (z) (−1) m ǫ (−1) p+n when Π = Π + ; R f (z) (−1) m ǫ −1 when Π = Π + and N f 1 dz = N f (z) (−1) m ǫ (−1) p+n when Π = Π + ; N f (z) (−1) m ǫ −1 when Π = Π + where d is a degree f , m = dim M , ǫ is a constant in C × , σ = (−1) n ,R ≥ exp(−h) > 0, where h = inf{h(g) | g ≃ f }. If 1 is not in the spectrum of D * , the radius R of convergence of R f (z) is R = 1 N ∞ (f ) = 1 exp h(f ) = 1 sp( D * ) . Theorem 7.11. Let f be a homeomorphism on an infra-solvmanifold Π\S of type (R) with an affine homotopy lift (d, D). Then |N f ((−1) n λ) (−1) p+n | = |L f (λ)| = τ (T f ; p * E) −1 when Π = Π + ; |L f + (λ)L f (λ) −1 | = τ (T f ; p * E)τ (T f + ; p * + E) −1 when Π = Π + , where p is the number of real eigenvalues of D * which are > 1 and n is the number of real eigenvalues of D * which are < −1. Remark 7.12. One may formulate the above theorem also for the Reidemeister zeta function of a homeomorphism f on an infra-solvmanifold of type (R) . However it will be seen in Theorem 8.2 that in the case of R f (z) such a manifold must be an infra-nilmanifold. Remark 7.13. For any map f on an infra-solvmanifold of type (R), Theorem 1.2 states the relation between the Lefschetz numbers and the Nielsen numbers of iterates of f and Corollary 7.6 states the relation of the Nielsen numbers with the Reidemeister numbers of iterates of f when these are finite. Via these relations some of the arithmetic, analytic, and asymptotic properties of the sequences N (f n ) and R(f n ) can be determined from the corresponding properties of the sequence L(f n ). For the sequence L(f n ), all these properties were thoroughly discussed in [35,Sect. 3.1], see also [2]. 8. The Reidemeister zeta function is never defined for any homeomorphism of infra-solvmanifold of type (R), not an infra-nilmanifold Consider now as an example closed 3-manifolds with Sol-geometry. We refer to [31,Sec. 6] for details about the Reidemeister numbers on these manifolds. These are infra-solvmanifolds Π\Sol of type (R). Let Π 1 be a lattice of Sol: Π 1 = Γ A = a 1 , a 2 , τ | [a 1 , a 2 ] = 1, τ a i τ −1 = A(a i ) , where A is a 2 × 2-integer matrix of determinant 1 and trace > 2. Consider a homomorphism φ on Π 1 of type (III), i.e., φ is given by the form φ(a 1 ) = φ(a 2 ) = 1, φ(τ ) = a p 1 a q 2 τ r , r = ±1. Then it is shown in [31, Theorem 6.1] that R(φ) = |1 − r|. We can observe easily that φ n is also of type (III) and R(φ n ) = |1 − r n | for all n > 0. Hence R φ (z) = exp ∞ n=1 |1 − r n | n z n = 1 1−z when r = 0; 1− r |r| z 1−|r|z when |r| > 1. It can be seen also that if φ is not of type (III), then R(φ) = ∞ or R(φ 2 ) = ∞. Thus the associated Reidemeister zeta function is not defined. A similar phenomenon happens for the infra-solvmanifold Π ± \Sol. For the remaining infra-solvmanifolds Π 3 \Sol and Π 6 \Sol, it is shown that only trivial map has a finite Reidemeister number, which is 1. That is, only the trivial map defines the Reidemeister zeta function. The homomorphisms above are eventually commutative, and in fact, for every eventually commutative homomorphism the Reidemeister zeta function, whenever it is defined, is a rational function, see Theorem 9 and Theorem 10 in [20]. We will show now that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type (R), then the manifold must be an infra-nilmanifold. Recall the following R(f n ) = 1 [Π : Λ] ᾱ∈Π/Λ R(ᾱf n ). Assume now that f defines the Reidemeister zeta function. Then R(f n ) < ∞ for all n > 0. The above averaging formula implies that R(f n ) < ∞ for all n. By Theorem 7.2, we must have R(f n ) = N (f n ) = |L(f n )| > 0. Since L(f n ) = det(I −D n * ) = 0 for all n > 0 by [32,Theorem 3.1], this would imply that the differential D * of D has no roots of unity. By Proposition 8.1, S must be nilpotent. Remark 8.3. Let A be an Anosov diffeomorphism on an infra-nilmanifold. Then an iteration A n will be also an Anosov diffeomorphism for every n ≥ 1. The Reidemeister number of an Anosov diffeomorphism is always finite [10]. Hence the Reidemeister zeta R A (z) is well-defined. From Theorem 3.6 and Theorem 3.9 it follows that the Reidemeister zeta function R A (z) of an Anosov diffeomorphism on an infra-nilmanifold is a rational function with functional equation. It is known that a nilmanifold modelled on a free cstep nilpotent Lie group on r generators admits an Anosov diffeomorphism if and only if r > c [5]. Hence the Reidemeister zeta function of an Anosov diffeomorphism on such nilmanifold is well-defined if r > c and is a rational function with functional equation. 9. The Artin-Mazur zeta functions on infra-solvmanifolds of type (R) Let f be a continuous map on a topological space X. Then the Artin-Mazur zeta function of f is defined as follows: AM f (z) = exp ∞ n=1 F (f n ) n z n where F (f ) is the number of isolated fixed points of f . Remark 9.2. The above proposition is a straightforward generalization of [40, Proposition 1] from infra-nilmanifolds to infra-solvmanifolds of type (E). Further, the linear part of the affine map F need not be an automorphism. Proposition 9.1 is proved when the manifold is a special solvmanifold of type (E) and the map is induced by a homomorphism in Lemma 7.1, [51]. In fact, the converse is also proved. That is, every essential fixed point class consists of a single element. We will prove the converse of the proposition on infra-solvmanifolds of type (R). Then we have an averaging formula, [42,Theorem 4.2], N (f ) = 1 [Π : Λ] ᾱ∈Π/Λ N (ᾱ •f ). Assume that f has an essential fixed point class. The averaging formula tells that this essential fixed point class of f is lifted to an essential fixed point class of someᾱ •f . That is, there is α = (a, A) ∈ Π such that the fixed point class p ′ (Fix(α •f )) ofᾱ •f is essential (and so p(Fix(α •f )) is an essential fixed point class of f ). It suffices to show that the fixed point class p ′ (Fix(α •f )) consists of only one point. Let F = α •f = (a, A)(d, D) := (e, E) be the affine map on S, and let F =ᾱ •f . Then p ′ (Fix(F )) is essential and N (F ) = | det(I − E * )| = 0. Choose x ∈ Fix(F ) = Fix((e, E)). Then the left multiplication by x −1 , ℓ x −1 : y ∈ Fix((e, E)) → x −1 y ∈ Fix(E), is a bijection. Further, since exp : S → S is a diffeomorphism, it follows that Fix(E) ↔ fix(E * ) = ker(I − E * ). Since I − E * is invertible, we see that Fix(F ) and hence p ′ (Fix(F )) and p(Fix(F )) consist of a single element. By the main result in [50], if f is a map on an infra-solvmanifold of type (R) which is induced by an affine map and is homotopically periodic, then we have AM f (z) = N f (z) = L f (z) as N (f n ) = L(f n ). According to Theorem 10.3, if f is a virtually unipotent affine diffeomorphism on an infra-solvmanifold of type (R), then we still have AM f (z) = N f (z) = L f (z). Let g = ((x, y), t) ∈ Sol. Then it can be seen easily that τ g : Sol → Sol is given by , y), t) ∈ Sol | x = y = 0}. Fix g = ((0, 0), t) ∈ Sol Φ with t = 0 and consider (g, τ g −1 ) ∈ Aff(Sol). Then (g, τ g −1 ) centralizes Π + 2 and (g, τ g −1 ) induces an affine diffeomorphism f on Π + 2 \Sol given bȳ x →xḡ. Hence the affine diffeomorphism f is homotopic to the identity map. However f is not virtually unipotent since (τ g −1 ) * = Ad(g −1 ) is not virtually unipotent. Remark 10.2. Recall [46,Lemma 3.6], which states that if an affine diffeomorphism f on an infra-nilmanifold M is homotopic to a virtually unipotent affine diffeomorphism on M , then f is virtually unipotent. However, the above example shows that this statement is not true in general for infrasolvmanifolds of type (R). Namely, there is an affine diffeomorphism on an infra-solvmanifold of type (R) which not virtually unipotent, but is homotopic to a virtually unipotent affine diffeomorphism. Furthermore, in the above example, f ≃ id is a homotopically periodic map which is not virtually unipotent. Therefore [46,Proposition 3.11] is not true in general for infra-solvmanifolds of type (R). Note also that there is a unipotent affine diffeomorphism on the torus which is not homotopically periodic, see [46,Remark 3.12]. Consequently, on infra-nilmanifolds homotopically periodic maps are virtually unipotent maps. But on infra-solvmanifolds of type (R), there is no relation between homotopically periodic maps and virtually unipotent maps. Proof. Let M be an infra-solvmanifold of type (R) with holonomy group Φ. Then we can assume f is an affine diffeomorphism induced by an affine map (d, D) such that D * is virtually unipotent. This implies that (d, D) normalizes Π and hence it follows that D normalizes the holonomy group Φ. By the previous observation (2), since D * is virtually unipotent, so are all A * D * where A ∈ Φ and hence by [46, Gauss congruences for the Nielsen and Reidemeister numbers In number theory, the following Gauss congruence for integers holds: d|n µ(d) a n/d ≡ 0 mod n for any integer a and any natural number n. Here µ is the Möbius function. In the case of a prime power n = p r , the Gauss congruence turns into the Euler congruence. Indeed, for n = p r the Möbius function µ(n/d) = µ(p r /d) is different from zero only in two cases: when d = p r and when d = p r−1 . Therefore, from the Gauss congruence we obtain the Euler congruence a p r ≡ a p r−1 mod p r This congruence is equivalent to the following classical Euler's theorem: a ϕ(n) ≡ 1 mod n where (a, n) = 1. These congruences have been generalized from integers a to some other mathematical invariants such as the traces of all integer matrices A and the Lefschetz numbers of iterates of a map, see [47,61]: d|n µ(d) tr(A n/d ) ≡ 0 mod n,(8) tr(A p r ) ≡ tr(A p r−1 ) mod p r . A. Dold in [15] proved by a geometric argument the following congruence for the fixed point index of iterates of a map f on a compact ANR X and any natural number n d|n µ(d) ind(f n/d , X) ≡ 0 mod n, thus consequently d|n µ(d) L(f n/d ) ≡ 0 mod n (DL) by using Hopf theorem. These congruences are now called the Dold congruences. It is also shown in [47] (see also [61,Theorem 9]) that the above congruences (8), (9) . When d = ±1, then all R(f n ) < ∞ and so the congruences hold. When d = 1, the congruence for the Nielsen number is obviously true. We assume d = −1. So, N (f n ) = 2 for odd n and 0 for even n. For n = 2 · 3 2 · 5, we have d|n µ(d) N (f n/d ) = d|n d even µ(d) 2 = 2 µ(2) + µ(2 · 3) + µ(2 · 3 2 ) + µ(2 · 3 · 5) + µ(2 · 3 2 · 5) = 2 ((−1) + 1 + 0 + (−1) + 0) = −2 = 0 mod 2 · 3 2 · 5. Thus we have no congruence (DN). Next we consider the congruences (EN) and (ER). If d ≥ 0, then L(f n ) = 1 − d n = −N (f n ) = −R(f n ). The congruence (EL) L(f p r ) ≡ L(f p r−1 ) mod p r implies the other congruences (EN) and (ER). Assume d < 0. The congruence (EL) L(f p r ) ≡ L(f p r−1 ) mod p r is exactly 1 − d p r ≡ 1 − d p r−1 mod p r , which implies that d p r ≡ d p r−1 mod p r and so d p r ± 1 ≡ d p r−1 ± 1 mod p r . Thus the other congruences (EN) and (ER) hold true. In summary, (EN) and (ER) are true, but (DN) is not true. The congruence (DR) was previously known for automorphisms of almost polycyclic groups ( [23, p. 195]) and for all continuous maps only on nilmanifolds ( [20,Theorem 58]) provided that all Reidemeister numbers of iterates of the maps are finite. We generalize these on infra-solvmanifolds of type (R). Theorem 2 . 4 . 24Then Π 0 is a subgroup of Π of index 2, Π 0 is also an SBgroup and the corresponding infra-solvmanifold M 0 = Π 0 \S is a double covering of M = Π\S; the maps f, g lift to map f 0 , g 0 : M 0 → M 0 which have the same affine homotopy lifts (d, D), (e, E) as f and g. Proposition 3. 2 . 2Let f be a continuous map on an infra-nilmanifold Π\G with holonomy group Φ. Let f have an affine homotopy lift (d, D) and let φ : Π → Π be the homomorphism induced by f . Then We denote generators of H m (M ; Q) and H m (M + ; Q) by [M ] and [M + ], respectively. The above diagram shows that x * (π * ([M ])) = π * ([M ]), which induces that x * ([M + ]) = [M + ] as π * is injective, and hence x acts on H m (M + ; Q) trivially. In other words, H m (M + ; Q) = H m (M + ; Q) Π/Π + and π * : H m (M ; Q) → H m (M + ; Q) is an isomorphism. This implies that f and f + have the same degree. induced in the exterior algebra * R m := m ℓ=0 ℓ R m of G considered as the linear space R m . Theorem 4.1 (see also the proof of [48, Theorem 1.5]). Let M = Γ\S be a special solvmanifold of type (R) and let f be a continuous map on M with a Lie group homomorphism D : S → S as a homotopy lift. Then we have N ∞ (f ) = sp( D * ) provided that 1 is not in the spectrum of D * . Proof. We give a very elementary proof of this theorem. Compare with the proof of [48, Theorem 1.5] in which the authors are interested only in the case of positive topological entropy which excludes the case N ∞ (f ) = 1. we first consider the case where N (f n ) = 0 for all n. Corollary 4. 2 . 2Let D be a matrix with eigenvalues λ 1 , · · · , λ m , counted with multiplicities. Let L(D n ) = det(I − D n ). If 1 is not in the spectrum of D, then Growth (L(D n )) = sp( D). Theorem 4. 3 . 3Let f be a continuous map on an infra-solvmanifold of type (R) with an affine homotopy lift (d, D). Then we have N ∞ (f ) = sp( D * )provided that 1 is not in the spectrum of D * . Remark 4. 4 . 4The above theorem was proved when M is a special solvmanifold of type (R), see Theorem 4.1 and the proof of [48, Theorem 1.5]. In the paper [48], it is assumed that sp(D * ) > 1. Since N (f ) = | det(I − D * )|, 1 is not in the spectrum of D * if and only if f is not homotopic to a fixed point free map. Now, we provide an example of the asymptotic Nielsen numbers. Example 4. 5 .when r is odd and q == 0 . 50Let f : Π\R 2 → Π\R 2 be any continuous map on the Klein bottle Π\R 2 of type (r, ℓ, q). Recall from[38, Theorem 2.3] and its proof that r is odd or q = 0, and N (f n ) = |q n (1 − r n )| when r is odd and q = 0; |1 − r n | when q = 0, If |r| ≤ 1, then N (f n ) ≤ 2 and so N ∞ (f ) = 1; if |r| > 1 then log lim sup n→∞ N (f n ) 1/n = lim sup n→∞ 1 n log |1 − r n | = log |r|.Thus N ∞ (f ) = max{1, |r|}.Assume q = 0 and r is odd.If r = 1 then N (f n ) = 0 ⇒ N ∞ (|1 − r n | = log |q| when |r| ≤ 1, i.e., r = −1; log |qr| when |r| > 1.Thus N ∞ (f ) = 1 q = 0 and r = 1 max{1, |q|, |qr|} q = 0 and r = 1 is odd.On the other hand, since sp( D * ) is the largest modulus of an eigenvalue of D * , it follows that sp( D * ) = max{1, |r|, |q|, |qr|}.Hence:(1) If r = 1, then N ∞ (f ) = 1 and sp( D * ) = |q| ≥ 1 (since r = 1 is odd and so q = 0). (2) If r = 0 (even), then q must be 0 and so N ∞ (f ) = sp( D * ) = 1.(3) Otherwise, N ∞ (f ) = sp( D * ).We observe explicitly in this example that the condition that 1 is not in the spectrum of D * induce the identity N ∞ (f ) = sp( D * ). If q = 0 then sp(D * ) = |r| and so N ∞ (f ) = max{1, |r|} = sp( D * ). If q = 0 then r is odd and sp(D * ) = max{|r|, |q|} > 1; if sp(D * ) = |r| ≥ |q| then |r| > 1 and so N ∞ (f ) = |qr| = sp( D * ); if sp(D * ) = |q| ≥ |r| then |q| > 1 and |r| > 1 or r = −1 (because r cannot be 1) so N ∞ (f ) = |qr| = sp( D * ). log s n (ǫ, f ) h(f ) := lim sup ǫ→0 h(f, ǫ). for 0 0≤ i < n and let c n = c 0 . By the choice of δ and ǫ, f • c i ≃ c i+1 for all i, so f n • c 0 ≃ c n = c 0 . This means x, y in the same fixed point class of f n , contradicting the construction of E n . This inequality is remarkable in that it does not require smoothness of the map and provides a common lower bound for the topological entropy of all maps in a homotopy class. Let H * (f ) : H * (M ; R) → H * (M ; R) be a linear map induced by f on the total cohomology H * (M ; R) of M with real coefficients. Theorem 5. 2 . 2Let f be a continuous map on an infra-solvmanifold M of type (R) with an affine homotopy lift (d, D). If 1 is not in the spectrum of D * , then h(f ) ≥ log(sp(f )). Iff is the map on M induced by the affine map (d, D), then h(f ) ≥ h(f ) ≥ log sp(f ), h(f ) = log sp( D * ) = log N ∞ (f ) = log N ∞ (f ).Hencef minimizes the entropy in the homotopy class of f . Proof. Letf be the map on M induced by the affine map (d, D). Thus f is homotopic tof . By [42, Lemma 2.1], there is a special solvmanifold which regularly and finitely covers the infra-solvmanifold M so that f can be lifted tof on the solvmanifold. We also remark that the Lie group homomorphism τ d D induces a mapf on the solvmanifold so thatf lifts tō f ,f is homotopic to φ f , the linearization D * of f is also a linearization of the liftf , and the topological entropies of f,f and their liftsf ,f are the same, i.e., h(f ) = h(f ) and h(f ) = h(f ). Moreover, since the spectral radius is a homotopy invariant, sp(f ) = sp(f ) and sp(f ) = sp(f ). It is also known that sp(f ) ≤ sp(f ). See, for example, [49, Proposition 2]. Now observe that log sp( D * ) = log N ∞ (f ) (Theorem 4.3) = log N ∞ (f ) (homotopy invariance of N ∞ (·)) ≤ h(f ) (Lemma 5.1) = h(f ) (lift under a finite regular cover) ≤ log sp( D * ). The fundamental is the last inequality. It follows from the estimate of topological entropy of a C 1 self-map of a compact manifold M h(f ) ≤ lim sup n→∞ 1 n log sup x∈M || Df (x)|| ( D * ) ≥ log sp(f ) ([48, Theorem 2.1]) = log sp(f ) (homotopy invariance of sp(·)) Remark 5. 3 . 3If sp(D * ) ≤ 1, then sp( D * ) = 1 and so we have the obvious inequality h(f ) ≥ 0 = log sp( D * ) ≥ log sp(f ). Theorem 7 . 4 . 74Let M = Π\S be an infra-solvmanifold of type (R) with holonomy group Φ. Let f, g : M → M be continuous maps with affine homotopy lifts (d, D), (e, E) respectively. Then where A * , D * and E * induced by A, D and E are expressed with respect to a preferred basis of Π ∩ S and where σ : R → R ∪ {∞} is given by σ(0) = ∞ and σ(x) = |x| for all x = 0.Proof. Choose a fully invariant subgroup Λ ⊂ Γ := Π ∩ S of Π with finite index ([42, Lemma 2.1]). Then f, g lift to mapsf ,ḡ on the special solvmanifold Λ\S of type (R) and by [30, Corollary 1.3] we have R(f, g) = 1 [Π : Λ] ᾱ∈Π/Λ R(ᾱf ,ḡ).By Theorem 7.2, R(ᾱf ,ḡ) = σ N (ᾱf ,ḡ) for allᾱ ∈ Π/Λ. Corollary 7 . 5 . 75Let M = Π\S be an orientable infra-solvmanifold of type (R). Let f, g : M → M be continuous maps. If R(f, g) < ∞, then R(f, g) = N (f, g). Corollary 7. 6 . 6Let M = Π\S be an infra-solvmanifold of type (R) with holonomy group Φ. Let f : M → M be a continuous map with an affine homotopy lift (d, D). Theorem 7 . 7 . 77Let f be a continuous map on an infra-solvmanifold of type (R) with an affine homotopy lift (d, D). Assume N (f ) = |L(f )|. Then the Nielsen zeta function N f (z) is a rational function and is equal to p is the number of real eigenvalues of D * which are > 1 and n is the number of real eigenvalues of D * which are < −1. If |d| = 1 then ǫ = ±1. Theorem 7.10. Let f be a continuous map on an infra-solvmanifold of type (R) with an affine homotopy lift (d, Proposition 9. 1 ([ 40 , 140Proposition 1]). Let f be a continuous map on an infra-solvmanifold Π\S of type (E) induced by an affine map F : S → S. For any α ∈ Π, Fix(α • F ) is an empty set or path connected. Hence every nonempty fixed point class of f is path connected, and every isolated fixed point class forms an essential fixed point class.Proof. Let x, y ∈ Fix(α • F ). So, the affine map αF fixes x and y. Writingα • F = (d, D) ∈ S ⋊ Endo(S), we see that • (d, D)(x) = x ⇒ D(x) = d −1 x, • (d, D)(y) = y ⇒ D(y) = d −1 y, • (x, I) −1 (α•F )(x, I) = (x, I) −1 (d, D)(x, I) = (x −1 dD(x), D) = (1, D)and D fixes 1 and x −1 y. Since S is of type (E), exp : S → S is a diffeomorphism with inverse log. Let X = log(x −1 y) ∈ S. Then the 1-parameter subgroup {exp(tX) | t ∈ R} of S is fixed by the endomorphism D. Consequently, the affine map α•F fixes the 'line' connecting the points x and y.In particular, p(Fix(α • F )) is isolated {x} if and only if Fix(α•F ) is isolated {x}, where p : S → Π\S is the covering projection. Further, the index of the fixed point class p(Fix(α • F )) = {x} is det(I − dfx) = ± det(I − d(α • F ) x ) = ± det(I − D * ) where the second identity follows from the fact that x −1 (α•F )x = D. Since D fixes only the identity element of S, D * fixes only the zero element of S and so I − D * is nonsingular. Hence the fixed point class p(Fix(α • F )) is essential. Proposition 9 . 3 . 93Let f be a continuous map on an infra-solvmanifold of type (R) induced by an affine map. Then every essential fixed point class of f consists of a single element. Proof. Letf = (d, D) be the affine map on the connected, simply connected solvable Lie group S of type (R) which induces f : Π\S → Π\S. Then f induces a homomorphism φ : Π → Π. By [42, Lemma 2.1], we can choose a fully invariant subgroup Λ ⊂ Π ∩ S of Π with finite index. Hence φ(Λ) ⊂ Λ. This implies thatf induces a map f on Λ\S. Remark 9. 4 . 4In Propositions 9.1 and 9.3, we have shown that for any continuous map on an infra-solvmanifold of type (R) induced by an affine map the isolated fixed points of f are the essential fixed point classes of f . That is, F (f ) = N (f ). Similarly F (f n ) = N (f n ) for all n. Therefore, by Theorem 7.8 and Theorem 7.9 we have Theorem 9.5. Let f be a continuous map on an infra-solvmanifold of type (R) induced by an affine map. Then AM f (z) = N f (z), i.e., AM f (z) is a rational function with functional equation. 10 . 10The Nielsen numbers of virtually unipotent maps on infra-solvmanifolds of type (R) A square matrix is unipotent if all of its eigenvalues are 1. A square matrix is called virtually unipotent if some power of it is unipotent. Let M = Π\S be an infra-solvmanifold of type (R). Let f : M → M be a continuous map with an affine homotopy lift (d, D) ∈ Aff(S). Then f is homotopic to the diffeomorphism on M induced by the affine map (d, D), called an affine diffeomorphism. If, in addition, D * is virtually unipotent then we say that f is a virtually unipotent map. Now we observe the following: (1) A matrix is virtually unipotent if and only if all of its eigenvalues have absolute value 1, see [59, Lemma 11.6]. (2) Let Φ be a finite subgroup of GL(n, R) and let D ∈ GL(n, R) normalize Φ. If D is virtually unipotent, then for all A ∈ Φ, AD is virtually unipotent, see [46, Lemma 3.2]. Example 10.1. Consider the 3-dimensional Lie group Sol = R 2 ⋊ σ R, where σ(t) = e t 0 0 e −t . τ g : ((u, v), s) → (e t u − e s x + x, e −t v − e −s y + y), s),and Ad(g) : sol → sol is given basis of sol. Hence Ad(g) is not virtually unipotent unless t = 0. Now consider the infra-solvmanifold Π + 2 \Sol. Sol Φ = {((x Theorem 10 . 3 . 103If f is a virtually unipotent map on an infra-solvmanifold of type (R), then L(f ) = N (f ). Lemma 4.2], det(I − A * D * ) ≥ 0. Using the averaging formula [42, Theorem 4.3], we obtain N (f ) = 1 |Φ| A∈Φ | det(I − A * D * )| = 1 |Φ| A∈Φ det(I − A * D * ) = L(f ). = and (DL) are equivalent. For example, (8) ⇒ (DL) follows easily by the following observation: Let A i be an integer matrix obtained from the homomorphism f i * : H i (X; Q) → H i (X; Q). L(f p r−1 ) mod p r . Now we shall consider the following congruences for the Nielsen numbers and Reidemeister numbersd|n µ(d) R(f n/d ) ≡ 0 mod n, (DR) R(f p r ) ≡ R(f p r−1 ) mod p r , (ER) d|n µ(d) N (f n/d ) ≡ 0 mod n, (DN) N (f p r ) ≡ N (f p r−1 ) mod p r (EN)and find the relations between them and the conditions on spaces, groups and/or on maps for which the congruences hold true.Example 11.1. Let f be a map on an infra-solvmanifold of type (R) which is homotopically periodic or virtually unipotent. Then N (f n ) = L(f n ) for all n > 0. The congruence (DL) immediately implies the congruences (DN) and (EN). Example 11.2. Let f : S 2 ∨ S 4 → S 2 ∨ S 4 be the map considered in Example 3.1. ThenL(f ) = N (f ) = 0, L(f k ) = 2 + (−2) k , N (f k ) = 1 ∀k > 1, R(f k ) = 1 ∀k ≥ 1.Thus we have no congruence (DN) and we have nice congruences (DR) and (DL). Example 11 . 3 . 113Let f be a map on the circle S 1 of degree d. Then N (f n ) = |L(f n )| = |1 − d n |(= R(f n ) when d = ±1) Theorem 11 . 4 . 114Let f be any continuous map on an infra-solvmanifold of type (R) such that all R(f n ) are finite. Then we have d|n µ(d) R(f n/d ) = d|n µ(d) N (f n/d ) ≡ 0 mod n for all n > 0. Proof. We define P n (f ) = the set of isolated periodic points of f with period n, P d (f ) = P d (f ) − k|d P k (f ) = the set of isolated periodic points of f with least period d. Then we have P n (f ) = d|n P d (f ) or #P n (f ) = d|n #P d (f ). By the Möbius inversion formula when all terms are finite, we have #P n (f ) = d|n µ(d) #P n/d (f ). is a double covering of M = Π\G; the map f lifts to a map f + : M + → M + which has the same affine homotopy lift (d, D) as f . If D * has no eigenvalues of modulus > 1, then for any A ∈ Φ, A = A 1 and in this case we take Π + = Π.Now, a main result, Theorem 4.4, of [11] is the following: Theorem 1.2 ([11],Theorem 4.4, when Π = Π + see also proof of Theorem 3.5). Let f be a continuous map on an infra-nilmanifold Π\G with an affine homotopy lift (d, D). Then the Nielsen numbers of f k are Theorem 2.3 (Compare with [12, Theorem 6.1]). Let f and g be continuous maps on an orientable infra-solvmanifold of type (R) with cyclic holonomy groupsee [12, Lemma 6.3]. From [31, Theorem 4.5], immediately we have: Proposition 8.1 ([3, Ex. 21(b), p. 97],[58, Proposition 3.6]). Let σ be a Lie algebra automorphism. If none of the eigenvalues of σ is a root of unity, then the Lie algebra must be nilpotent.Theorem 8.2. If the Reidemeister zeta function R f (z) is defined for a homeomorphism f on an infra-solvmanifold M of type (R), then M is an infra-nilmanifold.Proof. Let f be a homeomorphism on an infra-solvmanifold M = Π\S of type (R). By[42, Theorem 2.2], we may assume that f has an affine map as a homotopy lift. By [42, Lemma 2.1], there is a special solvmanifold N = Λ\S which covers M finitely and on which f has a liftf , which is induced by a Lie group automorphism D on the solvable Lie group S. From [30, Corollary 1.3], we have an averaging formula for Reidemeister numbers: Acknowledgments. The first author is indebted to the Institut des Hauteś Etudes Scientifiques (Bures-sur-Yvette) for the support and hospitality and For, f k (f (x)) = f (x) ⇒ f -orbits, each of length n. So, when #P n (f ) is finite, it is a multiple of n. Let M be an infra-solvmanifold of type (R) and let f be a map on M . Since we are working with the Nielsen numbers and the Reidemeister numbers of iterates of f , we may assume that f is induced by an affine map on S. Assume R(f n ) < ∞. On the other hand, if x ∈ P n (f ) then f (x) ∈ P n (f ). then N (f n ) = R(f n ) > 0 by Corollary 7.6. By Remark 9.4, N (f n ) is the number of isolated periodic points of f with period nOn the other hand, if x ∈ P n (f ) then f (x) ∈ P n (f ). For, f k (f (x)) = f (x) ⇒ f -orbits, each of length n. So, when #P n (f ) is finite, it is a multiple of n. Let M be an infra-solvmanifold of type (R) and let f be a map on M . Since we are working with the Nielsen numbers and the Reidemeister num- bers of iterates of f , we may assume that f is induced by an affine map on S. Assume R(f n ) < ∞; then N (f n ) = R(f n ) > 0 by Corollary 7.6. By Remark 9.4, N (f n ) is the number of isolated periodic points of f with period n; N (f n ) = #P n (f ). N (f n ) = #P n (f ). Consequently, what we have shown is that if all R(f n ) < ∞, then. Consequently, what we have shown is that if all R(f n ) < ∞, then Let f be a map on an infra-solvmanifold of type (R) which is homotopic to an affine diffeomorphism induced by an affine map. Corollary 11.5.. d, D)Corollary 11.5. Let f be a map on an infra-solvmanifold of type (R) which is homotopic to an affine diffeomorphism induced by an affine map (d, D). If D * has no eigenvalue that is a root of unity, then all R(f n ) are finite. Hence the Gauss congruences (DR) for the Reidemeister and (DN) for the Nielsen numbers hold true. If D * has no eigenvalue that is a root of unity, then all R(f n ) are finite. Hence the Gauss congruences (DR) for the Reidemeister and (DN) for the Nielsen numbers hold true. This follows from a straightforward generalization of. Proposi- tion 4.3infra-nilmanifolds to infra-solvmanifolds of type (R). 10Proof. This follows from a straightforward generalization of [10, Proposi- tion 4.3] from infra-nilmanifolds to infra-solvmanifolds of type (R). Recall that f induces a homomorphism φ : Π → Π given by φ(α) • (d, D) = (d, D) • α for all α ∈ Π. That is, φ = τ (d,D). Let M = Π\S be the infra-solvmanifold of type (R) with holonomy group Φ. This implies that (d, D) normalizes Π and hence D normalizes Φ. So AD n normalizes Φ for all A ∈ Φ and all n. Assume that R(f n ) = ∞. By Corollary 7.6, there exists A ∈ Φ such that A * D n * has eigenvalue 1. By [46, Lemma 3.2], D n * = A −1 * (A * D n * ) is virtually unipotent. Thus D * is virtually unipotent, a contradictionLet M = Π\S be the infra-solvmanifold of type (R) with holonomy group Φ. Recall that f induces a homomorphism φ : Π → Π given by φ(α) • (d, D) = (d, D) • α for all α ∈ Π. That is, φ = τ (d,D) . This implies that (d, D) normalizes Π and hence D normalizes Φ. So AD n normalizes Φ for all A ∈ Φ and all n. Assume that R(f n ) = ∞. By Corollary 7.6, there exists A ∈ Φ such that A * D n * has eigenvalue 1. By [46, Lemma 3.2], D n * = A −1 * (A * D n * ) is virtually unipotent. Thus D * is virtually unipotent, a contradiction. Example 11.6. Let f be an Anosov diffeomorphism on an infra-nilmanifold. By [10, Lemma 4.2], f has an affine homotopy lift (d, D) with hyperbolic D * . From the above corollary, the Gauss congruences (DR) and (DN) hold trueExample 11.6. Let f be an Anosov diffeomorphism on an infra-nilmanifold. By [10, Lemma 4.2], f has an affine homotopy lift (d, D) with hyperbolic D * . From the above corollary, the Gauss congruences (DR) and (DN) hold true. It is known in [29] that f is topologically conjugate to an expanding map on an infra-nilmanifold. Thus we can assume that M is an infra-nilmanifold and f is a map induced by an affine map (d, D), where all the eigenvalues of D * are of modulus > 1. Since (d, D) satisfies the conditions of Corollary 11.5, all R(f n ) are finite and so the congruences (DR) and (DN). Let f : M → M be an expanding smooth map on a closed smooth manifold. 1120, Example 11. hold true. On the other hand, by [56], the set Fix(f n ) of fixed points of the expandingExample 11.7 ([20, Example 11], [41]). Let f : M → M be an expanding smooth map on a closed smooth manifold. It is known in [29] that f is topologically conjugate to an expanding map on an infra-nilmanifold. Thus we can assume that M is an infra-nilmanifold and f is a map induced by an affine map (d, D), where all the eigenvalues of D * are of modulus > 1. Since (d, D) satisfies the conditions of Corollary 11.5, all R(f n ) are finite and so the congruences (DR) and (DN) hold true. On the other hand, by [56], the set Fix(f n ) of fixed points of the expanding The Nielsen numbers of maps of nil-manifolds. D V Anosov, 40Uspekhi Mat. Naukin RussianD. V. Anosov, The Nielsen numbers of maps of nil-manifolds, Uspekhi Mat. Nauk, 40 (1985), 133-134 (in Russian); . English Transl, Russian Math. Surveys. 40English transl.: Russian Math. Surveys, 40 (1985), 149-150. The behavior of the index of periodic points under iterations of a mapping. I K Babenko, S A Bogatyǐ, Izv. Akad. Nauk SSSR Ser. Mat. 55Russian. translation in Math. 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Lefschetz formula for flows, The Lefschetz centennial conference, Part III. D Fried, Contemp. Math. 58III, Amer. Math. SocD. Fried, Lefschetz formula for flows, The Lefschetz centennial conference, Part III (Mexico City, 1984), 19-69, Contemp. Math., 58, III, Amer. Math. Soc., Providence, RI, 1987. D Gonçalves, Coincidenc Reidemeister classes on nilmanifolds and nilpotent fibrations. 83D. Gonçalves, Coincidenc Reidemeister classes on nilmanifolds and nilpotent fibra- tions, Topology Appl., 83 (1998), 169-186. Nilmanifolds are Jiang-type spaces for coincidences, Forum Math. D Gonçalves, P Wong, 13D. Gonçalves and P. Wong, Nilmanifolds are Jiang-type spaces for coincidences, Fo- rum Math., 13 (2001), 133-141. Groups of polynomial growth and expanding maps. M Gromov, Inst. HautesÉtudes Sci. 53M. Gromov, Groups of polynomial growth and expanding maps, Inst. HautesÉtudes Sci., 53 (1981), 53-73. The R∞ property for crystallographic groups of Sol. K Y Ha, J B Lee, preprintK. Y. Ha and J. B. 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Ivanov, Entropy and the Nielsen numbers. Dokl. Akad. Nauk SSSR 265 (2) (1982), 284-287 (in Russian); . English Transl, Soviet Math. Dokl. 26English transl.: Soviet Math. Dokl. 26 (1982), 63-66. Homotopy methods in topological fixed and periodic points theory. J Jezierski, W Marzantowicz, Topological Fixed Point Theory and Its Applications. DordrechtSpringer3J. Jezierski and W. Marzantowicz, Homotopy methods in topological fixed and peri- odic points theory, Topological Fixed Point Theory and Its Applications, 3, Springer, Dordrecht, 2006. Lectures on Nielsen fixed point theory. B Jiang, Contemporary Math. 14Amer. Math. SocB. Jiang, Lectures on Nielsen fixed point theory, Contemporary Math., 14, Amer. Math. Soc., Providence, R.I., 1983. The entropy conjecture, Smooth dynamical dystems. A B Katok, Mir PublishingMoscowin RussianA. B. Katok, The entropy conjecture, Smooth dynamical dystems (in Russian), Mir Publishing, Moscow 1977, 182-203. Computation of the Nielsen type numbers for maps on the Klein bottle. H J Kim, J B Lee, W S Yoo, J. Korean Math. Soc. 45H. J. Kim, J. B. Lee and W. S. Yoo, Computation of the Nielsen type numbers for maps on the Klein bottle, J. Korean Math. Soc., 45 (2008), 1483-1503. Averaging formular for Nielsen numbers. S W Kim, J B Lee, K B Lee, Nagoya Math. J. 178S. W. Kim, J. B. Lee and K. B. Lee, Averaging formular for Nielsen numbers, Nagoya Math. J., 178 (2005), 37-53. The Nielsen numbers of homotopically periodic maps of infranilmanifolds. S Kwasik, K B Lee, J. London Math. Soc. 382S. Kwasik and K. B. Lee, The Nielsen numbers of homotopically periodic maps of infranilmanifolds, J. London Math. Soc. (2), 38 (1988), 544-554. Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds. J B Lee, K B Lee, J. Geom. Phys. 56J. B. Lee and K. B. Lee, Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds, J. Geom. Phys., 56 (2006), 2011-2023. Averaging formula for Nielsen numbers of maps on infrasolvmanifolds of type (R). J B Lee, K B Lee, Nagoya Math. J. 196J. B. Lee and K. B. Lee, Averaging formula for Nielsen numbers of maps on infra- solvmanifolds of type (R), Nagoya Math. J., 196 (2009), 117-134. Maps on infra-nilmanifolds. K B Lee, Pacific J. Math. 168K. B. Lee, Maps on infra-nilmanifolds, Pacific J. Math., 168 (1995), 157-166. Rigidity of almost crystallographic groups. K B Lee, F Raymond, Contemporary Math. Amer. Math. Soc. 44K. B. Lee and F. Raymond, Rigidity of almost crystallographic groups, Contemporary Math. Amer. Math. Soc., 44 (1985), 73-78. On the rationality of the Nielsen zeta function. L Li, Adv. in Math. (China). 23L. Li, On the rationality of the Nielsen zeta function, Adv. in Math. (China), 23 (1994), 251-256. The Nielsen numbers of virtually unipotent maps on infra-nilmanifolds. W Malfait, Forum Math. 13W. Malfait, The Nielsen numbers of virtually unipotent maps on infra-nilmanifolds, Forum Math., 13 (2001), 227-237. Finding periodic points of a map by use of a k-adic expansion. W Marzantowicz, P M Przygodzki, Discrete Contin. Dyn. Syst. 5W. Marzantowicz and P. M. Przygodzki, Finding periodic points of a map by use of a k-adic expansion, Discrete Contin. Dyn. Syst., 5 (1999), 495-514. Entropy conjecture for continuous maps of nilmanifolds. W Marzantowicz, F Przytycki, Israel J. Math. 165W. Marzantowicz and F. Przytycki, Entropy conjecture for continuous maps of nil- manifolds, Israel J. Math., 165 (2008), 349-379. Estimates of the topological entropy from below for continuous self-maps on some compact manifolds. W Marzantowicz, F Przytycki, Discrete Contin. Dyn. Syst. 21W. Marzantowicz and F. Przytycki, Estimates of the topological entropy from below for continuous self-maps on some compact manifolds, Discrete Contin. Dyn. Syst., 21 (2008), 501-512. Nielsen numbers of homotopically periodic maps on infrasolvmanifolds. C K Mccord, Proc. Amer. Math. Soc. 120C. K. McCord, Nielsen numbers of homotopically periodic maps on infrasolvmani- folds, Proc. Amer. Math. Soc., 120 (1994), 311-315. C K Mccord, Lefschetz and Nielsen coincidenc numbers on nilmanifolds and solvmanifolds II. 75C. K. McCord, Lefschetz and Nielsen coincidenc numbers on nilmanifolds and solv- manifolds II, Topology Appl., 75 (1997), 81-92. P Penninckx, Fixed point theory and coincidence theory for infra-nilmanifolds. Leuven, BelgiumKatholieke Universiteit LeuvenPh.D. thesisP. Penninckx, Fixed point theory and coincidence theory for infra-nilmanifolds, Ph.D. thesis, Katholieke Universiteit Leuven, Leuven, Belgium, December 2009. Fel'shtyn, The Nielsen zeta function. V B Pilyugina, A L , Funktsional. Anal. i Prilozhen. 19in RussianV. B. Pilyugina and A. L. Fel'shtyn, The Nielsen zeta function, Funktsional. Anal. i Prilozhen., 19 (1985) 61-67 (in Russian); . English Transl, Functional Anal. Appl. 19English transl.: Functional Anal. Appl., 19, (1985) 300-305. An upper estimation for topological entropy. F Przytycki, Invent. Math. 59F. Przytycki, An upper estimation for topological entropy, Invent. Math., 59 (1980), 205-213. Almost flat manifolds. E A Ruh, J. Differential Geom. 171E. A. Ruh, Almost flat manifolds, J. Differential Geom., 17 (1982), no. 1, 1-14. Endomorphism of compact differentiable manifolds. M Shub, Amer. J. Math. 91M. Shub, Endomorphism of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199. Dynamical systems, filtrations and entropy. M Shub, Bull. Amer. Math. Soc. 80M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41. Differentiable dynamical systems. S Smale, Bull. Amer. Math. Soc. 73S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747- 817. I Stewart, D Tall, Algebraic number theory and Fermat's last theorem. Ltd., Natick, MAthird edition, A K PetersI. Stewart and D. Tall, Algebraic number theory and Fermat's last theorem, third edition, A K Peters, Ltd., Natick, MA, 2002. Reidemeister zeta function for group extensions. P Wong, J. Korean Math. Soc. 38P. Wong, Reidemeister zeta function for group extensions, J. Korean Math. Soc., 38 (2001), 1107-1116. Geometriya, Topologiya i Matematicheskaya Fizika. I (in Russian). A V Zarelua, Proc. Steklov Inst. Math. 263Tr. Mat. Inst. SteklovaA. V. Zarelua, On congruences for the traces of powers of some matrices, Tr. Mat. Inst. Steklova, 263 (2008), 85-105, Geometriya, Topologiya i Matematicheskaya Fizika. I (in Russian); translation in Proc. Steklov Inst. Math., 263 (2008), 78-98. address: felshtyn@ihes.fr, fels@wmf.univ.szczecin.pl Department of mathematics. Instytut Matematyki, Uniwersytet Szczecinski, Szczecin, Poland and Institut des HautesÉtudes Scientifiques, Le Bois-Marie 35, route de Chartres 91440. Bures-sur-Yvette, France E-mail15Sogang UniversitySeoul 121-742, KOREA E-mail address: jlee@sogang.ac.krInstytut Matematyki, Uniwersytet Szczecinski, ul. Wielkopolska 15, 70- 451 Szczecin, Poland and Institut des HautesÉtudes Scientifiques, Le Bois-Marie 35, route de Chartres 91440 Bures-sur-Yvette, France E-mail address: felshtyn@ihes.fr, fels@wmf.univ.szczecin.pl Department of mathematics, Sogang University, Seoul 121-742, KOREA E-mail address: jlee@sogang.ac.kr

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{'abstract': 'We prove the rationality, the functional equations and calculate the radii of convergence of the Nielsen and the Reidemeister zeta functions of continuous maps on infra-solvmanifolds of type (R). We find a connection between the Reidemeister and Nielsen zeta functions and the Reidemeister torsions of the corresponding mapping tori. We show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type (R), then this manifold is an infra-nilmanifold. We also prove that a map on an infra-solvmanifold of type (R) induced by an affine map minimizes the topological entropy in its homotopy class and it has a rational Artin-Mazur zeta function. Finally we prove the Gauss congruences for the Reidemeister and Nielsen numbers of any map on an infra-solvmanifolds of type (R) whenever all the Reidemeister numbers of iterates of the map are finite. Our main technical tool is the averaging formulas for the Lefschetz, the Nielsen and the Reidemeister numbers on infra-solvmanifolds of type (R).', 'arxivid': '1303.0784', 'author': ['Alexander Fel&apos;shtyn ', 'Jong Bum Lee '], 'authoraffiliation': [], 'corpusid': 119610752, 'doi': '10.1016/j.topol.2014.12.003', 'github_urls': [], 'n_tokens_mistral': 42489, 'n_tokens_neox': 37301, 'n_words': 21388, 'pdfsha': '7ca4465933f9a9eb88ee9973cb2a4153221ff5f7', 'pdfurls': ['https://arxiv.org/pdf/1303.0784v4.pdf'], 'title': ['THE NIELSEN AND REIDEMEISTER NUMBERS OF MAPS ON INFRA-SOLVMANIFOLDS OF TYPE (R)', 'THE NIELSEN AND REIDEMEISTER NUMBERS OF MAPS ON INFRA-SOLVMANIFOLDS OF TYPE (R)'], 'venue': []}

arxiv

Improving the Generalizability of Trajectory Prediction Models with Frenét-Based Domain Normalization Luyao Ye Zikang Zhou Jianping Wang Improving the Generalizability of Trajectory Prediction Models with Frenét-Based Domain Normalization Predicting the future trajectories of robots' nearby objects plays a pivotal role in applications such as autonomous driving. While learning-based trajectory prediction methods have achieved remarkable performance on public benchmarks, the generalization ability of these approaches remains questionable. The poor generalizability on unseen domains, a wellrecognized defect of data-driven approaches, can potentially harm the real-world performance of trajectory prediction models. We are thus motivated to improve models' generalization ability instead of merely pursuing high accuracy on average. Due to the lack of benchmarks for quantifying the generalization ability of trajectory predictors, we first construct a new benchmark called argoverse-shift, where the data distributions of domains are significantly different. Using this benchmark for evaluation, we identify that the domain shift problem seriously hinders the generalization of trajectory predictors since state-of-the-art approaches suffer from severe performance degradation when facing those out-of-distribution scenes. To enhance the robustness of models against domain shift problem, we propose a plug-and-play strategy for domain normalization in trajectory prediction. Our strategy utilizes the Frenét coordinate frame for modeling and can effectively narrow the domain gap of different scenes caused by the variety of road geometry and topology. Experiments show that our strategy noticeably boosts the prediction performance of the state-of-the-art in domains that were previously unseen to the models, thereby improving the generalization ability of datadriven trajectory prediction methods. I. INTRODUCTION The task of trajectory prediction is one of the indispensable components in safety-critical robotic applications, e.g., autonomous driving and robot obstacle avoidance. Given objects' past trajectories and the associated scene context, such as high-definition (HD) map, the goal of trajectory prediction is to predict objects' future movements and thereby enable safe motion planning of robots. Recent research in trajectory prediction has witnessed the huge success of deep learning. With their strong capability of fusing heterogeneous information in the scene, deep learning approaches have dominated the public benchmarks for trajectory prediction [1], [2], [3], [4]. However, whether these data-driven models can be generalized to out-of-distribution (OOD) scenes is still undetermined. As a well-known issue of learning-based approaches, the performance depends heavily on the distribution of training data and is prone All to be affected by the distribution shift problem. Namely, these models usually have satisfactory accuracy in scenes frequently appearing in training data but may have trouble making correct predictions when facing those less frequent ones. For example, a trajectory predictor fully trained on the highway dataset can perform well on straight roads that were unseen before but is very likely to fail when tested on roundabouts due to the different data distributions on straight roads and roundabouts. Collecting sufficient data in all domains for training is not an affordable or feasible solution. If a model cannot make reliable predictions in unseen situations, catastrophic accidents may happen in the real world. For this reason, there is an urgent need to enhance the generalization ability of trajectory prediction models. In daily traffic scenarios, traffic participants do not move in free space but need to obey traffic rules, e.g., driving in lanes or walking on the sidewalk. These traffic rules are mostly conveyed by the map. Therefore, many advanced trajectory prediction approaches [5], [6], [7] focus on modeling the interactions between objects and HD maps to assist trajectory prediction. However, the geometry and topology of the map elements (e.g., the curvature of lanes) vary dramatically in different scenes, which brings the distribution shift problem and makes it difficult for the model to generalize in OOD scenes [8]. To address the above issues, we propose a domain normalization method, termed as Frenét+, for eliminating the difference among scenes by utilizing the Frenét frame [9]. This method explicitly moderates the distribution shift problem caused by the diversity of HD maps and enables the trajectory prediction models to focus more on domain-independent features (e.g., motion patterns of objects and social interactions among traffic participants) rather than overfitting the training data by memorizing the domainspecific map features. Specifically, we calculate the relative coordinates of the target object with respect to the centerline and use the relative coordinates for modeling. Here, we use the Frenét coordinate (i.e., the arc length and the perpendicular offset of the centerline) to represent the relative position of the target object. As shown in Fig. 1, converting to the Frenét coordinate has a significant advantage of reducing the difference among the road shapes of different traffic scenes. We expect that existing trajectory prediction models combined with this domain normalization technique will be able to perform almost identically well on the seen and unseen domains. To verify the existence of the domain shift problem and the effectiveness of our method, we first propose an automatic domain split schema based on a clustering algorithm to construct a new benchmark named argoverse-shift for model evaluation. After splitting different domains, we divide these domains into training set, seen validation set, and unseen validation set. Then, we evaluate several state-of-the-art models on this benchmark and observe that their performance on the unseen validation set is much worse than that on the seen validation set. After integrating with our domain normalization approach, the performance of these models on unseen domains is substantially improved. The key contributions are summarised as follows: • We propose an automatic domain split schema and construct a new benchmark for evaluating the generalization ability of trajectory prediction models against the distribution shift problem. • We design validation experiments to explicitly quantify the generalizability of learning-based trajectory prediction models using our benchmark. • We introduce a plug-and-play strategy for domain normalization based on the Frenét frame to help the model recover from the distribution shift problems. Experiments show that this strategy noticeably enhances the prediction performance of state-of-the-art methods in the unseen domains. II. RELATED WORK A. Trajectory prediction Trajectory prediction is a classic problem of autonomous driving and has been widely studied in recent years. Early approaches to trajectory prediction are only based on historical trajectories of the ego vehicle and neighborhoods. While models are rapidly updating, using the LSTM network [10], [11], [12], Convolutional Neural Network [13], Generative Adversarial Networks [14], [15], [16] and graph neural networks [5], [17], [18], all these models neglect the influence of map information on trajectory prediction, so as to make it difficult to breakthrough in terms of prediction accuracy. Thanks to the development of High Definition Map and the release of new trajectory prediction datasets [1], [3], [2], recent works focus on capturing scene representation from HD map in order to improve the performance. Some suggested using the image learning ability of CNN to represent traffic scenes on the map [19], [20]. Another solution has rasterized map elements from the HD map as model inputs [21], [22], [20], [23], [24], [25]. While raster map representation is popular, this method was replaced by vectorized map data due to its efficiency. Vectorized methods [5], [6], [7] learn the relationships among entities in the scene as a set of vectorized entities with semantic and geometric features. Despite the good performance achieved by these methods, researchers have paid limited attention to the generalizability of trajectory prediction models to new domains. Our work proposes new solutions and evaluation benchmarks for this problem. B. Coping with distribution shift in autonomous driving Prior work by Angelos et al. [26] highlight the necessity of out-of-training-distribution scene detection in autonomous driving. They proposed a robust imitative planning method for detecting distribution shift and generating a safe plan. They also provided online supervision to efficiently query expert guidance for a safe course of action when extreme uncertainty. Another recent work demonstrated that sixty percent of existing scenes could be modified to make trajectory prediction models fail [8]. They presented a scene generation model to provide richer and enough scenes for model training. To further evaluate the generalizability of trajectory prediction models, Thomas et al. [27] studied the performance of baseline models across four different datasets, and used the heap map to measure model uncertainty. However, no direct and effective method has been proposed yet to enhance the generalizability of trajectory prediction models. In contrast to the work mentioned above, we present a practical and effective solution to mitigate the effect of distribution shift in trajectory prediction, which does not require manual supervision, is model-independent, and can be directly used in current trajectory prediction models. C. Frenét frame The Frenét frame based on the Frenet-Serret formulas [28] locally describes one point on a curve, a moving coordinate system determined by the curvature and the tangent line along the curve. In autonomous driving, many studies [9], [29], [30], [31] have used the Frenét frame for safe and optimal motion planning. They assumed that the centerline was the ideal path along the free road. Therefore, they chose the centerline as the reference path and solved the motion planning problem in Frenét coordinates rather than Cartesian coordinates. The Frenét coordinates use the arc length of the centerline and the perpendicular offset to indicate the relative position of the points on the trajectory with respect to the centerline. Inspired by the Frenét frame method of motion planning, we apply Frenét frame to alleviate the impact of distribution shift in trajectory prediction. Finding the reference path in a complex scene and figuring out the projection of the given point are two critical challenges to transferring trajectory from Cartesian to Frenét frame. We proposed the Frenét+ strategy to solve the above problems. III. PRELIMINARY STUDY In this section, we first present an automatic domain split schema since very few cross-domain datasets and benchmarks are available. Then, we demonstrate that the distribution shift problem exists in the state-of-the-art trajectory prediction models using our constructed benchmark. A. Domain split schema In this work, we define the domain as a cluster of similar samples. For example, tracks sampled from the straight road and bend can be regarded as two domains. Dividing the dataset into multiple domains needs to ensure that the data within each domain has unique features as much as possible. In other words, the gap between the data distribution of different domains should be large. In this way, the partitioned domains are more conducive to verifying the distribution shift problem. However, it is a challenge to find out the domain boundaries precisely. Designing the boundaries manually is not only a large workload but also easily influenced by individual subjectivity, leading to inconsistent split criteria. For this reason, we propose a clustering-based automatic domain split schema. Specifically, we first perform feature extraction for each record in the dataset. We pre-define 21 features, such as the lane deflection angle, the differences of lane coordinates and the lane boundary. We extract these 21 features for j-th record to formulate a feature vector d j ∈ R 21 . This 21dimensional vectors are then downscaled to 2-dimensionŝ d j ∈ R 2 using the PCA algorithm [32]. Experiments show that the 2-dimensions feature is good enough to characterize a sample and also convenient for visualization. Intuitively, these 2-dimensional vectors have a stronger characterization capability. The K-Means algorithm [33] is then used to cluster these 2-dimensional vectors. In this way, each record is assigned with a cluster and we believe these clusters can be regarded as domains. Based on the clustering results, we retrace the corresponding data and match them with the corresponding domains. We further plot the distribution between each pair of divided domains to investigate the overlap among them. Fig. 2 shows no overlap between each domain pair and suggests that the split schema achieves the expectation. B. Argoverse-shift benchmark We construct a benchmark, referred to as argoverse-shift, to evaluate the generalization ability of trajectory prediction models against the distribution shift problem. Following the domain split schema proposed in Section III-A, we partition Argoverse dataset [3] into ten domains. We take the first seven domains as seen domains and the three left domains as unseen domains. The training set and validation set are sampled from all seen domains with a ratio of 8:2. All of the unseen domains are taken to formulate the test set. The detailed statistics are shown in Table I. On the one hand, the new cross-domain dataset, argoverse-shift, can be used to verify the existence of the distribution shift problem in the trajectory prediction task, and on the other hand, it can be used as a new benchmark to evaluate the trajectory prediction model's generalizability. To verify that current trajectory prediction models suffer from the distribution shift problem, we select several stateof-the-art models and observe whether their performances degrade when the train and test sets are taken from different domains. Experiment results show the performance of welltrained models decreases in the unseen domains. The analysis and implementation details for the domain shift verification experiment are described in Section VI. IV. PROBLEM DEFINITION We take X ∈ X to denote the input sampled from the input space X and Y ∈ Y to denote the output from the output space Y. As recent works included many features into consideration, e.g., neighbors' coordinates and HD map, trajectory predication is not necessarily a self-regression task, i.e., X ̸ = Y. We suppose that the data in a given dataset S can be divided into M independent domains, i.e., S = {D 1 , · · · , D M }, where D i = {(X i j , Y i j )} ni j=1 denotes the i-th domain. The distributions between each pair of domains are different. The first K domains of S are taken as the seen domains S seen = {D 1 , · · · , D K } that models can access during the training, and the rest are taken as the unseen domains S unseen = {D K+1 , · · · , D M } that are used to simulate the new scenes appeared on the road. The training set is sampled from seen domains, i.e., S train ∈ S seen , and the validation set used during training is also from seen domain and does not overlap with the training set, i.e., S val ∈ {S seen \S train }. The test set is sampled from the unseen domains S test ∈ S unseen . The goal of domain generalization in trajectory prediction task is to learn a robust and generalizable prediction function h : X → Y from the seen domains to achieve a minimum prediction error on the unseen test domain [34]: min h E (X,Y )∈Stest [L(h(X), Y )],(1) where E is the expectation and L is the loss function. V. FRENÉT+ STRATEGY In this section, we first show the strategy to select the reference path. Then we illustrate the pipeline that transfers a trajectory point from the Cartesian frame to the Frenét frame based on the selected reference path. Finally, we provide the solution to determine the projections for those trajectory points located in the non-differential areas. A. The selection of the reference path In complex scenes, such as the intersection, there are multiple centerlines, as shown in Fig. 3, which increases the difficulty of finding the appropriate reference path. To solve this problem, we proposed a method to determine a proper centerline as the reference path based on the vehicle's historical trajectory. Specifically, we first calculate the Euclidean distance between the vehicle's historical trajectory and each centerline and take the reciprocal of the distance as the similarity. S 1 j = [ 1 n n t=1 ∥x t − x j t ∥ 2 ] −1 ,(2) where x t ∈ R 2 and x t ∈ R 2 are the coordinates of the trajectory and corresponding projection on the centerline j at the time step t. Intuitively, the closer centerline would be a better choice of the reference path. Next, we compare the shape similarity between centerlines and historical trajectory. Specifically, we translate candidate centerlines towards the vehicle and then calculate the Euclidean distance. The translation direction and length are defined by the current location of the vehicle with its projections on centerlines: ∆x j = x T − x j T ,(3) where T indicates the current time step. Then the shape similarity is defined as: After that, we take the average of the S 1 j and S 2 j . Finally, we choose the centerline j * with the largest value as the reference line. S 2 j = [ 1 n n t=1 ∥x t − ( x j t + ∆x j )∥ 2 ] −1 .(4)j * = arg max j (S 1 j + S 2 j ).(5) B. Coordinate transfer Suppose that a reference path is composed of m segments and saved as a list of coordinate points following the order of the direction of the trajectory, i.e., [p 1 , · · · , p m+1 ]. The projection of the vehicle on the reference path at the time step t falls on the J-th segment, i.e., the segment bounded with the points p J and p J+1 . Based on these two points, we can easily calculate the expression of the line where the J-th segment is located, i.e., y = k J x + b J . Then the projection of the trajectory point x t = (x t , y t ) at the time step t can be represented as: x t = ( x t , y t ), where, x t = k J (y t − b J ) + x t (k J ) 2 + 1 , y t = k J x t + b J .(6) The offset of the trajectory point about the reference path, i.e., the d coordinate in Frenét Frame, can be derived from the distance between the trajectory point and the projection: d t = I t · ∥ x t − x t ∥ 2 , I t = −1, if −−−−→ p J p J+1 · −−→ p J x t < 0, 1, if −−−−→ p J p J+1 · −−→ p J x t ≥ 0.(7) The I t is the indicator which indicates the side of the trajectory point relative to the reference path in the right-hand system. We initially set the first point of the reference path as the starting point. Then the arc length s of the trajectory point in the Frenét frame can be expressed as the sum of some segments: s t = J j=2 ∥p j − p j−1 ∥ + ∥ x t − p J ∥.(8) We take the difference between s t and s 1 as the final arc length, i.e., s t ← − (s t − s 1 ). In this way, the projection of the first trajectory point is set as the starting point. Therefore, the arc length of the first trajectory point s 1 in the Frenét frame is 0. So far, we present the whole process to convert a trajectory point from the Cartesian coordinate (x t , y t ) to the Frenét coordinate (s t , d t ). This process is unrelated to model design and can be easily adapted to most current trajectory prediction models. It can be set between the data pipe and the model, i.e., the coordinate transformation is performed before feeding trajectory into the model. C. Projections on the non-differentiable area Another challenge is to find the correct projections on the non-differentiable area of the reference line. From the engineering perspective, centerlines are often stored as a list of coordinate points. Connecting these points in order gives a series of line segments. In this case, the centerline is not a smooth curve but a polyline. Therefore, not all points have projections on the reference path. As shown in Fig. 4, we cannot find a projection for points in the red area since it is non-differential at joint points. We represent the red area in Fig. 4 at the joint (x * , y * ) as: p = {(x, y)| − 1 k 1 < y − y * x − x * < − 1 k 2 },(9) where k 1 > k 2 are slopes of segments that intersect at the joint (x * , y * ). We artificially set the projections of the trajectory points (x, y) ∈p to the nearest endpoints on the reference line. Moreover, we set the angle bisector as the vertical direction at the joint point: y = 1 2 ( 1 k 1 + 1 k 2 )(x * − x) + y * .(10) In this way, multiple points in the area can be converted to the same Frenét coordinate. It introduces extra errors when converting them back to the Cartesian coordinate frame. However, we experimentally verified that the error is less than 10 −4 meters on average, which has a negligible effect on the final results. Fig. 3 shows that our Frenét+ strategy can find the proper reference path from a complex scene and get the correct projections on the non-differentiable centerlines. VI. EXPERIMENTS In this section, we conduct comparative experiments by combining Frenét+ strategy with baseline models to demonstrate the effectiveness of our proposed strategy. We also perform result analysis and visualize several prediction results to give an insight into domain shift problem on trajectory prediction. A. Experimental settings The experiments are conducted on the argoverse-shift dataset. Following the same setting as Argoverse motion forecasting challenge [3], we require models to predict the position of the agents in the future 3 seconds, given the initial 2 seconds observations. Baselines: We take NN+map [3], LSTM ED+map [3], WIMP [35], LaneGCN [5], HiVT [7], the five representative state-of-the-art trajectory prediction models, as baselines. Metrics: We employ three standard metrics for trajectory prediction, including the Minimum Average Displacement Error (minADE), Minimum Final Displacement Error (minFDE) and Miss Rate (MR). Models predict six trajectories, and we report the best result with minimum errors. We train each baseline on the training set with the best practice of the hyperparameters reported in the original paper and select the best parameter group via the validation set. After training, we separately evaluate the model on the test set, i.e., the unseen domains, and the validation set, i.e., the seen domains, for comparison. We then apply the Frenét+ strategy to each model. We train and re-evaluate them, following the same process. The comparison results are reported in Table II. Table II shows the performance of the five models on the seen domain and unseen domain, respectively, and their performance after combining the Frenét+ strategy. A smaller value means better performance. Due to changes in the dataset volume and split scheme, the results of baseline models are slightly different from the performance reported in the original paper. The value in parentheses is the performance deterioration on unseen domains relative to the performance on seen domains. B. Result analysis Domains shift problem verification: From the top half part of Table II, the performance of well-trained models decreased on the unseen domains in terms of all evaluation metrics. The result illustrates that the domain shift problem exists exactly in current models. For those naive models, e.g., NN + MAP, this problem has a more serious negative impact. By comparison, LaneGCN is more robust against domain shift. It has the slightest drop in performance on the unseen domain. Because LaneGCN takes into account a large amount of information about the relative positions of centerlines, their predictions are less susceptible to topographic changes. Frenét+ effectiveness evaluation: By comparing the upper and lower parts of Table II, we find that our Frenét+ strategy gives a significant improvement across all baselines on unseen domains in terms of all metrics. Though it is still not as good as the results on seen domains, the difference is not significant. Frenét+ brings more improvements to those simple models, like NN + MAP, since strong models are more robust in design. We also find a slight decrease in the performance of the model with Frenét+, compared with the original one, on seen domains, e.g., the MinADE of HiVT increased from 0.7642 to 0.7756. We believe this is the necessary cost to improve generalization ability of baseline models, since we make the model focuses on general features rather than domain-specific details during training in order to obtain better generalizability. C. Visualization of prediction results and error distribution In Fig. 5, we visualize several prediction results of HiVT compared with Frenét+ HiVT. In the four scenes, the predicted trajectories by HiVT deviated from the ground truth and even crossed three lanes in the third scene, which is abnormal driving behavior. With the help of Frenét+ strategy, the predicted trajectory is corrected by the reference path and closer to the ground truth. It shows that the Frenét+ strategy provides a strong reference for models and has a significant effect on normalized abnormal prediction. As for the error distribution in the Frenét frame and Cartesian frame, it is obvious that they are exactly the same when the reference path is a straight line. We show a case with a curve reference path in Fig. 6. Compared with Cartesian frame, the error distribution in Frenét frame is not a standard circle. When the projection falls on the part of the reference path with a large gradient, the changes of the loss are more dramatic. The third picture shows the difference between loss in the Frenét frame and Cartesian frame. Though the error distributions are not the same in these two systems, the difference is limited. Hence, we claim that training models in the Frenét frame will not introduce significant optimization gaps, and the optimization objectives, i.e., the 0-error point, are the same. VII. CONCLUSIONS In this paper, we have introduced a new benchmark called argoverse-shift to verify that the domain shift problem does exist in data-driven trajectory prediction models. Then, we have proposed a Frenét-based strategy, Frenét+, to enhance the robustness of models against domain shift. Our approach can diminish the variation of trajectory coordinates across domains by exploiting the local coordinates of trajectory waypoints relative to the lane centerlines. Experiments and visualization results show that the Frenét+ strategy significantly mitigates the domain shift problem and makes stateof-the-art models generalize better on unseen domains. In the near future, we plan to explore the domain shift problem on more datasets, such as Waymo Open Dataset [2] and NuScenes Dataset [36]. Fig. 1 . 1The solid black line represents the road boundary, the dotted gray line represents the centerline, and the orange arrow represents the vehicle trajectory. Fig. 2 . 2The 10 by 10 scatter plots matrix of domains overlaps. The cross of i-th row j-th column represents the overlap scatter between i-th domain and j-th domain. For each single scatter plot, the points in the same color belong to the same domain. The scatter plots on the principal diagonal show the points distribution of corresponding domains. Fig. 3 . 3Trajectories with their projections. The solid black lines with points are centerlines that appeared in this scene. The centerline in red is selected as the reference path. The blue points are the trajectory points of the vehicle. The orange points are the projections on the reference path. The gray lines connect the trajectory points with their corresponding projections. Fig. 4 . 4Find the correct projections on the non-differentiable area. The red points are trajectory points; the black points are the projections of the trajectory points on the reference path, and the blue line is the angle bisector. Fig. 5 . 5Visualized trajectory prediction results of HiVT and Frenét+ HiVT. The green line is the ground truth, the red line is the predicted trajectories by HiVT and the yellow line is predicted by Frenét+ HiVT. Fig. 6 . 6Contour maps of error distribution in Frenét frame, compared with Cartesian frame. The black point indicates the true trajectory point, i.e., the point with 0 error. The red curve is the reference path. The lighter the color, the less the error, and vice versa. authors are with the Department of Computer Science, City University of Hong Kong, Hong Kong SAR, and City University of Hong Kong Shenzhen Research Institute, Shenzhen, China. Emails: {luyaoye2c, zikanzhou2-c}@my.cityu.edu.hk; jianwang@cityu.edu.hk. This work was partially supported by Hong Kong Research Grant Council under GRF 11200220, Science and Technology Innovation Committee Foundation of Shenzhen under Grant No. JCYJ20200109143223052. TABLE I IARGOVERSE-SHIFT DATASET STATISTICSSubset Domain ID Ratio (rel.) Volume Train 0, 1, 2, 3, 4, 5, 6 58.35% 143,202 Validation 0, 1, 2, 3, 4, 5, 6 14.59% 35,800 Test 7, 8, 9 27.06% 66,412 All 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 100% 245,414 TABLE II THE IIQUALITATIVE RESULTS OF 5 BASELINE MODELS IN SEEN AND UNSEEN DOMAINS, COMPARED WITH FRENÉT+ MODELSModel Seen Domains Unseen Domains minADE ↓ minFDE ↓ MR ↓ minADE ↓ minFDE ↓ MR ↓ NN + MAP 0.6342 1.3887 0.1515 1.9689 (+210.45%) 3.7502 (+170.05%) 0.5501 (+263.10%) LSTM ED + MAP 2.0870 4.4180 0.6485 2.2622 (+8.39%) 4.7286 (+7.03%) 0.6830 (+5.32%) WIMP 0.7507 1.1189 0.1092 0.8311 (+10.71%) 1.2525 (+11.94%) 0.1364 (+24.91%) LaneGCN 0.7152 1.0974 0.1065 0.7653 (+7.01%) 1.1554 (+5.29%) 0.1148 (+7.79%) HiVT 0.7642 1.2081 0.1263 0.8595 (+12.47%) 1.3836 (+14.53%) 0.1505 (+19.16%) Frenét+ NN + MAP 0.8284 1.7193 0.2114 0.9984 (+20.52%) 2.0223 (+17.62%) 0.2625 (+24.17%) Frenét+ LSTM ED +MAP 2.0918 4.4296 0.6467 2.1058 (+0.67%) 4.4537 (+0.54%) 0.6498 (+0.48%) Frenét+ WIMP 0.7596 1.1263 0.1167 0.7718 (+1.61%) 1.1475 (+1.88%) 0.1183 (+1.37%) Frenét+ LaneGCN 0.7218 1.1039 0.1064 0.7330 (+1.55%) 1.1215 (+1.59%) 0.1091 (+2.54%) Frenét+ HiVT 0.7756 1.2370 0.1344 0.7882 (+1.62%) 1.2602 (+1.88%) 0.1402 (+4.32%) Interaction dataset: An international, adversarial and cooperative motion dataset in interactive driving scenarios with semantic maps. 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Stanciulescu, and F. Moutarde, "Uncertainty estimation for cross-dataset performance in trajectory prediction," 2022. [Online]. Available: https://arxiv.org/abs/2205. 07310 Journal de mathématiques pures et appliquées. F Frenet, Sur les courbes a double courbureF. Frenet, "Sur les courbes a double courbure." Journal de mathématiques pures et appliquées, pp. 437-447, 1852. On-line trajectory generation for safe and optimal vehicle motion planning. D Althoff, M Buss, A Lawitzky, M Werling, D Wollherr, Autonomous Mobile Systems. SpringerD. Althoff, M. Buss, A. Lawitzky, M. Werling, and D. Wollherr, "On-line trajectory generation for safe and optimal vehicle motion planning," in Autonomous Mobile Systems 2012. Springer, 2012, pp. 99-107. Timeoptimal motion planning for n-dof robot manipulators using a pathparametric system reformulation. R Verschueren, N Van Duijkeren, J Swevers, M Diehl, 2016 American Control Conference (ACC). IEEER. Verschueren, N. van Duijkeren, J. Swevers, and M. Diehl, "Time- optimal motion planning for n-dof robot manipulators using a path- parametric system reformulation," in 2016 American Control Confer- ence (ACC). IEEE, 2016, pp. 2092-2097. Baidu apollo em motion planner. H Fan, F Zhu, C Liu, L Zhang, L Zhuang, D Li, W Zhu, J Hu, H Li, Q Kong, arXiv:1807.08048arXiv preprintH. Fan, F. Zhu, C. Liu, L. Zhang, L. Zhuang, D. Li, W. Zhu, J. Hu, H. Li, and Q. Kong, "Baidu apollo em motion planner," arXiv preprint arXiv:1807.08048, 2018. Genetic k-means algorithm. K Krishna, M N Murty, IEEE Transactions on Systems, Man, and Cybernetics. 293Part B (Cybernetics)K. Krishna and M. N. Murty, "Genetic k-means algorithm," IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 29, no. 3, pp. 433-439, 1999. Principal component analysis. H Abdi, L J Williams, Wiley interdisciplinary reviews: computational statistics. 24H. Abdi and L. J. Williams, "Principal component analysis," Wiley interdisciplinary reviews: computational statistics, vol. 2, no. 4, pp. 433-459, 2010. Generalizing to unseen domains: A survey on domain generalization. J Wang, C Lan, C Liu, Y Ouyang, T Qin, W Lu, Y Chen, W Zeng, P Yu, IEEE Transactions on Knowledge and Data Engineering. J. Wang, C. Lan, C. Liu, Y. Ouyang, T. Qin, W. Lu, Y. Chen, W. Zeng, and P. Yu, "Generalizing to unseen domains: A survey on domain generalization," IEEE Transactions on Knowledge and Data Engineering, 2022. What-if motion prediction for autonomous driving. S Khandelwal, W Qi, J Singh, A Hartnett, D Ramanan, arXiv:2008.10587arXiv preprintS. Khandelwal, W. Qi, J. Singh, A. Hartnett, and D. Ramanan, "What-if motion prediction for autonomous driving," arXiv preprint arXiv:2008.10587, 2020. nuscenes: A multimodal dataset for autonomous driving. H Caesar, V Bankiti, A H Lang, S Vora, V E Liong, Q Xu, A Krishnan, Y Pan, G Baldan, O Beijbom, Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. the IEEE/CVF conference on computer vision and pattern recognitionH. Caesar, V. Bankiti, A. H. Lang, S. Vora, V. E. Liong, Q. Xu, A. Krishnan, Y. Pan, G. Baldan, and O. Beijbom, "nuscenes: A multimodal dataset for autonomous driving," in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2020, pp. 11 621-11 631.

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{'abstract': "Predicting the future trajectories of robots' nearby objects plays a pivotal role in applications such as autonomous driving. While learning-based trajectory prediction methods have achieved remarkable performance on public benchmarks, the generalization ability of these approaches remains questionable. The poor generalizability on unseen domains, a wellrecognized defect of data-driven approaches, can potentially harm the real-world performance of trajectory prediction models. We are thus motivated to improve models' generalization ability instead of merely pursuing high accuracy on average. Due to the lack of benchmarks for quantifying the generalization ability of trajectory predictors, we first construct a new benchmark called argoverse-shift, where the data distributions of domains are significantly different. Using this benchmark for evaluation, we identify that the domain shift problem seriously hinders the generalization of trajectory predictors since state-of-the-art approaches suffer from severe performance degradation when facing those out-of-distribution scenes. To enhance the robustness of models against domain shift problem, we propose a plug-and-play strategy for domain normalization in trajectory prediction. Our strategy utilizes the Frenét coordinate frame for modeling and can effectively narrow the domain gap of different scenes caused by the variety of road geometry and topology. Experiments show that our strategy noticeably boosts the prediction performance of the state-of-the-art in domains that were previously unseen to the models, thereby improving the generalization ability of datadriven trajectory prediction methods.", 'arxivid': '2305.17965', 'author': ['Luyao Ye ', 'Zikang Zhou ', 'Jianping Wang '], 'authoraffiliation': [], 'corpusid': 258959543, 'doi': '10.48550/arxiv.2305.17965', 'github_urls': [], 'n_tokens_mistral': 13388, 'n_tokens_neox': 11708, 'n_words': 7259, 'pdfsha': 'eb98999ed2b0b057157a5171559073614c6b55dd', 'pdfurls': ['https://export.arxiv.org/pdf/2305.17965v1.pdf'], 'title': ['Improving the Generalizability of Trajectory Prediction Models with Frenét-Based Domain Normalization', 'Improving the Generalizability of Trajectory Prediction Models with Frenét-Based Domain Normalization'], 'venue': []}

arxiv

Competing charge and magnetic order in the candidate centrosymmetric skyrmion host EuGa Al Appleton Laboratory-STFC ISIS facility OX11 0QXRutherford, Chilton, DidcotUnited Kingdom A M Vibhakar Diamond Light Source Ltd Harwell Science and Innovation Campus OX11 0DEDidcotOxfordshireUnited Kingdom Appleton Laboratory-STFC ISIS facility OX11 0QXRutherford, Chilton, DidcotUnited Kingdom D D Khalyavin Appleton Laboratory-STFC ISIS facility OX11 0QXRutherford, Chilton, DidcotUnited Kingdom J M Moya Applied Physics Graduate Program Smalley-Curl Institute Rice University 77005HoustonTexasUSA Department of Physics and Astronomy Rice University 77005HoustonTexasUSA P Manuel Appleton Laboratory-STFC ISIS facility OX11 0QXRutherford, Chilton, DidcotUnited Kingdom F Orlandi Appleton Laboratory-STFC ISIS facility OX11 0QXRutherford, Chilton, DidcotUnited Kingdom S Lei Applied Physics Graduate Program Smalley-Curl Institute Rice University 77005HoustonTexasUSA Department of Physics and Astronomy Rice University 77005HoustonTexasUSA E Morosan Applied Physics Graduate Program Smalley-Curl Institute Rice University 77005HoustonTexasUSA Department of Physics and Astronomy Rice University 77005HoustonTexasUSA A Bombardi Diamond Light Source Ltd Harwell Science and Innovation Campus OX11 0DEDidcotOxfordshireUnited Kingdom Department of Physics Clarendon Laboratory University of Oxford Parks RoadOX1 3PUOxfordUK Competing charge and magnetic order in the candidate centrosymmetric skyrmion host EuGa (Dated: April 18, 2023) Eu(Ga1-x Alx )4 are centrosymmetric systems that have recently been identified as candidates to stabilise topologically non-trivial magnetic phases, such as skyrmion lattices. In this Letter, we present a high-resolution resonant x-ray and neutron scattering study on EuGa2Al2 that provides new details of the complex coupling between the electronic ordering phenomena. Our results unambiguously demonstrate that the system orders to form a spin density wave with moments aligned perpendicular to the direction of the propagation vector, and upon further cooling, a cycloid with moments in the ab plane, in contrast to what has been reported in the literature. We show that concomitant with the onset of the spin density wave is the suppression of the charge order, indicative of a coupling between the localised 4f electrons and itinerant electron density. Furthermore we demonstrate that the charge density wave order breaks the four-fold symmetry present in the I4/mmm crystal structure, thus declassifying these systems as square-net magnets. arXiv:2304.07903v1 [cond-mat.str-el] I. INTRODUCTION The formation of skyrmion lattices (skLs) in materials with centrosymmetric symmetry has prompted an interest in the scientific community to pinpoint the mechanisms stabilising skyrmions formation in the absence of Dzyaloshinskii-Moriya exchange (DM) interactions [1][2][3][4][5][6][7]. The absence of dominant sources of anisotropy and the presence of weakly competing exchange interactions is common to many of the centrosymmetric skyrmion hosts, however the exact mechanism that leads to their formation is not yet understood. Thus far several mechanisms have been proposed, such as geometrical frustration [2,3], Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions mediating a four-spin interaction in itinerant electronic square-net systems [8] and more recently magnetic dipolar interactions [9]. Many centrosymmetric skyrmion hosts are intermetallic, and Gd-based, for instance GdPd 2 Si 3 [10,11] Gd 3 Ru 4 Al 12 [12] and GdRu 2 Si 2 [13,14]. More recently members of the Eu(Ga 1x Al x ) 4 series have been identified as candidate skyrmion hosts [15]. For instance the end member of this series, EuAl 4 , develops two different skLs under an applied magnetic field [16]. Furthermore, unique to the Eu(Ga 1x Al x ) 4 series for x = 0.5 and 1 is the development of a charge density wave (CDW) in zero field. EuGa 2 Al 2 , the focus of this Letter, is electronically and structurally similar to EuAl 4 and GdRu 2 Si 2 ; it is a rare-earth intermetallic that is expected to mediate magnetic exchange via long range RKKY interactions, the magnetic ions Gd 3+ and Eu 2+ are isoelectronic with spin only moments (L = 0, J = 7/2) [13,16,17]. Furthermore all three materials have been found to crystallise with I4/mmm symmetry, where the magnetic ions form square nets in the ab plane, which are not expected to support any geometrical frustration, and that are coupled along c to form a three dimensional network of magnetic ions. The tetragonal symmetry of the crystal structure is thought to allow for the formation of multiple magnetic modulation vectors, that may then develop a double-Q square skyrmion lattice [13]. A topological hall effect analysis of the Hall resistivity in EuGa 2 Al 2 suggests that a non-coplanar spin texture is stabilised when a magnetic field between 1.2 T < H < 1.8 T is applied parallel to the c axis, and for temperatures below ∼ 7 K, hinting at the existence of a topologically non-trivial magnetic phase [15]. Despite its electronic and structural similarity to EuAl 4 and GdRu 2 Si 2 , EuGa 2 Al 2 orders with different magnetic and electronic structures in zero field [18] [16,19], indicating it has different underlying electronic interactions. EuGa 2 Al 2 develops a CDW below T CDW = 50 K [15], and three distinct magnetic phases below T 1 = 19.5 K, T 2 = 15 K and T 3 = 11 K, labelled the AFM1, AFM2 and AFM3 phases respectively [15,17]. Building a theoretical model that can describe the formation of skyrmion lattices in the absence of DM exchange necessitates an accurate experimental determination of the electronic ordering phenomena in such systems and across their phase diagram. To this end, we performed neutron powder diffraction (NPD) and resonant x-ray scattering (RXS) experiments on high-quality single crystals of EuGa 2 Al 2 that demonstrates the following. Firstly the onset of the CDW order broke the fourfold symmetry that was present in the I4/mmm crystal structure, and stabilised orthorhombic domains with either Immm(0, 0, g)s00) or F mmm(0, 0, g)s00) symme-try. Secondly the onset of the magnetic order below T 1 suppressed the CDW order, indicating the two electronic order parameters were in competition. Thirdly, we rigorously demonstrate that EuGa 2 Al 2 formed a SDW with moments oriented perpendicular to the direction of the propagation vector in the AFM1 phase and a cycloid with moments in the ab plane in the AFM3 phase, in contrast to what has been reported in the literature [15]. In the I4/mmm symmetry the Eu 2+ ions, Wyckoff position 2a, form two-dimensional square layers in the ab plane, and neighbouring Eu layers, which are separated along c, are translated by ( 1 2 , 1 2 , 1 2 ) owing to the I-centering that relates them. The Al and Ga ions sit between neighbouring Eu layers and are ordered across Wyckoff positions 4d and 4e, where Al fully occupies the 4d site and Ga the 4e site [17]. Single crystals of EuGa 2 Al 2 were grown in accordance with Ref. 15 and 17. RXS measurements were performed on the I16 beamline at Diamond Light Source [20]. An as grown 1 × 1 × 0.5 mm 3 single crystal sample was fixed onto a Cu sample holder, and mounted onto a six-circle kappa goniometer. The (00l) was specular, and the (0k0) direction was used as an azimuthal reference. The incident energy was tuned to 6.97 keV, the Eu L 3 edge. Further details of the experimental set-up are given in the Sec. S1 of the Supplemental Material (SM). At room temperature the crystal was indexed using the published I4/mmm space group [17]. As the sample was cooled a number of distinct changes to the crystal structure were observed. Below 50 K satellite peaks appeared that were indexed with propagation vector k = (0,0,τ ), τ ∼ 0.125, which were identified to originate in a CDW, consistent with reports in the literature [15]. Owing to the high reciprocal space resolution provided by the I16 beamline it was possible to observe for the first time subtle changes to the crystal structure as we cooled through the CDW and magnetic phase transitions. We observed an elongation of the structural Bragg reflections in the hk plane below T CDW , indicating a loss to the four-fold symmetry that was present at 300 K. The possible subgroups of the displacive representation of the incommensurate CDW propagation vector, k = (0,0,τ ), consistent with a loss to the four-fold symmetry, are the orthorhombic space groups Immm(0, 0, g)s00 and F mmm(0, 0, g)s00. Each of the two orthorhombic subgroups would produce a distinct splitting of (1,1,l) type reflections, as illustrated in Sec. S2 of the SM. For instance Fig. 1(a) shows the (1,1,8), which begins to split close to the (1,1,0) direction as the temperature approaches T CDW , consistent with the presence of domains of Immm(0, 0, g)s00 symmetry. Upon moving to different positions on the sample we observed a splitting of the (1,1,8) that was consistent with the presence of both orthorhombic domains, Sec. S2 of the SM. This finding demonstrates that EuGa 2 Al 2 is not a square-net magnet, similar to EuAl 4 [21]. Furthermore it implies the symmetry of the magnetically ordered phases cannot be higher than monoclinic and each orthorhombic domain could further split into two magnetic domains. Below T 1 we measured several magnetic reflections that indexed with propagation vector, k = (α,0,0) or k = (0,β,0), where α and β varied between 0.196 and 0.188 depending on the temperature, Fig. 2. Sudden changes to these satellite reflections appeared at T 1 , T 2 and T 3 , confirming the presence of the magnetically ordered phases previously reported in the literature [15], Fig. 2(a). The α and β satellites were observed at a position in the sample representative of a single orthorhombic domain, demonstrating that the orthorhombic domains, of which there can be four, were further split into two magnetic domains. No intensity was observed at reflections where h + k + l = 2n + 1, n ∈ Z, nor α and β magnetic satellites centred about them, implying the Icentring relating the Eu1 and Eu2 ions was not broken in any of the three magnetic phases, as shown in Fig. S5 of the SM. We describe the temperature dependent changes observed for the (α,0,8) and (0,β,8) magnetic satellites belonging to a single orthorhombic domain measured upon warming. For the (α,0,8) reflection a single peak, labelled α 3 , was observed in the AFM3 phase, Fig. 2(b). The intensity of this peak decreased to zero as the system was warmed to the AFM1 phase. At T 2 , a second peak labelled α 1 appeared, such that the α 1 and α 3 peaks coexisted over a temperature interval of 0.5 K. The intensity of the α 1 peak was maximised at T 2 before steadily decreasing to zero as the system was warmed to the paramagnetic phase. A similar dependence was observed for (0,β,8), Fig. 2(b), with the exception that the coexistence of the β 1 and β 3 peaks occurred over a wider temperature interval of 4 K. This suggests that the two peaks, α 1 /β 1 and α 3 /β 3 are representative of the AFM1 and AFM3 phases respectively, and originate in two competing magnetic phases. Note that the difference in the coexistence region of the two magnetic domains can depend on microscopic characteristics such as their size. We also followed the evolution of the propagation vector and intensity of the (1,1,8+τ ) CDW reflection through the magnetic phase transitions to establish the coupling between these phenomena. The intensity of the CDW almost halved, while the value of τ steadily increased between T 1 and T 2 , indicating that it was competing with the magnetic order. Note that the temperature dependence of over thirty different CDW reflections were collected as the crystal was warmed through the magnetically ordered phases, shown in Sec. S2 of the SM, which all showed the same behaviour. The electrons that give rise to the CDW are itinerant and expected to originate from the Al ions [22], while the electrons responsible for the magnetic order are the localised 4f electrons on the Eu sites. If the onset of the magnetic order polarises the itinerant electronic density that gives rise to the CDW, it is feasible that that it could destabilise the CDW order and thus cause its suppression. Indeed theoretically it has been shown that the spin of the conduction electrons tends to align with the underlying local moment texture in such itinerant magnets [4]. Below T 2 we also observed a change to the relative intensity of different CDW reflections, Fig. 1(b), indicating the structure of the CDW is changing. We note that for EuAl 4 , a CDW with orthorhombic symmetry with a similar propagation vector, k = (0, 0, 0.1781(3))) was observed, where the CDW order was not thought to originate in a simple nesting of the Fermi surface [21,22]. Using magnetic symmetry analysis we identified nine different magnetic structures that the system could adopt, shown in Sec. S3 of the SM. The magnetic structures differ according to the moment direction adopted by the Eu ions, and also according to whether they are collinear (SDW) or non-collinear (helix and cycloid). As the resonant magnetic x-ray scattering (RMXS) cross section for the σπ channel is dependent on the dot product between the incident wavevector,k i , and the magnetic interaction vector the moment direction can be obtained by rotating the crystal relative to these vectors and measuring the scattered signal (azimuthal scan). can be used to determine if the magnetic structure is non-collinear, as explained in Sec. S4 of the SM. The difference in the scattered intensity between the CR and CL light was zero across the majority of azimuthal values measured, Fig. 3(c), indicating the AFM1 phase was collinear. The best fit to the azimuthal dependencies collected on the (±α,0,8) and (0,±β,8) reflections measured with all four incident polarisation of light, details of which are given in Sec. S4 of the SM and shown for the (0,β,8) in Fig. 3, was a SDW with moments aligned perpendicular to the direction of the propagation vector, consistent with the observations above. This magnetic structure solution transforms by a single irrep, which is consistent with the PM to AFM1 phase transition being second order in nature. The adoption of the moments perpendicular to the direction of the propagation vector is similar to that observed for EuAl 4 [23] and Gd 2 PdSi 3 [9], suggesting an anisotropy term that may be common to many of the centrosymmetric skyrmion hosts. At 7 K, in the AFM3 phase, we found that the magnetic structure of the α and β magnetic domains was non-collinear, owing to a difference in the magnetic intensity measured with CR and CL light, as shown for the (0,β,8) in Fig. 4(a). A determination of moment direction using an azimuthal scan was not possible owing to the appearance of additional peaks that changed significantly with sample position, and hence azimuth. Given that a non-collinear magnetic structure would split each magnetic domain into a further two magnetic domains related by inversion symmetry, the appearance of the additional peaks was likely owing to the now quite complex domain structure. As such we employed the use of NPD to determine the ground state magnetic ordering. Several single crystal samples of EuGa 2 Al 2 produced from the same growth were finely crushed to produce a 0.95 g polycrystalline sample, which was measured using the time-of-flight diffractometer WISH at ISIS [24]. The NPD data shown in Fig. 4(c) was collected at 1.5 K, deep into the AFM3 phase, and a Rietveld refinement of the data conclusively showed that the magnetic structure is a cycloid with moments in the ab plane. Details of the Rietveld refinement are given in Sec. S5 of the SM. The sudden changes to the propagation vector and intensity of the magnetic satellites at T 2 , shown in Fig. 2, is consistent with this phase transition being first order in nature. We question whether there are three distinct magnetically ordered phases, given that the AFM2 phase appears to be a coexistence of the AFM1 and AFM3 phases, Fig. 2(a). Furthermore a region of phase coexistence is common following a first order phase transition. We suggest that the signature of the phase transition in the specific heat capacity at T 3 reported in Ref. 17 may be related to the onset of a structural phase transition as indicated by small changes to the relative intensity of the CDW reflections observed between T 2 and T 3 , Fig. 1, indicating a possible rearrangement of the CDW structure. While the change to the magnetic susceptibility at T 3 , Ref. 17, may be related to a change in the magnetic domain pattern caused by a complete suppression of the AFM1 phase and the creation of inversion domains. We propose that if the CDW structure does transform below T 2 it may cause a change to the itinerant electronic density, which in turn may modify the RKKY interactions, proving a possible mechanism by which the cycloid m||ab plane is stabilised. In conclusion our study of EuGa 2 Al 2 has shown that the onset of the CDW order breaks the four-fold symmetry present in the I4/mmm crystal structure stabilising orthorhombic domains, which demonstrates EuGa 2 Al 2 is not a square-net magnet. We find that the single crystal samples of EuGa 2 Al 2 are composed of magnetic domains described by propagation vector (α,0,0) or (0,β,0), which order in the AFM1 phase by a SDW with moments perpendicular to the direction of the propagation vector. We observed a suppression of the CDW order as the system ordered to form the SDW, suggesting the two electronic ordering phenomena were in competition, which in turn implies that the localised 4f electrons and itinerant electronic density are coupled. Finally our results show that the ground state magnetic structure was a cycloid with moments in the ab plane. Our findings map the zero-field magnetic and structural phases of EuGa 2 Al 2 , revealing more complexity than was previously discovered, and demonstrating the requirement for high-resolution scattering studies to elucidate the true nature of the complex ordering present in such candidate centrosymmetric skyrmion hosts. FIG. 1 . 1RXS data collected on (a) the (1,1,8) charge reflection between 300 K and 7 K. (b) the (0,1,5+τ ) and (2,0,8+τ ) CDW reflections between 7 K and 60 K. These data were collected using incident σ polarised light. FIG. 2 . 2(a) Projection of the (1,1+β,8) reflection in the hk plane collected in the PM, AFM1, AFM2 and AFM3 phases. Temperature dependence of (b) the normalised integrated intensity and (c) propagation vectors of the magnetic satellites representative of the AFM3 phase, (α3,0,8) & (0,β3,8), and the AFM1 phase, (α1,0,8) & (0,β1,8). (d) Temperature dependence of the normalised integrated intensity and propagation vector of the (1,1,8+τ ) CDW reflection. These data were collected using incident σ polarised light. FIG. 3 . 3Azimuthal scans collected in (a) πσ + ππ channels (b) σπ channel and (c) the CRσ + CRπ and CLσ + CLπ channels for the (0,β,8) reflection. The lines represent the simulated azimuthal dependencies for the different magnetic structure solutions. Details of the simulations are presented in Sec. S4 of the SM. The unfilled markers represent the normalised data. Fig. 3 3shows the azimuthal scan collected on the (0,β,8) in the σπ channel at 17 K in the AFM1 phase. Maximums in the intensity of the scattered signal were measured at ψ = 90°and ψ = -90°, whenk i was parallel to the a-axis, therefore a component of the magnetic moment was oriented along a. Furthermore as the diffracted intensity was zero at ψ of 0, 180°, and -180°, a b or c axis component to the magnetic moment was not present. Measurements with incident circular lightFIG. 4. (a) RMXS data collected on (0,β,8) magnetic satellite with CR and CL incident light. (b) Subtracted NPD data at 1.5 K (25 K data was subtracted) showing only magnetic reflections. Data is given by the unfilled red circles, the fit to the data using a circular cycloid with moments in the ab plane by the solid black line, and their difference by the solid blue line. The green ticks mark the position of the magnetic reflections in d-spacing. The inset shows the subtracted data fit using a cycloid with moments in the ac plane. (c) Experimentally determined magnetic structure of EuGa2Al2 in the AFM1 and AFM3 phases projected in the ab plane. The black arrows represent the moments on the Eu ions, and the Ga and Al ions are denoted by the circles. The I4/mmm crystallographic unit cell is shown by the solid black line. 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{'abstract': 'Eu(Ga1-x Alx )4 are centrosymmetric systems that have recently been identified as candidates to stabilise topologically non-trivial magnetic phases, such as skyrmion lattices. In this Letter, we present a high-resolution resonant x-ray and neutron scattering study on EuGa2Al2 that provides new details of the complex coupling between the electronic ordering phenomena. Our results unambiguously demonstrate that the system orders to form a spin density wave with moments aligned perpendicular to the direction of the propagation vector, and upon further cooling, a cycloid with moments in the ab plane, in contrast to what has been reported in the literature. We show that concomitant with the onset of the spin density wave is the suppression of the charge order, indicative of a coupling between the localised 4f electrons and itinerant electron density. Furthermore we demonstrate that the charge density wave order breaks the four-fold symmetry present in the I4/mmm crystal structure, thus declassifying these systems as square-net magnets. arXiv:2304.07903v1 [cond-mat.str-el]', 'arxivid': '2304.07903', 'author': ['Al \nAppleton Laboratory-STFC\nISIS facility\nOX11 0QXRutherford, Chilton, DidcotUnited Kingdom\n', 'A M Vibhakar \nDiamond Light Source Ltd\nHarwell Science and Innovation Campus\nOX11 0DEDidcotOxfordshireUnited Kingdom\n\nAppleton Laboratory-STFC\nISIS facility\nOX11 0QXRutherford, Chilton, DidcotUnited Kingdom\n', 'D D Khalyavin \nAppleton Laboratory-STFC\nISIS facility\nOX11 0QXRutherford, Chilton, DidcotUnited Kingdom\n', 'J M Moya \nApplied Physics Graduate Program\nSmalley-Curl Institute\nRice University\n77005HoustonTexasUSA\n\nDepartment of Physics and Astronomy\nRice University\n77005HoustonTexasUSA\n', 'P Manuel \nAppleton Laboratory-STFC\nISIS facility\nOX11 0QXRutherford, Chilton, DidcotUnited Kingdom\n', 'F Orlandi \nAppleton Laboratory-STFC\nISIS facility\nOX11 0QXRutherford, Chilton, DidcotUnited Kingdom\n', 'S Lei \nApplied Physics Graduate Program\nSmalley-Curl Institute\nRice University\n77005HoustonTexasUSA\n\nDepartment of Physics and Astronomy\nRice University\n77005HoustonTexasUSA\n', 'E Morosan \nApplied Physics Graduate Program\nSmalley-Curl Institute\nRice University\n77005HoustonTexasUSA\n\nDepartment of Physics and Astronomy\nRice University\n77005HoustonTexasUSA\n', 'A Bombardi \nDiamond Light Source Ltd\nHarwell Science and Innovation Campus\nOX11 0DEDidcotOxfordshireUnited Kingdom\n\nDepartment of Physics\nClarendon Laboratory\nUniversity of Oxford\nParks RoadOX1 3PUOxfordUK\n'], 'authoraffiliation': ['Appleton Laboratory-STFC\nISIS facility\nOX11 0QXRutherford, Chilton, DidcotUnited Kingdom', 'Diamond Light Source Ltd\nHarwell Science and Innovation Campus\nOX11 0DEDidcotOxfordshireUnited Kingdom', 'Appleton Laboratory-STFC\nISIS facility\nOX11 0QXRutherford, Chilton, DidcotUnited Kingdom', 'Appleton Laboratory-STFC\nISIS facility\nOX11 0QXRutherford, Chilton, DidcotUnited Kingdom', 'Applied Physics Graduate Program\nSmalley-Curl Institute\nRice University\n77005HoustonTexasUSA', 'Department of Physics and Astronomy\nRice University\n77005HoustonTexasUSA', 'Appleton Laboratory-STFC\nISIS facility\nOX11 0QXRutherford, Chilton, DidcotUnited Kingdom', 'Appleton Laboratory-STFC\nISIS facility\nOX11 0QXRutherford, Chilton, DidcotUnited Kingdom', 'Applied Physics Graduate Program\nSmalley-Curl Institute\nRice University\n77005HoustonTexasUSA', 'Department of Physics and Astronomy\nRice University\n77005HoustonTexasUSA', 'Applied Physics Graduate Program\nSmalley-Curl Institute\nRice University\n77005HoustonTexasUSA', 'Department of Physics and Astronomy\nRice University\n77005HoustonTexasUSA', 'Diamond Light Source Ltd\nHarwell Science and Innovation Campus\nOX11 0DEDidcotOxfordshireUnited Kingdom', 'Department of Physics\nClarendon Laboratory\nUniversity of Oxford\nParks RoadOX1 3PUOxfordUK'], 'corpusid': 258180207, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8943, 'n_tokens_neox': 7673, 'n_words': 4457, 'pdfsha': '033c8859fd6da6f8d2d3ac88b551071da29e9321', 'pdfurls': ['https://export.arxiv.org/pdf/2304.07903v1.pdf'], 'title': ['Competing charge and magnetic order in the candidate centrosymmetric skyrmion host EuGa', 'Competing charge and magnetic order in the candidate centrosymmetric skyrmion host EuGa'], 'venue': []}

arxiv

Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics 4 Jan 2012 Xiuxiong Chen Song Sun Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics 4 Jan 2012 We prove that constant scalar curvature Kähler metric "adjacent" to a fixed Kähler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T picture described by Donaldson. We prove that the Calabi flow near a cscK metric exists globally and converges uniformly to a cscK metric in a polynomial rate. Viewed in fixed a Kähler class, the Calabi flow is also shown to be asymptotic to a smooth geodesic ray at infinity. This latter fact is also interesting in the finite dimensional case, where we show that the downward gradient flow of the Kempf-Ness function in a semi-stable orbit is asymptotic to the direction of optimal degeneration. Introduction The Kempf-Ness theorem relates complex quotient to symplectic reduction. Suppose a compact connected group G acts on a compact Kähler manifold X. We assume the action preserves the Kähler structure, with a moment map µ : X → g * . Then the action extends to a holomorphic action of the complexified group G C . Under proper hypothesis the notion of stability could be defined. Then the Kempf-Ness theorem says that as sets: X ss /G C ≃ µ −1 (0)//G. To be more precise, (1). A G C -orbit is poly-stable if and only if it contains a zero of the moment map. The zeroes within it form a unique G orbit. (2). A G C -orbit is semi-stable if and only if its closure contains a zero of the moment map. We call such a zero a de-stabilizer of the original G C orbit. The de-stabilizers all lie in the unique poly-stable orbit in the closure of the original orbit. In Kähler geometry according to S. Donaldson([D1])(see also [Fu]) the problem of finding cscK (constant scalar curvature Kähler) metrics formally fits into a similar picture. However the spaces involved are infinite dimensional. Given a compact Kähler manifold (M, ω, J), denote by G the group of Hamiltonian diffeomorphisms of (M, ω) and by J the space consists of almost complex structures on M which are compatible with ω. J admits a natural Kähler structure which is invariant under the action of G. The moment map is given by the Hermitian scalar curvature. The complexification of G may not exist, since G is infinite dimensional. Nevertheless, it still makes sense talking about the orbits of G C -it is simply the leaf of the foliation obtained by complexifying the infinitesimal actions of G. Then the G C leaf of an integrable complex structure can be viewed as a principal G-bundle over the Kähler class [ω]. Thus an analogue to the Kempf-Ness theorem should relate the stability of the leaves to the existence of cscK metrics in the corresponding Kähler class. This was made more precise as the Yau-Tian-Donaldson conjecture(see [Th]). The notion of "stability" in this case is the so-called "K-stability", see [Ti1], [D5]. There are also other related notion of stability, see for example [RT], [Pa], etc. Note that the Kempf-Ness theorem consists of both the existence and uniqueness part. It is known that the existence of cscK metrics implies various kinds of stability, however the converse is fairly difficult, due to the appearance of fourth order non-linear P.D.E's. Recently Donaldson([D6]) proved a general result that the conjecture is true for toric surfaces. The uniqueness part corresponding to the poly-stable case is known by Theorem 1.1. (Donaldson[D3], Chen-Tian [CT]) Constant scalar Kähler metric in a fixed Kähler class, if exists, is unique up to holomorphic isometry. structures on a fixed manifold is in general not Hausdorff. As a simple example, we can consider the blown-up of P 2 at three points p 1 , p 2 , and p 3 . The underlying differential manifold is fixed, and a choice of the three points defines a complex structure. A choice of three points in a general position gives rise to the same complex structure, while a choice of three points on a line provides an example of an adjacent complex structure. It follows theorem 1.3 that Theorem 1.6. Let (M, ω, J) be a Kähler manifold. Assume [ω] is integral. Suppose there are two csc Kähler structures (ω 1 , J 1 ) and (ω 2 , J 2 ) both adjacent to the Kähler class of (ω, J), then they are isomorphic. Corollary 1.7. Let (M, J) be a Fano manifold. Suppose there are two complex structures J 1 and J 2 both adjacent to J and both admitting Kähler-Einstein metrics, then (M, J 1 ) and (M, J 2 ) are bi-holomorphic. Remark 1.8. After finishing this paper, we learned that our theorem 1.6 and corollary 1.7 partially confirmed a conjecture of G. Tian([Ti2]) in the case of constant scalar curvature Kähler metric. The main technical ingredient in the proof of the above theorems is to obtain some C 0 bound. We shall study the asymptotic behavior of the Calabi flow near a cscK metric. The global existence and convergence are established by using the Lojasiewicz inequality which controls the gradient of a real analytic function near a critical point. Suppose now we have two cscK metrics adjacent to a fixed Kähler class, then there are two Calabi flows in the neighborhoods of the corresponding cscK metrics. Since the Calabi flow decreases geodesic distance, we get a bound on the two Calabi flows in terms of geodesic distance. It is not known whether this bound implies C 0 bound automatically. Here we get around this difficulty by showing that the previous Calabi flow is asymptotic to a smooth geodesic ray. This involves a local study of the infinite dimensional Hamiltonian action of G, which is the main technical part of this paper. We shall first look at the analogous finite dimensional problem. Finally we are able to derive C 0 bound for the two parallel geodesic rays. The organization of this paper is as follows. In section 2, we review Donaldson's infinite dimensional moment map picture in Kähler geometry, and recall some known results for our later use. In section 3, we state the Lojasiewicz inequality and "Lojasiewicz arguments" for the gradient flow of a real analytic function. In section 4, we prove that in the finite dimensional case, the Kempf-Ness flow for a semi-stable point is asymptotic to a rational geodesic ray. In section 5, we study the stability of the Calabi flow near a cscK metric when the complex structure is deformed. In section 6, we generalize the arguments in section 4 to the infinite dimensional setting by considering the "reduced" Calabi flow. In section 7, the relative C 0 bound for two smooth parallel geodesic rays tamed by bounded geometry is derived. In section 8, we prove the main theorems. In Section 9, we shall discuss some further problems related to this study. The appendix contains the proof of the technical lemmas used in sections 4 and 6. Acknowledgements: This paper was essentially finished in the October of 2009 during a conference in honor of Simon Donaldson at Northwestern University. With admiration, we want to dedicate this modest paper to him for his teaching of Kähler geometry to the first author in the last 12 years. Part of this work was done while both authors were visiting Stony Brook. We wish to thank both the department of Mathematics and the Simons Center for Geometry and Physics for their generous hospitality. We also thank Professors Blaine Lawson, Claude Lebrun, and Gang Tian for their interest in this work. The second author would also like to thank Joel Fine, Sean Paul and Zhan Wang for interesting discussions. Both authors are partially supported by an NSF grant. The space of Kähler structures Here we review the infinite dimensional moment map picture discovered by Fujiki([Fu]) and Donaldson([D1]). Let (M, ω, J 0 ) be a compact Kähler manifold. Denote by J the space of almost complex structures on M which are compatible with ω, and by J int the subspace of J consisting of integrable almost complex structures compatible with ω. Then J is the space of smooth sections of an Sp(2n)/U (n) bundle over M , so it carries a natural Kähler structure. Indeed, there is a global holomorphic coordinate chart if we use the ball model of the Siegel upper half space in the usual way. J 0 determines a splitting T M ⊗ C = T 1,0 ⊕ T 0,1 such that ω induces a positive definite Hermitian inner product on T 1,0 , then J could be identified with the space Ω 0,1 S (T 1,0 ) = {µ ∈ Ω 0,1 (T 1,0 )|A(µ) = 0, Id −μ • µ > 0}, where A is the composition Ω 0,p (T 1,0 ) → Ω 0,p (T * 0,1 ) → Ω 0,p+1 . An element µ corresponds to an almost complex structure J whose corresponding (1, 0) tangent space consists of vectors of the form X −μ(X)(X ∈ T 1,0 ). J int is a subvariety of J cut out by quadratic equations: N (µ) =∂µ + [µ, µ] = 0. Denote by G the group of Hamiltonian diffeomorphisms of (M, ω). Its Lie algebra is C ∞ 0 (M ; R). G will be the infinite dimensional analogue of a compact group, though the exponential map is not locally surjective for G. G acts naturally on J , keeping J int invariant. A. Fujiki [Fu] and S. Donaldson([D1]) independently discovered that the G action has a moment map given by the Hermitian scalar curvature functional S − S 1 , which can be viewed as an element in (C ∞ 0 (M ; R)) * through the L 2 inner product with respect to the measure dµ = ω n . When J is integrable S(J) is simply the Riemannian scalar curvature of the Riemannian metric induced by ω and J. We say J 0 ∈ J is cscK if J 0 is integrable and (ω, J 0 ) has constant scalar curvature. So in the symplectic theory we are naturally lead to consider cscK metrics. In the complex story, we need to look at G C . Since G is infinite dimensional, there may not exist a genuine complexification G C . Nevertheless, we can still define the G C leaf of an integral complex structure J 0 , as follows. The infinitesimal action of G at a point J ∈ J is given by D J : C ∞ 0 (M ; R) → Ω 0,1 S (T 1,0 ); φ →∂ J X φ . This operator can be naturally complexified to an operator from C ∞ 0 (M ; C) = C ∞ 0 (M ; R) ⊕ √ −1C ∞ 0 (M ; R) to Ω 0,1 S (T 1,0 ) . Then a complex structure J is on the G C leaf of J 0 if there is a smooth path J t ∈ J int such thatJ t lies in the image of D Jt . G acts on the leaf naturally and the quotient is the space of Kähler metrics cohomologous to [ω] J 0 . So the latter could be viewed as "G C /G". We define the space of Kähler potentials H = {φ ∈ C ∞ (M ; R)|ω + √ −1∂∂φ > 0}. Then H/R is formally the "dual" symmetric space of G. This was made more precise by Mabuchi([M1]), Semmes([Se]) and Donaldson([D2]). Define a Weil-Petersson type Riemannian metric on H by (ψ 1 , ψ 2 ) φ = M ψ 1 ψ 2 dµ φ for ψ 1 , ψ 2 ∈ T φ H. It can be shown that the Riemannian curvature tensor is co-variantly constant and the sectional curvature is non-positive. A path φ(t) in H is a geodesic if it satisfies the equation φ(t) − |∇ φ(t)φ (t)| 2 φ(t) = 0. The first author( [Ch1]) proved the existence of a unique C 1,1 geodesic connecting any two points in H, and consequently that H is a metric space with the distance given by the length of the C 1,1 geodesics. It is proved in [CC] that under this metric H is non-positively curved in the sense of Alexanderov. So far the best regularity for the Dirichlet problems of the geodesic equation was obtained by Chen-Tian([CT]). The initial value problem for the geodesic equation is in general not well-posed. But by the non-positiveness of the curvature of H, there should be lots of geodesic rays in H. In [Ch3], the first author proved the following general theorem which we shall use later: Theorem 2.1. Given a smooth geodesic ray φ(t) in H which is tamed by a bounded geometry, there is a unique relative C 1,1 geodesic ray ψ(t) emanating from any point ψ in H such that |φ(t) − ψ(t)| C 1,1 ≤ C. Remark 2.2. For the precise definition of "tameness" we refer to [Ch3]. But we point out that this is merely a technical condition imposed on the behavior of φ(t) at infinity so that the analysis on non-compact manifolds work. In our later applications where the geodesic ray φ(t) arises naturally from a test configuration with smooth total space, this assumption is always satisfied. Defintion 2.3. Two geodesic rays φ(t) and ψ(t) in H are said to be parallel if d H (φ(t), ψ(t)) ≤ C. Hence it is clear by definition that if |φ(t) − ψ(t)| C 0 ≤ C, then φ and ψ are parallel. Analogous to the finite dimensional Kempf-Ness setting, there is a relevant functional E defined on H, called the Mabuchi K-energy. It is the anti-derivative of the following closed one-form: dE φ (ψ) = − M (S(φ) − S)ψdµ φ .(1) So the norm square of the gradient of E is the Calabi energy: Ca(φ) = M (S(φ) − S) 2 dµ φ . By a direct calculation, along a smooth geodesic φ(t), we have d 2 dt 2 E(φ(t)) = M |D tφ (t)| 2 dµ φ(t) ≥ 0. According to [Ch2], E can be extended to a continuous function on all C 1,1 potentials in H. However, it is not clear why E is still convex. The first author proved some weak versions of convexity. In the case when [ω] is integral, we gave simplified proofs in [CS] using quantization(See also [Be]). We recall them for our later purpose. Lemma 2.4. ( [Ch3], [CS]). Given any φ 0 , φ 1 ∈ H, we have Ch3], [CS]) Given any φ 0 , φ 1 ∈ H, let φ(t) be the C 1,1 geodesic connecting them. Then the derivatives of E(φ(t)) at the end-points are well-defined and they satisfy the following inequality: E(φ 1 ) − E(φ 0 ) ≤ Ca(φ 1 ) · d(φ 0 , φ 1 ). Lemma 2.5. ([d dt | t=0 E(φ(t)) ≤ d dt | t=1 E(φ(t)) . This lemma implies that Lemma 2.6. ( [CC]) The Calabi flow on H decreases geodesic distance. Lojasiewicz inequality In this section we recall Lojasiewicz's theory for the structure of a real analytic function. The following fundamental structure theorem for real analytic functions is well-known: Theorem 3.1. (Lojasiewicz inequality) Suppose f is a real analytic function defined in a neighborhood U of the origin in R n . If f (0) = 0 and ∇f (0) = 0, then there exist constants C > 0, and α ∈ [ 1 2 , 1), and shrinking U if necessary, depending on n and f , such that for any x ∈ V , it holds that |∇f (x)| ≥ C · |f (x)| α .(2) This type of inequality is crucial in controlling the behavior of the gradient flow. If α = 1 2 , then we get exponential convergence. If α > 1 2 , then we can obtain polynomial convergence: Corollary 3.2. Suppose f is a non-negative real-analytic function defined in a neighborhood U of the origin in R n with f (0) = 0. Then there exists a neighborhood V ⊂ U of the origin such that for any x 0 ∈ V , the downward gradient flow of f : d dt x(t) = −∇f (x(t)), x(0) = x 0 . converges uniformly to a limit x ∞ ∈ U with f (x ∞ ) = 0. Moreover, we have the following estimate: 1. f (x(t)) ≤ C · t − 1 2α−1 ; 2. d(x(t), x(∞)) ≤ C · t − 1−α 2α−1 , where we assume the Lojasiewicz exponent α > 1 2 . Proof. The proof is quite standard, and we call it "Lojasiewicz arguments" for later reference. Denote V δ = {x ∈ R n ||x| ≤ δ}, and fix δ > 0 small so that inequality (2) holds for x ∈ V δ . In our calculation the constant C may vary from line to line. If x(t) ∈ V δ for t ∈ [0, T ] , then we compute d dt f 1−α (x(t)) = −(1 − α) · f −α (x(t)) · |∇f (x(t))| 2 ≤ −C · |ẋ(t)|, thus for any T > 0, T 0 |ẋ(t)|dt ≤ 1 C · f 1−α (x 0 ). For any ǫ ≤ δ 2 small, we choose δ 2 ≤ δ small such that f (x) ≤ (C · ǫ) 1 1−α for x ∈ V δ 2 , and δ 1 = min{ǫ, δ 2 }, then the flow initiating from any point x 0 ∈ V δ 1 will stay in V 2ǫ . So the Lojasiewicz inequality holds for all x(t). Now d dt f 1−2α (x(t)) = −(1 − 2α) · f −2α (x(t)) · |∇f (x(t))| 2 ≥ (2α − 1) · C 2 , so f (x(t)) ≤ C · t − 1 2α−1 . For any T 1 ≤ T 2 , we get d(x(T 1 ), x(T 2 )) ≤ T 2 T 1 |ẋ(t)|dt ≤ C · T 1 − 1−α 2α−1 . Therefore we obtain polynomial convergence and the required estimates. Finite dimensional case 4.1 Kempf-Ness theorem Let (M, ω, J) be a Kähler manifold and assume there is an action of a compact connected group G on M which preserves the Kähler structure. Let µ be the corresponding moment map. This induces a holomorphic action of the complexified group G C . Then the Kempf-Ness theorem relates the complex quotient by G C to the symplectic reduction by G( [DK]). Theorem 4.1. (Kempf-Ness)A G C -orbit contains a zero of the moment map if and only if it is poly-stable. It is unique up to the action of G. A G C -orbit is semi-stable if and only if its closure contains a zero of the moment map; this zero is in the unique poly-stable orbit in the closure of the original orbit. In this paper we are only interested in the uniqueness problem. We will first give a proof in the finite dimensional case, using an analytic approach. An essential ingredient in the proof of the Kempf-Ness theorem is the existence of a function E, called the Kempf-Ness function. Given a point x ∈ M , one can define a one-form α on G C as: α g (R g ξ) = − µ(g.x), Jξ , where R g is the right translation by g and ξ ∈ g C . It is easy to check that α is closed and invariant under the left G-action. Then α is the pull back of a closed one-formᾱ from G C /G. It is well known that G C /G is always contractible, so α gives rise to a function E, up to an additive constant. Notice if the G action is linearizable, this coincides with the usual definition given by the logarithm of the length of a vector on the induced line bundle. It is a standard fact that E is geodesically convex, i.e.ᾱ is monotone along geodesics in G C /G. The critical points of E consist exactly of the zeroes of µ in the given G C orbit. So any G C orbit contains at most one zero of the moment map, up to the action of G. In the semi-stable case, we consider the function f (x) = |µ(x)| 2 on M , and its downward gradient flow x(t). The flow line is tangent to the G C orbit and the induced flow in G C /G is exactly the downward gradient flow of E. We call either flow the Kempf-Ness flow. As we will see more explicitly later, a theorem of Duistermaat( [Le]) says that for x(0) close to a zero of µ, the flow x(t) converges polynomially fast to a limit in µ −1 (0). Now suppose x is semi-stable, and x 1 , x 2 are two poly-stable points in G C .x. W.L.O.G, we can assume µ(x 1 ) = µ(x 2 ) = 0. Take y 1 , y 2 ∈ G C .x such that y i is close to x i . Then the gradient flows x i (t) converges to a point z i ∈ µ −1 (0) near x i . Denote by γ i (t) the corresponding flow in G C /G. Since the gradient flow of a geodesically convex function decreases the geodesic distance, d(γ 1 (t), γ 2 (t)) is uniformly bounded. By compactness, we conclude that z 1 and z 2 must be in the same G C orbit and by the uniqueness in the poly-stable case, we see that z 1 and z 2 must lie in the same G orbit. By choosing y i arbitrarily close to x i , we conclude that x 1 and x 2 are in the same G orbit. The above argument proves the uniqueness of the poly-stable orbits in the closure of a semi-stable orbit. There are technical difficulties to extend this argument to the infinite dimensional setting, due to the loss of compactness. As a result, we need to investigate more about the gradient flow in the γ(t) χ(t) s Figure 1: a curve asymptotic to a geodesic ray finite dimensional case. What we shall show next is that for a semi-stable point, the gradient flow is asymptotic to an "optimal" geodesic ray at infinity. Defintion 4.2. We say a curve γ(t)(t ∈ [0, ∞)) in a simply-connected nonpositively curved space is asymptotic to a geodesic ray χ(t) if for any fixed s > 0, d(γ t (s), χ(s)) tends to zero as t tends to ∞, where γ t is the geodesic connecting χ(0) and γ(t) which is parametrized by arc-length. In other words, χ(t) is the point in the sphere at infinity induced by γ(t) as t → ∞(see figure 1). It follows from the definition that any two geodesic rays χ 1 (t) and χ 2 (t) that are both asymptotic to a given curve γ(t) must be parallel, i.e. d(χ 1 (t), χ 2 (t)) is uniformly bounded. Standard case Let (V, J 0 , g 0 ) be an n dimensional unitary representation of a compact connected Lie group G, so we have a group homomorphism: G → U (n). V is then a representation of the complexified group G C . Denote by Ω 0 the induced Kähler form on V . It is easy to see that the G action always has a moment map µ : V → g * ≃ g, where we have identified g with g * by fixing an invariant metric. It is defined as (µ(v), ξ) = 1 2 Ω 0 (ξ.v, v).(3) For any v ∈ V , denote the infinitesimal action of G at v by L v : g → V ; ξ → ξ.v, then it is easy to see that µ(v) = 1 2 L * v (J 0 v). L v can also be viewed as a map from g C to V , and then µ(v) = − 1 2 JL * v v. Now consider the function f : V → R; v → |µ(v)| 2 , whose downward gradient flow equation is: d dt v = −∇f (v) = −J 0 L v (µ(v)).(4) Since f is a homogeneous polynomial, and thus real analytic, the Lojasiewicz inequality holds for f , i.e. there exists constant C > 0 and α ∈ [ 1 2 , 1), such that for v close to zero, |∇f (v)| ≥ C · |f (v)| α . The previous Lojasiewicz arguments show that for v close to 0, the flow (4) starting from v will converge polynomially fast to a critical point of f . From now on we assume 0 de-stabilizes v, i.e. 0 ∈ G C .v. Thus the gradient flow (4) converges to the origin by the uniqueness in the previous section. Since everything is homogeneous, we can study the induced flow on P(V ). The action of G is then holomorphic and Hamiltonian with respect to the Fubini-Study metric on P(V ), with moment mapμ : P(V ) → g. It is then easy to see thatμ ([v]) = µ(v) |v| 2 . Letf = |μ| 2 , then we can study the downward gradient flow off on P(V ): d ds [v] = −∇f ([v]) = −J 0 L [v] (μ([v])).(5) Let π : V → P(V ) be the quotient map, then clearly π * (∇f (v)) = |v| 2 ∇f ([v]). So the flow (5) is just a re-parametrization of the image under π of the flow (4): if v(t) satisfies (4), then [v(s)] satisfies (5), with ds dt = |v(t)| 2 . Sincê f is also real analytic, the flow [v(s)] converges polynomially fast to a unique limit [v] ∞ . Lemma 4.3.μ ([v] ∞ ) = 0 . Proof. Otherwise [v] is semi-stable with respect to the action of G C on P(V ), thus the corresponding Kempf-Ness function log |g.v| 2 is bounded below on G C . This contradicts the assumption that 0 ∈ G C .v. Thus we know that µ(v(s)) |v(s)| 2 =μ([v] ∞ ) + O(s −γ )(γ > 0) is bounded away from zero when s is large enough. So for t sufficiently large, we have |∇f (v(t))| 4 ≥ C · |f (v(t))| 3 . The Lojasiewicz arguments then ensure that v(t) actually converges to 0 in the order O(t − 1 2 ). So we obtain s ≤ C · log t. Now since the gradient flow of f is tangent to the G C orbit, it can also be viewed as a flow on G C /G. This is given by a path γ(t) = [g(t)], where g(t) ∈ G C satisfiesġ (t)g(t) −1 = −Jµ(g(t).v), and the re-parameterized path corresponding to (5) iṡ g(s)g(s) −1 = −Jμ(g(s). [v]), and d ds γ(s) = −Jμ([v] ∞ ) + O(s −γ ). In the following we shall use the re-parameterized version as | d ds γ(s)| has a lower bound as s → ∞ which makes it more convenient to analyze the asymptotic behavior. Theorem 4.4. γ is asymptotic to a geodesic ray χ in G C /G. Moreover, the direction of γ is conjugate toμ ([v]∞) |μ([v]∞)| under the adjoint action of G. Proof. We already knowγ(s) is getting close toμ([v] ∞ ), but this is not sufficient to conclude that γ is asymptotic to a geodesic ray with direction µ([v] ∞ ). We shall analyze this more carefully, by elementary geometry. First it is easy to see that |γ(s)| = |L * [v](s) L [v](s)μ ([v](s))|, where L [v](s) is the infinitesimal action of g at [v](s). Since [v](s) → [v] ∞ as s → ∞, by corollary 3.2 we get ∞ t |γ(s)|ds ≤ C ∞ t |L [v](s)μ ([v](s))|ds = C ∞ t |∇f (s)|ds ≤ C · t −β , where β = 1−α 2α−1 > 0. Notice that here α is the exponent appearing in the Lojasiewicz inequality forf , not the original f . From the above we know lim s→∞ |γ(s)| = |μ([v] ∞ )| > 0, so if we parameterize γ by arc-length and denote the resulting path by γ(u), then we have |¨ γ(u)| = |γ(s)| −2 |γ(s) − γ(s),γ(s) |γ(s)| 2γ (s)| ≤ C · |γ(s)|. Therefore ∞ t |¨ γ(u)|du ≤ C · t −β , Now for any u > 0, let γ u (v)(v ∈ [0, 1) be the geodesic in G C /G connect- ing γ(0) and γ(u). Denote by L u (v)(v ∈ [0, u]) the distance between γ(v) and γ u (v). Then L u (0) = L u (u) = 0 and a standard calculation of the second variation of length(using the non-positivity of the sectional curvature of G C /G) gives d 2 dv 2 L u (v) ≥ −|¨ γ(v)|. Now define the function f u (v) = v 0 ∞ w |¨ γ(r)|drdw − v u u 0 ∞ w |¨ γ(r)|drdw. Then it is well-defined by the decay of |¨ γ|, and f u (0) = f u (u) = 0 and d 2 dv 2 f u (v) = −|¨ γ(v)|. Thus by maximum principle L u (v) ≤ f u (v) for all u > 0 and v ∈ [0, u]. Fix v we see sup u L u (v) ≤ v 0 ∞ w |¨ γ(r)|drdw ≤ C · v 1−β . Moreover, for any u 2 > u 1 >> 1, by comparison argument the angle between γ u 1 and γ u 2 is bounded by d( γ u 1 (u 1 ), γ u 2 (u 1 ))/u 1 = L u 2 (u 1 )/u 1 , which is controlled by C·u β−1 1 . Thus we conclude that the direction of γ u is converging uniformly to some limit direction and so γ(and thus γ) is asymptotic to a geodesic ray χ starting from γ(0). Now for any s > 0 by the same way we get a geodesic ray χ s starting from γ(s) which is asymptotic to γ. So the rays χ s are all asymptotic to each other and one could easily see that they are all parallel, and thenχ s (0) are all conjugate to each other under the action of G. On the other hand, if we denote by γ s,t (u)(u ∈ [0, 1]) the geodesic connecting γ(s) and γ(t) for s < t, then again by second variation, d dt γ(t) |γ(t)| ,γ s,t (1) |γ s,t (1)| ≥ −C |γ(t)| |γ(t)| ≥ −C|γ(t)|. So we get γ(t) |γ(t)| ,γ s,t (1) |γ s,t (1)| ≥ 1 − t s |γ(u)|du ≥ 1 − C · s −β . We knowγ(t) = Jμ([v] ∞ ) + O(t −α ), and as t → ∞ up to the adjoint action of G we haveγ s,t (1) |γ s,t (1)| →χ(s). So let s → ∞ we seeχ(0) is conjugate toμ ([v]∞) |μ([v]∞)| under the adjoint action of G. From the proof of the above theorem we see that χ(s) also degenerates v to the origin since the path v(t) is of order O(t − 1 2 ) = O(e −C·s ). By Kempf([Ke]) and Ness([Ne]), the directionμ([v] ∞ ) is indeed rational, i.e. it generates an algebraic one-parameter subgroup λ : C * → G C . Moreover, the directionμ([v] ∞ ) is the unique(up to the adjoint action of G) optimal direction for v in the sense of Kirwan([Ki])) and Ness([Ne]). Linear Case Now we suppose G acts linearly on (V = C n , Ω, J 0 ) where J 0 is the standard complex structure on C n and Ω is a real-analytic symplectic form compatible with J 0 . Then the action has a real-analytic moment map µ with µ(0) = 0. µ is not necessarily standard but the Lojasiewicz inequality still holds for f = |µ| 2 . Suppose 0 ∈ G C .v, then the downward gradient flow v(t) of f (v) = |µ(v)| 2 converges to the origin polynomially fast. Letv(t) be the downward gradient flow off (v) = |μ(v)| 2 , whereμ is the moment map for the linearized G action on (V = T 0 V, Ω 0 , J 0 ). By the arguments in the previous section,v(t) converges to zero in the order O(t − 1 2 ) and the corresponding flowγ(t) is asymptotic to a rational geodesic ray χ(t). Let γ(t) in G C /G be the flow corresponding to v(t), we want to show γ(t) is also asymptotic to χ(t). It suffices to bound the distance L(t) between γ(t) andγ(t). Let ψ t (s)(s ∈ [0, 1]) be the geodesic connecting γ(t) andγ(t), then d dt L(t) = 1 L(t) ψ (1),μ(v(t)) − 1 L(t) ψ (0), µ(v(t)) = 1 L(t) ( ψ (1), µ(v(t)) − ψ (0), µ(v(t)) ) + 1 L(t) ψ (1),μ(v(t)) − µ(v(t)) ≤ |μ(v(t)) − µ(v(t))|, where we used the fact that the Kempf-Ness function is geodesically convex. To estimate the last term, notice since the G action is linear, we have for any ξ ∈ g µ(v), ξ = µ(0) + 1 0 d dt µ(tv)dt, ξ = 1 0 Ω tv (ξ.tv, v)dt = 1 2 Ω 0 (ξ.v, v)dt + O(|v| 3 ) = μ(v), ξ + O(|v| 3 ). From the previous secion we knowv(t) = O(t − 1 2 ), so we obtain d dt L(t) ≤ C · t − 3 2 , and so L(t) is uniformly bounded. Therefore, we conclude the following theorem: Theorem 4.5. Suppose G acts Hamiltonian linearly on (V, Ω, J 0 ), with the moment map given by µ. Suppose also a vector v 0 is de-stabilized by the origin. Let v(t) be the downward gradient flow of |µ| 2 emanating from v, then v(t) converges to 0 in the order O(t − 1 2 ). Let γ(t) be the corresponding flow in G C /G, then there exists a geodesic ray χ in G C /G, which is asymptotic to γ. Moreover, χ is rational. General Case In general we need to linearize the problem, using the Marle-Guillemin-Sternberg normal form. Let (M, ω, J, G, µ) be a real analytic Hamiltonian G-action on a real analytic Kähler manifold. Choosing a bi-invariant metric on g we can identify g with g * . Suppose x ∈ M is a zero of µ. Let G 0 be the isotropy group of x and g 0 be its Lie algebra. The bi-invariant product on g allows a G 0 invariant splitting: g = g 0 ⊕ m. Notice g.x ⊂ (g.x) ω . Denote by N the orthogonal complement of g.x ⊕ Jg.x in T x M , then N is G 0 -invariant and the linear G 0 action on N has a canonical moment map µ N : N → g 0 . Let Y = G × G 0 (m × N ), then G acts naturally on Y on the left. Lemma 4.6. (Marle-Guillemin-Sternberg [GS], [OR]) There exists a symplectic form ω defined in a neighborhood U of [e, 0, 0] in Y , under which the G action is Hamiltonian with a moment map given by µ : U → g; [g, ρ, v] → Ad * g (µ N (v) + ρ). There exists a local G equivariant symplectic diffeomorphism Φ : Y → M which respects the moment maps, and satisfies Φ([e, 0, 0]) = x, Φ * J − J 0 = O(r 2 )) on N and Φ * J = J 0 at [e, 0, 0]. Here J 0 is the canonical G-invariant almost complex structure on Y induced by J, which will be more explicit in the proof. Moreover, we can take Φ to be real analytic if everything we start with is so. The only new feature here is the control on the complex structure. The proof of this theorem is a bit technical and will be deferred to the appendix. From now on we will work on (U, Ω 0 , J) where we also denote by J the pullback Φ * J. Theorem 4.7. Suppose y ∈ U is de-stabilized by x, then the Kempf-Ness flow y(t) of |µ| 2 converges to y ∞ ∈ G.x polynomially fast. Moreover the corresponding flow γ(t) in G C /G is asymptotic to a geodesic ray χ(t) which is rational and also degenerates y to y ∞ . Remark 4.8. Here we could define χ(t) as the "optimal" degeneration of y, generalizing the usual definition in the linear case. To prove the theorem, we study the function f = |µ| 2 on U . By definition, f ([g, ρ, v]) = |ρ| 2 + |µ N (v)| 2 , ∇f ([g, ρ, v]) = J[L g ρ, ad µ N (v) ρ, µ N (v).v], Since f is real analytic, we have for some α ∈ [ 1 2 , 1) that |∇f | ≥ C · |f | α . Therefore y(t) converges to a zero y ∞ of µ polynomially fast. By uniqueness, y ∞ ∈ G.x. Without loss of generality, we will assume y ∞ = x from now on, and we shall distinguish between two cases. In the first case we assume G 0 = G, then m = 0, and we are essentially reduced to the linear case. What we obtain is a Kähler manifold (U ⊂ N, Ω 0 , J). We just need to holomorphically linearize the G action: Lemma 4.9. There exits a G-equivariant holomorphic embedding Φ : (V ⊂ T 0 U, J 0 ) ֒→ (U, J); 0 → x. Proof. Shrinking U if necessaray, we can first choose a holomorphic embed- ding Ψ : (U, J) ֒→ (T 0 U, J 0 ); x → 0. Again Shrinking U if necessary, definê Ψ : (U, J) → (T 0 U, J 0 ); y → 1 |G| G g −1 · Ψ(g.y)dµ, where µ is a Harr measure on G. ThenΨ is holomorphic, and dΨ x = dΨ x , soΨ is an embedding near x. Then we can just take Φ =Ψ −1 . Now using Φ we can work on (V 1 , Ω = Φ * Ω 0 , J 0 ) with a linear Hamiltonian of G, and the linear theory in the previous section applies to conclude the theorem in this case. In the second case we assume G 0 is a proper subgroup of G. We will try to reduce to the first case. It is easy to see that the G 0 action on Y is also Hamiltonian, with a moment mapμ equal to the orthogonal projection of µ to g x . Therefore,μ ([g, ρ, v] ) = Ad * g µ N (v). Denote by G C 0 the isotropy group of x. Lemma 4.10. G C 0 is the complexification of G 0 (hence is reductive). Proof. This lemma is well-known. In the Lie algebra level, we just need to show if ξ.x + Jη.x = 0 for some ξ, η ∈ g, then ξ.x = η.x = 0. This follows easily from the definition of the moment map: ω(η.x, Jη.x) = (dµ(Jη.x), η) = (dµ(Jη.x + ξ.x), η) − (Ad * ξ µ(x), η) = 0. Hence η.x = 0 and ξ.x = 0. Lemma 4.11. We can choose a point in the G C orbit of y, denoted byŷ, so that x de-stabilizesŷ for the group G 0 . Proof. It suffices to findŷ in the G C orbit of y such that x lies in the closure of G C 0 .ŷ. To do this, we first choose an arbitrary holomorphic map Ψ : T x M → M with Ψ(0) = x and dΨ(0) = Id. As before we can linearize the action so that Ψ is G 0 -equivariant. T x M has a C-linear decomposition T x M = g C .x ⊕ N, where N is as before the orthogonal complement of g C .x = g.x ⊗ C = g.x ⊕ J 0 (g.x). Then we define Φ : G C × G C 0 N → M ; [(g, v)] → g.Ψ(v). This is a local diffeomorphism around [(Id, 0)]. So for any y close to x, there is a unique (g, v) ∈ G C ×N which is close to [(Id, 0)] such that y = g.Ψ(Id, v). Letŷ = Ψ(Id, v). We claim x ∈ G C 0 .ŷ. Notice that the Kempf-Ness flow y(t) converges to x, so this gives rise to a smooth family (g(t), v(t)) with y(t) = g(t).Ψ(Id, v(t)). Letŷ(t) = Ψ(Id, v(t)). Since y(t) all lie in the same G C orbit, so areŷ(t). Thus all v(t) lie in the G C 0 orbit of v, and lim t→∞ v(t) = 0. Therefore, x ∈ G 0 .ŷ. Letỹ(t) be the downward gradient flow of f withỹ(0) =ŷ, andŷ(t) be the downward gradient flow off = |μ| 2 withŷ(0) =ŷ. Letγ(t) and γ(t) be the corresponding path in G C /G and G C 0 /G 0 respectively. Then the previous linear theory tells thatŷ(t) converges to x in the order O(t − 1 2 ) and γ(t) is asymptotic to a rational geodesic ray χ(t) with the same degeneration limit. On the other hand G C 0 /G 0 is naturally a totally geodesic submanifold of G C /G, and next we will prove that the distance betweenγ(t) andγ(t) in G C /G is uniformly bounded. We denote by ψ t (s)(s ∈ [0, 1]) the geodesic in G C /G connectingγ(t) and γ(t), and L(t) the length of ψ t , then it is easy to see that d dt L(t) = 1 L(t) (µ(y(t)),ψ t (0)) − 1 L(t) (μ(ŷ(t)),ψ t (1)) = 1 L(t) (µ(y(t)),ψ t (0)) − 1 L(t) (µ(ŷ(t)),ψ t (1)) + 1 L(t) (µ(ŷ(t)) −μ(ŷ(t)),ψ t (1)) ≤ |µ(ŷ(t)) −μ(ŷ(t))|, where again we have used the convexity of the Kempf-Ness function. In our situation, µ −μ = Ad * g ρ. Here g(t) are ρ(t) are uniquely determined by the choice at t = 0 if we requireρ(t) ∈ m and g(t) −1ġ (t) ∈ m. Now at y(t) = [g(t), ρ(t), v(t)], we have ∇f = J.([0, ad * µ N (v) ρ, µ N (v).v]) = [ad * µ N (v) ρ, 0, J 0 · (µ N (v).v)] + (J − J 0 )ad * µ N (v) ρ + (J − J 0 )µ N (v).v. Therefore, | d dt ρ(ŷ(t))| = |Π m (∇f )| ≤ C · |J − J 0 ||µ N (v)||ρ| + C · d(ŷ(t), x) 2 |µ N (v).v|) ≤ C · (t − 3 2 |ρ| + t − 5 2 ). Since ρ(∞) = 0, we first get |ρ(t)| ≤ C · t − 1 2 . Then plug back into the previous inequality and repeat to obtain d dt ρ(x(t)) ≤ C · t − 5 2 , and then |ρ(x(t))| ≤ C · t − 3 2 . So L(t) ≤ t 1 s − 3 2 ds + C ≤ C. Therefore L(t) is uniformly bounded. By definition, we see thatγ(t) is also asymptotic to the geodesic ray χ(t). Now the original γ(t) is also asymptotic to χ(t) again because that the Kempf-Ness flow in G C /G decreases the geodesic distance. Then it is easy to see that χ(t) has the same degeneration limit as γ(t). So this completes the proof of theorem 4.7. Stability of the Calabi flow We first recall the definition of the Calabi flow. It is an infinite dimensional analogue of the previously mentioned Kempf-Ness flow. Let (M, ω, J 0 ) be a Kähler manifold. As before, we have the group G acting on J and preserves J int . The action of G on J has a moment map given by the Hermitian scalar curvature functional S − S : J → C ∞ 0 (M ; R). Its norm is called the Calabi functional: Ca(J) = M (S(J) − S) 2 dµ ω . The gradient of Ca under the natural metric on J is given by ∇Ca(J) = 1 2 JD J S(J). 2 The Calabi flow is the downward gradient flow of Ca on J int . Its equation is given by d dt J(t) = − 1 2 J(t)D J(t) S(J(t)).(6) As in the finite dimensional space, the Calabi flow can be lifted to G C /G, which in this case is just the space of Kähler metrics H J = {φ ∈ C ∞ 0 (M ; R)|ω + √ −1∂ J∂J φ > 0}. The equation reads: d dt φ(t) = S(φ(t)) − S.(7) By (1), this is also the downward gradient flow of the Mabuchi functional E. The two equations (6) and (7) are essentially equivalent: Lemma 5.1. Any solution of (7) naturally gives rise to a solution of (6); any solution J(t) of (6) induces a solution of (7), if J(t) all lie in J int . Proof. Given a path φ(t) ∈ H, we consider the time-dependent vector fields X(t) = − 1 2 ∇ φ(t)φ (t). Let f t be the family of diffeomorphisms generated by X(t). Then f * t (ω + √ −1∂∂φ(t)) = ω. Let J(t) = f * t J. Then d dt J(t) = − 1 2 J(t)D J(t)φ (t). This proves the first half of the lemma. For the second half, if J(t) is a solution to (6). We again consider the vector fields X(t) = 1 2 ∇ J(t) S(J(t)) and the induced diffeomorphisms f t . Then f * t J(t) = J(0) since J(t) ∈ J int , and f * t ω = ω + √ −1dJ(0)dφ(t), with d dt φ(t) = S(φ(t)) − S. Equation (6) is not parabolic, due to the G invariance. But (7) is parabolic and we have the following estimates: Lemma 5.2. (see [CH2]) Suppose there are constants C 1 , C 2 > 0 such that along the Calabi flow: ∂φ ∂t = S − S φ(0) = φ 0 ,(8) we have ||Rm(g(t))|| L ∞ (g(t)) ≤ C 1 , and the Sobolev constant of g(t) is bounded by C 2 for all t ∈ [0, T ), then for any l > 0, and t ∈ [1, T ), we have ||∇ l t Rm(g(t))|| L ∞ (g(t)) ≤ C, where C > 0 depends only C 1 , C 2 , l, n. The Calabi flow equation in the form (7) was first proposed by E. Calabi([Ca1], [Ca2]) to find extremal metrics in a fixed Kähler class. The short time existence was established by ). They also proved the global existence assuming Ricci curvature bound. The equation (6) also has its own advantage. Namely, when the space H does not admit any cscK metric, the solution of equation (7) must diverge when t → ∞. However, it is still possible that the corresponding J(t) still converges in the bigger ambient space J . In this section we are interested in the Calabi flow (6) starting from an integrable complex structure in a neighborhood of a cscK metric. We shall prove the following theorem: Theorem 5.3. Suppose J 0 ∈ J is cscK. Then there exists a small C k,λ (k ≫ 1) neighborhood U of J 0 in J int , such that the Calabi flow J(t) starting from any J ∈ U exists globally and converges polynomially fast to a cscK metric J ∞ ∈ J in C k,λ topology. Up to a Hamiltonian diffeomorphism we can assume J ∞ is smooth, then the convergence is also in C ∞ . Remark 5.4. When J lies on the leaf of J 0 , i.e. the corresponding Kähler metrics are in the same Kähler classes, this was proved in [CH1] and the convergence is indeed exponential. In general, the convergence is exponential if and only if J 0 and J ∞ are on the same G C leaf. Remark 5.5. There are also studies of stabiliy of other geometrical flows (such as Kähler-Ricci flow) in Kähler geometry when the complex structure is deformed, see for example [CLW], [TZ2]... We believe the idea in this section could also apply to other settings. In a sequel to this paper( [SW]), the second author and Y-Q. Wang proved a similar stability theorem for the Kähler-Ricci flow on Fano manifolds. We should mention that two alternative approaches in the study of the stability of Kähler-Ricci flow have been announced by C.Arezzo-G. La Nave and G. Tian-X. Zhu. In general this type of stability result is based on a very rough a priori estimate of the length of the flow and the parabolicity. Here the key ingredient is the following Lojasiewicz type inequality which yields the required a priori estimate. Theorem 5.6. Suppose J 0 ∈ J int is cscK, then there exists a L 2 k (k ≫ 1) neighborhood U of J 0 in J int and constants C > 0, α ∈ [ 1 2 , 1) such that for any J ∈ U , the following inequality holds: ||D J S(J)|| L 2 ≥ C · ||S(J) − S|| 2α L 2 ,(9) where D J φ =∂ J X φ +X φ .N J . When J is integrable, D J φ =∂ J X φ is the Lichnerowicz operator. Remark 5.7. The Lojasiewicz inequality was first used by L. Simon([Si]) in the study of convergence of parabolic P.D.E's. Råde([Ra]) used Simon's idea to study the convergence of the Yang-Mills flow on two or three dimensional manifold. It also appeared in the study of asymptotic behavior in Floer theory in [D4]. Here we follow [Ra] closely. We begin the proof by reducing the problem to a finite dimensional one and then use Lojasiewicz's inequality(theorem 3.1). To simplify the notation, we assume the function spaces appearing below consist of normalized functions, i.e. functions with average zero. We have the elliptic complex at J 0 (see [FS]): L 2 k+2 (M ; C) D 0 −→ T J 0 J = L 2 k (Ω 0,1 S (T 1,0 ))∂ 0 −→ L 2 k−1 (Ω 0,2 S (T 1,0 )), where Ω 0,p S (T 1,0 ) is the kernel of the operator A in section 3. So we have an L 2 orthogonal decomposition: Ω 0,1 S (T 1,0 ) = ImD 0 ⊕ KerD * 0 . On the other hand, the infinitesimal action of the gauge group G is just the restriction of D 0 to L 2 k+2 (M ; R), which we denote by Q 0 . Since J 0 is cscK, D * 0 D 0 is a real operator. Thus Im(D 0 ) = D 0 (L 2 k+2 (M ; R)) ⊕ D 0 (L 2 k+2 (M ; √ −1R)) is an L 2 orthogonal decomposition, so L 2 k (Ω 0,1 S (T 1,0 )) = ImQ 0 ⊕ KerQ * 0 , where explicitly, Q * 0 µ = ReD * 0 µ. Now as in section 2 we identify a L 2 k neighborhood of J 0 with an open set in the Hilbert space L 2 k (Ω 0,1 S (T 1,0 )). By the implicit function theorem, any integrable complex structure J = J 0 + µ ∈ J int with ||µ|| L 2 k small is in the G orbit of an integrable complex structure J 0 + ν with ν ∈ KerQ * 0 and ||ν|| L 2 k small. Since both sides of (9) are invariant under the action of G, it suffices to prove it for µ ∈ KerQ * 0 . We still need to fix another gauge so that the problem becomes elliptic. Recall that J int is the subvariety of J cut out by the equation: N (µ) =∂ 0 µ + [µ, µ] = 0. We would like to linearize this space to Ker∂ 0 . Let W = KerQ * 0 ∩ Ker∂ 0 . Consider the operator Φ : (W ∩L 2 k (Ω 0,1 S (T 1,0 )))×(Im∂ 0 ∩L 2 k+1 (Ω 0,2 S (T 1,0 ))) → Im∂ 0 ∩L 2 k−1 (Ω 0,2 S (T 1,0 )) by sending (µ, α) to the orthogonal projection to Im∂ 0 of N (µ +∂ * 0 α). Since the linearization DΦ 0 (ν, β) =∂ 0∂ * 0 β whose second component is an isomorphism, by the implicit function theorem, for any ν ∈ W ∩ L 2 k (Ω 0,1 S (T 1,0 )) with ||ν|| L 2 k small, there exists a unique α = α(ν) ∈ Im∂ 0 ⊂ L 2 k+1 (Ω 0,2 S (T 1,0 )) with ||α|| L 2 k+1 small such that µ = ν +∂ * 0 α satisfies Φ(µ) = 0. Furthermore, we have ||α(ν)|| L 2 k+1 ≤ C · ||ν|| 2 L 2 k . Define a map L from B ǫ 1 (W ∩ L 2 k (Ω 0,1 S (T 1,0 ))) to KerQ * 0 ∩ L 2 k (Ω 0,1 S (T 1,0 )) by sending ν to µ, then L is real analytic and a neighborhood of J 0 in J int ∩ KerQ * 0 ∩ L 2 k (Ω 0,1 S (T 1,0 )) is contained in the image of L. Moreover we have that for all ν ∈ B ǫ 1 W ∩ L 2 k (Ω 0,1 S (T 1,0 )) and λ ∈ W ∩ L 2 l (Ω 0,1 S (T 1,0 ))(for any l ≤ k), c l · ||λ|| L 2 l ≤ ||(DL) ν (λ)|| L 2 l ≤ C l · ||λ|| L 2 l ,(10) and c l · ||λ|| L 2 l ≤ ||(DL) * ν (DL) ν (λ)|| L 2 l ≤ C l · ||λ|| L 2 l .(11) To be explicit, the differential of α at ν is given by (Dα) ν (λ) = (DΦ) L(ν) (0, −) −1 • (DΦ) L(ν) (λ, 0). So if we denote µ = L(ν) and β = (Dα) ν (λ), then β satisfies: ∂ 0∂ * 0 β + Π Im∂ 0 [µ,∂ * 0 β] =∂ 0 λ + Π Im∂ 0 [µ, λ] = Π Im∂ 0 [µ, λ]. Thus by ellipticity we obtain for ν small that ||(Dα) ν (λ)|| L 2 l+1 ≤ C · ||ν|| L 2 k · ||λ|| L 2 l .(12) (10) follows from (12) and similarly we can prove (11). Now consider the Hilbert space W ∩ L 2 k (Ω 0,1 S (T 1,0 )) with the constant L 2 metric defined by J 0 . Define the functional Ca on on a small neighborhood of the origin in W ∩ L 2 k (Ω 0,1 S (T 1,0 )) by pulling back Ca through L, i.e. Ca(ν) = 1 2 Ca(L(ν)) = 1 2 (S(L(ν)) − S) 2 ω n . It is easy to see that δ λ S(L(ν)) = 2ImD * L(ν) ((DL) ν (λ)) So the gradient is ∇ Ca = (DL) * ν (JD L(ν) S(L(ν))). We first prove that in a neighborhood of 0 in W , ||∇ Ca(ν)|| L 2 ≥ C · ( Ca(ν)) α .(13) The linearization of the gradient is the Hessian: H 0 := δ · ∇ Ca : L 2 k (W ) → L 2 k−4 (W ); λ → 2J 0 D 0 D * 0 λ. H 0 is an elliptic operator, so it has a finite dimensional kernel W 0 consisting of smooth elements, and W has the following decomposition: W = W 0 ⊕ W ′ , where H 0 restricts to invertible operators from L 2 k (W ′ ) to L 2 k−4 (W ′ ). So there exists a c > 0, such that for any µ ′ ∈ W ′ , we have ||H 0 (µ ′ )|| L 2 k−4 ≥ C · ||µ ′ || L 2 k . By the implicit function theorem, for any µ 0 ∈ W 0 with ||µ 0 || L 2 3 small, there exists a unique element µ ′ = G(µ 0 ) ∈ W ′ with ||µ ′ || L 2 k small, such that ∇ Ca(µ 0 + µ ′ ) ∈ W 0 . Moreover the map G : B ǫ 1 W 0 → B ǫ 2 W ′ is real analytic. Now consider the function f : W 0 → R; µ 0 → Ca(µ 0 + G(µ 0 )). By construction, this is a real analytic function. For any µ 0 ∈ W 0 , it is easy to see that ∇f (µ 0 ) = ∇ Ca(µ 0 + G(µ 0 )) ∈ W 0 . Now we shall estimate the two sides of inequality (13) separately. For any µ ∈ W with ||µ|| L 2 k ≤ ǫ, we can write µ = µ 0 + G(µ 0 ) + µ ′ , where µ 0 ∈ W 0 , µ ′ ∈ W ′ , and ||µ 0 || L 2 k ≤ c · ||µ|| L 2 k , ||G(µ 0 )|| L 2 k ≤ c · ||µ|| L 2 k , ||µ ′ || L 2 k ≤ c · ||µ|| L 2 k . For the left hand side of (13), we have: ∇ Ca(µ) = ∇ Ca(µ 0 + G(µ 0 ) + µ ′ ) = ∇ Ca(µ 0 + G(µ 0 )) + 1 0 δ µ ′ ∇ Ca(µ 0 + G(µ 0 ) + sµ ′ )ds = ∇f (µ 0 ) + δ µ ′ ∇ Ca(0) + 1 0 (δ µ ′ ∇ Ca(µ 0 + G(µ 0 ) + sµ ′ ) − δ µ ′ ∇ Ca(0))ds The first two terms are L 2 orthogonal to each other. For the second term we have ||δ µ ′ ∇ Ca(0)|| 2 L 2 = ||H 0 (µ ′ )|| 2 L 2 ≥ C · ||µ ′ || 2 L 2 4 . For the last term, we have ||δ µ ′ ∇ Ca(µ 0 + G(µ 0 ) + sµ ′ ) − δ µ ′ ∇ Ca(0)|| ≤ C · ||µ|| L 2 k ||µ ′ || L 2 4 ≤ C · ǫ · ||µ ′ || L 2 4 . Therefore, we have ||∇ Ca(µ)|| 2 L 2 ≥ |∇f (µ 0 )| 2 L 2 + C · ||µ ′ || 2 L 2 4 .(14) For the right hand side of (13), we have Ca(µ) = Ca(µ 0 + G(µ 0 ) + µ ′ ) = Ca(µ 0 + G(µ 0 )) + 1 0 ∇ Ca(µ 0 + G(µ 0 ) + sµ ′ )µ ′ ds = f (µ 0 ) + ∇f (µ 0 )µ ′ + 1 0 1 0 δ µ ′ ∇ Ca(µ 0 + G(µ 0 ) + stµ ′ )µ ′ dtds = f (µ 0 ) + H 0 (µ ′ )µ ′ + 1 0 1 0 (δ µ ′ ∇ Ca(µ 0 + G(µ 0 ) + stµ ′ ) − δ µ ′ ∇ Ca(0))µ ′ dtds So Ca(µ) ≤ |f (µ 0 )| L 2 + C · ||µ ′ || 2 L 2 4 .(15) Now we apply the Lojasiewicz inequality to f , and obtain that |∇f (µ 0 )| L 2 ≥ C · |f (µ 0 )| α , for some α ∈ [ 1 2 , 1). Together with (14) and (15) we have proved (13). To prove (9), we need to compare ||∇Ca(L(ν))|| L 2 and ||∇ Ca(ν)|| L 2 , i.e. we want ||(DL) * ν (D L(ν) S(L(ν)))|| L 2 ≤ C · ||D L(ν) S(L(ν))|| L 2 .(16) We can take L 2 decomposition D L(ν) S(L(ν)) = (DL) ν λ + β, where λ ∈ W and β ∈ Ker(DL) * ν . So we just need to prove ||(DL) * ν (DL) ν λ|| L 2 ≤ C · ||(DL) ν λ|| L 2 for any λ. This follows from (10) and (11). Now we follow the Lojasiewicz arguments. Suppose we have a Calabi flow J(t) along an integral leaf staying in a L 2 k neighborhood of J 0 , then by (6) d dt Ca(J) 1−α = −(1 − α)Ca(J) −α ||∇Ca(J)|| 2 L 2 (t) ≤ −C · ||∇Ca(J)|| L 2 (t) . Thus t 0 ||J || L 2 (s) ds = t 0 ||∇Ca(J(s))|| L 2 (s) ds ≤ C · Ca(J(0)) 1−α .(17) So we get L 2 length estimate for the Calabi flow in terms of the initial Calabi energy. For γ slightly bigger than α, we have for β = 2 − γ α < 1, d dt Ca(J) 1−γ = −(1 − γ)Ca(J) −γ ||∇Ca(J)|| 2 L 2 (t) ≤ −C · ||∇Ca(J)|| β L 2 (t) . So for β ∈ (2 − 1 α , 1) we have t 0 ||J(s)|| β L 2 (s) ds = t 0 ||∇Ca(J(s))|| β L 2 (s) ds ≤ C(β) · Ca(J(0)) 1−(2−β)α . (18) Also we have polynomial decay: d dt Ca(t) 1−2α ≥ C > 0, so Ca(J(t)) ≤ C · (t + 1) − 1 2α−1 .(19) Now we define U δ k = {J ∈ C k,λ (J int ) | ||µ J || C k,λ ≤ δ}, where again we identify J close to J 0 with µ J ∈ Ω 0,1 S (T 1,0 ). Notice that if δ ≪ 1, then for any tensor ξ, the C k,λ J norms defined by (J, ω) are equivalent for any J ∈ U δ k . We omit the subscript J if J = J 0 . Also for k sufficiently large, the Sobolev constant is uniformly bounded in U δ k . Theorem 5.8. Suppose J 0 is a cscK metric in J int . Then there exist δ 2 > δ 1 > 0, such that for any J(0) ∈ U δ 1 k , the Calabi flow J(t)(t > 0) starting from J(0) will stay in U δ 2 k all the time. Proof. Choose δ > 0 such that the previous a priori estimates hold in U δ k . If suffices to prove that there exists δ 1 < δ 2 < δ such that for any Calabi flow J(t) with J(0) ∈ U δ 1 k , if J(t) ∈ U δ k for t ∈ [0, T ), then J(T ) ∈ U δ 2 k . By lemma 5.2, for t ≥ 1 and l, we have ||Rm(J(t))|| C l,λ t ≤ C(l). Now fix β ∈ (2 − 1 α , 1), for any p, there is an N (p)(independent of t ≥ 1), such that the following interpolation inequality holds ||J (t)|| L 2 p (t) ≤ C(p) · ||J (t)|| β L 2 (t) · ||D J S(J)|| 1−β L 2 N(p) (t) ≤ C(p) · ||J(t)|| β L 2 (t) , So by (18) we have T 1 ||J (t)|| L 2 p (t) dt ≤ C(p)·Ca(J(1)) 1−(2−β)α ≤ C(p)·Ca(J(0)) 1−(2−β)α ≤ C(p)·ǫ(δ 1 ). Since the Sobolev constant is uniformly bounded in U δ k , we obtain for any l, T 1 ||J (t)|| C l,λ t dt ≤ C(l) · ǫ(δ 1 ). Therefore, ||J(T ) − J(1)|| C k,λ ≤ T 1 ||J(t)|| C k,λ dt ≤ ǫ(δ 1 ). By the finite time stability of the Calabi flow, we have ||J(1) − J 0 || C k,λ = ǫ(δ 1 ). Thus ||J(T ) − J 0 || C k,λ ≤ ǫ(δ 1 ). Now choose δ 2 = δ 2 , and ǫ(δ 1 ) ≤ δ 2 , then the theorem is concluded. From theorem 5.8, we know the Calabi flow exists globally in C k,λ and thus by sequence converges to J ∞ in C k,β for β < α. Now again by the Lojasiewicz arguments we see the limit must be unique and the convergence is in a polynomial rate in C k,λ . Now we assume that J ∞ = J 0 is smooth. Then we can prove smooth convergence. We first use the ellipticity to obtain a priori estimates in U δ k for k ≫ 1. Any µ ∈ U δ k satisfies the following elliptic system:    ImD * 0 µ = S(µ) + O(||µ|| 2 L 2 2 ), ReD * 0 µ = Q * 0 (µ), ∂µ + [µ, µ] = 0.(20) So we have the following a priori estimate: ||µ|| C l+2,α ≤ C · (||µ|| C l,λ + ||S(µ)|| C l,λ + ||Q * 0 (µ)|| C l,λ ).(21) From the proof of theorem 5.8, we know that ||µ(t)|| C k,λ and ||S(µ(t))|| C k,λ are uniformly bounded. Since ||Q * 0 (µ(t))|| C k,λ ≤ ∞ t ||Q * 0 (μ(s))|| C k,λ s ds ≤ ǫ(Ca(J(s))) is bounded, we obtain ||µ(t)|| C k+2,α bound, so we can derive smooth convergence by bootstrapping argument. This finishes the proof of theorem 5.3. Theorem 5.3 has its own interest. This yields a purely analytical proof of an extension of a theorem due to Chen [Ch4] and Székelyhidi [Sz]. This is inspired by an observation of Tosatti [To]. In particular, we do not require the Kähler class to be integral. Theorem 5.9. ( [Ch4]) For any J ∈ U , the Mabuchi functional E on the space of Kähler metrics compatible with J is bounded below, and the lower bound is achieved by the infimum along the Calabi flow initiating from J. Proof. From the proof of theorem 5.3 we know the Calabi flow J(t) ∈ J int starting from J converges to a limit J ∞ with estimate Ca(J(t)) ≤ C · (t + 1) − 1 2α−1 . By lemma 5.1, this is equivalent to the Calabi flow φ(t) in the space of Kähler metrics compatible with J. Then E(φ(t)) = E(φ(0))− t 0 Ca(φ(s))ds ≥ E(φ(0))−C· 2α − 1 2α − 2 ·[1−(t+1) 2α−2 2α−1 ] ≥ −C ′ . For any other Kähler potential φ, we have by lemma 2.4 that E(φ) ≥ E(φ(t)) − Ca(φ(t)) · d(φ, φ(t)). Since d(φ, φ(t)) ≤ d(φ, φ(0))+d(φ(0), φ(t)) ≤ C+ t 0 Ca(φ(s))ds ≤ C·[1+(t+1) 4α−3 4α−2 ], we have E(φ) ≥ lim inf t→∞ E(φ(t)) − C · (t + 1) − 1 4α−2 · [1 + (t + 1) 4α−3 4α−2 ] = lim t→∞ E(φ(t)) is bounded below. Reduced Calabi flow In this section we shall discuss a reduced finite dimensional problem. The usual Kuranishi method provides a local slice as follows. Assume J 0 is cscK. We have as before the following elliptic complex: C ∞ 0 (M ; C) D 0 −→ T J 0 J = Ω 0,1 S (T 1,0 )∂ 0 −→ Ω 0,2 S (T 1,0 ). Let 0 = D 0 D * 0 + (∂ *0∂ 0 ) 2 , and H 1 = Ker 0 . Let G be the isotropy group of J 0 , which is the group of Hamiltonian isometries of (M, ω, J 0 ), with Lie algebra g = KerD 0 ∩ C ∞ 0 (M ; R). By the classical Matsushima-Lichnerowicz theorem, KerD 0 is the complexification g C of g, and so the complexification G C of G is a subgroup of the group of holomorphic transformations of (M, J 0 ), with Lie algebra g C = KerD 0 . Then the linear G action on H 1 extends to an action of G C . For convenience, we include a proof of the following standard fact. (2). If v 1 and v 2 in B are in the same G C orbit and Φ(v 1 ) is integrable, then Φ(v 2 ) is integrable, and Φ(v 1 ) and Φ(v 2 ) are in the same G C leaf. Conversely, if Φ(v) is integrable and (dΦ) v (u) is tangent to the G C leaf at Φ(v), then u is tangent to the G C orbit at v. (3). Any integrable J sufficiently close to J 0 lies in the G C leaf of some element in the image of Φ. Proof. We can identify any J close to J 0 with an element µ in Ω 0,1 S (T 1,0 ), and J is integrable if and only if N (µ) =∂ 0 µ + [µ, µ] = 0. We can first choose a G-equivariant holomorphic embedding Ψ from a ball B in Ω 0,1 S (T 1,0 ) into J with dΨ 0 = Id, by using the same "average trick" as in the proof of lemma 4.9. Let V = {µ ∈ Ω 0,1 S (T 1,0 )|D * 0 µ = 0}, and U = {µ ∈ Ω 0,1 S (T 1,0 )|N (µ) = 0, D * 0 µ = 0}. Denote by G the Green operator for 0 and H : Ω 0,1 S (T 1,0 ) → H 1 the orthogonal projection. Then for any µ ∈ U , we have µ = G 0 µ + Hµ = −G∂ * 0∂ 0∂ * 0 [µ, µ] + Hµ. Define a G-equivariant map F : Ω 0,1 S (T 1,0 ) → Ω 0,1 S (T 1,0 ); µ → µ + G∂ * 0∂0∂ * 0 [µ, µ], where both spaces are endowed with the Sobolev L 2 k norm. Its derivative at 0 is the identity map, so by the implicit function theorem, there is an inverse holomorphic map F −1 : V 1 (⊂ Ω 0,1 S (T 1,0 )) → V 2 (⊂ Ω 0,1 S (T 1,0 )). Let Q be restriction of F −1 on B = V 1 ∩ H 1 and Φ be the composition Φ : B → J ; v → Ψ • Q(v). Since H 1 consists of smooth elements, the image of Φ also consists of smooth elements. Now we check Φ is the desired map. For any v ∈ B, we have D * 0 Q(v) = −D * 0 G∂ * 0∂0∂ * 0 [Q(v), Q(v)] = 0, and N (Q(v)) = −∂ 0 G∂ * 0∂ 0∂ * 0 [Q(v), Q(v)]+[Q(v), Q(v)] = G(∂ * 0∂ 0 ) 2 [Q(v), Q(v)]−H[Q(v), Q(v)]. So N (Q(v)) = 0 if and only if H[Q(v), Q(v)] = 0, as in [Ku]. Therefore a neighborhood of 0 in U is an analytic set contained in the image of Q. Since both Ψ and F are G-equivariant and holomorphic, the first part of (2) is true. Following [Sz], we define a map P from a neighborhood of (J 0 , 0) in J × C ∞ 0 (M ; C) to J as follows. Given µ ∈ Ω 0,1 S (T 1,0 ) representing an element in J close to J 0 , and φ = φ 1 + √ −1φ 2 ∈ C ∞ 0 (M ; C) small. There is a family of Hamiltonian diffeomorphsms f t witḣ f t = X φ 1 . Denote J 1 = f * 1 J. Since ω φ = ω + √ −1dJ 1 dφ 2 is isotopic to ω through the path ω t = (1 − t)ω + tω φ 2 . Then there is a canonical path of diffeomorphisms g t such that g * t ω t = ω. Now g * 1 J 1 is the image under Ψ of an element µ 1 ∈ Ω 0,1 S (T 1,0 ). Then define P (µ, φ) = GD * 0 µ 1 . Then P is a smooth function from L 2 k (V ) × L 2 k (M ; C) to the orthogonal complement L 2 k (A 0 ) of g C in L 2 k (M ; C). It is easy to calculate the derivative of P at (J 0 , 0) is (DP ) 0 (ν, ψ) = GD * 0 ν + GD * 0 D 0 ψ. The derivative with respect to the second variable is surjective with a finite dimensional kernel 0×g C . Thus by implicit function theorem, any integrable complex structure close to J 0 lies in the G C leaf of an element in U , and thus is contained in the G C leaf of the image of Φ. So (3) is proved. It suffices to prove the last statement in (2). Suppose µ = Φ(v), and ν = (dΦ) v (u) is tangent to the G C leaf, i.e ν = D µ φ for some complex valued function φ. Then DP (µ,0) (0, φ) = 0. On the other hand, the kernel of DP (µ,0) (0, −) has the same dimension as dim g C if µ is sufficiently close to zero. Thus, φ ∈ g C and u is tangent to the G C orbit of v. By [D1] the action of G on J has a moment map given by the scalar curvature functional µ = S − S : J → C ∞ 0 (M ; R). The downward gradient flow of |µ| 2 is just the Calabi flow. Now we reduce this flow to a finite dimensional flow. Note G as a subgroup of G acts on J with induced moment mapμ = Π g (S − S). It is the L 2 projection of µ to g with respect to the natural volume form. We can consider the gradient flow of |μ| 2 , whose equation reads d dt J = − 1 2 JD Jμ (J).(22) If we have a solution to equation (22) such that J t is integrable for all t ∈ [0, T ], then we can translate it to a flow in H given by d dt φ = Π f * t g (S(φ) − S),(23) where f t is the family of diffeomorphism satisfying d dt f t = − 1 2 J t X S(Jt) , and the projection is taken with respect to the volume form of f * t ω. We will study the relation between this flow and the Calabi flow later on. Let us call the flow (22) or (23) the reduced Calabi flow. It is the gradient flow of the norm squared of the moment map of a finite dimensional compact group action. Now we can pull back the Kähler structure on J to B, denoted by (Ω,J). By the previous lemma, we know G acts on (B,Ω,J ) holomorphically and isometrically, with moment mapμ equal to Φ * μ . We can then study the reduced Calabi flow on a finite dimensional ambient space B. Let J be an integrable complex structure J close to J 0 such that the Calabi flow J(t) converges to J 0 . Suppose J 0 is not in the G C leaf of J. By property (3) in lemma 6.1, we can smoothly perturb J(t) toJ(t) in the G C orbit such that J(t) = Φ(v(t)) for v(t) → 0 ∈ B. SinceJ(t) is tangent to the G C leaf, by property (2) in lemma 6.1, we see thatv(t) is tangent to the G C orbit. So v is de-stabilized by 0 in B under the G C action. By our previous study of the finite dimensional case, the reduced Calabi flow starting from v exists for all time and converges to 0 in the order O(t − 1 2 ), and the corresponding flowĴ (t) in G C /G is asymptotic to a rational geodesic ray χ which also degenerate v to zero. We can view χ as a geodesic ray in H as well, so the reduced Calabi flow in H is asymptotic to a smooth geodesic ray with the same degeneration limit. This needs a bit more clarification. First of all, for any element g in G C , one can choose a path g(t) in G C with g(0) equal to identity and g(1) = g. Then we have d dt g(t) · g(t) −1 = ξ(t) + √ −1η(t). We can choose a path h(t) in G with h(0) being identity, such that d dt (h(t)g(t)) ∈ √ −1g. This is equivalent to d dt h(t) · h(t) −1 + h(t)ξ(t)h(t) −1 = 0. Now we define a map F from an open set in G C /G to H as follows. This open set is a geodesic convex open set U in G C /G such that [g].v still lies in the previously constructed Kuranishi slice. Let v(t) = g(t).v, and J(t) = Φ(v(t)). Then J(t) are all integrable and d dt J(t) = −(D J(t) ξ(t) + J(t)D J(t) η(t)), where ξ(t) and η(t) are viewed as functions on M through the inclusion g ⊂ C ∞ 0 (M ; R). Choose an isotopy of Hamiltonian diffeomorphisms f t such that d dt f t = X ξ(t) . Then J (t) = f * t J(t) satisfies d dt J (t) = J(t)D J (t) η(t), where η(t) = f * t η(t). In fact, J(t) = Φ(h(t)g(t) .v). Then by Lemma 5.1 if we choose an isotopy of diffeomorphisms k t with d dt k t = −∇ J(t) η(t), then k * t J(t) = J, and k * t ω = ω t = ω + √ −1∂∂φ(t). We define F ([g]) to be φ(1). Of course we need to show this is well-defined, it suffices to show the definition is independent of the path chosen in G C /G. Since G C /G is always simply connected, we only to show it is invariant under based homotopy. Fo this, we choose a two parameter family g s,t in G C such that g s,0 is equal to identity, and g s,1 = g. Correspondingly we have h(s, t) in G with h(s, 0) equal to identity. Let g s,t = h s,t · g s,t , then we have ∂ ∂t g s,t · g −1 s,t = √ −1η(s, t) ∈ √ −1g. Also we have ∂ ∂s g s,t · g −1 s,t = ξ(s, t) + √ −1ζ(s, t) ∈ g ⊕ √ −1g. So we have the relation √ −1 ∂ ∂s η(s, t) = ∂ ∂t ξ(s, t) + √ −1 ∂ ∂t ζ(s, t) + [ √ −1η(s, t), ξ(s, t) + √ −1ζ(s, t)]. In particular ∂ ∂s η(s, t) = ∂ ∂t ζ(s, t) + [η(s, t), ξ(s, t)]. Also ξ(s, 0) = ζ(s, 0) = ξ(s, 1) = ζ(s, 1) = 0. Let J s,t = Φ(g s,t .v), and f s,t be the two parameter family of diffeomorphisms obtained by fixing s and integrate along the t direction as before. In particular, f (s, 0) is equal to identity for all s. We compute ∂ ∂s ∂ ∂t f * s,t ω = − ∂ ∂s f * s,t dJ s,t dη(s, t) = − ∂ ∂s dJdf * s,t η(s, t). We have ∂ ∂s f * s,t η(s, t) = f * s,t ( ∂ ∂s η(s, t) + L Js,t∇s,tξ(s,t)−∇s,tζ(s,t) η(s, t)) = f * s,t ( ∂ ∂s η(s, t) + {ξ(s, t), η(s, t)} − ∇ s,t ζ(s, t), ∇ s,t η(s, t) ) = f * s,t ( ∂ ∂t ζ(s, t) − ∇ s,t ζ(s, t), ∇ s,t η(s, t) )) = ∂ ∂t F * s,t ζ(s, t). Thus ∂ ∂s | t=1 f * s,t ω = −dJd( 1 0 ∂ ∂t f * s,t ζ(s, t)dt) = −dJd(f * s,1 ζ(s, 1)) = 0. Thus the map F depends only on the point [g], not on the path chosen. So F is a well-defined smooth map. From this it is clear that F is a local isometric embedding, in particular, the image is totally geodesic. Thus we have proved that the reduced Calabi flow in H is asymptotic to a smooth geodesic ray with the same degeneration limit. By Section 4.2 this geodesic ray is indeed rational, i.e. extends to a C * action. Then it follows from arguments in [Sz] that χ is tamed by a smooth test configuration, so it is tamed by a bounded geometry in the sense of [Ch3]. To prove that the Calabi flow is asymptotic to the reduced Calabi flow, we need to generalize lemma 4.6 to the infinite dimensional case. Then by the same argument as before, together with lemma 2.5 that the Mabuchi functional is weakly convex, one can show Lemma 6.2. LetĴ(t) be the reduced Calabi flow as before andφ(t) be the corresponding flow in H. Then for any Calabi flow path φ(t) ∈ H, we have for all t that d(φ(t),φ(t)) ≤ C. The proof will be given in the appendix. Combining all these we arrive at the following theorem: Theorem 6.3. Let (M, ω 0 , J 0 ) be a csc Kähler manifold. Let J be a complex structure in J close to J 0 and the Calabi flow starting from J converges to J 0 at the infinity. Suppose J 0 is not in the G C leaf of J. Then there is a smooth geodesic ray φ(t) in the space of Kähler metrics H ω,J which is tamed by bounded geometry and degenerates J to J 0 in the space J . Furthermore, φ(t) is asymptotic to the Calabi flow with respect to the Mabuchi-Semmes-Donaldson metric in the sense of definition 4.2. Relative Bound for parallel Geodesic rays It is well-known that in a Riemannian manifold with non-positive curvature, the distance between two geodesics is a convex function. In this section we first justify this property for the infinite dimensional space H. Lemma 7.1. Let φ 1 (t) and φ 2 (t) be two C 1,1 geodesics in H, then d(φ 1 (t), φ 2 (t)) is a convex function of t. Proof. . We first assume both geodesics are C ∞ . Let γ ǫ (t, s) be the ǫgeodesic connecting γ 1 (t) and γ 2 (t)(see [Ch1]), then d 2 dt 2 L(γ ǫ (t)) = 1 0 1 |γ ǫ,s | {|γ ⊥ ǫ,ts | 2 − R(γ ǫ,s , γ ǫ,t )}ds + 1 |γ ǫ,s | γ ǫ,s , γ ǫ,tt | 1 0 − 1 0 γ ǫ,ss , γ ǫ,tt |γ ǫ,s | + γ ǫ,s , γ ǫ,ss γ ǫ,s , γ ǫ,tt |γ ǫ,s | 3 ds Along the ǫ-geodesics, we have |γ ǫ,ss | = 1 0 (φ ǫ,ss − ∇ φǫ,s φ ǫ,s ) 2 ω n φǫ ≤ C(t) √ ǫ, where C(t) is uniformly bounded if t varies in a bounded interval. Also |γ ǫ,tt | ≤ C(t), and |γ ǫ,s | → L t , uniformly for s ∈ [0, 1] and t bounded. Therefore, we have d 2 dt 2 L(γ ǫ (t)) ≥ −C(t) √ ǫ, so for any a ≤ b, L ǫ (ta + (1 − t)b) ≤ tL ǫ (a) + (1 − t)L ǫ (b) + C √ ǫ(t − a)(b − t). Let ǫ → 0, L(ta + (1 − t)b) ≤ tL(a) + (1 − t)L(b). So L(t) is still a convex function, and the argument of the lemma yields the same conclusion. In the general case we need to define the distance between two C 1,1 potentials, which is just the infimum of the length of all C 1,1 paths connecting the two points. Clearly the distance between any two points is always nonnegative. Now we assume φ 1 and φ 2 are C 1,1 but φ i (0) and φ i (1) are smooth, we want to prove for t ∈ [0, 1], L(t) ≤ (1 − t)L(0) + tL(1).(24) To prove this, choose a δ-geodesic φ i δ approximating φ i with endpoints fixed. Let φ ǫ,δ (t, s) be the geodesic connecting φ 1 δ (t) and φ 2 δ (t), and L ǫ,δ (t) be its length. Then similar calculation shows that d 2 dt 2 L ǫ,δ (t) ≥ −C √ δ − C(δ, t) √ ǫ, So L ǫ,δ (t) ≤ (1 − t)L ǫ,δ (0) + tL ǫ,δ (1) + 1 2 (C √ δ + C(δ, t) √ ǫ)t(1 − t). Let ǫ → 0, we have L δ (t) ≤ (1 − t)L δ (0) + tL δ (1) + C √ δ. Let δ → 0, we get the desired inequality. So the theorem is true in this case. If φ i (0) and φ i (1) are not assumed to be smooth, we can approximate them weakly in C 1,1 by smooth potentials φ ǫ i (0), φ ǫ i (1) respectively. Let φ ǫ i (t) be the geodesic connecting φ ǫ i (0) and φ ǫ i (1). Then we know d(φ ǫ 1 (t), φ ǫ 2 (t)) is a convex function. By maximum principle for the Monge-Ampère equations, we know |φ ǫ i (t) − φ i (t)| C 0 ≤ max(|φ ǫ i (0) − φ i (0)| C 0 , |φ ǫ i (1) − φ i (1)| C 0 ). Hence |φ ǫ i (t) − φ i (t)| C 0 → 0, in particular, d(φ ǫ i (t), φ i (t)) → 0. Therefore, d(φ ǫ 1 (t), φ ǫ 2 (t) ) converges uniformly to d(φ 1 (t), φ 2 (t)). So the latter is also convex. Lemma 7.2. If φ 1 is in H(i.e. φ 1 is smooth and ω 1 is positive) and φ 2 is C 1,1 , then d(φ 1 , φ 2 ) = 0 if and only if φ 1 = φ 2 . Proof. We can choose C ∞ potential φ ǫ 2 converging to φ 2 weakly in C 1,1 as ǫ → 0. Then by [Ch1], d(φ 1 , φ ǫ 2 ) ≥ max( φ 1 ≥φ ǫ 2 (φ 1 − φ ǫ 2 )ω n φ 1 , φ ǫ 2 ≥φ 1 (φ ǫ 2 − φ 1 )ω n φ ǫ 2 ) Let ǫ → 0, we get d(φ 1 , φ 2 ) ≥ max( φ 1 ≥φ 2 (φ 1 − φ 2 )ω n 1 , φ 2 ≥φ 1 (φ 2 − φ 1 )ω n 2 ). So if d(φ 1 , φ 2 ) = 0, then φ 1 ≥φ 2 (φ 1 − φ 2 )ω n 1 = 0, and φ 2 ≥φ 1 (φ 2 − φ 1 )ω n 2 = 0. The first equation implies φ 1 ≤ φ 2 . The second equation implies that φ 2 >φ 1 ω n 2 = 0. Let Ω = {x ∈ M |φ 2 (x) > φ 1 (x)}. Then by Stokes' formula, Ω ω n 1 = Ω ω n 1 − ω n 2 = Ω √ −1∂∂(φ 1 − φ 2 ) · n−1 j=0 ω j 1 ∧ ω n−1−j 2 = ∂Ω √ −1∂(φ 1 − φ 2 ) · n−1 j=0 ω j 1 ∧ ω n−1−j 2 = 0. So Ω is empty. Thus φ 1 = φ 2 . Corollary 7.3. Let φ 1 be a geodesic ray tamed by bounded geometry(see [Ch3]), and φ 2 another geodesic ray parallel to φ 1 with φ 2 (0) smooth. Then φ 1 − φ 2 has a uniform relative C 1,1 bound(with respect to ω φ 1 ). Proof. By [Ch3], there is a C 1,1 geodesic ray φ 3 emanating from φ 2 (0) such that |φ 3 (t) − φ 1 (t)| C 1,1 φ 1 ≤ C. Thus d(φ 2 (t), φ 3 (t)) is uniformly bounded. Since φ 2 (0) = φ 3 (0), by lemma 7.1, d(φ 2 (t), φ 3 (t)) = 0. Lemma 7.2 then implies φ 2 (t) = φ 3 (t). So |φ 2 (t) − φ 1 (t)| C 1,1 φ 1 ≤ C. Corollary 7.4. Let γ 1 (t) and γ 2 (t) be two smooth paths in H with d(γ 1 (t), γ 2 (t)) uniformly bounded. Suppose φ(t) is a smooth geodesic ray in H asymptotic to γ 1 , then it is also asymptotic to γ 2 . Proof. Let γ i (t, s) be the geodesic connecting φ(0) and γ i (t) parametrized by arc-length. Fix s, by assumption, d(γ 1 (t, s), φ(s)) → 0 as t → ∞. So in particular, d(φ(0), γ 1 (t)) → ∞. Suppose d(γ 1 (t), γ 2 (t)) ≤ C. Choose T large enough so that d(φ(0), γ 1 (T )) ≫ s+C. Then d(γ 1 (T, T −C), γ 2 (T, T −C)) ≤ 4C. By lemma 7.1, as T → ∞, d(γ 1 (T, s), γ 2 (T, s)) ≤ s T · 4C → 0. By definition, φ(t) is asymptotic to γ 2 . Similarly we can prove Corollary 7.5. Let γ(t) be a smooth path in H which is asymptotic to two smooth geodesic rays φ 1 (t) and φ 2 (t). Then φ 1 and φ 2 are parallel, i.e. d(φ 1 (t), φ 2 (t)) is uniformly bounded. If we assume one of them is tamed by bounded geometry, say φ 1 then by corollary 7.3, |φ 1 (t) − φ 2 (t)| C 1,1 φ 1 ≤ C. Proof of the main theorems Now we proceed to prove the main theorems. Lemma 8.1. Suppose g i is a sequence of Riemmanian metrics on a manifold M . If there are two sequences f i and h i of diffeomorphism of M such that f * i g i → g 1 , and h * i g i → g 2 in C ∞ , then f i • h −1 i converges by subsequence to a diffeomorphism f in C ∞ with f * g 2 = g 1 . The proof is standard using compactness. We omit it here. Corollary 8.2. The quotient J /G is Hausdorff in the C ∞ topology. χ 1 (t) φ 1 (t) φ 2 (t) χ 2 (t)f * i (J, ω φ i ) → (J 1 , ω 1 ) and h * i (J, ω ψ i ) → (J 2 , ω 2 ) in the C ∞ topology. If |φ i − ψ i | C 0 ≤ C, then |φ i − ψ i | C k ω φ i is bounded for all k, and there is a subsequence k i such that f −1 k i • h k i converges in C ∞ to a diffeomorphism f with f * J 1 = J 2 and f * ω 1 = ω 2 + √ −1∂ J 2∂ J 2 φ. The proof is quite standard now, given the volume estimates in [CH1]. We will omit it here. Proof. (of theorem 1.3). We may assume J 1 and J 2 are not in the G C leaf of J, the proof in the other case is similar. We proceed by contradiction. Suppose J 1 and J 2 were not in the same G orbit. Then by corollary 8.2 we can assume there are disjoint G invariant neighborhoods U 1 , U 2 of J 1 , J 2 respectively. Pick J ′ i in the intersection of U i with G C leaf of J. Now by theorem 5.3, we know that the Calabi flow J i (t) starting from J ′ i exits globally and converges to J i (∞) ∈ U i . So J 1 (∞) and J 2 (∞) are not in the same G-orbit either. By theorem 6.3, the corresponding Calabi flow φ i (t) in the space of Kähler metrics is asymptotic to a smooth geodesic ray which also degenerates some otherĴ i to J i (∞). Since J ′ 1 and J ′ 2 are both in the G C leaf of J, we can pull everything back to J and then we have two Calabi flows φ i (t) each asymptotic to a smooth geodesic ray χ i (t) tamed by bounded geometry. By [CC], d(φ 1 (t), φ 2 (t)) is decreasing, so by corollary 7.4, φ 1 (t) is also asymptotic to χ 2 (t). By corollary 7.3 and corollary 7.5 |χ 1 (t) − χ 2 (t)| C 1,1 ≤ C. So lemma 8.3 implies, there is no Kähler collapsing, and there is a diffeomorphism f with f * J 1 (∞) = J 2 (∞), and f * ω = ω+ √ −1∂∂φ. Since(f * ω, J 2 (∞)) and (ω, J 2 (∞) are both csc Kähler structures in the same Kähler class, by theorem 1.1, there is a diffeomorphism h with h * J 2 (∞) = J 2 (∞) and h * f * ω = ω, so (f • h) * (ω, J 1 (∞)) = (ω, J 2 (∞)). Contradiction. Proof. (of theorem 1.6). Suppose f * i (ω φ i , J) → (ω 1 , J 1 ), and h * i (ω ψ i , J) → (ω 2 , J 2 ). Since [ω] is integral, we see that [f * i ω φ i ] = [ω 1 ] for i large enough, so we can further assume that f * i ω φ i = ω 1 , and h * i ω ψ i = ω 2 . Then we can follow the proof of theorem 1.3. Proof. (of corollary 1.7). Suppose f * i J → J 1 . Since c 1 (J 1 ) > 0, we have c 1 (f * i J) > 0, and we can choose a sequence of Kähler metrics ω i in c 1 (J) such that f * i ω i → ω 1 . Then we can apply theorem 1.6. Further Discussions There are also some further interesting questions. Problem 9.1. A general notion of optimal degenerations and its relation to the Calabi flow. Generalize the theorem to the uniqueness of some "canonical" objects in the closure, allowing the occurrence of singularities. On the other hand, by the Yau-Tian-Donaldson conjecture, one would like to know if there is a direct algebraic-geometric counterpart of Theorem 1.3, i.e. whether a K-polystable adjacent Kähler structure is unique. Problem 9.2. Quantization approach( [D4], [Fi]). In the case of discrete automorphism group, Donaldson [D4] proved the existence of cscK metric implies asymptotic Chow stability. Theorem 1.1 in this case follows immediately. It looks like one can use the finite dimensional Kempf-Ness theorem to deal with Theorem 1.3 also. However, this can not be straightforward. The reason is that for an adjacent cscK Kähler structure whose underlying complex structure is different from the original one, the automorphism group can not be finite; and it is known that the existence of cscK metric(or even KE metric) does not necessarily imply asymptotic Chow poly-stability, see the recent counter-example in [OSY], [DZ]. It seems to the authors that more delicate work is required to proceed by the quantization method. Problem 9.3. It follows from our result that Tian's conjecture in [Ti2] is likely to hold for cscK metrics(The original conjecture allows mild singularities). In the case of general extremal metrics, we might need to modify the statement in Tian's conjecture a bit. This can be easily seen in the corresponding finite dimensional analogue. In that case any gradient flow can be reversed and we can get critical points in the limit along both directions of the flow. Clearly they are not in the same G orbit and therefore "adjacent" critical point is not necessarily unique. In our infinite dimensional case, the naive uniqueness also fails for adjacent extremal metrics. Such examples were already implicit in Calabi's seminal paper [Ca2]. Namely, we consider the blown up of P 2 at three distinct points p 1 , p 2 and p 3 (denoted by Bl p 1 ,p 2 ,p 3 (P 2 )), then by [APS], the class π * [ω F S ] − ǫ 2 ([E 1 ] + [E 2 ] + [E 3 ]) contains extremal metrics for ǫ small enough. If p 1 , p 2 and p 3 are in general position(i.e. they do not lie on a line), then Bl p 1 ,p 2 ,p 3 (P 2 ) are all biholomorphic and by [Ca2] the classes π * [ω F S ] − ǫ 2 ([E 1 ] + [E 2 ] + [E 3 ]) have vanishing Futaki invariant thus the extremal metrics are cscK. If p 1 , p 2 and p 3 lie on a line, Calabi pointed out in [Ca2] that there is no cscK metric due to the Lichnérowicz-Matsushima theorem. It is easy to see that for a fixed Kähler class π * [ω F S ] − ǫ 2 ([E 1 ] + [E 2 ] + [E 3 ]) , the extremal metrics in the case p 1 , p 2 and p 3 lie on a line are adjacent to the cscK metrics in the case p 1 , p 2 , p 3 are in general position. So we can find proper extremal metrics even adjacent to cscK metrics. C. Lebrun also pointed out to us another example, where we can look at the Hirzebruch surfaces F 2n of even degree. If n > m, then with appropriate polarization, F 2n is adjacent to F 2m , while in [Ca1], Calabi explicitly constructed extremal metrics in any Kähler classes. The problem where the uniqueness fails can be seen from the fact that our proof depends on the Calabi flow in an essential way. Since the Calabi flow can only detect de-stabilizing extremal metrics, we might want to consider only the uniqueness of de-stabilizing(i.e. energy minimizing) extremal metrics, as a modification of Tian's conjecture. This idea of de-stabilizing extremal metrics has already been implicitly discussed in [Ch3]. Problem 9.4. The integrality assumption in theorem 1.6 is just for fixing the symplectic form. It seems possible to remove this assumption. A Marle-Guillemin-Sternberg normal form In this appendix, we shall give a proof of the Marle-Guillemin-Sternberg normal form theorem for a Hamiltonian group action in the finite dimensional case(lemma 4.6). We shall also consider an infinite dimensional case for our purpose(lemma 6.2). We suppose that there is a compatible complex structure, which in general we can not standardize without some "errors". A.1 Model case We first look at a prototype. Suppose ω is a Kähler metric defined in a neighborhood of 0 in C n . Then we can not trivialize both the complex structure and the symplectic structure simultaneously, however, we can make either of them standard, with appropriate control on the other. First, it is easy choose a holomorphic coordinate such that ω = ω 0 + O(|z| 2 ), where ω 0 is the standard symplectic form on C n . In this way the complex structure is made standard, while the error on the symplectic form is quadratic. Now we denote α = ω − ω 0 . Let f t : z → tz be the contraction map. Then α = f * 1 α − f * 0 α = dθ, where θ = 1 0 f * t (X α)dt, and X = z ∂ ∂z +z ∂ ∂z . So θ = O(|z| 3 ). Let ω t = (1 − t)ω + tω 0 , then φ * t ω t = ω 0 , where φ t is the isotopy generated by the vector fields Y t satisfying Y t ω t = −θ. Thus, Y t = O(|z| 3 ) and so φ t (z) = z + O(|z| 3 ), and φ * t J 0 − J 0 = O(|z| 2 ) . In this way the symplectic structure is standard, with an quadratic error on the complex structure. A.2 Proof of lemma 4.5 Suppose a compact group G acts on a Kähler manifold (M, Ω, J) with moment map µ, and z 0 is a zero of the moment map, but not fixed by the whole group G. We denote by G 0 the isotropy group of z 0 and g 0 its Lie algebra. We also fix an Ad G -invariant metric on g. Now consider Y = G× G 0 (m ⊕ N ). Here N is the orthogonal complement of g.z 0 in (g.z 0 ) ω 0 , and m is the orthogonal complement of g 0 in g. We identify ρ ∈ m withρ ∈ J 0 · (g.z 0 ) through ρ, η = Ω 0 (ρ, X η ). This also induces an identification between m and g.z 0 which is different from the one coming from the action. G 0 acts on (N, Ω N = Ω 0 | N ) linearly with a natural moment map µ N . Y is in fact the symplectic quotient of G × (g 0 ⊕ m ⊕ N ) ≃ T * G × N by G 0 . The induced symplectic form on Y is given explicitly by(see [OR]) Ω [g,ρ,v] ((L g ξ 1 , ρ 1 , v 1 ), (L g ξ 2 , ρ 2 , v 2 )) := ρ 2 + d v µ N (v 2 ), ξ 1 − ρ 1 + d v µ N (v 1 ), ξ 2 + ρ + µ N (v), [ξ 1 , ξ 2 ] +Ω 0 (X ξ 1 , X ξ 2 ) + Ω 0 (v 1 , v 2 ) = ρ 2 + d v µ N (v 2 ), ξ 1 − ρ 1 + d v µ N (v 1 ), ξ 2 + ρ, [ξ 1 , ξ 2 ] + Ω 0 (v 1 , v 2 ) + µ N (v), [ξ 1 , ξ 2 ] = Ω 0 (X 1 , X 2 ) + ρ, [ξ 1 , ξ 2 ] + ( d v µ N (v 2 ), ξ 1 − d v µ N (v 1 ), ξ 2 ) + µ N (v), [ξ 1 , ξ 2 ] , where we identify T g G with g through left translation, and X i = X ξ i +α i +v i is viewed as a tangent vector at z 0 . The G action on Y is Hamiltonian with moment map:μ : Y → g; [g, ρ, v] → Ad * g (µ N (v) + ρ). To prove lemma 4.6, we need to trace the proof of the relative Darboux theorem. Since Ω is Kähler, we can choose holomorphic coordinates on a neighborhood V of z 0 such that Ω−Ω 0 = O(r 2 ). Let exp z 0 be the exponential map with respect to the metric induced from J and Ω. Then we have exp z 0 (ρ + v) = z 0 + ρ + v + O(r 3 ). Consider the map exp : G × G 0 (m ⊕ N ) → M ; (ξ, ρ, v) → e ξ .exp z 0 (ρ + v). This is a diffeomorphism from a G-invariant neighborhood U of G × 0 to a neighborhood V of G.z 0 . Indeed, its derivative at [e, 0, 0] is given by dexp z 0 : m ⊕ m ⊕ N → T z 0 M = m ⊕ m ⊕ N ; (ξ, ρ, v) → (L(ξ), ρ, v), where we have made use of the identification (25), and L : m → m is the the automorphism such that (L(ξ), η) = g 0 (X ξ , X η ) for any ξ, η ∈ m. Denote Ω ′ = exp * Ω and J ′ = exp * J, then we have J ′ (0,0,0) (ξ, ρ, v) = (L −1 (ρ), −L(ξ), J 0 · v). We can extend J ′ to an almost complex structureJ defined on Y . On V , denote by (z,z) the coordinates for N , x for g.z 0 and y for W = J 0 · (g.z 0 ). The tangent space at z 0 is naturally identified with V . Let ( ∂ ∂v , ∂ ∂v ), and ∂ ∂ρ be the vector fields on U corresponding to ∂ ∂z , ∂ ∂z and ∂ ∂y (on m ⊕ N ) respectively and ∂ ∂ξ the vector fields induced by left translation of ∂ ∂x ∈ T z 0 V . These vector fields could also be viewed as vector fields on V through the map exp. Then at [e, ρ, v] we have ∂ ∂v = ∂ ∂z + O(r 2 ); ∂ ∂v = ∂ ∂z + O(r 2 ); ∂ ∂ρ = ∂ ∂y + O(r 2 ); L ∂ ∂ξ = ∂ ∂x + ξ.y + ξ.z + O(r 2 ). Now it is easy to see that Ω( ∂ ∂z , ∂ ∂z ) = Ω ′ ( ∂ ∂z , ∂ ∂z ) + O(r 2 ), Ω( ∂ ∂z , ∂ ∂z ) = Ω ′ ( ∂ ∂z , ∂ ∂z ) + O(r 2 ) = O(r 2 ); Ω( ∂ ∂z , ∂ ∂y ) = Ω ′ ( ∂ ∂z , ∂ ∂y ) + O(r 2 ) = O(r 2 ); Ω( ∂ ∂y , ∂ ∂y ) = Ω ′ ( ∂ ∂y , ∂ ∂y ) + O(r 2 ) = O(r 2 ). andΩ ( ∂ ∂z , ∂ ∂x ) =Ω( ∂ ∂v + O(r 2 ), L ∂ ∂ξ − ξ.y − ξ.z + O(r 2 )) = Ω ′ ( ∂ ∂z , ∂ ∂x ) + O(r); similarly,Ω ( ∂ ∂y , ∂ ∂x ) = Ω ′ ( ∂ ∂y , ∂ ∂x ) + O(r) ; Ω( ∂ ∂x , ∂ ∂x ) =Ω(L ∂ ∂ξ − ξ.y − ξ.z + O(r 2 ), L ∂ ∂ξ − ξ.y − ξ.z + O(r 2 )) = Ω ′ ( ∂ ∂x , ∂ ∂x ) + O(r). Therefore, we obtain: α = Ω ′ −Ω = O(r 2 )(dzdz+dzdy+dzdy+dydy)+O(r)(dzdx+dzdx+dydx+dxdx). Now let f t : (g, ρ, v) → (g, tρ, tv), then X t =ḟ t = tρ ∂ ∂ρ + tv ∂ ∂v + tv ∂ ∂v = ty ∂ ∂y + tz ∂ ∂z + tz ∂ ∂z + O(r 2 ). We have α = dθ, with θ = 1 0 f * t (X t α)dt = 1 0 (ty ∂ ∂y + tz ∂ ∂z + tz ∂ ∂z + O(r 2 )) [O(r 2 )(dzdz + dzdy + dzdy + dydy) +O(r)(dzdx + dzdx + dydx + dxdx)]dt = O(r 2 )dx + O(r 3 ), where the estimate is valid at [e, ρ, v]. Let Ω t = (1 − t)Ω + tΩ ′ , then φ * t Ω t =Ω, whereφ t = Y t satisfies Y t Ω t = θ. Since Ω t = Ω 0 +O(r 2 )(dzdz+dzdy+dzdy+dydy)+O(r)(dzdx+dzdx+dydx+dxdx). So at [e, ρ, v], we have Y t = O(r 2 ) ∂ ∂y + O(r 3 ) = O(r 2 ) ∂ ∂ρ + O(r 3 ). Since Y t is G-invariant, this is also true at [g, ρ, v] for g close to Id. Thus the integral curve of Y t satisfies v t = v 0 + O(r 3 0 ); ρ t = ρ 0 + O(r 2 0 ). Therefore, (φ * t J ′ ) ∂ ∂v = φ −1 t * J ′ ((φ t ) * ∂ ∂v ) = φ −1 t * J ′ ( ∂ ∂v +O(r 2 )) = φ −1 t * J( ∂ ∂z +O(r 2 )) =J ∂ ∂v +O(r 2 ), and similarly (φ * t J ′ ) ∂ ∂v =J ∂ ∂z + O(r 2 ). Let Φ = φ 1 , then Φ * Ω ′ =Ω. We get the required estimate that Φ * J ′ −J = O(r), and Φ * J ′ · X −J · X = O(r 2 )|X|, for X ∈ N . Hence lemma 4.6 is proved. A.3 Proof of lemma 6.2 Now we proceed to our infinite dimensional problem, following the same route as in the finite dimensional setting. However, there are a few more technical issues, as we shall see below. Suppose (M, ω, J 0 ) is a csc Kähler manifold. Then the relevant group G is the group of Hamiltonian diffeomorphisms of (M, ω), which acts on the space J of almost complex structures compatible with ω. Here in order to apply the implicit function theorem, we shall put C ∞ topology on these infinite dimensional objects which makes them into tame Fréchet spaces( [Ha]). J inherits a natural (weak) Kähler structure (Ω, I) from the original Kähler manifold M . The action of G preserves the Kähler structure and has a moment map given by the Hermitian scalar curvature functional m(J) = S(J)−S. Denote by G the identity component of the holomorphic isometry group of (M, ω, J 0 ). Let g and g 0 the Lie algebra of G, G respectively. Then we have an L 2 orthogonal decomposition g = g 0 ⊕ m, where m is the image of Q * = ReD * . We want to show that a neighborhood V of J 0 in J is G-equivariantly Hamiltonian diffeomorphic to a neighborhood U in Y = G × G (m ⊕ N ), where G acts adjointly on g by f.φ = f * φ. N is the orthogonal complement of the image of D in Ω 0,1 (T 1,0 ), and G acts on N by pulling back: g.µ = g * µ. This action is Hamiltonian with moment map given by m N : N → g 0 ; (m N (v), ξ) = 1 2 Ω(ξ.v, v). Similar to the finite dimensional case we can define a (weak) symplectic form on U . The left G action on Y is Hamiltonian with moment map given bỹ m : [g, ρ, v] = g * (ρ + m N (v)). The exponential map Ψ on J with respect to the natural Riemannian metric is well defined by fiber-wise exponential map of the symmetric space Sp(2n)/U (n), and it is easy to see that it is a local tame embedding of a neighborhood of the origin in Ω 0,1 S (T 1,0 ) into J . Using the local holomorphic coordinate chart of J , the Kähler form satisfies Ω µ = Ω 0 + O(|µ| 2 ). It is also clear that Ψ(µ) = µ + O(|µ| 3 ). Here the norms on both sides could be taken to be the same. Now we can define a map Φ : U → J ; [g, ρ, v] → g * Ψ(ρ + v). Lemma A.1. G is a smooth tame Lie group. Proof. We first prove it is a smooth tame space. We can identify a Hamiltonian diffeomorphism H with an exact Lagrangian graph G H in M = M ×M , i.e. G H = {(x, H(x))|x ∈ M }. Here M is endowed with a canonical symplectic form ω ′ = π * 1 ω − π * 2 ω, where π i is the projection map to the i-th factor. A Lagrangian graph is called exact if it can be deformed by exact Lagrangian isotopies to the identity. We can construct local charts for G as follows. Given any H ∈ G, by Weinstein's Lagrangian neighborhood theorem( [We]), we can choose a symplectic diffeomorphism between a tubular neighborhood U of G H in M and a tubular neighborhood V of 0 section in the cotangent bundle T * M . Then locally any Hamiltonian diffeomorphism close to H is represented by the graph of an exact one-form, i.e. the differential of some real valued function on M . So locally U can be identified with an open subset of C ∞ 0 (M ; R). Thus G is modelled on C ∞ 0 (M ; R). Now we check the transition function is smooth tame. In our case locally between any two charts there is a symplectic diffeomorphism of the cotangent bundle F : T * M → T * M which is identity on the zero section. Then the induced transition map is smooth tame, by observing that the C k distance between the graph of exact one-forms dφ 1 and dφ 2 is equivalent to the C k+1 distance between φ 1 and φ 2 . Similary we can prove that the group multiplication and inverses are both smooth tame. Since the finite dimensional group G acts smooth tame and freely on G × (m ⊕ N ), we know that Y = G × G (m ⊕ N ) is a tame space with a smooth tame Gaction. Proof. It is clear by definition that the map is smooth and tame. The k-th derivative of Φ is tame of degree k +1. To apply Hamilton's implicit function theorem, we need to study the derivative of Φ near [Id,0,0]. At δ = [g, ρ, v], we denote µ = Φ(δ). Then we have D δ Φ : m ⊕ m ⊕ N → Ω 0,1 S (T 1,0 ); [φ, ψ, u] → (Id −μ) • (I − µ •μ) −1 • [Q µ φ + g * DΨ| ρ+v ( √ −1D 0 ψ + u)] • (Id − µ) −1 . To find the inverse to D δ , we need to first decompose Ω 0,1 S (T 1,0 ) into the direct sum of DΨ −1 • ImQ µ | m and KerQ * 0 with estimate. This can be done using elliptic theory. We can obtain that ν = (DΨ) −1 • Q µ φ + √ −1Q 0 ψ + η, where η ∈ KerD * 0 . Take the map P µ : ν → (φ, √ −1Q 0 ψ + η). Then it is smooth tame again by elliptic estimates. Since the inverse of D δ Φ is the combination of P with some other smooth tame operator, it is also smooth tame. Then we can apply the Nash-Moser implicit function theorem( [Ha]) to conclude the lemma. As in the finite dimensional case, there is a canonically defined (weak) symplectic form on U given bỹ Ω [g,ρ,v] ((L g ξ 1 , ρ 1 , v 1 ), (L g ξ 2 , ρ 2 , v 2 )) := ρ 2 + d v µ N (v 2 ), ξ 1 − ρ 1 + d v µ N (v 1 ), ξ 2 + ρ + µ N (v), [ξ 1 , ξ 2 ] + Ω 0 (v 1 , v 2 ) = (D * 0 D 0 ξ 1 , ρ 2 ) − (D * 0 D 0 ρ 1 + d v µ N (v 1 ) − [D * 0 D 0 ρ + µ N (v), ξ 1 ], ξ 2 ) + Ω 0 (v 1 , v 2 ) + (ξ 1 , d v µ N (v 2 )) By the above lemma we can pull back the symplectic form Ω and the complex structure I to U , denoted by Ω ′ and I ′ respectively. There is also a canonical almost complex structure I 0 on U defined by I 0 : m ⊕ m ⊕ N → m ⊕ m ⊕ N ; (ξ, ρ, v) → ((D * 0 D 0 ) −1 ρ, −D * 0 D 0 ξ, I(0)(v)). It is easy to see that I ′ = I 0 at [Id, 0, 0]. Proposition A.3. There are neighborhoods U i , V i (i = 1, 2)(U 2 ⊂ U 1 ) of [Id, 0, 0] in Y and two G-equivariant smooth tame maps Σ 1 : U 1 → V 1 , Σ 2 : V 2 → U 2 , which fixes the G-orbit of [Id, 0, 0] such that Σ 1 • Σ 2 equal to the identity and such that Σ * 1Ω = Ω ′ , Σ * 2 Ω ′ =Ω, and for any X ∈ N , and [g, ρ, v] ∈ V 2 , (DΣ 1 ) • I ′ • (DΣ 2 )(X) − I 0 (X) = O(r 2 ) · |X|, and at [Id, 0, 0], (DΣ 1 ) • I ′ • (DΣ 2 ) = I 0 . Here the estimate is only in the tame sense, i.e. the norm on the left hand side might be weaker than that on the right, r is the norm of [g, ρ, v]. Proof. The idea of the proof is the same as the finite dimensional case. The main difficulty is to show the existence of solutions to the involved O.D.E's in infinite dimension. Once this is established, then everything else will follow formally. First we have (µ = Φ([Id, ρ, v])) Ω ′ [g,ρ,v] ((L g ξ 1 , ρ 1 , v 1 ), (L g ξ 2 , ρ 2 , v 2 )) = Ω µ (D µ ξ 1 + dΦ * ( √ −1D 0 ρ 1 + v 1 ), D µ ξ 2 + dΦ * ( √ −1D 0 ρ 2 + v 2 )) = −Im(D µ ξ 1 + dΦ * ( √ −1D 0 ρ 1 + v 1 ), D µ ξ 2 + dΦ * ( √ −1D 0 ρ 2 + v 2 )) L 2 = (−ImD * µ D µ ξ 1 − ReD * µ • dΦ * (D 0 ρ 1 − √ −1v 1 ), ξ 2 ) +(ReD * 0 • (dΦ * ) t (D µ ξ 1 ) + ImD * 0 • (dΦ * ) t dΦ * (D 0 ρ 1 − √ −1v 1 ), ρ 2 ) +Ω 0 ((dΦ * ) t (D µ ξ 1 + dΦ * ( √ −1D 0 ρ 1 + v 1 )), v 2 ) As in the finite dimensional case, we need to solve an O.D.E. Let Ω t = (1 − t)Ω + tΩ ′ . The isotopies f t : [g, ρ, v] → [g, tρ, tv] gives rise to timedependent vector field X t (f t ([g, ρ, v])) = [0, ρ, v]. We first need to solve another time-dependent vector field Y t through the following relation: Ω t [g,ρ,v] (Y t , Z) = 1 0 (Ω ′ −Ω) [g,sρ,sv] ((0, ρ, v), f s * Z)ds Notice that Y t is G-invariant. So we can assume g = Id. Let Y t = (ξ 1 , ρ 1 , v 1 ) and Z = (ξ 2 , ρ 2 , v 2 ). By choosing Z arbitrarily, we get the following system of equations: −tImD * µ D µ ξ 1 − tReD * µ • dΦ * (D 0 ρ 1 − √ −1v 1 ) −(1 − t)D * 0 D 0 ρ 1 − (1 − t)d v µ N (v 1 ) + (1 − t)[D * 0 D 0 ρ + µ N (v), ξ 1 ] + 1 0 ReD * µs • dΦ s * (sD 0 ρ + √ −1sv) + D * 0 D 0 (sρ) − d v µ N (sv)ds ∈ g 0(27)tReD * 0 • (dΦ * ) t D µ ξ 1 + tImD * 0 • (dΦ * ) t • (dΦ * )(D 0 ρ 1 − √ −1v 1 ) +(1 − t)D * 0 D 0 ξ 1 − 1 0 ImD * 0 • (dΦ s ) t * • (dΦ s ) * (sD 0 ρ − √ −1sv)ds ∈ g 0 (28) t(dΦ * ) t • (D µ ξ 1 + dΦ * ( √ −1D 0 ρ 1 + v 1 )) + (1 − t)v 1 + (1 − t)(d v µ N ) * (ξ 1 ) − 1 0 (dΦ s * ) t • dΦ s * (s √ −1D 0 ρ + sv) − svds ∈ ImD 0(29) Since Ω ′ andΩ are both non-degenerate, this system admits a (unique) weak solution. Then applying elliptic regularity, the solutions are smooth. Next we shall prove that there are two neighborhoods N 1 , N 2 of 0 in m × N , and a smooth tame map F from N 1 to C ∞ ([0, 1], m × N ) such that the time 1 evaluation of the image of F is a smooth tame map from N 1 to N 2 and for any (ρ, v) ∈ N 1 ,    d dt F t (ρ, v) = (ρ 1 (t), v 1 (t)), t ∈ [0, 1] ; F 0 (ρ, v) = (ρ, v). To prove this claim, we shall exploit Hamilton's implicit function theorem again. Define a map H : C ∞ ([0, 1], m × N ) → (m × N ) × C ∞ ([0, 1], m × N ) which sends (ρ(t), v(t)) to (ρ(0), v(0)) × (ρ(t) − ρ 1 (t),v(t) − v 1 (t)). It is clear that H is a smooth tame map and H(0) = 0. We shall show that for x = (ρ(t), v(t)) close to zero, the derivative of H at x is invertible and its inverse is smooth tame. Let δx = (ρ(t),ṽ(t)), then the derivative of H along δx is given by (ρ(0),ṽ(0)) × (ρ − δρ 1 (ρ), (v − δv 1 (ṽ)). So the invertibilty of dH is equivalent to the solvability of the Cauchy problem of the following linear system along (ρ(t), v(t)):    d dt (α, u) = (δρ 1 (α), δv 1 (u)) + (β, q), t ∈ [0, 1] (α(0), u(0)) = (ρ(0),ṽ(0)). Thus we need to linearize equations (27) and (29). As a result, we get the following           α (t) = A 1 (ρ(t), v(t))α(t) + A 2 (ρ(t), v(t))u(t) + β(t), u(t) = B 2 (ρ(t), v(t))α(t) + C 2 (ρ(t), v(t))α(t) + B 0 (ρ(t), v(t))u(t) + q(t), (α(0), u(0)) = (ρ(0),ṽ(0)), where A i , B i , C i are pseudo-differential operators of order i whose coefficients depend on (ρ(t), v(t)). Let w(t) = u(t)−B 2 (ρ(t), v(t))α(t). Then the systems of equations for (α(t), w(t)) become symmetric hyperbolic, for which the Cauchy problem is always solvable with estimates, see [AG]. From the proof we can check that the solution depends tamely on (ρ(t), v(t)), (β(t), q(t)) and the initial condition (ρ(0),ṽ(0)). So by Hamilton's implicit function theorem H has a local smooth tame inverse. Let F = H −1 (−, 0) and the claim is then proved. Now for any [g, ρ, v] close to [Id, 0, 0], we obtain a path (ρ(t), v(t)) = F t (ρ, v). Then we can solve the O.D.Eġ(t) = L g(t) ξ 1 (t), where ξ 1 (t) is determined by (ρ(t), v(t)). [g(t), ρ(t), v(t)] is then an integral curve of Y t by the G-invariancy. Now we define Σ 2 : V 2 → U 2 ; [g, ρ, v] → [g 1 , F 1 (ρ, v)]. Then from the previous arguments we know that Σ 2 is smooth tame and fixes G.[Id, 0, 0]. Moreover, Σ * Ω ′ =Ω. It follows from equations (27), (28), (29) that we have a tame estimate |v 1 (t)| ≤ C · (|ρ(t)| + |v(t)|) 3 . Since |(v(t), ρ(t))| ≤ C · |(v(0), ρ(0))|, we obtain |v 1 (t) − v 1 (0)| ≤ C · (|ρ(0)| + |v(0)|) 3 . By symmetry, we can obtain the map Σ 1 . Then one can check that the required estimates hold. 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R Thomas, arxiv: math/0512411Notes on GIT and symplectic reduction for bundles and varieties. R. Thomas. Notes on GIT and symplectic reduction for bundles and varieties , arxiv: math/0512411. Kähler-Einstein metrics with positive scalar curvature. G Tian, Invent. Math. 130G. Tian. Kähler-Einstein metrics with positive scalar curvature, In- vent. Math. 130 (1997), 1-39. Extremal metrics and geometric stability. G Tian, Special issue for S. S. Chern. Houston J. Math. 282G. Tian. Extremal metrics and geometric stability. Special issue for S. S. Chern. Houston J. Math. 28 (2002), no. 2, 411-432. The K-energy on small deformations of constant saclar curvature Kähler manifolds. V Tosatti, arxiv: math/1010.1859V. Tosatti. The K-energy on small deformations of constant saclar cur- vature Kähler manifolds. arxiv: math/1010.1859. A new holomorphic invariant and uniqueness of Kähler-Ricci solitons. G Tian, X-H Zhu, Comment. Math. Helv. 772G. Tian, X-H. Zhu. A new holomorphic invariant and uniqueness of Kähler-Ricci solitons. Comment. Math. Helv., 77 (2002), no. 2, 297-325. Perelman's W-functional and stability of Kähler-Ricci flow. G Tian, X-H Zhu, arxiv: math/0801.3504G. Tian, X-H.Zhu. Perelman's W-functional and stability of Kähler- Ricci flow, arxiv: math/0801.3504. Symplectic manifolds and their Lagrangian submanifolds. A Weinstein, Advances in Math. 6A. Weinstein. Symplectic manifolds and their Lagrangian submanifolds. Advances in Math. 6 (1971), 329-346. . X X , 480Department of Mathematics, University of Wisconsin-MadisonX.X.C. Department of Mathematics, University of Wisconsin-Madison, 480 . Lincoln Drive, U S A Madison, Xxchen@math, Stony Brook, NY 11794, U.SDepartment of Mathematics, Stony Brook Universitywisc.edu; New address. A./xiu@math.sunysb.eduLincoln Drive, Madison, WI 53706, U.S.A./ xxchen@math.wisc.edu; New address: Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, U.S.A./xiu@math.sunysb.edu. New address: Department of Mathematics. Imperial College, London SW7 2AZ, U.K.New address: Department of Mathematics, Imperial College, London SW7 2AZ, U.K./s.sun@imperial.ac.uk.

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{'abstract': 'We prove that constant scalar curvature Kähler metric "adjacent" to a fixed Kähler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T picture described by Donaldson. We prove that the Calabi flow near a cscK metric exists globally and converges uniformly to a cscK metric in a polynomial rate. Viewed in fixed a Kähler class, the Calabi flow is also shown to be asymptotic to a smooth geodesic ray at infinity. This latter fact is also interesting in the finite dimensional case, where we show that the downward gradient flow of the Kempf-Ness function in a semi-stable orbit is asymptotic to the direction of optimal degeneration.', 'arxivid': '1004.2012', 'author': ['Xiuxiong Chen ', 'Song Sun '], 'authoraffiliation': [], 'corpusid': 115177135, 'doi': '10.4007/annals.2014.180.2.1', 'github_urls': [], 'n_tokens_mistral': 41204, 'n_tokens_neox': 36122, 'n_words': 21313, 'pdfsha': '73893db02713bbe391f43f5fdc60ea2aac086683', 'pdfurls': ['https://arxiv.org/pdf/1004.2012v2.pdf'], 'title': ['Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics', 'Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics'], 'venue': []}

arxiv

Correlated Stochastic Knapsack with a Submodular Objective Subrata Mitra 3 Aug 2022 Sheng Yang Adobe Research Adobe Research Northwestern University Northwestern University Samir Khuller Adobe Research Adobe Research Northwestern University Northwestern University Sunav Choudhary Adobe Research Adobe Research Northwestern University Northwestern University Adobe Research Adobe Research Adobe Research Northwestern University Northwestern University Kanak Mahadik Adobe Research Adobe Research Northwestern University Northwestern University Correlated Stochastic Knapsack with a Submodular Objective Subrata Mitra 3 Aug 20222012 ACM Subject Classification Theory of computation → Rounding techniquesTheory of com- putation → Stochastic approximationTheory of computation → Stochastic control and optimizationTheory of computation → Submodular optimization and polymatroids Keywords and phrases Stochastic Knapsack, Submodular Optimization, Stochastic Optimization We study the correlated stochastic knapsack problem of a submodular target function, with optional additional constraints. We utilize the multilinear extension of submodular function, and bundle it with an adaptation of the relaxed linear constraints from Ma [Mathematics of Operations Research, Volume 43(3), 2018] on correlated stochastic knapsack problem. The relaxation is then solved by the stochastic continuous greedy algorithm, and rounded by a novel method to fit the contention resolution scheme (Feldman et al. [FOCS 2011]). We obtain a pseudo-polynomial time (1 − 1/ √ e)/2 0.1967 approximation algorithm with or without those additional constraints, eliminating the need of a key assumption and improving on the (1−1/ 4 √ e)/2 0.1106 approximation by Fukunaga et al. [AAAI 2019]. Introduction The knapsack problem is one of the most celebrated frameworks to model profit maximization with limited resources. Though well understood in its basic form, many new variants were proposed to model and target more complicated problems. One line of variants take randomness into consideration. Such randomness may appear on item sizes only, on item profits only, or on both in a correlated fashion. A significant body of work [4,11,23,28,29] connects knapsack problem with the field of stochastic optimization, greatly broadening the spectrum of knapsack problems while introducing various challenges for theoretical analysis. Another line of variants model diminishing returns in the profit, leading to the field of submodular optimization [30,15,6,8,13,14], which enjoys tremendous popularity both in theory and in practice. The two lines of work are connected together into stochastic submodular optimization, another fruitful field [3,21,10,16,19,18,22,34]. In this work, we follow this line, and consider a correlated stochastic knapsack problem with a submodular target function. We arrived at this problem when modeling the spot scheduling problem in Yang et al. [33] (see details in Appendix B). A slight variant of the final problem was first proposed in Fukunaga et al. [17], trying to model "performance-dependent costs of items" in stochastic submodular optimization. This problem turns out to be a very powerful framework that applies to several other real world applications, like recommendation systems [35,1], and batch-mode active learning [25]. Formal Problem Statement There are n items, each takes a random size i ∈ N with probability p i (size i ), and gets a reward that corresponds to its size. In other words, for each item i, there is a reward function R i : N → [M ], such that r i = R i (size i ). (For simplicity, we define [n] to be the set {0, 1, 2, . . . , n}, and M a positive integer that upper bounds the maximum reward.) We assume each R i to be non-decreasing, i.e., the larger an item, the more reward it deserves. We are given a budget B ∈ N for the total size of items, and wish to extract as much profit as possible. The total profit is a lattice-submodular function 1 f : [M ] n → R + on the rewards of included items, and we wish to maximize its expectation 2 . Items are put in the knapsack one by one. As soon as an item is put in the knapsack, its reward and size are revealed. We halt when the knapsack overflows (not collecting the last item's reward), and proceed to add another item otherwise. We consider adaptive policies, i.e., we can choose an item to include, observe its realized size, and make further decisions based on the realized size. At first, only the reward function and the size distribution of items are known. When the policy includes an item i, its size i is realized, and so is its reward r i = R i (size i ). In this work, we only consider adaptive policies without cancellation, i.e., the policy can make its decision based on all the realizations it has seen so far, and the inclusion of an item is irrevocable. For a vector q ∈ [M ] n , let Pr γ [q] denote the probability that we get outcome q when running policy γ. Note this probability is with respect to the randomness both in the state of items and in the policy γ. value obtained by γ. Our aim is to find a policy γ that maximizes f avg (γ). We say γ is an α-approximation policy if f avg (γ) ≥ αf avg (γ * ) for any policy γ * . In addition to all the above, we further require that the chosen set of items S be an independent set of a partition matroid 3 I = {I k } k∈ [K] . This is without loss of generality as we can put each item in a separate partition, and every subset of items is valid. The additional constraint allows us to impose conflicts between items, which is needed for the modeling in Yang et al. [33]. More importantly, it is also crucial if we are to allow the attempt to include an item that could possibly overflow the knapsack, a case unsolved and left as open problem in Fukunaga et al. [17] (see details in Section 1.3). This partition matroid is also used to ensure the correctness of our approach based on a time-indexed LP. Our Contributions We present a pseudo-polynomial time algorithm for the correlated stochastic knapsack problem with a submodular target function. It computes an adaptive policy for this problem which is guaranteed to achieve (1 − 1/ √ e)/2 0.1967 of the optimal solution on expectation. It improves on the (1 − 1/ 4 √ e)/2 0.1106 approximation algorithm from Fukunaga et al. [17]. Furthermore, we eliminate one key assumption in Fukunaga et al. [17] which does not allow the inclusion of any item which could possible overflow the budget. Eliminating An Assumption in Previous Work In Fukunaga et al. [17], the authors considered a slightly different problem. They made two assumptions, and we managed to eliminate one of them. The first assumption states that larger size means larger reward for every particular job. This assumption is reasonable for general problems and remains crucial in our analysis. The second assumption states that we will never select an item which could overflow the budget, given the realization of selected items 4 . However, for many cases, selecting such an item is a desirable choice since additional value is obtained with high probability. If we are unlucky and the size goes beyond the remaining budget, we either receive a partial value, or do not get any value at all. Our Techniques If the target function is linearly additive, this problem becomes the correlated stochastic knapsack problem. For this problem, Gupta et al. [23] gave an 1/8 approximation algorithm for adaptive policies based on LP relaxation. The approximation ratio was improved to 1/(2 + ) by Ma [29], via a different time indexed LP formulation and a more sophisticated rounding scheme. Fukunaga et al. [17] extends the 1/8 approximation algorithm, and achieve a (1 − 1/ 4 √ e)/2 approximation for a case with submodular target function. This is achieved via a combination of the stochastic continuous greedy algorithm [2] (for getting a fractional solution), and the contention resolution scheme [14] (for rounding). A natural idea for improvement is to take ingredients from the 1/(2 + ) algorithm by Ma [29]. While the LP can be easily adapted, its rounding exhibits complicated dependencies that can be hard to analyze. We also have no luck with a direct application of the contention resolution scheme [9,13,14], a powerful technique in submodular optimization. The difficulties come in two folds. First, the original scheme is based on FKG inequality, which requires an independently rounded solution (possibly invalid) to start with. Any attempt to enforce our partition matroid at this step will break the whole scheme. This invalid solution is later fixed by ignoring some items from the rounded solution, and we need a way to impose the additional partition matroid constraint. Second, the ignoring step needs a critical "monotone" property. At a high level, the property says that the more items you choose, the lower the probability every other items will be selected (See Section 5.1 for a rigours definition). While this may seem trivially true for any reasonable algorithm, it is not. In particular, it does not hold for Ma's algorithm [29], due to its complicated dependencies. The first difficulty is not hard: if we happen to pick two items from the same partition, we just throw the later one out. Unfortunately, this makes the second obstacle even harder. The second obstacle is overcome by designing a brand-new rounding scheme which allows the direct analysis on the correlated probability of events. In order to achieve the aforementioned monotone property, we insert phantom items to block some "time slots" even when no item is there to conflict with. Such phantom items may be of independent interest for other applications of the contention resolution scheme. This alternate way of achieving monotonicity simultaneously free our analysis from one assumption mentioned in Section 1.3, which was needed in Fukunaga et al. [17] in their proof of the monotone property. A factor of (1 − 1/ √ e) is lost for the continuous optimization part, and another factor of 2 is lost for rounding, leading to our (1 − 1/ √ e)/2 0.1967 approximation algorithm. Other Related Works Stochastic Knapsack Problem The stochastic version of the knapsack problem has long been studied. Kleinberg et al. [27], and Goel and Indyk [20] consider the stochastic version to maximize profit that will overflow the budget with probability at most p. However, they assume deterministic profits and special size distributions. Dean et al. [12] relax the limit on size and allow arbitrary distributions for item sizes. They investigate the gap between non-adaptive policies (the order of items to insert is fixed) and adaptive policies (allowed to make dynamic decision based on the realized size of items) and give a polynomial-time non-adaptive algorithm that approximates the optimal adaptive policy within a factor of 1/4 in expectation. They also give an adaptive policy that approximates within a factor of 1/(3 + ) for any constant > 0. Bhalgat et al. [4] improves on this and give a bi-criteria (1 − ) algorithm by relaxing the budget by (1 + ). Dean et al. [11] show that if correlation between size and reward is allowed, the problem would be PSPACE-hard. Gupta et al. [23] considered the case where the size and reward of an item can be arbitrarily correlated, and give an 1/8 approximation. Li and Yuan [28] improved on this and get a 1/(2 + ) approximation with correlations and cancellation when fraction of extra space is allowed. This was further improved by Ma [29], who gets the same approximation ratio but without the budget augmentation requirement. Submodular Maximization Nemhauser et al. [30] studied the problem of maximizing a monotone submodular function subject to a cardinality constraint and gave the standard greedy (1 − 1/e)-approximation algorithm. For the case with a matroid constraint, Fisher et al. [15] showed that the standard greedy algorithm gives a 1/2-approximation. This was improved to (1 − 1/e) by Calinescu et al. [6], via the continuous greedy algorithm, which was originally developed by Calinescu et al. [5] for the submodular welfare problem. In this algorithm, the target function is relaxed via an exponential multilinear-extension. Though exponential, this version can be approximately solved to arbitrary precision in polynomial time. The fractional solution is then rounded via pipage rounding [5,6,32] or other rounding schemes [8]. In order to generalize the problem for other constraints and non-monotone submodular functions, a general rounding framework contention resolution scheme was proposed [6,13,14]. In this framework, the rounding step happens in two phases, an independent rounding phase followed by a pruning phase, where the second phase ensures an upper bound on the probability that an element is pruned. One line of stochastic submodular optimization [3] assumes items have stochastic states, and would like to maximize a monotone submodular function on the stochastic states, under constraints on the set of chosen items. In other words, the constraints only depend on the selection of items, but not on the stochastic states of them. This is a generalization of the stochastic knapsack problem where the size of items are deterministic. Various settings of this problem are investigated by a series of follow-up works [21,10,16,19,18,22,34]. Asadpour and Nazerzadeh [2] considers the maximization of a monotone lattice-submodular function. In this problem, each selected item has a stochastic state (a non-negative real number). The target function accepts a vector of such numbers, and satisfies lattice-submodularity (defined in Section 3). In their problem, only the states are stochastic, while the matroid constraint is on the set of selected items. Fukunaga et al. [17] pushed one step further and allowed the constraints to be dependent on the state of items, but limited the set of states to be non-negative integers. Preliminary We start with some notations. The description of the spot scheduling problem [33] is delayed to Appendix B, together with its modeling and reduction to the problem we consider. In Section 3.1, we explain how we reduce and manage to eliminate one critical assumption in the previous work by Fukunaga et al. [17]. Given two d dimensional vectors u, v ∈ [n] d , we write u ≤ v if the inequality holds coordinate wise, i.e. ∀i ∈ [d], u(i) ≤ v(i). Similarly, u ∨ v and u ∧ v are defined coordinate wise: (u ∨ v)(i) = max{u(i), v(i)}, (u ∧ v)(i) = min{u(i), v(i)}. Consider a base set [n], a matroid is defined to be an independent set I ⊆ 2 n . This independent set needs to contain ∅, and if A ∈ I, so is every A ⊆ A. Furthermore, if A, B ∈ I and |A| > |B|, then there exists an element x ∈ A \ B such that B ∪ x is in I. Particularly, for a partition matroid {I k } k∈[K] where I i ∩ I j = ∅, ∀i = j, its independent set I is {S|∀k, S ∩ I k ≤ 1}. A function f : 2 d → R is submodular if for every A, B ⊆ [d], f (A) + f (B) ≥ f (A ∪ B) + f (A ∩ B). An equivalent definition is that for every A ⊆ B ⊆ [d] and e ∈ [d], f (A ∪ {e}) − f (A) ≥ f (B ∪ {e}) − f (B). This definition is generalized to a domain of [n] d , where function f : [n] d → R + is called lattice-submodular if f (u) + f (v) ≥ f (u ∧ v) + f (u ∨ v) holds for all u, v ∈ [n] d . Note that the lattice-submodularity does not imply the property called DR-submodularity, which is the diminishing marginal returns along the direction of χ i for each i ∈ I, where χ i ∈ {0, 1} n , and only the i-th coordinate is 1. That is, f (u + χ i ) − f (u) ≥ f (v + χ i ) − f (v) does not necessarily hold for all u, v ∈ [n] d such that u ≤ v and i ∈ [d] even if f is lattice-submodular. Function f : [n] d → R + is called monotone if f (u) ≤ f (v) for all u ≤ v. Reduction and Eliminating an Assumption In order to eliminate the second assumption mentioned in Section 1.3, we introduce the notion of a "size cap". For each item i and a size cap b, we define an item (i, b), where p (i,b) (s) = π i (s) when s < b; p (i,b) (s) = s ≥s π i (s ) when s = b; and 0 otherwise. The new reward function is exactly R (i,b) (·). We will be using a time-indexed LP formulation following Ma [29]. Instead of making a decision at each time step, we do it at each remaining size level. When there is enough room, we take item i itself into consideration. If the remaining size b is small, we are not able to get more reward for an item than when it has a size of b. Therefore, instead of trying to include the original item i, we include item (i, b), which is item i with size cap b. Obviously, we can include each item at most once. To achieve this, we impose a partition matroid {I i } i∈[K] on the items, where I i = {(i, b)|∀b}. For the remainder of this paper, we view each (i, b) as an item, and the conflict between them is captured by the partition matroid. Continuous Optimization Phase Like most submodular maximization problems, our algorithms consists of two phases, a continuous optimization phase and a rounding phase. In this section, we describe the former. Target Function Given a lattice-submodular function f : [M ] n → R + and a distribution q S of elements in set S ⊆ [n], we define a set-submodular functionf : 2 n → R + , wheref (S) := E r∼q S [f (r) ]. This f is guaranteed to be a monotone set-submodular function (See proof in [2]). Suppose the final selected (random) set is S, the value we are interested in would be E[f (S)]. Letx be a vector, wherex(i) denotes the probability that item i is in S. Using the well-established multi-linear extension, we defineF : 2 n → R + , whereF (x) = S⊆[n] i∈Sx i i / ∈S (1 −x i )f (S) . This is the target function we are maximizing. Evaluating the functionF can take exponential time, but it can be approximated within a multiplicative factor of (1 + ) for any constant > 0. For simplicity, we assumeF (x) can be evaluated exactly in this paper, which is standard in the literature (e.g. see [6]). Stochastic Knapsack Exponential Constraints The exponential and polynomial constraints onx are adapted from Ma [29]. A group of exponential sized constraints describes the problem exactly. They are then relaxed to have a polynomial size, losing a factor of 2. For ease of notation, we follow Ma [29] and view an stochastic item i as an equivalent Markovian bandit, a special one that can force us to keep pulling it for a certain period of time. We use state u i (k, s) to indicate that arm i has been pulled k times, and the corresponding item has size s. From its initial state ρ i , a single pull would decide the size s of this job, and move to state u i (1, s) respectively. We are then forced to keep pulling this arm (we will be using arm and item interchangeably) for the next s − 1 steps, and the last of such pulls moves us to its termination state ∅ i , and we can pull a new arm. Denote the probability of moving from state u to state v with p u,v . After the first pull of item i, it moves to state u i (1, s) (having size s) with probability p ρi,ui(1,s) = p i (s). Therefore, if k < s, a pull will transit it to state u i (k + 1, s) with probability p ui(k,s),ui(k+1,s) = 1. Otherwise, when k = s, transit to state ∅ i with probability p ui(k,s),∅i = 1, and we are allowed to pull a new arm. Let π be a vector representing a joint state/node, where π i denotes the state on item i. Let S i = {u i ( * , * )} ∪ {ρ i , ∅ i } for all i ∈ [n] , the set of all states for arm i, andS = S 1 × · · · × S n , the set of all possible (maybe invalid) joint states. Let S = {π ∈S|∃i = j, π i / ∈ {ρ i , ∅ i }, π j / ∈ {ρ j , ∅ j }}, the set of states where at least two arms are in the middle of processing at the same time, and S = {π ∈S|π i = ρ i and π j = ρ j , i, j ∈ I k for some k}, the set of states where some conflicting arms (due to the partition matroid) have been started. Define S :=S \ (S ∪ S ), which is the set of all valid states. Let I(π) = {i|π i = ∅ i }, the set of arms that could be played from state π. Let π u denote the joint node where the component corresponding to u is replaced by u (note u can correspond to only one component). Let y π,t be the probability that we are at state π at time t, and z π,i,t the probability that we pulled arm i at time t, when the current state was π. Recall B is the total budget, we have the following basic constraints. i∈I(π) z π,i,t ≤ y π,t π ∈ S, t ∈ [B] (1) z π,i,t = y π,t π ∈ S, i : π i ∈ S i \ {ρ i , ∅ i }, t ∈ [B] (2) z π,i,t ≥ 0 π ∈ S, i ∈ [n], t ∈ [B](3) Let A i = {π ∈ S : π i / ∈ {ρ i , ∅ i }}, the joint node when arm i is in the middle of processing. Note A i and A j are disjoint for i = j. We call arm i the active arm. Let A = i∈[n] A i , the set of all states where some arm is active. For a state π ∈ S, let P(π) denote the subset of S that would transit to π with no play, which could happen when some arms turned inactive automatically: if π / ∈ A, then P(π) = {π} ∪ ( i / ∈I(π) {π u |u ∈ S i \ {ρ i }}; if π ∈ A, then P(π) = ∅. Suppose u corresponds to coordinate i, define Par(u) = {v ∈ S i : p v,u > 0}, the nodes that have a positive probability of transitioning to u. Then y-variables are updated as follows: y (ρ 1 ,...,ρn),0 = 1 (4) yπ,0 = 0, π ∈ S \ {(ρ1, · · · , ρn)} (5) yπ,t = π ∈P(π) y π ,t−1 − n i∈I(π ) z π ,i,t−1 t > 0, π ∈ S \ A (6) yπ,t = ρ i ∈Par(π i ) n π ∈P(π ρ i ) z π ,i,t−1 · pρ i ,π i , t > 0, i ∈ [n], π ∈ Ai, πi ∈ {ui(1, * )} (7) yπ,t = u∈Par(π i ) zπu,i,t−1 · pu,π i , t > 0, i ∈ [n], π ∈ Ai, πi / ∈ {ui(1, * )}(8) Equation (6) updates y π,t for π / ∈ A, i.e. joint nodes with no active arms. Such a joint node π can only come from a no-play from a joint node in P(π). Equations (7) and (8) update y π,t for π ∈ A. To get to the joint node π, we must have played arm i in previous step(s). In Equation (7), we consider the case if π i is one of u i (1, * ). We were at ρ i right before, so it is possible that in the last step, we switched to π ρi from some joint node in P(π ρi ) without playing an arm. In Equation (8), we consider other cases, in which arm i was played at time t − 1. These equations guarantee that at each time step, y * ,t form a distribution, i.e. π∈S y π,t = 1. Combining this with Equation (1), we get π∈S i∈I(π) z π,i,t ≤ 1, ∀t ∈ [B]. Equations (1)-(8) form the exponential constraints. We also need to relate these constraints withx (recallx(i) is the probability that item i is included):x(i) = t u∈Si π∈S:πi=u z π,i,t , which is the last missing piece for our exponential program, denoted as ExpP. Stochastic Knapsack Polynomial Constraints Obviously, we cannot solve this exponential program directly in polynomial time. In order to solve it, we relax the exponential program by disassemble the joint distribution of items. Let s u,t be the probability that arm i is on node u at the beginning of time t. Let x u,t be the probability that we pull an arm on node u at time t. Suppose S = i S i , we have the following constraints between x u,t and s u,t . xu,t ≤ su,t u ∈ S, t ∈ [B] (9) xu,t = su,t u ∈ i∈[n] Si \ {ρi, ∅i}, t ∈ [B](10)xu,t ≥ 0 u ∈ S, t ∈ [B] (11) u∈S xu,t ≤ 1 t ∈ [B](12) We also need constraints (13) for the partition matroid of arms (recall I k is a partition), and the state transition constraints (14)- (16). i∈I k sρ i ,0 ≤ 1, ∀I k sρ i ,0 ≥ 0, i ∈ [n](13)su,0 = 0 u ∈ S \ {ρ1, · · · , ρn}(14)sρ i ,t = sρ i ,t−1 − xρ i ,t−1 t > 0, i ∈ [n](15)su,t = v∈Par(u) xv,t−1 · pv,u t > 0, u ∈ S \ {ρ1,...,ρ n }(16) Relating these constraints withx:x(i) = t u∈Si x u,t , we get the polynomial program PolyP. For any program P ∈ {PolyP, ExpP}, let OPT P denote its optimal value. Relating between the Exponential and the Polynomial Constraints This was given in Ma [29], and we re-state for completes without proof. The direction from ExpP to PolyP is trivial. Theorem 1 (reformation of Lemma 2.3 from Ma [29] ). Given a feasible solution {z π,i,t }, {y π,t } to ExpP, we can construct a solution to PolyP with the same objective value by setting x u,t = π∈S:πi=u z π,i,t , s u,t = π∈S:πi=u y π,t for all i ∈ [n], u ∈ [0, 1], t ∈ B. Thus, the feasible region of PolyP is a projection of that of ExpP onto a subspace and OPT ExpP ≤ OPT PolyP . For the other direction, we construct a solution {z π,i,t , y π,t } of ExpP from a solution {x u,t , s u,t } of PolyP, which obtains half its objective value. It will satisfy π∈S:πi=u z π,i,t = xu,t 2 i ∈ [n], u ∈ S i , t ∈ [B]. We define specific {z π,i,t , y π,t } over B iterations. On iteration t: Compute y π,t for all π ∈ S. Defineỹ π,t = y π,t if π / ∈ A, andỹ π,t = y π,t − a∈A z π,i,t if π ∈ A i for some i ∈ [n] (if π ∈ A i , then {z π,i,t : a ∈ A} is already set in a previous iteration). For all i ∈ [n], define f i,t = π∈S:πi=ρiỹ π,t . For all i ∈ [n], π ∈ S such that π i = ρ i , and a ∈ A, set z a π,i,t =ỹ π,t · 1 2 · xρ i,t fi,t . For all i ∈ [n], π ∈ S such that π i = ρ i and π j ∈ {ρ j , φ j } for j = i, define g π,i,t = π ∈P(π) z π ,i,t . For all i ∈ [n], u ∈ S i \ {ρ i }, π ∈ S such that π i = u, and a ∈ A, set z a π,i,t+depth(u) = g π ρ i ,i,t · (x a u,t+depth(u)) )/x ρi,t . Solve the Continuous Optimization Problem In order to solve PolyP, we follow Fukunaga et al. [17] and use the Stochastic Continuous Greedy algorithm. This algorithm maximizes the multi-linear extension G of a monotone set-submodular function g over a solvable downward-closed polytope. A polytope P ⊆ [0, 1] N is considered solvable if we can find an algorithm to optimize linear functions over it, and downward-closed if x ∈ P and 0 ≤ y ≤ x imply y ∈ P. In our case, P is solvable due to its linearity, and that solving a linear program falls in polynomial time. Note P is down-monotone. The algorithm involves a controlling parameter called stopping time. For a stopping time 0 < b ≤ 1, the algorithm outputs a solution x such that O(n 3 δ)) max y∈Q G(y), where n is the size of the set over which g is defined and δ is the step size used in the algorithm. Here P is assumed to include the characteristic vector of every singleton set. x/b ∈ P, while G(x) ≥ (1 − e −b − Theorem 2 (reformation of Theorem 3 from Fukunaga et al. [17]). If the stochastic continuous greedy algorithm with stopping time b = 1/2 ∈ (0, 1] and step size δ = o(|I| −3 ) is applied to program PolyP, then the algorithm outputs a solution x ∈ bP such thatF (x) ≥ (1 − e −b − o(1))f avg (π * ) 0.3935f avg (π * ) for any adaptive policy π * . Rounding Phase Now that we have a fractional solution x, we proceed to round it to an integral policy (notice the fractional solution has already been scaled by a factor of 2). We need a variant of the contention resolution scheme that was introduced as a general framework for designing rounding algorithms that maximizes expected submodular functions ( [9,13,14]). The variant is an extension from a set submodular function to a lattice-submodular function, first introduced in Fukunaga et al. [17]. We include its definition here for self-containment. . We have the following definition for a α-CRS, its monotonicity, and one key property. Contention Resolution Scheme Definition 3 (α-Contention Resolution Scheme (α-CRS)). A mapping ψ : [B] n → F is an α-CRS with respect to q if it satisfies: 1. ψ(v)(i) ∈ {v(i), 0} for each i ∈ [n]; 2. if v ∼ q, then Pr[ψ(v)(i) = j|v(i) = j] ≥ α holds for all i ∈ I and j ∈ B. The probability is based on randomness both in v and in ψ when ψ is randomized. Definition 4 (monotone α-CRS). An α-CRS ψ is considered monotone, if, for each u, v ∈ [B] n such that u(i) = v(i) and u ≤ v, Pr[ψ(u)(i) = u(i)] ≥ Pr[ψ(v)(i) = v(i)] holds. The probability is based only on the randomness of ψ. Lemma 5 (Theorem 4 from Fukunaga et al. [17]). If ψ is a monotone α-CRS with respect to q, then E v∼q [f (ψ(v))] ≥ αE v∈q [f (v)]. Rounding Algorithm To fit in the contention resolution scheme, we need to first round everything independently. This means for each pair (i, t), item i is scheduled at time t with probability x ρi,t . Now we have a set R = {(i, t)} of proposed item time pairs. We sort the set according to t, and include the items one by one. Intuitively, for a pair (i, t), we will only include item i if time t is available and does not invalidate the solution, i.e. each item is scheduled at most once, and at most one item from each partition. After including it in our solution, we get its realized size, and mark the corresponding time slots unavailable. The main problem of this naive approach is that it does not exhibit monotonicity, which is a subtle but critical requirement for a CRS. To fix it, we schedule phantom item i even when we cannot fit it. We simulate its inclusion, and sample its size size i should it be included. We also mark those time slots corresponding to this phantom item unavailable, even when they are actually unoccupied. This seemingly wasteful step ensures that the rounding scheme is monotone. The final rounding algorithm is described in Algorithm 1. Mark time slots from t to t + s i unavailable; The remaining of this section is devoted to proving the following theorem, which combined with Theorem 2 leads to our main result. Theorem 6. Let π denote Algorithm 1, and x denote the solution we get from PolyP. Then f avg (π) ≥F (x)/2. To prove Theorem 6, we define two mappings σ(·) and ω(·), where the first corresponds (roughly) to the step that maps x to I in Algorithm 1, and ω(·) corresponds to the mapping (CRS) from set I to the final output. The mapping σ(x) receives a real vectorx ∈ [0, 1] n and returns a random vector v ∈ [B] n . From each partition I k , we pick at most one i, each i ∈ I k is picked with probabilityx(i). If it is picked, the i-th component v(i) independently takes value j with probability p i (j), and 0 otherwise, which happens with probability 1 − j p i (j). This captures the construction of set I (only the item part, note Pr[σ(x)(i) > 0] = Pr[∃t, s.t.(i, t) ∈ I]) , together with the random outcome of the item. The mapping ω(·) maps v ∈ [B] n to w ∈ [B] n . To mimic Algorithm 1, we first assign time value t(i) to each component v(i), according to x ρi,t . Based on t(i), we form a precedence ordering ≺ between i after random tie breaking (a random tie breaking is crucial). Then, we set ω(v)(i) = 0 if there exists a component j ≺ i such that t(j) ≤ t(i) < t(j) + v(i), and w(v)(i) = v(i) otherwise. We can observe that given input x, Algorithm 1 outputs exactly ω(σ(x)) if the random realized sizes of items are the same. In order to prove Theorem 6, we need the following two lemmas. The first, whose proof in Fukunaga [17], corresponding to the independent rounding step, and the second corresponding to the CRS step. Lemma 7 is trivially true by the definition ofF (x), which is the common starting point of contention resolution scheme. We first prove ω is a 1/2-CRS. Proof of Lemma 8. Recall there are two properties needed for an α-CRS. The first property is obviously correct due to the definition of ω(·). The second property needs to prove Pr[ω(v)(i) = j|v(i) = j] ≥ 1/2. In the language of the rounding algorithm, let Drop i,t denotes the event (respect to the randomness in ω and v) that we drop the pair (i, t). It is the same as proving Pr[Drop i,t |item i is selected at time t] ≤ 1 2 . Due to the way we round the solution, item i may be included more than once (at different times), and more than one item from the same partition may be included. Consider an item j at time t (maybe the same as i) that could affect the pruning of item i at time t. Define (j, t ) ≺ (i, t) if t < t, or t = t and j ≺ i. It is clear that (j, t ) will affect (i, t) if and only if (j, t ) ≺ (i, t) We slightly abuse notation, and let Drop i,t (j) denote the probability that the item j can causes the drop out of item i if a copy of it is scheduled at time t. Note this does not depend on whether item i is scheduled on t or not. We have: Lemma 10. Drop i,t (j) ≤ 1 2 u∈{∅j }∪{uj ( * , * )} x u,t + 1 2 x ρj ,t . Proof of Lemma 10. There are two cases and we bound the probability of dropping in each case. Case 1. j belongs to the same partition as i, Case 2. j belongs to a different partition. For the first case, the probability that it makes (i, t) invalid is Drop i,t (j) ≤ 1 2 (s ρj ,0 − s ρj ,t ) + Pr[item j is considered before i] · 1 2 x ρj ,t ≤ 1 2 u∈{∅j }∪{uj ( * , * )} x u,t + 1 2 x ρj ,t . The first term is the probability that at least one item j is scheduled before time t. Note this is actually an union bound due to our independent rounding. The second term is the probability that it is scheduled at time t, but will invalidate i since j ≺ i. The second equality comes from the fact that if item j is scheduled some time before t, then it must be at some state at time t that is not the starting state ρ j . In other words, either the end state ∅ j or some transient state u j ( * , * ). For the second case, fix j, it can only drop i if it marked time slot t unavailable. The probability is Drop i,t (j) ≤ 1 2 t−1 t =1 x ρj ,t · Pr[size j ≥ t − t ] + Pr[item j is considered before i] · 1 2 x ρj ,t ≤ 1 2 t−1 t =1 x ρj ,t · Pr[size j ≥ t − t ] + 1 2 x ρj ,t . The first term is a summation of all the possible starting point of job j, times the probability that it will mark time slot t unavailable. Note this is also a union bound since there can be more than one copy of item j due to independent rounding. The second term is the probability that item j is also scheduled at time t, but is considered before i, i.e. j ≺ i, which marks time slot t unavailable for i. We focus on the first term, t−1 t =1 x ρj ,t · Pr[size j ≥ t − t ] = t−1 t =1 B−t τ =t−t x ρj ,t Pr[size j = τ ] = t−1 t =1 B−t τ =t−t x uj (1,τ ),t +1 = t−1 t =1 B−t τ =t−t x uj (t−t ,τ ),t ≤ u∈{∅j }∪{uj ( * , * )} x u,t . The last inequality holds because the index set of the summation on the left is a subset of that on the right. Therefore, the total probability that item i is blocked by any item is upper bounded by the union bound: Drop i,t = j) Drop i,t (j) ≤ 1 2 j u∈{∅j }∪{ui( * , * )} x u,t + 1 2 j∈[n] x ρj ,t ≤ 1 2 j∈[n] u∈{∅j }∪{ui( * , * )} x u,t + 1 2 j∈[n] x ρj ,t ≤ 1 2 (1 − j∈n x ρj ,t ) + 1 2 j∈[n] x ρj ,t = 1 2 . Lastly, we show ω is monotone in Appendix A.1. With everything ready, we can now prove Theorem 6, which combined with Theorem 2 leads to the main claim. Proof of Theorem 6. The output r of Algorithm 1 satisfies E[f (r)] = E[f (ω(σ(x)))], and its feasibility is guaranteed by the algorithm. By Lemma 8 and Lemma 9, ω is a monotone 1/2-CRS with respect to q, where q is the probability defined in Lemma 8. Moreover, σ(x) ∼ q holds. By Lemma 5, E[f (ω(σ(x)))] ≥ E[f (σ(x))]/2. Using Lemma 7, we get f avg (π) = E[f (r)] = E[f (ω(σ(x)))] ≥ E[f (σ(x))]/2 ≥F ((x))/2. Conclusion We consider the well studied correlated stochastic knapsack problem, generalizing its target function with submodularity to capture diminishing returns. An extra partition matroid constraint is added to generalize it and resolve an open question raised in a previous work to eliminate an assumption. We also make improvement on the approximation ratio. There is still a gap of 2 comparing to the variant with linear target function and we leave it as an open problem to close the gap. matroid while doing the first rounding step, leading to dependency between items. This seemingly convenient step actually breaks the correctness of contention resolution scheme, which is built on FKG inequality and intrinsically needs an independent rounding step. We fixed the issue by replacing it with a true independent rounding step, and fix the solution to fit the partition matroid later on. While this breaks the symmetry between items, the gap of 2 turns out to be large enough to fix everything. Check the use of union bound in Case 1 for the proof of Lemma 10 for details. (1 − Pr[i ≺ i|t(i) = t]x ρ i ,t )   · i ∈I t−1 t =0 (1 − x ρ i ,t ) . This is a summation over all possible time slot that the first copy of item i is scheduled. For simplicity, we define the following: a i,t = B t=1 x ρi,t t−1 τ =1 (1 − x ρi,τ ) b i,t,u = i / ∈I t t =t−u(i ) (1 − Pr[i ≺ i|t(i) = t]x ρ i ,t ) c i,t = i ∈I t−1 t =0 (1 − x ρ i ,t ) which re-writes Pr[ω(u)(i) = j] as B t=1 a i,t b i,t,u c i,t . The part in the first large bracket (b i,t,u ) is the probability that non of the items in a different partition prunes item i at time t. The part in the second large bracket (c i,t ) is that for items in the same partition. Such multiplication of probability is only possible due to the phantom items and the independence they brought. to be separate instances. For a spot instance i, a job can run on it for a period time before interruption, which follows a given distribution independent of each other. Let π i,s be the probability that instance i costs exactly s dollars before it gets interrupted. When we schedule a job j on an instance i, we can also specify a budget cap. Let R (j,i) (s) denotes the progress of job j achieves when s dollars have been spent, before the last check point, e.g. the number of trained epochs. Notice the function R (j,i) (·) is monotone, i.e. R (j,i) (s) ≤ R (j,i) (s ) if s ≤ s . In practice, when we schedule job j onto instance i, it cannot start training immediately. Some processing time is wasted on environment setup and checkpoint restoration, which does not count towards progress. This is captured by setting R (j,i) (s) = 0 if s dollars is not enough to finish the first epoch. When a job gets interrupted, we can reschedule it on a different instance, starting from the latest checkpoint. The total utility model this as a submodular function. With a given budget B, we would like to maximize the total expected utility of all jobs. B.1 Reduction The reduction from the scheduling problem to the final knapsack problem is as follows. For each job j, instance i and budget cap b, we define an item (j, i, b), where p (j,i,b) (s) = π i,s when s < b; p (j,i,b) (s) = s ≥s π i,s when s = b; and 0 otherwise. The new reward function is exactly R (j,i,b) (·). Notice a job may be scheduled on multiple instances sequentially due to interruptions, but for each instance, only a single job can be scheduled on it, and a specific budget cap can be chosen, we further impose a partition matroid {I i } i∈ [K] on the items, where I i = {(j, i, b)|∀j, ∀b}. A contention resolution scheme (CRS) accepts a pairwise independently rounded solution which may violate some constraints, and fixes it without losing too much on expectation. Let f : [B] n → R + be a monotone lattice-submodular function and the probability distribution q i : [B] → [0, 1] on [B] be given for each i ∈ {1, . . . , n}}. We write v ∼ q if v ∈ [B] n is a random vector such that, for each i ∈ {1, . . . , n}, the corresponding component v(i) is determined independently as j ∈ [B] with probability q i (j). This is the independently rounded solution we feed into a CRS. Let F ⊆ [B] n be a downward-closed subset of [B] n , and let α ∈ [0, 1] Algorithm 1 1Rounding Algorithm 1 foreach pair (i, t) do 2 Sample (i, t) with probability x ρi,t , and gets ∅ otherwise; 3 if not get ∅ then I ← I ∪ {(i, t)} ; 4 Sort I according to a non-decreasing ordering of t, break ties uniformly at random; 5 C = 0, S = ∅, mark all times slots available; 6 for (i, t) ∈ I do 7 if time slot t is available and item i does not violate constraints then 8 Include item i and observe s i ; 9 else 10 Simulate including item i, and observe s i ; Lemma 7 . 7E[f (σ(x))] ≥F (x) holds for any x ∈ P . Lemma 8. ω is a 1/2-CRS with respect tox. Lemma 9. The 1/2-CRS ω is monotone. Lemma 9 . 9The 1/2-CRS ω is monotone. Proof of Lemma 9. Suppose vectors u, v ∈ [B] n satisfies u ≤ v, and u(i) = v(i) = j > 0. We only need to show Pr[ω(u)(i) = j] ≥ Pr[ω(v)(i) = j],where randomness is with respect to the choice of time and ordering. Let I denote the partition that includes item i. Let f avg (γ) denote q∈[M ] n Pr γ [q]f (q), i.e., the average objective 1 See definition of partition matroid, submodular and lattice-submodular in Section 3. 2 Let S ⊆ [n], we sample a vector q ∈ [M ] n as follows. Each component q(i) is sampled independently.For i ∈ S, Pr[r i = R i (s)] = p i (s); for i / ∈ S, r i = 0 with probability 1. Denote this distribution as q S . Then the objective is to select a (random) set S ⊆ I of items that maximizes E θ∼qS [f (θ)] subject to i∈S size i ≤ B. See definition of partition matroid, submodular and lattice-submodular in Section 3.4 For example, suppose we are left with a remaining budget of 20 at some time, and all items have a 0.001 probability of size 21. What this assumption suggests is that none of the items are allowed to be selected. Known as spot instances by Amazon Web Services (AWS), low-priority VMs by Microsoft Azure, preemptible instances by Google Cloud and transient virtual machines/servers in some literature. We refer to all such revocable computing instances as spot instances. AcknowledgementWe would like to thank the reviewers for comments on a previous version of this draft, who found a flaw in the algorithm and its analysis. The old algorithm imposes the partitionWe wish to prove that Pr[ω(u)(i) = j] ≥ Pr[ω(v)(i) = j], and we instead prove that coordinately a i,t b i,t,u c i,t ≥ a i,t b i,t,v c t ≥ 0 always holds. In factB Scheduling ML Jobs on Cloud Spot InstancesCloud Computing Instance Characteristics: Demands for cloud resources display large fluctuations across time and availability zones[7,31]. During times of low actual demand, cloud vendors make unused resources available to user as cheaper entities, a.k.a. spot instances 5 , that may be interrupted, so they can take them back when demands surge. In practice, spot instances are often available at up to 70%-90% discounts compared to their on-demand equivalences[24,26], making them an economical option if interruptions can be handled. Machine Learning Characteristics: An ML training algorithm is usually an iterative algorithm (each iteration is also known as an epoch) and it produces a better estimate of the model parameters with each iteration, usually with diminishing marginal returns. If a long-running ML training job is interrupted prematurely, model parameter estimates from the latest successfully completed iteration are still a valid model instance, hence interruptions can be tolerated with adequate planning. We try to model and answer the following question from a theoretical perspective:How can ML training jobs be scheduled and executed on interruptible but relatively inexpensive spot instances to increase their cost efficiency?Now we give a rigid definition of the spot scheduling problem. For justification of the modeling, please refer to Yang et al.[33]. N jobs need to be scheduled on M instances, where the instances may have different CPU/RAM configurations, i.e. have different speeds for various jobs, or come from different available zones, i.e. have different interruption patterns. Each instance has a finite supply, and without loss of generality, we assume different copies Fair and balanced: Learning to present news stories. Amr Ahmed, Choon Hui Teo, S V N Vishwanathan, Alex Smola, 10.1145/2124295.2124337Proceedings of the Fifth ACM International Conference on Web Search and Data Mining, WSDM '12. the Fifth ACM International Conference on Web Search and Data Mining, WSDM '12New York, NY, USAAssociation for Computing MachineryAmr Ahmed, Choon Hui Teo, S.V.N. Vishwanathan, and Alex Smola. Fair and balanced: Learning to present news stories. 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{'abstract': 'We study the correlated stochastic knapsack problem of a submodular target function, with optional additional constraints. We utilize the multilinear extension of submodular function, and bundle it with an adaptation of the relaxed linear constraints from Ma [Mathematics of Operations Research, Volume 43(3), 2018] on correlated stochastic knapsack problem. The relaxation is then solved by the stochastic continuous greedy algorithm, and rounded by a novel method to fit the contention resolution scheme (Feldman et al. [FOCS 2011]). We obtain a pseudo-polynomial time (1 − 1/ √ e)/2 0.1967 approximation algorithm with or without those additional constraints, eliminating the need of a key assumption and improving on the (1−1/ 4 √ e)/2 0.1106 approximation by Fukunaga et al. [AAAI 2019].', 'arxivid': '2207.01551', 'author': ['Sheng Yang \nAdobe Research\nAdobe Research\nNorthwestern University\nNorthwestern University\n\n', 'Samir Khuller \nAdobe Research\nAdobe Research\nNorthwestern University\nNorthwestern University\n\n', 'Sunav Choudhary \nAdobe Research\nAdobe Research\nNorthwestern University\nNorthwestern University\n\n', 'Adobe Research \nAdobe Research\nAdobe Research\nNorthwestern University\nNorthwestern University\n\n', 'Kanak Mahadik \nAdobe Research\nAdobe Research\nNorthwestern University\nNorthwestern University\n\n'], 'authoraffiliation': ['Adobe Research\nAdobe Research\nNorthwestern University\nNorthwestern University\n', 'Adobe Research\nAdobe Research\nNorthwestern University\nNorthwestern University\n', 'Adobe Research\nAdobe Research\nNorthwestern University\nNorthwestern University\n', 'Adobe Research\nAdobe Research\nNorthwestern University\nNorthwestern University\n', 'Adobe Research\nAdobe Research\nNorthwestern University\nNorthwestern University\n'], 'corpusid': 250265102, 'doi': '10.48550/arxiv.2207.01551', 'github_urls': [], 'n_tokens_mistral': 18940, 'n_tokens_neox': 16608, 'n_words': 9871, 'pdfsha': '4562dc1193525f1aefc150023c8ea0ba598dfbcb', 'pdfurls': ['https://export.arxiv.org/pdf/2207.01551v2.pdf'], 'title': ['Correlated Stochastic Knapsack with a Submodular Objective Subrata Mitra', 'Correlated Stochastic Knapsack with a Submodular Objective Subrata Mitra'], 'venue': []}

arxiv

MINIMAL INDUCED SUBGRAPHS OF THE CLASS OF 2-CONNECTED NON-HAMILTONIAN WHEEL-FREE GRAPHS 3 Jan 2023 Aristotelis Chaniotis Zishen Qu Sophie Spirkl MINIMAL INDUCED SUBGRAPHS OF THE CLASS OF 2-CONNECTED NON-HAMILTONIAN WHEEL-FREE GRAPHS 3 Jan 2023arXiv:2204.07671v2 [math.CO] Given a graph G and a graph property P we say that G is minimal with respect to P if no proper induced subgraph of G has the property P . An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph H, a graph G is H-free if G has no induced subgraph isomorphic to H. The main motivation for this paper originates from a theorem of Duffus,Gould, and Jacobson (1981), which characterizes all the minimal connected graphs with no Hamiltonian path. In 1998, Brousek characterized all the clawfree HC-obstructions. On a similar note, Chiba and Furuya(2021), characterized all (not only the minimal) 2-connected non-Hamiltonian {K 1,3 , N 3,1,1 }free graphs. Recently, Cheriyan, Hajebi, and two of us (2022), characterized all triangle-free HC-obstructions and all the HC-obstructions which are split graphs. A wheel is a graph obtained from a cycle by adding a new vertex with at least three neighbors in the cycle. In this paper we characterize all the HC-obstructions which are wheel-free graphs. † We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number RGPIN-2020-03912]. Cette recherche aété financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de référence RGPIN-2020-03912]. Introduction In this paper we consider finite undirected graphs without loops or multiple edges. Let G be a graph and let X ⊆ V (G). We denote by G[X] the induced subgraph of G with vertex set X, and by G \ X the graph G[V (G) \ X]. Given two graphs H and G, the graph G is H-free if it has no induced subgraph isomorphic to H. Let X ⊆ V (G). The set X is a cutset if G \ X is not connected, and if X is a cutset of cardinality k, then X is a k-cutset. The set X is a clique if it is a set of pairwise adjacent vertices. If X is both a clique and a cutset, then X is a clique cutset. Let F ⊆ E(G). The set F is an edge-cutset if the graph (V (G), E(G) \ F ) is not connected. An edge-cutset of cardinality k is a k-edge-cutset. Let v ∈ V (G) and let X ⊆ V (G). We denote by N X (v) the set of neighbors of v in X, by d(v) the degree of the vertex v in G, and by ∆(G) the maximum degree of a vertex in G. Given two graphs G 1 and G 2 , we denote by G 1 ∪ G 2 the graph (V (G 1 ) ∪ V (G 2 ), E(G 1 ) ∪ E(G 2 )). A Hamiltonian path (resp. Hamiltonian cycle) in a graph G is a (not necessarily induced) subgraph H of G which is a path (resp. cycle), and V (H) = V (G). A graph is Hamiltonian if it has a Hamiltonian cycle and non-Hamiltonian otherwise. Following the notation of [1], we say that a graph H is an HP-obstruction if H is connected, has no Hamiltonian path, and every induced subgraph of H either equals H, is not connected, or has a Hamiltonian path. Analogously, a graph H is an HC-obstruction if H is 2-connected, has no Hamiltonian cycle, and every induced subgraph of H either equals H, is not 2-connected, or is Hamiltonian. Below we state some results which involve specific graphs not defined in the present paper. Our first motivation for this paper originates from the following result of Duffus, Gould, and Jacobson [2], which characterizes all the graphs which are HP-obstructions. Theorem A ( [2], see also [3,Theorem 2.9]). There are exactly two HP-obstructions: the claw and the net. In 1998, Brousek [4] obtained a complete characterization of the claw-free HCobstructions. On a similar note, Chiba and Furuya [5], characterized all (not only the minimal) 2-connected non-Hamiltonian {K 1,3 , N 3,1,1 }-free graphs. Cheriyan, Hajebi, and two of us [1] characterized all HC-obstructions which are split graphs and all HC-obstructions which are triangle-free graphs. Theorem B ( [1]). The snare and all n-novae for n ≥ 2 are HC-obstructions. Moreover, these are the only HC-obstructions which are split graphs. Theorem C ( [1]). All thetas, triangle-free closed thetas, and triangle-free wheels are HC-obstructions, and they are the only HC-obstructions which are triangle-free. We remark that the analogous problem with respect to the induced minor relation has been fully resolved by Ding and Marshall [6], who obtained a complete characterization of the minimal, with respect to the induced minor relation, 2connected non-Hamiltonian graphs. A wheel is a graph obtained from a cycle C by adding a new vertex (not in V (C)) which has at least three neighbors in V (C). We remark that in several papers about wheels (for example, in [7]), the cycle C is required to have at least four vertices and thus the above definition is non-standard: for example, a four-vertex complete graph is a wheel according to our definition. In this paper we give a complete characterization of the HC-obstructions which are wheel-free graphs. Since, by Theorem C, all triangle-free wheels are HCobstructions, this is equivalent to characterizing all HC-obstructions which do not contain a wheel and contain a triangle. We proceed with some definitions in order to state our main result. Given two vertices, u and v, a (u, v)-path is a path which has the vertices u and v as its ends. Given two sets of vertices, A and B, an (A, B)-path is a (u, v)-path, where u ∈ A and v ∈ B. Let G be a graph, let K be the complete graph on V (G), and let F be a subset of E(K) such that F ∩ E(G) = ∅. We denote by G + F the graph (V (G), E(G) ∪ F ). Given a positive integer k, we denote by [k] the set of positive integers {1, . . . , k}. Let K and L be two disjoint triangles on {k 1 , k 2 , k 3 } and {l 1 , l 2 , l 3 } respectively. For each i ∈ [3] let P i be a (k i , l i )-path, such that the paths P 1 , P 2 , P 3 are vertexdisjoint. Let P := K ∪ L ∪ ( i∈ [3] P i ). If for each i ∈ [3] the path P i has length at least two, then P is a prism, otherwise P is a short prism. Let l be a vertex, such that l / ∈ V (K). For each i ∈ [3] let P i be a (k i , l)-path of length at least two, such that the paths P 1 , P 2 , P 3 are internally vertex-disjoint. Then the graph K ∪ ( i∈ [3] P i ) is a pyramid. Let a and b be two vertices. For each i ∈ [3] let P i be an (a, b)-path of length at least two, such that the paths P 1 , P 2 , P 3 are internally vertex-disjoint. The graph i∈ [3] P i is a theta. Figure 2. Top row from left to right: A prism, a theta and a pyramid. Bottom row from left to right: A prism + , pyramid + , theta + . Squiggly edges represent paths of length at least two. A prism + /pyramid + /theta + is a graph obtained from a prism/pyramid/theta by selecting one or more of the three paths involved in the definition of the prism/pyramid/theta and adding for each of the selected paths an edge which joins its ends. (In the case of a theta + , we add at most one edge between a and b.) A 3-path-configuration is a graph isomorphic to a prism, a pyramid, a theta, a prism + , a pyramid + or a theta + . We refer to 3-path-configuration as a 3PC. Again, we note that our definitions differ from standard literature (such as [7]): usually, only prisms, pyramids, and thetas are referred to as 3PCs, and usually short prisms, as well as pyramids with a path of length one, are included. Note that the latter are wheels according to our definition. In this paper we prove the following theorem: Theorem 1. All 3PCs are HC-obstructions. Moreover, these are the only HCobstructions which are wheel-free graphs. Wheel-free graphs In this section we prove Theorem 1. In [1] it is proved that all thetas are HC-obstructions, by the exact same argument it follows that all thetas + are HCobstructions. Lemma 1 ([1]). All thetas and thetas + are HC-obstructions. Lemma 2. All prisms, pyramids, prisms + , pyramids + are HC-obstructions. Proof. The fact that none of these graphs has a cutvertex can be easily checked. Also it can be easily checked that for each of these graphs, each of its proper induced subgraphs is either not 2-connected or Hamiltonian. Let us prove that each of the graphs in the statement of the lemma is non-Hamiltonian. Let G be a prism/prism + /pyramid/pyramid + as in the definition in Section 1. Let suppose towards a contradiction that G has a Hamiltonian cycle C. Then since for each i ∈ [3], the path P i has at least one internal vertex which has degree two in G, the cycle C contains as subgraph the path P i and if G has an edge which joins the ends of P i , then clearly C does not contain this edge. Since C is 2-regular, it follows that it contains at most one of the edges of G[k 1 , k 2 , k 3 ]. Now one of the vertices k 1 , k 2 , k 3 has degree one in C, which is a contradiction. By Lemma 1 and Lemma 2, we get the following: Corollary 1. All 3PCs are HC-obstructions. In view of Corollary 1, in order to prove Theorem 1, it suffices to prove the following: Theorem 2. If G is a 2-connected, wheel-free graph which has no induced subgraph isomorphic to a 3PC, then G is Hamiltonian. Below we state some notions and results that we need for the proof of Theorem 2. If H is a graph, then the line graph of H is the graph G whose vertices are the edges of H and two vertices of G are adjacent if and only if the corresponding edges of H share a vertex. Given a graph H we denote its line graph by L(H). The following is well known: Lemma 3. Let H be a graph and J be a subgraph of H. Then L(J) is an induced subgraph of L(H). Following the notation of [7], we say that a graph G is an only-prism graph if it is (theta, wheel, pyramid)-free. A short pyramid is a graph as in the definition of the pyramid in Section 1 with the difference that there exists exactly one i ∈ [3] such that P i has length one. We remark that our definitions of a wheel and a pyramid are sligthly different from the corresponding definitions in [7]. In particular, in [7] the complete graph on four vertices is not considered a wheel and a short pyramid is considered as a pyramid, whereas we consider it to be a wheel. Overall, however, only-prism graphs in our sense are also only-prism graphs as defined in [7]. Let G be a graph and C be a cycle which is a subgraph of G. A chord of C is an edge e such that both the ends of e are vertices of C and e / ∈ E(C). A cycle C is chordless if C has no chords and a graph G is chordless if every cycle of G is chordless. Theorem 3 ([7]). If G is an only-prism graph, then either G is the line graph of a triangle-free chordless graph, or G admits a clique cutset. We will use the following definitions from [8]. A graph G is 2-sparse if every edge of G is incident to at least one vertex of degree at most two. A proper 2cutset of a connected graph G is a pair of non-adjacent vertices u, v such that V (G) can be partitioned into non-empty sets X, Y and {u, v} such that there is no edge between X and Y , and with the property that each of the graphs G[X ∪ {u, v}] and G[Y ∪ {u, v}] contains a (u, v)-path, and neither of the graphs G[X ∪ {u, v}] and G[Y ∪ {u, v}] is a path. We say that (X, Y, u, v) is a split of this proper 2-cutset. Theorem 4 ( [8,9]). If H is a 2-connected chordless graph, then either H is 2sparse or H admits a proper 2-cutset. Given a graph H, a subdivision of H is a graph obtained from H by replacing each edge uv of H by a (u, v)-path of length at least one in such a way that none of the paths that we add has an internal vertex in V (H) or in one of the other paths we add. A We are now ready to prove Theorem 2. model of a graph H in a graph G is a collection (A h ) h∈V (H) of disjoint subsets of V (G) such that G[A h ] is connected for all h ∈ V (H), Proof of Theorem 2. Let us assume towards a contradiction that the theorem does not hold. Let G be a minimal, with respect to the number of its vertices, counterexample for the theorem. Claim 1. Let {u, v} ∈ E(G). Then d(u) ≥ 3 or d(v) ≥ 3. Proof of Claim 1. Let {u, v} ∈ E(G). Since G is 2-connected it follows that d(u) ≥ 2 and d(v) ≥ 2. Let us suppose towards a contradiction that d(u) = d(v) = 2 and let u ′ be the unique vertex of the set N G (u) \ {v} and let v ′ be the unique vertex of the set N G (v) \ {u}. LetG be the graph obtained from G by contracting the edge {u, v}, and let v uv ∈ V (G) be the vertex formed by the contraction. We claim thatG is 2-connected. Suppose not, and let w ∈ V (G) be a cutvertex ofG. If w ∈ V (G) \ {v uv }, then w is a cutvertex of G, which is a contradiction. Thus, w = v uv and hence both v and u, are cutvertices of G which is again a contradiction. We claim thatG has no Hamiltonian cycle. Suppose not, and let C be a Hamiltonian cycle inG. Let C ′ := C \ v uv and let C ′′ : = (V (C ′ ) ∪ {u, v}, E(C ′ ) ∪ {{u, u ′ }, {u, v}, {v, v ′ }}) . Then, C ′′ is a Hamiltonian cycle in G, which is a contradiction. ThusG has no Hamiltonian cycle. We claim thatG has no induced subgraph which is isomorphic to a 3PC. Suppose not, and let H be such an induced subgraph ofG. Since by our assumptions H is not an induced subgraph of G, it follows that v uv ∈ V (H). Since v uv has degree two inG, and since H is 2-connected, we deduce that v uv has degree two in H. Thus, v uv is an internal vertex of a path of length at least two of H. Let H ′ := H\v uv and let H ′′ : = (V (H ′ )∪{u, v}, E(H ′ )∪{{u, u ′ }, {u, v}, {v, v ′ }}) . Then, H ′′ is an induced subgraph of G which is isomorphic to a 3PC, which is a contradiction. HenceG has no induced subgraph isomorphic to a 3PC. By the above it follows thatG is a counterexample for Theorem 2, which is a contradiction because |V (G)| = |V (G)| − 1 < |V (G)|, and G is a minimum counterexample. G, say G 1 , G 2 , such that |V (G 1 )|, |V (G 2 )| < |V (G)|, V (G 1 ) ∩ V (G 2 ) = {u, v}, V (G 1 ) ∪ V (G 2 ) = G, G 1 is a (u, v) -path of length two or a triangle and G 2 is neither a path nor a cycle. Proof of Claim 2. We first prove that G \ {u, v} has exactly two connected connected components. Suppose not, and let G 1 , G 2 and G 3 be three connected components of G \ {u, v}. Since G is 2-connected, neither of the vertices u, v is a cutvertex, and thus for each i ∈ [3], each of u, v has at least one neighbor in G i . For each i ∈ [3], let P i be a shortest (u, v)-path with all its internal vertices in G i and such that P i does not use the edge {u, v} if it is present in G. P i has length at least two, and G[V (P 1 ) ∪ V (P 2 ) ∪ V (P 3 )] is either a theta + or a theta (depending on whether {u, v} ∈ E(G) or not), which is a contradiction. Thus G \ {u, v} has exactly two connected components. Let . We claim that there exists i ∈ [2], such that the graph H ′ i is neither a (u, v)-path, nor a cycle. Suppose not, then either {u, v} / ∈ E(G) and the graph G is a cycle, and hence a Hamiltonian graph; or {u, v} ∈ E(G), in which case the subgraph of G which we obtain if we delete the edge {u, v} is a Hamiltonian cycle of G. Both outcomes contradict the fact that G is a non-Hamiltonian graph. We claim that there exists i ∈ [2], such that the graph H ′ i is either a (u, v)-path or a cycle. Suppose not. For each i ∈ [2] let P i be a shortest (u, v)-path in H ′ i which does not use the edge {u, v} if it is present in G. Then |V (P i )| < |H ′ i |. The graphs H ′′ 1 := G[V (H ′ 1 ) ∪ V (P 2 )] and H ′′ 2 := G[V (H ′ 2 ) ∪ V (P 1 )] are 2-connected induced subgraphs of G and |V (H ′′ 1 )| < |V (G)| and |V (H ′′ 2 )| < |V (G)|. Thus, by the minimality of G as a counterexample, it follows that for each i ∈ [2], the graph H ′′ i has a Hamiltonian cycle C i since every vertex in V (P 3−i ) \ {u, v} has degree 2 in H ′′ i , it follows that E(P 3−i ) ⊆ E(C i ) for i ∈ {1, 2}. Let C := (C 1 \ P 2 )∪(C 2 \ P 1 ) . Then C is a Hamiltonian cycle of G, which is a contradiction. Without loss of generality we may assume that H ′ 1 is either a (u, v)-path, or a cycle and that H ′ 2 is neither a path nor a cycle. Since {u, v} is a cutset, we have that |V (H ′ 1 )| ≥ 3. We claim that |V (H ′ 1 )| = 3. Suppose not. Then there exist v 1 , v 2 ∈ V (G 1 ), such that {v 1 , v 2 } ∈ E(G) and d(v 1 ) = d(v 2 ) = 2, which contradicts Claim 1. The graphs G 1 and G 2 are induced subgraphs of G, which witness that the claim holds. G 1 is a triangle and G 2 is neither a path nor a cycle. In this case we call the edge {u, v} a special edge of the graph G, and the unique (u, v)-path of length two in G 1 the corresponding path of this special edge. Corollary 2. If {u, v} is a clique cutset of size two of G, then there exist induced subgraphs of G, say G 1 , G 2 , such that |V (G 1 )|, |V (G 2 )| < |V (G)|, V (G 1 )∩V (G 2 ) = {u, v}, V (G 1 ) ∪ V (G 2 ) = V (G), Proof of Corollary 2. Immediate by Claim 2. Let G ′ be the induced subgraph of G that we obtain if for each special edge of G, we delete the unique internal vertex of its corresponding path. In claims 3, 4, 5 and 6, we prove some properties of the graph G ′ . Claim 3. Let X be a cutset in G ′ . Then X is a cutset in G. Proof of Claim 4. By Claim 3, the definition of G ′ , and the fact that G is 2connected, we conclude that G ′ has neither cutvertex, nor clique cutset of size two. Also, since G ′ is wheel-free, it follows that it does not contain an induced subgraph isomorphic to the complete graph on 4 vertices and thus G ′ has no clique cutset of size four or more. Proof of Claim 3. Let us suppose towards a contradiction that G \ X is connected. Let G ′ 1 , G ′ 2 be two connected components of G ′ \ X. Let u ∈ V (G ′ 1 ) and v ∈ V (G ′ 2 ). Let P be an induced (u, v)-path in G. Let x ∈ V (P ) \ V (G ′ ). Then N V (G) (x) is a clique, so |N V (P ) (x)| ≤ 1 as P is induced. But then x ∈ {u, v}, a contradiction. We claim that G ′ has no clique cutset of size three. Suppose not and let {u, v, w} be a clique cutset of G ′ . Let G ′ 1 , G ′ 2 be two connected components of G ′ \ {u, v, w}. Since G ′ has no clique cutset of size two, it follows that for each i ∈ [2] each of u, v and w, has at least one neighbor in G ′ i . For each i ∈ [2], let P i be a shortest (u, v)-path which does not use the edge {u, v}, and with all its internal vertices in G ′ i . Then G[V (P i )] is a cycle and w / ∈ V (P i ). We claim that for each i ∈ [2] no internal vertex of P i is adjacent with the vertex w. Suppose not. Then G ′ [V (P i ) ∪ {u, v, w}] is a wheel, which is a contradiction. It follows that G ′ [V (P 1 ) ∪ V (P 2 ) ∪ {u, v, w}], is a theta + , which is a contradiction. Thus, G ′ has no clique cutset of size three, and hence G ′ has no clique cutset. Proof of Claim 6. Let us assume towards a contradiction that G ′ has a Hamiltonian cycle, say C ′ , which contains all the special edges of G. For every special edge {x, y} of G, let us denote by P xy its corresponding path. Let C be the subgraph of G obtained from C ′ as follows: For every special edge {x, y} of G, we delete from C ′ the edge {x, y} and we add the path P xy . Then, C is a Hamiltonian cycle of G, which contradicts the fact that G is a non-Hamiltonian graph. Proof of Claim 7. By our assumptions for G, and the definition of G ′ , the graph G ′ is an only-prism graph. By Claim 4, G ′ has no clique cutset. Thus, by Theorem 3, there exists a triangle-free chordless graph H, such that G ′ = L(H). Since G ′ is K 4 -free, it follows that ∆(H) ≤ 3. In Claims 8-12, we prove some properties of the graph H. Claim 8. H is 2-connected. Proof of Claim 8. Since G ′ is connected and G ′ = L(H), it follows that H is connected. Let us assume towards a contradiction that H is not 2-connected. Let v ∈ V (H) be a cutvertex of H. If every edge of H is incident with v, then G ′ is a complete graph. Since G ′ is 2-connected and K 4 -free, it follows that G ′ is isomorphic to K 3 , contrary to Claim 6. Now let X be a component of H \ {v} containing an edge e not incident with v. Let K be the set of vertices in G ′ that correspond to edges of H of the form {u, v} where u ∈ X. Then K is a clique cutset in G ′ , contrary to Claim 4. Claim 9. If F is a 2-edge-cutset in H, then the graph H \ F has exactly two connected components, one of which is either a single vertex or a single edge. Proof of Claim 9. Follows immediately by Claim 5 and by the fact that G ′ = L(H). Claim 10. The graph H does not admit a proper 2-cutset, and H is 2-sparse. Proof of Claim 10. By Theorem 4, it suffices to prove that H does not admit a proper 2-cutset. Suppose for a contradiction that H admits a proper 2-cutset, and let (X, Y, u, v) be a split of this proper 2-cutset. Since u and v each have degree at most three in H by Claim 7, it follows that there is a vertex u ′ ∈ V (H) such that either N (u) ∩ X = {u ′ } or N (u) ∩ Y = {u ′ }. Likewise, there is a vertex v ′ ∈ V (H) such that either N (v) ∩ X = {v ′ } or N (v) ∩ Y = {v ′ }. It follows that {uu ′ , vv ′ } is a two-edge cutset in H. By Claim 9, it follows that H \ {uu ′ , vv ′ } has exactly two connected components, say X ′ and Y ′ , and that X ′ is a clique and |X ′ | ≤ 2. Since X, Y = ∅ and H \ {uu ′ , vv ′ } contains no (X, Y )-path, we may assume that X ⊆ X ′ and Y ⊆ Y ′ . It follows that |X| ≤ 2. If u has two neighbors in X, then u ′ ∈ Y and hence X ′ contains u and its two neighbors, contrary to the fact that |X ′ | ≤ 2; and likewise for v. It follows that u, v both have exactly one neighbor in X. If u and v have a common neighbor x in X, then either |X| = 1 and H[X ∪ {u, v}] is a path, contrary to fact that {u, v} is a proper 2-cutset; or x is a cutvertex of H, contrary to Claim 8. It follows that |X| = 2, and so X = X ′ and u ′ , v ′ ∈ X. But now H[X ∪ {u, v}] is a four-vertex path with vertices u, u ′ , v ′ , v in this order, contrary to the fact that {u, v} is a proper 2-cutset. Claim 11. The graph H has no K 4 -minor. Proof of Claim 11. Let us assume towards a contradiction that H has a K 4 -minor. Then, by Lemma 4, it follows that H has a subgraph K which is a subdivision of K 4 . Thus, there exist four vertices of K, say {v 1 , v 2 , v 3 , v 4 }, and a set of six internally vertex-disjoint paths P := {P i,j : i, j ∈ [4] and i < j and P i,j is a (v i , v j )-path in K}, such that K = P. Since H is 2-sparse, it follows that each path in P has length at least two. We claim that each path in P has length exactly two. Suppose not. Without loss of generality, we may assume that P 1,2 has length at least three. Let J be the subgraph of K (and thus of H) obtained by deleting from K the internal vertices of the path P 3,4 . Then, the graph L(J) is a prism. By Lemma 3, since J is a subgraph of H the graph L(J) is an induced subgraph of G ′ and thus G ′ has a prism as an induced subgraph, a contradiction. Thus, each path in P has length exactly two. We claim that H = K. Suppose not. Then, since H is connected, there exists an edge in E(H) \ E(K) which is incident to a vertex v ∈ V (K). Since ∆(H) ≤ 3, we have that v / ∈ {v 1 , v 2 , v 3 , v 4 }. Thus there exists a path P ∈ P such that v is the unique internal vertex of P . Then, d H (v) = 3, and thus both the two edges of P which are incident with the vertex v are not incident with a vertex of degree at most two, which contradicts that, by Claim 10, the graph H is 2-sparse. Hence, H = K and thus G ′ = L(K). We claim that G has no special edges. Suppose not. Let e be a special edge of G and let P e be the corresponding path of the special edge e. Either both the endpoints of e lie in one of the four triangles of G ′ or the endpoints of e lie in two different triangles of G ′ . Suppose that the former case holds. Without loss of generality we may assume that e = {k 1 , k 2 }. Then G[V (P e ) ∪ {k 3 , l 1 , l 3 , n 3 , n 2 }] is a theta + , which is a contradiction. Hence, there exist two triangles of G ′ such that special edge e joins these two triangles. Without loss of generality we may assume that e = {k 1 , l 1 }. Then G[V (P e ) ∪ {k 1 , k 2 , k 3 } ∪ {l 1 , l 2 , l 3 } ∪ {m 1 , m 2 , n 3 , n 2 }] is a prism + , which is a contradiction. Thus G has no special edges and hence G = G ′ = L(K). Let C := l 1 l 3 n 3 n 1 n 2 k 2 k 1 k 3 m 2 m 3 m 1 l 2 l 1 (see Figure 3). Then C is a Hamiltonian cycle in G, which is a contradiction. Hence, H has no K 4 -minor. Since H is 2-connected, G ′ = L(H), and G ′ has no Hamiltonian cycle which uses all of its edges, it follows that H has at least four vertices and H contains a vertex of degree at least three. Thus, since by Claim 11, H has no K 4 -minor, it follows from Lemma 5 that H has at least one vertex of degree two. Let c be such a vertex and let e c 1 and e c 2 be the two edges of H which are incident to c. For each i ∈ [2] let P i be a minimum-length path which has as one end the vertex c, contains the edge e c i , and the other end of P i is a vertex of degree three. The paths P 1 , P 2 exist, because H is 2-connected and not a cycle. Let a, b be the degree-three ends of P 1 and P 2 respectively. We claim that a = b. Suppose not. Then, since P 1 ∪ P 2 is a cycle, and H is not a cycle, the vertex a = b is a cutvertex of H, contradicting Claim 8. Thus, a = b. We claim that the sum of the lengths of P 1 and P 2 is at most three. Suppose not, and let e a and e b be the edges of P 1 and P 2 which are incident to a and b, respectively. Then {e a , e b } is a 2-edge-cutset of H and the graph H \ {e a , e b } has at least two components, each of which contains at least two edges, contradicting Claim 9. This proves our claim. Let C be the set of the connected components of H \ {a, b}. Since ∆(H) ≤ 3 and H is 2-connected, it follows that either |C| = 2 or |C| = 3. We claim that there exist no two vertex-disjoint (A, B)-paths in H 2 . Suppose not, and let Q 1 , Q 2 be such paths. Since H 2 is connected, there exists a (V (Q 1 ), V (Q 2 ))path. Let Q 3 be a minimum length (V (Q 1 ), V (Q 2 ))-path in H 2 . Then no internal vertex of Q 3 lies in V (Q 1 ) ∪ V (Q 2 ). Let K := H 1 ∪ Q 1 ∪ Q 2 ∪ Q 3 + {{a, a 1 }, {a, a 2 }, {b, b 1 }, {b, b 2 }}.. Let F := {{d, d A }, {b, b ′ }}, where b ′ is the neighbor of b in P 2 . Then H \ F has two connected components, say F 1 , F 2 , where F 1 contains a and its neighbors, except possibly d, and {{b, b 1 }, {b, b 2 }} ⊆ E(F 2 ), which contradicts Claim 9. Thus |C| = 3. Claim 13. G ′ is a short prism. Proof of Claim 13. By Claim 12, we have that the graph H \ {a, b} has three connected components. Let C 1 , C 2 and C 3 be the three components of H \ {a, b} where c ∈ V (C 1 ). For each i ∈ [3], let e a i and e b i be the edge of H which has as one end the vertex a or b, respectively, and its other end lies in C i . Since, by Claim 9, for each i ∈ [3] there exists at least one component of H \ {e a i , e b i } which is either a single vertex or a single edge, it follows that each of the components C 2 and C 3 , is either a single vertex or a single edge. Recall that C 1 is a path by construction, shown to have length at most 1. Since H has no cutvertex, it follows that there are only four possibilities for the graph H, depending on how many of the components C 1 , C 2 , C 3 consist of a single vertex (see Figure 4). If each of C 1 , C 2 , C 3 contains two vertices, then G ′ is a prism, a contradiction. So at least one of the components C 1 , C 2 , C 3 contains only one vertex, and G ′ is a short prism. Since G ′ is a short prism, let {k 1 , k 2 , k 3 }, {l 1 , l 2 , l 3 }, P 1 , P 2 , P 3 as in the definition of the short prism in Section 1. We claim that no edge of a triangle of G ′ is a special edge of G. Suppose not. Without loss of generality we may assume that {k 1 , k 2 } ∈ E(G ′ ) is a special edge of G. Let P k1k2 be the corresponding path of the edge {k 1 , k 2 }. Then G[V (P k1k2 ) ∪ V (P 1 ) ∪ V (P 2 ) ∪ {k 3 }] is a theta + (see Figure 5), which contradicts the fact that G is (theta + )-free. We claim that there exists at least one path in the set {P 1 , P 2 , P 3 } of length one such that the unique edge of this path is not a special edge of G. Suppose not, let Q ⊆ V (G) be the set of the internal vertices of the corresponding paths of all P i where P i is length one. Then, G[Q ∪ {k 1 , k 2 , k 3 } ∪ {l 1 , l 2 , l 3 } ∪ P 1 ∪ P 2 ∪ P 3 ] is a prism + , which contradicts the fact that G is (prism + )-free. Hence, without loss of generality we may assume that P 1 has length one and the unique edge of P 1 is not a special edge. It follows that every special edge of G ′ is contained in E(P 2 ) ∪ E(P 3 ). Let C ′ := (P 2 ∪ P 3 ∪ ({k 1 , l 1 }, ∅)) + {{k 1 , k 2 }, {k 1 , k 3 }, {l 1 , l 2 }, {l 1 , l 3 }}. Then, C ′ is a Hamiltonian cycle of G ′ which contains all the special edges of G, contradicting Claim 6. This concludes the proof. Figure 1 . 1An example of a wheel. Claim 2 . 2Let {u, v} be a 2-cutset of G. Then there exist induced subgraphs of H 1 , H 2 12be the two connected components of G \ {u, v} and let H ′ 1 := G[V (H 1 ) ∪ {u, v}] and H ′ 2 := G[V (H 2 ) ∪ {u, v}] Claim 4 . 4The graph G ′ does not admit a clique cutset. Claim 5 . 5Let {u, v} be a 2-cutset of G ′ . Then G ′ \ {u, v} has exactly two connected components, and one of these is a single vertex.Proof of Claim 5. Immediate by Claim 2 and Claim 3. Claim 6 . 6The graph G ′ does not have a Hamiltonian cycle which contains all the special edges of G. Claim 7 . 7There exists a triangle-free chordless graph H, with ∆(H) ≤ 3, such that G ′ = L(H). Figure 3 . 3Proof of Claim 11: The Hamiltonian graph L(K). Let {k 1 , k 2 , k 3 }, {l 1 , l 2 , l 3 }, {m 1 , m 2 , m 3 } and {n 1 , n 2 , n 3 }, be the four cliques of size three of the graph G ′ . Without loss of generality we may assume that: {{k 3 , m 2 }, {m 1 , l 2 }, {m 3 , n 1 }, {k 1 , l 1 }, {l 3 , n 3 }, {n 2 , k 2 }} ⊆ E(G ′ ). Claim 12 . 12|C| = 3. Proof of Claim 12. Let us suppose towards a contradiction that |C| = 2. Let H 1 , H 2 be the two connected components of H \ {a, b}, where c ∈ V (H 1 ). Since H is 2-sparse and the vertices a, b have degree three in H, it follows that {a, b} / ∈ E(H). Since d(a) = d(b) = 3 and {a, b} / ∈ E(H), it follows that each of a, b has two neighbors in H 2 . Let a 1 , a 2 and b 1 , b 2 be the neighbors of a and b respectively in H 2 . Let A := {a 1 , a 2 } and B := {b 1 , b 2 }. By Menger's theorem (see, for example [12]), applied in the component H 2 , either there exist two vertexdisjoint (A, B)-paths in H 2 or there exists a vertex separating A from B in H 2 . Figure 4 . 4Proof of Claim 13: The graph H is one of the graphs illustrated above. Figure 5 . 5A theta + in G in case one of the triangles of G ′ contains a special edge. and for every edge e = hh ′ ∈ E(H), there is at least one edge between A h and A h ′ in G. We say that the graph G contains H as a minor (or contains an H-minor ) if G contains a model of H. The following is well known (see, for example,[10], 1.7).Lemma 4. Let G and H be graphs such that ∆(H) ≤ 3 and G contains H as a minor. Then G has a subdivision of H as a subgraph.Lemma 5 ([11]). 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Trotignon, On graphs with no induced subdivision of K 4 , Journal of Combinatorial Theory, Series B 102 (4) (2012) 924-947. R Diestel, Graph Theory: 5th edition, Springer Graduate Texts in Mathematics. Springer-Verlag© Reinhard DiestelR. Diestel, Graph Theory: 5th edition, Springer Graduate Texts in Mathematics, Springer- Verlag, © Reinhard Diestel, 2017. Topology of series-parallel networks. R J Duffin, Journal of Mathematical Analysis and Applications. 102R. J. Duffin, Topology of series-parallel networks, Journal of Mathematical Analysis and Applications 10 (2) (1965) 303-318. . K Menger, Zur Allgemeinen Kurventheorie, Fundamenta Mathematicae. 10K. Menger, Zur allgemeinen Kurventheorie, Fundamenta Mathematicae 10 (1927) 96-115.

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{'abstract': 'Given a graph G and a graph property P we say that G is minimal with respect to P if no proper induced subgraph of G has the property P . An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph H, a graph G is H-free if G has no induced subgraph isomorphic to H. The main motivation for this paper originates from a theorem of Duffus,Gould, and Jacobson (1981), which characterizes all the minimal connected graphs with no Hamiltonian path. In 1998, Brousek characterized all the clawfree HC-obstructions. On a similar note, Chiba and Furuya(2021), characterized all (not only the minimal) 2-connected non-Hamiltonian {K 1,3 , N 3,1,1 }free graphs. Recently, Cheriyan, Hajebi, and two of us (2022), characterized all triangle-free HC-obstructions and all the HC-obstructions which are split graphs. A wheel is a graph obtained from a cycle by adding a new vertex with at least three neighbors in the cycle. In this paper we characterize all the HC-obstructions which are wheel-free graphs. † We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number RGPIN-2020-03912]. Cette recherche aété financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de référence RGPIN-2020-03912].', 'arxivid': '2204.07671', 'author': ['Aristotelis Chaniotis ', 'Zishen Qu ', 'Sophie Spirkl '], 'authoraffiliation': [], 'corpusid': 248228070, 'doi': '10.1016/j.disc.2022.113289', 'github_urls': [], 'n_tokens_mistral': 12005, 'n_tokens_neox': 10856, 'n_words': 7081, 'pdfsha': 'ab1cddbb7556becf31baec5ebcf196f6222bb1be', 'pdfurls': ['https://export.arxiv.org/pdf/2204.07671v2.pdf'], 'title': ['MINIMAL INDUCED SUBGRAPHS OF THE CLASS OF 2-CONNECTED NON-HAMILTONIAN WHEEL-FREE GRAPHS', 'MINIMAL INDUCED SUBGRAPHS OF THE CLASS OF 2-CONNECTED NON-HAMILTONIAN WHEEL-FREE GRAPHS'], 'venue': []}

arxiv

A Hybrid Quantum-Classical Hamiltonian Learning Algorithm 24 Aug 2021 Youle Wang Center for Quantum Software and Information University of Technology Sydney 2007NSWAustralia Institute for Quantum Computing Baidu Research 100193BeijingChina Guangxi Li Center for Quantum Software and Information University of Technology Sydney 2007NSWAustralia Institute for Quantum Computing Baidu Research 100193BeijingChina Xin Wang Institute for Quantum Computing Baidu Research 100193BeijingChina A Hybrid Quantum-Classical Hamiltonian Learning Algorithm 24 Aug 2021 Hamiltonian learning is central to studying complex many-body physics and the certification of quantum devices and simulators. How to learn the Hamiltonian in general with near-term quantum devices is a challenging problem. In this paper, we develop a hybrid quantum-classical Hamiltonian learning algorithm to tackle this problem. By transforming the Hamiltonian learning problem to an optimization problem using the Jaynes' principle, we employ a gradient-descent method to give the solution and could reveal the interaction coefficients from the system's Gibbs state measurement results. In particular, the computation of the gradients relies on the Hamiltonian spectrum and the log-partition function. Hence, as the main subroutine, we develop a variational quantum algorithm to extract the Hamiltonian spectrum and utilize convex optimization to output the log-partition function. We also apply the importance sampling technique to circumvent the resource requirements for dealing with large-scale Hamiltonians. As a proof of principle, we demonstrate the effectiveness of our algorithm by conducting numerical experiments for randomly generated Hamiltonians and many-body Hamiltonians of theoretical and practical interest.To establish the results, our method for minimizing the free energy depends on two critical steps. First, we choose a suitable PQC with enough expressiveness and train it to learn the eigenvectors of the Hamiltonian and output the corresponding eigenvalues. Second, we combine the post-training PQC with the classical methods for convex optimization to find the global minimum of the free energy. Next, we utilize the post-training PQC and the optimizer of the convex optimization to compute the gradients. Furthermore, we theoretically analyze the estimation precision of the gradients. We also show the efficiency of loss evaluation and gradients estimation by the importance sampling technique when the underlying Hamiltonian is large.As the proof of principle, we study the effectiveness of our algorithm for Hamiltonian learning by conducting numerical experiments for randomly generated Hamiltonians and several many-body Hamiltonians. To generate random Hamiltonians, we choose Pauli tensor products E ℓ from the set {X, Y, Z, I} ⊗n at random, with n ranging from 3 to 5. The target interaction coefficients are chosen via a uniform distribution over [−1, 1]. The tested many-body Hamiltonians consist of Ising, XY -spin, and Heisenberg models, where size also varies from 3 to 5 qubits. For these Hamiltonians, we test our algorithm for different parameters β and µ with different lengths. As a result, the numerical results show that the target interaction coefficients can be estimated with high precision. In these experiments, our algorithm learns all eigenvalues of Hamiltonians. Moreover, we show the effectiveness by partially learning few smallest eigenvalues of Ising Hamiltonians. In particular, the circuit depth of used PQC could be significantly reduced. Finally, we also generalize the experiments to larger Ising Hamiltonians with 6/7 qubits.Next, we summarize the contribution of this paper and all mentioned results above. 1. We propose a hybrid quantum-classical Hamiltonian learning framework based on the fundamental properties of free energy, which mainly consists of the following two subroutines: log-partition function estimation and stochastic variational quantum eigensolver (SVQE). 2. The main subroutine is the log-partition function estimation algorithm, which combines the SVQE with the classical convex optimization to minimize the free energy. 3. We also propose a feasible scheme for learning the spectrum of the many-body Hamiltonian by integrating variational quantum algorithms with the importance sampling technique. 4. We demonstrate our algorithm's validity by numerical simulations on several random Hamiltonians and many-body Hamiltonians (e.g., Ising model, XY model, and Heisenberg model). Organization. The remaining paper proceeds as follows. In Sec. II, we formally define the problems we studied in this work; In Sec. III, we present the main results, including the Hamiltonian learning algorithm, and its main subroutines logpartition function estimation, stochastic variational quantum eigensolver, and gradient estimation; In Sec. IV, we describe the experimental settings and provide numerical results to demonstrate the efficacy of our algorithm; Lastly, we conclude the paper in Sec. V. Proofs and more discussions are presented in the Supplementary Material. I. INTRODUCTION Hamiltonian learning is an important task in studying quantum physics systems and the experimental realization of quantum computers. For instance, it can predict the quantum system's locality to describe the effective interactions between particles, which plays a crucial role in quantum technology, such as quantum lattice models [1], quantum simulation [2], and adiabatic quantum computation [3]. Moreover, with recent experimental advances in tools for studying complex interacting quantum systems [4], it is becoming more and more essential to learn the dynamics of complicated physical systems, which can predict the evolution of any initial state governed by the Hamiltonian. Another critical utility is relevant to the verification of quantum devices and simulators towards building fault-tolerant quantum computers [5] since certifying that the engineered Hamiltonian matches the theoretically predicted models will always be an indispensable step in developing high-fidelity quantum gates [6]. Hamiltonian of many-body physics is often characterized by some parameters, which describe the interactions between the particles. Technically, a many-body Hamiltonian is composed of polynomially many local Pauli operators, i.e., H = m ℓ=1 µ ℓ E ℓ ,(1) where µ = (µ 1 , . . . , µ m ) ∈ [−1, 1] m , and {E ℓ } m ℓ=1 are n-qubit local Pauli operators, with m = O(poly(n)). Despite the number of these parameters µ in general scales polynomially in the system's size, it is pretty challenging to learn these parameters. Classically characterizing the system's Hamiltonian via tomography would require resources that exponentially scale in the system's size [7]. Other than tomography, there are methods [8][9][10][11][12] that cost polynomially many resources while requiring the ability to simulate the dynamics of the system, which is classically intractable. Moreover, it is difficult to perform quantum simulation as a large amount of low-decoherence and fully-connected qubits are required, which are not available on NISQ devices [13]. The major goal of this paper is to learn the many-body Hamiltonians using a trusted NISQ device. For this purpose, we exploit the variational quantum algorithms (VQAs) that have been gaining popularity in many areas [14][15][16][17][18][19][20][21][22][23]. VQAs are a class of hybrid quantum-classical algorithms that are expected to be implementable on NISQ devices. The main process is to optimize a certain loss function via parameterized quantum circuits (PQCs). In particular, the loss function depending on parameters of the circuit is evaluated on quantum devices, and then the parameters are updated using gradient-based methods classically. As for Hamiltonian learning, we take advantage of the strategy proposed recently in [24], which allows recovering parameters µ from the measurement results of a quantum Gibbs state ρ β = e −βH / Tr(e −βH ), i.e., e ℓ = Tr(ρ β E ℓ ) for all ℓ = 1, . . . , m. It has been shown that solving the optimization problem below suffices to complete the Hamiltonian learning task. µ = argmin ν log Z β (ν) + β m ℓ=1 ν ℓ e ℓ .(2) Here, Z β (ν) = Tr(e −β m ℓ=1 ν ℓ E ℓ ) denotes the partition function, parameterized by ν = (ν 1 , ..., ν m ) ∈ [−1, 1] m , and β denotes the inverse temperature of the system. In this paper, we propose a hybrid quantum-classical algorithm to perform the Hamiltonian learning task, whose aim is to recover the interaction coefficients µ from the measurement results {e ℓ } m ℓ=1 . The main idea is to solve the optimization problem in Eq. (2) by a gradient-descent method and compute the corresponding gradients utilizing variational quantum algorithms. The challenge of our approach is to compute the log-partition function log Z β (ν) and its gradient since computing partition function is #P-hard [25,26]. To overcome this challenge, we accordingly develop a method based on the relation between the log-partition function and the system's free energy. In general, suppose the state of the system is ρ, then the free energy is given by F (ρ) = Tr(Hρ)−β −1 S(ρ), where S(ρ) is the von Neumann entropy. The relation states that the global minimum of F (ρ) is proportional to the log-partition function, i.e., log Tr(e −βH ) = −β min ρ F (ρ). (3) II. PROBLEM STATEMENT In this paper, the goal of Hamiltonian learning is to learn the interaction coefficients µ from the measurement results of a quantum Gibbs state. We assume that the Hamiltonian to be learned H is composed of local Pauli operators {E ℓ } m ℓ=1 , and the measurements corresponding to {E ℓ } m ℓ=1 are performed on the Gibbs state ρ β = e −βH / Tr(e −βH ) at an inverse temperature β. The measurement results are denoted by {e ℓ } m ℓ=1 , given by e ℓ = Tr(ρ β E ℓ ), ∀ℓ ∈ [m].(4) Recently, there are many methods proposed to efficiently obtain measurement results {e ℓ } m ℓ=1 [27][28][29]. We, therefore, assume the measurement results {e ℓ } m ℓ=1 have been given previously and focus on learning interaction coefficients from them. Formally, we define the Hamiltonian learning problem (HLP) as follows: Definition 1 (HLP) Consider a many-body Hamiltonian with a decomposition given in Eq. (1), where |µ ℓ | ≤ 1 for all ℓ = 1, ..., m. Suppose we are given measurement results {e ℓ } m ℓ=1 of the quantum Gibbs state ρ β , then the goal is to find an estimate µ of µ such that where · ∞ means the maximum norm. µ − µ ∞ ≤ ǫ,(5) To solve the HLP, we adopt a strategy that is proposed recently in Ref. [24], which transforms HLP into an optimization problem by using the Jaynes' principle (or maximal entropy principle) [30]. This strategy is to find a quantum state with the maximal entropy from all states whose measurement results under {E ℓ } m ℓ=1 match {e ℓ } m ℓ=1 . max ρ S(ρ)(6) s.t. Tr(ρE ℓ ) = e ℓ , ∀ℓ = 1, ..., m ρ > 0, Tr(ρ) = 1. It has been shown in [30] that the optimal state is of the following form: σ = exp(−β m ℓ=1 µ * ℓ E ℓ ) Tr(exp(−β m ℓ=1 µ * ℓ E ℓ )) .(7) Here, state σ is a quantum Gibbs state of a Hamiltonian with interaction coefficients µ * = (µ * 1 , ..., µ * m ). As a result, Ref. [24] shows that coefficients of σ is the target interaction coefficients, i.e., µ * = µ. Moreover, Ref. [24] also points out an approach for obtaining µ * that is to solve the dual optimization problem in Eq. (2). To this end, we develop a gradient-descent method to solve the problem in Eq. (2). A flowchart for illustration is shown in Figure 1. Clearly, the main obstacle is to compute the corresponding gradients of the objective function, which involves computing the partition function. Then, we formalize the gradient estimation problem below. Definition 2 (Gradient estimation) Given a Hamiltonian parameterized by coefficients ν, i.e., H(ν) = m ℓ=1 ν ℓ E ℓ , let L(ν) be the objective function L(ν) = log Z β (ν) + β m ℓ=1 ν ℓ e ℓ ,(8) where Z β (ν) = Tr(e −βH(ν) ). Then the goal is to estimate the gradient ∇L(ν) with respect to ν. The following sections are devoted to solving HLP and the Gradient estimation problem. III. MAIN RESULTS This section presents the main results of this paper. Specifically, we first discuss the core idea and outline the framework for computing the log-partition function in Sec. III A. In Sec. III B, we provide a variational quantum algorithm for learning the eigenvectors of Hamiltonians. Based on the results in Sec III A-III B, we then proceed to give the gradient estimation procedure in Sec. III C. Last, Sec. III D provides the main algorithm, the hybrid quantum-classical Hamiltonian learning algorithm (HQHL). A. Log-partition function estimation Here, we consider computing the log-partition function log Z β (ν). Motivating our method is the relationship between the logpartition function and free energy. Recall that free energy of the system being state ρ is given by F (ρ) = Tr(H(ν)ρ)−β −1 S(ρ), assuming the parameterized Hamiltonian is H(ν) = m ℓ=1 ν ℓ E ℓ . Then the relation states that log Z β (ν) = −β min ρ F (ρ).(9) As shown in Eq. (9), it is natural to minimize the free energy to obtain the value of log Z β (ν). However, it is infeasible to directly minimize the free energy on NISQ devices since performing entropy estimation with even shallow circuits is difficult [31]. To deal with this issue, we choose an alternate version of Eq. (9): log Z β (ν) = −β min p N j=1 p j · λ j + β −1 N j=1 p j log p j ,(10) where λ = (λ 1 , ..., λ N ) is the vector of eigenvalues of H(ν), and p = (p 1 , ..., p N ) represents an N -dimensional probability distribution, with N = 2 n the Hamiltonian's dimension. Please note that proofs for Eqs. (9)-(10) are provided in the supplementary file. Thus, optimizing the R.H.S of Eq. (10) could obtain the desired quantity and avoid the von Neumann entropy estimation simultaneously, assuming eigenvalues of the Hamiltonian H(ν) is given previously. As a result, our task is reduced to solve the following optimization program based on the equality in Eq. (10): min p C(p)(11)s.t. N j=1 p j = 1 p j ≥ 0, ∀j = 1, . . . , N where C(p) = N j=1 p j · λ j + β −1 N j=1 p j log p j .(12) The optimization program in Eq. (11) is a typical convex optimization program. In the context of convex optimization, there are many classical algorithms to solve the optimization program, such as the interior-point method [32], ellipsoid method [33], cutting-plane method [34], and random walks [35], etc. For example, we consider using the cutting plane method [36,37], which requires the membership and evaluation procedures [38]. Concerning the program in Eq. (11), the membership procedure determines whether a point belongs to the set of probability distributions, and the evaluation procedure takes in a probability distribution p and returns the value C(p) with high accuracy. Clearly, it is easy to determine whether the given point is a probability distribution while challenging to efficiently evaluate the function value. Thus, we provide a procedure to solve the convex optimization problem as well as overcome this challenge at the same time in Algorithm 1. In Algorithm 1, we compute the log-partition function using a classical convex optimization method. For this purpose, we first show the construction process of evaluation procedure. That is, given a point p, find an estimate for C(p). We assume we are given a parameterized quantum circuit U (θ) that can learn eigenvectors of the Hamiltonian H(ν). In our approach, the U (θ) is combined with the importance sampling technique (cf. lines [3][4][5][6][7][8] to deal with the large-sized Hamiltonians. Specifically, i) we sample T D indices according to the distribution p (cf. line 4); ii) we evaluate the eigenvalues associated with the sampled indices (cf. lines 5-6); iii) we take the average over T (cf. line 7) and the median over D (cf. line 8) to evaluate the function value C(p) with high accuracy and success probability. Eventually, with the evaluation procedure and the membership procedure, the global minimum of C(p) could be obtained via the cutting plane method [36][37][38]. Finally, based on the relationship between log Z β (ν) and C(p * ) (cf. Eq. (10)), we could derive the log-partition function value. Here p * denotes the optimal distribution of the optimization in Eq. (10)). Algorithm 1 Log-partition function estimation Remark 1 Notice that a crucial gadget in Algorithm 1 is the PQC U (θ), which we have assumed to be accessible. To complement the assumption, we provide a procedure for extracting eigenvalues in the next section, Stochastic variational quantum eigensolver. In particular, we will prsent a variational quantum algorithm for learning the eigenvectors of the parameterized Hamiltonians. Now we discuss the cost of applying Algorithm 1. As the efficiency of Algorithm 1 mainly relies on the cost of the evaluation procedure, we only discuss it here. Suppose we have access to Hamiltonian H(ν)'s eigenvalues λ, then the objective function C(p) can be effectively evaluated. Recall that C(p) contains two parts N j=1 p j · λ j and β −1 N j=1 p j log p j . On the one hand, the latter value can be computed immediately since p is stored on classical devices. On the other hand, value N j=1 p j · λ j can be regarded as an expectation of the probability p, where value λ j is sampled with probability p j . Notably, the total cost for estimating C(p) is dominated by the number of samples. Then we analyze the number of required samples for loss evaluation in Proposition 1. Proposition 1 For any constant β > 0 and parameterized Hamiltonian H(ν) = m ℓ=1 ν ℓ E ℓ with E ℓ ∈ {X, Y, Z, I} ⊗n and ν ∈ R m , suppose we are given access to a parameterized quantum circuit U (θ) that can prepare H(ν)'s eigenvectors, then the objective function C(p) can be computed up to precision ǫ with probability larger than 2/3 by taking T = O(m ν 2 2 /ǫ 2 ) samples. Furthermore, the probability can be improved to 1 − η costing an additional multiplicative factor of D = O(log(1/η)). Sketch of Proof. In general, the expectation can be approximated by the sample mean according to Chebyshev's inequality. Specifically speaking, the expectation can be estimated up to precision ǫ with high probability (e.g., larger than 2/3) by taking O(Var/ǫ 2 ) samples, where Var denotes the variance of the distribution. Here, the number of samples is T = O(m ν 2 2 /ǫ 2 ), since the variance is bounded by the squared spectral norm of H(ν), which is less than √ m ν 2 . Furthermore, Chernoff bounds allow improving success probability to 1 − η at an additional cost of a multiplicative factor of D = O(log(1/η)). ⊓ ⊔ As shown in Proposition 1, our evaluation method is computationally efficient, since the number of samples scales polynomially with the number of qubits. Hence Algorithm 1 could be applied to compute the partition function of the parameterized Hamiltonian, given the suitable PQC U (θ). B. Stochastic variational quantum eigensolver This section discusses learning the eigenvectors of the parameterized Hamiltonian H(ν) using variational quantum algorithms and the importance sampling technique. First, we outline the algorithm in Algorithm 2 and then discuss the fundamental theory. Second, we circumvent the cost for coping with large-scaled Hamiltonians by the importance sampling technique. We also analyze the cost of loss evaluation in the algorithm. To incorporate variational quantum algorithms, we utilize the variational principle of Hamiltonian's eigenvalues. That is, Hamiltonian's eigenvalues majorize the diagonal elements, and the dot function with an increasingly ordered vector is Schur concave [40]. A similar idea has already been discussed in [41]. In contrast, our method learns the full spectrum of the Hamiltonian. We define a function M (θ) over all parameters θ of the circuit. M (θ) = N j=1 q j · ψ j |U † (θ)H(ν)U (θ)|ψ j ,(13) where q = (q 1 , ..., q N ) is a probability distribution such that q 1 < q 2 < ... < q N , and notations |ψ 1 , . . . , |ψ N denote the computational basis. Suppose that PQC U (θ) has enough expressiveness, then U (θ)|ψ j could learn the j-th eigenvector of the Hamiltonian H(ν) with suitable parameters. Particularly, M (θ) will reach the global minimum when all eigenvectors are Algorithm 2 Stochastic variational quantum eigensolver (SVQE) Require: Parameterized quantum circuit U (θ), Hamiltonian H(ν), and weights q; Ensure: Optimal PQC U (θ); 1: Set number of iterations I and l = 1; 2: Set integers T and D; 3: Set learning rate r θ ; 4: Set probability distribution q; 5: Sample T D integers k 1 1 ,. . .,k 1 T ,. . . ,k D 1 ,. . .,k D T according to q; 6: Prepare computational states |ψ k 1 1 , . . . , |ψ k 1 T , . . ., |ψ k D 1 , . . ., |ψ k D T ; 7: while l ≤ I do 8: Compute value ψ k s j |U † (θ)H(ν)U (θ)|ψ k s j for all j = 1,. . .,T and s = 1,. . .,D; 9: Compute averages: learned. In other words, we use the PQC U (θ) to learn eigenvectors via finding the global minimum of M (θ) over all parameters θ. aves = 1 T T j=1 ψ k s j |U † (θ)H(ν)U (θ)|ψ k Remark 2 Choosing a suitable U (θ) is critical to many variational quantum algorithms as well as our Algorithm 2. With enough expressibility, training the PQC U (θ) would allow us to exactly or approximately learn the solution to the certain problem. The expressibility of PQCs has been recently studied in [42]. Throughout this paper, we assume the used PQC U (θ) is able to learn well the eigenvectors of Hamiltonians H(ν) for arbitrary ν. Remark 3 In the learning process, we employ a gradient-based method to update the parameters θ iteratively. In each iteration, the corresponding gradients are computed via the parameter shift rule [39], which outsources the gradient estimation to the loss evaluation. As this is similar to other variational quantum algorithms, we omit the details of gradient computation. For details of gradient derivation, please refer to the proof of Proposition 3 in [22]. Notice that for large Hamiltonians, the loss M (θ) may consist of exponentially many terms, which would be a huge burden to the loss evaluation. However, we could employ the importance sampling technique to circumvent this issue. To this end, M (θ) is taken as an expectation of the distribution q. Hence, M (θ) is to be estimated by the sample mean. Notably, the cost of loss evaluation is dominated by the number of samples, which is why we call our method stochastic variational quantum eigensolver (SVQE). Our algorithm with importance sampling for minimizing M (θ) is depicted in Algorithm 2. In the following, we analyze the sample complexity in the loss evaluation. 1]. Given any constants ǫ > 0, η ∈ (0, 1), β > 0, the objective function M (θ) in SVQE can be estimated up to precision ǫ with probability at least 1 − η, costing T D samples with T = O(m ν 2 2 /ǫ 2 ) and D = O(log(1/η)). Besides, the total number of measurements is given below: Proposition 2 Consider a Hamiltonian H(ν) = m ℓ=1 ν ℓ E ℓ with Pauli operators E ℓ ∈ {X, Y, Z, I} ⊗n and constants ν ℓ ∈ [−1,O mT D ν 2 1 (n + log(m/η)) ǫ 2 .(14) Sketch of Proof. The number of samples is determined by the accuracy ǫ and Hamiltonian H(ν). By Chebyshev's inequality, estimating M (θ) up to precision ǫ with high probability requires T = O(m ν 2 2 /ǫ 2 ) samples, since the variance is bounded by the spectral norm, which is less than √ m ν 2 . Meanwhile, the expectation value ψ j |U † (θ)H(ν)U (θ)|ψ j is evaluated by measurements. We compute the expectation value of the observable H(ν) by measuring each Pauli operator E ℓ separately, since there are only m = O(poly(n)) Pauli operators. ⊓ ⊔ Remark 4 Other methods for computing expectation value of Hamiltonians can be found in Ref. [43,44], where importance sampling is employed to sample Pauli operator E l of the Hamiltonian. Remark 5 In the context of quantum algorithms, there are many proposed methods for learning the low-lying eigenvectors of the Hamiltonian and diagonalizing Hamiltonian. Some known quantum algorithms for Hamiltonian diagonalization are based on quantum fast Fourier transform [45], which may be too costly for NISQ computers and thus not suitable for our purpose. Recently, there have already been some works on finding ground and excited eigenstates of the Hamiltonian with NISQ devices, i.e., variational quantum eigensolvers [19,22,41,[46][47][48][49][50]. They maybe employed to learn eigenvectors in the Hamiltonian learning framework. C. Gradient estimation Recall that we employ a gradient-based method to do the optimization in the Hamiltonian learning (cf. Figure 1). We use the tools developed in Sec. III A-III B to derive the gradient estimation procedure. Usually, with the estimated gradient, parameters are updated in the following way: ν ← ν − r∇L(ν),(15) where r is the learning rate. The expression of the gradient is given below. ∇L(ν) = ∂L(ν) ∂ν 1 , ..., ∂L(ν) ∂ν m .(16) Furthermore, the explicit formula of each partial derivative is given in [24]: ∂L(ν) ∂ν ℓ = ∂ ∂ν ℓ log Z β (ν) + βe ℓ = −β Tr(ρ β (ν)E ℓ ) + βe ℓ ,(17) where ρ β (ν) = e −βH(ν) /Z β (ν) represents the Gibbs state associated with the parameterized Hamiltonian H(ν). According to the second equality in Eq. (17), preparing Gibbs state ρ β (ν) is likely to be necessary to the gradient estimation, which is quite challenging [21,[51][52][53][54]. However, we provide a procedure for gradient estimation without preparing the Gibbs state ρ β (ν) in Algorithm 3. We use the post-training PQC U (θ) and the optimal distribution p * (cf. Algorithm 1) from Sec. III A-III B, respectively. And the component of the gradient can be computed in the sense that Algorithm 3 Gradient estimation ∂L(ν) ∂ν ℓ ≈ −β N j=1 p * j · ψ j |U † (θ)E ℓ U (θ)|ψ j + βe ℓ .(18) The validity of the relation in Eq. (18) is proved in Proposition 3. Proposition 3 (Correctness) Consider a parameterized Hamiltonian H(ν) and its Gibbs state ρ β (ν). Suppose the U (θ) from SVQE (cf. Algorithm 2) and p * from log-partition function estimation procedure (cf. Algorithm 1) are optimal. Define a density operator ρ * β as follows: ρ * β = N j=1 p * j · U (θ) |ψ j ψ j | U † (θ),(19) Algorithm 4 Hybrid quantum-classical Hamiltonian learning algorithm (HQHL) Require: Pauli operators {E ℓ } m ℓ=1 , constants {e ℓ } m ℓ=1 , and β; Ensure: An estimate for target coefficients ν; 1: Initialize coefficients {ν ℓ } m ℓ=1 ; 2: Set number of iterations I and l = 1; 3: Set parameterized quantum circuit U (θ); 4: Set learning rate r; 5: while l ≤ I do 6: Set Hamiltonian H(ν) = m ℓ=1 ν ℓ E ℓ ; 7: Train U (θ) by SVQE with H(ν); 8: Derive a probability p * by performing log-partition function estimation with U (θ) and β; 9: Compute gradient ∇L(ν) by gradient estimation with U (θ), p * , and β; 10: Update coefficients ν ← ν − r∇L(ν); 11: Set l ← l + 1; 12: end while 13: return the final coefficients ν. where {|ψ j } denote the computational basis. Denote the estimated eigenvalues by λ, where λ j = ψ j |U † (θ)H(ν)U (θ)|ψ j for all j = 1, . . . , N . Then, ρ * β is an approximation of ρ β (ν) in the sense that D(ρ * β , ρ β (ν)) ≤ 2β max E p * [| λ − λ|], E p * [| λ − λ|] .(20) where D(·, ·) denotes the trace distance, λ represent H(ν)'s true eigenvalues, p * is the distribution corresponding to λ, i.e., λ j = e −βλj / l e −βλ l , and E p * [| λ − λ|] = N j=1 p * j | λ j − λ j |, E p * [| λ − λ|] = N j=1 p * j | λ j − λ j |.(21) Note that the quantity in Eq. (18) contains an expectation of distribution p * , then the partial derivative ∂L(ν) ∂ν ℓ is estimated by the sample mean. Specifically, we first randomly select the computational basis vectors |ψ j complying with distribution p * and then compute the associated eigenvalues via U (θ). The detailed procedure of sampling and estimate computation is laid out in Algorithm 3. The number of required samples is analyzed in Proposition 4. Proposition 4 (Sample complexity) Given ǫ > 0 and η ∈ (0, 1), Algorithm 3 can compute an estimate for the gradient ∇L(ν) up to precision ǫ with probability larger than 1 − η. Particularly, the overall number of samples is KD = O(β 2 log(2m/η)/ǫ 2 ) with K = O(β 2 /ǫ 2 ) and D = O(log(2m/η)). Besides, the total number of measurements is O(KD · mβ 2 (n + log(m/η))/ǫ 2 ). The proofs for Propositions 3-4 are deferred to the supplementary file. To validate the gradient estimation, we show that the average of the overall errors determines the accuracy of the gradient estimation. For this purpose, Proposition 3 shows that matrix ρ * β is an approximation of the desired density matrix ρ β (ν). Specifically, the trance distance between ρ * β and ρ β (ν) is dependent on the averaged errors E p * [| λ − λ|] and E p * [| λ − λ|]. Here, notation | λ − λ| denotes the difference between estimated eigenvalue and the associated real eigenvalue. p * and p * are probability distributions, corresponding to λ and λ, respectively. In particular, it implies that learning several low-lying eigenvectors with high accuracy may lead to a high-precision estimate of the gradient. We numerically verify this feature in Sec. IV C. Moreover, Proposition 4 shows the feasibility of our approach as the number of measurements scales polynomially in parameters n, 1/ǫ, and β. D. Hamiltonian learning algorithm Eventually, we present our hybrid quantum-classical algorithm for Hamiltonian learning (HQHL) in Algorithm 4. The main idea of HQHL is to find the target interaction coefficients by a gradient-descent method (cf. Figure 1). Thus, HQHL's main process is to compute the gradient of the objective function. Specifically, we take Pauli operators {E ℓ } m ℓ=1 , {e ℓ } m ℓ=1 , and β as input. Then we initialize the coefficients by choosing ν from [−1, 1] m uniformly at random. Next, we compute the gradient of the objective function L(ν) by Algorithm 3. Then update the coefficients by choosing a suitable learning rate r and using the estimated gradient. In consequence, after repeating the training process sufficiently many times, the final coefficients are supposed to approximate the target coefficients ν. Notably, the learning process is in the "while" loop of HQHL. In the loop, the subroutine SVQE (cf. Sec. III B) is first called to learn Hamiltonian's eigenvectors and eigenvalues. Here, we choose a suitable parameterized quantum circuit U (θ) and train it to prepare the eigenvectors of the Hamiltonian H(ν). Afterwards, we enter the process of the log-partition function estimation (cf. Sec. III A). It first exploits the U (θ) to output the estimated eigenvalues of the parameterized Hamiltonian H(ν) and then computes the objective function L(ν). We would obtain a probability distribution p * that consists of eigenvalues of the associated Gibbs state ρ β (ν) = e −βH(ν) /Z β (ν). Lastly, we exploit the resultant results (post-training circuit U (θ) and distribution p * ) to compute the gradients following the procedure in Algorithm 3 and update the coefficient ν accordingly (cf. Eq. (15)). IV. NUMERICAL RESULTS In this section, we conduct numerical experiments to verify the correctness of our algorithm. Specifically, we consider recovering interactions coefficients of several Hamiltonians, including randomly generated Hamiltonians and many-body Hamiltonians. To ensure the performance of the algorithm, we choose a PQC (shown in Fig. 2) and set the circuit with enough expressibility. When testing our algorithm, we first use SVQE to learn the full spectrum of Hamiltonians, where size of the Hamiltonian varies from 3 to 5. In SVQE, weights q consists of a normalized sequence of arithmetic sequence. For instance, when n = 3, q = (1, 2, 3, . . . , 8)/S 3 , where S 3 = 8 l=1 l. Furthermore, in order to reduce quantum resources, we also partially learn the few smallest eigenvalues of the selected Ising models and derive estimates for coefficients up to precision 0.05. With fewer eigenvalues to be learned, the depth of the used PQC is significantly reduced. Rz(θ0,0,0) Ry(θ0,0,1) Rz(θ0,0,2) • Rz(θ1,0,0) Ry(θ1,0,1) Rz(θ1,0,2) · · · Rz(θ0,1,0) Ry(θ0,1,1) Rz(θ0,1,2) • Rz(θ1,1,0) Ry(θ1,1,1) Rz(θ1,1,2) · · · Rz(θ0,2,0) Ry(θ0,2,1) Rz(θ0,2,2) • Rz(θ1,2,0) Ry(θ1,2,1) Rz(θ1,2,2) · · · Rz(θ0,3,0) Ry(θ0,3,1) Rz(θ0,3,2) • Rz(θ1,3,0) Ry(θ1,3,1) Rz(θ1,3,2) · · · ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ×D A. Random Hamiltonian models This section shows the effectiveness of our algorithm with random Hamiltonians from three aspects: different β, different numbers of µ (# µ) and a different number of qubits (# qubits). In the experimental setting, we randomly choose Pauli tensor products E ℓ from {X, Y, Z, I} ⊗n and target coefficients µ by a uniform distribution over [−1, 1]. Specifically, we first vary the values of β by fixing the number of µ and the number of qubits to explore our method's sensitivity to temperature. We similarly vary the number of µ and the number of qubits by fixing other hyper-parameters to explore our method's scalability. The actual values of these hyper-parameters sampled/chosen in each trial are concluded in Table I. In addition, the deep, D, of the PQC U (θ) is set according to the size of Hamiltonian. As number of qubits ranges from n = 3 to n = 5, the depth D is set to be 10, 20, 40, respectively. In Table I 0.3408I ⊗ Y ⊗ X − 0.6384Y ⊗ X ⊗ Z − 0.4988I ⊗ Z ⊗ Z.(22) Other Hamiltonians to be tested are represented in a similar fashion. The results for these three aspects are illustrated in Fig. 3. We find that all curves converge to the values close to 0 in less than ten iterations, which shows our method is effective. In particular, our method works for different β means that it is robust to temperature. And the results for the different number of µ and qubits reveals our method's scalability to a certain extent. 3 3]]", indicates there are three E l 's and each has three qubits with the corresponding Pauli tensor product. Here "0,1,2,3" represent "I, X, Y, Z" respectively. For example, for the first sample, the corresponding Hamiltonian is taken as H=0.3408 ·I ⊗ Y ⊗ X -0.6384 ·Y ⊗ X ⊗ Z -0.4988 ·I ⊗ Z ⊗ Z. B. Quantum many-body models Here, we demonstrate the performance of our algorithm for quantum many-body models. Specifically, we consider the onedimensional nearest-neighbor Ising model, XY model, and Heisenberg model. These many-body models are described by the Hamiltonians shown below: (Ising model) H 0 = J 0 n l=1 Z l Z l+1 + h 0 n l=1 X l ,(23)(XY model) H 1 = J 1 n l=1 (X l X l+1 + Y l Y l+1 ),(24)(Heisenberg model) H 2 = J 2 n l=1 (X l X l+1 + Y l Y l+1 + Z l Z l+1 ) + h 2 n l=1 Z l ,(25) Many-body # qubits # µ β LR µ models Ising model where periodic boundary conditions are assumed (i.e., X n+1 = X 1 , Y n+1 = Y 1 , and Z n+1 = Z 1 ). Coefficient J is the coupling constant for the nearest neighbor interaction, and h represents the external transverse magnetic field. The experimental parameters are concluded in Table II. We consider the models with a different number of qubits, varying from n = 3 to n = 5. The inverse temperature is set as β = 1. The coefficients J 0 , J 1 , J 2 and h 0 , h 2 are sampled uniformly from a uniform distribution on [-1,1]. We also employ the parameterized quantum circuit U (θ) in Fig. 2 for the SVQE. And the depth of U (θ) is also set as D = 10, 20, 40 for different n. Moreover, the numerical results are shown in Fig. 3, which imply our method is applicable to recover quantum many-body Hamiltonians. C. Numerical results using fewer eigenvalues of Ising Hamiltonians Notice that we use a PQC U (θ) with deep depths to learn the full spectrum of small-sized Hamiltonians in Secs. IV A-IV B, which may be beyond the capacity of NISQ devices. However, this section demonstrates the efficacy of HQHL in learning the Ising Hamiltonians using a circuit with reduced depth, where few eigenvalues (instead of the full spectrum) are learned. In particular, only halved circuit depths are needed for Hamiltonians with 3-5 qubits, given in Table II. Furthermore, the performance on n = 6 and n = 7-qubit Ising models, given below, is tested as well. The presented results imply the potential efficacy of our approach for larger Hamiltonians. To reduce the number of eigenvalues to be learned, we tune the weights q of the SVQE such that the U (θ) can output several smallest eigenvalues. For instance, five eigenvalues are learned for 4 & 5-qubit Ising Hamiltonians, and four eigenvalues are learned for 3-qubit Ising Hamiltonians. As a result, the circuit depth of the used U (θ) is significantly reduced. For example, we only use depth D = 20 to learn the coefficients with precision 0.05 for 5-qubit Ising models. While, in Sec. IV B, we use the depth D = 40. Moreover, we find out that using a circuit with 35 depths suffices to learn well the 6-qubit Ising model, where SVQE only learns six eigenvalues. Using the circuit with 40 depths could also reach a precision of 0.05 for the 7-qubit Ising Hamiltonian. The details of parameters setting (weights, depth, learning rate, etc.) are given in Table III. Besides, the experimental results are depicted in Figure 4. H = 0.1981 n l=1 Z l Z l+1 + 0.7544 n l=1 X l .(26) V. CONCLUSION We have proposed a hybrid quantum-classical Hamiltonian learning algorithm that employs a gradient-descent method to find the desired interaction coefficients. We achieve this purpose by unifying the variational quantum algorithms (VQAs) with the strategy proposed in [24]. To this end, we develop several subroutines: log-partition function estimation, stochastic variational quantum eigensolver (SVQE), and gradient estimation. In SVQE, we propose a method to learn the full/partial spectrum of the Hamiltonian and use the importance sampling to circumvent the resources in the loss evaluation. In the log-partition function, we propose a method that combines the parameterized quantum circuits and convex optimization to find the global minimum of the free energy as well as compute the log-partition function. In gradient estimation, we present a procedure to compute the gradient of the objective function costing polynomially many resources. Finally, we conduct numerical experiments to demonstrate the effectiveness of our approach with randomly generated Hamiltonians and selected many-body Hamiltonians. In consequence, we show that learning the full spectrum of Hamiltonians in the learning process could produce high-precision estimates of the desired interaction coefficients. Moreover, we also show that partially learning several smallest eigenvalues of Ising Hamiltonians could derive estimates up to a precision of 0.05. Overall, this paper develops a concrete near-term quantum algorithm for Hamiltonian learning and demonstrates the effectiveness as well, which has potential applications in quantum device certification, quantum simulation, and quantum machine learning. We believe our approach would shed lights on near-term quantum applications. For example, SVQE might enrich the VQE family in the fields of molecules and materials. Moreover, as many problems in computer science can be framed as partition function problems (e.g., counting coloring), our method may contribute to these fields as well. Furthermore, it is reasonable to explore our algorithm's applications in quantum machine learning [55], quantum error correction [6], and tomography [56]. Consider a Hamiltonian H ∈ C N ×N and a constant β > 0, then the system's free energy is given by F (ρ) = Tr(Hρ) − β −1 S(ρ). Recall the fact [57] that S(ρ) ≤ − N j=1 ρ jj log ρ jj ,(S1) where ρ jj are the diagonal elements of quantum state ρ. Using this fact, for any state ρ, we can find a lower bound on free energy in the sense that F (ρ) ≥ Tr(Hρ) + β −1 N j=1 ρ jj log ρ jj .(S2) On the other hand, let U be a unitary such that H = U ΛU † , where Λ = diag(λ 1 , ..., λ N ) is a diagonal matrix. Let ρ = diag(ρ 11 , ..., ρ N N ) be the diagonal matrix consisting of ρ's diagonal elements and let σ = U † ρU . It is easy to verify that Tr(Hρ) = Tr(Λσ). Furthermore, taking this relation into Eq. (S2)'s right hand side, we can find that F (ρ) ≥ Tr(Λσ) − β −1 S(σ).(S3) Notice that Eq. (S3)'s right-hand side is equal to F ( ρ), then we have F (ρ) ≥ F ( ρ).(S4) The inequality in Eq. (S4) shows that free energy's global optimum is commuting with the Hamiltonian H. According to the above discussion, we can rewrite the optimization program of finding free energy's minimal value as follows min ρ F (ρ) = min p N j=1 λ j p j + β −1 N j=1 p j log p j ,(S5) where p represents an arbitrary probability distribution. Eq. (S5)'s right-hand side can be solved using the Lagrange multiplier method, and the optimum is given below: p * := 1 Z (e −βλ1 , ..., e −βλN ),(S6) with Z := N j=1 e −βλj . Finally, the equalities in Eqs. (9)-(10) can be proved by taking p * into Eq. (S5)'s right-hand side and computing the minimal value. Appendix B: Proof for Proposition 1 Lemma 5 For any parameterized Hamiltonian H(ν) = m ℓ=1 ν ℓ E ℓ with E ℓ ∈ {X, Y, Z, I} ⊗n , we have H(ν) ≤ √ m· ν 2 . (S1) where · denotes the spectral norm and · 2 is the ℓ 2 -norm. Proof Let U be the unitary that diagonalizes the Hamiltonian H(ν), and then we can use the following form to represent H(ν). H(ν) = N j=1 λ j · U |ψ j ψ j | U † ,(S2) where |ψ 1 , ..., |ψ N are the computational basis. Typically, each eigenvalue is represented as follows: λ j = ψ j |U † H(ν)U |ψ j (S3) = m ℓ=1 ν ℓ ψ j |U † E ℓ U |ψ j (S4) Then, applying the Cauchy-Schwarz inequality leads to an upper bound on each eigenvalue: (λ j ) 2 ≤ m ℓ=1 (ν ℓ ) 2 · m ℓ=1 ( ψ j |U † E ℓ U |ψ j ) 2 . (S5) Meanwhile, recalling that all E ℓ are Pauli matrix tensor product, we can obtain an upper bound below: (λ j ) 2 ≤ m m ℓ=1 (ν ℓ ) 2 .(S6) Ranging j in {1, ..., N } in Eq. (S6), the maximal eigenvalue is upper bounded by √ m ν 2 , validating the claim. Proposition 1 For any parameterized Hamiltonian H(ν) = m ℓ=1 ν ℓ E ℓ with E ℓ ∈ {X, Y, Z, I} ⊗n and ν ∈ R m and constant β > 0, suppose we are given access to a parameterized quantum circuit U (θ) that can learn H(ν)'s eigenvectors, then the objective function C(p) can be computed up to precision ǫ with probability larger than 2/3 by taking T = O(m ν 2 2 /ǫ 2 ) samples. Furthermore, the probability can be improved to 1 − η costing an additional multiplicative factor of O(log(1/η)). Proof Since the expression N j=1 p j λ j is regarded as an expectation, then we can estimate it by the sample mean with high accuracy and probability. To be specific, let X denote a random variable that takes value λ j with probability p j . Then, this expression can be written as E[X] = N j=1 p j λ j .(S7) Furthermore, recall Chebyshev's inequality, then we have Pr |X − E[X]| ≤ ǫ ≥ 1 − Var[X] T ǫ 2 .(S8) whereX = 1 T (X 1 + X 2 + ...+ X T ) and Var[X] is the variance of X. Technically, we can set large T to increase the probability. Here, we only need to choose T such that Var[X] T ǫ 2 = 2 3 . (S9) Note that the second moment E[X 2 ] bounds the variance Var[X]. Meanwhile, the second moment of X is bounded by the squared spectral norm of H, shown below. E[X 2 ] = N j=1 p j (λ j ) 2 (S10) ≤ N j=1 p j H(ν) 2 (S11) = H(ν) 2 .(S12) The inequality is due to the fact that each eigenvalue is less than the spectral norm. Apply Lemma 5, then we will obtain an bound on T : T = 3Var[X] 2ǫ 2 ≤ 3E[X 2 ] 2ǫ 2 ≤ 3m ν 2 2 2ǫ 2 .(S13) Lastly, according to the Chernoff bound, we can boost the probability to 1 − η for any η > 0 by repeatedly computing the sample mean O(log(1/η)) times and taking the median of all sample means. (S1) Let p * be the global optimal point of G(p), that is, for any probability distribution p, we have G( p * ) ≤ G(p). Meanwhile, suppose p * is the global optimal point of C(p). Then, we have |G( p * ) − C(p * )| ≤ max E p * [| λ − λ|], E p * [| λ − λ|] ,(S2) where E p * [| λ − λ|] = N j=1 p * j | λ j − λ j |,(S3)E p * [| λ − λ|] = N j=1 p * j | λ j − λ j |. (S4) Proof Since functions C(p) and G(p) reach their global minimums at points p * and p * respectively, then we have C( p * ) ≥ C(p * ), (S5) G( p * ) ≤ G(p * ).(S6) Besides, we also have another relation: |C(p) − G(p)| = N j=1 p j |( λ j − λ j )|,(S7) where · ∞ denotes the maximum norm. Combining the above inequalities, we have the following result: C(p * ) ≤ C( p * ) ≤ G( p * ) + E p * [| λ − λ|] ≤ G(p * ) + E p * [| λ − λ|] ≤ C(p * ) + E p * [| λ − λ|] + E p * [| λ − λ|].(S8) Then the inequality in Eq. (S2) is proved. Proposition 3 (Correctness) Consider a parameterized Hamiltonian H(ν) and its Gibbs state ρ β (ν). Suppose the U (θ) from SVQE (cf. Algorithm 2) and p * from log-partition function estimation procedure (cf. Algorithm 1) are optimal. Define a density operator ρ * β as follows: ρ * β := N j=1 p * j · U (θ) |ψ j ψ j | U † (θ).(S9) where {|ψ j } denote the computational basis. Denote the estimated eigenvalues by λ, where λ j = ψ j |U † (θ)H(ν)U (θ)|ψ j . Then, ρ * β is an approximate of ρ β (ν) in the sense that D(ρ * β , ρ β (ν)) ≤ 2β max E p * [| λ − λ|], E p * [| λ − λ|] .(S10) where D(·, ·) denotes the trace distance, λ represent H(ν)'s true eigenvalues. Proof Recalling the expressions of C(p * ) and G( p * ) in Eqs. (12)-(S1), it is easy to verify the following inequalities: F (ρ β (ν)) = C(p * ), (S11) F (ρ * β ) = G( p * ). where F denotes the free energy, i.e., F (ρ) = Tr(Hρ) − β −1 S(ρ). Using the result in Lemma 2, we will obtain the following inequality. |F (ρ * β ) − F (ρ β (ν))| = |G( p * ) − C(p * )| ≤ max E p * [| λ − λ|], E p * [| λ − λ|] .(S13) In the meanwhile, a property of the free energy says that F (ρ * β ) = F (ρ β (ν)) + β −1 S(ρ * β ρ β (ν)). (S14) where S(ρ * β ρ β (ν)) is the relative entropy. Rewriting the above equation as follows: F (ρ * β ) − F (ρ β (ν)) = β −1 S(ρ * β ρ β (ν)). (S15) Combining the relations in Eqs. (S13) and (S15), we obtain the following inequality: S(ρ * β ρ β (ν)) ≤ β max E p * [| λ − λ|], E p * [| λ − λ|] .(S16) Lastly, according to Pinsker's inequality, the above inequality immediately leads to a bound on the trace distance between ρ β and ρ * β in the sense that D(ρ * β , ρ β (ν)) ≤ 2S(ρ * β ρ β (ν)) ≤ 2β max E p * [| λ − λ|], E p * [| λ − λ|] .(S17) The the claimed is proved. Require: Parameterized quantum circuit U (θ), Hamiltonian H(ν), constant β; Ensure: An estimate for log Z β (ν); 1: # Evaluation procedure construction 2: Take probability distribution p as input; 3: Set integer T and D; 4: Sample T D integers t 1 1 , ..., t 1 T , ..., t D 1 , ..., t D T according to p; 5: Prepare computational states |ψ t 1 1 , ..., |ψ t 1 T , ..., |ψ t D 1 , . . ., |ψ t D T ; 6: Compute approximate eigenvalues: λ t s j = ψ t s j |U † (θ)H(ν)U (θ)|ψ t s j for all j = 1, . . . , T and s = 1, . . . , D; 7: Compute averages: aves = 1 T T j=1 λ t s j for all s = 1, ..., D; 8: Take the median value C(p) ← median(λave 1 , ..., λave D ) + β −1 N j=1 p j log p j ; 9: # Membership procedure construction 10: Construct a membership procedure; 11: # Convex optimization solution 12: Compute the function's global minimum value C(p * ) and the optimal point p * via the cutting plane method. 13: return value −βC(p * ) and the final point p * . s j for all s = 1, ..., D; 10: Let M (θ) ← median(ave 1 , ..., ave D ); 11: Use M (θ) to compute the gradient ∇ by parameter shift rules [39]; 12: Update parameters θ ← θ − r θ ∇; 13: Set l ← l + 1; 14: end while 15: return the final U (θ). Require: Post-training circuit U (θ), Pauli operators {E ℓ } m ℓ=1 , optimal p * , and constants β and {e ℓ } m ℓ=1 ; Ensure: Gradient estimate ∇L(ν); 1: Set ℓ = 1; 2: Set integer K and D; 3: Sample K integers l 1 1 , ..., l 1 K , ..., l D 1 , ..., l D K , according to p * ; 4: Prepare computational states |ψ l 1 1 ,. . .,|ψ l 1 K ,. . .,|ψ l D 1 ,. . ., |ψ l D K ; 5: while ℓ ≤ m do 6: Compute value ψ l s j |U † (θ)E ℓ U (θ)|ψ l s j for j = 1, .., K and s = 1, ..., D; 7: Calculate averages: aves = 1 K K j=1 ψ l s j |U † (θ)E ℓ U (θ)|ψ l s j for all s = 1, ..., D; 8: Take the median value: s ℓ = −β · median(ave 1 , . . . , ave D ) + βe ℓ ; 9: Set ℓ ← ℓ + 1; 10: end while 11: return vector (s 1 , ..., sm). FIG. 2 : 2The selected quantum circuit U (θ) for stochastic variational quantum eigensolver (SVQE). Here, D represents circuit depth. Parameters θ are randomly initialized from a uniform distribution in [0, 2π] and updated via gradient descent method. FIG. 3 : 3The curves in (a), (b), (c) represent the infinity norm of the error of µ with different β, different number of µ, and different number of qubits, respectively. In (d), (e), (f), the curves represent the infinity norm of the error of µ for different many-body Hamiltonians with the number of qubits varies from 3 to 5. The numbers on the line represent the values of the last iteration. These numbers close to 0 indicate that our algorithm is effective. h2 = −0.2385] TABLE II: Hyper-parameters setting for many-body models. For each Hamiltonian model, the number of qubits varies from 3 to 5, and the number of µ is determined by the number of Pauli operators. "LR" denotes learning rate. The values of µ are sampled uniformly in the range of [−1, 1]. : Parameters setting for HQHL. The script index means the length of the tuple, e.g., ()8 indicates the tuple consists of 8 entries. The notation 0, . . . means the entries following 0 are all zeros as well. Notation #λ means the number of eigenvalues we learned. Please note that we omit the β = 1 in the table. Let λ = ( λ 1 , ..., λ N ) denote the estimated eigenvalues from SVQE and define a function G(p) as follows: log p j . , Hamioltonian is represented by a tuple. Each number 0, 1, 2, 3 corresponds to matrices I, X, Y, Z, respectively. µ denotes the interaction coefficients to be learned. For instance, [[0 2 1] [2 1 3] [03 3]] means that the Hamiltonian consists of three Pauli operators, where each term represents a Pauli operator, e.g., [0 2 1] means I ⊗ Y ⊗ X. Then, the parameters in the top second row represents the following Hamiltonian. TABLE I : IHyper-parameters setting. The number of qubits (# qubits) varies from 3 to 5, and the number of µ (# µ) from 3 to 6. β is chosen as 0.3, 1, 3. "LR" denotes learning rate. The values of µ are sampled uniformly in the range of [-1, 1]. The term, likes "[[0 2 1] [2 1 3] [0 Supplementary MaterialAppendix A: Proofs for Eqs. (S1)Proof First, we rewrite the value ψ|U † H(ν)U |ψ as follows:Second, we count the required number of measurements to estimate the value ψ|U † E ℓ U |ψ up to precision ǫ/ ν 1 with probability at least 1 − η/m, where · 1 denotes the ℓ 1 -norm. Since the Pauli operator, E ℓ , has eigenvalues ±1, we can partition E ℓ 's eigenvectors into two sets, corresponding to positive and negative eigenvalues, respectively. For convenience, we call the measurement outcome corresponding to eigenvalue 1 as the positive measurement outcome and the rest as the negative measurement outcome. We define a random variable X in the sense that log(m/η)/ǫ 2 ). Lastly, for ψ|U † H(ν)U |ψ , the estimate's maximal error is ν 1 ·ǫ/ ν 1 = ǫ. By union bound, the overall failure probability is less than m · η/m = η. Thus, the claim is proved.Proposition 2 Consider a parameterized HamiltonianBesides, the total number of measurements is given below:Proof Let Y denote a random variable that takes value ψ j |U † (θ)H(ν)U (θ)|ψ j with probability q j , then the objective function M (θ) can be rewritten asBy Chebyshev's inequality, the expectation can be computed by taking enough samples of Y and averaging them. Note that the variance of Y determines the number of samples, and the absolute value Y is less than the spectral norm H(ν) , i.e., |Y | ≤ H(ν) . Along with Lemma 5, it is easy to see that the required number of Y 's samples for obtaining an estimate with error ǫ/2 and probability larger than 2/3 is T = O(m ν 2 2 /ǫ 2 ). Furthermore, by Chernoff bounds, the probability can be improved to 1 − η/2 at an additional cost of multiplicative factor of D = O(log(1/η)).On the other hand, each sample Y 's value has to be determined by performing the measurement. Since |ψ j is a computational basis, hence Y can take at most 2 n different values. To ensure the probability for estimating E[Y ] larger than 1−η, the probability of each estimate ψ j |U † (θ)H(ν)U (θ)|ψ j only needs to be at least 1 − η/2 n+1 . By union bound, the overall failure probability is at most η/2 + η · T D 2 n+1 < η (For large Hamiltonians, the number of samples T D can be significantly less than dimension 2 n ). Besides, according to Lemma 1, ψ j |U † (θ)H(ν)U (θ)|ψ j 's estimate within accuracy ǫ/2 and probability 1−η/2 n+1 requires a sample complexity of O(m ν 2 1 (n + log(m/η))/ǫ 2 ). Thus, the overall number of measurements is the product of the number of samples Proof Let Z ℓ denote the random variable that takes value ψ j |U † (θ)E ℓ U (θ)|ψ j with probability p * j , for all ℓ = 1, ..., m. Then we haveThus partial derivative can be computed in the following wayIt implies that the estimate's error can be set as ǫ/β to ensure the gradient's maximal error less than ǫ. Next, we determine the number of samples such that the overall failure probability for estimating the gradient is less than δ. Since the gradient has m partial derivatives, corresping to E[Z ℓ ], thus it suffices to estimate each with probability larger than 1 − δ/m. Meanwhile, each mean E[Z ℓ ] can be computed by sampling. Notice that all |Z ℓ | ≤ 1, by Chebyshev's inequality, then it suffices to take K = O(β 2 /ǫ 2 ) samples to compute an estimate for each E[Z ℓ ] with precision ǫ/2β and probability larger than 2/3. Furthermore, by Chernoff bounds, the probability can be improved to 1 − η/2m at an additional cost of multiplicative factor of D = O(log(2m/η)). It is worth pointing out that, for each variable Z ℓ , the samples are taken according to the same probability distribution p * , thus it is natural to use the sampled states |ψ t s j (cf. Algorithm 3) to compute all means E[Z ℓ ]. Then the total number of samples is KD = O(β 2 log(m/η)/ǫ 2 ).On the other hand, each value ψ j |U † (θ)E ℓ U (θ)|ψ j in Eq. (S1) has to be computed by performing the measurement. Note that there are 2 n values ψ j |U † (θ)E ℓ U (θ)|ψ j in all. To ensure the mean estimate's failure probability less than η/2m, it suffices to suppress each value's failure probability to η/2 n+1 m. Following the same discussion in Lemma 1, the estimate for value ψ j |U † (θ)E ℓ U (θ)|ψ j can be computed up to precision ǫ/2β using O(β 2 log(2 n+1 m/η)/ǫ 2 ) measurements.Regarding the failure probability, by union bound, the overall failure probability is at most m·(η/2m+KD·η/2 n+1 m), where KD is the number of samples KD = O(β 2 log(m/η)/ǫ 2 ). Especially, for larger Hamiltonians, the number of measurements is usually less than the dimension 2 n . 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{'abstract': "Hamiltonian learning is central to studying complex many-body physics and the certification of quantum devices and simulators. How to learn the Hamiltonian in general with near-term quantum devices is a challenging problem. In this paper, we develop a hybrid quantum-classical Hamiltonian learning algorithm to tackle this problem. By transforming the Hamiltonian learning problem to an optimization problem using the Jaynes' principle, we employ a gradient-descent method to give the solution and could reveal the interaction coefficients from the system's Gibbs state measurement results. In particular, the computation of the gradients relies on the Hamiltonian spectrum and the log-partition function. Hence, as the main subroutine, we develop a variational quantum algorithm to extract the Hamiltonian spectrum and utilize convex optimization to output the log-partition function. We also apply the importance sampling technique to circumvent the resource requirements for dealing with large-scale Hamiltonians. As a proof of principle, we demonstrate the effectiveness of our algorithm by conducting numerical experiments for randomly generated Hamiltonians and many-body Hamiltonians of theoretical and practical interest.To establish the results, our method for minimizing the free energy depends on two critical steps. First, we choose a suitable PQC with enough expressiveness and train it to learn the eigenvectors of the Hamiltonian and output the corresponding eigenvalues. Second, we combine the post-training PQC with the classical methods for convex optimization to find the global minimum of the free energy. Next, we utilize the post-training PQC and the optimizer of the convex optimization to compute the gradients. Furthermore, we theoretically analyze the estimation precision of the gradients. We also show the efficiency of loss evaluation and gradients estimation by the importance sampling technique when the underlying Hamiltonian is large.As the proof of principle, we study the effectiveness of our algorithm for Hamiltonian learning by conducting numerical experiments for randomly generated Hamiltonians and several many-body Hamiltonians. To generate random Hamiltonians, we choose Pauli tensor products E ℓ from the set {X, Y, Z, I} ⊗n at random, with n ranging from 3 to 5. The target interaction coefficients are chosen via a uniform distribution over [−1, 1]. The tested many-body Hamiltonians consist of Ising, XY -spin, and Heisenberg models, where size also varies from 3 to 5 qubits. For these Hamiltonians, we test our algorithm for different parameters β and µ with different lengths. As a result, the numerical results show that the target interaction coefficients can be estimated with high precision. In these experiments, our algorithm learns all eigenvalues of Hamiltonians. Moreover, we show the effectiveness by partially learning few smallest eigenvalues of Ising Hamiltonians. In particular, the circuit depth of used PQC could be significantly reduced. Finally, we also generalize the experiments to larger Ising Hamiltonians with 6/7 qubits.Next, we summarize the contribution of this paper and all mentioned results above. 1. We propose a hybrid quantum-classical Hamiltonian learning framework based on the fundamental properties of free energy, which mainly consists of the following two subroutines: log-partition function estimation and stochastic variational quantum eigensolver (SVQE). 2. The main subroutine is the log-partition function estimation algorithm, which combines the SVQE with the classical convex optimization to minimize the free energy. 3. We also propose a feasible scheme for learning the spectrum of the many-body Hamiltonian by integrating variational quantum algorithms with the importance sampling technique. 4. We demonstrate our algorithm's validity by numerical simulations on several random Hamiltonians and many-body Hamiltonians (e.g., Ising model, XY model, and Heisenberg model). Organization. The remaining paper proceeds as follows. In Sec. II, we formally define the problems we studied in this work; In Sec. III, we present the main results, including the Hamiltonian learning algorithm, and its main subroutines logpartition function estimation, stochastic variational quantum eigensolver, and gradient estimation; In Sec. IV, we describe the experimental settings and provide numerical results to demonstrate the efficacy of our algorithm; Lastly, we conclude the paper in Sec. V. Proofs and more discussions are presented in the Supplementary Material.", 'arxivid': '2103.01061', 'author': ['Youle Wang \nCenter for Quantum Software and Information\nUniversity of Technology Sydney\n2007NSWAustralia\n\nInstitute for Quantum Computing\nBaidu Research\n100193BeijingChina\n', 'Guangxi Li \nCenter for Quantum Software and Information\nUniversity of Technology Sydney\n2007NSWAustralia\n\nInstitute for Quantum Computing\nBaidu Research\n100193BeijingChina\n', 'Xin Wang \nInstitute for Quantum Computing\nBaidu Research\n100193BeijingChina\n'], 'authoraffiliation': ['Center for Quantum Software and Information\nUniversity of Technology Sydney\n2007NSWAustralia', 'Institute for Quantum Computing\nBaidu Research\n100193BeijingChina', 'Center for Quantum Software and Information\nUniversity of Technology Sydney\n2007NSWAustralia', 'Institute for Quantum Computing\nBaidu Research\n100193BeijingChina', 'Institute for Quantum Computing\nBaidu Research\n100193BeijingChina'], 'corpusid': 232092492, 'doi': '10.1007/s11432-021-3382-2', 'github_urls': [], 'n_tokens_mistral': 23446, 'n_tokens_neox': 20345, 'n_words': 11396, 'pdfsha': '66a3f38f5b3d034f4fac61d095f836a90f2ebb36', 'pdfurls': ['https://arxiv.org/pdf/2103.01061v2.pdf'], 'title': ['A Hybrid Quantum-Classical Hamiltonian Learning Algorithm', 'A Hybrid Quantum-Classical Hamiltonian Learning Algorithm'], 'venue': []}

arxiv

DECOMPOSABLE (5, 6)-SOLUTIONS IN ELEVEN-DIMENSIONAL SUPERGRAVITY 19 Oct 2021 Hanci Chi Ioannis Chrysikos Eivind Schneider DECOMPOSABLE (5, 6)-SOLUTIONS IN ELEVEN-DIMENSIONAL SUPERGRAVITY 19 Oct 2021 We present decomposable (5,6)-solutions M 1,4 ×M 6 in eleven-dimensional supergravity by solving the bosonic supergravity equations for a variety of non-trivial flux forms. Many of the bosonic backgrounds presented here are induced by various types of null flux forms on products of certain totally Ricci-isotropic Lorentzian Walker manifolds and Ricci-flat Riemannian manifolds. These constructions provide an analogue of the work in [CG20], where similar computations were made for decomposable (6,5)-solutions. We also present bosonic backgrounds that are products of Lorentzian Einstein manifolds with negative Einstein constant (in the "mostly plus" convention) and Riemannian Kähler-Einstein manifolds with positive Einstein constant. This conclusion generalizes a result of C. N. Pope et al.[PN89]concerning the appearance of six-dimensional Kähler-Einstein manifolds in eleven-dimensional supergravity. In this setting we construct infinitely many nonsymmetric decomposable (5, 6)-supergravity backgrounds, by using the infinitely many Lorentzian Einstein-Sasakian structures with negative Einstein constant on the 5-sphere, known from the work of C. P. Boyer et al.[BGM06].ContentsLemma 2.2. Let X = M p × M q be a product of two pseudo-Riemannian manifolds (M ,g) and (M, g) of dimensions p, q, and lets, s be the number of negative eigenvalues ofg, g, respectively. Let us denote by ⋆, ⋆ p , ⋆ q the Hodge operator on (X, h =g + g), ( M ,g), and (M, g), respectively. Then, for anyα ∈ Ωk( M ), and β ∈ Ω k (M ) the following hold:With the help of Lemma 2.2 we compute ⋆ F and d ⋆ F for F being of the form (1.2). We obtain Motivation. The five established ten-dimensional superstring theories (Type I, Type IIA, Type IIB, heterotic SO(32) and heterotic E 8 × E 8 ) provide frameworks for uniting quantum theory and general relativity. By string dualities such as T-duality, a unique eleven-dimensional superstring theory, called M-theory, unites these five theories. As a result, eleven-dimensional supergravity theory, viewed as a low-energy limit of M-theory, has attracted much attention during the last half century (see for example [DNP86,Wi95,D99,GSW12,T14]). The Lagrangian of eleven-dimensional supergravity was proposed in [CJS78]. The fields in the theory is a Lorentzian metric h, a closed 4-form F (called the flux form), and a Majorana spinor Ψ. They are defined on an eleven-dimensional manifold X and are subject to the equations of motion determined by the Lagrangian. A special class of supergravity solutions are those with vanishing fermionic part, Ψ = 0. In this case the equations of motion reduce to a simpler set of equations, involving only h and F, which we call the bosonic supergravity equations. This set of equations closely resembles the Einstein-Maxwell equations in four dimensions. Solutions to the bosonic supergravity equations are called bosonic supergravity backgrounds. The bosonic backgrounds include the special class of eleven-dimensional Ricci-flat Lorentzian manifolds for which F = 0. Finding bosonic supergravity backgrounds is an important task, and the literature on bosonic supergravity backgrounds is vast. Several geometrical tools and constructions have been used for finding them, including manifolds with special holonomy or special G-structures, irreducible symmetric spaces, compactifications, Killing superalgebras, certain ansatzes on h and F, and other. We refer to some representative works [PT95, BB96, F00, F01, BDS02, GP03, MS03, HM05, Ts06, F07, F13, GP15, MFS18, FeGP19,BF21], and the reader can find more references therein. Supersymmetries also play an important role in these investigations. The maximally supersymmetric bosonic backgrounds are, in addition to flat Minkowski space, the Freund-Rubin backgrounds (AdS) 7 × S 4 and (AdS) 4 × S 7 ( [FR80]) and a particular pp-wave (see for example [FP03]). These are locally homogeneous, something which is true for all backgrounds admitting more than half of the maximal amount of supersymmetries ( [FH12]). Other well-known examples are the M2-brane and the M5-brane ( [DS91,Gu92]) whose near-horizon geometries are the Freund-Rubin backgrounds (see also [F98, S09]). These have exactly half of the maximal amount of supersymmetries. On the other side of the spectrum, with respect to the number of supersymmetries admitted, we have bosonic backgrounds such as (AdS) 5 × CP 3 which admit no supersymmetry (see [PN89]). Outline. In this article we search for bosonic supergravity backgrounds that are products of an oriented Lorentzian manifold ( M 1,4 ,g) and an oriented Riemannian manifold (M 6 , g), with flux form F ∈ Ω 4 (X) of the type F = ϕα +β ∧ ν +γ ∧ δ +̟ ∧ ǫ +ψθ , where the ith term is the product of an (5 − i)-form on M 1,4 and an (i − 1)-form on M 6 , for i = 1, . . . , 5. For various flux forms of the above type, we write down the corresponding simplified form of the bosonic supergravity equations and find particular solutions to these equations. Our work can be considered as a natural continuation of [CG20], where products of six-dimensional Lorentzian manifolds and five-dimensional Riemannian manifolds are treated in a similar way. We begin by describing the general constraints that appear due to the bosonic supergravity equations (which we split up into the closedness condition, the Maxwell equation and the supergravity Einstein equation). For a general 4-form F of the above form, the resulting system is still quite complicated. See for example Proposition 2.3 for the Maxwell equation and the equations (3.3), (3.4), (3.5) for the supergravity Einstein equation. In order to obtain a more tractable system of equations, we specify F even further by letting three or four of its terms vanish. Then, as in [CG20], the constraints which occur due to the Maxwell equation in combination with the closedness condition are simplified (see Proposition 2.4), and the same applies for the supergravity Einstein equation (Proposition 3.3). It is worth mentioning that the form of F can impose non-trivial restrictions on the geometry of M 1,4 or M 6 (see Corollary 3.5). For example, for F =α the supergravity Einstein equation implies that (M 6 , g) is an Einstein manifold, while for F = θ the Einstein equation implies that ( M 1,4 ,g) is an Einstein manifold. In both cases, the scalar curvature of (X, h) is constant (Corollary 3.8). In order to find explicit solutions of eleven-dimensional bosonic supergravity, we follow two approaches. First, we examine the case when F is composed of null forms. In this case, the bosonic supergravity equations simplify significantly, as shown in Proposition 4.1 and Theorems 4.2, 4.3, 4.4, 4.5, 4.8 and 4.9. Moreover, the supergravity Einstein equation requires (M 6 , g) to be Ricci-flat (Proposition 4.1). In addition, for a bosonic supergravity background (X 1,10 = M 1,4 × M 6 , h = g + g, F =̟ ∧ ǫ), where̟ ∈ Ω 1 ( M 1,4 ) is null, we see in Corollary 4.7 that (X, h) is totally Ricciisotropic, as an analog of [CG20,Cor. 4.10]. To find concrete solutions to the bosonic supergravity equations, we follow [CG20] and assume that the Lorentzian part ( M 1,4 ,g) is a (special type of) Walker metrics, since these come equipped with a distribution of null lines, from which non-trivial null flux forms can be built. Propositions 5.4, 5.6, 5.8, 5.9, 5.11, 5.13 and 5.16 concern nonsymmetric bosonic backgrounds that are direct products of a Ricci-isotropic Lorentzian Walker manifold and a Ricci-flat Riemannian manifold. These products have special holonomy properties and can potentially support supersymmetries (see also [F00]). We explicitly illustrate these results by examples involving five-dimensional pp-waves, while an investigation of supersymmetries will be left for a forthcoming work. In the second approach, we study some cases where the Riemannian part (M 6 , g, ω) is a Kähler manifold. In this case, we do not assume that F is null, but rather that it is related to the Kähler form ω. In particular, we consider the cases F =γ ∧ δ with δ = ω and F = θ = c ⋆ 6 ω, where c is a constant. We see that if the flux form is given by F =γ ∧ δ and γ 2 g is not constant, then the supergravity Einstein equation forces M 6 to be a Ricci-flat almost Hermitian manifold (Corollary 3.5), so in Proposition 6.2 we write down the bosonic supergravity equations for the case when M 6 is a Kähler manifold. For the case where the flux form is given by F = c ⋆ 6 ω, Proposition 6.4 says that the bosonic supergravity equations are satisfied, if and only if both ( M 1,4 ,g) and (M 6 , g) are Einstein with Einstein constants 1 6 c 2 and − 1 6 c 2 , respectively. Thus ( M 1,4 ,g) has positive scalar curvature, while (M 6 , g) has negative scalar curvature. 1 For instance, the symmetric spaces CP 3 and Gr + (2, 5) endowed with their respective (unique) homogeneous Kähler-Einstein metrics can be used to obtain some of the decomposable symmetric supergravity backgrounds presented in [F13]. Proposition 6.4 generalizes a result presented in [PN89] which involved bosonic backgrounds of the form (AdS) 5 × M 6 , where M 6 is a compact Kähler manifold. We discuss some possible candidates for the Einstein manifold ( M 1,4 ,g), other than (AdS) 5 . In particular, we are based on negative Sasakian geometries and use for instance the infinitely many different Lorentzian Einstein-Sasakian structures with negative Einstein constant (in the "mostly plus" convention) described on the 5-sphere S 5 by Boyer et al. [BGM06]. In this way we result with infinitely many new bosonic non-symmetric decomposable (5, 6)-solutions in eleven-dimensional supergravity given by S 5 ×M 6 , where M 6 is any six-dimensional (de Rham irreducible) Kähler-Einstein manifold with positive scalar curvature (also in the "mostly plus" convention). Note that all such solutions that are based on the same Kähler-Einstein manifold M 6 have equal flux forms. Other such examples can be obtained by using the connected sum ♯k(S 2 × S 3 ), since this manifold also admits Lorentzian Einstein-Sasakian metrics for any integer k ≥ 1. We should finally mention that the above conclusion fails if M 6 is a six-dimensional (strictly) nearly Kähler manifold, since in this case the Kähler form ω is not closed, so the 4-form F indicated above cannot serve as a flux form (see the final section). As a consequence, and in a line with the conclusion pointed out in [PN89] for (AdS) 5 × M 6 , bosonic solutions of the form M 1,4 × M 6 , where M 1,4 is a Lorentzian Einstein manifold and M 6 is a compact Kähler-Einstein manifold, are not expected to admit supersymmetries. Essentially, this is because in dimension 6 smooth spin manifolds admitting real Killing spinors are exhausted by nearly Kähler manifolds, see for example [BFGK91], and see also [GMSW04] for the classification of eleven-dimensional superymmetric supergravity solutions containing (AdS) 5 . The paper is structured as follows. In Section 1 we lay out the framework that will be used throughout the paper and establish some notation. We introduce the eleven-dimensional bosonic supergravity equations, and write down the ansatz of the general flux form, which we use in this paper. The bosonic supergravity equations corresponding to this ansatz are computed in Sections 2 and 3, respectively. There, we also investigate the form of the equations after further simplification of the flux form, and state some general consequences of the equations. As mentioned above, the bosonic supergravity equations are simpler when the flux form is composed of null forms, and in Section 4 we present some general results for such flux forms. Next, in Section 5 we apply these results to Ricci-isotropic Lorentzian Walker manifolds and produce several explicit examples of decomposable (5, 6)-supergravity backgrounds. In Section 6, we drop the requirement that F is null and analyze the appearance of Kähler-Einstein manifolds and of negative Einstein-Sasakian geometries in our decomposable (5, 6)-solutions. Preliminaries In this work we study connected eleven-dimensional Lorentzian manifolds of the form X 1,10 = M 1,4 × M 6 , where ( M 1,4 ,g) is a five-dimensional connected oriented Lorentzian manifold and (M 6 , g) is a sixdimensional connected oriented Riemannian manifold. Our aim is to present on such products a systematic examination of the bosonic supergravity equations, i.e. of the following system of field equations (see for example [F01,AℓCT19] ):    d F = 0 , d ⋆ F = 1 2 F ∧ F , Ric h (X, Y ) = − 1 2 X F, Y F h + 1 6 h(X, Y ) F 2 h . (1.1) Here, the Lorentzian metric on X 1,10 is the product metric h =g+g and ⋆ : Ω k (X 1,10 ) → Ω 11−k (X 1,10 ) is the Hodge-star operator on (X 1,10 , h), defined by α ∧ ⋆β = α, β h vol X , where vol X = vol M ∧ vol M denotes the volume form on (X 1,10 , h). We also have F 2 h = F, F h . The bosonic field F is a global 4-form on X 1,10 , called the flux form which, together with the Lorentzian metric h, form the bosonic sector of eleven-dimensional supergravity. We will refer to the three conditions appearing in (1.1) as the closedness condition, the Maxwell equation, and the supergravity Einstein equation, respectively. Triples (X 1,10 , h, F) solving this system of equations are called bosonic supergravity backgrounds. Remark 1.1. In this paper we apply the "mostly minus" convention. That is, the signature for h is (+, −, . . . , −) and hence g is a negative definite Riemannian metric on M . Recall that for any two k-forms ω and φ, we have ω, φ h = 1 k! 1≤iα,j β ≤11 ω i 1 ...i k φ j 1 ...j k h i 1 j 1 . . . h i k j k Then, for a k-form ω ∈ Ω k (M 6 ) the sign of ω 2 h = ω 2 g is equal to that of (−1) k . Note that this choice of convention makes the right-hand-side of the last equation of (1.1) different from how it usually appears in the literature, by a minus sign. Before we discuss closed 4-forms F ∈ Ω 4 cl (X 1,10 ) on X 1,10 , let us decompose the tangent space V := T x X ≃ R 1,10 of X 1,10 at a point x ∈ X as V = L 1,4 ⊕ E 6 , where we identify L with the five-dimensional Minkowski tangent space of M 1,4 and E with the six-dimensional Euclidean tangent space of M 6 . Then, one has an orthogonal decomposition Λ 4 V = Λ 4 R 1,10 = Λ 4 L (Λ 3 L ∧ Λ 1 E) (Λ 2 L ∧ Λ 2 E) (Λ 1 L ∧ Λ 3 E) Λ 4 E. In this paper, we consider global differential 4-forms F ∈ Ω 4 (X), given by F = ϕα +β ∧ ν +γ ∧ δ +̟ ∧ ǫ +ψθ , (1.2) for someα ∈ Ω 4 ( M ) ,β ∈ Ω 3 ( M ) ,γ ∈ Ω 2 ( M ) ,̟ ∈ Ω 1 ( M ) ,ψ ∈ C ∞ ( M ) , ϕ ∈ C ∞ (M ) , ν ∈ Ω 1 (M ) , δ ∈ Ω 2 (M ) , ǫ ∈ Ω 3 (M ) , θ ∈ Ω 4 (M ) . We will show that this class of 4-forms is large enough to allow for a variety of non-trivial bosonic supergravity backgrounds. Note that the difference between (1.2) and a general 4-form on X 1,10 is, firstly, that a general 4-form may have more terms taking values in each of the subspaces Λ i L∧Λ 4−i E and, secondly, that each term can be multiplied by a function on X 1,10 . The closedness condition and the Maxwell equation We begin by writing down the closedness condition and the Maxwell equation on (X 1,10 = M 1,4 × M 6 , h =g + g) for the 4-form F given by (1.2). The closedness condition can be found by computing d F and comparing terms of similar type, a procedure which gives the following. Lemma 2.1. The 4-form F defined by (1.2) is closed if and only if the following system is satisfied:    ϕdα = 0 ,γ ∧ dδ + d̟ ∧ ǫ = 0 , α ∧ dϕ + dβ ∧ ν = 0 ,̟ ∧ dǫ − dψ ∧ θ = 0 , dγ ∧ δ −β ∧ dν = 0 ,ψdθ = 0 . In particular, we notice that F given by (1.2) is closed in the case thatα,β,γ,̟,ψ, and ϕ, ν, δ, ǫ, θ are closed on their respective manifolds. Before studying the Maxwell equation, we recall some basic useful formulas (see also [AℓCT19,CG20]). Notice that d ⋆ 6 ϕ = 0 since ⋆ 6 ϕ is a 6-form on a six-dimensional manifold. Similarly, we have d ⋆ 5ψ = 0. We also compute 1 2 F ∧ F = ϕα ∧̟ ∧ ǫ + ϕψα ∧ θ +β ∧γ ∧ δ ∧ ν +β ∧̟ ∧ ǫ ∧ ν +ψβ ∧ θ ∧ ν +ψγ ∧ θ ∧ δ +γ ∧̟ ∧ ǫ ∧ δ + 1 2γ ∧γ ∧ δ ∧ δ . After collecting the terms according to the subspace Λ i L ∧ Λ 4−i E in which they take values, we get the following proposition. Proposition 2.3. The Maxwell equation on the Lorentzian manifold (X 1,10 = M 1,4 ×M 6 , h =g+g) with 4-form F given by (1.2) is equivalent to the following system of equations: Type (2, 6) d ⋆ 5α ∧ ⋆ 6 ϕ + ⋆ 5β ∧ d ⋆ 6 ν =ψγ ∧ θ ∧ δ , Type (3, 5) d ⋆ 5β ∧ ⋆ 6 ν − ⋆ 5γ ∧ d ⋆ 6 δ =ψβ ∧ θ ∧ ν +γ ∧̟ ∧ ǫ ∧ δ , Type (4, 4) d ⋆ 5γ ∧ ⋆ 6 δ + ⋆ 5̟ ∧ d ⋆ 6 ǫ =ψ ϕα ∧ θ +β ∧̟ ∧ ǫ ∧ ν + 1 2γ ∧γ ∧ δ ∧ δ , Type (5, 3) d ⋆ 5̟ ∧ ⋆ 6 ǫ − ⋆ 5ψ ∧ d ⋆ 6 θ = ϕα ∧̟ ∧ ǫ +β ∧γ ∧ δ ∧ ν . Of course, the system of equations given in Lemma 2.1 and Proposition 2.3 is significantly simplified when some terms of F vanish. Let us list some of the important cases and write down the corresponding equations. Proposition 2.4. Consider the Lorentzian manifold (X 1,10 = M 1,4 × M 6 , h =g + g) as above. Then the following hold: (1) The 4-form F ∈ Ω 4 (X) defined by F = ϕα (2.1) satisfies the Maxwell equation and the closedness condition if and only if ϕ is constant andα is closed and coclosed: dα = d ⋆ 5α = 0 . (2) The 4-form F ∈ Ω 4 (X) defined by F =β ∧ ν (2.2) satisfies the Maxwell equation and the closedness condition if and only ifβ and ν are closed and coclosed: dβ = d ⋆ 5β = 0 , d ν = d ⋆ 6 ν = 0 . (3) The 4-form F ∈ Ω 4 (X) defined by F =γ ∧ δ (2.3) satisfies the Maxwell equation and the closedness condition if and only if dγ = d δ = d ⋆ 6 δ = 0 , d ⋆ 5γ ∧ ⋆ 6 δ =γ ∧γ ∧ δ ∧ δ 2 . Ifγ ∧γ = 0, then the last equation implies d ⋆ 5γ = 0, and it puts no additional constraints on δ. If γ ∧γ is nonzero, the last condition is equivalent to d ⋆ 5γ = κγ ∧γ , κ ⋆ 6 δ = δ ∧ δ 2 , for some constant κ ∈ R. (4) The 4-form F ∈ Ω 4 (X) defined by F =̟ ∧ ǫ (2.4) satisfies the Maxwell equation and the closedness condition if and only if̟ and ǫ are closed and coclosed: d̟ = d ⋆ 5̟ = 0 , d ǫ = d ⋆ 6 ǫ = 0 . (5) The 4-form F ∈ Ω 4 (X) defined by F =ψθ (2.5) satisfies the Maxwell equation and the closedness condition if and only ifψ is constant and θ is closed and coclosed: d θ = d ⋆ 6 θ = 0 . (6) The 4-form F ∈ Ω 4 (X) defined by F = ϕα +β ∧ ν (2.6) satisfies the Maxwell equation and the closedness condition if and only if dα = d ⋆ 5β = d ν = 0 , d ϕ = κν , dβ = −κα , d ⋆ 5α = −λ ⋆ 5β , d ⋆ 6 ν = λ ⋆ 6 ϕ (2.7) for some constants κ, λ ∈ R. The last four conditions imply ⋆ 6 d ⋆ 6 d ϕ = κλϕ , ⋆ 5 d ⋆ 5 dβ = κλβ . (7) The 4-form F ∈ Ω 4 (X) defined by F =̟ ∧ ǫ +ψθ (2.8) satisfies the Maxwell equation and the closedness condition if and only if d θ = d̟ = d ⋆ 6 ǫ = 0 , dψ = κ̟ , d ǫ = κθ , d ⋆ 5̟ = λ ⋆ 5ψ , d ⋆ 6 θ = λ ⋆ 6 ǫ (2.9) for some constants κ, λ ∈ R. The last four conditions imply ⋆ 5 d ⋆ 5 dψ = κλψ , ⋆ 6 d ⋆ 6 d ǫ = −κλǫ . (8) The 4-form F ∈ Ω 4 (X) defined by F = ϕα +ψθ (2.10) satisfies the Maxwell equation and the closedness condition if and only if either ϕα = 0 orψθ = 0. Thus, this case reduces to case (5) or case (1), respectively. (9) The 4-form F ∈ Ω 4 (X) defined by F =β ∧ ν +̟ ∧ ǫ (2.11) satisfies the Maxwell equation and the closedness condition if and only if dβ = d ν = d̟ = d ǫ = 0 , d ⋆ 6 ν = d ⋆ 5β = d ⋆ 5̟ = 0 , ⋆ 5̟ ∧ d ⋆ 6 ǫ =β ∧̟ ∧ ǫ ∧ ν . If ǫ ∧ ν = 0, then the last equation implies d ⋆ 6 ǫ = 0. If ǫ ∧ ν is nonzero, then the last equation is equivalent to d ⋆ 6 ǫ = κǫ ∧ ν , κ ⋆ 5̟ =β ∧̟ , for some constant κ ∈ R. Proof. All the cases are direct consequences of Lemma 2.1 and Proposition 2.3, i.e. the closedness condition and Maxwell equation for F of the form (1.2). We show the calculations for (6) in detail. For F = ϕα +β ∧ ν, the first three equations of Lemma 2.1 reduce to ϕ dα = 0 ,α ∧ d ϕ + dβ ∧ ν = 0 ,β ∧ d ν = 0 , while the last three equations hold automatically. In particular, we see that dα = 0 and d ν = 0 (assuming ϕ = 0 andβ = 0). We also have d ϕ = κν and dβ = −κα for some constant κ ∈ R. The first two equations of Proposition 2.3 reduce to d ⋆ 5α ∧ ⋆ 6 ϕ + ⋆ 5β ∧ d ⋆ 6 ν = 0 and d ⋆ 5β ∧ ⋆ 6 ν = 0, respectively, while the last two hold automatically. This implies d ⋆ 5β = 0, d ⋆ 6 ν = λ ⋆ 6 ϕ and d ⋆ 5α = −λ ⋆ 5β . All together, we obtain (2.7). The other cases are treated similarly. Remark 2.5. In comparison with the examination of (6, 5)-decomposable supergravity backgrounds presented in [CG20], i.e. Lorentzian manifolds of the form Y = M 1,5 × M 5 , we see that the system of the closedness condition and the Maxwell equation are very similar, although non-identical. In particular, a comparison of our Proposition 3.3 with [CG20, Prop. 2.5] shows that when the 4-form F is determined via one of the cases (1), (2), or (4)-(8), then we obtain very similar constraints. On the other hand, the cases (3) and (9) are quite different. For example, in our case X = M 1,4 ×M 6 the Maxwell equation contains the new termψγ ∧θ ∧δ. On the other hand, it does not contain the term corresponding toα ∧γ ∧ δ, which appears in the Maxwell equation for Y = M 1,5 × M 5 . For certain flux forms, imposing topological restrictions on M 6 may result in additional conditions on F. For example, let F = ϕα+β∧ν be a 4-form satisfying the Maxwell equation and the closedness condition. We assume thatα,β, ν are nonzero differential forms, and also that the function ϕ is nonzero at every point in M 6 . Proposition 2.4 implies that d ⋆ 6 ν = λ ⋆ 6 ϕ or, equivalently, d ⋆ 6 ν = λϕ vol M . Now, assume that M 6 is a closed manifold. We will see that this implies λ = 0. If λ = 0, then there exists a constant K > 0 such that |λϕ| ≥ K. Thus K vol(M ) = K M vol M ≤ M |λϕ| vol M . The function λϕ is either positive at each point in M 6 or negative at each point. Therefore, by Stokes' theorem we deduce that for a connected closed manifold M 6 the right-hand-side is (up to an overall sign) equal to M λϕ vol M = M d ⋆ 6 ν = ∂M ⋆ 6 ν = 0 . This implies vol(M ) = 0, a contradiction. In particular, if ϕ is a nonzero constant, we get the following statement (after absorbing the constant ϕ intoα). Proposition 2.6. Let (X 1,10 = M 1,4 × M 6 , h =g + g, F =α +β ∧ ν) be an eleven-dimensional bosonic supergravity background. If M 6 is closed, then λ = 0 in equation (2.7), i.e. d ⋆ 5α = 0 and d ⋆ 6 ν = 0. A similar phenomenon occurs for the flux form F =̟ ∧ ǫ + θ. Proposition 2.7. Let (X 1,10 = M 1,4 × M 6 , h =g + g, F =̟ ∧ ǫ + θ) be an eleven-dimensional bosonic supergravity background. If M 1,4 is closed, then λ = 0 in equation (2.9), i.e. d ⋆ 5̟ = 0 and d ⋆ 6 θ = 0. Proof. By assumption, we haveψ = 1 in (2.9) which implies d ⋆ 5̟ = λ ⋆ 5 1 = λ vol M . We have λ vol( M ) = M λ vol M = M d ⋆ 5̟ = ∂ M ⋆ 5̟ = 0 , where the last equality follows since M 1,4 has been assumed to be closed. This implies λ = 0. Before we treat the supergravity Einstein equation, let us consider one more special case for which the Maxwell equation significantly simplifies. Namely, assume thatψθ = 0 and thatα,β,γ,̟ share a common factorω ∈ Ω 1 ( M 1,4 ), meaning thatα =ω ∧α,β =ω ∧β,γ =ω ∧γ,̟ =̟ω. Sincẽ ω is a 1-form, we haveω ∧ω = 0. Therefore the right-hand-sides in Proposition 2.3 vanish. We summarize this in a proposition that we will take advantage of in Section 5.6. Proposition 2.8. Consider the Lorentzian manifold (X 1,10 = M 1,4 × M 6 , h =g + g) with 4-form F = ϕα +β ∧ ν +γ ∧ δ +̟ ∧ ǫ and assume thatα =ω ∧α,β =ω ∧β,γ =ω ∧γ,̟ =̟ω for a non-trivial 1-formω ∈ Ω 1 ( M 1,4 ) andα ∈ Ω 3 ( M 1,4 ),β ∈ Ω 2 ( M 1,4 ),γ ∈ Ω 1 ( M 1,4 ),̟ ∈ C ∞ ( M 1,4 ). Then, the Maxwell equation is equivalent to the following system of equations: d ⋆ 5α ∧ ⋆ 6 ϕ + ⋆ 5β ∧ d ⋆ 6 ν = 0 , d ⋆ 5β ∧ ⋆ 6 ν − ⋆ 5γ ∧ d ⋆ 6 δ = 0 , d ⋆ 5γ ∧ ⋆ 6 δ + ⋆ 5̟ ∧ d ⋆ 6 ǫ = 0 , d ⋆ 5̟ ∧ ⋆ 6 ǫ = 0 . In particular, Proposition 2.8 shows that ifα,β,γ,̟, ν, δ, ǫ are coclosed on their respective manifolds and ifα,β,γ,̟ share a common factorω as above, then F = ϕα +β ∧ ν +γ ∧ δ +̟ ∧ ǫ satisfies the Maxwell equation. The supergravity Einstein equation In this section we present the supergravity Einstein equation for an oriented Lorentzian manifold of the form X = M 1,4 × M 6 , endowed with the product metric h =g + g and the 4-form F defined by (1.2). We recall that the supergravity Einstein equation has the form Ric h (X, Y ) = − 1 2 X F, Y F h + 1 6 h(X, Y ) F 2 h , (3.1) where X, Y are vector fields on X 1,10 . Note that Lemma 2.2 implies the following: Lemma 3.1. Let F be the 4-form on (X 1,10 = M 1,4 × M 6 , h =g + g) defined by (1.2). Then, F 2 h = ϕ 2 α 2 g + β 2 g ν 2 g + γ 2 g δ 2 g + ̟ 2 g ǫ 2 g +ψ 2 θ 2 g . (3.2) Since X 1,10 = M 1,4 × M 6 is a direct product of pseudo-Riemannian manifolds we have Ric h (X, Y ) = Ric g (X, Y ) , ∀ X, Y ∈ Γ(T M 6 ) , Ric h (X,Ỹ ) = Ricg(X,Ỹ ) , ∀X,Ỹ ∈ Γ(T M 1,4 ) , Ric h (X,Ỹ ) = 0 , ∀ X ∈ Γ(T M 6 ),Ỹ ∈ Γ(T M 1,4 ) . This lets us split the equation (3.1) into three parts. By using (3.2) we obtain explicitly each of the parts of the supergravity Einstein equation. In particular: For any X, Y ∈ Γ(T M 6 ) we compute Ric h (X, Y ) = ϕ 2 α 2 g 6 g(X, Y ) + ν 2 g 6 g(X, Y ) − 1 2 ν(X)ν(Y ) β 2 g + δ 2 g 6 g(X, Y ) − 1 2 X δ, Y δ g γ 2 g + ǫ 2 g 6 g(X, Y ) − 1 2 X ǫ, Y ǫ g ̟ 2 g + θ 2 g 6 g(X, Y ) − 1 2 X θ, Y θ g ψ 2 . . (3.3) For anyX,Ỹ ∈ Γ(T M 1,4 ) we obtain Ric h (X,Ỹ ) = α 2 g 6g (X,Ỹ ) − 1 2 X α, Y α g ϕ 2 + β 2 g 6g (X,Ỹ ) − 1 2 X β ,Ỹ β g ν 2 g + γ 2 g 6g (X,Ỹ ) − 1 2 X γ,Ỹ γ g δ 2 g + ̟ 2 g 6g (X,Ỹ ) − 1 2̟ (X)̟(Ỹ ) ǫ 2 g +ψ 2 θ 2 g 6g (X,Ỹ ). (3.4) Finally, for any X ∈ Γ(T M 6 ) andỸ ∈ Γ(T M 1,4 ) we get the following condition: 0 = Ric h (X,Ỹ ) = 1 2 ϕν(X) β ,Ỹ α g − γ ∧ (X δ), (Ỹ β ) ∧ ν h + ̟ ∧ (X ǫ), (Ỹ γ) ∧ δ h −ψ̟(Ỹ ) X θ, ǫ g . (3.5) As a summary, we state the following: Regarding the supergravity Einstein equation for the various special cases of F discussed in Proposition 2.4, we present the following result (all equations below hold for general vector fields X, Y on M 6 andX,Ỹ on M 1,4 , which for brevity we will not repeat). Proposition 3.3. Consider the Lorentzian manifold (X 1,10 = M 1,4 × M 6 , h =g + g). ( 1) The 4-form F ∈ Ω 4 (X) defined by F =α (3.6) satisfies the Einstein condition if and only if the following equations hold: Ric h (X, Y ) = α 2 g 6 g(X, Y ) , Ric h (X,Ỹ ) = α 2 g 6g (X,Ỹ ) − 1 2 X α, Y α g . (3.7) (2) The 4-form F ∈ Ω 4 (X) defined by F =β ∧ ν (3.8) satisfies the Einstein condition if and only if the following equations hold: Ric h (X, Y ) = ν 2 g 6 g(X, Y ) − 1 2 ν(X)ν(Y ) β 2 g , Ric h (X,Ỹ ) = β 2 g 6g (X,Ỹ ) − 1 2 X β ,Ỹ β g ν 2 g . (3.9) (3) The 4-form F ∈ Ω 4 (X) defined by F =γ ∧ δ (3.10) satisfies the Einstein condition if and only if the following equations hold: Ric h (X, Y ) = δ 2 g 6 g(X, Y ) − 1 2 X δ, Y δ g γ 2 g , Ric h (X,Ỹ ) = γ 2 g 6g (X,Ỹ ) − 1 2 X γ,Ỹ γ g δ 2 g . (3.11) (4) The 4-form F ∈ Ω 4 (X) defined by F =̟ ∧ ǫ (3.12) satisfies the Einstein condition if and only if the following equations hold: Ric h (X, Y ) = ǫ 2 g 6 g(X, Y ) − 1 2 X ǫ, Y ǫ g ̟ 2 g , Ric h (X,Ỹ ) = ̟ 2 g 6g (X,Ỹ ) − 1 2̟ (X)̟(Ỹ ) ǫ 2 g . (3.13) (5) The 4-form F ∈ Ω 4 (X) defined by F = θ (3.14) satisfies the Einstein condition if and only if the following equations hold: Ric h (X, Y ) = θ 2 g 6 g(X, Y ) − 1 2 X θ, Y θ g , Ric h (X,Ỹ ) = θ 2 g 6g (X,Ỹ ) . (3.15) (6) The 4-form F ∈ Ω 4 (X) defined by F = ϕα +β ∧ ν (3.16) satisfies the Einstein condition if and only if the following equations hold: Ric h (X, Y ) = ϕ 2 α 2 g 6 g(X, Y ) + ν 2 g 6 g(X, Y ) − 1 2 ν(X)ν(Y ) β 2 g , Ric h (X,Ỹ ) = α 2 g 6g (X,Ỹ ) − 1 2 X α,Ỹ α g ϕ 2 + β 2 g 6g (X,Ỹ ) − 1 2 X β ,Ỹ β g ν 2 g , 0 = ϕ ν(X) β ,Ỹ α g . (3.17) (7) The 4-form F ∈ Ω 4 (X) defined by F =̟ ∧ ǫ +ψ θ (3.18) satisfies the Einstein condition if and only if the following equations hold: Ric h (X, Y ) = ǫ 2 g 6 g(X, Y ) − 1 2 X ǫ, Y ǫ g ̟ 2 g + θ 2 g 6 g(X, Y ) − 1 2 X θ, Y θ g ψ 2 , Ric h (X,Ỹ ) = ̟ 2 g 6g (X,Ỹ ) − 1 2̟ (X)̟(Ỹ ) ǫ 2 g +ψ 2 θ 2 g 6g (X,Ỹ ) , 0 =ψ̟(Ỹ ) X θ, ǫ g . (3.19) (8) The 4-form F ∈ Ω 4 (X) defined by F =α + θ (3.20) satisfies the Einstein condition if and only if the following equations hold: Ric h (X, Y ) = α 2 g 6 g(X, Y ) + θ 2 g 6 g(X, Y ) − 1 2 X θ, Y θ g , Ric h (X,Ỹ ) = α 2 g 6g (X,Ỹ ) − 1 2 X α, Y α g + θ 2 g 6g (X,Ỹ ) . (3.21) (9) The 4-form F ∈ Ω 4 (X) defined by F =β ∧ ν +̟ ∧ ǫ (3.22) satisfies the Einstein condition if and only if the following equations hold: Ric h (X, Y ) = ν 2 g 6 g(X, Y ) − 1 2 ν(X)ν(Y ) β 2 g + ǫ 2 g 6 g(X, Y ) − 1 2 X ǫ, Y ǫ g ̟ 2 g , Ric h (X,Ỹ ) = β 2 g 6g (X,Ỹ ) − 1 2 X β ,Ỹ β g ν 2 g + ̟ 2 g 6g (X,Ỹ ) − 1 2̟ (X)̟(Ỹ ) ǫ 2 g . (3.23) Proof. For each case, the equations involving the Ricci tensor follow directly from the equations (3.3), (3.4) and (3.5). Note that we have simplified the cases (1) and (5) by setting ϕ andψ equal to 1, since we are only interested in solutions to the supergravity Einstein equation that also satisfy the Maxwell equation and the closedness condition (recall by Lemma 2.1 that the closedness condition implies that ϕ andψ are constant in these particular cases). We see that the supergravity Einstein equation simplifies significantly when the form of F is further specified as above. In fact, the special form of many of the equations in Proposition 3.3 leads to some particular consequences that we will now investigate. Let (x, x) denote a general point on M 1,4 × M 6 . For each equation in Proposition 3.3 we observe that the left-hand-side depends either onx or x (but not both), while the right-hand-side is either of the form f 1 (x)g 1 (x) or f 1 (x)g 1 (x) + f 2 (x)g 2 (x) for every pair of vector fields. From this we draw some conclusions about f i and g i , which are functions on M 1,4 and M 6 , respectively. They are based on the following simple observation. Lemma 3.4. Assume that r(x) = f 1 (x)g 1 (x) (for everyx ∈ M 1,4 and every x ∈ M 6 ). Then either f 1 is identically equal to zero or g 1 is constant. Assume that r(x) = f 1 (x)g 1 (x) + f 2 (x)g 2 (x) and that none of the functions f 1 and f 2 are identically equal to zero (if one of them is, then the situation is the same as above). Then either g 1 and g 2 are both constant, or f 1 (x) = Cf 2 (x) for some C ∈ R \ {0} and g 2 (x) = −Cg 1 (x) + D for some constant D ∈ R. In the latter case we have r(x) = Df 2 (x). Notice that a similar statement holds after switching x withx. The application of these statements to the equations of Proposition 3.3 results in the following corollary. Corollary 3.5. Assume that (X 1,10 = M 1,4 × M 6 , h =g + g, F) is a solution of the supergravity Einstein equations (3.1). (1) If F =α, then (M 6 , g) is Einstein with Einstein constant α 2 g /6. (4) If F =̟ ∧ ǫ, then ǫ 2 g is constant and negative. (5) If F = θ, then ( M 1,4 ,g) is Einstein with Einstein constant θ 2 g /6 > 0. (6) If F = ϕα +β ∧ ν, then β 2 g and α 2 g are constant. (7) If F =̟ ∧ ǫ +ψθ, then ǫ 2 g and θ 2 g are constant. (2) If F =β ∧ ν, then β 2 g is constant. (3) If F =γ ∧ δ and γ 2 g is not constant, then we have g(X, Y ) = 3 δ 2 g X δ, Y δ g for all X, Y ∈ Γ(T M 6 ), Proof. Statements (1) and (5) are obvious. We prove the rest of them. (2) When Y = X, the first equation of (3.9) reduces to Ric h (X, X) = ν 2 g 6 X 2 g − 1 2 ν(X) 2 β 2 g , which must hold for every X ∈ Γ(T M 6 ). By Lemma 3.4, either β 2 g is constant, or ν 2 g 6 X 2 g − 1 2 ν(X) 2 = 0 . Let X be a nonzero vector field in the kernel of ν, that is ν(X) = 0. Since g is negative definite, the functions ν 2 g and X 2 g are not identically zero. Therefore β 2 g must be constant. (3) If F =γ ∧ δ, we have from the first equation of (3.11) that Ric h (X, Y ) = δ 2 g 6 g(X, Y ) − 1 2 X δ, Y δ g γ 2 g . If γ 2 g is not constant, it follows from Lemma 3.4 that g(X, Y ) = 3 δ 2 g X δ, Y δ g . Consequently, (M 6 , g) is Ricci-flat. Since δ is a 2-form, δ 2 g is positive. Using the definition of ω we can write the condition as g(X, Y ) = X ω, Y ω g . If ω(X, Z) = 0 for every Z, then g(X, Y ) = 0 for every Y , so non-degeneracy of g implies non-degeneracy of ω. This allows us to define an almost complex structure J via ω(X, Y ) = g(JX, Y ), since we then get (4) ForỸ =X the second equation of (3.13) reduces to J j i J k j = ω ia g aj ω jb g bk = −g aj ω ia ω bj g bk = −Id k i ,Ric h (X,X) = ̟ 2 g 6g (X,X) − 1 2̟ (X) 2 ǫ 2 g . If ̟ 2 g = 0, letX be a vector with the property̟(X) = 0. If ̟ 2 g is different from 0, letX be theg-dual of the 1-form̟. Then we obtain the equation Ric h (X,X) = − ̟ 4 g ǫ 2 g /3 . In both cases we see that ǫ 2 g is constant due to Lemma 3.4, and it is negative since g is negative definite on 3-forms. (6) To see that α 2 g is constant, let us assume that X is given by X j = 1 3 g ij ν i . Then, the first equation of (3.17) reduces to Ric h (X, Y ) = ϕ 2 α 2 g 6 g(X, Y ) + ν 2 g 6 g(X, Y ) − 1 2 ν(X)ν(Y ) β 2 g = ϕ 2 α 2 g 6 + ν 2 g 6 − 1 2 ν(X) β 2 g ν(Y ) = ϕ 2 α 2 g 6 ν(Y ) , since ν(X) = ν 2 g /3. By choosing Y such that ν(Y ) = 3g(X, Y ) = 0, we see that α 2 g is constant by Lemma 3.4. It is then clear that also β 2 g must be constant. (7) The proof is similar to that of (6). Remark 3.6. The sign of the Einstein constant depends on the signature convention. For example, if the manifold ( M 1,4 ,g) in point (5) has positive Einstein constant, it will have negative Einstein constant in the "mostly plus" convention. Remark 3.7. For point (3) in Corollary 3.5 we observe the following. If (X 1,10 , h, F =γ ∧ δ) satisfies the closedness condition, then d δ = 0. If δ 2 g is constant, this implies that ω is closed and so (M 6 , g, J, ω) must be an almost Kähler manifold. From taking the trace of the Einstein equation, it follows that the scalar curvature Scal h of a bosonic supergravity background (X 1,10 , h, F) satisfies the relation Scal h = 1 6 F 2 h (see for example [FP03,H16]). For the cases (1) and (5) in Corollary 3.5 we obtain that F 2 h is constant. Corollary 3.8. Assume that (X 1,10 = M 1,4 × M 6 , h =g + g, F) is a solution of the supergravity Einstein equations (3.1). If F =α, then Scal h = 1 6 α 2 g is constant. If F = θ, then Scal h = 1 6 θ 2 g is constant and positive. General theorems regarding flux forms composed of null forms We continue our investigation of manifolds of the form X 1,10 = M 1,4 × M 6 , endowed with the product metric h =g + g and the 4-form F given in (1.2). Sinceg is a Lorentzian metric, there exist differential forms on M 1,4 which are null. Recall that a k-form ω ∈ Ω k ( M 1,4 ) is called null if ω, ω g = 0. In this section we will show that if F is composed of such forms, then the supergravity Einstein equation simplifies. These results are of particular relevance whenever M 1,4 comes equipped with a distribution of null lines. Indeed, this is the case for example if M 1,4 is a Walker manifold, or a Kundt spacetime. The case when M 1,4 is a Walker manifold will be further explored in Section 5 in a way analogous to what was done in [CG20]. The following proposition concerns a particular type of null flux forms and is a direct consequence of the relations (3.3) and (3.4). Ric h (X,Ỹ ) = − 1 2 X F,Ỹ F h = − 1 2 X α,Ỹ α g ϕ 2 + X β ,Ỹ β g ν 2 g + X γ,Ỹ γ g δ 2 g +̟(X)̟(Ỹ ) ǫ 2 g , 0 = ϕν(Z) β ,Ỹ α g − γ ∧ (Z δ), (Ỹ β ) ∧ ν h + ̟ ∧ (Z ǫ), (Ỹ γ) ∧ δ h for anyX,Ỹ ∈ Γ(T M 1,4 ) and any Z ∈ Γ(T M 6 ). One notable consequence is that the component (M 6 , g) of bosonic supergravity backgrounds of the type described in Proposition 4.1 is required to be Ricci-flat. Moreover, by combining Proposition 4.1 with previous results concerning the Maxwell equation and the closedness condition, we arrive at the following general statements. Theorem 4.2. Consider the Lorentzian manifold X 1,10 = M 1,4 × M 6 with metric h =g + g and the 4-form F =α, whereα is null. Then (X 1,10 , h, F) is a bosonic supergravity background if and only if M 6 is Ricci-flat,α is closed and co-closed on M 6 , and Ric h (X,Ỹ ) = − 1 2 X α,Ỹ α g , ∀X,Ỹ ∈ Γ(T M ). (4.1) Note that if F = ϕα, then Proposition 2.4 implies that ϕ is constant, and it can thus be absorbed intoα. Thus, above the condition F =α is considered without loss of generality. d ⋆ 5γ ∧ ⋆ 6 δ = 1 2γ ∧γ ∧ δ ∧ δ, Ric h (X,Ỹ ) = − 1 2 X γ,Ỹ γ g δ 2 g , ∀X,Ỹ ∈ Γ(T M ). (4.3) Note that the first condition of (4.3) is satisfied ifγ is co-closed and eitherγ ∧γ = 0 or δ ∧ δ = 0. Observe that the equation (4.4) implies that ǫ 2 g is constant, and without loss of generality one may assume that it is equal to −1 (by absorbing the constant into̟). Let us also recall the following definition. Definition 4.6. A Lorentzian manifold (X, h) is called totally Ricci-isotropic if the Ricci endomorphism ric h : T X → T X corresponding to Ric h satisfies the relation h(ric h (X), ric h (Y )) = 0 , ∀ X, Y ∈ Γ(T X) . In Theorem 4.5 we observe that a bosonic supergravity background (X 1,10 = M 1,4 × M 6 , h = g + g), with flux form F =̟ ∧ ǫ and̟ null, is totally Ricci-isotropic. This is essentially the same claim as [CG20, Cor. 4.10], and the proof is similar. Corollary 4.7. Let̟ ∈ Ω 2 ( M 1,4 ) be null. Then, a bosonic supergravity background (X 1,10 = M 1,4 × M 6 , h =g + g, F =̟ ∧ ǫ) is totally Ricci-isotropic. Next, we state two results concerning flux forms of the form ϕα +β ∧ ν andβ ∧ ν +̟ ∧ ǫ, respectively, whereα,β and̟ are null. dα = d ν = 0 , d ϕ = κν , dβ = −κα , d ⋆ 5β = 0 , d ⋆ 5α = −λ ⋆ 5β , d ⋆ 6 ν = λ ⋆ 6 ϕ for some constants κ, λ ∈ R and Ric h (X,Ỹ ) = − 1 2 X α,Ỹ α g ϕ 2 − 1 2 X β ,Ỹ β g ν 2 g , ∀X,Ỹ ∈ Γ(T M ), 0 = β ,X α g , ∀X ∈ Γ(T M ). Theorem 4.9. Consider the Lorentzian manifold X 1,10 = M 1,4 × M 6 with metric h =g + g and the 4-form F =β ∧ ν +̟ ∧ ǫ, whereβ and̟ are null. Then (X 1,10 , h, F) is a bosonic supergravity background if and only if M 6 is Ricci-flat, dβ = d ν = d̟ = d ǫ = 0 , d ⋆ 6 ν = d ⋆ 5β = d ⋆ 5̟ = 0 , ⋆ 5̟ ∧ d ⋆ 6 ǫ =β ∧̟ ∧ ǫ ∧ ν , and Ric h (X,Ỹ ) = − 1 2 X β ,Ỹ β g ν 2 g − 1 2̟ (X)̟(Ỹ ) ǫ 2 g , ∀X,Ỹ ∈ Γ(T M ). Notice that the last equation in Proposition 4.1, the one coming from equation (3.5), is satisfied automatically in Theorems 4.2 -4.9. The only exception is Theorem 4.8 where the consequence β ,X α g = 0 gives additional constraints. Let us also pose the following: Proof. This is a simple consequence of the relation Scal h = 1 6 F 2 h and the fact that the flux form F is null for all the backgrounds presented above. Bosonic supergravity backgrounds for which M 1,4 is a Ricci-isotropic Walker manifold In order to construct explicit examples of bosonic supergravity backgrounds for which the flux form is composed of null forms, as treated in the previous section, we assume in this section that M 1,4 is a Lorentzian Walker manifold. Lorentzian Walker manifolds admit a parallel distribution of isotropic lines which we will use to build the Lorentzian part of the flux form F. With the aim to further simplify the supergravity Einstein equations, we follow [CG20] and will work with special type of totally Ricci-isotropic Walker manifolds, defined below. In Sections 5.2 -5.5 we consider the simplest type of flux forms, namely those of the formα,β ∧ ν,γ ∧ δ, and̟ ∧ ǫ. In Section 5.6 we unify these results by considering the more general flux form ϕα +β ∧ ν +γ ∧ δ +̟ ∧ ǫ, under the additional condition that the eight involved differential formsα,β,γ,̟, ϕ, ν, δ, ǫ are closed and coclosed on their respective manifolds. Finally, in Section 5.7 we consider the flux form ϕα +β ∧ ν, without the strict assumption of closedness and coclosedness on each of its components. 5.1. Ricci-isotropic Walker manifolds and null forms. Let us recall the definition of a Lorentzian Walker manifold. Definition 5.1. A Lorentzian Walker manifold is a Lorentzian manifold that admits a parallel distribution of isotropic (or null) lines. Next we focus on the five-dimensional case. If ( M 1,4 ,g) is a Lorentzian Walker manifold, then it is locally diffeomorphic to a product R × N 3 × R of manifolds with coordinates u, x = (x 1 , x 2 , x 3 ) and v, respectively, on which the metric takes the form g = 2 d u d v + ρ + 2A d u + H d u 2 . (5.1) 1-forms on N 3 , and H = H(u, x, v) is a smooth function on M 1,4 (see [Wa50,G10,GL10,CG20]). In these coordinates, the distribution spanned by ∂ v consists of isotropic lines and it is parallel, since Here, ρ = ρ ij (u, x) d x i d x∇g∂ v = 1 2 H v ∂ v ⊗ d u, where ∇g denotes the Levi-Civita connection with respect tog. Observe also that d u = ∂ v g is null, that is, d u, d u g = 0. Following [CG20], we will further assume the following: ∂ v H = 0 , A i = 0 for any i = 1, 2, 3 , ρ is a family of Ricci-flat metrics . (5. 2) The first condition implies ∇g∂ v = 0 and consequently ∇g d u = 0. Under these assumptions, the Ricci tensor is significantly simplified to Ricg = − 1 2 ∆ ρ (H) d u 2 , (5.3) where ∆ ρ (H) = 3 i,j,k=1 ρ ij ∂ x i ∂ x j H − Γ k ij ∂ x k H is the Laplace-Beltrami operator of the metric ρ applied to H (see [G10]). It follows that the Ricci endomorphism ricg is null, i.e. ricg, ricg g = 0. As in [CG20], we shall slightly abuse the terminology and call Walker metrics satisfying the conditions (5.2) Ricci-isotropic Walker metrics, referring to the property that the image of the Ricci endomorphism related to the Walker metric is totally null (see Definition 4.6). Note that since the 1-form d u is null, we can use it to build other null differential forms on M 1,4 . In particular, if we use d u to construct a null 4-form F, we may take advantage of the results found in the previous section. By Proposition 4.1 we know that if (X 1,10 = M 1,4 × M 6 , h =g + g, F) is a bosonic supergravity background, then the Ricci tensor of M 1,4 satisfies the equation Ric h (X,Ỹ ) = − 1 2 X α,Ỹ α g ϕ 2 + X β ,Ỹ β g ν 2 g + X γ,Ỹ γ g δ 2 g +̟(X)̟(Ỹ ) ǫ 2 g . When ( M 1,4 ,g) is a Ricci-isotropic Walker manifold of the type described above, the right-hand-side of this equation must be the same type of tensor as in (5.3). The following lemma shows that this happens whenα,β,γ,̟ are of the form d u ∧ ω(u), for some ω(u) ∈ Ω k (N 3 ). Here, the notation indicates that the differential forms on N 3 are parametrized by u. 2 Lemma 5.2. Let ω(u) ∈ Ω k (N 3 ) be a k-form and letg = 2 d u d v + ρ + H(u, x) d u 2 be a metric on M 1,4 = R × N 3 × R. Then X (d u ∧ ω(u)),Ỹ (d u ∧ ω(u)) g = a 1 a 2 ω(u), ω(u) g , where a 1 =X d u and a 2 =Ỹ d u. In particular, the expression vanishes, unless bothX d u and Y d u are nonzero. Proof. We have d u, d u g = 0 and d u, d x i g = 0, which implies d u ∧ ω 1 , ω 2 g = 0 for every k-form ω 2 and every (k − 1)-form ω 1 on R × N 3 . LetX = a 1 ∂ u + 3 i=1 b i 1 ∂ x i + c 1 ∂ v andỸ = a 2 ∂ u + 3 i=1 b i 2 ∂ x i + c 2 ∂ v . Then, forω = d u ∧ ω(u) we compute X ω,Ỹ ω g = a 1 ω(u) − d u ∧ ( 3 i=1 b i 1 ∂ x i ω(u)), a 2 ω(u) − d u ∧ ( 3 i=1 b i 2 ∂ x i ω(u)) g = a 1 a 2 ω(u), ω(u) g . An important subclass of Ricci-isotropic Lorentzian Walker metrics, which we may use in our study to construct explicit examples of bosonic supergravity backgrounds, consists of the so-called pp-waves (see [F00, L06] for details). Locally, in five dimensions such manifolds are of the form Proof. Sinceα does not depend on v, we have dα = 0. The condition d ⋆ 5α = 0 is equivalent to ∂ x i (f ) = 0 for i = 1, 2, 3. From Lemma 5.2 we see that X α,Ỹ α g = 0 for every pairX,Ỹ on which d u 2 vanishes. The statement then follows from Theorem 4.2 since we get (5.1), with A = 0, ρ = −(d x 1 ) 2 − (d x 2 ) 2 − (d x 3 ) 2 and ∂ v H = 0, and so topologically M 1,4 = R × R 3 × R ∼ = R 5 . In particular, we have ∆ ρ H = − 3 i=1 H x i x i .Ricg = 1 2 f 2 d u 2 , which by (5.3) is equivalent to ∆ ρ H = −f 2 . Example 5.5. For an explicit example, let (M 6 , g) be a Ricci-flat Riemannian manifold and let ( M 1,4 ,g = 2 d u d v − 3 i=1 (d x i ) 2 + H d u 2 ) be a five-dimensional pp-wave. Since ∆ ρ H = − 3 i=1 H x i x i , the equation ∆ ρ H = −f 2 is satisfied when H = 1 6 f (u) 2 3 i=1 (x i ) 2 . Thus, with this choice of H, (X 1,10 = M 1,4 × M 6 , h =g + g, F = f (u) d u ∧ d x 1 ∧ d x 2 ∧ d x 3 ) is an eleven-dimensional bosonic supergravity background. 5.3. Results concerning the flux form F =β ∧ ν. Let us now consider a Ricci-flat Riemannian manifold M 6 = R × Σ with metric g = − d t 2 − µ, where µ is a positive definite metric on the five-dimensional manifold Σ. Proposition 5.6. Let (M 6 = R × Σ, g) be a Ricci-flat Riemannian manifold with metric g = − d t 2 − µ and let ( M 1,4 = R × N 3 × R,g = 2 d v d u + ρ + H d u 2 ) be a Walker manifold with ∂ v (H) = 0 and ρ u-independent and Ricci-flat. Set ν = d t andβ = d u ∧ ω for a closed and coclosed 2-form ω on N 3 . Then (X 1,10 = M 1,4 × M 6 , h =g + g, F =β ∧ ν) is a bosonic supergravity background if and only if ∆ ρ H = − ω 2 ρ . Proof. It is clear thatβ is closed, and that ν is closed and coclosed. It follows from ⋆ 5β = ⋆ 2 d u ∧ ⋆ ρ ω = − d u ∧ ⋆ ρ ω thatβ is coclosed. We also see that X β ,Ỹ β g = 0 for every pair X,Ỹ on which d u 2 vanishes. Thus it follows from Theorem 4.3 that the Ricci tensor is given by Ric h = − 1 2 ω 2 g ν 2 g d u 2 . This equivalent to ∆ ρ H = − ω 2 g = − ω 2 ρ , since ν 2 g = −1. Example 5.7. Let ( M 1,4 = R × N 3 × R,g = 2 d v d u − 3 i=1 (d x i ) 2 + H d u 2 ) be a pp-wave with H = 1 6 3 i=1 (x i ) 2 . Let (M 6 , g) and ν be as described in the proposition above and set β = d u ∧ d x 1 ∧ d x 2 . Then (X 1,10 = M 1,4 × M 6 , h =g + g, F =β ∧ ν) is an eleven-dimensional bosonic supergravity background. 5.4. Results concerning the flux form F =γ ∧ δ. Letγ = d u ∧ ζ for a 1-form ζ on N 3 and assume that M 6 is a Calabi-Yau manifold and that δ is its Kähler form. Proof. We use Theorem 4.4. Sinceγ ∧γ = 0, the Maxwell equation and closedness condition are satisfied whenγ and δ are closed and coclosed. Since δ is the Kähler form, it is closed and coclosed, and the same holds forγ. Since (M 6 , g, δ) is a Calabi-Yau manifold, it is Ricci-flat. We also see that X γ,Ỹ γ g = 0 for every pairX,Ỹ on which d u 2 vanishes. It follows from (4.3) that Ric h = − 1 2 ζ 2 g δ 2 g d u 2 , which is equivalent to (5.4) (note that the function δ 2 g is constant since it is the Kähler form). This proves our claim. 5.5. Results concerning the flux form F =̟ ∧ ǫ. Proposition 5.9. Let (M 6 , g) be a Riemannian Ricci-flat manifold and ǫ a closed and coclosed 3-form on M 6 . Let ( M 1,4 = R × N 3 × R,g = 2 d v d u + ρ + H d u 2 ) be a Walker manifold with ρ u-independent and Ricci-flat and ∂ v (H) = 0. Set̟ = d u. Then (X 1,10 = M 1,4 ×M 6 , h =g +g, F = ̟ ∧ ǫ) is a bosonic supergravity background if and only if ǫ 2 g is constant and ∆H = ǫ 2 g . Proof. We use Theorem 4.5. It is clear that̟ = d u is both closed and coclosed. We havẽ ̟(X)̟(Ỹ ) = 0 for every pairX,Ỹ on which d u 2 vanishes. Thus, by (4.4) the Ricci tensor takes the form Ric h = − 1 2 ǫ 2 g d u 2 , or, equivalently, ∆ ρ H = ǫ 2 g . Since the left-hand-side of this equation is a function on M 1,4 , ǫ 2 g must be constant. This completes the proof. We illustrate Proposition 5.9 with the following example: Example 5.10. LetM 1,4 be a pp-wave with metric g = 2 d v d u − 3 i=1 (d x i ) 2 − E 2 6 3 i=1 (x i ) 2 d u 2 , let (M 6 , g) be a Riemannian Ricci-flat manifold and let ǫ be a closed and coclosed 3-form on M 6 with ǫ 2 g = −E 2 constant. Then (X 1,10 = M 1,4 ×M 6 , h =g+g, F = d u∧ǫ) is an eleven-dimensional bosonic supergravity background. 5.6. Results concerning the flux form F = ϕα +β ∧ ν +γ ∧ δ +̟ ∧ ǫ. In this section, we unify the previous four cases by considering a flux form F = ϕα +β ∧ ν +γ ∧ δ +̟ ∧ ǫ wherẽ α = d u ∧α(u) ,β = d u ∧β(u) ,γ = d u ∧γ(u) ,̟ = d u ∧̟(u) , withα(u) ∈ Ω 3 (N 3 ),β(u) ∈ Ω 2 (N 3 ),γ(u) ∈ Ω 1 (N 3 ), and̟(u) ∈ C ∞ (N 3 ), respectively. Recall that the notation indicates that the differential forms on N 3 are parametrized by u. Since dim N 3 = 3, we haveα(u) = f (u, x) vol ρ for some function f ∈ C ∞ (R × N 3 ), where vol ρ is the (in general u-dependent) volume form with respect to the metric ρ. In this case, the Maxwell equations (Proposition 2.3) simplify significantly: all right-hand-sides vanish due to d u ∧ d u = 0, and we obtain the equations in Proposition 2.8. It is easily seen that both the closedness condition and the Maxwell equation are satisfied in the particular case thatα,β,γ,̟ and ϕ, ν, δ, ǫ are closed and coclosed on their respective manifolds. Let us remark that the closedness of ϕ implies its constancy, and we can without loss of generality assume that it is equal to 1. Finally, notice that if ω = d u ∧ω(u) for someω(u) ∈ Ω k (N 3 ), then closedness of ω with respect to the exterior derivative on M 1,4 is equivalent to closedness ofω(u) with respect to the exterior derivative d N on N 3 : d ω = − d u ∧ dω(u) = − d u ∧ d Nω (u) . A similar statement can be made for coclosedness of ω: d ⋆ 5 ω = 0 if and only if d N ⋆ ρω (u) = 0. Proposition 5.11. Consider the 4-form F = d u ∧ (α(u) +β(u) ∧ ν +γ(u) ∧ δ +̟(u)ǫ), wherê α(u) ∈ Ω 3 (N 3 ) ,β(u) ∈ Ω 2 (N 3 ) ,γ(u) ∈ Ω 1 (N 3 ) ,̟(u) ∈ C ∞ (N 3 ) , ν ∈ Ω 1 (M 6 ) , δ ∈ Ω 2 (M 6 ) , ǫ ∈ Ω 3 (M 6 ) ,∆ ρ H = α(u) 2 ρ + β (u) 2 ρ ν 2 g + γ(u) 2 ρ δ 2 g +̟(u) 2 ǫ 2 g . Proof. It is clear that the closedness condition in Lemma 2.1 is satisfied, and so is the Maxwell equation (Proposition 2.8). Thus, the condition for being a bosonic supergravity background boils down to the supergravity Einstein equation, which by Proposition 4.1 consists of the following system of equations:    Ric h (X,Ỹ ) = − 1 2 X α,Ỹ α g + X β ,Ỹ β g ν 2 g + X γ,Ỹ γ g δ 2 g +̟(X)̟(Ỹ ) ǫ 2 g , 0 = ϕν(Z) β ,Ỹ α g − γ ∧ (Z δ), (Ỹ β ) ∧ ν h + ̟ ∧ (Z ǫ), (Ỹ γ) ∧ δ h . The first equation reduces to ∆ ρ H = −2 Ric h (∂ u , ∂ u ) = α(u) 2 ρ + β (u) 2 ρ ν 2 g + γ(u) 2 ρ δ 2 g +̟(u) 2 ǫ 2 g due to (5.3) and Lemma 5.2, while the second one holds automatically. Thus we get our claim. Notice that sinceα(u) = f (u, x) vol ρ , we have d Nα (u) = 0. We also see that α(u) 2 ρ = −f 2 , and the condition d N ⋆ ρα (u) = 0 is equivalent to condition ∂ x i (f ) = 0 for i = 1, 2, 3, which we recognize from Proposition 5.4. Notice that closedness of̟(u) on N 3 means that̟(u) is a function of u only. The propositions 5.4, 5.6, 5.8, 5.9 can now be viewed as corollaries of Proposition 5.11. Moreover, their corresponding examples are special cases of the following more general example. Example 5.12. Let (M 6 , g) be a Ricci-flat Riemannian manifold and assume that there exist differential forms ν ∈ Ω 1 (M 6 ), δ ∈ Ω 2 (M 6 ), ǫ ∈ Ω 3 (M 6 ), which are closed and coclosed, satisfying ν 2 g = −1, δ 2 g = 1, and ǫ 2 g = −1, respectively. Let M 1,4 be a five-dimensional pp-wave with metricg = 2 d u d v − 3 i=1 (d x i ) 2 + f 1 (u) 2 − f 2 (u) 2 + f 3 (u) 2 − f 4 (u) 2 6 3 i=1 (x i ) 2 d u 2 , and setα(u) = f 1 (u) vol ρ ,β(u) = f 2 (u) d x 1 ∧ d x 2 ,γ(u) = f 3 (u) d x 1 ,̟(u) = f 4 (u). If F = d u ∧ (α(u) +β(u) ∧ ν +γ(u) ∧ δ +̟(u)ǫ) = d u ∧ (f 1 (u) d x 1 ∧ d x 2 ∧ d x 3 + f 2 (u) d x 1 ∧ d x 2 ∧ ν + f 3 (u) d x 1 ∧ δ + f 4 (u)ǫ) then (X 1,10 = M 1,4 × M 6 , h =g + g, F) is an eleven-dimensional bosonic supergravity background. Notice that when all but one of the functions f i vanish, then the example reduces to ones previously considered. We can specify the example even more by considering M 6 = R 3 × Σ with metric g = −(d y 1 ) 2 − (d y 2 ) 2 − (d y 3 ) 2 − ν, where ν is a Ricci-flat positive definite metric on a threedimensional manifold Σ, and ν = d y 3 , δ = d y 2 ∧ d y 3 , ǫ = d y 1 ∧ d y 2 ∧ d y 3 . 5.7. Results concerning the flux form F = ϕα +β ∧ ν. Now, setα = d u ∧ f (u, x) vol ρ and β = d u ∧ ω(u), where f ∈ C ∞ (R × N 3 ) is a function, ω(u) ∈ Ω 2 (N 3 ) is a 2-form depending smoothly on the parameter u, and f, ω(u), ν, ϕ are nonzero. Then we get the following statement regarding bosonic supergravity backgrounds with flux form F = ϕα +β ∧ ν. Proposition 5.13. Let (M 6 , g) be a Riemannian Ricci-flat manifold, ϕ a function on M 6 satisfying ⋆ 6 d ⋆ 6 d ϕ = κλϕ and set ν = 1 κ d ϕ, for a nonzero constant κ. Let also ( M 1, 4 = R × N 3 × R,g = 2 d v d u + ρ + H d u 2 ) be a Walker manifold with ρ Ricci-flat and ∂ v (H) = 0. Set as aboveα = d u ∧ f vol ρ andβ = d u ∧ ω(u) with ω(u) = − 1 λ ⋆ ρ d N f ,∆ ρ H = ω(u) 2 ρ ν 2 g − f 2 ϕ 2 . Proof. The proof is mainly based on Theorem 4.8. The Maxwell equation and the closedness condition reduce to d N ⋆ ρ ω(u) = 0, d N ω(u) = κf vol ρ , d N f = λ ⋆ ρ ω(u), d ν = 0, d ϕ = κν, d ⋆ 6 ν = λ ⋆ 6 ϕ. These equations are satisfied due to the definitions of ν and ω(u) and the two differential equations ⋆ 6 d ⋆ 6 d ϕ = κλϕ , ⋆ ρ d N ⋆ ρ d N f = −κλf constraining them. Next, the supergravity Einstein equation consists of the following system of equations:    Ric h (X,Ỹ ) = − 1 2 X α,Ỹ α g ϕ 2 − 1 2 X β ,Ỹ β g ν 2 g , ∀X,Ỹ ∈ Γ(T M ) , 0 = β ,X α g , ∀X ∈ Γ(T M ) . By Lemma 5.2 and by assuming that ( M 1,4 ,g) is of the form (5.1) with ρ Ricci-flat and ∂ v (H) = 0, we see that the only nonzero component of the Ricci tensor has the form Ric h (∂ u , ∂ u ) = − 1 2 f 2 ϕ 2 vol ρ 2 g + ω(u) 2 g ν 2 g = 1 2 f 2 ϕ 2 − ω(u) 2 g ν 2 g , while forα andβ of the chosen form the equation β ,X α g = 0 holds automatically. But then, by (5.3), it turns out that the above relation is equivalent to the condition ∆ ρ H = ω(u) 2 g ν 2 g −f 2 ϕ 2 , which proves our claim. Remark 5.14. By assumption the function f 2 is nonzero, and ω(u) 2 g = ω(u) 2 ρ > 0, since ρ is positive definite on 2-forms. By Lemma 3.4 we see that either ϕ and ν 2 g are both constant, or f 2 = C ω(u) 2 g for some nonzero constant C, which can only be positive. In the latter case we obtain ν 2 g = Cϕ 2 + D , for some constant D. In this equation the left-hand-side is negative. For this reason we look for an example involving trigonometric functions. Example 5.15. Let (M 6 = S 1 ×R 5 , g) be a flat Riemannian manifold with coordinates y 1 , . . . , y 6 and metric g = −(d y 1 ) 2 − 6 i=2 (d y i ) 2 . Set ϕ = sin(y 1 ) , ν = 1 κ d ϕ = 1 κ cos(y 1 ) d y 1 . Consider also a pp-wave ( M 1,4 ,g) with metricg = 2 d u d v − 3 i=1 (d x i ) 2 + H(u, x) d u 2 . Set f = exp(x 1 ) and ω = κ ⋆ ρ d f = −κ exp(x 1 ) d x 2 ∧ d x 3 . In terms of the notation in Proposition 5.13, we have λ = −1/κ. Now, the supergravity Einstein condition gives ∆ ρ H = ω 2 g ν 2 g − f 2 ϕ 2 = − exp(2x 1 ) . This equation is satisfied if, for example, H = 1 4 exp(2x 1 ). Thus (X 1,10 = M 1,4 × M 6 , h = g +g, F = ϕα +β ∧ ν) is a bosonic supergravity background with these choices of ϕ, ν, f, ω, H, and then the flux form F reads as F = exp(x 1 ) d u ∧ d x 2 ∧ d x 3 ∧ sin(y 1 ) d x 1 + cos(y 1 ) d y 1 . Let us now consider the case where d ϕ = 0 (and κ = 0). By absorbing the constant intoα, we can without loss of generality assume that ϕ = 1. As long as λ = 0, the 2-form ω(u) is determined by f via the relation ω(u) = − 1 λ ⋆ ρ d N f . This implies at once d N ⋆ ρ ω(u) = 0. The equation d N ω(u) = 0 is then equivalent to d N ⋆ ρ d N f = 0, or ⋆ ρ d N ⋆ ρ d N f = 0. With this simplification we obtain the following statement. Proposition 5.16. Let (M 6 , g) be a Riemannian Ricci-flat manifold endowed with a closed 1-form ν ∈ Ω 1 (M ) satisfying ⋆ 6 d ⋆ 6 ν = λ, for some constant λ = 0. Let also ( M 1,4 = R × N 3 × R,g = 2 d v d u + ρ + H d u 2 ) be a Walker manifold with ρ Ricci-flat and ∂ v (H) = 0, and assume that f is a smooth function on R × N 3 , such that ⋆ ρ d N ⋆ ρ d N f = 0 . Setα = d u ∧ f vol ρ ,β = d u ∧ ω(u), with ω(u) = − 1 λ ⋆ ρ d N f . Then (X 1,10 = M 1,4 × M 6 , h = g + g, F =α +β ∧ ν ) is a bosonic supergravity background if and only if ∆ ρ H = ω(u) 2 ρ ν 2 g − f 2 . Consequently, ν 2 g must be constant for such a bosonic supergravity background. Recall that by Proposition 2.6, the six-dimensional Riemannian manifold M 6 appearing in Proposition 5.16 must be non-closed. Notice that examples 5.12 and 5.15 can easily be modified to make M 6 a compact manifold, for example a flat six-dimensional torus. This is not the case for the following example. Example 5.17. Consider the Riemannian manifold M 6 = (−L, L)×R 5 with metric g = −(d y 1 ) 2 − 6 i=2 (d y i ) 2 . We set λ = 1 and ν = −y 1 d y 1 + L 2 − (y 1 ) 2 d y 2 , so that ν 2 g = −L 2 . Let ( M 1,4 ,g) be a pp-wave, that isg = 2 d u d v − 3 i=1 (d x i ) 2 + H(u, x) d u 2 . If we set f = x 1 and ω = − ⋆ ρ d N f = d x 2 ∧ d x 3 , then the Einstein equation is given by ∆ ρ H = ω 2 ρ ν 2 g − f = −L 2 − (x 1 ) 2 . This equation is satisfied when, for example, H = 1 12 (x 1 ) 4 + L 2 2 (x 1 ) 2 . Thus (X 1,10 = M 1,4 × M 6 , h = g + g, F =α +β ∧ ν) is a bosonic supergravity background with these choices of ν, f, ω, H. In this case, the flux form is given by F = d u ∧ d x 2 ∧ d x 3 ∧ (x 1 d x 1 − y 1 d y 1 + L 2 − (y 1 ) 2 d y 2 ). Bosonic backgrounds involving Kähler manifolds and non-null flux forms In this section we treat flux forms of type F =γ ∧ δ and F = θ for the case when (M 6 , g) is a Kähler manifold with Kähler form ω. We will assume that the part of F taking values in T M 6 is given by ω. More precisely, we consider the cases for which δ = ω and θ = c ⋆ 6 ω for some nonzero constant c ∈ R. 6.1. Results concerning F =γ ∧ δ. Inspired by Corollary 3.5, we consider bosonic supergravity backgrounds with F =γ ∧ δ where γ 2 g is not (necessarily) constant, and hence not null. The following theorem follows directly from the bosonic supergravity equations (Proposition 2.4 and Proposition 3.3). Theorem 6.1. Consider the Lorentzian manifold X 1,10 = M 1,4 × M 6 with metric h =g + g and a 4-form F =γ ∧ δ. Then (X 1,10 , h, F) is a bosonic supergravity background if and only if δ is closed and coclosed,γ is closed and satisfies d ⋆ 5γ ∧ ⋆ 6 δ =γ ∧γ ∧ δ ∧ δ 2 , and the following equations hold: Ric h (X, Y ) = δ 2 g 6 g(X, Y ) − 1 2 X δ, Y δ g γ 2 g , Ric h (X,Ỹ ) = γ 2 g 6g (X,Ỹ ) − 1 2 X γ,Ỹ γ g δ 2 g . (6.1) Now, Corollary 3.5 tells us that if γ 2 g is not constant, then M 6 is a Ricci-flat almost Hermitian manifold. If we let M 6 be a Kähler manifold with Kähler form δ, we get the following statement. Proof. Let us first mention that the Kähler form δ is closed and coclosed, and that δ 2 g = 3. The equation d ⋆ 5γ ∧ ⋆ 6 δ =γ ∧γ∧δ∧δ 2 is in this case equivalent to d ⋆ 5γ =γ ∧γ because of the the identity ⋆ 6 δ = 1 2 δ ∧ δ (6.3) on the Kähler form. The first of the equations (6.1) is satisfied since M 6 is Kähler while the second reduces to (6.2). This completes the proof. This construction illustrates the well-known fact that Ricci-flat Kähler manifolds and, more specifically, Calabi-Yau manifolds play an important role as components of bosonic supergravity backgrounds. 6.2. Results concerning the flux form F = θ and Kähler-Einstein metrics. By Corollary 3.5, we know that (X 1,10 = M 1,4 × M 6 , h =g + g, F = θ) is a solution of the supergravity Einstein equations, only if θ 2 g is constant and ( M 1,4 ,g) is an Einstein manifold with positive Einstein constant θ 2 g /6 (in our "mostly minus" convention). By also taking into account Propositions 2.4 and 3.3, we obtain the following theorem. Theorem 6.3. Consider the Lorentzian manifold X 1,10 = M 1,4 × M 6 with metric h =g + g and a 4-form F = θ ∈ Ω 4 (M 6 ). Then (X 1,10 , h, F) is a bosonic supergravity background if and only if θ is closed and coclosed, θ 2 g is constant, ( M 1,4 ,g) is an Einstein manifold with Einstein constant the sense that M 1,4 is here allowed to be any Lorentzian Einstein manifold with positive Einstein constant (in the "mostly minus" setup), not only (AdS) 5 . In order to get other bosonic supergravity backgrounds from Corollary 6.5, we need Lorentzian Einstein manifolds ( M 1,4 ,g) with positive scalar curvature that are different from (AdS) 5 . The class of Lorentzian Einstein-Sasakian manifolds provides us with many candidates. Lorentzian Einstein-Sasakian structures were studied in [B00,Bo03,BL04] under the geometric perspective of Killing and twistor spinors on Lorentzian manifolds. Such manifolds admit a cone characterization and, in the simply connected case, a spin structure ([Bo03, Lem. 12]), as in the Riemannian case. Moreover, there is an analogue of the well-known construction of Riemannian Einstein-Sasakian manifolds (see for example [BFGK91]), which provides Lorentzian Einstein-Sasakian manifolds in terms of circle bundles over Kähler-Einstein manifolds of negative scalar curvature ([Bo03, Lem. 14]). Lorentzian Einstein-Sasakian manifolds with positive (in the "mostly minus" convention) Einstein constant also occur within the framework of η-Einstein Sasakian geometry, and we refer to [BGM06] for many details and notions that we omit. A particularly important result, from our perspective, is that every negative Sasakian manifold M 2n+1 admits a Lorentzian Einstein-Sasakian structure with positive Einstein constant 2n, in the "mostly minus" convention ([BGM06, Cor. 24]). In particular, the 5-sphere S 5 admits infinitely many different Lorentzian Einstein-Sasakian structures, with Einstein constant 4 (all of them are inhomogeneous). Each of these can be used as the five-dimensional Lorentzian manifold M 1,4 in Corollary 6.5, providing us with infinitely many decomposable backgrounds (carrying the same flux form). Example 6.7. Let ( M 1,4 ,g) be the 5-sphere S 5 with an Einstein metric coming from any of the infinitely many Lorentzian Einstein-Sasakian structures mentioned above with Einstein constant 4, and let (M 6 , g, ω) be any Einstein Kähler manifold with Einstein constant −4. Then the triple ( M 1,4 × M 6 ,g + g, F = 2 √ 2 ⋆ 6 ω) is a bosonic supergravity background. The connected sum ♯k(S 2 × S 3 ) also admits Lorentzian Einstein-Sasakian metrics for any integer k ≥ 1 (see [G11,BGM06]), giving another class of Lorentzian Einstein-Sasakian manifolds that can potentially be used as ingredients in bosonic supergravity backgrounds. Example 6.7 highlights the appearance of Lorentzian Einstein-Sasakian geometries in supergravity, and more applications of such structures in eleven-dimensional supergravity are described in [FS15]. For further details on the applications of five-dimensional Lorentzian manifolds in certain supergravity theories, the reader may consult the recent work [BF21] and the references therein. Remark 6.8. Some of the bosonic backgrounds of the type appearing in Corollary 6.5 are symmetric. For example, if M 1,4 = (AdS) 5 and M 6 is one of the symmetric spaces CP 3 = SU(4)/ U(3) or Gr + (2, 5), endowed with their respective homogeneous Kähler-Einstein metrics, then we obtain decomposable symmetric backgrounds (see also [F13,Sect. 4.4]). To construct decomposable, homogeneous but non-symmetric, (5, 6)-supergravity backgrounds, we may use the full flag manifold F = SU(3)/T max , see for example [S99] for the corresponding Kähler-Einstein metrics. Also, in [OS17] the reader can find families of non-supersymmetric bosonic backgrounds with non-relativistic symmetry based on products involving (AdS) 5 . Finally, it is worth mentioning that (6.3) also holds for a strictly nearly Kähler manifold, and θ 2 g = c 2 ω 2 g is a constant as well, see [MNS05, Cor. 2.7]. However, for a strictly nearly Kähler manifold, the Kähler form is not closed. Thus, for this case the 4-form F chosen above is not coclosed, and can not serve as a flux form. General theorems regarding flux forms composed of null forms 15 5. Bosonic supergravity backgrounds for which M 1,4 is a Ricci-isotropic Walker manifold 17 5.1. Ricci-isotropic Walker manifolds and null forms 17 5.2. Results concerning the flux form F =α 19 5.3. Results concerning the flux form F =β ∧ ν 20 5.4. Results concerning the flux form F =γ ∧ δ 20 5.5. Results concerning the flux form F =̟ ∧ ǫ 20 5.6. Results concerning the flux form F = ϕα +β ∧ ν +γ ∧ δ +̟ ∧ ǫ 21 5.7. Results concerning the flux form F = ϕα +β ∧ ν 22 6. Bosonic backgrounds involving Kähler manifolds and non-null flux forms 24 6.1. Results concerning F =γ ∧ δ 25 6.2. Results concerning the flux form F = θ and Kähler-Einstein metrics 25 References 28 Introduction which implies that (M 6 , g, ω, J) is a Ricci-flat almost Hermitian manifold with almost complex structure J defined by ω(X, Y ) = g(JX, Y ), and Kähler form ω = where the last equality follows from g(X, Y ) = X ω, Y ω g . Moreover, g(JX, JY ) = ω(JX, J 2 Y ) = ω(JX, −Y ) = ω(Y, JX) = g(Y, X) = g(X, Y ) and consequently (M 6 , g, J) is an almost Hermitian manifold. Proposition 4 . 1 . 41Consider the Lorentzian manifold X 1,10 = M 1,4 × M 6 with metric h =g + g and the 4-form F, given by (1.2). Assume thatα,β,γ,̟ are null and moreover thatψ = 0. Then, (X 1,10 , h, F) satisfies the supergravity Einstein equations if and only if (M 6 , g) is Ricci-flat and Theorem 4. 3 . 3Consider the Lorentzian manifold X 1,10 = M 1,4 × M 6 with metric h =g + g and the 4-form F =β ∧ ν, whereβ is null. Then (X 1,10 , h, F) is a bosonic supergravity background if and only if M 6 is Ricci-flat,β and ν are closed and co-closed (on M and M , respectively), and Ric h (X,Ỹ ) = − 1 2 X β ,Ỹ β g ν 2 g , ∀X,Ỹ ∈ Γ(T M ). (4.2) Theorem 4.4. Consider the Lorentzian manifold X 1,10 = M 1,4 × M 6 with metric h =g + g and the 4-form F =γ ∧ δ, whereγ is null. Then (X 1,10 , h, F) is a bosonic supergravity background if and only if M 6 is Ricci-flat,γ and δ are closed (on M and M , respectively), δ is co-closed on M and the following two equations hold: Theorem 4. 5 . 5Consider the Lorentzian manifold X 1,10 = M 1,4 × M 6 with metric h =g + g and the 4-form F =̟ ∧ ǫ, where̟ is null. Then (X 1,10 , h, F) is a bosonic supergravity background if and only if M 6 is Ricci-flat,̟ and ǫ are closed and co-closed (on M and M , respectively), and Ric h (X,Ỹ ) = − 1 2̟ (X)̟(Ỹ ) ǫ 2 g , ∀X,Ỹ ∈ Γ(T M ). (4.4) Theorem 4 . 8 . 48Consider the Lorentzian manifold X 1,10 = M 1,4 × M 6 with metric h =g + g and the 4-form F = ϕα +β ∧ ν, whereα andβ are null. Then (X 1,10 , h, F) is a bosonic supergravity background if and only if M 6 is Ricci-flat, Corollary 4 . 10 . 410All bosonic supergravity backgrounds appearing in this section have vanishing scalar curvature. j is a family of Riemannian metrics on N 3 (parametrized by u and of signature (−, −, −)), A = A i (u, x) d x i is a family of Remark 5 . 3 . 53Walker manifolds provide examples of indefinite metrics that exhibit various geometric aspects (see for example[Br00,GiP08,GL10] for the Lorentzian version of such manifolds). For instance, the pp-waves form one of the simplest and well-known classes of Lorentzian Walker manifolds. On the other hand, (totally) Ricci-isotropic Lorentzian manifolds are known to be important in holonomy theory of indefinite metrics, and their Ricci tensor attains a simplified expression ([G10]). Due to this special holonomy feature, Ricci-isotropic Lorentzian Walker manifolds have many natural applications in supergravity theories, see for instance [BCH08, CGHP08, Gi09, Br00, F00, CG20].In the remainder of this section, we apply the results from the previous sections to the case where ( M 1,4 ,g) is a Lorentzian Walker manifold satisfying (5.2).5.2. Resultsconcerning the flux form F =α. Let us consider the non-trivial 4-form F =α = d u ∧ f (u, x) vol ρ , where vol ρ denotes the (in general u-dependent) volume form of the metric ρ on N 3 . Proposition 5.4. Let (M 6 , g) be a Ricci-flat Riemannian manifold and let ( M 1,4 ,g = 2 d u d v + ρ + H d u 2 ) be a Walker manifold with ρ Ricci-flat and ∂ v (H) = 0. Then (X 1,10 = M 1,4 × M 6 , h = g + g, F = f (x, u) d u ∧ vol ρ ) is a bosonic supergravity background if and only if ∂ x i (f ) = 0 for i = 1, 2, 3 and ∆ ρ H = −f 2 . Proposition 5 . 8 . 58Consider a six-dimensional Calabi-Yau manifold (M 6 , g, δ), where δ denotes the Kähler form, and a five-dimensional Walker manifold ( M 1,4 = R×N 3 ×R,g = 2 d v d u+ρ+H d u 2 ) with ρ u-independent and Ricci-flat, and ∂ v (H) = 0. Setγ = d u ∧ ζ for some closed and coclosed 1-form ζ on N 3 . Then (X 1,10 = M 1,4 × M, h =g + g, F =γ ∧ δ) is a bosonic supergravity background if and only if ∆ ρ H = ζ 2 ρ δ 2 g . (5.4) are closed and coclosed on N 3 and M 6 , respectively. Let also ( M 1,4 ,g) be a Walker metric of the form (5.1) with ρ Ricci-flat, A = 0, and ∂ v (H) = 0. Then, (X 1,10 = M 1,4 × M 6 , h = g +g, F) is a bosonic supergravity background if and only if (M 6 , g) is Ricci-flat and for a nonzero constant λ, and assume that f satisfies ⋆ ρ d N ⋆ ρ d N f = −κλf . Then (X 1,10 = M 1,4 × M 6 , h =g + g, F = ϕα +β ∧ ν) is a bosonic supergravity background if and only if Proposition 6 . 2 . 62Let (M 6 , g, δ) be a Kähler manifold and let ( M 1,4 ,g) be a Lorentzian manifold. Assume also thatγ is a closed 2-form on M 1,4 . Then (X 1,10 = M 1,4 × M 6 , h = g +g, F =γ ∧ δ) is a bosonic supergravity background if and only if (M 6 , g) is Ricci-flat, d ⋆ 5γ =γ ∧γ and Ric h (X, Theorem 3.2. Consider the manifold X 1,10 = M 1,4 × M 6 with the product metric h =g + g wherẽ g is a Lorentzian metric on M 1,4 and g a Riemannian metric on M 6 , and let F be the 4-form definedby (1.2). Then the eleven-dimensional supergravity Einstein equation (3.1) decomposes into the equations (3.3), (3.4) and (3.5). We should mention that in this paper we use the "mostly minus" convention for the Loretnzian metric h, and thus a Riemannian metric g is viewed as a negative definite metric. Remember that the metric ρ in general also depends on u even though our notation does not emphasize this. Acknowledgements: H.C. thanks NSFC for partial support under grants No. 11521101 and No. 12071489. I.C. and E.S. acknowledge full support by Czech Science Foundation via the project GAČR No. 19-14466Y. It is also our pleasure to thank A. Galaev (UHK) for helpful discussions.Note that ⋆ 6 is an isometry and we have c 2 ω 2 g = θ 2 g . Hence (6.4) together with the second equation of (3.15) reduces to the following system of equations:2g (X,Ỹ ) .(6.5)This completes our proof.By Corollary 3.8 the bosonic background (X 1,10 = M 1,4 × M 6 , h =g + g, F = θ = c⋆ 6 ω) discussed above must have constant positive scalar curvature given by Scal h = 1 6 θ 2 g = c 2 2 . Moreover, we see that ( M 1,4 ,g) has positive scalar curvature while (M 6 , g) has negative scalar curvature.Corollary 6.5. Let ( M 1,4 ,g) be a Lorentzian Einstein manifold with Einstein constant 1 2 c 2 and let (M 6 , g, ω) be an Einstein Kähler manifold with with Einstein constant − 1 2 c 2 . Then the triple ( M 1,4 × M 6 ,g + g, F = c ⋆ 6 ω) is a bosonic supergravity background.Remark 6.6. Several bosonic backgrounds of the type appearing in Corollary 6.5 are already known. In[PN89]it was shown that (AdS) 5 × M 6 , where (M 6 , g, ω) is a Kähler manifold, defines a bosonic background with flux form F = 1 2 cω ∧ ω (which is proportional to ⋆ 6 ω), if and only if (M 6 , g) is Einstein with negative scalar curvature (in "mostly minus" setting). In particular, the equations (4) and (5) in[PN89]are the same as (6.5), up to a scalar and signature convention. Thus Corollary 6.5 can be viewed as a slight generalization of the backgrounds found in[PN89], in Decomposable (4, 7) solutions in 11-dimensional supergravity. D Alekseevsky, I ; A Chrysikos, Taghavi-Chabert, Class. 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(cited on p. 2) School of Mathematics, Sun Yat-sen University, 135 Xingang Xi Road, Haizhu District, Guangzhou, Guangdong, China Email address: chihc@mail.sysu.edu.cn Faculty of Science, University of Hradec Králové, Rokitanského 62, Hradec Králové 50003, Czech Republic Email address: ioannis.chrysikos@uhk.cz Faculty of Science, University of Hradec Králové, Rokitanského 62, Hradec Králové 50003, Czech Republic Email address: eivind.schneider@uhk.cz

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{'abstract': 'We present decomposable (5,6)-solutions M 1,4 ×M 6 in eleven-dimensional supergravity by solving the bosonic supergravity equations for a variety of non-trivial flux forms. Many of the bosonic backgrounds presented here are induced by various types of null flux forms on products of certain totally Ricci-isotropic Lorentzian Walker manifolds and Ricci-flat Riemannian manifolds. These constructions provide an analogue of the work in [CG20], where similar computations were made for decomposable (6,5)-solutions. We also present bosonic backgrounds that are products of Lorentzian Einstein manifolds with negative Einstein constant (in the "mostly plus" convention) and Riemannian Kähler-Einstein manifolds with positive Einstein constant. This conclusion generalizes a result of C. N. Pope et al.[PN89]concerning the appearance of six-dimensional Kähler-Einstein manifolds in eleven-dimensional supergravity. In this setting we construct infinitely many nonsymmetric decomposable (5, 6)-supergravity backgrounds, by using the infinitely many Lorentzian Einstein-Sasakian structures with negative Einstein constant on the 5-sphere, known from the work of C. P. Boyer et al.[BGM06].ContentsLemma 2.2. Let X = M p × M q be a product of two pseudo-Riemannian manifolds (M ,g) and (M, g) of dimensions p, q, and lets, s be the number of negative eigenvalues ofg, g, respectively. Let us denote by ⋆, ⋆ p , ⋆ q the Hodge operator on (X, h =g + g), ( M ,g), and (M, g), respectively. Then, for anyα ∈ Ωk( M ), and β ∈ Ω k (M ) the following hold:With the help of Lemma 2.2 we compute ⋆ F and d ⋆ F for F being of the form (1.2). We obtain', 'arxivid': '2110.10084', 'author': ['Hanci Chi ', 'Ioannis Chrysikos ', 'Eivind Schneider ', 'Hanci Chi ', 'Ioannis Chrysikos ', 'Eivind Schneider '], 'authoraffiliation': [], 'corpusid': 119333091, 'doi': '10.1088/1361-6382/ab0615', 'github_urls': [], 'n_tokens_mistral': 35006, 'n_tokens_neox': 29309, 'n_words': 16680, 'pdfsha': '6103b1a192ca9cd4cfd0e1c1e016340b028fbf72', 'pdfurls': ['https://arxiv.org/pdf/2110.10084v1.pdf'], 'title': ['DECOMPOSABLE (5, 6)-SOLUTIONS IN ELEVEN-DIMENSIONAL SUPERGRAVITY', 'DECOMPOSABLE (5, 6)-SOLUTIONS IN ELEVEN-DIMENSIONAL SUPERGRAVITY', 'DECOMPOSABLE (5, 6)-SOLUTIONS IN ELEVEN-DIMENSIONAL SUPERGRAVITY', 'DECOMPOSABLE (5, 6)-SOLUTIONS IN ELEVEN-DIMENSIONAL SUPERGRAVITY'], 'venue': []}

arxiv

New methodology to design advanced MR-IR- UWB communication system A Lecointre D Dragomirescu R Plana New methodology to design advanced MR-IR- UWB communication system -1- A new model is proposed giving the channel capability of a MB-IR-UWB system versus the number of subband and the duty cycle. The architecture simulated shows data rate ranging from 1,434 Gbits/s to 0.9 Gbits/s for 16 to 10 subbands and duty cycle ranging from 20% to 12%.Introduction: simulated shows data rate ranging from 1,434 Gbits/s to 0.9 Gbits/s for 16 to 10 subbands and duty cycle ranging from 20% to 12%. Introduction: In the field of wireless sensor, one issue deals with the availability of wireless architectures featuring optimized performances in terms of power consumption and bit rate and lowering interference issues that has motivated the emergence of impulse radio ultra wideband medium (IR-UWB) [ (1/α max -1) / d spread <= Bs. (7) Where n is the number of subbands which maximizes the channel capacity, d spread is the delay spread of the considered channel, Bs is the number of subbands for optimizing data rate and Bt is the entire bandwidth used by the IR-UWB system. The duty cycle that is a good indicator of the power consumption is defined by the following expression : α = T p / (T p + d spread )(8) Equations (3) model is proposed giving the channel capability of a MB-IR-UWB system versus the number of subband and the duty cycle. The architecture Fig. 1 = n x ( 1 11shows the IR-UWB channel capacity for binary modulation versus bandwidth in the case of channel delay spread of 10, 50, and 100 ns with a required SNR greater than 3 dB.Fig. 1illustrates the existence of a delay spread asymptote at 1/d spread which limits the IR-UWB channel capacity. Near to the delay spread limitation, a bandwidth increase has a very low impact that motivates the use of multiband impulse ultra wide band technique.-3-MB-IR-UWB channel capacity optimization: Most of the time, MB-IR-UWB implementations, such as MB-OOK [5], use 500 MHz subbands and explain this bandwidth by the difficulty to generate an UWB pulse over 7.5 GHz, the reduction of data converter performance requirements, or the good behavior regarding flexibility for adaptation to local regulation. This widely used subband bandwidth of 500 MHz can also be explained by the UWB definition of the Federal Communications Commission (FCC), and by the implementation hardware constraints which limit the number of subbands and thus their size. However with the progress of the technology and continuous cost reduction, it could be possible to investigate different architectures and then opening some degree of freedom for designers that were not authorized before. Assuming implementation limitations, such as the maximum number of subbands n max (which is imposed by the number of mixers, the complexity and cost of the transceiver) , the maximum bandwidth of a subband Bs max (which depends on the pulse generators and data converters performances), the total available bandwidth for the system Bt max (which is imposed by the local regulation) and the maximum authorized duty cycle α max of this discontinuous emission technique (IR-UWB), we express the bandwidth of each subband Bs and the number of subbands n by an optimization problem approach of the MB-IR-UWB capacity: max [C MB-IR-UWB (Bs,n)] Figure captions :Fig. 2 captions2Fig. 1 IR-UWB channel capacity versus bandwidth (required SNR > Simulations of the MB-IR-UWB channel capacity (2.b), duty cycle (2.b), number of subband (2.a) and subband channel capacity (2.a) versus subband bandwith using the model developed. Figure 1 Theoretical background: In a low power and consequently low complexity context, IR-UWB techniques are suited as physical layer[1][2]. Assuming a receiver without equalization and binary modulation schemes to meet both simplicity of implementation, high data rate and low power consumption, the IR-UWB symbol duration is determined by the addition of the channel delay1] and more recently multiband IR-UWB (MB-IR-UWB) [2] aiming to have reconfigurable and flexible architecture that could be adapted following the application. However despite their attractive capabilities already demonstrated, there is a lack concerning a methodology aiming to design a MB-IR-UWB system under different requirements. This paper proposes an original approach to determine system design rules that will be useful to implement advanced communication architectures for wireless sensor network. -2- spread and the pulse duration [3]. From the Shannon relation defining the channel capacity C of an ideal band-limited channel B with additive white Gaussian noise (AWGN) interference [4] C(bits/s) = B.log 2 (1+P s /(B.N 0 )) = B.log 2 (1+SNR) (1) where P s is the signal power, SNR is the signal to noise ratio, and N 0 the noise spectral density, we can obtain the IR-UWB channel capacity. The expression of the latter includes parameters such as delay spread for dealing with time variant multipath channel. Since no inter-symbol interference (ISI) is assumed (to avoid equalization [3]) and binary modulation is imposed, the IR- UWB channel capacity can be expressed by only considering the temporal resolution defined by the channel delay spread d spread , and the IR-UWB pulse duration T p : C IR-UWB (bits/s) = 1 / ( T p + d spread ) (2). to (7) indicate that implementation and channel constraints impact the achievable data rate of the MB-IR-UWB system. The optimization problem has a solution only if implementation constraints permit an achievable value for Bs. Bs must be located between Bs min and Bs max . Bs min is defined by n max , α max , d spread , and Bt:If Bs max is greater than Bs min then the problem has a solution. The value of Bs, which optimizes the MB-IR-UWB capacity under implementation constraints, is reached when the number of subband is the highest and when Bs is maximum for this optimized number of subbands.Bs min = max [ Bt / n max ; (1/α max -1) / d spread ] (9) Using the model and the methodology developed, we have performed simulations and the results are summarized in Figure 2. For these simulations, we have used a channel delay spread of 9 ns, a total available bandwidth Bt max of 7.5 GHz (FCC regulation), a Bs max of 750 MHz, α max equal to 20%, and n max equal to 30. Bs max, α max, n max, are arbitrary chosen here to accommodate the hardware implementation constraints. This channel delay spread is extracted from the IEEE 802.15.4a UWB channel model in the case of industrial line of sight environment [6]. First of all, we can observe that the MB-IR-UWB capacity ranges from 1.434 Gbits/s to 0.967 Gbits/s when the subband bandwith varies from 444,4 MHz to 750 MHz. It has to be outlines that the maximum data rate (1434 Mbits/s) is achieved for the largest number of subband (16) and the locally largest subband bandwidth (464 Mhz). This maximum corresponds also to the locally lowest power consumption (since the duty cycle is at its locally lowest value 19.3 %). Our model also provides the subband channel capacity that varies from 88 Mbits/s to 97Mbits/s when the subband ranges from 444,4MHz to 750 MHz. Finally, the simulations give indication on the duty cycle that is obtained versus the subband bandwidth ranging from 20% to 12% versus subband bandwidth. Perspectives and conclusions: This paper proposes an original approach to calculate the capabilities of MB-IR-UWB channel and to exploit this approach to design the future communication architecture that will be implemented within wireless sensor network where it is often necessary to accommodate both high data rate and low power consumption. The model proposes new design rules involving channel capacity, duty cycle versus number of subband and subband bandwidth that allow designing a MB-IR-UWB architecture featuring data rate ranging from 1.434 Gb/s to 1 Gb/s with number of subband ranging from 16 to 10 and duty cycle varying from 20% to 12% over a 9ns delay spread UWB channel. This open the way of a new generation of software radio with enhanced capabilities. AIELLO G. R., and ROGERSON G. D.: 'Ultra-Wideband Wireless Systems', IEEE Microwave Magazine 2003, vol. 4, issue 2. 3 PAQUELET, S.. AUBERT, L-M. and UGUEN, B. : 'An Impulse Radio Asynchronous Transceiver for High Data Rates', IEEE Joint UWBST & IWUWBS 2004, Conference on Ultra Wideband Systems and Technologies, International Workshop on UWB Systems. 4 PROAKIS, J. G. : 'Digital Communications', McGraw-Hill, Fourth Edition, 2001, pp. 13-15, pp. 386-387. 5 AUBERT, L-M.: 'Mise en place d'une couche physique pour les futures systèmes de radiocommunications hauts débits UWB', Thesis of IETR of Rennes, France, 2005. 6 MOLISCH, A. F. et al. : 'IEEE 802.15.4a channel model -final report'. IEEE 802.15.4A Task Group. Authors' affiliations:A. Lecointre, D. Dragomirescu and R. Plana (University of Toulouse, LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse Cedex 4, FRANCE) alecoint@laas.fr, daniela@laas.fr, plana@laas.fr. UWB Theory and Applications. I Oppermann, M Hamalainen, Linatti , J , WileyOPPERMANN, I., HAMALAINEN, M., and LINATTI, J.: 'UWB Theory and Applications', Wiley, 2004, pp. 1-8.

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{'abstract': 'A new model is proposed giving the channel capability of a MB-IR-UWB system versus the number of subband and the duty cycle. The architecture simulated shows data rate ranging from 1,434 Gbits/s to 0.9 Gbits/s for 16 to 10 subbands and duty cycle ranging from 20% to 12%.Introduction:', 'arxivid': '1002.0576', 'author': ['A Lecointre ', 'D Dragomirescu ', 'R Plana '], 'authoraffiliation': [], 'corpusid': 953932, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2742, 'n_tokens_neox': 2372, 'n_words': 1461, 'pdfsha': 'ef6d534bffdd28909775a5eda11dfa81ca66b200', 'pdfurls': ['https://arxiv.org/pdf/1002.0576v1.pdf'], 'title': ['New methodology to design advanced MR-IR- UWB communication system', 'New methodology to design advanced MR-IR- UWB communication system'], 'venue': []}

arxiv

High-Order Mesh Morphing for Boundary and Interface Fitting to Implicit Geometries Jorge-Luis Barrera Tzanio Kolev Ketan Mittal Vladimir Tomov High-Order Mesh Morphing for Boundary and Interface Fitting to Implicit Geometries High-orderImplicit meshingMesh morphingr-adaptivityFinite elementsTMOP We propose a method that morphs high-orger meshes such that their boundaries and interfaces coincide/align with implicitly defined geometries. Our focus is particularly on the case when the target surface is prescribed as the zero isocontour of a smooth discrete function.Common examples of this scenario include using level set functions to represent material interfaces in multimaterial configurations, and evolving geometries in shape and topology optimization. The proposed method formulates the mesh optimization problem as a variational minimization of the sum of a chosen mesh-quality metric using the Target-Matrix Optimization Paradigm (TMOP) and a penalty term that weakly forces the selected faces of the mesh to align with the target surface. The distinct features of the method are use of a source mesh to represent the level set function with sufficient accuracy, and adaptive strategies for setting the penalization weight and selecting the faces of the mesh to be fit to the target isocontour of the level set field. We demonstrate that the proposed method is robust for generating boundary-and interface-fitted meshes for curvilinear domains using different element types in 2D and 3D. racy and favorable scaling on modern architectures [1,2,3,4]. A vital component of these methods is high-order computational meshes for discretizing the geometry. Such meshes are essential for achieving optimal convergence rates on domains with curved boundaries/interfaces, symmetry preservation, and alignment with the key features of the flow in moving mesh simulations [5,6,7]. To fully benefit from high-order geometry representation, however, one must be able to control the quality and adapt the properties of a high-order mesh. Two common requirements for mesh optimization methods are (i) to fit certain mesh faces to a given surface representation, and (ii) to perform tangential node movement along a mesh surface. This paper is concerned with these two requirements, in the particular case when the surface representation is a discrete (or implicit) function. Common examples of this scenario include use of level set functions to represent curvilinear domains as a combination of geometric primitives in Constructive Solid Geometry (CSG) [8], material interfaces in multimaterial configurations [9], and evolving geometries in shape and topology optimization [10,11,12], amongst other applications. Boundary conforming high-order meshes are typically generated by starting with a conforming linear mesh that is projected to a higher order space before the mesh faces are curved to fit the boundary [13,14,15,16,17,18,19]. In context of boundary fitting, a closely related work is Rangarajan's method for tetrahedral meshes [20] where an immersed linear mesh is trimmed, projected to a surface defined using a point-cloud, and smoothed to generate a high-order boundary fitted mesh. Other related approaches are Mittal's distance function-based approach for tangential relaxation during optimization of initially fitted highorder meshes [21] and the DistMesh algorithm where Delaunay triangulations are aligned to implicitly defined domains using force balance. Note that we are interested in generating boundary fitted meshes using mesh morphing because it offers a way to use existing finite element framework for adapting meshes to the problem of interest. An alternative is classic mesh generation, which is usually a pre-processing step, and has been an area of interest to the meshing community for decades; summarizing different mesh generation techniques is beyond the scope of this work and the reader is referred to [22,23,24] for a review on this topic. For generating interface fitted meshes, existing methods mainly rely on topological operations where the input mesh is split across the interface to generate an interface conforming mesh [25,26]. Some exceptions are Ohtake's method for adapting linear triangular surface meshes to align with domains with sharp features [27] and Le Goff's method for aligning meshes to interfaces prescribed implicitly using volume fractions [28]. Barring [29,20,30,27,28], existing methods mainly rely on an initial conforming meshing for boundary fitting and on topological operations such as splitting for interface fitting. We propose a boundary and interface fitting method for high-order meshes that is algebraic and seldom requires topological operations, extends to different element types (quadrilaterals/triangles/hexahedra/tetrahedra) in 2D and 3D, and works for implicit parameterization of the target surface using discrete finite element (FE) functions. We formulate the implicit meshing challenge as a mesh optimization problem where the objective function is the sum of a chosen mesh-quality metric using the Target-Matrix Optimization Paradigm (TMOP) [31,32] and a penalty term that weakly forces nodes of selected faces of a mesh to align with the target surface prescribed as the zero level set of a discrete function. Additionally, we use an adaptive strategy to choose element faces/edges for alignment/fitting and set the penalization weight, to ensure robustness of the method for nontrivial curvilinear boundaries/interfaces. We also introduce the notion of a source mesh that can be used to accurately represent the level set with sufficient accuracy. This mesh is decoupled from the mesh being optimized, which allows to represent the domain of interest with higher level of detail. This approach is crucial for cases where the target boundary is beyond the domain of the mesh being optimized or the input mesh does not have sufficient resolution around the zero level set. The remainder of the paper is organized as follows. In Section 2 we review the basic TMOP components and our framework to represent and optimize high-order meshes. The technical details of the proposed method for surface fitting and tangential relaxation are described in Section 3. Section 4 presents several academic tests that demonstrate the main features of the methods, followed by conclusions and direction for future work in Section 5. Preliminaries In this section, we describe the key notation and our prior work that is relevant for understanding our newly developed boundary-and interface-fitting method. Discrete Mesh Representation In our finite element based framework, the domain Ω ∈ R d , d = {2, 3}, is discretized as a union of N E curved mesh elements, each of order p. To obtain a discrete representation of these elements, we select a set of scalar basis functions {w i } Np i=1 , on the reference element E. For example, for tensor-based elements (quadrilaterals in 2D, hexahedra in 3D), we have N p = (p + 1) d , and the basis spans the space of all polynomials of degree at most p in each variable, denoted by Q p . These pth-order basis functions are typically chosen to be Lagrange interpolation polynomials at the Gauss-Lobatto nodes of the reference element. The position of an element E in the mesh M is fully described by a matrix x E of size d × N p whose columns represent the coordinates of the element control points (also known as nodes or element degrees of freedom). Given x E and the positionsx of the reference elementĒ, we introduce the map Φ E :Ē → R d whose image is the geometry of the physical element E: x(x) = Φ E (x) ≡ Np i=1 x E,iwi (x),x ∈Ē, x = x(x) ∈ E,(1) where x E,i denotes the i-th column of x E , i.e., the i-th node of element E. Throughout the manuscript, x will denote the position function defined by (1), while bold x will denote the global vector of all node coordinates. Geometric Optimization and Simulation-Based r−adaptivity with TMOP The input of TMOP is the user-specified transformation matrix W , from reference-spacē E to target element E t , which represents the ideal geometric properties desired for every mesh point. Note that after discretization, there will be multiple input transformation matrices W -one for every quadrature point in every mesh element. The construction of this transformation is guided by the fact that any Jacobian matrix can be written as a composition of four geometric components: W = ζ [volume] R [rotation] Q [skewness] D [aspect ratio] . (2) A detailed discussion on the construction of matrices associated with these geometric components and on how TMOP's target construction methods encode geometric information into the target matrix W is given by Knupp in [33]. Various examples of target construction for different mesh adaptivity goals are given in [31,34,35]. Using (1), the Jacobian of the mapping Φ E at any reference pointx ∈Ē from the reference-space coordinates to the current physical-space coordinates is A(x) = ∂Φ E ∂x = Nw i=1 x E,i [∇w i (x)] T .(3) In this manuscript, we assume that all the elements in the initial mesh are not inverted, i.e. det(A) > 0 ∀x ∈Ē. Combining (3) and (2), the transformation from the target coordinates to the current physical coordinates (see Fig. 1) is T = AW −1 .(4) With the target transformation W defined in the domain, we next specify a mesh quality metric µ(T ) that compares the transformations A and W in terms of the geometric parameters of interest. For example, µ 2 = |T | 2 2τ − 1 is a shape metric 3 that depends on the skewness and aspect ratio components, but is invariant to orientation/rotation and volume. Here, |T | and τ are the Frobenius norm and determinant of T , respectively. Similarly, µ 77 = 1 2 (τ − 1 τ ) 2 is a size metric that depends only on the volume of the element. We also have shape+size metrics such as µ 80 = γµ 2 + (1 − γ)µ 77 that depend on volume, skewness and aspect ratio, but are invariant to orientation/rotation. Note that the mesh quality metrics are defined such that they evaluate to 0 for an identity transformation, i.e. µ(T ) = 0 when T = I (A = W ). This allows us to pose the mesh optimization problem as minimization of µ(T ), amongst other advantages [36]. The quality metric µ(T ) is used to define the TMOP objective function for r−adaptivity F (x) = E∈M F E (x E ) = E∈M Et µ(T (x))dx t ,(5) where F is a sum of the TMOP objective function for each element in the mesh (F E ), and E t is the target element corresponding to the element E. In (5), the integral is computed as E∈M Et µ(T (x t ))dx t = E∈M xq∈Et w q det(W (x q )) µ(T (x q )),(6) where M is the current mesh with N E elements, w q is the quadrature weight, and the position x q is the image of the reference quadrature point locationx q in the target element E t . Optimal node locations are determined by minimizing the TMOP objective function (5). This is performed by solving ∂F (x)/∂x = 0 using the Newton's method where we improve the mesh degrees-of-freedom (nodal positions) as x k+1 = x k − αH −1 (x k )J (x k ).(7) Here, x k refers to the nodal positions at the k-th Newton iteration during r−adaptivity, α is a scalar determined by a line-search procedure, and H(x k ) and J (x k ) are the Hessian (∂ 2 F (x k )/∂x j ∂x i ) and the gradient (∂F (x k )/∂x i ), respectively, associated with the TMOP objective function. The line-search procedure requires that α is chosen such that (MINRES) method with Jacobi preconditioning; more sophisticated preconditioning techniques can be found in [38]. Additionally, the optimization solver iterations (7) are done until the relative l 2 norm of the gradient of the objective function with respect to the current and original mesh nodal positions is below a certain tolerance ε, i.e., |J (x)|/|J (x 0 )| ≤ ε. We set ε = 10 −10 for the results presented in the current work. F (x k+1 ) < 1.2F (x k ), |J (x k+1 )| < 1.2|J (x k )|, Using the approach described in this section, we have demonstrated r−adaptivity with TMOP for geometry and simulation-driven optimization; see Fig. 2 for example of highorder mesh optimization for a turbine blade using W = I with a shape metric. The resulting optimized mesh has elements closer to unity aspect ratio and skewness closer to π/2 radians in comparison to the original mesh, as prescribed by the target W = I. Boundary & Interface Fitting Our goal for boundary and interface fitting is to enable alignment of a selected set of mesh nodes to some target surface of interest prescribed as the zero isocontour of a smooth signeddiscrete level set function, σ(x). interface that is to be aligned to the circular interface. To effect alignment with the zero isocontour of σ(x), we modify the TMOP objective function (5) as: Here, F σ is a penalty-type term that depends on the penalization weight w σ , the set of nodes S to be aligned to the level set (e.g., the mesh nodes discretizing the material interface in F (x) = E∈M Et µ(T (x))dx t Fµ + w σ s∈S σ 2 (x s ) Fσ . (8) (a) (b) (c)= F µ + F σ , produces a balance between mesh quality and surface fitting. Note that all nodes are treated together, i.e., the nonlinear solver makes no explicit separation between volume nodes of M and the nodes s ∈ S ⊆ M set for fitting. Furthermore, as there is no pre-determined unique target position for each node of S, the method naturally allows tangential relaxation along the interface of interest, so that mesh quality can be improved while maintaining a good fit to the surface. Figure 3(c) shows an example of a triangular mesh fit to a circular interface using (8). In this example, we use the shape metric µ 2 with equilateral targets and a constant penalization weight, w σ = 10 3 . The first step in our method is to use a suitable strategy for representing the level set function with sufficient accuracy (Section 3.1). Next, we determine the set of nodes S that will be aligned to the zero level set of σ. For boundary fitting, S depends on the elements located on the boundary of interest. For interface fitting, S is the set of nodes shared between elements with different fictitious material indicators, and we describe our approach for setting the material indicators of elements in Section 3.2. Finally, we set the penalization weight w σ such that an adequate fit is achieved to the target surface while optimizing the mesh with respect to the quality metric µ (Section 3.3). The adaptive strategy for setting w σ requires us to modify the line-search and convergence criterion of the Newton's method (Section 3.4). For completeness, the derivatives of F σ are discussed in Section 3.5. Using various examples, we demonstrate in Section 4 that our method extends to both simplices and hexahedrals/quadrilaterals of any order, and is robust in adapting a mesh interface and/or boundary to nontrivial curvilinear shapes. Level Set Representation The following discussion is related to the case when σ(x) is a discrete function so that its values and derivatives can't be computed analytically. Then the first step in our implicit meshing framework is to ensure that the level set function σ(x) is defined with sufficient accuracy. A drawback of discretizing σ(x) on the mesh being optimized, M, is that the resulting mesh will be sub-optimal in terms of mesh quality and interface/boundary fit if (1). Since M B is independent of M, we can choose M B based on the desired accuracy; the maximum error in representing the implicit geometry discretely is bounded by the element size of the background mesh at the location of the zero level set. Thus, we use adaptive nonconforming mesh refinements [39] around the zero level set of σ(x B ), as shown in Fig. 4(b). Using a source mesh for σ(x B ), however, requires transfer of the level set function and its derivatives from M B to the nodes S ∈ M prior to each Newton iteration. This transfer between the source mesh and the current mesh is done using gslib, a high-order interpolation library [40]: σ(x) = I(x, x B , σ(x B )), ∂σ(x) = I(x, x B , ∂σ(x B )), ∂ 2 σ(x) = I(x, x B , ∂ 2 σ(x B )),(9) where I represent the interpolation operator that depends on the current mesh nodes (x), the source mesh nodes (x B ), and the source function σ(x B ) or its gradients. A detailed description of how high-order functions can be transferred from a mesh to an arbitrary set of points in physical space using gslib is described in Section 2.3 of [41]. While surfaces such as the circular interface in Fig. 3 and sinusoidal boundary in Fig. 4 are straightforward to define using smooth level-set functions, more intricate domains are typically defined as a combination of non-smooth functions for different geometric primitives, which are not well suited for our penalization-based formulation (8). Consider for example a two-material application problem in Fig. 5(a) where the domain is prescribed as a combina- tion of geometric primitives for a circle, rectangle, parabola, and a trapezium. The resulting step function G(x B ) is 1 inside one material and -1 inside the other material. For such cases, we start with a coarse background mesh and use adaptive mesh refinement around the zero level set of G(x B ), see Fig 5(b). Then we compute a discrete distance function using the p-Laplacian solver of [42], Section 7, from the zero level set of G(x B ), see Fig 5(c). The advantage of using this distance function in (8) is that it is (i) generally smoother and (ii) maintains the location of the zero level set. (a) (b) (c) Setting S for Interface Fitting For interface fitting, the set S contains the nodes that are used to discretize the material interface in the mesh. As demonstrated in this section, the accuracy of the fitting to the target surface using (8) depends on the combination of the (i) mesh topology around the interface, and (ii) the shape of the target surface. Thus, for a given initial mesh and implicit interface, it is important to choose the fictious material indicator of elements such that the resulting material interface is compatible with the target surface. Consider for example the triangular mesh shown in Figure 6(a) which must be aligned to a circular level set. A naive approach for setting the material indicators is to partition the mesh into two fictitious materials based on the level-set function sampled at the set Q of quadrature points inside each element. For example, the material indicator η E for element E can be set using the integral of σ(x) as: η E =      0, if E σ(x) ≥ 0 1, otherwise ,(10) or using the maximum norm of σ(x) as: η E =      0, if max q∈Q |σ(x q )| ≥ 0 1, otherwise. .(11) Using such approaches can lead to elements that have multiple adjacent faces as a part of the material interface of the mesh (highlighted in red). When the mesh deforms for fitting, this results in sub-optimal Jacobians in the elements at the vertex shared by the (highlighted) adjacent faces. This is evident in Figure 6(b) where the material indicator is set using (10). The fundamental issue here is that whenever adjacent faces of an element are aligned to a level set, the resulting mesh quality and fit can be sub-optimal depending on the complexity of the target shape/geometry. To address this issue, we update the material indicator for each element in the mesh as: (a) (b) (c) (d)η E =              η E , if N E,M <= 1 1 − η E , else if N E,M = N E,F − 1 η E , (optionally) mark E for splitting otherwise ,(12) where N E,M is the number of faces of element E that are part of the material interface, and N E,F is the total number of faces of element E. Note that (12) is formulated as a two pass approach where we first loop over all elements with η E = 0 and then with η E = 1. This approach ensures that conflicting decisions are not made for adjacent elements surrounding the interface. The first condition in (12) increased number of degrees-of-freedom, but lead to a better mesh quality. Figure 7 shows an example of a comparison of a quadrilateral mesh fit to the circular level set using (10) and (12). Note that conforming splits can also be done for triangles, if needed, by connecting each of its vertex to the centroid of the triangle. For tetrahedra and hexahedra, conforming splits independent of adjacent elements are not yet possible, and we are currently exploring nonconforming refinement strategies. Nonetheless, if such a situation arises where multiple adjacent faces of an element are marked for fitting, the proposed method will still align the mesh the best it can under the constraint of a prescribed threshold on minimum Jacobian in the mesh (15). Adaptive Penalization Weight Recall that the balance between mesh quality and node fitting is controlled by the penalization weight w σ in (8). Numerical experiments show that use of a constant w σ requires tweaking on a case-by-case basis, and can result in a sub-optimal fit if w σ is too small, in which case the objective function is dominated by the mesh quality metric term, or if w σ is too large, in which case the conditioning of the Hessian matrix is poor. Figure 8 demon- Figure 8 shows that as we increase w σ from 1 to 10 4 , the fitting error decreases. However, the error worsens if we increase w σ further. To address this issue we use an adaptive approach for setting w σ where we monitor |σ| S,∞ , and scale w σ by a user-defined constant (α σ = 10 by default) if the relative decrease in the maximum nodal fitting error between subsequent Newton iterations is below a prescribed threshold ( ∆σ = 0.001 by default). That is, w k+1,σ =      α σ · w k,σ if |σ| k,S,∞ −|σ| k+1,S,∞ |σ| k+1,S,∞ < ∆σ w k,σ otherwise ,(13) where we use the subscript k to denote a quantity at the kth Newton iteration. Figure 8 shows that this adaptive approach for setting w σ significantly improves the quality of the mesh fit to the desired level set. Updating the value of w σ changes the definition of the objective function (8), which requires some modifications of the line search and convergence criterion of the Newton iterations to achieve overall convergence, compared to the constant w σ case; details will follow in Section 3.4. Nevertheless, our numerical tests suggest that this impact is negligent in comparison to the improvement of the fitting error. Convergence & Line-Search Criterion Recall that in the general TMOP approach, the line-search and convergence criteria for the Newton's method are based on the magnitude and the derivatives of the objective function, see Section 2.2. In the penalization-based formulation (8), the current criteria do not suffice because the magnitude and derivatives of the objective function depend on the penalization weight w k,σ , which can change between subsequent Newton iterations due to (13). We modify our line-search criteria by adding two additional inequalities, namely, α in (7) is chosen to ensure: |σ| k+1,S,∞ < 1.2 |σ| k,S,∞ ,(14) min det(A(x k+1 )) > 0.001 · min det(A(x 0 )) . The inequality (14) prevents sudden jumps of the fitting error, and the scaling factor 1.2 has been chosen empirically. The constraint (15) is mostly applicable in regimes when w σ is big enough to make the quality term F µ effectively inactive. Such regimes represent a situation when one is willing to sacrifice mesh quality for more accurate fitting. When the quality term F µ is relatively small, it may be unable to prevent the appearance of infinitesimally small positive Jacobians; the constraint (15) is used to alleviate this situation. The convergence criterion is also modified to utilize the fitting error, i.e., the Newton's method is used until |σ| S,∞ is below a certain user-specified threshold ( σ = 10 −5 by default). We also use an optional convergence criterion based on a limit on the maximum number of consecutive Newton iterations through which the penalization weight w σ is adapted using (13); this limit is N σ = 10 by default. This latter criterion avoids excessive computations in cases where the mesh topology does not allow the fitting error to reduce beyond a certain limit. Derivatives As our default choice for nonlinear optimization is the Newton's method, we must compute first and second order derivatives of F µ and F σ with respect to the mesh nodes. The definition of the derivatives of F µ is given in Sections 3.3 and 3.4 of [43], and here we focus ∂F σ (x) ∂x a,i = 2ω σ s∈S σ(x s ) ∂σ(x s ) ∂x a ∂x a (x s ) ∂x a,i = 2ω σ s∈S σ(x s ) ∂σ(x s ) ∂x aw i (x s ), ∂ 2 F σ (x) ∂x b,j ∂x a,i = 2ω σ s∈S ∂σ(x s ) ∂x b ∂σ(x s ) ∂x a + 2ω σ ∂ 2 σ(x s ) ∂x b ∂x a w i (x s )w j (x s ), a, b = 1 . . . d, i, j = 1 . . . N x .(16) The above formulas require the spatial gradients of σ at the current positions {x s } s∈S of the marked nodes. These gradients can be closed-form expressions, when σ is prescribed analytically, or the gradients are obtained from the background mesh M B (see Section 3.1), when σ is a discrete function. Algorithm/Summary In this section, we summarize our penalization-based method for boundary and interface fitting with TMOP. The inputs to our method are the active/current mesh M that is defined through the global vector x of nodal positions, a user-selected target construction option to form W as in (2), a mesh quality metric µ, a source/background mesh with nodal coordinates x B along with the level set function σ(x B ) as explained in Section 3.1, initial penalization weight w σ , parameters for adaptive penalization-weight (α σ and ∆σ as in Section 3.3), and parameters for convergence criterion σ and N σ as in Section 3.4. Algorithm 1 summarizes our penalization-based method where we use subscript k = 0 . . . N opt to denote different quantities at the kth Newton iteration. Algorithm 1: Implicit Meshing Input: x, µ, x B , σ(x B ), α σ , ∆σ , w σ , σ , N opt Output: x s 1 n σ := 0, k = 0 ,x 0 = x 2 Determine S, the set of nodes for fitting (Section 3.2) 3 σ(x 0 ) = I(x 0 , x B , σ(x B )) 4 while |σ| k,S,∞ > σ && n σ < N σ && k < N opt do 5 W i = I for each quadrature point i [33]. Numerical Results In this section, we demonstrate the main properties of the method using several examples. The presented tests use W = I as the target matrix, and the following shape metrics: µ 2 = |T | 2 2τ − 1, µ 303 = |T | 2 3τ 2/3 − 1,(17) where |T | and τ are the Frobenius norm and determinant of T , respectively. Both metrics are polyconvex in the sense of [44,45], i.e., the metric integral F µ in (5) theoretically has a minimizer. Exploring the effect of the smoothness of σ(x) on the convexity of the objective function (8) will be the subject of future studies. Our implementation utilizes MFEM, a finite element library that supports arbitrarily high-order spaces and meshes [3]. This implementation is freely available at https://mfem. org. Fitting to a Spherical Interface As a proof of concept, we adapt 3rd order hexahedral (hex) and tetrahedral (tet) meshes to align to a spherical surface; see Figure 9 tetrahedral mesh, the minimum Jacobian decreases from 4 × 10 −3 to 4.6 × 10 −6 and in the hexahedral mesh, the minimum Jacobian decreases from 1.2×10 −3 to 1.5×10 −5 during mesh optimization as the mesh deforms to align to the target surface; in both cases the minimum Jacobian appears near the fitted faces. The decrease in the Jacobian at the interface is expected, with a lower bound on the Jacobian imposed by (15) to ensure that the mesh stays valid for desired finite element computations. Note that the hexahedral mesh requires more iterations as compared to the tetrahedral mesh partly because there are multiple elements with adjacent faces marked for fitting, which requires more work from the adaptive weight mechanism to force those mesh faces to align with the spherical interface. Boundary and Interface Fitting for Geometric Primitive-Based Domains in 3D This example demonstrates that the proposed fitting method can be applied not only to internal interfaces, but also to high-order boundaries. The target domain is represented as a combination of simple geometric primitives, namely, an intersection of a cube (side=0.5, centered around x c = (0.5, 0.5, 0.5)) with a sphere (radius=0.3, centered around x c ) that has three Cartesian-aligned cylinders (radius=0.15, length=0.5) removed from it. Figure 10 shows the CSG tree with geometric primitives and the Boolean operations that are used to construct the target geometry. Using the approach outlined in Section 3.1, we combine the geometric primitives on a third-order source mesh that is adaptively refined five times around the zero level set of the target domain. We then compute the distance function on this background mesh using the p−Laplacian solver [42]. This distance function is used as the level set function in (8). Our numerical experiments showed that using a third order source mesh with five adaptive refinements was computationally cheaper than using a lower order mesh with more adaptive refinements to obtain the same level of accuracy for capturing the the curvilinear boundary using the distance function. The input mesh to be fit for this problem is a uniform second-order 128 × 128 × 128 Cartesian-aligned mesh for Ω ∈ [0, 1] 3 . This mesh is trimmed by using (10) and removing all elements with η E = 0. The minimum Jacobian in the trimmed mesh is 4.8 × 10 −7 . The trimmed mesh is optimized using (8) where the nodes on the boundary are set for alignment. colored golden, with the mesh morphed to align to the indents highlighted at the interface. Interface Fitting for Shape Optimization Application This example serves to demonstrate the applicability of the proposed approach to setup the initial multimaterial domain to be used in a shape optimization problem. Figure 14 shows the cross section of a tubular reactor that consists of a highly conductive metal (red ) and a low conductivity heat generating region (blue). The design optimization problem is formulated such that the energy production in the system is maximized while keeping the overall volume of the red region constant. To achieve this, the shape of the red subdomain is morphed using a gradient-based approach in our in-house design optimization framework that requires an interface fitted mesh as an input. This initial fitted mesh is generated using the method described in this manuscript. Exploiting the cyclic symmetry, we discretize a portion of the domain via a triangular mesh. Since this initial mesh does not need to align with the material interface, it can be conveniently generated using an automatic mesh generator [46]. We then use an approach similar to the previous section for interface fitting where the multimaterial domain is realized as a combination of geometric primitives (annulus, parabola, and trapezium, as highlighted in Figure 14). The finite element distance function from the interface is computed using the p-Laplacian solver of [42], Section 7, and used as the level-set function σ(x) in (8). high quality interface fitted meshes with minimal user intervention, and is currently being used for similar 3D shape optimization applications that will be presented in future work. Conclusion & Future Work We have presented a novel method to morph and align high-order meshes to the domain of interest. We formulate the mesh optimization problem as a variational minimization of the sum of a chosen mesh-quality metric and a penalty term that weakly forces the selected faces of the mesh to align with the target surface. The penalty-based formulation makes the proposed method suitable for adoption in existing mesh optimization frameworks. There are three key features of the proposed method that enable its robustness. First, a source mesh is used to represent the level set function with sufficient accuracy when the mesh being morphed does not have enough resolution or is beyond the target surface (Section 3.1). Second, an adaptive approach is proposed for setting the fictitious material indicators in the mesh to ensure that the resulting material interface can align to the target surface for interface fitting (Section 3.2). Finally, an adaptive approach for setting the penalization weight is developed to eliminate the need for tuning the penalization weight on a case-bycase basis (Section 3.3). Numerical experiments demonstrate that the proposed method is effective for generating boundary-and interface-fitted meshes for nontrivial curvilinear geometries. In future work, we will improve the method by developing mesh refinement strategies for hexahedral and tetrahedral meshes, which are required when the mesh topology limits the fit of a mesh to the target surface (Section 3.2). We will also explore ways for aligning meshes to domains with sharp features [27,47], as we currently assume that the level set function σ used in (8) is sufficiently smooth around its zero level set. Finally, we will also look to optimize our method and leverage accelerator-based architectures by utilizing partial assembly and matrix-free finite element calculations [43]. Figure 1 : 1Schematic representation of the major TMOP matrices. Figure 2 : 2and that the determinant of the Jacobian of the transformation at each quadrature point in the mesh is positive, det(A(x k+1 )) > 0. These line-search constraints have been tuned using many numerical experiments and have demonstrated to be effective in improving mesh quality. For Newton's method, we solve the problem H(x k )∆x = J (x k ) using a Krylov subspace method such as the Minimum Residual (a) Original and (b) optimized 4th order meshes for a turbine blade. Figure 3 ( 3a) and (b) show a simple example of a circular interface represented using a level set function and a triangular mesh with multimaterial Figure 3 : 3(a) Level set function σ(x), (b) a Cartesian mesh with material interface nodes to be aligned to the zero level set of σ(x), and (c) the optimized mesh. Fig. 3 ( 3b)), and the level set function σ(x), evaluated at the positions x s of all nodes s ∈ S. The F σ term penalizes the nonzero values of σ(x s ) for all s ∈ S. Minimizing this term represents weak enforcement of σ(x s ) = 0, only for the nodes in S, while ignoring the values of σ for the nodes outside S. Minimizing the full nonlinear objective function, F Figure 4 : 4(a) the mesh does not have enough resolution to represent σ(x) with sufficient accuracy, especially near the zero level set, or (b) the target level set is outside the initial domain of the mesh. For the latter, it's impossible to compute the necessary values and derivatives of the level set function accurately at the boundary nodes that we wish to fit, once the mesh moves outside the initial domain; seeFigure 4(a) for an example.To address these issues, we introduce the notion of a background/source mesh (M B ) for discretizing σ(x B ), where x B represents the positions of the source mesh nodes, Using a background mesh to fit discretely prescribed domain boundaries. (a) Initial unfitted mesh and target boundary curve (orange). (b) The boundary curve is prescribed implicitly as the zero level set of a discrete function on an adaptively refined nonconforming background mesh. Figure 5 : 5Multimaterial domain for a target application. (a) Domain modeled using geometric primitives with G(x B ) = 1 (orange) and −1 (blue), (b) background mesh with adaptive mesh refinements around the zero level set of G(x B ), and (c) distance function calculated on the background mesh [42] to be used as σ(x)in(8). Figure 6 : 6(a) Triangular mesh with material indicators of elements using (10) and the (b) optimized mesh.(c) Mesh with material indicators of elements using(12) and the resulting (d) optimized mesh. Figure 7 : 7keeps the original material indicator of an element when one of its faces is to be aligned to the level set. The second condition switches the material indicator of an element if all but one of the faces of an element is to be aligned, thus resulting in only one of its faces to be aligned after the switch. These two conditions are usually sufficient to guarantee that at most one face per element is set for alignment in triangular meshes.Figures 6(c, d) show an example of material interface that results from (12), which avoids the issue of sub-optimal Jacobian at the vertex shared by adjacent faces. In quadrilateral elements, when 1 < N E,M < N E,F − 1 (i.e. N E,M = 2), we can optionally (a) Quadrilateral mesh with material indicators using the strategy in (10) and the (b) optimized mesh. (c) Mesh material indicators set using (12) with conforming split introduced for elements with 2 faces marked for fitting, and (d) the resulting optimized mesh. do conforming splits on each element to bisect the vertex connecting adjacent faces that have been marked for fitting. This approach results in elements that have only 1 face marked for alignment, as long as the elements resulting from split keep the material indicator of the original element. Conforming mesh refinements increase the computational cost due to Figure 8 : 8Impact of w σ on the surface fitting error.strates how the maximum surface fitting error varies for a uniform quad mesh fit to a circular interface (recallFigure 7(a, b)) for different fixed values of w σ . Here, we define the maximum surface fitting error as the maximum value of the level set function evaluated at the nodes s ∈ S: |σ| S,∞ := max s∈S |σ(x s )|. on the derivatives of F σ . Let the FE position function bex = (x 1 . . . x d ) T where d is the space dimension; each component can be written as x a (x) = i x a,iwi (x), where x = x(x),see Section 2.1. Then the Newton's method solves for the full vector x = (x 1,1 . . . x 1,Nx , x 2,1 . . . x 2,Nx . . . x d,Nx ) T that contains the positions of all mesh nodes. The formulas for the first and second derivatives of F σ are the following: 6 H 6(x k )∆x = J (x k ) → solve using MINRES (Section 2.2). 7 x k+1 = x k − α∆x, with α determined using line-search (Section 3.k+1 ) = I(x k+1 , x B , σ(x B )) (a) and (b). The domain is a unit-sized cube, Ω ∈ [0, 1] 3 , and the level set function σ representing the sphere is defined such that its zero isosurface is located at a distance of 0.3 from the center of the domain (x c = (0.5, 0.5, 0.5)).Although this level set is simple enough to be defined analytically, the presented computations represent and use σ as a discrete finite element function. Additionally, no background mesh is used in this example, i.e., x B = x.The initial hex mesh is an 8 × 8 × 8 Cartesian-aligned mesh. The material indicators are setup such that there are a total of 32 elements that have more than one face marked for fitting. The tet mesh is obtained by taking a 4 × 4 Cartesian-aligned hex mesh and splitting each hex into 24 tetrahedra sharing a vertex at the center of the cube (4 tets-perhex face). The material indicators are setup such that all faces marked for fitting belong to different elements. The optimized meshes, shown in Fig. 9(c) and (d), have a maximum error of O(10 −10 ) at the interface with respect to the zero level set, which is achieved in 76 and 44 Newton iterations for the hexahedral and tetrahedral mesh, respectively. In the (a) (b) (c) (d) Figure 9 : 9Initial (a) hexahedral and (b) tetrahedral meshes showing the interfaces for the 3D surface fitting tests. Optimized (c) hexahedral and (d) tetrahedral mesh obtained using (8). Figure 10 : 10CSG tree with geometric primitives used to define the target geometry. Here, denotes the geometric intersection operator, denotes the union operation, and \ denotes the exclusion operator. Figure 11 : 11(a) Adaptively refined background mesh used to model the target domain, and (b) level set function computed using the distance from the zero level set of the geometric primitive-based geometry. Figure 11 ( 11a) shows a slice-view of the background mesh along with the zero iso-surface of the level set function, and Fig. 11(b) shows a slice-view of the level set function computed as the distance from the zero level set of the geometric primitive-based description of the domain. Figure 12 showsFigure 12 : 1212the input and the optimized mesh, where the achieved fitting error is |σ| S,∞ = O(10 −6 ) with the minimum Jacobian at the boundary decreasing to 4.8 × 10 −10 . Other 3D applications of interest that are currently leveraging the proposed method are shown next. Figure 13(a) shows an example of one of the boundary-fitted meshes obtained for simulation and design of lattice structures in the context of additive manufacturing. (a) Initial trimmed mesh and (b) final fitted mesh for the primitive-based geometry test. Figure 13 (Figure 13 : 1313b) shows an example of one of the interface-fitted meshes obtained for analyzing the impact of fluid flow on parameterized surfaces. Here, the target multimaterial domain consists of two concentric shells with parameterized locations and sizes for indents at the material interface. Note, the example shown inFig. 13(b) consists of uniformly spaced indents of the same size. The inner shell is colored red and the outer shell is translucent and (a) Boundary-fitted mesh generated using a geometric-primitive based description for an Octet truss. (b) Interface-fitted mesh for a multimaterial domain with concentric spherical shells with uniformly spaced indents of same size at the material interface. The inner shell is colored red and the outer shell is translucent and colored golden. Figure 14 : 14Reactor domain to be meshed for shape optimization, along with the symmetric portion and its primitive decomposition. The internal material interfaces must be fitted in the final mesh. Figure 15 : 15(a) Original mesh and (b) interface fitted mesh for the reactor design problem. Figure 15 ( 15a) shows the input non-fitted mesh, colored by the adaptively assigned material indicator, such that there is at most one face per triangle marked for fitting.Figure 15(b) shows the adapted mesh which aligns with the material interface of the domain. The maximum interface fitting error of the optimized mesh is O(10 −10 ), and the minimum Jacobian in the mesh decreases from 10 −7 to 2 × 10 −8 at the nodes along the interface due to mesh deformation enforcing alignment to the target surface. Finally, Figures 16(a)-(d) show the initial interface-fitted domain and the shape optimized domain along with the corresponding temperature (Kelvin) fields. 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Fischer, GSLIB: sparse communication library [Software], https://github.com/ gslib/gslib (2017). 10 Nonconforming Schwarz-spectral element methods for incompressible flow. K Mittal, S Dutta, P Fischer, 104237. 10Computers & Fluids. 191K. Mittal, S. Dutta, P. Fischer, Nonconforming Schwarz-spectral element methods for incompressible flow, Computers & Fluids 191 (2019) 104237. 10 On variational and PDE-based distance function approximations. A G Belyaev, P.-A Fayolle, Computer Graphics Forum. 3422Wiley Online LibraryA. G. Belyaev, P.-A. Fayolle, On variational and PDE-based distance function approx- imations, in: Computer Graphics Forum, Vol. 34, Wiley Online Library, 2015, pp. 104-118. 10, 11, 20, 22 Accelerating high-order mesh optimization using finite element partial assembly on GPUs. J.-S Camier, V Dobrev, P Knupp, T Kolev, K Mittal, R Rieben, V Tomov, arXiv:2205.127211624arXiv preprintJ.-S. Camier, V. Dobrev, P. Knupp, T. Kolev, K. Mittal, R. Rieben, V. Tomov, Accel- erating high-order mesh optimization using finite element partial assembly on GPUs, arXiv preprint arXiv:2205.12721 (2022). 16, 24 Polyconvex potentials, invertible deformations, and thermodynamically consistent formulation of the nonlinear elasticity equations. V A Garanzha, Computational Mathematics and Mathematical Physics. 509V. A. Garanzha, Polyconvex potentials, invertible deformations, and thermodynamically consistent formulation of the nonlinear elasticity equations, Computational Mathemat- ics and Mathematical Physics 50 (9) (2010) 1561-1587. 17 Variational method for untangling and optimization of spatial meshes. V Garanzha, L Kudryavtseva, S Utyuzhnikov, Journal of Computational and Applied Mathematics. 269V. Garanzha, L. Kudryavtseva, S. Utyuzhnikov, Variational method for untangling and optimization of spatial meshes, Journal of Computational and Applied Mathematics 269 (2014) 24-41. 17 Cubit geometry and mesh generation toolkit 15.1 user documentation. T D Blacker, S J Owen, M L Staten, W R Quadros, B Hanks, B W Clark, R J Meyers, C Ernst, K Merkley, R Morris, SNL-NM). Sandia National Lab; Albuquerque, NM (United States22Tech. rep.T. D. Blacker, S. J. Owen, M. L. Staten, W. R. Quadros, B. Hanks, B. W. Clark, R. J. Meyers, C. Ernst, K. Merkley, R. Morris, et al., Cubit geometry and mesh gen- eration toolkit 15.1 user documentation, Tech. rep., Sandia National Lab.(SNL-NM), Albuquerque, NM (United States) (2016). 22 Implicit shock tracking using an optimizationbased high-order discontinuous Galerkin method. M J Zahr, A Shi, P.-O Persson, Journal of Computational Physics. 41024M. J. Zahr, A. Shi, P.-O. Persson, Implicit shock tracking using an optimization- based high-order discontinuous Galerkin method, Journal of Computational Physics 410 (2020) 109385. 24

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{'abstract': 'We propose a method that morphs high-orger meshes such that their boundaries and interfaces coincide/align with implicitly defined geometries. Our focus is particularly on the case when the target surface is prescribed as the zero isocontour of a smooth discrete function.Common examples of this scenario include using level set functions to represent material interfaces in multimaterial configurations, and evolving geometries in shape and topology optimization. The proposed method formulates the mesh optimization problem as a variational minimization of the sum of a chosen mesh-quality metric using the Target-Matrix Optimization Paradigm (TMOP) and a penalty term that weakly forces the selected faces of the mesh to align with the target surface. The distinct features of the method are use of a source mesh to represent the level set function with sufficient accuracy, and adaptive strategies for setting the penalization weight and selecting the faces of the mesh to be fit to the target isocontour of the level set field. We demonstrate that the proposed method is robust for generating boundary-and interface-fitted meshes for curvilinear domains using different element types in 2D and 3D.', 'arxivid': '2208.05062', 'author': ['Jorge-Luis Barrera ', 'Tzanio Kolev ', 'Ketan Mittal ', 'Vladimir Tomov '], 'authoraffiliation': [], 'corpusid': 256615164, 'doi': '10.1016/j.cad.2023.103499', 'github_urls': [], 'n_tokens_mistral': 17155, 'n_tokens_neox': 15066, 'n_words': 9427, 'pdfsha': '6cbe9ef9119b08510e8f18d4b41f95861c299cea', 'pdfurls': ['https://export.arxiv.org/pdf/2208.05062v2.pdf'], 'title': ['High-Order Mesh Morphing for Boundary and Interface Fitting to Implicit Geometries', 'High-Order Mesh Morphing for Boundary and Interface Fitting to Implicit Geometries'], 'venue': []}

arxiv

Reconfigurable Intelligent Surfaces 2.0: Beyond Diagonal Phase Shift Matrices 4 Apr 2023 Student Member, IEEEHongyu Li Member, IEEEShanpu Shen Student Member, IEEEMatteo Nerini Fellow, IEEEBruno Clerckx Reconfigurable Intelligent Surfaces 2.0: Beyond Diagonal Phase Shift Matrices 4 Apr 20231Index Terms-Beyond diagonal reconfigurable intelligent sur- facefull space coveragegroup-connectedmodes/architectures Reconfigurable intelligent surface (RIS) has been envisioned as a promising technique to enable and enhance future wireless communications due to its potential to engineer the wireless channels in a cost-effective manner. Extensive research attention has been drawn to the use of conventional RIS 1.0 with diagonal phase shift matrices, where each RIS element is connected to its own load to ground but not connected to other elements. However, the simple architecture of RIS 1.0 limits its flexibility of manipulating passive beamforming. To fully exploit the benefits of RIS, in this paper, we introduce RIS 2.0 beyond diagonal phase shift matrices, namely beyond diagonal RIS (BD-RIS). We first explain the modeling of BD-RIS based on the scattering parameter network analysis and classify BD-RIS by the mathematical characteristics of the scattering matrix, supported modes, and architectures. Then, we provide simulations to evaluate the sum-rate performance with different modes/architectures of BD-RIS. We summarize the benefits of BD-RIS in providing high flexibility in wave manipulation, enlarging coverage, facilitating the deployment, and requiring fewer resolution bits and scattering elements. Inspired by the benefits of BD-RIS, we also discuss potential applications of BD-RIS in various wireless systems. Finally, we list key challenges in modeling, designing, and implementing BD-RIS in practice and point to possible future research directions for BD-RIS.Index Terms-Beyond diagonal reconfigurable intelligent surface, full space coverage, group-connected, modes/architectures.II. MODELING AND CLASSIFICATION OF BD-RISIn this section, we introduce the model of BD-RIS based on the scattering parameter network analysis, and classify BD-RIS based on different modes and architectures.A. BD-RIS ModelIn general, an M -element RIS is a passive device modeled as M antennas connected to an M -port reconfigurable impedance network[5]. The M -port reconfigurable impedance network is constructed by reconfigurable passive components and mathematically characterized by the scattering matrix Φ ∈ C M×M . The scattering matrix generally describes the scattering characteristics of the M -port reconfigurable impedance network regardless of specific circuit designs, I. INTRODUCTION Wireless networks for the first five generations have been operated by catering the uncontrollable wireless environment through various sophisticated designs at the transmitter/receiver. For beyond 5G and 6G, however, wireless networks are expected to have manipulations of both transmitter/receiver and wireless environment, thanks to the emergence of a promising technique, namely reconfigurable intelligent surface (RIS) [1], [2]. RIS consists of numerous passive reconfigurable scattering elements so that it can manipulate the wireless environment and thus enhance the spectrum and energy efficiency of the wireless network [3]. The advantages of RIS have been demonstrated in various wireless systems, such as enabling integrated sensing and communication and improving power relaying [4]. However, most existing works focus on using a simple RIS model with diagonal phase shift matrix, here referred to as RIS 1.0, where each RIS element is connected to its own reconfigurable impedance without inter-element connections. More specifically, there are two limitations of conventional RIS 1.0: 1) It can only control the phase of incident signal, which limits capability for manipulating passive beamforming and thus degrades the performance. 2) It only enables the signal reflection towards the same side, which limits the coverage. To address these limitations of RIS 1.0 and further enhance the performance gain of RIS, in this paper, we branch out to RIS 2.0 by introducing inter-element connections at the expense of additional circuit complexity, whose mathematical model is not limited to be diagonal matrices. We refer to this RIS 2.0 as beyond diagonal RIS (BD-RIS). We start from the BD-RIS modeling through scattering parameter network analysis. Then, we classify the BD-RIS based on the characteristics of the BD-RIS matrix, the supported modes, and the architectures, and categorize the existing BD-RIS works accordingly. Next, we consider a BD-RIS aided multi-user wireless communication system and evaluate the achievable sum-rate performance with different modes/architectures of BD-RIS. We summarize the benefits of BD-RIS such as high flexibility in wave manipulation and full-space coverage. Inspired by the benefits of BD-RIS, we look ahead to potential applications of BD-RIS in future wireless networks, such as enabling energy-efficient power relay in the power grid, and assisting wireless sensing in vehicular networks. We also discuss key challenges and future work of BD-RIS. Finally, we conclude this paper. which relates the voltage of incident waves and reflected waves from the M ports. As per the microwave network theory, for passive reconfigurable impedance network, the scattering matrix should satisfy Φ H Φ I M , which denotes I M − Φ H Φ is positive semi-definite. Particularly, when the reconfigurable impedance network is lossless, we have a unitary constraint for the scattering matrix, that is the power of reflected waves is equal to that of the incident waves. It should be noted that the characteristics of the scattering matrix is associated with the circuit topology of the M -port reconfigurable impedance network. In this sense, in conventional RIS 1.0, each port is connected to its own reconfigurable impedance without any connection across ports, referred to as single-connected RIS in [5], which yields a diagonal scattering matrix. However, in BD-RIS, part of/all the ports are connected to each other so that the scattering matrix is not limited to be diagonal. In the following subsection, we will classify BD-RIS by the characteristics of scattering matrix, supported modes, and architectures. B. BD-RIS Classification We establish a three-layer RIS classification tree as shown in Fig. 1, where each layer is explained in detail as below. The first layer is classified by the characteristics of the scattering matrix Φ. 1) Block Diagonal Matrix: In this category, the M antennas are uniformly divided into G groups and antennas within the same group are connected to each other while those across groups are not connected. We refer to this category as group-connected RIS [5] and the corresponding scattering matrix Φ is a block diagonal matrix with each block being unitary, which enables manipulating not only the phase but also the magnitude of incident waves and thus a better performance than the conventional RIS 1.0. Particularly, when there is only one group G = 1, i.e. all the M antennas are connected to each other, it is referred to as fullyconnected RIS [5], which results in a unitary scattering matrix. Besides, the conventional RIS 1.0, i.e. single-connected RIS, can be regarded as a special case of group-connected RIS with M groups, which has a diagonal scattering matrix. 2) Permuted Block Diagonal Matrix: In this category, the grouping strategy, that is how the M antennas are grouped, for the group-connected RIS is adaptive to the channel state information (CSI), which is thus referred to as dynamically group-connected RIS. The resulting scattering matrix is a permuted block diagonal matrix [6], which provides higher flexibility in beam control than the fixed group-connected RIS. 3) Non-Diagonal Matrix: In this category, antennas are linked in pairs through phase shifters so that the signal impinging on one antenna is purely reflected from another antenna, which results in an asymmetric non-diagonal scattering matrix [7] and a higher power gain than conventional RIS 1.0. The second layer is classified by the modes supported by RIS, including reflective, hybrid, and multi-sector modes as detailed in the following. 1) Reflective Mode: In this mode, signals impinging on one side of the RIS are reflected toward the same side, yielding a half-space coverage. To support the reflective mode, all the M antennas of RIS are placed towards the same direction as shown in Fig. 2(a). Mathematically, the RIS with reflective mode is characterized by the matrix Φ with a unitary constraint. 2) Hybrid Mode: In this mode, signals impinging on one side of the RIS can be partially reflected toward the same side and partially transmitted toward the opposite side, yielding a whole space coverage. The RIS with hybrid mode is also known as simultaneous transmitting and reflecting RIS (STAR-RIS) or intelligent omni-surface (IOS) [8]. To support the hybrid mode, each two antennas with uni-directional radiation pattern are back to back placed to form one cell, and are connected to a 2-port fully-connected reconfigurable impedance network [9] as shown in Fig. 2(c), so that each antenna in one cell respectively covers half space to achieve full-space coverage. Mathematically, the RIS with hybrid mode is characterized by two matrices, Φ r ∈ C M 2 × M 2 and Φ t ∈ C M 2 × M 2 , which satisfy that Φ H r Φ r + Φ H t Φ t = I M 2 . 3) Multi-Sector Mode: This mode is a generalization of hybrid mode. In this mode, the full space is divided into L sectors (L ≥ 2) and signals impinging on one sector of RIS can be partially reflected toward the same sector and partially scattered toward the other L − 1 sectors. To support the multi- sector mode, in each cell there are L antennas placed at each edge of an L-sided polygon, with each antenna having a unidirectional radiation pattern covering 1/L space, and the L antennas are connected to an L-port fully-connected reconfigurable impedance network, as shown in Fig. 2(e). Hence, the multi-sector mode can cover the full space while providing higher performance gains than the hybrid mode, thanks to the use of higher-gain antennas with narrower beamwidth covering 1/L space. Mathematically, the RIS with multi-sector mode is characterized by L matrices, Φ l ∈ C M L × M L , l = 1, . . . , L, which satisfy L l=1 Φ H l Φ l = I M L . The third layer is classified by the inter-cell architecture, i.e. how the cells are connected to each other, in BD-RIS with hybrid/multi-sector modes. Analogous to the first layer in RIS classification tree, here we have cell-wise single/group/fullyconnected architectures, where the resulting Φ r and Φ t for hybrid mode or Φ l ∀l for multi-sector mode are diagonal/block diagonal/full matrices, respectively. In [9], it is shown that the cell-wise group/fully connected architecture has a better performance than the cell-wise single connected architecture, i.e. the STAR-RIS/IOS. To further enhance the performance, we have cell-wise dynamically group-connected architecture, where the inter-cell grouping strategy is adaptive to CSI and the resulting Φ r and Φ t for hybrid mode or Φ l ∀l for multisector mode are permuted block diagonal matrices [6]. C. Unified Architectures and Modes It is worthwhile highlighting that the BD-RIS with different modes and architectures are realized by group-connected reconfigurable impedance network together with different antenna array arrangements. To get insights into the essence of BD-RIS with different modes/architectures, three examples are illustrated in Fig. 2, including 1) a BD-RIS with reflective mode and group-connected architecture, 2) a BD-RIS with hybrid mode and cell-wise group-connected architecture, and 3) a BD-RIS with multi-sector mode and cell-wise singleconnected architecture. From Figs. 2(b), (d), and (f), we can find these three BD-RISs have the same circuit topology of reconfigurable impedance network but different antenna array arrangements, which results in different modes and inter-cell architectures. For clarity, we summarize the circuit complexity, that is the required number of reconfigurable impedance components, of BD-RIS with nine different modes/architectures in Table I. Combining Fig. 3 and Table I, we can observe that the performance enhancement of BD-RIS with different modes and architectures is achieved at the expense of increasing circuit complexity, which also induces additional optimization complexity due to the increasing number of variables. III. PERFORMANCE EVALUATION FOR BD-RIS In this section, we evaluate the performance of BD-RIS with different modes and architectures. To that end, we consider a BD-RIS aided multiuser multiple input single output (MU-MISO) system, where a 4-antenna transmitter serves 4 singleantenna users with the aid of BD-RIS. The 4 users are located at one side for BD-RIS with reflective mode, while they are located at four corners for BD-RIS with hybrid and multi-sector modes. The transmit precoder and BD-RIS are jointly optimized to maximize the sum-rate of the MU-MISO system as detailed in [9], [10]. Fig. 3 shows the sum-rate performance versus the number of BD-RIS antennas for the BD-RIS with nine different modes and architectures. In Fig. 3, we consider a typical sub-6GHz narrowband scenario with carrier frequency 2.4 GHz. In this scenario, the direct link between the transmitter and users is assumed to be blocked. The distance between the transmitter and the BD-RIS is set as 100 m. The distance between the BD-RIS and users is set as 10 m [4]. Results for other settings can be found in [9], [11], [12]. Channels from the transmitter to the BD-RIS and from BD-RIS to users are modeled as a combination of small-scale fading and large-scale fading. Specifically, the small-scale fading components follow the Rician fading with Rician factor 5 dB. As per the results in [5], [9], decreasing the Rician factor can further enhance the performance of BD-RIS with group/fully-connected architectures. The large-scale fading components are related to the BD-RIS antenna gains and path loss, which are modeled and calculated based on [10]. Transmit power is set as P = 30 dBm. The noise power at each user is set as −80 dBm. The number of groups G for all three modes is fixed to 4. We make the following observations. First, under the reflective mode, BD-RIS with group/fullyconnected architectures always achieves better performance than conventional RIS 1.0 due to the more general constraint of the BD-RIS matrix. Second, with the same cell-wise architecture, the BD-RIS with multi-sector mode always outperforms that with hybrid mode, even though the number of antennas covering each user for the former case is reduced compared to the latter. This is because the BD-RIS antennas with multi-sector mode has narrower beamwidth compared to those with hybrid mode, and thus provide higher gains. More interestingly, multi-sector BD-RIS with cell-wise single-connected architecture outperforms the hybrid BD-RIS with all three inter-cell architectures. This finding implies that with proper antenna array arrangements of BD-RIS, a reduced circuit complexity can achieve both satisfactory performance and full-space coverage. Third, for all three modes, the sum-rate achieved by BD-RIS with (cell-wise) fully/group-connected architectures grows faster with M than that with single-connected architecture. This phenomenon can be explained by Table I, which indicates that the circuit complexity of BD-RIS grows linearly with M for single-connected architecture, but grows quadratically with M for group/fully-connected architectures: The higher the circuit complexity, the more the number of non-zero elements of BD-RIS matrices, and thus the higher the flexibility of passive beamforming. IV. BENEFITS AND POTENTIAL APPLICATIONS OF BD-RIS We have shown the pronounced benefits of BD-RIS compared to conventional RIS 1.0 in the example of MU-MISO system in Section III. In this section, we summarize the key benefits of BD-RIS and discuss potential applications of BD-RIS in various wireless systems as illustrated in Fig. 4. A. Benefits of BD-RIS 1) High Flexibility in Wave Manipulation: Compared with the conventional RIS 1.0 which can only manipulate the phase of incident wave, the BD-RIS has higher flexibility in manipulating both the magnitude and phase, which further boosts the performance in various wireless systems. Results in [5], [12] show that BD-RIS with reflective mode and group/fullyconnected architectures increases the received power by up to 62% compared to conventional RIS 1.0; results in [9] show that hybrid BD-RIS with group/fully-connected architectures achieves up to 75% higher sum-rate than STAR-RIS. 2) Full-Space Coverage: Compared with conventional RIS 1.0 which can only cover half-space, the BD-RIS utilizing appropriate group-connected reconfigurable impedance network Step-up Transformer Step-up Transformer Step-down Transformer Step and antenna array arrangement can support the hybrid and multi-sector modes to realize full-space coverage [9], [10]. Moreover, the multi-sector mode can provide high channel gain and thus effectively extend the communication range for full-space coverage [10]. 3) Facilitating Deployments: BD-RIS with hybrid and multi-sector modes facilitates practical deployments. Benefiting from the full-space coverage, the locations of the BD-RIS could be more flexible than conventional RIS 1.0. 4) Low Complexity in Resolution Bit Number: When considering RIS with discrete values, BD-RIS is shown to achieve a better performance than conventional RIS 1.0 with fewer resolution bits [11], due to the high flexibility of reconfigurable impedance network. Specifically, results in [11] show that to achieve satisfactory performance close to the continuous-value case, four resolution bits are required in conventional RIS 1.0, but only one resolution bit is sufficient in fully-connected BD-RIS with reflective mode. Such reduction of resolution bits is beneficial for implementation of BD-RIS. 5) Low Complexity in Element Number: As the BD-RIS, especially with multi-sector mode, greatly enhances the performance in various wireless networks, given the same performance requirement, the required BD-RIS element number can be effectively reduced. Results in [10] show that a 6-sector BD-RIS can maintain the same sum-rate as a 3-sector BD-RIS with a number of elements reduced by 20%. This benefit lowers the RIS complexity, cost, and form factor. B. Potential Applications of BD-RIS 1) Wireless Power Relay/Transfer: One promising application of BD-RIS is to deploy it in the power grid to relay wireless power. The power grid is an electricity system which is generally used to carry power from a few central generators to numerous users/customers/devices. Specifically, the power grid consists of the power generation, the transmission grid which moves the up-stepped power over long distances to substations, and the distribution grid which delivers the downstepped power to serve users [13]. In Fig. 4 we provide a diagram of employing BD-RIS in the power distribution grid to relay wireless power. With proper power levels, suitable deployments and locations of BD-RIS, the BD-RIS could act as a passive energy-efficient and low-cost power relay, which provides better relay performance than conventional RIS 1.0 while realizing wide coverage. 2) Wireless Communications: Another interesting application of BD-RIS is to enable flexible and scalable integrated access and backhaul (IAB) [14]. IAB is one of the promising techniques for 5G networks, where the operator can use part of the radio resources for wireless backhauling while providing the existing cellular services in the same node. Fig. 4 illustrates the BD-RIS assisted IAB, where the BD-RIS can be flexibly deployed in the IAB system to not only assist the wireless backhauling between the macrocell and picocells, but also the wireless access between picocells and users. Specifically, the wireless backhauling usually have complicated propagation environments and various obstacles, e.g. trees and high buildings as shown in Fig. 4. BD-RIS with full space coverage and high gain performance can be easily incorporated into real environments to bypass the obstacles and assist/enhance the wireless backhaul. Meanwhile, wireless access, especially in millimeter wave or Terahertz wireless frequencies, usually has sparse and highly-directional channels, suffers from high path loss, and is vulnerable to blockages. In this case, BD-RIS is more appealing in providing highly-directional beams to align with low-rank channels, compensate for the severe path loss, and enlarge coverage. 3) Wireless Sensing: BD-RIS can also be deployed to boost the wireless sensing performance, such as improving the target detection accuracy and reducing the parameter estimation error, for targets enjoying line of sight (LoS) links. More importantly, for those complicated propagation environments without LoS links between the radar and targets, such as the vehicle networks, BD-RIS enables wireless sensing and enlarges coverage by creating effective LoS links. 4) Integrated Wireless Power Transfer, Communications, and Sensing: In addition to stand-alone wireless power transfer, communications, and sensing, BD-RIS can also be used to assist integrated systems, such as simultaneous wireless information and power transfer as shown in Fig. 4, which helps to increase the output power level while enhancing the information transfer, or integrated sensing and communication to enable better communication and sensing performance. Not limited to these applications, BD-RIS can be applied in all the conventional RIS 1.0 enabled systems, but with higher flexibility and better performance in architecture design, beam manipulation, and deployment than conventional RIS 1.0. V. CHALLENGES AND FUTURE WORK OF BD-RIS While the BD-RIS has benefits compared with conventional RIS 1.0, there exist challenges in designing and implementing BD-RIS for practical wireless networks, which shed light on future research directions for BD-RIS. In this section, we list four challenges and future work from the perspectives of hardware implementation, RF impairments, channel estimation, and wideband modeling as follows. A. Hardware Implementation The hardware implementation of BD-RIS is a fundamental issue. Currently, only RIS with either transmissive or reflective mode [15] and STAR-RIS/IOS [8] have been implemented, which indicates the physical availability of modeling RIS as antenna arrays connected to reconfigurable impedance network, while the comprehensive hardware implementation of BD-RIS with different architectures is still on its way. In general, as per the model in Section II, an M -element BD-RIS consists of two parts, which can be implemented as follows. 1) M -Antenna Array: For the reflective mode, we can use the conventional uniform linear or planar antenna array. For the hybrid mode, we need to place each two antennas with unidirectional radiation pattern (e.g. patch antenna) back to back to form a cell and then arrange all the cells in a uniform array. Furthermore, for the multi-sector mode, we need to place each L antennas with narrow beamwidth at each edge of an L-side polygon to form a cell and arrange the cells in a uniform array. 2) M -Port Reconfigurable Impedance Network: As shown in Section II-C, the group-connected reconfigurable impedance network is the key to implement the BD-RIS with different modes and architecture. We can utilize tunable inductance and capacitance, e.g. varactors, to construct the group-connected reconfigurable impedance network as per the circuit topology shown in Section II-C, so that the continuous value BD-RIS can be implemented. Alternatively, we can use PIN diodes as switches to reconfigure the impedance network to implement discrete value BD-RIS. However, as the group size increases, the circuit complexity and cost also increases. Hence, it is challenging but worthwhile to achieve good trade-off between performance and circuit complexity/cost for BD-RIS architecture design, hardware implementation, and prototyping to verify its superiority compared to conventional RIS 1.0. B. RF Impairments Most existing designs for RIS 2.0 focus on idealized models with perfect matching and no mutual coupling of antennas, and lossless impedance components with continuous values. However, those idealized assumptions do not always hold in practical scenarios, thereby generating the following twofold challenges. 1) When mismatching and mutual coupling exist in practical scenarios, the channel model will no longer be a linear function of Θ. This issue will complicate the beamforming design, which has never been investigated in the existing works. 2) When using PIN diodes to implement the discrete value BD-RIS, it is challenging to design discrete values of the BD-RIS matrix. In the recent work [11], a potential direction for the codebook design of group/fully-connected BD-RIS with reflective mode has been provided. Nevertheless, investigating discrete value BD-RIS design with hybrid/multisector modes and different architectures still remains an open problem. Therefore, one of the meaningful reseach directions is to develop efficient beamforming design approaches while accurately capturing the RF impairments of BD-RIS. C. Channel Estimation The pronounced performance gain brought by the BD-RIS requires accurate CSI. For conventional RIS 1.0, there are two channel estimation strategies: 1) Semi-passive channel estimation by equipping a few low-power RF chains to the RIS to enable the pilot transmission/reception; 2) Pure passive channel estimation by estimating the cascaded transmitter-RIS-user channels with pre-defined RIS patterns, which characterize the variation of RIS matrix during the training period. The first channel estimation strategy is still available for the proposed BD-RIS but at the expense of additional power consumption due to the introduced RF chains. The second strategy may not be feasible for BD-RIS since most existing BD-RIS designs require the CSI for separate transmitter-RIS and RIS-user channels. Therefore, it is important to develop new channel estimation strategies with reduced power consumption for BD-RIS in the near future. D. Wideband BD-RIS Modeling The current BD-RIS model is only for narrowband communication. When it comes to wideband communications, the modeling of BD-RIS should take into account the frequency response of the reconfigurable impedance network. Specifically, each reconfigurable component of the impedance network is frequency dependent, where the frequency response is determined by the circuit designs. Consequently, the resulting BD-RIS matrices at different frequencies are dependent on each other, which will complicate the wideband BD-RIS design. To tackle the frequency dependent BD-RIS matrices and simplify the wideband BD-RIS design, a possible solution is to 1) analyze and fit the relationship between amplitudes/phase shifts of BD-RIS matrices and frequencies based on practical and specific circuits and 2) consider the wideband BD-RIS design based on the fitted frequency dependent BD-RIS model. VI. CONCLUSION In this paper, we depart from conventional RIS 1.0 with diagonal phase shift matrices and branch out to RIS 2.0 (BD-RIS) with beyond diagonal scattering matrices. Specifically, we model and classify the BD-RIS based on fundamental circuit topologies of reconfigurable impedance network. In addition, we highlight the benefits of BD-RIS with different modes/architectures in providing high flexibility in wave manipulation, achieving full-space coverage, flexibility in various deployments, and low complexity in resolution bit and element numbers of the impedance network. Potential applications, challenges, and future work of BD-RIS are also discussed and summarized. As BD-RIS is a brand-new advance in RIS technology that remains unexplored from various perspectives, it is hoped that this paper could offer a useful and stimulating guide on future research directions of BD-RIS. ( Corresponding author: Shanpu Shen.) H. Li and M. Nerini are with the Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: {c.li21,m.nerini20}@imperial.ac.uk). S. Shen is with the Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (e-mail: sshenaa@connect.ust.hk). B. Clerckx is with the Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.K. and with Silicon Austria Labs (SAL), Graz A-8010, Austria (e-mail: b.clerckx@imperial.ac.uk). Fig. 1 . 1RIS classification tree. Fig. 2 . 2RISs with the same circuit topologies of reconfigurable impedance network while supporting different modes. (a) RIS with reflective mode and (b) group-connected architecture; (c) RIS with hybrid mode and (d) cell-wise group-connected architecture; (e) RIS with multi-sector mode and (f) cell-wise single-connected architecture. Fig. 3 . 3Sum-rate versus the number of BD-RIS antennas. Left: BD-RIS with reflective mode; right: BD-RIS with hybrid/multi-sector modes. Fig. 4 . 4Potential applications of BD-RIS. ① BD-RIS works as a passive relay in the power grid; ② BD-RIS insists wireless power transfer; ③ BD-RIS enables wireless backhaul and access; ④ BD-RIS insists millimeter wave (mmWave)/Terahertz (THz) communications; ⑤ BD-RIS insists wireless sensing; ⑥ BD-RIS insists simultaneous wireless information and power transfer; ⑦ BD-RIS enables integrated sensing and communication. TABLE I CIRCUIT ICOMPLEXITY OF BD-RIS WITH NINE MODES/ARCHITECTURES † M : number of RIS elements; G: number of groups for reconfigurable impedance network; L: number of sectors for multi-sector BD-RIS.Mode Architecture (Inter-Cell) Cell-Wise Cell-Wise Cell-Wise Single- Group- Fully- Connected † Connected † Connected † Reflective M ( M G + 1) M 2 (M + 1) M 2 Hybrid 3 2 M Multi-Sector (L + 1) M 2 Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead. M Di Renzo, A Zappone, M Debbah, M.-S Alouini, C Yuen, J De Rosny, S Tretyakov, IEEE J. Sel. Areas Commun. 3811M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen, J. De Rosny, and S. Tretyakov, "Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead," IEEE J. Sel. Areas Commun., vol. 38, no. 11, pp. 2450-2525, 2020. Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network. Q Wu, R Zhang, IEEE Commun. Mag. 581Q. Wu and R. Zhang, "Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network," IEEE Commun. Mag., vol. 58, no. 1, pp. 106-112, 2019. Reconfigurable intelligent surfaces for wireless communications: Overview of hardware designs, channel models, and estimation techniques. M Jian, G C Alexandropoulos, E Basar, C Huang, R Liu, Y Liu, C Yuen, Intelligent and Converged Networks. 31M. Jian, G. C. Alexandropoulos, E. Basar, C. Huang, R. Liu, Y. Liu, and C. Yuen, "Reconfigurable intelligent surfaces for wireless communi- cations: Overview of hardware designs, channel models, and estimation techniques," Intelligent and Converged Networks, vol. 3, no. 1, pp. 1-32, 2022. Intelligent reflecting surface-aided wireless communications: A tutorial. Q Wu, S Zhang, B Zheng, C You, R Zhang, IEEE Trans. Commun. 695Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, "Intelligent reflecting surface-aided wireless communications: A tutorial," IEEE Trans. Commun., vol. 69, no. 5, pp. 3313-3351, 2021. Modeling and architecture design of reconfigurable intelligent surfaces using scattering parameter network analysis. S Shen, B Clerckx, R Murch, IEEE Trans. Wireless Commun. 212S. Shen, B. Clerckx, and R. Murch, "Modeling and architecture design of reconfigurable intelligent surfaces using scattering parameter network analysis," IEEE Trans. Wireless Commun., vol. 21, no. 2, pp. 1229-1243, 2021. A dynamic grouping strategy for beyond diagonal reconfigurable intelligent surfaces with hybrid transmitting and reflecting mode. H Li, S Shen, B Clerckx, arXiv:2210.02499arXiv preprintH. Li, S. Shen, and B. Clerckx, "A dynamic grouping strategy for beyond diagonal reconfigurable intelligent surfaces with hybrid transmitting and reflecting mode," arXiv preprint arXiv:2210.02499, 2022. Reconfigurable intelligent surfaces relying on non-diagonal phase shift matrices. Q Li, M El-Hajjar, I A Hemadeh, A Shojaeifard, A Mourad, B Clerckx, L Hanzo, IEEE Trans. Veh. Technol. Q. Li, M. El-Hajjar, I. A. Hemadeh, A. Shojaeifard, A. Mourad, B. Clerckx, and L. Hanzo, "Reconfigurable intelligent surfaces relying on non-diagonal phase shift matrices," IEEE Trans. Veh. Technol., 2022. Intelligent omni-surfaces: Simultaneous refraction and reflection for full-dimensional wireless communications. H Zhang, B Di, IEEE Commun. Surveys & Tutorials. H. Zhang and B. Di, "Intelligent omni-surfaces: Simultaneous refraction and reflection for full-dimensional wireless communications," IEEE Commun. Surveys & Tutorials, 2022. Beyond diagonal reconfigurable intelligent surfaces: From transmitting and reflecting modes to single-, group-, and fully-connected architectures. H Li, S Shen, B Clerckx, IEEE Trans. Wireless Commun. H. Li, S. Shen, and B. Clerckx, "Beyond diagonal reconfigurable intelli- gent surfaces: From transmitting and reflecting modes to single-, group- , and fully-connected architectures," IEEE Trans. Wireless Commun., 2022. Beyond diagonal reconfigurable intelligent surfaces: A multisector mode enabling highly directional full-space wireless coverage. arXiv:2209.00301arXiv preprint--, "Beyond diagonal reconfigurable intelligent surfaces: A multi- sector mode enabling highly directional full-space wireless coverage," arXiv preprint arXiv:2209.00301, 2022. Discrete-value group and fully connected architectures for beyond diagonal reconfigurable intelligent surfaces. M Nerini, S Shen, B Clerckx, arXiv:2110.00077v3arXiv preprintM. Nerini, S. Shen, and B. Clerckx, "Discrete-value group and fully connected architectures for beyond diagonal reconfigurable intelligent surfaces," arXiv preprint arXiv:2110.00077v3, 2021. Optimal group and fully connected design for beyond diagonal reconfigurable intelligent surfaces. arXiv:2211.06117arXiv preprint--, "Optimal group and fully connected design for beyond diagonal reconfigurable intelligent surfaces," arXiv preprint arXiv:2211.06117, 2022. Smart grid-The new and improved power grid: A survey. X Fang, S Misra, G Xue, D Yang, IEEE Commun. Surveys & Tutorials. 144X. Fang, S. Misra, G. Xue, and D. Yang, "Smart grid-The new and improved power grid: A survey," IEEE Commun. Surveys & Tutorials, vol. 14, no. 4, pp. 944-980, 2011. On integrated access and backhaul networks: Current status and potentials. C Madapatha, B Makki, C Fang, O Teyeb, E Dahlman, M.-S Alouini, T Svensson, IEEE Open J. Commun. Society. 1C. Madapatha, B. Makki, C. Fang, O. Teyeb, E. Dahlman, M.-S. Alouini, and T. Svensson, "On integrated access and backhaul networks: Current status and potentials," IEEE Open J. Commun. Society, vol. 1, pp. 1374- 1389, 2020. mmwall: A reconfigurable metamaterial surface for mmwave networks. K W Cho, M H Mazaheri, J Gummeson, O Abari, K Jamieson, Proc. 22nd Int. Workshop on Mobile Computing Systems and Applications. 22nd Int. Workshop on Mobile Computing Systems and ApplicationsK. W. Cho, M. H. Mazaheri, J. Gummeson, O. Abari, and K. Jamieson, "mmwall: A reconfigurable metamaterial surface for mmwave net- works," in Proc. 22nd Int. Workshop on Mobile Computing Systems and Applications, 2021, pp. 119-125.

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{'abstract': 'Reconfigurable intelligent surface (RIS) has been envisioned as a promising technique to enable and enhance future wireless communications due to its potential to engineer the wireless channels in a cost-effective manner. Extensive research attention has been drawn to the use of conventional RIS 1.0 with diagonal phase shift matrices, where each RIS element is connected to its own load to ground but not connected to other elements. However, the simple architecture of RIS 1.0 limits its flexibility of manipulating passive beamforming. To fully exploit the benefits of RIS, in this paper, we introduce RIS 2.0 beyond diagonal phase shift matrices, namely beyond diagonal RIS (BD-RIS). We first explain the modeling of BD-RIS based on the scattering parameter network analysis and classify BD-RIS by the mathematical characteristics of the scattering matrix, supported modes, and architectures. Then, we provide simulations to evaluate the sum-rate performance with different modes/architectures of BD-RIS. We summarize the benefits of BD-RIS in providing high flexibility in wave manipulation, enlarging coverage, facilitating the deployment, and requiring fewer resolution bits and scattering elements. Inspired by the benefits of BD-RIS, we also discuss potential applications of BD-RIS in various wireless systems. Finally, we list key challenges in modeling, designing, and implementing BD-RIS in practice and point to possible future research directions for BD-RIS.Index Terms-Beyond diagonal reconfigurable intelligent surface, full space coverage, group-connected, modes/architectures.II. MODELING AND CLASSIFICATION OF BD-RISIn this section, we introduce the model of BD-RIS based on the scattering parameter network analysis, and classify BD-RIS based on different modes and architectures.A. BD-RIS ModelIn general, an M -element RIS is a passive device modeled as M antennas connected to an M -port reconfigurable impedance network[5]. The M -port reconfigurable impedance network is constructed by reconfigurable passive components and mathematically characterized by the scattering matrix Φ ∈ C M×M . The scattering matrix generally describes the scattering characteristics of the M -port reconfigurable impedance network regardless of specific circuit designs,', 'arxivid': '2301.03288', 'author': ['Student Member, IEEEHongyu Li ', 'Member, IEEEShanpu Shen ', 'Student Member, IEEEMatteo Nerini ', 'Fellow, IEEEBruno Clerckx '], 'authoraffiliation': [], 'corpusid': 255546539, 'doi': '10.48550/arxiv.2301.03288', 'github_urls': [], 'n_tokens_mistral': 9490, 'n_tokens_neox': 8454, 'n_words': 5314, 'pdfsha': '01123ed93a217f99a9e14f54f2fab8cee55cba02', 'pdfurls': ['https://export.arxiv.org/pdf/2301.03288v2.pdf'], 'title': ['Reconfigurable Intelligent Surfaces 2.0: Beyond Diagonal Phase Shift Matrices', 'Reconfigurable Intelligent Surfaces 2.0: Beyond Diagonal Phase Shift Matrices'], 'venue': []}

arxiv

Measures of imaginarity and quantum state order 27 Feb 2023 Qiang Chen School of Mathematical Sciences Hebei Normal University 050024ShijiazhuangChina Ting Gao School of Mathematical Sciences Hebei Normal University 050024ShijiazhuangChina Fengli Yan College of Physics Hebei Key Laboratory of Photophysics Research and Application Hebei Normal University 050024ShijiazhuangChina Measures of imaginarity and quantum state order 27 Feb 2023 Complex numbers are widely used in both classical and quantum physics, and play an important role in describing quantum systems and their dynamical behavior. In this paper we study several measures of imaginarity of quantum states in the framework of resource theory, such as the measures based on l1 norm, and convex function, etc. We also investigate the influence of the quantum channels on quantum state order for a single-qubit. * I. INTRODUCTION Quantum resource theory provides a method for exploring the properties of quantum systems [1,2]. In this theory the resource of the quantum system is quantified by an operational method and the information processing tasks which can be realized are determined by the resource consumed. For example, in the resource theory of entanglement, the quantization of entanglement [3][4][5][6][7] and a series of applications of entanglement, such as quantum key distribution [8][9][10][11][12][13][14], quantum teleportation [15,16], quantum direct communication [17][18][19][20], quantum secret sharing [21,22] have been provided. In recent years, researchers proposed many resource theories, such as resource theories of coherence [23][24][25], asymmetry [26], quantum thermodynamics [27], nonlocality [28], superposition [29], etc. In addition, people also have developed applicable quantities in mathematical framework of resource theory [30]. One feature of quantum mechanics is the use of imaginary numbers. Although imaginary numbers are used to describe the motion of an oscillatory in classical physics, they play a very important role in quantum mechanics, because the wave functions of quantum system all involve complex numbers [31]. Consider, for example, the polarization density matrix of a single photon in the {|H , |V } basis, where |H and |V express the horizontal polarization, and vertical polarization, respectively. As a matter of fact, the imaginary numbers in the density matrix cause the rotation of the electric field vector. Based on this phenomenon, Hickey and Gour [32] came up with imaginarity resource theory. In this theory, the density matrix with imaginary elements is defined as resource state, otherwise as free state. Hickey and Gour [32] also defined the largest class of free operations. For the special physical constraints, some free operations are obtained, and then the theoretical framework of imaginarity resource is established. In this framework, several measures of imaginarity are given, and a state conversion condition for the pure states of a single qubit is discussed. Furthermore, in 2021, Wu et al [33,34] proposed the robustness measure of imaginarity, and gave the transformation condition of states of a single qubit under free operation. In this paper, we investigate several measures of imaginarity in the framework of resource theory. The rest of this paper is organized as follows. In Sec. II, we review some concepts including the real states, the free operations and measures of imaginarity. In Sec. III, we mainly study whether the measures of imaginarity based on l p norm, p−norm, and convex roof extended are good measures in the framework of the resource theory. The influence of the quantum channels on quantum state order for a single-qubit is discussed in Sec. IV. II. BACKGROUND A. Theoretical framework of imaginarity resource Suppose {|j } d−1 j=0 is a fixed basis in a d-dimensional Hilbert space H. We use D(H) to denote the set of density operators acting on H. In fact, a quantum state is descibed by a density operator ρ in D(H). The theoretical framework of imaginarity resource [32] consists of three ingredients: real states (free states), free operations and measures of imaginarity. They are defined as follows. Real state [32][33][34]: In a fixed basis {|j } d−1 j=0 , if quan- tum state ρ = jk ρ jk |j k|(1) satisfies each ρ jk ∈ R, we call ρ a real state (free state). Here R is the set of real numbers. We denote the set of all real states by F . In other words, the density matrices of free states are real with respect to a fixed basis. Free operation [32]: Let Λ be a quantum operation with Kraus operators {K j }, and ρ be a density operator, Λ[ρ] = j K j ρK † j . We say that Λ is a free operation (real quantum operation) if i|K j |l ∈ R(2) for arbitrary j and i, l ∈ {0, 1, · · · , d − 1}. Measures of imaginarity [33]: A measure of imaginarity is a function M : D(H) → [0, ∞) such that 1. M (ρ) = 0 if and only if ρ ∈ F . 2. M (ε(ρ)) ≤ M (ρ), where ε is a free operation. Condition 2 is also called monotonic. It is easy to observe that this theory is basis dependent, the real states do not possess any resource, and the free operations can not generate resources from real states. III. MEASURES OF IMAGINARITY Let us begin to discuss the quantization of imaginarity, which plays a very important role in determining the resources of a given quantum state. Two measures of imaginarity of the quantum state ρ have been proposed in [32]. They are the measure of imaginarity based on the 1−norm, M (ρ) = min σ∈F ρ − σ 1 = 1 2 ρ − ρ T 1 ,(3) where ρ T denotes the transposition of density matrix ρ, A 1 = Tr[(A † A) 1/2 ] is the 1−norm of matrix A [23] , and the robustness of imaginarity R(ρ) = min σ∈D(H) {s ≥ 0 : sσ + ρ 1 + s ∈ F }.(4) The geometric measure of imaginarity for pure states |ψ is [33] M g (|ψ ) = 1 − max |φ ∈F | φ|ψ | 2 .(5) Next we discuss several important distance-based imaginarity functions. Consider the function constructed based on the l p norm. The l p norm of a matrix A [23] is defined as A lp = { ij |A ij | p } 1/p .(6) Specially we can define the function based on the l 1 norm as M l1 (ρ) = min σ∈F ρ − σ l1 ,(7) where ρ is an arbitrary quantum state, σ ∈ F is real quantum state. Then one can obtain the following result. Theorem 1. M l1 (ρ) = i =j |Im(ρ ij )|, and M l1 (ρ) is a measure of imaginarity for free operations being all real operations within complete positivity trace-preserving (CPTP) quantum operations, where Im(ρ ij ) represents the imaginary part of the matrix element ρ ij . Proof. Firstly, we prove M l1 (ρ) = i =j |Im(ρ ij )|. Obviously, each quantum state ρ = (ρ ij ) in a ddimensional Hilbert space can be written as ρ = (ρ ij ) = (a ij + ib ij ), where a ij , b ij are real numbers, and when i = j, then b ij = 0 holds. The real state σ = (σ ij ) = (c ij ) with c ij being real numbers. Hence ρ − σ l1 = |(a 11 − c 11 )| + |(a 22 − c 22 )| + · · · + |(a dd − c dd )| + 2|(a 12 − c 12 ) + b 12 i| + 2|(a 13 − c 13 ) + b 13 i| + · · · + 2|(a 1d − c 1d ) + b 1d i| + 2|(a 23 − c 23 ) + b 23 i| + 2|(a 24 − c 24 ) + b 24 i| + · · · + 2|(a 2d − c 2d ) + b 2d i| + 2|(a (d−1)d − c (d−1)d ) + b (d−1)d i| = ij (a ij − c ij ) 2 + b 2 ij .(8) Clearly, the minimum of ρ − σ l1 occurs at c ij = a ij . That is, when σ = Re(ρ), one gets M l1 (ρ) = i =j |Im(ρ ij )|.(9) Here Re(ρ) stands for the real part of ρ. It means that Re(ρ) is the closest real state of ρ. New we demonstrate that the function M l1 (ρ) is a measure of imaginarity of quantum state ρ. Obviously, for an arbitrary quantum state ρ, we have M l1 (ρ) = i =j |Im(ρ ij )| ≥ 0.(10) For a real quantum state ρ we can easily derive M l1 (ρ) = 0 by Eq. (10). When the function M l1 (ρ) = 0, one has |Im(ρ ij )| = 0. It implies that the matrix elements of quantum state ρ are real numbers. Hence quantum state ρ is real. After that, we want to show that M l1 (ρ) is monotonic under an arbitrary real operation within CPTP. Assume ε is the real operation within CPTP, ρ and σ are two density operators, according to the definition of l 1 norm [23], we have ε(ρ) − ε(σ) l1 ≤ ρ − σ l1 .(12) Evidently, a quantum state ρ can be written as ρ = ρ R + iρ I , where ρ R = 1 2 (ρ + ρ T ), ρ I = 1 2i (ρ − ρ T ) . It is not difficult to observe that ρ R is real symmetric, ρ I is real antisymmetric, and Trρ R = Tr[ 1 2 (ρ + ρ T )] = 1 2 [Tr(ρ) + Tr(ρ T )] = 1,(13)ψ|ρ R |ψ = 1 2 ψ|ρ|ψ + 1 2 ψ|ρ T |ψ ≥ 0.(14) Therefore ρ R is the real density matrix. According to the l 1 norm of the matrix is contracted under CPTP, one can obtain M l1 (ε(ρ)) = inf σ∈F ε(ρ) − σ l1 ≤ ε(ρ) − ε(ρ R ) l1 = ε(ρ R + iρ I ) − ε(ρ R ) l1 ≤ ρ − ρ R l1 = iρ I l1 = M l1 (ρ).(15) Thus we arrive at that the function M l1 (ρ) is a measure of imaginarity for free operations being all real operations within CPTP quantum operations. The proof of Theorem 1 has been completed. However, for the functions induced by the l p norm or p−norm [24] we have the following conclusion. Theorem 2. For any quantum state ρ in a d- dimensional Hilbert space, when p > 1, both the func- tion M lp (ρ ⊗ I d ) and function M p (ρ ⊗ I d ) induced by the l p norm and p−norm respectively, do not satisfy monotonicity under all real operations within CPTP mappings. Proof. It is not difficult to observe that for a particular real state ρ 1 = |0 0|,(16) there exists a real operation Λ which transforms the quantum state ρ 2 = I d(17) to the quantum state ρ 1 . Here I is the d-dimensional identity operator, the Kraus operators of the real op- eration Λ are {K i = |0 i − 1|}, and {K i } satisfy d i=1 K † i K i = I. We choose the real operationΛ, whose Kraus operators are {K i = I ⊗ K i }. Clearly, {K i } satisfy iK † iK i = I ′ , where I ′ is the identity operator of the direct product space. Then we have M lp (Λ[ρ ⊗ I d ]) = M lp (ρ ⊗ |0 0|) = { ij |Im(ρ ⊗ |0 0|) ij | p } 1/p = M lp (ρ) > M lp (ρ ⊗ I d ).(18) Here Im(ρ⊗ |0 0|) ij represents the imaginary part of the matrix element (ρ⊗|0 0|) ij . The above inequality takes advantage of the following results M lp (ρ ⊗ I d ) = { ij |Im(ρ ⊗ I d ) ij | p } 1/p = d 1 p −1 M lp (ρ) < M lp (ρ).(19) Obviously, Eq. (18) indicates that when p > 1, function M lp does not satisfy the condition M lp (ε(ρ)) ≤ M lp (ρ) for arbitrary free operation ε and quantum state ρ. That is when p > 1, function M lp can not be regarded as a measure of imaginarity. For a matrix A, its p-norm A p is defined as [Tr(A + A) p 2 ] 1 p . When p > 1, for p-norm induced func- tion M p (ρ) = min σ∈F ρ − σ p ,(20) we have M p (Λ[ρ ⊗ I d ]) = M p (ρ ⊗ |0 0|) = M p (ρ) > M p (ρ ⊗ I d ).(21) The inequality above can be derived from M p (ρ ⊗ I d ) ≤ min σ∈F ρ ⊗ I d − σ ⊗ I d p = min σ∈F (ρ − σ) ⊗ I d p = min σ∈F ρ − σ p I d p = M p (ρ) I d p < M p (ρ).(22) Thus we have demonstrated that when p > 1, the function M p violates monotonicity under all real operations within CPTP mappings. Hence Theorem 2 is true. Let us discuss the measure of imaginarity based on relative entropy. In resource theory of coherence, coherence measure C r (ρ) based on relative entropy satisfies the axiomatic condition of coherence measure [2], and its expression being similar to coherence distillation [25] is C r (ρ) = S(∆ ′ (ρ)) − S(ρ),(23) where ∆ ′ is the decoherence operation and S(ρ) stands for Von Neumann entropy of quantum state ρ. Similar to resource theory of coherence, here we need an operator ∆. Definition 1. The mathematical operator ∆ is de- fined by ∆(ρ) = 1 2 (ρ + ρ T ),(24) where ρ is any quantum state. Evidently, ∆ is just a simple mathematical operator, rather than a free operation. The relationship between ∆(ρ) and quantum real operation satisfying the physically consistent condition [32] can be stated as the following theorem. Theorem 3. Let ε be a real operation within CPTP. If ε satisfies the condition of physical consistency. Then for any quantum state ρ, we have ε(∆(ρ)) = ∆(ε(ρ)).(25) Proof. For any quantum state ρ, because the real operation ε is linear, hence one has ε[∆(ρ)] = ε[ 1 2 (ρ + ρ T )] = 1 2 [ε(ρ) + ε(ρ T )] = 1 2 [ε(ρ) + ε(ρ) T ] = ∆(ε(ρ)),(26) where the third equality of the above equation is obtained from the condition of physical consistency [32]. Therefore Theorem 3 holds. The quantum relative entropy between quantum states ρ and σ is usually taken as [35] S(ρ σ) = Tr[ρ log 2 ρ] − Tr[ρ log 2 σ]. The relative entropy of imaginarity of a quantum state ρ is defined as [36] M r (ρ) = min σ∈F S(ρ σ). Then the relative entropy function M r (ρ) can be reexpressed as [36] M r (ρ) = S(∆(ρ)) − S(ρ). Theorem 4. For any qubit pure state |ψ , the measure of imaginarity based on the relative entropy satisfies M r (|ψ ) ≤ M l1 (|ψ ),(30) the equality holds if M l1 (|ψ ) = 1. Proof. Choose a qubit pure state |ψ = α|0 + β|1 , where α, β are complex numbers and satisfy |α| 2 +|β| 2 = 1. Assume that α = c + di, β = e + f i, and H(x) = −x log 2 x − (1 − x) log 2 (1 − x). It is not difficult to obtain M r (|ψ ) = H(λ 1 ),(31) where In addition to the above measures of imaginarity, there exist other measures. Next, based on the measure of imaginarity of pure states, we will give a measure of imaginarity of mixed quantum states by convex roof extended [38]. λ 1 = 1 + 1 − 4(cf − de) 2 2 .(32)According to H(x) ≤ 2 x(1 − x) [37], we have M r (|ψ ) = H(λ 1 ) ≤ 2 λ 1 (1 − λ 1 ) = 2 1 + 1 − 4(cf − de) 2 2 × 1 − 1 − 4(cf − de) 2 2 = 2 1 4 − 1 4 (1 − 4(cf − de) 2 ) = 2 (cf − de) 2 = 2|cf − de| = M l1 (|ψ ).(33)p i M (|ψ i )(34) is a measure of imaginarity of mixed state ρ if M (ρ) is a convex function. Here {p i , |ψ i } is the decomposition of quantum state ρ, and {p i } is a probability distribution, namely, ρ = i p i |ψ i ψ i |. Proof. According to the definition of the function M (ρ), when M (ρ) = 0, obviously we can obtain that quantum state ρ is a real one. Conversely, if ρ is a real state, there is a real decomposition ρ = i p i |ψ i ψ i | such that M (|ψ i ) = 0. So M (ρ) = 0. Next, we will prove that the function M (ρ) is monotonic. For any quantum state ρ, we take the best decomposition of quantum state ρ, expressed as ρ = k p k |ψ k ψ k |, then one has M (ρ) = k p k M (|ψ k ).(35) Assume {K j } is the set of Kraus operators of a real operation, c jk = ψ k |K T j K j |ψ k , q j = TrK j ρK T j , then j q j M ( K j ρK T j q j ) = j q j M ( k p k K j |ψ k ψ k |K T j q j ) = j q j M ( k p k c jk q j × K j |ψ k ψ k |K T j c jk ) ≤ j,k p k c jk M ( K j |ψ k ψ k |K T j c jk ) ≤ k p k M (|ψ k ) = M (ρ),(36) where first inequality is true because M (ρ) is a convex function. Thus we demonstrate that the function M (ρ) is monotonic. So Theorem 5 holds. IV. INFLUENCE OF QUANTUM CHANNEL ON QUANTUM STATE ORDER In this section, we mainly investigate the ordering of quantum states based on the measure of imaginarity after passing through a real channel. The main real channels involved are amplitude damping channel, phase flip channel, and bit flip channel. We restate the definition of the ordering of quantum states as follows [39][40][41]. M A (ρ 1 ) ≤ M A (ρ 2 ) ⇔ M B (ρ 1 ) ≤ M B (ρ 2 )(37) is true, then the measures M A and M B are said to be of the same order, if the above relation is not satisfied, the measures M A and M B are considered to be of different order. We only discuss the ordering of quantum states in the case of a single qubit. In a fixed reference basis, the state of a single-qubit can always be written as ρ = 1 2 (I + r · σ) = 1 2 (I + tn · σ) = 1+tnz 2 t(nx−iny) 2 t(nx+iny) 2 1−tnz 2 ,(38) where σ is the Pauli vector, t = r ≤ 1, n = (n x , n y , n z ) = 1 t r is a unitary vector. It is easy to obtain that the measures of imaginarity of quantum state ρ M l1 (ρ) = t|n y |, M r (ρ) = H( 1 2 + t 1 − n 2 y 2 ) − H( 1 + t 2 ).(39) Now let's consider the monotonicity of these functions. One can easily obtain ∂M r (ρ) ∂|n y | = t 2 · −|n y | 1 − n 2 y log 2 1 − t 1 − n 2 y 1 + t 1 − n 2 y ≥ 0. (41) ∂M r (ρ) ∂t = 1 2 log 2 1 + t 1 − t + 1 − n 2 y 2 log 2 1 − t 1 − n 2 y 1 + t 1 − n 2 y . (42) Because the function f (x) = x log 2 1−tx 1+tx (0 ≤ x ≤ 1) is decreasing monotonically, so we have ∂M r (ρ) ∂t = 1 2 log 2 1 + t 1 − t + n 2 x + n 2 z 2 log 2 1 − t n 2 x + n 2 z 1 + t n 2 x + n 2 z ≥ 1 2 log 2 1 + t 1 − t + 1 2 log 2 1 − t 1 + t ≥ 0. (43) Therefore, M r (ρ) is monotonic increasing about the independent variables |n y | and t. Evidently M l1 (ρ) is also monotonic increasing about the independent variables |n y | and t. Thus we have the following conclusion. Proposition 1. The measure M l1 (ρ) and the measure M r (ρ) are of the same order for qubit quantum states. It is well known that the quantum channel can change the quantum state, furthermore it can affect the quantum state order also. For a measure of quantum states, we define the influence of quantum channel on quantum state order as follows. M (ρ 1 ) ≤ M (ρ 2 ) ⇔ M (ε(ρ 1 )) ≤ M (ε(ρ 2 )) (44) holds, then we say the quantum channel ε does not change the quantum state order; otherwise we say the quantum state order is changed by the quantum channel ε. Next, we discuss the influence of a quantum channels on the ordering of qubit quantum states when one chooses a measure of imaginarity. Firstly we study the case of the bit flip channel ε and imaginarity measure M r (ρ). Here the quantum state of the qubit is stated as Eq. (38), the bit flip channel ε is expressed by the real Kraus operators {K 0 = √ pI, K 1 = √ 1 − pσ x }, where p ∈ [0, 1], σ x is the Pauli operator. Proposition 2. Suppose one chooses M r (ρ) as the measure of imaginarity, then the quantum state order does not change after a single-qubit goes through a bit flip channel. Proof. The state of the qubit system after passing through the bit flip channel ε is ε(ρ) = K 0 ρK † 0 + K 1 ρK † 1 = 1+tnz (2p−1) 2 tnx−itny(2p−1) 2 tnx+itny(2p−1) 2 1−tnz(2p−1) 2 ,(45) where ρ is expressed by Eq. (38). It is easy to derive that M r (ε(ρ)) = H( 1 + t n 2 x + (2p − 1) 2 n 2 z 2 ) − H( 1 + t n 2 x + (2p − 1) 2 (1 − n 2 x ) 2 ). (46) Obviously, M r (ε(ρ)) contains four parameters t, p, n x , n z . We can easily get ∂M r (ε(ρ)) ∂|n z | = t 2 · (2p − 1) 2 |n z | n 2 x + (2p − 1) 2 n 2 z log 2 1 − t n 2 x + (2p − 1) 2 n 2 z 1 + t n 2 x + (2p − 1) 2 n 2 z ≤ 0. (47) By using the monotonically increasing property of f (x) = 1 x log 2 1 + tx 1 − tx , (0 ≤ x ≤ 1),(48) we have ∂M r (ε(ρ)) ∂|n x | = t 2 · |n x | n 2 x + (2p − 1) 2 n 2 z · log 2 1 − t n 2 x + (2p − 1) 2 n 2 z 1 + t n 2 x + (2p − 1) 2 n 2 z − t 2 · |n x |[1 − (2p − 1) 2 ] n 2 x + (2p − 1) 2 n 2 y + (2p − 1) 2 n 2 z · log 2 1 − t n 2 x + (2p − 1) 2 n 2 z + (2p − 1) 2 n 2 y 1 + t n 2 x + (2p − 1) 2 n 2 z + (2p − 1) 2 n 2 y ≥ t 2 · |n x |(2p − 1) 2 n 2 x + (2p − 1) 2 n 2 z log 2 1 − t n 2 x + (2p − 1) 2 n 2 z 1 + t n 2 x + (2p − 1) 2 n 2 z . (49) So when n x ≤ 0, we have ∂Mr(ε(ρ)) ∂nx ≥ 0. Because M r (ε(ρ)) is an even function of the variable n x , we can conclude that M r (ε(ρ)) is monotonic decreasing function of variable |n x |, i.e. ∂M r (ε(ρ)) ∂|n x | ≤ 0.(50) The partial derivative of M r (ε(ρ)) with respect to t is ∂M r (ε(ρ)) ∂t = n 2 x + (2p − 1) 2 n 2 z 2 log 2 1 − t n 2 x + (2p − 1) 2 n 2 z 1 + t n 2 x + (2p − 1) 2 n 2 z + n 2 x + (2p − 1) 2 n 2 z + (2p − 1) 2 n 2 z 2 · log 2 1 + t n 2 x + (2p − 1) 2 n 2 z + (2p − 1) 2 n 2 y 1 − t n 2 x + (2p − 1) 2 n 2 z + (2p − 1) 2 n 2 y ≥ 0.(51) So the measure M r (ε(ρ)) is a monotonically decreasing function with respect to the variable |n x |, |n z |, and a monotonically increasing function with respect to the variable t. On the other hand, we can obtain that ∂M r (ρ) ∂|n x | = ∂M r (ρ) ∂|n y | ∂|n y | ∂|n x | = ∂M r (ρ) ∂|n y | ∂ 1 − n 2 x − n 2 z ∂|n x | = ∂M r (ρ) ∂|n y | −|n x | 1 − n 2 x − n 2 z .(52) By using Eq.(41), one gets ∂M r (ρ) ∂|n x | ≤ 0.(53) Similarly, we have ∂M r (ρ) ∂|n z | ≤ 0.(54) Combining Eqs. (43), (47), (50), (51), (53), and (54), one arrives at that the quantum state order does not change after a single-qubit goes through a bit flip channel. Thus Proposition 2 is true. Proposition 3. Assume we choose M l1 (ρ) as the measure of imaginarity, then the quantum state order does not change after a single-qubit goes through a bit flip channel. Proof. By using Eq.(45) we have M l1 (ε(ρ)) = t|(2p − 1)n y |. (55) Considering the above equation and Eq. (39), it is not difficult to obtain that when we choose M l1 (ρ) as the measure of imaginarity, the quantum state order does not change after a single-qubit goes through a bit flip channel. This implies that Proposition 3 holds. Now let us investigate the case that when the imaginarity measure M r (ρ) has been choosed, and the quantum channel is the phase flip channel Λ. Here the quantum state of the qubit is stated as Eq. (38), the phase flip channel Λ is expressed by the real Kraus operators K 0 = √ pI, K 1 = √ 1 − p|0 0|, K 2 = √ 1 − p|1 1|, 0 ≤ p ≤ 1. For this case we will prove the following proposition. Proposition 4. Suppose we choose M r (ρ) as the measure of imaginarity, then the quantum state order does not change after a single-qubit goes through a phase flip channel. Proof. After a qubit passes through a phase flip channel, the quantum state can be written as Λ(ρ) = K 0 ρK † 0 + K 1 ρK † 1 + K 2 ρK † 2 = 1+tnz 2 tp(nx−iny) 2 tp(nx+iny) 2 1−tnz 2 .(56) One can easily deduce M r (Λ(ρ)) = H( 1 + t n 2 z + p 2 n 2 x 2 ) − H( 1 + t n 2 z + p 2 (1 − n 2 z ) 2 ). Proof. By using Eq.(56), we have M l1 (Λ(ρ)) = tp|n y |. By considering Eq.(39) and above equation one can easily see that Proposition 5 holds. Next we discuss the influence of an amplitude damping channel on the ordering of quantum states. Here an amplitude damping channel Γ is expressed by the real Kraus operators {K 0 = |0 0| + √ 1 − p|1 1|, K 1 = √ p|0 1|, 0 ≤ p ≤ 1}. We will prove the following result. Proposition 6. When qubit state ρ satisfies n z ≤ 0, if one chooses M r (ρ) as the measure of imaginarity, then the quantum state order does not change after a singlequbit goes through an amplitude damping channel Proof. For a qubit state stated by Eq. (38), the amplitude damping channel leads it to Γ(ρ) = K 0 ρK † 0 + K 1 ρK † 1 = 1+tnz 2 + p(1−tnz ) 2 √ 1−pt(nx−iny) 2 √ 1−pt(nx+iny) 2 (1−p)(1−tnz ) 2 .(62) One can easily obtain the measure of imaginarity based on relative entropy M r (Γ(ρ)) = H( 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 n 2 x 2 ) − H( 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) 2 ). (63) Therefore, we get the partial derivatives ∂M r (Γ(ρ)) ∂t = [p + tn z (1 − p)]n z (1 − p) + t(1 − p)n 2 x 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 n 2 x × log 2 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 n 2 x 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 n 2 x + [p + tn z (1 − p)]n z (1 − p) + t(1 − p)(n 2 x + n 2 y ) 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 (n 2 x + n 2 y ) × log 2 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 (n 2 x + n 2 y ) 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 (n 2 x + n 2 y ) ≥ 0; (64) ∂M r (Γ(ρ)) ∂|n x | = (1 − p)t 2 |n x | 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 n 2 x × log 2 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 n 2 x 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 n 2 x ≤ 0; (65) ∂M r (Γ(ρ)) ∂n z = [p + tn z (1 − p)]t(1 − p) 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 n 2 x × log 2 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 n 2 x 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 n 2 x + [p + tn z (1 − p)]t(1 − p) − (1 − p)t 2 n z 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) × log 2 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) .(66) By using the monotonically increasing property of f (x) = 1 x log 2 1 + x 1 − x , (0 ≤ x ≤ 1),(67) and 0 ≤ n 2 x ≤ 1 − n 2 z , then we have ∂M r (Γ(ρ)) ∂n z ≥ Min [p + tn z (1 − p)]t(1 − p) 2 [p + tn z (1 − p)] 2 × log 2 1 − [p + tn z (1 − p)] 2 1 + [p + tn z (1 − p)] 2 + [p + tn z (1 − p)]t(1 − p) − (1 − p)t 2 n z 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) × log 2 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) , (1 − p)t 2 n z 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) × log 2 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) .(68)Let A = [p + tn z (1 − p)]t(1 − p) 2 [p + tn z (1 − p)] 2 × log 2 1 − [p + tn z (1 − p)] 2 1 + [p + tn z (1 − p)] 2 + [p + tn z (1 − p)]t(1 − p) − (1 − p)t 2 n z 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) × log 2 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) ,(69)B = (1 − p)t 2 n z 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) × log 2 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) . (70) A − B = [p + tn z (1 − p)]t(1 − p) 2 [p + tn z (1 − p)] 2 × log 2 1 − [p + tn z (1 − p)] 2 1 + [p + tn z (1 − p)] 2 + [p + tn z (1 − p)]t(1 − p) 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) × log 2 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) . (71) So when p + tn z (1 − p) ≥ 0, we have A ≥ B; when p + tn z (1 − p) ≤ 0, we have A ≤ B. In the situation p + tn z (1 − p) ≤ 0, one gets ∂M r (Γ(ρ)) ∂n z ≥ A = −t(1 − p) 2 log 2 1 − [p + tn z (1 − p)] 2 1 + [p + tn z (1 − p)] 2 + t(1 − p) 2 p(1 − tn z ) [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) × log 2 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) ≥ 0.(72) On the other hand, in the case p + tn z (1 − p) ≥ 0 we have ∂M r (Γ(ρ)) ∂n z ≥ B = (1 − p)t 2 n z 2 [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) × log 2 1 − [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) 1 + [p + tn z (1 − p)] 2 + (1 − p)t 2 (1 − n 2 z ) . (73) So when n z ≤ 0 and p + tn z (1 − p) ≥ 0, we have ∂Mr(Γ(ρ)) ∂nz ≥ 0, that is, if n x , t, p are fixed and satisfy n z ≤ 0 and p+tn z (1−p) ≥ 0, then the function M r (Γ(ρ)) is monotonically increasing with respect to the variables n z . Combining Eqs. (43), (53), (54), (64), (65), (72), (73), we arrive at that when qubit state ρ satisfies n z ≤ 0, if one chooses M r (ρ) as the measure of imaginarity, then the quantum state order does not change after a singlequbit goes through an amplitude damping channel. Thus we have demonstrated Proposition 6. Proposition 7. When we take M l1 (ρ) as the measure of imaginarity, then the quantum state order does not change after a single-qubit goes through an amplitude damping channel. Proof. By using Eq.(62) we can easily deduce the measure of imaginarity M l1 (Γ(ρ)) = t 1 − p|n y |. (74) By using Eq.(39) and above equation one can easily obtain that Proposition 7 is true. V. CONCLUSION In summary, we study the measures of imaginarity in the framework of resource theory and the quantum state order after a quantum system passes through a real channel. We define functions based on l 1 norm and the convex roof extended, and show that they are the measures of imaginarity. The relationships between relative entropy of imaginarity M r (ρ) and the imaginarity measure M l1 (ρ) based on l 1 norm for the single-qubit pure state ρ is investigated. We also prove that the functions based on l p norm and p−norm are not the measures of imaginarity. Moreover, we demonstrate that the measure M l1 (ρ) and the measure M r (ρ) are of the same order for qubit quantum states and discuss the influences of the bit flip channel, phase damping channel and amplitude flip channel on single-qubit state order, respectively. Thus we have proved that for a qubit pure state |ψ ,M r (|ψ ) ≤ M l1 (|ψ ) is true.Clearly, when M l1 (|ψ ) = 1, one has |cf − de| = 1 2 . So λ 1 = 1 2 , which induces 1 = H(λ 1 ) = M r (|ψ ). This fact shows that M l1 (|ψ ) = M r (|ψ ), if M l1 (|ψ ) = 1. So Theorem 4 holds. Theorem 5 . 5If M (|ψ ) is a measure of imaginarity of pure state |ψ , then the convex roof extended M (ρ) = min {pi,|ψi } i Definition 2 . 2Let M A and M B be two measures of imaginarity. For arbitrary two quantum states ρ 1 and ρ 2 , if the following relationship Definition 3 . 3Let M be a measure of imaginarity and ε be a quantum channel. For arbitrary two quantum states ρ 1 and ρ 2 , if ACKNOWLEDGMENTSThe partial derivatives are ∂M r (Λ(ρ)) ∂t = p 2 n 2 x + n 2 z 2 log 2 1 − t n 2 z + p 2 n 2 x 1 + t n 2 z + p 2 n 2(60) Here the first inequality in Eq.(60) comes from the fact that when t is fixed, 1x log 2 1−tx 1+tx is monotonically decreasing function with respect to x and n 2 z +n 2 x ≤ 1; the second inequality in Eq.(60) is based on that when n z , t are fixed,is monotonically decreasing function with respect to p 2 .By Eqs. 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{'abstract': 'Complex numbers are widely used in both classical and quantum physics, and play an important role in describing quantum systems and their dynamical behavior. In this paper we study several measures of imaginarity of quantum states in the framework of resource theory, such as the measures based on l1 norm, and convex function, etc. We also investigate the influence of the quantum channels on quantum state order for a single-qubit. *', 'arxivid': '2210.14443', 'author': ['Qiang Chen \nSchool of Mathematical Sciences\nHebei Normal University\n050024ShijiazhuangChina\n', 'Ting Gao \nSchool of Mathematical Sciences\nHebei Normal University\n050024ShijiazhuangChina\n', 'Fengli Yan \nCollege of Physics\nHebei Key Laboratory of Photophysics Research and Application\nHebei Normal University\n050024ShijiazhuangChina\n'], 'authoraffiliation': ['School of Mathematical Sciences\nHebei Normal University\n050024ShijiazhuangChina', 'School of Mathematical Sciences\nHebei Normal University\n050024ShijiazhuangChina', 'College of Physics\nHebei Key Laboratory of Photophysics Research and Application\nHebei Normal University\n050024ShijiazhuangChina'], 'corpusid': 253116678, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 16714, 'n_tokens_neox': 14257, 'n_words': 8045, 'pdfsha': '517aa64e8f54d12a75d7e1ef785041e690c04097', 'pdfurls': ['https://export.arxiv.org/pdf/2210.14443v2.pdf'], 'title': ['Measures of imaginarity and quantum state order', 'Measures of imaginarity and quantum state order'], 'venue': []}

arxiv

Interpretable and Fine-Grained Visual Explanations for Convolutional Neural Networks Jörg Wagner joerg.wagner3@de.bosch.com Jan Mathias Köhler Tobias Gindele Leon Hetzel Jakob Thaddäus Wiedemer Sven Behnke behnke@cs.uni-bonn.de Jörg Wagner Bosch Center for Artificial Intelligence (BCAI) Germany University of Bonn Germany Jan Mathias Köhler Bosch Center for Artificial Intelligence (BCAI) Germany Tobias Gindele Bosch Center for Artificial Intelligence (BCAI) Germany Leon Hetzel Bosch Center for Artificial Intelligence (BCAI) Germany Jakob Thaddäus Wiedemer Bosch Center for Artificial Intelligence (BCAI) Germany Sven Behnke University of Bonn Germany Interpretable and Fine-Grained Visual Explanations for Convolutional Neural Networks The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp. 9097-9107; the final, published version of this paper is available on IEEE Xplore. To verify and validate networks, it is essential to gain insight into their decisions, limitations as well as possible shortcomings of training data. In this work, we propose a post-hoc, optimization based visual explanation method, which highlights the evidence in the input image for a specific prediction. Our approach is based on a novel technique to defend against adversarial evidence (i.e. faulty evidence due to artefacts) by filtering gradients during optimization. The defense does not depend on human-tuned parameters. It enables explanations which are both fine-grained and preserve the characteristics of images, such as edges and colors. The explanations are interpretable, suited for visualizing detailed evidence and can be tested as they are valid model inputs. We qualitatively and quantitatively evaluate our approach on a multitude of models and datasets. Introduction Convolutional Neural Networks (CNNs) have proven to produce state-of-the-art results on a multitude of vision benchmarks, such as ImageNet [34], Caltech [12] or Cityscapes [9] which led to CNNs being used in numerous real-world systems (e.g. autonomous vehicles) and services (e.g. translation services). Though, the use of CNNs in safety-critical domains presents engineers with challenges resulting from their black-box character. A better understanding of the inner workings of a model provides hints for improving it, understanding failure cases and it may reveal shortcomings of the training data. Additionally, users generally trust a model more when they understand its decision process and are able to anticipate or verify outputs [30]. To overcome the interpretation and transparency disadvantage of black-box models, post-hoc explanation meth- * contributed while working at BCAI. We additionally thank Volker Fischer, Michael Herman, Anna Khoreva for discussions and feedback. Figure 1: Fine-grained explanations computed by removing irrelevant pixels. a) Input image with softmax score p(c ml ) of the most-likely class. Our method tries to find a sparse mask (c) with irrelevant pixels set to zero. The resulting explanation (b), i.e.: 'image × mask', is optimized in the image space and, thus, can directly be used as model input. The parameter λ is optimized to produce an explanation with a softmax score comparable to the image. ods have been introduced [53,35,42,49,32,17,11]. These methods provide explanations for individual predictions and thus help to understand on which evidence a model bases its decisions. The most common form of explanations are visual, image-like representations, which depict the important pixels or image regions in a human interpretable manner. In general, an explanation should be easily interpretable (Sec. 4.1). Additionally, a visual explanation should be class discriminative and fine-grained [35] (Sec. 4.2). The latter property is particularly important for classification tasks in the medical [20,18] domain, where fine structures (e.g. capillary hemorrhages) have a major influence on the classification result (Sec. 5.2). Besides, the importance of different color channels should be captured, e.g. to uncover a color bias in the training data (Sec. 4.3). Moreover, explanations should be faithful, meaning they accurately explain the function of the black-box model [35]. To evaluate the faithfulness (Sec. 5.1), recent work [35,32,7] introduce metrics which are based on model predictions of explanations. To be able to compute such metrics without having to rely on proxy measures [35], it is beneficial to employ explanation methods which directly generate valid model inputs (e.g. a perturbed version of the image). A major concern of optimization based visual explanation methods is adversarial evidence, i.e. faulty evidence generated by artefacts introduced in the computation of the explanation. Therefore, additional constraints or regularizations are used to prevent such faulty evidence [17,11,14]. A drawback of these defenses are added hyperparameters and the necessity of either a reduced resolution of the explanation or a smoothed explanation (Sec. 3.2), thus, they are not well suited for displaying fine-grained evidence. Our main contribution is a new adversarial defense technique which selectively filters gradients in the optimization which would lead to adversarial evidence otherwise (Sec. 3.2). Using this defense, we extend the work of [17] and propose a new fine-grained visual explanation method (FGVis). The proposed defense is not dependend on hyperparameters and is the key to produce fine-grained explanations ( Fig. 1) as no smoothing or regularizations are necessary. Like other optimization-based approaches, FGVis computes a perturbed version of the original image, in which either all irrelevant or the most relevant pixels are removed. The resulting explanations (Fig 1 b) are valid model inputs and their faithfulness can, thus, be directly verified (as in methods from [17,14,6,11]). Moreover, they are additionally fine-grained (as in methods from [35,38,48,42]). To the best of our knowledge, this is the first method to be able to produce fine-grained explanations directly in the image space. We evaluate our defense (Sec. 3.2) and FGVis (Sec. 4 and 5) qualitatively and quantitatively. Related Work Various methods to create explanations have been introduced. Thang et al. [50] and DU et al. [13] provide a survey of these. In this section, we give an overview of explanation methods which generate visual, image-like explanations. Backpropagation Based Methods (BBM). These methods generate an importance measure for each pixel by backpropagating an error signal to the image. Simonyan et al. [38], which build on work of Baehrens et al. [5], use the derivative of a class score with respect to the image as an importance measure. Similar methods have been introduced in Zeiler et al. [48] and Springenberg et al. [42], which additionally manipulate the gradient when backpropagating through ReLU nonlinearities. Integrated Gradients [43] additionally accumulates gradients along a path from a base image to the input image. SmoothGrad [40] and VarGrad [1] visually sharpen explanations by combining multiple explanations of noisy copies of the image. Other BBMs such as Layer-wise Relevance Propagation [4], DeepLift [37] or Excitation Backprop [49] utilize top-down relevancy propagation rules. BBMs are usually fast to compute and produce fine-grained importance/relevancy maps. However, these maps are generally of low quality [11,14] and are less interpretable. To verify their faithfulness it is necessary to apply proxy measures or use pre-processing steps, which may falsify the result. Activation Based Methods (ABM). These approaches use a linear combination of activations from convolutional layers to form an explanation. Prominent methods of this category are CAM (Class Activation Mapping) [53] and its generalizations Grad-CAM [35] and Grad-CAM++ [7]. These methods mainly differ in how they calculate the weights of the linear combination and what restrictions they impose on the CNN. Extensions of such approaches have been proposed in Selvaraju et al. [35] and Du et al. [14], which combine ABMs with backpropagation or perturbation based approaches. ABMs generate easy to interpret heat-maps which can be overlaid on the image. However, they are generally not well suited to visualize fine-grained evidence or color dependencies. Additionally, it is not guaranteed that the resulting explanations are faithful and reflect the decision making process of the model [14,35]. Perturbation Based Methods (PBM). Such approaches perturb the input and monitor the prediction of the model. Zeiler et al. [48] slide a grey square over the image and use the change in class probability as a measure of importance. Several approaches are based on this idea, but use other importance measures or occlusion strategies. Petsiuk et al. [32] use randomly sampled occlusion masks and define importance based on the expected model score over masks. LIME [33] uses a super-pixel based occlusion strategy and a surrogate model to compute importance scores. Further super-pixel or segment based methods are introduced in Seo et al. [36] and Zhou et al. [52]. The so far mentioned approaches do not need access to the internal state or structure of the model. Though, they are often quite time consuming and only generate coarse explanations. Other PBMs generate an explanation by optimizing for a perturbed version of the image [11,17,14,6]. The perturbed image e is defined by e = m · x + (1 − m) · r, where m is a mask, x the input image, and r a reference image containing little information (Sec. 3.1). To avoid adversarial evidence, these approaches need additional regularizations [17], constrain the explanation (e.g. optimize for a coarse mask [6,17,14]), introduce stochasticity [17], or utilize regularizing surrogate models [11]. These approaches generate easy to interpret explanations in the image space, which are valid model inputs and faithful (i.e. a faithfulness measure is incorporated in the optimization). Our method also optimizes for a perturbed version of the input. Compared to existing approaches we propose a new adversarial defense technique which filters gradients during optimization. This defense does not need hyperparameters which have to be fine-tuned. Besides, we optimize each pixel individually, thus, the resulting explanations have no limitations on the resolution and are fine-grained. Explaining Model Predictions Explanations provide insights into the decision-making process of a model. The most universal form of explanations are global ones which characterize the overall model behavior. Global explanations specify for all possible model inputs the corresponding output in an intuitive manner. A decision boundary plot of a classifier in a low-dimensional vector space, for example, represents a global explanation. For high-dimensional data and complex models, it is practically impossible to generate such explanations. Current approaches therefore utilize local explanations 1 , which focus on individual inputs. Given one data point, these methods highlight the evidence on which a model bases its decisions. As outlined in Sec. 2, the definition of highlighting depends on the used explanation method. In this work, we follow the paradigm introduced in [17] and directly optimize for a perturbed version of the input image. Such an approach has several advantages: 1) The resulting explanations are interpretable due to their imagelike nature; 2) Explanations represent valid model inputs and are thus testable; 3) Explanations are optimized to be faithful. In Sec. 3.1 we briefly review the general paradigm of optimization based explanation methods before we introduce our novel adversarial defense technique in Sec. 3.2. Perturbation based Visual Explanations Following the paradigm of optimization based explanation methods, which compute a perturbed version of the image [17,14,6,11], an explanation can be defined as: Explanation by Preservation: The smallest region of the image which must be retained to preserve the original model output (i.e. minimal sufficient evidence). Explanation by Deletion: The smallest region of the image which must be deleted to change the model output. To formally derive an explanation method based on this paradigm, we assume that a CNN f cnn is given which maps an input image x ∈ R 3×H×W to an output y x = f cnn (x; θ cnn ). The ouput y x ∈ R C is a vector representing the softmax scores y c x of the different classes c. Given an input image x, an explanation e * c T of a target class c T (e.g. the most-likely class c T = c ml ) is computed by removing either relevant (deletion) or irrelevant, not supporting c T , information (preservation) from the image. Since it is not possible to remove information without replacing it, and we do not have access to the image generating process, we have to use an approximate removal operator [17]. A common approach is to use a mask based operator Φ, which computes a weighted average between the image x and a reference image r, using a mask m c T ∈ [0, 1] 3×H×W : e c T = Φ(x, m c T ) = x · m c T + (1 − m c T ) · r. (1) Common choices for the reference image are constant values (e.g. zero), a blurred version of the original image, Gaussian noise, or sampled references of a generative model [17,14,6,11]. In this work, we take a zero image as reference. In our opinion, this reference produces the most pleasing visual explanations, since irrelevant image areas are set to zero 2 ( Fig. 1) and not replaced by other structures. In addition, the zero image (and random image) carry comparatively little information and lead to a model prediction with a high entropy. Other references, such as a blurred version of the image, usually result in lower prediction entropies, as shown in Sec. A3.1. Due to the additional computational effort, we have not considered model-based references as proposed in Chang et al. [6]. In addition, a similarity metric ϕ(y c T x , y c T e ) is needed, which measures the consistency of the model output generated by the explanation y c T e and the output of the image y c T x with respect to a target class c T . This similarity metric should be small if the explanation preserves the output of the target class and large if the explanation manages to significantly drop the probability of the target class [17]. Typical choices for the metric are the cross-entropy with the class c T as a hard target [24] or the negative softmax score of the target class c T . The similarity metric ensures that the explanation remains faithful to the model and thus accurately explains the function of the model, this property is a major advantage of PBMs. Using the mask based definition of an explanation with a zero image as reference (r = 0) as well as the similarity metric, a preserving explanation can be computed by: e * c T = m * c T · x, m * c T = arg min mc T {ϕ(y c T x , y c T e ) + λ · m c T 1 }. (2) We will refer to the optimization in Eq. 2 as the preservation game. Masks ( Fig. 2 / b2) 3 generated by this game are sparse (i.e. many pixels are zero / appear black; enforced by minimizing m c T 1 ) and only contain large values at most important pixels. The corresponding explanation is computed by multiplying the mask with the image (Fig. 2 / c2). Alternatively, we can compute a deleting explanation using: e * c T = m * c T · x, m * c T = arg max mc T {ϕ(y c T x , y c T e ) + λ · m c T 1 }. (3) This optimization will be called deletion game henceforward. Masks ( Fig. 2 / b1) generated by this game contain mainly ones (i.e. appear white; enforced by maximizing m c T 1 in Eq. 3) and only small entries at pixels, which provide the most prominent evidence for the target class. The colors in a mask of the deletion game are complementary to the image colors. To obtain a true-color representation analogous to the preservation game, one can alternatively visualize the complementary mask ( Fig. 2 / d1): m * c T = (1 − m * c T ) . A resulting explanation of the deletion game, as defined in Eq. 3, is visualized in Fig. 2 / c1. This explanation is visually very similar to the original image as only a few pixels need to be deleted to change the model output. In the remaining of the paper for better visualization, we depict a modified version of the explanation for the deletion game: ẽ * c T = x · (1 − m * c T ) . This explanation has the same properties as the one of the preservation game, i.e. it only highlights the important evidence. We observe that the deletion game generally produces sparser explanations compared to the preservation game, as less pixels have to be removed to delete evidence for a class than to maintain evidence by preserving pixels. To solve the optimization in Eq. 2 and Eq. 3, we utilize Stochastic Gradient Descent and start with an explanation e 0 c T = 1 · x identical to the original image (i.e. a mask initialized with ones). As an alternative initialization of the masks, we additionally explore a zero initialization m 0 c T = 0. In this setting the initial explanation contains no evidence towards any class and the optimization iteratively has to add relevant (generation game) or irrelevant, not supporting the class c T , information (repression game). The visualizations of the generation game are equivalent to those of the preservation game, the same holds for the deletion and repression game. In our experiments the deletion game produces the most fine-grained and visually pleasing explanations. Compared to the other games it usually needs the least amount of optimization iterations since we start with m 0 c T = 1 and comparatively few mask values have to be changed to delete the evidence for the target class. A comparison and additional characteristics of the four optimization settings (i.e. games) are included in Sec. A3.5. Defending against Adversarial Evidence CNNs have been proven susceptible to adversarial images [45,19,27], i.e. a perturbed version of a correctly classified image crafted to fool a CNN. Due to the computational similarity of adversarial methods and optimization based visual explanation approaches, adversarial noise is also a concern for the latter methods and one has to ensure that an explanation is based on true evidence present in the image and not on false adversarial evidence introduced during optimization. This is particularly true for the generation/repression game as their optimization start with m 0 c T = 0 and iteratively adds information. [17] and [11] showed the vulnerability of optimization based explanation methods to adversarial noise. To avoid adversarial evidence, explanation methods use stochastic operations [17], additional regularizations [17,11], optimize on a low-resolution mask with upsampling of the computed mask [17,14,6], or utilize a regularizing surrogate model [11]. In general, these operations impede the generation of adversarial noise by obscuring the gradient direction in which the model is susceptible to false evidence, or by constraining the search space for potential adversarials. These techniques help to reduce adversarial evidence, but also introduce new drawbacks: 1) Defense capabilities usually depend on human-tuned parameters; 2) Explanations are limited to being low resolution and/or smooth, which prevents fine-grained evidence from being visualized. A novel Adversarial Defense. To overcome these drawbacks, we propose a novel adversarial defense which filters gradients during backpropagation in a targeted way. The basic idea of our approach is: A neuron within a CNN is only allowed to be activated by the explanation e c T if the same neuron was also activated by the original image x. If we regard neurons as indicators for the existence of features (e.g. edges, object parts, . . . ), the proposed constraint enforces that the explanation e c T can only contain features which exist at the same location in the original image x. By ensuring that the allowed features in e c T are a subset of the features in x it prevents the generation of new evidence. This defense technique can be integrated in the introduced explanation methods via an optimization constraint: 0 ≤ h l i (e c T ) ≤ h l i (x), if h l i (x) ≥ 0, 0 ≥ h l i (e c T ) ≥ h l i (x), otherwise,(4) where h l i is the activation of the i-th neuron in the l-th layer of the network after the nonlinearity. For brevity, the index i references one specific feature at one spatial position in the activation map. This constraint is applied after all nonlinearity-layers (e.g. ReLU-Layers) of the network, besides the final classification layer. It ensures that the absolute value of activations can only be reduced towards values representing lower information content (we assume that zero activations have the lowest information as commonly applied in network pruning [22]). To solve the optimization with subject to Eq. 4, one could incorporate the constraints via a penalty function in the optimization loss. The drawback is one additional hyperparameter. Alternatively, one could add an additional layerh l i after each nonlinearity which ensures the validity of Eq. 4: h l i (e c T ) = min(bu, max(bl, h l i (e c T ))), bu = max(0, h l i (x)), bl = min(0, h l i (x)),(5) where h l i (e c T ) is the actual activation of the original nonlinearity-layer andh l i (e c T ) the adjusted activation after ensuring the bounds bu, bl of the original input. For instance, for a ReLU nonlinearity, the upper bound bu is equal to h l i (x) and the lower bound bl is zero. We are not applying this method as it changes the architecture of the model which we try to explain. Instead, we clip gradients in the backward pass of the optimization, which lead to a violation of Eq. 4. This is equivalent to adding an additional clipping-layer after each nonlinearity which acts as the identity in the forward pass and uses the gradient update of Eq. 5 in the backward pass. When backpropagating an error-signalγ l i through the clipping-layer, the gradient update rule for the resulting error γ l i is defined by: γ l i =γ l i · [h l i (e c T ) ≤ bu] · [h l i (e c T ) ≥ bl],(6) where [ · ] is the indicator function and bl, bu the bounds computed in Eq. 5. This clipping only affects the gradients of the similarity metric ϕ(· , ·) which are propagated through the network. The proposed gradient clipping does not add hyperparameters and keeps the original structure of the model during the forward pass. Compared to other adversarial defense techniques ( [11], [17], [6]), it imposes no constraint on the explanation (e.g. resolution/smoothness constraints), enabling fine-grained explanations. Validating the Adversarial Defense. To evaluate the performance of our defense, we compute an explanation for a class c A for which there is no evidence in the image (i.e. it is visually not present). We approximate c A with the least-likely class c ll considering only images which yield very high predictive confidence for the true class p(c true ) ≥ 0.995. Using c ll as the target class, the resulting explanation method without defense is similar to an adversarial attack (the Iterative Least-Likely Class Method [27]). A correct explanation for the adversarial class c A should be "empty" (i.e. grey), as seen in Fig. 3 b, top row, when using our adversarial defense. If, on the other hand, the explanation method is susceptible to adversarial noise, the optimization procedure should be able to perfectly generate an explanation for any class. This behavior can be seen in Fig. 3 c, top row. The shown explanation for the adversarial Model No Defense Defended VGG16 [39] 100.0 % 0.2 % AlexNet [26] 100.0 % 0.0 % ResNet50 [23] 100.0 % 0.0 % GoogleNet [44] 100.0 % 0.0 % Table 1: Ratio how often an adversarial class c A was generated, using the generation game with no sparsity loss on VGG16 with and without our defense. class (c A : limousine) contains primarily artificial structures and is classified with a probability of 1 as limousine. We also depict the explanation of the predicted class (c pred : agama). The explanation with our defense results in a meaningful representation of the agama (Fig. 3 b, bottom row); without defense ( Fig. 3 c / d, bottom row) it is much more sparse. As there is no constraint to change pixel values arbitrarily, we assume the algorithm introduces additional structures to produce a sparse explanation. A quantitative evaluation of the proposed defense is reported in Tab. 1. We generate explanations for 1000 random ImageNet validation images and use a class c A as the explanation target 4 . To ease the generation of adversarial examples, we set the sparsity loss to zero and only use the similarity metric which tries to maximize the probability of the target class c A . Without an employed defense technique, the optimization is able to generate an adversarial explanation for 100% of the images. Applying our defense (Eq. 6), the optimization nearly never was able to do so. The two adverarial examples generated in VGG16 have a low confidence, so we assume that there has been some evidence for the chosen class c A in the image. Our proposed technique is thus well suited to defend against adversarial evidence. Qualitative Results Implementation details are stated in Sec. A2. Interpretability Comparison of methods. Using the deletion game we compute mean explanation masks for GoogleNet and compare these in Fig. 5 with state-of-the-art methods. Our method delivers the most fine-grained explanation by deleting important pixels of the target object. Especially explanations b), f), and g) are coarser and, therefore, tend to include background information not necessary to be deleted to change the original prediction. The majority of pixels highlighted by FGVis form edges of the object. This cannot be seen in other methods. The explanations from c) and d) are most similar to ours. However, our masks are computed to directly produce explanations which are viable network inputs and are, therefore, verifiable -The deletion of the highlighted pixels prevents the model from correctly predicting the object. This statement does not necessarily hold for explanations calculated with methods c) and d). Architectural insights. As first noted in [31] explanations using backpropagation based approaches show a gridlike pattern for ResNet. In general, [31] demonstrate that the network structure influences the visualization and assume that for ResNet the skip connections play an important role in their explanation behavior. As shown in Fig 6 this pattern is also visible in our explanations to an even finer degree. Interestingly, the grid pattern is also visible to a lesser extent outside the object. A detailed investigation of this phenomenon is left for future research. See A3.4 for a comparison of explanations between models. Class Discriminative / Fine-Grained Visual explanation methods should be able to produce class discriminative (i.e. focus on one object) and finegrained explanations [35]. To test FGVis with respect to these properties, we generate explanations for images containing two objects. The objects are chosen from highly different categories to ensure little overlapping evidence. In Fig. 4, we visualize explanations of three such images, computed using the deletion game and GoogleNet. Additional results can be found in Sec. A3.2. FGVis is able to generate class discriminative explanations and only highlights pixels of the chosen target class. Even partially overlapping objects, as the elkhound and ball in Fig. 4, first row, or the bridge and schooner in Fig. 4, Figure 4: Explanation masks for images with multiple objects computed using the deletion game and GoogleNet. FGVis produces class discriminating explanations, even when objects partially overlap. Additionally, FGVis is able to visualize fine-grained details down to the pixel level. [17], c) Gradient [38], d) Guided Backprop [42] , e) Contrastive Excitation Backprop [49], f) Grad-CAM [35], g) Occlusion [48], h) FGVis (ours). The masks of all reference methods are based on work by [17]. Due to our detailed and sparse masks, we plot them in a larger size. masks (b, d) show a grid-like pattern, as also observed in [31] for ResNet50. third row, are correctly discriminated. One major advantage of FGVis is its ability to visualize fine-grained details. This property is especially visible in Fig 4, second row, which shows an explanation for the target class fence. Despite the fine structure of the fence, FGVis is able to compute a precise explanation which mainly contains fence pixels. Investigating Biases of Training Data An application of explanation methods is to identify a bias in the training data. Especially for safety-critical, highrisk domains (e.g. autonomous driving), such a bias can lead to failures if the model does not generalize to the real world. Learned objects. One common bias is the coexistence of objects in images which can be depicted using FGVis. In Sec. A3.3, we describe such a bias in ImageNet for sports equipment appearing in combination with players. Learned color. Objects are often biased towards specific colors. FGVis can give a first visual indication for the importance of different color channels. We investigate if a VGG16 model trained on ImageNet shows such a bias using the preservation game. We focus on images of school buses and minivans and compare explanations ( Fig. 7; all correctly predicted images in Fig. A6 and A8). Explanations of minivans focus on edges, not consistently preserving the color compared to school buses with yellow dominating those explanations. This is a first indication for the importance of color for the prediction of school buses. To verify the qualitative finding, we quantitatively give an estimation of the color bias. As an evaluation we swap each of the three color channels BGR to either RBG or GRB and calculate the ratio of maintained true classifications on the validation data after the swap. For minivans 83.3% (averaged over RBG and GRB) of the 21 correctly classified images keep their class label, for school buses it is only 8.3% of 42 images. For 80 ImageNet classes at least 75% of images are no longer truly classified after the color swap. We show the results for the most and least affected 19 classes and minivan / school bus in Tab. A3. To the best of our knowledge, FGVis is the first method used to highlight color channel importance. Quantitative Results Faithfulness of Explanations The faithfulness of generated visual explanations to the underlying neural network is an important property of explanation methods [35]. To quantitatively compare the faithfulness of methods, Petsiuk et al. [32] proposed causal metrics which do not depend on human labels. These metrics are not biased towards human perception and are thus well suited to verify if an explanation correctly represents the evidence on which a model bases its prediction. We use the deletion metric [32] to evaluate the faith- fulnes of explanations generated by our method. This metric measures how the removal of evidence effects the prediction of the used model. The metric assumes that an importance map is given, which ranks all image pixels with respect to their evidence for the predicted class c ml . By iteratively removing important pixels from the input image and measuring the resulting probability of the class c ml a deletion curve can be generated, whose area under the curve AUC is used as a measure of faithfulness (Sec. A4.1). In Tab. 2, we report the deletion metric of FGVis, computed on the validation split of ImageNet using different models. We use the deletion game to generate masks m ml , which determine the importance of each pixel. A detailed description of the experiment settings as well as additional figures, can be found in Sec. A4.1. FGVis outperforms the other explanation methods on both models by a large margin. This performance increase can be attributed to the ability of FGVis to visualize fine-grained evidence. All other approaches are limited to coarse explanations, either due to computational constraints or due to the used measures to avoid adversarial evidence. The difference between the two model architectures can most likely be attributed to the superior performance of ResNet50, resulting in on average higher softmax scores over all validation images. Method ResNet50 VGG16 Grad-Cam [35] 0.1232 0.1087 Sliding Window [48] 0.1421 0.1158 LIME [33] 0.1217 0.1014 RISE [32] 0.1076 0.0980 FGVis (ours) 0.0644 0.0636 Visual explanation for medical images We evaluate FGVis on a real-world use case to identify regions in eye fundus images which lead a CNN to classify the image as being affected with referable diabetic retinopathy (RDR). Using the deletion game we derive a weaklysupervised approach to detect RDR lesions. The setup, used network, as well as details on the disease and training data are described in A4.2. To evaluate FGVis, the DiaretDB1 dataset [25] is used containing 89 fundus images with different lesion types, ground truth marked by four experts. To quantitatively judge the performance, we compare in Tab. 3 the image level sensitivity of detecting if a certain lesion type is present in an image. The methods [54,28,21,29] use supervised approaches on image level without reporting a localization. [51] propose an unsupervised approach to extract salient regions. [18] use a comparable setting to ours applying CAM [53] in a weakly-supervised way to highlight important regions. To decide if a lesion is detected, [18] suggest an overlap of 50% between proposed regions and ground truth. As our explanation masks are fine-grained and the ground truth is coarse, we compare using a 25% overlap and for completeness report a 50% overlap. It is remarkable that FGVis performs comparable or outperforms fully supervised approaches which are designed to detect the presence of one lesion type. The strength of FGVis is especially visible in detecting RSD, as these small lesions only cover some pixels in the image. In Fig. A21 we show fundus images, ground truth and our predictions. Method H HE SE RSD Zhou et al. [54] 94.4 --Liu et al. [28] -83.0 83.0 -Haloi et al. [21] 96.5 --Mane et al. [29] ---96.4 Zhao et al. [51] 98.1 --Gondal et al. [18] 97 Conclusion We propose a method which generates fine-grained visual explanations in the image space using on a novel technique to defend adversarial evidence. Our defense does not introduce hyperparameters. We show the effectivity of the defense on different models, compare our explanations to other methods, and quantitatively evaluate the faithfulness. Moreover, we underline the strength in producing class discriminative visualizations and point to characteristics in explanations of a ResNet50. Due to the fine-grained nature of our explanations, we achieve remarkable results on a medical dataset. Besides, we show the usability of our approach to visually indicate a color bias in training data. The supplementary material provides details, additional results, and further comparisons. A1. Defending against Adversarial Evidence Our method produces explanations based on evidence in the image and suppresses hallucination of adversarial evidence. Without our adversarial defense the optimization can produce an explanation for any class (i.e. even for a class visually not present in the image). To illustrate this differently to the experiment reported in Sec. 3.1 (Tab. 1 and Fig. 3), we show an alternative version of the evaluation, only using a black image as input. Fig. A1 shows an explanation for the adversarial class iguana with and without defense. For Tab. A1 we create explanations for each of the 998 ImageNet classes, using always the same black input image. We omit the predicted class of the black image and the class of the starting condition (image · zero mask). Without defense an explanation can always be generated due to hallucination of adversarial evidence. The results are comparable to the evaluation in the main paper. A2. Implementation Details Unless otherwise specified, the explanations are computed for the most-likely class using SGD with a learning rate of 0.1, running for 500 iterations. To improve optimization and avoid instabilities, we initialize the masks m with noise sampled for each pixel from a uniform distribution U(a, b). with U(0, 0.01) for the generation and repression game and U(0.99, 1) for the preservation and deletion game. We normalize the gradient using its maximum value to avoid large changes of individual mask pixels. For the similarity metric ϕ(·, ·) we use the cross-entropy * contributed while working at BCAI. We additionally thank Volker Fischer, Michael Herman, Anna Khoreva for discussions and feedback. Table A1: How often an adversarial class could be generated from a black image averaged over 998 ImageNet classes (generation game, λ = 0). for the generation and preservation game and the negative probability for the deletion and repression game. When computing an explanation for the most-likely class, we use a line-search for the parameter λ to determine its optimal value. Unless otherwise noted, we iteratively use 13 equally spaced λ values between 10 −4 and 10 −10 and stop when the resulting most-likely class of e ml shifts (deletion and repression game) or achieves the highest probability among all classes (preservation and generation game). We use images of the ImageNet [26] validation set and pre-trained model weights. A comparison of resulting masks for different learning rates and λ values for GoogleNet computed with the deletion game are shown in Fig. A2. A higher λ value causes sparser masks due to a higher weighting of the sparsity invoking part m C T 1 within the loss function (Eq. 2 and Eq. 3). Especially for higher λ values, the resulting masks are rather independent of the chosen learning rate of the SGD optimization. A3. Qualitative Results A3.1. Entropy of Reference Images FGVis computes explanations e c T by optimizing for a perturbed version of the input image x. The perturbation is modelled via a removal operator Φ [17,14,6,11], which computes a weighted average between the image x and a reference image r, using a mask m C T : e c T = Φ(x, m c T ) = x · m c T + (1 − m c T ) · r. (7) A good reference image r should carry little information and lead to a model prediction with a high entropy, meaning, ideally all classes are assigned the same softmax score (see 'Maximum (1000 classes)' in Tab. A2 for the resulting maximum entropy). To compare references, we report their entropy for different models in Tab. A2. For all models except GoogleNet the zero image reference has the highest entropy. Interestingly, for the zero image reference, the more recent architectures (GoogleNet, ResNet50) have a lower entropy. This indicates that these architectures do not assign a roughly equally distributed softmax score to all classes (as AlexNet or VGG16). As expected, an increasing noise level σ n for a Gaussian noise image as well as a decreasing standard deviation of the Gaussian blur filter σ b reduces the entropy. Only GoogleNet does not fully follow this characteristic. For comparison, we report the entropy for 1000 random ImageNet validation images for the different models. Due to the high entropy as well as the low computational effort of a zero reference image, we choose this reference for FGVis. A3.2. Class Discriminative / Fine-Grained In Fig. A3 and Fig. A4 we show additional explanation masks for images containing two distinct objects. The objects are chosen from highly different categories to ensure little overlapping evidence. The explanations are computed using the deletion game, which generates the most pleasing class-discriminative explanations, and GoogleNet. Note that FGVis discriminates well even if the two objects partially overlap. The figures additionally highlight the ability of FGVis to generate fine-grained explanations. To determine λ we use for the most-likely class the strategy as described in Sec. A2. For the second class λ is optimized to significantly drop the softmax score of this class. A3.3. Investigating Biases of Training Data Learned objects. The coexistence of objects in images often results in a learned bias. In Fig. A5, we visualize such a bias for GoogleNet trained on ImageNet. Sports equipment like hockey pucks or ping-pong balls frequently appear in combination with players. This bias is learned by the neural network and results in explanations that also contain pixels belonging to the players. Without deleting these pixels, the deletion game is not able to shift the class of the images. Learned color. We quantitatively verify the color bias reported in Sec. 4.3 and show the 19 classes of ImageNet which are most and least affected by swapping the color in Tab. A3. We swap each of the three color channels BGR to either RBG or GRB and calculate the ratio of maintained true classifications on the validation data after the swap. Fig. A6 shows explanations for the class school bus computed using the preservation game for VGG. The yellow color, also visible in the original images (Fig. A7), is dominant in most of the explanations. Fig. A8 shows explanations for the class minivan computed using the preservation game for VGG. Table A2: Entropy of reference images r for different models. The entropy is averaged over 1000 random instances of each reference image. Gaussian noise images are generated by independently sampling for each pixel from a Gaussian distribution with zero-mean and a standard deviation of σ n . The blurred ImageNet images are computed using a Gaussian blur filter with a standard deviation of σ b . For all random references we report the mean ± standard deviation of the entropy. white or grey cars (original images in Fig. A9) the visible color in the explanation is reduced to a greenish-blue color. 32 % · · · · · · · · · · · · · · · · · · 779 school bus 42 8.33 % 9.52 % 7.14 % · · · · · · · · · · · · · · · · · · 656 minivan 21 83.33 % 71.43 % 95.24 % · · · · · · · · · · · · · · · · · · 528 dial telephone, A3.4. Comparison of Networks In Fig. A10 and Fig. A11 we compare the mask and explanation for four network architectures (GoogleNet, VGG16, AlexNet, ResNet50) using the deletion game. Respectively, in Fig. A12 and Fig. A13 we use the preservation game for the same comparison. For all settings the explanations of ResNet50 and VGG16 are more dense, meaning more pixels have to be deleted/preserved to change/preserve the class prediction. This could be an indicator that these models are more robust, though, a detailed explanation would require further research. Besides, the grid-like pattern for the explanations from ResNet50, described in Sec. 4.1 are visible. The importance of the color to classify the school bus (described in Sec. 4.3) can be seen in Fig. A13. For VGG16 we have observed that the pixels at the image edge are in many cases highlighted in the explanations. Furthermore, VGG16 shows pronounced edges in the explanation compared to the other networks. A3.5. Comparison of Games In Fig. A14 and A15 the different game types (see Sec. 3.1) are visually compared for GoogleNet. The resulting explanations for the repression and deletion game are qualitatively similar. The similarity among the two games is due to both using the same optimization with only a different starting condition m = 0 for the repression vs. m = 1 for the deletion game. The same observation holds for the generation / preservation game. The explanations of the repression and deletion game are more sparse compared to the generation / preservation game. This is most likely due to the fact that only small parts of the image need to be suppressed to change the model output (e.g. shifting one breed of dog to another), though, to evoke a certain model output one needs to create sufficient amount of evidence for this class. During the optimization only class pixels containing evidence towards the target class need to be changed for the generation and deletion game. After optimization most of the mask values stay zero for the generation game and one for the deletion game. The optimized masks are thus similar to its starting conditions. Vice versa, the opposite holds for the preservation and repression game. A3.6. Further Examples In Fig. A16, A17, A18, and A19 further explanations computed using FGVis are shown. A4. Quantitative Results A4.1. Faithfulness of Explanations To evaluate the faithfulness of our approach, we use the deletion metric of Petsiuk et al. [32]. This metric measures how the removal of evidence affects the prediction of the used model. The metric assumes that an importance map is given, which ranks all image pixels with respect to their evidence for the predicted class c ml (i.e. the most-likely class). We use the mean mask (see Sec. A3.5) as the pixel-wise importance map. The mean mask is computed for all images in the ImageNet validation dataset using the deletion game with a learning rate of 0.3 and a line-search to determine the λ value. We iteratively use 4 equally spaced λ values between 10 −7 and 10 −10 and stop when y c T e < 0.02 · y c T x , where y c T e is the softmax score of class c T given the explanation and y c T x the corresponding score given the image. Using the importance map, the deletion curve is generated by successively removing pixels from the input image according to their importance and measuring the resulting probability of the class c ml (see Fig. A20c). The removed pixels are set to zero, as proposed in Petsiuk et al. [32]. The fraction of removed pixels is increased in increments of 0.25% for the first 100 steps and in increments of 1% for the remaining 75 steps. In Fig. A20b, we visualize for an example image the binary masks used to successively set pixels to zero. For a clearer illustration, we reduced the number of deletion steps in this figure. The deletion metric is computed by measuring the area under the curve AUC of the deletion curve (see Fig. A20c) using the trapezoidal rule. A4.2. Visual Explanation for Medical Images Background of the disease: As people with diabetes have a high prevalence for RDR [47], a frequent retinal screening is recommended and deep learning algorithms have been successfully developed to classify fundus images ( [8], [20], [3], [46]). The black box character of these algorithms can be reduced by visual explanation techniques as shown in [18]. Of the publicly available 88,702 images [15] from Eye-PACS [10], we us 80% for training and 20% for validation for a classifier with binary outcome (referable diabetic retinopathy (RDR) vs. non-RDR) which is later used for the weakly-supervised localization. We use a similar setup as in [18] to train the binary classifier (RDR vs. non-RDR). Training was conducted with the same implementation settings as described in [18] using an adopted version of the CNN architecture proposed by [16] for classifying retinal images. We use leaky ReLUs as non-linearities and include batch normalization. The DiaretDB1 dataset [25] used to evaluate the weaklysupervised localization is a dataset of 89 color fundus images collected at the Kuopio University Hospital, Finland. All images have a resolution of 1500x1152 pixels and are scaled to the input dimension of the model. The dataset is ground truth marked by four experts. As proposed in [25] we consider pixels as lesions if at least three experts have agreed. We use FGVis with a fixed λ = 10 −10 and a learning rate of 0.25 stopping if the softmax score for RDR falls below 10% with a maximum of 500 iterations. In Fig. A21 retinal images overlaid with the ground truth (top row) are compared to our prediction (bottom row). To be consistent with [18] the masks m are binarized for better visualization and to be able to quantitatively report the sensitivity (see Tab. 3). Values greater or equal than 4% of the maximum are set to one, the remaining pixels to zero. The predicted pixels in the fine-grained masks m map to the ground truth. Note that FGVis detects these pixels as they are the important ones to be deleted to reduce the softmax score for RDR. A medical expert would also look at mutations in the optic disk or blood vessels which additionally are an indicator for the disease [41]. These mutations are also highlighted by our method. They are not labelled in the ground truth markings leading to visual false positives (FPs). The strength of FGVis to visualize fine-grained structures can be seen in the detection of red small dots (microaneurysm) which are the earliest sign of diabetic retinopathy [2]. As these often merely cover some pixels in the image, it is hard to detect them (zooming in Fig. A21 is necessary to spot these). Figure A20: The deletion curve (c) is computed by successively deleting pixels (b) from the image according to their importance and measuring the resulting probability of the class c ml . Figure A21: Weakly-supervised localization results on DiaretDB1 images. The top row shows fundus images, the bottom row our detection. All images are overlaid with ground truth markings in green (hard exudates), blue (soft exudates), orange (hemorrhages), red (red small dots). Though the network was trained in a weakly-supervised way given only the image label, most of the regions highlighted by FGVis fall within the ground truth markings. Note that mutations in the optic disk or blood vessels are an indicator for the disease [41] but these are not covered by the ground truth markings leading visually to false positives. FGVis highlights part of the blood vessels and optic disks. Figure 2 : 2Visualization types calculated for VGG using deletion / preservation game. For the repression / generation game the same characteristics hold. Subscript c T ommited to ease readability. a) Input image. b) Mask obtained by the optimization. Colors in a deletion mask are complementary to the image colors. c) Explanation directly obtained by the optimization. d) Complementary mask with a true-color representation for the deletion game. e) Explanation highlighting the important evidence for the deletion game. f) Mean mask: mask / comp. mask averaged over colors. -To underline important evidence, we use e for the explanation of the preservation / generation game andẽ for the deletion / repression game. Figure 3 : 3Explanations computed for the adversarial class limousine and the predicted class agama using the generation game and VGG16 with and without our adversarial defense. An adversarial for class limousine can only be computed without the defense. d) Mean mask enhanced by a factor of 7 to show small adversarial structures. Figure 5 : 5Comparison of mean explanation masks: a) Image, b) BBMP Figure 6 : 6Visual explanations computed using the deletion game for ResNet50. The Figure 7 : 7Explanations computed using the preservation game for VGG16. Explanations of the class minivan focus on edges, hardly preserving the color, compared to the class school bus, with yellow dominating the explanations. Figure A1 : A1Explanation for the adversarial class iguana starting from a black image. An adversarial can only be computed without defense (generation game, GoogleNet). Mean masks are enhanced by a factor of 10. Fig ) Masks of class otter. Figure A2 : A2Comparison of resulting masks for different learning rates (lr) and λ values computed using the deletion game and GoogleNet. Figure A3 : A3Explanation masks for images with multiple objects computed using the deletion game and GoogleNet. FGVis produces class discriminative explanations, even when objects partially overlap. Note that objects not belonging to either class, e.g. the rug in the top row, the blue sign on the chainlink fence, or the window in the bottom row vanish in the explanation. Additionally, FGVis is able to visualize fine-grained details down to the pixel level. Figure A4 : A4Explanation masks for images with multiple objects computed using the deletion game and GoogleNet. FGVis produces class discriminative explanations, even when objects partially overlap. This is especially visible in the last row where the tennis balls are almost all removed in the explanation mask for the class strainer. Figure A5 : A5Visual explanations computed using the deletion game for GoogleNet. For both classes (hockey puck and pingpong ball) the explanation method has to additionally delete pixels of the players and the Figure A6 : A6Explanations computed using the preservation game for VGG for the class school bus. Figure A7 : A6 Figure A8 : A7A6A8Input images for the explanations in Fig. Explanations computed using the preservation game for VGG for the class minivan. Figure A9 : A9Input images for the explanations inFig. A8 Figure A10 : A10Masks and explanations computed using the deletion game for different networks. Figure A11 : A11Masks and explanations computed using the deletion game for different networks. Figure A12 : A12Masks and explanations computed using the preservation game for different networks. Figure A13 : A13Masks and explanations computed using the preservation game for different networks. Figure A14 : A14Explanations and masks computed using the different games for GoogleNet. For the repression and deletion game the complementary masks (1 − m) are plotted to have true-color representations (see Sec. A3.4). Figure A15 : A15Explanations and masks computed using the different games for GoogleNet. For the repression and deletion game the complementary masks (1 − m) are plotted to have true-color representations (see Sec. A3.4). Figure A16 : A16Explanation masks computed using the repression game for VGG16. Figure A17 : A17Explanation masks computed using the repression game for VGG16. Figure A18 : A18Explanation masks computed using the preservation game for ResNet50. Figure A19 : A19Explanation masks computed using the preservation game for ResNet50. Table 2 : 2Deletion metric computed on the ImageNet validation dataset (lower is better). The results for all reference methods were taken from Petsiuk et al.[32]. Table 3 : 3Image level sensitivity in % (higher is better) for four different lesions H, HE, SE, RSD: Hemorrhages, Hard Exudates, Soft Exudates and Red Small Dots. The original color of the car is not consistently preserved. Especially forReference image r AlexNet GoogleNet VGG16 ResNet50 Zero image 6.90 4.08 6.31 5.09 Gaussian noise image (σ n = 8) 5.11 ± 0.16 4.62 ± 0.16 5.59 ± 0.09 4.56 ± 0.14 Gaussian noise image (σ n = 32) 2.61 ± 0.29 4.67 ± 0.22 4.38 ± 0.23 4.07 ± 0.30 Blurred ImageNet image (σ b = 5) 3.67 ± 1.12 3.15 ± 1.31 4.08 ± 1.43 2.38 ± 1.58 Blurred ImageNet image (σ b = 10) 4.56 ± 0.88 4.09 ± 1.08 4.83 ± 0.86 3.22 ± 1.25 ImageNet image 1.73 ± 1.43 1.09 ± 1.14 1.06 ± 1.22 0.67 ± 0.91 Maximum (1000 classes) 6.91 6.91 6.91 6.91 table tennis bat/ice-hockey stick to shift the prediction of the model. This clearly highlights a bias of the data towards images which contain a puck/ball, a player and sports equipment.ID Class name #Images Avg. RBG, GRB RBG GRB 168 redbone 31 0.00 % 0.00 % 0.00 % 964 potpie 28 0.00 % 0.00 % 0.00 % 159 Rhodesian ridgeback 35 0.00 % 0.00 % 0.00 % 930 French loaf 27 0.00 % 0.00 % 0.00 % 234 Rottweiler 42 1.19 % 0.00 % 2.38 % 214 Gordon setter 36 1.39 % 2.78 % 0.00 % 963 pizza, pizza pie 35 1.43 % 2.86 % 0.00 % 950 orange 35 1.43 % 2.86 % 0.00 % 184 Irish terrier 33 1.52 % 0.00 % 3.03 % 962 meat loaf, meatloaf 29 1.72 % 3.45 % 0.00 % 984 rapeseed 47 2.13 % 4.26 % 0.00 % 211 vizsla, Hungarian pointer 35 2.86 % 2.86 % 2.86 % 11 goldfinch, Carduelis carduelis 48 3.12 % 0.00 % 6.25 % 934 hotdog, hot dog, red hot 40 3.75 % 2.50 % 5.00 % 218 Welsh springer spaniel 39 3.85 % 2.56 % 5.13 % 191 Airedale, Airedale terrier 37 5.41 % 5.41 % 5.41 % 163 bloodhound, sleuthhound 18 5.56 % 5.56 % 5.56 % 961 dough 15 6.67 % 0.00 % 13.33 % 263 Pembroke, Pembroke Welsh corgi 41 7.32 % 7.32 % 7. Table A3: Ratio of maintained true classifications on the validation data of ImageNet after swapping color channels for the most and least affected 19 classes and minivan / school bus. Each of the three color channels BGR are swapped to either RBG or GRB. The class ID, class name, number of truly classified images before the color swap (#Images) and percentage of maintained classification after the swap for the average over RBG or GRB and each swap individually are reported. Most color-dependent classes are redbone or potpie. Most color-independent classes zebra or electric fan.dial phone 36 95.83 % 91.67 % 100.00 % 866 tractor 37 95.95 % 91.89 % 100.00 % 572 goblet 26 96.15 % 96.15 % 96.15 % 47 African chameleon, Chamaeleo chamaeleon 40 96.25 % 95.00 % 97.50 % 302 ground beetle, carabid beetle 27 96.30 % 96.30 % 96.30 % 463 bucket, pail 27 96.30 % 96.30 % 96.30 % 717 pickup, pickup truck 28 96.43 % 100.00 % 92.86 % 178 Weimaraner 44 96.59 % 93.18 % 100.00 % 669 mosquito net 44 96.59 % 97.73 % 95.45 % 661 Model T 46 96.74 % 97.83 % 95.65 % 769 rule, ruler 36 97.22 % 100.00 % 94.44 % 771 safe 40 97.50 % 97.50 % 97.50 % 829 streetcar, tram, tramcar, trolley, ... 41 97.56 % 97.56 % 97.56 % 713 photocopier 44 97.73 % 100.00 % 95.45 % 916 web site, website, internet site, site 47 97.87 % 100.00 % 95.74 % 423 barber chair 31 98.39 % 96.77 % 100.00 % 190 Sealyham terrier, Sealyham 39 98.72 % 97.44 % 100.00 % 340 zebra 47 100.00 % 100.00 % 100.00 % 545 electric fan, blower 37 100.00 % 100.00 % 100.00 % For the sake of brevity, we will use the term explanations as a synonym for local explanations throughout this work. 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Yitian Zhao, Yalin Zheng, Yifan Zhao, Yonghuai Liu, Zhili Chen, Peng Liu, Jiang Liu, International Conference on Medical Image Computing and Computer-Assisted Intervention. SpringerYitian Zhao, Yalin Zheng, Yifan Zhao, Yonghuai Liu, Zhili Chen, Peng Liu, and Jiang Liu. Uniqueness-driven saliency analysis for automated lesion detection with applications to retinal diseases. In International Conference on Medical Im- age Computing and Computer-Assisted Intervention, pages 109-118. Springer, 2018. 8 . Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, Antonio Torralba, arXiv:1412.6856Object Detectors Emerge in Deep Scene CNNs. arXiv e-printsBolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba. Object Detectors Emerge in Deep Scene CNNs. arXiv e-prints, page arXiv:1412.6856, Dec 2014. 2 Learning deep features for discriminative localization. Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, Antonio Torralba, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba. Learning deep features for discrimi- native localization. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 2921-2929, 2016. 1, 2, 8 Automatic hemorrhage detection in color fundus images based on gradual removal of vascular branches. Lei Zhou, Penglin Li, Qi Yu, Yu Qiao, Jie Yang, IEEE International Conference on Image Processing (ICIP). Lei Zhou, Penglin Li, Qi Yu, Yu Qiao, and Jie Yang. Au- tomatic hemorrhage detection in color fundus images based on gradual removal of vascular branches. In IEEE Interna- tional Conference on Image Processing (ICIP), pages 399- 403, 2016. 8

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{'abstract': 'To verify and validate networks, it is essential to gain insight into their decisions, limitations as well as possible shortcomings of training data. In this work, we propose a post-hoc, optimization based visual explanation method, which highlights the evidence in the input image for a specific prediction. Our approach is based on a novel technique to defend against adversarial evidence (i.e. faulty evidence due to artefacts) by filtering gradients during optimization. The defense does not depend on human-tuned parameters. It enables explanations which are both fine-grained and preserve the characteristics of images, such as edges and colors. The explanations are interpretable, suited for visualizing detailed evidence and can be tested as they are valid model inputs. We qualitatively and quantitatively evaluate our approach on a multitude of models and datasets.', 'arxivid': '1908.02686', 'author': ['Jörg Wagner joerg.wagner3@de.bosch.com ', 'Jan Mathias Köhler ', 'Tobias Gindele ', 'Leon Hetzel ', 'Jakob Thaddäus Wiedemer ', 'Sven Behnke behnke@cs.uni-bonn.de ', 'Jörg Wagner \nBosch Center for Artificial Intelligence (BCAI)\nGermany\n\nUniversity of Bonn\nGermany\n', 'Jan Mathias Köhler \nBosch Center for Artificial Intelligence (BCAI)\nGermany\n', 'Tobias Gindele \nBosch Center for Artificial Intelligence (BCAI)\nGermany\n', 'Leon Hetzel \nBosch Center for Artificial Intelligence (BCAI)\nGermany\n', 'Jakob Thaddäus Wiedemer \nBosch Center for Artificial Intelligence (BCAI)\nGermany\n', 'Sven Behnke \nUniversity of Bonn\nGermany\n'], 'authoraffiliation': ['Bosch Center for Artificial Intelligence (BCAI)\nGermany', 'University of Bonn\nGermany', 'Bosch Center for Artificial Intelligence (BCAI)\nGermany', 'Bosch Center for Artificial Intelligence (BCAI)\nGermany', 'Bosch Center for Artificial Intelligence (BCAI)\nGermany', 'Bosch Center for Artificial Intelligence (BCAI)\nGermany', 'University of Bonn\nGermany'], 'corpusid': 197465391, 'doi': '10.1109/cvpr.2019.00931', 'github_urls': [], 'n_tokens_mistral': 23376, 'n_tokens_neox': 20155, 'n_words': 12476, 'pdfsha': 'c397b1e8ebf48e9bf7f2630f030c000568947756', 'pdfurls': ['https://arxiv.org/pdf/1908.02686v1.pdf'], 'title': ['Interpretable and Fine-Grained Visual Explanations for Convolutional Neural Networks', 'Interpretable and Fine-Grained Visual Explanations for Convolutional Neural Networks'], 'venue': []}

arxiv

Production of light pseudoscalars in external electromagnetic fields by the Schwinger mechanism Typeset using REVT E X 1 9912485v3 23 Apr 2002 J A Grifols Grup de Física Teòrica and IFAE Universitat Autònoma de Barcelona 08193Bellaterra, BarcelonaSpain Eduard Massó Grup de Física Teòrica and IFAE Universitat Autònoma de Barcelona 08193Bellaterra, BarcelonaSpain Subhendra Mohanty Physical Research Laboratory Ahmedabad -380 009NavrangpuraIndia Production of light pseudoscalars in external electromagnetic fields by the Schwinger mechanism Typeset using REVT E X 1 9912485v3 23 Apr 2002arXiv:hep-ph/ We calculate the probability of the decay of external inhomogeneous electromagnetic fields to neutral pseudoscalar particles that have a coupling to two photons. We also point out that our estimate for axion emission in a previous paper was incorrect. I. INTRODUCTION The Schwinger mechanism is a non-perturbative process by which an infinite number of zero frequency photons can decay into electron-positron pairs [1]. In this paper we show that this mechanism can be generalized to study the production of other kinds of light particles from intense electromagnetic (EM) fields. The light particle that we consider is a pseudoscalar (PS) having a coupling to two photons. In Section II we derive the formula for the decay of classical background fields into PS particles. This is achieved by integrating out the particle fields from the total Lagrangian to obtain the effective action of the classical background fields. The imaginary part of the effective Lagrangian is related to the probability of decay of classical background fields into particles. In Section III, we derive, from the usual coupling of the PS to two photons, the specific interaction Lagrangian that should be used in the general formalism of Section II in order to account for vacuum decay into PS. For static EM fields, we show that a necessary condition is that the fields are inhomogeneous. In Sections IV, V, and VI, we explicitly calculate the PS production in a variety of situations. Specifically we consider a dipole magnetic field, a cylindrical capacitor, and a spherical capacitor. A final section is devoted to the conclusions. II. DECAY OF CLASSICAL BACKGROUND FIELDS INTO PARTICLES We start with the action for the pseudoscalar φ (mass m) coupled to the background E and B fields of the general form S[φ, E, B] = d 4 x 1 2 φ(x) −∂ 2 − m 2 + f (x) φ(x)(1) where f (x) is some scalar function of E and B fields. From (1) we obtain the effective action for the background E and B fields formally as e iS ef f [E,B] = Dφ e iS[φ,E,B](2) The effective Lagrangian for the E and B fields can be related to the Green's function of φ in external E and B fields as follows. Differentiate (2) by m 2 i ∂S ef f [E, B] ∂m 2 = − Dφ φ 2 e iS[φ,E,B] Dφ e iS[φ,E,B] = − 1 2 d 4 x G (x, x; E, B) = − 1 2 d 4 x d 4 p (2π) 4 G (p; E, B)(3) The effective Lagrangian of the background fields is therefore formally given by the expression L ef f [E, B] = i 2 dm 2 d 4 p (2π) 4 G (p; E, B)(4) The probability of external E and B fields to decay into quanta of φ is related to the imaginary part of L ef f as follows P = 1 − 0 e iS ef f [E,B] 0 = 1 − exp − 2 Im d 3 x dt L ef f [E, B](5) In the case that this probability is small, we can write the probability density w (per unit volume and unit time) approximately as w = 2 ImL ef f [E, B](6) We now give the general procedure for obtaining the effective action of the background fields by calculating the Green's function of φ in background E and B fields following the method of Duff and Brown [2]. The effective Lagrangian can be calculated by this method if the background fields contained in f (x) in the interaction Lagrangian L I (x) = 1 2 f (x)φ 2 (x)(7) can be expanded in a Taylor series near some reference pointx. Expanding f (x) near x =x, f (x) = α(x) + β µ (x)(x −x) µ + γ 2 µν (x)(x −x) µ (x −x) ν + ... (8) α(x) = f (x), β µ (x) = ∂f ∂x µ x=x , γ 2 µν (x) = 1 2 ∂ ∂x µ ∂f ∂x ν x=x The equation for the Green's function for the φ field is given by ∂ 2 x + m 2 − α − β µ (x −x) µ − γ 2 µν (x −x) µ (x −x) ν G(x,x) = δ 4 (x −x)(9) In momentum space (x −x) µ → −i ∂ ∂p µ(10) and the equation for the Green's function in momentum space is −p 2 + m 2 − α + iβ µ ∂ ∂p µ + γ 2 µν ∂ ∂p µ ∂ ∂p ν G(p) = 1(11) We choose as an ansatz for the solution G(p) the form G(p) = i ∞ 0 ds e −is(m 2 −iǫ) e ipµA µν pν +B µ pµ+C(12) where A(s), B(s) and C(s) are to be determined. They must satisfy the boundary condition in the case of vanishing external fields, i.e. when α, β, γ → 0 A µν → s g µν , B µ → 0, C → 0(13) and in this limit we should obtain G(p) = i ∞ 0 ds e −ism 2 +isp 2 = 1 m 2 − p 2(14) i.e., the free particle Green's function. To solve for A, B, and C we insert ansatz (12) in (11). We have i ∞ 0 ds − p 2 + m 2 − α + iβ · (2iA · p + B) + (2ip · A + B) · γ 2 · (2iA · p + B) + 2i tr(γ 2 · A) exp −ism 2 + ip · A · p + B · p + C = 1(15) Equation (15) has the general form ∞ 0 ds g(s) e −h(s) = 1 (16) whose solution is g(s) = ∂h(s) ∂s(17) with h(0) = 0 and h(∞) = ∞. Using the form of the solution (17) for equation (15) i − p 2 + m 2 − α + iβ · (2iA · p + B) + (2ip · A + B) · γ 2 · (2iA · p + B) + 2i tr(γ 2 · A) = im 2 − ip · ∂A ∂s · p − ∂B ∂s · p − ∂C ∂s(18) and comparing equal powers of p on both sides we get the following linear differential equations for A, B, and C, ∂A ∂s = 1 + 4A · γ 2 · A ∂B ∂s = 2iA · β + 4A · γ 2 · B ∂C ∂s = iα + β · B − iB · γ 2 · B + 2 tr(γ 2 · A)(19) The solutions of these equation which satisfy the boundary conditions (13) are given by A = 1 2 γ −1 · tan(2γs) (20) B = − i 2 γ −2 · [1 − sec(2γs)] · β(21)C = iαs − 1 2 tr [ln cos(2γs)] + i 8 β · γ −3 · [tan(2γs) − 2γs] · β(22) These A, B and C determine G(p) when substituted in (12). The effective Lagrangian is obtained by substituting this G(p) in (4) and carrying out the integration over m 2 , L ef f = − i 2 ∞ 0 ds s d 4 p (2π) 4 exp −ism 2 + ip · A · p + B · p + C (23) The Gaussian integral may be evaluated using d 4 p exp {ip · A · p + B · p} = − iπ 2 (detA) − 1 2 exp i 4 B · A −1 · B(24) where detA is the determinant of the matrix A µ ν . Using now (20-22), we have L ef f = − 1 32π 2 ∞ 0 ds s 3 e −is(m 2 −α) det 2γs sin 2γs 1 2 e il(s)(25) where l(s) = 1 4 β · γ −3 · [tan(γs) − γs] · β(26) The coefficients of the Taylor expansion of the background fields (8) determine the effective action on integrating out the quantum field φ. In particular, an imaginary part of L ef f may be non-zero depending on the signs of the eigenvalues of the γ 2 matrix. When this occurrs, we have a non-zero probability (6) that the external EM fields decay in PS particles. To the effective Lagrangian in (25) we should add subtractions to render it finite at s = 0. When this is done, we have that in the limit β → 0, γ 2 → 0, the effective Lagrangian L ef f → 0, as it should be. In the Appendix A we illustrate the method for the familiar case of production of charged scalar fields in a constant electric field. The formulae (20-22,25) differ by some signs and factors of i from the solutions displayed in reference [3]. There, we presented the formulae for the case that f (x) had only spatial variation and therefore β and γ 2 had only i = 1, 2, 3 indices. In [3] we used the metric (+, +, +) while in the present paper we consider spatial as well temporal variation and use the metric (+, −, −, −). This introduces some changes in intermediate formulae but of course the final results we get in the present paper are identical with the final results we got in [3]. III. EFFECTIVE EM-FIELDS -PS PAIR INTERACTIONS In this Section, we show that a coupling of a pseudoscalar to two photons induces an interaction L I that may lead to PS production in a background of EM fields. The generic pseudoscalar-two-photon interaction (see fig.1) can be written as L φγγ = 1 8 gφǫ µνρσ F µν F ρσ(27) We should mention that in the special case where the PS is an axion the coupling g is related to the mass m of the axion. We need to evaluate the loop diagram of the type shown in fig.2 with infinite number of zero-frequency photon external legs. The imaginary part of this diagram gives the probability for the decay of the external electromagnetic field. To calculate this diagram, we first evaluate the process φA → φA, where A is an external photon. We use iL φγγ from (27) in momentum space, 1 4 g φǫ µνρσ k µ A ν F ρσ (28) The two-photon two-PS interaction is then obtained contracting the internal photon legs, 4 1 4 g φ 2 ǫ µνρσ k µ F ρσ −ig νν ′ k 2 ǫ µ ′ ν ′ ρ ′ σ ′ (−k µ ′ ) F ρ ′ σ ′(29) The factor of 4 in equation (29) is for the four possible ways of joining the photon legs. Due to the presence of the k 2 term in the denominator, the effective coupling (29) is non-local. However, when we calculate the effective action for the external EM field the momentum k is integrated over. One can therefore make use of the identity d 4 k k µ k µ ′ g(k 2 ) = d 4 k g µµ ′ k 2 4 g(k 2 )(30) to simplify (29). Thus, we can reduce the effective two PS-two photon interaction to a local interaction vertex. Back in configuration space, it is given by L I = − 1 4 g 2 φ 2 F µν F µν = 1 2 g 2 φ 2 (E 2 − B 2 )(31) (see fig.3). With the interaction Lagrangian (31) we can go back to the formalism of Section II and calculate the probability density. We can readily identify f (x) in (7), f (x) = g 2 (E 2 − B 2 )(32) In order to have a non trivial L ef f , one needs non-zero second derivatives of the EM fields as they appear in expression (32). As we said in Section I, depending on the sign of the corresponding γ 2 matrix we may have PS production. We illustrate it in some simple physical situations in the following sections. IV. PRODUCTION OF PSEUDOSCALARS IN DIPOLE MAGNETIC FIELDS In a static dipole magnetic field the PS-pair -EM interaction is given by L I = − 1 2 g 2 B 2 (r)φ 2 = − 1 2 g 2 B 2 0 z 6 0 3z 2 + r 2 4r 8 φ 2(33) where B 0 is the field strength at a point r 0 = (0, 0, z 0 ) on the z-axis. We have now f ( r) = −g 2 B 2 0 z 6 0 3z 2 + r 2 4r 8(34) Expanding B 2 (r) near the point r 0 L = 1 2 φ − ∂ 2 − m 2 + f ( r 0 ) + ∂f ∂x i r= r 0 (x i − x i0 ) (35) + 1 2 ∂ 2 f ∂x i ∂x j r= r 0 (x i − x i0 )(x j − x j0 ) φ + ...(36) we find that the coefficients of the Taylor expansion are given by α = f ( r 0 ) = −g 2 B 2 0 ≡ α m (37) (β) i = ∂f ∂x i r= r 0 = (0, 0, 6g 2 B 2 0 z −1 0 )(38) and (γ 2 ) ij = 1 2 ∂ 2 f ∂x i ∂x j r= r 0 = 3g 2 B 2 0 4z 2 0    5 0 0 0 5 0 0 0 −28    ≡    a 2 m 0 0 0 a 2 m 0 0 0 −b 2 m   (39) Therefore we find using the notation and formalism of Section II that the effective action on integrating out the PS field is given by L ef f = − 1 32π 2 ∞ 0 ds s 3 e −is(m 2 −αm) 2a m s sinh 2a m s 2b m s sin 2b m s e ilm(s) (40) with l m (s) = λ m (b m s − tan b m s) (41) λ m = 9g 4 B 4 0 z 2 0 1 b 3 m = 3 7 √ 21 gB 0 z 0(42) The imaginary part of the expression (40) can be performed by enclosing the simple poles at s = −inπ(2a m ) −1 , n = 0, 1, ..., with a contour from below. We get Im L ef f = 1 8π 5 2 a 3 2 m b 1 2 m ∞ n=1 (−1) n+1 C (m) n e −nπ/2ηm (43) C (m) n = n − 3 2 sinh n b m a m π − 1 2 el m (44) with η m = a m m 2 − α m = √ 15 2 m 2 z 0 gB 0 + z 0 gB 0 −1 (45) andl m = λ m n b m a m π 2 − tanh n b m a m π 2 (46) In (44) and (46) we can put b m /a m = 28/5. The main contribution to the above integral comes from the n = 1 term and we find that w is given by the expression w = 0.036 g 2 B 2 0 z 2 0 e 2.72λm e −π/2ηm(47) The probability w of field decay is extremely suppressed for realistic parameters. To illustrate this, let us choose a mass m and a coupling g consistent with the axion window: m ∼ 10 −3 eV g ∼ 10 −13 GeV −1(48) Also, let us choose B 0 = 1 Tesla z 0 = 10 cm(49) With these values, we get η m ∼ 10 −19(50) and since η m appears in the exponential in (47), we see that the probability w is indeed extremely suppressed. For a dipole magnetic field to be unstable and decay into axions, one needs η m > ∼ 1, but this would correspond either to unrealistic values for the external field parameters (49) or to excluded values for the axion mass and coupling (48). For non-axion models, g and m are not related (still there are restrictions on these parameters, see ref. [4]). One could have η m ∼ 1 by tuning g and m. Imposing that the field (49) does not decay into pseudoscalars leads to the constraint m 10 −12 eV 2 > ∼ g 10 −13 GeV −1(51) V. PRODUCTION OF PSEUDOSCALARS IN A CYLINDRICAL CAPACITOR The modulus of the electric field inside a cylindrical capacitor whose axis lies along the z-axis depends only on ρ = (x 2 + y 2 ) E(ρ) = λ 2π 1 ρ(52) with λ the linear electric charge density. The bilinear interaction term (31) is L I = 1 2 g 2 E 2 (ρ)φ 2 (x) = 1 2 g 2 c 1 ρ 2 φ 2 (x)(53) where g c ≡ λg/2π. The corresponding function f (x) is f (ρ) = g 2 c 1 ρ 2(54) Expanding the fields near some reference point (x 0 , y 0 , z 0 ) with ρ 0 = (x 2 0 + y 2 0 ) 1 2 L = 1 2 φ   −∂ 2 − m 2 + f (ρ 0 ) + ∂f ∂ρ ρ=ρ 0 (ρ − ρ 0 ) + 1 2 ∂ 2 f ∂ρ 2 ρ=ρ 0 (ρ − ρ 0 ) 2   φ + ...(55) It can be written as in (8) with α = f (ρ 0 ) ≡ α c (56) (β) i = f ′ ρ 0 (x 0 , y 0 )(57) and (γ 2 ) ij = f ′ 2ρ 3 0 y 2 0 −x 0 y 0 −x 0 y 0 x 2 0 + f ′′ 2ρ 2 0 x 2 0 x 0 y 0 x 0 y 0 y 2 0 (58) where primes denote derivatives with respect to ρ taken at ρ = ρ 0 . In the above formulae, the spatial indices run over 1, 2. Introducing the explicit form of f , we get α c = g 2 c ρ 2 0 (59) (β) i = − 2g 2 c ρ 4 0 (x 0 , y 0 )(60) and the γ 2 matrix (58) reads (γ 2 ) ij = g 2 c ρ 6 0 −y 2 0 + 3x 2 0 4x 0 y 0 4x 0 y 0 −x 2 0 + 3y 2 0(61) Next, we diagonalise γ 2 by rotating the coordinates with an orthogonal matrix. For example, we can use (O) ij = 1 ρ 0 x 0 y 0 −y 0 x 0(62) In diagonal form we have (γ 2 D ) ij = g 2 c ρ 4 0 3 0 0 −1 ≡ a 2 c 0 0 −b 2 c (63) We need β in the diagonal basis given by β D = O · β. We get (β D ) i = −2g 2 c ρ −3 0 , 0(64) The expression for L ef f (25) finally reads L ef f = − 1 32π 2 ∞ 0 ds s 3 e −is(m 2 −αc) 2a c s sinh 2a c s 2b c s sin 2b c s e ilc(s)(65) where l c (s) = λ c (a c s − tanh a c s) (66) λ c = g 4 c ρ −6 0 a −3 c = g c 3 √ 3(67) We are not able to perform the integration in (65) by the procedure of extending s to the complex plane, as we have done in section IV. The reason is the presence of essential singularities contained in l c (s). (For a discussion of the implications of essential singularities in the context of QED pair production calculations at finite temperature, see ref. ( [5]. ) We shall calculate the integral (65) numerically. We make the change of variables x = (m 2 − α c − λ c a c ) s(68) so that w = 2 Im L ef f = 1 16π 2 1 (m 2 − α c − λ c a c ) 2 I c(69) where I c = ∞ 0 dx x 3 sin(φ c )   2η c x sinh 2η c x 2η c x/ √ 3 sin(η c x/ √ 3) + 2 9 x 2 − 1  (70) We have introduced the necessary subtractions and defined η c = a c m 2 − α c − λ c a c(71) and φ c = x + λ c tanh η c x(72) As it stands, I c depends on the two adimensional parameters λ c and η c , which reflect its dependence on both the strength of the interaction and on the mass of the scalar particles. In order to explore numerically this two-dimensional space we should focus on those regions for which the results make physical sense. This is even more so because a blind computation of the integral leads easily to wild fluctuations due to the violent oscillations of the integrand for large domains of parameter space. Since the electric field should be inhomogeneous on scales of the order of a particle's Compton wavelength, we have E −1 dE dρ 0 m −1 > 1(73) In our case, this means ρ 0 m < 1(74) Moreover, we should restrict our survey to subcritical conditions, i.e. conditions such that we find ourselves below the point where catastrophical pair-production starts in the earnest and vacuum breakdown occurs. The system will be subcritical whenever the field energy stored in a volume of size m −3 , and which is converted into a PS pair, is at most of the order of twice the scalar particle rest mass. A crude back of an envelope estimate gives, λ 2 c /m 2 ρ 2 0 < ∼ O(1)(75) Using both restrictions as a rough guide, we perform the numerical integration of I c as a function of η −1 c for η c and λ c in the ballpark of the values required by (74) which implies a large negative value of the exponent in (76) that suppresses field decay. VI. PRODUCTION OF PSEUDOSCALARS IN A SPHERICAL CAPACITOR The modulus of the electric field inside a spherical capacitor depends only on r = | r |, E(r) = Q 4π 1 r 2 (79) where Q is the electric charge. The bilinear interaction term is then L I = 1 2 g 2 E 2 (r)φ 2 (x) = 1 2 g 2 s 1 r 4 φ 2 (x)(80) where g s = Qg/4π. The corresponding function f (x) depends only on r, f (r) = g 2 s 1 r 4(81) Expanding the fields near some reference point with (x 0 , y 0 , z 0 ) with modulus r 0 L = 1 2 φ −∂ 2 − m 2 + f (r 0 ) + ∂f ∂r r=r 0 (r − r 0 ) + 1 2 ∂ 2 f ∂r 2 r=r 0 (r − r 0 ) 2 φ + ...(82) In Cartesian coordinates it can be written as in (8) with α = f (r 0 ) ≡ α s (83) (β) i = f ′ r 0 (x 0 , y 0 , z 0 )(84) and (γ 2 ) ij = f ′ 2r 3 0    y 2 0 + z 2 0 −x 0 y 0 −x 0 z 0 −x 0 y 0 x 2 0 + z 2 0 −y 0 z 0 −x 0 z 0 −y 0 z 0 x 2 0 + y 2 0    + f ′′ 2r 2 0    x 2 0 x 0 y 0 x 0 z 0 x 0 y 0 y 2 0 y 0 z 0 x 0 z 0 y 0 z 0 z 2 0   (85) where primes denote derivatives with respect to r taken at r = r 0 . Introducing the form of f (x), we get α s = g 2 s r 4 0(86) and (β) i = − 4g 2 s r 6 0 (x 0 , y 0 , z 0 )(87) and (γ 2 ) ij = 2g 2 s r 8 0    5x 2 0 − y 2 0 − z 2 0 6x 0 y 0 6x 0 z 0 6x 0 y 0 −x 2 0 + 5y 2 0 − z 2 0 6y 0 z 0 6x 0 z 0 6y 0 z 0 −x 2 0 − y 2 0 + 5z 2 0   (88) The γ 2 matrix (88) can be diagonalised by rotating the coordinates with an orthogonal matrix. For example, we can use (O) ij = 1 r 0 d 0    −z 0 r 0 0 x 0 r 0 −x 0 y 0 d 2 0 −y 0 z 0 x 0 d 0 y 0 d 0 z 0 d 0    (89) where d 0 = x 2 0 + z 2 0 . In diagonal form we have (γ 2 D ) ij = 2g 2 s r 6 0    −1 0 0 0 −1 0 0 0 5    ≡    −b 2 s 0 0 0 −b 2 s 0 0 0 a 2 s   (90) One must also use β in the diagonal basis given by β D = O · β. We get (β D ) i = 0, 0, −4g 2 c r −5 0(91) The expression for L ef f (25) is given in this case by L ef f = − 1 32π 2 ∞ 0 ds s 3 e −is(m 2 −αs) As in the precedent section, we cannot perform the integration in (92) by extending s to the complex plane since again there are essential singularities and we do a numerical integration. The procedure is very similar. It is convenient to change variables, x = (m 2 − α s − λ s a s ) s(95) so that w = 2 Im L ef f = 1 16π 2 1 (m 2 − α s − λ s a s ) 2 I s(96) where I s = ∞ 0 dx x 3 sin(φ s ) 2η s x sinh 2η s x 2η s x/ √ 5 sin(η s x/ √ 5) + 1 5 x 2 − 1(97) We have introduced the necessary substractions and defined η s = a s m 2 − α s − λ s a s(98) and φ s = x + λ s tanh η s x(99) Here we follow a similar strategy as before to pinpoint the relevant parameter space. We get similar restrictions: r 0 m < 1 and λ 2 s /r 2 0 m 2 < ∼ O(1)(100) In figs.6 and 7 we display I s as a function of η −1 s , for a couple of values of λ s . Again, we see that the approximate behaviour is that of a decreasing exponential, I s = Aη 2 s e −k/ηs(101) with positive constants A and k (that depend on λ s ). When one considers realistic values for the field and PS parameters it turns out that the probability of field breakdown is extremely suppressed as it happens in the cases that have been analyzed in the precedent sections, namely the dipole magnetic field and the cylindrical capacitor. VII. CONCLUSIONS AND FINAL REMARKS In the presence of strong external fields, the physical vacuum breaks down because particle-antiparticle pairs are being pumped out of it at the expense of field energy. The case of a strong uniform electric field spontaneously creating electron-positron pairs is the best known (QED) example for this phenomenon. Such process is of a non-perturbative nature and the QED case has been solved exactly by Schwinger and others [1]. Their solution however does not include the backreaction on the external field exerted by the presence of the produced e + e − pairs. Clearly, creation of pairs requires the supply of mass energy and kinetic energy which must be furnished by the external field. A balanced energy budget is therefore only possible through a corresponding reduction of the energy stored in the field. Because electrons and positrons carry charge they will fly to the external sources of the field and thus the field (and hence its energy) will diminish. So, unless from the outside the field is restored, the pair production process cannot be indefinitely sustained. If nothing is done from the outside a catastrophic breakdown of the initially strong (critical) electric field will inevitably follow. In the present paper we dealt with pseudoscalar particles. Pseudoscalars are fundamental ingredients of many completions of Particle Physics models. Examples run from axions to superlight partners of gravitinos. In the previous sections we have derived the probability for pair production of PS in electric and magnetic fields. Contrary to the QED case mentioned above, constant fields do not cause the disruption of the vacuum. Field gradients are necessary for the phenomenon to occur. Hence, we studied PS pair production in inhomogeneous fields. We have calculated the probability in a general case and based our computation on an effective action formalism formulated by Brown and Duff. We then have applied the general formulae to a few specific cases: PS production in a magnetic dipole field and between the plates of a charged capacitor (either cylindrical or spherical). Again, backreaction was ignored and therefore adequate boundary conditions were implicitly assumed that take into account the fact that pairwise creation of PS requires field energy to be depleted. In the three cases studied, we found that our probability shows the non-perturbative behaviour w ∼ exp(−const × m 2 /g) expected for subcritical fields. Finally, we should point out that in a previous paper [3] we erroneously estimated axion emission in the Coulomb field of an atomic nucleus. This result is incorrect because we overlooked the question of appropriate boundary conditions that guarantee energy conservation and which are clearly not met in this microscopic system. ACKNOWLEDGMENTS We would like to thank G. Raffelt for pointing out a conceptual inconsistency in a previous version of this work and for the illuminating discussions that followed. We also thank the referees for their valuable comments, especially for raising the question concerning essential singularities and for drawing our attention to reference [5]. Two of us (J.A.G. and E.M.) have partial support from the CICYT Research Project AEN99-0766. APPENDIX A: DECAY OF A CONSTANT ELECTRIC FIELD INTO CHARGED SCALARS We start with the equation for the Green's function G(p) −(p − eA) 2 + m 2 G(p) = 1 −p 2 + m 2 + e(A µ p µ + p µ A µ ) − e 2 A 2 G(p) = 1 (A1) We assume constant E and B fields. The vector potential can be choosen as A µ = − 1 2 F µν x ν → i 2 F µν ∂ ∂p ν (A2) When inserted in (A1), one gets an equation of the form given in (11), except for a term F µ ν p µ ∂ ∂p ν G(p)(A3) that leads to an expression containing F µ ν p µ p α A αµ (A4) The antisymmetry of F and the fact that F and A commute makes this term vanish. Then our equation is To this expression for L ef f one should add a subtraction to make it finite at s = 0. When this is done, L ef f → 0 when eE → 0. The probability of scalar production can now be calculated using (5). The integral can be calculated by contour integration by closing the real axis with a contour on the negative imaginary plane. This contour encloses poles at s = −inπ/eE which contribute to the integral. The final result for the constant electric field decay probability density is w = αE 2 2π 2 ∞ n=1 (−1) n+1 n 2 exp − nπm 2 eE (A10) which coincides with the well-known formula found in textbooks [6]. and (75). The results for two values of λ c are displayed in figs.4 and 5. The curves are accurately fit by an analytic expression of the form, I c = Aη 2 c e −k/ηc (76) with A and k depending on λ c . (A and k are positive.) Equation (76) has a form that closely resembles the classical Schwinger result and characterizes a typical non-perturbative process.The electric field would break down into pseudoscalars if the exponent in (76) becomes small. Again we choose the values consistent with the axion window (48) and for the field parameters we take E 0 ∼ 10 4 s (s) = λ s (a s s − tanh a s s) −p 2 ( 2+ m 2 + e 2 4 F µν F µ ρ ∂ 2 ∂p ν ∂p ρ G(p) = 1(A5)and with our definitions in (8), β work out the special case of a constant E field. We have (γ 2 ) νρ = − We have chosen the z-direction as the direction of E). The eigenvalues of γ FIG. 1 . 1PS-two-photon interaction. FIG. 2. Loop diagram showing infinite number of photon external legs. FIG. 3 .FIG. 4 .FIG. 5 .FIG. 6 .FIG. 7 . 34567Two PS-two photon interaction. I c as a function of 1/η c for the value λ c = 0.1. Dotted line: I c obtained by numerical integration of (70). Full line: I c as given by (76) with A = 0.28 and k = 1Same as fig.4 for λ c = 0.4 and A = 0.43 and k = 2I s as a function of 1/η s for the value λ s = 0.2. Dotted line: I s obtained by numerical integration of (97). Full line: I s given by (101) with A = 0.30 and k = 1Same as fig.6 for λ s = 0.5 and A = 0.35 and k = 1.92 J Schwinger, ; W Greiner, B Müller, J Rafelski, Quantum Electrodynamics of Strong Fields. Springer-Verlag82664J. Schwinger, Phys. Rev. 82 (1951) 664; for early work look in W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields, Springer-Verlag (1985). . M R Brown, M J Duff, Phys. Rev. 112124M. R. Brown and M. J. Duff, Phys. Rev. D11 (1975) 2124. . J A Grifols, E Massó, S Mohanty, K V Shajesh, Phys. Rev. 6097701J. A. Grifols, E. Massó, S. Mohanty and K.V. Shajesh, Phys. Rev. D60 (1999) 097701. . E Massó, R Toldrà, hep-ph/9503293Phys. Rev. D. 521755E. Massó and R. Toldrà, Phys. Rev. D 52, 1755 (1995) [hep-ph/9503293]; . J A Grifols, E Massó, R Toldrà, astro- ph/9606028Phys. Rev. Lett. 772372J. A. Grifols, E. Massó and R. Toldrà, Phys. Rev. Lett. 77, 2372 (1996) [astro- ph/9606028]. W Dittrich, H Gies, Probing the Quantum Vacuum: Perturbative Effective Action Approach in Quantum Electrodynamics and its Application. Springer-VerlagW. Dittrich and H. Gies, Probing the Quantum Vacuum: Perturbative Effective Action Approach in Quantum Electrodynamics and its Application, Springer-Verlag (2000). J B Itzykson, Zuber, Quantum Field Theory. McGraw-Hill CoSee e.g. C. Itzykson and J.B. Zuber, Quantum Field Theory, McGraw-Hill Co. (1980).

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{'abstract': 'We calculate the probability of the decay of external inhomogeneous electromagnetic fields to neutral pseudoscalar particles that have a coupling to two photons. We also point out that our estimate for axion emission in a previous paper was incorrect.', 'arxivid': 'hep-ph/9912485', 'author': ['J A Grifols \nGrup de Física Teòrica and IFAE\nUniversitat Autònoma de Barcelona\n08193Bellaterra, BarcelonaSpain\n', 'Eduard Massó \nGrup de Física Teòrica and IFAE\nUniversitat Autònoma de Barcelona\n08193Bellaterra, BarcelonaSpain\n', 'Subhendra Mohanty \nPhysical Research Laboratory\nAhmedabad -380 009NavrangpuraIndia\n'], 'authoraffiliation': ['Grup de Física Teòrica and IFAE\nUniversitat Autònoma de Barcelona\n08193Bellaterra, BarcelonaSpain', 'Grup de Física Teòrica and IFAE\nUniversitat Autònoma de Barcelona\n08193Bellaterra, BarcelonaSpain', 'Physical Research Laboratory\nAhmedabad -380 009NavrangpuraIndia'], 'corpusid': 6525751, 'doi': '10.1103/physrevd.65.055004', 'github_urls': [], 'n_tokens_mistral': 10358, 'n_tokens_neox': 8682, 'n_words': 5725, 'pdfsha': '63247f8d33be7f88c38d2c656df38aea2c2c9402', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ph/9912485v3.pdf'], 'title': ['Production of light pseudoscalars in external electromagnetic fields by the Schwinger mechanism Typeset using REVT E X 1', 'Production of light pseudoscalars in external electromagnetic fields by the Schwinger mechanism Typeset using REVT E X 1'], 'venue': []}

arxiv

Testing CPT invariance by using C-even neutral-meson-antimeson correlated states 12 Dec 2000 12 December 2000 G V Dass Physics Department Institut für Theoretische Physik Indian Institute of Technology Powai 400076BombayIndia W Grimus Universität Wien Boltzmanngasse 5A-1090WienAustria L Lavoura Técnica de Lisboa Centro de Física das Interacções Fundamentais Instituto Superior Técnico Universidade 1049-001LisboaPortugal Testing CPT invariance by using C-even neutral-meson-antimeson correlated states 12 Dec 2000 12 December 2000PACS numbers: 1130Er, 1320-v We consider the decays of a correlated neutral-meson-antimeson state with Cparity +1. We show that there is CPT noninvariance in the mixing of the neutral mesons if, for any two decay modes f and g, the decay rate has a component R A which is antisymmetric under the interchange of the decay times t 1 and t 2 . In particular, one may cleanly extract the CPT-noninvariance parameter with the help of R A by using opposite-sign dilepton events.PACS numbers: 11.30.Er, 13.20.-v * For discussions of other CPT-noninvariant observables, see for instance the recent reviews in Ref.[8], and the references cited therein. The system formed by a spin-0 flavoured neutral meson M 0 and its antimesonM 0 (where M may be either K, D, B d , or B s ) is experimentally interesting for testing the symmetries CP, T, and CPT [1]. We may recall that CP violation was first observed in the K 0 -K 0 system [2], and is now being actively searched for in the B 0 d -B 0 d system [3]; there are also observations of T violation [4] and tests of CPT invariance [5] in the K 0 -K 0 system, although their interpretation remains controversial [6]. We introduce the usual propagation eigenstates |M H = p H |M 0 + q H |M 0 and |M L = p L |M 0 − q L |M 0 .(1) These states get multiplied by probability amplitudes exp (−iλ H t) and exp (−iλ L t), respectively, for nonzero proper time t. Here, λ H = m H − i 2 Γ H and λ L = m L − i 2 Γ L ,(2) where m H , m L , Γ H , and Γ L are real. Defining [7] θ = q H /p H − q L /p L q H /p H + q L /p L and q p = q H q L p H p L ,(3) one finds that both the real and the imaginary parts of θ are in principle measurable and violate both CP and CPT; * while the phase of q/p is convention-dependent and, therefore, unphysical. On the other hand, the magnitude of q/p is measurable and signals T and CP violation in the mixing when it differs from 1. It is convenient to define the functions g ± (t) = 1 2 e −iλ H t ± e −iλ L t .(4) Then, the probability amplitudes a(t), b(t),ā(t), andb(t) for, respectively, the transitions M → M, M →M ,M →M , andM → M are [7,9] a(t) = g + (t) − θg − (t) , b(t) = q p √ 1 − θ 2 g − (t) , a(t) = g + (t) + θg − (t) , b(t) = p q √ 1 − θ 2 g − (t) .(5) In the experimental study of the neutral-meson-antimeson systems it is interesting to use correlated meson-antimeson states of the form |M ± = 1 √ 2 |M 0 ( k) ⊗ |M 0 (− k) ± |M 0 ( k) ⊗ |M 0 (− k) ,(6) in which one of the mesons flies in one direction (denoted by the momentum k) and the other one flies in the opposite direction. Indeed, the DAΦNE experiment at Frascati will use correlated states of the form |K − , while the Belle and BaBar Collaborations are working with states |B d− . The correlated states |M ± have C-parity ±1 and are produced from the decay of certain spin-1 resonances like the Φ and Υ(4S). At e + e − colliders, states of the type |M − are preferred since the produced resonances almost always have quantum numbers J P C = 1 −− . Possible ways of experimentally producing states of the type |M + have nevertheless been discussed in the literature, see for instance Ref. [10]. It has recently been shown [7] that it is difficult or even impossible to extract θ from experiments with states of the experimentally preferred form |M − alone. This happens because, in the semileptonic decays of |M − , the effects of CPT noninvariance are entangled with those of violations of the ∆F = ∆Q rule-where F means flavour, i.e., F may be either S, C, or B. Thus, it is difficult to know whether one is really measuring violations of CPT invariance or one is measuring violations of that rule. For instance, in Ref. [11] CPT invariance has been used for checking the rule ∆B = ∆Q, whereas in Ref. [12] θ has been constrained by assuming the validity of this rule. The purpose of this note is to show that θ can neatly be determined by using the opposite-sign dilepton decays of the symmetrical correlated state |M + , which has C-parity +1. As a matter of fact, tests of CPT invariance using states |M + do not necessarily require the use of any specific final states-like the semileptonic decays-and the signal for CPT noninvariance is independent of assumptions about the decay amplitudes to the particular final states that one may wish to use. However, if one wants to extract the actual value of the CPT-violating parameter θ, then it is better to use opposite-sign semileptonic decays. Let us consider the event in which the meson of |M + with momentum k decays into f at time t 1 , while the meson with momentum − k decays into g at time t 2 . The rate of such an event is R (f, t 1 ; g, t 2 ) = 1 2 |A (f, t 1 ; g, t 2 )| 2 ,(7) where A (f, t 1 ; g, t 2 ) = a (t 1 ) A f + b (t 1 )Ā f b (t 2 ) A g +ā (t 2 )Ā g ) + b (t 1 ) A f +ā (t 1 )Ā f a (t 2 ) A g + b (t 2 )Ā g ,(8) where A f is the amplitude for the decay |M 0 → |f ,Ā f is the amplitude for |M 0 → |f , and similarly for the decays into g. One may use Eqs. (5), together with g + (t 1 ) g + (t 2 ) + g − (t 1 ) g − (t 2 ) = g + (t 1 + t 2 ) , g + (t 1 ) g − (t 2 ) + g − (t 1 ) g + (t 2 ) = g − (t 1 + t 2 ) ,(9) to show that the amplitude A given in Eq. (8) can be written as A (f, t 1 ; g, t 2 ) = p q √ 1 − θ 2 [g − (t 1 + t 2 ) − 2θg − (t 1 ) g − (t 2 )] A f A g + q p √ 1 − θ 2 [g − (t 1 + t 2 ) + 2θg − (t 1 ) g − (t 2 )]Ā fĀg + g + (t 1 + t 2 ) − 2θ 2 g − (t 1 ) g − (t 2 ) A fĀg +Ā f A g +θ [g + (t 1 ) g − (t 2 ) − g − (t 1 ) g + (t 2 )] A fĀg −Ā f A g .(10) Notice that, when θ = 0, the amplitude A is not only symmetric under the interchange t 1 ↔ t 2 but, as a matter of fact, it is a function only of the sum t 1 + t 2 [9,13]. If, however, the CPT-noninvariance parameter θ is nonzero, then A is not any more a function only of t 1 + t 2 and, indeed, it acquires a term-the one in the last line of Eq. (10)-antisymmetric under t 1 ↔ t 2 . Then, the decay rate can be written as the sum of a symmetric component R S (f, t 1 ; g, t 2 ) and an antisymmetric component R A (f, t 1 ; g, t 2 ), R S (f, t 1 ; g, t 2 ) = 1 2 [R (f, t 1 ; g, t 2 ) + R (f, t 2 ; g, t 1 )] , R A (f, t 1 ; g, t 2 ) = 1 2 [R (f, t 1 ; g, t 2 ) − R (f, t 2 ; g, t 1 )] ,(11) and a nonzero R A arises only if CPT invariance does not hold for neutral-meson mixing treated in the usual fashion. We conclude that, if experimentally one finds R A = 0, then this signals the presence of CPT violation. If, on the other hand, R A is not significantly different from zero, then this may allow to put a bound on θ. We want to stress that a measurable CPT-violating asymmetry exists for any two different final states f = g provided A fĀg −Ā f A g = 0; no other assumptions about the amplitudes A f , A g ,Ā f , andĀ g are needed-in particular, about their behaviour under the transformations CP, CPT, and T. As an illustration, let us consider the particular case of opposite-sign dilepton events [14], i.e., inclusive semileptonic decays with f = Xℓ + ν ℓ and g =Xℓ −ν ℓ . We shall use the simple notation ℓ + for f and ℓ − for g. We want to show that, in this particular case, it is possible, at least in principle, to explicitly derive the value of θ from the observation of the time dependence of R A . Allowing for transitions which violate the ∆F = ∆Q rule, we introduce the rephasing-invariant quantities [7] λ + = q pĀ + A + andλ − = p q A − A − ,(12) where A + ≡ A ℓ + ,Ā − ≡Ā ℓ − , and so on. As usual, we assume that the quantitities θ, λ + , andλ − , which describe 'unexpected' physics, are small, and we confine ourselves to the first order in these quantities. If we do this, then the amplitude A is given by A = A +Ā− g + (t 1 + t 2 ) + λ + +λ − g − (t 1 + t 2 ) + θ [g + (t 1 ) g − (t 2 ) − g − (t 1 ) g + (t 2 )] .(13)R S ℓ + , t 1 ; ℓ − , t 2 = 1 8 |A + | 2 Ā − 2 e −Γ H t + + e −Γ L t + + 2 e −Γt + cos (∆mt + ) +2 e −Γ H t + − e −Γ L t + Re λ + +λ − + 4 e −Γt + sin (∆mt + ) Im λ + +λ −(14) and R A ℓ + , t 1 ; ℓ − , t 2 = 1 2 |A + | 2 Ā − 2 e −Γt + × Re θ sinh ∆Γt 1 2 cos (∆mt 2 ) − cos (∆mt 1 ) sinh ∆Γt 2 2 + Im θ cosh ∆Γt 1 2 sin (∆mt 2 ) − sin (∆mt 1 ) cosh ∆Γt 2 2 ,(15) where t + = t 1 + t 2 . One may note that violations of the ∆F = ∆Q rule appear only in R S , while the CPT-noninvariance parameter θ appears only in R A , when we take into account 'small' physics to first order only. A glance at Eq. (15) confirms that both the real and the imaginary part of θ can be determined using this method. However, in practice that determination depends on the values of ∆m and ∆Γ. If we use M = B d , we must set ∆Γ ≈ 0 in Eq. (15). Then, only Im θ is measurable and the time dependence is exp [−Γ (t 1 + t 2 )] [sin (∆mt 1 ) − sin (∆mt 2 )]. On the other hand, for M = K, † where Γ H ≪ Γ L , we may choose t 1 and t 2 such that t 1 ∼ 1/Γ L ≪ t 2 . Equation (15) then yields R A ℓ + , t 1 ; ℓ − , t 2 ≃ 1 4 |A + | 2 Ā − 2 e −Γt 1 −Γ H t 2 [Re θ cos (∆mt 1 ) − Im θ sin (∆mt 1 )] ,(16) which is practically independent of t 2 as long as t 2 ≪ 1/Γ H . In conclusion, in this note we have discussed the possibility of using the states |M + for probing CPT invariance in the mixing of neutral mesons. We have shown that this is feasible for A fĀg −Ā f A g = 0, because then the decay rate R (f, t 1 ; g, t 2 ) has a component which is antisymmetric with respect to the interchange t 1 ↔ t 2 if and only if CPT invariance does not hold. This conclusion is independent of the final states f and g, or any assumptions thereon. We have, in particular, considered the case of opposite-sign dilepton events; then, in the decay rate the effects of CPT violation appear separated from the effects of violation of the ∆F = ∆Q rule. It is then possible to cleanly extract the CPT-noninvariance parameter θ, at least when the values of ∆m and ∆Γ are not too unfavourable. We believe that, although experimentally it might be hard to work with states |M + , the cleanness with which they probe CPT invariance might make the effort worthwhile. With the usual definitions ∆m = m H − m L , ∆Γ = Γ H − Γ L , and Γ = (Γ H + Γ L ) /2, we obtain for the decay rate the results Acknowledgement L.L. thanks João P. Silva for an enlightening discussion on the production of the states |M + , and for reading the manuscript. For a general theoretical treatment, see. G C Branco, L Lavoura, J P Silva, CP violation. OxfordOxford University PressFor a general theoretical treatment, see G. C. Branco, L. Lavoura, and J. P. Silva, CP violation (Oxford University Press, Oxford, 1999). . J H Christenson, J W Cronin, V L Fitch, R Turlay, Phys. Rev. Lett. 13138J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Phys. Rev. Lett. 13 (1964) 138. † The usual long-lived neutral kaon K L is our heavier state M H , and the usual short-lived neutral kaon K S corresponds to the lighter state M L. † The usual long-lived neutral kaon K L is our heavier state M H , and the usual short-lived neutral kaon K S corresponds to the lighter state M L . For the latest results. T Affolder, CDF Collaboration ; BaBar CollaborationHitlin at ICHEP. Osaka, Japan6172005to be published in the proceedings (hepex/0011024For the latest results, see CDF Collaboration, T. Affolder et al., Phys. Rev. D 61 (2000) 072005; BaBar Collaboration, talk presented by D. G. Hitlin at ICHEP 2000, Osaka, Japan, 27 July-2 August 2000, to be published in the proceedings (hep- ex/0011024); hep- ex/0010008Aihara at ICHEP. Osaka, Japanto be published in the proceedingsBelle Collaboration, talk presented by H. Aihara at ICHEP 2000, Osaka, Japan, 27 July-2 August 2000, to be published in the proceedings (hep- ex/0010008). . A Angelopoulos, CPLEAR CollaborationPhys. Lett. B. 44443CPLEAR Collaboration, A. Angelopoulos et al., Phys. Lett. B 444 (1998) 43. . R Carosi, NA31 CollaborationPhys. Lett. B. 237303NA31 Collaboration, R. Carosi et al., Phys. Lett. B 237 (1990) 303; . A Angelopoulos, CPLEAR Collaboration ; CPLEAR CollaborationPhys. Lett. B. 444297Phys. Lett. BCPLEAR Collab- oration, A. Angelopoulos et al., Phys. Lett. B 444 (1998) 52; CPLEAR Collaboration, A. Apostolakis et al., Phys. Lett. B 456 (1999) 297. . L Lavoura, Mod. Phys. Lett. A. 71367L. Lavoura, Mod. Phys. Lett. A 7 (1992) 1367; hep-ph/9909205. A K Rougé ; P, Kabir, Phys. Lett. B. 459335A. Rougé, hep-ph/9909205; P. K. Kabir, Phys. Lett. B 459 (1999) 335. . L Lavoura, J P Silva, Phys. Rev. D. 6056003L. Lavoura and J. P. Silva, Phys. Rev. D 60 (1999) 056003. . I I Bigi, hep-ph/0011231I. I. Bigi, hep-ph/0011231; . R Bluhm, hep-ph/0011272R. Bluhm, hep-ph/0011272. . M Kobayashi, A I Sanda, Phys. Rev. Lett. 693139M. Kobayashi and A. I. Sanda, Phys. Rev. Lett. 69 (1992) 3139. . Z.-Z Xing, D.-S Du, Phys. Lett. B. 276511Z.-Z. Xing and D.-S. Du, Phys. Lett. B 276 (1992) 511; . J R Fry, T Ruf, preprint CERN-PPE/94-20J. R. Fry and T. Ruf, preprint CERN-PPE/94-20 (1994); F E Close, G J Gounaris, The second DAΦNE physics handbook. L. Maiani, G. Pancheri, and N. Paver (SIS-Pubblicazioni dei Laboratori di FrascatiItalyII323Z.-Z. XingF. E. Close and G. J. Gounaris, in The second DAΦNE physics handbook, eds. L. Maiani, G. Pancheri, and N. Paver (SIS-Pubblicazioni dei Laboratori di Frascati, Italy, 1995), Vol. II, p. 681; Z.-Z. Xing, Phys. Lett. B 463 (1999) 323. . G V Dass, K V L Sarma, Phys. Rev. Lett. 721573erratum ibid. 72G. V. Dass and K. V. L. Sarma, Phys. Rev. Lett. 72 (1994) 191; erratum ibid. 72 (1994) 1573. . K Ackerstaff, OPAL CollaborationZ. Phys. C. 76401OPAL Collaboration, K. Ackerstaff et al., Z. Phys. C 76 (1997) 401; . K Belle Collaboration, Abe, hep-ex/0011090Belle Collabo- ration, K. Abe et al., hep-ex/0011090. . Z.-Z Xing, Phys. Rev. D. 53204Z.-Z. Xing, Phys. Rev. D 53 (1996) 204. . Z.-Z Xing, Phys. Lett. B. 450202Z.-Z. Xing, Phys. Lett. B 450 (1999) 202.

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{'abstract': 'We consider the decays of a correlated neutral-meson-antimeson state with Cparity +1. We show that there is CPT noninvariance in the mixing of the neutral mesons if, for any two decay modes f and g, the decay rate has a component R A which is antisymmetric under the interchange of the decay times t 1 and t 2 . In particular, one may cleanly extract the CPT-noninvariance parameter with the help of R A by using opposite-sign dilepton events.PACS numbers: 11.30.Er, 13.20.-v * For discussions of other CPT-noninvariant observables, see for instance the recent reviews in Ref.[8], and the references cited therein.', 'arxivid': 'hep-ph/0012131', 'author': ['G V Dass \nPhysics Department\nInstitut für Theoretische Physik\nIndian Institute of Technology Powai\n400076BombayIndia\n', 'W Grimus \nUniversität Wien\nBoltzmanngasse 5A-1090WienAustria\n', 'L Lavoura \nTécnica de Lisboa Centro de Física das Interacções Fundamentais Instituto Superior Técnico\nUniversidade\n1049-001LisboaPortugal\n'], 'authoraffiliation': ['Physics Department\nInstitut für Theoretische Physik\nIndian Institute of Technology Powai\n400076BombayIndia', 'Universität Wien\nBoltzmanngasse 5A-1090WienAustria', 'Técnica de Lisboa Centro de Física das Interacções Fundamentais Instituto Superior Técnico\nUniversidade\n1049-001LisboaPortugal'], 'corpusid': 18558418, 'doi': '10.1088/1126-6708/2001/02/044', 'github_urls': [], 'n_tokens_mistral': 5527, 'n_tokens_neox': 4747, 'n_words': 2786, 'pdfsha': '20c98175ba7e5c9b5bebba540db16ea18fc07a83', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ph/0012131v1.pdf'], 'title': ['Testing CPT invariance by using C-even neutral-meson-antimeson correlated states', 'Testing CPT invariance by using C-even neutral-meson-antimeson correlated states'], 'venue': []}

arxiv

Construction and Testing of a Common Mode Choke for Cryogenic Detector Pre-Amplifiers 8 Mar 2023 Mathias Richerzhagen European Southern Observatory (ESO) Karl-Schwarzschild-Str. 285748Garching bei MünchenGermany Joshua Hopgood European Southern Observatory (ESO) Karl-Schwarzschild-Str. 285748Garching bei MünchenGermany Construction and Testing of a Common Mode Choke for Cryogenic Detector Pre-Amplifiers 8 Mar 2023common-mode chokelarge instrumentsgrounding issuedetector systemcryogenic electronicsnanocrys- talline ferrite material Common-mode choke inductors are useful tools for resolving grounding issues in large detector systems. Using inductive components on cryogenic pre-amplifier boards has so far been prevented by the poor low-temperature performance of common ferrite materials such as NiZn and MnZn. Recently developed nanocrystalline and amorphous ferrite materials promise improved performance up to the point where using magnetics at liquid mitrogen temperatures becomes feasible. This research applies the work of Yin et al. 1 on characterizing ferrite materials by constructing and testing a common mode choke inductor for use on detector pre-amplifiers for the ELT first generation instruments. Introduction Common mode chokes are a well-understood instrument for improving immunity to electromagnetic disturbances as described among many in Ref. [2]. As passive components consisting of two windings on a common ferromagnetic core, chokes provide high impedance for common mode signals while having significantly lower impedance to differential signals. These properties make them efficient at preventing the coupling of common mode disturbance signals into electronic circuits through power or data lines, thus reducing the occurrence of a class of artifacts summarized as "Grounding Issues" during integration and testing. In cryogenic detector systems, the use of common mode chokes on power and signal inputs may be especially advantageous since the detector and associated pre-amplifier board are typically located far away from other control electronics (i.e. the detector controller). For the ELT first generation instruments METIS, 3 HARMONI 4 and MICADO 5 the warm control electronics cabinet and cold pre-amplifier are separated by cables of up to 5m length. Ideally, differential power and signal inputs of a cryogenic pre-amplifier would be common mode filtered at the pre-amplifier. According to Ref. [6], the use of common mag-1 netic core materials is not feasible in cryogenic systems due to the significant loss in magnetic permeability of the core at liquid nitrogen temperatures. Recently-developed nanocrystalline and amorphous core materials have been shown to perform much better at low temperatures as evaluated in Ref. [1]. This research aims at applying the results of Ref. [1] to construct a common mode choke inductor suitable for cryogenic detector systems and comparing its performance to classic NiZn or MnZn based cores. The inductor is characterized in terms of its impedance over frequency curve and not its magnetic properties for easier use in electronics engineering design documentation. Method Construction Four common mode choke samples are constructed on commercially available toroidal cores. Samples are constructed using nanocrystalline and traditional NiZn based core materials and placed on a test coupon made from perforated circuit board. For the NiZn core, a larger core is split in half to achieve a comparable magnetic cross section to the nanocrystalline core. Windings are added manually using a segmented winding pattern as shown in Figure 1. A bifilar winding pattern could be chosen as an alternative, but the segmented technique allows for some spatial separation between the two windings making the effect of an insulation failure less severe since a windingto-winding short circuit is less likely. The resulting inductor samples are shown in Figure 2 with construction parameters listed in Table 1. The test samples are connected to a vector network analyser (OMICRON Lab -Bode 100) configured in one-port impedance measurement mode as shown in Figure 3 and Figure 4 to mea- The supported frequency range is up to 40MHz. For the cold temperature test, the entire sample is manually submersed in liquid nitrogen as shown in Figure 5 until steady state is reached. Testing Results Test results are presented as Bode plots of impedance magnitude over frequency in Figure 6 since these plots are often found in the data sheets of commercial common mode chokes which allows comparison with other parts. It is observed that at room temperature, all samples show the typical impedance curve of common mode choke inductors. In differential mode the impedance is clearly lower than in common by the parasitic winding capacitance is present. In common mode operation, the nanocrystalline chokes exhibit higher impedance than the traditional core materials. This is in line with expectations compared to commercially available nanocrystalline inductors. The performance advantage is offset by higher cost. At cryogenic temperature it becomes apparent that the common mode impedance of the traditional NiZn cores degrades until the point where it is almost indistinguishable from the differential mode impedance. This indicates that the core material is no longer effective, confirming the result stated in Ref. [1]. The nanocrystalline cores maintain performance except for an insignificant decrease in impedance. It is also observed that the DC resistance visible in the low frequency range of the differential impedance curves decreases with temperature, as expected. Discussion The experiments show that common mode choke inductors based on commercially-available nanocrystalline core materials are suitable for operation at liquid nitrogen temperatures since the core material maintains its magnetic properties at cold temperature. Commercial inductors based on similar core materials may be considered for application in detector control systems operating at cryogenic temperatures. The tested common mode chokes are most suitable for decoupling the preamp power supply lines due to the chosen core size and winding wire gauge. It may be advantageous to also commonmode filter signal inputs in the future. At the time of writing, no small-size nanocrystalline cores were available for purchase that allow manual construction of sufficiently small data line common mode chokes. With the cooperation of a commercial magnetics manufacturer, it may be possible to adapt smaller signal line common mode chokes, possibly in surface mount technology form factors, to nanocrystalline core materials. Due to the general performance advantage of nanocrystalline cores even at room temperature it is expected that magnetic components using this technology will soon become more widely available. Long-term reliability remains to be demonstrated, but it is promising that none of the cores or windings were damaged by repeated immersion in liquid nitrogen during this work. In general, it is advisable to cool down ferrite materials slowly due to the risk of cracking. To prevent contamination of the cryostat in case a core does fracture it is also advisable to use epoxy coated cores to keep fragments from separating. Before inclusion in a scientific detector system, other parameters of the chokes such as winding resistance, DC bias current, thermal effects, and self-resonance need to be characterised. Consideration of these parameters should be part of the standard design procedure for a common mode choke and are expected to be unlikely to change at cold temperatures based on the test results presented here. Fig 1 1Segmented Winding Common Mode Choke Fig 2 2Common Mode Choke Samples Fig 3 3Common Mode Test Setup Fig 4 Fig 5 45Differential Liquid Nitrogen Immersion Setup sure common mode and differential impedance over frequency both at room temperature (approximately 293K) and liquid nitrogen temperature (77K at standard pressure). The chosen vector network analyser is capable of directly plotting the impedance magnitude trace over frequency. Fig 6 6Measurement Results mode since magnetic flux induced by both windings cancels out. A clear resonance peak caused Table 1 1Common Mode Choke Sample ParametersSample Core Windings A ½ Würth 74270113 NiZn 2x 6 Windings AWG22 Magnet Wire B VAC L2009-W914 Nanocrystalline 2x 6 Windings AWG22 Magnet Wire C ½ Würth 74270113 NiZn 2x 12 Windings AWG26 Magnet Wire D VAC L2009-W914 Nanocrystalline 2x 12 Windings AWG26 Magnet Wire He received his Engineering Diploma from RWTH Aachen University in 2012. His current research includes development of the detector controller for the ELT as well as some work in cryoelectronics.Biographies and photographs of the other authors are not available.Tables 1 Common Mode Choke Sample Parameters 8List of FiguresList of Characterization of inductor magnetic cores for cryogenic applications. S Yin, M Mehrabankhomartash, A J Cruz, 2021 IEEE Energy Conversion Congress and Exposition (ECCE). S. Yin, M. Mehrabankhomartash, A. J. Cruz, et al., "Characterization of inductor magnetic cores for cryogenic applications," in 2021 IEEE Energy Conversion Congress and Exposition (ECCE), 5327-5333 (2021). Electromagnetic compatibility. J J Goedbloed, Translation of: Electromagnetische compatibiliteit. New YorkPrentice HallJ. J. Goedbloed, Electromagnetic compatibility, Prentice Hall, New York (1992). Translation of: Electromagnetische compatibiliteit. METIS: the mid-infrared E-ELT imager and spectrograph. B R Brandl, M Feldt, A Glasse, Ground-based and Airborne Instrumentation for Astronomy. B. R. Brandl, M. Feldt, A. Glasse, et al., "METIS: the mid-infrared E-ELT imager and spectro- graph," in Ground-based and Airborne Instrumentation for Astronomy V, S. K. Ramsay, I. S. H Mclean, Takami, 9147, 914721, International Society for Optics and Photonics. McLean, and H. Takami, Eds., 9147, 914721, International Society for Optics and Photonics, SPIE (2014). HARMONI: first light spectroscopy for the ELT: instrument final design and quantitative performance predictions. N A Thatte, I Bryson, F Clarke, 11447, 114471W, International Society for Optics and Photonics. C. J. Evans, J. J. Bryant, and K. MotoharaSPIEGround-based and Airborne Instrumentation for AstronomyN. A. Thatte, I. Bryson, F. Clarke, et al., "HARMONI: first light spectroscopy for the ELT: instrument final design and quantitative performance predictions," in Ground-based and Air- borne Instrumentation for Astronomy VIII, C. J. Evans, J. J. Bryant, and K. Motohara, Eds., 11447, 114471W, International Society for Optics and Photonics, SPIE (2020). MICADO: first light imager for the E-ELT. R Davies, J Schubert, M Hartl, 9908, 99081Z, International Society for Optics and Photonics. C. J. Evans, L. Simard, and H. TakamiSPIEGround-based and Airborne Instrumentation for AstronomyR. Davies, J. Schubert, M. Hartl, et al., "MICADO: first light imager for the E-ELT," in Ground-based and Airborne Instrumentation for Astronomy VI, C. J. Evans, L. Simard, and H. Takami, Eds., 9908, 99081Z, International Society for Optics and Photonics, SPIE (2016). Review of power electronics components at cryogenic temperatures. H Gui, R Chen, J Niu, IEEE Transactions on Power Electronics. 355H. Gui, R. Chen, J. Niu, et al., "Review of power electronics components at cryogenic temper- atures," IEEE Transactions on Power Electronics 35(5), 5144-5156 (2020). Mathias Richerzhagen is a detector electronics engineer at the European Southern Observatory. Mathias Richerzhagen is a detector electronics engineer at the European Southern Observatory.

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{'abstract': 'Common-mode choke inductors are useful tools for resolving grounding issues in large detector systems. Using inductive components on cryogenic pre-amplifier boards has so far been prevented by the poor low-temperature performance of common ferrite materials such as NiZn and MnZn. Recently developed nanocrystalline and amorphous ferrite materials promise improved performance up to the point where using magnetics at liquid mitrogen temperatures becomes feasible. This research applies the work of Yin et al. 1 on characterizing ferrite materials by constructing and testing a common mode choke inductor for use on detector pre-amplifiers for the ELT first generation instruments.', 'arxivid': '2303.04541', 'author': ['Mathias Richerzhagen \nEuropean Southern Observatory (ESO)\nKarl-Schwarzschild-Str. 285748Garching bei MünchenGermany\n', 'Joshua Hopgood \nEuropean Southern Observatory (ESO)\nKarl-Schwarzschild-Str. 285748Garching bei MünchenGermany\n', 'Mathias Richerzhagen \nEuropean Southern Observatory (ESO)\nKarl-Schwarzschild-Str. 285748Garching bei MünchenGermany\n', 'Joshua Hopgood \nEuropean Southern Observatory (ESO)\nKarl-Schwarzschild-Str. 285748Garching bei MünchenGermany\n'], 'authoraffiliation': ['European Southern Observatory (ESO)\nKarl-Schwarzschild-Str. 285748Garching bei MünchenGermany', 'European Southern Observatory (ESO)\nKarl-Schwarzschild-Str. 285748Garching bei MünchenGermany', 'European Southern Observatory (ESO)\nKarl-Schwarzschild-Str. 285748Garching bei MünchenGermany', 'European Southern Observatory (ESO)\nKarl-Schwarzschild-Str. 285748Garching bei MünchenGermany'], 'corpusid': 255711589, 'doi': '10.1117/1.jatis.9.1.010501', 'github_urls': [], 'n_tokens_mistral': 3085, 'n_tokens_neox': 2622, 'n_words': 1708, 'pdfsha': '16182695c714731d7ef40d156cfe84e644ed2b1d', 'pdfurls': ['https://export.arxiv.org/pdf/2303.04541v1.pdf'], 'title': ['Construction and Testing of a Common Mode Choke for Cryogenic Detector Pre-Amplifiers', 'Construction and Testing of a Common Mode Choke for Cryogenic Detector Pre-Amplifiers', 'Construction and Testing of a Common Mode Choke for Cryogenic Detector Pre-Amplifiers', 'Construction and Testing of a Common Mode Choke for Cryogenic Detector Pre-Amplifiers'], 'venue': []}

arxiv

Published as a conference paper at ICLR 2023 QAID: QUESTION ANSWERING INSPIRED FEW-SHOT INTENT DETECTION Asaf Yehudai IBM Israel Research Lab ♢ Hebrew University of Jerusalem ♢ ♣ IBM Israel Research Lab ♢ Hebrew University of Jerusalem Matan Vetzler IBM Israel Research Lab ♢ Hebrew University of Jerusalem Yosi Mass yosimass@boazc.il.ibm.com IBM Israel Research Lab ♢ Hebrew University of Jerusalem ♢ IBM Israel Research Lab ♢ Hebrew University of Jerusalem Koren Lazar IBM Israel Research Lab ♢ Hebrew University of Jerusalem Doron Cohen doronc@boazc.il.ibm.com IBM Israel Research Lab ♢ Hebrew University of Jerusalem ♢ IBM Israel Research Lab ♢ Hebrew University of Jerusalem Boaz Carmeli IBM Israel Research Lab ♢ Hebrew University of Jerusalem Published as a conference paper at ICLR 2023 QAID: QUESTION ANSWERING INSPIRED FEW-SHOT INTENT DETECTION Intent detection with semantically similar fine-grained intents is a challenging task. To address it, we reformulate intent detection as a question-answering retrieval task by treating utterances and intent names as questions and answers. To that end, we utilize a question-answering retrieval architecture and adopt a two stages training schema with batch contrastive loss. In the pre-training stage, we improve query representations through self-supervised training. Then, in the finetuning stage, we increase contextualized token-level similarity scores between queries and answers from the same intent. Our results on three few-shot intent detection benchmarks achieve state-of-the-art performance. INTRODUCTION Intent detection (ID) is the task of classifying an incoming user query to one class from a set of mutually-exclusive classes, a.k.a. intents (Wang et al., 2014;Schuurmans & Frasincar, 2019;Liu et al., 2019a). This ability is a cornerstone for task-oriented dialogue systems as correctly identifying the user intent at the beginning of an interaction is crucial to its success. However, labeled data is required for training and manual annotation is costly. This calls for sample efficient methods, gaining high accuracy with minimal amounts of labeled data. Recent works tackling few-shot ID have relied on large-scale pre-trained language models, such as BERT (Devlin et al., 2018). These works leverage task-adaptive training and focus on pre-training a model on a large open-domain dialogue corpus and fine-tuning it for ID classification (Mehri et al., 2020;Wu et al., 2020a;Casanueva et al., 2020;Zhang et al., 2021a). Alternative approaches tried to learn query representation based on query-to-query matching (henceforth, Match-QQ systems) Mass et al., 2020;Mehri et al., 2021). ; Mass et al. (2020) adopt pairwise-encoding systems with cross-attention to deploy K-Nearest-Neighbor (K-NN) (Fix & Hodges, 1989) classification schema where training queries are fully utilized for both training and inference stages. Nevertheless, those methods' downside is the processing time combined with the difficulty of scaling to large number of intents (Liu et al., 2021c). The need to efficiently compare an incoming query to a large set of possible answers resides at the core of any question answering (QA) retrieval system (henceforth, Match-QA systems) (Karpukhin et al., 2020). Recently, Khattab & Zaharia (2020) introduced ColBERT, which allows faster training and inference by replacing the cross-attention mechanism used by Match-QQ systems Mass et al., 2020;Nogueira & Cho, 2019) with a fast contextualized token-level similarity mechanism dubbed late interaction. In this work, we present a Question Answering inspired Intent Detection system, named QAID. We start by formulating the ID task as a question-answering retrieval task by treating the utterances and the intent names as queries and answers, respectively. This reformulation allows us to introduce valuable additional signal from the intent names. Then, we adapts the efficient architecture of Col-BERT while replacing its triplet function loss with batch contrastive loss which was proven to be more robust (Khosla et al., 2020) and performs well in various tasks (Gunel et al., 2021;Gao et al., 2021a), including ID classification (Zhang et al., 2021b). In contrast to ColBERT which compares a query to a pair of positive and negative documents, we also include queries as positive examples, and so we compare the queries both to their answers and to other queries from the same intent. This allows QAID to represent similarly both queries and answers of the same intent. Therefore, our training method assumes the settings of both Match-QQ and Match-QA. In inference, QAID relies on the token-level similarity (late interaction) mechanism between incoming query and all intent names for its predictions (Khattab & Zaharia, 2020). Our contribution is thus threefold. (1) We show that few-shot intent detection can be successfully handled by QA systems when letting the intent name play the role of the answer. (2) We show how intent detection architectures can benefit from recent advancements in supervised batch contrastive training and late-interaction scores. (3) We report state-of-the-art results on three few-shot intent detection benchmarks. METHOD Our method addresses the few-shot intent detection task, in which we have C defined intents and the task is to classify an incoming user query, q, into one of the C classes. In our formulation, upon getting a new user query q, we need to retrieve the most suited intent name. We set balanced K-shot learning for each intent (Mehri et al., 2020;Casanueva et al., 2020;, i.e., the training data containing K examples per intent 1 . In the following section, we describe the structure of our QAID framework and its training stages. First, in Section 2.1 we elaborate on the different components of QAID. Then, in Section 2.2 we present the two training stages: the self-supervised contrastive pre-training in 2.2.1 and the supervised batch contrastive fine-tuning in 2.2.2. Lastly, in Section 2.3 we briefly touch on our decision to formulate ID as a question retrieval task. REPRESENTATION LEARNING FRAMEWORK The main components of our framework are: • Data Augmentation module, Aug(⋅). For each input query, q, we generate two random augmentations,q = Aug(q), each of which represents a different view of the input, q. For our augmentation we use the combination of two simple and intuitive 'corruption' techniques (Gao et al., 2021a;Wu et al., 2020b;; (i) randomly masking tokens from q (Devlin et al., 2018); (ii) dropping a small subset of neurons and representation dimensions. Technique (i) is done before passing the query to the encoder and technique (ii) is done in the forward propagation through the encoder model. • Encoder model, Enc(⋅), which maps a query q, consisting q 1 , ..., q m tokens, to Enc(q) ∈ R m×D E , where D E is the embedding dimension. In our experiments, it is either 768 or 1024. • Projection layer, P roj(⋅), a single linear layer that maps vectors of dimension D E to vectors of dimension D P = 128, followed by normalization to the unit hypersphere. • Token-level score, Score(⋅, ⋅), given two queries u = (u 1 , ..., u m ) and v = (v 1 , ..., v l ), the relevance score of u regarding to v, denoted by Score(u, v), is calculated by the late interaction between their bags of projected contextualized representations, i.e z(u) = P roj(Enc(u)). Namely, the sum of the maximum token-wise cosine similarity of their projected representations (Khattab & Zaharia, 2020). Equation 1 shows the formulation of this score. Score(u, v) = i∈[m] max j∈[l] z(u) i ⋅ z(v) j(1) TWO-STAGE CONTRASTIVE TRAINING In both stages, given a batch of input samples Q = (q 1 , .., q n ), we first apply Aug followed by the encoding and projection layer, denoted by the z(⋅) function as described in the last section, and so we have X Q = z(Aug(Q)) ∈ R 2n×D P , where the 2n is a result of the two random augmentations we applied to each query. In the self-supervised training, each two augmented queries are the only positive examples for each other while all other queries are negative. In the supervised training phase, we also run the same process with the same encoder on the corresponding intent names, A = (a 1 , ..., a n ), resulting in X A = z(Aug(A)) ∈ R 2n×D P . X A together with X Q forms a training batch of 4n instances. In this supervised setting, all queries and intent names of the same intent are positive to each other while all others are negative. PRE-TRAINING We use a task-adaptive pre-training stage to overcome the few-shot constraint, as done by most goaloriented dialogue works (Mehri et al., 2020;. Our pre-training aims to facilitate domain adaptation by two early-stage objectives: (1) Incorporate token-level domain knowledge into the model (Mehri et al., 2020;; (2) Adopt queries' representations to the dialog domain through data augmentation techniques and self-supervised contrastive training (Zhang et al., 2021b;2022a). Practically, in a training batch containing 2n augmented queries, let t ∈ [2n] be the index of an arbitrary augmented query. Then in the self-supervised contrastive learning stage, the loss takes the following form: L self = − t∈[2n] log exp(Score(q t , q J(t) )/τ ) ∑ a∈A(t) exp(Score(q t , q a )/τ )(2) where t is called the anchor/pivot, J(t) is the index of the second augmented sample deriving from the same source sample (a.k.a positive), A(t) = {[2n] \ t}, A(t) \ J(t) are called the negative and τ ∈ R + is a scalar temperature parameter that controls the penalty to negative queries (see step (A) in Figure 1). Masked language modeling as an auxiliary loss: In addition to the self-supervised contrastive training, we pre-train also on the masked language modeling (MLM) task (Taylor, 1953), to further adjust sentence-level representation to the domain of the data. Moreover, this improves lexical-level representation which is essential for token-level similarity. Hence we define L mlm as the average cross-entropy loss over all masked tokens. The overall loss for the pre-training phase is L P T = L self + λL mlm , where λ is a controllable hyperparameter. FINE-TUNING At the fine-tuning stage, we only have a limited number of examples for each intent, and intents may be semantically similar, making the classification task difficult. To address the data scarcity, we utilize the explicit intent names as unique examples that serve as answers in our QA retrieval framework. This is similar to recent works that leverage label names for zero-shot or few-shot text classification (Meng et al., 2020;Basile et al., 2021). The intent name is usually assigned by a domain expert when designing a goal-oriented dialogue system. As such, it provides a semantic description of the intent that aims to discriminate it from other intents. Consequently, intent names may provide a signal that is otherwise difficult to extract from a small set of queries. Additionally, we utilize the pre-trained model designed for queries' representations and continue fine-tuning it on the few-shot examples with the permanent representation learning technique (Khosla et al., 2020) of supervised batch contrastive training (see step (B) in Fig. 1). In that way, each time our model pulls together two queries from the same class it simultaneously also pulls their intent names closer together as well. Formally our supervised contrastive loss has the form: L sup = t∈[4n] −1 |P (t)| p∈P (t) log exp(Score(q t , q p )/τ ) ∑ a∈A(t) exp(Score(q t , q a )/τ )(3) In this formulation, q may represent either an augmented query or its intent name, and since each instance has four views: two augmented queries and two augmented intent names, we have a total of 4n samples. A(t) = {[4n] \ t}; P (t) is the group of all samples that are positive to q t and is defined as all the augmented queries or intent names derived from the same label as q t . Besides the supervised contrastive loss we also train with a classification loss, L clss , and an MLM loss, L mlm . In total, the fine-tuning loss is L F T = L sup + λ class L class + λ mlm L mlm , where λ class and λ mlm are controllable hyperparameters. Indexing and Inference. After fine-tuning, we index the embeddings of all candidate answers (a.k.a intent names) outputted from the encoder into Faiss (Johnson et al., 2021), a library for large-scale vector-similarity search. see Khattab & Zaharia (2020) for more details. Then, at inference time, we compare the incoming query representation with all answers in the index and retrieve the most similar (see step (C) and (D) in Fig. 1). WHY FORMULATE THE TASK AS A QUESTION-ANSWERING RETRIEVAL TASK? For a few-shot intent detection model to be practical, it must be computationally efficient in both training and inference phases, while possibly handling a large number of intents (Qi et al., 2020). Here we are leveraging the recent success of dense passage retrieval for question answering that was shown to perform well and efficiently (Karpukhin et al., 2020;Khattab & Zaharia, 2020). Those systems can handle a large number of candidate answers by using (1) a dual-encoder framework with light comparison instead of the computationally demanding pairwise encoding, (2) an indexation that enables a large-scale fast retrieval. EXPERIMENTAL SETUP DATASETS We experiment with three widely studied few-shot intent detection datasets which represent the intent detection (ID) part of DialoGLUE benchmark. 2 These datasets present challenging few-shot ID tasks with fine-grained intents that are semantically similar. Moreover, they facilitate the comparison with recent state-of-the-art baselines. Clinc150 (Larson et al., 2019) contains 22,500 personal assistance queries classified into 150 intents across 10 domains. Banking77 (Casanueva et al., 2020) contains 13,242 online banking queries classified into 77 finegrained intents in a single domain. HWU64 (Liu et al., 2019b) MODELS AND BASELINES We experiment with RoBERTa-base and RoBERTa-large encoders from the Hugging Face transformers library 3 with ColBERT architecture. 4 In the self-supervised pre-training stage, we utilize the training and validation sets of the six ID datasets from Zhang et al. (2021b); Mehri et al. (2020;. For a fair evaluation, we exclude the test sets from our pre-training following the observation of Zhang et al. (2021b). We fix the number of embeddings per query at m = 32 same as Khattab & Zaharia (2020). For the MLM loss, we follow the masking strategy of Devlin et al. (2018). We also use this masking strategy to augment our input queries in addition to the representation-level augmentation through the encoder build in 10% dropout. For the contrastive loss, we use the implementation of Khosla et al. (2020) 5 . For the late interaction score, we normalize the score by the number of tokens in the summation. We train our encoder for 20 epochs with a batch size of 64, a learning rate of 1e −5 , a temperature parameter τ of 0.07 (same as Khosla et al. (2020)) and λ = 0.1 as recommended in the literature. For the fine-tuning stage, we train our models on 5-and 10-shot splits as available within the public dataset distribution. We train our model for 10 epochs starting from the pre-trained model checkpoint. We set the batch size to 32, where queries and answers are encoded separately by the same encoder. Answers are truncated by the longest one in the batch same as Khattab & Zaharia (2020). We apply the same masking schema to queries and answers as done in the pre-training. We set the temperature to 0.07. We also set λ class and λ mlm to 0.1 and 0.05, respectively, making L sup the main summand in L F T . We used those parameters as they were recommended in the literature and shown to perform best in hyperparameter tuning. Following previous works, we ran the experiments with five different seeds and report the average accuracy. For our Faiss Index 6 implementation, we use IVFScalarQuantizer ("InVerted File with Scalar Quantizer."). To improve memory efficiency, every embedding is represented by 8 bytes. In our work, we use the full retrieval option that effectively retrieves all candidate answers. In cases of many intents, one can deploy fast retrieval of top candidate answers. BASELINES We start by categorizing the baselines by three main representation and prediction methodologies. A Classifier architecture learns both query representation and a linear classification head via crossentropy loss. The classification head contains a single vector per class that is implicitly trained to represent the class and enable prediction at inference time. A Match-QQ architecture learns query representation via query-to-query similarity matching as it learns to increase the similarity between embeddings of queries from the same class while simultaneously decreasing the similarity between queries from disparate classes. During inference, an input query is matched against a large collection of training queries, and the intent in which its queries are the most similar to the input is predicted. A Match-QA architecture learns to match queries to answers. The model learns to increase the similarity between queries and their corresponding answers while simultaneously decreasing the similarity between queries and other answers. At inference time, an incoming query is matched against all possible answers, and the most similar is predicted. In these terms, the pretraining of QAID is based on Match-QQ, its fine-tuning involves both Match-QQ and Match-QA, and its prediction is based on Match-QA. We experimented with prediction methods that include also Match-QQ, but found it preform slightly worse than inference with only Match-QA. We will elaborate more in Section §4.1. BASELINE MODELS We compare our approach and results against strong baseline models reported in the literature. In the rest of the section, we discuss these models in more detail and align them with the paradigms mentioned above. Notably, some baseline models mix and match components across architectures. • Classifier: We fine-tune RoBERTa-base encoder with a feed-forward classification head as our classifier baseline. • ColBERT (Khattab & Zaharia, 2020): Contextualized Late Interaction over BERT (ColBERT) is a state-of-the-art passage search and retrieval system. ColBERT provides a Match-QA baseline and is the basis for the QAID architecture. For training we use 20 triplets (query, pos answer, neg answer) for each query, with hard negatives, namely, we run the query against all answers using bm25 (Robertson & Zaragoza, 2009) and select the negatives from the most similar answers. • DNNC : Discriminative Nearest Neighbor Classification (DNNC) model is trained to find the best-matched example from the training set through similarity matching (Match-QQ). The model conducts data augmentation during training and boosts performance by pre-training on three natural language inference tasks. • CPFT (Zhang et al., 2021b): Contrastive Pre-training and Fine-Tuning (CPFT) is a two-stage intent-detection architecture. During the first stage, the model learns with a self-supervised contrastive loss on a large set of unlabeled queries. In the second stage, the model learns with supervised contrastive loss to pull together query representation from the same intent (Match-QQ). The inference is done via a classification head that is added and trained during the second stage. • SetFit (Tunstall et al., 2022): Sentence Transformer Fine-tuning (SetFit) is a two-stage method for training a Sentence Transformer model specifically for few-shot classification tasks. In the first stage, the encoder undergoes fine-tuning using triplet loss, and in the second stage, the classification head is trained. Table 2 lists the results on the three datasets described in Section 3.1. QAID with RoBERTa-base achieved the best results across all datasets and shots. Notably, increasing the model size from RoBERTa-base to RoBERTa-large resulted in additional significant improvement across all datasets. RESULTS For 5-shot, QAID with RoBERTa-base improves over CPFT, which achieved the best results reported so far, by more than 4 points on the BANKING77 dataset which is translated to 30.64% in error rate reduction (ERR). Similarly, QAID achieves ERR of 8.5% and 18.9% over CPFT for the CLINC150 and HWU64 respectively. We attribute our improvement to three key differences between our method and CPFT. (1) Our problem formulation kept the class representation the same during training and inference. In other words, we didn't train a classification layer for inference. (2) Incorporating answers as data points contributes an additional discriminating signal. (3) The token-level late interaction score is shown to perform better as our ablation experiments demonstrate in Section 4.1. Moreover, our standard deviations (std) are consistently lower than those of DNNC and CPFT with the highest std of 0.15 and average std of 0.07. We believe the reason for the low std in the results is the combination of batch contrastive loss with data augmentation and the fine-grained late-interaction similarity score. Accordingly, our improvements are significant according to an unpaired t-test with a p-value of 1e −4 . An additional advantage of our method is its efficiency. QAID pre-training run-time takes about two hours and it has to run only once for all of our targets. Our fine-tuning takes only ten minutes on one NVIDIA V100 GPU, compared to three hours of fine-tuning of DNNC. Another important aspect of our results is the effect of scaling from RoBERTa-base to RoBERTa-large, which resulted in significant improvements in both 5 and 10shot scenarios across all datasets, aligned with results showing larger models generalize better from small data (Bandel et al., 2022). Moreover, in some cases scaling the model was more beneficial than additional examples. Namely, RoBERTa-large in 5-shot surpasses RoBERTa-base in 10-shot. ABLATION TESTS In this section, we describe several ablation studies demonstrating the importance of our method components and the main factors that contribute to our improvement over ColBERT. We present our ablation results in Table 3. We start by analyzing the improving effect of the pretraining (PT) and batch contrastive training on ColBERT. We can see that both stages boost the performances considerably across all settings. It is noticeable that the pre-training (row ColBERT with PT) improves more in the 5-shot than in the 10-shot setting with deltas of 4.59 and 2.00 points on average, respectively. This result is consistent with the observation that a model with better query representation is essential when only a few examples are available . Batch contrastive training (row ColBERT with batch contrastive) improves performance in most settings, with an average improvement of 5.74 points over ColBERT. We attribute this improvement to two major enhancements that batch contrastive training introduces. The first is the shift from a model that learns only Match-QA to a model that learns both Match-QA and Match-QQ. The second is the improved technique of batch contrastive loss over triplet loss that allow to process many positive and negative examples at once and intrinsic ability to perform hard positive/negative mining (Khosla et al., 2020). We note that this change has a minor effect on the training time as it relies on representations calculated in the batch and has no effect on the inference time. In addition, we study some modifications of QAID to understand their effect. To further investigate the effect of batch contrastive we train QAID with N-pairs loss (row QAID -N-pairs loss) with in-batch negatives, a widely used loss in retrieval models, e.g. DPR and RocketQA (Karpukhin et al., 2020;Qu et al., 2021). In this setting, each query has one positive example, which in our case is the intent name, and multiple irrelevant (negative) examples, either different queries or intent names. This method differs from our supervised batch contrastive loss which allows many positive examples. Our results show that replacing QAID supervised batch contrastive loss with N-pairs loss leads to a decrease of more than 1 point on average. These results support our claim that retrieval models can benefit from adopting supervised batch contrastive loss. When we conducted fine-tuning training with and without auxiliary tasks (MLM and classification losses), we found that the auxiliary tasks increased QAID accuracy by 0.43 on average. Interestingly, the increase was more pronounced as the dataset contained fewer domains, 0.72, 0.48, and 0.10 of average improvement on Banking77, Clinc150, and HWU64, respectively. We also examine the performance of our method without pre-training, row QAID w/o PT. Results indicate that our method achieves an average improvement of 3.53 points compared to CPFT without pre-training (CPFT w/o PT) in Table 3. This result emphasizes the superiority of our proposed method as a fine-tuning method. Additionally, to better understand the role of the data augmentation module in our training, we conduct an experiment where we did not apply the data augmentation module. Results, QAID w/o data augmentation, show that by augmenting the data we improve the results by about a third of a point on average. In the 5-shot setting, the improvement is about 0.75 points on average, and in the 10-shot setting, the effect is inconsistent. Those results can indicate that data augmentation is more beneficial where less data is available. We experiment with replacing the similarity score in QAID with the cosine similarity of the CLS token (row QAID -Cosine Similarity) instead of the token-level late interaction (Khattab & Zaharia, 2020) (row QAID). We can see that using the token-level late interaction achieves higher results across all datasets and shots. We ascribe this improvement to the fine-grained nature of the late-interaction score that enables detailed token-level comparison. This score presents an efficient alternative to the costly cross-attention scoring that most Match-QQ methods use. Finally, we discuss our inference method. We experiment with indexing and predicting based on both queries and answers, i.e., using Match-QQ and Match-QA in the inference stage as we do in training. Inference based only on Match-QA achieve slightly better (0.07) results on average, with an average improvement of 0.18 and 0.17 on Banking77 and Clinc150, respectively, and an average decrease of 0.14 on HWU64. These results indicate that our training method achieves answer representation that reflects the distribution of both training queries and answers. Therefore, allowing a more efficient inference that relies only on the answers' representations. Task adaptive pre-training is a common strategy to face the data scarcity problem in few-shot intent detection classification. Predominant approach for task adaptive pre-training models leverages selfsupervised mask language modeling training on large dialogues datasets (a few hundred million dialogues) and on the domain itself to tackle few-shot intent detection (Casanueva et al., 2020;Mehri et al., 2020;. Shnarch et al. (2022) showed that unsupervised clustering helps better than MLM pre-training. DNNC pre-trains their system on annotated pairs from natural language inference (NLI) leveraging BERT (Devlin et al., 2018) pairwise encoding. Then they model intent detection as a kNN problem with k = 1 where the pre-trained NLI model learns to predict the similarity score for pair of queries. However, this model is computationally expensive as it fully utilized training examples in both training and inference. The CPFT work, suggested by Zhang et al. (2021b), shares some similarities with our approach. when the two main ones are: the two-stage training process and the use of batch contrastive loss. However, we have several key differences where we extend upon their method. Those differences make our method more effective and efficient, especially when the task involves a large number of intents. Firstly, we reformulate the few-shot ID classification task as a retrieval task. To that end, we adopt an efficient Dual-Encoder-based retrieval architecture -ColBERT (Khattab & Zaharia, 2020), and a late-interaction similarity score. Moreover, we treat the intent names as the answers we wish to retrieve. Secondly, we adjust the training method to learn both Match-QQ and Match-QA similarities. Finally, we adopt a retrieval-based inference based on the similarity between the incoming query and the intent names, therefore we are not required to train an additional classification head. Zhang et al. (2022b) design two stage training with batch contrastive loss and add explicit regularization loss directing the feature space towards isotropy. They report high 5-way few-shot results on the same benchmarks we use. Nevertheless, when evaluating their model accuracy the results are much lower than ours (about 16% and 9% lower on Banking77 and HWU64 respectively). INTENT NAME The idea of exploiting class names was proposed in the setting of zero and few-shot classification by a few past works (Meng et al., 2020;Yin et al., 2019;Basile et al., 2021). Yin et al. (2019) propose to formulate text classification tasks as a textual entailment problem (Dagan et al., 2005). This mapping enables using a model trained on natural language inference (NLI) as a zero-shot text classifier for a wide variety of unseen downstream tasks (Gera et al., 2022). Zhong et al. (2021) map the classification tasks to a question-answering format, where each class is formulated as a question and given as a prompt, and the decoder probabilities of the "Yes" and "No" tokens correspond to a positive or negative prediction of the class. In our work, we cast the classification problem to the task of question answering retrieval and treat a much larger number of classes than these works tackle, which is usually up to twenty. BATCH CONTRASTIVE LEARNING Batch contrastive training was shown to achieve improved representation and perform better than contrastive losses such as triplet, max-margin, and the N-pairs loss (Khosla et al., 2020). Gunel et al. (2021); Gao et al. (2021b) suggest incorporating batch contrastive learning to train the encoder in natural language processing tasks. Gao et al. (2021b) designed a simple contrastive learning framework through dropout augmentation. They trained on NLI data to achieve state-of-the-art results on unsupervised and full-shot supervised semantic textual similarity (STS) tasks (Agirre et al., 2012;2015;2016). Liu et al. (2021a) suggest MirrorBERT, a self-supervised framework with two types of random data augmentation: randomly erase or mask parts of the texts during tokenization, and representation-level augmentation through built-in encoder dropout. We differ from those works as we target the few-shot intent detection task. Moreover, we adjust the late interaction score from Khattab & Zaharia (2020) to achieve cross-attention-like similarity scores. We also showed that class names can serve as an additional augmentation that can be the base for inference prediction. CONCLUSIONS In this paper, we present QAID, Question Answering inspired Intent Detection system, that models the few-shot ID classification as a question-answering retrieval task, where utterances serve as questions and intent names as answers. We train QAID with a two-stage training schema with batch contrastive loss. Results show that replacing ColBERT triplet loss with batch contrastive loss leads to a considerable improvement. We assert that a contributing factor to this effect is the shift to learning Match-QQ and Match-QA representations. We leave for further research to investigate this effect on retrieval tasks. Moreover, our results show that incorporating token-level similarity scores in contrastive loss outperforms the common cosine similarity score without a notable increase in training time. We encourage future research to utilize this type of contrastive loss in other tasks and investigate its effect. Finally, our results on three few-shot ID benchmarks show that QAID achieves state-of-the-art performance. Figure 1 : 1Schematic illustration of our method. Hat and bar represent augmentation, the subscript is a ruining index, superscript is the class index. (A) Self-supervised contrastive training with data augmentation to enhance quires' representations. (B) Supervise contrastive fine-tuning to learn queryto-query and query-to-answer similarity. (C) Indexing the answers into Faiss index. (D) Compare incoming query against all answers in the index and predict the most similar one. • USE+CONVERT(Casanueva et al., 2020): USE(Yang et al., 2019) is a large multilingual dualencoder model pre-trained in 16 languages. CONVERT(Casanueva et al., 2020) is an intent detection model with dual encoders that are pre-trained on 654 million (input, response) pairs from Reddit.• CONVEBERT(Mehri et al., 2020): a BERT-base model which has been trained on a large opendomain dialogue corpus. CONVEBERT achieved improvements over a vanilla BERT architecture and state-of-the-art results on a few task-oriented dialogue tasks.• CONVEBERT+Combined(Mehri et al., 2021): a CONVEBERT-base model trained to improve similarity matching of training examples, i.e., Match-QQ. Additionally, the model trains with observers for transformer attention and conducts task-adaptive self-supervised learning with mask language modeling (MLM) on the intent detection datasets. Combined represents the best MLM+Example+Observers setting in the referenced paper. contains 11,106 personal assistant queries classified into 64 intents across 21 different domains.Table 1reports the data splits of each dataset.Table 1: Data statistics of the three intent detection datasets from DialoGLUE.Dataset #Train #Vaild #Test #Intents #Domains CLINC150 15000 3000 4500 150 10 BANKING77 8622 1540 3080 77 1 HWU64 8954 1076 1076 64 21 Table 2 : 2Accuracy results on three ID datasets in 5-shot and 10-shot settings. Baseline results are from the original papers except for RoBERTa Classifier, ColBERT and SetFit. Table 3 : 3Accuracy results of our ablation experiments.5 RELATED WORK 5.1 FEW-SHOT INTENT DETECTION In the rest of the paper we refer to the intent examples as queries and use intents and classes interchangeably For readily use: https://github.com/jianguoz/Few-Shot-Intent-Detection/tree/main/Datasets https://github.com/huggingface/transformers 4 https://github.com/stanford-futuredata/ColBERT/tree/colbertv1 5 https://github.com/HobbitLong/SupContrast/blob/master/losses.py 6 https://github.com/facebookresearch/faiss ACKNOWLEDGEMENTSWe thank Leshem Choshen and Ariel Gera for their helpful feedback as we pursued this research. SemEval-2012 task 6: A pilot on semantic textual similarity. Eneko Agirre, Daniel Cer, Mona Diab, Aitor Gonzalez-Agirre, *SEM 2012: The First Joint Conference on Lexical and Computational Semantics. 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Benchmarking commercial intent detection services with practice-driven evaluations. Haode Qi, Lin Pan, Atin Sood, Abhishek Shah, Ladislav Kunc, Mo Yu, Saloni Potdar, arXiv:2012.03929arXiv preprintHaode Qi, Lin Pan, Atin Sood, Abhishek Shah, Ladislav Kunc, Mo Yu, and Saloni Potdar. Bench- marking commercial intent detection services with practice-driven evaluations. arXiv preprint arXiv:2012.03929, 2020. RocketQA: An optimized training approach to dense passage retrieval for open-domain question answering. 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Journalism quarterly, 30(4):415-433, 1953. Moshe Wasserblat, and Oren Pereg. Efficient few-shot learning without prompts. Lewis Tunstall, Nils Reimers, Unso Eun Seo Jo, Luke Bates, Daniel Korat, Lewis Tunstall, Nils Reimers, Unso Eun Seo Jo, Luke Bates, Daniel Korat, Moshe Wasserblat, and Oren Pereg. Efficient few-shot learning without prompts, 2022. Concept-based short text classification and ranking. Fang Wang, Zhongyuan Wang, Zhoujun Li, Ji-Rong Wen, Proceedings of the 23rd ACM International Conference on Conference on Information and Knowledge Management. the 23rd ACM International Conference on Conference on Information and Knowledge ManagementFang Wang, Zhongyuan Wang, Zhoujun Li, and Ji-Rong Wen. Concept-based short text classifica- tion and ranking. In Proceedings of the 23rd ACM International Conference on Conference on Information and Knowledge Management, pp. 1069-1078, 2014. Pre-trained natural language understanding for task-oriented dialogue. Chien-Sheng Wu, C H Steven, Richard Hoi, Caiming Socher, Xiong, Tod-Bert, 10.18653/v1/2020.emnlp-main.66Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP). the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)Association for Computational LinguisticsOnline, November 2020aChien-Sheng Wu, Steven C.H. Hoi, Richard Socher, and Caiming Xiong. TOD-BERT: Pre-trained natural language understanding for task-oriented dialogue. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 917-929, Online, Novem- ber 2020a. Association for Computational Linguistics. doi: 10.18653/v1/2020.emnlp-main.66. URL https://aclanthology.org/2020.emnlp-main.66. Clear: Contrastive learning for sentence representation. Zhuofeng Wu, Sinong Wang, Jiatao Gu, Madian Khabsa, Fei Sun, Hao Ma, abs/2012.15466ArXiv. Zhuofeng Wu, Sinong Wang, Jiatao Gu, Madian Khabsa, Fei Sun, and Hao Ma. 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Wenpeng Yin, Jamaal Hay, Dan Roth, 10.18653/v1/D19-1404Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing. the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language ProcessingHong Kong, ChinaAssociation for Computational LinguisticsWenpeng Yin, Jamaal Hay, and Dan Roth. Benchmarking zero-shot text classification: Datasets, evaluation and entailment approach. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Nat- ural Language Processing (EMNLP-IJCNLP), pp. 3914-3923, Hong Kong, China, November 2019. Association for Computational Linguistics. doi: 10.18653/v1/D19-1404. URL https: //aclanthology.org/D19-1404. Effectiveness of pre-training for few-shot intent classification. Haode Zhang, Yuwei Zhang, Li-Ming Zhan, Jiaxin Chen, Guangyuan Shi, Xiao-Ming Wu, Albert Lam, arXiv:2109.05782arXiv preprintHaode Zhang, Yuwei Zhang, Li-Ming Zhan, Jiaxin Chen, Guangyuan Shi, Xiao-Ming Wu, and Albert Lam. Effectiveness of pre-training for few-shot intent classification. arXiv preprint arXiv:2109.05782, 2021a. Fine-tuning pre-trained language models for few-shot intent detection: Supervised pre-training and isotropization. Haode Zhang, Haowen Liang, Yuwei Zhang, Li-Ming Zhan, Xiao-Ming Wu, Xiaolei Lu, Albert Y S Lam, NAACL. Haode Zhang, Haowen Liang, Yuwei Zhang, Li-Ming Zhan, Xiao-Ming Wu, Xiaolei Lu, and Albert Y. S. Lam. Fine-tuning pre-trained language models for few-shot intent detection: Supervised pre-training and isotropization. In NAACL, 2022a. Fine-tuning pre-trained language models for few-shot intent detection: Supervised pre-training and isotropization. Haode Zhang, Haowen Liang, Yuwei Zhang, Liming Zhan, Xiao-Ming Wu, Xiaolei Lu, Albert Y S Lam, Haode Zhang, Haowen Liang, Yuwei Zhang, Liming Zhan, Xiao-Ming Wu, Xiaolei Lu, and Albert Y. S. Lam. Fine-tuning pre-trained language models for few-shot intent detection: Supervised pre-training and isotropization, 2022b. URL https://arxiv.org/abs/2205.07208. Discriminative nearest neighbor few-shot intent detection by transferring natural language inference. Jian-Guo Zhang, Kazuma Hashimoto, Wenhao Liu, Chien-Sheng Wu, Yao Wan, S Philip, Richard Yu, Caiming Socher, Xiong, arXiv:2010.13009arXiv preprintJian-Guo Zhang, Kazuma Hashimoto, Wenhao Liu, Chien-Sheng Wu, Yao Wan, Philip S Yu, Richard Socher, and Caiming Xiong. Discriminative nearest neighbor few-shot intent detection by transferring natural language inference. arXiv preprint arXiv:2010.13009, 2020. Few-shot intent detection via contrastive pre-training and fine-tuning. 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{'abstract': 'Intent detection with semantically similar fine-grained intents is a challenging task. To address it, we reformulate intent detection as a question-answering retrieval task by treating utterances and intent names as questions and answers. To that end, we utilize a question-answering retrieval architecture and adopt a two stages training schema with batch contrastive loss. In the pre-training stage, we improve query representations through self-supervised training. Then, in the finetuning stage, we increase contextualized token-level similarity scores between queries and answers from the same intent. Our results on three few-shot intent detection benchmarks achieve state-of-the-art performance.', 'arxivid': '2303.01593', 'author': ['Asaf Yehudai \nIBM Israel Research Lab ♢\nHebrew University of Jerusalem\n\n', '♢ ♣ \nIBM Israel Research Lab ♢\nHebrew University of Jerusalem\n\n', 'Matan Vetzler \nIBM Israel Research Lab ♢\nHebrew University of Jerusalem\n\n', 'Yosi Mass yosimass@boazc.il.ibm.com \nIBM Israel Research Lab ♢\nHebrew University of Jerusalem\n\n', '♢ \nIBM Israel Research Lab ♢\nHebrew University of Jerusalem\n\n', 'Koren Lazar \nIBM Israel Research Lab ♢\nHebrew University of Jerusalem\n\n', 'Doron Cohen doronc@boazc.il.ibm.com \nIBM Israel Research Lab ♢\nHebrew University of Jerusalem\n\n', '♢ \nIBM Israel Research Lab ♢\nHebrew University of Jerusalem\n\n', 'Boaz Carmeli \nIBM Israel Research Lab ♢\nHebrew University of Jerusalem\n\n'], 'authoraffiliation': ['IBM Israel Research Lab ♢\nHebrew University of Jerusalem\n', 'IBM Israel Research Lab ♢\nHebrew University of Jerusalem\n', 'IBM Israel Research Lab ♢\nHebrew University of Jerusalem\n', 'IBM Israel Research Lab ♢\nHebrew University of Jerusalem\n', 'IBM Israel Research Lab ♢\nHebrew University of Jerusalem\n', 'IBM Israel Research Lab ♢\nHebrew University of Jerusalem\n', 'IBM Israel Research Lab ♢\nHebrew University of Jerusalem\n', 'IBM Israel Research Lab ♢\nHebrew University of Jerusalem\n', 'IBM Israel Research Lab ♢\nHebrew University of Jerusalem\n'], 'corpusid': 257353677, 'doi': '10.48550/arxiv.2303.01593', 'github_urls': ['https://github.com/jianguoz/Few-Shot-Intent-Detection/tree/main/Datasets', 'https://github.com/huggingface/transformers', 'https://github.com/stanford-futuredata/ColBERT/tree/colbertv1', 'https://github.com/HobbitLong/SupContrast/blob/master/losses.py', 'https://github.com/facebookresearch/faiss'], 'n_tokens_mistral': 17436, 'n_tokens_neox': 14796, 'n_words': 8199, 'pdfsha': 'c27210604ad9626e4cf928c5073678493dd9c8fe', 'pdfurls': ['https://export.arxiv.org/pdf/2303.01593v2.pdf'], 'title': ['Published as a conference paper at ICLR 2023 QAID: QUESTION ANSWERING INSPIRED FEW-SHOT INTENT DETECTION', 'Published as a conference paper at ICLR 2023 QAID: QUESTION ANSWERING INSPIRED FEW-SHOT INTENT DETECTION'], 'venue': []}

arxiv

Distribution of particles and bubbles in turbulence at small Stokes number Itzhak Fouxon 11 Oct 2011 Raymond Sackler School of Physics and Astronomy Tel-Aviv University 69978Tel-AvivIsrael Beverly Sackler School of Physics and Astronomy Tel-Aviv University 69978Tel-AvivIsrael Distribution of particles and bubbles in turbulence at small Stokes number Itzhak Fouxon 11 Oct 2011 The inertia of particles driven by the turbulent flow of the surrounding fluid makes them prefer certain regions of the flow. The heavy particles lag behind the flow and tend to accumulate in the regions with less vorticity, while the light particles do the opposite. As a result of the long-time evolution, the particles distribute over a multi-fractal attractor in space. We consider this distribution using our recent results on the steady states of chaotic dynamics. We describe the preferential concentration analytically and derive the correlation functions of density and the fractal dimensions of the attractor. The results are obtained for real turbulence and are testable experimentally. PACS numbers: 47.55.Kf, 47.10.Fg, 05.45.Df, 47.53.+n Recently the problem of distribution of inertial particles in turbulence received much attention of the researchers[1][2][3][4][5][6][7][8][9][10][11][12]. This is largely thanks to the breakthrough in the theoretical understanding of the Lagrangian motion of particles in the flow that occurred lately[13]. While the understanding of the behavior of particles that have negligible inertia and follow the flow is quite complete by now[13], the understanding of the behavior of inertial particles is still insufficient. This is while the subject has an extremely wide range of applications: the flows of fluids are typically turbulent and often laden with external particles. Theoretical advancement was made mainly for the case of small Stokes number, where the inertia is weak and the particles "almost" follow the flow. Even in this limit of small Stokes numbers, the particles' distribution is highly non-trivial. Particles' deviations from the surrounding flow accumulate with time, bringing particles to a strange attractor in space. This attractor is multi-fractal and the only theoretical result obtained so far for the real turbulent flow was the derivation of the correlation codimension[4]. Here a result obtained for real turbulence is a result obtained without modeling turbulence and expressed in terms of the (unknown) statistical properties of turbulence. Since the statistics of turbulence is largely unknown[14], then to obtain such a result one needs to make universal predictions on particles' behavior in the flow independent of the details of the statistics of that flow.In this Letter we provide the complete description of the distribution of particles in real turbulence at small Stokes numbers, describing both the correlation of the particles' density with the surrounding flow and the statistics of the singular density on the attractor. We give a number of predictions that are testable experimentally.The idea that particles' inertia leads to inhomogeneous spatial distribution dates back to the seminal paper by Maxey[2]. It was observed that due to inertia heavy particles are pushed out of the vortices and hence they will not distribute uniformly in the flow, like the inertialess particles. However the quantitative description of the correlations between the locations of particles and of vortices stayed unaddressed. Note the distribution of vorticity in turbulence is random and dynamical, while the distribution of particles reflects its cumulative effect over time. There is a residual correlation that we describe by an integral relation holding in the steady state.We find the spectrum of fractal dimensions of the attractor. We show that while the correlation dimension is different from the dimension of space, the fractal or similarity dimension[15]is equal to the space dimension. In contrast the information dimension is different from the spatial dimension and it equals the Kaplan-Yorke dimension. In turn, the correlation codimension equals twice the Kaplan-Yorke codimension which constitutes a prediction allowing direct testing in the laboratory.The analysis is based on the recent finding of a universal description for the steady state density of the weakly compressible dynamical systems[16]. The particles' motion, though governed by Newton's law, admits an effective description in terms of a velocity field in space. Inertia is described by a small compressible correction to the incompressible velocity of the background turbulent flow. This correction leads to a small disbalance of trajectories going in and out of space regions, which accumulates over a long time to a big effect. Thus compressibility is a singular perturbation which treatment was performed in[16]. For a mixing incompressible velocity the evolution of a small volume of particles makes it dense in space. The volume's coarse-graining over an arbitrarily small scale covers all the available space, which volume is assumed finite. When a small compressible component is added to the velocity, the coarse-graining of the evolved volume over an arbitrarily small scale does not cover the whole space any longer. However the coarsegraining over a small but finite scale, that tends to zero with compressibility, already covers the whole volume.The analysis assumes the single-particle approximation where one neglects the interaction between the particles and their back reaction on the flow. We consider a small spherical particle with the radius a and the material density ρ p suspended in a fluid with the density ρ and the kinematic viscosity ν. The fluid flow u(t, x) is assumed to be incompressible. The Newton law governing the evolution of the particle's position x(t) and the particle's The inertia of particles driven by the turbulent flow of the surrounding fluid makes them prefer certain regions of the flow. The heavy particles lag behind the flow and tend to accumulate in the regions with less vorticity, while the light particles do the opposite. As a result of the long-time evolution, the particles distribute over a multi-fractal attractor in space. We consider this distribution using our recent results on the steady states of chaotic dynamics. We describe the preferential concentration analytically and derive the correlation functions of density and the fractal dimensions of the attractor. The results are obtained for real turbulence and are testable experimentally. Recently the problem of distribution of inertial particles in turbulence received much attention of the researchers [1][2][3][4][5][6][7][8][9][10][11][12]. This is largely thanks to the breakthrough in the theoretical understanding of the Lagrangian motion of particles in the flow that occurred lately [13]. While the understanding of the behavior of particles that have negligible inertia and follow the flow is quite complete by now [13], the understanding of the behavior of inertial particles is still insufficient. This is while the subject has an extremely wide range of applications: the flows of fluids are typically turbulent and often laden with external particles. Theoretical advancement was made mainly for the case of small Stokes number, where the inertia is weak and the particles "almost" follow the flow. Even in this limit of small Stokes numbers, the particles' distribution is highly non-trivial. Particles' deviations from the surrounding flow accumulate with time, bringing particles to a strange attractor in space. This attractor is multi-fractal and the only theoretical result obtained so far for the real turbulent flow was the derivation of the correlation codimension [4]. Here a result obtained for real turbulence is a result obtained without modeling turbulence and expressed in terms of the (unknown) statistical properties of turbulence. Since the statistics of turbulence is largely unknown [14], then to obtain such a result one needs to make universal predictions on particles' behavior in the flow independent of the details of the statistics of that flow. In this Letter we provide the complete description of the distribution of particles in real turbulence at small Stokes numbers, describing both the correlation of the particles' density with the surrounding flow and the statistics of the singular density on the attractor. We give a number of predictions that are testable experimentally. The idea that particles' inertia leads to inhomogeneous spatial distribution dates back to the seminal paper by Maxey [2]. It was observed that due to inertia heavy particles are pushed out of the vortices and hence they will not distribute uniformly in the flow, like the inertialess particles. However the quantitative description of the correlations between the locations of particles and of vortices stayed unaddressed. Note the distribution of vorticity in turbulence is random and dynamical, while the distribution of particles reflects its cumulative effect over time. There is a residual correlation that we describe by an integral relation holding in the steady state. We find the spectrum of fractal dimensions of the attractor. We show that while the correlation dimension is different from the dimension of space, the fractal or similarity dimension [15] is equal to the space dimension. In contrast the information dimension is different from the spatial dimension and it equals the Kaplan-Yorke dimension. In turn, the correlation codimension equals twice the Kaplan-Yorke codimension which constitutes a prediction allowing direct testing in the laboratory. The analysis is based on the recent finding of a universal description for the steady state density of the weakly compressible dynamical systems [16]. The particles' motion, though governed by Newton's law, admits an effective description in terms of a velocity field in space. Inertia is described by a small compressible correction to the incompressible velocity of the background turbulent flow. This correction leads to a small disbalance of trajectories going in and out of space regions, which accumulates over a long time to a big effect. Thus compressibility is a singular perturbation which treatment was performed in [16]. For a mixing incompressible velocity the evolution of a small volume of particles makes it dense in space. The volume's coarse-graining over an arbitrarily small scale covers all the available space, which volume is assumed finite. When a small compressible component is added to the velocity, the coarse-graining of the evolved volume over an arbitrarily small scale does not cover the whole space any longer. However the coarsegraining over a small but finite scale, that tends to zero with compressibility, already covers the whole volume. The analysis assumes the single-particle approximation where one neglects the interaction between the particles and their back reaction on the flow. We consider a small spherical particle with the radius a and the material density ρ p suspended in a fluid with the density ρ and the kinematic viscosity ν. The fluid flow u(t, x) is assumed to be incompressible. The Newton law governing the evolution of the particle's position x(t) and the particle's velocity v(t) is assumed to have the form dv dt = γ d dt u[t, x(t)] − v − u[t, x(t)] τ ,(1) where γ = 3ρ/(ρ + 2ρ p ) and τ = a 2 /(3νγ) is the Stokes time. Thus we assume that all the forces besides the added mass and the drag can be neglected [2,10]. With no loss we set the total volume and the mass equal to one, so the spatial average of the particles' density n obeys n = 1. We set β = γ − 1 so the particle's velocity relative to the flow w(t) ≡ v(t) − u[t, x(t)] obeys dw dt = − w τ + β d dt u[t, x(t)].(2) The parameter β = 2(ρ − ρ p )/(ρ + 2ρ p ) changes from −1 for heavy particles to 2 for light ones. After transients w(t) = β t −∞ exp − t − t ′ τ d dt ′ u[t ′ , x(t ′ )]dt ′ . (3) We assume τ is much smaller than the smallest timescale of turbulence, which is the viscous time-scale t η , so the Stokes number St ≡ τ /t η ≪ 1. Then we can substitute the derivative in the integrand by its value at t ′ = t so v(t) ≈ u + µ [∂ t u + (u · ∇)u] with µ ≡ βτ = 2a 2 (ρ−ρ p )/(9νρ) . Thus at St ≪ 1 the particle's velocity is determined uniquely by its position x(t) in space and one can introduce the particle's velocity field v(t, x) x(t) = v [t, x(t)] , v ≡ u + µ [∂ t u + (u · ∇)u] . (4) In the zero inertia limit St → 0 the particles follow the incompressible mixing flow of turbulenceẋ = u [t, x(t)] and in the steady state they are uniformly distributed in space, so their steady state density n s equals one. This behavior is characteristic of small dye particles. However, at a small but finite St, the small correction v − u gives the particles' velocity field a finite compressibility [2] w ≡ ∇ · v = −µφ = 0, φ = ω 2 − s 2 ,(5) so the constant is no longer a solution to the continuity equation ∂ t n + ∇ · (nv) = 0. Above s 2 = s ij s ij and ω 2 = a ij a ij , where s ij is the symmetric (strain) and a ij is the antisymmetric (vorticity) parts of velocity gradient ∂ j u i = s ij + a ij . The field φ(x) is positive in the regions dominated by vorticity and negative in the regions dominated by the strain, and it will be called below the indicator, indicating whether x is in a vortex. It follows from the Navier-Stokes equations that φ equals the Laplacian of the turbulent pressure, φ = ∇ 2 p. Eq. (5) shows that heavy particles β < 0 are repelled from vortices (here and below "vortex" is used qualitatively), while the light ones β > 0 are attracted. This is the generalization of the familiar fact that a heavy particle in a centrifuge is pushed out to the boundary. Turbulence can be considered as a dynamically changing spatial distribution of vorticity, so heavy particles tend to accumulate on the boundaries between the vortices, cf. [1,4,11], forming a singular density supported on these boundaries. This accumulation however is insignificant during the life-time of a single vortex and the ultimate singular distribution of particles in space n s forms from the long-time combined action of many uncorrelated vortices. Still one expects a residual correlation between the distributions of vorticity and particles, to find which we consider the steady state density n s . One expects n s can be obtained by letting an arbitrary initial condition n 0 in the remote past n(t = −T ) = n 0 evolve for infinite time, T → ∞, according to the continuity equation. Starting from the uniform initial distribution we obtain the steady state density n s (x) = lim T →∞ n(T ), n(T ) = exp − 0 −T w[t, q(t, x)]dt ,(6) provided a condition of decay of correlations [16], that should hold in our case, is met. For the cross-correlation of density and vorticity F (x) = φ(0)n s (x) we find F (x)= φ(0, 0) exp µ 0 −∞ φ[t, q(t, x)]dt (7) = ∂ α ln exp αφ(0, 0) + µ 0 −∞ φ[t, q(t, x)]dt | α=0 , where we used the conservation of the mean density n(t) = const. Applying the cumulant expansion [17], taking derivative of the series and setting α = 0 we find F (x) = µ 0 −∞ dt φ(0, 0)φ[t, q(t, x)] c + O(St 2 ).(8) The above formula is the same as one would obtain by expanding the exponent in Eq. (7) and keeping the lowest order term in τ , with one important difference. In Eq. (8) one has the second order cumulant or dispersion, that one would not get by the series expansion of the exponent, cf. [2]. This difference is essential as without the cumulant the integral in Eq. (8) diverges: w[t, q(t, x)] = (1 − β)τ φ[t, q(t, x)] is equal to a non-zero sum of Lyapunov exponents, see below. To leading order in St one can substitute q(t, x) in Eq. (8) by X(t, x) ∂ t X(t, x) = u[t, X(t, x)], X(0, x) = x.(9) where X(t, x) are Lagrangian trajectories of u. One finds φ(0)n s (x) = µ 0 −∞ dt φ(0, 0)φ[t, X(t, x)] ,(10) where we can already omit the cumulant since by incompressibility w[t, X(t, x)] = w(t, x) , while w(t, x)dx = ∇ · v(t, x) = 0 by the boundary conditions. Since there is no degeneracy, the non-negative spectrum of the Laplacian of pressure in the Lagrangian frame φ[t, X(t, x)] is strictly positive at zero frequency E(0) = ∞ −∞ φ(0, 0)φ[t, X(t, 0)] > 0.(11) The single-point correlation φn s ≡ φ(x)n s (x)dx equals µE(0)/2 and it gives the integral of φ where each region weighted by the number of particles in it (12) where we used the definitions of φ and µ. For heavy particles, ρ p > ρ, the answer is negative giving a measure of the extent to which the particles favor regions with negative φ. For light particles, ρ p < ρ the answer is positive measuring their favoring of vortices. The above integral steady state relation holds at any t. ω 2 (x)−s 2 (x) n s (x)dx = a 2 (ρ−ρ p )(9νρ) −1 E(0), The quantity E(0) appeared first in [4], where it was shown to determine the correlation dimension of the particles' attractor in space. This quantity is increased by the intermittency of turbulence and it can be estimated as t −3 η f (Re) where f (Re) is a growing function of the Reynolds number Re that grows as a power [4,5]. We show E(0) determines all the fractal dimensions. We observe that Eq. (4) is a weakly dissipative dynamical system, defined as the dynamics for which the potential part of v is much smaller than the solenoidal one. The statistics of the steady state density of such systems was shown recently to allow for a complete and universal description [16]. The application to our case gives the following results. The motion of particles in space is chaotic and is characterized by the Lyapunov exponents [19]. To the lowest order in St the exponents are equal to the Lyapunov exponents λ i of the turbulent flow u. However, the value of the sum of the Lyapunov exponents λ + i that determines the logarithmic rate of growth of the volumes forward in time, λ + i (x) ≡ lim t→∞ t −1 ln det ∇ j q i (t, x),(13) is zero for u, so the leading order approximation demands the account of the correction v − u. This is also the case of the sum of the Lyapunov exponents λ − i of the backward-in-time flow that determines the density [16] lim T →∞ T −1 ln n(0, x, T ) = λ − i (x).(14) For turbulence λ ± i is expected to be the same for all x with the possible exception of a set of points with zero volume. The results of [16,18] give λ ± i ≈ − 1 2 ∞ −∞ w(0, 0)w[t, X(t, 0)] = − µ 2 E(0) 2 . Thus for all initial points, with the possible exception of a set of points with zero volume, the infinitesimal volumes decay to zero in the limit of infinite evolution time, while the steady state density n s is zero except for a set of points with zero volume. Due to conservation of mass ndx, we conclude that n s has δ−function type singularities on its support. This support is the strange attractor -the multifractal set in space that is approached by the particles' trajectories at large times. We now find the Kaplan-Yorke codimension C KY of the attractor. At weak compressibility the definition [20] of C KY reduces [16] to C KY = λ + i /λ + 3 which gives to the leading order C KY = µ 2 E(0) 2|λ 3 | = 2a 4 (ρ−ρ p ) 2 E(0) 81ν 2 ρ 2 |λ 3 | ,(15) where the third Lyapunov exponent λ 3 determines the rate of exponential separation of X(t, x) back in time. We have |λ 3 | ∼ t −1 η and C KY ∼ β 2 St 2 f (Re), cf. [4,7]. The probability for two particles to be at the distance x is described by the pair-correlation function n s (0)n s (x) . Substituting for n s the expression from Eq. (6) and using the cumulant expansion [16] one finds n s (0)n s (x) = exp µ 2 g(x) where the structure function g(x) depends only on the statistics of turbulence g(x) ≡ 0 −∞ dt 1 dt 2 φ[t 1 , X(t 1 , 0)]φ[t 2 , X(t 2 , x)] . (16) The above is valid if the higher order terms in the cumulant expansion are negligible [16]. The Kolmogorov theory (KT) estimate would give the validity condition St ≪ 1, while the account of intermittency changes the condition to h(Re)St ≪ 1 where h(Re) is expected to be a slowly growing function of Re, cf. [5]. The function g(x) has a universal behavior [16] at small x that gives n s (0)n s (x) = (η/x) 2CKY , x ≪ η,(17) where η ∼ (νt η ) 1/2 is the Kolmogorov scale of turbulence [4]. Thus the correlation codimension equals 2C KY . Remarkably, the structure function determines all the correlation functions of n s . Generalization of the calculation of n s (0)n s (x) gives the log-normal statistics [16] n s (x 1 )n s (x 2 )..n s (x N ) = exp   µ 2 i>j g(x i −x j )   ,(18) The density n s does not have physical meaning and we consider the coarse-grained density n l , m l (x) ≡ |x ′ −x|<l n s (x ′ )dx ′ , n l (x) ≡ 3m l (x)/(4πl 3 ), For any St > 0 the fluctuations of n l are large for a sufficiently small l. On the other hand, by lim St→0 n 2 l = 1 one sees that for any fixed l > 0 the fluctuations of n l are small for a sufficiently small St. The coarse-grained density is uniform over scales which minimal value vanishes with St. Thus turbulence effect on the particles depends on the observer's resolution l: at 2C KY ln (η/l) 1 segregation holds, while 2C KY ln (η/l) ≪ 1 -mixing. This is how mixing works effectively for particles on a multifractal. Segregation may also bring physical effects [4]. At St ≪ 1 there is a scale L ≪ η over which the density is almost uniform. We note that m l (t = 0, x) is equal to the mass contained in the preimage of the ball time t ago, which is an ellipsoid around q(−t, x) with the largest axis growing as l exp[|λ 3 t|]. At t * = |λ 3 | −1 ln(L/l) the ellipsoid has the scale over which the density is uniform, so the mass contained in it is just its volume 4πl 3 exp[− 0 −t * w[t ′ , q(t ′ , x)]dt ′ ]/3 and we find n l (x) = exp −µ 0 −|λ3| −1 ln(L/l) φ[t ′ , q(t ′ , x)]dt ′ ,(20) see [16] for details. The smallness of µ brings the expected conclusion that the statistics of n l is log-normal n ρ l = (η/l) CKY ρ(ρ−1) ,(21) cf. Eq. (18). The spectrum of the fractal dimensions D(α) ≡ lim l→0 ln m α−1 l n s /[(α − 1) ln l] involves the average with n s , rather than the spatial average [6,15]. To find it consider n α−1 l n s = lim T →∞ exp[−α 0 −t * ω[t, q(t, r)]dt − −t * −T ω[t, q(t, r)]dt] . Due to St ≪ 1 the contribution of time-intervals with length t η is negligible and we may substitute the upper limit in the last integral by −t * − t η which allows to perform independent averaging exp[−α 0 −t * ω[t, q(t, r)]dt − −t * −tη −T ω[t, q(t, r)]dt] ≈ exp[−α 0 −t * ω[t, q(t, r)]dt] exp[− −t * −tη −T ω[t, q(t, r)]dt] . However the last average is equal to one by the conservation of mean density, so n α−1 l n s = n α l and D(α) = 3 − C KY α.(22) Our results generalize to the two-dimensional case, where they can be compared with [6]. Working out the small compressibility limit reproduces our answer. Returning to the three-dimensional case, we observe that the fractal dimensions are close to 3 (we do not consider α ≫ 1) The fractal dimension of the attractor D(0) coincides with the space dimension 3, which is somewhat counter-intuitive since the volume of the attractor is zero. The information dimension D(1) is equal to the Kaplan-Yorke dimension. At St ≪ 1 density inhomogeneities are absent in the inertial range [14]. Then Eq. (17) is a complete description. In contrast, at St ∼ 1 the inertial range inhomogeneities are important [9] and Eq. (16), extended to hold asymptotically at St ∼ 1, gives a unique access to the inhomogeneities. In KT g(x) depends only on x and the mean energy dissipation ǫ so n s (0)n s (x) = exp Cµ 2 ǫ 2/3 x −4/3 . This prediction describes correctly the model of v decorrelated in time [3,7,8]. However, for turbulence, simulations [9] show ln n s (0)n s (x) ∝ x −10/3 at moderate Re where KT is expected to work. Noting g(x) ∼ τ 2 x ∂ 4 x [p(x)−p(0)] 2 , where τ x is the relevant timescale, we suggest the difference has the same origin as the deviations of the pressure scaling from KT [21]. A central result is the analytic description of the preferential concentration by Eq. (12). A single number E(0) completely characterizes the influence of turbulence on the log-normal statistics of density at r ≪ η. Lognormality arises because the steady state density is the cumulative result of the creation of inhomogeneities by many uncorrelated vortices, each of which creates but weak inhomogeneity. The fractal structure at scale l forms relatively fast -within the characteristic time-scale |λ 3 | −1 ln(η/l). The predictions are testable. The author is grateful to J. Bec, M. Cencini, G. Falkovich, K. Gawedzki, J. Kurchan, A. Leshansky, R. Vilela, and M. Wilkinson, for discussions. This work was supported by COST Action MP0806. PACS numbers: 47.55.Kf, 47.10.Fg, 05.45.Df, 47.53.+n where m l (x) is the mass in a small ball. The limits l → 0 and St → 0 do not commute ( n 2 l ∼ (η/l) . M R Maxey, J J Riley, Phys. Fluids. 26883M. R. Maxey and J. J. Riley, Phys. Fluids 26, 883 (1983). . M R Maxey, J. Fluid Mech. 174441M. R. Maxey, J. Fluid Mech. 174, 441 (1987). . E Balkovsky, G Falkovich, A Fouxon, Phys. Rev. Lett. 862790E. Balkovsky, G. Falkovich and A. Fouxon, Phys. Rev. Lett. 86, 2790 (2001). . G Falkovich, A Fouxon, M Stepanov, Nature. 419151G. Falkovich, A. Fouxon and M. Stepanov, Nature 419, 151 (2002). . G Falkovich, A Pumir, Phys. Fluids. 1647G. Falkovich and A. Pumir, Phys. Fluids 16, L47 (2004). . J Bec, K Gawedzki, P Horvai, Phys. Rev. Lett. 92224501J. Bec, K. Gawedzki, and P. Horvai, Phys. Rev. Lett. 92, 224501 (2004). . J Chun, D L Koch, S L Rani, A Ahluwalia, L R Collins, J. Fluid Mech. 536219J. Chun, D. L. Koch, S. L. Rani, A. Ahluwalia, and L. R. Collins, J. Fluid Mech. 536, 219 (2005). . J Bec, M Cencini, R Hillerbrand, Phys. Rev. E. 7525301J. Bec, M. Cencini, and R. Hillerbrand Phys. Rev. E 75, 025301 (2007). . J Bec, L Biferale, M Cencini, A Lanotte, S Musacchio, F Toschi, Phys. Rev. Lett. 9884502J. Bec, L. Biferale, M. Cencini, A. Lanotte, S. Musacchio, and F. Toschi, Phys. Rev. Lett. 98, 084502 (2007). . E Calzavarini, M Cencini, D Lohse, F , Toschi , Phys. Rev. Lett. 10184504E. Calzavarini, M. Cencini, D. Lohse, and F, Toschi, Phys. Rev. Lett. 101, 084504 (2008). . P Olla, Phys. Rev. E. 8116305P. Olla, Phys. Rev. E 81, 016305 (2010). . M Wilkinson, B Mehlig, K Gustavsson, Europhys. Lett. 8950002M. Wilkinson, B. Mehlig and K. Gustavsson, Europhys. Lett. 89, 50002 (2010). . G Falkovich, K Gawedzki, M Vergassola, Rev. Mod. Phys. 73913975G. Falkovich, K. Gawedzki, and M. Vergassola, Rev. Mod. Phys. 73, 913975 (2001). U Frisch, Turbulence: The Legacy of A. N. Kolmogorov. Cambridge Univ. PressU. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge Univ. Press, 1995. Hentschel and I. Procaccia, Phys. D. H G , 8435H. G. E. Hentschel and I. Procaccia, Phys. D 8, 435 (1983). . I Fouxon, arXiv:1110.1625I. Fouxon, arXiv:1110.1625. S.-K Ma, Statistical Mechanics. World Scientific Publishing CompanyS.-K. Ma, Statistical Mechanics, World Scientific Pub- lishing Company (1984). . G Falkovich, A Fouxon, N , J. Phys. 650G. Falkovich and A. Fouxon, N. J. Phys. 6, 50 (2004). . V I Oseledets, Trans. Mosc. Math. Soc. 19197V. I. Oseledets, Trans. Mosc. Math. Soc. 19, 197 (1968). Functional Differential Equations and Approximations of Fixed Points. J L Kaplan, J A Yorke, H. O. Peitgen and H. O. WaltherSpringer204J. L. Kaplan and J. A. Yorke, Functional Differential Equations and Approximations of Fixed Points, Ed. H. O. Peitgen and H. O. Walther, Springer, 204 (1979). . T Gotoh, D Fukayama, Phys. Rev. Lett. 863775T. Gotoh and D. Fukayama, Phys. Rev. Lett. 86, 3775 (2001).

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{'abstract': 'The inertia of particles driven by the turbulent flow of the surrounding fluid makes them prefer certain regions of the flow. The heavy particles lag behind the flow and tend to accumulate in the regions with less vorticity, while the light particles do the opposite. As a result of the long-time evolution, the particles distribute over a multi-fractal attractor in space. We consider this distribution using our recent results on the steady states of chaotic dynamics. We describe the preferential concentration analytically and derive the correlation functions of density and the fractal dimensions of the attractor. The results are obtained for real turbulence and are testable experimentally. PACS numbers: 47.55.Kf, 47.10.Fg, 05.45.Df, 47.53.+n Recently the problem of distribution of inertial particles in turbulence received much attention of the researchers[1][2][3][4][5][6][7][8][9][10][11][12]. This is largely thanks to the breakthrough in the theoretical understanding of the Lagrangian motion of particles in the flow that occurred lately[13]. While the understanding of the behavior of particles that have negligible inertia and follow the flow is quite complete by now[13], the understanding of the behavior of inertial particles is still insufficient. This is while the subject has an extremely wide range of applications: the flows of fluids are typically turbulent and often laden with external particles. Theoretical advancement was made mainly for the case of small Stokes number, where the inertia is weak and the particles "almost" follow the flow. Even in this limit of small Stokes numbers, the particles\' distribution is highly non-trivial. Particles\' deviations from the surrounding flow accumulate with time, bringing particles to a strange attractor in space. This attractor is multi-fractal and the only theoretical result obtained so far for the real turbulent flow was the derivation of the correlation codimension[4]. Here a result obtained for real turbulence is a result obtained without modeling turbulence and expressed in terms of the (unknown) statistical properties of turbulence. Since the statistics of turbulence is largely unknown[14], then to obtain such a result one needs to make universal predictions on particles\' behavior in the flow independent of the details of the statistics of that flow.In this Letter we provide the complete description of the distribution of particles in real turbulence at small Stokes numbers, describing both the correlation of the particles\' density with the surrounding flow and the statistics of the singular density on the attractor. We give a number of predictions that are testable experimentally.The idea that particles\' inertia leads to inhomogeneous spatial distribution dates back to the seminal paper by Maxey[2]. It was observed that due to inertia heavy particles are pushed out of the vortices and hence they will not distribute uniformly in the flow, like the inertialess particles. However the quantitative description of the correlations between the locations of particles and of vortices stayed unaddressed. Note the distribution of vorticity in turbulence is random and dynamical, while the distribution of particles reflects its cumulative effect over time. There is a residual correlation that we describe by an integral relation holding in the steady state.We find the spectrum of fractal dimensions of the attractor. We show that while the correlation dimension is different from the dimension of space, the fractal or similarity dimension[15]is equal to the space dimension. In contrast the information dimension is different from the spatial dimension and it equals the Kaplan-Yorke dimension. In turn, the correlation codimension equals twice the Kaplan-Yorke codimension which constitutes a prediction allowing direct testing in the laboratory.The analysis is based on the recent finding of a universal description for the steady state density of the weakly compressible dynamical systems[16]. The particles\' motion, though governed by Newton\'s law, admits an effective description in terms of a velocity field in space. Inertia is described by a small compressible correction to the incompressible velocity of the background turbulent flow. This correction leads to a small disbalance of trajectories going in and out of space regions, which accumulates over a long time to a big effect. Thus compressibility is a singular perturbation which treatment was performed in[16]. For a mixing incompressible velocity the evolution of a small volume of particles makes it dense in space. The volume\'s coarse-graining over an arbitrarily small scale covers all the available space, which volume is assumed finite. When a small compressible component is added to the velocity, the coarse-graining of the evolved volume over an arbitrarily small scale does not cover the whole space any longer. However the coarsegraining over a small but finite scale, that tends to zero with compressibility, already covers the whole volume.The analysis assumes the single-particle approximation where one neglects the interaction between the particles and their back reaction on the flow. We consider a small spherical particle with the radius a and the material density ρ p suspended in a fluid with the density ρ and the kinematic viscosity ν. The fluid flow u(t, x) is assumed to be incompressible. The Newton law governing the evolution of the particle\'s position x(t) and the particle\'s', 'arxivid': '1110.2262', 'author': ['Raymond \nSackler School of Physics and Astronomy\nTel-Aviv University\n69978Tel-AvivIsrael\n', 'Beverly \nSackler School of Physics and Astronomy\nTel-Aviv University\n69978Tel-AvivIsrael\n', 'Raymond \nSackler School of Physics and Astronomy\nTel-Aviv University\n69978Tel-AvivIsrael\n', 'Beverly \nSackler School of Physics and Astronomy\nTel-Aviv University\n69978Tel-AvivIsrael\n'], 'authoraffiliation': ['Sackler School of Physics and Astronomy\nTel-Aviv University\n69978Tel-AvivIsrael', 'Sackler School of Physics and Astronomy\nTel-Aviv University\n69978Tel-AvivIsrael', 'Sackler School of Physics and Astronomy\nTel-Aviv University\n69978Tel-AvivIsrael', 'Sackler School of Physics and Astronomy\nTel-Aviv University\n69978Tel-AvivIsrael'], 'corpusid': 40560416, 'doi': '10.1103/physrevlett.108.134502', 'github_urls': [], 'n_tokens_mistral': 8551, 'n_tokens_neox': 7440, 'n_words': 4760, 'pdfsha': 'd04d6a21c38135014b12ab2665746b4974389549', 'pdfurls': ['https://arxiv.org/pdf/1110.2262v1.pdf'], 'title': ['Distribution of particles and bubbles in turbulence at small Stokes number Itzhak Fouxon', 'Distribution of particles and bubbles in turbulence at small Stokes number Itzhak Fouxon', 'Distribution of particles and bubbles in turbulence at small Stokes number Itzhak Fouxon', 'Distribution of particles and bubbles in turbulence at small Stokes number Itzhak Fouxon'], 'venue': []}

arxiv

ImageNet Challenging Classification with the Raspberry Pi: An Incremental Local Stochastic Gradient Descent Algorithm Thanh-Nghi Do dtnghi@ctu.edu.vn College of Information Technology Can Tho University 92000-CanthoVietnam UMI UMMISCO 209 (IRD/UPMC Sorbonne University Pierre and Marie Curie University -Paris 6France ImageNet Challenging Classification with the Raspberry Pi: An Incremental Local Stochastic Gradient Descent Algorithm ImageNet classification · Incremental local SGD · Rasp- berry Pi With rising powerful, low-cost embedded devices, the edge computing has become an increasingly popular choice. In this paper, we propose a new incremental local stochastic gradient descent (SGD) tailored on the Raspberry Pi to deal with large ImageNet ILSVRC 2010 dataset having 1,261,405 images with 1,000 classes. The local SGD splits the data block into k partitions using kmeans algorithm and then it learns in the parallel way SGD models in each data partition to classify the data locally. The incremental local SGD sequentially loads small data blocks of the training dataset to learn local SGD models. The numerical test results on Imagenet dataset show that our incremental local SGD algorithm with the Raspberry Pi 4 is faster and more accurate than the state-of-the-art linear SVM run on a PC Intel(R) Core i7-4790 CPU, 3.6 GHz, 4 cores. Introduction The efficient image classification algorithms allow to find what we are looking for in very large amount of images produced by internet users. The image classification task automatically categorizes the image into one of predefined classes. It consists of two key stages: the feature extraction and the machine learning scheme. The classical approaches [1,7,9,12,13,22,34,30,38] proposed to use popular handcrafted features such as the scale-invariant feature transform (SIFT [23,24]), the bag-of-words model (BoW) and then to train Support Vector Machines (SVM [36]) to classify images. Recent convolutional neural networks (CNN [21]), deep neural networks including VGG19 [33], ResNet50 [16], Inception v3 [35], Xception [4] aim to learn visual features from images and the classifier in an unified algorithm to efficiently classify images. These deep networks achieve the prediction correctness more over 70% for ImageNet challenging dataset [5,6]. In this paper, we use the pre-trained deep learning network Inception v3 [35] to extract invariant features from images. Followed which, we develop the new incremental local SGD algorithm tailored on the Raspberry Pi for classifying very large ImageNet ILSVRC 2010 dataset. To overcome the main memory limit, the incremental local SGD sequentially loads small data blocks of large trainset to learn local SGD models. The local SGD uses kmeans algorithm [25] to split the training data block into k data partitions and then it learns in the parallel way SGD models in each data partition to classify the data locally. The numerical test results on ImageNet dataset show that our incremental local SGD algorithm on the Raspberry Pi 4 (Broadcom BCM2711, Quad core Cortex-A72 (ARM v8) 64-bit SoC @ 1.5GHz, 4GB RAM) is faster and more accurate than the stateof-the-art linear SVM such as LIBLINEAR [14] run on a PC (Intel(R) Core i7-4790 CPU, 3.6 GHz, 4 cores and 32 GB RAM). The incremental local SGD classifies ImageNet dataset having 1,261,405 images in 2048 deep features into 1,000 classes with an accuracy of 75.61% in 2 hours and 9.48 minutes using the Raspberry Pi 4. The remainder of this paper is organized as follows. Section 2 briefly presents the incremental local SGD algorithm. Section 3 shows the experimental results before conclusions and future works presented in section 4. Incremental local stochastic gradient descent Let us consider a classification task with the dataset D = [X, Y ] consisting of m datapoints X = {x 1 , x 2 , . . . , x m } in the n-dimensional input space R n , having corresponding labels Y = {y 1 , y 2 , . . . , y m } being {c 1 , c 2 , . . . , c p }. Stochastic gradient descent The stochastic gradient descent (SGD) algorithm tries to find p separating planes for p classes (denoted by normal vectors w 1 , w 2 , . . . , w p ∈ R n ) in which the plane w i separates the class c i from the rest. This is accomplished through the unconstrained problem (1). min Ψ (w p , [X, Y ]) = λ 2 w p 2 + 1 m m i=1 L(w p , [x i , y i ])(1) where the errors are measured by L(w p , [x i , y i ]) = max{0, 1 − y i (w p .x i )} and a positive constant λ is to control the regularization strength ( w p 2 ). Studies in [2,32] illustrate that the (SGD) algorithm solves the unconstrained problem (1) by updating w on T epochs with a learning rate η. For each epoch t, the SGD uses a single datapoint (x i , y i ) randomly in the mini batch B i to compute the sub-gradient ∇ t Ψ (w p , [x i , y i ]) and update w p as follows: w p = w p − η∇ t Ψ (w p , [x i , y i ])(2) The SGD is a simple yet efficient algorithm for large-scale learning due to the computational complexity corresponding to O(mnp) (linear in the number of training datapoints m). In recent last years, it rises powerful, low-cost embedded devices. For example, the Raspberry Pi 4 (Broadcom BCM2711, Quad core Cortex-A72 (ARM v8) 64-bit SoC @ 1.5GHz, 4GB RAM) is only 55 USD. This leads an increasingly popular choice for machine learning and IoT projects, as illustration in [18,19,20,27], image classification on IoT edge devices [26], MobileNet family tailored for Raspberry Pi [15], running AlexNet on Raspberry Pi [17]. Nevertheless, it is intractable to train the SGD model with the Raspberry Pi for ImageNet challenging ILSVRC 2010 problem having 1,261,405 images with 1,000 classes since it requires at least 16 GB RAM for loading the training dataset and the high computational cost. Our investigation aims to reduce the training time and the required main memory of the SGD algorithm, being tailored on the Raspberry Pi. Incremental local stochastic gradient descent As illustration in Fig. 1, our proposed local SGD (denoted by kSGD) uses kmeans [25] to divide the training data block into k partitions and then it learns local SGD models in data partitions in parallel way on multi-core computers. Algorithms 1 and 2 describe the k local SGD learning stage and prediction, respectively. The k local SGD algorithm not only reduces the training complexity of the full SGD and but also allows to parallelize the training task of k local SGD models on multi-core computers. Partition D into 3 clusters with k-means D1 D2 D3 c1, lSGD1 = SGD(D1, θ) c2, lSGD2 = SGD(D2, θ) c3, lSGD3 = SGD(D3, θ) Let us to illustrate the complexity of the k local SGD algorithm. Splitting the full training dataset with m datapoints in n dimensions and p classes into k balanced clusters leads the cluster size being about m k and the number of classes in a cluster scaling pω k < p. The training complexity of a local SGD 1 is O( m k n pω k ). Therefore, the complexity of parallel training k local SGD models on a P -core processor is O( m P k ωnp). This illustrates that parallel learning k local SGD models is P k ω times faster than the global SGD training (O(mnp)). Studies in [3,10,37] point out the trade-off between the capacity of the local learning algorithm and the complexity. The large value of k reduces significant training time of kSGD and making a very low generalization capacity. The small value of k improves the generalization capacity but also increasing the training time. To overcome the main memory limit of the Raspberry Pi, we propose to train local SGD models in the incremental fashion. The full training set D is split into T small blocks {D 1 , D 2 , . . . , D T }. The incremental local SGD (denoted by Inc-kSGD in Algorithm 3) sequentially loads data block D t to learn local SGD models. The prediction of a new datapoint x (in algorithm 4) is the majority vote among classification resultsŷ 1 ,ŷ 2 , . . . ,ŷ T } obtained by T kSGD models. The incremental local SGD can be explained by training an ensemble of local SGD models. Experimental results We are interested in the assessment of the incremental local SGD (Inc-kSGD) algorithm with the Raspberry Pi for handling ImageNet challenging dataset. Therefore, it needs to evaluate the performance in terms of training time and classification correctness. Software programs We implemented the Inc-kSGD in Python using library Scikit-learn [29]. The full SGD algorithm is already implemented in Scikit-learn. We would like to compare with the best state-of-the-art linear SVM algorithm, LIBLINEAR [14] implemented in C/C++ (the parallel version on multi-core computers with OpenMP [28]) and the full SGD algorithm. Our Inc-kSGD trains the ensemble of local SGD models with the Raspberry Pi 4 (RPi4) Raspbian Bulleyes, Broadcom BCM2711, Quad core Cortex-A72 (ARM v8) 64-bit SoC @ 1.5GHz, 4GB RAM. LIBLINEAR, full SGD learn classification models on a machine (PC) Linux Fedora 32, Intel(R) Core i7-4790 CPU, 3.6 GHz, 4 cores and 32 GB main memory. ImageNet challenging dataset Experimental results are evaluated on ImageNet challenging ILSVRC2010 dataset [5,6] with 1,261,405 images and 1,000 classes which is the most popular visual classification benchmark [4,5,6,7,8,9,11,12,13,16,33,35,38]. We propose to use pre-trained Inception v3 [35] described in Fig. 2 to extract 2,048 invariant features from images (getting the last AvgPool layer). ImageNet dataset is randomly divided into training set (1,009,124 datapoints) and testing set (252,281 datapoints) with random guess 0.1% due to 1,000 classes. Tuning parameter For training linear SVM models, it needs to tune the positive constant C in SVM algorithms for keeping the trade-off between the margin size and the errors. We use the cross-validation (hold-out) protocol to find-out the best value C = 100, 000. LIBLINEAR uses L2-regularized Logistic Regression that is very closed to the softmax classifier used in deep learning networks, e.g. Inception v3 [35]. Training dataset is split into 8 blocks (the block size 127,000 requires about 2GB RAM) for inc-kSGD learning. The parameter k local SGD models (number of clusters) of kSGD is set to 300 so that each cluster has about 500 datapoints. The idea gives a trade-off between the generalization capacity and the computational cost. Furthermore, the number of epochs and learning rate eta of SGD are 50 and 0.001, respectively. Due to the Raspberry Pi 4 (Broadcom BCM2711, Quad core Cortex-A72) and the PC (Intel(R) Core i7-4790 CPU, 4 cores) used in the experimental setup, the number of threads is setting to 4 for all training tasks. Classification results We obtain classification results of IncklSGD, full SGD and LIBLINEAR in Table 1, Fig. 3 and Fig. 4. The fastest training algorithm is in bold-faced and the second one is in italic. The same presentation format is accorded to performance in terms of classification accuracy, demanded memory size. Given the differences in implementation, including the programming language (C++ versus Python), computer (PC Intel(R) Core i7-4790 CPU, 4 cores, 32 GB RAM versus Raspberry Pi 4 Broadcom BCM2711, Quad core Cortex-A72, 4GB RAM), the comparison of training time is not really fair. But our Inc-kSGD achieves interesting results. In the comparison of training time among algorithms, we can see that the Inc-kSGD with the Raspberry Pi is fastest training algorithm. Our Inc-kSGD with the Raspberry Pi is 75.79 times faster than LIBLINEAR with the PC. The full SGD is 54.74 times faster than LIBLINEAR. The full SGD on the PC is 1.38 times longer than the Inc-kSGD on the Raspberry Pi. In terms of overall accuracy, the Inc-kSGD gives the highest accuracy in the classification. The comparison, algorithm by algorithm, shows that the superiority of Inc-kSGD on LIBLINEAR corresponds to 1.95%. Inc-kSGD also improves 1.16% compared to the full SGD. Our Inc-kSGD training algorithm requires about 2 GB RAM against at least 30 GB RAM being used by the full SGD and LIBLINEAR. The classification results show that our Inc-kSGD algorithm is efficient for handling such large-scale multi-class datasets with the Raspberry Pi. Conclusion and future works We have presented the new incremental local SGD (Inc-kSGD) algorithm tailored on the Raspberry Pi to handle ImageNet challenging ILSVRC 2010 dataset having 1,261,405 images with 1,000 classes. The Inc-kSGD trains an ensemble of local SGD models by sequentially loading small data blocks for learning local SGD models. The local SGD uses kmeans algorithm to split the data block into k partitions and then it learns in the parallel way SGD models in each data partition to classify the data locally. The numerical test results on ImageNet challenging dataset show that our Inc-kSGD algorithm with the Raspberry Pi 4 is 75.79 times faster than the state-of-the-art LIBLINEAR on the PC with an improvement of 1.95% accuracy. In the near future, we will develop the distributed implementation for the incremental local SGD algorithm on an in-memory cluster-computing platform with the Raspberry Pis. Algorithm 1 : 1k local SGD algorithm (kSGD) input : training dataset D number of local models k parameters of SGD θ output: kSGD-model (k local SGD models) 1 begin 2 /*k-means performs the data clustering on D;*/ 3 creating k clusters denoted by D1, D2, . . . , D k and 4 their corresponding centers c1, c2, . . . kSGD-model = {(c1, SGD1), (c2, SGD2), . . . , (c k , SGD k )} 11 end Fig. 1 : 1Three local SGD model (kSGD) Algorithm 2 : 2Prediction of a new individual x with kSGD model input : a new datapoint x kSGD-model = {(c1, SGD1), (c2, SGD2), . . . , (c k , SGD k )} output: predicted classŷ 1 begin 2 /* find the closest cluster based on the distance between x and cluster centers c1, c2, . . . , c k */ 3 cNN = arg minc distance(x, c) 4 /* the class of x is predicted by the local SGD model SGDNN corresponding to cNN */ 5ŷ = predict(x, SGDNN ) : Incremental k local SGD (Inc-kSGD) input : training dataset D = {D1, D2, . . . , DT } number of local models k parameters of SGD θ output: inc-kSGD-model (ensemble of kSGD models) -kSGD-model = {kSGD1, kSGD2, . . . , kSGDT } 7 end Algorithm 4: Prediction of a new individual x with inc-kSGD model input : a new datapoint x inc-kSGD-model = {kSGD1, kSGD2, . . . , kSGDT } output: predicted classŷ 1 begin 2 for t ← 1 to T do 3ŷt = predict(x, kSGDt) 4 end 5ŷ = majority-vote{ŷ1,ŷ2, . . . ,ŷT } 6 return predicted classŷ 7 end Fig. 2 : 2Architecture of Inception v3 Fig. 3 : 3Training time (min) Fig. 4 : 4Overall classification accuracy Table 1 : 1Overall classification accuracy Our Inc-kSGD classifies ImageNet dataset in 129.48 minutes with 75.61% accuracy. The full SGD achieves 74.45% accuracy with 179.29 minutes in the training time. LIBLINEAR takes 9, 813.58 minutes for training the classification model with 73.66% accuracy.No Algorithm Language Machine Demanded memsize (GB) Time (min) Accuracy (%) 1 Inc-kSGD Python RPi4 2 129.48 75.61 2 Full-SGD Python PC 30 179.29 74.45 3 LIBLINEAR C/C++ PC 30 9,813.58 73.66 It must be noted that the complexity does not include the minibatch k-means[31] used to partition the full dataset. Acknowledgments This work has received support from the College of Information Technology, Can Tho University. We would like to thank very much the Big Data and Mobile Computing Laboratory. Scene classification via pLSA. A Bosch, A Zisserman, X Munoz, Proceedings of the European Conference on Computer Vision. the European Conference on Computer VisionBosch, A., Zisserman, A., Munoz, X.: Scene classification via pLSA. 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{'abstract': 'With rising powerful, low-cost embedded devices, the edge computing has become an increasingly popular choice. In this paper, we propose a new incremental local stochastic gradient descent (SGD) tailored on the Raspberry Pi to deal with large ImageNet ILSVRC 2010 dataset having 1,261,405 images with 1,000 classes. The local SGD splits the data block into k partitions using kmeans algorithm and then it learns in the parallel way SGD models in each data partition to classify the data locally. The incremental local SGD sequentially loads small data blocks of the training dataset to learn local SGD models. The numerical test results on Imagenet dataset show that our incremental local SGD algorithm with the Raspberry Pi 4 is faster and more accurate than the state-of-the-art linear SVM run on a PC Intel(R) Core i7-4790 CPU, 3.6 GHz, 4 cores.', 'arxivid': '2203.11853', 'author': ['Thanh-Nghi Do dtnghi@ctu.edu.vn \nCollege of Information Technology Can\nTho University\n92000-CanthoVietnam\n\nUMI UMMISCO 209 (IRD/UPMC\nSorbonne University\nPierre and Marie Curie University -Paris 6France\n'], 'authoraffiliation': ['College of Information Technology Can\nTho University\n92000-CanthoVietnam', 'UMI UMMISCO 209 (IRD/UPMC\nSorbonne University\nPierre and Marie Curie University -Paris 6France'], 'corpusid': 247597311, 'doi': '10.48550/arxiv.2203.11853', 'github_urls': [], 'n_tokens_mistral': 9198, 'n_tokens_neox': 7772, 'n_words': 4328, 'pdfsha': 'f7faeead94e57d80638930a0a83a89b07235b4ec', 'pdfurls': ['https://export.arxiv.org/pdf/2203.11853v3.pdf'], 'title': ['ImageNet Challenging Classification with the Raspberry Pi: An Incremental Local Stochastic Gradient Descent Algorithm', 'ImageNet Challenging Classification with the Raspberry Pi: An Incremental Local Stochastic Gradient Descent Algorithm'], 'venue': []}

arxiv

Quantum Algorithm for the Shortest Superstring Problem ⋆ 26 Dec 2021 Kamil Khadiev Institute of Computational Mathematics and Information Technologies Kazan Federal University 35Kremlyovskaya, KazanRussia Zavoisky Physical-Technical Institute FRC Kazan Scientific Center of RAS KazanRussia Carlos Manuel Bosch Machado Institute of Computational Mathematics and Information Technologies Kazan Federal University 35Kremlyovskaya, KazanRussia Quantum Algorithm for the Shortest Superstring Problem ⋆ 26 Dec 2021quantum algorithms· shortest superstring· strings· DNA as- sembly In this paper, we consider the "Shortest Superstring Problem"(SSP) or the "Shortest Common Superstring Problem"(SCS). The problem is as follows. For a positive integer n, a sequence of n strings S = (s 1 , . . . , s n ) is given. We should construct the shortest string t (we call it superstring) that contains each string from the given sequence as a substring. The problem is connected with the sequence assembly method for reconstructing a long DNA sequence from small fragments. We present a quantum algorithm with running time O * (1.728 n ). Here O * notation does not consider polynomials of n and the length of t.Keywords: quantum algorithms· shortest superstring· strings· DNA assembly. Introduction In this paper, we consider the "Shortest Superstring Problem"(SSP) or the "Shortest Common Superstring Problem"(SCS). The problem is as follows. For a positive integer n, a sequence of n strings S = (s 1 , . . . , s n ) is given. We should construct the shortest string t (we call it superstring) that contains each string from the given sequence as a substring. The problem is connected with the sequence assembly method for reconstructing a long DNA sequence from small fragments [28]. There are two types of sequence assemble problems. The first one is the Reference-guided genome assembly method that constructs an existing long DNA string from the sequence S. For the problem, we should know the string t apriori and check whether we can construct it from S. The second one is de-novo assembly; in this problem, we do not have the string t and should construct it using all strings from S. The Shortest Superstring Problem is used as one of the main tools for solving de-novo assembly problems [5]. The problem has applications in other areas such as virology and immunology (the SSP models ⋆ A part of the reported study is funded by RFBR according to the research project No.20-37-70080. The research is funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project No. 0671-2020-0065. the compression of viral genome); the SSP can be used to achieve data compression; in scheduling ( solutions can be used to schedule operations in machines with coordinated starting times), and others. It is known that the Shortest Superstring Problem is an NP-hard problem [24,25,26,31]. Researchers interested in approximation algorithm, the best known algorithm is [16]. At the same time, researchers are interested in exact solutions also. The algorithm based on [4,14] have O * (2 n ) running time. If we have a restriction on a length of the strings s i , then there are better algorithms. If a length of strings s i is at most 3, then the there is an algorithm [11] with running time O * (1.443 n ). For a constant c, if a length of strings s i is at most c, then there is a randomized algorithm [12] with running time O * (2 (1−f (c))c ) where f (c) = 1/(1 + 2c 2 ). We investigate the problem from the quantum computing [29,2] point of view. There are many problems where quantum algorithms outperform the best-known classical algorithms. Some of them can be founded here [32,15,19,23,18,10]. Problems for strings are examples of such problems [17,20,30,9,1,27]. One of the most important performance metrics in this regard is query complexity; and we investigate problems using this metric for complexity. Researchers investigated quantum algorithms for different problems that can be represented as an assembly of a long string t from the collection of strings S if t is known a priori [22,21]. We present a quantum algorithm for SSP with running time O * (1.728 n ). Here O * notation does not consider a log factor; in other words, polynomials of n and the length of t. The algorithm is based on Grover's search algorithm [13] and the Dynamic programming approach for a Boolean cube [3,4,14]. As far as we know, our algorithm is the first quantum algorithm for SSP. The structure of this paper is the following. Section 2 contains preliminaries. We present the algorithm in Section 3. Section 4 concludes the paper. Preliminaries Let us consider a string u = (u 1 , . . . , u m ). Let u[i, j] be a substring (u i , . . . , u j ) for 1 ≤ i ≤ j ≤ m. Let |u| be a length of a string u. Shortest Superstring Problem The problem is as follows. For positive integers n and k, a sequence of n strings S = (s 1 , . . . , s n ) is given. The length of each string |s i | = k for i ∈ {1, . . . , n}. We should construct the shortest string t (we call it superstring), i.e. |t| is the minimal possible such that each s i is substring of t for i ∈ {1, . . . , n}. In there words, for each i ∈ {1, . . . , n} there are 1 ≤ ℓ, r ≤ |t| such that t[ℓ, r] = s i . Quantum Query Model We use the standard form of the quantum query model. Let f : D → {0, 1}, D ⊆ {0, 1} N be an N variable function. An input for the function is x = (x 1 , . . . , x N ) ∈ D where x i ∈ {0, 1} for i ∈ {1, . . . , N }. We are given oracle access to the input x, i.e. it is realized by a specific unitary transformation usually defined as |i |z |w → |i |z + x i (mod 2) |w where the |i register indicates the index of the variable we are querying, |z is the output register, and |w is some auxiliary work-space. It can be interpreted as a sequence of control-not transformations such that we apply inversion operation (X-gate) to the second register that contains |z in a case of the first register equals i and the variable x i = 1. We interpret the oracle access transformation as N such controlled transformations for each i ∈ {1, . . . , N }. An algorithm in the query model consists of alternating applications of arbitrary unitaries independent of the input and the query unitary, and a measurement in the end. The smallest number of queries for an algorithm that outputs f (x) with a probability that is at least 2 3 on all x is called the quantum query complexity of the function f and is denoted by Q(f ). We refer the readers to [29,2] for more details on quantum computing. In this paper's quantum algorithms, we refer to the quantum query complexity as the quantum running time. We use modifications of Grover's search algorithm [13,6] as quantum subroutines. For these subroutines, time complexity is more than query complexity for additional log factor. Algorithm We discuss our algorithm in this section. Firstly, let us reformulate the problem in the Graph form. Let us construct a full directed weighted graph G = (V, E) by the sequence S. A vertex v i corresponds to the string s i for i ∈ {1, . . . , n}. Set of vertexes is V = (v 1 , . . . , v n ). The weight of an edge between two vertexes v i and v j is the length of maximal overlap for s i and s j . Formally, w(i, j) = max{z : s i [k−z+1, k] = s j [1, z]}. We can see that any path that visits all vertexes exactly once represents a superstring. Let the weight of the path be the sum of weights of all edges the belongs to a path. The path that visits all vertexes exactly once and has maximal weight represents the shortest superstring. Let P = (v i1 , . . . , v i ℓ ) be a path. Let a weight of the path P be w( P ) = w(v i1 , v i2 ) + · · · + w(v i ℓ−1 , v i ℓ ), let |P | = ℓ. Let us present two procedures: -ConstructTheGraph(S) constructs the graph G = (V, E) by S. The implementation of the procedure is presented in Algorithm 1. -ConstructSuperstringByPath(P ) constructs the target superstring by a path P in the graph G = (V, E). Implementation of the procedure is presented in Algorithm 2. Let us consider a function L : 2 V × V × V → R where 2 V is the set of all subsets of V . The function L is such that L(S, v, u) is the maximum of all weights of paths that visit all vertexes of S exactly ones, start from the vertex v, and finish in the vertex u. If there is no such path, then we assume that L(S, v, u) = −∞. Algorithm 1 Implementation of ConstructTheGraph(S) for S = (s 1 , . . . , s n ), |s i | = k. , u) is the path that visit all vertexes of S exactly once, starts from the vertex v, finishes in the vertex u and has maximal weight. In other words for P = F (S, u, v) we have w(P ) = L(S, u, v). V = (v 1 , . . . , v n ) for i ∈ {1, . . . , n} do for j ∈ {1, . . . , n} do if i = j then maxOverlap ← 0 for z ∈ {1, . . . , k} do if s i [n − z + 1, n] = s j [1, z] then maxOverlap ← z end if end for E ← E ∪ (v i , v j ) w(v i , v j ) ← maxOverlap end if end for end for return (V, E) Algorithm 2 Implementation of ConstructSuperstringByPath(P ) for P = (v i1 , . . . , v i ℓ ). t = s i 1 for j ∈ {2, . . . , ℓ} do t ← t • s i j [w(v i j−1 , v i j−1 ) + 1, k] ⊲ Here • is the concatenation operation. end for return t Let the function F : 2 V × V × V → V * be such that F (S, v We assume, that L({v}, v, v) = 0 and F ({v}, v, v) = (v) for any v ∈ V by definition. Let us discuss the properties of the function. Property 1. Suppose S ⊂ V, v, u ∈ V , an integer k ≤ |S|. The function L is such that L(S, v, u) = max S ′ ⊂S,|S ′ |=k,y∈S ′ (L(S ′ , v, y) + L((S\S ′ ) ∪ {y}, y, u)) and F (S, u, v) is the path that is concatenation of corresponding paths. Proof. Let P 1 = F (S ′ , v, y) and P 2 = F ((S\S ′ ) ∪ {y}, y, u). The path P = P 1 • P 2 belongs to S ′ , starts from v and finishes in u, where • means concatenation of paths excluding the duplication of common vertex v. Because of definition of L, we have L(S, v, u) ≥ w(P ). Assume that there is a path T = (v i1 , . . . , v i ℓ ) such that w(T ) = L(S, v, u) and w(T ) > w(P ). Let us select S ′′ such that |S ′′ | = k, S ′′ ⊂ S and there is j < |T | such that R 1 = v i1 , . . . , v ij ∈ S ′′ and R 2 = v ij , v ij+1 , . . . , v i ℓ ∈ S ′′ \{v j }. Then w(R 1 ) ≤ w(P 1 ) and w(R 2 ) ≤ w(P 2 ) by definition of L. Therefore, w(R) = w(R 1 ) + w(R 2 ) − 1 ≤ w(P 1 ) + w(P 2 ) − 1 = w(P ) . We obtain a contradiction with assumption. As a corollary we obtain the following result: (L(S\{u}, v, y) + w(y, u)) . and F (S, u, v) is the path that is the corresponding path. Using this idea, we construct the following algorithm. Step 1. Let α = 0.055. We classically compute L(S, v, u) and F (S, v, u) for all |S| ≤ (1 − α) n 4 and v, u ∈ S Step 2. Let V 4 ⊂ V be such that |V 4 | = n 4 . Then, we have L(V 4 , u, v) = max Vα⊂V4,|Vα|=(1−α)n/4,y∈Vα (L(V α , v, y) + L((V 4 \V α ) ∪ {y}, y, u)) . Let V 2 ⊂ V be such that |V 2 | = n 2 . Then, we have L(V 2 , u, v) = max V4⊂V2,|V4|=n/4,y∈V4 (L(V 4 , v, y) + L((V 2 \V 4 ) ∪ {y}, y, u)) . Finally, L(V, u, v) = max V2⊂V,|V2|=n/2,y∈V2 (L(V 2 , v, y) + L((V \V 2 ) ∪ {y}, y, u)) . We can compute L(V, u, v) and corresponding F (V, u, v) using three nested procedures for maximum finding. As such procedure, we use Durr-Hoyer [7,8] quantum minimum finding algorithm. Note that the error probability for the Durr-Hoyer algorithm is at most 0.1. So, we use the standard boosting technique to decrease the total error probability to constant by O(n) repetition of the maximum finding algorithm in each level. Let us present the implementation of Step 1. Assume that I(u) is the sequence of neighbor vertexes for u. Let us present a recursive function GetL(S, v, u) for S ⊂ V, u, v ∈ V with cashing that is Dynamic Programming approach in fact. The function is based on Corollary 1. Algorithm 3 GetL(S, v, u). if v = u and S = {v} then ⊲ Initialization L({v}, v, v) ← 0 F ({v}, v, v) ← (v) end if if L(S, v, u) is not computed then weight ← −∞ path ← () for y ∈ I(u) do if y ∈ S\{u} and GetL(S\{u}, v, y) + w(y, u) > weight then weight ← L(S\{u}, v, y) + w(y, u) path ← F (S\{u}, v, y) ∪ u end if end for L(S, v, u) ← weight F (S, v, u) ← path end if return L(S, v, u) Let QMax ((x 1 , . . . , x N )) be the implementation of the quantum maximum finding algorithm [7] for a sequence x 1 , . . . , x N . The most nested quantum maximum finding algorithm for some V 4 ⊂ V, |V 4 | = n 4 and u, v ∈ V is QMax((L(V α , v, y) + L((V 4 \V α ) ∪ {y}, y, u) : V α ⊂ V 4 , |V α | = (1 − α) n 4 , y ∈ V α )) The middle quantum maximum finding algorithm for some V 2 ⊂ V, |V 2 | = n 2 and u, v ∈ V is QMax((L(V 4 , v, y) + L((V 2 \V 4 ) ∪ {y}, y, u) : V 4 ⊂ V 2 , |V 4 | = n/4, y ∈ V 4 )) Algorithm 4 Step1. for S ∈ 2 V such that |S| ≤ (1 − α) n 4 do for v ∈ V do for u ∈ V do if v ∈ S and u ∈ S then GetL(S, v, u)⊲ We are computing L(S, v, u) and F (S, v, u) but we are not needing these results at the moment. We need it for Step 2. end if end for end for end for Note that |V 4 | = n/4 and |V 2 \V 4 | = n/4. We use the invocation of QMax (the most nested quantum maximum finding algorithm) instead of L(V 4 , v, y) and L(V 2 \V 4 , y, u). The final quantum maximum finding algorithm for some u, v ∈ V is QMax((L(V 2 , v, y) + L((V \V 2 ) ∪ {y}, y, u) : V 2 ⊂ V, |V 2 | = n/2, y ∈ V 2 )) Note that |V 2 | = n/2 and |V \V 2 | = n/2. We use the invocation of QMax (the middle quantum maximum finding algorithm) instead of L(V 2 , v, y) and L((V \V 2 ) ∪ {y}, y, u). The procedure QMax returns not only the maximal value, but the index of the target element. Therefore, by the "index" we can obtain the target paths using F function. So, the result path is P = P 1 • P 2 , where P 1 is the result path for L(V 2 , v, y) and P 2 is the result path for L((V \V 2 ) ∪ {y}, y, u). P 1 = P 1,1 • P 1,2 , where P 1,1 is the result path for L(V 4 , v, y) and P 1,2 is the result path for L((V 2 \V 4 ) ∪ {y}, y, u). By the same way we can construct P 2 = P 2,1 • P 2,2 . P 1,1 = P 1,1,1 • P 1,1,2 , where P 1,1,1 is the result path for L(V α , v, y) and P 1,1,2 is the result path for L((V 4 \V α ) ∪ {y}, y, u). Note, that this values were precomputed classically on Step 1, and were have stored them in F (V α , v, y) and F ((V 4 \V α ) ∪ {y}, y, u) respectively. By the same way we can construct P 1,2 = P 1,2,1 • P 1,2,2 , P 2,1 = P 2,1,1 • P 2,1,2 , P 2,2 = P 2,2,1 • P 2,2,2 . The final Path is P = P 1 • P 2 = (P 1,1 • P 1,2 ) • (P 2,1 • P 2,2 ) = (P 1,1,1 • P 1,1,2 ) • (P 1,2,1 • P 1,2,2 ) • (P 2,1,1 • P 2,1,2 ) • (P 2,2,1 • P 2,2,2 ) Let us present the final algorithm as Algorithm 5. The complexity of the algorithm is presented in the following theorem. Algorithm 5 Algorithm for SSP. (V, E) ← ConstructTheGraph(S) Step1() weight ← −∞ path ← () for v ∈ V do for u ∈ V do currentW eight ← QMax((L(V2, v, y) + L((V \V2) ∪ {y}, y, u) : V2 ⊂ V, |V2| = n/2, y ∈ V2)) if weight < currentW eight then weight ← currentW eight path ← (P 1,1,1 • P 1,1,2 ) • (P 1,2,1 • P 1,2,2 ) • (P 2,1,1 • P 2,1,2 ) • (P 2,2,1 • P 2,2,2 ) end if end for end for t ← ConstructSuperstringByPath(path) return t Theorem 1. Algorithm 5 solves LTP with O * (1.728 n ) running time and constant bounded error. Proof. The correctness of the algorithm follows from the above discussion. Let us present an analysis of running time. Complexity of Step 1 (Classical preprocessing) is (1−α) n 4 i=1 O * i (1 − α) n 4 = O * (1.728 n ). Complexity of Step 2 (Quantum part) is complexity of three nested Durr-Hoyer maximum finding algorithms. Due to [7,13,8], the complexity is O * m n/2 · n/2 n/4 · n/4 αn/4 = O * (1.728 n ). Complexity of ConstructTheGraph is O(n 2 k) and complexity of ConstructSuperstringByPath is O(n). We invoke ConstructTheGraph, Step 1, Step 2 and ConstructSuperstringByPath sequentially. Therefore, the total complexity is the sum of complexities for these steps. So, the total complexity is O * (1.728 n ). Only Step 2 has an error probability. The most nested invocation of the Durr-Hoyer algorithm has an error probability 0.1. Let us repeat it 2n times and choose the maximal value among all invocations. The algorithm has an error only if all invocations have an error. Therefore, the error probability is 0.1 2n = 100 −n . Let us consider the middle Durr-Hoyer algorithm's invocation. The probability of success is the probability of correctness of maximum finding and the probability of input correctness, i.e., the correctness of all the nested Durr-Hoyer algorithm's invocations. It is 0.9 · (1 − 100 −n ) γ , where γ = n/2 n/4 ≥ 0.8, for enough big n. So, the error probability is at most 0.2. Let us repeat the middle Durr-Hoyer algorithm 2n times and choose the maximal value among all invocations. Similar to the previous analysis, the error probability is 0.2 2n = 25 −n . Therefore, the total success probability that is the final Durr-Hoyer algorithm's success probability is the following one. Therefore, the total error probability is at most 0.2. Conclusion We present a quantum algorithm for the SSP or SCS problem. It works faster than existing classical algorithms. 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{'abstract': 'In this paper, we consider the "Shortest Superstring Problem"(SSP) or the "Shortest Common Superstring Problem"(SCS). The problem is as follows. For a positive integer n, a sequence of n strings S = (s 1 , . . . , s n ) is given. We should construct the shortest string t (we call it superstring) that contains each string from the given sequence as a substring. The problem is connected with the sequence assembly method for reconstructing a long DNA sequence from small fragments. We present a quantum algorithm with running time O * (1.728 n ). Here O * notation does not consider polynomials of n and the length of t.Keywords: quantum algorithms· shortest superstring· strings· DNA assembly.', 'arxivid': '2112.13319', 'author': ['Kamil Khadiev \nInstitute of Computational Mathematics and Information Technologies\nKazan Federal University\n35Kremlyovskaya, KazanRussia\n\nZavoisky Physical-Technical Institute\nFRC Kazan Scientific Center of RAS\nKazanRussia\n', 'Carlos Manuel ', 'Bosch Machado \nInstitute of Computational Mathematics and Information Technologies\nKazan Federal University\n35Kremlyovskaya, KazanRussia\n'], 'authoraffiliation': ['Institute of Computational Mathematics and Information Technologies\nKazan Federal University\n35Kremlyovskaya, KazanRussia', 'Zavoisky Physical-Technical Institute\nFRC Kazan Scientific Center of RAS\nKazanRussia', 'Institute of Computational Mathematics and Information Technologies\nKazan Federal University\n35Kremlyovskaya, KazanRussia'], 'corpusid': 245501985, 'doi': '10.1117/12.2624618', 'github_urls': [], 'n_tokens_mistral': 9349, 'n_tokens_neox': 8310, 'n_words': 4697, 'pdfsha': '8fda64b43a76cd5215e44c65308c7f0a2aee2621', 'pdfurls': ['https://arxiv.org/pdf/2112.13319v1.pdf'], 'title': ['Quantum Algorithm for the Shortest Superstring Problem ⋆', 'Quantum Algorithm for the Shortest Superstring Problem ⋆'], 'venue': []}

arxiv

Quantum Eigenvector Continuation for Chemistry Applications Carlos Mejuto-Zaera Scuola Internazionale Superiore di Studi Avanzati (SISSA) Trieste TS Italy Alexander F Kemper Department of Physics North Carolina State University 27695RaleighNorth CarolinaUSA Quantum Eigenvector Continuation for Chemistry Applications (Dated: May 2, 2023) A typical task for classical and quantum computing in chemistry is finding a potential energy surface (PES) along a reaction coordinate, which involves solving the quantum chemistry problem for many points along the reaction path. Developing algorithms to accomplish this task on quantum computers has been an active area of development, yet finding all the relevant eigenstates along the reaction coordinate remains a difficult problem, and determining PESs is thus a costly proposal. In this paper, we demonstrate the use of a eigenvector continuation -a subspace expansion that uses a few eigenstates as a basis -as a tool for rapidly exploring potential energy surfaces. We apply this to determining the binding PES or torsion PES for several molecules of varying complexity. In all cases, we show that the PES can be captured using relatively few basis states; suggesting that a significant amount of (quantum) computational effort can be saved by making use of already calculated ground states in this manner. A typical task for classical and quantum computing in chemistry is finding a potential energy surface (PES) along a reaction coordinate, which involves solving the quantum chemistry problem for many points along the reaction path. Developing algorithms to accomplish this task on quantum computers has been an active area of development, yet finding all the relevant eigenstates along the reaction coordinate remains a difficult problem, and determining PESs is thus a costly proposal. In this paper, we demonstrate the use of a eigenvector continuation -a subspace expansion that uses a few eigenstates as a basis -as a tool for rapidly exploring potential energy surfaces. We apply this to determining the binding PES or torsion PES for several molecules of varying complexity. In all cases, we show that the PES can be captured using relatively few basis states; suggesting that a significant amount of (quantum) computational effort can be saved by making use of already calculated ground states in this manner. I. INTRODUCTION A central motive of quantum physics and chemistry is the accurate determination of the low lying energy eigenstates of a Hamiltonian describing a system of interest. Whether for finding the ground state of a highly degenerate spin system, such as in spin liquids, or for studying a pathway for a particular chemical reaction, one inevitably uses the eigenstates and expectation values computed with them. Unfortunately, finding the energies and eigenstates is a computationally challenging problem, lying in either the NP-hard (classical) or QMA (quantum) complexity class [1]. A number of classical and quantum algorithms for finding low-lying eigenstates have been developed [2][3][4][5][6][7][8][9][10][11][12][13][14], addressing specific difficulties of the problem, but an approach working for all systems and computational demands is so far non-existing. A common instance of the Hamiltonian eigenstate problem concerns the situation in which one requires the ground and/or excited states as a function of some Hamiltonian parameter(s) λ. For example, such a parameter could be a reaction coordinate in a chemical process, or an interaction strength coefficient when investigating the phase transitions in a spin model in condensed matter physics. In either case, the quantities of interest lie along some path through parameter space. Given that finding the eigenstates of a single Hamiltonian is already complex, it is easy to imagine that doing so with consistent accuracy along a Hamiltonian path is a daunting endeavor. In some reasonably common cases, this problem can be simplified. In particular, unless there is a symmetry-protected level crossing, the eigenvectors and eigenvalues along a given Hamiltonian path are continuous [15]. This allows making use of previously computed eigenstates at a set of parameters Λ = {λ 1 , . . . , λ N } * cmejutoz@sissa.it † akemper@ncsu.edu along the path as an efficient subspace basis in which to represent the Hamiltonian at some new / ∈ Λ. As long as N is not exponentially large, this subspace is a much smaller problem to handle than the full diagonalization at , and can thus be performed classically at negligible computational cost. This approach, first introduced in the context of nuclear physics [15], is named "eigenvector continuation" (EC), and it has been further extended to condensed matter physics [16,17] and quantum computing [18]. Since it requires computing eigenstates at a limited number of points, EC may be particularly fruitful in situations where doing so is computationally expensive. Quantum computing may be such a case, as the currently viable algorithms for finding the ground stateincluding the variational quantum eigensolver (VQE), adiabatic state preparation (ASP), or quantum approximate adiabatic optimization (QAOA) -are expensive hybrid quantum-classical algorithms that are difficult to converge. And yet, to compute e.g. the binding curve of a molecule [19][20][21][22][23], or the phase diagram of a quantum magnet [24,25], a new iterative loop is started at each new parameter point. While the initial guess for the loop may be improved, no further information is carried forward between iterations. Given the high cost of each eigenstate calculation, being able to reuse previously obtained eigenvectors could be a great advantage [16][17][18]. The main strategy is thus to perform the costly, exact eigenstate determination in only a small number of parameter points, and then reconstruct the eigenstates in the full path using a reduced, effective Hamiltonian representation in the basis of these selected points. EC has been succesfully employed to model Hamiltonians for solid state systems, with potential applications in the scope of computational chemistry mostly unexplored. In this work we address this issue and demonstrate that EC can be readily applied to computing the binding curves of a number of chemical compounds, with the eye towards applying this to quantum computing. We study singly-bonded (F 2 , HF) doubly-bonded (H 2 CO, O 2 ) and triply-bonded (N 2 , CO) molecules, as well as more strongly correlated examples (C 6 H 8 -torsion, Cr 2 ). Within the context of these molecules, we investigate the use of EC and details of its implementation, particularly the special considerations that are unique to the ab initio setting. II. GLOSSARY We summarize the main choices of notation and nomenclature used throughout the paper in the following list. Point in parameter space λ, A vector containing all values defining a single point in the parameter space of the system studied. In this paper, these correspond to all the relative atomic coordinates describing the molecular geometry. In a spin Hamiltonian, these would correspond to the spin-spin couplings. Local atomic orbitals (AOs) {φ} a The basis for the quantum chemistry problem at a particular atomic configuration λ. These are typically not orthogonal. Overlap matrix between the training state vectors C ij := v i |v j . Atomic Metric g ab A matrix of overlap integrals between two sets of local AO bases {φ} a and {φ} b for two different atomic configurations λ a and λ b . Note that a and b are a label for the matrix, and not the matrix indices. EC eigenstate |ṽ (n) Approximation of the n-th eigenstate of the Hamiltonian at within the EC representation. III. EIGENVECTOR CONTINUATION FOR AB INITIO CALCULATIONS The basic goal of Eigenvector Continuation (EC), also referred to as the reduced basis method (RBM) in the linear algebra community [16][17][18]26], is accessing the lowest energy solution of a family of time-independent Schrödinger equations which share a parametrized Hamiltonian H H |v (n) = E (n) |v (n) .(1) Given H , the aim is to access the energies E (0) (as well as other observables) of the ground state wave functions |v (0) in some subset of the parameter phase space. This is to be done without actually undertaking the exponentially expensive exact solution of Eq. (1) for all parameter points of interest. Instead, the aim is to approximate the ground states for an arbitrary parameter choice inside the region of interest as a linear combination of a small number of selected parameter points λ i ∈ Λ. Hence, after the exact ground state wave functions |v (0) i are determined, the problem shifts to finding a set of expansion coefficients a i ( ) such that |v (0) ≈ |ṽ (0) = i∈Λ a i ( ) |v (0) i .(2) These coefficients can be variationally optimized by solving the corresponding generalized eigenvalue equation H |ṽ (0) =Ẽ (0) C |ṽ (0)(3) where the Hamiltonian and overlap matrix elements are H ij ( ) = v (0) i |H |v (0) j , C ij = v (0) i |v (0) j .(4) In the above equation, C ij is in general not the identity matrix since the states |v An accurate representation of the ground states at all of interest can be achieved with a judicious choice of a small number of expansion points λ i , which we will refer to as training points or EC points [15,18]. Such a compact representation can be of great value for phase space screening of a Hamiltonian, e.g. to characterize the existing phases and transitions. For how to perform this "judicious" choice of training points, we refer to the existing literature, such as the residue estimation method presented in Ref. [16,17]; however, we note that a common ingredient in these approaches is the natural assumption that all Hamiltonians in the phase space of parameters λ share a single Hilbert space. This is not generally true in computational chemistry, as we discuss in detail below. Within the context of quantum computation, the EC scheme allows a natural and potentially attractive approach to investigate the phase diagram of complex systems, where solving the Schrödinger Equation (1) on the full set of parameter points to a desired accuracy is prohibitively expensive. Indeed, if preparing the expansion states |v (0) i on a quantum register is feasible, one can then measure the Hamiltonian and overlap matrix elements in Eq. (4) and solve the generalized eigenvalue problem classically. This strategy has been successfully demonstrated on simple spin and chemical models in Ref. [18]. Of particular interest would be the application of EC to problems in ab initio computational chemistry, where the phase space studied can be a parameterized chemical reaction. However, a particular complication arises in the ab initio context that needs to be addressed: the fact that the Hamiltonians for different parameter points λ i will in general live, for realistic applications, in different Hilbert spaces. A. The Hilbert space problem in ab initio computational chemistry When considering EC in the context of quantum chemistry, a particular complication that arises is the fact that the atomic basis is not necessarily consistent for the set of training points. For example, consider the typical task of finding a binding energy curve of a diatomic molecule. At each separation R, the atomic orbitals are centered at different points in space, which has to be handled in computing the EC Hamiltonian (H) and overlap (C) matrices -we discuss this procedure in detail in this section. A similar issue arises in performing EC in finite volume calculations where the volume is not consistent [27]. To set the stage for our discussion, we first outline the quantum chemistry process to obtaining a ground state for a correlated problem (see Fig. 1). Each training point λ i comes with a set of atomic orbitals {φ} i . The initial step of finding the ground state of the interacting problem |v (0) i is usually to solve the Hartree-Fock problem. This is done solving the generalized eigenvalue problem determined by the Fock-matrix (F ) and overlap (S) for that set of atomic orbitals (AOs). This yields a rotation matrix U which mixes the AOs into a set of molecular orbitals (MOs). In turn, these MOs can be used as single-particle orbital basis to define a Fock space of many-electron states in their occupation number representation. In principle, one can then exactly solve the problem by projecting the Hamiltonian operator into this basis of Fock states, resulting in the so-called full config- v r G Z m G r u L 2 z u 7 d f O j h s 6 C R T D H 2 W i E S 1 Q q p R c I m + 4 U Z g K 1 V I 4 1 B g M x z e T f 3 m E y r N E / l g R i k G M e 1 L H n F G j Z X q f r d U d i v u D G S Z e D k p Q 4 5 a t / T V 6 S U s i 1 E a J q j W b c 9 N T T C m y n A m c F L s Z B p T y o a 0 j 2 1 L J Y 1 R B + P Z o R N y a p U e i R J l S x o y U 3 9 P j G m s 9 S g O b W d M z U A v e l P x P 6 + d m e g m G H O Z Z g Y l m y + K M k F M Q q Z f k x 5 X y I w Y W U K Z 4 v Z W w g Z U U W Z s N k U b g r f 4 8 j J p n F e 8 q 8 p l / a J c v c 3 j K M A x n M A Z e H A N V b i H G v j A A O E Z X u H N e X R e n H f n Y 9 6 6 4 u Q z R / A H z u c P t G 2 M 4 w = = < / l a t e x i t > U < l a t e x i t s h a 1 _ b a s e 6 4 = " k i v D 0 D w u D Z h 8 H p X o L c X + F q I y C U Q = " > A A A B 8 H i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s y I r 2 X R j c s K 9 i H t U D K Z T B u a Z I Y k I 5 S h X + H G h S J u / R x 3 / o 3 p d B b a e i B w O O d c c u 8 J E s 6 0 c d 1 v p 7 S y u r a + U d 6 s b G 3 v 7 O 5 V 9 w / a O k 4 V o S 0 S 8 1 h 1 A 6 w p Z 5 K 2 D D O c d h N F s Q g 4 7 Q T j 2 5 n f e a J K s 1 g + m E l C f Y G H k k W M Y G O l x z 6 3 0 R A P 2 K B a c + t u D r R M v I L U o E B z U P 3 q h z F J B Z W G c K x 1 z 3 M T 4 2 d Y G U Y 4 n V b 6 q a Y J J m M 8 p D 1 L J R Z U + 1 m + 8 B S d W C V E U a z s k w b l 6 u + J D A u t J y K w S Y H N S C 9 6 M / E / r 5 e a 6 N r P m E x S Q y W Z f x S l H J k Y z a 5 H I V O U G D 6 x B B P F 7 K 6 I j L D C x N i O K r Y E b / H k Z d I + q 3 u X 9 Y v 7 8 1 r j p q i j D E d w D K f g w R U 0 4 A 6 a 0 A I C A p 7 h F d 4 c 5 b w 4 7 8 7 H P F p y i p l D + A P n 8 w e + 7 5 B j < / l a t e x i t > i Training point Atomic Orbitals Molecular Orbitals Active Space Ground State < l a t e x i t s h a 1 _ b a s e 6 4 = " Y 1 h J i z H l d N 0 m H g w N 5 0 b c h S + 3 n + E = " > A A A B 6 H i c b V D L S g N B E O z 1 G e M r 6 t H L Y B A 8 h V 3 x d Q x 6 8 Z i A e U C y h N l J b z J m d n a Z m R X C k i / w 4 k E R r 3 6 S N / / G S b I H T S x o K K q 6 6 e 4 K E s G 1 c d 1 v Z 2 V 1 b X 1 j s 7 B V 3 N 7 Z 3 d s v H R w 2 d Z w q h g 0 W i 1 i 1 A 6 p R c I k N w 4 3 A d q K Q R o H A V j C 6 m / q t J 1 S a x / L B j B P 0 I z q Q P O S M G i v V 6 7 1 S 2 a 2 4 M 5 B l 4 u W k D D l q v d J X t x + z N E J p m K B a d z w 3 M X 5 G l e F M 4 K T Y T T U m l I 3 o A D u W S h q h 9 r P Z o R N y a p U + C W N l S x o y U 3 9 P Z D T S e h w F t j O i Z q g X v a n 4 n 9 d J T X j j Z 1 w m q U H J 5 o v C V B A T k + n X p M 8 V M i P G l l C m u L 2 V s C F V l B m b T d G G 4 C 2 + v E y a 5 x X v q n J Z v y h X b / M 4 C n A M J 3 A G H l x D F e 6 h B g 1 g g P A M r / D m P D o v z r v z M W 9 d c f K Z I / g D 5 / M H r l 2 M 3 w = = < / l a t e x i t > Q < l a t e x i t s h a 1 _ b a s e 6 4 = " v O z M u 3 M 0 Y A h H u h e b 2 D F F u s X a G Z I = " > A A A B 6 H i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K o n 4 d S x 6 8 d i C a Q t t K J v t p F 2 7 2 Y T d j V B K f 4 E X D 4 p 4 9 S d 5 8 9 + 4 b X P Q 1 g c D j / d m m J k X p o J r 4 7 r f z s r q 2 v r G Z m G r u L 2 z u 7 d f O j h s 6 C R T D H 2 W i E S 1 Q q p R c I m + 4 U Z g K 1 V I 4 1 B g M x z e T f 3 m E y r N E / l g R i k G M e 1 L H n F G j Z X q f r d U d i v u D G S Z e D k p Q 4 5 a t / T V 6 S U s i 1 E a J q j W b c 9 N T T C m y n A m c F L s Z B p T y o a 0 j 2 1 L J Y 1 R B + P Z o R N y a p U e i R J l S x o y U 3 9 P j G m s 9 S g O b W d M z U A v e l P x P 6 + d m e g m G H O Z Z g Y l m y + K M k F M Q q Z f k x 5 X y I w Y W U K Z 4 v Z W w g Z U U W Z s N k U b g r f 4 8 j J p n F e 8 q 8 p l / a J c v c 3 j K M A x n M A Z e H A N V b i H G v j A A O E Z X u H N e X R e n H f n Y 9 6 6 4 u Q z R / A H z u c P t G 2 M 4 w = = < / l a t e x i t > U Training point < l a t e x i t s h a 1 _ b a s e 6 4 = " B S b Q M X 7 8 p i g 0 w V N q H d V X r g 1 u N o M = " > A A A B 8 H i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s y I V Z d F N y 4 r 2 I e 0 Q 8 l k M m 1 s k h m S j F C G f o U b F 4 q 4 9 X P c + T e m 7 S y 0 9 U D g c M 6 5 5 N 4 T J J x p 4 7 r f T m F l d W 1 9 o 7 h Z 2 t r e 2 d 0 r 7 x + 0 d J w q Q p s k 5 r H q B F h T z i R t G m Y 4 7 S S K Y h F w 2 g 5 G N 1 O / / U S V Z r G 8 N + O E + g I P J I s Y w c Z K D z 1 u o y H u P / b L F b f q z o C W i Z e T C u R o 9 M t f v T A m q a D S E I 6 1 7 n p u Y v w M K 8 M I p 5 N S L 9 U 0 w W S E B 7 R r q c S C a j + b L T x B J 1 Y J U R Q r + 6 R B M / X 3 R I a F 1 m M R 2 K T A Z q g X v a n 4 n 9 d N T X T l Z 0 w m q a G S z D + K U o 5 M j K b X o 5 A p S g w f W 4 K J Y n Z X R I Z Y Y W J s R y V b g r d 4 8 j J p n V W 9 i 2 r t 7 r x S v 8 7 r K M I R H M M p e H A J d b i F B j S B g I B n e I U 3 R z k v z r v z M Y 8 W n H z m E P 7 A + f w B w H O Q Z A = = < / l a t e x i t > j < l a t e x i t s h a 1 _ b a s e 6 4 = " Z H v e X B g R q b 5 3 4 A R 6 V b 4 Q L Z x 3 r p Y = " > A A A B 7 X i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y K r 2 P Q i 8 c I 5 g H J G m Y n s 8 k k s z P L z K w Q l v y D F w + K e P V / v P k 3 T j Z 7 0 M S C h q K q m + 6 u I O Z M G 9 f 9 d g o r q 2 v r G 8 X N 0 t b 2 z u 5 e e f + g q W . Note that in this diagonalization, one typically does not need to consider an overlap matrix, since the MOs are typically orthonormal. W i C G 0 Q y a V q B 1 h T z g R t G G Y 4 b c e K 4 i j g t B W M b 2 d + 6 4 k q z a R 4 M J O Y + h E e C B Y y g o 2 V m o P H l I 2 m v X L F r b o Z 0 D L x c l K B H P V e + a v b l y S J q D C E Y 6 0 7 n h s b P 8 X K M M L p t N R N N I 0 x G e M B 7 V g q c E S 1 n 2 b X T t G J V f o o l M q W M C h T f 0 + k O N J 6 E g W 2 One complexity arises in the EC when the inner product must be taken between two vectors |v (0) i and |v (0) j that arose from distinct sets of atomic orbitals. This is already clear from the overlap matrix element C ij at the Hartree-Fock level. Neglecting other complexities for now, the overlap between MOs |α(λ i ) and |β(λ j ) that arise from the different Hilbert spaces at training points λ i and λ j is given by α(λ i )|β(λ j ) = mn (U i ) * n,α (U j ) β,m φ i n |φ j m = mn (U i ) * n,α (U j ) β,m (g ij ) nm ,(5) where in the last line we have introduced the metric g ij between the two training points i and j, which is a matrix containing the inner product between the two sets of atomic orbitals (c.f. Fig. 1). This already suggests that in following the EC strategy, some care will need to be taken to account for this difference in the orbital basis between different training points. An additional step, matching the orbitals of different training points, will be necessary to evaluate the expectation values in Eq. (4). To the above considerations concerning the overlap of the AOs and MOs between different geometries one has to add an additional complication which arises frequently in realistic ab initio calculations: the notion of an active space. Even after the massive dimensionality decrease from the uncountable real-space basis, required to describe continuous space, to the finite number of AOs, the exponential scaling of the many-body Hilbert space as a function of the number of orbitals and electrons makes it computationally impossible to perform all-orbital, all-electron calculations except in the smallest molecules with the most modest basis sets. In all other cases, one typically restricts the post-HF (meanfield) determination of correlation effects to a subset of all orbitals, i.e. those orbitals deemed to be the most relevant for the electronic properties of the system. These are typically chosen to be the first N o orbitals around the Fermi-level (the HOMO/LUMO frontier in the single reference description) containing the first N e electrons in the mean-field reference determinant. These N o orbitals with N e electrons constitute the so called active space. An effective Hamiltonian for the active space can be formulated, in which all occupied orbitals outside the active space appear only as a constant shift in energy and as modified one-body terms. Post-HF correlated methods can then be applied to the active space alone, and additionally feedback correlation effects between the active space and the non-active orbitals can also be taken into account at different levels [2,8]. This notion of active space adds another layer of inconsistency between the training FCI vectors on each geometry: Since the simplest way to define active space orbitals is in reference to the mean-field orbitals, and these change between each geometry, there is no guarantee that any given subset of them (such as the active space) span the same region of the one-body Hilbert space at each geometry. In an extreme example, if the mean-field orbitals close to the Fermi-level have a completely different AO character between two given parameters, then the FCI vectors obtained from the corresponding effective Hamiltonians will be essentially orthogonal. These notions of orbital matching has to be included in the evaluation of the matrix elements in Eq. (4), where a transformation between the now active orbital basis in {|v (0) i , |v (0) j } and H( ) has to be performed. If the mismatch between the active orbital basis between the EC points and the target point is large, then this transformation will result in a reduction of the norm of the training vectors in the new basis. This is detrimental to the information contained in the overlap matrix, and thus to the conditioning of the generalized eigenvalue problem, although the issue can in principle be remedied by incorporating more training points to faithfully model the parameterized ground states in the parameter range of interest. In order to minimize this effect, it is necessary to en-sure that the active spaces in the parameter range to be studied with EC are spanned by AOs of the same character.This can be achieved by choosing large active spaces, such that all the relevant AOs for all parameter points will always be included; alternatively, one can choose the nature of the active space orbitals by more systematic means than proximity to the Fermi-level, e.g. using complete active space self-consistent field (CASSCF) orbital optimization (cf. Ch. 12 in Ref. [2]). In the result section below, we exemplify the first strategy for weakly-correlated molecules and employ the second for the strongly correlated Cr 2 . On a molecular torsion example, we will show a case in which this orbital mismatch is harder to solve, and consequently an increased number of training points is needed to cover the full parameter space of interest. B. Possible orbital matchings As discussed above (Fig. 1), finding the EC Hamiltonian and overlap matrices involves a local rotation from atomic orbitals (AOs) to molecular orbitals (MOs) U , a local rotation from MOs to FCI eigenvectors Q, and an inner product between two separate sets of atomic orbitals (which we encode in the metric g). In principle, to capture the proper inner products, each of these must be taken into account; in practice, however, it can be beneficial to neglect one or more of these. In Fig. 2 we show the results of applying eigenvector continuation to the dissociation of F 2 using up to 4 training points and different orbital matching strategies. We compare the binding energy to the Hartree-Fock (HF) results, as well as a reference CAS-FCI result. The calculations are performed in the cc-pVDZ basis set, with an (8o, 14e) active space. The panels show three possible orbital matching approaches: 1. Full rotation: The H and C matrices are determined as discussed above, incorporating the U and Q rotations, as well as the metric. 2. No metric: The U and Q rotations from the FCI vectors to the atomic orbitals are kept, but the metric is neglected. 3. No rotation: The FCI vectors are treated entirely without reference to their origin. The U and Q rotations are neglected, as well as the metric. Somewhat counter-intuitively, the best results are obtained when the metric is neglected, and the method still works when the metric and rotations are neglected (although with limited success). On the other hand, the notionally correct calculation which incorporates the rotations and the metric performs quite poorly. The poor behavior of the calculation with full rotations can be understood by considering the metric. In our calculations we use localized atomic orbitals; while their highly localized nature is desirable from a quantum chemistry perspective, it also leads to a rapidly decaying metric. In essence, the overlap between atomic bases at different training point R tends exponentially to zero as R increases. We illustrate this in Fig. 3 where we plot the vector norm of one of a training state at R = 1.5Å in the basis corresponding to a range of R. When the metric is included, the norm drops and nearly vanishes for R > 2.5Å. The inner product is thus not well captured, and the eigenvector continuation fails. When the metric is neglected, however, eigenvector continuation performs quite well. In particular, when the local rotations U and Q are kept, 3 training points are sufficient to obtain the full binding energy curve. Intuitively, this can be understood as follows. The Q and U rotations describe how the final FCI eigenvector is composed of the molecular orbitals, and how the molecules orbitals are composed of the atomic orbitals. In other words, the FCI eigenvector at a given parameter is a vector in the space spanned by the basis of atomic orbitals at that parameter point. As the atomic separation is varied the FCI eigenvector rotates in the space spanned by the local AOs. However, it is in fact irrelevant that the local atomic basis is now shifted in real space. For EC, it suffices that the FCI eigenvector expressed in its own local basis can be spanned by the training points in their own local basis. This information is encoded in Q and U , and thus keeping those is sufficient. Putting this together with the issues with the metric, we conclude that neglecting the metric is a better choice than keeping it. In Fig. 3, we show that the previous issues with the vanishing overlap due to the metric do not arise here. Finally, we can choose to neglect all rotations, and treat the FCI eigenvector as a vector divorced from any basis information. Here, a more straightforward linear algebra perspective is insightful. The FCI eigenvector simply needs to be spanned by a sufficient number of linearly independent basis vectors; the basis vectors need to be sufficiently expressive in order to be able to orthogonalize the ground state with respect to any other states in the subspace. Thus, this method works, but a larger number of training states may be required. Fig. 2 shows that the 4 number of training points considered here are not sufficient to achieve agreement with the reference FCI result. IV. ANALYZING THE PERFORMANCE OF EIGENVECTOR CONTINUATION In this section we turn to the analysis of the reliability of EC as a compact and accurate approximation to characterize the ground state of ab initio molecular problems in simple one-dimensional parameter spaces. Arguably the most relevant parameter entering the molecular Hamiltonian, within the Born-Oppenheimer approximation, is the molecular geometry. The electronic energy eigenvalues as a function of the nuclear positions are commonly referred to as potential energy surfaces (PES). Hence, we can reformulate our goal as the study of how many EC points are necessary for accurately reconstructing one-dimensional PES in a few cases of chemical interest, namely stretching and torsion of covalent bonds. While a one-dimensional PES for a particular molecule and bond is a fundamentally well defined target, the different approximations typically invoked in a computational chemistry calculation limit the ultimate accuracy of even a hypothetical and exact full configuration interaction (FCI) simulation. Indeed, choices like the atomic basis set, the single-particle orbitals and the correlated active space all affect the FCI reference, and a careful analysis of the convergence of the observables of interest with respect to these factors is a necessary step in an electronic structure investigation. However, these considerations fall beyond the scope of this work, as we concern ourselves with examining how well EC can reproduce a given FCI reference with a small number of training points. We will thus choose a single, reasonable, but by no means final, FCI reference for each molecular case study, but make no claims as to its ultimate relevance towards the accurate description of the real physical sys-tem. We compensate this simplification by examining molecular examples of different degree of electronic correlation and computational complexity, in order to keep the validity of our conclusions as broad as possible. A brief description of the FCI references for all molecular systems follows, with a subsequent presentation and discussion of the numerical results. All calculations were performed using the PYSCF package for electronic structure [28][29][30]. A. Molecular Systems Bond Stretching of Weakly Correlated Molecules The majority of the PESs studied in this work fall under the category of bond stretching of "weakly correlated" molecules. By this, we mean that the nature of the electronic correlation in the equilibrium geometry is well captured by single-reference methods. Nonetheless, in the bond stretching process the ground state naturally becomes multi-reference (to some degree strongly-correlated), making the accurate description of bond dissociation energies a challenge for effective single-particle theories even in these comparatively simple molecules. In addition, the study of bond stretching is of relevance to the quantum computing community, where the bond stretching and dissociation problem is a drosophila [11,13,[31][32][33][34][35]. We consider bonds of different chemical character. We take into account the common heuristic distinction of single, double and triple bonds derived mostly from Lewis structures, and distinguish between symmetric and asymmetric bonds, i.e. bonds between chemically equivalent and inequivalent atoms. As examples of single bonds, we perform EC calculations for F 2 and HF, while we consider O 2 and the CO bond in H 2 CO for the double bond category; N 2 and CO are our triple bond representatives. For all these systems, we used a cc-pVDZ basis set, in which at each geometry we perform a restricted Hartree-Fock (HF) calculation. For the asymmetric bond stretchings, in order to generate smooth PES, all possible spin states within restricted open-shell HF were considered, and the lowest in energy for each bond length was used as the molecular orbital basis for the subsequent FCI calculations. Even in this small molecules and moderate basis set, performing FCI on all electrons and orbitals is computationally prohibitive on a single processor. Hence, we performed instead complete active space (CAS) calculations including the 2p and 2s atomic orbital manifolds involved in the bond breaking. We summarize the active space sizes in Tab. I. We considered bond lengths up to 3.5 times the FCI equilibrium bond length. For all the 6 molecules presented in Fig. 4, the PES curve obtained by EC is in good agreement with the FCI reference. To ease the comparison between different molecules, we rescale the bond length axis by the FCI equilibrium distance of each molecule, and the en- ergy axis by the bonding energy, taking the energy at 3.5 times the equilibrium bond length as the dissociated asymptote. For each molecule, we present the minimal number of EC training points that leads to an acceptable result compared to FCI. None of the molecules require a particularly large number, but some variation does exist between the molecules. We note in particular that F 2 , N 2 and CO exhibit better agreement; while the other EC PES curves have some departure from the FCI result, these could be readily improved by the addition of more EC points (c.f. Fig. 2). In the case of F 2 , it is remarkable that just 3 points are enough to recover the full PES faithfully. These can be interpreted as the three distinct physical regions in the bond dissociation process: the bound region, the dissociated region, and the Coulson-Fisher point where a mean-field description would start breaking translational and/or spin symmetry. While for all other bonds shown in Fig. 4 more than 3 training points are needed, these typically agglomerate around the Coulson-Fisher region, where the system is arguably more strongly correlated. In this sense, a qualitative relationship can be established between the variability of the eigenstate character in a bond-length region and the number of EC points needed to sample that zone accurately. This matches well the observations using EC in spin models [16][17][18]. We note that there are some unusual kinks in the PES curves for the asymmetric bond breakings, which is due to the limitations of the FCI calculations, rather than an artifact introduced by EC. Cr2 and Bond Torsion To test the performance of EC for intrinsically strongly correlated molecules, we consider the bond stretching of a Cr 2 dimer, where we used a cc-pVTZ-dk basis set. Besides the restricted HF calculation at each bond length, a further orbital optimization was performed at the complete active space self-consistent field (CASSCF) level of theory, with a (12o, 12e) active space. The multireference orbital optimization was necessary to obtain a homogeneous 3d and 4s orbital character in the active space through the bond dissociation. The (12o, 12e) CASSCF energy served then also as FCI reference for the EC. Fig. 5 shows the results of the EC calculation for 2 and 3 training points. As was observed in the weakly correlated molecules, in Cr 2 as well a sparse sampling of the bound, dissociated and Coulson-Fisher regions is sufficient to recover the full PES. Considering all the bond stretching examples, it is remarkable that the EC scheme seems to be fairly insensitive to the chemical nature of the PES modelled. Indeed, regardless of the chemical complexity of the bond, represented by single, double, triple, symmetric, asymmetric and correlated bonds, as well as the computational complexity of the FCI reference, based on either RHF or CASSCF orbitals, the EC representation shows a rela-tively homogeneous convergence in terms of the training points. A handful (up to five) points along the bond stretching, typically including at least the bound, dissociated and Coulson-Fisher region, are enough to obtain a visually accurate representation of the ground state PES along the full reaction path. Finally, we considered the bond torsion of transhexatriene around the central CC double bond. This PES was evaluated in the cc-pVDZ basis, using a minimal active space including all π orbitals, namely (6o, 6e), on restricted HF orbitals. The φ = 0 • geometry was taken from Ref. [36]. The orbital mismatch problem is more severe in this case as the rotation mixes the p-orbital manifold. By rotating around the bond, the atomic p z -orbitals of one half of the molecule become eventually completely orthogonal to the p z orbitals of the other half, and consequently the AO character of the frontier MOs changes drastically from 0 torsion to 90 • . As a result, the PES requires a larger number of training points (7) to capture the full surface properly. Nonetheless, this is still modest sampling with which to recover the full PES. B. Choosing the EC training points As mentioned in Sec. III, how to judiciously choose the EC training points to maximize the compactness of the approximation without compromising accuracy has been discussed by Herbst et al. [16], who suggest the use of a residual estimate for determining what points to add. In essence, given a previous EC approximation, the next point to be chosen is the one that maximizes the additional information in the basis as measured by the accuracy of the EC eigenvalue at that point. Here, we briefly exemplify how the residue estimate proposed therein satisfactorily adapts to the aim of achieving "chemical accuracy" in ab initio simulations. By chemical accuracy, one refers to a maximal error of 1.6 mHa of a computational estimate with the true or reference value. When controlled experimental results are not available, one often uses as reference a computational result from a more accurate theoretical model. For our purposes, we can use the FCI reference to determine the error of our EC results at each molecular geometry. If the average error in the energy of a given PES curve calculated via EC with m training points (m-EC) is above chemical accuracy, a natural choice for the m+1th training point is to pick a molecular geometry in the region of maximal deviation. The FCI reference is in general not available, and it is necessary to obtain an estimate of the error using exclusively the data available within the EC calculation. Following Ref. [16], one can evaluate the residue of the EC approximation at each geometry of interest. Given a geometry , for which the m-EC simulation provides with a ground state wave function approximant |v r (m) = H |v (0),m −Ẽ (0),m |v (0),m 2 .(6) This residue r (n) can be written in terms of the Hamiltonian and overlap matrices, the eigenvector from the generalized eigenvalue problem in Eq. (3), and the matrix elements of the squared Hamiltonian in the EC training basis ( H 2 ) ij = v (0) i |H 2 |v (0) j . The only additional cost on top of the EC calculation is thus the measurement of the squared Hamiltonian matrix elements. In Fig. 7, we present the residues of the EC calculations for the bond stretchings in Fig. 4, keeping the same number of training points. We compare these to the error in the energy with respect to the FCI energies in the corresponding active spaces. As can be seen in Fig. 7, the residue estimate closely follows the actual error with respect to FCI, and thus offers an effective indicator to choose the EC training points in order to ultimately reach chemical accuracy with respect to the FCI reference. Of course, this does not guarantee an excellent agreement with experiment, as several approximations enter the FCI reference chosen in each case. However, the compactification offered by the EC approximation enables FCI-quality results within a small fraction of the cost of actually performing an FCI-level calculation (be it using a classical or quantum algorithm) at each point on the potential energy surface. C. Accessing excited states with EC In principle, the EC formalism is not limited to the approximation of ground states. As long as more than one training point is used, the generalized eigenvalue problem in Eq. (3) will have multiple eigenvectors, some of which could be accurate approximations of some excited state in the full Hamiltonian. Indeed, there are two scenarios in which it is natural to expect EC to provide a compact representation of excited states. First, consider Hamiltonians that have avoided level crossings, such that the ground and first excited states switch character continuously across some path in the parameter space. In this case, using only ground states as training points along a path through the level crossing should also potentially result in an acceptable representation of the first excited state. Second, there is no fundamental need to use exclusively FCI ground states to build the EC training sets. If excited states are used, this should produce an EC approximation targeting the corresponding excited state PES. Simultaneously, in the presence of the aforementioned level crossings, having a mixed EC training set containing ground and excited states can lead to accurate representations for both. Here we consider these different possibilities in the example of the F 2 dimer. We present excited state PES for the F 2 molecule in the cc-pVDZ basis, with a (8e, 14o) active space in Fig. 8. The FCI surfaces, shown as grey lines, do not show a level crossing between the ground and first excited state along the dimer bond dissociation. Nonetheless, these states become degenerate in the dissociation limit, and hence a complete decoupling between both states is not obvious a priori. Fig. 8 shows the results for three different EC calculations in three panels. For all of these, the training points were obtained from the same molecular geometries, matching those already shown in the upper left panel of Fig. 4. In the left-most panel of Fig. 8, we compare all three eigenstates obtained from the effective Hamiltonians of a ground state based EC to the first few lying FCI eigenstates. While the exact ground state PES is perfectly reproduced, as was already discussed in Fig. 4 A similar picture occurs in the middle panel of Fig. 8, where the three training points in the EC simulation were all first excited states. Consequently, a relatively faithful approximation of the FCI first excited state PES is obtained, although a noticeable deviation appears at ∼ 2.2Å through an artificial, i.e. not present in the FCI reference, avoided crossing with the second excited state. Similarly to the first case, neither of the two higher lying excited states from the EC calculation approximate any of the FCI PESs well. Surprisingly, all the EC curves in this case have some degree of bonding character (a minimum), even when all excited states in the corresponding energy window are dissociating, including the one used to obtain the training points. Still, since there is no overlap with the results from the EC calculation using ground state training vectors, it seems that the ground state and first excited eigenstate manifolds are mostly decoupled. To further confirm this, an EC calculation was performed using 6 training points in the same 3 molecular geometries, shown in the right-most panel of Fig. 8. For each of the 3 bond lengths, the ground and first excited states at the FCI level were used as independent training vectors for the EC simulation. The three PES already obtained in the EC calculation using only ground state training points (cf. left panel in Fig. 8) are found again in this larger simulation. However, the three PES that would correspond to the EC calculation based on excited states only (cf. middle panel in Fig. 8) are significantly changed. The lowest in energy of the three -the overall first excited state of the EC simulation -follows the exact result better than in the middle panel, missing the deviation at ∼ 2.2Å. Furthermore, the next excited state PES is significantly shallower, closer to the expected non-binding behavior. Finally, the highest excited state among the central three is now pushed in energy much closer to the excited state manifold at ∼ 0.8 Ha, better justifying its bound character. Despite these noticeable improvements, we still find that the only PES that accurately reproduces the FCI reference are those which are sampled explicitly, namely the FCI ground and first excited state in this case. V. CONCLUSIONS The spirit of the eigenvector continuation (EC) approach is proposing low-dimensional effective models to accurately reproduce targeted eigenstates of a parameterized Hamiltonian in some region of the parametric phase space. This is done by sampling a small number of points in said region, i.e. performing a computationally expensive, accurate determination of the eigenstates of interest in these few points, and then using their information to reconstruct the eigenstates inexpensively in the rest of phase space. The computation at the training points may be exact full configuration interaction (FCI) when feasible [16], based on highly accurate matrix-product state (MPS) Ansätze [17], or even the result of a quantum computation [18] for systems beyond the current reach of classical approaches. With a modest number of training points, the accurate results of comparable quality to these expensive methods can be recovered in the full parameter phase space at a fraction of the computational complexity. This becomes especially attractive for studying chemical reactions, which involves the accurate determination of the potential energy surfaces (PES) of ground and excited states along the reaction coordinates. Therefore, here we have investigated the applicability and effectiveness of EC in the ab initio setting. One of the major hurdles in applying EC to ab initio quantum chemistry is the mismatch in basis that arises from disparate molecular geometries for the subspace basis point, which we discussed extensively in the text. One significant conclusion from this work is that parts of this mismatch may be entirely neglected; specifically, the mismatch between the most basic level of the calculations, i.e. the atomic orbital overlap between different training states. After doing so, we have shown that the PES can be captured with remarkably few subspace basis vectors for a number of chemically distinct molecules single, double and triple bonds between chemically equivalent and inequivalent atoms in weakly correlated molecules, bond stretching of the intrinsically strongly correlated Cr 2 , and the bond torsion of trans-hexatriene around the central CC double bond. The associated error as compared to the FCI reference calculations is quite low. Several aspects of the results that go beyond simple ground state manifolds are worth highlighting. First, EC can correctly handle level crossings in the ground state spectrum in chemical molecules, as long as training points are chosen on both sides of the crossing; this extends to any situation where multiple orthogonal subsectors are of interest. Second, we have shown that the use of eigenvector continuation is not limited to the ground state. Excited state manifolds can also be captured by inclusion of representatives of the excited states into the subspace. We exemplify this in F 2 by sampling with excited states instead of ground states. In short, eigenvector continuation is a promising tool for ab initio calculations in any situation where the eigenstates are difficult to obtain. This is in particular true on quantum computers, where finding ground states is a primary target and yet remains elusive; the current state of the art is plagued with issues in the optimization. It is thus quite difficult to find a ground state, and when this feat is accomplished, it should be used to maximum effect. Eigenvector continuation is one way to achieve this goal. FIG. 1 . 1Diagrammatic illustration of the flow from atomic orbitals to the ground state in quantum chemistry. The metric g ij connects the atomic orbital bases belonging to each training point λi,j. The U and Q matrices are rotations that diagonalize the Hartree-Fock and FCI Hamiltonians, respectively.uration interaction (FCI) Hamiltonian H F CI . The FCI Hamiltonian is diagonalized via another rotation matrix Q in the exponentially large Fock space to finally obtain the ground state |v(0) i FIG. 2 . 2Potential energy surfaces (PES) for the F2 dissociation using the cc-pvDZ basis and an (8o, 14e) active space. Solid lines show the Hartree-Fock (grey) and complete active space (CAS) FCI results, while the discontinuous lines present eigenvector continuation (EC) results with a different number of training points. The training points are shown as red, square markers and are labeled with numbers, representing the order in which they were included in the EC calculation. Results are shown for 3 types of orbital matchings: one ignoring all orbital rotations (leftmost), one ignoring the metric, but including all molecular orbital rotation factors (center), and finally one including all effects of the change in atomic orbital (AO) basis between geometries (rightmost). FIG. 4 . 4Bond stretching potential energy surfaces (PES) for small molecules, comparing FCI and eigenvector continuation (EC). The x-axis is the bond length rescaled with the equilibrium value for the given molecule, and the y-axis is the ground state energy rescaled by the minimum value and shifted by the large distance asymptotic (i.e. the bond energy). Symmetric bonds are shown in the upper row, while asymmetric bonds are found in the lower row. FIG. 5 . 5Potential energy surface (PES) for Cr2 dimer in cc-pvTZ-dk basis. The FCI results correspond to a CASSCF (12o, 12e) calculation. Shown are eigenvector continuation (EC) for two different numbers of training points. ( 0 )FIG. 6 . 06,m with estimate energyẼ (0),m , the residue is defined as Potential energy surface (PES) for hexatriene in the cc-pvDZ basis as a function of the torsion angle φ around the central C-C double bond. φ = 0 • corresponds to the trans configuration, φ = 180 • to cis. The FCI results correspond to a complete active space (6o, 6e) calculation involving only the π orbital manifold. Shown are eigenvector continuation (EC) for three different numbers of training points, always symmetrically chosen around φ = 180 • . FIG. 7 . 7Two measures of the error on the potential energy surfaces (PES) for the bond stretching examples in Fig 4. The exact error with respect to FCI is shown with square markers, and the residue estimate in Eq (6) with round markers. The approximate residue follows the exact error closely. Fock operator in the basis local AOs at a particular λ a . Plays the role of the Hamiltonian in the non-linear, generalized eigenvalue problem of the Hartree-Fock approximation.Overlap integrals of a single local AO basis set at a particular λ a .The (orthogonal) molecular orbitals found by solving the Hartree-Fock generalized eigenvalue equation.The unitary rotation matrix that diagonalizes the Hartree-Fock Hamiltonian, and rotates from AOs to MOs.Full configuration interaction Hamiltonian in the basis of molecular orbitals (MOs) at a particular λ a .The unitary rotation matrix that diagonalizes the FCI Hamiltonian.n-th eigenstate of the FCI Hamiltonian at a particular training point λ i .Hamiltonian matrix element evaluated with the training state vectors H ij ( ) := v i |H |v j .Fock operator F a Local overlap S a Local molecular orbitals (MOs) Local orbital rotation matrix U FCI Hamiltonian H F CI a FCI rotation matrix Q EC training vector |v (n) i EC overlap H ij ( ) EC overlap C ij , the excited states of the effective EC Hamiltonian do not match well any of the exact excited state PESs. Moreover, these ECFIG. 8. Results of eigenvector continuation (EC) for excited state potential energy surfaces (PES) in F2 using the cc-pVDZ basis set. The FCI PES in the (8o, 14e) active space for the first few excited states are presented in grey. Results are shown for three different EC simulations. Left panel: EC with 3 training points using always FCI ground state vectors. Middle panel: EC with 3 training points using always FCI first excited state vectors. Right panel: EC with 6 training points in 3 different geometries, using both the ground state and 1st excited state of the FCI Hamiltonian in each point.excited state PESs come after the first group of closelying FCI excited state PESs, appearing at ∼ 0.8 Ha above the ground state. This suggests that the subspace that captures the ground state PES could be orthogonal to the first bundle of excited states.2 3 4 0.05 0.00 0.05 0.10 0.15 E E 0 (R ) [Ha] 3 EC GS 0.6 0.8 1.0 1.2 2 3 4 F-F Bond Length [Å] 3 EC 1st Exc FCI EC EC points 2 3 4 6 EC GS + 1st Exc ACKNOWLEDGEMENTSCMZ acknowledges financial support from the European Research Council (ERC), under the European Union's Horizon 2020 research and innovation programme, Grant agreement No. 692670 "FIRSTORM". AFK acknowledges financial support from the US National Science Foundation under grant no. NSF DMR-1752713Appendix A: Equilibrium geometries for H 2 CO and trans-hexatrieneThe equilibrium geometry used for H 2 CO in this paper was optimized using the PYSCF interface to PyBerny[37]at the restricted Hartree-Fock level, using the cc-pVDZ basis. The obtained geometry is presented in Tab II. 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{'abstract': 'A typical task for classical and quantum computing in chemistry is finding a potential energy surface (PES) along a reaction coordinate, which involves solving the quantum chemistry problem for many points along the reaction path. Developing algorithms to accomplish this task on quantum computers has been an active area of development, yet finding all the relevant eigenstates along the reaction coordinate remains a difficult problem, and determining PESs is thus a costly proposal. In this paper, we demonstrate the use of a eigenvector continuation -a subspace expansion that uses a few eigenstates as a basis -as a tool for rapidly exploring potential energy surfaces. We apply this to determining the binding PES or torsion PES for several molecules of varying complexity. In all cases, we show that the PES can be captured using relatively few basis states; suggesting that a significant amount of (quantum) computational effort can be saved by making use of already calculated ground states in this manner.', 'arxivid': '2305.00060', 'author': ['Carlos Mejuto-Zaera \nScuola Internazionale Superiore di Studi Avanzati (SISSA)\nTrieste TS\nItaly\n', 'Alexander F Kemper \nDepartment of Physics\nNorth Carolina State University\n27695RaleighNorth CarolinaUSA\n'], 'authoraffiliation': ['Scuola Internazionale Superiore di Studi Avanzati (SISSA)\nTrieste TS\nItaly', 'Department of Physics\nNorth Carolina State University\n27695RaleighNorth CarolinaUSA'], 'corpusid': 258426211, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19575, 'n_tokens_neox': 17451, 'n_words': 12178, 'pdfsha': '924bb4d3679f724efda4ca356d39c74b8797ad9a', 'pdfurls': ['https://export.arxiv.org/pdf/2305.00060v1.pdf'], 'title': ['Quantum Eigenvector Continuation for Chemistry Applications', 'Quantum Eigenvector Continuation for Chemistry Applications'], 'venue': []}

arxiv

Asymptotics for Push on the Complete Graph 24th March, 2020 Rami Daknama Ludwig-Maximilians-Universität München Konstantinos Panagiotou Ludwig-Maximilians-Universität München Simon Reisser Ludwig-Maximilians-Universität München Asymptotics for Push on the Complete Graph 24th March, 2020 We study the popular randomized rumour spreading protocol push. Initially, a node in a graph possesses some information, which is then spread in a round based manner. In each round, each informed node chooses uniformly at random one of its neighbours and passes the information to it. The central quantity to investigate is the runtime, that is, the number of rounds needed until every node has received the information.The push protocol and variations of it have been studied extensively. Here we study the case where the underlying graph is complete with n nodes. Even in this most basic setting, specifying the limiting distribution of the runtime as well as determining related quantities, like its expectation, have remained open problems since the protocol was introduced.In our main result we describe the limiting distribution of the runtime. We show that it does not converge, and that it becomes, after the appropriate normalization, asymptotically periodic both on the log 2 n as well as on the ln n scale. In particular, the limiting distribution converges only if we restrict ourselves to suitable subsequences of N, where simultaneously log 2 n − log 2 n → x and ln n − ln n → y for some fixed x, y ∈ [0, 1). On such subsequences we show that the expected runtime is log 2 n + ln n + h(x, y) + o(1), where h is explicitly given and numerically | sup h − inf h| ≈ 2 · 10 −4 .This "double oscillatory" behaviour has its origin in two key ingredients that were also implicit in previous works: first, an intricate discrete recursive relation that describes how the set of informed nodes grows, and second, a coupon collector problem with batches of size n that takes the lead when the protocol is almost finished. Rounding in the recursion introduces the periodicity on the log 2 n scale -as it is the case in many discrete systems -and rounding in the batched problem is the source of the second periodicity.X n = log 2 n + ln n + o(ln n).arXiv:2003.10762v1 [math.PR] 24 Mar 2020Moreover, they obtained bounds for (very) large deviations of X n from its expectation. In[20], Pittel improved upon the results in[12], in particular, he showed that for any f :The currently most precise result in this context was obtained by Doerr and Künnemann [7], who considered in great detail the distribution of X n . They showed that X n can be stochastically bounded (from both sides) by coupon collector type problems. This gives a lot of control regarding the distribution of X n , and it allowed them to derive, for example, very sharp bounds for tail probabilities. Apart from that, it enabled them to consider related quantities, as for example the expectation of X n . Among other results, their bounds on the distribution of X n imply that log 2 n + ln n − 1.116 ≤ E[X n ] ≤ log 2 n + ln n + 2.765, (1.1) which pins down the expectation up to a constant additive term. Besides on complete graphs, push has been extensively studied on several other graph classes. For example, Erdős-Rényi random graphs [9, 10], random regular graphs and expander graphs[11,18,5]. More general bounds that only depend on some graph parameter have also been derived, e.g. the diameter [9], graph conductance[17,3,4,13]and node expansion[4,22,15,14].Results In order to state our main result we need some definitions first. SetAs we will see later, the function g describes, for a wide range of the parameters, the evolution of the number of uninformed nodes; in particular, if at the beginning of some round there are xn uninformed nodes, then at the end of the same round there will be (roughly) g(x)n uninformed nodes, and after i rounds there will be (roughly) g (i) (x)n uninformed nodes. This fact is not new -at least for bounded i -and has been observed long ago, see for example[20,Lem. 2]. For x ∈ R define the function2) whose actual meaning will become clear later. We will show that the double limit exists, so that this indeed defines a function c : R → R. Moreover, we will show that c is continuous and periodic with period 1, that is, if we write {x} = x − x then c(x) = c({x}), and that (numerically) | sup c − inf c| ≈ 10 −9 , cf.Figure 1. The Gumbel distribution will play a prominent role in our considerations. We say that a real valued random variable G follows a Gum(α) distribution with parameter α ∈ R, G ∼ Gum(α), if for all x ∈ RFinally, let γ denote the Euler-Mascheroni constant. With all these ingredients we can now state our main result, which specifies -see also below -the distribution of the runtime of push on the complete graph.This lemma does not look completely innocent, and it actually has striking consequences. It readily implies the following result, which establishes that the limiting distribution X n is periodic both on the log 2 n and on the ln n scale. In order to formulate it, we need a version of the Gumbel distribution where we restrict ourselves to integers only. More specifically, we say that a random variable G follows a discrete Gumbel distribution, G ∼ dGum(α), if the domain of G is Z and P [G ≤ k] = e −e −k−α , k ∈ Z. Introduction We consider the well-known and well-studied rumour spreading protocol Push. It has applications in replicated databases [6], multicast [1] and blockchain technology [19]. Push operates on graphs and proceeds in rounds as follows. In the beginning, one node has a piece of information. In subsequent rounds each informed node chooses a neighbour independently and uniformly at random and informs it. For a graph G = (V, E) with |V | = n and a node v ∈ V we denote by X(G, v) the (random) number of rounds needed to inform all nodes, where at the beginning of the first round only v knows the information. We call X(G, v) the runtime (on G with start node v). The most basic case, and the one that we study here, is when G is the complete graph K n . Since in that case the initially informed node makes no difference, we will abbreviate X(K n , v) = X n for any starting node v. Related Work There are several works studying the runtime of push on the complete graph. The first paper considering this protocol is by Frieze and Grimmett [12], who showed that with high probability (whp), that is, with probability 1 − o(1) as n → ∞, that Theorem 1.2. Let x, y ∈ [0, 1) and (n i ) i∈N be a strictly increasing sequence of natural numbers, such that log 2 n i − log 2 n i → x and ln n i − ln n i → y as i → ∞. Then in distribution, as i → ∞ X ni − log 2 n i + ln n i → dGum(−x − y − c(x)). Some remarks are in place. First, it is a priori not obvious (at least it was not to us) that subsequences as required in the theorem indeed exist. They do, and the fundamental reason for this is that real numbers can be approximated arbitrarily well by rational numbers; we include a short proof of the existence in the Appendix. Second, it is a priori not clear that x + c(x) is not constant for x ∈ [0, 1). If it was constant, Theorem 1.2 would imply that the limiting distribution of X n is periodic on the ln n scale only. Although we didn't manage to prove that x + c(x) is not constant, we have stong numerical evidence that it indeed is not so. In particular, as we shall also see later, the double limit in the definition of c converges exponentially fast and thus it is not difficult to obtain accurate estimates for it and explicit error bounds. We leave it as an open problem to study the behavior of c more accurately. Our next result addresses moments of X n . Bounds given in previous works, for example in [7], guarantee that X n − log 2 n − ln n and all integer powers of it are uniformly integrable. This allows us to conclude that the expectation and all of its moments also converge. Theorem 1. 3. Let x, y ∈ [0, 1) and (n i ) i∈N be a strictly increasing sequence of natural numbers, such that log 2 n i − log 2 n i → x and ln n i − ln n i → y as i → ∞. Then for all k ∈ N, as i → ∞ E X ni − ( log 2 n i + ln n i ) k → E dGum(−x − y − c(x)) k . For x, y ∈ [0, 1) and a strictly increasing sequence of natural numbers (n i ) i∈N such that {log 2 n i } → x and {ln n i } → y Theorem 1.3 immediately implies that, as i → ∞, E X ni = log 2 n i + ln n i + h(x, y) + o(1), where we abbreviated h(x, y) = E dGum(−x − y − c(x)) − x − y, cf. and using Theorem 1.3, consequently Var[X ni ] = E dGum(−x − y − c(x)) 2 − E dGum(−x − y − c(x)) 2 + o(1). To determine the expectation and variance of the runtime we need to consider various moments of the discrete Gumbel distribution. To this end, let X be an integer valued random variable with finite kth moment, then E X k = ∈Z k P [X = ] = ∈Z k P [X ≤ ] − P [X ≤ − 1] , and therefore, for all α ∈ R and k ∈ N, E dGum(α) k = ∈Z k e −e − −α − e −e − −α+1 . This sum converges exponentially fast, both for → ∞ and → −∞, and thus allows for effective numerical treatment. In summary, improving (1.1), we get for all n ∈ N the numerical bounds log 2 n + ln n + 1.18242 ≤ E[X n ] ≤ log 2 n + ln n + 1.18263, as inf 0≤x,y≤1 h(x, y) = 1.18242 . . . , sup 0≤x,y≤1 h(x, y) = 1.18262 . . . and 1.7277 ≤ Var[X n ] ≤ 1.7289. These numerical bounds are (essentially) best possible, see also Figure 2. Higher moments can be estimated similarly. Let us close this section with a final remark on the function c defined in (1.2), as this might be helpful in future works. This function is defined as the limit of a sequence in two parameters a, b; the main reason for this is its combinatorial origin, which will become apparent in the proofs. However, all that is actually important is that b is large enough, in the sense that the difference b − a → ∞ as a → ∞. So, if we write h for an integer function that diverges to infinity, then we could define d(x) = −x + lim a→∞,a∈N h(a) + ln g (a+h(a)) (1 − 2 −a−x ) . Then c(x) = d(x) (which we state without proof, as we do not need it here), and c can be represented as a limit of an (one-dimesional) sequence. Outline In the next section we give an outline of the proof of our main results. At the beginning of the rumour spreading process push is dominated by an exponential growth of the informed nodes (Lemma 2.2). For the main part, where most nodes get informed, it closely follows a deterministic recursion (Lemma 2.1) and at the end it is described by a coupon collector type problem (Lemma 2.3). Based on these lemmas we give the rigorous proof of our claims in Section 3. The proof to these three important lemmas can also be found there, in Subsections 3.3-3.5. Subsections 3.6 and 3.7 contain all other proofs. Further Notation Unless stated otherwise, all asymptotic behaviour in this paper is for n → ∞. Consider a graph G = (V, E). For t ∈ N 0 (= N ∪ {0}) we denote by I t ⊆ V Proof Overview Let us start the proof of Lemma 1.1 about the distribution of the runtime of push on K n with a simple observation, that is more or less explicit also in previous works. Note that as long as the total number of pushes performed is o( √ n), then whp no node will be informed twice -this is a simple consequence of the famous birthday paradox. That is, whp as long as |I t | = o( √ n), every node in I t will inform a currently uninformed node and thus |I t+1 | = 2|I t |. In particular, whp |I t0 | = 2 t0 , where t 0 := 0.49 · log 2 n . (2.1) Soon after round t 0 things get more complicated. We continue with a definition. Apart from the functions g (i) defined in the previous section, we will also need the following functions. Set f = f (1) : [0, 1] → [0, 1], x → 1 − e −x (1 − x) and f (i) : [0, 1] → [0, 1], f (i) = f • f (i−1) , i ≥ 2. Some elementary properties of f are: f is strictly increasing and concave, and f (b) (x) → 1 as b → ∞ for all x ∈ (0, 1]. Moreover, f (i) (x) = 1 − g (i) (1 − x) for all x ∈ [0, 1] and i ∈ N. It is also not difficult to establish, see also [20] and Lemma 3.5 below, that f captures the behavior of the expected number of informed nodes after one round of the protocol. Moreover, |I t+1 | is typically close to f (|I t |/n)n. Here we will need a more explicit qualitative control of how |I t | behaves, since our aim is to specify the limiting distribution. We show the following statement, which implies that if we start in round t 0 (set T = t 0 in that lemma) then whp for all succeeding rounds t 0 + t the number of informed nodes is close to f (t) (|I t0 /n|)n. Lemma 2.1. Let 0 < c < 0.49 and T ≥ c log 2 n. Then P T t∈N0 |I T +t | − f (t) (|I T |/n) n ≤ n 1−c/4 = 1 − O(n −c 2 /10 ). Thus, the key to understanding |I t | is to understand how f behaves when iterated very many times. Note that when the number of informed nodes is xn for some very small x, then the e −x term in the definition of f can be approximated by 1 − x and therefore f ( x) ≈ 1 − (1 − x) 2 ≈ 2x. This crude estimate suggests that the number of informed nodes doubles every round as long as there are only few informed nodes, and we know already that the doubling is perfect if xn = o( √ n). Our next lemma actually shows that the doubling continues to be almost perfect, as long as the total number of nodes is not close to n. Lemma 2.2. Let a, T ∈ N be such that 2 −a < 0.1 and T ≤ 0.49 · log 2 n . Set t 1 := log 2 n − a. Then 2 t1 − f (t1−T ) 2 T /n n ≤ 2 −2a+1 n. Combining the previous lemmas we have thus established that for any a ∈ N with 2 −a < 0.1 whp (1 − 2 −a+2 ) · 2 t1 ≤ |I t1 | ≤ 2 t1 , t 1 := log 2 n − a. (2.2) Here we can think of a being very large (but fixed) and then the two bounds are very close to each other; in particular, |I t1 | ≈ 2 log 2 n −a and thus I t contains a linear number of nodes. Up to that point we have studied the behaviour of the process up to time t 1 . Next we perform another b steps, where again b is fixed. Applying Lemma 2.1 once more and using that f (b) (x) is increasing and is less than 1 for x < 1 yields with room to spare that whp 1 − n − 1 /6 f (b) (1 − 2 −a+2 )2 t1 /n ≤ n −1 |I t2 | ≤ 1 + n − 1 /6 f (b) 2 t1 /n , t 2 := t 1 + b. (2.3) In essence, this says that if we write x = log 2 n − log 2 n = {log 2 n}, then (we begin getting informal and obtain that) |I t2 | ≈ f (b) 2 t1 /n n = f (b) 2 −a−x n, where t 2 = log 2 n − a + b. In particular, choosing a priori b large enough makes the fraction |I t2 |/n arbitrarily close to 1, that is, almost all nodes except for a tiny fraction are informed. All in all, up to time t 2 we have very fine control of the number of informed nodes, and we also see how the quantity {log 2 n} slowly sneaks in. After time t 2 the behavior changes once more. In this regime there is an interesting connection to the well-known Coupon Collector Problem (CCP), which was also exploited in [7]. In order to formulate the connection, note that the number of pushes that are needed to inform one uninformed node, having N informed nodes, is (in distribution) equal to the number of coupons needed to draw the (N + 1)st distinct coupon. The CCP is very well understood, and it is a classic result that, appropriately normalized, the number of coupons tends to a Gumbel distribution. However, translating the number of required pushes to the number of rounds -the quantity we are interested in -is not straightforward. In particular, the number of pushes in one round depends on the current number of informed nodes. On the other hand, after round t 2 there are n − o(n) informed nodes, so that we may hope to approximate the remaining number of rounds with n −1 times the number of coupons in the CCP. The next lemma establishes the precise bridge between the two problems. There, for two sequences of random variables (X n ) n∈N and (Y n ) n∈N we write X n Y n if there is a function h : N → R + with h = o(1) such that P [X n ≥ x] ≤ P [Y n ≥ x] + h(n) for all n ∈ N, x ∈ R; X n Y n is defined with "≥" instead of "≤". Lemma 2.3. Let G ∼ Gum(γ), b > 2a ∈ N and assume that · n ≤ |I log 2 n −a+b | ≤ u · n for some , u ∈ [0, 1). Then X n − log 2 n + a − b ln n + ln 1 u − 1 + γ+ and X n − log 2 n + a − b ln n + ln 1 − 1 + ln e − e + 1 + γ + G . Note that the previous discussion guarantees that , u in Lemma 2.3 are very close to 1 and very close to each other. So, the term ln( /(e − e + 1)) is very close to 0. We obtain that in distribution X n − log 2 n + a − b ≈ ln n + ln 1 u − 1 + γ + G , where u = f (b) 2 −a−x . and equivalently with x = log 2 n − log 2 n X n ≈ log 2 n + ln n − a + b + ln g (b) (2 −a−x ) − x + γ + G . (2.4) Here we now encounter the mysterious function c from (1.2). The next lemma collects some important properties of it that will turn out to be very helpful. Lemma 2.4. The function c(x) = lim a→∞,a∈N lim b→∞,b∈N −a + b + ln g (b) (1 − 2 −a−x ) − x is well-defined, continuous and periodic with period 1. With all these facts at hand, the proof of Lemma 1.1 is completed by considering the random variable on the right-hand side of (2.4); in particular, the dependence on y = ln n − ln n arises naturally. The complete details of the proof, which is based on Lemmas 2.1-2.3 and follows the strategy outlined here can be found in Section 3 (together with the proofs of the lemmas). As described in the introduction, apart from the limiting distribution we are interested in the asymptotic expectation of the runtime. A key ingredient towards the proof of Theorem 1.3 is uniform integrability, which can be shown by using the distributional bounds from [7]. Uniform integrability is a sufficient condition that convergence in distribution also implies convergence of the means. Lemma 2.5 (uniform integrability). Let k ∈ N and set Y n := X n − log 2 n − ln n . Then Y k n is uniformly integrable, that is lim N →∞ sup n∈N E |Y n | k 1 |Y n | k > N = 0. Proof of the Main Result In this section we complete the proof of Lemma 1.1 outlined in Section 2. Afterwards we give the (short) proofs for Theorems 1.2 and 1.3. Proof of Lemma 1.1 As the outline was indeed rigorous until (2.3) we take the proof up from there. Choose the quantities a, b ∈ N such that 2a < b and recall that t 1 = log 2 n − a. Set furthermore for brevity = 1 − n − 1 /6 f (b) (1 − 2 −a+2 )2 t1 /n and u = 1 + n − 1 /6 f (b) 2 t1 /n . Then (2.3) states that, for t 2 = log 2 n − a + b, ≤ n −1 |I t2 | ≤ u, and Lemma 2.3 yields, for Y n = X n − log 2 n + a − b, that Y n ln n + ln 1 − 1 + ln e − e + 1 + γ + G and Y n ln n + ln 1 u − 1 + γ + G . The next lemma establishes that both , u tend to 1 as a gets large, and moreover that the difference ln (1/ − 1) − ln (1/u − 1) can be made arbitrarily small. Its proof can be found in Subsection 3.7. Furthermore, lim a→∞ sup n∈N | ln(1 − ) − ln(1 − u)| = 0. Thus, as n → ∞, ln(1 − u) = ln 1 − f (b) 2 t1 /n + o(1) = ln g (b) 1 − 2 −a−{log 2 n} + o(1). Let ε > 0. Lemma 3.1 readily implies that there are a 0 , n 0 ∈ N such that for all a > a 0 and n > n 0 , Y n ln n + ln g (b) 1 − 2 −a−{log 2 n} + γ + G − ε and similarly also Y n ln n + ln g (b) 1 − 2 −a−{log 2 n} + γ + G + ε . Lemma 2.4 guarantees that there is an a 1 ≥ a 0 such that for all a ≥ a 1 ln g (b) 1 − 2 −a−{log 2 n} − a + b − (c({log 2 n}) + {log 2 n}) ≤ ε. Thus for all a > a 1 and n > n 0 X n log 2 n + ln n + c({log 2 n}) + γ + G − 2ε , as well as X n log 2 n + ln n + c({log 2 n}) + γ + G + 2ε . Thus we are left with getting rid of the ε terms in the previous equations. The following lemma accomplishes exactly that and therefore implies the claim of Lemma 1.1. Its proof can be found in Subsection 3.7. Lemma 3.2. Let h : N → R + and G ∼ Gum(γ). Then ∀ε > 0 : X n h(n) + G + ε =⇒ X n h(n) + G . The respective statement also holds for " ". Proof of Theorems 1.2 and 1.3 Proof of Theorem 1.2. Recall that {z} = z − z , z ∈ R. Let (n i ) i∈N be a strictly increasing subsequence of N such that {log 2 n i } → x and {ln n i } → y. Substituting k = log 2 n i + ln n i + 1 + t for any t ∈ Z we get that P G + log 2 n i + ln n i + γ + c({log 2 n i }) ≥ k = P G + log 2 n i + ln n i + γ + c({log 2 n i }) ≥ log 2 n i + ln n i + 1 + t = P G + {log 2 n i } + {ln n i } + γ + c({log 2 n i }) > t = P G + {log 2 n i } + {ln n i } + γ + c({log 2 n i }) > t . Thus using Lemma 1.1, Lemma 2.4 and Lemma 3.2 we get that, as i → ∞, sup t∈Z P X ni ≥ log 2 n i + ln n i + 1 + t − P G + x + y + γ + c(x) > t = o(1). Using the distribution function of G ∼ Gum(γ) we get P X ni ≥ log 2 n i + ln n i + 1 + t i→∞ −→ 1 − exp − exp (−t + x + y + c(x)) , that is, P X ni ≤ log 2 n i + ln n i + t i→∞ −→ P (dGum(−x − y − c(x)) ≤ t). Next we prove Theorem 1.3. Proof of Theorem 1.3. Lemma 2.5 states that X n − log 2 n − ln n k is uniformly integrable and Theorem 1. 2 established its convergence in distribution to dGum(−x − y − c(x)) k . Together this implies E X n − log 2 n − ln n k → E dGum(−x − y − c(x)) k . Proof of Lemma 2.1 The number of informed nodes, |I t |, fulfils a so-called self-bounding property, for reference see [2]. One striking consequence thereof is the following bound. Lemma 3.3 ([5] ). For any t ∈ N, Var |I t+1 | | I t ≤ E |I t+1 | | I t . This bound on the variance and Chebychev's inequality ensure that the number of informed nodes is highly concentrated around its expectation as soon as enough nodes are informed. Moreover, even stronger concentration results are possible, as self-bounding functions admit exponential concentration inequalities, see e.g. [2]. Here, Chebychev is sufficient for our application. Lemma 3.4. Let 0 < c ≤ 1, let t 0 ∈ N and assume that |I t0 | ≥ n c . For t ∈ N and ε > 0 let C t denote the event that |I t+1 | − E t [|I t+1 |] ≤ (E t [|I t+1 |]) 1/2+ε . Then P t0   t≥t0 C t   = 1 − O n −cε . Proof. From [7, Corollary 3.2] it is known that for any r > 0 P X n ≥ log 2 n + ln n + 2.188 + r ≤ 2e −r . Thus it suffices (with lots of room to spare) to show P t0   t0≤t≤log 2 n C t   = O n −3cε/2 . (3.1) By using Chebychev's inequality and Lemma 3.3, P t C t = P t |I t+1 | − E t [|I t+1 |] > E t [|I t+1 |] 1/2+ε ≤ Var[|I t+1 |] E t [|I t+1 |] 1+2ε ≤ E t [|I t+1 |] −2ε . Since E t [|I t+1 |] ≥ |I t+1 | ≥ |I t0 | the claim follows from (3.1) and the union bound. Lemma 3.5 establishes a connection between the expected value of |I t+1 | and our previously defined function f , see below Equation 2.1. This has also been observed (though not so precise) in [20] and we include a quick proof for completeness. Lemma 3.5. Let t ∈ N and n ≥ 3. Then f (|I t |/n)n ≤ E t |I t+1 |] ≤ f (|I t |/n)n + 5. Proof. Each uninformed node u ∈ U t remains uninformed if all |I t | informed nodes do not push to u. Since all these events are independent, we obtain that the probability that u remains uninformed is (1 − 1/(n − 1)) |It| . Thus by linearity of expectation E t [|I t+1 |] = |I t | + n − |I t | 1 − 1 − 1 n − 1 |It| = n − n − |I t | 1 − 1 n − 1 |It| . For a lower bound we use 1 − x ≤ e −x and get E t [|I t+1 |] ≥ n − n − |I t | e −|It|/(n−1) ≥ n − n − |I t | e −|It|/n = f (|I t |/n)n. For an upper bound we use 1 − x ≥ e −x−x 2 for all x ≤ 1/2 E t [|I t+1 |] ≤ n − n − |I t | e −|It|/(n−1)−|It|/(n−1) 2 ≤ n − n − |I t | e −|It|/n exp − 2|I t | (n − 1) 2 and again using 1 − x ≤ e −x E t [|I t+1 |] ≤ n − n − |I t | e −|It|/n 1 − 2|I t | (n − 1) 2 ≤ f (|I t |/n)n + 5. Lemma 3.6 is an auxiliary result that we use in the proof of Lemma 2.1. It shows that f is concave and has decreasing derivative on the interval [0, 1], the stated property is a direct consequence. and therefore, as a direct consequence of the mean value theorem, we have Lemma 3.6. Let 0 < x 1 ≤ x 2 < 1. Then |f (x 2 ) − f (x 1 )| ≤ (2 − x 1 )e −x1 (x 2 − x 1 ). Proof. It is f (x) = (2−x)e −|f (x 2 ) − f (x 1 )| ≤ (x 2 − x 1 ) max x∈[x1,x2] f (x) = (2 − x 1 )e −x1 (x 2 − x 1 ). We state a simple corollary for later reference. Having these lemmas as ingredients we can prove the main result of this subsection. Lemma 3.5 shows that the expectation of |I t+1 | is given by f (|I t |/n)n and Lemma 3.4 shows that |I t+1 | is closely concentrated around its expectation in (nearly) all rounds. To then prove that |I t+τ | is close to f (τ ) (|I t |/)n for any τ ∈ N we need to make sure that the errors in the concentration and the approximation of the expectation are not blown up by repeated applications of f . We will show that f can indeed increase the error in each step by a factor that can be as large as √ 2, but luckily this only happens when |I t+τ | = o(n) and thus the accumulated error will remain small (as |I t | nearly doubles in this regime). Proof of Lemma 2.1. Let 0 < ε < c/10, and assume, with foresight, that n ≥ n 0 , where n 0 satisfies the inequalities √ 2 + 10n −ε 0 < √ 2 + ε and n c 0 ≥ 25. As T ≥ c log 2 n and because of (2.1), that is, |I t | = 2 t for all t ≤ 0.49 log 2 n , we have |I T | ≥ n c . Consequently we can apply Lemma 3.4 and thus get with probability 1 − O(n −cε ) |I t+1 | − E t [|I t+1 |] ≤ E t [|I t+1 |] 1/2+ε , for all t ≥ T . (3.2) For the rest of this proof we assume that (3.2) holds. Set α T +t = f (t) (|I T |/n), t ∈ N 0 . We will first argue that |I t | − α t n ≤ α 1/2+ε t n 1/2+2ε √ 2 + ε t−T =: d t . (3.3) for all t ≥ T such that d t ≤ n 1−ε . Note that this is obviously true for t = T . For the induction step we argue that |I t+1 | − α t+1 n ≤ α 1/2+ε t+1 n 1/2+2ε √ 2 + ε t+1−T = d t+1 . (3.4) To see this, we use Lemma 3.5, (3.2) and the fact that |I t+1 | ≤ 2|I t | (in this order) to obtain the bound |I t+1 | − f (|I t |/n)n ≤ |I t+1 | − E t [|I t+1 |] + 5 ≤ (2|I t |) 1/2+ε + 5. Then we apply Lemma 3.6 to estimate the difference of f (|I t |/n) and α t+1 = f (α t ), and infer from (3.3), using e x ≤ 1 + 2x for all 0 ≤ x ≤ 1, that f (|I t |/n)n − α t+1 n ≤ |I t | − α t n 2 − min{α t , |I t |/n} e − min{αt,|It|/n} ≤ d t 2 − α t + d t /n e − min{αt,|It|/n} ≤ d t 2 − α t e −αt+dt/n + d 2 t /n ≤ d t (2 − α t )e −αt + 5d 2 t /n. All in all we have argued that for all t such that d t ≤ n 1−ε |I t+1 | − α t+1 n ≤ |I t+1 | − f (|I t |/n)n + f (|I t |/n)n − α t+1 n ≤ (2|I t |) 1/2+ε + 5 + d t (2 − α t )e −αt + 5d 2 t /n ≤ 2(α t n + d t ) 1/2+ε + 5 + d t (2 − α t )e −αt + 5d t n −ε . Our assumptions on ε and n imply that d 1/2+ε t ≤ d t n −ε . Moreover, α T n ≥ n c ≥ 25, and thus |I t+1 | − α t+1 n ≤ 3(α t n) 1/2+ε + d t (2 − α t )e −αt + 7d t n −ε ≤ d t (2 − α t )e −αt + 10d t n −ε . (3.5) To understand (3.5) consider the auxiliary function H(x) = f (x) x − f (x) √ 2 = 1 − (1 − x)e −x x − (2 − x)e −x √ 2 . As (1 − x)e −x = 1 − 2x + O(x 2 ) as x → 0 we have that lim x→0 (1 − (1 − x)e −H = 1 2 2(1 − x)e −x x 2 −1/2 + (3 − x)e −x √ 2 ≥ 0 for x ≤ 1. Therefore H(x) ≥ 0 for all 0 ≤ x ≤ 1 and consequently, using α t+1 > α t , α t+1 α t 1/2+ε ≥ α t+1 α t 1/2 ≥ (2 − α t )e −αt √ 2 . Since d t = α 1/2+ε t n 1/2+2ε √ 2 + ε t−T , applying the previous bound to (3.5) implies (3.4) for all n ≥ n 0 , that is, all n such that √ 2 + 10n −ε < √ 2 + ε. This completes the induction step and the proof of (3.3) is completed. Actually our arguments yield also the following statement, which is stronger than (3.3) when there are "many" informed nodes. In particular, for all t ∈ N such that (2 − α t )e −α t < 1 − ε Equation (3.5) also yields for all n ≥ n 0 |I T +t | − α T +t n ≤ d t ⇒ |I T +t +1 | − α T +t +1 n ≤ d t , meaning that the absolute error does not increase any more after round t . (Actually the error decreases by a factor of at least ε after that round, but we do not need this.) To complete the proof we show that we can choose t such that d t ≤ n 1−c/4 and (2 − α t )e −α t < 1 − ε. To this end, consider T = log 2 n − 4 − T and applying Lemma 2.2 to α T yields α T +T = f (T ) (|I T |/n) ≥ f ( log 2 n −4−T ) 2 T /n ≥ 2 −4 1 − 2 −8+1 and furthermore, a simple computation yields that α T +T +5 ≥ 3/4. Thus (2 − α T +T +5 )e −α T +T +5 ≤ (2 − 3/4)e −3/4 < 1 − ε and we set t := T + 5. Moreover, d t ≤ n 1/2+2ε √ 2 + ε t ≤ (2 + ε)n 1/2+2ε+(1−c) log 2 (2+ε)/2 . Note that log 2 (2 + ε) ≤ 1 + ε. Plugging this into the exponent yields that if ε < c/10 and n is large enough then d t ≤ n 1−c/4 (≤ n 1−ε ), as claimed. Proof of Lemma 2.2 We begin with showing the basic inequality 2x(1 − x) ≤ f (x) ≤ 2x. (3.6) To see this, note that e −x ≤ 1 − x + x 2 /2 for x ∈ [0, 1] and so f (x) = 1 − e −x (1 − x) ≥ 1 − 1 − x + x 2 2 (1 − x) ≥ x 2 − 3 2 x ≥ 2x − 2x 2 , which establishes the first inequality in (3.6). The other inequality follows directly from the simple bound e −x ≥ 1 − x. Let us write z 0 = 2 t0 /n and z i = f (z i−1 ) = f (i) (z 0 ); we want to bound z t1−t0 , where t 1 = log 2 n − a and t 0 ≤ 0.49 log 2 n . Clearly z i ≤ 2 i z 0 , which shows the upper bound in Lemma 2.2. Using (3.6) we obtain by induction z i ≥ 2 i z 0 · i−1 j=0 (1 − 2 j z 0 ), i ∈ N. Further, using the bound 1 − x ≥ e −x−x 2 /2(1−x) , valid for any x ∈ [0, 1) we obtain z i ≥ 2 i z 0 · exp    −z 0 0≤j<i 2 j − z 2 0 0≤j<i 4 j 2(1 − 2 j z 0 )    Note that our assumptions guarantee that 2 t1−t0 z 0 = 2 −a < 0.1, and so for any 1 ≤ i ≤ t 1 − t 0 z i ≥ 2 i z 0 · exp −2 i z 0 − (2 i z 0 ) 2 ≥ 2 i z 0 · (1 − 2 −a − 2 −2a ). Finally note that 1 − y − y 2 ≥ 1 − 2y for any y ∈ [0, 1], and so the last term is bounded by 2 i z 0 · (1 − 2 −2a+1 ), which coincides with the lower bound claimed in Lemma 2.2. 2 i x 1 − 2 i x − 2 2i x 2 ≤ f (i) (x) ≤ 2 i x. Proof of Lemma 2.3 A main tool in the fortcoming proof is the following result, which states that a sum of normalized independent geometric random variables converges to a Gumbel distributed random variable. Theorem 3.9 ([8]). Let T 1 , . . . , T n−1 be independent random variables such that T i ∼ Geo((n − i)/(n − 1)) for 1 ≤ i < n. Then, in distribution n −1 1≤i<n T i − E[T i ] → Gum(γ). Unfortunately we can not apply directly Theorem 3.9 to our setting, as we will have to deal with a sum of independent geometric random variables that are not normalized with the 'correct' factor n −1 . However, the next well-known statement assures that if the error is small enough we will still converge to the same limiting distribution Theorem 3.10 (Slutsky's Theorem, see, e.g., [23, p. 19]). Let (X n ) n∈N , (Y n ) n∈N and (Z n ) n∈N be sequences of real-valued random variables. Suppose that X n → X in distribution and that there are constants a, b ∈ R such that Y n → a and Z n → b in probability. Then Y n X n + Z n → aX + b in distribution. We now show a more general version of Theorem 3.9 that is applicable to our setting. Lemma 3.11. Let T 1 , . . . , T n−1 be independent random variables such that T i ∼ Geo((n − i)/(n − 1)) for 1 ≤ i < n. Let furthermore ε > 0 and s : N → [1, n] be a function such that s(n − i) ≥ 1 − o(1) (n − c · i) for any positive integer i < εn. Then, in distribution (1−ε)n≤i<n T i − E[T i ] s(i) → Gum(γ). Proof. Let D i = T i − E[T i ] be the centralised version of T i . Then (1−ε)n≤i<n D i s(i) = 1≤i<n D i n − 1≤i<(1−ε)n D i n +   (1−ε)n≤i<n D i s(i) − (1−ε)n≤i<n D i n   . A direct applicaition of Theorem 3.9 guarantees that the first sum converges to Gum(γ) in distribution. To complete the proof it is sufficient to argue that in probability 1≤i<(1−ε)n D i n → 0 and (1−ε)n≤i<n D i s(i) − D i n → 0, (3.7) from which the claim in the lemma follows immediately from Theorem 3.10. Since the D i 's are centralised E   1≤i<(1−ε)n D i n   = 0, and using that Var [T i ] = (n − 1)(i − 1) /(n − i) 2 for all i < n Var   1≤i<(1−ε)n D i n   = 1≤i<(1−ε)n Var[T i ] n 2 = 1≤i<(1−ε)n 1 n 2 (n − 1)(i − 1) (n − i) 2 ≤ 1≤i<(1−ε)n 1 (εn) 2 = o(1). Thus Chebychev's inequality directly implies that 1≤i<(1−ε)n D i n → 0 in probability. It remains to treat the second term in (3.7). We compute the variance as before Var   (1−ε)n≤i<n D i s(i) − D i n   = (1−ε)n≤i<n 1 s(i) − 1 n 2 (n − 1)(i − 1) (n − i) 2 ≤ 1≤i≤εn 1 s(n − i) − 1 n 2 n 2 i 2 . However, this is also o(1), as s(n − i) ≥ 1 − o(1) (n − c · i) for all integers i ≤ εn by assumption, and therefore 0 ≤ 1 s(i) − 1 n ≤ 1 (1 + o(1))(n − c · i) − 1 n = (1 + o(1)) c · i + o(n) n 2 , i ≤ εn. In summary we have shown that Var   (1−ε)n≤i<n D i s(i) − D i n   = o(1) and clearly E   (1−ε)n≤i<n D i s(i) − D i n   = 0. Thus Chebychev's inequality implies also the second statement in (3.7) and the proof is complete. A further ingredient that we shall exploit is the following fact. If a sequence of random variables X n → X in distribution with distribution functions F n → F and if F is continuous everywhere, then the convergence of F n to F is even uniform. . For each n ∈ N let X n be a real-valued random variable with distribution function F n . Assume that X n → X in distribution. If X has continuous distribution function F , then lim n→∞ sup x∈R |F n (x) − F (x)| = 0. We need one more auxiliary lemma that gives an upper bound on the informed nodes when going one round backwards in order to later convert the number of Coupons into the number of rounds that are needed to finish the protocol. Appropriately, Lemma 3.4 assures that in all rounds the number of informed nodes is tightly concentrated around its expectation, which in turn is described by f , thus applying f −1 will give a good bound. Lemma 3.13. Let t 0 ∈ N and 0 < ε < 1/6. Let C t be the event that |I t+1 | − E t [|I t+1 |] ≤ (E t [|I t+1 |]) 1/2+ε , as given in Lemma 3.4. Then for n large enough the event t≥t0 C t implies for all t ≥ t 0 |I t | ≥ 1 − n −1/3 · e · |I t+1 | − (1 − 1/e)n . Proof. Lemma 3.5 and C t together give that |I t+1 | ≤ E t [|I t+1 |]| + (E t [|I t+1 |]) 1/2+ε = f (|I t |/n)n + o n 2/3 . Using the definition of f (x) = 1 − (1 − x)e −x and that |I t | ≤ n for all t we get that |I t+1 | ≤ n − e −|It|/n (n − |I t |) + o n 2/3 ≤ (1 − 1/e)n + |I t |/e + o n 2/3 . Rearranging yields the claimed statement. Let us briefly outline the proof of Lemma 2.3. We have already shown bounds for the number of informed nodes after log 2 n − a + b rounds in (2.3). Starting from these bounds we will use the Coupon Collector Problem to compute the number of pushes that are needed to inform all remaining uninformed nodes. This will yield sums of independent geometric random variables (one summand for each uninformed node). Using Lemma 3.13 we will translate these numbers of pushes into numbers of rounds, which results in an almost correctly normalised sum of geometric random variables that Lemma 3.11 assures to converge to a Gumbel distribution. We will end up with upper and lower bounds to the distribution function of push. Proof of Lemma 2.3. In this proof we will establish a connection between the Coupon Collector Problem and the behavior of push. Let v ∈ V be the node that was initially informed. Instead of every informed node choosing one of its neighbours uniformly at random, we now assume that it samples one node in V \ {v} uniformly at random. This defines an equivalent model, as for all u ∈ V the probability to choose any specific node in V \ {u, v} does not change (it equals 1/(n − 1) in both models) and choosing u or v makes no difference for the distribution of the set of informed nodes. Thus push is the same as drawing coupons out of a pool of n − 1 different coupons, but doing so in batches with size being the number of distinct coupons already collected plus one, the 'plus one' representing the initially informed node v. It is widely known and easy to see that assuming 1 ≤ i ≤ n − 1 coupons (including v) have already been collected, then T i ∼ Geo n − i n − 1 , 1 ≤ i ≤ n − 1. (3.8) describes the number of coupons one needs to draw in order to draw the next, (i+1)st new, distinct coupon. Thus in order to collect all n coupons one needs to draw n−1 i=1 T i coupons, where the summands are independent random variables. However, we are not particularly interested in the total number of coupons drawn, but in the number of batches needed. If a batch has size s ≤ n − 1, then this batch is worth s coupons, or vice versa, each coupon drawn in this batch is worth 1/s batches. Thus we need to estimate the size of the batch that contained all coupons that were needed to draw the (i + 1)st distinct coupon, or if these coupons were contained in multiple batches, then we bound all those involved -we call these batches the batches that are linked to i + 1. Let L i be the smallest and U i the largest size of a batch linked to the (i + 1)st coupon. Then certainly U i ≤ i, as at the time that the (i + 1)st distinct coupon gets collected there are obviously at most i distinct collected coupons. Using our assumption · n ≤ |I log 2 n −a+b | ≤ u · n we thus obtain     n−1 i= un T i U i     ≤ X n − log 2 n − a + b ≤     n−1 i= n T i L i     . (3.9) Abbreviating Y n = X n − ( log 2 n − a + b) and recalling that U i ≤ i yields Y n ≥     n−1 i= un T i U i     =     n−1 i= un T i i     . As the T i are independent and geometrically distributed, we can apply Lemma 3.11 and for G ∼ Gum(γ) we obtain with Theorem 3.12 sup k∈Z P n−1 i= un T i − E[T i ] i ≥ k − P [G ≥ k] = o(1) and therefore     n−1 i= un T i i     =     n−1 i= un E[T i ] i + n−1 i= un T i − E[T i ] i         n−1 i= un 1 i(1 − i/n) + G     . The partial fraction decomposition i(1 − i/n) −1 = (n − i) −1 + i −1 allows us to simplify     n−1 i= un 1 i(1 − i/n) + G     =     n−1 i= un 1 n − i + n−1 i= un 1 i + G     =     n− un i=1 1 i + n−1 i= un 1 i + G     . Expressing these partial harmonic sums using the asymptotic expansion for the nth harmonic number 1≤k≤n k −1 = H n = ln n + γ + O(1/n) (3.10) we get, using Lemma 3.2, Y n ln(n − un) + γ + ln n + γ − ln(un) − γ + G + O(1/n) = ln n + ln n(1 − u) un + γ + G + O(1/n) ln n + ln(1/u − 1) + γ + G . We now look at the upper bound in (3.9). For all n ≤ i ≤ n − 1 we specify an appropriate bound for L i . To obtain it, assume that t is the round in which the ith vertex was informed. Then all batches that are linked to the (i + 1)st coupon have size at least |I t |, i.e. L i ≥ |I t |, as the (i + 1)st distinct coupon is drawn after the ith distinct coupon, i.e., it cannot be drawn in any round t < t. However, we do not know |I t |, but we certainly can say that |I t+1 | ≥ i. So, Lemma 3.13, guarantees that whp |I t | ≥ 1 − n −1/3 · e · i − (1 − 1/e)n for all i ∈ { n , . . . , n − 1}. (Note that t = t(i) in that statement.) In particular, whp L i ≥ |I t | ≥ 1 − n −1/3 · (n − e · (n − i)) for all i ∈ { n , . . . , n − 1}. Let C be the event that Lemma 3.13 conditions on, that is that |I t | (for all t ∈ N) is closely concentrated around its expectation. Let k ∈ N and B =        n−1 i= n T i L i     ≥ k    . Then P (B) = P (C ∩ B) + o(1) and as {C ∩ B ≥ k} ⇒        n−1 i= n T i (1 − n −1/3 )(n − e · (n − i))     ≥ k    we get, recalling Y n = X n − ( log 2 n − a + b), that Y n ≤     n−1 i= n T i L i         n−1 i= n T i (1 − n −1/3 )(n − e · (n − i))     . Again applying Lemma 3.11 and Theorem 3.12, for G ∼ Gum(γ) and c = e, we obtain Y n     n−1 i= n E[T i ] (1 − n −1/3 )(n − e · (n − i)) + n−1 i= n T i − E[T i ] (1 − n −1/3 )(n − e · (n − i))         1 + O(n −1/3 ) n−1 i= n 1 (n − e · (n − i))(1 − i/n) + G     . Let c = 1 − 1/e. Using that (n − e · (n − i))(1 − i/n) −1 = (n − i) −1 + (i − cn) −1 gives Y n     1 + O(n −1/3 )   n−1 i= n 1 n − i + n−1 i= n 1 i − cn   + G     . Using index shifts, the asymptotic expansion for the harmonic number (3.10) and Lemma 3.2 yields Y n     1 + O(n −1/3 )   n− n i=1 1 i + n−1− cn i= n − cn 1 i   + G + o(1)     ln n + ln(1/ − 1) − ln(1/ ) + γ + ln 1 − c − c + G . Proof of Lemma 2.4 In this subsection we investigate the double limit lim a→∞,a∈N lim b→∞,b∈N −a + b + ln g (b) (1 − 2 −a−x ) − x where g(x) = xe x−1 . We will show that this limit exists and defines a continuous function c(x). It being periodic with period 1 is an immediate consequence of substituting a → a + 1 in the limit. A similar proof would also yield that c is continuously differentiable, but we only need continuity in the proof of our main theorem. Before we actually prove Lemma 2.4 let us state two auxiliary statements first. In Definition 3.14, we quantify "exponentially fast convergence" and in Lemma 3.15 we state some simple properties. Definition 3.14 (Exponentially fast convergence). Let (a n ) n∈N be a real-valued sequence and let c ∈ (0, 1). If there is an n 0 ∈ N such that for all n ≥ n 0 we have |a n+1 | < c|a n |, then we say that a n converges exponentially fast to zero at rate c with start number n 0 . Lemma 3.15. a) Let c ∈ (0, 1) and let (a n ) n∈N be a real-valued sequence that converges exponentially fast to zero at rate c. Then n≥1 a n converges absolutely. b) Let c ∈ (0, 1), n 0 ∈ N and let (h n ) n∈N denote a sequence of functions with h n : [0, 1] → R such that for any x ∈ [0, 1] the sequence (h n (x)) n∈N converges exponentially fast to zero at rate c with start number at most n 0 . h(x) − n j=1 h j (x) = 0. Proof. a) is elementary. We prove b). Let ε > 0. We show that there is an n 1 ∈ N such that for all n ≥ n 1 and for all x ∈ [0, 1] it holds n j=1 h j (x) − h(x) < ε. For n ≥ n 0 it is n j=1 h j (x) − h(x) = ∞ j=n+1 h n (x) ≤ ∞ j=n+1 |h n (x)| ≤ |a n0 | ∞ j=n+1 c j = |a n0 | c n+1 1 − c which implies that an n 1 as required exists. Proof of Lemma 2.4. We show first, that for a fixed and any x ∈ [0, 1] the limit lim b→∞,b∈N b + ln g (b) 1 − 2 −a−x exists and the convergence is uniform. Inductively we get b + ln g (b) 1 − 2 −a−x = b + ln g (b−1) 1 − 2 −a−x + g (b−1) 1 − 2 −a−x − 1 = 1 + ln 1 − 2 −a−x − 2 −a−x + b−1 j=1 g (j) 1 − 2 −a−x (3.11) which, according to Lemma 3.15 a), converges for b → ∞ because g (j) (1 − 2 −a−x ) converges exponentially fast to zero at rate at most exp(−2 −a−1 ) < 1 and start number 1 for j → ∞ in the sense of Definition 3.14. For x ∈ [0, 1], according to Lemma 3.15 b), the convergence is even uniform with respect to x. By the Uniform Limit Theorem we thus showed that γ a (x) = −a + j≥1 g (j) (1 − 2 −a−x ) is continuous for a ∈ N. (3.12) To complete the proof we show that the sequence of continuous functions (γ a ) a∈N converges uniformly. But first we make an observation. Let a > a ∈ N and x ∈ [0, 1], then, using g(x) = 1 − f (1 − x), γ a (x) = −a + j≥1 g (j) (1 − 2 −a −x ) = −a + a −a j=1 g (j) 1 − 2 −a −x + ∞ j≥1 g (j) g (a −a) 1 − 2 −a −x = −a − a −a j=1 f (j) 2 −a −x + j≥1 g (j) 1 − f (a −a) 2 −a −x . Furthermore, we can bound the repeated application of f using Corollary 3.8 and therefore 0 ≤ a −a j=1 f (j) 2 −a −x ≤ 2 −a−x+1 and 2 −a−x 1 − 2 −a−x − 2 −2a−2x ≤ f (a −a) 2 −a −x ≤ 2 −a−x . Thus there is x ∈ [0, 1] such that |x − x | ≤ 2 −a and γ a (x) = γ a (x ) + O 2 −a . With this at hand we show uniform convergence of (γ a ) a∈N . In particular, for any 0 < ε < 1/8 we will show that there is some N ∈ N such that sup x∈[0,1] |γ a (x) − γ a (x)| ≤ ε for all a > a > N . To achieve this we use our previous observation and obtain that sup x∈[0,1] |γ a (x) − γ a (x)| ≤ sup x∈[0,1],|x−x |≤2 −a |γ a (x) − γ a (x )| + O 2 −a = sup x∈[0,1],|x−x |≤2 −a j≥1 g (j) 1 − 2 −a−x − g (j) 1 − 2 −a−x + O 2 −a . We bound this sum by splitting it into three parts. There is M 1 ∈ N such that for any a > M 1 there is N 1 ∈ N (N 1 depending on a and ε) such that ε ≤ f (N1) 2 −a−1 ≤ f (N1+1) 2 −a ≤ 8ε. (3.13) That is, N 1 is the number of iterations such that f (N1) 2 −a ≈ ε, in particular N 1 ≤ a, as f (a) 2 −a−1 ≥ 1/8 by Corollary 3.8 and the fact that f is increasing. Furthermore, using again that g (j) (1 − 2 −a−x ) converges exponentially fast to zero with rate at most exp(−2 −a−1 ) < 1 for j → ∞, there is c ∈ N depending only on ε such that for N 2 := N 1 + c 0 ≤ sup x∈[0,1] j≥N2 g (j) 1 − 2 −a−x ≤ ε for all a > M 1 . (3.14) Then, abbreviating h (j) = g (j) 1 − 2 −a−x − g (j) 1 − 2 −a−x , we can write ∞ j=1 g (j) 1 − 2 −a−x − g (j) 1 − 2 −a−x = N1 j=1 h (j) + N2 j=N1+1 h (j) + j>N2 h (j) . (3.15) In the rest of the proof estimate these sums individually, starting with the first one. Again using (3.11) and f (x) = 1 − g(1 − x) we have as a → ∞ N1 j=1 g (j) 1 − 2 −a−x − g (j) 1 − 2 −a−x = ln g (N1) 1 − 2 −a−x − ln g (N1) 1 − 2 −a−x + O(2 −a ) = ln 1 − f (N1) 2 −a−x − ln 1 − f (N1) 2 −a−x + O(2 −ah (j) = N2 j=N1+1 f (j) 2 −a−x − f (j) 2 −a−x ≤ 2 −a−x − 2 −a−x N2 j=N1+1 2 j . Thus, as N 1 ≤ a and N 2 = N 1 + c, where c depends on ε only, and our assumption |x − x | ≤ 2 −a there is M 2 ≥ M 1 such that N2 j=N1+1 h (j) ≤ 2 2 −a − 1 · 2 −a · N2 j=N1+1 2 j ≤ 2 2 −a − 1 · 2 c+1 ≤ ε for all a > M 2 . (3.17) In summary, (3.15) gives sup x∈[0,1] γ a (x) − γ a (x) ≤ sup x∈[0,1],|x−x |≤2 −a N1 j=1 h (j) + N2 j=N1+1 h (j) + j>N2 h (j) + O 2 −a . and for a > M 2 > M 1 , applying (3.16), (3.17) and (3.14) yields the uniform convergence of (γ a ) a∈N . Other Proofs In this subsection we complete the rigorous treatment of our main theorems and give the last two remaining proofs. First we prove Lemma 2.5, which states that X n − log 2 n − ln n is uniformly integrable. P X n ≤ r ≤ P log 2 n − 1 + C n ( n/2 ) n ≤ r , where C n ( n/2 ) is the number of rounds a coupon collector needs to draw the last n/2 out of n coupons. These two bounds together with common deviation bounds for the coupon collector problem imply, see e.g. [8], that P [Y n / ∈ log 2 n + ln n ± (r + 5)] ≤ 4e −r . Using this inequality we get that for any N ∈ N E |Y n | k 1 |Y n | k > N ≤ t≥ k √ N (t + 5) k 4e −t , which implies the claim. We close the section with the proof of Lemmas 3.1 and 3.2. Proof of Lemma 3.1. First we observe that the (1 − n −1/6 ) error term in the definition of , u is negligible as is factors out as a small additional term. Thus it suffices to consider = f (b) (L) and u = f (b) (U ) where L = 1 − 2 −a+2 2 −a−x and U = 2 −a−x for some x ∈ (0, 1]. We assume that a ≥ 3. We start by showing an analogue to Corollary 3.7 but concerning g. For all r ≥ s ∈ [0, 1], using 1 − x ≤ e −x , g(r − s) = (r − s)e r−s−1 ≥ re r−s−1 − se r−1 ≥ g(r) − s(1 + r)e r−1 and consequently g (i) (r − s) ≥ g (i) (r) − s (1 + r)e r−1 i for all r ≥ s ∈ [0, 1) and i ∈ N. (3.18) This completes our preparations. In order to show that (1 − )/(1 − u) → 1 as a → ∞ we argue that and u are very close together and approach 1 as a (and b > 2a) gets big. We start by bounding the distance between and u. Applying Corollary 3.7 to U = L + 2 −2a−x+2 we get that f (a) (U ) = f (a) L + 2 −2a−x+2 ≤ f (a) (L) + 2 −a−x+2 (3.19) and Corollary 3.8 bounds f (a−1) (U ) from below with 2 −x−1 1 − 2 −x−1 − 2 −2x−2 ≥ 1/8, thus f (a+2) (U ) ≥ 1/2, and therefore we get using the monotonicity of f 1 2 ≤ f (a+2) (U ) ≤ f (a+3) (L) ≤ f (a+3) (U ) (3.19) ≤ f (a+3) (L) + 2 −a−x+5 .(3.20) We switch our focus to g. Observe that z := 3e −1/2 /2 < 1 and, using (3.18), g (b−a−3) 1 − f (a+3) (L) − 2 −a−x+5 ≥ g (b−a−3) 1 − f (a+3) (L) − 2 −a−x+5 · z b−a−3 . This implies, using (3.20) and the previous equation, that Next we show that u, approach 1. Using g(x) = 1 − f (1 − x), (3.20), g being increasing and g(x) ≤ xe −1/2 for all x ≤ 1/2 (in that order), we get g (b−a−3) 1 − f (a+3) (U ) ≥ g (b−a−3) 1 − f (a+3) (L) − 2 −a−x+5 · z b−g (b) (1 − L) = g (b−a−3) 1 − f (a+3) (L) ≤ g (b−a−3) 1 2 ≤ 1 2 e −(b−a−3)/2 , b > a + 3. Moreover, using that f (x) ≤ 2x and g(x) ≥ x/e, g (b) (1 − U ) = g (b−a) 1 − f (a) (U ) ≥ 1 − 2 −x e −(b−a) . Thus, these two bounds together give −a) for all b > a + 3. We just showed that u, → 1 as a (and b) tends to infinity. This yields that ln u, ln and ln /(e − e + 1) tend to 0, leaving us with the term ln (1 − )/(1 − u) . The fact U ≤ f (L) (and so f (b−2) (U ) ≤ f (b−1) (L)) implies that 1 − 1 2 e −(b−a−3)/2 ≤ f (b) (L) ≤ f (b) (U ) ≤ 1 − 1 − 2 −x e −(b1 − 1 − u = g (b) 1 − L g (b) 1 − U = exp g (b−1) (1 − L) − 1 · exp g (b−2) (1 − L) − 1 exp g (b−1) (1 − U ) − 1 · exp g (b−2) (1 − U ) − 1 · g (b−2) 1 − L g (b−2) 1 − U ≤ exp g (b−2) (1 − L) − 1 exp g (b−1) (1 − U ) − 1 · g (b−2) 1 − L g (b−2) 1 − U . Applying the same estimate to the latter fraction inductively we get for any c ∈ N Using e x ≤ 1 + 2x, x ∈ [0, 1] this yields the bounds 1 − 1 − u ≤ exp g (c) (1 − L) − 1 exp g (b−1) (1 − U ) − 1 · g (c) 1 − L g (c) 1 − U ≤ exp g (c) (1 − L) · g (c) 1 − L g (c) 1 − U .1 ≤ 1 − 1 − u ≤ 1 + √ 2 −a+4 1 + 2 −x+5 e 1 − 2 −x · z a ln 2−2 for all a ∈ N. Therefore, as 0 < z < 1 we obtain 1− 1−u → 1, and consequently ln (1 − )/(1 − u) → 0, as a → ∞. Lemma 3.2 states that disturbing a Gumbel distributed random variable by a small amount does not significantly alter its distribution. Proof of Lemma 3.2. Observe that h(n) + G ± ε = h(n) + G is equivalent to G ∈ j − h(n) − ε, j − h(n) + ε for some j ≥ 1. But as G is absolutely continuous, for any δ > 0 we can choose ε small enough such that P   G ∈ j≥1 j − h(n) − ε, j − h(n) + ε   ≤ δ. Figure 1 : 1The function c(x) − c(0), c(0) ≈ 0.105, plotted for values of x between 0 and 2. The periodic nature of the function and its small amplitude are evident. Figure 2 Figure 2 : 22for a visualization of h. Similarly, to obtain an expression for the variance of the runtime, see thatVar[X ni ] = Var X ni − ( log 2 n i + ln n i ) = E X ni − ( log 2 n i + ln n i ) 2 − E X ni − ( log 2 n i + ln n i ) Let (ni) i∈Nbe a sequence of natural numbers such that {log 2 ni} → x and {ln ni} → y for x, y ∈ [0, 1). The left figure shows the function h(x, y) (appearing in the expectation of Xn i ) for values of x and y between 0 and 1. The right figure shows Var[Xn i ] as a function of x, y. Lemma 3. 1 . 1For , u defined as above, where b x and f (x) = (x−3)e −x ; in particular, f is monotonically decreasing and takes only positive values on [x 1 , x 2 ]. Furthermore max x∈[x1,x2]f (x) = (2 − x 1 )e −x1 Corollary 3. 7 . 7Let i ∈ N and r, s ∈ [0, 1/2]. Then f (i) (r + s) ≤ f (i) (r) + 2 i s. x )/x = 2 and thus lim x→0 H(x) = 0. Furthermore is H an increasing function on the interval [0, 1] as, Corollary 3 . 8 . 38For all x ∈ [0, 1] and i ∈ N Theorem 3 . 12 ( 312Polya's Theorem, [21, Theorem 1]) a− 3 3, and therefore, as1 − f (b) (L) ≥ 1 − f (b) (U ) = g (b−a−3) 1 − f (a+3) (U ) , |u − | = |f (b) (U ) − f (b) (L)| ≤ 2 −a−x+5 z b−a−3 → 0 as a → ∞, b − a → ∞.(3.21) Set c = a(1 + ln 2) . Using(3.21) and (3.22), where we set b = c, we obtain (for large enough a) that g (c) (1 − L) − g (c) (1 − U ) ≤ 2 −a−x+5 z c−a−3 as well as f (c) U ≤ 1 − 1 − 2 −x e −(c−a) and f (c) (L) ≥ 1 − e −(2 −x )e −c+a . the set of informed nodes at the end of round t; in particular |I 0 | = 1. Analogously we write U t = V \I t for the set of uninformed nodes. For an event A, we sometimes write P A [·] instead of P [· | A] to denote the conditional probability and we write E A [·] = E[· | A]. If we condition on I t , then we also abbreviate P [· | I t ] = P t [·] and E[· | I t ] = E t [·]. ) . )By our choice of N 1 , see(3.13), and the elementary inequalities z/(1 + z) ≤ ln(1 + z) ≤ z for all z > −1 this yields the upper bound −a ) for all a > M 1 .(3.16)We continue with the second sum in (3.15). Corollary 3.7 yieldssup x∈[0,1] N1 j=1 h (j) ≤ ε + 8ε 1 + 8ε + O(2 N2 j=N1+1 Proof of Lemma 2.5. Doerr and Künnemann show in [7, Cor. 3.2 and Thm. 4.1] that for all r ∈ N P X n ≥ log 2 n + ln n + 2.188 + r ≤ 2e −r and A Existence of SubsequenceLet x, y ∈ [0, 1]. In this section we show that there is an unbounded sequence of natural numbers (n i ) i∈N such that log 2 n i − log 2 n i → x and ln n i − ln n i → y as i → ∞. To this end, set z = y − x ln 2. According to a Theorem of Kronecker, see e.g.[16,Thm. 440], for all i ∈ N, there are p i , q i ∈ N such thatActually even more is true: there are infinitely many p i , q i ∈ N that solve (A.1). To see this, assume that there are only finitely many, then there is k, ∈ N such that k ln 2 = + z, otherwise there would be some i ∈ N where (A.1) has no solution. However, according to a Theorem of Hurwitz, see e.g.[16,Thm. 193], there are infinitely many r j , s j ∈ N such that r j ln 2 − s j ≤ r −2 j . But then r j ln 2 − s j = (r j + k) ln 2 − (s j + ) − z ≤ r −2 j , a contradiction, thus there are infinitely many solutions to (A.1). We continue with that equation, which we can restate, as i → ∞,Taking the exponential on both sides thus yields, as i → ∞, 2 qi+x = e pi+y+O(i −1 ) .Set n i = 2 qi+x for all i ∈ N, where we choose q i such that q i ≥ i from the infinitely many solutions to (A.1). Then n i ∈ N for all i ∈ N and log 2 n i − log 2 n i = x + O 2 −i as well as ln n i − ln n i = y + O i −1 .Thus the subsequence of natural numbers that is induced by log 2 n i − log 2 n i → x and ln n i − ln n i → y is non-empty and unbounded. . 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Epidemic algorithms for replicated database maintenance. A Demers, D Greene, C Hauser, W Irish, J Larson, S Shenker, H Sturgis, D Swinehart, D Terry, Proceedings of the sixth annual ACM Symposium on Principles of distributed computing. the sixth annual ACM Symposium on Principles of distributed computingACMA. Demers, D. Greene, C. Hauser, W. Irish, J. Larson, S. Shenker, H. Sturgis, D. Swinehart, and D. Terry. Epidemic algorithms for replicated database maintenance. In Proceedings of the sixth annual ACM Symposium on Principles of distributed computing, pages 1-12. ACM, 1987. Tight analysis of randomized rumor spreading in complete graphs. B Doerr, M Künnemann, Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO). the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)SIAMB. Doerr and M. Künnemann. Tight analysis of randomized rumor spreading in complete graphs. In 2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combi- natorics (ANALCO), pages 82-91. SIAM, 2014. On a classical problem of probability theory. P Erdős, A Rényi, Magyar Tud. Akad. Mat. Kutató Int. Közl. 6P. Erdős and A. Rényi. On a classical problem of probability theory. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6, pages 215-220, 1961. Randomized broadcast in networks. U Feige, D Peleg, P Raghavan, E Upfal, Random Structures & Algorithms. 14U. Feige, D. Peleg, P. Raghavan, and E. Upfal. Randomized broadcast in networks. Random Structures & Algorithms, 1(4):447-460, 1990. Reliable Broadcasting in Random Networks and the Effect of Density. N Fountoulakis, A Huber, K Panagiotou, INFOCOM 2010. 29th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies. San Diego, CA, USAN. Fountoulakis, A. Huber, and K. Panagiotou. Reliable Broadcasting in Random Networks and the Effect of Density. In INFOCOM 2010. 29th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies, 15-19 March 2010, San Diego, CA, USA, pages 2552-2560, 2010. Rumor spreading on random regular graphs and expanders. Approximation, Randomization, and Combinatorial Optimization. N Fountoulakis, K Panagiotou, Algorithms and Techniques. N. Fountoulakis and K. Panagiotou. Rumor spreading on random regular graphs and ex- panders. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 560-573, 2010. The shortest-path problem for graphs with random arclengths. A M Frieze, G R Grimmett, Discrete Applied Mathematics. 101A. M. Frieze and G. R. Grimmett. The shortest-path problem for graphs with random arc- lengths. Discrete Applied Mathematics, 10(1):57-77, 1985. Tight bounds for rumor spreading in graphs of a given conductance. G Giakkoupis, 28th International Symposium on Theoretical Aspects of Computer Science. Dortmund, GermanySTACS 2011G. Giakkoupis. Tight bounds for rumor spreading in graphs of a given conductance. In 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011, March 10-12, 2011, Dortmund, Germany, pages 57-68, 2011. Tight Bounds for Rumor Spreading with Vertex Expansion. G Giakkoupis, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014. the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014Portland, Oregon, USAG. Giakkoupis. Tight Bounds for Rumor Spreading with Vertex Expansion. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 801-815, 2014. Rumor spreading and vertex expansion. G Giakkoupis, T Sauerwald, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012Kyoto, JapanG. Giakkoupis and T. Sauerwald. Rumor spreading and vertex expansion. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1623-1641, 2012. An introduction to the theory of numbers. G H Hardy, E M Wright, Oxford University PressG. H. Hardy, E. M. Wright, et al. An introduction to the theory of numbers. Oxford University Press, 1979. Fast Distributed Algorithms for Computing Separable Functions. D Mosk-Aoyama, D Shah, IEEE Trans. Information Theory. 547D. Mosk-Aoyama and D. Shah. Fast Distributed Algorithms for Computing Separable Func- tions. IEEE Trans. Information Theory, 54(7):2997-3007, 2008. Randomized Rumour Spreading: The Effect of the Network Topology. K Panagiotou, X Pérez-Giménez, T Sauerwald, H Sun, Combinatorics, Probability & Computing. 242K. Panagiotou, X. Pérez-Giménez, T. Sauerwald, and H. Sun. Randomized Rumour Spread- ing: The Effect of the Network Topology. Combinatorics, Probability & Computing, 24(2):457- 479, 2015. Revisiting Asynchronous Rumor Spreading in the Blockchain Era. C Patsonakis, M Roussopoulos, 2019 IEEE 25th International Conference on Parallel and Distributed Systems (ICPADS). C. Patsonakis and M. Roussopoulos. Revisiting Asynchronous Rumor Spreading in the Blockchain Era. In 2019 IEEE 25th International Conference on Parallel and Distributed Systems (ICPADS), pages 284-293, Dec 2019. On Spreading a Rumor. B Pittel, SIAM J. Appl. Math. 471B. Pittel. On Spreading a Rumor. SIAM J. Appl. Math., 47(1):213-223, 1987. Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem. G Pólya, Mathematische Zeitschrift. 83G. Pólya. Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Mo- mentenproblem. Mathematische Zeitschrift, 8(3):171-181, 1920. Rumor Spreading and Vertex Expansion on Regular Graphs. T Sauerwald, A Stauffer, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011. the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011San Francisco, California, USAT. Sauerwald and A. Stauffer. Rumor Spreading and Vertex Expansion on Regular Graphs. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 462-475, 2011. Approximation Theorems of Mathematical Statistics. R Serfling, Wiley Series in Probability and Statistics. WileyR. Serfling. Approximation Theorems of Mathematical Statistics. Wiley Series in Probability and Statistics. Wiley, 2009.

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{'abstract': 'We study the popular randomized rumour spreading protocol push. Initially, a node in a graph possesses some information, which is then spread in a round based manner. In each round, each informed node chooses uniformly at random one of its neighbours and passes the information to it. The central quantity to investigate is the runtime, that is, the number of rounds needed until every node has received the information.The push protocol and variations of it have been studied extensively. Here we study the case where the underlying graph is complete with n nodes. Even in this most basic setting, specifying the limiting distribution of the runtime as well as determining related quantities, like its expectation, have remained open problems since the protocol was introduced.In our main result we describe the limiting distribution of the runtime. We show that it does not converge, and that it becomes, after the appropriate normalization, asymptotically periodic both on the log 2 n as well as on the ln n scale. In particular, the limiting distribution converges only if we restrict ourselves to suitable subsequences of N, where simultaneously log 2 n − log 2 n → x and ln n − ln n → y for some fixed x, y ∈ [0, 1). On such subsequences we show that the expected runtime is log 2 n + ln n + h(x, y) + o(1), where h is explicitly given and numerically | sup h − inf h| ≈ 2 · 10 −4 .This "double oscillatory" behaviour has its origin in two key ingredients that were also implicit in previous works: first, an intricate discrete recursive relation that describes how the set of informed nodes grows, and second, a coupon collector problem with batches of size n that takes the lead when the protocol is almost finished. Rounding in the recursion introduces the periodicity on the log 2 n scale -as it is the case in many discrete systems -and rounding in the batched problem is the source of the second periodicity.X n = log 2 n + ln n + o(ln n).arXiv:2003.10762v1 [math.PR] 24 Mar 2020Moreover, they obtained bounds for (very) large deviations of X n from its expectation. In[20], Pittel improved upon the results in[12], in particular, he showed that for any f :The currently most precise result in this context was obtained by Doerr and Künnemann [7], who considered in great detail the distribution of X n . They showed that X n can be stochastically bounded (from both sides) by coupon collector type problems. This gives a lot of control regarding the distribution of X n , and it allowed them to derive, for example, very sharp bounds for tail probabilities. Apart from that, it enabled them to consider related quantities, as for example the expectation of X n . Among other results, their bounds on the distribution of X n imply that log 2 n + ln n − 1.116 ≤ E[X n ] ≤ log 2 n + ln n + 2.765, (1.1) which pins down the expectation up to a constant additive term. Besides on complete graphs, push has been extensively studied on several other graph classes. For example, Erdős-Rényi random graphs [9, 10], random regular graphs and expander graphs[11,18,5]. More general bounds that only depend on some graph parameter have also been derived, e.g. the diameter [9], graph conductance[17,3,4,13]and node expansion[4,22,15,14].Results In order to state our main result we need some definitions first. SetAs we will see later, the function g describes, for a wide range of the parameters, the evolution of the number of uninformed nodes; in particular, if at the beginning of some round there are xn uninformed nodes, then at the end of the same round there will be (roughly) g(x)n uninformed nodes, and after i rounds there will be (roughly) g (i) (x)n uninformed nodes. This fact is not new -at least for bounded i -and has been observed long ago, see for example[20,Lem. 2]. For x ∈ R define the function2) whose actual meaning will become clear later. We will show that the double limit exists, so that this indeed defines a function c : R → R. Moreover, we will show that c is continuous and periodic with period 1, that is, if we write {x} = x − x then c(x) = c({x}), and that (numerically) | sup c − inf c| ≈ 10 −9 , cf.Figure 1. The Gumbel distribution will play a prominent role in our considerations. We say that a real valued random variable G follows a Gum(α) distribution with parameter α ∈ R, G ∼ Gum(α), if for all x ∈ RFinally, let γ denote the Euler-Mascheroni constant. With all these ingredients we can now state our main result, which specifies -see also below -the distribution of the runtime of push on the complete graph.This lemma does not look completely innocent, and it actually has striking consequences. It readily implies the following result, which establishes that the limiting distribution X n is periodic both on the log 2 n and on the ln n scale. In order to formulate it, we need a version of the Gumbel distribution where we restrict ourselves to integers only. More specifically, we say that a random variable G follows a discrete Gumbel distribution, G ∼ dGum(α), if the domain of G is Z and P [G ≤ k] = e −e −k−α , k ∈ Z.', 'arxivid': '2003.10762', 'author': ['Rami Daknama \nLudwig-Maximilians-Universität München\n\n', 'Konstantinos Panagiotou \nLudwig-Maximilians-Universität München\n\n', 'Simon Reisser \nLudwig-Maximilians-Universität München\n\n', 'Rami Daknama \nLudwig-Maximilians-Universität München\n\n', 'Konstantinos Panagiotou \nLudwig-Maximilians-Universität München\n\n', 'Simon Reisser \nLudwig-Maximilians-Universität München\n\n', 'Rami Daknama \nLudwig-Maximilians-Universität München\n\n', 'Konstantinos Panagiotou \nLudwig-Maximilians-Universität München\n\n', 'Simon Reisser \nLudwig-Maximilians-Universität München\n\n'], 'authoraffiliation': ['Ludwig-Maximilians-Universität München\n', 'Ludwig-Maximilians-Universität München\n', 'Ludwig-Maximilians-Universität München\n', 'Ludwig-Maximilians-Universität München\n', 'Ludwig-Maximilians-Universität München\n', 'Ludwig-Maximilians-Universität München\n', 'Ludwig-Maximilians-Universität München\n', 'Ludwig-Maximilians-Universität München\n', 'Ludwig-Maximilians-Universität München\n'], 'corpusid': 214623092, 'doi': '10.1016/j.spa.2021.03.008', 'github_urls': [], 'n_tokens_mistral': 24178, 'n_tokens_neox': 21388, 'n_words': 12953, 'pdfsha': 'f52ef5fdb2185fefefbfb681002ba3ff6559bf15', 'pdfurls': ['https://arxiv.org/pdf/2003.10762v1.pdf'], 'title': ['Asymptotics for Push on the Complete Graph', 'Asymptotics for Push on the Complete Graph', 'Asymptotics for Push on the Complete Graph', 'Asymptotics for Push on the Complete Graph', 'Asymptotics for Push on the Complete Graph', 'Asymptotics for Push on the Complete Graph'], 'venue': []}

arxiv

STen: Productive and Efficient Sparsity in PyTorch Andrei Ivanov Department of Computer Science Department of Computer Science ETH Zürich Switzerland Nikoli Dryden Department of Computer Science ETH Zürich Switzerland Tal Ben-Nun Department of Computer Science ETH Zürich Switzerland Saleh Ashkboos Department of Computer Science ETH Zürich Switzerland Torsten Hoefler ETH Zürich Switzerland STen: Productive and Efficient Sparsity in PyTorch Sparsity, Deep Learning As deep learning models grow, sparsity is becoming an increasingly critical component of deep neural networks, enabling improved performance and reduced storage. However, existing frameworks offer poor support for sparsity. Specialized sparsity engines focus exclusively on sparse inference, while general frameworks primarily focus on sparse tensors in classical formats and neglect the broader sparsification pipeline necessary for using sparse models, especially during training. Further, existing frameworks are not easily extensible: adding a new sparse tensor format or operator is challenging and time-consuming. To address this, we propose STen, a sparsity programming model and interface for PyTorch, which incorporates sparsity layouts, operators, and sparsifiers, in an efficient, customizable, and extensible framework that supports virtually all sparsification methods. We demonstrate this by developing a high-performance grouped : sparsity layout for CPU inference at moderate sparsity. STen brings high performance and ease of use to the ML community, making sparsity easily accessible. Layout Layout Layout Sparsifiers Output Outputs Streamed Blocked Materialized Implementation defined by input/output sparsity layout Keep-all Random fraction Scalar threshold ...Figure 1: Overview of the STen sparsity programming model.supported. If there is no implementation for a sparse operator or sparsity layout, STen automatically converts to a supported implementation to ease development. This allows STen to provide a productive and efficient playground for sparsity in PyTorch, allowing researchers to pursue novel ideas without significant framework engineering overhead. To demonstrate this flexibility, we develop a novel grouped : sparsity layout ( : : , §5), building on existing : formats[48,76], for fast sparse inference at moderate sparsity levels. We also include a suite of standard sparsifiers.STen delivers competitive performance for both sparse inference and training without sacrificing its flexibility ( §6). We evaluate our : : sparsity layout for sparse inference against state-of-the-art inference engines, where we are up to 3.9×faster than DeepSparse [36] using unstructured sparsity for sparse-dense GEMM. We also show end-to-end performance for BERT BASE [12], where sparse : : models recover the original accuracy and are 3.2× faster than a dense PyTorch version. Finally, we showcase the productivity STen brings. It supports torchvision's [59] entire suite of classification models and common HuggingFace transformer models [69] out-of-the-box, and we illustrate how it is simple to sparsify existing models, run sparse fine-tuning, and integrate additional sparsity layouts. STen brings both high performance and ease of use to the machine learning community, enabling users to easily leverage sparsity, engineers to develop optimized sparse formats and operators, and researchers to quickly explore new sparsification methods. INTRODUCTION Deep learning models are growing voraciously, and require ever growing amounts of compute and memory [51,63,67], as larger models trained for longer continue to offer improvements [28,33]. Indeed, the largest models (e.g., [4,9,57]) do not fit in a single GPU for inference, much less training. To address this, sparsity has emerged as a major research and engineering direction [19,27]. Sparsity is widely used to reduce storage requirements and improve performance during inference. More recently, there has been interest in sparsity during training, including both sparse fine-tuning for large models and fully sparse training from scratch. Many frameworks have begun to integrate support for sparsity. PyTorch's [53] torch.sparse includes COO and CSR tensors and a limited set of operations; TensorFlow's [1] tf.sparse similarly supports COO tensors. SparseML and DeepSparse [36] provide recipes and implementations for sparsity algorithms, and a highly-tuned sparse inference engine for CPUs, respectively. Other frameworks, such as TVM [18] and TFLite [43], also support sparse inference. However, sparsity is typically "tacked on" to general deep learning frameworks: existing frameworks offer only limited support for sparsity, and lack a complete and flexible sparsification pipeline. Frameworks like TensorFlow and PyTorch primarily support sparse tensors, and do not offer general support for sparse operations (e.g., torch.sparse supports only 13 of 44 linear algebra operations, of which four support backpropagation, and hundreds of other operations are also not supported). They also lack native support for sparsification operations that can efficiently produce sparse tensors (e.g., without materializing intermediates). Further, at sparsities common in deep learning (50-95%), although the supported classical sparse matrix formats reduce storage (COO, CSR), they often perform worse than dense implementations. It is also challenging to add support for additional formats, especially if automatic differentiation is to be supported. While blocked formats (e.g., ELL, BCSR) support efficient implementations by calling dense kernels for each block [20], they restrict where nonzeros can be placed and can limit the information preserved after sparsification. Such implementations (e.g., [37,50]) also typically support only a handful of customized operations. Finally, many implementations use masks to emulate sparsity by zeroing out elements; while valuable for research, this offers no storage or performance improvements and the masking is typically implemented by hand. Thus, existing frameworks offer limited productivity and efficiency improvements for sparsity, and do not provide a clear path toward supporting a broad use of sparsity. This is especially acute for researchers working at the forefront of sparsity, who want to rapidly iterate on novel sparsity formats, fast implementations, and sparsification methods, yet are stymied by frameworks. To address this, we first propose a new programming model for sparsity in PyTorch ( §3), overviewed in Fig. 1. It consists of three components: sparsity layouts for tensors; operators, which provide implementations for computations with any combination of sparsity layouts for input and output tensors; and sparsifiers, which are applied to operator outputs to compute a new sparse tensor. Sparsifiers are further classified as streaming, blocking, or materializing, based on the number and structure of outputs they require. Our model supports the vast majority of sparsification approaches, and enables them to be implemented efficiently; for example, threshold pruning is a streaming sparsifier, and a highperformance implementation could be inlined into operators. We provide an implementation of this model, STen 1 , in PyTorch ( §4). STen provides a comprehensive sparsification pipeline and supports fast sparse inference as well as sparse GPU training using masking, for all PyTorch operators. STen is fully extensible, and a user or developer can easily implement additional sparsity layouts, operators, or sparsifiers. Backpropagation is transparently SPARSIFICATION BACKGROUND Sparsity occurs when some values in a tensor are zero. When many values are zero, it can be more efficient to only store the nonzero values, saving space. It can also be more efficient to skip computations involving zeros, as the output is already known. Hence, exploiting sparsity, if done correctly, can reduce memory and compute requirements. It can also reduce data movement, a particular bottleneck in deep learning [32]. The sparsity of a tensor is defined as the ratio of the number of zeros to the size of the tensor; the number of nonzeros is referred to as . In general, for sparsity to pay off in performance, a tensor must be sufficiently sparse [24,74]; however, the sparsity necessary to achieve improvements will vary depending on the workload, sparsity structure, hardware architecture, and implementation. We focus on sparsity in deep learning, where sparsities in the range of 50-95% are common (see [27] for a comprehensive overview). Sparsity typically occurs in DL workflows in three regimes: (1) Sparse inference, where a network is already sparse. (2) Sparse fine-tuning, where sparsity is induced in a pretrained dense network through a retraining process. (3) Sparse training, where a sparse network is trained from scratch, possibly with the sparsity changing over time. We distinguish between fine-tuning, which is typically a relatively short process on a pretrained network, and training from scratch, typically a much longer process. There are many works that aim to accelerate sparse inference in particular (e.g., [18,36,73]). Such sparse models are usually constructed by using sparsifiers to prune the weights of an existing dense model in a sparse fine-tuning stage. Both sparse finetuning and training typically use dense tensors in combination with masking to emulate sparsity, especially as the sparsity pattern may change during training. Fully sparse training remains an open problem, although there has been recent progress on this, including iterative pruning [17,77] and training from scratch (e.g., [10,15,16]). Beyond sparsity in weights, other tensors in a network may also be sparse. This includes activations (e.g., through dropout [64], ReLUs [47], or other methods [31,36]) and gradients (common in communication compression, e.g., [13,44,65]). Sparsity is induced using sparsifiers, which select the values to prune (i.e., set to zero). A simple, yet powerful, class of sparsifiers which we consider is magnitude pruning [17,77], which prunes small values on the assumption that they will not change the output significantly and so are unimportant. There are several varieties of magnitude pruning. One-shot magnitude pruning prunes a network to a desired sparsity in a single step, after which the network may be fine-tuned. Iterative magnitude pruning alternates between pruning and fine-tuning stages while gradually increasing the sparsity level. The pruning may also be local or global; the former prunes each layer to the desired sparsity, while the latter considers the entire network. This can allow the pruning to better allocate parameters throughout the network. Another variant is layer-wise magnitude pruning, which prunes layers one at a time in sequential order, fine-tuning after each. Numerous formats exist for efficiently representing sparsity, although the level and structure of the sparsity influences which formats yield speedups [56]. Frameworks like PyTorch support formats such as Compressed Sparse Row (CSR), which represents nonzeros using a matrix of column and row indices; and Coordinate Offset (COO), which stores nonzeros together with their absolute offset. Approaches like ELLPACK (ELL) and block CSR (BCSR) more efficiently store blocked data. Recently, specialized formats for DL workloads have been developed, such as : [48,76], where each group of elements has nonzeros (e.g., 2:4 ). THE STEN PROGRAMMING MODEL We now introduce our programming model for sparsity; see Fig. 1 for an overview. The core components of the model are sparsity layouts ( §3.1), operators ( §3.2), and sparsifiers ( §3.3), which together can represent sparse operations and sparsification. Here we describe the high-level concepts and interface, and discuss implementation details in §4. Although our focus is on PyTorch, we expect the STen model to be portable to other frameworks. The key aim of this model is to support a complete and performant sparsity pipeline with extensive customizability for researchers, both those working on sparsity and on performance. Sparsity Layouts Sparsity layouts augment the typical memory layout of a tensor by annotating the sparsity format used to store the data. The user can specify any sparsity layout, including classic formats like CSR, CSC, and COO; blocked formats like ELL and BCSR; specialized formats like : ; or dense tensor masking; plus any associated parameters (e.g., and ). A sparsity layout is assigned to a tensor as follows: x = sten.torch_tensor_to_csr(sparsifier, x) Further, custom sparsity formats can easily be added as well: Programmers simply define the to_dense operation and one sparsifier (see §3.3) to enable sparse/dense conversion for the format. If in-place operations, which directly modify an existing tensor, are to be supported, an additional implementation needs to be defined for STen's SameFormatSparsifier to handle sparsification (see §4). This contrasts with existing frameworks, where adding support for new sparsity layouts is a complex and time-consuming process that often involves significant additions to the framework "core". STen's flexibility enables researchers to rapidly prototype and evaluate novel sparsity layouts and performance engineers to easily tune the layouts of individual tensors for maximum performance. Operators Operators are any function (e.g., matrix multiplication), with any number of input and output tensors (we ignore non-tensor arguments for simplicity). We allow the input and output tensors to have any sparsity layout, and an operator may have different implementations for each combination of tensor layouts for maximum performance. Calling a built-in operator is simple: a = sten.torch_tensor_to_csr(sparsifier, torch.randn(4, 4)) # Sparse b = torch.randn(4, 4) # Dense c = torch.mm(a, b) # Dispatched to sparse(CSR)*dense->dense mm implementation To produce sparse output, an operator must be associated with a sparsifier. We discuss such sparse operators in §3.3. Beyond this, there are no restrictions or additional requirements for operators. As it can be infeasible to provide implementations for every combination, several approaches to simplify this are provided (see §4.4). A pattern matching system for supported input/output sparsity layout combinations can dispatch to a single underlying implementation. Otherwise, the interface falls back to a dense implementation with masks and issues a warning. This enables incremental optimization by performance engineers, who can decide which operators it is most profitable to provide sparse implementations for. It also allows users to rapidly explore the effects of sparsity on models (albeit with a performance penalty). Sparsifiers Sparsifiers decide which output values to keep. Each output tensor of an operator is associated with a sparsifier. A sparsifier can be thought of as a special kind of operator, and may include additional inputs, which may delay its application until they are ready (e.g., gradients for first-order sparsification). Sparsifiers may also produce output in a different sparsity layout than what the associated operator outputs. We use the term sparse operator to refer to the combination of an operator and sparsifier. We classify sparsifiers as one of streaming, blocking, or materializing, depending on the amount of data they need before they can begin to produce output. Streaming sparsifiers decide whether to prune each output value in a single pass before writing to the output tensor. Op Stream Select Blocking sparsifiers require a small set of output values to decide which ones to drop. Op Block Select Materializing sparsifiers require the operator to fully store all values before pruning can be performed. Select Op Materialize Different types of sparsifiers support different optimizations. E.g., streaming sparsifiers could be fused into their associated operator; this may also be possible with blockin sparsifiers. Sparsifiers may also be used standalone to convert dense tensors to sparse. Table 1 lists example sparsifiers and their characteristics. The trivial keep-all sparsifier preserves all produced values and is the default for dense tensors. It is not limited to dense tensors, however: the sum of two sparse tensors with a keep-all sparsifier produces a new sparse tensor with nonzero values given by the union of the nonzeros of the inputs. A random fraction sparsifier drops values with a fixed probability, while a scalar threshold sparsifier drops them if they are less than a fixed threshold. Per-block fraction drops the smallest proportion of values within fixed blocks of elements. Scalar fraction drops the smallest portion of the values (i.e., magnitude pruning) and block-wise fraction drops entire blocks with the smallest combined absolute magnitude. Finally, more advanced complex weight sparsifiers require additional information such as the loss or gradients. All of these sparsifiers could be supported by STen. Further, these examples are not exhaustive, and STen can support nearly any sparsifier. Declaring a new sparsifier is straightforward in STen: class RandomFractionSparsifier: def __init__(self, fraction): self.fraction = fraction @sten.register_sparsifier_implementation( sparsifier=RandomFractionSparsifier, inp=torch.Tensor, out=CscTensor) def dense_to_csc_random_fraction(sparsifier, tensor, grad_fmt=None): # Implementation for dense->CSC sparsification We now discuss sparse operators, which associate a sparsifier with an operator to enable it to produce sparse output. For maximum performance and flexibility, a sparse operator requires an output format, which consists of an inline sparsifier, a temporary sparsity layout, an external sparsifier, and the output sparsity layout. Operator Inline sparsifier External sparsifier Temporary layout Output layout The inline sparsifier is inlined into the operator and produces output in the temporary layout, which is materialized before being sparsified by the external sparsifier, which produces the final output. Streaming or blocking sparsifiers are good candidates for implementation as inline sparsifiers, to benefit from inlining, whereas materializing sparsifiers are typically implemented as external sparsifiers. One typically does not need both an inline and external sparsifier; however, highly optimized implementations of two-pass sparsifiers may use an inline sparsifier for the first pass and an external sparsifier for the second pass. Sparse gradients are similarly specified for operators. Defining a sparse operator in STen simply requires specifying the original operator and output format: Constructing Sparse Models With the core components of our sparsity programming model in place, we now describe the process for building sparse models. We consider two cases: constructing a model from scratch and sparsifying an existing model. In short, a sparse model is set up by providing a list of tensors and the desired sparsity layout for each. Given this, STen can initialize tensors and dispatch operators to appropriate implementations. If an operator implementation is not available, the user can provide one, convert tensors to a supported sparsity layout, or fall back to a dense implementation. In this model, all tensors are used as operator inputs, outputs, or both; for simplicity we will ignore sparsity for model inputs and outputs. Tensors that are input-only during forward propagation are typically weights. All other tensors are used as both inputs and outputs and occur within the computation graph (e.g., activations); we refer to these as intermediate tensors. In practice in most frameworks (including PyTorch), intermediate tensors do not exist until runtime, so their sparsity layout is instead defined by the operator that produces them. Constructing a sparse model from scratch is similar to the typical process in PyTorch, but tensors and operators in its computational graph are annotated with specific sparsity layouts and, if needed, sparsifiers. For example, one can construct a simple sparse linear layer as follows: To sparsify existing dense weights, or load sparse weights, we need only the desired sparsity layout and sparsifier. We first trivially convert sparsifiers to a materializing version if needed. Then the weights are sparsified and subsequent operator calls will use the sparse version. For iterative sparsification methods, the sparsity can be adjusted as sparse fine-tuning proceeds. Intermediate tensors are sparsified at runtime, as they do not exist in advance. For training, error signals (gradients of the loss w.r.t. layer output) and gradients may also be sparse, and can have independent sparse layouts and sparsifiers from their associated forward pass tensors. These are marked using a similar API: sb.set_weight_grad and sb.set_interm_grad. We treat these identically to intermediate tensors in the forward pass. Note that during training, weight tensors are no longer input-only, as gradient updates are applied to them. This is not a significant change from the user perspective, and mainly implies that materializing sparsifiers may be less efficient for weights and that sparsifying on-the-fly with the gradient update operator may be faster. STEN IMPLEMENTATION We now discuss the implementation of STen in PyTorch. STen builds upon and extends existing PyTorch infrastructure to support sparsity in a familiar manner. Figure 2 illustrates how STen modifies the computation graph to support the interface specified in §3. Depending on the location in the graph of the tensor(s) to be sparsified (i.e., intermediate or weight), the semantics of the forward pass differ. Intermediate tensors require both inline and external sparsifiers, as detailed in §3.3. Weight tensors only require an initialization sparsifier. During the forward pass, the Add operator incorporates sparsifiers. However, during the backward pass, sparsifiers need to be integrated into the MatMul operator to provide sparsified gradients for the weights. Given that the tensor and its gradient may have different formats, the operator that normally performs inplace modifications of the weights must now calculate the updated weights into a new tensor. The new tensor is sparsified using the SameFormatSparsifier to maintain the same format it had before. Creating an STen Model There are two mechanisms for creating an STen model: constructing a model from scratch using sparse operators, or replacing weights or operators in an existing model with sparse versions ( §3.4). In either case, models still follow the standard torch.nn.Module style. However, operations are automatically dispatched to sparse implementations by STen when sparse input tensors or operators are used ( §4.4), so code can be written using standard operators. Incorporating sparse weights into a model is straightforward, as PyTorch Parameters are easily accessible and modifiable. Replacing operators with sparse versions or sparsifying intermediate tensors is more challenging, as operators may not exist as standalone objects (e.g., torch.nn.functional methods), and intermediate tensors only exist at runtime. We use the torch.fx symbolic tracer to obtain the complete computation graph of the network, and then replace operators with STen wrappers that use our operator dispatcher. Unlike weights, intermediate tensors are sparsified by associating a sparsifier with the operator that outputs them. Sparse Tensors Adding new sparse tensor formats to PyTorch is challenging, especially given that we want STen to support arbitrary sparsity layouts. While PyTorch supports custom Tensor types [58], none of the approaches are suitable for STen. The key issue is that one cannot separate tensor metadata and values in the context of PyTorch's autograd engine, the core of which is written in C++. As it is currently implemented, the C++ component of autograd expects that gradient shapes are the same as that of the corresponding parameters and that the parameters have all their data materialized. This does not hold with STen, as the tensors may have different sparsity layouts and we do not wish to materialize them at each iteration. PyTorch's own sparse tensors avoid this issue by being directly supported within the autograd engine. This is also not suitable for STen, as we support any user-provided format from Python; rebuilding PyTorch would be infeasible for users and modifying its core time-consuming for developers. We work around this by wrapping all user-provided sparse tensors into a single-element dense PyTorch tensor. This ensures Py-Torch's autograd can call the correct backward operator implementations when needed, without adding overhead or requiring modifications to PyTorch. The STen interface ensures the dummy tensors behave like tensors of the correct shape, so this is entirely hidden from users and all operations function as expected. Sparsifiers and Sparsity Layouts STen imposes no restrictions on the implementation of sparsity layouts or sparsifiers, which may leverage existing libraries, customoptimized implementations, or pure Python implementations (for research and prototyping). The API is designed with the assumption that the user declares classes that store the desired metadata (e.g., sparsity level, block size) and then registers implementations for particular input layouts. Registered implementations are treated as a black box for maximum flexibility. STen relies on the implementer to manage the internal structure and metadata of these objects. For sparsity layouts in particular, dense values and location metadata (e.g., indices of nonzeros) may be stored. Sparse operator implementations ( §4.4) are then expected to call optimized (e.g., native C++ or CUDA) implementations and pass relevant data. We encourage implementers to rely on PyTorch's memory management primitives (e.g., storing nonzeros in dense PyTorch tensors) to simplify code and benefit from memory management optimizations. Sparse Operators and Dispatch STen's operator dispatch is the key component that ties sparsity layouts, operators, and sparsifiers together. Sparse operators may be implemented in any way desired (see §4.3), and the dispatch mechanism will call the appropriate implementation for the given combination of input and output sparsity layouts and sparsifiers. Further, STen will also intercept calls to standard PyTorch operations using sparse inputs and dispatch them to specialized implementations. Fig. 3 provides an overview of the STen dispatcher, while Fig. 4 details its integration with PyTorch. To customize implementations for a specific sparse tensor class in STen, the most straightforward way is to override the method or attribute (e.g., shape) by defining it with the same name as in the corresponding dense tensor. STen will search for the user class implementation first before proceeding to other dispatch cases. We use PyTorch's standard tensor extension mechanisms to handle calls to simple operations (e.g., torch.add, torch.mm) using sparse matrices. If sparse implementations are not found, the inputs are automatically converted to dense fallbacks. More complicated operations, such as in-place modifications (e.g., add_) or views are handled similarly, although STen's default implementations may be pessimistic, necessitating the resparsification of the original tensor in the inplace fallback implementation. There is an additional global route to STen's sparse dispatcher, which is used for operators that are not supported by PyTorch's tensor extension mechanism. This is primarily to support external libraries that provide extensions in the form of native, compiled implementations (e.g., Nvidia Apex [49]). While such libraries are implemented via autograd function extensions, PyTorch does not catch calls to their implementations with custom tensor types. To support such libraries, we provide a patching API that redirects any Python function (including native built-ins) into our dispatcher when one of their arguments is a sparse tensor. Users need only request such patching for non-standard extension libraries, as STen supports common ones by default. STen does not allow the use of tensor attributes that trigger inplace modifications on the applied tensor. This restriction is in place to prevent potential interactions with PyTorch's implementation details, such as autograd functionality. STen manages such operations internally within its priority dispatcher for core functionality. The priority operators are given precedence 1 and are looked up first. The dispatcher section, which is aware of operator semantics (inplace or not) and its type (function, method, or attribute), handles all the remaining operators 2 . When execution reaches STen's dispatcher, STen uses information about the original operator, input and output sparsity layouts, and sparsifiers to look up the sparse implementation in a global registry. This is done independently for forward and backward passes, allowing separate implementations to be chosen. The lookup simply involves a hash table search based on a canonicalized list of input and output sparsity layouts and sparsifiers, a search which can easily be rapidly pruned using the layouts. If no implementation is found, STen can attempt to convert sparse tensors to alternative sparsity layouts and rerun the dispatch lookup. Conversion is only attempted when STen can guarantee that it is lossless, to prevent any information loss; generally this means it only attempts conversion to formats such as CSR. Finally, if no other implementation is found, STen will fall back to a dense implementation by converting sparse tensors to dense tensors with masks and applying standard dense operator implementations. Note that, depending on the hardware and implementations, a dense implementation may outperform unoptimized sparse implementations, so users may prefer this to other conversions. Hence, STen automatically supports sparse implementations of all PyTorch operators (albeit with some performance overhead when conversion is needed, see §6). STen does not implement its own sparse operators, and relies on users (or downstream libraries) to register their own implementations. There are no restrictions on implementations, which may leverage existing libraries or custom implementations. Support for common sparse operators (e.g., linear layers) using the implementations in torch.sparse and scipy.sparse, as well as our new : : format, are provided by default. Additionally, the operators are custom PyTorch operators and interface with standard PyTorch features such as JIT compilation. We emphasize that a user is not expected to provide optimized implementations for all combinations of sparsity layouts, sparsifiers, hardware, and the like, and indeed this is likely to be infeasible. Rather, STen enables users to implement the key operators which their particular problem requires, while STen's dispatcher will use conversion or dense implementations for any unsupported operators, allowing rapid prototyping and incremental optimization. Backpropagation STen is fully integrated with PyTorch's autograd engine. We leverage a custom autograd extension (via torch.autograd.Function), which lets PyTorch manage calling the correct backward implementation. Custom implementations of backward operators can be provided, and STen will fall back to dense versions (with conversion) if implementations are not available. If a custom forward implementation is provided for an operator, we require that a backward pass implementation be provided as well if backpropagation is to be supported (as the default implementation may expect different preserved states than what the custom one provides). Distributed Training STen natively supports distributed data-parallel training using Py-Torch's DistributedDataParallel module. This is handled internally by converting sparse gradient tensors to dense before utilizing native collective operations, and finally converting the tensors back. We implement this using STen's operator patching mechanism. As tensor conversion may be expensive for some formats, we support optimized converters. In particular, for structured sparsity formats, we avoid unnecessary conversions when the nonzero locations of the initial and replacement tensors match. This naturally appears during gradient synchronization, as tensors have the same nonzeros on every processor at the beginning of training, and the user can optionally ensure the sparsity pattern remains fixed or changes slowly. These optimizations are beneficial because it can be cheaper to convert from sparse to dense formats and compare the nonzero layout than to convert from dense to sparse. Future work in STen could allow it to support fully sparse gradient communication (e.g., [13,40,44,60,65]) or more general communication compression [66] for additional performance. GROUPED : SPARSITY ( : : ) To illustrate the flexibility of STen, we develop a novel sparsity layout, a grouped : format that enables fast CPU inference at moderate sparsity. The format is an extension of existing : sparsity formats [48,76] with the addition that each nonzero pattern is repeated times, forming a group, for easier mapping to vector operations for performance. Additionally, we combine groups into chunks with all combinations of nonzeros in fixed order. For flexibility in the encoding, we permit reordering the blocks of elements within each chunk, and so store an index encoding the original location of each block. This means that, even with one group, we impose more structure than in standard : . We refer to this as : : sparsity, and illustrate it in Fig. 5. Below we discuss its implementation and performance considerations, and we evaluate accuracy and performance in §6. : : Implementation The design of the : : sparsity layout is heavily influenced by performance considerations. Figure 6 illustrates our : : sparsedense GEMM kernel. We implement high-level optimizations such as caching, tiling, and parallelism following OpenBLAS [68,71], and utilize an optimized microkernel to implement the complete GEMM algorithm. 1 Data is first loaded from sparse values, then 2 broadcast into vector registers. Chunks, which fix the order of sparsity permutations, allow kernels to avoid branches based on the sparsity structure. 3 The corresponding dense data is then loaded using indirect loads from specific rows of , 4 and a fused multiplyadd (FMA) is performed. We provide implementations using both AVX2 and AVX-512 vector operations. Finally, when a permutation boundary is reached, 5 the result is stored in . 6 -10 perform the next iteration. By using fixed group sizes, we can efficiently unroll iterations over groups and map them to vector registers. The permutation order in chunks is selected so the nonzero pattern between adjacent groups differs in only one location, so that we need save and initialize only one vector register. Overall, this implementation delivers high performance even at moderate sparsities. The choice of and influence the efficiency of the implementation. In particular, as the number of permutations in a chunk grows, the implementation may be less efficient (e.g., due to limited instruction cache space). Nevertheless, we can efficiently support sparsity between 50 and 95%. Constructing : : Sparse Tensors We now propose algorithms for converting dense tensors to : : sparsity layouts on both CPU and GPU. When converting, we attempt to preserve elements to maintain the highest overall magnitude of the tensor. More formally, for a dense tensor , we wish to find arg maxˆ∥ˆ∥ whereˆis in : : format and ∥ · ∥ is a matrix norm (we use 1 ). Performance is critical, as the primary use of these conversions is sparsifying weights after gradient updates during training. While this could be formulated as an integer programming problem with unknowns representing the specific choice of sparsity pattern for each block, we found this to be too slow, and instead utilize fast approximations. To sparsify on CPUs, we first compute the total magnitude of the preserved elements for each column of a chunk with all permutations of nonzero patterns. With a fixed : : format, there are 2 such columnwise magnitudes. This list is then sorted and processed from highest to lowest. We use a specific nonzero pattern for a column in the dense tensor only if this column was not yet selected and the group corresponding to the pattern is not yet full. To sparsify on GPUs, we use a variant of this algorithm that assigns each column of a chunk to a separate GPU thread. Initially, columns are arbitrarily assigned to groups. Each thread iterates over the columns in the other sparsity groups and attempts to exchange the nonzero pattern it is assigned with an alternative nonzero pattern. If such a swap improves the overall magnitude for the pair of columns, it is performed atomically. This continues until no changes are made. The opposite conversion, from : : to dense is simpler, and requires a single iteration over the values, reordering their location according to the stored index. Integration with STen Finally, we show key details for implementing : : sparsity in PyTorch with STen. We first declare a new sparsity layout ( §3.1). class GroupedNMTensor: def __init__(self, val, idx, nm_strides): self.val, self.idx, self.nm_strides = val, idx, nm_strides @staticmethod def from_dense(tensor, n, m, sparse_dim, group_size, group_dim): # Call optimized dense->n:m:g conversion routine val, idx, nm_strides = dense_to_grouped_n_m( tensor, n, m, sparse_dim, group_size, group_dim) return GroupedNMTensor(val, idx, nm_strides) def to_dense(self): # Call optimized n:m:g->dense conversion routine return grouped_n_m_to_dense(self.nm_strides, self.val, self.idx) For sparse fine-tuning and training, we also make use of masked dense tensors, implemented via FixedMaskTensor. We next define the sparsifier and register implementations ( §3.3). Computed on BERT-base-uncased layer 8 attention query projection weights Unstructured n:m n:m:g g=16 n:m:g g=4 n:m:g g=1 Blocked b=2 Blocked b=4 Blocked b=8 Figure 7: Trade-off between sparsity structure and accuracy for unstructured, : , : : , and blocked sparsity. class GroupedNMSparsifier: def __init__(self, n, m, g): self.n, self.m, self.g = n, m, g @sten.register_sparsifier_implementation( sparsifier=GroupedNMSparsifier, inp=torch.Tensor, out=FixedMaskTensor) def dense_to_grouped_n_m(sparsifier, tensor, grad_fmt=None): ... # Optimized sparsifier implementation @sten.register_sparsifier_implementation( sparsifier=sten.SameFormatSparsifier, inp=torch.Tensor, out=FixedMaskTensor) def same_format_grouped_n_m_and_fixed(sparsifier, tensor, grad_fmt=None): . .. # Optimized sparsifier implementation We now define implementations for a sparse linear operator ( §3.2) using : : . These definitions are used for sparse training with masked dense tensors; a similar approach defines operators natively using the : : format for sparse inference. This suffices to enable full use of : : sparsity within PyTorch for inference and training. Linear operators will use masked dense tensors during training, with conversions implemented with our optimized conversion routines, and our sparse-dense GEMM kernel during inference. All other operators will use masked dense implementations through STen's dispatcher fallbacks. EVALUATION Here we first consider the performance and accuracy of our : : sparsity layout, and then showcase the productivity of STen by implementing multiple different sparsifiers. All experiments used PyTorch [53] 1.12 and CUDA 11.4. CPU implementations for : : sparsity were compiled with GCC 8.4.1. : : : Performance and Accuracy We now evaluate our : : sparsity layout. First, we study how well it preserves tensors compared to sparsity layouts with more or less structure. We then show that it can prune BERT BASE [12] with little loss in accuracy. Finally, we evaluate the performance of our pruned BERT models for sparse inference. Step Figure 8: BERT BASE training loss with : : sparse pruning. : : structure. Generally, there is a trade-off in pruning: adding structure to sparsity can improve performance, but may lower accuracy, as it is more restrictive. We therefore compare : : with varying group size to less (unstructured and : ) and more (blocked) structured sparsity on individual tensors. To do this, we compare the energy, a novel metric we define to be the ratio ∥ˆ∥ 1 /∥ ∥ 1 , where is the original tensor andˆis the pruned version. The energy ranges between 0 and 1, and captures the intuition that we want to preserve larger-magnitude values. In Fig. 7, we show the energy for a weight tensor from the Hug-gingFace [69] bert-base-uncased model. The trends are nearly identical for other layers, as well as layers from ResNet-50 [25]. Unstructured sparsity preserves the maximum possible energy, followed by : . Our : : sparsity achieves almost the same energy as : when = 16, with quality decreasing slightly with smaller . This is because increasing group size tends to be less restrictive: as chunks are larger, it is easier to find more : patterns. Blocked sparsity performs the worst. Hence, we can see that : : sparsity preserves energy nearly as well as : sparsity, while its additional structure can be taken advantage of for performance. Sparse fine-tuning. We now evaluate the quality of : : sparsity by sparsifying a dense, pretrained BERT BASE model (Hugging-Face bert-base-uncased) using iterative layer-wise magnitude pruning. Each layer is sparsified, and then the model is fine-tuned following standard BERT pretraining until the model recovers its original loss. This process continues until either all 72 layers are sparsified or the loss ceases to diminish. In the latter case, the remaining weights are left dense, making the overall sparsity smaller than the per-layer target. We use the Wikipedia and BookCorpus datasets, a constant learning rate of 5 · 10 −4 , and a global effective batch size of 4096 with gradient accumulation. We use three RTX3090 GPUs (local batch size 16); when all layers are sparse, each step takes a median of 42 s (compared to 18 s when dense). Figure 8 shows the training loss over the course of pruning and reports the final test masked language modeling accuracy. During training using the 1:10:4 format, we reduced the loss to 10 −4 after step 48,000, where the spike in the loss curve occurred. Our sparse models are able to maintain good performance, although fine-tuning for greater sparsity requires more training iterations. In Fig. 9, we show the overheads added by masked sparse training with STen. During training, GPU kernels are faster than the CPU kernels used for inference, so STen dispatch overheads are more noticeable, although further optimization could mitigate this. As : : is optimized for CPU inference and we make use of masking during training, there is necessarily some overhead compared to dense training. In most iterations during training, the sparsity mask is fixed as the sparsity pattern changes slowly, which reduces sparsification overhead. Recomputing the mask (e.g., when sparsity increases) is more expensive for formats with complex constraints. Distributed training is another source of overheads as tensors on different model copies may have different masks due to independent sparsification. To synchronize weights, collective operations on sparse tensors need to be performed, which we implement by converting tensors to dense, exchanging their values, and rerunning sparsification with an optimized conversion path (see §4.6). We conducted an evaluation of the overheads associated with distributed sparse training by performing weak scaling across varying numbers of GPUs, while keeping the mini-batch size fixed at 8. The experiments were conducted on 128 nodes of the Piz Daint supercomputer, utilizing the Intel Xeon E5-2690v3 CPU (2.60GHz), and one NVIDIA Tesla P100 GPU with 16GB of memory per node. Scaling efficiency, computed as the ratio of training runtime on 1 and 128 GPUs, went from 40% (0.52 s on 1 GPU vs 1.32 s on 128 GPUs) for dense training to 30% (1.02 s on 1 GPU vs 3.37 s on 128 GPUs) for masked sparse training. Despite the conservative handling of sparse tensors by requiring conversion to dense and resparsification on every step, STen exhibits less than 10% of weak scaling overhead in total. Sparse inference. We first evaluate the performance of our : : sparse-dense GEMM ( §5.1), and compare it with DeepSparse [36] (v1.1.0) on a sparse unstructured tensor. Figure 10 shows results for a 768×3072×4096 GEMM from a BERT BASE feed-forward layer run on an AMD EPYC 7742 64-core processor (3.3 GHz) and an Intel Xeon Gold 6130 16-core processor (2.8 GHz). Our : : implementation is faster than DeepSparse at every sparsity level. Overall, we see that the added structure in : : sparsity enables significant performance gains: up to 3.9× on AMD and 1.5× on Intel. We next show end-to-end sparse inference with BERT BASE using the same configurations as above in Fig. 11. We use batch size 8 and sequence length 512. We also compare with TVM [5] (v0.10.0) using either an unstructured or block pruned tensor. We tuned using AutoTVM [6] but observed it did not tune sparse operators. We see that STen improves performance over the dense baseline by up to 3.2×, with a similar speedup pattern as in the sparsedense GEMM. STen also matches or outperforms TVM up to 90% sparsity, but performs worse than DeepSparse or blocked TVM at higher sparsity. This is due to framework overheads in PyTorch; Fig. 11 breaks down our performance into STen and PyTorch runtime. While inference engines are able to eliminate these overheads through data layout transformations and operator fusion, doing so is more challenging in a general DL framework, and we leave such optimizations to future work. Nevertheless, STen is faster than PyTorch, while retaining the productivity of a full DL framework. Indeed, STen enabled the rapid prototyping and testing of the : : format, which could subsequently be integrated into inference engines to benefit from their optimizations. STen Productivity While §3 has shown that STen's interface is intuitive, we now demonstrate this practically by implementing several existing sparsifiers. As our goal is to showcase productivity, we conduct a simple evaluation in which we fine-tune a model to 50% sparsity using unstructured one-shot, iterative, and layer-wise magnitude pruning. We consider the programming effort to implement the sparsifiers and associated sparse training. We emphasize that we do not aim to outperform state-of-the-art pruning methods in this study. We use a Wide ResNet-16-8 [75] trained on CIFAR10 [34], which consists of 11.0 M parameters. Our dense model is trained following Zagoruyko and Komodakis [75], with batch size 512 for 200 epochs on four 16 GB V100 GPUs. We apply one-shot pruning and then fine-tune following the same schedule. Iterative pruning begins at 10% sparsity, then iteratively fine-tunes for 75 epochs before increasing the sparsity by 10%. Finally, layer-wise pruning prunes each layer and fine-tunes for 30 epochs, starting from the first layer. We use one V100 GPU for fine-tuning, with batch size 128. Figure 12 shows training curves for this process and Table 2 reports the final test accuracies. Each method is able to approximately recover the dense accuracy, although one-shot pruning performs slightly worse. Sparse training iterations are about 18% slower than a dense iteration (median of 0.274 s verus 0.233 s per batch), showing the overhead of sparsity is low. Layer-wise magnitude Figure 12: Training loss across sparsifiers for pruning a pretrained Wide ResNet-16-8 on CIFAR10 to 50% sparsity. Implementing this sparsification is straightforward with STen, and required no modification to the existing dense training loop, which is simply used with altered training schedules. Table 2 shows the number of lines of code added to implement sparse fine-tuning. The sparsification setup consists of general components, including the implementation of magnitude pruning and a masked tensor which stores the pruning mask. Sparse models are constructed using STen's SparsityBuilder ( §3.4). This handles initial sparsification and storage, converting tensor types, and ensures that weights are re-sparsified after gradient updates. Given this infrastructure, one-shot, iterative, and layer-wise magnitude pruning are then implemented using different different training and re-sparsification schedules, which requires only a few additional lines of code. More generally, STen supports sparsifying a broad range of models out-of-the-box, including torchvision's [59] entire suite of image classification models, covering twenty model families and over a hundred pretrained models. Adding support for sparse fine-tuning to torchvision's training code required only 25 lines of code. RELATED WORK There has been extensive work on sparsity in deep learning [27], and in scientific and high-performance computing in general [2]. Here we focus on key related work for deep learning sparsity systems. Systems for sparsity. General deep learning frameworks, such as TensorFlow [1] and PyTorch [53] provide basic support for sparse tensors and operations; more specialized frameworks, such as TVM [5,18], TFLite [43], and DeepSparse [36] aim to accelerate sparse inference. Many works target sparsity for specific kernels or networks, rather than providing a complete pipeline [7,14,20,22,26,35,37,42,46,50,52,72,74]; these could be incorporated into STen for further performance improvements. Another branch of work focuses on hardware acceleration for sparsity [11]. Graph neural network systems also extensively leverage sparsity (e.g., [3,29,30]) Other model compression and acceleration approaches. There are many other approaches to model compression beyond sparsity; these include quantization [21], parameter sharing (e.g., [38,55,62]), distillation [23], and factorization (e.g., [54,70]). Such methods are typically orthogonal and can be applied with sparsity for further improvements (e.g., [8,41]). DISCUSSION We introduced STen, an interface for productive and efficient sparsity in PyTorch. STen is highly extensible and customizable, making it easy to explore sparsity. Adding sparsity layouts or sparsifiers is simple, as is providing custom, optimized implementations for sparse operators. Further, our new : : sparsity layout provides performance competitive with highly-optimized sparse inference frameworks. With STen, sparsity "just works", typically taking only a handful of lines of code and allowing existing training code to be reused as-is. This combination of features makes STen an engine to help drive forward sparsity research and development within the broader machine learning community, allowing users to tackle problems in sparsity that previously would have been out of reach. There are many potential future directions for study. In particular, developing fully sparse training methods, where dense tensors are never materialized, is a major open problem [27] that STen allows researchers to make progress on. We also see developing improved systems for sparsity as a major direction, especially for sparse training, where automatic optimizations (e.g., fusing sparsifiers) are less explored and potentially very valuable. # Output is CSC, sparsified w/ RandomFractionSparsifier sparse_add = sten.sparsified_op( orig_op=torch.add, out_fmt=[ (sten.KeepAll(), torch.Tensor, RandomFractionSparsifier(0.5), CscTensor)], grad_out_fmt=tuple([...])) # Similar to out_fmt # Call the operator c = sparse_add(a, b) Figure 2 : 2Example sparsification of a model with STen, originally an Add followed by a MatMul (left). STen constructs sparse forward (center) and backward (right) passes.# Create sparse version of original model: snet = sb.get_sparse_model(net) Figure 4 : 4Overview of STen's dispatcher in PyTorch. Figure 5 : 5Example illustrating one chunk of our new grouped : sparsity format ( : : ) with = 2, = 4, and = 3. Figure 6 : 6Sparse-dense GEMM kernel for 3:6: sparsity format. Figure 9 : 9Masked training overheads for unstructured, : , and : : formats. Fixed sparsification keeps the same nonzero mask, while new sparsification recomputes the mask. Figure 10 : 10Median runtime of our : : sparse-dense GEMM compared with the DeepSparse optimized inference engine. Figure 11 : 11Median sparse BERT BASE inference latency. ACKNOWLEDGMENTS This work has received funding from the European High Performance Computing Joint Undertaking (JU) under grant agreements No. 955513 (MAELSTROM) and No. 101034126 (EU-Pilot). T.B.N. was supported by the Swiss National Science Foundation (Ambizione Project #185778). The experiments were conducted utilizing the resources of the Swiss National Supercomputing Centre. Table 1 : 1Sparsifier types and examples, number of passes mode over a tensor, memory requirements ( total nonzeros, block size when blocking), and sparsifier type. Some complex weight sparsifiers could be implemented more efficiently than with materialization.Sparsifier Examples Passes Memory Type Keep-all Sparse add 1 O (1) Streaming Random fraction Dropout [64] 1 O (1) Streaming Scalar threshold ReLU [47] 1 O (1) Streaming Per-block fraction : [48, 76] 2 O ( ) Blocking Scalar fraction Magnitude [17, 77] 2 O ( ) Materializing Block-wise fraction Block magnitude [39] 2 O ( ) Materializing Complex weight sparsifiers Movement [61], ℓ 0 [45], ≥ 1 O ( ) Materializing and others [27] Sparsifying an existing model requires marking a subset of the model's tensors as sparse. While this is straightforward for weights, sparsifying intermediate tensors is more challenging. Unlike when building a model from scratch, we cannot mark operators as sparse: this would require modifying or rewriting the original model definition, a significant overhead for the user. Doing so may also break compatibility with existing saved model checkpoints, complicating sparse fine-tuning. To address this, STen supports tracing an existing model to identify its intermediate tensors and operators (see §4.1), which can then be marked with the desired sparsity layouts. As an example:class SparseLinear(torch.nn.Module): def __init__(self, in_f, out_f, wsparsity): super().__init__() self.w = sten.SparseParameterWrapper( dense_to_csc_random_fraction( RandomFractionSparsifier(wsparsity), torch.randn(in_f, out_f), # Output format for gradients: (sten.KeepAll(), torch.Tensor, RandomFractionSparsifier(wsparsity), CscTensor))) def forward(self, x): return torch.matmul(x, self.w) # Calls sparse impl # Sparsify a parameter: sb = sten.SparsityBuilder() sb.set_weight( 'net.weight', # Traced name initial_sparsifier=RandomFractionSparsifer(0.9), out_format=CscTensor) # Sparsify an intermediate activation: sb.set_interm( 'net.gelu', # Traced name inline_sparsifier=RandomFractionSparsifier(0.9), tmp_format=CscTensor, external_sparsifier=KeepAll(), out_format=CscTensor) Intermediate Add MatMul S3' D D2 D1 S3 D D5 S4 Add MatMul D D2 D1 D3 D D5 D4 Intermediate Weight Weight D4 Intermediate Add MatMul S3 D D2 D1 S3'' D D5 S4'' Weight S4 inline inline init ext inline ext ext Forward Dense Forward Sparse Backward Sparse D: Dense S: Sparse same Sparsifier: Initial, Inline, External, or SameFormat Equivalent tensors match by color STen DispatcherFigure 3: Overview of STen's sparse dispatch engine.Sparse operators Sparsify existing NN sten.SparsityBuilder torch.fx tracing Implementation registry torch.mm: CSC*dense➔dense torch.mm: CSC*CSC➔CSC (RandomFraction) … Custom implementation Sparsifier lookup Masked dense fallback Implementation lookup Model construction Intercepted call dispatch entry point functions receiving sparse argumetns __torch_function__ functions manually sparsified by user sparsified_op Tensor attributes PyTorch functions Tensor methods with input modification no input modification Not allowed User class Inplace fallback User class STen Dispatch Inplace fallback User class STen Dispatch Dense fallback external libraries accepting tensors patched libraries distributed training weight sync DDP all_reduce .grad .backward .data .detach .register_hook ... 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Zhuliang Yao, Shijie Cao, Wencong Xiao, Chen Zhang, Lanshun Nie, Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). the AAAI Conference on Artificial Intelligence (AAAI)Zhuliang Yao, Shijie Cao, Wencong Xiao, Chen Zhang, and Lanshun Nie. 2019. Balanced sparsity for efficient dnn inference on gpu. In Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). Zihao Ye, Ruihang Lai, Junru Shao, Tianqi Chen, Luis Ceze, arXiv:2207.04606Sparse-TIR: Composable Abstractions for Sparse Compilation in Deep Learning. arXiv preprintZihao Ye, Ruihang Lai, Junru Shao, Tianqi Chen, and Luis Ceze. 2022. Sparse- TIR: Composable Abstractions for Sparse Compilation in Deep Learning. arXiv preprint arXiv:2207.04606 (2022). Scalpel: Customizing DNN pruning to the underlying hardware parallelism. Jiecao Yu, Andrew Lukefahr, David Palframan, Ganesh Dasika, Reetuparna Das, Scott Mahlke, ACM SIGARCH Computer Architecture News. 45Jiecao Yu, Andrew Lukefahr, David Palframan, Ganesh Dasika, Reetuparna Das, and Scott Mahlke. 2017. Scalpel: Customizing DNN pruning to the underlying hardware parallelism. ACM SIGARCH Computer Architecture News 45, 2 (2017). Wide Residual Networks. Sergey Zagoruyko, Nikos Komodakis, British Machine Vision Conference (BMVC). Sergey Zagoruyko and Nikos Komodakis. 2016. Wide Residual Networks. In British Machine Vision Conference (BMVC). Learning N:M fine-grained structured sparse neural networks from scratch. Aojun Zhou, Yukun Ma, Junnan Zhu, Jianbo Liu, Zhijie Zhang, Kun Yuan, Wenxiu Sun, Hongsheng Li, International Conference on Learning Representations. ICLRAojun Zhou, Yukun Ma, Junnan Zhu, Jianbo Liu, Zhijie Zhang, Kun Yuan, Wenxiu Sun, and Hongsheng Li. 2021. Learning N:M fine-grained structured sparse neural networks from scratch. 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{'abstract': "As deep learning models grow, sparsity is becoming an increasingly critical component of deep neural networks, enabling improved performance and reduced storage. However, existing frameworks offer poor support for sparsity. Specialized sparsity engines focus exclusively on sparse inference, while general frameworks primarily focus on sparse tensors in classical formats and neglect the broader sparsification pipeline necessary for using sparse models, especially during training. Further, existing frameworks are not easily extensible: adding a new sparse tensor format or operator is challenging and time-consuming. To address this, we propose STen, a sparsity programming model and interface for PyTorch, which incorporates sparsity layouts, operators, and sparsifiers, in an efficient, customizable, and extensible framework that supports virtually all sparsification methods. We demonstrate this by developing a high-performance grouped : sparsity layout for CPU inference at moderate sparsity. STen brings high performance and ease of use to the ML community, making sparsity easily accessible. Layout Layout Layout Sparsifiers Output Outputs Streamed Blocked Materialized Implementation defined by input/output sparsity layout Keep-all Random fraction Scalar threshold ...Figure 1: Overview of the STen sparsity programming model.supported. If there is no implementation for a sparse operator or sparsity layout, STen automatically converts to a supported implementation to ease development. This allows STen to provide a productive and efficient playground for sparsity in PyTorch, allowing researchers to pursue novel ideas without significant framework engineering overhead. To demonstrate this flexibility, we develop a novel grouped : sparsity layout ( : : , §5), building on existing : formats[48,76], for fast sparse inference at moderate sparsity levels. We also include a suite of standard sparsifiers.STen delivers competitive performance for both sparse inference and training without sacrificing its flexibility ( §6). We evaluate our : : sparsity layout for sparse inference against state-of-the-art inference engines, where we are up to 3.9×faster than DeepSparse [36] using unstructured sparsity for sparse-dense GEMM. We also show end-to-end performance for BERT BASE [12], where sparse : : models recover the original accuracy and are 3.2× faster than a dense PyTorch version. Finally, we showcase the productivity STen brings. It supports torchvision's [59] entire suite of classification models and common HuggingFace transformer models [69] out-of-the-box, and we illustrate how it is simple to sparsify existing models, run sparse fine-tuning, and integrate additional sparsity layouts. STen brings both high performance and ease of use to the machine learning community, enabling users to easily leverage sparsity, engineers to develop optimized sparse formats and operators, and researchers to quickly explore new sparsification methods.", 'arxivid': '2304.07613', 'author': ['Andrei Ivanov \nDepartment of Computer Science\nDepartment of Computer Science\nETH Zürich\nSwitzerland\n', 'Nikoli Dryden \nDepartment of Computer Science\nETH Zürich\nSwitzerland\n', 'Tal Ben-Nun \nDepartment of Computer Science\nETH Zürich\nSwitzerland\n', 'Saleh Ashkboos \nDepartment of Computer Science\nETH Zürich\nSwitzerland\n', 'Torsten Hoefler \nETH Zürich\nSwitzerland\n'], 'authoraffiliation': ['Department of Computer Science\nDepartment of Computer Science\nETH Zürich\nSwitzerland', 'Department of Computer Science\nETH Zürich\nSwitzerland', 'Department of Computer Science\nETH Zürich\nSwitzerland', 'Department of Computer Science\nETH Zürich\nSwitzerland', 'ETH Zürich\nSwitzerland'], 'corpusid': 258179995, 'doi': '10.48550/arxiv.2304.07613', 'github_urls': [], 'n_tokens_mistral': 24478, 'n_tokens_neox': 21559, 'n_words': 12290, 'pdfsha': 'd4d461288e76c7b6a36ffd6b66c26815bc7bd8e6', 'pdfurls': ['https://export.arxiv.org/pdf/2304.07613v1.pdf'], 'title': ['STen: Productive and Efficient Sparsity in PyTorch', 'STen: Productive and Efficient Sparsity in PyTorch'], 'venue': []}

arxiv

SoK: Evaluating Privacy and Security Concerns of Using Web Services for the Disabled Population Alisa Zezulak alisa.zezulak@du.edu InSpirit Lab University of Denver Colorado, Emails Faiza Tazi faiza.tazi@du.edu InSpirit Lab University of Denver Colorado, Emails Sanchari Das sanchari.das@du.edu InSpirit Lab University of Denver Colorado, Emails SoK: Evaluating Privacy and Security Concerns of Using Web Services for the Disabled Population Index Terms-Disabled PopulationPrivacy and SecurityWeb ServicesLiterature Review The online privacy and security of the disabled community is a complex field that has implications for every user who navigates web services. While many disciplines have separately researched the disabled population and their online privacy and security concerns, the overlap between the two is very high but under-researched. Moreover, a complex relationship exists between the disabled population and web services where the interaction depends on several web service developmental factors, including usability and accessibility. To this aid, we explored this intersection of privacy and security of web services as perceived by the disabled community through previous studies by conducting a detailed systematic literature review and analysis of 63 articles. Our findings encompassed several topics, including how the disabled population navigates around authentication interfaces, online privacy concerns, universal design practices, and how security methods such as CAPTCHAs can be improved to become more accessible and usable for people of all needs and abilities. We further discuss the gap in the current research, including solutions such as the universal implementation of inclusive privacy and security tools and protocols. I. INTRODUCTION The Covid-19 pandemic has necessitated people worldwide to adapt to new ways of doing things [1]. With billions of people forced to conduct their daily activities online, including attending school, working from home, grocery shopping, banking, and other critical tasks [2,3,4,5,6,7], the move to a fully digital world has been an inconvenience for some. Unfortunately, this drastic shift to online services has left many behind, particularly those who rely on usable, accessible, and inclusive services [8,9,10,11,12]. While the vulnerabilities of the disabled population have always existed, this sudden move to digital services has exacerbated existing problems [8,13], including privacy and security since vulnerable populations cannot use privacy and security tools and protocols successfully due to the disparities in usability and accessibility levels. Furthermore, these tools often fail to meet the specific requirements of the disabled population, even in fundamental areas such as authentication techniques [11,14,15]. Along with the usability and accessibility concerns, there are many data security and privacy concerns present, such as critical data access, smart home technology data usage, and inadequate authentication protocols. Additionally, the disabled population uses medical technology more than their nondisabled counterparts, but many of these tools and protocols are not accessible to users with different needs and abilities [16]. This makes accessing personal health records, and user accounts difficult for many users. Furthermore, the disabled population faces many difficulties online relating to authentication methods such as CAPTCHAs [17,18]. Most CAPTCHAs require a user to enter an alphanumeric code, which can be difficult or impossible for visually impaired users. This raises questions about if privacy and security tools are designed with different user populations in mind. To provide a comprehensive understanding of the research undertaken in this area, we conducted a systematic literature review of 2, 352 research articles on the privacy and security of web services and the disabled populations. We screened these articles by title, abstract, and full text, selecting 63 papers that focused on the privacy and security of web services as they relate to the disabled population. We then conducted a detailed thematic analysis of these papers, uncovering valuable solutions to address some privacy and security concerns of the disabled population. However, our analysis also revealed significant gaps in the research, highlighting the need for future work in this area. As far as we know, this is the first Systematization of Knowledge (SoK) paper to focus on the privacy and security challenges faced by the disabled community while accessing web services. II. RELATED WORK While still a relatively new and developing field, a growing collection of literature focuses on the privacy and security of people with disabilities using web services. A. Differing Tool Usage Perceptions: Web Services Both on and offline, the general population and disabled population have vastly different needs and abilities. As technology advances, many adults increasingly use online services such as banking, social media, email, and healthcare [19,20,21,22,23,24,25]. As a result of this increase in technology use, many of these users have privacy and security concerns related to web services and how their data is being used [26,27,28]. While these web services can benefit users greatly, researchers such as Mentis et al. have found that they also create various privacy and security risks for vulnerable populations. In addition, many adults who use these services have mild cognitive impairment and other disabilities that make it difficult to understand the implications of sharing personal information online, the importance of password management, and recognizing scams [29,30,31,32,33,34]. While these web services should make technology more accessible to all users, our SOK demonstrates that we need to perform an in-depth study to understand the needs of understudied populations. B. Privacy and Security Concerns When trying to understand more about how tool usage differs amongst these populations, the topic of authentication and CAPTCHA completion was at the forefront of six [17,35,36,37,38,39] research papers. Authentication protocols are a hallmark of online privacy and security [40,41,42,43], necessary for all users to complete to gain access to their accounts or personal information. However, some authentication methods, such as CAPTCHAs, can be difficult or impossible for disabled users to complete since they rely heavily on visual outputs [44,45,46,47]. Therefore, Fuglerud et al. proposed a talking mobile one-time-password client that would provide users with both auditory and visual outputs [36]. This tool creates an environment where various types of users can complete authentication mechanisms without being overlooked based on their needs or abilities. However, our research reveals a scarcity of authentication tools and designs tailored to address the requirements of disabled populations. III. METHODS Through this study, we aim to answer the following research questions (RQs): • RQ1: What are the privacy and security concerns related to the disabled community when interacting with web services? • RQ2: How can CAPTCHAs/authentication be improved to protect the privacy and security of people with disabilities for online communication? • RQ3: How can universal design, design for privacy, and inclusive privacy and security be implemented in different web services? To answer these questions, our literature review included several steps: (i) database search, (ii) title screening, (iii) duplicate removal, (iv) abstract screening,(v) full-text screening, and (vi)thematic analysis. Papers were included if they meet the following criteria: (1) Published in a peer-reviewed publication, (2) Published in English, (3) Technology discussed focuses on privacy and/or security of web services, (4) Target population includes a significant portion of individuals with disabilities. The exclusion criteria includes: (1) The technology discussed in the research work was not used primarily by people with disabilities, (2) The papers did not include a direct discussion of the privacy and security of users with disabilities for web services, (3) The paper was an abstract, poster, work-in-progress, or otherwise not a full paper, (4) The full-text of the papers were not available even after searching through multiple databases or after contacting the authors. Our methodology was adapted from prior works by Stowell et al. [48], Das et al. [49], Tazi et al. [50,51], Noah and Das [52], and Shrestha et al. [53,54]. A. Database Search and Title Screening We conducted our search by exploring five digital databases, namely:IEEE Xplore 1 , SSRN 2 , Google Scholar 3 , Science Direct 4 , and ACM Digital Library 5 . The data collection spanned from May to July 2021 and included any paper published before July 2021. We collected 14 papers from IEEE Xplore, 3 papers from SSRN, 1000 papers from Google Scholar, 991 papers from Science Direct, and 344 papers from ACM Digital Library. The keyword search for IEEE Xplore, SSRN, and Science Direct was "disability + privacy + security," and the "research articles" filter was applied. For ACM Digital Library, the keyword search used was "disability" AND "privacy," AND "security" with the "full text" filter applied. We used the Publish or Perish [55] software to review Google Scholar articles. The keyword search used in Publish or Perish was "privacy and security" + "disabled people." This search was limited to 1000 results by the software. We reviewed a total of 2, 352 article titles from all five databases. A paper was at this point deemed pertinent if the title discussed web services for people with disabilities, including those with specific impairments like visual, hearing, or motor impairments. Additionally, the title was required to describe a study investigating privacy and security concerns of using web services for the disabled population or the usage of web services in general about privacy or security. A paper was also only considered if it met the inclusion requirements. After duplicate removal, our corpus consisted of 138 articles. B. Abstract and Full Text Screening We manually reviewed the abstracts of all 138 papers in the research collection for relevance to our RQs. We removed 27 papers during abstract screening, leaving 111 papers for full-text screening. On these 111 papers, we conducted a full-text screening where we reviewed the methods, findings, analysis, and discussions. After the full-text screening, 63 relevant papers remained for the detailed thematic analysis. C. Data Extraction and Thematic Analysis For all 63 papers remaining in our corpus, we extracted quantitative and qualitative findings to assess the web services' privacy and security perspectives on the disabled populationfocused research conducted by prior studies. The extracted data included population samples, user experience, study design characteristics, and type of technology used (web services for our research). The results, discussion, and conclusion data from each paper were analyzed and coded according to themes identified by the first and third authors. The inter-coder reliability score for the coding was 89.4%. In places where the two authors could not agree, the second author decided. A random sample of 12 papers was taken where the abstracts, methods, results, and discussions were reviewed. This resulted in themes such as: • Type of disability: visual impairments, Down Syndrome, cognitive disabilities • Type of participant: some studies include both disabled and non-disabled people, while other studies include only disabled people • Difficulty using authentication interfaces • CAPTCHA completion can be hard or impossible for those who are blind, have low vision, or have a learning disability (dyslexia, ADHD.) The remaining papers were then evaluated by going through each and generating a complete codebook. This process yielded a codebook that consists of 33 overarching codes, which were themed into seven overarching themes including," Authentication Interface Issues ", " Privacy Concerns as Reasons for Non-Use "," Critical Data Access "," Online Vulnerability "," Solutions to authentica "," Universal Design "and" Usability of Security Tools and Protocols ". IV. FINDINGS AND DISCUSSIONS In this section, we report on our findings while addressing the RQs mentioned in the previous section. A. RQ1: Privacy and Security Concerns of Disabled People for Web Services Our first research question addresses the privacy and security concerns of people with disabilities when interacting with web services. We addressed this RQ by analyzing the different papers within the themes related to this specific research question which are four, namely:" Authentication Interface Issues ", " Privacy Concerns as Reasons for Non-Use "," Critical Data Access "," Online Vulnerability ". Table I provides the snapshot of the distribution of the papers which cater to RQ1. In the following subsections, we will provide more details about these themes. 1) Authentication Interface Issues: Authentication is a basis of security standards and protocols for web services. While CAPTCHA completion and authentication steps are often easy for non-disabled users, the disabled population faces countless difficulties accessing their online services. While analyzing papers on security concerns for people with disabilities, we found that issues with authentication interfaces were a common theme discussed. We found underlying sub-themes, such as difficulty using authentication due to technical hindrances and how each disability can affect a user's capability to complete authentication mechanisms. Four papers from the 63 in our corpus [13,17,37,38] relating to this category. One such paper discusses the success of CAPTCHA completion depending on the disabilities; for most non-disabled users, CAPTCHA completion and other forms of authentication are an almost unnoticeable part of using web services. However, users with any level of disability or impairment can find these same tasks to be difficult or impossible, as Helkala explains [17]. Through their work, Helkala explores how users with vastly different disabilities like Parkinson's disease, dyslexia, vision impairment, and upper extremity disabilities all experience different issues with CAPTCHA completion based on their abilities. In addition, this research raises important questions about how current authentication methods, such as static PIN codes, textual passwords, and onetime codes, can be altered better to fit different populations' needs and abilities. Another equally important code within this theme is the difficulty of using authentication due to technical hindrances; these difficulties discussed were at the conceptual and adoption levels. This was detailed by Bayor et al. in their research analyzing interest in using social media amongst users with intellectual disabilities. Their findings suggest that a lack of accessible authentication methods for disabled users often hinders this interest. The authors also note that voice search, auto-login, and password retrieval protocols could be alreadyexisting solutions for this user population [13] 2) Privacy Concerns as Reasons for Non-Use: In reviewing research papers on the privacy and security concerns of the disabled population when using web services, we found that an overwhelming majority of users cited privacy concerns as reasons for non-use. Every user wants their account and data to be protected from social media sites to healthcare technology. Some of the most prevalent sub-themes related to non-use were found in connection to medical technology in smart homes and concerns about health information technology used frequently by people with disabilities. If a user feels that their health information needs to be adequately protected, it was found that they often choose not to use the service at all. There are 27 papers related to this theme, as detailed in table I. One such paper analyzes the privacy and security concerns of disabled people regarding medical technology used in smart homes. Ziefle et al. researched the attitudes of disabled users towards a video-based monitoring system in the smart home environments of elderly or disabled people. They found that users would only feel comfortable with this system in their homes if strict privacy protocols were followed, including anonymity in transferring medical data, password protection, discretion, and avoidance of stigmatization [64]. Furthermore, many health information technologies are becoming popular amongst users, especially smartphone apps and websites that access medical data. Onyeaka et al. discuss how it may be difficult for some user populations, such as those with disabilities or mental health conditions, to use these smartphone apps and websites. The researchers found that many users with disabilities would withhold crucial medical information from their healthcare providers because of privacy and security concerns about how their data was being used by the healthcare apps and websites [88]. Concerns exist that these privacy and security issues could lead to further stigmatization and non-use by the disabled population. 3) Critical Data Access: We classified papers within " Critical Data Access" if they discuss data sharing, specifically medical data, and the privacy and security concerns of disabled people over their critical data. Through these papers, we determine that users have privacy and security concerns related to sharing personal health records with caretakers, healthcare providers, insurance companies, researchers, and governments. In particular, many people with disabilities feel there are privacy trade-offs in emergency situations when they do not have control over who has access to their personal medical data. Seven papers from our corpus were included in this theme [81,82,83,84,85,86,87]. One of these papers; Beach et al. discuss how technology aimed at enhancing independent living for people with disabilities is a growing field. However, there are still a lot of privacy and security concerns to consider. This is particularly relevant because the researchers found that users with disabilities are significantly more accepting of the sharing and recording personal medical information than nondisabled people [82]. This raises concerns about how disabled people are more at risk of privacy and security failures than their non-disabled counterparts. On the other hand, Solanas et al. propose m-Carer, a smart mobile device that monitors patients' movements. The researchers hope to provide a way to track and find disabled users who become lost, disoriented, or need emergency medical attention [81]. Although this new technology could help users in emergencies, it raises concerns about patient privacy invasions and how the tracking data is stored and transmitted. 4) Online Vulnerability: we classified papers that examine online vulnerabilities, particularly those that affect individuals with disabilities, as " Online Vulnerability". More than 22% of the papers in our corpus fall under this theme, making it a prevalent one. [8,29,88,89,90,91,92,93,94,95,96,97,98,99]. Many disabled users are unaware of the everchanging nature of online privacy and security issues, and must rely on the assistance of a caregiver or family member to safeguard themselves. This raises concerns about the tradeoffs between autonomy and privacy when disabled people use digital services. According to Chalghoumi et al., many disabled users are unaware of technology and web services' privacy and security concerns. The researchers found that the opinions of caregivers and family members of the disabled participant were significantly influential on the user's behavior toward online privacy [99]. This raises questions regarding how much of a disabled user's web services experience can be autonomous if caretakers substantially impact them. B. RQ2: Improving CAPTCHA/authentication The second RQ focuses on how CAPTCHAs/authentication can be improved to protect the privacy and security of people with disabilities when using web services. Some disabled users can find authentication completion impossible and are consequently unable to access their accounts. Six papers [35,100,101] from our corpus focus on solutions to improving authentication and CAPTCHAs. Table II provides the snapshot of the distribution of these papers. Theme Number of Papers Solutions to authentication/CAPTCHA Issues 3 (4.76%) [35,100,101] Some papers relating to this theme provided the solution to authentication problems; one such solution is using passtones instead of passwords, as researched by Brown and Doswell. Rather than remembering alphanumeric sequences, Brown and Doswell propose a password alternative where users would remember a sequence of sounds [100]. The researchers explain how this tool has already been implemented using photos, but using auditory passwords would improve the experience of users with visual disabilities. While explicitly a solution for visually impaired users, this solution could be widely implemented and used by people of all different needs and abilities. Similarly, accessible password managers are another solution to issues with authentication that many users face. Barbosa et al. describe their implementation of UniPass, an accessible password manager for visually impaired users on a smart device. This tool includes features such as reading prompts and messages aloud, buttons and other graphical elements are avoided, and the device vibrates to signify the need for user input [101]. The researchers found that password managers are a promising solution for the difficulties visually impaired users face with authentication mechanisms. A different way to enhance the authentication experience of disabled users when interacting with web services is Spoken CAPTCHA. Shirali-Shahreza et al. discuss how most CAPTCHA methods currently only use visual patterns, making it impossible for blind users to complete them. The researchers propose a new CAPTCHA method, Spoken CAPTCHA, where users would hear a short sound clip asking them to say a word. The user will then respond in a speech file that can be checked not to be computer generated [35]. This solution focuses on the visually impaired population and provides a way to improve authentication methods for all types of users. C. RQ3: Universal Design, Design for Privacy, and Inclusive Privacy and Security in Web Services The third RQ focuses on how universal design, design for privacy, and inclusive privacy and security can be implemented in different web services. These inclusive concepts provide design tools and protocols to make web services more accessible for various user populations, regardless of needs and abilities. We have gleaned two themes pertaining to this research question," Universal Design "and" Usability of Security Tools and Protocols ". Table III 1) Universal Design: The Universal Design concept describes how the design of all products and environments should be usable by all people without the need for adaptation or specialized design. Inclusive privacy and security and privacy by design are closely related to the overarching theme of universal design. Six papers [102,103,104,105,106,107] were included in this theme. These papers discuss the current privacy and security protocols that are most widely used and why they do not consider the needs and abilities of underserved populations such as children, older adults, people with disabilities, and people from non-Western populations. Wang et al. discuss the implementation of inclusive privacy and security tools, and protocols would prioritize the design of mechanisms that are inclusive to people with various characteristics, abilities, needs, and values [103]. Similarly, we considered papers on privacy by design and how designers and technologies must put inclusive privacy and security tools/protocols at the forefront of their design. One of the most practical ways these designers can implement privacy by design is to increase digital citizen awareness surrounding consent for data processing and usage. O'Connor et al. discuss how users must have the information they need to make informed decisions about how their data is being used [105]. 2) Usability of Security Tools and Protocols: The usability and accessibility of security tools and protocols are essential to the overarching theme of universal design. While the previous theme describes the theory of universal design, this theme explores implementations of the theory. The two papers related to this theme [36,108] present inclusive password management and two-factor authentication solutions for various user populations across two related papers. Password protection is a hallmark of online security tools and protocols. However, complicated authentication procedures to access web services can be cumbersome, especially for people with disabilities or the elderly. According to Fuglerud et al., a secure and accessible multi-modal authentication method using a onetime password client could solve this problem. Users with impairments affecting their ability to complete authentication steps now have access to auditory and visual outputs from the password client [36]. This allows all users equal access to password management tools and protocols. The second paper by Han et al. describes how current 2FA solutions all require some form of user effort, with can negatively impact the experience of disabled users or the elderly. Therefore, the researchers propose a new type of mobile 2FA, Proximity-Proof, that does not require user interactions and defends against the powerful man-in-the-middle attack [108]. According to the authors, Proximity-Proof is as secure as other 2FA methods and provides innovative ways for 2FA techniques to become more usable and accessible for all users. V. FUTURE WORK AND LIMITATION In this paper, we conducted a systematic analysis to evaluate the research articles and peer-reviewed papers published in the field of security and privacy of web services for the disabled population. We collected papers from five digital databases and limited the papers to ones available in English. As such we might have missed papers not available in these databases. However, our extensive literature review provides a detailed overview of the current research on security and privacy of web services for the disabled population. And while this gives a broad understanding of the current research and methods used, there is limited in-depth research on individual user groups within the disabled population. For example, five of the six papers relating to solutions for authentication issues were only solutions for visually impaired users. Future analyses of privacy and security concerns of the disabled population can provide valuable research into more specific subsections of the population, such as those with cognitive disabilities, mental illnesses, and different types of physical impairments. VI. CONCLUSION For many disabled users, information technology and web services can be a way to enhance their autonomy and discover new interests or communities. However, disability can make the internet a challenging place, seeing as many disabled people have trouble writing, reading, and comprehending text information, making it hard for them to understand and use basic security and privacy measures such as passwords and passwords CAPTCHAs. To that regard, we conducted a systematic literature review on 63 papers focused on the privacy and security of web services for the disabled population. Our findings reveal valuable solutions to privacy and security concerns of the disabled population, focused on universal design and inclusive privacy and security methods. Universal design, in particular, provides a way to create inclusive, accessible, and usable tools and protocols to protect the privacy and security of both the disabled and general populations online. These solutions would address issues such as authentication improvement, critical data access, online vulnerability, and usability of tools and protocols. However, our findings reveal gaps in the current research, such as a lack of implementation of these universal design methods and how solutions must focus on more subsections of the disabled population. VII. ACKNOWLEDGEMENT We would like to thank the Inclusive Security and Privacy focused Innovative Research in Information Technology (In-SPIRIT) Laboratory at the University of Denver. This research has been funded by the Faculty Research Fund (FRF) at the University of Denver. Any opinions, findings, conclusions, or recommendations expressed in this material are solely those of the authors and not of the organization or the funding agency. THE DISTRIBUTION OF PAPERS ACROSS THEMES ANSWERING THE RQ1Themes Number of Papers Authentication Interface Issues 4 (6.35%) [13, 17, 37, 38] Privacy Concerns as Rea- sons for Non-Use 27 (42.86%) [16, 39, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80] Critical Data Access 7 (11.11%) [81, 82, 83, 84, 85, 86, 87] Online Vulnerability 14 (22.22%) [8, 29, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99] TABLE I TABLE II THE IIDISTRIBUTION OF PAPERS ACROSS THEMES ANSWERING THE RQ2 provides the snapshot of the distribution of the papers which caters to the RQ3.TABLE III THE DISTRIBUTION OF PAPERS ACROSS THEMES ANSWERING THE RQ3Theme Number of Papers Universal Design 6 (9.53%) [102, 103, 104, 105, 106, 107] Usability of Security Tools and Protocols 2 (3.17%) [36, 108] https://ieeexplore.ieee.org/Xplore 2 https://www.ssrn.com 3 https://scholar.google.com/ 4 https://www.sciencedirect.com/ 5 https://dl.acm.org/ Covid-19 and digital inequalities: Reciprocal impacts and mitigation strategies. 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{'abstract': 'The online privacy and security of the disabled community is a complex field that has implications for every user who navigates web services. While many disciplines have separately researched the disabled population and their online privacy and security concerns, the overlap between the two is very high but under-researched. Moreover, a complex relationship exists between the disabled population and web services where the interaction depends on several web service developmental factors, including usability and accessibility. To this aid, we explored this intersection of privacy and security of web services as perceived by the disabled community through previous studies by conducting a detailed systematic literature review and analysis of 63 articles. Our findings encompassed several topics, including how the disabled population navigates around authentication interfaces, online privacy concerns, universal design practices, and how security methods such as CAPTCHAs can be improved to become more accessible and usable for people of all needs and abilities. We further discuss the gap in the current research, including solutions such as the universal implementation of inclusive privacy and security tools and protocols.', 'arxivid': '2302.13261', 'author': ['Alisa Zezulak alisa.zezulak@du.edu \nInSpirit Lab\nUniversity of Denver\nColorado, Emails\n', 'Faiza Tazi faiza.tazi@du.edu \nInSpirit Lab\nUniversity of Denver\nColorado, Emails\n', 'Sanchari Das sanchari.das@du.edu \nInSpirit Lab\nUniversity of Denver\nColorado, Emails\n'], 'authoraffiliation': ['InSpirit Lab\nUniversity of Denver\nColorado, Emails', 'InSpirit Lab\nUniversity of Denver\nColorado, Emails', 'InSpirit Lab\nUniversity of Denver\nColorado, Emails'], 'corpusid': 257219328, 'doi': '10.48550/arxiv.2302.13261', 'github_urls': [], 'n_tokens_mistral': 21190, 'n_tokens_neox': 18074, 'n_words': 11144, 'pdfsha': '2a3f0f1c8840191c559ea68f153fcb19f2571c3e', 'pdfurls': ['https://export.arxiv.org/pdf/2302.13261v1.pdf'], 'title': ['SoK: Evaluating Privacy and Security Concerns of Using Web Services for the Disabled Population', 'SoK: Evaluating Privacy and Security Concerns of Using Web Services for the Disabled Population'], 'venue': []}

arxiv

A comment on the "A unified Bayesian inference framework for generalized linear models" Jiang Zhu jiangzhu16@zju.edu.cn. is with Ocean College Zhejiang University Jiang Zhu is with Ocean College Zhejiang University A comment on the "A unified Bayesian inference framework for generalized linear models" April 10, 2019 DRAFT arXiv:1904.04485v1 [eess.SP] 9 Apr 2019GLMSLMexpectation propagationMMSEMAP The recent work "A unified Bayesian inference framework for generalized linear models"[2]shows that the GLM can be solved via iterating between the standard linear module (SLM) (running with standard Bayesian algorithm) and the minimum mean squared error (MMSE) module. The proposed framework utilizes expectation propagation and corresponds to the sum-product version [1]. While in[1], a max-sum GAMP is also proposed. What is their intrinsic relationship? This comment aims to answer this. I. MAX-SUM GAMP According to [1], the output scalar estimation functions of sum-product GAMP (for MMSE estimation) are [1] g out (p, y, τ p ) = (ẑ 0 −p)/τ p , z 0 = E[z|p, y, τ p ], y ∼ p(y|z), z ∼ N (p, τ p ). and −g out (p, y, τ p ) = (τ p − var(z|p, y))/τ 2 p . While for the max-sum GAMP (for MAP estimation), the output scalar estimation functions are g out (p, y, τ p ) = (ẑ 0 −p)/τ p ,(3)z 0 = argmax z F out (z,p, y, τ p ). and −g out (p, y, τ p ) = f out (ẑ 0 , y)/(τ p f out (ẑ 0 , y) − 1),(4) where F out (z,p, y, τ p ) f out (z, y) − (z −p) 2 2τ p , f out (z, y) log p(y|z). For MAP and MMSE,ẑ 0 is found via MAP or MMSE methods. Note that the output function (3) of max-sum GAMP is basically the same as the output function (1) of sum-product GAMP. We now show that the output function (4) of max-sum GAMP can also be written in the form of sum-product GAMP (2). To calculate −g out (p, y, τ p ) (4) for max-sum GAMP, we refer to the sum-product GAMP methods. By using Laplace approximation aroundẑ 0 [5], var MAP (z|p, y) is calculated as 1/var MAP (z|p, y) = −F out (ẑ 0 ,p, y, τ p ) = −f out (ẑ 0 , y) + 1/τ p .(6) Substituting (6) in (4) and eliminating f out (ẑ 0 , y), we obtain −g out (p, y, τ p ) = (τ p − var MAP (z|p, y))/τ 2 p ,(7) which has the same form as (2). It has shown that the sum-product GAMP can be decomposed as SLM and MMSE module [2], as shown in Fig. 1 It is shown in [2] that z ext A =p, v ext A = τ p , z ext B =ỹ, v ext B =σ 2 . For the AWGN channel, the output scalar estimation functions of GAMP 1 are [1] g out (p,ỹ, τ p ) = (ỹ −p)/(σ 2 + τ p ),(8)y = z + N (0,σ 2 ), z ∼ N (p, τ p ). and −g out (p,ỹ, τ p ) = 1/(σ 2 + τ p ). According to expectation propagation (EP),ỹ andσ 2 is calculated as [2] 1 σ 2 + 1 τ p = 1 var MAP (z|p, y) , (10a) ỹ σ 2 +p τ p =ẑ 0 var MAP (z|p, y) .(10b) Substituting (10) in (8) and (9) and eliminatingσ 2 andỹ, one obtains (3) and (7). As a result, the sum-product GAMP can be decomposed as SLM and MAP module shown in 2. II. CONCLUSION This note reveals the difference between max-sum GAMP and sum-product GAMP. Specifically, maxsum GAMP uses the MAP and Laplace approximation to calculate the MAP estimate and variance of z, while sum-product GAMP performs the MMSE and calculates the MMSE estimate and variance of z. For both max-sum GAMP and sum-product GAMP, EP is used to update the messages [3,4]. . In the following, we show that max-sum GAMP can be decomposed as SLM and MAP module.Fig. 1. A unified Bayesian inference framework proposed in[2]. It is shown that utilizing the unified inference framework, many standard Bayesian inference algorithm can be extended to solve the GLM.April 10, 2019 DRAFT module A SLM x y module B   y Ax w MMSE ext A z ext A v ext B z ext B v   2 ,diag( )  w σ module A SLM x y module B   y Ax w MAP ext A z ext A v ext B z ext B v   2 ,diag( )  w σ Fig. 2. A variant of the unified Bayesian inference framework [2]. Here MAP is used instead of MMSE in module B. Sum-product GAMP and max-sum GAMP are the same in this setting. April 10, 2019 DRAFT Generalized approximate message passing for estimation with random linear mixing. S Rangan, arXiv:1010.5141arXiv preprintS. Rangan, "Generalized approximate message passing for estimation with random linear mixing," arXiv preprint, arXiv:1010.5141, 2012. A unified Bayesian inference framework for generalized linear models. X Meng, S Wu, J Zhu, IEEE Signal Process. Lett. 253X. Meng, S. Wu, and J. Zhu, "A unified Bayesian inference framework for generalized linear models," IEEE Signal Process. Lett., vol. 25, no. 3, pp. 398-402, 2018. A family of algorithms for approximate Bayesian inference. T Minka, Mass. Inst. Technol. Dept. of Electrical Engineering and Computer SciencePh.D. dissertationT. Minka, "A family of algorithms for approximate Bayesian inference," Ph.D. dissertation, Dept. of Electrical Engineering and Computer Science, Mass. Inst. Technol., Cambridge, MA, USA, 2001. An expectation propagation perspective on approximate message passing. X Meng, S Wu, L Kuang, J Lu, IEEE Signal Process. Lett. 228X. Meng, S. Wu, L. Kuang, and J. Lu, "An expectation propagation perspective on approximate message passing," IEEE Signal Process. Lett., vol. 22, no. 8, pp. 1194-1197, Aug. 2015. D J C Mackay, information Theory, Inference and Learning Algorithms. Cambridge University PressD. J. C. MacKay, information Theory, Inference and Learning Algorithms, Cambridge University Press, 2003.

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{'abstract': 'The recent work "A unified Bayesian inference framework for generalized linear models"[2]shows that the GLM can be solved via iterating between the standard linear module (SLM) (running with standard Bayesian algorithm) and the minimum mean squared error (MMSE) module. The proposed framework utilizes expectation propagation and corresponds to the sum-product version [1]. While in[1], a max-sum GAMP is also proposed. What is their intrinsic relationship? This comment aims to answer this.', 'arxivid': '1904.04485', 'author': ['Jiang Zhu jiangzhu16@zju.edu.cn. \nis with Ocean College\nZhejiang University\n\n', 'Jiang Zhu \nis with Ocean College\nZhejiang University\n\n'], 'authoraffiliation': ['is with Ocean College\nZhejiang University\n', 'is with Ocean College\nZhejiang University\n'], 'corpusid': 104292468, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2124, 'n_tokens_neox': 1805, 'n_words': 966, 'pdfsha': '811a1473d900a474d560a072d6ab3cb8473247ff', 'pdfurls': ['https://arxiv.org/pdf/1904.04485v1.pdf'], 'title': ['A comment on the "A unified Bayesian inference framework for generalized linear models"', 'A comment on the "A unified Bayesian inference framework for generalized linear models"'], 'venue': []}

arxiv

Higher-genus corrections to black-string solution 4 Mar 1996 M Z Iofa Nuclear Physics Institute Moscow State University 119899MoscowRussia Higher-genus corrections to black-string solution 4 Mar 1996arXiv:hep-th/9603008v1 One-string-loop (torus topology) corrections to black-string backgrounds corresponding to gauged SL(2, R) × R/R WZW model are calculated using β-function equations derived from string-loop-corrected effective action. Loop-corrected backgrounds are used to calculate ADM mass of the black string. * Introduction Recently much attention have recieved solutions of the gauged Wess-Zumino-Witten (WZW) G/H-models which provide conformal field theories interpreted as describing geometries of black holes, black strings, etc. [1,2,3,4,5,6,7,8]. These solutions satisfy O(α ′ ) β-function equations 1 , but there are also conformally exact results valid in all orders in α ′ [8,10,11]. However, usually these solutions are discussed for conformal field theories defined on manifolds of topology of the sphere, i.e. at the tree level of string-loop expansion. Some time ago, in papers [12], string-loop corrections to the tree-level solutions were discussed for 2D black-hole solution of gauged SL(2, R)/U(1) WZW model. It was noted that in bosonic string theory, as a result of regularization of divergent integrals over the moduli, there appear imaginary corrections to the lowest genus solutions. Possible imaginary corrections to the mass of black hole could be interpreted as a manifestation of quantum instability of solution (stable at the tree level). However, in 2D theories the question of modular divergences is somewhat ambiguous because in this case modular divergences are absent. 2D theory can be considered as a limit from D > 2-dimensional models which can have modular divergences, but in this setting the problem requires more careful analysis. At present, a large variety of gauged WZW models was investigated which yield solutions interpreted as backgrounds of string theories in dimensions D ≥ 3 . In this paper, starting from loop-corrected renormalized string effective action (EA), we calculate loop corrections to tree-level backgrounds for the gauged SL(2, R) × R/R WZW model [2] which is the first one from the set of SL(2, R) × R N /R models [3,8] associated with D = (N + 2)-dimensional backgrounds. Asymptotics of backgrounds are used to calculate string-loop-corrected ADM mass of the black string. After fixing in sect.2 some notations, in sect.3 we introduce basic formulas of Tseytlin's approach to construction of string-loop-corrected EA. In sect.4 general expressions are applied to the one-loop case, i.e. for the torus topology. In sect.5 we calculate asymptotics of background solutions to β-function equations. In sect.6 these asymptotics are used to calculate loop-corrected ADM black-string mass. In sect.7 we discuss duality for the loopcorrected solutions of β-equations. Sect.8 contains concluding remarks and discussion. 2. Gauged WZW models provide a natural framework for Lagrangian realization of coset models and form a bridge between conformal field theories and σ-model description of strings propagating in nontrivial backgrounds [1,13]. For the gauged SL(2, R) × R/R WZW model, after setting the axial gauge and integrating out nonpropagating fields, in the limit of large central extension parameter k, in the leading order in 1/k, one obtains the action [2] I = k 4π d 2 z − 1 − 1 + λ r ∂t∂t + 1 − λ r ∂x∂x + 1 4 ∂r∂r (r − λ)(r − 1 − λ) + + λ 1 + λ (1 − 1 + λ r ) ∂x∂t −∂x∂t + 1 k √ hR (2) (h)Φ(r)    .(1) With identification 1 k → α ′ , where α ′ is the string constant in dimensionless units, the action (1) can be interpreted as the action for the closed string propagating in 3D spacetime equipped with the metric ds 2 = G µν dx µ dx ν = − 1 − 1 + λ r dt 2 + 1 − λ r dx 2 + 1 4 dr 2 (r − λ)(r − 1 − λ)(2) antisymmetric tensor gauge field B tx = λ 1 + λ 1 − 1 + λ r(3) and dilaton Φ = 1 2 (a − ln r). Backgrounds (2)-(4) are solutions of equations of motion derived from the O(α ′ ) part of string effective action (EA) [2,3,9] S = a 0 d D x |G| e −2Φ Λ − α ′ 2 R + 4D 2 Φ − 4(DΦ) 2 + O(α ′2 )(5) for D = 3. Here Λ = D − 26 3 ;R = R − H 2 12 Equations of motion following from (5) are equivalent to conditions of Weyl invariance of the theory with the σ-model action I = 1 4πα ′ d 2 z √ h (G µν h ab + B µν ε ab √ h )∂ a x µ ∂ a x ν + α ′ R (2) Φ(x)(6) where h ab is the world-sheet metric, and can be symbolically written as [14,15] β i (ϕ i ) = 0. (ϕ i = G µν , B µν , Φ)(7) 3. In closed bosonic string theory, the full string EA is obtained as the renormalized generating function of massless string amplitudes S(ϕ R ) =Ẑ R (ϕ R ) =Ẑ(ϕ(ǫ), ǫ) where the bare fields ϕ(ǫ) are expressed as perturbative expansion in ln ǫ with the coefficients depending on renormalized fields ϕ R . Generating functionẐ(ϕ(ǫ), ǫ) is constructed as the sum of contributions from all the genera χ = 2 − n Z(ϕ(ǫ), ǫ) = ∂ ∂ ln ǫ ∞ n=0Z n (ϕ(ǫ), ǫ)(8) where n is the number of handles of the world-sheet surface [17,18]. Herē Z n = [dµ(τ, ǫ)] n Z n(9) and Z n = a n D[x, h] ′ | det G|e −I is the partition function obtained by integration over all (functional) variables except for the moduli τ of the world-sheet metric h. The basic assumption aboutẐ(ϕ(ǫ), ǫ) is that it is perturbatively renormalizable both in α ′ and string-loop expansions [17,18]. An important aspect of this property is that all divergences (modular and ultraviolet) are regularized in a universal way by the same cutoff parameter ǫ. The derivative with respect to ln ǫ takes care of the properly accounted divergent volume of the Möbius group. An explicit realization of such regularization is provided by Schottky parametrization [19,20,21,18] of the extended moduli space. In this parametrization, a surface of genus χ = 2 − n is mapped on the complex plane C 2 with n pairs of holes with the pairwise identified boundaries. On the complex plane C acts the group SL(2, C). If the corresponding Möbius symmetry is not fixed, then the volume of the group SL(2, C) enters the amplitudes as the universal divergent factor 3 . Fixing 3 complex parameters of the group SL(2, C), one reduces the number of independent moduli to 3n − 3. Divergences of the amplitudes can appear either if positions of several vertex operators (punctures) tend to each other or/and if the holes from the handles shrink to a point. In Schottky parametrization, all the divergences are universally regularized by introducing the "minimal distance" ǫ which enters propagators as well as integration measure over the moduli. Partition function Z n has the following form [17] Z n = a n e χΛ 2 ln ǫ d D x |G| e −χΦ 1 + α ′ b (n) 1R + b (n) 2 D 2 Φ + . . . ,(10) where b (n) 1 = π V d 2 z √ h G(z, z ′ ) − π d 2 z √ h ∇ a ∇ ′a G(z, z ′ ) G(z, z ′ ) − (∇ a G(z, z ′ )) 2 z=z ′ b (n) 2 = − 1 4 d 2 z √ h R (2) (h)G(z, z) (11) V = d 2 z √ h. Here G(z, z ′ ) is the regularized propagator on the world sheet of genus χ = 2 − n with a metric h (regularized Green function of scalar Laplacian). The coefficients b (n) 1,2 contain logarithmically divergent parts which appear from the limit of coinciding arguments in the propagators b (n) 1 = 1 2 ln ǫ +b (n) 1 ; b (n) 2 = (n − 1) ln ǫ +b (n) 2 ,(12) and are defined up to transformations of the finite partsb G µν → G µν + α ′ (a 1 R µν + a 2 G µν R + a 3 D µ D ν Φ + a 4 G µν D 2 Φ + . . .) + . . . ,(13)Φ → Φ + α ′ b 1 R + . . . , B µν → B µν + α ′ c 1 D λ H λ µν + . . . which do not change the massless sector of the (tree) string theory S-matrix [17,18]. Renormalized partition function Z R n (ϕ R ) is obtained by substituting expressions for bare fields in terms of renormalized fields. In the leading order in α ′ ln ǫ one has ϕ i = ϕ i R − β i (ϕ R ) ln ǫ/µ + . . . ..(14) 4. The tree-level (topology of sphere) generating functionalZ 0 contains no integration over the moduli. At the one-string-loop level (topology of torus), the "extended" moduli space is parametrized by three complex parameters ξ, η and k. The measure on the "extended" moduli space is dµ 1 = d 2 ξd 2 η |ξ − η| 4 [d 2 k].(15) Here |ξ − η| has the meaning of the distance between the centers of the holes from the handle on the complex plane C 4 . In parametrization k = e 2πiτ , the measure [d 2 k] is given by [d 2 τ ] = d 2 τ τ 2 2 τ 2 |η(τ )| 4 − D−2 2 ,(16) where η(τ ) is the Dedekind η-function η(τ ) = k 1 24 ∞ 1 (1 − k m ). Note that k ∼ e −2πτ 2 as τ 2 → ∞ . Propagator on the complex plane C with two discs from the handle removed is G(z 1 , z 2 ) = − 1 4π ln |z 1 − z 2 | 2 + ǫ 2 ∞ m=1 (1 − λk m )(1 − λ −1 k m ) (1 − k m ) 2 + (ln |λ|) 2 2πτ 2 ,(17) where λ = (z 1 − ξ)(z 2 − η)(z 1 − η) −1 (z 2 − ξ) −1 . Singularity at ξ = η in the measure (15) is regularized by the same cutoff as in the propagator (17): |ξ − η| 2 → |ξ − η| 2 + ǫ 2 . Performing integrations in the formulas (11), one obtains the generating functionalZ 1 in the form [17,18] Z 1 = a 1 d D y |G| [d 2 τ ] ln ǫ 1 + α ′ 2R ln ǫ + α ′ (b (1) 1R + b (1) 2 (DΦ) 2 ) + O(α ′ 2 ) .(18) Here the first logarithmic factor appears from integration over the moduli, the second one is due to ultraviolet divergences in the propagators at the coinciding arguments. All constant factors are included in a 1 . 5 Collecting tree-level and one-loop contributins, one haŝ Z(ϕ(ǫ), ǫ) = a 0 e Λ ln ǫ d D x |G| e −2Φ Λ − α ′ 2 R + 4(DΦ) 2 + O(α ′2 ) + (19) a 1 [d 2 τ ] d D x |G| [1 + α ′R ln ǫ + α ′ (b (1) 1R + b (1) 2 (DΦ) 2 ) + O(α ′ 2 )] 4 To be precise, the measure (14) can be used to calculate N ≥ 3-point amplitudes. To calculate N ≤ 3-point amplitudes and, in particular, the vacuum functionalZ 1 some modifications are required [18] yielding the final expression (18). 5 Substituting propagator (17) in expressions (11), we obtain thatb (1) 1 =b(1)2 = 0. However, nonzero termsb(1) 1,2 can be generated by transformations (13). Renormalized generating functionalẐ R is obtained by substituting expressions for bare fields in terms of renormalized ones and taking into account additional terms from stringloop divergences (cf. with (13)) [17,18]. ϕ i = ϕ i R − β i (ϕ R )α ′ ln ǫ/µ + δβ i (ϕ R )α ′ ln ǫ/µ + . . .(20) Additional terms δβ i (ϕ R ) are obtained from the requirement to cancel string-loop ln ǫ terms in (18) and are equal to δβ i = ρe 2Φ R d 4 , 1 2 g R µν , B R µν ,(21)where ρ = 2a 1 /a 0 [d 2 τ ] . Substituting (20) in (19), we obtain the expression for the renormalized generating functionalẐ R which includes contributions from sphere and torus topologiesẐ R = a 0 d D x |G|e −2Φ Λ − α ′ 2 R + 4(DΦ) 2 + ρ 2 e 2Φ 1 + α ′R ln µ + α ′ (b (1) 1R + b (1) 2 (DΦ) 2 ) + . . .(22) (henceforth we omit the subscript R). It is seen that the formal effective parameter of string-loop expansion is ρe 2Φ . Using the freedom in the choice of reparametrization of fields (13) the terms α ′R ln µ + α ′ (b (1) 1R + b(1) 2 (DΦ) 2 ) can be set to zero. Adding to the action (22) the total derivative 2D 2 (e −2Φ ) to have the same tree-level part of the action as in (5), one finally obtains the renormalized EA S = a 0 d D x |G|e −2Φ Λ − α ′ 2 R + 4D 2 Φ − 4(DΦ) 2 + ρ 2 e 2Φ .(23) 5. Our next aim is to find string-loop corrections to tree-level solutions of eqs. (6). Variation of EA (22) yields the following equations of motion: β G µν + δβ G µν =R µν + 2D µ D ν Φ + ρ 2α ′ G µν e 2Φ = 0,(24)β Φ =R − 2Λ α ′ − 4(DΦ) 2 + 4(D 2 Φ) = 0,(25)β B µν = D λ ΦH λµν = 0.(26) Tree-level backgrounds (2)-(4) depend on a single parameter r and provide an example of the "rolling moduli" solution [26] to non-linear eqs. (7). In the following, having in view calculation of the mass of the black string, we shall be interested in finding asymptotics of solutions to nonlinear equations (24)-(26) at r → ∞. In this limit, we can linearize the system (24)-(26) and solve it explicitly. Solving the tree-level eqs. (24)-(26) (with ρ = 0), one obtains the "rolling" solution for the metric and dilaton (together with the corresponding solution for the antisymmetric tensor) of the form ds 2 = − sinh 2 γz 2 λ + cosh 2 γz 2 dt 2 + cosh 2 γz 2 λ + cosh 2 γz 2 dx 2 + dz 2 ,(27)Φ = 1 2 (a − ln cosh 2 γz 2 ) where γ 2 = 2|Λ| α ′ . Taking γ 2 = 4 (α ′ in dimensionless units), and introducing new variable r by the relation r = λ + cosh 2 z one obtains the solution (2)-(4) of the gauged WZW model. The metric and dilaton (27) are asymptotic to the flat-space solution (η µν , Φ 0 ), where Φ 0 = 1 2 (a − γ|z|). Flatspace solution can be considered as the limiting form of the "rolling" solution (27) as α ′ → 0 (γ → ∞) 6 . To solve loop-corrected equations, let us introduce new variable z by the relation r = λ + cosh 2 γz 2 + f (ρ, z), where the function f is chosen so that in new variables the zz component of the metric is again equal to unity. Asymptotically as z → ∞, f (z) = O(ρz n e −γ|z| ) with some n. As in the case ρ = 0, we are looking for solution for the metric and dilaton asymptotic to the flat-space solution. Writing the metric and dilaton as G µν = η µν + h µν Φ = Φ 0 + ϕ, where h µν and ϕ are of order O(ρz n e −γ|z| ) and linearizing the equations about the flatspace solution, we obtain h ′′ tt − 2Φ 0 ′ h ′ tt + ρ α ′ e 2Φ 0 = 0 (28) h ′′ xx − 2Φ 0 ′ h ′ xx − ρ α ′ e 2Φ 0 = 0 h ′′ tt − h ′′ xx + 4ϕ ′′ + ρ α ′ e 2Φ 0 = 0 (29) h ′′ µν − 2Φ 0 ′ h ′ µν = 0 (µ = ν).(30) Here primes stand for derivatives with respect to z. The term H 2 µν is asymptotically of order O(e −2γ|z| ) and can be neglected as a small correction to the leading terms. Integrating eqs. (28) we get h ′ tt = e 2Φ 0 − ρ α ′ z + c t (31) h ′ xx = e 2Φ 0 ρ α ′ z + c x(32) and h ′ µν = e 2Φ 0 c µν(33) The constants c t and c x are assumed to be independent of ρ and can be defined by taking the limit ρ = 0 and comparing with the tree-level solution. In the same way, assuming that the constants c µν are ρ-independent and comparing (33) with the tree-level solution, we set c µν = 0. Integrating eq. (29) we get h ′ tt − h ′ xx + 4ϕ ′ + ρ α ′ dze 2Φ 0 = const. 6 Dilaton Φ 0 is composed from two branches of solutions to the flat-space dilaton equation γ 2 4 = (Φ 0 ′ ) 2 . In the following, this results in "constants" having the sgnz factor. Note that we are interested only in asymptotic region of large |z| where, up to exponentially small terms, one can use a smooth approximation of Φ 0 , for example, Φ from (27). Adjusting the const to have asymptotically vanishing solution, we obtain h ′ tt − h ′ xx + 4ϕ ′ = − ρ α ′ γ sgnze a−γ|z|(34) On the other hand, linearizing eq. (25) about the flat-space solution, we have h ′′ tt − h ′′ xx − 2Λ α ′ − 4(Φ 0 ′ ) 2 − 8Φ 0 ′ ϕ ′ + 4ϕ ′′ − 8Φ 0 ′ (h ′ tt − h ′ xx ) = 0.(35) Noting that − 2Λ α ′ − 4(Φ 0 ′ ) 2 =d dz − 2Φ 0 ′ (h ′ tt − h ′ xx + 4ϕ ′ ) = 0 and solved as h ′ tt − h ′ xx + 4ϕ ′ = ce 2Φ 0(36) Comparing (34) and (36), we see that both forms of solution are equivalent up to exponentially small corrections if we make the identification c = − ρ α ′ γ sgnz. 6. Having obtained loop-corrected asymptotics of backgrounds, we can calculate stringloop-corrected mass of black string. In standard gravity interacting with matter, for a class of metrics which asymptotically sufficiently quickly approach the flat-space metric, the total energy of a field configuration can be defined in the framework of canonical approach [22,23,24] 7 . The total energy is defined as the value of the hamiltonian taken on the shell of zero constraints {Ψ} = 0. The resulting expression is of the form of space integral over the total derivative E = − 1 κ D d D−1 x∂ i ( |G D−1 |f i )| {Ψ}=0 ,(37) where f i = G lm,k (E il E km − E ik E lm ) Here G ik is the spacial (D − 1)-dimensional part of the metric, G D−1 = det G ik , and E ik G kl = δ i l . For solutions with G 0i = 0 this formula is simplified because in this case E ik = G ik . In dilatonic gravity, separating the α ′ dependence in a 0 = N(α ′ ) −D/2 and introducing the D-dimensional gravitation constant 1 κ D = N 2 (α ′ ) − D−2 2 , the action (22) is written as S = − 1 κ D d D y |G|e −2Φ R − 4(DΦ) 2 + 4D 2 Φ − 2Λ α ′ + ρ α ′ e 2Φ .(38) For the genus zero part of the action (ρ = 0), for solutions which sufficiently rapidly approach the vacuum solution (η µν , Φ 0 ), calculations similar to those in the standard gravity give [25] E = − 1 κ D d D−1 x∂ i e −2Φ |G D−1 |(f i − 4G ik ∂ k ϕ) | {Ψ}=0 ,(39) Here we again assumed that G 0k = 0. The expression (39) is valid also for the action (38) containing the one-string-loop correction, because the latter contributes only to the potential part of the action (38). In the case D = 3, for solutions sufficiently rapidly approaching the vacuum configuration, the divergence in the integrand in (39) is asymptotically equal to e −2Φ 0 (h ′ xx − 4ϕ ′ ) ′ . Coordinate x asymptotically measures distances along the string and z is the transverse coordinate. Substituting the asymtotic expression for the black string solution (34) in (39) we obtain the mass of the black string per unit length E = −N(α ′ ) 1/2 e −2Φ 0 h ′ tt − ρ α ′ γ | z→∞ .(40) For the tree-level solution (ρ = 0), the expression (40) reproduces the the mass of the black string calculated in ref. [2]: E| ρ=0 = 8N √ k (1 + λ)e −a where we substituted α ′ = 1 k . If string-loop correction is taken into account, h ′ tt contains a term linear in z, and expression (40) diverges, the divergence being proportional to ρ. This means that, modifying the tree-level bosonic string action by one-loop corrections, one cannot define finite ADM mass for the black-string solution. If one defines the energy by subtracting the infinite part which is proportional to the mixing parameter ρ, one again obtains an expression independent of ρ. 7. It is well known that the 3D black-string solution is dual to the spherically-symmetric 3D black hole solution [27,28,29,30]. To have the dual solution in the standard form, solution (2)-(4) is written as ds 2 = − 1 − r 2 + r 2 dt 2 + 1 − r 2 − r 2 dx 2 + 1 − r 2 + r 2 −1 1 − r 2 − r 2 −1 l 2 dr 2 r 2 (41) Bxt = r + r − r 2 , Φ = 1 2 (a − ln r 2 ). In new variables t = al(x −t), ϕ = a(r 2 +t − r 2 −x ), a = (r 2 + − r 2 − ) −1/2 it takes the form ds 2 = − M − J 2 4r 2 dt 2 + 2 l dtdϕ + 1 r 2 dϕ 2 + r 2 l 2 − M + J 2 4r 2 −1 dr 2 (42) B ϕt = − J 2r 2 , Φ = 1 2 (a − ln r 2 ). where M = r 2 + + r 2 − l 2 , J = 2r + r − l Dual (with respect to the variable ϕ) transformation of the fields (42) gives the solution ds 2 = − M − r 2 l 2 dt 2 + Jdtdϕ + r 2 dϕ 2 + r 2 l 2 − M + J 2 4r 2 −1 dr 2 (43) B ϕt = − r 2 l 2 , Φ = 0. The metric ds 2 is the black-hole solution in 3D Einstein gravity [27,28,29,30]. The fields (43) are solutions of equations of motion for the action (5) with the cosmological constant Λ = − 2α ′ l 2 . Duality transformation [31] leaves the form of equations of motion unchanged. Let us consider duality transformations in the theory with the loop-corrected action (23). Requiring that equations of motion (24)-(26) do not change their functional form under duality transformations (cf. [31]) ϕ i = ϕ(φ) + ρδϕ(φ), we have β i (ϕ) + ρδβ i (ϕ)| ϕ i =ϕ(φ)+ρδϕ(φ) = β i (φ) + ρδβ i (φ). Keeping terms linear in ρ, we obtain δβ + δϕ j ∂β i ∂ϕ j | ϕ i =ϕ i (φ) = δβ i (φ) The number of equations on the functions δϕ i is equal the number of the functions δϕ i . Finding the functions δϕ i , we obtain the duality transformations which leave the functional form of the loop-corrected β-equations unchanged. 8. Conclusions and discussion. In this paper, starting from string-loop-corrected renormalized EA, we calculated onestring-loop corrections to black-string backgrounds which, on one hand, are obtained from the gauged WZW model, and, on the other hand, are solutions to O(α ′ ) β-function equations derived from tree-level EA. Although final calculations were performed for an example of SL(2, R) × R/R WZW model, our discussion was quite general: all the expressions can be written in D-dimensional form and applied to the case of a general SL(2, R) × R N /R model. It was found that backgrounds acquire corrections of order ρ [d 2 τ ], where ρ is parameter accounting for an admixture of genus-one contribution to the tree-level part of EA. From (15) it follows that for all SL(2, R) × R N /R models and, in particular, for the 3D black-string solution, the integral over the moduli is exponentially divergent. This divergence is the well-known tachyonic divergence in bosonic string theory, which is absent in superstring theory. In paper [32] this problem was discussed in the framework of fermionic string theory and it was argued that in this theory there appear no terms in EA which could give divergent corrections to tree-level result. Using the one-string-loop-corrected backgrounds obtained by solving the β-equations, we calculated the ADM mass of the black string. It appeared that the result is divergent, the divergence being proportional to the mixing parameter ρ. Redefining the energy (mass) by subtracting the infinite part, one obtains an expression independent of ρ. Thus, in the present case, the conjecture of ref. [12] about the imaginary string-loop corrections to the tree-level mass does not work. Although our calculations were restricted to one-loop contributions, higher-order corrections can be discussed as well. Loop-corrected EA has the following structure: Z =Ẑ 0 +Ẑ 1 +Ẑ 2 + . . . = a 0 d D x √ Ge −Φ (−2Λ + α ′ R + . . .) + a 1 d D x √ G(1 + . . .)+ (44) a 2 d D x √ Ge Φ (1 + . . .) + . . . (here a i include integrals over the moduli). It is seen that higher-genus corrections are accompanied by the factors e χΦ 2 and are exponentially suppressed at spatial infinity for the black-string solution in question. Thus, in any finite order in string-loop expansion, corrections from higher topologies will not contribute to the ADM mass. For the SL(2, R)/R model, loop corrections can be calculated in the same way as above. However, in this case, because of 2D relation R µν = 1 2 G µν R, β-equations are much simpler and can be easily solved exactly. In notations of [33] we have Φ = Qη 2 G µν = diag[−g(η), g(η) −1 ] where g(η) is now solution of loop-corrected equation g ′′ = Qg ′ + ρ α ′ e Qη(45) of the form g(η) = 1 + ae Qη + ρη α ′ Q e Qη . Note that in this case, as in 3D theory there appear the term O(ηe Qη ). The relation Q 2 = |Λ| does not recieve loop corrections and is the same as at the tree level. Since in 2D theory there is no tachyon in the spectrum, integration measure over the moduli contains no exponential factors. 0 is the equation for the vacuum dilaton, equation (35) can be rewritten in a form See, however, paper[9] where backgrounds of SL(2, R) × R/R model were shown to satisfy β-function equations in O(α ′ 2 ) approximation. More exactly, a surface is mapped on compactification of C i.e. on the 2-sphere[18].3 This is true for n ≥ 3-point amplitudes. It should be mentioned that in the absence of a preferred asymptotic frame, the notions of ADM energy and mass are nonunique. As usual, we introduce energy as a quantity conjugate to variable t. Acknowledgements. I would like to thank V.Belokurov, R.Metsaev and I.Tyutin for useful discussions. . E Witten, Phys. Rev. 44314E.Witten, Phys. Rev. D44 (1991) 314. . J H Horne, G T Horowitz, Nucl. Phys. 368444J.H.Horne and G.T.Horowitz, Nucl. Phys. B368 (1992) 444. . P Ginsparg, F Quevedo, Nucl. Phys. 385527P.Ginsparg and F.Quevedo, Nucl. Phys. B385 (1992) 527. . P Horava, Phys. Lett. 378101P.Horava, Phys. Lett. B378 (1992) 101. . C Nappi, E Witten, Phys. Lett. 293309C.Nappi and E.Witten, Phys. Lett. B293 (1992) 309. . M Crescimanno, Mod. Phys. Lett. 7489M.Crescimanno, Mod. Phys. Lett. A7 (1992) 489. . E S Fradkin, V Ya, Linetsky, Phys. Lett. 27773E.S.Fradkin and V.Ya.Linetsky, Phys. Lett. B277 (1992) 73. . I Bars, K Sfetsos, Phys. Lett. 277844Phys. Rev.I.Bars and K.Sfetsos, Phys. Lett. B277 (1992) 269. Phys. Rev. D46 (1992) 4510, Phys. Rev. D48 (1993) 844. . K Sfetsos, A A Tseytlin, Phys. Rev. 492933K.Sfetsos and A.A.Tseytlin, Phys. Rev. D49 (1994) 2933. . R Dijkgraaf, H Verlinde, E Verlinde, Nucl. Phys. 371269R.Dijkgraaf, H.Verlinde and E.Verlinde, Nucl. Phys. B371 (1992) 269. . A A Tseytlin, CERN-TH.6804/93ReportA.A.Tseytlin, Report CERN-TH.6804/93 . J Ellis, N E Mavromatos, D V Nanopoulos, Phys. Lett. 276246J.Ellis, N.E.Mavromatos and D.V.Nanopoulos, Phys. Lett. B276 (1992) 56, B278 (1992) 246. . I Bars, D Nemeschansky, Nucl. Phys. 34889I.Bars and D.Nemeschansky, Nucl. Phys. B348 (1991) 89. . C G Callan, D Friedan, E Martinec, M J Perry, Nucl. Phys. 262593C.G.Callan, D.Friedan, E.Martinec and M.J.Perry, Nucl. Phys. B262 (1985) 593. . A A Tseytlin, Nucl. Phys. 294383A.A.Tseytlin, Nucl. Phys. B294 (1987) 383. . A A Tseytlin, Phys. Lett. 208221A.A.Tseytlin, Phys. Lett. B208 (1988) 221. . A A Tseytlin, Phys. Lett. 223165A.A.Tseytlin, Phys. Lett. B223 (1989) 165. . A A Tseytlin, Int. Journ. Mod. Phys. 5589A.A.Tseytlin, Int. Journ. Mod. Phys. A5 (1990) 589. . C Lovelace, Phys. Lett. 32703C.Lovelace, Phys. Lett. B32 (1970) 490,703. . V , Nuovo Cim. 2321V.Alessandrini, Nuovo Cim. 2A (1971) 321. . M Kaku, L P Yu, Phys. Rev. 32992M.Kaku and L.P.Yu, Phys. Rev. D3 (1971) 2992. . T Regge, C Teitelboim, Ann. Phys. 88286T.Regge and C.Teitelboim, Ann. Phys. 88 (1974) 286. . L D Faddeev, Uspekhi Fiz, Nauk. 136435L.D.Faddeev, Uspekhi Fiz. Nauk. 136 (1982) 435. D M Gitman, I V Tyutin, Quantization of Fields with Constraints. Springer-VerlagD.M.Gitman and I.V.Tyutin, Quantization of Fields with Constraints. Springer- Verlag, 1990. . M Z Iofa, 9602023M.Z.Iofa, hep-th 9602023. . M Mueller, Nucl. Phys. 33737M.Mueller, Nucl. Phys. B337 (1990) 37. . M Banados, C Teitelboim, J Zanelli, Phys. Rev. Lett. 691849M.Banados, C.Teitelboim and J.Zanelli, Phys. Rev. Lett. 69 (1992) 1849. . M Banados, M Henneaux, C Teitelboim, J Zanelli, Phys. Rev. Lett. 691849M.Banados, M.Henneaux, C.Teitelboim and J.Zanelli, Phys. Rev. Lett. 69 (1992) 1849. . D Horne, G Horowitz, Steif , Phys. Rev. Lett. 68568D.Horne, G.Horowitz and Steif, Phys. Rev. Lett. 68 (1992) 568. . G Horowitz, D Welch, Phys. Rev. Lett. 71328G.Horowitz and D.Welch, Phys. Rev. Lett. 71 (1993) 328. . T Buscher, Phys. Lett. 194466T.Buscher, Phys. Lett. B194 (1987) 59, B201 (1988) 466. . V V Belokurov, M Z Iofa, Mod. Phys. Lett. 10575V.V.Belokurov and M.Z.Iofa, Mod. Phys. Lett. A10 (1995) 575. . G Mandal, A M Sengupta, S R Wadia, Mod. Phys. Lett. 61685G.Mandal, A.M.Sengupta and S.R.Wadia, Mod. Phys. Lett. A6 (1991) 1685.

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{'abstract': 'One-string-loop (torus topology) corrections to black-string backgrounds corresponding to gauged SL(2, R) × R/R WZW model are calculated using β-function equations derived from string-loop-corrected effective action. Loop-corrected backgrounds are used to calculate ADM mass of the black string. *', 'arxivid': 'hep-th/9603008', 'author': ['M Z Iofa \nNuclear Physics Institute\nMoscow State University\n119899MoscowRussia\n'], 'authoraffiliation': ['Nuclear Physics Institute\nMoscow State University\n119899MoscowRussia'], 'corpusid': 119102126, 'doi': '10.1142/s0217732397000868', 'github_urls': [], 'n_tokens_mistral': 10440, 'n_tokens_neox': 8896, 'n_words': 4981, 'pdfsha': '8f041e260fd03bf8e6cf95bc987309e822ba3ea8', 'pdfurls': ['https://arxiv.org/pdf/hep-th/9603008v1.pdf'], 'title': ['Higher-genus corrections to black-string solution', 'Higher-genus corrections to black-string solution'], 'venue': []}

arxiv

Compton Scattering on Black Body Photons Sep 1996 Lowell S Brown Department of Physics University of Washington 98195SeattleWashington Ronald S Steinke Department of Physics University of Washington 98195SeattleWashington Compton Scattering on Black Body Photons Sep 1996arXiv:hep-ph/9606303v2 27 We examine Compton scattering of electrons on black body photons in the case where the electrons are highly relativistic, but the center of mass energy is small in comparison with the electron mass. We derive the partial lifetime of electrons in the LEP accelerator due to this form of scattering in the vacuum beam pipe and compare it with previous results. I. INTRODUCTION Vacuum beam pipes of modern particle accelerators closely approach the ideal limit of a pipe completely devoid of gas molecules. However, even an ideal vacuum beam pipe in a laboratory at room temperature is filled with photons having an energy distribution given by Planck's law. Some time ago, Telnov 1 noted that the scattering of electrons on these black body photons could be a significant mechanism for the depletion of the beam. This scattering of the electrons in the Large Electron Positron collider at CERN (LEP) on the black body radiation has been detected [2][3][4] . There is a long history of theoretical investigations on the scattering of high-energy electrons on black body photons, centering around the role this plays as an energy loss mechanism for cosmic rays, which is summarized by Blumenthal and Gould 5 . More recently, Domenico 6 and Burkhardt 7 have considered this effect for the LEP experiments and the consequent limit on the beam lifetime by using numerical Monte Carlo methods. In view of the intrinsic interest of the problem of high-energy electron scattering on black body photons, we believe that it is worthwhile to present here a simplified calculation of the effect. We compute the total cross section analytically. The cross section as a function of the energy loss -which is the important quantity for the beam lifetime -is also done analytically except for a final straightforward numerical integration. Our calculations use relativistic invariant methods, and are thus of some pedagogical interest. Analytic computations can be performed because the problem involves two small dimensionless parameters. On the one hand, the electron of mass m has a very large laboratory energy E and it is ultrarelativistic, as characterized by the parameter m 2 /E 2 . (We use natural units in which the velocity of light c = 1, Planck's constanth = 1, and Boltzmann's constant k = 1, so that temperature is measured in energy units.) At LEP, m 2 /E 2 ≈ 10 −10 . We shall neglect terms of order m 2 /E 2 . On the other hand, the temperature T of the black body radiation is very small in comparison with the electron mass m. Thus, although the electron is ultrarelativistic, the energy in the center of mass of the electron-photon system is still small in comparison with the electron mass. The head-on collision of a photon of energy T with an relativistic electron of energy E produces, with the neglect of the electron mass, the squared center-of-mass energy 4ET . We shall use the dimensionless parameter (which gives an average value) s = 2ET /m 2 . (1.1) At LEP, s ≈ 10 −2 . Thus it is a good approximation to use the non-relativistic limit in the center of mass, with the relativistic Compton cross section replaced by its constant, nonrelativistic Thomson limit. To assess this approximation, we shall also compute the first corrections in s. In the next section we use simple relativistic techniques to compute the total cross section for the scattering of an ultrarelativistic electron on the Planck distribution of black body photons. The third section describes the more detailed calculation needed for the cross section in which the electron loses an energy greater than ∆E. If the energy loss ∆E in an electron-photon collision is too large, the electron's motion falls outside of the acceptance parameters of the machine. At LEP this happens when ∆E/E is greater than about 1%. As we shall see, this means that even if the beam were in a perfect vacuum, it would decay with a half life of about two days. The vacuum in the LEP accelerator is so good that the beam scattering of the black body photons is, in fact, the primary mechanism for beam loss when the machine is run with a single beam. Scattering on the residual gas in the beam pipe gives a considerably longer half life of about six days. 8 When the machine is run in the usual mode with two beams for e + e − collision experiments, beam-beam collisions reduce the beam half life to about 14 hours. 8 II. TOTAL SCATTERING RATE In the general case of an electron scattering off some photon distribution, the scattering rateΓ in the electron's rest frame may be computed using the formulā Γ = (d 3k ) (2π) 3f (k)σ(k) ,(2.p µ = m dz µ dτ , (2.3) where z µ (τ ) is the world line of this particle, its space-time position as a function of proper time, and m is the electron mass. This gives the familiar time-dilation formula dt dτ = dz 0 dτ = p 0 m = E m , (2.4) where E is the electron's total energy. Thus the scattering rate Γ in the lab frame may be easily evaluated using Γ = dn dt = m E dn dτ . (2.5) In the non-relativistic limit, σ may be replaced with the Thomson cross section, σ T = 8πr 2 0 /3, where r 0 = e 2 /4πm is the classical electron radius. Since this is independent of k, the scattering rate may be rewritten as Γ 0 = σ Tj 0 , (2.6) where j µ = (d 3k ) (2π) 3k μ k 0f (k) (2.7) is the photon number flux four vector. Sincep µ /m = (1 , 0) in the electron's rest frame, we may write this leading approximation, denoted with a 0 subscript, as dn dτ 0 = −σ Tj µp µ m , (2.8) with the minus sign arising from our Lorentz metric convention in which the metric has the signature (−, +, +, +). The result (2.8) is now in an invariant form which holds in any frame. With a thermal photon distribution in the lab frame, f (k) = 2 e ω/T − 1 , (2.9) where ω = k 0 is the photon energy, the photon number distribution is isotropic, and so only the number density component j 0 is nonvanishing. Thus, in the lab frame, Γ 0 = m E σ T j 0 p 0 m = σ T j 0 . (2.10) The lab photon number density obtained from integrating (2.7) with the distribution (2.9) is the familiar result j 0 = 2ζ(3) π 2 T 3 , (2.11) in which ζ(3) = 1.202 . . . is the Riemann zeta function. Thus, Γ 0 is given by Γ 0 = 2ζ(3) π 2 T 3 σ T . (2.12) The first order relativistic correction to this result is obtained with the use of the corrected cross section σ = σ T 1 + 2pk m 2 . (2.13) Note that the product kp = k µ p µ of the two four-momenta is negative with our metric. Because σ is no longer independent of k, the corresponding form of Eq. (2.10) is Γ 1 = m E σ T (d 3 k) (2π) 3 −kp k 0 m f (k) 1 + 2pk m 2 = − 1 E σ T j µ p µ + 2 m 2 T µν p µ p ν , (2.14) where T µν = (d 3 k) (2π) 3 k µ k ν k 0 f (k) (2.15) is the stress-energy tensor of the photons. Due to the isotropy of the thermal photons in the lab frame, T µν has no off diagonal components, and it is also traceless because the photon is massless, k µ k µ = 0. Therefore, in the lab frame, T µν p µ p ν = E 2 + 1 3 |p 2 | T 00 = 4 3 E 2 1 − m 2 4E 2 T 00 . (2.16) The m 2 /E 2 term is very small, and it may be neglected. Integrating over the photon distribution in Eq. (2.15) gives the well-known black body energy density T 00 = 6ζ(4) π 2 T 4 , (2.17) where ζ(4) = π 4 /90 = 1.082 . . .. This yields the corrected scattering rate Γ 1 = 2ζ(3) π 2 T 3 σ T 1 − 4s ζ(4) ζ(3) . (2.18) For the temperature in the LEP beam pipe we take T = 291 K = 0.0251 eV, which is about room temperature. This gives the leading rate Γ 0 = 9.98 × 10 −6 s −1 corresponding to the mean life τ 0 = 1/Γ 0 = 28 hr. A typical LEP beam energy E = 46.1 GeV is just above half the Z 0 mass -within the width, but on the high side of resonance curve. Together with the previous value of the temperature, this gives s = 0.00886, and the first-order corrected rate Γ 1 = 9.66 × 10 −6 s −1 , which is about 3% smaller than the leading rate. This gives a mean life τ 1 = 1/Γ 1 = 29 hr. III. RATE WITH ENERGY LOSS The calculation of the scattering rate in which the electron loses an energy greater than ∆E is facilitated by going back to the basic formula 9 that expresses the rate in terms of Lorentz invariant phase space integrals, an energy-momentum conserving δ function, and the square of the scattering amplitude |T | 2 . The total electron scattering rate as observed in the lab frame reads Γ = 1 2E (d 3 k) (2π) 3 1 2ω f (k) (d 3 k ′ ) (2π) 3 1 2ω ′ (d 3 p ′ ) (2π) 3 1 2E ′ (2π) 4 δ (4) (k ′ + p ′ − k − p)|T | 2 ,(3.1) where p and p ′ are the initial and final electron four momenta, k and k ′ the initial and final photon four momenta, with E = p 0 , E ′ = p ′ 0 , ω = k 0 , ω ′ = k ′ 0 the time components of these four vectors. Except for the initial factor of 1/2E which is the lab energy of the initial electron and which converts the invariant proper time into the lab time, the right-hand side of this expression is a Lorentz invariant. The problem proves to be greatly simplified if the integrals are evaluated in the rest frame of the electron, because Compton scattering of a photon on an electron at rest has a very simple nonrelativistic limit. This complicates the initial photon distribution, but, if we introduce a four vector β µ , whose time component in the lab frame is one over the temperature of the photon distribution and whose spatial components are zero in the lab frame, the distribution in an arbitrary frame still has the simple form f (k) = 2 e −β µ kµ − 1 . (3.2) From the definition of β µ , −β µ β µ = 1/T 2 and −β µ p µ = E/T , because multiplication by β µ selects out the time component in the lab frame. In the electron rest frame, β µ therefore takes on the value β µ = E T m , − p T m ,(3.3) where E and p are taken in the lab frame. We have not yet taken into account the lower bound on the electron energy loss in the lab frame. Because multiplication by β µ selects the time component in the lab frame, this limit may be instituted by the inclusion of an "energy loss" step function in the integrand of Eq. (3.1), θ −β µ (p µ − p ′ µ ) − ∆E T ,(3.4) where the 1/T in the second term has been included to compensate for the factor of 1/T which the first term has picked up by being multiplied by β µ . Using the identity (d 3 p ′ ) (2π) 3 1 2E ′ = (d 4 p ′ ) (2π) 4 θ(E ′ − m)2πδ(p ′2 + m 2 ) ,(3.5) the final electron four momentum p ′ may be integrated over to leave Γ(∆E) = 1 2E (d 3 k) (2π) 3 1 2ω f (k) (d 3 k ′ ) (2π) 3 1 2ω ′ θ(ω − ω ′ )2πδ(−2mω + 2mω ′ − 2kk ′ ) θ −βk ′ − ∆E T − βk |T | 2 . (3.6) We do the k ′ integral in spherical coordinates and take the z axis to be parallel to k, with θ the angle between these two vectors. The angle θ is the photon scattering angle in the electron rest frame, and − kk ′ = ωω ′ (1 − cos θ) . (3.7) The δ function can now be solved for ω ′ to yield δ(−2mω + 2mω ′ − 2kk ′ ) = 1 2(m + ω(1 − cos θ)) δ ω ′ − mω m + ω(1 − cos θ) ,(3.8) which requires that ω ′ < ω and thus makes the θ(ω − ω ′ ) step function redundant. The scattered photon energy ω ′ given by the δ function is, of course, just the Compton energy. To deal with the "energy loss" step function, we note that the Lorentz transformation from the lab frame to the initial electron rest frame turns the lab frame isotropic black body photon distribution into a very narrow pencil in the electron rest frame in which we are now working. Thus the initial photon distribution is sharply peaked about the average value (k µ /ω) = β µ /β 0 . (3.9) Hence we can approximate − βk ′ ≃ β 0 ω (−kk ′ ) = β 0 ω ′ (1 − cos θ) . (3.10) To verify this and assess the order of accuracy, we define the average more precisely by X = (d 3 k) (2π) 3 1 2ω f (−kβ) X (d 3 k) (2π) 3 1 2ω f (−kβ) . (3.11) Then, by virtue of the relativistic invariance of this definition, k µ k ν = β µ β ν − 1 4 β 2 g µν A(β 2 ) , (3.12) since β λ is the only four vector available and k µ k µ = 0. Remembering that ω = k 0 , this presents the squared fluctuation about the average as k µ − β µ ω β 0 k ν − β ν ω β 0 ω 2 = B g µν − β ν β 0 g µ0 − β µ β 0 g ν0 + β µ β ν (β 0 ) 2 , (3.13) where B = −β 2 4(β 0 ) 2 + β 2 ≃ m 2 4E 2 . (3.14) Thus the deviations away from our approximation may be neglected because they involve the very small quantity m 2 /E 2 . Using this approximation for −k ′ β simplifies the "energy loss" step function to θ ω ′ − ∆E β 0 T (1 − cos θ) = θ ω ′ − ∆E E m 1 − cos θ , (3.15) where the −βk in the original step function has been neglected because it is much less than ∆E/T . Inserting the value of ω ′ given by the energy-conserving δ function (3.8) into the step function gives θ mω m + ω(1 − cos θ) − ∆E E m 1 − cos θ = θ ω − ∆E E ′ m 1 − cos θ , (3.16) where on the right hand side we have solved for ω and defined E ′ = E − ∆E , (3.17) which is the maximum final electron energy in the lab frame. We perform the k integral in spherical coordinates, with the polar angle χ taken to be the angle between k and β β β, so that − kβ = ω(β 0 − |β β β| cos χ) . (3.18) We rewrite the angular integral for k in terms of an integration over kβ by noting the limits − kβ < ω(β 0 + |β β β|) ≃ 2ωβ 0 = 2ωE mT , (3.19) and − kβ > ω(β 0 − |β β β|) = ωβ 2 β 0 + |β β β| ≃ ωm 2T E . (3.20) Thus, with the neglect of order m 2 /E 2 corrections, and remembering that |β β β| ≃ E/mT , 1 −1 d cos χ = mT ωE 2ωE/mT ωm/2T E d(−kβ) . (3.21) We shall do the −kβ integral last, due to its dependence on the initial photon distribution. In order to interchange the order of the ω and −kβ integrations, we note that the lower limit on −kβ, − kβ > ωm 2T E ,(3.22) gives the upper bound on ω, ω < 2T E(−kβ) m = s(−kβ)m . (3.23) The upper limit on −kβ, − kβ < 2ωE mT ,(3.24) gives the lower bound on ω, ω > (−kβ)mT 2E = s(−kβ)m m 2 4E 2 . (3.25) In view of the extreme smallness of m 2 /E 2 , we may replace this lower limit by ω = 0. Hence, switching the order of integration gives We perform this reversal of integrals, do the two trivial azimuthal integrals, and do the ω ′ integral using the δ function to obtain Γ(∆E) = m 2 T 16E 2 (2π) 3 ∞ 0 d(−kβ)f (−kβ) 1 −1 d cos θ s(−kβ)m 0 dω ω [m + ω(1 − cos θ)] 2 |T | 2 θ ω − ∆E E ′ m 1 − cos θ . (3.27) To work out the integrals which appear here, it is convenient to first introduce the appropriate, dimensionless variables, x = −kβ , z = 1 − cos θ , ν = ω s(−kβ)m ,(3.28) and define u = ∆E 2sE ′ . (3.29) With this new notation, we have Γ(∆E) = T 3 4m 2 (2π) 3 ∞ 0 dx x 2 f (x) 2 0 dz 1 0 dν ν (1 + sνxz) 2 |T | 2 θ ν − 2u xz . (3.30) The final step function provides the lower limit ν = 2u/xz. This lower limit must not exceed the upper limit ν = 1. Hence we must have condition z > 2u/x on the z integration. But again, this must not exceed the upper limit z = 2. Thus x > u, and imposing all these limits gives Γ(∆E) = T 3 4m 2 (2π) 3 ∞ u dx x 2 f (x) 2 2u/x dz 1 2u/xz dν ν (1 + sνxz) 2 |T | 2 . (3.31) This will be evaluated in the nonrelativistic limit, keeping first order corrections in s. The exact squared amplitude differs from its nonrelativistic limit |T | 2 = 2e 4 (1 + cos 2 θ) = 2e 4 (2 − 2z + z 2 ) (3.32) by corrections of order ωω ′ /m 2 . These corrections involve s 2 and are thus negligible. To first order in s, 1 (1 + sνxz) 2 ≃ 1 − 2sνxz . (3.33) The z and ν integrations are now straightforward. We express the result as Γ(∆E) = Γ 0 F 0 (u) − 4s ζ(4) ζ(3) F 1 (u) , (3.34) where Γ 0 is the approximate total scattering rate from Eq. (2.12). The straightforward integrations give F 0 (u) = 1 2ζ(3) ∞ u dx e x − 1 x 2 − 3ux + 3u 2 ln x u + 2u 3 x ,(3.35) and 10 F 1 (u) = 1 6ζ(4) ∞ u dx e x − 1 x 3 − 9 2 u 2 x − u 3 + 6u 3 ln x u + 9u 4 2x . (3.36) It can be seen that in the ∆E → 0 limit, F 0 (0) = F 1 (0) = 1, so Γ(∆E) reduces to the result (2.18) for Γ 1 . At this stage, one must resort to numerical calculations to evaluate the integrals. However, analytic calculations of the energy weighted moments of the distribution can still be performed. The simplest of these is the average energy loss observed in the lab frame, E −E ′ . Because Γ(∆E) describes the rate due to all scattering events where E −E ′ > ∆E, this average value may be computed by E − E ′ = − ∞ 0 d∆E∆E d d∆E Γ(∆E) Γ 1 , (3.37) where Γ 1 is the total scattering rate including the first correction in s given by Eq. (2.18). Changing variables to u and integrating by parts gives (3.39) E − E ′ = ∞ 0 du Γ(∆E) Γ 1 d∆E du ,(3. Inserting the expressions for Γ(∆E) and Γ 1 into the integral and expanding in powers of s gives E − E ′ = 2sE ∞ 0 du F 0 (u) − 4s uF 0 (u) − ζ(4) ζ(3) F 0 (u) + ζ(4) ζ(3) F 1 (u) + O(s 2 ) . (3.40) Inserting the expressions for the F 's from Eq. (3.35) and Eq. (3.36) and interchanging the order of the x and u integrals, the integrals may be evaluated analytically, and we find that E − E ′ = 2sE ζ(4) ζ(3) 1 + s 4 ζ(4) ζ(3) − 63 5 ζ(5) ζ(4) + O(s 2 ) = 1.80 sE 1 − 8.5 s . (3.41) To check that no mistakes have been made in our calculation of F 0 (u) and temperature, our value of the beam lifetime for ∆E = 0.012E with his parameters is reduced from 68 to 65 hours. Burkhardt finds 83 hours. (To compare with Domenico, we note that modifying his result of 90 hours by the 4% change in T 3 produces 86 hours.) Burkhardt and Kleiss 8 also state that the average fractional energy loss E − E ′ /E is 1.1% for this value of s, but our result (3.41) gives the larger value 1.5% corresponding to our shorter beam lifetime. Again, we we can only state that we do not understand the reason for these discrepancies. ACKNOWLEDGMENTS We should like to thank J. Rothberg for making us aware of the electron scattering on the photons in the LEP beam pipe. We would also like to thank H. Burkhardt FIG. 1 . 1F 1 (u) given in Eq. (3.35) and Eq. (3.36), we have independently evaluated the average energy loss E − E ′ starting from Eq. (3.6) and only making the small s approximation towards the end of the calculation. We find the same result with this different method. The integrals in the definitions (3.35) and (3.36) of the functions F 0 (u) and F 1 (u) have been calculated numerically, and the results are displayed in Fig. 1 and Fig. 2. As a check on this numerical result, we have used it to evaluate the integrals in Eq. (3.40) numerically, and the results agree with the analytic expression (3.41) to within 0.2 percent. We may compare our calculations with the those of Domenico 6 who employed a Monte Carlo method. He used the values E = 46.1 GeV and T = 291 K (which we have previously employed) that give s = 0.0089. He also took ∆E = 0.012E which places u = 0.69. Numerical integration gives F 0 (0.69) = 0.44 and F 1 (0.69) = 0.83, and from these values we calculate a mean beam lifetime of 64 hours to zeroth order in the nonrelativistic limit, and of 68 hours when the first order relativistic corrections are included. This is to be compared with Domenico's value of 90 hours for the same input parameters. We do not understand the reason for this discrepancy.We may also compare our results with those of H. Burkhardt 7 who also used a Monte Carlo method. Burkhardt uses parameters that are slightly different than those used by Domenico, namely E = 45.6 GeV and T = 295 K. These parameters yield the same s = 0.0089. However, since the overall rate scales as T 3 , which is 4% larger with Burkhardt's The dimensionless F 0 (u) defined in Eq.(3.35) as a function of the dimensionless variable u defined in Eq. (3.29). for informing us of relevant literature and C. Woll for checking much of our calculations. This work was supported, in part, by the U.S. Department of Energy under Grant. No. DE- Scattering of electrons on thermal radiation photons in electron-positron storage rings. V I Telnov, Nucl. Instr. and Meth. 260V. I. Telnov, "Scattering of electrons on thermal radiation photons in electron-positron storage rings," Nucl. Instr. and Meth. A260, 304-308 (1987). Scattering of high energy electrons off thermal photons. B Dehning, A C Melissinos, F Perrone, C Rizzo, G Holtey, Phys. Lett. B. 249B. Dehning, A. C. Melissinos, F. Perrone, C. Rizzo and G. von Holtey, "Scattering of high energy electrons off thermal photons," Phys. Lett. B 249, 145-148 (1990). Scattering of thermal photons by a 46 GeV positron beam at LEP. C Bini, G Zorzi, G Diambrini-Palazzi, G Di Cosimo, A Di Domenico, P Gauzzi, D Zanello, Phys. Lett. B. 262C. Bini, G. De Zorzi, G. Diambrini-Palazzi, G. Di Cosimo, A. Di Domenico, P. Gauzzi and D. Zanello, "Scattering of thermal photons by a 46 GeV positron beam at LEP," Phys. Lett. B 262, 135-138 (1991). Fast measurement of luminosity at LEP by detecting the single bremsstrahlung photons. C Bini, G Zorzi, G Diambrini-Palazzi, G Di Cosimo, A Di Domenico, P Gauzzi, D Zanello, Nucl. Instr. and Meth. 306C. Bini, G. De Zorzi, G. Diambrini-Palazzi, G. Di Cosimo, A. Di Domenico, P. Gauzzi and D. Zanello, "Fast measurement of luminosity at LEP by detecting the single bremsstrahlung photons," Nucl. Instr. and Meth. A306, 467-473 (1991). Bremsstrahlung, synchrotron radiation, and Compton scattering of high-energy electrons traversing dilute gases. G R Blumenthal, R J Gould, Rev. Mod. Phys. 42G. R. Blumenthal and R. J. Gould, "Bremsstrahlung, synchrotron radiation, and Compton scattering of high-energy electrons traversing dilute gases," Rev. Mod. Phys 42, 237-270 (1970). Inverse Compton scattering of thermal radiation at LEP and LEP-200. A D Domenico, Part. Accel. 39A. D. Domenico, "Inverse Compton scattering of thermal radiation at LEP and LEP- 200," Part. Accel. 39, 137-146 (1992). Monte Carlo Simulation of Beam Particles and Thermal Photons. H Burkhardt, CERN/SL Note 93-73 (OP). Internal Note, unpublishedH. Burkhardt, "Monte Carlo Simulation of Beam Particles and Thermal Photons", CERN/SL Note 93-73 (OP), 1993 (Internal Note, unpublished). Beam Lifetimes in LEP. See, H Example, R Burkhardt, Kleiss, Proc. 4th. 4thSee, for example, H. Burkhardt and R. Kleiss, "Beam Lifetimes in LEP", Proc. 4th . Eur. Part. Acc. Conf. EPAC. V. Suller and Ch. Petit-Jean-GenazIIEur. Part. Acc. Conf. EPAC, London 1994, Eds. V. Suller and Ch. Petit-Jean-Genaz, Vol II, pp. 1353-1355. L S See, Brown, Quantum Field Theory. Cambridge Univ. PressSection 3.4.See, for example, L. S. Brown, Quantum Field Theory (Cambridge Univ. Press, 1992), Section 3.4. 35) can be obtained from an integration of Eq. (2.42) of Blumenthal and Gould 5. The result for F 0 (u) given in Eq. but the result for F 1 (u) in Eq. (3.36) appears to be newThe result for F 0 (u) given in Eq. (3.35) can be obtained from an integration of Eq. (2.42) of Blumenthal and Gould 5 , but the result for F 1 (u) in Eq. (3.36) appears to be new.

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{'abstract': 'We examine Compton scattering of electrons on black body photons in the case where the electrons are highly relativistic, but the center of mass energy is small in comparison with the electron mass. We derive the partial lifetime of electrons in the LEP accelerator due to this form of scattering in the vacuum beam pipe and compare it with previous results.', 'arxivid': 'hep-ph/9606303', 'author': ['Lowell S Brown \nDepartment of Physics\nUniversity of Washington\n98195SeattleWashington\n', 'Ronald S Steinke \nDepartment of Physics\nUniversity of Washington\n98195SeattleWashington\n'], 'authoraffiliation': ['Department of Physics\nUniversity of Washington\n98195SeattleWashington', 'Department of Physics\nUniversity of Washington\n98195SeattleWashington'], 'corpusid': 17792903, 'doi': '10.1119/1.18512', 'github_urls': [], 'n_tokens_mistral': 8181, 'n_tokens_neox': 6994, 'n_words': 4413, 'pdfsha': 'c10daa2502b77bb87c095e4974d0e1c72118da99', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ph/9606303v2.pdf'], 'title': ['Compton Scattering on Black Body Photons', 'Compton Scattering on Black Body Photons'], 'venue': []}

arxiv

Some recent results on singular p-Laplacian equations 6 Jul 2022 Umberto Guarnotta umberto.guarnotta@unipa.it Roberto Livrea roberto.livrea@unipa.it Salvatore A Marano marano@dmi.unict.it Dipartimento di Matematica e Informatica Dipartimento di Matematica e Informatica Università di Palermo Via Archirafi 3490123PalermoItaly Università di Catania Viale A. Doria 695125CataniaItaly Some recent results on singular p-Laplacian equations 6 Jul 2022quasi-linear elliptic equationgradient dependencesingular termentire solutionstrong solution AMS Subject Classification: 35-0235J6235J7535J92 A short account of some recent existence, multiplicity, and uniqueness results for singular p-Laplacian problems either in bounded domains or in the whole space is performed, with a special attention to the case of convective reactions. An extensive bibliography is also provided. Introduction When studying quasi-linear elliptic systems in the whole space and with singular, possibly convective, reactions, a natural preliminary step is looking for the previous literature on equations of the same type, which we have done in the latest years. At first, this obviously led us to investigate singular p-Laplacian Dirichlet problems as      −∆ p u = h(x, u, ∇u) in Ω, u > 0 in Ω, u = 0 on ∂Ω,(1.1) where 1 < p < ∞, the symbol ∆ p denotes the p-Laplace operator, namely ∆ p u := div(|∇u| p−2 ∇u), Ω is a bounded domain in R N , N ≥ 3, with smooth boundary ∂Ω, while h ∈ C 0 (Ω × R + × R N ) satisfies lim t→0 + h(x, t, ξ) = ∞. If p = 2 then various special (chiefly non-convective) cases of (1.1) have been thoroughly studied (see Subsection 3.1). Both surveys [1,2,3] and a monograph [4], besides many proceeding papers, are already available. The main purpose of Section 3 below is to provide a short account on some recent existence, multiplicity or uniqueness results for p = 2 and the relevant technical approaches. Let us also point out [5,6,7]. The work [5] treats a singular p(x)-Laplacian Robin problem, while [6,7] are devoted to singular (p, q)-Laplacian equations with Neumann and Robin boundary conditions, respectively; cf. [8] too. Section 4 aims at performing the same as regards singular p-Laplacian problems in the whole space. So, it deals with situations like      −∆ p u = h(x, u, ∇u) in R N , u > 0 in R N , u(x) → 0 as |x| → ∞. (1.2) To the best of our knowledge, except [4], even when p = 2 and h does not depend on ∇u, there are no surveys concerning (1.2). Hence, this probably represents the first contribution. Both sections are divided into four parts. The first is a historical sketch of the case p = 2. The next two treat existence, multiplicity, and uniqueness in the non-convective case. The fourth is devoted to singular problems with convection. Since the literature on (1.1)-(1.2) is by now very wide and our knowledge is limited, significant works may have been not mentioned here, something of which we apologize in advance. Moreover, for the sake of brevity, we did not treat singular parabolic boundary-value problems and refer the reader to [2,9,10,11,12]. Basic notation Let X(Ω) be a real-valued function space on a nonempty measurable set Ω ⊆ R N . If u 1 , u 2 ∈ X(Ω) and u 1 (x) < u 2 (x) a.e. in Ω then we simply write u 1 < u 2 . The meaning of u 1 ≤ u 2 , etc. is analogous. Put X(Ω) + := {u ∈ X(Ω) : u ≥ 0} . The symbol u ∈ X loc (Ω) means that u : Ω → R and u⌊ K ∈ X(K) for all nonempty compact subset K of Ω. Given 1 < r < N, define r ′ := r r − 1 , r * := Nr N − r . Let us next recall the notion and some relevant properties of the so-called Beppo Levi space D 1,r 0 (R N ), addressing the reader to [13, Chapter II] for a complete treatment. Set D 1,r := z ∈ L 1 loc (R N ) : |∇z| ∈ L r (R N ) and denote by R the equivalence relation that identifies two elements in D 1,r whose difference is a constant. The quotient setḊ 1,r , endowed with the norm u 1,r := ˆR N |∇u(x)| r dx 1/r , turns out complete. Write D 1,r 0 (R N ) for the subspace ofḊ 1,r defined as the closure of C ∞ 0 (R N ) under · 1,r , namely D 1,r 0 (R N ) := C ∞ 0 (R N ) · 1,r . D 1,r 0 (R N ), usually called Beppo Levi space, is reflexive and continuously embeds in L r * (R N ), i.e., D 1,r 0 (R N ) ֒→ L r * (R N ). (2.1) Consequently, if u ∈ D 1,r 0 (R N ) then u vanishes at infinity, meaning that the set {x ∈ R N : |u(x)| ≥ ε} has finite measure for any ε > 0. Let Ω be a bounded domain in R N , N ≥ 3, with smooth boundary ∂Ω, let a : Ω → R + 0 be nontrivial measurable, and let γ > 0. The simplest singular elliptic Dirichlet problem writes as      −∆u = a(x)u −γ in Ω, u > 0 in Ω, u = 0 on ∂Ω. (3.1) Since the pioneering papers [14,15,16,17,18], a wealth of existence, uniqueness or multiplicity, and regularity results concerning (3.1) have been published. We refer the reader to the monograph [4] as well as the surveys [1,2] for an exhaustive account. Roughly speaking, four basic questions can be identified: • find the right conditions on the datum a. Usually, a ∈ L q (Ω) with q ≥ 1 is enough for existence. However, starting from the works [19,20], the case when a is a bounded Radon measure took interest. • consider non-monotone singular terms. This is a difficult task, mainly when we want to guarantee uniqueness of solutions. • insert convective terms on the right-hand side. For equations driven by the Laplacian, good references are [4, Section 9] and [21]. Otherwise, cf. [22,23,24,25]. • substitute the Laplacian with more general elliptic operators. Obviously, a first attempt might be considering equations driven by the p-Laplacian, and this section aims to provide a short account of the nowadays literature. However, further possibly non-homogeneous operators have been considered; see, e.g., [14,15,26,27,28,29,20,30,31]. Incidentally, we recall that (3.1) stems from important applied questions, as the study of heat conduction in electrically conducting materials [32], chemical heterogeneous catalysts [33], and non-Newtonian fluids [34]. Existence and multiplicity Consider the model problem      −∆ p u = a(x)u −γ + λf (x, u) in Ω, u > 0 in Ω, u = 0 on ∂Ω, (3.2) where a : Ω → R + 0 denotes a nonzero measurable function, γ, λ > 0, while f : Ω × R + 0 → R satisfies Carathéodory's conditions. Let us stress that, here, the parameter λ multiplies the non-singular term. In 2006, Perera and Silva investigated (3.2) under the assumptions below, where f is allowed to change sign. (a 1 ) There exist ϕ 0 ∈ C 1 0 (Ω) + andq > N such that aϕ −γ 0 ∈ Lq(Ω). (a 2 ) With appropriate δ, c 1 > 0 one has f (x, t) ≥ −c 1 a(x) in Ω × [0, δ]. (a 3 ) To every M > 0 there correspond h ∈ L 1 (Ω) and c 2 > 0 such that −h(x) ≤ f (x, t) ≤ c 2 ∀ (x, t) ∈ Ω × [0, M]. (a 4 ) With appropriate q ∈]1, p * [ and c 3 > 0 one has f (x, t) ≤ c 3 (t q−1 + 1) in Ω × R + 0 . (a 5 ) There are t 0 > 0 and µ > p such that 0 < µˆt 0 f (x, τ )dτ ≤ tf (x, t) ∀ (x, t) ∈ Ω × [t 0 , +∞[ . They seek distributional solutions to (3.2), i.e., functions u ∈ W 1,p 0 (Ω) such that u > 0 and Proofs employ perturbation arguments and variational methods, previously introduced in [36]. An immediate but hopefully useful consequence of Theorem 3.1 is the next Corollary 3.2. Let (a 1 ) be fulfilled. Suppose f does not depend on x and, moreover, f (t) ≥ 0 in a neighborhood of zero once ess inf Ω a = 0. Then, for every λ > 0 sufficiently small, the problem Ω |∇u| p−2 ∇u · ∇ϕdx =ˆΩ au −γ ϕdx +ˆΩ f (·, u)ϕdx ∀ ϕ ∈ C ∞ 0 (Ω).− ∆ p u = a(x)u −γ + λf (u) in Ω, u > 0 in Ω, u = 0 on ∂Ω (3.3) possesses a distributional solution. Further results concerning (3.3) can be found in Aranda-Godoy [37], where a continuous non-increasing function g(u) takes the place of u −γ and, from a technical point of view, fixed point theorems for nonlinear eigenvalue problems are exploited. The case λ = 0 in (3.3) was well investigated by Canino, Sciunzi, and Trombetta [38], with a special attention to uniqueness (see the next section). Here, given u ∈ W 1,p loc (Ω), u = 0 on ∂Ω def ⇐⇒ u ≥ 0 and (u − ε) + ∈ W 1,p 0 (Ω) ∀ ε > 0. Theorem 3.3 ([38], Theorem 1.3). Let λ = 0. If γ ≥ 1 and a ∈ L 1 (Ω) then (3.3) admits a distributional solution u ∈ W 1,p loc (Ω) such that ess inf K u > 0 for any compact set K ⊆ Ω. Moreover, u 1+(γ−1)/p ∈ W 1,p 0 (Ω). If 0 < γ < 1 then (3.3) has a solution u ∈ W 1.p 0 (Ω) in each of the following cases: • 1 < p < N and a ∈ L m (Ω), with m := p * 1−γ ′ . • p = N and a ∈ L m (Ω) for some m > 1. • p > N and a ∈ L 1 (Ω). The proof of this result relies on a technique previously introduced in [29] for the semi-linear case. It employs truncation and regularization arguments. The work [39] contains a version of Theorem 3.3 for the so-called Φ-Laplacian. A more general problem patterned after − ∆ p u = µ u −γ in Ω, u > 0 in Ω, u = 0 on ∂Ω, (3.4) where µ denotes a non-negative bounded Radon measure on Ω while γ ≥ 0, is thoroughly studied in [40]; see also [41] and the references therein. Finally, as regards Problem (3.2) again, the papers [42,43,44,45] do not require Ambrosetti-Rabinowitz's condition (a 5 ), while [46] establishes the existence of at least three weak solutions. Moreover, a possibly nonhomogeneous elliptic operator is considered in [44], but λ = 1. The nice paper [47] investigates the problem      −∆ p u = λu −γ + u q−1 in Ω, u > 0 in Ω, u = 0 on ∂Ω,(3.5) where 0 < γ < 1 and 1 < p < q < p * . It should be noted that, here, contrary to above, the parameter λ multiplies the singular term. Combining known variational methods with a C 1,α (Ω)-regularity result [ Then there is Λ > 0 such that (3.5) has: • at least two ordered solutions in C 1 (Ω) for every λ ∈ ]0, Λ[, • at least one solution in C 1 (Ω) when λ = Λ, and • no solutions once λ > Λ. The case q = p * is also studied and it is shown that γ < 1 is a reasonable sufficient (and likely optimal) condition to get C 1 (Ω)-solutions of (3.5). If p = 2 and, roughly speaking, a ≡ −1 while f does not depend on u then Problem (3.2) was fruitfully studied in [48]. We end this section by pointing out two very recent works, namely [49], which deals with possibly non-monotone singular reactions (see also [50,51], essentially based on sub-super-solution methods) and [31], devoted to singular equations driven by the (p, q)-Laplace operator u → ∆ p u + ∆ q u. Uniqueness Surprisingly enough, if p = 2, uniqueness of solutions looks a difficult matter, even for the model problem      −∆ p u = a(x)u −γ in Ω, u > 0 in Ω, u = 0 on ∂Ω. (3.6) As observed in [38], this is mainly caused by the fact that, in general, solutions do not belong to W 1,p 0 (Ω) once γ ≥ 1. The paper [38] provides two different results. The first one (Theorem 1.4) holds in star-shaped domains, while the other is the following Theorem 3.5 ([38], Theorem 1.5). Assume that either γ ≤ 1 and a ∈ L 1 (Ω) or γ > 1 and • a ∈ L m (Ω) for some m > N p if 1 < p < N, • a ∈ L m (Ω) with m > 1 when p = N, and • a ∈ L 1 (Ω) if p > N. Then (3.6) possesses a unique distributional solution. We next point out that, for γ ≤ 1, Theorem 3.4 of [40] establishes the uniqueness of renormalized solutions to (3.4). The situation becomes quite clear when p = 2 and one seeks sufficiently regular solutions. Denote by ϕ 1 a positive eigenfunction corresponding to the first eigenvalue λ 1 of the problem −∆u = λu in Ω, u = 0 on ∂Ω. Theorem 3.6 ([17], Theorems 1-2). Let p = 2 and let a ∈ C 0,α (Ω) be positive. Then (3.6) has a unique solution u ∈ C 2,α (Ω) ∩ C 0 (Ω). Moreover, • there exist c 1 , c 2 > 0 such that c 1 ϕ 2/(1+γ) 1 ≤ u ≤ c 2 ϕ 2/(1+γ) 1 in Ω, • u ∈ H 1 0 (Ω) ⇐⇒ γ < 3, and • γ > 1 =⇒ u ∈ C 1 (Ω). See also the nice paper [52]. As regards weak solutions, one has Another uniqueness case occurs when γ > 1. Equations with convective terms Consider the problem      −∆ p u = f (x, u, ∇u) + g(x, u) in Ω, u > 0 in Ω, u = 0 on ∂Ω,(3.7) where p < N while f : Ω × R + 0 × R N → R + 0 and g : Ω × R + → R + 0 satisfy Carathéodory's conditions. In 2019, Liu, Motreanu, and Zeng established the existence of solutions u ∈ W 1,p 0 (Ω) to (3.7) under the hypotheses below, where λ 1 stands for the first eigenvalue of −∆ p in W 1,p 0 (Ω). (h 1 ) There exist c 0 , c 1 , c 2 > 0 such that c 1 + c 2 λ 1−1/p 1 < λ 1 and f (x, t, ξ) ≤ c 0 + c 1 t p−1 + c 2 |ξ| p−1 ∀ (x, t, ξ) ∈ Ω × R + 0 × R N . (h 2 ) g(x, ·) is non-increasing on (0, 1] for all x ∈ Ω and g(·, 1) ≡ 0. (h 3 ) With appropriate θ ∈ int(C 1 0 (Ω) + ),q > max{N, p ′ }, and ε 0 > 0, the map x → g(x, εθ(x)) belongs to Lq(Ω) for any ε ∈ (0, ε 0 ). [54]. It represents a natural generalization of (a 1 ) in Section 3.2. 7) has a solution u ∈ int(C 1 0 (Ω) + ). We think worthwhile to sketch the main ideas of the proof. For every fixed w ∈ C 1 0 (Ω), an intermediate problem, where ∇w replaces ∇u in f (x, u, ∇u) and the singular term remains unchanged, is considered. The authors construct a positive sub-solution u ∈ int(C 1 0 (Ω) + ) independently of w and show the existence of a solution greater than u. If S(w) denotes the set of such solutions then, via suitable properties of the multi-function w → S(w), it is proved that the map Γ, which assigns to each w the minimal element of S(w), is completely continuous. Now, Leray-Schauder's alternative principle applied to Γ yields a solution u ∈ int(C 1 0 (Ω) + ) to (3.7). The recent paper [24], partially patterned after [55], treats the Robin problem Condition (h 3 ) was previously introduced by Faraci and Puglisi        −div A(∇u) = f (x, u, ∇u) + g(x, u) in Ω, u > 0 in Ω, ∂u ∂ν A + βu p−1 = 0 on ∂Ω, (3.8) where A : R N → R N denotes a continuous strictly monotone map having suitable properties, which basically stem from Lieberman's nonlinear regularity theory [56] and Pucci-Serrin's maximum principle [57]. By the way, the conditions on A include classical non-homogeneous operators as, e.g., the (p, q)-Laplacian. Moreover, β is a positive constant while ∂ ∂ν A indicates the co-normal derivative associated with A. If p = 2 then a uniqueness result is also presented; cf. [24,Theorem 4.2]. The special case A(ξ) := |ξ| p−2 ξ, g(x, t) := t −γ for some 0 < γ < 1, and β = 0 (which reduces (3.8) to a Neumann problem) has been investigated in [8] without imposing any global growth condition on t → f (x, t, ξ). Instead, a kind of oscillatory behavior near zero is taken on. For such an f , the work [25] establishes the existence of a solution u ∈ C 1 0 (Ω) to the parametric problem      −div A(∇u) = f (x, u, ∇u) + λu −γ in Ω, u > 0 in Ω, u = 0 on ∂Ω, provided λ > 0 is small enough. Finally, the very recent paper [58] treats Φ-Laplacian equations with strongly singular reactions perturbed by gradient terms. 4 Problems on the whole space 4.1 The case p = 2 Let N ≥ 3, let a : R N → R + 0 be nontrivial measurable, and let γ > 0. The simplest singular elliptic problem in the whole space writes as −∆u = a(x)u −γ in R N , u > 0 in R N . (4.1) Sometimes it is also required that u(x) → 0 as |x| → ∞. Since the pioneering papers [59,60,61,62], some existence and uniqueness results concerning (4.1) have been published. We refer the reader to the monograph [4] for a deep account. Roughly speaking, four basic questions can be identified: • find the right hypotheses on a. Usually, a ∈ C 0,α loc (R N ) + as well aŝ • replace u −γ with a function f (u) such that lim t→0 + f (t) = ∞. This was done in [63,64] for decreasing f . Later on, also non-monotone singular reactions were fruitfully treated [65,66,67]. • put convective terms on the right-hand side. For equations driven by the Laplacian, a good reference is [4, Section 9.8]; cf. in addition [68,69]. • generalize the left-hand side of the equation. The case of a second-order uniformly elliptic operator is treated in [70,27], while [71] deals with u → −∆u + c(x)u, where c ∈ L ∞ loc (R N ) + . The equation of Problem (4.1) arises in the boundary-layer theory of viscous fluids [72,73,74] and is called Lane-Emden-Fowler equation. Its importance in scientific applications has by now been widely recognized; see, e.g., [75]. Existence and multiplicity To the best of our knowledge, the first paper treating singular p-Laplacian equations on the whole space is that of Goncalves and Santos [76], published in 2004. The authors consider the problem      −∆ p u = a(x)f (u) in R N , u > 0 in R N , u(x) → 0 as |x| → ∞, (4.2) where a ∈ C 0 (R N ) + is radially symmetric while f ∈ C 1 (R + , R + ), and assume that: • a radially symmetric solution u ∈ C 1 (R N ) ∩ C 2 (R N \ {0}) when p < N. (a 6 ) the function t → f (t) t p−1 is non-increasing on R + . (a 7 ) lim inf t→0 + f (t) > 0 as well as lim t→∞ f (t) t p−1 = 0. (a 8 ) if Φ(r) := max |x|=r a(x), r > 0, then 0 <ˆ∞ 1 [rΦ(r)] 1 p−1 dr < ∞ for 1 < p ≤ 2, 0 <ˆ∞ 1 r (p−2)N+1 p−1 Φ(r) dr < ∞ for p > 2. • no radially symmetric solution in C 1 (R N ) ∩ C 2 (R N \ {0}) if p ≥ N. The proof exploits fixed point arguments, the shooting method, and subsuper-solution techniques. One year later, Covei [77] did not assume a radially symmetric but locally Hölder continuous and positive, replaced conditions (a 6 )-(a 7 ) with those below, and obtained similar results. See also [78], where the asymptotic behavior of solutions is described. (a ′ 6 ) The function t → f (t) (t+β) p−1 turns out decreasing on R + for some β > 0. (a ′ 7 ) lim t→0 + f (t) t p−1 = ∞ and f (t) ≤ c for any t large enough. The work [79] treats the parametric problem −∆ p u = a(x)u −γ + λb(x)u q−1 in R N , u > 0 in R N ,(4.3) where 1 < p < N, 0 < γ < 1, λ > 0, max{p, 2} < q < p * , and the coefficients fulfill It may be of interest to point out that this result is proved by combining sub-super-solution methods with the mountain pass theorem for continuous functionals. • at least one solution u ∈ C 1 (R N ) for every 0 ≤ λ < λ * . Moreover, u(x) → 0 as |x| → ∞. a ∈ L p * p * −(1−γ) (R N ) + , a ≡ 0, b ∈ L p * p * −q (R N ), b > 0. (4.4)     −∆ p w = M(x) in R N , w > 0 in R N , w(x) → 0 as |x| → ∞ • no solution once λ > λ * . This result was next generalized under various aspects by the same author and Rezende [82]; cf. also [80]. Finally, infinite semi-positone problems, i.e., lim t→0 + f (t) = −∞, were fruitfully investigated in [83]. Precisely, given a ∈ L ∞ (R N ) and f ∈ C 0 (R + ), consider the problem      −∆ p u = λa(x)f (u) in R N , u > 0 in R N , u(x) → 0 as |x| → ∞,(4.7) where λ > 0, 1 < p < N. The following conditions will be posited. (a 9 ) There exists γ ∈ ]0, 1[ such that lim t→0 + t γ f (t) = c 0 ∈ R − . (a 10 ) lim t→∞ f (t) = ∞ but lim t→∞ f (t) t p−1 = 0. (a 11 ) inf |x|=r a(x) > 0 for all r > 0 and 0 < a(x) < C 0 |x| σ in R N \ {0} with suitable C 0 > 0, σ > N + γ N −p p−1 . Theorem 4.6 ([83], Theorem 1.4). If (a 9 )-(a 11 ) hold and λ is sufficiently large then (4.7) has a solution in C 1,α loc (R N ). Uniqueness As far as we know, uniqueness has been addressed only in [76,Remark 1.2] and [77, Section 2] under the key assumption (a ′ 6 ) above. The arguments of both papers rely on a famous result by Diaz and Saa [84]. Theorem 1.3 of [85] contains a nice idea to achieve uniqueness for singular problems in exterior domains. Equations with convective terms To the best of our knowledge, there is only one paper concerning singular quasi-linear elliptic equations in the whole space and with convective terms, namely [86]. It treats the problem −div A(∇u) = f (x, u) + g(x, ∇u) in R N , 8) where N ≥ 2 and 1 < p < N. The differential operator u → div A(∇u) is as in (3.8), while f : R N × R + → R + 0 and g : R N × R N → R + 0 fulfill Carathéodory's conditions. Moreover, (4.9) and g(x, ξ) ≤ k(x)|ξ| r in R N × R N , with k ∈ L 1 (R N ) ∩ L θ (R N ). u > 0 in R N ,(4.lim inf t→0 + f (x, t) > 0 uniformly with respect to x ∈ B σ (x 0 ), f (x, t) ≤ h(x)t −γ in R N × R + , where h ∈ L 1 (R N ) ∩ L η (R N ), (4.10) Here, x 0 ∈ R N , σ ∈]0, 1[, γ ≥ 1, r ∈ [0, p − 1[, as well as η > (p * ) ′ , θ > 1 (p * ) ′ − r p −1 . (4.11) Theorem 4.7 ([86], Theorem 1.2). Under (4.9)-(4.11), there exists a distributional solution u ∈ W 1,p loc (R N ) to (4.8) such that ess inf K u > 0 for every compact set K ⊆ R N . To prove this result, the authors first solve some auxiliary problems, obtained by shifting the singular term and working in balls, via sub-supersolution techniques. A compactness result, jointly with a fine local energy estimate on super-level sets of solutions, then yields the conclusion. Theorem 3.1 ([35], Theorems 1.1-1.2). Let (a 1 )-(a 3 ) be satisfied. Then Problem (3.2) admits a distributional solution for every λ > 0 small. If, in addition, (a 4 )-(a 5 ) hold true then a further distributional solution exists by decreasing λ when necessary. Theorem 3. 7 7([53], Theorem 3.1). Suppose p = 2 and a ∈ L 1 (Ω). Then (3.6) admits at most one solution belonging to H 1 0 (Ω). Theorem 3. 8 8([53], Theorem 1.3). If p = 2, γ > 1, and a ∈ L 1 (Ω) then (3.6) possesses at most one solution u ∈ H 1 loc (Ω) such that u (γ+1)/2 ∈ H 1 0 (Ω). Theorem 3.9 ([55], Theorem 25). Let (h 1 )-(h 3 ) be satisfied. Then (3. x) dr < ∞ (cf. condition (a 8 ) below) guarantee both existence and uniqueness of solutions u ∈ C 2,α loc (R N ). Theorem 4. 1 1([76], Theorem 1.1). Under (a 6 )-(a 8 ), Problem (4.2) admits: Theorem 4. 2 2([79], Theorem 1.2). If (4.4) holds then there exists Λ > 0 such that (4.3) possesses • at least two solutions in D 1,p 0 (R N ) for every λ ∈ ]0, Λ[, • at least one solution belonging to D 1,p 0 (R N ) when λ = Λ, and • no solutions once λ > Λ. Remark 4 . 3 .. 4 . 434If b ≡ 0 then Problem (4.3) reduces to a well-known one, very important in scientific applications; cf. [80, Remark 2.2]. A meaningful case occurs when a, b : R N → R + 0 turn out nonzero locally Hölder continuous functions. In fact, define M(x) := max{a(x), b(x)}, x ∈ R N . Suppose that p < N, the functions a, b : R N → R + 0 are nontrivial and locally Hölder continuous, while (a 8 ) holds with M in place of a. Then the problem solution w M ∈ C 1,α loc (R N ) for suitable α ∈ ]0,1[. 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R. Acad. Sci. Paris Sér. I Math. 305 (1987), 521-524. Analysis of positive solutions for classes of quasilinear singular problems on exterior domains. M Chhetri, P Drábek, R Shivaji, Adv. Nonlinear Anal. 6M. Chhetri, P. Drábek, and R. Shivaji, Analysis of positive solutions for classes of quasilinear singular problems on exterior domains, Adv. Nonlinear Anal. 6 (2017), 447-459. Strongly singular convective elliptic equations in R N driven by a non-homogeneous operator. L Gambera, U Guarnotta, 10.3934/cpaa.2022088Comm. Pure Appl. Anal. L. Gambera and U. Guarnotta, Strongly singular convective elliptic equations in R N driven by a non-homogeneous operator, Comm. Pure Appl. Anal., doi:10.3934/cpaa.2022088.

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{'abstract': 'A short account of some recent existence, multiplicity, and uniqueness results for singular p-Laplacian problems either in bounded domains or in the whole space is performed, with a special attention to the case of convective reactions. An extensive bibliography is also provided.', 'arxivid': '2207.02490', 'author': ['Umberto Guarnotta umberto.guarnotta@unipa.it ', 'Roberto Livrea roberto.livrea@unipa.it ', 'Salvatore A Marano marano@dmi.unict.it ', '\nDipartimento di Matematica e Informatica\nDipartimento di Matematica e Informatica\nUniversità di Palermo\nVia Archirafi 3490123PalermoItaly\n', '\nUniversità di Catania\nViale A. Doria 695125CataniaItaly\n'], 'authoraffiliation': ['Dipartimento di Matematica e Informatica\nDipartimento di Matematica e Informatica\nUniversità di Palermo\nVia Archirafi 3490123PalermoItaly', 'Università di Catania\nViale A. Doria 695125CataniaItaly'], 'corpusid': 250311136, 'doi': '10.1515/dema-2022-0031', 'github_urls': [], 'n_tokens_mistral': 17855, 'n_tokens_neox': 14752, 'n_words': 7981, 'pdfsha': '152e00d9ec9ee92c0b0f5f28b8e672fb2ca25ac6', 'pdfurls': ['https://arxiv.org/pdf/2207.02490v1.pdf'], 'title': ['Some recent results on singular p-Laplacian equations', 'Some recent results on singular p-Laplacian equations'], 'venue': []}

arxiv

A deep learning approach to using wearable seismocardiography (SCG) for diagnosing aortic valve stenosis and predicting aortic hemodynamics obtained by 4D flow MRI Mahmoud E Khani Department of Radiology Feinberg School of Medicine Northwestern University 60611ChicagoIL Ethan M I Johnson Department of Radiology Feinberg School of Medicine Northwestern University 60611ChicagoIL Aparna Sodhi Ann & Robert H. Lurie Children's Hospital 60611ChicagoIL Joshua D Robinson Department of Radiology Feinberg School of Medicine Northwestern University 60611ChicagoIL Ann & Robert H. Lurie Children's Hospital 60611ChicagoIL Department of Pediatrics Feinberg School of Medicine Northwestern University 60611ChicagoIL Cynthia K Rigsby Department of Radiology Feinberg School of Medicine Northwestern University 60611ChicagoIL Ann & Robert H. Lurie Children's Hospital 60611ChicagoIL Department of Pediatrics Feinberg School of Medicine Northwestern University 60611ChicagoIL Bradly D Allen Department of Radiology Feinberg School of Medicine Northwestern University 60611ChicagoIL Michael Markl Department of Radiology Feinberg School of Medicine Northwestern University 60611ChicagoIL Department of Biomedical Engineering McCormick School of Engineering Northwestern University 60208EvanstonIL A deep learning approach to using wearable seismocardiography (SCG) for diagnosing aortic valve stenosis and predicting aortic hemodynamics obtained by 4D flow MRI In this paper, we explored the use of deep learning for the prediction of aortic flow metrics obtained using 4D flow MRI using wearable seismocardiography (SCG) devices. 4D flow MRI provides a comprehensive assessment of cardiovascular hemodynamics, but it is costly and time-consuming. We hypothesized that deep learning could be used to identify pathological changes in blood flow, such as elevated peak systolic velocity Vmax in patients with heart valve diseases, from SCG signals. We also investigated the ability of this deep learning technique to differentiate between patients diagnosed with aortic valve stenosis (AS), non-AS patients with a bicuspid aortic valve (BAV), non-AS patients with a mechanical aortic valve (MAV), and healthy subjects with a normal tricuspid aortic valve (TAV). In a study of 77 subjects who underwent same-day 4D flow MRI and SCG, we found that the Vmax values obtained using deep learning and SCGs were in good agreement with those obtained by 4D flow MRI. Additionally, subjects with TAV, BAV, MAV, and AS could be classified with ROC-AUC values of 92%, 95%, 81%, and 83%, respectively. This suggests that SCG obtained using low-cost wearable electronics may be used as a supplement to 4D flow MRI exams or as a screening tool for aortic valve disease. Introduction Magnetic resonance imaging (MRI) is a crucial tool in the clinical evaluation of cardiovascular diseases. Phase contrast MRI (PC-MRI), specifically four-dimensional (4D) flow MRI, has become a routine technique for assessing the functional changes in cardiovascular blood flow in patients with heart valve, aortic, or pulmonary diseases [1][2][3][4][5][6][7][8][9][10][11][12][13]. 4D flow MRI provides a comprehensive evaluation of the temporal and spatial evolution of cardiovascular hemodynamics by acquiring time-resolved, three-dimensional (x-y-z) measurements of blood velocity with 3-directional velocity encoding [14][15][16][17]. Additionally, various cardiac flow metrics such as peak systolic velocity, regurgitation fraction, and wall shear stress can be retrospectively obtained from the measured blood velocities [18][19][20][21][22][23][24][25][26][27][28][29]. Despite the development of efficient 4D flow protocols for clinical applications [30][31][32][33][34][35][36][37], this technique is still considered to be costly and time-consuming due to the advanced MR sequences and computational demands of the 4D flow analysis. To address this, a preliminary evaluation of aortic flow dynamics using a cost-effective and efficient method could be valuable in diagnosing abnormalities prior to prescribing a comprehensive cardiac MRI. This study aims to investigate the utility of a wearable seismocardiography (SCG) device to predict aortic flow metrics similar to those obtained using 4D flow MRI and diagnose aortic valve pathologies. SCG signals are non-invasive measures of cardiac vibrations obtained at the chest surface [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. These vibrations are associated with heart mechanical activities such as cardiac contractions, valve closures, and changes in blood momentum [49]. For example, the atrial systole has been shown to result in a low-frequency SCG wave component, while the ventricular systole produces a high-amplitude wave, and the first and second heart sounds generate two high-frequency vibrations [43]. Additionally, simultaneous SCG and electrocardiogram (ECG) acquisition has revealed that the timing of various physiological events, such as the opening and closure of the mitral and aortic valves and their rapid filling or ejection, correspond to various features in the SCG waveforms [45]. However, SCG pulses are highly susceptible to inter-subject variations due to differences in factors such as body mass index, gender, age, and other demographic and health attributes, making it difficult to derive consistent clinical inferences [53]. Additionally, the location of the SCG device on the chest can impact the shape of the recorded waveforms. Lastly, SCG vibrations have relatively low amplitudes and are easily contaminated by subject motion artifacts or respiratory movements, which can lead to misinterpretation of diagnostic features [43]. The purpose of this study was to investigate the relationships between chest vibrations measured by SCG and cardiac flow metrics obtained using 4D flow MRI using a new deep learning approach. Our technique used ECG-synchronized scalograms of the SCG signals as input to a convolutional neural network (CNN) model for the regression or classification of cardiac parameters. In addition, we used a multi-layer perceptron (MLP) based on demographic attributes such as body mass index, gender, and age to account for inter-subject variations in SCG. We hypothesized that this deep learning approach could be used to infer pathological changes in blood flow, such as an increased peak systolic velocity (Vmax) in patients with aortic valve disease, from SCG pulses. Additionally, because the presence of flow abnormalities such as elevated Vmax are indicative of aortic valve complications, we hypothesized that this technique could be used to diagnose various aortic valve conditions, such as bicuspid or mechanical aortic valves (BAV and MAV) and aortic stenosis (AS) and differentiate them from healthy subjects with normal tricuspid aortic valves (TAV). Study cohort The study was approved by the Institutional Review Board (IRB) and informed consent was obtained from all participants. We recruited 46 healthy subjects (20 females, age: 45.9 ± 17.2 years) with no known history of cardiovascular disease and 31 patients with aortic valve disease (6 females, age: 32.6 ± 20.9 years). Among the patients, 1 had a normal tricuspid aortic valve (TAV), 21 had bicuspid aortic valves (BAV), and 9 had mechanical valve implants. In addition, 12 patients were diagnosed with varying degrees of aortic stenosis (AS): 2 were mild, 6 were moderate, and 4 were severe. 4D flow MRI acquisition and analysis 4D flow MRI was performed using a spatial resolution of 1-3 mm 3 , time resolution of 30-40 ms, and velocity encoding (venc) of 150-375 cm/s on a 1.5T or 3T scanner. The MRI was performed during free breathing with navigator respiration control and prospective or retrospective gating and covered the full volume of the thoracic aorta. Pre-processing of the 4D flow data included correction for eddy currents, velocity antialiasing, and application of a deep learning tool to automatically derive a 3D segmentation of the thoracic aorta [54]. This segmentation was used to mask the 4D flow velocity data, and the ascending aorta (AAo) was manually delineated from its root to the branching vessels by placing a plane proximate to the branching vessels, perpendicular to the centerline. As shown in Fig. 1(a), peak systolic velocity Vmax in the AAo was computed from a volumetric analysis of all voxels in the AAo, with outlier-rejection [55]. All 4D flow MRI analyses were performed using a custom code in MATLAB (MathWorks, Natick, MA, USA). SCG analysis We used a custom-designed, wearable cardiac sensor ( Fig. 1(b)) incorporating a MEMS accelerometer (1 kHz sampling rate, 2 µs/m 2 sensitivity) to acquire chest accelerations in three directions immediately before MRI. The SCG device was placed on the sternum of subjects while they were in the supine position. These acceleration recordings were beatby-beat time-referenced to the R-waves within simultaneously acquired ECG signals. Figure 1(c) shows an example of the gating ECG signal, while Fig. 1(d-f) show the accelerations recorded in the three directions using the accelerometer over a 5 s period. At each heartbeat, one SCG pulse was calculated using the net magnitude of the accelerations along the three directions., i.e., scg i ( ) = √acc , 2 ( ) + acc , 2 ( ) + acc , 2 ( )(1) where subscript represents the heartbeat number, and subscripts , , and represent the direction of the measured accelerations. Figure 1(g) shows how the gating of SCG signals was performed using the ECG recordings. In this way, multiple SCG signals, each corresponding to one R-R period in ECG, one example shown in Fig. 1(h), were measured and processed for each subject. It should be noted that the duration of measurement was not the same for all subjects, so the number of SCG signals obtained varied for different subjects. Signal conditioning The following steps were used to condition SCG signals in MATLAB. First, high-pass filtering was applied to remove low-frequency artifacts using a minimum-order elliptic filter with an infinite impulse response (IIR) function. The filter had a stop-band attenuation of 60 dB, a stop-band frequency of 8.4 Hz, a pass-band frequency of 10 Hz, and a pass-0 R a e band ripple of 0.1. A Blackman window was used to taper the transition of boundary points into zero. Wavelet denoising was performed to improve the signal-to-noise ratio (SNR) of the signals. A symlet a elet ith four anishing moments ('sym4') as used, and the denoising method employed was the false discovery rate (FDR) with a hard thresholding rule. The noise estimate used was level-dependent, and the number of levels of decomposition was calculated as the integer part of log 2 , where is the number of signal points. Figure 2(a) shows examples of raw and denoised SCG signals. After denoising the SCG signals, we used the Signal Quality Index (SQI) to identify and remove any outliers. The SQI is a measure of the quality of the signals and is calculated based on the distance of each SCG from a reference signal. This step helped ensure that the resulting signals were of high quality and accurately represented the subject's physiology. Detailed information about the SQI can be found in reference [56]. Briefly, for each subject, the SQI of the th SCG signal, ( ), was calculated based on its distance from ̂( ), given by, SQI i = exp (− ( ,̂) ℒ( ,̂) ) ,(2) where ̂( ) represents the point-wise average of all SCGs for the subject, ( ,̂) represents the distance between and ̂, and ℒ( ,̂) represents the length of and ̂. Following the approach described in [56], we used dynamic time warping (DTW) to calculate ( ,̂). The DTW algorithm was used to find the minimum Euclidean distance between two SCG signals, even if they were of different lengths. This distance, denoted as ℒ( ,̂), was calculated by stretching or compressing the signals to match their lengths. A small distance in indicates a high-quality signal, while a large distance indicates a poor-quality signal. To determine which signals were considered high quality, the SQI scores were calculated for each signal. The signals were then ranked in order of their SQI scores, and the top 95% of signals were selected as the "good" signals. Any signals with an SQI below the 5% threshold were considered outliers and were removed from further processing. This can be seen in Fig. 2(b-c), which shows examples of good SCG signals and the outliers for a single subject. The outlier signals are highly fluctuating and do not resemble a typical SCG. Time-frequency representation -SCG scalograms To obtain time-frequency representations of the remaining SCG signals, we used the continuous wavelet transform (CWT). We defined a CWT filter bank composed of 48 filters per octave using the analytic Morse wavelet. The filter amplitudes were normalized so that the peak magnitude of all passbands was equal to 2. The highest passband frequency was designed such that the magnitude falls to half the peak value at the Nyquist frequency. The absolute value of the CWT coefficients produced a scalogram for each SCG. Figure 2(d-e) show an example of a denoised SCG and its corresponding scalogram. It can be seen that the two main features in the SCG pulse, which occur between 0 and 0.5 seconds and between 0.5 and 1 second, appear at the same locations in the scalogram. In addition, the frequency components of these features, which lie between 10 and 100 Hz, are shown on the y-axis. These frequency values were directly estimated from their corresponding wavelet scales. All the scalograms were converted to 256×256 images using bicubic interpolation, with each interpolated pixel representing the weighted average of the pixels in its nearest 4×4 neighborhood. This allows for easy visualization and analysis of the time-frequency characteristics of the SCG signals. Figure 3(a) shows the deep learning model used for the regression of peak systolic velocity. The SCG scalograms were used as input data for a convolutional neural network (CNN) model to predict Vmax in the AAo. The CNN architecture consisted of a stack of five convolutional blocks, with each block containing a sequence of Conv2D, ReLU (rectified linear unit) activation, batch normalization, and max pooling layers. The Conv2D layer creates multiple kernels to be convolved with the input data and produces output tensors. The kernel weights and bias vectors are adjusted during backpropagation. For all convolutional blocks, the kernel size, stride, and padding parameters were set to 3×3, 1, , (b-c), the none-outlier and outlier pulses identified based on SQI scores for a healthy subject, (d) an example SCG pulse composed of two main features, (e) the scalogram of the signal in (d) representing the absolute value of its CWT coefficients. and "same", respectively. The ReLU activation function outputs its input if it is positive and outputs zero otherwise. This adds non-linearity to the features learned by the model. The batch normalization layer normalizes each batch of data using its mean and standard deviation to facilitate backpropagation. Finally, the max pooling layer outputs the largest number within each 2×2 window of its input tensor to extract increasingly generic features by down-sampling. These layers work together to allow the CNN to efficiently learn and extract useful features from the SCG scalograms. The output of the last convolutional block was flattened and connected to two fully-connected dense units, with the first unit containing a sequence of dense, ReLU activation, batch normalization, and dropout layers. The second unit only contained dense and ReLU activation layers. Deep learning To incorporate demographic attributes of the subjects (weight, height, age, and sex) in the deep learning model, we created a multilayer perceptron (MLP) with two fullyconnected dense layers. All three continuous attributes were min-max normalized, and the participants' gender was one-hot encoded (i.e. female = 1, male = 0) before being fed into the MLP. The outputs of the MLP and CNN, which both have four nodes, were concatenated and fed into a single dense layer followed by a linear activation unit. This final layer regresses the desired flow metric, such as Vmax peak systolic velocity, using the combined information from both the demographic attributes and the time-frequency characteristics of the SCG signals. The same model was also used to classify the subjects into different valve conditions: non-AS TAV, non-AS BAV, non-AS MAV, and AS. However, as it is shown by Fig. 3(b) for the classification task, the number of nodes in the final dense layer was equal to the eight g Height m Age y e f m eight g Height m Age y e f m number of valve condition categories, and a SoftMax activation function was used to produce the probabilities of the output belonging to each class. This allows the model to accurately classify the subjects based on their SCG signals and demographic attributes. In both regression and classification models, the training parameters of MLP and CNN were optimized simultaneously to minimize the loss function, which was calculated as the mean percentage error between the predicted and true velocities for the regression task, and the categorical cross entropy for the classification task. The models were implemented using Tensorflow 2.6 and Keras libraries in Python 3.8 and were run on a GPU node (equipped with a 40 GB Tesla A100 GPU) within the Northwestern University Quest Computing Cluster. To evaluate the performance of the models, we used a leave-subject-out cross-validation approach. This means that for each iteration, we used 80% of the available SCG pulses (N = 6249) for training the model and reserved the remaining 20% for testing. Importantly, the SCG pulses belonging to each subject were only included in either the training or test set, but not both, in order to avoid any potential bias in the performance metrics. We repeated this process for 10 iterations, randomly selecting different subjects to be included in the training and test sets at each trial. This allowed us to verify that the reported performance metrics were not influenced by any specific patterns in the distribution of the training and test sets. For each iteration, the model was trained for 150 epochs. To assess the agreement between the values obtained by the DNN and 4D flow, we performed correlation and Bland-Altman analyses. The linear relationship between the two sets of values was determined using Pearson correlation coefficient, and the limits of agreement (LOA) were calculated using the Bland-Altman method. In Bland-Altman plots, each sample is represented by the mean of the two measurement techniques on the xaxis, and the difference between the two techniques on the y-axis. The mean difference is an estimate of the bias, the standard deviation (SD) of the difference measures the random fluctuations around the mean, and the LOA is defined as 1.96×SD. If the mean difference is significantly different from 0, based on a one-sample t-test (significance level = 0.05), this indicates the presence of a fixed bias. It is important to note that for each subject, there were multiple SCG scalograms recorded at different heartbeats. The DNN model assigned a Vmax value to each scalogram, and we used the average Vmax value over all scalograms for each subject as the representative value for that subject. To investigate the performance of the classification model, we used receiver operating characteristic (ROC) curves. The ROC curves were constructed using the probabilities assigned by the model to each observation belonging to a particular valve condition group. An ROC curve plots the true positive rate (TPR) against the false positive rate (FPR) at different thresholds chosen from the predicted probabilities, and a higher ROC-AUC (area under the curve) indicates a better performance by the model. We also used the confusion matrix obtained by testing the classification model. In a confusion matrix, the diagonal elements represent the number of true positives ( ) in each category, while the off-diagonal elements at each row and column respectively show the number of false negatives ( ) and false positives ( ). In addition, we calculated and compared the precision ( + ) and recall ( + ) rates in diagnosing each valve condition. Results Prediction of Vmax The peak systolic velocities predicted by our deep learning model were in good agreement with the velocities obtained using 4D flow MRI, as demonstrated by the low mean squared error of 0.2 m/s across ten random trials. Figure 4 Figure 4(d) shows a confusion matrix, which was used to evaluate the model's performance on the test set in one random trial. The rows of the matrix represent the ground-truth categories, while the columns represent the categories predicted by the model. The precision and recall rates indicate the model's ability to correctly classify subjects in each of the four classes. The precision rates for the four classes were 93%, 82%, 83%, and 49%, while the recall rates were 81%, 94%, 68%, and 86%. Classification of aortic valve condition Discussion and conclusion In this study, we evaluated the effectiveness of using deep learning and SCG scalograms to predict aortic peak systolic velocity and diagnose patients with various aortic valve complications. Our deep learning model accurately predicted peak systolic velocities, with MSE = 0.2 m/s and r = 0.76 (p<0.01), compared to velocities obtained using 4D flow MRI. Despite the imbalanced distribution of subjects with different types of aortic valve pathologies, we achieved high precision rates (over 82% for all categories except non-MAV) and recall rates (over 81% for all categories except AS) in the diagnosis of these conditions. Our results suggest that deep learning and SCG have significant potential as a substitute or screening tool for more advanced imaging techniques such as 4D flow MRI. Deep learning models have been previously applied to extract various types of information from SCG signals. For example, a CNN model was developed to continuously identify R-peaks from SCG signals with high sensitivity, and it was demonstrated that heart rate variability indices obtained using this model from SCG signals were in good agreement with those obtained from ECG signals [57]. A study of 36 healthy subjects showed that deep learning could accurately map SCG signal segments to whole-body ballistocardiograms [51]. A U-Net-based cascaded framework was also proposed for estimating respiratory rate from ECG and SCG signals [58]. Other research has investigated the use of machine learning and deep learning for detecting heartbeats and heartbeat rates from SCG signals [59,60]. In contrast to these previous works, this paper examines the correlation between wearable SCG pulses and cardiac peak systolic velocity, whose ground truth was determined using comprehensive 4D flow MRI. This study therefore expands the potential use of SCG for diagnosing cardiac abnormalities based on blood velocity, rather than just heart rate beats. One potential application of the technique described in this work is to improve the accuracy of estimating aortic velocity for PC-MRI. A crucial parameter that needs to be specified before performing PC-MRI is the venc threshold [61]. This parameter should be set to capture the highest expected velocity within the vessel of interest while maintaining a sufficient velocity-to-noise ratio. Although the venc value is crucial for proper performance of the PC pulse sequence, it is often estimated because its optimal value is not known in advance, and sometimes a study needs to be repeated using different vencs to obtain the optimal results. The technique presented in this work could potentially be used to obtain a reliable estimation of the optimal venc value before imaging, reducing MR imaging time and costs. A major limitation of the current study was the imbalanced distribution of subjects. In predicting peak systolic velocity, the majority of study participants were healthy, resulting in only a small percentage (about 20%) of training set samples having a velocity greater than 2 m/s. This made it more difficult for the model to accurately predict higher peak velocities in the test set, as shown in Fig. 4(a) where the correlation with ground-truth values was weaker for higher velocities compared to lower ones. In classifying valve conditions, the non-TAV group had 46 subjects, while the non-BAV, non-MAV, and AS groups each had only 10, 9, and 12 subjects, respectively. This may have contributed to the less accurate predictions of the model in diagnosing non-MAV and AS categories compared to non-TAV subjects. Furthermore, while it would have been valuable to further stratify AS severity (e.g., mild, moderate, and severe), doing so would likely result in even smaller numbers of subjects in each group. Therefore, it is necessary to expand the patient cohort to include more subjects with higher aortic velocities and different severities of AS in future studies. Figure 1 . 1(a) the derivation of Vmax by AI-based 3D-segmentation of the thoracic aorta within the full 4D flow MRI measurements and manual delineation of AAo, (b) the SCG device, (c) an example of ECG signal, (d-f) acceleration signals measured in , , and directions, respectively, (g) gating of SCG to R-waves within simultaneously acquired ECG signals, (h) an example SCG pulse measured over one heartbeat. Figure 2 . 2(a) comparison of a raw and denoised SCG signal Figure 3 . 3The mixture DNN model composed of a MLP and a CCN to (a) predict the aortic Vmax and (b) classify the subjects into different valve conditions: non-AS TAV, non-AS BAV, non-AS MAV, and AS, based on the demographic attributes of the subjects and scalograms of the SCG signals. (a) shows the average DNN-predicted Vmax values versus the velocities obtained by 4D flow MRI for the test subjects. As mentioned earlier, we used the average Vmax value over all scalograms for each subject as the representative value. The figure shows a strong linear correlation between the estimated and measured Vmax values (y = 0.89x, r = 0.76, p ≪ 0.01). Additionally, Figure 4(b) shows the Bland-Altman plot for the Vmax values obtained by the DNN model and 4D flow MRI measurements. This plot indicates a low, non-significant bias (-0.08 m/s, p = 0.18) and moderate limits of agreement (±0.86 m/s). Figure 4 ( 4c) shows the results of a classification model that was trained to identify four different types of subjects based on their SCG scalograms and demographic attributes: non-AS TAV, non-AS BAV, non-AS MAV, and AS. The model was evaluated using receiver operating characteristic (ROC) curves, which plot the true positive rate (recall) against the false positive rate. The solid lines on the ROC curves represent the mean performance of the model over 10 random iterations, while the error regions show the standard deviation of the performance across the iterations. 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{'abstract': 'In this paper, we explored the use of deep learning for the prediction of aortic flow metrics obtained using 4D flow MRI using wearable seismocardiography (SCG) devices. 4D flow MRI provides a comprehensive assessment of cardiovascular hemodynamics, but it is costly and time-consuming. We hypothesized that deep learning could be used to identify pathological changes in blood flow, such as elevated peak systolic velocity Vmax in patients with heart valve diseases, from SCG signals. We also investigated the ability of this deep learning technique to differentiate between patients diagnosed with aortic valve stenosis (AS), non-AS patients with a bicuspid aortic valve (BAV), non-AS patients with a mechanical aortic valve (MAV), and healthy subjects with a normal tricuspid aortic valve (TAV). In a study of 77 subjects who underwent same-day 4D flow MRI and SCG, we found that the Vmax values obtained using deep learning and SCGs were in good agreement with those obtained by 4D flow MRI. Additionally, subjects with TAV, BAV, MAV, and AS could be classified with ROC-AUC values of 92%, 95%, 81%, and 83%, respectively. This suggests that SCG obtained using low-cost wearable electronics may be used as a supplement to 4D flow MRI exams or as a screening tool for aortic valve disease.', 'arxivid': '2301.02130', 'author': ['Mahmoud E Khani \nDepartment of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL\n', 'Ethan M I Johnson \nDepartment of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL\n', "Aparna Sodhi \nAnn & Robert H. Lurie Children's Hospital\n60611ChicagoIL\n", "Joshua D Robinson \nDepartment of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL\n\nAnn & Robert H. Lurie Children's Hospital\n60611ChicagoIL\n\nDepartment of Pediatrics\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL\n", "Cynthia K Rigsby \nDepartment of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL\n\nAnn & Robert H. Lurie Children's Hospital\n60611ChicagoIL\n\nDepartment of Pediatrics\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL\n", 'Bradly D Allen \nDepartment of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL\n', 'Michael Markl \nDepartment of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL\n\nDepartment of Biomedical Engineering\nMcCormick School of Engineering\nNorthwestern University\n60208EvanstonIL\n'], 'authoraffiliation': ['Department of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL', 'Department of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL', "Ann & Robert H. Lurie Children's Hospital\n60611ChicagoIL", 'Department of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL', "Ann & Robert H. Lurie Children's Hospital\n60611ChicagoIL", 'Department of Pediatrics\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL', 'Department of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL', "Ann & Robert H. Lurie Children's Hospital\n60611ChicagoIL", 'Department of Pediatrics\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL', 'Department of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL', 'Department of Radiology\nFeinberg School of Medicine\nNorthwestern University\n60611ChicagoIL', 'Department of Biomedical Engineering\nMcCormick School of Engineering\nNorthwestern University\n60208EvanstonIL'], 'corpusid': 255440641, 'doi': '10.48550/arxiv.2301.02130', 'github_urls': [], 'n_tokens_mistral': 14991, 'n_tokens_neox': 12460, 'n_words': 7346, 'pdfsha': '7589747734120c3709c87b5a349f85cb1153ea7f', 'pdfurls': ['https://export.arxiv.org/pdf/2301.02130v1.pdf'], 'title': ['A deep learning approach to using wearable seismocardiography (SCG) for diagnosing aortic valve stenosis and predicting aortic hemodynamics obtained by 4D flow MRI', 'A deep learning approach to using wearable seismocardiography (SCG) for diagnosing aortic valve stenosis and predicting aortic hemodynamics obtained by 4D flow MRI'], 'venue': []}

arxiv

0:46 WSPC/INSTRUCTION FILE mufmuf4 Jun 2023 June 2, 2023 Tomi Koivisto Institute of Physics National Institute of Chemical Physics and Biophysics Laboratory of Theoretical Physics University of Tartu W. Ostwaldi 1, Rävala pst. 1050411, 10143Tartu, TallinnEstonia, Estonia 0:46 WSPC/INSTRUCTION FILE mufmuf4 International Journal of Geometric Methods in Modern Physics Jun 2023 June 2, 2023Foundations of Cosmologythe Problem of TimeGauge theory of Spacetime and GravityDark Energy and Dark MatterCosmological InflationPregeometric theory of Physics This proceeding is an introduction to cosmological applications of the Lorentz gauge theory. It provides the ingredients for a unique, though yet tentative ΛCDM theory cosmology. The emergence of spacetime is described by the spontaneous symmetry breaking called here the khronogenesis. Space is then associated with the field strength of the antiself-dual gauge potential, and gravity is associated with the self-dual field strength. In the cosmological setting, khronogenesis seems to predict inflation. It is shown that the Lorentz gauge theory allows the consistent description of spin currents which could have important roles in cosmological phenomenology. Introduction In a discussion at the workshop Metric Affine Frameworks for Gravity at Tartu 2022 it was suggested that in a fundamental theory of spacetime and gravitation we should not presuppose a Metric but instead describe its emergence in terms of more elementary objects, whilst gauge theories based on symmetries under Affine transformations are known to describe the material dynamics on Hamiltonian lattice or continuum fields in spacetime yet may not furnish the most elementary Framework to describe the dynamics of spacetime, Gravity. These proceedings are an introduction to the Lorentz gauge theory, a new pregeometric framework for gravity and cosmology [40]. In pregeometric theories, the metric can arise as a composite object constructed from fundamental fermions [1,3,38,39]. Such a theory can be formulated without any reference to the metric, and accommodate the ground state wherein the spacetime metric vanishes. Thus, there exists a ground state that describes the absence of spacetime rather than a given reference spacetime. In contrast to conventional gauge theories of gravity (see e.g. [8] for some of the seminal papers), the Lorentz gauge theory is not formulated on an affine bundle, but is based on a different approach akin to parameterised field theories [29]. The theory introduced by Z lośnik et al is based on the complexified Lorentz group. It is not necessary to dwell on the foundational importance of Lorentz symmetry. However, the complexification demands some justification from physics and not only from mathematical convenience. The justification is chirality. Matter, from macroscopic objects (like us) to its most elementary constituents (Weyl fermions) has chiral features, and we may ask should not the spacetime and the gravitational interaction reflect this property of matters. It can be incorporated by complexification, when working in tensor representations. Having the Lorentz-covariant derivative operator D we can, in the complexified theory, introduce its self-dual and antiself-dual projections ± D. In the spinor representation, these are just the projections that act on the left-and the right-handed Weyl spinors, respectively a . (The definitions of the projections are recalled below in 1.1, and the exterior algebra notation below in 1.2.) Now, we can deduce the action. A field needs to be introduced into the theory, since only topological invariants can be constructed from the operator D alone. We consider a field φ a in the fundamental representation. Again, the more fundamental formulation would take place in the spinor representation s.t. φ a =ψγ a ψ, but working with tensors is convenient and suffices for our present purposes. Demanding that the global translation symmetry is preserved (though not gauged), the action is determined I = i 2 φ a D + (DD)Dφ a + L M (Dφ a , ψ, + Dψ) ,(1) up to a boundary term that can be fixed by matching the conserved charges with the observables [7] (and of course up to a specification of the matter source L M we have included for generality). It turns out that the φ a plays the role of a clock field, for which reason it was dubbed the khronon. When the clock doesn't tick, Dφ a = 0, time doesn't flow. Due to the global symmetry we demanded, this is gauge-equivalent to φ = 0, and it is this trivial solution we identify as the pregeometric ground state. It has been often contemplated if a universe could appear ex nihilo, where the nihil could mean a quantum field theory vacuum, a spaceless or a boundary-free geometry, or something else, e.g. [36,37,19,18]. Since the pregeometric ground state in our theory offers a candidate for "the nothing", a question we shall begin to explore is whether the action principle (1) alone might determine some of the boundary conditions for the universe, and in particular, explain its (hypothetically inflationary) beginnings. Thus, the focus of these proceedings is on cosmology. The main results of the following sections, from the perspective of cosmology, are listed below. • In section 2 we introduce the khronogenesis, emergence of space and time via a spontaneous Lorentz symmetry breaking. • Section 3 shows that the cosmology of theory (1) is viable without any dark matters in L M , and points out a duality relating the Λ and the CDM. • In section 4 we discover that the ground state can spontaneously yet smoothly begin to inflate into our hot big bang universe. • Section 5 introduces spin currents. They could be significant in early universe phenomenology, and potentially resolve the H 0 tension. The only new (very simple) solutions are in sections 4 and 5. Many of the derivations and clarifications in the preceding sections haven't been published elsewhere either. In the final section 6 we point out some of the calculations that should be tackled next, and dicuss some of the new possibilities for cosmological model-building. Lorentz algebra The algebra so(4, C) has 2 invariants: η ab and ǫ abcd . Our convention is η ab = (−1, 1, 1, 1), and ǫ 0123 = 1. Consider a bivector X ab in the algebra. Its ⋆-dual is defined as ⋆X ab = 1 2 ǫ ab cd X cd ⇒ ⋆ ⋆ X ab = −X ab .(2) We can also define the (anti)self-dual projections, ± X ab = 1 2 (1 ∓ i⋆) X ab = 1 2 δ a c δ d b ∓ i 2 ǫ ab cd X cd ⇒ ⋆ ± X ab = ±i ± X ab . (3) These are indeed projections since X ab = + X ab + − X ab , (4a) ± ( ± X ab ) = ± X ab , (4b) ∓ ( ± X ab ) = 0 . (4c) It is useful to note properties of the products ± X ab Y ab = X ab ± Y ab = ± X ab ± Y ab , (5a) ± X ab ∓ Y ab = 0 , (5b) ǫ abcd ± X ab Y cd = ǫ abcd X ab ± Y cd = ǫ abcd ± X ab ± Y cd ,(5c)ǫ abcd ± X ab ∓ Y cd = 0 ,(5d) which follow immediately from the definition of the projection. The ± split basically realises the isomorphism so(4, C) = su(2, C) × su(2, C). In the latter form, we have 2 decoupled algebras with the 2 invariants: δ IJ and ǫ IJK , where the indices I, J, K take the values 1,2,3. We will be using also this form of the theory. Exterior algebra We denote p-forms with bold symbols if p > 0. The antisymmetric wedge product is used explicitly. For a p-form p and a q-form q, we have p ∧ q = (−1) qp q ∧ p. To set up a Lorentz gauge theory, we introduce the Lorentz gauge potential 1form ω ab , implicit in the so(4, C)-covariant exterior derivative D. We note that D(p∧q) = Dp∧q+(−1) p p∧Dq. We can perform the ± decomposition of the gauge potential ω ab = + ω ab + − ω ab . The exterior derivative + D is then covariant only wrt self-dual Lorentz transformations, and the − D is covariant only wrt to antiself-dual transformations. However, the projection of bivectors is Lorentz-invariant, since the symbols η ab and ǫ abcd are. In particular, the field strength of the gauge potential, R ab = dω ab + ω a b ∧ ω cb = + R ab + − R ab ,(6) can be split into the 2 projections which are the field strengths of the respective 2 projections of the gauge potentials (only after we have moved towards spacetime geometry in section 2.3 we begin refer to the Lorentz gauge potentials also as connections). As one quickly checks using the Poincaré lemma d 2 = 0, we have D 2 X a = R a b X b for a Lorentz vector X a . Similarly, for a bivector X ab we obtain D 2 X ab = −2R [a c X b]c , and if X ab is a bivector p-form, we write D 2 X ab = −2R [a c ∧ X b]c , etc. A conventional tool in spacetime geometry is the coframe e a . We define the 4-volume element as ⋆1 = 1 4! ǫ abcd e a ∧ e b ∧ e c ∧ e d ,(7a) and also the 3-form basis ⋆e a = 1 3! ǫ a bcd e b ∧ e c ∧ e d ,(7b) will be useful. The 2-form basis e a ∧ e b is a bivector (since it is antisymmetric) and therefore the rules in 1.1 apply to it as well. It is now straightforward to show that e.g. e a ∧ ⋆e b = −η ab ⋆ 1, and e a ∧ e b ∧ ⋆(e c ∧ e d ) = −2δ [a c δ b] d ⋆ 1. Lorentz gauge theory We begin the study of Lorentz gauge theory (1) by stating its EoM's (equations of motion) in 2.1. The spontaneous symmetry breaking giving rise to space and time is demonstrated straightaway in 2.2. The simple example of flat Minkowski spacetime is already a non-trivial Lorentz gauge field configuration with dynamical field strength. In 2.3 the structure of the theory is clarified by establishing the relations of the Lorentz gauge field strengths and more conventional geometrical objects, such as the curvatures of the metrical Levi-Civita connection or the selfdual Ashtekar connection. In 2.4 the dynamical equations are put into a convenient 1+3 form. Field equations For generality, we have included matter sources for some fields ψ in the action (1). The variation of lagrangian 4-form L M wrt the field ψ produces their Euler-Lagrange EoMs. The variations of L M wrt the fundamental fields can be parameterised in terms of the two 3-forms t a and O ab = O [ab] as δL M δφ a = −Dt a ,(8a)δL M δω ab = −φ [a t b] + O ab . (8b) The t a is the material energy-momentum 3-form and O ab is the material angularmomentum 3-form, which we may call more briefly the energy current and the spin current, respectively. They are the sources in the gravitational field equations we obtain from (1), D(i + R a b ∧ Dφ b − t a ) = 0 , (9a) i 2 D + (Dφ [a ∧ Dφ b] ) = iφ [a + R b] c ∧ Dφ c − φ [a t b] + O ab .(9b) The 1 st equation shows a covariantly closed 3-form. Let us just call this 3-form M a , i + R a b ∧ Dφ b − t a = M a where DM a = 0 .(10) Using this in the 2 nd field equation (9b) the system is i + R a b ∧ Dφ b = t a + M a , (11a) i 2 D + (Dφ [a ∧ Dφ b] ) = φ [a M b] + O ab .(11b) The first of these equations may look familiar. The LHS would become the Einstein 3-form if we could identify (up to i) the self-dual field strength with a metric curvature 2-form, and identify the 1-form Dφ a with the coframe of the metric tensor g s.t. g = η ab Dφ a ⊗ Dφ b . Khronogenesis The simplest solution to the theory (1) without matter sources is φ a = 0. The gauge potential ω ab is then completely arbitrary. Due to the existence of this totally symmetric solution of the theory, it can be regarded as a pre-geometric theory of gravity and spacetime. Neither of these is postulated a priori, but they can emerge in a symmetry-broken phase of the theory. Let us assume that φ a φ a < 0. A time-like expectation value of the field breaks the Lorentz symmetry down to the rotational symmetry. For convenience, we may then adopt the gauge φ a = φ(x)δ a 0 wherein the 0 th axis is aligned with the field. It will turn out that the 0 th component φ can then interpreted as a clock function, for which reason the field φ a is called the khronon scalar. To see how time and space is constructed in such a symmetry-broken phase, let us first consider the most basic case, the flat-metric spacetime which is the background usually postulated in standard quantum field theory. For simplicity, we pick coordinates s.t. the clock function φ = t is the time coordinate. Then Dφ 0 = dφ = dt, so the time component of the Minkowski coframe is reproduced correctly. The spatial components of the coframe, Dφ I = dφ I + ω I 0 φ 0 = ω I 0 t are now proportional to the electric components of the gauge potential. The spatial coframe of the Minkowski space would be dx I . Thus we require the non-vanishing gauge potential ω I 0 = t −1 dx I . Is this a pure gauge potential? To compute the field strength of the potential, we need also its magnetic components which are not determined by the background geometry alone. The magnetic components ω I J we can solve from the field equations. We choose the integration form M a = 0 to vanish at (10) and assume no sources t a = 0. Then (11a) reduces to + R a b ∧ Dφ b = 0, which is satisfied if the self-dual field strength vanishes. Thus, we can set the self-dual gauge potential to vanish. Then ⋆ω ab = −iω ab , implying that the magnetic components of the potential are given by the electric components as ω IJ = iǫ IJ K ω K 0 . To summarise, the Minkowski vacuum is supported by the field configuration φ = t , ω I 0 = t −1 dx I , ω I J = it −1 ǫ I JK dx K .(12) This simple but non-trivial configuration is described in gauge-invariant terms by the field strength R I0 = t −2 dt ∧ dx I + iǫ I JK dx J ∧ dx K ,(13a)R IJ = t −2 −iǫ IJ K dt ∧ dx K + 2dx I ∧ dx J .(13b) Space and time, even the case of Minkowski, requires the anti-selfdual gauge field strength. In this sense, spacetime in our theory not only has but is geometry. Reference frames in gravitational theories can be described in terms of tetrad components. Now we may compose a coframe 1-form e a from the derivative of the symmetry-breaking field e a = Dφ a . If this represents an invertible matrix, there exist the quartet of vectors @ a s.t. @ a e b = δ b a , and these four vectors @ a can then play the role of tetrads. They do this for example in the Minkowski configuration (12), but the theory (1) remains well defined in generic configurations wherein the coframe composed as Dφ a can be degenerate. A criterion for a configuration to describe a spacetime is that there exists a non-degenerate tetrad. Only then can we define conventional spacetime tensor such as e.g. the 4-index Riemann curvature tensor @ b (@ a R ab ). Another invariant characterisation of geometry is the torsion 2-form T a defined as the derivative of the coframe, T a = De a . In conventional models of gravity, torsion is independent of the curvature. However, now we find the relation T a = De a = DDφ a = R a b φ b .(14) By taking further the derivative of this relation, we obtain DT a = R a b ∧ e b , which is a geometric Bianchi identity satisfied in any gravity theory, but the relation (14) is peculiar to the Lorentz gauge theory. For the Minkowski solution (12) the relation implies that T 0 = 0 ,(15a)T I = t −1 dt ∧ dx I + iǫ I JK dx J ∧ dx K .(15b) Both the curvature and the torsion of the Minkowski solution exist in the antiself dual sector of the theory. We have R ab = − R ab , and also T a = − T a . In the asymptotic future, the khronon grows without a bound and the geometry (13,15) fades away. There is no asymptotic past, but a singularity at φ = t → 0. Thus, in fact this geometry could not have been created by khronogenesis. The lesson is that a transition "from no space to flat space" is impossible. We have to study the theory in more depth and consider a bit more elaborated model. A viable class of khronogenetic spacetimes will then be arrived at in 4.2. Kinematics It can be useful to look in more detail at the geometric structure of the theory and see how it gives rise to the standard description of general relativity in terms of the e I -compatible torsion-free connectionω I J = ǫ I JKω K , and the extrinsic curvature 1-form K I = @ I (L @ a φ a h) computed from the canonical spatial metric h = δ IJ e I ⊗ e J . It can be deduced that the connection in the time gauge assumes the form [40] ω ab = 0 e I /φ −e I /φ ǫ IJ K 2A K + ie K /φ ,(16) where we have denoted A I = 1 2 (ω I − iK I ). The self-dual and the anti-self-dual parts of this connection are + ω ab = 0 iA I −iA I ǫ IJ K A K , (17a) − ω ab = 0 −iA I + e I /φ iA I − e I /φ ǫ IJ K A K + ie K /φ . (17b) Then the curvature can be written as R ab = 0 T I /φ −T I /φ ǫ IJ K 2F K + iT K /φ .(18) We have denoted F I = dA I − ǫ I JK A J ∧ A K .(19a) The connection A I corresponds to the Ashtekar connection and F I is its curvature [4,34]. The latter can also be expressed in terms of the metric curvatureR IJ and the metric-covariant derivativeD as F I = 1 4 ǫ I JK R JK + K J ∧ K K − i 2D K I . (19b) In the decomposition of (18), the self-dual part involves only these standard ingredients, + R ab = 0 iF I −iF I ǫ IJ K F K ,(20a)− R ab = 0 −iF I + T I /φ iF I − T I /φ ǫ IJ K F K + iT I /φ . (20b) The expression for torsion, generalising (15) is, T 0 = 0 ,(21a)T I = de I + e I ∧ d log φ − ǫ I JK 2A J + ie J /φ ∧ e K .(21b) The presence of the torsion 2-form distinguishes the + R a b and − R a b as independent of each other. As we found in the previous subsection, the − R a b sector is excited even in the flat Minkowski background spacetime. In this subsection, we have not used the EoM's but only looked at the kinematic structure. Dynamics We have now considered two decompositions of the Lorentz gauge field strength. It can be splitted into the self-dual and antiself-dual field strengths ± R ab , but an alternative split in the time gauge was made in terms of the two Lorentz 3vector 2-forms, the curvature F I and the torsion T I , given by equations (19) and (21b), respectively. Exploiting the latter decomposition, we find that the gaugefixed action reduces to I = ie 0 ∧ e I + 2 ⋆ e 0 ∧ e I ∧ F I + L M ,(22a) and can be further massaged into the remarkably simple form I = 1 2 φF I ∧ T I + L M . (22b) This looks like the su(2, C) × su(2, C) version of the theory. We can also rewrite the field equations (11), as F I ∧ e I = M 0 + t 0 ,(23a)F I ∧ e 0 + iǫ I JK F J ∧ e K = −M I − t I ,(23b)T I ∧ e 0 − iǫ I JK T J ∧ e K = iφM I + 2iO 0I .(23c) The coframe field can be considered as the short-hand notation for e 0 = dφ, e I = φω I 0 . Assuming that − (O ab + φ [a M b] ) = 0, this is the full set of field equations. Otherwise, the 2 nd torsion equation, T [I ∧ e J] + i 2 ǫ IJ K T K ∧ e 0 = −2iO IJ ,(23d) is not redundant with (23c), but their combination will impose the latter constraint. 3. The Λ-space κ and the CDM-time φ In general, we have deduced that the 3-form M a obeys DM a = 0 ,(24a)M K = −φ −1 2O 0 K − iǫ IJK O IJ .(24b) In the next subsection 3.1 we will solve these equations and show that cosmology in the Lorentz gauge theory can be viable without dark matter. Then, in 3.2 we introduce another 3-form κ conjugated to a cosmological constant, and discover a cosmic duality between the "local time" measured by φ a and the "global time" measured by κ [14]. The CDM The properties of the 3-form M a can be deduced from its EoM (24) in 3 steps. 3.1.1. The M a is aligned with the khronon φ a . To begin with, in (1) it is assumed that material sources do not have nonminimal couplings to the antiself-dual connection. Then equation (24b) dictates that M I = 0, so that M a as a Lorentz vector has only the possible non-zero time-like component. We could deduce this without gauge-fixing by starting from (11b), and by setting the non-minimal matter coupling to vanish, − O ab = 0. Then it follows that also − φ [a M b] = 0, from which it follows that φ [a M b] = 0, so that M a ∼ φ a , which in the time gauge is just the statement M I = 0. The point is that we could relax the assumptions − O ab = 0 and φ [a M b] = 0 only together but not separately. The antiself-dual matter hypermomentum and the geometrical spin current must cancel each other, if they are non-zero. This might be an interesting possibility to explore, but in these proceedings we stick to the assumption − O ab = 0. 3.1.2. The M a is a 3-space volume form. We have now reduced the four 3-form components of M a to just one 3-form M 0 . As it arises from the integration of the gravitational field equations, a priori we would have to specify its four independent components in order to evolve the dynamical system. However, upon closer look at the structure of the theory, the number of the required initial conditions can be further reduced. Now, the spatial components of the conservation equation (24a) give DM I = dM I − ω 0 I ∧ M 0 − ω J I ∧ M J = −φ −1 e I ∧ M 0 = 0 ,(25) where we used the previous step M I = 0 and the definition of the composite coframe e a = Dφ a . The condition shows that the 3-form must be pure spatial volume in the coframe basis, in other words it is determined by a single function, as M 0 = 1 2 ρ D ⋆ e 0 ,(26) where we call the function ρ D and the factor 1/2 is just conventional. conservation equation (24a) shows that the energy is constant, DM 0 = dM 0 − ω I 0 ∧ M I = 1 2v 0Ṁ (dt ∧ dx ∧ dy ∧ dz) = M ′ 2v 0 √ det h ⋆ 1 = 0 . (27) The prime denotes the reparameterisation-invariant time derivative M ′ = ∂M/∂φ =Ṁ /φ wrt the khronon. A region of space V is associated with an energy V M 0 which does not change as the region V expands, contracts or changes its shape. This is exactly how ideal dust would behave. In particular, material CDM would consist of massive particles which dilute in an expanding universe such that the energy density is simply inversely proportional to the volume. The fluid approximation would break down when probing so small scales that the collisions and other possible interactions between the particles would have to be taken into account. This distinguishes material and the geometrical realisations of CDM. In the present case, the fluid description of CDM is not approximate but exact. The M is not the mass of a particle contained in a volume v 0 but the mass of the space that spans a volume v 0 . In summary, we have proven that (given that − O ab = 0 as follows from (1)), the 3-form M a introduces only one integration constant into the solutions of the theory, and this constant determines the density of an effective ideal dust which interacts only gravitationally. Therefore it is an obvious candidate for the missing mass of the universe. The Λ It is well-known that in unimodular gravity the cosmological constant Λ appears as a constant of integration. Exploited in general relativity since the 1910's, the unimodular device continues to find interesting new applications in current research, e.g. [24,17,31]. Covariant formulations of unimodular gravity [21] reveal that the 3space is a carrier of information about time [6] cf. Misner volume time, and below we show that this is compatible with the Lorentz gauge theory. The action formulation we consider is I Λ = Λ (dκ − ⋆1) ,(28) where the new fields are the scalar Λ and the 3-form κ which is the Lagrange multiplier that sets the constancy of the cosmological constant. We note that in an alternative formulation, the dark matter candidate 3-form M 0 as well can be understood as the Lagrange multiplier (which determines the coframe as the derivative of the khronon). The EoMs for the 2 fields in the action (1) are dΛ = 0 ,(29a)dκ = ⋆1 .(29b) We shall integrate the previous equation over a 4-volume W bounded by two Cauchy surfaces V 1 and V 2 , W ⋆1 = W dκ = V2 κ − V1 κ .(30) Thus, the Λ-conjugated time between V 2 and V 1 is the invariant spacetime volume enclosed by these hypersurfaces. On the other hand, the invariant lapse of the CDMconjugated khronon time ∆φ = φ 2 − φ 1 is determined by another fundamental field of the theory. We conclude that there is a relation between the two concepts of time and therefore only one of them can be chosen arbitrarily. An explicit example will be checked below at (46) in 4.2. Is there a duality between the conjugates of the two concepts of time as well? Can we determine the energy scales of the integration constants Λ and M from 1 st principles, or at least fix one of them given the other? It is at this point that the answers are "yet tentative" as the abstract disclaimed. Perhaps, the local and the global views of time could be related to what Dirac called the atomic units and the Einstein units, in the context of a profound idea on the connection between the cosmological evolution and the constants of Nature known as the large number hypothesis [13], which though has not yet found its precise and viable mathematical expression. In the similar way khronogenesis gives rise to the constant c, the speed of light, the breaking of the de Sitter to the Lorentz can give rise to the gravitational constant [28]. Finally, the Planck constant could be the result of the reduction of the conformal into the de Sitter symmetry. The 3-form κ can be related to the kairon scalar field κ a = * κ Dφ a which is the dual of our symmetry-breaking field φ a in a SO(6, C) extension of the SO(4, C) Lorentz gauge theory [26]. The duality suggests Λκ ∼ M a φ a which leads to the solution of the κ ∼ Λ −1 M t * dt, in a coordinate system with time t. It is plausible that the two fundamental scales of cosmology could be predicted in a more complete theory containing the Lorentz gauge theory. Cosmology We shall now consider the cosmology of the Lorentz gauge theory. In 4.1 we construct the isotropic and homogeneous geometry, and in 4.2 we derive the dynamical equations and study their exact solution in a simple toy model. The purpose is to demonstrate that the Lorentz gauge theory can provide a completion of the standard inflationary hot big bang cosmology. Isotropic and homogeneous kinematics The point of departure is the generic isotropic and homogeneous, spatially flat Ansatz for the fundamental fields, the Lorentz gauge potential 1-form ω a b and the khronon scalar field φ a (for a spatially curved case, see [29]). The generic Ansatz is given 3 independent functions of time, φ a = δ a 0 φ(t) ,(31a)ω I 0 = A(t)dx I ,(31b)ω I J = iB(t)ǫ I JK dx K .(31c) We may split the connection also as ± ω I 0 = 1 2 (A ∓ B) dx I ,(32a)± ω I J = i 2 (∓A + B) ǫ I JK dx K .(32b) The coframe field composed from these fields, e 0 =φdt ,(33a)e I = φAdx I ,(33b) shows how the Friedmann-Lemaître geometry emerges from the fields in (31). The lapse function n(t) =φ is time derivative of the khronon. The scale factor a(t) = φA is obtained by multiplying the khronon b with the component of the gauge potential A. The role of the component B we have to deduce from the dynamical equations. First we shall look at the gauge-invariant characterisation of the geometry in terms of the field strengths. The field strength (6) for the Ansatz (31) is R I 0 =Ȧdt ∧ dx I − iABǫ I JK dx J ∧ dx K ,(34a)R I J = iḂǫ I JK dt ∧ dx K + A 2 + B 2 dx I ∧ dx J .(34b) From this we obtain, according to (18), the 2-forms F I = − i 2 Ȧ −Ḃ dt ∧ dx I + 1 4 (A − B) 2 ǫ I JK dx J ∧ dx K ,(35a)T I = φ Ȧ dt ∧ dx I − iABǫ I JK dx J ∧ dx K . (35b) It is a useful cross-check to verify that F I is the Ashtekar curvature according to (19). The metric spin connection in cosmology is given bẙ ω I 0 = a ′ dx I ,(36a)ω I J = 0 ,(36b) as one readily deduces from (33). On the other hand, from (17a) and (32a) we see that now the extrinsic curvature is K I = (A − B)dx I = 2iA I .(36c) With the simple expressions (36) plugged into the formulae (19) both of them result in (35a), so the cross-check is passed and we may continue to the field equations. Isotropic and homogeneous dynamics Since we have introduced various different connections along the way, we can consider the different torsions wrt these connections. For the cosmological Ansatz (31), the torsion of the self-dual and the antiself-dual, the Levi-Civita and the Ashtekar connections are, respectively, ± T I = φȦ + 1 2φ (A ± B) dt ∧ dx I + i 2 φA (±A − B) ǫ I JK dx J ∧ dx K , (37a) T I = 0 ,(37b)T I = φȦ +φA dt ∧ dx I + i 2 φA (A − B) ǫ I JK dx J ∧ dx K . (37c) It is illuminating to revisit the field equations (11) in their fully covariant form before adapting them to cosmology in the time gauge. Exploiting the geometric identity (14), we see that the 2 equations can be rewritten as i + D + T a = t a + M a ,(38a)iDB ab = O ab . (38b) This makes the structure of the theory quite transparent: the self-dual torsion is the excitation whose flux is sourced by energy momentum, and the (proto)area element B ab = + Dφ [a ∧ Dφ b] /2 is the excitation sourced by angular momentum, B 0I = 1 4 φA φ dt ∧ dx I + i 2 φAǫ I JK dx J ∧ dx K ,(39a)B IJ = − 1 4 φA φ ǫ IJ K dt ∧ dx K + 1 2 φAdx I ∧ dx J . (39b) In section (2.2) we saw that in the Minkowski limit the self-dual torsion will vanish, consistently with our interpretation of + T I as the material energy excitation. Eq. (37b) shows that when there is no expansion,ȧ = 0, either A = B must be the same constant or otherwise A = 0. As it is clear from 2.2; the angular excitation B ab does not necessarily vanish in the absence of material sources. To complete the cosmological system, we need to specify the matter sources. We assume an isotropic and homogeneous perfect fluid. Such a source is completely determined by its energy density ρ M , pressure p M and angular momentum Ω, and these may only depend on time. To first recover the standard Friedmann equations, we consider the case of vanishing angular momentum, Ω = 0. Then O ab = 0 and the energy current is determined as t 0 = 1 2 ρ M ⋆ e 0 ,(40a)t I = − 1 2 p M ⋆ e I ,(40b) Plugging these sources into the above equations (38) using the torsion of the selfdual connection (37b) and the area excitation (39), or alternatively plugging the sources into the equations (23) formulated in terms of the 2-forms which are given by (35), we arrive at the 3 equations 3 (A − B) 2 Aφ = M v −1 o + ρ M (Aφ) 3 ,(41a)2 Ȧ −Ḃ Aφ + (A − B) 2φ = −p (Aφ) 2φ ,(41b)Ȧ φ + Bφ Aφ = 0 . (41c) The system is trivially solved by the ground state φ = 0, as well as the Minkowski solution A = 0 which implies also B = 0, and therefore we will assume Aφ = 0. Eq.(53c) is then solved by B = −A ′ φ, and plugging this into (53a) gives the 1 st Friedmann equation 3aa ′ = M v −1 0 + a 3 ρ M ⇒ 3H 2 = ρ D + ρ M ,(42a) and (53b) yields the 2 nd Friedmann equation 2aa ′′ + (a ′ ) 2 = −a 2 p M ⇒ 2H ′ + 3H 2 = −p M . (42b) As expected, we recover the standard cosmological dynamics in general relativity. The H in (42) is the reparameterisation-invariant expansion rate, H =ȧ na =ȧ φa = (log a) ′ ,(43) which is why the lapse function n and the rateṅ/n do not explicitly enter into the equations. It is well-known that cosmological trajectories are generically traced back to a singularity as one extrapolates them back in time (e.g. [20,10]). The question arises whether the fundamental fields of the Lorentz gauge theory could remain well-behaved in the limit that the composite scale factor a = φA → 0 vanishes, and the expansion rate H of this composite scale factor and its higher time derivatives such as H ′ hit infinity. If it makes sense to talk about the beginning of the universe, the only natural beginning is the symmetric phase φ = 0. Anything else would be something rather than nothing. It seems that the connection coefficients A and thus also B could smoothly evolve across φ = 0 without any obvious obstacles. However, the invariant description of the gauge field dynamics is in terms of the field strengths. In the case at hand, recovering the standard cosmological solution in general relativity for a = 0, the 2-forms (35) read F I = − i 2 a ′′ dφ ∧ dx I + 1 4 (a ′ ) 2 ǫ I JK dx J ∧ dx K ,(44a)T I = a ′ − a φ dφ ∧ dx I + iaǫ I JK dx J ∧ dx K .(44b) Let us assume that the energy density of the primordial universe is dominated by a source with some constant equation of state w = p/ρ. The solution to the Friedmann equations (42) then determines us the scalings of the coefficients of the above 2-forms, a ∼ φ 2 3(1+w) , (45a) a ′ ∼ a/φ ∼ φ −(1+3w) 3(1+w) ,(45b)(a ′ ) 2 ∼ φ −2(1+3w) 3(1+w) ,(45c)a ′′ ∼ φ −2(2+3w) 3(1+w) .(45d) Whilst (45a,45b,45c) stay finite as φ → 0 for accelerating equation of state, −1 < w < −1/3 (effectively, violation of the strong energy condition), (45d) requires the stronger condition −1 < w < −2/3. The scaling (45d) describes the electric component of the self-dual field strength (44a). The Lorentz gauge theory thus predicts inflation from khronogenesis since a decelerating universe cannot appear from nothing. The action is not only stationary δI = 0 but realises the density-free boundary condition L| φ=0 = 0 since ground state does not contribute to the action c . The point H = ∞ where the metric curvatureR a b diverges in general relativity and other conventional formulations of gravity is understood as the totally symmetric phase in the ground state φ = 0 of the Lorentz gauge theory which is consistent with any gauge field strength R a b . The density and the pressure are of course divergent ρ M ∼ p M ∼ φ −2 , but they are not directly observable nor fundamental fields of the theory, but quantities derived by dividing physical charges by spatial volumes. The matter Lagrangian is expected to scale as L M ∼ φ −2w/(1+w) , and thus smoothly disappears as we wind the clock back to φ = 0. The constant-w toy model thus predicts not only something rather than nothing, but something material in an initially inflating background. In particular, as seen already from the solution we derived in 2.2, time cannot begin in an empty flat space. Note that a bouncing scale factor is not suggested, since that would require w < −1 (or the violation of the weak energy condition). To close this section, we check the how cosmic dual time evolves according to considerations in 3.2. It is easy to solve (30) using (45). If we consider a unit chunk of space until the khronon time φ, the LHS gives the spacetime volume ⋆1 = 1 + w 3 + w φ 3+w 1+w v 0 .(46a) On the other hand, the LHS gives, assuming only a homogeneous and isotropic 3-form κ, κ = φ 2 1+w κv 0 ⇒ κ = 1 + w 3 + w φ . (46b) c Therefore the action is stationary wrt arbitrary variations of the fields [27]. Thus the conclusions do not hinge on Dirichlet or any other boundary conditions at the initial time φ = 0. Nothing changes if the integration limit is extended towards φ → −∞, since there is no action there, L = 0. Thus, the kairon time κ is simply proportional to the khronon time φ in the case of the constant-w solution. The de Sitter solution w = −1 is a special case for which κ is the constant κ = 1/3H. Cosmology with spin Material spin currents, being the Noether currents corresponding to Lorentz symmetry, are of a paramount interest in Lorentz gauge theory. It is not obvious that the generalised Friedmann equations (41) are consistent with nontrivial dynamics in the presence of a spin current Ω. Its presence will presumably modify the energy conservation of the matter sources, besides modifying the gravitational dynamics. Below in 5.1 we derive the general consistency conditions for spinning fluids in the Lorentz gauge theory by studying its Noether identities, and then in 5.2 we apply the conditions to the case of a perfect spinning fluid in isotropic and homogeneous cosmology. A class of exact solutions is presented. Spin and energy conservation We had parameterised the variation of the matter action by introducing the currents in (8). More precisely, when the matter fields ψ obey their Euler-Lagrange equations, we have δL M = d −t a δφ a + δψ ∧ ∂L M ∂Dψ + δφ a Dt a + δω ab ∧ φ [a t b] − O ab .(47) The density L M is taken to be invariant δ λ L M = 0 under the infinitesimal Lorentz transformation with parameters λ a b , δ λ φ a = λ a b φ b , (48a) δ λ ω a b = −Dλ a b .(48b) We may consider parameters λ a b s.t. that they vanish at the boundary, in which case we can neglect the symplectic piece in (47) as well as another boundary term which arises from a partial integration after inserting (48) in (47). We then see that the Lorentz transformation is an invariance of the density L M if DO ab = Dφ [a ∧ t b] .(49) This identity holds even off-shell since the field equations were not used in the derivation. We have reproduced the result that the divergence of the spin tensor is the antisymmetric energy tensor. The matter action should also be invariant under infinitesimal diffeomorphisms parameterised by a vector ξ. The symmetry is exact only up to a boundary term δ ξ L M = ξ L M , but the boundary term is not relevant for the Noether identity. Diffeomorphisms in Lorentz gauge theory generate spacetime geometry from the fundamental fields. The latter are the khronon and the gauge potential and the former, as we have seen, is constructed in terms of the (co)frame and the curvature. So, δ ξ φ a = ξ Dφ a ,(50a)δ ξ ω a b = ξ R a b .(50b) We adapt the generic variation (47) to this case (50) and again neglect a boundary term. Choosing the vector ξ = @ a to be one of the four "legs" of the Vierbein, the resulting Noether identity assumes the form Dt a = −@ a R bc ∧ φ [b t c] − O bc .(51a) Matter fields without spin (or nonminimal couplings) are described by energy currents without divergence wrt the metric connectionDt a = 0, regardless of the gravity theory [25]. It is useful to recover this from our result (51a). For this purpose, let us decompose the gauge potential ω a b =ω a b + C a b in terms of the Levi-Civita connectionω a b ∧ e b = −de a and the contorsion C a b ∧ e b = T a . Using this decomposition we can rewrite (51a) as Dt a = −(@ a C b c )t b ∧ Dφ c + @ a R bc ∧ O bc .(51b) By taking into account the previous Noether identity (49) we can rewrite this in yet another formD t a = (@ a C bc )DO bc + @ a R bc ∧ O bc .(51c) which makes it manifest that the usual covariant conservation law for material energy currents only has to be generalised for spinning matters in the Lorentz gauge theory. Isotropic and homogeneous spin fluid We are now ready to derive the spin fluid conservation equations in cosmology. In particular, we consider the Ansatz (40) for the energy current, and now take into account also the spin current d O 0I = 1 2 Ω ⋆ e I ,(52a)O IJ = − i 2 Ωǫ IJ K ⋆ e K .(52b) We note that the cosmological symmetry would allow to consider the 2 independent functions Ω andΩ for the 2 sets of components of the spin current (52). However, d The cosmological implications of Weyssenhoff spinning perfect fluids have been investigated in the context of Poincaré gauge theories of gravity, e.g. [16,33,11]. The Weyssenhoff fluid obeys the so called Frenkel condition (see though e.g. [9,22] for alternatives), with the interpretation that the spin reduces to a pure rotation in the rest frame of the fluid. The equivalent form of the cosmological Ansatz (52), not subject to the Frenkel condition O ab @a Dφ 0 = 0, has also been considered in Poincaré gauge theory cosmology, e.g. [30,2]. the field equation would enforceΩ = Ω since they only are compatible with a self-dual spin current O ab = + O ab . This we assumed already in the Ansatz (52) because it is the consequence of the minimal coupling of matter fields ψ to the self-dual connection in the action (1). The generalised Friedmann equations (41) now read, in terms of the gaugeinvariant time variable, 3 (A − B) 2 a = M v −1 0 + ρ M a 3 ,(53a)2 (A ′ − B ′ ) a + (A − B) 2 = −p M a 2 ,(53b)(log A) ′ a + B = Ωa .(53c) Before using these field equations, we will consider the spin current (52) in light of the off-shell Noether identities. First, it is easy to see that there is now antisymmetric energy source for the divergence of the spin, since according (40) we have that t a ∼ ⋆e a and therefore e [a ∧ t b] ∼ e [a ∧ ⋆e b] = −η [ab] ⋆ 1 = 0 .(54) The LHS of the (49) also consistently vanishes as it is easy to check using (52). Thus, the Noether identity resulting from the Lorentz invariance is trivially satisfied in the isotropic and homogeneous setting. We then compute the diffeomorphism invariance Noether identity (51). There identity has 1+3 Lorentz components, but the 3 space components vanish trivially and we can focus on the 1 time component. We begin with the LHS of (51c), recalling the metric connection from (36), and arrive at Dt 0 = 1 2 [ρ ′ M + 3H (ρ M + p M )] ⋆ 1 .(55a) We have already deduced that the 1 st term in the RHS of (51c) vanishes (@ 0 C bc )DO bc = −(@ 0 C b c )t b ∧ Dφ c = 0 ,(55b) and do not actually need the contorsion coefficients C I 0 = (A − aH)dx I and C I J = iBǫ I JK dx K in this computation. The 2 nd term in the RHS of (51c) we obtain by plugging in (34,52) into −@ 0 R bc ∧ O bc = − 3 a (A ′ − B ′ ) Ω ⋆ 1 .(55c) Combining the results (55) we obtain ρ ′ M + 3H (ρ M + p M ) = − 6 a (A ′ − B ′ ) Ω ,(56a) from the time component of the diffeomorphism Noether identity (51). Let us now go on shell. Taking the time derivative of the 1 st Friedmann eq. (53a) and then using the 2 nd Friedmann eq. (53b) we arrive at ρ ′ M + 2H + A − B a ρ M + 3 (A − B) a p M = 0 ,(56b) where the spin function Ω is not explicitly involved. However, we can yet use the 3 rd Friedmann eq. (53c) to solve (A − B)/a = H − Ω, and the above equation becomes ρ ′ M + 3H (ρ M + p M ) = (ρ M + 3p M ) Ω .(56c) Using the 1 st and 2 nd Friedmann equations to rewrite the RHS of (56a) in terms of the matter density and pressure, we confirm that all the forms (56) are onshell equivalent. Thus, this is the consistent generalisation of the matter continuity equation in the presence of cosmological spin fluid. The same result has been derived previously [30]. The generalised Friedmann equations (41) are reduced to 3 (H − Ω) 2 = ρ D + ρ M ,(57a)2 (H ′ − Ω ′ ) + (3H − Ω) (H − Ω) = −p M ,(57b) or equivalently, in terms of an effective energy density ρ eff and effective pressure p eff , 3H 2 = ρ eff + ρ D , where ρ eff = ρ M + 6HΩ − 3Ω 2 ,(58a)2H ′ + 3H 2 = −p eff , where p eff = p M − 2Ω ′ − 4HΩ + Ω 2 ,(58b) To solve these equations, the properties of the fluid source have to be specified. Now this requires the 2 equation of state parameters, to determine the spin Ω as well as the pressure p M . The former has the dimension M and the latter has the dimension M 4 , and since the only dimensional quantities we have at hand are H and ρ M , the apparently natural form of the equations of state would be Ω = αH and p M = w M ρ M with some dimensionless parameters α and w M . The simplest assumption is that both these 2 parameters are constant. Assuming the spin fluid effective energy dominates a 3 v o ρ eff > M , we can then immediately solve the equations, e.g. by integrating (56), and find that the cosmological expansion corresponds to the effective equation of state w eff = ρ eff p eff = w M (1 − α) − α 3 .(59) If we consider the very earliest moments of the universe, the fields are expected to be in the radiation-like w M = 1/3 phase. Then any α > 1 leads to an accelerating universe, α = 2 corresponding to the de Sitter -like phase with w eff = 2. This raises the possibility of a completely new kind inflation [16], not driven by the pressure exerted by a hypothetical slowly-rolling scalar field but by the spin currents of ordinary matter fields which, hypothetically, become significant at extremely high energies. This could explain not only the inflationary beginning of the universe from khronogenesis but also the smooth recovery of the standard hot big bang cosmological evolution. (In the conventional scalar field models, one has to device an exit from inflation and an accompanying so called reheating process which then transforms the scalar field into ordinary matter fields.) The Ω introduces novel possibilities also in the context of dark energy cosmologies. The discrepancy between the Hubble evolution as deduced from the source density and from the actual one, as characterised by (59), suggests that the effect of Ω on the expansion rate could address the H 0 tension e , the statistically significant discrepancy of the expansion rate normalisations as inferred from observations at larger and at smaller redshifts within the standard ΛCDM model [12]. For example, a spinning dark energy could in principle account for the dark matter phenomenology w M = 0 but nevertheless predict w eff ≈ −0.7 in the present universe. Again, the condition for acceleration is α > 1. Discussion To assess the viability of the scenarios we have discussed in these proceedings, it is necessary to study the effects of spinning fluids on cosmological perturbations. At the level of perturbations, further parameters or assumptions will be required to determine the evolution of the source fluid, now for the spin as well as for the pressure. In section 5.2 we exploited a simple parameterisation at the background level to arrive at exact solutions, and this approach can be extended to uncover the phenomenological impact of spin currents in the cosmic microwave background radiation and in the large-scale structure formation. However, from the perspective of the Lorentz gauge theory, the most interesting approach is to develop the Lagrangian formulation of fields with spin currents. Novel phenomenology may result without invoking any new exotic ingredients, since indeed all the elementary fields of the standard model of particle physics have spin (with one exception if the Higgs is supposed to be elementary). In conventional gauge theories of gravity, spinors are well-known to induce a 4-fermion interaction via axial torsion [32], but the implications of the fermionic spin in the Lorentz gauge theory remain to be explored. A consistent coupling of spinor matters ψ to the selfdual connection [5] would result in O ab ∼ψ(γ c γ [a γ b] ψ − +γ [a γ b] γ c ψ + )⋆(Dφ c ), where γ a are elements of the Clifford algebra γ (a γ b) = −η ab . Also, since the standard model gauge fields have spin 1, they can be associated with spin currents. This was found to be possible in Gallagher's pregeometric 1 st order Yang-Mills theory, in the presence of an effective vacuum excitation [15]. Finally, we recall that even the immaterial 3-form M a may have a role in supporting spin currents when the assumption we made in section 3.1.1 is relaxed. There is an intriguing relation between the latter two effects. The CDM candidate 3-form M a is the gravitational analogy of vacuum polarisation in Yang-Mills theories. Relaxing the assumption of section 3.1.1 results simultaneously in the analogy of vacuum magnetisation in the gravitational sector, and the possible spin currents of the gauge fields in the Yang-Mills sector. In such a phase of the theory, the 3-form M a no longer describes ideal dust. Khronogenesis in this phase could therefore be followed by the usual ∼60 e-folds of inflation without diluting the energy in the 3-form M a to completely negligible (which would be the result for an ideal dust energy). Some such mechanism is called for if aiming to unify both the suggested rationale for the initial conditions of the universe and the suggested ingredients for a ΛCDM theory into a more complete paradigm of cosmology based on the Lorentz gauge theory. 3.1. 3 . 3The M a describes conserved energy. Now it is already clear that if interpreted as some effective material energymomentum current, the 3-form M a describes a pure energy current in the sense that it does not contain an effective pressure component. Taking a pull-back to a spatial hypersurface, it is clear that M 0 in (26) can be interpreted as an energy of a 3-space volume element, and we can write M 0 = (2v 0 ) −1 M dx ∧ dy ∧ dz, where M is the mass of a unit coordinate volume v 0 . Note that M/(v o ρ D ) = √ det h can be identified as the determinant of the 3-space metric. 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{'abstract': 'This proceeding is an introduction to cosmological applications of the Lorentz gauge theory. It provides the ingredients for a unique, though yet tentative ΛCDM theory cosmology. The emergence of spacetime is described by the spontaneous symmetry breaking called here the khronogenesis. Space is then associated with the field strength of the antiself-dual gauge potential, and gravity is associated with the self-dual field strength. In the cosmological setting, khronogenesis seems to predict inflation. It is shown that the Lorentz gauge theory allows the consistent description of spin currents which could have important roles in cosmological phenomenology.', 'arxivid': '2306.00963', 'author': ['Tomi Koivisto \nInstitute of Physics\nNational Institute of Chemical Physics and Biophysics\nLaboratory of Theoretical Physics\nUniversity of Tartu\nW. Ostwaldi 1, Rävala pst. 1050411, 10143Tartu, TallinnEstonia, Estonia\n'], 'authoraffiliation': ['Institute of Physics\nNational Institute of Chemical Physics and Biophysics\nLaboratory of Theoretical Physics\nUniversity of Tartu\nW. Ostwaldi 1, Rävala pst. 1050411, 10143Tartu, TallinnEstonia, Estonia'], 'corpusid': 258999431, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 18466, 'n_tokens_neox': 15927, 'n_words': 10399, 'pdfsha': '5f23e46e7bc7ad9cbbd31f78e874aa3a16db7f04', 'pdfurls': ['https://export.arxiv.org/pdf/2306.00963v1.pdf'], 'title': ['0:46 WSPC/INSTRUCTION FILE mufmuf4', '0:46 WSPC/INSTRUCTION FILE mufmuf4'], 'venue': ['International Journal of Geometric Methods in Modern Physics']}

arxiv

Magnetic field measurement from the Davis-Chandrasekhar-Fermi method employed with Atomic Alignment 1-?? (2023 Parth Pavaskar Deutsches Elektronen-Synchrotron DESY Platanenallee 615738ZeuthenGermany Institut für Physik und Astronomie Universität Potsdam Karl-Liebknecht-Str. 24-25, Haus 2814476PotsdamGermany Huirong Yan Deutsches Elektronen-Synchrotron DESY Platanenallee 615738ZeuthenGermany Institut für Physik und Astronomie Universität Potsdam Karl-Liebknecht-Str. 24-25, Haus 2814476PotsdamGermany Jungyeon Cho Department of Astronomy and Space Science Chungnam National University DaejeonKorea Magnetic field measurement from the Davis-Chandrasekhar-Fermi method employed with Atomic Alignment MNRAS 0001-?? (2023Accepted XXX. Received YYY; in original form ZZZPreprint 1 May 2023 Compiled using MNRAS L A T E X style file v3.0plasmas -polarization -methods: numerical -ISM: magnetic fields -(magnetohydrodynamics) MHD The Davis-Chandrasekhar-Fermi (DCF) method is widely employed to estimate the mean magnetic field strength in astrophysical plasmas. In this study, we present a numerical investigation using the DCF method in conjunction with a promising new diagnostic tool for studying magnetic fields: the polarization of spectral lines resulting from the atomic alignment effect. We obtain synthetic spectro-polarimetry observations from 3D magnetohydrodynamic (MHD) turbulence simulations and estimate the mean magnetic field projected onto the plane of the sky using the DCF method with Ground-State-Alignment (GSA) polarization maps and a modification to account for the driving scale of turbulence. We also compare the method to the classical DCF approach using dust polarization observations. Our observations indicate that the modified DCF method correctly estimates the plane-of-sky projected magnetic field strengths for sub-Alfvénic turbulence using a newly proposed correction factor of ∈ 0.35 − 0.75. We find that the field strengths are accurately obtained for all magnetic field inclination and azimuth angles. We also observe a minimum threshold for the mean magnetic field inclination angle with respect to the line of sight, ∼ 16 • , for the method. The magnetic field dispersion traced by the polarization from the spectral lines is comparable in accuracy to dust polarization, while mitigating some of the uncertainties associated with dust observations. The measurements of the DCF observables from the same atomic/ionic line targets ensure the same origin for the magnetic field and velocity fluctuations and offer a possibility of tracing the 3D direction of the magnetic field. INTRODUCTION The interstellar medium (ISM) has been extensively studied in the past due to its importance in a wide range of astrophysical phenomena. One particularly crucial aspect of the ISM is the interstellar magnetic fields, which significantly influence the dynamics of the plasma. In addition, the magnetic fields impact several processes, including but not limited to plasma turbulence (Goldreich & Sridhar 1995;Cho & Vishniac 2000;Cho & Lazarian 2003), star formation (Crutcher 2012;McKee & Ostriker 2007;Fissel et al. 2016), stellar feedback, cosmic-ray transport and acceleration (Schlickeiser 2002;Yan & Lazarian 2002, 2004, accretion disk dynamics, astrophysical jets, and the chemical evolution of the galaxy (see, e.g. Ge et al. 2016). Therefore, accurately measuring the interstellar magnetic fields and their contributions to these processes is crucial in developing consistent theories. However, this measurement is not trivial. On length scales shorter than the coherence scale of interstellar magnetic fields, the total field can be decomposed into two components; the (global) mean field with a preferential direction and the (local) turbulent field. While there are methods that utilize polarization information to measure the magnetic fields, e.g., the Davis-★ E-mail: huirong.yan@desy.de Chandrasekhar-Fermi method (Davis 1951;Chandrasekhar & Fermi 1953, hereinafter the DCF method) and the Polarization-Intensity gradient method (Koch et al. 2012), their probes typically rely on the polarization of emission/absorption arising from magnetically aligned dust. Although widely accepted as the conventional polarization diagnostic, dust alignment measurements may not be completely accurate owing to a number of uncertainties and inconsistencies (see, e.g. Reissl et al. 2014). For instance, an obvious caveat with the conventional DCF method is the utilization of measurements of the line-of-sight (LOS) velocity and polarization from separate targets, i.e., the Doppler shift of spectral lines and the polarization of aligned dust emission or absorption. While modifications have been made to the DCF method to improve its accuracy (Heitsch et al. 2001;Falceta-Gonçalves et al. 2008;Hildebrand et al. 2009;Houde et al. 2009;Cho & Yoo 2016;Federrath 2016;Skalidis & Tassis 2021), this inconsistency is typically not addressed in the studies, making it necessary for other methods to be developed to be used in complement with the current techniques to trace the magnetic fields. Several past studies have shown that in the presence of anisotropic optical pumping, the alignment of angular momenta of atoms and ions in the plasma can lead to the polarization of atomic spectral lines (Yan & Lazarian 2006, 2007, 2012Shangguan & Yan 2013;Zhang & Yan 2018). The UV or optical pumping by an anisotropic radiation field can cause uneven population distribution on the ground/metastable states and align the angular momenta of the atoms. In the presence of an external uniform magnetic field, the atoms are realigned owing to the fast magnetic precession. The resultant spectral lines from the aligned states are thus polarized toward the magnetic field. This effect, named atomic alignment or Ground-State-Alignment (GSA), is a powerful diagnostic in the study of the magnetic fields in the ISM. Both 3D direction and tomography can be retrieved by GSA (Yan & Lazarian 2012;Yan et al. 2019). More recently, polarized absorption lines from thr ground-state have been identified in a Post-AGB binary system 89Her, giving the observational confirmation of the applicability of the GSA effect (Zhang et al. 2020). In this paper, we present a study that utilizes polarization observations and line width measurements from the same spectral lines to measure the magnetic field strength. We employ 3D simulations of magnetohydrodynamic (MHD) turbulence to obtain synthetic polarization observations arising from the Ground-State-Alignment (GSA) effect, which we then combine with the Davis-Chandrasekhar-Fermi (DCF) method to estimate the plane-of-sky (POS) projected magnetic field strength. We compare our new technique to the traditional DCF method, including cases of non-perfectly aligned polarized spectral lines. Our work is organized as follows: we provide a brief explanation of the DCF method and the GSA effect in §2 and §3, respectively. In §4, we describe the simulation setup and the numerical methods used in this study. In §5, we present our observations and results. Finally, we summarize our work in §6. THE MODIFIED DCF METHOD The DCF method (Davis 1951;Chandrasekhar & Fermi 1953) is one of the most commonly used techniques for measuring the magnetic fields in a wide range of astrophysical systems, including molecular clouds, HII regions, and the interstellar medium in general. The method is based on the assumption that the magnetic field in a given region is in a state of equipartition with the turbulent motion of the gas inside it (Chandrasekhar & Fermi 1953). According to the method, the strength of the mean magnetic field projected onto the POS is given by: 0,pos = √︁ 4¯v los(1) where¯is the mean density, los is the velocity dispersion along the LOS and is the dispersion in the angle between the turbulent and the mean magnetic fields projected on the POS. This angle dispersion is typically measured as the dispersion in the observed polarization vectors, while the LOS velocity fluctuations can be measured from the widths of optically thin emission lines. The constant is a correction factor usually taken to be ∼ 0.5 (Heitsch et al. 2001;Ostriker et al. 2001) or lower (Liu et al. 2021). The expression is derived from the condition that in Alfvénic turbulence, there exists an equipartition between the kinetic and magnetic energy densities, i.e. the root-mean-square (rms) fluctuations of the velocity and the magnetic field are related. In addition to Alfvénic (incompressible) turbulence, the DCF method also assumes that the velocity and magnetic field fluctuations are isotropic, and that the turbulent magnetic field energy is much smaller than the global B-field energy. The POS-projected observables are always LOS-integrated, resulting in intrinsic limitations of the observed signals due to LOS averaging effects (Zweibel 1990;Myers & Goodman 1991). This effect depends on the number of individual turbulent eddies along Figure 1. The geometry of our numerical setup. B 0 represents the mean magnetic field, and the LOS is fixed in the Z direction, with the X-Y plane representing the POS. The star symbol represents the source of anisotropic radiation, which is considered to be coming from an infinite distance and is parallel. The angle between the magnetic field and the LOS is denoted by , while the angle between the projection of the magnetic field on the POS and the X axis is denoted by . The angle 0 follows the same logic for the radiation field direction with respect to the LOS. The angle between the magnetic field and radiation field is denoted by . the LOS. The error is typically seen as an exaggerated alignment or ordering of polarization vectors, meaning that the polarization angles do not give accurate approximations of . The error usually leads to an underestimation of or an overestimation of the measured field strength 0,pos in the DCF method. Cho & Yoo (2016) (hereafter CY16) found that this overestimation is roughly equal to a factor of √ N ≈ √︁ L los /L f , where N is the number of independent eddies along the LOS, L los is the length of the system along the LOS, and L f is the driving scale of turbulence. They proposed a modified DCF method to account for the averaging effects, given by: 0,pos = √︁ 4¯c (2) where c is the normalized velocity centroid of the optically thin line, and the modified correction factor ∼ 0.7 − 1.0. The velocity centroid is defined at the th LOS by c,i = ∫ los ( los ) los ∫ ( los ) los .(3) is the optically thin emission line profile for the LOS. Since it has been shown by CY16 that the modification makes the DCF method invariant to the turbulence driving scale, we shall use the modified method (equation 2) for all the following numerical tests involving the DCF technique in this paper. GROUND STATE ALIGNMENT In a typical ISM region where radiation sources, such as massive stars, are embedded in the diffuse plasma, atoms and ions in the plasma are continuously excited through optical pumping. When radiation excitation dominates, the occupation of the atoms/ions is determined by the optical pumping rate. In the case of anisotropic radiation, the net angular momentum in the photons is transferred to the atoms. If the collisional excitation rate is significantly lower than the radiative excitation rate, the angular momentum transfer causes the atoms to align along the direction of the incident radiation at the rate of the radiative pumping. Furthermore, if the Larmor frequency is larger than the radiative pumping rate in the presence of an external magnetic field, the atoms will be realigned due to fast magnetic precession. This condition can realistically be fulfilled in the diffuse ISM. For micro-Gauss scale magnetic fields in the diffuse medium, the atoms can only be aligned in their ground and/or metastable states. The magnetic realignment (parallel or perpendicular to the B field) depends on the angle between the mean magnetic field and the radiation field direction, , and the resulting degree of polarization also varies with the magnetic field inclination, (the angle between the magnetic field and the LOS). As a result, information on the direction of the magnetic field is encoded in the polarization arising from the aligned atoms and ions. In the case of absorption from the atoms aligned in their ground or metastable states, the polarization direction directly traces the magnetic field in the plane of the sky (Yan & Lazarian 2006, 2012. For atoms with fine structures, submillimeter fine-structure transitions are also polarized in the same manner (Yan & Lazarian 2008). 1 . For a background unpolarized pumping source, the GSA effect will only produce linearly polarized lines. The degree of polarization for transitions from 1 to 2 for both absorption and fine structure emission lines is given by Yan & Lazarian (2006 ) = 1.5 2 0 ( 1 , ) sin 2 2 1 2 √ 2 + 2 0 ( 1 , ) (1 − 1.5 sin 2 ) 2 1 2(4) where and are the polar coordinates of the magnetic field vector (see Fig. 1). The alignment parameter 2 0 ≡ 2 0 / 0 0 , is the normalized dipole component of the ground state density matrix, where 2,0 0 are the irreducible density matrices. The parameter 2 1 ≡ {1, 1, 2; 1 , 1 , }/{1, 1, 0; 1 , 1 , } is determined by the atomic structure (see Yan & Lazarian 2012). The sign of 2 0 determines the orientation of the polarization vector with respect to the magnetic field. A positive polarization degree means a parallel orientation, while a negative polarization degree indicates a perpendicular orientation. This sign change or flipping of the polarization vector orientation happens at a specific = 54.7 • , (180 − 54.7) • , also known as the Van Vleck angle (Van Vleck 1925;House 1974). In real observations, this leads to the magnetic field being mapped with a 90 • degeneracy (VV degeneracy from here onward). In principle, this degeneracy can be broken if more than two lines are identified in the observations. NUMERICAL METHOD The numerical method used in the study is divided into two parts: the generation of synthetic polarization maps, and the analysis of the maps using the modified DCF method described in §2. In this work, we calculated 3D MHD turbulence simulations with spatial grids of 512 3 pixels. The set of sub-Alfvénic simulations ranging from = 0.26 to = 0.8 is generated using the high-order finite-difference PENCIL-code 2 . Turbulence is driven solenoidally with an isothermal equation of state i.e = 2 where is the 1 See review by Yan & Lazarian (2012) for the list of absorption, emission as well as fine structure lines and their maximum polarization fractions. 2 http://pencil-code.nordita.org density and is the sound speed. The solenoidal (divergence-free) forcing ensures that the energy fraction of the incompressible Alfvén mode dominates over the compressible magnetosonic (fast and slow) modes in the turbulence. The details of all the simulations used in this work are given in Table 1. The geometry of the numerical setup is shown in Fig. 1. We fix the LOS along the Z direction of the simulation box so that the POS is the X-Y plane. The incoming radiation is considered to be parallel and originating from an external source in the X-Z plane. Calculation of synthetic Stokes maps The line polarization to be simulated from the GSA effect depends on the directions of the radiation field and the local magnetic field. Without loss of generality, we choose the [C II] 157 m (C + ) fine structure emission line for our synthetic observations. Zhang & Yan (2018) have shown that C + can reach high maximum polarization (up to almost 30%). Moreover, C + is commonly observed in the diffuse ISM. The degree of polarization arising from the GSA effect for the C + line for different mean field inclinations ( ) is shown in Fig. 2. As is evident, the sign of the polarization fraction changes at the VV angle (shown by the vertical dotted lines), which means that the polarization vector is aligned parallel to the magnetic field direction in the range = (54.7 • , 125.3 • ), and perpendicular for other inclinations. To get the synthetic polarization, we first obtain the and at each grid point relative to the local magnetic field direction, and calculate the total polarization degree ( , ) using the transition equation (4). Next, we calculate the local Stokes parameters and at the grid points as follows = ( , ) cos 2 (5) = ( , ) sin 2(6) where and are the local density and the local magnetic field azimuth angle, respectively. Typically in the previous studies of the modified DCF method (CY16; Yoon & Cho 2019), the dust grains responsible for the polarized emission are assumed to be perfectly aligned with the external magnetic field. While this is done for the sake of simplicity in calculations, this assumption itself can cause errors in magnetic field measurement. Owing to the 90 • VV degeneracy in the polarization of spectral lines by the GSA effect, such an assumption is not straightforward for our case. Consequently, we want to make sure that this assumption does not have a significant impact on the estimation of the magnetic field strength. For this reason, we generate two kinds of line polarization Stokes maps, one of the realistic scenario where the local Stokes parameters are weighted by the quantitative polarization fractions (first term on the right hand side of equations (5) and (6)) using equation (4), and one with perfectly aligned atoms where the polarization fraction is assumed to be 1. To simulate the LOS averaging in observations, we integrate the Stokes parameters along the LOS (which is the Z direction) to get the observed Stokes parameters = ∫ (7) = ∫(8) Since the background source is unpolarized, lines arising from the GSA effect will be linearly polarized i.e. the Stokes V = 0. After the integration, we obtain 2D Stokes maps with the line averaged Stokes parameters Q and U. For the purpose of comparison with the conventional DCF approach utilizing dust polarization measurements, we also generate synthetic dust polarization maps following the method from Fiege & Pudritz (2000) (see also Zweibel 1996;Heitsch et al. 2001). While previous numerical studies involving the DCF method have tested the applicability of the technique in various systems like the ISM and star forming regions, the effect of the orientation of the mean magnetic field, and especially the LOS inclination, is usually neglected. For studies that use dust polarization as a measure of the local field dispersion, it is common practice to assume a mean field aligned with the POS (Ostriker et al. 2001;Padoan et al. 2001). While it is helpful to consider such a case to simplify calculations and calibrate the methods, it rarely reflects the real astrophysical environments. We examine all the possible geometries of the system in our study. This is achieved by generating a range of synthetic polarization maps while rotating the simulation box each time such that the mean magnetic field vector B 0 scans across the entire solid angle, covering all possible orientations, i.e. ∈ [0, ], ∈ [0, 2 ], with respect to the radiation field. The rotation of the simulation box is performed using the Euler 3D rotation algorithm 3 similar to the one adopted in Figure 3. An example of the rms normalized mean field strength measured using the modified DCF method at different orientations of the mean magnetic field vector. Top: Since the plot is in spherical coordinates, a 2D representation can be confusing and difficult to interpret. Bottom: We wrap the 2D heat map around a sphere such that every point on the sphere corresponds to a magnetic field strength measured when the mean magnetic field vector is pointing in that direction. The red and black arrows on the sphere represent the direction of the radiation field and the LOS, respectively. Lazarian et al. (2018) and Yuen et al. (2018). Moreover, the polar angle of radiation field source 0 is changed from 0 to /2 in six equal steps to check the effect of the LOS inclination of the radiation field on the observed polarization signals. The range of polarization maps allow us to employ the DCF method using atomic-line polarization while studying the accuracy of the technique as a function of three distinct parameters, namely the Alfvén Mach number ( ) , the magnetic field direction ( and ) and the radiation direction ( 0 ). B-field estimation using DCF analysis With the 2D polarization maps, we can now apply the modified DCF method analysis to estimate the mean magnetic field strength using equation (2). Since the term √︁ 4¯is normalised to 1 in the MHD simulations, we require the two observables, i.e., the dispersion of the LOS velocity centroids and the local magnetic field dispersion on the POS , to obtain the mean field strength. The centroids are calculated at each LOS using equation (3) where obs is the POS spatial resolution. The POS local magnetic field dispersion can be estimated in the synthetic observations by measuring the deviation in the polarization vectors. The linear polarization fraction and angle can be recovered on each grid point (LOS) on the Stokes maps using = √︁ 2 + 2 (10) = 1 2 tan −1 2 ,(11) where tan −1 2 is the 2-argument arc-tangent function. To convert the Stokes maps into polarization maps, we transform and into polarization vectors for each LOS. We can then use the minimum variance method to estimate the mean polarization direction 0 . This is done to simulate real polarization observations, where the direction of the projected mean magnetic field is not necessarily known 4 . The method involves computing the variance in the polarization vectors around arbitrary unit vectors in the range (0 • , 180 • ). The direction of the unit vector that has the minimum variance in the polarization vectors is chosen as the mean polarization direction, i.e 0 = arg min ∈ (0, ) 2 ( , )(12) where is the polarization vector for the i ℎ line of sight, is an arbitrary vector in the range (0, ), and 2 ( , ) is the variance of the polarization vectors around the vector . The angle dispersion is then simply computed as = ( , 0 ). Finally, we can substitute these measures in equation (2) to obtain the POS projected mean magnetic field strength. Since we calculate the Stokes maps for all orientations of the mean magnetic field in the − space, we can use the DCF analysis to estimate the B-field strengths as a function of and ( and being the polar and azimuth angle, respectively). This is shown with a heat map in the top panel of Fig. 3 with an example simulation. Since the plot is in spherical coordinates of the mean magnetic field vector, a 2D heat map does no represent the dependency accurately. We wrap the values around a sphere (bottom panel of Fig. 3) to make the plot more intuitive, such that the colorbar value at each point on the sphere shows the B-field strength estimated by the DCF method when the mean magnetic field vector in the system points in that direction. This approach provides a clearer representation of the effect of the B-field direction on the DCF method and helps to better convey the results. Thus, while the sphere itself does not mean anything in the geometry of the setup, it allows us to easily visualize the performance of the DCF method as a function of the mean magnetic field inclination and orientation. The line of sight (LOS) and the direction of the incoming parallel radiation is also depicted by black and red arrows, respectively. Consequently, the angle 4 while circular statistics are typically used to compute the dispersion from polarization maps (Fisher et al. 1993), we notice that for sub-Alfvénic turbulence (with < 1), the difference between the angle dispersion calculated using the circular standard deviation and the minimum variance method is negligible. between the radiation arrow and an arrow pointing to any point on the sphere gives us the mean for the corresponding geometry. RESULTS AND DISCUSSION We divide the results section into three parts. Initially, we study the influence of the Alfvén Mach number on the estimated field strength using DCF with GSA polarization. In the second part, we examine how accurately the technique works in different magnetic field orientations. Lastly, we study the performance of our technique when varying the direction of the radiation source relative to the LOS. Dependence on the Alfvén Mach number In order to study the influence of the Alfvén Mach number, we consider the DCF estimated field strengths for all simulations, covering range in the sub-Alfvénic regime from 0.26 to 0.8. For this purpose, we average the B-field strengths over all possible mean B-field orientations. However, we choose to exclude all the cases where the magnetic field inclination with the LOS is smaller than a threshold angle < 16 • from the averages, since the DCF method has intrinsic limitation at small magnetic field inclination angles (see §5.2 for further discussion). The radiation source is fixed in the POS i.e 0 = 90 • . For comparison, we utilize both the perfectly aligned and realistic GSA polarization maps (see § 4.1) to measure the magnetic field dispersion in the DCF method. Lastly, we normalize the measured field strengths with the POS projected rms magnetic field strength from the simulations, which is the ideal value that we aim to measure. Fig 4 shows the normalized field strengths measured using the modified DCF method utilizing the two types of synthetic GSA polarization observations (ideal and realistic alignment, see § 4.1) for all numerical simulations used in this study. In both cases for perfect and realistic atomic alignment, the measured field strengths seem consistent for low , with a rise in values as increases. Apart from a noticeable difference in the error bounds at high , the values for perfect and non-perfect alignment are relatively similar in the sub-Alfvénic regime. Overall, the B-field predictions from the realistic GSA technique are slightly higher than the ideal counterpart for different . This distribution can be explained through the intrinsic assumptions considered in the DCF method, which requires that the turbulent B-field energy is much smaller than the mean B-field energy. As increases in the turbulence, the turbulent field energy increases relative to the mean B-field energy. As a result, it is less likely that this assumption is satisfied. The highly turbulent plasma can also lead to abrupt changes in the mean field orientation along the LOS, which can contribute to high LOS averaging errors. From a general perspective, the method seems to typically overestimate the magnetic field strength by a factor of 1.3 to 2.5 in the sub-Alfvénic regime with similar error-bar spreads. Following their modification to the DCF method, CY16 proposed a correction factor = 0.7 − 1 based on the variation in their measured B-field strengths. However, the general trend in Fig 4 indicates that the correction factor in the modified DCF equation (2) should be a function of when utilised with polarization from atomic alignment, rather than a constant correction as is typically considered for DCF using dust polarization. While determining the exact dependency of the on for our technique requires further investigation into the method, it is apparent that the factor is lower than CY16. We propose a new correction factor ∈ 0.35−0.75 in case of sub-Alfvénic turbulence. In principle, our proposed correction factor is similar to the correction factor used in the conventional DCF method ( in equation (1), typically taken ∼ 0.5). Although it should be noted that CY16 only considered turbulence with ∼ 0.6 in their simulations, which could have influenced the resulting correction factor in their work. Influence of the magnetic field orientation on the DCF method To investigate how the estimation from the DCF technique is affected by the mean magnetic field direction, we examine the B-field strengths for all magnetic field orientations in 3D (in the − space). Fig. 5 shows the B-field strength estimations from the DCF analysis with polarization from realistic GSA using a correction factor = 0.5 for the simulations with = 0.66, 0.4, 0.26, shown from top to bottom. The colorbar in Fig. 5 shows the ratio between the DCF-measured B-field and the actual B-field after the POS projection. As described in section 4.2 (Fig. 3), the values at each point on the sphere show the magnetic field strength when the mean magnetic field vector points in that direction in the geometry given by Fig. 1, i.e. the measured field strength as a function of and . 0 is fixed at 90 • in all tests, and the LOS and the direction of the radiation field are shown using the black and red arrows, respectively. As can be seen from the general distribution on the sphere, the observed mean field strengths are consistent with the rms values at most and with two noticeable exceptions: underestimations near the LOS and in ring-like regions around the radiation field direction and near the poles of the spheres (low ). The underestimation in the ring-like region is due the the VV effect, which is an intrinsic property of the atomic alignment process (see § 3, andLazarian (2006, 2008) for a detailed description). When the angle between the mean magnetic field and the radiation direction is close to the VV angle, the fluctuations of the local magnetic field cause the sign of the GSA polarization fraction to change from point to point. This results in the local polarization vectors flipping abruptly between parallel and perpendicular alignment relative to the neighboring grid-points. A large number of such flips along the LOS can cause the magnetic field to be traced with significant inaccuracy after the LOS averaging. As a result, the dispersion in the polarization vectors is large, causing the B-field prediction from DCF to be underestimated. Since majority of the error from the VV effect arises due to the LOS integration of the polarization signal, it can be difficult to recognize the contribution of the VV effect in real observations if the geometry of the system is unknown. However, it is worth mentioning that the condition for the VV degeneracy, that the angle between the radiation field and the magnetic field ≈ 54.7 • , is a rare and special case of the system orientation that is unlikely to occur in majority of realistic astrophysical environments. Consequently, the phase volume of geometries where the observed polarization is affected by the VV degeneracy is limited. In principle, it is also possible to account for the VV effect by analysing the polarization signals in position-positionvelocity (PPV) space and employing for e.g. a nearest neighbor filter to remove the LOS with large fluctuations in polarization vectors that do not correspond to large fluctuations in velocity. Such an approach, however, is not trivial and will be studied in detail in the future. At small , i.e. when the LOS and the mean magnetic field are close to being aligned, the projection of the mean field on the the POS is close to zero. Consequently, the polarization signals trace the turbulent field instead of the uniform field in the projection frame. Since the DCF method relies on the assumption that the dispersion in the polarization vectors is equal to the dispersion in the uniform field direction, this results in an intrinsic limitation of the DCF method at small inclination angles. In a detailed study and discussion regarding the inclination angle dependence in the DCF method, Falceta-Gonçalves et al. (2008) observed that the DCF method heavily underestimates the field strength as approaches 0 • due to projection effects. More recently, Lazarian et al. (2022) proposed a modification to the DCF method to account for the inclination angle projection effect, given by 0,pos = √︁ 4¯v los 1 sin (13) where = in this study. Essentially, the modification accounts for the difference between the strength of the projected magnetic field in the 2D plane and the total magnetic field strength through the correction factor sin . We took the correction for the projection effect into account by normalising the values of magnetic field to the rms 0,pos instead of 0 . It is worth mentioning that with the developments of new techniques which are capable of estimating the inclination angle of the mean magnetic field (see, e.g., Yuen et al. 2023;Malik et al. 2023), the mean field strength can be measured using equation (13). Lazarian et al. (2022) also showed that the method is only accurate at inclination angles larger than a minimum threshold, which they measured to be a function of , given by (4 tan −1 ( / √ 3)). Ensuring if the B-field inclination of the system is larger than this threshold condition can be particularly challenging with real observations, as it is notoriously difficult to estimate the of astrophysical plasma, even if the inclination angle is known. While we do see an dependence in our results in the form of increasing field strengths as we go to higher , we find that it is not as strong as their measurement. Instead, we propose a minimum threshold independent of for all our simulations. Accordingly, we only consider orientations with > 16 • to make sure the projection effects do not influence the DCF estimate averages. Significance of the radiation field direction in GSA The results presented in the previous section were limited to the special case of 0 = 90 • , i.e. the external radiation source fixed in the POS. For the sake of completeness, we change the location of the radiation source in the X-Z plane and perform the DCF analysis on the generated GSA polarization synthetic maps to check the effect of the radiation field direction on the method. Fig. 6, which uses the simulation with = .26 and a correction factor = 0.5, shows the distribution of field strengths in − space for different 0 . It is evident that the change in 0 changes the location of the VV regions, which is expected. However, the changing radiation source does not influence the DCF estimates at geometries in which the system is not affected by VV degeneracy (i.e. ≠ VV angle). Although the method cannot resolve the magnetic fields in the VV region in its current state and requires some modifications, the VV orientation in itself is a special case geometry. Therefore, we expect the DCF method to work accurately with polarization from spectral lines regardless of the location of the radiation source, as long as the geometry of the system in real observations does not correspond to this special VV case. In addition, the correction factor of ∈ 0.35 − 0.75 discussed in section 5.1 applies to our method irrespective of the radiation source geometry. Comparison to dust polarization method In most studies that employ the DCF method or its variations, both historically and presently, dust polarization observations are used to calculate the polarization angle dispersion. To examine how the DCF method utilizing polarization from atomic alignment compares to the classical dust approach, we simulated the synthetic polarization for both the mechanisms and use the DCF analysis to estimate the magnetic fields. The comparison is shown through the measured B-field strengths averaged over all magnetic field orientations versus in Fig. 7. The averaging and normalization is performed similarly to Fig 4 (see § 5.1). For the purpose of a balanced comparison, we consider only perfectly aligned atoms and dust grains. The comparison is also shown in the − space for three different values of in Fig. 8, where the GSA method uses the correction factor = 0.5 as calculated in this work (see § 5.1), while the dust alignment method uses = 0.8 as is suggested for dust polarization by CY16. From Fig. 7, it is clear that in the sub-Alfvénic regime, the DCF technique using atomic GSA polarization measures the B-field strength with similar precision compared to the dust polarization method. The spread in errors also seem to be consistent for the two methods. From Fig. 8, it can be seen that the only difference in the two methods is the ring-like VV region for the GSA measurements, which is absent in the dust polarization. Although the VV regions can explain the lower averages for GSA in Fig. 7, the estimations from the two approaches outside the VV regions appear to be highly consistent with each other. Despite the fact that dust polarization does not suffer from the VV degeneracy, it is important to note that the observations of dust maps are usually accompanied by their own uncertainties. Previous observations have shown that the dust grains are asymmetrical and align with the magnetic field lines along their shortest axis due to radiative torques (Davis & Greenstein 1949, 1951Cho & Lazarian 2005;Lazarian 2007;Andersson et al. 2015), which means that realistic dust alignment traces the magnetic filed direction with a 90 • flip as well. Since the physical properties such as size and shape of the individual dust grains vary in the diffuse interstellar medium, the efficiency of the radiative alignment is different (Draine & Weingartner 1996, 1997Lazarian & Hoang 2007). As a result, smaller grains which might not be aligned with the magnetic field, also contribute to the observed polarization signals. Especially in low density plasmas, the continuum dust polarization measurements can suffer from decreased signal-to-noise ratios due to low polarization fractions. In addition to physical properties, variation in the chemical composition of the dust also contributes to the unreliable measurements in the observations. Another challenge with dust polarization that can lead to inaccuracies, particularly in the DCF analysis, arises from the fact that optical/IR continuum observations are used for the polarization dispersion measurements, while velocity dispersion is obtained using optically thin spectral lines. While it is generally assumed that information from these sources originate in the same region in the magnetized medium, it may not always be true. It is straightforward that when the dispersions in velocity and polarization angles are calculated from two different layers along the LOS in the plasma, the DCF method does not give a correct estimate for the B-field strength at either of those layers. This particular uncertainty, as well as those arising from the diversity of the sizes, shapes and compositions of dust particles can be averted by using GSA observations, in which the same polarized atomic line can be used to gain information about the velocity and magnetic field fluctuations (polarization angle dispersion). In addition, GSA could facilitate a new avenue in magnetic field diagnostics, namely the 3D tomography of the magnetic field in the ISM. In principle, this can be achieved by performing the DCF analysis using velocity slices, i.e, thin wavelength intervals or segments of the line profiles to get information about the magnetic field strength and orientation in the PPV space. However, this will require further numerical and observational studies, which we will address in the future. Testing the CY16 method for atomic line polarization As a motivation for the modification to the DCF method, CY16 showed that the DCF method is affected by the driving scale of the turbulence, and that the conventional DCF method overestimates the POS field strength by a factor proportional to the ratio of the LOS scale and the driving scale of turbulence (∼ √︁ los / ). We decided to check the efficiency of the modified DCF method while using atomic line polarization instead of dust polarization which was used in CY16. For this purpose, we utilized two separate simulations with similar and , but different . The details are given in Table 2. Since the simulation box length is normalized to unity, the driving scale of turbulence (1/ ) for the simulation d_040 is larger by a factor of 5 than the simulation k_024. According to CY16, a discrepancy of ∼ √︁ los / in the conventional DCF method would lead to the POS mean field strength measured from k_024 to be overestimated than that of d_040 by a factor of √ 5 ≈ 2.3. We use the modified DCF method with line polarization from GSA to measure the POS field strengths for the two simulations for different magnetic field orientations, and plotted the averages over ( , ) against the 0 in Fig 9. The B-field strength estimations at low 0 show no significant difference, while even as 0 approaches 90 • , the largest difference seen in the two simulations is by a factor of ∼ 1.4. This is direct evidence that the modified DCF method from CY16 corrects for the averaging effects from multiple eddies along the LOS, even when used with polarization from atomic alignment, and regardless of the radiation source orientation. SUMMARY In this paper, we have employed the modified Davis-Chandrasekhar-Fermi method proposed by Cho & Yoo (2016) along with synthetic polarization observations arising due to the Ground State Alignment effect (Yan & Lazarian 2006 in simulated magnetized plasma. Using 3D MHD turbulence simulations with varying plasma properties and geometries, we demonstrate the compatibility of the polarization observations of the GSA effect with conventional techniques like the DCF method and its variations. The method differs from the traditional DCF method by measuring the dispersion of the mean magnetic field direction through polarized spectral lines instead of continuum dust polarization measurements. The paper adopted the [C II] fine structure emission line without loss of generality. The method can be readily applied to archived and new spectro-polarimetry data covering wide wavelength ranges from UV to sub-millimeter (Yan & Lazarian 2012). The results from the numerical investigation of the method can be summarized as follows: • The modified DCF method using polarization maps from atomic ground state alignment gives consistently accurate estimates of the POS projected mean magnetic field strengths in the ISM. We propose a correction factor of ∈ 0.35 − 0.75 for sub-Alfveńic turbulence. • The strength of the projected magnetic field in the plane of sky is obtained for all magnetic field inclination angles. We identify a minimum threshold angle for the magnetic field inclination with the line-of-sight of = 16 • below which the DCF method does not trace the magnetic fields accurately. • The DCF method utilizing polarization measurements from atomic alignment is equally accurate as the conventional method utilizing dust polarization observations, while avoiding the uncertainties accompanied by dust alignment such as variations in grain size, shape and chemical composition. • The spectro-polarimetry combined with spectrometry from the same atomic/ionic lines not only improves the accuracy of the DCF method by ensuring the same origin for the magnetic field and velocity fluctuations, but can also potentially trace the 3D direction and strength of the local magnetic field. • The modified DCF method from Cho & Yoo (2016) successfully accounts for the correction to the conventional DCF method due to the driving scale of turbulence irrespective of the polarization source. It is also invariant to the geometry of the local radiation source in case of atomic alignment by GSA. In this study, we present a novel diagnostic for tracing the magnetic field fluctuations through atomic alignment, which can be used in conjunction with the DCF method to estimate the POS-projected mean magnetic field strength in the ISM. We demonstrate that our method improves the accuracy of the conventional DCF approach while taking into account the differences in atomic and dust polarization approaches. Figure 8. A visual representation of the difference in field strengths measured at different and for DCF using GSA (left) and dust (right) polarization. A correction factor of = 0.5 and = 0.8 is used for GSA and , respectively following CY16 and this work. The projection on the X-Z and isometric planes is shown. The normalization is similar to Fig. 5.1. Red arrows depict the radiation field direction for GSA. Black arrows show the LOS. Figure 9. The figure shows the normalised mean magnetic field strengths from DCF averaged over all magnetic field orientations versus the angle between the radiation source and the LOS ( 0 ). Simulation d_040 with driving wavenumber = 2 is shown in blue, while k_024 with = 10 is shown in orange. Figure 2 . 2This figure shows the computed degree of polarization versus for the fine structure emission line [C II] 157 m. The colors represent different . The positive and negative polarization fractions indicate parallel and perpendicular alignment to the magnetic field, respectively. The Van Vleck angles (54.7 • , (180-54.7) • ), at which the transition takes place are marked by vertical dotted lines. Figure 4 . 4The POS mean magnetic field strength for different Alfvén Mach numbers (data cubes) computed with (orange) and without (blue) assuming polarization from perfectly aligned atoms. The field strength is averaged over all values in the − space, excluding the low inclination region at < 16 • . The values are normalised with the POS projected rms magnetic field in the simulation box, such that a value of 1 represents the ideal measurement in the above plot. Radiation source direction is fixed at 0 = 90 • i.e. in the POS. Figure 5 . 5The normalized POS mean magnetic field strength observed using the modified DCF method from polarization observations arising from GSA using a correction factor = 0.5. The four columns are used to show the 3D distribution in − space (which is done by rotating the numerical cube) in the X-Y, Y-Z, X-Z and isometric planes. The rows show three different Alfvén Mach numbers (from the top, = 0.66, 0.40 and 0.26). The black and red arrows represent the LOS and the radiation field direction, respectively. Figure 6 . 6An example showing normalised field strengths measured in the − space for different 0 using the simulation d_070 ( = 0.26) and = 0.5. For each 0 , the X-Z (first and third rows) and the isometric projections (second and fourth rows) are shown. The color bar normalization is similar toFig. 5.1. Red and black arrows depict the radiation field direction and the LOS, respectively. Figure 7 . 7Comparison between field strengths estimated by the DCF method utilizing polarization from perfectly aligned dust (blue) and GSA (orange). 0 is fixed at 90 • for the GSA measurements. Table 1 . 1Descriptions of MHD simulation cubes. The Alfvén velocity is in code units i.e. in units of 1/ √︁ 4¯. The Alfvén and sonic Mach numbers are given by / and / , respectively, where is the rms velocity and is the sound speed.Name Grid Alfvén Alfvén Mach Sonic Mach size velocity ( ) number ( ) number ( ) d_024 512 3 0.24 0.80 1.68 d_030 512 3 0.30 0.66 1.98 d_040 512 3 0.40 0.50 2.00 d_050 512 3 0.50 0.40 2.00 d_070 512 3 0.70 0.26 1.82 d_080 512 3 0.80 0.20 1.65 Table 2 . 2Descriptions of MHD simulation cubes with different driving scale . The driving wavenumber is in units of / , where is the driving scale of turbulence and is the size of the simulation box along one axis.Name d_040 k_024 Resolution 512 3 512 3 Alfvén velocity ( ) 0.40 0.12 Alfvén Mach number ( ) 0.50 0.50 Sonic Mach number ( ) 2.00 2.50 Driving wavenumber ( ) 2 10 Pavaskar et al. MNRAS 000, 1-?? (2023) https://www.github.com/doraemonho/LazRotationDev ACKNOWLEDGEMENTSWe would like to acknowledge the referee for the constructive suggestions. We acknowledge DESY (Zeuthen, Germany), a member of the Helmholtz Association HGF, and the University of Potsdam for the support and the resources to make this work possible. We are grateful to Heshou Zhang and Bolu Feng for their contributions. We would also like to thank Ka Ho Yuen, Sunil Malik and Thiem Hoang for the helpful discussions.DATA AVAILABILITYThe data involved in this work will be shared upon reasonable request to the corresponding author. . 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{'abstract': 'The Davis-Chandrasekhar-Fermi (DCF) method is widely employed to estimate the mean magnetic field strength in astrophysical plasmas. In this study, we present a numerical investigation using the DCF method in conjunction with a promising new diagnostic tool for studying magnetic fields: the polarization of spectral lines resulting from the atomic alignment effect. We obtain synthetic spectro-polarimetry observations from 3D magnetohydrodynamic (MHD) turbulence simulations and estimate the mean magnetic field projected onto the plane of the sky using the DCF method with Ground-State-Alignment (GSA) polarization maps and a modification to account for the driving scale of turbulence. We also compare the method to the classical DCF approach using dust polarization observations. Our observations indicate that the modified DCF method correctly estimates the plane-of-sky projected magnetic field strengths for sub-Alfvénic turbulence using a newly proposed correction factor of ∈ 0.35 − 0.75. We find that the field strengths are accurately obtained for all magnetic field inclination and azimuth angles. We also observe a minimum threshold for the mean magnetic field inclination angle with respect to the line of sight, ∼ 16 • , for the method. The magnetic field dispersion traced by the polarization from the spectral lines is comparable in accuracy to dust polarization, while mitigating some of the uncertainties associated with dust observations. The measurements of the DCF observables from the same atomic/ionic line targets ensure the same origin for the magnetic field and velocity fluctuations and offer a possibility of tracing the 3D direction of the magnetic field.', 'arxivid': '2304.10665', 'author': ['Parth Pavaskar \nDeutsches Elektronen-Synchrotron DESY\nPlatanenallee 615738ZeuthenGermany\n\nInstitut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-25, Haus 2814476PotsdamGermany\n', 'Huirong Yan \nDeutsches Elektronen-Synchrotron DESY\nPlatanenallee 615738ZeuthenGermany\n\nInstitut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-25, Haus 2814476PotsdamGermany\n', 'Jungyeon Cho \nDepartment of Astronomy and Space Science\nChungnam National University\nDaejeonKorea\n'], 'authoraffiliation': ['Deutsches Elektronen-Synchrotron DESY\nPlatanenallee 615738ZeuthenGermany', 'Institut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-25, Haus 2814476PotsdamGermany', 'Deutsches Elektronen-Synchrotron DESY\nPlatanenallee 615738ZeuthenGermany', 'Institut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24-25, Haus 2814476PotsdamGermany', 'Department of Astronomy and Space Science\nChungnam National University\nDaejeonKorea'], 'corpusid': 258291640, 'doi': '10.1093/mnras/stad1237', 'github_urls': ['https://www.github.com/doraemonho/LazRotationDev'], 'n_tokens_mistral': 16087, 'n_tokens_neox': 13258, 'n_words': 8635, 'pdfsha': 'faca678affe64aabb9f4273a50ef3880ae1d7028', 'pdfurls': ['https://export.arxiv.org/pdf/2304.10665v2.pdf'], 'title': ['Magnetic field measurement from the Davis-Chandrasekhar-Fermi method employed with Atomic Alignment', 'Magnetic field measurement from the Davis-Chandrasekhar-Fermi method employed with Atomic Alignment'], 'venue': ['MNRAS']}

arxiv

Artificial Wrestling: A Dynamical Formulation of Autonomous Agents Fighting in a Coupled Inverted Pendula Framework Jun 2014. MOVIC2014 Katsutoshi Yoshida yoshidak@cc.utsunomiya-u.ac.jp Shigeki Matsumoto Yoichi Matsue Department of Mechanical and Intelligent Engineering Utsunomiya University 7-1-2 Yoto321-8585Utsunomiya, TochigiJapan Yuhara Mfg Co 1256, 329-1311Ltd, Ujiie, Sakura, TochigiJapan Artificial Wrestling: A Dynamical Formulation of Autonomous Agents Fighting in a Coupled Inverted Pendula Framework No.X, XXXXXJun 2014. MOVIC201410.1299/XXX.X.1]Multiagent SystemCompetitive ProblemIntelligent ControlNonlinear DynamicsReachable Set We develop autonomous agents fighting with each other, inspired by human wrestling. For this purpose, we propose a coupled inverted pendula (CIP) framework in which: 1) tips of two inverted pendulums are linked by a connection rod, 2) each pendulum is primarily stabilized by a PD-controller, 3) and is additionally equipped with an intelligent controller. Based on this framework, we dynamically formulate an intelligent controller designed to store dynamical correspondence from initial states to final states of the CIP model, to receive state vectors of the model, and to output impulsive control forces to produce desired final states of the model. Developing a quantized and reduced order design of this controller, we have a practical control procedure based on an off-line learning method. We then conduct numerical simulations to investigate individual performance of the intelligent controller, showing that the performance can be improved by adding a delay element into the intelligent controller. The result shows that the performance depends not only on quantization resolutions of learning data but also on delay time of the delay element. Finally, we install the intelligent controllers into both pendulums in the proposed framework to demonstrate autonomous competitive behavior between inverted pendulums. Introduction Wrestling seems to be composed artificially of two mechanical agents maintaining their balance, coupled via mechanical interactions such as contact, connection, collision, etc., and equipped with intelligent controllers competitive with each other. In this paper, we develop a simple model to create such competitive agents. For this purpose, we propose a coupled inverted pendula (CIP) framework in which: 1) tips of two inverted pendulums are linked by a connection rod, 2) each pendulum is primarily stabilized by a PD-controller, 3) and is additionally equipped with an intelligent controller that individually generates a series of impulsive internal forces to achieve its own desired final states based on knowledge of correspondence from initial states to the final states. In general, multiple agents can exhibit competitive and cooperative dynamics when sharing common resources and environments. Historically, early mathematical insights into such mutual interactions seem to have appeared in the filed of mathematical ecology (Hofbauer and Sigmund, 1998) in which population dynamics of different species sharing a common environment is described by a system of coupled nonlinear differential equations such as the Lotka-Volterra equation. Contrary to the ecosystem in which the medium of interaction is given by environments, in our CIP framework, the medium of interaction is given by a mechanical structure. In our previous study (Yoshida and Ohta, 2008), we have already demonstrated that the CIP model, even without intelligent controllers, can produce competitive dynamics comparable to that in the ecosystem such as coexistence and dominance by assigning competitive meanings to the stable equilibriums of the CIP model. Although quite similar mechanical models have been considered in the field of multiple manipulator systems (Hsu, 1993;Nakamura et al., 1987;Panwar et al., 2012), they have only focused on cooperative dynamics because of their aim at developing coordinated motions in those systems. In our study mentioned above (Yoshida and Ohta, 2008), each pendulum is PD-controlled to be bistable at the top and bottom dead points such that the coupled system produces quadra-stability. In absence of additional inputs, this system converges into one of the four stable positions (equilibriums) depending upon the initial conditions. We then applied a single impulsive force to one of the pendulums to generate switching behavior from a given stable position to a desired stable position and considered it as a prototype of fighting-like behavior. In this prototype, however, the behavior is exactly determined by the initial position and strength of the impulse because of uniqueness of solution of differential equation. Therefore, it is quite hard to say that this first prototype is comparable to the wrestling players who seek how to generate internal forces to achieve desired final positions in autonomous ways. In order to build such autonomous agents fighting with each other, a certain intelligent motion controller is required. On such controllers, extensive research has been conducted in the field of multi-robot systems (Maravall et al., 2013;Pagello et al., 1999). The major issue in this field appears to be how to obtain cooperative group dynamics of robots both in algorithm-based approaches (Stone and Veloso, 2000) and in differential-equation-based approaches (Hsu, 1993;Nakamura et al., 1987;Panwar et al., 2012). On the other hand, competitive group behavior seems to have been studied mainly based on algorithm-based approaches. For example, Nelson et al. (2004) studied an evolutionary controller to investigate a form of reinforcement learning that makes use of competitive tournaments of games (robot capture the flag) played by individuals in a population of neural controllers. Moreover, Wu et al. (2013) developed rule or knowledge-based techniques to analyze strategy in robot soccer game. In this way, in contrast to our approach (Yoshida and Ohta, 2008), differential-equation-based techniques are not always essential in these studies because they seek step-by-step algorithms predicting free space determined by other robots. In this paper, we first introduce the CIP framework to describe the competitive behavior in differential-equationbased manners. Next, we develop a competitive intelligent controller that receives state vectors of the CIP model and outputs impulsive control forces to produce desired final states. After evaluating individual performance of the intelligent controller and investigating how to improve the performance, we will demonstrate autonomous competitive behavior between two inverted pendulums equipped with the proposed intelligent controllers. Coupled inverted pendula framework Coupled inverted pendula In order to create wrestler-like mechanical agents maintaining their balance while being coupled mechanically with each other, we consider a CIP model (Yoshida and Ohta, 2008) as shown in Fig. 1. Each inverted pendulum consists of a cart moving along the horizontal floor (Y = 0) and a simple pendulum rotating about a point on the cart. For simplicity, a common physical specification is given to both of the pendulums where m θ is a mass and r is a length of the pendulum, and m x is a mass of the cart. Linking the tips of the pendulums with a viscoelastic connection rod of length w, we obtain the CIP model in Fig. 1 where T i is an input torque on θ i , f i is a reaction force acting on the tip of ith pendulum, and k w and c w are a spring coefficient and a viscous friction coefficient of the rod respectively. We assume that a mass of the rod is negligible. As configuration of this linkage is uniquely determined by the four variables: horizontal displacements of the carts x 1 , x 2 and slant angles of the pendulums θ 1 , θ 2 , the dynamics of this linkage is described by the eight-dimensional state vector: x := (x T 1 , x T 2 ) T , x i := x i ,ẋ i , θ i ,θ i T (i = 1, 2),(1) where A T denotes the transpose of a matrix A. According to Lagrangian mechanics and assuming viscous friction forces c xẋi and c θθi on x i and θ i respectively, we obtain equations of motion (EOM) of the CIP model in Fig. 1 as follows:          (m x + m θ )ẍ i + (m θ r cos θ i )θ i − m θ rθ 2 i sin θ i = −c xẋi + (1, 0) f i , (m θ r cos θ i )ẍ i + (m θ r 2 )θ i − m θ gr sin θ i = −c θθi + r(cos θ i , − sin θ i ) f i + T i (i = 1, 2),(2) whereẊ := dX/dt. Reaction force from the connection rod We calculate the reaction force, say p, from the connection rod as shown in Fig. 2. The displacement vector w from the left-hand tip to the right-hand tip of pendulums is given by w =       w X w Y       := X 2 − X 1 , X i =       X i Y i       =       x i + r sin θ i r cos θ i       (i = 1, 2),(3) and the length of rod is expressed as w = w = {(x 2 − x 1 ) + r(sin θ 2 − sin θ 1 )} 2 + {r(cos θ 2 − cos θ 1 )} 2(4) with the time derivativeẇ = (ẇ X w X +ẇ Y w Y )/w. Then, we model viscoelasticity of the connection rod as p = p := −k w (w − w 0 ) − c wẇ ,(5) where w 0 is a natural length of the connection rod. As the force vector p is parallel to the displacement vector w, we have the reaction force: p = (p/w)w.(6) Substituting p into the EOM (2) through, f 1 = −p, f 2 = p or f i = (−1) i p ,(7) we obtain an analytic expression of the CIP model shown in Fig. 1 via the viscoelastic connection. Modeling floor In the previous study (Yoshida and Ohta, 2008), the pendulum can fall down freely to the bottom dead point, in other word, there was no floor in the previous CIP model. In that case, both of forward and backward falling motions converge to the same equilibrium of the model so that orbital information is required in order to detect the direction of falling. Not only to simplify the detection process but also to develop more realistic simulator of wrestling, we introduce the floor model to the CIP model in the following manner. Based on penalty methods (Moore and Wilhelms, 1988), we first model a normal force R i from the floor (Y = 0) acting on the tip of ith pendulum as R i = U(−Y i ){−k f Y i − c fẎi },(8) where Y i is a height of the ith tip from the floor in (3), U( · ) is a unit step function, and k f , c f are viscoelastic parameters representing property of the reaction. In practice, in order to avoid numerical errors, we approximate the step function with a sigmoid function differentiable, defined by U σ (s) := 1 + exp(−σs) −1 ,(9) where lim σ→∞ U σ (s) = U(s) holds. Furthermore, a Coulomb friction force F i acting on the ith tip from the floor can be expressed as F i = −µR i sgn(Ẋ i ),(10) where µ is a friction coefficient,Ẋ i is a relative horizontal velocity of the ith tip from the floor, and sgn( · ) is a unit signum function whose smooth approximation can be given by sgn(s) ≈ sgn σ (s) := 2U σ (s) − 1. Therefore, the CIP model via the viscoelastic connection in (7) on the floor can be obtained by substituting f i = (−1) i p + (F i , R i ) T (i = 1, 2)(11) into the EOM in (2). Standing control with falling The CIP framework in absence of intelligent controllers is completed by giving dynamical meanings of winning and losing to states of the CIP model. To this end, we begin with developing a feedback controller by which each inverted pendulum on the floor forms three stable equilibriums: θ i = 0 for standing or winning and θ i = ±π/2 for falling or losing. This can be done by introducing a feedback controller in the following form: T i = u pd i := trap α (θ i ; ∆θ){−K p θ i − K dθi } (i = 1, 2),(12) where trap α (θ; ∆θ) := U α (θ + ∆θ) · U α (−θ + ∆θ)(13) is a smooth trapezoidal function of unit height and centered at θ = 0 made of a product of the sigmoid function in (9), ∆θ > 0 is a half width of the trapezoidal shape, and the shape is getting steeper as α increases. It follows from the deadband characteristics in (12) that u pd i simply acts as a PD controller within the limit |θ i | < ∆θ while it rapidly cuts off the output outside of the interval. Therefore, appropriate setting of the gains K p , K d make it possible for the ith pendulum to be stabilized about the standing position θ i = 0 while to be falling down to the floor when |θ i | exceeds the given limit ∆θ. It is worthy to note that in the field of gerontology and related fields, human standing (or falling) limits comparable to the threshold ∆θ have been measured by the functional reach test (Duncan et al., 1990) in which the difference between arm length and maximal forward reach of human subjects is measured to evaluate risk of falls of them. CIP framework In what follows, we compare the inverted pendulums in this model to wrestler-like agents maintaining their standing balance. Since each agent with the standing control in (12) has the three stable equilibriums, a pair of the agents being coupled with each other under the suitable conditions can produce 3 × 3 = 9 stable equilibriums: ω i := lim t→∞ x(t) = x 1 , 0,θ 1 , 0,x 2 , 0,θ 2 , 0 T (i = 1, · · · , 9),(14) as shown in Fig. 3 when equating horizontal translations of final positionx 1 ,x 2 without loss of generality. Namely, the components x 1 (t), x 2 (t) of the solution x(t) of (2) are not stable asymptotically but neutrally because no restoring forces on x 1 (t), x 2 (t) are assumed by definition. It also should be noted that due to the penalty method in (8), the gravity force makes the equilibriumθ i on the floor slightly exceed the floor, i.e., |θ i | − π/2 > 0, but we formally denote θ i =θ i = ±π/2 because this slight exceedance only affects almost converged states and does not change the correspondence from the initial to final states. We then attach competitive meanings to the nine equilibriums as listed in Fig. 3 in which the agent that remains standing is regarded as a winner. Eventually, we have the CIP framework composed of the set of (A) and (B): (A) The CIP model: the system of equations defined in (2), (11) and (12). (B) The win-loss matrix: the competitive interpretation of the nine equilibriums defined in Fig. 3. Intelligent controller Based on the CIP framework in Section 2.5, we develop an intelligent controller (IC) to produce desired final positions in Fig. 3 from given initial states of CIP by generating certain impulsive forces. Problem setting and requirements According to the definition of the state vector (1), the CIP model in (2) can be expressed as an eight-dimensional dynamical system: x = (x T 1 , x T 2 ) T inx = f (x, T), x(0) = x 0 , T := (T 1 , T 2 ) T(15) that can be divided into a pair of four-dimensional subsystems:         ẋ 1 = f 1 (x 1 , x 2 , T 1 ), x 2 = f 2 (x 2 , x 1 , T 2 ).(16) We introduce the IC by adding an intelligent control input u ic i to the torque T i of CIP model as T := u pd + u ic = u pd 1 , u pd 2 T + u ic 1 , u ic 2 T ,(17) where u pd i is the standing control input already given in (12). In the present study, the input-output relationship around u ic i is designed as shown in Fig. 4, in which u ic i receives all the state vectors x 1 , x 2 and outputs a series of impulsive forces given by u ic i (t) := N j=1 P i I ∆τ (t − t j i ),(18) where I ∆τ (t) =          (∆τ) −1 (0 ≤ t < ∆τ), 0 (otherwise)(19) is a rectangular function of unit area of width ∆τ ≪ 1, P i is an angular impulse of the input torque u ic i (t), and {t 1 i , · · · , t N i } is a series of rise time satisfying t 1 i < t 2 i < · · · < t N i , max j,k |t j i − t k i | ≥ τ G ≥ ∆τ,(20) where τ G is a relaxation time to avoid overlapped outputs. In practical implementation, the rise times t 1 i , · · · , t N i are supposed to be determined sequentially by a real time architecture described in Fig. 5, which is composed of three components, a classifier C, a selector S J , and an impulse generator G. Classifier C We define the classifier C as a function from a state vector x = (x T 1 , x T 2 ) T at the time t = t 0 , say x(t 0 ) = ξ 0 , to an index number ν of equilibrium ω ν . The function C takes the value C(ξ 0 ) = ν if a solution of the following system: x = f (x, T), x(t 0 ) = ξ 0 , T := u pd +       P 1 I ∆τ (t − t 0 ) 0      (21) converges to the equilibrium ω ν . From this definition in which a single impulse at t = t 0 is applied only on the left-hand agent, it is implied that the classifier C is valid only in absence of additional inputs, in other words, it can fail to return correct equilibriums for more general cases of input as in (18) where both agents can produce impulsive forces for their own decisions. Despite that, we will proceed with a discussion to build a first prototype of artificial wrestling. Consider the transition operator of a solution of (21) as and define a set of the initial state ξ 0 approaching the equilibrium ω i as x(t) := φ t (ξ 0 , T),(22)Φ i := ξ 0 ∈ R 8 lim t→∞ φ t (ξ 0 , T) = ω i .(23) The set Φ i is generally called a basin of attraction (Lhommeau et al., 2011) or a reachable set (Bayadi et al., 2013). It follows from uniqueness of solution of initial value problem in (21) that the reachable set in (23) satisfies, Φ i ∩ Φ j = ∅ (i j).(24) Thus, the classifier C is obtained as a single-valued function: C ξ 0 := ν if ξ 0 ∈ Φ ν .(25) In Section 3.5, we will discuss a numerical approximation of C because explicit expressions of C( · ) are hardly obtainable from nonlinear systems such as (15) and also (21). Selector S J In the competitive problem in Fig. 3, some of the equilibriums are selected depending upon strategies of the agent considered. Such a selection process can be modeled by a selector S J given by δ = S J (ν) :=          1 (ν ∈ J ⊂ {1, · · · , 9}), 0 (otherwise),(26) where ν = C(x) is an output of the classifier and J is a given subset of indices of the equilibriums ω 1 , · · · , ω 9 . For example, the trajectory x(t) in (21) starting from an initial state ξ 0 satisfying (S J • C)(ξ 0 ) := S J C(ξ 0 ) = 1 for J = {2, 3}(27) converges one of the two equilibriums ω 2 and ω 3 . Impulse generator G The impulse generator G is designed to receive the binary signal δ(t) = S J (ν(t)) from the selector and output the unit impulse G(t) as shown in Fig. 6, which is composed of a two-input AND gate and two timer functions T I and T G . The timer T I produces a unit impulse as G(t) = T I (t) := I ∆τ (t − t r ),(28) and the timer T G cuts off the binary signal δ(t) = S J (ν(t)) for a given relaxation time τ G discussed in (20) as T G (t) :=          0 (t r < t < t r + τ G ), 1 (otherwise),(29) where t r is the rise time from 0 to 1 of the Boolean productδ(t) = S J (ν(t))∧T G (t). Note that although the signalδ(t) can be pulses of infinitesimal width in this continuous time expression, this problem will never arise in discrete time applications with digital computers. Numerical approximation of the classifier C Explicit expression of C(x) is hardly obtainable from nonlinear systems such as (15). It seems that there are two types of solutions: solving (15) in process or making a numerical table of the mapping C : x → ν in advance. We take the latter approach in the following manner. For practical applications, we introduce a linear measurement equation: y = Hx, y ∈ R M , x ∈ R 8 ,(30) where M ≤ 8 and H is a M × 8 matrix of rank M, and define a reduced-order reachable set in the following sense: Ψ i = H(Φ i ) := η 0 ∈ R M ξ 0 = h(η 0 ) ∈ R 8 , lim t→∞ φ t (ξ 0 , T) = ω i ,(31) where x ′ = h(y) ( x in general) is a certain inverse of y = Hx. An actual example of h will be given in Section 3.6. Firstly, we take a M-dimensional cubic region D of measuring range within a direct sum of the reachable sets as 9 i=1 Ψ i ⊃ D := [a 1 , b 1 ] × · · · × [a M , · · · , b M ](32) where represents a direct sum (disjoint union) and [a, b] denotes an interval. We divide it into a direct sum of uniform subcubes D i as D = D i i ∈ I = [1, · · · , m 1 ] × · · · × [1, · · · , m M ] ,(33) where I is a space of M-dimensional indices and m j is the number of subcubes in the jth direction. We then define center points of the subcubes y i ∈ D i (i ∈ I) whose jth component is given by (y i ) j := a j + (i) j − 1 2 b j − a j m j ,(34) where (u) j denotes the jth component of a vector u. The setup above allows us to build a numerical method as follows: ( 1 ) As an offline learning procedure, the mappingC: C(i) := ν if lim t→∞ φ t h(y i ), T = ω ν ,(35a) is numerically stored by solving (21) from ξ 0 = h(y i ). ( 2 ) Within the IC in process, the classifier C is quantized by C * as C(x) ≈ C * (x) :=C(i) for i such that Hx ∈ D i . (35b) In this method, accuracy of the classifier C * depends on the dimension of measurement M, the size and placement of measuring range [a j , b j ], and the resolution of quantization of reachable set m j ( j = 1, · · · , M). In summary, we have obtained the IC for the left-hand agent as a composed function of the quantized classifier C * , the selector S J , and the impulse generator G, which results in the closed-loop form: u ic 1 = u ic 1 x(t); J := P 1 · (G • S J • C * ) x(t) .(36) If M = 8 and τ G is sufficiently large, it is implied that the solution x(t) in (21) starting from any initial states in D undergoes an impulsive force at the time t = t 0 that u ic 1 decides autonomously and that it converges to the stable equilibriums specified by J, under the resolution limit, min j (m j ) → ∞. Reduced-order design of measurement for rigid connection The connection rod can behave like a rigid rod if the viscoelastic parameters k w , c w are sufficiently large. For the rest of this paper, we restrict our problem to this nearly rigid connection. Although human wrestling involves more flexible interactions between agents, this allows us substantially to reduce the computational efforts as follows. In this case, dependency between displacements and velocities of the left-hand and right-hand carts is imposed in the following form: x 2 = x 2 (x 1 , θ 1 , θ 2 ) = x 1 − r(sin θ 2 − sin θ 1 ) + w 2 0 − r 2 (cos θ 2 − cos θ 1 ) 2 , x 2 =ẋ 2 (ẋ 1 , θ 1 ,θ 1 , θ 2 ,θ 2 ) =ẋ 1 − r(θ 2 cos θ 2 −θ 1 cos θ 1 ) + r 2 (cos θ 2 − cos θ 1 )(θ 2 sin θ 2 −θ 1 sin θ 1 ) where w = w 0 ≫ 2r is a constant length of the rigid rod. Then, a loss-less state feedback to IC can be done by the following measurement: w 2 0 − r 2 (cos θ 2 − cos θ 1 ) 2 ,(37b)y = Hx = (x 1 ,ẋ 1 , θ 1 ,θ 1 , θ 2 ,θ 2 ) T , H = e (6) 1 , e (6) 2 , e (6) 3 , e (6) 4 , o (6) , o (6) , e (6) 5 , e (6) 6 ,(38a) where e (d) i , o (d) denote the ith standard basis vector and the zero vector in Euclidean space R d respectively. A corresponding inverse satisfying the rigid constraint can be defined by h(y) := H + y + x 2 (x 1 , θ 1 , θ 2 )e (8) 5 +ẋ 2 (ẋ 1 , θ 1 ,θ 1 , θ 2 ,θ 2 )e (8) 6 ,(38b) where H + is the Moore-Penrose pseudoinverse of H. This measurement reduces computational efforts to obtain the quantized reachable sets ofC in (35a) h(y) := H + y + x 1 e (8) 1 +ẋ 1 e (8) 2 + x 2 (x 1 , θ 1 , θ 2 )e (8) 5 +ẋ 2 (ẋ 1 , θ 1 ,θ 1 , θ 2 ,θ 2 )e (8) 6 with x 1 =ẋ 1 = 0. Although this measurement suffers complete loss of information about the cart motion x 1 ,ẋ 1 (and x 2 ,ẋ 2 via (37)), it reduces the computational efforts into O(m 4 ). In this paper, we employ this four-dimensional measurement in priority to reducing the computational efforts. Moreover, in this four-dimensional measurement, the controller u ic 1 (x; J) originally designed for the left-hand agent can symmetrically be reused for the right-hand agent by a transformation: u ic 2 (x; J ′ ) := u ic 1 (x ′ ; J), P 2 := −P 1 , x ′ := −(x T 2 , x T 1 ) T ,(40) where J ′ = {4, 7} for J = {2, 3} due to the transpose of the 3 × 3 matrix of ω i in Fig. 3. Numerical investigation We conduct numerical experiments to evaluate performance of the four-dimensional IC of the measurement in (39). For simplicity, in what follows, we set the resolution m j of quatization to a common value m for all j. The parameter values used in the following examples are listed in Table 1. The physical dimensions m θ , m x and r are roughly collected from the commercially available inverted pendulum (ZMP INC., 2011). For numerical integration, the fourth-order Runge-Kutta-Gill method is employed with the time step ∆t listed in Table 2. Individual performance of IC In order to investigate individual performance of IC, we first consider impulse responses of the CIP model in (15) equipped with the IC in the left-hand only given by T = u pd + u ic + u = u pd +       u ic 1 x(t); J 1 0       +       v(t) 0       , v(t) := QI ∆τ (t), J 1 := {2, 3},(41) where u pd is the standing controller in (12), Q is an impulse of initial disturbance, u ic 1 is the four-dimensional IC developed in (39), and I ∆τ (t) is the unit impulse function defined in (19). In the following numerical examples, we assume the maximal strength of impulse Q max = 0.06 so that v(t) cannot produce switching motions from the trivial initial state x(0) = o (8) + w 0 e (8) 5 (or ω 1 ) to the other stable states in order to avoid trivial switching motions. Following this assumption, the region of measurement D is taken as listed in Table 2 so that it circumscribes at least all the trajectories y(t) = Hx(t) for the maximal disturbance Q = Q max . We also set the strength of impulse of IC to a common value P 1 = −P 2 = Q max to avoid the trivial switching motions mentioned above. Note that under the conditions of k w and c w listed in Table 1 and for Q ≤ Q max , dynamic change of length of the connection rod in (41) is limited to |w(t) − w 0 |/w 0 < 0.01 so that the measurement in Section 3.6 is expected to work. Solving (15) with (41) numerically from a trivial initial state x(0) = o (8) + w 0 e (8) 5 for a given Q, we have the corresponding final position ω ν and obtain correspondence from Q to ν as plotted in Fig. 7 for the resolution m = 50 where Q is taken at N Q = 100 points with uniform increment over the interval [0, Q max ]. The small circles represent the results of ν(Q) in presence of the IC's outputs and the cross marks represent those in absence of the outputs due to weak v(t). To evaluate the performance, we define a success rate in the following form: E = E(J) := N J /(N Q − N 0 ) (0 ≤ E ≤ 1),(42) where N J is the number of points on Q at which lim t→∞ x(t) = ω ν for all ν ∈ J and N 0 is the number of points in absence of the IC's outputs. The success rate of the result for m = 50 shown in Fig. 7 is calculated as E = 0.165. Also, the rate for m = 100 can be obtained as E = 0.305 in the same manner. Therefore, it appears that our IC has low performance. Performance improvement with a delay element One of the reasons of the low performance above can be explained in Fig. 8. The colored areas represent the quantized reachable sets: Ψ * ν := D i C (i) = ν, i ∈ I (ν = 1, · · · , 9),(43) plotted on the hyper plane that contains the point x(t 1 ) on the phase trajectory at which the IC's output occurs (indicated by the cross mark). The resolutions of quantization of reachable sets are m = 50 for Fig. 8 (a) and m = 1000 for Fig. 8 (b). In the low resolution case in Fig. 8 (a), it can numerically be clarified that x(t 1 ) belongs to the reachable set Ψ * 2 (in red) of ω 2 while x(t) (t > t 1 ) actually converges to ω 4 ( ω 2 ). Such misclassification can be refined by the high resolution as shown in Fig. 8 (b). In this case, the point x(t 1 ) is primly classified into Ψ * 4 (in blue) of ω 4 . Although in theory, taking a sufficiently large resolution m provides nearly exact reachable sets, it greatly enlarges computational efforts. Another approach to reduce the misclassification is to replace the quantized reachable set Ψ * i in (43) of ω i with a subset Ψ • i := Ψ * i −∆Ψ * i where ∆Ψ * i ⊂ Ψ * i is a set of border points neighboring the other sets Ψ * j ( j i). However, extracting the border points is not necessarily easy because the reachable sets sometimes exhibit nested structures as discussed by one of the authors (Yoshida, 2009). Actually, in Fig. 8 (b), quite narrow region of Ψ * 1 (in white) appears between Ψ * 2 (in red) and Ψ * 4 (in blue). Therefore, let us take yet another approach by replacing the IC with a delayed IC (DIC) in the following form: u dic i x(t); J, τ d := u ic i x(t − τ d ); J (i = 1, 2),(44) where τ d is a delay time. It seems reasonable to expect that a trajectory about to crossing a course-grained border of a reachable set reaches the true border soon. Figure 9 shows the success rate E corresponding to this replacement given by T = u pd +        u dic 1 x(t); J 1 , τ d 0        + u, J 1 := {2, 3},(45) where E is averaged over the two types of initial disturbance: u(t) = v(t), 0 T and 0, v(t) T , and the other procedures of obtaining E are the same as in Fig. 7. In Fig. 9, the triangles and the circles represent E as functions of the delay time τ d of DIC in (44) Competitive behavior In this final section, we install the DICs into both sides of controllers as T = u pd +        u dic 1 x(t); J 1 , τ d 1 u dic 2 x(t); J 2 , τ d 2        + u, J 1 := {2, 3}, J 2 := {4, 7},(46) where u pd is the standing controller in (12), u dic 1 and u dic 2 is the four-dimensional DIC in (44) through (40), and u(t) = v(t), 0 T , 0, v(t) T are initial impulsive disturbances of strength Q as shown in (41). The competitive meanings of J 1 and J 2 are shown in Fig. 3. Figure 10 shows a competitive behavior between the normal IC: u dic 1 x(t); J 1 , 0 = u ic 1 x(t); J 1 for m = 100 and the DIC with optimal delay time: u dic 2 x(t); J 2 , 0.0045 for m = 50. The time responses in Fig. 10 (a) are obtained by solving (15) with (46) from x(0) = o (8) +w 0 e (8) 5 under the conditions listed in Table 1 and Table 2 by applying the initial disturbance u(t) = v(t), 0 T with Q = 0.0186. The same solution is plotted in Fig. 10 (b) as a motion of the CIP mechanism. For convenience, we refer to the former IC as "IC100" and the later as "DIC50". It is shown in Fig. 10 that the ICs above generate the impulsive control forces autonomously to make the CIP system drop into ω 2 . According to the competitive interpretation in Fig. 3 inspired by some kind of wrestling match, it can be -π/2 0 π/2 0 3 6 9 12 15 said that "the left agent wins by pulling the right agent." Figure 11 shows the result of competition. The plots are obtained in the same manner as those in Fig. 7, except that in Fig. 11, the open and filled circles represent the results for u(t) = v(t), 0 T and 0, v(t) T respectively, and that the cross marks represent the results in which neither of IC100 and DIC50 produces its own output. Similar to the individual case in (42), we define the success rate of this competition as P 2 0 P 1 θ i u i dic t i =1 i =2(E i = E(J i ) := N J i /(N Q − N 0 ) (0 ≤ E i ≤ 1) (i = 1, 2),(47) where N Q = 100 × 2 is the number of all plots in Fig. 11, N 0 is the number of trials in absence of the IC's outputs (cross marks), and the definition of N J i is the same as that of N J in (42). From the result in Fig. 11, the success rates are obtained as E 1 = 66/167 ≈ 0.395 for IC100 and E 2 = 96/167 ≈ 0.575 for DIC50. Therefore, it is shown that at least based on the present definition of competition and success rate, the performance improvement using time delay is more effective than that doubling the quantization resolution without the time delay. Conclusion In this paper, we discussed a competitive problem in which mechanical agents are fighting with each other and formulated it as the set of: (A) the nonlinear dynamical model with nine stable equilibriums and (B) the matrix describing competitive interpretation of these equilibriums. Based on this framework, we proposed a competitive IC that receives the state vector and output the impulsive forces to make the competitor fall down. Developing a quantized and reduced order design of the controller, we derived a practical control procedure along with an off-line learning method. To investigate performance of the controller in individual use and also in competition use, we conducted numerical experiments and obtained the following results: • The individual performance of IC depends on resolutions of the quantized reachable sets. • The individual performance of IC can be improved by adding the delay element into the IC. • To improve the competitive performance of IC, adding the delay element may become more effective than refining the resolutions. In future work, we plan to investigate a further order reduction of measurement based on time delayed embedding methods and to improve classification accuracy by applying machine learning techniques. We also plan to conduct competitions between humans and the proposed ICs. Moreover, application of position control to the carts will be considered to investigate competitive problems in a bounded area. Fig. 1 1Coupled inverted pendula model via viscoelastic connection.Fig. 2Reaction force p from the connection rod. Fig. 3 3Competitive interpretation of the equilibriums.Fig. 4Architecture of the proposed control system to generate competitive motions. Fig. 5 5Intelligent controller.Fig. 6Impulse generator. x viscous coefficient along x 0.01 Ns/m c θ viscous coefficient about θ 0.01 Ns/m w 0 natural length of connection rod 1 m k w spring coefficient of connection rod 5000 N/m c w viscous coefficient of connection rod 50 Ns/m k f spring coefficient of floor 500 N/m c f viscous coefficient of floor 10 Ns/m µ Coulomb friction coefficient of floor 0 σ steepness of step function 10 6 α steepness of trapezoidal function 25 K p proportional gain of standing control 1 K d derivative gain of standing control 0.01 ∆θ threshold of standing control π/6 rad from O(m 8 ) to O(m 6 ) with respect to the resolution of quantization m. In the following numerical examples, we perform a further reduction of order given by y = Hx = (θ 1 ,θ 1 , θ 2 ,θ 2 ) T , H = o (4) , o (4) , e (4) 1 , e (4) 2 , o (4) , o (4) , e (4) 3 , e (4) 4 , Fig. 7 Fig. 8 Fig. 9 789Index of final state ω ν as a function of the strength of initial disturbance Q for the resolution m = 50. Misclassification of reachable sets. Colored areas are quantized reachable sets for (a) m=50 and (Success rate E as functions of the delay time τ d of DIC in (44). Fig. 11 11E 1 = 0.395 (ic; m = 100) E 2 = 0.575 (dic; m = 50)v=(v(t),0) v=(0,v(t))Index of final state ω ν as functions of Q for IC100 vs DIC50 (τ d 2 = 0.0045). Table Table 2 2Parameter setting of the four-dimensional IC and numerical integration.Parameters Values Q max maximal strength of impulse 0.06 Nms P 1 strength of impulse of the 1st IC = Q max P 2 strength of impulse of the 2nd IC = −Q max ∆τ width of impulse = ∆t τ G relaxation time of impulse generator = ∆t D region of measurement [−0.13, 0.43] × [−3.28, 10.58] ×[−0.35, 0.31] × [−3.80, 5.15] ∆t step size of numerical integration 5 × 10 −4 s Free parameters m resolution of numerical classifier (m j := m for all j) Q strength of impulsive disturbance τ d delay time of delay element for m = 50 and 100 respectively. It is clearly seen that the functions E(τ d ) are nearly concave down. The maximal values are E = 0.392 at τ d = 0.0045 and E = 0.683 at τ d = 0.0025 for m = 50 and 100 respectively. Therefore, the DIC roughly doubles the success rate, namely, 0.392/0.165 ≈ 2.38 for m = 50 and 0.683/0.305 ≈ 2.24 for m = 100. AcknowledgmentThe authors appreciate the feedback offered by Dr. Munehisa Sekikawa. Characterizing the reachable set for a spacecraft with two rotors. R Bayadi, R N Banavar, D E Chang, Systems & Control Letters. 626Bayadi, R., Banavar, R. N. and Chang, D. E., Characterizing the reachable set for a spacecraft with two rotors, Systems & Control Letters, Vol.62, No.6 (2013), pp.453-460. Functional reach: A new clinical measure of balance. P W Duncan, D K Weiner, J Chandler, S Studenski, Journal of Gerontology. 456Duncan, P. W., Weiner, D. K., Chandler, J. and Studenski, S., Functional reach: A new clinical measure of balance, Journal of Gerontology, Vol.45, No.6 (1990), pp.M192-M197. J Hofbauer, K Sigmund, Evolutionary Games and Population Dynamics. Cambridge University PressHofbauer, J. and Sigmund, K., Evolutionary Games and Population Dynamics (1998), Cambridge University Press. Coordinated control of multiple manipulator systems. P Hsu, IEEE Transactions on Robotics and Automation. 94Hsu, P., Coordinated control of multiple manipulator systems, IEEE Transactions on Robotics and Automation, Vol.9, No.4 (1993), pp.400-410. Capture basin approximation using interval analysis. M Lhommeau, L Jaulin, L Hardouin, International Journal of Adaptive Control and Signal Processing. 253Lhommeau, M., Jaulin, L. and Hardouin, L., Capture basin approximation using interval analysis, International Journal of Adaptive Control and Signal Processing, Vol.25, No.3 (2011), pp.264-272. Coordination of communication in robot teams by reinforcement learning. D Maravall, J De Lope, R Domínguez, Robotics and Autonomous Systems. 617Maravall, D., de Lope, J. and Domínguez, R., Coordination of communication in robot teams by reinforcement learning, Robotics and Autonomous Systems, Vol.61, No.7 (2013), pp.661-666. Collision detection and response for computer animation. M Moore, J Wilhelms, SIGGRAPH Computer Graphics. 224Moore, M. and Wilhelms, J., Collision detection and response for computer animation, SIGGRAPH Computer Graphics, Vol.22, No.4 (1988), pp.289-298. Mechanics of coordinative manipulation by multiple robotic mechanisms. Y Nakamura, K Nagai, T Yoshikawa, Proceedings of 1987 IEEE International Conference on Robotics and Automation. 1987 IEEE International Conference on Robotics and Automation4Nakamura, Y., Nagai, K. and Yoshikawa, T., Mechanics of coordinative manipulation by multiple robotic mechanisms, Proceedings of 1987 IEEE International Conference on Robotics and Automation, Vol.4 (1987), pp.991-998. Evolution of neural controllers for competitive game playing with teams of mobile robots. A Nelson, E Grant, T Henderson, Robotics and Autonomous Systems. 463Nelson, A., Grant, E. and Henderson, T., Evolution of neural controllers for competitive game playing with teams of mobile robots, Robotics and Autonomous Systems, Vol.46, No.3 (2004), pp.135-150. Cooperative behaviors in multi-robot systems through implicit communication. E Pagello, A Dangelo, F Montesello, F Garelli, C Ferrari, Robotics and Autonomous Systems. 291Pagello, E., DAngelo, A., Montesello, F., Garelli, F. and Ferrari, C., Cooperative behaviors in multi-robot systems through implicit communication, Robotics and Autonomous Systems, Vol.29, No.1 (1999), pp.65-77. Adaptive neural controller for cooperative multiple robot manipulator system manipulating a single rigid object. V Panwar, N Kumar, N Sukavanam, J Borm, Applied Soft Computing. 121Panwar, V., Kumar, N., Sukavanam, N. and Borm, J., Adaptive neural controller for cooperative multiple robot manipula- tor system manipulating a single rigid object, Applied Soft Computing, Vol.12, No.1 (2012), pp.216-227. Multiagent systems: A survey from a machine learning perspective. P Stone, M Veloso, Autonomous Robots. 83Stone, P. and Veloso, M., Multiagent systems: A survey from a machine learning perspective, Autonomous Robots, Vol.8, No.3 (2000), pp.345-383. Analysis of strategy in robot soccer game. J Wu, V Snášel, E Ochodková, J Martinovič, V Svatoň, A Abraham, Neurocomputing. 109Wu, J., Snášel, V., Ochodková, E., Martinovič, J., Svatoň, V. and Abraham, A., Analysis of strategy in robot soccer game, Neurocomputing, Vol.109 (2013), pp.66-75. Fractal dependence on initial conditions of coupled inverted pendula model of competition and cooperation. K Yoshida, Journal of System Design and Dynamics. 36Yoshida, K., Fractal dependence on initial conditions of coupled inverted pendula model of competition and cooperation, Journal of System Design and Dynamics, Vol.3, No.6 (2009), pp.966-974. Coupled inverted pendula model of competition and cooperation. K Yoshida, H Ohta, Journal of System Design and Dynamics. 23Yoshida, K. and Ohta, H., Coupled inverted pendula model of competition and cooperation, Journal of System Design and Dynamics, Vol.2, No.3 (2008), pp.727-737. . Zmp Inc, Wheel, ZMP INC., e-nuvo WHEEL (2011), http://www.zmp.co.jp/e-nuvo/en/.

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{'abstract': 'We develop autonomous agents fighting with each other, inspired by human wrestling. For this purpose, we propose a coupled inverted pendula (CIP) framework in which: 1) tips of two inverted pendulums are linked by a connection rod, 2) each pendulum is primarily stabilized by a PD-controller, 3) and is additionally equipped with an intelligent controller. Based on this framework, we dynamically formulate an intelligent controller designed to store dynamical correspondence from initial states to final states of the CIP model, to receive state vectors of the model, and to output impulsive control forces to produce desired final states of the model. Developing a quantized and reduced order design of this controller, we have a practical control procedure based on an off-line learning method. We then conduct numerical simulations to investigate individual performance of the intelligent controller, showing that the performance can be improved by adding a delay element into the intelligent controller. The result shows that the performance depends not only on quantization resolutions of learning data but also on delay time of the delay element. Finally, we install the intelligent controllers into both pendulums in the proposed framework to demonstrate autonomous competitive behavior between inverted pendulums.', 'arxivid': '1405.7178', 'author': ['Katsutoshi Yoshida yoshidak@cc.utsunomiya-u.ac.jp ', 'Shigeki Matsumoto ', 'Yoichi Matsue ', '\nDepartment of Mechanical and Intelligent Engineering\nUtsunomiya University\n7-1-2 Yoto321-8585Utsunomiya, TochigiJapan\n', '\nYuhara Mfg Co\n1256, 329-1311Ltd, Ujiie, Sakura, TochigiJapan\n'], 'authoraffiliation': ['Department of Mechanical and Intelligent Engineering\nUtsunomiya University\n7-1-2 Yoto321-8585Utsunomiya, TochigiJapan', 'Yuhara Mfg Co\n1256, 329-1311Ltd, Ujiie, Sakura, TochigiJapan'], 'corpusid': 10233567, 'doi': '10.1299/mej.14-00518', 'github_urls': [], 'n_tokens_mistral': 13165, 'n_tokens_neox': 11365, 'n_words': 7372, 'pdfsha': 'f0d05037615524de28cbb513ca766851c999fef2', 'pdfurls': ['https://arxiv.org/pdf/1405.7178v2.pdf'], 'title': ['Artificial Wrestling: A Dynamical Formulation of Autonomous Agents Fighting in a Coupled Inverted Pendula Framework', 'Artificial Wrestling: A Dynamical Formulation of Autonomous Agents Fighting in a Coupled Inverted Pendula Framework'], 'venue': []}

arxiv

Improving Model Generalization by Agreement of Learned Representations from Data Augmentation Rowel Atienza rowel@eee.upd.edu.ph University of the Philippines Electrical and Electronics Engineering Institute 1101Diliman, Quezon CityPhilippines Improving Model Generalization by Agreement of Learned Representations from Data Augmentation Data augmentation reduces the generalization error by forcing a model to learn invariant representations given different transformations of the input image. In computer vision, on top of the standard image processing functions, data augmentation techniques based on regional dropout such as CutOut, MixUp, and CutMix and policy-based selection such as AutoAugment demonstrated state-of-the-art (SOTA) results. With an increasing number of data augmentation algorithms being proposed, the focus is always on optimizing the input-output mapping while not realizing that there might be an untapped value in the transformed images with the same label. We hypothesize that by forcing the representations of two transformations to agree, we can further reduce the model generalization error. We call our proposed method Agreement Maximization or simply AgMax. With this simple constraint applied during training, empirical results show that data augmentation algorithms can further improve the classification accuracy of ResNet50 on ImageNet by up to 1.5%, WideResNet40-2 on CIFAR10 by up to 0.7%, WideResNet40-2 on CIFAR100 by up to 1.6%, and LeNet5 on Speech Commands Dataset by up to 1.4%. Experimental results further show that unlike other regularization terms such as label smoothing, AgMax can take advantage of the data augmentation to consistently improve model generalization by a significant margin. On downstream tasks such as object detection and segmentation on PascalVOC and COCO, AgMax pre-trained models outperforms other data augmentation methods by as much as 1.0mAP (box) and 0.5mAP (mask). Code is available at https://github.com/roatienza/agmax. Introduction To achieve state-of-the-art (SOTA) performance, data augmentation plays a crucial role in both supervised and self-supervised model training. In computer vision, image E(x; θ) σ(z) x z p(ŷ) "cat" Figure 1. In supervised learning, a model E is trained to find the optimal parameters θ * . Input data augmentation improves the performance by forcing E to learn invariant representations z under image transformation such as removing a random square region as done in CutOut. x is a labelled positive sample. E(x 1 ; θ) σ(z 1 ) x 1 z 1 p(ŷ 1 ) "cat" E(x 2 ; θ) σ(z 2 ) Agreement 2 positive samples x 2 z 2 θ p(ŷ 2 ) "cat" Same label Figure 2. In supervised learning with AgMax, we impose an additional constraint that representations z1 and z2 must also agree. The two parallel models are just one and the same and share the same set of parameters θ. x1 and x2 are 2 positive samples with the same label. processing functions such as rotation, translation, cropping, flipping, and color distortion improve model generalization. In recent years, a strong interest in new data augmentation techniques has emerged because of the significant improvement in model performance compared to baseline scores. Instead of just applying random image processing operations, policy-based methods such as AutoAugment (AA) [5], FastAugment [30], RandAugment [6], Adversarial Au-toAugment [53], and PBA [22] [7] 76.2 ± 0.0 76.5 ± 0.1 77.1 ± 0.0 MixUp [51] 76.5 ± 0.1 76.7 ± 0.1 77.6 ± 0.1 CutMix [49] 76.3 ± 0.0 76.4 ± 0.1 77.4 ± 0.0 AA [5] 76.2 ± 0.1 76.2 ± 0.1 77.1 ± 0.1 CutOut+AA [5] 75.7 ± 0.1 75.7 ± 0.1 76.6 ± 0.1 MixUp+AA 75.9 ± 0.0 76.5 ± 0.1 77.1 ± 0.1 CutMix+AA 75.5 ± 0.1 75.5 ± 0.1 77.0 ± 0.1 model overfitting. Regional dropouts or techniques based on direct image alteration such as CutOut [7], RICAP [42], CutMix [49], GridMask [4] and MixUp [51] improve model performance by forcing the model to learn invariant representations. With an increasing number of data augmentation algorithms being introduced, as shown in Figure 1 the focus is always on optimizing the input-output mapping. We hypothesize that there might be an untapped value between representations of transformed inputs with the same label. Basic intuition tells us that under different transformations such as in Figure 2, two inputs with the same label should agree on which representations that a model learns. These two inputs are called positive samples since they have the same label but two different transformations. A classifier receiving two positive samples of a cat must learn to extract the minimum common set of representations such as the presence of whiskers, fur, sharp eyes, short nose, etc. We call this function Agreement. In this paper, we use mutual information (MI) to estimate Agreement. We performed ablation studies to demonstrate that other agreement functions such as MSE, KL-divergence and cross-entropy (CE) are also effective. Using a common evaluation protocol, experimental results indicate that our proposed method AgMax improves the performance of almost all model-dataset configurations. On ResNet50 trained on ImageNet for 90 epochs, as shown in Table 1, AgMax consistently outperforms Label Smoothing [41] especially under heavy data augmentation. In a bigger evaluation landscape, the results in Tables 2 and 3 demonstrate the consistent improvement in generalization for different models and datasets due to AgMax. On downstream tasks such as object detection and segmentation, a ResNet50 model pre-trained with AgMax outperforms its counterpart pre-trained model by as much as 1.0mAP on bounding box and 0.5mAP on segmentation mask. Related Work To achieve state-of-the-art (SOTA) performance, data augmentation plays a crucial role in both supervised and self-supervised model training. Data augmentation belongs to a bigger field of study called regularization. The objective of regularization is to improve model generalization by modifying the network structure during training, augmenting the train dataset, modifying the loss function or modifying the model training algorithm. For example, dropout [40] randomly drops neural network units during training to mimic data and network perturbations. As a result, a model improves its test performance. In deep CNNs, instead of dropping feature maps, noise injection or substitution such as Stochastic Depth [23], Shake-Shake [10], DropBlock [12], DropPath [27], SpatialDropout [44] and Shake-Drop [48] are used. Related to dropout is the regional dropout. Instead of dropping neural network units, a certain region of the input is removed, mixed or blended. In effect, regional dropout augments the training dataset by exposing a model to extreme input data transformations. CutOut [7], RICAP [42], CutMix [49], GridMask [4] and MixUp [51] belong to this category. Before the regional dropout methods were proposed, data augmentation was achieved by basic input data transformations. In computer vision, padding, random cropping, translation, rotation, horizontal flipping and color distortion are commonly used. Recently, these standard data augmentation techniques have been supplanted by a more structured learned policy in order to arrive at an optimal recipe of data transformation functions. In computer vision, Au-toAugment [5], FastAugment [30], Adversarial AutoAugment [53], RandAugment [6] and PBA [22] have been proposed. Among the available data augmentation methods, regional dropouts, policy-based, and gradient augmentation have demonstrated state-of-the-art results. Improving model generalization can also be achieved by modifying the loss function. Label smoothing [41] and weight decay [15] belong to this category. In this paper, the idea is to take advantage of the huge amount of data generated by augmentation methods. If an image undergoes data augmentation to produce 2 new images, the resulting representations a model learns must agree since both inputs belong to the same category. This idea bears resemblance to representations matching in selfsupervised learning such as BYOL [14] and DINO [2] where the predictions of teacher(target) and student(online) networks on 2 positive samples (2 views of the same image) are reinforced by similarity learning. The similarity function could be MSE in BYOL or cross-entropy (CE) function in DINO. The teacher and student networks could be similar in architecture but have different sets of parameters. The teacher network parameters are exponential moving average of the online parameters. The main difference of Ag- CIFAR10[25] CIFAR100 ImageNet [38] WideResNet [50] ResNet50 [ Table 2. Evaluation landscape showing model accuracy of different data augmentation algorithms with and without AgMax. Underscore is the best performing configuration without AgMax. Bold is the best performing method for all configurations. The absolute percentage increase in accuracy due to AgMax is enclosed in parentheses. Standard data augmentation algorithm is defined in the Experimental Results section. Max is it uses one network for both predictions. Instead of maximizing a similarity function, AgMax maximizes the agreement using mutual information. In this paper, we validate our hypothesis on 4 commonly used data augmentation algorithms: 1 policy-based approach and 3 regional dropout techniques. We chose Au-toAugment for the policy-based data augmentation given that its policy is publicly available. FastAugment, Adversarial AutoAugment, and RandAugment are built on top of the key ideas of AutoAugment. For regional dropout, we used CutOut, MixUp, and CutMix. These methods have gained mainstream use and achieved state-of-the-art results. Furthermore, their code implementations are publicly available for reproducibility. Improving Generalization by Agreement With reference to Figure 2, we hypothesize that the representations of two images derived by applying a data augmentation method on an image must agree for a model to further improve its classification accuracy. In this paper, we propose that the Agreement is the amount of shared information between the two views of the same image. Therefore, on top of the classification loss function, maximizing the mutual information between the two representations could improve the model generalization. The total loss function can be expressed as: L = L CE + λL M I .(1) λ is the weight of the MI loss function. For the case of discrete random variables such as in image classification, mutual information is expressed as: I(z 1 ; z 2 ) = z1,z2 P (z 1 , z 2 ) log P (z 1 , z 2 ) P (z 1 )P (z 2 ) .(2) In other words, MI is the KL-divergence between the joint and product of marginal probabilities of z 1 and z 2 . In recent years, several neural MI estimators have been proposed [1,45,21,24,33]. Invariant Information Clus- tering (IIC) [24] proposed a method to estimate Equation 2. For a given dataset or batch of size n, the joint probability matrix P ∈ R C×C can be computed as: P = 1 n n i=1 Φ(x 1,i ) · Φ(x 2,i ) ,(3) where x 1,i and x 2,i are two transformed versions of the same image x i . Φ(x) = σ(E(x)) = sof tmax(z) ∈ [0, 1] C . This can be interpreted as the distribution of z over C classes formally given as P (z = c|x, θ)) = Φ c (x). The marginal distributions P c1 = P (z 1 = c 1 ) and P c2 = P (z 2 = c 2 ) can be obtained by summing the rows and columns of P respectively. Each element of P is the joint probability P c1c2 = P (z 1 = c 1 , z 2 = c 2 ). Since P c1c2 = P c2c1 , the matrix P must be symmetric. Ensuring a symmetric P is done by P = P +P 2 . Using the joint and marginal probabilities, the mutual information loss is computed as: L M I = −I(z 1 ; z 2 ) = − C c1=1 C c2=1 P c1c2 ln P c1c2 P c1 · P c2 .(4) A 2-layer MLP network can also be used to estimate the joint distribution. The MLP network is trained using the objective functions: p(z 1 , z 2 ) − → p(z 1 , z 1 ) and p(z 1 , z 2 ) − → p(z 2 , z 2 ) since both features refer to the same class and so is the joint distribution. Note that to estimate the MI loss, only pairs of positive samples or one-to-one mapping is needed. Therefore, Ag-Max works even for small batch sizes. This is different from constrastive learning that requires a positive and many negative samples or one-to-many mapping. Contrastive learning needs large batch sizes (e.g. 4,096 and up) to work. This has a huge negative implication on GPU memory requirements. Agreement by Mutual Information In this section, we attempt to find a possible explanation on why MI provides a good Agreement function using the Maximum a Posteriori (MAP) principle: θ * = argmax θ log p(θ|D) = argmax θ log p(θ|x, y). (5) When applied to deep neural networks (DNNs) as shown in Figure 1, θ is the model parameters while θ * represents the maximal point estimate for a given dataset D = {x, y}. Using Bayes' Theorem and with the constant term dropped, the conditional probability in Equation 5 can be rewritten as: θ * = argmax θ [log p(y|x, θ) + log p(x|θ)p(θ)] . (6) In a given model, a backbone network encodes the input into a latent variable, z = E(x; θ 1 ). A decoder decides what output is generated,ŷ = D(z; θ 2 ). Collectively, θ = {θ 1 , θ 2 }. If z is taken from the last layer before the non-parametric softmax prediction ofŷ = σ(z), then θ = {θ 1 , ∅} andŷ is a good proxy of z. During supervised training, the goal is to estimate the empirical distribution p(y|x) using a parameterized distribution p(ŷ|x, θ) as modelled by the encoder-decoder. This is done by minimizing a distance function such as the Kullback-Leibler divergence function D KL (p(y|x) p(ŷ|x, θ)). In supervised classification problems, this is equivalent to minimizing the categorical cross-entropy loss function, L CE = −E p(y|x) log p(ŷ|x, θ). As shown in Figure 2, given two positive samples of x, we can reformulate MAP as: θ * = argmax θ 1 2 2 i=1 E log p(y|x i , θ)+ 1 2 E log p(z 1 , z 2 |x 1 , x 2 , θ) p(z 1 |x 1 , θ)p(z 2 |x 2 , θ) + 1 2 E log p(x 1 , x 2 , θ) ,(7) since Equation 6 can be rewritten as: θ * = argmax θ [log p(y|x, θ) + log p(z, x, θ) − log p(z|x, θ)] . (8) Given two data augmentations for a given input: θ * = argmax θ 1 2 2 i=1 log p(y|x i , θ)+ 1 2 log p(z 1 , x 1 , θ)p(z 2 , x 2 , θ) − 1 2 2 i=1 log p(z i |x i , θ) ,(9) Assuming independence, the second term in Equation 9 can be expressed as: log p(z 1 , x 1 , θ)p(z 2 , x 2 , θ) = log p(z 1 , z 2 , x 1 , x 2 , θ).(10) With dataset sampling, Equation 9 can be rewritten as Equation 7. The second term in Equation 7 resembles the mutual information (MI): I(z 1 ; z 2 |x 1 , x 2 , θ). Maximizing the MI between the representations of x 1 and x 2 regularizes the choice of model parameters θ. In Figure 2 for example, the network is encouraged to find the representations of two views of the same cat, such that the shared information between z 1 and z 2 is maximized. This could be finding the common features such as presence of whiskers, fur, sharp eyes, short nose, etc. The third term in Equation 7 can be rewritten as 1 2 E log p(θ) since the model parameters can be assumed to be independent from the input distribution. This term can be represented by L2 weight regularization. The first term of Equation 7 can be optimized by the standard cross entropy loss function as discussed earlier in this section. Therefore, the total loss function can be written as Equation 1. Experimental Results To validate our hypothesis, we first reproduced different data augmentation results. Then, we applied AgMax to verify if the model generalization will improve. To arrive at a fair comparison, we looked for the common evaluation protocols among the data augmentation algorithms under study starting with datasets and encoders or backbone networks. CIFAR10, CIFAR100 and Ima-geNet datasets are commonly used. We included Speech Commands Dataset as evaluated by MixUp. Except for CutOut, all methods used ResNet50 as the backbone network on ImageNet. We included WideResNet28-10 and WideResNet40-2 as used by AutoAugment, RandAugment, and MixUp on CIFAR10, and CIFAR100. Lastly, we However, an analysis of published experiments revealed that it is difficult to make a fair comparison of scores produced by different data augmentation algorithms due to a lack of consistent evaluation protocol. For example, on Im-ageNet dataset, the ResNet50 model was trained for epochs ep=90 and 200, lr=0.4, bs=1, 024 in MixUp, and ep=300, lr=0.1, bs=256 in CutMix. It leads to an unfair comparison since the performance of ResNet varies with the number of epochs, batch size, and learning rate settings. To address these issues, we formulated a common training condition. Then, we reproduced the reported scores. In the following subsections, we discuss the details of the uniform experimental setup that we used for each model and dataset. Whenever possible, we implemented the settings in the published literature or official code implementations. A further examination of published experiments showed that policy-based methods are seldom evaluated with complementary regional dropout algorithms. For example, Au-toAugment has demonstrated that it can achieve better results with CutOut but its use with other regional dropout algorithms was not fully exploited. In the Published Results section of Table 2, only 58% of the evaluation space has data. In our experiments, combinations of complementary data augmentation algorithms were also examined. This enabled us to see the big picture of the evaluation space. To arrive at the results in Tables 2 and 3, all models were trained from scratch using random seeds for at least 3 times with and without AgMax. The best test scores in each run were averaged for the evaluation reporting. We used the default parameter initialization in PyTorch [34] but we observed improvement in performance if a higher entropy Gaussian distribution is used in AgMax. For the MI loss function, λ = −1 in our experiments. From Tables 2 and 3 and Figure 3, we make the following general observations: 1. AgMax consistently improves the performance of all data augmentation algorithms either as a standalone or in combination with other methods. For ResNet50 (90 epochs ImageNet in Figure 3) and LeNet5 (30 epochs Speech Commands in Table 3), only AgMax achieves significant generalization improvements. 2. There is no data augmentation algorithm, separately or in combination with AutoAugment and/or AgMax that can outperform all other methods in all datasets, models, and training conditions. This means that there is no single superior data augmentation method among the techniques that we evaluated. 3. AutoAugment improves the performance of baseline and regional dropout algorithms. Exceptions are on ResNet50 at 90 epochs and on CutMix at 270 epochs. Both configurations were not explored in their original papers. 4. Similar to policy-based methods, AgMax can be applied as an add-on regularizer to improve model generalization. This simplifies the overall optimization process. CIFAR10 and CIFAR100 Both CIFAR10 and CIFAR100 datasets [25] have 60,000 real-world color images of size 32×32 pixels. Both datasets have train-test split of 50,000-10,000. CIFAR10 has 10 classes while CIFAR100 is made of 100 classes. All regularization methods were evaluated after training WideResNet28-10 and WideResNet40-2 models [50] with an initial learning rate of 0.1 for 200 epochs using cosine learning rate decay, batch size of 128, and SGD optimizer with weight decay of 5e −4 . The standard data augmentation is made of random cropping of 32 × 32 pixels with padding size of 4 pixels, random horizontal flipping, and normalization. CutOut size is 16 × 16 pixels. MixUp has α=1. CutMix probability is 0.5 with Beta distribution of α=1. Table 2 shows that 2 out of 4 model and dataset configurations, CutMix+AA+AgMax is the top performing. The rest is split between CutOut+AA+AgMax and MixUp+AA+AgMax. ImageNet The ILSVRC ImageNet dataset [38] has 1.2M real-world color images made of 1,000 classes for training. The validation set has 50,000 color images. The standard image processing includes resizing, cropping to 224 × 224 pixels, random horizontal flipping, color jitter, lighting, and normalization. All data augmentation methods were evaluated after training a ResNet50 network for 90 and 270 epochs. The initial learning rate is 0.1. The optimizer is SGD with a weight decay of 1e −4 . The learning rate is reduced by a factor of 0.1 at 30, 60, and 80 epochs during the 90 epochs training and 75, 150, and 225 epochs during the 270 epochs training. Batch size is 256. CutOut size is 112 × 112 pixels as used in CutMix paper. MixUp has α=0.2. CutMix probability is 1.0 with Beta distribution of α=1. Table 2 and Figure 3 show that when ResNet50 is trained for 90 epochs, MixUp+AgMax is the best performing method while CutMix+AA+AgMax is the top performing algorithm for 270 epochs. When only one regularizer is combined with the standard data augmentation, only Ag-Max has a significant positive gain in accuracy and therefore has the highest performance in 90 epochs. Speech Commands Dataset Google Speech Commands Dataset [47] contains 64,727 30-class utterances from 1,881 speakers. Each single-word utterance such as yes, no, up, or down is about one-second long. The standard audio signal preprocessing include random amplitude, pitch, and speed adjustment, stretching, time shifting, and addition of background noise. The audio signals are converted into 32 × 32 mel spectrogram input data. Table 3 data augmentation methods were evaluated after training LeNet5 [28] and VGG11 [39] for 30 epochs. The initial learning rate is 0.003 and adjusted by a factor of 0.1 after every 10 epochs for LeNet5 and 15 epochs for VGG11. The optimizer is Adam with a weight decay of 5e −4 for LeNet5 and 1e −4 for VGG11. The settings in CIFAR10/100 for CutOut, MixUp, and CutMix are used. Table 3 shows that CutOut+AgMax is the best performing algorithm for LeNet5 and a tie between MixUp+AgMax and CutMix+AgMax for VGG11. Except for AgMax, we observed that when used as a standalone regularizer all data augmentation algorithms fail to improve the generalization of LeNet5. Robustness Robustness has been increasing in importance as we deploy deep learning models on safety-critical applications. Although the regularization methods that we evaluated are not consciously optimized for robustness, it is worth knowing how our different configurations perform under data corruption and adversarial attack. We evaluated our ResNet50 models trained for 270 epochs using a comprehensive data corruption suite called ImageNet-C [19]. For the adversarial white-box attack, we subjected our trained models under Fast Gradient Sign Method (FGSM) [13]. The mean Corruption Error (mCE) in Table 4 is normalized with respect to AlexNet [26] performance on Im-ageNet as proposed by Hendrycks et al. [19]. In Table 4, the configurations that are most robust against data corruption have one method in common -MixUp. This confirms the MixUp robustness study by Zhang et al. [52]. In many cases, AutoAugment and AgMax further improve Table 5. ImageNet-C robustness leaderboard with a ResNet50 backbone. Standalone indicates whether the method is a combination of techniques or a single method. Clean Error is the classification error on the uncorrupted validation set. the corruption robustness of MixUp. The results show that while configurations with CutMix have low generalization errors, they perform poorly in terms of corruption robustness. To get an idea on how the best performing method MixUp+AA+AgMax fares in comparison with algorithms that are optimized for robustness, we borrowed the online leaderboard of Hendrycks et al. [19] as shown in Table 5. The performance of MixUp+AA+AgMax is competitive with the SOTA. also exhibiting robustness against FGSM. On the average, MixUp+AA+AgMax is also the best performing against adversarial attack. Slower Training Normalized to Standard model training time, AgMax incurs a performance penalty of about 1.7× mainly due to the computation of the agreement between 2 positive samples. Other Agreement Functions Other agreement functions can be used in place of MI. Table 7 shows the comparison among MI, MSE, KLdivergence and CE on CIFAR100. In self-supervised learning, BYOL uses MSE while CE is the loss function in DINO. KL-divergence is also utilized since it is a good distance measure between two probability distributions. Table 7 shows that MSE is the best performing agreement function on Standard, CutOut, CutMix and AA. KL excels on MixUp. MI is the top performing method on MixUp+AA and CutMix+AA and has the highest overall performance. Table 8 shows that an ablation study on ResNet50, comparable results among MI, MSE and KL on MixUp and Cut-Mix are observed. However, KL has a significantly lower performance on both MixUp+AA and CutMix+AA. Table 9 shows that other SOTA models benefit from Ag-Max. Large model such as ResNet101 gains +3.1% with CutMix+AA+AgMax. For RegNet [35] and EfficientNet [43], the use of AgMax improves the model accuracy. How- ever, unlike in previous models, CutMix and AA perform poorly. Other SOTA Models Object Detection and Instance Segmentation Using the MMDetection framework [3], the performance of our pre-trained ResNet50 models on object detection and instance segmentation tasks on both PascalVOC [9,8] and MS COCO datasets [31] can be evaluated. Table 10 shows that the backbone models pre-trained with AgMax consistently outperforms models without it. Similar to the results in robustness, models pre-trained with MixUp exhibit the best results in both detection and segmentation. Training Faster R-CNN [36] and Mask R-CNN [16] models uses the default MMDetection configurations with gradient clipping and 1× schedule. Conclusion AgMax is a simple regularization technique that maximizes the agreement between the predictions of two positive samples. Empirical results demonstrated significant gains in performance on classification, object detection and segmentation. Acknowledgement This work was funded by the UP ECWRG 2019-2020, CHED-PCARI and ERDT-FRDG. Thanks to CNL and PCARI-PRIME: Roel Ocampo and Vladimir Zurbano, for server hosting. Figure 3 . 3Top-1% accuracy of different regularizer configurations on ImageNet using ResNet50 trained for 90 epochs. trained VGG11 and LeNet5 on Speech Commands Dataset as done in MixUp. carefully select a recipe of image operations to generate new input data to minimizeData with Label with Augmentation Baseline Smoothing AgMax Standard 76.4 ± 0.1 76.8 ± 0.1 76.9 ± 0.1 CutOut Table 1 . 1Top-1% accuracy of ResNet50 trained for 90 epochs on ImageNet using different data augmentation methods with Label Smoothing or AgMax. Standard data augmentation is random hor- izontal flipping, color jitter and lighting. Table 3. Evaluation landscape showing Top-1% model accuracy of different data augmentation configurations on Speech Commands Dataset. AutoAugment is not included since there is no publicly available policy for Speech Commands Dataset.Speech Commands [47] LeNet5 [28] VGG11 [39] Data 30 epochs Augmentation test val test val Published Results Standard 89.7 90.2 95.4 95.0 CutOut[7] - - - - MixUp[51] 89.2 89.9 96.6 96.1 CutMix[49] - - - - without AgMax (Reproduced Results) Standard 89.9 90.0 96.3 96.0 CutOut[7] 89.0 89.2 96.5 96.1 MixUp[51] 89.4 89.4 96.5 96.2 CutMix[49] 87.1 87.5 96.4 96.2 with AgMax Standard 90.2(0.3) 90.0(0.0) 96.4(0.1) 96.1(0.1) CutOut 90.4(1.4) 90.0(0.8) 96.5(0.0) 96.1(0.0) MixUp 89.4(0.0) 89.6(0.2) 96.8(0.3) 96.3(0.1) CutMix 88.8(1.7) 89.3(1.8) 96.7(0.3) 96.4(0.2) Gauss Shot Impulse Defocus Glass Motion Zoom Snow Frost Fog Bright Contrast Elastic Pixel JPEGData Noise↓ Blur↓ Weather↓ Digital↓ Augmentation mCE↓ without AgMax Standard 74.8 71.8 73.3 76.5 79.2 91.0 82.3 80.9 74.3 73.0 61.6 57.8 65.2 89.1 71.7 73.9 CutOut 75.2 74.3 76.4 80.8 77.3 91.2 79.1 79.8 75.5 74.0 63.3 57.1 64.7 87.9 73.4 73.8 MixUp 69.9 65.0 68.7 70.1 76.8 90.2 78.7 77.6 67.6 62.7 54.9 53.9 54.5 87.6 68.7 71.9 CutMix 75.0 74.5 76.8 81.8 79.0 92.3 81.2 78.2 73.5 73.8 62.0 55.9 64.8 88.9 68.9 73.9 AA 72.6 66.6 67.1 71.5 77.6 88.8 78.1 83.2 72.9 73.2 60.3 54.2 60.2 91.0 71.9 72.5 CutOut+AA 72.2 67.6 68.6 72.0 75.4 90.0 78.6 81.8 73.2 74.0 59.3 54.2 58.6 92.6 68.3 68.5 MixUp+AA 67.6 59.8 60.6 63.8 74.9 86.9 74.9 78.2 66.4 64.7 53.3 52.3 53.3 88.6 69.0 67.1 CutMix+AA 72.2 66.4 67.0 72.5 76.7 91.5 77.5 78.1 73.3 72.8 58.8 53.3 60.1 91.1 71.7 72.3 with AgMax Standard 74.9 72.4 74.6 77.9 78.5 89.9 81.3 82.4 75.0 73.3 62.3 56.3 64.3 89.7 70.8 74.3 CutOut 74.9 72.7 75.1 78.6 78.4 91.8 79.5 79.9 74.5 74.5 61.7 55.8 63.6 89.5 74.6 73.6 MixUp 69.9 65.0 68.7 69.0 76.8 90.4 77.8 77.3 67.8 61.2 54.9 54.4 56.3 89.0 67.9 72.1 CutMix 75.3 74.5 76.7 80.4 78.2 91.7 79.0 77.3 74.4 74.2 62.0 56.2 64.8 89.6 74.9 76.1 AA 73.7 68.7 69.6 73.9 79.3 88.7 79.4 83.5 73.8 73.3 62.2 55.5 59.4 92.5 72.0 74.3 CutOut+AA 73.3 68.4 69.0 74.6 77.1 90.6 78.2 83.6 75.0 74.5 61.3 54.1 58.8 92.5 71.7 70.8 MixUp+AA 67.1 57.2 58.5 60.8 75.8 89.6 76.2 78.7 64.4 60.2 52.1 51.9 52.5 90.4 69.1 69.8 CutMix+AA 72.9 70.8 71.5 75.4 75.8 91.9 77.8 78.6 72.2 72.9 58.1 52.8 58.9 90.0 74.0 72.9 Table 4. Evaluation landscape showing corruption robustness of ResNet50 model trained for 270 epochs using different regularizer config- urations. mCE is Top-1% mean Corruption Error. Stand- Clean Method Year alone? mCE↓ Error↓ DeepAugment+AugMix[18] 2020 No 53.6 24.2 Assemble-ResNet50[29] 2020 No 56.5 17.9 ANT (3 × 3)[37] 2020 Yes 63.0 23.9 BlurAfterConv [46] 2020 Yes 64.9 21.2 AugMix[20] 2020 Yes 65.3 22.5 MixUp+AA+AgMax - No 67.1 21.4 Stylized ImageNet[11] 2019 Yes 69.3 25.1 Patch Uniform[32] 2019 Yes 74.3 24.5 Baseline - N/A 76.7 23.8 Table 6 6shows that increasing the adversary strength on the validation set, AgMax generally improves the robustness of all pre-trained models by a wide margin. Similar to corruption robustness results, configurations that have good performance have MixUp in common. CutMix isTable 6. Pre-trained (270 epochs) ResNet50 model Top-1% accuracy after FGSM attack with increasing strength, .Data FGSM↑ Augmentation = 0.1 = 0.3 = 0.5 without AgMax Standard 24.9 13.4 8.0 CutOut 24.6 12.8 7.4 MixUp 31.8 21.0 15.1 CutMix 34.5 20.4 13.0 AA 29.3 20.4 15.3 CutOut+AA 28.9 20.2 14.4 MixUp+AA 35.0 26.4 20.7 CutMix+AA 37.0 27.2 20.3 with AgMax Standard 28.5 16.6 10.5 CutOut 26.2 14.3 9.0 MixUp 35.0 23.4 16.4 CutMix 37.5 22.3 13.5 AA 28.1 19.9 14.9 CutOut+AA 32.0 22.1 14.4 MixUp+AA 35.6 28.1 23.5 CutMix+AA 37.7 28.0 21.4 Table 7 . 777.5 77.3 77.4 79.2 80.0 78.9 79.2 79.5 78.7 79.7 79.3 80.0 80.1 80.0 80.0 79.6 79.0 79.0 81.1 81.2 81.1 80.9 80.7 80.1 80.5 80.5 81.3 80.8 81.2 80.9 Top-1% accuracy on CIFAR100 of WideResNet40-2 trained for 200 epochs using different agreement functions: MI, MSE, KL-divergence and cross-entropy (CE). MSE KL MI MSE KL MI MSE KL MI MSE KL 77.5 77.4 77.7 77.4 77.5 77.6 77.1 76.9 76.1 77.0 76.6 74.5Table 8. Top-1% accuracy on ImageNet of ResNet50 trained for 90 epochs using different agreement functions: MI, MSE and KLdivergence.Table 9. Top-1% accuracy on ImageNet using ResNet101 (RN101) for 270 epochs, RegNetX200MF (RX200), RegNetY400MF (RY400) and EfficientNet-B0 (EfNB0) all for 100 epochs as done in[35]. Legend: Standard (Stan), AgMax-MI (AM), AutoAugment (AA) and CutMix (CM). EfNB0-AM is AgMax-MSE. to Standard model, CutMix, MixUp, CutOut and AutoAugment have minimal impact on the training time.Standard CutOut MixUp CutMix MI MSE KL CE MI MSE KL CE MI MSE KL CE MI MSE KL CE 77.4 AA CutOut+AA MixUp+AA CutMix+AA MI MSE KL CE MI MSE KL CE MI MSE KL CE MI MSE KL CE 79.2 MixUp CutMix MixUp+AA CutMix+AA MI Model Stan AM CM CM+AA CM+AA+AM RN101 78.1 79.2 79.8 80.7 81.2 RX200 68.8 70.1 65.6 63.5 65.5 RY400 74.3 75.0 71.4 70.6 71.8 EfNB0 75.3 75.7 72.3 71.7 71.9 Relative Table 10. mAP on object detection and segmentation tasks using ResNet50-FPN backbone trained for 270 epochs with Top-1% accuracy indicated for reference. CO is CutOut. MU is MixUp. CM is CutMix. 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{'abstract': 'Data augmentation reduces the generalization error by forcing a model to learn invariant representations given different transformations of the input image. In computer vision, on top of the standard image processing functions, data augmentation techniques based on regional dropout such as CutOut, MixUp, and CutMix and policy-based selection such as AutoAugment demonstrated state-of-the-art (SOTA) results. With an increasing number of data augmentation algorithms being proposed, the focus is always on optimizing the input-output mapping while not realizing that there might be an untapped value in the transformed images with the same label. We hypothesize that by forcing the representations of two transformations to agree, we can further reduce the model generalization error. We call our proposed method Agreement Maximization or simply AgMax. With this simple constraint applied during training, empirical results show that data augmentation algorithms can further improve the classification accuracy of ResNet50 on ImageNet by up to 1.5%, WideResNet40-2 on CIFAR10 by up to 0.7%, WideResNet40-2 on CIFAR100 by up to 1.6%, and LeNet5 on Speech Commands Dataset by up to 1.4%. Experimental results further show that unlike other regularization terms such as label smoothing, AgMax can take advantage of the data augmentation to consistently improve model generalization by a significant margin. On downstream tasks such as object detection and segmentation on PascalVOC and COCO, AgMax pre-trained models outperforms other data augmentation methods by as much as 1.0mAP (box) and 0.5mAP (mask). Code is available at https://github.com/roatienza/agmax.', 'arxivid': '2110.10536', 'author': ['Rowel Atienza rowel@eee.upd.edu.ph \nUniversity of the Philippines Electrical and Electronics Engineering Institute\n1101Diliman, Quezon CityPhilippines\n'], 'authoraffiliation': ['University of the Philippines Electrical and Electronics Engineering Institute\n1101Diliman, Quezon CityPhilippines'], 'corpusid': 239049853, 'doi': '10.1109/wacv51458.2022.00398', 'github_urls': ['https://github.com/roatienza/agmax.'], 'n_tokens_mistral': 18221, 'n_tokens_neox': 15373, 'n_words': 8020, 'pdfsha': '9d7a6d0d92ec1ae3d84d7eaaa8c7bad13452de6b', 'pdfurls': ['https://arxiv.org/pdf/2110.10536v1.pdf'], 'title': ['Improving Model Generalization by Agreement of Learned Representations from Data Augmentation', 'Improving Model Generalization by Agreement of Learned Representations from Data Augmentation'], 'venue': []}

arxiv

Effect of charge doping on the electronic structure, orbital polarization, and structural distortion in nickelate superlattice 4 Jun 2013 Heung-Sik Kim Department of Physics Korean Advanced Institute of Science and Technology 305-701DaejeonKorea Myung Joon Han Department of Physics Korean Advanced Institute of Science and Technology 305-701DaejeonKorea KAIST Institute for the NanoCentury KAIST 305-701DaejeonKorea Effect of charge doping on the electronic structure, orbital polarization, and structural distortion in nickelate superlattice 4 Jun 2013 Using first-principles density functional theory calculations, we investigated the effect of charge doping in a LaNiO 3 /SrTiO 3 superlattice. The detailed analysis based on two different simulation methods for doping clearly shows that the electronic and structural properties change in a systematic way that the orbital polarization (i.e. relative occupation of two Ni-e g orbitals) is reduced and the Ni to apical oxygen distance enlarged as the number of doped electrons increases. Also, the rotation angles of the NiO 6 /TiO 6 octahedra strongly and systematically depend on the doping so that the angle γ gradually decreases whereas the α and β increase as a function of electron doping. Further analysis shows that the electron (hole) doping can play a similar role with the compressive (tensile) strain for the octahedral rotations. Our results not only suggest a possible way to control the orbital and structural property by means of charge doping, but also provide useful information to understand the experiments under various doping situations such as oxygen vacancy. PACS numbers: 75.25.Dk, 75.70.Cn, 73.21.Cd 1 arXiv:1306.0713v1 [cond-mat.str-el] I. INTRODUCTION Recent advances in the atomic-scale growth technique of transition metal oxide (TMO) heterostructures have created considerable research interest [1,2]. In TMO, multiple degrees of freedom (i.e., charge, spin, orbital, lattice) are coupled to each other, often creating novel material characteristics such as high-temperature superconductivity and colossal magneto resistence [3]. By making artificial heterostructures of TMO, it is possible to control those degrees of freedom and band structures, and therefore create or design new 'correlated electron' properties. Previous TMO superlattice studies [4][5][6][7][8] have shown that many unexpected material phenomena can be realized at the TMO heterointerface, such as magnetism and superconductivity [9][10][11][12][13]. Combined with other degrees of freedom in TMO, charge doping can play a significant role in determining material properties. Sometimes extra charges are introduced in an unexpected and uncontrolled way. For example, oxygen vacancy often drives a TMO system to have fairly different material characteristics (e.g., an insulating material to be metallic) [14][15][16]. Cation inter-mixing can also be important as it introduces a different local ionic potential to the nearby atoms (e.g., inter-mixing of Sr 2+ and La 3+ ). On the other hand, it is also possible to control the amount of extra charges by chemical doping or electric field, for example. Considering all these possibilities, it is important to understand the effect of charge doping in TMO heterostructures. In particular, the relation between rotation of metal-oxygen octahedra and charge doping has never been investigated in a systematic way. In this paper, we examine the effect of doping on nickelate superlattices which are being actively studied nowadays [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Taking LaNiO 3 /SrTiO 3 as a prototype example [28][29][30][31], we performed first-principles density functional theory (DFT) calculations and examined the electronic structure, orbital polarization, magnetism, and structural distortion as a function of doping. To simulate charge doping, two different approaches were adapted, namely, the rigid band shift method and the supercell calculation with a couple of Sr 2+ /La 3+ ratios. The systematic changes are found in these physical properties: As the number of extra electrons increases, the orbital polarization (i.e., the relative occupation of d x 2 −y 2 to d 3z 2 −r 2 ) is reduced, and the Ni to apical oxygen distance enhanced. The octahedra rotation angles also exhibit a systematic dependence on charge doping. Our results suggest a possibility to control the structural property as well as the electronic structure by doping, and provides useful information to understand the experiments under the various types of doping situations. After presenting computation details in Sec. II, we discuss the doping effect with no oxygen octahedral rotation in Sec. III. In Sec. IV, the possible NiO 6 and TiO 6 cage rotations are taken into account. After discussing further issues in Sec. V, a summary is given in Sec. VI. In the appexdix, we briefly discuss the electronic structure changes as a function of doping, which provides an electronic origin of the systematic changes found in the other physical quantities. II. COMPUTATION DETAILS For the band structure calculations, we used DFT within local spin density approximation (LSDA) [32] and the projector-augmented wave (PAW) method [33] as implemented in the Vienna ab initio simulation package [34]. We adopted a plane-wave energy cutoff of 400 eV with a 5 × 7 × 7 k-point sampling on the Monkhorst-Pack grid. To incorporate the effect of correlations, the so-called simplified version of rotationally invariant LSDA+U as suggested by Dudarev et al. [35] was used with the effective U ≡ U −J varying from 0 to 5 eV while we mainly present the U =3 eV results considering the literature values [21,22,27]. Structural optimization was performed with a force criterion of 1 meV/Å . During this process, a ferromagnetic order, suggested to be the ground state in recent DFT calculations [21,22,27], was set as the initial magnetic configuration. However, we also note the unresolved issue regarding the spin ground state predicted by DFT-based methods [21,25,27]. Fig. 1 Fig. 1(c) and (d) respectively. These two will be denoted as LaSr 3 -and La 3 Sr-cell, respectively, hereafter. Note that, LaSr 3 -and La 3 Sr-cell calculations correspond to the −0.50e/Ni-and +0.50e/Ni-cell, respectively, in terms of electron doping. The orbital polarization, representing the relative occupations in the two Ni-e g orbitals, can be defined as [26]: P eg = n d x 2 −y 2 − n d 3z 2 −r 2 n d x 2 −y 2 + n d 3z 2 −r 2 ,(1)n i = F b f i ( )d ,(2) where we choose b = −3.5 eV (with the Fermi energy F = 0) to capture the occupations on the valence d-orbital complexes. We found that our conclusions were unchanged even when we used the values of b down to −10 eV. III. RESULTS WITHOUT OCTAHEDRAL ROTATION In this section, we present the calculation results that are obtained from the structures with no rotational distortion. In these calculations, due to the high symmetry of the initial geometry, the undistorted structures are maintained after the relaxation process. The results are meaningful to understand the system in further details and can be relevant to the experimental situation in which the rotational distortion modes are suppressed for some reason, such as substrate strain, although a recent study of LNO/STO indicates the possible rotations [30]. This feature is presumably attributted to the U -dependence of d 3z 2 −r 2 -orbital occupations which will be discussed further in the following subsection. A. Structural changes B. Electronic structure, orbital polarization, and magnetism First we note that the doped charges reside mostly in the Ni-e g bands and the empty Ti-t 2g is located well above ∼2 eV. In case of La 3 Sr-/LaSr 3 -cell calculations, it is found that a small fraction of Ti-t 2g bands touches the Fermi level at U =0 and is pushed far away by U Ni . From the shape of Ni-e g projected density of states (PDOS), schematically shown in Fig. 3(a), it is expected that a major portion of the doped charges will go into d 3z 2 −r 2 orbitals and the orbital polarization changes accordingly (for the full details of the PDOS, see Fig. 6 in Appendix). As clearly seen in Fig. 3(b), the orbital polarization gradually decreases as the number of doped electron increases for both U = 0 and 3 eV. This feature is also confirmed by the La 3 Sr-/LaSr 3 -cell calculations (open symbols in Fig. 3(b)) and can be understood well from the PDOS features in Fig. 3(a). Notably, the calculated orbital polarization is similar for both cases despite the differences in the two computation methods. The PDOS calculated by the two computation methods are found to be quite similar as shown in Appendix (Fig. 6). This behavior of orbital polarization can be consistent with the result of d Ni−Oap in the sense that the more occupations in d 3z 2 −r 2 (i.e., the smaller P eg ) corresponds to the longer the Jahn-Teller-or breathing-type distortion of NiO 6 octahedra is obtained in some cases. A. Structural changes The rotation pattern of NiO 6 and TiO 6 octahedra also exhibits a systematic dependence on charge doping. Fig. 4(a) shows the evolution of NiO 6 rotation angles as a function of doping (U = 0 eV). In the case of uniform background doping (filled symbols), the antiferrodistortive angle γ (see the inset of Fig. 4(b)) is gradually reduced from −6 [37]. Our results suggest that, due to the strong charge-orbital-lattice coupling, not only epitaxial strain but also doping can be used to control the octahedral rotation pattern in perovskite superlattices or thin films. While the systematic trend of rotation is maintained in the finite U calculations, the rotation angles are significantly enhanced as clearly seen in Fig. 4(b). Also, further structural changes are introduced. In the undoped case, the Jahn-Teller type distortion of NiO 6 is stabilized (Fig. 4(c)). The long (a l ) and short (a s ) Ni-O bond lengths are presented in Table I. At the doping level between 0.0 and +0.25e/Ni, the structural transition from (a 0 a 0 c − )-to (a − a − c + )-phase and the breathing-type distortion is found to occur (Fig. 4(d)) so that the two Ni sites become inequivalent. The Ni-O bond lengths for these two different Ni-sites (denoted as Ni (1) and Ni (2)) are presented in Table I. Simultaneously with the charge disproportionation, the d Ni−Oap / d Ni−O in -ratio is significantly enhanced for Ni (2) (see Fig. 5(a)). Owing to the large d Ni−Oap for both Ni(1) and Ni (2), the (a 0 a 0 c − )-type rotational pattern becomes unstable and the tilting pattern of the (a − a − c + )-type is stabilized in response to the out-of-plane lattice mismatch and the 'effective' compressive strain. B. Electronic structure, orbital polarization, and magnetism The decreasing feature of P eg discussed in Sec. III ( without octahedral rotation) is found to be enhanced by rotational distortion. Fig. 5(b) summarizes our calculation results, where Ni (1) and Ni(2) are distinguished due to the charge disproportionation. P eg decreases as the number of doped electron increases for both U =0 and 3 eV, indicating that the extra electrons mainly occupy the d 3z 2 −r 2 orbital, rather than d x 2 −y 2 . This feature can also be found in PDOS (see Appendix, Fig. 8). While the general decreasing trend of P eg is found for both Ni(1) and Ni (2), some deviation is also noted at the doping level higher than +0.25e ( Fig. 5(b)). The PDOS analysis shows that, as more electrons are introduced, the Ni(2)-d x 2 −y 2 occupation is reduced, while the two e g orbitals of Ni(1) are occupied with equal amounts of electrons (see Appendix, Fig. 8). This feature is also reflected in the Fig. 5(a)), P eg (Fig. 5(b)), and magnetizations (Fig. 5(c)). d Ni−Oap / d Ni−O in -ratio ( Note that magnetism occurs even in U = 0 eV, possibly due to the d have a local magnetic moment [17] while the ordering type is still not clearly resolved [21,25,27]. Our result seems to support the idea that the moment is related to the enhanced Fig. 4 are used for a l , a 1 , a s , a 2 the metal-oxygen-metal network. However, we note that the two different approaches, which incorporate the charge doping in our calculations, produce consistent results regarding the change of the electronic structure, orbital polarization and structural property. Also, the orbital polarization is insensitive to some degrees of structural difference. These findings strongly suggest that the overall conclusions presented in this study are quite relevant to various doping situations in experiments, in spite of the limitation of simulation methods. We emphaize that our results can provide useful information to understand the experiment. For example, the further distortion of rotated oxygen octahedra caused by doping implies that more oxygen vacancies are not necessarily leading the system to be more metallic because further rotation can simulatneously make the system be less metallic due to the enlarged effective U/t parameter. Also, our prediction of polarization dependence as a function of electron doping can be tested in experiments, for example by changing the oxygen partial pressure in the pulsed laser deposition process. In this Appendix, we present the electronic structure change as a function of doping, which provides further information to understand the doping dependence of our system and is closely related to the other physical quantities discussed above. Fig. 6 shows the PDOS in the structure with no octahedral rotation. First of all, we note that the electronic structure difference between rigid band shift and LaSr 3 /La 3 Sr-cell calculation is not significant. This point also holds for the results of rotated structures ( Fig. 7 and Fig. 8). It is therefore consistent with our finding that two different approaches predict the same features regarding the orbital occupation and structural properties as discussed above. The overall shape of the Ni-e g PDOS is actually consistent with the schematic picture in Fig. 2(a), vindicating our discussion in Sec. III.B based on this picture. IV). Compared to the rotation-free results, due to the rotations, additional splittings are introduced in the Ni-e g states as clearly shown in Fig. 8. Also, in the presence of U , electron doping induces charge disproportionation (see the fifth, sixth, and seventh rows of Fig. 8(b) and (c)). From the PDOS results and the data in Fig. 5, it can be suggested that Ni(1) and Ni(2) (Fig. 8(b) and (c)) move closer to the d 8 -and d 7 -configuration in the electron-doped regime, respectively. (1) and Ni (2), respectively. In this case, the out-of-plane c-lattice parameters are also different and denoted as c 1 and c 2 , corresponding to Ni(1) and Ni (2) Fig. 1 1shows the unit cell structure of the (LaNiO 3 ) 1 /(SrTiO 3 ) 1 superlattice used in this study. As shown in Fig. 2 ( 2a)-(e) summarizes the structural change induced by doping. The main features are as follows: (i) The distance between Ni and apical oxygen (d Ni−Oap ) increases as the number of doped electrons increases. As clearly shown in Fig. 2(e), this increasing trend is evident for both U = 0 and 3 eV. It is also noted that both the LaSr 3 /La 3 Sr-cell and rigid band shift calculations predict the consistent results regarding this increasing trend. In the case of LaSr 3 -/La 3 Sr-cell calculations, due to the lowered symmetry along the outof-plane direction, two inequivalent d Ni−Oap distances are obtained, corresponding to two open symbols at the same doping level of ±0.5e in Fig. 2(e). The calculated values of d Ni−Oap by LaSr 3 -/La 3 Sr-cell deviate from the corresponding values of the rigid band shift by ∼0.01-0.04Å (compare the filled and open symbols at ±0.5e), which reflects the intrinsic difference between the two computation methods. (ii) As schematically shown in Fig. 2(a), La and Sr cations move toward the NiO 2 plane as more electrons are introduced. From the electrostatic point of view, the excess electrons mainly doped into Ni-e g orbitals create an attractive force for La 3+ and Sr 2+ , which can be responsible for these movements. (iii) In LaSr 3 /La 3 Sr-cell calculations, the lowered lattice symmetry allows the oxygen atoms in the NiO 2 plane to deviate from their original positions. The in-plane and out-of-plane oxygen displacements are illustrated in Fig. 2(c) and (d), respectively, and these displacements are ∼0.05-0.09Å .The additional ionic movement caused by U is found to be a few percent. That is, d Ni−Oap is slightly enhanced further by U . As shown in the inset ofFig. 2(e), d Ni−Oap in the case of +0.50e/Ni can be increased up to ∼2%. It is interesting to note that, at a fixed doping level of +0.5e/Ni, d Ni−Oap gradually increases as U increases (with some deviation at 5 eV). d Ni−Oap from the point of view of Coulomb repulsion between the apical oxygen p z and Nid 3z 2 −r 2 electrons. Note that the lattice degree of freedom couples to the orbital occupancy and polarization so that by electron (hole) doping, d Ni−Oap increases (decreases) and P eg is enhanced (reduced). Another intriguing feature is the behavior of magnetization of e gorbitals upon doping shown inFig. 3(c). While the paramagnetic solution is always stable in U = 0 calculations, in the intermetiate strength of 2 < U ≤ 3 eV, Ni magnetic moment can be induced either in the electron-doped (U = 2 and 3 eV) or in the hole-doped regime (U = 3 eV). The schematic shape of PDOS suggests, in this range of U , that the Stoner mechanism may be responsible for the magnetism as the PDOS peaks of d 3z 2 −r 2 and d x 2 −y 2 state get closer to the Fermi level. Note that, at higher U =5 eV such doping-dependent behavior disappears and the magnetization is gradually enhanced with more doped electrons.IV. RESULTS WITH OCTAHEDRAL ROTATIONIn this section, the rotational degree of freedom is taken into account. In combination with it, the doping effect on the structural, electronic, orbital, and magnetic properties is examined. As the starting configurations for the structural optimization, we adopted four different unit cells having (a 0 a 0 c − )-, (a − a − c 0 )-, and (a − a − c ± )-type rotations (following Glazer notation as defined in Ref.22, 36, and 37, where the positive and negative signs mean the ferro-and antiferro-distortive rotation, respectively) and performed the relaxation calculations with the symmetry enforced. The structures obtained by this process are further optimized with the symmetry constraint turned off. Without the constraint, the oxygen cage shape can deviate from the ideal one. Therefore, we present the averaged values of the bond lengths and the rotation angles. Also, owing to the removal of the structural constraints, • to − 2 2• as the electron doping level increases while the angle α and β are enhanced from −2 • to −4 • . A similar feature is found in the LaSr 3 -/La 3 Sr-cell calculation (open symbols) although the tendency is less clear because we have only two points. In the hole-doped region (from −0.50e to 0.00e), α and β are basically unchanged while γ exhibits the same decreasing feature as in the electron-doped region (from 0.00e to 0.50e). To understand this rotation pattern we performed the analysis on the distance between Ni and in-plane oxygen (d Ni−O in ) relative to d Ni−Oap . As summarized in Fig. 5(a) it was found that the d Ni−Oap /d Ni−O in ratio gradually increases as the electron doping level increases. In the sense that the larger d Ni−Oap /d Ni−O in ratio corresponds to a more compressively strained situation, the electron (hole) doping plays a similar role with the compressive (tensile) strain. Therefore, the rotation pattern can be understood as an adaptation of the NiO 6 and TiO 6 octahedra in response to this effective strain that comes in due to the change of the d Ni−Oap / d Ni−O in -ratio under the fixed lattice constants, which is consistent with previous studies of d 0 -perovskite systems under real strain x 2 −y 2 22bandwidth narrowing (see Appendix, Fig. 8) induced by the octahedral rotation. The spin order is antiferromagnetic at +0.50e. Not surprisingly, the finite U calculation also predicts the magnetic solution. As reported recently by Boris et al., LaNiO 3 -based superlattices can d 3z 2 2−r 2 component at the Fermi level in the electron-doped case combined with the narrowed bandwidth. The fact that doping can induce the local magnetic moment add an interesting aspect to the discussion regarding the metal-insulator and magnetic transition in rare-earth nickelates [3, 38-45]. V. DISCUSSION The doping simulated by the rigid band shift cannot be same as what happens in real experimental situations such as oxygen vacancy. For example, oxygen vacancy not only introduces effective electron dopings, but also the distortion of local structure by disconnecting Similar doping-induced structural changes may happen in other oxide superlattices with d-orbital degrees of freedom, such as the LaTiO 3 /LaAlO 3 system. One may also speculate the possibility of rich phases from the LaNiO 3 /LaTiO 3 superlattices, where the two independent orbital degrees of freedom from the Ti-t 2g and Ni-e g can interact through corner-sharing coupling of the NiO 6 and TiO 6 octahedra. Such materials as well as other relevant systems may have substantial importances to deserve further theoretical investigation.VI. SUMMARYThe effect of charge doping on the electronic, orbital and structural properties in LaNiO 3 /SrTiO 3 has been investigated using first-principles density functional theory calculation in which doping was simulated with two different methods, namely, rigid band shift and the supercell calculation. The results clearly show the systematic dependence of these physical properties on doping. As more electrons are introduced, the orbital polarization is gradually reduced and the Ni to apical oxygen distance increases. These features are found in both structural phases with and without octahedral rotations. Remarkably, the rotation angles of the NiO 6 /TiO 6 octahedra are also found to strongly depend on doping so that the angle γ gradually decreases, whereas α and β increase as a function of electrondoping. Further analysis shows that the electron (hole) doping can play a similar role with the compressive (tensile) strain for the octahedral rotations. Our results suggest a possible way to control the orbital and structural property by means of charge doping and provide useful information to understand the experiments under various doping situations, such as oxygen vacancy. ACKNOWLEDGMENTS MJH thanks Michel van Veenendaal for fruitful discussion. This work was supported by the National Institute of Supercomputing and Networking / Korea Institue of Science and Technology Information with supercomputing resources including technical support (KSC-2013-C2-005). Appendix A: Projected density of states (PDOS) Fig. 7 7and 8 show the PDOS from the structures with octahedral rotations in which the Jahn-Teller or breathing type distortion is incorporated (main results discussed in Sec. FIG. 1 . 1(Color online) (a) The atomic structures used in the calculations. The black thin lines show the unit cell. (b)-(d) The arrangement of La and Sr ions are shown in (b) (LaNiO 3 ) 1 /(SrTiO 3 ) 1 , (c) (La 0.5 NiO 3 ) 1 /(Sr 1.5 TiO 3 ) 1 , (LaSr 3 -cell) and (d) (La 1.5 NiO 3 ) 1 /(Sr 0.5 TiO 3 ) 1 (La 3 Sr-cell) FIG. 2 .FIG. 3 . 23(Color online) Schematic pictures illustrating the displacement of (a) La, Sr and (b) O atoms upon electron doping in the case of rigid band shift calculations. (c) In-plane and (d) out-of plane displacement of the oxygen ions in the case of LaSr 3 /La 3 Sr-cells. (e) The calculated d Ni−Oap as a function of doping. The circles and squares represent the results of U =0 and 3 eV calculations, respectively. Filled and empty symbols correspond to the results from rigid band shift and LaSr 3 /La 3 Sr-cells, respectively. The inset shows the calculated d Ni−Oap as a function of U in the case of +0.50e/Ni. (Color online) (a) Schematic PDOS of Ni d x 2 −y 2 -and d 3z 2 −r 2 -orbital, (b) the calculated orbital polarization, and (c) the magnetic moment (per a Ni atom) as a function of doping. The filled and empty symbols correspond to the results from rigid band shift and LaSr 3 /La 3 Sr-cells, respectively. FIG. 4 . 4(Color online) (a)-(b) The calculated rotation angle changes for (a) U =0 and (b) 3 eV as a function of doping. The filled and open symbols represent rigid band shift and the LaSr 3 /La 3 Srcell calculation, respectively. The angle α, β, γ are defined in the inset of (b). Following the Glazer notation, the positive and negative signs mean the ferro-and antiferro-distortive rotation, respectively. (c)-(d) Schematic figures illustrating (c) the Jahn-Teller-distorted structure (with no doping and U =3 eV) and (d) the breathing type distortion (+0.25 and +0.50 doping case with U =3.0 eV). In (c) the longer and shorter Ni-O bonds are denoted by a l and a s , respectively, and the c-lattice parameter is unchanged. In (d), two inequivalent Ni sites are denoted as Ni FIG. 5 . 5(Color online) The calculation result of (a) d Ni−Oap / d Ni−O in ratio, (b) orbital polarization P eg , and (c) Ni-magnetic moment as a function of electron doping. The filled and open symbols represent the rigid band shift and the LaSr 3 /La 3 Sr-cell calculation, respectively. Note that for the doping level higher than +0.25e, and with U =3 eV, there are two different Ni-sites due to the charge disproportionation. FIG. 6 .FIG. 7 .FIG. 8 . 678(Color online) The change of Ni-d and Ti-d PDOS as a function of doping. (a)-(b) Ni-t 2g , Ni-e g , and Ti-d-orbitals are presented by blue, red, and light blue lines, respectively. (c)-(d) Nid x 2 −y 2 and d 3z 2 −r 2 states are represented by blue and red lines. (a) and (c) are the results with U =0 eV, and (b) and (d) with U =3 eV. The first and last rows show the results from LaSr 3 -and La 3 Sr-cell calculations, respectively. The second, third, and fourth rows correspond to rigid band shift calculation with −0.50e/Ni, 0.00e/Ni, and +0.50e/Ni doping, respectively. (Color online) PDOS of the Ni-t 2g , Ni-e g and Ti-d states in the presence of NiO 6 and TiO 6 octahedra rotations. (a) U =0 (b) U =3 for Ni(1), and (c) U =3 for Ni(2). The first to fifth (last) rows correspond to the results from the LaSr 3 -cell, −0.50e/Ni, 0.00e/Ni, and +0.50e/Ni, and La 3 Sr-cell, respectively. (Color online) PDOS of the Ni-d x 2 −y 2 and d 3z 2 −r 2 states in the presence of NiO 6 and TiO 6 octahedra rotations. (a) U =0 (b) U =3 for Ni(1), and (c) U =3 for Ni(2). The first to fifth (last) rows correspond to the results from the LaSr 3 -cell, −0.50e/Ni, 0.00e/Ni, and +0.50e/Ni, and La 3 Sr-cell, respectively. 732Å. 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{'abstract': 'Using first-principles density functional theory calculations, we investigated the effect of charge doping in a LaNiO 3 /SrTiO 3 superlattice. The detailed analysis based on two different simulation methods for doping clearly shows that the electronic and structural properties change in a systematic way that the orbital polarization (i.e. relative occupation of two Ni-e g orbitals) is reduced and the Ni to apical oxygen distance enlarged as the number of doped electrons increases. Also, the rotation angles of the NiO 6 /TiO 6 octahedra strongly and systematically depend on the doping so that the angle γ gradually decreases whereas the α and β increase as a function of electron doping. Further analysis shows that the electron (hole) doping can play a similar role with the compressive (tensile) strain for the octahedral rotations. Our results not only suggest a possible way to control the orbital and structural property by means of charge doping, but also provide useful information to understand the experiments under various doping situations such as oxygen vacancy. PACS numbers: 75.25.Dk, 75.70.Cn, 73.21.Cd 1 arXiv:1306.0713v1 [cond-mat.str-el]', 'arxivid': '1306.0713', 'author': ['Heung-Sik Kim \nDepartment of Physics\nKorean Advanced Institute of Science and Technology\n305-701DaejeonKorea\n', 'Myung Joon Han \nDepartment of Physics\nKorean Advanced Institute of Science and Technology\n305-701DaejeonKorea\n\nKAIST Institute for the NanoCentury\nKAIST\n305-701DaejeonKorea\n'], 'authoraffiliation': ['Department of Physics\nKorean Advanced Institute of Science and Technology\n305-701DaejeonKorea', 'Department of Physics\nKorean Advanced Institute of Science and Technology\n305-701DaejeonKorea', 'KAIST Institute for the NanoCentury\nKAIST\n305-701DaejeonKorea'], 'corpusid': 118794974, 'doi': '10.1103/physrevb.91.235102', 'github_urls': [], 'n_tokens_mistral': 13036, 'n_tokens_neox': 11202, 'n_words': 6371, 'pdfsha': 'c259e7e345c8e58f7e72e24b49e8bae9e27702e0', 'pdfurls': ['https://arxiv.org/pdf/1306.0713v2.pdf'], 'title': ['Effect of charge doping on the electronic structure, orbital polarization, and structural distortion in nickelate superlattice', 'Effect of charge doping on the electronic structure, orbital polarization, and structural distortion in nickelate superlattice'], 'venue': []}

arxiv

THETA LIFTS AND LOCAL MAASS FORMS 24 Sep 2012 Kathrin Bringmann ANDBen Kane Maryna Viazovska THETA LIFTS AND LOCAL MAASS FORMS 24 Sep 2012 The first two authors and Kohnen have recently introduced a new class of modular objects called locally harmonic Maass forms, which are annihilated almost everywhere by the hyperbolic Laplacian operator. In this paper, we realize these locally harmonic Maass forms as theta lifts of harmonic weak Maass forms. Using the theory of theta lifts, we then construct examples of (non-harmonic) local Maass forms, which are instead eigenfunctions of the hyperbolic Laplacian almost everywhere. Introduction and statement of results In [7], a new class of modular objects was introduced. These functions, known as locally harmonic Maass forms, satisfy negative weight modularity and are annihilated almost everywhere by the hyperbolic Laplacian (see Section 2 for the relevant definitions), mirroring harmonic weak Maass forms. Recent interest in harmonic weak Maass forms initiated with their systematic treatment by Bruinier and Funke [13]. Following their appearance in the theory of mock theta functions due to Zwegers [34], it has been shown that harmonic weak Maass forms have applications ranging from partition theory (for example [2,4,6,9,11]) and Zagier's duality [33] relating "modular objects" of different weights (for example [10]) to derivatives of L-functions (for example [14,15]). They also arise in mathematical physics, as recently evidenced in Eguchi, Ooguri, and Tachikawa's [16] investigation of moonshine for the largest Mathieu group M 24 . The main difference between locally harmonic Maass forms and harmonic weak Maass forms is that there are certain geodesics along which locally harmonic weak Maass forms are not necessarily real analytic and may even exhibit discontinuities. In this paper, we realize the locally harmonic Maass forms studied in [7] as theta lifts of harmonic weak Maass forms. Theta lifts form connections between different types of modular objects and the regularization of Harvey-Moore [19] and Borcherds [3] allow one to extend their definitions to previously divergent theta integrals. In particular, the Shimura lift [26] was realized as a theta lift by Niwa [24]. Borcherds [3] later placed this into the framework of a larger family of theta lifts. Following his work, theta lifts have more recently appeared in a variety of applications including generalized Kac-Moody algebras [18] and the arithmetic of Shimura varieties [15]. To expound upon one example, Katok and Sarnak [21] used theta lifts to relate the central value of the L-series of a Maass cusp form to the Fourier coefficients of corresponding Maass cusp forms under the Shimura lift. This extended a famous result of Waldspurger [30] proving that the central value of the L-function of an integral weight Hecke eigenform is proportional to the square of a coefficient of its half-integral weight counterpart under the Shintani lift. Tunnell [28] later exploited this link to express the central value of the L-function of an elliptic curve in terms of the coefficients of a theta function associated to a Q z := 1 y a|z| 2 + bx + c . Using this notation, Shintani's theta function projected into Kohnen's plus space equals (1.2) Θ(z, τ ) := y −2k v 1 2 D∈Z Q∈Q D Q(z, 1) k e −4πQ 2 z v e 2πiDτ . The function Θ (−z, τ ) transforms like a modular form of weight k + 1 2 in τ and weight 2k in z (see Proposition 3.2 (1)). Integrating the D-th weight k + 1 2 (holomorphic) Poincaré series against Θ yields f k,D . One can use Borcherds's [3] aforementioned regularized version f, g reg of the Petersson inner product (see Section 2 for a definition) to extend the utility of the Shimura lift (realized as Niwa's [24] theta lift) to weak Maass forms. To be more precise, for a weight k + 1 2 weak Maass form H with eigenvalue λ s := s − k 2 − 1 4 1 − s − k 2 − 1 4 under the hyperbolic Laplacian ∆ k+ 1 2 , we define the theta lift Φ k (H)(z) := H, Θ (z, ·) reg . 2 By choosing an appropriate input, this lift leads to the natural generalization (1.3) f k,s,D (z) := Q∈Q D Q(z, 1) −k ϕ s Dy 2 |Q (z, 1)| 2 of f k,D . Here, for 0 < w ≤ 1 and Re(s) ≥ k 2 + 1 4 , using the usual 2 F 1 notation for Gauss's hypergeometric function, we define ϕ s (w) := Γ s + k 2 − 1 4 D k 2 + 1 4 6Γ(2s) (4π) k 2 − 1 4 w s− k 2 − 1 4 2 F 1 s + k 2 − 1 4 , s − k 2 − 1 4 ; 2s; w , which is easily seen to be a constant when s = k 2 + 1 4 . Note that for Re(s) > k 2 + 1 4 , the Euler integral representation of the 2 F 1 (see (4.3)) yields ϕ s (w) = Γ s + k 2 − 1 4 D k 2 + 1 4 w s− k 2 − 1 4 6Γ s + k 2 + 1 4 Γ s − k 2 − 1 4 (4π) k 2 − 1 4 1 0 (1 − t) s+ k 2 − 3 4 t s− k 2 − 5 4 (1 − wt) −s− k 2 + 1 4 dt. In order to obtain the functions f k,s,D , we apply the theta lift Φ k to the D-th Poincaré series P k+ 1 2 ,s,D (see (2.12)) of weight k + 1 2 with eigenvalue λ s under ∆ k+ 1 2 in Kohnen's plus space. In the special case that s = k 2 + 1 4 , this Poincaré series is precisely the classical cuspidal Poincaré series and f k, k 2 + 1 4 ,D is essentially f k,D because ϕ k 2 + 1 4 is a constant. We next show that in general the functions f k,s,D are local Maass forms with exceptional set given by the closed geodesics (1.4) E D := z = x + iy ∈ H : ∃a, b, c ∈ Z, b 2 − 4ac = D, a |z| 2 + bx + c = 0 . Theorem 1.1. Suppose that s ∈ C satisfies Re(s) ≥ k 2 + 1 4 and D > 0 is a discriminant. Then the following hold. (1) The function f k,s,D is a local Maass form of weight 2k and eigenvalue 4λ s under ∆ 2k with exceptional set E D . Moreover, (1.5) f k, k 2 + 1 4 ,D = 2 2k−3 3(2k − 1) (4πD) 3 4 − k 2 f k,D , which is a cusp form. (2) The theta lift Φ k maps weight k + 1 2 weak Maass forms with eigenvalue λ s under ∆ k+ 1 2 to weight 2k local Maass forms with eigenvalue 4λ s under ∆ 2k . In particular, the image of the D-th Poincaré series under the theta lift Φ k equals Φ k P k+ 1 2 ,s,D = f k,s,D . Remark. The function f k,s,D is continuous for every Re(s) ≥ k 2 + 1 4 , but whenever λ s = 0 there exist points along E D along which f k,s,D is not differentiable. In particular, one should note the astonishing fact that while the functions are not differentiable for λ s = 0, the case λ s = 0 yields a (holomorphic) cusp form by (1.5). We now investigate the general properties of the theta lift. Let T p and T 2 p denote the Hecke operators of integral and half-integral weight, respectively (see (2.3) and (2.4)). We next show that the theta lift commutes with the Hecke operators. (1) For every weight k + 1 2 weak Maass form H with eigenvalue λ s with Re(s) ≥ k 2 + 1 4 Φ k (H) 2k T p = Φ k H k+ 1 2 T p 2 . 3 (2) If Re(s) ≥ k 2 + 1 4 and s = k 2 + 1 4 , then the lift Φ k is injective on the space of weak Maass forms with eigenvalue λ s under ∆ k+ 1 2 . We next describe a theta lift which parallels the construction of Shintani [27] and Niwa [24] in negative weight. Define the following theta function (1.6) Θ * (z, τ ) := v k D∈Z Q∈Q D Q z Q(z, 1) k−1 e − 4π|Q(z,1)| 2 v y 2 e −2πiDτ . The function Θ * transforms like a modular form of weight 3 2 − k in τ and weight 2 − 2k in z (see Proposition 3.2 (2)). Similar to the positive weight case, for a weak Maass form H of weight 3 2 − k, we define the theta lift by Φ * 1−k (H)(z) := H, Θ * (−z, ·) reg . Since the space of weak Maass forms is spanned by the Poincaré series P 3 2 −k,s,D (defined in (2.12)), it suffices to consider their image under the theta lifting. This leads to the definition (1.7) F 1−k,s,D (z) := Q∈Q D sgn (Q z ) Q(z, 1) k−1 ϕ * s Dy 2 |Q(z, 1)| 2 , where, for 0 < w ≤ 1 and s ∈ C with Re(s) ≥ k 2 − 3 4 , we define ϕ * s (w) := Γ s + k 2 − 1 4 (4πD) 3 4 − k 2 12 √ πΓ(2s) w k 2 − 3 4 +s 2 F 1 s − k 2 + 1 4 , s + k 2 − 3 4 ; 2s; w . The Euler integral representation (4.3) again implies that ϕ * s (w) = Γ s + k 2 − 1 4 (4πD) 3 4 − k 2 12 √ πΓ s + k 2 − 3 4 Γ s − k 2 + 3 4 w k 2 − 3 4 +s 1 0 t s+ k 2 − 7 4 (1 − t) s− k 2 − 1 4 (1 − wt) −s+ k 2 − 1 4 dt. In the special case that s = k 2 + 1 4 , a change of variables yields the locally harmonic Maass form F 1−k,D (z) := 1 12ψ(1) (4πD) 3 4 − k 2 Q∈Q D sgn (Q z ) Q (z, 1) k−1 ψ Dy 2 |Q (z, 1)| 2 , investigated in [7]. Here ψ (v) := 1 2 β v; k − 1 2 , 1 2 is a special value of the incomplete β-function, which is defined for r, s ∈ C satisfying Re (r), [7], the first two authors and Kohnen introduced the functions F 1−k,D and showed that they transform like weight 2 − 2k modular forms and are locally harmonic in every neighborhood of H which does not intersect E D . More generally, the functions F 1−k,s,D are local Maass forms with exceptional set E D . (2) The theta lift Φ * 1−k maps weight 3 Re (s) > 0 by β (v; s, r) := v 0 u s−1 (1 − u) r−1 du. In(1.8) Φ * 1−k P 3 2 −k,s,D = F 1−k,s,D . Remarks. (1) The functions F 1−k,s,D are never continuous. That is to say, for every s and D satisfying the conditions of Theorem 1.3, there exist points along E D for which F 1−k,s,D exhibits discontinuities. (2) Although F 1−k,D is never continuous, one may add a piecewise polynomial function to obtain a real analytic function. The polynomial in question is related to the period polynomial of f k,D and was thoroughly investigated in [7]. (3) In the omitted case k = 1 and λ s = 0, Hövel [20] has constructed locally harmonic Maass forms via a theta lift. The relationship with the Shimura and Shintani lifts as well as its geometric interpretation were further investigated there. (4) The regularized theta lifts considered here should also have a geometric interpretation. One expects that their images represent cohomology classes of geodesic cycles as currents. We again turn to the general properties of this theta lift. In particular, it also commutes with the Hecke operators. Theorem 1.4. Suppose that s ∈ C satisfies Re(s) ≥ k 2 − 3 4 . The following hold. (1) For every weak Maass form H of weight 3 2 − k in Kohnen's plus space with eigenvalue λ s under ∆ 3 2 −k , one has (1.9) Φ * 1−k (H) 2−2k T p = Φ * 1−k H 3 2 −k T p 2 . (2) The lift Φ * 1−k is injective on the space of weak Maass forms with eigenvalue λ s under ∆ 3 2 −k . Remark. In [7], it was shown that the functions F 1−k,D satisfy relations under the Hecke operators which seemed to imply a natural connection to weight 3 2 − k objects. This is explained by the relation (1.9) between integral and half-integral weight Hecke operators. The images F 1−k,s,D and f k,s,D under the two theta lifts considered in this paper are related through the antiholomorphic differential operator ξ κ := 2iy κ ∂ ∂z . Theorem 1.5. Suppose that k > 0 is an even integer, D is a positive discriminant, and s ∈ C satisfies Re(s) ≥ k 2 + 1 4 . (1) For every z / ∈ E D , we have that (1.10) ξ 2−2k (F 1−k,s,D (z)) = 2 s − 3 4 + k 2 f k,s,D (z). (2) For z / ∈ E D , we have that (1.11) ξ 2k (f k,s,D (z)) = 2 s − k 2 − 1 4 F 1−k,s,D (z). 5 Theorem 1.5 states that for s ≥ k 2 + 1 4 the following commutative diagram holds: P 3 2 −k,s,D ξ 3 2 −k Φ * 1−k / / F 1−k,s,D ξ 2−2k s − 3 4 + k 2 P k+ 1 2 ,s,D 2Φ k / / ξ k+ 1 2 2 s − 3 4 + k 2 f k,s,D ξ 2k −λ s P 3 2 −k,s,D 4Φ * 1−k / / −4λ s F 1−k,s,D Denote the d-th Shimura [26] lift by S d and P κ,D := P κ, k 2 + 1 4 ,D . In the special case that s = k 2 + 1 4 (see Corollary 9 of [23] for the constant multiple of S 1 ), the diagram becomes the following: P 3 2 −k,D ξ 3 2 −k Φ * 1−k / / F 1−k,D ξ 2−2k k − 1 2 P k+ 1 2 ,D 2Φ k 3 −1 2 −k S 1 / / 2 2k−3 3 (4πD) 3 4 − k 2 f k,D Remarks. (1) The above diagram extends work of Bruinier and Funke [13] and Hövel [20] in the case of O(2, 1) to higher weight. (2) By applying (6.1) (used to obtain (1.10)) to s-derivatives of weak Maass forms, one could also obtain links between modular objects known as sesquiharmonic forms [5]. These functions map to weakly holomorphic modular forms under the hyperbolic Laplacian. The paper is organized as follows. In Section 2, we recall the theory of weak Maass forms and give a formal definition of local Maass forms. Section 2.1 is devoted to the properties of the regularized inner product. The modularity properties of the theta functions are enunciated in Section 3, where we derive a number of interrelations between the theta functions through differential operators. The image of Φ k (Theorem 1.1 (2)) is determined in Section 4, while Section 5 is devoted to the image of Φ * 1−k (Theorem 1.3 (2)) and the injectivity of the lift (Theorem 1.4 (2)). In Section 6, Theorem 1.5 is established and the relationship between f k,s,D and F 1−k,s,D is then used to conclude Theorems 1.1 (1) and 1.3 (1). Finally, Section 7 concludes the paper with a discussion of the Hecke operators and the injectivity of Φ k (Theorems 1.2 and 1.4 (1)). Basic facts on weak and local Maass forms In this section, we recall the basic definitions necessary to describe the modular objects and the theta lifts used in this paper. We first define the regularized inner product used in the definitions of Φ and Φ * . In order to understand the relationship between lifts in different spaces, we then define the Hecke operators, which act formally on any translation invariant function. We then recall Kohnen's plus space and weak Maass forms, upon which we apply our theta lifts. The next subsection is devoted to constructing Poincaré series which span these spaces of weak Maass forms. Following this, we give the definition of local Maass forms, which are the focus of this paper. Thoughout this section, κ ∈ 1 2 Z and we set Γ := SL 2 (Z) whenever κ ∈ Z, while Γ := Γ 0 (4) if κ ∈ 1 2 Z \ Z. 2.1. Regularized inner products and Hecke operators. For T > 0, denote the truncated fundamental domain for SL 2 (Z) by (2.1) F T := τ ∈ H : |u| ≤ 1 2 , |τ | ≥ 1, v ≤ T . For a finite index subgroup Γ ⊆ SL 2 (Z) we further define F T (Γ) := γ∈Γ\ SL 2 (Z) γF T . In particular, we set F T (4) := F T (Γ 0 (4)). For two functions G and H satisfying weight κ modularity for the group Γ, we define, whenever the limit exists, the regularized inner product G, H reg := 1 [SL 2 (Z) : Γ] lim T →∞ F T (Γ) G(τ )H(τ )v κ dudv v 2 . We use the following lemma, which follows by standard arguments using Stokes's Theorem. Lemma 2.1. Suppose that F , G : H → C are real analytic functions that satisfy F | 2−κ γ = F and G| κ γ = G for all γ ∈ Γ. Then (2.2) F T (Γ) ξ 2−κ (F (τ )) G(τ ) v κ−2 du dv + F T (Γ) ξ κ (G(τ )) F (τ ) v −κ du dv = − ∂F T (Γ) F (τ )G(τ ) dτ . A number of important operators are Hermitian with respect to the regularized inner product. One such class of operators is the Hecke operators. Suppose that F is a function satisfying weight κ modularity and write its Fourier expansion as F (τ ) = n∈Z a v (n)e 2πinu . If κ ∈ Z (resp. κ ∈ 1 2 Z \ Z), then for a prime p, the Hecke operator T p (resp. T p 2 ) is defined by F κ T p (τ ) := n∈Z a v (pn) + p κ−1 a v n p e 2πinu , (2.3) F κ T p 2 (τ ) := n∈Z a v p 2 n + p κ− 3 2 (−1) κ− 1 2 n p a v (n) + p 2κ−2 a v n p 2 e 2πinu . (2.4) We apply the regularized inner product to (half-integral weight) weak Maass forms, which we define in the following subsection. Weak Maass forms. When κ ∈ 1 2 Z\Z, we are interested in weight κ real analytic modular forms on Γ in Kohnen's plus space. This means that the Fourier expansions are supported on the coefficients n satisfying (−1) κ− 1 2 n ≡ 0, 1 (mod 4). We use pr to denote the projection operator (see Section 2.3 of [22]) into Kohnen's plus space. It is useful to recall that if F is modular in Kohnen's plus space for Γ, then its Fourier expansions at the cusps 0 and 1 2 are determined by the expansion at i∞ (see [22] for a proof in the holomorphic case). Like the Hecke operators, the projection operator pr is Hermitian with respect to the regularized inner product, i.e., (2.5) G pr, H reg = G, H pr reg . The real analytic modular forms of particular interest for this paper are weak Maass forms. A good background reference for weak Maass forms is [13]. Recall that we write τ = u + iv throughout. For κ ∈ 1 2 Z, the weight κ hyperbolic Laplacian is defined by ∆ κ := ∆ κ,τ := −v 2 ∂ 2 ∂u 2 + ∂ 2 ∂v 2 + iκv ∂ ∂u + i ∂ ∂v . It is related to the operator ξ κ = ξ κ,τ := 2iv κ ∂ ∂τ through ∆ κ = −ξ 2−κ • ξ κ . In order to define weak Maass forms, we require (2.6) M κ,s (t) := |t| − κ 2 M κ 2 sgn(t), s− 1 2 (|t|) , where M µ,s− 1 2 is the usual M -Whittaker function. For Re (s ± µ) > 0 and v > 0, we have the integral representation (2.7) M µ,s− 1 2 (v) = v s e v 2 Γ(2s) Γ (s + µ) Γ (s − µ) 1 0 t s+µ−1 (1 − t) s−µ−1 e −vt dt. In the special case that µ = s, we have (2.8) M µ,s− 1 2 (v) = e − v 2 v s . Furthermore, as v → ∞, the Whittaker function satisfies the following asymptotic behavior for µ = s: (2.9) M µ,s− 1 2 (v) ∼ Γ(2s) Γ(s − µ) e v 2 v −µ . We move on to the definition of weak Maass forms. For s ∈ C a weak Maass form F : H → C of weight κ for Γ with eigenvalue λ = s − κ 2 1 − s − κ 2 is a real analytic function satisfying: (1) For every γ ∈ Γ, one has F | κ γ = F , where | κ denotes the usual weight κ slash-operator. (2) One has ∆ κ (F ) = λF. There are analogous conditions at the other cusps of Γ. In the case that κ ∈ 1 2 Z \ Z, we then project the Poincaré series into Kohnen's plus space, defining (2.12) P κ,s,m := P κ,s,Γ 0 (4),m pr . In the special cases that s = 1 − κ 2 or s = κ 2 , the resulting Poincaré series is harmonic. For D = 0, the positive and negative weight Poincaré series are related to each other via (2.13) ξ κ (P κ,s,D ) = s − κ 2 P 2−κ,s,D . Local Maass forms. Mirroring the definition of weak Maass forms, for κ ∈ 2Z, λ ∈ C, and a measure zero set E, we call a function F a weight κ local Maass form with eigenvalue λ and exceptional set E if F satisfies the following: (1) For every γ ∈ SL 2 (Z), one has F| κ γ = F (2) For every τ / ∈ E there exists a neighborhood around τ for which F is real analytic and ∆ κ (F)(τ ) = λF(τ ). (3) For τ ∈ E one has F(τ ) = 1 2 lim r→0 + (F (τ + ir) + F (τ − ir)) . (4) The function F exhibits at most polynomial growth as v → ∞. Examples of locally harmonic Maass forms (those with eigenvalue 0) are given in [7] as "quadratic form Poincaré series." In this paper, we give further examples of local Maass forms via theta lifts. Indefinite theta functions In this section we collect several important properties of the theta functions (1.2) and (1.6). The modularity properties of these indefinite theta functions follow by a result of Vignéras [29]. To state these, we define the Euler operator E := n i=1 w i ∂ ∂w i . As usual, we denote the Gram matrix associated to a nondegenerate quadratic form q on R n by A. The Laplacian associated to q is then defined by ∆ := ∂ ∂w , A −1 ∂ ∂w . Here ·, · denotes the usual inner product on R n . Theorem 3.1 (Vignéras). Suppose that n ∈ N, q is a nondegenerate quadratic form on R n , L ⊂ R n is a lattice on which q takes integer values, and p : R n → C is a function satisfying the following conditions: 9 (i) The function f (w) := p(w)e −2πq(w) times any polynomial of degree at most 2 and all partial derivatives of f of order at most 2 are elements of L 2 (R n ) ∩ L 1 (R n ). (ii) For some λ ∈ Z, the function p satisfies E − ∆ 4π p = λp. Then the indefinite theta function v − λ 2 w∈L p w √ v e 2πiq(w)τ is modular of weight λ + n 2 for Γ 0 (N ) and character χ · χ λ −4 , where N and χ are the level and character of q and χ −4 is the unique primitive Dirichlet character of conductor 4. Remark. Note that the definition of the character given in Vignéras [29] differs to that given by Shimura [26] by a factor of χ λ −4 . We adopt Shimura's notation here. Applying Theorem 3.1 to Θ and Θ * yields their modularity properties (see [8] for details). Proposition 3.2. (1) The function Θ (−z, τ ) transforms like a modular form of weight k + 1 2 in Kohnen's plus space on Γ 0 (4) in τ and weight 2k on SL 2 (Z) in z. (2) The function Θ * transforms like a modular form of weight 3 2 − k in Kohnen's plus space on Γ 0 (4) in τ and weight 2 − 2k on SL 2 (Z) in z. The following lemma is the key relation needed to establish a link between the functions f k,s,D and F 1−k,s,D . The correspondence is formed through a relation betwen the respective differential operators in τ and z on Θ and Θ * , mirroring an important connection formed in [13]. ξ k+ 1 2 ,τ (Θ (z, τ )) = −iy 2−2k ∂ ∂z Θ * (−z, τ ), (3.1) ξ 3 2 −k,τ (Θ * (−z, τ )) = −iy 2k ∂ ∂z Θ (z, τ ) . (3.2) Proof: We first prove (3.1). We compute that ∂ ∂z Θ * (−z, τ ) equals D∈Z Q∈Q D Q(−z, 1) k−1 e − 4πv y 2 |Q(−z,1)| 2 e −2πiDτ ∂ ∂z Q −z − 4πQ −z v ∂ ∂z |Q(−z, 1)| 2 y 2 . We then use (3.3) |Q(z, 1)| 2 = Q 2 z y 2 + Dy 2 and y 2 ∂ ∂z Q −z = i 2 Q(−z, 1) to obtain −iy 2−2k ∂ ∂z Θ * (−z, τ ) = 1 2 y −2k v k D∈Z Q∈Q D Q(−z, 1) k e −4πQ 2 −z v e −2πiDτ 1 − 8πQ 2 −z v . We similarly compute the action of ξ k+ 1 2 ,τ on Θ. A straightforward calculation yields ξ k+ 1 2 ,τ (Θ (z, τ )) = We move on to proving (3.2). Since Q z ∈ R, a direct calculation, mirroring the proof of (3.1) and using (3.4), yields 1 2 y −2k v k D∈Z Q∈Q D Q(z, 1) k e −4πQ 2 z v e −2πiDτ 1 − 8πQ 2 z v .ξ 3 2 −k,τ (Θ * (−z, τ )) = v 1 2 D∈Z Q∈Q D Q z Q(z, 1) k−1 e −4πQ 2 z v e 2πiDτ k − 4πv y 2 |Q(z, 1)| 2 . We next obtain (3.2) by showing that −iy 2k ∂ ∂z Θ (z, τ ) equals − iv 1 2 y 2 D∈Z Q∈Q D Q(z, 1) k−1 e −4πQ 2 z v e 2πiDτ k ∂ ∂z y −2 Q (z, 1) − 8πQ z y −2 Q (z, 1) v ∂ ∂z Q z = v 1 2 D∈Z Q∈Q D Q z Q(z, 1) k−1 e −4πQ 2 z v e 2πiDτ k − 4πv y 2 |Q(z, 1)| 2 , where in the last line we have used y 2 ∂ ∂z Q z = i 2 Q(z, 1) and y 2 ∂ ∂z y −2 Q (z, 1) = iQ z . The following lemma relates the regularized inner products in positive and negative weight through the ξ-operator. Proof: Note that all of the regularized integrals exist, as will be shown in the proofs of Theorem 1.1 (2) and 1.3 (2). We begin with the proof of (3.5) and abbreviate P := P k+ 1 2 ,s,D . By Lemma 2.1, we have ξ k+ 1 2 (P ) , Θ * (−z, ·) reg + P, ξ 3 2 −k (Θ * (−z, ·)) reg = − 1 6 lim T →∞ ∂F T (4) P (τ )Θ * (−z, τ )dτ , provided that the limit exists. Hence our goal is to show that the limit on the right hand side is zero. A standard argument reduces this claim to showing that (3.7) lim T →∞ 1 0 P (u + iT )Θ * (−z, u + iT ) du = 0 as well as vanishing of similar integrals around the other cusps of Γ 0 (4). However, since both P and Θ * are in Kohnen's plus space, the vanishing of the corresponding integrals at the other cusps may be reduced to showing that (3.7) vanishes. In order to prove (3.7), we first recall the growth condition (2.11) and note that Θ * (−z, u + iT ) decays exponentially as T → ∞. Indeed, using (3.3), one can show that for fixed z ∈ H the quadratic form R T = 0, where R T := M k+ 1 2 ,s (4πDT ) T k Q∈Q D Q −z Q(−z, 1) k−1 e − 4π|Q(−z,1)| 2 T y 2 e 2πDT . However, the asymptotic behavior for the Whittaker function coming from (2.8) and (2.9) yields M k+ 1 2 ,s (4πDT ) ≪ k,s,D e 2πDT T −k− 1 2 . Using (3.3), we may hence bound R T ≪ k,s,D T − 1 2 Q∈Q D Q −z Q(−z, 1) k−1 e −4πQ 2 −z T . Since z / ∈ E D (and hence −z / ∈ E D ), Q 2 −z > 0 for every Q ∈ Q D and hence R T exhibits exponential decay as T → ∞. This concludes (3.7), yielding (3.5). The proof of (3.6) follows analogously. Image of the theta lift Φ k In this section, we introduce a spectral parameter in the classical Shintani lift. Proof of Theorem 1.1 (2): In order to compute the regularized inner product, we use a method of Zagier [32]. He defined a regularization which he used for functions which grow at most polynomially, but the method may be extended to the functions of interest here, as we now describe. We first define H T := γ∈SL 2 (Z) γF T = γ∈Γ 0 (4) γF T (4). We first use (2.5) together with the fact that Θ = Θ| pr to compute Then the usual unfolding argument yields P k+ 1 2 ,s,D , Θ(z, ·) reg = 1 6 lim T →∞ Γ∞\H T ψ D,k+ 1 2 (s; τ ) Θ (z, τ )v k+ 1 2 dudv v 2 . We now rewrite H T = τ ∈ H Im(τ ) ≤ T c≥1 a∈Z (a,c)=1 S a c (T ), where S a c (T ) is the disc of radius 1 2c 2 T tangent to the real axis at a c . Hence, we have where Hence, the change of variables τ → γτ , together with the modularity of Θ in Proposition 3.2, yields I 2 (T ) = − 1 6 ∞ T ∞ −∞ Θ (−z, τ ) (cτ + d) k+ 1 2 ψ D,k+ 1 2 (s; γτ ) Im (γτ ) k+ 1 2 dudv v 2 . Using the facts that Im (γτ ) = v |cτ +d| 2 and Θ is translation invariant, the integral may be rewritten as − 1 6 ∞ T 1 0 Θ (−z, τ ) ∞ n=−∞ ψ D,k+ 1 2 (s; τ ) k+ 1 2 γ 1 n 0 1 v k+ 1 2 dudv v 2 . Taking the sum over all a, c with 4 | c > 0, the inner sum precisely evaluates as P k+ 1 2 ,s,Γ 0 (4),D − ψ D,k+ 1 2 (s; τ ) . Comparing the polynomial growth in (2.11) with the exponential decay of Θ (−z, τ ) towards i∞, one concludes that the limit T → ∞ vanishes. A similar argument shows that the contribution to I 2 (T ) coming from 4 ∤ c also vanishes as T → ∞. This yields (1.8). 5. Image of the theta lift Φ * 1−k We next compute the image of Φ * 1−k with the method from Section 4. Proof of Theorem 1.3 (2): Following the argument in the proof of Theorem 1.1 (2), we may reduce the theorem to evaluating (5.1) 1 6 lim T →∞ T 0 1 0 ψ D, 3 2 −k (s; τ ) Θ * (−z, τ )v 3 2 −k dudv v 2 − 1 6 lim T →∞ c≥1 a (mod c) (a,c)=1 S a c (T ) ψ D, 3 2 −k (s; τ ) Θ * (−z, τ )v 3 2 −k dudv v 2 . Using the argument from before, the second summand vanishes. We use (3.3) to rewrite the exponential in the theta series as b 2 − 4ac u + iv 2Q 2 −z + b 2 − 4ac . Therefore, evaluating the integral over u and then making the change of variables Q → Q (as defined before (3.4)), it suffices to compute (5.2) 1 6 (4πD) 1 4 − k 2 Γ(2s) −1 Q∈Q D Q z Q(z, 1) k−1 I Dy 2 |Q(z, 1)| 2 , where I(w) := ∞ 0 M 3 2 −k,s (−v) e v 2 v − 1 2 e −vw −1 dv.I(w) = Γ(2s) Γ s − k 2 + 3 4 Γ s + k 2 − 3 4 1 0 t s+ k 2 − 7 4 (1 − t) s− k 2 − 1 4 ∞ 0 v s+ k 2 − 5 4 e −v t−1+w −1 dvdt = Γ(2s)Γ s + k 2 − 1 4 Γ s − k 2 + 3 4 Γ s + k 2 − 3 4 w s+ k 2 − 1 4 1 0 (1 − t) s+ k 2 − 7 4 t s− k 2 − 1 4 (1 − wt) −s− k 2 + 1 4 dt. We again employ the Euler integral representation (4.3) to show that I(w) = Γ s + k 2 − 1 4 w s+ k 2 − 1 4 2 F 1 s + k 2 − 1 4 , s − k 2 + 3 4 ; 2s; w . We then rewrite the hypergeometric function by using the Euler transform 2 F 1 (A, B; C; w) = (1 − w) C−A−B 2 F 1 (C − A, C − B; C; w) to yield I(w) = Γ s + k 2 − 1 4 (1 − w) − 1 2 w s+ k 2 − 1 4 2 F 1 s − k 2 + 1 4 , s + k 2 − 3 4 ; 2s; w . Finally, we conclude that (5.2) equals (1.8) by using (3.3) to rewrite |Q z | in terms of Dy 2 |Q(z,1)| 2 . We next establish the injectivity of the lift. Proof of Theorem 1.4 (2): Since the Poincaré series P 3 2 −k,s,D span the space of weak Maass forms and are linearly independent (which can be seen by comparing their principal parts), it is enough to show that the functions F 1−k,s,D are linearly independent. This follows by proving that any linear combination F := n j=1 a j F 1−k,s,D j with a j not all zero exhibits discontinuities and is hence nonzero. Comparing the sets E D j of geodesics defined in (1.4) implies the result. Relation between positive and negative weight local Maass forms In this section we relate f k,s,D and F 1−k,s,D . Proof of Theorem 1.5: We prove (1.10) by establishing that for P := P 3 2 −k,s,D and z / ∈ E D , one has (6.1) ξ 2−2k Φ * 1−k (P ) (z) = 2Φ k ξ 3 2 −k (P ) (z). We first use (3.6) and then (3.1) to obtain for z / ∈ E D = iy 2−2k ∂ ∂z P, Θ * (−z, ·) reg . Since (6.3) ξ κ (G(z)) = 2iy κ ∂ ∂z G(z), we conclude (6.1) from (6.2). We now apply Theorem 1.3 (2), (6.1), (2.13), and finally Theorem 1.1 (2) to yield ξ 2−2k (F 1−k,s,D (z)) = ξ 2−2k Φ * 1−k P 3 2 −k,s,D (z) = 2Φ k ξ 3 2 −k P 3 2 −k,s,D This concludes the proof of (1.10). We next prove (1.11). Denoting P := P k+ 1 2 ,s,D , we use (3.5) to conclude that for z / ∈ E D We then employ (3.2) and (6.3) to obtain (6.5) Φ * 1−k ξ k+ 1 2 (P ) (z) = iy 2k ∂ ∂z P, Θ (z, ·) reg = 1 2 ξ 2k (Φ k (P )(z)) . Combining this with Theorem 1.3 (2), (2.13), and Theorem 1.1 (2) yields We are now ready to prove Theorem 1.1 (1) and Theorem 1.3 (1). Proof of Theorem 1.1 (1): Note that Θ (z, τ ) = Θ (−z, −τ ) . s − k 2 − 1 4 F 1−k,s,D (z) = s − k 2 − 1 4 Φ * 1−k P 3 2 −k, Hence f k,s,D is modular of weight 2k by Proposition 3.2. The functions f k,s,D are continuous since for Re(C) > Re(A+B), the hypergeometric function Theorem 1 . 3 . 13Suppose that k is even, D > 0 is a discriminant, and s ∈ C satisfies Re(s) ≥ k 2 − 3 4 . Then the following hold. (1) The function F 1−k,s,D is a local Maass form of weight 2 − 2k with eigenvalue 4λ s under ∆ 2−2k and exceptional set E D . ( 3 ) 3There exist a 1 , . . . , a N ∈ C for whichF (τ ) −N m=1 a m M κ,s (4π sgn(κ)mv) e 2πim sgn(κ)u = O v 1−Re(s)− κ 2 . 8 2. 3 . 83Poincaré series. One builds explicit examples of weak Maass forms by constructing Poincaré series [17]. For m ∈ Z \ {0}, ∞ := {± ( 1 n 0 1 ) : n ∈ Z}, is also an eigenfunction under ∆ κ with the same eigenvalue. Moreover, the space of weight κ weak Maass forms with this eigenvalue is spanned by such Poincaré series. The Poincaré series satisfies the growth condition (2.11) P κ,s,Γ,m (τ ) − ψ sgn(κ)m,κ (s; τ ) = O v 1−Re(s)− κ 2 . Lemma 3 . 3 . 33For every integer k ≥ 1, one has Lemma 3 . 4 . 34Suppose that D > 0 is a discriminant and z / ∈ E D . Then for every s with Re(s) definite on the lattice of all binary quadratic forms Q = [a, b, c] ∈ Q D . After evaluating the integral over u, one reduces (3.7) to showing that lim T →∞ s,D , Θ(z, ·) reg = P k+ 1 2 ,s,Γ 0 (4),D pr, Θ(z, ·) reg = P k+ 1 2 ,s,Γ 0 (4),D , Θ(z, ·) reg . P k+ 1 2 1,s,D , Θ(z, ·) reg = lim T →∞ (I 1 (T ) + I 2 (T )) , =P − P, −iy 2−2k ∂ ∂z Θ * (−z, ·) (τ )Θ * (−z, τ ) 2k (f k,s,D (z)) . Inserting the definition (2.6) of M 3 2 −k,s and the integral representation (2.7) of the M -Whittaker function, we evaluate(5.3) F 1 (A, B; C; w), and hence ϕ s (w), is continuous for w ≤ 1. This implies condition (3). For z / ∈ E D , (1.11) and (1.10) imply that AcknowledgementsThe authors thank Jan Bruinier for suggesting to investigate the connection between F 1−k,D and theta lifts and for fruitful discussion. The authors also thank Jens Funke for helpful comments which aided the exposition.We first consider I 1 (T ). Evaluating the integral over u and using(3.3), we obtain (4.2) lim T →∞where for 0 < w < 1 we defineIn the case that s = k 2 + 1 4 , we insert the definition (2.6) of M k+ 1 2 ,s (v) and then substitute the integral representation (2.7) of the M -Whittaker function when Re(s) > k 2 . The change of variables t → 1 − t yieldsWe then rewrite the integral using the Euler integral representation for 2 F 1 (see(15.3.1)in[1]), given for Re(C) > Re(B) > 0 and |w| < 1 byInserting this into (4.2) shows that limTo conclude (1.8), it remains to show that I 2 (T ) vanishes as T → ∞. We first assume that 4 | c and choose γ = a b c d ∈ Γ 0 (4). A direct calculation shows thatA straightforward calculation shows that f k,s,D (z) grows at most polynomially as y → ∞. Finally, one uses (4.4) and the duplication formula for the Γ-function to conclude (1.5).Remark. The non-differentiability of f k,s,D follows by using (1.11) and then proving that the functions F 1−k,s,D are not continuous. Computational evidence indicates that f k,s,D (z) decays exponentially as y → ∞.Proof of Theorem 1.3 (1): Noting thatProposition 3.2 implies that F 1−k,s,D is modular of weight 2 − 2k. The proof that F 1−k,s,D is an eigenfunction under ∆ 2−2k with eigenvalue 4λ s follows by (1.10) and (1.11) precisely as in the proof of Theorem 1.1 (1).16In order to show condition(3)in the definition of local Maass forms, we first note that ϕ * s (w) is continuous for 0 < w ≤ 1. The locally uniform convergence of the sum allows us to pull the limit r → 0 + of F 1−k,s,D (z ± ir) into each term. DefineBy Lemma 5.1 of[7], there are only finitely many Q ∈ B z . Note that sgn (Q z ) = sgn (Q z±ir ) for r sufficiently small and Q / ∈ B z , while for Q ∈ B z one has sgn (Q z+ir ) = − sgn (Q z−ir ) .Hence, since the terms of F 1−k,s,D (z) with Q ∈ B z vanish,A direct calculation shows that F 1−k,s,D (z) grows at most polynomially as y → ∞.Remark. To show that F 1−k,s,D exhibits discontinuities along the set E D , one computessimilarly as in the proof of Theorem 1.3(1). It is shown to be nonzero by using Gauss's summation formula to conclude that ϕ * s (1) = 0. If D is not a square and Re(s) ≥ k 2 + 1 4 , then computational evidence indicates that F 1−k,s,D is bounded as y → ∞.Hecke operatorsIn this section, we consider the action of the Hecke operators on the theta lifts. Proof of Theorem 1.4 (1): Since the Poincaré series span the space of weight32 − k weak Maass forms, it suffices to compute the action of the Hecke operators on Poincaré series. 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Sarnak, Heegner points, cycles, and Maass forms, Israel J. Math. 84 (1984), 193-227. Modular forms of half-integral weight on Γ0(4). W Kohnen, Math. Ann. 248W. Kohnen, Modular forms of half-integral weight on Γ0(4), Math. Ann. 248 (1980), 249-266. Values of L-series of modular forms at the center of the critical strip. W Kohnen, D Zagier, Invent. Math. 64W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175-198. Modular forms of half integral weight and the integral of certain theta-functions. S Niwa, Nagoya Math. J. 56S. Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Math. J. 56 (1974), 147-161. Einheitliche Begründung der Vollständigkeitssätze für die Poincaréschen Reihen von reeller Dimension bei beliebigen Grenzkreisgruppen von erster Art. H Petersson, Abh. Math. Semin. Univ. Hambg. 14H. Petersson, Einheitliche Begründung der Vollständigkeitssätze für die Poincaréschen Reihen von reeller Dimension bei beliebigen Grenzkreisgruppen von erster Art, Abh. Math. Semin. Univ. Hambg. 14 (1941), 22-60. On modular forms of half integral weight. G Shimura, Ann. of Math. 97G. Shimura, On modular forms of half integral weight, Ann. of Math. 97 (1973), 440-481. On construction of holomorphic cusp forms of half integral weight. T Shintani, Nagoya Math. J. 58T. Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83-126. A classical Diophantine problem and modular forms of weight 3 2. J , Invent. Math. 72J. Tunnell, A classical Diophantine problem and modular forms of weight 3 2 , Invent. Math. 72 (1983), 323-334. M.-F Vignéras, Série theta des formes quadratiques indéfinies in Modular functions of one variable VI. Springer627M.-F. Vignéras, Série theta des formes quadratiques indéfinies in Modular functions of one variable VI, Springer Lecture Notes 627 (1977), 227-239. Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J.-L Waldspurger, J. Math. Pures et Appl. 60J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures et Appl. 60 (1981), 375-484. Modular forms associated to real quadratic fields. D Zagier, Invent. Math. 30D. Zagier, Modular forms associated to real quadratic fields, Invent. Math. 30 (1975), 1-46. The Rankin-Selberg method for automorphic functions which are not of rapid decay. D Zagier, J. Fac. Sci. Tokyo. 28D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Tokyo 28 (1982), 415-438. Traces of singular moduli in Motives, Polylogarithms and Hodge Theory, Part I, International Press Lecture Series. D Zagier, F. Bogomolov and L. KatzarkovInternational PressD. Zagier, Traces of singular moduli in Motives, Polylogarithms and Hodge Theory, Part I, International Press Lecture Series (Eds. F. Bogomolov and L. Katzarkov), International Press (2002), 211-244. Mock theta functions. S Zwegers, Utrecht UniversityPh.D. thesisS. Zwegers, Mock theta functions, Ph.D. thesis, Utrecht University (2002). Weyertal 86-90, 50931 Cologne, Germany E-mail address: kbringma@math.uni-koeln. 86-90de Mathematical Institute50931Cologne, Germany E-mail addressMathematical Institute, University of Cologne ; University of CologneMathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany E-mail address: kbringma@math.uni-koeln.de Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany E-mail address: bkane@math.uni-koeln.de

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{'abstract': 'The first two authors and Kohnen have recently introduced a new class of modular objects called locally harmonic Maass forms, which are annihilated almost everywhere by the hyperbolic Laplacian operator. In this paper, we realize these locally harmonic Maass forms as theta lifts of harmonic weak Maass forms. Using the theory of theta lifts, we then construct examples of (non-harmonic) local Maass forms, which are instead eigenfunctions of the hyperbolic Laplacian almost everywhere.', 'arxivid': '1209.5163', 'author': ['Kathrin Bringmann ', 'ANDBen Kane ', 'Maryna Viazovska '], 'authoraffiliation': [], 'corpusid': 54938624, 'doi': '10.4310/mrl.2013.v20.n2.a2', 'github_urls': [], 'n_tokens_mistral': 16061, 'n_tokens_neox': 13890, 'n_words': 8142, 'pdfsha': '8fff814a3317e667ca613712b8c723915c5e969a', 'pdfurls': ['https://arxiv.org/pdf/1209.5163v1.pdf'], 'title': ['THETA LIFTS AND LOCAL MAASS FORMS', 'THETA LIFTS AND LOCAL MAASS FORMS'], 'venue': []}

arxiv

Automatic Heteronym Resolution Pipeline Using RAD-TTS Aligners Jocelyn Huang jocelynh@nvidia.com Evelina Bakhturina ebakhturina@nvidia.com Oktai Tatanov Nvidia Axb Research Automatic Heteronym Resolution Pipeline Using RAD-TTS Aligners Index Terms: grapheme-to-phonemetext-to-speechhet- eronym disambiguation Grapheme-to-phoneme (G2P) transduction is part of the standard text-to-speech (TTS) pipeline. However, G2P conversion is difficult for languages that contain heteronyms -words that have one spelling but can be pronounced in multiple ways. G2P datasets with annotated heteronyms are limited in size and expensive to create, as human labeling remains the primary method for heteronym disambiguation. We propose a RAD-TTS Aligner-based pipeline to automatically disambiguate heteronyms in datasets that contain both audio with text transcripts. The best pronunciation can be chosen by generating all possible candidates for each heteronym and scoring them with an Aligner model. The resulting labels can be used to create training datasets for use in both multi-stage and end-to-end G2P systems. Introduction Modern text-to-speech (TTS) models can learn pronunciations from raw text input and its corresponding audio data, but in languages such as English, phonemes provide more precise pronunciation information than graphemes. As a result, many TTS systems use phonemic input during training to directly access and correct pronunciations for new vocabulary at inference time. One of the hardest problems for grapheme-to-phoneme (G2P) systems is the resolution of heteronyms, i.e., words that have a single spelling but different pronunciations. For example, "read" in "I will read the book" vs. "She read her project last week". Some heteronyms, such as "bass", have multiple pronunciations with the same part of speech, and they need to be disambiguated based on semantic context. In this work, we focus on the heteronym disambiguation task and propose a pipeline for labeling heteronyms in training data for both multi-stage and end-to-end (E2E) G2P models. Some multi-stage G2P systems [1,2] use a set of rules for heteronym disambiguation, but high-quality rule-based systems require expert knowledge and are difficult to scale and maintain. An alternative machine learning approach for heteronym disambiguation is to treat this task as a part-of-speech tagging or a classification problem [3,4]. Emerging E2E G2P systems use sentence-level training data [5,6] and aim to handle outof-vocabulary (OOV) and heteronyms in a single pass. Neural multi-stage and E2E solutions for heteronym disambiguation require labeled data where heteronyms appear in context, but unfortunately, there is a dearth of such data. * *Equal contribution. † † Work done while at NVIDIA. Due to the domain expertise required for labeling phonemes, G2P datasets are few and far between. In datasets like TIMIT [7] and The Buckeye Speech Corpus [8], phoneme transcriptions of audio are provided along with grapheme transcriptions. In TIMIT, transcriptions were human-verified, but the number of unique sentences is too small to train a G2P model. The Buckeye Speech Corpus consists of around 26 hours of conversational speech that was transcribed and phonemically labeled. Since the phoneme labels were automatically generated from the audio, the labels are noisy and sometimes contain alignment errors despite some corrections made by human research assistants, which makes the dataset more unreliable for G2P training. To our knowledge, the Wikipedia Homograph Data [4] (WikiHomograph) is the only open-source dataset with a sufficient number of samples to train a neural model for heteronym disambiguation. WikiHomograph is a text-only dataset where each sample is an entire sentence with a labeled heteronym. Unfortunately, this dataset does not contain a comprehensive list of English homographs. Moreover, some pronunciations in the WikiHomograph set of heteronyms are significantly underrepresented, leading to class imbalance [9]. For example, the corpus contains multiple sentences with the noun form of the heteronyms "desert", "addict" and "subject" and no samples with the verb forms. The WikiHomograph dataset was annotated by linguists, and manual annotation remains the mainstream method of data creation. In addition, some preprocessing is required to train an E2E G2P model on the WikiHomograph dataset, as only the target homograph is labeled in each example sentence. [6] uses CMUdict [10] to label known words while dropping sentences with OOV words. As a heteronym data augmentation technique, Nishiyama et al. [11] introduced a method to match each sense of a heteronym to a synonymous word with a unique pronunciation and to substitute the heteronym for its synonym in a text corpus. This method requires a large textual database for queries, as well as expert knowledge and evaluators to confirm that the resulting sentences are correct. As the method was applied to Japanese heteronyms, there is no available data for English. Other relevant methods of heteronym resolution and verification include the morphological rewriting rules [12] and the context-dependent phone-based HMMs that use acoustic features [13]. [14] skips the phoneme representation in lieu of passing graphemes into a language model to generate its text representation. We plan to add these to our paper to address this broader context. We propose an automatic heteronym disambiguation approach that can generate examples for underrepresented or missing heteronym forms. Our proposed pipeline annotates speech data with heteronym phoneme labels automatically. The … ənd ˈðɛɹˌfɔɹ ˈfɑɹ pleasanter ənd ˈiziɝ ˈtu read. Figure 2: A comparison between the L2 distance matrices between the aligned text and audio embeddings when disambiguating the word "read" from the entry: "... and therefore far pleasanter and easier to read". Values shown correspond to the audio frames that were aligned with each text token, and the average distance is taken across this diagonal to find the overall score for a given pronunciation; the rest of the values are disregarded. The average embedding distances for /ôEd/ and /ôid/ are 452.9 and 403.3, respectively. The latter one would be picked, as it is closer to the audio embedding across the aligned frames. labeled sentences can then be used in conjunction with dictionary lookups for unambiguous known words and "<unk>" tokens for OOV words to create training data for neural G2P or heteronym classification models without human labeling. To get target phonemic labels for heteronyms, we train the RAD-TTS Aligner [15] on transcribed audio data. Then we use the Aligner to score possible heteronym pronunciation options and choose the one that matches the corresponding audio best. To evaluate the quality of generated data, we train a BERT-based classification model and E2E ByT5 G2P model. The results show that the proposed data augmentation technique improves heteronym disambiguation accuracy for both models. We release code 1 and all aligner-generated and hand-checked data for heteronym disambiguation model training. Heteronym resolution pipeline We propose using a RAD-TTS Aligner [15] model to automatically select correct heteronym forms. The RAD-TTS Aligner [15] is a speech-to-text alignment model based on the alignment mechanism introduced in RAD-TTS [16], which allows for easy visualization and human-understandable scores when comparing candidate pronunciations. The Aligner takes a mix of graphemes and phonemes as input: phonemes for known unambiguous words and graphemes for ambiguous or OOV words. It learns to align text tokens and audio frame encodings using the L2 distance between the representations, generating a 1 https://github.com/NVIDIA/NeMo soft alignment that can be converted to a hard alignment using the Viterbi algorithm. These hard alignments between text tokens and audio frames can be used in tandem with the predicted L2 distance matrix in order to determine the distances between a token encoding and each of its corresponding audio frames' encodings. Thus, given a word T consisting of N input tokens t1, ..., tN , where token ti has been aligned with Mi audio frames a (i) 1 , ..., a (i) M i out of audio A, the average distance, Davg, between a word and the audio can be found as: Davg T, A = N i=1 M i j=1 L2(enc ti, enc a (i) j ) N i=1 Mi(1) In essence, the average distance between a word and its acoustic form is a sum of distances between its constituent tokens and their aligned audio frames, divided by the number of audio frames corresponding to the word. We can use these distances to disambiguate heteronyms with an audio sample. Figure 1 shows the proposed automatic phoneme-labeling process for generating sentences with disambiguated heteronyms for sentence-level G2P model training. We first convert known unambiguous words to their phonetic pronunciations with dictionary lookups. This work uses the CMUdict training split defined in [17]. OOV words are left as graphemes. Next, we generate multiple candidates by substituting the heteronym with each possible phonemic form in the dictionary. Then, we pass each candidate along with the corresponding audio file through a trained Aligner model to automatically label heteronyms by picking the pronunciation whose phoneme encodings are closer on average to the audio encodings, i.e., smallest Davg. Figure 2 shows an example of the alignments and distances for two potential pronunciations of "read" from an entry that ends "and therefore far pleasanter and easier to read." Using this method, we can disambiguate all known heteronyms in our speech dataset. Finally, we mask out OOV words with a special masking token, "<unk>", and force the G2P model to produce the same masking token as a phonetic representation during training. During inference, the model generates phoneme predictions for OOV words without emitting the masking token as long as this token is not included in the grapheme input. To control the quality of the disambiguated data, we propose thresholding with a confidence score that represents how much closer the best candidate pronunciation is to the audio. Specifically, the score is a normalized difference between the chosen candidate's L2 distance versus the least likely candidate's L2 distance. The confidence score of disambiguation is found by taking the difference between the highest and lowest L2 distances over all the candidates, then dividing it by the average between the highest and lowest L2 distances. For the example in Figure 2, this would be (452.9 − 403.3)/(452.9 + 403.3)/2) = 0.116. The higher the score, the more likely it is for the disambiguation to be correct. We can now remove any samples with disambiguations that have confidence scores lower than the desired threshold. Once heteronym disambiguations have been performed, the sentences can then be converted to phonemes for use in sentence-level G2P training. As before, we use a dictionary lookup for known unambiguous words, and now we can replace heteronyms with the disambiguated phoneme form. Samples with OOV words can either be dropped, or OOV labels can be replaced with an unk token for training. Aligner training and dataset generation We use the LJSpeech [18] and Hi-Fi TTS [19] (speakers 9017 and 12787) datasets to generate G2P data with disambiguated heteronyms, and train one Aligner model per speaker. Speaker 9017's data contains 57.8 hours and its Aligner model was trained for 250 epochs, speaker 12787 contains 27.1 hours and its Aligner model was trained for 400 epochs, and the LJSpeech model was trained for 1000 epochs on 22.8 hours of data. All models were trained on a single RTX 8000 GPU using the Adam optimizer, a learning rate of 0.001, and a weight decay of 1e-6. A Cosine Annealing scheduler was used, with a minimum learning rate of 5e-5 and a warmup ratio of 0.35. For disambiguation, sentences without heteronyms were discarded. Aligner-disambiguated training sets of speakers 9017, 12787, and LJSpeech were compiled into the Aug set. We also created subsets of the data by filtering out samples where the Aligner confidence score was below a threshold value: Aug-0.01% consists of samples with a confidence score of at least 0.01%; similarly for thresholds of 0.02% and 0.03%. For each augmented subset, we created a "balanced" version that aims to equalize the number of occurrences of each heteronym form in the combined WikiHomograph and Aug. training data to mitigate model bias (Table 1). Evaluation To assess the quality of heteronym resolution with the Aligner model, we hand-checked sentences from LJSpeech dev set, which contains 26 heteronyms. The LJSpeech Aligner model chose the grammatically correct candidate 23 times. However, two of the grammatically incorrect selections accurately reflected the pronunciation of the speaker. We also performed limited human evaluation of the heteronym labels derived from the Hi-Fi TTS speaker 9017 model for textit"read" and "subject". Out of 179 occurrences of the word "read," (87 /ôid/, 92 /ôEd/), the Aligner model picked the correct form 176 times (an accuracy of 98.3%), with only three errors. However, it performs poorly on heteronym "subject", which has two forms that sound similar: /s@b"dZEkt/ and /"s@bdZIkt/. This can be mitigated by confidence thresholding, as seen in Table 2. We conclude that the Aligner model is highly dependent on the enunciation and pronunciation of the speaker, and is prone to error if the audio is noisy or if the speaker mispronounces a heteronym. It also tends to have trouble with heteronyms that have forms that sound similar, but this can be mitigated by confidence thresholding. We also manually verified heteronyms from the dev and test sets of the selected Hi-Fi TTS speakers. We then combined these samples with some proprietary sentences to create a test set that covers most of the heteronym forms missing from the evaluation set of the WikiHomograph dataset. This dataset (hereafter Hard-eval) contains 195 sentences and is used to evaluate the effect of the Aug data on the G2P models' performance. To perform automatic quality estimation, we train a token classification BERT-based [20] heteronym disambiguation model on the WikiHomograph dataset. The model takes a sentence as input, and then for every word, it selects a heteronym option out of the available dictionary forms. The model handles multiple heteronyms simultaneously. We mask irrelevant forms to disregard the model's predictions for non-ambiguous words. E.g., given the input "The Poems are simple to read and easy to comprehend." the model scores possible 'read present' and 'read past' options for the word "read". We finetuned our model from pre-trained "bert-base-cased" 2 model for ten epochs on a 16GB GPU with batch size 32, the AdamW optimizer, a learning rate of 5e-5, a WarmupAnnealing scheduler, and a weight decay of 0.01. Table 3 summarizes experiments with the BERT classification model trained on WikiHomograph data and various amounts of Aligner-generated data. The results are the averages of 3 runs. The highest accuracy was achieved on WikiHomograph and Hard-eval sets with "non-balanced 0.02" aligner data augmentation, 99.07% and 91.04%, respectively. Performance on the balanced set is more consistent on the WikiHomograph set (99+%) and slightly below the best result. Non-balanced data augmentation leads to better results in the Hard-eval set than the performance with balanced data augmentation, 90+% vs. about 89%. We hypothesize that this is because the augmented data provides more non-Wikipedia examples with a vocabulary closer to the Hard-eval set. A confidence threshold of at least 0.01% is recommended as it provides a higher quality of the augmented data; see the performance drop from 86.64% to 83.02% if no thresholding is used. The heteronym disambiguation task has a low tolerance towards errors as these errors propagate down to the text-to-speech pipeline. Using higher Aligner threshold values reduces the number of the augmented samples but assures a higher quality of the training data. To check the validity of our sentence-level labeling pipeline on E2E G2P models, we follow [5] and [17] and train a sentence-level ByT5 model G2P model. The training data for our E2E G2P model consists of CMUdict [10] and WikiHomograph with various amounts of Aligner augmented data. We used the same CMUdict split proposed in [17] for labeling known words and "<unk>" token for OOV words. We finetuned our model from pre-trained "google/byt5-small" 3 model for five epochs on eight 16GB GPUs with batch size 8, the AdamW optimizer, a learning rate of 1e-3, a WarmupAnneal-3 https://huggingface.co/google/byt5-small ing scheduler, and a weight decay of 0.01. Experiments with E2E ByT5 model (Table 4) second the positive effect of the data augmentation while keeping the phoneme error rate (PER) on CMUDict test nearly the same. PER measures the generation capabilities of E2E G2P models. Conclusions In this paper, we propose a data augmentation method that can automatically disambiguate heteronyms to generate data for sentence-level G2P model training. This data labeling technique can be used to balance out existing heteronym forms in gold standard data, add new heteronyms without manual labeling, or simply create more training data as labeled heteronym data is scarce. The proposed method is also controllable using confidence threshold filtering, depending on whether a particular application may need more data with potentially lower quality, or high confidence labels at the cost of the number of samples generated. Additionally, we introduce a masking token that opens the door to sentence-level G2P model training without human annotation. We show through human evaluation and experimentation that the resulting automatically-generated data improves the performance of both BERT classification and E2E G2P systems. We hope that this method will help to remedy this lack of data both for more robust training and for more informative evaluation. 1 . 1Replace known unambiguous words with phoneme forms from dictionary: … ənd ˈðɛɹˌfɔɹ ˈfɑɹ pleasanter ənd ˈiziɝ ˈtu ˈɹɛd.2. For sentences with heteronyms: generate sentence with all possible heteronym forms 3. Score candidate pronunciations with context using the Aligner 5. Mask remaining OOV words 4. Select sentence with the minimum score … ənd ˈðɛɹˌfɔɹ ˈfɑɹ pleasanter ənd ˈiziɝ ˈtu ˈɹid. … ənd ˈðɛɹˌfɔɹ ˈfɑɹ <unk> ənd ˈiziɝ ˈtu ˈɹid. … ənd ˈðɛɹˌfɔɹ ˈfɑɹ pleasanter ənd ˈiziɝ ˈtu ˈɹid. [403.3] … ənd ˈðɛɹˌfɔɹ ˈfɑɹ pleasanter ənd ˈiziɝ ˈtu ˈɹɛd. [452.9] Input sentence: …. and therefore far pleasanter and easier to read … ənd ˈðɛɹˌfɔɹ ˈfɑɹ pleasanter ənd ˈiziɝ ˈtu ˈɹid. Figure 1: Data labeling pipeline for sentence-level G2P model training includes the following steps: 1) Input text. 2) Replace known unambiguous words with phoneme forms from the dictionary. 3) For sentences with heteronyms: generate sentences with all possible heteronym forms. 4) Score candidate pronunciations with context using the Aligner. 5) Select a sentence with the minimum score. 6) Mask remaining OOV words. 722 723 724 725 726 727 728 729 730 731 Audio Frame ˈɹ ɛ d Ph neme 393 385 442 46ɛ 529 545 484 455 426 4ɹ8 ˈɹɛd Candidate ˈ Embedding L2 Distance Matrix 722 723 724 725 726 727 728 729 73ɹ 73ɛ 732 733 734 735 736 737 738 739 74ɹ 74ɛ 742 743 744 745 746 747 748 Audi Frame ˈɹ i d Ph neme 38ɛ 383 42ɹ 4ɛ8 485 5ɹ8 43ɛ 395 376 349 348 367 395 43ɹ 46ɹ 442 423 4ɛɹ 395 394 4ɹ9 4ɛ2 4ɹ2 4ɹ5 38ɛ 334 337 ˈɹid Candidate ˈ Embedding L2 Distance Matrix Table 1 : 1Number of aligner-generated samples added depending on the confidence threshold values and balancing strategy.Threshold 0.00% 0.01% 0.02% 0.03% Num samples (bal) 1230 794 620 572 Num samples (non bal) 3883 2939 2286 1805 Table 2 : 2True positives and false positives of each pronunciation of "subject" as predicted by the speaker 9017 Aligner with various confidence thresholds."Subject" Eval /s@b"dZEkt/ (v.) /"s@bdZIkt/ (adj./n.) Total TP FP TP FP Threshold: 0.00% 1 30 48 0 79 Threshold: 0.01% 1 5 25 0 31 Threshold: 0.02% 1 1 13 0 15 Threshold: 0.03% 0 0 4 0 4 Table 3 : 3Accuracy on the heteronym disambiguation task of the BERT-based heteronym classification model on WikiHomograph and Hard evaluation sets depending on the amount and quality of the Aligner-generated augmented data.Training data Threshold Accuracy, % WikiH Hard WikiHomograph - 98.70 86.64 0.00% 98.99 89.63 + Aligner data 0.01% 98.97 90.88 (no balance) 0.02% 99.07 91.04 0.03% 98.97 90.09 0.00% 98.97 83.02 + Aligner data 0.01% 99.05 89.47 (balance) 0.02% 99.03 89.00 0.03% 99.03 89.46 Table 4 : 4Evaluationof ByT5 E2E G2P model on heteronym disambiguation task (accuracy on WikiHomograph and Hard set) and on OOV (PER on CMUdict test split) depending on the Aligner-augmented data. Training data Thres- Accuracy, % CMUdict hold WikiH Hard PER, % WikiH + CMU - 95.42 79.72 8.62 0.00% 95.42 83.02 8.24 + Aligner 0.01% 96.10 85.85 8.97 (not balanced) 0.02% 95.79 82.08 8.47 0.03% 95.79 83.02 8.06 https://huggingface.co/bert-base-cased . K Park, J Kim, 2K. Park and J. Kim, "g2pE," https://github.com/Kyubyong/g2p, 2019. eSpeak NG. R H Dunn, R. H. Dunn, "eSpeak NG," https://github.com/espeak-ng/ espeak-ng, 2007. Homograph disambiguation in text-to-speech synthesis," in Progress in speech synthesis. D Yarowsky, SpringerD. Yarowsky, "Homograph disambiguation in text-to-speech syn- thesis," in Progress in speech synthesis. Springer, 1997, pp. 157- 172. Improving homograph disambiguation with supervised machine learning. K Gorman, G Mazovetskiy, V Nikolaev, LREC. K. Gorman, G. Mazovetskiy, and V. Nikolaev, "Improving ho- mograph disambiguation with supervised machine learning," in LREC, 2018. T5G2P: Using Textto-Text Transfer Transformer for Grapheme-to-Phoneme Conversion. M Řezáčková, J Švec, D Tihelka, InterspeechM.Řezáčková, J.Švec, and D. Tihelka, "T5G2P: Using Text- to-Text Transfer Transformer for Grapheme-to-Phoneme Conver- sion," in Interspeech, 2021. SoundChoice: Grapheme-to-Phoneme Models with Semantic Disambiguation. A Ploujnikov, M Ravanelli, InterspeechA. Ploujnikov and M. Ravanelli, "SoundChoice: Grapheme-to- Phoneme Models with Semantic Disambiguation," in Interspeech, 2022. DARPA TIMIT acoustic phonetic continuous speech corpus cdrom. J S Garofolo, L F Lamel, W M Fisher, J G Fiscus, D S Pallett, N L Dahlgren, J. S. Garofolo, L. F. Lamel, W. M. Fisher, J. G. Fiscus, D. S. Pallett, and N. L. Dahlgren, "DARPA TIMIT acoustic phonetic continuous speech corpus cdrom," 1993. The buckeye corpus of speech: updates and enhancements. E Fosler-Lussier, L Dilley, N R Tyson, M A Pitt, IN-TERSPEECH. E. Fosler-Lussier, L. Dilley, N. R. Tyson, and M. A. Pitt, "The buckeye corpus of speech: updates and enhancements." in IN- TERSPEECH, 2007. [Online]. Available: http://dblp.uni-trier.de/ db/conf/interspeech/interspeech2007.html#Fosler-LussierDTP07 Homograph disambiguation with contextual word embeddings for TTS systems. M Nicolis, V Klimkov, Speech Synthesis Workshop. SSW2021M. Nicolis and V. Klimkov, "Homograph disambiguation with contextual word embeddings for TTS systems," in Speech Syn- thesis Workshop (SSW 11), 2021. The CMU Pronouncing Dictionary. "The CMU Pronouncing Dictionary," http://www.speech.cs.cmu. edu/cgi-bin/cmudict/, 2017. Dataset construction method for word reading disambiguation. K Nishiyama, K Yamamoto, H Nakajima, Pacific Asia Conference on Language, Information and Computation. K. Nishiyama, K. Yamamoto, and H. Nakajima, "Dataset con- struction method for word reading disambiguation," in Pacific Asia Conference on Language, Information and Computation, 2018. Natural language processing in a japanese text-to-speech system for written-style texts. K Matsuoka, E Takeishi, H Asano, R Ichii, Y Ooyama, Proceedings of IVTTA '96. Workshop on Interactive Voice Technology for Telecommunications Applications. IVTTA '96. Workshop on Interactive Voice Technology for Telecommunications ApplicationsK. Matsuoka, E. Takeishi, H. Asano, R. Ichii, and Y. Ooyama, "Natural language processing in a japanese text-to-speech system for written-style texts," in Proceedings of IVTTA '96. Workshop on Interactive Voice Technology for Telecommunications Applica- tions, 1996, pp. 33-36. Heteronym verification for mandarin speech synthesis. H Lu, Z Ling, S Wei, Y Hu, L Dai, R Wang, 6th International Symposium on Chinese Spoken Language Processing. H. Lu, Z. Ling, S. Wei, Y. Hu, L. Dai, and R. Wang, "Heteronym verification for mandarin speech synthesis," in 2008 6th Inter- national Symposium on Chinese Spoken Language Processing, 2008, pp. 1-4. Mixer-tts: nonautoregressive, fast and compact text-to-speech model conditioned on language model embeddings. O Tatanov, S Beliaev, B Ginsburg, ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing. IEEEICASSPO. Tatanov, S. Beliaev, and B. Ginsburg, "Mixer-tts: non- autoregressive, fast and compact text-to-speech model condi- tioned on language model embeddings," in ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2022, pp. 7482-7486. One TTS alignment to rule them all. R Badlani, A Łańcucki, K J Shih, R Valle, W Ping, B Catanzaro, ICASSP. R. Badlani, A. Łańcucki, K. J. Shih, R. Valle, W. Ping, and B. Catanzaro, "One TTS alignment to rule them all," in ICASSP, 2022. Rad-tts: Parallel flow-based TTS with robust alignment learning and diverse synthesis. K J Shih, R Valle, R Badlani, A Lancucki, W Ping, B Catanzaro, ICML Workshop on Invertible Neural Networks, Normalizing Flows, and Explicit Likelihood Models. K. J. Shih, R. Valle, R. Badlani, A. Lancucki, W. Ping, and B. Catanzaro, "Rad-tts: Parallel flow-based TTS with robust alignment learning and diverse synthesis," in ICML Workshop on Invertible Neural Networks, Normalizing Flows, and Explicit Likelihood Models, 2021. ByT5 model for massively multilingual grapheme-to-phoneme conversion. J Zhu, C Zhang, D Jurgens, InterspeechJ. Zhu, C. Zhang, and D. Jurgens, "ByT5 model for massively multilingual grapheme-to-phoneme conversion," in Interspeech, 2022. K Ito, L Johnson, The LJ Speech Dataset. K. Ito and L. Johnson, "The LJ Speech Dataset," https://keithito. com/LJ-Speech-Dataset/, 2017. Hi-Fi Multi-Speaker English TTS Dataset. E Bakhturina, V Lavrukhin, B Ginsburg, Y Zhang, InterspeechE. Bakhturina, V. Lavrukhin, B. Ginsburg, and Y. Zhang, "Hi-Fi Multi-Speaker English TTS Dataset," in Interspeech, 2021. BERT: Pre-training of deep bidirectional transformers for language understanding. J Devlin, M.-W Chang, K Lee, K Toutanova, Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, "BERT: Pre-training of deep bidirectional transformers for language un- derstanding," in "Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies", 2019.

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{'abstract': 'Grapheme-to-phoneme (G2P) transduction is part of the standard text-to-speech (TTS) pipeline. However, G2P conversion is difficult for languages that contain heteronyms -words that have one spelling but can be pronounced in multiple ways. G2P datasets with annotated heteronyms are limited in size and expensive to create, as human labeling remains the primary method for heteronym disambiguation. We propose a RAD-TTS Aligner-based pipeline to automatically disambiguate heteronyms in datasets that contain both audio with text transcripts. The best pronunciation can be chosen by generating all possible candidates for each heteronym and scoring them with an Aligner model. The resulting labels can be used to create training datasets for use in both multi-stage and end-to-end G2P systems.', 'arxivid': '2302.14523', 'author': ['Jocelyn Huang jocelynh@nvidia.com ', 'Evelina Bakhturina ebakhturina@nvidia.com ', 'Oktai Tatanov ', 'Nvidia ', 'Axb Research '], 'authoraffiliation': [], 'corpusid': 257232771, 'doi': '10.48550/arxiv.2302.14523', 'github_urls': ['https://github.com/NVIDIA/NeMo', 'https://github.com/Kyubyong/g2p,', 'https://github.com/espeak-ng/'], 'n_tokens_mistral': 8503, 'n_tokens_neox': 7614, 'n_words': 4101, 'pdfsha': '22d9dc6b254d1d6584688f7da4767edc795fd46a', 'pdfurls': ['https://export.arxiv.org/pdf/2302.14523v1.pdf'], 'title': ['Automatic Heteronym Resolution Pipeline Using RAD-TTS Aligners', 'Automatic Heteronym Resolution Pipeline Using RAD-TTS Aligners'], 'venue': []}

arxiv

Statistical Modeling of Atmospheric Mean Temperature in sub−Sahel West Africa O A * Falaiye Department of Physics Faculty of Physical Sciences University of Ilorin Nigeria Y M Sukam Department of Physics Faculty of Physical Sciences University of Ilorin Nigeria Department of Physics Faculty of Science Federal University Lafia Nigeria Abimbola O J Statistical Modeling of Atmospheric Mean Temperature in sub−Sahel West Africa West Africaatmospheric mean temperatureprecipitable water vapor Atmospheric mean temperature T m , is a vital parameter in the evaluation of precipitable water vapor (PWV) through the analysis of GPS signal, it is therefore important to have a good way of evaluation of Tm for the eventual accurate evaluation of PWV using GPS. Simple statistical models exist for various regions of the world for the evaluation of T m using surface temperature T s , in the form T m = aT s + b where a and b are constants. For West Africa, where atmospheric data is usually very scarce, there is a minimal attempt at finding a statistical model for T m , as a function of T s . In this work attempt has been made to find such a model using data from the Climate Monitoring Satellite Facilities (CM-SAF) of the European Meteorological Satellites (EUMETSAT). The model derived was found to compare well with that obtained using radiosonde data with root-mean-square error of 1.189 and mean-biased error of 0.0952 between the two models. I. Introduction Preciptable water vapor (PWV) is an important atmospheric parameter essential in both weather and climatic prediction. Knowledge of the variability of PWV is also very important in astronomy as water vapor interact with the incoming electromagnetic waves in the atmosphere. GPS meteorology offers a real-time continuous measurement of PWV. From the GPS data, it is possible to estimate the zenith total delay (ZTD). The zenith hydrostatic delay (ZHD), which is the delay error caused by the dry component of the atmosphere could be estimated using models such as the Saastamoinen [7] model. The delay error introduced by the water vapor in the atmosphere, called the zenith wet delay (ZWD) is calculated thus: The ZWD is usually transformed into the PWV using the transformation equation as given below ZWD = ZTD − ZHD(1)PWV = Π × ZWD(2) The transformation constant Π, is given by Π = 10 −6 k 3 T m + k 2 ρ v R v −1(3) here, ρ v is the liquid water density = 1000 kgm −3 , R v is water vapor gas constant = 461.524 JK −1 kg −1 , k 3 = 377600 K 2 hPa, k 2 = 22. 1 KhPa −1 and T m is called weighted atmospheric mean temperature determination of which is very crucial to accurate transformation of ZWD into the PWV. The weighted mean atmospheric temperature T m , could be defined as T m = p 1 p 2 Td(lnp) p 1 p 2 d(lnp) = ∞ z e T dz ∞ z e T 2 dz(4) where p 1 and p 2 are pressures at two different layers of the atmosphere. T is the ambient temperature. e is the vapor pressure while z is the height above the ground. From Eq. 4. T m could be estimated from the radiosonde data taken at different layers of the atmosphere. However, such radiosonde data are usually not readily available across the globe, especially Africa, hence several attempts have been made to statistically relate T m with the surface temperature T s which is readily available as a basic meteorological parameter at any weather observing station. Bevis et al. [1] used 8700 radiosonde profiles from 13 stations in the United States of America (USA) to obtain a statistical model of the form T m = aT s + b. The Bevis et al. [1] model was adopted in many studies (e.g., Raju et al. [2]; Fernandez et al. [3]; Musa et al. [4]; Abimbola et al. [5]) across the globe despite the fact that it was obtained for the USA region. Mendes et al. [6] and Solbrig [8], using radiosonde data, obtained similar linear relation for Germany, Raju et al. [2] for Indian sub-continent, Shoji [9] for Japan and Isioye et al. [10] for West Africa. Meanwhile, Schuler et al. [11], using a numerical weather prediction data, obtained a model which is more global in outlook relating T m to T s as shown in Eq. 5 For ease of analysis of the data sets and climatic consideration, the study locations were further divided into (1) Hinterland, that is those locations in the interior of the study region and (2) Coastal, that is those regions close to the Atlantic Ocean. These divisions are shown in Table 1. 3 T m = 0.65T s + 86.9(5) III. Results and Discussion The derived empirical equations relating the mean atmospheric temperature T m to the surface temperature T s for each of the locations considered in this work within the West African region are shown in Table 2. All the stations considered show good linear correlations between T m and T s Except for Conakry with a below average coefficient of determination. It will be observed from T m = 0.617T s + 108.49 R 2 = 0.709(9) Isioye et. al. [10] using radiosonde data covering 2009 to 2013 for West Africa obtained a corresponding statistical model as given in Eq. 10: T m = 0.5743T s + 116.60 R 2 = 0.436(10) It will be noted that Eq. 9 shows a better statistical performance than Eq. 10, though the two statistical models could be observed to be quite similar. A statistical comparison of Eq. 9 with Eq. 10 yields root-mean-square error (RMSE) of 1.189 and mean-biased error (MBE) of 0.0953, 6 Falaiye et al.: Weighted Atmospheric Mean Temperature further showing that the two statistical models are comparable. Satellite data has wider spatiotemporal coverage than radiosonde data hence, the better coefficient of determination R 2 , observed for Eq. 9 as compared to Eq. 10. IV. Conclusion Using satellite data from the Satellite Application Facility (CM-SAF) of the EUMETSAT a suitable statistical model has been derived to estimate weighted atmospheric mean temperature T m . The statistical model derived is a simple linear model of T m as a function of surface temperature T s . The derived model was compared with a similar statistical model which had earlier been derived from the radiosonde data in West Africa. The correspondence between the two models were found to be significant. Figure 1 . 1Map of West Africa showing Locations of study area (Encyclopeadia Britannica, 2018). 850 hPa and level6 (surface) = 1000 hPa). are the specific humidity q m (g/kg) and the ambient temperature T (K). The vapor pressure e (in hPa) was estimated from the specific humidity using: v4.3.1), open source software from NOAA was used for the processing of the data while Jupyter notebook, an environment for running Python code, was used for data plotting, curve fittings and other statistical analysis.Goodness of fit and correlational analysis of the derived empirical equations were done using coefficient of determination (R 2 ), mean-biased error (MBE) and root-mean-square error (RMSE). Figure 2 .Figure 3 .Figure 4 234Coastal Region data plot and linear fit.T m = 0.60T s + 112.73 R 2 = Hinterland Region data plot and linear fit. T m = 0.72T s + 77.89 R 2 = shows a plot of all the combined data for West Africa. The statistical model derived from Figure 4 is given in Eq. 9 with the corresponding coefficient of determination: Figure 4 . 4A combined West African data plot and linear fit. Falaiye et al.: Weighted Atmospheric Mean Temperature1 arXiv:1901.02342v1 [physics.ao-ph] 27 Dec 2018 the aim of this work to find a suitable statistical model for the evaluation of atmospheric mean temperature T m in West African region.This study was conducted for locations in West African region as shown inFigure 1. The selected locations in West Africa for this research work are: Dakar (14.76 o N Latitude, 17.36 o W Longitude), Conakry (9.64 o N Latitude, 13.57 o W Longitude), Bamako (12.63 o N Latitude, 8.00 o W Longitude), Abidjan (5.36 o N Latitude, 4.00 o W Longitude), Niamey (13.51 o N Latitude, 12.12 o E Longitude), and Abuja (90.07 o N Latitude, 7.39 o E Longitude).II. Methodology i. Study Area Table 1 . 1Division of the Study Area into Hinterland and Coastal Regions.Station Country Longitude Latitude Period of data Dakar Coastal Region Senegal 17.366 o W 14.765 o N 2004 − 2016 Conakry Guinea 13.578 o W 09.641 o N 2004 − 2016 Abidjan Cote d'ivoire 04.008 o W 05.360 o N 2004 − 2016 Bamako Hinter- land Region Mali 08.003 o W 12.639 o N 2004 − 2016 Niamey Niger 12.125 o E 13.512 o N 2004 − 2016 Abuja Nigeria 07.399 o E 09.077 o N 2004 − 2016 ii. Data Analysis The data used in this work was obtained from the archive of the Climate Monitoring Satel- lite Application Facilities (CM-SAF) of the European Meteorological Satellites (EUMETSAT) at https://www.cmsaf.eu. The data was obtained as a monthly average for a time period spanning from 2004 to 2016 in Network Common Data Format (netCDF). The data obtained at six different pressure levels (level1 = 200 hPa; level2 = 300 hPa; level3 = 500 hPa; level4 = 700 hPa; level5 = Table 2 2that the coastal region generally does not show good linear correlation between T m and T s ,whereas the hinterland region shows very good linear correlation: also it was observed that the 4 Table 2 . 2Statistics for each of the stations in West Africa.respectively. For these figures empirical equations Eq. 7 and Eq. 8 were derived for the coastal and hinterland respectively. It will be observed again fromFigures 2 and 3as well as the derived Eqs. 7 and 8 that the model fit to the hinterland data has better correlation than the fit to the coastal data.Location Empirical Model T m = aT s + b Coefficient of Determination Name a b R 2 Dakar Coastal Region 0.57 123.15 0.575 Conakry 1.19 63.28 0.442 Abidjan 0.87 32.68 0.741 Bamako Hinter- land Region 0.68 89.96 0.646 Niamey 0.81 52.23 0.763 Abuja 0.85 40.88 0.902 ambient surface temperature at the coast is generally much lower than that at the hinterland of West Africa. Figures 2 and 3 show the combined plot and linear fit to the dataset of coastal and hinterland AcknowledgmentThis work was done using data from EUMETSATâȂŹs Satellite Application Facility on Climate Monitoring (CM SAF). DOI: 10.5676/EUM_SAF_CM/WVT_ATOVS/V001. GPS Meteorology: Remote Sensing of Atmospheric Water Vapor using the Global Positioning System. M Bevis, S Businger, T A Herring, C Rocken, R A Anthes, R H Ware, 10.1029/92JD01517Journ. Geophy. Res. 97Bevis, M., Businger, S., Herring, T. A., Rocken, C., Anthes, R. A. and Ware R. H. (1992). GPS Meteorology: Remote Sens- ing of Atmospheric Water Vapor using the Global Positioning System. Journ. Geophy. Res., 97. doi:10.1029/92JD01517. Empirical Model for Mean Temperature for Indian Zone and Estimation of Precipitable Water Vapor from ground based GPS Measurements. C S Raju, K Saha, B V Thampi, K Parameswaran, Annal of Geophysics. 25Raju, C. S., Saha, K., Thampi, B. V. and Parameswaran, K. (2007). Empirical Model for Mean Temperature for Indian Zone and Estimation of Precipitable Water Vapor from ground based GPS Measurements. Annal of Geophysics, 25, 1935-1948. Observation of Long range, Near-side Angular Correlations in Proton-proton Collisions at the LHC. W Fernandez, V Khachatryan, A Tumasyan, Journal of High Energy Physics. 91Fernandez, W., KhaChatryan, V., Tumasyan, A. (2010). Observation of Long range, Near-side Angular Correlations in Proton-proton Collisions at the LHC Journal of High Energy Physics, 2010:91. GPS Meteorology in a low Latitude Region: Remote Sensing of Atmospheric Vapor over the Malaysian Peninsula. T A Musa, S Amir, R Othman, S Ses, K Omar, K Abdullah, S Lim, C Rizos, Journal of Atmospheric Physics. 73Musa, T. A., Amir, S., Othman, R., Ses, S., Omar, K., Abdullah, K., Lim, S. and Rizos, C. (2011). GPS Meteorology in a low Latitude Region: Remote Sensing of Atmospheric Vapor over the Malaysian Peninsula. Journal of Atmospheric Physics, 73:2410-2422. Estimation of precipitable water vapor in Nigeria using NIGNET GNSS/GPS, NCEP-DOE Reanalysis II and surface meteorological data. O J Abimbola, O A Falaiye, J Omojola, 10.21315/jps2017.28.2.2J. Phys. Sci. 282Abimbola, O. J., O. A. Falaiye and J. Omojola (2017). Estimation of precipitable water vapor in Nigeria using NIGNET GNSS/GPS, NCEP-DOE Reanalysis II and surface meteorological data. J. Phys. Sci., 28(2), 19 âȂŞ 29. doi: 10.21315/jps2017.28.2.2. An Evaluation of the Accuracy of Models of the Determination of the Weighted Mean Temperature of the Atmosphere. V B Mendes, G Prates, L Santos, R B Langley, Proceedings of ION, National Technical Meeting. ION, National Technical MeetingAnaheim, CA2.2. Pacific Hotel DisneylandMendes, V. B., Prates, G., Santos, L. and Langley, R. B. (2000). An Evaluation of the Accuracy of Models of the Determination of the Weighted Mean Temperature of the Atmosphere. Proceedings of ION, National Technical Meeting, Jan. 26-28, 2000,.2.2. Pacific Hotel Disneyland, Anaheim, CA. Atmospheric correction for the troposphere and stratosphere in radio ranging of satellites. J Saastamoinen, W Henriksen, 10.1029/GM015p0247American Geophysical Union15The use of artificial satellites for geodesy, geophysics monograph seriesSaastamoinen, J. (1972). Atmospheric correction for the troposphere and stratosphere in radio ranging of satellites. In Henriksen, W. et al. The use of artificial satellites for geodesy, geophysics monograph series, vol. 15. Washinghton DC: American Geophysical Union, 247, https://doi.org/10.1029/GM015p0247. Untersuchungenuber die Nutzungnumerischer Wettermodellezur wasser dampfbestimmungmitHilfe des Global Positioning System. P Solbrig, GermanyInstitude of Geodesy and Navigation, University FAF MunichDiploma ThesisSolbrig, P. (2000). Untersuchungenuber die Nutzungnumerischer Wettermodellezur wasser dampfbestimmungmitHilfe des Global Positioning System. Diploma Thesis, Institude of Geodesy and Navigation, University FAF Munich,Germany, Feb. 2000. Accurate Estimation of Precipitable Water Vapor using Ground based GPS Observation Network and its data assimilation into a mesoscale numerical weather prediction model. Y Shoji, Kyoto UniversityPhD ThesisShoji, Y. (2010). Accurate Estimation of Precipitable Water Vapor using Ground based GPS Observation Network and its data assimilation into a mesoscale numerical weather prediction model. PhD Thesis, Kyoto University. Modelling weighted mean temperature in the West African region: implications for GNSS meteorology. O A Isioye, L Combrinck, J Botai, 10.1002/met.1584Meteorol. Appl. 23Isioye, O. A., L. Combrinck and J. Botai (2016). Modelling weighted mean temperature in the West African region: implications for GNSS meteorology. Meteorol. Appl. 23, 614 âȂŞ 632. doi: 10.1002/met.1584. A global analysis of the mean atmospheric temperature for GPS water vapor estimation. T Schuler, A Posfay, G W Hein, R Biberger, Inproc. 14th Int. Tech. Meeting Satell. âȂŞ C5: 558 Atmospheric Effects. 10559Schuler, T., Posfay, A., Hein, G. W. and Biberger, R. (2001). A global analysis of the mean atmospheric temperature for GPS water vapor estimation. Inproc. 14th Int. Tech. Meeting Satell. âȂŞ C5: 558 Atmospheric Effects, 10NGPS, P.559.

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{'abstract': 'Atmospheric mean temperature T m , is a vital parameter in the evaluation of precipitable water vapor (PWV) through the analysis of GPS signal, it is therefore important to have a good way of evaluation of Tm for the eventual accurate evaluation of PWV using GPS. Simple statistical models exist for various regions of the world for the evaluation of T m using surface temperature T s , in the form T m = aT s + b where a and b are constants. For West Africa, where atmospheric data is usually very scarce, there is a minimal attempt at finding a statistical model for T m , as a function of T s . In this work attempt has been made to find such a model using data from the Climate Monitoring Satellite Facilities (CM-SAF) of the European Meteorological Satellites (EUMETSAT). The model derived was found to compare well with that obtained using radiosonde data with root-mean-square error of 1.189 and mean-biased error of 0.0952 between the two models.', 'arxivid': '1901.02342', 'author': ['O A * Falaiye \nDepartment of Physics\nFaculty of Physical Sciences\nUniversity of Ilorin\nNigeria\n', 'Y M Sukam \nDepartment of Physics\nFaculty of Physical Sciences\nUniversity of Ilorin\nNigeria\n\nDepartment of Physics\nFaculty of Science\nFederal University Lafia\nNigeria\n', 'Abimbola O J '], 'authoraffiliation': ['Department of Physics\nFaculty of Physical Sciences\nUniversity of Ilorin\nNigeria', 'Department of Physics\nFaculty of Physical Sciences\nUniversity of Ilorin\nNigeria', 'Department of Physics\nFaculty of Science\nFederal University Lafia\nNigeria'], 'corpusid': 119371367, 'doi': '10.1016/j.sciaf.2019.e00254', 'github_urls': [], 'n_tokens_mistral': 5103, 'n_tokens_neox': 4283, 'n_words': 2415, 'pdfsha': '22b36607d0313990dfaa0a3748450875b8a99514', 'pdfurls': ['https://arxiv.org/pdf/1901.02342v1.pdf'], 'title': ['Statistical Modeling of Atmospheric Mean Temperature in sub−Sahel West Africa', 'Statistical Modeling of Atmospheric Mean Temperature in sub−Sahel West Africa'], 'venue': []}

arxiv

Mass spectra of four quark states in hidden charm sector 17 Feb 2014 (Dated: February 18, 2014) Smruti Patel Department of Physics Sardar Patel University Vallabh VidyanagarINDIA Manan Shah Department of Physics Sardar Patel University Vallabh VidyanagarINDIA P C Vinodkumar Department of Physics Sardar Patel University Vallabh VidyanagarINDIA Mass spectra of four quark states in hidden charm sector 17 Feb 2014 (Dated: February 18, 2014) Masses of the low lying four quark states in the hidden charm sector (cqcq; q ∈ u, d) are calculated within the framework of a non-relativistic quark model. The four body system is considered as two two-body systems such as diquark-diantiquark (q1q2 −q3q4) tetraquark states and di-mesonic (q1q2 − q3q4) molecular states. Here, Cornell type potential has been used for describing the two body interactions among q − q,q −q, q −q, qq −qq and qq − qq, with appropriate string tensions. Our present anylysis suggests the following exotic states, X(3823), Zc(3900), X(3915), X(3940), Zc(4025), ψ(4040), Z1(4050) and X(4160) as di-mesonic molecular states, while Zc(3885) and Y (4140) as the diquark-diantiquark tetraquark state. We have been able to assign the J P C values for many of the recently observed exotic states according to their structure. I. INTRODUCTION Over the past decade, the family of Exotic states has become more and more abundant due to the experimental development. It is a topic of current interest full of opportunities and challenges for theorists as well as experimentalists to reveal the internal mechanisms originating from these novel and complicated states. Many exotic states in the charm sector with cc content have been discovered by Belle [1] and BESIII [2] and others which provide challenges to theorists studying hadron spectroscopy. With the experimental progress, theorists have paid more attention to these observations by proposing different explanations. Due to the asymptotic property of the QCD, study of the hadron physics have to concern about the nonperturbative effect which is difficult in quantum field theory. It has been realized early on that quark models and QCD sustain a much richer pattern of different multi-quark and/or color network configurations, beyond the "non-exotic" standard qq mesons and qqq baryons. There are growing evidences for the existence of exotic mesons containing both heavy and light quark-antiquark pairs i.e. ccqq. In the past few years, the experimental observations of so many X, Y and Z states have stimulated the study of exotic states greatly as they have induced a pre Gell-Mann like situation in our knowledge of the hadron spectroscopy. Definite conclusions have not yet been reached about the internal structure of newly observed four quark systems. Models to accommodate the exotic states have been proposed over the last decades. Different attempts have been made for the interpretation of the internal structure of the exotic hadronic states of four quark system. These four quark state can in principle be composed of diquark-diantiquark (a tetraquark), a loosely bound state of two mesons (a molecular state), gluballs or qq pair with gluons (hybrids). Here, we confine ourselves to a model of four quark exotic states with a diquarkdiantiquark model and a di-mesonic molecular state. The * fizix.smriti@gmail.com † mnshah09@gmail.com ‡ p.c.vinodkumar@gmail.com study of such structures is important from the point of view of understanding interaction among hadrons at different energy scales related to their formation of bound states as well as decay processes. These interactions provide useful information to study fundamental problems of QCD such as color confinement.The first papers suggesting the existence of tetraquark configurations were given by [3,4], within the MIT bag model with color spin interaction. In the beginning, light flavor tetraquark states were predicted. Later on, Weinstein and Isgur [5,6] extended this tetraquark picture into a variety of quark models. This means that tetraquarks with heavy quarks can also exist. In the last year, few exotic mesons in the charm sector were discovered. These exotic states can be the candidates for the tetraquark states. Moreover, these states can be dimesonic bound molecular states as was predicted more than twenty years ago [7]. The calculation for the tetraquark state cqcq which was performed by Dias [8] suggests that the newly discovered Z c (3900) can be a tetraquark state. Another interesting possible interpretation of the Z c (3900) proposed by Hong et al., is that it can be a molecular state of 1/( √ 2)(DD * + D * D ) resulting from the binding of two meson molecules [9]. Thus, its interpretation as a tetraquark state or a di-mesonic molecular state remains unresolved. Here, we briefly review recent results for the masses of heavy tetraquarks in the framework of a non-relativistic quark model based on the potential approach. We use the diquark-diantiquark and the Di-meson approximation to reduce a complicated four-body problem to the subsequent more simple two two-body problems. II. THEORETICAL FRAMEWORK In this paper we shall take a different path and investigate different ways in which the experimental data can be reproduced. There are many methods to estimate the mass of a hadron, among which phenomenological potential model is a fairly reliable one. Non-relativistic interaction potential we have used here is the Cornell potential consists of a central term V(r) which is being just a sum of the Coulomb(vector) and linear confining(scalar) parts given by V (r) = V V + V S = k s α s r + σr (1) k s = −4/3 f or qq = −2/3 f or qq orqq (2) Different degenerate exotic states can be calculated by including spin-dependent part of the usual one gluon exchange potential. The potential description extended to spin dependent interactions results in three types of interaction terms such as the spin-spin , the spin-orbit and the tensor part that are to be added to the discussed leading non-relativistic description. Accordingly, the spindependent part V SD is given by V SD = V SS 1 2 (S(S + 1) − S 1 (S 1 + 1) − S 2 (S 2 + 1)) +V LS 1 2 (J(J + 1) − S(S + 1) − L(L + 1)) +V T 12 (S 1 .r)(S 2 .r) r 2 − 1 3 (S 1 .S 2 )(3) The spin-orbit term containing V LS and tensor term containing V T describe the fine structure of the states, while the spin-spin term containing V SS proportional to 2S 1 .S 2 gives the hyperfine splitting. The coefficient of these spin-dependent terms of Eq.(3) can be written in terms of the vector and scalar parts of static potential V(r) as V ij LS (r) = 1 2M i M j r + 3 dV V dr − dV S dr (4) V ij T (r) = 1 6M i M j r + 3 d 2 V V dr 2 − 1 r dV S dr (5) V ij SS (r) = 1 3M i M j r ∇ 2 V V = 16πα s 9M i M j δ 3 (r)(6) Where M i , M j corresponds to the masses and r is relative co-ordinate of the two body system under consideration. Our main aim is to interpret the four quark state structure in two different schemes: (1) Tetraquark picture in which four quark state is considered as a diquark-diantiquark and (2)Molecular picture in which the four quark state as a di-mesonic molecular state. In this case, one uses the fact that the motion of the quarks that comprise the exotic state is nonrelativistic to assume that they move in a static potential, much like nonrelativistic models of the hydrogen atom. In both the pictures, we have treated the four particle system as two-two body system and the same form of the two body interaction potential discussed above but with different potential strengths, k s and σ of Eq. (1) as assumed. II.1. Four quark state as Diquark-Diantiquark system (Tetraquark) We have calculated the masses of heavy tetraquarks considered as the bound states of a heavy-light diquark and diantiquark. We discuss the spectra in the framework of a non-relativistic hamiltonian including chromomagnetic spin-spin interactions between the quarks (antiquarks) within a diquark(di-antiquark). We calculate the masses of ground state as well as excited heavy tetraquarks with hidden charm diquark-diantiquark (cq− cq) picture. The mass of diquark (diantiquark) is obtained by numericallly solving the Schrödinger equation with the potential given by Eq.(1) and incorporating the spin interaction described by equation (3) perturbatively. Further, the same procedure is adopted to compute the binding energy of the diquark-diantiquark bound system but with a different potential strength σ of Eq. (1) In the diquark-diantiquark model, the masses of the diquark/diantiquark system and tetraquark states are given by: M d = m q1 + m q2 + E d + V SD (7) md = mq 3 + mq 4 + Ed + V SD (8) M d−d = m d + md + E dd + V SD(9) Where m 1,2 and m 3,4 represents the masses of quarks, antiquarks respectively. In the present paper, d andd represents diquark and diantiquark respectively. While E d , Ed, E dd are the energy eigen values of the diquark, diantiquark and diquark-diantiquark system respectively. The spin-dependent potential (V SD ) part of the hamiltonian described by Eq.(3) has been treated perturbatively. II.2. Four quark state as di-mesonic molecular system In the past thirty five years, theorists have been studying whether two charmed mesons can be bound into a molecular state, because the presence of the heavy quarks lowers the kinetic energy while the interaction between two light quarks could still provide a strong enough attraction. Voloshin and Okun studied the interaction between a pair of charmed mesons and proposed the possibilities of the molecular states involving charmed quarks [10]. In the present molecular framework, masses of meson molecules are determined by employing Coulomb plus Linear potential between heavy quark and light antiquark and vice-versa. Here, we have taken various combinations of spin and orbital angular momentum. We have considered the total spin J 1 and J 2 of the two mesons as spins S 1 and S 2 and these spins couple to J 12 together with relative orbital motion L 12 presents the total spin J of the di-mesonic system. In the di-mesonic picture, we consider the masses of the quark-antiquark states as M 1 = m q1 + mq 2 + E q1q2(10)M 2 = m q3 + mq 4 + E q3q4(11) to describe the two mesons and the di-mesonic molecular mass as M = M 1 + M 2 + E M12 + V SD(12) Where M 1 , M 2 are two meson masses. Here, E q1q2 , E q3q4 represent the binding energy of the quarkantiquark constituting the mesons and E M12 is the binding energy of the di-mesonic system. The interaction between the di-mesons are also assumed to be of the same form as given by Eq.(1) except that the string tension is assumed to be different. III. RESULTS AND DISCUSSIONS The masses of the low lying hidden charm four quarks system as diquark-diantiquark (tetraquark) state as well as di-mesonic molecular states have been computed. For four quark system (cqcq; q ∈ u, d), we have used: m u = m d =0.323 GeV, m c =1.486 GeV and the string tension (potential parameter) σ=0.030 GeV 2 for diquark-diantiquark interaction and 0.018 GeV 2 Table 6 as the summary of the present study. Finally, we find it interesting to compare our results with the newly discovered exotic charm states. For example, soon after the Z + c (3900) observation, the BES-III reported the observation of three more charge states: Z + c (4025) [11], Z + c (4020) [12] and Z + c (3885) [13]. Here, we have been able to identify the X(3823), Z c (3900), X(3915), X(3940), Z c (4025), X(4160) and ψ(4040) resonances as di-mesonic molecular state, while Y (4140) as diquark-diantiquark tetraquark state. Though the parity of states X(3823) and Z c (3900) (J=1 and C=-1) is experimentally unknown, our predictions suggest Z c (3900) as 1 +− state while X(3823) state as 1 −− . There is a still question regarding the structure of two states Z c (3900) and Z c (3885) that whether they are two different states or the same state. Recently, BES-III group reported that Z c (3885) may have 1 + quantum number and if so then it can be in a S wave or/and a D wave [13]. Our present study predicts Z c (3885) as a diquark-diantiquark tetraquark state with J P C =1 +− ( 3 S 1 ). Although the J P quantum numbers of Z c (4025) still remain to be determined experimentally, it is assumed to have spin parity J P = 1 + by BES-III group [11]. This assignment for Z c (4025) is in agreement with the interpretation of this state to be dimesonic molecular state having J P C = 1 +− . Present identification of ψ(4040) as a di-mesonic molecular state with J P C = 1 −− is in accordance with what was suggested by De Rujula, Georgi and Glashow [14]. The state Z 1 (4050) is close to the interpretations of both diquark-diantiquark system and di-mesonic molecular state having same positive parity but with different J values. Thus, state Z 1 (4050) still needs more experimental confirmation of its J value. From our present prediction for Z 1 (4050), we suggest assignment provided by the experiment, then it can be interpreted as a molecular state only if its parity is positive. However, its experimental confirmation is awaited. Finally, We believe that strong experimental efforts aimed at determining the spin parity of the exotic states are required for understanding the status of many of the newly observed exotic states. IV. ACKNOWLEDGMENTS The work is part of Major research project NO. F. 40-457/2011(SR) funded by UGC, INDIA. center of weight massTABLE II. Mass spectra in tetraquark picture for L d =1 and Ld=0 (In GeV).S d L d Sd Ld J d Jd J J P C 2S+1 XJ Mcw VSS di-mesonic interaction. The string tension σ for intra q − q, q −q andq-q are assumed as having same interaction strength 0.015 GeV 2 . Various combinations of the orbital and spin excitations have been considered. The results obtained in both the cases are tabulated in Tables 1 to 5. Selected states for known experimental exotic states are identified and their J P C values are assigned. Their interpretations are shown in -0.0044 -0.058 3.974 3 3 ++ 7 P3 4.036 0.0009 -0.0011 0.018 4.054 4 4 ++ 7 P4 0.0033 -0.038 4.002 TABLE I . IMass spectra in tetraquark picture for L d =0 and Ld=0 (In GeV). TABLE III . IIIMass spectra in molecular picture for L1=0 and L2=0 (In GeV). TABLE TABLE V . Vthat its spin parity is 4 ++ /1 +− if it is a tetraquark state or 2 +− /3 ++ as a molecular state. Up to now, the interpretation of the state X(4160) and X(3915) is still unclear. The state X(3915) clearly has C=+, but J P remains to be determined. T Branz et al.[15] predicted this state as a molecular state. But we have four possibilities for this state X(3915): it can be either one of the 0 ++ /2 ++ /3 +− molecular state or 1 +− tetraquark state. If we follow experimental clue for C=+, then it could be a 0 ++ /2 ++ molecular state. For X(4160), we have predicted that it can be a either 2 +− molecular state or 1 +− /1 −+ tetraquark state. For Y (4140), we are having four different possible states 0 −+ , 1 −− , 2 −+ , 3 −− in the energy range 4.136 − 4.146 GeV as diquark -diantiquark states while only 1 ++ state in the di-mesonic molecular model. As per C=+1Mass spectra in molecular picture for L1=1 and L2=1 (In GeV). S1 L1 S2 L2 J1 J2 J12 L12 J J P C 2S+1 XJ Mcw VSS VLS VT Mass 0 1 0 1 1 1 0 0 0 −+ 1 S0 -0.076 0 0 3.960 1 0 1 1 −− 3 S1 4.037 -0.038 0 0 3.998 2 2 2 −+ 5 S2 0.038 0 0 4.075 0 1 1 1 +− 1 P1 4.143 -0.0010 0 -0.0093 4.133 1 1 0 0 ++ 3 P0 -0.0022 -0.018 4.122 1 1 ++ 3 P0 4.143 -0.0005 -0.0011 -0.0046 4.137 2 2 ++ 3 P2 0.0011 -0.010 4.134 2 1 1 1 +− 5 P1 -0.0033 -0.029 4.112 2 2 +− 5 P2 4.143 0.0005 -0.0011 0.010 4.153 3 3 +− 5 P3 0.0012 -0.014 4.131 1 1 0 1 0 1 1 0 1 1 −− 3 S1 4.037 0 0 0 4.037 0 0 0 −+ 1 S0 -0.076 0 0 3.960 1 1 1 0 1 1 −− 3 S1 4.037 -0.038 0 0 3.998 2 2 2 −+ 5 S2 0.038 0 0 4.075 1 1 1 −− 3 S1 -0.1149 0 0 3.922 2 1 2 0 2 2 −+ 5 S2 4.037 -0.038 0 0 3.998 3 3 3 −− 7 S3 0.076 0 0 4.113 TABLE VI . VIComparison of some predicted states with experimental results (In GeV) Present Experiment State Mass J P C 2S+1 XJ Model Mass J P C. X(3823) 3.823 1 −− 3 S1 Mole 3.823 ± 0.0019[16] ? ?− Zc(3885) 3.882 1 +− 3 S1 TQ 3.883 ±0.0015 ±0.0042 [13] 1 +? Zc(3900) 3.897 1 +− 5 P1 Mole 3.899 ±0.0036 ±0.0049 [1, 2] ? ?− X(3915) 3.917 3 +− 5 P3 Mole 3.917 ± 0.0027 0/2 ?+ 3.919 2 ++ 3 P2 Mole 3.910 1 +− 3 S1 TQ X(3940) 3.935 1 +− 1 P1 Mole 3.942 +0.009 −0.008 [17] ? ?+ 3.939 2 +− 5 P2 Mole Y (4008) 3.998 1 −− 3 S1 Mole 4.008 +0.121 −0.049 [18] 1 −− 3.997 1 −− 3 D1 Mole Zc(4025) 4.023 0 ++ 3 P0 Mole 4.026 ±0.0026 ±0.0037 [11] 1 +? 4.026 1 +− 1 P1 Mole 4.027 2 ++ 3 P2 Mole ψ(4040) 4.037 1 −− 3 S1 Mole 4.039 ± 0.001 [19] 1 −− 4.038 1 ++ 3 P1 Mole Z1(4050) 4.049 4 ++ 7 P4 TQ 4.051 1 +− 5 P1 TQ 4.051 +0.024 −0.043 [20, 21] ? ?? 4.046 2 +− 5 P2 Mole 4.054 3 ++ 7 P3 Mole Y (4140) 4.136 1 −− 3 P0 TQ 4.142 1 +− 5 P3 TQ 4.143 ± 0.003 [22] ? ?+ 4.145 1 −− 1 P1 TQ 4.145 1 −+ 3 P2 TQ 4.137 1 ++ 3 P1 Mole X(4160) 4.153 2 +− 5 P2 Mole 4.156 +0.029 −0.025 [17] ? ?+ 4.164 1 −− 1 P1 TQ 4.159 1 −+ 3 P1 TQ . Z Q Liu, Belle CollaborationPhys. Rev. Lett. 110252002Z. Q. Liu et al. (Belle Collaboration), Phys. Rev. Lett. 110, 252002 (2013). . M Ablikim, BESIII CollaborationPhys. Rev. Lett. 110252001M. Ablikim et al.,(The BESIII Collaboration), Phys. Rev. Lett. 110, 252001 (2013). . R L Jaffe, Phys. Rev. D. 15281R. L. Jaffe, Phys. Rev. D 15, 281 (1977). . R L Jaffe, Phys. Rev. D. 15267R. L. Jaffe, Phys. Rev. D 15, 267 (1977). . J D Weinstein, N Isgur, Phys. Rev. D. 15588J. D. Weinstein and N. Isgur, Phys. Rev. D 15, 588 (1983). M ole-molecular picture. M ole-molecular picture. . T Q]-Tetraquark Picture, T Q]-tetraquark picture. . J D Weinstein, N Isgur, Phys. Rev. D. 412236J. D. Weinstein and N. Isgur, Phys. Rev. D 41, 2236 (1990). . N A Tornqvist, Z. Phys. C. 61525N. A. Tornqvist, Z. Phys. C 61, 525 (1994). . J M Dias, F S Navarra, M Nielsen, Phys. Rev. D. 8016004J. M. Dias, F. S. Navarra, and M. Nielsen, Phys. Rev. D 80, 016004 (2013). . Hong-Wei Ke, Eur. Phys. J. C. 732561Hong-Wei Ke et al., Eur. Phys. J. C 73, 2561 (2013). . M B Voloshin, BESIII CollaborationL B Okun, BESIII Collaboration; M Ablikim, BESIII CollaborationarXiv:1308.2760JETP Lett. 23333M.B. Voloshin, L.B. Okun, JETP Lett. 23, 333 (1976) [11] M. Ablikim et al. [BESIII Collaboration], arXiv:1308.2760. . M Ablikim, BESIII CollaborationarXiv:1309.1896M. Ablikim et al. [BESIII Collaboration], arXiv:1309.1896. . M Ablikim, BESIII CollaborationPhysics review letters. 11222001M. Ablikim et al. [BESIII Collaboration], Physics review letters 112, 022001(2014). . A De Rujula, H Georgi, S L Glashow, Phys. Rev. Lett. 38317A. De Rujula, H. Georgi, S.L. Glashow, Phys. Rev. Lett. 38, 317 (1977). . T Branz, Phys. Rev D. 8054019T Branz et al, Phys. Rev D 80, 054019 (2009). . V Bhardwaj, Belle CollaborationarXiv:1304.3975Phys. Rev. Lett. hep-exV. Bhardwaj et al. [Belle Collaboration], accepted for publication in Phys. Rev. Lett, arXiv:1304.3975 [hep-ex]. . P Pakhlov, Belle CollaborationPhys. Rev. Lett. 100202001P. Pakhlov et al. (Belle Collaboration), Phys. Rev. Lett. 100, 202001 (2008). . C Z Yuan, Belle CollaborationPhys. Rev. Lett. 99182004C. Z. Yuan et al. (Belle Collaboration), Phys. Rev. Lett. 99, 182004 (2007). . J Beringer, Particle Data GroupPhys. Rev. D. 8610001J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012). . R Mizuk, Belle CollaborationPhys. Rev. D. 7872004R. Mizuk et al. (Belle Collaboration), Phys. Rev. D 78, 072004 (2008). . J P Lees, BABAR CollaborationarXiv:1111.5919hep-exJ. P. Lees et al. (BABAR Collaboration), arXiv:1111.5919 [hep-ex]. . T Aaltonen, CDF CollaborationPhys. Rev. Lett. 102242002T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 102, 242002 (2009).

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{'abstract': 'Masses of the low lying four quark states in the hidden charm sector (cqcq; q ∈ u, d) are calculated within the framework of a non-relativistic quark model. The four body system is considered as two two-body systems such as diquark-diantiquark (q1q2 −q3q4) tetraquark states and di-mesonic (q1q2 − q3q4) molecular states. Here, Cornell type potential has been used for describing the two body interactions among q − q,q −q, q −q, qq −qq and qq − qq, with appropriate string tensions. Our present anylysis suggests the following exotic states, X(3823), Zc(3900), X(3915), X(3940), Zc(4025), ψ(4040), Z1(4050) and X(4160) as di-mesonic molecular states, while Zc(3885) and Y (4140) as the diquark-diantiquark tetraquark state. We have been able to assign the J P C values for many of the recently observed exotic states according to their structure.', 'arxivid': '1402.3974', 'author': ['Smruti Patel \nDepartment of Physics\nSardar Patel University\nVallabh VidyanagarINDIA\n', 'Manan Shah \nDepartment of Physics\nSardar Patel University\nVallabh VidyanagarINDIA\n', 'P C Vinodkumar \nDepartment of Physics\nSardar Patel University\nVallabh VidyanagarINDIA\n'], 'authoraffiliation': ['Department of Physics\nSardar Patel University\nVallabh VidyanagarINDIA', 'Department of Physics\nSardar Patel University\nVallabh VidyanagarINDIA', 'Department of Physics\nSardar Patel University\nVallabh VidyanagarINDIA'], 'corpusid': 118477654, 'doi': '10.1140/epja/i2014-14131-9', 'github_urls': [], 'n_tokens_mistral': 7674, 'n_tokens_neox': 6282, 'n_words': 3440, 'pdfsha': '896fa53aec15ff5afbce86faf41ce4b1a9e3bfa0', 'pdfurls': ['https://arxiv.org/pdf/1402.3974v3.pdf'], 'title': ['Mass spectra of four quark states in hidden charm sector', 'Mass spectra of four quark states in hidden charm sector'], 'venue': []}

arxiv

Conservation laws with nonlocality in density and velocity and their applicability in traffic flow modelling 24 Apr 2023 April 25, 2023 Jan Friedrich friedrich@igpm.rwth-aachen.de Institute of Applied Mathematics RWTH Aachen University 52064AachenGermany Department of Mathematics Friedrich-Alexander University of Mannheim 68131MannheimGermany Department Mathematik § Friedrich-Alexander Universität Erlangen-Nürnberg, Competence Unit for Scientific Computing, Martensstr. 5a Universität Erlangen-Nürnberg Cauerstr. 1191058, 91058Erlan-gen, ErlangenGermany, Germany Simone Göttlich goettlich@uni-mannheim.de Alexander Keimer alexander.keimer@fau.de Lukas Pflug lukas.pflug@fau.de Conservation laws with nonlocality in density and velocity and their applicability in traffic flow modelling 24 Apr 2023 April 25, 2023 In this work we present a nonlocal conservation law with a velocity depending on an integral term over a part of the space. The model class covers already existing models in literature, but it is also able to describe new dynamics mainly arising in the context of traffic flow modelling. We prove the existence and uniqueness of weak solutions of the nonlocal conservation law. Further, we provide a suitable numerical discretization and present numerical examples. Modelling equations In recent years, the mathematical analysis on nonlocal conservation laws [1,2,9,18] has drawn increased attention. These nonlocal models are capable to describe a variety of applications such as traffic flow modelling [3,5,7,8,15,21,23,24], supply chains [13,17,22], sedimentation processes [4,28], pedestrian dynamics [11], particle growth [26,29], crowd dynamics and population modelling [12,25] as well as opinion formation [20,27]. By a nonlocal conservation law we mean a conservation law, in which the velocity depends on a space dependent integral term of some quantity of interest. In the literature, there are two main approaches: on the one hand the averaging is done over the density which is then used to determine the velocity, e.g. [4,5,9]; on the other hand, the averaging is done directly over the velocity [14,15,28]. Both approaches can be found in applications for traffic flow and sedimentation. Here, we want to consider a unifying approach, incorporating both averaging into one equation. Consider that 1 , 2 : R → R are given velocities, we study the nonlocal conservation law + 1 * 2 ( ) = 0 ( , ) ∈ (0, ) × R (0, ) = 0 ( ) ∈ R(1) with * 2 ( ) ( , ) ≔ ∫ ∞ ( − ) 2 ( ( , )) d . with 0 denoting the initial density and a weight function appearing in the nonlocal term. As previously stated, the equation entails also the pure nonlocality in the velocity ( 1 ≡ Id) and the pure nonlocality in the density ( 2 ≡ Id). Let us explain the idea behind the proposed model (1) for traffic flow in greater detail: Drivers adapt their velocity according to a mean of some quantity of interest. This quantity is computed out of the density by the function 2 . Then, the function 1 transforms the quantity of interest into a velocity. In particular, for 2 ≡ Id the quantity of interest is the density and 1 a suitable velocity function such that we obtain the averaging over the density, see [5]. If 2 transforms the density directly into a velocity, we choose 1 ≡ Id and obtain the averaging over the velocity as proposed in [15]. But besides these two choices the model (1) is capable to describe much more effects. We will present examples in section 4. Existence and uniqueness of solutions In this section, we shortly discuss the well-posedness, i.e. the existence and uniqueness of solutions. We will also dwell on the maximum principle, a quite important property as it tells that the time dependent density is always bounded between a minimal and maximal density which stems from the initial density on the road. For traffic applications this property is crucial to have. We start with an existence and uniqueness result on small time horizon and require for now the following assumptions: Assumption 2.1 (Minimal assumptions on the involved input data). We assume that • 0 ∈ ∞ (R; R ≥0 ) ∩ (R), • 1 , 2 ∈ 1,∞ loc (R), • ∈ (R >0 ; R ≥0 ) with (R) ≔ ∈ 1 loc (R) : | | (R) < ∞ and (R) ≔ 1 (R) + supp ∈ ∞ c (R): ∞ (R) ≤1 ∫ R ′ ( ) ( ) d , ∈ 1 loc (R). As we will talk about weak solutions, we just mention that we mean with weak solutions the canonical definition (see for instance [18,Definiton 2.13]). Theorem 2.2 (Existence and uniqueness on a small time horizon). Let assumption 2.1 hold. Then, there exists ∈ R >0 so that (1) admits a unique weak solution ∈ [0, ]; L 1 (R) ∩ ∞ (0, ); (R) . Proof. The proof consists of applying a fixed-point approach on the nonlocal term and the solution as follows. Assume that we know the solution to the balance law , we can plug it into the nonlocal operator and have * 2 ( ) ( , ) = ∫ ∞ ( − ) 2 ( ( , )) d from which we can compute the "velocity" of the conservation law as ( , ) ↦ → 1 * 2 ( ) ( , ) , ( , ) ∈ (0, ) × R.(2) Assuming that has the postulated regularity [0, ]; L 1 (R) ∩ ∞ (0, ); (R) , we can observe that the velocity is Lipschitz-continuous. We can thus invoke characteristics and state the solution once more as ( , ) = 0 ( ( , ; 0)) 2 ( , ; 0),(3) where is the solution to the characteristics, i.e. ( , ; ) = + ∫ 1 * 2 ( ) ( , ( , ; )) d , ( , , ) ∈ (0, ) × R × (0, ). As depends on the solution , the identity in eq. (3) is actually a fixed-point problem in the solution . By means of Banach's fixed-point theory we can then prove that a unique solution to eq. (3) exists. The uniqueness carries over from the fixed-point argument. As can be seen, the choice of topology is crucial to obtain a unique solution and the small time horizon guarantees that the fixed-point mapping is a contraction (involving the properties of the characteristics). We do not detail this further, but just mention that these approaches have been used in several publications [3], often not in the solution but in the nonlocal term [18]. Having established the well-posedness of solutions, we can restrict the velocities 1 , 2 in a reasonable way -from an application point of view -so that a maximum principle holds which itself implies the existence and uniqueness of solutions on any arbitrary time horizon. Theorem 2.3 (Maximum principle/existence of solutions on finite time horizon). Let assumption 2.1 hold and assume in addition that ′ 1 ≦ 0 ∧ ′ 2 ≧ 0 ∨ ′ 1 ≧ 0 ∧ ′ 2 ≦ 0 on ess-inf ∈R 0 ( ), ess-sup ∈R 0 ( ) monotonically decreasing.(4) Then, for every ∈ R >0 there is a unique weak solution ∈ [0, ]; 1 (R) ∩ ∞ ((0, ); (R)) of the nonlocal conservation law in eq. (1), and the following maximum principle holds: ess-inf ∈R 0 ( ) ≤ ( , ) ≤ ess-sup ∈R 0 ( ) ( , ) ∈ (0, ) × R a.e..(5) Proof. We only sketch the proof and assume that the solution to the conservation law with nonlocality as in eq. (1) is smooth and compactly supported (this can be obtained by a classical approximation argument, we do not detail here). Then, we can assume that we are at a spatial location˜ ∈ R, where the maximum is attained. Recalling the differential equation which is due to the higher regularity now satisfied in its strong version, we obtain for the time derivative at˜ and any given time ∈ [0, ] ( ,˜ ) = − ′ 1 ( * 2 ( ( , ·)) (˜ )) 2 * 2 ( ( , ·)) (˜ ) ( ,˜ ) (6) − 1 * 2 ( ( , ·)) (˜ ) 2 ( ,˜ ) .(7) Recalling that˜ was one of the maximal points, we know that 2 ( ,˜ ) = 0 and get = − ′ 1 ( * 2 ( ( , ·)) (˜ )) 2 * 2 ( ( , ·)) (˜ ) ( ,˜ ) ≥0 .(8) Let us distinguish two cases: • Assume that 1 is monotonically increasing (i.e., ′ 1 ≧ 0) and 2 monotonically decreasing (i.e., ′ 2 ≦ 0) as well as monotonically decreasing, we can continue the computations and have . (8) = − ′ 1 ( * 2 ( ( , ·)) (˜ )) ≤0 2 * 2 ( ( , ·)) (˜ ) ≥0 ( ,˜ ) ≥0 Thus, the time derivative at the maximal points is negative, resulting in the mentioned upper bounds on the solution. • Assume that 1 is monotonically decreasing (i.e., ′ 1 ≧ 0) and 2 monotonically increasing (i.e., ′ 2 ≦ 0) we can argue similarly to obtain again the upper bound on the solution. For the lower bound, one can consider the time derivative at a minimal point and obtain that the derivative is then nonnegative. Finally, having uniform bounds as in eq. (5), the solution can be extended to any finite time horizon as the velocity in eq. (2) will remain uniformly Lipschitz-continuous with a Lipschitz-constant independent of the time considered. Thus, the typical clustering in time argument can be applied leading to the existence and uniqueness of weak solutions on any finite time horizon. Remark 2.4 (Reasonability for the sign restriction in eq. (4) and ). The sign restrictions on the velocities are quite reasonable as one of them should be a decreasing function with regard to the traffic density while, when both of them would be decreasing or increasing, the composed velocity as in eq. (2) would not be decreasing with regard to the traffic density, something nonphysical from the point of view of traffic. The assumption on the nonlocal kernel acting at the position ∈ R from to possibly ∞ and being according to eq. (4) monotonically increasing can be understood from a traffic's perspective as follows: The velocity of the current density is only adjusted based on what is ahead in traffic, and the further away the traffic is, the less impact (or even no impact in the case that is compactly supported) the traffic information will have. This is in line with previous assumptions on the kernel having been explored in [1,2,3,5,9,15,18,21]. Remark 2.5 (Monotonicity preserving dynamics under more restrictive velocity 1 ). Another interesting fact worth mentioning is the monotonicity preserving dynamics, provided that eq. (4) holds and in addition ′′ 1 ≤ 0, thus necessitating 1 ∈ 2,∞ loc (R). Consider again smooth solutions and assume for now (only for reasons of simplicity) that we have a piecewise constant kernel, i.e. * 2 ( ) ( , ) = 1 ∫ + 2 ( ( , )) d , ( , ) ∈ [0, ] × R (the estimate can of course be made for general monotonically decreasing kernels of sufficient regularity). Then, recall eqs. (6) to (7) and assume that the initial datum is monotonically increasing and that we are at a time ∈ [0, ] so that for the first time there exists˜ ∈ R with ( , ) =˜ = 0, i.e., a point where the monotonicity might break. Then, we manipulate terms as follows: − , ( , ) = 1 2 ′′ 1 (. . .) 2 ( ( , + )) − ( , ) 2 ( , ) + 1 ′ 1 (...) ′ 2 ( ( , + )) ( + ) − ′ 2 ( ( , )) ( ) ( , ) + 1 2 ′ 1 (...) 2 ( ( , + )) − 2 ( ( , ))) ( , ) + 1 . . . ( , ) and evaluate now at =˜ so that ( , ) ≥ 0 ∀ ∈ R, ( , ) =˜ = 0, 2 ( , ) =˜ = 0 (thus assuming ) is minimal at =˜ ) 2 ( ,˜ ) = − 1 2 ′′ 1 (...) 2 ( ( ,˜ + )) − 2 ( ( ,˜ )) 2 ( ,˜ ) − 1 ′ 1 (...) ′ 2 ( ( ,˜ + )) 2 ( ,˜ + ) − ′ 2 ( ( ,˜ )) 2 ( ,˜ ) ( ,˜ ) − 1 2 ′ 1 (...) ( 2 ( ( ,˜ + )) − 2 ( ( ,˜ ))) 2 ( ,˜ ) using that ′′ 1 ≤ 0 and that ( , ) =˜ = 0 as well as the sign restrictions on ′ 1 , ′ 2 = − ′ 1 (...) ′ 2 ( ( ,˜ + )) ≥0 2 ( ,˜ + ) ( ,˜ ) ≥0 ≥ − ′ 1 (...) ′ 2 ( ( ,˜ + )) ≥0 2 ( ,˜ ) ( ,˜ ) ≥0 = 0. Thus, the solution remains monotonically increasing for all times. For monotonically decreasing initial datum, we would require ′′ 1 ≧ 0, and could then establish similarly that the solutions then remains decreasing. This assumption is somewhat in line with observations in [19,Theorem 4.13 & Theorem 4.18], where exactly the same assumption can be found for the simpler case 2 ≡ Id. Choosing 1 ≡ Id, we obtain on the other hand no restrictions on 2 to preserve monotonicity which is in line with [14,Theorem 5.1]. Numerical discretization To construct an approximate solution we adapt the approaches presented in [15,16] to the setting in eq. (1). In particular, we rely on a Godunov-type scheme. Hence, we discretize space and time by an equidistant grid with the step sizes Δ ∈ R >0 in space and Δ ∈ R >0 in time, such that = Δ with ∈ N describes the time mesh and = Δ , ∈ Z the cell centres of the space mesh with the cell interfaces − 1 2 and + 1 2 . The finite volume approximation Δ is given by Δ ( , ) = for ( , ) ∈ [ , +1 ) × − 1 2 , + 1 2 and we approximate the initial data by 0 = 1 Δ ∫ + 1 2 − 1 2 0 ( ) , ∈ Z.(9) Following [15] the scheme is given by +1 = − − −1 −1 with ≔ Δ Δ ,(10) with the nonlocal term . To compute this term numerically we need to restrict the support of the kernel on an interval [0, ] with > 0. For a spatial step size of Δ , must be chosen such that ∫ ∞ d = O (Δ ) to maintain first order accuracy. If the support of the kernel is already compact, we simply choose as the supremum of the support. Then, the nonlocal term is approximated for ≔ ⌊ /Δ ⌋ by ≔ 1 −1 =0 2 ( + +1 ) with = ∫ ( +1)Δ Δ ( ) .(11) The weights need to be computed exactly and the CFL condition is given by ≤ 1 0 ′ 1 ∞ ( 2 ) ′ 2 ∞ (Id) ess-sup ∈R 0 ( ) + 1 ∞ ( 2 ) ,(12) where we use the notation · ∞ ( ) ≔ · ∞ ( (essinf ∈R ( 0 ( )),esssup ∈R ( 0 ( )))) for simplicity. We now prove that the numerical discretization fulfills the same maximum principle as the analytical solution. Proof. Following closely the proof of [15, Theorem 3.1] it turns out that we only need to consider the difference between two nonlocal velocities. If they satisfy −1 − ≤ ′ 1 ∞ ( 2 ) ′ 2 ∞ (Id) 0 ( − ), then we obtain +1 ≤ by following the corresponding steps in [15, Theorem 3.1], using the CFL condition (12) and the scheme (10). For ∈ Z and a fixed ∈ N with > 0 we obtain −1 − = 1 −1 =0 2 ( + ) − 1 −1 =0 2 ( + +1 ) = ′ 1 ( ) −1 =1 ( − −1 ) 2 ( + ) − −1 2 ( + ) + 0 2 ( ) with ∈ R appropriatly chosen (it exists thanks to the mean value theorem), ≤ ′ 1 ( ) −1 =1 ( − −1 ) 2 ( ) − −1 2 ( ) + 0 2 ( ) which holds since ≤ −1 for = 1, . . . , − 1 and because of the signs of ′ 1 and ′ 2 in (4), = ′ 1 ( ) 0 2 ( ) − 2 ( ) ≤ ′ 1 ∞ ( 2 ) ′ 2 ∞ (Id) 0 ( − ), where we use again the signs of ′ 1 and ′ 2 given by eq. (4). Analogously, we can prove −1 − ≥ ′ 1 ∞ ( 2 ) ′ 2 ∞ (Id) 0 ( − ) which can be used to prove the lower bound. We note that following similar steps as in [15], it is possible to derive bounded variation estimates on the approximate solution such that the convergence of the scheme against a weak solution can be obtained. We do not go into details here. . We are interested in the approximate solution at the time = 0.5 for a spatial discretization given by Δ = 10 −3 and a time grid size of Δ = Δ /(3 0 + 1). We note that the CFL condition is slightly stricter than the one given by (12), but it allows to choose the same CFL condition in all simulations. In the first example, drivers might not be able to perceive the true density on a road. They only estimate the observed density ahead of them and base their velocity on this estimation. Hence, 2 expresses the estimated density in dependence of the true density. A possible choice is 2 ( ) = + (1 − ) for ∈ [−1, 1] \ {0}, i.e. underestimation of the density for < 0 and overestimation for > 0. Further, if 0 ∈ [0, 1], it follows ′ 2 ≥ 0. Here, 1 is the velocity function to determine the velocity out of the estimated density with ′ 1 ≤ 0, e.g. 1 ( ) = 1− 2 . Figure 1 shows the approximate solutions for different values of . In particular, the case = 0 is the nonlocal in density model, in which the drivers have perfect knowledge of the density. It can be seen that an underestimation of the velocity results in a higher density while for overestimation the density is lower. Further, in the case of underestimation the density is located further downstream. Numerical examples Even more interesting is the case that drivers consider a mixture of relative velocity and relative density for their movement. Hence, for a suitable velocity function (meaning ′ ≤ 0) to estimate the velocity and ∈ [0, 1], which expresses the preference to adapt more according to the density or the velocity, a quantity of interest might be for different values of and 2 ( ) as in (13) with ( ) = 1 − 2 , max = max = 1 and 1 ( ) = (1 − ) 2 . Interestingly, we can see that if more preference is given to the velocity, the front end of the traffic jam moves faster, the peak of the traffic jam decreases, but at the back end of the traffic jam the density is higher. Remark 4.1. We note that the monotonicity property in remark 2.5 can be seen in fig. 1 and fig. 2. In particular, in the first example ′′ 1 < 0 holds and hence the monotonicity is only kept for the increasing part, see fig. 1, and in the second example due to ′′ 1 > 0 the montonicity is kept on the decreasing part. Conclusion We have presented results on the well-posedness of a nonlocal conservation law incorporating both approaches of space averaging, i.e., mean density and mean velocity. Numerical examples demonstrate the performance of the model. Future work may include studying the behavior of the model when the kernel function tends to the Dirac delta. This question has been intensively studied in the literature for the case Theorem 3. 1 ( 1Discrete version of the maximum principle). Given assumption 2.1 and the conditions in eq. (4), for a given initial datum 0 , ∈ Z as in eq. (9) with = min ∈Z 0 and = max ∈Z 0 , the scheme (10)-(11) fulfills under the CFL condition (12) ≤ ≤ , ∈ Z, ∈ N. Figure 1 : 1Approximate solution of (1) at = 0.5 with 2 ( ) = + (1 − ) and 1 ( ) = 1 − 2 . Let us present two examples which can be described by eq. (1) and give a suitable interpretation from a traffic modelling point of view. During this section we set ( ) Figure 2 : 2Approximate solution of (1) at = 0.5 with 2 as in (13) and 1 ( ) = (1 − ) 2 . ( ) = max + (1 − ) 1 − ( ) max .(13)Again 1 transforms this quantity into a velocity.Figure 2displays the approximate solutions ≡ Id and 1 ≡ Id, see e.g.[2,6,10,14,19] and the references therein, but so far not for the general case of eq. (1). Nonlocal systems of conservation laws in several space dimensions. A Aggarwal, R M Colombo, P Goatin, SIAM J. Numer. Anal. 532A. Aggarwal, R.M. Colombo, and P. Goatin. Nonlocal systems of conservation laws in several space dimensions. SIAM J. Numer. Anal., 53(2):963-983, 2015. On the numerical integration of scalar nonlocal conservation laws. P Amorim, R M Colombo, A Teixeira, ESAIM Math. Model. Numer. Anal. 491P. Amorim, R.M. Colombo, and A. Teixeira. On the numerical integration of scalar nonlocal conservation laws. ESAIM Math. Model. Numer. Anal., 49(1):19-37, 2015. Modeling multilane traffic with moving obstacles by nonlocal balance laws. A Bayen, J Friedrich, A Keimer, L Pflug, T Veeravalli, SIAM J. Appl. Dyn. Syst. 212A. Bayen, J. Friedrich, A. Keimer, L. Pflug, and T. Veeravalli. Modeling multilane traffic with moving obstacles by nonlocal balance laws. SIAM J. Appl. Dyn. Syst., 21(2):1495- 1538, 2022. On nonlocal conservation laws modelling sedimentation. F Betancourt, R Bürger, K H Karlsen, E M Tory, Nonlinearity. 243855F. Betancourt, R. Bürger, K.H. Karlsen, and E.M. Tory. On nonlocal conservation laws modelling sedimentation. Nonlinearity, 24(3):855, 2011. Well-posedness of a conservation law with non-local flux arising in traffic flow modeling. S Blandin, P Goatin, Numer. Math. 1322S. Blandin and P. Goatin. Well-posedness of a conservation law with non-local flux arising in traffic flow modeling. Numer. Math., 132(2):217-241, 2016. Entropy admissibility of the limit solution for a nonlocal model of traffic flow. A Bressan, W Shen, Commun. Math. Sci. 195A. Bressan and W. Shen. Entropy admissibility of the limit solution for a nonlocal model of traffic flow. Commun. Math. Sci., 19(5):1447-1450, 2021. Micro-macro limit of a nonlocal generalized Aw-Rascle type model. F Chiarello, J Friedrich, P Goatin, S Göttlich, SIAM J. Appl. Math. 804F. Chiarello, J. Friedrich, P. Goatin, and S. Göttlich. Micro-macro limit of a nonlocal generalized Aw-Rascle type model. SIAM J. Appl. Math., 80(4):1841-1861, 2020. Non-local multi-class traffic flow models. F A Chiarello, P Goatin, Netw. Heterog. Media. 14371F. A. Chiarello and P. Goatin. Non-local multi-class traffic flow models. Netw. Heterog. Media, 14:371, 2019. Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel. F A Chiarello, P Goatin, ESAIM Math. Model. Numer. Anal. 521F.A. Chiarello and P. Goatin. Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel. ESAIM Math. Model. Numer. Anal., 52(1):163-180, 2018. Nonlocal traffic models with general kernels: singular limit, entropy admissibility, and convergence rate. M Colombo, G Crippa, E Marconi, L V Spinolo, Arch. Ration. Mech. Anal. 247218M. Colombo, G. Crippa, E. Marconi, and L. V. Spinolo. Nonlocal traffic models with general kernels: singular limit, entropy admissibility, and convergence rate. Arch. Ration. Mech. Anal., 247(2):18, 2023. A class of nonlocal models for pedestrian traffic. R M Colombo, M Garavello, M Lécureux-Mercier, Math. Models Methods Appl. Sci. 22041150023R.M. Colombo, M. Garavello, and M. Lécureux-Mercier. A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci., 22(04):1150023, 2012. Nonlocal crowd dynamics models for several populations. R M Colombo, M Lécureux-Mercier, Acta Math. Sci. 321R.M. Colombo and M. Lécureux-Mercier. Nonlocal crowd dynamics models for several populations. Acta Math. Sci., 32(1):177-196, 2012. Analysis of a conservation law modeling a highly re-entrant manufacturing system. J.-M Coron, M Kawski, Z Wang, Disc. Contin. Dyn. Syst. Ser. B. 144J.-M. Coron, M. Kawski, and Z. Wang. Analysis of a conservation law modeling a highly re-entrant manufacturing system. Disc. Contin. Dyn. Syst. Ser. B, 14(4):1337-1359, 2010. Conservation laws with nonlocal velocity-the singular limit problem. J Friedrich, S Göttlich, A Keimer, L Pflug, arXiv:2210.12141arXiv preprintJ. Friedrich, S. Göttlich, A. Keimer, and L. Pflug. Conservation laws with nonlocal velocity-the singular limit problem. arXiv preprint arXiv:2210.12141, 2022. A godunov type scheme for a class of LWR traffic flow models with non-local flux. J Friedrich, O Kolb, S Göttlich, Netw. Heterog. Media. 13531J. Friedrich, O. Kolb, and S. Göttlich. A godunov type scheme for a class of LWR traffic flow models with non-local flux. Netw. Heterog. Media, 13:531, 2018. Numerical schemes for a class of nonlocal conservation laws: a general approach. J Friedrich, S Sudha, S Rathan, Netw. Heterog. Media. to appearJ. Friedrich, S. Sudha, and S. Rathan. Numerical schemes for a class of nonlocal conser- vation laws: a general approach. Netw. Heterog. Media, to appear. Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. M Gugat, A Keimer, G Leugering, Z Wang, Netw. Heterog. Media. 104M. Gugat, A. Keimer, G. Leugering, and Z. Wang. Analysis of a system of nonlo- cal conservation laws for multi-commodity flow on networks. Netw. Heterog. Media, 10(4):749-785, 2015. Existence, uniqueness and regularity results on nonlocal balance laws. A Keimer, L Pflug, J. Differential Equations. 263A. Keimer and L. Pflug. Existence, uniqueness and regularity results on nonlocal balance laws. J. Differential Equations, 263:4023-4069, 2017. On approximation of local conservation laws by nonlocal conservation laws. 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Keimer, M. Singh, and T. Veeravalli. Existence and uniqueness results for a class of nonlocal conservation laws by means of a lax-hopf-type solution formula. J. Hyperbolic Differ. Equ., 17(04):677-705, 2020. Nonlocal multi-scale traffic flow models: analysis beyond vector spaces. P E Kloeden, T Lorenz, Bull. Math. Sci. 63P.E. Kloeden and T. Lorenz. Nonlocal multi-scale traffic flow models: analysis beyond vector spaces. Bull. Math. Sci., 6(3):453-514, 2016. Thresholds for shock formation in traffic flow models with nonlocal-concaveconvex flux. Y Lee, J. Differential Equations. 2661Y. Lee. Thresholds for shock formation in traffic flow models with nonlocal-concave- convex flux. J. Differential Equations, 266(1):580 -599, 2019. Nonlocal hyperbolic population models structured by size and spatial position: Well-posedness. T Lorenz, Discrete Contin. Dyn. Syst. Ser. B. 2484547T. Lorenz. Nonlocal hyperbolic population models structured by size and spatial position: Well-posedness. 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{'abstract': 'In this work we present a nonlocal conservation law with a velocity depending on an integral term over a part of the space. The model class covers already existing models in literature, but it is also able to describe new dynamics mainly arising in the context of traffic flow modelling. We prove the existence and uniqueness of weak solutions of the nonlocal conservation law. Further, we provide a suitable numerical discretization and present numerical examples.', 'arxivid': '2302.12797', 'author': ['Jan Friedrich friedrich@igpm.rwth-aachen.de \nInstitute of Applied Mathematics\nRWTH Aachen University\n52064AachenGermany\n\nDepartment of Mathematics\nFriedrich-Alexander\nUniversity of Mannheim\n68131MannheimGermany\n\nDepartment Mathematik\n§ Friedrich-Alexander Universität Erlangen-Nürnberg, Competence Unit for Scientific Computing, Martensstr. 5a\nUniversität Erlangen-Nürnberg\nCauerstr. 1191058, 91058Erlan-gen, ErlangenGermany, Germany\n', 'Simone Göttlich goettlich@uni-mannheim.de ', 'Alexander Keimer alexander.keimer@fau.de ', 'Lukas Pflug lukas.pflug@fau.de '], 'authoraffiliation': ['Institute of Applied Mathematics\nRWTH Aachen University\n52064AachenGermany', 'Department of Mathematics\nFriedrich-Alexander\nUniversity of Mannheim\n68131MannheimGermany', 'Department Mathematik\n§ Friedrich-Alexander Universität Erlangen-Nürnberg, Competence Unit for Scientific Computing, Martensstr. 5a\nUniversität Erlangen-Nürnberg\nCauerstr. 1191058, 91058Erlan-gen, ErlangenGermany, Germany'], 'corpusid': 257206002, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8820, 'n_tokens_neox': 7699, 'n_words': 4392, 'pdfsha': '8fec84748212577935dc362c10c00d39d6ce0dea', 'pdfurls': ['https://export.arxiv.org/pdf/2302.12797v2.pdf'], 'title': ['Conservation laws with nonlocality in density and velocity and their applicability in traffic flow modelling', 'Conservation laws with nonlocality in density and velocity and their applicability in traffic flow modelling'], 'venue': []}

arxiv

Strain Induced Enhanced Photocatalytic Activities in Layered Two Dimensional C 2 N/MoS 2 Heterostructure: A Meta-GGA Study 10 Jan 2023 Soumendra Kumar Das School of Physical Sciences National Institute of Science Education and Research (NISER) Bhubaneswar HBNI Khurda-752050JatniOdishaIndia Lokanath Patra Department of Mechanical Engineering University of California Santa Barbara 93106CAUSA Prasanjit Samal psamal@niser.ac.in School of Physical Sciences National Institute of Science Education and Research (NISER) Bhubaneswar HBNI Khurda-752050JatniOdishaIndia Pratap K Sahoo pratap.sahoo@niser.ac.in School of Physical Sciences National Institute of Science Education and Research (NISER) Bhubaneswar HBNI Khurda-752050JatniOdishaIndia Strain Induced Enhanced Photocatalytic Activities in Layered Two Dimensional C 2 N/MoS 2 Heterostructure: A Meta-GGA Study 10 Jan 2023Submitted to: 2D Mater.2Photocatalytic water splittingDFTVan der Waals heterostructuresstrainType-II Band alignment The improved photocatalytic water splitting using 2D materials has technological importance for economically viable renewable energy. The present study focuses on the effect of uniaxial, biaxial, and vertical strain on the energy gap and band edge positions of C 2 N/MoS 2 van der Waals heterostructures through first-principles density functional theory using PBE and SCAN functionals. The calculations establish that SCAN functional provides comparatively much better results as compared to the PBE for the band gap and band alignment study. The heterostructure exhibits a type-II band alignment which is beneficial for the efficient separation of charge carriers. For a good photocatalyst, the band edge positions should straddle the water redox potentials. It is observed that for both compressive and tensile vertical strain, the water redox potential values lie within the valence band maximum (VBM) and conduction band minimum (CBM) of the heterostructure. On the other hand, for uniaxial and biaxial strain, the system can be used as a useful photocatalyst only for larger compressive strain, whereas for tensile strain, the energy gap between VBM and CBM keeps on decreasing and lie within the water oxidation/reduction potential. Our study also establishes that the meta-GGA SCAN functional shows similar results as compared to the computationally expensive hybrid HSE functionals. The present work can be extremely useful for experimentalists to design artificial heterostructure devices for better performance in photocatalytic water splitting. Introduction The emergence of photocatalytic water splitting has been a successful technology to meet the demand for the energy crisis and environmental pollution created by our fastgrowing economy. The development of high-performance photo-catalytic materials to create hydrogen by using solar energy has been a serious focus of research for many years [1,2]. The key factor for achieving highly efficient photocatalysts (PCs) is that the band gap should be larger than the water redox potentials. More specifically, the conduction band minimum (CBM) of the PCs should be above the H + /H 2 potential and the valence band maximum (VBM) should be below H 2 O/O 2 potential simultaneously, thus requiring a minimum band gap of 1.23 eV [3]. In addition, literature reports have established the importance of co-catalysts for boosting the electron-hole separation and improving the reaction kinetics [4]. Under such circumstances, two-dimensional materials like graphene, hexagonal boron nitride (h-BN) mono layers, transition metal dichalcogenides (TMDCs), C 3 N 4 , C 2 N, etc, have created a lot of interest, in meeting the demand, because of their novel electronic, thermal and optoelectronic properties. In particular, MoS 2 has a direct band gap (2 eV), and high carrier mobility in the form of a single monolayer, which makes it an important candidate for photocatalytic and photovoltaic applications [5,6]. Similarly, the porous C 2 N monolayer is found to be a direct band gap semiconductor with a gap of 1.96 eV [7]. Zhao et al. have adopted a 2D/2D polymeric Z-scheme heterostructure by using a pair of ultrathin g-C 3 N 4 nanosheets in order to provide H 2 -and O 2 -evolving photocatalysts through the strategy of electrostatic self-assembly. Using Pt and Co(OH) 2 as co-catalysts, the heterostructure achieved a solar-to-hydrogen efficiency of 1.16 % which originates due to the formation of direct Z-scheme charge transfer pathway through the interface between H 2 -and O 2 -evolving components [8]. It has been suggested that the use of C 2 N and MoS 2 can be highly efficient for photocatalytic study and also can be complementary to the use of graphene and h-BN [9]. Despite the extensive use of C 2 N and MoS 2 , there are some challenges as well for the application of these materials for photocatalytic study. The charge distribution of the valence band maximum (VBM) and conduction band minimum (CBM) states for these systems are not well separated in space resulting in reduced light absorbing efficiency because of the recombination of the photoinduced electrons and holes [7,10]. Therefore, attempts have been made to use van der Waals heterostructures to fix the issues. The electronic properties of C 2 N/InSe heterostructure are found to be greatly affected by vertical strain and electric field. Without any electric field, the heterostructure possesses a type-II band alignment with an indirect band gap of 1.34 eV at an equilibrium interlayer distance of 3.325Å. Application of an electric field or a change in interlayer distance results in a transition from type-II to type-I band alignment and indirect to direct band gap in this heterostructure [11]. Band gap and band offset engineering at C 2 N/MSe 2 (M = Mo, W) interface have established that the heterostructure possesses a narrow indirect band gap with type-II band alignment which is favourable for the photogenerated electron-hole pairs. The application of vertical strain and electric field strongly modulate the magnitude of band gap values and band offsets, but the type-II band alignment nature remains preserved [12]. Strain engineering plays a significant role in tuning the electronic properties and photocatalytic performance of 2D heterostructures. Wang et al. have studied the effect of strain on the electronic structure of C 2 N/MTe (M= Ga, In) through DFT calculations which exhibit excellent optical properties with good structural stability. It was observed that for C 2 N/GaTe heterostructure, the exciton-Bohr radius remains unaffected by the application of strain. On the other hand, compressive strain reduces the exciton-Bohr radius in C 2 N/InTe system. The power conversion efficiency shows an increase up to 22.1% for C 2 N/GaTe with 4% strain and 19.8% for C 2 N/GaTe heterostructure with 6% strain [13]. Han et al. reported that the catalytic efficiency of C 2 N/SiH heterojunction can be effectively adjusted by the application of -2% and +4% biaxial strain for the hydrogen evolution reaction (HER) and oxygen evolution reaction (OER), respectively [14]. The Cs 3 Bi 2 I 9 /C 2 N heterostructure exhibits a charge distribution across the whole structure due to the difference in the work function between the two monolayers and a charge transfer at the interface due to the formation of an internal electric field [15]. The photocatalytic study in MoS 2 /ZnO heterostructure shows an indirect band gap with type-II band alignment, a significant built-in potential of 7.42 eV, and a valence band offset of 1.23 eV across the interface. The photogenerated carriers are localized in different layers and can effectively generate hydrogen energy. In contrast, the MoSe 2 /ZnO heterostructure possesses a type-I band alignment with a direct band gap of 1.80 eV, built-in potential around 3.64 eV, and a valence band offset of 0.34 eV [16]. Despite the extensive studies on C 2 N based van der Waals heterostructures, significant progress has not been achieved both theoretically and experimentally. As a typical case, reports on the C 2 N/MoS 2 heterostructures are very scarce and a systematic investigation of the physical properties of this system is still lacking. In this article, we report the effect of vertical, uniaxial, and biaxial strain on the photocatalytic water splitting performance of C 2 N/MoS 2 van der Waals heterostructures through firstprinciples electronic structure calculations. Results and Discussion The schematic of the crystal structure of C 2 N monolayer, MoS 2 layer, and C 2 N/MoS 2 heterostructure is illustrated in Fig. 1. In C 2 N monolayer, twelve C-N bonds are alternately connected to the six C-C bonds in such a way that the periodic holes are present without any atom in the middle of the lattice. The in-plane bond length is estimated to be around 1.43 and 1.47Å for the C-C bond and 1.34Å for the C-N bond, respectively which is also close to the previously reported values [17]. The optimized lattice constant using PBE functional is estimated to be 8.33Å and 3.19Å for the C 2 N and MoS 2 monolayer, respectively which is in reasonable agreement with the reported experimental and theoretical results [18,19]. The heterostructure is constructed by making a supercell of (3×3×1) for C 2 N and (8×8×1) for MoS 2 layer, respectively. The corresponding lattice mismatch between the two monolayers is around 1.9% which is within the acceptable range. The heterostructure after ionic relaxation results in a corrugated structure (Fig. 1d), very similar to silicene. The initial van der Waals gap (3.5Å) between the layers reduces to 3.373Å with an optimized lattice constant of 25.47Å (Fig. 1c, d). The calculated electronic structure of free-standing C 2 N monolayer, MoS 2 using the PBE functional are given in Fig. 2a,b. The band dispersion for both the material indicates the presence of valence band maximum (VBM) and conduction band minimum (CBM) at the same k value in the Brillouin zone. In addition, for both structures, the VBM remains close to the Fermi level as compared to the CBM. This confirms that both C 2 N and MoS 2 monolayers are p-type direct band gap semiconductors and the calculated energy gap values are estimated to be 1.735 eV and 1.645 eV, respectively. It is interesting to note that the energy band dispersion in C 2 N monolayer is relatively flat where as MoS 2 shows highly dispersive bands both in VB and CB regions. Hence, it is expected that the MoS 2 system will have a relatively low electron and hole effective mass along the high symmetry path (Γ-M-K-Γ) as compared to that of C 2 N monolayer. Therefore, the electron and hole mobilities of MoS 2 would be larger than that of the C 2 N monolayer, since the effective mass is inversely proportional to the carrier mobility. Figure 2e illustrates the electronic structure of C 2 N/MoS 2 heterostructure which preserves the direct band gap behaviour of the isolated monolayers. The VBM is situated close to the Fermi level suggesting that the charge carriers are of p-type. However, the value of the energy gap is comparatively reduced and becomes 1.353 eV. We note that the ideal band gap for a semiconductor to use more visible light is around 1.5 eV [20]. It is well known that the PBE functional severely underestimates the energy gap of the system. Again calculations involving the hybrid density functional like HSE-06, B3LYP, etc are highly computationally expensive which is beyond the computational resources available to us. Therefore we have further investigated the systems with SCAN meta-GGA functional. The obtained results are consistent with the experiment as well as previously reported simulated results. The contribution of different orbitals at the band edges is evident from the total and orbital projected density of states analysis as given in Fig. 2b-f. From Fig. 2b, it is clear that the valence band of C 2 N is dominated by the C '2p' states with a significant contribution from N '2p' states as well. The CBM minimum is also populated by the hybridization of the 'p' states of C and N atoms. The partial DOS analysis of MoS 2 establishes that the VBM is highly dominated by Mo '4d' states with a relatively less population than S '3p' states. On the other hand, the CBM is mainly populated by Mo '4d' states. The projected DOS for the heterostructure indicates that the VBM is populated by the states from MoS 2 and the CBM is dominated by the C 2 N. It has been established that strain plays a significant role in efficiently tuning the electronic, optical, and photocatalytic properties of van der Waals heterojunctions. In this section, we studied the effect of vertical strain by changing the interlayer distance between the C 2 N and MoS 2 monolayers. However, we would like to mention that experimentally, the interlayer distance can be effectively varied by changing pressure with a scanning tunneling microscopy tip [21], through vacuum thermal annealing [22], inserting a dielectric BN layer inside the van der Waals gap of the heterostructure [23], using diamond anvil cells [24]. The vertical strain (∆D) can be defined as ∆D = d−d 0 d 0 × 100, where d 0 and d are the interlayer separation between C 2 N and MoS 2 under equilibrium and strained configurations, respectively. The effect of vertical strain on the band gap of the heterostructure is given in Fig. 3a. It is observed that with an increase in tensile strain along the 'z' direction, the band gap value shows an increasing trend and exhibits a linear variation. Instead, when the compressive strain between the layer is increased, the band gap decreases and follows a linear relationship with the direction of applied strain. The band evolution is further analysed through the straindependent projected density of states (PDOS) as shown in supporting information S1. It is clear that the compressive strain increases the population of Mo '4d' states for the VBM near the Fermi level. The CBM is mainly composed of a mixture of 'p' states of C 2 N and MoS 2 . With an increase in tensile strain both Mo '4d' and C 2 N, MoS 2 'p' states move away from the Fermi level resulting in a larger band gap. The band gap calculation is further analysed using the meta-GGA SCAN functional which indicates an enhanced band gap value as compared to the PBE result for the entire range of compressive and tensile strains considered. The band gap variation for the compressive strain shows a nearly linear behaviour consistent with the PBE result. Similarly, for the tensile configuration, the energy gap exhibits an increase in value up to 3% strain and then remains constant with further expansion in interlayer separation. The present heterostructure is also investigated with the application of uniaxial and biaxial strain in order to study the influence of in-plane orbital overlapping. The PBE result for the band gap evolution as a function of uniaxial strain is represented in Fig. 3b. It is interesting to note that the band gap remains nearly linear for larger compressive strain. As the compressive strain decreases from -5% to -1% the band gap shows a mild increase in value. The application of tensile strain along the uniaxial direction shows a different trend as compared to the compressive one. The band gap shows a systematic decrease in value with the increase in tensile strain from 0 to +5%. The reduction in band gap may be due to the decrease in orbital overlapping between the 'p' states of C and N and the '4d' states of Mo. The corresponding partial density of state (See supporting information Figure S2a, b) suggests that tensile strain increases the DOS near the Fermi level for Mo '4d' and C, N 'p' states. Similarly, the band gap study was also executed by applying biaxial strain from -5% to +5% range. The system shows a completely different behaviour as compared to the result for vertical strain. The linear variation of the band gap for the compressive strain region remains preserved as similar to the case of uniaxial strain with a linear variation with a slowly decreasing trend. However, if we apply tensile strain then the band gap decreases much more rapidly. The corresponding PDOS analysis indicates that for -5% compressive strain, the VBM is mainly composed of C and N '2p' states, and CBM is populated by 'p' states of N atoms. When the strain increases from -5% to +5%, Mo '4d' states become the highest occupied band near the Fermi level. The CBM comes closer to the Fermi level and resulting in the reduced band gap. The band gap calculation is further analysed using SCAN functional for the uniaxial and biaxial strain configuration which provides an enhanced band gap value as compared to the PBE functional. However, the trend in variation of the band gap remains almost the same (Fig. 3b, c). The essential requirement for any material for efficient photocatalytic water splitting application is to identify appropriate band edge positions relative to the hydrogen evolution reaction (HER) and oxygen evolution reaction (OER) potentials of the water. The CBM of the material should lie above the reduction reaction potential and the VBM should lie below the oxidation reaction potential of water. To estimate the band edge positions relative to the oxidation and reduction potential, the absolute energy position of the CBM and VBM are calculated with respect to the vacuum level. The vacuum level for the material is obtained by calculating the local potential and then taking the average of the constant potential region. In the present study, we have investigated the photocatalytic activity of the C 2 N/MoS 2 heterostructure by applying strain in the uniaxial, biaxial, and vertical directions. We note that the reduction potential (E H + /H 2 ) and oxidation potential (E O 2 /H 2 O) of water with respect to vacuum level are -4.44 and -5.67 eV, respectively [25]. The band alignment is then calculated by plotting the appropriate band edge position with respect to the vacuum level as a function of strain. The calculated band edge position of the CBM and VBM under the influence of vertical strain, using PBE functional is presented in Fig. 4a. It was observed that without applying any strain, the heterostructure exhibits band edge positions within the water redox potentials of the water. This result indicates that the sample is not useful to carry out photocatalytic water splitting. The sample is subjected to compressive (tensile) vertical strain by reducing (increasing) the interlayer separation between the C 2 N and MoS 2 monolayers, respectively. It was observed that by applying compressive strain, the band edge positions got improved as compared to the zero strain case. However, both the CBM and VBM lie within the water redox potential values. Applying tensile strain, the VBM and CBM move closer to the water redox potential values but still lie below them. Therefore the PBE result indicates that the application of -5 to +5% vertical strain does not enhance the band edge positions to be useful for photocatalytic water splitting. However, we would like to mention that PBE functional severely underestimates the band gap and band edge positions. Therefore the calculations are further executed using meta-GGA SCAN functional (Fig. 4b). The result indicates a significant improvement as compared to that of the PBE functional. It was observed that, without applying any strain, the VBM comes below the oxidation reaction potential, whereas the CBM moves up but still lies below the reduction potential value. The application of compressive strain up to -2% positions the VBM well below the oxidation potential of water but the CBM still lies below the reduction potential. On further increase in the compressive strain from -3 to -5%, both the CBM and VBM lie outside the range of water redox potentials hence enhancing the photocatalytic activities of the sample. When the heterostructure is subjected to vertical tensile strain from +1% to +5%, both the CBM and VBM straddle the water redox potential range hence enhancing the photocatalytic response under visible light. There fore we conclude that the present heterostructure exhibits efficient charge separation for photocatalytic water splitting under tensile strain and larger compressive strain. The sample is further subjected to uniaxial and biaxial strain to analyse the photocatalytic performance. The uniaxial strain is applied by increasing or compressing the in-plane lattice constant along the X-direction keeping the Y value fixed. Similarly, the biaxial strain is applied by increasing or decreasing the X-Y lattice constant values simultaneously. It is to be noted that the interlayer separation is kept at its equilibrium value i.e 3.373Å. The detailed graphical representation of band alignment under uniaxial strain is given in the Supporting information S3 a. It was observed that for PBE functionals the CBM and VBM both lie within the redox potential range for the entire range of uniaxial strain from -5 to +5%, thus indicating that uniaxial strain does not produce a significant improvement to be useful for photocatalytic water splitting. To get a better result, the strain calculation is repeated for the heterostructure using SCAN functional and is given in Supporting information S3b. It was observed that the application of larger uniaxial strain (-3 to -5%) puts the CBM and VBM outside the water redox potential range. But the tensile strain reduces the band edge separation even below the unstrained case. The VBM lies below the oxidation potential value but the CBM finds its position below the reduction potential. Therefore we found that the sample can be useful for photocatalytic water splitting for larger compressive uniaxial strain whereas the tensile strain does not produce an enhancement in the photocatalytic response. The present C 2 N/MoS 2 heterostructure is also investigated under biaxial strain using both PBE and SCAN functionals (See Supporting information S4). The results are quite similar to that of the uniaxial strain. Like previous results, PBE functional does not produce a better result for the photocatalytic activity. A larger compressive strain puts the CBM above the reduction potential but the VBM lies above the oxidation potential of water, thus not providing a useful result for our purpose. Using SCAN functional, it was observed that from -2 to -5% compressive strain both the CBM and VBM straddle outside the water redox potential. On the other hand, tensile strain decreases the band edges position and put the CBM below the reduction potential of water. Therefore, the heterostructure can be used for photocatalytic studies for larger uniaxial and biaxial compressive strains. In order to illustrate the charge transfer process between the C 2 N and MoS 2 monolayers during the formation of the heterostructure, the charge density difference is executed using PBE functional. The charge density difference is calculated by subtracting the charge density of the individual C 2 N and MoS 2 monolayers Fig. 5b. In all our charge density figures, the cyan color indicates the charge depletion and the yellow color represents the charge accumulation process. From Fig. 5b, we observe that the charge accumulation mainly occurs in the interface region and partially in MoS 2 monolayers, whereas most of the charge depletion occurs in the top C 2 N and bottom MoS 2 . The change in charge density under vertical strain is given in Fig. 5. From the Fig.5a, we observe that with an increase in compressive vertical strain by 5%, there is an equal proportion of charge accumulation and depletion close to the MoS 2 layer whereas the charge depletion mainly occurs in the region close to C 2 N layer. The charge density given in cyan color near the C atom in the C 2 N monolayer indicates a charge depletion. Similarly, the charge density given in yellow color near the S atom in the MoS 2 layer indicates a charge accumulation. Hence this observation indicates that there is a charge transfer from the S atom of MoS 2 to the C atom of C 2 N monolayer. The intensification of the charge transfer process under large compressive strain indicates an enhanced interaction between the two monolayers. The strain-induced charge transfer in C 2 N based heterostructures is consistent with other literature reports [26,27,28]. When the interlayer separation in increases to achieve a strain around +5%, the charge depletion from the C 2 N layer almost disappears and most of the charge accumulation occurs in the interface region close to MoS 2 . In other words, there is no appreciable charge redistribution close to the MoS 2 layer. This may be the influence of the van der Waals gap between the two monolayers which varies as the structure moves from a compressive strain state to a tensile strain state. The effect of uniaxial compressive and tensile strain on the charge transfer process is illustrated in supplementary information Fig. S5. It indicates that there is no appreciable change in the charge density difference under uniaxial compressive and tensile strain (up to 5%). On the other hand application of biaxial strain indicates a significant influence on the charge density under compressive and tensile strains. From the Supplementary information Fig. S6, we observe that as the system is subjected to -5% compressive strain the charge depletion occurs in the C atom of the top layer. With system changes from large compression to large tensile strain state, the depletion near the C atom increases indicating more charge transfer from the S atom of MoS 2 to the C atom of C 2 N layer. Therefore we observe that both vertical and biaxial strain affects the charge transfer process significantly as compared to the uniaxial strain state. Conclusion In summary, we have studied the photocatalytic performance of C 2 N/MoS 2 heterostructure as a function of uniaxial, biaxial, and vertical strain configuration through first-principles DFT calculations. The unstrained heterostructure possesses a direct band gap of 1.35 eV, where the VBM is populated by Mo 'd' states and the CBM is contributed by 'C' and 'N' 'p' states. The calculated position of CBM and VBM of individual monolayers indicates that the heterostructure possesses a type-II band alignment which is beneficial for charge separation across the two monolayers and preventing their recombination process. The SCAN functional provides better results for the band gap and band edge positions calculation as compared to that of the PBE result. Using the C 2 N/MoS 2 heterostructure as a prototype, we have also found that the meta-GGA SCAN functional shows similar results as compared to the computationally expensive hybrid HSE functionals. The estimated CBM and CBM position as a function of vertical strain indicates that the water reduction and oxidation potential values lie within the band gap region with respect to the vacuum level for larger compressive and tensile strain. In contrast, the system exhibits good photocatalytic performance only for larger compression for uniaxial and biaxial strain states whereas the tensile strain reduces the separation between the VBM and CBM within the water redox potential value. The charge density difference indicates a significant charge transfer for vertical and biaxial configuration as compared to the uniaxial state. The present study can help researchers to reduce the computational cost by considering the meta-GGA functionals over HSE for electronic structure calculations of similar systems. Our calculation will also be extremely useful for designing artificial strained heterostructure for the experimental community for better device application for photocatalytic water splitting. Computational Methods The first principles electronic structure calculations were performed using density functional theory (DFT) with projector augmented-wave method [29] as implemented in the Vienna ab initio simulation package (VASP) [30]. The Perdew-Burke-Ernzerhof (PBE) [31] parametrization-based generalized gradient approximation (GGA) was chosen for the exchange-correlation functional. To accurately describe the interaction between the layered structures, we have included the van der Waals correction method (DFT-D2) presented by Grimme [32]. Initially, the gap between C 2 N and MoS 2 monolayers is kept at 3.5Å which is further optimized before running the electronic structure calculation. As a benchmark, the system is further studied using strongly constrained and appropriately normed (SCAN) meta-GGA functionals to get more accurate results as compared to the PBE functional and also to be consistent with the reported experimental data. A vacuum layer of 20Å was selected along the 'Z' direction to avoid interaction between the adjacent layers. The plane wave expansion cut-off was chosen to be 520 eV. The Brillouin zone integration was performed using a Γ-centered (9×9×1) k-mesh for the structural relaxation and electronic structure calculation of the isolated C 2 N and MoS 2 monolayers. For the C 2 N/MoS 2 heterostructure containing 354 atoms, the geometry optimization and electronic structure calculation are performed using a Γ centered (2×2×1) k-point samplings. All the structures are allowed for relaxation to get the optimized atomic position until the total forces acting on each atom are less than 0.02 eV/Å. Data availability statement The additional data that support the findings of this article are available in the Supplementary Information. Figure 1 . 1Schematic representation of the top view of isolated (a) C 2 N monolayer, (b) MoS 2 monolayer, (c) side view and (d) top view of C 2 N/MoS 2 heterostructure. The schematic symbols of the C, N, Mo, and S atoms are indicated as blue, red, green, and brown colors, respectively. Figure 2 . 2Electronic band structure of (a) C 2 N, (b) MoS 2 monolayer, (c) C 2 N/MoS 2 heterostructure, (d-f) The total and projected density of states for the corresponding system calculated using PBE functional. The dashed line in each figure indicates the Fermi level. Figure 3 . 3Band gap tuning of C 2 N/MoS 2 heterostructure under (a) vertical, (b) uniaxial, and (c) biaxial strain using PBE and SCAN functional Figure 4 . 4Band edge position of C 2 N/MoS 2 heterostructure as a function of vertical strain with respect to vacuum potential using (a) PBE and (b) SCAN functional. The horizontal solid lines represent the water redox potentials Figure 5 . 5Charge density difference of C 2 N/MoS 2 heterostructure as a function of vertical (a) -5% compressive (b) 0% (c) +5% tensile strain from the C 2 N/MoS 2 heterostructure.The charge density difference for the unstrained heterostructure is illustrated in AcknowledgmentsThe authors would like to thank the National Institute of Science Education and Research (NISER), Department of Atomic Energy, Government of India, for funding the research work through project number RIN-4001. The authors acknowledge the high-performance computing facility at NISER.Data availability statementThe additional data that support the findings of this article are available in the Supplementary Information.Conflict of interestThe authors have no conflicts to disclose. . X Song, G Wei, J Sun, C Peng, J Yin, X Zhang, Y Jiang, H Fei, Nat. Catal. 3Song X, Wei G, Sun J, Peng C, Yin J, Zhang X, Jiang Y and Fei H 2020 Nat. Catal. 3 1027-1033 . 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G Wang, J Chang, W Tang, W Xie, Y S Ang, J. Phys. D: Appl. Phys. 55293002Wang G, Chang J, Tang W, Xie W and Ang Y S 2022 J. Phys. D: Appl. Phys. 55 293002 . P Blöchl, Phys. Rev. B. 5017953Blöchl P E 1994 Phys. Rev. B 50 17953 . G Kresse, J Furthmüller, Phys. Rev. B. 5411169Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169 . J P Perdew, Burke K Ernzerhof, M , Phys. Rev. Lett. 773865Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 . S Grimme, J. Comput. Chem. 27Grimme S 2006 J. Comput. Chem. 27 1787-1799

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{'abstract': 'The improved photocatalytic water splitting using 2D materials has technological importance for economically viable renewable energy. The present study focuses on the effect of uniaxial, biaxial, and vertical strain on the energy gap and band edge positions of C 2 N/MoS 2 van der Waals heterostructures through first-principles density functional theory using PBE and SCAN functionals. The calculations establish that SCAN functional provides comparatively much better results as compared to the PBE for the band gap and band alignment study. The heterostructure exhibits a type-II band alignment which is beneficial for the efficient separation of charge carriers. For a good photocatalyst, the band edge positions should straddle the water redox potentials. It is observed that for both compressive and tensile vertical strain, the water redox potential values lie within the valence band maximum (VBM) and conduction band minimum (CBM) of the heterostructure. On the other hand, for uniaxial and biaxial strain, the system can be used as a useful photocatalyst only for larger compressive strain, whereas for tensile strain, the energy gap between VBM and CBM keeps on decreasing and lie within the water oxidation/reduction potential. Our study also establishes that the meta-GGA SCAN functional shows similar results as compared to the computationally expensive hybrid HSE functionals. The present work can be extremely useful for experimentalists to design artificial heterostructure devices for better performance in photocatalytic water splitting.', 'arxivid': '2301.03809', 'author': ['Soumendra Kumar Das \nSchool of Physical Sciences\nNational Institute of Science Education and Research (NISER) Bhubaneswar\nHBNI\nKhurda-752050JatniOdishaIndia\n', 'Lokanath Patra \nDepartment of Mechanical Engineering\nUniversity of California Santa Barbara\n93106CAUSA\n', 'Prasanjit Samal psamal@niser.ac.in \nSchool of Physical Sciences\nNational Institute of Science Education and Research (NISER) Bhubaneswar\nHBNI\nKhurda-752050JatniOdishaIndia\n', 'Pratap K Sahoo pratap.sahoo@niser.ac.in \nSchool of Physical Sciences\nNational Institute of Science Education and Research (NISER) Bhubaneswar\nHBNI\nKhurda-752050JatniOdishaIndia\n'], 'authoraffiliation': ['School of Physical Sciences\nNational Institute of Science Education and Research (NISER) Bhubaneswar\nHBNI\nKhurda-752050JatniOdishaIndia', 'Department of Mechanical Engineering\nUniversity of California Santa Barbara\n93106CAUSA', 'School of Physical Sciences\nNational Institute of Science Education and Research (NISER) Bhubaneswar\nHBNI\nKhurda-752050JatniOdishaIndia', 'School of Physical Sciences\nNational Institute of Science Education and Research (NISER) Bhubaneswar\nHBNI\nKhurda-752050JatniOdishaIndia'], 'corpusid': 255570122, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 10553, 'n_tokens_neox': 9168, 'n_words': 6169, 'pdfsha': '12e98a4eb9cc8ba2336b1d527e3c70b857424406', 'pdfurls': ['https://export.arxiv.org/pdf/2301.03809v1.pdf'], 'title': ['Strain Induced Enhanced Photocatalytic Activities in Layered Two Dimensional C 2 N/MoS 2 Heterostructure: A Meta-GGA Study', 'Strain Induced Enhanced Photocatalytic Activities in Layered Two Dimensional C 2 N/MoS 2 Heterostructure: A Meta-GGA Study'], 'venue': []}

arxiv

Impact of NOMA on Age of Information: A Grant-Free Transmission Perspective Fellow, IEEEZhiguo Ding Life Fellow IEEE Fellow, IEEERobert Schober Life Fellow IEEE H Vincent Poor Life Fellow IEEE Impact of NOMA on Age of Information: A Grant-Free Transmission Perspective arXiv:2211.13773v2 [cs.IT] 3 May 2023 1 The aim of this paper is to characterize the impact of non-orthogonal multiple access (NOMA) on the age of information (AoI) of grant-free transmission. In particular, a low-complexity form of NOMA, termed NOMA-assisted random access, is applied to grant-free transmission in order to illustrate the two benefits of NOMA for AoI reduction, namely increasing channel access and reducing user collisions.Closed-form analytical expressions for the AoI achieved by NOMA assisted grant-free transmission are obtained, and asymptotic studies are carried out to demonstrate that the use of the simplest form of NOMA is already sufficient to reduce the AoI of orthogonal multiple access (OMA) by more than 40%.In addition, the developed analytical expressions are also shown to be useful for optimizing the users' transmission attempt probabilities, which are key parameters for grant-free transmission.Alternatively, massive multi-input multi-output (MIMO) can also be applied to support grantfree transmission by using the excessive spatial degrees of freedom offered by massive MIMORecently, the application of non-orthogonal multiple access (NOMA) to grant-free transmission has received significant attention due to the following two reasons. First, the NOMA principle is highly compatible, and the use of NOMA can significantly improve the reliability and spectral efficiency of random access and massive MIMO based grant-free protocols [8]-[11]. Second, more importantly, the use of NOMA alone is sufficient to support grant-free transmission. For example, NOMA-based grant-free transmission has been proposed in [12], where a Bayesian learning based scheme has been designed to ensure successful multi-user detection, even if the number of active grant-free users is unknown. The principle of NOMA can also be used to develop so-called semi-grant-free transmission protocols, where the bandwidth resources which would be solely occupied by grant-based users are released for supporting multiple grant-free users in a distributed manner [13]. In addition, NOMA-based grant-free transmission has also been shown to be robust and efficient in various communication scenarios, such as satellite communication networks, secure Internet of Things (IoT), intelligent reflecting surface (IRS) networks, marine communication systems, etc., see [14]-[17].The aim of this paper is to characterize the impact of NOMA on the performance of grantfree transmission with respect to a recently developed new performance metric, termed the age of information (AoI) [18]-[20]. In particular, the AoI describes the freshness of data updates collected in the network, and is an important metric to measure the success of the 6G services, including umMTC and euRLLC. We note that most existing works have focused on the impact of NOMA on grant-based networks [21]-[25]. For example, for two-user grant-based networks, the capability of NOMA to reduce the AoI has been shown to be related to the spectral efficiency gain of NOMA over orthogonal multiple access (OMA)[26]. To the authors' best knowledge, there is only a single existing work which applied NOMA to reduce the AoI of grant-free transmission[27], where the strong assumption that the base station estimates all users' channel state information (CSI) was made.In this paper, the impact of NOMA on the AoI of grant-free transmission is investigated from the perspective of performance analysis, which is different from the existing work focusing on resource allocation[28]. In particular, a low-complexity form of NOMA, which was originally termed NOMA-assisted random access[29]and recently also termed ALOHA with successive, and E{·} denotes the expectation.Remark 1: We note that the AoI expressions in (4) of this paper and[35,Eq. (3)] are consistent.The reason why there is an extra factor of 1 2 in [35, Eq. (3)] is that the users' instantaneous AoI was assumed to be discrete-valued in[35], instead of continuous-valued as in this paper. I. INTRODUCTION Grant-free transmission is an important enabling technique to support the sixth-generation (6G) services, including ultra massive machine type communications (umMTC) and enhanced ultra-reliable low latency communications (euRLLC) [1], [2]. Unlike conventional grant-based transmission, grant-free transmission enables users to avoid multi-step signalling and directly transmit data signals together with access control information, which can reduce the system overhead significantly, particularly for scenarios with massive users requiring short-package transmission. Grant-free transmission can be realized by applying the random access protocols developed for conventional computer networks, such as ALOHA random access [3], [4]. interference cancellation (SIC) [30], [31], is adopted in order to illustrate the two benefits of NOMA for AoI reduction, namely increasing channel access and reducing user collisions. The key element of the proposed performance analysis is the modelling of the channel competition among the grant-free users as a Markov chain, which is different from the performance analysis for grant-based NOMA networks [21]- [23], [26]. The main contributions of this paper are twofold: • Analytical expressions for the AoI achieved by NOMA assisted grant-free transmission are obtained, by rigorously characterizing the state transition probabilities of the considered Markov chain. We note that by using NOMA-assisted random access, the base station creates multiple preconfigured receive signal-to-noise ratio (SNR) levels, which makes NOMAassisted grant-free transmission similar to multi-channel ALOHA. As a result, the calculation of the state transition probabilities for the NOMA case is more challenging than that for the OMA case, which can be viewed as single-channel ALOHA. By exploiting the properties of the considered Markov chain and also the characteristics of SIC, closed-form expressions for the state transition probabilities are developed for NOMA assisted grant-free transmission. • Valuable insights regarding the relative performance of NOMA and OMA assisted grant-free transmission are also obtained. For example, for the case where users always have updates to deliver, asymptotic expressions are developed to demonstrate that the use of NOMA can almost halve the AoI achieved by OMA, even if the simplest form of NOMA is implemented. In addition, the optimal choices of the users' transmission probabilities for random access with NOMA and OMA are obtained and compared. This study reveals that NOMA-assisted grant-free transmission is fundamentally different from multi-channel ALOHA due to the use of SIC. Furthermore, simulation results are provided to verify the developed analytical expressions and also demonstrate that the use of NOMA can significantly reduce the AoI frame i … … t TS 1 i+2 TS N i+2 TS N i+1 TS 1 i+1 TS 1 i TS N i t 1 i t N i t N i+1 t 1 i+1 t 1 i+2 t N i+2 frame i + 1 frame i + 2 A. Data Generation Models For the considered grant-free transmission scenario, each user tries to deliver one update to the base station in each time frame. When the users' updates are generated depends on which of the following two data generation models is used [21]. 1) Generate-at-request (GAR): For GAR, the base station requests all users to simultaneously generate their updates at the beginning of each time frame. GAR is applicable to many important IoT applications, such as structural health monitoring and autonomous driving. 2) Generate-at-will (GAW): For GAW, a user's update is generated right before its transmit time slot. GAW has been commonly used in the AoI literature, since it ensures that the delivered updates are freshly generated. In this paper, GAR will be focused on due to the following two reasons. First, the AoI expression for grant-free transmission for GAW can be straightforwardly obtained from that for GAR, as shown in the next section. Second, if there are retransmissions within one time frame, GAW requires a user to repeatedly generate updates, and hence, causes a higher energy consumption than GAR. For grant-based transmission, this increase in energy consumption is not severe since the number of retransmissions in one frame is small [21]. However, in the grant-free case, a user might have to carry out a large number of retransmissions due to potential collisions, which means that GAW can cause a significantly higher energy consumption than GAR. For more sophisticated random access schemes, e.g., irregular repetition slotted ALOHA (IRSA) [32]- [34], which allows a user to choose a subset of time slots for transmission, the principle of NOMA can be also applied as an add-on, e.g., a group of users, instead of a single user, share the same subset of time slots. B. Channel Access Modes Prior to TS n i , assume that j users have successfully delivered their updates to the base station. For grant-free transmission, each of the remaining M − j users will independently make a transmission attempt with the same transmit power, denoted by P 2 , and the same transmission probability, denoted by P TX . P TX can be based on a fixed choice, or be state-dependent, i.e., [35]. It is assumed that the base station can inform the users about the outcome of their updates via a dedicated control channel. P j = 1 M −j There are three possible events which cause an update failure for a user: i) the user does not make a transmission attempt; ii) a collision occurs, i.e., there are more than one concurrent transmissions; iii) an outage occurs due to the user's weak channel condition, i.e., log(1 + P |h i,n m | 2 ) ≤ R, where R denotes the user's target data rate, and h i,n m denotes U m 's channel gain in TS n i . In this paper, all users are assumed to have the identical target data rates, and their channel gains are assumed to be independent and identically complex Gaussian distributed with zero mean and unit variance. 2) Non-orthogonal Multiple Access (NOMA): With NOMA, a user can still succeed in its updating, even if multiple users choose the same time slot. Recall that the principle of NOMA can be implemented in different forms. For the purpose of illustration, a particular form of NOMA, termed NOMA-assisted random access, is adopted to reduce the AoI of grant-free transmission [29]. In particular, prior to transmission, the base station configures K receive SNR levels, denoted by P 1 ≥ · · · ≥ P K . If U m chooses P k during TS n i , it will scale its transmitted signal by P k |h i,n m | 2 . 3 The base station carries out SIC by decoding the signal delivered at SNR level, P k , before decoding the one at P k+1 , 1 ≤ k ≤ K − 1. The SNR levels are preconfigured to guarantee the success of SIC, i.e., the following conditions need to be satisfied: log 1 + P k 1 + (M − 1)P k+1 = R, 1 ≤ k ≤ K − 1,(1) and log (1 + P K ) = R, which means P K = 2 R − 1 and P k = 2 R − 1 (1 + (M − 1)P k+1 ), where the noise power is assumed to be normalized to one. We note that the condition in (1) is stricter than the condition log 1 + P k 1+ K i=k+1 P i = R, and ensures the success of SIC, even if 2 Because the noise power is assumed to be normalized to one, P can also be interpreted as the transmit SNR. 3 One benefit of this form of NOMA is that the base station does not need to estimate all users' CSI for implementing SIC, an assumption commonly required in, e.g., [27], [28]. For the adopted form of NOMA, the users are assumed to have access to their own CSI only. In practice, this CSI assumption can be realized by asking the base station to broadcast pilot signals at the beginning of a time slot, where the users can perform channel estimation individually. one user chooses P k and the remaining users choose the SNR level which contributes the most interference, i.e., P k+1 . We further note that the case in which all M − 1 remaining users choose P k+1 is the worst case, since some users may choose SNR levels other than P k+1 or even decide not to transmit at all. Again assume that there are j users which have successfully sent their updates to the base station prior to TS n i . Each of the remaining M − j users will first randomly choose an SNR level with equal probability, denoted by P k = 1 K , and independently make a transmission attempt with probability P TX . For illustrative purposes, assume that U m is among the M − j remaining users, and chooses P k . The possible events which cause U m 's update to fail are listed as follows: • The user does not make an attempt for transmission; • The receive SNR level chosen by the user is not feasible due to the user's transmit power budget, i.e., P k is not feasible for U m in TS n i if P k |h i,n m | 2 > P ; • Another user also chooses P k , which leads to a collision at P k and hence a failure at the k-th stage of SIC; • Prior to the k-th stage of SIC, SIC has already been terminated due to one or more failures in the previous SIC stages. We note that for both the OMA and NOMA cases, a user keeps re-sending its update to the base station until either the user succeeds or the time frame is finished. C. AoI Model AoI is an important performance metric for quantifying the freshness of the updates delivered to the base station. We note that for the considered grant-free scenario, all the users experience the same AoI. Therefore, without loss of generality, U 1 's instantaneous AoI at time t is focused on and defined as follows [18]: ∆(t) = t − T (t),(2) where T (t) denotes the generation time of U 1 's freshest update successfully delivered to the base station. U 1 's average AoI of the considered network is given bȳ ∆ = lim T ∆ →∞ 1 T ∆ T ∆ 0 ∆(t)dt.(3) The AoI achieved by OMA and NOMA assisted grant-free transmission will be analyzed in the following section. Q j frame i frame i + 1 frame i + 2 t n+1 i t 1 i t 1 i+2 Sj−1 Y j Sj−1 t l+1 i+2 Sj Fig. 2. Illustration of AoI evolution with grant-free transmission for GAR. III. AOI OF GRANT-FREE TRANSMISSION As discussed in the previous section, U 1 is treated as the tagged user and its AoI will be focused on in this section, without loss of generality. For the example shown in Fig • X j : The number of frames between the (j − 1)-th and the j-th successful updates. An example of X j = 2 is shown in Fig. 2. We note that in the literature of random access, S j is termed the service delay and Y j is termed the inter-departure time [35]. By using the aforementioned metrics, for GAR, U 1 's averaged AoI is given bȳ ∆ = lim J→∞ J j=1 Q j J j=1 Y j = lim J→∞ J j=1 S j−1 Y j + 1 2 Y 2 j J j=1 Y j = E{S j−1 Y j } E{Y j } + E{Y 2 j } 2E{Y j } ,(4) where J denotes the total number of successful updates, Q j denotes the area of the shaded region shown in Fig. 2, E{Y 2 j } = lim Remark 2: For GAW, the user's averaged AoI is given bȳ ∆ GAW = lim J→∞ J j=1 T Y j + 1 2 Y 2 j J j=1 Y j = T + E{Y 2 j } 2E{Y j } ,(5) which is simpler than the AoI expression in (4). Therefore, the analytical results developed for GAR are straightforwardly applicable to the case for GAW. A. Generic Expressions for E{S j−1 Y j }, E{Y j }, and E{Y 2 j } As shown in (4), the AoI is a function of E{S j−1 Y j }, E{Y j }, and E{Y 2 j }, and generic expressions for these metrics have been derived in [35], and will be briefly introduced in this subsection. In particular, the considered grant-free transmission can be modelled by a Markov chain with M + 1 states, denoted by s k , 0 ≤ k ≤ M. In particular, s k , 0 ≤ k ≤ M − 1, denotes the transient state, where k users, other than U 1 , have successfully delivered their updates to the base station. s M means that U 1 has successfully delivered its update to the base station. Define the state transition probability from s j to s i by P j,i , 0 ≤ i, j ≤ M. Build an M × M matrix, denoted by P, whose element in the (i + 1)-th column and the (j + 1)-th row is P j,i , 0 ≤ i, j ≤ M − 1. Furthermore, build an M × 1 vector, denoted by p, whose (j + 1)-th element is P j,M . Once P and p are available, E{S j−1 Y j }, E{Y j }, and E{Y 2 j } can be obtained as follows. Denote by Z the number of time slots required by U 1 to successfully deliver its update to its base station. Then, the probability mass function (pmf) of Z is given by P (Z = n) = s T 0 P n−1 M p, n = 1, 2, · · · ,(6)where s 0 = 1 0 1×(M −1) T denotes the initial probability vector and 0 m×n denotes an all-zero m × n matrix. Therefore, the probability that U 1 cannot complete an update within one frame is given by P fail = P (Z > N) = s T 0 P N M 1, where 1 denotes an M × 1 all-one vector. Therefore, the pmf of access delay, S j , can be written as follows: P(S j = nT ) = P (Z = n) 1 − P fail = s T 0 P n−1 M p 1 − s T 0 P N M 1 ,(7)for 1 ≤ n ≤ N, which means E{S j } = T N n=1 n s T 0 P n−1 M p 1−s T 0 P N M 1 and E{S 2 j } = T 2 N n=1 n 2 s T 0 P n−1 M p 1−s T 0 P N M 1 . Similarly, the pmf of X j is given by P(X j = n) = P n−1 fail (1 − P fail ),(8) which means that E{X j } = 1 1−P fail and E{X 2 j } = 1+P fail (1−P fail ) 2 . The expressions of E{X j } and E{X 2 j } can be used to evaluate E{Y j } and E{Y 2 j }, since E{Y j } = T NE{X j } and E{Y 2 j } = N 2 T 2 E X 2 j + 2E S 2 j − 2E {S j } 2 . Furthermore, E{S j }, E{S 2 j }, and E{Y j } can be used to evaluate E{S j−1 Y j } which can be expressed as follows: E{S j−1 Y j } =E{S j }E{Y j } − E{S 2 j } + E{S j } 2 , where the last step follows by the fact that S j−1 and Y j − (NT − S j−1 ) are independent. As discussed above, the crucial step to evaluate the AoI is to find P and p, which depends on the used multiple access schemes. B. OMA-Based Grant-Free Transmission The state transition probabilities for the OMA case can be straightforwardly obtained, as shown in the following. With OMA, a single user can be served in each time slot, which means that the number of successful users after one time slot can be increased by one at most. Therefore, most of the state transition probabilities in matrix P are zero, except for P j,j , P j,j+1 , and P j,M , 0 ≤ j ≤ M − 1. In particular, P j,j denotes the probability of the event that no user succeeds, and is given by [35] P j,j = 1 − (M − j)P TX e − ǫ P (1 − P TX ) M −j−1 ,(9) where ǫ = 2 R − 1. P j,j+1 denotes the probability of the event that a single user, but not U 1 , succeeds and is given by P j,j+1 = (M − j − 1)P TX e − ǫ P (1 − P TX ) M −j−1 .(10) Furthermore, the j-th element of p, denoted by P j,M , is given by P j,M = P TX e − ǫ P (1 − P TX ) M −j−1 .(11) C. NOMA-Based Grant-Free Transmission The benefit of using NOMA is that more users can be admitted simultaneously than for OMA. In particular, with NOMA, the number of successful users after one time slot can be increased by K at most, whereas the number of successful users was no more than 1 for OMA. This means that the non-zero state transition probabilities in matrix P include P j,j , P j,j+i , and P j,M , 0 ≤ j ≤ M − 1, 1 ≤ i ≤ K and j + i ≤ M − 1. The analysis of the state transition probabilities for the NOMA case is more challenging than that for the OMA case, mainly due to the application of SIC. For example, a collision at SNR level P k can prevent all those users, which choose SNR level P i , i > k, from being successful. The following lemma provides a high-SNR approximation for the state transition probabilities. Lemma 1. At high SNR, the state transition probability, P j,j , 0 ≤ j ≤ M − 1, can be approximated as follows: P j,j ≈1 − M −j m=1 M − j m P m TX (1 − P TX ) M −j−m (12) × K k=1 mP K (1 − kP K ) m−1 , the state transition probability, P j,j+1 , 0 ≤ j ≤ M − 2, can be approximated as follows: P j,j+1 ≈(M − j)P TX (1 − P TX ) M −j−1 M − j − 1 M − j KP K + M −j m=2 M − j m P m TX (1 − P TX ) M −j−m K−1 k=1 M − j − 1 M − j mP K (13) (1 − kP K ) m−1 − K κ=k+1 (m − 1)P K (1 − κP K ) m−2 , and the state transition probability, P j,j+i , 0 ≤ j ≤ M − 3 and 2 ≤ i ≤ min{M − 1 − j, K}, can be approximated as follows: P j,j+i ≈ M − j i P i TX (1 − P TX ) M −j−i K−i+1 k 1 =1 K k 2 =k 1 +i−1 M − j − i M − j P i K k 2 − k 1 − 1 i − 2 i p=1 p + M −j m=i+1 M − j m P m TX (1 − P TX ) M −j−m × K−i k 1 =1 K−1 k 2 =k 1 +i−1 M − j − i M − j P i K k 2 − k 1 − 1 i − 2 i−1 p=0 (m − p) × (1 − k 2 P K ) m−i − K κ=k 2 +1 (m − i)P K (1 − κP K ) m−i−1 .(14) Proof. See Appendix A. Once the transition probability matrix P is obtained, the elements of p can be obtained straightforwardly by applying P1 + p = 1, where recall that 1 denotes an M × 1 all-one vector. The closed-form analytical expressions shown in Lemma 1 allow the evaluation of the impact of NOMA on the AoI without carrying out intensive Monte Carlo simulations. However, the expressions of the state transition probabilities shown in Lemma 1 are quite involved, which makes it difficult to obtain insights about the performance difference between OMA and NOMA. For this reason, the special case of K = 2 and N = 1 is focused on in the remainder of this section. K = 2 means that there are two SNR levels, i.e., the base station needs to carry out two-stage SIC only, which is an important case in practice due to its low system complexity. N = 1 implies that there is one time slot in each frame, i.e., in each time slot, all users have updates to deliver and hence always participate in contention. For this case, the following lemma provides the optimal choice for the transmission probability P TX . Lemma 2. For the special case of K = 2 and N = 1, the optimal choice for the transmission probability P TX is given by P * TX = η M ,(15) for M → ∞ and P → ∞, where η is the root of the following equation: 1 − η 2 e − η 2 + 1 − η 2 2 e −η = 0. Proof. See Appendix B. Remark 3: Note that 1 − η 2 e − η 2 + 1 − η 2 2 e −η = 0 is not related to M, which means that η is not a function of M. By applying off-shelf root solvers, the exact value of η can be straightforwardly obtained as follows: η ≈ 1.6646. By using Lemma 2, the AoI performance difference between NOMA and OMA is analyzed in the following proposition. Proposition 1. For the special case of K = 2 and N = 1, for M → ∞ and P → ∞, the ratio between the AoI achieved by NOMA and OMA is given bȳ ∆ N ∆ O ≈ 2e η−1 η e η 2 + 1 + η 2 ,(16) where∆ N and∆ O denote the AoI achieved by NOMA and OMA, respectively. Proof. See Appendix C. Remark 4: Recall that η ≈ 1.6646, which means that the ratio in Proposition 1 is∆ N ∆ O ≈ 0.5653, i.e., the use of NOMA can almost halve the AoI achieved by OMA. We note that this significant performance gain is achieved by using NOMA with two SNR levels only, i.e., the simplest form of NOMA. By implementing NOMA with more than two SNR levels, the performance gain of NOMA over OMA can be further increased, as shown in the next section. Remark 5: As shown in the proof for Proposition 1, the optimal choice of P TX for OMA is 1 M . This is expected as explained in the following. With M users competing for access in a single channel, the use of a transmission probability of 1 M is reasonable. By using the same rationale, one might expect that 2 M should be optimal for the NOMA transmission probability with two SNR levels. However, Lemma 2 shows that the optimal value of P TX is 1.6646 M , which is a more conservative choice for transmission than 2 M . The reason for this are potential SIC errors. In particular, although there are two channels (or two SNR levels, P 1 and P 2 ), a collision at P 1 causes SIC to immediately terminate, which means that P 2 can no longer be used to serve any users, i.e., the number of the effective channels is less than 2. can be used for NOMA. In fact, the simulation results presented in the next section show that this choice is sufficient to realize a significant performance gain of NOMA over OMA. We note that this choice of P TX depends on M only, unlike the the state-dependent choice of P TX used for OMA, which is P TX = 1 M −j [35]. Therefore, an important direction for future research is to find a more sophisticated statedependent choice of P TX for NOMA-assisted grant-free transmission. Remark 7: Proposition 1 implies that the application of NOMA is particularly beneficial for grant-free transmission, compared to its application to grant-based transmission. Recall that in grant-based networks, one important benefit of using NOMA for AoI reduction is that a user can be scheduled to transmit earlier than with OMA [21]. However, for a user which has already been scheduled to transmit early in OMA, the impact of NOMA on the user' AoI can be insignificant, particularly at high SNR. Unlike grant-based networks, Proposition 1 shows that in grant-free networks, the use of NOMA can reduce the AoI of OMA by more than 40%, and this significant performance gain applies for all users in the network. IV. SIMULATION RESULTS In this section, simulation results are presented to demonstrate the AoI achieved by the considered grant-free transmission schemes and to also verify the developed analytical results. In Fig. 3, the impact of the number of users on the average AoI achieved by the considered grant-free transmission schemes is investigated. As can be seen from the figure, the AoI achieved with NOMA is significantly lower than that of OMA. In addition, Fig. 3 demonstrates that the performance gain of NOMA over OMA increases as the number of users, M, grows. This observation can be explained by using Proposition 1 which states that, for K = 2, N = 1, M → ∞ and P → ∞,∆ N ≈ 0.5653∆ O , or equivalently∆ N −∆ O ≈ 0.4347∆ O . Because increasing M increases∆ O , the performance gain of NOMA over OMA also increases as the number of users grows. Therefore, the use of NOMA is particularly important for grant-free transmission with a massive number of users, an important use case for 6G. Between the two choices of P TX , the adaptive choice yields a better AoI than the fixed choice. For the two subfigures in Fig. 3, different numbers of SNR levels, K, are used. By comparing the two subfigures, one can observe that the AoI achieved by the NOMA scheme can be reduced by increasing the number of SNR levels. This is because the use of more SNR levels makes user collisions less likely to happen. Fig. 3 also demonstrates the accuracy of the AoI expressions developed in Lemma 1. In Fig. 4, the impact of the number of time slots in each frame, N, on the average AoI achieved by the two considered grant-free transmission schemes is studied. As can be seen from the figure, the use of NOMA can always realize lower AoI than OMA, regardless of the choices of N. An interesting observation from Fig. 4 is that a small increase of N, e.g., from 1 to 5, can reduce the AoI. This is because the likelihood for users to deliver their updates to the base station is improved if there are more time slots in each frame. However, after N ≥ 10, further adding more time slots in each frame increases the AoI, which can be explained with the following example. Assume that U 1 can always successfully update its base station in the first time slot of each frame. For this example, U 1 's AoI is simply the length of one time frame, and hence its AoI is increased if there are more time slots in one frame. As discussed in the previous section, the special case with K = 2 and N = 1 is important in practice, and hence the AoI realized by the OMA and NOMA assisted grant-free transmission schemes is investigated in Fig. 5. In particular, the figure shows that the performance gain of NOMA over OMA is particularly large at low SNR. This is a valuable property in practice since most AoI sensitive applications, such as IoT and sensor networks, are energy constrained and In Fig. 7, the AoI of grant-free transmission is shown as a function of the transmission probability, P TX , and the figure demonstrates that the choices of P TX are crucial to the AoI performance of grant-free transmission. Furthermore, Fig. 7 shows that NOMA assisted grantfree transmission always yields a smaller AoI than the OMA case, if the same values for the transmission probability are used for both of the schemes. As shown in Lemma 2 and the proof for Proposition 1, P TX = 1 M is optimal for OMA, and P TX = η M is optimal for NOMA in the case with K = 2 and N = 1. Fig. 7 verifies the optimality of these choices of P TX , since the minimal AoIs achieved by the fixed choices of P TX match perfectly with the AoIs realized by the optimal choices of P TX . In Fig. 8, the performance of the considered OMA and NOMA grant-free schemes are compared by using the following AoI ratio,∆ N ∆ O . For the special case of N = 1 and K = 2, Proposition 1 predicts that this ratio is 0.5653 for large M, which is confirmed by Fig. 8. If K is fixed, i.e., K = 2, an increase of N does not change the ratio significantly, particularly in the case of large M. By introducing more SNR levels, i.e., increasing K, the AoI ratio can be further reduced, which means that the performance gain of NOMA over OMA can be increased by introducing more SNR levels. This is expected since increasing K reduces the likelihood of user collisions and ensures that users can update the base station earlier. For all the previous simulation results, GAR has been considered, which means that a user's update is generated at the beginning of a time frame, instead of at the beginning of a time slot as for GAW. In Fig. 9, the AoI achieved by the considered grant-free transmission schemes for the two different data generation models is illustrated. As can be seen from the figure, for both GAR and GAW, NOMA assisted grant-free transmission always outperforms the OMA based scheme. In addition, the figure shows that the AoI realized by the considered schemes for GAW is smaller than that for GAR, because, for GAW, each update is generated right before its delivery time, i.e., there is no service delay S j . We also note that the difference between the AoI for GAW and GAR is not significant, but the use of GAW can cause a higher energy consumption than GAR, since GAW requires a user to re-generate an update for each retransmission. V. CONCLUSIONS In this paper, the impact of NOMA on the AoI of grant-free transmission has been investigated by applying a particular form of NOMA, namely NOMA-assisted random access. By modelling grant-free transmission as a Markov chain and accounting for SIC, closed-form analytical expressions for the AoI achieved by NOMA assisted grant-free transmission have been obtained, and asymptotic studies have been carried out to demonstrate that the use of the simplest form of NOMA is already sufficient to reduce the AoI of OMA by more than 40%. In addition, the developed analytical results have also been shown useful for optimizing the users' transmission probabilities, P TX , which is crucial for performance maximization of grant-free transmission. In this paper, concise and insightful analytical results have been developed for the special case of N = 1. An important direction for future research is the development of similar insightful results for the general case of N ≥ 1. We also note that the use of NOMA may reduce a user's energy consumption by avoiding the possible large number of retransmissions needed for OMA; however, a user that chooses a high receive SNR level, e.g., P 1 , may consume more energy than in OMA. Therefore, another important direction for future research is to study how to realize a balanced tradeoff between energy efficiency and AoI reduction. VI. ACKNOWLEDGEMENTS The authors thank Dr. Jinho Choi for his kind suggestions about the implementation of NOMA assisted random access. APPENDIX A PROOF FOR LEMMA 1 The proof is divided into three parts to evaluate P j,j , P j,j+1 , and P j,j+i , i ≥ 2, respectively. Throughout the proof, the high SNR assumption is made, which ensures that all the SNR levels, P k , 1 ≤ k ≤ K, are feasible for each user, i.e., transmission failures are due to user collisions only. The users' channel gains in different time slots are assumed to be independent and identically complex Gaussian distributed with zero mean and unit variance. A. Evaluating P j,j To find the expression for P j,j , assume that j users have successfully delivered their updates to the base station. Therefore, each of the remaining M − j users independently makes an attempt to transmit with the probability, P TX , at a randomly chosen SNR level. P j,j is the probability of the event that none of the M − j users succeeds. Define E P k |j as the event that given M − j remaining users, a user successfully updates the base station by using the k-th SNR level, P k , and no user succeeds at P i , i < k. The reason to include the constraint that no user succeeds at P i , i < k, in the definition of E P k |j is to ensure that E P k |j and E Pp|j , k < p, are uncorrelated. For example, the event that U i succeeds by using P 2 and U j succeeds by using P 3 belongs to E P 2 |j only, and is not included in E P 3 |j . Therefore, the probability P j,j can be expressed as follows: P j,j =1 − P E P 1 |j ∪ · · · ∪ E P K |j (17) =1 − K k=1 P E P k |j . Further define E m|j as the event that among the M − j remaining users, there are m active users which make the transmission attempts, and define E P k |m as the event that among the m active users, a single user successfully updates the base station by using the k-th SNR level, P k , and no user chooses P i , i < k, which means P j,j =1 − K k=1 M −j m=1 M − j m P E m|j P E P k |m .(18) By using the transmission attempt probability of P TX , the probability, E m|j , can be obtained as follows: P(E m|j ) = P m TX (1 − P TX ) M −j−m .(19) Without loss of generality, assume that U i is one of the m active users which make transmission attempts. The probability for the event that U i chooses P k , and no user chooses P i , i < k, is given by P K (1 − kP K ) m−1 ,(20) where (1 − kP K ) is the probability of the event that a user which is not U i cannot choose P p , 1 ≤ p ≤ k. Therefore, P E P k |j,m can be approximated as follows: P E P k |m ≈ mP K (1 − kP K ) m−1 ,(21) since each of the m active users can be the successful user with the equal probability, where the high SNR assumption is used, i.e., all SNR levels are feasible to each user and only the errors caused by user collisions are considered. By combining (18), (19), and (21), P j,j can be expressed as follows: P j,j ≈1 − M −j m=1 M − j m P m TX (1 − P TX ) M −j−m (22) × K k=1 mP K (1 − kP K ) m−1 . B. Evaluating P j,j+1 Recall that P j,j+1 is also conditioned on the assumption that j users have successfully updated the base station, and P j,j+1 is the probability of the event that there is a single successful update from a user which cannot be U 1 . Define E 1 P k |j as the event that given M − j remaining users, a single user, other than U 1 , successfully updates the base station by using the k-th SNR level. At high SNR, the following two conclusions can be made regarding E 1 P k |j . On the one hand, due to the feature of SIC, E 1 P k |j implies that no user chooses P i , i < k, which can be shown by contradiction. Assume that U p chooses P i . If U p is the only user choosing P i , U p becomes an additional successful user, which contradicts the assumption that there is a single successful user. If multiple users choose P i , a collision occurs and SIC needs to be terminated at P i , which contradicts the assumption that a 20 successful update happens at P k . On the other hand, E 1 P k |j does not exclude the event that an SNR level, P i , i > k, is chosen by a user; however, E 1 P k |j does imply that if P i , i > k, is chosen, a collision must happen at this SNR level, otherwise there will be an additional successful user. We also note that E 1 P k |j is different from E P k |j for the following two reasons. First, E P k |j does not exclude the event that the successful user is U 1 . Second, for E P k |j , it is still possible for a user to succeed at SNR levels, P i , i > k. By using E 1 P k |j , the probability P j,j+1 can be expressed as follows: P j,j+1 =P(E 1 L 1 |j ∪ · · · ∪ E 1 L K |j ) = K k=1 P(E 1 L k |j ),(23) where the last step follows by the fact that the E 1 P k |j , 1 ≤ k ≤ K, are uncorrelated. Similar to E P k |m , define E 1 P k |m as the event that among the m active users, a single user, other than U 1 , successfully updates the base station by using the k-th SNR level. By using E m|j and E 1 P k |m , P j,j+1 can be expressed as follows: P j,j+1 = K k=1 M −j m=1 M − j m P E m|j P E P 1 k |m .(24) The analysis of P E P 1 k |m is challenging. First, we assume that m ≥ 2, i.e., there are more than one active users. For illustrative purposes, assume that U 2 is an active user. Denote by E + 2k the event that U 2 succeeds by using P k , and P E + 2k is given by P E + 2k ≈ P K (1 − kP K ) m−1 ,(25) where the high SNR approximation is used, and the reason to have (1 − kP K ) m−1 is that the k highest SNR levels are no longer available to the other m − 1 active users. Because m ≥ 2, k ≤ K − 1, i.e., U 2 cannot succeed by using P K , which can be explained by using a simple example with U 2 and U 3 being the active users (m = 2). As discussed previously, if U 2 chooses P K , U 3 has to choose P p , p > K, which is not possible. Furthermore, denote by E − 2k the event that U 2 succeeds by using P k , and there is at least one additional user which succeeds by using P i , i ≥ k + 1. P E − 2k is given by P E − 2k ≈ P K K κ=k+1 (m − 1)P K (1 − κP K ) m−2 ,(26) which is obtained in a similar manner as P E P k |m in (21). By using E + 2k and E − 2k , the probability for the event that among the m active users, only U 2 succeeds by using P k , denoted by E 2k , is given by P (E 2k ) =P E + 2k − P E − 2k ≈ P K (1 − kP K ) m−1 − K κ=k+1 (m − 1)P K (1 − κP K ) m−2 . (27) Intuitively, P E P 1 k |m should be simply P E P 1 k |m = (m − 1)P (E 2k ), i.e., including (m − 1) cases corresponding to U i , 2 ≤ i ≤ m. If U 1 is one of the active users, indeed, P E P 1 k |m = (m − 1)P (E 2k ). However, if U 1 is not an active user, P E P 1 k |m = mP (E 2k ). Therefore, by taking into account the fact that U 1 might not be an active user, P E P 1 k |m can be expressed as follows: P E P 1 k |m ≈ m M − j (m − 1) + M − j − m M − j (m) P (E 2k ) (28) = M − j − 1 M − j mP K (1 − kP K ) m−1 − K κ=k+1 (m − 1)P K (1 − κP K ) m−2 , for the case m ≥ 2. For the case m = 1, i.e., there is a single active user, P E P 1 k |m is simply zero if this user is U 1 . Otherwise, P E P 1 k |m = P K , i.e., P k is selected by the active user. Therefore, for m = 1, P E P 1 k |m is given by P E P 1 k |m ≈ M − j − 1 M − j P K .(29) By combining (24), (28) and (29), the probability, P j,j+1 , can be obtained as shown in the lemma. C. Evaluating P j,j+i , i ≥ 2 We note that for P j,j+1 , there is a single successful update, and P j,j+1 has been analyzed by first specifying which SNR level, i.e., P k , is used for this successful update. The same method could be applied to analyze P j,j+i , i ≥ 2; however, the resulting expression can be extremely complicated if the number of the used SNR levels is large. Instead, the feature of SIC can be used to simplify the analysis of P j,j+i . Consider the following example with i = 3 and K = 8. Assume that the following three SNR levels, P 2 , P 4 , and P 5 , are used by the successful users. The key observation for simplifying the performance analysis is that those SNR levels prior to P 2 and between the chosen ones are not selected by any users, e.g., no user chooses P 1 and P 3 . This observation can be explained by taking P 3 as an example. If this SNR level has been selected, a collision must happen at this level, otherwise there should be an additional successful user. However, if a collision does happen at P 3 , SIC needs to be terminated in the third SIC stage, and hence no successful update can happen at P 4 , which contradicts to the assumption that P 4 is used by a successful user. As a result, no one chooses P 3 . By using this observation, we note that only two SNR levels are significant to the analysis of P j,j+i , namely the highest and the lowest SNR levels, which are denoted by P k 1 and P k 2 , respectively. For the aforementioned example, P k 1 = P 2 and P k 2 = P 5 . Define E i P k |j as the event that given M − j remaining users, i users which are not U 1 successfully update the base station, where P k 1 and P k 2 are the highest and lowest used SNR levels. By using E i P k |j , the probability P j,j+i can be expressed as follows: P j,j+i = K−i+1 k 1 =1 K k 2 =k 1 +i−1 P(E i L k |j ) = K−i+1 k 1 =1 K k 2 =k 1 +i−1 M m=1 M − j m P E m|j P E P i k |m ,(30) where E P i k |m is defined similar to E i P k |j with the assumption that there are m active users. In order to find P E P i k |m , again, we first focus on the case m ≥ i + 1, i.e., there are more than i active user. Define E ik as the probability for the particular event that among the m active users, U 2 succeeds by using P k 1 , U j succeeds by using P k 1 +j−2 , 3 ≤ j ≤ i, and U i+1 succeeds by using P k 2 . Similar to P (E 2k ), P (E ik ) can be approximated at high SNR as follows: P (E ik ) ≈ P i K (1 − k 2 P K ) m−i − K κ=k 2 +1 (m − i)P K (1 − κP K ) m−i−1 ,(31) where the term, (1 − k 2 P K ) m−i , is due to the fact that the remaining m − i active users can choose P i , i > k 2 , only, and the second term in the bracket is obtained similar to P E P k |m in (21). Following steps similar to those to obtain P E P 1 k |m in (28), P E P i k |m can be obtained from P (E ik ) as follows: P E P i k |m ≈P (E ik ) k 2 − k 1 − 1 i − 2 m M − j (m − 1)(m − 2) · · · (m − i)+ repetitions if the tagged user is one of the m active users (32) M − j − m M − j m(m − 1) · · · (m − i + 1) repetitions if the tagged user is not one of the m active users =P (E ik ) k 2 − k 1 − 1 i − 2 M − j − i M − j m · · · (m − i + 1), for the case m ≥ i + 1, where k 2 −k 1 −1 i−2 is the number of the possible choices for the (i − 2) SNR levels which are between P k 1 and P k 2 . The special case m = i means that there are m active users and each of the active users is a successful user. Therefore, the probability in (31), P (E ik ), is simply P i K , and hence for the case m = i, P E P i k |m can be expressed as follows: P E P i k |m ≈ P i K k 2 − k 1 − 1 i − 2 M − j − i M − j i · · · 1.(33) By combining (32) and (33), the expression of P j,j+i can be obtained as shown in the lemma. Therefore, the proof for the lemma is complete. APPENDIX B PROOF FOR LEMMA 2 The proof comprises first simplifying the state transition probabilities, then developing an asymptotic expression of the AoI, and finally finding the optimal choice of P TX . A. Simplifying the State Transition Probabilities For the case of N = 1 and K = 2, only the following transition probabilities, P 0,0 , P 0,1 , P 0,2 , need to be focused on. In particular, the expression of P 0,0 can be simplified as follows: P 0,0 ≈1 − M m=1 M m P m TX (1 − P TX ) M −m K k=1 mP K (1 − kP K ) m−1 (1) =1 − M m=1 M m P m TX (1 − P TX ) M −m mP K (1 − P K ) m−1 − MP TX (1 − P TX ) M −1 P K (b) =1 − M m=1 M m P m TX (1 − P TX ) M −m mP m K − MP TX (1 − P TX ) M −1 P K , where step (a) follows by the fact that (1 − KP K ) m−1 = 0 only if m = 1, and step (b) follows by P K = 1 − P K for K = 2. By using the properties of the binomial coefficients, P 0,0 can be simplified as follows: P 0,0 ≈1 − MP K P TX M −1 i=0 M − 1 i (P K P TX ) i (1 − P TX ) M −1−i − MP TX (1 − P TX ) M −1 P K (34) =1 − MP K P TX (P K P TX + 1 − P TX ) M −1 − MP TX (1 − P TX ) M −1 P K . Similarly, for the case of K = 2, P 0,1 can be first written as follows: P 0,1 ≈MP TX (1 − P TX ) M −1 M − 1 M KP K + M m=2 M m P m TX (1 − P TX ) M −m K−1 k=1 M − 1 M mP K (1 − kP K ) m−1 − K κ=k+1 (m − 1)P K (1 − κP K ) m−2 .(35) Define τ (m) = K−1 k=1 M −1 M mP K [(1 − kP K ) m−1 − K κ=k+1 (m − 1)P K (1 − κP K ) m−2 ] . Note that in the expression of τ (m), k ≥ 1, and hence κ ≥ 2. For the case of K = 2, 1 − 2P K = 0. By using this observation, τ (m) can be simplified as follows: τ (m) = m(M − 1) M P K (1 − P K ) m−1 ,(36) for m > 2, and τ (2) = m M (m − 1) + M − m M m P K [(1 − P K ) − P K ] = 0.(37) By using the simplified expression of τ (m), P 0,1 can be simplified as follows: P 0,1 ≈MP TX (1 − P TX ) M −1 M − 1 M KP K (38) + M m=3 M m P m TX (1 − P TX ) M −m m(M − 1) M P K (1 − P K ) m−1 =(M − 1)P TX (1 − P TX ) M −1 + (M − 1) M m=3 M − 1 m − 1 P m TX (1 − P TX ) M −m P m K , where the last step follows by the fact that 1 − P K = P K . Finally, P 0,2 can rewritten as follows: P 0,2 ≈ M 2 P i TX (1 − P TX ) M −2 K−1 k 1 =1 K k 2 =k 1 +1 M − i M P 2 K k 2 − k 1 − 1 0 1 p=0 (2 − p)(39)+ M m=3 M m P m TX (1 − P TX ) M −m K−2 k 1 =1 K−1 k 2 =k 1 +1 M − i M P 2 K k 2 − k 1 − 1 0 1 p=0 (m − p) × (1 − k 2 P K ) m−2 − K κ=k 2 +1 (m − 2)P K (1 − κP K ) m−3 (a) = M 2 P 2 TX (1 − P TX ) M −2 2 M − 2 M P 2 K = (M − 1)! (M − 3)! P 2 TX (1 − P TX ) M −2 P 2 K , where step (a) follows by employing the properties of the binomial coefficients. B. Asymptotic Studies of AoI By using the above transition probabilities, the probability for the event that U 1 cannot complete an update within one frame, P fail , can be simplified as follows: where x = P TX P K . By using the properties of binomial coefficients, τ 0 can be rewritten as follows: P fail ≈P (Z > N) = s T 0 P N M 1 = 2 j=0 P 0,j(40)τ 0 =x(M − 1) M i=2 M − 1 i (1 − P TX ) M −i−1 x i (41) =(1 − P TX + x) M −1 (M − 1)x − (1 − P TX ) M −1 (M − 1)x − (M − 1)(1 − P TX ) M −2 (M − 1)x 2 . By using the simplified expression of τ 0 , P fail can be rewritten as follows: The simplified expression of P fail can be used to facilitate the asymptotic studies of the AoI. For the case of N = 1, the pmf of the access delay can be obtained from P fail as follows: P(S j = T ) = s T 0 P 0 M p 1 − s T 0 P M 1 = p 0,M p 0,M = 1,(42) where p 0,M is the first element of p. As a result, E{S j } and E{S 2 j } can be obtained as follows: E{S j } = T 1 n=1 n s T 0 P n−1 M p 1 − s T 0 P N M 1 = T,(43) and E{S 2 j } = T 2 1 n=1 n 2 s T 0 P n−1 M p 1 − s T 0 P N M 1 = T 2 ,(44) which are expected since S j becomes a deterministic parameter for this special case. Furthermore, recall that the inter-departure time Y j can be rewritten as follows: Y j = (NT − S j−1 ) + (X j − 1)NT + S j . Therefore, for the case of N = 1, the expectation of Y j can be simplified as follows: E{Y j } =E{(NT − S j−1 ) + (X j − 1)NT + S j }(46) =T NE{X j } = T N 1 1 − P fail , which is obtained by using the fact that X j follows the geometric distribution. Similarly, the expectation of Y 2 j can be simplified as follows: E{Y 2 j } =N 2 T 2 E X 2 j + 2E S 2 j − 2E {S j } 2 = N 2 T 2 1 + P fail (1 − P fail ) 2 + 2T 2 − 2T 2 .(47) Finally, for the case of N = 1, the averaged AoI can be expressed as follows: ∆ N = E{S j }E{Y j } − E{S 2 j } + E{S j } 2 E{Y j } + E{Y 2 j } 2E{Y j } = T E{Y j } − T 2 + T 2 E{Y j } + E{Y 2 j } 2E{Y j } (48) =T + N 2 T 2 1+P fail (1−P fail ) 2 2T N 1 1−P fail ≈ T + NT (2 − f (P TX )) 2f (P TX ) , where f (P TX ) = (P TX P K + 1 − P TX ) M −1 P TX P K + (1 − P TX + (M − 1)P TX P K )(1 − P TX ) M −2 P TX P K . C. Finding the Optimal Choice of P TX The considered AoI minimization problem can be expressed as follows: min 0≤P TX ≤1∆ N . It is challenging to show whether∆ N is a convex function of P TX , given the complex expression of∆ N . However, it can be shown that∆ N first decreases and then increases as P TX grows, as shown in the following. The first order derivative of∆ N with respect to P TX is given by ∆ N ′ ≈ NT 2 −f ′ (P TX ) f (P TX ) − 2f ′ (P TX ) ′ − f (P TX )f ′ (P TX ) f 2 (P TX ) = − NT 2 2f ′ (P TX ) f 2 (P TX ) , which shows that the monotonicity of∆ N is decided by the sign of f ′ (P TX ) only. In the following, we will first show that f ′ (P TX ) = 0 has a single root for 0 ≤ P TX ≤ 1. With some straightforward algebraic manipulations, f ′ (P TX ) can be expressed as follows: Z.Ding and H. V. Poor are with the Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544, USA. Z. Ding is also with the School of Electrical and Electronic Engineering, the University of Manchester, Manchester, UK (email: zhiguo.ding@manchester.ac.uk, poor@princeton.edu). R. Schober is with the Institute for Digital Communications, Friedrich-Alexander-University Erlangen-Nurnberg (FAU), Germany (email: robert.schober@fau.de). compared to OMA, particularly for the case of low transmit SNR and a massive number of grant-free users. II. NOMA ASSISTED GRANT-FREE TRANSMISSION Consider a grant-free communication network with M users, denoted by U m , 1 ≤ m ≤ M, communicating with the same base station. Assume that each time frame comprises N time slots, each of duration of T seconds, where the n-th time slot of the i-th frame is denoted by TS n i , and the starting time of TS n i is denoted by t n i , 1 ≤ n ≤ N and i ≥ 1, as shown in Fig. 1. … Fig. 1 . 1Considered slotted time frame structure. . 2 , 2U 1 successfully sends its updates to the base station in TS i n of frame i and TS i+2 l of frame i + 2, but fails in frame i + 1. For the AoI analysis, the following metrics are required: • S j : The time duration between the generation time and the receive time of the j-th successful update. For the example shown in Fig. 2, S j−1 = nT and S j = lT . • Y j : The time duration between the (j−1)-th and the j-th successful updates. For the example shown in Fig. 2, Y j = (N − n)T + NT + lT . Remark 6 : 6Motivated by the results shown in Lemma 2 and Proposition 1, for the general case of K > 2, a simple choice of P TX = min 1, K M Fig. 3 . 3Impact of the number of users on the average AoI achieved with OMA and NOMA assisted grant-free transmission for GAR. T = 6, P = 20 dB, R = 0.5 BPCU, and N = 8. For the fixed choice of PTX, PTX = 0.05, and for the adaptive choice of PTX, PTX = min 1, K M for NOMA, and PTX = 1 M −j for OMA, i.e., a state-dependent choice is used for OMA as discussed in Section II-B. Fig. 4 .Fig. 5 . 45Impact of the number of time slots in each frame on the average AoI achieved by the OMA and NOMA assisted grant-free transmission schemes for GAR. K = 4, T = 6, P = 20 dB, R = 0.5 BPCU, M = 8, and the adaptive choices for PTX are used. AoI performance achieved by the two grant-free transmission schemes for the special case with K = 2 and N = 1,where T = 6, and the optimal choices of PTX are used. Fig. 6 . 6Illustration of the accuracy of the developed high-SNR analytical results. K = 2, N = 1, T = 6, and the optimal choices of PTX are used. operate in the low SNR regime. Fig. 5(a) also demonstrates that for the case with small R, even if the SNR is low, i.e., −5 and 0 dB, the difference between the analytical and simulation results is negligible. This property of the developed analytical results is particularly important, given the fact that, for many important applications of grant-free transmission, such as IoT and uMTC, the users' target data rates are indeed small. The aforementioned conclusions are also confirmed by Fig. 6, where the AoI is shown as a function of the transmit SNR. In particular, the developed analytical results provide accurate estimates in the medium-to-high SNR regions regardless of the choices of R. Fig. 7 . 7Illustration of the impact of PTX on the AoI achieved by the considered grant-free schemes for GAR. T = 6, P = 20 dB, R = 0.5 BPCU, K = 2, N = 1, and M = 8. Fig. 8 . 8Impact of the number of users on the ratio between the AoI achieved with the NOMA and OMA, i.e.,∆ N ∆ O , for GAR. T = 6, P = 20 dB, R = 0.5 BPCU, and the optimal choices of PTX are used. Fig. 9 . 9AoI achieved by the considered grant-free transmission schemes for different data generation models. N = 8, T = 6, R = 0.5 BPCU, and the adaptive choices of PTX are used. =1 − M(x + 1 − P TX ) M −1 x − M(1 − P TX ) M −1 x + 2(M − 1)(1 − P TX ) − P TX ) M −2 x 2 , P fail ≈1 − M(x + 1 − P TX ) M −1 x − M(1 − P TX ) M −1 x + 2(M − 1)(1 − P TX ) M −1 x + (1 − P TX + x) M −1 (M − 1)x − (1 − P TX ) M −1 (M − 1)x − (M − 1)(1 − P TX ) M −2 (M − 1)x 2 + (M − 1)(M − 2)(1 − P TX ) M −2 x 2 =1 − (P TX P K + 1 − P TX ) M −1 P TX P K − (1 − P TX + (M − 1)P TX P K )(1 − P TX ) M −2 P TX P K . fM− 2 + 1 − 21′ (P TX ) = 1 − M P TX 2 1 − P TX 2 2P TX − M (M − 3) P 2 TX 2 (1 − P TX ) M−3 . 1 ) 1Orthogonal Multiple Access (OMA): Based on OMA, a user's transmission can be successful only if it solely occupies the bandwidth resource block, i.e., a time slot. A simple example of OMA based grant-free transmission is slotted ALOHA, as described in the following 1 . In this paper, a simple slotted ALOHA scheme is considered, where each user can use any of the time slots in the frame.1 . When M → ∞, the two roots can be approximated as follows:TX , which is bounded as follows:A key observation from(51)is that the upper and lower bounds on P * TX are of the same order of 1 M . Therefore, P * TX can be expressed as P *By using this expression for P * TX , f ′ (P * TX ) = 0 can be expressed as follows:In order to find an explicit expression of P * TX , we note that f x (x) = 1 − a x x can be approximated at x → ∞ as follows: Therefore, for M → ∞, (52) can be approximated as follows:where the value of η can be straightforwardly obtained by applying off-the-shelf root solvers. It is important to point out that (54) has a single root, which means that P * TX = η M is the single root of f ′ (P TX ) = 0. Therefore, P * TX = η M is the optimal choice of P TX to minimize∆ N , and the proof is complete.Based on Lemma 2, the optimal choice for the transmission probability is given by P * TX = η M . 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{'abstract': "The aim of this paper is to characterize the impact of non-orthogonal multiple access (NOMA) on the age of information (AoI) of grant-free transmission. In particular, a low-complexity form of NOMA, termed NOMA-assisted random access, is applied to grant-free transmission in order to illustrate the two benefits of NOMA for AoI reduction, namely increasing channel access and reducing user collisions.Closed-form analytical expressions for the AoI achieved by NOMA assisted grant-free transmission are obtained, and asymptotic studies are carried out to demonstrate that the use of the simplest form of NOMA is already sufficient to reduce the AoI of orthogonal multiple access (OMA) by more than 40%.In addition, the developed analytical expressions are also shown to be useful for optimizing the users' transmission attempt probabilities, which are key parameters for grant-free transmission.Alternatively, massive multi-input multi-output (MIMO) can also be applied to support grantfree transmission by using the excessive spatial degrees of freedom offered by massive MIMORecently, the application of non-orthogonal multiple access (NOMA) to grant-free transmission has received significant attention due to the following two reasons. First, the NOMA principle is highly compatible, and the use of NOMA can significantly improve the reliability and spectral efficiency of random access and massive MIMO based grant-free protocols [8]-[11]. Second, more importantly, the use of NOMA alone is sufficient to support grant-free transmission. For example, NOMA-based grant-free transmission has been proposed in [12], where a Bayesian learning based scheme has been designed to ensure successful multi-user detection, even if the number of active grant-free users is unknown. The principle of NOMA can also be used to develop so-called semi-grant-free transmission protocols, where the bandwidth resources which would be solely occupied by grant-based users are released for supporting multiple grant-free users in a distributed manner [13]. In addition, NOMA-based grant-free transmission has also been shown to be robust and efficient in various communication scenarios, such as satellite communication networks, secure Internet of Things (IoT), intelligent reflecting surface (IRS) networks, marine communication systems, etc., see [14]-[17].The aim of this paper is to characterize the impact of NOMA on the performance of grantfree transmission with respect to a recently developed new performance metric, termed the age of information (AoI) [18]-[20]. In particular, the AoI describes the freshness of data updates collected in the network, and is an important metric to measure the success of the 6G services, including umMTC and euRLLC. We note that most existing works have focused on the impact of NOMA on grant-based networks [21]-[25]. For example, for two-user grant-based networks, the capability of NOMA to reduce the AoI has been shown to be related to the spectral efficiency gain of NOMA over orthogonal multiple access (OMA)[26]. To the authors' best knowledge, there is only a single existing work which applied NOMA to reduce the AoI of grant-free transmission[27], where the strong assumption that the base station estimates all users' channel state information (CSI) was made.In this paper, the impact of NOMA on the AoI of grant-free transmission is investigated from the perspective of performance analysis, which is different from the existing work focusing on resource allocation[28]. In particular, a low-complexity form of NOMA, which was originally termed NOMA-assisted random access[29]and recently also termed ALOHA with successive, and E{·} denotes the expectation.Remark 1: We note that the AoI expressions in (4) of this paper and[35,Eq. (3)] are consistent.The reason why there is an extra factor of 1 2 in [35, Eq. (3)] is that the users' instantaneous AoI was assumed to be discrete-valued in[35], instead of continuous-valued as in this paper.", 'arxivid': '2211.13773', 'author': ['Fellow, IEEEZhiguo Ding \nLife Fellow\nIEEE\n', 'Fellow, IEEERobert Schober \nLife Fellow\nIEEE\n', 'H Vincent Poor \nLife Fellow\nIEEE\n'], 'authoraffiliation': ['Life Fellow\nIEEE', 'Life Fellow\nIEEE', 'Life Fellow\nIEEE'], 'corpusid': 254017648, 'doi': '10.48550/arxiv.2211.13773', 'github_urls': [], 'n_tokens_mistral': 21416, 'n_tokens_neox': 19287, 'n_words': 12771, 'pdfsha': '92dc255d89dfd3704a78721966c557f7e77363d2', 'pdfurls': ['https://export.arxiv.org/pdf/2211.13773v2.pdf'], 'title': ['Impact of NOMA on Age of Information: A Grant-Free Transmission Perspective', 'Impact of NOMA on Age of Information: A Grant-Free Transmission Perspective'], 'venue': []}

arxiv

A numerical scheme for the quantile hedging problem March 5, 2019 Cyril Bénézet Jean-François Chassagneux Christoph Reisinger A numerical scheme for the quantile hedging problem March 5, 2019Quantile hedgingBSDEsmonotone approximation schemes We consider the numerical approximation of the quantile hedging price in a non-linear market. In a Markovian framework, we propose a numerical method based on a Piecewise Constant Policy Timestepping (PCPT) scheme coupled with a monotone finite difference approximation. We prove the convergence of our algorithm combining BSDE arguments with the Barles & Jakobsen and Barles & Souganidis approaches for non-linear equations. In a numerical section, we illustrate the efficiency of our scheme by considering a financial example in a market with imperfections. Introduction In this work, we study the numerical approximation of the quantile hedging price of a European contingent claim in a market with possibly some imperfections. The quantile hedging problem is a specific case of a broader class of approximate hedging problems. It consists in finding the minimal initial endowment of a portfolio that will allow the hedging a European claim with a given probability p of success, the case p " 1 corresponding to the classical problem of (super)replication. This approach has been made popular by the work of Föllmer and Leukert [19] who provided a closed form solution in a special setting. The first PDE characterisation was introduced by [8] in a possibly incomplete market setting with portfolio constraints. Various extensions have been considered since this work: to jump dynamics [25]; to the Bermudan case [6] and American case [16]; to a non-Markovian setting [7,15]; and to a finite number of quantile constraints [9]. Except for [6,9], all the aforementioned works are of a theoretical nature. The lack of established numerical methods for these problems is a clear motivation for our study. We now present in more detail the quantile hedging problem and the new numerical method we introduce and study in this paper. On a complete probability space pΩ, F, Pq, we consider a d-dimensional Brownian motion pW t q tPr0,T s and denote by pF t q tPr0,T s its natural filtration. We suppose that all the randomness comes from the Brownian motion and assume that F " F T . Let µ : R d Ñ R d , σ : R d Ñ M d pRq, where M d pRq is the set of dˆd matrices with real entries, f : r0, T sˆR dˆRˆRd Ñ R be Lipschitz continuous functions, with Lipschitz constant L. For pt, x, yq P r0, T sˆR dˆR and ν P H 2 , which denotes the set of predictable squareintegrable processes, we consider the solution pX t,x , Y t,x,y,ν q to the following stochastic differential equations: X s " x`ż s t µpX u q du`ż s t σpX u q dW u , Y s " y´ż s t f pu, X u , Y u , ν u q du`ż s t ν u dW u , s P rt, T s. In the financial applications we are considering, X will typically represent the log-price of risky assets, the control process ν is the amount invested in the risky assets, and the function f is non-linear to allow to take into account some market imperfections in the model. A typical financial example, which will be investigated in the numerical section, is the following: Example 1.1. The underlying diffusion X is a one-dimensional Brownian motion with constant drift µ P R and volatility σ ą 0. There is a constant borrowing rate R and a lending rate r with R ě r. In this situation, the function f is given by: f pt, x, y, zq "´ry´σ´1µz`pR´rqpy´σ´1zq´. The quantile hedging problem corresponds to the following stochastic control problem: for pt, x, pq P r0, T sˆR dˆr 0, 1s find vpt, x, pq :" inf ! y ě 0 : Dν P H 2 , P´Y t,x,y,ν T ě gpX t,x T q¯ě p . (1.1) The main objective of this paper is to design a numerical procedure to approximate the function v by discretizing an associated non-linear PDE first derived in [8]. A key point in the derivation of this PDE is to observe that the above problem can be reformulated as a classical stochastic target problem by introducing a new control process representing the conditional probability of success. To this end, for α P H 2 , we denote P t,p,α s :" p`ż s t α s dW s , t ď s ď T , and by A t,p the set of α such that P t,p,α P r0, 1s. The problem (1.1) can be rewritten as vpt, x, pq :" inf ! y ě 0 : Dpν, αq P pH 2 q 2 , Y t,x,y,ν T ě gpX t,x T q1 tP t,p,α T ą0u ) (see Proposition 3.1 in [8] for details). In our framework, the above singular stochastic control problem admits a representation in terms of a non-linear expectation, generated by a Backward Stochastic Differential Equation (BSDE), vpt, x, pq " inf αPA t,p Y α t (1.2) where pY α , Z α q is the solution to Y α s " gpX t,x T q1 tP t,p,α T ą0u`ż T s f ps, X s , Y α s , Z α s q ds´ż T s Z α s dW s , t ď s ď T . The article [7] justifies the previous representation and proves a dynamic programming principle for the control problem in a general setting. In the Markovian setting, this would lead naturally to the following PDE for v in r0, T qˆR dˆp 0, 1q: "´B t ϕ`sup aPR d F a pt, x, ϕ, Dϕ, D 2 ϕq " 0 " (1.3) where for pt, x, yq P r0, T sˆR dˆR`, q :"ˆq x q p˙P R d`1 and A :"ˆA xx A xp A xp J A pp˙P S d`1 , A xx P S d , denoting Ξ :" pt, x, y, q, Aq, we define F a pΞq :"´f pt, x, y, zpx, q, aqq´Lpx, q, A, aq , (1.4) with zpx, q, aq :" q x σpxq`q p a , (1.5) Lpx, q, A, aq :" µpxq J q x`1 2 Tr " σpxqσpxq J A xx ‰`| a| 2 2 A pp`aJ σpxq J A xp . (1. 6) The PDE formulation in (1.3) is not entirely correct as the supremum part may degenerate and it would require using semi-limit relaxation to be mathematically rigourous. We refer to [8], where it has been obtain in a more general context. We shall use an alternative PDE formulation to this "natural" one (1.3), which we give at the start of Section 2. Moreover, the value function v continuously satisfies the following boundary conditions in the p-variable: vpt, x, 0q " 0 and vpt, x, 1q " V pt, xq on r0, T sˆp0, 8q d , (1.7) where V is the super-replication price of the contingent claim with payoff gp¨q. It is also known that v has a discontinuity as t Ñ T . By definition, the terminal condition is R dˆr 0, 1s Q px, pq Þ Ñ gpxq1 pą0 P R`, but the values which are continuously attained are obtained by convexification [8], namely vpT´, x, pq " pgpxq on R dˆr 0, 1s, (1.8) and we shall work with this terminal condition at t " T from now on. To design the numerical scheme to approximate v, we use the following strategy: 1. Bound and discretise the set where the controls α take their values. 2. Consider an associated Piecewise Constant Policy Timestepping (PCPT) scheme for the control processes . 3. Use a monotone finite difference scheme to approximate in time and space the PCPT solution resulting from 1. & 2. The approximation of controlled diffusion processes by ones where policies are piecewise constant in time was first analysed by [23]; in [24], this procedure is used in conjunction with Markov chain approximations to diffusion processes to construct fully discrete approximation schemes to the associated Bellman equations and to derive their convergence order. An improvement to the order of convergence from [23] was shown recently in [22] using a refinement of Krylov's original, probabilistic techniques. Using purely viscosity solution arguments for PDEs, error bounds for such approximations are derived in [3], which are weaker than those in [23] for the control approximation scheme, but improve the bounds in [24] for the fully discrete scheme. In [27], using a switching system approximation introduced in [3], convergence is proven for a generalised scheme where linear PDEs are solved piecewise in time on different meshes, and the control optimisation is carried out at the end of time intervals using possibly non-monotone, higher order interpolations. An extension of the analysis in [27] to jump-processes and non-linear expectations is given in [17]. Our first contribution is to prove that the approximations built in step 1. and 2. above are convergent for the quantile hedging problem, which has substantial new difficulties compared to the settings considered in the aforementioned works. For this we rely heavily on the comparison theorem for the formulation in (2.1) and we take advantage of the monotonicity property of the approximating sequences. The main new difficulties come from the non-linear form of the PDE including unbounded controls, and in particular the boundaries in the p-variable. To deal with the latter especially, we rely on some fine estimates for BSDEs to prove the consistency of the scheme including the strong boundary conditions (see Lemma 2.2 and Lemma 2.3). Our second contribution is to design the monotone scheme in step 3. and to prove its convergence. The main difficulties come here from the non-linearity of the new term from the driver of the BSDE in the gradient combined with the degeneracy of the diffusion operator given in (1.6), and again the boundedness for the domain in p. In particular, a careful analysis of the consistency of the boundary condition is needed (see Proposition 3.4). To the best of our knowledge, this is the first numerical method for the quantile hedging problem in this non-linear market specification. In the linear market setting, using a dual approach, [6] combines the solution of a linear PDE with Fenchel-Legendre transforms to tackle the problem of Bermudan quantile hedging. Their approach cannot be directly adapted here due to the presence of the non-linearity. The dual approach in the non-linear setting would impose some convexity assumption on the f parameter and would require to solve fully non-linear PDEs. Note that here f is only required to be Lipschitz continuous in py, zq. We believe that an interesting alternative to our method would be to extend the work of [5] to the non-linear market setting we consider here. The rest of the paper is organised as follows. In Section 2, we derive the control approximation and PCPT scheme associated with items 1. & 2. above and prove their convergence. In Section 3, we present a monotone finite difference approximation which is shown to convergs to the semi-discrete PCPT scheme. In Section 4, we present numerical results for a specific application and analyse the observed convergence. Finally, the appendix contains some of the longer, more technical proofs and collects useful background results used in the paper. Notations diagpxq is the diagonal matrix of size d, whose diagonal is given by x. Let us denote by S the sphere in R d`1 of radius 1 and by D the set of vectors η P S such that their first component η 1 " 0. For a vector η P SzD, we denote η 5 :" 1 η 1 pη 2 , . . . , η d`1 q J P R d . By extension, we denote, for Z Ă SzD, Z 5 :" tη 5 P R d | η P Zu . We denote by BC :" L 8 pr0, T s, C 0 pR dˆr 0, 1sqq, namely the space of functions u that are essentially bounded in time and continuous with respect to their space variable. The convergence in C 0 pr0, T sˆR d q considered here is the local uniform convergence. Convergence of a discrete-time scheme In this section, we design a Piecewise Constant Policy Timestepping (PCPT) scheme which is convergent to the value function v defined in 1.2. Following [5], it has been shown in [10], that the function v is equivalently a viscosity solution of the following PDE (see Theorems 3.1 and 3.2 in [10]): Hpt, x, ϕ, B t ϕ, Dϕ, D 2 ϕq " 0 (2.1) in p0, T qˆR dˆp 0, 1q, where H is a continuous operator HpΘq " sup ηPS H η pΘq ,(2.2) where for pt, x, y, bq P r0, T sˆR dˆR`ˆR , q :"ˆq x q p˙P R d`1 and A :"ˆA xx A xp A xp J A pp˙P S d`1 , and Θ :" pt, x, y, b, q, Aq, we define H η pΘq " pη 1 q 2´´b´f pt, x, y, zpx, q, η 5 qq´Lpx, q, A, η 5 q¯, for η P SzD . (2.3) Recall also the definition of L and z in (1.5) and (1.6). This representation and its properties are key in the proof of convergence. Loosely speaking, it is obtained by "compactifying" the set t1uˆR d to the unit sphere S. A comparison theorem is shown in Theorem 3.2 in [10]. As partially stated in the introduction, we will work under the following assumption: pHq(i) The functions b, σ are L-Lipschitz continuous and g is bounded and L g -Lipschitz continuous. (ii) The function f is measurable and for all t P r0, T s, f pt,¨,¨,¨q is L-Lipschitz continuous. For all pt, x, zq P r0, T sˆR dˆRd , the function y Þ Ñ f pt, x, y, zq is decreasing. Moreover, f pt, x, 0, 0q " 0. (2.4) Under the above Lipschitz continuity assumption, the mapping SzD Q η Þ Ñ H η pΘq P R extends continuously to S by setting, for all η P D, H η pΘq "´1 2 A pp , see Remark 3.1 in [10]. Remark 2.1. (i) In pHq(ii), the monotonicity assumption is not a restriction, as in a Lipschitz framework, the classical transformationṽpt, x, pq :" e λt vpt, x, pq for λ large enough allows to reach this setting; see Remark 3.3 in [10] for details. (ii) The condition (2.4) is a reasonable financial modelling assumption: It says that starting out in the market with zero initial wealth and making no investments will lead to a zero value of the wealth process. (iii) Since f is decreasing and g is bounded, it is easy to see that |V | 8 ď |g| 8 , where V is the super-replication price. Discrete set of control In order to introduce a discrete-time scheme which approximates the solution v of (2.1)-(1.8), we start by discretizing the set of controls S. Let pR n q ně1 be an increasing sequence of closed subsets of SzD such that ď ně1 R n " S . For n ě 1, let v n : r0, T sˆR dˆr 0, 1s Ñ R be the unique continuous viscosity solution of the following PDE: H n pt, x, ϕ, B t ϕ, Dϕ, D 2 ϕq " 0,(2.Proposition 2.1. The functions v n converge to v in C 0 pr0, T sˆR d q. Proof. 1. For n 1 ă n, we observe that v n 1 is a super-solution of (2.6) as R n 1 Ă R n . Using the comparison result of Proposition 6.1, we obtain that v n 1 ě v n . Similarly, using the comparison principle ( [10], Theorem 3.2), we obtain that v n ě v, for all n ě 1. For all pt, x, pq P r0, T sˆp0, 8q dˆr 0, 1s, let: vpt, x, pq " lim jÑ8 sup " v n ps, y, qq : n ě j and }ps, y, qq´pt, x, pq} ď 1 j * , (2.8) vpt, x, pq " lim jÑ8 inf " v n ps, y, qq : n ě j and }ps, y, qq´pt, x, pq} ď 1 j * . (2.9) From the above discussion, recalling that v 1 and v are continuous, we have v 1 ě v ě v ě v , which shows that v and v satisfy the boundary conditions (1.7)-(1.8). In order to prove the theorem, it is enough to show that v is a viscosity subsolution of (2.1) and v is a viscosity supersolution (which follows similarly and is therefore omitted). The comparison principle ( [10], Theorem 3.2) then implies that v " v " v, and it follows from [12], Remark 6.4 that the convergence v n Ñ v as n Ñ 8 is uniform on every compact set. Using Theorem 6.2 in [1], we obtain that v is a subsolution to Hpt, x, ϕ, B t ϕ, Dϕ, D 2 ϕq " 0 on p0, T qˆp0, 8q dˆp 0, 1q , where HpΘq " lim jÑ8 inf " H n pΘ 1 q : n ě j and }Θ´Θ 1 } ď 1 j * . In the next step, we prove that H " H, which concludes the proof of the proposition. 2. Let us denote by P n : S ÑS n the closest neighbour projection on the closed setS n . From (2.5), we have that lim nÑ8 P n pηq " η, for all η P S. We also have that HpΘq " H η˚p Θq for some η˚P argmax ηPS H η pΘq as S is compact. Let us now introduce η n :" P n pη˚q and by continuity of H, we have H ηn pΘq Ñ HpΘq . We also observe that H ηn pΘq ď H n pΘq ď HpΘq . This proves the convergence H n pΘq Ò HpΘq, for all Θ. As H is continuous, we conclude by using Dini's Theorem that the convergence is uniform on compact subsets, leading to H " H. l The PCPT scheme From now on, we fix n ě 1 and R n the associated discrete set of control. For pt, x, yq P r0, T sˆR dˆR`, q P R d`1 and A P S d`1 , denoting Ξ :" pt, x, y, q, Aq, we define F n pΞq " sup aPR 5 n F a pΞq with F a pΞq :"´f pt, x, y, zpx, q, aqq´Lpx, q, A, aq . Following the proof of Corollary 6.1 in the appendix, we easily observe that v n is also the unique viscosity solution tó B t ϕ`F n pt, x, ϕ, Dϕ, D 2 ϕq " 0 on r0, T qˆR dˆp 0, 1q (2.10) with the same boundary conditions (1.7)-(1.8). The above PDE is written in a more classical way and we will mainly consider this form in the sequel. Let us observe in particular that K :" R 5 n is a discrete subset of R d , such that (2.10) appears as a natural discretisation of (1.3) and will be simpler to study. To approximate v n , we consider an adaptation of the PCPT scheme in [24,3], and especially [17], to our setting, as described below. For κ P N˚, we consider grids of the time interval r0, T s: π " t0 ": t 0 㨨¨ă t k 㨨¨ă t κ :" T u, and denote |π| :" max 0ďkďκ pt k`1´tk q. For 0 ď t ă s ď T , a P K and a continuous φ : R dˆr 0, 1s Ñ R, we denote by S a ps, t, φq : rt, ssˆR dˆr 0, 1s Ñ R the unique solution of B t ϕ`F a pt, x, ϕ, Dϕ, D 2 ϕq " 0 on rt, sqˆR dˆp 0, 1q, (2.11) ϕps, x, pq " φpx, pq on R dˆr 0, 1s, (2.12) ϕpr, x, 0q " B 0 pt, s, φqpr, xq, ϕpr, x, 1q " B 1 pt, s, φqpr, xq on rt, sqˆR d . (2.13) The function B p pt, s, φq for p P t0, 1u is solution tó B t ϕ`F 0 pr, x, ϕ, Dϕ, D 2 ϕq " 0 on rt, sqˆR d , (2.14) with terminal condition B p pt, s, φqpr, xqps, xq " φpx, pq. The solution to the PCPT scheme associated with the grid π is then the function v n,π : r0, T sˆR dˆr 0, 1s such that Spπ, t, x, p, v n,π pt, x, pq, v n,π q " 0, (2.15) where for a grid π, pt, x, p, yq P r0, T sˆR dˆr 0, 1sˆR`and a function u P BC, Spπ, t, x, p, y, uq " " y´min aPK S a ptπ , tπ , uptπ ,¨qq pt, x, pq if t ă T, y´ĝpxqp otherwise, (2.16) with tπ :" inftr P π | r ą tu and tπ :" suptr P π | r ď tu. (2.17) We will drop the subscript π for brevity whenever we consider a fixed mesh. Let us observe that the function v n,π can be alternatively described by the following backward algorithm: 1. Initialisation: set v n,π pT, x, pq :" gpxqp, x P R dˆr 0, 1s. 2. Backward step: For k " κ´1, . . . , 0, compute w k,a :" S a pt k , t k`1 , v n,π pt k`1 ,¨qq and set v n,π p¨q :" inf aPK w k,a . (i) At p " 0, the terminal condition is φpT, xq " 0 (recall that vpT, x, pq " gpxq1 tpą0u ), and this propagates through the backward iteration, so that v n,π pt, x, 0q " 0 for all pt, xq P r0, T sˆR d . (ii) At p " 1, the terminal condition is φpT, xq " gpxq and the boundary condition is thus given by v n,π pt, x, 0q " V pt, xq for all pt, xq P r0, T sˆR d , where V is the super-replication price. The main result of this section is the following. Theorem 2.1. The function v n,π converges to v n in C 0 as |π| Ñ 0. Proof. 1. We first check the consistency with the boundary condition. Letâ P K andŵ be the (continuous) solution of B t ϕ`Fâpt, x, ϕ, Dϕ, D 2 ϕq " 0 on r0, T qˆR dˆp 0, 1q (2.19) with boundary condition vpt, x, pq " pV pt, xq on r0, T sˆR dˆt 0, 1u Ť tT uˆR dˆr 0, 1s. By backward induction on π, one gets that v n,π ďŵ . (2.20) Indeed, we have v n,π pT,¨q "ŵpT,¨q. Now if the inequality is true at time t k , k ě 1, we have, using the comparison result for (2.19), recalling Proposition 6.1, that w k,â pt,¨q ďŵpt,¨q for t P rt k´1 , t k s , and thus a fortioriŵpt,¨q ě v n,π pt,¨q, for t P rt k´1 , t k s. We also obtain that v n,π p¨q ě v n p¨q (2.21) by backward induction. Indeed, we have v n,π pT,¨q " v n pT,¨q. Assume that the inequality is true at time t k , k ě 1. We observe that w k,a is a supersolution of (2.6), namely the PDE satisfied by v n . By the comparison result, this implies that w k,a pt,¨q ě v n pt,¨q, for t P rt k´1 , t k s. Taking the infimum over a P K yields then (2.21). Since v n ď w ď w ďŵ , (2.22) where wpt, x, pq " lim sup pt 1 ,x 1 ,p 1 ,|π|qÑpt,x,p,0q v n,π pt 1 , x 1 , p 1 q and w " lim inf pt 1 ,x 1 ,p 1 ,|π|qÑpt,x,p,0q v n,π pt 1 , x 1 , p 1 q, we obtain that w and w satisfy the boundary conditions (1.7)-(1.8). 2. We prove below that the scheme is monotone, stable and consistent, see Proposition 2.2, Proposition 2.3 and Proposition 2.4 respectively. Combining this with step 1. and Theorem 2.1 in [4] then ensures the convergence in C 0 of v n,π to v n as |π| Ñ 0. l Remark 2.3. We prove the following properties by a combination of viscosity solution arguments and, mostly, BSDE arguments, where they appear more natural. It should be possible to derive these results purely using PDE arguments using similar main steps as in [3] . Proposition 2.2 (Monotonicity). Let u ě v for u, v P BC, pt, x, pq P r0, T sˆR dˆr 0, 1s, y P R. We have: Spπ, t, x, p, y, uq ď Spπ, t, x, p, y, vq. (2.23) Proof. Let t ă T, x P R d , p P r0, 1s. By definition of v n,π , recalling (2.18), it is sufficient to prove that, for any a P K, we have: S a pt`, t´, upt`,¨qqpt, x, pq ě S a pt`, t´, vpt`,¨qqpt, x, pq. (2.24) with t`, t´defined in (2.17) . But this follows directly from the comparison result given in Proposition 6.1. l We now study the stability of the scheme. We first show that the solution of the scheme v n,π is increasing in its third variable. This is not only an interesting property in its own right which the piecewise constant policy solution inherits from the solution to the original problem (1.1), but it also allows us to obtain easily a uniform bound for v n,π , namely the boundary condition at p " 1. Lemma 2.1. The scheme (2.16) has the property, for all t P r0, T s and x P R d : v n,π pt, x, qq ď v n,π pt, x, pq if 0 ď q ď p ď 1. (2.25) Proof. We are going to prove the assertion by induction on k P t0, . . . , κu. For t " T " t κ and every x P R d , we have px, pq Þ Ñ v n,π pT, x, pq :" gpxqp, which is an increasing function of p. Let 1 ď k ă κ´1. Assume now that v n,π pt, x,¨q is an increasing function for all t ě t k`1 and x P R d . We show that v n,π pt, x,¨q is also increasing for t P rt k , t k`1 q and x P R d . Let 0 ď q ď p ď 1. By the definition of v n,π in (2.18), it is sufficient to show that for each a P K, we have, for pt, xq P r0, T sˆR d , w k,a pt, x, qq ď w k,a pt, x, pq . From Lemma 6.1(i) in the appendix, these two quantities admit a probabilistic representation with two different random terminal times τ q " infts ě t : P t,q,a s P t0, 1uu^t k`1 , (2.26) τ p " infts ě t : P t,p,a s P t0, 1uu^t k`1 . (2.27) However, using Lemma 6.1(ii), we can write probabilistic representations with BSDEs with terminal time t k`1 : we have that S a pt k`1 , t k , w π pt k`1 ,¨qqpt, x, pq "Ỹ t,x,p,a t , wherẽ Y t,x,p,a t is the first component of the solution of the following BSDE: Y s " v n,π pt k`1 , X t,x t k`1 ,P t,p,a t k`1 q`ż t k`1 s f pu, X t,x u , Y u , Z u q du´ż t k`1 s Z u dW u , (2.28) whereP t,p,a is the process defined by: P t,p,a s " p`ż s t a1 tuďτ p u dW u , (2.29) and a similar representation holds for S a pt k`1 , t k , w π pt k`1 ,¨qqpt, x, qq. It remains to show that v n,π pt k`1 , X t,x t k`1 ,P t,p,a t k`1 q ě v n,π pt k`1 , X t,x t k`1 ,P t,q,a t k`1 q. (2.30) If this is true, the classical comparison theorem for BSDEs (see e.g. Theorem 2.2 in [18]), concludes the proof. First, we observe that P t,p,a τp ě P t,q,a τp . On tτ p " T u, (2.30) holds straightforwardly by the induction hypothesis. On tτ p ă T u, if P t,p,a τp " 1 then P t,p,a T " 1 and (2.30) holds by induction hypothesis, as P t,q,a T ď 1; if P t,p,a τp " 0 then a fortiori P t,q,a τp " 0 and P t,p,a T " P t,q,a T " 0, which concludes the proof. l Proof. For any π and any pt, x, pq P r0, T sˆR dˆr 0, 1s, we have v n,π pt, x, pq ď v n,π pt, x, 1q " V pt, xq. l To prove the consistency of the scheme, we will need the two following lemmata. Lemma 2.2. For 0 ď τ ď t ď θ ď T , ξ P R, and φ P C 8 pr0, T sˆR dˆr 0, 1sq, the following holds |S a pτ, θ, φpθ,¨q`ξqpt,¨q´S a pτ, θ, φpθ,¨qqpt,¨q´ξ| 8 ď C|θ´t||ξ| . Proof. We denote w " S a pτ, θ, φp¨qq andw " S a pτ, θ, φp¨q`ξq. Using Lemma 6.1, we have that, for pt, x, pq P rτ, θsˆR dˆr 0, 1s, wpt, x, pq " Y t andwpt, x, pq "Ŷ t where pY, Zq and pŶ ,Ẑq are solutions to, respectively, Y r " φpX t,x θ ,P t,p,a θ q`ż T r f ps, X t,x s , Y s , Z s q ds´ż θ r Z s dW s , t ď r ď θ , Y r " φpX t,x θ ,P t,p,a θ q`ξ`ż T r f ps, X t,x s ,Ŷ s ,Ẑ s q ds´ż θ rẐ s dW s , t ď r ď θ . Denoting Γ :" Y`ξ and f ξ pt, x, y, zq " f pt, x, y´ξ, zq, one observes then that pΓ, Zq is the solution to Γ r " φpX t,x θ ,P t,p,a θ q`ξ`ż T r f ξ ps, X t,x s , Γ s , Z s q ds´ż θ r Z s dW s , t ď r ď θ . Let ∆ :" Γ´Ŷ , δZ " Z´Ẑ and δf s " f ξ ps, X t,x s , Γ s , Z s q´f ps, X t,x s , Γ s , Z s q, for s P rt, θs. We then get ∆ r :" ż θ r`f ps, X t,x s , Γ s , Z s q´f ps, X t,x s , Y s , Z s q`δf s˘d s´ż θ r δZ s dW s . Classical energy estimates for BSDEs [18,11] lead to E « sup rPrt,θs |∆ r | 2 ff ď CE "ż θ t |∆ s δf s | ds  . (2.31) Next, we compute ż θ t |∆ s δf s | ds ď 1 2C sup sPrt,θs |∆ s | 2`2 Cˆż θ t |δf s | ds˙2 . Combining the previous inequality with (2.31), we obtain E « sup rPrt,θs |∆ r | 2 ff ď 4C 2 E «ˆż θ t |δf s |˙2 ff . Using the Lipschitz property of f , we get from the definition of f ξ , |δf s | ď Lξ , which eventually leads to E « sup rPrt,θs |∆ r | 2 ff ď C|θ´t| 2 ξ 2 (2.32) and concludes the proof. l Lemma 2.3. Let 0 ď τ ă θ ď T and φ P C 8 pr0, T sˆR dˆr 0, 1sq. For pt, x, pq P rτ, θqˆR dˆp 0, 1q, φpt, x, pq´S a pτ, θ, φpθ,¨qqpt, x, pq´pθ´tqG a φpt, x, pq " opθ´tq . where G a φpt, x, pq :"´B t φpt, x, pq`F a pt, x, p, φ, Dφ, D 2 φq. Proof. We first observe that S a pτ, θ, φp¨qqpt, x, pq " Y t , where pY a , Z a q is solution to Y r " Φ θ`ż θ r f ps, X t,x s , Y s , Z s q ds´ż θ r Z s dW s with, for t ď s ď θ, Φ s " φps, X t, x s , P t,p,α s q and α :" a1 r0,τ s . By a direct application of Ito's formula, we observe that Φ r " Φ θ´ż θ r tB t φ`L α φups, X t,x s , P t,p,α s q ds´ż θ r Z s dW s , t ď r ď θ , where Z s :" zpX t,x s , Dφps, X t,x s , P t,p,a s q, α s q, t ď s ď θ. For ease of exposition, we also introduce an "intermediary" process pŶ ,Ẑq as the solution toŶ r " Φ θ`ż θ r f ps, X t,x s , Φ s , Z s q ds´ż θ rẐ s dW s , t ď r ď θ . Now, we computê Y t´Φt`p θ´tqG a φpt, x, pq " E "ż θ t`t B t φps, X t,x s , P t,p,a s q´B t φpt, x, pqu`tF a φps, X t,x s , P t,p,a s q´F a φpt, x, pqu˘ds  . Using the smoothness of φ, the Lipschitz property of f and the following control E " |X t,x s´x |`|P t,p,α s´p | ‰ ď C a |θ´t| 1 2 , (2.33) we obtain |Ŷ t´Φt`p θ´tqG a φpt, x, pq| ď C a,φ pθ´tq 3 2 . (2.34) We also haveŶ r´Φr " ż θ r G a φps, X t,x s , P t,p,α s q ds´ż θ r pẐ s´Zs q dW s Applying classical energy estimates for BSDEs, we obtain E « sup rPrt,θs |Ŷ r´Φr | 2`ż θ t |Ẑ s´Zs | 2 ds ff ď CE «ˆż θ t |G a φps, X t,x s , P t,p,α s | ds˙2 ff ď C a,φ pθ´tq 2 , (2.35) where for the last inequality we used the smoothness of φ and the linear growth of f and σ. We also observe that Y r´Yr " ż θ r tδf s`f ps, X t,x s ,Ŷ s ,Ẑ s q´f ps, X t,x s , Y s , Z s qu ds´ż θ r tẐ s´Zs u dW s , where δf s :" f ps, X t,x s , Φ s , Z s q´f ps, X s ,Ŷ s ,Ẑ s q, for t ď s ď θ. Once again, from classical energy estimates [18,11], we obtain |Ŷ t´Yt | 2 ď CE «ˆż θ t δf s ds˙2 ff . Using the Cauchy-Schwarz inequality and the Lipschitz property of f , |Ŷ t´Yt | 2 ď Cpθ´tqE « sup rPrt,θs |Ŷ r´Φr | 2`ż θ t |Ẑ s´Zs | 2 ds ff . This last inequality, combined with (2.35), leads to |Ŷ t´Yt | ď Cpθ´tq 3 2 . The proof is concluded by combining the above inequality with (2.34). l Finally, we can prove the following consistency property. Proposition 2.4 (Consistency). Let φ P C 8 pr0, T sˆR dˆr 0, 1sq. For pt, x, pq P r0, T qR dˆp 0, 1q,ˇˇˇ1 tπ´t S`π, t, x, p, φpt, x, pq`ξ, φp¨q`ξ˘`B t φ´F n pt, x, p, φ, Dφ, D 2 φqˇˇˇˇÑ 0 (2.36) as p|π|, ξq Ñ 0. Proof. We first observe that by Lemma 2.2, it is sufficient to prov졡ˇ1 tπ´t S`π, t, x, p, φpt, x, pq, φp¨q˘`B t φ´F n pt, x, p, φ, Dφ, D 2 φqˇˇˇˇÝÑ |π|Ó0 0 We have thaťˇˇˇ1 tπ´t S`π, t, x, p, φpt, x, pq, φp¨q˘`B t φ´F n pt, x, p, φ, Dφ, D 2 φqˇˇˇ"ˇˇˇˇ1 tπ´t tφpt, x, pq´min aPK S a`tπ , tπ , φptπ ,¨q˘pt, x, pqu´max aPK G a pt, x, pqφˇˇˇď max aPKˇ1 tπ´t tφpt, x, pq´S a`tπ , tπ , φptπ ,¨q˘pt, x, pqu´G a φˇˇˇˇ. The proof is then concluded by applying Lemma 2.3. l To conclude this section, let us observe that we obtain the following result, combining Proposition 2.1 and Theorem 2.1 . Corollary 2.1. In the setting of this section, assuming (H), the following holds lim nÑ8 lim |π|Ó0 v n,π " v . Remark 2.4. An important question, from numerical perspective, is to understand how to fix the parameters n and π in relation to each other. The theoretical difficulty here is to obtain a precise rate of convergence for the approximations given in Proposition 2.1 and Theorem 2.1, along the lines of the continuous dependence estimates with respect to control discretisation in [21,17], and estimates of the approximation by piecewise constant controls as in [23,22]. To answer this question in our general setting is a challenging task, extending also to error estimates for the full discretisation in the next section, which is left for further research. 3 Application to the Black-Scholes model: a fully discrete monotone scheme The goal of this section is to introduce a fully implementable scheme and to prove its convergence. The scheme is obtained by adding a finite difference approximation to the PCPT procedure described in Section 2.2. Then in Section 4, we present numerical tests that demonstrate the practical feasibility of our numerical method. From now on, we will assume that the log-price process X is a one-dimensional Brownian motion with drift, for pt, xq P r0, T sˆR: X t,x s " x`µps´tq`σpW s´Wt q, s P rt, T s, (3.1) with µ P R and σ ą 0. This restriction to Black-Scholes is not essential, as the main difficulty and nonlinearities are already present in this case and the analysis technique can be extended straightforwardly to more general monotone schemes in the setting of more complex SDEs for X. We take advantage of the specific dynamics to design a simple to implement numerical scheme, which also simplifies the notation. We shall moreover work under the following hypothesis. Assumption 3.1. The coefficient µ is non-negative. Remark 3.1. This assumption is introduced without loss of generality in order to alleviate the notation in the scheme definition. We detail in Remark 3.2(ii) how to modify the schemefor non-positive drift µ. The convergence properties are the same. We now fix n ě 1, R n the associated discrete set of controls (see Section 2.1). We denote K " R 5 n assuming that 0 R K and recall that v n is the solution to (2.10). We consider the grid π " t0 ": t 0 㨨¨ă t k 㨨¨ă t κ :" T u on r0, T s and approximate v n by a PCPT scheme, extending Section 2.2. The main point here is that we introduce a finite difference approximation for the solution S a p¨q, a P K to (2.11)-(2.13). This approximation, denoted by S a δ p¨q for a parameter δ ą 0, will be specified in Section 3.1 below. For δ ą 0 and a P K, each approximation S a δ p¨q is defined on a spatial grid G a δ :" δZˆΓ a δ Ă Rˆr0, 1s. (3.2) where Γ a δ is a uniform grid of r0, 1s, with N a δ`1 points and mesh size 1{N a δ . A typical element of G a δ is denoted px k , p l q :" pkδ, l{N a δ q, and an element of 8 pG a δ q is u k,l :" upx k , p l q, for all k P Z and 0 ď l ď N a δ . For 0 ď t ă s ď T , and ϕ : δZˆr0, 1s Ñ R a bounded function, we have that S a δ ps, t, ϕq P 8 pG a δ q. In order to define our approximation of v n , it is not enough to replace S a p¨q in the minimisation (2.16), or similarly (2.18), by S a δ p¨q, as the approximations are not defined on the same grid for the p-variable. (The flexibility of different grids will be important later on.) We thus have to consider a supplementary step which consists in a linear interpolation in the p-variable. Namely any mapping u P 8 pG a δ q is extended into I a δ puq : δZˆr0, 1s Ñ R by linear interpolation in the second variable: if u P 8 pG a δ q, k P Z and p P rp l , p l`1 q with 0 ď l ă N a δ , I a δ px k , pq " p l`1´p p l`1´pl u k,l`p´p l p l`1´pl u k,l`1 , and obviously I a δ px k , 1q " u k,N a δ . The solution to the numerical scheme associated with π, δ is then v n,π,δ : πˆδZˆr0, 1s Ñ R satisfying p Spπ, δ, t, x, p, v n,π,δ pt, x, pq, v n,π,δ q " 0, (3.3) where, for any 0 ď t P π, x P δZ, p P r0, 1s, y P R`and any bounded function u : πˆδZˆr0, 1s Ñ R: p Spπ, δ, t, x, p, y, uq " " y´min aPK I a δ pS a δ ptπ , t k , uptπ ,¨qqq pt k , x, pq if k ă κ, y´gpxqp otherwise,(3.4) where tπ " infts P π : s ě tu. Alternatively, the approximation v n,π,δ is defined by the following backward induction: 1. Initialisation: set v n,π,δ pT, x, pq :" gpxqp, x P R dˆr 0, 1s. 2. Backward step: For k " κ´1, . . . , 0, compute w k,a δ :" S a δ pt k , t k`1 , v n,π,δ pt k`1 ,¨qq and set, for px, pq P δZˆr0, 1s, v n,π,δ pt k , x, pq :" inf aPK I a δ pw k,a δ qpt k , x, pq . (3.5) Before stating the main convergence result of this section, see Theorem 3.1 below, we give the precise definition of S a δ p¨q using finite difference operators. Finite difference scheme definition and convergence result Let 0 ď t ă s ď T, δ ą 0, ϕ : δZˆr0, 1s Ñ R. We set h :" s´t ą 0. For a P K, we will describe the grid G a δ " δZˆΓ a δ Ă δZˆr0, 1s and the finite difference scheme used to define S a δ . First, we observe that for the model specification of this section, (2.10) can be rewritten as sup aPKˆ´D a ϕ´µ∇ a ϕ´σ 2 2 ∆ a ϕ´f pt, x, ϕ, σ∇ a ϕq˙" 0, (3.6) with: ∇ a ϕ :" B y ϕ`a σ B p ϕ, (3.7) ∆ a ϕ :" B 2 yy ϕ`2 a σ B 2 yp ϕ`a 2 σ 2 B 2 pp ϕ, (3.8) D a ϕ :" B t ϕ´a σ µB p ϕ. (3.9) Exploiting the degeneracy of the operators ∇ a and ∆ a in the direction pa,´σq, we construct Γ a δ so that the solution to (3.6) is approximated by the solution of an implicit finite difference scheme with only one-directional derivatives. To take into account the boundaries p " 0, p " 1, we set where a ‰ 0. We have N a δ " σ{δ|apa, δq|. We finally set: Γ a δ :" We now define the finite difference scheme. To use the degeneracy of the operators ∇ apa,δq and ∆ apa,δq in the direction papa, δq,´σq, we define the following finite difference operators, for v " pv k,l q kPZ,0ďlďN a δ " pvpx k , p l qq kPZ,0ďlďN a δ P 8 pG a δ q and w " pw k q kPZ P 8 pkZq: ∇ a δ v k,l :" 1 2δ`v k`1,l`sgnpaq´vk´1,l´sgnpaq˘, ∇ δ w k :" 1 2δ pw k`1´wk´1 q , ∇ à ,δ v k,l :" 1 δ`v k`1,l`sgnpaq´vk,l˘, ∇`, δ w k :" 1 δ pw k`1´wk q , ∆ a δ v k,l :" 1 δ 2`v k`1,l`sgnpaq`vk´1,l´sgnpaq´2 v k,l˘, ∆ δ w k :" 1 δ 2 pw k`1`wk´1´2 w k q . Let θ ą 0 a parameter to be fixed later. We define, for pt, x, y, q, q`, Aq P r0, T sˆR 5 . F pt, x, y, q, Aq :"´µq´σ 2 2 A´f pt, x, y, σqq, and (3.13) p F pt, x, y, q, q`, Aq :"´µq`´ˆσ 2 2`θ δ 2 h˙A´f pt, x, y, σqq . (3.14) Now, S a δ ps, t, ϕq P 8 pG a δ q is defined as the unique solution to (see Proposition 3.1 below for the well-posedness of this definition) Spk, l, v k,l , ∇ a δ v k,l , ∇ à ,δ v k,l , ∆ a δ v k,l , ϕq " 0, (3.15) v k,0 " v k , v k,N a δ " v k ,(3.16) where, for k P Z, 0 ă l ă N a δ , pv, v`, v´q P R 3 , and any bounded function u : δZr 0, 1s Ñ R: Spk, l, v, q, q`, A, uq " v´u px k , p a pp l qq`h p F pt, kδ, v, q, q`, Aq, (3.17) with, for p P r0, 1s, p a ppq :" p´µ apa, δq σ h, (3.18) and where pv k q kPZ (resp. pv k q kPZ ) is the solution to S b pk, v k , ∇ δ v k , ∇`, δ v k , ∆ δ v k , pϕ k q kPZ q " 0, (3.19) presp. S b pk, v k , ∇ δ v k , ∇`, δ v k , ∆ δ v k , pϕ k q kPZ q " 0q (3.20) with ϕ k " ϕpkδ, 0q (resp. ϕ k " ϕpkδ, N a δ q) and, for k P Z, pv, v`, v´q P R 3 , u P 8 pZq: S b pk, v, q, q`, A, uq " v´u k`h p F pt, kδ, v, q, q`, Aq. (3.21) Remark 3.2. (i) Here, as stated before, we have assumed µ ě 0. If the opposite is true, one has to consider ∇ á pδqv k,l :" 1 δ`v k,l´vk´1,l´sgnpaq˘( resp. ∇´pδqw k :" 1 δ pw k´wk´1 q) instead of ∇ à ,δ v k,l (resp. ∇`, δ w k ), in the definition of S a δ ps, t, ϕq (resp. v k , v k ), to obtain a monotone scheme. (ii) For the nonlinearity f , we used the Lax-Friedrichs scheme [13,17], adding the term θpv``v´´2vq term in the definition of p F to enforce monotonicity. We now assume that the following conditions on the parameters are satisfied: δ ď 1, (3.22) hL 2δ ď θ ă 1 4 , (3.23) µh ď δ ď M h,(3.24) for a constant M ą 0. Under these conditions, we prove that S a δ ps, t, ϕq is uniquely defined, and that it can be obtained by Picard iteration. Proof. First, v P pδZq (resp. v P pδZq) is uniquely defined by (3.19) (resp. (3.20)), see Proposition 6.2. We consider the following map: 8 pG a δ q Ñ 8 pG a δ q, v Þ Ñ ψpvq, where ψpvq is defined by, for k P Z and l P t1, . . . , N a´1 u: ψpvq k,l " 1 1`h δ µ`σ 2 h δ 2`2 θ pϕ pkδ, p a pp l qq` (3.25) h δ µv k`1,l`sgnpaq`σ 2 2 h δ 2 pv k`1,l`sgnpaq`vk´1,l´sgnpaq qh f´t´, kδ, v k,l , σ 2δ pv k`1,l`sgnpaq´vk´1,l´sgnpaq q¯`θpv k`1,l`sgnpaq`vk´1,l´sgnpaq q¯, ψpvq k,0 " v k , ψpvq k,Na " v k . (3.26) Notice that v is a solution to (3.15)- (3.16) if and only if v is a fixed point of ψ. It is now enough to show that ψ maps 8 pG a δ q into 8 pG a δ q and is contracting. If v P 8 pG a δ q, by boundedness of ϕ, v and v, it is clear that ψpvq is bounded. If v 1 , v 2 P 8 pG a δ q 2 , we have, for all k P Z and 1 ď l ď N a´1 : |ψpv 1 q k,l´ψ pv 2 q k,l | ď h δ µ`σ 2 h δ 2`2 θ`hL`h L δ 1`h δ µ`σ 2 h δ 2`2 θ |v 1´v2 | 8 .(3.27) Since δ ď 1 by assumption (3.22), one has hL`h L δ ď 2 hL δ ď 4θ, thus: |ψpv 1 q´ψpv 2 q| 8 ď 4θ`h δ µ`σ 2 h δ 2`2 θ 1`h δ µ`σ 2 h δ 2`2 θ |v 1´v2 | 8 . (3.28) Since 4θ ă 1 by assumption (3.23) and the function x Þ Ñ 4θ`x 1`x is increasing on r0, 8q with limit 1 when x Ñ`8, this proves that ψ is a contracting mapping. l For this scheme, we have the following strong uniqueness result: Proposition 3.2. Let ϕ 1 , ϕ 2 : δZˆr0, 1s Ñ R two bounded functions satisfying ϕ 1 ď ϕ 2 on δZˆr0, 1s. 1. (Monotonicity) For all k P Z, 1 ď l ď N a , pv, q, q`, Aq P R 4 , we have: Spk, l, v, q, q`, A, ϕ 2 q ď Spk, l, v, q, q`, A, ϕ 1 q. (3.29) 2. (Comparison theorem) Let pv 1 , v 2 q P 8 pG a δ q 2 satisfy, for all k P Z and 1 ď l ď N a δ´1 : Spk, l, v 1 k,l , ∇ a δ v 1 k,l ,∇ à ,δ v 1 k,l , ∆ a δ v 1 k,l , ϕ 2 q ď Spk, l, v 2 k,l , ∇ a δ v 2 k,l , ∇ à ,δ v 2 k,l , ∆ a δ v 2 k,l , ϕ 2 q (3.30) v 1 k,0 ď v 2 k,0 , (3.31) v 1 k,N a δ ď v 2 k,N a δ . (3.32) Then v 1 ď v 2 . 3. We have S a δ ps, t, ϕ 1 q k,l ď S a δ ps, t, ϕ 2 q k,l for all k P Z and 0 ď l ď N a δ . Proof. Let ϕ 1 , ϕ 2 as stated in the proposition. 1. We have, for k P Z and 0 ă l ă N a δ : Spk, l, v, q, q`, A, ϕ 2 q´Spk, l, v, q, q`, A, ϕ 1 q " pϕ 1´ϕ2 q px k , p a pp l qq ď 0. 2. We assume here that a ą 0. For k P Z, let M k " max 0ďlďN a δ pv 1 k`l,l´v 2 k`l,l q ă 8 (if a ă 0, we have to consider max 0ďlďN a δ pv 1 k´l,l´v 2 k´l,l )). We want to prove that M k ď 0 for all k. Assume to the contrary that there exists k P Z such that M k ą 0. Then there exists 0 ď l ď N a δ such that v 1 k`l,l´v 2 k`l,l " M k ą 0. (3.33) First, we have v 1 k,0 ď v 2 k,0 and v 1 k`N a δ ,N a δ ď v 2 k`N a δ ,N a δ . Thus 0 ă l ă N a δ . Moreover, using (3.30), re-arranging the terms, using the fact that f is nonincreasing with respect to its third variable and Lipschitz-continuous, by (3.33), p1`2θqM k ď hL 2δˇˇv 2 k`l`1,l`1´v 1 k`l`1,l`1ˇ´θ pv 2 k`l`1,l`1´v 1 k`l`1,l`1 qh L 2δˇˇv 2 k`l´1,l´1´v 1 k`l´1,l´1ˇ´θ pv 2 k`l´1,l´1´v 1 k`l´1,l´1 q. (3.34) For j P tl´1, l`1u, we observe that hL 2δ |v 2 k`j,j´v 1 k`j,j |´θpv 2 k`j,j´v 1 k`j,j q ďˆh L 2δ`θ˙M k . (3.35) Indeed, if v 2 k`j,j ě v 1 k`j,j then hL 2δ |v 2 k`j,j´v 1 k`j,j |´θpv 2 k`j,j´v 1 k`j,j q "ˆh L 2δ´θ˙p v 2 k`j,j´v 1 k`j,j q ď 0, since hL 2δ ď θ. Otherwise, if v 2 k`j,j ă v 1 k`j,j hL 2δ |v 2 k`j,j´v 1 k`j,j |´θpv 2 k`j,j´v 1 k`j,j q "ˆh L 2δ`θ˙p v 1 k`j,j´v 2 k`j,j q ďˆh L 2δ`θ˙M k . Inserting (3.35) into (3.34), we get p1`2θqM k ď 2ˆh L 2δ`θ˙M k . (3.36) Thus,ˆ1´h L δ˙M k ď 0, (3.37) which is a contradiction to M k ą 0 since hL δ ď 2θ ă 1 2 . 3. Let v i " S a δ ps, t, ϕ i q for i " 1, 2. Since ϕ 1 ď ϕ 2 and ϕ 1 ď ϕ 2 , we get by Proposition 6.2 that v 1 k,0 ď v 2 k,0 and v 1 k,N a δ ď v 2 k,N a δ for all k P Z. By monotonicity, we get, for all k P Z and 0 ă l ă N a δ , Spk, l, v 1 k,l , ∇ a δ v 1 k,l ,∇ à ,δ v 1 k,l , ∆ a δ v 1 k,l , ϕ 2 q ď Spk, l, v 1 k,l , ∇ a δ v 1 k,l , ∇ à ,δ v 1 k,l , ∆ a δ v 1 k,l , ϕ 1 q Moreover, Spk, l, v 1 k,l , ∇ a δ v 1 k,l ,∇ à ,δ v 1 k,l , ∆ a δ v 1 k,l , ϕ 1 q " Spk, l, v 2 k,l , ∇ a δ v 2 k,l , ∇ à ,δ v 2 k,l , ∆ a δ v 2 k,l , ϕ 2 q " 0 So that, Spk, l, v 1 k,l , ∇ a δ v 1 k,l ,∇ à ,δ v 1 k,l , ∆ a δ v 1 k,l , ϕ 2 q ď Spk, l, v 2 k,l , ∇ a δ v 2 k,l , ∇ à ,δ v 2 k,l , ∆ a δ v 2 k,l , ϕ 2 q and the proof is concluded applying the previous point. l We last give a refinement of the comparison theorem, which will be useful in the sequel. Proposition 3.3. Let u : δZˆr0, 1s Ñ R be a bounded function, and let v 1 , v 2 P 8 pG a δ q. Assume that, for all k P Z and 0 ă l ă N a δ , we have Spk, l, v 1 k,l , ∇ a δ v 1 k,l ,∇ à ,δ v 1 k,l , ∆ a δ v 1 k,l , uq ď 0 ď Spk, l, v 2 k,l , ∇ a δ v 2 k,l , ∇ à ,δ v 2 k,l , ∆ a δ v 2 k,l , uq. Then: where Cph, δq :" lnˆ1`h δ µ`σ 2 h δ 2`2 θ`h L 2δ h δ µ`σ 2 h δ 2`2 θ`h L 2δδ 2 . (3.39) Moreover, Cph, δq ě 1 pµ`L 2 qM`2θM 2˘h2`σ2 h´M 2 2σ 4 . (3.40) Remark 3.4. (i) To prove the consistency of the scheme, we define in Lemma 3.2 smooth functions w˘so that pw˘px k , p l qq P l 8 pG a δ q satisfy S ě 0 or S ď 0, but we cannot use the comparison theorem as the values at the boundary cannot be controlled. The previous proposition will be used in Lemma 3.3 to show that the difference between w˘and the linear interpolant of a solution of S " 0 is small. (ii) The coefficient exp´´4 apa,δq σ 2 Cph, δqlpN a δ´l¯t hat appears in the first equation of the previous proposition shows that the dependance on the boundary values decays exponentially with the distance to the boundary. This was to be expected and was already observed in similar situations, see for example Lemma 3.2 in [3] for Hamilton-Jacobi-Bellman equations. We now can state the main result of this section. Theorem 3.1. The function v n,π,δ converges to v n uniformly on compact sets, as |π|, δ Ñ 0 satisfying conditions (3.22)-(3.24) for all h " t i`1´ti , where π " t0 " t 0 ă t 1 㨨¨ă t κ " T u. We prove below that the scheme is monotone (see Proposition 3.5), stable (see Proposition 3.6), consistent with (2.10) in r0, T qˆRˆp0, 1q (see Proposition 3.7) and with the boundary conditions (see Proposition 3.4). The theorem then follows by identical arguments to [4]. Proof of Theorem 3.1 We first show that the numerical scheme is consistent with the boundary conditions. For any discretisation parameters π, δ, we define V π,δ : πˆδZ Ñ R as the solution to the following system: S b pk, v j k , ∇ δ v j k , ∇`, δ v j k , ∆ δ v j k , v j`1 k q " 0, k P Z, 0 ď j ă κ (3.41) v κ k " gpx k q, k P Z,(3.42) where v j k :" vpt j , x k q for 0 ď j ď κ and k P Z. We set pU π,δ q j k :" ∇ δ pV π,δ q j k " 1 2δ ppV π,δ q j k`1´p V π,δ q j k´1 q. We recall from Proposition 6.3 that V π,δ and U π,δ are bounded, uniformly in π, δ, and, by [4], that V π,δ converges to V uniformly on compact sets as |π| Ñ 0 and δ Ñ 0. Proposition 3.4. There exists constants K 1 , K 2 , K 3 ą 0 such that, for all discretisation parameters π, δ with |π| small enough, we have, for pt j , x k , pq P πˆδZˆr0, 1s: pV π,δ pt j , x k q´K 1 pT´t j q ď v n,π,δ pt j , x k , pq ď pV π,δ pt j , x k q`K 1 pT´t j q, pV π,δ pt j , x k q´p1´e´K 2 p qp1´e´K 2 p1´pq q ď v n,π,δ pt j , x k , pq ďpV π,δ pt j , x k q`p1´e´K 2 p qp1´e´K 2 p1´pq q. Proof. We only prove, by backward induction, the lower bounds, while the proof of the upper bounds is similar. We need to introduce first some notation. For 0 ď j ď κ, k P δZ and 0 ď l ď N a δ , we set V j k :" V π,δ pt j , x k q and U j k :" U π,δ pt j , x k q. For P t0, 1u, we define: wpt j , x k , pq :" pV j k´ cpt j , pq, (3.43) with cpt j , pq :" K 1 pT´t j q`p1´ qp1´e´K 2 p qp1´e´K 2 p1´pq q, (3.44) and w j k,l " wpt j , x k , p l q, c j l " cpt j , p l q, p l P Γ a δ . The proof now procedes in two steps. 1. First, we have wpT, x k , pq ď pV π,δ pT, x k q " pgpx k q " v n,π,δ pT, x k , pq on δZˆr0, 1s. Suppose that, for 0 ď j ă κ, on δZˆr0, 1s, we have wpt j`1 , x k , pq ď v n,π,δ pt j`1 , x k , pq. We want to prove on δZˆr0, 1s wpt j , x k , pq ď v n,π,δ pt j , x k , pq. Since w is convex in p, wpt j , x k ,¨q ď I a δ p w j k,¨q on r0, 1s. By definition, we have v n,π,δ pt j , x k , pq " min aPK I a δ pS a δ pt j`1 , t j , v n,π,δ pt j`1 ,¨qqqpx k , pq, we are thus going to prove w j k,l ď S a δ pt j`1 , t j , v n,π,δ pt j`1 ,¨qqpt j , x k , p l q (3.45) for all a P K and all k P Z, 0 ď l ď N a δ . For a P K, by induction hypothesis, wpt j`1 ,¨q ď v n,π,δ pt j`1 ,¨q, so if we are able to get S b pk, w k , ∇ δ w k , ∇`, δ w k , ∆ δ w k , w j`1 k,0 q ď 0, k P Z, (3.46) S b pk, w k , ∇ δ w k , ∇`, δ w k , ∆ δ w k , w j`1 k,N a δ q ď 0, k P Z, (3.47) Spk, l, w k,l , ∇ a δ w k,l , ∇ à ,δ w k,l , ∆ a δ w k,l , wpt j`1 ,¨qq ď 0, k P Z, 0 ă l ă N a δ ,(3.48) where w j k " wpt j , x k , 0q, w j k " wpt j , x k , 1q, we obtain that (3.45) holds true by the comparison result in Proposition 3.2, which concludes the proof. We now proceed with the proof of (3.46), (3.47) and (3.48). 2.a Now, observe that w j k "´ K 1 pT´t j q, for k P Z. We have, since f pt j , x k , 0, 0q " 0 and f is non-increasing in its third variable, S b pk, w k , ∇ δ w k , ∇`, δ w k , ∆ δ w k , w j`1 k,0 q "´ Kh´hf pt j , x k ,´ KpT´t j q, 0q ď 0. 2.b We have that w j k " V j k´ K 1 pT´t j q, for k P Z. Since f pt j , x k , V j k´ K 1 pT´t j q, U j k q ě f pt j , x k , V j k , U j k q , and by definition of V π,δ : S b pk, w k , ∇ δ w k , ∇`, δ w k , ∆ δ w k , w j`1 k,N a δ q "´ Kh`S b pk, V j k , ∇ δ V j k , ∇`, δ V j k , ∆ δ V j k q ď´ Kh ď 0. 2.c We now prove (3.48). Let k P Z, 0 ă l ă N a δ . We have, by definition (3.17) of S: Spk, l, w j k,l ,∇ a δ w j k,l , ∇ à ,δ w j k,l , ∆ a δ w j k,l , wpt j`1 ,¨qq " w j k,l´ wpt j`1 , x k , p a pp l qq h p F pt, kδ, w j k,l , ∇ a δ w j k,l , ∇ à ,δ w j k,l , ∆ a δ w j k,l q ď´ c j k,l`µ apa, δq σ hV j`1 k` cpt j`1 , x k , p a pp l qq p l h p F pt, x k , V j k , U j k , ∇`, δ V j k , ∆ δ V j k q h p F pt, x k , p l V j k , ∇ a δ w j k,l , ∇ à ,δ w j k,l , ∆ a δ w j k,l q, where we have used (3.41) and f pt, x k , w j k,l , σ∇ a δ w j k,l q ě f pt, x k , p l V j k , σ∇ a δ w j k,l q. By adding˘p l hf pt j , x k , p l V j k , σ∇ a δ w j k,l q, using the Lipschitz continuity of f and ∇ a δ w j k,l " p l U j k`a pa, δq 2σ pV j k`1`V j k´1 q`1 2δ´ c j l´sgnpaq´ c j l`sgnpaq¯, we get, by definition (3.14) of p F , Spk, l, w k,l ,∇ a δ w j k,l , ∇ à ,δ w j k,l , ∆ a δ w j k,l , w j pt j`1 ,¨qq ďh apa, δq σ µpV j`1 k´V j k`1 q´hσapa, δqU j k´2 θ apa, δq σ δ 2 U j k 2hLp l p1´p l qpV j k`| U j k |q`hL |apa, δq| 2σ pV j k`1`V j k´1 qˆ c j l´ cpt j`1 , p a pp l qq´µh∇ à ,δ c j l´ˆσ 2 2 h`θδ 2˙∆a δ c j lḣ L|∇ a δ c j l |. Since |apa, δq| ď maxt|a|, a P Ku ď n and V and U are bounded uniformly in h, δ (see Proposition 6.3 in the appendix), there exists a constant K n,θ,M,L ą 0 such that h apa, δq σ µpV j`1 k´V j k`1 q´hσapa, δqU j k´2 θ apa, δq σ δ 2 U j k 2hLp l p1´p l qpV j k`| U j k |q`hL |apa, δq| 2σ pV j k`1`V j k´1 q ď hK n,θ,M,L . When " 1, the terms of the last three lines all vanish except the first one, and c j l´c pt j`1 , p l´µ apa,δq σ hq " K 1 h. Thus we get: Spk, l, w k,l ,∇ a δ w k,l , ∇ à ,δ w k,l , ∆ a δ w k,l , wpt j`1 ,¨qq ď hp´K 1`Kn,θ,M,L q. Hence, chosing K 1 large enough gives the result. We now deal with the case " 0. By Taylor expansions of c around pt j , p l q, we get: Spk, l, w k,l , ∇ a δ w j k,l , ∇ à ,δ w j k,l , ∆ a δ w j k,l , w j pt j`1 ,¨qq ď hK n,θ,M,L`h L |apa, δq| σ |B p cpt j , p l q|`hB t cpt j , p l q`h apa, δq 2 2 B 2 pp cpt j , p l q`hεph; K 2 q, with lim hÑ0 εph; K 2 q " 0. By definition of c, we get, for h 0 ą 0 to be fixed later on and h P r0, h 0 s: Spk, l, w k,l ,∇ a δ w j k,l , ∇ à ,δ w j k,l , ∆ a δ w j k,l , w j pt j`1 ,¨qq ďh " K n,θ,M,L`K2 L |apa, δq| σ e´K 2 p l`K 2 L apa, δq σ e´K 2 p1´p l q K 2 2 apa, δq 2 2 e´K 2 p l´K 2 2 apa, δq 2 2 e´K 2 p1´p l q`| εph; K 2 q|  ď h " max hPr0,h 0 s |εph; K 2 q|`K n,θ,M,L`K2 |apa, δq|pe´K 2 p l`e´K2 p1´p l q qˆL σ´| apa, δq| 2 K 2˙ . To conclude, one can choose K 2 large enough so that K n,θ,M,L`K2 |apa, δq|pe´K 2 p lè´K 2 p1´p l q qp L σ´| apa,δq 2 K 2 q ď´η ă 0, and then consider h 0 ą 0 small enough so that |εph; K 2 q| ď η for h P r0, h 0 s. l Proposition 3.5 (Monotonicity). Let π be a grid of r0, T s and δ ą 0 satisfying (3.22)- (3.24). Let y P R, 0 ď k ď κ, j P Z and p P r0, 1s, and let U, V : πˆδZˆr0, 1s Ñ R be two bounded functions such that U ď V. Then: p Spπ, δ, k, j, p, y, Uq ě p Spπ, δ, k, j, p, y, Vq. (3.49) Proof. The result is clear for k " κ. If k ă κ, it is sufficient to show that: I a δ pS a δ pt k`1 , t k , Upt k`1 ,¨qqq ď I a δ pS a δ pt k`1 , t k , Vpt k`1 ,¨qqq, for all a P K, recalling (3.4). This is a consequence of the comparison result in Proposition 3.2 and the monotonicity of the linear interpolator. l We now prove the stability of the scheme. Here, in contrast to Lemma 2.1, we are not able to prove that the solution of the scheme is increasing in p. However, due to the boundedness of the terminal condition, we obtain uniform bounds for v n,π,δ . Proposition 3.6 (Stability). For all π and δ ą 0, there exists a unique solution v n,π,δ to (3.4), which satisfies: 0 ď v n,π,δ ď |g| 8 on πˆδZˆr0, 1s. (3.50) Proof. We prove the proposition by backward induction. First, since v n,π,δ is a solution to (3.4), v n,π,δ pT, x, pq " pgpxq on δZˆr0, 1s, and we have 0 ď v n,π,δ pT, x, pq ď |g| 8 for all px, pq P δZˆr0, 1s. Let 0 ď j ď κ´1 and assume that v n,π,δ pt k ,¨q is uniquely determined for k ą j, and that 0 ď v n,π,δ pt j`1 ,¨q ď |g| 8 . Since v n,π,δ is a solution to (3.4), we have v n,π,δ pt j , x, pq " min aPK I a δ pS a δ pt j`1 , t j , v n,π,δ pt j`1 ,¨qqq, and for each a P K, S a δ pt j`1 , t j , v n,π,δ pt j`1 ,¨qq is uniquely determined by Proposition 3.1, so v n,π,δ pt j ,¨q is uniquely determined. Next, we show that, for all k P Z and 0 ď l ď N a : 0 ď S a δ pt j`1 , t j , v n,π,δ pt j`1 ,¨qq ď |g| 8 . Then it is easy to conclude that 0 ď v n,π,δ pt j ,¨q ď e LT |g| 8 on Rˆr0, 1s, by properties of the linear interpolation and the minimisation. First, it is straightforward thatǔ defined byǔ k,l " 0 for all k P Z and 0 ď l ď N a satisfiesǔ " S a δ pt j`1 , t j , 0q. The comparison theorem gives 0 ď S a δ pt j`1 , t j , v n,π,δ pt j`1 ,¨qq, since 0 ď v n,π,δ pt j`1 ,¨q. To obtain the upper bound, we notice thatû defined byû k,l :" |g| 8 for all k P Z and 0 ď l ď N a satisfies Spk, l,û k,l ,∇ a δûk,l , ∇ à ,δûk,l , ∆ a δûk,l ,ûq "´hf pt j , x k ,û, 0q ě´hf pt j , x k , 0, 0q ě 0. Hence the comparison result in Proposition 3.2 yields S a δ pt j`1 , t j , v n,π,δ pt j`1 ,¨qq ď |g| 8 . l We now prove the consistency. The proof requires several lemmata. First, we show that the perturbation induced by the change of controls vanishes as δ Ñ 0. Moreover, there exists c ą 0 such that for all a P K and δ ą 0, |apa, δq| ě c ą 0. Proof. By definition of N a δ , pN a δ´1 q |a| σ δ ă 1 " N a δ |apa, δq| σ δ ď N a δ |a| σ δ, thus |a|´| a| N a δ ă |apa, δq| ď |a|. Also, we observe |a| N a δ " |a| r σ |a|δ s ď |a| σ |a|δ " a 2 δ σ ď n 2 σ δ, which concludes the proof of (3.51). By (3.10), we have: N a δ ď σ |a|δ`1 ď σ a m δ`1 ď σ cδ where a m " mint|a| : a P Ku and c ą 0 is independant of a, δ. Now, by (3.11), we get: |apa, δq| " σ δN a δ ě cδσ δσ " c. l Last, we give explicit supersolutions and subsolutions satisfying appropriate conditions. Let 0 ď t ă s ď T , δ ą 0 and a P K be fixed. For ą 0, we set f pt, x, y, νq :" pf pt,¨,¨,¨q˚ρ qpt, y, νq :" ż RˆRˆR f pt, x´u, y´z, ν´ηqρ pu, z, ηq du dz dη, where˚is the convolution operator and, for ą 0, ρ pxq :" ´3 ρpx{ q with ρ : R 3 Ñ R is a mollifier, i.e. a smooth function supported on r´1, 1s 3 satisfying ş R ρ " 1. We set F pt, x, y, q, Aq "ˆ1 2 σ 2´µ˙q´σ 2 2 A´f pt, x, y, σqq. Remark 3.5. Since f is L-Lipschitz continuous with respect to its three last variables, we have |f ´f | 8 ď L . The lengthy proof of the following lemma by insertion is given in the appendix. Lemma 3.2. Let 0 ď t ă s ď T, ϕ P C 8 b pRˆR, Rq, a P K. We set h :" s´t. Let ą 0 such that Ñ 0 and δ 2 Ñ 0 as h Ñ 0, observing (3.24). Then there exist bounded functions S a,δ ps, t, ϕq : δZˆr0, 1s Ñ R of the form S a,δ , ps, t, ϕqpx, pq "ϕpx, p a ppqq (3.52) hF pt, x, ϕpx, p a ppqq, ∇ apa,δq ϕpx, p a ppqq, ∆ apa,δq ϕpx, p a ppqqq C ϕ,n ph, q, where p a is defined in (3.18), and where C ϕ,n ph, q ą 0 satisfies Cϕ,nph, q h Ñ 0 as h Ñ 0, such that w˘:" pS a,δ , ps, t, ϕqpx k , p l qq kPZ,0ďlďN a δ P 8 pG a δ qq satisfy Spk, l, wk ,l , ∇ a δ wk ,l , ∇ à ,δ wk ,l , ∆ a δ wk ,l q ě 0, (3.53) Spk, l, wḱ ,l , ∇ a δ wḱ ,l , ∇ à ,δ wḱ ,l , ∆ a δ wḱ ,l q ď 0, (3.54) for all k P Z and 0 ă l ă N a δ . Furthermore, for all x P δZ, S a,δ , ps, t, ϕqpx,¨q P C 2 pr0, 1s, Rq, and |B 2 pp S a,δ , ps, t, ϕq| 8 ď Cϕphq 2 for some constant C ϕ phq ą 0 independent of . Lemma 3.3. Let 0 ď t ă s ď T, δ ą 0, a P K, ϕ P C 8 b pRˆRq be fixed. Let h " s´t, k P Z, x k P δZ, p P p0, 1q, and assume that h is sufficiently small so that p P rp 1 , p N a δ´1 s, observing (3.24). Let ą 0 such that Ñ 0 and δ 2 Ñ 0 as h Ñ 0. Then we have: S a,δ , ps, t, ϕqpx k , pq´I a δ pS a δ ps, t, ϕqqpx k , pq ď C 1 ϕ,n ph, q, (3.55) I a δ pS a δ ps, t, ϕqqpx k , pq´S a,δ , ps, t, ϕqpx k , pq ď C 1 ϕ,n ph, q, (3.56) where C 1 ϕ,n ph, q ą 0 satisfies C 1 ϕ,n ph, q h Ñ 0, as h Ñ 0 and where the functions S a,δ , ps, t, ϕq are introduced in Lemma 3.2. Proof. We prove the first identity, the second one is similar. Set w :" S a δ ps, t, ϕq and w´:" S a,δ , ps, t, ϕq. By definition of w and by (3.54), one can apply Proposition 3.3. For all k P Z and 0 ă l ă N a δ : wḱ ,l´w k,l ď Be´4 apa,δq 2 σ 2 Cph,δqlpN a δ´l q ď Be´4 apa,δq 2 σ 2 Cph,δqpN a δ´1 q , (3.57) with B " |pw, 0´w¨, 0 q`| 8`| pw, N a δ´w¨, N a δ q`| 8 and Cph, δq is defined in (3.39). By Lemma 3.1, there exists a constant c ą 0 such that |apa, δq| ě c. In addition, using (3.40), we get: B h e´4 Cph,δq apa,δq 2 σ pN a δ´1 q ď B h e´4 c 2 σ 2˜1 p pµ`L 2 qM`2θM 2 q h 2`σ2 h´M 2 2σ 4" Be 4 c 2 M 2 2σ 6 e´4 c 2 σ 2 1 p pµ`L 2 qM`2θM 2 q h 2`σ2 h h Ñ 0, as h Ñ 0. Now, let p P rp 1 , p N a δ´1 q and k P Z. By definition of I a δ , one has: I a δ pS a δ ps, t, ϕqqpx k , pq " λw k,l`p 1´λqw k,l`1 , (3.58) where p P rp l , p l`1 q with 0 ă l ă N a δ´1 , and λ " p l`1´p p l`1´pl . Thus: S a,δ , ps, t, ϕqpx k , pq´I a δ pwqpx k , pq " S a,δ , ps, t, ϕqpx k , pq´I a δ pw´qpx k , pq`I a δ pw´qpx k , pq´I a δ pwqpx k , pq (3.59) " S a,δ , ps, t, ϕqpx k , pq´I a δ pw´qpx k , pq`λpwḱ ,l´w k,l q`p1´λqpwḱ ,l`1´w k,l`1 q. The two last terms are controlled using (3.57), and, by properties of linear interpolation of the function p Þ Ñ S a,δ , pt`, t´, ϕqpx k , pq P C 2 pr0, 1s, Rq with |B 2 pp S a,δ , pt`, t´, ϕq| 8 ď Cϕphq 2 (recall the previous Lemma) the first term is of order δ 2 2 " ophq since (3.24) is in force and δ 2 Ñ 0. l Lemma 3.4. For 0 ď t ă s ď T such that Lps´tq ď 1, ξ ą 0, ϕ : δZˆr0, 1s Ñ R a bounded function, the following holds for all a P K: S a δ ps, t, ϕq`ξ´Lps´tqξ ď S a δ ps, t, ϕ`ξq ď S a δ ps, t, ϕq`ξ, where L is the Lipschitz constant of f . Proof. Let v " S a δ ps, t, ϕq, w " v`ξ´Lps´tqξ. Since v satisfies (3.15), we have, for k P Z and 0 ă l ă N a δ , Spk, l, w k,l , ∇ a δ w k,l , ∇ à ,δ w k,l , ∆ a δ w k,l , ϕ`ξq "´Lps´tqξ ps´tq pf pt, x k , v k,l , ∇ a δ v k,l q´f pt, x k , w k,l , ∇ a δ v k,l qq . Since f is non-increasing in its third variable and Lipschitz continuous, we get: Spk, l, w k,l , ∇ a δ w k,l , ∇ à ,δ w k,l , ∆ a δ w k,l , ϕ`ξq ď 0. The same computation with l " 0 or l " N a δ and S b instead of S gives S b pk, l, w k,l , ∇ δ w k,l , ∇`, δ w k,l , ∆ δ w k,l , ϕ`ξq ď 0, and the comparison theorem given in Proposition 6.2 gives w k,l ď S a δ ps, t, ϕ`ξq k,l for k P Z and l P t0, N a δ u. The comparison result from Proposition 3.2 gives the first inequality of the lemma. The second one is proved similarly. l Proposition 3.7 (Consistency). Let ϕ P C 8 b pr0, T sˆRˆR, Rq, pt, x, pq P r0, T qˆRp 0, 1q. We have, with the notation in (3.6):ˇˇˇ1 t j`1´tj p Spπ, δ, t j , x k , q, ϕpt j , x k , qq`ξ, ϕ`ξq sup aPK r´D a ϕpt, x, pq`F pt, x, ϕpt, x, pq, ∇ a ϕpt, x, pq, ∆ a ϕpt, x, pqqsˇˇˇˇÑ 0, as δ, |π| Ñ 0 satisfying (3.22)-(3.24), πˆδZˆr0, 1s Q pt j , x k , qq Ñ pt, x, pq, ξ Ñ 0. Proof. Let ϕ, j, k, p, l as in the statement of the Proposition. Without loss of generality, we can consider π, δ, t j , x k , q such that, for all a P K: 0 ď p a pqq ď 1, Since ϕ is smooth and pt k , x j , p l q Ñ pt, x, pq, we have | sup aPK r´D a ϕpt, x, pq`F pt, x, p, ϕpt, x, pq, ∇ a ϕpt, x, pq, ∆ a ϕpt, x, pqqś sup aPK r´D a ϕpt j , x k , p l q`F pt j , x k , p l , ϕpt j , x k , p l q, ∇ a ϕpt j , x k , p l q, ∆ a ϕpt j , x k , p l qqqs | Ñ 0. Thanks to Lemma 3.4, it suffices to prove:ˇˇˇ1 t j`1´tj p Spπ, δ, t j , x k , q, ϕpt j , x k , qq, ϕq sup aPK p´D a ϕpt j , x k , qq`F pt j , x k , ϕpt j , x k , qq, ∇ a ϕpt j , x k , qq, ∆ a ϕpt j , x k , qqqqˇˇˇˇÑ 0, as |π| Ñ 0 and πˆδZˆp0, 1q Q pt j , x k , qq Ñ pt, x, pq. Let ą 0 such that Ñ 0 and δ 2 Ñ 0. Using | inf´sup | ď sup |¨´¨|, addingˆ1 t j`1´tj ϕpt j`1 , x k , p a pqqq F pt j , x k , ϕpt j`1 , x k , p a pqqq, ∇ apa,δq ϕpt j`1 , x k , p a pqqq, ∆ apa,δq ϕpt j`1 , x k , p a pqqqqā nd using Lemma 3.1, it is enough to show that, for all a P K,ˇˇˇ1 t j`1´tj pϕpt j`1 , x k , p a pqqq´I a δ pS a δ pt j`1 , t j , ϕpt j`1 ,¨qqqpx k , qqq (3.60) F pt j , x k , ϕpt j`1 , x k , p a pqqq, ∇ apa,δq ϕpt j`1 , x k , p a pqqq, ∆ apa,δq ϕpt j`1 , x k , p a pqqqqˇˇÑ 0. The proof is concluded using the equality |¨| " maxp¨,´¨q and the two following inequalities, obtained by Lemma 3.3, and by definition (3.53)-(3.54) of S a,δ , : 1 t j`1´tj pϕpt j`1 , x k , p a pqqq´I a δ pS a δ pt j`1 , t j , ϕpt j`1 ,¨qqqpx k , qqq F pt j , x k , ϕpt j`1 , x k , p a pqqq, ∇ apa,δq ϕpt j`1 , x k , p a pqqq, ∆ apa,δq ϕpt j`1 , x k , p a pqqqq ď 1 t j`1´tj pϕpt j`1 , x k , p a pqqq´S a,δ , pt j`1 , t j , ϕpt j`1 ,¨qqqpx k , qq`opt j`1´tj qq F pt j , x k , ϕpt j`1 , x k , p a pqqq, ∇ apa,δq ϕpt j`1 , x k , p a pqqq, ∆ apa,δq ϕpt j`1 , x k , p a pqqqq, and F pt j , x k , ϕpt j`1 , x k , p a pqqq, ∇ apa,δq ϕpt j`1 , x k , p a pqqq, ∆ apa,δq ϕpt j`1 , x k , p a pqqqq 1 t j`1´tj pϕpt j`1 , x k , p a pqqq´I a δ pS a δ pt j`1 , t j , ϕpt j`1 ,¨qqqpx k , qqq ď F pt j , x k , ϕpt j`1 , x k , p a pqqq, ∇ apa,δq ϕpt j`1 , x k , p a pqqq, ∆ apa,δq ϕpt j`1 , x k , p a pqqqq 1 t j`1´tj pϕpt j`1 , x k , p a pqqq´S a,δ , pt j`1 , t j , ϕpt j`1 ,¨qqqpx k , qq`opt j`1´tj qq. l Numerical studies We now present a numerical application of the previous algorithm. Model We keep the notation of the previous section: the process X is a Brownian motion with drift. In this numerical example, the drift of the process Y is given by the following functions: f 1 pt, x, y, zq :"´σ´1µz, and f 2 pt, x, y, zq :"´σ´1µz`Rpy´σ´1zq´, where, for x P R, x´" maxp´x, 0q denotes the negative part of x. The function f 1 corresponds to pricing in a linear complete Black & Scholes market. It is well-known that there are explicit formulae for the quantile hedging price for a vanilla put or call, see [19]. In both cases, we compute the quantile hedging price of a put option with strike K " 30 and maturity T " 1, i.e. gpxq " maxpK´exppxq, 0q. The parameters of X are σ " 0.25 and µ " 0.01875 (this corresponds to a parameter b " 0.05 in the dynamics of the associated geometric Brownian motion, where µ " b´σ 2 {2). In the rest of this section, we present some numerical experiments. First, using the non-linear driver f 2 , we observe the convergence of v π,δ towards v n for a fixed discrete control set, and we estimate the rate of convergence with respect to δ. Second, we show that the conditions (3.22) to (3.24) are not only theoretically important, but also numerically. Last, we use the fact that the analytical solution to the quantile hedging problem with driver f 1 is known (see [19]) to assess the convergence (order) of the scheme more precisely. We observe that a judicious choice of control discretisation, time and space discretisation leads to convergence of v π,δ to v. However, the unboundedness of the optimal control as t Ñ T leads to expensive computations. The scheme obtained in the previous section deals with an infinite domain in the x variable. In practice, one needs to consider a bounded interval rB 1 , B 2 s and to add some boundary conditions. Here, we choose B 1 " logp10q and B 2 " logp45q, and the approximate Dirichlet boundary values for vpt, B i , pq are the limits lim v th pt, x, pq as x Ñ 0 or x Ñ`8, where v th is the analytical solution obtained in [19] for the linear driver f 1 . Since the non-linearity in f 2 is small for realistic parameters (we choose R " 0.05 in our tests), it is expected that the prices are close (see also [20]). Furthermore, we will consider values obtained for points pt, x, pq with x far from to the boundary. In this situation, the influence of our choice of boundary condition should be small, as noticed for example in Proposition 3.3. This was studied more systematically, for example, in [2]. Convergence towards v n with the non-linear driver In this section, we consider the non-linear driver f 2 defined above, where there is no known analytical expression for the quantile heding price. We now fix a discrete control set, and we compute the value function v π,δ for various discretisation parameters π, δ satisfying (3.22) to (3.24). We consider the following control set with 22 controls: r´2, 2s X Z´t0u 2˙ďˆr´3 , 3s X Z´t0u 3˙( 4.1) " t´2,´1.5, . . . , 1.5, 2u Y "´3 ,´3`1 3 , . . . , 3´1 3 , 3 * , and δ P t0.05, 0.03, 0.005u. For a fixed δ, we set h " Cδ with C :" minp1, 2 θ L , 1 |σ 2´µ | q, θ " 1 5 and L " |µ|`R, so that (3.22) to (3.24) are satisfied. We get the graphs shown in Figure 1 for the function p Þ Ñ v π,δ pt, x, pq, where pt, xq " p0, 30q, p0, 37q. We observe, while not proved, that the numerical approximation always gives an upper bound for v n , which is itself greater than the quantile hedging price v. This is a practically useful feature of this numerical method. The scheme preserves a key feature of the exact solution, namely that the quantile hedging price is 0 exactly for p below a certain threshold, depending on t, x. This is a consequence of the diffusion stencil ∆ a δ respecting the degeneracy of the diffusion operator ∆ a in (3.8), which acts only in direction p1, pq, and by the specific construction of the meshes. In Table 1, we show some discretisation parameters obtained by this construction with selected values of δ. Here, N x is the number of points for the x-variable, N c the number of controls, and N p the total number of points for the p variable (i.e., for all meshes combined). Moreover, a max (resp. a min ) is the greatest (resp. smallest) control obtained, using the modification of the control set (4.1) as described in Section 3.1. Also recall that different meshes are applied in each step of the PCPT algorithm for different a, hence we also report N p pa max q (resp. N p pa min q), the number of points for the p variable for the control a max (resp. a min ). With our choice of parameters, we have h " δ, so the number of time steps is always 1 δ . δ N t N x N c N p a max N p pa max q a CFL conditions Using the same discrete control set as above, we now fix h " 0.1 and compute v π,δ for δ chosen as above. The conditions (3.22) to (3.24) are then not satisfied anymore. First, while π is coarse, we observe that the computational time to get v π,δ pt j ,¨q from v π,δ pt j`1 ,¨q is larger. In fact, since the conditions are not satisfied anymore, the results of Proposition 3.1 are not valid anymore. While convergence to a fixed point is still observed, many more Picard iterations are needed. For example, for δ " 0.005 and h " 0.1, we observe that 3000 Picard iterations are needed, while in the example where (3.22) to (3.24) were satisfied, 250 iterations sufficed to obtain convergence (with a tolerance parameter of 10´5). The second observation is that, while we observe convergence to some limit (at least with our choice of δ: it might start to diverge for smaller δ, as seen for the case δ fixed and varying h below), it is not the limit observed in the previous subsection. We show in Figure 2 the difference between the solution obtained with δ " 0.005, h " 0.1, and δ " 0.005, h " Cδ. When the conditions are not met, we are dealing with a nonmonotone scheme, and convergence to the unique viscosity solution of the PDE, which equals the value function of the stochastic target problem, is not guaranteed. Conversely, when δ is fixed and we vary h, the situation is different. There is no issue with the Picard iterations, as the conditions needed for Proposition 3.1 are still satisfied. The issue here is that the consistency hypothesis is not satisfied, and convergence is not observed: when h is too close to 0, the value v π,δ gets bigger, as seen in Figure 3. Here, δ is fixed to 0.05 and h goes from 0.025 to 1.2ˆ10´5. Convergence to the analytical solution with linear driver We now consider the linear driver f 1 . In that case, the quantile hedging price can be found explicitly (see [19]). For each pt, x, pq, the optimal control can also be computed explicitly: αpt, x, pq " 1 a 2πpT´tq expˆ´N´1 ppq 2 2˙, where N is the cumulative distribution function of the standard normal distribution. In particular, if the uniform grid π " t0, h, . . . , κh " T u is fixed, one obtains that the optimal controls are contained in the interval r0, 1 ? 2πh s. On the other hand, if δ is fixed, one sees from (3.11) that the greatest control one can reach (with a non-trivial grid for the p variable) is σ 2δ . We set our parameters as follows: we first choose n ě 2, we pick δ such that σ nδ ě 1 ? 2πCδ , and we set h " Cδ. It is easy to see that δ is proportional to n´2. We now pick the controls in t σ mδ , m ě nu to obtain K n :" ta i :" σ m i δ , i " 1, . . . , M u as follows: let m 1 " n so that a 1 " a n max " σ nδ . If m 1 , . . . , m i are constructed, we set m i`1 " inftm ě m i , σ m i δ´σ mδ ě 1 n u and a i`1 " σ m i`1 δ . If m i`1 ă n´1, then we set M " i`1 and we are done. In Figure 4, we observe convergence towards the quantile hedging price. Moreover, Figure 5 demonstrates that the pointwise error, here for pt, x, pq " p0, 30, 0.8q, has a convergence rate of about 1.3 with respect to n in the construction described previously. Last, in Table 2, we report the values of δ and a max obtained for different choices of n. Conclusions We have introduced semi-discrete and discrete schemes for the quantile hedging problem, proven their convergence, and illustrated their behaviour in a numerical test. The scheme, based on piecewise constant policy time-stepping, has the attractive feature that semi-linear PDEs for individual controls can be solved independently on adapted meshes. In the example of the Black-Scholes dynamics this had the effect that in spite of the degeneracy of the diffusion operator it was possible to construct on each mesh a local scheme, i.e. one where only neighbouring points are involved in the discretisation. This does not contradict known results on the necessity of nonlocal stencils for monotone consistent schemes in this degenerate situation (see e.g. [26]), because of the superposition of different highly anisotropic meshes to arrive at a scheme which is consistent overall. A more accurate scheme could be constructed by exploiting higher order, limited interpolation in the p-variable, such as in [27]. It should be possible to deduce convergence from the results of this paper and the properties of the interpolator using the techniques in [28]. 2 ,0 q`| 8`| pv 1 ,N a δ´v 2 ,N a δ q`| 8 . By the comparison theorem, it is enough to show that w P 8 pG a δ q defined by w k,l :" v 2 k,l`B eplq (6.1) satisfies w k,0 ě v 1 k,0 , w k,N a δ ě v 1 k,N a δ and Spk, l, w k,l , ∇ a δ w k,l , ∇ à ,δ w k,l , ∆ a δ w k,l , uq ě 0, for all k P Z and 0 ă l ă N a δ . The boundary conditions are easily checked: if k P Z and l P t0, N a u, we have, since ep0q " epN a q " 1: w k,l " v 2 k,l`B ě v 2 k,l`p v 1 k,l´v 2 k,l q`ě v 1 k,l . For k P Z, 1 ď l ď N a δ´1 , we prove Spk, l, w k,l , ∇ a δ w k,l , ∇ à ,δ w k,l , ∆ a δ w k,l , uq ě 0. By definition (3.17), inserting˘hf pt´, e kδ , v 2 k,l , 1 2δ pw k`1,l`sgnpaq´wk´1,l´sgnpaq qq, since Spk, l, v 2 k,l , ∇ a δ v 2 k,l , ∇ à ,δ v 2 k,l , ∆ a δ v 2 k,l , uq ě 0 and since f is non-increasing with respect to its third variable and Lipschitz continuous with respect to its fourth variable, we have: We have |epl`sgnpaqq´epl´sgnpaqq| ď 1´e ‹ , thus: Spk, l, w k,l , ∇ a δ w k,l , ∇ à ,δ w k,l , ∆ a δ w k, Spk, l, w k,l , ∇ a δ w k,l , ∇ à ,δ w k,l , ∆ a δ w k,l , uq ( 1 pµ`L 2 qδh`σ 2 h`2θδ 2´1 2`pµ`L 2 qh`σ 2 h δ`2 δθ˘2 ě 1 pµ`L 2 qM`2θM 2˘h2`σ2 h´1 2´pp1`2θqµ`L 2 qh`σ 2 M¯2 ě 1 pµ`L 2 qM`2θM 2˘h2`σ2 h´M 2 2σ 4 . Proof. [Lemma 3.2] We show the result for S a,δ , , the proof is similar for S a,δ , . For k P Z and 0 ď l ď N a δ , let z k,l :" ϕpx k , p a pp l qq´hF pt, x k , ϕpx, p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qq, ∆ apa,δq ϕpx k , p a pp l qqq. (6.6) F pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qq, ∆ apa,δq ϕpx k , p a pp l qqq ě´C φ,n ph, q h . We split the sum into three terms: A " p F pt, x k , z k,l , ∇ a δ z k,l , ∇ à ,δ z k,l , ∆ a δ z k,l q p F pt, x k , ϕpx k , p a pp l qq, ∇ a δ ϕpx k , p a pp l qq, ∇ à ,δ ϕpx k , p a pp l qq, ∆ a δ ϕpx k , p a pp l qqq, B " p F pt, x k , ϕpx k , p a pp l qq, ∇ a δ ϕpx k , p a pp l qq, ∇ à ,δ ϕpx k , p a pp l qq, ∆ a δ ϕpx k , p a pp l qqq F pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qq, ∆ apa,δq ϕpx k , p a pp l qqq, C " F pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qq, ∆ apa,δq ϕpx k , p a pp l qqq F pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qq, ∆ apa,δq ϕpx k , p a pp l qqq. First, we have C " f pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qqq´f pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qqq ě´|f ´f | 8 . Secondly, by (3.13)-(3.14), we have, B "´θ δ 2 h ∆ a δ ϕpx k , p a pp l qq µp∇ apa,δq ϕpx k , p a pp l qq´∇ à ,δ ϕpx k , p a pp l qqq`σ 2 2 p∆ apa,δq ϕpx k , p a pp l qq´∆ a δ ϕpx k , p a pp l qqq pf pt, x k , ϕpx k , p a pp l qq, σ∇ apa,δq ϕpx k , p a pp l qqq´f pt, x k , ϕpx k , p a pp l qq, σ∇ a δ ϕpx k , p a pp l qqqq ě´θ δ 2 h |∆ a δ ϕpx k , p a pp l qq| µ|∇ apa,δq ϕpx k , p a pp l qq´∇ à ,δ ϕpx k , p a pp l qqq´σ 2 2 |∆ apa,δq ϕpx k , p a pp l qq´∆ a δ ϕpx k , p a pp l qq| σL|∇ apa,δq ϕpx k , p a pp l qq´∇ a δ ϕpx k , p a pp l qq| The first term goes to 0 since δ 2 h Ñ 0 as h Ñ 0 and ∆ a δ ϕpx k , p a pp l qq is bounded. The last three terms go to 0 by Taylor expansion and Lemma 3.1, since ϕ is smooth. Finally, by (3.13)-(3.14), using the linearity of the discrete differential operators and (6.6), and since f is Lipschitz-continuous, we have, A ě´hµ|∇ à ,δ F pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qq, ∆ apa,δq ϕpx k , p a pp l qqq| hp σ 2 2`θ δ 2 h q|∆ a δ F pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qq, ∆ apa,δq ϕpx k , p a pp l qqq| Lh|F pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qq, ∆ apa,δq ϕpx k , p a pp l qqq| Lσh|∇ a δ F pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qq, ∆ apa,δq ϕpx k , p a pp l qqq|. We can show that each term goes to 0 as h Ñ 0. By example: h σ 2 2 |∆ a δ F pt, x k , ϕpx k , p a pp l qq, ∇ apa,δq ϕpx k , p a pp l qq, ∆ apa,δq ϕpx k , p a pp l qqq| ě h σ 2 2 µ|∆ a δ ∇ apa,δq ϕpx k , p a pp l qq| h σ 4 4 |∆ a δ ∆ apa,δq ϕpx k , p a pp l qq| h σ 2 2 |∆ a δ f pt, x k , ϕpx k , p a pp l qq, σ∇ a δ ϕpx k , p a pp l qqq|. The first two terms go to 0 with h since |∆ a δ ∇ apa,δq ϕpx k , p a pp l qq| and |∆ a δ ∆ apa,δq ϕpx k , p a pp l qq| are bounded, by smoothness of ϕ and by Lemma 3.1. We can control the derivatives of f : px, pq Þ Ñ f pt, x, ϕpx, pq, σϕpx, pqq with respect to : for any α " pα 1 , α 2 q P N 2 , we have |D α f | 8 ď C ϕ,α α 1`α2 ,(6.7) for a constant C ϕ,α ą 0. By the triangle inequality and Taylor expansion, we get: h σ 2 2 |∆ a δ f pt, x k , ϕpx k , p a pp l qq, σ∇ a δ ϕpx k , p a pp l qqq| ě´h σ 2 2 |p∆ a δ´∆ apa,δq qf pt, x k , ϕpx k , p a pp l qq, σ∇ a δ ϕpx k , p a pp l qqq| h σ 2 2 |∆ apa,δq f pt, x k , ϕpx k , p a pp l qq, σ∇ a δ ϕpx k , p a pp l qqq| ě´C 1 h σ 2 2 δ 2 4´C 2 h σ 2 2 1 2 , where C 1 , C 2 ą 0, and this quantity goes to 0 by our choice of . Last, the smoothness of S a,δ , is straightforward by (3.52) and the control on its second derivative with respect to p is obtained by (6.7). l Representation and comparison results For pt, x, yq P r0, T sˆR dˆR`, q :"ˆq x q p˙P R d`1 and A :"ˆA xx A xp A xp J A pp˙P S d`1 , A xx P S d , denoting Ξ :" pt, x, y, q, Aq, we define, recalling (1.4)-(1.5)-(1.6), FpΞq " sup aPR 5 F a pΞq with F a pΞq :"´f pt, x, y, zpx, q, aqq´Lpx, q, A, aq , where R Ă SzD with a finite number of elements. Proposition 6.1. Let 0 ď τ ă θ ď T and u 1 (resp. u 2 ) be a lower semi-continuous super-solution (resp. upper semi-continuous sub-solution) with polynomial growth, of B t ϕ`Fpt, x, ϕ, Dϕ, D 2 ϕq " 0 on rτ, θqˆR dˆp 0, 1q (6.8) with u 1 ě u 2 on rτ, θsˆR dˆt 0, 1u Ť tθuˆR dˆr 0, 1s, then u 1 ě u 2 on rτ, θsˆR dˆr 0, 1s. Corollary 6.1. There exists a unique continuous solution w to (6.8) or equivalently sup ηPR H η pt, x, ϕ, B t ϕ, Dϕ, D 2 ϕq " 0 on rτ, θqˆR dˆp 0, 1q (6.9) satisfying wp¨q " Ψp¨q on rτ, θsˆR dˆt 0, 1u Ť tθuˆR dˆr 0, 1s, where Ψ P C 0 Proof. This is a direct application of the comparison principles. The equivalence between (6.8) and (6.9), comes from the fact that H η pΘq and´b´F η 5 pΞq have the same sign. l Lemma 6.1. (i) Let a P R d and w a be the unique solution tó B t ϕ`F a pt, x, ϕ, Dϕ, D 2 ϕq " 0 on rτ, θqˆR dˆp 0, 1q (6.10) satisfying w a p¨q " Ψp¨q on rτ, θsˆR dˆt 0, 1u Ť tθuˆR dˆr 0, 1s, where Ψ P C 0 . Then it admits the following probabilistic representation: w a pt, x, pq " Y t , (6.11) where Y is the first component of the solution pY, Zq to the following BSDE with random terminal time Y¨" ΨpT , X t,x T , P t,p,a T q`ż T f ps, X t,x s , Y s , Z s q ds´ż T Z s dW s , (6.12) with T :" infts ě t : P t,p,a Proof. (i) The probabilistic representation is proved in [14]. Note that uniqueness to the PDE comes from the previous lemma in the special case where R is reduced to one element. (ii) Let A :" tT " θu, B :" tT ă θ, P p,a . 2 . 2In our setting, we can easily identify the boundary values (of the scheme): Proposition 2. 3 ( 3Stability). The solution to scheme (2.16) is bounded. : j " 0, . . . , N a δ *.(3.12) Remark 3. 3 . 3Since µh ď δ, we have |µ apa,δq σ h| ď |apa,δq| σ δ, which ensures that from (3.18), p a pp l q P r0, 1s for all 0 ă l ă N a δ . Proposition 3.1. For every bounded function ϕ : δZˆr0, 1s Ñ R, there exists a unique solution to (3.15)-(3.16). Lemma 3. 1 . 1For all a P K, a and apa, δq have the same sign, and: Figure 1 : 1The functions v π,δ pt, x,¨q, t " 0 and x P t30, 37u. Figure 2 :Figure 3 : 23Comparison of v 0.1,0.005 p0, 37,¨q and v Cδ,0.005 p0, 37,¨q. Comparison of v h,0.05 p0, 30,¨q for some h. Figure 4 : 4v n p0, 30,¨q and vp0, 30,¨q for n " 3, 5, 8. Figure 5 : 5Convergence rate estimation of v n p0, 30, 0.8q to vp0, 30, 0.8q and log-log plot. Table 2 : 2Discretisation parameters for selected values of n. Proof. [Proposition 3.3] For ease of notation, we set, eplq :" e´4Cph,δqlpN a´l q , e ‹ :" min6 Appendix 6.1 Proofs apa,δq 2 σ 2 xPr0,N a δ s epxq " epN a δ {2q " e´a pa,δq 2 σ 2 Cph,δqpN a δ q 2 " e´C ph,δq δ 2 , B :" |pv 1 ,0´v and one can easily check that this is the case with our choice of Cph, δq. It remains to prove (3.40). Since lnp1`xq ą x´x4) ě B "ˆ1`µ h δ`σ 2 h δ 2`2 θ`h L 2δ˙e ‹´µ h δ´σ 2 h δ 2´2 θ´h L 2δ  . It is thus enough to havê 1`µ h δ`σ 2 h δ 2`2 θ`h L 2δ˙e ‹´µ h δ´σ 2 h δ 2´2 θ´h L 2δ ě 0, (6.5) 2 2 for all x ą 0, we have, by (3.24): Cph, δq ą 1 δ 2˜1 µ h δ`σ 2 h δ 2`2 θ`h L 2δ´1 2 1 µ h δ`σ 2 h δ 2`2 θ`h L 2δ˘2" AcknowledgementsThis work was partially funded in the scope of the research project "Advanced techniques for non-linear pricing and risk management of derivatives" under the aegis of the Europlace Institute of Finance, with the support of AXA Research Fund.Proposition 6.2 (Comparison theorem). Let 0 ď t ă s ď T, δ ą 0, h " s´t such that (3.22)-(3.23)-(3.24) is satisfied. Let pu 1 , u 2 , v 1 , v 2 q P 8 pδZq 4 such that u 1 k ď u 2 k for all k P Z.1. For all k P Z, pv, ∇, ∇`, ∆q P R 4 :2. Assume that, for all k P Z:Then v 1 k ď v 2 k for all k P Z.3. Assume that, for all k P Z:The proof is similar to the proof of Proposition 3.2 and is ommited. lFor all k P Z, let U π,δ q k :". Then:1. pV π,δ , U π,δ q P 8 pδZq 2 and their bound is independant of π, δ.2. V π,δ converges uniformly on compacts sets to V , the super-replication price of the contingent claim with payoff g.Proof.We only show the first point, the second one is obtained by applying the arguments of[4], after proving monotonicity, stability and consistency following the steps of Subsection 3.2. Since g is bounded, it is easy to show that V π,δ is also bounded independently of π, δ, and the proof is similar to the proof of Proposition 3.6. Since g is Lipschitz-continuous, we get that U π,δ pT,¨q is bounded. Using the Lipschitzcontinuity of f , one deduces easily that U π,δ is a solution ofgpx k`1 q´gpx k´1 q 2δ P r´L, Ls, k P Z.Again, comparison theorems can be proved, and it is now enough to show that there exists pu, uq P 8 pπˆδZq 2 which are bounded uniformly in π, δ such thatu κ k ď´L, u κ k ě L, k P Z. We deal with u only, we obtain similar results for u. One can easily show that u j :" 1´pL`1q ś κ k"j`1 1 1´h k L , where h k :" t k´tk´1 , satisfies the requirements. 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A backward dual representation for the quantile hedging of Bermudan options. Bruno Bouchard, Géraldine Bouveret, Jean-François Chassagneux, SIAM Journal on Financial Mathematics. 71Bruno Bouchard, Géraldine Bouveret, and Jean-François Chassagneux. A back- ward dual representation for the quantile hedging of Bermudan options. SIAM Journal on Financial Mathematics, 7(1):215-235, 2016. BSDEs with weak terminal condition. The Annals of Probability. Bruno Bouchard, Romuald Elie, Antony Réveillac, 43Bruno Bouchard, Romuald Elie, and Antony Réveillac. BSDEs with weak terminal condition. The Annals of Probability, 43(2):572-604, 2015. Stochastic target problems with controlled loss. Bruno Bouchard, Romuald Elie, Nizar Touzi, SIAM Journal on Control and Optimization. 485Bruno Bouchard, Romuald Elie, and Nizar Touzi. Stochastic target problems with controlled loss. SIAM Journal on Control and Optimization, 48(5):3123-3150, 2009. A stochastic target approach for P&L matching problems. 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{'abstract': 'We consider the numerical approximation of the quantile hedging price in a non-linear market. In a Markovian framework, we propose a numerical method based on a Piecewise Constant Policy Timestepping (PCPT) scheme coupled with a monotone finite difference approximation. We prove the convergence of our algorithm combining BSDE arguments with the Barles & Jakobsen and Barles & Souganidis approaches for non-linear equations. In a numerical section, we illustrate the efficiency of our scheme by considering a financial example in a market with imperfections.', 'arxivid': '1902.11228', 'author': ['Cyril Bénézet ', 'Jean-François Chassagneux ', 'Christoph Reisinger '], 'authoraffiliation': [], 'corpusid': 67877088, 'doi': '10.1137/19m1267477', 'github_urls': [], 'n_tokens_mistral': 37930, 'n_tokens_neox': 34107, 'n_words': 18610, 'pdfsha': 'da8555cca5c6b4288f10daaa762b831610f3ecdf', 'pdfurls': ['https://arxiv.org/pdf/1902.11228v1.pdf'], 'title': ['A numerical scheme for the quantile hedging problem', 'A numerical scheme for the quantile hedging problem'], 'venue': []}

arxiv

Parameterized Complexity of Problems in Coalitional Resource Games * 3 May 2011 January 20, 2013 Rajesh Chitnis rchitnis@cs.umd.edu Department of Computer Science University of Maryland at College Park USA Mohammadtaghi Hajiaghayi hajiagha@cs.umd.edu Department of Computer Science University of Maryland at College Park USA Vahid Liaghat vliaghat@cs.umd.edu Department of Computer Science University of Maryland at College Park USA Parameterized Complexity of Problems in Coalitional Resource Games * 3 May 2011 January 20, 20131 Coalition formation is a key topic in multi-agent systems. Coalitions enable agents to achieve goals that they may not have been able to achieve on their own. Previous work has shown problems in coalition games to be computationally hard. Wooldridge and Dunne (Artificial Intelligence 2006) studied the classical computational complexity of several natural decision problems in Coalitional Resource Games (CRG) -games in which each agent is endowed with a set of resources and coalitions can bring about a set of goals if they are collectively endowed with the necessary amount of resources. The input of coalitional resource games bundles together several elements, e.g., the agent set Ag, the goal set G, the resource set R, etc. Shrot, Aumann and Kraus (AAMAS 2009) examine coalition formation problems in the CRG model using the theory of Parameterized Complexity. Their refined analysis shows that not all parts of input act equal -some instances of the problem are indeed tractable while others still remain intractable.We answer an important question left open by Shrot, Aumann and Kraus by showing that the SC Problem (checking whether a Coalition is Successful) is W[1]-hard when parameterized by the size of the coalition. Then via a single theme of reduction from SC, we are able to show that various problems related to resources, resource bounds and resource conflicts introduced by Wooldridge et al are 1. W[1]-hard or co-W[1]-hard when parameterized by the size of the coalition.2. para-NP-hard or co-para-NP-hard when parameterized by |R|.3. FPT when parameterized by either |G| or |Ag| + |R|. Introduction Coalitions In multi-agent systems (MAS), where each agent has limited resources, the formation of coalitions of agents is a very powerful tool [6]. Coalitions enable agents to accomplish goals they may not have been able to accomplish individually. As such, understanding and predicting the dynamics of coalitions formation, e.g., which coalitions are more beneficial and/or more likely to emerge, is a question of considerable interest in multi-agent settings. Unfortunately, a range of previous studies have shown that many of these problems are computationally complex [7,8]. Nonetheless, as noted by Garey and Johnson [4], hardness results, such as NP-completeness, should merely constitute the beginning of the research. NP-hardness just indicates that a general solution for all instances of the problem most probably does not exist. Still, efficient solutions for important sub-classes may well exist. Formal Model of Coalition Resource Games The framework we use to model coalitions is the CRG model introduced in [8], defined as follows. The model contains a non-empty, finite set Ag = {a 1 , . . . , a n } of agents. A coalition, typically denoted by C, is simply a set of agents, i.e., a subset of Ag. The grand coalition is the set of all agents, Ag. There is also a finite set of goals G. Each agent i ∈ Ag is associated with a subset G i of the goals. Agent i is satisfied if at least one member of G i is achieved, and unsatisfied otherwise. Achieving the goals requires the expenditure of resources, drawn from the total set of resource types R. Achieving different goals may require different quantities of each resource type. The quantity req(g, r) denotes the amount of resource r required to achieve goal g. It is assumed that req(g, r) is a non-negative integer. Each agent is endowed certain amounts of some or all of the resource types. The quantity en(i, r) denotes the amount of resource r endowed to agent i. Again, it is assumed that en(i, r) is a non-negative integer. Formally, a Coalition Resource Game Γ is a (n + 5)-tuple given by Γ = Ag, G, R, G 1 , G 2 , . . . , G n , en, req where: • Ag = {a 1 , a 2 , . . . , a n } is the set of agents • G = {g 1 , g 2 , . . . , g m } is the set of possible goals • R = {r 1 , r 2 , . . . , r t } is the set of resources • For each i ∈ Ag, G i is a subset of G such that any of the goals in G i would satisfy i but i is indifferent between the members of G i • en : Ag × R → N ∪ {0} is the endowment function • req : G × R → N ∪ {0} is the requirement function The endowment function en extends to coalitions by summing up endowments of its members as en(C, r) = ∑ i∈C en(i, r) ∀r ∈ R The requirement function req extends to sets of goals by summing up requirements of its members as req(G ′ , r) = ∑ g∈G ′ req(g, r) ∀r ∈ R Figure 1: Results of Shrot et al. [5] A set of goals G ′ satisfies agent i if G i ∩ G ′ = / 0 and satisfies a coalition C if it satisfies every member of C. A set of goals G ′ is feasible for coalition C if that coalition is endowed with sufficient resources to achieve all goals in G ′ , i.e., for all r ∈ R we have req(G ′ , r) ≤ en(C, r). Finally we say that a coalition C is successful if there exists a non-empty set of goals G ′ that satisfies C and is feasible for it. In general, we use the notation succ(C) = {G ′ | G ′ ⊆ G, G ′ = / 0 and G ′ is successful for C}. The CRG models many real-world situations like the virtual organizations problem [1] and voting domains. Problem Definitions and Previous Work Problems Related to Coalition Formation Shrot et al. [5] considered the following four problems related to coalitions. The results from Shrot et al. [5] are summarized in Figure 1. In this work we consider the problems which were defined by Wooldridge et al. [8] but were not considered by Shrot et al. [5]. We define these problems in detail in the following sections. NECESSARY RESOURCE (NR) Instance: A CRG Γ, coalition C and resource r Question: Is req(G ′ , r) > 0 ∀ G ′ ∈ succ(C)? 6. STRICTLY NECESSARY RESOURCE (SNR) Instance: A CRG Γ, coalition C and resource r Question: Is succ(C) = / 0 and ∀ G ′ ∈ succ(C) we have req(G ′ , r) > 0? 7. (C, G 0 , r)-OPTIMALITY (CGRO) Instance: A CRG Γ, coalition C, goal set G 0 ∈ succ(C) and resource r Question: Is req(G ′ , r) ≥ req(G 0 , r) ∀ G ′ ∈ succ(C)? R-PARETO EFFICIENT GOAL SET (RPEGS) Instance: A CRG Γ, coalition C and a goal set G 0 Question: Is G 0 R-Pareto Efficient for coalition C? 9. SUCCESSFUL COALITION WITH RESOURCE BOUND (SCRB) Instance: A CRG Γ, coalition C and a resource bound b Question: Does ∃ G 0 ∈ succ(C) such that G 0 respects b? 10. CONFLICTING COALITIONS (CC) Instance: A CRG Γ, coalitions C 1 ,C 2 and a resource bound b Question: If ∀ G 1 ∈ succ(C 1 ) and ∀ G 2 ∈ succ(C 2 ) we have cgs(G 1 , G 2 , b)? Parameterized Complexity We now provide a brief introduction to the key relevant concepts from the theory of parameterized complexity. The definitions in this section are taken from [3] and [2]. The core idea of parameterized complexity is to single out a specific part of the input as the parameter and ask whether the problem admits an algorithm that is efficient in all but the parameter. In most cases the parameter is simply one of the elements of the input (e.g., the size of the goal set), but it can actually be any computable function of the input: Definition 3.1. Let Σ be a finite alphabet. 1. A parametrization of Σ * is a mapping κ : Σ * → N that is polynomial time computable. A parameterized problem (over Σ) is a pair (Q, κ) consisting of a set Q ⊆ Σ * of strings over Σ and a parameterization κ of Σ * . As stated, given a parameterized problem we seek an algorithm that is efficient in all but the parameter. This is captured by the notion of fixed parameter tractability, as follows: Definition 3.2. A parameterized problem (Q, κ) is fixed parameter tractable (FPT) if there exist an algorithm A, a constant α, and a computable function f , such that A decides Q in time f (κ(x))|x| α . Thus, while the fixed-parameter notion allows inefficiency in the parameter κ(x), by means of the function f , it requires polynomial complexity in all the rest of the input. In particular, a problem that is FPT is tractable for any bounded parameter value. While the core aim of parameterized complexity is to identify problems that are fixed-parameter tractable, it has also developed an extensive complexity theory, allowing to prove hardness results, e.g., that certain problems are (most probably) not FPT. To this end, several parameterized complexity classes have been defined. Two of these classes are the class W [1] and the class para-NP. We will formally define these classes shortly, but the important point to know is that there is strong evidence to believe that both classes are not contained in FPT (much like NP is probably not contained in P). Thus, W[1]-hard and para-NP-hard problems are most probably not fixed-parameter tractable. The class W [1] can be defined by its core complete problem, defined as follows: SHORT NONDETERMINISTIC TURING MACHINE COMPUTATION Instance: A single-tape, single-head non-deterministic Turing machine M, a word x and an integer k Question: Is there a computation of M on input x that reaches the accepting state in at most k steps? Parameter: k Note that this definition is analogous to that of NP, with the addition of the parameter k. The class para-NP is defined as follows : Establishing hardness results most frequently requires reductions. In parameterized complexity, we use FPT-reduction, defined as follows: Definition 3.5. Let (Q, κ) and (Q ′ , κ ′ ) be parameterized problems over the alphabets Σ and Σ ′ respectively. An FPT-reduction (FPT many-to-one reduction) from (Q, κ) to (Q ′ , κ ′ ) is a mapping R : Σ * → (Σ ′ ) * such that: 1. For all x ∈ Σ * we have x ∈ Q ⇔ R(x) ∈ Q ′ . 2. R is computable in time f (κ(x))|x| α for some constant α and an arbitrary function f . There is a computable function g : N → N such that κ ′ (R(x)) ≤ g(κ(x)) for all x ∈ Σ * . Point (1) simply states that R is indeed a reduction. Point (2) says that it can be computed in the right amount of time -efficient in all but the parameter. Point (3) states that the parameter of the image is bounded by (a function of) that of the source. This is necessary in order to guarantee that FPT-reductions preserve FPT-ness, i.e. with this definition we obtain that if (Q, κ) reduces to (Q ′ , κ ′ ) and (Q ′ , κ ′ ) ∈ FPT then (Q, κ) is also in FPT. Our Results & Techniques We consider problems regarding resources bounds and resource conflicts which were shown to be computationally hard in Wooldridge et al. ( [8]) but were not considered in Shrot et al. [5]. We also solve three open questions posed in Shrot et al. by showing that 1. SC parameterized by |C| is W[1]-hard 2. ESCK parameterized by |Ag| + |R| is FPT 3. ESCK parameterized by |R| is para-NP-hard We study the complexity of NR, SNR, CGRO, RPEGS, SCRB and CC problems when parameterized by natural parameters |G|, |C|, |R| and |Ag| + |R|. We also give a general integer program which with slight modifications for each problem shows that these problems are FPT when parameterized by |G| or |Ag| + |R| (except CC parameterized by |Ag| + |R| which is open). We note that Shrot et al. showed that SC parameterized by |R| is para-NP-hard. We complete this hardness result by showing that SC parameterized by |C| is W[1]-hard and thus answer their open question. Using these hardness results and via a single theme of parameter preserving reductions we show that hardness results for all of the above problems when parameterized by |R| and |C|. We also show that Theorem 3.2 of Shrot et al. [5] is false -which claims that ESCK is FPT when parameterized by |G|. We give a counterexample to their proposed algorithm and show that the problem is indeed para-NP-hard. These results help us to understand the role of various components of the input and identify which ones actually make the input hard. Since all the problems we considered remain intractable when parameterized by |C| or |R|, there is no point in trying to restrict these parameters. On the other hand, most of the problems are FPT when parameterized by |G| or |Ag| + |R| and thus we might enforce this restriction in real-life situations to ensure the tractability of these problems. SC ESCK NR,CGRO,RPEGS SNR SCRB CC NPC NPC co-NPC D p -complete NPC co-NPC |G| FPT FPT p-NP-hard FPT FPT FPT FPT |C| W[1]-hard W[1]-hard co-W[1]-hard W[1]-hard co-W[1]-hard co-W[1]-hard |R| p-NP-hard p-NP-hard co-pNP-hard pNP-hard co-pNP-hard co-pNP-hard |Ag| + |R| FPT FPT FPT FPT FPT ? We summarize all the results in Figure 2. The results from [8] are in green, from [5] in black and our results are in red color. We use the abbreviations NPC for NP-complete, and pNP for para-NP. Problems Left Open in Shrot et al. [5] First we show that SC parameterized by |C| is W[1]-hard. Theorem 5.1. SC is W[1]-hard when parameterized by |C|. Proof. We prove this by reduction from Independent Set (parameterized by size of independent set) which is a well-known W[1]-complete problem. Let H = (V, E) be a graph with V = {x 1 , . . . , x n } and E = {e 1 , . . . , e m }. Let k be a given integer. We also assume that H has no isolated points as we can just add those points to the independent set and decrease the parameter appropriately. We build a CRG Γ as follows: Γ = Ag, G, R, G 1 , G 2 , . . . , G k , en, req where • Ag = {c 1 , . . . , c k } • G i = {g 1 i , . . . , g n i } for all i ∈ [k] • G = k i=1 G i • R = {r 1 , . . . , r m } • For all i ∈ [k], j ∈ [m] , en(c i , r j ) = 1 • For all i ∈ [k], j ∈ [m] and ℓ ∈ [n], we have req(g ℓ i , r j ) = k if e j and x ℓ are incident in H and req(g ℓ i , r j ) = 0 otherwise We claim that H has an independent set of size k if and only if the grand coalition Ag is successful in Γ. Suppose INDEPENDENT SET answers YES, i.e., H has an independent set of size k say I = {x β 1 , . . . , x β k }. Consider the goal set given by G ′ = {g β 1 1 , . . . , g β k k }. Clearly G ′ satisfies Ag as g β i i ∈ G i for all i ∈ [k] . Now consider any edge e j ∈ E(H). Let λ be the number of vertices from I incident on e j . Clearly 2 ≥ λ but as I is independent set we have 1 ≥ λ . Now, for every j ∈ [m] we have req(G ′ , r j ) = kλ ≤ k = en(Ag, r j ). Thus G ′ is feasible for Ag. Summing up, G ′ is successful for Ag and hence SC answers YES for C = Ag. Suppose now that SC answers YES for C = Ag. Let G ′′ = / 0 be successful for Ag. Claim is that both g β i and g β j cannot be in G ′′ if i = j. To see this, let e ℓ be any edge incident on x β (we had assumed earlier that graph has no isolated vertices). Then req(G ′′ , r ℓ ) ≥ req(g β i , r ℓ ) + req(g β i , r ℓ ) = 2k > k = en(Ag, r ℓ ) which contradicts the fact that G ′′ is successful for Ag. Since G i 's are disjoint and G ′′ is successful (hence also satisfiable) for Ag, we know that G ′′ contains at least one goal from each G i . Also we have seen before that g β i , g γ j ∈ G ′′ and i = j implies that β = γ. From each G i we pick any goal that is in G ′′ . Let us call this as G ′ = {g β 1 1 , . . . , g β k k }. We know that β i = β j when i = j. We claim that I = {x β 1 , . . . , x β k } is an independent set in H. Suppose not and let e l be an edge between x β i and x β j for some i, j ∈ [k]. Then req(G ′′ , r ℓ ) ≥ req(G ′ , r ℓ ) ≥ req(g β i i , r ℓ ) + req(g β j j , r ℓ ) = k + k > k = en(Ag, r ℓ ) which contradicts the fact that G ′′ is successful for Ag. Thus I is an independent set of size k in H and so INDEPENDENT SET also answers YES. Note that |Ag| = k, |G| = nk, |R| = m and so this reduction shows that the SC problem is W[1]-hard. We note that the SC problem can be solved in O(|G| |C| × |R|) time (since we only need to check the subsets of size at most |C| of G) and thus SC parameterized by |C| is not para-NP-hard. Now we answer the only remaining open problem by Shrot et al. by showing that ESCK parameterized by |R| is para-NP-hard. Theorem 5.2. Checking whether there exists a successful coalition of size k (ESCK) is para-NP-hard when parameterized by |R|. Proof. We prove this by reduction from SC which was shown to be para-NP-hard with respect to the parameter |R| in Theorem 3.8 of [5]. Let (Γ,C) be a given instance of SC. We consider an instance (Γ ′ , k) of ESCK • Ag ′ = C • R ′ = R • G ′ i = G i for all i ∈ C • k = |C| We claim that SC answers YES if and only if ESCK answers YES. Suppose SC answers YES, i.e., C is a successful coalition in Γ. In Γ ′ we just remove all agents not belonging to C from Γ. All the resources and the en and req functions carry over. So C is a successful coalition for Γ ′ also. But we had chosen k = |C| and so ESCK answers YES. Suppose that ESCK answers YES. So there exists a successful coalition of size k in Γ ′ . But Ag ′ = C and we had chosen k = |C| and so the only coalition of size k in Γ ′ is the grand coalition C = Ag ′ . As ESCK answered YES we know that C is successful in Γ ′ . So it is also successful in Γ and so SC also answers YES. Note that |Ag ′ | = k, |G ′ | = |G|, |R ′ | = |R| and so this reduction shows that the ESCK problem is para-NPhard. Problems Related to Resources For a coalition C, we recollect the notation we use: succ(C) = {G ′ | G ′ ⊆ G ; G ′ = / 0 and G ′ both satisfies C and is feasible for it}. In this section we show hardness results for three different problems related to resources. Necessary Resource (NR) The idea of a necessary resource is similar to that of a veto player in the context of conventional coalition games. A resource is said to be necessary if the accomplishment of any set of goals which is successful for the coalition would need a non-zero consumption of this resource. Thus if a necessary resource is scarce then the agents possessing the resource become important. We consider the NECESSARY RESOURCE problem: Given a coalition C and a resource r answer YES if and only if req(G ′ , r) > 0 for all G ′ ∈ succ(C). NR was shown to be co-NP-complete in Wooldridge et al. [8]. We note that if C is not successful, then NR vacuously answers YES. We give a reduction from SC to NR. Lemma 6.1. Given an instance (Γ,C) of SC we can construct an instance (Γ ′ ,C ′ , r ′ ) of NR such that SC answers YES iff NR answers NO. Proof. We keep everything the same except R ′ = R ∪ {r ′ }. We extend the en and req functions to r ′ by en(i, r ′ ) = 1 for all i ∈ Ag and req(g, r ′ ) = 0 for all g ∈ G. Now claim is that SC answers YES iff NR answers NO. Suppose SC answers YES. So ∃ G ′ = / 0 such that G ′ ∈ succ Γ (C). Now C = / 0 and so en(C, r ′ ) > 0 = req(G ′ , r ′ ) and thus G ′ ∈ succ Γ ′ (C). But req(G ′ , r ′ ) = 0 and so NR answers NO. Suppose NR answers NO. So succ Γ ′ (C) = / 0 as ∃ G ′ ∈ succ Γ ′ (C) such that G ′ = / 0 and req(G ′ , r ′ ) = 0. Now Γ ′ is obtained from Γ by only adding a new resource and so clearly G ′ ∈ succ Γ (C). Thus SC will answer YES. Strictly Necessary Resource (SNR) The fact that a resource is necessary does not mean that it will be used. Because the coalition in question can be unsuccessful and hence the resource is trivially necessary. So we have the STRICTLY NECESSARY RESOURCE problem: Given a coalition C and a resource r answer YES if and only if succ(C) = / 0 and ∀ G ′ ∈ succ(C) we have req(G ′ , r) > 0. SNR was shown to be strongly D p -complete in Wooldridge et al. [8]. To prove the parameterized hardness results, we give a reduction from SC to SNR. Lemma 6.3. Given an instance (Γ,C) of SC we can construct an instance (Γ ′ ,C ′ , r ′ ) of SNR such that SC answers YES iff SNR answers YES. Proof. We keep everything the same except R ′ = R ∪ {r ′ }. We extend the en and req functions to r ′ by en(i, r ′ ) = |G| for all i ∈ Ag and req(g, r ′ ) = 1 for all g ∈ G. Now claim is that SC answers YES iff SNR answers YES. We first show that succ Γ (C) = succ Γ ′ (C). As Γ ′ is obtained from Γ by just adding one resource and keeping everything else the same, we have succ Γ ′ (C) ⊆ succ Γ (C). Now let G 0 ∈ succ Γ (C). Any coalition has at least one member and hence at least one |G| endowment of resource r ′ . But req(G 0 , r ′ ) = |G 0 | ≤ |G| ≤ en(C, r ′ ) and so G 0 ∈ succ Γ ′ (C). Summing up we have succ Γ (C) = succ Γ ′ (C). Suppose SC answers YES. This implies succ Γ (C) = / 0. So succ Γ ′ (C) = succ Γ (C) = / 0. For every G 0 ∈ succ Γ ′ (C), req(G 0 , r ′ ) = |G 0 | > 0 as G 0 = / 0. Therefore SNR answers YES Suppose SNR answers YES. So succ Γ ′ (C) = / 0 as otherwise SNR would have said NO. Hence succ Γ (C) = succ Γ ′ (C) = / 0 and SC so answers YES. (C, G 0 , r)-Optimality (CGRO) We may want to consider the issue of minimizing usage of a particular resource. If satisfaction is the only issue, then a coalition C will be equally happy between any of the goal sets in succ(C). However in practical situations we may want to choose a goal set among succ(C) which minimizes the usage of some particular costly resource. Thus we have the (C, G 0 , r)-OPTIMALITY problem: Given a coalition C, resource r and a goal set G 0 ∈ succ(C) answer YES if and only if req(G ′ , r) ≥ req(G 0 , r) for all G ′ ∈ succ(C). CGRO was shown to be strongly co-NP-complete in Wooldridge et al. [8]. To prove the parameterized hardness results, we give a reduction from SC to CGRO. Lemma 6.5. Given an instance (Γ,C) of SC we can construct an instance (Γ ′ ,C ′ , G 0 , r ′ ) of CGRO such that SC answers YES iff CGRO answers NO. Proof. Define G ′ = G ∪ {g ′ }, R ′ = R ∪ {r ′ } and C ′ = C. We extend the en to r ′ as follows: en(i, r ′ ) = 1 for all i ∈ C and en(i, r ′ ) = 0 if i / ∈ C. We extend req to g ′ and r ′ as follows: req(g ′ , r ′ ) = |C|, req(g ′ , r) = 0 for all r ∈ R and req(g, r ′ ) = 0 for all g ∈ G. Let G 0 = {g ′ }. Now claim is that SC answers YES iff CGRO answers NO. Suppose SC answers YES. So, ∃ G 1 ∈ succ Γ (C). Claim is that G 1 ∈ succ Γ ′ (C) because en(C, r ′ ) = |C| > 0 = req(G 1 , r ′ ) as G 1 ⊆ G. Note also that G 0 = {g ′ } ∈ succ Γ ′ (C) as en(C, r ′ ) = |C| = req(G 0 , r ′ ) and for every r ∈ R, en(C, r) ≥ 0 = req(G 0 , r). Therefore req(G 1 , r ′ ) = 0 < |C| = req(G 0 , r ′ ) and hence CGRO answers NO. Suppose CGRO answers NO. So ∃ G 1 ∈ succ Γ ′ (C) such that req(G 1 , r ′ ) < req(G 0 , r ′ ) = |C|. Claim is g ′ / ∈ G 1 otherwise req(G 1 , r ′ ) ≥ req(g ′ , r ′ ) = |C|. So G 1 ⊆ G and we already had G 1 ∈ succ Γ ′ (C). Therefore G 1 ∈ succ Γ (C) and so SC answers YES. Theorem 6.6. The parameterized complexity status of (C, G 0 , r)-Optimality is as follows : • FPT when parameterized by |G| • co-W[1]-hard when parameterized by |C| • co-para-NP-hard when parameterized by |R| Proof. When parameterized by |G|, we consider all 2 |G| subsets of G. For each subset, we can check in polynomial time if it is a member of succ(C) and if it requires atleast req(G 0 , r ′ ) quantity of resource r ′ where G 0 and r ′ are given in the input. The other two claims follow from Lemma 6.5, Theorem 3.8 in Shrot et al., and Theorem 5.1. Problems Related to Resource Bounds R-Pareto Efficient Goal Set (RPEGS) We use the idea of Pareto Efficiency to measure the optimality of a goal set w.r.t the set of all resources. In our model we say that a goal set G ′ is R-Pareto Efficient w.r.t a coalition C if no goal set in succ Γ (C) requires at most as much of every resource and strictly less of some resource. More formally we say that a goal set G ′ is R-Pareto Efficient w.r.t a coalition C if and only if ∀ G ′′ ∈ succ Γ (C), ∃ r 1 ∈ R : req(G ′′ , r 1 ) < req(G ′ , r 1 ) ⇒ ∃ r 2 ∈ R : req(G ′′ , r 2 ) > req(G ′ , r 2 ) We note that G ′ is not necessarily in succ(C). Thus we have the R-PARETO EFFICIENT GOAL SET problem: Given a coalition C and a goal set G 0 answer YES if and only if G 0 is R-Pareto Efficient w.r.t C. Wooldridge et al. [8] show that RPEGS is strongly co-NP-complete. To prove the parameterized hardness results, we give a reduction from SC to RPEGS. Proof. Define R ′ = R ∪ {r ′ }, G ′ = G ∪ {g ′ } and C ′ = C. We extend the en to r ′ as follows: en(i, r ′ ) = |G| for all i ∈ C and en(i, r ′ ) = 0 if i / ∈ C. We extend req to r ′ as follows: req(g, r ′ ) = |C| for all g ∈ G; req(g ′ , r ′ ) = |G| · |C| + 1 and req(g ′ , r) = ∞ for all r ∈ R. Let G 0 = {g ′ }. Now claim is that SC answers YES iff RPEGS answers NO. We first show that succ Γ (C) = succ Γ ′ (C). Let G 1 ∈ succ Γ ′ (C). Then claim is that g ′ / ∈ G 1 because otherwise for all r ∈ R we have req(G 1 , r) ≥ req(g ′ , r) = ∞ > |G| · |C| = en(C, r). Also claim is that any goal set G 2 in succ Γ (C) also is in succ Γ ′ (C). All other things carry over from Γ and we have additionally that req(G 2 , r ′ ) = |G 2 | · |C| ≤ |G| · |C| = en(C, r ′ ) as G 2 ⊆ G. Hence we have succ Γ (C) = succ Γ ′ (C). Suppose SC answers YES, i.e., ∃ G 1 ∈ succ Γ (C). As succ Γ (C) = succ Γ ′ (C) we have G 1 ∈ succ Γ ′ (C). Now for every r ∈ R, req(G 1 , r) < ∞ = req(G 0 , r). Also req(G 1 , r ′ ) = |G 1 | · |C| ≤ |G| · |C| < |G| · |C| + 1 = req(G 0 , r ′ ). Therefore G 0 requires strictly more of every resource in R ′ than G 1 and hence RPEGS answers NO. Suppose RPEGS answers NO. Claim is that succ Γ ′ (C) = / 0 otherwise it would have answered YES vacuously. As succ Γ (C) = succ Γ ′ (C) we have succ Γ (C) = / 0 and hence SC answers YES. Successful Coalition with Resource Bound (SCRB) In real-life situations we typically have a bound on the amount of each resource. A resource bound is a function b : R → N with the interpretation that each coalition has at most b(r) quantity of resource r for every r ∈ R. We say that a goal set G 0 respects a resource bound b w.r.t. a given CRG Γ iff ∀ r ∈ R we have b(r) ≥ req(G 0 , r). Thus we have the SUCCESSFUL COALITION WITH RESOURCE BOUND problem: Given a coalition C and a resource bound b answer YES if and only if ∃ G 0 ∈ succ(C) such that G 0 respects b. Wooldridge et al. [8] show that SCRB is strongly NP-complete. To prove the parameterized hardness results, we give a reduction from SC to SCRB. Proof. Define R ′ = R ∪ {r ′ } and C ′ = C. Let b be a vector with |R ′ | components whose first |R ′ | − 1 entries are 1 and the last entry is |C| − 1, i.e., b = {1, 1, . . . , 1, 1, |C| − 1}. We extend the en to r ′ as follows: en(i, r ′ ) = |G| for all i ∈ C and en(i, r ′ ) = 0 if i / ∈ C. We extend req to r ′ as follows: req(g, r ′ ) = |C| for all g ∈ G. Now the claim is that SC answers YES if and only if SCRB answers NO. Suppose SC answers YES. So, there exists G 0 = / 0 such that G 0 ∈ succ Γ (C). In Γ ′ we have en(C, r ′ ) = |G| · |C| ≥ req(G 0 , r ′ ) as G 0 ⊆ G and req(g, r ′ ) = |C| for all g ∈ G. Thus G 0 ∈ succ Γ ′ (C) and so SCRB cannot vacuously answer YES. Now, for any G ′′ ∈ succ Γ ′ (C) such that G ′′ = / 0 we have req(G ′′ , r ′ ) ≥ |C| > |C| − 1 = b(r ′ ). This means that no goal set in the non-empty set succ Γ ′ (C) respects b which implies that SCRB answers NO. Suppose SCRB answers NO. So ∃ G 0 ∈ succ Γ ′ (C) such that G 0 = / 0 and G 0 respects b. As Γ ′ was obtained from Γ by adding a resource and keeping everything else same, we have G 0 ∈ succ Γ (C) and hence SC answers YES. Problems Related to Resource Conflicts Conflicting Coalitions (CC) When two or more coalitions desire to use some scarce resource, it leads to a conflict in the system. This issue is a classic problem in distributed and concurrent systems. In our framework we say that two goal sets are in conflict w.r.t a resource bound if they are individually achievable within the resource bound but their union is not. Formally a resource bound is a function b : R → N with the interpretation that each coalition has at most b(r) quantity of resource r for every r ∈ R. We say that a goal set G 0 respects a resource bound b w.r.t. a given CRG Γ if and only if ∀ r ∈ R we have b(r) ≥ req(G 0 , r). We denote by cgs (G 1 , G 2 , b) the fact that G 1 and G 2 are in conflict w. r.t b. Formally, cgs(G 1 , G 2 , b) is defined as respects (G 1 , b) ∧ respects(G 2 , b) ∧ ¬respects(G 1 ∪ G 2 , b). Thus we have the CONFLICTING COALITIONS problem: Given coalitions C 1 ,C 2 and a resource bound b answer YES if and only if ∀ G 1 ∈ succ(C 1 ) and ∀ G 2 ∈ succ(C 2 ) we have cgs (G 1 , G 2 , b). Wooldridge et al. [8] show that CC is strongly co-NP-complete. To prove the parameterized hardness results, we give a reduction from SC to CC. Proof. Define R ′ = R ∪ {r ′ } and C ′ 1 = C = C ′ 2 . Let b be a vector with |R ′ | components whose first |R ′ | − 1 entries are ∞ and the last entry is |G| · |C|, i.e., b = {∞, ∞, . . . , ∞, ∞, |G| · |C|}. We extend the en to r ′ as follows: en(i, r ′ ) = |G| for all i ∈ C and en(i, r ′ ) = 0 if i / ∈ C. We extend req to r ′ as follows: req(g, r ′ ) = |C| for all g ∈ G. Now the claim is that SC answers YES if and only if CC answers NO. First we claim that succ Γ (C) = succ Γ ′ (C). We built Γ ′ from Γ by just adding one resource and so clearly succ Γ ′ (C) ⊆ succ Γ (C). Now let G ′′ ∈ succ Γ (C). Then req(G ′′ , r ′ ) = |G ′′ | · |C| ≤ |G| · |C| = en(C, r ′ ) and G ′′ ∈ succ Γ ′ (C). Summarizing we have our claim. Suppose SC answers YES. So, there exists G 0 = / 0 such that G 0 ∈ succ Γ (C). As succ Γ (C) = succ Γ ′ (C) we have G 0 ∈ succ Γ ′ (C). As C ′ 1 = C = C ′ 2 the cgs condition fails for G 1 = G 0 = G 2 and so CC answers NO. Suppose CC answers NO. Claim is that succ Γ ′ (C) = / 0. If not then succ Γ ′ (C ′ 1 ) = / 0 = succ Γ ′ (C ′ 2 ) and in fact CC would have vacuously answered YES. But succ Γ (C) = succ Γ ′ (C) and so succ Γ (C) = / 0. Thus SC answers YES. The Parameter |Ag| + |R| : Case of Bounded Agents plus Resources Considering the results in previous sections, we can see that even in the case that size of coalition or number of resources is bounded the problem still remains computationally hard. So a natural question is what happens if we have a bound on |Ag| + |R| ? Can we do better if total number of agents plus resources is bounded? Shrot et.al [5] show that by this parameterization the problems SC, MAXC and MAXSC have FPT algorithms and they left the corresponding question for the ESCK open. We will generalize the integer program given in Theorem 3.1 of [5], to give a FPT algorithm for the open problem of Existence of Successful Coalition of size k (ESCK). Then by using a similar approach we will design FPT algorithms for the four other problems (NR, SNR, CGRO, SCRB) considered in this paper. The integer program we define is a satisfiability problem (rather than an optimization problem). It consists of a set of constraints, and the question is whether there exists an integral solution to this set. Consider the following integer program (which we will name as IP): ∀i ∈ Ag : ∑ g∈G i x g ≥ y i (1) ∀r ∈ R : ∑ g∈G x g × req(g, r) ≤ ∑ i∈Ag y i × en(i, r)(2) ∀g ∈ G : x g ∈ {0, 1} ∀i ∈ Ag : y i ∈ {0, 1} In this setting, y i = 1, for each i ∈ Ag, represents the situation that the agent i is participating in the coalition and x g = 1, for each g ∈ G, represents the situation that goal g is achieved. The first constraint guarantees that any participating agent has at least one of his goals achieved. The second constraint ensures that the participating agents have enough endowment to achieve all of the chosen goals. It is clear that any solution for this integer program is a coalition of agents and a successful set of goals for that coalition. The above integer program has |Ag|+ |R| constraints and in Flum and Grohe [3] it is shown that checking feasibility of INTEGER LINEAR PROGRAMMING is FPT in the number of constraints or in the number of variables. Now for each of our problems we will add some constraints to get new integer programs which solve those problems. Theorem 9.1. Checking whether there is a Successful Coalition of size k (ESCK) is FPT when parameterized by |Ag| + |R|. Proof. For ESCK, the general integer program given above needs only one additional constraint: We have to ensure that exactly k number of agents will be selected. Therefore adding the constraint ∑ i∈Ag y i = k gives us the integer program for the problem ESCK. The number of constraints, i.e., |Ag| + |R| + 1 for this integer program is |Ag| + |R| + 1 and as INTEGER LINEAR PROGRAMMING is FPT w.r.t number of variables or constraints we have that ESCK parameterized by |Ag| + |R| is FPT. In the problems NR, SNR and CGRO the coalition C is always given. So we will change the variables y i 's to constants where y i = 1 if i ∈ C and 0 otherwise. We call this new integer program a Fixed Coalition Integer Program (FCIP). The coalition C is successful if and only if FCIP is satisfiable. Theorem 9.2. Checking whether the Resource r is Needed for a Coalition C to be Successful (NR) is FPT when parameterized by |Ag| + |R|. Proof. We start with the integer program FCIP. The answer to NR is YES, if and only if in any successful subset of goals, there is at least one goal g with req(g, r) > 0. So we just need to check and see if the coalition is successful by only using the goals which do not need the resource r. Therefore in FCIP, for all goals g ∈ G where req(g, r) > 0 we will set the variable x g to zero. Now the answer to NR is YES iff the resulting integer program is not satisfiable. Note that the number of constraints is still same as previously -|Ag| + |R|. As INTEGER LINEAR PROGRAMMING is FPT wrt number of variables or constraints we have that NR parameterized by |Ag| + |R| is FPT. ∑ g∈G x g × req(g, r) ≤ ∑ i∈Ag y i × en(i, r) (2) ∀g ∈ G : x g ∈ {0, 1} ∀i ∈ Ag : y i ∈ {0, 1} ∀i ∈ Ag : ∑ g∈G i X g ≥ Y i(3) ∀r ∈ R : ∑ g∈G X g × req(g, r) ≤ ∑ i∈Ag Y i × en(i, r) (4) ∀g ∈ G : X g ∈ {0, 1} ∀i ∈ Ag : Y i ∈ {0, 1} In the first sub-program, we set y i = 1 if i ∈ C 1 and 0 otherwise. Then this sub-program finds a goal set G 1 ∈ succ Γ (C 1 ). The second sub-program is similar. Now we add the resource bound conditions : ∀ r ∈ R ∑ g∈G x g × req(g, r) ≤ b(r) ∀ r ∈ R ∑ g∈G X g × req(g, r) ≤ b(r) ∃ r ∈ R s.t. ∑ g∈G x g + X g − x g · X g × req(g, r) > b(r) The first two conditions state that both G 1 and G 2 respect b and the third condition says that G 1 ∪ G 2 does not respect b. However the above program is quadratic due to the last constraint and there is no known result about fixed parameter tractability for quadratic integer programs. Hence we leave open the question about status of CC parameterized by |Ag| + |R|. Revisiting ESCK Parameterized by |G| Shrot et al. [5] show in Theorem 3.2 of their paper that ESCK parameterized by |G| is FPT. We first show their proposed FPT algorithm is wrong by giving an instance when their algorithm gives incorrect answer. Then we show that in fact the problem is para-NP-hard via a reduction from the independent set problem. Counterexample to the Algorithm Given in Theorem 3.2 of Shrot et al. [5] The algorithm is as follows: 1. For each G ′ ⊆ G • Let C ′ be set of all agents satisfied by G ′ • If |C ′ | = k , go to 1. • If G ′ is feasible for C ′ , return TRUE We also found a bug in Theorem 3.2 of [5] which claimed that ESCK parameterized by |G| is FPT. We give a counterexample to their algorithm and in fact show that the problem is para-NP-hard. Then for some problems related to resources, resource bounds and resource conflicts like NR, SNR, CGRO, RPEGS, SCRB and CC we have results when parameterized by various natural parameters like |G|, |C|, |R| and |Ag| + |R| (only CC parameterized by |Ag| + |R| is left open). These results help us to understand better the role of the various components of the input and identify exactly the ones which make the input hard. Since all the problems are known to be FPT when parameterized by |G| and all of them except CC are known to be FPT when parameterized by |Ag| + |R| we know that our problems are tractable when the goal set is small. With this knowledge we can even want to enforce this restriction in real-life situations as much as possible. On the other hand we know that all the problems we considered remain intractable when parameterized by |C| or |R| and hence there is no point in trying to restrict size of coalition or number of resources as it does not make the computation faster The study of problems arising in coalitions of agents in multi-agents systems using the parameterized complexity paradigm was initiated by Shrot et al. [5] In this paper we have tried to take a further step in this direction which we believe is still unexplored. There are various (classically) computationally hard problems which need to be better analyzed through the rich theory of parameterized complexity. Both in Shrot et al. [5] and this paper only the CRG model has been considered. In CRG model the status of CC parameterized by |Ag| + |R| is left open. Alternatively one might consider other natural parameters like |Ag| or try to examine other models like the QCG model [7] through parameterized complexity analysis. Acknowledgments We would like to thank Yuk Hei (Tom) Chan, Dana Nau and Kanthi Sarpatwar for helpful discussions. Definition 3. 3 . 3The class W[1] contains all parameterized problems FPT-reducible (defined hereunder) to Short-Nondeterministic-Turing-Machine-Computation. Definition 3.4. A parameterized problem (Q, κ) is in para-NP if there exists a non-deterministic Turing machine M, constant α and an arbitrary computable function f , such that for any input x, M decides if x ∈ Q in time ≤ |x| α f (κ(x)). Figure 2 : 2Summary of results Theorem 6. 2 . 2The parameterized complexity status of Necessary Resource is as follows :• FPT when parameterized by |G|• co-W[1]-hard when parameterized by |C| • co-para-NP-hard when parameterized by |R| Proof. When parameterized by |G|, we consider all 2 |G| subsets of G. For each subset, we can check in polynomial time if it is a member of succ(C) and if it requires non-zero quantity of the resource given in the input. The other two claims follow from Lemma 6.1, Theorem 3.8 in Shrot et al., and Theorem 5.1. Lemma 7. 1 . 1Given an instance (Γ,C) of SC we can construct an instance (Γ ′ ,C ′ , G 0 ) of RPEGS such that SC answers YES iff RPEGS answers NO. Lemma 7. 3 . 3Given an instance (Γ,C) of SC we can construct an instance (Γ ′ ,C ′ , b') of SCRB such that SC answers YES if and only if SCRB answers NO. Theorem 7. 4 . 4The parameterized complexity status of Successful Coalition With Resource Bound (SCRB) is as follows: • FPT when parameterized by |G| • co-W[1]-hard when parameterized by |C| • co-para-NP-hard when parameterized by |R| Proof. When parameterized by |G|, we consider all 2 |G| subsets of G. For each subset,we can check in polynomial time if it is a member of succ(C) and if it requires non-zero quantity of the resource given in the input. The other two claims follow from Lemma 7.3, Theorem 3.8 in Shrot et al., and Theorem 5.1. Theorem 8. 2 . 2The parameterized complexity status of Conflicting Coalitions (CC) is as follows : • FPT when parameterized by |G| • co-W[1]-hard when parameterized by |C| • co-para-NP-hard when parameterized by |R| Proof. When parameterized by |G|, we consider all 2 |G| choices for G 1 and G 2 . Given a choice (G 1 , G 2 ) we can check in polynomial time if G 1 and G 2 are members of succ(C 1 ) and succ(C 2 ) respectively. Also we can check the condition cgs(G 1 , G 2 , b) in polynomial time. The other two claims follow from Lemma 8.1, Theorem 3.8 in Shrot et al., and Theorem 5.1. Question: Is C successful and every proper superset of C not successful?1. SUCCESSFUL COALITION (SC) Instance: A CRG Γ and a coalition C Question: Is C successful? 2. EXISTS A SUCCESSFUL COALITION OF SIZE k (ESCK) Instance: A CRG Γ and an integer k Question: Does there exist a successful coalition of size exactly k? 3. MAXIMAL COALITION (MAXC) Instance: A CRG Γ and a coalition C Question: Is every proper superset of C not successful? 4. MAXIMAL SUCCESSFUL COALITION (MAXS) Instance: A CRG Γ and a coalition C Proof. When parameterized by |G|, we consider all 2 |G| subsets of G. For each subset, we can check in polynomial time if it is a member of succ(C) and if it requires non-zero quantity of the resource given in the input.The other two claims follow from Lemma 6.3, Theorem 3.8 in Shrot et al., and Theorem 5.1.Theorem 6.4. The parameterized complexity status of Strictly Necessary Resource is as follows : • FPT when parameterized by |G| • W[1]-hard when parameterized by |C| • para-NP-hard when parameterized by |R| Proof. When parameterized by |G|, we consider all 2 |G| subsets of G. For each subset, we can check in polynomial time if it is a member of succ(C) and if it shows that G 0 is not R-Pareto Efficient.The other two claims follow from Lemma 7.1, Theorem 3.8 in Shrot et al., and Theorem 5.1.Theorem 7.2. The parameterized complexity status of R-Pareto Efficient Goal Set is as follows : • FPT when parameterized by |G| • co-W[1]-hard when parameterized by |C| • co-para-NP-hard when parameterized by |R| Theorem 9.3. Checking whether the Resource r is Strictly Needed for a Coalition C to be Successful (SNR)is FPT when parameterized by |Ag| + |R|. ∀i ∈ Ag :∑ g∈G i x g ≥ y i (1) ∀r ∈ R : Proof. We start with the integer program FCIP. Since SNR answers NO when the coalition is not successful, we should first check if the coalition is successful. Therefore we will check the answer to FCIP. If it is not satisfiable, then the answer for SNR would be NO. But if FCIP is satisfiable, i.e., succ(C) = / 0, then we just need to check and see if the coalition is successful by only using the goals which do not need the resource r. Again with the same approach as the proof of Theorem 9.2, for all goals g ∈ G where req(g, r) > 0 we will set the variable x g to zero. Now the answer to SNR is YES iff the resulting IP is not satisfiable. Note that the number of constraints is still same as previously -|Ag| + |R|. As INTEGER LINEAR PROGRAMMING is FPT w.r.t number of variables or constraints we have that SNR parameterized by |Ag| + |R| is FPT.Theorem 9.4. Checking whether the successful goal set G 0 has optimal usage of Resource r for a Coalition C (CGRO) is FPT when parameterized by |Ag| + |R|.Proof. We start with the integer program FCIP. The limit on usage of resource r is req(G 0 , r). Let β = req(G 0 , r) be the limit. So the answer for CGRO is YES iff there is no successful set of goals G ′ with req(G ′ , r) < β . So by adding the constraint ∑ g∈G x g × req(g, r) < β to FCIP, the answer for CGRO would be YES iff the resulting IP is not satisfiable. Note that the number of constraints |Ag| + |R| + 1. As INTEGER LINEAR PROGRAMMING is FPT w.r.t number of variables or constraints we have that CGRO parameterized by |Ag| + |R| is FPT.Theorem 9.5. Checking whether a given coalition C is Successful by respecting the Resource Bound b (SCRB) is FPT when parameterized by |Ag| + |R|.Proof. We start with the integer program FCIP. Now the limit on usage of any resource r ∈ R is b(r). So for every resource r ∈ R we will bound its usage by adding the constraint ∑ g∈G x g × req(g, r) ≤ b(r) to FCIP. Now the answer for SCRB would be YES if and only if the resulting integer program is satisfiable. Note that number of constraints now is |Ag| + 2|R| and INTEGER LINEAR PROGRAMMING is FPT w.r.t number of variables or constraints we have that SCRB is FPT w.r.t |Ag| + 2|R| and hence wrt |Ag| + |R|.Theorem 9.6. Checking whether a given goal set G 0 is R-Pareto Efficient (RPEGS) is FPT when parameterized by |Ag| + |R|.Proof. As in the proof of Theorem 9.5, set the variables y i = 1 if i ∈ C and 0 otherwise. The answer for the problem is NO if there exists a successful G ′ such that there is a resource r ∈ R such that req(G ′ , r) < req(G 0 , r) and for every other resource r ′ ∈ R we have req(G ′ , r ′ ) ≤ req(G 0 , r ′ ). Since G 0 is given, req(G 0 , r) is a constant. So we can write |R| IPs, such that in the integer program for the resource r, we have the constraint req(G ′ , r) < req(G 0 , r), and |R| − 1 constraints req(G ′ , r ′ ) ≤ req(G 0 , r ′ ), one for each resource r ′ = r. Now the answer for RPEGS would be YES iff all |R| integer programs are not satisfiable. Note that the number of constraints in each of the integer programs is |Ag| + 2|R| and INTEGER LINEAR PROGRAMMING is FPT w.r.t number of variables or constraints we have that RPEGS is FPT w.r.t |Ag| + 2|R| and hence w.r.t |Ag| + |R|. Now we give an integer quadratic program for the CC problem : each resource, each goal requires 0 of each resource, and G i = G for all agents i ∈ Ag. Thus each coalition is successful and ∀ G ′ ⊆ G we have C ′ = Ag which means that |C ′ | = |Ag| > k and so the algorithm answers NO while the correct answer is YES. Indeed by reducing Independent Set to a CRG instance with |G| = 1, we prove the following theoremeach resource, each goal requires 0 of each resource, and G i = G for all agents i ∈ Ag. Thus each coalition is successful and ∀ G ′ ⊆ G we have C ′ = Ag which means that |C ′ | = |Ag| > k and so the algorithm answers NO while the correct answer is YES. Indeed by reducing Independent Set to a CRG instance with |G| = 1, we prove the following theorem. Theorem 10.1. ESCK parameterized by |G| is para-NP-hard. Theorem 10.1. ESCK parameterized by |G| is para-NP-hard. } , E = {e, We prove this by reduction from INDEPENDENT SET to a CRG with |G| = 1. Let H = (V, E) be a given graph and let k be the given parameter. Let V = {v 1. We build an instance (Γ, k) of ESCK where • Ag = {a 1Proof. We prove this by reduction from INDEPENDENT SET to a CRG with |G| = 1. Let H = (V, E) be a given graph and let k be the given parameter. Let V = {v 1 , v 2 , . . . , v n } and E = {e 1 , e 2 , . . . , e m }. We build an instance (Γ, k) of ESCK where • Ag = {a 1 , a 2 , . . . , a n } . . . • R = {r 1 , R 2, • R M } • G = {g} • G I = G ∀ I ∈ Ag, Req, r j ) = k − 1 ∀ j ∈ [m• R = {r 1 , r 2 , . . . , r m } • G = {g} • G i = G ∀ i ∈ Ag • req(g, r j ) = k − 1 ∀ j ∈ [m] H has an independent set of size k say I = {v β 1 , v β 2 , . . . , v β k }. Consider the following coalition of size k: C = {a β 1 , a β 2 , . . . , a β k }. Clearly the goal set {g} is satisfying for C. Also, as I is independent set the number of vertices from I incident on any r j is atmost 1. So ∀ j ∈ [m] we have req(g, r j ) = k − 1 ≤ en(C, r j ) and so {g} is feasible for C which means that C is successful coalition. As |C| = k we have that ESCK answers YES. Suppose that ESCK answers YES. So there exists a successful coalition of size k in Γ say C = {a β 1. • en(a i , r j ) = 0 if v i and e j are incident and 1 otherwise We now claim that INDEPENDENT SET answers YES if and only if ESCK answers YES. Suppose INDEPENDENT SET answers YES, i.e. Consider the set of vertices I = {v β 1 , v β 2 , . . . , v β k } in V . We claim that it is an independent set. v β l ∈ I. Then we have en(C, r j ) ≤ k − 2 < k − 1 = req(g, r j ) which contradicts the fact that C is successful (Since G = {g} the only possible goal set is {g}• en(a i , r j ) = 0 if v i and e j are incident and 1 otherwise We now claim that INDEPENDENT SET answers YES if and only if ESCK answers YES. Suppose INDEPENDENT SET answers YES, i.e., H has an independent set of size k say I = {v β 1 , v β 2 , . . . , v β k }. Consider the following coalition of size k: C = {a β 1 , a β 2 , . . . , a β k }. Clearly the goal set {g} is satisfying for C. Also, as I is independent set the number of vertices from I incident on any r j is atmost 1. So ∀ j ∈ [m] we have req(g, r j ) = k − 1 ≤ en(C, r j ) and so {g} is feasible for C which means that C is successful coalition. As |C| = k we have that ESCK answers YES. Suppose that ESCK answers YES. So there exists a successful coalition of size k in Γ say C = {a β 1 , a β 2 , . . . , a β k }. Consider the set of vertices I = {v β 1 , v β 2 , . . . , v β k } in V . We claim that it is an independent set. Suppose not and let e j be an edge joining v β i and v β l such that v β l , v β l ∈ I. Then we have en(C, r j ) ≤ k − 2 < k − 1 = req(g, r j ) which contradicts the fact that C is successful (Since G = {g} the only possible goal set is {g}). Therefore I is an independent set and as |I| = k we have that INDEPENDENT SET answers YES. Note that as |G| = 1 in our CRG Γ and INDEPENDENT SET is known to be NP-hard we have that ESCK. parameterized by |G| is para-NP-hardTherefore I is an independent set and as |I| = k we have that INDEPENDENT SET answers YES. Note that as |G| = 1 in our CRG Γ and INDEPENDENT SET is known to be NP-hard we have that ESCK parameterized by |G| is para-NP-hard. SC parameterized by |C| is W[1]-hard 2. ESCK parameterized by |Ag| + |R| is FPT 3. ESCK parameterized by |R| is para-NP-hard. SC parameterized by |C| is W[1]-hard 2. ESCK parameterized by |Ag| + |R| is FPT 3. ESCK parameterized by |R| is para-NP-hard Complexity of constructing solutions in the core based on synergies among coalitions. V Conitzer, T Sandholm, Artificial Intelligence. 1706-7V. Conitzer, T. Sandholm, Complexity of constructing solutions in the core based on synergies among coalitions, Artificial Intelligence 170 (6-7) (2006) 607-619. Parameterized complexity for the skeptic. R Downey, Proc. 18th IEEE Annual Conference on Computational Complexity. 18th IEEE Annual Conference on Computational ComplexityR. Downey, Parameterized complexity for the skeptic, in: In Proc. 18th IEEE Annual Conference on Computational Complexity, 2003. Parameterized Complexity Theory (Texts in Theoretical Computer Science. J Flum, M Grohe, An EATCS Series). Springer-Verlag New York, IncJ. Flum, M. Grohe, Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series), Springer-Verlag New York, Inc., 2006. Computers and Intractability: A Guide to the Theory of NP-Completeness. M R Garey, D S Johnson, M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, 1979. Easy and hard coalition resource game formation problems: a parameterized complexity analysis. T Shrot, Y Aumann, S Kraus, AAMAST. Shrot, Y. Aumann, S. Kraus, Easy and hard coalition resource game formation problems: a parame- terized complexity analysis, in: AAMAS (1), 2009. An Introduction to Multiagent Systems. M Wooldridge, WileyChichester, UK2nd ed.M. Wooldridge, An Introduction to Multiagent Systems, 2nd ed., Wiley, Chichester, UK, 2009. M Wooldridge, P E Dunne, On the computational complexity of qualitative coalitional games. 158M. Wooldridge, P. E. Dunne, On the computational complexity of qualitative coalitional games, Artifi- cial Intelligence 158 (1) (2004) 27-73. On the computational complexity of coalitional resource games. M Wooldridge, P E Dunne, Artificial Intelligence. 17010M. Wooldridge, P. E. Dunne, On the computational complexity of coalitional resource games, Artificial Intelligence 170 (10) (2006) 835-871.

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{'abstract': 'Coalition formation is a key topic in multi-agent systems. Coalitions enable agents to achieve goals that they may not have been able to achieve on their own. Previous work has shown problems in coalition games to be computationally hard. Wooldridge and Dunne (Artificial Intelligence 2006) studied the classical computational complexity of several natural decision problems in Coalitional Resource Games (CRG) -games in which each agent is endowed with a set of resources and coalitions can bring about a set of goals if they are collectively endowed with the necessary amount of resources. The input of coalitional resource games bundles together several elements, e.g., the agent set Ag, the goal set G, the resource set R, etc. Shrot, Aumann and Kraus (AAMAS 2009) examine coalition formation problems in the CRG model using the theory of Parameterized Complexity. Their refined analysis shows that not all parts of input act equal -some instances of the problem are indeed tractable while others still remain intractable.We answer an important question left open by Shrot, Aumann and Kraus by showing that the SC Problem (checking whether a Coalition is Successful) is W[1]-hard when parameterized by the size of the coalition. Then via a single theme of reduction from SC, we are able to show that various problems related to resources, resource bounds and resource conflicts introduced by Wooldridge et al are 1. W[1]-hard or co-W[1]-hard when parameterized by the size of the coalition.2. para-NP-hard or co-para-NP-hard when parameterized by |R|.3. FPT when parameterized by either |G| or |Ag| + |R|.', 'arxivid': '1105.0707', 'author': ['Rajesh Chitnis rchitnis@cs.umd.edu \nDepartment of Computer Science\nUniversity of Maryland at College Park\nUSA\n', 'Mohammadtaghi Hajiaghayi hajiagha@cs.umd.edu \nDepartment of Computer Science\nUniversity of Maryland at College Park\nUSA\n', 'Vahid Liaghat vliaghat@cs.umd.edu \nDepartment of Computer Science\nUniversity of Maryland at College Park\nUSA\n'], 'authoraffiliation': ['Department of Computer Science\nUniversity of Maryland at College Park\nUSA', 'Department of Computer Science\nUniversity of Maryland at College Park\nUSA', 'Department of Computer Science\nUniversity of Maryland at College Park\nUSA'], 'corpusid': 8750578, 'doi': '10.1609/aaai.v25i1.7887', 'github_urls': [], 'n_tokens_mistral': 17007, 'n_tokens_neox': 15841, 'n_words': 9940, 'pdfsha': '909bd025955e77340cf2ed89964501e046f7cfb9', 'pdfurls': ['https://arxiv.org/pdf/1105.0707v1.pdf'], 'title': ['Parameterized Complexity of Problems in Coalitional Resource Games *', 'Parameterized Complexity of Problems in Coalitional Resource Games *'], 'venue': []}

arxiv

Transverse-Momentum-Dependent Wave Functions of Pion from Lattice QCD Min-Huan Chu School of Physics and Astronomy Shanghai Key Laboratory for Particle Physics and Cosmology Key Laboratory for Particle Astrophysics and Cosmology (MOE) INPAC Shanghai Jiao Tong University 200240ShanghaiChina Yang Yuanqing Scientific Computering Center Tsung-Dao Lee Institute Shanghai Jiao Tong University 200240ShanghaiChina Jin-Chen He School of Physics and Astronomy Shanghai Key Laboratory for Particle Physics and Cosmology Key Laboratory for Particle Astrophysics and Cosmology (MOE) INPAC Shanghai Jiao Tong University 200240ShanghaiChina Department of Physics University of Maryland 20742College ParkMDUSA Jun Hua Guangdong Provincial Key Laboratory of Nuclear Science Institute of Quantum Matter South China Normal University 510006GuangzhouChina Guangdong-Hong Kong Joint Laboratory of Quantum Matter Southern Nuclear Science Computing Center South China Normal University 510006GuangzhouChina Jian Liang Guangdong Provincial Key Laboratory of Nuclear Science Institute of Quantum Matter South China Normal University 510006GuangzhouChina Guangdong-Hong Kong Joint Laboratory of Quantum Matter Southern Nuclear Science Computing Center South China Normal University 510006GuangzhouChina Xiangdong Ji Department of Physics University of Maryland 20742College ParkMDUSA Andreas Schäfer Institut für Theoretische Physik Universität Regensburg D-93040RegensburgGermany Hai-Tao Shu Institut für Theoretische Physik Universität Regensburg D-93040RegensburgGermany Yushan Su Department of Physics University of Maryland 20742College ParkMDUSA Ji-Hao Wang Institute of Theoretical Physics CAS Key Laboratory of Theoretical Physics Chinese Academy of Sciences 100190BeijingChina School of Fundamental Physics and Mathematical Sciences Hangzhou Institute for Advanced Study UCAS 310024HangzhouChina Wei Wang School of Physics and Astronomy Shanghai Key Laboratory for Particle Physics and Cosmology Key Laboratory for Particle Astrophysics and Cosmology (MOE) INPAC Shanghai Jiao Tong University 200240ShanghaiChina Southern Center for Nuclear-Science Theory (SCNT) Institute of Modern Physics Guangdong Province Chinese Academy of Sciences 516000HuizhouChina Yi-Bo Yang Institute of Theoretical Physics CAS Key Laboratory of Theoretical Physics Chinese Academy of Sciences 100190BeijingChina School of Fundamental Physics and Mathematical Sciences Hangzhou Institute for Advanced Study UCAS 310024HangzhouChina International Centre for Theoretical Physics Asia-Pacific Beijing/HangzhouChina Jun Zeng School of Physics and Astronomy Shanghai Key Laboratory for Particle Physics and Cosmology Key Laboratory for Particle Astrophysics and Cosmology (MOE) INPAC Shanghai Jiao Tong University 200240ShanghaiChina School of Physical Sciences University of Chinese Academy of Sciences 100049BeijingChina Jian-Hui Zhang School of Science and Engineering The Chinese University of Hong Kong 518172ShenzhenChina Center of Advanced Quantum Studies Department of Physics Beijing Normal University 100875BeijingChina Qi-An Zhang School of Physics Beihang University 102206BeijingChina Transverse-Momentum-Dependent Wave Functions of Pion from Lattice QCD (Lattice Parton Collaboration (LPC)) We present a first lattice QCD calculation of the transverse-momentum-dependent wave functions (TMDWFs) of the pion using large-momentum effective theory. Numerical simulations are based on one ensemble with 2+1+1 flavors of highly improved staggered quarks action with lattice spacing a = 0.121 fm from the MILC Collaboration, and one with 2 +1 flavor clover fermions and tree-level Symanzik gauge action generated by the CLS Collaboration with a = 0.098 fm. As a key ingredient, the soft function is first obtained by incorporating the one-loop perturbative contributions and a proper normalization. Based on this and the equal-time quasi-TMDWFs simulated on the lattice, we extract the light-cone TMDWFs. The results are comparable between the two lattice ensembles and a comparison with phenomenological parametrization is made. Our studies provide a first attempt of ab initio calculation of TMDWFs which will eventually lead to crucial theory inputs for making predictions for exclusive processes under QCD factorization.Introduction: The light-front wave functions (LFWFs) are an important quantity for hadrons in particle physics. They characterize the nonperturbative structure of hadrons, and enter the prediction of a wide variety of measurable observables through quantum chromodynamics (QCD) factorization. While searching for new physics beyond the standard model (SM) requires a dedicated study of high-energy processes at colliders, this goal can partially be achieved by investigating low-energy processes, among which the flavor-changingneutral-current (FCNC) in a heavy quark system is an ideal probe[1]. A key input of calculating the SM contributions to the FCNC are LFWFs, including We present a first lattice QCD calculation of the transverse-momentum-dependent wave functions (TMDWFs) of the pion using large-momentum effective theory. Numerical simulations are based on one ensemble with 2+1+1 flavors of highly improved staggered quarks action with lattice spacing a = 0.121 fm from the MILC Collaboration, and one with 2 +1 flavor clover fermions and tree-level Symanzik gauge action generated by the CLS Collaboration with a = 0.098 fm. As a key ingredient, the soft function is first obtained by incorporating the one-loop perturbative contributions and a proper normalization. Based on this and the equal-time quasi-TMDWFs simulated on the lattice, we extract the light-cone TMDWFs. The results are comparable between the two lattice ensembles and a comparison with phenomenological parametrization is made. Our studies provide a first attempt of ab initio calculation of TMDWFs which will eventually lead to crucial theory inputs for making predictions for exclusive processes under QCD factorization. Introduction: The light-front wave functions (LFWFs) are an important quantity for hadrons in particle physics. They characterize the nonperturbative structure of hadrons, and enter the prediction of a wide variety of measurable observables through quantum chromodynamics (QCD) factorization. While searching for new physics beyond the standard model (SM) requires a dedicated study of high-energy processes at colliders, this goal can partially be achieved by investigating low-energy processes, among which the flavor-changingneutral-current (FCNC) in a heavy quark system is an ideal probe [1]. A key input of calculating the SM contributions to the FCNC are LFWFs, including the collinear distribution amplitudes (LCDAs) and the transverse-momentum-dependent wave functions (TMD-WFs). LFWFs in fact play an essential role in lightfront quantization. In particular, the parton distribution functions can be expressed in terms of the square of the TMDWFs [2,3]. The TMDWFs are characterized by physics at distance scale of a fermi or equivalently momentum scale of a few hundred MeV, which are similar to the confinement scale. Experimental mappings and theoretical computations of these distributions may help to reveal the nature of non-perturbative phenomena such as confinement and chiral symmetry breaking in QCD. Although TMDWFs describe important aspects of the arXiv:2302.09961v1 [hep-lat] 20 Feb 2023 three-dimensional structure of hadrons, they have never been studied in the literature from the first principles of QCD with systematic approximation. Similar with transverse momentum dependent parton distribution functions (TMDPDFs), it is nontrivial to present a rigorous definition of TMDWFs [4]. A key difficulty resides in the rapidity divergences that show up in regularizing the soft contributions from a collinear constituent [5]. Therefore, most applications of TMD factorization to hard exclusive processes have adopted phenomenological models to parametrize the TMDWFs [6][7][8], which inevitably introduce uncontrollable systematic uncertainties and challenge the precision tests of the SM and probes for new physics. Large-momentum effective theory (LaMET) [9, 10] develops a novel way to extract parton physics from the lattice QCD calculations through expansion in large hardon momentum (see [11] for a review and many references therein). For TMDWFs, the calculation requires the knowledge on the so-called soft function, which incorporates the effects of soft gluon radiation from colored collinear particles from two opposite light-like directions [12,13]. It was recently discovered that the soft function can be determined by calculating a largemomentum-transfer form factor of a light meson and quasi TMDWFs on the lattice [14,15], which removes the obstacle in calculating the TMDWFs from the lattice QCD [11,16]. In this Letter, we report a first lattice QCD calculation of the pion TMDWFs using LaMET. The calculation is performed on two lattice ensembles with three hadron momenta up to 2.63 GeV. We obtain the soft function by incorporating the one-loop perturbative contributions and a proper normalization. Based on this, we present first results for the physical TMDWFs. Comparable behaviors between the two lattice ensembles are found and a comparison with the phenomenological model is shown. Theoretical Framework: The TMDWF Ψ ± (x, b ⊥ , µ, ζ) provides the momentum distribution between the quark and antiquark in its leading Fock state. The superscript "±" denotes that in Ψ ± Wilson lines will approach the positive and negative infinity along the lightcone direction. x denotes the momentum fraction in longitudinal direction, and b ⊥ is the Fourier conjugate of transverse momentum. In addition, TMDWFs also depend on the renormalization scale µ and the rapidity scale ζ. LaMET allows to access the TMDWF Ψ ± by simulating an equal-time quasi-TMDWFΨ ± defined in Euclidean space. The relation between them follows the factorization formula [15,16]: where ζ z = (2P z ) 2 . S I (b ⊥ , µ) denotes the intrinsic soft function, K (b ⊥ , µ) is the Collins-Soper kernel and has been calculated on the lattice in [17][18][19]. H ± (x, ζ z , µ) represents a perturbative matching kernel. At one-loop level it is given by [16,20]: Ψ ± (x, b ⊥ , µ, ζ z ) S 1 2 I (b ⊥ , µ) = H ± (x, ζ z , µ) e 1 2 K(b ⊥ ,µ) ln ∓ζ z +i ζ Ψ ± (x, b ⊥ , µ, ζ) + O Λ 2 QCD /(x 2 ζ z ), M 2 /(P z ) 2 , 1/(b 2 ⊥ ζ z ) (1)q (zn z + b ⊥n ⊥ ) q(0)n ⊥ tn z (L + z,0) (−L,0) 2L + | z | b ⊥ P zH ± (x, ζ z , µ) = 1 + α s C F 4π − 5π 2 6 − 4 + l ± +l ± − 1 2 l 2 ± +l 2 ± ,(2)where l ± = ln[(−x 2 ζ z ± i )/µ 2 ] andl ± = ln[(−x 2 ζ z ± i )/µ 2 ]. x andx = 1 − x are the momentum fractions of quark and antiquark. Power corrections in LaMET factorization are generically suppressed by factors Λ 2 QCD /(x 2 ζ z ), M 2 / (P z ) 2 , 1/(b 2 ⊥ ζ z ) . In Euclidean lattice, the equal-time quasi-TMDWF in momentum spaceΨ ± (x, b ⊥ , µ, ζ z ) can be constructed with a large P z meson-to-vacuum matrix element of a nonlocal billinear operator for the pseudoscalar meson: Ψ ± (x, b ⊥ , µ, ζ z ) = lim L→∞ 1 −if π P z dzP z 2π e ixzP z × 0 |q (zn z + b ⊥n⊥ ) γ t γ 5 U c± q(0)| π (P z ) Z E (2L + |z|, b ⊥ , µ)Z O (1/a, µ) ,(3) where we choose γ t γ 5 to project onto the leading-twist TMDWF. The staple-shaped Wilson line between the quark fields U c± is required as: U c± = U † z (zn z + b ⊥n⊥ ; −L ± )U ⊥ (L ±nz + zn z ; b ⊥ ) × U z (0n z ;L ± + z),(4) where U µ (x; l) ≡ U µ (x, x + ln µ ) andL ± ≡ ±max(L, L ∓ z), see Fig. 1. L is the length of path-ordered Euclidean Wilson lines along the z-direction which will take the L → ∞ limit. But in lattice calculation, one can adopt a sufficiently large L. Based on the discussion in [18], we adopt L 0.7 fm in our lattice simulation. The bare matrix element in the numerator in Eq. (3) contains both pinch pole singularity and linear divergence which can be removed by the Wilson loop Z E (2L + |z|, b ⊥ , µ) [15]. The logarithmic divergences arising from the endpoints of the Wilson line need an additional quark Wilson line vertex renormalization factor Z O (1/a, µ). A straightforward way to determine Z O is to evaluate the quotient of the renormalized quasi-TMDWF calculated on the lattice in the small b ⊥ region and the quasi-TMDWF perturbatively calculated in MS scheme, as discussed in [21]. In practice, we adopt Z O = {0.917(2), 0.903(2)} for MILC and CLS ensembles, for details see the Supplemental Material [22]. Lattice simulation: We use one ensemble of the HYPsmeared clover valence fermions action on 2+1+1 flavors of highly improved staggered sea quarks (HISQ) [23] generated by MILC [24] at the lattice spacing a = 0.121 fm, and one ensemble of 2+1 flavors clover fermions generated by the CLS Collaboration at a = 0.098 fm with the unitary valence fermion action. The rest of the simulation setups are collected in Table. I. To improve the signal-to-noise ratio, we adopt hypercubic (HYP) smeared fat links [25] for the staple-shaped gauge link U c± , and generate the Coulomb gauge fixed wall source propagators S w to build correlation functions. To access the large-momentum limit, we employ three different hadron momenta P z = 2π/n s × {4, 5, 6} = {1.72, 2.15, 2.58} GeV for the MILC ensemble and P z = 2π/n s × {6, 8, 10} = {1.58, 2.11, 2.64} GeV for the CLS ensemble. To determine the quasi-TMDWF, one can construct the non-local two point correlation function as follows: C ± 2 (L, z, b ⊥ , t, P z ) = x e iP z x·nz S † w ( x + zn z + b ⊥n⊥ , t)U c± S w ( x, t) (5) Due to the limited L in lattice simulation discussed in Eq. (4), we adopt (z > 0) for C + 2 and (z < 0) for C − 2 in numerical practice, while the remainder can be obtained by isospin symmetry. Such a symmetry behavior in quasi-TMDWF for ±z have been numerically shown in [18]. The ground-state contribution to the quasi-TMDWF can be extracted by the following two-state fit parametrization: 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Re[Ψ − (x, b ⊥ , µ, ζ z )], P z =2.15 GeV b ⊥ =2a b ⊥ =3a b ⊥ =4a 0.0 0.2 0.4 0.6 0.8 1.0 x −0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Im[Ψ − (x, b ⊥ , µ, ζ z )], P z =2.15 GeV b ⊥ =2a b ⊥ =3a b ⊥ =4aC ± 2 (L, z, b ⊥ , t, P z ) C ± 2 (L, z = 0, b ⊥ = 0, t, P z ) =Ψ ±,0 (z, b ⊥ , ζ z , L) 1 + c 0 (z, b ⊥ , P z , L)e −∆Et 1 + c 1 e −∆Et ,(6) whereΨ ±,0 (z, b ⊥ , ζ z , L) is the bare quasi-TMDWF in coordinate space, while c 0,1 and ∆E are free parameters accounting for excited state contamination. In the large t limit, this contamination is suppressed exponentially, which gives the possibility to extract the quasi-TMDWF through a one-state parametrization. Based on the comparison of one-and two-state fits in Supplemental Material [22], we find that the one-state fit gives a more stable result which will be used in the following analysis. Numerical results: After renormalization by Wilson loop Z E and quark Wilson line vertex correction Z O referring to Eq. (3), the quasi-TMDWF in coordinate space can be obtained straightforwardly. As discussed in a hybrid scheme [26], in the Fourier transformation, a = 1/b * ⊥ varying in the range b * ⊥ ∈ 1/ √ 2, √ 2 b ⊥ . The label ± in S lat,1 loop± represents the lattice results extracted byΨ ± . brute-force truncation at finite z will introduce unphysical oscillations. To avoid these oscillations, we adopt an analytical extrapolation at large light front (LF) distance (λ = zP z ) for quasi-TMDWF in coordinate space: Ψ(z, b ⊥ , µ, ζ z ) = f (b ⊥ ) k 1 (−iλ) d + e iλ k 2 (iλ) d e − λ λ 0 ,(7) where k 1,2 , d are free parameters, λ 0 denotes a large distance parameter [26,27], and the complex parameter f (b ⊥ ) describes the behavior in transverse direction. After extrapolation and Fourier transformation we get the results shown in Fig. 2 for the real part (upper panel) and the imaginary part (lower panel) of the quasi-TMDWF in momentum space at P z = 2.15 GeV on MILC ensemble. For b ⊥ = 1a, there might be sizable discretization effects, and thus we show only the quasi-TMDWF with b ⊥ = {2, 3, 4}a. As can be seen from this figure, the real part decreases slowly with the increasing b ⊥ , while the imaginary part increases rapidly with b ⊥ . Unlike the one dimensional quasi distribution amplitude in [27], the quasi-TMDWF has a sizable nonzero imaginary part. According to the LaMET factorization in Eq. (1), apart from the quasi-TMDWF, one requires the intrinsic soft function and Collins-Soper (CS) evolution kernel to obtain the TMDWF. In recent years, the CS kernel has been determined on the lattice [17][18][19]. A recent analysis on MILC ensemble at 0.121 fm that includes the oneloop perturbative contributions can be found in Ref. was performed in [28,29]. Inspired by a detailed theoretical analysis on normalization condition and twist combination of the form factor in [20], we present the intrinsic soft function in Fig. 3 that is based on the one-loop matching kernel. As can be seen from this figure, the intrinsic soft functions extracted byΨ + andΨ − on MILC ensemble are consistent with each other, which is in line with the expectation that the intrinsic soft function is universal. The result obtained fromΨ − on CLS ensemble is similar but decreases more slowly than the MILC results. A potential reason for this difference might be the discretization effects, which will be further investigated in future work. Our lattice results have similar b ⊥ dependence as one-loop perturbative result in the MS scheme [30] in both the small and large b ⊥ regions. However, it is necessary to point out that the one-loop perturbative result might be unreliable at large b ⊥ . Together with the quasi-TMDWF, one-loop intrinsic soft function and CS kernel, the TMDWF can be obtained through a perturbative matching, see Eq.(1). In (1), the endpoint region suffers from sizable higher power corrections. With a rough estimation [31] λ 10, we conclude that the shaded regions (x < 0.1 and x > 0.9) cannot be reliably controlled in LaMET at present. In Fig. 5, we show a comparison of TMDWFs Ψ ± at the momentum fraction x = 0.5 on MILC ensemble and CLS ensemble with a phenomenological model [32], which factorizes TMDWF into longitudinal and transverse momentum distributions. The TMDWFs decay with increasing b ⊥ , which is consistent with the phenomenological model. However, the phenomenological parametrization only contains the real parts and does not include the difference of Wilson line directions in Eq. (4). The non-zero imaginary part may introduce additional complexity to phenomenological applications which has not been discussed in previous analyses. Ψ(x =0.5, b ⊥ ) phenomenology model Re[Ψ + (x, b ⊥ )] on MILC Re[Ψ − (x, b ⊥ )] on MILC Re[Ψ − (x, b ⊥ )] on CLS Our numerical results are based on different discretizations and lattice spacings, thus their difference can be considered as an estimate of the discretization error before further studies at smaller lattice spacings. Besides, our lattice simulations are performed on pion mass around 670 MeV, which is far from the physical point. Therefore, our results are still subject to large systematic uncertainties, and future calculations with smaller lattice spacings and lighter quark masses can significantly improve them. Summary: We present a first lattice calculation of the transverse momentum dependent wave function of the pion. Numerical simulations are conducted on two ensembles by the MILC and CLS collaborations. The linear and logarithmic divergences are cancelled by Wilson loop and quark Wilson line vertex correction. The extrapolation strategy for quasi-TMDWF in coordinate space follows the hybrid scheme. The final results of TMDWFs extracted from both two ensembles have a consistent b ⊥ dependence, with some differences at small b ⊥ which would come from discretization errors. These results provide a first attempt of ab initio calculation for TMDWFs which will eventually lead to crucial theory inputs for making predictions for exclusive processes under QCD factorization. Acknowledgement: We thank the CLS Collaboration for sharing the ensembles used to perform this study. We thank Wolfgang Söldner for valuable discussions on the X650 ensemble. This work is supported in part by Natural Science FIG. 6. One-and two-state fit for C R+ 2 (L = 6a, z, b ⊥ , P z , t) at {L, z, b ⊥ , P z } = {6a, 3a, 4a, 2.15 GeV} on MILC ensemble. The red/blue curve corresponds to the two-state fits of the real/imaginary part of C R+ 2 , and the color labeled horizontal bars are the extracted ground-state contributions from oneand two-state fits. Fits for two point functions In lattice simulations, quasi-TMDWFs can be extracted from the two-point correlation functions C ± 2 as shown in Eq. (6) of the main text. C ± 2 consists of ground-state contributions and excited-state contaminations, thus one can adopt a two-state fit to separate the ground-state contribution to quasi-TMDWF. In addition, when t becomes large, the excited-state contamination decreases exponentially, so a one-state analysis also allows to extract the ground-state contribution. Fig. 6 shows a comparison of one-and two-state fits for the renormalized C R+ 2 at {L, z, b ⊥ } = {6, 3, 4}a on MILC ensemble, in which they give consistent results. However, the stability of a two-state fit relies strongly on the size of the excited-state contribution, which requires very high precision lattice data. Therefore we employ a one-state fit in the range t ≥ 3a to provide reasonable and stable uncertainties in the following analysis. Renormalization The linear divergence and pinch pole singularity in quasi-TMDWFs can be removed by the Wilson loop Z E (L E , b ⊥ ), in which L E ≡ 2L + |z|. In lattice simulations, the statistical uncertainty of Z E for large L E and b ⊥ is out of control as shown in Fig. 7. Fortunately, the self-energy corrections and gluon exchanging effect introduce linear divergence as exponential form proportional to L E [14,33,34]. Therefore one can adopt an extrapolation for the Wilson loop by a two-state fit via the following equation: Z E (L E , b ⊥ ) on MILC b ⊥ =0.12fm b ⊥ =0.24fm b ⊥ =0.36fm b ⊥ =0.48fm b ⊥ =0.6fmZ E (L E , b ⊥ ) = c 0 (b ⊥ )e −E(b ⊥ )L E 1 + c 1 (b ⊥ )e −∆E(b ⊥ )L E . (8) As shown in Fig. 7, taking the result on MILC ensemble as an example, one can see that the extrapolated data is in line with the original data. Moreover, to cancel the logarithmic divergence arising from the endpoints of the Wilson lines, we need an additional quark Wilson line vertex renormalization factor Z O . According to Ref. [21], this factor Z O can be computed from the quotient of renormalized quasi-TMDWF in the rest frame calculated on the lattice and perturbatively: Z O (1/a, µ) =Ψ ±,0 (z 0 , b ⊥0 , ζ z = 0, L) Z E (2L + |z 0 |, b ⊥0 , µ)ψ MS (z 0 , b ⊥0 , µ) .(9) Ψ ±,0 denotes the bare quasi-TMDWF in coordinate space. The perturbative quasi-TMDWFψ MS in the MS scheme in the denominator has been recently calculated in [21]. In our analysis, we adopt a short distance region for z 0 and b ⊥0 matching with perturbative calculation. We look for a window of b ⊥0 where both discretization effects and higher twist contaminations are negligible. In such a window, Z O should have only a mild dependence on b ⊥0 . These dependences are investigated in Fig. 8. As shown in the figure, Z O (z 0 = 0a, b ⊥0 = 2a) and Z O (z 0 = 0a, b ⊥0 = 3a) reach a b ⊥0 window for both MILC and CLS ensembles. Such a window is also visible at z 0 = 1a, but becomes invisible as z 0 increases. Z O (z 0 = 0a, b ⊥0 = 0a) may suffer from discretization effects since the b ⊥0 → 0 and a → 0 limits do not commute [26]. Thus, Z O is taken as {0.917(2), 0.903(2)} for MILC and CLS ensembles, which is the average of Z O (z 0 , b ⊥0 , µ = 2 GeV) on MILC b ⊥0 =0.12fm b ⊥0 =0.24fm b ⊥0 =0.36fm b ⊥0 =0.48fmZ O (z 0 , b ⊥0 , µ = 2 GeV) on CLS b ⊥0 =0.1fm b ⊥0 =0.2fm b ⊥0 =0.3fm b ⊥0 =0.4fmZ O (z 0 = 0a, b ⊥0 = 2a) and Z O (z 0 = 0a, b ⊥0 = 3a). Extrapolation for quasi-TMDWF In this Letter, we perform an analytical extrapolation for quasi-TMDWFs in coordinate space to remove unphysical oscillations at large quasi-LF distance (λ = zP z ) as shown in Eq. (7) of the main text. In lattice simulations, we use the following parametrization for real and imaginary parts: Re[Ψ ± (z, b ⊥ , µ, ζ z )] = [k 1 (b ⊥ ) cos( πd 2 ) − k 2 (b ⊥ )f (b ⊥ ) sin( πd 2 )] e − λ λ 0 λ d , Im[Ψ ± (z, b ⊥ , µ, ζ z )] = [k 1 (b ⊥ )f (b ⊥ ) cos( πd 2 ) + k 2 (b ⊥ ) sin( πd 2 )] e − λ λ 0 λ d .(10) The parameter d accounts for the geometric attenuation, and the trigonometric function terms represent the periodic changes. Since we have parametrized the transversemomentum-dependence part by a multiplicative complex number for each b ⊥ , the parameter d no longer depends on b ⊥ . Therefore we perform a joint fit with b ⊥ = {1, 2, 3, 4, 5}a (a = 0.121 fm) on MILC ensemble and b ⊥ = {1, 2, 3, 4, 5, 6}a (a = 0.098 fm) on CLS ensemble. As shown in Fig. 9, the uncertainty of the lattice data grows rapidly for large λ. The extrapolated data matches the original ones and control the uncertainties well. We adopt λ L = z L P z ≈ 10 to distinguish the original data (λ < λ L ) and the extrapolated ones (λ > λ L ). To estimate the modification effects of the extrapolation form, we take additional two extrapolation cases λ L = (z L ± 1)P z , and treat the average of their differences as an estimate of the systematic uncertainty. Collins-Soper kernel on CLS ensemble We calculate the Collins-Soper kernel on CLS ensemble in this work inspired by [18]. The result is shown in Fig. 10, in which previous lattice QCD and perturbative calculations are also included as a comparison. As discussed in [18], the dominant systematic uncertainty at small b ⊥ comes from the imaginary part of the matching kernel. Large momentum extrapolation and TMDWF In lattice simulations, we adopt an extrapolation to infinite P z for TMDWFs with the following equation: Ψ ± (P z ) = Ψ ± (P z → ∞) + A (P z ) 2 .(11) The extrapolations are performed on MILC ensemble with P z = {1.72, 2.15, 2.58} GeV and on CLS ensemble with P z = {1.58, 2.1, 2.63} GeV, as shown in Fig. 11. The difference between the largest P z and P z → ∞ is taken as one of the systematic uncertainties. Two systematic uncertainties for TMDWFs are considered in our results. As previously illustrated, one is from large λ extrapolation of quasi-TMDWFs, the other is from infinite P z extrapolation. The whole uncertainty is the quadratic summation: σ all = σ 2 stt + σ 2 λ L + σ 2 P z lim ,(12) where σ stt represents the statistical uncertainty, σ P z lim is from infinite momentum extrapolation, and σ λ L corresponds to the difference between extrapolation with λ L = (z L ± 1)P z and λ L = z L P z . In Fig. 12, we show the final results of TMDWFs containing statistical and systematic uncertainties, the TMDWFs Ψ ± on MILC ensemble (left and central panels) and Ψ − on CLS ensemble (right panel). As one can see from those figures, the real parts of TMDWFs Ψ ± on both MILC and CLS ensembles decrease with increasing b ⊥ . While the real part of Ψ − on CLS in small b ⊥ (b ⊥ < 0.3 fm) is smaller compared with Ψ − on MILC, which might be caused by discretization effects. The imaginary parts of TMDWFs in all three cases increase with b ⊥ and become stable for b ⊥ > 0.36 fm. In addition, for large b ⊥ (b ⊥ ≥ 0.48 fm), the imaginary part of Ψ − converges, while for Ψ + it does not. The reason for this different behaviour is that in the hard kernels H ± (x, ζ z , µ) in Eq.(2) of the main text, the logarithmic term ln(−x 2 ζ 2 ±i ) has a different sign of imaginary part for Ψ ± . These results approach the infinite P z limit with ζ = (6 GeV) 2 and µ = 2 GeV. FIG. 1 . 1Illustration of quasi-TMDWF in coordinate space with a staple-shaped Wilson line inside. The green and red double lines represent the Wilson lines inΨ + (z, b ⊥ , µ, ζ z ) and Ψ − (z, b ⊥ , µ, ζ z ). A corresponding staple-shaped Wilson loop ZE(2L+|z|, b ⊥ , µ) is constructed to cancel the linear and cusp divergences. FIG. 2 . 2The real part (upper panel) and the imaginary part (lower panel) of the quasi-TMDWF in momentum space, with hadron momentum P z = 2.15 GeV on MILC ensemble. FIG. 3 . 3The one-loop intrinsic soft function as a function of b ⊥ . The grey band corresponds to the one-loop perturbative result in the MS scheme and the band is obtained by µ0 [18], while on CLS ensemble at 0.098 fm the result is given in the Supplemental Material[22].The intrinsic soft function can be determined from the quasi-TMDWF and the form factor of a pseudoscalar meson. The calculation for tree level intrinsic soft functionIm[Ψ − (x, b ⊥ )] on MILCb⊥ =0.12 fm b⊥ =0.24 fm b⊥ =0.36 fm b⊥ =0.48 fm b⊥ =0.6 fm FIG. 4. The real parts (upper panel) and the imaginary parts (lower panel) of the TMDWF on MILC ensemble. The TMDWF results approach the infinite P z limit with rapidity scale ζ = (6 GeV) 2 and renormalization scale µ = 2 GeV. FIG. 5 . 5Comparison of the transverse momentum distribution in our results with {ζ, µ} = {(6 GeV) 2 , 2 GeV} and phenomenological model at x = 0.5. Fig. 4 , 4we show the real parts (upper panel) and the imaginary parts (lower panel) of TMDWF Ψ − calculated on MILC ensemble. Results in this figure contain both statistical and systematic uncertainties, where the systematic ones come from the large λ extrapolation and the infinite momentum extrapolation [22]. The renormalization scale is chosen as µ = 2 GeV and the rapidity scale as ζ = (2P + ) 2 = (6 GeV) 2 . As can be seen from the figure, the real part of the TMDWF decreases as b ⊥ increases, while the imaginary part first increases and stabilizes for b ⊥ > 0.36 fm. The imaginary part shows a weaker dependence on b ⊥ than the real part. From LaMET factorization in Eq. Foundation of China under grant No. U2032102, 12125503, 12205106, 12175073, 12222503, 12293062, 12147140, 12205180, 12047503, 12005130. The computations in this paper were run on the Siyuan-1 cluster supported by the Center for High Performance Computing at Shanghai Jiao Tong University, and Advanced Computing East China Sub-center. J.H and J.L are also supported by Guangdong Major Project of Basic and Applied Basic Research No. 2020B0301030008, the Science and Technology Program of Guangzhou No. 2019050001. Y.B.Y is also supported by the Strategic Priority Research Program of Chinese Academy of Sciences, Grant No. XDB34030303 and XDPB15. J.H.Z. is supported in part by National Natural Science Foundation of China under grant No. 11975051. J.Z. is also supported by the China Postdoctoral Science Foundation under Grant No. 2022M712088. A.S., H.T.S, W.W, Y.B.Y and J.H.Z are also supported by a NSFC-DFG joint grant under grant No. 12061131006 and SCHA 458/22. FIG. 7 . 7Extrapolation of the Wilson loop ZE (LE, b ⊥ ) at b ⊥ = {1, 2, 3, 4, 5}a on MILC ensemble. FIG. 8 . 8The renormalization factor ZO (see Eq.(9)) measured at several selected b ⊥0 on MILC ensemble (upper panel) and on CLS ensemble (lower panel). FIG. 9 . 9Extrapolation of the quasi-TMDWF at large λ, via a joint fit at different b ⊥ using Eq.(10). The fits are conducted from the start point of shaded green band to the largest λ. The data beyond the vertical dashed line are replaced by the fit results. In the panel we show the results on MILC ensemble. Ψ − (x, b ⊥ )] on CLS P z =1.58GeV P z =2.1GeV P z =2.63GeV P z =limitFIG. 11. The figures display the P z dependence of TMDWFs at b ⊥ = 2a on MILC ensemble (upper panel) and b ⊥ = 3a on CLS ensemble (lower panel) and their comparison with the infinite P z limit. Im[Ψ − (x, b ⊥ )] on CLS b⊥ =0.1 fm b⊥ =0.2 fm b⊥ =0.3 fm b⊥ =0.4 fm b⊥ =0.5 fm b⊥ =0.59 fm FIG. 12. The left two figures are for real (upper panel) and imaginary parts (lower panel) of the TMDWF Ψ + on MILC ensemble, and the central two correspond to Ψ − on MILC ensemble. The right two ones correspond to Ψ − on CLS ensemble. TABLE I . IThe numerical simulation setup. On each ensemble, we put 8/4 source slices in time direction. Ensemble a(fm) n 3 s × nt m sea π m val π Measure a12m310 0.121 24 3 × 64 310 MeV 670 MeV 1053×8 X650 0.098 48 3 × 48 333 MeV 662 MeV 911×4 FIG.10. The blue data points display our CS kernel results on CLS ensemble, which contain statistical and systematic uncertainties shown as inner error bars and outer ones. 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{'abstract': 'We present a first lattice QCD calculation of the transverse-momentum-dependent wave functions (TMDWFs) of the pion using large-momentum effective theory. Numerical simulations are based on one ensemble with 2+1+1 flavors of highly improved staggered quarks action with lattice spacing a = 0.121 fm from the MILC Collaboration, and one with 2 +1 flavor clover fermions and tree-level Symanzik gauge action generated by the CLS Collaboration with a = 0.098 fm. As a key ingredient, the soft function is first obtained by incorporating the one-loop perturbative contributions and a proper normalization. Based on this and the equal-time quasi-TMDWFs simulated on the lattice, we extract the light-cone TMDWFs. The results are comparable between the two lattice ensembles and a comparison with phenomenological parametrization is made. Our studies provide a first attempt of ab initio calculation of TMDWFs which will eventually lead to crucial theory inputs for making predictions for exclusive processes under QCD factorization.Introduction: The light-front wave functions (LFWFs) are an important quantity for hadrons in particle physics. They characterize the nonperturbative structure of hadrons, and enter the prediction of a wide variety of measurable observables through quantum chromodynamics (QCD) factorization. While searching for new physics beyond the standard model (SM) requires a dedicated study of high-energy processes at colliders, this goal can partially be achieved by investigating low-energy processes, among which the flavor-changingneutral-current (FCNC) in a heavy quark system is an ideal probe[1]. A key input of calculating the SM contributions to the FCNC are LFWFs, including', 'arxivid': '2302.09961', 'author': ['Min-Huan Chu \nSchool of Physics and Astronomy\nShanghai Key Laboratory for Particle Physics and Cosmology\nKey Laboratory for Particle Astrophysics and Cosmology (MOE)\nINPAC\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nYang Yuanqing Scientific Computering Center\nTsung-Dao Lee Institute\nShanghai Jiao Tong University\n200240ShanghaiChina\n', 'Jin-Chen He \nSchool of Physics and Astronomy\nShanghai Key Laboratory for Particle Physics and Cosmology\nKey Laboratory for Particle Astrophysics and Cosmology (MOE)\nINPAC\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nDepartment of Physics\nUniversity of Maryland\n20742College ParkMDUSA\n', 'Jun Hua \nGuangdong Provincial Key Laboratory of Nuclear Science\nInstitute of Quantum Matter\nSouth China Normal University\n510006GuangzhouChina\n\nGuangdong-Hong Kong Joint Laboratory of Quantum Matter\nSouthern Nuclear Science Computing Center\nSouth China Normal University\n510006GuangzhouChina\n', 'Jian Liang \nGuangdong Provincial Key Laboratory of Nuclear Science\nInstitute of Quantum Matter\nSouth China Normal University\n510006GuangzhouChina\n\nGuangdong-Hong Kong Joint Laboratory of Quantum Matter\nSouthern Nuclear Science Computing Center\nSouth China Normal University\n510006GuangzhouChina\n', 'Xiangdong Ji \nDepartment of Physics\nUniversity of Maryland\n20742College ParkMDUSA\n', 'Andreas Schäfer \nInstitut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n', 'Hai-Tao Shu \nInstitut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n', 'Yushan Su \nDepartment of Physics\nUniversity of Maryland\n20742College ParkMDUSA\n', 'Ji-Hao Wang \nInstitute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Fundamental Physics and Mathematical Sciences\nHangzhou Institute for Advanced Study\nUCAS\n310024HangzhouChina\n', 'Wei Wang \nSchool of Physics and Astronomy\nShanghai Key Laboratory for Particle Physics and Cosmology\nKey Laboratory for Particle Astrophysics and Cosmology (MOE)\nINPAC\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nSouthern Center for Nuclear-Science Theory (SCNT)\nInstitute of Modern Physics\nGuangdong Province\nChinese Academy of Sciences\n516000HuizhouChina\n', 'Yi-Bo Yang \nInstitute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Fundamental Physics and Mathematical Sciences\nHangzhou Institute for Advanced Study\nUCAS\n310024HangzhouChina\n\nInternational Centre for Theoretical Physics\nAsia-Pacific\nBeijing/HangzhouChina\n', 'Jun Zeng \nSchool of Physics and Astronomy\nShanghai Key Laboratory for Particle Physics and Cosmology\nKey Laboratory for Particle Astrophysics and Cosmology (MOE)\nINPAC\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n', 'Jian-Hui Zhang \nSchool of Science and Engineering\nThe Chinese University of Hong Kong\n518172ShenzhenChina\n\nCenter of Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n', 'Qi-An Zhang \nSchool of Physics\nBeihang University\n102206BeijingChina\n'], 'authoraffiliation': ['School of Physics and Astronomy\nShanghai Key Laboratory for Particle Physics and Cosmology\nKey Laboratory for Particle Astrophysics and Cosmology (MOE)\nINPAC\nShanghai Jiao Tong University\n200240ShanghaiChina', 'Yang Yuanqing Scientific Computering Center\nTsung-Dao Lee Institute\nShanghai Jiao Tong University\n200240ShanghaiChina', 'School of Physics and Astronomy\nShanghai Key Laboratory for Particle Physics and Cosmology\nKey Laboratory for Particle Astrophysics and Cosmology (MOE)\nINPAC\nShanghai Jiao Tong University\n200240ShanghaiChina', 'Department of Physics\nUniversity of Maryland\n20742College ParkMDUSA', 'Guangdong Provincial Key Laboratory of Nuclear Science\nInstitute of Quantum Matter\nSouth China Normal University\n510006GuangzhouChina', 'Guangdong-Hong Kong Joint Laboratory of Quantum Matter\nSouthern Nuclear Science Computing Center\nSouth China Normal University\n510006GuangzhouChina', 'Guangdong Provincial Key Laboratory of Nuclear Science\nInstitute of Quantum Matter\nSouth China Normal University\n510006GuangzhouChina', 'Guangdong-Hong Kong Joint Laboratory of Quantum Matter\nSouthern Nuclear Science Computing Center\nSouth China Normal University\n510006GuangzhouChina', 'Department of Physics\nUniversity of Maryland\n20742College ParkMDUSA', 'Institut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany', 'Institut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany', 'Department of Physics\nUniversity of Maryland\n20742College ParkMDUSA', 'Institute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina', 'School of Fundamental Physics and Mathematical Sciences\nHangzhou Institute for Advanced Study\nUCAS\n310024HangzhouChina', 'School of Physics and Astronomy\nShanghai Key Laboratory for Particle Physics and Cosmology\nKey Laboratory for Particle Astrophysics and Cosmology (MOE)\nINPAC\nShanghai Jiao Tong University\n200240ShanghaiChina', 'Southern Center for Nuclear-Science Theory (SCNT)\nInstitute of Modern Physics\nGuangdong Province\nChinese Academy of Sciences\n516000HuizhouChina', 'Institute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina', 'School of Fundamental Physics and Mathematical Sciences\nHangzhou Institute for Advanced Study\nUCAS\n310024HangzhouChina', 'International Centre for Theoretical Physics\nAsia-Pacific\nBeijing/HangzhouChina', 'School of Physics and Astronomy\nShanghai Key Laboratory for Particle Physics and Cosmology\nKey Laboratory for Particle Astrophysics and Cosmology (MOE)\nINPAC\nShanghai Jiao Tong University\n200240ShanghaiChina', 'School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina', 'School of Science and Engineering\nThe Chinese University of Hong Kong\n518172ShenzhenChina', 'Center of Advanced Quantum Studies\nDepartment of Physics\nBeijing Normal University\n100875BeijingChina', 'School of Physics\nBeihang University\n102206BeijingChina'], 'corpusid': 257038505, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 15137, 'n_tokens_neox': 12106, 'n_words': 6386, 'pdfsha': 'fefa0f1ba17e2af7f65a90fe57d95ae36bcb565a', 'pdfurls': ['https://export.arxiv.org/pdf/2302.09961v1.pdf'], 'title': ['Transverse-Momentum-Dependent Wave Functions of Pion from Lattice QCD', 'Transverse-Momentum-Dependent Wave Functions of Pion from Lattice QCD'], 'venue': []}

arxiv

The equivariant Ehrhart theory of polytopes with order-two symmetries 14 Feb 2023 Oliver Clarke Akihiro Higashitani Max Kölbl The equivariant Ehrhart theory of polytopes with order-two symmetries 14 Feb 2023arXiv:2209.00755v2 [math.CO] We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant H * -polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycle graphs and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period one and coincides with the Ehrhart function of a lattice polytope. * Oliver Clarke is an overseas researcher under Postdoctoral Fellowship of Japan Society for the Promotion of Science (JSPS). Introduction Ehrhart theory is the enumerative study of the lattice points of polytopes and their dilations [3,Section 3]. Let P ⊆ R d be a polytope. The Ehrhart function L P (m) = |mP ∩ Z d | with m ∈ Z ≥0 counts the number of lattice points of mP . If P is a lattice polytope then L P (m) is a polynomial called the Ehrhart polynomial of P . More generally, if P is a rational polytope, i.e. its vertices have rational coordinates, then L P (m) becomes a quasipolynomial. We say that Q(t) = c d (t)t d + · · · + c 1 (t)t + c 0 (t) is a quasipolynomial if c 0 , . . . , c d are periodic functions in t and define the period of Q(t) to be the least common multiple of the periods of c 0 , . . . , c d . Note that a usual polynomial can be regarded as a quasipolynomial with period one. The data of L P is expressed as a power series Ehr(P, t) = m≥0 L P (m)t m called the Ehrhart series. If P is a rational polytope, then Ehr(P, t) = h * P (t) (1 − t N ) d+1 for some polynomial h * P (t) ∈ Z[t] called the h * -polynomial of P . The value of N is the denominator of P , which is defined as the smallest positive integer ℓ such that ℓP is a lattice polytope. It is well known that the denominator of P is divisible by the period of L P (t). While all lattice polytopes have period one, the converse is not true. A rational polytope P whose Ehrhart quasipolynomial has period one is said to be a pseudo-integral polytope or PIP. We will see examples of PIPs in Section 4. In [9], Stapledon introduces a generalisation of Ehrhart theory to study polytopes that exhibit symmetries. The theory has connections to toric geometry, representation theory, and mirror symmetry. Suppose that the polytope P is invariant (up to translation) under the action of a finite group G acting linearly on the lattice by a representation ρ : G → GL(Z d ). The equivariant Ehrhart series Ehr ρ (P, t) ∈ R(G) [[t]] is a power series in t with coefficients in the representation ring R(G). The series Ehr ρ (P, t) can be thought of as a union of the Ehrhart series of fixed sub-polytopes of P . Explicitly, for each g ∈ G, the power series Ehr ρ (P, t)(g) ∈ Z [[t]] is the Ehrhart series for the sub-polytope of P fixed by g. In particular, Ehr ρ (P, t)(1 G ) = Ehr(P, t) recovers the original Ehrhart series. See Remark 2.2. The analogue for the h * -polynomial in equivariant Ehrhart theory is the equivariant H * -series denoted H * [t] = i≥0 H * i t i ∈ R(G) [[t]]. In older literature, it is also denoted as ϕ [t]. In general, this series is not a polynomial. However, one of the central questions in equivariant Ehrhart theory is to determine when H * [t] is a polynomial and how to interpret its coefficients H * i ∈ R(G). Recall that for a lattice polytope P , the coefficients of its h * -polynomial are non-negative [8]. The equivariant analogue for non-negativity is effectiveness. We say that H * [t] is effective if each of its coefficients H * i is a non-negative integer sum of irreducible representations. In this paper we study two new families of polytopes; the symmetric edge polytopes of the cycle graph under the induced action of the automophism group of the graph, and rational crosspolytopes under the action of coordinate reflections. We describe the fixed polytopes in each case, which are related to rational cross-polytopes. We compute the equivariant Ehrhart series in each case to verify the effectiveness conjecture. In particular, in Example 4.4 we see that PIP need not satisfy the effectiveness conjecture if the assumption that P is a lattice polytope is dropped. Outline. In Section 2 we introduce the necessary preliminaries with the aim of fixing the main setup for equivariant Ehrhart theory. In Section 2.1, we recall some basics of representation theory of finite groups, in particular the representation ring R(G) which serves as the coefficient ring for the equivariant Ehrhart series. In Section 2.2, we fix our notation for actions of groups on lattices and describe their affine lattices. In Section 2.3, we recall the main setup of equivariant Ehrhart theory. In Remark 2.3, we also give an alternative but equivalent setup. In the next sections, we analyse two families of symmetric polytopes. In Section 3, we consider the symmetric edge polytopes of cycle graphs under symmetry induced by the dihedral group acting on the graph. We prove Theorems 3.4 and 3.6 which show that Conjecture 1.1 holds in the following respective cases. Firstly, for the cycle graph with a prime number of vertices under the action of its full automorphism group and secondly for any cycle graph with at least three vertices under the action of a reflection. In Section 4, we consider a family of rational cross-polytopes under the group of coordinate reflections. We prove Theorem 4.6 which computes the equivariant H * -series for all polytopes we consider. We note that this family contains rational polytopes with non-effective polynomial H * -series; see Example 4.4. Acknowledgements. We would like to express our gratitude to the anonymous reviewers for paying a great deal of attention to this paper and supplying many helpful comments and suggestions. Preliminaries Equivariant Ehrhart theory concerns the study of polytopes and their lattice points under a given group action. In this section we introduce the necessary preliminaries and fix the main setup following [9]. We begin with some background on the representation theory of finite groups [4,6]. Representations of groups Let G be a finite group and K a field. A finite dimensional K-representation of G is a homomorphism ρ : G → GL(V ) from G to the group of invertible linear maps of an n-dimensional K-vector space V . Fixing a basis for V identifies ρ(g) with an n × n matrix, for each g ∈ G. Equivalently, a representation is a module V for the group ring KG where g ∈ G ⊆ KG acts via the linear map ρ(g). The character of ρ is the function χ : G → K defined by the trace χ(g) = tr(ρ(g)). We say that a representation is irreducible if it contains no proper G-invariant subspaces, indecomposable if it cannot be written as a non-trivial direct sum of representations, and semisimple if it is a direct sum of irreducible representations. The representation ring R(G) is the set of formal differences of isomorphism classes of representations of G. The addition and multiplication structure of R(G) are given by direct sums and tensor products respectively. Given a KG-module V , we write [V ] for its isomorphism class in R(G). So given [V ] and [W ] in R(G) we have [V ] + [W ] = [V ⊕ W ] and [V ] · [W ] = [V ⊗ K W ]. In this paper, we work with representations defined over R. In this case Maschke's Theorem holds, so all representations are semisimple. In particular, all indecomposable representations are irreducible and any representation is a direct sum of irreducible representations. Therefore, R(G) is a free abelian group generated by the irreducible representations of G. Since the isomorphism class of a representation is determined uniquely by its character, we identify elements of R(G) with Z-linear combinations of characters. Permutation representations. Suppose G acts on a finite set S. Then the action induces a so-called permutation representation constructed as follows. Let V be the vector space over some field K with basis {e s : s ∈ S}. We define the permutation representation ρ : G → GL(V ) by its action on the basis ρ(g)(e s ) = e g(s) . Each matrix ρ(g) is a permutation matrix, hence the character of the representation is given by χ(g) = |{s ∈ S : g(s) = s}|. We say that a KG-module V is a permutation representation if it is isomorphic to a permutation representation. Group actions on lattices Let M ∼ = Z n+1 be a lattice with a distinguished basis and G a finite group. We say that G acts on M if there is a homomorphism ρ : G → GL n+1 (Z) from G to the group of invertible (n + 1) × (n + 1) matrices with entries in Z. Note, this action extends naturally to the vector space M R = M ⊗ Z R. Assume that G fixes a lattice point e ∈ M \{0}. We proceed to describe how M decomposes into a disjoint union of G-invariant affine lattices. By assumption M has a basis, so we denote by ·, · : M × M → Z the standard inner-product. We construct a new inner-product by averaging over the group: u, v G := 1 |G| g∈G ρ(g)u, ρ(g)v ∈ Q. Using the above inner-product, we observe two important properties about the orthogonal space e ⊥ ⊆ M R . Firstly, we have that e ⊥ is G-invariant, which follows from the fact that ρ(g)u, ρ(g)v G = u, v G for all u, v ∈ M R and g ∈ G. Secondly, we may choose a basis for e ⊥ that lies in M , since u, v G ∈ Q for all u, v ∈ M . It follows that the lattice N generated by e ⊥ ∩ M and e has rank n + 1. Therefore, N is a finite index subgroup of M and we write [M : N ] for the index. We define the affine space (M i ) R and the affine lattice M i at height i ∈ Z as follows: (M i ) R = i [M : N ] e + e ⊥ and M i = (M i ) R ∩ M. Since e ⊥ and M are G-invariant, we have that M i is G-invariant for each i ∈ Z. Note that M = i∈Z M i is a disjoint union and for each v ∈ M i we have v + M j = M i+j . Example 2.1. Let G = {1 , σ} ≤ S 4 be a subgroup of the symmetric group on four letters with σ = (1, 2)(3, 4). The permutation representation ρ maps σ to the permutation matrix ρ(σ) =     0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0     ∈ GL 4 (R). In particular, this matrix lies in GL 4 (Z), hence G preserves the lattice M = Z[e 1 , e 2 , e 3 , e 4 ]. Notice that e = e 1 + e 2 + e 3 + e 4 is fixed by the action of G. We compute a basis F that decomposes ρ(σ) as a block diagonal matrix: F =            1 1 1 1     ,     1 −1 0 0     ,     1 0 −1 0     ,     1 0 0 −1            and ρ(σ) F =     1 0 0 0 0 −1 −1 −1 0 0 0 1 0 0 1 0     . The orthogonal lattice M 0 is the 3-dimensional lattice generated by F \{e}. Observe that the sublattice N = Z[F ] has index 4 inside M . Therefore, the affine lattice M 1 = ( 1 4 e + (M 0 ) R ) ∩ M is equal to the lattice affinely generated by {e 1 , e 2 , e 3 , e 4 }. Main setup Let M ∼ = Z n+1 be a lattice with a distinguished basis and G a finite group that acts on M by ρ : G → GL n+1 (Z). Assume that there is a lattice point e ∈ M \{0} fixed by G. Let P ⊆ (M 1 ) R be a rational G-invariant polytope. For each non-negative integer m ∈ Z ≥0 , we obtain a permutation representation of the lattice points mP ∩M ⊆ M m and denote by χ mP its character. The equivariant Ehrhart series is an element of the ring of formal power series R(G)[[t]] given by: Ehr ρ (P, t) = m≥0 χ mP t m = H * [t] det[I − t · ρ] = H * [t] (1 − t) det[I − t · ρ| M 0 ] where H * [t] ∈ R(G)[[t]] is the equivariant H * -series. The denominator det[I − t · ρ] denotes the formal alternating sum n+1 i=0 [Λ i M R ](−t) i ∈ R(G)[t] , where Λ i M R is the i-th alternating power of the representation M R . If the character of the above alternating sum is evaluated at an element g ∈ G, then the resulting polynomial is equal to det[I − t · ρ(g)] where I is the identity matrix, see [9, Lemma 3.1]. By assumption, M R = e R ⊕ (M 0 ) R is a G-invariant decomposition of M R . So, for each g ∈ G, we may write ρ(g) = [1] ⊕ ρ(g)| M 0 as a block diagonal matrix, hence det[I − t · ρ(g)] = (1 − t) det[I − t · ρ(g)| M 0 ]. Remark 2.2. The equivariant Ehrhart series and H * -series are a generalisation of the usual Ehrhart series and h * -polynomial. If the equivariant Ehrhart series is evaluated at the identity element, then each character χ mP (1 G ) is equal to the number of lattice points of mP . Since det[ I − t · ρ(1 G )] = (1 − t) n+1 , it follows that the equivariant Ehrhart series evaluated at 1 G is equal to the classical Ehrhart series Ehr(P, t). The equivariant Ehrhart series contains all the data about the Ehrhart series for fixed sub- polytopes of P . Let M g R = {x ∈ M R : g(x) = x} be the subspace of M R fixed by g ∈ G. For each m ≥ 0 and g ∈ G, the value χ mP (g) is the number of lattice points of mP fixed by g. Equivalently, χ mP (g) is the number of lattice points in the m-th dilate of the fixed polytope P g = P ∩ M g R . Therefore, the evaluation of the equivariant Ehrhart series at g ∈ G is the Ehrhart series Ehr(P g , t). Remark 2.3. The setup may be equivalently defined by fixing: a group action ρ| M 0 of G on a lattice M 0 ∼ = Z n ; a rational polytope P ⊆ (M 1 ) R , where M 1 ∼ = Z n is a lattice of the same rank; and a lattice-preserving isomorphism between (M 1 ) R and (M 0 ) R , which induces an action of G on P . We require that, for each g ∈ G, the polytope g(P ) = (−v g ) + P differs from P only by a translation v g ∈ M 0 . So, for all g, h ∈ G we have that (gh)(P ) + v gh = P = g(P ) + v g = g(h(P ) + v h ) + v g = (gh)(P ) + g(v h ) + v g , hence v gh = g(v h ) + v g . We recover the original setup by taking e ∈ |G| · P ⊆ (M |G| ) R to be any G-invariant lattice point of the |G|th dilate of P . Explicitly, for all g ∈ G, we require g(e) + |G| · v g = e. For example, such a point can always be constructed from any lattice point p ∈ P by summing over the group: e = g∈G (g(p) + v g ). We define M to be the lattice generated by M 0 and M 1 where M 0 is a lattice that contains the origin and M 1 is the affine lattice at height 1 such that the orthogonal projection of (M 1 ) R onto (M 0 ) R sends 1 |G| e ∈ (M 1 ) R to 0 ∈ M 0 and differs from the lattice-preserving isomorphism by a translation. Concretely, we may take M = Z × M 0 ∼ = Z n+1 and define the action of G on M by the matrix ρ(g) = 1 0 v g ρ| M 0 (g) . Note that ρ is indeed a group homomorphism. That is, for all g and h in G we have ρ(g)ρ(h) = 1 0 v g ρ| M 0 (g) 1 0 v h ρ| M 0 (h) = 1 0 g(v h ) + v g ρ| M 0 (gh) = ρ(gh) since g(v h ) + v g = v gh . Let λ ∈ Z >0 be the smallest positive integer such that λ |G| e is a lattice point. The value of λ coincides with the index of the sublattice N in M from the original setup. Example 2.4 (Continuation of Example 2.1). Recall G = {1, σ} ≤ S 4 , with σ = (1, 2)(3, 4), acting by a permutation representation on M = Z 4 . Let P = Conv{e 1 , e 2 , e 3 , e 4 } ⊆ (M 1 ) R be a G-invariant 3-dimensional simplex. The permutation character χ mP counts the number of lattice points of mP ⊆ M m fixed by each g ∈ G. Explicitly, we have χ mP (1) = m + 3 3 and χ mP (σ) = m 2 + 1 if 2 | m, 0 otherwise. Computing the equivariant Ehrhart series, we have m≥0 χ mP (1)t m = 1 (1 − t) 4 and m≥0 χ mP (σ)t m = 1 (1 − t 2 ) 2 . For each g ∈ G, we observe that the equivariant Ehrhart series is given by σ →   −1 −1 −1 0 0 1 0 1 0   . Let P = Conv{0, e 1 , e 2 , e 3 } and notice that σ(P ) = (−e 1 ) + P , hence the above map defines a valid setup. This setup is equivalent to the setup in Example 2.4, which can be seen as follows. By averaging the vertex 0 ∈ P over G, we obtain the G-invariant point e = 1 2 e 1 , verified by the fact that e = σ(e) + e 1 . We define the lattice M = Z[e 0 , e 1 , e 2 , e 3 ] and identify the affine sublattice of M containing P with the affine span of {e 0 + e 1 , e 0 + e 2 , e 0 + e 3 }. In particular, the polytope P is identified in M R as Conv{e 0 , e 0 + e 1 , e 0 + e 2 , e 0 + e 3 }. The action of G on P extends to an action of G on M given by σ →     1 0 0 0 1 −1 −1 −1 0 0 0 1 0 0 1 0     . The point e in M R is identified with e 0 + 1 2 e 1 which spans a 1-dimensional G-invariant subspace. Observe that the vertices of P ⊆ M R are a basis for the lattice M . Rewriting the action of G in terms of this basis identifies it with Example 2.4. Effectiveness of the equivariant H * -series. We say that the equivariant H * -series H * [t] = i≥0 H * i t i ∈ R(G)[[t]] is effective if each H * i ∈ R(G) is the isomorphism class of a representation of G. In other words, H * i is a non-negative sum of irreducible representations of G. One of the main problems in equivariant Ehrhart theory is to understand when H * [t] is effective. Conjecture 2.6 ([9, Conjecture 12.1]). Let G be a finite group that acts on a lattice and P a G-invariant lattice polytope. Let Y be the toric variety with ample line bundle L associated to P . Then the following are equivalent: (1) L admits a G-invariant section that defines a non-degenerate hypersurface of Y , (2) H * [t] is effective, (3) H * [t] is a polynomial. It is well known that (1) ⇒ (2) ⇒ (3), see [9], and a counterexample has been constructed by Santos and Stapledon [5,Theorem 1.2] showing that (2) (1) and (3) (1). It is currently open whether (3) ⇒ (2). Symmetric edge polytopes of cycle graphs In this section we consider symmetric edge polytopes coming from cycle graphs and show that Conjecture 1.1 holds for the action of the dihedral group (Theorem 3.4) if the cycle graphs have prime order, and for the action of the two element subgroups of the dihedral group (Theorem 3.6) for cycle graphs of any order. We begin by fixing the setup for this section. Then, we consider the fixed polytopes of certain symmetric edge polytopes. Lastly we conclude the section with the statements and proofs of the main theorems. Let Γ = (V, E) be an undirected graph and Z |V | a lattice whose basis elements e v are associated to the vertices v ∈ V . Then the symmetric edge polytope P Γ ⊂ R |V | associated to Γ is defined as follows: P Γ = Conv {±(e v − e w ) : {v, w} ∈ E} . Throughout this section, we shall consider the automorphism group of Γ, denoted Aut(Γ). One sees that Aut(Γ) naturally induces a permutation representation ρ Γ on R |V | , which leaves P Γ invariant. We focus on the case when Γ is the cycle graph C d for some integer d ≥ 3. In this case, Aut(C d ) ∼ = D 2d = r, s | s 2 = r d = (sr) 2 = 1 is the dihedral group of order 2d. We identify D 2d with the automorphism group of C d . We fix the generator s ∈ D 2d , in the presentation of the group, to be a reflection that fixes the fewest number of vertices of C d . Let if d is odd, then v 0 = w 0 is the unique fixed vertex of s; if d is even, then v 0 and w 0 are neighbours; and r is the rotation that maps w 0 to w 1 (see Figure 1). Studying the equivariant Ehrhart theory of P d := P C d under the action of D 2d involves understanding the Ehrhart series of the individual sub-polytopes P g d fixed by the individual elements g ∈ D 2d . Let us begin with the trivial element 1 ∈ D 2d . Proposition 3.1 ([7, Theorem 3.3]). The Ehrhart series of P d is given by Ehr(P d , t) = h (d) 0 + h (d) 1 t + · · · + h (d) d−1 t d−1 (1 − t) d where: h (d) 0 = 1; for 1 ≤ j ≤ ⌊ d 2 ⌋, we have h (d) j = (−1) j j i=0 (−2) i d i d − 1 − i j − i = 2 d−1 if d is odd and j = d−1 2 , h (d−1) j−1 + h (d−1) j otherwise; and for each d 2 < j < d, the coefficients are h For odd cycle graphs C 2ℓ+1 , all reflections in D 4ℓ+2 are conjugate and so the corresponding fixed polytopes are unimodularly equivalent. Hence, it suffices to compute the fixed polytope for a single reflection, say s ∈ D 4ℓ+2 . Proposition 3.2. Let ℓ ≥ 1 be an integer. The fixed sub-polytopes P s 2ℓ+1 and P s 2ℓ+2 are unimodularly equivalent to the cross-polytope of dimension ℓ dilated by the factor 1 2 and their Ehrhart series are given by (d) j = h (d) d−1−j . v 0 w 0 w 1 v 1 w (d−2)/2 v (d−2)/2 s s r r v 0 = w 0 w 1 v 1 w (d−1)/2 v (d−1)/2Ehr(P s 2ℓ+1 , t) = Ehr(P s 2ℓ+2 , t) = (1 + t 2 ) ℓ (1 − t)(1 − t 2 ) ℓ . Proof. We start by giving a full description of the vertices of P s 2ℓ+1 and P s 2ℓ+2 . Each s-orbit is given by {v i , w i } for each 0 ≤ i ≤ ℓ. Note, the s-orbit that is an edge of C 2ℓ+1 is {v ℓ , w ℓ } ∈ E, while those of C 2ℓ+2 are {v 0 , w 0 } and {v ℓ , w ℓ }. The s-orbits of the vertices of P 2ℓ+1 and P 2ℓ+2 are hence given by {±(e w i − e w i+1 ), ±(e v i − e v i+1 )} as well as {e w ℓ − e v ℓ , e v ℓ − e w ℓ }. In the case of P 2ℓ+2 , we have the additional vertex {e w 0 − e v 0 , e v 0 − e w 0 }. By Lemma 5.4 in [9], P s 2ℓ+1 (resp. P 2ℓ+2 ) is given by the convex hull of points of the form p∈I p |I| where I is an s-orbit of the vertices of P s 2ℓ+1 (resp. P 2ℓ+2 ). The orbits {e w 0 − e v 0 , e v 0 − e w 0 } and {e w ℓ − e v ℓ , e v ℓ − e w ℓ } correspond to the origin and do not contribute to the description of P s 2ℓ+1 (resp. P 2ℓ+2 ). The remaining orbits yield P s 2ℓ+1 = P s 2ℓ+2 = Conv ± 1 2 (e v i + e w i − e v i+1 − e w i+1 ) : 0 ≤ i ≤ ℓ − 1 . One can see that the points {e v i + e w i − e v i+1 − e w i+1 } form a lattice basis for the fixed subspace (note that it does not matter whether v 0 and w 0 are identical or not), and with respect to that basis, P s 2ℓ+1 is unimodularly equivalent to the cross-polytope of dimension ℓ dilated by the factor (1−t)(1−t 2 ) , the result follows by induction on ℓ. Remark 3.3. For even cycle graphs C 2ℓ+2 , there is another type of reflection: one that fixes two antipodal vertices. For such a reflection sr ∈ D 4ℓ+4 , the fixed sub-polytope P sr 2ℓ+2 cannot be studied using the same method as in Proposition 3.2. The one-element sr-orbits are {v 0 } and {w ℓ } and the other orbits are {v i , w i−1 }. By a similar argument as above, the vertices of the sub-polytope P sr 2ℓ+2 are of the form ± 1 2 (e v i + e w i−1 − e v i+1 − e w i ) for i = 1, . . . , ℓ − 1, ± 1 2 (e v 1 + e w 0 − 2e v 0 ) and ± 1 2 (e v ℓ + e w ℓ−1 − 2e w ℓ ). For ℓ = 1, this is unimodularly equivalent to a dilated square containing the origin in its interior. For ℓ > 1, one can cut through the points ±(e v 1 + e w 0 − 2e v 0 ) and ±(e v 1 + e w 0 ), which yields a subpolytope of 2P sr 2ℓ+2 containing the origin and four of its vertices. Again, this is unimodularly equivalent to a square containing the origin. Hence, P sr 2ℓ+2 is not unimodularly equivalent to a dilated cross-polytope. We have computed the invariant polytopes of the symmetric edge polytope fixed by reflections of D 2d . The remaining conjugacy classes are the rotations. For odd d, the irreducible characters of D 2d are determined by the following table: 1 r k sr k ψ 1 1 1 1 ψ 2 1 1 −1 χ j 2 2 cos 2jkπ d 0 . where j ranges from 1 to d−1 2 and k ranges from 1 to d. In general, the fixed polytope P r k d with respect to a rotation r k is very difficult to compute directly. Not only does the description of the vertices of P r k d depend on the cycle decomposition of the permutation action of r k on the basis vectors of R |V | , but also on the adjacency of these vertices in the cycle graph. However, the rotation r k ∈ D 2d , where k and d are coprime, does not fix any vertex of C d . Therefore, the induced action on P d fixes only the origin, whose Ehrhart series is simply a geometric series Ehr({0}, t) = 1 + t + t 2 + · · · = 1 1−t . This yields the following result when d is prime. Theorem 3.4. Let p ≥ 3 be a prime number. The H * -series H * (p) of P p with respect to the action of the dihedral group D 2p is a polynomial of degree p − 1 and its coefficients H * (p),j are given by H * (p),j = 1 2p (h (p) j − 1 + p(g (p) j + 1))ψ 1 + (h (p) j − 1 − p(g (p) j − 1))ψ 2 + (2h (p) j − 2)χ if j is even, (p + h (p) j − 1)ψ 1 + (p + h (p) j − 1)ψ 2 + (2h (p) j − 2)χ if j is odd. where h (p) j follows the notation from Proposition 3.1, g (p) j := (p−1)/2 j/2 , and χ = j χ j . In particular, H * (p) is effective. To prove Theorem 3.4, we require the following technical lemma. j = (−1) j j i=0 (−2) i d i d − 1 − i j − i . Then the inequality h (d) j ≥ d · (g (d) j − 1) + 1 holds. Proof. In the case of j = 0, the statement follows because h (d) 0 = g (d) 0 = 1. Hence, we let 0 < j ≤ d−1 2 . In particular, we have d ≥ 5. We start by observing the recurrence relations g (d) j = g (d−2) j−2 + g (d−2) j and h (d) j ≥ h (d−2) j−2 + 2h (d−2) j−1 + h (d−2) j for 0 < j ≤ d−1 2 and g (d) 0 = h (d) 0 = 1. The inequality for h (d) j is an equality if j < (d − 1)/2. If j = d−1 2 then we get h (d) d−1 2 = 4h (d−2) d−3 2 > 2h (d−2) d−3 2 + 2h (d−2) d−5 2 = h (d−2) d−5 2 + 2h (d−2) d−3 2 + h (d−2) d−1 2 . For j > 0, we prove the statement by induction on odd d. Assume h (d) j > d · (g (d) j − 1) + 1. Then, by the recurrences, we have: h (d+2) j ≥ h (d) j−2 + 2h (d) j−1 + h (d) j > d(g (d) j−2 − 1) + 1 + 2h (d) j−1 + d(g (d) j − 1) + 1 = d(g (d+2) j − 2) + 2 + 2h (d) j−1 . At the same time, we can write: (d + 2)(g (d+2) j − 1) + 1 = d(g (d+2) j − 2) + 2 + 2(g (d) j−2 + g (d) j ) + d − 3. Hence it remains to prove that h (d) j−1 ≥ g (d) j−2 + g (d) j + d−3 2 . Here, by our assumption, we let j := 2k and d := 2n + 1, where k ≥ 1, n ≥ 2 and 2k ≤ n. Since h (d) ℓ ≥ d−1 ℓ holds for any ℓ, we get the following inequalities: h (d) j−1 = h (2n+1) 2k−1 ≥ 2n 2k − 1 ≥ 2n k ≥ n + 1 k + n − 1 = n k + n k − 1 + n − 1 = g (d) j−2 + g (d) j + d − 3 2 . This concludes the proof. Proof of Theorem 3.4. For the reflection s, we obtain det(I − t · ρ p (s)) = det          1 − t 0 0 · · · 0 1 −t −t 1 0 · · · 0 0 1 −t −t 1 · · · . . . . . . . . . . . .          = (1 − t)(1 − t 2 ) p−1 2 . For the rotation r, note that p is odd, so we get det(I − t · ρ p (r)) = 1 + (−t) p = 1 − t p . Since p is a prime number, recall that the rotation r, and any power r k with 1 ≤ k ≤ p − 1, fixes only the origin. That is P r k p = {0}, and so Ehr(P r p , t) = 1 1−t . Using this and the description of the Ehrhart series in Propositions 3.1 and 3.2, we obtain: H * (p) [t](1) = h (p) 0 + h (p) 1 t + · · · + h (p) p−1 t p−1 , H * (p) [t](s) = (1 + t 2 ) p−1 2 , H * (p) [t](r) = 1 − t p 1 − t = 1 + t + · · · + t p−1 , where h (p) j are the values specified in Proposition 3.1. Consider now the character of the regular module RD 2p , which is given by ψ 1 + ψ 2 + 2 j χ j . It is well known that this character evaluates to zero at every element of D 2p except at 1 where it evaluates to 2p. Hence, we deduce that the composite character χ = j χ j , obtained by adding together all irreducible two-dimensional characters of D 2p , is given by: 1 r k sr k χ p − 1 −1 0 . The coefficients H * (p),j of the H * -series are given by H * (p),j = 1 2p (h (p) j − 1 + p(g (p) j + 1))ψ 1 + (h (p) j − 1 − p(g (p) j − 1))ψ 2 + (2h (p) j − 2)χ if j is even, (p + h (p) j − 1)ψ 1 + (p + h (p) j − 1)ψ 2 + (2h (p) j − 2)χ if j is odd. It remains to show that these quantities are non-negative integers. Non-negativity follows from Lemma 3.5 and integrality follows immediately from the fact that H * [t] is an element of R(D 2p )[[t]]. In the last part of this section, we study the equivariant Ehrhart theory of the order 2 subgroups associated to the reflections described in Proposition 3.2. Fix the subgroup S 2 = {1, s} of D 2d . We write χ 1 and χ 2 for the trivial and non-trivial characters of S 2 respectively. Theorem 3.6. Let d ≥ 3 be an integer and let ℓ = ⌊d/2⌋ and b ∈ {0, 1} be integers such that d = 2ℓ + 1 + b. The equivariant H * -series of P d under the action of S 2 , denoted H * (d) [t], is a polynomial of degree d − 1 and its coefficients H * (d),j are given by H * (d),j = 1 2 (h (d) j + g (d) j )χ 1 + (h (d) j − g (d) j )χ 2 . where h (d) j follows the notation from Proposition 3.1 and g (d) j are the coefficients of the polynomial (1 + t) b (1 + t 2 ) ℓ := g (d) 0 + g (d) 1 t + · · · + g (d) d−1 t d−1 . In particular, H * (d) [t] is effective. Proof. By a similar argument to the proof of Theorem 3.4, we obtain det(I − t · ρ d (s)) = (1 − t) 1−b (1 − t 2 ) ℓ+b . By the description of Ehr(P d , t) in Proposition 3.2, we have: H * (d) [t](1) = h (d) 0 + h (d) 1 t + · · · + h (d) d−1 t d−1 , H * (d) [t](s) = (1 + t) b (1 + t 2 ) ℓ = g (d) 0 + g (d) 1 t + · · · + g (d) d−1 t d−1 . For the coefficients H * (d),j of the H * -series, we obtain H * (d),j = 1 2 (h (d) j + g (d) j )χ 1 + (h (d) j − g (d) j )χ 2 . It remains to show that H * (d) is effective, for which it suffices to show that h (d) j ≥ g (d) j . If d is odd, this follows directly from Lemma 3.5. If d is even, we start with the case where j is also even. We can use that in this case, g (d) j = g (d−1) j , giving us h (d) j ≥ h (d−1) j−1 + h (d−1) j ≥ h (d−1) j ≥ g (d−1) j = g (d) j . For the case where j is odd, we may assume without loss of generality that j ≤ ℓ − 1. In this case, we use g (d) j = g (d) j−1 and the fact that H * (d) (1) is unimodal, to conclude h (d) j ≥ h (d) j−1 ≥ g (d) j−1 = g (d) j . So we have shown that H * (d) [t] is effective, completing the proof. Rational cross-polytopes Let k, d ∈ Z be positive integers with k odd and d ≥ 2. Throughout this section we consider the polytope P (k, d) = Conv ±e 1 , . . . , ±e d−1 , ± k 2 e d ⊆ M R ∼ = R d . In this section we prove Theorem 4.6 which gives a complete description of the equivariant H *series of P (k, d) under the action of a reflection group. We observe in Example 4.4 that a rational analogue of Conjecture 1.1 does not hold for rational polytopes with period one. The Ehrhart series Ehr(P (k, d), t) has the following explicit description. Ehr(P (k, d), t) = (1 − t) Ehr([k/2, −k/2], t) (1 + t) d−1 (1 − t) d = (1 + (k − 1)t + kt 2 )(1 + t) d−2 (1 − t) d+1 . In the following, we will refer to (1+(k−1)t+kt 2 )(1+t) d−2 byh P (k,d) . We denote by G = {1, σ} the group of order two. We fix its two irreducible characters: the trivial character χ 1 and non-trivial character χ 2 . Fix some index i ∈ [n]. We let G act on the lattice Z[e 1 , . . . , e d ] by a coordinate reflection σ(e i ) = −e i and σ(e j ) = e j for all j ∈ [n]\{i}. Proposition 4.2. If i ∈ {1, 2, . . . , d − 1}, then H * [t] = χ 1 ·h P (k,d) (t). Proof. The reflection σ acts on P (k, d) by the diagonal matrix A = Diag(1, . . . , 1, −1, 1, . . . , 1) where −1 appears in position i. Therefore, we may compute det(I − tA) = (1 − t) d−1 (1 + t). We proceed by taking cases on d; either d = 2 or d > 2. Fix d = 2. In this case, the fixed polytope P (k, 2) σ is a line segment [k/2, −k/2] and so its Ehrhart series is Ehr(P (k, 2) σ , t) = 1 + (k − 1)t + kt 2 (1 − t)(1 − t 2 ) =h P (k,2) (t) (1 − t) det(I − tA) . On the other hand, the identity element e ∈ G acts by the identity matrix I and so det(I − tI) = (1 − t) 3 . Clearly, this fixes the entire polytope P (k, d), so its Ehrhart series is given by Ehr(P (k, 2), t) = 1 + (k − 1)t + kt 2 (1 − t) 3 =h P (k,2) (t) (1 − t) det(I − tI) . And so we have that H * [t] = χ 1 ·h P (k,2) (t) and we are done for the case d = 2. Next, let d > 2. The fixed polytope P (k, d) σ is equal to P (k, d − 1) in a one-dimension-higher ambient space, and so, by Proposition 4.1, its Ehrhart series is given by Ehr(P (k, d) σ , t) = (1 + (k − 1)t + kt 2 )(1 + t) d−3 (1 − t) d (1 + t) (1 + t) =h P (k,d) (t) (1 − t) det(I − tA) . On the other hand the identity element e ∈ G fixes the entire polytope P (k, d) and so its Ehrhart series is Ehr(P (k, d), t) =h P (k,d) (t) (1 − t) d+1 =h P (k,d) (t) (1 − t) det(I − tI) . And so it follows that H * [t] = χ 1 ·h P (k,d) (t) and we are done for the case d > 2. Proposition 4.3. If i = d, then H * [t] = d j=0 (a j χ 1 + b j χ 2 )t j where a j = d − 2 j + 1 2 (k + 1) d − 1 j − 1 and b j = 1 2 (k − 1) d − 1 j − 1 − d − 2 j − 1 and n k is defined to be zero if k < 0 or k > n. Proof. The identity e ∈ G acts by the identity matrix I, hence det(I − tI) = (1 − t) d . So, by Proposition 4.1, we have Ehr(P (k, d), t) = (1 + (k − 1)t + kt 2 )(1 + t) d−2 (1 − t) d+1 = (1 + (k − 1)t + kt 2 )(1 + t) d−2 (1 − t) det(I − tI) . On the other hand, the reflection acts by the diagonal matrix A = Diag(1, . . . , 1, −1) hence det(I − tA) = (1 − t) d−1 (1 + t). Observe that the fixed polytope P (k, d) σ is a (d − 1)-dimensional cross-polytope, therefore we have Ehr(P (k, d) σ , t) = (1 + t) d−1 (1 − t) d = (1 + t) d (1 − t) det(I − tA) . Write H * [t] = d j=0 (a j χ 1 + b j χ 2 )t j for some a j and b j . By evaluating H * [t] at each group element g ∈ G, we have H * [t](g) = Ehr(P (k, d) g , t)(1 − t) det(I − tρ(g)). It follows that a j + b j = d−2 j + (k − 1) d−2 j−1 + k d−2 j−2 , a j − b j = d j = d−2 j + 2 d−2 j−1 + d−2 j−2 for j ∈ {0, 1, . . . , d} where n k is defined to be zero if k < n or k > n. By solving this, we obtain the desired conclusion. The group G = {1, σ} acts by a coordinate reflection: σ(e 2 ) = −e 2 and σ(e 1 ) = e 1 . The equivariant Ehrhart H * -series is H * [t] = χ 1 + (χ 1 − χ 2 )t + χ 1 t 2 . In particular, H * [t] is polynomial but not effective since χ 1 − χ 2 is not the character of a representation of G. Remark 4.5. Consider the dilate of the polytope 2P (1, 2) with the same group action as in Example 4.4. In this case the equivariant H * -series is given by H * [t] = χ 1 · (1 + 4t + 3t 2 ) = χ 1 ·h 2P (k,d) (t). The example P (1, 2) does not extend to an example of a lattice polytope since all lattice points of P (1, 2) are fixed by G. However, if G is a non-trivial group acting non-trivially on a full dimensional lattice polytope, then at least one lattice point of P is not fixed by G. Concretely, we can say the following about two dimensional polytopes. Suppose that G is the group of order 2 and irreducible characters χ 1 and χ 2 . Assume G acts on a 2-dimensional lattice M and let P be a G-invariant lattice polytope with a polynomial equivariant H * -series given by H * [t] = χ 1 + (aχ 1 + bχ 2 )t + cχ 1 t 2 for some a, b, c ∈ Z. By Corollary 6.7 in [9], H * [t] is effective. Moreover, since χ 1 corresponds to a trivial permutation representation and χ 1 + χ 2 corresponds to the regular representation, which is a permutation representation as well, the linear coefficient of H * [t] is itself a permutation representation if a ≥ b ≥ 0. To see that this is satisfied, one first should notice that 2P σ is a lattice polytope by Corollary 5.4 in [9] and so it is either a line segment or a point whose vertices have coordinates lying in 1 2 Z. If P σ is a non-lattice point, then the result follows from a simple computation. So, by Lemma 7.3 in [9] and our assumption that H * [t] is a polynomial, we only need to consider the case where P σ contain a lattice point. So, it follows that P σ is unimodular equivalent to a line segment [v, w] ⊆ R with v, w ∈ 1 2 Z. By taking cases on whether v or w lie in Z we can show that the Ehrhart series has the form Ehr(P σ , t) = 1 + rt + st 2 (1 − t)(1 − t 2 ) with r, s ≥ 0. Evaluating H * [t] at σ and comparing coefficients gives us a − b = r ≥ 0 Let G = (Z/2Z) d = σ 1 , σ 2 , . . . , σ d be the group of coordinate reflections of R d . Explicitly, for each i, j ∈ {1, 2, . . . , d} we have σ i (e i ) = −e i and σ i (e j ) = e j if i = j. Let χ 1 denote the trivial character of G and χ 2 denote the character satisfying χ 2 (σ d ) = −1 and χ 2 (σ i ) = 1 for all i ∈ {1, 2, . . . , d − 1}. The polytope P (k, d) is invariant under G. By Propositions 4.2 and 4.3 it follows that the equivariant H * -series H * [t] of P is a polynomial whose coefficients are integer multiples of χ 1 and χ 2 . Moreover, we obtain the following result. a j = d − 2 j + 1 2 (k + 1) d − 1 j − 1 and b j = 1 2 (k − 1) d − 1 j − 1 − d − 2 j − 1 and n k is defined to be zero if k < 0 or k > n. Example 4.7. Given two subsets A and B that lie in orthogonal subspaces of R N , we denote by A ⊕ B = Conv(A ∪ B) ⊆ R N the free sum of A and B. Let d ≥ 3 and k = 1. The polytope P (k, d) has Ehrhart series Ehr(P (1, d), t) = (1 + t + t 2 + t 3 )(1 + t) d−3 (1 − t) d+1 . We note that this coincides with the Ehrhart series of the lattice polytope Q d ⊆ R 3 × R d−3 given by Q d = Conv{e 1 , e 2 , e 3 , −e 1 − e 2 − e 3 } ⊕ [−1, 1] ⊕(d− 3) . By a result of Stapledon [9, Proposition 6.1], the equivariant H * -series of the simplex S = Conv{e 1 , e 2 , e 3 , −e 1 − e 2 − e 3 } is always effective. If a group G = {1, σ} acts on Q d with an action that factors σ(x, y) = (σ| R 3 (x), σ| R d−3 (y)) such that σ| R d−3 acts by a coordinate reflection, then the equivariant H * -series of Q d is (1 + t) d−3 times the H * -series of S, meaning that it is effective. On the other hand, if we take the polytope P (1, d) with respect to the action of G = {1, σ} given by σ(e d ) = −e d and σ(e i ) = e i for all i ∈ {1, . . . , d − 1} then the equivariant H * -series is not effective. t·ρ(g)] . Therefore, the equivariant H * -series is a polynomial given by H * [t] = 1. Example 2 . 5 . 25Following the alternative setup in Remark 2.3, let G = {1, σ} be the group with two elements that acts on a rank 3 lattice M 0 = Z[e 1 , e 2 , e 3 ] by the map ρ d := ρ C d : D 2d → GL(R d ) denote the associated permutation representation. From now on, we label the vertices of C d with {v 0 , . . . , v ⌈(d−2)/2⌉ , w 0 , . . . , w ⌈(d−2)/2⌉ }, where w 0 = v 0 if d is odd, so that: (v 0 , v 1 , . . . , v ⌈(d−2)/2⌉ ) and (w 0 , w 1 , . . . , w ⌈(d−2)/2⌉ ) are distinct paths in C d ; for each 0 ≤ i ≤ ⌈(d − 2)/2⌉ the s-orbits are {v i , w i }; Figure 1 : 1The vertex labelings for even (left) and odd (right) cycle graphs and the action of the generators of the dihedral group. 2 . 2By [2, Theorem 1.4] and the fact that the Ehrhart series of the interval [ Lemma 3 . 5 . 35Let d ≥ 3 be an odd integer and let 0 ≤ j ≤ Proposition 4. 1 ( 1An application of [2, Theorem 1.4]). For each k odd and d ≥ 2 we have Theorem 4. 6 . 6With the setup above, we have H * [t] = d j=0 (a j χ 1 + b j χ 2 )t j where Conjecture 1.1 (Effectiveness conjecture [9, Conjecture 12.1]). Fix the main setup from Section 2.3 and let P be a lattice polytope. The equivariant H * -series H * [t] is a polynomial if and only if H * [t] is effective. It is known that if H * [t] is effective, then it is a polynomial. However the converse is currently open. The equivariant Ehrhart theory for certain families of polytopes is well studied. In each of the following examples the effectiveness conjecture has been verified: simplices [9, Proposition 6.1], for which the coefficients of H * [t] are permutation representations; the hypercube [9, Section 9] under its full symmetry group; the permutahedron [1] under the symmetric group; graphic zonotopes of the path graph [5, Section 3.1] with the Z/2Z action; hypersimplices [5, Theorem 3.57] with the symmetric group action. The equivariant Ehrhart theory of the permutahedron. Federico Ardila, Mariel Supina, Andrés R Vindas-Meléndez, Proc. Amer. Math. Soc. 14812Federico Ardila, Mariel Supina, and Andrés R. Vindas-Meléndez. The equivariant Ehrhart theory of the permutahedron. Proc. Amer. Math. Soc., 148(12):5091-5107, 2020. Lattice-point generating functions for free sums of convex sets. Matthias Beck, Pallavi Jayawant, Tyrrell B Mcallister, J. Combin. Theory Ser. A. 1206Matthias Beck, Pallavi Jayawant, and Tyrrell B. McAllister. Lattice-point generating functions for free sums of convex sets. J. Combin. Theory Ser. A, 120(6):1246-1262, 2013. Computing the continuous discretely. Matthias Beck, Sinai Robins, Springer61Matthias Beck and Sinai Robins. Computing the continuous discretely, volume 61. Springer, 2007. Representation theory of finite groups and associative algebras. W Charles, Irving Curtis, Reiner, American Mathematical Soc356Charles W Curtis and Irving Reiner. Representation theory of finite groups and associative algebras, volume 356. American Mathematical Soc., 1966. Sophia Elia, Donghyun Kim, Mariel Supina, arXiv:2205.05900Techniques in equivariant ehrhart theory. Sophia Elia, Donghyun Kim, and Mariel Supina. Techniques in equivariant ehrhart theory. arXiv:2205.05900, 2022. Character theory of finite groups. I Martin Isaacs, Courier Corporation. 69I Martin Isaacs. Character theory of finite groups, volume 69. Courier Corporation, 1994. Smooth fano polytopes whose ehrhart polynomial has a root with large real part. Hidefumi Ohsugi, Kazuki Shibata, Discrete & Computational Geometry. 473Hidefumi Ohsugi and Kazuki Shibata. Smooth fano polytopes whose ehrhart polynomial has a root with large real part. Discrete & Computational Geometry, 47(3):624-628, 2012. Decompositions of rational convex polytopes. Richard P Stanley, Combinatorial Mathematics, Optimal Designs and Their Applications. J. SrivastavaElsevier6Richard P. Stanley. Decompositions of rational convex polytopes. In J. Srivastava, editor, Combinatorial Mathematics, Optimal Designs and Their Applications, volume 6 of Annals of Discrete Mathematics, pages 333-342. Elsevier, 1980. Equivariant ehrhart theory. Alan Stapledon, Advances in Mathematics. 2264Alan Stapledon. Equivariant ehrhart theory. Advances in Mathematics, 226(4):3622-3654, 2011. Authors' addresses:

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{'abstract': 'We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant H * -polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycle graphs and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period one and coincides with the Ehrhart function of a lattice polytope. * Oliver Clarke is an overseas researcher under Postdoctoral Fellowship of Japan Society for the Promotion of Science (JSPS).', 'arxivid': '2209.00755', 'author': ['Oliver Clarke ', 'Akihiro Higashitani ', 'Max Kölbl '], 'authoraffiliation': [], 'corpusid': 252070633, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 15754, 'n_tokens_neox': 14074, 'n_words': 8530, 'pdfsha': '7bf4d6ed14e13ce9efe28e997f22c5995998acf6', 'pdfurls': ['https://export.arxiv.org/pdf/2209.00755v2.pdf'], 'title': ['The equivariant Ehrhart theory of polytopes with order-two symmetries', 'The equivariant Ehrhart theory of polytopes with order-two symmetries'], 'venue': []}

arxiv

An entropy-controlled objective chip for reflective confocal microscopy with subdiffraction-limit resolution Jun He Department of Optics and Optical Engineering University of Science and Technology of China 230026HefeiAnhuiChina Dong Zhao Department of Optics and Optical Engineering University of Science and Technology of China 230026HefeiAnhuiChina Hong Liu Institute of Materials Research and Engineering Agency for Science Technology and Research (A*STAR) 2 Fusionopolis Way#08-03, 138634InnovisSingapore, Singapore Jinghua Teng Institute of Materials Research and Engineering Agency for Science Technology and Research (A*STAR) 2 Fusionopolis Way#08-03, 138634InnovisSingapore, Singapore Cheng-Wei Qiu c.q.chengwei.qiu@nus.edu.sg Department of Electrical and Computer Engineering National University of Singapore 4 Engineering Drive 3117576Singapore, Singapore Kun Huang huangk17@ustc.edu.cn Department of Optics and Optical Engineering University of Science and Technology of China 230026HefeiAnhuiChina An entropy-controlled objective chip for reflective confocal microscopy with subdiffraction-limit resolution # J. H. and D. Z. contributed equally to this work. * Corresponding authors: K. H. ( Planar lenses with optimized but disordered structures can focus light beyond the diffraction limit. However, these disordered structures have inevitably destroyed wide-field imaging capability, limiting their applications in microscopy. Here we introduce information entropy S to evaluate the disorder of an objective chip by using the probability of its structural deviation from standard Fresnel zone plates. Inspired by the theory of entropy change, we predict an equilibrium point S0=0.5 to balance wide-field imaging (theoretically evaluated by the Strehl ratio) and subdiffraction-limit focusing. To verify this, a NA=0.9 objective chip with a record-long focal length of 1 mm is designed with S=0.535, which is the nearest to the equilibrium point among all reported planar lenses. Consequently, our fabricated chip can focus light with subdiffraction-limit size of 0.44λ and image fine details with spatial frequencies up to 4000 lp/mm in experiment. These unprecedented performances enable ultracompact reflective confocal microscopy for superresolution imaging. Introduction Since the first microscope was invented in 1595 by a Dutch father-son team Hans and Zacharias Janssen 1 , optical objectives have developed rapidly with improved performance in aberration correction, field of view, magnification, and numerical aperture, except its imaging resolution being limited by diffraction of light 2 . Based on refractive optics 3 , traditional objectives need multiple elements with carefully designed curvatures and air intervals for better imaging, thus leading to a bulky volume. The situation of objectives has been sustained for over 400 years until subwavelength-thick metalenses were reported with ultracompact volume in 2016 4 . However, highaspect-ratio dielectric nanobricks in the metalenses exhibit high efficiency under normal incidence only with a small tolerance of tilting angle 5 , resulting in large spatial-frequency details of objects being rejected by the metalenses operating in an imaging mode 6,7 . Consequently, the imaging resolutions of the metalenses and their related scanning confocal microscopies (SCMs) are diffraction-limited to 0.51λ/NA (λ is the wavelength of light and NA is the numerical aperture of the metalens) 8 . Efforts to enhance the resolving power of objectives have also been made by using planar superoscillation lenses [9][10][11][12] and supercritical lenses [13][14][15] that realize subwavelength focal spots by optimizing the destructive or constructive interference between multiple diffracting beams 7, [16][17][18] . Since only the focusing properties of these planar lenses are designed without considering the capability of direct wide-field imaging, superoscillation and supercritical lenses are usually used as optical probes in an SCM 7 , where an additional refraction-based objective is mandatorily required to collect the transmitted light through the objects. This leads to the fact that all these planar-lensbased SCMs must operate in transmission mode and are only valid for objects sitting on a transparent substrate that introduces spherical aberration, which requires collection objectives with coverslip collars for correction 2 . Without involving these issues, reflective SCMs are, therefore, more popular for noninvasive and in vivo imaging of various specimens 19 . Despite the strong requirements from applications, it is still difficult to demonstrate reflective planar-lens-based SCMs with better resolving power than commercial SCMs, due to the lack of high-performance planar objectives that possess dual functionalities of focusing and imaging beyond the diffraction limit. The challenges to desigining such a planar objective are twofold. First, designing a planar objective for subdiffraction-limit focusing generally leads to an irregularly distributed phase or amplitude 18 . However, such structural disorder is not preferred in its imaging counterpart, where the analytical phase or amplitude is preferred for the constructive reconstruction of objects 7 . Such a dilemma is a fundamental barrier to demonstrating planar objectives with subdiffraction-limit resolution in both focusing and imaging. Second, the diameter of the planar objectives should be sufficiently large to suppress the diffraction effect of collected light by the planar objectives (when working in the imaging mode) for better image formation. Correspondingly, the number of fine structures in the planar objectives is extremely large because of the short wavelength in the visible spectrum, thereby increasing the technical difficulty in the design and fabrication of planar objectives. To overcome these challenges, we propose a disorder-controlled objective chip that functionally integrates a binary-phase Fresnel zone plate (FZP) and a weakly perturbed few-ring phase mask into a single ultrathin element. By introducing the concept of information entropy, we theoretically predict that an objective chip with its entropy at S0=0.5 can maintain the imaging and superfocusing properties simultaneously. Using deep-ultraviolet (DUV) lithography, the fabricated objective chip experimentally exhibits a focal spot of 0.44λ (below the Rayleigh criterion of 0.51λ/NA=0.57λ) without strong sidebands and the capability of imaging fine objects with spatial frequency of 4000 lp/mm. Benefiting from this, an ultracompact reflective SCM is built with an imaging resolution (center-to-center) of 200 nm at λ=405 nm and a record-long working distance of 1 mm, superior to the state-of-the-art SCMs. Disorder of objective chip Since standard FZPs have analytical complex transmission, i.e., = (where and are the amplitude and phase modulation, respectively), they can realize wide-field but diffraction-limit imaging. For an objective chip, its complex transmission ℎ = deviates irregularly from that of an FZP, thus creating undesired disorder for imaging purposes. Assuming that the deviations are ∆ = ℎ − for amplitude and ∆ = ℎ − for phase, we can rewrite the transmission of the objective chip as ℎ = ( + ∆ ) ( +∆ ) = (1 + ∆ ⁄ ) ∆ = • ∆ , where the deviated transmission ∆ = (1 + ∆ ⁄ ) ∆ introduces optical aberration in imaging but offers more degrees of freedom for subdiffraction-limit focusing. Thus, for an arbitrary objective chip, its complex transmission contains a standard imaging part (i.e., FZP) and an additional aberration part ∆ . In the design of an objective chip, the item ∆ is fundamentally important in building the connection between imaging and focusing. Considering that ∆ is distributed spatially in an irregular way, we first investigate its statistical property of such deviation by defining a dimensionless parameter-deviation probability 1 = ∑ 1 | ∫ ∆• +1 ∫ ∆ • +1 | −1 =0 ,(1) where ∆= ∆ with a maximum value of ∆ = 1 for the amplitude deviation, ∆= ∆ with a maximum value of ∆ = for the phase deviation, the objective chip is divided into minimum diffraction subunits (i.e., zones in the FZP) with their boundaries of = √ + ( ) 2 4 ⁄ (m is the index of the m-th zone, λ is the wavelength and f is the focal length of the lens), M is the total number of zones contained in the FZP, and the modulus is used to ensure the non-negative probability. Because the standard FZPs have pure amplitude (i.e., = 0 in for binaryamplitude FZPs) or phase (i.e., is a constant in for binary-phase FZPs) modulation, we only need to calculate ΔA or Δφ in Eq. (1) for most objective chips. According to its definition, the deviation probability p1 ranges from 0 to 1, leaving a probability p2=1-p1 for the unchanged part. It behaves like an information channel with binary values, where the entropy = − ∑ 2 2 =1 is usually used to evaluate the disorder of this information system 20 . Similarly, based on our defined probability p1 and p2, we can also calculate the disorder of an objective chip by using the information entropy S. When p1=0 or p1=1, the corresponding S equals zero, which means high certainty without any disorder. It agrees with the real cases that both objective chips with p1=0 and p1=1 refer to standard FZPs, where ∆ = 1. When p1=0.5, the entropy S=1, which implies the highest disorder because half of the zones are reversed randomly. Although high disorder offers large degrees of freedom for optical super-focusing, it also destroys the imaging capability of the objective chip due to optical aberration dominated by its random ∆ . Thus, we infer that the entropy S is symmetric about p1=0.5, where the peak is located. At the side of 0≤ p1 ≤0.5, the entropy S increases monotonically from 0 to 1. High certainty at S=0 is helpful in widefield imaging but high disorder at S=1 is required to realize super-focusing. To obtain a good balance between imaging and super-focusing, we introduce a thermodynamic analog that the change of entropy for an isolated system originates from outside work or heat transfer 21 . In our case, a virtual work W is assumed to govern the change ΔS of entropy by following a straightforward relationship ∝ . The change of entropy from S=0 to S0 requires the virtual work W1, while the change from S0 to S=1 requires virtual work W2. We suggest that, when both virtual works are equal, i.e., W1=W2, good balance between imaging and super-focusing is achieved with an equality 1 = 2 (i.e., 0 − 0 = 1 − 0 ), leading to the equilibrium point S0=0. 5. Interestingly, at this equilibrium point, its relative deviation probability p1 equals 0.11, which is smaller than the middle point p1=0. 25 of the interested range 0≤ p1 ≤0.5. This implies that the entropy S is more sensitive to the intrinsic disorder of the objective chip than the deviation probability p1, thereby indicating the rationality of the proposed equilibrium point S0=0.5. Strehl ratio and focal size of the objective chip To quantitatively investigate the imaging and focusing properties of an objective chip with different disorders, a binary-phase objective chip (Fig. 1a) with a focal length f=1mm and NA=0.9 (implying its diameter of 4.13 mm) is exemplified here to enhance the optical efficiency in both focusing and imaging. Compared with its corresponding binary-phase FZP, this proposed objective chip has only the phase deviation of Δφ because of ΔA=0, leaving ∆ = ∆ . This implies that any binary-phase objective chip can be taken as a combination of an analytical FZP and an additional few-ring phase (i.e., ∆ ) mask, as sketched in Fig. 1b. Since the phase ∆ in the few-ring mask introduces optical aberration, we evaluate its influence on imaging quality by using the relative Strehl ratio (SR) 3 = ℎ (0,0, = ) (0,0, = ) = |∫ ∫ 0 ( , )• ( +∆ ) ( ) 3 ( −1) | 2 |∫ ∫ 0 ( , )• ( ) 3 ( −1) | 2 ,(2) where the incident electric field E0 is taken as unity in this work, the wavenumber k=2π/λ, R 2 =(u- x) 2 +(v-y) 2 +z 2 , r 2 =x 2 +y 2 , tanφ=y/x, x and y are the Cartesian coordinates at the initial plane of the objective chip, u=v=0 and z=f are used to obtain the on-axis intensity at the focal plane, and is fixed once the FZP is given. In Eq. (2), the on-axis intensity of the FZP, having the same modulation (phase or amplitude) type as that of the designed objective chip, is used as the denominator to avoid the influence of the focusing efficiency of the FZP with different modulation types. Therefore, Eq. (2) defines a relative Strehl ratio, which is more useful in evaluating the optical aberration of imaging systems. For a binary-phase objective chip with its deviation probability p1 (only 0≤ p1 ≤0.5 is considered in the following because the entropy S is symmetric about p1=0.5), its relative Strehl ratio can be approximated as = (1 − 2 0 1 √ ⁄ ) 2 , where the on-axis intensity 0 of each zone plate ranges from a0min=0.87 to a0max=2, see the detailed derivations in Supplementary Section 1. Thus, for a given p1, we can analytically obtain the range of SR: 1) = (1 − 2 0 1 √ ⁄ ) 2 for 0 ≤ 1 ≤ √ (2 0 ) ⁄ and = 0 for √ (2 0 ) ⁄ ≤ 1 ≤ 0.5 ; 2) = (1 − 2 0 1 √ ⁄ ) 2 for 0 ≤ 1 ≤ √ ( 0 + 0 ) ⁄ and = (1 − 2 0 1 √ ⁄ ) 2 for √ ( 0 + 0 ) ⁄ ≤ 1 ≤ 0.5. By visualizing SRmin and SRmax, Figure 1c illustrates the correlation between the Strehl ratio and the information entropy S by using an intermediate parameter p1. With increasing S, both SRmin and SRmax decrease, implying large aberration and poor imaging quality. However, its increased range (ΔSR= SRmax-SRmin) indicates high uncertainty of SR, which has echoes of high disorder for a large S. Therefore, a small S with low disorder is preferred in imaging, which needs a large SR. Similarly, its focal spot is also controlled by the entropy S or p1. Although it is difficult to derive the spot size analytically, we obtain its upper (rmax) and lower (rmin) boundaries numerically (see Supplementary Section 1), as illustrated in Fig. 1c. For the larger entropy S, the spot size is valued in the wider range, which offers more opportunities to realize super-focusing. Hence, a large entropy S with high disorder is required in super-focusing. These results reveal that imaging and super-focusing have completely opposite requirement for the entropy S of an objective chip, which doubly confirms that the entropy S0=0.5 leads to good balance. Design of objective chip Since the FZP contained in the objective chip is given with analytically described structures, we only need to optimize its deviation part ∆ = ∆ , which refers to a few-ring phase mask (see Compared with other reported planar lenses 9,13,14,[22][23][24][25] , the achieved entropy in our objective chip is the closest to the equilibrium point S0=0.5 (see Fig. 1d), implying its advantages in balancing optical imaging and super-focusing. It exhibits the fundamental difference of our objective chip from other lenses with their entropies approaching 1. Furthermore, the relative SR=0.45 of our objective chip is also the highest among those lenses, which theoretically predicts the best imaging capability. The optimized focal spot has a lateral FWHM (i.e., full width at half maximum) of 180 nm (0.44λ, below the Rayleigh criterion of rRC=0.51λ/NA=0.57λ), which is slightly larger than the superoscillation criterion rSOC=0.358λ/NA=0.398λ (in terms of FWHM) 26 to avoid strong sidebands in a superoscillatory spot. The simulated electric fields near the focal plane are provided with experimental results for a better comparison, as shown later. Fabrication of objective chip The possibility of employing semiconductor processes to fabricate planar flat optics will ultimately allow mass production of flat optics at a low cost and push this technique for wide market adoption. As discussed above, our design strategy makes low-cost and fast DUV lithography feasible for fabricating a 4-mm-diameter objective chip. Under the conditions of λ=405 nm, f=1 mm and NA=0.9 in this work, Δrmin is only 225 nm, which is within the capability of commercial DUV lithography (with a critical dimension of 200 nm). This key feature will greatly facilitate the future manufacturing of our objective chips by reducing the cost compared with other flat lenses with critical dimensions smaller than 200 nm that would require much more costly 12-inch immersion lithography 27 . We note that E-beam lithography is able to write the pattern at high resolution; however, it is not the technique for large-scale fabrication due to its speed limit and the inevitable stitching error (with a small writing square of several tens of micrometers). The fabrication details of our objective chip are provided in the Methods. The inset of Fig. 1a shows the microscopic image of our fabricated objective chip, where different colors arise from optical scattering of daylight. To reveal the fine details, we show scanning-electron-microscopy (SEM) images of the objective chip in Fig. 2a, where both simulated and experimental widths from the center to the outermost boundary are also provided with a maximum deviation of <75 nm. The possible reason comes from insufficient exposure time of photoresist under DUV radiation, which can be solved by increasing the exposure time. Using a profilometer, we characterize the groove depth of ~530 nm (see the insert in Fig. 2a) that yields a phase modulation of ~1.23π, which is larger than the ideal value of π due to overetching issues. Nevertheless, we emphasize that such a deviation of 0.23π in phase modulation leads to a theoretical decrement of only ~2.6% in the focusing efficiency of the objective chip, see the simulation details in the Methods. Subdiffraction-limit focusing by objective chip To verify the focusing capability of the objective chip, we first measure the diffraction field near the focus of the objective chip under the illumination of collimated circularly polarized light by using a 0.95-NA objective lens, see the experimental details in Supplementary Section 3. Figure 2b shows the x-z and y-z cross-section of the measured light intensity, which reveals a record-long focal length of 1 mm (compared with the previous planar diffractive lenses 9,14,15,22,28,29 ). The axial depth of focus (DOF) is extended from the simulated ~500 nm to the experimental ~5500 nm, which is caused by the fabrication error of the groove width in the objective chip (see Fig. 2a). Such a long DOF offers good tolerance to sample alignment in an SCM 13 . Fortunately, the focusing spot size of the objective chip is less influenced by the weak phase perturbation from the imperfect zones, as confirmed by the good agreement between the simulated and experimental line-scanning intensity profiles (Fig. 2c). To quantitatively evaluate the focusing effect, we present the experimental FWHM of the focal spot size along the propagation of light in Fig. 2d, indicating the varying FWHM from 170 nm to 210 nm (tightly close to the simulated 180 nm). Compared with the Rayleigh criterion of 0.51λ/NA, these achieved focal spots confirm the subdiffraction-limit focusing capability of the proposed objective chip. Due to its supercritical feature with lateral FWHMs above the superoscillation criterion 26 , the focusing spots have no strong sideband, as observed in Supplementary Video 1, which dynamically records the focusing process near the focal plane. The focusing efficiency, defined as the ratio of the focused power (experimentally filtered by a 150 μm-diameter pinhole at the focal plane) to the total power incident on the objective chip, is measured to be 12.3% (see the measurement details in Supplementary Section 4), which is lower than its theoretical efficiency of 18.7% (see its calculation in the Methods), as shown in Fig. 2e. This discrepancy in efficiency is attributed to incomplete constructive interference of light diffracted from two neighboring zones because of the insufficient etching widths. Although our achieved efficiency is not as high as those of traditional objectives and metalenses, it still exhibits significant enhancement in comparison with those of amplitude-type planar diffractive lenses 9,14,15,22,29 . Direct wide-field imaging by objective chip To investigate its wide-field imaging, a knife-edge object (see its microscopic image at the leftbottom of Fig. 2f) is located at a distance of z=1.2f from the objective chip 30 These experimental results have confirmed that our objective chip has sufficient imaging ability to collect high spatial frequencies from fine details of objects, which is superior to all previous superoscillation 9, 10,12,22,28,29 and supercritical lenses [13][14][15] . Objective-chip-based reflective SCM A high-resolution reflective SCM (Fig. 3a, see its working principle in the Methods) has been built successfully due to both features (i.e., subdiffraction-limit focusing and direct wide-field imaging) of our objective chip. First, the enhanced focusing efficiency allows more light to illuminate the object in a reflective mode, which underpins subsequent collection and detection of reflected light. Figure 3b shows the experimental signals detected by the photomultiplier tube (PMT) and charge-coupled device (CCD) when the nano-objects are moved longitudinally near the focal plane of z=1 mm. It reveals that the PMT signal reaches its maximum for the in-focus (i.e., Δz=0) nano-objects and decreases gradually with the increment of the out-of-focus distance |Δz|, which is doubly checked by the CCD images (see the inserts in Fig Complex nano-objects can also be imaged with a high resolution by using our SCM. Figure 4a shows the SEM image of a dolphin (composed of 50 nm-width curves) with a total size of 8 μm×8 μm. Due to their limited resolutions, both coherent BF microscopy and traditional SCM can map only rough contours of the dolphin but lose fine details, such as the eye and tail (Figs. 4b and 4c). In comparison, our SCM can clearly resolve all these fine details ( In summary, we have proposed information entropy to evaluate the disorder of an optimized planar lens. The suggested equilibrium point S0=0.5 is used to guide the quick design of a 1-mm focal-length, high-NA and low-cost objective chip with efficiency-enhanced subdiffraction-limit focusing and direct wide-field imaging. These advantages open the way to demonstrate compact and high-resolution reflective SCM with planar lenses, which will greatly benefit from optical to biomedical imaging. Methods Design and optimization details. To avoid the creation of additional finer structures when combining the FZP and the few-ring mask, each radius ρn (n=0, 1, 2, … , N) in the N-ring mask is valued within the radii (i.e., rm) of belts in the FZP. To highlight these selected radii in the zone plates, we label them = , which means that the n th ring of the few-ring mask has the same radius as the Mn th belt of the zone plate. Although it decreases the degree of freedom to design the few-ring mask, significant benefits are achieved in simplifying the optimization and fabricating the sample. Benefiting from this design strategy, we can express the electric field of light focused by the objective chip as ℎ ( , , ) = ∑ (−1) [∑ (−1) = +1 = ] −1 =0 = ∑ ∑ (−1) + = +1 = −1 =0 = ∑ ∑ (−1) + ∫ ∫ 0 ( , ) ( ) 2 ( − 1 ) 2 0 +1 = +1 = −1 =0 ,(3) where Am is the electric field of light diffracting from the m th belt in the FZP, the wavenumber k=2π/λ, R 2 =(x-v) 2 +(y-u) 2 +z 2 , r 2 =x 2 +y 2 , tanφ=y/x, x and y are the spatial coordinates at the plane of the objective chip, x, y and z stand for the spatial position of interest. Considering the nonparaxial propagation of light in such a high-NA objective chip, we calculate Am by using the rigorous Rayleigh-Sommerfeld diffraction integral without any approximation 7,44 Since no new belt appears in this strategy, the electric field Am can be calculated ahead of optimization and then stored in a database, thus enhancing the design speed. The particle swarm algorithm 45 is used to optimize the N-1 parameters, see the details in Supplementary Section 2. In our design, N=5 is employed with 4 unknown parameters, which can be determined with the values of M1=283, M2=850, M3=1046 and M4=1258 by carrying out 500 iterations in ~6 minutes in a personal computer (Intel Core i5-7500 CPU 3.40 GHz, 32G RAM). In each iteration, the optimization algorithm contains 20 populations, each of which stands for one design of objective chip. If our design strategy with the pre-calculated database is not used, we can roughly estimate its time cost of 3.8×10 4 (=3.8×20×500) hours to finish the design by running 500 iterations, because it will take ~3.8 hours to calculate the focal field of a single objective chip by numerically integrating all the zones with Rayleigh-Sommerfeld diffraction under the same computation environment. Thus, our design strategy accelerates the optimization by a factor of 3.8×10 5 . In our designed objective chip, the phase-reversed zones contain two parts: 1) from m=284 to m=850 and 2) from m=1047 to by considering an actual phase difference of ∆ as ℎ ( ) = ∑ ∑ •∆ • 1+(−1) + 2 = +1 = =4 =0 ,(4)ℎ = ∫ ∫ ℎ ( ) 2 0 0 ,(5) where the intensity ℎ ( ) = | ℎ ( )| 2 . Similarly, we can acquire the energy flux of the standard binary-phase zone plate in the same area with = ∫ ∫ ( ) 2 0 0 , where = | ( )| 2 = |∑ (−1) =6391 =0 | 2 . Finally, the theoretical focusing efficiency of the objective chip can be evaluated as Work principle of objective-chip-based reflective SCM. A schematic diagram of the objectivechip-based reflective SCM is given in Fig. 3a. The confocal configuration consists of our objective chip and two tube lenses (TL1 and TL2), where their focal planes are conjugated with that of the objective chip. The objective chip is illuminated by a collimated light beam with a wavelength λ=405 nm. ℎ = • ℎ ,(6) To increase the signal-to-noise ratio of the entire system, we suppress the light reflected from Supplementary Fig. 5b for a better observation. is calculated by using the average intensity along the long side of the red box in the insert of (e)). To evaluate its resolving power, the spatial coordinate y is scaled down by its magnification of 5. The experimental ESF is fitted by an error function, the deviation of which outputs the line spread function (LSF). (h) Retrieved modulation transfer function (MTF, solid-circle curve) of the objective chip by using the Fourier transform of the achieved LSF in (g). The diffraction limit (dashed curve) is also provided for a better comparison. . . Section 1. Mathematical basis for Strehl ratio and focal size of an objective chip To reveal the relationship between the information entropy S and optical properties of an objective chip, we investigate its Strehl ratio and focal size under different deviation probability p1. For a given p1, the relative Strehl ratio and focal size change because the locations of the deviated zones in the objective chip are different. It means that the relative Strehl ratio and focal size have a certain range with the fixed minimum and maximum values, which can be determined mathematically by using diffraction properties of each zone. First, we derive the minimum and maximum Strehl ratios. Because the binary-phase objective chip is reported here with the modulation phase of 0 and π, we can directly use its complex modulation of 1 and -1, respectively. In our design strategy, the objective chip is functionally divided into a binary-phase FZP and an N-ring phase mask. The phase of π in the N-ring phase mask realizes the reversal (from 1 to -1, or from -1 to 1) of the complex modulation. Assuming that the odd and even rings have the phase of 0 and π respectively, it means that the electric fields contributed by the SRmax are shown in Supplementary Fig. S1a, exhibiting good agreement with their analytical values. Their slight deviations come from the approximations made during its derivation. Therefore, these results have confirmed that the analytical SRmin and SRmax give a good prediction for the range of Strehl ratio. Second, the focal size of an objective chip with different p1 can also be predicted by using Eq. (S1). As shown in Eq. (S1), the electric field of the objective chip is taken as the coherent super- the case of rmin, the discrepancy between the prediction and simulation increases with the increment of p1, which is caused by the large error of the approximation in Eq. (S5). In fact, for a large p1, the disorder in the objective chip is higher than the predicted one in Eq. (S5), so that the minimum focal spot has more choices with a larger range than that predicted by Eq. (S5). Despite this, our prediction in Eq. (S5) shows the same decreasing tendency when p1 increases, thereby confirming the validity of the predicted focal sizes. Section 2. Optimization of the objective chip According to the design strategy described in the main text, we implement the optimization of the objective chip with four key steps, as discussed below. where m=0, 1, 2,…, M, the wavelength λ=405 nm, the focal length f =1mm. In our design, the total number M of belts in this BPFZP is M=6391 and the radius of BPFZP is ~2.0648mm, which yields a numerical aperture (NA) of 0.9, as shown in Supplementary Fig. 2. The binary phase is employed to enhance optical efficiency of the objective chip. Calculating the focal field of each belt in the BPFZP Benefiting from our design strategy, no new ring is created during our optimization because all the radii can be described by Eq. (S6). It means that the focal field of light from each zone can be calculated ahead of optimization and then stored in a database, so that we directly revisit the relative focal field during the optimization. Thus, the time cost will be significantly shorten. Since Optimizing the 5-ring phase mask To optimize the detailed structures of the 5-ring phase mask, we use the well-matured particle swarm optimization (PSO) algorithms that have been used frequently to design various lenses, especially for super-oscillation and super-critical lenses. Considering the limited number of the fewring mask, the optimization will be implemented with its standard version of the PSO algorithm. For a 5-ring mask, the dimension of particle D is 5. In our algorithm, the size or population of the particle is 20. The details and flowchart of PSO are shown in Supplementary Fig. 4. Since the electric fields (Ar and Az) have been calculated in advance, the calculation of fitness of each particle in iteration can be finished quickly, as shown below. Supplementary Fig. 4. The detailed flowchart of our built PSO. To correlate the particle parameters [ 1 , 2 , 3 , 4 , 5 ] with the unknown structures of the designed objective chip, we define the relative NA at the each boundary of the 5-ring phase mask by using = ∑ =1 ∑ 5 =1 • 0 ,(S8) where n is the ring number of the few-ring phase mask, NA0=0.9 is used in this work. From Eq. (S8), one can induce that NA5=NA0, which means that the outer boundary of the 5-th ring in the phase mask refers to the maximum radius of 2.046 mm. The five parameters in each particle are related with the difference of NA between two neighboring rings. In this definition, we have built the oneto-one relationship between the particle parameters and the structures of the objective chip. In addition, such a definition will offer full degree of freedom to go through all the possible solutions because each xi can be valued within 0<xi<ꝏ. The normalization factor of 1 ∑ 5 =1 ⁄ is quite helpful to make the maximum value of NA0, thus enabling each particle to yield a physically meaningful objective chip. Thus, the universal properties of design an objective chip are maintained in our definition, which is an important step to implement this optimization. In one iteration, each xi is updated with the PSO algorithm (as described in the dashed rectangle of Supplementary Fig. 4). Based on the optimized xi, we derive the NA parameters in Eq. (S8), from which we find each in the designed objective chip by using the equation √ 2 + 2 ⁄ = (or = • √1 − 2 ⁄ ). To avoid the creation of additional finer structures when combining the zone plate and 5-ring mask, the optimized is approximated by the closet rm, which is labelled as = . Thus, we can find all the , hereby fixing the geometric structures of the few-ring phase mask in each iteration. According to the updated and the well-built database, we can calculate the electric fields of objective chip at two positions of interest: ( , = ) = ∑ ∑ (−1) + ( ) = +1 +1 = =4 =0 , ( = 0, ) = ∑ ∑ (−1) + ( ) = +1 = =4 =0 .(S9) From Eq. (S9), we can obtain the relative intensity = | ( , = )| 2 and = | ( = 0, )| 2 . Then, two root-mean-square error (RMSE) between ideal and simulated patterns are calculated as 1 = √ ( − ) 2 2 = √ ( − ) 2 ,(S10) where the ideal radial-position intensity =| 0 ( )| 2 , 0 is the zero-order Bessel function of the first kind, k=2π/λ is wave vector, the radial position r is valued between 0 and λ with an interval of 0.01λ (i.e., the sampling number is Nr=101), the ideal longitudinal-position intensity To show the difference from the previous planar lenses, we provide the entropy, relative Strehl ratio and focal size of various reported lenses [1][2][3][4][5][6][7] in Supplementary Fig. 5b, which is an extension of Fig. 1d in the main text for a better observation. From both figures, we can conclude that the entropy S of our proposed objective chip is the closet to the equilibrium point S0=0.5, implying the good balance between imaging and super-focusing. Note that, although Ref. 3 has nearly identical focal size and relative Strehl ratio to ours, its entropy S is much higher than S=0.535 in our objective chip, implying that its imaging capability is poor. For the other lenses, their entropy approaches 1, which is a natural result when a lens is designed only for super-focusing with high disorder. Therefore, the information entropy is a good measurement of evaluating the imaging and super-focusing capabilities in a straightforward way. To characterize the focusing capability of objective chip, we use the experimental setup as shown in Supplementary Fig. 6. The beam from a λ=405 nm laser is reshaped by a beam expander (consisting of L1, L2, and a 25 μm-diameter pinhole), yielding a fundamental Gaussian beam with the diameter of ~1 mm. To generate circularly polarized illumination, a linear polarizer and quarterwave plate are employed for obtaining the highly axisymmetric focal spot. Then, a sub-diffractionlimit focal spot generated by objective chip is projected by using the imaging system (composed of a 100× objective with 0.95 NA and a lens L3) onto the CCD camera. = − ( − ) 2 2 , = 1−√ , = 0 = 0.9 , The focusing process of this objective chip is recorded dynamically in Supplementary Video 1, which presents the focused spot near the focal plane. Such a video shows the same intensity profiles that are similar to those in Fig. 2b of main text. All these experimental results have confirmed the good focusing with our proposed objective chip. The focusing efficiency of the objective chip is measured by using the experimental setup sketched in Supplementary Fig. 7. To obtain quasi-plane-wave illumination, we reshape a λ=405 nm laser into a ~1 mm-diameter fundamental Gaussian beam by using a beam expander (containing L1, L2, and a 25 μm-diameter pinhole PH1). A circular polarizer composed of a linear polarizer (LP) and a quarter-wave plate (QWP) is used to convert the polarization of the incident beam into the circular polarization, which is helpful to achieve the circular focal spot under the high-NA focusing condition. To remove the background light from high-diffraction-order rings diffracted by the pinhole PH1, an iris aperture with its transmission area slight larger than the entrance of the objective chip is utilized here to keep the nearly uniform incident light. The second pinhole PH2 with a 150 μm diameter is placed at focal plane of objective chip to select the focused power, which is recorded as I1 by using a power meter. Due to the high NA of 0.9 in the objective chip, the divergence angle of light passed through PH2 is large. Therefore, the power-meter is located close to the PH2 for complete collection of all focused light. By removing the objective chip and the PH2 simultaneously, we can measure the total power of the incident beam, as recorded as I0. Finally, we obtain the experimental focusing efficiency of objective chip, i.e., = 1 2 = 12.3%, which is tightly consistent with the theoretical 16.9%. Although such an experimental efficiency is not so high compared with the dielectric metasurfaces, we will show its capability in optical imaging and focusing for highquality confocal microscopy. To measure the MTF of the objective chip, a self-made setup presented in Supplementary Fig. 8 is implemented to demonstrate its imaging properties in a transmission mode for a better experimental operation. Note that, the configuration of transmission or reflection will not influence the imaging properties of the objective chip. Similarly, a fundamental Gaussian beam with circular polarization is obtained by using a beam expander and a circular polarizer, and then works as the illumination beam of a knife-edge object (a 140 nm-thick Cr film coated on quartz substrate), which is mounted on 3-dimensional piezo stage (PI). To directly image the knife-edge, the objective chip mounted on a mechanical stage is placed behind the knife-edge with its structure side closer to the knife-edge for the collection of the transmitted light. To adjust the object distance d1, we move the imaging system to see the knife-edge and the structure surface of the objective chip respectively and record their corresponding positions z1 and z2. Thus, the object distance can be evaluated roughly by using 1 = | 1 − 2 |. We tune the position of objective chip so that the object distance d1=1.2f (f=1 mm is the focal length of objective chip) is achieved, which yielding the relative imaging distance of d2=6f (see Fig. 2f in the main text). The captured image is shown in the insert of Fig. 2f in the main text. To show its imaging process of the knife edge, we provide a dynamic video (see Supplementary Video 2) that records the out-of-focus and in-focus images by moving the axial position of knife-edge near d1=1.2f and simultaneously fixing the position of the imaging system. These experimental results clearly show the good imaging ability of our proposed objective chip. To test its field of view, we use a larger object of "USTC" that has a horizontal length of 310 μm, see Supplementary Fig. 8b. The object "USTC" is placed at z=1.2f, which means its magnification of 5X. The resulting image is shown in Supplementary Fig. 8c, which shows the clear image with the slightly blurred horizontal edges. It means that a bigger object with its dimension larger than 310 μm cannot be imaged with clear edges. This indicates that the field of view is 310 μm×310 μm at the magnification of 5X. These results show the wide-field image capability of our developed objective chip. Section 6. Roles of collection objective in scanning confocal microscopy To highlight the importance of the collection objective in a scanning confocal microscopy, we implement the numerical simulation of the imaging processes by using the theory of scanning confocal microscopy 8 . When the scanned object is not an infinitesimal point, the imaging resolution of scanning confocal microscopy is determined by the NA of collection objective. In our simulations, the theoretically focused spot by using our objective chip is taken as the focal field h1 of the condenser lens while the PSF of collection objective is the well-known Airy spot ℎ 2 = 1 ( ) , where NA is the numerical aperture the collection objective, k=2π/λ is the wave number, λ is the wavelength and r is the radial coordinate. The transmission T of the nano-objects is taken as its original pattern without considering the light-structure interaction, for the simplicity of the entire simulation process. For a certain scanning position (xs, ys), the electric field at the pinhole plane (i.e., the imaging plane of the collection objective) can be written as (ℎ 1 • )⨂ℎ 2 , where ⨂ stands for the convolution operation. After selected by the pinhole with a circular aperture, the total power is taken as the value of image at this scanning position (xs, ys). Thus, if all the positions in the object are scanned, we can achieve the final image. All the codes are built in a Matlab software. Supplementary Fig. 8 show the simulated images by using collection objectives with different NAs scanning images will be distorted seriously with a bad imaging resolution. In comparison, the imaging resolution is high if NA≥0.7, which suggests the smallest NA of 0.7 to achieve highresolution imaging in a SCM. Therefore, the NA of collection objective is also critical for highquality imaging of real objects in SCM. Moreover, the resolution reduces with the increment of slit width, because the contrast of image is getting lower. This result is consistent with the fact that the fixed 200nm-CTC-distance double slits will not be resolved when the double silt merges into one slit with the continuing increment of slit width. Supplementary Fig. 10 shows the simulated images that reveal the different resolution in various microscopies. As observed in our experiment, the coherent bright-field microscopy cannot resolve any double slit, suggesting its low resolving power. But, the traditional SCM by using refraction-based objective can only resolve the double slits with the CTC distance large than 240 nm, which agrees with our experimental results in Fig. 3 of main text. To show the advantages of our proposed SCMs, we give a detailed comparison with the previously reported SCMs by presenting their various parameters. One can find that we have proposed the first planar-lens-based reflective SCM and the ultra-long working distance at the level of millimeter. More importantly, we can use industrial DUV lithography to fabricate all these objective chips in a mass-product and low-cost way, which pushes this objective chip towards practical applications. Supplementary Fig. 11 shows the images of our fabricated objective chips, which validates the feasibility for industrial mass production. Supplementary Fig. 1b ) 1b. Considering that the ideal equilibrium point S0=0.5 has a small deviation probability, only 5 rings are used to avoid introducing high disorder. In our design, all radii (n=1, 2, …, 5) of these 5 rings are chosen from the radii of zones in the FZP, which means that no new finer structure is created when combining the FZP and this 5-ring phase mask. Such a design strategy has twofold significance. First, it greatly enhances the speed of optimization. Because all the structural details of the objective chip are given by the radii rm of the FZP, we can calculate the electric field of each zone ahead of optimization and then store it as a database for quick reading during the design, thus avoiding repeated calculation. For example, to design our objective chip with 6387 zones, it takes only 6 mins to run 500 iterations by using the particle swarm algorithm in a personal computer, enhancing the speed by a factor of ~3.8×10 5 (see the calculation details in the Methods). Second, it allows large-scale and low-cost fabrication of our objective chip because its smallest feature 4 = 1258 and 5 = 6391 , which yields p1=(567+212)/6391=0.122 and entropy S=0.535. . After illuminating the knife-edge object, the transmitted light is collected by our objective chip. According to the imaging formula of a lens, we can roughly estimate its imaging distance of 6f at the other side of the objective chip, thereby exhibiting an imaging magnification of 5×. The recorded image of the knife-edge object is shown at the bottom-right panel ofFig. 2f, which reveals a well-defined boundary at the edge. A dynamic video that records its imaging process by tuning the axial position of such a knifeedge object is provided in Supplementary Video 2, which is captured in a homemade measurement setup (seeSupplementary Fig.8and Supplementary Section 5).The modulation transfer function (MTF) of this objective chip is characterized by using the linescanning intensity across the edge in the image. To decrease the experimental error, we employ a mean of the line-scanning intensity (with its spatial dimension scaled down by a factor of its magnification M=5) at the red-rectangle region in the image ofFig. 2fto recover the line spread function (LSF) of this imaging configuration. After fitting the line-scanning intensity with an error function (taken as a convolution between a Gaussian function and a jump function), we carry out the deconvolution of the fitted error function, yielding the retrieved LSF(Fig. 2g). Using a Fourier transformation of the retrieved LSF, we finally obtain its MTF(Fig. 2h), which indicates a cut-off frequency of 4000 lp/mm. This implies an imaging resolution of 250 nm, which corresponds to an effective NA of 0.83 (evaluated by using 0.51λ/NAeff=250 nm) for imaging. Compared with the previous metalens with a cut-off spatial frequency of 2000 lp/mm 4 , our objective chip achieves a twofold enhancement in resolving power when operating in the mode of direct wide-field imaging. . 3b and Supplementary Video 3). The FWHM of the PMT measured intensity profile is ~5 μm, which agrees with the experimental DOF of 5.5 μm (see Figs. 2b and 2d). This result confirms that our objective chip can efficiently focus the incident beam and collect the reflected light. Second, a powerful imaging ability with an effective NA of 0.83 is required to enhance the practical imaging resolution of an SCM. Theoretically, we have already shown that the resolution of an SCM is less influenced by the NA of the collection objective 7 , which, however, is valid only for infinitesimal point objects. For real objects with finite sizes ranging from tens to hundreds of nanometers, the NA of the collection objective should be larger than 0.7 for a better resolution in an SCM, see the simulated proofs in Supplementary Section 6. Due to these two features of our objective lens mentioned above, the subdiffraction-limit focusing and the high-resolution imaging enable a reflective SCM. More experimental details about the scanning imaging are provided in the Methods.To test its resolution, we provide the imaging results of 50 nm-width and 2 -length double slits with center-to-center (CTC) distances ranging from 190 nm to 270 nm. As shown inFig. 3c, these slits are etched on a 140 nm-thick chromium film on a quartz substrate. Using a 0.9 NA objective for a fair comparison, the coherent bright-field microscope cannot resolve these double slits(Fig.3d)while conventional reflective SCM can only resolve the double slits with CTC distances larger than 240 nm(Fig.3e). In contrast, our objective-chip-based reflective CSM has an enhanced resolution so that double slits with a CTC distance of 200 nm can be distinguished with a valley of intensity in the image(Fig.3f), where the distortion is caused by mechanical variation of the sample. The qualitative comparisons among their line-scanning intensity profiles(Fig. 3g)doubly verify an imaging resolution of 200 nm achieved by our SCM. In addition, all these experimental results regarding scanning images are confirmed by our simulations (see Supplementary Section 7 andSupplementary Fig. 10) with the theory of SCM 7, 31, 32 . Fig. 4d) with a narrower line width(Fig. 4e). Furthermore, two lines with a CTC distance of 225 nm (see the dashed-red lines at the lower rows in Figs. 4b-4d) in the tail can be distinguished only by using our SCM. The low contrast of intensity in the image for our SCM comes from the relatively lower focusing efficiency of the objective chip in comparison with the traditional objective. However, it has little influence on the resolution and clarity of the image, as observed inFig. 4d.DiscussionAmong all the planar-lens-based SCMs, our current SCM has the advantages of eliminating bulky objectives, a millimeter-level working distance, reflection-mode operation, working for both transparent and non-transparent substrates, and a competitive resolution of 0.49λ, as shown inSupplementary Table 1. For commercial objectives, pursuing a high NA and long working distance simultaneously leads to an increment in the diameters of optical elements and the accompanying optical aberrations that need large-scale nonspherical surfaces for correction 3 , thereby yielding extremely high costs in both the fabrication and design of elements. This issue does not exist in our objective chip, where the smallest feature of λ/2 will not change with increasing NA and focal length, so the same fabrication tools and design methods reported in this work can also be used to develop more advanced objective chips with even larger NAs and longer focal lengths.Our objective chip has a fabrication cost of ~$42 dollars (estimated by the total price of 300 chips in an 8-inch quartz wafer, seeSupplementary Fig. 11), which is ~100 times cheaper than the price of commercial objectives. This objective chip has a volume of 4 mm×4 mm×0.5 mm, which indicates a shrinking factor of 4300 (~3.6 orders of magnitude) compared with traditional objectives (ZEISS, EC Epiplan-Neufluar 100×, NA 0.9, M27).Developing this reflective configuration makes a truly important step to push the technology of planar-lens-based SCMs towards practical applications, because many samples have opaque substrates that are incompatible with all the previous planar-lens-based SCMs. Note that the objects used in this work are made in a high-reflectivity metal film, which helps to enhance the imaging contrast. If the difference between the optical reflection of the object and its surrounding backgroundis not obvious, one should increase the optical efficiency of the objective chip and the sensitivity of the optical detector. The focusing efficiency of the objective chip can be enhanced further if multilevel phase elements 33 or high-efficiency dielectric metasurfaces 4, 5, 34-43 are used. The detector can also be updated to the single-photon level for a better recording of collected photons by our objective chip, enabling the characterization of less-reflective biological tissues and cells even in a living body. . Although thousands of belts are included in this objective chip, only N-1 variables (i.e., M1, M2, …, MN-1 because M0=0 and MN=6391) are unknown in Eq. (3) because all rm are given. Since the phase jump of π occurs at the Mn th belt (n=1, 2, …, N-1), both the Mn th and (Mn+1) th belts are combined into one, leading to the belt number of M-(N-1) in the final objective chip. m=1258, resulting in p1=(567+213)/6391=0.122 and p2=1-p1=0.878. According to the definition of information entropy, we have S=0.535, which is tightly close to the equilibrium point S0=0.5. Fabrication details. The designed objective chip is fabricated through deep-ultraviolet (DUV, Nikon S204) exposure process. The quartz substrate is first deposited with 200 nm-thick aluminum using a physical vapour deposition (PVD) system (AMAT Endura). Then, a 300 nm-thick positive photoresist (UV135) is coated and baked. Subsequently, the photoresist is patterned using the DUV lithography. After development, the aluminum film without photoresist is etched sufficiently by an inductively coupled plasma (ICP) etching system (LAM 9600). Thus, the patterns of the objective chip are transferred into the aluminum film after removing the residual photoresist. Next, using the aluminum film as masking layer, the quartz substrate is etched for a designed thickness by an inductively coupled plasma-reactive ion etching (ICP-RIE) system (Oxford, Plasma Pro System100 ICP380). Finally, the aluminum hard mask is removed by Tetramethylammonium Hydroxide (TMAH, 2.5%) solvent, yielding the expected phase-type objective chip. Theoretical efficiency of the objective chip. To obtain the theoretical efficiency of the objective chip, we first calculate the electric field of the m th belt in the zone plate at the focal plane (ignoring the influence from the width error of etched belts). For the sake of convenient simulation, we calculate the one-dimensional field along the radial direction within the range of λ (starting from r=0), where the focused light is concentrated. To evaluate the experimental error, we update Eq. (3) where M0=0, M1=283, M2=850, M3=1046, M4=1258, and M5=6391, ∆ = 2 ( − 1) ℎ is the phase difference between etched and unetched zones in the objective chip at a wavelength of λ=405 nm, the refraction index of the quartz substrate is n=1.47 and the experimental etching depth ℎ =530 nm (referring to ∆ = 1.23 ). To further evaluate the energy flux in the circular area with a radius of λ at the focal plane of the objective chip, we employ the expression: Figure 1 . 1the back-surface (i.e., the bare-quartz side without any structure) of the objective chip by utilizing two orthogonal linear polarizers (LP1 and LP2) and a quarter-waveplate (QWP) thin film, as sketched inFig. 3a. Because both LP1 and LP2 have orthogonal transmission directions, the reflected light from the back-surface of the objective chip is blocked efficiently, leaving a very weak background with a four-lobe pattern (see the insert inFig. 3b). It is important to note that such a four-lobe pattern has a dark center, where the signal light reflected by the objects is located, thus leading to spatial separation between the noise and signal. The 60 μm-thick QWP thin film (with an angle of 45 degrees between its fast axis and the transmission directions of both LPs) is adhered to the structure side of the objective chip to obtain circular polarization (CP), which enables us to realize a circularly symmetric focal spot for isotropic scanning of the image. Moreover, the reflected CP signal light from nano-objects passes through the QWP thin film again and is converted into linear polarization with its direction aligned to the transmission direction of LP2. Therefore, the second linear polarizer (LP2) can block the noise light reflected from the back surface of the objective chip and transmit the signal light efficiently, thus increasing the signal-to-noise ratio of the PMT signals.After tuning the signal, we coarsely move the objective chip mounted on the electric stage toward the nano-objects. When the nano-objects are close to the focal plane, the PMT signals behave like that shown inFig. 3b, having a peak when the nano-objects are in focus. At the same time, the recorded pattern at the CCD becomes the smallest. Due to the conjugation relationship between the objects and the PMT, we can adjust the PMT to collect an optical signal at the focal plane of TL2 filtered by a 10 μm-diameter pinhole. Then, we finely move the nano-objects mounted on the 3D piezo stage to the focal plane by observing the signal collected by the PMT. When the PMT signal reaches its maximum (seeFig. 3b), we assume that the nano-objects are at the focal plane of the objective chip. Finally, the nano-objects can be scanned at the focal plane, and the signal collected by the PMT can be recorded simultaneously to complete the scanning image.Experimental details about the scanning imaging. The 3D piezo stage (PI-545.3R8S) and the controller (PI-E727) are integrated into a single device with a scanning resolution of ~1 nm. The PMT has the module of the WiTec 3000R series. We utilize the LabVIEW language to control the movement of the 3D piezo stage and read the signal collected by the PMT with a DAQ card (NI USB-6000, 12-bit, sampling rate: 10 Ks/s) simultaneously. ) is employed here as the pinhole.In the experiment testing the imaging resolution, the scanning range of double slits is 3 μm×1 μm with 100×100 pixels, which takes ~15 minutes to finish one image. The scanning speed can beupdated further by using a high-speed stage and digital-analog converter. The 2.4 μm×0.6 μm range of scanning images in Figs. 3(c)-(f) is employed to fully cover the objects. In the experiment for imaging complex nano-objects, the scanning range is 9 μm×9 μm with 150×150 sampling points, which takes ~34 minutes. Only the 8 μm×8 μm range is shown in Figs. 4(b)-(d) to highlight more details of the images. Acknowledgement K.H. thanks the CAS Project for Young Scientists in Basic Research (Grant No.YSBR-049), the National Natural Science Foundation of China (Grant No. 12134013), the National Key Research and Development Program of China (No. 2022YFB3607300), the CAS Pioneer Hundred Talents Program, and support from the University of Science and Technology of China's Centre for Micro and Nanoscale Research and Fabrication. J.T. thanks the A*STAR AME IRG program (Grant No. A2083c0058). Author contributions K. H., J. T. and C. Q. conceived the idea. J. H. and K. H. performed the simulations. D. Z. and H. L. prepared and fabricated optical samples. J. H. and D. Z. built up the experimental setup and performed the characterization. J. H., K. H., C. Q. and J. T. wrote the manuscript. K. H., J. T., and C. Q. supervised the overall project. All authors discussed the results, carried out the data analysis and commented on the manuscript. Working principle of the bifunctional objective chip. (a) Sketch of the objective chip with both focusing and imaging functionalities. (b) Design principle of the objective chip composed of a FZP and a few-ring phase with weak disorder. (c) Simulated relative Strehl ratio and spot size under different deviation probabilities of 0≤ p1 ≤0.35. The information entropy S is also provided together as a function of p1. The correlation among the Strehl ratio (blue), p1 and S yields a threedimensional plotting that visualizes the underlying link between the Strehl ratio and S straightforwardly. Similarly, the relationship between spot size (yellow) and S is also shown with a three-dimensional configuration. Both three-dimensional drawings are projected to the longitudinal plane of S=0 for a better observation, where the Rayleigh criterion (RC) and super-oscillation criterion (SOC) are shown in red-dashed lines to distinguish the subdiffraction-limit focusing. (d) Entropy S, relative Strehl ratio and focal size of other reported planar lenses that provide the structural parameters (which are used to output p1 for calculating entropy S) in their publications.For a fair comparison, all the focal sizes are normalized to the Rayleigh diffraction limit of rRC=0.51λ/NA (in terms of FWHM). The solid circles denote these three parameters of all these reported lenses (distinguished by colors), while the hollow circles are their projections on different two-dimensional planes for a clear demonstration. Its extended version with more data is provided in Figure 2 . 2Focusing and imaging properties of our objective chip. (a) Simulated (curves) and experimental (dots) widths of belts at the different regions of our objective chip. Inset: SEM images of the different regions by addressing the corresponding zone numbers. The etched depth (left-bottom) around the center region of the objective chip is measured experimentally by using a profilometer. (b) Optical field near the focal plane of the objective chip. Cross sections of the measured intensity profiles are shown in the left panel, while the right panel shows a comparison between the simulated and experimental on-axis line-scanning intensity profiles. (c) Simulated (curve) and experimental (dots and asterisks) line-scanning intensity profiles at the focal plane of objective chip. Their 2-dimensional intensity profiles in the region of 1.2 μm×1.2 μm are provided in the insert. (d) Lateral (squares for the x direction and triangles for the y direction) FWHMs of the measured spot near the focal plane (i.e., Δz=z-f=0). RC: Rayleigh criterion (0.51λ/NA); SOC: Superoscillation criterion (0.358λ/NA). (e) Simulated (curve) and experimental (triangle) efficiency when the phase difference Δφ between two neighboring etched and unetched parts changes from 0 to 2π. (f) Sketch for wide-field imaging by using our objective chip. The object and image distances are 1.2f and 6f (the focal length f=1mm) respectively, yielding a magnification of 5×. Such a magnification is chosen to avoid optical aberration, while it is enough to demonstrate the capability of collecting light with high spatial frequencies. Inserts: the knife-edge object (left, captured by using a reflective microscope that generates bright chromium film and dark quartz substrate) and its image (right) taken by using our objective chip. (g) Experimental edge spread function (ESF, which Figure 3 . 3Reflective scanning confocal microscopy based on our objective chip. (a) Sketch of the optical setup of the objective-chip-based reflective SCM. LP: linear polarizer; BS: beam splitter; QWP: quarter wave plate; L: lens; PH: pinhole; PMT: photomultiplier tube. (b) Detected signals (PMT, curve) and images (CCD, inserts) when the nano-objects are scanned with the out-of-focus distance Δz=z-f. In the CCD images, the dashed circles denote the position of the focused signal light. In the PMT signals, the nonzero background (~0.1) is caused by the incompletely suppressed light (i.e., the four-lobe patterns) reflected from the back surface of the objective chip. (c-f) Double slits (c, SEM) and their images by using coherent bright-field microscopy (d), traditional RSCM (e) and objective-chip-based RSCM (f). The CTC distances of double slits Ⅰ to Ⅵ are 190 nm, 200 nm, 220 nm, 240 nm, 250 nm and 270 nm, respectively. The height and width of each slit are 2 μm and 50 nm respectively. Scale bar: 300 nm. (g) Line-scanning intensity profiles of images by using different microscopies. Figure 4 . 4Imaging complex nano-objects by using different microscopies. (a) SEM image of a "dolphin" object composed of 50 nm-width curves. (b-d) Images (upper row) by using coherent bright-field microscopy (b), traditional RSCM (c) and objective-chip-based RSCM (d). Their zoomed-in images with more details are shown in the lower row. (e) The line-scanning profiles of the imaged dolphins (along the colored lines of the upper low in b-d) by using coherent bright-field microscopy (green circles), traditional RSCM (triangles) and objective-chip-based RSCM (red squares). These line-scanning data are fitted by using a Gaussian shape to guide the eyes. .... 5 Section 3. Experimental characterization of focusing properties of the objective chip . 10 Section 4. Measuring the focusing efficiency of the objective chip .... 11 Section 5. Measuring modulation transfer function (MTF) of the objective chip . 12 Section 6. Roles of collection objective in scanning confocal microscopy . 13 Section 7. Simulated images by using different microscopies for comparison . 15 where. 5 . 5zones in the even rings are removed from those of the standard FZP and then are used to interfere constructively with those of the zones in the odd rings. For the objective chip containing a N-ring phase mask, Eq. (3) describing the total electric fields of our objective chip can be rewritten as a set of the indices of all the phase-reversed zones (i.e., contained in the even rings of the N-ring phase mask). For our design strategy used in this work, we can obtain the number of the set by using M•p1, where M is the total number of zones in the corresponding FZP and p1 is the deviation probability (see Eq. (1) in the main text) of the phase-reversed zones. Considering its university, Eq. (S1) is valid for all binary-phase planar lenses. After substituting Eq. (S1) into Eq. (2) of main text, the relative Strehl ratio can be EFZP(0, 0, f) is in phase with a0=(-1) m Am(0, 0, f). Meanwhile, for different m, the item a0 is also in phase with each other and nearly a constant with a slow variation from a0min=0.87 to a0max=2, which can be numerically calculated by using the rigorous Rayleigh-Sommerfeld diffraction integral. Due to their feature of slow variation, the on-axis intensity 0 from all the zones with the zone indices ∈ are assumed to be identical. Thus, 0.87≤a0≤2. Eq. (S3) reveals the direct link between SR and p1 (which is a key parameter to evaluate the disorder of a planar lens). Based on it, we can evaluate the range of SR by determining its minimum and maximum values. By carrying out straightforward derivations, we have the minimum value Note that the analytical SRmin and SRmax depend on only the deviation probability p1, and are therefore valid for various objective chips with different N. Supplementary Fig. 1. Analytical and simulated Strehl ratio (a) and focal size (b) under the different deviation probability p1. The analytical Strehl ratios are determined by using Eq. (S3), which the analytical focal sizes are obtained by numerically solving Eqs. (S4) and (S5). To verify the analytical SRmin and SRmax, we have simulated the range of SR by using the proposed objective chip with a 5-ring phase mask. The limited N=5 of the phase mask allows us to go through all possible solutions quickly without any optimization because all the Am can be calculated ahead. By controlling the number and position of the phase-reversed zones in the 5-few ring phase mask, the simulations are implemented within the range of 0 ≤ 1 ≤ 0.35 with an interval of 0.01, which is enough here because low disorder 1 is important to develop the objective chip with good balance between imaging and super-focusing. The simulated SRmin and position of the electric fields from all the zones. Diffraction behavior of each zone is important in predicting the focal size of the objective chip. Because the width of each zone is small, the diffraction field from each zone is mainly determined by its focusing angle between the outmost boundary of each zone and optical axis. When the focusing angle is large, the relative focal spot size of diffraction field from one zone is small; vice versa. For an objective chip, its maximum focal spot is achieved when the contribution from the outermost zones is small, where the electric field at phases of the outermost zones are reversed with a deviation probability p1. From Eq. (S4), we numerically predict the maximum focal size, which depends on only p1. In contrast, when the phases of the inner zones are reversed, the minimum focal size can be predicted by using the electric field at the focal plane Eqs.(S4) and (S5), we can calculate the maximum and minimum focal spots, as shown in the solid lines in Supplementary Fig. S1b. Because both Eqs. (S4) and (S5) have no limitation about the number N, we use our proposed objective chip with a 5-ring phase mask to verify the predicted minimum (rmin) and maximum (rmax) focal size. Similarly, we go through all the possible solutions by changing the number and position of the phase-reversed zone in the second and fourth rings of the few-ring phase mask, which can be implemented together with the above calculation of the Strehl ratio. The simulated rmin and rmax are provided in Supplementary Fig. 1b. Both the predicted and simulated rmax agree with each other for the interested range of 0 ≤ 1 ≤ 0.35. However, for 2. 1 . 1Determining the radius of each belt in a standard zone plate Supplementary Fig. 2. Sketch of standard zone plate with 6391 belts. The imaging parameters are λ=405 nm, f=250 μm, which indicates a radius of 2 mm.Considering the non-paraxial feature of this objective chip, a rigorous formula of standard binary phase Fresnel zone plate (BPFZP) should be used to calculate the radius rm of the m th we need to evaluate the lateral focal size and the longitudinal depth of focus, both focal fields along the radial (i.e., r) and longitudinal (i.e., z) direction are calculated ahead, where the positons of interest are: 1) the lateral positions 0≤r≤λ at the focal plane z=1000 μm; 2) the longitudinal positions 950 μm≤z≤1050 μm at the on-axis position r=0. By using Rayleigh-Sommerfeld diffraction theory without any approximation for high accuracy, we calculate the focal fields of each belt along the lateral and longitudinal positions, and store them in two matrices (i.e., Ar and Az, see Supplementary Figs. 3a and 3b) respectively. According to the electric field stored in the 1 th , 3000 th and 6391 th column of Ar and Az, the normalized intensity of diffraction field at two target positions of corresponding belts of BPFZP are exemplified in Supplementary Figs. 3c and 3d, respectively. To show its convenience, the focal fields at two target positions for a BPFZP can be calculated as ) is the radial-position electric field of light diffracting from the m th belt in the zone plate and saved in the m th column of Ar, ( ) is the longitudinal-position electric field of light diffracting from the m th belt in the zone plate and stored in the m th column of Az. According to this definition, ( )= ( )=0 when m=0. Based on these database, we can calculate any focal field of the objective chip only if the phase of each zone is given. Supplementary Fig. 3. Data preparation for optimization. The Ar (a) and Az (b) database show the way of saving the pre-calculated data in our optimization. The normalized line intensity at the focal plane (c) and on the optical axis (d) of the 1 th , 3000 th and 6391 th belt of BPFZP. the longitudinal position z is valued between 950 μm and 1050 μm with an sampling interval of 0. 1 μm (i.e., the sampling number is Nr=1001). The RMSE1 and RMSE2 are used to evaluate the electric fields at the radial and longitudinal positions. Based on them, we can build the cost function by using CF=C•RMSE1 +RMSE2, where the positive constant C can be adjusted according to any special requirement. In this work, C=2 is used to realize the sub-diffraction-limit focusing. After 500 iterations in the PSO algorithm, we finally obtain the M1=283, M2=850, M3=1046, M4=1258 and M5=6391, and the additional phase mask can be sketched in Supplementary Fig. 5. The combination between standard zone plate (Supplementary Fig. 2) and additional phase masks (Supplementary Fig. 5a) forms the objective chip. The simulated intensity profiles at the longitudinal and radial positions are provided in Figs. 2b and 2c of main text, respectively. Supplementary Fig. 5. Design results of a 5-ring phase mask. (a) Phase profile of the optimized few-ring mask. The ideal phase difference Δφ between two neighboring rings is π. (b) Entropy S, relative Strehl ratio and focal size of other reported planar lenses with the structural parameters (which is used to output p1 for calculating entropy S) provided in their corresponding publications. This figure is an extension of Fig. 1d in the main text for a better observation due to the overlay of data. Section 3 . 3Experimental characterization of focusing properties of the objective chip Supplementary Fig. 6. Experimental setup for characterizing the focusing capability of the fabricated objective chip. The beam expander consists of lenses L1 and L2 (with the focal lengths of 25.4 mm and 300 mm respectively) and a 25μm-diameter pinhole. The imaging system consists of 100× objective with 0.95 NA, a lens L3 (with a focal length 400 mm) and a CCD camera. LP: linear polarizer. QWP: quarter-wave plate; Obj. chip: objective chip; Section 4 . 4Measuring the focusing efficiency of the objective chip Supplementary Fig. 7. The schematic diagram of the experimental setup for measuring the focusing efficiency of the objective chip. The beam expander is composed of two lenses L1 and L2 (with their focal lengths of 25.4 mm and 300 mm, respectively) and a 25 μm-diameter pinhole (PH1). LP: linear polarizer; QWP: quarter-wave plate; Ape.: aperture; Obj. chip: objective chip; PH2: pinhole with 150 μm diameter. Section 5 . 5Measuring modulation transfer function (MTF) of the objective chip Supplementary Fig. 8. Sketch for the experimental setup to measure MTF of the objective chip when working in an imaging mode. (a) The beam expander is made of two lenses L1 and L2 (with their focal lengths of 25.4 mm and 300 mm respectively) and a 25 μm-diameter pinhole (PH). The imaging system consists of a 20× objective, a lens (with a focal length of 400 mm) and a CCD camera. LP: linear polarizer; QWP: quarter-wave plate; Obj. chip: objective chip; d1 and d2 are the object and image distances, respectively. (b-c) Experimental measurement of field of view by using a large object "USTC" (b) with its horizontal length of 310 μm. The relative image is shown in (c). Supplementary Fig. 9 . 9Simulated images of double slits by using collection objectives with different NAs. (a) Double slits with the center-to-center distances of 200 nm and the silt widths ranging from 10nm to 100nm. The length of slits are 2 μm. (b) Simulated scanning images under different NAs (0.5-0.9) and slit widths (10nm-100nm). (c) The line-scanning profiles of these slit images for the collection objectives with different NAs.Section 7. Simulated images by using different microscopies for comparison Supplementary Fig. 10. Simulated images of double slits with different CTC distances. (a) Sketch of double slits (top row) and their simulated images by using coherent bright field microscope (second row), traditional RSCM (third row) and objective-chip-based RSCM (forth row). The height and width of slit are 2 μm and 50nm, respectively. The CTC distances of double slits are 190 nm, 200 nm, 220 nm, 240 nm, 250 nm and 270 nm, respectively. Image Size: 2.4 μm×0.6 μm. Scale bars: 300 nm. (b) The line-scanning profiles of these simulated images by addressing the CTC distances of double slits. By using the imaging theory, we provide a numerical simulation of images by using different microscopies. To simulate the imaging results from different types of microscopes, these double slits with the fixed 50-nm width and varied CTC distance are employed as the objects for keep the consistence with our experimental cases. To simulate the images of doubles slits by coherent brightfield microscope, the convolution between the objects and an Airy spot profile 1 ( ) with 0.9 NA is employed. Similarly, the simulation of traditional RSCM and objective-chip-based RSCM is implemented according to the theory of SCM, as mentioned in Ref 8 . Both PSFs of condenser lens and collector lens in the traditional RSCM are the Airy spots with the amplitude of 1 ( ) , where NA=0.9 for a fair comparison. For our objective-chip-based RSCM, the PSF of condenser lens is the simulated electric field of our objective chip and the PSF of collector lens is Airy spot 1 ( ) with a measured NA of 0.83. Table of Contents ofSection 1. Mathematical basis for Strehl ratio and focal size of an objective chip . 2 Section 2. Optimization of the objective chip Table 1 . 1A comparison among far-field label-free SCMs based on planar lensesReference Material Thickness of lens Type of lens λ Ambient Medium Working Mode Working Distance Substrate Center To Center Distance Rogers et al. 9 Al 100 nm Amplitude 640 nm oil Transmission 10.3 μm quartz 315 nm(0.492 ) Qin et al. 10 Cr 100 nm Amplitude 405 nm air Transmission 55 μm quartz 228 nm(0.563 ) Chen et al. 11 TiO2 600 nm Phase 532 nm oil Transmission 125 μm quartz 400 nm(0.752 ) Yuan et al. 12 Gold 100 nm Amplitude 800 nm air Transmission 10 μm quartz 320 nm(0.4 ) Wang et al. 13 MoS2 10 nm Amplitude 450 nm air Transmission 20 μm sapphire 200 nm(0.444 ) This work SiO2 527 nm Phase 405 nm air Reflection 1000 μm quartz 200 nm(0.494 ) Competing financial interestsThe authors declare no competing financial interests.Supplementary Figure 11. 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{'abstract': 'Planar lenses with optimized but disordered structures can focus light beyond the diffraction limit. However, these disordered structures have inevitably destroyed wide-field imaging capability, limiting their applications in microscopy. Here we introduce information entropy S to evaluate the disorder of an objective chip by using the probability of its structural deviation from standard Fresnel zone plates. Inspired by the theory of entropy change, we predict an equilibrium point S0=0.5 to balance wide-field imaging (theoretically evaluated by the Strehl ratio) and subdiffraction-limit focusing. To verify this, a NA=0.9 objective chip with a record-long focal length of 1 mm is designed with S=0.535, which is the nearest to the equilibrium point among all reported planar lenses. Consequently, our fabricated chip can focus light with subdiffraction-limit size of 0.44λ and image fine details with spatial frequencies up to 4000 lp/mm in experiment. These unprecedented performances enable ultracompact reflective confocal microscopy for superresolution imaging.', 'arxivid': '2303.11528', 'author': ['Jun He \nDepartment of Optics and Optical Engineering\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n', 'Dong Zhao \nDepartment of Optics and Optical Engineering\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n', 'Hong Liu \nInstitute of Materials Research and Engineering\nAgency for Science Technology and Research (A*STAR)\n2 Fusionopolis Way#08-03, 138634InnovisSingapore, Singapore\n', 'Jinghua Teng \nInstitute of Materials Research and Engineering\nAgency for Science Technology and Research (A*STAR)\n2 Fusionopolis Way#08-03, 138634InnovisSingapore, Singapore\n', 'Cheng-Wei Qiu c.q.chengwei.qiu@nus.edu.sg \nDepartment of Electrical and Computer Engineering\nNational University of Singapore\n4 Engineering Drive 3117576Singapore, Singapore\n', 'Kun Huang huangk17@ustc.edu.cn \nDepartment of Optics and Optical Engineering\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n'], 'authoraffiliation': ['Department of Optics and Optical Engineering\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina', 'Department of Optics and Optical Engineering\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina', 'Institute of Materials Research and Engineering\nAgency for Science Technology and Research (A*STAR)\n2 Fusionopolis Way#08-03, 138634InnovisSingapore, Singapore', 'Institute of Materials Research and Engineering\nAgency for Science Technology and Research (A*STAR)\n2 Fusionopolis Way#08-03, 138634InnovisSingapore, Singapore', 'Department of Electrical and Computer Engineering\nNational University of Singapore\n4 Engineering Drive 3117576Singapore, Singapore', 'Department of Optics and Optical Engineering\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina'], 'corpusid': 257636858, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 28311, 'n_tokens_neox': 24270, 'n_words': 15161, 'pdfsha': '1057fb606c883415cb17cdf7d7d9426749bca98d', 'pdfurls': ['https://export.arxiv.org/pdf/2303.11528v1.pdf'], 'title': ['An entropy-controlled objective chip for reflective confocal microscopy with subdiffraction-limit resolution', 'An entropy-controlled objective chip for reflective confocal microscopy with subdiffraction-limit resolution'], 'venue': []}

arxiv

Comment on "Witnessed entanglement and the geometric measure of quantum discord" 9 Jan 2013 Swapan Rana Physics and Applied Mathematics Unit Indian Statistical Institute 203 B T RoadKolkataIndia Preeti Parashar Physics and Applied Mathematics Unit Indian Statistical Institute 203 B T RoadKolkataIndia Comment on "Witnessed entanglement and the geometric measure of quantum discord" 9 Jan 2013(Dated: December 11, 2013) In a recent article [Phys. Rev. A 86, 024302 (2012)], the authors have derived some hierarchy relations between geometric discord and entanglement (measured by negativity and its square). We point out that these results are incorrect and give analytic counterexamples. We also discuss briefly the reason for such violations.PACS numbers: 03.67. Mn, 03.65.Ud We start with the definition of GD and negativity from Ref.[1]. For an m ⊗ n (m ≤ n) state ρ, they have definedwhere Ω is the set of zero-discord (or classical-quantum) states given by ξ = p i |i i| ⊗ ρ i and X (p) is the Schatten p norm given by X (p) = {Tr[X † X] p/2 } 1/p . The negativity was defined bywhere ρ T A is the partial transposition of ρ with respect to A and W is any optimal entangled witness. Attempting to prove a conjecture made in [2], they have claimed (Eq. 17 therein) that all bipartite states satisfyWe first observe a typo that the two definitions of negativity in Eq. (2) are not equal, as the quantity in Eq. (2b) is the sum of absolute values of negative eigenvalues of ρ T A whereas that in Eq. (2a) is just double of it.As a first gap in their derivation (which could be taken as another typo, though), we note that they have not normalized D (2) to have maximum value unity, as has been done in the original paper[2]. As a result, if we take Eq. (2a) as definition of negativity, Eq.(3)is not necessarily satisfied even by the two-qubit maximally entangled state (any one of the Bell states has D (2) = 1/2 whereas N = 1). The importance of normalization could be found in Refs.[3,4]. Now we will try to remove all these (possible) typos and show that the relation (3), whether normalized or not, is always violated by some states.Let us first consider the case when D (2) is normalized (taking D = m m−1 D (2) ) and Eq. (2a) is taken as the definition of negativity. Then it becomes the original conjecture (D ≥ N 2 /(m − 1) 2 ) made in Ref.[2], which we have refuted recently in Ref.[5]. Note that the normalizing factor * swapanqic@gmail.com † parashar@isical.ac.in m/(m − 1) > 1 and hence this case also includes the case when D (2) is not normalized and Eq. (2a) is taken as the definition of negativity. Now we will give an analytic example to show that there are states violating even the weaker relation [6]Consider the m ⊗ m Werner state given bywhere F = |k l| ⊗ |l k| and set m = 8, z = −1. Now, if we consider the matrix form of ρ w (in computational basis) as the state of a 2 ⊗ 32 system, the left hand side of Eq. (4) becomes 1/49 while the right hand side becomes 25/784. Though we have used the formula for D (2) developed in Ref.[7] (which is exact for 2 ⊗ n states), a measurement in computational basis will yield the result. Any value of z ∈ [−1, −34/43) will also work well. We note that for large enough n, the Refs.[3,8]give enough intuition for violation of this relation. Nonetheless, the analytic counterexample makes it more explicit. The authors of Ref.[1] also proposed to take D (1) as a proper measure of geometric discord and derived the hierarchy relation (Eq. 27 therein)However, as can be seen easily, this result is also not correct. It is well known that the trace distance satisfies ρ − σ (1) ≤ 2, for all ρ and σ [9]. Hence we must have D (1) ≤ 2 whereas N can take value up to (m − 1)/2 (for example consider the Bell state in m ⊗ m). Taking the identity matrix as the classicalquantum state (need not be optimal), we see that the relation is violated by 4 ⊗ 4 Bell states.As has been pointed out in Ref.[3], the violation for D (2) stems from the fact that the Hilbert-Schmidt norm is not monotone-D (2) could be increased or decreased by adding or removing a factorized local ancilla. We would like to mention that the trace norm being monotone, does not suffer from this problem. Thus the proposal of taking D (1) as a good measure, is interesting and might be worth investigating. However, as we pointed out here, establishing any interrelation should be done carefully. Another point of concern regarding the use of the trace norm is that its analytic calculation is very difficult In a recent article [Phys. Rev. A 86, 024302 (2012)], the authors have derived some hierarchy relations between geometric discord and entanglement (measured by negativity and its square). We point out that these results are incorrect and give analytic counterexamples. We also discuss briefly the reason for such violations. We start with the definition of GD and negativity from Ref. [1]. For an m ⊗ n (m ≤ n) state ρ, they have defined D (p) (ρ) = min ξ∈Ω ρ − ξ p (p) (1) where Ω is the set of zero-discord (or classical-quantum) states given by ξ = p i |i i| ⊗ ρ i and X (p) is the Schatten p norm given by X (p) = {Tr[X † X] p/2 } 1/p . The negativity was defined by N(ρ) = ρ T A (1) − 1 (2a) = max{0, − min 0≤W T A ≤I Tr(Wρ)} (2b) where ρ T A is the partial transposition of ρ with respect to A and W is any optimal entangled witness. Attempting to prove a conjecture made in [2], they have claimed (Eq. 17 therein) that all bipartite states satisfy D (2) ≥ N 2 (m − 1) 2(3) We first observe a typo that the two definitions of negativity in Eq. (2) are not equal, as the quantity in Eq. (2b) is the sum of absolute values of negative eigenvalues of ρ T A whereas that in Eq. (2a) is just double of it. As a first gap in their derivation (which could be taken as another typo, though), we note that they have not normalized D (2) to have maximum value unity, as has been done in the original paper [2]. As a result, if we take Eq. (2a) as definition of negativity, Eq. (3) is not necessarily satisfied even by the two-qubit maximally entangled state (any one of the Bell states has D (2) = 1/2 whereas N = 1). The importance of normalization could be found in Refs. [3,4]. Now we will try to remove all these (possible) typos and show that the relation (3), whether normalized or not, is always violated by some states. Let us first consider the case when D (2) is normalized (taking D = m m−1 D (2) ) and Eq. (2a) is taken as the definition of negativity. Then it becomes the original conjecture (D ≥ N 2 /(m − 1) 2 ) made in Ref. [2], which we have refuted recently in Ref. [5]. Note that the normalizing factor * swapanqic@gmail.com † parashar@isical.ac.in m/(m − 1) > 1 and hence this case also includes the case when D (2) is not normalized and Eq. (2a) is taken as the definition of negativity. Now we will give an analytic example to show that there are states violating even the weaker relation [6] m m − 1 D (2) ≥ N 2 (m − 1) 2 = λ i <0 λ i (ρ T A ) 2 (m − 1) 2(4) Consider the m ⊗ m Werner state given by ρ w = m − z m 3 − m I + mz − 1 m 3 − m F, z ∈ [−1, 1](5) where F = |k l| ⊗ |l k| and set m = 8, z = −1. Now, if we consider the matrix form of ρ w (in computational basis) as the state of a 2 ⊗ 32 system, the left hand side of Eq. (4) becomes 1/49 while the right hand side becomes 25/784. Though we have used the formula for D (2) developed in Ref. [7] (which is exact for 2 ⊗ n states), a measurement in computational basis will yield the result. Any value of z ∈ [−1, −34/43) will also work well. We note that for large enough n, the Refs. [3,8] give enough intuition for violation of this relation. Nonetheless, the analytic counterexample makes it more explicit. The authors of Ref. [1] also proposed to take D (1) as a proper measure of geometric discord and derived the hierarchy relation (Eq. 27 therein) D (1) ≥ N(6) However, as can be seen easily, this result is also not correct. It is well known that the trace distance satisfies ρ − σ (1) ≤ 2, for all ρ and σ [9]. Hence we must have D (1) ≤ 2 whereas N can take value up to (m − 1)/2 (for example consider the Bell state in m ⊗ m). Taking the identity matrix as the classicalquantum state (need not be optimal), we see that the relation is violated by 4 ⊗ 4 Bell states. As has been pointed out in Ref. [3], the violation for D (2) stems from the fact that the Hilbert-Schmidt norm is not monotone-D (2) could be increased or decreased by adding or removing a factorized local ancilla. We would like to mention that the trace norm being monotone, does not suffer from this problem. Thus the proposal of taking D (1) as a good measure, is interesting and might be worth investigating. However, as we pointed out here, establishing any interrelation should be done carefully. Another point of concern regarding the use of the trace norm is that its analytic calculation is very difficult and hence the main spirit of usual geometric discord will be lost. Note added: After submission of this comment to journal, we found an erratum posted on arXive [10] in which the authors have tried to fix the errors, based on our criticism. Though the erratum is out of purview of the present comment, we would like to point out that some further modifications are needed. First of all, there is no state ρ for which n = d − 1 holds; so the relation must be a strict inequality. Although the relation [D (2) > N 2 /(d −1)] will then be correct for NPT states (the PPT states trivially satisfy equality), an important point is that it can no longer be used to compare geometric discord and entanglement. We note that, both N and N 2 are entanglement monotones, but N/d (in general N/ f (d)) is not an entanglement monotone, as it might be increased with removal (or addition) of local ancillary systems. This observation applies equally well to the interrelation between D (1) and N/d. PACS numbers: 03.67.Mn, 03.65.Ud . T Debarba, T O Maciel, R O Vianna, 10.1103/PhysRevA.86.024302Phys. Rev. A. 8624302T. Debarba, T. O. Maciel and R. O. Vianna, Phys. Rev. A 86, 024302 (2012). . D Girolami, G Adesso, 10.1103/PhysRevA.84.052110Phys. Rev. A. 8452110D. Girolami and G. Adesso, Phys. Rev. A 84, 052110 (2011). . M Piani, 10.1103/PhysRevA.86.034101Phys. Rev. A. 8634101M. Piani, Phys. Rev. A 86, 034101 (2012). . E Chitambar, 10.1103/PhysRevA.86.032110Phys. Rev. A. 8632110E. Chitambar, Phys. Rev. A 86, 032110 (2012). . S Rana, P Parashar, 10.1103/PhysRevA.86.030302Phys. Rev. A. 8630302S. Rana and P. Parashar, Phys. Rev. A 86, 030302(R) (2012). This relation is not well-justified for comparison, because the left hand side varies from 0 to 1 whereas the right hand side from 0 to 1/4 (i.e., not normalized). Note that a violation of Eq. (4) automatically implies a violation of EqThis relation is not well-justified for comparison, because the left hand side varies from 0 to 1 whereas the right hand side from 0 to 1/4 (i.e., not normalized). Note that a violation of Eq. (4) automatically implies a violation of Eq. (3). . S Rana, P Parashar, 10.1103/PhysRevA.85.024102Phys. Rev. A. 8524102S. Rana and P. Parashar, Phys. Rev. A 85, 024102 (2012). . T Tufarelli, D Girolami, R Vasile, S Bose, G Adesso, arXiv:1205.0251T. Tufarelli, D. Girolami, R. Vasile, S. Bose and G. Adesso, arXiv:1205.0251 being a norm, must satisfy the triangular inequality: ρ − σ (1) ≤ ρ (1) + σ (1) = 2. For many properties and interpretations of the trace norm, we refer to A. N K Gilchrist, M A Langford, Nielsen, 10.1103/PhysRevA.71.062310Phys. Rev. A. 71162310The trace norm . (1) , being a norm, must satisfy the tri- angular inequality: ρ − σ (1) ≤ ρ (1) + σ (1) = 2. For many properties and interpretations of the trace norm, we refer to A. Gilchrist, N. K. Langford and M. A. Nielsen, Phys. Rev. A 71, 062310 (2005). . T Debarba, T O Maciel, R O Vianna, T. Debarba, T. O. Maciel and R. O. Vianna arXiv 1207.1298v3.

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{'abstract': 'In a recent article [Phys. Rev. A 86, 024302 (2012)], the authors have derived some hierarchy relations between geometric discord and entanglement (measured by negativity and its square). We point out that these results are incorrect and give analytic counterexamples. We also discuss briefly the reason for such violations.PACS numbers: 03.67. Mn, 03.65.Ud We start with the definition of GD and negativity from Ref.[1]. For an m ⊗ n (m ≤ n) state ρ, they have definedwhere Ω is the set of zero-discord (or classical-quantum) states given by ξ = p i |i i| ⊗ ρ i and X (p) is the Schatten p norm given by X (p) = {Tr[X † X] p/2 } 1/p . The negativity was defined bywhere ρ T A is the partial transposition of ρ with respect to A and W is any optimal entangled witness. Attempting to prove a conjecture made in [2], they have claimed (Eq. 17 therein) that all bipartite states satisfyWe first observe a typo that the two definitions of negativity in Eq. (2) are not equal, as the quantity in Eq. (2b) is the sum of absolute values of negative eigenvalues of ρ T A whereas that in Eq. (2a) is just double of it.As a first gap in their derivation (which could be taken as another typo, though), we note that they have not normalized D (2) to have maximum value unity, as has been done in the original paper[2]. As a result, if we take Eq. (2a) as definition of negativity, Eq.(3)is not necessarily satisfied even by the two-qubit maximally entangled state (any one of the Bell states has D (2) = 1/2 whereas N = 1). The importance of normalization could be found in Refs.[3,4]. Now we will try to remove all these (possible) typos and show that the relation (3), whether normalized or not, is always violated by some states.Let us first consider the case when D (2) is normalized (taking D = m m−1 D (2) ) and Eq. (2a) is taken as the definition of negativity. Then it becomes the original conjecture (D ≥ N 2 /(m − 1) 2 ) made in Ref.[2], which we have refuted recently in Ref.[5]. Note that the normalizing factor * swapanqic@gmail.com † parashar@isical.ac.in m/(m − 1) > 1 and hence this case also includes the case when D (2) is not normalized and Eq. (2a) is taken as the definition of negativity. Now we will give an analytic example to show that there are states violating even the weaker relation [6]Consider the m ⊗ m Werner state given bywhere F = |k l| ⊗ |l k| and set m = 8, z = −1. Now, if we consider the matrix form of ρ w (in computational basis) as the state of a 2 ⊗ 32 system, the left hand side of Eq. (4) becomes 1/49 while the right hand side becomes 25/784. Though we have used the formula for D (2) developed in Ref.[7] (which is exact for 2 ⊗ n states), a measurement in computational basis will yield the result. Any value of z ∈ [−1, −34/43) will also work well. We note that for large enough n, the Refs.[3,8]give enough intuition for violation of this relation. Nonetheless, the analytic counterexample makes it more explicit. The authors of Ref.[1] also proposed to take D (1) as a proper measure of geometric discord and derived the hierarchy relation (Eq. 27 therein)However, as can be seen easily, this result is also not correct. It is well known that the trace distance satisfies ρ − σ (1) ≤ 2, for all ρ and σ [9]. Hence we must have D (1) ≤ 2 whereas N can take value up to (m − 1)/2 (for example consider the Bell state in m ⊗ m). Taking the identity matrix as the classicalquantum state (need not be optimal), we see that the relation is violated by 4 ⊗ 4 Bell states.As has been pointed out in Ref.[3], the violation for D (2) stems from the fact that the Hilbert-Schmidt norm is not monotone-D (2) could be increased or decreased by adding or removing a factorized local ancilla. We would like to mention that the trace norm being monotone, does not suffer from this problem. Thus the proposal of taking D (1) as a good measure, is interesting and might be worth investigating. However, as we pointed out here, establishing any interrelation should be done carefully. Another point of concern regarding the use of the trace norm is that its analytic calculation is very difficult', 'arxivid': '1301.1961', 'author': ['Swapan Rana \nPhysics and Applied Mathematics Unit\nIndian Statistical Institute\n203 B T RoadKolkataIndia\n', 'Preeti Parashar \nPhysics and Applied Mathematics Unit\nIndian Statistical Institute\n203 B T RoadKolkataIndia\n'], 'authoraffiliation': ['Physics and Applied Mathematics Unit\nIndian Statistical Institute\n203 B T RoadKolkataIndia', 'Physics and Applied Mathematics Unit\nIndian Statistical Institute\n203 B T RoadKolkataIndia'], 'corpusid': 118471693, 'doi': '10.1103/physreva.87.016301', 'github_urls': [], 'n_tokens_mistral': 3949, 'n_tokens_neox': 3430, 'n_words': 2166, 'pdfsha': '2b1fd5897c09c91189aa01ad643a04e4fb988e80', 'pdfurls': ['https://arxiv.org/pdf/1301.1961v1.pdf'], 'title': ['Comment on "Witnessed entanglement and the geometric measure of quantum discord"', 'Comment on "Witnessed entanglement and the geometric measure of quantum discord"'], 'venue': []}

arxiv

Unlocking the synergy between CMB spectral distortions and anisotropies 11 Jan 2022 Hao Fu h.fu@soton.ac.uk Department of Physics and Astronomy University of Southampton Highfield CampusSO17 1BJUK Matteo Lucca mlucca@ulb.ac.be Service de Physique Théorique Université Libre de Bruxelles C.P. 225B-1050BrusselsBelgium Silvia Galli gallis@iap.fr UMR 7095 Sorbonne Université CNRS Institut d'Astrophysique de Paris 98 bis bd Arago75014ParisFrance Elia S Battistelli Sapienza University of Rome Piazzale Aldo Moro 500185RomeItaly Deanna C Hooper Service de Physique Théorique Université Libre de Bruxelles C.P. 225B-1050BrusselsBelgium Julien Lesgourgues Institute for Theoretical Particle Physics and Cosmology (TTK) RWTH Aachen University D-52056AachenGermany Nils Schöneberg Institute for Theoretical Particle Physics and Cosmology (TTK) RWTH Aachen University D-52056AachenGermany Unlocking the synergy between CMB spectral distortions and anisotropies 11 Jan 2022Prepared for submission to JCAPULB-TH/20-06 TTK-20-18 Measurements of the cosmic microwave background (CMB) spectral distortions (SDs) will open a new window on the very early universe, providing new information complementary to that gathered from CMB temperature and polarization anisotropies. In this paper, we study their synergy as a function of the characteristics of the considered experiments. In particular, we examine a wide range of sensitivities for possible SD measurements, spanning from FIRAS up to noise levels 1000 times better than PIXIE, and study their constraining power when combined with current or future CMB anisotropy experiments such as Planck or LiteBIRD plus CMB-S4. We consider a number of different cosmological models such as the ΛCDM, as well as its extensions with the running of the scalar spectral index, the decay or the annihilation of dark matter (DM) particles. While upcoming CMB anisotropy experiments will be able to decrease the uncertainties on inflationary parameters such as A s and n s by about a factor 2 in the ΛCDM case, we find that an SD experiment 100 times more sensitive than PIXIE (comparable to the proposed Super-PIXIE satellite) could potentially further contribute to constrain these parameters. This is even more significant in the case of the running of the scalar spectral index. Furthermore, as expected, constraints on DM particles decaying at redshifts probed by SDs will improve by orders of magnitude even with an experiment 10 times worse than PIXIE as compared to CMB anisotropies or Big Bang Nucleosynthesis bounds. On the contrary, DM annihilation constraints will not significantly improve over CMB anisotropy measurements. Finally, we forecast the constraints obtainable with sensitivities achievable either from the ground or from a balloon. Introduction The cosmic microwave background (CMB) is a remarkably rich source of information about early universe physics. In particular during the last decades significant effort has been devoted to the investigation of the CMB temperature and polarization anisotropies, including also the latest data release of the ESA Planck mission [1] (from now on referred to as Planck), which allowed to build a very accurate all-sky map and the power spectrum of the CMB. Another very promising and not yet fully explored complementary cosmological probe is given by CMB spectral distortions (SDs) (see e.g. [2,3] for recent reviews). According to their spectral shape, these can be classified as a combination of a temperature-shift, a chemical potential µ-type, a Compton y-type, and a residual distortion (see e.g. [4] for more details). Indeed, studying distortions of the CMB black body (BB) spectrum allows us to constrain several energy release mechanisms occurring at redshifts z 2 × 10 6 . A variety of such astrophysical and cosmological processes are expected within the Standard Model of cosmology, while others, if present at all, could be caused by any exotic mechanism injecting energy in the CMB photon bath [5][6][7][8][9][10]. However, although major progress has been made in the theoretical and numerical prediction of SDs in the last years, exactly three decades have passed since the development of experiments that measured the CMB frequency spectrum, such as the COBRA [11] and FIRAS [12,13] missions, which measured the CMB spectrum to be that of a pure BB up to uncertainties in the brightness intensity of ∆I ν /I ν 10 −5 . Today, thanks to new technologies, much more accurate SD measurements are feasible and could easily achieve sensitivities three or four orders of magnitude better than those of FIRAS, as for instance in the case of spectrometers like PRISTINE, PIXIE [14], PRISM [15], Super-PIXIE [16] and Voyage 2050 [17], as well as with imaging telescopes as recently proposed in [18]. Therefore, precise forecasts of the constraining power of different SD observational setups are needed, and addressing this necessity is precisely the goal of this paper. In particular, as a first systematic effort in this direction, we consider a series of experimental designs assuming an idealised foreground removal to investigate the maximum amount of information that could be extracted from SDs on the early universe for a variety of different cosmological models. In this sense, our results are not meant to realistically estimate the reach of different experimental setups (which are themselves anyway idealized to a large extent in our treatment), but rather to study and compare the dependence of the cosmological constraints on the sensitivity of those experiments. For our forecasts we use the latest versions of the Boltzmann solver class [19] and the Monte Carlo Markov Chain (MCMC) sampler Mon-tePython [20,21] as described in [3], which now allow us to include SD experiments in combination with CMB anisotropy surveys. This paper extends some previous work by [3] in mainly two directions. On the one hand, we consider several additional SD experiment designs in order to study the minimum sensitivity required to improve constraints on specific cosmological scenarios with respect to the ones provided by CMB anisotropies. On the other hand, we marginalize over the contribution from late-time sources of SDs such as clusters of galaxies, while [3] only assumed SDs produced in the early universe. Indeed, within this work, we forecast the sensitivity of various possible SD experiments other than just PIXIE and PRISM to the cosmological parameters that describe possible sources of energy release. For different cosmological scenarios and a few assumptions concerning future CMB anisotropy experiments, we show how the interplay between CMB SD and anisotropy data could break degeneracies between many of the cosmological parameters involved, especially in the case of the most futuristic SD configurations. This level of synergy between the two probes was out of reach for the low-sensitivity SD configurations considered in previous works, like e.g. [3]. We also present for the first time some MCMC forecasts for experiments attempting to measure SDs from the ground, and we show the improvement that might be achieved with future measurements from the stratosphere or from space. Basing ourselves on existing concept studies for ground-based experiments such as COSMO [22] and BISOU (see e.g. [17] for additional discussions), we consider the case of an experiment with two frequency bands dictated by the favorable atmospheric window and sensitivity-limited by the photon background from the atmosphere and the CMB. This represents a first attempt to estimate the potential constraining power of these experiments on cosmological parameters. For all of the considered cases we account for the fact that primordial y distortions will be hardly distinguishable from the contribution of late-time y sources such as cosmic reionization, the intracluster medium (ICM) of groups and clusters of galaxies, and the intergalactic medium (IGM) between halos [23]. We effectively model these contributions by marginalizing over the optical depth and mean temperature of the dominant late-time source, the ICM. Moreover, we also marginalize over the uncertainty on the monopole temperature T 0 . However, we assume a perfect removal of some major potential sources of contamination, such as galactic and extra-galactic foregrounds. Also, when considering ground-based or balloonborne experiments, we also neglect the contribution of atmospheric brightness fluctuations (while we do include the increase in photon noise due to the atmospheric emission). This paper is organized as follows. In Section 2 we provide an overview on the energy injection mechanisms considered in this work. In Section 3 we describe the specifics of the considered experiments and in Section 4 we describe the method used to forecast their constraining power on cosmological parameters. In Section 5 we display the results of our investigation. Finally, a summary of the results together with additional discussions is given in Section 6. Energy release scenarios In our analysis we consider different energy injection mechanisms that can lead to SDs in the CMB energy spectrum focusing exclusively on the "primordial" contributions (i.e., those that take place prior to recombination, neglecting late-time effects such as the Sunyeav-Zeldovich effect which are instead marginalized over as explained in Sec. 4.2). Among these processes, we focus on the standard ΛCDM ones, such as the dissipation of acoustic waves and adiabatic cooling, as well as the annihilation and decay of relic particles. These cases are particularly interesting as they encompass a variety of non-standard scenarios. For instance, in the case of acoustic wave dissipation it is possible to test many alternative inflationary models via the dependence on the primordial power spectrum (see e.g. [24][25][26]). In the case of decaying relic particles (see e.g. [27][28][29]), the discussion on the required detector sensitivity can be extended to the evaporation of primordial black holes (see e.g. [28,30]) due to the similar injection histories (see e.g. Figures 4 and 5 of [3]). Other possible sources of SDs include interactions between dark matter (DM) and Standard Model particles (see e.g. [31][32][33]), but since results for one model do not necessarily generalize, we leave the analysis of these different models for future work. Although the theory underlying these processes is already widely documented in the literature, in this section we provide a brief overview for the sake of completeness. The adopted notation is based on the work of [3], where the interested reader can find more in-depth discussions. Other related works are e.g. [4,10]. The ΛCDM model and running of the spectral index In the ΛCDM model, we consider two main sources of SDs: the damping of acoustic waves and the baryon adiabatic cooling. When the primordial energy density perturbations enter the horizon, pressure gradients form and cause pressure waves. The oscillation of these waves (referred to hereafter as acoustic waves) is affected by dissipation, which causes damping at small scales and creates distortions in the CMB frequency spectrum [6,34]. The type of distortions that can be generated depends on the epochs of the damping, while the intensity of the signal depends on the amplitude of the damped wave. It is important to specify that the acoustic wave damping is not an energy injection to the CMB radiation field, but rather a redistribution of the radiation field energy (and hence referred to specifically asQ non −inj , whereQ is the heating rate). The distortion generated by the damping of the CMB small-scale fluctuations depends on the amplitude and the shape of the primordial power spectrum at scales 1 Mpc −1 ≤ k ≤ 2 × 10 4 Mpc −1 [10,35,36]. An accurate approximation for the effective heating rate and the curvature power spectrum of scalar perturbations is given by [37,38] Q non −inj = 4A 2 ρ γ ∂ z k −2 D ∞ k min k 4 dk 2π 2 P R (k)e −2k 2 /k 2 D , (2.1) where A is a normalization factor, k D is the photon damping scale [39,40] and P R (k) is the dimensionless primordial power spectrum of curvature fluctuations. Alternative approximations can be found e.g. in [41]. On the other hand, baryon adiabatic cooling is caused by the fact that the temperature of photons cools due to the expansion of the universe as T γ ∝ (1 + z), while baryons cool faster, T e ∝ (1+z) 2 . Thus, photons transfer energy to baryons when strictly coupled to them in the early universe to maintain equilibrium. This subtraction of energy from the photon field causes distortions in the CMB spectrum, which partially cancel out those due to heating (see e.g. [10]). Within this work, we will consider the standard ΛCDM case, as well as a minimal extension of the standard primordial power spectrum including the running of the spectral index, i.e. P R (k) = 2π 2 A s k −3 k k 0 ns−1+ 1 2 nrun ln(k/k 0 ) , (2.2) where A s , n s and n run are the amplitude, the power index and its running, respectively, while k 0 is the pivot scale, which we assume to be k 0 = 0.05 Mpc −1 . The running is set to zero when considering the ΛCDM case. Overall, the final set of free parameters involved in the ΛCDM scenario is {ω b , ω cdm , 100θ s , ln(10 10 A s ), n s , τ reio }, (2.3) while for the ΛCDM+n run case is {ω b , ω cdm , 100θ s , ln(10 10 A s ), n s , τ reio } + n run ,(2.4) with ω b and ω cdm as the physical baryon and DM densities respectively, θ s as the angular scale of sound horizon at last scattering and τ reio as the reionization optical depth. Using the Planck ΛCDM best-fit model [1], one can predict the expected amplitudes of the y and µ distortions within the ΛCDM model as being 3.6 × 10 −9 and 1.9 × 10 −8 , respectively. The single contribution from the dissipation of acoustic waves is y 4.1 × 10 −9 and µ 2.3 × 10 −8 , while for the adiabatic cooling of baryons one has y −5.2 × 10 −10 and µ −3.3 × 10 −9 . Of the ΛCDM parameters listed in Equation (2.3), only four (ω b , ω cdm , A s and n s ) effectively influence the shape of the SD signal, and can thus be constrained with SDs. To show the impact of the single parameters, in the left panel of Figure 1 we vary each of them by 1% with respect to the Planck best fits (keeping all the others fixed) and display the corresponding variation of the SD signal. The parameter that affects the total signal the most is the spectral index n s , which induces variations of the order of ∆(∆I)/∆I ∼ 10∆n s , as can be seen from the lower panel of the figure. This is because even a small change of this quantity strongly influences the amplitude of the power spectrum at scales much smaller than the pivot scale (k 0.05 Mpc −1 ) and, therefore, the amount of acoustic dissipation at high redshifts. On the other hand, changing the amplitude A s by 1% results in a variation of the signal by 1%, as expected and evident from the constant green line in the lower panel of the figure. The impact of ω b is due to its role in the definition of the damping scale The top panel shows the difference ∆(∆I) = ∆I var − ∆I fid (in logarithmic scale) between the total SD spectrum ∆I var obtained by changing a cosmological parameter by 1% while keeping all the other fixed, and the fiducial ΛCDM total distortion spectrum ∆I fid . As a reference, we also show the ΛCDM fiducial primordial SD spectrum (∆I fid ) in dashed gray. The bottom panel shows the same as the top panel, but in relative units. These plots show that a 1% change in n s produces the largest SD variation, of the order of ∼ 10%, followed by A s , which induces a ∼ 1% variation. Right panel : Same as in the left panel but for the extensions of the ΛCDM model considered within this work. k D of Equation (2.1). A higher baryon density ensures a tighter coupling of electrons and photons, thus moving k D to higher values, i.e. to smaller scales. The impact of this effect is very mild on the SD signal, of the order of 0.1%. Finally, since a (minor) part of the SD signal is produced during matter domination, where the expansion of the universe is mostly determined by ω cdm , the DM energy density plays an almost negligible role in the final shape of ∆I, inducing variations of the order of 0.001%. The right panel of Figure 1 shows the impact on the SD signal of the running of the scalar spectral index, together with a few other extensions of the ΛCDM model considered in this paper. A running of n run = −0.01, compatible with current limits, generates a variation of the SD signal of the order of 10%, similar to the modifications produced by n s . Annihilating and decaying relic particles In addition to the contribution to SDs from the ΛCDM model, it is also interesting to consider exotic energy injection scenarios. As representative cases, here we consider the annihilation and the decay of relic particles. Since in these scenarios the intensity of the distortion signal directly depends on the particle physics nature of the DM, an eventual SD measurement could be very interesting from a model building perspective. When exploring these models, their effects add to the ones described for the ΛCDM case. Firstly, we consider a self-annihilating DM particle χ and its antiparticleχ. We allow such particles to possibly coexist with another fully stable cold DM species, and we call f frac the annihilating DM fraction. Adopting the parametrization of [42,43], the energy injection rate predicted by this model takes the forṁ Q = ρ cdm (z) 2 p ann ,(2.5) where p ann = f frac f eff σv M χ (2.6) represents the annihilation efficiency and contains information on the mass M χ and fractional density f frac of the particle, on the thermally averaged cross-section σv of the annihilation process, as well as on the energy injection efficiency f eff . We adopt here an on-the-spot approximation, i.e. the energy emission and absorption are assumed to happen at the same redshift. Similarly to the previous scenario, in this case we have a 6+1 parameter extension of the ΛCDM model, with the set of parameters: {ω b , ω cdm , 100θ s , ln(10 10 A s ), n s , τ reio } + p ann . (2.7) Assuming the largest value allowed by Planck 1 for p ann , i.e. p ann = 3.3 × 10 −28 cm 3 /(s GeV), and the best-fit value of the other parameters [1], we obtain values for the y and µ parameters in the order of 7.1 × 10 −11 and 5.5 × 10 −10 , respectively. On the other hand, the energy release rate in the case of decaying relic particles can be parametrized according to [27] aṡ Q = ρ cdm (z)f frac f eff Γ dec e −Γ dec t , (2.8) where f frac is the fraction of decaying DM and Γ dec is the particle decay width. Due to the exponential factor, the decay process starts to be inefficient as soon as the age of the universe is comparable to the lifetime of the particle, τ dec ∼ Γ −1 dec . For simplicity, we will assume that f eff = 1 and only keep f frac as a free parameter. Note that in principle the two are completely degenerate, and thus the constraint on f frac depends on the specific decay channel considered and therefore on the value of f eff ≤ 1. As in the case of annihilating DM particles, we assume an on-the-spot approximation (for a possible generalization of these approximations see e.g. [28,29]). Differently than for the previous heating mechanisms, in the case of DM decay we have a 6+2 parameter extension: {ω b , ω cdm , 100θ s , ln(10 10 A s ), n s , τ reio } + f frac , Γ dec . (2.9) For parameter values such as f frac = 1 × 10 −5 and Γ dec = 1 × 10 −10 1/s, which are well within FIRAS bounds, one obtains y ∼ 1.2 × 10 −7 and µ ∼ 9.6 × 10 −7 . Note, however, that the relative amplitude of y and µ distortions generated by DM decay strongly depends on the particle's lifetime. The right panel of Figure 1 shows the impact on the SD signal of DM annihilation and decay. A DM annihilation process with p ann = 3.3 × 10 −28 cm 3 /(s GeV), compatible with current limits, generates a variation of the SD signal of the order of 2%, while a DM decay with f frac = 1 × 10 −5 and Γ dec = 1 × 10 −10 1/s produces variations of a factor of order 50, making it the most impactful process considered within this work. Experimental setups Within this work, we consider several experimental configurations spanning a variety of different sensitivities and possible observational environments. In this section we provide an overview of their most relevant characteristics. Table 4 in [13]. For the PIXIE variants, we use the same binning as PIXIE, indicated by the gray vertical lines. For illustrative purposes, here we refer to PIXIE10 as PRISM. PIXIE and its variants The Primordial Inflation Explorer (PIXIE) is a mission proposed to NASA aimed at probing the nature of primordial inflation with measurements of the CMB B-modes linear polarization caused by such an inflationary epoch [14]. However, the experiment was not only designed for B-modes, but also for the observation of CMB SDs. PIXIE consists of a polarizing Fourier Transform Spectrometer (FTS) that synthesizes ∼ 400 frequency channels, ranging from 30 GHz to 6 THz, with a photon noise equal to ∆I noise 5 × 10 −26 W/m 2 /Hz/sr (more details on how this value is obtained, such as the integration time and the number of detectors assumed, can be found in [4] and Appendix E.1 of [3]). For the purposes of this work, and as also commonly done in the literature, we assume for PIXIE a constant channel frequency resolution of ∆ν = 15 GHz in the range 30 GHz − 1 THz and uncorrelated noise for each resulting bin (64 in total). Hereafter, we will refer to this setup as PIXIE. Furthermore, we consider several other experimental configurations with identical frequency binning as PIXIE, but with different sensitivities ranging from 1000 times smaller to 1000 times larger noise than that of PIXIE 2 . Hereafter, we will refer to these configurations as PIXIE variants, and label the cases with better sensitivity as PIXIE10, PIXIE100 and PIXIE1000 (for configurations with 10, 100 and 1000 times improved sensitivity, respectively). In Figure 2 we show the sensitivity of these experimental concepts, along with those of PIXIE and FIRAS, together with the primordial ΛCDM signal (which is what we will mostly focus on within this work) in red for reference. As a remark, note that the Polarized Radiation Imaging and Spectroscopy Mission (PRISM), a mission proposed to ESA for the investigation of early universe physics [15], is predicted to reach sensitivities roughly one order of magnitude better than those of PIXIE (∆I noise 6 × 10 −27 W/m 2 /Hz/sr), and more accurate performances are reported in Table 2 of [15]. Assuming, therefore, an experimental configuration with the same frequency binning as PIXIE and 10 times better sensitivity, i.e. ∆I noise 5 × 10 −27 W/m 2 /Hz/sr, it is in principle possible to compare the results obtained for PIXIE10 to those within the reach of PRISM. Similarly, it is also possible to place the recently proposed Super-PIXIE mission [16] and its upgraded Voyage 2050 version [17] in the sensitivity range between PIXIE10 and PIXIE100, although for these cases the improved low-frequency sensitivity band of the former configurations does not allow for a direct comparison. Experiments in different environments We also consider three additional instrumental configurations for possible future experiments, hereafter referred to as ground, (stratospheric-) balloon and satellite configurations. This analysis is aimed at testing the feasibility of an SD detection with a ground-based experiment, and at showing the level of improvement that could be achieved by further investing in a balloon-borne or satellite experiment. For these three experimental configurations, we have assumed that the instrument measures the signal laying within two frequency ranges centered at 150 GHz and 220 GHz, each covering a bandwidth corresponding to 25% of the central value, and with realistic optics and atmosphere emission. The particular choice made for the frequency intervals is dictated by the favorable atmospheric windows for the ground-based configuration (see e.g. [44] and references therein) and applied also to the other setups to ease the comparison between environments. In other words, for the balloon and satellite configurations we use the same frequency settings to isolate the impact of just improving the sensitivity of the experiment on the considered parameters. We assume that the experiments are limited by the radiation photon noise on the detectors, which is in turn dominated by the atmosphere for the groundbased experiment, by the residual atmosphere and by the optics emission for the balloon, and by the CMB for the satellite configuration. The noise equivalent power (NEP) for these configurations has been calculated taking into account the photon noise from the optics and the atmosphere emission only, although for ground-based operations one should also account for the fluctuations of the atmosphere (see e.g. [45]). Concerning the photon noise from foregrounds, in particular galactic dust, we estimate that the contribution is negligible for the considered configurations (with contributions up to an order of 5 × 10 −20 − 5 × 10 −18 W/Hz 1/2 for both frequency windows), especially considering that these experiments would observe patches of the sky with low galactic emission. Photon noise levels have been derived considering the three different experimental setups. For the satellite configuration we assumed a cold optics configuration and calculated the photon noise level arising from the CMB itself including both photon (shot) noise and the bunching component (see e.g. [46]). For the ground-based configuration, we accounted for the atmospheric emission using am 3 [47], a radiative transfer tool developed for microwave to sub-millimeter wavelengths atmospheric emission, which accounts for realistic antarctic conditions [44]. In the balloon-borne configuration, we have accounted for both the residual atmospheric emission (see e.g. [48]) and the optics emission. For the latter, we have assumed mirrors, lenses and a window with 1% emissivity. Table 1. Instrumental specifications of PIXIE [14], and three other possible future experimental configurations centered in two frequency bands. The second column represents the frequency range explored by each experiment, while the third column represents the bandwidth of each frequency channel matching the atmospheric windows. The fourth column represents the corresponding detector's noise equivalent power (NEP) and the last column the final sensitivity. In this paper, we also consider experiments with NEP (and thus sensitivities) 10, 100 or 1000 better or worse than PIXIE. . The gray area represents the sensitivity of PIXIE. The blue, orange and green error bars represent respectively the sensitivity of the configurations of an experiment from the ground, balloon and satellite, all with seven channels, 5 GHz distant, centered in two frequency bands (150 GHz and 220 GHz) with 25% width each. The error bars representing the balloon experiment noise are plotted in the real value of the frequency, while for the satellite and ground cases there is an offset of ±1 GHz for the sake of graphical clearness. The red line represents the predicted ΛCDM distortion signal assuming Planck bestfits. The final detector sensitivity for each single frequency channel has been derived and calculated from the NEP according to Equation (3.3) in [14], ∆I noise = NEP/ τ /2 AΩ∆ν(α f ) , (3.1) where τ is the integration time, ∆ν the frequency bandwidth, α its absorptivity, the source emissivity and f the transmissivity of the optics. In our calculation, we have assumed a detector with αf = 0.4, = 1, twenty modes of the radiation and integrated over 1 year mission time. We have assumed a diffraction limited detector model, i.e. theétendue AΩ = N λ 2 being constant in each observed band, where N is the radiation modes and λ the smallest wavelength in each band, corresponding to 0.18 cm and 0.12 cm respectively. In Table 1 we report the value of the different photon noises within each frequency band. These are also illustrated in Figure 3. From this figure it is already possible to notice that neither a ground-based nor a balloon-borne experiment will be able to detect the minimal primordial distortion expected in the ΛCDM model. A PRISM-like experiment will be barely able to do so, as also confirmed by the results reported in Section 5. Methodology In the previous sections we have introduced some well-studied representative examples of models causing an energy injection in the CMB photon field. For the numerical evaluation of the many cosmological quantities involved, such as the CMB power spectra, we use the Boltzmann solver class 4 [19,49]. In particular, we use the latest version of the code which allows the calculation of energy injection rates and SD spectra, as described in [3], that will be released very soon as v3.0. Furthermore, to forecast the constraints on cosmological parameters we use the MCMC sampler MontePython 5 [20,21]. The implementation of the specific likelihoods required for SD experiments are also described in [3]. In the following subsections, we will report some of the details related to the implementation of the SD modelling and forecasting in class and MontePython. The class implementation In full generality, the SD signal can be parametrized as ∆I tot = ∆I T + ∆I y + ∆I µ + ∆I R + ∆I reio + ∆I fg , (4.1) where ∆I T represents temperature shifts, ∆I y , ∆I µ and ∆I R the Compton y, the chemical potential µ and the residual distortions, ∆I reio the contribution from late-time SD sources (such as cosmic reionization, the ICM or IGM), and ∆I fg all foreground contaminations. According to this formalism, the evaluation of the SD signal is performed using class, which calculates the heating history and the distortion signal for a specific energy release scenario and experimental setting, making use of the Green's function approximation introduced in [50] for fast computations. In this approximation, once the cosmological model has been defined, the SD intensity can be linearized as ∆I(ν, z 0 ) = ∞ z 0 G th (ν, z ) dQ(z )/dz ρ γ (z ) dz ,(4.2) where ρ γ (z) is the energy density of the CMB photons and G th ν, z = G(ν)J g z + Y(ν)J y z + M(ν)J µ z + R ν, z (4.3) is the Green's function of the SD signal. Here, G, Y and M are the temperatureshift, the y-type and the µ-type spectral shapes, respectively, and J g , J y and J µ are the corresponding branching ratios. Finally, R (ν, z ) represents the residual distortions which are not captured by the other ones. This decomposition of ∆I in Green's function and heating rate presents the important advantage of shifting the whole model dependence of the final signal into the shape of Q, while G th is completely model independent. Additional details on the assumptions on which this expansion is based are provided e.g. in [4,50,51] and [3] (see Section 3.2.1 therein). For the calculation of the different heating rates, we follow the prescriptions described in Section 2. In our analysis we included only the dissipation of acoustic waves and the adiabatic cooling of baryons and electrons as the ΛCDM prediction. The contributions from the CMB dipole and from the cosmic recombination radiation (CRR) have been neglected. Note, however, that both of these effects are predicted to have only a minor impact on the final result (see e.g. Figure 1 of [36]). Dedicated forecasts focused on the role of the CRR can be found in [52]. Moreover, we rely on the publicly available Green's functions of the CosmoTherm repository 6 , computed following the method first introduced in [50] and distributed together with class 3.0. There, all frequency-dependent components of G th are discretized. Therefore, the spectral shapes G, Y, M and R are fixed vectors in frequency space, with frequency range and resolution depending on the considered experiment. To determine the branching ratios J g , J y and J µ , the total Green's function G th is then projected onto these vectors. However, since these vectors are not orthogonal, the projection is not unambiguous 7 (see Section 3.2.2 of [3]), i.e. the branching ratios depend on the choice of the projection procedure, as well as on the frequency range and resolution of the experiment considered. In this formalism, we define the experimentally determined µ and y parameters as µ = ∞ z 0 J µ z dQ(z )/dz ρ γ (z ) dz and y = ∞ z 0 J y z dQ(z )/dz ρ γ (z ) dz . (4.4) Note that in this definition, µ and y correspond to the amplitudes of the spectral shapes M and Y, respectively, that one obtains from a least-square fit of the total ∆I(x) in Eq. (4.2) for given a frequency range and resolution (encoded by the branching ratios J ). These amplitudes can thus slightly differ from the µ and y quantities one can theoretically define (see e.g. [9]). However, the two should agree in the limit of infinite frequency range and resolution. 8 Since the heating rate dQ(z)/dz depends on the cosmological parameters as described in Section 2, these measured µ and y parameters are then derived quantities. 6 CosmoTherm: Website 7 Even beyond frequency coverage, channel spacing and sensitivity there exists an intrinsic degeneracy within the Gram-Schmidt orthogonalization process (which is employed in [50]). Consider, for example, the vectors v1 = (1, 1) and v2 = (1, −2). Then, depending on the order of the Gram-Schmidt process we obtain (up to normalization) u1 = (1, 1) and u2 = (1, −1) or u 2 = (1, −2) and u 1 = (2, 1), which are not equivalent, and would give different amplitudes for the components 1 and 2. Additionally, note also that there are alternatives beyond the Gram-Schmidt orthogonoalization employed in [50]. 8 Since the the branching ratios Jµ and Jy depend on frequency resolution and range of a given SD experiment as well as on the projection procedure, the µ and y parameters inferred from Equation (4.4) can vary according to these possible choices. Instead, one could choose to display the true physical y and µ parameters, which would correspond also to those measured by an experiment with infinitely precise frequency coverage. However, this would not be a perfectly fair comparison, as a given experiment's bounds will depend on the finite precision of its frequency resolution and the corresponding fitting of the SD amplitudes. This is consistent with the idea that a given experiment will not be able to perfectly differentiate between the different contributions to the total signal, such as the y, µ, or residual distortion, which are usually fitted to the total signal as in the case of FIRAS [13]. Their uncertainties are thus determined by the combined power of the experiments used to constrain the cosmological parameters of the considered model, which in this paper are CMB anisotropy and SD experiments. Therefore, the uncertainties on the µ and y parameters reported in Section 5 do not reflect the constraining power of a given SD experiment alone on these spectral shapes (see Section 5.1 for additional details), which are however commonly available in the original papers such as [13] for FIRAS and [14] for PIXIE. To illustrate this point, we compare three experimental configurations featuring different frequency ranges and resolutions: FIRAS and PIXIE (defined according to Section 3), as well as an "ideal" detector with much wider frequency range and higher resolution with respect to the other two, i.e. {ν min , ν max , ∆ν} = {1GHz, 2THz, 1GHz}, which will be referred to as IDEAL. The left panel of Figure 4 shows the G, Y and M spectral shapes, together with the range of frequencies probed by each of the three configurations. The figure shows that IDEAL covers the whole frequency space where the shapes are non-zero, while both PIXIE and FIRAS are insensitive to the low-frequency tail of the curves. As a consequence, the branching ratios are different for each experimental configuration, as shown in the right panel of Figure 4. There, the branching ratio of the µ distortion is enhanced for FIRAS with respect to PIXIE, while the y branching ratio is reduced. A consequence of this is clearly visible, for instance, in the right panel of Figure 7 of [3], where the FIRAS contour is shifted towards lower values of y and higher values of µ with respect to PIXIE-like configurations (this effect is not noticeable in Figure 6 because of the marginalization over late-time SD sources performed in this work) 9 . Interestingly, although the difference of coverage between PIXIE and IDEAL in the low energy tail is substantial, the deviation of PIXIE from the ideal setup is minor, suggesting that PIXIE-like frequency arrays are already close to optimal for the purpose of disambiguation of the µ and y distortion shapes, and considering the limited foreground model described in Section 4.2. In any case, the dependence of the amplitude of the µ and y parameters on the exper-ω b ω cdm 100θ s ln(10 10 A s ) n s τ reio 0.022377 0.1201 1.0411 3.0447 0.9659 0.0543 Table 2. Values of the cosmological parameters used for the fiducial model [1], which assumes a ΛCDM scenario. imental configuration is not a problem, since the final observable is the total SD spectrum, which is the weighted sum of all shapes and which is thus independent of the details of the projection. We base our results solely on this total SD spectrum. However, as already pointed out in Section 2.2.1 and Figure 3 of [4] as well as in Section 4.1 of [3], this means that the same heating rate history can result in different measured distortion amplitudes 10 depending on the experiment and projection choices considered. The MontePython implementation In order to constrain the cosmological parameters involved in each model considered within this work, we perform a number of MCMC scans using the MontePython code [21]. We produce synthetic mock data for all of the CMB anisotropy and SD experiments considered, using as a fiducial the ΛCDM model with best-fit cosmological parameters from the Planck mission [1] reported explicitly in Table 2. We use simulated data, rather than real data, also when considering completed missions such as Planck or FIRAS. This ensures that all of the experiments we include in our forecasts share exactly the same fiducial model. The experimental specifications used to simulate CMB anisotropy experiments such as Planck, LiteBIRD [53] and CMB-S4 [54,55] are detailed in Section 3 and Table 5 of [56]. The specifications used for SD experiments such as FIRAS and PIXIE are detailed in Section 3.3 of [3]. To discuss the detection capability of variants of the PIXIE experiment, as detailed in Section 3.1, or of ground-based, balloon-borne and satellite-based future experiments, as described in Section 3.2, we implement likelihoods analogously to those of FIRAS and PIXIE. For each energy release scenario introduced in Section 2, we then forecast the constraints on the relevant cosmological parameters that different combinations of CMB anisotropy and SD experiments will be able to achieve, with particular regard to their detectability and corresponding uncertainties. In particular, we perform forecasts for the constraints on the 6 ΛCDM parameters, the running of the scalar spectral index and the parameters describing the DM annihilation or decay scenario. In forecasting the capability of an SD experimental concept, we do not include the presence of foreground contamination such as galactic thermal dust, cosmic infrared background, synchrotron, free-free, integrated CO, and anomalous microwave emission (see e.g. [57] for a detailed overview of foreground effects on SD detection). These can deteriorate the sensitivity of the overall signal by a factor of order 10 -depending on the priors on the foreground parameters (see e.g. Tables 3 and 4 of [57] for a quantitative evaluation). However, we do account for the contribution from the SZ effect in the ICM, which is the dominant late-time source of distortions due to its much higher temperature with respect to reionization and the IGM. This mainly contributes to the y signal, and it is therefore strongly degenerate with early-universe phenomena which also produce y distortions. We also marginalize over the experimental uncertainty ∆T on the monopole temperature T 0 , as explained in Section 3.3 of [3]. This deteriorates the precision of the final SD signal because of the degeneracies that the temperature shifts included in ∆I T share with the y and µ distortions. To marginalize over the contribution from the ICM, we use the implementation of the thermal (relativistic and non-relativistic) and kinematic SZ effects (kSZ) [58] already present in class and based on [59] (see Equations (2.54)-(2.58) of [3] and references therein for more details on the class implementation). Since this effect is predominantly proportional to the optical depth of the electron plasma ∆τ and the electron temperature T e [23,[59][60][61][62][63][64], to marginalize over late-time sources we leave ∆τ and T e free to vary, while keeping other quantities related to the SZ effect fixed, such as the average cluster velocity with respect to the line of sight β . In this parametrization ∆τ is the mean optical depth of the ICM, which is different from τ reio , the optical depth of reionization. For the temperature of the electron plasma we adopt the fiducial value of T e = 1.3 keV, while the optical depth is set to provide a value of y ∼ 10 −6 consistent with [23]. Furthermore, importantly, out of the aforementioned contributions to the SZ effect the kSZ effect is expected to be strongly suppressed when averaged over the full sky due to the different peculiar velocities of the various clusters. However, to account for the impact of a possibly incomplete sky coverage, we conservatively set β equal to the dispersion of the cluster velocity distribution described in [65], β = 1/300. In fact, since the kSZ corresponds at leading order to a temperature shift with amplitude proportional to β ∆τ , it is degenerate with the temperature shift parameter ∆T , degrading the constraints of the latter when β is non-zero. However, this degeneracy can be partially lifted by the kSZ higher order contributions, which can be captured by the setups discussed in Section 3.1. We find using the aforementioned assumptions that ∆T /T 0 3 × 10 −6 , 5 × 10 −8 , 5 × 10 −9 , 6 × 10 −10 and 1 × 10 −10 for FIRAS, PIXIE, PIXIE10, PIXIE100 and PIXIE1000, respectively, almost independently from the cosmological model assumed. For the case of PIXIE1000, for which the impact of the marginalization is the strongest, taking the more realistic value of β = 0 reduces the uncertainty on ∆T by a factor of roughly 10, while it has a negligible impact on cosmological parameters. For the experimental setups considered in Section 3.2, we find that the limited frequency range of those experimental settings would not allow us to sufficiently break the degeneracy between ∆T and ∆τ . Therefore, in this case we can not consider the impact of this degeneracy and simply choose to set the sky-average velocity to its expected value β = 0, which yields ∆T /T 0 1 × 10 −7 , 2.7 × 10 −8 and 5.8 × 10 −9 for the ground-based, balloon and satellite missions, respectively. Finally, we remark again that these sensitivities to ∆T are conservative estimates and dependent on our choice of the fiducial values. A dedicated analysis with more focus on the nuisance parameters (including also ∆τ and T e ) is left for future work, while here we mainly concentrate only on the cosmological parameters. Results In this section, we apply the numerical framework presented in Section 4 to the different energy release scenarios described in Section 2, and we discuss the most relevant results. ΛCDM First of all, we focus our attention on the standard ΛCDM model and the consequences of including SDs together with completed or upcoming CMB anisotropies experiments. It is already known that current proposed missions such as PIXIE are not sensitive enough to improve constraints on the vanilla ΛCDM model [57] 11 -although, obviously, they offer an extraordinary test of the model over scales that are completely different from the ones currently probed. We thus tested whether a more sensitive mission (in absence of foregrounds, which are however included in [57]) could have a larger impact. The results are reported in the upper section of Table 3. First, we find that for an SD experiment such as PIXIE10 (i.e. comparable to PRISM) still no sizable improvement in the bounds on the cosmological parameters is present with respect to Planck alone. However, a futuristic experiment such as PIXIE1000 combined with Planck could, in principle, improve the bounds on A s and n s by a factor of ∼ 1.2 and ∼ 4, respectively, with respect to Planck alone. We also find that adding PIXIE1000 to LiteBIRD+CMB-S4 would improve the constraint on n s by roughly a factor of 4 with respect to LiteBIRD+CMB-S4 alone. These are the most affected ΛCDM parameters, since they are the ones with the largest impact on SDs through the effect of anisotropy dissipation, as discussed above. As a comparison, upcoming CMB anisotropy experiments alone, such as LiteBIRD combined with CMB-S4, are expected to improve the constraints on these parameters by a factor of ∼ 2 with respect to Planck alone. We further find that the improvement in A s from an SD experiment would also lift the known degeneracy between A s and τ reio , improving the constraint on the latter by roughly a factor of 1.4. While a PIXIE1000 experiment is currently very optimistic, the results presented in this section show that an SD experiment with perfectly controlled systematics and futuristic sensitivity could, in principle, improve the constraints even on just the ΛCDM model, albeit by a small amount. Finally, as a remark, note that the SD y and µ parameters listed in Table 3 have been calculated from the cosmological parameters estimated from the MCMC as explained in Section 4.1. Their errors are not to be considered as the error on a direct measurement of the distortion signal, since they are derived from the uncertainties on the cosmological parameters, so that y and µ can have much smaller uncertainties than what they would have if evaluated considering the corresponding SD mission alone. Therefore, these results translate to model-dependent constraints on y and µ, and only when the SD experiments become sensitive at a level comparable to the given CMB anisotropy mission their combination can effectively improve constraints also on cosmological parameters. Running of the spectral index In the second scenario that we consider, we investigate the impact that SDs might have on an extension of the ΛCDM model involving the running of the spectral index, as described in Section 2.1. In Figure 5 Here y and µ are derived quantities inferred from other cosmological parameters, which are tightly constrained by the combination of CMB anisotropies+SDs. As a consequence, y and µ have much smaller uncertainties than what they would have if evaluated from SDs alone (see text for additional discussions). Note also that the constraints from Planck are calculated via mock likelihoods rather than from the real data. While this makes only a small difference on most of the parameters, it provides a tighter constraint on τ reio with respect to published results, which, however, does not impact our discussion. reported in the lower section of Table 3. For similar and complementary discussions see e.g. Figure 3 of [17] and the related text as well as [25]. The results displayed in Figure 5 allow a number of interesting considerations. First of all, it is clear that the addition of FIRAS to Planck does not improve the bounds on n run at all, while the addition of PIXIE does so only marginally. However, we find that an experiment 10 times more sensitive than PIXIE, such as PRISM, could provide a major improvement on these constraints. When combined with either Planck or LiteBIRD+CMB-S4, PRISM would straighten the bounds by a factor of 3 and 1.7, respectively, with respect to CMB anisotropy data alone, providing a constraint of σ(n run = 2 × 10 −3 ) for Planck+PIXIE10 compared to σ(n run = 6.5 × 10 −3 ) for Planck alone, or σ(n run = 1.5 × 10 −3 ) for LiteBIRD+CMB-S4+PIXIE10 compared to σ(n run = 3.5 × 10 −3 ) for LiteBIRD+CMB-S4 12 . This is potentially very interesting, since such combinations of experiments would start probing values of the running in agreement with the simplest inflationary models, predicting |n run | ∼ (n s − 1) 2 ∼ 10 −3 [66]. In the presence of foregrounds, which are expected to worsen the sensitivity of SDs to cosmological parameters by a factor of approximately 10 [57], this expectation would still be within the reach of a PIXIE1000-like mission. In Table 4 we also checked the impact of only letting n run vary while fixing all the other cosmological parameters to the best-fit values from [1] for the test-case of PIXIE1000. In this case, the constraint on n run would improve by more than one order of magnitude Planck PIXIE1000 Planck+PIXIE1000 σ(10 3 n run ) 3.8 0.018 0.019 σ(10 9 y) -0.0010 0.0011 σ(10 8 µ) -0.0015 0.0015 with respect to the combination LiteBIRD+CMB-S4+PIXIE1000 when marginalizing over all parameters. This highlights the importance of performing these forecasts in a consistent way by marginalizing over all of the parameters which impact the SD signal, and by breaking degeneracies by combining with other probes, in our case CMB anisotropies. Furthermore, in Figure 6 we also show the expected posterior distributions of the power spectrum parameters (left panel), and of the derived y and µ parameters (right panel) for a selection of the configurations listed in Table 3. It is interesting to notice that when adding an extremely sensitive SD experiment such as PIXIE1000 to a CMB anisotropy experiment such as Planck, a degeneracy appears between the power spectra parameters, in particular A s and n s , which was absent in the similar Figure 7 of [3]. This is due to the fact that the SD experiment becomes sensitive enough to set competitive constraints on the power spectrum amplitude with respect to the anisotropy experiment. However, it cannot completely disentangle a larger amplitude of the overall spectrum A s from a larger power at very small scales due to an increase in n s , thus imposing a negative correlation between the two parameters. Dark matter annihilation We now focus on the annihilating DM particle scenario, described by the six ΛCDM parameters plus the annihilation efficiency p ann . As in previous cases, we perform forecasts Table 5. Same as in Table 3 but for the annihilating DM case (see Equation (2.7)). For p ann we show the 95% C.L. upper limit in units of cm 3 /(s GeV). for the cases of Planck and LiteBIRD+CMB-S4 combined with SD experiments of different sensitivities. Marginalized errors on each parameter are reported in Table 5. We find, however, that the addition of SD experiments -even up to 1000 times the sensitivity of PIXIE -only improves the bound on p ann by roughly a factor of 1.5 with respect to Planck or LiteBIRD plus CMB-S4 alone. Therefore, CMB SDs are not a particularly suitable tool to probe the annihilation of relic particles, as already suggested by previous works (see e.g. [35]), even when considering the most futuristic SD missions 13 . This is mostly due to two facts. First of all, DM annihilation produces secondary ionizing and Lyman-α photons which can significantly affect the recombination history of the universe, thus leaving a strong imprint on CMB anisotropies. Conversely, such an energy injection would produce a number density of extra photons that would be negligible compared to that of CMB photons, and would result in negligible SDs. Secondly, as can be seen e.g. in Figure 4 of [3], the energy injection due to DM annihilation is nearly constant at redshifts z > 10 4 . Since the same is also true for the dissipation of acoustic waves (see again Figure 4 of [3]), the respective effects of p ann and A s on SDs are partly degenerate. Thus p ann alone cannot be efficiently constrained by SD experiments. Dark matter decay Unlike in the case of DM annihilation, SD observations are an extremely powerful constraining tool in the context of decaying DM. This is because they allow to probe a range of DM lifetimes lying outside of the reach of other complementary probes such as CMB anisotropies. In order to precisely evaluate the extent of the constraining power of SDs, we perform forecasts on the parameters influencing the decay of DM particles, i.e. the decaying DM fraction f frac and its lifetime expressed in terms of the decay width Γ dec ≡ 1/τ dec , by consid- Figure 7. Forecasted 95% C.L. upper limits on the decaying DM fraction f frac for three different particle lifetimes, for Planck combined with PIXIE variants (solid lines). As a reference, the current bounds from FIRAS are shown as horizontal dotted lines and the BBN bounds from [28] as horizontal dashed lines. Note that, differently from what is done in Figure 5, here we do not show the LiteBIRD+CMB-S4+PIXIE/variants bounds, as they do not differ from the Planck+PIXE/variants already shown. ering a 6 + 2 extension of the ΛCDM model including {f frac , Γ dec }. To efficiently perform the MCMC runs, we slice the parameter space along Γ dec for different discrete values and sample the remaining 6 + 1 parameter space -otherwise the non-convex topology of the posterior distribution in the f frac − Γ dec plane would slow down the convergence of the chains and highly increase the CPU time [3]. In Figure 7 we show the resulting uncertainties on f frac for three different particle decay rates (Γ dec = 10 −8 , 10 −9 and 10 −10 1/s) as a function of the sensitivity of the experiment (solid lines). The same quantities are also listed in Table 6. For reference, in the figure we also display the corresponding FIRAS predictions as dotted lines and the current BBN bounds for the different Γ dec as given in [28]. The results shown in Figure 7 rely on the combination of future SD experiments with Planck data. CMB anisotropies are know to be directly sensitive to DM decay only when the lifetime is larger than O(10 12 s), such that the released energy may affect the thermal history of the universe around the time of recombination. Still, for the smaller lifetimes considered here, the inclusion of future CMB anisotropy data could in principle strengthen the bounds on {f frac , Γ dec } through the reduction of parameter degeneracies. We performed dedicated runs with Planck replaced by LiteBIRD+CMB-S4 and found that this is not the case. The reason is that the parameters {f frac , Γ dec }, that are probed only by SD experiments for Γ dec ≥ 10 −10 1/s, do not appear to be degenerate with any other ΛCDM parameter. A key conclusion of this section is that a future SD mission with a sensitivity one order of magnitude worse than PIXIE would already be able to set stronger bounds on the decaying DM fraction than BBN (true even accounting for the drop in sensitivity expected due to the presence of foregrounds). This can be quantitatively seen comparing the solid (SDs) and the dashed (BBN) lines in Figure 7. This means that, even using current technology, SD experiments could be by far the best available probe of the thermal history prior to Table 6. Same as in Table 3 but for the decaying DM case (see Equation (2.9)) for different DM lifetimes. For f frac we show the 95% C.L. upper limit. For the Planck alone case, which is not sensitive to the lifetimes considered here, we just report for reference the ΛCDM constraints already listed in Table 3. recombination. This is even more true considering that, as already argued in [3], we are reaching the maximum amount of information that can be extracted from BBN observables, and therefore significant improvements in the corresponding sensitivities are not expected. As a final remark, note that in Table 6 the constraints on several ΛCDM parameters also improve once more accurate SD data are included. This shows that even in the case of DM decay, there is enough information in the SD signal to still improve the constraints on multiple parameters such as ln(10 10 A s ) and n s . Feasibility in different environments In the previous sections we explored the synergies that future SD experiments could potentially have with CMB anisotropy experiments. However, most of the experimental configurations considered in the previous forecasts assume observations from a satellite in a very wide range of frequencies (see Section 3.1). Realistically, such experiments will be preceded by pathfinders from the ground or from a balloon which will allow us to test the required technologies. For such forerunners, the atmosphere represents a formidable contaminant [67]. First, photon absorption in water vapor limits the observable frequency windows. Second, the average brightness of the sky increases the level of photon noise in the detectors. Third, the dependence of this brightness over time and direction introduces further systematics and compromises the efficiency of noise subtraction. In this section, we explore the ability of a ground-based or balloon-borne experiment to measure an SD signal. We consider the experimental configurations described in Section 3.2. As mentioned there, we take into account two of the main limitations induced by the atmosphere, i.e. narrower frequency bands and an increased photon noise due to the average brightness of the sky. However, we do not take into account the third and potentially most problematic source of contamination, the fluctuations in sky brightness. Strategies to mitigate the corresponding uncertainties are in development (see e.g. [45]), and we leave the analysis of their impact for future work. Here we study the case of the ΛCDM model, in which SDs are produced by the dissipation of acoustic waves and baryon cooling. For this model, the ground-based or balloon-borne experiments considered here are not sensitive enough to set competitive bounds on {A s , n s } compared to CMB anisotropy data. However, we should stress again that such new SD bounds, although not very strong, would provide independent information on the primordial power spectrum at scales k ∼ 1 − 10 4 Mpc −1 , and thus give some precious insight on possible deviations from a power-law or running primordial spectrum on these scales (see Figure 8 for a graphical overview). Therefore, in this Section we will only use SD experiments to set a constraint on A s , while assigning Gaussian priors derived from the Planck experiment to all other parameters. Furthermore, for these runs we will also marginalize over the scattering optical depth ∆τ and the electron temperature T e , as discussed in Section 4.2, in order to account for uncertainties on the SD signal caused by the ICM, which provides the largest late-time contribution to the total SD signal [23]. Indeed, the level of this signal will not be accurately known prior to such experiments. On the contrary, this will constitute one of their main targets. However, as in the previous sections, here we also neglect the deterioration of the signal due to the presence of galactic and extra-galactic foregrounds. Although a precise estimate of the impact that such foregrounds would have on the experimental configurations considered in this section is not present in the literature, we expect them to degrade the final sensitivity to the SD signal by a factor larger than 10 (which was derived for PIXIE [57]) because of the limited frequency array. We leave a more detailed analysis of this aspect for future work. Our results for a ground-based, balloon and satellite configuration are presented in Table 7. In order to quantify the impact of marginalizing over late-time sources we also quote the results obtained without marginalizing over late-time sources for the satellite configuration. As clearly shown in the table, a detection of the ICM signal (y ∼ 10 −6 ) would be in the reach of a balloon experiment (with σ(y) 2 × 10 −8 ), while the primordial ΛCDM signal, although with a low significance, would only be possible with an SD mission from space (with σ(A s ) 2 × 10 −9 ). A ground-based experiment would instead only be able to set an upper bound on A s and y, as has been the case for FIRAS. However, in order to truly appreciate the constraining power of the experimental configurations discussed in this Section, it is useful to compare them with current constraints from other types of observations. For this purpose, we show in Figure 8 the most up-to-date constraints on the primordial power spectrum shape, coming from a variety of complementary probes at different scales. In particular, we compare the bounds imposed by Planck at large scales with those from several SD observations at small scales. While the former are inferred from Planck CMB anisotropy data (similarly to Figure 24 of [68]), the latter . Current and forecasted constraints on the primordial power spectrum shape. As a reference, the black solid line shows the power-law ΛCDM prediction (assuming Planck best-fitting values of A s and n s ). The orange region corresponds to the 2σ uncertainty on the primordial power spectrum inferred from Planck CMB anisotropy data (similarly to Figure 24 of [68]). For simplicity, the Planck constraint has been cut at 1 Mpc −1 . The dashed blue area represents the region of parameter space that can be excluded by the non-observation of Ultra-Compact Mini Halos (UCMH) [69]. The green and pink regions correspond to the area of parameter space probed respectively by FIRAS (y < 1.5 × 10 −5 and µ < 9 × 10 −5 at 2σ, [13]) and PIXIE (y 4 × 10 −9 and µ 2 × 10 −8 at 2σ, [14]). The curves have been calculated according to the analytical approximations presented in [38] (see in particular Equations (24)-(25) therein). As the approximations are only valid for k > 1 Mpc −1 , the contours are sharply cut a that scale. For a graphical comparison, similar contours have already been shown e.g. in Figure 4 of [24]. Finally, the red curves represent the regions potentially probed at 2σ by the ground-based (solid), balloon (dashed) and satellite (dot-dashed) configurations considered in this work. have been calculated according to the analytical approximations presented in [38] (see in particular Equations (24)-(25) therein). The red lines represent the different environments considered in this section, while for the sensitivities of FIRAS and PIXIE we refer to the results obtained in [13] and [14], respectively 14 . The shape of the SD bounds follows the extrapolated ΛCDM prediction up to k values of the order of 25 Mpc −1 . In this range, the bounds match the uncertainties on A s reported in Table 7. For higher wavenumbers the constraining power of SDs rapidly deteriorates because of the exponential drop in the production of µ distortions at redshifts of the order of 10 6 (proportionally to the visibility function, see e.g. [34,70], as well as Figure 2 of [4]). Focusing now on the red lines representing the configurations listed in Table 7, it becomes clear from Figure 8 that even the least sensitive of the considered setups would improve by up to two orders of magnitude the current most stringent bounds on the primordial power spectrum at scales higher than k 1 Mpc −1 . The improvement extends over four decades in k space and, although with a reduced significance, can be expected to still be present even after the inclusion of galactic and extra-galactic foregrounds. This clearly shows that a future SD mission, even from the ground, would greatly contribute to our understanding of the inflationary epoch at scales yet unexplored by any other cosmological probe. Finally, Table 7 also shows the impact of not marginalizing over the late-time effects for the satellite case. In this case, it becomes possible to use the information stored in the y distortion to constrain the early universe. In the ΛCDM case, this would give an opportunity to tighten the constraint on A s by a factor of 10. Conclusion The recent advent of precision cosmology has enabled us to test the ΛCDM model and its extensions with an unprecedented level of accuracy. In particular, CMB anisotropies have provided very stringent constraints on a wide range of cosmological models. In order to further explore and test these scenarios, complementary measurements are required. In this paper we studied the possibility of combining CMB anisotropy experiments with CMB SD measurements, extending the work of [3]. SDs are predicted to exist even within the standard ΛCDM scenario and can in principle help to constrain its free parameters. For instance, SDs provide unique information on the shape of the primordial scalar power spectrum at scales much smaller than those probed by CMB anisotropies. Furthermore, SDs are also sensitive to more exotic models which modify the thermal history of the universe. Therefore, they can constrain energy injection phenomena which happened at times much earlier than the epoch of recombination, such as the early decay of DM particles. Within this work, we investigated the amount of information that will be possible to extract from the combination of CMB anisotropies and SD measurements, even in very futuristic SD configurations. To this purpose, based on the setups of proposed CMB SD experiments such as PIXIE, we explored the constraining power of combining current and up-coming CMB anisotropy missions -such as Planck or LiteBIRD plus CMB-S4 -with SD experiments with different sensitivities. This allowed us to present constraints on some of the most interesting parameters which impact SDs (such as the running of the scalar spectral index), while consistently marginalizing over the uncertainties on other cosmological parameters. Indeed, parameters such as the baryon or the DM density would be almost completely degenerate in an analysis of SD experimental data alone, but can be strongly constrained by CMB anisotropy experiments. These forecasts marginalize over late-time sources of SDs such as the ICM. However, they assume the perfect removal of other galactic and extra-galactic foregrounds. For the ΛCDM case, we find that an experiment 1000 times more sensitive than PIXIE, comparable to the proposed Voyager 2050 mission, would in principle be able to improve the constraints on A s and n s inferred from LiteBIRD+CMB-S4 data alone, albeit by a small amount. On the other hand, we find that for the running of the spectral index an experiment 10 (resp. 1000) times more sensitive than PIXIE combined with LiteBIRD+CMB-S4 could improve the constraints on n run by a factor of almost 2 (9) with respect to the anisotropy data alone. Moreover, DM annihilation constraints would only improve by up to a factor of 1.4 with respect to anisotropy experiments alone even with an SD experiment 1000 times more sensitive than PIXIE. On the contrary, when exploring models of DM decay with lifetimes shorter than the age of universe at the time of recombination, SD experiments with sensitivities already 10 times worse than PIXIE could provide constraints stronger than current CMB and BBN measurements. Finally, we discuss a few short-term SD experiments, which could be considered as pathfinders for more futuristic configurations. In particular, we consider a ground-based and a balloon configuration, which we then compare with a similar setup sent to space, i.e. a satellite. Also in this case, even if the sensitivities are not competitive with current CMB anisotropy missions, SDs can provide important insights to our understanding of the inflationary epoch. In fact, although a direct detection of the primordial ΛCDM SD signal would be out of reach, even a measurement from the ground could potentially set the currently strongest constraints on the amplitude of the primordial power spectrum at scales between 1 Mpc −1 and 10 4 Mpc −1 . In addition to that, the detection of the late-time SD signal (possible already with a balloon) would also considerably help in understanding the epoch of reionization and structure formation. Again, these are optimistic forecasts, since foreground cleaning as well as atmospheric brightness fluctuations will represent a formidable source of contamination for these measurements, calling for more accurate and realistic forecasts in the future. Overall, with respect to [3], we explored a much larger range of experimental configurations, from ground-based to very futuristic satellite experiments, exploring their complementarity with CMB anisotropies and explicitly accounting for the impact of marginalizing over the contribution of the ICM. Furthermore, we calculated constraints for the ΛCDM model, which in futuristic scenarios could potentially be interesting. In conclusion, SDs are a potentially competitive cosmological tool. Although a direct detection of the ΛCDM SD signal is still missing, we clearly showed in this work that their constraining power ranges from greatly improving existing constraints to probing yet unexplored cosmological scales relevant, for instance, for our understanding of inflation and DM. Figure 1 . 1Left panel : Impact of a variation in the ΛCDM parameters on the primordial SD signal. Figure 2 . 2Total primordial SD signal expected within the ΛCDM scenario (red line) in absolute values, with positive (negative) values shown with a solid (dashed) line. The signal includes the contribution of the adiabatic cooling of baryons and small-scale acoustic wave dissipation computed according to Equation (2.1), but does not include foregrounds. We compare this signal to the sensitivity of the experimental setups considered within this work. For the sensitivity of FIRAS we employ the same frequency binning as in Figure 3 . 3Final sensitivity for different experimental configurations (in units of Jy/sr = 10 −26 W/m/sr/Hz −1 ) Figure 4 . 4Left panel : SD shapes assuming an instantaneous energy injection during the y-era (red line), during the µ-era (green line) and at early times (blue line), same as inFigure 1of[3]. The vertical red, green and blue bands represent the frequency range covered by FIRAS, PIXIE and an ideal detector with improved characteristics referred to as IDEAL. Right panel : Comparison between the branching ratios of the different SD components for the three aforementioned configurations. Figure 5 . 5Forecasted 1σ uncertainties on the running of the scalar spectral index n run for different experimental setups, as a function of the PIXIE/variants sensitivity expressed in units of the PIXIE sensitivity, δI/δI PIXIE . The solid red curve shows results for Planck+PIXIE/variants, while the solid blue one shows the case of LiteBIRD+CMB-S4+PIXIE/variants. As a reference, we also show the constraints obtained with Planck (red dashed), as well as the forecasts for LiteBIRD+CMB-S4 (blue dashed). The constraints of the Planck+FIRAS combination are not explicitly shown as they nearly perfectly overlap with the Planck alone case. Figure 6 . 61D posterior distributions and 2D contours (68% and 95% C.L.) for the primordial power spectrum amplitude A s , the spectral index n s and the running n run (left panel) and the derived y and µ parameters (right panel) for different experimental configurations. Figure 8 8Figure 8. Current and forecasted constraints on the primordial power spectrum shape. As a reference, the black solid line shows the power-law ΛCDM prediction (assuming Planck best-fitting values of A s and n s ). The orange region corresponds to the 2σ uncertainty on the primordial power spectrum inferred from Planck CMB anisotropy data (similarly to Figure 24 of [68]). For simplicity, the Planck constraint has been cut at 1 Mpc −1 . The dashed blue area represents the region of parameter space that can be excluded by the non-observation of Ultra-Compact Mini Halos (UCMH) [69]. The green and pink regions correspond to the area of parameter space probed respectively by FIRAS (y < 1.5 × 10 −5 and µ < 9 × 10 −5 at 2σ, [13]) and PIXIE (y 4 × 10 −9 and µ 2 × 10 −8 at 2σ, [14]). The curves have been calculated according to the analytical approximations presented in [38] (see in particular Equations (24)-(25) therein). As the approximations are only valid for k > 1 Mpc −1 , the contours are sharply cut a that scale. For a graphical comparison, similar contours have already been shown e.g. in Figure 4 of [24]. Finally, the red curves represent the regions potentially probed at 2σ by the ground-based (solid), balloon (dashed) and satellite (dot-dashed) configurations considered in this work. ν [ νGHz] ∆ν [GHz] NEP [W/Hz 1/2 ] ∆I noise [W/m 2 /Hz/sr]PIXIE 30 -1005 15 7.0 × 10 −17 5.0 × 10 −26 SATELLITE 135 -165 5 1.0 × 10 −17 3.8 × 10 −26 195 -245 5 1.0 × 10 −18 8.4 × 10 −27 BALLOON 135 -165 5 2.0 × 10 −17 7.6 × 10 −26 195 -245 5 2.5 × 10 −17 2.1 × 10 −25 GROUND 135 -165 5 6.5 × 10 −17 2.5 × 10 −25 195 -245 5 1.0 × 10 −16 8.4 × 10 −25 we show the resulting forecasted uncertainties on the running of the spectral index for the CMB anisotropy experiments Planck or LiteBIRD+CMB-S4 combined with the SD experiments FIRAS, PIXIE or PIXIE variants. The corresponding numerical values areTable 3. Forecasted 1σ uncertainties on cosmological parameters for the ΛCDM model (top) and the ΛCDM+running of the spectral index (bottom) for different CMB anisotropy experiments (Planck or LiteBIRD+CMB-S4, LiteBIRD is indicated as "LB" in the table) combined with SD experiments.ΛCDM Planck +FIRAS +PIXIE +PIXIE10 +PIXIE100 +PIXIE1000 σ(100ω b ) 0.015 0.017 0.016 0.015 0.015 0.015 σ(ω cdm ) 0.0013 0.0012 0.0013 0.0012 0.00089 0.00090 σ(100θ s ) 0.00035 0.00032 0.00033 0.00035 0.00033 0.00034 σ(τ reio ) 0.0046 0.0046 0.0044 0.0044 0.0039 0.0034 σ(ln(10 10 A s )) 0.0086 0.0086 0.0087 0.0085 0.0083 0.0074 σ(n s ) 0.0039 0.0037 0.0039 0.0034 0.0013 0.00092 σ(10 9 y) - 1461 1.0 0.081 0.010 0.0019 σ(10 8 µ) - 0.073 0.075 0.065 0.015 0.0015 LB+CMB-S4 σ(100ω b ) 0.0034 - 0.0030 0.0033 0.0030 0.0026 σ(ω cdm ) 0.00027 - 0.00026 0.00028 0.00023 0.00020 σ(100θ s ) 0.000086 - 0.000087 0.000087 0.000080 0.000084 σ(τ reio ) 0.0019 - 0.0020 0.0021 0.0017 0.0015 σ(ln(10 10 A s )) 0.0033 - 0.0035 0.0037 0.0032 0.0031 σ(n s ) 0.0016 - 0.0014 0.0014 0.00074 0.00038 σ(10 9 y) - - 0.92 0.080 0.0097 0.0012 σ(10 8 µ) - - 0.028 0.029 0.013 0.0015 ΛCDM + running of the spectral index Planck +FIRAS +PIXIE +PIXIE10 +PIXIE100 +PIXIE1000 σ(100ω b ) 0.016 0.016 0.016 0.016 0.015 0.015 σ(ω cdm ) 0.0013 0.0013 0.0013 0.0013 0.0013 0.0011 σ(100θ s ) 0.00035 0.00034 0.00034 0.00034 0.00033 0.00033 σ(τ reio ) 0.0049 0.0045 0.0047 0.0047 0.0044 0.0035 σ(ln(10 10 A s )) 0.0092 0.0094 0.0095 0.0089 0.0087 0.0077 σ(n s ) 0.0038 0.0039 0.0038 0.0038 0.0035 0.0027 σ(10 3 n run ) 6.5 6.4 5.9 2.0 0.85 0.52 σ(10 9 y) - 1509 0.86 0.11 0.011 0.0023 σ(10 8 µ) - 0.53 0.47 0.15 0.014 0.0016 LB+CMB-S4 σ(100ω b ) 0.0038 - 0.0036 0.0033 0.0030 0.0029 σ(ω cdm ) 0.00029 - 0.00027 0.00026 0.00028 0.00023 σ(100θ s ) 0.000085 - 0.000085 0.000087 0.000087 0.000086 σ(τ reio ) 0.0021 - 0.0020 0.0020 0.0019 0.0018 σ(ln(10 10 A s )) 0.0036 - 0.0037 0.0035 0.0034 0.0032 σ(n s ) 0.0017 - 0.0.0014 0.0015 0.0014 0.0012 σ(10 3 n run ) 2.6 - 2.3 1.5 0.41 0.30 σ(10 9 y) - - 0.84 0.092 0.012 0.0016 σ(10 8 µ) - - 0.18 0.11 0.019 0.0015 Table 4 . 4Forecasted1σ uncertainties on the running of the spectral index by keeping the ΛCDM parameters fixed to Planck best-fitting values. -0.022 -0.00268 0.0166 nrun 2.05 2.11 2.17 10 +9 As 0.954 0.966 0.978 ns 2.05 2.11 2.17 10 +9 As -0.022 -0.00268 0.0166 nrun Planck+PIXIE Planck+PIXIE1000 LiteBIRD+CMB-S4+PIXIE LiteBIRD+CMB-S4+PIXIE1000 0.74 2.1 3.46 GROUND BALLOON SATELLITE SATELLITEnoreio σ(10 9 A s )Table 7. Forecasted 2σ upper bounds and 1σ uncertainties on the power spectrum amplitude, as well as on the y and µ parameters, for the experimental setups in different environments. Note that we have set Gaussian priors derived from the Planck experiment on all the cosmological parameters except for A s . The fiducial value of the primordial amplitude parameter is A s = 2.106 × 10 9 . The configurations labeled SATELLITEnoreio refer to the same setup as SATELLITE, but without marginalizing over the contribution of reionization and late-time sources.< 137 < 47.1 2.2 0.23 σ(10 9 y) < 2616 20.6 15.8 0.43 σ(10 8 µ) < 145 < 50.3 2.3 0.25 The units adopted in[1] are cm 3 /(s GeV). A useful conversion is 1 × 10 −28 cm 3 /(s GeV) = 6.2 × 10 −25 m 3 /(s J). Specifically, three configurations with better sensitivity than PIXIE and three with worse sensitivity. am: Atmospheric Model, Website CLASS: the Cosmic Linear Anisotropy Solving System, Website 5 MontePython 3: Website We re-iterate that this is a result of considering the y and µ parameters a given experiment would measure, not those generically predicted by the given model. Of course, a given experiment might attempt to correct for the known biases on the measured y and µ parameters in order to infer the underlying physical y and µ parameters. Nevertheless, since in this work we focus solely on the constraints on the involved cosmological parameters, we leave such an analysis for future work. However, we remind the reader that PIXIE will be able to detect the SD signal from late-time sources[14], which we do not discuss in this Section. Note that this considerable improvement is due to the fact that the pivot scale has been set very close to the recombination scale, so that the presence of SDs constitute a second reference point very far from that scale, thus greatly improving the measurement. Shifting the pivot scale to higher k values would invert the role of anisotropies and SDs, and the relative improvement on the measurement of nrun. Note, however, that our setup relies on the assumption that the annihilation rate stays constant over the whole history of the universe. SDs might still be valuable to test a possible time dependence of the annihilation rate. We do not consider the values presented in Sec. 5.1 because they are mainly driven by the inclusion of Planck for FIRAS and PIXIE, and we refrain from showing more realistic forecasts such as the ones presented in[57] which also include foregrounds (while the ones we employ based on[14] do not) for sake of consistency with the other configurations we display. AcknowledgementsWe thank Jens Chluba for reading the manuscript and providing very helpful suggestions. HF acknowledges support from the IAP for the completion of his Master's work and supplying computational resources. ML is supported by the "Probing dark matter with neutrinos" -25 - arXiv:1807.062091807.06209Planck 2018 results. VI. Cosmological parameters, arXiv e-prints. Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv e-prints (Jul, 2018) arXiv:1807.06209, [1807.06209]. J Chluba, 1903.04218Spectral Distortions of the CMB as a Probe of Inflation, Recombination, Structure Formation and Particle Physics: Astro2020 Science White Paper. 51184J. Chluba et al., Spectral Distortions of the CMB as a Probe of Inflation, Recombination, Structure Formation and Particle Physics: Astro2020 Science White Paper, Bull. Am. Astron. Soc. 51 (2019) 184, [1903.04218]. 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{'abstract': 'Measurements of the cosmic microwave background (CMB) spectral distortions (SDs) will open a new window on the very early universe, providing new information complementary to that gathered from CMB temperature and polarization anisotropies. In this paper, we study their synergy as a function of the characteristics of the considered experiments. In particular, we examine a wide range of sensitivities for possible SD measurements, spanning from FIRAS up to noise levels 1000 times better than PIXIE, and study their constraining power when combined with current or future CMB anisotropy experiments such as Planck or LiteBIRD plus CMB-S4. We consider a number of different cosmological models such as the ΛCDM, as well as its extensions with the running of the scalar spectral index, the decay or the annihilation of dark matter (DM) particles. While upcoming CMB anisotropy experiments will be able to decrease the uncertainties on inflationary parameters such as A s and n s by about a factor 2 in the ΛCDM case, we find that an SD experiment 100 times more sensitive than PIXIE (comparable to the proposed Super-PIXIE satellite) could potentially further contribute to constrain these parameters. This is even more significant in the case of the running of the scalar spectral index. Furthermore, as expected, constraints on DM particles decaying at redshifts probed by SDs will improve by orders of magnitude even with an experiment 10 times worse than PIXIE as compared to CMB anisotropies or Big Bang Nucleosynthesis bounds. On the contrary, DM annihilation constraints will not significantly improve over CMB anisotropy measurements. Finally, we forecast the constraints obtainable with sensitivities achievable either from the ground or from a balloon.', 'arxivid': '2006.12886', 'author': ['Hao Fu h.fu@soton.ac.uk \nDepartment of Physics and Astronomy\nUniversity of Southampton\nHighfield CampusSO17 1BJUK\n', 'Matteo Lucca mlucca@ulb.ac.be \nService de Physique Théorique\nUniversité Libre de Bruxelles\nC.P. 225B-1050BrusselsBelgium\n', "Silvia Galli gallis@iap.fr \nUMR 7095\nSorbonne Université\nCNRS\nInstitut d'Astrophysique de Paris\n98 bis bd Arago75014ParisFrance\n", 'Elia S Battistelli \nSapienza University of Rome\nPiazzale Aldo Moro 500185RomeItaly\n', 'Deanna C Hooper \nService de Physique Théorique\nUniversité Libre de Bruxelles\nC.P. 225B-1050BrusselsBelgium\n', 'Julien Lesgourgues \nInstitute for Theoretical Particle Physics and Cosmology (TTK)\nRWTH Aachen University\nD-52056AachenGermany\n', 'Nils Schöneberg \nInstitute for Theoretical Particle Physics and Cosmology (TTK)\nRWTH Aachen University\nD-52056AachenGermany\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nUniversity of Southampton\nHighfield CampusSO17 1BJUK', 'Service de Physique Théorique\nUniversité Libre de Bruxelles\nC.P. 225B-1050BrusselsBelgium', "UMR 7095\nSorbonne Université\nCNRS\nInstitut d'Astrophysique de Paris\n98 bis bd Arago75014ParisFrance", 'Sapienza University of Rome\nPiazzale Aldo Moro 500185RomeItaly', 'Service de Physique Théorique\nUniversité Libre de Bruxelles\nC.P. 225B-1050BrusselsBelgium', 'Institute for Theoretical Particle Physics and Cosmology (TTK)\nRWTH Aachen University\nD-52056AachenGermany', 'Institute for Theoretical Particle Physics and Cosmology (TTK)\nRWTH Aachen University\nD-52056AachenGermany'], 'corpusid': 245383200, 'doi': '10.1088/1475-7516/2021/12/050', 'github_urls': [], 'n_tokens_mistral': 31443, 'n_tokens_neox': 25899, 'n_words': 15901, 'pdfsha': '77abc0ee812af0ae2d07c8bd9dc4a37db2c7a086', 'pdfurls': ['https://arxiv.org/pdf/2006.12886v2.pdf'], 'title': ['Unlocking the synergy between CMB spectral distortions and anisotropies', 'Unlocking the synergy between CMB spectral distortions and anisotropies'], 'venue': []}

arxiv

Optimal Local Thresholds for Distributed Detection in Energy Harvesting Wireless Sensor Networks 5 Nov 2018 Ghazaleh Ardeshiri gh.ardeshiri@knights.ucf.edu University of Central Florida Azadeh Vosoughi Senior Member, IEEEHassan Yazdani h.yazdani@knights.ucf.edu University of Central Florida Optimal Local Thresholds for Distributed Detection in Energy Harvesting Wireless Sensor Networks 5 Nov 2018 We consider a wireless sensor network, consisting of K heterogeneous sensors and a fusion center (FC), that is tasked with solving a binary distributed detection problem. Each sensor is capable of harvesting and storing energy for communication with the FC. For energy efficiency, a sensor transmits only if the sensor test statistic exceeds a local threshold θ k , its channel gain exceeds a minimum threshold, and its battery state can afford the transmission. Our proposed transmission model at each sensor is motivated by the channel inversion power control strategy in the wireless communication community. Considering a constraint on the average energy of transmit symbols, we study the optimal θ k 's that optimize two detection performance metrics: (i) the detection probability PD at the FC, assuming that the FC utilizes the optimal fusion rule based on Neyman-Pearson optimality criterion, and (ii) Kullback-Leibler distance (KL) between the two distributions of the received signals at the FC conditioned by each hypothesis. Our numerical results indicate that θ k 's obtained from maximizing the KL distance are near-optimal. Finding these thresholds is computationally efficient, as it requires only K one-dimensional searches, as opposed to a K-dimensional search required to find the thresholds that maximize PD. I. INTRODUCTION The designs of wireless sensor networks to perform the task of distributed detection are often based on the conventional battery-powered sensors, leading into designs with a short lifetime, due to battery depletion [1], [2], [3]. Recently, energy harvesting, which can collect energy from renewable resources in ambient environment (e.g., solar, wind, and geothermal energy) has attracted much attention [4], [5]. Energy harvesting technology in wireless sensor networks promises a selfsustainable system with a lifetime that is not limited by the lifetime of the conventional batteries [2], [6], [7]. In this paper, we consider the distributed detection of a known signal using a wireless network with K energy harvesting sensors and a fusion center (FC). Each sensor makes a noisy observation, corrupted by both additive and multiplicative observation noises. Each sensor applies an energy detector, to compare its test statistic against a local decision threshold θ k (to be optimized), and transmits only if the test statistic exceeds θ k , its channel gain exceeds a minimum threshold ζ k , and its battery state can afford transmission. Given our transmission and battery state models, our goal is to investigate the optimal θ k 's that optimize the detection performance metric, subject to average transmit symbol energy constraint. The paper organization follows: in Section II we present our system model, including our transmission and battery state models. In Section III we derive the optimal fusion rule and its corresponding detection and false alarm probabilities P D , P F , we provide two approximate expressions for the total Kullback-Leibler (KL) distance KL tot at the FC, and we discuss finding ζ k 's based on the average transmit symbol energy constraint. Section IV illustrates our numerical results on optimizing θ k 's based on maximizing P D and KL tot , and our concluding remarks. II. OUR SYSTEM MODEL AND PROBLEM STATEMENT We consider a distributed binary hypothesis testing problem where K sensors and a FC are tasked with solving a binary hypothesis testing problem. The particular detection problem we focus on is determining the presence or absence of a known scalar signal A (see Fig.??). Let x k denote the local observation at sensor k during an observation period. We assume the following signal model H 1 : x k = Ag k + w k , H 0 : x k = w k(1) where w k and g k are additive and multiplicative observation noises, respectively. We assume w k ∼ N (0, σ 2 w k ), g k ∼ N (0, γ g k ) and all observation noises are independent over time and among K sensors. During each observation period, sensor k takes N samples of x k to measure the received signal energy and applies an energy detector to make a binary decision, i.e., sensor k decides whether or not signal A is present. Let d k denote the binary decision of sensor k, where d k = 0 and d k = 1, respectively, correspond to H 0 and H 1 . The test statistic for sensor k is Λ k = 1 N N n=1 |x k,n | 2 ≷ d k =1 d k =0 θ k (2) where θ k is local decision threshold to be optimized. For the signal model in (1), conditioned on each hypothesis x k is Gaussian, that is, x k |H 0 ∼ N 0, σ 2 w k and x k |H 1 ∼ N Aγ g k , σ 2 w k . The test statistic Λ k in (2) has non-central Chi-square distribution [7] as given below H 1 : Λ k ∼ χ 2 N (η k ), H 0 : Λ k ∼ χ 2 N(3) where η k = A 2 E{g 2 k,n } = A 2 γ g k is the non-centrality parameter. Using (3), the false-alarm probability P f k and detection probability P d k can be derived as following P f k = Pr(Λ k > θ k |H 0 ) = Γ N/2, N θ k σ 2 w k Γ (N/2)(4)P d k = Pr(Λ k > θ k |H 1 ) = Q N/2 √ η k σ w k , √ N θ k σ w k (5) where Γ(n) is the gamma function, Γ(n, x) = ∞ x t n−1 e −t dt is the upper incomplete gamma function, Q n (a, b) = ∞ b x( x a ) n−1 exp( x 2 +a 2 −2 )I n−1 (ax) dx is the generalized Marcum-Q function, and I n−1 (·) is modified Bessel function of order n − 1 [8]. We assume each sensor is able to harvest energy from the environment and stores this harvested energy in a battery that has the capacity of storing at most K units of energy. As shown in Fig. ??, the sensors communicate with the FC through orthogonal fading channels with channel gains |h k |'s that are independent and have Rayleigh distribution with parameters γ h k . The sensors employ on-off keying (OOK) signaling for communication, where a d k = 1 decision at sensor k is conveyed at the cost of spending one or more energy units and a d k = 0 decision is conveyed through a no-transmission with no energy cost. We assume that only sending a message costs units of energy, and the energy of making the observation and processing is negligible. The number of energy units spent to convey a d k = 1 decision depends on the quality of the channel gain |h k | and the battery state of sensor k. Motivated by the channel-inversion power control strategy developed in the wireless communication community [9] we try to compensate for the fading and let the number of energy units spent to convey a d k = 1 decision be (roughly) inversely proportional to |h k | (i.e., a smaller |h k | corresponds to a larger number of energy units), albeit if the battery has sufficient number of stored energy units. To avoid the battery depletion when |h k | is too small, we impose an extra constraint inspired by the channel truncation technique in the channel-inversion power control strategy [9], to ensure that a d k = 1 decision is conveyed only if |h k | exceeds a minimum threshold ζ k (choice of ζ k will be discussed later). Let t indicate the index of the observation period and b k,t denote the battery state of sensor k in the observation period t. Let u k,t represent the sensor output corresponding to the observation period t. Based on the above explanations, we define u k,t as u k,t = ⌈ λ |h k | ⌉ Λ k > θ k , b k,t > ⌈ λ |h k | ⌉, |h k | 2 > ζ k 0 Otherwise (6) where λ is a power regulation constant (that depends on the battery structure). We use the round function ⌈.⌉ toward +∞, to ensure that u k,t is a discrete symbol and the energy of this symbol is equal to the number of consumed energy units to convey d k = 1. The constraint Λ k > θ k in (6) comes directly from (2). We assume the average energy of the transmitted symbol u k,t is constrained, i.e., P av k = E{⌈ λ |h k | ⌉ 2 u k = ⌈ λ |h k | ⌉}, where the expectation is taken with respect to |h k |. We model b k,t in (6) as the following b k,t = min b k,t−1 − ⌈ λ |h k | ⌉I u k,t−1 + Ω k,t , K(7) where b k,t−1 is the battery state of the previous observation period and Ω k,t ∈ {0, 1} is a binary random variable, indicating whether or not sensor k harvests one unit of energy. We assume Ω k,t is a Bernoulli random variable, with Pr(Ω k,t = 1) = p e , where p e depends on the harvesting structure. This assumption is repeatedly used in the literature (see [10] and references therein). The indicator function I u k,t−1 in (7) is defined as I u k,t−1 = 1 u k,t−1 > 0 0 Otherwise (8) In the remaining, we focus on one observation period and we drop the subscript t from the battery state b k,t and the sensor output u k,t . Given our system model description above, our goal is to investigate the optimal local decision thresholds θ k 's in (2) that optimizes the detection performance metric. III. OPTIMIZING LOCAL DECISION THRESHOLDS We consider two detection performance metrics to find the optimal θ k 's: (i) the detection probability at the FC, assuming that the FC utilizes the optimal fusion rule based on Neyman-Pearson optimality criterion, and (ii) the KL distance between the two distributions of the received signals at the FC conditioned on hypothesis H 0 , H 1 . In Section III-A we derive the optimal fusion rule and the expressions for the detection and false alarm probabilities P D , P F at the FC. In Section III-B we derive two approximate expressions for the KL distance at the FC. In Section III-C we discuss the choice of the threshold ζ k in (6). A. Optimal LRT Fusion Rule and P D , P F Expressions The received signal at the FC from sensor k is y k = h k u k + n k , where the additive communication channel noise n k ∼ N 0, σ 2 n k . The likelihood ratio at the FC is [11] ∆ LRT = log f (y 1 , ..., y K |H 1 ) f (y 1 , ..., y K |H 0 ) = K k=1 log u k f (y k |u k , H 1 ) Pr (u k |H 1 ) u k f (y k |u k , H 0 ) Pr (u k |H 0 )(9) in which we use the fact that, given H i the received signals at the FC are independent, i.e., f (y 1 , ..., y K |H i ) = K k=1 f (y k |H i ). Examining (9), we note given u k , y k and H i are independent and hence f (y k |u k , H i ) = f (y k |u k ) for i = 0, 1. Also, given u k , y k is Gaussian, i.e., y k | u k =0 ∼ N 0, σ 2 n k and y k | u k =⌈ λ |h k | ⌉ ∼ N ⌈ λ |h k | ⌉h k , σ 2 n k . The probabilities Pr(u k |H 1 ), Pr(u k |H 0 ) in (9) are Pr u k = ⌈ λ |h k | ⌉ H 1 = Pr Λ k > θ k , b k > ⌈ λ |h k | ⌉, |h k | 2 > ζ k H 1 = Pr Λ k > θ k |H 1 Pr b k > ⌈ λ |h k | ⌉ Pr |h k | 2 > ζ k = P d k ρ k q k = α k (10) Pr u k = ⌈ λ |h k | ⌉|H 0 = Pr Λ k > θ k H 0 Pr b k > ⌈ λ |h k | ⌉ Pr |h k | 2 > ζ k = P f k ρ k q k = β k(11) where P f k , P d k are given in (4), (5), ρ k = Pr(b k > ⌈ λ |h k | ⌉) and q k = Pr(|h k | 2 > ζ k ) = exp(−ζ k /γ h k ). Assuming b k in (7) is a stationary random process, one can compute the cumulative distribution function (CDF) and the probability mass function (pmf) of b k in terms of K, p e , γ h k . Fig.2(a) shows CDF of b k for K = 20 and p e = 0.5, 0.75, 0.82, and Fig.2(b) depicts pmf of b k for K = 50 and p e = 0.8. For our numerical results in Section IV we use pmf of b k to find ρ k in (10) and (11). Combing all, we can rewrite ∆ LRT as the following [12] ∆ LRT = K k=1 log α k f (y k |u k = ⌈ λ |h k | ⌉) + (1−α k )f (y k |u k = 0) β k f (y k |u k = ⌈ λ |h k | ⌉) + (1−β k )f (y k |u k = 0) = K k=1 log α k exp − (y k −⌈ λ |h k | ⌉h k ) 2 2σ 2 n k +(1 − α k )exp − y 2 k 2σ 2 n k β k exp − (y k −⌈ λ |h k | ⌉h k ) 2 2σ 2 n k +(1 − β k )exp − y 2 k 2σ 2 n k In low SNR regime as σ 2 n k → ∞ taking a logarithm from ∆ LRT and using the approximations e −x ≈ 1 − x and log(1 + x) for small x, we can simplify alarm and detection probabilities P F , P D at the FC are ∆ LRT to ∆ LRT ≈ −T k + K k=1 ν k y k where T k = K k=1 ⌈ λ |h k | ⌉ 2 h 2 k (α k −β k )/2σ 2 n k and ν k = ⌈ λ |h k | ⌉h k (α k − β k )/σ 2 n k .P F = Pr (∆ LRT > τ |H 0 ) = Q τ − µ ∆|H0 σ ∆|H0 (12) P D = Pr (∆ LRT > τ |H 1 ) = Q Q −1 (a)σ ∆|H0 + µ ∆|H0 − µ ∆|H1 σ ∆|H1(13) where µ ∆|Hi = −T k + K k=1 ν k µ y k |Hi , σ 2 ∆|Hi = K k=1 ν 2 k σ 2 y k |Hi , i = 0, 1 µ y k |H0 = ⌈ λ |h k | ⌉h k β k , σ 2 y k |H0 = ⌈ λ |h k | ⌉ 2 h 2 k β k (1−β k )+σ 2 n k µ y k |H1 = ⌈ λ |h k | ⌉h k α k , σ 2 y k |H1 = ⌈ λ |h k | ⌉ 2 h 2 k α k (1−α k )+σ 2 n k The threshold τ is determined from the constraint on P F ≤ a in terms of a. We note that P D expression depends on all our optimization variables θ k 's through α k , β k 's in µ ∆|Hi and σ 2 ∆|Hi . B. KL Expression Let KL tot denote the KL distance between the two distributions f (y 1 , ..., y K |H 1 ) and f (y 1 , ..., y K |H 0 ) at the FC. Since f (y 1 , ..., y K |H i ) = K k=1 f (y k |H i ), we have KL tot = K k=1 KL k where KL k by definition is [13] KL k = y k f (y k |H 1 ) log f (y k |H 1 ) f (y k |H 0 ) dy k(14) We note that the distributions f (y k |H i ), i = 0, 1 are Gaussian mixtures and thus KL k in (14) does not have a general closedform expression [14] and approximations must be made. One can approximate KL k in (14) by the KL distance of two Gaussian distributions with the means µ y k |H0 , µ y k |H1 , and the variances σ 2 y k |H0 and σ 2 y k |H1 , respectively, i.e., KL k can be approximated as [15] KL k ≈ 1 2 log( σ 2 Another approximation for KL k in (14) can be found using the low SNR regime approximation in Section III-A, as the following y k |H0 σ 2 y k |H1 ) + σ 2 y k |H1 − σ 2 y k |H0 + (µ y k |H1 − µ y k |H0 ) 2 2σ 2 y k |H0(15)KL k ≈ c k (β k −α k ) c k π 2σ 2 n k (1 − α k )(Q( y k σn k ) − 0.5) +α k Q( y k −c k σn k ) +α k exp (c k −y k ) 2 −2σ 2 n k + (1−α k ) exp −y 2 k 2σ 2 n k(16) where c k = ⌈ λ |h k | ⌉h k . Different from P D expression that depends on all θ k 's, KL tot is decoupled such that KL k depends on θ k only through α k , β k 's in µ y k |Hi and σ 2 y k |Hi . C. Choosing Threshold ζ k in (6) We find ζ k in (6) via solving the constraint P av k = E{⌈ λ |h k | ⌉ 2 u k = ⌈ λ |h k | ⌉}. Recall h k has Rayleigh distribution. After some algebraic manipulations we obtain P av k = α k ∞ i=1 (i + 1) e −1 γ h k max ζ k , λ 2 i+1 −e −λ 2 iγ h k u λ 2 i −ζ k (17) where u[.] is the step function and α k is given in (10). Note α k depends on ζ k through q k . Although there is no explicit expression for ζ k , for our numerical results in Section IV we use (17) to find ζ k given P av k via the interpolation technique. IV. SIMULATION RESULTS AND CONCLUSIONS In this section, we numerically (i) find θ k 's which maximize P D in (13). Finding θ k 's in this case requires K-dimensional search, as K grows the computational complexity grows exponentially; (ii) θ k 's which maximize KL tot = K k=1 KL k , using the KL k approximations in (15), (16). Finding θ k in this case requires only one dimensional search and is computationally very efficient. We then compare P D evaluated at the θ k 's obtained from maximizing P D (refer to as scheme I in the plots), with P D evaluated at the θ k 's obtained from maximizing KL tot (refer to as scheme II in the plots). Our simulation parameters are K = 3, A = 1, N = 100, λ = 1, γ h = [1.5, 0.8, 1.4], γ g = [1.3, 2, 0.9] and σ 2 n = [0.9, 1.2, 0.8]. Note that sensors are heterogeneous, in the sense that their statistical information parameters are different. Given P av k = P av we first obtain numerically ζ k 's using (17), where ζ k 's are still different since α k 's are different. Fig. 3 plots P D versus P F , where for each P F we evaluate P D using θ k 's which maximize KL tot , based on the KL k approximations in (15) and (16). The fixed parameters in Fig. (3) are K = 20 units, p e = 0.75, P av = 1 dB. This figure shows that, these two approximations have similar P D − P F behavior. Therefore, in the remaining figures, we use the KL k approximation in (15). Fig. 4 depicts P D versus P F for K = 20 units, p e = 0.75, P av = 1 dB. To plot Fig. 4, for each P F we evaluate P D using θ k 's that maximize P D (scheme I) and KL tot (scheme II). Comparing schemes I and II in Fig. 4, we observe that these schemes perform very closely, indicating that using θ k 's that are obtained from maximizing KL tot are near-optimal. In Fig. 4, we also compare schemes I and II for the special case where we assume all sensors employ the same local threshold θ k = θ. For this special case, finding θ maximizing P D or KL tot only needs one dimensional search. The performance gap between each scheme and its corresponding special case indicates that when sensors are heterogeneous, it is advantageous to use different local thresholds according to sensors' statistics (i.e., γ h k , γ g k , σ n k ). Fig. 5 plots P D versus P av for K = 20 units, p e = 0.75, P F = 0.5. As expected, P D increases as P av increases. The reason is as P av increases ζ k 's decrease, and sensors can afford to transmit even when their channel gains are weaker. Fig. 6 illustrates P D versus K for p e = 0.8, P av = 1 dB, P F = 0.5. As expected, P D increases as K increases and it saturates after certain K, since P D is not limited by the battery size anymore and instead is limited by the sensors' statistics. Comparing schemes I and II and their corresponding special cases in Figs. 5 and 6, we make similar observations to those in Fig. 4. In summary, we studied a distributed detection problem in a wireless network with K heterogeneous energy harvesting sensors and investigated the optimal local decision thresholds for given transmission and battery state models. Our numerical results indicate that the thresholds obtained from maximizing the KL distance are near-optimal. Finding these thresholds is computationally very efficient, as it requires only K onedimensional searches, as opposed to a K-dimensional search required to find the thresholds that maximize the detection probability. ACKNOWLEDGMENT This research is supported by NSF under grant 1341966. Fig. 1 : 1Our System model Fig. 2 : 2(a) CDF of b k for K = 20 and pe = 0.5, 0.75, 0.82, (b) pmf of b k for K = 50 and pe = 0.8. Fig. 3 : 3Given a threshold τ , the optimal likelihood ratio test (LRT) is ∆ LRT ≷ PD vs. PF , K = 20, pe = 0.75, Pav = 1dB. Fig. 4 :Fig. 5 : 45PD vs. PF , K = 20, pe = 0.75, Pav = 1 dB. PD vs. Pav , K = 20, pe = 0.75, PF = 0.5. Fig. 6 : 6PD vs. K, pe = 0.8, Pav = 1 dB, PF = 0.5. Energy harvesting sensor nodes: Survey and implications. S Sudevalayam, P Kulkarni, IEEE Communications Surveys Tutorials. 133443461S. Sudevalayam and P. Kulkarni, "Energy harvesting sensor nodes: Survey and implications," IEEE Communications Surveys Tutorials, vol. 13, no. 3, pp. 443461, Third 2011. 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Jaldn, "Decentralized detection in energy harvesting wireless sensor networks," in 2016 24th European Signal Processing Conference (EUSIPCO), Aug 2016, pp. 567571. Optimal energy management policies for energy harvesting sensor nodes. V Sharma, U Mukherji, V Joseph, S Gupta, IEEE Transactions on Wireless Communications. 9413261336V. Sharma, U. Mukherji, V. Joseph, and S. Gupta, "Optimal energy man- agement policies for energy harvesting sensor nodes," IEEE Transactions on Wireless Communications, vol. 9, no. 4, pp. 13261336, April 2010. I S Gradshteyn, I M Ryzhik, Table of integrals, series, and products. 7th ed.I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., 2007. A Goldsmith, Wireless Communications. Cambridge University PressA. Goldsmith, Wireless Communications. Cambridge University Press, 2005. Aging Aware Random Channel Access for Battery-Powered Wireless Networks. R Valentini, M Levorato, F Santucci, IEEE Wireless Communications Letters. 5R. Valentini, M. Levorato and F. Santucci, "Aging Aware Random Chan- nel Access for Battery-Powered Wireless Networks," in IEEE Wireless Communications Letters, vol. 5, no. 2, pp. 176-179, April 2016. Decision fusion rules in multihop wireless sensor networks. Y Lin, B Chen, P K Varshney, IEEE Transactions on Aerospace and Electronic Systems. 412475488Y. Lin, B. Chen, and P. K. Varshney, "Decision fusion rules in mul- tihop wireless sensor networks," IEEE Transactions on Aerospace and Electronic Systems, vol. 41, no. 2, pp. 475488, April 2005. Channel aware sensor selection in distributed detection systems. H R Ahmadi, A Vosoughi, 2009 IEEE 10th Workshop on Signal Processing Advances in Wireless Communications. 7175H. R. Ahmadi and A. Vosoughi, "Channel aware sensor selection in distributed detection systems," in 2009 IEEE 10th Workshop on Signal Processing Advances in Wireless Communications, June 2009, pp. 7175 Sensor selection for hypothesis testing in wireless sensor networks: a kullback-leibler based approach. D Bajoiv, B Sinopoli, J Xavier, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference. the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference16591664D. Bajoiv, B. Sinopoli, and J. Xavier, "Sensor selection for hypothesis testing in wireless sensor networks: a kullback-leibler based approach," in Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, Dec 2009, pp. 16591664. Optimal Power Allocation for Distributed Detection Over MIMO Channels in Wireless Sensor Networks. X Zhang, H V Poor, M Chiang, IEEE Transactions on Signal Processing. 56X. Zhang, H. V. Poor and M. Chiang, "Optimal Power Allocation for Distributed Detection Over MIMO Channels in Wireless Sensor Networks," in IEEE Transactions on Signal Processing, vol. 56, no. 9, pp. 4124-4140, Sept. 2008. Fusion of decisions transmitted over fading channels in wireless sensor networks. B Chen, R Jiang, T Kasetkasem, P K Varshney, Conference Record of the Thirty-Sixth Asilomar Conference on Signals, Systems and Computers. 2B. Chen, R. Jiang, T. Kasetkasem, and P. K. Varshney, "Fusion of decisions transmitted over fading channels in wireless sensor networks," in Conference Record of the Thirty-Sixth Asilomar Conference on Signals, Systems and Computers, 2002., vol. 2, Nov 2002, pp. 1184 1188 vol.2.

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{'abstract': "We consider a wireless sensor network, consisting of K heterogeneous sensors and a fusion center (FC), that is tasked with solving a binary distributed detection problem. Each sensor is capable of harvesting and storing energy for communication with the FC. For energy efficiency, a sensor transmits only if the sensor test statistic exceeds a local threshold θ k , its channel gain exceeds a minimum threshold, and its battery state can afford the transmission. Our proposed transmission model at each sensor is motivated by the channel inversion power control strategy in the wireless communication community. Considering a constraint on the average energy of transmit symbols, we study the optimal θ k 's that optimize two detection performance metrics: (i) the detection probability PD at the FC, assuming that the FC utilizes the optimal fusion rule based on Neyman-Pearson optimality criterion, and (ii) Kullback-Leibler distance (KL) between the two distributions of the received signals at the FC conditioned by each hypothesis. Our numerical results indicate that θ k 's obtained from maximizing the KL distance are near-optimal. Finding these thresholds is computationally efficient, as it requires only K one-dimensional searches, as opposed to a K-dimensional search required to find the thresholds that maximize PD.", 'arxivid': '1811.01909', 'author': ['Ghazaleh Ardeshiri gh.ardeshiri@knights.ucf.edu \nUniversity of Central Florida\n\n', 'Azadeh Vosoughi Senior Member, IEEEHassan Yazdani h.yazdani@knights.ucf.edu \nUniversity of Central Florida\n\n'], 'authoraffiliation': ['University of Central Florida\n', 'University of Central Florida\n'], 'corpusid': 67874946, 'doi': '10.1109/globalsip.2018.8646364', 'github_urls': [], 'n_tokens_mistral': 8250, 'n_tokens_neox': 7081, 'n_words': 4613, 'pdfsha': 'fa864b131608173137fa87544ed04aa58c684229', 'pdfurls': ['https://arxiv.org/pdf/1811.01909v1.pdf'], 'title': ['Optimal Local Thresholds for Distributed Detection in Energy Harvesting Wireless Sensor Networks', 'Optimal Local Thresholds for Distributed Detection in Energy Harvesting Wireless Sensor Networks'], 'venue': []}

arxiv

Relating the Newman-Penrose constants to the Geroch-Hansen multipole moments 25 Aug 2009 Thomas Bäckdahl t.backdahl@qmul.ac.uk School of Mathematical Sciences University of London Mile End RoadE1 4NSLondonQueen MaryEngland Relating the Newman-Penrose constants to the Geroch-Hansen multipole moments 25 Aug 2009 In this paper, we express the Newman-Penrose constants in terms of the Geroch-Hansen multipole moments for stationary spacetimes. These expressions are translation-invariant combinations of the multipole moments up to quadrupole order, which do not normally vanish. Introduction The Newman-Penrose (NP) constants were defined by Newman and Penrose in [12]. They are quantities defined on the null-infinities, and turn out to be conserved under time translations. Even though they have been studied for a long time, their meaning is still not fully understood. Lately, it has been disputed whether the NP constants are zero for stationary spacetimes or not. For the Kerr solution they are zero [3]. In fact, it has been shown that they are zero for all algebraically special stationary spacetimes [15]. The NP constants have also been calculated for a wide set of examples [4,7,11]. The original paper [12] by Newman and Penrose gives expressions of the NP constants in terms of multipole moments. It is unclear, however, how these moments were defined, if they are coordinate independent and if different moments can be specified independently. The Geroch-Hansen multipole moments have these properties, but were defined later [8,9]. Therefore, this paper is intended to clearly settle the matter by expressing the NP constants in terms of the Geroch-Hansen multipole moments. These multipole moments also give a possibility of physical interpretation. The Geroch-Hansen multipole moments can be freely specified under a simple convergence condition. That is, for any given choice of multipoles, satisfying the convergence condition, there is a unique stationay spacetime with these multipole moments. This was shown in [2] for the stationary axisymmetric case. Recently, Herberthson [10] showed this for the general static case using results of Friedrich [6]. The result of Frierdich states that for static spacetimes one can freely specify null data under a convergence condition. These null data are related to the multipole moments, but the relation is fairly complicated. The results by Friedrich have been extended to the stationary case by Aceña [1]. Hopefully, the results by Herberthson can also be extended to the stationary case, but for now it is still an open problem. For the static case, one could establish the relation between the NP constants and the Geroch-Hansen multipole moments, using the results of Friedrich and Kánnár [7], but it will not give the general stationary case. One would also need to be careful with the translation between formalisms. Therefore, the original definition of the multipole moments, and the asymptotic expansions of Wu and Shang [15] are used in this paper. Throughout this paper we use abstract index notation. For coordinate expressions we sometimes omit the indices, and use the short hand notation dxdy = (dx) (a (dy) b) . Tetrad expressions In this paper, we will use series expansions of stationary spacetimes in Bondi-Sachs coordinates (u, r, ζ,ζ). Expressed in standard angular coordinates, the complex angle ζ = e iφ cot θ 2 . The differential operators ð,ð are defined as in equation (4.15.117) in [13], for the complex stereographic coordinates ζ,ζ, i.e. ðf = 1 + ζζ √ 2 ∂f ∂ζ + s ζ √ 2 f,ðf = 1 + ζζ √ 2 ∂f ∂ζ − sζ √ 2 f (1) where s is the spin-weight of f . Observe that this differs slightly from the operator usually used for the θ, φ coordinates, due to a different choice of spinframe. The corresponding spin-weighted spherical harmonics are then given by s Y j,m = (2j + 1)(j + s)!(j − s)!(j + m)!(j − m)!ζ j−m ζ j+s (−1) m 2 √ π(1 + ζζ) j × min(j−m,j+s) r=max(0,s−m) (−ζζ) −r r!(j − m − r)!(j + s − r)!(r + m − s)! (2) where −j ≤ s ≤ j, −j ≤ m ≤ j. We take the following expansion of the null tetrad from [15], using Ψ 0 2 =Ψ 0 2 . l a = ∂ ∂r , n a = ∂ ∂u + − 1 2 − Ψ 0 2 r +ð Ψ 0 1 + ðΨ 0 1 6r 2 −ð 2 Ψ 0 0 + ð 2Ψ0 0 24r 3 − |Ψ 0 1 | 2 12 +ð 2 Ψ 1 0 + ð 2Ψ1 0 120 r −4 + O(r −5 ) ∂ ∂r + 1 + ζζ 6 √ 2r 3 Ψ 0 1 − 1 + ζζ 12 √ 2r 4ð Ψ 0 0 + O(r −5 ) ∂ ∂ζ + 1 + ζζ 6 √ 2r 3Ψ 0 1 − 1 + ζζ 12 √ 2r 4 ðΨ 0 0 + O(r −5 ) ∂ ∂ζ , m a = − Ψ 0 1 2r 2 +ð Ψ 0 0 6r 3 +ð Ψ 1 0 24r 4 + O(r −5 ) ∂ ∂r + 1 + ζζ 6 √ 2r 4 Ψ 0 0 + O(r −5 ) ∂ ∂ζ + 1 + ζζ √ 2r + O(r −5 ) ∂ ∂ζ ,(3) where the expansions of the Weyl curvature are Ψ 0 = Ψ 0 0 r 5 + Ψ 1 0 r 6 + O(r −7 ), Ψ 3 = Ψ 2 3 r 4 + Ψ 3 3 r 5 + Ψ 4 3 r 6 + O(r −7 ), Ψ 1 = Ψ 0 1 r 4 + Ψ 1 1 r 5 + Ψ 2 1 r 6 + O(r −7 ), Ψ 4 = Ψ 4 4 r 5 + Ψ 5 4 r 6 + O(r −7 ),(4)Ψ 2 = Ψ 0 2 r 3 + Ψ 1 2 r 4 + Ψ 2 2 r 5 + Ψ 3 2 r 6 + O(r −7 ). We find that for stationary spacetimes, the timelike Killing vector field, can be expressed as t a = T l a + n a +Ām a + Am a , where T and A were computed in [15] from the Killing equations, and found to be T = 1 2 + Ψ 0 2 r −ð Ψ 0 1 + ðΨ 0 1 6r 2 +ð 2 Ψ 0 0 + ð 2Ψ0 0 24r 3 +ð 2 Ψ 1 0 + ð 2Ψ1 0 120r 4 − |Ψ 0 1 | 2 12r 4 + O(r −5 ), A = − Ψ 0 1 6r 2 +ð Ψ 0 0 12r 3 +ð Ψ 1 0 40r 4 + O(r −5 ).(5) The metric and quotient metric Expressed in terms of the coordinate basis, the Killing vector is t a = ∂ ∂u + O(r −5 ) ∂ ∂r + O(r −5 ) ∂ ∂ζ + O(r −5 ) ∂ ∂ζ .(6) For further calculations, we need expansions of the metric components. The contravariant metric is given by g ab = 2l (a n b) − 2m (amb) . Matrix inversion then gives the covariant metric g ab = 1 + 2Ψ 0 2 r −1 − 1 3 (ðΨ 0 1 + ðΨ 0 1 )r −2 + 1 12 (ð 2 Ψ 0 0 + ð 2Ψ0 0 )r −3 + O(r −4 ) du 2 + 4 √ 2Ψ 0 1 3(1 + ζζ) r −1 − ðΨ 0 0 √ 2(1 + ζζ) r −2 + O(r −3 ) dudζ + 4 √ 2Ψ 0 1 3(1 + ζζ) r −1 −ð Ψ 0 0 √ 2(1 + ζζ) r −2 + O(r −3 ) dudζ + 2dudr(7)+ 2Ψ 0 0 3(1 + ζζ) 2 r −1 + O(r −2 ) dζ 2 + 2Ψ 0 0 3(1 + ζζ) 2 r −1 + O(r −2 ) dζ 2 + − 4 (1 + ζζ) 2 r 2 + O(r −2 ) dζdζ. The norm λ = t a t a = 2T − 2AĀ is λ = 1 + 2Ψ 0 2 r −1 − 1 3 (ðΨ 0 1 + ðΨ 0 1 )r −2 + 1 12 (ð 2 Ψ 0 0 + ð 2Ψ0 0 )r −3 + O(r −4 ) (8) Furthermore, the twist ω a = −ε abcd t b ∇ c t d has a potential ω, which is defined via ∇ a ω = ω a and ω → 0 as r → ∞. Observe that the sign convention alternates throughout the literature. A change of the sign corresponds to complex conjugation of the multipole moments. From the metric we compute ( ∂ ∂r ) a ω a = 2i 3 (ðΨ 0 1 −ðΨ 0 1 )r −3 − i 4 (ð 2Ψ0 0 −ð 2 Ψ 0 0 )r −4 + O(r −5 ).(9) An integration then yields ω = − i 3 (ðΨ 0 1 −ðΨ 0 1 )r −2 + i 12 (ð 2Ψ0 0 −ð 2 Ψ 0 0 )r −3 + O(r −4 ).(10) The equations for the other components are then satisfied due to the vacuum field equations. Now consider a conformal compactification V of the 3-manifold of trajectories of t a with metric h ab = Ω 2 (−λg ab + t a t b ). We want to choose Ω such that we can add a point Λ (the infinity point) such that h ab extends smoothly to Λ. We also demand Ω = 0, D a Ω = 0, D a D b Ω = 2h ab at Λ,(11) where D a is the covariant derivative on h ab . The following choice of conformal factor turns out to be adequate: Ω = (r −1 − Ψ 0 2 r −2 + 11 8 (Ψ 0 2 ) 2 r −3 ) 2 .(12) The coefficients are chosen so as to make the limit of the Ricci tensor of h ab to vanish. The coordinates r, ζ,ζ will naturally induce coordinates on V . With a slight abuse of notation we will use the same name for the induced coordinates. Note that r will be a radial coordinate on V for large r. Unfortunately, the components of the metric h ab will not extend smoothly to Λ in the Cartesian coordinates corresponding to the coordinates (R = r −1 , ζ,ζ). Therefore, we need better coordinates to verify that our choice of conformal factor is good. 1 One way to find good coordinates is to compute harmonic coordinates. Hence, we will use asymptotically Euclidian harmonic coordinates (x, y, z). For computational purposes, we also use the corresponding spherical coordinates with complex stereographic angles. Thus, x = ρ η +η 1 + ηη , y = −iρ η −η 1 + ηη , z = ρ ηη − 1 1 + ηη .(13) A fairly straightforward computation gives us the new coordinates expressed in terms of the old ones: ρ = r −1 − Ψ 0 2 r −2 + 5 4 (Ψ 0 2 ) 2 r −3 + O(r −4 ), η = ζ − √ 2 6 (1 + ζζ)Ψ 0 1 r −2 + O(r −3 ).(14) The conformal metric and the conformal factor are then found to be h ab = dx 2 + dy 2 + dz 2 + O(ρ 3 ), Ω =ρ 2 + 1 4 (Ψ 0 2 ) 2 ρ 4 + O(ρ 5 ).(15) Now we easily see that Ω → 0 when ρ → 0; thus, ρ = 0 will now represent the infinity Λ on our 3-manifold. The smoothness of h ab and the conditions (11) can now be easily verified. The Ricci tensor R ab of h ab is R ab = O(ρ). Geroch-Hansen multipole moments Define the complex potential P = 1 − λ − iω (1 + λ + iω) √ Ω .(16) This potential as well as the choice of sign in the definition of the twist is taken from [5]. There are many different possible choices of potential, but large classes of potentials do produce the same moments [14]. The Geroch-Hansen multipole moments [8,9] are given by the limits of P a1...an = C D a1 P a2...an − (n − 1)(2n − 3) 2 R a1a2 P a3...an ,(17) as one approaches Λ. Here C[·] represents the totally symmetric and trace-free part. Hence, with monopole (mass) M , dipole C a , and quadrupole Q ab expressed in Cartesian coordinates, we by definition have lim ρ→0 P = M lim ρ→0 P a = C x dx + C y dy + C z dz lim ρ→0 P ab = Q xx dx 2 + Q yy dy 2 − (Q xx + Q yy )dz 2 + 2Q xy dxdy + 2Q xz dxdz + 2Q yz dydz.(18) Under a translation Ω ′ = Ω(1 + xT x + yT y + zT z ) the dipole will transform like C ′ j = C j − 1 2 M T j , while the quadrupole will transform like Q ′ xx = Q xx − 2T x C x + T y C y + T z C z − 1 4 M −2T x 2 + T y 2 + T z 2 , Q ′ yy = Q yy + T x C x − 2T y C y + T z C z − 1 4 M T x 2 − 2T y 2 + T z 2 , Q ′ xy = Q xy − 3 2 T x C y − 3 2 T y C x + 3 4 M T x T y , Q ′ xz = Q xz − 3 2 T x C z − 3 2 T z C x + 3 4 M T x T z , Q ′ yz = Q yz − 3 2 T y C z − 3 2 T z C y + 3 4 M T y T z .(19) We expand Ψ 0 0 , Ψ 0 1 and Ψ 0 2 in terms of spin-weighted spherical harmonics: Ψ 0 0 = 2 m=−2 A m2 Y 2,m = √ 5 A −2 + 2ζA −1 + √ 6ζ 2 A 0 + 2ζ 3 A 1 + ζ 4 A 2 2 √ π(1 + ζζ) 2 , Ψ 0 1 = 1 m=−1 B m1 Y 1,m = − √ 3 B −1 + √ 2ζB 0 + ζ 2 B 1 2 √ π(1 + ζζ) , Ψ 0 2 = C.(20) Here C is real, B m and A m are complex. A series expansion of the potential yiels P = − C + 2ηB −1 + √ 2(ηη − 1)B 0 − 2ηB 1 √ 24π(1 + ηη) ρ − √ 5(η 2η2 − 4ηη + 1)A 0 4 √ 6π(1 + ηη) 2 ρ 2 − √ 5(η 2 A −2 +η(ηη − 1)A −1 − η(ηη − 1)A 1 + η 2 A 2 ) 4 √ π(1 + ηη) 2 ρ 2 + 3C 3 8 ρ 2 + O(ρ 3 )(21) One then easily obtains the multipole moments by changing to Cartesian coordinates and taking limits: lim ρ→0 P = − C, lim ρ→0 P a = lim ρ→0 D a P = √ 6 12 √ π (B −1 − B 1 )dx − i √ 6 12 √ π (B −1 + B 1 )dy + √ 3 6 √ π B 0 dz, lim ρ→0 P ab = lim ρ→0 (D a D b P − 1 3 h ab D c D c P ) = √ 5 24 √ π ( √ 6A 0 − 3A 2 − 3A −2 )dx 2 + √ 5 24 √ π ( √ 6A 0 + 3A 2 + 3A −2 )dy 2 + i √ 5 4 √ π (−A 2 + A −2 )dxdy − √ 30 12 √ π A 0 dz 2 + √ 5 4 √ π (A 1 − A −1 )dxdz + i √ 5 4 √ π (A 1 + A −1 )dydz.(22) Newman-Penrose constants By comparing the limits (18) and (22), one can conclude that A −2 = −2 π 5 (Q xx − Q yy + 2iQ xy ), B −1 = √ 6π(C x + iC y ), A −1 = −4 π 5 (Q xz + iQ yz ), B 0 = 2 √ 3πC z , A 0 = 2 6π 5 (Q xx + Q yy ), B 1 = √ 6π(−C x + iC y ),(23)A 1 = 4 π 5 (Q xz − iQ yz ), C = −M, A 2 = 2 π 5 (−Q xx + Q yy + 2iQ xy ). The NP constants {G m } can then be computed from G m = 2π 0 π 0 Ψ 1 0 2 Y 2,m sin θdθdφ = 2π 0 π 0 ( 10 3 Ψ 0 1 − 5Ψ 0 2 Ψ 0 0 ) 2 Y 2,m sin θdθdφ. (24) Here the spin-weighted spherical harmonics are as in the definition (2). For the integration, the variables are changed to (θ, φ) via ζ = e iφ cot θ 2 . Observe that we do not change the spin frame to be adapted to the new coordinates. Expansions of the integrands can, in principle, be taken from [15] eq (51), but they do use a different spin-frame in that section the paper, hence it is easier to redo the calculations than translating the result. The integration gives G −2 = −2 √ 5π(3C 2 y − 3C 2 x + M Q xx − M Q yy + 2iM Q xy − 6iC x C y ), G −1 = −4 √ 5π(iM Q yz − 3C x C z − 3iC y C z + M Q xz ), G 0 = 2 √ 30π(−C 2 x − C 2 y + 2C 2 z + M Q xx + M Q yy ),(25)G 1 = −4 √ 5π(iM Q yz + 3C x C z − 3iC y C z − M Q xz ), G 2 = −2 √ 5π(3C 2 y − 3C 2 x + M Q xx − M Q yy − 2iM Q xy + 6iC x C y ). As expected, this is the same form as in the original paper by Newman and Penrose [12], i.e., linear combinations of dipole squared and monopole times quadrupole. From the translation rules (19), it is easy to see that the NP constants are invariant under translations. Hence, they are independent of the choice of conformal factor. As the NP constants are expansion coefficients for spin-weighted spherical harmonics, they will depend on the spin-frame though. For the axisymmetric case, we see that G −2 = G −1 = G 1 = G 2 = 0 and G 0 = 2 √ 30π(2C 2 z − M Q zz ), where Q zz = −2Q xx = −2Q yy is the zz-component of the quadrupole. We can conclude that the NP constants are, in general, not zero, but for some important solutions they are. For instance, the Kerr solution has C z = iM a, Q zz = −2Q xx = −2Q yy = −2M a 2 , and all other components of C a and Q ab are zero. This yields the well-known fact that all NP constants are zero for the Kerr solution. In fact, they are zero for all stationary, algebraically special solutions [15]. For the computation of the multipole moments, we actually do not need better coordinates, but to verify smoothness, we do. AcknowledgementsThis work was supported by the Wenner-Gren foundations. Thanks to Juan A. Valiente Kroon, for helpful discussions. I would also like to thank Lars Andersson for asking about the relation between multipole moments and Newman-Penrose constants. Convergent Null Data Expansions at Space-Like Infinity of Stationary Vacuum Solutions. A E Aceña, Ann. Henri Poincaré. 10Aceña, A. E., Convergent Null Data Expansions at Space-Like Infinity of Stationary Vacuum Solutions, Ann. Henri Poincaré 10 (2009), 275-337. Axisymmetric stationary solutions with arbitrary multipole moments. T Bäckdahl, Class. Quantum Grav. 24Bäckdahl, T., Axisymmetric stationary solutions with arbitrary multi- pole moments, Class. Quantum Grav. 24 (2007), 2205-2215. Light cone structure near null infinity of the Kerr metric. S Bai, Z Cao, X Gong, Y Shang, X Wu, Y K Lau, Phys. Rev. D. 7544003Bai, S., Cao, Z., Gong, X., Shang, Y., Wu, X. and Lau, Y. K., Light cone structure near null infinity of the Kerr metric, Phys. Rev. D 75 (2007), 044003. Conserved quantities in a black hole collision. S Dain, J A Valiente-Kroon, Class. Quantum Grav. 19Dain, S. and Valiente-Kroon, J. A., Conserved quantities in a black hole collision, Class. Quantum Grav. 19 (2002), 811-816. Multipole moments of axisymmetric systems in relativity. G Fodor, C Hoenselaers, Z Perjés, J. Math. Phys. 30Fodor, G., Hoenselaers, C. and Perjés, Z., Multipole moments of axisymmetric systems in relativity, J. Math. Phys. 30 (1989), 2252-2257. Static Vacuum Solutions from Convergent Null Data Expansions at Space-Like Infinity. H Friedrich, Ann. Henri Poincaré. 8Friedrich, H., Static Vacuum Solutions from Convergent Null Data Ex- pansions at Space-Like Infinity, Ann. Henri Poincaré 8 (2007), 817-884. Bondi-type systems near spacelike infinity and the calculations of the Newman-Penrose constants. H Friedrich, J Kánnár, J. Math. Phys. 41Friedrich, H. and Kánnár, J., Bondi-type systems near spacelike infin- ity and the calculations of the Newman-Penrose constants, J. Math. Phys. 41 (2000), 2195-2232. Multipole Moments. II. Curved Space. R Geroch, J. Math. Phys. 11Geroch, R., Multipole Moments. II. Curved Space, J. Math. Phys. 11 (1970), 2580-2588. Multipole moments of stationary spacetimes. R O Hansen, J. Math. Phys. 15Hansen, R. O., Multipole moments of stationary spacetimes, J. Math. Phys. 15 (1974), 46-52. Static spacetimes with prescribed multipole moments. M Herberthson, arXiv:0906.4247v1Gerochgr-qcHerberthson, M., Static spacetimes with prescribed multipole moments; a proof of a conjecture by Geroch, arXiv:0906.4247v1 [gr-qc]. Boost-rotation symmetric type D radiative metrics in Bondi coordinates. R Lazkoz, J A Valiente-Kroon, Phys. Rev. D. 6284033Lazkoz, R. and Valiente-Kroon, J. A., Boost-rotation symmetric type D radiative metrics in Bondi coordinates, Phys. Rev. D 62 (2000), 084033. New conservation laws from zero rest-mass fields in asymptotically flat space-time. E T Newman, R Penrose, Proc. Roy. Soc. A. 305Newman, E. T. and Penrose, R., New conservation laws from zero rest-mass fields in asymptotically flat space-time, Proc. Roy. Soc. A. 305 (1968), 175-204. Spinors and space-time. R Penrose, W Rindler, Cambridge University Press1CambridgePenrose, R. and Rindler, W., Spinors and space-time, volume 1, Cam- bridge: Cambridge University Press, (1984). The multipole structure of stationary spacetimes. W Simon, R Beig, J. Math. Phys. 24Simon, W. and Beig, R., The multipole structure of stationary space- times, J. Math. Phys. 24 (1983), 1163-1171. On Newman-Penrose constants of stationary spacetimes. X Wu, Y Shang, Class. Quantum Grav. 24Wu, X. and Shang, Y., On Newman-Penrose constants of stationary spacetimes, Class. Quantum Grav. 24 (2007), 679-690.

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{'abstract': 'In this paper, we express the Newman-Penrose constants in terms of the Geroch-Hansen multipole moments for stationary spacetimes. These expressions are translation-invariant combinations of the multipole moments up to quadrupole order, which do not normally vanish.', 'arxivid': '0905.3292', 'author': ['Thomas Bäckdahl t.backdahl@qmul.ac.uk \nSchool of Mathematical Sciences\nUniversity of London\nMile End RoadE1 4NSLondonQueen MaryEngland\n'], 'authoraffiliation': ['School of Mathematical Sciences\nUniversity of London\nMile End RoadE1 4NSLondonQueen MaryEngland'], 'corpusid': 5228279, 'doi': '10.1088/0264-9381/26/17/175021', 'github_urls': [], 'n_tokens_mistral': 7074, 'n_tokens_neox': 5993, 'n_words': 3527, 'pdfsha': '7309c807a1cfba6fa2745fb8495f85538580eb07', 'pdfurls': ['https://arxiv.org/pdf/0905.3292v2.pdf'], 'title': ['Relating the Newman-Penrose constants to the Geroch-Hansen multipole moments', 'Relating the Newman-Penrose constants to the Geroch-Hansen multipole moments'], 'venue': []}

arxiv

Multi-Black Rings on Eguchi-Hanson Space 14 Apr 2008 Shinya Tomizawa 3.*e-mail:tomizawa@sci.osaka-cu.ac.jp Department of Mathematics and Physics Graduate School of Science Osaka City University 3-3-138 Sugimoto558-8585SumiyoshiOsakaJapan Multi-Black Rings on Eguchi-Hanson Space 14 Apr 2008(Dated: April 15, 2008)arXiv:0802.0741v2 [hep-th] We construct new supersymmetric multi-black ring solutions on the Eguchi-Hanson base space as solutions of the five-dimensional minimal supergravity. The space-time has an asymptotically locally Euclidean time slice, i.e., it has the spatial infinity with the topology of the lens space L(2; 1) = S 3 /Z 2 . The configurations of black rings are restricted by the requirement of the absence of a Dirac-Misner string everywhere outside horizons. Especially, in the case of two black rings, the solutions have the limit to a pair of rotating black holes with the horizon topology of S I. INTRODUCTION One of striking features of asymptotically flat black holes in five dimensions is that they admit event horizons with non-spherical topology [1, 2,3] in contrast to four dimensions [4,5]. The black ring solutions with horizon topology S 1 × S 2 , which rotate in the S 1 direction, were found by Emparan and Reall as solutions to the five-dimensional vacuum Einstein equations [6]. This is the first example of asymptotically flat black hole solutions with such non-spherical horizon topology. Remarkably, within some range of the parameters, there are a black hole [7] and two black rings for the same values of the mass and the angular momentum, which means the violation of the black hole uniqueness [8,9,10,11,12,13,14,15,16] known in four dimension (See Ref. [17,18] about the discussion on the uniqueness of a black hole and a black ring solution.). Subsequently, by using solitonic techniques, another black ring solutions were found. The black ring solutions with a rotating two sphere were found by Mishima and Iguchi [19] by the Bäcklund transformation (they were independently found by Figueras [20]), and moreover, ones with two angular momenta were constructed by Pomeransky and Sen'kov [21] by using the inverse scattering method [22,23,24,25,26,27,28,29,30,31,32,33,34]. Elvang and Figureas generated a black Saturn solution, which describes a spherical black hole surrounded by a black ring [35]. Furthermore, Iguchi and Mishima also generated a black di-ring solution [36]. An orthogonal black di-ring solution was also constructed [37,38]. Based on the classification of the solutions in the five-dimensional minimal supergravity [39], in addition to a black hole solution [40], several supersymmetric black ring solutions have been constructed. The point is that they have been constructed on any four dimensional hyper-Kähler base spaces, especially, the Gibbons-Hawking base space. Elvang et.al. found the first supersymmetric black ring solutions with asymptotic flatness on a four dimensional Euclid space [41]. They also presented more general supersymmetric black ring solutions with three charges [42]. Gauntlett and Gutowski constructed concentric multi-black ring solutions on the base space [43] and also generalized these solutions to black ring solutions with multiple charges [44]. The supersymmetric black ring solutions with a compactified extra dimension on the Taub-NUT base space were constructed [45,46,47,48]. See Refs. [49] about the detail review of black ring solutions. So far, most of people have much attention to asymptotically flat black hole solutions. In four dimension, asymptotic flatness, which in general means the spacetime approaches a Minkowski spacetime at the infinity, is expected to be a good idealization of an isolated system. In higher dimensions, such an asymptotic Minkowski spacetime is considered to be realized if black holes are small enough compared with tension of the brane or the curvature radius of the bulk or the size of extra dimensions. However, in higher dimensional spacetimes, the asymptotic flatness admits a variety of rich structures in the sense that the curvatures vanish at the infinity. In fact, higher dimensional black holes admit a variety of asymptotic structures, although the spacetimes become flat at the infinity. Kaluza-Klein black hole solutions [50,51,52,53,54,55,56,57,58] have the spatial infinity with a compactified extra dimension, i.e., the spacetime approaches a twisted S 1 bundle over a four-dimensional Minkowski spacetime. Black hole solutions on the Eguchi-Hanson space [59] have the spatial infinity of topologically various lens spaces L(2n; 1) = S 3 /Z 2n (n:natural number). Since the latter black hole spacetimes have asymptotically locally Euclidean timeslice, the asymptotic structures of the spacetimes are locally isometric to a five-dimensional Minkowski spacetime. The supersymmetric black ring solutions with non-trivial asymptotic structure were constructed in Ref. [60]. However, the properties of these black object solutions are considerably different from that of the black hole in asymptotic Minkowski spacetimes. For instance, the Kaluza-Klein black holes [53,54] and the black holes on the Eguchi-Hanson space [59] can have the horizon of lens spaces in addition to S 3 . The location of the black rings on the Eguchi-Hanson is restricted unlike the black ring on the Euclid space [41]. In the five-dimensional Einstein-Maxwell theory with a positive cosmological constant, black hole solutions on the Euclid base space [61], the Taub-NUT base space [62] and the Eguchi-Hanson base space [63] were also constructed. In general, these black hole solutions are dynamical. In particular, two-black hole solution on the Eguchi-Hanson space describes a non-trivial coalescence of black holes. In Refes. [63,64], the authors compared the twoblack hole solution on the Eguchi-Hanson space with the two-black holes solution on the Euclid space [61], and discussed how the coalescence of five-dimensional black holes depends on the asymptotic structure of the spacetime. Two black holes with the topology of S 3 coalesce into a single black hole with the topology of the lens space L(2; 1) = S 3 /Z 2 in the case of Eguchi-Hanson space, while two black holes with the topology of S 3 coalesce into a single black hole with the topology of S 3 in the Euclid case. When the action has the Chern-Simon term, black holes can rotate [65,66]. There is no reason to restrict ourselves to the spacetime which asymptotes to the Minkowski spacetime at the infinity. For example, in the context of Kaluza-Klein theory, in the presence of a lot Kaluza-Klein monopoles, the spatial infinity has a topological structure of a lens space L(n, 1) = S 3 /Z n . Hence in a spacetime with such a non-trivial spatial infinity, it is an important issue to study the properties of black hole and black ring solutions. The end of this article is to generalize the supersymmetric black ring solutions on the Eguchi-Hanson base space in Ref. [60] to multi-black ring solutions and to investigate the features of the multi-black solutions. The construction of the solutions is based on the discussion in Ref. [39,60]. We study what the possible configuration of the black rings is. The existence of more than one nut, which means an isolated fixed point of the action of one parameter family of an isometry, make it difficult to construct black ring solutions since Misner strings appears on the nuts. A flat space and the Euclidean self-dual Taub-NUT space has a single nut, while the Eguchi-Hanson space has two nut. However, as discussed in Ref. [60], if we impose the reflection symmetry on the locations and the shapes of black rings, we can also construct multi-black ring solutions on the Eguchi-Hanson space. The remainder of this article is organized as follows. In Sec.II, we give a brief review in Ref. [39]. In Sec.II B, we construct new multi-black ring solutions on the Eguchi-Hanson space, which is the results based on the previous work [60]. In Sec.III, we discuss the properties of these solutions, in particular, asymptotic structure, possible configurations of black rings, and the limit to black holes. We investigate the limit of the two-black ring solutions to two-black hole solutions. In Sec.IV, we summarize our results and give some discussions. II. MULTI-BLACK RING SOLUTIONS A. Construction of solutions The bosonic sector of the five-dimensional minimal supergravity is the Einstein-Maxwell theory with a Chern-Simon term. Following the reference [39], all supersymmetric solution of five-dimensional minimal supergravity have a non-spacelike Killing vector field. In a region where the Killing vector field ∂ t is timelike, the metric and the gauge potential are given by ds 2 = −H −2 (dt + ω) 2 + Hds 2 B , A = √ 3 2 [H −1 (dt + ω) − β],(1) respectively. where the base space ds 2 B is a metric of an arbitrary hyper-Kähler space. In this article, we choose the Eguchi-Hanson space as the base space. The scalar function H, one-forms ω and β on B are given by ∆H = 4 9 (G + ) 2 , dG + = 0, dβ = 2 3 G + .(2) Here, △ is the Laplacian on B and the two-form G + is the self-dual part of the one-form H −1 ω, i.e., G + := 1 2 H −1 (dω + * dω),(3) where (G + ) 2 := 1 2 G mn G mn and * is the Hodge dual operator on B. As is shown in Appendix A, the metric of the Eguchi-Hanson space in the Gibbons-Hawking coordinates is given by ds 2 EH = H k (dr 2 + r 2 dΩ 2 S 2 ) + H −1 k a 8 dψ + ϕ 2 ,(4) where dΩ 2 S 2 = dθ 2 + sin 2 θdφ 2 . The harmonic function on the three-dimensional Euclid space H k takes the form of [54] H k = a 8 1 ∆ a 1 + 1 ∆ a 2 .(5) The one-form ϕ is determined by the equation, rot ϕ = grad H k , and it is explicitly written as ϕ = a 8 (r cos θ + a 1 ) ∆ a 1 + (r cos θ + a 2 ) ∆ a 2 dφ. Here, the functions ∆ a i (i = 1, 2) are defined as ∆ a i = r 2 + 2a i r cos θ + a 2 i with a 1 = −a 2 = −a. The coordinates run the ranges 0 < r, 0 ≤ θ < π, 0 ≤ φ < 2π and 0 ≤ ψ < 4π. ∂ ψ is a Killing vector field with closed orbits on the base space and it has fixed points at the point sources of the harmonic function H k . Such a fixed point of a Killing vector field is called a nut. It should be noted that if Eq.(5) is replaced by a harmonic function with a single nut, i.e., H k = a 8 1 ∆ a 1 ,(7) the metric (4) coincides with that of the four-dimensional Euclid space. The functions H and the one-form ω can be solved explicitly if the Killing vector field ∂ ψ is also a Killing vector field of the full five-dimensional spacetime, i.e., H and ω are independent of ψ and furthermore, if the one-forms ω and β can be written in the form β = β 0 a 8 dψ + ϕ +β,(8)ω = ω 0 a 8 dψ + ϕ +ω.(9) Then, under these assumptions, the functions H, ω 0 and β 0 are written as H = H −1 k K 2 + L,(10)ω 0 = H −2 k K 3 + 3 2 H −1 k KL + M,(11)β 0 = H −1 k K,(12) where K, L and M are another harmonic functions on a three dimensional Euclid space. The one-formsω andβ are determined by the equations dω = * H k dM − MdH k + 3 2 (KdL − LdK) ,(13)dβ = − * dK,(14) where * denotes the Hodge dual on the three dimensional Euclid space. Assume that all point sources of three harmonic functions K, L and M are also located on the z-axis of the three dimensional Euclid space, i.e., K = k 0 + i k i ∆ R i ,(15)L = l 0 + i l i ∆ R i ,(16)M = m 0 + i m i ∆ R i ,(17) with ∆ R i := r 2 + R 2 i + 2R i r cos θ,(18) where these harmonic functions have point sources at r = R i := (0, 0, −R i ). Then, substituting Eqs. (15), (16) and (17) into Eq. (13), (14) and integrating them, we obtain the explicit forms of the one-formsω =ω φ dφ andβ =β φ dφ as follows ω φ = a 8 n i=1 m i a 1 (r cos θ + R i ) + r(r + R i cos θ) (a 1 − R i )∆ a 1 ∆ R i + a 8 n i=1 m i a 2 (r cos θ + R i ) + r(r + R i cos θ) (a 2 − R i )∆ a 2 ∆ R i + n i,j≥1,i =j 3 2 k i l j R i (r cos θ + R j ) + r(r + R j cos θ) (R i − R j )∆ R i ∆ R j + n i=1 3 2 k 0 l i − 3 2 l 0 k i r cos θ + R i ∆ R i − a 8 2 i=1 m 0 r cos θ + a i ∆ a i + C ω ,(19)β φ = − i k i r cos θ + R i ∆ R i + C β ,(20) where C ω and C β are arbitrary constants. We put k 0 = 0 so that the metric component does not diverge at the spatial infinity. We can always put C β = 0 from the freedom of the gauge of the potential A. The tt-component of the metric behaves as g tt ≃ −l −2 0 for r → ∞. To fix the normalization of the timelike Killing vector field at the infinity, we put l 0 = 1. As will be mentioned in the next subsection, C ω and m i are determined by the requirement for the absence of Misner strings for given k i and l i . B. Determination of m i and C ω In this article, we construct multi-black ring solutions on the Eguchi-Hanson space such that there is no Dirac-Misner string everywhere in the space-time. The existence would yield closed timelike curves if one impose that there is no conical singularity. In our solutions the conditionsω φ (θ = 0) = 0 andω φ (θ = π) = 0 assure the absence of Misner strings. Whether there exist multi-black ring solutions is essentially determined by the conditions for the absence of Dirac-Misner strings, namely, the existence of the parameters C ω , m 0 , · · · , m n satisfying the conditionsω φ (θ = 0) =ω φ (θ = π) = 0. For example, in the case of concentric black rings on a flat base space [43], the number of these independent conditions amounts to n + 1 since the z-axis on the three-dimensional Euclid space in the Gibbons-Hawking coordinate are divided into n + 2 intervals by a point source at a 1 of the harmonics H k and N point sources at R 1 , · · · , R n . In general, the conditionsω φ (θ = 0) =ω φ (θ = π) = 0 for each interval give rise to n + 2 independent equations to C ω , m 0 , · · · , m n satisfying all equations. Therefore, since it is assured that there exist these parameters, the locations of black rings are arbitrary. In contrast, in the case of the Eguchi-Hanson base space, the situation changes. Since there are two nuts located at point sources a 1 and a 2 , the number of the intervals on the z-axis increase by one. Namely, the number of independent equations exceeds that of the parameters. Thus, in general, it is impossible to put black rings at arbitrary positions on the z-axis unlike concentric black ring solutions on the flat space. However, it should be noted that if we impose reflection symmetry on the location of the point sources R 1 , · · · , R n on the z-axis, i.e., if we choose the parameters such that m 1 = m 2 , · · · , m N = m 2N , k 1 = k 2 , · · · , k N = k 2N , l 1 = l 2 , · · · , l N = l 2N R 1 = −R 2 , · · · , R N = −R 2N(21) if n is even (n = 2N for a positive integer N), and m 1 = m 2 , · · · , m N −1 = m 2N −2 , k 1 = k 2 , · · · , k N −1 = k 2N −2 , l 1 = l 2 , · · · , l N −1 = l 2N −2 , R 1 = −R 2 , · · · , R N −1 = −R 2N −2 , R 2N −1 = 0 (22) if n is odd (n = 2N − 1), the black rings can be located on arbitrary places on the z-axis except r = a 1 , a 2 since the number of independent equations coincides with that of the parameters. Next we consider black ring solutions with the parameters satisfying (21) or (22). Black rings on an S 2 -bolt For simplicity, we assume that all (2N −1)-black rings are located on an S 2 -bolt. Under the conditions (22), we assume that 0 < −R 1 = R 2 < · · · < −R N −1 = R 2N −2 < a, R 2N −1 = 0. From Eqs.ω φ (θ = 0) =ω φ (θ = π)=0 , the parameters C ω , and m k (k = 0, · · · , 2N − 1) are obtained as C ω = 0,(23)m 0 = − 6 a k 2N −1 + 2 N −1 k=1 k 2k−1 ,(24)m 2i−1 = 6 1 − R 2 2i−1 a 2 k 2i−1 − 2 i−1 k=1 R 2i−1 D i,k R 2 2k−1 − R 2 2i−1 + 2 N −1 k=i+1 R 2k−1 D k,i R 2 2i−1 − R 2 2k−1 − D N,i R 2i−1 ,(25)m 2N −1 = 6 k 2N −1 + 2 N −1 k=1 D N,k R 2k−1 ,(26) where D p,q := k 2p−1 l 2q−1 −k 2q−1 l 2p−1 and i = 1, · · · , N −1. On the other hand, if all 2N-black rings are on an S 2 -bolt, assuming that 0 < −R 1 = R 2 < · · · < −R N = R 2N < a under the conditions (21), we obtain the parameters C ω and m k (k = 0, · · · , 2N) as C ω = 0,(27)m 0 = − 12 a N k=1 k 2k−1 ,(28)m 2i−1 = 6 1 − R 2 2i−1 a 2 k 2i−1 − i−1 k=1 2R 2i−1 D i,k R 2 2k−1 − R 2 2i−1 + N k=i+1 2R 2k−1 D k,i R 2 2i−1 − R 2 2k−1 ,(29)m 2N −1 = 6 1 − R 2 2N −1 a 2 k 2N −1 − N −1 k=1 2R 2N −1 D N,k R 2 2k−1 − R 2 2N −1 .(30) Black rings outside an S 2 -bolt Next we assume that all black rings are located outsides an S 2 -bolt. In the case of (2N −1)black rings, under the assumption a < −R 1 = R 2 < · · · < −R N −1 = R 2N −2 , R 2N −1 = 0 and the conditions (22), Eqs.ω φ (θ = 0) =ω φ (θ = π) = 0 determine the parameters C ω and m k (k = 0, · · · , 2N − 1) as follows C ω = 0,(31)m 0 = − 6 a 2 N −1 k=1 k 2k−1 + k 2N −1 ,(32)m 2i−1 = −6 1 − R 2 2i−1 a 2 × ak 2i−1 R 2i−1 + i−1 k=1 2aD i,k R 2 2i−1 − R 2 2k−1 − 2a R 2i−1 N −1 k=i+1 R 2k−1 D i,k R 2 2i−1 − R 2 2k−1 − aD N,i R 2 2i−1 ,(33)m 2N −1 = 6 2 N −1 k=1 D N,k R 2k−1 + k 2N −1 .(34) In the case of 2N-black rings, assuming that a < −R 1 = R 2 < · · · < −R N = R 2N under (21), we obtain the parameters C ω and m k (k = 0, · · · , 2N) as C ω = 0,(35)m 0 = − 12 a N k=1 k 2k−1 ,(36)m 2i−1 = −6 1 − R 2 2i−1 a 2 × i−1 k=1 2aD i,k R 2 2i−1 − R 2 2k−1 + ak 2i−1 R 2i−1 − 2a R 2i−1 N k=i+1 R 2k−1 D i,k R 2 2i−1 − R 2 2k−1 ,(37)m 2N −1 = −6 1 − R 2 2N −1 a 2 N −1 k=1 2aD N,k R 2 2N −1 − R 2 2k−1 + ak 2N −1 R 2N −1 ,(38) where i = 1, · · · , N − 1. Black rings on and outside an S 2 -bolt Finally, we consider the case where there exists black rings on and outside an S 2 -bolt in the presence of more than two black rings. For instance, in the case of odd black rings, i.e., 0 < −R 1 = R 2 < · · · < −R l = R 2l < a < −R l+1 = R 2(l+1) < · · · < −R N −1 = R 2N −2 , R 2N −1 = 0, the parameters C ω , m k = m 2k (k = 1, · · · , l) are given by Eqs.(23)- (25), and mk = m 2k (k = l + 1, · · · , N − 1) and m 2N −1 are given by Eqs.(33)-(34). In the case of even black rings, i.e., 0 < −R 1 = R 2 < · · · < −R l = R 2l < a < −R l+1 = R 2(l+1) < · · · < −R N = R 2N , the parameters C ω , m k = m 2k (k = 1, · · · , l) and mk = m 2k (k = l + 1, · · · , N) are given by Eqs. (27)- (29) and Eqs.(37)- (38), respectively. C. A single black ring In the special case of n = 1, this solution coincides with a single black ring solution constructed in the previous article [60]. Here we give the short review about the previous work. The parameters C ω m 0 and m 1 are given by C ω = 0, m 0 = − 6k 1 a , m 1 = 6k 1 .(39) Then the metric takes the following form ds 2 = −H −2 dt + ω 0 a 8 dψ + ϕ φ dφ +ω φ dφ 2 +H H k (dr 2 + r 2 dΩ 2 S 2 ) + H −1 k a 8 dψ + ϕ φ dφ 2 ,(40) where dΩ 2 S 2 = dθ 2 + sin 2 θdφ 2 . The coordinates r, ψ, φ, θ run the ranges r > 0, 0 ≤ ψ ≤ 4π, 0 ≤ φ ≤ 2π, 0 ≤ θ ≤ π.(41) The five functions H k , H, ω 0 ,ω φ and ϕ φ are given by H k = a 8 1 ∆ a + 1 ∆ −a ,(42)H = 1 + l 1 r + 8k 2 1 ∆ a ∆ −a ar 2 (∆ a + ∆ −a ) ,(43)ω 0 = 2k 1 − 3 a + 3 r + 6(l 1 + r)∆ a ∆ −a ar 2 (∆ a + ∆ −a ) + 32k 2 1 ∆ 2 a ∆ 2 −a a 2 r 3 (∆ a + ∆ −a ) 2 ,(44)ω φ = 3k 1 4∆ a ∆ −a [(r + a)(∆ −a − ∆ a ) + ((r + a)(∆ −a + ∆ a ) − 2∆ −a ∆ a ) cos θ],(45)ϕ φ = a 8 a(∆ −a − ∆ a ) + r(∆ a + ∆ −a ) cos θ ∆ a ∆ −a .(46) It is noted that our solutions have three independent parameters l 1 , k 1 and a, where k 1 and l 1 are related to the dipole charge q of the black ring and the total electric charge Q e by k 1 = −q/2 and al 1 = 4G 5 Q e /( √ 3π) − q 2 , and a is the radius of the S 2 -bolt on the Eguchi-Hanson space. To avoid the existence of CTCs (closed timelike curves) outside the event horizon, we impose the following conditions on these parameters k 1 < 0, l 1 > −4k 1 .(47) It should be noted that we can choose the origin of the three-dimensional Euclid space E 3 in the Gibbons-Hawking coordinate such that that a 1 = −a 2 = (0, 0, a) and R 1 = 0 without loss of generality. For example, if we shift the origin so that a 1 = 0, we need change ∆ −a , ∆ a and r in Eqs.(42)-(45) into r, ∆ 2a and ∆ a , respectively, and moreover we need replace r cos θ, ∆ −a and ∆ a in Eq.(46) with r cos θ + a, r and ∆ 2a , respectively. As shown in the previous work, the solutions have the same two angular momentum components and the asymptotic structure on timeslices is asymptotically locally Euclidean. The S 1 -direction of the black ring is along the equator on a S 2 -bolt on the Eguchi-Hanson space since the horizon is located at r = 0. There are isometries acting on the S 2 -bolt on the Eguchi-Hanson space. Using these isometries, we can set these two nuts on other poles on the S 2 -bolt. Therefore we can put a black ring along the other equator. Namely, we can construct a black ring along arbitrary equators on the S 2 bolt. D. Two black rings Now we consider the solutions with a pair of black rings located on an S 2 -bolt (0 < −R 1 = R 2 < a). In this case, the parameters C ω , m 0 and m 1 are given by C ω = 0, m 0 = − 12k 1 a , m 1 = m 2 = 6k 1 1 − R 2 1 a 2 .(48) Then the metric functions take the following form H = 1 + l 1 1 ∆ R 1 + 1 ∆ −R 1 + k 2 1 H −1 k 1 ∆ R 1 + 1 ∆ −R 1 2 ,(49)ω 0 = − 12k 1 a − 6k 1 R 2 1 a 2 1 ∆ R 1 + 1 ∆ −R 1 + k 3 1 H −2 k 1 ∆ R 1 + 1 ∆ −R 1 3 + 3k 1 2 H −1 k (1 + 4H k ) 1 ∆ R 1 + 1 ∆ −R 1 + 3k 1 l 1 2 H −1 k 1 ∆ R 1 + 1 ∆ −R 1 2 ,(50)ω φ = 3ak 1 4 1 − a 2 R 2 1 − r 2 + r(R 1 − a) cos θ − aR 1 (a + R 1 )∆ −a ∆ R 1 + r 2 − r(R 1 + a) cos θ + aR 1 (R 1 − a)∆ −a ∆ −R 1 + r 2 + r(R 1 + a) cos θ + aR 1 (a − R 1 )∆ a ∆ R 1 + r 2 + r(−R 1 + a) cos θ − aR 1 (a + R 1 )∆ a ∆ −R 1 + 3(k 1 l 2 − k 2 l 1 ) 4R 1 r 2 − R 2 1 ∆ R 1 ∆ −R 1 + 3k 1 2 − r cos θ + R 1 ∆ R 1 − r cos θ − R 1 ∆ −R 1 + r cos θ − a ∆ −a + r cos θ + a ∆ a .(51) As will be explained later, the necessary and sufficient conditions for the absence of CTCs outside the horizons of two black rings are k 1 < 0, l 1 > −4k 1 1 − R 2 1 a 2 .(52) Next, we consider the solutions with a pair of black rings outside an S 2 -bolt (a < −R 1 = R 2 ). The parameters C ω , m 0 and m 1 are given by m 0 = − 12k 1 a , m 1 = m 2 = − 6k 1 a R 1 1 − R 2 1 a 2 .(53) Then the metric functions are written in the form H = 1 + l 1 1 ∆ R 1 + 1 ∆ −R 1 + k 2 1 H −1 k 1 ∆ R 1 + 1 ∆ −R 1 2 ,(54)ω 0 = − 12k 1 a + 6k 1 R 1 a − a R 1 1 ∆ R 1 + 1 ∆ −R 1 + k 3 1 H −2 k 1 ∆ R 1 + 1 ∆ −R 1 3 + 3k 1 l 1 2 H −1 k 1 ∆ R 1 + 1 ∆ −R 1 2 + 3k 1 2 H −1 k 1 ∆ R 1 + 1 ∆ −R 1 ,(55)ω φ = − 3a 2 k 1 4R 1 1 − a 2 R 2 1 − r 2 + r(R 1 − a) cos θ − aR 1 (a + R 1 )∆ −a ∆ R 1 + r 2 − r(R 1 + a) cos θ + aR 1 (R 1 − a)∆ −a ∆ −R 1 + r 2 + r(R 1 + a) cos θ + aR 1 (a − R 1 )∆ a ∆ R 1 + r 2 + r(−R 1 + a) cos θ − aR 1 (a + R 1 )∆ a ∆ −R 1 + 3(k 1 l 2 − k 2 l 1 ) 4R 1 r 2 − R 2 1 ∆ R 1 ∆ −R 1 + 3k 1 2 − r cos θ + R 1 ∆ R 1 − r cos θ − R 1 ∆ −R 1 + r cos θ − a ∆ −a + r cos θ + a ∆ a .(56) The conditions for the absence of CTCs outside the horizons of two black rings are k 1 < 0, l 1 > 4k 1 a R 1 R 2 1 a 2 − 1.(57) As will be shown later, in both two black ring solutions, the horizons are located at r = ±R 1 . It is noted that in addition to three parameters k 1 , l 1 and a, both of two black ring solutions have an additional parameter R 1 . This fact means that unlike a single black ring solution, the black rings can be located at arbitrary places on the z-axis in the Gibbons-Hawking coordinate as far as two black rings are located so that they have reflection symmetry about the origin in the Gibbons-Hawking coordinates. III. PROPERTIES A. Asymptotic structure To study the asymptotic structure of the solutions, we introduce a new coordinater 2 := ar. The asymptotic form of the metric forr → ∞ becomes ds 2 ≃ −dt 2 + dr 2 +r 2 4 dψ 2 + cos θdφ 2 + dθ 2 + sin 2 θdφ 2 ,(58) where it is noted that ther constant surface can be regarded as a Hopf bundle, i.e., a twisted S 1 bundle over a S 2 base space. If dψ/2 is replaced by dψ, this coincides with the metric of the five-dimensional Minkowski spacetime. In other word, the twisted fiber in Eq.(58) has half of the periodicity of S 3 , which implies that the time slices is asymptotically locally Euclidean, i.e., the spatial infinity has the topological structure of the lens space L(2; 1) = S 3 /Z 2 . Here we use new angular variablesφ := (2φ + ψ)/4,ψ := (−2φ + ψ)/4 and Θ := θ/2. Then the asymptotic form can be rewritten as ds 2 ≃ −dt 2 + dr 2 +r 2 (dΘ 2 + cos 2 Θdφ 2 + sin 2 Θdψ 2 ). The total mass and the total angular momenta with respect to ∂φ and ∂ψ are obtained as M ADM = √ 3 2 Q e = 3π 8G   4 i k i 2 + a i l i   ,(60)Jφ = Jψ = − π 4G   4 i k i 3 + 3 2 a i k i i l i + a 2 4 i m i   .(61) The mass and the electric charge satisfy the BPS condition. Two angular momenta are equal in contrast to the concentric multi-black ring solutions on a flat space [43,44]. B. Near-Horizon and regularity Here we investigate the near-horizon geometry of our solutions. From the reflection symmetry of black rings, we consider only the neighborhood of the point sources r = R 2i (i = 1, 2, . . . ). Let us shift a origin of the three-dimensional Euclid space E 3 in the Gibbons- Hawking coordinate so that one of nuts, r = a 1 is located on the origin, i.e., a 1 = 0. We define new coordinates (x, y,φ,ψ) as r = −(a + R 2i ) x + y x − y , cos θ = −1 + 2 1 − x 2 y 2 − x 2 , φ =φ −ψ, ψ =φ +ψ.(62) As will be explained below, y → ∞ corresponds to event horizons. Moreover, let us introduce new coordinates (z, ζ) defined by y = − P z , x = cos ζ,(63) where P is some constant with dimension of length. Since the metric is apparently singular at z = 0, we introduce new coordinates dt = dv − 2 k=0 (B k /z k )dz, dφ = dφ ′ − 1 l=0 (C l /z l ) and dψ = dψ ′ − 1 l=0 (C l /z l ), where the constants B k (k = 0, 1, 2) and C l (l = 0, 1) are suitably chosen: the constants B 2 , C 1 and B 0 are chosen to cure the divergences 1/z in gψ ′ z , 1/z 2 and 1/z in g zz , respectively; the constants C 0 and B 0 are determined so that g zz = O(z) for z → 0. In the neighborhood of the 2i-th point source on the S 2 -bolt, i.e., r ≃ R 2i (a > |R 2i |), the metric behaves as ds 2 ≃ g (0) vz dvdz + 2g (0) zφ ′ dzdφ ′ + 2g (0) zψ ′ dzdψ ′ + a 2 (3l 2 2i − 8k 2i m 2i ) 64k 2 2i dφ 2 2 + k 2 2i dζ 2 + sin 2 ζdφ 2 1 ,(64) where the angular coordinates φ 1 and φ 2 are defined as φ 1 =φ ′ −ψ ′ =φ −ψ = φ and φ 2 =φ ′ +ψ ′ . They run the ranges of 0 ≤ φ 1 ≤ 2π and 0 ≤ φ 2 ≤ 4π, respectively. On the other hand, near the 2i-th point source outside the S 2 -bolt, r ≃ R 2i (a < |R 2i |), the metric behaves as ds 2 ≃ g (0) vz dvdz + 2g (0) zφ dzdφ ′ + 2g (0) zψ dzdψ ′ + a 2 (3l 2 2i − 8k 2i m 2i ) 16k 2 2i dψ ′2 + k 2 2i dζ 2 + sin 2 ζdφ ′2 ,(65) whereφ ′ andψ ′ run the ranges of 0 ≤φ ′ ≤ 2π and 0 ≤ψ ′ ≤ 2π, respectively. Since the explicit form of g (0) vz , g zφ ′ and g (0) zψ ′ are unimportant, we do not write it here. Since the Killing vector field V = ∂ v is null at z = 0 and furthermore V µ dx µ = g (0) vz dz, the hypersurface z = 0 is a Killing horizon, whose spatial topology is S 1 × S 2 . It is noted that the S 2 of a black rings on an S 2 -bolt and a black ring outside an S 2 -bolt have the radius of l 2i := a (3l 2 2i − 8k 2i m 2i )/(16k 2 2i ) and the S 1 of them have the radius of k 2i . Since the metric is analytic at the horizons and the nuts, there is no curvature singularity on and outside the event horizons. C. Absence of CTCs From the near horizon geometry, it is necessary that for each i the following inequality are satisfied 3l 2 i − 8k i m i > 0, k i < 0,(66) which is the condition that there is no CTC in the neighborhood of the horizons of all black rings. To ensure the absence of CTCs everywhere outside the horizons, we must demand that the spatial part of the metric is positive definite. This metric is positive-definite if and only if the following two-dimensional matrix is positive-definite M =   A −C −C B   ,(67) where A, B and C are given by A = H 3 H −1 k − ω 2 0 , B = H 3 H k r 2 sin 2 θ −ω 2 φ , C = ω 0ωφ .(68) Therefore, noting that AB − C 2 = H 3 H k (AH 2 k r 2 sin 2 θ −ω 2 φ ), we obtain the condition M > 0 ⇐⇒ A > 0, AB − C 2 > 0 (69) ⇐⇒ AH 2 k r 2 sin 2 θ −ω 2 φ > 0.(70) As a result, to show that the necessary conditions (66) are also sufficient for the absence of CTCs everywhere outside the horizons, it is enough to prove that D := AH 2 k r 2 sin 2 θ−ω 2 φ > 0 there. In the previous work, we showed this in the case of a single black ring. We can also confirm numerically that this is also achieved for the special cases, i.e., the cases of two black rings. It is difficult to show that in general, these inequalities are also sufficient conditions for CTCs to vanish outside all horizons. D. Black hole limit Now we consider the limit of our solutions to black hole solutions. For simplicity, we concentrate on two black rings located on or outside an S 2 -bolt. Taking the limit R 1 → a 1 (R 2 → a 2 ), we can obtain a pair of black holes on Eguchi-Hanson space: ds 2 = −H −2 dt + ω 0 a 8 dψ + ϕ φ dφ 2 + Hds 2 EH ,(71) where in the limit, the functions H, ω 0 and ϕ φ take the form of H = 1 + 8k 2 1 + al 1 a 1 ∆ −a + 1 ∆ a ,(72)ω 0 = 4k 1 16k 2 1 + 3al 1 a 2 1 ∆ −a + 1 ∆ a ,(73)ϕ φ = a 8 r cos θ − a ∆ −a + r cos θ + a ∆ a .(74) Note that in this limit, the functions H and ω 0 becomes harmonic functions on a threedimensional Euclid space. Also note that the absence of a Dirac-Misner string is assured sinceω φ → 0 in this limit. From the refection symmetry about z = 0 on the z-axis in E 3 , we analyze only one of two point sources r = (0, 0, a). We set up new spherical coordinates (r, θ, φ) centered on this point. Then the metric written in the coordinates is singular at r = 0 since g rr behaves as O(1/r 2 ) in the neighborhood of this point. To eliminate this divergence, we introduce new coordinates (v, ψ ′ ) defined by dt = dv + f (r)dr, dψ = dψ ′ + g(r)dr, where the functions f (r) and g(r) are given by f (r) = l 1 6k 2 1 + al 1 2 √ 2r 2 + 192k 4 1 + 12k 2 1 l 1 (4a + l 1 ) + al 2 1 (3a + 2l 1 ) 4 √ 2al 1 6k 2 1 + al 1 r ,(75)g(r) = √ 2k 1 (16k 2 1 + 3al 1 ) al 1 6k 2 1 + al 1 − k 1 (16k 2 1 + 3al 1 )(192k 4 1 + 6k 2 1 l 1 (8a − l 1 ) + a(3a − l 1 )l 2 1 ) √ 2a 2 l 3 1 6k 2 1 + al 1 3 .(76) Then, near the point source of r = 0, the metric behaves as ds 2 ≃ − 2(8k 2 1 + al 1 ) 2 √ 2l 1 6k 2 1 + al 1 drdv + k 2 1 + al 1 8 (dθ 2 + sin 2 θdφ 2 ) + k 2 1 + al 1 8 − k 1 (16k 2 1 + 3al 1 ) 2(8k 2 1 + al 1 ) 2 (dψ + (1 + cos θ)dφ) 2 + O(r).(77) In this coordinate system (v, r, θ, φ, ψ ′ ), all of the metric components take finite values and are analytic. Hence the coordinate system is well-defined around this point. Since the Killing vector field V := ∂ v becomes null at r = 0 and it is hypersurface orthogonal, i.e., V µ ∝ dr, the hypersurface r = 0 is a Killing horizon. From the metric on the point source, we find that the spatial topology of the horizon is a squashed S 3 . Therefore, in this limit, the solutions describe a pair of rotating black holes located at the north pole and the south pole on the S 2 -bolt. The angular velocities of the horizon vanish, although the total angular momenta evaluated at the spatial infinity do not vanish. Especially, if we restrict ourselves to the vanishing angular momentum case, the solutions exactly coincide with the black hole solutions in the case of equal masses obtained in Ref. [54]. IV. SUMMARY AND DISCUSSIONS In this article, we have constructed supersymmetric multi-black ring solutions on the Eguchi-Hanson base space as solutions of the five-dimensional minimal supergravity. The basic idea for constructing solutions is based on the programs to classify supersymmetric solutions of five-dimensional N = 1 supergravity in Ref. [39]. We have also investigated the properties of our black ring solutions for the configuration of more than one black rings. The space-time has the asymptotically locally Euclidean time slices, i.e., it has the spatial infinity with its topology the lens space L(2, 1) = S 3 /Z 2 . If we assume all the point sources of the harmonics K, L and M are put on the z-axis in the Gibbons-Hawking space, the black ring solutions can be constructed. However, the configuration of the black rings are restricted so that they have the reflection symmetry about z = 0. This results from the requirement of the absence of a Dirac-Misner string everywhere outside horizons. We have found that in the case of two black rings, the solutions have the limit to a pair of black holes with a squashed S 3 . Furthermore, in the vanishing limit of angular momenta, the solutions coincide with the black hole solutions with equal masses which was constructed in the previous work [54]. In this article, assuming that the Killing vector field ∂ ψ of the Eguchi-Hanson space is also H k = i Q i ∆ a i ,(78) where Q i (i = 1, 2, · · · ) are constants. If all Q i take the same value a/8, the space is regular. If not so, conical singularities appear at nuts. There seems to be no solution tõ ω φ (θ = 0) =ω φ (θ = π) = 0 even if we impose some symmetry on the configurations and the arrangements of black rings. Hence there is no black ring solution in such a spacetime. This fact might suggest that in a spacetime with a lot of nuts, black rings cannot be produced. It is interesting to see how the areas of black rings depend on the values of R 1 for the same masses and the same angular momenta. For simplicity, we consider the configuration of two black rings on a S 2 -bolt with its radius a fixed. From Eq.(29), the parameters m i (i = 0, 1, 2) are given by m 0 = − 12k 1 a , m 1 = m 2 = 6k 1 1 − R 2 1 a 2 .(79) Hence the total area of the two black rings is given by A 2Ring = −2ak 1 π 2 3l 2 1 − 48k 2 1 1 − R 2 1 a 2 .(80) The mass and the angular momenta depend on the parameters k 1 , l 1 and R 1 as M ∝ 8k 2 1 + al 1 , J ∝ −k 1 16k 2 1 + 3al 1 + 3 2 (a 2 − R 2 1 ) ,(81) respectively. For fixed asymptotic charges M and J, the parameters k 1 and l 1 are specified as functions of R 1 . The partial derivative of M and J with respect to R 1 are computed as Solving these equations, we obtain the derivatives of k 1 and l 1 as follows 0 = ∂M ∂R 1 ∝ 16k 1 ∂k 1 ∂R 1 M,J +a ∂l 1 ∂R 1 M,J ,(82)∂k 1 ∂R 1 M,J = −2k 1 R 1 R 2 1 − a 2 − 2al 1 ,(84)∂l 1 ∂R 1 M,J = 32k 2 1 R 1 a(R 2 1 − a 2 − 2al 1 ) .(85) Therefore, the partial derivative of the total area of two black rings with respect to R 1 is given by ∂A 2Rings ∂(−R 1 ) M,J = 4 √ 3π 2 k 2 1 R 1 (24(a 2 − R 2 1 )k 2 1 − a 2 l 2 1 ) (a 2 − R 2 1 + 2al 1 ) a 2 k 2 1 l 2 1 − 16(a 2 − R 2 1 )k 4 1 . For example, we consider the case where two black rings are put near the nuts at the north pole and the south pole on an S 2 -bolt, i.e., −R 1 = R 2 ≃ a. The right hand side of Eq.(86) is positive. Hence we should not expect the transition from black holes into black rings to occur. Next, we arrange that two black rings are located in the neighborhood of the equator on the S 2 -bolt, i.e., −R 1 = R 2 ≃ 0. In the case of −4k 1 < l 1 < −2 √ 6k 1 it can take a negative value. Therefore we should not expect two black rings near the equator of the S 2 -bolt to spontaneously coalesce and change into a single black ring. where a are a constant, 0 ≤θ ≤ π, 0 ≤φ ≤ 2π/ and 0 ≤ψ ≤ 2π. The Eguchi-Hanson space has an S 2 -bolt at r = a, where the Killing vector field ∂/∂ψ vanishes. In order to clarify the relationship between the Gibbons-Hawking coordinate and the metric (A1), we introduce the coordinates as follows [67], r = a r 4 a 4 − sin 2θ , tan θ = 1 − a 4 r 4 tanθ, φ =ψ, ψ = 2φ. 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{'abstract': 'We construct new supersymmetric multi-black ring solutions on the Eguchi-Hanson base space as solutions of the five-dimensional minimal supergravity. The space-time has an asymptotically locally Euclidean time slice, i.e., it has the spatial infinity with the topology of the lens space L(2; 1) = S 3 /Z 2 . The configurations of black rings are restricted by the requirement of the absence of a Dirac-Misner string everywhere outside horizons. Especially, in the case of two black rings, the solutions have the limit to a pair of rotating black holes with the horizon topology of S', 'arxivid': '0802.0741', 'author': ['Shinya Tomizawa 3.*e-mail:tomizawa@sci.osaka-cu.ac.jp \nDepartment of Mathematics and Physics\nGraduate School of Science\nOsaka City University\n3-3-138 Sugimoto558-8585SumiyoshiOsakaJapan\n'], 'authoraffiliation': ['Department of Mathematics and Physics\nGraduate School of Science\nOsaka City University\n3-3-138 Sugimoto558-8585SumiyoshiOsakaJapan'], 'corpusid': 119249414, 'doi': '10.1088/0264-9381/25/14/145014', 'github_urls': [], 'n_tokens_mistral': 18495, 'n_tokens_neox': 15344, 'n_words': 9141, 'pdfsha': 'baa8f969e891e6e5451935a63833321f8fb776d9', 'pdfurls': ['https://arxiv.org/pdf/0802.0741v2.pdf'], 'title': ['Multi-Black Rings on Eguchi-Hanson Space', 'Multi-Black Rings on Eguchi-Hanson Space'], 'venue': []}

arxiv

Thermodynamic Limit and Dispersive Regularisation in Matrix Models 4 Apr 2019 Costanza Benassi Department of Mathematics Physics and Electrical Engineering Northumbria University Newcastle Newcastle upon Tyne United Kingdom Antonio Moro Department of Mathematics Physics and Electrical Engineering Northumbria University Newcastle Newcastle upon Tyne United Kingdom Thermodynamic Limit and Dispersive Regularisation in Matrix Models 4 Apr 2019(Dated: April 5, 2019) We show that Hermitian Matrix Models support the occurrence of a new type of phase transition characterised by dispersive regularisation of the order parameter near the critical point. Using the identification of the partition function with a solution of a reduction of the Toda hierarchy, known as Volterra system, we argue that the singularity is resolved via the onset of a multi-dimensional dispersive shock described by an integrable flow in the space of coupling constants. This analysis explains the origin and mechanism leading to the emergence of chaotic behaviours observed in M 6 matrix models and extends its validity to even nonlinearity of arbitrary order. Random Matrix Models, originally introduced to describe spectra of heavy nuclei, became a universal paradigm for modelling complex phenomena. They naturally arise in connection with different areas of mathematics and physics, from quantum field theory to the theory of integrable systems [1][2][3]. A celebrated conjecture of Witten [4], proven by Kontsevich [5], established the identification of the free energy of 2D quantum gravity and the tau-function of a particular solution of the Korteweg-de Vries hierarchy. Thereafter, similar relations between specific matrix models on Hermitian, Unitary and Symplectic ensembles and integrable hierarchies have been discovered (see e.g. [6][7][8][9] and references therein). Furthermore, extensive studies of properties of matrix models partition functions unravelled intriguing connections between the theory of integrable systems, statistical mechanics, quantum field theory, algebraic and enumerative geometry [4,6,[8][9][10][11]. For the sake of simplicity, we focus on a case of Hermitian Matrix Models with even nonlinear interaction terms and their connection with the Toda hierachy, but our considerations can be in principle extended to other matrix ensembles. We also note that asymptotic properties of partition functions in the thermodynamic limit of one matrix models with even and odd nonlinearity and their connection with the Toda hierarchy have been previously considered in [13,14]. A key point is that the sequence of partition functions Z n for the one-matrix model of n × n matrices can be expressed in terms of the tau-function of the one dimensional Toda chain restricted to the even times of the hierarchy and where the index n labels points on the positive semi-axis of the chain. Identification of the Toda system with the matrix model is based on a one to one correspondence between the coupling constants of the model and the times of the hierarchy. The partition function Z n for fixed coupling constants is therefore specified by the state of the n−th particle of the chain at the corresponding times. Most importantly, the dynamics of the even hierarchy is uniquely specified by the initial conditions that are given in terms of the partition function of the free model, i.e. where all coupling constants vanish. In this respect, the model is simpler than the case of 2D gravity studied in [4] where the initial condition is specified by additional symmetries that are compatible with the hierarchy, namely the Virasoro constraints [6,11]. In his pioneering work [15], Jurkiewicz observed that a natural order parameter can be introduced, using orthogonal polynomial decompositions and combinatorial considerations [16]. Such order parameter develops, in the thermodynamic limit, a singularity that is regularised by apparently chaotic oscillations. Rigorous proof of the occurrence of asymptotic oscillations of the partition function has been found in [17,18]. We argue that the oscillations are the result of the dispersive regularisation of the shock in the continuum limit of the Toda-Volterra system [19][20][21]. The chaotic phase is therefore interpreted as a dispersive shock propagating through the chain in the continuum/thermodynamic limit. In this regime, the natural order parameter is given by interpolation of Flaschka's coordinates and its behaviour in the space of coupling constants is described by a solution of a scalar integrable hierarchy of hydrodynamic type. The considerations above outline the following scenario: when a thermodynamic system undergoes a phase transition, some specific quantities, the order parameters, develop singularities. In the context of conservation laws of hydrodynamic type, when a singularity (hydrodynamic catastrophe) occurs, viscosity and dispersion underpin two different mechanisms of regularisation of such singularity. In presence of small viscosity the solution develops a sharp but smooth wavefront [22]. If small viscosity is replaced by small dispersion, when the wavefront approaches the point of gradient catastrophe the dispersion induces initially small oscillations that further evolve into a dispersive shock [23][24][25]. In classical mean field fluid and spin models phase transitions are associated to classical shocks of order parameters in the space of thermodynamic parameters [26][27][28]. In this work we show that the chaotic behaviour observed in [15] is indeed a phase transition where the order parameter develops a singularity that is resolved via dispersion rather than viscosity as in classical spin models. This observation paves the way to a classification programme of phase transitions based on the normal forms of the differential identities satisfied by the free energy and order parameters. arXiv:1903.11473v2 [math-ph] 4 Apr 2019 Hermitian Matrix models. We study the Hermitian Matrix Model defined by the partition function Z n (t) = Hn e H(M ) dM,(1) where M are Hermitian matrices of order n, H(M ) = Tr (−M 2 /2 + ∞ j=1 t 2j M 2j ) is the Hamiltonian, with t = {t 2j } j≥1 the coupling constants, and dM is the Haar measure in the space of Hermitian Matrices H n . Based on a classical result by Weil [29], the partition function (1) is proportional to an integral over the eigenvalues of the matrix M , that is Z n (t) = c n τ n (t) where c n is a constant and τ n (t) = 1 n! R n ∆ n (λ) 2 n i=1 e H(λi) dλ i(2) with ∆ n (λ) = 1≤i<j≤n (λ i − λ j ) denoting the Vandermonde determinant. A theorem by Adler and van Moerbeke [6] implies that the quantity (2) can be interpreted as a tau-function of the Toda hierarchy restricted to the even flows ∂L ∂t 2k = 1 2 L 2k s , L k = 1, 2, . . . .(3) with L the tridiagonal symmetric Lax matrix of the form L =      0 b 1 0 0 . . . b 1 0 b 2 0 . . . 0 b 2 0 b 3 . . . . . . . . . . . . . . . . . .     (4) where b i = τ i+1 τ i−1 /τ 2 i and L 2k s denotes the skewsymmetric part of the matrix L 2k (see e.g. [6]). The solution of interest is specified by the initial conditions b i (0) = √ n obtained via a direct calculation of Gaussian integrals for the quantities τ n (0) = (2π) n/2 n j=1 j!/n! . We note that the Lax matrix of the type (4), originally considered in [12], and more recently in [13], corresponds to a reduction of the even Toda hierarchy known as Volterra hierarchy. Incidentally, we mention that the model with odd nonlinearities is different from the present case and its relation with the Toda hierarchy has been considered in [14]. We observe that the hierarchy (3) can be written in the form ∂B n ∂t 2k = B n (V (2k) n+1 − V (2k) n−1 ) k = 1, 2, . . .(5) where B n = b 2 n and V (2k) n are suitable functions of the variables B n . For instance, the first three non-trivial flows are given by V (2) n = B n , V (4) n = V (2) n V (2) n−1 + V (2) n + V (2) n+1 , V (6) n = V (2) n V (2) n−1 V (2) n+1 + V (4) n−1 + V (4) n + V (4) n+1 . We conjecture that the required solution to the above reduction of the even Toda hierarchy is given by the recursive formula (string equation) n = B n − ∞ j=1 2j t 2j V (2j) n .(6) We proved that Eq. (6) gives the exact solution of the equations (5) for t 2 , t 4 , . . . , t 10 , hence the conjecture. Eq. (6) allows to evaluate the order parameter of the M 2q model for arbitrary q and generalises the formula obtained by Jurkiewicz for q = 3 [15,30]. We analyse the Matrix Model in the large n (thermodynamic) regime via the continuum limit of the solution (6) of the reduced Toda hierarchy. Introducing the scale given by a suitable large integer N and the rescaled variables u n = B n /N , T 2k = N k−1 t 2k , Eq. (6) reads as follows n N = u n − ∞ j=1 2jT 2j W 2j n(7) where W 2j n = V (2j) n /N j . We then define the interpolating function u(x) such that u n = u(x) for x = n/N and u n±1 = u(x ± ) with the notation = 1/N . Using this substitution in the equations (7) and expanding in Taylor series for small , at the leading order we have the polynomial equation in u of the form Ω := −x + (1 − 2T 2 )u − 12T 4 u 2 − 60T 6 u 3 + · · · = 0 (8) where the dots denote terms with higher times T 2k with k > 3. Formula (8) can be viewed as a solution of the Hopf hierachy of PDEs u T 2k = C k u 2k−1 u x obtained from the leading order of continuum limit expansion of the Volterra hierarchy (5). It is well known [22] that the generic solution of the Hopf hierarchy develops singular behaviour for finite value of the "time" variables T 2k . In the following, we study these singularities and their relation with the occurrence of phase transitions. Eq. (7) is expected to reproduce quasi-trivial deformations of the Hopf hierarchy and the behaviour near the singularity to be universally described by a solution of the fourth order analogue of the Painlevé I equation [31,32]. Dispersive regularisation. We illustrate the general phenomenology by considering the particular case T 2k = t 2k = 0 for all k > 3 so that T 2 , T 4 and T 6 are the only non zero coupling constants. This choice allows for a simple but sufficiently general analysis demonstrating as chaotic behaviours observed in [30] correspond to a type of phase transition comprised by a dispersive shock of the order parameters. The shock occurs as a dispersive regularisation mechanism of a particular solution of the hierarchy (5) in the continuum limit. In Fig. 1 we compare the order parameter u(x) obtained as solution of the recurrence equation (7) and the limit equation (8). Values T 2 , T 4 and T 6 are chosen in such a way that the solution of the cubic equation (8) shows that the two solutions fully overlap for sufficiently small , but, as shown in Fig. 1b, a relevant deviation is observed in the vicinity of the point of gradient catastrophe of the solution to Eq. (8). We observe that equation (8) provides, for the above choices of coupling constants, the condition for extremising the free energy functional of density F [u] = −xu + 1 2 (1 − 2T 2 ) u 2 − 4T 4 u 3 − 15T 6 u 4 . (9) In particular, local minima and maxima depend on the signature of the discriminant ∆(x, T 2 , T 4 , T 6 ) of the cubic equation (8). If ∆ > 0 the free energy has two local minima and one local maximum, if ∆ < 0 the free energy presents one local minimum only. The set in the space of parameters such that ∆ = 0 corresponds to the critical set where a phase transition occurs. The analysis of the signature of ∆ shows that different scenarios need to be considered depending on whether the coefficients of the cubic equation (8) are negative or positive. Necessarily, in order to ensure convergence of the integral (1), it is T 6 < 0. Hence, we have four distinct cases, depending on the signs of the coefficient 1 − 2T 2 and −12T 4 in Eq. (8). Fig. 2 illustrates a case where T 2 < 1/2 and T 4 > 0. A similar analysis can be carried out for the remaining cases. This choice corresponds to the case analysed in [15,30], hence it allows for a direct comparison. In Fig. 2a we plot the set ∆ = 0 in the x-T 6 plane for a given choice of T 2 and T 4 . The convex sector corresponds to the region where the equation of state (8) admits three real solutions that correspond to the stationary points of the free energy density plotted in Fig. 2b. We compare the free energy for a given value of x and two different values of T 6 . For T 6 = −0.0051 the difference of the values of the free energy density at its local minima is particularly pronounced compared with the case T 6 = −0.0067. Figs. 2c and 2d show a comparison between the cubic solution (8) and the exact solution (7) for different values of T 6 within the convex region shown in Fig. 2a where the solution of (8) is multivalued. Both figures demonstrate the onset of a dispersive shock wave. This behaviour is intriguing as, unlike classical statistical mechanical sys-tems, e.g. magnetic and fluid models [33], the order parameter u(x) develops oscillations in conjunction with the existence of additional stationary points for the free energy such as unstable and metastable states. Fig. 3 shows two examples where T 2 > 1/2, with T 4 < 0 ( Fig. 3a and Fig. 3b) and T 4 > 0 ( Fig. 3c and Fig. 3d). In both cases the solution to Eq. (8) is three-valued but one root associated to a local minimum is negative and therefore does not correspond to a state of the system (by definition u ≥ 0). However, two concurrent states, although of different nature, one stable and one unstable, underlie a dispersive shock. Notice that for x > 0 solution to Eq. (8) has one non negative branch only. Nonetheless, u(x) develops a dispersive shock profile at positive x, although this is originated by a catastrophe located at x < 0. In both scenarios the solution to Eq. (8) is multivalued with two non negative branches for a small interval of negative values of x. It is therefore natural to study cases where the cubic solution is multivalued but only one branch is positive and therefore only one solution corresponds to a state that is accessible by the system. Such a case is shown in Fig. 4, where the the solution of the recurrence equation (7) converges to the cubic solution and no dispersive shocks occur. The above analysis suggests that the dispersive regularisation in the form of a dispersive shock of the order parameter is related to the existence of accessible (meta-)stable/unstable states. In particular, the behaviour of the order parameter, specifi- cally the form of the envelope, appears to be highly sensitive to the choice of the parameters T 2k . For instance, Fig. 2c, 2d, 3b and 3d show a dispersive shock whose envelope displays very distinctive features which require further investigations. A detailed study of this intriguing behaviour will require the construction of the asymptotic genus expansion of the solution (7) and Whitham's modulation theory for solutions of Eq. (5). We also point out that the rich phenomenology described reflects the fact that the dispersive shock given by the solution (7) is an intrinsic multidimensional object arising form the simultaneous solution of equations of the hierarchy (5). is single valued.Fig. 1a FIG. 1 : 1Comparison of the order parameter evaluated using Eq. (7) and Eq. (8) at T2 = 0, T4 = 0.1. In Fig. 1a T6 = −0.01 and ∆ < 0 for all x. In Fig. 1b T6 = −0.008 and ∆ = 0 at x = 5/18 0.28. FT2 = 0 ,T2 = 0 ,FIG. 2 :T2 = 1 ,T2 0021T2 = 0, T4 = 0.1, x = 0.22 T4 = 0.1, T6 = -0.0067, ϵ = 0.01 T4 = 0.1, T6 = -0.0051, ϵ = 0.01 In all figures T2 = 0 and T4 = 0.1. Fig. 2a: critical set ∆ = 0 in the x-T6 plane. The dashed lines correspond to the specific values of T6 and x analysed in the following figures. Fig. 2b: free energy for different values of T6 at x = 0.22. Fig.s 2c and 2d: comparison of the order parameter evaluated by using Eq. (7) and Eq. (8) at T6 = −0.0067 and T6 = −0.0051 respectively. T4 = 0.25, T6 = -= 1, T4 = 0.25, T6 = -0.25, ϵ = 0.01 (d) FIG. 3: In all figures T2 = 1 and T6 = −0.25. Fig. 3a and Fig. 3c: multivalued solution of Eq. (8) for T4 = −0.25 and T4 = 0.25 respectively. Fig. 3b and Fig. 3d: comparison of the order parameter evaluated by using Eq. (7) and Eq. (8) for T4 = −0.25 and T4 = 0.25 respectively. FIG. 4 : 4In all figures T2 = 0.25, T4 = −1, T6 = −0.5. Fig. 4a: solution of Eq. (8). Fig. 4b: comparison of the solution of Eq. (7) with the non negative branch of the solution to Eq. (8). Acknowledgements. This work is dedicated to the memory of Boris Dubrovin (1950-2019) whose magnificent scientific foresight and generosity have inspired the original ideas of this research. 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R Baxter, Academic PressR. Baxter Exactly Solved Models in Statistical Mechanics, Academic Press (1982).

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{'abstract': 'We show that Hermitian Matrix Models support the occurrence of a new type of phase transition characterised by dispersive regularisation of the order parameter near the critical point. Using the identification of the partition function with a solution of a reduction of the Toda hierarchy, known as Volterra system, we argue that the singularity is resolved via the onset of a multi-dimensional dispersive shock described by an integrable flow in the space of coupling constants. This analysis explains the origin and mechanism leading to the emergence of chaotic behaviours observed in M 6 matrix models and extends its validity to even nonlinearity of arbitrary order.', 'arxivid': '1903.11473', 'author': ['Costanza Benassi \nDepartment of Mathematics\nPhysics and Electrical Engineering\nNorthumbria University Newcastle Newcastle upon Tyne\nUnited Kingdom\n', 'Antonio Moro \nDepartment of Mathematics\nPhysics and Electrical Engineering\nNorthumbria University Newcastle Newcastle upon Tyne\nUnited Kingdom\n'], 'authoraffiliation': ['Department of Mathematics\nPhysics and Electrical Engineering\nNorthumbria University Newcastle Newcastle upon Tyne\nUnited Kingdom', 'Department of Mathematics\nPhysics and Electrical Engineering\nNorthumbria University Newcastle Newcastle upon Tyne\nUnited Kingdom'], 'corpusid': 85530573, 'doi': '10.1103/physreve.101.052118', 'github_urls': [], 'n_tokens_mistral': 7731, 'n_tokens_neox': 6572, 'n_words': 4053, 'pdfsha': 'a29944452c98c9103cd67fecac7a0f8af532c0bf', 'pdfurls': ['https://arxiv.org/pdf/1903.11473v2.pdf'], 'title': ['Thermodynamic Limit and Dispersive Regularisation in Matrix Models', 'Thermodynamic Limit and Dispersive Regularisation in Matrix Models'], 'venue': []}

arxiv

A Framework for Control Channels Applied to Reconfigurable Intelligent Surfaces 29 Mar 2023 Member, IEEE, Victor CroisfeltFabio Saggese Department of Electronic Systems Department of Informatics and Telecommunications, National and Kapodistrian Aalborg University AalborgDenmark Student Member, IEEERadosław Kotaba Department of Electronic Systems Department of Informatics and Telecommunications, National and Kapodistrian Aalborg University AalborgDenmark Student Member, IEEEKyriakos Stylianopoulos Department of Electronic Systems Department of Informatics and Telecommunications, National and Kapodistrian Aalborg University AalborgDenmark Senior Member, IEEEGeorge C Alexandropoulos Department of Electronic Systems Department of Informatics and Telecommunications, National and Kapodistrian Aalborg University AalborgDenmark Fellow, IEEEPetar Popovski petarp@es.aau.dk.k Department of Electronic Systems Department of Informatics and Telecommunications, National and Kapodistrian Aalborg University AalborgDenmark G C Stylianopoulos University of Athens Panepistimiopolis Ilissia15784AthensGreece Alexandropoulos University of Athens Panepistimiopolis Ilissia15784AthensGreece A Framework for Control Channels Applied to Reconfigurable Intelligent Surfaces 29 Mar 20231 2Index Terms Reconfigurable intelligent surfacescontrol channelprotocol designperformance analysis The research on Reconfigurable Intelligent Surfaces (RISs) has dominantly been focused on physicallayer aspects and analyses of the achievable adaptation of the propagation environment. Compared to that, the questions related to link/MAC protocol and system-level integration of RISs have received much less attention. This paper addresses the problem of designing and analyzing control/signaling procedures, which are necessary for the integration of RISs as a new type of network element within the overall wireless infrastructure. We build a general model for designing control channels along two dimensions: i) allocated bandwidth (in-band and out-of band) and ii) rate selection (multiplexing or diversity). Specifically, the second dimension results in two transmission schemes, one based on channel estimation and the subsequent adapted RIS configuration, while the other is based on sweeping through predefined RIS phase profiles. The paper analyzes the performance of the control channel in multiple communication setups, obtained as combinations of the aforementioned dimensions. While necessarily simplified, our analysis reveals the basic trade-offs in designing control channels and the associated communication algorithms. Perhaps the main value of this work is to serve as a framework for subsequent design and analysis of various system-level aspects related to the RIS technology.In[19]and [20], a detailed protocol for RIS-aided communication system was presented tackling the initial access problem. It was showcased that, despite the configuration control overhead, the RIS brings notable performance benefits allowing more UEs to access the network on average.However, it was assumed that the RIS control is perfect, which can be hardly true in practice.4The effect of retransmission protocols in RIS-aided systems for cases of erroneous transmissions was studied in[21], although the presented methodology assumed perfect control signals.There has been a significant discussion regarding the comparison of RISs and conventional amplify-and-forward relays[1]. While the distinction between them can sometimes be blurred[22], one way to make a clear distinction is the use of the flow of control and data through the communication layers [23,Fig. 2]. Those considerations set the basis for the definition of the control channel options in this paper.B. ContributionsThe main objective of this work is to develop a framework for designing and analyzing the control channel (CC) in RIS-aided communication systems. The number of actual CC designs is subject to a combinatorial explosion, due to the large number of configurable parameters in the system, such as frame size or feedback design. Clearly, we cannot address all these designs in a single work, but what we are striving for is to get a simple, yet generic, model for analyzing the impact of CCs that captures the essential design trade-offs and can be used as a framework to analyze other, more elaborate, CC designs.We build generic CC models along two dimensions. The first dimension is related to how the CC interacts with the bandwidth used for data communication. An out-of-band CC (OB-CC) uses communication resources that are orthogonal to the ones used for data communication.More precisely, OB-CC exerts control over the propagation environment, but is not affected by this control. Contrary to this, an in-band CC (IB-CC) uses the same communication resources as data communication. This implies that the IB-CC decreases the number of degrees of freedom for transmission of useful data, thereby decreasing the spectral efficiency (SE) of the overall system. Furthermore, the successful transmission of the control messages toward the RIS is dependent on its phase profile. For instance, an unfavorable RIS configuration may cause blockage of the IB-CC and transmission of further control messages, impacting the overall system performance.The second dimension is built along the traditional diversity-multiplexing trade-off in wireless communication systems. In a diversity transmission, the data rate is predefined and the sender hopes that the propagation environment is going to support that rate. If this is not the case, then, an outage occurs. To reflect this paradigm in an RIS setup, we consider a transmission setup in which the RIS sweeps through different configurations and the BS tries to select the one that is likely to support the predefined data rate. In a multiplexing transmission, the data rate is adapted I. INTRODUCTION Reconfigurable Intelligent Surfaces (RISs) constitute a promising technology that in recent years has received significant attention within the wireless communication research community [1]. The main underlying idea is to electronically tune the reflective properties of an RIS in order to manipulate the phase, amplitude, and polarization of the incident electromagnetic waves [2]. This results in the effect of creating a propagation environment that is, at least partially, controlled [3]. RISs can be fabricated with classical antenna elements controlled through switching elements or, more advanced, can be based on matematerials with tunable electromagnetic properties [4]. In the context of 6G wireless systems, the RIS technology has been identified as one of the cost-effective solutions to address the increasing demand for higher data rates, reduced latency, and better coverage. In particular, an RIS can improve the received signal strength and reduce interference by directing signals to intended receivers and away from non-intended ones; this leads to applications aiming increased communication security [5] and/or reduced electromagnetic field exposure [6]. RISs can also extend the coverage of wireless communication systems by redirecting the signals to areas that are difficult to reach using conventional means. In terms of challenges related to the RIS technology, the dominant part of the literature concerning RIS-aided communication systems deals with physical-layer (PHY) aspects. Recent studies have explored physics-based derivation of channel models, extending plane wave expansions beyond the far-field approximation [7]. While many papers have investigated the potential benefits of RIS-assisted systems in terms of spectral and energy efficiency (see, e.g., [8]), others have concentrated on optimizing the RIS configuration alone or jointly with the beamforming at the base station (BS). On the other hand, several works have focused on designing and evaluating channel estimation (CE) methods in the presence of RISs, either relying on the cascaded endto-end channel when dealing with reflective RISs [9], or focusing on the individual links using simultaneous reflecting and sensing RISs [10]. The latter design belongs to the attempts to minimize the RIS reconfiguration overhead, which can be considerably large due to the expected high numbers of RIS elements [11] or hardware-induced non linearities [8]. A different research direction bypasses explicit channel estimation and relies on beam sweeping methods [12]. Accordingly, within each coherent channel block, the RIS is scheduled to progressively realize phase configurations from a predefined codebook, for the end-to-end system to discover the most suitable reflective beamforming pattern [13]- [16]. The beams are practically optimized for different purposes [17], possibly comprising hierarchical structures [15], [18]. Within the existing research literature, the questions related to link/MAC protocol and systemlevel integration of RISs have received much less attention as compared to PHY topics. Specifically, the aspects related to control/signaling procedures have been largely neglected, despite the fact that those procedures are central to the integration of RISs as a new type of network element within existing wireless infrastructure. In this regard, it is important to study the RIS control from two angles. First, as an enabler of the new features that come with the RIS technology and a component that ensures its proper operation in general. Assessing this aspect requires looking into the required performance of the control channel in terms of, for example, rate, latency, and, not the least, reliability. Second, control procedures introduce an overhead in the system, such that it is important to characterize the trade-off between spending more time and resources on the auxiliary procedures (such as more robust channel estimation or optimization of the RIS reflection pattern) versus data transmission. This paper aims to fill this important knowledge gap, relevant to both theory and practice of RIS-aided communications, by providing a systematic analysis of the control architecture options and the associated protocols. A. Related literature One of the first works focusing on fast RIS programmability [13] presented a multi-stage configuration sweeping protocol. By tasking the RIS to dynamically illuminate the area where a user equipment (UE) is located, a downlink transmission protocol, including sub-blocks of UE localization, RIS configuration, and pilot-assisted end-to-end channel estimation, was introduced in [14]. In [15], a fast near-field alignment scheme for the RIS phase shifts and the transceiver beamformers, relying on a variable-width hierarchical RIS phase configuration codebook, was proposed. Very recently, in [16], the overhead and challenges brought by the RIS network integration were discussed. It was argued that the reduced overhead offered by codebook-based RIS configuration schemes, is beneficial to the overall system performance. Nevertheless, the required control information that need to be exchanged for those schemes was not investigated. to the actual channel conditions; however, this incurs more signaling for channel estimation. In an RIS-aided setup, the multiplexing transmission corresponds to a case in which the RIS configuration is purposefully configured to maximize the link signal-to-noise ratio (SNR) and the data rate is chosen accordingly. This paper analyzes the CC performance and impact in several communication setups, obtained as combinations of the aforementioned dimensions. In doing so, we have necessarily made simplifying assumptions, such as the use of a frame of a fixed length in which the communication takes place. This is especially important when analyzing a CC performance since any flexibility will affect the design of the CC. For instance, if a frame has a flexible length that is dependent on the current communication conditions, then, this flexibility can only be enabled through specific signaling over the CC, including encoding of control information and feedback 1 . Paper Outline: In Section II, the system model is described, while Section III describes the paradigms of communication, focusing on the general description of the methods and the description of the obtained SNR, SE, and (eventually) outage generated by the method itself without accounting for potential control error. Section IV describes firstly how to take into account the errors in the CCs, which generate further outages/reduction of throughput performance. Then, the methods presented in the previous section are analyzed from the control perspective. In Section V, the performance of the studied communication paradigms is evaluated, while Section VI concludes the paper. Notation: Lower and upper case boldface letters denote vectors and matrices, respectively; the Euclidean norm of x is x ; and denotes the element-wise product. P(e) is the probability that event e occurs; CN (µ, R) is the complex Gaussian distribution with mean µ and covariance matrix R, Exp(λ) is the exponential distribution with mean value 1/λ. E[·] is the expected value, a is the nearest lower integer of a, and j √ −1. II. SYSTEM MODEL Let us consider the simple uplink (UL) scenario depicted in Fig. 1 γ = ρ u σ 2 b |φ T (h d g d )| 2 = ρ u σ 2 b |φ T z d | 2 ,(1) where h d ∈ C N is the data channel from the UE to the RIS, while g d ∈ C N defines the one from the RIS to the BS. For simplicity of notation, we define the equivalent channel as z d = (h d g d ) ∈ C N . The UE transmit power is ρ u and σ 2 b is the noise power at the BS radio frequency (RF) chain. In the remainder of the paper, we assume that the BS knows the values of ρ u and σ 2 b : the transmit power is usually determined by the protocol or set by the BS itself; the noise power can be considered static for a time horizon longer than the coherence time and hence estimated previously through standard estimation techniques, e.g., [24]. 2 For the sake of simplicity and following the standard practice in literature, we consider an ideal RIS to show the theoretical performance achievable by the system at hand. We expect that more realistic models addressing attenuation, mutual coupling, and non-linear effects would reduce the overall performance [8]. 3 The narrowband assumption of the channel is considered to simplify the analysis done throughout the paper in order to focus on the timing of the operations needed in the control and data channels to successfully perform a RIS-aided wireless transmission, as specified in Sects. III and IV. Nevertheless, the system aspects of the control channel can be carried over to a wideband or orthogonal frequency-division multiplexing (OFDM) case. the UE control messages reflected by the RIS reach the BS and vice-versa, we consider that the RIS makes use of a wide beamwidth configuration, termed control (ctrl) configuration. Wide beamwidth radiation patterns generally offer increased robustness in terms of outage probabilities when low data rates are needed [25], which makes them an ideal choice for ctrl configurations. Without loss of generality, we assume that the RISC loads the ctrl configuration anytime it is in an idle state. In other words, if the RISC has not been triggered to load other configurations, the ctrl configuration is loaded. In those cases, the UE-CC channel is described by h cu = φ T ctrl (h c g c ) = φ T ctrl z c ,(2) where φ ctrl is the ctrl configuration and h c ∈ C N and g c ∈ C N are the UE-RIS and RIS-UE CCs, respectively. The equivalent end-to-end channel is z c = h c g c ∈ C N . Herein, we do not focus on designing the ctrl configuration, whose design can be based on other works (e.g., see the configuration design proposed in [15]). Instead, we assume that the above channel in (2) is Gaussian distributed as h cu ∼ CN (0,λ u ) withλ u being a term accounting for the (known) large-scale fading dependent on the ctrl configuration. Hence, the SNR measured at the UE is Γ u = ρ b σ 2 u |h cu | 2 ∼ Exp 1 λ u ,(3) where λ u = ρ bλu σ 2 u denotes the average SNR at the UE, being ρ b the BS transmit power and σ 2 u the UE's RF chain noise power. c) RIS-CC: We assume that the RIS-CC is narrowband having central frequency f r , bandwidth B r , and channel coefficient denoted as h cr ∈ C. To obtain simple analytic results, we assume that the BS-RISC coefficient can be modeled as h cr ∼ CN (0,λ r ) whereλ r accounts for the large-scale fading, assumed known. Therefore, the SNR measured at the RISC results Γ r = ρ b σ 2 r |h cr | 2 ∼ Exp 1 λ r ,(4) where λ r = ρ bλr σ 2 r denotes the average SNR at the RISC receiver, being σ 2 r the noise power at its RF chain. Remark that this channel can be either: i) IB-CC meaning that the physical resources employed by the UE-DC are overlapped by the one used by the RIS-CC, i.e., f r = f d , B r ≥ B d ; or ii) OB-CC, where the physical resources of the control channel are not consuming degrees of freedom from the data-transmission resources, thereby simulating a cabled connection between the decision maker and the RISC. In the case of OB-CC, we further assume that the RIS-CC is an error-free channel with feedback capabilities, i.e., λ r → ∞. This is reasonable to assume, as, once it is decided to use OB-CC, the system designer has a large pool of reliable options. III. RIS-AIDED COMMUNICATION PARADIGMS In this section, we first describe a structure and building blocks of a general RIS-aided communication paradigm that is applicable to a large number of potential systemic and algorithmic realizations. Then, we use the general paradigm to describe two particular transmission strategies and their respective systemic and algorithmic modules. We analyze their performance in terms of the expected SNR and SE while describing the errors eventually occurring during their operation. A. General paradigm structure Throughout the remainder of the paper, we assume that the system operates based on frames with duration, τ , shorter than the channel coherence time, i.e., the channel coefficients are assumed to be constant -or change negligibly -over the duration of the frame. The channel coherence time is considered to be estimated beforehand and hence known at the beginning of each frame. Within each frame, a RIS-aided communication paradigm can be divided into four main phases, namely "Setup", "Algorithmic", "Acknowledgement", and "Payload", described in the following. We note that there could be access algorithms in which some of these phases may not be present; however, the mentioned four phases set a basis for a sufficiently general framework that can be used, in principle, to design other schemes where some of the steps are merged or omitted. Setup: The communication procedure starts with the Setup phase; it is typically initiated by the BS and relies on control signaling to notify the RIS and the scheduled UE about the start of the new round of transmissions, i.e., a frame. It is assumed that the RISC loads the ctrl configuration at the beginning of this phase. The duration of this phase is denoted with τ set < τ and it depends on the type of available CC. Although not considered in this paper, the Setup phase can also incorporate the random access phase as an intermediate step where the scheduling and resource allocation needs of the UEs are determined. Algorithmic: After the Setup is performed, the Algorithmic phase starts. In general, this phase encompasses all the processes and computations that are needed to optimize the data transmission taking place later on. This phase has a duration τ alg < τ that depends on the choice of the employed communication paradigm. Apart from the evaluation of an appropriate RIS configuration enabling the data transmission, other objectives could be to determine the transmission parameters for the UE, and/or BS beamforming, etc. To tackle these objectives, some form of sensing of the wireless environment is required, typically enabled by the transmission of pilot sequences. The specifications of those pilots, i.e. their number, design, whether they are transmitted in the UL or downlink (DL), whether feedback is available, etc., are implementationand system-defined. The computational nodes of the system (usually, the BS) use the collected pilot signals and invoke pre-defined algorithms to determine the aforementioned transmission parameters to be used in the Payload phase. The outcome of these algorithms might be affected by different types of algorithmic errors that might prevent the system to perform as expected, and thus should be taken into account when analyzing the overall performance. Acknowledgement: The Acknowledgement phase starts once the Algorithmic phase ends; during this phase, the RIS configuration chosen needs to be communicated to the RISC, which in turn commands the RIS to load the specified phase shifts. Additionally, some further control signaling may occur between the BS and the UE as a final check before the data transmission, for example, to set the modulation and coding scheme (MCS). It is implied that the RISC loads the ctrl configuration at the beginning of this phase. Similar to the Setup phase, the Acknowledgement phase duration τ ack < τ , depends on the type of CC used. Payload: The payload phase is the last one, during which the actual data transmission takes place, which is in the UL for this work. This phase spans a duration τ pay < τ until the end of the channel coherence frame. This phase may or may not include the feedback at the end; this aspect is not considered in this paper. In the following subsections, we describe two state-of-the-art paradigms for RIS-aided communications employing different approaches for the Algorithmic phase. The first is the optimization based on channel estimation (OPT-CE), which follows a standard multiplexing transmission: the UE's channel state information (CSI) is evaluated at the BS, which then uses this information to compute the RIS' optimal configuration and the corresponding achievable data rate. The BS sends the optimal configuration to the RISC, which loads it to the RIS surface, while the UE is instructed to transmit the data using the stipulated MCS. The second approach is the codebookbased beam sweeping (CB-BSW), which was formally defined as a communication paradigm in [16], but already used in previous works (e.g., [19]). This paradigm resembles the concept of diversity transmission. Here the BS does not spend time figuring out the best configuration to improve the quality of the UE-DC and does not tune the transmission rate; it instead applies a best-effort strategy, as in every diversity-oriented transmission. Specifically, the BS instructs the RISC to sweep through a set of predefined configurations -the configuration codebook -and hopes that at least one will satisfy a target key performance indicator (KPI) a priori specified for the transmission (e.g., a minimum SNR to support a predefined rate). Fig. 2 shows the data exchange diagrams of the two paradigms comprised of CC messages, configuration loading, processing operations, and data transmission. Based on these, a detailed description of the two paradigms is given in the following sections. The details on the design and reliability of the messages being sent through the CCs are given in Sect. IV. B. Optimization based on channel estimation (OPT-CE) For this communication paradigm, the BS needs to obtain the CSI for the UE in order to optimize the RIS configuration. The necessary measurements can be collected through the transmission of pilot sequences from the UE. During the Setup phase, the BS informs the other entities that the procedure is starting: the UE is informed through the UE-CC to prepare to send pilots. To solve the indeterminacy of the N path CE problem because of the presence of the RIS [26], the RIS is instructed to sweep through a common codebook of configurations during the Algorithmic phase called channel estimation codebook and denoted as C CE ⊆ C. between the configurations in the channel estimation codebook, the BS needs to send only a single control message to the RISC since the RIS sweeps following the order stipulated by the codebook. During the Algorithmic phase, the UE sends replicas of its pilot sequence subject to different RIS configurations to let each of them experience a different propagation environment. After a sufficient number of samples is received, the BS can estimate the CSI and compute the optimal RIS's configuration. Then, the Acknowledgement phase starts, in which the BS informs the UE over the UE-CC to start sending data by using the ctrl configuration 4 and, subsequently, the BS informs the RISC over the RIS-CC to load the optimal configuration. Finally, the Payload phase takes place. Performance Analysis: We now present the CE procedure and analyze its performance in connection to the cardinality of the employed codebook C CE = |C CE |. The method employed can be seen as a simplification of the method proposed in [26]. Let us start with the pilot sequence transmission and processing. We denote a single pilot sequence as ψ ∈ C p , spanning p symbols and having ψ 2 = p. Every time a configuration from the codebook is loaded at the RIS, the UE sends a replica of the sequence ψ towards the BS. When configuration c ∈ C CE is loaded, the following signal is received at the BS: y T c = √ ρ u φ T c z d ψ T +w T c ∈ C 1×p ,(5) where φ c denotes the phase-shift profile vector of the configuration c ∈ C CE , ρ u is the transmit power, andw c ∼ CN (0, σ 2 b I p ) is the additive white gaussian noise (AWGN). The received symbol is then correlated with the pilot sequence and normalized by √ ρ u p, obtaining y c = 1 √ ρ u p y T c ψ * = φ T c z d + w c ∈ C,(6) where w c ∼ CN (0, σ 2 b pρu ) is the resulting AWGN 5 . The pilot transmissions and the processing in eq. (6) are repeated for all configurations, ∀c ∈ C CE . The resulting signal y ∈ C C CE = [y 1 , y 2 , . . . , y C CE ] T can be written in the following form: y = Θ T z d + w,(7) where Θ = [φ 1 , φ 2 , . . . , φ C CE ] ∈ C N ×C CE is the matrix containing all the configurations em- ployed and w = [w 1 , . . . , w C CE ] T ∈ C C CE ×1 is the noise term. For the sake of generality, we will assume that there is no prior information about the channel distribution at the BS. Therefore, we cannot estimate separately h d and g d , but only the equivalent channel z d . It is possible to show that a necessary (but not sufficient) condition to perfectly estimate the channel coefficients is that C CE ≥ N [26]. Indeed, we want to have a linearly independent set of equations, which can be obtained by constructing the configuration codebook for CE to be at least rank N . As an example, we can use the discrete Fourier transform (DFT) matrix, i.e., [Θ] n,i = e −j2π (n−1)(c−1) C CE , n = {1, . . . , N }, c ∈ C CE , with Θ * Θ T = C CE I N . Considering that the parameter vector of interest is deterministic, the least-squares estimate [28] yieldŝ z d = 1 C CE Θ * y = z d + n,(8) where n ∼ CN (0, σ 2 b pρuC CE I N ) and whose performance is proportional to 1/C CE . Based on the estimated equivalent channel, the BS can obtain the configuration φ that maximizes the instantaneous SNR of the typical UE as follows 5 The consideration of dividing the pilot transmission over configurations over small blocks of p symbols basically serves three purposes: i) from the hardware point of view, it might be difficult to change the phase-shift profile of a RIS within the symbol time, ii) to reduce the impact of the noise, and iii) to have the possibility of separating up to p UE's data streams, if the pilots are designed to be orthogonal to each other [27]. φ = max φ |φ Tẑ d | 2 φ 2 = N ,(9) which turns out to provide the intuitive solution of setting (φ ) n = e −j∠(ẑ d )n , ∀n = {1, . . . , N }. Finally, the UL estimated SNR at the BS resultŝ γ CE = ρ u σ 2 b |φ T ẑ d | 2 .(10) Based on the estimated SNR, the SE of the data communication can be adapted to be the maximum achievable, i.e., η CE = log 2 (1 +γ CE ).(11) Algorithmic errors: For the OPT-CE paradigm, a communication outage occurs in the case of an overestimation error, i.e., if the selected SE η CE is higher than the actual channel capacity leading to an unachievable communication rate [29]. The probability of this event is p ae = P [η CE = log(1 +γ CE ) ≥ log 2 (1 + γ CE )] ,(12) where γ CE = ρu σ 2 b |φ T z d | 2 is the actual SNR at the BS. Eq. (12) translates to the condition p ae = P [γ CE ≥ γ CE ] = P |φ T z d + φ T n| 2 ≥ |φ T z d | 2 .(13) A formal analysis of eq. (13) depends on the channel model of z d , and, hence, on making a prior assumption on the scenario at hand. To keep the analysis general, we resort to numerical methods to evaluate the impact of the OPT-CE algorithmic error. C. Codebook-based beam sweeping (CB-BSW) In the Setup phase, the BS commands the start of a new frame by signaling to the RIS and the UE. During the Algorithmic phase, a beam sweeping process and the configuration selection are performed. The beam-sweeping process comprises the UE sending reference signals, while the BS commands the RIS to change its configuration at regular time periods accordingly to a set of common configurations, labeled as the beam-sweeping codebook and denoted by the symbol C CB ⊆ C. The BS receives the reference signals that are used to measure the UE's KPI. Again, a single BS control message received by the RISC is enough to trigger the whole sweeping process. At the end of the beam-sweeping process, the BS selects a configuration satisfying the target KPI. During the Acknowledgment phase, the BS informs the UE over the UE-CC to prepare to send data by using the ctrl configuration and informs the RISC through the RIS-CC to load the selected configuration. Finally, the Payload phase takes place. We consider that the beam-sweeping process occurring in the Algorithmic phase may make use of i) a fixed or ii) a flexible frame structure. The former is based on a fixed number of configurations in the beam-sweeping codebook: the beam-sweeping process ends after the last configuration in the codebook is loaded. The latter allows stopping the beam-sweeping earlier, as soon as a KPI value measured is above the target one. Enabling the flexible structure method requires that the BS makes the KPI measurements on-the-fly; moreover, resources on the UE-CC need to be reserved to promptly communicate to the UE to stop sending pilot sequences when the target KPI is met, modifying the organization of the overall frame. A detailed description of these differences and the impact on the CCs design are given in Sect. IV. Performance analysis: In order to study the beam sweeping performance, let us assume that the target KPI is a target SNR γ 0 measured at the BS from the average received signal strength (RSS). Therefore, in this case we have a fixed SE defined a priori given by η CB = log 2 (1 + γ 0 ),(14) and we aim to find a configuration from the codebook that supports such SE. Let us then analyze the system by starting from the pilot sequence transmission and processing. As before, every pilot sequence consists of p symbols 6 . Once again, we denote a single sequence as ψ ∈ C p having ψ 2 = p. After the RIS has configuration c ∈ C CB loaded, the UE sends a replica of the sequence ψ; the following signal is obtained at the BS: y T c = √ ρ u φ T c z d ψ T +w T c ∈ C 1×p ,(15) which has the same formulation of eq. (5), where φ c now denotes the configuration c ∈ C CB . The received signal is then correlated with the pilot symbol and normalized by p, obtaining y c = 1 p y T c ψ * = √ ρ u φ T c z d + w c ∈ C,(16) where w c ∼ CN (0, σ 2 b p ) is the resulted AWGN. The SNR provided by the configuration can be estimated by taking the absolute square of the sample and dividing it by the (known) noise variance asγ 6 The pilot sequences for OPT-CE and CB-BSW can be different and have different lengths. In practice, they should be designed and optimized for each of those approaches, which is beyond the scope of this paper. The same notation to denote the length of the pilot sequence in both paradigms is used for simplicity. c = |y c | 2 σ 2 b = ρ u σ 2 b |φ T c z d | 2 γc +2 √ ρ u σ 2 b φ T c z d w c + |w c | 2 σ 2 b ,(17) where |w c | 2 /σ 2 b ∼ Exp(p). It is worth noting that the estimated SNR is affected by the exponential error generated by the noise, but also by the error of the mixed product between the signal and the noise, whose probability distribution function (pdf) depends on the pdf of z d . Based on eq. (17), we can select the best configuration c ∈ C CB providing the target KPI. In the following, we discuss the selection of the configuration for the two different frame structures. a) Fixed frame: When the frame has a fixed structure, the sweeping procedure ends after the RIS sweeps through the whole codebook. In this case, we can measure the KPIs for all the configurations in the codebook. Then, the configuration selected for the payload phase is set to be the one achieving the highest estimated SNR among the ones satisfying the target KPI γ 0 , that is, c = arg max c∈C CB {γ c |γ c ≥ γ 0 }.(18) As before, if no configuration achieves the target KPI, the communication is not feasible and we run into an outage event. b) Flexible frame: When the frame has a flexible structure, the end of the sweeping process is triggered by the BS when the measured KPI reaches the target value. A simple on-the-fly selection method involves testing if the estimated SNR is greater than the target γ 0 ; i.e., after eq. (17) is obtained for configuration c ∈ C CB , we set c = c ⇐⇒γ c ≥ γ 0 . As soon as c is found, the BS communicates to both RIS and UE that the Payload phase can start, otherwise, the sweeping process continues until a configuration is selected. In case the whole codebook C CB is tested and no configuration satisfies condition (19), the communication is not feasible and we run into an outage event. Algorithmic errors: For the CB-BSW paradigm, a communication outage occurs when no configuration in the beam sweeping codebook satisfies the target KPI, and in the case of overestimation error, which now occurs if the selected configuration provides an actual SNR lower than the target one, knowing that the estimated SNR was higher. These events are mutually exclusive, and hence their probability results in p ae = P [γ c ≤ γ 0 |γ c > γ 0 ] + P [γ c ≤ γ 0 , ∀c ∈ C CB ] = P γ c − γ 0 ≤ |w c | 2 σ 2 b + 2 √ ρ u σ 2 b φ T c z d w c + P [γ 1 ≤ γ 0 , . . . ,γ C CB ≤ γ 0 ] ,(20) where γ c = ρu σ 2 b |φ T c z d | 2 is the actual SNR, andγ c − γ 0 > 0. By applying Chebychev inequality, the overestimation probability term in (20) can be upper bounded by P γ c − γ 0 ≤ |w c | 2 σ 2 b + 2 √ ρ u σ 2 b φ T c z d w c ≤ E |w c | 2 σ 2 b + 2 √ ρu σ 2 b φ T c z d w c γ c − γ 0 = p −1 γ c − γ 0 .(21) From eq. (21), we infer that the higher the gap betweenγ c and γ 0 , the lower the probability of error. The CB-BSW employing the fixed structure generally has a higher value ofγ c − γ 0 than the one with the flexible structure due to the use of the arg max operator to select the configuration c . Therefore, the fixed structure is generally more robust to overestimation errors. On the other hand, the evaluation of the probability that no configuration in the beam sweeping codebook satisfies the target KPI requires the knowledge of the cumulative density function (CDF) of the estimated SNR, whose analytical expression is channel model dependent and generally hard to obtain. Also in this case, we resort to numerical simulations to evaluate the impact of the CB-BSW algorithmic errors. D. Trade-offs in different paradigms The two RIS-aided communication paradigms can be seen as a generalization of the fixed rate and adaptive rate transmission approaches. Essentially, the SE of the OPT-CE is adapted to the achievable rate under the optimal configuration (see eq. (11)) obtaining the so-called multiplexing transmission, while the SE of the CB-BSW is set a priori according to the target KPI (see eq. (14)) obtaining a diversity transmission. Comparing eqs. (11) and (14) under the same environmental conditions, we have that η CB ≤ η CE ,(22) where the price to pay for the higher SE of the OPT-CE paradigm is the increased overhead. Indeed, for the OPT-CE, an accurate CSI is needed for a reliable rate adaptation, which generally translates into a higher number of sequences to be transmitted by the UE compared to CB-BSW. Furthermore, after the pilot transmission, additional time and processing are required to determine the optimal configuration of the RIS. As a consequence, the SE of data transmission alone cannot be considered a fair metric of comparison, as it does not take into account the overheads generated by the communication paradigms. In the next section, we will introduce the impact of the control channel and give the main metric of the comparison. IV. IMPACT OF THE CONTROL CHANNELS In this section, we define a performance metric that simultaneously measures the performance of a RIS-aided communication scheme and the impact of the CCs over it. We then further characterize the terms regarding the overhead and the reliability of the CCs for the particular paradigms discussed in Section III. A. Performance evaluation: Utility function We start by defining a utility function measuring the communication performance by taking into account a) the overhead and error of the communication paradigms and b) the reliability of the CCs. Regarding overhead and errors of the paradigms, we consider the goodput metric defined as R(τ pay , η) = (1 − p ae ) τ pay τ B d η,(23) where 1 − p ae represents the probability that no algorithmic error occurs; η = η CE in (11) or (14) if OPT-CE or CB-BSW is employed, respectively; τ pay is the duration of the payload phase, and τ is the overall duration of a frame. The overhead time is the time employed by the Setup, Algorithmic and Acknowledgement phases, being denoted as τ set , τ alg , and τ ack , respectively. Accordingly, the payload time can be written as η = η CB inτ pay = τ − τ set − τ alg − τ ack .(24) While the overall frame length is fixed, the overhead time depends on the paradigm of communication, being a function of: the duration of a pilot, τ p , and the number of replicas transmitted; Regarding the reliability of the CCs, we denote as P = P u + P r the total number of control packets needed to let a communication paradigm work, where P u and P r are the numbers of control packets intended for the UE and the RISC, respectively. Whenever one of such packets is erroneously decoded or lost, an event of erroneous control occurs. We assume that these events are independent of each other (and of the algorithmic errors), and we denote the probability of erroneous control on the packet i toward entity k ∈ {u, r} as p (k) i , with i ∈ {1, . . . , P k } and k ∈ {u, r}. Erroneous controls may influence the overhead time and the communication performance: the RIS phase-shift profile may change in an unpredictable way leading to a degradation of the performance, or worse, letting the data transmission fails. While losing a single control packet may be tolerable depending on its content, here, we assume that all the control packets need to be received correctly in order to let the communication be successful. In other words, no erroneous control event is allowed. Consequently, the probability of correct control results p cc = k∈{u,r} P k i=1 (1 − p (k) i ).(25) We are interested to show the average performance of the analyzed communication paradigms. According to the considerations made so far, the goodput is a discrete random variable having value given by (23) if correct control occurs, while it is 0 otherwise. Therefore, the average performance can be described by the following utility function: U (τ pay , r) = E k,i [R(τ pay , γ)] = p cc (1 − p ae ) 1 − τ set + τ alg + τ ack τ B d η.(26) In the following subsections, we analyze the terms involved in eqs. (26) describing the difference between the transmission paradigms and particularizing the analysis for the different CCs. the UE that the OPT-CE procedure has started. If an IB-CC is employed, this is followed by the transmission of the SET-R packet to the RISC notifying the start of the procedure, and a consequent TTI for feedback is reserved to notify back to the BS if the SET-R packet has been received. If an OB-CC is employed, no TTI needs to be reserved because the SET-R and its feedback are scheduled at the same time as the SET-U packet but on different resources. This is compliant with the assumption of error-free CC made on the definition of RIS-OB-CC in Sect. II. Accordingly, the Setup phase duration is B. Overhead evaluation τ set =      T, OB-CC, 3T, IB-CC.(27) 2) Acknowledgement phase: The time needed to acknowledge the UE and the RISC follows the setup phase: after the optimization has run, an acknowledgment (ACK-U) packet spanning one TTI is sent to the UE notifying it to prepare to send the data; then, if a IB-CC is present, a TTI is used to send an RIS acknowledgment (ACK-R) packet containing the information of which configuration to load during the Payload phase; a further TTI is reserved for feedback. In the Setup phase, if a OB-CC is present, no TTI needs to be reserved because the ACK-R and its feedback are scheduled at the same time as the SET-U packet but on different resources. Remark that the τ s guard period must be considered by the UE when transmitting the data, to avoid data being disrupted during the load of the configuration employed in the Payload. For simplicity of evaluation, we insert the guard period in the overall Acknowledgement phase duration, resulting in τ ack = τ set + τ s 3) Algorithmic phase: This phase comprises the process of sending pilot sequences and the consequent evaluation of the configuration for the transmission. Regardless of the paradigm, each pilot sequence spans an entire TTI, but the switching time of the configuration must be taken into account as a guard period. Therefore, the actual time occupied by a pilot sequence is τ p ≤ T − τ s and the number of samples p of every pilot sequence results p = T − τ s T n ,(29) where T n is the symbol period in seconds. Assuming that the TTI and the symbol period are fixed, the UE is able to compute the pilot length if it is informed about the guard period. On the other hand, the overall duration of the Algorithmic phase depends on the paradigm employed. a) OPT-CE: In this case, the Algorithmic phase starts with C CE TTIs; at the beginning of each of these TTIs, the RISC loads a different configuration, while the UE transmits replicas of the pilot sequence. After all the sequences are received, the CE process at the BS starts, followed by the configuration optimization. The time needed to perform the CE and optimization processes depends on the algorithm, as well as the available hardware. To consider a generic case, we denote this time as τ A = AT . b) CB-BSW fixed frame structure: Similarly to the previous case, the Algorithmic phase starts with C CB TTIs, at the beginning of which the RISC loads a different configuration, and the UE transmits replicas of the pilot sequence. After all the sequences are received, the BS selects the configuration as described in Sect. III-C. The time needed to select the configuration is considered negligible, and hence the Acknowledgement phase may start in the TTI after the last pilot sequence is sent. c) CB-BSW flexible frame structure: In this case, the number of TTIs used for the beam sweeping process is not known a priori and it depends on the measured SNR. However, to allow the system to react in case the desired threshold is reached, a TTI is reserved for the transmission of the ACK-U after each TTI used for pilot transmission. Hence, the number of TTIs needed is 2c − 1, where 0 < c ≤ C CB is a random variable. According to the previous discussion, the Algorithmic phase duration is τ alg =            (C CE + A)T, OPT-CE, C CB T, CB-BSW fixed frame structure, (2c − 1)T, CB-BSW flexible frame structure. (30) C. Reliability evaluation In this section, we evaluate the reliability of the control packets. The content of each control packet depends on the kind of control packet considered and on the communication paradigms employed, as we will describe throughout the section. Without loss of generality, we assume that the i-th control packet toward entity k comprises a total of b (k) i informative bits. Accordingly, we can express the probability of error of a single packet by means of an outage event, obtaining where τ (k) i is the time employed by the transmission of the i-th control packet intended for entity k ∈ {u, r}, and B c is the CC transmission bandwidth. According to the assumption on the channel distribution, eq. (31) can be rewritten as p (k) i = Pr log (1 + Γ k ) ≤ b (k) i τ (k) i B c , k ∈ {u, r}, i = {1, 2}(31)ID control preamble b ID 1 b (k) i − 1 − b ID b (k) ip (k) i = 1 − exp − 1 λ k 2 b (k) i /τ (k) i /Bc − 1 .(32) Plugging eq. (32) into (25), the correct control probability for the paradigms under tests results p cc = exp 1 λ u 2 − 2 i=1 2 b (u) i /τ (u) i /Bc exp 1 λ r 2 − 2 i=1 2 b (r) i /τ (r) i /Bc .(33) In the following, we evaluate the time occupied by the transmission of the control packets τ i , i.e., the time in which informative bits can be sent without risk to be disrupted, depends on the RIS switching time. As already discussed in Sect. IV-B, a guard period of τ s seconds must be considered if the RISC loads a new configuration in that TTI. Following the frame structure of Fig. 3, the SET-R and ACK-R packets can use the whole TTI, while the SET-U packets need the guard period. Note that the ACK-U control packet does not employ the guard period under the OPT-CE, as long as the time employed by the optimization process is at least a TTI, i.e., A ≥ 1. For the CB-BSW paradigm, the guard period is needed. As a consequence, the useful time of the packets intended for the UE results τ (u) 1 = T − τ s , τ (u) 2 =      T − τ s , CB-BSW, T, OPT-CE,(34) while the useful time of the packets intended for the RISC results τ (r) 1 = τ (r) 2 = T.(35) 2) Control packet content: In this part, we evaluate the minimum number of informative bits b (k) i of each control packet. Without loss of generality, we can assume a common structure for all the control packets, comprising a control preamble and control payload parts as depicted in b (k) i = b ID + 1 +                    b frame + b guard + b conf , k = u, i = 1, (SET-U), b frame + C CE b conf , k = r, i = 1, (SET-R), b SE , k = u, i = 2, (ACK-U), N b quant , k = r, i = 2, (ACK-R).(36b (k) i = b ID + 1 +                    b frame + b guard + b conf , k = u, i = 1, (SET-U), b frame + C CB b conf , k = r, i = 1, (SET-R), 0, k = u, i = 2, (ACK-U), b conf , k = r, i = 2, (ACK-R).(37) V. NUMERICAL RESULTS In this section, we show the performance evaluation of the communication paradigms under study. The parameters set for the simulations are given in Table I [30]. For the OPT-CE paradigm, the channel estimation codebook C CE is designed from the DFT, as described in Sect. III-B. For the sake of simplicity, the same configurations are used in the beam sweeping codebook C CB . In particular, the codebook used by the CB-BSW with flexible frame structure is C fle CB = C CE , while the one used by the CB-BSW with fixed frame structure uses one every three configurations, to take advantage of the possible lower overhead. In the following, we divide the results into two parts: the evaluation of the paradigms performance under error-free CCs, and the investigation of the impact of CCs reliability. A. Paradigms performance (error-free CCs) To summarize, the OPT-CE paradigm is inherently more robust to algorithmic errors than the CB-BSW paradigm. To provide a fair comparison between the paradigms, we evaluate the optimal target SNR γ 0 used as relevant KPI for the CB-BSW paradigm. Fig. 6 shows the average goodput R achieved as a function of the target SNR, under different kinds of CC and for different values of τ . We note that the optimal γ 0 depends on the frame structure chosen while it does not depend on the kind of CC, the latter influencing slightly the achievable goodput under the same target SNR. Moreover, the duration τ influences negligibly the optimal γ 0 in the flexible structure, being approx. 13.8 dB for τ = 30 ms and approx. 12.4 dB for τ = 90 ms for the flexible frame structure, while approx. 10.9 dB for any value of τ for the fixed frame structure. We remark that the selection of the target KPI is also scenario dependent, and hence this procedure should be performed during the deployment of the RIS. Using the optimal target SNR, we now compare the performance of the communication paradigms. Fig. 7a shows the average goodput as a function of the overall frame duration. Again, the impact of the kind of CC on the average goodput is negligible. The main advantage of the CB-BSW approach is the possibility of providing a non-null transmission rate even in to bep cc = 0.99, and we study the minimum average SNRs λ u and λ r providing such reliability following the control packet content given in Table I. Fig. 9a shows the achieved p cc for the OB-CC as a function of λ u only, according to the assumption of error-free RIS-CC in the OB-CC case. With this kind of CC, the probabilities of correct control achieved by OPT-CE and CB-BSW have negligible differences, and aλ u = 10.5 dB is enough to provide the target correct control probability in both cases. Fig. 9b shows the p cc as a heatmap function of λ r and λ u for the IB-CC case. Note that only the region of λ u and λ r providing at least the targetp cc is colored, while the white part of the heatmap represents the SNRs values not satisfying the target reliability. The minimum value of λ r and λ u needed are also given in the figure. It is worth noting that the OPT-CE needs higher SNRs than CB-BSW due to the higher information content of the control packet of the former. Finally, we remark that the performance provided by the CCs should be accounted for simultaneously to achieve the desired control reliability. VI. CONCLUSIONS In this paper, we proposed a general framework of four phases -Setup, Algorithmic, Ac- the performance considering a utility function that takes into account the overhead generated by the various phases of the paradigms, the possible errors coming from the Algorithmic phase, and the impact of losing control packets needed for signalling purposes. Moreover, we particularized the performance evaluation for two kinds of CCs connecting the decision maker and the RISC -IB-CC and OB-CC -, showcasing the minimum performance needed to obtain the desired target control reliability. While some oversimplification has necessarily been introduced, we believe that the proposed framework can be used to include the control operations into the communication performance evaluation for various scenarios of interest. For example, the framework can be applied to multi-user wideband/OFDM communications by accounting for the subcarrier allocation of the different control and payload messages. Differently from the cases studied in this paper, the Algorithmic phase should also consider the resource allocation process, whose output should be signaled to the UEs through a specific design of the Acknowledgement phase. Other potential control and algorithmic designs can be addressed by using the proposed framework, taking care of omitting, merging, or repeating some of the general phases to meet the design requirements. Fig. 1 : 1Scenario of interest: RIS extends the coverage of the BS, which has a blocked link to the UE. During data transmission, the RISC loads a configuration aiming to achieve a certain communication performance; during control signaling, the RISC loads a wide beamwidth configuration to deliver low-rate control packets to the UE. b) UE-CC: This narrowband channel operates on central frequency f u and bandwidth B u and is assumed to be a wireless IB-CC, meaning that the the physical resources employed for the UE-CC overlaps the one used for the UE-DC, i.e., f u = f d , and B u ≥ B d . To ensure that Fig. 2 : 2Data exchange diagram of the two RIS-aided communication paradigms. Signals traveling through RIS-CC, UE-CC, and UE-DC are represented by solid red, solid blue and solid black lines, respectively. RISC to RIS commands are indicated with dashed black lines. BS operations are in monospaced font. i the optimization time, τ A ; the time to control the RIS, composed of the time employed for the transmission of the control packets to the UE (RISC), τ ), and the time needed by the RIS to switch configuration, τ s . Fig. 3 : 3Following the description given in Sect. III, we show the frame structures of the communication paradigms under study inFig. 3, where the rows represent the time horizon of the packets travelling on the different channels (first three rows) and the configuration loading time at the RISC (last row). The time horizon is obtained assuming that all the operations span multiple numbers of transmission time intervals (TTIs), each of duration of T seconds, where τ /T ∈ N is the total number of TTIs in a frame. At the beginning of each TTI, if the RISC loads a new configuration, the first τ s seconds of data might be lost, due to the unpredictable response of the channel during this switching time. When needed, we will consider a guard period of τ s seconds in the overhead evaluation to avoid data disruption. Remember that the RISC loads the widebeam crtl configuration any time it is in an idle state, i.e., at the beginning of the Setup and Acknowledgement phases (see the "RISC" row ofFig. 3).From Fig. 3, we can note that the overhead generated by Setup and Acknowledgement phases is communication paradigm independent 7 , while it is CC dependent. Indeed, all the paradigms Frame structure for the communication paradigms under study. Packets colored in blue and in yellow have DL and UL directions, respectively. Remark that SET-R (ACK-R) packet and its feedback are sent at the same time as the SET-U (ACK-U) but on different resources if OB-CC is present. make use of P = P u + P r = 4 control packets, P u = 2 control packets sent on the UE-CC and P r = 2 on RIS-CC. Nevertheless, the kind of RIS-CC employed can reduce the time employed for the communication of those packets. On the other hand, the Algorithmic phase is CC independent and communication paradigm dependent being designed to achieve the goal of the communication paradigm itself. In the following, the overhead evaluation is performed for the various cases of interest.1) Setup phase: This phase starts with the SET-U control packet sent on the UE-CC, informing Fig. 4 : 4General control packet structure comprising a preamble and a payload part. number of informative bits contained in the control packets b (k) i . 1) Useful time for control packets: Following the data frame, each control packet spans an entire TTI. However, the useful time τ (k) Fig. 4 . 4The preamble comprises b ID bits representing the unique identifier (ID) of the destination entity in the network, and a single bit flag specifying if the packet is a SET or an ACK one. From the preamble, the UE (RISC) can understand if the control packet is meant to be decoded and how to interpret the control payload. Accordingly, the remaining number of bits, b(k) i − 1 − b ID , depends on the control payload, which, in turn, depends on the kind of control packet considered and on the communication paradigms employed. a) OPT-CE: To initialize the overall procedure at the UE, the payload of the SET-U packet must contain the length of the frame τ , the cardinality of the set C CE , and the guard time τ s . To simplify the data transmission, the frame duration can be notified through an (unsigned) integer of b frame containing the number of total TTIs τ /T set for the frame. In the same manner, we can translate the guard time into an unsigned integer representing the number of guard symbols τ s /T n so as to send b guard bits. Finally, another integer of b conf bits can be used to represent the cardinality C CE and to notify it to the UE. Remark that the minimum b conf = log 2 (C) ,where C is the total number of configurations stored in the common codebook. Similarly, the payload of the SET-R packets needs to contain the information of the length of the frame τ , and the set of configuration C CE to switch through. Also, in this case, encoding the data as integers may reduce the number of informative bits to transmit. For the frame length, the same b frame bits of the SET-U packet are used. To encode the information of the set to be employed, b conf bits are used to identify a single configuration in the common codebook, and thus C CE b conf needs to be transmitted to the RISC, one per wanted configuration. Regarding the Acknowledgement phase, we can assume that the payload of the ACK-U contains only the chosen SE of the communication r CE . This can be encoded in similar manner the MCS is encoded for the 5G standard[30]: a table of predefined values indexed by b SE bits. On the other hand, the payload of the ACK-R must contain the optimal phase-shift profile φ , that is, a value of the phase-shift for each element. Without loss of generality, we can denote as b quant the number of bits used to control each element, i.e., the level of quantization of the RIS[4]. Hence, the overall number of informative bits is the number of elements to control times the level of quantization, i.e., N b quant . To summarize, the packets length results ) b) CB-BSW: The payload of the Setup packets follows the same scheme used for the OPT-CE paradigm. The SET-U packet contains the length of the frame τ , the cardinality of the set C CB , and the guard time τ s translated to (unsigned) integer of b frame , b guard and b conf bits, respectively.The payload of the SET-R packets contains the information of the length of the frame τ , and the set of configuration C CB to switch through, encoded in (unsigned) integers of b frame and C CB b conf bits, respectively. Instead, the Acknowledgement contains different information. In particular, the payload of the ACK-U is empty, according to the fixed rate transmission used by this paradigm.The payload of the ACK-R contains the configuration c chosen, encoded by the same b conf bits representing an index in the common codebook. To summarize, the packets length results 8 , if not otherwise specified. The scenario is tested through Monte Carlo simulations. With respect to the scenario described in Sect. II, we consider that the BS and RIS positions x b and x r = (0, 0, 0) T are kept fixed, while the UE position, x u , changes at every simulation according to a uniform distribution having limits (−D/2, 0, 0) T and (D/2, D, −D) T . In this section, when referring to average performance, we implicitly assume averaging over different UE positions and noise realizations. The DC channel coefficients are evaluated considering the line-of-sight (LoS) component of h d and g d following Fig. 5 Fig. 5 : 55shows the CDF of the actual and estimated SNR to give some insight on the impact of the possible algorithm errors. From the figure, it can be inferred that the impact of the noise on the SNR estimation is generally negligible for the OPT-CE paradigm. This finding is justified by the fact that the power of the noise influencing the measurement is proportional to CDF of the actual and estimated SNR for the communication paradigms under study. ) R vs γ0, τ = 30 ms. (b) R vs γ0, τ = 60 ms.(c) R vs γ0, τ = 90 ms. Fig. 6 : 6Analysis of the target SNR for the CB-BSW paradigm. Note the different scale in Fig. 6a. presence of a lower coherence block (< 60 ms), while the OPT-CE needs a longer time horizon to obtain the CSI and perform the Payload phase (≥ 60 ms). On the other hand, as long as the time horizon is sufficiently long (τ ≥ 75 ms), the OPT-CE paradigm outperforms the CB-BSW ones. In Fig. 7b, we show the CDF of the goodput for τ = 60 ms. For this frame duration, the kind of CC influences the performance of the OPT-CE paradigm, while its impact is less predominant on the CB-BSW performance. As expected, the IB-CC provides worse performance due to its increased overhead. Nevertheless, remark that for the CB-BSW approximately half of the transmissions has null goodput because of algorithmic errors, while the OPT-CE provide a non-null goodput for all values, corroborating the results of Fig. 5. ) R vs τ , OB-CC (b) CDF of R, τ = 60 ms. Fig. 7 :Fig. 8 : 78Analysis of the goodput performance. B. Impact of the CCs reliabilityFig. 8 shows the average utility (26) as a function of the erroneous control probability 1 − p cc , for τ = 60 ms and for both kinds of CC. The results of the transmission paradigms are in line with the one presented inFig. 7. The CC reliability influences significantly the performance when 1 − p cc ≤ 0.05, i.e., p cc ≤ 0.95. To consider a conservative case, we set a target reliability Analysis of the utility function vs. erroneous control probability. knowledgement, Payload -to evaluate RIS-aided communication performance, addressing the impact of control and signaling procedures in a generic scenario. Employing this framework, we detailed the data exchange and the frame structure diagram for two different communication paradigms employed in RIS-aided communications, namely OPT-CE and CB-BSW. We analyzed OB-CC: pcc vs λu.(b) IB-CC: pcc vs λu, λr. Fig. 9 : 9Impact of the CC SNR to the reliability. , which consists of a single-antenna BS, a single-antenna UE, and a fully-reflective RIS. The RIS has N elements equally spaced on a planar surface. Each RIS element is able to change the phase shift of an incoming wave by ϕ n ∈ [0, 2π], ∀n ∈ N 2 . We denote as φ = [e jϕ 1 , . . . , e jϕ N ] T ∈ C N the vector representing a particular configuration of the phase shifts of the elements at a given time. ARIS controller (RISC) is in charge of loading different configurations to the RIS surface. TheRISC is equipped with a look-up table containing a set of predefined configurations C, |C| = C; a copy of this look-up table is stored in the BS, so that the BS can send control signals to instruct the RISC to load a configuration already stored in C. Therefore, the set C is the socalled common codebook of configurations. Remark that the BS can also issue a command to load a configuration not present in the common codebook by sending the explicit phase-shift values for each RIS element.To study the impact of the control signals on communication, we focus on characterizing three narrowband 3 wireless channels: a) the UE-data channel (DC), where the UE sends payload data to the BS, b) the UE-CC, in which the BS and the UE can share control messages to coordinate their communication, and c) the RIS-CC that connects the BS to the RISC so that the former can control the operation of the latter.Fig. 1illustrates the channels further detailed below.a) UE-DC: This narrowband channel operates with frequency f d and bandwidth B d . The UL SNR over the data channel is To change send payload re-load configuration c ⋆ 3. Acknowledgement phase . . . (b) CB-BSW paradigm.UE RIS BS RISC Setup phase initiate sending pilots initiate beam sweeping load configuration 1 load configuration 2 . . . . . . pilots pilots load configuration CCE pilots prepare to send the data load configuration φ⋆ Algorithmic phase: channel estimation & optimize the transmission 1. 2. 4. collects pilots channel estimation configuration opt. collects pilots collects pilots send payload time channel estimation configuration opt. load configuration φ⋆ Payload phase: data transmission . . . 3. Acknowledgement phase (a) OPT-CE paradigm. UE RIS BS RISC Setup phase initiate sending pilots initiate beam sweeping 1. load configuration 1 load configuration 2 . . . . . . time pilots pilots KPI not satisfied KPI not satisfied load configuration CCB KPI satisfied pilots Prepare to send the data re-load configuration c ⋆ Algorithmic phase: beam sweeping 2. Payload phase 4. TABLE I : ISimulation parameters. the model of [31, Sect. II]. Note that the RIS-OB-CC uses a different operating frequency and bandwidth w.r.t. the DC ones, while, for the IB-CC, we have f r = f d and B r = 5B d . In any case, the UE-CC has operating frequency f u = f d and bandwidth B u = 5B d . The overall frame duration τ set reflects the coherence time of the channel: a low τ represents a high mobility environment with low coherence time, and vice versa. The TTI duration is set according to the half of the subframe duration in the OFDM 5G NR standardScenario Communication Scenario side D 20 m DC frequency f d 3 GHz BS position x b (25, 5, 5) T m Bandwidth of DC B d 180 kHz RIS element spacing d ν/f d /2 UE transmit power ρu, ρ b 24 Number of RIS elements N 100 UE/RIS noise power σ 2 u , σ 2 r −94 dBm Paradigms Control packet content Codebooks cardinality CCE, C fix CB , C fle CB N , N/3 , N ID bits b ID 8 Overall frame duration τ [10, 200] ms TTIs bits b frame 16 TTI duration T 0.5 ms Guard period bits b guard 16 Guard period τs 50 µs Configuration selection bits b conf 8 Pilot sequence length p 1 SE table bits b SE 6 CE duration in TTIs A 5 RIS quantization level bits b quant 2 1/C CE , where C CE ≥ N as described in Section III-B. On the other hand, when employing the CB-BSW paradigm, the measurement of the SNR is done separately per each configuration employed, and the resulting noise has a higher impact on the estimation. Moreover, we can also observe that the SNR of the CB-BSW extends to very low values, while the minimum target KPI needs to be set to provide a non-negligible and supportable SE (at least higher than -13 dB to have the minimum SE of the 5G NR standard of 0.0586[32, Table 5.1.3.1-3]). Therefore, whatever reasonable target KPI is chosen will result in a relatively high outage probability. The simulation code for the paper is available at https://github.com/lostinafro/ris-control We remark that the ctrl configuration is automatically loaded after the Algorithmic phase ends, due to the idle state of the RIS.Another approach is loading the optimal configuration evaluated in the Algorithmic phase also to send the control information toward the UE; nevertheless, the RISC needs to be informed previously by a specific control message by the BS. We do not consider this approach to keep the frame structure of the two paradigms similar and the analysis complexity at the minimum, simplifying the presentation of Sect. IV. The reliability of the control packets exchanged is still dependent on the communication paradigm, see Section IV-C. In the Table, ν is the speed of light Reconfigurable intelligent surfaces for energy efficiency in wireless communication. C Huang, IEEE Trans. Wireless Commun. 188C. Huang et al., "Reconfigurable intelligent surfaces for energy efficiency in wireless communication," IEEE Trans. Wireless Commun., vol. 18, no. 8, pp. 4157-4170, Aug. 2019. Intelligent reflecting surface aided wireless communications: A tutorial. Q Wu, IEEE Trans. Commun. 695Q. Wu et al., "Intelligent reflecting surface aided wireless communications: A tutorial," IEEE Trans. Commun., vol. 69, no. 5, pp. 3313-3351, May 2021. Reconfigurable, intelligent, and sustainable wireless environments for 6G smart connectivity. 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Singh et al., "Fast beam training for RIS-assisted uplink communication," arXiv preprint arXiv:2107.14138, 2021. Intelligent reflecting surface-enhanced ofdm: Channel estimation and reflection optimization. B Zheng, R Zhang, IEEE Wireless Commun. 9B. Zheng and R. Zhang, "Intelligent reflecting surface-enhanced ofdm: Channel estimation and reflection optimization," IEEE Wireless Commun., vol. 9, pp. 518-522, 2020. Low-to-zero-overhead IRS reconfiguration: Decoupling illumination and channel estimation. V Jamali, IEEE Commun. Lett. 164V. Jamali et al., "Low-to-zero-overhead IRS reconfiguration: Decoupling illumination and channel estimation," IEEE Commun. Lett., vol. 16, no. 4, pp. 932-936, Apr. 2022. Near-field hierarchical beam management for RIS-enabled millimeter wave multi-antenna systems. G C , Proc. IEEE SAM. IEEE SAMTrondheim, Norway, 2022G. C. Alexandropoulos et al., "Near-field hierarchical beam management for RIS-enabled millimeter wave multi-antenna systems," in Proc. IEEE SAM, Trondheim, Norway, 2022, pp. 460-464. Codebook-based solutions for reconfigurable intelligent surfaces and their open challenges. J An, arXiv:2211.05976arXiv preprintJ. An et al., "Codebook-based solutions for reconfigurable intelligent surfaces and their open challenges," arXiv preprint arXiv:2211.05976, 2022. Performance of RIS-aided nearfield localization under beams approximation from real hardware characterization. M , arXiv:2303.15176arXiv preprintM. Rahal et al., "Performance of RIS-aided nearfield localization under beams approximation from real hardware characterization," arXiv preprint arXiv:2303.15176, 2023. Hierarchical codebook-based beam training for RIS-assisted mmwave communication systems. J Wang, IEEE Trans. Commun. early accessJ. Wang et al., "Hierarchical codebook-based beam training for RIS-assisted mmwave communication systems," IEEE Trans. Commun., early access, 2023. A random access protocol for RIS-aided wireless communications. V Croisfelt, Proc. IEEE SPAWC. IEEE SPAWC2022V. Croisfelt et al., "A random access protocol for RIS-aided wireless communications," in Proc. IEEE SPAWC, 2022. Random access protocol with channel oracle enabled by a reconfigurable intelligent surface. V Croisfelt, arXiv:2210.04230arXiv preprintV. Croisfelt et al., "Random access protocol with channel oracle enabled by a reconfigurable intelligent surface," arXiv preprint arXiv:2210.04230, 2022. Performance analysis of PHY layer for ris-assisted wireless communication systems with retransmission protocols. S Hao, H Zhang, J. Comp. Inf. Sciences. 348Part AS. Hao and H. Zhang, "Performance analysis of PHY layer for ris-assisted wireless communication systems with retransmission protocols," J. Comp. Inf. Sciences, vol. 34, no. 8, Part A, pp. 5388-5404, 2022. Active reconfigurable intelligent surface-aided wireless communications. R Long, IEEE Trans. Wireless Commun. 208R. 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Channel estimation for intelligent reflecting surface assisted multiuser communications: Framework, algorithms, and analysis. Z Wang, IEEE Trans. Wireless Commun. 1910Z. Wang et al., "Channel estimation for intelligent reflecting surface assisted multiuser communications: Framework, algorithms, and analysis," IEEE Trans. Wireless Commun., vol. 19, no. 10, pp. 6607-6620, 2020. Massive MIMO networks: Spectral, energy, and hardware efficiency. E Björnson, Foundations Trends Signal Process. 113-4E. Björnson et al., "Massive MIMO networks: Spectral, energy, and hardware efficiency," Foundations Trends Signal Process., vol. 11, no. 3-4, pp. 154-655, 2017. S M Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice HallS. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall, 1997. Communication in the presence of noise. C Shannon, Proceedings of the IRE. 371C. Shannon, "Communication in the presence of noise," Proceedings of the IRE, vol. 37, no. 1, pp. 10-21, 1949. (TR) 21.915Study on New Radio (NR) access technology. 102019Technical Report3rd Generation Partnership Project (3GPP). version 15.0.03GPP, "Study on New Radio (NR) access technology," 3rd Generation Partnership Project (3GPP), Technical Report (TR) 21.915, 10 2019, version 15.0.0. MARISA: A self-configuring metasurfaces absorption and reflection solution towards 6G. A , Proc. IEEE INFOCOM. IEEE, 2022. IEEE INFOCOM. IEEE, 2022A. Albanese et al., "MARISA: A self-configuring metasurfaces absorption and reflection solution towards 6G," in Proc. IEEE INFOCOM. IEEE, 2022, pp. 250-259. NR; Physical layer procedures for data. 3GPP. 2142022Technical specification (TS) 38. version 15.0.03GPP, "NR; Physical layer procedures for data," 3rd Generation Partnership Project (3GPP), Technical specification (TS) 38.214, 10 2022, version 15.0.0.

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{'abstract': 'The research on Reconfigurable Intelligent Surfaces (RISs) has dominantly been focused on physicallayer aspects and analyses of the achievable adaptation of the propagation environment. Compared to that, the questions related to link/MAC protocol and system-level integration of RISs have received much less attention. This paper addresses the problem of designing and analyzing control/signaling procedures, which are necessary for the integration of RISs as a new type of network element within the overall wireless infrastructure. We build a general model for designing control channels along two dimensions: i) allocated bandwidth (in-band and out-of band) and ii) rate selection (multiplexing or diversity). Specifically, the second dimension results in two transmission schemes, one based on channel estimation and the subsequent adapted RIS configuration, while the other is based on sweeping through predefined RIS phase profiles. The paper analyzes the performance of the control channel in multiple communication setups, obtained as combinations of the aforementioned dimensions. While necessarily simplified, our analysis reveals the basic trade-offs in designing control channels and the associated communication algorithms. Perhaps the main value of this work is to serve as a framework for subsequent design and analysis of various system-level aspects related to the RIS technology.In[19]and [20], a detailed protocol for RIS-aided communication system was presented tackling the initial access problem. It was showcased that, despite the configuration control overhead, the RIS brings notable performance benefits allowing more UEs to access the network on average.However, it was assumed that the RIS control is perfect, which can be hardly true in practice.4The effect of retransmission protocols in RIS-aided systems for cases of erroneous transmissions was studied in[21], although the presented methodology assumed perfect control signals.There has been a significant discussion regarding the comparison of RISs and conventional amplify-and-forward relays[1]. While the distinction between them can sometimes be blurred[22], one way to make a clear distinction is the use of the flow of control and data through the communication layers [23,Fig. 2]. Those considerations set the basis for the definition of the control channel options in this paper.B. ContributionsThe main objective of this work is to develop a framework for designing and analyzing the control channel (CC) in RIS-aided communication systems. The number of actual CC designs is subject to a combinatorial explosion, due to the large number of configurable parameters in the system, such as frame size or feedback design. Clearly, we cannot address all these designs in a single work, but what we are striving for is to get a simple, yet generic, model for analyzing the impact of CCs that captures the essential design trade-offs and can be used as a framework to analyze other, more elaborate, CC designs.We build generic CC models along two dimensions. The first dimension is related to how the CC interacts with the bandwidth used for data communication. An out-of-band CC (OB-CC) uses communication resources that are orthogonal to the ones used for data communication.More precisely, OB-CC exerts control over the propagation environment, but is not affected by this control. Contrary to this, an in-band CC (IB-CC) uses the same communication resources as data communication. This implies that the IB-CC decreases the number of degrees of freedom for transmission of useful data, thereby decreasing the spectral efficiency (SE) of the overall system. Furthermore, the successful transmission of the control messages toward the RIS is dependent on its phase profile. For instance, an unfavorable RIS configuration may cause blockage of the IB-CC and transmission of further control messages, impacting the overall system performance.The second dimension is built along the traditional diversity-multiplexing trade-off in wireless communication systems. In a diversity transmission, the data rate is predefined and the sender hopes that the propagation environment is going to support that rate. If this is not the case, then, an outage occurs. To reflect this paradigm in an RIS setup, we consider a transmission setup in which the RIS sweeps through different configurations and the BS tries to select the one that is likely to support the predefined data rate. In a multiplexing transmission, the data rate is adapted', 'arxivid': '2303.16797', 'author': ['Member, IEEE, Victor CroisfeltFabio Saggese \nDepartment of Electronic Systems\nDepartment of Informatics and Telecommunications, National and Kapodistrian\nAalborg University\nAalborgDenmark\n', 'Student Member, IEEERadosław Kotaba \nDepartment of Electronic Systems\nDepartment of Informatics and Telecommunications, National and Kapodistrian\nAalborg University\nAalborgDenmark\n', 'Student Member, IEEEKyriakos Stylianopoulos \nDepartment of Electronic Systems\nDepartment of Informatics and Telecommunications, National and Kapodistrian\nAalborg University\nAalborgDenmark\n', 'Senior Member, IEEEGeorge C Alexandropoulos \nDepartment of Electronic Systems\nDepartment of Informatics and Telecommunications, National and Kapodistrian\nAalborg University\nAalborgDenmark\n', 'Fellow, IEEEPetar Popovski petarp@es.aau.dk.k \nDepartment of Electronic Systems\nDepartment of Informatics and Telecommunications, National and Kapodistrian\nAalborg University\nAalborgDenmark\n', 'G C Stylianopoulos \nUniversity of Athens\nPanepistimiopolis Ilissia15784AthensGreece\n', 'Alexandropoulos \nUniversity of Athens\nPanepistimiopolis Ilissia15784AthensGreece\n'], 'authoraffiliation': ['Department of Electronic Systems\nDepartment of Informatics and Telecommunications, National and Kapodistrian\nAalborg University\nAalborgDenmark', 'Department of Electronic Systems\nDepartment of Informatics and Telecommunications, National and Kapodistrian\nAalborg University\nAalborgDenmark', 'Department of Electronic Systems\nDepartment of Informatics and Telecommunications, National and Kapodistrian\nAalborg University\nAalborgDenmark', 'Department of Electronic Systems\nDepartment of Informatics and Telecommunications, National and Kapodistrian\nAalborg University\nAalborgDenmark', 'Department of Electronic Systems\nDepartment of Informatics and Telecommunications, National and Kapodistrian\nAalborg University\nAalborgDenmark', 'University of Athens\nPanepistimiopolis Ilissia15784AthensGreece', 'University of Athens\nPanepistimiopolis Ilissia15784AthensGreece'], 'corpusid': 257804661, 'doi': '10.48550/arxiv.2303.16797', 'github_urls': ['https://github.com/lostinafro/ris-control'], 'n_tokens_mistral': 20668, 'n_tokens_neox': 18515, 'n_words': 12715, 'pdfsha': '1e5d5848f2655b08067073aa1c8d53aeaa69ddb2', 'pdfurls': ['https://export.arxiv.org/pdf/2303.16797v1.pdf'], 'title': ['A Framework for Control Channels Applied to Reconfigurable Intelligent Surfaces', 'A Framework for Control Channels Applied to Reconfigurable Intelligent Surfaces'], 'venue': []}

arxiv

Diffusion in Curved Tube 20 Sep 2011 Naohisa Ogawa Hokkaido Institute of Technology 006-8585SapporoJapan Diffusion in Curved Tube 20 Sep 2011(Dated: September 21, 2011)arXiv:1109.0590v2 [math-ph]numbers: 0560Cd0240Hw0240Ma8240Ck * ogawanao@hitacjp Particle diffusion in a curved tube embedded in R 3 is considered. We find the diffusion coefficient depends on tube's curvature. Diffusion coefficient is obtained in ǫ (radius of tube) expansion.Physical interpretation of curvature dependent diffusion coefficient is given. I. INTRODUCTION In the previous paper on surface diffusion with thickness embedded in R 3 established the curvature dependence of surface diffusion [1]. This theory might be applied to the molecular diffusion in lipid bilayer or applied to the reaction diffusion system [2], [3]. However, the physical meaning of curvature dependence was not clear. To make clear this point we consider an one dimension less system, that is, the diffusion in tube embedded in R 3 . The quantum mechanical version to this problem is given in [4], and higher dimensional extension is given in [5]. (This quantum mechanical problem was originally discussed by many authors [6], [7] to solve the ordering problem of quantum mechanics on curved space.) In the case of thin tube (string) in R 3 , the dimensional difference is two and there appears additional degree of freedom to rotate around the central axis directions, so called torsion that makes calculation bit complicated. However the "local equilibrium condition" (explained later) makes the effect of torsion disappear, and we have the curvature dependent diffusion equation. The physical reason of its curvature dependence is discussed precisely in last section. II. FRENET-SERET EQUATIONS Let us consider a line specified by x(s),(1) where the parameter s is the length of line. The unit tangent vector is defined by e 1 ≡ d x(s) ds .(2) Another (normal) unit vector e 2 is defined by equation. d e 1 (s) ds = κ e 2 , | e 2 |= 1.(3) κ is called the curvature of this line in R 3 . The reason is the following. We can set a circle which tangents to this curved line at point x(s). From figure 1 we can easily find two relations. where R is the radius of this circle. Then we find the equation (3) with identifying κ = 1/R. e 1 (s + ds) − e 1 (s) = e 2 dθ,(4)ds = R dθ,(5) Next we introduce another independent unit vector e 3 by e 3 = e 1 × e 2 .(6) Then we can identify the derivation of e 2 generally such as, d e 2 ds = α e 1 + τ e 3 ,(7) where α and τ are unknown some functions. But α can be calculated as follows. α = e 1 · d e 2 ds = − e 2 · d e 1 ds = −κ.(8) So we can replace (7) as d e 2 ds = −κ e 1 + τ e 3 . Function τ has a geometrical meaning. From the equation τ = d e 2 ds · e 3 ,(10) we see τ means the rotation of normal vector e 2 around e 1 (figure 2). So this function is called torsion. We further take a derivative of e 3 . d e 3 ds = β e 1 + γ e 2 ,(11) with unknown function β and γ. Both of them can be obtained as follows. β = d e 3 ds · e 1 = − d e 1 ds · e 3 = −κ e 2 · e 3 = 0.(12) and γ = d e 3 ds · e 2 = − d e 2 ds · e 3 = −τ.(13) So we obtain d e 3 ds = −τ e 2 .(14) We call these three equations (3), (9), and (14) as Frenet-Seret equations, and we have two geometrical quantities, curvature κ and torsion τ . III. METRIC IN TUBE We re-identify the one dimensional diffusion on curved line as the limitation process from three dimensional diffusion. We set the curved tube with radius ǫ in three dimensional Euclidean space R 3 . Our particles can move only in this tube. We look for the form of diffusion equation in the limit of small ǫ. The coordinates we use hereafter is the followings. (See FIG.3) X is the Cartesian coordinate in R 3 . x is the Cartesian coordinate which specifies only the points on center line. q i is the normal coordinate in a direction of e i . (Small Latin indices i, j, k, · · · runs from 2 to 3.) s is the length parameter along the center line. Further by using the normal unit vector e i , we can identify any points in tube by s, q 2 , q 3 by the following equation [4] [5]. X(s, q 2 , q 3 ) = x(s) + q i e i (s),(15)where 0 ≤| q |≤ ǫ with | q |= (q 2 ) 2 + (q 3 ) 2 . FIG. 3. Coordinate From this relation we can obtain the curvilinear coordinate system in tube (⊂ R 3 ) by the coordinate q µ = (s, q 2 , q 3 ), and metric G µν . (Hereafter Greek indices µ, ν, · · · runs from 1 to 3 with s = q 1 .) G µν = ∂ X ∂q µ · ∂ X ∂q ν .(16) Each part of G µν is the following. G 11 = 1 − 2κq 2 + (κ 2 + τ 2 )(q 2 ) 2 + τ 2 (q 3 ) 2 ,(17)G 12 = −τ q 3 ,(18)G 13 = τ q 2 ,(19)G 23 = 0,(20)G 22 = G 33 = 1.(21) Compared to the previous paper, we have non zero off-diagonal elements. This comes from the fact that dimensional difference between outer space R 3 and inner space R 1 is two. Now the following relations follow. G ≡ det(G µν ) = (1 − κq 2 ) 2 .(22) The inverse metric is given as, G µν = 1 (1 − κq 2 ) 2 ×      1 τ q 3 −τ q 2 τ q 3 (1 − κq 2 ) 2 + (τ q 3 ) 2 −τ 2 q 2 q 3 −τ q 2 −τ 2 q 2 q 3 (1 − κq 2 ) 2 + (τ q 2 ) 2      . IV. DIFFUSION FIELD IN TUBE Let us denote 3 dimensional diffusion field as φ (3) , and Laplacian as ∆ (3) . Then we have the equation with normalization condition ∂φ (3) ∂t = D∆ (3) φ (3) ,(23)1 = φ (3) (q 1 , q 2 , q 3 ) √ G d 3 q,(24) where D is the diffusion constant, and G ≡ det(G µν ). Our aim is to construct the effective one dimensional diffusion equation from 3D equation above. ∂φ (1) ∂t = D∆ (ef f ) φ (1) ,(25)1 = φ (1) (s)ds,(26) where φ (1) is the one dimensional diffusion field, and ∆ (ef f ) is unknown effective 1D diffusion operator which might not be equal to simple 1D Laplace Beltrami operator d 2 /ds 2 . From two normalization conditions, we obtain 1 = φ (3) (q 1 , q 2 , q 3 ) √ G d 3 q, = [ dq 2 dq 3 (φ (3) √ G)] ds, = φ (1) (s) ds. Therefore we obtain the relation, φ (1) (s) = φ (3) √ Gdq 2 dq 3 .(27) We multiply √ G to equation (23) and integrate by q 2 , q 3 , then we obtain ∂φ (1) ∂t = D ( √ G∆ (3) )φ (3) dq 2 dq 3 .(28) From the form of Laplace Beltrami operator ∆ (3) = G −1/2 ∂ ∂q µ G 1/2 G µν ∂ ∂q ν , our diffusion equation has form ∂φ (1) ∂t = D ( ∂ ∂q µ G 1/2 G µν ∂ ∂q ν φ (3) )dq 2 dq 3 .(29) Here we suppose the "local equilibrium condition" such as, ∂φ (3) ∂q i = 0, i = 2, 3.(30) Because in direction of e 2 , e 3 , equilibrium would be holds in a short time δt ∼ ǫ 2 /D. When our observation is given in time scale t satisfying t >> δt, we can always assume local equilibrium condition (30). Then we have instead, ∂φ (1) ∂t = D ( ∂ ∂q µ G 1/2 G µ1 ∂ ∂s φ (3) )dq 2 dq 3 = D ∂ ∂s ( G 1/2 G 11 dq 2 dq 3 ) ∂ ∂s φ (3) + D{ ∂ ∂q i (G 1/2 G i1 )dq 2 dq 3 } ∂ ∂s φ (3) .(31) From (27) and (30) we also have φ (3) = φ (1) N , N = √ Gdq 2 dq 3 = πǫ 2 .(32) So we obtain ∂φ (1) ∂t = D πǫ 2 ∂ ∂s ( G 1/2 G 11 dq 2 dq 3 ) ∂ ∂s φ (1) + D πǫ 2 { ∂ ∂q i (G 1/2 G i1 )dq 2 dq 3 } ∂ ∂s φ (1) .(33) The explicit calculation gives the integral. ∂ ∂q i (G 1/2 G i1 )dq 2 dq 3 = 0,(34) and G 1/2 G 11 dq 2 dq 3 = dq 2 dq 3 1 − κq 2 = πǫ 2 {1 + ( κǫ 2 ) 2 + O(ǫ 4 )}.(35) FIG. 4. Curved point Then we come to the result. ∂φ (1) ∂t = ∂ ∂s D ef f (s) ∂ ∂s φ (1) ,(36) with the definition of effective diffusion coefficient, D ef f = D(1 + ( κǫ 2 ) 2 + O(ǫ 4 )).(37) The static solution has the form, φ (1) = C 1 + C 2 s 0 ds ′ D ef f (s ′ ) .(38) V. PHYSICAL INTERPRETATION We find the relation that, D ef f = D < 1 1 − κq 2 >,(39) where < · · · >= 1 πǫ 2 dq 2 dq 3 · · · . Then we find the simple interpretation. Let us consider the point P on tube where the curvature is κ. We chose two sections near P, and discuss about the length connecting these two sections. See FIG.4. At the coordinate q 2 , the length between two section is given by ∆s ′ = (1 − κq 2 )∆s. Therefore the curvature dependence of effective diffusion coefficient is given by the mean value of rate of length i.e., < ∆s/∆s ′ >. Next we consider the physical meaning. The diffusion coefficient is proportional to the mobility by Einstein's relation. The mobility is also proportional to the conductivity with When the flux is straight with length ∆s , each length is the same ∆s = s 1 = s 2 = · · · , and so we obtain D stra = α N j=1 ∆σ j s j = α ∆s ( N j=1 ∆σ j ) = ασ ∆s , where σ = j ∆σ j . In the case of FIG.4, we have D bent = α N j=1 ∆σ j s j = α dq 1 dq 2 (1/κ − q 2 )∆θ . (We consider the diffusion only in s direction since we used the local equilibrium condition.) So we obtain the rate D bent D stra = 1 σ dq 1 dq 2 ∆s (1/κ − q 2 )∆θ = 1 σ dq 1 dq 2 1 − κq 2 . Then we obtain the result. D bent =< 1 1 − κq 2 > D stra . On the other hand as we have shown as R = 0 case in the previous paper [1], the curved surface with one direction has no curvature and other has finite curvature has anomalous diffusion coefficient such as, D ef f ∼ = (1 + ǫ 2 κ 2 /12)D. This relation also can be explained in this physical interpretation. In this case we consider the cross section as in FIG.6. < 1 1 − κq 2 > = 1 W ǫ ǫ/2 −ǫ/2 dq 2 W 0 dq 1 1 1 − κq 2 = 1 ǫ ǫ/2 −ǫ/2 dq 2 {1 + κq 2 + (κq 2 ) 2 + · · · } = 1 + ǫ 2 κ 2 /12 + · · · .(40) FIG. 1 1FIG. 1. Curvature FIG. 5 . 5Flux of Tubes FIG. 6. another cross section unit length and unit cross section. In the following we calculate the rate of conductivity between bent and straight for the same length and same cross section flux. The conductivity of flux consist of N infinitesimal thin tubes is proportional to the sum of each conductivity of tubes. Each conductivity proportionals to the cross section ∆σ j and inversely proportionals to the length s j . See FIG.5. So we have . N Ogawa, Phys. Rev. E. 8161113N. Ogawa, Phys. Rev. E. 81 (2010) 061113. . A M Turing, Phil. Trans. R. Soc. London B. 237A. M. Turing, Phil. Trans. R. Soc. London B 237 (1952) 37-72; Models of Biological Pattern Formation. H Meinhardt, Academic PressLondonH. Meinhardt, Models of Biological Pattern Formation., Academic Press, London (1982); J D Murray, Mathematical Biology. New YorkSpringer2nd ed.J. D. Murray, Mathematical Biology. 2nd ed. Springer, New York (1989). . H Shoji, Y Iwasa, S Kondo, J. Theor. Biol. 224H. Shoji, Y. Iwasa and S. Kondo, J. Theor. Biol. 224 (2003) 339-350; . H Shoji, Y Iwasa, ; H Shoji, Y Iwasa, A Mori, S Kondo, J. Theor. Biol. 237J. Theor. Biol.H. Shoji and Y. Iwasa, J. Theor. Biol. 237 (2005) 104-116, H. Shoji, Y. Iwasa, A. Mori and S. Kondo, J. Theor. Biol. 214 (2002) 549-561; . H Shoji, Y Iwasa, M Hirata, T Watanabe, S Hioki, Developmental Dynamics. 226H. Shoji, Y. Iwasa, M. Hirata, T. Watanabe and S. Hioki, Developmental Dynamics, 226 (2003) 627-633. . S Takagi, T Tanzawa, Prog. Theor. Physics. 87561S. Takagi and T. Tanzawa, Prog. Theor. Physics. 87 (1992) 561. . K Fujii, N Ogawa, Prog. Theor. Phys. 89575K. Fujii and N. Ogawa, Prog. Theor. Phys. 89 (1993) 575. . R C T Da Costa, Phys. Rev. A. 23R. C. T. da Costa, Phys. Rev. A. 23 (1981) 1982; . J Tolar, Group Theoretical Methods in Physics. 313268SpringerLecture Notes in PhysicsJ. Tolar, 1988 Lecture Notes in Physics 313,Group Theoretical Methods in Physics, (Springer, Berlin, Heidelberg) 268. . N Ogawa, K Fujii, K P Kobushkin, Prog. Theor. Phys. 83894N. Ogawa, K. Fujii, and K. P. Kobushkin, Prog. Theor. Phys. 83 (1990) 894; . N Ogawa, K , N. Ogawa, K. . N M Fujii, K P Chepilko, Kobushkin, Prog. Theor. Phys. 851189Fujii, N. M. Chepilko, and K. P. Kobushkin, Prog. Theor. Phys. 85 (1991) 1189; . N Ogawa, Prog. Theor. Phys. 87513N. Ogawa, Prog. Theor. Phys. 87 (1992) 513.

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{'abstract': "Particle diffusion in a curved tube embedded in R 3 is considered. We find the diffusion coefficient depends on tube's curvature. Diffusion coefficient is obtained in ǫ (radius of tube) expansion.Physical interpretation of curvature dependent diffusion coefficient is given.", 'arxivid': '1109.0590', 'author': ['Naohisa Ogawa \nHokkaido Institute of Technology\n006-8585SapporoJapan\n'], 'authoraffiliation': ['Hokkaido Institute of Technology\n006-8585SapporoJapan'], 'corpusid': 119634192, 'doi': '10.1016/j.physleta.2013.07.054', 'github_urls': [], 'n_tokens_mistral': 4711, 'n_tokens_neox': 3999, 'n_words': 2321, 'pdfsha': 'db35e22da72226a07f12aa8f87805d0dc3ecee7d', 'pdfurls': ['https://arxiv.org/pdf/1109.0590v5.pdf'], 'title': ['Diffusion in Curved Tube', 'Diffusion in Curved Tube'], 'venue': []}

arxiv

Checking Zenon Modulo Proofs in Dedukti * 2015 Raphaël Cauderlier raphael.cauderlier@inria.frpierre.halmagrand@inria.fr Cnam -Inria Paris France Pierre Halmagrand Cnam -Inria Paris France Checking Zenon Modulo Proofs in Dedukti * EPTCS 186201510.4204/EPTCS.186.7c R. Cauderlier & P. Halmagrand This work is licensed under the Creative Commons Attribution License. Dedukti has been proposed as a universal proof checker. It is a logical framework based on the λΠcalculus modulo that is used as a backend to verify proofs coming from theorem provers, especially those implementing some form of rewriting. We present a shallow embedding into Dedukti of proofs produced by Zenon Modulo, an extension of the tableau-based first-order theorem prover Zenon to deduction modulo and typing. Zenon Modulo is applied to the verification of programs in both academic and industrial projects. The purpose of our embedding is to increase the confidence in automatically generated proofs by separating untrusted proof search from trusted proof verification. * This work has received funding from the BWare project (ANR-12-INSE-0010) funded by the INS programme of the French National Research Agency (ANR). Introduction Program verification using deductive methods has become a valued technique among formal methods, with practical applications in industry. It guarantees a high level of confidence regarding the correctness of the developed software with respect to its specification. This certification process is generally based on the verification of a set of proof obligations, generated by deductive verification tools. Unfortunately, the number of proof obligations generated may be very high. To address this issue, deductive verification tools often rely on automated deduction tools such as first-order Automated Theorem Provers (ATP) or Satisfiability Modulo Theories solvers (SMT) to automatically discharge a large number of those proof obligations. For instance, Boogie is distributed with the SMT Z3 [4] and the Why3 platform with Alt-Ergo [16]. After decades of constant work, ATP and SMT have reached a high level of efficiency and now discharge more proof obligations than ever. At the end, many of these program verification tools use their corresponding ATP or SMT as oracles. The main concern here is the level of confidence users give to them. These programs are generally large software, consisting of dozens of thousands of lines of code, and using some elaborate heuristics, with some ad hoc proof traces at best, and with a simple "yes or no" binary answer at worst. A solution, stated by Barendregt and Barendsen [3] and pursued by Miller [19] among others, relies on the concept of proof certificates. ATP and SMT should be seen as proof-certificate generators. The final "yes or no" answer is therefore left to an external proof checker. In addition, Barendregt and Barendsen proposed that proof checkers should satisfy two principles called the De Bruijn criterion and the Poincaré principle. The former states that proof checkers have to be built on a light and auditable kernel. The latter recommends that they distinguish reasoning and computing and that it should not be necessary to record pure computational steps. Relying on an external proof checker to verify proofs strongly increases the trust we give them, but it also provides a common framework to express proofs. A profit made by using this common framework is the possibility to share proofs coming from different theorem provers, relying on different proof systems. But nothing comes for free, and using the same proof checker does not guarantee in general that we can share proofs because formulae and proofs can be translated in incompatible ways. Translation of proofs must rely on a shallow embedding in the sense proposed by Burel [10]: it reuses the features of the target language. It does not introduce new axioms and constants for logical symbols and inference rules. Connectives and binders of the underlying logic of ATP are translated to their corresponding connectives and binders in the target language. In addition, a shallow embedding preserves the computational behavior of the original ATP and the underlying type system of the logic. In this paper, we present a shallow embedding of Zenon Modulo proofs into the proof checker Dedukti, consisting of an encoding of a typed classical sequent calculus modulo into the λΠ-calculus modulo (λΠ ≡ for short). Zenon Modulo [13] is an extension to deduction modulo [15] of the first-order tableau-based ATP Zenon [8]. It has also been extended to support ML polymorphism by implementing the TFF1 format [5]. Dedukti [7] is a proof checker that implements λΠ ≡ , a proof language that has been proposed as a proof standard for proof checking and interoperability. This embedding is used to certify proofs in two different projects: FoCaLiZe [17], a programming environment to develop certified programs and based on a functional programming language with object-oriented features, and BWare [14], an industrial research project that aims at providing a framework for the automated verification of proof obligations coming from the B method [1]. The main benefit of Zenon Modulo and Dedukti relies on deduction modulo. Deduction modulo is an extension of first-order logic that allows reasoning modulo a congruence relation over propositions. It is well suited for automated theorem proving when dealing with theories since it turns axioms into rewrite rules. Using rewrite rules during proof search instead of reasoning on axioms lets provers focus on the challenging part of proofs, speeds up the tool and reduces the size of final proof trees [12]. Zenon was designed to support FoCaLiZe as its dedicated deductive tool and to generate proof certificates for Coq. Extension to deduction modulo constrains us to use a proof checker that can easily reason modulo rewriting. Dedukti is a good candidate to meet this specification. A previous embedding of Zenon Modulo proofs into Dedukti, based on a ¬¬ translation [13], was implemented as a tool to translate classical proofs into constructive ones. This tool has the benefit to be shallower since it does not need add the excluded middle as an axiom into the target logic defined in Dedukti, but in return this transformation may be very time-consuming [12] and was not scalable to large proofs like those produced in BWare. The closest related work is the shallow embedding of resolution and superposition proofs into Dedukti proposed by Burel [10] and implemented in iProver Modulo [9]. Our embedding is close enough to easily share proofs of Zenon Modulo and iProver Modulo in Dedukti, at least for the subset of untyped formulae. The first contribution presented in this paper consists in the encoding into λΠ ≡ of typed deduction modulo and a set of translation functions into λΠ ≡ of theories expressed in this logic. Another contribution of this paper is the extension to deduction modulo and types of the sequent-like proof system LLproof which is the output format of Zenon Modulo proofs. The latter contribution is the embedding of this proof system into λΠ ≡ and the associated translation function for proofs coming from this system. This paper is organized as follows: in Sec. 2, we introduce typed deduction modulo; in Sec. 3, we present λΠ ≡ , its proof checker Dedukti, and a canonical encoding of typed deduction modulo in λΠ ≡ ; Sec. 4 introduces the ATP Zenon Modulo, the proof system LLproof used by Zenon Modulo to output proofs; and the translation scheme implemented as the new output of Zenon Modulo; finally, in Sec. 5, we present some examples and results to assess our implementation. Typed Deduction Modulo The Poincaré principle, as stated by Barendregt and Barendsen [3], makes a distinction between deduction and computation. Deduction may be defined using a set of inference rules and axioms, while computation consists mainly in simplification and unfolding of definitions. When dealing with axiomatic theories, keeping all axioms on the deduction side leads to inefficient proof search since the proof-search space grows with the theory. For instance, proving the following statement: fst(a, a) = snd(a, a) where a is a constant, and fst and snd are defined by: ∀x, y. fst(x, y) = x ∀x, y. snd(x, y) = y and with the reflexivity axiom: ∀x. x = x using a usual automated theorem proving method such as tableau, will generate some useless boilerplate proof steps, whereas a simple unfolding of definitions of fst and snd directly leads to the formula a = a. Deduction modulo was introduced by Dowek, Hardin and Kirchner [15] as a logical formalism to deal with axiomatic theories in automated theorem proving. The proposed solution is to remove computational arguments from proofs by reasoning modulo a decidable congruence relation ≡ on propositions. Such a congruence may be generated by a confluent and terminating system of rewrite rules (sometimes extended by equational axioms). In our example, the two definitions may be replaced by the rewrite rules: fst(x, y) −→ x snd(x, y) −→ y And we obtain the following equivalence between propositions: (fst(a, a) = snd(a, a)) ≡ (a = a) Reasoning with several theories at the same time is often necessary in practice. For instance, in the BWare project, almost all proof obligations combine the theory of booleans, arithmetic and set theory. In this case, we have to introduce an expressive enough type system to ensure that an axiom about booleans, for instance ∀x. x = true ∨ x = false, will not be used with a term that has another type. An input format for ATP called TFF1 [5] has been proposed recently by Blanchette and Paskevich to deal with first-order problems with polymorphic types. We propose to extend this format to deduction modulo. We now introduce the notion of typed rewrite system, extending notations of Dowek et al. [15]. In the following, FV(t) stands for the set of free variables of t where t is either a TFF1 term or a TFF1 formula. Definition (Typed Rewrite System) A term rewrite rule is a pair of TFF1 terms l and r together with a TFF1 typing context ∆ denoted by l −→ ∆ r, where FV(r) ⊆ FV(l) ⊆ ∆. It is well-typed in a theory T if l and r can be given the same type A in T using ∆ to type free variables. A proposition rewrite rule is a pair of TFF1 formulae l and r together with a typing context ∆ denoted by l −→ ∆ r, where l is an atomic formula and r is an arbitrary formula, | ¬ϕ | ϕ 1 ∧ ϕ 2 | ϕ 1 ∨ ϕ 2 | ϕ 1 ⇒ ϕ 2 | ϕ 1 ⇔ ϕ 2 (logical connectors) | e 1 = τ e 2 (term equality) | P(τ 1 , . . . , τ m ; e 1 , . . . , e n ) (predicate) | ∀x : τ. ϕ(x) | ∃x : τ. ϕ(x) (term quantifiers) | ∀ type α : type. ϕ(α) | ∃ type α : type. ϕ(α) (type quantifiers) Context ∆ ::= / 0 (empty context) | ∆, x : τ (declaration) Theory T ::= / 0 (empty theory) | T , T/m (m-ary type constructor declaration | T , f : Π α. τ → τ (function declaration) | T , P : Π α. τ → o (predicate declaration) | T , name : ϕ (axiom) | T , l −→ ∆ r (rewrite rule) Figure 1: Syntax of TFF1 ≡ and where FV(r) ⊆ FV(l) ⊆ ∆. It is well-typed in a theory T if both l and r are well-formed formulae in T using ∆ to type free variables. A typed rewrite system is a set R of proposition rewrite rules along with a set E of term rewrite rules. Given a rewrite system RE , the relation = RE denotes the congruence generated by RE . It is well-formed in a theory T , if all its rewrite rules are well-typed in T . The notion of TFF1 theory can be extended with rewrite rules; we call the resulting logic TFF1 ≡ . Its syntax is given in Fig. 1. Dedukti The λΠ-calculus [2] is the simplest Pure Type System featuring dependent types. It is commonly used as a logical framework for encoding logics [18]. The λΠ-calculus modulo, presented in Fig. 2, is an extension of the λΠ-calculus with rewriting. The λΠ-calculus modulo (abbreviated as λΠ ≡ ) has successfully been used to encode many logical systems (Coq [6], HOL, iProver Modulo [10], FoCaLiZe) using shallow embeddings. In λΠ ≡ , conversion goes beyond simple β-equivalence since it is extended by a custom rewrite system. When this rewrite system is both strongly normalizing and confluent, each term gets a unique (up to α-conversion) normal form and both conversion and type-checking become decidable. Dedukti is an implementation of this decision procedure. Burel [10] defines two encodings of deduction modulo in Dedukti: a deep encoding |ϕ| in which logical connectives are simply declared as Dedukti constants and a shallow encoding ϕ := prf |ϕ| using a decoding function prf for translating connectives to their impredicative encodings. In Sec. 3.1 and Sec. 3.2, we extend these encodings to TFF1 ≡ . Syntax s ::= Type | Kind t ::= x | t t | λx : t.t | Πx : t.t | s ∆ ::= / 0 | ∆, x : t Γ ::= / 0 | Γ, x : t | Γ, t ֒→ ∆ t Well-formdness (Empty) / 0 ⊢ Γ ⊢ Γ ⊢ A : s x / ∈ Γ (Decl) Γ, x : A ⊢ Γ, ∆ ⊢ l : A Γ, ∆ ⊢ r : A Γ, ∆ ⊢ A : Type FV(r) ⊆ FV(l) ⊆ ∆ (Rew) Γ, l ֒→ ∆ r ⊢ Typing Γ ⊢ (Sort) Γ ⊢ Type : Kind Γ ⊢ x : A ∈ Γ (Var) Γ ⊢ x : A Γ ⊢ t 1 : Πx : A.B(x) Γ ⊢ t 2 : A (App) Γ ⊢ t 1 t 2 : B(t 1 ) Γ ⊢ A : Type Γ, x : A ⊢ t : B(x) Γ, x : A ⊢ B(x) : s (Abs) Γ ⊢ λx : A.t(x) : Πx : A.B(x) Γ ⊢ A : Type Γ, x : A ⊢ B(x) : s (Prod) Γ ⊢ Πx : A.B(x) : s Γ ⊢ t : A Γ ⊢ B : s A ≡ βΓ B (Conv) Γ ⊢ t : B Deep Embedding of Typed Deduction Modulo in Dedukti In Fig. 3, for each symbol of our first-order typed logic, we declare its corresponding symbol into λΠ ≡ . In λΠ ≡ , types cannot be passed as arguments (no polymorphism) so we have to translate TFF1 ≡ types as Dedukti terms. The Dedukti type of translated TFF1 ≡ types is type and we can see an inhabitant of type as a Dedukti type thanks to the term function. In Fig. 4, we define a direct translation of TFF1 ≡ in Dedukti. It is correct in the following sense: • if the theory T is well-formed in TFF1 ≡ , then |T | ⊢. • if τ is a well-formed TFF1 ≡ type in a theory T , then |T | ⊢ |τ| : type. • if t is a TFF1 ≡ term of type τ in a theory T , then |T | ⊢ |t| : term |τ|. • if ϕ is a well-formed TFF1 ≡ formula in a theory T , then |T | ⊢ |ϕ| : Prop. From Deep to Shallow Following Burel [10], we add rewrite rules defining the decoding function prf in Fig. 5 using the usual impredicative encoding of connectives. This transforms our deep encoding of TFF1 ≡ into a shallow encoding in which all connectives are defined by the built-in constructions of λΠ ≡ . This encoding is better suited for sharing proofs with other ATP because it is less sensible to small modifications of the logic. Any proof found, for example, by iProver Modulo is directly usable as an (untyped) proof in the shallow encoding. Primitive Types Prop : Type prf : Prop → Type type : Type term : type → Type term Primitive Connectives ⊤ : Prop ⊥ : Prop ¬-: Prop → Prop -∧ -: Prop → Prop → Prop -∨ -: Prop → Prop → Prop -⇒ -: Prop → Prop → Prop -⇔ -: Prop → Prop → Prop ∀ --: Πα : type.(term α → Prop) → Prop ∀ type -: (type → Prop) → Prop ∃ --: Πα : type.(term α → Prop) → Prop ∃ type -: (type → Prop) → Prop -= --: Πα : type.term α → term α → Prop|x| := x | f (τ 1 , . . . , τ m ; e 1 , . . . , e n )| := f |τ 1 | . . . |τ m | |e 1 | . . . |e n | Translation Function for Formulae |⊤|:= ⊤ |⊥|:= ⊥ |¬ϕ| := ¬|ϕ| |ϕ 1 ∧ ϕ 2 |:= |ϕ 1 | ∧ |ϕ 2 | |ϕ 1 ∨ ϕ 2 | := |ϕ 1 | ∨ |ϕ 2 | |ϕ 1 ⇒ ϕ 2 |:= |ϕ 1 | ⇒ |ϕ 2 | |ϕ 1 ⇔ ϕ 2 | := |ϕ 1 | ⇔ |ϕ 2 | |e 1 = τ e 2 |:= |e 1 | = |τ| |e 2 | |∀x : τ. ϕ| := ∀ |τ| (λx : term |τ| . |ϕ|) |∃x : τ. ϕ| := ∃ |τ| (λx : term |τ| . |ϕ|) |∀ type α : type. ϕ| := ∀ type (λα : type . |ϕ|) |∃ type α : type. ϕ| := ∃ type (λα : type . |ϕ|) |P(τ 1 , . . . , τ m ; e 1 , . . . , e n )| := P |τ 1 | . . . |τ m | |e 1 | . . .|τ 1 | → . . . → term |τ n | → Prop |T , name : ϕ| := |T | , name : prf |ϕ| |T , l −→ ∆ r| := |T | , |l| ֒→ ∆ |r| Zenon Modulo Zenon Modulo [13] is an extension to deduction modulo [15] of the first-order tableau-based automated theorem prover Zenon [8]. It has also been improved to deal with typed formulae and TFF1 input files. In this paper, we focus on the output format of Zenon Modulo. After finding a proof using its tableau-based proof-search algorithm [8], Zenon translates its proof tree into a low level format called LLproof, which is a classical sequent-like proof system. This format is used for Zenon proofs before their automatic translation to Coq. LLproof is a one-sided sequent calculus with explicit contractions in every inference rule, which is close to an upside-down non-destructive tableau method. prf ⊤ ֒→ ΠP : Prop. prf P → prf P prf ⊥ ֒→ ΠP : Prop. prf P prf (¬A) ֒→ prf A → prf ⊥ prf (A ∧ B) ֒→ ΠP : Prop. (prf A → prf B → prf P) → prf P prf (A ∨ B) ֒→ ΠP : Prop. (prf A → prf P) → (prf B → prf P) → prf P prf (A ⇒ B) ֒→ prf A → prf B prf (A ⇔ B) ֒→ prf ((A ⇒ B) ∧ (B ⇒ A)) prf (∀ τ P) ֒→ Πx : term τ. prf (P x) prf (∀ type P)֒→ Πα : type. prf (P α) prf (∃ τ P) ֒→ ΠP : Prop. (Πx : term τ. prf (P x) → prf P) → prf P prf (∃ type P)֒→ ΠP : Prop. (Πα : type. prf (P α) → prf P) → prf P prf (x = τ y) ֒→ ΠP : (term τ → Prop). prf (P x) → prf (P y) We present in Figs. 6 and 7 the new proof system LLproof ≡ , an adaptation of Zenon output format LLproof [8] to deduction modulo and TFF1 typing. Normalization and deduction steps may interleave anywhere in the final proof tree. This leads to the introduction of the congruence relation = RE inside rules of Figs. 6 and 7: if the formula P is in normal form (with respect to RE ), we denote by [P] any formula congruent to P modulo = RE . Extension of LLproof to TFF1 typing leads to the introduction of four new rules for quantification over type variables ∃ type , ¬∀ type , ∀ type and ¬∃ type , and also to introduce some type information into Closure and Quantifier-free Rules other rules dealing with equality or quantification. For instance, equality of two closed terms t and u, both of type τ, is denoted by t = τ u. For predicate and function symbols, we first list types, then terms, separated by a semi-colon. Finally, last difference regarding rules presented in [8] is the removal of rules "definition" and "lemma". Zenon Modulo, unlike Zenon, does not need to explicitly unfold definitions and the lemma constructions have been removed. ⊥ Γ, [⊥] ⊢ ⊥ ¬⊤ Γ, [¬⊤] ⊢ ⊥ Ax Γ, [P], [¬P] ⊢ ⊥ Γ, P ⊢ ⊥ Γ, ¬P ⊢ ⊥ Cut Γ ⊢ ⊥ = Γ, [t = τ t] ⊢ ⊥ Sym Γ, [t = τ u], [u = τ t] ⊢ ⊥ Γ, ¬¬P, P ⊢ ⊥ ¬¬ Γ, [¬¬P] ⊢ ⊥ Γ, P ∧ Q, P, Q ⊢ ⊥ ∧ Γ, [P ∧ Q] ⊢ ⊥ Γ, P ∨ Q, P ⊢ ⊥ Γ, P ∨ Q, Q ⊢ ⊥ ∨ Γ, [P ∨ Q] ⊢ ⊥ Γ, P ⇒ Q, ¬P ⊢ ⊥ Γ, P ⇒ Q, Q ⊢ ⊥ ⇒ Γ, [P ⇒ Q] ⊢ ⊥ Γ, P ⇔ Q, ¬P, ¬Q ⊢ ⊥ Γ, P ⇔ Q, P, Q ⊢ ⊥ ⇔ Γ, [P ⇔ Q] ⊢ ⊥ Γ, ¬(P ∧ Q), ¬P ⊢ ⊥ Γ, ¬(P ∧ Q), ¬Q ⊢ ⊥ ¬∧ Γ, [¬(P ∧ Q)] ⊢ ⊥ Γ, ¬(P ∨ Q), ¬P, ¬Q ⊢ ⊥ ¬∨ Γ, [¬(P ∨ Q)] ⊢ ⊥ Γ, ¬(P ⇒ Q), P, ¬Q ⊢ ⊥ ¬ ⇒ Γ, [¬(P ⇒ Q)] ⊢ ⊥ Γ, ¬(P ⇔ Q), ¬P, Q ⊢ ⊥ Γ, ¬(P ⇔ Q), P, ¬Q ⊢ ⊥ ¬ ⇔ Γ, [¬(P ⇔ Q)] ⊢ ⊥Γ, ∃x : τ. P(x), P(c) ⊢ ⊥ ∃ Γ, [∃x : τ. P(x)] ⊢ ⊥ Γ, ¬∀x : τ. P(x), ¬P(c) ⊢ ⊥ ¬∀ Γ, [¬∀x : τ. P(x)] ⊢ ⊥ where c : τ is a fresh constant Γ, ∀x : τ. P(x), P(t) ⊢ ⊥ ∀ Γ, [∀x : τ. P(x)] ⊢ ⊥ Γ, ¬∃x : τ. P(x), ¬P(t) ⊢ ⊥ ¬∃ Γ, [¬∃x : τ. P(x)] ⊢ ⊥ where t : τ is any closed term Special Rules ∆, t 1 = τ ′ 1 u 1 ⊢ ⊥ . . . ∆, t n = τ ′ n u n ⊢ ⊥ Pred Γ, [P(τ 1 , . . . , τ m ; t 1 , . . . , t n )], [¬P(τ 1 , . . . , τ m ; u 1 , . . . , u n )] ⊢ ⊥ where ∆ = Γ ∪ {P(τ 1 , . . . , τ m ; t 1 , . . . , t n ), ¬P(τ 1 , . . . , τ m ; u 1 , . . . , u n )} ∆, t 1 = τ ′ 1 u 1 ⊢ ⊥ . . . ∆, t n = τ ′ n u n ⊢ ⊥ Fun Γ, [ f (τ 1 , . . . , τ m ; t 1 , . . . , t n ) = τ f (τ 1 , . . ., τ m ; u 1 , . . . , u n )] ⊢ ⊥ where ∆ = Γ ∪ { f (τ 1 , . . . , τ m ; t 1 , . . . , t n ) = τ f (τ 1 , Translation of Zenon Modulo Proofs into λΠ ≡ We present in Fig. 8 a deep embedding of LLproof ≡ into λΠ ≡ . We declare a constant for each inference rule, except for special rules Pred and Fun which have a dependency on the arity n of their underlying predicate and function. Fortunately, they can be expressed with the following Subst inference rule which corresponds to the substitution in a predicate P of a subterm t : τ ′ by another u : τ ′ : Γ, P( τ; t), t = τ ′ u ⊢ ⊥ Γ, P( τ; t), P( τ; u) ⊢ ⊥ Subst Γ, P( τ; t) ⊢ ⊥ Zenon Modulo Rules R ⊥ : prf ⊥ → prf ⊥ R ¬⊤ : prf (¬⊤) → prf ⊥ R Ax : ΠP : Prop. prf P → prf (¬P) → prf ⊥ R Cut : ΠP : Prop. (prf P → prf ⊥) → (prf (¬P) → prf ⊥) → prf ⊥ R = : Πα : type. Πt : term α. prf (t = α t) → prf ⊥ R S ym : Πα : type. Πt, u : term α. prf (t = α u) → prf (u = α t) → prf ⊥ R ¬¬ : ΠP : Prop. (prf P → prf ⊥) → prf (¬¬P) → prf ⊥ R ∧ : ΠP, Q : Prop. (prf P → prf Q → prf ⊥) → prf (P ∧ Q) → prf ⊥ R ∨ : ΠP, Q : Prop. (prf P → prf ⊥) → (prf Q → prf ⊥) → prf (P ∨ Q) → prf ⊥ R ⇒ : ΠP, Q : Prop. (prf (¬P) → prf ⊥) → (prf Q → prf ⊥) → prf (P ⇒ Q) → prf ⊥ R ⇔ : ΠP, Q : Prop. (prf (¬P) → prf (¬Q) → prf ⊥) → (prf P → prf Q → prf ⊥) → prf (P ⇔ Q) → prf ⊥ R ¬∧ : ΠP, Q : Prop. (prf (¬P) → prf ⊥) → (prf (¬Q) → prf ⊥) → prf (¬(P ∧ Q)) → prf ⊥ R ¬∨ : ΠP, Q : Prop. (prf (¬P) → prf (¬Q) → prf ⊥) → prf (¬(P ∨ Q)) → prf ⊥ R ¬⇒ : ΠP, Q : Prop. (prf P → prf (¬Q) → prf ⊥) → prf (¬(P ⇒ Q)) → prf ⊥ R ¬⇔ : ΠP, Q : Prop. (prf (¬P) → prf Q → prf ⊥) → (prf P → prf (¬Q) → prf ⊥) → prf (¬(P ⇔ Q)) → prf ⊥ R ∃ : Πα : type. ΠP : (term α → Prop). (Πt : term α. (prf (P t) → prf ⊥)) → prf (∃ α P) → prf ⊥ R ∀ : Πα : type. ΠP : (term α → Prop). Πt : term α. (prf (P t) → prf ⊥) → prf (∀ α P) → prf ⊥ R ¬∃ : Πα : type. ΠP : (term α → Prop). Πt : term α. (prf (¬(P t)) → prf ⊥) → prf (¬(∃ α P) → prf ⊥ R ¬∀ : Πα : type. ΠP : (term α → Prop). (Πt : term α. (prf (¬(P t)) → prf ⊥)) → prf (¬∀ α P) → prf ⊥ R ∃ type : ΠP : (type → Prop). (Πα : type. (prf (P α) → prf ⊥)) → prf (∃ type P) → prf ⊥ R ∀ type : ΠP : (type → Prop). Πα : type. (prf (P α) → prf ⊥) → prf (∀ type f ) → prf ⊥ R ¬∃ type : ΠP : (type → Prop). Πα : type. (prf (¬(P α)) → prf ⊥) → prf (¬(∃ type P)) → prf ⊥ R ¬∀ type : ΠP : (type → Prop). (Πα : type. (prf (¬(P α)) → prf ⊥)) → prf (¬(∀ type P)) → prf ⊥ R S ubst : Πα : type. ΠP : (term α → Prop). Πt, u : term α. (prf (t = α u) → prf ⊥) → (prf (P u) → prf ⊥) → prf (P t) → prf ⊥ Figure 8: LLproof ≡ in λΠ ≡ The special rules Pred and Fun can be easily decomposed into n applications of the Subst rule. For instance, for a binary predicate P, from (we omit to repeat the context Γ) Fig. 9, we present the translation function for LLproof ≡ sequents and proofs into λΠ ≡ . Let us present a simple example. We want to translate this proof tree: where Π P and Π Q are respectively proofs of sequents Γ, P ⊢ ⊥ and Γ, Q ⊢ ⊥, and where we annotate rule names with its parameters. Then, by applying the translation procedure of Figs. 4 and 9, we obtain the Dedukti term Π 1 t 1 = τ ′ u 1 ⊢ ⊥ Π 2 t 2 = τ ′′ u 2 ⊢ ⊥ Pred P( τ; t 1 , t 2 ), ¬P( τ; u 1 , u 2 ) ⊢ ⊥ we obtain Π 1 t 1 = τ ′ u 1 ⊢ ⊥ Π 2 t 2 = τ ′′ u 2 ⊢ ⊥ Ax P( τ; u 1 , u 2 ) Subst P( τ; u 1 , t 2 ) Subst P( τ; t 1 , t 2 ), ¬P( τ; u 1 , u 2 ) ⊢ ⊥ InΠ := Π P Γ, P ∨ Q, P ⊢ ⊥ Π Q Γ, P ∨ Q, Q ⊢ ⊥ ∨(P,Q) Γ, P ∨ Q ⊢ ⊥R ∨ |P| |Q| (λx P : prf |P| . |Π P |) (λx Q : prf |Q| . |Π Q |) x P∨Q where the notation |x| means the translation of x into λΠ ≡ , and x P is a variable declared of type prf |P|. We then check that Π is a proof of the sequent Γ, P ∨ Q ⊢ ⊥ in a TFF1 ≡ theory T , by checking that |T | , |Γ, P ∨ Q| ⊢ |Π| : prf ⊥ in λΠ ≡ . More generally, for any LLproof ≡ proof Π and any sequent Γ ⊢ ⊥, we check that Π is a proof of Γ ⊢ ⊥ by checking the λΠ ≡ typing judgment |T | , |Γ| ⊢ |Π| : prf ⊥. Shallow Embedding of LLproof ≡ The embedding of LLproof ≡ presented in Fig. 8 can also be lifted to a shallow embedding. In Fig. 13 of Appendix A, we present rewrite rules that prove all constants corresponding to LLproof ≡ inference rules into the logic presented in Sec. 3. This has been written in Dedukti syntax and successfully checked by Dedukti (see the file modulogic.dk distributed with the source code of Zenon Modulo 1 ). The only remaining axiom is the law of excluded middle. This shows the soundness of LLproof ≡ relatively to the consistency of the logic of Sec. 3. Experimental Results Zenon Modulo helps to automatically discharge proof obligations in particular in the two projects Fo-CaLiZe [17] and BWare [14]. We present in this section some examples of theories, and simple related properties, that are handled successfully by Zenon Modulo, and its translation to Dedukti. Declarations bool/0 false : bool true : bool ∼ : bool → bool -&& -: bool → bool → bool -|| -: bool → bool → bool if --then -else -: Πα. bool → α → α → α Rewrite rules true && a −→a true || a −→true a && true −→a a || true −→true false && a −→false false || a −→a a && false −→false a || false −→a a && a −→a a || a −→a a && (b && c) −→(a && b) && c a || (b || c) −→(a || b) || c ∼ true −→false a && (b || c) −→(a && b) || (a && c) ∼ false −→true (a || b) && c −→(a && c) || (b && c) ∼ (∼ a) −→a ∼ (a || b) −→(∼ a) && (∼ b) if α true then t else e −→t ∼ (a && b) −→(∼ a) || (∼ b) if α false then t else e −→e Extension deduction rules Γ, ¬P(true) ⊢ ⊥ Γ, ¬P(false) ⊢ ⊥ Ext(bool-case-¬∀, P) Γ, [¬∀b : bool. P(b)] ⊢ ⊥ Γ, P(true) ⊢ ⊥ Γ, P(false) ⊢ ⊥ Ext(bool-case-∃, P) Γ, [∃b : bool. P(b)] ⊢ ⊥ Application to FoCaLiZe FoCaLiZe is a framework for specifying, developing and certifying programs. The specification language is first-order logic and proofs can be discharged to Zenon or Zenon Modulo. The FoCaLiZe compiler produces both a regular program written in OCaml and a certificate written either in Coq or in Dedukti (but only the Dedukti output can be used from Zenon Modulo). In FoCaLiZe, specifications usually rely a lot on the primitive type bool so it is important that Zenon Modulo deals with booleans efficiently. In order to prove all propositional tautologies, it is enough to add the following rules for reasoning by case on booleans (together with truth tables of connectives): Γ, ¬P(true) ⊢ ⊥ Γ, ¬P(false) ⊢ ⊥ Ext(bool-case-¬∀, P) Γ, [¬∀b : bool. P(b)] ⊢ ⊥ Γ, P(true) ⊢ ⊥ Γ, P(false) ⊢ ⊥ Ext(bool-case-∃, P) Γ, [∃b : bool. P(b)] ⊢ ⊥ However, we get a much smaller proof-search space and smaller proofs by adding common algebraic laws as rewrite rules. In Fig. 10, we define a theory of booleans in TFF1 ≡ . This theory handles idempotency and associativity of conjunction and disjunction but not commutativity because the rule λx 1 : prf(¬(∀ bool (λx : term bool. ∀ bool (λy : term bool. x && y = bool y && x)))). R bool-case-¬∀ (λx : term bool. ∀ bool (λy : term bool. x && y = bool y && x)) (λx 2 : prf(¬(∀ bool (λy : term bool. y = bool y))). R ¬∀ bool (λy : term bool. y = bool y) (λa : term bool. λx 3 : prf(a = bool a). R = bool a x 3 ) x 2 ) (λx 4 : prf(¬(∀ bool (λy : term bool. false = bool false))). R ¬∀ bool (λy : term bool. false = bool false) (λa : term bool. Application to Set Theory The BWare project is an industrial research project that aims to provide a framework to support the automated verification of proof obligations coming from the development of industrial applications using the B method [1]. The B method relies on a particular set theory with types. In the context of the BWare project, this typed set theory has been encoded into WhyML, the native language of Why3 [16]. To call Zenon Modulo, Why3 translates proof obligations and the B theory into TFF1 format. If it succeeds in proving the proof obligation, Zenon Modulo produces a proof certificate containing both the theory and the term, following the model presented in Fig. 12. The BWare project provides a large benchmark made of 12,876 proof obligations coming from industrial projects. The embedding presented in this paper allowed us to verify with Dedukti all the 10,340 proof obligations that are proved by Zenon Modulo. Let us present a small subset of this set theory, and a simple example of LLproof ≡ proof produced by Zenon Modulo. The theory consists of three axioms that have been turned into rewrite rules. We define constructors: a type constructor set, the membership predicate ∈, equality on sets = set , the empty set / 0 and difference of sets −. For readability, we use an infix notation and let type parameters of functions and predicates in subscript. We want to prove the property ∀ type α : type. ∀s : setα. s − α s = setα / 0 α set-: type → type -∈ --: Πα : type. (term setα) → (term setα) → Prop -= set--: Πα : type. (term setα) → (term setα) → Prop / 0 -: Πα : type. (term setα) -− --: Πα : type. (term setα) → (term setα) → (term setα) s = setα t ֒→ ∀ (α) (λx : (term α). x ∈ α s ⇔ x ∈ α t) x ∈ α / 0 α ֒→ ⊥ x ∈ α s − α t ֒→ x ∈ α s ∧ x ∈ α t Goal : prf (¬(∀ type (λα : type. (∀ (setα) (λs : (term setα). s − α s = setα / 0 α )))) → prf ⊥) [] Goal ֒→ λx 1 : prf (¬(∀ type (λα : type. (∀ (setα) (λs : (term setα). s − α s = setα / 0 α ))))). R ¬∀ type (λα : type. (∀ (setα) λs : (term setα). s − α s = setα / 0 α )) (λτ : type. λx 2 : prf (¬(∀ (setτ) (λs : (term setτ). s − τ s = setτ / 0 τ ))). R ¬∀ (setτ) (λs : (term setτ). s − τ s = setτ / 0 τ ) (λc 1 : (term setτ). λx 3 : prf (c 1 − τ c 1 = setτ / 0 τ ). R ¬∀ (τ) (λx : (term τ). (x ∈ τ c 1 − τ c 1 ) ⇔ (x ∈ τ / 0 τ )) (λc 2 : (term τ). λx 4 : prf (¬((c 2 ∈ τ c 1 − τ c 1 ) ⇔ (c 2 ∈ τ / 0 τ ))). R ¬⇔ (c 2 ∈ τ c 1 − τ c 1 ) (c 2 ∈ τ / 0 τ ) (λx 5 : prf (¬(c 2 ∈ τ c 1 − τ c 1 )). λx 6 : prf (c 2 ∈ τ / 0 τ ). R ⊥ x 6 ) (λx 7 : prf (c 2 ∈ τ c 1 − τ c 1 ). λx 8 : prf (¬(c 2 ∈ τ / 0 τ )). R ∧ (c 2 ∈ τ c 1 ) (c 2 ∈ τ c 1 ) (λx 9 : prf (c 2 ∈ τ c 1 ). λx 9 : prf (c 2 ∈ τ c 1 ). R Ax (c 2 ∈ τ c 1 ) x 8 x 9 ) x 7 ) x 4 ) x 3 ) x 2 ) x 1 )set-/1 -∈ --:Πα. setα → setα → Prop -= set--:Πα. setα → setα → Prop / 0 -:Πα. setα -− --:Πα. setα → setα → setα s = setα t−→∀x : α. x ∈ α s ⇔ x ∈ α t x ∈ α / 0 α −→⊥ x ∈ α s − α t−→ x ∈ α s ∧ x ∈ α t The LLproof proof tree generated by Zenon Modulo is (we omit to repeat context Γ): ⊥ ¬(c 2 ∈ τ c 1 − τ c 1 ), c 2 ∈ τ / 0 τ ⊢ ⊥ Ax c 2 ∈ τ c 1 , c 2 ∈ τ c 1 ⊢ ⊥ ∧ c 2 ∈ τ c 1 − τ c 1 , ¬(c 2 ∈ τ / 0 τ ) ⊢ ⊥ ¬ ⇔ ¬((c 2 ∈ τ c 1 − τ c 1 ) ⇔ (c 2 ∈ τ / 0 τ )) ⊢ ⊥ ¬∀ ¬(c 1 − τ c 1 = setτ / 0 τ ) ⊢ ⊥ ¬∀ ¬(∀s : setτ. s − τ s = setτ / 0 τ ) ⊢ ⊥ ¬∀ type ¬(∀ type α : type. ∀s : setα. s − α s = setα / 0 α ) ⊢ ⊥ We obtain the proof certificate of Fig. 12 checkable by Dedukti, using the file modulogic.dk, and that is successfully checked. Conclusion We have presented a shallow embedding of Zenon Modulo proofs into Dedukti. For this encoding, we have needed to embed into λΠ ≡ an extension to deduction modulo of the underlying logic of the TFF1 format, denoted by TFF1 ≡ . We then defined LLproof ≡ , the extension to TFF1 ≡ of the proof system LLproof, which is the output format of Zenon Modulo. Finally, we have embedded LLproof ≡ into λΠ ≡ by giving the translation function for proofs. This embedding is shallow in the sense that we have reused the features of the target language and have not declared new constants for connectives and inference rules. The only axiom that we have added is the law of excluded middle. This embedding has helped us to verify a large set of proof obligations coming from two different projects. FoCaLiZe can now benefit from deduction modulo to improve program verification when dealing with theories. In BWare, this work allowed us to certify all the proofs generated by Zenon Modulo. Our work is closely related to the embedding of iProver Modulo proofs into Dedukti [10]. The two main differences are the assumption of the excluded middle and the extension of the logic to deal with ML-style polymorphism. Because these shallow encodings are close, we could easily share proofs of untyped formulae with iProver Modulo. We do not have to trust the full implementation of Zenon Modulo but only the translation of TFF1 ≡ problems to λΠ ≡ discussed in Sec. 3 and, of course, Dedukti. In the case of FoCaLiZe, we go even further by using an external translator, Focalide [11]. Hence Zenon Modulo requires no confidence in that context. As future work, we want export this model. To achieve that, deduction tools must be able to read Dedukti in addition to write some. This model improves the confidence on automated deduction tools because it is no more possible to introduce inconsistency inside a proof certificate. In addition, in case of the verification of several formulae, it should be possible to inject terms coming from different tools inside the same Dedukti file. A first experiment with Zenon Modulo and iProver Modulo in FoCaLiZe would be an interesting proof of concept. λH 10 : (prf (P ⇒ Q) → prf (Q ⇒ P) → prf Z). H 10 (λH 11 : prf P. H 9 ) (λH 12 : prf Q. H 8 )))) [α : type, P : term α → Prop] R ∃ α P ֒→ λH 1 : (t : term α → prf (P t) → prf ⊥). λH 2 : prf (∃ α P). H 2 ⊥ H 1 [α : type, P : term α → Prop, t : term α] R ∀ α P t ֒→ λH 1 : (prf (P t) → prf ⊥). λH 2 : prf (∀ α P). H 1 (H 2 t) [α : type, P : term α → Prop, t : term α] R ¬∃ α P t ֒→ λH 1 : (prf (¬(P t)) → prf ⊥). λH 2 : prf (¬(∃ α P)). H 1 (λH 4 : prf (P t). H 2 (λZ : Prop. λH 3 : (x : term α → prf (P x) → prf Z). H 3 t H 4 )) [α : type, P : term α → Prop] R ¬∀ α P ֒→ λH 1 : (t : term α → prf (¬(P t)) → prf ⊥). λH 2 : prf (¬(∀ α P)). H 2 (λt : term α. NNPP (P t) (H 1 t)) [P : type → Prop] R ∃ type P ֒→ λH 1 : (α : type → prf (P α) → prf ⊥). λH 2 : prf (∃ type P). H 2 ⊥ H 1 [P : type → Prop, α : type] R ∀ type P α ֒→ λH 1 : (prf (P α) → prf ⊥). λH 2 : prf (∀ type P). H 1 (H 2 α) [P : type → Prop, α : type] R ¬∃ type P α ֒→ λH 1 : (prf (¬(P α)) → prf ⊥). λH 2 : prf (¬(∃ type P)). H 1 (λH 4 : prf (P α). H 2 λZ : Prop. λH 3 : (β : type → prf (P β) → prf Z). H 3 α H 4 ) [P : type → Prop] R ¬∀ type P ֒→ λH 1 : (α : type → prf (¬(P α)) → prf ⊥). λH 2 : prf (¬(∀ type P)). H 2 (λα : type. NNPP (P α) (H 1 α)) [α : type, P : term α → Prop, t 1 : term α, t 2 : term α] R S ubst α P t 1 t 2 ֒→ λH 1 : (prf (t 1 = α t 2 ) → prf ⊥). λH 2 : (prf (P t 2 ) → prf ⊥). The deep embedding of LLproof ≡ presented in Sec. 4.1 is well-typed with respect to the deep em- Figure 2 : 2The λΠ-calculus modulo Figure 3 : 3Dedukti Declarations of TFF1 ≡ Symbols Translation Function for Types |α| := α |T (τ 1 , . . . , τ m )| := T |τ 1 | . . . |τ m | Translation Function for Terms |∆, x : τ| := |∆| , x : term |τ| Translation Function for Theories | / 0|:= Γ 0 where Γ 0 is the Dedukti context of Fig. 3 |T , T/m|:= |T | , T : m times type → . . . → type → type |T , f : Π(α 1 , . . . , α m ). (τ 1 , . . . , τ n ) → τ| := |T | , f : Πα 1 : type. . . . Πα m : type. term |τ 1 | → . . . → term |τ n | → |τ| |T , P : Π(α 1 , . . . , α m ). (τ 1 , . . . , τ n ) → o|:= |T | , P : Πα 1 : type. . . . Πα m : type. Figure 4 : 4Translation Functions from TFF1 ≡ to λΠ ≡ Figure 5 : 5Shallow Definition of Logical Connectives in Dedukti Figure 6 : 6LLproof ≡ Inference Rules of Zenon Modulo (part 1) Figure 7 : 7. . . , τ m ; u 1 , . . . , u n )} ∆, H 11 , . . . , H 1m ⊢ ⊥ . . . ∆, H n1 , . . . , H nq ⊢ ⊥ Ext(name, args,C 1 , . . . ,C p , H 11 , . . . , H nq ) Γ, [C 1 ], . . . , [C p ] ⊢ ⊥ where ∆ = Γ ∪ {C 1 , . . . ,C p } LLproof ≡ Inference Rules of Zenon Modulo (part 2) Figure 9 : 9Translation Functions for LLproof ≡ Proofs into λΠ ≡ Figure 10 : 10A TFF1 ≡ Theory of Booleans λx 5 : 1 Figure 11 : 5111prf(false = bool false).R = bool false x 5 ) x 4 )x Proof Certificate for Commutativity of Conjunction in Dedukti a && b ֒→ b && a would lead to a non terminating rewrite system; therefore, commutativity is a lemma with the following proof:= a = bool a ⊢ ⊥ ¬∀ ¬∀y : bool. y = bool y ⊢ ⊥ = false = bool false ⊢ ⊥ ¬∀ ¬∀y : bool. false = bool false ⊢ ⊥ Ext(bool-case-¬∀) ¬∀x, y : bool. x && y = bool y && x ⊢ ⊥The translation of this proof in Dedukti is shown inFig. 11. Figure 12 : 12Proof Certificate for a B Set Theory Property in Dedukti with the theory: λH 3 : prf (P ∨ Q). H 3 ⊥ H 1 H 2 [P : Prop, Q : Prop] R ⇒ P Q ֒→ λH 1 : (prf (¬P) → prf ⊥). λH 2 : (prf Q → prf ⊥). λH 3 : prf (P ⇒ Q). H 1 (Contr P Q H 3 H 2 ) [P : Prop, Q : Prop] R ⇔ P Q ֒→ λH 1 : (prf (¬P) → prf (¬Q) → prf ⊥). λH 2 : (prf P → prf Q → prf ⊥). λH 3 : prf (P ⇔ Q). H 3 ⊥ (λH 4 : (prf P → prf Q). λH 5 : (prf Q → prf P). (H 1 (Contr P Q H 4 (λH 6 : prf Q. (H 2 (H 5 H 6 )) H 6 ))) (λH 7 : prf Q. (H 2 (H 5 H 7 )) H 7 )) [P : Prop, Q : Prop] R ¬∧ P Q ֒→ λH 1 : (prf (¬P) → prf ⊥). λH 2 : (prf (¬Q) → prf ⊥). λH 3 : prf (¬(P ∧ Q)). H 1 (λH 5 : prf P. H 2 (λH 6 : prf Q. H 3 (λZ : Prop. λH 4 : (prf P → prf Q → prf Z). H 4 H 5 H 6 ))) [P : Prop, Q : Prop] R ¬∨ P Q ֒→ λH 1 : (prf (¬P)prf (¬Q) → prf ⊥).λH 2 : prf (¬(P ∨ Q)). H 1 (Contr P (P ∨ Q) (λH 3 : prf P. λZ : Prop. λH 4 : (prf P → prf Z). λH 5 : (prf P → prf Z). H 4 H 3 ) H 2 ) (Contr Q (P ∨ Q) (λH 6 : prf Q. λZ : Prop. λH 7 : (prf P → prf Z). λH 8 : (prf Q → prf Z). H 8 H 6 ) H 2 ) [P : Prop, Q : Prop] R ¬⇒ P Q ֒→ λH 1 : (prf P → prf (¬Q) → prf ⊥). λH 2 : prf (¬(P ⇒ Q)). H 2 (λH 3 : prf P. (H 1 H 3 ) (λH 4 : prf Q. H 2 (λH 5 : prf P. H 4 )) Q) [P : Prop, Q : Prop] R ¬⇔ P Q ֒→ λH 1 : (prf (¬P) → prf (¬Q)). λH 2 : (prf P → prf (¬¬Q)). λH 3 : prf (¬(P ⇔ Q)). (λH 4 : prf (¬P). H 3 (λZ : Prop. λH 5 : (prf (P ⇒ Q) → prf (Q ⇒ P) → prf Z). H 5 (λH 6 : prf P. H 4 H 6 Q) (λH 7 : prf Q. H 1 H 4 H 7 P))) (λH 8 : prf P. H 2 H 8 (λH 9 : prf Q. H 3 (λZ : Prop. λH 3 :Figure 13 : 313prf (P t 1 ). H 1 (λH 4 : prf (t 1 = α t 2 ). H 2 (H 4 P H 3 )) Shallow Embedding of LLproof into Dedukti Translation Function for Sequents|[ϕ 1 ], . . . , [ϕ n ] ⊢ ⊥| := x ϕ 1 : prf |ϕ 1 |, . . . , x ϕ n : prf |ϕ n | Translation Function for Proofs Π 1 ∆, H 11 , . . . , H 1m ⊢ ⊥ . . . Rule |Arg 1 | . . . |Arg r | (λx H 11 : prf |H 11 | . . . . .λx H 1m : prf |H| 1m . |Π 1 |) . . . (λx H n1 : prf |H n1 | . . . . .λx H nq : prf H nq . |Π n |) x C 1 . . . x C pΠ n ∆, H n1 , . . . , H nq ⊢ ⊥ Rule(Arg 1 , . . . , Arg r ) Γ,C 1 , . . . ,C p ⊢ ⊥ := R A Appendix: Shallow Embedding of LLproof ≡ System into DeduktiLaw of Excluded Middle and LemmasExMid(P : Prop) : ΠZ : Prop. (prf P → prf Z) → (prf (¬P) → prf Z) → prf Z NNPP(P : Prop) : prf (¬¬P) → prf P := λH 1 : prf (¬¬P). ExMid P P (λH 2 : prf P. H 2 ) (λH 3 : prf (¬P). H 1 H 3 P) Contr(P : Prop, Q : Prop) : (prf (P ⇒ Q) → prf (¬Q → ¬P)) := λ H 1 : prf (P ⇒ Q). λH 2 : prf (¬Q). λH 3 : prf P. H 2 (H 1 H 3 ) LLproof Inference Rules [] R ⊥ ֒→ λH : prf ⊥. H [] R ¬⊤ ֒→ λH 1 : prf (¬⊤). H 1 (λZ : Prop. λH 2 : prf Z. H 2 ) [P : Prop] R Ax P ֒→ λH 1 : prf P. λH 2 : prf (¬P). H 2 H 1 [α : type, t : term α] R = α t ֒→ λH 1 : prf (t = α t). H 1 (λz : (term α → Prop). λH 2 : prf (z t). H 2 ) [α : type, t : term α, u : term α] R S ym α t u ֒→ λH 1 : prf (t = α u). λH 2 : prf (u = α t). H 2 (λz : (term α → Prop). λH 3 : prf (z u). H 1 (λx : term α. (z x) ⇒ (z t)) (λH 4 : prf (z t). H 4 ) H 3 ) [P : Prop] R Cut P ֒→ λH 1 : (prf P → prf ⊥). λH 2 : (prf (¬P) → prf ⊥). H 2 H 1 [P : Prop] R ¬¬ P ֒→ λH 1 : (prf P → prf ⊥). λH 2 : prf (¬¬P). H 2 H 1 [P : Prop, Q : Prop] R ∧ P Q ֒→ λH 1 : (prf P → prf Q → prf⊥).λH 2 : prf (P ∧ Q). H 2 ⊥ H 1 [P : Prop, Q : Prop] R ∨ P Q ֒→ λH 1 : (prf P → prf⊥). λH 2 : (prf Q → prf ⊥). https://www.rocq.inria.fr/deducteam/ZenonModulo/ bedding of typed deduction modulo presented in Sec. 3.1. Using the shallow embedding presented in Sec. 3.2, we can prove all the rules declared inFig. 8by rewriting the R rule symbols using only one axiom: the law of excluded middle. These proofs are listed inFig. 13where the symbol is used to delimit proofs. Jean-Raymond Abrial, 10.1017/CBO9780511624162The B-Book, Assigning Programs to Meanings. Cambridge University PressJean-Raymond Abrial (1996): The B-Book, Assigning Programs to Meanings. Cambridge University Press, doi:10.1017/CBO9780511624162. Hendrik Pieter Barendregt, 10.1017/CBO9781139032636Lambda calculus with types. Cambridge University PressHendrik Pieter Barendregt, Wil Dekkers & Richard Statman (2013): Lambda calculus with types. Cambridge University Press, doi:10.1017/CBO9781139032636. Autarkic Computations in Formal Proofs. Henk Barendregt, &amp; Erik Barendsen, 10.1023/A:1015761529444Journal of Automated Reasoning (JAR). 28Henk Barendregt & Erik Barendsen (2002): Autarkic Computations in Formal Proofs. Journal of Automated Reasoning (JAR) 28, doi:10.1023/A:1015761529444. Boogie: A modular reusable verifier for object-oriented programs. Mike Barnett, -Yuh Evan Bor, Robert Chang, Bart Deline, M Jacobs &amp; K Rustan, Leino, 10.1007/11804192_17Formal Methods for Components and Objects. SpringerMike Barnett, Bor-Yuh Evan Chang, Robert DeLine, Bart Jacobs & K Rustan M Leino (2006): Boogie: A modular reusable verifier for object-oriented programs. In: Formal Methods for Components and Objects, Springer, doi:10.1007/11804192_17. TFF1: The TPTP Typed First-Order Form with Rank-1 Polymorphism. 10.1007/978-3-642-38574-2_29doi:10. 1007/978-3-642-38574-2_29Conference on Automated Deduction (CADE). Jasmin Christian Blanchette & Andrei PaskevichSpringer7898Jasmin Christian Blanchette & Andrei Paskevich (2013): TFF1: The TPTP Typed First-Order Form with Rank-1 Polymorphism. In: Conference on Automated Deduction (CADE), LNCS 7898, Springer, doi:10. 1007/978-3-642-38574-2_29. CoqInE: translating the calculus of inductive constructions into the λΠ-calculus modulo. Mathieu Boespflug, &amp; Guillaume Burel, Proof Exchange for Theorem Proving (PxTP). Mathieu Boespflug & Guillaume Burel (2012): CoqInE: translating the calculus of inductive constructions into the λΠ-calculus modulo. In: Proof Exchange for Theorem Proving (PxTP). The λΠ-Calculus Modulo as a Universal Proof Language. Mathieu Boespflug, Quentin Carbonneaux &amp; Olivier, Hermant, Proof Exchange for Theorem Proving (PxTP). Mathieu Boespflug, Quentin Carbonneaux & Olivier Hermant (2012): The λΠ-Calculus Modulo as a Uni- versal Proof Language. In: Proof Exchange for Theorem Proving (PxTP). Zenon: An Extensible Automated Theorem Prover Producing Checkable Proofs. Richard Bonichon, David Delahaye &amp; Damien, Doligez, 10.1007/978-3-540-75560-9_13Logic for Programming Artificial Intelligence and Reasoning (LPAR). Springer4790Richard Bonichon, David Delahaye & Damien Doligez (2007): Zenon: An Extensible Automated Theo- rem Prover Producing Checkable Proofs. In: Logic for Programming Artificial Intelligence and Reasoning (LPAR), LNCS/LNAI 4790, Springer, doi:10.1007/978-3-540-75560-9_13. Experimenting with Deduction Modulo. Guillaume Burel, 10.1007/978-3-642-22438-6_14Conference on Automated Deduction (CADE), LNCS/LNAI 6803. SpringerGuillaume Burel (2011): Experimenting with Deduction Modulo. In: Conference on Automated Deduction (CADE), LNCS/LNAI 6803, Springer, doi:10.1007/978-3-642-22438-6_14. A Shallow Embedding of Resolution and Superposition Proofs into the λΠ-Calculus Modulo. Guillaume Burel, International Workshop on Proof Exchange for Theorem Proving (PxTP). Guillaume Burel (2013): A Shallow Embedding of Resolution and Superposition Proofs into the λΠ-Calculus Modulo. In: International Workshop on Proof Exchange for Theorem Proving (PxTP). . Raphaël Cauderlier, Raphaël Cauderlier: Focalide. https://www.rocq.inria.fr/deducteam/Focalide. Proof Certification in Zenon Modulo: When Achilles Uses Deduction Modulo to Outrun the Tortoise with Shorter Steps. David Delahaye, Damien Doligez, Frédéric Gilbert, Pierre Halmagrand &amp; Olivier, Hermant, International Workshop on the Implementation of Logics (IWIL). David Delahaye, Damien Doligez, Frédéric Gilbert, Pierre Halmagrand & Olivier Hermant (2013): Proof Certification in Zenon Modulo: When Achilles Uses Deduction Modulo to Outrun the Tortoise with Shorter Steps. In: International Workshop on the Implementation of Logics (IWIL). David Delahaye, Damien Doligez, Frédéric Gilbert, Pierre Halmagrand &amp; Olivier, Hermant, 10.1007/978-3-642-45221-5_20doi:10.1007/ 978-3-642-45221-5_20Zenon Modulo: When Achilles Outruns the Tortoise using Deduction Modulo. In: Logic for Programming Artificial Intelligence and Reasoning (LPAR). Springer8312David Delahaye, Damien Doligez, Frédéric Gilbert, Pierre Halmagrand & Olivier Hermant (2013): Zenon Modulo: When Achilles Outruns the Tortoise using Deduction Modulo. In: Logic for Pro- gramming Artificial Intelligence and Reasoning (LPAR), LNCS/ARCoSS 8312, Springer, doi:10.1007/ 978-3-642-45221-5_20. The BWare Project: Building a Proof Platform for the Automated Verification of B Proof Obligations. David Delahaye, Catherine Dubois, Claude Marché &amp; David Mentré, 10.1007/978-3-662-43652-3_26Abstract State Machines, Alloy, B, VDM, and Z (ABZ). SpringerDavid Delahaye, Catherine Dubois, Claude Marché & David Mentré (2014): The BWare Project: Building a Proof Platform for the Automated Verification of B Proof Obligations. In: Abstract State Machines, Alloy, B, VDM, and Z (ABZ), LNCS, Springer, doi:10.1007/978-3-662-43652-3_26. Gilles Dowek, Thérèse Hardin, Claude Kirchner, 10.1023/A:1027357912519Theorem Proving Modulo. 31Gilles Dowek, Thérèse Hardin & Claude Kirchner (2003): Theorem Proving Modulo. Journal of Automated Reasoning (JAR) 31, doi:10.1023/A:1027357912519. Jean-Christophe Filliâtre &amp; Andrei Paskevich, 10.1007/978-3-642-37036-6_8European Symposium on Programming (ESOP). Why3 -Where Programs Meet ProversJean-Christophe Filliâtre & Andrei Paskevich (2013): Why3 -Where Programs Meet Provers. In: European Symposium on Programming (ESOP), doi:10.1007/978-3-642-37036-6_8. Thérèse Hardin, François Pessaux, Pierre Weis &amp; Damien, Doligez, FoCaLiZe reference manual. Thérèse Hardin, François Pessaux, Pierre Weis & Damien Doligez (2009): FoCaLiZe reference manual. A Framework for Defining Logics. Robert Harper, Furio Honsell &amp; Gordon Plotkin, 10.1145/138027.138060Journal of the ACM. 40Robert Harper, Furio Honsell & Gordon Plotkin (1993): A Framework for Defining Logics. Journal of the ACM 40, doi:10.1145/138027.138060. A proposal for broad spectrum proof certificates. Dale Miller, 10.1007/978-3-642-25379-9_6Certified Programs and Proofs (CPP). SpringerDale Miller (2011): A proposal for broad spectrum proof certificates. In: Certified Programs and Proofs (CPP), Springer, doi:10.1007/978-3-642-25379-9_6.

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{'abstract': 'Dedukti has been proposed as a universal proof checker. It is a logical framework based on the λΠcalculus modulo that is used as a backend to verify proofs coming from theorem provers, especially those implementing some form of rewriting. We present a shallow embedding into Dedukti of proofs produced by Zenon Modulo, an extension of the tableau-based first-order theorem prover Zenon to deduction modulo and typing. Zenon Modulo is applied to the verification of programs in both academic and industrial projects. The purpose of our embedding is to increase the confidence in automatically generated proofs by separating untrusted proof search from trusted proof verification. * This work has received funding from the BWare project (ANR-12-INSE-0010) funded by the INS programme of the French National Research Agency (ANR).', 'arxivid': '1507.08719', 'author': ['Raphaël Cauderlier raphael.cauderlier@inria.frpierre.halmagrand@inria.fr \nCnam -Inria Paris\nFrance\n', 'Pierre Halmagrand \nCnam -Inria Paris\nFrance\n'], 'authoraffiliation': ['Cnam -Inria Paris\nFrance', 'Cnam -Inria Paris\nFrance'], 'corpusid': 1472806, 'doi': '10.4204/eptcs.186.7', 'github_urls': [], 'n_tokens_mistral': 19359, 'n_tokens_neox': 16869, 'n_words': 8783, 'pdfsha': '13f3a9e4abda695f71346c8c38dc3877d1c34279', 'pdfurls': ['https://arxiv.org/pdf/1507.08719v1.pdf'], 'title': ['Checking Zenon Modulo Proofs in Dedukti *', 'Checking Zenon Modulo Proofs in Dedukti *'], 'venue': ['EPTCS']}

arxiv

Quenched Heavy-Light Decay Constants UKQCD Collaboration R M Baxter Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland S P Booth Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland K C Bowler Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland S Collins Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland D S Henty Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland R D Kenway Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland D G Richards Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland H P Shanahan Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland J N Simone Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland A D Simpson Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland B E Wilkes Department of Physics Physics Department The University of Edinburgh EH9 3JZEdinburghScotland A K Ewing The University SO9 5NHSouthamptonUK L Lellouch The University SO9 5NHSouthamptonUK C T Sachrajda The University SO9 5NHSouthamptonUK H Wittig The University SO9 5NHSouthamptonUK Quenched Heavy-Light Decay Constants UKQCD Collaboration arXiv:hep-lat/9308020v1 31 Aug 1993 Edinburgh Preprint: 93/526 Southampton Preprint SHEP 92/93-24 PACS numbers: 12.38.G, 14.40 We present results for heavy-light decay constants, using both propagating quarks and the static approximation, in O(a)-improved, quenched lattice QCD. At β = 6.2 on a 24 3 ×48 lattice we find f D = 185+ 2 − 2 and f Bs /f B = 1.22 + 4 − 3 , in good agreement with earlier studies. From the static theory we obtain f stat B = 253 + 16 − 15 + 105− 14 MeV. We also present results from a simulation at β = 6.0 on a 16 3 × 48 lattice, which are consistent with those at β = 6.2. In order to study the effects of improvement, we present a direct comparison of the results using both the Wilson and the improved action at β = 6.0. Introduction The leptonic decay constants, f P , of pseudoscalar mesons composed of a heavy and a light quark play an important rôle in weak-interaction phenomenology. In particular f B , or more [2]- [5]. The experimental bound is f D < 290 MeV [6]. strictly f B √ B B (where B B , In the heavy-quark limit the scaling law for the decay constant of a heavy-light pseudoscalar meson is f P √ M P ∼ constant (up to mild logarithmic corrections). Lattice simulations using heavy-quark masses in the charm region indicate that there are large corrections to this scaling law (of order 40% at the charm quark mass, decreasing to about 15% at the mass of the bottom quark) [2,3,4]. The value of the decay constant of the B meson deduced from these simulations is in the region of 180 MeV. The conclusion that there are violations of the scaling law is supported by the large value for f P √ M P deduced from simulations obtained using a static (i.e infinitely-massive) heavy quark [5,7,8,9,10,11,12]. The important results and conclusions quoted above were obtained from simulations in which the mass of the heavy quark is large in lattice units (up to about a half). One may therefore worry that discretisation errors significantly contaminate the results. In this paper we present the results for decay constants of heavy-light mesons computed using the O(a)-improved lattice action proposed by Sheikholeslami and Wohlert [13], with which the discretisation errors in operator matrix elements (and hence in the computed values of the decay constants) can be reduced from O(m Q a) to O(α s m Q a), where m Q is the mass of the heavy quark [14]. This formal reduction in discretisation errors provides an important check on the stability of results and conclusions obtained with Wilson fermions. The results presented in this paper were obtained from two simulations of quenched QCD, using the Sheikholeslami-Wohlert (SW) or "clover" fermion action for the quarks (see Subsection 1.1 below). Our main results come from a simulation on a 24 3 × 48 lattice at β = 6.2, for which 60 gauge field configurations were generated. Details of this simulation and the determination of the values of the Wilson hopping parameter corresponding to the chiral limit, κ crit , and to the mass of the strange quark have been presented in ref. [15]. The heavy quarks have masses in the region of the charm quark mass and we study the behaviour of the decay constants with the mass of the heavy quark. Interpolating to the mass of the charm quark itself, and extrapolating the results to the mass of the b quark, we find that our best results for the decay constants of the B and D mesons are f D = 185 + 4 − 3 (stat) + 42 − 7 (syst) MeV(1) f B = 160 + 6 − 6 + 53 − 19 MeV (2) f Ds f D = 1.18 + 2 − 2(3) f Bs f B = 1.22 + 4 − 3 .(4) The details of this calculation and a complete set of results are presented in Section 2. The second simulation is on a 16 3 × 48 lattice at β = 6.0, using 36 configurations. The results, which are consistent with those mentioned above, are presented in Section 3. In order to study the effects of improvement on the calculation of heavy-light decay constants, we have repeated the computation for both the Wilson and SW actions using a subset of 16 of these configurations. The results and a discussion are presented in Subsection 3.2. We have also computed f B in the static approximation (in which contributions of O(1/m b ) are neglected). A discussion of the calculation and of the results is presented in Section 4. The result from the simulation at β = 6.2, on 20 of the 60 configurations, is f stat B = 253 + 16 − 15 (stat) + 105 − 14 (syst) MeV,(5) and the result at β = 6.0 on all 36 configurations is f stat B = 286 + 8 − 10 + 67 − 42 MeV.(6) Finally, Section 5 contains our conclusions. Improved Action and Operators The SW action is S SW F = S W F − i κ 2 x,µ,νq (x)F µν (x)σ µν q(x),(7) where S W F is the Wilson action: S W F = x q(x)q(x) − κ µ q(x)(1 − γ µ )U µ (x)q(x +μ) +q(x +μ)(1 + γ µ )U † µ (x)q(x) . (8) The decay constants of heavy-light pseudoscalar and vector mesons are computed using lattice axial and vector currents as interpolating operators. In order to obtain O(a)-improved matrix elements we use "rotated" operators [14]: Q(x)(1 + 1 2 γ· ← D ) Γ (1 − 1 2 γ· → D ) q(x),(9) where Γ is one of the Dirac matrices (either γ µ γ 5 or γ µ ), and Q and q represent the fields of the heavy and light quark respectively. In the static effective theory, in which the heavy propagator is expressed in terms of the link variables [16], it is sufficient to rotate the light-quark fields only [17], i.e. to use the operatorsQ (x) Γ (1 − 1 2 γ· → D ) q(x),(10) in order to eliminate the O(a)-discretisation errors. Renormalisation Constants Z V and Z A In order to determine the physical values of the decay constants from those obtained in lattice simulations using the interpolating operators in eq. (9), it is necessary to know the corresponding renormalisation constants. These are defined by requiring that Z A A latt µ and Z V V latt µ are the correctly normalised currents, where the superscript "latt" denotes that the operator is a lattice current. These renormalisation constants have been calculated at one-loop order in perturbation theory for the SW action with rotated operators [18]: Z V = 1 − 0.10g 2(11)Z A = 1 − 0.02g 2 .(12) In this paper they are evaluated using the "boosted" coupling suggested in ref. [19]; specifically, we use g 2 = 6/(β u 4 0 ), where u 0 is a measure of the average link variable, for which we take u 0 = (8κ crit ) −1 . It has been suggested [19,20] that the use of such an effective coupling, rather than the bare lattice coupling, resums some of the large higher-order corrections, and in particular some of the tadpole diagrams. Using the measured values of κ crit from our simulations, we obtain Z A ≃ 0.97 (0.96) and Z V ≃ 0.83 (0.82) for the simulation at β = 6.2 (6.0). In a recent non-perturbative determination of these renormalisation constants, obtained by requiring that the correctly-normalised currents obey the continuum Ward Identities, it was found that Z V = 0.824(2) and Z A = 1.09(3) [21]. These results were obtained from a simulation at β = 6.0 for one value of the quark mass. It remains to be checked that the results are independent of the quark mass and insensitive to small variations in β. For this reason, we use the perturbative values, given above, throughout the paper. We note, however, that the non-perturbative value of Z A may be larger by about 15%. In ref. [15] we obtained f π /m ρ = 0.138 + 6 − 9 , using the perturbative value of Z A . A larger value of Z A , such as Z A = 1.09, would bring this result closer to the physical value of 0.172. However, we also observed that f K /f π , which does not require Z A , was in very good agreement with the experimental value, and therefore we quote values for the ratios f D /f π and f B /f π in the following sections. The normalisation of the axial current in the static effective theory is discussed in Section 4. Error Estimation Statistical errors are obtained from a bootstrap procedure [22]. This involves the creation of 1000 bootstrap samples from the original set of N configurations by randomly selecting The latter correlations are preserved by using the same sequence of bootstrap samples at each quark mass. When extrapolating our results to the chiral limit and physical meson masses, the correlation matrix for the fitted quantities is estimated from the full bootstrap ensemble. All quoted statistical errors are obtained from the central 68% of the corresponding bootstrap distributions [23]. We attempt to quantify the systematic error arising from the uncertainty in the value of the lattice spacing, a, determined from properties of light hadrons, and from the string ten-sion [15]. The differences between results obtained using our central value for a −1 [GeV] and our upper and lower estimates are quoted as systematic uncertainties in the final estimates for decay constants in physical units. Hereafter, where we quote two errors, the first is statistical and the second is systematic. Extended Interpolating Operators In order to isolate the ground state in correlation functions effectively, it is useful to use extended (or "smeared") interpolating operators for the mesons. In particular, in the static theory it has been found to be essential to use smeared operators in order to obtain any signal for the ground state [24]. In this study we use gauge-invariant Jacobi smearing on the heavy-quark field (described in detail in ref. [25]), in which the smeared field, Q S ( x, t), is defined by Q S ( x, t) ≡ x ′ J(x, x ′ )Q( x ′ , t),(13) where J(x, x ′ ) = N n=0 κ n S ∆ n (x, x ′ )(14) and ∆(x, x ′ ) = 3 i=1 {δ x ′ , x−î U † i ( x −î, t) + δ x ′ , x+î U i ( x, t)}.(15) Wuppertal smearing [26], which uses the operator (1 − κ S ∆) −1 as the kernel of the smearing, corresponds to N = ∞, provided that κ S is sufficiently small to guarantee convergence. Following the discussion in ref. [25], we choose κ S = 0.25 and use the parameter N to control the smearing radius, defined by r 2 ≡ x | x| 2 |J(x, 0)| 2 x |J(x, 0)| 2 .(16) The values of N and r used in each of the calculations below will be quoted in the corresponding sections. 2 Decay Constants from the Simulation at β = 6.2 In this section we present the results obtained for the decay constants of heavy-light mesons from our simulation on 60 configurations of a 24 3 × 48 at β = 6.2, using the SW action in the C QR J 1 J 2 (t) = x 0|J Q 1 (x)J † R 2 (0)|0(17) where J 1 and J † 2 are interpolating operators which can annihilate or create the pseudoscalar or vector meson being studied. The labels Q, R denote whether a local (L) or smeared (S) interpolating operator is being used. In this simulation we use Jacobi smearing with N = 75, corresponding to a smearing radius of r = 5.2. The decay constants are obtained from the matrix elements of the local operators, which are determined by computing both the C SS and C LS correlation functions. In order to determine the decay constant, it is necessary to know the value of the lattice spacing in physical units. This can be done by relating the lattice measurements of some dimensionful quantity to its physical value, e.g. the mass of a light hadron or f π . Among the other frequently-used choices are the string tension, √ K, and the 1P − 1S mass splitting in charmonium. Using m ρ to set the scale in our study of light hadrons [15] we found a −1 (m ρ ) = 2.7(1) GeV, and a mass spectrum in physical units which was close to experimental values. Furthermore, our determination of the string tension [23] gave a −1 = 2.73(5) GeV. Encouraged by the consistency of these results, we use a −1 = 2.7 GeV.(18) However, the study described in ref. [15] showed that the measurement of the pion decay constant gave a higher value for the scale, i.e. a −1 (f π ) = 3.4 + 2 − 1 GeV, using the perturbative value for Z A . In order to get an estimate of the systematic uncertainties in the final numbers, we evaluate all our results using the central value of a −1 (f π ) as well, and quote the difference as the upper systematic error on decay constants. The lower systematic error is obtained from the uncertainty of −0.1 GeV in a −1 (m ρ ). In an attempt to reduce the systematic errors associated with the value of the renormalisation constant of the axial current, we also compute f D,B /f π , and determine f D,B by using the physical value of f π . Decay Constants of Pseudoscalar Mesons In order to determine the pseudoscalar decay constants, we start by fitting the two-point correlation function C SS P P (t) ≡ x 0|P S ( x, t)P †S (0)|0 → Z 2 P S 2M P exp(−M P L t /2) cosh (M P (L t /2 − t)) ,(19) where P is the pseudoscalar density, Z P S = 0|P S (0)|P and L t is the temporal extent Table 1. The values obtained by linear extrapolation to the chiral limit for the light quark are also tabulated. At large values of t, the ratio of correlation functions C LS AP (t) C SS P P (t) → 0|A L 4 (0)|P 0|P S (0)|P tanh (M P (L t /2 − t))(20) is used to extract the pseudoscalar decay constant, where A 4 is the temporal component of the axial current. The ratio is fitted in the range 15 ≤ t ≤ 22 with the pseudoscalar mass M P (in each bootstrap sample) constrained to its value extracted from fits to eq. (19). In Fig. 1 we plot the ratio of correlators together with the fit to eq. (20) as a function of t. Using the value of Z P S obtained from the fits of eq. (19), the matrix elements of the local axial current are obtained. Although there are other ways of determining these matrix elements, we find that the ratios in eq. (20) give the most precise determination. In Table 2 we present the results for the decay constants (in lattice units) of the pseudoscalar mesons for the twelve κ h -κ l combinations, as well as the values obtained by linearly extrapolating the results to the chiral limit. We also tabulate the results for the quantity f P √ M P which, in the heavy-quark limit, is independent of the mass of the heavy quark (except for a mild logarithmic dependence). κ h κ l M P M V 0.14144 0.924 + 3 − 1 0.944 + 4 − 2 0.121 0.14226 0.900 + 3 − 2 0.920 + 4 − 3 0.14262 0.890 + 4 − 3 0.909 + 5 − 4 κ crit 0.875 + 4 − 3 0.894 + 6 − 4 0.14144 0.822 + 3 − 1 0.847 + 4 − 2 0.125 0.14226 0.799 + 3 − 2 0.823 + 4 − 3 0.14262 0.789 + 4 − 2 0.811 + 5 − 4 κ crit 0.773 + 5 − 2 0.797 + 6 − 4 0.14144 0.715 + 3 − 1 0.745 + 4 − 2 0.129 0.14226 0.691 + 3 − 2 0.721 + 4 −(κ l → κ crit = 0.14315). κ h κ l f P /Z A f P √ M P /Z A 1/(f V Z V ) 0.14144 0.086 + 1 − 1 0.083 + 2 − 1 0.124 + 2 − 3 0.121 0.14226 0.079 + 1 − 1 0.075 + 2 − 1 0.116 + 3 − 3 0.14262 0.076 + 2 − 2 0.071 + 2 − 2 0.111 + 3 − 3 κ crit 0.071 + 2 − 1 0.066 + 2 − 1 0.105 + 4 − 4 0.14144 0.086 + 1 − 1 0.078 + 1 − 1 0.141 + 3 − 3 0.125 0.14226 0.079 + 1 − 1 0.070 + 1 − 1 0.132 + 3 − 3 0.14262 0.076 + 2 − 2 0.067 + 2 − 2 0.128 + 4 − 4 κ crit 0.071 + 2 − 1 0.062 + 2 − 1 0.123 + 4 − 4 0.14144 0.085 + 1 − 1 0.071 + 1 − 1 0.163 + 3 − 3 0.129 0.14226 0.078 + 1 − 1 0.065 + 1 − 1 0.155 + 3 − 3 0.14262 0.075 + 2 − 1 0.062 + 1 − 1 0.151 + 4 − 4 κ crit 0.071 + 2 − 1 0.057 + 2 − 1 0.146 + 5 − 5 0.14144 0.082 + 1 − 1 0.063 + 1 − 1 0.193 + 3 − 4 0.133 0.14226 0.076 + 1 − 1 0.057 + 1 − 1 0.186 + 3 − 4 0.14262 0.073 mesons. Also shown are the results for the combination f P √ M P which in the heavy-quark limit is independent of the heavy-quark mass (up to mild logarithmic corrections). We start the discussion of our results with the behaviour of the pseudoscalar decay constants as a function of the mass of the meson, with all dimensionful quantities given in lattice units. + 1 − 1 0.055 + 1 − 1 0.183 + 5 − 5 κ crit 0.069 + 1 − 1 0.051 + 1 − 1 0.179 + 5 − 5 In the heavy-quark limit, the quantity f P √ M P scales like f P M P = const. × [α s (M P )] −2/β 0 , M P −→ ∞.(21) In order to detect possible deviations from this scaling law we plot in Fig. 2 the quantity 1 Φ(M P ) ≡ (α s (M P )/α s (M B )) 2/β 0 Z −1 A f P M P (22) as a function of 1/M P . We approximate α s (M) by α s (M) = 2π β 0 log(M/Λ QCD )(23) where we take Λ QCD = 200 MeV, and β 0 = 11 − 2 3 n f , with n f = 0 in the quenched approximation. From the figure we see thatΦ(M P ) increases as the mass of the heavy quark is increased (in agreement with the behaviour found using the Wilson action for the quarks [2,3,4]). In order to quantify this behaviour, we fitΦ(M P ) to either a linear or quadratic function of 1/M P :Φ (M P ) = A 1 − B M P(24) orΦ (M P ) = C 1 − D M P + E M 2 P .(25) We have performed these fits twice; once using the values of f P √ M P for all four values of κ h , and once using those for only the smallest three κ h 's (i.e. for the heaviest three heavy-quark masses). The results of the fits are given in Table 3. We find that the non-scaling corrections are of O(30%) for f D and O(10%) for f B , in agreement with previous results obtained using Wilson fermions [2,3,4]. From the quadratic fit to the data at all four heavy-meson masses we find, in physical units, Ca −3/2 = 0.45 + 2 − 2 + 19 − 3 GeV 3/2 Da −1 = 0.84 + 11 − 8 + 22 − 3 GeV Ea −2 = 0.28 + 7 − 9 + 16 − 2 GeV 2 .(26) The second error in eq. (26) corresponds solely to the uncertainty in the scale. It should be mentioned that ignoring the residual logarithmic dependence of f P √ M P on M P makes the slope more pronounced. However it is clear from Fig. 2 and Table 3 that the logarithmic corrections to the scaling law can by no means account for the observed slope in f P √ M P . We use the parameters of the fits in Table 3 to make our predictions for the values of the decay constants f D and f B . The results corresponding to the four fits are presented in Taking the results from the quadratic fit using all four κ h values we find: f D = 185 + 4 − 3 + 42 − 7 MeV(27) f B = 160 + 6 − 6 + 53 − 19 MeV(28) where we have included the uncertainty of 11 MeV from the extrapolations in the systematic error quoted for f B . We take the results presented in equations (27) and (28) as our best estimates of the decay constants of the D and B mesons. In ref. [3] it was found useful to use the pion decay constant, f π , to set the scale in the computations of the decay constants of heavy-light mesons. By calculating f D /f π and f B /f π it may be expected that some of the systematic errors cancel, since, in particular, the ratios are independent of Z A . Our results for the decay constants obtained in this way are, as expected, close to the upper systematic error margins in Table 4. We find f D f π × 132 MeV = 232 + 12 − 5 MeV(29) f B f π × 132 MeV = 201 Finally in this subsection, we present our results for f Ds and f Bs . These are obtained by interpolating the measured values of the decay constants given in Table 2 to κ l = κ s = 0.1419 [15]. The extrapolations in the heavy-quark masses are done as above. We find + 12 − 8 MeV.(30)+ 1 − 1f Ds f D = 1.18 + 2 − 2 (31) f Bs f B = 1.22 + 4 − 3 .(32) In physical units we obtain f Ds = 212 + 4 − 4 + 46 − 7 MeV(33) f Bs = 194 + 6 − 5 + 62 − 9 MeV.(34) Recently the first measurement of f Ds has been made by the WA75 collaboration [27], who found f Ds = (232 ± 45 ± 20 ± 48 )MeV. Our result is in good agreement with the measured value, and also with previous lattice calculations using Wilson fermions [3,5]. Decay Constants of Vector Mesons In this subsection we present our results for the decay constants of heavy-light vector mesons. These are defined by 0|V µ |V ≡ ǫ µ M 2 V f V = Z V 0|V L µ (0)|V ,(35) where |V represents a state containing a vector meson V , with mass M V , polarisation vector ǫ µ and decay constant f V . V L µ denotes the local lattice vector current, defined in eq. (9), with Γ = γ µ , which has to be multiplied by the renormalisation constant Z V . The vector mass M V is extracted from fits to the correlator C SS V V ≡ 3 j=1 x 0|V S j ( x, t)V S j (0)|0 → − 3Z 2 V S 2M V exp(−M V L t /2) cosh(M V (L t /2 − t)),(36) where V S j is the jth spatial component of the smeared vector operator and Z V S is defined through 0|V S j (0)|V = ǫ j Z V S .(37) Fitting timeslices 14 ≤ t ≤ 23, symmetrized, we obtain the vector meson masses shown in Table 1. In order to extract the matrix element of the local vector current we fit the ratio C LS V V (t) C SS V V (t) −→ − 3 j=1 0|V L j (0)|V ǫ * j 3Z V S(38) to a constant in the fitting interval 15 ≤ t ≤ 23. The results for 1/f V for the twelve κ h − κ l combinations are presented in Table 2 for f −1 V Z −1 V are now interpolated to the D * mass using a quadratic fit to the data at all four values of κ h , giving, f −1 D * = 0.110 + 5 − 5 + 36 − 5 . (39) This is slightly below, but still compatible with, earlier studies (e.g. [3]) when the systematic error is taken into account. This result remains unaltered if a linear fit is used instead of a quadratic one. A Test of the Heavy Quark Symmetry In the heavy-quark limit, the decay constants of heavy-light pseudoscalar and vector mesons are related by [28] Fig. 3, and we display our results in Table 5. The perturbative values of Z A and Z V are used. U(M) ≡ f V f P M = 1 + 8 3 α s (M) 4π + O(1/M) ,(40) The fact thatŨ(∞) is around one in Table 5 provides support for our parametrisations, in eqs. (24) and (25), of the non-scaling behaviour of the decay constants for finite heavy-quark masses. − 5 respectively. Using the mass of the ρ meson to determine the value of the lattice spacing, we find a −1 = 2.0 + 3 − 2 GeV, whilst using f π we find a −1 = 2.1 + 2 − 1 GeV. These two results are compatible, and below we will use the value a −1 = 2.0 + 3 − 2 GeV (42) to convert the results from lattice to physical units. We have computed the heavy-light correlation functions as series in κ h (the hopping-parameter expansion [29]), thus enabling us to obtain the decay constants at any value of the mass of the heavy quark, without explicitly computing the heavy-quark propagators. The decay constants are obtained by fitting to eq. (19) and eq. (20), over the range 12 ≤ t ≤ 18 for both fits. We employ the Jacobi smearing algorithm with N = 50, corresponding to a smearing radius of r = 4.2. In an attempt to improve our understanding of the discretisation errors, we have also com- Pseudoscalar Decay Constants In Fig. 4 we plot the chirally-extrapolated values ofΦ(M P ) as a function of 1/M P , for 11 values of the heavy-quark mass. We fit the points corresponding to the five lightest meson masses (for which m Q = 1/2(1/κ h − 1/κ crit ) < 0.7, as was the case at β = 6.2) to eq. (25), and this is shown as the solid curve in the figure. For the coefficients of the fit we find: The figure also shows the results obtained with the Wilson action, but using the normalisation √ 1 − 6κ for the quark fields [5,19,30], whereκ = u 0 κ and u 0 = 1/(8κ crit ). It has been suggested that this normalisation may absorb some of the discretisation errors [19], and indeed the corresponding results agree remarkably with those obtained using the SW action. This agreement provides considerable motivation for a theoretical study to investigate whether there is any formal connection between the ansatz above and the improvement programme initiated by Symanzik [31]. C = 0.18 + 3 − 3 ; D = Comparison of Results Using Wilson and SW Actions f B in the Static Limit An alternative and complementary approach to heavy-quark physics using lattice QCD was proposed by Eichten [16]. This technique is based on an expansion of the heavy-quark propagator in inverse powers of the quark mass. In practice, one keeps just the leading term, given by (at zero velocity) S Q ( x, t; 0, 0) = θ(t)e −m Q t 1 + γ 4 2 + θ(−t)e m Q t 1 − γ 4 2 δ (3) ( x)P 0 (t, 0),(51) where P 0 (t, 0) is the product of links from ( 0, t) to the origin, for example for t > 0, P 0 (t, 0) = U † 4 ( 0, t − 1)U † 4 ( 0, t − 2) · · · U † 4 ( 0, 0).(52) At sufficiently large times x A 4 ( x, t)A † 4 (0) → f 2 P M P 2 e −M P t ,(53) where A µ is the improved axial current of eq. (10) with Γ = γ µ γ 5 . Since the only dependence on m Q in eq. (53) arises through the exponential factor in eq. (51), we deduce the scaling law We compute the two correlation functions, C SS and C LS , defined by 2 C SS (t) = x 0|A S 4 ( x, t)A †S 4 ( 0, 0)|0 → (Z S ) 2 e −∆E t (55) C LS (t) = x 0|A L 4 ( x, t)A †S 4 ( 0, 0)|0 → Z L Z S e −∆E t ,(56) where ∆E is the (unphysical) difference between the mass of the meson and the bare mass of the heavy quark. The matrix element of the local operator A L 4 is obtained from the two correlation functions C SS and C LS as follows. By fitting C SS (t) to the functional form given in eq. (55) we obtain Z S (and ∆E). At sufficiently large times the ratio C LS (t)/C SS (t) → Z L /Z S , so that Z L can be determined. In view of the difficulty in isolating the ground state in correlation functions using the static effective theory, we have compared results obtained with different numbers of iterations of the Jacobi smearing algorithm [33]. For N less than about 80 the plateaus do not start until at least t = 7. In this paper we present our results obtained with N = 110 and N = 140, corresponding to r = 5.9 and 6.4 respectively, where plateaus begin as early as t = 4 and hence statistical errors are smaller. In Fig. 6(a) we show the effective masses obtained from C SS (t), and in Fig. 6(b) the ratio C LS (t)/C SS (t), both at κ l = 0.14226 and N = 140. Excellent plateaus are obtained, giving us confidence that the ground state has indeed been isolated. In Table 7 we present the results for ∆E, (Z S ) 2 , Z L /Z S and Z L at all three values of κ l , from fits over the range 5 ≤ t ≤ 11, without symmetrization in Euclidean time. ∆E is obtained from the fit to C SS (t) for N = 140; consistent values are obtained for N = 110. Extrapolating the results for Z L in Table 7 to the chiral limit and to the mass of the strange quark we find: When matching the static lattice theory to the full theory at a scale m b , the factor required is [35]: Z L = 0.124 + 8 − 7 at κ l = κ crit(57)Z stat A = Z A 1 + α s (a −1 ) 3π 3 2 log a 2 m 2 b − 2 .(61) Z A , relating the axial current in the static lattice theory to the static continuum one, has been calculated in perturbation theory for the SW action [34,17]: Z A = 1 − 0.127 g 2 ≃ 0.79.(62) The value of 0.79 on the right hand side of eq. (62) was estimated using the boosted coupling at β = 6.2. For the remaining factor in eq. (61), we take m b = 5 GeV, α s given by eq. (23) with n f = 0, and Λ QCD = 200 MeV, yielding a number close to one (note that this is insensitive to small changes in m b These results at β = 6.0 are consistent with those at β = 6.2 presented in eqs. (65), (66) and (67). + 4 − 3 .(70) The results plotted in Fig. 6(b) for the ratio C LS (t)/C SS (t) appear to have considerably smaller errors, and a clearer plateau, than in some recent studies [8,10], in spite of our limited statistics. We attribute this to the fact that we use the C LS correlation function in which the smearing is performed at the source, rather than the C SL correlation function in which the smearing is performed at the sink 3 . Of course, with sufficiently many configurations, the results are independent of this choice. However in the C LS correlation function, the heavy-quark propagator is sampled at many spatial points, whereas in the C SL correlation function only the heavy-quark propagator at x = 0 contributes. Thus it seems plausible that the statistical errors are considerably reduced using the C LS correlation function. To check this hypothesis, we have computed the ratios C LS (t)/C SS (t) and C SL (t)/C SS (t) at β = 6.0, with κ l = 0.1432. The results are plotted in Fig. 7, and indeed confirm that there is an enormous improvement in precision when the correlation function C LS is used. We believe that this, rather than the different method of smearing, is the reason for the relatively poor plateaus in ref. [10]. Discussion We A better way of determining the consistency of the static and propagating results is to include the static result in the quadratic fit. Such a fit using the full correlation matrix at β = 6.0 gives a χ 2 /dof of 1.5. This is still acceptable, and provides further evidence that using rotated operators with the SW action gives a sensible normalisation for propagating heavy-quark fields. From this fit we obtain f B = 220 + 6 − 7 + 40 − 27 MeV, which is 44 MeV higher than that obtained from the propagating points alone. In Table 8 we present the results for f stat B obtained by other groups, together with our values. Although at β = 6.0 the values of Z L found by all groups and for both actions are in broad agreement, the different treatment of systematics leads to the spread of results in f stat B . It has been suggested that f stat B decreases as a → 0 [12]. However, the agreement of the results obtained with the Wilson and SW actions at β = 6.0, together with consistency between our results at β = 6.0 and β = 6.2, suggests that the discretisation errors are small. Conclusions In this paper we have carried out an extensive study of the decay constants of heavy-light mesons using the SW action for the quarks. The use of the SW action confirms the large, effects. It has been suggested that such effects may largely be absorbed by the use of a different normalization [5]. We find that such a normalization yields results in agreement with those obtained using the SW action with rotated operators and the √ 2κ normalization for the quark fields. Our best estimates of f D and f B are f D = 185 the B parameter of B 0 -B 0 mixing, is expected to be close to one), is one of the principal unknown quantities needed for the determination of the CP -violating phase in the Standard Model, as well as other properties of weak decays. Lattice QCD offers the opportunity for a non-perturbative computation of the operator matrix elements which are necessary for the determination of the decay constants and B parameters. During the last few years there have been several lattice computations of the decay constants of "heavy-light" pseudoscalar (and vector) mesons. The results for the decay constant of the D meson, obtained using the Wilson action for the quarks, are in the region of 200 MeV (using a normalisation for which f π =132 MeV). For example, in his 1989 review S. Sharpe quoted [1] f D ≃ 180 ± 25(stat) ± 30(syst) MeV as his summary of the lattice results. More recent simulations with Wilson fermions also give results in this range N configurations per sample (with replacement). Correlators are fitted for each bootstrap sample by minimising χ 2 . During the fits, correlations among different timeslices are taken into account, whereas correlations among different values of the quark mass are neglected. quenched approximation. The computations are performed for four different values of the mass of the heavy quark, corresponding to κ h = 0.121, 0.125, 0.129 and 0.133, and for three values of the mass of the light quark, corresponding to κ l = 0.14144, 0.14226 and 0.14262. The mass of the charm quark corresponds approximately to κ h = 0.129. The value of the hopping parameter corresponding to the mass of the strange quark is κ s = 0.1419 + 1 − 1 and the critical value is κ crit = 0.14315 + 2 − 2 [15]. The decay constants are determined by computing two-point correlation functions of the form of the lattice. This correlation function gives the best determination of the masses of the heavy-light pseudoscalars. Symmetrizing in Euclidean time, the fitting range was chosen to be 13 ≤ t ≤ 22 for all three values of the light-quark mass. Good plateaus in the effective mass were observed and stable fits obtained. The values of the masses of the pseudoscalar and vector mesons for the twelve κ h -κ l combinations are presented in Figure 1 : 1The ratio of correlators defined in eq. (20) plotted versus t for κ h = 0.129, κ l = 0.14262. The curve represents the fit using timeslices15-22. Figure 2 : 2The data forΦ(M P ) plotted against the inverse meson mass. The open symbols denote points with κ l < κ crit , whereas full symbols denote those extrapolated to κ crit . The solid line represents the linear fit to the chirally-extrapolated points using the three heaviest meson masses, whereas the dashed curve results from a quadratic fit to all four. where we take the heavy mass scale, M, to be the spin-averaged meson mass, M = (M P + 3M V )/4. In order to test the predicted behaviour of U(M), we take the chirally-extrapolated values for both the pseudoscalar and vector decay constants, and fit U (M) ≡ U(M)a linear or quadratic function of 1/M. The data together with the fits are shown in Figure 3 : 3The quantityŨ(M) plotted against the inverse spin-averaged mass. Linear and quadratic fits are represented by the solid and dashed curves, respectively. Also shown are the statistical errors of the extrapolation to the infinite mass limit. 3 Decay Constants from the Simulation at β = 6.0 In this section we describe the results of a computation of the decay constants using the SW fermion action at β = 6.0 on a 16 3 × 48 lattice. These results were obtained using 36 configurations, with light-quark masses corresponding to κ l = 0.1432, 0.1440 and 0.1445. The corresponding light-light pseudoscalar and vector meson masses, and pseudoscalar decay constants, all in lattice units, are presented in puted the decay constants for the Wilson action at one value of the light-quark mass, using a subset of 16 of the 36 configurations. The comparison of the results for the two actions is presented in Subsection 3.2. . 16 of the configurations, we have computed the decay constants for the Wilson fermion action, again using the hopping-parameter expansion. We compute light-quark propagators at a single value of the hopping parameter, κ W l = 0.155, corresponding to a pseudoscalarmeson mass of 0This was chosen to match the SW pseudoscalar-meson mass of 0.31 + 2 − 1 obtained at κ SW l = 0.144 on the same set of configurations. Figure 4 : 4The chirally-extrapolated data forΦ(M P ) at β = 6.0 plotted against the inverse meson mass. The solid curve represents a quadratic fit to the points denoted by circles. Points represented by diamonds are not included in the fit. The dashed curves are the 68% confidence bounds on the fit. In Fig. 5 we plot f P √ M P as a function of 1/M P . Using the conventional normalisation of √ 2κ for the quark fields, we see a clear divergence between the results for the two actions for m Q > 0.7; the Wilson results turn over and decrease. However, we note that uncorrelated χ 2 fits of the Wilson points, at the lightest few meson masses, to eqs. (24) and (25) would yield coefficients of the 1/M P term broadly consistent with previous Wilson analyses. Figure 5 : 5f P √ M P for both the Wilson and SW actions. Diamonds denote points obtained with the Wilson action in the conventional normalisation, √ 2κ, whereas squares denote points normalised by √ 1 − 6κ. Results using the SW action are represented by circles. The solid curves are quadratic fits in 1/M P to f P √ M P for the Wilson action, with fields normalized by √ 1 − 6κ, and for the SW action. The dashed curve is to guide the eye. that f P √ M P is independent of the heavy-quark mass. Matching the result from the Heavy Quark Effective Theory with that in the full theory introduces the logarithmic corrections in eq. (21). The full scaling law is of the form f P M P = const. (α s (M P )) −2/β 0 (1 + O(α s )) + O(1/M P ) . (54) The objective of lattice computations is to determine the constant. We refer to the value of f B obtained using eq. (54), but dropping the O(1/M B ) corrections, as f stat B . 7 : 7Values of ∆E, (Z S ) 2 , Z L /Z S and Z L at the three value of κ l . ∆E is obtained from the fit to C SS (t).4.1 Results at β = 6.2 We now report on a computation of f stat B at β = 6.2. The results presented here were obtained using a subset of 20 of the 60 configurations discussed in Section 2, at the three values of the light-quark mass. The values of f stat B and f stat Bs were determined by extrapolating the results to κ crit and κ s respectively. ZFigure 6 : 6L = 0.140 + 7 − 6 at κ l = κ s (58) (a) The effective mass obtained from C SS (t), and (b) the ratio C LS (t)/C SS (t) at β = 6.2, κ l = 0.14226 and N = 140. The solid lines represent fits from 5 ≤ t ≤ 11.when obtained using smeared interpolating operators with N interpolating operators with N = 140. Figure 7 : 7(a) The ratio C SL (t)/C SS (t) and (b) C LS (t)/C SS (t) plotted against t for β = 6.0, κ l = 0.1432, and N = 50 iterations used in the Jacobi smearing algorithm. begin with a comparison of the static and propagating results. Because we do not yet have static results for the full set of configurations at β = 6.2, we focus on a comparison at β = 6.0. In Fig. 8 we plot our results for the scaling quantity f P √ M P (α(M P )/α(M B )) 6/33 from the simulation at β = 6.0 as a function of 1/M P (in lattice units), together with our result for f stat B √ M B . The quadratic fit which we used to obtain our estimate for f B in Subsection 3.1 gives an intercept at 1/M P = 0 which is about 25% and two standard deviations below the static result; a similar discrepancy is observed at β = 6.2. There are a number of possible reasons for this, e.g. uncertainties in the renormalisation constants (which are different for the static and propagating quarks), residual discretisation errors in the simulation of the propagating quarks, and uncertainties in the various extrapolations. negative O(1/M P ) corrections to the scaling law f P √ M P ∼ constant at the mass of the charm quark and the significant corrections at the mass of the b quark, previously observed with the Wilson action. However, from our comparison of results for the Wilson and SW actions at β = 6.0, we observe clear evidence that the √ 2κ normalisation of the Wilson quark fields fails for large quark mass. This failure is presumably due to large O(m Q a) propagating quarks at β = 6.2. Our analysis at β = 6.0 yields entirely consistent results. The latter analysis also suggests that including the static result in the fits is likely to increase the value of f B by around 40 MeV. Figure 8 : 8Z AΦ (M P ) at β = 6.0 from the simulation with propagating quarks (open symbols) and the static theory (cross). The dashed curve is the fit to the open circles, with the parameters of eq. (43); the square is the intercept at 1/M P = 0. The solid curve is the fit with the static point included. The most urgent extension of this work is to determine the B parameter for B 0 -B 0 mixing, since it is the combination f B √ B B which is directly relevant for phenomenological studies of the mixing and of CP -violation. A recent simulation with Wilson fermions found B B = Table 1 : 1Masses (in lattice units) of the pseudoscalar and vector mesons for the twelve κ h -κ l combinations at β = 6.2 on a 24 3 × 48 lattice. Also presented are the values obtained by linear extrapolation to the chiral limit Table 2 : 2The decay constants (in lattice units) of the pseudoscalar and vector Table 4 . 4From this table it is clear that there is a further systematic uncertainty in f B of about 11 MeV from extrapolating using either linear or quadratic fits. In contrast to this, since we interpolate to m D , the results for f D are very stable. It should be emphasised thatchoosing a different value for Λ QCD (e.g. Λ QCD = 250 MeV), or for the anomalous dimension (e.g., by taking n f = 4), changes the results by only about 1 MeV. Table 3 : 3Values of the parameters of the linear and quadratic fits to thebehaviour of the pseudoscalar decay constants with the mass of the mesons (as defined in the text). Linear Fit Quadratic Fit f D f B f D f B 4 κ h 's 185 + 4 − 3 + 45 − 7 149 + 5 − 3 + 52 − 7 185 + 4 − 3 + 42 − 7 160 + 6 − 6 + 53 − 8 3 κ h 's 186 + 4 − 3 + 41 − 7 154 + 5 − 4 + 53 − 8 185 + 4 − 3 + 42 − 7 160 + 7 − 7 + 54 − 7 Table 4 : 4Values of the decay constants f B and f D in MeV, corresponding to the linear and quadratic fits. , together with those obtained after extrapolation to the chiral limit. The chirally-extrapolated valuesM Linear Fit Quadratic Fit ∞ 1.02 + 5 − 4 1.09 + 7 − 8 (M B + 3 M * B )/4 0.93 + 4 − 3 0.96 + 4 − 5 (M D + 3 M * D )/4 0.77 + 2 − 2 0.77 + 2 − 2 Table 5 : 5The quantityŨ (M) obtained from linear and quadratic fits. Table 6 . 6The values of the hopping parametercorresponding to the chiral limit and the strange quark mass are κ crit = 0.14556 + 6 − 6 and Table 6 : 6Masses of light-light pseudoscalar and vector mesons, and the pseu-doscalar decay constants at β = 6.0. κ s = 0.1437 + 4 Table when using those with N = 140. The systematic errors quoted arise from the uncertainty in the scale. We take the results in equations (65) and (66) as our best values, and these giveWe have performed a similar analysis on the 36 configurations at β = 6.0, using Jacobi smearing with N = 50 and 150, corresponding to r = 4.2 and 6.2 respectively. The results obtained using the two smearing radii are consistent, and our best results are those at N = 50 for which Z L = 0.211). Thus Z stat A = 0.79 also. We find f stat B = 266 + 17 − 15 + 110 − 14 Z stat A 0.79 MeV (63) f stat Bs = 300 + 14 − 13 + 125 − 16 Z stat A 0.79 MeV, (64) when using the interpolating operators with N = 110, and f stat B = 253 + 16 − 15 + 105 − 14 Z stat A 0.79 MeV (65) f stat Bs = 287 + 14 − 13 + 119 − 15 Z stat A 0.79 MeV (66) for the ratio: f stat Bs f stat B = 1.14 + 4 − 3 . (67) 4.2 Results at β = 6.0 + 6 − 7 , yielding f stat B = 286 + 8 − 10 + 67 − 42 Z stat A 0.78 MeV (68) f stat Bs = 323 + 14 − 14 + 75 − 47 Z stat A 0.78 MeV. (69) f stat Bs f stat B = 1.13 Ref.Action β a −1 [GeV] Z stat A f stat B [MeV] [11] Wilson 5.9 1.75 0.79 319 ± 11 [8] Wilson 6.0 2.0 ± 0.2 0.8 310 ± 25 ± 50 [9] Wilson 6.0 2.2 ± 0.2 0.8 366 ± 22 ± 55 [10] Wilson 6.0 2.11 ± 0.05 ± 0.10 0.8 350 ± 40 ± 30 [10] SW 6.0 2.05 ± 0.06 0.89 370 ± 40 This Work SW 6.0 2.0 + 3 − 2 0.78 286 + 8 − 10 + 67 − 42 This Work SW 6.2 2.7 + 7 − 1 0.79 253 + 16 − 15 + 105 − 14 [5] Wilson 6.3 3.21 ± .09 ± .17 0.69 235 ± 20 ± 21 [12] Wilson 5.74, 6.0, 6.26 1.12, 1.88, 2.78 0.71(8) 230 ± 22 ± 26 Table 8 : 8Compilation of lattice results for f statB 1.16 ± 0.07[3], and it is important to confirm this result with the improved action.grateful to Mike Brown of Edinburgh University Computing Service, and to Arthur Trew of EPCC, for provision and maintenance of service on the Meiko i860 Computing Surface and the Thinking Machines CM-200. We wish to thank Brian Murdoch for access to a Digital 7640 AXP ("Alpha") system placed at the University of Edinburgh Computing Service by Digital Equipment Corporation for field test. The normalization factor α s (M B ) −2/β0 is convenient when comparing these results with those obtained in the static theory. In principle the behaviour of the correlation functions in eqs. (55) and (56) is given by a cosh (as in eq.(19)), however the contribution of the backward-propagating meson is negligible in the time intervals we will be considering. In both refs.[8] and[10] it was in fact the C SL , and not the C LS , correlation function which was computed[36]. Acknowledgements We are grateful to G. Martinelli for many helpful discussions. CTS . S R Sharpe, Nucl. Phys. B (Proc. Suppl.). 17146S.R. Sharpe, Nucl. Phys. B (Proc. Suppl.) 17 (1991) 146 . C R Allton, Nucl. Phys. B (Proc. Suppl.). 20504C.R. Allton et al., Nucl. Phys. B (Proc. Suppl.) 20 (1991) 504 . A Abada, Nucl. Phys. 376172A. 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B(Proc.Suppl.) 30 (1993) 433 . C , PSI-PR-92-27PSI PreprintC. Alexandrou et al., PSI Preprint PSI-PR-92-27 (1992) . B Sheikholeslami, R Wohlert, Nucl. Phys. 259572B. Sheikholeslami and R. Wohlert, Nucl. Phys. B259 (1985) 572 . G Heatlie, Nucl. Phys. 352266G. Heatlie et al., Nucl. Phys. B352 (1991) 266 . C R Allton, UKQCD CollaborationEdinburgh University Preprint. 93UKQCD Collaboration, C.R. Allton et al., Edinburgh University Preprint 93/524 (1993) . E Eichten, Nucl. Phys. B (Proc. Suppl.). 4170E. Eichten, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 170 . A Borrelli, C Pittori, Nucl. Phys. 385502A. Borrelli and C. Pittori, Nucl. Phys. B385 (1992) 502 . A Borrelli, R Frezotti, E Gabrielli, C Pittori, CERN preprint TH-6587A. Borrelli, R. Frezotti, E. Gabrielli and C. Pittori, CERN preprint TH-6587 (1992) . G P Lepage, P B Mackenzie, Nucl. Phys. B (Proc.Suppl.). 20173Fermilab preprint PUB-91/355-T(revisedG.P. Lepage and P.B. Mackenzie, Nucl. Phys. B (Proc.Suppl.) 20 (1992) 173; Fermilab preprint PUB-91/355-T(revised) (1992) . P B Mackenzie, Nucl. Phys. B (Proc. Suppl.). 3035P.B.Mackenzie, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 35 . G Martinelli, S Petrarca, C T Sachrajda, A Vladikas, Phys. Lett. 311241G. Martinelli, S. Petrarca, C.T. Sachrajda and A. Vladikas, Phys. Lett. B311 (1993) 241 . B Efron, SIAM Review. 21460B. Efron, SIAM Review 21 (1979) 460 Edinburgh University Preprint 92/507 (1992), to be published in. C R Allton, UKQCD CollaborationNucl. Phys. B. UKQCD Collaboration, C.R. Allton et al., Edinburgh University Preprint 92/507 (1992), to be published in Nucl. Phys. B . Ph, Boucaud, Phys. Lett. 220219Ph. Boucaud et al., Phys. Lett. 220B (1989) 219 . C R Allton, UKQCD CollaborationPhys. Rev. 475128UKQCD Collaboration, C.R. Allton et al., Phys. Rev. D47 (1993) 5128 . S Güsken, Nucl. Phys. B (Proc. Suppl.). 17266Phys. Lett.S. Güsken et al., Nucl. Phys. B (Proc. Suppl.) 17 (1990) 361; Phys. Lett. 227B (1989) 266 . S Aoki, WA75 CollaborationCERN preprint PPE/92-157WA75 Collaboration, S. Aoki et al., CERN preprint PPE/92-157 (1992) . M B Voloshin, M A Shifman, Sov. J. Nucl. Phys. 47801Yad. Fiz.M.B. Voloshin and M.A. Shifman, Yad. Fiz. 47 (1988) 801 [Sov. J. Nucl. Phys. 47 (1988) 511]; . H D Politzer, M B Wise, Phys. Lett. 206504Phys. Lett.H.D. Politzer and M.B. Wise, Phys. Lett. B206 (1988) 681; Phys. Lett. B208 (1988) 504; . M Neubert, Phys. Rev. 461076M. Neubert, Phys. Rev. D46 (1992) 1076; . X Ji, M J Musolf, Phys , X. Ji and M.J. Musolf, Phys. . Lett. 257409Lett. B257 (1991) 409; . D J Broadhurst, A G Grozin, Phys. Lett. 274421D.J. Broadhurst and A.G. Grozin, Phys. Lett. 274B (1992) 421 . D S Henty, UKQCD CollaborationR D Kenway, UKQCD CollaborationPhys. Lett. 289408UKQCD Collaboration, D.S. Henty and R.D. Kenway, Phys. Lett. B289 (1992) 408 . M Lüscher, Commun. Math. Phys. 54283M. Lüscher, Commun. Math. Phys. 54 (1977) 283 K Symanzik, ; R Schrader, R Seiler, D A Uhlenbrock, Springer Lect, Mathematical problems in theoretical physics. 15347K. Symanzik, in Mathematical problems in theoretical physics, ed. R. Schrader, R. Seiler and D.A. Uhlenbrock, Springer Lect. Notes Phys. 153 (1982) 47; . Nucl. Phys. 226205Nucl. Phys. B226 (1983) 187, 205 . J N Simone, UKQCD CollaborationNucl. Phys. B (Proc. Suppl.). 30461UKQCD Collaboration, J.N. Simone, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 461 . B E Wilkes, University of EdinburghPh.D. thesis. in preparationB.E. Wilkes, Ph.D. thesis, University of Edinburgh (1993), in preparation . O F Hernandez, B R Hill, Phys. Lett. 289417O.F. Hernandez and B.R. Hill, Phys. Lett. B289 (1992) 417 . E Eichten, B R Hill, Phys. Lett. 234511E. Eichten and B.R. Hill, Phys. Lett. B234 (1990) 511; . Phys. Lett. 240193Phys. Lett. B240 (1990) 193 . G Martinelli, private communicationG. Martinelli, private communication

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{'abstract': 'We present results for heavy-light decay constants, using both propagating quarks and the static approximation, in O(a)-improved, quenched lattice QCD. At β = 6.2 on a 24 3 ×48 lattice we find f D = 185+ 2 − 2 and f Bs /f B = 1.22 + 4 − 3 , in good agreement with earlier studies. From the static theory we obtain f stat B = 253 + 16 − 15 + 105− 14 MeV. We also present results from a simulation at β = 6.0 on a 16 3 × 48 lattice, which are consistent with those at β = 6.2. In order to study the effects of improvement, we present a direct comparison of the results using both the Wilson and the improved action at β = 6.0.', 'arxivid': 'hep-lat/9308020', 'author': ['R M Baxter \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'S P Booth \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'K C Bowler \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'S Collins \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'D S Henty \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'R D Kenway \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'D G Richards \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'H P Shanahan \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'J N Simone \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'A D Simpson \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'B E Wilkes \nDepartment of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland\n', 'A K Ewing \nThe University\nSO9 5NHSouthamptonUK\n', 'L Lellouch \nThe University\nSO9 5NHSouthamptonUK\n', 'C T Sachrajda \nThe University\nSO9 5NHSouthamptonUK\n', 'H Wittig \nThe University\nSO9 5NHSouthamptonUK\n'], 'authoraffiliation': ['Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'Department of Physics\nPhysics Department\nThe University of Edinburgh\nEH9 3JZEdinburghScotland', 'The University\nSO9 5NHSouthamptonUK', 'The University\nSO9 5NHSouthamptonUK', 'The University\nSO9 5NHSouthamptonUK', 'The University\nSO9 5NHSouthamptonUK'], 'corpusid': 119362995, 'doi': '10.1103/physrevd.49.1594', 'github_urls': [], 'n_tokens_mistral': 18070, 'n_tokens_neox': 14878, 'n_words': 9246, 'pdfsha': 'befdcabfd16d112521c05bbc6b5bd8e5da285d7f', 'pdfurls': ['https://export.arxiv.org/pdf/hep-lat/9308020v1.pdf'], 'title': ['Quenched Heavy-Light Decay Constants UKQCD Collaboration', 'Quenched Heavy-Light Decay Constants UKQCD Collaboration'], 'venue': []}

arxiv

Frame-dragging effect in the field of non rotating body due to unit gravimagnetic moment 21 Feb 2018 (Dated: February 23, 2018) Alexei A Deriglazov Depto. de Matemática, ICE Universidade Federal de Juiz de Fora MGBrazil Walberto Guzmán Ramírez Depto. de Matemática, ICE Universidade Federal de Juiz de Fora MGBrazil Frame-dragging effect in the field of non rotating body due to unit gravimagnetic moment 21 Feb 2018 (Dated: February 23, 2018) Nonminimal spin-gravity interaction through unit gravimagnetic moment leads to modified Mathisson-Papapetrou-Tulczyjew-Dixon equations with improved behavior in the ultrarelativistic limit. We present exact Hamiltonian of the resulting theory and compute an effective 1 c 2 -Hamiltonian and leading post-Newtonian corrections to the trajectory and spin. Gravimagnetic moment causes the same precession of spin S as a fictitious rotation of the central body with angular momentum J = M m S. So the modified equations imply a number of qualitatively new effects, that could be used to test experimentally, whether a rotating body in general relativity has null or unit gravimagnetic moment. The manifestly generally covariant Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations [1][2][3][4][5][6] are widely used in general relativity to describe a rotating test body in pole-dipole approximation. In the current literature (see [7][8][9][10][11][12] and references therein), they usually appear in the form given by Dixon ∇P µ = − 1 4 θ µνẋ ν , ∇S µν = 0 ,(1) where θ µν = R µναβ S αβ is the gravitational analogy of the electromagnetic field strength F µν [7,13]. (Our spintensor S µν is twice of that of Dixon. Besides, in the last equation we omitted the term 2P [µẋν] , which does not contribute in 1 c 2 -approximation we are interested in the present work. Concerning other notation, see the footnote 1 .) Together with the spin supplementary condition 2 S µν P ν = 0, MPTD equations prescribe the evolution of both trajectory and spin of the body in 1/c 2 -approximation. Starting from the pioneer works, MPTD equations were considered as a Hamiltonian-type system. Following this spirit in the recent work [15], we explicitly * Electronic address: alexei.deriglazov@ufjf.edu.br † Electronic address: wguzman@cbpf.br 1 Our variables are taken in arbitrary parametrization τ , theṅ x µ = dx µ dτ and the covariant derivative is ∇ω µ = dω µ dτ + Γ µ αβẋ α ω β . The square brackets mean antisymmetrization, ω [µ π ν] = ω µ π ν − ω ν π µ . We often miss the four-dimensional indexes and use the notationẋ µ Nµνẋ ν =ẋNẋ, N µ νẋ ν = (Nẋ) µ , ω 2 = gµν ω µ ω ν , µ, ν = 0, 1, 2, 3, sign gµν = (−, +, +, +). Suppressing the indexes of three-dimensional quantities, we use bold letters. The tensor of Riemann curvature is R σ λµν = ∂µΓ σ λν − ∂ν Γ σ λµ + Γ σ βµ Γ β λν − Γ σ βν Γ β λµ . 2 While the Lagrangian and Hamiltonian formalisms dictate [14] the condition (2), in the multipole approach there is a freedom in the choice of a spin supplementary condition, related with the freedom to choose a representative point of the body [3,4,6]. Different conditions lead to the same results for observables in 1 c 2 -approximation, see [5,36,42]. realized this idea by constructing the minimal interaction os spin with gravity in the Lagrangian of vector model of spinning particle, and showed that this indeed leads to MPTD equations in the Hamiltonian formalism. This allowed us to study ultrarelativistic limit in exact equations for the trajectory of MPTD particle in the laboratory time. Using the Landau-Lifshitz (1 + 3)decomposition [16] we observed that, unlike a geodesic equation, the MPTD equations lead to the expression for three-acceleration which contains divergent terms as v → c [13]. Therefore it seems interesting to find a generalization of MPTD equations with improved behavior in the ultrarelativistic regime. This can be achieved, if we add a nonminimal spin-gravity interaction through gravimagnetic moment κ [13]. κ = 0 corresponds to the MPTD equations. The most interesting case turns out to be κ = 1 (gravimagnetic body). Keeping only the terms, which may contribute in the leading post-Newtonian approximation, this gives the modified equations (among other equations, see below) ∇P µ = − 1 4 θ µνẋ ν − √ −ẋ 2 32mc (∇ µ θ σλ )S σλ , ∇S µν = √ −ẋ 2 4mc θ [µ α S ν]α .(3) Comparing (3) with (1), we see that unit gravimagnetic moment yields quadratic in spin corrections to MPTD equations in 1 c 2 -approximation. Both acceleration and spin torque of gravimagnetic body have reasonable behavior in ultrarelativistic limit [13]. In the present work we study the modified equations and the corresponding effective Hamiltonian in the regime of small velocities, and compute 1 c 2 -corrections due to the extra-terms appeared in (3). In Schwarzschild and Kerr space-times, the modified equations predict a number of qualitatively new effects, that could be used to test experimentally, whether a rotating body in general relativity has null or unit gravimagnetic moment. Let us briefly describe the variational problem which implies the modified equations (3). In the vector model of spin presented in [17], the configuration space consist of the position of the particle x µ (τ ), and the vector ω µ (τ ) attached to the point x µ (τ ). Minimal interaction with gravity is achieved by direct covariantization of the free action, initially formulated in Minkowski space. That is we replace η µν → g µν , and usual derivative of the vector ω µ by the covariant derivative:ω µ → ∇ω µ . The nonminimal spin-gravity interaction through the gravimagnetic moment κ can be thought as a deformation of original metric: g µν → σ µν = g µν + κR α µ β ν ω α ω β , with the resulting Lagrangian action [13] S = − dτ (mc) 2 − α ω 2 × √ −ẋN KσNẋ − ∇ωN KN ∇ω + 2λẋN KN ∇ω. (4) We have denoted K = (σ − λ 2 g) −1 , where λ is the only Lagrangian multiplier in the theory. The matrix N µν ≡ g µν − ωµων ω 2 is a projector on the plane orthogonal to ω: N µν ω ν = 0. The parameter α determines the value of spin, in particular, α = 3 2 4 corresponds to the spin onehalf particle. In the spinless limit, ω µ = 0 and α = 0, Eq. (4) reduces to the standard Lagrangian of a point particle, −mc −g µνẋ µẋν . The action (4) is manifestly invariant under general-coordinate transformations as well as under reparametrizations of the evolution parameter τ . Besides, there is one more local symmetry, which acts in the spin-sector and called the spin-plane symmetry: the action remains invariant under rotations of the vectors ω µ and π µ = ∂L ∂ω µ in their own plane [18]. Being affected by the local transformation, these vectors do not represent observable quantities. But their combination (5) is an invariant quantity, which represents the spintensor of the particle. In Eq. (5), we decomposed the spin-tensor into three-dimensional spin-vector S = 1 2 (S 23 , S 31 , S 12 ), and dipole electric moment [19] S µν = 2(ω µ π ν − ω ν π µ ) = (S i0 = D i , S ij = 2ǫ ijk S k ),D i . For the general-covariant and spin-plane invariant variables x µ , P µ = p µ − Γ β αµ ω α π β and S µν (here p µ = δS δẋ µ ), the Hamiltonian equations of motion of the theory (4) acquire especially simple form when κ = 1. In 1 c 2approximation, we obtained the equations (3), accompanied by the Hamiltonian equation for x µ ,ẋ µ = √ −ẋ 2 mc P µ , the latter can be identified with velocity-momentum relation implied by MPTD equations [13]. Besides the dynamical equations, the variational problem (4) implies the mass-shell constraint T ≡ P 2 + κ 16 θ µν S µν + (mc) 2 = 0,(6) and the spin-sector constraints P ω = 0 , P π = 0, ωπ = 0 and π 2 − α ω 2 = 0. Their meaning becomes clear if we consider their effect over the spin-tensor. The secondclass constraints P ω = 0 and P π = 0 imply the spin supplementary condition (2), while the remaining firstclass constraints fix the value of square of the spin-tensor, S µν S µν = 8α. The equations imply that only two components of spin-tensor are independent, as it should be for an elementary spin one-half particle. The mass-shell constraint (6) look like that of a spinning particle with gyromagnetic ratio g, P 2 − eg c F µν S µν + (mc) 2 = 0. In view of this similarity, the interaction constant κ has been called gravimagnetic moment [7,20]. Although the vector model of spin has been initially developed to describe an elementary particle of spin onehalf, it can be adopted to study a rotating body in general relativity. The action (4) with κ = 0 implies MPTD equations, and the only difference among two formalisms is that values of momentum and spin are conserved quantities of MPTD equations, while in the vector model they are fixed by constraints. In summary [13], to study the class of trajectories of a body with √ −P 2 = k and S 2 = β, we can use our spinning particle with m = k c and α = β 8 . Although the post-Newtonian approximation can be obtained by direct computations on the base of equations of motion, we prefer to work with an approximate Hamiltonian. This gives a more transparent picture of nonminimal interaction, in particular, display strong analogy with a spinning particle with magnetic moment in electromagnetic background. We could consider a Hamiltonian corresponding to either Poisson or Dirac brackets. We work with Dirac bracket 3 for the second-class constraints P ω = 0 and P π = 0, since in this case the relativistic Hamiltonian acquires the conventional form H rel = λ 2 T . According to the procedure described in [21], exact Hamiltonian for dynamical variables x(t), p(t) and S(t) as functions of the coordinate time t = x 0 c is H = −cp 0 , where the conjugated momentum p 0 is a solution to the mass-shell constraint (6). Solving the constraint, we obtain H = c −g 00 (mc) 2 + γ ij P i P j + 1 16 (θS)− cπ µ Γ µ 0ν ω ν + cg 0i g 00 P i ,(7) where γ ij = g ij − g 0i g 0j g 00 . Let us consider a stationary, asymptotically flat metric of a spherical body with mass M and angular momentum J. In the post-Newtonian approximation 4 1 c 4 , this reads [26,31] ds 2 = −1 + 2GM c 2 r − 2G 2 M 2 c 4 r 2 (dx 0 ) 2 − 4G ǫ ijk J j x k c 3 r 3 dx 0 dx i + 1 + 2GM c 2 r dx i dx i .(8) 3 The Dirac bracket turns the spinning particle into intrinsically noncommutative theory. This could manifest itself in various applications [22][23][24]. In particular, our Hamiltonian differs from those suggested by other groups, for instance [25]. They have been compared in [15]. 4 We omitted 1 c 4 -term in g ij as it does not contribute into the quantities under interest in 1 c 2 -approximation. To obtain the effective Hamiltonian, we expand all quantities in (7) in series up to 1 c 2 -order. To write the result in a compact form, we introduce the vector po- tential A J = 2G c [J × r r 3 ] for the gravitomagnetic field B J = [∇ × A J ] = 2G c 3(J·r)r−J r 3 , produced by rotation of central body (we use the conventional factor 2G c , different from that of Wald [31]. In the result, our B J = 4B W ald ). Besides we define the vector potential A S = M m G c [S × r r 3 ] of fictitious gravitomagnetic field B S = [∇ × A S ] = M m G c 3(S·r)r−S r 3 due to rotation of a gyroscope, as well as the extended momentum Π ≡ p + m c (A J + 2A S ). With these notation, 1 c 2 -Hamiltonian becomes similar to that of spinning particle in a magnetic field 5 H = c −g 00 (mc) 2 + g ij Π i Π j + 1 2c (B J + B S ) · S. (9) Note that the Hamiltonian c √ −g 00 (mc) 2 + g ij p i p j corresponds to the usual Lagrangian L = −mc −g µνẋ µẋν describing a spinless particle propagating in the Schwarzschild metric g µν . So, the approximate Hamiltonian (9) can be thought as describing a gyroscope orbiting in the field of Schwarzschild space-time and interacting with the gravitomagnetic field B J + B S . Effective Hamiltonian for MPTD equations turns out to be less symmetric: it is obtained from (9) excluding the term 1 2c (B S · S), while keeping the potential A S in Π. Hence the only effect of nonminimal interaction is the deformation of gravitomagnetic field of central body according to the rule B J → B J + B S .(10) Eq. (9) together with the Dirac brackets, also taken in 1 c 2 -approximation, gives us Hamiltonian equations of motion for x(t), p(t) and S(t). Excluding from them the momentum p, we obtain acceleration and spin precession of gravimagnetic particle in 1 c 2 -approximation. Total acceleration of gravimagnetic particle in 1 c 2approximation reads (herer = r/|r|) a = − M G r 2r + 4GM c 2 r 2 (r · v)v − GM c 2 r 2 v 2r + 4G 2 M 2 c 2 r 3r + 1 c (B J + B S ) × v + GM mc 2 r 3 [S × v + 3(S · (r × v))r] − 1 2mc ∇([B J + B S ] · S).(11) The first and second lines in (11) come from the first term of the effective Hamiltonian (9), while the last line comes from the second term of (9). The new term due to gravimagnetic moment is − 1 2mc ∇(B S · S). As it should be expected, other terms coincide with those of known from analysis of MPTD equations [27][28][29][30][31][32][33][34][35][36]42]. The first term in (11) represents the standard limit of Newtonian gravity and implies an elliptical orbit. The next three terms represent an acceleration in the plane of orbit and are responsible for the precession of perihelia [26][27][28]. The term 1 c B J × v represents the acceleration due to Lense-Thirring rotation of central body [29,30,32], while the remaining terms describe the influence of the gyroscopes spins on its trajectory. The gravitational dipole-dipole force 1 2mc ∇(B J · S) has been computed by Wald [31]. The new contribution due to nonminimal interaction, 1 2mc ∇(B S · S), is similar to the Wald term and is of the same (or less) magnitude. In a co-moving frame, the effective Hamiltonian (9) implies precession of spin dS dt = [Ω × S] with angular velocity vector Ω = 3GM 2c 2 r 2 [r × v] + 1 2c B J + 1 c B S .(12) The geodetic precession (first term in (12)) comes from the first term of effective Hamiltonian (9), while the frame-dragging precession (second term in (12)) is produced by the term 1 2c (B J · S). So they are the same for both gravimagnetic and MPTD particle. They have been computed by Schiff [36], and measured during the Stanford Gravity Probe B experiment [37]. The last term in (12) appears only for the gravimagnetic particle and depends on gyroscopes spin S. Hence, two gyroscopes with different magnitudes and directions of spin will precess around different rotation axes. Then the angle between their own rotation axes will change with time in Schwarzschild or Kerr space-time. Since the variation of the angle can be measured with high precision, this effect could be used to find out whether a rotating body has unit or null gravimagnetic moment. Comparing the last two terms in (12), we conclude that the precession of spin S due to gravimagnetic moment is equivalent to that of caused by rotation of the central body with momentum J f ict = M m S. Effective Hamiltonian for the case of the non rotating central body (Schwarzschild metric) is obtained from (9) by setting A J = B J = 0. We conclude that, due to the term 1 2c B S · S, spin of gravimagnetic particle will experience frame-dragging precession with angular velocity 1 c B S even in the field of non rotating central body, see (12). To estimate the relative magnitude of spin torques due to B J and B S , we represent them in terms of angular velocities. Assuming that the two bodies are spinning spheres of uniform density, we write J = I 1 ω 1 and S = I 2 ω 2 , where ω i is angular velocity and I i = (2/5)m i r 2 i is the moment of inertia. Then the last two terms in (12) read Ω f d = 2Gm 1 r 2 1 5c 2 r 3 3 [ω 1 + ρ 2 ω 2 ] ·r r − (ω 1 + ρ 2 ω 2 ) ,(13) where ρ ≡ r 2 /r 1 . Note that Ω f d does not depend on mass of the test particle. The ratio ρ 2 is extremely small for the case of Gravity Probe B experiment, so the MPTD and gravimagnetic bodies are indistinguishable in this experiment. For a system like Sun-Mercury ρ 2 ∼ 10 −5 . For a system like Sun-Jupiter ρ 2 ∼ 10 −2 . The two torques could have a comparable magnitudes in a binary system with stars of the same size [38][39][40] (then ρ = 1), but one of them much heavier than the other (neutron star or white dwarf). Then our approximation of a central field is reasonable, and according to Eq. (13), the framedragging effect due to gravimagnetic moment becomes comparable with the Schiff frame-dragging effect. To compare the two effects in a binary system with arbitrary masses, we need to go beyond the central-field approximation. Probably, this case can be reduced to the central-field approximation following the procedure [41][42][43]. Here the square root should be expanded up to 1 c 2 -order. Acknowledgments We thank Wenbin Lin for helpful discussions. The work of AAD has been supported by the Brazilian foundation CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico -Brasil). WGR thanks CAPES for the financial support (Programm PNPD/2011). . M Mathisson, Acta Phys. Polon. 61011Gen. Rel. Grav.M. Mathisson, Acta Phys. Polon. 6 (1937) 163; Republi- cation: Gen. Rel. Grav. 42, 1011 (2010). . V A Fock, J. Phys. USSR. 181V. A. Fock, J. Phys. USSR, 1, 81 (1939). . A Papapetrou, Proc. Roy. Soc. Lond. A. 209248A. Papapetrou, Proc. Roy. Soc. Lond. A 209, 248 (1951). . 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{'abstract': 'Nonminimal spin-gravity interaction through unit gravimagnetic moment leads to modified Mathisson-Papapetrou-Tulczyjew-Dixon equations with improved behavior in the ultrarelativistic limit. We present exact Hamiltonian of the resulting theory and compute an effective 1 c 2 -Hamiltonian and leading post-Newtonian corrections to the trajectory and spin. Gravimagnetic moment causes the same precession of spin S as a fictitious rotation of the central body with angular momentum J = M m S. So the modified equations imply a number of qualitatively new effects, that could be used to test experimentally, whether a rotating body in general relativity has null or unit gravimagnetic moment.', 'arxivid': '1802.08079', 'author': ['Alexei A Deriglazov \nDepto. de Matemática, ICE\nUniversidade Federal de Juiz de Fora\nMGBrazil\n', 'Walberto Guzmán Ramírez \nDepto. de Matemática, ICE\nUniversidade Federal de Juiz de Fora\nMGBrazil\n'], 'authoraffiliation': ['Depto. de Matemática, ICE\nUniversidade Federal de Juiz de Fora\nMGBrazil', 'Depto. de Matemática, ICE\nUniversidade Federal de Juiz de Fora\nMGBrazil'], 'corpusid': 119520581, 'doi': '10.1016/j.physletb.2018.01.063', 'github_urls': [], 'n_tokens_mistral': 8298, 'n_tokens_neox': 6805, 'n_words': 4010, 'pdfsha': '8709e4e0aeda832607fd9498c72ce2179f0e2b18', 'pdfurls': ['https://arxiv.org/pdf/1802.08079v1.pdf'], 'title': ['Frame-dragging effect in the field of non rotating body due to unit gravimagnetic moment', 'Frame-dragging effect in the field of non rotating body due to unit gravimagnetic moment'], 'venue': []}

arxiv

The Gauss-Dirichlet Orbit Number 20 Sep 2008 The Gauss-Dirichlet Orbit Number 20 Sep 2008 To the memory of Pierre KaplanIn [1] (Section 91) Dirichlet attaches a nonnegative integer C(d, m) to each pair (d, m) where d is an integer congruent to 0 or 1 mod 4 and m is a nonzero integer, and expresses, in some particular cases, C(d, m) in terms of Jacobi symbols. The construction was already implicit in Article 169 of Gauss's Disquisitiones Arithmeticae (called "Disquisitiones" henceforth), and can be described as follows.Let F be the (nonempty) set of those quadratic formswhere a, b, c are integers satisfying b 2 − 4ac = d, and denote again by f the corresponding function from Z 2 to Z. Let S be a representative system for the orbits of the natural action of G := SL(2, Z) on F ; observe that the stabilizer SO(f ) of f ∈ S in G acts on f −1 (m); and let C f be the cardinality of the orbit set. Then C(d, m) is the sum of the C f when f runs over S:Clearly C(d, m) doesn't depend on the choice of S. In [3] (Theorem 204) Landau enlarges the validity domain of Dirichlet's formula by replacing Jacobi symbols with Kronecker symbols. The integer C(d, m) is computed for d nonsquare and m arbitrary by Huard, Kaplan and Williams in [2] (Theorem 9.1), and by Sun and Williams in [5] (Theorems 4.1 and 4.2). In [5] the authors make the crucial observation that C(d, m) is multiplicative in m. They prove this multiplicativity for nonsquare d, but it holds for all d, as follows immediately from Article 169 of the Disquisitiones. In [4] Rudnick gives a definition of C(d, m) which is clearer than Dirichlet's. Here we compute C(d, m) in full generality. Thank you to Keith Matthews, Zeév Rudnick and Kenneth Williams for their friendly and efficient help. Gauss's Observation. For f in S, let f −1 (m) ′ be the set of those elements of f −1 (m) which have coprime coordinates; and put T := {t ∈ Z/2mZ | t 2 ≡ d mod 4m}.Then there is a unique map ϕ f from f −1 (m) ′ to T such that for any z in f −1 (m) ′ and any g in G mapping (1, 0) to z we have f g = [m, n, ℓ], ϕ f (z) = n mod 2m. Moreover the ϕ f induce a bijection f ∈S SO(f )\f −1 (m) ′ ∼ −→ T. In particular C(d, m) is finite. Moreover we have C(d, mn) = C(d, m)C(d, n) whenever m and n are coprime -property which we express by saying that the function C(d, ?) is multiplicative. Here is a proof of Gauss's Observation. Put T := {t ∈ Z | t 2 ≡ d mod 4m}. Define, for f in S, the map ϕ f from f −1 (m) ′ to T as follows. Let z be in f −1 (m) ′ . There is a g in G which maps (1, 0) to z. Choose such a g, note that f g is equal to [m, n, ℓ] for some (n, ℓ) in T × Z, and define ϕ f (z) as n. Define the map ψ from T to f −1 (m) ′ as follows. Let n be in T . Then there is a unique pair (ℓ, f ) in Z × S for which [m, n, ℓ] is equal to f g for some g in G. Choose such a g, note that g(1, 0) is in f −1 (m) ′ , and define ψ(n) as g(1, 0). One checks that the map ϕ : f −1 (m) ′ → T induced by the ϕ f doesn't depend on the choices made to define the ϕ f , that the map ψ : T → SO(f )\f −1 (m) ′ induced by ψ doesn't depend on the choices made to define ψ, that ϕ factors through a map ϕ : SO(f )\f −1 (m) ′ → T, that ψ factors through a map ψ : T → SO(f )\f −1 (m) ′ , and that ϕ and ψ are inverse. Of course there are many verifications to make, but they are straightforward. Obviously Gauss couldn't use exactly this language, but the substance of the above argument is clearly contained in Article 169 of the Disquisitiones. Let us fix d and let m vary. To prove the multiplicativity of C(d, m) it suffices to prove that of the cardinality of T . We have the following lemma. Let α and µ be arithmetic functions defined on the nonzero integers. Assume µ is multiplicative, 2α(x) = µ(4x) for all x, and µ(4) = 2. Then α is multiplicative. Now take the cardinality of T as α(m), and take the number of square roots of d mod x as µ(x) -a multiplicative function by the Chinese Remainder Theorem. Definition of K(d, m). Fix an integer d congruent to 0 or 1 mod 4. Let χ be the map from the positive integers to the integers characterized by the following properties: In particular K(d, ?) is multiplicative. • χ(1) = 1, χ(p) = 0 if p is a prime dividing d, • χ(p) is the Legendre symbol ( d p ) if p is an odd prime not dividing d, • χ(2) = 1 if d ≡ 1 mod 8, χ(2) = −1 if d ≡ 5 mod 8, • χ(mn) = χ(m)χ(n) for all m, n. Claims. If d and m are coprime then C(d, m) = K(d, m). We clearly have C(d, ±1) = 1. By multiplicativity it only remains to compute C(d, m) when m is a power of a prime factor of d. Fix such a prime factor p, and write B(d, n) for C(d, p n ), where n is a nonnegative integer. As B(d, 0) and B(d, 1) are equal to 1 (easy verification), we henceforth assume n ≥ 2. Putting k := ⌊n/2⌋ + 1 we get B(0, n) = (p k − 1)/(p − 1). Let p be odd and d a nonzero multiple of p. Write d as p δ D with δ > 0 and D prime to p. As we clearly have B(d, n) = B(0, n) -which is already computed -for n ≤ δ, we can assume n > δ. Recall that d is the square of a p-adic integer if and only if δ is even and the Legendre symbol ( D p ) equals 1. Put k := ⌈(n−δ)/2⌉. Then B(d, n) is equal to B(0, n−2k)+2kp δ/2 if d is the square of a p-adic integer, and to B(0, n − 2k) otherwise. Consider now the case p = 2. Let δ be a positive integer. Then B(d, n) is equal to 2 δ − 1 if n is odd, and to 2 δ+1 − 1 if n is even. Assume d = 2 2δ+1 D with D odd. Then B(d, n) is equal to B(0, n) if n ≤ 2δ − 1, and to 2 δ − 1 if n ≥ 2δ. Assume d = 2 2δ D with D ≡ 3 mod 4. Then B(d, n) is equal to B(0, n) if n ≤ 2δ − 2, and to 2 δ − 1 if n ≥ 2δ − 2. Assume d = 2 2δ D with D ≡ 5 mod 8. Then B(d, n) is equal to B(0, n) if n ≤ 2δ − 2. Suppose n ≥ 2δ − 1.Assume d = 2 2δ D with D ≡ 1 mod 8, i.e. d is the square of a 2-adic integer. Then B(d, n) is equal to B(0, n) if n ≤ 2δ − 2. For k ≥ 0 we have B(d, 2δ − 1 + 2k) = 2 δ (2k + 1) − 1, B(d, 2δ + 2k) = 2 δ+1 (k + 1) − 1. For n > 0 0define the Kronecker symbol by ( d n ) := χ(n), and for all nonzero integer m put Corollary 1 . 1The functions C(d, ?) and K(d, ?) coincide if and only if d has no odd square factor and d is not an even square mod 16. Corollary 2. The function n → C(d, p n ) is bounded if and only if d is not the square of a p-adic integer.Lemma. For d, n as above, let 2B ′ (d, n) be the number of roots of the congruencex 2 ≡ d mod 4p n .In particular, if p > 2, then B ′ (d, n) is the number of roots of x 2 ≡ d mod p n . Moreover we haveB(d, n) = 0≤j≤⌊n/2⌋ B ′ (d, n − 2j).Proofs. Gauss's Observation, along with Articles 104 and 105 of the Disquisitiones, implies the Lemma and the Claims.In the following two examples S reduces to a singleton. See the Disquisitiones or[1] for a proof of this fact. The other statements follow from Corollary 1. Let m be a nonzero integer. Example 1 . 1Let d be −4. We can take S := {f } with f := X 2 + Y 2 . We have | SO(f )| = 4, and the number of representations of m as a sum of two squares is four times M − N, where M is the number of divisors of m congruent to 1 mod 4, whereas N is the number of divisors of m congruent to −1 mod 4. Example 2. Let d be 8. We can take S := {f } with f := X 2 − 2Y 2 . Then C(8, m) is the number of divisors of m congruent to ±1 mod 8 minus the number of divisors of m congruent to ±3 mod 8. Vorlesungenüber Zahlentheorie, 1863. Peter Dirichlet, Gustav, Lectures on Number Theory. AMS translation)Dirichlet, Peter Gustav. Vorlesungenüber Zahlentheorie, 1863. ("Lectures on Number Theory", AMS translation). The Chowla-Selberg formula for genera. James G Huard, Pierre ; Kaplan, Williams, S Kenneth, Acta arithmetica. 73Huard, James G.; Kaplan, Pierre; Williams, Kenneth S. The Chowla-Selberg formula for genera, Acta arithmetica 73 (1995), 271-301. Elementary Number Theory. Edmund Landau, AMSLandau, Edmund. Elementary Number Theory. AMS 1999. The mass formula for binary quadratic forms. Zeev Rudnick, Rudnick, Zeev. The mass formula for binary quadratic forms. http://www.math.tau.ac.il/∼rudnick/courses/adv numth.html . Zhi-Hong Sun, Sun, Zhi-Hong; On the number of representations of n by ax 2 + bxy + cy 2. Kenneth S Williams, Acta Arith. 1222Williams, Kenneth S. On the number of representations of n by ax 2 + bxy + cy 2 . Acta Arith. 122 (2006), no. 2, 101-171. http://mathstat.carleton.ca/∼williams/papers/papers.html Yves Pierre-, Gaillard gauss-dirichlet.orbit.number.080910, Wed Sep 10 08:55:21 CEST. Pierre-Yves Gaillard gauss-dirichlet.orbit.number.080910, Wed Sep 10 08:55:21 CEST 2008.

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{'abstract': 'To the memory of Pierre KaplanIn [1] (Section 91) Dirichlet attaches a nonnegative integer C(d, m) to each pair (d, m) where d is an integer congruent to 0 or 1 mod 4 and m is a nonzero integer, and expresses, in some particular cases, C(d, m) in terms of Jacobi symbols. The construction was already implicit in Article 169 of Gauss\'s Disquisitiones Arithmeticae (called "Disquisitiones" henceforth), and can be described as follows.Let F be the (nonempty) set of those quadratic formswhere a, b, c are integers satisfying b 2 − 4ac = d, and denote again by f the corresponding function from Z 2 to Z. Let S be a representative system for the orbits of the natural action of G := SL(2, Z) on F ; observe that the stabilizer SO(f ) of f ∈ S in G acts on f −1 (m); and let C f be the cardinality of the orbit set. Then C(d, m) is the sum of the C f when f runs over S:Clearly C(d, m) doesn\'t depend on the choice of S. In [3] (Theorem 204) Landau enlarges the validity domain of Dirichlet\'s formula by replacing Jacobi symbols with Kronecker symbols. The integer C(d, m) is computed for d nonsquare and m arbitrary by Huard, Kaplan and Williams in [2] (Theorem 9.1), and by Sun and Williams in [5] (Theorems 4.1 and 4.2). In [5] the authors make the crucial observation that C(d, m) is multiplicative in m. They prove this multiplicativity for nonsquare d, but it holds for all d, as follows immediately from Article 169 of the Disquisitiones. In [4] Rudnick gives a definition of C(d, m) which is clearer than Dirichlet\'s. Here we compute C(d, m) in full generality. Thank you to Keith Matthews, Zeév Rudnick and Kenneth Williams for their friendly and efficient help. Gauss\'s Observation. For f in S, let f −1 (m) ′ be the set of those elements of f −1 (m) which have coprime coordinates; and put T := {t ∈ Z/2mZ | t 2 ≡ d mod 4m}.Then there is a unique map ϕ f from f −1 (m) ′ to T such that for any z in f −1 (m) ′ and any g in G mapping (1, 0) to z we have', 'arxivid': '0809.1918', 'author': [], 'authoraffiliation': [], 'corpusid': 6287432, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 3281, 'n_tokens_neox': 2848, 'n_words': 1722, 'pdfsha': '408e11c352378f24a243359a5bf24f63a9eb0d01', 'pdfurls': ['https://arxiv.org/pdf/0809.1918v2.pdf'], 'title': ['The Gauss-Dirichlet Orbit Number', 'The Gauss-Dirichlet Orbit Number'], 'venue': []}

arxiv

Spectrum-aware Multi-hop Task Routing in Vehicle-assisted Collaborative Edge Computing Member, IEEEYiqin Deng Senior Member, IEEEHaixia Zhang Member, IEEEXianhao Chen Fellow, IEEEYuguang Fang Spectrum-aware Multi-hop Task Routing in Vehicle-assisted Collaborative Edge Computing 1 Multi-access edge computing (MEC) is a promising technology to enhance the quality of service, particularly for low-latency services, by enabling computing offloading to edge servers (ESs) in close proximity. To avoid network congestion, collaborative edge computing has become an emerging paradigm to enable different ESs to collaboratively share their data and computation resources. However, most papers in collaborative edge computing only allow one-hop offloading, which may limit computing resource sharing due to either poor channel conditions or computing workload at ESs one-hop away. By allowing ESs multi-hop away to also share the computing workload, a multihop MEC enables more ESs to share their computing resources. Inspired by this observation, in this paper, we propose to leverage omnipresent vehicles in a city to form a data transportation network for task delivery in a multi-hop fashion. Here, we propose a general multi-hop task offloading framework for vehicle-assisted MEC where tasks from users can be offloaded to powerful ESs via potentially multi-hop transmissions. Under the proposed framework, we develop a reinforcement learning based task offloading approach to address the curse of dimensionality problem due to vehicular mobility and channel variability, with the goal to maximize the aggregated service throughput under constraints on end-to-end latency, spectrum, and computing resources. Numerical results demonstrate that the proposed algorithm achieves excellent performance with low complexity and outperforms existing benchmark schemes.Index Terms-Computation offloading, Collaborative edge computing, Vehicular networks, Multi-hop service request routing, Deep reinforcement learning (DRL). I. INTRODUCTION Multi-access edge computing (MEC) has been identified as a promising architecture for computing services that aims to provide real-time or low latency services to end-users located in close proximity [1], [2]. One of the primary techniques utilized in MEC is computation offloading, which enables computing tasks to be processed locally or offloaded to an edge server (ES) based on the availability of local computing resources and transmission conditions [3]. This approach is beneficial for mobile devices (MDs) that are typically limited in terms of their computing capability, storage capacity, and battery power [4], [5]. Moreover, through effective computation offloading in MEC, end-to-end (e2e) latency for emerging capability-demanding or latency-sensitive applications can be drastically reduced, ultimately providing high quality-ofservices to end users [6], [7]. To optimize resource utilization and efficiency, extensive research efforts have been dedicated to addressing computation offloading in MEC [8]- [13]. Previous research efforts have primarily focused on computation offloading and resource optimization that directly associate users with ESs within a user's communication range (i.e., one-hop away ESs), considering computing resources and/or communication resource optimization at a single ES [8]- [10]. However, this approach may fall short in practice due to the lack of coordination among ESs, which are required for better load balancing [14]. Despite research efforts attempting to enable resource-constrained ESs to collaborate in processing computation-intensive tasks to achieve workload balancing, the current literature is primarily concerned with one-hop offloading between MDs and ESs [11]- [13]. Such an approach incurs an implicit assumption, whereby communication resources available at the ESs one-hop away are sufficient for uploading complex computing tasks. This suppositional approach may not always be effective in reality. Specifically, the single-hop offloading approaches may not work well under resource-constrained scenarios. For example, when considering MEC-enabled surveillance video analytics for public safety applications in smart cities [15], where a large volume of high-resolution videos should be transported from street cameras to distributed MEC servers for processing, task uploading and/or computing may fail when the spectrum/computing resources are insufficient to serve the service demands at the spot. To enhance resource utilization, a data transportation network is needed for task delivery from end users to appropriate edge servers with available computing and spectrum resources via potentially multi-hop delivery [16]. Therefore, to optimize resource utilization and efficiency, future research efforts in MEC systems ought to explore better coordination and collaboration between MDs and ESs to address computation offloading issues, which will lead to improved throughput performance, better workload balancing, and ultimately, better resource utilization [17]. Aiming at small computing latency while avoiding network congestion, Dai et al. [18] recently proposed a cooperative offloading framework in device-to-device (D2D)-assisted MEC networks, where both ESs and idle MDs enable offloading services for computing-intensive industrial tasks. Here, each user delivers offloading service for at most one neighbor MD to avoid queueing latency as the communication coverage by D2D links is small. Chukhno et al. [19] claimed that reliability in public safety services can be achieved via multi-hop relaying, which is considered to be one of the key technologies facilitating enhanced system performance in future 5G+ systems. For example, it allows establishing direct connections between devices in scenarios outside the coverage area, thus ensuring first responders with the connectivity they need, especially in hazardous situations. As discussed in [20], the Third Generation Partnership Project (3GPP) has already identified new study and work items for New Radio (NR) Vehicle-to-Everything (V2X) side-link (SL) communication within Release 17, among which the concept of MD relaying has been proposed to extend the coverage range. Different from utilizing a single relay, which is referred to in 3GPP as a single-hop NR SL-based relay, forward compatibility for multi-hop relay support in a future release will be taken into account [20]. These standardization progresses demonstrate the importance and feasibility of relaying in vehicular networks, which can naturally be leveraged for computing task offloading. Motivated by the performance enhancement brought by such multi-hop D2D transmissions, in our prior work [21], we have explored vehicle-assisted multi-hop transmissions to balance the computing workload at ESs under a simple scenario with one MD and multiple ESs. In this paper, we propose to employ vehicles ubiquitously available in a city to form a data transportation network, which could facilitate multi-hop task offloading between a user and the associated ES. Due to the omnipresence of vehicles, this approach is economically sound because no additional fixed relays are needed. Moreover, thanks to the short device-vehicle and inter-vehicle distances, MDs and vehicles can employ short-range multi-hop transmissions with low transmit power, thereby causing less interference and improving network-wide spectrum reuse [22]. When extending to a more general case, new challenges will arise. One critical issue in multi-hop offloading is to make a trade-off between the communication overhead and the computing capability to satisfy quality-of-service (QoS) requirements [23], [24], which will increase the complexity of the task offloading problem. Such an issue is further complicated by the dynamic nature of network topology due to vehicular mobility [25], [26]. To fill in this gap, this paper investigates the problem of spectrum-aware task offloading in vehicle-assisted multihop edge computing. The challenges are fourfold. First, the vehicular network environment is highly complicated and dynamic, which can hardly be captured by an accurate and mathematically solvable model. Thus, traditional task offloading methods are not suitable under this scenario. Second, dynamic task routing decisions are jointly made with taskserver assignments, which is more challenging than traditional routing with predetermined source and destination nodes. Third, the network-wide tradeoff between the communication and computing workloads further complicates the task offloading problem. At last, typical solutions based on queueing theory may not work well for the situation involving multi-hop routing and e2e QoS guarantees because a few strong assumptions (e.g., task arrivals at every source node and intermediate node follow a Poisson process) underpinning the analytical results may not hold and many problems in multipoint to multi-point queueing networks still remain open [27]. To tackle the above challenges, we first propose a general multi-hop task offloading framework for a vehicle-assisted MEC network, where different e2e paths can be simultaneously established between users and remote ESs via multi-hop transmissions by utilizing different groups of relay vehicles and the destination ES. This gives rise to a new task routing design problem in which the selected vehicles, the target ES, and their routing paths for different users need to be jointly optimized to balance the workloads in terms of both communication and computing to maximize the processed task size over the whole system. Although the system capacity can be enhanced by coordinating the network-wide resources, it is hard to guarantee service reliability with the uncertainty of vehicular trajectories. To deal with this issue, we resort to the multi-agent deep deterministic policy gradient (MAD-DPG) method, a deep reinforcement learning (DRL), which is capable of addressing issues with high dimensional states and huge action spaces [28]. To this end, we present a novel and highly effective MADDPG-based task offloading approach for a multi-hop MEC by learning network dynamics. Note that our approach is not restricted to vehicle-aided MEC and it can be easily extended to other multi-hop MEC systems facing similar challenges. Our main contributions can be summarized as follows. • We are the first to present the framework of multihop task offloading by coordinating the resources in a multi-hop multi-edge multi-user MEC system. Under the proposed framework, we formulate a throughput maximization problem subject to both communication and computing resource constraints and e2e latency requirements. • To solve the original optimization problem, we reconstruct a Markov decision process (MDP) based formulation for the task offloading decision-making under uncertainty of network topology. • We resort to a model-free DRL method, i.e., MADDPG, to find an effective task offloading solution under dynamic spectrum and computing resource constraints by learning the undetermined model via interactions with a vehicleassisted MEC environment. The remainder of this paper is organized as follows. In Section II, we present the related works. Section III describes the system model and problem formulation. In Section IV and V, we present the preliminaries for MADDPG and the MADDPG-based task offloading scheme, respectively. Section VI presents simulation results, and Section VI concludes this paper. II. RELATED WORK Most existing works on computation offloading in MEC focus on single-hop offloading from MDs to ESs. They can be roughly divided into two categories according to whether the cooperation between ESs is involved: resource optimization for an MEC with a single ES [8]- [10] or cooperative MEC over multiple ESs [11], [12]. In this section, we will first review the research status according to the above two categories, and then survey the related works on multi-hop task offloading from MDs to ESs. A. Resource optimization for an MEC with a single ES Cao et al. [8] proposed computation partitioning, dispatching, and scheduling algorithms for 5G-based edge computing systems, under the assumption that there is plenty of spectrum bandwidth, to support the data transmissions between MDs and an ES, which could parallelize computing tasks and fully utilize the computing resources at both the ES and MDs. Based on the observations that a considerable amount of data should be pre-stored and asymmetric spectrum bandwidth is required for uplink and downlink transmissions to support many emerging services (e.g., Augmented Reality (AR) services) at ESs, Poularakis et al. [9] studied the joint optimization of service placement and computation offloading for MEC networks with storage, computation, and communication constraints. While the optimization problem here is probably the most general one to minimize the computing workload offloaded to the centralized cloud under the above system consideration, they did not consider queueing at ESs, which is commonly encountered in practical systems. In [10], Deng et al. proposed a scheme to maximize the task completion ratio (throughput) in MEC under e2e latency constraints by using a tandem queue model to characterize the joint resource allocation of communications and computing. They also considered the stochasticity of involved processes, e.g., task arrivals, random channels, and varying computing power. However, this paper only focuses on a single MD and single ES scenario. It is also observed that all these works only study the scenarios that the computing tasks can directly be transmitted to the destination ES within the MD's communication range (i.e., one-hop) at one single ES without considering the cooperation among ESs. B. Cooperative MEC over multiple ESs By exploiting cooperation among ESs, tasks that arrive at one ES can be either processed locally or partially/fully offloaded to powerful ESs via backbone or backhaul links to enhance the quality of experience (QoE). In [11], Li et al. proposed an online cooperative offloading mechanism to optimize the decision of task admission and scheduling among ESs with the objective to minimize the long-term system cost by considering full offloading (i.e., binary offloading). In [12], Li et al. extended the cooperative computing framework in MEC to vehicular networks by considering challenges in computing result delivery due to the uncertainty of vehicular mobility. To address the complexity resulting from the dynamic network topologies in MEC-enabled vehicular networks, they proposed a location-aware offloading and computing strategy to coordinate ESs with partial offloading (i.e., computing at multiple ESs in parallel). However, they still assume that backbone/backhaul links have plenty of bandwidth, and hence will not pose any constraints on communications between ESs. It is also observed that all these works enable resourceconstrained ESs to help each other in processing computationintensive tasks, thereby enhancing computing workload balancing and resource utilization in MEC systems. C. Multi-hop task offloading between MDs and ESs The aforementioned research works generally make an implicit assumption that MDs can only offload tasks to ES one-hop away, which significantly restricts the solution space and limits resource sharing. For example, when a nearby server is overwhelmed with its computing, it is natural to offload a MD's task to other servers potentially unreachable by one-hop communications, or when too many MDs at one-hop away ESs are excessive, there is no spectrum used to offload data to one-hop away ESs, while there may exist multi-hop path connecting to multi-hop away ESs. In either case, multi-hop offloading may be leveraged to increase resource sharing and load balancing. As far as we know, [29] and [21] are probably the most related works tackling vehicle-assisted multi-hop task offloading as done in this paper. In [29], Hui et al. designed a request relay mechanism for MEC-enabled vehicular networks to reduce the cost of the relay service by taking the dynamic traffic conditions and the reputation of vehicles into consideration. However, they merely considered the limited transmission ranges of vehicles and ESs while ignoring resource constraints and QoS requirements. In [21], Deng et al. proposed a load-balanced relay mechanism for MEC in which the relay vehicle and destination ES are jointly determined according to the queueing status at an MD and traffic status, significantly enhancing the system performance. Nevertheless, Their work just considered a simple case with a single MD where the complicated task routing between multiple MDs and destinations is not involved. Different from these works, this paper intends to employ vehicles as relays for computing task delivery by taking advantage of the mobility and spectrum opportunities in vehicular environments. To deal with the "curse of dimensionality" arising from large-scale vehicular networks, we use DRL to find the multi-hop task routing paths. III. SYSTEM MODEL AND PROBLEM FORMULATION This section describes the proposed multi-hop task offloading framework in vehicle-assisted MEC. A. Multi-hop task offloading framework in vehicle-assisted MEC As shown in Fig. 1, we consider an MEC system with multiple MDs and ESs, where N distributed vehicles are deployed to assist in the communication from MDs to J remote ESs. Without loss of generality, we assume that i) MDs in this system can be either pedestrians on the roadside or passengers in the vehicle, and ii) tasks from MDs can either be offloaded to the ES within their communication range or offloaded to a remote ES via multi-hop transmission path assisted by vehicles. We consider a time-slotted system t ∈ T = {0, 1, 2, · · ·, T }. Suppose that each MD is associated with at most one ES at any given time while each ES can serve multiple MDs via proper user scheduling. As such, we focus on the design of a routing path for tasks generated from a set of MDs in one given time slot. For convenience, we denote the sets of MDs, vehicles, and ESs as I = {1, 2, · · ·, I}, N = {1, 2, · · ·, N }, and J = {1, 2, · · ·, J}, respectively. MD i may fail to access the service of an MEC server within its deadline D i because i) computing workload at the surrounding ESs is excessive or ii) the ES is out of its communication range or iii) the channel condition between MD i and the ES is poor or transmission channel between MD i and its surrounding ESs are excessively busy. Thus, it is possible that MD i seeks help from vehicles on the road to relay its data to an appropriate ES multi-hop away, improving the system capacity. We assume that a global controller has global knowledge of the network dynamics and makes offloading decisions for all users in a centralized manner. For example, we could take software-defined networking (SDN) design approach to implement our proposed MEC systems. To conclude, a multi-hop MEC service session for a task includes the following four steps while we ignore the procedure of result returning since the size of results in many practical applications (e.g., object detection results) is relatively small. 1) Offloading: when a computing task is generated at MD, it selects a relay vehicle within its communication range and offloads the computing data of the task to the vehicle. 2) Relaying: after a vehicle receives the computing data from the MD, it transfers data across vehicles on the road by choosing an appropriate route to the destination ES. 3) Uploading: when the relay vehicle arrives in the communication range of the destination ES, it uploads the carried data to the ES. 4) Computing: after the computing data is fully offloaded, the destination ES can process the computing task and send back the result to the MD after finishing the computing. Let p i ∈ P denote the route from MD i to a destination ES, where P is the set of feasible routes and it consists of none (i.e., one-hop transmission from MD i to the ES) or multiple vehicles (i.e., multi-hop routing) and one ES. Let a i,pi (t) = {0, 1} denote the offloading decision for MD i ∈ I at time slot t. Thus, we have pi∈P a i,pi (t) ≤ 1, ∀i, t.(1) According to the propagation model in 3GPP standards [12], the path loss between a transmitter and a receiver with distance d (km) can be computed as: Ψ(d) =40 1 − 4 × 10 −3 H log 10 d − 18 log 10 H + 21 log 10 f + 80(dB),(2) where H and f are the antenna height in meter and the carrier frequency in MHz, respectively. The distance between node a and b is denoted as D a,b . Thus, from the Shannon capacity theorem, the data rate between node a and b can be expressed as: R a,b (t) = B log 2 1 + P * 10 −Ψ(D a,b )/10 σ 2 ,(3) where σ 2 denotes the power of the Gaussian noise in the channel (e.g., the user-to-vehicle channel, the vehicle-to-vehicle (V2V) channel, or the vehicle-to-infrastructure channel), P represents the node's transmit power, and B represents the spectrum bandwidth used by the MD. Suppose the size of the data generated by MD i in time slot t is W i (t), we have the latency when delivering the data from MD i to the first relay vehicle n f irst : L trans i,n f irst (t) = a i,p W i (t) R i,n f irst (t) .(4) Moreover, the latency when relaying the data between two adjacent vehicles in the route p i , e.g., vehicles n and n , is: L trans n,n (t) = a i,pi W i (t) R n,n (t) .(5) Accordingly, the latency when uploading the data from the last relay vehicle n last to the destination ES j is: L trans n last ,j (t) = a i,p W i (t) R n last ,j (t) .(6) Let P v denote the set of adjacent vehicle pairs, e.g., vehicle pair (n, n ), in the route p i . Thus, the latency when forwarding the data from MD i to the destination ES j can be expressed as: L trans i,pi (t) = L trans i,n f irst (t) + (n,n )∈P v L trans n,n (t) + L trans n last ,j (t). (7) After the task from MD i is delivered and other tasks in the queue has been completed, the task can be processed by the dedicated ES. The latency when accomplishing task i generated in time slot t on ES j can be formulated as L comp i,pi (t) = κa i,pi (t)W i (t) C j ,(8) where κ is the computation cycle per bit data. Besides, the queueing latency of task from MD i at ES j is defined as the latency for finishing the uncompleted tasks offloaded in previous time slots {1, 2, · · ·, t − 1}, which can be formulated as follows: L queue i,pi (t) = max    i ∈I/i L comp i ,pi (t − 1) − , 0    ,(9) where is the duration of a time slot. Given the transmission latency, computing latency, and queueing latency, the e2e service latency for MD i can be formulated as follows: L i,pi (t) = L trans i,pi (t) + L comp i,pi (t) + L queue i,pi (t).(10) B. Problem formulation Our objective is to find a task routing policy α with the goal to maximize the aggregated throughput for the vehicle-assisted multi-hop MEC system while guaranteeing the end-to-end latency requirements from MDs. The optimization objective is thus formulated as max α t∈T i∈I pi∈P a i,pi (t)W i (t) · 1 {Li,p i (t)≤Di}(11) where 1 {Li,p i (t)≤Di} is the indicator function whose value takes 1 when the e2e service latency requirement of MD i is satisfied, or 0 otherwise. Note that L i,pi (t) is calculated after the task generated by user i at time slot t is accomplished in the current or the future time slot. Moreover, the objective function in (11) represents the total size of the tasks completed with latency requirements during the considered time duration T . With the optimization objective in (11), we have to take into account communication and computing resource constraints. To solve problem (11), we are facing three nontrivial challenges. First, it is hard to express the e2e latency in a closed form. As we mentioned in Section I, the queueing theory as a typical tool to address the e2e latency may not fit the problem of multi-hop task offloading since the task arrival assumption at every node fails to hold for the classical queueing models with multi-point to multi-point to have close form solutions. Second, it is infeasible to directly solve problem (11) by the traditional optimization method since it is a mixed-integer non-linear optimization problem. Third, even though we can model the system as a Markov decision process, it is hard to overcome the curse of dimensionality in terms of both state space and action space. Taking the action variable as an example, i.e., a i,pi (t), there exists I × N × J decisions to make in one time slot. Moreover, it is not efficient to use the traditional queueing theory to handle the problem with multiple outputs while the task offloading policy for each MD is given simultaneously in problem (11). Based on the above analysis, we resort to the deep deterministic policy gradient (DDPG) method to address these challenging issues since DDPG is not only good at solving optimization problems with large action spaces, but also has a good convergence performance as demonstrated in [30]. IV. PRELIMINARIES FOR DEEP DETERMINISTIC POLICY GRADIENT A. MDP-based Task Offloading Model To solve problem (11) with DDPG, we first model it as an Markov decision process (MDP) (S, A, P, R), where S and A are the sets of system states and actions, respectively, and P and R are the functions of state transition and reward, respectively. The specific definitions are given below. State space: The design of state space is to reflect the status of the considered system completely and informatively. Therefore, we build the state space S consisting of vehicle status, server status, and system workload. Vehicle status provides the information of the feasible relay vehicles and the channel states among vehicles. Server status includes the computing capability of ESs and the available bandwidth. System workload provides the information of the amount of input data from MDs and the number of queueing tasks at ESs. Action space: Based on the observed state, the actions can be chosen from the feasible action space A in each time slot whose element represents the routing path for each MD. Transition probability: Transition probability in MDP represents the probability that the system state moves from the current state s to the next state s when action a is taken, i.e., P a ss = P{s |(s, a)}. Reward: In an MDP, the reward is related to both state and action. When an action, e.g., a task scheduling policy, is selected under the current state, the corresponding reward will be received from the system, i.e., R a s = E{R|(s, a)}. In the considered problem, the reward function can be set according to the objective function (11). For the MDP, π(s, a) : S × A → [0, 1] is set to a policy that gives the probability of taking action a when in the state s. To obtain the expected long-term discounted reward, the value function Q of state s by taking policy π is Q(s, π) = E t∈T γ t R a s (t) ,(12) where γ ∈ [0, 1) is a discounting factor. By maximizing the value function across different states, we can obtain the optimal task scheduling policy π * : π * (s, a) = arg max s P(s |(s, a)) [R(s, a) + γQ(s , π * )] . B. Deep Deterministic Policy Gradient For the optimization in (13), the traditional dynamic programming cannot find the optimal policy as we have no knowledge about the transition probability P in the considered system. Therefore, we resort to model-free reinforcement learning, i.e., deep deterministic policy gradient (DDPG) [30], to learn the model via the interactions between agents and the environment. In DDPG, there are a total of four networks: the Actor, the Critic, and the corresponding target networks for the Actor and Critic, respectively. The target networks can be regarded as time-delayed copies of their original networks that slowly track the learned networks, which will significantly enhance the stability of learning. The specific functions of these four neural networks are as follows. 1) Actor network: The Actor network is in charge of the iterative update of policy network parameters and the direct maps from the current state to the current action. In this way, it interacts with the vehicle-assisted multi-hop MEC environment to generate the next state and reward. 2) Actor target network: The Actor target network outputs the next optimal action according to the next state sampled in the experience replay. The network parameters in the Actor target network are periodically copied from the Actor network. 3) Critic network: The Critic network is responsible for the iterative update of the parameters in the value network and calculating the current Q value. 4) Critic target network: The Critic target network calculates Q value according to the next state-action. The network parameters in the Critic target network are periodically copied from the Critic network. The above two target networks have "soft"-updates based on main networks, i.e., the target networks only update a small part based on the current network, to improve the stability of learning. That is, θ ← τ θ + (1 − τ )θ ,(14)w ← τ w + (1 − τ )w ,(15) where 0<τ 1 is the update frequency for the parameters in actor target network (θ) and critic target network (w). To improve the exploration capability and thus avoid getting stuck in a local optimum, DDPG typically adds noise (N t ) to the action (π θ (s)) produced by the actor network to get a new action, i.e., a = π θ (s) + N t . The loss functions for the critic network and the actor network are respectively defined as L(w) = 1 m m z=1 (y z − Q(φ(S z ), A z , w)) 2 ,(17) and L(θ) = − 1 m m z=1 Q(s, a, θ), z = 1, 2, · · ·, m,(18) where m is the number of samples (including eigenvector of state φ(S z ), and action A z ) from Replay Buffer D, y z is the target value of Q, V. MADDPG-BASED TASK ROUTING IN VEHICLE-ASSISTED MEC In this section, we elaborate how to leverage DDPG to solve our task routing problem. Here we exploit Multi-Agent Deep Deterministic Policy Gradient (MADDPG) to cope with the curse of dimensionality in the multi-hop task offloading optimization problem. To tackle the problem efficiently, the problem is decomposed and each MD acts as an agent to maximize the amount of the completed tasks (task throughput). A. Multi-Agent Deep Deterministic Policy Gradient Although DDPG can adapt to the environment of multidimensional actions, it is difficult for a single super-agent to learn large-scale decentralized policies whose action space grows exponentially with the number of participants [31]. MADDPG is an intuitive extension to the DDPG algorithm under a multi-agent system by decomposing a single monolithic agent into multiple simpler agents to reduce the dimensionality of the state and action spaces and thus overcome the scalability issue. In MADDPG, each agent makes the most suitable decision for itself, and multiple agents can achieve the common goal through cooperation. In this paper, we take advantage of MADDPG to train multiple agents for the optimization of multi-hop task offloading in vehicle-assisted MEC. B. State Space, Action Space and Reward Function 1) State Space: The state observed by MD i at time slot t is defined as s i (t) = {s w i (t), s s (t), s n (t), s l (t)},(19) where s w i (t) represents the number of tasks of MD i arriving in time slot t, s s (t) denotes the indexes of the selected ESs for all MDs at time slot t, s n (t) denotes the number of MDs which select the same ES in time slot t, and s l (t) denotes the remaining task size in the buffer at each ES in time slot t. 2) Action Space: In the system, every MD has to decide the serving ES. Thus, the action of user i at time slot t is expressed as a i (t) = {a s i (t)},(20) where a s i (t) is the index of the ES selected by MD i. Besides, let A s i (t) denote the set of the selection actions by the feasible destinations. Therefore, action a i (t) is valid if a s i (t) ∈ A s i (t). Note that the routing path from MD i to its destination ES will be uniquely determined if the ES is selected in MADDPG. For example, in this paper, we use the shortest path in terms of the travel distance between an MD and the associated ES. 3) Reward Function: Since each MD intends to maximize its completed tasks while meeting the required e2e latency, the immediate reward is represented as r i (t) = c i (t),(21) where c i (t) is defined as the total size of accomplished tasks in time slot t within the deadline, including tasks generated in the current time slot and those queued in the buffer. Note that the choice of the reward function is to approximately maximize the objective function defined in (11), i.e., the number of tasks accomplished with latency requirements in the long run. The gained reward depends on the action of an MD, i.e., the MD gets an immediate reward r i (t) given observed state s i (t) and action a i (t) in time slot t. Each MD aims at learning the MD N s I a I Actor TargetActor Loss TargetCritic Critic Vehicle-assisted MEC environment ¼ ¼ ¼ ¼ ¼ ¼ Fig. 2: The framework for an multi-agent deep deterministic policy gradient for task routing in vehicle-assisted MEC. optimal policy which maximizes the long-term reward, which is given by R i (t) = max E T −1 k=0 γ k r i (k + t) ,(22) where T is the number of consecutive time slots for calculating the long-term reward and 0<γ <1 is the discounting factor for determining the importance of the immediate reward and future rewards, where a smaller γ means that more importance is given to the immediate reward. Fig.2 illustrates the framework of MADDPG with two main procedures: i) using the global information to train the critic network, which is different from the traditional DDPG algorithm; and ii) using the local information to execute the actor network. Suppose that there are I agents (corresponding to the MDs in our system) in the vehicle-assisted MEC environment, in which we have two assumptions: i) the policy of each MD depends only on its own observed state, ii) the environment is unknown, and thus the reward for each agent and the next state after taking an action is unpredictable, which can only be acquired through the feedback from the environment. C. The Training and Execution of MADDPG Global training for the Critic network: In the training of MADDPG, the actor network selects an action according to the current state, and then the critic network can calculate a Q value according to the state-action pair as feedback to the action. The critic network is trained based on the estimated Q value and the actual Q value, and the actor network updates the policy based on the feedback from the critic network. To speed up the learning process of an MD, the input to the critic network for training includes both its own observation and the observations (e.g., the states and actions) of other agents in the environment. The parameters in the critic network are updated by minimizing the loss function based on Eq. (17). Local execution for the actor network: When each MD is fully trained, each actor network outputs appropriate actions according to its own state without the observed information from other MDs. The parameters in the actor network are updated using gradient descent according to the loss function based on Eq. (18). The training algorithm is summarized in Algorithm 1. We omit the introduction to Algorithm 1 due to the page limit. Please refer to [32] for more information. VI. PERFORMANCE EVALUATION We have conducted extensive studies to evaluate and compare the performance of the proposed MADDPG-based task routing in vehicle-assisted MEC with other benchmark solutions. The simulation experiments have been carried out on a ThinkPad X1 Carbon with a 4.7 GHz 12-Core Intel Core i7-1260P processor. The performance evaluation has been performed for two performance metrics: average throughput (task completion rate) and success rate of the algorithms. A. Simulation Settings 1) Simulation Parameters: We consider a road network shown in Fig. 3, where four ESs (i.e., black circles) are deployed as indicated in the figure, and multiple vehicles (blue rectangles) are moving following traffic rules, i.e., subject to speed limits, safe distance, and traffic lights. The considered vehicle is 4 m in length and the safe distance between vehicles should be no less than 4 m. Besides, the speed limit in the considered road network is 60 km/h. The MDs' positions are randomly initialized in the road network at the beginning and fixed during the simulation. For each MD, the computing tasks w i ← τ w i + (1 − τ )w i θ i ← τ θ i + (1 − τ )θ i are generated following a Poisson process and the task size is randomly distributed in the range of [2, 5] × 10 5 Kbits. In the simulation, we use β to represent the probability of generating tasks for each MD in each time slot. Similar to [33], we consider a fair spectrum allocation rule among links in which the total bandwidth is proportionally allocated according to the size of transmitted tasks. Other simulation parameter settings are given in Table I. We evaluate the performance within a duration of 100 seconds. 2) MADDPG Hyperparameters: The actor network is a four- For the critic network, the input includes the actions produced by the actor network and the states. There are two hidden layers for the states and one hidden layer for the actions before these two inputs are concatenated, after which there are two fully connected hidden layers, each layer with 256 units and activated by ReLu functions. Finally, the critic network has an output layer to calculate the Q value for the given stateaction pair, with no activation. Other hyperparameters used for training MADDPG can be found in Table II. B. Compared Methods The proposed MADDPG-based task offloading method (referred to as MADDPG in the following part) is compared with the following algorithms. 1) Single-hop: A direct task offloading from an MD to its ES via one-hop transmission is adopted considering both communication and computing resources, e.g., a game theoretic task offloading algorithm [10]. Once the workload exceeds the capacity of servers one-hop away, tasks will not be admitted into the system. 2) Multi-hop+Greedy: A particular case of multi-hop task offloading with a naive solution. Once the workload exceeds the capabilities of local ESs, the remaining tasks will be delivered to ESs within the shortest distance via multi-hop transmission from the MD to the target ES. We use two metrics to evaluate the effectiveness of the proposed MADDPG: the average service throughput (average Average throughput (KB) 10 6 Single-hop Multi-hop+Greedy MADDPG (a) Impact of task arrivals (when I = 20, N = 4, J = 4). 12 14 16 18 20 Number of MDs C. Evaluation of the average throughput In Fig. 4, we compare the average throughput of Singlehop, Multi-hop+Greedy, and MADDPG. Fig. 4a demonstrates that the average throughput of MADDPG climbs up as the task arrival rate increases. The reason is as follows. When the task arrivals from MDs or the number of ESs is large, the number of successfully completed tasks become large. Fig. 4b demonstrates that the average throughput of MAD-DPG increases as the number of MDs increases when the number of MDs is in the range of [10,20]. Fig. 4c shows that MADDPG achieves a relatively stable average throughput when the number of vehicles varies from 2 to 10. The reason is that MADDPG is capable of adaptively making multi-hop task offloading decisions by interacting with the environment. Fig. 4d shows that the average throughput of MADDPG increases with the number of ESs. This is because more ESs provides more resources for processing tasks in the system. Moreover, we observe that MADDPG always has much higher average throughput than Single-hop and Multi-hop+Greedy. That is because MADDPG achieves load balancing via multihop transmissions between MDs and ESs, while Single-hop can only exploit ESs one-hop away and Multi-hop+Greedy may cause computing overload and network congestion by blindly selecting ESs. Fig. 5a shows the success rate of three task offloading mechanisms against the task arrivals. We observe that tasks generated by MDs can be almost completed by MADDPG when the task arrival rate is small. In addition, Single-hop and Multi-hop+Greedy have unstable performance due to highly dynamic network environments, e.g., randomly distributed MDs and frequently moving vehicles. In Fig. 5b, we observe that tasks can be almost accomplished when the number of MDs varies from 10 to 20. Fig. 5c demonstrates that the success rate of three task offloading mechanisms against the number of vehicles. Fig. 5d shows the success rate of three task offloading mechanisms against the number of ESs. We can see that the success rate of MADDPG is approaching 1 as the number of ESs increases while Single-hop and Multi-hop+Greedy have much lower success rate. This is because MADDPG enables MDs to adaptively select destination ESs while Single-hop can only utilize local ES resources. Besides, Multi-hop+Greedy may incur selection conflicts of destination ESs, which causes performance degradation to some extent. Another observation is that MADDPG keeps the highest success rate among the three schemes given any task arrival rate, the number of MDs, the number of vehicles, and the number of ESs, respectively. D. Evaluation of the success rate VII. CONCLUSION By allowing edge servers multi-hop away to share the computing workload, the multi-hop MEC enables more edge servers to share their computing resources. In this paper, we have proposed such a novel multi-hop task offloading approach for MEC systems with the assistance of vehicles to enhance the system capacity while satisfying users' e2e latency requirements. In a highly dynamic and complicated system, we have taken into account several practical factors, such as vehicular mobility, spectrum availability, and computing capabilities, to obtain the association between users and edge servers potentially multi-hop away, a scenario rarely considered before. By employing multi-agent reinforcement learning, each end user acts as an agent to efficiently and adaptively make offloading policy achieve high aggregated throughput subject to its end-to-end latency requirement and resource limitations in a time-varying vehicular network. Extensive simulations have demonstrated that the proposed MADDPGbased task offloading scheme can increase the number of completed tasks while providing latency guarantee through adaptive load balancing among edge servers possibly multihop away achieved by running effective reinforcement learning mechanisms. Fig. 1 : 1A multi-hop task offloading framework for a vehicle-aided MEC system. Fig. 3 : 3The simulation scenario. 17 agent i, select action a i = π θi (s i ) + N t w.r.t the current policy and exploration6 Execute actions a = (a 1 , · · ·, a I ) and obtain rewards r and new state s from the environment 7 Store (s, a, r, s ) in Replay Buffer D random minibatch of m samples (s z , a z , r z , s z ), z = 1, 2, · · ·, m, from D11Set y z i = r z i + γQ π i (s z , a 1 , · · ·, a i · ··, a I )| ai=π i (s z i ) 12 U pdateDDP G 13 Procedure: U pdateDDP G 14 Update critic network by minimizing the loss function for w Q(φ(S z ), A z , w i )) 2Update actor network by minimizing the loss function for θ Update target network parameters for each MD i 18 of the number of MDs (when β = 0.5, N = 4, J = 4). Impact of the number of vehicles (when β = 0.5, I = 20, J = 2). Impact of the number of ESs (when β = 0.5, I = 20, N = 4). Fig. 4 : 4Evaluation of the average throughput throughput) and the average success rate (success rate). Average throughput is the average size of completed tasks from MDs during T time slots. Success rate is the average ratio of the completed tasks (including the newly arrived and the buffered at ESs) to the generated tasks in each time slot. Fig. 5 5shows the success rate comparison between MAD-DPG with Single-hop and Multi-hop+Greedy. of task arrivals (when I = 20, N = 4, J = 4). of the number of MDs (when β = 0.5, N = 4, J = 4). Impact of the number of vehicles (when β = 0.5, I = 20, J = 2). Impact of the number of ESs (when β = 0.5, I = 20, N = 4). Fig. 5 : 5Evaluation of the success rate This work was supported in part by the Project of International Cooperation and Exchanges NSFC under Grant No. 61860206005 and in part by the Joint Funds of the NSFC under Grant No. U22A2003. (Corresponding author: Haixia Zhang) Yiqin Deng and Haixia Zhang are with the Shandong Key Laboratory of Wireless Communication Technologies, Jinan, Shandong, 250061, China, and also with the School of Control Science and Engineering, Shandong University, Jinan, Shandong, 250061, China (email: yiqin.deng@email.sdu.edu.cn; haixia.zhang@sdu.edu.cn). Xianhao Chen is with the Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong, China (email: xchen@eee.hku.hk). Yuguang Fang is with the Department of Computer Science, City University of Hong Kong, Hong Kong, China (email: my.fang@cityu.edu.hk). Initialize a Gaussian noise N for action explorationReceive initial state sAlgorithm 1: Multi-Agent Deep Deterministic Policy Gradient for Task Routing in Vehicle-assisted MEC Input: the number of arriving tasks for each MD, the locations of MDs, vehicles, and ESs, e2e latency requirements for MDs, computing capabilities of ESs, spectrum resources Output: task routing policy 1 for each episode do 2 3 TABLE I : Ilayer neural network with two fully connected hidden layers, each with 256 units and activated by sigmoid functions. The number of units of the input and the output layers are equal to the number of states and the number of actions, respectively.Simulation Parameters Parameter Value Coverage Radius of ESs 200 m Height of antenna 1.5 m Carrier frequency 2800 MHz Computation complexity 1200 CPU cycles/bits Bandwidth 5 MHz Computing rate at ESs [1, 2, 3, 4] × 10 7 cycles/s Transmit power of MD 1 W Power of the Gaussian noise 5 × 10 −13 W Discounting factor 0.99 TABLE II: Multi-Agent Deep Deterministic Policy Gradient Hyperparameters Parameter Value Replay buffer size 10 5 Minibatch size 64 Gaussian noise, N (µ, σ 2 ) (0.15, e −2 ) Learning rate of critic network 0.002 Learning rate of actor network 0.001 Update frequency of target network 0.005 Actions at the edge: Jointly optimizing the resources in multi-access edge computing. Y Deng, X Chen, G Zhu, Y Fang, Z Chen, X Deng, IEEE Wireless Commun. 292Y. Deng, X. Chen, G. Zhu, Y. Fang, Z. Chen, and X. 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R Lowe, Y I Wu, A Tamar, J Harb, O Pieter Abbeel, I Mordatch, NIPS. R. Lowe, Y. I. Wu, A. Tamar, J. Harb, O. Pieter Abbeel, and I. Mordatch, "Multi-agent actor-critic for mixed cooperative-competitive environ- ments," NIPS, Jan. 2017. Deep reinforcement learning-based spectrum allocation in integrated access and backhaul networks. W Lei, Y Ye, M Xiao, IEEE Trans. Cogn. Commun. Netw. 63W. Lei, Y. Ye, and M. Xiao, "Deep reinforcement learning-based spectrum allocation in integrated access and backhaul networks," IEEE Trans. Cogn. Commun. Netw., vol. 6, no. 3, pp. 970-979, Sep. 2020. She received her B.S. degree in project management from Hunan Institute of Engineering. 2014, and her M.S. degree in software engineering and her Ph.D. degree in computer science and technology from. Xiangtan, China; Changsha, China; GainesvilleYiqin Deng is currently a Postdoctoral Research Fellow with the School of Control Science and Engineering, Shandong University ; Central South Universityrespectively. She was a visiting researcher at the University of Florida. from 2019 to 2021. Her research interests includeYiqin Deng is currently a Postdoctoral Research Fellow with the School of Control Science and Engineering, Shandong University. She received her B.S. degree in project management from Hunan Institute of Engineering, Xiangtan, China, in 2014, and her M.S. degree in software engineering and her Ph.D. degree in computer science and technology from Central South University, Changsha, China, in 2017 and 2022, respectively. She was a visiting researcher at the University of Florida, Gainesville, from 2019 to 2021. Her research interests include Munich University of Technology, as an Academic Assistant. From 2016 to 2017, she was a Visiting Professor with the University of Florida, USA. She is currently a Distinguished Professor with Shandong University. Her current research interests include industrial Internet of Things (IIoT), resource management, mobile edge computing, and smart communication technologies. Edge/Fog computing, Internet of Vehicles, and Resource Management. Haixia Zhang (M'08-SM'11) received the B.E. degree from the Department of Communication and Information Engineering. IEEE Wireless Communication Letters, and China CommunicationsGuilin University of Electronic Technology, China ; Shandong University, ChinaDr. Zhang serves on editorial boards of the IEEE Transactions on Wireless Communications. She has been serving as TPC member, session chair. invited speaker and keynote speaker for conferencesEdge/Fog computing, Internet of Vehicles, and Resource Management. Haixia Zhang (M'08-SM'11) received the B.E. degree from the Department of Communication and Information Engineering, Guilin University of Elec- tronic Technology, China, in 2001, and the M.Eng. and Ph.D. degrees in communication and informa- tion systems from the School of Information Science and Engineering, Shandong University, China, in 2004 and 2008, respectively. From 2006 to 2008, she was with the Institute for Circuit and Signal Processing, Munich University of Technology, as an Academic Assistant. From 2016 to 2017, she was a Visiting Professor with the University of Florida, USA. She is currently a Distinguished Professor with Shandong University. Her current research inter- ests include industrial Internet of Things (IIoT), resource management, mobile edge computing, and smart communication technologies. Dr. Zhang serves on editorial boards of the IEEE Transactions on Wireless Communications, IEEE Wireless Communication Letters, and China Communications. She has been serving as TPC member, session chair, invited speaker and keynote speaker for conferences. Kong. He obtained the Ph.D. degree from the University of Florida in 2022, and received the B.Eng. degree from Southwest Jiaotong University in 2017. His research interests include wireless networking and machine learning. Department of Electrical and Electronic Engineering, the University of HongXianhao Chen is currently an assistant professor with theXianhao Chen is currently an assistant professor with the Department of Electrical and Electronic Engineering, the University of Hong Kong. He obtained the Ph.D. degree from the University of Florida in 2022, and received the B.Eng. degree from Southwest Jiaotong University in 2017. His research interests include wireless networking and machine learning. He joined the Department of Electrical and Computer Engineering at University of Florida in 2000 as an assistant professor, then was promoted to associate professor in 2003, full professor in 2005, and distinguished professor in 2019, respectively. Since 2022, he has been the Chair Professor of Internet of Things with Department of Computer Science at City University of Hong Kong. Dr. Fang received many awards including the US NSF CAREER Award, US ONR Young Investigator Award. the Best Paper Award from IEEE ICNP (2006), and 2010-2011 UF Doctoral Dissertation Advisor/Mentoring Award. He was the Editor-in-Chief of IEEE Transactions on Vehicular Technology (2013-2017) and IEEE Wireless Communications. MS degree from Qufu Normal University ; Case Western Reserve UniversityCISTC Technical Recognition Award (2015), and WTC Recognition Award. and has served on several editorial boards of premier journals. He also served as the Technical Program Co-Chair of IEEE INFOCOM'2014. He is a fellow of ACM, IEEE, and AAASYuguang Fang (S'92, M'97, SM'99, F'08) received an MS degree from Qufu Normal University, China in 1987, a PhD degree from Case Western Reserve University in 1994, and a PhD degree from Boston University in 1997. He joined the Department of Electrical and Computer Engineering at University of Florida in 2000 as an assistant professor, then was promoted to associate professor in 2003, full professor in 2005, and distinguished professor in 2019, respectively. Since 2022, he has been the Chair Professor of Internet of Things with Department of Computer Science at City University of Hong Kong. Dr. Fang received many awards including the US NSF CAREER Award, US ONR Young Investigator Award, 2018 IEEE Vehicular Technology Out- standing Service Award, IEEE Communications Society AHSN Technical Achievement Award (2019), CISTC Technical Recognition Award (2015), and WTC Recognition Award (2014), the Best Paper Award from IEEE ICNP (2006), and 2010-2011 UF Doctoral Dissertation Advisor/Mentoring Award. He was the Editor-in-Chief of IEEE Transactions on Vehicular Technology (2013-2017) and IEEE Wireless Communications (2009-2012) and has served on several editorial boards of premier journals. He also served as the Technical Program Co-Chair of IEEE INFOCOM'2014. He is a fellow of ACM, IEEE, and AAAS.

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{'abstract': 'Multi-access edge computing (MEC) is a promising technology to enhance the quality of service, particularly for low-latency services, by enabling computing offloading to edge servers (ESs) in close proximity. To avoid network congestion, collaborative edge computing has become an emerging paradigm to enable different ESs to collaboratively share their data and computation resources. However, most papers in collaborative edge computing only allow one-hop offloading, which may limit computing resource sharing due to either poor channel conditions or computing workload at ESs one-hop away. By allowing ESs multi-hop away to also share the computing workload, a multihop MEC enables more ESs to share their computing resources. Inspired by this observation, in this paper, we propose to leverage omnipresent vehicles in a city to form a data transportation network for task delivery in a multi-hop fashion. Here, we propose a general multi-hop task offloading framework for vehicle-assisted MEC where tasks from users can be offloaded to powerful ESs via potentially multi-hop transmissions. Under the proposed framework, we develop a reinforcement learning based task offloading approach to address the curse of dimensionality problem due to vehicular mobility and channel variability, with the goal to maximize the aggregated service throughput under constraints on end-to-end latency, spectrum, and computing resources. Numerical results demonstrate that the proposed algorithm achieves excellent performance with low complexity and outperforms existing benchmark schemes.Index Terms-Computation offloading, Collaborative edge computing, Vehicular networks, Multi-hop service request routing, Deep reinforcement learning (DRL).', 'arxivid': '2304.07422', 'author': ['Member, IEEEYiqin Deng ', 'Senior Member, IEEEHaixia Zhang ', 'Member, IEEEXianhao Chen ', 'Fellow, IEEEYuguang Fang '], 'authoraffiliation': [], 'corpusid': 258180128, 'doi': '10.48550/arxiv.2304.07422', 'github_urls': [], 'n_tokens_mistral': 17782, 'n_tokens_neox': 15897, 'n_words': 10397, 'pdfsha': 'e94c1995017daa281f6dc5cffda9dc89cb84b1be', 'pdfurls': ['https://export.arxiv.org/pdf/2304.07422v1.pdf'], 'title': ['Spectrum-aware Multi-hop Task Routing in Vehicle-assisted Collaborative Edge Computing', 'Spectrum-aware Multi-hop Task Routing in Vehicle-assisted Collaborative Edge Computing'], 'venue': []}

arxiv

Center to limb variation of transition region Doppler shift in active regions 18 Feb 2023 Abhishek Rajhans Inter-University Centre for Astronomy and Astrophysics Post Bag -4411007Ganeshkhind, PuneIndia Durgesh Tripathi Inter-University Centre for Astronomy and Astrophysics Post Bag -4411007Ganeshkhind, PuneIndia Vinay L Kashyap Center for Astrophysics | Harvard & Smithsonian 60 Garden St02138CambridgeMAUSA James A Klimchuk Solar Physics Laboratory NASA Goddard Space Flight Center Code 67120771GreenbeltMDUSA Avyarthana Ghosh Inter-University Centre for Astronomy and Astrophysics Post Bag -4411007Ganeshkhind, PuneIndia Tata Research Development and Design Center Tata Consultancy Services Ltd India Center to limb variation of transition region Doppler shift in active regions 18 Feb 2023(Received; Revised; Accepted)Draft version February 21, 2023 Typeset using L A T E X default style in AASTeX631 Studying Doppler shifts provides deeper insights into the flow of mass and energy in the solar atmosphere. We perform a comprehensive measurement of Doppler shifts in the transition region and its center-to-limb variation (CLV) in the strong field regions (|B| ≥ 50 G) of 50 active regions (ARs), using the Si IV 1394Å line recorded by the Interface Region Imaging Spectrometer(IRIS). To locate the ARs and identify strong field regions, we have used the magnetograms obtained by the Helioseismic and Magnetic Imager (HMI). We find that in strong field regions, on average, all the ARs show mean redshifts ranging between 4-11 km/s, which varies with ARs. These flows show a mild CLV, with sizable magnitudes at the limb and substantial scatter at the mid-longitude range. Our observations do not support the idea that redshifts in the lower transition region (T <∼ 0.1 MK) are produced by field-aligned downflows as a result of impulsive heating and warrant alternative interpretation, such as downflow of type-II spicules in the presence of a chromospheric wall created by cooler type-I spicules. INTRODUCTION The heating of the solar atmosphere continues to be a challenging problem. Though magnetic fields are known to be responsible, the exact mechanism for energy dissipation and the transport of mass and energy across different layers of the atmosphere remains elusive. A possible explanation is the heating of the solar corona by impulsive events (see for a review Klimchuk 2006). Impulsive heating results in the evaporation of chromospheric plasma along the loops into the corona, followed by draining and condensation. Hence, studying flows in different layers of the solar atmosphere sheds valuable insights into the heating and possible ways these layers may be coupled with each other. Observations show that the transition region has a ubiquitous presence of redshifts (downflows). Early observations from Orbiting Solar Observatory (OSO-8;Bruner 1977), the Naval Research Laboratory (NRL) normal incidence spectrograph onboard Skylab (S082-B), NRL High-Resolution Telescope and Spectrograph (HRTS; Bartoe & Brueckner 1975), and the Ultra-Violet Spectrometer and Polarimeter (UVSP; Woodgate et al. 1980) onboard the Solar Maximum Mission (SMM; Simnett 1981) show downflows in the range of 5-20 km s −1 in ultraviolet spectral emission lines from bright regions in the chromosphere and the transition region (also see Lemaire et al. 1978;Brueckner et al. 1980;Gebbie et al. 1980;Lites 1980;Brueckner 1981;Athay et al. 1982Athay et al. , 1983Dere 1982;Rottman et al. 1982;Brekke 1993;Achour et al. 1995). Moreover, transition region downflows in the range of 80-100 km s −1 have also been reported in small regions within active regions. However, due to their rare occurrence, these are considered to be associated with transients (Nicolas et al. 1982;Dere et al. 1984). Studies with similar scientific goals have also been performed using observations from Solar Ultraviolet Measurements of Emitted Radiation (SUMER; Wilhelm et al. 1995), the Coronal Diagnostic Spectrometer (CDS; Harrison et al. 1995) onboard SOlar and Heliospheric Observatory (SOHO; Domingo et al. 1995), EUV Imaging Spectrometer (EIS; Culhane et al. 2007) onboard Hinode (Kosugi et al. 2007), and Interface Region Imaging Spectrograph (IRIS;De Pontieu et al. 2014). Teriaca et al. (1999) used observations from SUMER to show downflows in the active regions to be ranging from ∼ 0 km s −1 at log[T(K)] = 4.3 to ∼ 15 km s −1 at log[T(K)] = 5.0. At log[T(K)] = 5.8 blueshifts ∼ 8 km s −1 are observed. Further studies on plasma flows were conducted using observations from EIS (see, e.g., Del Zanna 2008; Brooks & Warren 2009;Tripathi et al. 2009Tripathi et al. , 2012Dadashi et al. 2011;Gupta et al. 2015;Ghosh et al. 2017) in warm loops as well as moss regions (transition region counterpart of hot loops). Persistent downflows were reported across the range of temperature EIS observed, log T = 4.0 to 5.0. However, lower transition region spectral lines like O IV, O V, and Mg V were very weak in the observations (Young et al. 2007). A possible explanation for these downflows can be impulsive heating occurring in the solar corona (see, e.g., Klimchuk 2006;Reale 2014;Klimchuk 2015, for a review). In this scenario, the redshift is due to field-aligned downflows of cooling and draining plasma that were pushed up in the coronal loops due to chromospheric evaporation. If the flows have a random orientation relative to vertical, then they should show, on average, a center-to-limb variation (CLV) and vanish as one approaches the limb. However, Feldman et al. (1982) found almost no CLV observed in data from NRL onboard Skylab(S082-B). They tracked two active regions as they traversed across the solar disk to study the Doppler shifts in the temperature range log T = 4.7-5.0, and found the downflows to be in the range of 4-17 km s −1 . Moreover, the redshifts extended out to the limb. Klimchuk (1987Klimchuk ( , 1989 used UVSP data and found similar results in measuring Doppler shifts relative to the average over the full raster. The observation of persistent downflows was explained by Antiochos (1984) as a signature of field-aligned flows due to condensation. Moreover, to explain the absence of CLV and non-diminishing flows at the limb, Antiochos (1984) introduced the idea of a chromospheric well, which is formed due to the enhanced localized pressure due to impulsive heating. Under this scenario, the absence of CLV naturally arises due to projection effects. However, there is a drawback to this scenario. Under impulsive heating, field-aligned hydrodynamical simulations show downflows with much lower amplitude than those observed at similar temperatures. For example, the downflows in the Fe VIII line formed at an approximate temperature of 0.4 MK is ∼0.9 km s −1 (see, e.g., López Fuentes & Klimchuk 2018, 2022. Under the assumption of constancy of mass flux and pressure along a given flux tube in the transition region, the peak formation temperature of Si IV and Fe VIII (forming at different heights in the transition region) imply that the speed of downflows in Si IV line should be less than 0.1 km s −1 , which is about two orders of magnitude lower than observed velocities in the lower transition regions. Ghosh et al. (2019Ghosh et al. ( , 2021 studied the Doppler shift and non-thermal velocities in Si IV line and their CLV for a single active region as it traversed the central meridian. They used the IRIS instrument, which provides regular spectroscopic observations of the transition region in Si IV line, with an accuracy of about 1 km s −1 . Moreover, the presence of multiple spectral lines due to neutral and single ionized ions provides the best opportunity to perform wavelength calibration and measure and characterize flows in the transition region. Ghosh et al. (2019) found that the strong field regions of active regions (where magnetic field strengths are larger than 50 G) were redshifted by 5-10 km s −1 and showed evidence of some CLV but less than expected for nearly vertical flows. To mitigate the discrepancy between the observed redshifts and those obtained from hydrodynamical simulations, Ghosh et al. (2019) suggested that the downflows observed in the transition regions are very likely related to the downflow of type-II spicules. They proposed the idea of a chromospheric wall formed by cold spicules heated to a temperature of about 10 4 K in the vicinity of hot spicules, which get heated to 10 5 K. They argued that the optical depth of surrounding cold spicules is close to but less than unity, hence, allowing some center to limb variation in Si IV line. We note that Ghosh et al. (2019) performed the Doppler measurements for a single active region while it crossed the central meridian. Additionally, the coverage of radius vectors (fractional distance to the limb from the disk center) was limited. Only eight values were covered in the range of -0.8-0.9 (0 denotes the disk center, and +1(-1) are the maximum values of the radius vector in the eastern (western) limb). The study assumed that AR evolution does not affect how flow velocities and directions might change. Hence, it is important to study a wide range of active regions to improve the longitude range and characterize the actual behavior of flow velocities. In this work, instead of tracking active regions, we perform a snapshot study by observing Doppler shifts in different active regions at instants to check if the findings of Ghosh et al. (2019) for one active region are valid for an ensemble of active regions. This provides a statistically larger sample and better longitudinal coverage. In §2 we describe the data from different instruments used in this study. In §3 we describe the various procedures involved in analyzing data from different instruments, viz (i) wavelength calibration, (ii) coalignment of data from AIA-1600, HMI, and IRIS, (iii) identification of strong-field regions within the active regions, and (iv) computation of Doppler shifts in these regions and associated radius vector. We discuss the results for all active regions and their CLV in §4. We summarize in §5. OBSERVATIONS AND DATA To study the Doppler shifts, we have used IRIS observations. IRIS provides spectra and images with spatial resolutions varying between 0.33 ′′ and 0.4 ′′ and a cadence of up to 20 s for spectra and 10 s for images. The field of view (FOV) can extend to 175 ′′ ×175 ′′ . The spectra obtained allow us to resolve velocities of 1 km s −1 . IRIS records a pair of Si IV lines at 1393.78Å and 1402.77Å, with peak formation temperature 10 4.9 K. Under the optically thin conditions, the line at 1393.78Å is a factor of two stronger than that at 1402.77Å (Dere et al. (1996); Landi et al. (2013); see however, Gontikakis & Vial (2018); Tripathi et al. (2020)). Hence, following Ghosh et al. (2019), we use the line at 1393.78Å for our study. We have also used observations from Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) in 1600Å filter for co-alignment purposes. We aim to study the Doppler shifts in the two major polarities of the ARs. Hence to identify the two polarities, we have used the line of sight (LOS) magnetograms obtained from Helioseismic and Magnetic Imager (HMI; Schou et al. 2012a,b). To study the CLV of the Doppler shift, we have selected 50 active regions, listed in Table 1, observed at various locations covering the full range of longitudes. The upper panel of Figure 1 displays the location of all the active regions over the solar disk. The lower panel of Figure 1 shows AIA-1600Å image on 8 th of July 2014 with the black box showing the field of view of the IRIS raster corresponding to the exemplar active region (Case 39; the first row of Table 1) that is described in detail. DATA ANALYSIS AND RESULTS To measure the absolute Doppler shift, we need to perform wavelength calibration. Also, since HMI and IRIS observe the Sun from two different vantage points, a proper coalignment needs to be ensured. For this purpose, we coalign IRIS observations with those obtained using AIA 1600Å passband. Since AIA and HMI are both onboard the Solar Dynamics Observatory (SDO), the magnetograms can be readily coaligned with that of AIA. Once we obtain a calibrated Doppler map in Si IV and coaligned magnetograms, we identify the pixels associated with strong field areas of the active region and deduce the average Doppler shift. Here, we discuss the above-mentioned procedure in detail for an exemplar case of active region AR 12104. IRIS provided observations of this region from 23:35 UT on 7 th of July 2014 to 03:05 UT on 8 th of July 2014. The spatial extent of the corresponding IRIS raster extended from 490 to 630 arcseconds along the x-axis and -310 to -130 ′′ along the y-axis. The position of the raster for the exemplar case is shown with yellow box in Figure 1. Wavelength Calibration Wavelength calibration involves identifying average Doppler shifts in emission lines coming from neutral or singly ionized atoms, which are expected to be approximately at rest (Hassler et al. 1991). Such neutral or singly ionized atoms are present in the photosphere or chromosphere. There are multiple lines such as Fe II, O I, and S I present in IRIS spectral windows. Following Ghosh et al. (2019), we have used O I (1355.6Å) line for performing wavelength calibration, which is a mid-chromospheric line in which velocities are small, and rarely exceeding 1.5 km/s for O I line (Lin & Carlsson 2015). The average velocity can be assumed to be zero at the cost of finite random error. In this case, the spectrum should peak at the rest wavelength of the line. This should be the case in the ideal scenario because we expect atoms emitting these lines to be at rest. Any deviation in the peak of the spectrum from the rest wavelength should be due to instrumental effects, which need to be corrected. The average spectrum from a system of atoms at a finite non-zero temperature is Gaussian. However, directly fitting obtained spectrum with a Gaussian profile has limitations. The spectrum obtained by an instrument gives the average energy recorded in different wavelength bins, not the energy associated with the center of each bin. Even though, in the first approximation, the energy in the bin is associated with the central wavelength, it is valid only if the spectral line profile in the bin is linear. This certainly cannot be expected to be the case always. Consequently, to increase our accuracy in finding the line center, we have applied the method of Intensity Conserving Spline Fitting (ICSF) to the spectra using icsf.pro procedure (Klimchuk et al. 2016). It preserves the total intensity in each spectral bin and performs a spline fitting to account for the line profile variation within the wavelength bin. Finally, the spectrum obtained after the application of ICSF procedure is fitted with a Gaussian using eis auto fit.pro routine in solarsoft (Freeland & Handy 1998). In Figure 2, we plot the spectrum obtained in O I (panel a) and Si IV (panel b) lines. These spectra have been averaged over the full raster. The black asterisks denote the original spectrum obtained from IRIS level2 fits files. The red curve represents the spectrum obtained after applying ICSF correction, and the blue curve is the final Gaussian fit to the ICSF correction. The wavelength for Si IV is adjusted according to the difference between the laboratory rest wavelength of O I and its observed wavelength of peak intensity in the raster-averaged spectrum. For the exemplar case, we find the wavelength at which the raster-averaged spectrum of O I line peaks is 1355.5987Å, which is larger than the lab measurements of the rest wavelength, which is at 1355.5980Å as obtained from Sandlin et al. (1986) and Edlén (1943). The wavelength for Si IV is adjusted accordingly. Co-alignment of observations from IRIS, HMI, and AIA For the purpose of coalignment of IRIS and HMI, we consider AIA observations taken at 1600Å, as this is closest in temperature to that is recorded by IRIS in Si IV line. We first make a data cube of AIA 1600Å images and LOS magnetograms of the region of interest during the entire duration of the raster. All AIA images and HMI LOS magnetograms have been coaligned with the IRIS slitjaw image in 1330Å taken at the closest time. All the AIA images and LOS magnetograms in datacubes are then corrected for solar rotation with respect to the first AIA 1600Å image as the reference. We then create artificial AIA-1600 and HMI LOS magnetogram rasters corresponding to IRIS rasters to ensure proper coalignment. Figure 3 (left panel) displays the intensity map obtained in Si IV. The middle and right panel displays the AIA 1600Å image and the LOS magnetogram obtained by artificial rastering. The over-plotted contours correspond to 250 DN s −1 pix −1 in AIA 1600Å images. The excellent correspondence between the AIA contours on IRIS image and the magnetogram suggests a near-perfect coalignment of the data. Identification of active region and computing average Doppler shifts After coaligning the data from different instruments and ensuring that we select the same structures from different data, we identify the strong field areas inside the active regions. Klimchuk (1987) identified the pixels in which the magnitude of the magnetic field exceeded 100 G. Ghosh et al. (2019) on the other hand, used a magnitude of 50 G for the same purpose. Ghosh et al. (2019) noted that the precise value is unimportant because the magnetic field strength decays rapidly outside the strong field regions. Consequently, the contours of magnetic fields of ± 100 G or ± 50 G are not very different. Following Ghosh et al. (2019), here we have used contours of ±50 G to identify the strong field regions. We plot the velocity maps obtained in Si IV in the left panel of Figure 4. The over-plotted green and yellow contours correspond to ± 50 Gauss, respectively, obtained from the magnetograms shown in Figure 3.c. The right panel of Figure 4 shows he histogram of velocity in such pixels. The average Doppler shifts in the strong field regions is 7.44 ± 0.02 km s −1 . We estimate the uncertainties by accounting for random errors due to variations in velocities and central wavelength of O I across pixels identified as strong field regions. We have also taken into account the systematic errors due to an expected 3 mÅ [0.66 km/s] uncertainty in rest wavelength of O I line used for wavelength calibration and a 0.1 pix dispersion uncertainty for Si IV [0.56 km/s]. For calculating errors, both random and systematic, their components have been added in quadrature. These procedures are discussed in detail in Ghosh et al. (2019). While the cumulative random error for this active region is ± 0.02 km s −1 , the total systematic error, which is assumed to be the same for all the regions, is ≅ 0.9 km s −1 . Radius vector We need to compute the radius vector of observed active regions to study the CLV of Doppler shifts. Radius vector is defined as the sine of the angle between the LOS and local vertical (Klimchuk 1987). A value of zero corresponds to disk center whereas positive(negative) radius vector represents longitudes to the east(west) of the central meridian. We compute the radius vector of a given IRIS raster using its central pixel. If the central position of IRIS raster is [x,y] ′′ , the radius vector is computed as x 2 + y 2 R Sun (where R Sun is 959 ′′ ). We multiply it by ±1 for the east(west) limb. For the exemplar case under consideration, the radius vector is 0.64. Table 1. List of active regions studied. The file name of the IRIS rasters belonging to the different active regions studied is tabulated along with their radius vector (RV) and the average velocity ± in Si IV line (Vavg) in the pixels with strong magnetic fields(|B| ≥ 50 G). The error cumulative random errors are also listed along with Vavg. Index Fits file a RV Vavg Index Fits file RV Vavg 0 a The tabulated filename excludes the common part 'iris l2 ' on left and ' raster t000 r00000.fits' on right. CENTER TO LIMB VARIATION OF DOPPLER SHIFTS We carry out exactly the same analysis discussed above for 50 active regions listed in Table 1. These active regions have been randomly selected so as cover radius vectors over the whole disk. The name of the analyzed iris level2 fits files, their radius vector (RV), the mean velocity (V avg with random error) in its strong field regions are given in Table 1. These active regions have been arranged in ascending order of radius vectors. Case 39 corresponds to the exemplar case discussed in the previous section. Figure 5 plots the Doppler shifts as a function of radius vector. The black asterisks show the average Doppler shift in the strong field regions. We demonstrate the expected behavior of the line-of-sight Doppler shifts, v LOS = v 0 1 − r R 2 , where v 0 is the mean velocity corresponding to an active region at disk center, and RV= r R is the radius vector. The black dashed lines in Figure 5 show the trend for v 0 = 10 km s −1 , which matches the measured average Doppler shift in the strong field region of the active region closest to disk center (at RV= 0.01). The effect of systematic shifts from variations of ±0.9 km s −1 are shown as red and blue dashed curves. These curves show what kind of variation is expected if v 0 were the same for all active regions. In reality, the average flow velocity will vary across different active regions. However, note that the scatter in the data points is similar to the expected systematic variations, suggesting that the variation in v 0 across active regions is not much larger than ∼1 km s −1 . Figure 5 is thus an illustration of how the data differs from the expected variation. Some center-to-limb variation is clearly visible, consistent with the observations of Ghosh et al. (2019) based on following a single active region over time. Large departures from the expected trend are also seen as the RV approaches the limb, suggesting that an additional effect is responsible. Detailed modeling of these effects is beyond the scope of this work. SUMMARY AND DISCUSSION Here, we report on the most comprehensive measurements and analysis of AR Doppler shifts and CLV to date using IRIS spectral measurements of Si IV. For the purpose of co-alignment and to identify the strong field regions in active regions, we have used the observations from AIA and HMI, both onboard SDO. Similar to the results obtained by Feldman et al. (1976), Klimchuk (1987Klimchuk ( , 1989, and Ghosh et al. (2019), we find that in lower transition region emissions, active regions are predominantly red shifted with velocities ranging between 4-11 km s −1 . Moreover, the Doppler shifts show CLV, as was also reported by Ghosh et al. (2019). Note that the results obtained by Ghosh et al. (2019) was based on the tracking of a single active region AR 12641 as it crossed from the center towards the limb. Here, we have studied 50 active regions located at different locations across the solar disk. Ghosh et al. (2019) proposed that the lower transition region redshifts are not due to the draining of cooling coronal material but rather the main bodies of falling type II spicules. If these spicules have a random orientation relative to vertical, then their average redshift should exhibit the CLV expected of a vertical flow. To explain the weaker variation that is observed, Ghosh et al. proposed that absorption from interlaced cold type I spicules gives preferential weighting to type II spicules that are more closely aligned with the line of sight. These "selected" spicules have similar Doppler shift everywhere across the solar disk. A modest CLV occurs because the type I spicules are not totally opaque. Ghosh et al. (2019) described the type I spicules as providing a chromospheric wall, which is slightly different from the chromospheric well proposed by Antiochos (1984). We thank the referee for careful reading and constructive comments. This research is partly supported by the Max-Planck Partner Group on the Coupling and Dynamics of the Solar Atmosphere of MPS at IUCAA. AR acknowledges financial support from University Grants Commission in form of SRF. VLK acknowledges support from NASA Contract NAS8-03060 to the Chandra X-ray Center, and the hospitality of IUCAA during several visits. The work of JAK was supported by the Internal Scientist Funding Model (competitive work package program) at Goddard Space Flight Center. AIA and HMI data are onboard SDO (NASA mission). IRIS is a small explorer mission of NASA, developed and operated by LMSAL. Figure 1 . 1Top panel: Solar disk plotted with the red boxes showing the field of view (FOV) of all the IRIS rasters studied in the paper. Bottom panel: AIA-1600Å image taken on 8 th of July 2014. The black box shows the FOV of the IRIS raster of the exemplar case discussed in detail. Figure 2 . 2Spectrum obtained from IRIS in O I (panel a) and Si IV (panel b) line windows. These spectra have been averaged over the full raster, for the exemplar case. The black asterisks show the data obtained from IRIS level2 fits files. Red curves denote the ICSF corrected spectrum. The blue curves show the Gaussian fit to the ICSF corrected spectrum. Figure 3 . 3Intensity maps of emission in Si-IV line (left), artificial rasters of AIA-1600 (middle), and HMI LOS magnetogram (right). Contours of 250 DN s −1 pix −1 in AIA-1600Å filter are over-plotted. Figure 4 . 4[Left] Velocity maps in Si IV line. The green and yellow contours in the right panel are of +50 and -50 Gauss, respectively.[Right] Histogram of velocities in strong field regions of active regions (where |B| ≥ 50 G). The dotted vertical line corresponds to 0 km s −1 . 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{'abstract': 'Studying Doppler shifts provides deeper insights into the flow of mass and energy in the solar atmosphere. We perform a comprehensive measurement of Doppler shifts in the transition region and its center-to-limb variation (CLV) in the strong field regions (|B| ≥ 50 G) of 50 active regions (ARs), using the Si IV 1394Å line recorded by the Interface Region Imaging Spectrometer(IRIS). To locate the ARs and identify strong field regions, we have used the magnetograms obtained by the Helioseismic and Magnetic Imager (HMI). We find that in strong field regions, on average, all the ARs show mean redshifts ranging between 4-11 km/s, which varies with ARs. These flows show a mild CLV, with sizable magnitudes at the limb and substantial scatter at the mid-longitude range. Our observations do not support the idea that redshifts in the lower transition region (T <∼ 0.1 MK) are produced by field-aligned downflows as a result of impulsive heating and warrant alternative interpretation, such as downflow of type-II spicules in the presence of a chromospheric wall created by cooler type-I spicules.', 'arxivid': '2301.06723', 'author': ['Abhishek Rajhans \nInter-University Centre for Astronomy and Astrophysics\nPost Bag -4411007Ganeshkhind, PuneIndia\n', 'Durgesh Tripathi \nInter-University Centre for Astronomy and Astrophysics\nPost Bag -4411007Ganeshkhind, PuneIndia\n', 'Vinay L Kashyap \nCenter for Astrophysics |\nHarvard & Smithsonian\n60 Garden St02138CambridgeMAUSA\n', 'James A Klimchuk \nSolar Physics Laboratory\nNASA Goddard Space Flight Center\nCode 67120771GreenbeltMDUSA\n', 'Avyarthana Ghosh \nInter-University Centre for Astronomy and Astrophysics\nPost Bag -4411007Ganeshkhind, PuneIndia\n\nTata Research Development and Design Center\nTata Consultancy Services Ltd\nIndia\n'], 'authoraffiliation': ['Inter-University Centre for Astronomy and Astrophysics\nPost Bag -4411007Ganeshkhind, PuneIndia', 'Inter-University Centre for Astronomy and Astrophysics\nPost Bag -4411007Ganeshkhind, PuneIndia', 'Center for Astrophysics |\nHarvard & Smithsonian\n60 Garden St02138CambridgeMAUSA', 'Solar Physics Laboratory\nNASA Goddard Space Flight Center\nCode 67120771GreenbeltMDUSA', 'Inter-University Centre for Astronomy and Astrophysics\nPost Bag -4411007Ganeshkhind, PuneIndia', 'Tata Research Development and Design Center\nTata Consultancy Services Ltd\nIndia'], 'corpusid': 255942057, 'doi': '10.3847/1538-4357/acb4ed', 'github_urls': [], 'n_tokens_mistral': 13015, 'n_tokens_neox': 10201, 'n_words': 5680, 'pdfsha': '4c5dddcf780c2629082808a2bdd94937f037740a', 'pdfurls': ['https://export.arxiv.org/pdf/2301.06723v2.pdf'], 'title': ['Center to limb variation of transition region Doppler shift in active regions', 'Center to limb variation of transition region Doppler shift in active regions'], 'venue': []}

arxiv

Expanding Explainability Horizons: A Unified Concept-Based System for Local, Global, and Misclassification Explanations Fatemeh Aghaeipoor School of Computer Science Institute for Research in Fundamental Sciences (IPM) Dorsa Asgarian dorsaasgarian@gmail.com School of Computer Science Institute for Research in Fundamental Sciences (IPM) Mohammad Sabokrou School of Computer Science Institute for Research in Fundamental Sciences (IPM) Okinawa Institute of Science and Technology (OIST) Mohammad Sabokrou@oist Jp Expanding Explainability Horizons: A Unified Concept-Based System for Local, Global, and Misclassification Explanations Explainability of intelligent models has been garnering increasing attention in recent years. Of the various explainability approaches, concept-based techniques are notable for utilizing a set of human-meaningful concepts instead of focusing on individual pixels. However, there is a scarcity of methods that consistently provide both local and global explanations. Moreover, most of the methods have no offer to explain misclassification cases. To address these challenges, our study follows a straightforward yet effective approach. We propose a unified concept-based system, which inputs a number of super-pixelated images into the networks, allowing them to learn better representations of the target's objects as well as the target's concepts. This method automatically learns, scores, and extracts local and global concepts. Our experiments revealed that, in addition to enhancing performance, the models could provide deeper insights into predictions and elucidate false classifications. all share a common goal of investigating the group's contribution rather than the pixels' individual roles. As with the other ML methods, Concept-based techniques are developed in two primary directions, namely supervised and unsupervised. In the former, a set of user-provided concept examples must be available to score and detect. However, defining proper concepts and labels and the process of labeling itself might be impossible or costly [14,15]. In contrast, there are automatic concept learning/extracting methods, which attempt to identify the concepts without human supervision [16]. The common concept learning methods typically consist of two main modules: one to form the concepts and another to score them [17]. In vision applications, segmentation methods like SLIC [18] are commonly used to define the concepts, and scoring tools like TCAV (Testing with Concept Activation Vectors) [7] are employed to compute the concepts' importance scores. This study's contribution lies in the second module, where a recent study has shown that TCAV is a vulnerable method against the perturbed input samples [19,20]. Additionally, It may overestimate the concepts' importance, which makes it potentially unfaithful because of scoring some irrelevant concepts [21]. This prompted us to explore ways to leverage knowledge of trained networks and eliminate external scoring modules. We drew inspiration from the Outlier Exposure technique in deep anomaly detection, which uses an auxiliary dataset, entirely disjoint from test-time data, to teach the network better representations of anomaly cases [22]. From another perspective, similar to the feature-based explanation methods, the concepts can be extracted locally (per sample) [23,24] or globally (per class/set of samples) [25,26]. While many methods focus on either one, few approaches could provide a consistent framework for both local and global explanations simultaneously [27,28]. However, Combining these two strategies in a single approach enhances understanding of correct predictions and facilitates further investigation of misclassification cases. For the latter, consider a medical setting where an AI model is set up to predict a tumor for a given image. In this situation, users may be interested in understanding false positive/negative results, in which healthy cases are labeled as malignant and malignant instances are labeled as healthy cases [29,30]. This can be achieved by comparing the local concepts of the given image with the global concepts of the target class, enabling the users to identify which parts of the image misled the model into making such wrong predictions. We address these limitations by proposing UCBS, a Unified Concept-Based System that extracts different targets' concepts from adapted trained/fine-tuned networks. UCBS attempts to provide a unified framework for both local and global explanations while shedding light on instances of misclassification. Indeed, a number of super-pixelated samples of the target classes (as the auxiliary dataset) are fed into the networks, directing the training/fine-tuning process towards learning the contribution of the superpixels rather than individual pixels. Having the networks adapted, the local and global concepts are extracted. Experimental results confirmed intriguing findings for different target classes. The main contributions of this work can be summarised as follows: • Introducing a novel unified method for local and global concept learning, which, to the best of our knowledge, is the first unsupervised unified concept extraction framework applicable to image data. • Proposing a simple yet effective internal concept scoring tool, eliminating the need for external ones. • Applying the unified framework to directly explain misclassification cases, either the False Positive (FP) or False Negative (FN) predictions. Background and related works In the context of XAI, one essential perspective is the scope of the explainability methods, in which local and global explanations are taken into account. Local explanations are specific to a single or a small group of samples, e.g., one patient of interest. On the other hand, there are global explanations, which attempt to gain a general understanding of what the models predict for a set of examples or an entire class, e.g., a group of similar patients [21]. Specifically, in vision-based applications, the significance of the pixels is usually determined in the scope of a single image, and it is not straightforward to determine important pixels for a variety of images of a certain target class [31]. While the fact is that the practitioners are eager to make sense of the overall reasoning of the models in addition to the individual explanations [32,33]. [34] proposed a novel two-stage framework composed of a feature occlusion analysis and a mapping stage, which explains the model decisions in terms of the importance of category-wide concepts. Summit [35] proposed a dynamic platform to summarize and visualize the learned features of a DNN and their interaction. It introduced two summarization techniques, namely activation aggregation and neuron-influence aggregation. Despite these single-purpose works, having local and global approaches in a single framework can make explanations practical for comparing assessment tasks, especially in the concept-based approaches where we are not faced with the firmed pre-defined concepts. In unified frameworks [36,16,37], local and global explanations can be directly contrasted, highlighting the unique characteristics of each sample against the representative concepts of the understudied target class. In this context, there are few frameworks, and most of them extract and score discovered concepts in separate modules. For instance, [21] combined two powerful existing techniques, one local, namely Integrated Gradients, and one global, namely TCAV [7]. Ghorbani et al. also proposed ACE, Automated Concept-based Explanation, which aggregates relevant image segments across multiple input data to obtain global concepts [38]. ACE tends to provide global concept samples and has no straightforward suggestions for local interpretations of either false or true predictions. Moreover, these methods used TCAV to score the discovered concepts, which has its own fragility, as discussed. Proposed method UCBS is an automated concept-based system that is able to explain a certain target class either locally or globally. This method is developed using a set of super-pixelated images of the given target class feeding to the DNNs' training/fine-tuning process. Indeed, the superpixels are employed to shed more light on the learning process of the DNNs and assist in revealing the most influential part(s) of the images in certain predictions. The procedure of automated concept-based explanation methods is typically composed of two main stages: one to form/learn the concepts and second to score the importance of these concepts in making predictions. As the results of these two stages, local and global concepts can be extracted. In what follows, first, we present the algorithm for learning and scoring the concepts, and then we describe the extraction mechanisms of the concepts. It should be noted that while our emphasis is on fine-tuning pre-trained models to use in the post-training explanation methods, UCBS serves as a general solution that can be readily adapted for training the models from scratch. Notation Suppose that for the given target class c, there is a set of n input images as D c = {x c 1 , x c 2 , ..., x c i , ..., x c n }. Each of these input images can be segmented into k segments, i.e., for sample x c i , there is a set of segments as S c xi = {s c i,1 , s c i,2 , ..., s c i,k }. In addition, we consider the pre-trained DNN f θ , which will be adapted for a binary classification task one (target class) vs. all (other classes), i.e., f c θ : 1}). For the sake of simplicity, we drop the target index c in the remainder of the paper. x c i → p({0, Concept learning In Concept-based explanations, the concepts' contribution is meant rather than the pixels' individual roles. Particularly, in image data, the concepts are defined by groups of pixels that are called superpixels or segments [16]. UCBS creates these superpixels and feeds them into the learning procedure of a binary DNN classifier, i.e., the target class vs. the others (non-target). Indeed, as Fig 1-a shows, the classifier takes a set of super-pixelated images of the given target class along with the original target images and a set of random images as the non-target input data. It then starts to train or fine-tune the binary DNN classifier. To prepare the super-pixelated input data, a segmentation method is applied to a few images of the target class, resulting in a number of segments/superpixels for each image. After that, each segment is padded with zero value up to the original input size and labeled as the target class data. These segmented images are available in S xi , as notated. For the zero-pixel areas, the output of the convolution operation as well as the max or the average pooling, is zero. This prevents these regions' weights from being updated during the back-propagation operation (under some activation functions like relu or sigmoid) [39]. In fact, having only the weights associated with each segment updated assists in better learning of those regions. On the other hand, this effect, together with learning the non-segmented (original) images, makes the role of the most important parts of the images highlighted. For instance, imagine two segments in the class of tiger cat: one for a part of the ears and another for a part of the nature background. During the training process, on the one hand, the DNN learns these two segments are commonly associated with the tiger cat class and puts a certain concentration on them; on the other hand, in the learning of the complete images, the DNN learns that the similar parts of the former segment (ear) contribute more to identifying the tiger cat class than the background-like parts. In addition, the network observes the ear-like patterns more than the background-like data during the training process, either in the segmented or non-segmented images. The DNN aggregates these findings and assigns higher scores to the ear segments than to the background parts, and this is what we exactly need to discover the most important concepts. Concept scoring After adapting the DNN for a certain target class, UCBS takes advantage of this classifier for scoring the concepts. It indeed utilizes the knowledge of the adapted DNN rather than employing some external and fragile scoring tools like TCAV [20]. Each segment of an image can be treated as one concept. However, the concepts must be prioritized based on their contributions to the proper prediction of the given target class. To this end, the importance scores are defined. Considering the aforementioned framework, the importance score of segment s i,j ∈ S xi with respect to the target class c is assigned as follows: IS(s i,j ) = f ′,0 θ (s i,j )(1) where f ′,0 θ returns the pre-softmax value of the target class output. The raw output values were considered to preserve the original magnitude of the scores. Concept extraction Having scored the concepts, they can be used to either locally explain individual data points or globally explain a target class using an entire or a set of examples. These tasks are conducted using local and global concepts, respectively. In what follows, we describe how UCBS extracts these two sets of concepts. Local concepts In order to extract the local concepts of an (unseen) image given a certain target class, first, all the segments of it are passed through the corresponding network and the contributions of the segments are scored as described in the previous section. Then, the segments are sorted based on their importance scores, and the p top concepts of each image are selected as its final local concepts (notated as l i,j in Fig. 1-b). p is a user-defined parameter to provide different levels of complexity and accuracy, and our experiments showed that 3 is a well-set value to have the most influential parts of each image. Local concepts assist in a deeper understanding of the FP and FN predicted cases, discovering why the network misclassifies the input data inside/outside of a target class (See section 4.5). Global concepts Global concepts are used to obtain a general view of each target class and find out the top common concepts among the examples of that class. UCBS takes a set of random images of the given class and groups the top local concepts of these images based on their similarity, i.e., each group represents one global concept of that class. In detail, as Fig. 1-b illustrates, first, the top local concepts of the random target images are passed through the network and mapped to their embeddings (h(l i,j )). These values are then clustered using their Euclidean distances as an effective perceptual similarity measure [40]. Experiments and results This section presents a detailed discussion of the conducted experiments on the UCBS. First, the experimental framework is described, and next, the performance of UCBS with and without the inclusion of the super-pixelated samples is evaluated. Subsequently, we discuss the results of local and global concepts, as well as the application of UCBS in explaining misclassification cases. Experimental setup In order to establish UCBS 1 , we employed the widely-used ResNet model (34 layers) [41], pretrained on the ILSVRC2012 dataset (ImageNet) [42], and adapted it for the concept learning step. The target classes were randomly selected among the ImageNet classes (generally 100 classes to report the following measures). For each target class, we segmented 50 random images among the training set into 50 super-pixels using the SLIC segmentation method [18], resulting in a maximum of 2, 500 super-pixelated images for fine-tuning the networks, as discussed in section 3.2. The SLIC segmentation method was employed due to its speed and simplicity. The generated super-pixelated images, along with the original images and 1, 300 random images from the other classes, were used to fine-tune the models for a few epochs (50). In our experiments with the ImageNet dataset, this UCBS hyper-parameters setup proved sufficient, allowing the networks to learn the targets' concepts in addition to the original targets' objects. During the testing phase, three local concepts were extracted from each input image, and three global concepts were identified for every target class. The latter were obtained using the candidate local concepts of 20 unseen images. Finally, to determine the global concepts, we performed KMeans clustering with K = 5 and illustrated the three best representatives from each group. To the best of our knowledge, there is no unified framework extracting both local and global concepts automatically and without pre-provided labels. Consequently, as a pioneering approach, we were unable to compare the proposed method with similar works and set out to demonstrate the UCBS performances empirically and visually (More evaluations are available in the appendix). Performance evaluation Utilizing super-pixelated images in the fine-tuning process of the networks should not sacrifice performance. To investigate this, the discrimination capability of the models was assessed using the ImageNet validation set, once without the super-pixelated samples and once with them. We considered the validation set in three scenarios: 1) including only the target class images (val pure ), 2) including the target class images plus an equal number of the other class images (val equal ), and 3) including the target class images plus 1, 000 random images from other classes (val 1000 ). Table 1 reports the average loss and accuracy values of these three scenarios. The results clearly indicate that the networks' discrimination capabilities have not only been preserved but also enhanced across all the scenarios. This improvement can be attributed to the use of superpixelated images, which enable more effective learning of the target objects. That is, they progressively aid in concentrating on different parts of the target images, compelling the networks to learn the finer details of the targets' concepts in addition to the whole targets' objects. 2 . Furthermore, the super-pixelated samples assist the networks in learning the common and frequent concepts of each target class. As a result of these two features, the networks are directed to make predictions more precisely based on the importance and relevance of the targets' concepts rather than the individual pixels. This effect persists even when the number of images of other classes increases up to 1, 000 cases, implying that learning the targets' concepts enhances the networks' ability to better distinguish the non-target objects as well. Consequently, this approach can be regarded as a valuable tool for directing the networks to predict based on the targets' concepts and mitigating the influence of irrelevant factors such as background, possible biases, and so on. Local explanations Global explanations To illustrate the global concepts, UCBS was applied to the three aforementioned target classes, and mouth and nose (featuring cat whiskers). For police vans, it is evident that the network has primarily concentrated on the standalone police text logo, the police text logo positioned on the front of the vehicles (encompassing a section of the car bumper and headlight), and the van's tire, respectively. Regarding the revolver, UCBS clarifies that objects of this category can effectively be recognized using the concepts of the barrel (incorporating the front sight), the cylinder, and the trigger. Explaining misclassifications Explaining the network's functionality based on the most impactful concepts assists in the clarification of misclassification cases, either FP or FN ones. Fig. 4 presents one example of FP and FN for each of the three target classes. The first two rows pertain to the tiger cat class, where a cat image was wrongly classified and a dog image was mistakenly placed in the tiger cat class. Considering the local concepts of the former case, it becomes apparent that the network struggled to identify similarities to the global concepts of this class and primarily focused on irrelevant concepts. Conversely, the striped-shaped portions of the dog image and its front paws led the network to misclassify the object. The following two rows are associated with the police van class, where no police logo or vehicle parts were recognized in the first case, and parts of the windshield, bumper, headlight, and bus tire were detected in the FP case. These local concepts contributed to the incorrect classification of the objects. In the FN case of the revolver, the network failed to identify some influential parts of the object, as suggested by the global concepts. Additionally, in the FP case, the detected tube-like part of the object may have misled the network. 5 Discussion, limitations, and future works UCBS concentrates on human-understandable and identifiable concepts (e.g., "pointy ears", "police logo") rather than pixels to explain a specific target class. This approach eliminates the need for providing user-defined concepts and applying external concept scoring methods. We demonstrate the compatible comprehension of local and global concepts within a unified framework, which is particularly promising for clarifying false predictions. The provision of human-meaningful concepts is encouraging for (non-expert) end-user explanations. It is important to acknowledge that the limitations and drawbacks of superpixels are inherited by UCBS, potentially resulting in the creation of meaningless concepts. Moreover, automatic extraction of more abstract and complex concepts may prove challenging or necessitate additional post-processing and aggregation. Addressing these challenges presents an intriguing direction for future research, especially in identifying and surrounding the most influential concepts within bounding boxes, which could help mitigate these issues. In addition, in this study, the concept 3 class, one sample for FP and FN in each class = 6 rows Input sample Super-pixels Scores Top local concepts Tiger cat Tiger cat predicted as others Others predicted as tiger cat Police van Police van predicted as others Others predicted as police van Revolver Revolver predicted as others Others predicted as revolver learning and network adaptation process is designed as a binary classifier for each specific target class. Although only a few epochs are required for fine-tuning the networks, this approach may seem time-consuming when applied to multi-class datasets. Therefore, as the next step, we plan to apply super-pixelated images for the simultaneous concept learning of multiple target classes as a whole. Conclusion This work proposed UCBS to bridge the gap between local and global explanation techniques as well as to analyze misclassification cases. We took advantage of super-pixellating of the training data and incorporating them into the learning process. This resulted in not only improving the performances but also facilitating the identification and scoring of the most influential concepts. In practical applications, we are interested in obtaining a cognitively comprehensible rationale for the model's predictions, either for a single data point or a set of them. In the proposed UCBS, the former was achieved by ranking the concepts locally, and the latter was attained by tracking the concepts globally. Furthermore, by closely monitoring the local concepts and comparing them to the global ones, UCBS was effectively able to clarify the false predictions. Insights gained from this research may promote the safer use of AI models in vision-based applications within safety-critical fields such as healthcare or finance. Additionally, these insights may facilitate the design, development, and debugging process of the models, from the perspective of the developers. We selected the ResNet model [41] comprising 34 layers and pre-trained on the ImageNet dataset. This network has the classification head for a total of 1000 classes. To adapt this network for binary classification (fine-tuning purposes), we replaced the classification head with a binary one, as illustrated in Supp. Fig. 5. In what follows, we call the main network N et main and the adapted one N et a . The latter is indeed an auxiliary network (See Fig. 1 for details) that aids in explainability tasks and concept learning. This network comes into the scene on-demand to explain a specific target class among the existing ones. B. Insights into the number of required concepts To evaluate the sufficiency of a set of obtained concepts, we employed two importance measures that naturally belong to pixel-based methods and are adapted for concept-based areas. Smallest Sufficient Concepts (SSC) that seek the smallest set of concepts necessary for the accurate prediction of the target class, and Smallest Destroying Concepts (SDC) that refer to the smallest set of concepts whose removal will result in incorrect predictions [38,43]. These measures can be considered as local metrics, i.e., for input sample To demonstrate these measures, Supp. Fig. 6 is provided, and the local values of SSC and SDC and the images including/excluding the top concepts are provided for one instance of the tiger cat class. We passed these images through the N et a as well as N et main and reported the predictions below of each image. We did this experiment to evaluate the effectiveness of the obtained concepts to recognize the target images not only in a binary manner (target or others) but also among all the 1000 classes of the ImageNet. In the latter, a tiger cat image can wrongly be detected as a Conch, a Tabby, an Egyptian cat, a Persian cat, and so on. To provide a good representation of the local concepts, we set the value of p = 3. In this context, the number of top local concepts of target c is obtained as follows: in which D c is the validation set of this target. Although it is better to determine the values of p c per target class, we suggest defining hyper-parameter p as follows: p c = M ax SSC (D c ) , SDC (D c )(2)p = 1 |C| M ax c∈C SSC (D c ) , c∈C SDC (D c )(3) in which the average values are calculated across all the considered classes and utilized in the max operator to ensure that no useful concepts are lost. According to our experiments, which involve 100 selected classes from the ImageNet dataset, the average values of SSC and SDC are reported in Supp. Comparing the SSC measures of the two networks implies that explaining targets in a binary manner is more straightforward and requires fewer concepts than the multi-class approach, which necessitates a greater number of concepts (N et main ). In other words, taking Supp. Fig. 6 into account, it takes longer to ensure that a tiger cat is a tiger cat among a set of other specific classes, some of which are very similar to the tiger class (e.g., Tabby, Egyptian cat, and Persian cat), compared to understanding that a tiger cat is a tiger cat and not something else in general. On the other perspective, the SDC values demonstrate that N et main makes errors more quickly than N et a . In summary, the concepts extracted using auxiliary networks are both useful and important, as we generally require 16 such concepts for proper recognition of an image among the 1000 classes of ImageNet images. Despite these findings, we hypothesize that integrating these networks and feeding super-pixelated images into the main network will lead to lower numbers and eliminate the need for individual fine-tuning processes. Supplementary C. More discussion on limitations: Imperfectly super-pixelated inputs and Dataset biases Applying super-pixelating methods, even the multi-resolution ones, to obtain initial segments presents several limitations. These include the creation of large segments that contain multiple meaningful concepts or, conversely very small segments with no discernible concepts. Additionally, one single concept may be fragmented across several segments. Super-pixelating methods typically offer no solution for tracking entirely or partially abstract concepts such as brightness, daylight, and sun angle in the sunrise or sunset images. Supp. Fig. 7 illustrates some of these issues. Future works remain to see whether a sliding window that reflects the influence of the network's prediction on different parts of the images could overcome these issues. Another challenge encountered during our experiments is the possible dataset biases, where the extracted concepts are considerably influenced by the images available in the dataset. This results in concepts that might be deviated from human intuition. For instance, the most important concept for predicting mountain bike images is the driver rather than the bike itself (See Supp. Fig. 10). It turned out that most ImageNet mountain bike images mainly contain drivers. Another example involves the restaurant class, which contains images that differ from the more common ones. The UCBS concepts are not applicable in these instances, leading to unhelpful misclassification explanations, as Supp. Figure 1 : 1Remove axes, train #test, all same size,change other, add more train images CNN __ UCBS general schema. For the class of interest, (a) and (b) are executed successively. (a)The network is trained/fine-tuned as a binary classifier (target vs. others) using a set of super-pixelated images of the target class along with the original target images and a set of random images of the other classes. (b) For unseen images, the concepts are scored using the network and ranked by their importance scores. To identify the local concepts, the p top concepts of each image are selected, and for the global concepts, the embeddings of a set of candidate local concepts associated with a set of unseen images are clustered and the g top groups with their best samples are selected. In this way, similar concepts are grouped together as examples of a certain global concept. The clusters are then sorted based on their population, and the q top clusters are assigned as the final global concepts of the given class. To illustrate examples of each global concept, the densest areas of the clusters are selected. Indeed, clusters with the highest population are more prevalent and examples with the lowest cost represent the most similar cases of each concept. Fig. 2 2displays examples of local concepts for three target classes: tiger cat, police van, and revolver. Results of the first two rows suggest that (a portion of) the animals' faces and their body texture play a significant role in identifying the tiger cat objects. In the case of the police van examples, the figures illustrate that the network focuses on the police logo and various vehicle components, such as the tire, the windows, and the headlight. For the revolver, the extracted local concepts verify that the network can successfully recognize different parts of these objects, including the grip, the cylinder, and the barrel. Additional examples of local concepts can be found in the appendix. Fig. 3 3depicts their outputs. In this figure, the top three global concepts of each class have been indicated via the best three representative samples (having minimum cost) of each group. About the tiger cat class, UCBS proposes that the most crucial and general concepts for identifying such objects include the animal's (pointy) ears, the body texture (incorporating fur patterns), and a portion of their Figure 2 : 2Local concepts to explain individual images. For each class, there are two examples, displaying the original image, the segmented version, the complete scoring map, and the top three local concepts of each instance. Figure 3 : 3Global concepts to explain one certain target class. For each class, the top three global concepts are indicated via the best three representative samples of each group. Figure 4 : 4Explaining misclassification cases: two examples of FN and FP are represented for each target class, and the original image, the segmented version, the complete scoring map, and the scoring map of the top three local concepts are indicated for each example Supplementary Figure 5 : 5The classification and explainability networks x i , the values of SSC(x i ) and SDC(x i ) are calculated with the cardinality of the sets obtained from x i . From the global point of view, they are simply averaged over all the validation samples (D c ) of target class c and notated as SSC(D c ) and SDC(D c ). Figure 6 : 6Illustration of SSC and SDC for x i , an instance of tiger cat class. Figure 7 : 7The effect of imperfectly super-pixelated inputs: a) a large superpixel contains background b) Several concepts (different parts of the airplane) in one superpixel c) one concept (blue-red ambulance light) fragmented into two segments. Supplementary Figure 8 :Figure 9 :Figure 10 :Figure 11 :Figure 12 : 89101112Fig. 8 indicates. D. More examples of UCBS Results of local, global, and misclassification cases for six other ImageNet classes are provided in Supp. Figs. 9-14. Restaurant uncommon images and failure of misclassification explanations. Local concepts to explain individual images. For each class, there are two examples, displaying the original image, the segmented version, the complete scoring map, and the top three local concepts of each instance. Global concepts to explain one certain target class. For each class, the top three global concepts are indicated via the best three representative samples of each group. Explaining misclassification cases: two examples (if available) of FN and FP are represented for each target class, and the original image, the segmented version, the complete scoring map, and the scoring map of the top three local concepts are indicated for each example. Local concepts to explain individual images. For each class, there are two examples, displaying the original image, the segmented version, the complete scoring map, and the top three local concepts of each instance. Table 1 : 1The performance results, with and without feeding the super-pixelated samplesState val pure val equal val 1000 ACC(%) Loss ACC(%) Loss ACC(%) Loss Without 93.16 ± 6.09 0.37 93.36 ± 4.72 0.37 92.78 ± 4.30 0.38 With 95.93 ± 4.29 0.35 95.04 ± 4.19 0.36 94.80 ± 3.79 0.36 Table of the most important concepts from the images causes N et a to make incorrect predictions. For N et main , these values are 16.3 and 2.3, respectively.. 2. These values indicate that N et a requires a minimum of 3.6 concepts to distinguish target class images from unrelated ones. On the other hand, removing an average of 3.4 Table 2 : 2The average SSC and SDC values over all the considered target classes.SSC(D) SDC(D) p N et a 3.6 3.4 3 N et main 16.3 2.3 16 table Others predicted as pool table Others predicted as Pool table Supplementary Figure 14: Explaining misclassification cases: two examples (if available) of FN and FP are represented for each target class, and the original image, the segmented version, the complete scoring map, and the scoring map of the top three local concepts are indicated for each example. All of the code necessary to reproduce our experimental findings can be found after the reviewing process. 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Santos, and M. H. Abreu, "Interpreting deep machine learning models: an easy guide for oncologists," IEEE Reviews in Biomedical Engineering, 2021. A survey on explainable artificial intelligence (XAI): Toward medical XAI. E Tjoa, C Guan, IEEE transactions on neural networks and learning systems. 32E. Tjoa and C. Guan, "A survey on explainable artificial intelligence (XAI): Toward medical XAI," IEEE transactions on neural networks and learning systems, vol. 32, no. 11, pp. 4793- 4813, 2020. Why should I trust you? explaining the predictions of any classifier. M T Ribeiro, S Singh, C Guestrin, Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining. the 22nd ACM SIGKDD international conference on knowledge discovery and data miningM. T. Ribeiro, S. Singh, and C. Guestrin, "Why should I trust you? explaining the predictions of any classifier," in Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, pp. 1135-1144, 2016. What clinicians want: contextualizing explainable machine learning for clinical end use. S Tonekaboni, S Joshi, M D Mccradden, A Goldenberg, in Machine learning for healthcare conferenceS. Tonekaboni, S. Joshi, M. D. McCradden, and A. Goldenberg, "What clinicians want: contex- tualizing explainable machine learning for clinical end use," in Machine learning for healthcare conference, pp. 359-380, 2019. Efficient decompositional rule extraction for deep neural networks. M E Zarlenga, Z Shams, M Jamnik, 1st Workshop on eXplainable AI approaches for debugging and diagnosis (XAI4Debugging). NeurIPS2021M. E. Zarlenga, Z. Shams, and M. Jamnik, "Efficient decompositional rule extraction for deep neural networks," in 1st Workshop on eXplainable AI approaches for debugging and diagnosis (XAI4Debugging), NeurIPS, 2021. Towards global explanations of convolutional neural networks with concept attribution. W Wu, Y Su, X Chen, S Zhao, I King, M R Lyu, Y.-W Tai, IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). W. Wu, Y. Su, X. Chen, S. Zhao, I. King, M. R. Lyu, and Y.-W. Tai, "Towards global explana- tions of convolutional neural networks with concept attribution," in IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8649-8658, 2020. S ummit: Scaling deep learning interpretability by visualizing activation and attribution summarizations. F Hohman, H Park, C Robinson, D H P Chau, IEEE transactions on visualization and computer graphics. 261F. Hohman, H. Park, C. Robinson, and D. H. P. Chau, "S ummit: Scaling deep learning interpretability by visualizing activation and attribution summarizations," IEEE transactions on visualization and computer graphics, vol. 26, no. 1, pp. 1096-1106, 2019. From local explanations to global understanding with explainable AI for trees. S M Lundberg, G Erion, H Chen, A Degrave, J M Prutkin, B Nair, R Katz, J Himmelfarb, N Bansal, S.-I Lee, Nature machine intelligence. 21S. M. Lundberg, G. Erion, H. Chen, A. DeGrave, J. M. Prutkin, B. Nair, R. Katz, J. Himmelfarb, N. Bansal, and S.-I. Lee, "From local explanations to global understanding with explainable AI for trees," Nature machine intelligence, vol. 2, no. 1, pp. 56-67, 2020. A unified approach to interpreting model predictions. S M Lundberg, S.-I Lee, Advances in neural information processing systems. 30S. M. Lundberg and S.-I. Lee, "A unified approach to interpreting model predictions," Advances in neural information processing systems, vol. 30, 2017. Towards automatic concept-based explanations. A Ghorbani, J Wexler, J Y Zou, B Kim, Advances in Neural Information Processing Systems. 32A. Ghorbani, J. Wexler, J. Y. Zou, and B. Kim, "Towards automatic concept-based explanations," Advances in Neural Information Processing Systems, vol. 32, 2019. A survey on modern trainable activation functions. A Apicella, F Donnarumma, F Isgrò, R Prevete, Neural Networks. 138A. Apicella, F. Donnarumma, F. Isgrò, and R. Prevete, "A survey on modern trainable activation functions," Neural Networks, vol. 138, pp. 14-32, 2021. The unreasonable effectiveness of deep features as a perceptual metric. R Zhang, P Isola, A A Efros, E Shechtman, O Wang, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionR. Zhang, P. Isola, A. A. Efros, E. Shechtman, and O. Wang, "The unreasonable effectiveness of deep features as a perceptual metric," in Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 586-595, 2018. Deep residual learning for image recognition. K He, X Zhang, S Ren, J Sun, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionK. He, X. Zhang, S. Ren, and J. Sun, "Deep residual learning for image recognition," in Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016. Imagenet large scale visual recognition challenge. O Russakovsky, J Deng, H Su, J Krause, S Satheesh, S Ma, Z Huang, A Karpathy, A Khosla, M Bernstein, International journal of computer vision. 115O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, et al., "Imagenet large scale visual recognition challenge," Interna- tional journal of computer vision, vol. 115, pp. 211-252, 2015. Real-time image saliency for black box classifiers. P Dabkowski, Y Gal, Advances in neural information processing systems. 30P. Dabkowski and Y. Gal, "Real-time image saliency for black box classifiers," Advances in neural information processing systems, vol. 30, 2017.

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{'abstract': "Explainability of intelligent models has been garnering increasing attention in recent years. Of the various explainability approaches, concept-based techniques are notable for utilizing a set of human-meaningful concepts instead of focusing on individual pixels. However, there is a scarcity of methods that consistently provide both local and global explanations. Moreover, most of the methods have no offer to explain misclassification cases. To address these challenges, our study follows a straightforward yet effective approach. We propose a unified concept-based system, which inputs a number of super-pixelated images into the networks, allowing them to learn better representations of the target's objects as well as the target's concepts. This method automatically learns, scores, and extracts local and global concepts. Our experiments revealed that, in addition to enhancing performance, the models could provide deeper insights into predictions and elucidate false classifications.", 'arxivid': '2306.03531', 'author': ['Fatemeh Aghaeipoor \nSchool of Computer Science\nInstitute for Research in Fundamental Sciences (IPM)\n\n', 'Dorsa Asgarian dorsaasgarian@gmail.com \nSchool of Computer Science\nInstitute for Research in Fundamental Sciences (IPM)\n\n', 'Mohammad Sabokrou \nSchool of Computer Science\nInstitute for Research in Fundamental Sciences (IPM)\n\n\nOkinawa Institute of Science and Technology (OIST)\n\n', 'Mohammad Sabokrou@oist Jp '], 'authoraffiliation': ['School of Computer Science\nInstitute for Research in Fundamental Sciences (IPM)\n', 'School of Computer Science\nInstitute for Research in Fundamental Sciences (IPM)\n', 'School of Computer Science\nInstitute for Research in Fundamental Sciences (IPM)\n', 'Okinawa Institute of Science and Technology (OIST)\n'], 'corpusid': 259089318, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 13665, 'n_tokens_neox': 12195, 'n_words': 7992, 'pdfsha': '43ce5dec70004bb6780dd5bb04fa30d622f8d120', 'pdfurls': ['https://export.arxiv.org/pdf/2306.03531v1.pdf'], 'title': ['Expanding Explainability Horizons: A Unified Concept-Based System for Local, Global, and Misclassification Explanations', 'Expanding Explainability Horizons: A Unified Concept-Based System for Local, Global, and Misclassification Explanations'], 'venue': []}

arxiv

Cosmological Flow of Primordial Correlators Denis Werth UMR 7095 Sorbonne Université, Institut d'Astrophysique de Paris CNRS 98 bis bd Arago75014ParisFrance Lucas Pinol UMR 7095 Sorbonne Université, Institut d'Astrophysique de Paris CNRS 98 bis bd Arago75014ParisFrance Instituto de Física Teórica UAM-CSIC c/ Nicolás Cabrera 13-1528049Madrid Sébastien Renaux-Petel UMR 7095 Sorbonne Université, Institut d'Astrophysique de Paris CNRS 98 bis bd Arago75014ParisFrance Cosmological Flow of Primordial Correlators Correlation functions of primordial density fluctuations provide an exciting probe of the physics governing the earliest moments of our Universe. However, the standard approach to compute them is technically challenging. Theoretical predictions are therefore available only in restricted classes of theories. In this Letter, we present a complete method to systematically compute tree-level inflationary correlators. This method is based on following the time evolution of equal-time correlators and it accurately captures all physical effects in any theory. These theories are conveniently formulated at the level of inflationary fluctuations, and can feature any number of degrees of freedom with arbitrary dispersion relations and masses, coupled through any type of time-dependent interactions. We demonstrate the power of this approach by exploring the properties of the cosmological collider signal, a discovery channel for new high-energy physics, in theories with strong mixing and in the presence of features. This work lays the foundation for a universal program to assist our theoretical understanding of inflationary physics and generate theoretical data for an unbiased interpretation of upcoming cosmological observations. Correlation functions of primordial density fluctuations provide an exciting probe of the physics governing the earliest moments of our Universe. However, the standard approach to compute them is technically challenging. Theoretical predictions are therefore available only in restricted classes of theories. In this Letter, we present a complete method to systematically compute tree-level inflationary correlators. This method is based on following the time evolution of equal-time correlators and it accurately captures all physical effects in any theory. These theories are conveniently formulated at the level of inflationary fluctuations, and can feature any number of degrees of freedom with arbitrary dispersion relations and masses, coupled through any type of time-dependent interactions. We demonstrate the power of this approach by exploring the properties of the cosmological collider signal, a discovery channel for new high-energy physics, in theories with strong mixing and in the presence of features. This work lays the foundation for a universal program to assist our theoretical understanding of inflationary physics and generate theoretical data for an unbiased interpretation of upcoming cosmological observations. Introduction. Cosmology is about understanding time. Indeed, the physics governing the universe is deciphered through the time evolution of density perturbations, from the beginning of the Hot Big Bang to the late-time galaxy distribution. Remarkably, it is believed that these perturbations have emerged from quantum zero-point fluctuations during a period of accelerated expansion [1][2][3], i.e. cosmic inflation, providing the initial seeds for the subsequent evolution of cosmological structures [4][5][6][7][8]. Tracking down the cosmological flow of inflationary fluctuations thus connects the quantum laws of physics at a fundamental level to the largest observable scales. Yet, the correct theory of inflation remains unknown, and a central challenge is to decode it through the study of inflationary correlators, namely spatial correlation functions of the curvature perturbation field ζ(x, t). On large scales, current data have already well constrained the physics of inflation, especially at the linear level [9]. However, much information-e.g. the number of fields active during inflation, together with their mass spectra, spins, sound speeds, and how they interact-is encoded in non-linearities (see e.g. [10] for a recent review), of which the primary observable is the three-point correlator (bispectrum) (see [11] for current bounds). Computing equal-time correlators given an inflationary theory is a well-established procedure [12]. From first principles, calculations can be carried to arbitrary orders of perturbation theory. However, this program hides a daunting complexity: perturbation theory becomes intractable for realistic situations. The root of this difficulty resides in the challenge to track the detailed time evolution of the physics in the bulk of spacetime. Consequently, for technical reasons, most of the theoretical predictions have been derived under stringent assumptions, such as assuming weak mixing, perfect (or almost) scale invariance, large hierarchy of masses and couplings, and considering single-exchange diagrams [10]. There-fore, they do not cover the vast panorama of inflationary scenarios [13]. This can completely bias our interpretation of data and reveals the need to develop an approach that makes accurate predictions for all physically motivated inflationary theories. In this Letter and its companion paper [14], we present the cosmological flow, a systematic framework to compute primordial corrrelators. It is based on solving differential equations in time to track the evolution of primordial correlators through the entire spacetime during inflation (see Fig. 1), for theories formulated straight at the level of fluctuations. In contrast to previous works [15][16][17][18][19], our approach is not limited to particular background mechanisms. Instead, by adopting a philosophy that focuses on the study of correlators through effective field theories (EFT) of inflationary fluctuations, we are able to describe any theory. This allows us to obtain exact tree- This work offers new possibilities for the study of inflationary correlators. One of the motivations for studying them lies in the appreciation that inflation is a one-of-akind window to probe fundamental physics at the highest reachable energies, comparable to the Hubble scale H during inflation, which can be as high as 10 14 GeV. Information about new physics-e.g. the presence of heavy particles of masses m ∼ H-can be inferred from oscillatory patterns (or specific power law behaviors) present in the squeezed limit of the bispectrum [20][21][22]. 1 Our approach provides a systematic way to explore the characteristics of this cosmological collider signal in theories that are difficult to grasp analytically. First, as a heavy field weakly mixed with the curvature perturbation leads to the same cosmological collider frequency as a light but strongly mixed one, we show how complete predictions can be straightforwardly obtained in order to break theoretical degeneracies. We also present movies that display the cosmological flow of the bispectrum, therefore permitting to identify the characteristic times and scales at which the cosmological collider signal is building up. Second, we demonstrate that, due to the universal non-linearly realized boost symmetry, a time-dependent mixing-encompassing a wide range of inflationary models-leads to cosmological collider signals composed of modulated frequencies. X a X b X a X b X c time u a b u a b u a bc These results contradict some commonly held beliefs that the detection of such signals pinpoints the mass of a new particle. This highlights the importance of thoroughly studying signatures of all early-universe theories, as the cosmological flow approach enables one to do, for correctly interpreting cosmological data. More generally, our work paves the way for a far-reaching program of studying the phenomenology of inflationary correlators, shifting our focus from technical considerations to the unbiased exploration of the rich physics of inflation. Primordial Fluctuations. To begin, let us consider a set of bulk scalar degrees of freedom ϕ α (x, t), and p β (x, t) the corresponding conjugate momenta. For practical purposes, we gather these fields and momenta in a phase-space vector X a ≡ (ϕ α , p β ). 2 We denote the corresponding operators in Fourier space with sans serif indices X a (k, t). They verify the canonical commutation algebra [X a , X b ] = i ab . These inflationary fluctuations are described by a Hamiltonian H(ϕ α , p β ) which is a functional of the phase-space coordinates. Embracing an EFT point of view, it takes the form of a series expansion in powers of fluctuations H = 1 2! H ab X a X b + 1 3! H abc X a X b X c + . . . ,(1) where we adopt the extended Fourier summation convention for repeated indices to indicate a sum including integrals over Fourier modes. The tensors H ab , H abc , . . ., which can be taken symmetric without loss of generality, are arbitrary functions of time and momenta. This form of the Hamiltonian is completely general and captures all theories involving scalar degrees of freedom at the level of inflationary fluctuations. The fully non-linear equations of motion read dX a dt = i [H, X a ] = ac H cb X b + 1 2! ad H dbc X b X c + . . . = u a b X b + 1 2! u a bc X b X c + . . . ,(2) where the third line defines the tensors u a b , u a bc , . . . . Written in this form, it is clear that Eq. (2) encodes both the full classical evolution of X a and their quantum properties. Time Evolution of Primordial Correlators. We are interested in equal-time correlators of composite operators Ω| O(X a ) |Ω evaluated at time t, where |Ω is the vacuum of the full interacting theory. Because the dynamics governed by Eq. (2) cannot be solved exactly in full generality, we choose the quadratic Hamiltonian H 0 = 1 2! H ab X a X b to evolve the interaction-picture operators defined by X a ≡ U † X a U, thus resorting to a perturbative description of the interactions encoded in H I = H − H 0 [23,24]. This way, equal-time correlators are given by the well-known in-in formula [12] Ω| O(X a ) |Ω = 0| U O(X a ) U † |0 ,(3) where U =T exp(i t −∞(1−i ) H I (t ) dt ) withT = T † the anti-time ordering operator, and |0 is the vacuum of the free theory. The X a evolve with the full quadratic Hamiltonian. Working at tree-level and up to three-point correlators, one can expand the exponentials in Eq. (3) to obtain X a X b = 0| X a X b |0 , X a X b X c = 0| i 3! t −∞ dt H def X d X e X f , X a X b X c |0 .(4) For the three-point correlators, external operators are evaluated at the time t, and internal operators at time t . The simplicity of the cosmological flow lies in the ability to find, from first principles, a closed system of differential equations in time at the level of correlators [16]. This cosmological flow is given by d dt X a X b = u a c X c X b + u b c X a X c , (5a) d dt X a X b X c = u a d X d X b X c + u a de X b X d X c X e + (2 perms) .(5b) Eq. (5a) couples all two-point correlators, including those which contain conjugate momenta, and correctly capture all physical effects arising from quadratic operators in the theory. The structure of Eq. (5b) allows the flow of each kinematical configuration to be tracked separately. In [14], we introduce handy diagrammatic rules to derive the differential equations governing the time evolution of any tree-level n-point correlator. Considering the Bunch-Davies state, as we do in the following, the initial conditions for Eqs. (5a, 5b) can be readily derived analytically provided one initializes the correlators sufficiently in the far past, see [14] for explicit expressions. Naturally, the method equally works for any other state and is not restricted to inflation. Goldstone Description. We now apply this formalism to a concrete case. The spontaneous breaking of boost symmetry in cosmological backgrounds implies the unavoidable presence of a (canonically normalized) Goldstone boson π c (x, t) describing adiabatic fluctuations [25,26]. At linear order, the field π c is related to the curvature perturbation by ζ = −Hc 3/2 s f −2 π π c where c s is the propagation speed of π c and f 4 π ≡ 2M pl |Ḣ|c s is the symmetry breaking scale. Let us now consider an additional relativistic massive scalar field σ(x, t) with mass m, coupled to π c through the following interacting Lagrangian, commonplace in concrete realizations of inflation: L/a 3 = ρπ c σ + c 3/2 s ρ 2f 2 π (∂ i π c ) 2 a 2 σ + c 3/2 sρ f 2 π π cπc σ − 1 2Λπ 2 c σ − 1 2 απ c σ 2 − µσ 3 ,(6) where ρ, Λ, α and µ are-in general time-dependentcouplings. For the purpose of focusing on mixing interactions, we have omitted ever-present self-interactions of π c and have taken the decoupling limit where gravitational interactions vanish. Note that there is no a priori model-building requirement on the size of the dimensionless quadratic coupling ρ/H, and we allow this mixing parameter to be of order one or larger, which we call the strong mixing regime. A universal aspect is that this coupling ρ fixes both the quadratic interactioṅ π c σ and the cubic interactions (∂ i π c ) 2 σ and π cπc σ. This is a consequence of π c non-linearly realizing time diffeomorphisms, as these interactions are generated by the same operator ρ(t)δg 00 σ in the unitary gauge, after reintroducing the Goldstone boson t → t + π (see [14] for more details). After performing a Legendre transform, the found Hamiltonian can be arranged in the form (1) in terms of the phase-space vector X a = (π c , σ, p π , p σ ). The identification of the tensors u a b and u a bc defined in Eq. (2) follows simply, see [14]. Cosmological Colliders at Strong Mixing. At tree-level, each cubic interaction in Eq. (6) gives an independent contribution to the bispectrum, and therefore can be treated separately. Following standard conventions [27], we define the shape function S such that 3 ζ k1 ζ k2 ζ k3 ≡ (2π) 4 (k 1 k 2 k 3 ) 2 ∆ 4 ζ S(k 1 , k 2 , k 3 ) ,(7) where ∆ 2 ζ = k 3 2π 2 ζ k ζ −k is the dimensionless power spectrum of ζ. The squeezed limit k 3 k 1 k 2 is often described as a clean probe of the inflationary field content [20][21][22], notably with oscillations of the type S ∼ (k 3 /k 1 ) 1/2 cos(µ log(k 3 /k 1 ) + ϕ) revealing the spontaneous production, by the expansion of the universe, of a pair of scalar particles of masses m/H = µ 2 + 9/4. In Fig. (2), we compare these cosmological collider signals, 4 obtained by numerically solving Eqs. (5a, 5b), for different constant quadratic mixing strengths, and for each cubic interaction in (6). The most distinctive feature is that the frequency of the oscillations is set by µ 2 eff = m 2 eff /H 2 − 9/4, with m 2 eff = m 2 + ρ 2 playing the role of the effective mass for σ both at early and late times [14,[28][29][30]. The appearance of the quadratic mixing in m eff can be interpreted as the result of resumming an infinite number of quadratic mixing insertions in the propagators of both fields, when using an interaction scheme where quadratic mixings are treated perturbatively. At strong mixing, the propagation of σ is affected by the surrounding π c medium that interacts with it, leading to a self-energy correction. This is analogous to the electron self-energy correction in quantum electrodynamics due to its interaction with the photon [23,24], the difference being that in our case such resummation occurs at tree-level because Lorentz invariance is spontaneously broken. Focusing on a single property of the signal like its frequency, as is usually advertised, one could wrongly interpret a future detection as the discovery of a new weakly mixed massive particle, whereas this may well correspond to the signature of a light particle, albeit strongly mixed to the curvature perturbation. Our approach precisely allows one to break such degeneracies by providing complete predictions-covering the frequency, amplitude and phase of the cosmological colliders, as well as contaminations from equilateral shapes-for any theory. This enables one to explore the full range of possible signals without working under the lamppost of analytical tractability. 6) in the isosceles-triangle configuration k1 = k2, for cs = 0.1 and µ eff = 5 varying the quadratic mixing ρ/H = 0.1, 5, corresponding to weak and strong mixings, respectively. For illustration purposes and to highlight the cosmological collider signal frequency, we have normalized the amplitudes of the signals to unity in the equilateral configuration k1 = k2 = k3. We find that the amplitudes of the cosmological collider signals are non-monotonic as function of ρ, reaching a maximum for ρ ∼ m [14]. Moreover, the cosmological flow approach also enables one to reveal the dynamics of fluctuations, as we show in movies 5 displaying how the cosmological collider signals are built differently as inflation proceeds, in theories with weak and strong mixing. This exemplifies how our method provides a powerful guide for physicists to test their theoretical understanding. Cosmological Colliders with Features. Focusing on inflationary fluctuations, the background dynamics is encoded in the time dependence of H ab , H abc , . . .. Here we highlight the consequences of a time-dependent mixing on the cosmological collider signal. For definiteness, we consider ρ(t) = ρ 0 (a/a 0 ) −3/2 sin[ω c (t − t 0 )]. This is representative of, e.g. , background trajectories that undergo a sudden turn in field space, after which a massive field relaxes to its minimum subject to underdamped oscillations with frequency ω c . This feature is turned on smoothly at t 0 (a 0 = a(t 0 )) with a step of width ∆t, whose details are irrelevant for scales that cross the horizon after the feature, k k 0 ≡ a 0 H. We neglect any time variation of the Hubble parameter H, of the mass of σ, and of the cubic interaction strengths in the second line of (6), as our purpose is to concentrate on what is imposed by symmetries. For simplicity, we also set c s = 1 and focus first on weak mixing ρ 0 < H. The time-dependent mixing induces scale-dependent features in all correlation functions. For a fixed overall scale, the shape dependence of the cosmological collider depends on the cubic interaction that is considered. For those with constant strengths, the signal is conventional with frequency set by µ [31]. Instead, the cubic interactions dictated by the quadratic mixing are intrinsically timedependent and have a special status, withρ π cπc σ dominant for rapid oscillations with µ c ≡ ω c /H > 1. −4 −2 0 2 4 (k 1 /k 3 ) 2−ν S(k 1 , k 1 , k 3 ) ν = 3/2 ×10 −3 µ c = 5 µ c = 7 10 −2 10 −1 10 0 Figure 3. Shape dependence of the bispectrum dictated by the time-dependent mixing ρ(t), in the isosceles-triangle configuration k1 = k2 at a fixed scale log(k3/k0) = 2-so that the long wavelength mode exits the horizon two efolds after the location of the feature-for a massless field ν = 3/2 (top) and for a heavy field µ = 5 (bottom), varying the background frequency µc = 5, 7. We have chosen ρ0/H = 0.1 to respect the current bounds on the power spectrum [9] and ∆t = 0.8H −1 . k 3 /k 1 −4 −2 0 2 4 (k 1 /k 3 ) 2 S(k 1 , k 1 , k 3 ) µ = 5 We show in Fig. (3) the corresponding shape dependence of the bispectrum for a massless and heavy field σ, for different background frequencies µ c . We find new striking behaviors. First, the power-law scaling acquires an additional (k 3 /k 1 ) 3/2 suppression compared to the usual one [20]. This is directly related to the dilution of the feature in (a/a 0 ) −3/2 . Second, depending on whether the field σ is light (m/H ≤ 3/2) or heavy (m/H ≥ 3/2), the cosmological collider signal is either dictated by the background frequency µ c or presents modulated frequencies in µ ± µ c , respectively. In the squeezed limit, we find that the shape function takes the form S(k 1 , k 1 , k 3 ) =      k3 k1 2−ν A cos µ c log k3 k1 + ϕ ,(light) k3 k1 2 A + cos (µ + µ c ) log k3 k1 + ϕ + + A − cos (µ − µ c ) log k3 k1 + ϕ − , (heavy) (8) where µ = (m/H) 2 − 9/4 and ν = iµ. In [14], we give analytical expressions for the amplitudes A, A ± and the phases ϕ, ϕ ± as a function of k 3 /k 0 , capturing the scaledependence of the signal. Remarkably, we find numerically that the templates (8) hold in the strong mixing regime. These results provide a new way of probing the inflationary landscape through soft limits of cosmological correlators. Conclusions. The physics of inflation is phenomenologically rich and complex. In this Letter, we have presented a systematic framework for computing primordial correlators in any theory, focusing for definiteness on scalar degrees of freedom. Although massive fluctuations decay during inflation, their presence leaves a smoking-gun imprint in the squeezed limit of the bispectrum. Known as the cosmological collider signal, it offers a thrilling opportunity to identify new particles, in the same way as resonances for ground-based colliders. We demonstrated the power of our approach by computing cosmological collider signals in theories with strong and time-dependent mixings, and for various kinds of cubic interactions. These theoretically motivated scenarios present challenges that analytical methods are unable to address. The cosmological flow provides the means for exploring and understanding inflationary physics in full generality, bridging the gap between theories and observations. To ensure its accessibility, we will make our new computational approach available as an open-source numerical tool. As natural extensions, we plan to incorporate spinning fields and higher-order correlation functions, as well as consider loops. Figure 1 . 1Schematic diagram of the cosmological flow. The time evolution of inflationary correlators, given by Eqs. (5a, 5b), is tracked from initial quantum fluctuations in the infinite past to the end of inflation. All the theory dependence is left in the precise form of the tensors u a b and u a bc driving this flow. arXiv:2302.00655v1 [hep-th] 1 Feb 2023 level results in theories featuring an arbitrary number of fluctuating degrees of freedom with varied dispersion relations and masses, and coupled through any type of time-dependent interactions. Figure 2 . 2Squeezed limit of the shape functions for the four cubic interactions considered in Eq. ( More generally, smoking guns of new physics are encoded in particular soft limits of higher-order inflationary correlators.2 Following the conventions of[17], Greek indices α, β, γ, . . . run over fields, and Latin indices a, b, c, . . . run over phase-space coordinates, organised so that a block of field labels is followed by a block of momentum labels, in the same order. A prime on a correlator indicates that we have dropped the momentum conserving delta function (2π) 3 δ (3) (k 1 + k 2 + . . .).4 Even in squeezed configurations, the cosmological collider signal is altered by a residual background signal corresponding to EFT contributions after integrating out the heavy field. https://github.com/deniswerth/Cosmological-Collider-Flow Acknowledgements. We are grateful to Xingang Chen, Sadra Jazayeri, David . A H Guth, 10.1103/PhysRevD.23.347Phys. Rev. D. 23347A. H. Guth, Phys. Rev. D 23, 347 (1981). . A D Linde, 10.1016/0370-2693(82)91219-9Phys. Lett. B. 108389A. D. Linde, Phys. 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A Achúcarro, arXiv:2203.08128[astro-ph.COA. Achúcarro et al., (2022), arXiv:2203.08128 [astro- ph.CO]. . Y Akrami, Planck10.1051/0004-6361/201935891arXiv:1905.05697astro-ph.COAstron. Astrophys. 6419Y. Akrami et al. (Planck), Astron. Astrophys. 641, A9 (2020), arXiv:1905.05697 [astro-ph.CO]. . S Weinberg, 10.1103/PhysRevD.72.043514arXiv:hep-th/0506236Phys. Rev. D. 7243514S. Weinberg, Phys. Rev. D 72, 043514 (2005), arXiv:hep- th/0506236. Inflation and String Theory. D Baumann, L Mcallister, 10.1017/CBO9781316105733arXiv:1404.2601Cambridge Monographs on Mathematical Physics. Cambridge University Presshep-thD. Baumann and L. McAllister, Inflation and String The- ory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2015) arXiv:1404.2601 [hep-th]. . D Werth, L Pinol, S Renaux-Petel, D. Werth, L. Pinol, and S. Renaux-Petel, (2023). . D Seery, D J Mulryne, J Frazer, R H Ribeiro, 10.1088/1475-7516/2012/09/010arXiv:1203.2635JCAP. 0910astro-ph.COD. Seery, D. J. Mulryne, J. Frazer, and R. H. Ribeiro, JCAP 09, 010 (2012), arXiv:1203.2635 [astro-ph.CO]. . D J Mulryne, 10.1088/1475-7516/2013/09/010arXiv:1302.3842[astro-ph.COJCAP. 0910D. J. Mulryne, JCAP 09, 010 (2013), arXiv:1302.3842 [astro-ph.CO]. . M Dias, J Frazer, D J Mulryne, D Seery, 10.1088/1475-7516/2016/12/033arXiv:1609.00379JCAP. 1233astro-ph.COM. Dias, J. Frazer, D. J. Mulryne, and D. Seery, JCAP 12, 033 (2016), arXiv:1609.00379 [astro-ph.CO]. . D J Mulryne, J W Ronayne, 10.21105/joss.00494arXiv:1609.00381J. Open Source Softw. 3astro-ph.COD. J. Mulryne and J. W. Ronayne, J. Open Source Softw. 3, 494 (2018), arXiv:1609.00381 [astro-ph.CO]. . S Butchers, D Seery, 10.1088/1475-7516/2018/07/031arXiv:1803.10563JCAP. 0731astro-ph.COS. Butchers and D. Seery, JCAP 07, 031 (2018), arXiv:1803.10563 [astro-ph.CO]. . X Chen, Y Wang, 10.1088/1475-7516/2010/04/027arXiv:0911.3380JCAP. 0427hep-thX. Chen and Y. Wang, JCAP 04, 027 (2010), arXiv:0911.3380 [hep-th]. . 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Zhou, JCAP 01, 041 (2018), arXiv:1710.03054 [hep-th]. . X Chen, R Ebadi, S Kumar, 10.1088/1475-7516/2022/08/083arXiv:2205.01107JCAP. 0883hep-phX. Chen, R. Ebadi, and S. Kumar, JCAP 08, 083 (2022), arXiv:2205.01107 [hep-ph].

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{'abstract': 'Correlation functions of primordial density fluctuations provide an exciting probe of the physics governing the earliest moments of our Universe. However, the standard approach to compute them is technically challenging. Theoretical predictions are therefore available only in restricted classes of theories. In this Letter, we present a complete method to systematically compute tree-level inflationary correlators. This method is based on following the time evolution of equal-time correlators and it accurately captures all physical effects in any theory. These theories are conveniently formulated at the level of inflationary fluctuations, and can feature any number of degrees of freedom with arbitrary dispersion relations and masses, coupled through any type of time-dependent interactions. We demonstrate the power of this approach by exploring the properties of the cosmological collider signal, a discovery channel for new high-energy physics, in theories with strong mixing and in the presence of features. This work lays the foundation for a universal program to assist our theoretical understanding of inflationary physics and generate theoretical data for an unbiased interpretation of upcoming cosmological observations.', 'arxivid': '2302.00655', 'author': ["Denis Werth \nUMR 7095\nSorbonne Université, Institut d'Astrophysique de Paris\nCNRS\n98 bis bd Arago75014ParisFrance\n", "Lucas Pinol \nUMR 7095\nSorbonne Université, Institut d'Astrophysique de Paris\nCNRS\n98 bis bd Arago75014ParisFrance\n\nInstituto de Física Teórica UAM-CSIC\nc/ Nicolás Cabrera 13-1528049Madrid\n", "Sébastien Renaux-Petel \nUMR 7095\nSorbonne Université, Institut d'Astrophysique de Paris\nCNRS\n98 bis bd Arago75014ParisFrance\n"], 'authoraffiliation': ["UMR 7095\nSorbonne Université, Institut d'Astrophysique de Paris\nCNRS\n98 bis bd Arago75014ParisFrance", "UMR 7095\nSorbonne Université, Institut d'Astrophysique de Paris\nCNRS\n98 bis bd Arago75014ParisFrance", 'Instituto de Física Teórica UAM-CSIC\nc/ Nicolás Cabrera 13-1528049Madrid', "UMR 7095\nSorbonne Université, Institut d'Astrophysique de Paris\nCNRS\n98 bis bd Arago75014ParisFrance"], 'corpusid': 256459857, 'doi': None, 'github_urls': ['https://github.com/deniswerth/Cosmological-Collider-Flow'], 'n_tokens_mistral': 9237, 'n_tokens_neox': 7613, 'n_words': 4443, 'pdfsha': 'ef3932aae9aef2cc8ac8bf3096949cd5a167546e', 'pdfurls': ['https://export.arxiv.org/pdf/2302.00655v1.pdf'], 'title': ['Cosmological Flow of Primordial Correlators', 'Cosmological Flow of Primordial Correlators'], 'venue': []}

arxiv

Distribution of Normalized Zero-Sets of Random Entire Functions with Small Random Perturbation 11 Apr 2009 Weihong Yao whyao@sjtu.edu.cn Department of Mathematics Shanghai Jiaotong University 200240ShanghaiP. R. China Distribution of Normalized Zero-Sets of Random Entire Functions with Small Random Perturbation 11 Apr 2009arXiv:0904.1802v1 [math.CV]Hermitian holomorphic line bundlesrandom entire functionsPoincaré-Lelong formulacounting function In this paper, we extend our earlier result (see [Y-2008]) on the distribution of normalized zero-sets of random entire functions to random entire functions with small random perturbation. Introduction The well-known fundamental theorem of algebra states that for every complex polynomial P , the degree of P is equal to the number of zeros of P on the complex plane, counting multiplicities. This suggests that one can use the counting functions (the number of zeros) to measure the growth of P (i.e. the degree of P ). In 1929, Nevanlinna extended the polynomial theory to meromorphic functions on C (which can be viewed as holomorphic maps f : C → P 1 (C)), in which case the growth function of f is given by its characteristic function T f (r) for |z| < r. Geometrically, T f (r) is determined by the the area of the image of f (△(r)) in P 1 (C) with respect to the Fubini-Study metric. Similar to the polynomials case, Nevanlinna proved that for almost all a ∈ P 1 , N f (r, a) = T f (r) + O(1) (or more precisely the integral average of N f (r, a) with respect to a is T f (r)). The result of this type is called the First Main Theorem of Nevanlinna. Furthermore, Nevanlinna obtained a much deeper result (called the Second Main Theorem of Nevanlinna) which states that the sum of the difference T f (r) − N f (r, a j ), for any distinct a 1 , . . . , a q ∈ P 1 (C), cannot exceed (2 + ǫ)T f (r) for all r ∈ (0, +∞) except for a set of finite measure. The theory is now known as Nevanlinna theory or value distribution theory. Nevanlinna's theory was later extended by H. Cartan and L. Ahlfors to holomorphic curves. The proof of the fundamental theorem of algebra comes from the following observation: when we write P (z) = a n z n + Q n−1 (z), where n = deg P , then |Q n−1 (z)| < |a n z n | on |z| = r for r large enough, hence Rouché's theorem implies that the the zeros of P is the same as the zeros of a n z n . In other words, P (z) can be obtained from a n z n through a small perturbation by Q n−1 . Similarly, one can easily prove that the growth (characteristic function) of f is the same as f + g, the function obtained by small perturbation by g. (Here, by small perturbations we mean T g (r) = o(T f (r))). Problems of these types are called small perturbation problems or called problems of slowly moving targets. In 1983, Steinmetz successfully extended Nevanlinna's SMT to slowly moving targets, and in 1990, Ru-Stoll extended H. Cartan's theorem to slowly moving targets as well. Recently, Shiffman and Zelditch, in their series of papers, initiated the study of random value distribution theory. The theory is based on the following fundamental result of Hammersley: the zeros of random complex "Kac" polynomials f (z) = N j=0 a j z j (where the coefficients a j are independent complex Gaussian random variables of mean 0 and variance 1) tend to concentrate evenly about the unit circle as the degree N goes to the infinity. Shiffman and Zelditch extended the result to random polynomials of several complex variables, as well as random holomorphic sections of line bundles. Their method however largely relies on the use of kernel functions, thus the strong "orthonormal conditions" are unavoidable. In place of using kernel functions we propose a direct method to study the value distribution of random meromorphic functions (or maps). This method starts with the computation of mathematical expectations in the form of closed positive (1,1)-currents defined by normalized counting divisors, and is applicable to the much broader context of random holomorphic functions and more generally random meromorphic functions (maps). As a first step in this direction the author studied in [Y-2008] the distribution of the normalized zero-sets of random holomorphic functions G n (z) = n ν=0 ℓ j 1 =1 · · · ℓ jν =1 a j 1 ,··· ,jν f j 1 (z) · · · f jν (z), where f 1 (z), · · · , f ℓ (z) are fixed holomorphic functions on a domain Ω ∈ C, and coefficients a j 1 ,··· ,jν are independent complex Gaussian random variables with mean 0 and variance 1. More precisely, we studied the mean (mathematical expectation) Z (r, G n ) of the normalized counting divisor Z (r, G n ) of G n (z) on the punctured disk 0 < |z| < r (in the sense of distribution) which is given by Z (r, G n ) = 1 n Gn(z)=0, 0<|z|<r log r |z| δ z , where δ z is the Dirac function. We obtained the following result in [Y-2008]. Theorem A. Let C be the smooth (possibly non-closed) curve in C consisting of all the points z such that |f (z)| = ℓ j=1 |f j (z)| 2 1 2 = 1 and f ′ (z) = (f ′ 1 (z), ..., f ′ l (z)) = 0. Then the limit of E (Z (r, G n )) is equal to, in the sense of distribution, log r |z| times the sum of |f (z)| Ξ (|f (z)|) √ −1 2π ∂∂ log |f (z)| and the measure on C defined by the 1-form √ −1 2 ℓ j=1 f j (z)df j (z) − f j (z)df j (z) , where Ξ (x) =    2 x when x > 1 1 when x = 1 0 when x < 1. When ℓ = 1 and f 1 (z) = z, our theorem recovers the result of Hammersley. Theorem A can also be interpreted as an analogue of Nevanlinna's First Main Theorem, which states that the integral average of the counting function is equal to the growth (characteristic) function of f . We also note that the approach used in [Y-2008] is very different from the method of Shiffman-Zelditch. In place of the use of sophisticated results on kernel functions, we carry out a direct computation, which basically comes down to the computation of the following limit (see "Complex Version of Lemma on the Convergence of integrals as Distributions"), for w ∈ C ℓ , lim n→∞ 1 n √ −1 2π ∂∂ log n j=0 |w| 2j , in the sense of distribution. In this paper, we extend Theorem A to the random entire functions with small perturbation, similar to the moving target case in Nevanlinna's theory by an adaptation of the method and techniques of [Y-2008]. The main result of this paper is as follows: Main Theorem. Let Ω be an open subset of C, ℓ be a positive integer and f 1 (z), . . . , f ℓ (z) be holomorphic functions on Ω. For any z 0 ∈ Ω, let κ z 0 ,n , λ z 0 ,n , ξ z 0 ,n , η z 0 ,n , be four sequences of non-negative integers, 0 ≤ n < ∞. Assume that lim n→∞ κ z 0 ,n n = 0 , lim n→∞ λ z 0 ,n n = 0 , lim n→∞ ξ z 0 ,n n = 0 , lim n→∞ η z 0 ,n n = 0 . Let A z and B z be positive functions on Ω. For any nonnegative integer j, let g j (z) be a holomorphic function on Ω such that (i) g 0 (z) is nowhere zero on Ω, (ii) for each 0 ≤ j < ∞ and each point j ∈ Ω g j (z) ≤ (B z ) κ z,j 1 + f (z) λ z,j , (iii) for each point z ∈ Ω lim inf j→∞ (A z ) ξ z,j 1 + f (z) η z,j g j (z) is positive. Suppose furthermore that for each compact subset K ⊂ Ω there exists a positive constant C K such that for each z ∈ K and for each nonnegative integer j we have B z ≤ C K ; κ z,j , λ z,j ≤ C K For any positive integer n let G n (z) = n j=1 1≤ν 1 ,...,ν j ≤ℓ a ν 1 ,...,ν j g j (z)f ν 1 (z) · · · f ν j (z) be a random polynomial, where each coefficient a ν 1 ,...,ν j for 1 ≤ ν 1 , . . . , ν j ≤ ℓ and 0 ≤ j ≤ n is an indeterminate which satisfies the Gaussian distribution 1 π e −|z| 2 on C, with the convention that a 0 is the single indeterminate a ν 1 ,...,ν j , when j = 0. Let Z(G n ) be the normalized counting divisor of G n (z) on Ω (in the sense of distribution) given by Z(G n ) = √ −1 n Gn(z)=0, z∈Ω δ z , where δ z is the Dirac delta on C at the point z of C. Let E(Z(G n )) be the expectation of Z(G n ) which is defined by (aν 1 ,...,ν j )∈C N ℓ,n 1 n Gn(z)=0, z∈Ω δ z (aν 1 ,...,ν j )∈C N ℓ,n 1 π e −|aν 1 ···ν j | 2 √ −1 2 da ν 1 ,...,ν j ∧ da ν 1 ···ν j , where N ℓ,n = ℓ n+1 −1 ℓ−1 is the number of indeterminates a ν 1 ,...,ν j for 0 ≤ j ≤ n and 1 ≤ ν 1 , . . . , ν j ≤ ℓ. Let C be the smooth (possibly non-closed) curve in Ω consisting of all the points z of Ω such that f (z) = ℓ j=1 |f j (z)| 2 1 2 = 1 and f ′ (z) = (f ′ 1 (z), . . . , f ′ 1 (z)) = 0. Then E(Z(G n )) is equal to the sum of f (z) Ξ |f (z)| √ −1 2π ∂∂ log f (z) and the measure on C defined by the 1-form √ −1 2 ℓ j=1 f j (z)df j (z) − f j (z)df j (z) , where Ξ(x) =      2 x when x > 1 1 when x = 1 0 when x < 1 . Remarks In the formulation of the Main Theorem, by (iii) we impose at each point z ∈ Ω an asymptotic pointwise condition on a lower bound on |g j (z)|. It is natural to impose a condition in terms of lim inf, given that we want to allow the holomorphic functions g j to have zeros. On the other hand, in the computation of mathematical expectations of normalized counting divisors, some uniformity on compact subsets is required on asymptotic upper bounds on |g j (z)| in order to prove convergence of positive (1,1)-currents. Proof of the Main Theorem We first recall the following key lemmas in [Y-2008]. (2.1) Proposition [Y-2008] (Complex Version of Lemma on the Convergence of Integrals as Distributions). lim n→∞ 1 n √ −1 2π ∂∂ log n j=0 |z| 2j = rΞ (r) √ −1 2π ∂∂ log r + δ S 2ℓ−1 ∧ 1 r 2 √ −1 2 ℓ j=1 (z j dz j −z j dz j ) , is true, where δ S 2ℓ−1 denotes the 1-current on C ℓ defined by integration over S 2ℓ−1 =    z ∈ C ℓ |z| = ℓ j=1 |z j | 2 1 2 = 1    . By pulling back through f (z) = (f 1 (z), f 2 (z), · · · , f ℓ (z)) : Ω → C ℓ , the above proposition implies (2.2) Proposition [Y-2008]. Let Ω be a connected open subset of C and f (z) = (f 1 (z), f 2 (z), · · · , f ℓ (z)) : Ω → C ℓ be a nonconstant holomorphic function on Ω. Let C be the smooth (possibly non-closed) curve in Ω consisting of all the points z of Ω such that |f (z)| = 1 and f ′ (z) = (f ′ 1 (z), f ′ 2 (z), · · · , f ′ ℓ (z)) = 0, where |f (z)| = (|f 1 (z)| 2 + |f 2 (z)| 2 + · · · + |f ℓ (z)| 2 ) 1 2 . Then the following lim n→∞ 1 n √ −1 2π ∂∂ log n j=0 |f (z)| 2j = |f (z)| Ξ (|f (z)|) √ −1 2π ∂∂ log |f (z)| + δ S 2ℓ−1 ∧ 1 |f (z)| 2 √ −1 2 ℓ j=1 f j (z)df j (z) −f j (z)df j (z) , is true, where δ S 2ℓ−1 denotes the 1-current on C ℓ defined by integration over S 2ℓ−1 =    f (z) ∈ C ℓ |f (z)| = ℓ j=1 |f j (z)| 2 1 2 = 1    . (2.3) For the proof of the Main Theorem we will need to formulate a lemma on limits of certain potential functions. To start with define on the domain Ω ⊂ C the following subharmonic functions γ n = 1 n log(1 + |f | 2 + · · · + |f | 2n ). Define ϕ : Ω → R by ϕ(z) = log |f (z)| 2 if |f (z)| ≥ 1 ϕ(z) = 0 if |f (z)| ≤ 1 . In other words, ϕ(z) = max(0, log |f | 2 ) = log + |f | 2 . Then, we have Lemma 1 γ n (z) converges uniformly to ϕ(z) on Ω. As a consequence, √ −1∂∂γ n converges to √ −1∂∂ϕ as positive (1, 1)-currents on Ω. Proof. For each positive integer n define λ n : [0, ∞) → R by λ n (t) = 1 n log(1 + t + · · · + t n ). For 0 ≤ t ≤ 1 we have 0 ≤ λ n (t) ≤ 1 n log(n + 1). On the other hand, for t ≥ 1 we have log t = 1 n log(t n ) ≤ λ n (t) ≤ 1 n log (n + 1)t n = 1 n log(n + 1) + log t. Let λ : [0, ∞) → R be the monotonically increasing continuous function defined by λ(t) = log t for t ≥ 1; λ(t) = 0 for 0 ≤ t ≤ 1. Then, λ(t) ≤ λ n (t) ≤ 1 n log(n + 1) + λ(t). Thus, over [0, ∞), λ n (t) converges uniformly to λ(t). For the map f : Ω → C, γ n = 1 n log(1 + |f | 2 + · · · + |f | 2n ) = λ n (|f | 2 ), so that γ n converges uniformly to λ(|f | 2 ) = ϕ, and it follows that √ −1∂∂γ n converges to √ −1∂∂ϕ as positive (1, 1)-currents on Ω, as desired. (2.4) We proceed to give a proof of the Main Theorem. Proof. In the language of probability theory, (a j 1 ,··· ,jν ) 0≤ν≤n,1≤j 1 ≤ℓ,··· ,1≤jν ≤ℓ are independent complex Gaussian random variables of mean 0 and variance 1. Let N ℓ,n be the number of elements in (a j 1 ,··· ,jν ) 0≤ν≤n,1≤j 1 ≤ℓ,··· ,1≤jν ≤ℓ , which is N ℓ,n = 1 + ℓ + ℓ 2 + · · · + ℓ n . Let a 0 be the single indeterminate a j 1 ,··· ,jν when ν = 0. By Cauchy's integral formula (or the Poincaré-Lelong formula) ( * ) 1 n Gn(z)=0, z∈Ω δ z = √ −1 nπ ∂∂ log |G n (z)| on Ω, where δ z is the Dirac delta on C ℓ at the point z of C ℓ . We now consider the normalized counting divisor Z (G n ) of G n (z) on Ω (in the sense of distribution) which is given by Z (G n ) = 1 n Gn(z)=0, z∈Ω δ z . By ( * ), the expectation E (Z (G n )) of Z (G n ) is equal to (aj 1 ,··· ,jν )∈C N ℓ,n √ −1 nπ ∂∂ log |G n (z)| × (aj 1 ,··· ,jν )∈C N ℓ,n 1 π e −|a j 1 ···jν | 2 √ −1 2 da j 1 ···jν ∧ da j 1 ···jν . We introduce two column vectors a = [a j 1 ,··· ,jν ] 0≤ν≤n,1≤j 1 ≤ℓ,··· ,1≤jν ≤ℓ and v(z) = [g ν (z)f j 1 (z) · · · f jν (z)] 0≤ν≤n,1≤j 1 ≤ℓ,··· ,1≤jν ≤ℓ of N ℓ,n components each. Here we set f 0 (z) = 1. Then G n (z) is equal to the inner product a, v(z) = 0≤ν≤n,1≤j 1 ≤ℓ,··· ,1≤jν ≤ℓ a j 1 ,··· ,jν g ν (z)f j 1 (z) · · · f jν (z) of the two N ℓ,n -vectors a and v(z). The length of the N ℓ,n -vector v(z) is given by v(z) = 0≤ν≤n,1≤j 1 ≤ℓ,··· ,1≤jν ≤ℓ |g ν (z)| 2 |f j 1 (z)| 2 · · · |f jν (z)| 2 1 2 . Introduce the unit N ℓ,n -vector u(z) = 1 v(z) v(z) = 1 0≤ν≤n,1≤j 1 ,··· ,jν ≤ℓ |g ν (z)| 2 |f j 1 (z)| 2 · · · |f jν (z)| 2 1 2 [g ν (z)f j 1 (z) · · · f jν (z)] 0≤ν≤n,1≤j 1 ,··· ,jν ≤ℓ in the same direction as v(z). Then log |G n (z)| = log | a, v(z) | = log | a, v(z) u(z) | = log v(z) + log | a, u(z) | . Now E (Z (G n )) is equal to (aj 1 ,··· ,jν )∈C N ℓ,n √ −1 nπ ∂∂ (log v(z) + log | a, u(z) |) × (aj 1 ,··· ,jν )∈C N ℓ,n 1 π e −|a j 1 ···jν | 2 √ −1 2 da j 1 ···jν ∧ da j 1 ···jν . Let e 0 be the N ℓ,n -vector (e j 1 ,··· ,jν ) 0≤ν≤n,1≤j 1 ≤ℓ,··· ,1≤jν ≤ℓ whose only nonzero component is e 0 = 1. Here comes the key point of the whole argument. For fixed z, we integrate ( a j 1 ,··· ,jν )∈C N ℓ,n √ −1 nπ ∂∂ log | a, u(z) | × (aj 1 ,··· ,jν )∈C N ℓ,n 1 π e −|a j 1 ···jn | 2 √ −1 2 da j 1 ···jn ∧ da j 1 ···jn = (aj 1 ,··· ,jν )∈C N ℓ,n √ −1 nπ ∂∂ log | a, u(z) | 1 π N ℓ,n e − a 2 = (aj 1 ,··· ,jν )∈C N ℓ,n √ −1 nπ ∂∂ log | a, e 0 | 1 π N ℓ,n e − a 2 = (aj 1 ,··· ,jν )∈C N ℓ,n √ −1 nπ ∂∂ log |a 0 | 1 π N ℓ,n e − a 2 = √ −1 nπ ∂∂ a 0 ∈C (log |a 0 |) 1 π e −|a 0 | 2 which is equal to √ −1 nπ ∂∂A = 0 with A = a 0 ∈C (log |a 0 |) 1 π e −|a 0 | 2 , because A is a constant. Note that the equality ( a j 1 ,··· ,jν )∈C N ℓ,n √ −1 nπ ∂∂ log | a, u(z) | 1 π N ℓ,n e − a 2 = ( a j 1 ,··· ,jν )∈C N ℓ,n √ −1 nπ ∂∂ log | a, e 0 | 1 π N ℓ,n e − a 2 in the above string of equalities comes from the fact that for any fixed z ∈ C some unitary transformation of C N ℓ,n (which may depend on z) maps u(z) to e 0 and that e − a 2 is unchanged under any unitary transformation acting on a. Thus the limit of E (Z (G n )) as n → ∞ is equal to lim n→∞ (aj 1 ,··· ,jν )∈C N ℓ,n √ −1 nπ ∂∂ log v(z) 1 π N ℓ,n e − a 2 , which after integration over (a j 1 ,··· ,jν ) 0≤ν≤n,1≤j 1 ≤ℓ,··· ,1≤jν ≤ℓ is simply equal to lim n→∞ √ −1 nπ ∂∂ log v(z) = lim n→∞ 1 n   √ −1 2π ∂∂ log n k=0 |g k (z)| 2 ℓ j=1 |f j (z)| 2 k   From (2.1) Proposition, we have lim n→∞ 1 n   √ −1 2π ∂∂ log n k=0 ℓ j=1 |f j (z)| 2 k   is equal to the pullback by f of |w| Ξ (|w|) √ −1 2π ∂∂ log |w| + δ S 2ℓ−1 ∧ 1 |w| 2 √ −1 2 ℓ j=1 (w j dw j −w j dw j ) , where w ∈ C ℓ = (w 1 , · · · , w ℓ ) is variable in the target space of the map f = (f 1 , · · · , f ℓ ) : C → C ℓ . By computation lim n→∞ 1 n √ −1 2π ∂∂ log   n k=0 |g k (z)| 2 ℓ j=1 |f j (z)| 2 k   − lim n→∞ 1 n √ −1 2π ∂∂ log   n k=0 ℓ j=1 |f j (z)| 2 k   = lim n→∞ 1 n √ −1 2π ∂∂   log n k=0 |g k (z)| 2 ℓ j=1 |f j (z)| 2 k − log n k=0 ℓ j=1 |f j (z)| 2 k   = √ −1 2π ∂∂ lim n→∞ 1 n   log n k=0 |g k (z)| 2 ℓ j=1 |f j (z)| 2 k − log n k=0 ℓ j=1 |f j (z)| 2 k   = √ −1 2π ∂∂ lim n→∞ 1 n log    n k=0 |g k (z)| 2 ℓ j=1 |f j (z)| 2 k n k=0 ℓ j=1 |f j (z)| 2 k    . Without loss of generality, we may assume that for any z ∈ Ω, A z , B z ≥ 1. Granted this, replacing κ z,n by max κ z,0 , . . . , κ z,n , etc., without loss of generality we may assume that the four sequences κ z,n , λ z,n , ξ z,n and η z,n are non-decreasing sequences. By (iii) for every z ∈ Ω there exists a positive constant c z and a positive integer J(z) such that whenever j ≥ J(z) we have Similarly for the function n k=0 |f (z)| 2k 1 n we have max 1, |f (z)| 2 ≤ n k=0 |f (z)| 2k 1 n ≤ (n + 1) 1 n max 1, |f (z)| 2 . Finally, recalling that h n (z) 1 n =     n k=0 |g k (z)| 2 |f (z)| 2k n k=0 |f (z)| 2k     so that lim n→∞ h n (z) 1 n = 1 ; lim n→∞ log h n (z) 1 n = 0 . Under the assumptions of the Main Theorem write ϕ n = log n j=0 g j (z) 2 f (z) 2j 1 n . Then, log h n 1 n = ϕ n − γ n . Since γ n converges to ϕ = log + |f | 2 by Lemma 1 and log h n 1 n converges pointwise to 0, we conclude that ϕ n (z) converges to ϕ(z) for every z ∈ Ω. Clearly ϕ n and ϕ are continuous subharmonic functions on Ω. Moreover from ( †) we have for every z ∈ Ω ϕ n (z) ≤ 1 n log(n + 1) + 2κ z,n n log B(z) + λ z,n n log 4, and by assumption on any compact subset K ⊂ Ω, B z and the sequence of functions κ z,n n and λ z,n n are uniformly bounded from above by some constant c K for z ∈ K, and we conclude that the sequence of subharmonic functions ϕ n (z) ∞ n=0 are uniformly bounded from above on compact subsets. Finally, we make use of Lemma 2 below on the convergence of positive (1, 1) currents. Granting Lemma 2, the Main Theorem follows readily. The discussion below involves distributions on a domain in C. Denote by dλ the Lebesgue measure on C. Any locally integrable function s on Ω defines a distribution T s on Ω given by T s (ρ) = Ω sρ dλ for any smooth function ρ on Ω of compact support, and in what follows we will identify s with the distribution T s it defines. There is a standard process for smoothing distributions, as follows. Let χ be a nonnegative smooth function on C of support lying on the unit disk ∆ such that χ(e iθ z) = χ(z) for any z ∈ C and any θ ∈ R, and for any ǫ > 0 write χ ǫ (z) = χ z ǫ . For a distribution Q defined on some domain in C and for ǫ > 0 we write Q ǫ := Q * χ ǫ wherever the convolution is defined. We have the following elementary lemma on positive currents associated to subharmonic functions, for which a proof is included below for easy reference. Lemma 2 Let Ω ⊂ C be a plane domain. Suppose (ϕ n ) ∞ n=0 is a sequence of subharmonic functions on Ω such that ϕ n (z) are uniformly bounded from above on each compact subset K of Ω. Assume that ϕ n converges pointwise to some continuous (subharmonic) function ϕ. Then, lim n→∞ ϕ n = ϕ in L 1 loc (Ω). As a consequence, √ −1 ∂∂ϕ n converges to √ −1 ∂∂ϕ in the sense of currents. + |f (z)| 2λz,n |f (z)| 2n , so that n k=0 |g k (z)| 2 |f (z)| 2k ≤ max (n + 1)B 2κz,n z · 4 λz,n , (n + 1)B 2κz,n z 1 + |f (z)| 2λz,n |f (z)| 2n . AcknowledgmentsThe author would like to thank Professors Yum-Tong Siu and Ngaiming Mok for discussions and suggestions concerning this article. A ξ z,j z 1 + |f (z)| η z,j |g j (z)| ≥ c z .(Here and in what follows to streamline the notations we will write A ξ z,j z to mean (A z ) ξ z,j , etc.) For every z ∈ Ω we have n k=0 |g k (z)| 2 |f (z)| 2k ≥ max |g 0 (z)| 2 , |g n (z)| 2 |f (z)| 2n ≥ max |g 0 (z)| 2 , c 2 z A −2ξz,n zOn the other hand, when |f (z)| ≤ 1 we haveand, when |f (z)| ≥ 1 we haveFor the lower bound of h(z) 1 n we note thatwhere we have used the assumptions limwhere we have used the assumptions lim n→∞ κz,n n = lim n→∞ λz,n n = 0. Thus, for any z ∈ Ω we haveProof of Lemma 2. Let D = ∆(a; r) be any disk centred at a ∈ Ω of radius r > 0 such that D ⊂ Ω. We claim that the Lebesgue integrals ∆(a;r) |ϕ n | dλ are bounded independent of n. Without loss of generality, we may assume that ϕ ≤ 0 on D. By the sub-mean-value inequality for subharmonic functions we have ϕ n (a) ≤ 1 πr 2 ∆(a;r) ϕ n (ζ) dξ dη where ζ = ξ + √ −1η is the Euclidean coordinate of the variable of integration ζ, showing that the integral of −ϕ n over ∆(a; r) are bounded independent of n.Covering Ω by a countable and locally finite family of relatively compact open disks D, it follows that on any compact subset K ⊂ Ω the L 1 -norms of ϕ n | K are bounded independent of n. As a consequence, given any subsequence ϕ σ(n) of ϕ n , some subsequence ψ n := ϕ σ(τ (n)) of ϕ σ(n) must converge to a distribution S on Ω. We claim that any such a limit must be given by the (continuous) subharmonic function ϕ. As a consequence, ϕ n converges to ϕ in L 1 loc (Ω). Since ψ n converges to the distribution S, for any ǫ > 0, ϕ n,ǫ converges to the smooth function S ǫ as n tends to ∞. Since ψ n is subharmonic, ψ n,ǫ is monotonically decreasing as ǫ → 0 for each nonnegative integer n, and it follows readily that S ǫ is also monotonically decreasing as ǫ → 0. Hence, S is the limit as a distribution of the smooth functions S ǫ . Writing ψ(z) := lim ǫ →0 S ǫ (z), by the Monotone Convergence Theorem the distribution S is nothing other than the function ψ, which is in particular locally integrable. Since ψ n converges to S as distributions, we conclude that ϕ σ(n) = ψ n converges to ψ in L 1 loc (Ω), implying that ψ n converges pointwise to ψ almost everywhere on D. However, by assumption ψ n = ϕ σ(n) converges pointwise to ϕ, hence ϕ and ψ must agree almost everywhere on Ω. In particular, ϕ n must converge to ϕ in L 1 loc (Ω). The proof of Lemma 2 is complete. Critical points and supersymmetric vacua. II. Asymptotic and extremal metrics. Michael R Douglas, Bernard Shiffman, Steve Zelditch, J. Differential Geom. 723[DSZ-2006a] Michael R. Douglas, Bernard Shiffman, and Steve Zelditch, Critical points and supersymmetric vacua. II. Asymptotic and extremal metrics. J. Dif- ferential Geom. 72 (2006), no. 3, 381-427. Critical points and supersymmetric vacua. III. String/M models. Michael Douglas, Bernard Shiffman, Steve Zelditch, Comm. Math. Phys. 2653DSZ-2006b[DSZ-2006b] Michael Douglas, Bernard Shiffman, and Steve Zelditch, Critical points and supersymmetric vacua. III. String/M models. Comm. Math. Phys. 265 (2006), no. 3, 617-671. The zeros of a random polynomial. J M Hammersley, Proceedings of the. theJ M. Hammersley, The zeros of a random polynomial. In: Proceedings of the . Third Berkeley Symposium on Mathematical Statistics and Probability. IIUniversity of California PressThird Berkeley Symposium on Mathematical Statistics and Probability, 1954C1955, vol. II, 89C111, Berkeley-Los Angeles: University of California Press, 1956 L A Shepp, R J Vanderbei, The complex zeros of random polynomials. L.A. Shepp and R.J. Vanderbei, The complex zeros of random polynomials. . Trans. Am. Math. Soc. 347Trans. Am. Math. Soc. 347 (1995), 4365-4384. Distribution of zeros of random and quantum chaotic sections of positive line bundles. Bernard Shiffman, Steve Zelditch, Comm. Math. Phys. 2003SZ-1999[SZ-1999] Bernard Shiffman and Steve Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles. Comm. Math. Phys. 200 (1999), no. 3, 661-683. Distribution of Normalized Zero-Sets of Random Entire Functions. W Yao, arxiv.org/abs/0811.3365[Y-2008] W. Yao, Distribution of Normalized Zero-Sets of Random Entire Func- tions, arxiv.org/abs/0811.3365.

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{'abstract': 'In this paper, we extend our earlier result (see [Y-2008]) on the distribution of normalized zero-sets of random entire functions to random entire functions with small random perturbation.', 'arxivid': '0904.1802', 'author': ['Weihong Yao whyao@sjtu.edu.cn \nDepartment of Mathematics\nShanghai Jiaotong University\n200240ShanghaiP. R. China\n'], 'authoraffiliation': ['Department of Mathematics\nShanghai Jiaotong University\n200240ShanghaiP. R. China'], 'corpusid': 2974279, 'doi': '10.11650/tjm.15.2011.18', 'github_urls': [], 'n_tokens_mistral': 9670, 'n_tokens_neox': 8596, 'n_words': 4952, 'pdfsha': '9ca8fd9f56a154758491d7a7c32b0be5c008834f', 'pdfurls': ['https://arxiv.org/pdf/0904.1802v1.pdf'], 'title': ['Distribution of Normalized Zero-Sets of Random Entire Functions with Small Random Perturbation', 'Distribution of Normalized Zero-Sets of Random Entire Functions with Small Random Perturbation'], 'venue': []}

arxiv

Risk Sensitive Dead-end Identification in Safety-Critical Offline Reinforcement Learning Taylor W Killian twkillian@cs.toronto.edu University of Toronto Vector Institute Massachusetts Institute of Technology Imperial College London Sonali Parbhoo sparbhoo@imperial.ac.uk University of Toronto Vector Institute Massachusetts Institute of Technology Imperial College London Marzyeh Ghassemi mghassem@mit.edu University of Toronto Vector Institute Massachusetts Institute of Technology Imperial College London Risk Sensitive Dead-end Identification in Safety-Critical Offline Reinforcement Learning Published in Transactions on Machine Learning Research (01/2023) Massachusetts Institute of Technology CIFAR AI Chair, Vector Institute Reviewed on OpenReview: https: // openreview. net/ forum? id= oKlEOT83gI In safety-critical decision-making scenarios being able to identify worst-case outcomes, or dead-ends is crucial in order to develop safe and reliable policies in practice. These situations are typically rife with uncertainty due to unknown or stochastic characteristics of the environment as well as limited offline training data. As a result, the value of a decision at any time point should be based on the distribution of its anticipated effects. We propose a framework to identify worst-case decision points, by explicitly estimating distributions of the expected return of a decision. These estimates enable earlier indication of dead-ends in a manner that is tunable based on the risk tolerance of the designed task. We demonstrate the utility of Distributional Dead-end Discovery (DistDeD) in a toy domain as well as when assessing the risk of severely ill patients in the intensive care unit reaching a point where death is unavoidable. We find that DistDeD significantly improves over prior discovery approaches, providing indications of the risk 10 hours earlier on average as well as increasing detection by 20%. Introduction In complex safety-critical decision-making scenarios, being able to identify signs of rapid deterioration is crucial in order to proactively adjust a course of action, or policy. Consider the challenge of replacing an aging component within high-value manufacturing machinery. The longer one waits to replace this component, the efficiency of the process degrades until catastrophic failure at some unknown future time. However, the cost of temporarily stopping manufacturing to replace the component is non-trivial and the observed state of the system may not transparently signal when failure is imminent. Specifically, being aware of potential "worst-case" outcomes when choosing whether to delay repair is paramount to develop both safe and successful policies. Yet quantifying the worst-case outcomes in these and related circumstances among other safety critical domains-such as healthcare-is usually challenging as a result of unknown stochasticity in the environment, potentially changing dynamics, limited data, and the wide range of possible outcomes that might follow a sequence of decisions. By reliably providing an early indication of system failure to human operators, they would be enabled to intervene and make the necessary repairs in order to avoid system failure. Reinforcement learning (RL) is a natural paradigm to address sequential decision-making tasks in safetycritical settings, focusing on maximizing the cumulative effects of decisions over time (Sutton & Barto, 2018). In this paper, we propose a risk-sensitive decision-making framework positioned to serve as an early-warning system for dead-end discovery. Broadly, our framework may be thought of as a tool for thinking about risksensitivity in data-limited offline settings. Our contributions are as follows: (i) Unlike former approaches, we incorporate distributional estimates of the return (Bellemare et al., 2022) to determine when an observed state is at risk of becoming a dead-end from the expected worst-case outcomes over available decisions (Chow et al., 2015). (ii) We establish that our risk-estimation procedure serves as a lower-bound to the theoretical results underlying DeD (Fatemi et al., 2021), maintaining important characteristics for assessing when identifying dead-ends. As a result, we are able to detect and provide earlier indication of high-risk scenarios. (iii) By modeling the full distribution of the expected return, we construct a spectrum of risk-sensitivity when assessing dead-ends. We show that this flexibility allows for tunable risk estimation procedures and can be customised according to the task at hand. (iv) Finally, we provide empirical evidence that our proposed framework enables an earlier determination of high-risk areas of the state space on both a simulated environment and a real application within healthcare of treating patients with sepsis. Related Work Safe and Risk-sensitive RL A shortcoming of most approaches to offline RL is that they are designed to maximise the expected value of the cumulative reward of a policy. This assumes that the training data is sufficient to promote convergence toward an optimal policy. As a result they are unable to quantify the risk associated with a learnt policy to ensure that it acts in the intended way. The field of safe RL instead tries to learn policies that obtain good performance in terms of expected returns while satisfying some safety constraints during learning and/or deployment (Garcıa & Fernández, 2015), defined through a constrained MDP (CMDP). Several safe RL algorithms (Achiam et al., 2017;Berkenkamp et al., 2017;Alshiekh et al., 2018;Tessler et al., 2019;Xu et al., 2021;Yang et al., 2022;Polosky et al., 2022) have been developed that either i) transform the standard RL objective to include some form of risk or, ii) leverage external knowledge to satisfy certain safety constraints and quantify performance with a risk metric. However, safe RL assumes a priori knowledge of what unsafe regions are-through the definition of constraints whether implicitly through the environment or explicitly through agent behavior design-which is not always feasible in real-world safety-critical scenarios. Unlike these, we do not explicitly learn a policy, but learn a value function that conveys the risks inherent in making suboptimal decisions at inopportune times. Risk-sensitive RL instead focuses on learning to act in a dynamic environment, while accounting for risks that may arise during the learning process (Mihatsch & Neuneier, 2002), where high risk regions do not have to be known a priori. Unlike risk-neutral RL, these methods optimise a risk measure of the returns rather than the average or expected return. Among these, Fu et al. (2018) present a survey of policy optimization methods that consider stochastic formulations of the value function to ensure that certain risk constraints may be satisfied when maximising the expected return. Other approaches propose replacing the expected long-term reward used by most RL methods, with a risk-measure of the total reward such as the Conditional-Value-at-Risk (CVaR) (Chow et al., 2015;Stanko & Macek, 2019;Ying et al., 2022;Du et al., 2022) and develop a novel optimization strategy to minimize this risk to ensure safety all-the-time. Ma et al. (2021) adapt distributional RL frameworks (Bellemare et al., 2022) to offline settings and by penalizing the predicted quantiles of the return for out-of-distribution actions. While these methods may be used to learn a distribution of possible outcomes, they have not been used to identify dead-ends as we propose here. Unlike off-policy evaluation methods, we focus on estimating the risk associated with a policy in terms of the expected worst case outcomes. Specifically, we learn a distributional estimate of the future return of a policy using Implicit Quantile Networks (IQN) (Dabney et al., 2018), and integrate a conservative Q-learning (CQL) penalty (Kumar et al., 2020) into the loss to lower bound on the expected value of the policy. Nonstationary and Uncertainty-Aware RL Several works focus on explicitly modelling non-stationary dynamics in MDPs for decision-making that accounts for uncertainty over model dynamics. Among these, methods such as Chandak et al. (2020) focus on safe policy optimization and improvement in non-stationary MDP settings. Here, the authors assume that the non-stationarity in an MDP is governed by an exogenous process, or that past actions do not impact the underlying non-stationarity. Sonabend et al. (2020) use hypothesis testing to assess whether, at each state, a policy from a human expert would improve value estimates over a target policy during training to improve the target policy. More recently, Joshi et al. (2021) presented an approach for learning to defer to human expertise in nonstationary sequential settings based on the likelihood of improving the expected returns on a particular policy. Our work differs from these in that instead of focusing on optimizing a specific policy, we explicitly learn which types of behaviors to avoid using risk-sensitive distributional estimates of the future return, as opposed to a point estimate of the expectation of that distribution. RL in safety critical domains There are several works posed for uncertainty decomposition in applications such as healthcare. Specifically, Depeweg et al. (2018), decompose the uncertainty in bayesian neural networks to obtain an estimate of the aleatoric uncertainty for safety. Similarly, Kahn et al. (2017) use uncertainty-aware RL to guide robots to avoid collisions, while Cao et al. (2021) develop a domain-specific framework called Confidence-Aware RL for self-driving cars to learn when to switch between an RL policy and a baseline policy based on the uncertainty of the RL policy. Unlike these works, we propose a general purpose framework that can be applied to a number of safety-critical applications using risk-sensitive RL to provide an early warning of risk over possible future outcomes. Preliminaries As outlined above, we frame risk identification for safety critical decision making within a Reinforcement Learning (RL) context. We consider a standard episodic RL setting in an environment with non-stationary and stochastic dynamics where an agent determines actions a ∈ A after receiving a state representation s ∈ S of the environment, modeled as a Markov Decision Process (MDP) M = {S, A, T, R, γ}, where T (·|s, a) relates to the stochastic transition from state s given action a; R(s, a) is a finite, binary reward function that provides reward only at the terminal state of each episode and γ ∈ (0, 1] is a scalar discount factor. In offline safety critical settings, we assume that recorded actions are selected according to an unknown expert policy π(·|s), given the observed state s. The objective is to estimate the value of each action as the discounted sum of future rewards (e.g. the return) Z π (s, a) = ∞ t=0 γ t R(s t , a t ) where s 0 = s, a 0 = a, s t ∼ T (·|s t−1 , a t−1 ), and a t ∼ π(·|s t ). By characterizing the full probabilistic nature of how Z π (s, a) can be computed, it is used to represent the distribution of future return from the state s, executing action a. Distributional RL In challenging real-world scenarios, the consequences of a decision carry a measure of unpredictability. Standard approaches to RL seek to maximize the mean of this random return. In reality, complex phenomena in stochastic environment may fail to be accounted for, leading to rare but critical outcomes going ignored. To account for this, Distributional RL (Bellemare et al., 2022) has been introduced to model the full return distribution by treating the observed return from following a policy π and associated states as random variables when forming the Bellman equation: Z π (s, a) D = R(s, a) + γZ π (s , a ) The return distribution Z π (s, a) is most commonly represented in RL by the state-action value function Q π (s, a) which represents the expected future return. That is, Q π (s, a) = E[Z π (s, a)]. As the distribution is an infinite dimensional object, some approximations are needed for tractable estimation. Initially, the support of the distribution was discretized a priori over pre-defined categorical quantiles (Bellemare et al., 2017). More recently this approximation has been relaxed to a distribution of uniformly weighted particles, estimated with neural networks (Dabney et al., 2018), to implicitly represent these quantiles. Given the flexibility of these implicit quantile networks (IQN), they are well suited to define risk-aware decision criteria over value functions learned from real-world data where the anticipated return structure is unknown. As such, we build our proposed framework from IQN estimates of the state-action value function. Conservatism in offline RL An important consideration when learning from offline data with RL is avoiding value overestimation for actions not present in the data (Fujimoto et al., 2018;Bai et al., 2022). Prior work has attempted to choose a lower bound of approximated value functions (Fujimoto et al., 2018;Buckman et al., 2020), regularize policy learning by the observed behavior (Fujimoto et al., 2019;Wu et al., 2019;Kumar et al., 2019;Wang et al., 2020) or by directly regularizing the value estimates of the observed actions (Kumar et al., 2020;Jin et al., 2021). We utilize this last approach (termed conservative Qlearning; CQL) which resorts to minimizing the estimated values over the observed state-action distribution to enforce a conservative lower-bound of the value function. This is accomplished by simply adding a βweighted penalty term L CQL to the RL objective L RL . Thereby the optimzation objective becomes L RL + βL CQL where L CQL is chosen to be exponentially weighted average of Q-values for the chosen action (CQL(H) in Kumar et al. (2020)). This serves to additionally constrain the overestimation of actions not present in the dataset and has been shown to improve risk-averse performance with distributional RL (Ma et al., 2021). By increasing the value of β, the overall conservatism and thus risk-aversion is increased as the optimization of the estimated values is constrained further from the true value function. Risk estimation. Figure 1: Illustration of the determination of conditional value at risk (CVaR α ), with α = 0.1 We assume the return is bounded (e.g. E[|Z|] < ∞) with cumulative distribution function F(z) = P(Z ≤ z). When estimating the possible effects of a decision, we want to account for worst-case outcomes that occur with some level of confidence α ∈ (0, 1). The value-at-risk (VaR) with confidence α represents the α-quantile of the distribution Z: VaR α (Z) = min {z | α ≤ F(z)}. This quantile can then be used to determine the "expected worst-case outcome", or conditional value at risk (CVaR): CVaR α (Z) = 1 α E[(Z − VaR α (Z)) − ] + VaR α (Z) Published in Transactions on Machine Learning Research (01/2023) where (x) − = min(x, 0) is the negative part of x. We use the dual representation of CVaR (Artzner et al., 1999) which is formulated with a single expectation: CVaR α (Z) = min ξ∈U CVaR (α,P) E ξ [Z] where E ξ [Z] is the ξ-weighted expectation of Z within the α-quantile and U CVaR (α, P) is the portion of Z that falls below VaR α (Z). This establishes that: CVaR α (Z) ≤ E[Z](1) as α → 1, then U CVaR (α, P) encompasses all of Z and CVaR α (Z) → E[Z]. Thus, the CVaR is a lower-bound for value estimates derived through the expectation of the return distribution (e.g. the value function Q π ). Dead-end Discovery (DeD). As introduced by Fatemi et al. (2021) the DeD framework assures a notion of security when estimating whether an action will lead to a dead-end (see Eqt. 2). DeD constrains the scope of a given policy π if any knowledge exists about undesired outcomes. Formally, if at state s, action a transitions to a dead-end at the next state with probability P D (s, a) or the negative terminal state with probability F D (s, a) with a level of certainty λ ∈ [0, 1], then π must avoid a at s with the same certainty: P D (s, a) + F D (s, a) ≥ λ =⇒ π(s, a) ≤ 1 − λ.(2) Note that a dead-end may occur an indeterminate number of steps prior to the negative terminal condition. The defined notion of a dead-end is that once one is reached, all subsequent states are also dead-ends up to and including the negative terminal state. While P D , F D , and λ may not be able to be explicitly calculated, the DeD framework learns an estimate of the likelihood of transitioning to a dead-end as well as the reduction in likelihood of a positive outcome. This is done by constructing two independent MDPs M D and M R from the base environment MDP M focusing solely on negative and positive outcomes, respectively. DeD learns value approximations of each MDP, Q D (s, a) for negative outcomes and Q R (s, a) for positive outcomes (Q D ∈ [−1, 0] and Q R ∈ [0, 1] respectively). These value estimates enable the identification and confirmation of dead-ends and actions that lead to them through the relationship: −Q D (s, a) ≥ P D (s, a) + F D (s, a)(3) Then, the security condition is assured by π(s, a) ≤ 1 + Q D (s, a). In practice, the Q D and Q R functions are approximated with deep Q-networks (DQN) (called the D-and R-networks, respectively) in concert with empirically determined thresholds δ D and δ R to flag when actions or states have the risk of leading to dead-ends and should be avoided. The DeD framework determines an action a should be avoided when both Q D (s, a) ≤ δ D and Q R (s, a) ≤ δ R . A state s is said to be a dead-end if the median value over all actions falls below these thresholds. That is a dead-end is reached whenever both median(Q D (s, ·)) ≤ δ D and median(Q R (s, ·)) ≤ δ R . Our proposed distributional formulation of dead-end discovery uses these definitions, with slight adaptation to the risksensitive approach we use, allowing for the identification of both high-risk actions and states. However, in this paper we prioritize the identification of dead-end states, demonstrating that our proposed solution provides earlier identification. Risk-sensitive Dead-end Discovery While the DeD framework is promising for learning in offline safety-critical domains, it has limited risksensitivity by neglecting to model the full distribution of possible outcomes. We develop a risk-sensitive framework for dead-end discovery that conservatively models the full distribution of possible returns, driven by irreducible environment stochasticity. Our approach, DistDeD, utilizes distributional dynamic programming (Bellemare et al., 2022) to estimate the full distribution of possible returns while also limiting overestimation due to out-of-distribution actions by incorporating a CQL penalty (Kumar et al., 2020). c) The CVaR α is computed for each distribution and is then evaluated against the thresholds δ D and δ R . If both CVaR α (Z D ) and CVaR α (Z R ) fall below the respective thresholds for any action, then that action is recommended to be avoided. d) If the median over all actions falls below the thresholds for both distributions, then the state is said to be a dead-end. negative outcomes. R D returns −1 with any transition to a negative terminal state and is zero otherwise. R R returns +1 with any transition to a positive terminal state and is zero otherwise. We then approximate the distributional returns Z D and Z R of these separate MDPs independently, where the support of Z D is [−1, 0] and the support of Z R is [0, 1]. To quantify the risk of selecting an action a at state s, we consider the expected worst-case outcome-or conditional value at risk (CVaR)-of these return distributions. That is, we infer CVaR α (Z D (s, a)) and CVaR α (Z R (s, a)) for a chosen α ∈ (0, 1], which we consider to be a hyperparameter along with the choice of thresholds δ D and δ R . By using CVaR to determine the risk of approaching a dead-end, we effectively construct a lower-bound on the DeD value estimates (by virtue of Eqt. 1) which allows us to maintain the same theoretical framing. Since DeD is built around the expectation of the return: Q D (s, a) = E[Z D (s, a)]. Then, as CVaR α (Z D (s, a)) ≤ E[Z D (s, a)] we are assured that: −CVaR α (Z D (s, a)) ≥ −Q D (s, a) ≥ P D (s, a) + F D (s, a)(4) Thus, by bounding the estimates of entering a dead-end, we see that using CVaR satisfies the security condition: π(s, a) ≤ 1 + CVaR α (Z D (s, a)). Parallel results for Z R follow similarly. We choose to represent the distributions Z D and Z R for all states s and actions a using implicit Q-networks (IQN) (Dabney et al., 2018). To constrain the distributional estimates from overestimating the return for actions not present in the dataset, thus avoiding overconfidence, we train the IQN architectures with a conservative Q-learning (CQL) penalty (Kumar et al., 2020). CQL regularizes the distributional Bellman objective by minimizing the value of each action, which serves also to constrain overestimation of actions not present in the observed data. We weight this penalty by the hyperparameter β. An illustration of the DistDeD framework is included in Figure 2: a) If necessary 1 , observations are encoded into a state representation. b) The encoded state representations are then passed to independent IQN models to estimate Z D (s, ·) and Z R (s, ·) for each possible action. c) The CVaR is computed for each distribution and then evaluated against the thresholds δ D and δ R . Following the definition of dead-end discovery given in the previous section, if both CVaR(Z D ) and CVaR(Z R ) fall below the respective thresholds for any action, that action is recommended to be avoided. d) Furthermore, if the median over all actions falls below the thresholds for both distributions, then the state is said to be a dead-end. With the bounding provided by DistDeD, utilizing CVaR estimates of the inferred return distributions, we enable a more conservative and thereby risk-averse mechanism to determine whether a state s is at risk of being a dead-end. The level of risk-aversion, or conservatism, is jointly determined by the confidence level α, the weight of the CQL penalty β as well as the thresholds δ D and δ R . The level of conservatism within DistDeD depends on choices of all of these quantities. Since β directly affects the optimization process of the D-and R-Networks, we treat it as a hyperparameter. An investigation of the affect of increasing β can be found in Section A.4.3 in the Appendix. The choice of α, influencing the CVaR calculation, as well as the thresholds δ D and δ R can be tuned dependant on acceptable risk tolerances in the task when evaluating the trained D-and R-Networks. Choosing a smaller value for α constrains the CVaR evaluation of the estimated distributions to consider lower likelihood (and more adverse, by construction) outcomes, a form of increased conservatism. Smaller values of the thresholds increase the sensitivity of the risk determination of the framework. We demonstrate the effects of choosing different α values on the performance benefits of DistDeD in comparison to previous dead-end discovery approaches (Fatemi et al., 2021) across multiple settings of δ D and δ R in our experiments using real-world medical data in Section 6. Illustrative Demonstration of DistDeD We provide a preliminary empirical demonstration of the advantages seen by using our proposed DistDeD framework using the LifeGate toy domain (Fatemi et al., 2021). Here, the agent is to navigate around a barrier to a goal region while learning to avoid a dead-end zone which pushes the agent to the negative terminal edge after a random number of steps (See Figure 3). Empirical Comparison We aim to demonstrate the apparent advantages of our proposed DistDeD in comparison to the original DeD framework. For DeD, we model the Q D and Q R functions using the DDQN architecture (Hasselt et al., 2016) using two layers of 32-nodes with ReLU activations and a learning rate of 1e −3 . For DistDeD we utilize IQN architectures (Dabney et al., 2018) for both Z D and Z R using two layers of 32 nodes, ReLU activations and the same learning rate of 1e −3 . For each IQN model, we sample N, N = 8 particles from the local and target τ distributions while training and also weight the CQL penalty β = 0.1. When evaluating Z D and Z R , we select K = 1000 particles and set our confidence level to α = 0.1. (Fatemi et al., 2021). A) Evaluating returns from an initial state, B) evaluating returns from a more favorable location near the goal region. Notably, the CVaR estimate (the mean of the orange "worst-case distribution") is risk-sensitive and provides a lower bound of the expected value of the blue return distribution, while the value estimate of DeD (black dashed line) is far more optimistic. Here, we set δ D = −0.75 as a notional threshold (red dashed line). All approximate value functions (both expectational and distributional) were trained using 1 million randomly collected transitions from LifeGate. In Figure 3 we show the learned value estimates from the D-Networks for all actions available to the agent in select locations. We suppress the corresponding R-Network estimates for visual simplicity although they reflect qualitatively the same thing. For this demonstration we plot the full return distribution Z D (s, a), the α-quantile used to compute CVaR α (Z D (s, a)), the value estimate Q D (s, a) from the DeD, as well as a notional threshold δ D = −0.75. We see the inherent value of the distributional estimates used in DistDeD to determine which actions to avoid. Fig. 3(A) presents the returns at an initial state, from which encountering a dead-end is more common. Fig. 3(B) presents the estimated returns from a more favorable location near the goal region. As expected, the CVaR estimate, the mean of the orange "worst-case distribution", is a lower bound on the expected value of the full return distribution (plotted in blue). Notably, the value estimated using DeD (black dashed vertical line) is far more optimistic, since DeD only considers thresholded point-estimates of expected value. This provides evidence of the limitations of DeD, ignoring the full return distribution when estimating the value of available decisions. In Figure 4(A, B), we evaluate three pre-determined policies in LifeGate using both DeD and DistDeD. Two of the three policies attempt to navigate through the dead-end region of the environment. This construction is purposeful in order to indicate how reliably risk is flagged by each approach. The design of this experiment is to demonstrate the early-warning capability of DistDeD for those sub-optimal trajectories. In Figure 4(C) we evaluated 10,000 trajectories with stochastic execution of the two suboptimal policies and assess how many steps prior to entering the dead-end region that DistDeD and DeD raise alarm and recommend a change in policy. We assess the overall risk of each state s in a trajectory by averaging the median values of Q D (s, ·) and Q R (s, ·) − 1 (for DistDeD CVaR α (Z D (s, ·)) and CVaR α (Z R (s, ·)) − 1). If the averaged median value falls below the threshold δ D , an alarm is raised. We use the previously published value, δ D = −0.15 for DeD and choose δ D = −0.5 for DistDeD. These values were chosen empirically by attempting to minimize false-positives among a validation set of the data (see Section A.4.1 for more detail). DeD ( Fig. 4(A)) fails to adequately signal the risk of the two sub-optimal policies before they reach the dead-end region of the environment. In contrast, DistDeD ( Fig. 4(B)) appropriately flags the trajectories ahead of the dead-end region, allowing for correction if an overseeing agent is able to intervene. Fig. 4(C) quantifies this advantage, demonstrating that DistDeD provides an indication of risk, on average, 3 steps earlier. This result confirms the utility of modeling the full distribution of expected returns and using a more coherent estimation of risk, focused on expected worst case outcome. Figure 4: DistDeD's advantage when alerting that a trajectory is at risk of encountering a dead-end in the LifeGate domain. Three hand-designed policies (with two purposefully suboptimal) (shown in white) are evaluated using both DeD (A) and DistDeD (B), showing that DistDeD raises alarm earlier than DeD and in a manner that could alert a necessary change in policy before encountering a dead-end. 10000 stochastic executions of these suboptimal policies are then evaluated (C) using both approaches to understand the scope of how much earlier DistDeD raises a flag in comparison to DeD. Dotted lines show how raising alarms earlier leads to actions that could direct a patient's trajectory towards potential recovery (shown in blue). Assessing Medical Dead-ends with DistDeD Data We aim to identify medical dead-ends among a cohort of septic patients derived from the MIMIC-IV (Medical Information Mart for Intensive Care, v2.0) database (Johnson et al., 2020). This cohort comprises the recorded observations of 6,188 patients (5,352 survivors and 836 nonsurvivors), with 42 features, and 25 treatment choices (5 discrete levels for each of IV fluid and vasopressor), over time periods ranging between 12 and 72 hours. We aggregate each feature into hourly bins and fill missing values with zeros, keeping track of which features were actually observed with an appended binary mask. Missing features are implicitly accounted for when constructing state representations of a patient's health through time. Details about the exclusion and inclusion criteria used to define the construction of this patient cohort are contained in Section A.1 in the Appendix. State Construction As recommended by Killian et al. (2020) and implemented in DeD (Fatemi et al., 2021), we make use of a sequential autoencoder to construct fixed dimension state representations, embedding a history of recorded observation of a patient's health previous to each time step. This allows us to process partial and irregularly occurring observations through time, a characteristic of medical data. To do this, we use an online Neural Controlled Differential Equation (NCDE) (Morrill et al., 2021) for state construction as it naturally handles irregular temporal data. Additional information about the NCDE state construction can be found in Section A.2.1 in the Appendix. We define terminal conditions for each trajectory as whether the patient survives or succumbs to (within 48 hours of the final observation) their infection. There are no intermediate rewards aside from these terminal states. When a patient survives, the trajectory is given a +1 reward, where negative outcomes receive −1. D-and R-Networks The encoded state representations provided by the NCDE are provided as input to the D-and R -Networks to estimate the value (and risk of encountering a dead-end) of each state and all possible treatments. To form the DistDeD framework we use CQL (Kumar et al., 2020) constrained implementations of IQN (Dabney et al., 2018) to train each network, as discussed in Section 4 (details included in Appendix A.2.2). Training We train the NCDE for state construction as well as the IQN instantiations for the D-, and R-Networks in an offline manner. All models are trained with 75% of the data (4,014 surviving patients, 627 patients who died),validated with 5% (268 survivors, 42 nonsurvivors), and we report all results on the remaining held out 20% (1,070 survivors, 167 nonsurvivors). In order to account for the data imbalance between positive and negative outcomes, we follow a similar training procedure as DeD (Fatemi et al., 2021) where every sampled minibatch is ensured to contain a proportion of terminal transitions from non-surviving patient trajectories. This amplifies the training for conditions that lead to negative outcomes, ensuring that the D-and R-Networks are able to recognize scenarios that carry risk of encountering dead-ends. Specific details on the training of DistDeD can be found in Appendix A.2. 2 Experimental Setup By design, DistDeD is formulated to provide a more conservative and thereby earlier indication of risk. A secondary benefit of the design of DistDeD is that by adapting the risk tolerance level of the CVaR estimates (by selecting different values for α), we are provided a spectrum of value functions that could be used to assess whether a dead-end has been reached or is eminent. We therefore aim to execute a set of experiments that assess the extent at which these two points of improvement over DeD provide benefit. By establishing more conservative estimators with the IQN D-and R-Networks, we increase the occurrence of what could be identified as false positive indications of risk for patients whose health has not deteriorated to be a legitimate dead-end (e.g. patients who survive). We therefore need to assess the tradeoffs of increased "false-positives" against improved recall for indications of risk for patients who died. To perform this assessment we execute a set of experiments to quantitatively compare DistDeD to DeD when each approach is applied to the septic patient cohort outlined above. First, this entails measuring how much earlier DistDeD raises flags across a range of VaR α values (for a fixed set of thresholds δ D,R ). Second, we want to identify if DistDeD's variation-due to the choice of VaR α-introduces settings that perform worse than DeD when considering a full range of possible thresholds δ D,R . Finally, we aim to develop insight into the contributions of both the distributional and CQL additions to the DeD framework by considering ablations to DistDeD where each component is removed. Additional details of all experiments are contained in Section A.3 in the Appendix where there can also be found further experimental analyses in Section A.4, such as the effects of learning with reduced data (see Section A.4.4). Figure 5: The number of hours before patient death that DistDeD and DeD raise warning. Figure 6: The number of hours that DistDeD detects patient deterioration and first raises a flag before DeD. Results As outlined in Section 6.1 we highlight the importance of accounting for risk when thinking about dead-ends and validate the following aspects of DistDeD. First, we assess the performance of DistDeD by demonstrating how DistDeD can provide an earlier indication of risk in comparison to other baselines and notably, outperforms DeD across all settings. Second, we demonstrate the utility of having a tunable assessment of risk that allows for domain experts to easily apply and adapt our method to different contexts, hospital settings and illnesses. Finally, we show that including a CQL penalty in the DistDeD framework further improves performance in comparison to other baselines. DistDeD Provides Earlier Warning of Patient Risk We assess the ability of DistDeD to provide an early warning of patient risk in comparison to the original medical dead-ends framework, DeD. Figure 5 shows for non-survivors, the number of hours ahead of death that DistDeD raises a warning flag and how this changes with varying choices of VaR. In comparison to DeD, DistDeD is able to raise flags much earlier warning of up to 25 hours in advance across all values of VaR, thereby enabling timely intervention in safety-critical settings. To assess DistDeD's ability to raise flags in different contexts, we also compare how its performance varies across both surviving and non-surviving patients. These results are shown in Figure 6. In general we note that for both patient groups, DistDeD is able to detect patient deterioration and provide early warning of up to 20 hours in advance depending on the choice of VaR thresholds in comparison to DeD. The performance across both surviving and non-surviving patients is very similar. DistDeD Allows for a Tunable Assessment of Risk Note that because DistDeD explicitly uses the Value at Risk threshold parameter α to provide an assessment of risk, it can easily be adapted and tuned to various scenarios depending on how risk-averse a user would like to be. In addition, the choice of the thresholds δ D and δ R can be further adjusted to improve the precision of estimates of the risk of encountering a dead-end. For instance, in an ICU setting where timely intervention is crucial, a clinician may choose to adopt lower α and higher δ D & δ R threshold values to be more conservative such that flags may be raised earlier if necessary. In our experiments, we evaluate DistDeD and DeD over all possible settings of δ D and δ R to assess the sensitivity of those settings when computing the True Positive Rate (TPR) and False Positive Rate (FPR) of determining patient risk. We also continue to evaluate DistDeD over a range of CVaR α settings. Here, TPR corresponds to the percentage of non-survivor trajectories that are flagged, while FPR corresponds to the percentage of survivor trajectories that are flagged. Figure 7 shows a comparison of ROC curves derived from the DistDeD and DeD frameworks to exhibit how each balance the TPR and FPR tradeoff. For DistDeD we evaluated the TPR and FPR for a range of α values to identify whether there was a particular level of conservatism (or optimism) that would perform worse than DeD. However, we observe that DistDeD robustly outperforms DeD finding a higher TPR while having a low FPR in comparison, across all settings of α, δ D and δ R . Overall, having an tunable assessment of risk also enables a domain expert like a clinician balance the benefits of early warning with the risk of potential false positive indications of risk, where a patient at low-risk is potentially flagged. Moreover, a higher TPR counteracts an increased FPR when we are more conservative in the DistDeD framework. CQL Enhances DistDeD Performance In order to assess the individual contributions of implementing a distributional estimate of the risk of encountering a dead-end and constraining the values with CQL, we evaluate separate ablations to DistDeD by computing the area under the ROC curve derived from each approach. Figure 8 shows the performance comparison of DistDeD versus DeD and these two ablations that i) exclude a CQL penalty from the DistDeD framework and ii) incorporate a CQL penalty into the standard DeD framework. Overall, we see that the DistDeD framework outperforms the baselines in terms of AUC across varying levels of the VaR threshold. We summarize the findings with the maximum AUC of each approach in Discussion In this paper we have presented our justification, foundational evidence as well as our preliminary findings supporting the development of the DistDeD framework which incorporates a more complete notion of risk when identifying dead-ends in safety-critical scenarios. We do so by leveraging distributional dynamic programming to form estimates of the full return distribution from which we can calculate the expected worst-case outcome for each available action. This form of risk-estimation enables a more tangible decision surface for determining which actions to avoid and can be tuned according to the requirements or preferences set forward by human experts that may interact with the trained DistDeD models. Our DistDeD approach is based around risk-sensitive estimates of the expected worst-case outcome and is thereby contributes a conservative decision support framework. This framework is well suited for complex safety-critical situations where learning is completed in a fully offline manner. Limitations While DistDeD is a promising framework for decision support in safety-critical domains with limited offline data, there are certain core limitations. The techniques described in this paper have been explored in the context of discrete action spaces only. However in scenarios where continuous actions are featured, analyses with the DistDeD framework may have to be adapted to identify potential dead-ends. In addition, the method considers only cases where a binary reward signal is observed on the terminal state only. However, several applications may require us to account for intermediate and continuous outcomes as well. Moreover, the framework only explores a medical scenario where dead-ends are derived from a single condition whereas in reality, many concomitant conditions may exist, which contribute to and are associated with different dead-end regions. Finally, we do not make any causal claims about the impact of each action on the outcomes of interest. Future work may explore how to address some of these issues. In addition, we are currently in the process of applying DistDeD to real-world healthcare challenges in partnership with clinicians to further demonstrate its utility in that setting. We do however anticipate that DistDeD is widely useful for all safety-critical domains that may beset with limited offline data. Broader Impact This work serves as a proof of concept for identifying regions of risk in safety-critical settings, learning from offline data. While promising, it has not been thoroughly validated for immediate use in real environments. Despite the demonstrated utility of the DistDeD framework in healthcare problems, it should never be used in isolation to exclude patients from being treated, e.g., not admitting patients or blindly ignore treatments. The risk identification aspect of DistDeD demonstrated in this paper is to signal impending high-risk situations early enough so that the human decision maker has time to correct the course of action. This may help experts make better decisions and avoid circumstances that may lead to irrecoverably negative outcomes. The intention of our approach is to assist domain experts by highlighting possibly unanticipated risks when making decisions and is not to be used as a stand-alone tool nor as a replacement of a human operator. Misuse of this algorithmic solution could carry significant risk to the well-being and survival of critical systems and individuals placed in the care of the expert. The primary goal of this work is to improve upon the established DeD proof of concept, where high-risk situations can be avoided in context of a system's state (Fatemi et al., 2021). We present a distributional estimate of this risk profile which enables earlier detection of possible dead-ends as well as facilitating a tunable framework for adaptation to each individual task. In acute care scenarios, all decisions come with inherent risk profiles and potential harms. In this spirit, we endeavor to provide a flexible tool for clinical experts to gain an earlier indication when specific decisions or their patient's health state may carry a measure of outstanding risk. Author Contributions TK and SP conceived and designed the research questions as well as wrote the paper. TK extracted and processed the data, designed and executed the experiments, and performed the analyses. MG provided input on possible uses of the proposed framework in clinical settings, provided funding, and reviewed the paper prior to it being made public. A Appendix A.1 Sepsis Patient Cohort Details We use the MIMIC-IV (Medical Information Mart for Intensive Care; v2.0) database, sourced from the Beth Israel Deaconess Medical Center in Boston, Massachusetts Johnson et al. (2020). This database contains deidentified treatment records of patients admitted to critical care units (CCU, CSRU, MICU, SICU, TSICU). The database includes data collected from 76,540 distinct hospital admissions of patients over 16 years of age for a period of 12 years from 2008 to 2019 (inclusive). The MIMIC database has been used in many reinforcement learning for health care projects, including mechanical ventilation and sepsis treatment problems. There are various preprocessing steps that are performed on the MIMIC-IV database in order to obtain the cohort of patients and their relevant observables for sepsis cohort used in this study. To extract and process the data, we follow the approach used in (Fatemi et al., 2021). This includes all ICU patients over 18 years of age who have some presumed onset of sepsis (following the Sepsis 3 criterion) during their initial encounter in the ICU after admission, with a duration of at least 12 hours. We limited the observation window of each patient encounter from at most 24 hours before to at most 48 hours after presumed sepsis onset. We also constrained collection to include only those patients admitted to the Medical ICU (MICU) on that initial encounter. These criteria provide a cohort of 6,188 patients, among which there is an observed mortality rate of 13.5%, where mortality is determined by patient expiration within 48h of the final observation. Observations are processed and aggregated into hourly windows with treatment decisions (administering fluids, vasopressors, or both) discretized into 5 volumetric categories. All data is normalized to zero-mean and unit variance and missing values are zero-imputed with an binary mask appended to indicate which features were observed at each timestep. We report the 42 features used in the construction of this patient cohort in Table 2 with high-level statistics in Table 3. A.2 DistDeD Architecture Details In this section, we outline the motivation, design, and training of the various architectures used to formulate the DistDeD framework. To account for the irregularity and temporal dependence of the observations made with the medical data, we encode the data using an online Neural Controlled Differential Equation (NCDE). This provides a fixed dimensional state representation at each time step (here, aggregated by hour), to align with the frequency of treatment decisions. The encoded state representations are then used as input into the independent implicit quantile networks (IQN), used to represent the D-and R-Networks to estimate the risk of encountering a dead-end. Training We train the NCDE for state construction as well as the IQN instantiations for the D-, and R-Networks in an offline manner. All models are trained with 75% of the data (4,014 surviving patients, 627 patients who died),validated with 5% (268 survivors, 42 nonsurvivors), and we report all results on the remaining held out 20% (1,070 survivors, 167 nonsurvivors). In order to account for the data imbalance between positive and negative outcomes, we follow a similar training procedure as DeD (Fatemi et al., 2021) where every sampled minibatch is ensured to contain a proportion of terminal transitions from non-surviving patient trajectories. This amplifies the training for conditions that lead to negative outcomes, ensuring that the D-and R-Networks are able to recognize scenarios that carry risk of encountering dead-ends. 3 A.2.1 Neural Controlled Differential Equation Neural Differential Equations (NDEs; Chen et al. (2018)) have become a popular modeling framework for handling complex temporal data due to their flexibility and the ability to model data in continuous time. In particular, they are well matched for irregularly sampled (e.g. partially observed) data such as is common in healthcare. NDEs learn a continuous latent representation of the dynamics underlying the observed data in a fixed dimension representation; adapting for missingness, various periodic frequencies among features, as well as complex interactions between features (Kidger, 2022). These reasons provide a distinct motivation for using NDEs for processing fixed representations of healthcare data. In this work we use a variant of NDE, Neural Controlled Differential Equations (NCDE; Morrill et al. (2021)). NCDEs are designed to process irregular time series with a latent process that affects the evolution of the observed features. The particular variant of NCDE we use is designed to operate in online settings, only incorporating historical information to encode representations of the current time step. This is a departure from standard NDE methods that execute a forward-backward time of autoregression when representing the latent dynamics of the time series. By restricting ourselves to online types of processing, we honor the reality with which data is received in a healthcare setting which leads to a more realistic implementation. Otherwise, we may risk biasing the inference over missing data intervals using future observations. For specific algorithmic details of the NCDE, we refer the reader to Morrill et al. (2021). For our purposes, we train the NCDE as a continuous time autoencoder of the irregular patient observations. This provides a fixed dimension representation of a patient's state at hourly intervals, to match with the frequency of treatment decisions in our extracted data. Following procedures set forth by Morrill et al. (2021), we lightly pre-process the data with rectilinear interpolation (where each patient trajectory is han- dled independently) so as to signal when and where missing features occur. The NCDE is built around learning representations from an internal projection function (represented by a neural network), optimized using a differential equation solver. We fine-tuned the hyperparameters of this encoding function (most importantly the output embedding dimension) as well as optimization using Ax (Bakshy et al., 2018), an adaptive experimentation platform built on top of the BoTorch Bayesian Optimization library (Balandat et al., 2020). When training the NCDE, we found that using a fixed learning rate with the Adam optimizer (Kingma & Ba, 2014) performed best. After 100 Bayesian Optimization trials, we found the following hyperparameter settings to provide the best performing NCDE model. For the encoding neural network, we used 2 layers with 80 hidden units in each with ReLU activations. The output dimension of this encoding network was 55, which provided the state representations then used as input to the Reinforcement Learning models. For optimization, the best learning rate was 5e − 4 over 30 epochs. A.2.2 DistDeD Value Functions As outlined in Section 4, we construct two MDPs for estimating the return for negative and positive outcomes independently. With these independent learning objectives, we construct two independent value estimators based on Implicit Q-Networks (Dabney et al., 2018). For specific details on the development and training of these architectures, we refer the reader to the source literature. In summary, quantiles of the approximated distribution of return are approximated with sampled particles from a uniform distribution, which are then transformed with a learned projection function (represented by a neural network) to construct the implicit distribution. When training, a separate copy of the network parameters are kept as a target network to ensure more stable updates (following the double DQN strategy (Hasselt et al., 2016)), a number of samples K are drawn from each distribution (the one we're optimizing and the target distribution), then using a Wasserstein metric the projected particles are brought closer together. To constrain the value estimates of this distribution from overestimation for actions not in the dataset, we include a CQL penalty (Kumar et al., 2020) following Ma et al. (2021). There is a trade-off between maximizing the fit of the value distributions and the strength of the CQL regularization. We can modulate this by including a multiplicative weight β to the CQL penalty, which we treat as an additional hyperparameter. As done when training the NCDE state constructor, we optimized the hyperparameters of the IQN models used to represent Z D and Z R in DistDeD, the CQL penalty weight, and optimization parameters using the BoTorch Bayesian Optimization library Balandat et al. (2020) through the Ax API (Bakshy et al., 2018). The best performing hyperparameters used to define the IQN and CQL penalty were chosen after running 100 optimization trials. For the IQN, the projection neural network accepted a 55 dimensional input (from the NCDE), consisted of 2 layers with 16 hidden units in each, using ReLU activations. The number of samples K drawn each optimization step was set to 64. The target network parameters were updated after every 5 optimization steps using an exponentially-weighted moving average with parameter τ set to 0.005. By construction, the discount rate γ is set to 1. For the weighting of the CQL penalty, β = 0.035. For optimization, we used Adam (Kingma & Ba, 2014) with the best performing learning rate found to be 2e − 5 over 75 epochs of training. Of special note, the IQN architecture admits a family of risk-sensitive policies by constraining the space from which samples are drawn for the approximating distribution. The way the sampling space is constrained is called a distortion risk measure which influences the underlying value distributions to be more risk seeking or averse. By design, these distortion measures can be selected to influence the estimation of the implicit distributions over expected return. However, we chose not to bias the estimation of these distributions since we are deploying the IQN in an offline setting and cannot recover from a poor modeling choice through the acquisition of new data from the environment (following the optimism under uncertainty principle that guides much of RL). We therefore do not employ any distortion measures, evaluating the CVaR in a post-hoc manner so as to maximize the utility of the underlying distribution to represent the observed data. Computing the CVaR To compute the CVaR of Z D and Z R , we evaluate the fully trained IQN models using held out test data. The full distributions are then sampled using K test = 1000. We then sort the resultant estimated values for each particle and select the fraction of smallest values corresponding the the chosen value of α. For example, if α = 0.35, we would then take the smallest 350 values of the 1000 samples. The VaR α=0.35 would be the maximum value of this subsampled portion of the estimated distribution. Then, the CVaR α=0.35 would be the average value of the 350 sample subset. We use this quantity then to compare with the DistDeD thresholds δ D and δ R to determine whether or not an action should be avoided or whether the state is a dead-end as outlined in Section 4. A.3 Details of experimental setup This section lays out the details of the models used in all experiments as well as the relevant settings of each experiment presented in Section 5 and Section 6. We start by listing the important hyperparameters for both DeD and DistDeD with a description of their function in Table 4. Many of the components of each approach share the same description. We follow this description with a description of each experiment, listing relevant settings and parameters for the models used as well as analyses performed. DeD (Fatemi et al., 2021) DistDeD ( β the weight given to the CQL penalty during optimization A.3.1 Experimental Details for the Illustrative Demonstration In this subsection, we'll provide some additional details about the toy domain LifeGate which was introduced by Fatemi et al. (2021) and the empirical analyses done to compare DeD and DistDeD (all relevant parameters are contained in Table 5). In this domain, an agent is tasked with navigating to a goal region by making it's way around a barrier, while learning to avoid a "dead-end zone" at the right side of the barrier. In this zone, no matter what actions the agent takes it will be pushed to the right toward the negative terminal region after a random number of steps. Even in this simple domain, we found that prior dead-end discovery approaches (DeD) would overestimate the safety of actions around this dead-end zone (see Figure 3) and only raise a flag about the risk of an agent reaching a dead-end once it was squarely within this zone (see Figure 4). Using this toy domain, we set out to visualize the learned value distributions provided with DistDeD as an informative means to demonstrate it's utility broadly. To do so, we collected 1 million transitions using a random policy with the agent being initialized randomly all over the environment. With this data, we were able to then train the value functions for DeD and DistDeD using the DDQN (Hasselt et al., 2016) and IQN (Dabney et al., 2018) algorithms, respectively. Additionally, we applied a CQL penalty (Kumar et al., 2020) to the IQN training of DistDeD. Using the published δ D and δ R thresholds published by Fatemi et al. (2021) for DeD and those empirically derived for DistDeD (see Section A.4.1), which are included in Table 5, we could then quantitatively compare the performance of these two approaches in LifeGate. In the first experimental comparison (see Figure 3), we evaluated a large selection of states in the LifeGate domain choosing two to visualize the outputs of the value functions. For DeD, we simply recorded the value estimate provided by the D-and R-Networks. With DistDeD, we sampled 1000 points from the underlying quantile functions used to approximate the return distributions within the IQN from both the D-and R-Network. This allowed us to construct representative value distributions for each action. With each distribution, and a chosen α value for calculating the value-at-risk (VaR) and thereby the conditional value-at-risk (CVaR) we could then visualize what the estimated "worst-case value" of each action was. In the second and third experimental comparison using the LifeGate domain (see Figure 4), we evaluated three hand-designed policies used to collect trajectories. Two of these policies were purposefully made to be suboptimal (meaning that they would traverse through the dead-end zone) to demonstrate how early DistDeD would identify the risk of reaching a dead-end. We quantified this advantage by collection tenthousand additional trajectories, following these suboptimal policies, and recording the time either Q D and Q R or CVaR(Z D ) and CVaR(Z R ) violated their associated thresholds. We then subtracted the time the agent reached the dead-end zone from this recorded time. Using the 10,000 collected trajectories, we could then aggregate statistics about the time differential and how much earlier DistDeD signaled risk when compared to DeD. A.3.2 Experimental Details for Sepsis Treatment Evaluation All medical data used in this paper is derived from the MIMIC-IV database, as described in Section A.1. After filtering and data exploration, we ended up with 6,188 high quality trajectories of patients who developed Sepsis and were admitted to the intensive care unit. Approximately 13.5% of the trajectories end in the patient dying, reaching our defined negative terminal condition. More detailed statistics about the patient cohort can be found in Table 3. Since observations derived from electronic health records are irregular and sparse, we follow the previous literature applying RL to healthcare and learn a fixed-dimensional latent encoding of the data over time (Killian et al., 2020). We chose to model this encoder with a Neural Controlled Differential Equation (NCDE) (Morrill et al., 2021), trained via an objective to reconstruct the currently provided observation. Details about training the NCDE are given in Section A.2.1. Our best performing NCDE model took the 42 dimensional observations and projected them into a 55 dimensional latent state space, which was then used to train the D-and R-Networks for the value functions underlying DeD and DistDeD, details of which can be found in Section A.2.2. A table summarizing high-level parameters about the imposed MDP used to define the experiments in Section 6 can be found in Table 6. In Section 6.2.1, we determined a single set of thresholds for DistDeD following the same analysis done by Fatemi et al. (2021). We highlight how this is done in Section A.4.1. In essence, we plot sets of histograms (one for surviving patients, one for nonsurviving patients) of the computed CVaR for both the D-and R-Networks over all states for each time step, for all α values. We then attempted to select δ D and δ R that would separate the nonsurviving patient values from the surviving patient values, minimizing as many false positives (values from surviving patients that fall below the thresholds). Using these thresholds (for both DeD and DistDeD), we determine the first time step when the median of the CVaR values over all actions fell below the thresholds, for both D-and R-Networks respectively, which signifies a significant risk of the patient reaching a dead-end. In Figure 5, we measure how far ahead of patient death, in the case of non-surviving patients, this first flag is raised. The box plots are taken over the setting of CVaR risk threshold α. Figure 6 represents the spirit of our empirical analyses and and we compare the temporal difference between DistDeD and DeD directly for all patients. We see again that DistDeD provides earlier indication of patient health deterioration, particularly as lower values of α are selected. This analysis introduced an important question about the balance between early warning and increased false positives. We address the concern of increased false positives in Section 6.2.2 by defining the notions of true and false positive dead-end discovery and also investigate the range of performance acheived by selecting different values for the thresholds δ D and δ R . This also enabled us to establish a receiver operating characteristic (ROC) as a metric to holistically evaluate the performance of DistDeD vs. DeD. We evaluated each patient trajectory and aggregated the rate of nonsurviving patients having been correctly flagged by either DeD or DistDeD as well as the rate of surviving patients "wrongly" flagged. We construct the ROC curve by evaluating the sensitivity of the dead-end discovery process in each DistDeD and DeD by varying the thresholds δ D and δ R over 100 possible settings to provide a more complete picture of the performance of any method. Figure 7 shows this comparison, allowing us to see that DistDeD robustly outperforms DeD, regardless of the choice of α (each green curve corresponds to an independent setting of α). In Section 6.2.3, we repeat all of the above training and evaluation paradigms for two ablations of DistDeD by removing either the distributional component (essentially running DeD with a CQL penalty, which could be thought of as a separate baseline) or the CQL penalty. We present in Figure 8 the summary of this evaluation using the quantitative measure of Area under the ROC curve as a comparison. Table 1 takes the maximum AUC of each approach (picking the best configuration of α for DistDeD and DistDeD without CQL). Here, we conclude that DistDeD provides a 20% improvement over DeD using this AUC metric. A.4 Additional Experimental Analysis A.4.1 Preliminary selection of decision thresholds In Figure 9 we present a visual summary of how the thresholds δ D and δ R are empirically determined. Conceptually, we want to select thresholds that minimize the number of "false positives" that occur, meaning we don't want to unnecessarily flag trajectories arising from patients who ultimately survived. We plot the histograms of the assessed values for both non-surviving (blue) and surviving (green) patients for both the D-Network (top) and R-Network (bottom) using the validation set. To visualize how the estimated values change throughout the recorded trajectory (max 72 hours before termination) we also look at successive time periods when plotting the histograms. Unsurprisingly, as the trajectories near termination, the states from non-surviving patients have lower estimated value (being near to death). The choice of threshold is made δ D 100 settings w/in [-1,0] δ R 1+δ D δ R 1+δ D to provide as early of an separation of the estimated values between non-surviving and surviving patient as possible. While this approach carries some precedence, as it follows that done by Fatemi et al. (2021), but it's clear how tedious and in-exact this process is. This is what led to the analysis provided in Section 6.2.2, where we evaluated all possible settings of the thresholds when constructing the ROC curves in Figure 7. By providing the full information of possible precision and anticipated risk of false-positives, we enable the human expert to tune DistDeD according to the characteristics of the task. We suggest that this is a far superior approach to selecting the δ D and δ R . Table 7: Percentage of Patient Trajectories Missed. For DistDeD, δ D = −0.5, δ R = 0.5 A.4.2 DistDeD recovers risky trajectories overlooked by DeD While confirming the analysis underlying the results presented in Section 6.2.1, we were surprised to find that a significant number of non-surviving patient trajectories went undetected by DeD. In fact, nearly 60% of this high-risk subpopulation registered no indication from the prior dead-end discovery method. In Table 7 we present the proportion of trajectories (both non-suriving and surviving) where a flag is not raised for their duration, comparing between DeD and DistDeD. We also evaluated a range of α values used to calculate the CVaR of the estimated return distribution. We see that as α decreases, corresponding to a more conservative estimation of risk, that fewer non-surviving patient trajectories are missed at a cost of flagging more surviving patients. In concert with the results presented in Section 6.2.2, this table helps characterize the trade-off with early warning and the number of "false positives" that DistDeD provides. By providing this full range of options as an immediate consequence of the design of DistDeD, we empower the human decision maker to select the best setting of our proposed framework for their use-case. Published in Transactions on Machine Learning Research (01/2023) A.4.3 DistDeD performance suffers through an increase in conservatism By construction, DistDeD is more conservative than prior dead-end discovery approaches. This is achieved in two ways: a) the choice of value at risk threshold (α) and b) the weight (β) that the CQL penalty is given when optimizing the D-and R-Networks. We have demonstrated the tradeoff between high-levels of conservatism and performance by choosing a small value for the value-at-risk α in Figures 5,6,8 and Table 7. However the choice of β, which constrains value function learning, has a more significant impact on how much the value function is maximized with each gradient step. Increasing β increases the conservatism of the learning algorithm, increasing the gap between the constrained and true value functions, as described in Section 3. In all experiments presented in Sections 5 and 6, we treated β as a hyperparameter, tuned with the validation subset of our data. We leave the choice of α as a tunable parameter for the expert when evaluating the inferred value distributions, as described in Section A.2.2. However, to demonstrate the effect of increased conservatism on the performance of DistDeD we investigated the effect of setting β to larger values than those found through hyperparameter tuning. Specifically, we set β = {0.1, 0.2, 0.3, 0.4} and compare to the optimal DistDeD performance (with β = 0.35), DistDeD without the CQL penalty, and DeD in Figure 10. With increased values for β, this is a significant reduction in DistDeD performance where only a subsect of VaR thresholds surpass the performance of DeD. In fact, when β in greater than 0.2, DistDeD wholly underperforms DeD in identifying dead-ends. Additionally, when using larger values for β, the effect of choosing α for the value-at-risk is more pronounced as there is a wider range of performance as α varies when β is fixed. Figure 10: Demonstrating the effect of increased conservatism (increasing the CQL penalty weighting β) on DistDeD performance. Increasing the CQL weight serves to reduce the expressivity of the learned value distribution, constraining it further away from the true distribution. This corresponds to a reduction in performance of identifying dead-ends in the Septic patient population under consideration in this paper. A.4.4 DistDeD improves over DeD even in limited data settings The use of a more complex object to represent the value return in DistDeD raises natural questions about how performance degrades in low data regimes. While distributional RL has been shown to be robust to such reductions in training data Kumar et al., 2020), we evaluate DistDeD in comparison to DeD over a random subsampling of the training data. We ensure that the same proportion of positive to negative trajectories (e.g. derived from surviving and nonsurviving patients) is maintained when randomly sampling {10%, 25%, 50%, 75%} subsets of the training data. We then retrain the D-and R-Networks for both DeD and DistDeD with each subset and evaluate the trained networks using the same procedure and test dataset as used in Section 6. As demonstrated in Table 8 and Figure 11, we naturally see a reduction in test dead-end identification AUC. Yes, DistDeD's reduction in performance is not as sharp as DeD's while maintaining superiority over the prior approach. This confirms the findings in prior literature investigating the effect of low data regimes on the performance of distributional RL algorithms. While low data regimes are shown to affect top-line performance across all learning algorithms, the effect is not disproportionately seen among distributional RL algorithms. Figure 11: An investigation of the performance reduction of DistDeD and DeD when faced with limited training data. While a reduction in test performance is observed, DistDeD is more robust to the removal of training data, in comparison to DeD. The maximum AUC value for each setting of reduced training data is provided in Table 8. Figure 2 : 2Mirroring the construction of DeD, we instantiate two Markov Decision Processes (MDPs) M D and M R , derived from the original MDP M, γ = 1, with reward functions chosen to focus on either the positive or Distributional Dead-end Discovery (DistDeD) a) Observations are encoded (as needed) into a state representation and then b) passed to independent IQN models to estimate the distribution of returns (Z D and Z R ) for each possible action. Figure 3 : 3Demonstration of inherent value of using Z D (s, a) estimated with IQN and CVaR 0.1 (Z D (s, a)) in comparison to Q D (s, a) estimated with DDQN on the LifeGate toy domain Figure 7 : 7ROC curve comparison between CVaR α settings of DistDeD (green) and DeD (black), demonstrating DistDeD's robust improvement over DeD. Figure 8 : 8Evaluation of the CQL penalty in terms of area under the ROC curve (AUC), comparing DistDeD, DeD and two ablations to DistDeD. used to estimate the value of treatment options in relation to a negative terminal outcome Q D (s, a) DDQN Z D (s, a) IQN R-Network Neural Network used to estimate the value of treatment options in relation to a positive terminal outcome Q R (s, a) DDQN Z R (s, a) IQN γ Discount rate for training value functions, γ = 1 always δ D Threshold used to determine when to flag the values produced by the D-Network δ R Threshold used to determine when to flag the values produced by the R-Network R D (s, a) Reward function used in M D to train the D-Network. All positive terminal states have reward of 0, while negative terminal states have reward of -1. All other states receive a reward of 0. R R (s, a) Reward function used in M R to train the R-Network. All positive terminal states have reward of +1, while negative terminal states have reward of 0. All other states receive a reward of 0. N, K the number of samples drawn from the learned quantile function, differs between training (N ) and evaluation (K) Figure 9 : 9The evolution of estimated values using DistDeD over the course of the recorded patient trajectories (72 hours in total prior to termination), represented as histograms. The top row corresponds to the D-Network while the bottom row is derived from the R-Network. Estimated values from non-surviving patients are plotted in blue while those from surviving patients are plotted in green. Table 1 . 1Table 1: Comparison of AUC when considering each improvement to DeD, 1) incorporating the CQL penalty and 2) modeling the full distributions of the expected return. Values represented here for the distributional components represent the mean value over all settings of VaR α .In total, DistDeD (which Sebastian Thrun and Anton Schwartz. Issues in using function approximation for reinforcement learning. 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Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances in Neural Information Processing Systems, 2022. URL https://openreview.net/forum?id=22hMrSbQXzt. Chengyang Ying, Xinning Zhou, Hang Su, Dong Yan, Ning Chen, and Jun Zhu. Towards safe reinforcement learning via constraining conditional value-at-risk. arXiv preprint arXiv:2206.04436, 2022. ACM Computing Surveys (CSUR), 55(1):1-36, 2021. Table 2 : 2Patient features used for learning state representations for predicting future observationsAge Gender Weight (kg) Height Heart Rate Sys. BP Dia. BP Mean BP Respiratory Rate Body Temp (C) Glucose SO2 PaO2 PaCO2 FiO2 PaO2 / FiO2 Arterial pH Base Excess Chloride Calcium Potassium Sodium Lactate Hematocrit Hemoglobin Platelet White Blood Cells Albumin Anion Gap Bicarbonate (HCO3) PT PTT Glascow Coma Scale SpO2 BUN Creatinine INR Bilirubin SGOT (AST) SGPT (ALT) Urine Output Mech. Ventilation Table 3 : 3MIMIC-IV Sepsis Cohort Statistics: Median (25% -75% quantiles)Variable MIMIC (n = 6188) Variable MIMIC (n = 6, 188) Demographics Outcomes Age, years 68.0 (57.0.-80.0) Deceased 836 (13.51%) Age range, years Vasopressors administered 2241 (36.2%) 18-29 171 (2.8%) Fluids administered 6032 (97.5%) 30-39 269 (4.3%) Ventilator used 2201 (35.6%) 40-49 414 (6.7%) 50-59 911 (14.7%) 60-69 1426 (23%) 70-79 1332 (21.5%) 80-89 1207 (19.5%) ≥90 458 (7.4%) Gender Male 3251 (52.54%) Female 2937 (47.46%) Physical exam findings Temperature ( • C) 36.8 (36.6-37.3) Weight (kg) 75.7 (63.1-91. Table 4 : 4Listing of parameters and hyperparameters for dead-end discovery Table 5 : 5Experimental parameters for LifeGateDeD (Fatemi et al., 2021) DistDeD (this work) Q D (s, a) DDQN Z D (s, a) IQN 2 layers of 32 units 2 layers of 32 units Q R (s, a) DDQN Z R (s, a) IQN 2 layers of 32 units 2 layers of 32 units γ = 1 γ = 1 δ D -0.15 δ D -0.5 δ R 0.85 δ R 0.5 N 8 K 1000 α 0.1 β 0.1 # of datapoints 1e 6 # of training epochs 50 learning rate 1e −3 # of evaluation trajectories 10,000 dimension of state space S 2 dimension of action space A 5 Table 6 : 6Experimental parameters for Sepsis Treatment ExperimentsDeD (Fatemi et al., 2021) DistDeD (this work) Q D (s, a) DDQN Z D (s, a) IQN 2 layers of 64 units 2 layers of 16 units Q R (s, a) DDQN Z R (s, a) IQN 2 layers of 64 units 2 layers of 16 units Training epochs 100 Training epochs 75 γ = 1 γ = 1 δ D -0.15 δ D Experiment dependent δ R 0.85 δ R Experiment Dependent learning rate 1e −4 learning rate 2e −5 N 64 K 1000 α 50 settings linearly spaced along [0,1] β 0.035 NCDE Properties Number of training epochs 30 Encoder Neural Network 2 layers with 80 hidden units input dimension 42 output dimension 55 learning rate 5e −4 General MDP Properties # of patient trajectories 6, 188 minimum trajectory length 12 median trajectory length 42 maximum trajectory length 72 # of features 42 dimension of state space S 55 dimension of action space A 25 Experiments in Section 6.2.1 δ D -0.15 δ D -0.5 δ R 0.85 δ R 0.5 Experiments in Section 6.2.2 δ D 100 settings w/in [-1,0] Table 8 : 8Maximum AUC values evaluating DistDeD vs DeD performance when training in low-data regimes. When observations are irregular or partial All code for data extraction and preprocessing as well as for defining and training DistDeD models can be found at https://github.com/MLforHealth/DistDeD. All code for data extraction and preprocessing as well as for defining and training DistDeD models can be found at https://github.com/MLforHealth/DistDeD. AcknowledgmentsWe thank our many colleagues and friends who contributed to thoughtful discussions and provided timely advice to improve this work. Specifically, we appreciate the encouragement and enthusiasm provided by Vinith Suriyakumar, Haoran Zhang, Mehdi Fatemi, Will Dabney and Marc Bellemare. We are grateful for the feedback provided by Swami Sankaranarayanan, Qixuan Jin, Tom Hartvigsen, Intae Moon and the anonymous reviewers who helped improve the writing of the paper.This research was supported in part by Microsoft Research, a CIFAR AI Chair at the Vector Institute, a Canada Research Council Chair, and an NSERC Discovery Grant.Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute www.vectorinstitute. ai/#partners. Constrained policy optimization. Joshua Achiam, David Held, Aviv Tamar, Pieter Abbeel, International conference on machine learning. PMLRJoshua Achiam, David Held, Aviv Tamar, and Pieter Abbeel. Constrained policy optimization. In Interna- tional conference on machine learning, pp. 22-31. PMLR, 2017. An optimistic perspective on offline reinforcement learning. 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{'abstract': 'In safety-critical decision-making scenarios being able to identify worst-case outcomes, or dead-ends is crucial in order to develop safe and reliable policies in practice. These situations are typically rife with uncertainty due to unknown or stochastic characteristics of the environment as well as limited offline training data. As a result, the value of a decision at any time point should be based on the distribution of its anticipated effects. We propose a framework to identify worst-case decision points, by explicitly estimating distributions of the expected return of a decision. These estimates enable earlier indication of dead-ends in a manner that is tunable based on the risk tolerance of the designed task. We demonstrate the utility of Distributional Dead-end Discovery (DistDeD) in a toy domain as well as when assessing the risk of severely ill patients in the intensive care unit reaching a point where death is unavoidable. We find that DistDeD significantly improves over prior discovery approaches, providing indications of the risk 10 hours earlier on average as well as increasing detection by 20%.', 'arxivid': '2301.05664', 'author': ['Taylor W Killian twkillian@cs.toronto.edu \nUniversity of Toronto\nVector Institute Massachusetts Institute of Technology\nImperial College London\n\n', 'Sonali Parbhoo sparbhoo@imperial.ac.uk \nUniversity of Toronto\nVector Institute Massachusetts Institute of Technology\nImperial College London\n\n', 'Marzyeh Ghassemi mghassem@mit.edu \nUniversity of Toronto\nVector Institute Massachusetts Institute of Technology\nImperial College London\n\n'], 'authoraffiliation': ['University of Toronto\nVector Institute Massachusetts Institute of Technology\nImperial College London\n', 'University of Toronto\nVector Institute Massachusetts Institute of Technology\nImperial College London\n', 'University of Toronto\nVector Institute Massachusetts Institute of Technology\nImperial College London\n'], 'corpusid': 255825752, 'doi': '10.48550/arxiv.2301.05664', 'github_urls': ['https://github.com/MLforHealth/DistDeD.', 'https://github.com/MLforHealth/DistDeD.'], 'n_tokens_mistral': 25858, 'n_tokens_neox': 22679, 'n_words': 14709, 'pdfsha': '71446fc2294f5a70de6cf6fba9f8f4af0be4899e', 'pdfurls': ['https://export.arxiv.org/pdf/2301.05664v2.pdf'], 'title': ['Risk Sensitive Dead-end Identification in Safety-Critical Offline Reinforcement Learning', 'Risk Sensitive Dead-end Identification in Safety-Critical Offline Reinforcement Learning'], 'venue': []}